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+# UNVEILING THE MASK OF POSITION-INFORMATION PATTERN THROUGH THE MIST OF IMAGE FEATURES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent studies have shown that paddings in convolutional neural networks encode absolute position information which can negatively affect the model performance for certain tasks. However, existing metrics for quantifying the strength of positional information remain unreliable and frequently lead to erroneous results. To address this issue, we propose novel metrics for measuring and visualizing the encoded positional information. We formally define the encoded information as Position-information Pattern from Padding (PPP) and conduct a series of experiments to study its properties as well as its formation. The proposed metrics measure the presence of positional information more reliably than the existing metrics based on PosENet and tests in F-Conv. We also demonstrate that for any extant (and proposed) padding schemes, PPP is primarily a learning artifact and is less dependent on the characteristics of the underlying padding schemes.
+
+# 1 INTRODUCTION
+
+Padding, one of the most fundamental components in neural network architectures, has received much less attention than other modules in the literature. In convolutional neural networks (CNNs), zero padding is frequently used perhaps due to its simplicity and low computational costs. This design preference remains almost unchanged in the past decade. Recent studies (Islam\* et al., 2020; Islam et al., 2021b; Kayhan & Gemert, 2020; Innamorati et al., 2020) show that padding can implicitly provide a network model with positional information. Such positional information can cause unwanted side-effects by interfering and affecting other sources of position-sensitive cues (e.g., explicit coordinate inputs (Lin et al., 2022; Alsallakh et al., 2021a; Xu et al., 2021; Ntavelis et al., 2022; Choi et al., 2021), embeddings (Ge et al., 2022), or boundary conditions of the model (Innamorati et al., 2020; Alguacil et al., 2021; Islam et al., 2021a)). Furthermore, padding may lead to several unintended behaviors (Lin et al., 2022; Xu et al., 2021; Ntavelis et al., 2022; Choi et al., 2021), degrade model performance (Ge et al., 2022; Alguacil et al., 2021; Islam et al., 2021a), or sometimes create blind spots (Alsallakh et al., 2021a). Meanwhile, simply ignoring the padding pixels (known as no-padding or valid-padding) leads to the foveal effect (Alsallakh et al., 2021b; Luo et al., 2016) that causes a model to become less attentive to the features on the image border. These observations motivate us to thoroughly analyze the phenomenon of positional encoding including the effect of commonly used padding schemes.
+
+Conducting such a study requires reliable metrics to detect the presence of positional information introduced by padding, and more importantly, quantify its strength consistently. We observe that the existing methods for detecting and quantifying the strength of positional information yield inconsistent results. In Section 3, we revisit two closely related evaluation methods, PosENet (Islam\* et al., 2020) and F-Conv (Kayhan & Gemert, 2020). Our extensive experiments demonstrate that (a) metrics based on PosENet are unreliable with an unacceptably high variance, and (b) the Border Handling Variants (BHV) test in F-Conv suffers from unaware confounding variables in its design, leading to unreliable test results.
+
+In addition, we observe all commonly-used padding schemes actually encode consistent patterns underneath the highly dynamic model features. However, such a pattern is rather obscure, noisy, and visually imperceptible for most paddings (except zeros-padding), which makes recognizing and analyzing it difficult. Fortunately, we show that such patterns can be consistently revealed with a sufficient number of samples by defining an optimal padding scheme (see Section 2.1 and Figure 1).
+
+![](images/ab0eb0828e532ceff21f80209a0363116483e34653f04c92525481c6f69dd5c5.jpg)  
+Figure 1: Position-information Pattern from Padding (PPP). We propose a method that can consistently and effectively extract PPPs through the distributional difference between optimallypadded (gray-scale surfaces) and algorithmically-padded features (colored surfaces). The results show that the two distributions become distinguishable as the number of sample increases. Following the procedure in Section 2.2, we extract a clear view of PPP with the expectation of the pairwise differences between optimally-padded and algorithmically-padded features. We render each visualization in tilted view (first row) and top view (second row). The colors represent the magnitude (blue/cold/weak to green/warm/strong) at each pixel. The features are extracted at the 3rd layer of interest (Appendix A) from a randn-padded (Section 2.4) ResNet50 pretrained on ImageNet.
+
+We accordingly propose a new evaluation paradigm and develop a method to consistently detect the presence of the Position-information Pattern from Padding (PPP), which is a persistent pattern embedded in the model features to retain positional information. We present two metrics to measure the response of PPP from the signal-to-noise perspective and demonstrate its robustness and low deviation among different settings, each with multiple trials of training.
+
+To weaken the effect of PPP, in Section 2.4, we design a padding scheme with built-in stochasticity, making it difficult for the model to consistently construct such biases. However, our experiments show that the models can still circumvent the stochasticity and end up consistently constructing PPPs. These results suggest that a model likely constructs PPPs purposely to facilitate its training, rather than falsely or accidentally learning some filters that respond to padding features.
+
+With reliable PPP metrics, we conduct a series of experiments to analyze the characteristics of PPP in Section 4.1. Specifically, we analyze the formation of PPP throughout each model training process in Section 4.3. The results show PPPs are formed expeditiously at the early stage of model training, slowly but steadily strengthen through time, and eventually shaped in clear and complete patterns. These results show that a model intentionally develops and reinforces PPPs to facilitate its learning process. Moreover, we observe the PPPs of all pretrained networks are significantly stronger than those in their initial states. This indicates an unbiased training procedure is of great importance in resolving the critical failures caused by PPP in numerous vision tasks (Alsallakh et al., 2021a; Xu et al., 2021; Ge et al., 2022; Alguacil et al., 2021).
+
+# 2 OBSERVATIONS AND METHODOLOGY
+
+In this section, we first define symbols for expressing the functionality of paddings and define the optimal-padding scheme. We then give a formal definition of Position-information Pattern from Padding (PPP) and utilize the optimal-padding scheme to develop propose a method to capture PPP and measure its response with two metrics.
+
+# 2.1 OPTIMAL PADDING
+
+The process of capturing an image from the real world can be simplified into two steps: (a) 3D information of the environment is first projected onto an infinitely large 2D plane, and then (b) the camera determines resolution as well as field-of-view to form a digital image from such infinitely large and continuous 2D signals (Liu et al., 2019; Ravi et al., 2020). Let $S ^ { * } = \{ s _ { n } ^ { * } \} _ { n = 1 } ^ { N }$ be a collection of such infinitely large and continuous 2D signals, and the collection of 2D images captured by cameras at a spatial size $( h _ { n } , w _ { n } )$ be $S ^ { \prime } = \{ s _ { n } ^ { \prime } \} _ { n = 1 } ^ { \mathbb { N } }$ . A padding scheme can be used to generate a set of algorithmically-padded images $\hat { S } = \{ \hat { s } _ { n } \} _ { n = 1 } ^ { N }$ by a padding function $\rho$ :
+
+![](images/c0a25d4970af93cf15e96a1750f2a0448e16a47e0e5db0dcf5d02831db9d75cf.jpg)  
+Figure 2: Principal point shift. (a) The stride-2 Conv2d only pads on one side, causing the principal point shift (red squares) in earlier layers. (b) Such a shift requires careful margin correction while aligning algorithmically-padded and optimally-padded features (we describe the details of point shift in Appendix A). (c) The shift is visible in the feature space (marked with red and yellow boxes). (d) It is crucial to correct the principal point shift while measuring PPP. The PPP calculation involves pixel-wise distance functions, which are not robust to spatial shifts (Zhang et al., 2018).
+
+$$
+\hat { s } _ { n } [ i , j ] = \left\{ { \begin{array} { l l } { s _ { n } ^ { \prime } [ i , j ] = s ^ { * } [ i , j ] } & { { \mathrm { i f ~ } } 0 < i < h _ { n } { \mathrm { ~ a n d ~ } } 0 < j < w _ { n } , } \\ { \rho ( s _ { n } ^ { \prime } , i , j ) } & { { \mathrm { o t h e r w i s e } } , } \end{array} } \right.
+$$
+
+collection where $i$ and $\dot { S ^ { \dagger } } = \{ s _ { n } ^ { \dagger } \} _ { n = 1 } ^ { N }$ $j$ are indexes of a pixel in the spatial dimension. We define a theoretical optimally-padded with an optimal-padding function $\rho ^ { \dagger }$ by:
+
+$$
+\begin{array} { r } { s _ { n } ^ { \dagger } [ i , j ] = \left\{ { s _ { n } ^ { \prime } [ i , j ] } \atop { \rho ^ { \dagger } ( s _ { n } ^ { \prime } , i , j ) } \right. \ } & { = s ^ { * } [ i , j ] \quad \mathrm { i f ~ } 0 < i < h _ { n } \mathrm { ~ a n d ~ } 0 < j < w _ { n } , } \\ { s ^ { \dagger } _ { n } [ i , j ] = \left\{ { s _ { n } ^ { \prime } [ i , j ] } \atop { \rho ^ { \dagger } ( s _ { n } ^ { \prime } , i , j ) } \right. \ } & { = s ^ { * } [ i , j ] \quad \mathrm { o t h e r w i s e } . } \end{array}
+$$
+
+In practice, without curated data, the optimal-padding scheme described in Eq. 2 is difficult to achieve. We describe how we relax this constraint in Section 2.3
+
+# 2.2 POSITIONAL-INFORMATION PATTERN FROM PADDING
+
+Despite the previous literature discovered the existence of positional information caused by the model paddings, there is still no clear definition for such information, and lacks effective metrics to detect or quantify it. Ideally, an effective metric for such positional information should have two properties. First, it is a spatial pattern, it contributes distinctive information to different spatial locations. Its shape enables the network to develop and exploit the absolute positional information of each pixel, eventually leading to the unattended and undesirable effects in certain tasks (Lin et al., 2022; Alsallakh et al., 2021a; Xu et al., 2021; Ntavelis et al., 2022; Choi et al., 2021; Ge et al., 2022; Alguacil et al., 2021). Second, as it represents the positional information purely contributed by the padding, it is a constant pattern irrelevant to the image contents. We accordingly name it the Positional-information Pattern from Padding (PPP).
+
+Unfortunately, such a pattern shares space with image features, where the image features typically have very diverse appearances and high dimensionality. When these two signals interfere with each other, the appearance of PPP becomes extremely obscure and imperceptible in most cases (except zeros padding). Figure 1 shows if we visualize features sample-by-sample, there are no obvious differences between optimally-padded features (gray-scale surface) and algorithmically-padded features (colored surface). To address the issue, we show that, by assuming the interferences between PPP and image features to be random, its expectation over a large set of images will saturate to a constant bias and no longer hinder us from capturing PPP.
+
+Based on these observations and assumptions, we define PPP as the constant component independent of model inputs, and its presence is completely contributed by the existence of a padding scheme $\rho$ . Given $\hat { S }$ and a model $F ( \hat { s } ; \theta , \rho )$ , which $\theta$ is the model parameters and $\rho$ is a padding scheme applied to $F$ . Let the model feature extracted at $k$ -th layer be $f _ { n , k } = F _ { k } ( \hat { s } _ { n } ; \theta , \rho )$ , where $F _ { k }$ is the model from the first layer to the $k$ -th layer. The PPP at $k$ -th layer $( P P P _ { k } )$ ) can be formulated by:
+
+$$
+\begin{array} { r } { \mathsf { P P P } _ { k } \ = \ \underset { n } { \mathbb { E } } \left[ \textit { d } \big ( \begin{array} { l } { F _ { k } ( s _ { n } ^ { \dagger } ; \theta , \rho ^ { \dagger } ) , F _ { k } ( \hat { s } _ { n } ; \theta , \rho ) } \end{array} \big ) \ \right] \ , } \end{array}
+$$
+
+where $d ( \cdot , \cdot )$ can be any distance function. We use $\ell _ { 1 }$ distance in this work, and accordingly name the metric PPP-MAE.
+
+Pitfalls: feature misalignment. It is important to note that, some CNN components can cause serious feature misalignment while computing PPP and leads to erroneous results. A typical example is principal point shift, where the uneven padding in stride-2 convolution causes the center of features slightly drifted, as shown in Figure 2. Since the measurement of PPP requires perfect alignment, such a drift should be carefully considered while integrating PPP into new architectures. We discuss the issue along with other pitfalls in Appendix A and provide three detailed examples of correcting the principal point shifting.
+
+# 2.3 SIMULATED OPTIMAL PADDING
+
+In practice, it is impossible to gain access to $S ^ { * }$ for calculating the optimal padding $S ^ { \dagger }$ described in Eq. 2. But fortunately, given our goal in Eq. 3 is to analyze the model features within the $( h _ { n } , w _ { n } )$ region, $S ^ { * }$ is an overshoot of the data we actually required. Given a vision model $F ( \hat { s } ; \theta , \rho )$ trained at a field-of-view $( h _ { n } , w _ { n } )$ pixels, the receptive field of such vision model is $( h _ { m } , w _ { m } )$ pixels (we show the collection Apat where  pixels, $h _ { m } \gg h _ { n }$ and tion $w _ { m } \gg w _ { n }$ . Let an altern field implies $S ^ { \odot } = \{ s _ { n } ^ { \odot } \} _ { n = 1 } ^ { N }$ $( h _ { m } , w _ { m } )$ $F _ { k } ( s _ { n } ^ { \dagger } ; \theta , \mathbf { \bar { \rho } } )$ equals to $F _ { k } \big ( s _ { n } ^ { \odot } ; \theta , \rho ^ { \dagger } \big )$ for all $k$ .
+
+In other words, in terms of computing Eq. 3, $S ^ { \odot }$ is equivalent to $S ^ { * }$ within the finite $( h _ { n } , w _ { n } )$ region for a given model architecture. Therefore, we can simulate the procedure described in Eq. 1 and Eq. 2 using $S ^ { \odot }$ instead of $S ^ { \dagger }$ , as long as $\forall s _ { n } ^ { \odot } \in S ^ { \odot }$ the spatial size of $s _ { n } ^ { \odot }$ is strictly larger than $( h _ { m } , w _ { m } )$ .
+
+# 2.4 RANDN PADDING
+
+Most of the existing padding schemes (e.g., zeros, reflect, replicate, circular) exhibit certain consistent patterns that can be easily detected by some designed convolutional kernels. One may argue that the nature of easy detectability can be a root cause of encouraging the models to learn to rely on these obvious patterns. This motivates us to design an additional sampling-based padding scheme without any consistent patterns, namely randn (i.e., random normal) padding, which produces dynamical values from a normal distribution while following the local statistics. We first determine the maximal and minimal values of a sliding window (which can be easily achieved with max-pooling), use the average of them as a proxy mean $\mu _ { p }$ , and use the difference between the mean and the maximal value as a proxy standard deviation $\sigma _ { p }$ . For each padding location, we sample the padding value according to a normal distribution $\mathcal { N } ( \mu _ { p } , \sigma _ { p } ^ { 2 } )$ from the nearest sliding window. We include more implementation details in Appendix A.
+
+Aside from creating a pattern-less padding scheme with sampling, the design of randn padding is based on several factors. The sampled padding pixels are allowed to occasionally exceed the min/max bound of the sliding window. Without breaking the min/max bound can introduce detectable patterns in certain extreme cases, such as a gradient-like feature that has its maximal intensity at the top-left corner and minimal intensity at the bottom-right corner. We also design the padding scheme to follow the local distribution. The padding exhibits high entropy when the local variation is high, while degenerates to value repetition with imperceptible perturbations while padding a flat area. As such, not only do the padding pixels exhibit less pattern, but it also prevents the padding pixels from breaking the features in the border region. We later show that a model still deliberately and incredibly built up PPP over time even with such a sophisticated padding scheme.
+
+# 3 REVISITING PRIOR WORK
+
+In this section, we first reproduce two experiments from the prior art, which aim to assess positional information from paddings. We show several critical design issues in these experiments and discuss how these problems affect the drawn conclusions. Finally, we propose two additional experiments to quantify the amount of positional information embedded in the paddings.
+
+Table 1: Background color as a critical confounding variable in BHV test. We show that using a grey background similar to Figure 3 leads to discrepant results. The standard deviations are reported among 10 individual trials. We mark the best performance in green, and the worst two in red.   
+
+<table><tr><td rowspan="2">Padding</td><td rowspan="2">F-Conv?</td><td colspan="4">Black Background</td><td colspan="4">Grey Background</td></tr><tr><td>Similar (%)</td><td>Dissimilar (%)</td><td>Diff (%)</td><td>Inconsistency (%)</td><td>Similar (%)</td><td>Dissimilar (%)</td><td>Diff (%)</td><td>Inconsistency (%)</td></tr><tr><td rowspan="2">Zeros</td><td>N</td><td>99.83±0.00</td><td>3.21± 8.35</td><td>-87.68</td><td>95.81± 2.07</td><td>100.00± 0.00</td><td>4.96± 5.93</td><td>-95.04</td><td>97.85± 4.55</td></tr><tr><td>Y</td><td>89.24±0.98</td><td>89.24±0.98</td><td>0.00</td><td>18.02± 8.08</td><td>100.00±0.00</td><td>4.77± 6.52</td><td>-95.23</td><td>96.79±7.13</td></tr><tr><td rowspan="2">Circular</td><td>N</td><td>80.31±3.23</td><td>80.31± 3.23</td><td>0.00</td><td>34.25± 8.32</td><td>72.75± 0.96</td><td>72.75± 0.96</td><td>0.00</td><td>26.30± 5.55</td></tr><tr><td>Y</td><td>99.20±0.23</td><td>93.14±2.88</td><td>-6.06</td><td>18.48±3.55</td><td>98.26± 0.50</td><td>92.40±4.23</td><td>-5.87</td><td>28.67±6.18</td></tr><tr><td rowspan="2">Reflect</td><td>N</td><td>100.00±0.00</td><td>15.67±12.72</td><td>-84.33</td><td>91.18±13.19</td><td>100.00± 0.00</td><td>19.96±13.54</td><td>-80.04</td><td>90.33±11.95</td></tr><tr><td>Y</td><td>100.00±0.00</td><td>11.70±15.38</td><td>-88.30</td><td>97.33± 6.16</td><td>100.00±0.00</td><td>17.16±12.19</td><td>-82.84</td><td>98.13± 3.44</td></tr><tr><td rowspan="2">Replicate</td><td>N</td><td>100.00±0.00</td><td>43.39±11.42</td><td>-56.61</td><td>75.32± 8.20</td><td>100.00± 0.00</td><td>33.16± 6.42</td><td>-66.83</td><td>84.09± 6.47</td></tr><tr><td>Y</td><td>98.32±0.39</td><td>93.65± 1.36</td><td>-4.67</td><td>32.60± 4.97</td><td>97.17± 0.48</td><td>94.99± 1.20</td><td>-2.18</td><td>32.15± 5.11</td></tr><tr><td rowspan="2">Randn</td><td>N</td><td>100.00±0.00</td><td>10.31±12.56</td><td>-89.70</td><td>94.88± 5.55</td><td>99.97± 0.13</td><td>35.47±10.82</td><td>-64.50</td><td>83.59± 8.48</td></tr><tr><td>Y</td><td>100.00±0.00</td><td>20.80±14.15</td><td>-79.20</td><td>92.54±8.37</td><td>77.28±16.13</td><td>66.70±11.58</td><td>-10.59</td><td>45.70±20.62</td></tr><tr><td>No-pad</td><td>-</td><td>100.00±0.00</td><td>3.21± 8.35</td><td>-96.79</td><td>95.81± 2.07</td><td>100.00± 0.00</td><td>30.07± 4.06</td><td>-69.93</td><td>81.30± 2.44</td></tr></table>
+
+# 3.1 POSENET
+
+Islam et al. show zeros-padding provides CNN models positional information cues, and propose PosENet (Islam\* et al., 2020) to quantify the amount of positional information encoded within CNN features. A PosENet experiment involves several components: a pretrained CNN model $F$ , a shallow CNN $E _ { p e m }$ (i.e., position encoding module), an image dataset $X = \{ x _ { i } \} _ { i = 1 } ^ { N }$ to examine, and a constant target pattern $y$ (e.g., 2D Gaussian pattern). PosENet first extracts intermediate features at $k$ -th layer with $f _ { ( i , k ) } = F _ { k } ( x _ { i } )$ using the pretrained CNN, and then optimizes $E _ { p e m }$ to minimize $\mathbb { E } _ { i , k } [ | | E _ { p e m } ( f _ { ( i , k ) } ) - y | | _ { 2 } ]$ . Finally, the amount of positional information is quantified by the average Spearman’s correlation (SPC) and Mean Absolute Error (MAE) overall $E _ { p e m } ( f _ { ( i , k ) } )$ toward $y$ .
+
+A critical issue with PosENet is the use of an optimization-based metric. It is sensitive to hyperparameters with large variation. As shown in Table 2, for all the PosENet results, the standard deviation over five trials significantly dominates the differences between different types of paddings, and thus no definitive conclusions can be drawn. We also observed that PosENet can report NaN results in certain setups. Furthermore, PosENet quantifies the amount of positional information by the faithfulness of the final reconstruction. However, a better reconstruction does not have a clear relationship to measuring the strength and significance of positional information. For instance, PosENet sometimes shows responses to no-padding models, demonstrating it is a metric with an indefinite bias pending on the memorization ability of $E _ { p e m }$ . Moreover, optimizing for pattern reconstruction is highly dependent on the underlying data distribution, simply changing the evaluation data distribution without changing the model weights can drastically change the PosENet numerical magnitudes and the conclusions of which model embeds the strongest positional information.
+
+Another issue is that the no-padding scheme used in the $E _ { p e m }$ module in PosENet is known to have the foveal effect (Alsallakh et al., 2021b; Luo et al., 2016), where a model pays less attention to the information on the edge of inputs. Using such a padding scheme for detecting positional information from paddings, which is mostly concentrated on the edge of the feature maps, is less effective. This is an inevitable dilemma as PosENet aims to identify positional information from the padding of the pretrained $F$ , while applying any padding scheme to $E _ { p e m }$ introduces intractable effects between the paddings of the two models.
+
+# 3.2 F-CONV
+
+Kayhan et al. propose a full-padding scheme (F-Conv) (Kayhan & Gemert, 2020) and demonstrate it is more translational invariant than the alternatives. One of the critical results is on “border handling variants” (Exp 2 of (Kayhan & Gemert, 2020)), which we call it BHV test. The BHV test creates a toy dataset, where each image has a black background with a green square and a red square in the foreground. The task is to predict if the red square is on the left of the green square (class 1), or vice versa (class 2). In addition, Kayhan et al. intentionally adds a location bias such that both squares are located in the upper half of the image for class 1, and located in the lower half of the image for class 2. During testing, a “similar test” inherits the same bias, while a “dissimilar test” exchanges the bias (i.e., both squares are in the lower half of the image for class 1). As a truly translation-invariant CNN model should not be affected by the location bias, it should focus on the relation between the red and green squares and perform similarly on both tests. Since the experimental results show that F-Conv performs best on the dissimilar test, it is concluded that F-Conv is less sensitive to the location bias. The authors also conclude the circular padding performs worse due to the behavior of wrapping the pixels to the other side of the image, which leads to confusion between two classes.
+
+![](images/8eff9d62aa85c9765ac32f2d177447c962ee0762431f465e1b2d4ebf480d0341.jpg)  
+Figure 4: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples extracted at the 3rd layer-of-interest (Appendix A). The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to $[ 0 , 1 ]$ separately, therefore the colors between images are not comparable. More visualizations are presented in Appendix B.
+
+However, as shown in Figure 3, we find the experimental design does not consider a crucial confounding variable: the black background has a zero intensity, making zeros padding the optimal padding that perfectly follows the background distribution. In Table 1, we show that the dissimilar test is no longer in favor of F-Conv zeros after changing the background color to grey. We also show that F-Conv replicate and F-Conv circular perform best on the dissimilar test, which is different from the original observation.
+
+Finally, we report an additional inconsistency rate to show that the CNN architecture used in the BHV test actually has access to the absolute position of the squares. Given a random sample in class 1, we create a trajectory of samples by simultaneously moving the two squares to the bottom of the canvas and recording the CNNmodel prediction in all intermediate states. We label a trajectory to be inconsistent if the prediction of the CNN-model switches classes at any step of the trajectory. A CNN model with no access to the absolute-position information should have all trajectories maintaining consistent predictions, with $0 \%$ inconsistency. Table 1 shows the inconsistent ratio over 228 uniformly sampled trajectories, where all models maintain high inconsistency rates, even with a no-padding architecture. These results show that the CNN model used in the BHV test is not translation invariant. This can be attributed to that a CNN model has a large receptive field covering the whole experiment canvas, therefore capable of gradually constructing absolute coordinates for each input pixel. Note that we only show the design of the BHV test is not suitable for quantifying the amount of positional information exhibited in a CNN model. Such a conclusion does not imply that F-Conv cannot potentially improve the translation-invariant property of CNNs.
+
+![](images/911fffa05b9eedc6ab963d9f5c93aa9597332089a3ad2ea373b41482c80ba1b1.jpg)  
+Figure 3: The BHV test trains a binary classifier to predict the relative position of the two colored squares. It hypothesizes if the padding provides no positional information, the classifier will only focus on the relative position of the two squares. (Left) The black background is a confounding variable. (Right) Zeros padding no-longer pads optimum values after changing the background color.
+
+# 4 EXPERIMENTS AND ANALYSIS
+
+Datasets Since most vision models are trained on tasks for recognizing objects, an image collection containing a diverse object appearance is more suitable for the task. As mentioned in Section 2.3, evaluating PPP requires images at a large field-of-view, in practice, we collect images at $2 , 0 4 8 \times 2 , 0 4 8$ pixels, which is larger than the receptive field of all the models we tested. Due to the constraint of large field-of-view, we compute PPP on three datasets (all at $2 , 0 4 8 \times 2 , 0 4 8$ pixels): (a) 480 satellite images crawled from Google Map, (b) 1,024 images synthesized by InfinityGAN (Lin et al., 2022) trained with Flickr-Landscape dataset, and (c) 1,024 images synthesized by InfinityGAN trained with LSUN-Tower (Yu et al., 2015) dataset. We crop the images depending on the requested input image sizes and principal point shifts from each model (see Appendix A for details). We will release the script for collecting and composing these large images.
+
+Table 2: Comparing PosENet and our proposed PPP metrics. Most of the PosENet results are not distinguishable due to the high standard deviations. The standard deviation is computed by five different pretrained models for each test. The performance shows the accuracy (for classification) or weighted F-measure score (for saliency object detection). We use 2D Gaussian as PosENet reconstruction pattern, and PPP-MAE is measured at the 4th layer of interest. Here, (↑) indicates a higher value corresponds to stronger positional information or better performance on the task (vice versa for (↓)). For each group of pretrained models, we label the strongest positional information response with red, and the experiments within its standard deviation range with orange.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Padding</td><td rowspan="2">Eval Dataset</td><td colspan="2">PosENet</td><td rowspan="2">PPP-MAE(ours) (↑)</td><td rowspan="2">Performance (↑)</td></tr><tr><td>SPC (↑)</td><td>MAE(↓)</td></tr><tr><td rowspan="10">VGG-19</td><td>Zeros</td><td>GMap InfinityGAN-flickr</td><td>0.107±0.128 0.368±0.116</td><td>0.196±0.006 0.183±0.007</td><td>0.0176±0.0005 0.0163±0.0006</td><td rowspan="2">74.0972±0.0870</td></tr><tr><td></td><td>InfinityGAN-tower GMap</td><td>0.492±0.106</td><td>0.173±0.010</td><td>0.0179±0.0001</td></tr><tr><td rowspan="8">Circular</td><td></td><td>0.098±0.139</td><td>0.197±0.007</td><td>0.0158±0.0006</td><td rowspan="2">74.4716±0.0863</td></tr><tr><td>InfinityGAN-flickr</td><td>0.323±0.147</td><td>0.185±0.009</td><td>0.0137±0.0004</td></tr><tr><td rowspan="8">Reflect</td><td>InfinityGAN-tower</td><td>0.460±0.102</td><td>0.176±0.009</td><td>0.0184±0.0005</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>GMap InfinityGAN-flickr</td><td>0.109±0.139 0.343±0.132</td><td>0.196±0.007 0.185±0.008</td><td>0.0158±0.0002</td></tr><tr><td>InfinityGAN-tower</td><td></td><td>0.0146±0.0008 0.0168±0.0005</td><td>74.0516±0.0621</td></tr><tr><td>GMap</td><td>0.460±0.113 0.084±0.137</td><td>0.177±0.009</td><td>0.0144±0.0009 73.9964±0.1079</td></tr><tr><td>InfinityGAN-flickr</td><td>0.356±0.111</td><td>0.197±0.006 0.184±0.007</td><td>0.0128±0.0012</td></tr><tr><td></td><td></td><td>0.173±0.010</td><td></td></tr><tr><td rowspan="6"></td><td rowspan="2"></td><td>InfinityGAN-tower</td><td>0.498±0.111</td><td></td><td>0.0156±0.0006</td><td rowspan="2"></td></tr><tr><td>GMap</td><td>0.125±0.154</td><td>0.195±0.006</td><td>0.0182±0.0012</td></tr><tr><td rowspan="6">Randn</td><td>InfinityGAN-flickr</td><td>0.374±0.137</td><td>0.185±0.007</td><td>0.0167±0.0008</td><td rowspan="2">73.7716±0.0758</td></tr><tr><td>InfinityGAN-tower</td><td>0.421±0.161</td><td>0.181±0.010</td><td>0.0186±0.0012</td></tr><tr><td>GMap</td><td>0.001±0.239</td><td>0.204±0.013</td><td>0.0000±0.0000</td><td></td></tr><tr><td>InfinityGAN-flickr</td><td>0.303±0.192</td><td>0.187±0.012</td><td>0.0000±0.0000</td><td>62.0396±0.0830</td></tr><tr><td rowspan="8"></td><td>InfinityGAN-tower</td><td>0.516±0.139</td><td>0.172±0.014</td><td>0.0000±0.0000</td><td></td></tr><tr><td rowspan="8">Zeros</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GMap InfinityGAN-flickr</td><td>0.191±0.188 0.682±0.107</td><td>0.193±0.008 0.152±0.019</td><td>0.0162±0.0012 0.0137±0.0004</td><td>75.6856±0.0924</td></tr><tr><td>InfinityGAN-tower</td><td></td><td>0.144±0.017</td><td>0.0153±0.0013</td><td></td></tr><tr><td rowspan="10">Circular</td><td></td><td>0.721±0.077</td><td></td><td>0.0188±0.0016</td><td>76.1432±0.1026</td></tr><tr><td>GMap</td><td>0.398±0.115</td><td>0.197±0.007</td><td></td><td></td></tr><tr><td>InfinityGAN-flickr</td><td>0.628±0.084</td><td>0.159±0.013</td><td>0.0178±0.0005</td><td></td></tr><tr><td>InfinityGAN-tower</td><td>0.585±0.105</td><td>0.165±0.014</td><td>0.0189±0.0012</td><td></td></tr><tr><td rowspan="8"></td><td rowspan="2">Reflect</td><td></td><td></td><td></td><td></td><td rowspan="2"></td></tr><tr><td>GMap</td><td>0.197±0.185</td><td>0.192±0.008</td><td>0.0150±0.0004</td></tr><tr><td rowspan="9">Replicate</td><td>InfinityGAN-flickr</td><td>0.594±0.096</td><td>0.169±0.012</td><td>0.0134±0.0009</td><td rowspan="2">75.5068±0.1213</td></tr><tr><td>InfinityGAN-tower</td><td>0.667±0.087</td><td>0.153±0.016</td><td>0.0157±0.0002</td></tr><tr><td>GMap</td><td>0.249±0.192</td><td>0.189±0.009</td><td>0.0138±0.0003</td></tr><tr><td>InfinityGAN-flickr</td><td>0.700</td><td>0.147±0.018</td><td>0.0114±0.0003</td><td>75.6122±0.0911</td></tr><tr><td rowspan="8">Randn</td><td>InfinityGAN-tower</td><td>2±0.095</td><td></td><td>0.0142±0.0007</td><td></td></tr><tr><td></td><td>0.726±0.069</td><td>0.142±0.016</td><td></td><td></td></tr><tr><td>GMap</td><td>0.210±0.192</td><td>0.191±0.009</td><td>0.0147±0.0007</td><td>75.3076±0.1016</td></tr><tr><td rowspan="10"></td><td>InfinityGAN-flickr</td><td>0.566±0.100</td><td>0.171±0.011</td><td>0.0122±0.0011</td><td></td></tr><tr><td>InfinityGAN-tower</td><td>0.714±0.068</td><td>0.142±0.015</td><td>0.0153±0.0004</td><td></td></tr><tr><td></td><td></td><td></td><td>0.0049±0.0001</td><td></td></tr><tr><td>GMap InfinityGAN-flickr</td><td>0.156 ±0.212</td><td>0.201 ±0.017</td><td></td><td></td></tr><tr><td rowspan="8">Circular</td><td></td><td></td><td>0.184±0.012</td><td>0.0036±0.0001</td><td>0.6269±0.0015</td></tr><tr><td rowspan="8"></td><td></td><td>0.365±0.140</td><td></td><td></td><td></td></tr><tr><td>InfinityGAN-tower</td><td>0.449±0.120</td><td>0.179±0.013</td><td>0.0032±0.0001</td><td></td></tr><tr><td>GMap</td><td>0.011±0.209</td><td>0.207±0.014</td><td>0.0062±0.0001</td><td>0.6260±0.0009</td></tr><tr><td>InfinityGAN-flickr InfinityGAN-tower</td><td>0.329±0.133 0.398±0.115</td><td>0.187±0.012 0.182±0.011</td><td>0.0068±0.0001 0.0050±0.0002</td><td></td></tr><tr><td rowspan="2">Reflect</td><td></td><td></td><td></td><td></td></tr><tr><td>GMap</td><td>0.062±0.210 0.205±0.016</td><td></td><td>0.0053±0.0001</td></tr><tr><td rowspan="2"></td><td>InfinityGAN-flickr InfinityGAN-tower</td><td>0.322±0.133 0.396±0.125</td><td>0.188±0.013 0.183±0.013</td><td>0.0030±0.0001 0.0039±0.0001</td><td rowspan="2">0.6243±0.0022</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td rowspan="7"></td><td rowspan="2">Replicate</td><td>GMap</td><td></td><td>0.204±0.016</td><td>0.0043±0.0002</td><td rowspan="2">0.6255±0.0013</td></tr><tr><td>InfinityGAN-flickr</td><td>0.071±0.215 0.335±0.139</td><td></td><td>0.0023±0.0001</td></tr><tr><td rowspan="9"></td><td></td><td></td><td>0.186±0.012</td><td></td><td rowspan="2"></td></tr><tr><td>InfinityGAN-tower</td><td>0.409±0.120</td><td>0.182±0.012</td><td>0.0032±0.0001</td></tr><tr><td>GMap</td><td></td><td></td><td></td></tr><tr><td>InfinityGAN-flickr InfinityGAN-tower</td><td>0.002±0.244 0.228±0.173</td><td>0.202±0.009 0.197±0.010</td><td>0.0001±0.0000 0.0001±0.0000</td><td>0.2570±0.0022 0.0001±0.0000</td></tr><tr><td rowspan="8"></table>
+
+Table 3: Significant PPP gain from model training. We measure PPP-MAE on GMap with randomly initialized and fully trained models. The results show a consistent and significant increment of PPP is developed through the model training.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Pretrained</td><td colspan="5">Padding</td></tr><tr><td>Zeros</td><td>Circular</td><td>Reflect</td><td>Replicate</td><td>Randn</td></tr><tr><td rowspan="2">VGG-19</td><td>×</td><td>0.0132±0.0006</td><td>0.0000±0.0000</td><td>0.0000±0.0000</td><td>0.0000±0.0000</td><td>0.0000±0.0000</td></tr><tr><td>ImageNet</td><td>0.0176±0.0005</td><td>0.0158±0.0006</td><td>0.0158±0.0002</td><td>0.0144±0.0009</td><td>0.0182±0.0012</td></tr><tr><td rowspan="2">ResNet50</td><td>×</td><td>0.0052±0.0004</td><td>0.0032±0.0004</td><td>0.0018±0.0001</td><td>0.0015±0.0001</td><td>0.0020±0.0002</td></tr><tr><td>ImageNet</td><td>0.0162±0.0012</td><td>0.0188±0.0016</td><td>0.0150±0.0004</td><td>0.0150±0.0004</td><td>0.0147±0.0007</td></tr></table>
+
+# 4.1 VISUALIZING POSITION-INFORMATION PATTERN FROM PADDING (PPP)
+
+We start with visualizing PPP in Figure 4. All the visualizations are conducted at the 3rd layer of interest as detailed in Appendix A. We compute PPP using Eq. 3 and $\ell _ { 1 }$ norm as the distance metric, then average the resulting PPP in the channel dimension to generate a gray-scale image. Since the quantities are small and difficult to perceive, we normalize the gray-scale image to [0, 1] range, and thus the colors between images are not directly comparable.
+
+In all scenarios, PPP noticeably spreads out after being pretrained on ImageNet. In Table 3, the PPPMAE of the VGG19 and ResNet50 also reflects that the response of PPP is significantly strengthened after model training. That is, the model training has substantial effects on the construction of PPP. Although the formation of padding pattern is suggested to be mainly caused by the distributional difference between features and paddings (Alsallakh et al., 2021a), our results show that it only increases the response slightly, compared to the considerable PPP-MAE gain through training.
+
+Another intriguing observation is that, despite some variations in the detailed patterns, the overall structure of PPP remains similar. Regardless of padding minimum values with zero-padding (consider the features are processed with ReLU activation), randn-padding that can sometimes produce large quantities by chance, or the unbalanced initial state of ResNet50 caused by strided convolution (the first row of ResNet50 in Figure 4), all models tend to have the maximal PPP response in the corner of the features after fully trained. While the underlying mechanism causing such consistent preferences remains unknown, such preferences may be an important factor to consider in future model design.
+
+# 4.2 QUANTIFYING PPP AND COMPARING WITH POSENET
+
+Table 2 shows the measurements of PPP and PosENet on various architectures and padding schemes. We train five models for each setup and measure the standard deviation of these models. Our PPP-MAE has significantly lower standard deviations compared to PosENet, where the standard deviation of PosENet dominates the differences between padding variants, and thus the quantities from PosENet cannot provide sufficient information for any analysis. Evaluating the true mean of PosENet requires an even larger number of pretrained models, each requiring full training on the target dataset (e.g., ImageNet), which is impractical in reality. The main reason that PosENet has such a large variation is due to its optimization-based formulation, and thus the final quantities highly depend on the convergence of the PosENet training. In fact, we also observe a similar level of standard deviation even when the PosENet is measured on the same model for multiple trials. On the other hand, PPP is based on a closed-form formulation, and thus the variations are only introduced by the differences among the parameters of the pretrained models. Furthermore, PosENet often reports positive SPC responses from no-padding models, as shown in its large standard deviation. In contrast, PPP has zero response to no-padding models by definition, and therefore is less biased for measuring the positional information from padding.
+
+Although certain paddings seem to have slightly lower PPP-MAE than other paddings, in Table 3, we find the differences are not significant when comparing the extremely low PPP-MAE from most of the randomly initialized networks. In most cases, the network can effectively construct its PPP, even with the highly stochastic randn padding. The only exception seems to be the case of randn padding in the salient object detection (SOD) task, where the network fails to achieve a compatible performance with other paddings1. The results show that the model training plays an important role in the formation of PPP, and perhaps its contribution is much larger than which underlying padding scheme is being used. This motivates us to further analyze the PPP formulation during model training.
+
+![](images/7f1c9e44a46e7d7615b221277f4944764af9c97f2701bfe630341a9b9c42d699.jpg)  
+Figure 5: Chronological PPP. We quantify PPP every 10 epochs and plot its development in four different layer of depth (the rightmost layer is the one closest to model output). All curves consistently show a sudden surge at the early stage, and all the later layers are slowly but steadily gaining stronger PPP until the end of training. The shadow region represents standard deviations among 5 individual training episodes. The colors represent zeros, circular, reflect, replicate, and randn paddings.
+
+# 4.3 CHRONOLOGICAL PPP
+
+To understand the formulation of PPP through time, we snapshot checkpoints every 10 epochs for all training episodes. By measuring the PPP-MAE at all the checkpoints, we plot a chronological curve and monitor the progress of PPP. We train 5 individual models for each pair of model-padding setting and report the standard deviations, which demonstrates the significance of the trend.
+
+Figure 5 shows all models achieve a significant gain of PPP within the first 10 epochs in all intermediate layers. Most models continuously increase their PPP as training proceeds, especially in the fourth layer of interest, which is the last output from the convolutional layers before the final linear projection. Another interesting observation is that our randn padding, which is designed to be less easily detectable with built-in stochasticity, indeed shows less PPP built-up at the intermediate stages in certain layers. However, the network still adjusts the behavior and ends up forming complete PPPs at the fourth layer of interest in all scenarios. All these evidences show that the network builds PPP purposely as a favorable representation to assist its learning.
+
+# 5 CONCLUSION AND LIMITATIONS
+
+In this paper, we develop a reliable method for measuring PPP and conduct a series of analyses toward understanding the formation and properties of PPP. Through a large-scale study, we demonstrate that PPP is a representation that the network favorably develops as a part of its learning process, and its formation has weak connections to the underlying padding algorithm. We show that reliable PPP metrics are important steps for understanding the effects of PPPs in different tasks, and useful for measuring the effectiveness of future methods in debiasing PPP.
+
+However, an unfortunate and inevitable limitation of the PPP metrics is that their measure is biased by the model architecture and parameters. Since the PPP metrics are based on the distributional differences between the paired model outputs (i.e., optimal padding to algorithmic padding), different architecture and layers of depth exhibit different and intractable biases due to different interactions between PPP and model parameters. Such a bias makes PPP metrics less comparable while dissecting models with different architectures or parameter distributions (e.g., weight decay and weight normalization), which is important for studying the effect of architectural changes. However, this limitation is inevitable for any (and all existing) metric that attempts to measure PPP using the outputs of a model. We note future studies in measuring PPP without model inferences2 will be an important step toward tackling and understanding the property of PPP under different architectural choices.
+
+# REFERENCES
+
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+
+# Supplementary Material
+
+APPENDIX A IMPLEMENTATION DETAILS
+
+![](images/f4be03039849096030af3e78b6d7d901fb3b3e8f012e09ff2a3e21cfa0e50704.jpg)  
+Figure 6: The architecture for VGG19 and ResNet50 used in the paper. We mark the calculation of optimal padding in orange arrows and principal point in blue arrows. We label the layers of interest that are used in the paper. The red $\dagger$ indicates where a principal point shift is identified.
+
+# A.2 PPP FEATURE MISALIGNMENT
+
+There are several pitfalls in visualizing and quantifying PPP. We identify two critical pitfalls from the architectures we implemented. However, these may not be sufficient to cover all potential issues while integrated into other architectures. Therefore one must be alerted to any unusual behavior (e.g., Figure 2(d) in the main paper) throughout their implementation.
+
+Principal point shifting. Conv2d has a hidden behavior that few people are aware of, the operation is one-pixel skewed while applying a stride-two Conv2d on even-shaped features. To understand how the one-pixel shift happens, we first define the principal point of a feature map. We first define the principal point of the last feature map as the center pixel (note that we define it as the middle-point between the center-two pixels in case the last feature size is even). Then, we recursively define the principal point of the $( N - 1 )$ -th layer as the pixel that positions at the center of the Conv2d receptive field that mainly forms the principal point of the $N$ -th layer. In the case of optimally-padded features, the principal points in every layer are the center of the feature map. But, as shown in Figure 2(a), the principal point of algorithmically-padded features will have a one-pixel shift when a stride-2 convolution is applied to even-shaped features, which can be further amplified as more layers stack up. Such a skew causes the principal points of algorithmically-padded features shift several pixels away from the principal points of optimally-padded features. As PPP metrics use pixel-wise subtraction to distinguish the image content from PPP, the misalignment becomes a critical issue, since the image contents are no longer aligned and subtractable.
+
+In Figure 6, we show the procedure of calculating the principal point in blue arrows and marking the values impacted by principal point shift with red $\dagger$ . For the ResNet50 architecture, the principal point shift accumulates to $1 6 ( = 2 \bar { 2 } 4 / 2 - 9 6 )$ pixels in the early layers.
+
+Fortunately, such a displacement can be fixed by adding corrections to how we calculate the feature margins. As shown in Figure 2(b), the concept of the margin correction is to make the two principal points overlapping each other after adding the margin. In the example, the left-right margins are corrected to (209, 180) (instead of the more intuitive choice of (195, 194) or (194.5, 194.6)).
+
+We also show how the principal point shift visually looking like in Figure 2(c), notice the patterns have right-bottom shifted 16 pixels. As shown in Figure 2(d), failing to identify the principal point shift will result in checkerboard artifacts while calculating PPP, and adding correction eliminates the artifacts.
+
+Maxpooling misalignment. This is a hypothetical condition that may potentially happen but has not been observed in the three architectures we tested. Consider a case of a Maxpooling layer of window size 2 and stride 2, the sliding windows of each pooling operation have no overlap, therefore the initial index of the first sliding window solely determines the spatial location of all sliding windows. Accordingly, there is a chance that the initial condition of the optimally-padded features causes all of its sliding windows to be one-pixel misaligned to the algorithmically-padded features. Fortunately, the condition can be easily determined by calculating the top and left margins of the feature alignment (similar to the aforementioned principal point shift calculation). For the case of a Maxpooling layer of window size 2 and stride 2, the misalignment will not happen if the top and left margins are even numbers, and that is exactly the case for VGG19 and ResNet50, as shown in Figure 6.
+
+# A.3 RANDN PADDING
+
+A critical implementation detail is that such a padding scheme must be applied before activation functions. Since the paddings are based on the distribution within sliding windows, activation functions such as ReLU, which clamps all negative values, can discard a significant amount of information beforehand. Instead of the traditional use of padding-convolution-normalization-activation, we modify the order to convolution-normalization-padding-activation. Note that such a change of order does not affect the behavior or results of other padding schemes.
+
+# A.4 ACKNOWLEDGING OPEN-SOURCE CONTRIBUTORS
+
+Our implementation reuses codes from several open-source codebases, which greatly supports our development. The repositories used in the paper are F-Conv (Oskyhn, 2019), torchvision (Pytorch, 2016) and Pytorch-cifar (Kuangliu, 2017).
+
+# APPENDIX B MORE PPP VISUALIZATIONS
+
+![](images/30d63de2ca96d08fcb9c306ea6f49dee853045f2ae15c56e1bbe54f442d042e9.jpg)  
+Figure 7: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples. The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to $[ 0 , 1 ]$ separately, therefore the colors between images are not comparable.
+
+![](images/e4cd61435c542304b5c3cdb789a139ba20d665db70a7bc8695957868fac75dd7.jpg)  
+Figure 8: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples. The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to $[ 0 , 1 ]$ separately, therefore the colors between images are not comparable.
+
+![](images/575354cc406647ad7d0a0db777cb6d300e968b30e3073fc4c6fb00e81afc740b.jpg)  
+Figure 9: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples. The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to [0, 1] separately, therefore the colors between images are not comparable.

parse/dev/QDE5hzxVpS/QDE5hzxVpS_content_list.json

@@ -0,0 +1,1302 @@
+[
+    {
+        "type": "text",
+        "text": "UNVEILING THE MASK OF POSITION-INFORMATION PATTERN THROUGH THE MIST OF IMAGE FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent studies have shown that paddings in convolutional neural networks encode absolute position information which can negatively affect the model performance for certain tasks. However, existing metrics for quantifying the strength of positional information remain unreliable and frequently lead to erroneous results. To address this issue, we propose novel metrics for measuring and visualizing the encoded positional information. We formally define the encoded information as Position-information Pattern from Padding (PPP) and conduct a series of experiments to study its properties as well as its formation. The proposed metrics measure the presence of positional information more reliably than the existing metrics based on PosENet and tests in F-Conv. We also demonstrate that for any extant (and proposed) padding schemes, PPP is primarily a learning artifact and is less dependent on the characteristics of the underlying padding schemes. ",
+        "bbox": [
+            233,
+            265,
+            766,
+            431
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            452,
+            336,
+            467
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Padding, one of the most fundamental components in neural network architectures, has received much less attention than other modules in the literature. In convolutional neural networks (CNNs), zero padding is frequently used perhaps due to its simplicity and low computational costs. This design preference remains almost unchanged in the past decade. Recent studies (Islam\\* et al., 2020; Islam et al., 2021b; Kayhan & Gemert, 2020; Innamorati et al., 2020) show that padding can implicitly provide a network model with positional information. Such positional information can cause unwanted side-effects by interfering and affecting other sources of position-sensitive cues (e.g., explicit coordinate inputs (Lin et al., 2022; Alsallakh et al., 2021a; Xu et al., 2021; Ntavelis et al., 2022; Choi et al., 2021), embeddings (Ge et al., 2022), or boundary conditions of the model (Innamorati et al., 2020; Alguacil et al., 2021; Islam et al., 2021a)). Furthermore, padding may lead to several unintended behaviors (Lin et al., 2022; Xu et al., 2021; Ntavelis et al., 2022; Choi et al., 2021), degrade model performance (Ge et al., 2022; Alguacil et al., 2021; Islam et al., 2021a), or sometimes create blind spots (Alsallakh et al., 2021a). Meanwhile, simply ignoring the padding pixels (known as no-padding or valid-padding) leads to the foveal effect (Alsallakh et al., 2021b; Luo et al., 2016) that causes a model to become less attentive to the features on the image border. These observations motivate us to thoroughly analyze the phenomenon of positional encoding including the effect of commonly used padding schemes. ",
+        "bbox": [
+            174,
+            472,
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+            707
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Conducting such a study requires reliable metrics to detect the presence of positional information introduced by padding, and more importantly, quantify its strength consistently. We observe that the existing methods for detecting and quantifying the strength of positional information yield inconsistent results. In Section 3, we revisit two closely related evaluation methods, PosENet (Islam\\* et al., 2020) and F-Conv (Kayhan & Gemert, 2020). Our extensive experiments demonstrate that (a) metrics based on PosENet are unreliable with an unacceptably high variance, and (b) the Border Handling Variants (BHV) test in F-Conv suffers from unaware confounding variables in its design, leading to unreliable test results. ",
+        "bbox": [
+            174,
+            714,
+            825,
+            825
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In addition, we observe all commonly-used padding schemes actually encode consistent patterns underneath the highly dynamic model features. However, such a pattern is rather obscure, noisy, and visually imperceptible for most paddings (except zeros-padding), which makes recognizing and analyzing it difficult. Fortunately, we show that such patterns can be consistently revealed with a sufficient number of samples by defining an optimal padding scheme (see Section 2.1 and Figure 1). ",
+        "bbox": [
+            174,
+            833,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab0eb0828e532ceff21f80209a0363116483e34653f04c92525481c6f69dd5c5.jpg",
+        "image_caption": [
+            "Figure 1: Position-information Pattern from Padding (PPP). We propose a method that can consistently and effectively extract PPPs through the distributional difference between optimallypadded (gray-scale surfaces) and algorithmically-padded features (colored surfaces). The results show that the two distributions become distinguishable as the number of sample increases. Following the procedure in Section 2.2, we extract a clear view of PPP with the expectation of the pairwise differences between optimally-padded and algorithmically-padded features. We render each visualization in tilted view (first row) and top view (second row). The colors represent the magnitude (blue/cold/weak to green/warm/strong) at each pixel. The features are extracted at the 3rd layer of interest (Appendix A) from a randn-padded (Section 2.4) ResNet50 pretrained on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            99,
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+            344
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We accordingly propose a new evaluation paradigm and develop a method to consistently detect the presence of the Position-information Pattern from Padding (PPP), which is a persistent pattern embedded in the model features to retain positional information. We present two metrics to measure the response of PPP from the signal-to-noise perspective and demonstrate its robustness and low deviation among different settings, each with multiple trials of training. ",
+        "bbox": [
+            174,
+            488,
+            825,
+            558
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To weaken the effect of PPP, in Section 2.4, we design a padding scheme with built-in stochasticity, making it difficult for the model to consistently construct such biases. However, our experiments show that the models can still circumvent the stochasticity and end up consistently constructing PPPs. These results suggest that a model likely constructs PPPs purposely to facilitate its training, rather than falsely or accidentally learning some filters that respond to padding features. ",
+        "bbox": [
+            174,
+            564,
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+            635
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With reliable PPP metrics, we conduct a series of experiments to analyze the characteristics of PPP in Section 4.1. Specifically, we analyze the formation of PPP throughout each model training process in Section 4.3. The results show PPPs are formed expeditiously at the early stage of model training, slowly but steadily strengthen through time, and eventually shaped in clear and complete patterns. These results show that a model intentionally develops and reinforces PPPs to facilitate its learning process. Moreover, we observe the PPPs of all pretrained networks are significantly stronger than those in their initial states. This indicates an unbiased training procedure is of great importance in resolving the critical failures caused by PPP in numerous vision tasks (Alsallakh et al., 2021a; Xu et al., 2021; Ge et al., 2022; Alguacil et al., 2021). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 OBSERVATIONS AND METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            776,
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+            791
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first define symbols for expressing the functionality of paddings and define the optimal-padding scheme. We then give a formal definition of Position-information Pattern from Padding (PPP) and utilize the optimal-padding scheme to develop propose a method to capture PPP and measure its response with two metrics. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 OPTIMAL PADDING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            863,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The process of capturing an image from the real world can be simplified into two steps: (a) 3D information of the environment is first projected onto an infinitely large 2D plane, and then (b) the camera determines resolution as well as field-of-view to form a digital image from such infinitely large and continuous 2D signals (Liu et al., 2019; Ravi et al., 2020). Let $S ^ { * } = \\{ s _ { n } ^ { * } \\} _ { n = 1 } ^ { N }$ be a collection of such infinitely large and continuous 2D signals, and the collection of 2D images captured by cameras at a spatial size $( h _ { n } , w _ { n } )$ be $S ^ { \\prime } = \\{ s _ { n } ^ { \\prime } \\} _ { n = 1 } ^ { \\mathbb { N } }$ . A padding scheme can be used to generate a set of algorithmically-padded images $\\hat { S } = \\{ \\hat { s } _ { n } \\} _ { n = 1 } ^ { N }$ by a padding function $\\rho$ : ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/c0a25d4970af93cf15e96a1750f2a0448e16a47e0e5db0dcf5d02831db9d75cf.jpg",
+        "image_caption": [
+            "Figure 2: Principal point shift. (a) The stride-2 Conv2d only pads on one side, causing the principal point shift (red squares) in earlier layers. (b) Such a shift requires careful margin correction while aligning algorithmically-padded and optimally-padded features (we describe the details of point shift in Appendix A). (c) The shift is visible in the feature space (marked with red and yellow boxes). (d) It is crucial to correct the principal point shift while measuring PPP. The PPP calculation involves pixel-wise distance functions, which are not robust to spatial shifts (Zhang et al., 2018). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/480c3c378a0ccf1e0a03497ea0889718df90650e03a2a7cce12b624f0164928c.jpg",
+        "text": "$$\n\\hat { s } _ { n } [ i , j ] = \\left\\{ { \\begin{array} { l l } { s _ { n } ^ { \\prime } [ i , j ] = s ^ { * } [ i , j ] } & { { \\mathrm { i f ~ } } 0 < i < h _ { n } { \\mathrm { ~ a n d ~ } } 0 < j < w _ { n } , } \\\\ { \\rho ( s _ { n } ^ { \\prime } , i , j ) } & { { \\mathrm { o t h e r w i s e } } , } \\end{array} } \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            494,
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+            530
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "collection where $i$ and $\\dot { S ^ { \\dagger } } = \\{ s _ { n } ^ { \\dagger } \\} _ { n = 1 } ^ { N }$ $j$ are indexes of a pixel in the spatial dimension. We define a theoretical optimally-padded with an optimal-padding function $\\rho ^ { \\dagger }$ by: ",
+        "bbox": [
+            173,
+            531,
+            825,
+            560
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e31e8862a4da91509a7efd1fe329c440e30140f87dd82fd6a7eee6e4a2c2f411.jpg",
+        "text": "$$\n\\begin{array} { r } { s _ { n } ^ { \\dagger } [ i , j ] = \\left\\{ { s _ { n } ^ { \\prime } [ i , j ] } \\atop { \\rho ^ { \\dagger } ( s _ { n } ^ { \\prime } , i , j ) } \\right. \\ } & { = s ^ { * } [ i , j ] \\quad \\mathrm { i f ~ } 0 < i < h _ { n } \\mathrm { ~ a n d ~ } 0 < j < w _ { n } , } \\\\ { s ^ { \\dagger } _ { n } [ i , j ] = \\left\\{ { s _ { n } ^ { \\prime } [ i , j ] } \\atop { \\rho ^ { \\dagger } ( s _ { n } ^ { \\prime } , i , j ) } \\right. \\ } & { = s ^ { * } [ i , j ] \\quad \\mathrm { o t h e r w i s e } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In practice, without curated data, the optimal-padding scheme described in Eq. 2 is difficult to achieve. We describe how we relax this constraint in Section 2.3 ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 POSITIONAL-INFORMATION PATTERN FROM PADDING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            645,
+            586,
+            660
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Despite the previous literature discovered the existence of positional information caused by the model paddings, there is still no clear definition for such information, and lacks effective metrics to detect or quantify it. Ideally, an effective metric for such positional information should have two properties. First, it is a spatial pattern, it contributes distinctive information to different spatial locations. Its shape enables the network to develop and exploit the absolute positional information of each pixel, eventually leading to the unattended and undesirable effects in certain tasks (Lin et al., 2022; Alsallakh et al., 2021a; Xu et al., 2021; Ntavelis et al., 2022; Choi et al., 2021; Ge et al., 2022; Alguacil et al., 2021). Second, as it represents the positional information purely contributed by the padding, it is a constant pattern irrelevant to the image contents. We accordingly name it the Positional-information Pattern from Padding (PPP). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, such a pattern shares space with image features, where the image features typically have very diverse appearances and high dimensionality. When these two signals interfere with each other, the appearance of PPP becomes extremely obscure and imperceptible in most cases (except zeros padding). Figure 1 shows if we visualize features sample-by-sample, there are no obvious differences between optimally-padded features (gray-scale surface) and algorithmically-padded features (colored surface). To address the issue, we show that, by assuming the interferences between PPP and image features to be random, its expectation over a large set of images will saturate to a constant bias and no longer hinder us from capturing PPP. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on these observations and assumptions, we define PPP as the constant component independent of model inputs, and its presence is completely contributed by the existence of a padding scheme $\\rho$ . Given $\\hat { S }$ and a model $F ( \\hat { s } ; \\theta , \\rho )$ , which $\\theta$ is the model parameters and $\\rho$ is a padding scheme applied to $F$ . Let the model feature extracted at $k$ -th layer be $f _ { n , k } = F _ { k } ( \\hat { s } _ { n } ; \\theta , \\rho )$ , where $F _ { k }$ is the model from the first layer to the $k$ -th layer. The PPP at $k$ -th layer $( P P P _ { k } )$ ) can be formulated by: ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/760d08a16f7d0ce0a69b595865e543bed92b15474ff36e4ee6c02d57d9ce3162.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathsf { P P P } _ { k } \\ = \\ \\underset { n } { \\mathbb { E } } \\left[ \\textit { d } \\big ( \\begin{array} { l } { F _ { k } ( s _ { n } ^ { \\dagger } ; \\theta , \\rho ^ { \\dagger } ) , F _ { k } ( \\hat { s } _ { n } ; \\theta , \\rho ) } \\end{array} \\big ) \\ \\right] \\ , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            179,
+            668,
+            203
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $d ( \\cdot , \\cdot )$ can be any distance function. We use $\\ell _ { 1 }$ distance in this work, and accordingly name the metric PPP-MAE. ",
+        "bbox": [
+            176,
+            203,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Pitfalls: feature misalignment. It is important to note that, some CNN components can cause serious feature misalignment while computing PPP and leads to erroneous results. A typical example is principal point shift, where the uneven padding in stride-2 convolution causes the center of features slightly drifted, as shown in Figure 2. Since the measurement of PPP requires perfect alignment, such a drift should be carefully considered while integrating PPP into new architectures. We discuss the issue along with other pitfalls in Appendix A and provide three detailed examples of correcting the principal point shifting. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 SIMULATED OPTIMAL PADDING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            345,
+            436,
+            359
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In practice, it is impossible to gain access to $S ^ { * }$ for calculating the optimal padding $S ^ { \\dagger }$ described in Eq. 2. But fortunately, given our goal in Eq. 3 is to analyze the model features within the $( h _ { n } , w _ { n } )$ region, $S ^ { * }$ is an overshoot of the data we actually required. Given a vision model $F ( \\hat { s } ; \\theta , \\rho )$ trained at a field-of-view $( h _ { n } , w _ { n } )$ pixels, the receptive field of such vision model is $( h _ { m } , w _ { m } )$ pixels (we show the collection Apat where  pixels, $h _ { m } \\gg h _ { n }$ and tion $w _ { m } \\gg w _ { n }$ . Let an altern field implies $S ^ { \\odot } = \\{ s _ { n } ^ { \\odot } \\} _ { n = 1 } ^ { N }$ $( h _ { m } , w _ { m } )$ $F _ { k } ( s _ { n } ^ { \\dagger } ; \\theta , \\mathbf { \\bar { \\rho } } )$ equals to $F _ { k } \\big ( s _ { n } ^ { \\odot } ; \\theta , \\rho ^ { \\dagger } \\big )$ for all $k$ . ",
+        "bbox": [
+            173,
+            364,
+            825,
+            462
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In other words, in terms of computing Eq. 3, $S ^ { \\odot }$ is equivalent to $S ^ { * }$ within the finite $( h _ { n } , w _ { n } )$ region for a given model architecture. Therefore, we can simulate the procedure described in Eq. 1 and Eq. 2 using $S ^ { \\odot }$ instead of $S ^ { \\dagger }$ , as long as $\\forall s _ { n } ^ { \\odot } \\in S ^ { \\odot }$ the spatial size of $s _ { n } ^ { \\odot }$ is strictly larger than $( h _ { m } , w _ { m } )$ . ",
+        "bbox": [
+            174,
+            468,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 RANDN PADDING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            522,
+            338,
+            536
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Most of the existing padding schemes (e.g., zeros, reflect, replicate, circular) exhibit certain consistent patterns that can be easily detected by some designed convolutional kernels. One may argue that the nature of easy detectability can be a root cause of encouraging the models to learn to rely on these obvious patterns. This motivates us to design an additional sampling-based padding scheme without any consistent patterns, namely randn (i.e., random normal) padding, which produces dynamical values from a normal distribution while following the local statistics. We first determine the maximal and minimal values of a sliding window (which can be easily achieved with max-pooling), use the average of them as a proxy mean $\\mu _ { p }$ , and use the difference between the mean and the maximal value as a proxy standard deviation $\\sigma _ { p }$ . For each padding location, we sample the padding value according to a normal distribution $\\mathcal { N } ( \\mu _ { p } , \\sigma _ { p } ^ { 2 } )$ from the nearest sliding window. We include more implementation details in Appendix A. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Aside from creating a pattern-less padding scheme with sampling, the design of randn padding is based on several factors. The sampled padding pixels are allowed to occasionally exceed the min/max bound of the sliding window. Without breaking the min/max bound can introduce detectable patterns in certain extreme cases, such as a gradient-like feature that has its maximal intensity at the top-left corner and minimal intensity at the bottom-right corner. We also design the padding scheme to follow the local distribution. The padding exhibits high entropy when the local variation is high, while degenerates to value repetition with imperceptible perturbations while padding a flat area. As such, not only do the padding pixels exhibit less pattern, but it also prevents the padding pixels from breaking the features in the border region. We later show that a model still deliberately and incredibly built up PPP over time even with such a sophisticated padding scheme. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 REVISITING PRIOR WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            851,
+            423,
+            867
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first reproduce two experiments from the prior art, which aim to assess positional information from paddings. We show several critical design issues in these experiments and discuss how these problems affect the drawn conclusions. Finally, we propose two additional experiments to quantify the amount of positional information embedded in the paddings. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/4cfa9807e245de410ece3da272a9ed896dc8e24833a5a50d8c3bbc277aeff69f.jpg",
+        "table_caption": [
+            "Table 1: Background color as a critical confounding variable in BHV test. We show that using a grey background similar to Figure 3 leads to discrepant results. The standard deviations are reported among 10 individual trials. We mark the best performance in green, and the worst two in red. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Padding</td><td rowspan=\"2\">F-Conv?</td><td colspan=\"4\">Black Background</td><td colspan=\"4\">Grey Background</td></tr><tr><td>Similar (%)</td><td>Dissimilar (%)</td><td>Diff (%)</td><td>Inconsistency (%)</td><td>Similar (%)</td><td>Dissimilar (%)</td><td>Diff (%)</td><td>Inconsistency (%)</td></tr><tr><td rowspan=\"2\">Zeros</td><td>N</td><td>99.83±0.00</td><td>3.21± 8.35</td><td>-87.68</td><td>95.81± 2.07</td><td>100.00± 0.00</td><td>4.96± 5.93</td><td>-95.04</td><td>97.85± 4.55</td></tr><tr><td>Y</td><td>89.24±0.98</td><td>89.24±0.98</td><td>0.00</td><td>18.02± 8.08</td><td>100.00±0.00</td><td>4.77± 6.52</td><td>-95.23</td><td>96.79±7.13</td></tr><tr><td rowspan=\"2\">Circular</td><td>N</td><td>80.31±3.23</td><td>80.31± 3.23</td><td>0.00</td><td>34.25± 8.32</td><td>72.75± 0.96</td><td>72.75± 0.96</td><td>0.00</td><td>26.30± 5.55</td></tr><tr><td>Y</td><td>99.20±0.23</td><td>93.14±2.88</td><td>-6.06</td><td>18.48±3.55</td><td>98.26± 0.50</td><td>92.40±4.23</td><td>-5.87</td><td>28.67±6.18</td></tr><tr><td rowspan=\"2\">Reflect</td><td>N</td><td>100.00±0.00</td><td>15.67±12.72</td><td>-84.33</td><td>91.18±13.19</td><td>100.00± 0.00</td><td>19.96±13.54</td><td>-80.04</td><td>90.33±11.95</td></tr><tr><td>Y</td><td>100.00±0.00</td><td>11.70±15.38</td><td>-88.30</td><td>97.33± 6.16</td><td>100.00±0.00</td><td>17.16±12.19</td><td>-82.84</td><td>98.13± 3.44</td></tr><tr><td rowspan=\"2\">Replicate</td><td>N</td><td>100.00±0.00</td><td>43.39±11.42</td><td>-56.61</td><td>75.32± 8.20</td><td>100.00± 0.00</td><td>33.16± 6.42</td><td>-66.83</td><td>84.09± 6.47</td></tr><tr><td>Y</td><td>98.32±0.39</td><td>93.65± 1.36</td><td>-4.67</td><td>32.60± 4.97</td><td>97.17± 0.48</td><td>94.99± 1.20</td><td>-2.18</td><td>32.15± 5.11</td></tr><tr><td rowspan=\"2\">Randn</td><td>N</td><td>100.00±0.00</td><td>10.31±12.56</td><td>-89.70</td><td>94.88± 5.55</td><td>99.97± 0.13</td><td>35.47±10.82</td><td>-64.50</td><td>83.59± 8.48</td></tr><tr><td>Y</td><td>100.00±0.00</td><td>20.80±14.15</td><td>-79.20</td><td>92.54±8.37</td><td>77.28±16.13</td><td>66.70±11.58</td><td>-10.59</td><td>45.70±20.62</td></tr><tr><td>No-pad</td><td>-</td><td>100.00±0.00</td><td>3.21± 8.35</td><td>-96.79</td><td>95.81± 2.07</td><td>100.00± 0.00</td><td>30.07± 4.06</td><td>-69.93</td><td>81.30± 2.44</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.1 POSENET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Islam et al. show zeros-padding provides CNN models positional information cues, and propose PosENet (Islam\\* et al., 2020) to quantify the amount of positional information encoded within CNN features. A PosENet experiment involves several components: a pretrained CNN model $F$ , a shallow CNN $E _ { p e m }$ (i.e., position encoding module), an image dataset $X = \\{ x _ { i } \\} _ { i = 1 } ^ { N }$ to examine, and a constant target pattern $y$ (e.g., 2D Gaussian pattern). PosENet first extracts intermediate features at $k$ -th layer with $f _ { ( i , k ) } = F _ { k } ( x _ { i } )$ using the pretrained CNN, and then optimizes $E _ { p e m }$ to minimize $\\mathbb { E } _ { i , k } [ | | E _ { p e m } ( f _ { ( i , k ) } ) - y | | _ { 2 } ]$ . Finally, the amount of positional information is quantified by the average Spearman’s correlation (SPC) and Mean Absolute Error (MAE) overall $E _ { p e m } ( f _ { ( i , k ) } )$ toward $y$ . ",
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+        "type": "text",
+        "text": "A critical issue with PosENet is the use of an optimization-based metric. It is sensitive to hyperparameters with large variation. As shown in Table 2, for all the PosENet results, the standard deviation over five trials significantly dominates the differences between different types of paddings, and thus no definitive conclusions can be drawn. We also observed that PosENet can report NaN results in certain setups. Furthermore, PosENet quantifies the amount of positional information by the faithfulness of the final reconstruction. However, a better reconstruction does not have a clear relationship to measuring the strength and significance of positional information. For instance, PosENet sometimes shows responses to no-padding models, demonstrating it is a metric with an indefinite bias pending on the memorization ability of $E _ { p e m }$ . Moreover, optimizing for pattern reconstruction is highly dependent on the underlying data distribution, simply changing the evaluation data distribution without changing the model weights can drastically change the PosENet numerical magnitudes and the conclusions of which model embeds the strongest positional information. ",
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+        "type": "text",
+        "text": "Another issue is that the no-padding scheme used in the $E _ { p e m }$ module in PosENet is known to have the foveal effect (Alsallakh et al., 2021b; Luo et al., 2016), where a model pays less attention to the information on the edge of inputs. Using such a padding scheme for detecting positional information from paddings, which is mostly concentrated on the edge of the feature maps, is less effective. This is an inevitable dilemma as PosENet aims to identify positional information from the padding of the pretrained $F$ , while applying any padding scheme to $E _ { p e m }$ introduces intractable effects between the paddings of the two models. ",
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+    {
+        "type": "text",
+        "text": "3.2 F-CONV ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Kayhan et al. propose a full-padding scheme (F-Conv) (Kayhan & Gemert, 2020) and demonstrate it is more translational invariant than the alternatives. One of the critical results is on “border handling variants” (Exp 2 of (Kayhan & Gemert, 2020)), which we call it BHV test. The BHV test creates a toy dataset, where each image has a black background with a green square and a red square in the foreground. The task is to predict if the red square is on the left of the green square (class 1), or vice versa (class 2). In addition, Kayhan et al. intentionally adds a location bias such that both squares are located in the upper half of the image for class 1, and located in the lower half of the image for class 2. During testing, a “similar test” inherits the same bias, while a “dissimilar test” exchanges the bias (i.e., both squares are in the lower half of the image for class 1). As a truly translation-invariant CNN model should not be affected by the location bias, it should focus on the relation between the red and green squares and perform similarly on both tests. Since the experimental results show that F-Conv performs best on the dissimilar test, it is concluded that F-Conv is less sensitive to the location bias. The authors also conclude the circular padding performs worse due to the behavior of wrapping the pixels to the other side of the image, which leads to confusion between two classes. ",
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+    {
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+        "img_path": "images/8eff9d62aa85c9765ac32f2d177447c962ee0762431f465e1b2d4ebf480d0341.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples extracted at the 3rd layer-of-interest (Appendix A). The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to $[ 0 , 1 ]$ separately, therefore the colors between images are not comparable. More visualizations are presented in Appendix B. "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "However, as shown in Figure 3, we find the experimental design does not consider a crucial confounding variable: the black background has a zero intensity, making zeros padding the optimal padding that perfectly follows the background distribution. In Table 1, we show that the dissimilar test is no longer in favor of F-Conv zeros after changing the background color to grey. We also show that F-Conv replicate and F-Conv circular perform best on the dissimilar test, which is different from the original observation. ",
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+    {
+        "type": "text",
+        "text": "Finally, we report an additional inconsistency rate to show that the CNN architecture used in the BHV test actually has access to the absolute position of the squares. Given a random sample in class 1, we create a trajectory of samples by simultaneously moving the two squares to the bottom of the canvas and recording the CNNmodel prediction in all intermediate states. We label a trajectory to be inconsistent if the prediction of the CNN-model switches classes at any step of the trajectory. A CNN model with no access to the absolute-position information should have all trajectories maintaining consistent predictions, with $0 \\%$ inconsistency. Table 1 shows the inconsistent ratio over 228 uniformly sampled trajectories, where all models maintain high inconsistency rates, even with a no-padding architecture. These results show that the CNN model used in the BHV test is not translation invariant. This can be attributed to that a CNN model has a large receptive field covering the whole experiment canvas, therefore capable of gradually constructing absolute coordinates for each input pixel. Note that we only show the design of the BHV test is not suitable for quantifying the amount of positional information exhibited in a CNN model. Such a conclusion does not imply that F-Conv cannot potentially improve the translation-invariant property of CNNs. ",
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+        "image_caption": [
+            "Figure 3: The BHV test trains a binary classifier to predict the relative position of the two colored squares. It hypothesizes if the padding provides no positional information, the classifier will only focus on the relative position of the two squares. (Left) The black background is a confounding variable. (Right) Zeros padding no-longer pads optimum values after changing the background color. "
+        ],
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets Since most vision models are trained on tasks for recognizing objects, an image collection containing a diverse object appearance is more suitable for the task. As mentioned in Section 2.3, evaluating PPP requires images at a large field-of-view, in practice, we collect images at $2 , 0 4 8 \\times 2 , 0 4 8$ pixels, which is larger than the receptive field of all the models we tested. Due to the constraint of large field-of-view, we compute PPP on three datasets (all at $2 , 0 4 8 \\times 2 , 0 4 8$ pixels): (a) 480 satellite images crawled from Google Map, (b) 1,024 images synthesized by InfinityGAN (Lin et al., 2022) trained with Flickr-Landscape dataset, and (c) 1,024 images synthesized by InfinityGAN trained with LSUN-Tower (Yu et al., 2015) dataset. We crop the images depending on the requested input image sizes and principal point shifts from each model (see Appendix A for details). We will release the script for collecting and composing these large images. ",
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+    {
+        "type": "table",
+        "img_path": "images/2e1d22b2738a7f0c7749f2df3579deb711a209202701e159402b2d63649b0b52.jpg",
+        "table_caption": [
+            "Table 2: Comparing PosENet and our proposed PPP metrics. Most of the PosENet results are not distinguishable due to the high standard deviations. The standard deviation is computed by five different pretrained models for each test. The performance shows the accuracy (for classification) or weighted F-measure score (for saliency object detection). We use 2D Gaussian as PosENet reconstruction pattern, and PPP-MAE is measured at the 4th layer of interest. Here, (↑) indicates a higher value corresponds to stronger positional information or better performance on the task (vice versa for (↓)). For each group of pretrained models, we label the strongest positional information response with red, and the experiments within its standard deviation range with orange. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Padding</td><td rowspan=\"2\">Eval Dataset</td><td colspan=\"2\">PosENet</td><td rowspan=\"2\">PPP-MAE(ours) (↑)</td><td rowspan=\"2\">Performance (↑)</td></tr><tr><td>SPC (↑)</td><td>MAE(↓)</td></tr><tr><td rowspan=\"10\">VGG-19</td><td>Zeros</td><td>GMap InfinityGAN-flickr</td><td>0.107±0.128 0.368±0.116</td><td>0.196±0.006 0.183±0.007</td><td>0.0176±0.0005 0.0163±0.0006</td><td rowspan=\"2\">74.0972±0.0870</td></tr><tr><td></td><td>InfinityGAN-tower GMap</td><td>0.492±0.106</td><td>0.173±0.010</td><td>0.0179±0.0001</td></tr><tr><td rowspan=\"8\">Circular</td><td></td><td>0.098±0.139</td><td>0.197±0.007</td><td>0.0158±0.0006</td><td rowspan=\"2\">74.4716±0.0863</td></tr><tr><td>InfinityGAN-flickr</td><td>0.323±0.147</td><td>0.185±0.009</td><td>0.0137±0.0004</td></tr><tr><td rowspan=\"8\">Reflect</td><td>InfinityGAN-tower</td><td>0.460±0.102</td><td>0.176±0.009</td><td>0.0184±0.0005</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>GMap InfinityGAN-flickr</td><td>0.109±0.139 0.343±0.132</td><td>0.196±0.007 0.185±0.008</td><td>0.0158±0.0002</td></tr><tr><td>InfinityGAN-tower</td><td></td><td>0.0146±0.0008 0.0168±0.0005</td><td>74.0516±0.0621</td></tr><tr><td>GMap</td><td>0.460±0.113 0.084±0.137</td><td>0.177±0.009</td><td>0.0144±0.0009 73.9964±0.1079</td></tr><tr><td>InfinityGAN-flickr</td><td>0.356±0.111</td><td>0.197±0.006 0.184±0.007</td><td>0.0128±0.0012</td></tr><tr><td></td><td></td><td>0.173±0.010</td><td></td></tr><tr><td rowspan=\"6\"></td><td rowspan=\"2\"></td><td>InfinityGAN-tower</td><td>0.498±0.111</td><td></td><td>0.0156±0.0006</td><td rowspan=\"2\"></td></tr><tr><td>GMap</td><td>0.125±0.154</td><td>0.195±0.006</td><td>0.0182±0.0012</td></tr><tr><td rowspan=\"6\">Randn</td><td>InfinityGAN-flickr</td><td>0.374±0.137</td><td>0.185±0.007</td><td>0.0167±0.0008</td><td rowspan=\"2\">73.7716±0.0758</td></tr><tr><td>InfinityGAN-tower</td><td>0.421±0.161</td><td>0.181±0.010</td><td>0.0186±0.0012</td></tr><tr><td>GMap</td><td>0.001±0.239</td><td>0.204±0.013</td><td>0.0000±0.0000</td><td></td></tr><tr><td>InfinityGAN-flickr</td><td>0.303±0.192</td><td>0.187±0.012</td><td>0.0000±0.0000</td><td>62.0396±0.0830</td></tr><tr><td rowspan=\"8\"></td><td>InfinityGAN-tower</td><td>0.516±0.139</td><td>0.172±0.014</td><td>0.0000±0.0000</td><td></td></tr><tr><td rowspan=\"8\">Zeros</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GMap InfinityGAN-flickr</td><td>0.191±0.188 0.682±0.107</td><td>0.193±0.008 0.152±0.019</td><td>0.0162±0.0012 0.0137±0.0004</td><td>75.6856±0.0924</td></tr><tr><td>InfinityGAN-tower</td><td></td><td>0.144±0.017</td><td>0.0153±0.0013</td><td></td></tr><tr><td rowspan=\"10\">Circular</td><td></td><td>0.721±0.077</td><td></td><td>0.0188±0.0016</td><td>76.1432±0.1026</td></tr><tr><td>GMap</td><td>0.398±0.115</td><td>0.197±0.007</td><td></td><td></td></tr><tr><td>InfinityGAN-flickr</td><td>0.628±0.084</td><td>0.159±0.013</td><td>0.0178±0.0005</td><td></td></tr><tr><td>InfinityGAN-tower</td><td>0.585±0.105</td><td>0.165±0.014</td><td>0.0189±0.0012</td><td></td></tr><tr><td rowspan=\"8\"></td><td rowspan=\"2\">Reflect</td><td></td><td></td><td></td><td></td><td rowspan=\"2\"></td></tr><tr><td>GMap</td><td>0.197±0.185</td><td>0.192±0.008</td><td>0.0150±0.0004</td></tr><tr><td rowspan=\"9\">Replicate</td><td>InfinityGAN-flickr</td><td>0.594±0.096</td><td>0.169±0.012</td><td>0.0134±0.0009</td><td rowspan=\"2\">75.5068±0.1213</td></tr><tr><td>InfinityGAN-tower</td><td>0.667±0.087</td><td>0.153±0.016</td><td>0.0157±0.0002</td></tr><tr><td>GMap</td><td>0.249±0.192</td><td>0.189±0.009</td><td>0.0138±0.0003</td></tr><tr><td>InfinityGAN-flickr</td><td>0.700</td><td>0.147±0.018</td><td>0.0114±0.0003</td><td>75.6122±0.0911</td></tr><tr><td rowspan=\"8\">Randn</td><td>InfinityGAN-tower</td><td>2±0.095</td><td></td><td>0.0142±0.0007</td><td></td></tr><tr><td></td><td>0.726±0.069</td><td>0.142±0.016</td><td></td><td></td></tr><tr><td>GMap</td><td>0.210±0.192</td><td>0.191±0.009</td><td>0.0147±0.0007</td><td>75.3076±0.1016</td></tr><tr><td rowspan=\"10\"></td><td>InfinityGAN-flickr</td><td>0.566±0.100</td><td>0.171±0.011</td><td>0.0122±0.0011</td><td></td></tr><tr><td>InfinityGAN-tower</td><td>0.714±0.068</td><td>0.142±0.015</td><td>0.0153±0.0004</td><td></td></tr><tr><td></td><td></td><td></td><td>0.0049±0.0001</td><td></td></tr><tr><td>GMap InfinityGAN-flickr</td><td>0.156 ±0.212</td><td>0.201 ±0.017</td><td></td><td></td></tr><tr><td rowspan=\"8\">Circular</td><td></td><td></td><td>0.184±0.012</td><td>0.0036±0.0001</td><td>0.6269±0.0015</td></tr><tr><td rowspan=\"8\"></td><td></td><td>0.365±0.140</td><td></td><td></td><td></td></tr><tr><td>InfinityGAN-tower</td><td>0.449±0.120</td><td>0.179±0.013</td><td>0.0032±0.0001</td><td></td></tr><tr><td>GMap</td><td>0.011±0.209</td><td>0.207±0.014</td><td>0.0062±0.0001</td><td>0.6260±0.0009</td></tr><tr><td>InfinityGAN-flickr InfinityGAN-tower</td><td>0.329±0.133 0.398±0.115</td><td>0.187±0.012 0.182±0.011</td><td>0.0068±0.0001 0.0050±0.0002</td><td></td></tr><tr><td rowspan=\"2\">Reflect</td><td></td><td></td><td></td><td></td></tr><tr><td>GMap</td><td>0.062±0.210 0.205±0.016</td><td></td><td>0.0053±0.0001</td></tr><tr><td rowspan=\"2\"></td><td>InfinityGAN-flickr InfinityGAN-tower</td><td>0.322±0.133 0.396±0.125</td><td>0.188±0.013 0.183±0.013</td><td>0.0030±0.0001 0.0039±0.0001</td><td rowspan=\"2\">0.6243±0.0022</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"7\"></td><td rowspan=\"2\">Replicate</td><td>GMap</td><td></td><td>0.204±0.016</td><td>0.0043±0.0002</td><td rowspan=\"2\">0.6255±0.0013</td></tr><tr><td>InfinityGAN-flickr</td><td>0.071±0.215 0.335±0.139</td><td></td><td>0.0023±0.0001</td></tr><tr><td rowspan=\"9\"></td><td></td><td></td><td>0.186±0.012</td><td></td><td rowspan=\"2\"></td></tr><tr><td>InfinityGAN-tower</td><td>0.409±0.120</td><td>0.182±0.012</td><td>0.0032±0.0001</td></tr><tr><td>GMap</td><td></td><td></td><td></td></tr><tr><td>InfinityGAN-flickr InfinityGAN-tower</td><td>0.002±0.244 0.228±0.173</td><td>0.202±0.009 0.197±0.010</td><td>0.0001±0.0000 0.0001±0.0000</td><td>0.2570±0.0022 0.0001±0.0000</td></tr><tr><td rowspan=\"8\"></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7b6056a8371977e84bd6b707fbd7722a79673293f669541fd25be299565ac57a.jpg",
+        "table_caption": [
+            "Table 3: Significant PPP gain from model training. We measure PPP-MAE on GMap with randomly initialized and fully trained models. The results show a consistent and significant increment of PPP is developed through the model training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Pretrained</td><td colspan=\"5\">Padding</td></tr><tr><td>Zeros</td><td>Circular</td><td>Reflect</td><td>Replicate</td><td>Randn</td></tr><tr><td rowspan=\"2\">VGG-19</td><td>×</td><td>0.0132±0.0006</td><td>0.0000±0.0000</td><td>0.0000±0.0000</td><td>0.0000±0.0000</td><td>0.0000±0.0000</td></tr><tr><td>ImageNet</td><td>0.0176±0.0005</td><td>0.0158±0.0006</td><td>0.0158±0.0002</td><td>0.0144±0.0009</td><td>0.0182±0.0012</td></tr><tr><td rowspan=\"2\">ResNet50</td><td>×</td><td>0.0052±0.0004</td><td>0.0032±0.0004</td><td>0.0018±0.0001</td><td>0.0015±0.0001</td><td>0.0020±0.0002</td></tr><tr><td>ImageNet</td><td>0.0162±0.0012</td><td>0.0188±0.0016</td><td>0.0150±0.0004</td><td>0.0150±0.0004</td><td>0.0147±0.0007</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 VISUALIZING POSITION-INFORMATION PATTERN FROM PADDING (PPP) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We start with visualizing PPP in Figure 4. All the visualizations are conducted at the 3rd layer of interest as detailed in Appendix A. We compute PPP using Eq. 3 and $\\ell _ { 1 }$ norm as the distance metric, then average the resulting PPP in the channel dimension to generate a gray-scale image. Since the quantities are small and difficult to perceive, we normalize the gray-scale image to [0, 1] range, and thus the colors between images are not directly comparable. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In all scenarios, PPP noticeably spreads out after being pretrained on ImageNet. In Table 3, the PPPMAE of the VGG19 and ResNet50 also reflects that the response of PPP is significantly strengthened after model training. That is, the model training has substantial effects on the construction of PPP. Although the formation of padding pattern is suggested to be mainly caused by the distributional difference between features and paddings (Alsallakh et al., 2021a), our results show that it only increases the response slightly, compared to the considerable PPP-MAE gain through training. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Another intriguing observation is that, despite some variations in the detailed patterns, the overall structure of PPP remains similar. Regardless of padding minimum values with zero-padding (consider the features are processed with ReLU activation), randn-padding that can sometimes produce large quantities by chance, or the unbalanced initial state of ResNet50 caused by strided convolution (the first row of ResNet50 in Figure 4), all models tend to have the maximal PPP response in the corner of the features after fully trained. While the underlying mechanism causing such consistent preferences remains unknown, such preferences may be an important factor to consider in future model design. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 QUANTIFYING PPP AND COMPARING WITH POSENET ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the measurements of PPP and PosENet on various architectures and padding schemes. We train five models for each setup and measure the standard deviation of these models. Our PPP-MAE has significantly lower standard deviations compared to PosENet, where the standard deviation of PosENet dominates the differences between padding variants, and thus the quantities from PosENet cannot provide sufficient information for any analysis. Evaluating the true mean of PosENet requires an even larger number of pretrained models, each requiring full training on the target dataset (e.g., ImageNet), which is impractical in reality. The main reason that PosENet has such a large variation is due to its optimization-based formulation, and thus the final quantities highly depend on the convergence of the PosENet training. In fact, we also observe a similar level of standard deviation even when the PosENet is measured on the same model for multiple trials. On the other hand, PPP is based on a closed-form formulation, and thus the variations are only introduced by the differences among the parameters of the pretrained models. Furthermore, PosENet often reports positive SPC responses from no-padding models, as shown in its large standard deviation. In contrast, PPP has zero response to no-padding models by definition, and therefore is less biased for measuring the positional information from padding. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Although certain paddings seem to have slightly lower PPP-MAE than other paddings, in Table 3, we find the differences are not significant when comparing the extremely low PPP-MAE from most of the randomly initialized networks. In most cases, the network can effectively construct its PPP, even with the highly stochastic randn padding. The only exception seems to be the case of randn padding in the salient object detection (SOD) task, where the network fails to achieve a compatible performance with other paddings1. The results show that the model training plays an important role in the formation of PPP, and perhaps its contribution is much larger than which underlying padding scheme is being used. This motivates us to further analyze the PPP formulation during model training. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/7f1c9e44a46e7d7615b221277f4944764af9c97f2701bfe630341a9b9c42d699.jpg",
+        "image_caption": [
+            "Figure 5: Chronological PPP. We quantify PPP every 10 epochs and plot its development in four different layer of depth (the rightmost layer is the one closest to model output). All curves consistently show a sudden surge at the early stage, and all the later layers are slowly but steadily gaining stronger PPP until the end of training. The shadow region represents standard deviations among 5 individual training episodes. The colors represent zeros, circular, reflect, replicate, and randn paddings. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 CHRONOLOGICAL PPP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To understand the formulation of PPP through time, we snapshot checkpoints every 10 epochs for all training episodes. By measuring the PPP-MAE at all the checkpoints, we plot a chronological curve and monitor the progress of PPP. We train 5 individual models for each pair of model-padding setting and report the standard deviations, which demonstrates the significance of the trend. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows all models achieve a significant gain of PPP within the first 10 epochs in all intermediate layers. Most models continuously increase their PPP as training proceeds, especially in the fourth layer of interest, which is the last output from the convolutional layers before the final linear projection. Another interesting observation is that our randn padding, which is designed to be less easily detectable with built-in stochasticity, indeed shows less PPP built-up at the intermediate stages in certain layers. However, the network still adjusts the behavior and ends up forming complete PPPs at the fourth layer of interest in all scenarios. All these evidences show that the network builds PPP purposely as a favorable representation to assist its learning. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we develop a reliable method for measuring PPP and conduct a series of analyses toward understanding the formation and properties of PPP. Through a large-scale study, we demonstrate that PPP is a representation that the network favorably develops as a part of its learning process, and its formation has weak connections to the underlying padding algorithm. We show that reliable PPP metrics are important steps for understanding the effects of PPPs in different tasks, and useful for measuring the effectiveness of future methods in debiasing PPP. ",
+        "bbox": [
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+            636,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "However, an unfortunate and inevitable limitation of the PPP metrics is that their measure is biased by the model architecture and parameters. Since the PPP metrics are based on the distributional differences between the paired model outputs (i.e., optimal padding to algorithmic padding), different architecture and layers of depth exhibit different and intractable biases due to different interactions between PPP and model parameters. Such a bias makes PPP metrics less comparable while dissecting models with different architectures or parameter distributions (e.g., weight decay and weight normalization), which is important for studying the effect of architectural changes. However, this limitation is inevitable for any (and all existing) metric that attempts to measure PPP using the outputs of a model. We note future studies in measuring PPP without model inferences2 will be an important step toward tackling and understanding the property of PPP under different architectural choices. ",
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+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Supplementary Material ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A IMPLEMENTATION DETAILS ",
+        "bbox": [
+            176,
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+            534,
+            183
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/f4be03039849096030af3e78b6d7d901fb3b3e8f012e09ff2a3e21cfa0e50704.jpg",
+        "image_caption": [
+            "Figure 6: The architecture for VGG19 and ResNet50 used in the paper. We mark the calculation of optimal padding in orange arrows and principal point in blue arrows. We label the layers of interest that are used in the paper. The red $\\dagger$ indicates where a principal point shift is identified. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            214,
+            813,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 PPP FEATURE MISALIGNMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            434,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "There are several pitfalls in visualizing and quantifying PPP. We identify two critical pitfalls from the architectures we implemented. However, these may not be sufficient to cover all potential issues while integrated into other architectures. Therefore one must be alerted to any unusual behavior (e.g., Figure 2(d) in the main paper) throughout their implementation. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Principal point shifting. Conv2d has a hidden behavior that few people are aware of, the operation is one-pixel skewed while applying a stride-two Conv2d on even-shaped features. To understand how the one-pixel shift happens, we first define the principal point of a feature map. We first define the principal point of the last feature map as the center pixel (note that we define it as the middle-point between the center-two pixels in case the last feature size is even). Then, we recursively define the principal point of the $( N - 1 )$ -th layer as the pixel that positions at the center of the Conv2d receptive field that mainly forms the principal point of the $N$ -th layer. In the case of optimally-padded features, the principal points in every layer are the center of the feature map. But, as shown in Figure 2(a), the principal point of algorithmically-padded features will have a one-pixel shift when a stride-2 convolution is applied to even-shaped features, which can be further amplified as more layers stack up. Such a skew causes the principal points of algorithmically-padded features shift several pixels away from the principal points of optimally-padded features. As PPP metrics use pixel-wise subtraction to distinguish the image content from PPP, the misalignment becomes a critical issue, since the image contents are no longer aligned and subtractable. ",
+        "bbox": [
+            174,
+            193,
+            825,
+            386
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 6, we show the procedure of calculating the principal point in blue arrows and marking the values impacted by principal point shift with red $\\dagger$ . For the ResNet50 architecture, the principal point shift accumulates to $1 6 ( = 2 \\bar { 2 } 4 / 2 - 9 6 )$ pixels in the early layers. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Fortunately, such a displacement can be fixed by adding corrections to how we calculate the feature margins. As shown in Figure 2(b), the concept of the margin correction is to make the two principal points overlapping each other after adding the margin. In the example, the left-right margins are corrected to (209, 180) (instead of the more intuitive choice of (195, 194) or (194.5, 194.6)). ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We also show how the principal point shift visually looking like in Figure 2(c), notice the patterns have right-bottom shifted 16 pixels. As shown in Figure 2(d), failing to identify the principal point shift will result in checkerboard artifacts while calculating PPP, and adding correction eliminates the artifacts. ",
+        "bbox": [
+            174,
+            506,
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+            560
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Maxpooling misalignment. This is a hypothetical condition that may potentially happen but has not been observed in the three architectures we tested. Consider a case of a Maxpooling layer of window size 2 and stride 2, the sliding windows of each pooling operation have no overlap, therefore the initial index of the first sliding window solely determines the spatial location of all sliding windows. Accordingly, there is a chance that the initial condition of the optimally-padded features causes all of its sliding windows to be one-pixel misaligned to the algorithmically-padded features. Fortunately, the condition can be easily determined by calculating the top and left margins of the feature alignment (similar to the aforementioned principal point shift calculation). For the case of a Maxpooling layer of window size 2 and stride 2, the misalignment will not happen if the top and left margins are even numbers, and that is exactly the case for VGG19 and ResNet50, as shown in Figure 6. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 RANDN PADDING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "A critical implementation detail is that such a padding scheme must be applied before activation functions. Since the paddings are based on the distribution within sliding windows, activation functions such as ReLU, which clamps all negative values, can discard a significant amount of information beforehand. Instead of the traditional use of padding-convolution-normalization-activation, we modify the order to convolution-normalization-padding-activation. Note that such a change of order does not affect the behavior or results of other padding schemes. ",
+        "bbox": [
+            174,
+            751,
+            825,
+            834
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4 ACKNOWLEDGING OPEN-SOURCE CONTRIBUTORS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            851,
+            570,
+            866
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Our implementation reuses codes from several open-source codebases, which greatly supports our development. The repositories used in the paper are F-Conv (Oskyhn, 2019), torchvision (Pytorch, 2016) and Pytorch-cifar (Kuangliu, 2017). ",
+        "bbox": [
+            176,
+            877,
+            823,
+            920
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B MORE PPP VISUALIZATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            544,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/30d63de2ca96d08fcb9c306ea6f49dee853045f2ae15c56e1bbe54f442d042e9.jpg",
+        "image_caption": [
+            "Figure 7: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples. The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to $[ 0 , 1 ]$ separately, therefore the colors between images are not comparable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            176,
+            802,
+            857
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/e4cd61435c542304b5c3cdb789a139ba20d665db70a7bc8695957868fac75dd7.jpg",
+        "image_caption": [
+            "Figure 8: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples. The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to $[ 0 , 1 ]$ separately, therefore the colors between images are not comparable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            92,
+            802,
+            775
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/575354cc406647ad7d0a0db777cb6d300e968b30e3073fc4c6fb00e81afc740b.jpg",
+        "image_caption": [
+            "Figure 9: Visualization of Position-Information Pattern from Padding (PPP). The visualizations are calculated based on Eq. 3 over 480 GMap samples. The results show that the pretrained model significantly reinforces PPP compared to randomly initialized networks. Note that each image is normalized to [0, 1] separately, therefore the colors between images are not comparable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            97,
+            769,
+            832
+        ],
+        "page_idx": 14
+    }
+]

parse/dev/Uy6YEI9-6v/Uy6YEI9-6v.md

@@ -0,0 +1,353 @@
+# OBJECT-CENTRIC NEURAL SCENE RENDERING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present a method for composing photorealistic scenes from captured images of objects. Our work builds upon neural radiance fields (NeRFs), which implicitly model the volumetric density and directionally-emitted radiance of a scene from a collection of images. While NeRFs synthesize realistic pictures, they only model static scenes and are closely tied to specific imaging conditions. This property makes NeRFs hard to generalize to new scenarios, including new lighting or new arrangements of objects. Instead of learning a scene radiance field as a NeRF does, we propose to learn object-centric neural scattering functions (OSFs), a representation that models per-object light transport implicitly using a lighting- and view-dependent neural network. This enables rendering scenes even when objects or lights move, without retraining. Combined with a volumetric path tracing procedure, our framework is capable of rendering light transport effects including occlusions, specularities, shadows, and indirect illumination, both within individual objects and between different objects. We evaluate OSFs on synthetic and real world datasets, and on generalizing to new scene configurations. Learning OSFs leads to photorealistic, physically-accurate renderings of multi-object scenes.
+
+# 1 INTRODUCTION
+
+Synthesizing images of dynamic scenes is an important problem in computer vision and graphics, with applications in AR/VR and robotics (Savva et al., 2019; Xia et al., 2020). For synthetic scenes, a user typically designs a set of 3D objects separately, then composes them into scenes to be rendered with specified camera, material, and lighting parameters. While this traditional graphics approach allows for flexible scene compositions, it requires detailed models of geometry, lighting, materials, and cameras, which can be difficult to obtain for real-world scenes.
+
+To render real-world scenes without computer graphics models, recent works have explored using neural implicit methods (Lombardi et al., 2019; Sitzmann et al., 2019a;b). Most notably, Mildenhall et al. (2020) proposed neural radiance fields (NeRF), which achieve photorealistic quality by implicitly modeling the volumetric density and directional emitted radiance of a scene.
+
+However, as shown in Figure 1, NeRF cannot generalize beyond the scene it was trained on, because it assumes static scenes and fixed illumination and learns a radiance field, which estimates only the resulting radiance along a ray after all light transport has occurred in a scene. Thus, for dynamic scenes where lights and objects can move, a separate NeRF-based model is needed for each new scene configuration.
+
+![](images/8ed85615e21b0b053f778c5c06f54eb1cf4ef60465fc666bac5bef5e9879567b.jpg)  
+Figure 1: (a) NeRF. (b) Our method.
+
+To address this issue, we propose Object-Centric Neural Scattering Functions (OSFs) to synthesize dynamic scenes of objects learned from 2D images (Figure 2). We represent each object as a learned 7D scattering function with inputs $( x , y , z , \phi _ { i } , \theta _ { i } , \phi _ { o } , \theta _ { o } )$ , where $( x , y , z )$ is the spatial location, $( \phi _ { i } , \theta _ { i } )$ is the incoming light direction, and $\left( \phi _ { o } , \theta _ { o } \right)$ is the outgoing light direction. The function outputs the volumetric density as well as the fraction of light arriving from direction $( \phi _ { i } , \theta _ { i } )$ that scatters in outgoing direction $\left( \phi _ { o } , \theta _ { o } \right)$ .
+
+Each OSF models all light bounces (reflections) and occlusions (shadows) within an object. Since each object’s scattering function is a radiance transfer function rather than a radiance field, it is intrinsic to the object (independent of the scene it is in) and can be reused across different object placements and lighting conditions without retraining. We emphasize that because NeRFs are radiance fields, they cannot be composed, and cannot generalize beyond one scene. In contrast, we can render infinitely many scenes. We can build a library of OSFs trained independently for different objects to be composed into scenes with different object placements, camera, and lighting.
+
+![](images/fd921d3f24289566b23c0122c43b391f84f4457c5c3d2bcc99da20f473f6d700.jpg)  
+Figure 2: We propose an object-centric neural scene representation for image synthesis. Given a scene description (a), and a repository of neural object-centric scattering functions (OSF) trained independently from images and frozen for each object (b), we can compose the objects into scenes (c), and render photorealistic images as we move lights (d), cameras (e), and/or objects (f). Our framework is capable of rendering occlusions, specularities, shadows, and indirect illumination.
+
+To model light transport between objects, we integrate our implicit object functions with volumetric path tracing. Like NeRF, we evaluate the radiance and volumetric density at 5D samples along every primary ray to the camera and composite them with an over operator. However, unlike NeRF, we estimate the radiance for each 5D sample by integrating our 7D OSF across the 2D sphere of incoming light directions. We estimate the integral with Monte Carlo path tracing (Kajiya, 1986) to reproduce shadows and indirect illumination effects.
+
+Our key idea is to decompose the rendering problem into (i) a learned component (per-object asset creation), and (ii) a non-learned component (per-scene path tracing). The learned component models intra-object light transport (e.g., bounces from the seat of a chair to the back of the chair). The non-learned component handles inter-object light transport (e.g., bounces from a wall to a chair). Together, they model the full rendering equation (Kajiya, 1986) (except for occluders or light sources that intrude the object’s convex hull (Sloan et al., 2002)). Since only the inter-object light transport changes as objects and lights move, no re-training is required for different scene arrangements. Experimental results indicate that our method is capable of rendering images with novel scene compositions and lighting conditions better than alternative learned approaches.
+
+In summary, our contributions are:
+
+1. Learning Object-Centric Neural Scattering Functions (OSFs) that model intra-object light transport implicitly using a lighting- and view-dependent neural network.   
+2. Integrating implicitly learned object scattering functions with volumetric path tracing to model inter-object light transport.   
+3. A rendering algorithm that enables rendering scenes with moving objects, lights and cameras, using implicit functions.
+
+# 2 RELATED WORK
+
+Classical object-centric representations. Factoring light transport into intra- and inter-object illumination has a long history in traditional computer graphics (Dutre et al., 2018). In most cases, the motivation is to improve rendering efficiency by approximating intra-object lighting factors with simple transfer functions (e.g., linear) for simple radiance fields (e.g., spherical harmonics) derived from from computer graphics models, as in precomputed radiance transfer (PRT) (Sloan et al., 2002), ambient occlusion (Miller, 1994), or virtual walls (Arnaldi et al., 1994). In other cases, the motivation is to insert captured, real-world radiance fields into synthetic scenes, as in Light Field Transfer (Cossairt et al., 2008). These methods generally store the radiance field for objects in a discrete representation (e.g., a sampled 2D or 4D grid). As a result, they cannot reproduce accurate inter-object light transport, especially for objects with intersecting bounding volumes. In contrast, we focus on learning radiance transfer from images in order to model complex real-world scattering accurately, and utilize volumetric rendering techniques to account for inter-object illumination.
+
+Novel view synthesis. Traditional methods for synthesizing novel views of a scene from captured images include using Structure-From-Motion (Hartley & Zisserman, 2003) and bundle adjustment (Triggs et al., 1999) to predict a sparse point cloud and camera parameters of the scene. More recently, a number of learning-based novel view synthesis methods have been presented but require 3D geometry as inputs (Hedman et al., 2018; Thies et al., 2019; Meshry et al., 2019; Aliev et al., 2020; Martin-Brualla et al., 2018). Others use multiplane images as proxies for novel view synthesis, but their viewing ranges are limited to interpolated input views (Flynn et al., 2016; Zhou et al., 2018; Srinivasan et al., 2019; Mildenhall et al., 2019). Some works represent scenes as coarse voxel grids and use a CNN-based decoder for differentiable rendering, but lack view consistency due to the use of 2D convolutional kernels (Nguyen-Phuoc et al., 2018; 2019; 2020).
+
+Recently, volume rendering approaches have been used to render scenes represented as voxel grids that are more view-consistent (Lombardi et al., 2019; Sitzmann et al., 2019a). However, the rendering resolution of these methods are limited by the time and computational complexity of discretely sampled volumes. To address this issue, Neural Radiance Fields (NeRF) (Mildenhall et al., 2020) directly optimizes a continuous radiance field representation using a multi-layer perceptron. This allows synthesizing novel views of realistic images at an unprecedented level of fidelity. To make NeRF more efficient, Neural Sparse Voxel Fields (Liu et al., 2020) have been proposed as a sparse voxel octree variant of NeRF and demonstrate the ease of composing learned NeRFs with their voxel representation. See (Dellaert & Yen-Chen, 2020) for survey. While these implicit methods produce high-quality novel views of a scene, their models assume a static scene with fixed illumination. Our method enables synthesizing dynamic scenes with novel viewpoint, lighting, and object configurations.
+
+Relighting. Learning-based methods that relight images without explicit geometric reasoning have been proposed, but lack the ability to recover hard shadows (Sun et al., 2019; Xu et al., 2018; Zhou et al., 2019). Other works use geometric representations that facilitate shadowing computation, but require 3D geometry as input (Philip et al., 2019; Zhang et al., 2021; Oechsle et al., 2020; Rematas & Ferrari, 2020). Deep Reflectance Volumes (Bi et al., 2020b) reconstructs a voxelized representation of a scene and predict per-voxel BRDFs, but the fixed resolution of voxel grids limits the quality in the rendered images. Similarly, Neural Reflectance Fields (Bi et al., 2020a) predicts the parameters of a BRDF model, but demonstrate higher fidelity rendering by learning a continuous scene representation. However, Neural Reflectance Fields focuses on relighting single objects, and requires manual specification of the BRDF model. Parametric BRDF models are unable to handle complex scattering functions, including real-world scattering phenomena that are difficult to model. In contrast, our method is capable of learning all scattering functions, and can render multiple objects in dynamic scenes.
+
+# 3 PRELIMINARIES
+
+# 3.1 VOLUME RENDERING
+
+To render an image of a scene with arbitrary camera parameters, camera rays are sent into the scene, through each pixel on the image plane. The expected color of each pixel is computed as the radiance along each camera ray.
+
+Volume rendering is an approach for computing the radiance traveling along rays traced in a volume. Let ${ \pmb r } ( t ) = { \pmb x } _ { 0 } + \omega _ { o } t$ be a point along a ray $\mathbfit { \Delta } \mathbf { r }$ with origin $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ and direction $\omega _ { o }$ , where $t \in \mathbb { R }$ is a 1D location along the ray, and the $^ o$ in $\omega _ { o }$ denotes “outgoing” direction. For our purposes, we assume non-emissive and non-absorptive volumes. From Novak et al. (2018), the volume rendering ´ equation to compute the radiance $L ( \boldsymbol { x } _ { 0 } , \omega _ { o } )$ of the ray is defined as:
+
+$$
+L ( { \boldsymbol { x } } _ { 0 } , \omega _ { o } ) = \int _ { t _ { n } } ^ { t _ { f } } { \boldsymbol { \tau } } ( t ) { \boldsymbol { \sigma } } ( r ( t ) ) L _ { s } ( r ( t ) , \omega _ { o } ) d t , \quad { \mathrm { w h e r e } } \quad { \boldsymbol { \tau } } ( t ) = \exp { \left( - \int _ { t _ { n } } ^ { t } { \boldsymbol { \sigma } } ( r ( u ) ) d u \right) } ,
+$$
+
+where $t _ { n }$ and $t _ { f }$ are near and far integration bounds, $\sigma ( \pmb { r } ( t ) )$ denotes the volume density of point $\mathbf { } _ { \pmb { r } ( t ) }$ , and $\tau ( t )$ denotes the accumulated transmittance from $t _ { n }$ to $t$ . The term $L _ { s } ( \pmb { r } ( t ) , \pmb { \omega } _ { o } )$ is the light scattered at point $\mathbf { } _ { \mathbf { } } ^ { r ( t ) }$ along direction $\omega _ { o }$ , defined as the integral over all incoming light directions:
+
+$$
+L _ { s } ( \pmb { x } , \omega _ { o } ) = \int _ { S } L ( \pmb { x } , \omega _ { l } ) f _ { p } ( \pmb { x } , \omega _ { l } , \omega _ { o } ) d \omega _ { l } ,
+$$
+
+where $s$ is a unit sphere and $f _ { p }$ is a phase function that evaluates the fraction of light incoming from direction $\omega _ { l }$ at a point $_ { \textbf { \em x } }$ that scatters out in direction $\omega _ { o }$ . In NeRF, Mildenhall et al. (2020) assume
+
+fixed illumination and do not consider any form of Equation 2. We consider a more general form of the volume rendering equation that explicitly models light paths within and between objects. This is important for dynamic scenes, where lighting and objects can move with respect to one another.
+
+# 3.2 RAY MARCHING
+
+The continuous integrals in Equation 1 can be estimated with quadrature (Kniss et al., 2003; Max, 1995), as done in NeRF (Mildenhall et al., 2020). For each ray, stratified sampling is used to obtain $N$ samples $\{ t _ { i } \} _ { i = 1 } ^ { N }$ along the ray, where $t _ { i } \in [ t _ { n } , t _ { f } ]$ . The rendering equation is approximated by:
+
+$$
+L ( \boldsymbol { x } _ { 0 } , \omega _ { o } ) = \sum _ { i = 1 } ^ { N } \tau _ { i } \alpha _ { i } L _ { s } ( \boldsymbol { x } _ { i } , \omega _ { o } ) \quad \mathrm { w h e r e } \quad L _ { s } ( \boldsymbol { x } _ { i } , \omega _ { o } ) = \frac { 1 } { | \mathcal { L } | } \sum _ { l \in \mathcal { L } } L ( \boldsymbol { x } _ { i } , \omega _ { l } ) \rho _ { i } ^ { l } ,
+$$
+
+where $\begin{array} { r } { \tau _ { i } = \prod _ { j = 1 } ^ { i - 1 } ( 1 - \alpha _ { j } ) } \end{array}$ and $\alpha _ { i } = 1 - e ^ { - \sigma _ { i } \left( t _ { i + 1 } - t _ { i } \right) }$ . To compute the average over incoming light paths $L _ { s }$ , we discretize over the domain $s$ in Equation 2 by sampling a set of incoming light paths $\mathcal { L } = \{ l _ { 1 } , \ldots , l _ { K } \}$ , where $\pmb { \rho } _ { i } ^ { l } = f _ { p } ( \pmb { x } _ { i } , \omega _ { l } , \omega _ { o } ) \bar { \ } \in \ [ 0 , 1 ]$ , the fraction of light incoming from light path $\imath$ that is scattered in direction $\omega _ { o }$ .
+
+# 3.3 NEURAL RADIANCE FIELDS
+
+NeRF represents a continuous scene as a volumetric radiance field, approximated with a multilayer perceptron $F _ { \Theta }$ . The model $F _ { \Theta }$ takes spatial location $\pmb { x } = ( x , y , z )$ and viewing direction $\pmb { d } = ( \phi , \theta )$ as input, and outputs the density $\sigma$ and color $\boldsymbol { \mathbf { \mathit { c } } } = ( r , g , b )$ , where $r , g , b \in [ 0 , 1 ]$ . Frequency-based positional encoding (Rahaman et al., 2019; Vaswani et al., 2017) is applied to the inputs to better capture high-frequency variation in appearance and geometry.
+
+A hierarchical volume sampling procedure (Mildenhall et al., 2020; Levoy, 1990) is then employed to more efficiently allocate samples along each ray. This technique biases sample allocation to favor the visible parts of the scene that contribute the most to the final render, avoiding occluded or free space in the scene. NeRF simultaneously optimizes two radiance fields, where the sample weights $\tau _ { i } \cdot \alpha _ { i }$ from a coarse model are used to bias samples for a fine model. The $L _ { 2 }$ loss is used to optimize both models: $\begin{array} { r } { \sum _ { \pmb { r } \in \mathcal { R } } \| \hat { C } _ { c } ( \pmb { r } ) - C ( \pmb { r } ) \| _ { 2 } ^ { 2 } + \| \hat { C } _ { f } ( \pmb { r } ) - \check { C } ( \pmb { r } ) \| _ { 2 } ^ { 2 } } \end{array}$ , where $\mathcal { R }$ is the set of all camera rays, $\widehat { C } _ { c } ( \pmb { r } )$ and $\widehat { C } _ { f } ( \boldsymbol { r } )$ denote the radiance along ray $\mathbfit { \Delta } \mathbf { r }$ predicted by the coarse and fine models respectively, and $C ( \boldsymbol { r } )$ is the ground truth pixel color for $\pmb { r }$ .
+
+# 4 METHOD
+
+# 4.1 OBJECT-CENTRIC NEURAL SCATTERING FUNCTION
+
+We represent each object as a 7D object-centric neural scattering function (OSF), depicted in Figure 3a. For each object, we learn an implicit function $F _ { \Theta } \colon ( \pmb { x } , \omega _ { l } , \omega _ { o } )  ( \sigma , \pmb { \rho } )$ that receives a 3D point in the object coordinate frame, the incoming light direction, and the outgoing light direction, and predicts the volumetric density as well as fraction of incoming light that is scattered in the outgoing direction. $\Theta$ are learned weights that parameterize the neural network, $\pmb { x } = ( x , y , z )$ denotes the spatial location, $\omega _ { l } = \left( \phi _ { l } , \theta _ { l } \right)$ denotes the incoming light direction, $\omega _ { o } = ( \phi _ { o } , \theta _ { o } )$ denotes the outgoing light direction, $\sigma$ denotes the volumetric density, and $\rho = ( \rho _ { r } , \rho _ { g } , \rho _ { b } )$ denotes the fraction of light arriving at $_ { \textbf { \em x } }$ from direction $\omega _ { l }$ that is scattered and leaving in direction $\omega _ { o }$ . The final color of a point $_ { \textbf { \em x } }$ is the integral of $\rho$ multiplied by the incoming radiance over all incoming light directions in unit sphere $s$ (Equation 2). Following NeRF, we similarly apply positional encoding to our inputs $( \pmb { x } , \omega _ { l } , \omega _ { o } )$ and employ a hierarchical sampling procedure to recover higher quality appearance and geometry of learned objects.
+
+During training, we assume a single point light source with radiance of $( 1 , 1 , 1 )$ . This simplifies $L _ { s }$ from Equation 2 to $L _ { s } ( \pmb { x } , \omega _ { o } ) = L ( \pmb { x } , \omega _ { l } ) f _ { p } ( \pmb { x } , \omega _ { l } , \omega _ { o } ) = f _ { p } ( \pmb { x } , \omega _ { l } , \omega _ { o } )$ . To learn per-object NeRFs independent of object rotation and translation, the inputs to $F _ { \Theta }$ must be in the object’s canonical coordinate frame. Given a object transformation $T _ { i }$ for object $\mathbf { o } _ { i }$ , we apply $T _ { i } ^ { - 1 }$ to $( r , \omega _ { l } , \omega _ { o } )$ before feeding the inputs to the network.
+
+# 4.2 RENDERING MULTIPLE OSFS
+
+Once we have learned an OSF for each object, we aim at composing the learned objects into scenes.   
+An overview of our procedure is visually depicted in Figure 3b.
+
+![](images/66fc5a607c6b90609b949b4ac6d774a80f2f4bfc8041fa317f228a4221e631ec.jpg)
+
+(a) We represent each object as an object-centric neural scattering function (OSF), which models how light entering at a point $_ { \textbf { \em x } }$ on the object, from direction $\omega _ { l }$ where $\imath$ corresponds to a light path, undergoes multiple bounces within the object and exits along direction $\omega _ { o }$ with some fractional amount of light $\pmb { \rho }$ . We approximate the scattering function with a multilayer perceptron $F _ { \Theta }$ where $\Theta$ are learned weights that parameterize the neural network. Given a single point $_ { \textbf { \em x } }$ , an incoming light direction $\omega _ { l }$ , and an outgoing direction $\omega _ { o }$ , $F _ { \Theta }$ outputs the volume density $\sigma$ of that point, as well as the fraction of light arriving at $_ { \textbf { \em x } }$ from direction $\omega _ { l }$ that is scattered in direction $\omega _ { o }$ .
+
+(b) Our procedure for rendering an arbitrary scene consisting of multiple objects, light sources, and cameras. Given a set of objects, we compute direct illumination by shooting rays from each light source to each object (brown arrows). Shadows are computed by sending shadow rays back to each light source (purple arrow). The shadow ray from the desk is occluded by the mug, so the mug casts a shadow on the desk. We send secondary rays between objects to render indirect illumination effects, such as between the desk and the kettle (green and blue dashed arrows). Finally, rays are sent back to the camera to render the final image (dark blue arrows).
+
+![](images/03f3183f13c1fafe38d219921c954792e1c1c0cb4fd4256f42d25dcdb5dad07f.jpg)  
+Figure 3: Using our method (OSFs) to render: (a) single and (b) multiple objects.   
+Figure 4: Sampling procedure. (a) Scene with a camera, light source, and object bounding boxes. Primary rays are sent from the camera into the scene. Rays that do not intersect with objects are pruned. Of the intersecting rays, we sample points within intersecting regions. (b) Shadow rays from each sample are sent to the light source, and samples within intersecting regions are evaluated.
+
+Let $\mathcal { O } = \{ o _ { i } \} _ { i = 1 } ^ { N }$ be a set of $N$ objects we wish to render. For simplicity, we first describe the rendering process for each object $\mathbf { o } _ { i }$ , then explain the process to combine results across all objects to render the final scene. Let $\mathbf { } o _ { i } \in \mathcal { O }$ denote object $i$ with transformation $T _ { i } \in \mathbb { R } ^ { 4 \times 4 }$ and bounding box dimensions $D _ { i } \in \mathbb { R } ^ { 3 }$ . Further let $\mathbfit { \Delta } \mathbf { r }$ be a camera ray with origin $\boldsymbol { c } \in \mathbb { R } ^ { 3 }$ and direction $\omega _ { o } \in \mathbb { R } ^ { 3 }$ , which we define with parameters $\gamma = [ c , \omega _ { o } ] \in \mathbb { R } ^ { 6 }$ . Our goal is to compute $L ( c , \omega _ { o } )$ as described in Equation 3. We compute the ray-box intersection between the ray and the object to obtain near bound $t _ { n } ^ { i }$ and far bound $\bar { t } _ { f } ^ { i }$ such that $\textstyle r ( t _ { n } ^ { i } )$ and ${ \pmb r } ( t _ { f } ^ { i } )$ each intersect a box plane, as shown in Figure 4. Note thabetween ys thand do not intealong ray ect withto obtai $\mathbf { o } _ { i }$ are excl sample putatio, where $M$ pointsiven a $t _ { n } ^ { i }$ $t _ { f } ^ { i }$ $\mathbfit { \Delta } \mathbf { r }$ $X ^ { i } = \{ x _ { m } ^ { i } \} _ { m = 1 } ^ { M }$ $\pmb { X } ^ { i } \in \mathbb { R } ^ { M \times 3 }$ light source $\imath$ , we evaluate the object’s model $F _ { \Theta _ { i } } ( X ^ { i } , \omega _ { l } , \omega _ { o } )$ to obtain alpha values $\pmb { \alpha } ^ { i } \in \mathbb { R } ^ { M }$ and phase function values $\pmb { \rho } ^ { i } \in \mathbb { R } ^ { M \times 3 }$ .
+
+It is not always possible for a light ray from light source $\imath$ to reach the object $\mathbf { o } _ { i }$ . Any of the other objects in $\mathcal { O } ^ { \prime } = \{ o _ { j } \in \mathcal { O } \mid j \neq i \}$ in the scene may occlude the incoming light, casting a shadow on object $\mathbf { o } _ { i }$ . We compute shadows by sending a shadow ray $\boldsymbol { r } _ { m }$ from each of the $M$ samples in $X ^ { i }$ to the light source $\imath$ . Evaluating the shadow ray enables us to determine the amount of light blocked along the ray by other objects. We define the parameters of the M shadow rays as Γ ∈ RM×6.
+
+For each object $o _ { j } \in \mathcal { O } ^ { \prime }$ , we compute ray-box intersections between shadow rays $\Gamma$ and $o _ { j }$ ’s bounding box. This allows us to compute the amount of light traveling towards $\mathbf { o } _ { i }$ that is blocked by $o _ { j }$ . Similar to primary rays, we sample $M$ points along each shadow ray to obtain a set of points $\breve { \pmb { X } } ^ { j } \in \mathbb { R } ^ { M \times M }$ . We then evaluate the object model $F _ { \Theta _ { j } } ( \mathbf { X } ^ { j } )$ to obtain alpha values $\pmb { A } ^ { j } \in \mathbb { R } ^ { M \times M }$ . For each shadow ray $\boldsymbol { r _ { m } }$ , we combine samples $A _ { m } ^ { j }$ across the $N - 1$ objects in $\mathcal { O } ^ { \prime }$ by sorting according to sample distance to obtain alpha values $\pmb { A } _ { m } \in \mathbb { R } ^ { M ( N - 1 ) }$ . The fraction of unobstructed light traveling along the shadow ray $\mathbf { \Delta } _ { \mathbf { r } _ { m } }$ is computed as the transmittance:
+
+$$
+\tau _ { m } ^ { l } = \prod _ { n = 1 } ^ { M ( N - 1 ) } ( 1 - A _ { m n } ) .
+$$
+
+Thus, the adjusted incoming radiance from light source $\imath$ when accounting for occlusions is computed as $L _ { l } ( \mathbf { \bar { x } } _ { m } , \omega _ { l } ) = \tau _ { m } ^ { l } \bar { L } _ { l } ( \mathbf { x } _ { m } , \omega _ { l } )$ .
+
+We follow the scattering equation in Equation 2 and now consider all incoming light directions over the unit sphere $s$ . This accounts for secondary light rays traveling to an object $\mathbf { o } _ { i }$ indirectly from another object ${ \pmb O } _ { j }$ (indirect illumination). We approximate the integral over the unit sphere $s$ by sampling $K$ directions on the unit sphere uniformly at random. For each direction $\omega _ { k }$ randomly sampled for a point $_ { \textbf { \em x } }$ , we send a secondary ray $\mathbf { \nabla } r _ { k }$ from $_ { \textbf { \em x } }$ in direction $\omega _ { k }$ and evaluate the radiance $L ( x , \omega _ { k } )$ traveling along the ray. To compute the radiance of the secondary ray $L ( x , \omega _ { k } )$ , we employ the same technique used to compute the radiance of a primary ray $L ( c , \omega _ { o } )$ (described at the beginning of Section 4.2). The incoming radiance $L ( x , \omega _ { k } )$ is multiplied with the phase function value $\rho = f _ { p } ( { \pmb x } , \omega _ { { \pmb k } } , \omega _ { o } )$ to determine the outgoing radiance $\dot { L } ( { \pmb x } , { \pmb \omega } _ { o } )$ , where $\rho$ is evaluated using $F _ { \Theta _ { i } }$ . Note that this is possible due to the recursive nature of our formulation. Only secondary rays are described here (two bounces), but our method supports an arbitrary number of bounces.
+
+Rendering. We sample and evaluate all objects in $\mathcal { O }$ to obtain alpha values $\{ \alpha ^ { i } \} _ { i = 1 } ^ { N }$ and phase function values $\{ \rho ^ { i } \} _ { i = 1 } ^ { N }$ for a set of sampled points $\{ X ^ { i } \} _ { i = 1 } ^ { N }$ along ray $\pmb { r }$ . We sort the samples across all objects to produce a final set of $P = M \cdot N$ samples $\{ \pmb { x } _ { m } \} _ { m = 1 } ^ { P }$ , $\lbrace \alpha _ { m } \rbrace _ { m = 1 } ^ { P }$ , and $\{ \rho _ { m } \} _ { m = 1 } ^ { P }$ .
+
+Given light paths $\mathcal { L }$ containing both direct and indirect illumination, we render the final radiance of a ray with origin $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ and direction $\omega _ { o }$ with the following equation:
+
+$$
+L ( \boldsymbol { x } _ { 0 } , \omega _ { o } ) = \frac { 1 } { | \mathcal { L } | } \sum _ { l \in \mathcal { L } } \sum _ { m = 1 } ^ { P } \alpha _ { m } \rho _ { m } ^ { l } \tau _ { m } L _ { l } ( \boldsymbol { x } _ { m } , \omega _ { l } ) , \quad \mathrm { w h e r e } \quad \tau _ { m } = \prod _ { n = 1 } ^ { m - 1 } ( 1 - \alpha _ { n } ) ,
+$$
+
+and $L _ { l } ( \pmb { x } _ { m } , \pmb { \omega } _ { l } )$ is the radiance from light path $\imath$ arriving at point ${ \pmb x } _ { m }$
+
+Runtime. In total, the cost of rendering a single image with $N _ { \mathrm { p i x e l } }$ pixels and $N _ { \mathrm { o b j e c t } }$ objects is $\mathcal { O } ( P ^ { 2 } K N _ { \mathrm { p i x e l } } )$ . Note that $P$ is an upper bound on number of samples that need to be evaluated. In practice, a single ray often only intersects with at most one object in the scene, which means that the proposed rendering procedure is not significantly more expensive than the single object setting. We also note that compared to NRF Bi et al. (2020a) or traditional volumetric path tracing methods, OSF crucially does not require running path tracing within each object to simulate intra-object light bounces. This is because OSF learns the object-level scattering function that directly predicts the effects after all light bounces (reflections) and occlusions (shadows) within an object have occurred. Thus OSF is significantly faster than NRF which relies on simulating intra-object light bounces while querying its learned BRDF model.
+
+In our experiments, rendering a single image with a single OSF at a resolution of $2 5 6 \times 2 5 6$ takes roughly 3.7 seconds. While the computation cost is high, there are efforts to reduce the rendering speed of NeRF that are orthogonal to this work. For instance, KiloNeRF (Reiser et al., 2021) can easily adapted to this work by utilizing thousands of tiny MLPs instead of one single large MLP to represent each OSF to obtain 1-2 orders of magnitude speed up.
+
+# 5 EXPERIMENTS
+
+Datasets and evaluation metrics. We evaluate our approach on several image datasets:
+
+• FURNITURE-SINGLE: 15 objects rendered with random object pose, point light, and viewpoint.
+
+• FURNITURE-RANDOM: 25 dynamic scenes, each containing a random layout of multiple objects, point light, and viewpoint.   
+• FURNITURE-REALISTIC: Scenes containing realistic arrangements of objects in rooms.   
+• REAL-NRF: Real-world objects from Bi et al. (2020a), captured in a dark room under varying viewing and lighting directions.   
+• REAL-OUTDOOR: Real-world outdoor scenes from Mildenhall et al. (2020).
+
+For FURNITURE datasets, we use Blender’s Cycles path tracer (Blender Foundation, 1994) to render images at $2 5 6 \times 2 5 6$ resolution for different object arrangements, camera views, and lighting configurations. We report PSNR, SSIM (Wang et al., 2003), and LPIPS (Zhang et al., 2018) metrics.
+
+Baselines and ablations. We compare our method to the following baselines:
+
+1. o-NeRF: A variant of the NeRF model, but with one NeRF trained per object. When o-NeRFs are composed into scenes, they are rendered separately.   
+2. $\mathbf { 0 . N e R F + S }$ : An extension of o-NeRF with inter-object shadows; reduces the light arriving at each o-NeRF by the cumulative opacity of shadowing objects along the ray from the light (§4.2).
+
+These baselines represent what could be achieved by combining separately trained NeRFs into a scene. Of course, since o-NeRFs produce radiance fields (not scattering fields), we do not expect them to perform well in novel lighting environments or object placements.
+
+# 5.1 NOVEL LIGHTING
+
+In the first experiment, we investigate how OSF method handles novel lighting conditions.
+
+We train one model per object in FURNITURESINGLE. For each object model, we train on 400 images with randomized viewpoint and lighting, and test on 20 images of novel viewpoint and lighting. As can be seen in Figure 5, our method produces more accurate appearance of the objects in comparison to oNeRF when tested on novel illumination conditions. In particular, o-NeRF fails to predict self-shadows for the couch and chair correctly. Additionally, o-NeRF fails to disentangle viewpoint versus lighting-dependent appearance, producing incorrect shadows for the couch and chair, and fails to capture the specular details of the ottoman. Quantitative results can be found in Table 1.
+
+![](images/9b61f34eebb13a4a67b135de352952372729b021c23f20c296231ac3433af6da.jpg)  
+Figure 5: Novel lighting results.
+
+# 5.2 SCENE COMPOSITION
+
+In a second experiment, we conduct a scene composition task on FURNITURE-RANDOM, where multiple object models are combined into scenes in random pose, lighting, and viewpoint configurations. For this task, we use the same object models trained in Section 5.1. Results are shown in Table 1 and Figure 6. While not shown in the main text, results for FURNITURE-REALISTIC can be found in Appendix B.
+
+Table 1: Quantitative results for novel lighting (FURNITURE-SINGLE) and scene composition (FURNITURE-RANDOM). Rows denote different methods: our full model (OSF), a variant of NeRF where one NeRF is trained per object (o-NeRF), and o-NeRF with shadows $\left( { \bf { o - N e R F + S } } \right)$ ).   
+
+<table><tr><td>Dataset</td><td colspan="3">FURNITURE-SINGLE</td><td colspan="3">FURNITURE-RANDOM</td></tr><tr><td>Method</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>0-NeRF</td><td>33.22</td><td>0.980</td><td>0.021</td><td>12.17</td><td>0.690</td><td>0.280</td></tr><tr><td>0-NERF + S</td><td></td><td></td><td></td><td>14.70</td><td>0.697</td><td>0.267</td></tr><tr><td>OSF (Our Method)</td><td>44.07</td><td>0.998</td><td>0.002</td><td>19.02</td><td>0.793</td><td>0.135</td></tr></table>
+
+![](images/6fd2bebddfb99194e391f1326f1944add318e88954ead332f48bd7426a5bea82.jpg)  
+Figure 6: Scene composition results on FURNITURE-RANDOM. The models OSF, o-NeRF, and $_ { 0 - \mathrm { N e R F } + \mathrm { ~ S ~ } }$ are explained in $\ S 5$ . Compared to o-NeRF, our model (OSF) is able to disentangle lighting-dependent and view-dependent appearance and can render shadows.
+
+These results suggest that OSF outperforms all baselines and ablations, both quantitatively and qualitatively. As in the previous experiment (Section 5.1), we find that OSF reproduces object appearances and self-shadows more accurately than the baselines. The difference is especially apparent in the couches in scenes (a) and (b), where the couches predicted by o-NeRF are extremely dark. This is due to the fact that o-NeRF is unable to disentangle view-dependence appearance from lightdependent appearance, and simply interpolates the radiance field learned another different lighting configuration. Please note that OSF is able to model inter-object light transport effects by rendering shadows cast by one object onto another and on the ground plane. Plus, it is able to render indirect illumination of one object reflecting light onto another. For example, light reflected from the left wall causes the left of the couch and table in scenes (a) and (b) to be brighter. Neither of these lighting effects are present in the o-NeRF results.
+
+# 5.3 REAL-WORLD SCENES
+
+In this section we evaluate our method on real world objects and scenes from the REAL-NRF and REAL-OUTDOOR datasets. For these experiments, we train one OSF for each object in REAL-NRF and each scene in REALOUTDOOR.
+
+Figure 7 shows a comparison between ground truth, our method (OSF), and Neural Reflectance Fields (NRF) (Bi et al., 2020a). We show that OSF recovers stronger, more accurate specular highlights compared to NRF. OSF also produces more detailed appearances (see pony logo). This comparison demonstrates the main advantage of OSF: the ability to handle complex scattering functions.
+
+![](images/826048519b3a9c412be8dd529a100223ecefb879955bd4c18a74c8b6f81e36fc.jpg)  
+Figure 7: Comparison of OSF (ours) to Neural Reflectance Fields (NRF) (Bi et al., 2020a). OSF produces stronger, more accurate specular highlights on the legs (see zoomed view) and recovers more detailed appearances (see pony logo).
+
+For scene composition, the OSFs trained on each object are composed with a synthetic floor OSF in Figure 8 row (a). Our method is able to compute accurate shadows, such as the shadow cast by the pony onto the two other objects in the scene. The indirect reflections from the floor allow the shadowed objects to be slightly visible as shown in the “OSF” panel.
+
+Figure 8 rows (b) and (c) show results on inserting REAL-NRF objects into real outdoor scenes (REAL-OUTDOOR). Shadows and reflections are rendered with randomized lighting directions to approximate the environment lighting. Our method accurately renders occlusions between the inserted objects and the vase in Figure 8 row (c). Due to the compositional nature of OSFs, we are able to insert the learned pinecone from Figure 8 (b) into (c).
+
+In Figure 8, each column shows ablated versions of OSF to study the impact of computing shadows and indirect illumination with our path tracing algorithm. “No Shadows, No Indirect” represents a version of our model containing only direct illumination (without modeling inter-object lighting effects). We additionally show “No Indirect” and “Indirect Only” variants of our model which represent computing shadows and indirect illumination, respectively. As illustrated by Figure 8, our full model containing both shadows and indirect illumination effects is the most realistic. Additional results on real-world scenes, including complex shadows, can be found in Appendix A.
+
+![](images/6224a6f879de9a4e63c61617266d4a0c74591e17b19dbbdd9fbe529892d9da8a.jpg)  
+Figure 8: Real-world results. NRF (Bi et al., 2020a) and NeRF (Mildenhall et al., 2020) learn on individual static scenes or objects. In contrast, we compose real-world objects and scenes using OSFs. The objects are composed with a (a) synthetic floor and (b, c) real outdoor scenes from REALOUTDOOR. Columns show different ablated versions of our model: “No Shadows, No Indirect” which considers only direct illumination; “No Indirect” which includes both direct illumination and shadows; “Indirect Only” which considers only indirect illumination. Our OSFs show the most realistic renderings, with accurate shadows (e.g., pony shadowing the two other objects (row a) and indirect illumination (i.e., the ground and environment illuminating the objects).
+
+# 6 DISCUSSION
+
+We have proposed Object-Centric Neural Scattering Functions (OSFs), a method that enables composing objects captured only from photographs into photorealistic renderings of dynamic scenes. We demonstrated that decomposing a scene into implicit object functions that are view- and lightdependent enables reusabiliy of objects across scenes where objects, camera, and lighting can change. We presented a method for integrating our learned implicit functions with volumetric path tracing, and showed inter-object light transport effects such as shadow and indirect illumination for real-world objects where no computer graphics model is available. We believe our work is a step towards a graphics pipeline where real-world scenes are modeled by a composition of implicit functions to combine the flexibility of object-centric neural modeling with the photorealism of graphics rendering algorithms.
+
+There are a few main limitations to OSF. First, the computational complexity of our method is high, but there are several works tackling the orthogonal issue of improving NeRF efficiency (as discussed in Section 4.2) that can easily be applied to OSFs. Second, while learning intra-object light transport means that intra-object path tracing is not needed, this formulation assumes that at test time, there are no occluders or light sources that intrude the object’s convex hull (Sloan et al., 2002) (e.g., a person sitting in a chair). However, OSFs can still be rendered even if their bounding boxes are intersecting, as long as this assumption is not violated. Finally, acquiring datasets of real world objects with varying point light sources and viewpoints is challenging, but we hope that in the future such acquisition of real world datasets will become easier to capture and more widely available.
+
+# REPRODUCIBILITY STATEMENT
+
+We describe our method (Section 4) and experimental setup (Section 5) in detail to maximize reproducibility. We will release our code upon publication to facilitate future research.
+
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+
+# A REAL-WORLD SCENE COMPOSITION
+
+Different scene configurations of composed objects from REAL-NRF are shown in Figure 9. We show the effect of moving the light, camera, or objects. Notice how the the appearance and shadows of the objects are updated across different scene configurations. Also notice that even when parts of the palm tree object and the cartoon object are cast under the pony’s shadow, they do not appear completely dark due to the indirect illumination from the floor.
+
+Analyzing the effect of different numbers of indirect (secondary) rays per primary sample, Figure 10 shows the result. As can be seen from the figure, the noisiness of the indirect illumination render decreases as the number of samples increase. Results in this paper contain between one and five randomly sampled secondary ray for each primary ray sample.
+
+![](images/996c6ff928962faa2ace740e4440799a5fd225f1f4f5cf69e5fadb423a188b9a.jpg)  
+Figure 9: Composing real-world objects from REAL-NRF using our OSF method. We demonstrate the effect of moving the light, camera, or objects. Note how the appearance and shadows of the objects are updated across different scene configurations. Also notice that even when parts of the palm tree object and the cartoon object are cast under the pony’s shadow, they do not appear completely dark due to the indirect illumination from the floor.
+
+![](images/4bc96233e783024766f216d5f6bf446967dfdae14a4213e5ea5c67c5adc5280f.jpg)  
+Figure 10: Visualizing the effect of different numbers of indirect (secondary) rays $( N )$ per primary sample for our OSF model (the brightness of these images has been increased only for visualization purposes). Note that the noisiness of the render decreases as $N$ increases. We find that we are able to achieve relatively non-noisy results with approximately five samples.
+
+Single-object renderings from REAL-NRF are shown in Figure 11. The objects were captured in a dark room with a one-light-at-a-time setup. After training OSF on each object in this dataset, we are able to render the objects from novel viewpoints and lighting directions.
+
+![](images/bc10e31ff7c75cf65089b94716c57bdbbea0a20fce792cf26a6cbbf6e27a2617.jpg)  
+Figure 11: Learned OSFs on objects from REAL-NRF. The objects were captured in a dark room with a one-light-at-a-time setup. After training OSF on each object in this dataset, we are able to render the objects from novel viewpoints and lighting directions.
+
+# B ABLATION EXPERIMENTS
+
+![](images/3b4157b61022d2d3c8f075161056762b9abe59507469ae48618f30c5afdad8cb.jpg)  
+Figure 12: Ablation results on our OSF model.
+
+Figure 12 shows ablation results on FURNITURE-REALISTIC. We evaluate different variants of our model: “Direct Only” which considers only direct illumination; “Indirect Only” which considers only indirect illumination; “Direct $^ +$ Shadows” which includes both direct illumination and shadows. Our full model (OSF) shows the most realistic rendering, with accurate shadows and indirect illumination effects such as the left side of the couches and tables appearing brighter due to indirect lighting from the left wall. Note that the white area on the right of the images represent rays with zero density that are composited onto a white background (and therefore do not contribute indirect illumination to the scene).
+
+![](images/aac3b3e00f1a72e32e711f7f8082e45737aed705e332eed318b0455d108b3526.jpg)  
+Figure 13: Comparisons on scene composition on FURNITURE-REALISTIC. The models OSF, oNeRF, and $_ { 0 - \mathrm { N e R F } + \mathrm { ~ S ~ } }$ are explained in $\ S 5$ . Compared to o-NeRF, our model (OSF) is able to disentangle lighting-dependent appearance from view-dependent appearance for individual objects, and is able to render shadows cast by objects onto the ground correctly.
+
+# C COMPLEX ILLUMINATION
+
+In this experiment, we investigate how scenes composed of OSF objects can be rendered with complex illumination from an environment map.
+
+Specifically, we apply the combination of a point light source and the environment map shown in the top-left corner of Figure 14 to light one of our scenes in FURNITURE-REALISTIC. This simulates the appearance of the scene as if the scene were inserted into a complex lighting environment, which stresses the benefits of the OSF path tracing framework.
+
+For each OSF sample point, we project the equirectangular coordinates of the environment map into spherical coordinates, sample 20 directions on the unit sphere uniformly at random, evaluate the OSF function for each incoming direction, and integrate them outgoing radiance using Equation 5. Please note that a green-blue tint is slightly apparent in the scene rendering, due to the contribution of green and blue lighting from the environment map.
+
+![](images/74f46dde664bd5d47d66bf589c968aeb26ab69b2c53585a3a097ea99ebd89630.jpg)  
+Figure 14: Complex illumination results.
+
+# D IMPLEMENTATION DETAILS
+
+A flowchart of our method is shown in Figure 15.
+
+We approximate our model $F _ { \Theta }$ with a multilayer perception (MLP) with rectified linear activations. The predicted density $\sigma$ is view-invariant, while the scattering function value $\rho$ is dependent on the incoming and outgoing light directions. We use an eight-layer MLP with 256 channels to predict $\sigma$ , and a four-layer MLP with 128 channels to predict $\rho$ . For positional encoding, we use $W = 1 0$ to encode the position $_ { \textbf { \em x } }$ and $W = 4$ to encode the incoming and outgoing directions $( \omega _ { l } , \omega _ { o } )$ , where $W$ is the highest frequency level. To avoid $\rho$ from saturating in training, we adopt a scaled sigmoid (Brock et al., 2016) defined as $S ^ { \prime } ( \pmb { \rho } ) = \delta ( S ( \pmb { \rho } ) - 0 . 5 ) + 0 . 5$ with $\delta = 1 . 2$ . We use a batch size of 4,096 rays.
+
+![](images/e872a846f54ec153cb399e3329f55d9c4a07e4ff95330969d2e6e7aa06fc4226.jpg)  
+Figure 15: Flowchart of our method. See $\ S 4$ for more details.
+
+For synthetic datasets, we sample $N _ { c } = 6 4$ coarse samples and $N _ { f } = 1 2 8$ fine samples per ray. For real world datasets, we sample $N _ { c } = 6 4$ coarse samples and $N _ { f } = 6 4$ fine samples per ray. We use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001, $\beta _ { 1 } = 0 . 9$ , $\beta _ { 2 } = 0 . 9 9 9$ , and $\epsilon = 1 0 ^ { - 7 }$ .

parse/dev/Uy6YEI9-6v/Uy6YEI9-6v_content_list.json

@@ -0,0 +1,1891 @@
+[
+    {
+        "type": "text",
+        "text": "OBJECT-CENTRIC NEURAL SCENE RENDERING ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            398,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present a method for composing photorealistic scenes from captured images of objects. Our work builds upon neural radiance fields (NeRFs), which implicitly model the volumetric density and directionally-emitted radiance of a scene from a collection of images. While NeRFs synthesize realistic pictures, they only model static scenes and are closely tied to specific imaging conditions. This property makes NeRFs hard to generalize to new scenarios, including new lighting or new arrangements of objects. Instead of learning a scene radiance field as a NeRF does, we propose to learn object-centric neural scattering functions (OSFs), a representation that models per-object light transport implicitly using a lighting- and view-dependent neural network. This enables rendering scenes even when objects or lights move, without retraining. Combined with a volumetric path tracing procedure, our framework is capable of rendering light transport effects including occlusions, specularities, shadows, and indirect illumination, both within individual objects and between different objects. We evaluate OSFs on synthetic and real world datasets, and on generalizing to new scene configurations. Learning OSFs leads to photorealistic, physically-accurate renderings of multi-object scenes. ",
+        "bbox": [
+            233,
+            241,
+            764,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            484,
+            336,
+            500
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Synthesizing images of dynamic scenes is an important problem in computer vision and graphics, with applications in AR/VR and robotics (Savva et al., 2019; Xia et al., 2020). For synthetic scenes, a user typically designs a set of 3D objects separately, then composes them into scenes to be rendered with specified camera, material, and lighting parameters. While this traditional graphics approach allows for flexible scene compositions, it requires detailed models of geometry, lighting, materials, and cameras, which can be difficult to obtain for real-world scenes. ",
+        "bbox": [
+            174,
+            508,
+            825,
+            592
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To render real-world scenes without computer graphics models, recent works have explored using neural implicit methods (Lombardi et al., 2019; Sitzmann et al., 2019a;b). Most notably, Mildenhall et al. (2020) proposed neural radiance fields (NeRF), which achieve photorealistic quality by implicitly modeling the volumetric density and directional emitted radiance of a scene. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            654
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, as shown in Figure 1, NeRF cannot generalize beyond the scene it was trained on, because it assumes static scenes and fixed illumination and learns a radiance field, which estimates only the resulting radiance along a ray after all light transport has occurred in a scene. Thus, for dynamic scenes where lights and objects can move, a separate NeRF-based model is needed for each new scene configuration. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/8ed85615e21b0b053f778c5c06f54eb1cf4ef60465fc666bac5bef5e9879567b.jpg",
+        "image_caption": [
+            "Figure 1: (a) NeRF. (b) Our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            522,
+            667,
+            823,
+            772
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To address this issue, we propose Object-Centric Neural Scattering Functions (OSFs) to synthesize dynamic scenes of objects learned from 2D images (Figure 2). We represent each object as a learned 7D scattering function with inputs $( x , y , z , \\phi _ { i } , \\theta _ { i } , \\phi _ { o } , \\theta _ { o } )$ , where $( x , y , z )$ is the spatial location, $( \\phi _ { i } , \\theta _ { i } )$ is the incoming light direction, and $\\left( \\phi _ { o } , \\theta _ { o } \\right)$ is the outgoing light direction. The function outputs the volumetric density as well as the fraction of light arriving from direction $( \\phi _ { i } , \\theta _ { i } )$ that scatters in outgoing direction $\\left( \\phi _ { o } , \\theta _ { o } \\right)$ . ",
+        "bbox": [
+            174,
+            804,
+            825,
+            890
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Each OSF models all light bounces (reflections) and occlusions (shadows) within an object. Since each object’s scattering function is a radiance transfer function rather than a radiance field, it is intrinsic to the object (independent of the scene it is in) and can be reused across different object placements and lighting conditions without retraining. We emphasize that because NeRFs are radiance fields, they cannot be composed, and cannot generalize beyond one scene. In contrast, we can render infinitely many scenes. We can build a library of OSFs trained independently for different objects to be composed into scenes with different object placements, camera, and lighting. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd921d3f24289566b23c0122c43b391f84f4457c5c3d2bcc99da20f473f6d700.jpg",
+        "image_caption": [
+            "Figure 2: We propose an object-centric neural scene representation for image synthesis. Given a scene description (a), and a repository of neural object-centric scattering functions (OSF) trained independently from images and frozen for each object (b), we can compose the objects into scenes (c), and render photorealistic images as we move lights (d), cameras (e), and/or objects (f). Our framework is capable of rendering occlusions, specularities, shadows, and indirect illumination. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            102,
+            825,
+            204
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            314,
+            825,
+            383
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To model light transport between objects, we integrate our implicit object functions with volumetric path tracing. Like NeRF, we evaluate the radiance and volumetric density at 5D samples along every primary ray to the camera and composite them with an over operator. However, unlike NeRF, we estimate the radiance for each 5D sample by integrating our 7D OSF across the 2D sphere of incoming light directions. We estimate the integral with Monte Carlo path tracing (Kajiya, 1986) to reproduce shadows and indirect illumination effects. ",
+        "bbox": [
+            174,
+            390,
+            825,
+            473
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our key idea is to decompose the rendering problem into (i) a learned component (per-object asset creation), and (ii) a non-learned component (per-scene path tracing). The learned component models intra-object light transport (e.g., bounces from the seat of a chair to the back of the chair). The non-learned component handles inter-object light transport (e.g., bounces from a wall to a chair). Together, they model the full rendering equation (Kajiya, 1986) (except for occluders or light sources that intrude the object’s convex hull (Sloan et al., 2002)). Since only the inter-object light transport changes as objects and lights move, no re-training is required for different scene arrangements. Experimental results indicate that our method is capable of rendering images with novel scene compositions and lighting conditions better than alternative learned approaches. ",
+        "bbox": [
+            174,
+            481,
+            825,
+            607
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, our contributions are: ",
+        "bbox": [
+            174,
+            613,
+            400,
+            627
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Learning Object-Centric Neural Scattering Functions (OSFs) that model intra-object light transport implicitly using a lighting- and view-dependent neural network.   \n2. Integrating implicitly learned object scattering functions with volumetric path tracing to model inter-object light transport.   \n3. A rendering algorithm that enables rendering scenes with moving objects, lights and cameras, using implicit functions. ",
+        "bbox": [
+            174,
+            636,
+            825,
+            723
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            741,
+            344,
+            757
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Classical object-centric representations. Factoring light transport into intra- and inter-object illumination has a long history in traditional computer graphics (Dutre et al., 2018). In most cases, the motivation is to improve rendering efficiency by approximating intra-object lighting factors with simple transfer functions (e.g., linear) for simple radiance fields (e.g., spherical harmonics) derived from from computer graphics models, as in precomputed radiance transfer (PRT) (Sloan et al., 2002), ambient occlusion (Miller, 1994), or virtual walls (Arnaldi et al., 1994). In other cases, the motivation is to insert captured, real-world radiance fields into synthetic scenes, as in Light Field Transfer (Cossairt et al., 2008). These methods generally store the radiance field for objects in a discrete representation (e.g., a sampled 2D or 4D grid). As a result, they cannot reproduce accurate inter-object light transport, especially for objects with intersecting bounding volumes. In contrast, we focus on learning radiance transfer from images in order to model complex real-world scattering accurately, and utilize volumetric rendering techniques to account for inter-object illumination. ",
+        "bbox": [
+            173,
+            767,
+            825,
+            934
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Novel view synthesis. Traditional methods for synthesizing novel views of a scene from captured images include using Structure-From-Motion (Hartley & Zisserman, 2003) and bundle adjustment (Triggs et al., 1999) to predict a sparse point cloud and camera parameters of the scene. More recently, a number of learning-based novel view synthesis methods have been presented but require 3D geometry as inputs (Hedman et al., 2018; Thies et al., 2019; Meshry et al., 2019; Aliev et al., 2020; Martin-Brualla et al., 2018). Others use multiplane images as proxies for novel view synthesis, but their viewing ranges are limited to interpolated input views (Flynn et al., 2016; Zhou et al., 2018; Srinivasan et al., 2019; Mildenhall et al., 2019). Some works represent scenes as coarse voxel grids and use a CNN-based decoder for differentiable rendering, but lack view consistency due to the use of 2D convolutional kernels (Nguyen-Phuoc et al., 2018; 2019; 2020). ",
+        "bbox": [
+            173,
+            103,
+            825,
+            242
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, volume rendering approaches have been used to render scenes represented as voxel grids that are more view-consistent (Lombardi et al., 2019; Sitzmann et al., 2019a). However, the rendering resolution of these methods are limited by the time and computational complexity of discretely sampled volumes. To address this issue, Neural Radiance Fields (NeRF) (Mildenhall et al., 2020) directly optimizes a continuous radiance field representation using a multi-layer perceptron. This allows synthesizing novel views of realistic images at an unprecedented level of fidelity. To make NeRF more efficient, Neural Sparse Voxel Fields (Liu et al., 2020) have been proposed as a sparse voxel octree variant of NeRF and demonstrate the ease of composing learned NeRFs with their voxel representation. See (Dellaert & Yen-Chen, 2020) for survey. While these implicit methods produce high-quality novel views of a scene, their models assume a static scene with fixed illumination. Our method enables synthesizing dynamic scenes with novel viewpoint, lighting, and object configurations. ",
+        "bbox": [
+            174,
+            250,
+            825,
+            415
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Relighting. Learning-based methods that relight images without explicit geometric reasoning have been proposed, but lack the ability to recover hard shadows (Sun et al., 2019; Xu et al., 2018; Zhou et al., 2019). Other works use geometric representations that facilitate shadowing computation, but require 3D geometry as input (Philip et al., 2019; Zhang et al., 2021; Oechsle et al., 2020; Rematas & Ferrari, 2020). Deep Reflectance Volumes (Bi et al., 2020b) reconstructs a voxelized representation of a scene and predict per-voxel BRDFs, but the fixed resolution of voxel grids limits the quality in the rendered images. Similarly, Neural Reflectance Fields (Bi et al., 2020a) predicts the parameters of a BRDF model, but demonstrate higher fidelity rendering by learning a continuous scene representation. However, Neural Reflectance Fields focuses on relighting single objects, and requires manual specification of the BRDF model. Parametric BRDF models are unable to handle complex scattering functions, including real-world scattering phenomena that are difficult to model. In contrast, our method is capable of learning all scattering functions, and can render multiple objects in dynamic scenes. ",
+        "bbox": [
+            174,
+            419,
+            825,
+            598
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            607,
+            339,
+            622
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 VOLUME RENDERING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            630,
+            367,
+            645
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To render an image of a scene with arbitrary camera parameters, camera rays are sent into the scene, through each pixel on the image plane. The expected color of each pixel is computed as the radiance along each camera ray. ",
+        "bbox": [
+            176,
+            651,
+            823,
+            693
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Volume rendering is an approach for computing the radiance traveling along rays traced in a volume. Let ${ \\pmb r } ( t ) = { \\pmb x } _ { 0 } + \\omega _ { o } t$ be a point along a ray $\\mathbfit { \\Delta } \\mathbf { r }$ with origin $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ and direction $\\omega _ { o }$ , where $t \\in \\mathbb { R }$ is a 1D location along the ray, and the $^ o$ in $\\omega _ { o }$ denotes “outgoing” direction. For our purposes, we assume non-emissive and non-absorptive volumes. From Novak et al. (2018), the volume rendering ´ equation to compute the radiance $L ( \\boldsymbol { x } _ { 0 } , \\omega _ { o } )$ of the ray is defined as: ",
+        "bbox": [
+            174,
+            699,
+            825,
+            770
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/adc816bbe3f46d8eeec807f568417034d8aeacf164d731750571d8c8d1e52212.jpg",
+        "text": "$$\nL ( { \\boldsymbol { x } } _ { 0 } , \\omega _ { o } ) = \\int _ { t _ { n } } ^ { t _ { f } } { \\boldsymbol { \\tau } } ( t ) { \\boldsymbol { \\sigma } } ( r ( t ) ) L _ { s } ( r ( t ) , \\omega _ { o } ) d t , \\quad { \\mathrm { w h e r e } } \\quad { \\boldsymbol { \\tau } } ( t ) = \\exp { \\left( - \\int _ { t _ { n } } ^ { t } { \\boldsymbol { \\sigma } } ( r ( u ) ) d u \\right) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            773,
+            794,
+            810
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $t _ { n }$ and $t _ { f }$ are near and far integration bounds, $\\sigma ( \\pmb { r } ( t ) )$ denotes the volume density of point $\\mathbf { } _ { \\pmb { r } ( t ) }$ , and $\\tau ( t )$ denotes the accumulated transmittance from $t _ { n }$ to $t$ . The term $L _ { s } ( \\pmb { r } ( t ) , \\pmb { \\omega } _ { o } )$ is the light scattered at point $\\mathbf { } _ { \\mathbf { } } ^ { r ( t ) }$ along direction $\\omega _ { o }$ , defined as the integral over all incoming light directions: ",
+        "bbox": [
+            174,
+            814,
+            826,
+            856
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1e84c484c6439e5a3fc29823b1317dd2baa4c5f34569d2d762595f84a554013.jpg",
+        "text": "$$\nL _ { s } ( \\pmb { x } , \\omega _ { o } ) = \\int _ { S } L ( \\pmb { x } , \\omega _ { l } ) f _ { p } ( \\pmb { x } , \\omega _ { l } , \\omega _ { o } ) d \\omega _ { l } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            859,
+            647,
+            892
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $s$ is a unit sphere and $f _ { p }$ is a phase function that evaluates the fraction of light incoming from direction $\\omega _ { l }$ at a point $_ { \\textbf { \\em x } }$ that scatters out in direction $\\omega _ { o }$ . In NeRF, Mildenhall et al. (2020) assume ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "fixed illumination and do not consider any form of Equation 2. We consider a more general form of the volume rendering equation that explicitly models light paths within and between objects. This is important for dynamic scenes, where lighting and objects can move with respect to one another. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 RAY MARCHING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            152,
+            331,
+            167
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The continuous integrals in Equation 1 can be estimated with quadrature (Kniss et al., 2003; Max, 1995), as done in NeRF (Mildenhall et al., 2020). For each ray, stratified sampling is used to obtain $N$ samples $\\{ t _ { i } \\} _ { i = 1 } ^ { N }$ along the ray, where $t _ { i } \\in [ t _ { n } , t _ { f } ]$ . The rendering equation is approximated by: ",
+        "bbox": [
+            173,
+            175,
+            826,
+            218
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a955ba0532af3e18cce6b5d49629da2949b59db1c6359d6836a64e9ccb5d2c1c.jpg",
+        "text": "$$\nL ( \\boldsymbol { x } _ { 0 } , \\omega _ { o } ) = \\sum _ { i = 1 } ^ { N } \\tau _ { i } \\alpha _ { i } L _ { s } ( \\boldsymbol { x } _ { i } , \\omega _ { o } ) \\quad \\mathrm { w h e r e } \\quad L _ { s } ( \\boldsymbol { x } _ { i } , \\omega _ { o } ) = \\frac { 1 } { | \\mathcal { L } | } \\sum _ { l \\in \\mathcal { L } } L ( \\boldsymbol { x } _ { i } , \\omega _ { l } ) \\rho _ { i } ^ { l } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            226,
+            764,
+            271
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\tau _ { i } = \\prod _ { j = 1 } ^ { i - 1 } ( 1 - \\alpha _ { j } ) } \\end{array}$ and $\\alpha _ { i } = 1 - e ^ { - \\sigma _ { i } \\left( t _ { i + 1 } - t _ { i } \\right) }$ . To compute the average over incoming light paths $L _ { s }$ , we discretize over the domain $s$ in Equation 2 by sampling a set of incoming light paths $\\mathcal { L } = \\{ l _ { 1 } , \\ldots , l _ { K } \\}$ , where $\\pmb { \\rho } _ { i } ^ { l } = f _ { p } ( \\pmb { x } _ { i } , \\omega _ { l } , \\omega _ { o } ) \\bar { \\ } \\in \\ [ 0 , 1 ]$ , the fraction of light incoming from light path $\\imath$ that is scattered in direction $\\omega _ { o }$ . ",
+        "bbox": [
+            174,
+            280,
+            823,
+            339
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 NEURAL RADIANCE FIELDS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            347,
+            410,
+            361
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "NeRF represents a continuous scene as a volumetric radiance field, approximated with a multilayer perceptron $F _ { \\Theta }$ . The model $F _ { \\Theta }$ takes spatial location $\\pmb { x } = ( x , y , z )$ and viewing direction $\\pmb { d } = ( \\phi , \\theta )$ as input, and outputs the density $\\sigma$ and color $\\boldsymbol { \\mathbf { \\mathit { c } } } = ( r , g , b )$ , where $r , g , b \\in [ 0 , 1 ]$ . Frequency-based positional encoding (Rahaman et al., 2019; Vaswani et al., 2017) is applied to the inputs to better capture high-frequency variation in appearance and geometry. ",
+        "bbox": [
+            174,
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+            439
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A hierarchical volume sampling procedure (Mildenhall et al., 2020; Levoy, 1990) is then employed to more efficiently allocate samples along each ray. This technique biases sample allocation to favor the visible parts of the scene that contribute the most to the final render, avoiding occluded or free space in the scene. NeRF simultaneously optimizes two radiance fields, where the sample weights $\\tau _ { i } \\cdot \\alpha _ { i }$ from a coarse model are used to bias samples for a fine model. The $L _ { 2 }$ loss is used to optimize both models: $\\begin{array} { r } { \\sum _ { \\pmb { r } \\in \\mathcal { R } } \\| \\hat { C } _ { c } ( \\pmb { r } ) - C ( \\pmb { r } ) \\| _ { 2 } ^ { 2 } + \\| \\hat { C } _ { f } ( \\pmb { r } ) - \\check { C } ( \\pmb { r } ) \\| _ { 2 } ^ { 2 } } \\end{array}$ , where $\\mathcal { R }$ is the set of all camera rays, $\\widehat { C } _ { c } ( \\pmb { r } )$ and $\\widehat { C } _ { f } ( \\boldsymbol { r } )$ denote the radiance along ray $\\mathbfit { \\Delta } \\mathbf { r }$ predicted by the coarse and fine models respectively, and $C ( \\boldsymbol { r } )$ is the ground truth pixel color for $\\pmb { r }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 OBJECT-CENTRIC NEURAL SCATTERING FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We represent each object as a 7D object-centric neural scattering function (OSF), depicted in Figure 3a. For each object, we learn an implicit function $F _ { \\Theta } \\colon ( \\pmb { x } , \\omega _ { l } , \\omega _ { o } )  ( \\sigma , \\pmb { \\rho } )$ that receives a 3D point in the object coordinate frame, the incoming light direction, and the outgoing light direction, and predicts the volumetric density as well as fraction of incoming light that is scattered in the outgoing direction. $\\Theta$ are learned weights that parameterize the neural network, $\\pmb { x } = ( x , y , z )$ denotes the spatial location, $\\omega _ { l } = \\left( \\phi _ { l } , \\theta _ { l } \\right)$ denotes the incoming light direction, $\\omega _ { o } = ( \\phi _ { o } , \\theta _ { o } )$ denotes the outgoing light direction, $\\sigma$ denotes the volumetric density, and $\\rho = ( \\rho _ { r } , \\rho _ { g } , \\rho _ { b } )$ denotes the fraction of light arriving at $_ { \\textbf { \\em x } }$ from direction $\\omega _ { l }$ that is scattered and leaving in direction $\\omega _ { o }$ . The final color of a point $_ { \\textbf { \\em x } }$ is the integral of $\\rho$ multiplied by the incoming radiance over all incoming light directions in unit sphere $s$ (Equation 2). Following NeRF, we similarly apply positional encoding to our inputs $( \\pmb { x } , \\omega _ { l } , \\omega _ { o } )$ and employ a hierarchical sampling procedure to recover higher quality appearance and geometry of learned objects. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "During training, we assume a single point light source with radiance of $( 1 , 1 , 1 )$ . This simplifies $L _ { s }$ from Equation 2 to $L _ { s } ( \\pmb { x } , \\omega _ { o } ) = L ( \\pmb { x } , \\omega _ { l } ) f _ { p } ( \\pmb { x } , \\omega _ { l } , \\omega _ { o } ) = f _ { p } ( \\pmb { x } , \\omega _ { l } , \\omega _ { o } )$ . To learn per-object NeRFs independent of object rotation and translation, the inputs to $F _ { \\Theta }$ must be in the object’s canonical coordinate frame. Given a object transformation $T _ { i }$ for object $\\mathbf { o } _ { i }$ , we apply $T _ { i } ^ { - 1 }$ to $( r , \\omega _ { l } , \\omega _ { o } )$ before feeding the inputs to the network. ",
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+    {
+        "type": "text",
+        "text": "4.2 RENDERING MULTIPLE OSFS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Once we have learned an OSF for each object, we aim at composing the learned objects into scenes.   \nAn overview of our procedure is visually depicted in Figure 3b. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/66fc5a607c6b90609b949b4ac6d774a80f2f4bfc8041fa317f228a4221e631ec.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(a) We represent each object as an object-centric neural scattering function (OSF), which models how light entering at a point $_ { \\textbf { \\em x } }$ on the object, from direction $\\omega _ { l }$ where $\\imath$ corresponds to a light path, undergoes multiple bounces within the object and exits along direction $\\omega _ { o }$ with some fractional amount of light $\\pmb { \\rho }$ . We approximate the scattering function with a multilayer perceptron $F _ { \\Theta }$ where $\\Theta$ are learned weights that parameterize the neural network. Given a single point $_ { \\textbf { \\em x } }$ , an incoming light direction $\\omega _ { l }$ , and an outgoing direction $\\omega _ { o }$ , $F _ { \\Theta }$ outputs the volume density $\\sigma$ of that point, as well as the fraction of light arriving at $_ { \\textbf { \\em x } }$ from direction $\\omega _ { l }$ that is scattered in direction $\\omega _ { o }$ . ",
+        "bbox": [
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+            246,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(b) Our procedure for rendering an arbitrary scene consisting of multiple objects, light sources, and cameras. Given a set of objects, we compute direct illumination by shooting rays from each light source to each object (brown arrows). Shadows are computed by sending shadow rays back to each light source (purple arrow). The shadow ray from the desk is occluded by the mug, so the mug casts a shadow on the desk. We send secondary rays between objects to render indirect illumination effects, such as between the desk and the kettle (green and blue dashed arrows). Finally, rays are sent back to the camera to render the final image (dark blue arrows). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/03f3183f13c1fafe38d219921c954792e1c1c0cb4fd4256f42d25dcdb5dad07f.jpg",
+        "image_caption": [
+            "Figure 3: Using our method (OSFs) to render: (a) single and (b) multiple objects. ",
+            "Figure 4: Sampling procedure. (a) Scene with a camera, light source, and object bounding boxes. Primary rays are sent from the camera into the scene. Rays that do not intersect with objects are pruned. Of the intersecting rays, we sample points within intersecting regions. (b) Shadow rays from each sample are sent to the light source, and samples within intersecting regions are evaluated. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { O } = \\{ o _ { i } \\} _ { i = 1 } ^ { N }$ be a set of $N$ objects we wish to render. For simplicity, we first describe the rendering process for each object $\\mathbf { o } _ { i }$ , then explain the process to combine results across all objects to render the final scene. Let $\\mathbf { } o _ { i } \\in \\mathcal { O }$ denote object $i$ with transformation $T _ { i } \\in \\mathbb { R } ^ { 4 \\times 4 }$ and bounding box dimensions $D _ { i } \\in \\mathbb { R } ^ { 3 }$ . Further let $\\mathbfit { \\Delta } \\mathbf { r }$ be a camera ray with origin $\\boldsymbol { c } \\in \\mathbb { R } ^ { 3 }$ and direction $\\omega _ { o } \\in \\mathbb { R } ^ { 3 }$ , which we define with parameters $\\gamma = [ c , \\omega _ { o } ] \\in \\mathbb { R } ^ { 6 }$ . Our goal is to compute $L ( c , \\omega _ { o } )$ as described in Equation 3. We compute the ray-box intersection between the ray and the object to obtain near bound $t _ { n } ^ { i }$ and far bound $\\bar { t } _ { f } ^ { i }$ such that $\\textstyle r ( t _ { n } ^ { i } )$ and ${ \\pmb r } ( t _ { f } ^ { i } )$ each intersect a box plane, as shown in Figure 4. Note thabetween ys thand do not intealong ray ect withto obtai $\\mathbf { o } _ { i }$ are excl sample putatio, where $M$ pointsiven a $t _ { n } ^ { i }$ $t _ { f } ^ { i }$ $\\mathbfit { \\Delta } \\mathbf { r }$ $X ^ { i } = \\{ x _ { m } ^ { i } \\} _ { m = 1 } ^ { M }$ $\\pmb { X } ^ { i } \\in \\mathbb { R } ^ { M \\times 3 }$ light source $\\imath$ , we evaluate the object’s model $F _ { \\Theta _ { i } } ( X ^ { i } , \\omega _ { l } , \\omega _ { o } )$ to obtain alpha values $\\pmb { \\alpha } ^ { i } \\in \\mathbb { R } ^ { M }$ and phase function values $\\pmb { \\rho } ^ { i } \\in \\mathbb { R } ^ { M \\times 3 }$ . ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is not always possible for a light ray from light source $\\imath$ to reach the object $\\mathbf { o } _ { i }$ . Any of the other objects in $\\mathcal { O } ^ { \\prime } = \\{ o _ { j } \\in \\mathcal { O } \\mid j \\neq i \\}$ in the scene may occlude the incoming light, casting a shadow on object $\\mathbf { o } _ { i }$ . We compute shadows by sending a shadow ray $\\boldsymbol { r } _ { m }$ from each of the $M$ samples in $X ^ { i }$ to the light source $\\imath$ . Evaluating the shadow ray enables us to determine the amount of light blocked along the ray by other objects. We define the parameters of the M shadow rays as Γ ∈ RM×6. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each object $o _ { j } \\in \\mathcal { O } ^ { \\prime }$ , we compute ray-box intersections between shadow rays $\\Gamma$ and $o _ { j }$ ’s bounding box. This allows us to compute the amount of light traveling towards $\\mathbf { o } _ { i }$ that is blocked by $o _ { j }$ . Similar to primary rays, we sample $M$ points along each shadow ray to obtain a set of points $\\breve { \\pmb { X } } ^ { j } \\in \\mathbb { R } ^ { M \\times M }$ . We then evaluate the object model $F _ { \\Theta _ { j } } ( \\mathbf { X } ^ { j } )$ to obtain alpha values $\\pmb { A } ^ { j } \\in \\mathbb { R } ^ { M \\times M }$ . For each shadow ray $\\boldsymbol { r _ { m } }$ , we combine samples $A _ { m } ^ { j }$ across the $N - 1$ objects in $\\mathcal { O } ^ { \\prime }$ by sorting according to sample distance to obtain alpha values $\\pmb { A } _ { m } \\in \\mathbb { R } ^ { M ( N - 1 ) }$ . The fraction of unobstructed light traveling along the shadow ray $\\mathbf { \\Delta } _ { \\mathbf { r } _ { m } }$ is computed as the transmittance: ",
+        "bbox": [
+            173,
+            138,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc9c36456839b6704480f8d38b49d265cdc2b14956f2f43119b739cf280477e6.jpg",
+        "text": "$$\n\\tau _ { m } ^ { l } = \\prod _ { n = 1 } ^ { M ( N - 1 ) } ( 1 - A _ { m n } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            247,
+            588,
+            292
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus, the adjusted incoming radiance from light source $\\imath$ when accounting for occlusions is computed as $L _ { l } ( \\mathbf { \\bar { x } } _ { m } , \\omega _ { l } ) = \\tau _ { m } ^ { l } \\bar { L } _ { l } ( \\mathbf { x } _ { m } , \\omega _ { l } )$ . ",
+        "bbox": [
+            174,
+            297,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We follow the scattering equation in Equation 2 and now consider all incoming light directions over the unit sphere $s$ . This accounts for secondary light rays traveling to an object $\\mathbf { o } _ { i }$ indirectly from another object ${ \\pmb O } _ { j }$ (indirect illumination). We approximate the integral over the unit sphere $s$ by sampling $K$ directions on the unit sphere uniformly at random. For each direction $\\omega _ { k }$ randomly sampled for a point $_ { \\textbf { \\em x } }$ , we send a secondary ray $\\mathbf { \\nabla } r _ { k }$ from $_ { \\textbf { \\em x } }$ in direction $\\omega _ { k }$ and evaluate the radiance $L ( x , \\omega _ { k } )$ traveling along the ray. To compute the radiance of the secondary ray $L ( x , \\omega _ { k } )$ , we employ the same technique used to compute the radiance of a primary ray $L ( c , \\omega _ { o } )$ (described at the beginning of Section 4.2). The incoming radiance $L ( x , \\omega _ { k } )$ is multiplied with the phase function value $\\rho = f _ { p } ( { \\pmb x } , \\omega _ { { \\pmb k } } , \\omega _ { o } )$ to determine the outgoing radiance $\\dot { L } ( { \\pmb x } , { \\pmb \\omega } _ { o } )$ , where $\\rho$ is evaluated using $F _ { \\Theta _ { i } }$ . Note that this is possible due to the recursive nature of our formulation. Only secondary rays are described here (two bounces), but our method supports an arbitrary number of bounces. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Rendering. We sample and evaluate all objects in $\\mathcal { O }$ to obtain alpha values $\\{ \\alpha ^ { i } \\} _ { i = 1 } ^ { N }$ and phase function values $\\{ \\rho ^ { i } \\} _ { i = 1 } ^ { N }$ for a set of sampled points $\\{ X ^ { i } \\} _ { i = 1 } ^ { N }$ along ray $\\pmb { r }$ . We sort the samples across all objects to produce a final set of $P = M \\cdot N$ samples $\\{ \\pmb { x } _ { m } \\} _ { m = 1 } ^ { P }$ , $\\lbrace \\alpha _ { m } \\rbrace _ { m = 1 } ^ { P }$ , and $\\{ \\rho _ { m } \\} _ { m = 1 } ^ { P }$ . ",
+        "bbox": [
+            174,
+            488,
+            825,
+            534
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Given light paths $\\mathcal { L }$ containing both direct and indirect illumination, we render the final radiance of a ray with origin $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ and direction $\\omega _ { o }$ with the following equation: ",
+        "bbox": [
+            169,
+            539,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/33ad499470fb898f62dafe743d70cc55e69248eef6dc7c71cf748001ca755515.jpg",
+        "text": "$$\nL ( \\boldsymbol { x } _ { 0 } , \\omega _ { o } ) = \\frac { 1 } { | \\mathcal { L } | } \\sum _ { l \\in \\mathcal { L } } \\sum _ { m = 1 } ^ { P } \\alpha _ { m } \\rho _ { m } ^ { l } \\tau _ { m } L _ { l } ( \\boldsymbol { x } _ { m } , \\omega _ { l } ) , \\quad \\mathrm { w h e r e } \\quad \\tau _ { m } = \\prod _ { n = 1 } ^ { m - 1 } ( 1 - \\alpha _ { n } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            574,
+            767,
+            618
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "and $L _ { l } ( \\pmb { x } _ { m } , \\pmb { \\omega } _ { l } )$ is the radiance from light path $\\imath$ arriving at point ${ \\pmb x } _ { m }$ ",
+        "bbox": [
+            173,
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+            638
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Runtime. In total, the cost of rendering a single image with $N _ { \\mathrm { p i x e l } }$ pixels and $N _ { \\mathrm { o b j e c t } }$ objects is $\\mathcal { O } ( P ^ { 2 } K N _ { \\mathrm { p i x e l } } )$ . Note that $P$ is an upper bound on number of samples that need to be evaluated. In practice, a single ray often only intersects with at most one object in the scene, which means that the proposed rendering procedure is not significantly more expensive than the single object setting. We also note that compared to NRF Bi et al. (2020a) or traditional volumetric path tracing methods, OSF crucially does not require running path tracing within each object to simulate intra-object light bounces. This is because OSF learns the object-level scattering function that directly predicts the effects after all light bounces (reflections) and occlusions (shadows) within an object have occurred. Thus OSF is significantly faster than NRF which relies on simulating intra-object light bounces while querying its learned BRDF model. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, rendering a single image with a single OSF at a resolution of $2 5 6 \\times 2 5 6$ takes roughly 3.7 seconds. While the computation cost is high, there are efforts to reduce the rendering speed of NeRF that are orthogonal to this work. For instance, KiloNeRF (Reiser et al., 2021) can easily adapted to this work by utilizing thousands of tiny MLPs instead of one single large MLP to represent each OSF to obtain 1-2 orders of magnitude speed up. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets and evaluation metrics. We evaluate our approach on several image datasets: ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• FURNITURE-SINGLE: 15 objects rendered with random object pose, point light, and viewpoint. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• FURNITURE-RANDOM: 25 dynamic scenes, each containing a random layout of multiple objects, point light, and viewpoint.   \n• FURNITURE-REALISTIC: Scenes containing realistic arrangements of objects in rooms.   \n• REAL-NRF: Real-world objects from Bi et al. (2020a), captured in a dark room under varying viewing and lighting directions.   \n• REAL-OUTDOOR: Real-world outdoor scenes from Mildenhall et al. (2020). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For FURNITURE datasets, we use Blender’s Cycles path tracer (Blender Foundation, 1994) to render images at $2 5 6 \\times 2 5 6$ resolution for different object arrangements, camera views, and lighting configurations. We report PSNR, SSIM (Wang et al., 2003), and LPIPS (Zhang et al., 2018) metrics. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines and ablations. We compare our method to the following baselines: ",
+        "bbox": [
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+            247,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. o-NeRF: A variant of the NeRF model, but with one NeRF trained per object. When o-NeRFs are composed into scenes, they are rendered separately.   \n2. $\\mathbf { 0 . N e R F + S }$ : An extension of o-NeRF with inter-object shadows; reduces the light arriving at each o-NeRF by the cumulative opacity of shadowing objects along the ray from the light (§4.2). ",
+        "bbox": [
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+            268,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "These baselines represent what could be achieved by combining separately trained NeRFs into a scene. Of course, since o-NeRFs produce radiance fields (not scattering fields), we do not expect them to perform well in novel lighting environments or object placements. ",
+        "bbox": [
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+            337,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 NOVEL LIGHTING",
+        "text_level": 1,
+        "bbox": [
+            174,
+            387,
+            339,
+            401
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the first experiment, we investigate how OSF method handles novel lighting conditions. ",
+        "bbox": [
+            169,
+            409,
+            761,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We train one model per object in FURNITURESINGLE. For each object model, we train on 400 images with randomized viewpoint and lighting, and test on 20 images of novel viewpoint and lighting. As can be seen in Figure 5, our method produces more accurate appearance of the objects in comparison to oNeRF when tested on novel illumination conditions. In particular, o-NeRF fails to predict self-shadows for the couch and chair correctly. Additionally, o-NeRF fails to disentangle viewpoint versus lighting-dependent appearance, producing incorrect shadows for the couch and chair, and fails to capture the specular details of the ottoman. Quantitative results can be found in Table 1. ",
+        "bbox": [
+            174,
+            431,
+            480,
+            651
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b61f34eebb13a4a67b135de352952372729b021c23f20c296231ac3433af6da.jpg",
+        "image_caption": [
+            "Figure 5: Novel lighting results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            498,
+            434,
+            821,
+            625
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 SCENE COMPOSITION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            660,
+            366,
+            674
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In a second experiment, we conduct a scene composition task on FURNITURE-RANDOM, where multiple object models are combined into scenes in random pose, lighting, and viewpoint configurations. For this task, we use the same object models trained in Section 5.1. Results are shown in Table 1 and Figure 6. While not shown in the main text, results for FURNITURE-REALISTIC can be found in Appendix B. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b435aafad3795064f90397f06f15607a54f7e14a4339d7b11e4e654ecf06f25.jpg",
+        "table_caption": [
+            "Table 1: Quantitative results for novel lighting (FURNITURE-SINGLE) and scene composition (FURNITURE-RANDOM). Rows denote different methods: our full model (OSF), a variant of NeRF where one NeRF is trained per object (o-NeRF), and o-NeRF with shadows $\\left( { \\bf { o - N e R F + S } } \\right)$ ). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"3\">FURNITURE-SINGLE</td><td colspan=\"3\">FURNITURE-RANDOM</td></tr><tr><td>Method</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>0-NeRF</td><td>33.22</td><td>0.980</td><td>0.021</td><td>12.17</td><td>0.690</td><td>0.280</td></tr><tr><td>0-NERF + S</td><td></td><td></td><td></td><td>14.70</td><td>0.697</td><td>0.267</td></tr><tr><td>OSF (Our Method)</td><td>44.07</td><td>0.998</td><td>0.002</td><td>19.02</td><td>0.793</td><td>0.135</td></tr></table>",
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+        "img_path": "images/6fd2bebddfb99194e391f1326f1944add318e88954ead332f48bd7426a5bea82.jpg",
+        "image_caption": [
+            "Figure 6: Scene composition results on FURNITURE-RANDOM. The models OSF, o-NeRF, and $_ { 0 - \\mathrm { N e R F } + \\mathrm { ~ S ~ } }$ are explained in $\\ S 5$ . Compared to o-NeRF, our model (OSF) is able to disentangle lighting-dependent and view-dependent appearance and can render shadows. "
+        ],
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+    {
+        "type": "text",
+        "text": "These results suggest that OSF outperforms all baselines and ablations, both quantitatively and qualitatively. As in the previous experiment (Section 5.1), we find that OSF reproduces object appearances and self-shadows more accurately than the baselines. The difference is especially apparent in the couches in scenes (a) and (b), where the couches predicted by o-NeRF are extremely dark. This is due to the fact that o-NeRF is unable to disentangle view-dependence appearance from lightdependent appearance, and simply interpolates the radiance field learned another different lighting configuration. Please note that OSF is able to model inter-object light transport effects by rendering shadows cast by one object onto another and on the ground plane. Plus, it is able to render indirect illumination of one object reflecting light onto another. For example, light reflected from the left wall causes the left of the couch and table in scenes (a) and (b) to be brighter. Neither of these lighting effects are present in the o-NeRF results. ",
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+    {
+        "type": "text",
+        "text": "5.3 REAL-WORLD SCENES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we evaluate our method on real world objects and scenes from the REAL-NRF and REAL-OUTDOOR datasets. For these experiments, we train one OSF for each object in REAL-NRF and each scene in REALOUTDOOR. ",
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+    {
+        "type": "text",
+        "text": "Figure 7 shows a comparison between ground truth, our method (OSF), and Neural Reflectance Fields (NRF) (Bi et al., 2020a). We show that OSF recovers stronger, more accurate specular highlights compared to NRF. OSF also produces more detailed appearances (see pony logo). This comparison demonstrates the main advantage of OSF: the ability to handle complex scattering functions. ",
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+        "img_path": "images/826048519b3a9c412be8dd529a100223ecefb879955bd4c18a74c8b6f81e36fc.jpg",
+        "image_caption": [
+            "Figure 7: Comparison of OSF (ours) to Neural Reflectance Fields (NRF) (Bi et al., 2020a). OSF produces stronger, more accurate specular highlights on the legs (see zoomed view) and recovers more detailed appearances (see pony logo). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "For scene composition, the OSFs trained on each object are composed with a synthetic floor OSF in Figure 8 row (a). Our method is able to compute accurate shadows, such as the shadow cast by the pony onto the two other objects in the scene. The indirect reflections from the floor allow the shadowed objects to be slightly visible as shown in the “OSF” panel. ",
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+    {
+        "type": "text",
+        "text": "Figure 8 rows (b) and (c) show results on inserting REAL-NRF objects into real outdoor scenes (REAL-OUTDOOR). Shadows and reflections are rendered with randomized lighting directions to approximate the environment lighting. Our method accurately renders occlusions between the inserted objects and the vase in Figure 8 row (c). Due to the compositional nature of OSFs, we are able to insert the learned pinecone from Figure 8 (b) into (c). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8, each column shows ablated versions of OSF to study the impact of computing shadows and indirect illumination with our path tracing algorithm. “No Shadows, No Indirect” represents a version of our model containing only direct illumination (without modeling inter-object lighting effects). We additionally show “No Indirect” and “Indirect Only” variants of our model which represent computing shadows and indirect illumination, respectively. As illustrated by Figure 8, our full model containing both shadows and indirect illumination effects is the most realistic. Additional results on real-world scenes, including complex shadows, can be found in Appendix A. ",
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+    {
+        "type": "image",
+        "img_path": "images/6224a6f879de9a4e63c61617266d4a0c74591e17b19dbbdd9fbe529892d9da8a.jpg",
+        "image_caption": [
+            "Figure 8: Real-world results. NRF (Bi et al., 2020a) and NeRF (Mildenhall et al., 2020) learn on individual static scenes or objects. In contrast, we compose real-world objects and scenes using OSFs. The objects are composed with a (a) synthetic floor and (b, c) real outdoor scenes from REALOUTDOOR. Columns show different ablated versions of our model: “No Shadows, No Indirect” which considers only direct illumination; “No Indirect” which includes both direct illumination and shadows; “Indirect Only” which considers only indirect illumination. Our OSFs show the most realistic renderings, with accurate shadows (e.g., pony shadowing the two other objects (row a) and indirect illumination (i.e., the ground and environment illuminating the objects). "
+        ],
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have proposed Object-Centric Neural Scattering Functions (OSFs), a method that enables composing objects captured only from photographs into photorealistic renderings of dynamic scenes. We demonstrated that decomposing a scene into implicit object functions that are view- and lightdependent enables reusabiliy of objects across scenes where objects, camera, and lighting can change. We presented a method for integrating our learned implicit functions with volumetric path tracing, and showed inter-object light transport effects such as shadow and indirect illumination for real-world objects where no computer graphics model is available. We believe our work is a step towards a graphics pipeline where real-world scenes are modeled by a composition of implicit functions to combine the flexibility of object-centric neural modeling with the photorealism of graphics rendering algorithms. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are a few main limitations to OSF. First, the computational complexity of our method is high, but there are several works tackling the orthogonal issue of improving NeRF efficiency (as discussed in Section 4.2) that can easily be applied to OSFs. Second, while learning intra-object light transport means that intra-object path tracing is not needed, this formulation assumes that at test time, there are no occluders or light sources that intrude the object’s convex hull (Sloan et al., 2002) (e.g., a person sitting in a chair). However, OSFs can still be rendered even if their bounding boxes are intersecting, as long as this assumption is not violated. Finally, acquiring datasets of real world objects with varying point light sources and viewpoints is challenging, but we hope that in the future such acquisition of real world datasets will become easier to capture and more widely available. ",
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+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT ",
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+        "type": "text",
+        "text": "We describe our method (Section 4) and experimental setup (Section 5) in detail to maximize reproducibility. We will release our code upon publication to facilitate future research. ",
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+        "bbox": [
+            174,
+            694,
+            825,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A REAL-WORLD SCENE COMPOSITION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            516,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Different scene configurations of composed objects from REAL-NRF are shown in Figure 9. We show the effect of moving the light, camera, or objects. Notice how the the appearance and shadows of the objects are updated across different scene configurations. Also notice that even when parts of the palm tree object and the cartoon object are cast under the pony’s shadow, they do not appear completely dark due to the indirect illumination from the floor. ",
+        "bbox": [
+            174,
+            128,
+            825,
+            199
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Analyzing the effect of different numbers of indirect (secondary) rays per primary sample, Figure 10 shows the result. As can be seen from the figure, the noisiness of the indirect illumination render decreases as the number of samples increase. Results in this paper contain between one and five randomly sampled secondary ray for each primary ray sample. ",
+        "bbox": [
+            174,
+            205,
+            825,
+            261
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/996c6ff928962faa2ace740e4440799a5fd225f1f4f5cf69e5fadb423a188b9a.jpg",
+        "image_caption": [
+            "Figure 9: Composing real-world objects from REAL-NRF using our OSF method. We demonstrate the effect of moving the light, camera, or objects. Note how the appearance and shadows of the objects are updated across different scene configurations. Also notice that even when parts of the palm tree object and the cartoon object are cast under the pony’s shadow, they do not appear completely dark due to the indirect illumination from the floor. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            279,
+            821,
+            546
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/4bc96233e783024766f216d5f6bf446967dfdae14a4213e5ea5c67c5adc5280f.jpg",
+        "image_caption": [
+            "Figure 10: Visualizing the effect of different numbers of indirect (secondary) rays $( N )$ per primary sample for our OSF model (the brightness of these images has been increased only for visualization purposes). Note that the noisiness of the render decreases as $N$ increases. We find that we are able to achieve relatively non-noisy results with approximately five samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            659,
+            821,
+            849
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Single-object renderings from REAL-NRF are shown in Figure 11. The objects were captured in a dark room with a one-light-at-a-time setup. After training OSF on each object in this dataset, we are able to render the objects from novel viewpoints and lighting directions. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/bc10e31ff7c75cf65089b94716c57bdbbea0a20fce792cf26a6cbbf6e27a2617.jpg",
+        "image_caption": [
+            "Figure 11: Learned OSFs on objects from REAL-NRF. The objects were captured in a dark room with a one-light-at-a-time setup. After training OSF on each object in this dataset, we are able to render the objects from novel viewpoints and lighting directions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            154,
+            821,
+            739
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B ABLATION EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            423,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b4157b61022d2d3c8f075161056762b9abe59507469ae48618f30c5afdad8cb.jpg",
+        "image_caption": [
+            "Figure 12: Ablation results on our OSF model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            142,
+            823,
+            417
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 12 shows ablation results on FURNITURE-REALISTIC. We evaluate different variants of our model: “Direct Only” which considers only direct illumination; “Indirect Only” which considers only indirect illumination; “Direct $^ +$ Shadows” which includes both direct illumination and shadows. Our full model (OSF) shows the most realistic rendering, with accurate shadows and indirect illumination effects such as the left side of the couches and tables appearing brighter due to indirect lighting from the left wall. Note that the white area on the right of the images represent rays with zero density that are composited onto a white background (and therefore do not contribute indirect illumination to the scene). ",
+        "bbox": [
+            173,
+            458,
+            825,
+            570
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/aac3b3e00f1a72e32e711f7f8082e45737aed705e332eed318b0455d108b3526.jpg",
+        "image_caption": [
+            "Figure 13: Comparisons on scene composition on FURNITURE-REALISTIC. The models OSF, oNeRF, and $_ { 0 - \\mathrm { N e R F } + \\mathrm { ~ S ~ } }$ are explained in $\\ S 5$ . Compared to o-NeRF, our model (OSF) is able to disentangle lighting-dependent appearance from view-dependent appearance for individual objects, and is able to render shadows cast by objects onto the ground correctly. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            583,
+            823,
+            842
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C COMPLEX ILLUMINATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            423,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we investigate how scenes composed of OSF objects can be rendered with complex illumination from an environment map. ",
+        "bbox": [
+            169,
+            126,
+            823,
+            155
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we apply the combination of a point light source and the environment map shown in the top-left corner of Figure 14 to light one of our scenes in FURNITURE-REALISTIC. This simulates the appearance of the scene as if the scene were inserted into a complex lighting environment, which stresses the benefits of the OSF path tracing framework. ",
+        "bbox": [
+            174,
+            162,
+            493,
+            275
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For each OSF sample point, we project the equirectangular coordinates of the environment map into spherical coordinates, sample 20 directions on the unit sphere uniformly at random, evaluate the OSF function for each incoming direction, and integrate them outgoing radiance using Equation 5. Please note that a green-blue tint is slightly apparent in the scene rendering, due to the contribution of green and blue lighting from the environment map. ",
+        "bbox": [
+            174,
+            287,
+            493,
+            426
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/74f46dde664bd5d47d66bf589c968aeb26ab69b2c53585a3a097ea99ebd89630.jpg",
+        "image_caption": [
+            "Figure 14: Complex illumination results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            513,
+            162,
+            820,
+            401
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            441,
+            117
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A flowchart of our method is shown in Figure 15. ",
+        "bbox": [
+            176,
+            126,
+            496,
+            141
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We approximate our model $F _ { \\Theta }$ with a multilayer perception (MLP) with rectified linear activations. The predicted density $\\sigma$ is view-invariant, while the scattering function value $\\rho$ is dependent on the incoming and outgoing light directions. We use an eight-layer MLP with 256 channels to predict $\\sigma$ , and a four-layer MLP with 128 channels to predict $\\rho$ . For positional encoding, we use $W = 1 0$ to encode the position $_ { \\textbf { \\em x } }$ and $W = 4$ to encode the incoming and outgoing directions $( \\omega _ { l } , \\omega _ { o } )$ , where $W$ is the highest frequency level. To avoid $\\rho$ from saturating in training, we adopt a scaled sigmoid (Brock et al., 2016) defined as $S ^ { \\prime } ( \\pmb { \\rho } ) = \\delta ( S ( \\pmb { \\rho } ) - 0 . 5 ) + 0 . 5$ with $\\delta = 1 . 2$ . We use a batch size of 4,096 rays. ",
+        "bbox": [
+            173,
+            147,
+            826,
+            260
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/e872a846f54ec153cb399e3329f55d9c4a07e4ff95330969d2e6e7aa06fc4226.jpg",
+        "image_caption": [
+            "Figure 15: Flowchart of our method. See $\\ S 4$ for more details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            272,
+            823,
+            551
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "For synthetic datasets, we sample $N _ { c } = 6 4$ coarse samples and $N _ { f } = 1 2 8$ fine samples per ray. For real world datasets, we sample $N _ { c } = 6 4$ coarse samples and $N _ { f } = 6 4$ fine samples per ray. We use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001, $\\beta _ { 1 } = 0 . 9$ , $\\beta _ { 2 } = 0 . 9 9 9$ , and $\\epsilon = 1 0 ^ { - 7 }$ . ",
+        "bbox": [
+            173,
+            598,
+            826,
+            654
+        ],
+        "page_idx": 16
+    }
+]

parse/dev/fR-EnKWL_Zb/fR-EnKWL_Zb.md

@@ -0,0 +1,358 @@
+# QUADTREE ATTENTION FOR VISION TRANSFORMERS
+
+Shitao Tang1∗, Jiahui Zhang2∗, Siyu $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { 2 }$ , Ping Tan12
+
+1Simon Fraser University, 2Alibaba A.I. Lab shitaot@sfu.ca, zjhthu@gmail.com, siting.zsy@alibaba-inc.com, pingtan@sfu.ca
+
+# ABSTRACT
+
+Transformers have been successful in many vision tasks, thanks to their capability of capturing long-range dependency. However, their quadratic computational complexity poses a major obstacle for applying them to vision tasks requiring dense predictions, such as object detection, feature matching, stereo, etc. We introduce QuadTree Attention, which reduces the computational complexity from quadratic to linear. Our quadtree transformer builds token pyramids and computes attention in a coarse-to-fine manner. At each level, the top $K$ patches with the highest attention scores are selected, such that at the next level, attention is only evaluated within the relevant regions corresponding to these top $K$ patches. We demonstrate that quadtree attention achieves state-of-theart performance in various vision tasks, e.g. with $4 . 0 \%$ improvement in feature matching on ScanNet, about $50 \%$ flops reduction in stereo matching, $0 . 4 \substack { - 1 . 5 \% }$ improvement in top-1 accuracy on ImageNet classification, $1 . 2 \substack { - 1 . 8 \% }$ improvement on COCO object detection, and $0 . 7 \mathrm { - } 2 . 4 \%$ improvement on semantic segmentation over previous state-of-the-art transformers. The codes are available at https://github.com/Tangshitao/QuadtreeAttention.
+
+# 1 INTROCUTION
+
+Transformers can capture long-range dependencies by the attention module and have demonstrated tremendous success in natural language processing tasks. In recent years, transformers have also been adapted to computer vision tasks for image classification (Dosovitskiy et al., 2020), object detection (Wang et al., 2021c), semantic segmentation (Liu et al., 2021), feature matching (Sarlin et al., 2020), and stereo (Li et al., 2021), etc. Typically, images are divided into patches and these patches are flattened and fed to a transformer as word tokens to evaluate attention scores. However, transformers have quadratic computational complexity in terms of the number of tokens, i.e. number of image patches. Thus, applying transformers to computer vision applications requires careful simplification of the involved computation.
+
+To utilize the standard transformer in vision tasks, many works opt to apply it on low resolution or sparse tokens. ViT (Dosovitskiy et al., 2020) uses coarse image patches of $1 6 \times 1 6$ pixels to limit the number of tokens. DPT (Ranftl et al., 2021) up-samples low-resolution results from ViT to high resolution maps to achieve dense predictions. SuperGlue (Sarlin et al., 2020) applies transformer on sparse image keypoints. Focusing on correspondence and stereo matching applications, Germain et al. (2021) and Li et al. (2021) also apply transformers at a low resolution feature map.
+
+However, as demonstrated in several works (Wang et al., 2021c; Liu et al., 2021; Sun et al., 2021; Li et al., 2021; Shao et al., 2020), applying transformers on high resolution is beneficial for a variety of tasks. Thus, many efforts have been made to design efficient transformers to reduce computational complexity. Linear approximate transformers (Katharopoulos et al., 2020; Wang et al., 2020) approximate standard attention computation with linear methods. However, empirical studies (Germain et al., 2021; Chen et al., 2021) show those linear transformers are inferior in vision tasks. To reduce the computational cost, the PVT (Wang et al., 2021c) uses downsampled keys and values, which is harmful to capture pixel-level details. In comparison, the Swin Transformer (Liu et al.,
+
+![](images/12a037cdc35a63c32e1f970a6056546225f5c75ca22bd0f15cd2b6f15529093d.jpg)  
+Figure 1: Illustration of QuadTree Attention. Quadtree attention first builds token pyramids by down-sampling the query, key and value. From coarse to fine, quadtree attention selects top $K$ (here, $K = 2$ ) results with the highest attention scores at the coarse level. At the fine level, attention is only evaluated at regions corresponding to the top $K$ patches at the previous level. The query sub-patches in fine levels share the same top $K$ key tokens and coarse level messages, e.g., green and yellow sub-patches at level 2 share the same messages from level 1. We only show one patch in level 3 for simplicity.
+
+2021) restricts the attention in local windows in a single attention block, which might hurt longrange dependencies, the most important merit of transformers.
+
+Unlike all these previous works, we design an efficient vision transformer that captures both fine image details and long-range dependencies. Inspired by the observation that most image regions are irrelevant, we build token pyramids and compute attention in a coarse to fine manner. In this way, we can quickly skip irrelevant regions in the fine level if their corresponding coarse level regions are not promising. For example, as in Figure 1, at the 1st level, we compute the attention of the blue image patch in image A with all the patches in image B and choose the top $K$ (here, $K = 2$ ) patches which are also highlighted in blue. In the 2nd level, for the four framed sub-patches in image A (which are children patches of the blue patch at the 1st level), we only compute their attentions with the sub-patches corresponding to the top $K$ patches in image B at the 1st level. All the other shaded sub-patches are skipped to reduce computation. We highlight two sub-patches in image A in yellow and green. Their corresponding top $K$ patches in image B are also highlighted in the same color. This process is iterated in the 3rd level, where we only show the sub-sub-patches corresponding to the green sub-patch at the 2nd level. In this manner, our method can both obtain fine scale attention and retain long-range connections. Most importantly, only sparse attention is evaluated in the whole process. Thus, our method has low memory and computational costs. Since a quadtree structure is formed in this process, we refer to our method as QuadTree Attention, or QuadTree Transformer.
+
+In experiments, we demonstrate the effectiveness of our quadtree transformer in both tasks requiring cross attention, e.g. feature matching and stereo, and tasks only utilizing self-attention, e.g. image classification and object detection. Our method achieves state-of-the-art performance with significantly reduced computation, comparing to relevant efficient transformers (Katharopoulos et al. (2020); Wang et al. (2021c); Liu et al. (2021)). In feature matching, we achieve $6 1 . 6 \ : \mathrm { A U C } @ 2 0 ^ { \circ }$ in ScanNet (Dai et al., 2017), 4.0 higher than the linear transformer (Katharopoulos et al., 2020) but with similar flops. In stereo matching, we achieve a similar end-point-error as standard transformer, (Li et al., 2021) but with about $50 \%$ flops reduction and $40 \%$ memory reduction. In image classification, we achieve $8 4 . 0 \%$ top-1 accuracy in ImageNet (Deng et al., 2009), $5 . 7 \%$ higher than ResNet152 (He et al., 2016) and $1 . 0 \%$ higher than the Swin Transformer-S (Liu et al., 2021). In object detection, our QuadTree Attention $^ +$ RetinaNet achieves 47.9 AP in COCO (Lin et al., 2014), 1.8 higher than the backbone PVTv2 (Wang et al., 2021b) with fewer flops. In semantic segementation, QuadTree Attention improves the performance by $0 . 7 \mathrm { - } 2 . 4 \%$ .
+
+# 2 RELATED WORK
+
+Efficient Transformers. Transformers have shown great success in both natural language processing and computer vision. Due to the quadratic computational complexity, the computation of full attention is unaffordable when dealing with long sequence tokens. Therefore, many works design efficient transformers, aiming to reduce computational complexity (Katharopoulos et al., 2020; Choromanski et al., 2020; Shao et al., 2021; Wang et al., 2020; Lee et al., 2019; Ying et al., 2018). Current efficient transformers can be categorized into three classes. 1) Linear approximate attention (Katharopoulos et al., 2020; Choromanski et al., 2020; Wang et al., 2020; Beltagy et al., 2020; Zaheer et al., 2020) approximates the full attention matrix by linearizing the softmax attention and thus can accelerate the computation by first computing the product of keys and values. 2) Inducing point-based linear transformers (Lee et al., 2019; Ying et al., 2018) use learned inducing points with fixed size to compute attention with input tokens, thus can reduce the computation to linear complexity. However, these linear transformers are shown to have inferior results than standard transformers in different works (Germain et al., 2021; Chen et al., 2021). 3) Sparse attention, including Longformer (Beltagy et al., 2020), Big Bird (Zaheer et al., 2020), etc, attends each query token to part of key and value tokens instead of the entire sequence. Unlike these works, our quadtree attention can quickly skip the irrelevant tokens according to the attention scores at coarse levels. Thus, it achieves less information loss while keeps high efficiency.
+
+Vision Transformers. Transformers have shown extraordinary performance in many vision tasks. ViT (Dosovitskiy et al., 2020) applies transformers to image recognition, demonstrating the superiority of transformers for image classification at a large scale. However, due to the computational complexity of full attention, it is hard to apply transformers in dense prediction tasks, e.g. object detection, semantic segmentation, etc. To address this problem, Swin Transformer (Liu et al., 2021) restricts attention computation in a local window. Focal transformer (Yang et al., 2021) uses twolevel windows to increase the ability to capture long-range connection for local attention methods. Pyramid vision transformer (PVT) (Wang et al., 2021c) reduce the computation of global attention methods by downsampling key and value tokens. Although these methods have shown improvements in various tasks, they have drawbacks either in capturing long-range dependencies (Liu et al., 2021) or fine level attention (Wang et al., 2021c). Different from these methods, our method simultaneously capture both local and global attention by computing attention from full image levels to the finest token levels with token pyramids in one single block. Besides, the K-NN transformers (Wang et al., 2021a; Zhao et al., 2019) aggregate messages from top $K$ most similar tokens as ours, but they compute the attention scores among all pairs of query and key tokens, and thus still has quadratic complexity.
+
+Beyond self-attention, many tasks can largely benefit from cross attention. Superglue (Sarlin et al., 2020) processes detected local descriptors with self- and cross attention and shows significant improvement in feature matching. Standard transformers can be applied in SuperGlue because only sparse keypoints are considered. SGMNet (Chen et al., 2021) further reduces the computation by attending to seeded matches. LoFTR (Sun et al., 2021) utilizes linear transformer (Katharopoulos et al., 2020) on low-resolution feature maps to generate dense matches. For stereo matching, STTR (Li et al., 2021) applies self- and cross attention along epipolar lines and reduces the memory by gradient checkpointing engineering techniques. However, due to the requirement of processing a large number of points, these works either use linear transformers, which compromise performance, or a standard transformer, which compromises efficiency. In contrast, our transformer with quadtree attention achieves a significant performance boost compared with linear transformer or efficiency improvement compared with standard transformer. Besides, it can be applied to both self-attention and cross attention.
+
+# 3 METHOD
+
+We first briefly review the attention mechanism in transformers in Section 3.1 and then formulate our quadtree attention in Section 3.2.
+
+# 3.1 ATTENTION IN TRANSFORMER
+
+Vision transformers have shown great success in many tasks. At the heart of a transformer is the attention module, which can capture long-range information between feature embeddings. Given two image embeddings $\mathbf { X } _ { 1 }$ and $\mathbf { X } _ { 2 }$ , the attention module passes information between them. Selfattention is the case when $\mathbf { X } _ { 1 }$ and $\mathbf { X } _ { 2 }$ are the same, while cross attention covers a more general situation when $\mathbf { X } _ { 1 }$ and $\mathbf { X } _ { 2 }$ are different. It first generates the query $\mathbf { Q }$ , key $\mathbf { K }$ , and value $\mathbf { V }$ by the
+
+![](images/16f2691479fd7ec52d4b755609b61089c1ad77510396ee050e16edfc4cef4366.jpg)  
+⇤li i iFigure 2: Illustration of quadtree message aggregation for a query token $q _ { i }$ l+2. (a) shows the token ⇤l+1 ⇤iREFERENCESm3ipyramids and involved key/value tokens in each level. Attention scores are marked in the first two l+2levels for clarification, and the top $K$ REFERENCESscores are highlighted in red. (b) shows message aggregation i REFERENCESfor QuadTree-A architecture. The message is assembled from different levels along a quadtree. (c) REFERENCES shows message aggregation for QuadTree-B architecture. The message is collected from overlapping regions from different levels.
+
+following equation,
+
+$$
+\begin{array} { r } { \mathbf { Q } = \mathbf { W } _ { q } \mathbf { X } _ { 1 } , } \\ { \mathbf { K } = \mathbf { W } _ { k } \mathbf { X } _ { 2 } , } \\ { \mathbf { V } = \mathbf { W } _ { v } \mathbf { X } _ { 2 } , } \end{array}
+$$
+
+where $\mathbf { W } _ { q } , \ \mathbf { W } _ { k }$ and $\mathbf { W } _ { v }$ are learnable parameters. Then, it performs message aggregation by computing the attention scores between query and key as following,
+
+$$
+\mathbf { Y } = \operatorname { s o f t m a x } ( \frac { \mathbf { Q } \mathbf { K } ^ { T } } { \sqrt { C } } ) \mathbf { V } ,
+$$
+
+where $C$ 1is the embedding channel dimension. The above process has $O ( N ^ { 2 } )$ computational complexity, where $N$ is the number of image patches in a vision transformer. This quadratic complexity hinders transformers from being applied to tasks requiring high resolution output. To address this problem, PVT (Wang et al. (2021c)) downsamples $\mathbf { K }$ and $\mathbf { V }$ , while Swin Transformer (Liu et al. (2021)) limits the attention computation within local windows.
+
+# 3.2 QUADTREE ATTENTION
+
+In order to reduce the computational cost of vision transformers, we present QuadTree Attention. As the name implies, we borrow the idea from quadtrees, which are often used to partition a twodimensional space by recursively subdividing it into four quadrants or regions. Quadtree attention computes attention in a coarse to fine manner. According to the results at the coarse level, irrelevant image regions are skipped quickly at the fine level. This design achieves less information loss while keeping high efficiency.
+
+The same as the regular transformers, we first linearly project $\mathbf { X } _ { 1 }$ and $\mathbf { X } _ { 2 }$ to the query, key, and value tokens. To facilitate fast attention computation, we construct $L$ -level pyramids for query $\mathbf { Q }$ , key $\mathbf { K }$ , and value $\mathbf { V }$ tokens by downsampling feature maps. For query and key tokens, we use average pooling layers. For value tokens, average pooling is used for cross attention tasks and convolutionalnormalization-activation layers with stride 2 are used for self attention tasks if no special statement. As shown in Figure 1, after computing attention scores in the coarse level, for each query token, we select the top $K$ key tokens with the highest attention scores. At the fine level, query sub-tokens only need to be evaluated with those key sub-tokens that correspond to one of the selected $K$ key tokens at the coarse level. This process is repeated until the finest level. After computing the attention scores, we aggregate messages at all levels, where we design two architectures named as QuadTree-A and QuadTree-B.
+
+QuadTree-A. Considering the $i$ -th query token $\mathbf { q } _ { i }$ at the finest level, we need to compute its received message $\mathbf { m } _ { i }$ from all key tokens. This design assembles the full message by collecting partial messages from different pyramid levels. Specifically,
+
+$$
+\mathbf { m } _ { i } = \sum _ { 1 \leq l \leq L } \mathbf { m } _ { i } ^ { l } ,
+$$
+
+where $\mathbf { m } _ { i } ^ { l }$ indicates the partial message evaluated at level $l$ . This partial message $\mathbf { m } _ { i } ^ { l }$ assemble messages at the $l$ -th level from tokens within the region $\Omega _ { i } ^ { l }$ , which will be defined later. In this way, messages from less related regions are computed from coarse levels, while messages from highly related regions are computed in fine levels. This scheme is illustrated in Figure 2 (b), message $\mathbf { m } _ { i }$ is generated by assembling three partial messages that are computed from different image regions with different colors, which collectively cover the entire image space. The green region indicates the most relevant region and is evaluated at the finest level, while the red region is the most irrelevant region and is evaluated at the coarsest level. The region $\Omega _ { i } ^ { l }$ can be defined as $\Gamma _ { i } ^ { l } - \Gamma _ { i } ^ { l + 1 }$ , where the image region $\Gamma _ { i } ^ { l }$ corresponds to the top $K$ tokens at the level $l - 1$ . The regions $\Gamma _ { i } ^ { l }$ are illustrated in Figure 2 (c). The region $\Gamma _ { i } ^ { 1 }$ covers the entire image.
+
+The partial messages are computed as,
+
+$$
+\mathbf { m } _ { i } ^ { l } = \sum _ { j \in \Omega _ { i } ^ { l } } s _ { i j } ^ { l } \mathbf { v } _ { j } ^ { l } ,
+$$
+
+where $s _ { i j } ^ { l }$ is the attention score between the query and key tokens at level $l$ . Figure 2 (a) highlights query and key tokens involved in computing $\mathbf { m } _ { i } ^ { l }$ with the same color as $\Omega _ { i } ^ { l }$ . Attention scores are computed recursively,
+
+$$
+\begin{array} { r } { s _ { i j } ^ { l } = s _ { i j } ^ { l - 1 } t _ { i j } ^ { l } . } \end{array}
+$$
+
+Here, $s _ { i j } ^ { l - 1 }$ is the score of corresponding parent query and key tokens and $s _ { i j } ^ { 1 } = 1$ . The tentative attention score $t _ { i j } ^ { l }$ is evaluated according to Equation 1 among the $2 \times 2$ tokens of the same parent query token. For QuadTree-A, we use average pooling layers to downsample all query, key and value tokens.
+
+QuadTree-B. The attention scores $s _ { i j } ^ { l }$ in QuadTree-A are recursively computed from all levels, which makes scores smaller at finer levels and reduces the contributions of fine image features. Besides, fine level scores are also largely affected by the inaccuracy at coarse levels. So we design a different scheme, referred as QuadTree- $\mathbf { B }$ in this paper, to address this problem. Specifically, we compute $\mathbf { m } _ { i }$ as a weighted average of the partial messages from different levels,
+
+$$
+\mathbf { m } _ { i } = \sum _ { 1 \leq l \leq L } w _ { i } ^ { l } \mathbf { m } _ { i } ^ { l } ,
+$$
+
+where $w _ { i } ^ { l }$ is a learned weight. As shown in Figure 2 (c), the partial messages here overlap with each other, which are computed as,
+
+$$
+\mathbf { m } _ { i } ^ { l } = \mathrm { A t t e n t i o n } ( \mathbf { q } _ { i } ^ { l } , \mathbf { K } _ { \Gamma _ { i } ^ { l } } ^ { l } , \mathbf { V } _ { \Gamma _ { i } ^ { l } } ^ { l } ) ,
+$$
+
+where Attention is the attention message computation as Equation 1. Here, $\mathbf { K } _ { \Gamma _ { i } ^ { l } } ^ { l }$ and $\mathbf { V } _ { \Gamma _ { i } ^ { l } } ^ { l }$ are matrices formed by stacking all keys and values within the region $\Gamma _ { i } ^ { l }$ .
+
+Both QuadTree-A and QuadTree-B involve only sparse attention evaluation. Thus, our method largely reduces computational complexity. As analyzed in Appendix A.1, the computational complexity of our quadtree attention is linear to the number of tokens.
+
+<table><tr><td></td><td></td><td>AUC@5°</td><td>AUC@10°</td><td>AUC@20°</td></tr><tr><td rowspan="3">Others</td><td>ContextDesc + SGMNet(Chen et al. (2021))</td><td>15.4</td><td>32.3</td><td>48.8</td></tr><tr><td>SuperPoint + OANet (Zhang et al. (2019b))</td><td>11.8</td><td>26.9</td><td>43.9</td></tr><tr><td>SuperPoint + SuperGlue (Sarlin et al. (2020))</td><td>16.2</td><td>33.8</td><td>51.9</td></tr><tr><td rowspan="3">LoFTR-lite</td><td>DRC-Net (Li et al. (2020)) Linear Att. (LoFTR) (Katharopoulos et al. (2020))</td><td>7.7 16.1</td><td>17.9 32.6</td><td>30.5</td></tr><tr><td>PVT (Wang et al., 2021c)</td><td>16.2</td><td>32.7</td><td>49.0 49.2</td></tr><tr><td>QuadTree-A (ours,K = 8)</td><td>16.8</td><td>33.4</td><td>50.5</td></tr><tr><td rowspan="4">LoFTR</td><td>QuadTree-B (ours,K = 8)</td><td>17.4</td><td>34.4</td><td>51.6</td></tr><tr><td>Linear Att. (LoFTR)* (Sun et al. (2021), 64 GPUs)</td><td>22.1</td><td>40.8</td><td>57.6</td></tr><tr><td>Linear Att. (LoFTR) (Katharopoulos et al. (2020))</td><td>21.1</td><td>39.5</td><td>56.6</td></tr><tr><td>QuadTree-B (ours,K = 8) QuadTree-B* (ours,K = 16)</td><td>23.0</td><td>41.7</td><td>58.5</td></tr></table>
+
+Table 1: Results on feature matching. The symbol $\star$ indicates results cited from (Sun et al., 2021), where the model is trained with a batch size of 64 on 64 GPUs (a more preferable setting than ours). The symbol $^ *$ indicates we use the ViT (Dosovitskiy et al., 2020)-like architecture for transformer blocks. For PVT and our method, we replace the original linear attention in LoFTR with corresponding attentions.
+
+Multiscale position encoding. The computation of attention is permutation invariant to tokens, and thus positional information is missed. To address this problem, we adopt the locally-enhanced positional encoding (LePE) (Dong et al., 2021) at each level to design a multiscale position encoding. Specifically, for level $l$ , we apply unshared depth-wise convolution layers to value tokens $\mathbf { V } ^ { l }$ to encode the positional information.
+
+# 4 EXPERIMENT
+
+We experiment our quadtree transformer with four representative tasks, including feature matching, stereo, image classification, and object detection. The first two tasks require cross attention to fuse information across different images, while the latter two involve only self-attention. We implement our quadtree transformer using PyTorch and CUDA kernels. More implementation details are provided in Appendix B.
+
+# 4.1 CROSS ATTENTION TASKS
+
+# 4.1.1 FEATURE MATCHING
+
+Finding feature correspondence (Luo et al., 2019; DeTone et al., 2018) across different images is a precedent problem for many 3D computer vision tasks. It is typically evaluated by the accuracy of the camera pose estimated from the corresponding points. We follow the framework proposed in a recent state-of-the-art work LoFTR (Sun et al., 2021), which consists of a CNN-based feature extractor and a transformer-based matcher. We replace the linear transformer (Katharopoulos et al., 2020) in LoFTR with our quadtree transformer. Besides, we also implement a new version of LoFTR with the spatial reduction (SR) attention (Wang et al., 2021c) for additional comparison.
+
+Setting. We experiment on ScanNet (Dai et al., 2017) with 1,513 scans. In order to accelerate training, we design the LoFTR-lite setting, which uses half of the feature channels of LoFTR and 453 training scans. Ablation studies in section 4.3 are conducted in this setting. We train both LoFTRlite and LoFTR for 30 epochs with batch size 8. For quadtree transformer, we build pyramids of three levels with the coarsest resolution at $1 5 \times 2 0$ pixels. We set the parameter $K$ to 8 at the finest level, and double it at coarser levels. For the SR attention, we average pool the value and key tokens to the size $8 \times 8$ to keep similar memory usage and flops as our quadtree attention. More details are included in Appendix B.1.
+
+Results. Table 1 shows the AUC of camera pose errors1 under $( 5 ^ { \circ } , 1 0 ^ { \circ } , 2 0 ^ { \circ } )$ . We can see that the SR attention achieves similar results with linear transformer. In comparison, both QuadTree-A and QuadTree-B outperform linear transformer and SR attention by a large margin. Quadtree-B generally performs better than Quadtree-A. Quadtree-B has 2.6 and 1.9 improvements in terms of AUC $@ 2 0 ^ { \circ }$ over linear transformer on LoFTR-lite and LoFTR respectively. To further enhance the
+
+<table><tr><td></td><td>EPE (px)</td><td>IOU</td><td>Flops (G)</td><td>Mem. (MB)</td></tr><tr><td>GA-Net (Zhang et al., 2019a)</td><td>0.89</td><td>/</td><td>T</td><td>/</td></tr><tr><td>GWC-Net (Guo et al., 2019)</td><td>0.97</td><td>/</td><td>305</td><td>4339</td></tr><tr><td>Bi3D (Badki et al., 2020)</td><td>1.16</td><td>/</td><td>897</td><td>10031</td></tr><tr><td>STTR(Vanilla Transformer) (Li et al., 2021)</td><td>0.45</td><td>0.92</td><td>490</td><td>8507</td></tr><tr><td>QuadTree-B (ours,K = 6)</td><td>0.46</td><td>0.99</td><td>254 (52%)</td><td>5381 (63%)</td></tr></table>
+
+Table 2: Results of stereo matching. QuadTree-B achieves similar performance as STTR but with significantly lower flops and memory usage.
+
+results, we train a model with $K = 1 6$ and leverage a ViT (Dosovitskiy et al., 2020)-like transformer archtecture instead of the original one used in (Sun et al., 2021). This model achieves 4 improvements on $\mathbf { A U C } @ 2 0 ^ { \circ }$ over (Sun et al., 2021), where the LoFTR model is trained with a batch size of 64 with 64 GPUs, a more preferable setting leading to slightly better results than our linear transformer implementation shown in Table 1.
+
+# 4.1.2 STEREO MATCHING
+
+Stereo matching aims to find corresponding pixels on epipolar lines between two rectified images. The recent work STTR (Li et al., 2021) applies transformers to feature points between epipolar lines and achieves state-of-the-art performance. Note here, both self- and cross attention are applied along epipolar lines, pixels across different lines are not considered in the attention computation. We replace the standard transformer in STTR (Li et al., 2021) with our quadtree transformer.
+
+Setting. We experiment on the Scene Flow FlyingThings3D (Mayer et al., 2016) synthetic dataset, which contains 25,466 images with a resolution of $9 6 0 \times 5 4 0$ . We build pyramids of four levels to evaluate quadtree attention. While the STTR is applied to features of 1/3 of image resolution, we use feature maps of 1/2 of image resolution. More details about the network are included in Appendix B.2.
+
+Results. We report EPE (End-Point-Error) in non-occluded regions and IOU (Intersection-overUnion) for occlusion estimation in Table 2 as (Li et al., 2021). Computational complexity and memory usage are also reported. Compared with STTR based on the standard transformer, our quadtree transformer achieves similar EPE (0.45 px vs $0 . 4 6 ~ \mathrm { p x }$ ) and higher IOU for occlusion estimation, but with much lower computational and memory costs, with only $52 \%$ FLOPs and $63 \%$ memory consumption.
+
+# 4.2 SELF-ATTENTION TASK
+
+This section presents results on image classification and object detection. In the past, convolutional neural networks (CNNs) have dominated these tasks for a long time. Recently, vision transformers (Dosovitskiy et al., 2020; Liu et al., 2021; Wang et al., 2021c) show excellent potential on these problems, thanks to their capability in capturing long-range interactions. To compare our method with these vision transformers on image classification, we use the public codes of PVTv2 (Wang et al., 2021c) and replace all the spatial reduction attention with our quadtree attention. For object detection, we further apply a representative object detection framework, RetinaNet (Lin et al., 2017), which is a widely used single-stage object detector.
+
+# 4.2.1 IMAGE CLASSIFICATION
+
+Settings. We evaluate image classification on the ImageNet-1K dataset (Deng et al., 2009), which consists of 1.28M training images and 50K validation images from 1,000 categories. We build token pyramids with the coarsest level at a resolution of $7 \times 7$ and set $K = 8$ . We crop and resize the input images to $2 2 4 \times 2 2 4$ pixels and train the model with a mini-batch of 128. All models are trained for 300 epochs from scratch on 8 GPUs. All the other training settings are the same as in (Wang et al., 2021c). We build five different quadtree transformers at different complexity, named as b0, b1, b2, b3, b4. These models are gradually deeper and wider. More configuration details can be found in Appendix. B.3.
+
+Results. We provide the top-1 accuracy of various methods and network settings in Table 3. These results are grouped into five sections, each with several methods of similar network complexity, as indicated by the number of parameters. As shown in Table 3, QuadTree-B outperforms PVTv2 by $0 . 4 \% { - } 1 . 5 \%$ in top-1 accuracy with fewer parameters. Swin Transformer-S adopts local attention and is surpassed by our QuadTree-B-b2 by $1 . 0 \%$ in top-1 accuracy. This result proves that global information is important. In general, our quadtree transformer leverages both global information at the coarse level and local information at fine levels, and outperforms both PVTv2 and Swin Transformer.
+
+Table 3: Image classification results. We report top-1 accuracy on the ImageNet validation set.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Param (M)  Flops (G)  Top1 (%)</td></tr><tr><td rowspan=1 colspan=1>PVTv2-b0 (Wang et al., 2021b)QuadTree-A-bO (ours)QuadTree-B-bO (ours)</td><td rowspan=1 colspan=1>3.7         0.6        70.53.4          0.6        70.93.5         0.7        72.0</td></tr><tr><td rowspan=2 colspan=1>ResNet18 (He et al., 2016)PVTv1-Tiny (Wang et al.,2021c)PVTv2-b1 (Wang et al.,2021b)QuadTree-B-b1 (ours)</td><td rowspan=1 colspan=1>11.7        1.8        69.8</td></tr><tr><td rowspan=1 colspan=1>13.2        2.1         75.114.0        2.1         78.713.6        2.3         80.0</td></tr><tr><td rowspan=10 colspan=1>ResNet50 (He et al.,2016)ResNeXt50-32x4d (Xie et al., 2017)RegNetY-4G (Radosavovic et al.,2020)DeiT-Small/16 (Touvron et al., 2021)Swin-T (Liu et al.,2021)TNT-S (Han et al., 2021)CeiT (Yuan et al.,2021a)PVTv2-b2 (Wang et al.,2021c)Focal-T(Yang et al.,2021)QuadTree-B-b2 (ours)</td><td rowspan=1 colspan=1>25.1        4.1         76.4</td></tr><tr><td rowspan=1 colspan=1>25.0        4.3         77.6</td></tr><tr><td rowspan=1 colspan=1>21.0        4.0         80.0</td></tr><tr><td rowspan=1 colspan=1>22.1        4.6        79.9</td></tr><tr><td rowspan=1 colspan=1>29.0        4.5         81.3</td></tr><tr><td rowspan=1 colspan=1>23.8         5.2         81.3</td></tr><tr><td rowspan=1 colspan=1>24.2        4.5         82.0</td></tr><tr><td rowspan=1 colspan=1>25.4        4.0         82.0</td></tr><tr><td rowspan=1 colspan=1>29.1        4.9         82.2</td></tr><tr><td rowspan=1 colspan=1>24.2        4.5        82.7</td></tr><tr><td rowspan=6 colspan=1>ResNet101 (He et al.,2016)ResNeXt101-32x4d (Xie et al., 2017)RegNetY-8G (Radosavovic et al.,2020)CvT-21 (Wu et al., 2021)PVTv2-b3 (Wang et al., 2021c)Quadtree-B-b3 (ours)</td><td rowspan=1 colspan=1>44.7        7.9        77.4</td></tr><tr><td rowspan=1 colspan=1>44.2        8.0        78.8</td></tr><tr><td rowspan=1 colspan=1>39.0        8.0        81.7</td></tr><tr><td rowspan=1 colspan=1>32.0        7.1        82.5</td></tr><tr><td rowspan=1 colspan=1>45.2        6.9        83.2</td></tr><tr><td rowspan=1 colspan=1>46.3        7.8        83.7</td></tr><tr><td rowspan=6 colspan=1>ResNet152 (He etal.,2016)T2T-ViTt-24 (Yuan et al., 2021b)Swin-S (Liu et al., 2021)Focal-Small (Yang et al., 2021)PVTv2-b4 (Wang et al., 2021c)Quadtree-B-b4 (ours)</td><td rowspan=1 colspan=1>60.2         11.6       78.3</td></tr><tr><td rowspan=1 colspan=1>64.0        15.0       82.2</td></tr><tr><td rowspan=1 colspan=1>50.0        8.7        83.0</td></tr><tr><td rowspan=1 colspan=1>51.1        9.1        83.5</td></tr><tr><td rowspan=1 colspan=1>62.6        10.1       83.6</td></tr><tr><td rowspan=1 colspan=1>64.2        11.5       84.0</td></tr></table>
+
+Table 4: Object detection results on COCO val2017 with RetinaNet. We use PVTv2 backbone and replace the reduction attention with quadtree attention. ‘Flops’ is the backbone flops for input image size of $8 0 0 \times 1 , 3 3 3$ .   
+
+<table><tr><td></td><td>Flops (G)</td><td>AP</td><td>AP50</td><td>AP75</td><td>APs</td><td>APM</td><td>APL</td></tr><tr><td>PVTv2-b0 (Wang et al., 2021b) QuadTree-A-b0 (K=32,ours)</td><td>28.3 16.0</td><td>37.2 37.0</td><td>57.2 56.8</td><td>39.5 38.9</td><td>23.1 22.8</td><td>40.4 39.7</td><td>49.7 50.0</td></tr><tr><td>QuadTree-B-b0 (K=32,ours) ResNet18 (He et al.,2016)</td><td>16.5 38.6</td><td>38.4 31.8</td><td>58.7 49.6</td><td>41.1 33.6</td><td>22.5 16.3</td><td>41.7 34.3</td><td>51.6 43.2</td></tr><tr><td>PVTv1-Tiny (Wang et al.,2021c) PVTv2-b1 (Wang et al., 2021b) Quadtree-B-b1 (K=32,ours)</td><td>72.5 78.8 56.2</td><td>36.7 41.2 42.6</td><td>56.9 61.9 63.6</td><td>38.9 43.9 45.3</td><td>22.6 25.4 26.8</td><td>38.8 44.5 46.1</td><td>50.7 54.3 57.2</td></tr><tr><td>ResNet50 (He et al.,2016)</td><td>87.3</td><td>36.3</td><td>55.3</td><td>38.6</td><td>19.3</td><td>40.0</td><td>48.8</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ResNet101 (He et al.,2016)</td><td>166.3</td><td>38.5</td><td>57.8</td><td>41.2</td><td></td><td></td><td></td></tr><tr><td>ResNeXt101-32x4d (Xie et al., 2017)</td><td></td><td></td><td></td><td></td><td>21.4</td><td>42.6</td><td>51.1</td></tr><tr><td></td><td>170.2</td><td>39.9</td><td>59.6</td><td>42.7</td><td>22.3</td><td>44.2</td><td></td></tr><tr><td>PVTv1-small(Wang et al.,2021c)</td><td></td><td></td><td></td><td></td><td></td><td></td><td>52.5</td></tr><tr><td></td><td>139.8</td><td>36.7</td><td>56.9</td><td>38.9</td><td>25.0</td><td>42.9</td><td>55.7</td></tr><tr><td>PVTv2-b2 (Wang et al.,2021c)</td><td>149.1</td><td>44.6</td><td>65.6</td><td>47.6</td><td>27.4</td><td>48.8</td><td>58.6</td></tr><tr><td>QuadTree-B-b2 (K=32,ours) PVTv1-Medium (Wang et al., 2021c)</td><td>108.6</td><td>46.2</td><td>67.2</td><td>49.5</td><td>29.0</td><td>50.1</td><td>61.8</td></tr><tr><td>PVTv2-b3 (Wang et al.,2021b)</td><td>237.4</td><td>41.9</td><td>63.1</td><td>44.3</td><td>25.0</td><td>44.9</td><td>57.6</td></tr><tr><td></td><td>243.0</td><td>45.9</td><td>66.8</td><td>49.3</td><td>28.6</td><td>49.8</td><td>61.4</td></tr><tr><td>QuadTree-B-b3 (ours)</td><td>193.9</td><td>47.3</td><td>68.2</td><td>50.6</td><td>30.4</td><td>51.3</td><td>62.9</td></tr><tr><td>PVTv1-Large (Wang et al.,2021c) PVTv2-b4 (Wang et al.,2021b)</td><td>346.6</td><td>42.6</td><td>63.7</td><td>45.4</td><td>25.8</td><td>46.0</td><td>58.4</td></tr><tr><td></td><td>353.3</td><td>46.1</td><td>66.9</td><td>49.2</td><td>28.4</td><td>50.0</td><td>62.2</td></tr><tr><td>QuadTree-B-b4 (ours)</td><td>283.9</td><td>47.9</td><td>69.1</td><td>51.3</td><td>29.4</td><td>52.2</td><td>63.9</td></tr></table>
+
+Table 5: To fairly compare with Swin, PVT, Focal attention and our method, we replace the attention module in PVTv2-b0 with different types of attention and same position encoding method LePE and run image classification and object detection respectively.   
+
+<table><tr><td></td><td colspan="2">ImageNet-1K</td><td colspan="3">COCO (RetinaNet)</td></tr><tr><td></td><td>Flops (G) Top-1 (%)</td><td>Mem. (MB)</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>PVTv2 (Wang et al.,2021b) PVTv2+LePE (Dong et al., 2021)</td><td>0.6 70.5 0.6 70.9</td><td>574 574</td><td>37.2 37.6</td><td>57.2 57.8</td><td>39.5 39.9</td></tr><tr><td>Swin (Liu et al., 2021)</td><td>0.6 70.5</td><td>308</td><td>35.3</td><td>54.2</td><td>37.4</td></tr><tr><td>Swin+LePE Focal Attention (Yang et al., 2021)</td><td>0.6 70.7 0.7 71.6</td><td>308 732</td><td>35.8 37.5</td><td>55.3 57.6</td><td>37.7 39.5</td></tr><tr><td>Focal Attention+LePE</td><td>0.7 71.5</td><td>732</td><td>37.1</td><td>57.0</td><td>39.4</td></tr><tr><td>QuadTree-B</td><td>0.6 72.0</td><td>339</td><td>38.4</td><td>58.8</td><td>41.1</td></tr></table>
+
+# 4.2.2 OBJECT DETECTION
+
+Settings. We experiment on the COCO dataset. All models are trained on COCO train 2017 (118k images) and evaluated on val 2017 (5k images). We initialize the quadtree backbone with the weights pre-trained on ImageNet. We adopt the same setting as PVTv2, training the model with a batch size of 16 and AdamW optimizer with an initial learning rate of $1 \times 1 0 ^ { - \overline { { 4 } } }$ for 12 epochs. We use the standard metric average precision to evaluate our method.
+
+Results. We mainly compare our method with PVTv2, ResNet (He et al., 2016), and ResNeXt (Xie et al., 2017) using detection framework of RetinaNet (Lin et al., 2017), which are state-ofthe-art backbones for dense prediction. Table 4 lists the average precision of different methods and their backbone flops for images of resolution of $8 0 0 \times 1 , 3 3 3$ . Benefiting from the coarse to fine mechanism, a small $K$ is enough for our method. Thus, the computation can be reduced when using high resolution images. We can see that QuadTree-B achieves higher performance, but with much fewer flops than PVTv2. Our quadtree transformer also outperforms ResNet and ResNeXt. For example, QuadTree-B-b2 outperform ResNet101 and ResNeXt101-32x4d by 7.7 AP and 6.3 AP respectively with about $40 \%$ backbone flops reduction. We also show Mask-RCNN results (He et al., 2017) in Appendix. E.
+
+# 4.3 COMPARISON WITH OTHER ATTENTION MECHANISMS
+
+For a fair comparison with other attention mechanisms, we test these attention mechanisms under the same backbone and training settings. Specifically, we replace the original attention module in PVTv2-b0 with the attention method used in Swin Transformer and Focal Transformer. For more fair comparison, we adopt the same positional encoding LePE (Dong et al., 2021) to PVTv2, Swin and Focal transformer. As shown in Table 5, QuadTree attention obtain consistently better performance than Swin and PVTv2 in both classification task and detection task. Compared with focal attention, our method gets 0.9 higher AP in object detection, which might be because that QuadTree attention can always cover the whole images, while Focal attention only covers $1 / 6$ of the image in the first stage. More experiments on Swin-like architecture can be found in Appendix E.
+
+For cross attention tasks, we also provide visualization of attention score as shown in Fig.5 in Appendix E. Our method can attend to much more related regions than PVT (Wang et al., 2021b) and Linear attention (Katharopoulos et al., 2020).
+
+# 5 CONCLUSION
+
+We introduce QuadTree Attention to reduce the computational complexity of vision transformers from quadratic to linear. Quadtree transformers build token pyramids and compute attention in a coarse-to-fine manner. At each level, top $K$ regions with the highest attention scores are selected, such that in finer level, computation in irrelevant regions can be quickly skipped. Quadtree attention can be applied to cross attention as well as self-attention. It achieves state-of-the-art performance in various tasks including feature matching, stereo, image classification, and object detection.
+
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+
+# A APPENDIX
+
+# A.1 COMPLEXITY ANALYSIS
+
+In this section, we analyze the computational complexity of quadtree attention. Suppose the lengths of the query tokens, key tokens, and value tokens are all $H \times W$ . We build token pyramids of $L$ levels, the $\bar { l } ^ { t h }$ level has a token length of $\frac { H W } { 4 ^ { l - 1 } }$ . The flops of computing quadtree attention is,
+
+$$
+\begin{array} { l } { { \displaystyle { \mathrm { F l o p s } } = 2 ( H _ { 0 } ^ { 2 } W _ { 0 } ^ { 2 } + \sum _ { l = 2 } ^ { L - 1 } \frac { 4 K H W } { 4 ^ { l - 1 } } ) } } \\ { { \displaystyle ~ = 2 ( H _ { 0 } ^ { 2 } W _ { 0 } ^ { 2 } + \frac { 4 } { 3 } ( 1 - 4 ^ { 1 - L } ) K H W ) . } } \end{array}
+$$
+
+Here, $H _ { 0 }$ and $W _ { 0 }$ are the height and width of the coarsest level of token pyramids. Therefore, $H _ { 0 } ^ { 2 } W _ { 0 } ^ { 2 }$ is a constant and the computational complexity is $O ( K H W )$ . Since $K$ is a constant number, the complexity of quadtree attention is linear to the number of tokens.
+
+Table 6: Feature matching results on megadepth. Our method obtains better performance than other methods.   
+
+<table><tr><td></td><td>AUC@5°</td><td>AUC@10°</td><td>AUC@20°</td></tr><tr><td>DRC-Net (Li et al., 2020)</td><td>27.0</td><td>43.0</td><td>58.3</td></tr><tr><td>SuperPoint + SuperGlue (Sarlin et al., 2020)</td><td>42.2</td><td>61.2</td><td>76.0</td></tr><tr><td>LoFTR (Sun et al., 2021)</td><td>52.8</td><td>69.2</td><td>81.2</td></tr><tr><td>QuadTree-B (ours, K=16)</td><td>54.6</td><td>70.5</td><td>82.2</td></tr></table>
+
+# B ADDITIONAL EXPERIMENTS AND IMPLEMENTATION DETAILS
+
+# B.1 FEATURE MATCHING
+
+Implementation details. We train and evaluate the model in ScanNet (Dai et al., 2017), where 230M image pairs is sampled for training, with overlapping scores between 0.4 and 0.8. ScanNet provides RGB images, depth maps, and ground truth camera poses on a well-defined training and testing split. Following the same evaluation settings as Sarlin et al. (2020) and Sun et al. (2021), we evaluate our method on the 1,500 testing pairs from (Sarlin et al., 2020). For both trainig and testing, all images and depth maps are resized to $6 4 0 \times 4 8 0$ . Following (Sun et al., 2021), we compute the camera pose by solving the essential matrix from predicted matches with RANSAC. We report the AUC of the pose error at thresholds $( 5 ^ { \circ } , 1 0 ^ { \circ } , 2 0 ^ { \circ } )$ , where the pose error is defined as the maximum of angular error in rotation and translation. We only replace the coarse level transformer with quadtree attention.
+
+Results of megadepth. We show our results on Megadepth (Li & Snavely, 2018) in Table 6. We can see our method outperforms others by a large margin.
+
+# B.2 STEREO MATCHING
+
+Our network is based on the STTR (Li et al., 2021), where we replace the standard transformer with our quadtree transformer. The network consists of a CNN backbone which outputs feature maps of 1/2 image resolution, a quadtree transformer with both self- and cross attention, a regression head with optimal transport layers (Cuturi, 2013), and a context adjust layer to refine the disparity. Six self- and cross attention layers are used with 128 channels. We build pyramids with four levels for quadtree attention, and apply the Sinkhorn algorithm (Cuturi, 2013) for 10 iteration for optimal transport. We follow STTR to train the network, with 15 epochs of AdamW optimizer. OneCycle learning rate scheduler is used with a leaning rate of 6e-4 and a batch size of 8.
+
+# B.3 IMAGE CLASSIFICATION
+
+This paragraph introduces the details of PVTv2-b0, b1, b2, b3, b4. All these five networks have 4 stages. Each stage is dolutions for each stage are $\begin{array} { r } { \frac { H } { 4 } \times \frac { W } { 4 } , \frac { H } { 8 } \times \frac { W } { 8 } , \frac { H } { 1 6 } \times \frac { W } { 1 6 } } \end{array}$ ous sand $\frac { H } { 3 2 } \times \frac { W } { 3 2 }$ a stride of 2. The ferespectively, where $H$ re reand $W$ is the image height and width. For each stage, $M$ quadtree transformers are used with a channel number of $I$ and head number of $J$ . For the network PVTv2-b0, the parameters $M , I , J$ are set to [2, 2, 2, 2], [32, 64, 160, 256], $[ 1 , 2 , 5 , 8 ]$ at each stage respectively. For the network PVTv2-b1, the parameters $M$ , $I$ , $J$ are set to $[ 2 , 2 , 2 , 2 ]$ , [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. For PVTv2-b2, the parameters $M , I , J$ are set to $[ 3 , 4 , 6 , 3 ]$ , [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. For PVTv2- b3, the parameters $M$ , I , $J$ are set to [3, 4, 18, 3], [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. For PVTv2-b4, the parameters $M , I , J$ are set to [3, 8, 27, 3], [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively.
+
+# B.4 OBJECT DETECTION AND INSTANCE SEGMENTATION
+
+We show the object detection and instance segmentation results of Mask-RCNN (He et al., 2017) in Table 7 and Table 8 in different training settings. In Table 7, we train Mask-RCNN for 12 epoch and resize the image to $8 0 0 \times 1 3 3 3$ while In Table 8, we train the model for 36 epochs and resize the training images to different scales for data augmentation. We can see that the QuadTree attention obtains consistently better performance than other methods.
+
+Table 7: Object detection results on COCO val2017 with Mask-RCNN. We use PVTv2 backbone and replace the reduction attention with quadtree attention.   
+
+<table><tr><td></td><td>AP6</td><td>AP</td><td>AP75</td><td>APm</td><td>AP6</td><td>AP</td></tr><tr><td>PVTv2-b0 (Wang et al.,2021b) QuadTree-B-b0 (K=32,ours)</td><td>38.2 38.8</td><td>60.5 60.7</td><td>40.7 42.1</td><td>36.2 36.5</td><td>57.8 58.0</td><td>38.6 39.1</td></tr><tr><td>ResNet18 (He et al., 2016) PVTv1-Tiny (Wang et al.,2021c) PVTv2-b1 (Wang et al.,2021b) Quadtree-B-b1 (K=32,ours) ResNet50 (He et al., 2016)</td><td>34.0 36.7 41.8 43.5 38.0 40.4</td><td>54.0 59.2 64.3 65.6 58.6 61.1</td><td>36.7 39.3 45.9 47.6 41.4 44.2</td><td>31.2 35.1 38.8 40.1 34.4 36.4</td><td>51.0 56.7 61.2 62.6 55.1 57.7</td><td>32.7 37.3 41.6 43.3 36.7</td></tr><tr><td>PVTv2-b2 (Wang et al.,2021b) QuadTree-B-b2 (K=32,ours) PVTv1-Medium (Wang et al., 2021c) PVTv2-b3 (Wang et al.,2021b) QuadTree-B-b3</td><td>40.4 45.3 46.7 42.0 45.9 48.3</td><td>62.9 67.1 68.5 64.4 66.8 69.6</td><td>43.8 49.6 51.2 45.6 49.3 52.8</td><td>37.8 41.2 42.4 39.0 28.6 43.3</td><td>60.1 64.2 65.7 61.6 49.8 66.8</td><td>40.2 40.3 44.4 45.7 42.1 61.4 46.6</td></tr><tr><td>PVTv1-Large (Wang et al., 2021c) PVTv2-b4 (Wang et al.,2021b) QuadTree-B-b4</td><td>42.9 47.5 48.6</td><td>65.0 68.7 69.5</td><td>46.6 52.0 53.3</td><td>39.5 42.7 43.6</td><td>61.9 66.1 66.9</td><td>42.5 46.1 47.4</td></tr></table>
+
+Table 8: Object detection results on COCO val2017 with Mask-RCNN training with 36 epochs and multi-scale data argumentation strategy. We use PVTv2 backbone and replace the reduction attention with quadtree attention.   
+
+<table><tr><td></td><td>#Params</td><td>AP</td><td>AP50</td><td>AP75</td><td>APs</td><td>APM</td><td>APL</td></tr><tr><td>QuadTree-B-b0</td><td>23.4</td><td>42.4</td><td>64.5</td><td>45.9</td><td>38.9</td><td>61.6</td><td>41.6</td></tr><tr><td>QuadTree-B-b1</td><td>33.3</td><td>46.4</td><td>68.6</td><td>50.7</td><td>41.9</td><td>65.6</td><td>44.7</td></tr><tr><td>Swin-T (Liu et al., 2021) Focal-T (Yang et al., 2021)</td><td>47.8 48.8</td><td>46.0</td><td>68.1</td><td>50.3</td><td>41.6</td><td>65.1</td><td>44.9</td></tr><tr><td>QuadTree-B-b2</td><td>44.8</td><td>47.2 49.3</td><td>69.4 70.7</td><td>51.9 53.9</td><td>42.7 43.9</td><td>66.5 67.6</td><td>45.9 47.4</td></tr><tr><td>Swin-S (Liu et al., 2021)</td><td>69.1</td><td>48.5</td><td>70.2</td><td>53.5</td><td>43.3</td><td>67.3</td><td>46.6</td></tr><tr><td>Focal-S Yang et al. (2021)</td><td>71.2</td><td>48.8</td><td>70.5</td><td>53.6</td><td>43.8</td><td>67.7</td><td>47.2</td></tr><tr><td>QuadTree-B-b3</td><td>70.0</td><td>49.6</td><td>70.4</td><td>54.2</td><td>44.0</td><td>67.7</td><td>47.5</td></tr></table>
+
+![](images/89d11d209acd1c1fe2b07560fde2712e453f895f3126d55dc3944bf5a957a133.jpg)  
+Figure 3: Loss and AUC $@ 2 0 ^ { \circ }$ of image matching.
+
+![](images/3632a0b69b8ac8d2dfdc0794935b1ce26624e63c6be3b409a01fe41dc724a60e.jpg)  
+Figure 4: Loss and top 1 accuracy of image classification for PVTv2-b0 archtecture.
+
+<table><tr><td></td><td colspan="3">ImageNet</td><td colspan="3">COCO (RetinaNet)</td></tr><tr><td></td><td>Param. (M)</td><td>Flops (G)</td><td>Top1 (%)</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>Swin-T (Liu et al., 2021)</td><td>29</td><td>4.5</td><td>81.3</td><td>42.0</td><td></td><td>/</td></tr><tr><td>Focal-T (Yang et al., 2021)</td><td>29</td><td>4.9</td><td>82.2</td><td>43.7</td><td>1</td><td></td></tr><tr><td>Quadtree-B</td><td>30</td><td>4.6</td><td>82.2</td><td>44.6</td><td>65.8</td><td>47.7</td></tr></table>
+
+Table 9: Comparison under Swin-T settings in image classification and object detection.
+
+# C TRAINING LOSS
+
+Feature matching. We plot the training loss and validation performance for LoFTR-lite in Figure 3 for different efficient transformers, including spatial reduction (SR) transformer (Wang et al., 2021c), linear transformer (Katharopoulos et al., 2020), our Quadtree-A, and Quadtree-B transformers. We can see quadtree-B transformer obtains consistently lower training loss and higher performance over other three transformers. In addition, it is also noted that the spatial reduction (SR) transformer has lower training but worse AUC $@ 2 0 ^ { \circ }$ than QuadTree-A attention, which indicates that it cannot generalize well.
+
+Image classification. We also show traning and validation curve for image classification task with respective to different attentions in Fig. 4. Compared with Swin Transformer (Liu et al., 2021) and PVT (Wang et al., 2021c), the loss of Quadtree attention is consistently lower and the top 1 accuracy is higher.
+
+# D RUNNING TIME
+
+Currently, we only implement a naive CUDA kernel without many optimizations and it is not as efficient as the well-optimized dense GPU matrix operation. We test the running time of Retinanet under PVTv2-b0 architecture. For PVTv2-b0, The running time is 0.026s to forward one image and for Quadtree-b0, the running time is 0.046s for forwarding once. However, Quadtree-b0 has much lower memory usage than PVTv2-b0. Quadtree-b0 consumes about 339MB while PVTv2-b0 consumes about 574MB for one $8 0 0 \times 1 3 3 3$ image.
+
+# E ABLATIONS
+
+QuadTree-A vs QuadTree-B. QuadTree-B architecture consistently outperforms QuadTree-A in feature matching, image classification, and detection experiments. We analyze its reason as shown in Figure 5, where (d) and (e) show the attention score maps of QuadTree-A and QuadTree-B at different levels for the same point in the query image shown in (a). It is clear that the QuadTree-B has more accurate score maps, and is less affected by the inaccuracy in coarse level score estimation. We further visualize the attention scores of spatial reduction (SR) attention (Wang et al., 2021c) and linear transformer (Katharopoulos et al., 2020) in (b) and (c). We can see that SR attention and linear transformer attend the query token on large unrelated regions due to the loss of fine-grained information. In contrast, our quadtree transformer focus on the most relevant area.
+
+![](images/31e873488f53c6f34cce2e070891bb462d7027ed928708b3aff8b427aa8afe6b.jpg)  
+Figure 5: Score map visualization of different attention methods for one patch in the query image. The first row shows score maps of spatial reduction attention and linear attention. The second row shows score maps of QuadTree-A and QuadTree-B at different levels, and the left image is the coarsest level, while the right image is the finest level for both sub-figures.
+
+Table 10: Ablation on multiscale position encoding.   
+
+<table><tr><td colspan="3">ImageNet</td><td colspan="2">COCO (RetinaNet)</td></tr><tr><td></td><td>Flops (G)</td><td>Top 1(%)</td><td>AP AP50</td><td>AP75</td></tr><tr><td>Quadtree-B-b2</td><td>4.3</td><td>82.6</td><td>44.9 66.2</td><td>47.7</td></tr><tr><td>Quadtree-B-b2+MPE</td><td>4.3</td><td>82.7</td><td>46.2 67.2</td><td>49.5</td></tr></table>
+
+Comparison with Swin Transformer and Focal Transformer. We compare with Swin Transformer and Focal Transformer in Table. 9 using the released codes. We replace the corresponding attention in Swin Transformer with Quadtree-B attention. Our method obtains $0 . 9 \%$ higher top 1 accuracy than Swin Transformer and $2 . 6 \%$ higher AP in object detection. Compared with Focal transformer, quadtree attention achieve the same top 1 accuracy in classification with fewer flops, and $0 . 9 \%$ higher AP in object detection.
+
+Multiscale position encoding. We compare our method with or without multiscale position encoding (MPE). For Quadtree-B-b2 model, MPE can bring an improvement of 1.3 on object detection.
+
+Top $K$ numbers. Table 11 and Table 12 shows the performance of QuadTree-B architecture with different value of $K$ for object detection and feature matching respectively. The performance is improved when $K$ becomes larger and saturates quickly. This indicates only a few tokens with high attention scores should be subdivided in the next level for computing attentions.
+
+Table 11: The performance of QuadTree-B under different $K$ in object detection.   
+
+<table><tr><td></td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>K=1</td><td>37.3</td><td>57.2</td><td>39.4</td></tr><tr><td>K=8</td><td>38.0</td><td>58.2</td><td>40.4</td></tr><tr><td>K=16</td><td>38.4</td><td>58.7</td><td>41.1</td></tr><tr><td>K=32</td><td>38.5</td><td>58.8</td><td>41.1</td></tr></table>
+
+Table 12: The performance of QuadTree-B under different $K$ in feature matching   
+
+<table><tr><td></td><td>AUC@5°</td><td>AUC@10°</td><td>AUC@20°</td></tr><tr><td>K=1</td><td>15.7</td><td>32.3</td><td>48.9</td></tr><tr><td>K=4</td><td>16.2</td><td>33.3</td><td>50.8</td></tr><tr><td>K=8</td><td>17.4</td><td>34.4</td><td>51.6</td></tr><tr><td>K=16</td><td>17.7</td><td>34.6</td><td>51.7</td></tr></table>

parse/dev/fR-EnKWL_Zb/fR-EnKWL_Zb_content_list.json

@@ -0,0 +1,1923 @@
+[
+    {
+        "type": "text",
+        "text": "QUADTREE ATTENTION FOR VISION TRANSFORMERS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Shitao Tang1∗, Jiahui Zhang2∗, Siyu $\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { 2 }$ , Ping Tan12 ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Simon Fraser University, 2Alibaba A.I. Lab shitaot@sfu.ca, zjhthu@gmail.com, siting.zsy@alibaba-inc.com, pingtan@sfu.ca ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            242,
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+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers have been successful in many vision tasks, thanks to their capability of capturing long-range dependency. However, their quadratic computational complexity poses a major obstacle for applying them to vision tasks requiring dense predictions, such as object detection, feature matching, stereo, etc. We introduce QuadTree Attention, which reduces the computational complexity from quadratic to linear. Our quadtree transformer builds token pyramids and computes attention in a coarse-to-fine manner. At each level, the top $K$ patches with the highest attention scores are selected, such that at the next level, attention is only evaluated within the relevant regions corresponding to these top $K$ patches. We demonstrate that quadtree attention achieves state-of-theart performance in various vision tasks, e.g. with $4 . 0 \\%$ improvement in feature matching on ScanNet, about $50 \\%$ flops reduction in stereo matching, $0 . 4 \\substack { - 1 . 5 \\% }$ improvement in top-1 accuracy on ImageNet classification, $1 . 2 \\substack { - 1 . 8 \\% }$ improvement on COCO object detection, and $0 . 7 \\mathrm { - } 2 . 4 \\%$ improvement on semantic segmentation over previous state-of-the-art transformers. The codes are available at https://github.com/Tangshitao/QuadtreeAttention. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTROCUTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            529,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers can capture long-range dependencies by the attention module and have demonstrated tremendous success in natural language processing tasks. In recent years, transformers have also been adapted to computer vision tasks for image classification (Dosovitskiy et al., 2020), object detection (Wang et al., 2021c), semantic segmentation (Liu et al., 2021), feature matching (Sarlin et al., 2020), and stereo (Li et al., 2021), etc. Typically, images are divided into patches and these patches are flattened and fed to a transformer as word tokens to evaluate attention scores. However, transformers have quadratic computational complexity in terms of the number of tokens, i.e. number of image patches. Thus, applying transformers to computer vision applications requires careful simplification of the involved computation. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To utilize the standard transformer in vision tasks, many works opt to apply it on low resolution or sparse tokens. ViT (Dosovitskiy et al., 2020) uses coarse image patches of $1 6 \\times 1 6$ pixels to limit the number of tokens. DPT (Ranftl et al., 2021) up-samples low-resolution results from ViT to high resolution maps to achieve dense predictions. SuperGlue (Sarlin et al., 2020) applies transformer on sparse image keypoints. Focusing on correspondence and stereo matching applications, Germain et al. (2021) and Li et al. (2021) also apply transformers at a low resolution feature map. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, as demonstrated in several works (Wang et al., 2021c; Liu et al., 2021; Sun et al., 2021; Li et al., 2021; Shao et al., 2020), applying transformers on high resolution is beneficial for a variety of tasks. Thus, many efforts have been made to design efficient transformers to reduce computational complexity. Linear approximate transformers (Katharopoulos et al., 2020; Wang et al., 2020) approximate standard attention computation with linear methods. However, empirical studies (Germain et al., 2021; Chen et al., 2021) show those linear transformers are inferior in vision tasks. To reduce the computational cost, the PVT (Wang et al., 2021c) uses downsampled keys and values, which is harmful to capture pixel-level details. In comparison, the Swin Transformer (Liu et al., ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/12a037cdc35a63c32e1f970a6056546225f5c75ca22bd0f15cd2b6f15529093d.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of QuadTree Attention. Quadtree attention first builds token pyramids by down-sampling the query, key and value. From coarse to fine, quadtree attention selects top $K$ (here, $K = 2$ ) results with the highest attention scores at the coarse level. At the fine level, attention is only evaluated at regions corresponding to the top $K$ patches at the previous level. The query sub-patches in fine levels share the same top $K$ key tokens and coarse level messages, e.g., green and yellow sub-patches at level 2 share the same messages from level 1. We only show one patch in level 3 for simplicity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2021) restricts the attention in local windows in a single attention block, which might hurt longrange dependencies, the most important merit of transformers. ",
+        "bbox": [
+            173,
+            393,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unlike all these previous works, we design an efficient vision transformer that captures both fine image details and long-range dependencies. Inspired by the observation that most image regions are irrelevant, we build token pyramids and compute attention in a coarse to fine manner. In this way, we can quickly skip irrelevant regions in the fine level if their corresponding coarse level regions are not promising. For example, as in Figure 1, at the 1st level, we compute the attention of the blue image patch in image A with all the patches in image B and choose the top $K$ (here, $K = 2$ ) patches which are also highlighted in blue. In the 2nd level, for the four framed sub-patches in image A (which are children patches of the blue patch at the 1st level), we only compute their attentions with the sub-patches corresponding to the top $K$ patches in image B at the 1st level. All the other shaded sub-patches are skipped to reduce computation. We highlight two sub-patches in image A in yellow and green. Their corresponding top $K$ patches in image B are also highlighted in the same color. This process is iterated in the 3rd level, where we only show the sub-sub-patches corresponding to the green sub-patch at the 2nd level. In this manner, our method can both obtain fine scale attention and retain long-range connections. Most importantly, only sparse attention is evaluated in the whole process. Thus, our method has low memory and computational costs. Since a quadtree structure is formed in this process, we refer to our method as QuadTree Attention, or QuadTree Transformer. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In experiments, we demonstrate the effectiveness of our quadtree transformer in both tasks requiring cross attention, e.g. feature matching and stereo, and tasks only utilizing self-attention, e.g. image classification and object detection. Our method achieves state-of-the-art performance with significantly reduced computation, comparing to relevant efficient transformers (Katharopoulos et al. (2020); Wang et al. (2021c); Liu et al. (2021)). In feature matching, we achieve $6 1 . 6 \\ : \\mathrm { A U C } @ 2 0 ^ { \\circ }$ in ScanNet (Dai et al., 2017), 4.0 higher than the linear transformer (Katharopoulos et al., 2020) but with similar flops. In stereo matching, we achieve a similar end-point-error as standard transformer, (Li et al., 2021) but with about $50 \\%$ flops reduction and $40 \\%$ memory reduction. In image classification, we achieve $8 4 . 0 \\%$ top-1 accuracy in ImageNet (Deng et al., 2009), $5 . 7 \\%$ higher than ResNet152 (He et al., 2016) and $1 . 0 \\%$ higher than the Swin Transformer-S (Liu et al., 2021). In object detection, our QuadTree Attention $^ +$ RetinaNet achieves 47.9 AP in COCO (Lin et al., 2014), 1.8 higher than the backbone PVTv2 (Wang et al., 2021b) with fewer flops. In semantic segementation, QuadTree Attention improves the performance by $0 . 7 \\mathrm { - } 2 . 4 \\%$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            862,
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+            877
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Efficient Transformers. Transformers have shown great success in both natural language processing and computer vision. Due to the quadratic computational complexity, the computation of full attention is unaffordable when dealing with long sequence tokens. Therefore, many works design efficient transformers, aiming to reduce computational complexity (Katharopoulos et al., 2020; Choromanski et al., 2020; Shao et al., 2021; Wang et al., 2020; Lee et al., 2019; Ying et al., 2018). Current efficient transformers can be categorized into three classes. 1) Linear approximate attention (Katharopoulos et al., 2020; Choromanski et al., 2020; Wang et al., 2020; Beltagy et al., 2020; Zaheer et al., 2020) approximates the full attention matrix by linearizing the softmax attention and thus can accelerate the computation by first computing the product of keys and values. 2) Inducing point-based linear transformers (Lee et al., 2019; Ying et al., 2018) use learned inducing points with fixed size to compute attention with input tokens, thus can reduce the computation to linear complexity. However, these linear transformers are shown to have inferior results than standard transformers in different works (Germain et al., 2021; Chen et al., 2021). 3) Sparse attention, including Longformer (Beltagy et al., 2020), Big Bird (Zaheer et al., 2020), etc, attends each query token to part of key and value tokens instead of the entire sequence. Unlike these works, our quadtree attention can quickly skip the irrelevant tokens according to the attention scores at coarse levels. Thus, it achieves less information loss while keeps high efficiency. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Vision Transformers. Transformers have shown extraordinary performance in many vision tasks. ViT (Dosovitskiy et al., 2020) applies transformers to image recognition, demonstrating the superiority of transformers for image classification at a large scale. However, due to the computational complexity of full attention, it is hard to apply transformers in dense prediction tasks, e.g. object detection, semantic segmentation, etc. To address this problem, Swin Transformer (Liu et al., 2021) restricts attention computation in a local window. Focal transformer (Yang et al., 2021) uses twolevel windows to increase the ability to capture long-range connection for local attention methods. Pyramid vision transformer (PVT) (Wang et al., 2021c) reduce the computation of global attention methods by downsampling key and value tokens. Although these methods have shown improvements in various tasks, they have drawbacks either in capturing long-range dependencies (Liu et al., 2021) or fine level attention (Wang et al., 2021c). Different from these methods, our method simultaneously capture both local and global attention by computing attention from full image levels to the finest token levels with token pyramids in one single block. Besides, the K-NN transformers (Wang et al., 2021a; Zhao et al., 2019) aggregate messages from top $K$ most similar tokens as ours, but they compute the attention scores among all pairs of query and key tokens, and thus still has quadratic complexity. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Beyond self-attention, many tasks can largely benefit from cross attention. Superglue (Sarlin et al., 2020) processes detected local descriptors with self- and cross attention and shows significant improvement in feature matching. Standard transformers can be applied in SuperGlue because only sparse keypoints are considered. SGMNet (Chen et al., 2021) further reduces the computation by attending to seeded matches. LoFTR (Sun et al., 2021) utilizes linear transformer (Katharopoulos et al., 2020) on low-resolution feature maps to generate dense matches. For stereo matching, STTR (Li et al., 2021) applies self- and cross attention along epipolar lines and reduces the memory by gradient checkpointing engineering techniques. However, due to the requirement of processing a large number of points, these works either use linear transformers, which compromise performance, or a standard transformer, which compromises efficiency. In contrast, our transformer with quadtree attention achieves a significant performance boost compared with linear transformer or efficiency improvement compared with standard transformer. Besides, it can be applied to both self-attention and cross attention. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first briefly review the attention mechanism in transformers in Section 3.1 and then formulate our quadtree attention in Section 3.2. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 ATTENTION IN TRANSFORMER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Vision transformers have shown great success in many tasks. At the heart of a transformer is the attention module, which can capture long-range information between feature embeddings. Given two image embeddings $\\mathbf { X } _ { 1 }$ and $\\mathbf { X } _ { 2 }$ , the attention module passes information between them. Selfattention is the case when $\\mathbf { X } _ { 1 }$ and $\\mathbf { X } _ { 2 }$ are the same, while cross attention covers a more general situation when $\\mathbf { X } _ { 1 }$ and $\\mathbf { X } _ { 2 }$ are different. It first generates the query $\\mathbf { Q }$ , key $\\mathbf { K }$ , and value $\\mathbf { V }$ by the ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/16f2691479fd7ec52d4b755609b61089c1ad77510396ee050e16edfc4cef4366.jpg",
+        "image_caption": [
+            "⇤li i iFigure 2: Illustration of quadtree message aggregation for a query token $q _ { i }$ l+2. (a) shows the token ⇤l+1 ⇤iREFERENCESm3ipyramids and involved key/value tokens in each level. Attention scores are marked in the first two l+2levels for clarification, and the top $K$ REFERENCESscores are highlighted in red. (b) shows message aggregation i REFERENCESfor QuadTree-A architecture. The message is assembled from different levels along a quadtree. (c) REFERENCES shows message aggregation for QuadTree-B architecture. The message is collected from overlapping regions from different levels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "following equation, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/592525e73cd976e35a4c76832c8b81ec01480fcb915708e37be07a25633217c5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { Q } = \\mathbf { W } _ { q } \\mathbf { X } _ { 1 } , } \\\\ { \\mathbf { K } = \\mathbf { W } _ { k } \\mathbf { X } _ { 2 } , } \\\\ { \\mathbf { V } = \\mathbf { W } _ { v } \\mathbf { X } _ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { W } _ { q } , \\ \\mathbf { W } _ { k }$ and $\\mathbf { W } _ { v }$ are learnable parameters. Then, it performs message aggregation by computing the attention scores between query and key as following, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccaac8a1e49e48e2e580e1d9fc4f5947f7b8738642a0d1d64f0f37b11213e033.jpg",
+        "text": "$$\n\\mathbf { Y } = \\operatorname { s o f t m a x } ( \\frac { \\mathbf { Q } \\mathbf { K } ^ { T } } { \\sqrt { C } } ) \\mathbf { V } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            685
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $C$ 1is the embedding channel dimension. The above process has $O ( N ^ { 2 } )$ computational complexity, where $N$ is the number of image patches in a vision transformer. This quadratic complexity hinders transformers from being applied to tasks requiring high resolution output. To address this problem, PVT (Wang et al. (2021c)) downsamples $\\mathbf { K }$ and $\\mathbf { V }$ , while Swin Transformer (Liu et al. (2021)) limits the attention computation within local windows. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 QUADTREE ATTENTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to reduce the computational cost of vision transformers, we present QuadTree Attention. As the name implies, we borrow the idea from quadtrees, which are often used to partition a twodimensional space by recursively subdividing it into four quadrants or regions. Quadtree attention computes attention in a coarse to fine manner. According to the results at the coarse level, irrelevant image regions are skipped quickly at the fine level. This design achieves less information loss while keeping high efficiency. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The same as the regular transformers, we first linearly project $\\mathbf { X } _ { 1 }$ and $\\mathbf { X } _ { 2 }$ to the query, key, and value tokens. To facilitate fast attention computation, we construct $L$ -level pyramids for query $\\mathbf { Q }$ , key $\\mathbf { K }$ , and value $\\mathbf { V }$ tokens by downsampling feature maps. For query and key tokens, we use average pooling layers. For value tokens, average pooling is used for cross attention tasks and convolutionalnormalization-activation layers with stride 2 are used for self attention tasks if no special statement. As shown in Figure 1, after computing attention scores in the coarse level, for each query token, we select the top $K$ key tokens with the highest attention scores. At the fine level, query sub-tokens only need to be evaluated with those key sub-tokens that correspond to one of the selected $K$ key tokens at the coarse level. This process is repeated until the finest level. After computing the attention scores, we aggregate messages at all levels, where we design two architectures named as QuadTree-A and QuadTree-B. ",
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+        "page_idx": 3
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "QuadTree-A. Considering the $i$ -th query token $\\mathbf { q } _ { i }$ at the finest level, we need to compute its received message $\\mathbf { m } _ { i }$ from all key tokens. This design assembles the full message by collecting partial messages from different pyramid levels. Specifically, ",
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+    {
+        "type": "equation",
+        "img_path": "images/2c1398d13442b7526a2e4105a15824707e088a4dcc73830576be4a80ad1e6605.jpg",
+        "text": "$$\n\\mathbf { m } _ { i } = \\sum _ { 1 \\leq l \\leq L } \\mathbf { m } _ { i } ^ { l } ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { m } _ { i } ^ { l }$ indicates the partial message evaluated at level $l$ . This partial message $\\mathbf { m } _ { i } ^ { l }$ assemble messages at the $l$ -th level from tokens within the region $\\Omega _ { i } ^ { l }$ , which will be defined later. In this way, messages from less related regions are computed from coarse levels, while messages from highly related regions are computed in fine levels. This scheme is illustrated in Figure 2 (b), message $\\mathbf { m } _ { i }$ is generated by assembling three partial messages that are computed from different image regions with different colors, which collectively cover the entire image space. The green region indicates the most relevant region and is evaluated at the finest level, while the red region is the most irrelevant region and is evaluated at the coarsest level. The region $\\Omega _ { i } ^ { l }$ can be defined as $\\Gamma _ { i } ^ { l } - \\Gamma _ { i } ^ { l + 1 }$ , where the image region $\\Gamma _ { i } ^ { l }$ corresponds to the top $K$ tokens at the level $l - 1$ . The regions $\\Gamma _ { i } ^ { l }$ are illustrated in Figure 2 (c). The region $\\Gamma _ { i } ^ { 1 }$ covers the entire image. ",
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+        "type": "text",
+        "text": "The partial messages are computed as, ",
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+    {
+        "type": "equation",
+        "img_path": "images/abf507ac1b8b7f3082c8094c5d7f3655c7f32a9b525ae0d4c84644ee4ff50374.jpg",
+        "text": "$$\n\\mathbf { m } _ { i } ^ { l } = \\sum _ { j \\in \\Omega _ { i } ^ { l } } s _ { i j } ^ { l } \\mathbf { v } _ { j } ^ { l } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $s _ { i j } ^ { l }$ is the attention score between the query and key tokens at level $l$ . Figure 2 (a) highlights query and key tokens involved in computing $\\mathbf { m } _ { i } ^ { l }$ with the same color as $\\Omega _ { i } ^ { l }$ . Attention scores are computed recursively, ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f3006d084ba630d080876b61465a997930c69dc4175b5193d56768cfd4d87ed.jpg",
+        "text": "$$\n\\begin{array} { r } { s _ { i j } ^ { l } = s _ { i j } ^ { l - 1 } t _ { i j } ^ { l } . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here, $s _ { i j } ^ { l - 1 }$ is the score of corresponding parent query and key tokens and $s _ { i j } ^ { 1 } = 1$ . The tentative attention score $t _ { i j } ^ { l }$ is evaluated according to Equation 1 among the $2 \\times 2$ tokens of the same parent query token. For QuadTree-A, we use average pooling layers to downsample all query, key and value tokens. ",
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+        "type": "text",
+        "text": "QuadTree-B. The attention scores $s _ { i j } ^ { l }$ in QuadTree-A are recursively computed from all levels, which makes scores smaller at finer levels and reduces the contributions of fine image features. Besides, fine level scores are also largely affected by the inaccuracy at coarse levels. So we design a different scheme, referred as QuadTree- $\\mathbf { B }$ in this paper, to address this problem. Specifically, we compute $\\mathbf { m } _ { i }$ as a weighted average of the partial messages from different levels, ",
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+    {
+        "type": "equation",
+        "img_path": "images/12decfae9610916419637805301eb9039efe418d862cb53f20efcdef20ff1f0e.jpg",
+        "text": "$$\n\\mathbf { m } _ { i } = \\sum _ { 1 \\leq l \\leq L } w _ { i } ^ { l } \\mathbf { m } _ { i } ^ { l } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $w _ { i } ^ { l }$ is a learned weight. As shown in Figure 2 (c), the partial messages here overlap with each other, which are computed as, ",
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+        "type": "equation",
+        "img_path": "images/d2b830f9b0c41f6d9702a915372c0c77e8cb90b9d70afb2718ea5c826930a974.jpg",
+        "text": "$$\n\\mathbf { m } _ { i } ^ { l } = \\mathrm { A t t e n t i o n } ( \\mathbf { q } _ { i } ^ { l } , \\mathbf { K } _ { \\Gamma _ { i } ^ { l } } ^ { l } , \\mathbf { V } _ { \\Gamma _ { i } ^ { l } } ^ { l } ) ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where Attention is the attention message computation as Equation 1. Here, $\\mathbf { K } _ { \\Gamma _ { i } ^ { l } } ^ { l }$ and $\\mathbf { V } _ { \\Gamma _ { i } ^ { l } } ^ { l }$ are matrices formed by stacking all keys and values within the region $\\Gamma _ { i } ^ { l }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Both QuadTree-A and QuadTree-B involve only sparse attention evaluation. Thus, our method largely reduces computational complexity. As analyzed in Appendix A.1, the computational complexity of our quadtree attention is linear to the number of tokens. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9207dd5307409b91403d90d467a7eb782eb12b4a4ddb9a4f6f13ec3aa8ce2e15.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td>AUC@5°</td><td>AUC@10°</td><td>AUC@20°</td></tr><tr><td rowspan=\"3\">Others</td><td>ContextDesc + SGMNet(Chen et al. (2021))</td><td>15.4</td><td>32.3</td><td>48.8</td></tr><tr><td>SuperPoint + OANet (Zhang et al. (2019b))</td><td>11.8</td><td>26.9</td><td>43.9</td></tr><tr><td>SuperPoint + SuperGlue (Sarlin et al. (2020))</td><td>16.2</td><td>33.8</td><td>51.9</td></tr><tr><td rowspan=\"3\">LoFTR-lite</td><td>DRC-Net (Li et al. (2020)) Linear Att. (LoFTR) (Katharopoulos et al. (2020))</td><td>7.7 16.1</td><td>17.9 32.6</td><td>30.5</td></tr><tr><td>PVT (Wang et al., 2021c)</td><td>16.2</td><td>32.7</td><td>49.0 49.2</td></tr><tr><td>QuadTree-A (ours,K = 8)</td><td>16.8</td><td>33.4</td><td>50.5</td></tr><tr><td rowspan=\"4\">LoFTR</td><td>QuadTree-B (ours,K = 8)</td><td>17.4</td><td>34.4</td><td>51.6</td></tr><tr><td>Linear Att. (LoFTR)* (Sun et al. (2021), 64 GPUs)</td><td>22.1</td><td>40.8</td><td>57.6</td></tr><tr><td>Linear Att. (LoFTR) (Katharopoulos et al. (2020))</td><td>21.1</td><td>39.5</td><td>56.6</td></tr><tr><td>QuadTree-B (ours,K = 8) QuadTree-B* (ours,K = 16)</td><td>23.0</td><td>41.7</td><td>58.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Results on feature matching. The symbol $\\star$ indicates results cited from (Sun et al., 2021), where the model is trained with a batch size of 64 on 64 GPUs (a more preferable setting than ours). The symbol $^ *$ indicates we use the ViT (Dosovitskiy et al., 2020)-like architecture for transformer blocks. For PVT and our method, we replace the original linear attention in LoFTR with corresponding attentions. ",
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+    },
+    {
+        "type": "text",
+        "text": "Multiscale position encoding. The computation of attention is permutation invariant to tokens, and thus positional information is missed. To address this problem, we adopt the locally-enhanced positional encoding (LePE) (Dong et al., 2021) at each level to design a multiscale position encoding. Specifically, for level $l$ , we apply unshared depth-wise convolution layers to value tokens $\\mathbf { V } ^ { l }$ to encode the positional information. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We experiment our quadtree transformer with four representative tasks, including feature matching, stereo, image classification, and object detection. The first two tasks require cross attention to fuse information across different images, while the latter two involve only self-attention. We implement our quadtree transformer using PyTorch and CUDA kernels. More implementation details are provided in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "4.1 CROSS ATTENTION TASKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1.1 FEATURE MATCHING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finding feature correspondence (Luo et al., 2019; DeTone et al., 2018) across different images is a precedent problem for many 3D computer vision tasks. It is typically evaluated by the accuracy of the camera pose estimated from the corresponding points. We follow the framework proposed in a recent state-of-the-art work LoFTR (Sun et al., 2021), which consists of a CNN-based feature extractor and a transformer-based matcher. We replace the linear transformer (Katharopoulos et al., 2020) in LoFTR with our quadtree transformer. Besides, we also implement a new version of LoFTR with the spatial reduction (SR) attention (Wang et al., 2021c) for additional comparison. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Setting. We experiment on ScanNet (Dai et al., 2017) with 1,513 scans. In order to accelerate training, we design the LoFTR-lite setting, which uses half of the feature channels of LoFTR and 453 training scans. Ablation studies in section 4.3 are conducted in this setting. We train both LoFTRlite and LoFTR for 30 epochs with batch size 8. For quadtree transformer, we build pyramids of three levels with the coarsest resolution at $1 5 \\times 2 0$ pixels. We set the parameter $K$ to 8 at the finest level, and double it at coarser levels. For the SR attention, we average pool the value and key tokens to the size $8 \\times 8$ to keep similar memory usage and flops as our quadtree attention. More details are included in Appendix B.1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results. Table 1 shows the AUC of camera pose errors1 under $( 5 ^ { \\circ } , 1 0 ^ { \\circ } , 2 0 ^ { \\circ } )$ . We can see that the SR attention achieves similar results with linear transformer. In comparison, both QuadTree-A and QuadTree-B outperform linear transformer and SR attention by a large margin. Quadtree-B generally performs better than Quadtree-A. Quadtree-B has 2.6 and 1.9 improvements in terms of AUC $@ 2 0 ^ { \\circ }$ over linear transformer on LoFTR-lite and LoFTR respectively. To further enhance the ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9d3ff554edab578e23e68cac195a119ae69e1c33dc94106d6c09fd653195bec5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>EPE (px)</td><td>IOU</td><td>Flops (G)</td><td>Mem. (MB)</td></tr><tr><td>GA-Net (Zhang et al., 2019a)</td><td>0.89</td><td>/</td><td>T</td><td>/</td></tr><tr><td>GWC-Net (Guo et al., 2019)</td><td>0.97</td><td>/</td><td>305</td><td>4339</td></tr><tr><td>Bi3D (Badki et al., 2020)</td><td>1.16</td><td>/</td><td>897</td><td>10031</td></tr><tr><td>STTR(Vanilla Transformer) (Li et al., 2021)</td><td>0.45</td><td>0.92</td><td>490</td><td>8507</td></tr><tr><td>QuadTree-B (ours,K = 6)</td><td>0.46</td><td>0.99</td><td>254 (52%)</td><td>5381 (63%)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Results of stereo matching. QuadTree-B achieves similar performance as STTR but with significantly lower flops and memory usage. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "results, we train a model with $K = 1 6$ and leverage a ViT (Dosovitskiy et al., 2020)-like transformer archtecture instead of the original one used in (Sun et al., 2021). This model achieves 4 improvements on $\\mathbf { A U C } @ 2 0 ^ { \\circ }$ over (Sun et al., 2021), where the LoFTR model is trained with a batch size of 64 with 64 GPUs, a more preferable setting leading to slightly better results than our linear transformer implementation shown in Table 1. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 STEREO MATCHING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Stereo matching aims to find corresponding pixels on epipolar lines between two rectified images. The recent work STTR (Li et al., 2021) applies transformers to feature points between epipolar lines and achieves state-of-the-art performance. Note here, both self- and cross attention are applied along epipolar lines, pixels across different lines are not considered in the attention computation. We replace the standard transformer in STTR (Li et al., 2021) with our quadtree transformer. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Setting. We experiment on the Scene Flow FlyingThings3D (Mayer et al., 2016) synthetic dataset, which contains 25,466 images with a resolution of $9 6 0 \\times 5 4 0$ . We build pyramids of four levels to evaluate quadtree attention. While the STTR is applied to features of 1/3 of image resolution, we use feature maps of 1/2 of image resolution. More details about the network are included in Appendix B.2. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results. We report EPE (End-Point-Error) in non-occluded regions and IOU (Intersection-overUnion) for occlusion estimation in Table 2 as (Li et al., 2021). Computational complexity and memory usage are also reported. Compared with STTR based on the standard transformer, our quadtree transformer achieves similar EPE (0.45 px vs $0 . 4 6 ~ \\mathrm { p x }$ ) and higher IOU for occlusion estimation, but with much lower computational and memory costs, with only $52 \\%$ FLOPs and $63 \\%$ memory consumption. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 SELF-ATTENTION TASK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "This section presents results on image classification and object detection. In the past, convolutional neural networks (CNNs) have dominated these tasks for a long time. Recently, vision transformers (Dosovitskiy et al., 2020; Liu et al., 2021; Wang et al., 2021c) show excellent potential on these problems, thanks to their capability in capturing long-range interactions. To compare our method with these vision transformers on image classification, we use the public codes of PVTv2 (Wang et al., 2021c) and replace all the spatial reduction attention with our quadtree attention. For object detection, we further apply a representative object detection framework, RetinaNet (Lin et al., 2017), which is a widely used single-stage object detector. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Settings. We evaluate image classification on the ImageNet-1K dataset (Deng et al., 2009), which consists of 1.28M training images and 50K validation images from 1,000 categories. We build token pyramids with the coarsest level at a resolution of $7 \\times 7$ and set $K = 8$ . We crop and resize the input images to $2 2 4 \\times 2 2 4$ pixels and train the model with a mini-batch of 128. All models are trained for 300 epochs from scratch on 8 GPUs. All the other training settings are the same as in (Wang et al., 2021c). We build five different quadtree transformers at different complexity, named as b0, b1, b2, b3, b4. These models are gradually deeper and wider. More configuration details can be found in Appendix. B.3. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Results. We provide the top-1 accuracy of various methods and network settings in Table 3. These results are grouped into five sections, each with several methods of similar network complexity, as indicated by the number of parameters. As shown in Table 3, QuadTree-B outperforms PVTv2 by $0 . 4 \\% { - } 1 . 5 \\%$ in top-1 accuracy with fewer parameters. Swin Transformer-S adopts local attention and is surpassed by our QuadTree-B-b2 by $1 . 0 \\%$ in top-1 accuracy. This result proves that global information is important. In general, our quadtree transformer leverages both global information at the coarse level and local information at fine levels, and outperforms both PVTv2 and Swin Transformer. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/15b8faaa64b27c6237999a0952b4af48a58288f3cb6c99413534bf0d049b7041.jpg",
+        "table_caption": [
+            "Table 3: Image classification results. We report top-1 accuracy on the ImageNet validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Param (M)  Flops (G)  Top1 (%)</td></tr><tr><td rowspan=1 colspan=1>PVTv2-b0 (Wang et al., 2021b)QuadTree-A-bO (ours)QuadTree-B-bO (ours)</td><td rowspan=1 colspan=1>3.7         0.6        70.53.4          0.6        70.93.5         0.7        72.0</td></tr><tr><td rowspan=2 colspan=1>ResNet18 (He et al., 2016)PVTv1-Tiny (Wang et al.,2021c)PVTv2-b1 (Wang et al.,2021b)QuadTree-B-b1 (ours)</td><td rowspan=1 colspan=1>11.7        1.8        69.8</td></tr><tr><td rowspan=1 colspan=1>13.2        2.1         75.114.0        2.1         78.713.6        2.3         80.0</td></tr><tr><td rowspan=10 colspan=1>ResNet50 (He et al.,2016)ResNeXt50-32x4d (Xie et al., 2017)RegNetY-4G (Radosavovic et al.,2020)DeiT-Small/16 (Touvron et al., 2021)Swin-T (Liu et al.,2021)TNT-S (Han et al., 2021)CeiT (Yuan et al.,2021a)PVTv2-b2 (Wang et al.,2021c)Focal-T(Yang et al.,2021)QuadTree-B-b2 (ours)</td><td rowspan=1 colspan=1>25.1        4.1         76.4</td></tr><tr><td rowspan=1 colspan=1>25.0        4.3         77.6</td></tr><tr><td rowspan=1 colspan=1>21.0        4.0         80.0</td></tr><tr><td rowspan=1 colspan=1>22.1        4.6        79.9</td></tr><tr><td rowspan=1 colspan=1>29.0        4.5         81.3</td></tr><tr><td rowspan=1 colspan=1>23.8         5.2         81.3</td></tr><tr><td rowspan=1 colspan=1>24.2        4.5         82.0</td></tr><tr><td rowspan=1 colspan=1>25.4        4.0         82.0</td></tr><tr><td rowspan=1 colspan=1>29.1        4.9         82.2</td></tr><tr><td rowspan=1 colspan=1>24.2        4.5        82.7</td></tr><tr><td rowspan=6 colspan=1>ResNet101 (He et al.,2016)ResNeXt101-32x4d (Xie et al., 2017)RegNetY-8G (Radosavovic et al.,2020)CvT-21 (Wu et al., 2021)PVTv2-b3 (Wang et al., 2021c)Quadtree-B-b3 (ours)</td><td rowspan=1 colspan=1>44.7        7.9        77.4</td></tr><tr><td rowspan=1 colspan=1>44.2        8.0        78.8</td></tr><tr><td rowspan=1 colspan=1>39.0        8.0        81.7</td></tr><tr><td rowspan=1 colspan=1>32.0        7.1        82.5</td></tr><tr><td rowspan=1 colspan=1>45.2        6.9        83.2</td></tr><tr><td rowspan=1 colspan=1>46.3        7.8        83.7</td></tr><tr><td rowspan=6 colspan=1>ResNet152 (He etal.,2016)T2T-ViTt-24 (Yuan et al., 2021b)Swin-S (Liu et al., 2021)Focal-Small (Yang et al., 2021)PVTv2-b4 (Wang et al., 2021c)Quadtree-B-b4 (ours)</td><td rowspan=1 colspan=1>60.2         11.6       78.3</td></tr><tr><td rowspan=1 colspan=1>64.0        15.0       82.2</td></tr><tr><td rowspan=1 colspan=1>50.0        8.7        83.0</td></tr><tr><td rowspan=1 colspan=1>51.1        9.1        83.5</td></tr><tr><td rowspan=1 colspan=1>62.6        10.1       83.6</td></tr><tr><td rowspan=1 colspan=1>64.2        11.5       84.0</td></tr></table>",
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+        "table_caption": [
+            "Table 4: Object detection results on COCO val2017 with RetinaNet. We use PVTv2 backbone and replace the reduction attention with quadtree attention. ‘Flops’ is the backbone flops for input image size of $8 0 0 \\times 1 , 3 3 3$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Flops (G)</td><td>AP</td><td>AP50</td><td>AP75</td><td>APs</td><td>APM</td><td>APL</td></tr><tr><td>PVTv2-b0 (Wang et al., 2021b) QuadTree-A-b0 (K=32,ours)</td><td>28.3 16.0</td><td>37.2 37.0</td><td>57.2 56.8</td><td>39.5 38.9</td><td>23.1 22.8</td><td>40.4 39.7</td><td>49.7 50.0</td></tr><tr><td>QuadTree-B-b0 (K=32,ours) ResNet18 (He et al.,2016)</td><td>16.5 38.6</td><td>38.4 31.8</td><td>58.7 49.6</td><td>41.1 33.6</td><td>22.5 16.3</td><td>41.7 34.3</td><td>51.6 43.2</td></tr><tr><td>PVTv1-Tiny (Wang et al.,2021c) PVTv2-b1 (Wang et al., 2021b) Quadtree-B-b1 (K=32,ours)</td><td>72.5 78.8 56.2</td><td>36.7 41.2 42.6</td><td>56.9 61.9 63.6</td><td>38.9 43.9 45.3</td><td>22.6 25.4 26.8</td><td>38.8 44.5 46.1</td><td>50.7 54.3 57.2</td></tr><tr><td>ResNet50 (He et al.,2016)</td><td>87.3</td><td>36.3</td><td>55.3</td><td>38.6</td><td>19.3</td><td>40.0</td><td>48.8</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ResNet101 (He et al.,2016)</td><td>166.3</td><td>38.5</td><td>57.8</td><td>41.2</td><td></td><td></td><td></td></tr><tr><td>ResNeXt101-32x4d (Xie et al., 2017)</td><td></td><td></td><td></td><td></td><td>21.4</td><td>42.6</td><td>51.1</td></tr><tr><td></td><td>170.2</td><td>39.9</td><td>59.6</td><td>42.7</td><td>22.3</td><td>44.2</td><td></td></tr><tr><td>PVTv1-small(Wang et al.,2021c)</td><td></td><td></td><td></td><td></td><td></td><td></td><td>52.5</td></tr><tr><td></td><td>139.8</td><td>36.7</td><td>56.9</td><td>38.9</td><td>25.0</td><td>42.9</td><td>55.7</td></tr><tr><td>PVTv2-b2 (Wang et al.,2021c)</td><td>149.1</td><td>44.6</td><td>65.6</td><td>47.6</td><td>27.4</td><td>48.8</td><td>58.6</td></tr><tr><td>QuadTree-B-b2 (K=32,ours) PVTv1-Medium (Wang et al., 2021c)</td><td>108.6</td><td>46.2</td><td>67.2</td><td>49.5</td><td>29.0</td><td>50.1</td><td>61.8</td></tr><tr><td>PVTv2-b3 (Wang et al.,2021b)</td><td>237.4</td><td>41.9</td><td>63.1</td><td>44.3</td><td>25.0</td><td>44.9</td><td>57.6</td></tr><tr><td></td><td>243.0</td><td>45.9</td><td>66.8</td><td>49.3</td><td>28.6</td><td>49.8</td><td>61.4</td></tr><tr><td>QuadTree-B-b3 (ours)</td><td>193.9</td><td>47.3</td><td>68.2</td><td>50.6</td><td>30.4</td><td>51.3</td><td>62.9</td></tr><tr><td>PVTv1-Large (Wang et al.,2021c) PVTv2-b4 (Wang et al.,2021b)</td><td>346.6</td><td>42.6</td><td>63.7</td><td>45.4</td><td>25.8</td><td>46.0</td><td>58.4</td></tr><tr><td></td><td>353.3</td><td>46.1</td><td>66.9</td><td>49.2</td><td>28.4</td><td>50.0</td><td>62.2</td></tr><tr><td>QuadTree-B-b4 (ours)</td><td>283.9</td><td>47.9</td><td>69.1</td><td>51.3</td><td>29.4</td><td>52.2</td><td>63.9</td></tr></table>",
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+        "type": "text",
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+        "table_caption": [
+            "Table 5: To fairly compare with Swin, PVT, Focal attention and our method, we replace the attention module in PVTv2-b0 with different types of attention and same position encoding method LePE and run image classification and object detection respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">ImageNet-1K</td><td colspan=\"3\">COCO (RetinaNet)</td></tr><tr><td></td><td>Flops (G) Top-1 (%)</td><td>Mem. (MB)</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>PVTv2 (Wang et al.,2021b) PVTv2+LePE (Dong et al., 2021)</td><td>0.6 70.5 0.6 70.9</td><td>574 574</td><td>37.2 37.6</td><td>57.2 57.8</td><td>39.5 39.9</td></tr><tr><td>Swin (Liu et al., 2021)</td><td>0.6 70.5</td><td>308</td><td>35.3</td><td>54.2</td><td>37.4</td></tr><tr><td>Swin+LePE Focal Attention (Yang et al., 2021)</td><td>0.6 70.7 0.7 71.6</td><td>308 732</td><td>35.8 37.5</td><td>55.3 57.6</td><td>37.7 39.5</td></tr><tr><td>Focal Attention+LePE</td><td>0.7 71.5</td><td>732</td><td>37.1</td><td>57.0</td><td>39.4</td></tr><tr><td>QuadTree-B</td><td>0.6 72.0</td><td>339</td><td>38.4</td><td>58.8</td><td>41.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2.2 OBJECT DETECTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Settings. We experiment on the COCO dataset. All models are trained on COCO train 2017 (118k images) and evaluated on val 2017 (5k images). We initialize the quadtree backbone with the weights pre-trained on ImageNet. We adopt the same setting as PVTv2, training the model with a batch size of 16 and AdamW optimizer with an initial learning rate of $1 \\times 1 0 ^ { - \\overline { { 4 } } }$ for 12 epochs. We use the standard metric average precision to evaluate our method. ",
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+    },
+    {
+        "type": "text",
+        "text": "Results. We mainly compare our method with PVTv2, ResNet (He et al., 2016), and ResNeXt (Xie et al., 2017) using detection framework of RetinaNet (Lin et al., 2017), which are state-ofthe-art backbones for dense prediction. Table 4 lists the average precision of different methods and their backbone flops for images of resolution of $8 0 0 \\times 1 , 3 3 3$ . Benefiting from the coarse to fine mechanism, a small $K$ is enough for our method. Thus, the computation can be reduced when using high resolution images. We can see that QuadTree-B achieves higher performance, but with much fewer flops than PVTv2. Our quadtree transformer also outperforms ResNet and ResNeXt. For example, QuadTree-B-b2 outperform ResNet101 and ResNeXt101-32x4d by 7.7 AP and 6.3 AP respectively with about $40 \\%$ backbone flops reduction. We also show Mask-RCNN results (He et al., 2017) in Appendix. E. ",
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+    {
+        "type": "text",
+        "text": "4.3 COMPARISON WITH OTHER ATTENTION MECHANISMS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For a fair comparison with other attention mechanisms, we test these attention mechanisms under the same backbone and training settings. Specifically, we replace the original attention module in PVTv2-b0 with the attention method used in Swin Transformer and Focal Transformer. For more fair comparison, we adopt the same positional encoding LePE (Dong et al., 2021) to PVTv2, Swin and Focal transformer. As shown in Table 5, QuadTree attention obtain consistently better performance than Swin and PVTv2 in both classification task and detection task. Compared with focal attention, our method gets 0.9 higher AP in object detection, which might be because that QuadTree attention can always cover the whole images, while Focal attention only covers $1 / 6$ of the image in the first stage. More experiments on Swin-like architecture can be found in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "For cross attention tasks, we also provide visualization of attention score as shown in Fig.5 in Appendix E. Our method can attend to much more related regions than PVT (Wang et al., 2021b) and Linear attention (Katharopoulos et al., 2020). ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We introduce QuadTree Attention to reduce the computational complexity of vision transformers from quadratic to linear. Quadtree transformers build token pyramids and compute attention in a coarse-to-fine manner. At each level, top $K$ regions with the highest attention scores are selected, such that in finer level, computation in irrelevant regions can be quickly skipped. Quadtree attention can be applied to cross attention as well as self-attention. It achieves state-of-the-art performance in various tasks including feature matching, stereo, image classification, and object detection. ",
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+    {
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            702,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 COMPLEXITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            746
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze the computational complexity of quadtree attention. Suppose the lengths of the query tokens, key tokens, and value tokens are all $H \\times W$ . We build token pyramids of $L$ levels, the $\\bar { l } ^ { t h }$ level has a token length of $\\frac { H W } { 4 ^ { l - 1 } }$ . The flops of computing quadtree attention is, ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2b8a48608ca1c2ec17bb3072adb5ea33ebaa12931a522e1644deea40c55a6164.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathrm { F l o p s } } = 2 ( H _ { 0 } ^ { 2 } W _ { 0 } ^ { 2 } + \\sum _ { l = 2 } ^ { L - 1 } \\frac { 4 K H W } { 4 ^ { l - 1 } } ) } } \\\\ { { \\displaystyle ~ = 2 ( H _ { 0 } ^ { 2 } W _ { 0 } ^ { 2 } + \\frac { 4 } { 3 } ( 1 - 4 ^ { 1 - L } ) K H W ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            804,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here, $H _ { 0 }$ and $W _ { 0 }$ are the height and width of the coarsest level of token pyramids. Therefore, $H _ { 0 } ^ { 2 } W _ { 0 } ^ { 2 }$ is a constant and the computational complexity is $O ( K H W )$ . Since $K$ is a constant number, the complexity of quadtree attention is linear to the number of tokens. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/bb6dd539dae48ed351fe5f4fac16cfa187f15d0d963e425ec60ed1797205fa0b.jpg",
+        "table_caption": [
+            "Table 6: Feature matching results on megadepth. Our method obtains better performance than other methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AUC@5°</td><td>AUC@10°</td><td>AUC@20°</td></tr><tr><td>DRC-Net (Li et al., 2020)</td><td>27.0</td><td>43.0</td><td>58.3</td></tr><tr><td>SuperPoint + SuperGlue (Sarlin et al., 2020)</td><td>42.2</td><td>61.2</td><td>76.0</td></tr><tr><td>LoFTR (Sun et al., 2021)</td><td>52.8</td><td>69.2</td><td>81.2</td></tr><tr><td>QuadTree-B (ours, K=16)</td><td>54.6</td><td>70.5</td><td>82.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTS AND IMPLEMENTATION DETAILS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            239,
+            712,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.1 FEATURE MATCHING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            270,
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+            284
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Implementation details. We train and evaluate the model in ScanNet (Dai et al., 2017), where 230M image pairs is sampled for training, with overlapping scores between 0.4 and 0.8. ScanNet provides RGB images, depth maps, and ground truth camera poses on a well-defined training and testing split. Following the same evaluation settings as Sarlin et al. (2020) and Sun et al. (2021), we evaluate our method on the 1,500 testing pairs from (Sarlin et al., 2020). For both trainig and testing, all images and depth maps are resized to $6 4 0 \\times 4 8 0$ . Following (Sun et al., 2021), we compute the camera pose by solving the essential matrix from predicted matches with RANSAC. We report the AUC of the pose error at thresholds $( 5 ^ { \\circ } , 1 0 ^ { \\circ } , 2 0 ^ { \\circ } )$ , where the pose error is defined as the maximum of angular error in rotation and translation. We only replace the coarse level transformer with quadtree attention. ",
+        "bbox": [
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+            435
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Results of megadepth. We show our results on Megadepth (Li & Snavely, 2018) in Table 6. We can see our method outperforms others by a large margin. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2 STEREO MATCHING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            488,
+            352,
+            502
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our network is based on the STTR (Li et al., 2021), where we replace the standard transformer with our quadtree transformer. The network consists of a CNN backbone which outputs feature maps of 1/2 image resolution, a quadtree transformer with both self- and cross attention, a regression head with optimal transport layers (Cuturi, 2013), and a context adjust layer to refine the disparity. Six self- and cross attention layers are used with 128 channels. We build pyramids with four levels for quadtree attention, and apply the Sinkhorn algorithm (Cuturi, 2013) for 10 iteration for optimal transport. We follow STTR to train the network, with 15 epochs of AdamW optimizer. OneCycle learning rate scheduler is used with a leaning rate of 6e-4 and a batch size of 8. ",
+        "bbox": [
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+            515,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.3 IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            643,
+            383,
+            657
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This paragraph introduces the details of PVTv2-b0, b1, b2, b3, b4. All these five networks have 4 stages. Each stage is dolutions for each stage are $\\begin{array} { r } { \\frac { H } { 4 } \\times \\frac { W } { 4 } , \\frac { H } { 8 } \\times \\frac { W } { 8 } , \\frac { H } { 1 6 } \\times \\frac { W } { 1 6 } } \\end{array}$ ous sand $\\frac { H } { 3 2 } \\times \\frac { W } { 3 2 }$ a stride of 2. The ferespectively, where $H$ re reand $W$ is the image height and width. For each stage, $M$ quadtree transformers are used with a channel number of $I$ and head number of $J$ . For the network PVTv2-b0, the parameters $M , I , J$ are set to [2, 2, 2, 2], [32, 64, 160, 256], $[ 1 , 2 , 5 , 8 ]$ at each stage respectively. For the network PVTv2-b1, the parameters $M$ , $I$ , $J$ are set to $[ 2 , 2 , 2 , 2 ]$ , [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. For PVTv2-b2, the parameters $M , I , J$ are set to $[ 3 , 4 , 6 , 3 ]$ , [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. For PVTv2- b3, the parameters $M$ , I , $J$ are set to [3, 4, 18, 3], [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. For PVTv2-b4, the parameters $M , I , J$ are set to [3, 8, 27, 3], [64, 128, 320, 512], $[ 1 , 2 , 5 , 8 ]$ respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.4 OBJECT DETECTION AND INSTANCE SEGMENTATION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            828,
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+            842
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We show the object detection and instance segmentation results of Mask-RCNN (He et al., 2017) in Table 7 and Table 8 in different training settings. In Table 7, we train Mask-RCNN for 12 epoch and resize the image to $8 0 0 \\times 1 3 3 3$ while In Table 8, we train the model for 36 epochs and resize the training images to different scales for data augmentation. We can see that the QuadTree attention obtains consistently better performance than other methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/54562a0f07b634338a0e3fa712ced8306b0f504cddb6d00eb37fe0f169abc9fc.jpg",
+        "table_caption": [
+            "Table 7: Object detection results on COCO val2017 with Mask-RCNN. We use PVTv2 backbone and replace the reduction attention with quadtree attention. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AP6</td><td>AP</td><td>AP75</td><td>APm</td><td>AP6</td><td>AP</td></tr><tr><td>PVTv2-b0 (Wang et al.,2021b) QuadTree-B-b0 (K=32,ours)</td><td>38.2 38.8</td><td>60.5 60.7</td><td>40.7 42.1</td><td>36.2 36.5</td><td>57.8 58.0</td><td>38.6 39.1</td></tr><tr><td>ResNet18 (He et al., 2016) PVTv1-Tiny (Wang et al.,2021c) PVTv2-b1 (Wang et al.,2021b) Quadtree-B-b1 (K=32,ours) ResNet50 (He et al., 2016)</td><td>34.0 36.7 41.8 43.5 38.0 40.4</td><td>54.0 59.2 64.3 65.6 58.6 61.1</td><td>36.7 39.3 45.9 47.6 41.4 44.2</td><td>31.2 35.1 38.8 40.1 34.4 36.4</td><td>51.0 56.7 61.2 62.6 55.1 57.7</td><td>32.7 37.3 41.6 43.3 36.7</td></tr><tr><td>PVTv2-b2 (Wang et al.,2021b) QuadTree-B-b2 (K=32,ours) PVTv1-Medium (Wang et al., 2021c) PVTv2-b3 (Wang et al.,2021b) QuadTree-B-b3</td><td>40.4 45.3 46.7 42.0 45.9 48.3</td><td>62.9 67.1 68.5 64.4 66.8 69.6</td><td>43.8 49.6 51.2 45.6 49.3 52.8</td><td>37.8 41.2 42.4 39.0 28.6 43.3</td><td>60.1 64.2 65.7 61.6 49.8 66.8</td><td>40.2 40.3 44.4 45.7 42.1 61.4 46.6</td></tr><tr><td>PVTv1-Large (Wang et al., 2021c) PVTv2-b4 (Wang et al.,2021b) QuadTree-B-b4</td><td>42.9 47.5 48.6</td><td>65.0 68.7 69.5</td><td>46.6 52.0 53.3</td><td>39.5 42.7 43.6</td><td>61.9 66.1 66.9</td><td>42.5 46.1 47.4</td></tr></table>",
+        "bbox": [
+            227,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/66462dfe51801a95f7deb0903de5c8390d6d6ec7b7b162b542059cf955e2d665.jpg",
+        "table_caption": [
+            "Table 8: Object detection results on COCO val2017 with Mask-RCNN training with 36 epochs and multi-scale data argumentation strategy. We use PVTv2 backbone and replace the reduction attention with quadtree attention. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>#Params</td><td>AP</td><td>AP50</td><td>AP75</td><td>APs</td><td>APM</td><td>APL</td></tr><tr><td>QuadTree-B-b0</td><td>23.4</td><td>42.4</td><td>64.5</td><td>45.9</td><td>38.9</td><td>61.6</td><td>41.6</td></tr><tr><td>QuadTree-B-b1</td><td>33.3</td><td>46.4</td><td>68.6</td><td>50.7</td><td>41.9</td><td>65.6</td><td>44.7</td></tr><tr><td>Swin-T (Liu et al., 2021) Focal-T (Yang et al., 2021)</td><td>47.8 48.8</td><td>46.0</td><td>68.1</td><td>50.3</td><td>41.6</td><td>65.1</td><td>44.9</td></tr><tr><td>QuadTree-B-b2</td><td>44.8</td><td>47.2 49.3</td><td>69.4 70.7</td><td>51.9 53.9</td><td>42.7 43.9</td><td>66.5 67.6</td><td>45.9 47.4</td></tr><tr><td>Swin-S (Liu et al., 2021)</td><td>69.1</td><td>48.5</td><td>70.2</td><td>53.5</td><td>43.3</td><td>67.3</td><td>46.6</td></tr><tr><td>Focal-S Yang et al. (2021)</td><td>71.2</td><td>48.8</td><td>70.5</td><td>53.6</td><td>43.8</td><td>67.7</td><td>47.2</td></tr><tr><td>QuadTree-B-b3</td><td>70.0</td><td>49.6</td><td>70.4</td><td>54.2</td><td>44.0</td><td>67.7</td><td>47.5</td></tr></table>",
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/89d11d209acd1c1fe2b07560fde2712e453f895f3126d55dc3944bf5a957a133.jpg",
+        "image_caption": [
+            "Figure 3: Loss and AUC $@ 2 0 ^ { \\circ }$ of image matching. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/3632a0b69b8ac8d2dfdc0794935b1ce26624e63c6be3b409a01fe41dc724a60e.jpg",
+        "image_caption": [
+            "Figure 4: Loss and top 1 accuracy of image classification for PVTv2-b0 archtecture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            114,
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+            300
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/32558da942e2aa18ddf9cf70aa86410b2e75cd5babf7c61f94bdc4f6ae5a00e9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">ImageNet</td><td colspan=\"3\">COCO (RetinaNet)</td></tr><tr><td></td><td>Param. (M)</td><td>Flops (G)</td><td>Top1 (%)</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>Swin-T (Liu et al., 2021)</td><td>29</td><td>4.5</td><td>81.3</td><td>42.0</td><td></td><td>/</td></tr><tr><td>Focal-T (Yang et al., 2021)</td><td>29</td><td>4.9</td><td>82.2</td><td>43.7</td><td>1</td><td></td></tr><tr><td>Quadtree-B</td><td>30</td><td>4.6</td><td>82.2</td><td>44.6</td><td>65.8</td><td>47.7</td></tr></table>",
+        "bbox": [
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+            340,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 9: Comparison under Swin-T settings in image classification and object detection. ",
+        "bbox": [
+            210,
+            417,
+            785,
+            433
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C TRAINING LOSS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            458,
+            343,
+            474
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Feature matching. We plot the training loss and validation performance for LoFTR-lite in Figure 3 for different efficient transformers, including spatial reduction (SR) transformer (Wang et al., 2021c), linear transformer (Katharopoulos et al., 2020), our Quadtree-A, and Quadtree-B transformers. We can see quadtree-B transformer obtains consistently lower training loss and higher performance over other three transformers. In addition, it is also noted that the spatial reduction (SR) transformer has lower training but worse AUC $@ 2 0 ^ { \\circ }$ than QuadTree-A attention, which indicates that it cannot generalize well. ",
+        "bbox": [
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+            489,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Image classification. We also show traning and validation curve for image classification task with respective to different attentions in Fig. 4. Compared with Swin Transformer (Liu et al., 2021) and PVT (Wang et al., 2021c), the loss of Quadtree attention is consistently lower and the top 1 accuracy is higher. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D RUNNING TIME ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            671,
+            338,
+            688
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Currently, we only implement a naive CUDA kernel without many optimizations and it is not as efficient as the well-optimized dense GPU matrix operation. We test the running time of Retinanet under PVTv2-b0 architecture. For PVTv2-b0, The running time is 0.026s to forward one image and for Quadtree-b0, the running time is 0.046s for forwarding once. However, Quadtree-b0 has much lower memory usage than PVTv2-b0. Quadtree-b0 consumes about 339MB while PVTv2-b0 consumes about 574MB for one $8 0 0 \\times 1 3 3 3$ image. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E ABLATIONS ",
+        "text_level": 1,
+        "bbox": [
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+            307,
+            824
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "QuadTree-A vs QuadTree-B. QuadTree-B architecture consistently outperforms QuadTree-A in feature matching, image classification, and detection experiments. We analyze its reason as shown in Figure 5, where (d) and (e) show the attention score maps of QuadTree-A and QuadTree-B at different levels for the same point in the query image shown in (a). It is clear that the QuadTree-B has more accurate score maps, and is less affected by the inaccuracy in coarse level score estimation. We further visualize the attention scores of spatial reduction (SR) attention (Wang et al., 2021c) and linear transformer (Katharopoulos et al., 2020) in (b) and (c). We can see that SR attention and linear transformer attend the query token on large unrelated regions due to the loss of fine-grained information. In contrast, our quadtree transformer focus on the most relevant area. ",
+        "bbox": [
+            174,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/31e873488f53c6f34cce2e070891bb462d7027ed928708b3aff8b427aa8afe6b.jpg",
+        "image_caption": [
+            "Figure 5: Score map visualization of different attention methods for one patch in the query image. The first row shows score maps of spatial reduction attention and linear attention. The second row shows score maps of QuadTree-A and QuadTree-B at different levels, and the left image is the coarsest level, while the right image is the finest level for both sub-figures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            101,
+            805,
+            315
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/c7777220f5073e2fe910d97be7a14c00a59698403cfea47cef4101f02b4b1a94.jpg",
+        "table_caption": [
+            "Table 10: Ablation on multiscale position encoding. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">ImageNet</td><td colspan=\"2\">COCO (RetinaNet)</td></tr><tr><td></td><td>Flops (G)</td><td>Top 1(%)</td><td>AP AP50</td><td>AP75</td></tr><tr><td>Quadtree-B-b2</td><td>4.3</td><td>82.6</td><td>44.9 66.2</td><td>47.7</td></tr><tr><td>Quadtree-B-b2+MPE</td><td>4.3</td><td>82.7</td><td>46.2 67.2</td><td>49.5</td></tr></table>",
+        "bbox": [
+            254,
+            396,
+            741,
+            457
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            507,
+            825,
+            550
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Comparison with Swin Transformer and Focal Transformer. We compare with Swin Transformer and Focal Transformer in Table. 9 using the released codes. We replace the corresponding attention in Swin Transformer with Quadtree-B attention. Our method obtains $0 . 9 \\%$ higher top 1 accuracy than Swin Transformer and $2 . 6 \\%$ higher AP in object detection. Compared with Focal transformer, quadtree attention achieve the same top 1 accuracy in classification with fewer flops, and $0 . 9 \\%$ higher AP in object detection. ",
+        "bbox": [
+            174,
+            556,
+            825,
+            641
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Multiscale position encoding. We compare our method with or without multiscale position encoding (MPE). For Quadtree-B-b2 model, MPE can bring an improvement of 1.3 on object detection. ",
+        "bbox": [
+            176,
+            647,
+            823,
+            676
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Top $K$ numbers. Table 11 and Table 12 shows the performance of QuadTree-B architecture with different value of $K$ for object detection and feature matching respectively. The performance is improved when $K$ becomes larger and saturates quickly. This indicates only a few tokens with high attention scores should be subdivided in the next level for computing attentions. ",
+        "bbox": [
+            174,
+            683,
+            825,
+            739
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/660be27370bab90cce340aa842ea729c4ce73cbd4c6d183f7d23520a43da62b5.jpg",
+        "table_caption": [
+            "Table 11: The performance of QuadTree-B under different $K$ in object detection. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>K=1</td><td>37.3</td><td>57.2</td><td>39.4</td></tr><tr><td>K=8</td><td>38.0</td><td>58.2</td><td>40.4</td></tr><tr><td>K=16</td><td>38.4</td><td>58.7</td><td>41.1</td></tr><tr><td>K=32</td><td>38.5</td><td>58.8</td><td>41.1</td></tr></table>",
+        "bbox": [
+            218,
+            813,
+            442,
+            881
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/512f3b38c318e8103d761da3fab988a36b2c1b9931be1498fa29207ba9e46e10.jpg",
+        "table_caption": [
+            "Table 12: The performance of QuadTree-B under different $K$ in feature matching "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AUC@5°</td><td>AUC@10°</td><td>AUC@20°</td></tr><tr><td>K=1</td><td>15.7</td><td>32.3</td><td>48.9</td></tr><tr><td>K=4</td><td>16.2</td><td>33.3</td><td>50.8</td></tr><tr><td>K=8</td><td>17.4</td><td>34.4</td><td>51.6</td></tr><tr><td>K=16</td><td>17.7</td><td>34.6</td><td>51.7</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    }
+]

parse/dev/jUdZCcoOu3/jUdZCcoOu3_content_list.json

@@ -0,0 +1,1292 @@
+[
+    {
+        "type": "text",
+        "text": "RAPHAEL: Text-to-Image Generation via Large Mixture of Diffusion Paths ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zeyue Xue∗ The University of Hong Kong xuezeyue@connect.hku.hk ",
+        "bbox": [
+            250,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Guanglu Song∗ SenseTime Research songguanglu@sensetime.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Qiushan Guo The University of Hong Kong qsguo@cs.hku.hk ",
+        "bbox": [
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+            385,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Boxiao Liu SenseTime Research liuboxiao@sensetime.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zhuofan Zong SenseTime Research zongzhuofan@gmail.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yu Liu† ‡ SenseTime Research liuyuisanai@gmail.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ping Luo‡ The University of Hong Kong Shanghai AI Laboratory pluo@cs.hku.hk ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "“When one is painting one does not think.” ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "— Raffaello Sanzio da Urbino ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            501
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Text-to-image generation has recently witnessed remarkable achievements. We introduce a text-conditional image diffusion model, termed RAPHAEL, to generate highly artistic images, which accurately portray the text prompts, encompassing multiple nouns, adjectives, and verbs. This is achieved by stacking tens of mixtureof-experts (MoEs) layers, i.e., space-MoE and time-MoE layers, enabling billions of diffusion paths (routes) from the network input to the output. Each path intuitively functions as a “painter” for depicting a particular textual concept onto a specified image region at a diffusion timestep. Comprehensive experiments reveal that RAPHAEL outperforms recent cutting-edge models, such as Stable Diffusion, ERNIE-ViLG 2.0, DeepFloyd, and DALL-E 2, in terms of both image quality and aesthetic appeal. Firstly, RAPHAEL exhibits superior performance in switching images across diverse styles, such as Japanese comics, realism, cyberpunk, and ink illustration. Secondly, a single model with three billion parameters, trained on 1, 000 A100 GPUs for two months, achieves a state-of-the-art zero-shot FID score of 6.61 on the COCO dataset. Furthermore, RAPHAEL significantly surpasses its counterparts in human evaluation on the ViLG-300 benchmark. We believe that RAPHAEL holds the potential to propel the frontiers of image generation research in both academia and industry, paving the way for future breakthroughs in this rapidly evolving field. More details can be found on a webpage: https: //raphael-painter.github.io/§. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A parrot with a pearl earring, Vermeer style. ",
+        "bbox": [
+            215,
+            114,
+            287,
+            145
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A car playing soccer, digital art. ",
+        "bbox": [
+            215,
+            193,
+            297,
+            213
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A Pikachu with an angry expression and red eyes, with lightning around it, hyper realistic style. ",
+        "bbox": [
+            215,
+            246,
+            299,
+            308
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There are five cars in the street. ",
+        "bbox": [
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+            342,
+            299,
+            362
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Street shot of a fashionable Chinese lady in Shanghai, wearing black high­waisted trousers. ",
+        "bbox": [
+            215,
+            390,
+            300,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Moonlight Maiden, cute girl in school uniform, long white hair, standing under the moon, celluloid style, Japanese manga style. ",
+        "bbox": [
+            215,
+            455,
+            303,
+            525
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Half human, half robot, repaired human, human flesh warrior, mech display, man in mech, cyberpunk. ",
+        "bbox": [
+            214,
+            531,
+            302,
+            594
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/a2f8a3a6fc697dd6897623703a923c2f8825721e6032be3ac87ea412077c8f16.jpg",
+        "image_caption": [
+            "Figure 1: Comparisons of RAPHAEL with recent representative generators, Stable Diffusion XL [2], DeepFloyd, DALL-E 2 [3], and ERNIE-ViLG 2.0 [5]. They are given the same prompts, where the words that the human artists yearn to preserve within the generated images are highlighted in red. These images are not cherry-picked. We see that previous models often fail to preserve the desired concepts. For example, only the RAPHAEL-generated images precisely reflect the prompts such as “pearl earring, Vermeer”, “playing soccer”, “five cars”, “black high-waisted trouser”, “white hair, manga, moon”, and “sign, RAPHAEL”, while other models generate compromised results. Better zoom in $200 \\%$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            90,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A sign that says RAPHAEL. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recent advancements in text-to-image generators, such as Imagen [1], Stable Diffusion [2], DALLE 2 [3], eDiff-I [4], and ERNIE-ViLG 2.0 [5], have yielded remarkable success and found wide applications in computer graphics, culture and art, and the generation of medical and biological data. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite the substantial progress made in text-to-image diffusion models [1, 2, 3, 4, 5], there remains a pressing need for research to further achieve more precise alignment between text and image. As illustrated in Fig.1, existing models often fail to adequately preserve textual concepts within the generated images. This is primarily due to the reliance on a classic cross-attention mechanism for integrating text descriptions into visual representations, resulting in relatively coarse control of the diffusion process, and leading to compromised results. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To address this issue, we introduce RAPHAEL, a text-to-image generator, which yields images with superior artistry and fidelity compared to prior work, as demonstrated in Fig.2. RAPHAEL, an acronym that stands for “distinct image regions align with different text phases in attention learning”, offers an appealing benefit not found in existing approaches. ",
+        "bbox": [
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+            152,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we observe that different text concepts influence distinct image regions during the generation process [6], and the conventional cross-attention layer often struggles to preserve these varying concepts adequately in an image. To mitigate this issue, we employ a diffusion model stacking tens of mixture-of-experts (MoE) layers [7, 8], including both space-MoE and time-MoE layers. Concretely, the space-MoE layers are responsible for depicting different concepts in specific image regions, while the time-MoE layers focus on painting these concepts at different diffusion timesteps. ",
+        "bbox": [
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+            215,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This configuration leads to billions of diffusion paths from the network input to the output. Naturally, each path can act as a “painter” responsible for rendering a particular concept to an image region at a specific timestep. The result is a more precise alignment between text tokens and image regions, enabling the generated images that accurately represent the associated text prompt. This approach sets RAPHAEL apart from existing models and even sheds light on future studies of the explainability of the generation process. Additionally, we propose an edge-supervised learning module to further enhance the image quality and aesthetic appeal of the generated images. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Extensive experiments demonstrate that RAPHAEL outperforms preceding approaches, such as Stable Diffusion, ERNIE-ViLG 2.0, DeepFloyd, and DALL-E 2. (1) RAPHAEL exhibits superior performance in switching images across diverse styles, such as Japanese comics, realism, cyberpunk, and ink illustration. (2) RAPHAEL establishes a new state-of-the-art with a zero-shot FID-30k score of 6.61 on the COCO dataset. (3) RAPHAEL, a single model with three billion parameters trained on 1, 000 A100 GPUs, significantly surpasses its counterparts in human evaluation on the ViLG-300 benchmark. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The contributions of this work are three-fold: (i) We propose a novel text-to-image generator, RAPHAEL, which, through the implementation of several carefully-designed techniques, generates images that more accurately reflect textual prompts than previous works. (ii) We thoroughly explore RAPHAEL’s potential for switching images in diverse styles, such as Japanese comics, realism, cyberpunk, and ink illustration, and for extension using LoRA [9], ControlNet [10], and SR-GAN [11]. (iii) We will release a programming API for RAPHAEL to the public. We believe that RAPHAEL holds the potential to advance the frontiers of image generation in both academia and industry, paving the way for future breakthroughs in this rapidly evolving field. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Notation and Preliminary ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We present the necessary notations and the Denoising Diffusion Probabilistic Model (DDPM) [12] for text-to-image generation. Given a collection of $N$ images, denoted as $\\{ { \\mathbf { x } } _ { i } \\} _ { i = 1 } ^ { N }$ , the aim is to learn a generative model, $p ( \\mathbf { x } )$ , that is capable of accurately representing the underlying distribution. ",
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+        "text": "In forward diffusion, Gaussian noise is progressively introduced into the source images. At an arbitrary timestep $t$ , it is possible to directly sample from the Gaussian distribution following the $T$ -step noise schedule $\\{ \\alpha _ { t } \\} _ { t = 1 } ^ { T }$ , without iterative forward sampling. Consequently, the noisy image at√ √ timestep $t$ , denoted as $\\mathbf { x } _ { t }$ , can be expressed as $\\mathbf { x } _ { t } = \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\mathbf { x } _ { 0 } + \\sqrt { \\bar { \\alpha } _ { t } } \\epsilon _ { t }$ , where $\\textstyle { \\bar { \\alpha } } _ { t } = \\prod _ { i = 1 } ^ { t } \\alpha _ { i }$ . In this expression, $\\mathbf { x } _ { \\mathrm { 0 } }$ represents the source image, while $\\epsilon _ { t } \\sim \\mathcal { N } ( 0 , I )$ indicates the Gaussian noise at step $t$ . In the reverse process, a denoising neural network, denoted as $D _ { \\theta } ( \\cdot )$ , is employed to estimate the additive Gaussian noise. The optimization of this network is achieved by minimizing the loss function, $\\mathcal { L } _ { \\mathrm { d e n o i s e } } = \\mathbb { E } _ { t , \\mathbf { x } _ { 0 } , \\epsilon \\sim \\mathcal { N } ( 0 , I ) } \\bigg [ \\big \\| \\epsilon - D _ { \\theta } \\left( \\mathbf { x } _ { t } , t \\right) \\big \\| _ { 2 } ^ { 2 } \\bigg ] .$ . ",
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+        "type": "text",
+        "text": "By employing the Bayes’ theorem, it is feasible to iteratively estimate the image at timestep $t \\mathrm { ~ - ~ } 1$ through sampling from the posterior distribution, $\\dot { p _ { \\theta } } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } )$ . We have $\\begin{array} { r l } { \\mathbf { x } _ { t - 1 } } & { { } = } \\end{array}$ ",
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+            "Harvest of vegetables in a wooden box near the beds vegetables grow naturally, summer light background, backlight and sun rays, clean sharp focus. "
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+            "Chinese illustration, oriental landscape painting, above super wide angle, magical, romantic, detailed, colorful, multidimensional paper kirigami craft. "
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+            "Photography closeup portrait of an adorable rusty broken­down steampunk robot covered in budding vegetation, surrounded by tall grass, misty futuristic sci­fi forest environment. "
+        ],
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+        "image_caption": [
+            "A cute little matte low poly isometric Zelda Breath of the wild forest island, waterfalls, soft shadows, trending on Artstation, 3d render, monument valley, fez video game. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "The Goddess of high fashion, impressionistic line art, contrasting earth tones, vibrant, pen and ink illustration, ink splatter, abstract expressionism superimposed onto majestic space queen. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
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+        "image_caption": [],
+        "image_footnote": [],
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+    },
+    {
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+        "img_path": "images/074f263c149adaf34d2c244c4885de56e66ff45bace91eaf1ec3592af902df42.jpg",
+        "image_caption": [
+            "The Caped Crusader, Gotham skyline, rooftop, mysterious, powerful, nighttime, mixed media, expressionism, dark tones, high contrast, in the style of comic book artist Frank Miller, modern, gritty and textured, collage technique. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A beautiful woman dressed in a dress made of autumn leaves in the forest, photography, natural lighting, high detail. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
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+        "image_caption": [
+            "A wizard by Q Hayashida in the style of Dorohedoro for Elden Ring, with biggest most intricate sword, on sunlit battlefield, breath of the wild, striking illustration. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
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+        "image_caption": [
+            "Milkyway in a glass bottle, 4k, unreal engine, octane render. ",
+            "Figure 2: These examples show that RAPHAEL can generate artistic images with varying text prompts across various styles. The synthesized images have rich details and semantics. The prompts were written by human artists without cherry-picking. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\frac { 1 } { \\sqrt { \\alpha _ { t } } } \\left( \\mathbf { x } _ { t } - \\frac { 1 - \\alpha _ { t } } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } D _ { \\theta } \\left( \\mathbf { x } _ { t } , t \\right) \\right) + \\sigma _ { t } z } \\end{array}$ here $\\sigma _ { t }$ signifies the standard deviation of the newly injected noise into the image at each step, and $z$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In essence, the denoising neural network estimates the score function at varying time steps, thereby progressively recovering the structure of the image distribution. The fundamental insight provided by the DDPM lies in the fact that the perturbation of data points with noise serves to populate regions of low data density, ultimately enhancing the accuracy of estimated scores. This results in stable training and sampling. ",
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+    {
+        "type": "text",
+        "text": "U-Net with Text Prompts. The denoising network is commonly implemented using a U-Net [13] architecture, as depicted in Fig.8 in Appendix 7.3. To incorporate textual prompts (denoted by y) into the U-Net, a text encoder neural network, $E _ { \\theta } ( \\mathbf { y } )$ , is employed to extract the textual representation. ",
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+        "img_path": "images/3c4a95798c9118dcb4a57b39f0d32164fd6112a52cfdc1c66e8b2eef7752944a.jpg",
+        "image_caption": [
+            "Figure 3: Framework of RAPHAEL. (a) Each block contains four primary components including a selfattention layer, a cross-attention layer, a space-MoE layer, and a time-MoE layer. The space-MoE is responsible for depicting different text concepts in specific image regions, while the time-MoE handles different diffusion timesteps. Each block uses edge-supervised cross-attention learning to further improve image quality. (b) shows details of space-MoE. For example, given a prompt “a furry bear under sky”, each text token and its corresponding image region (given by a binary mask) are directed through distinct space experts, i.e., each expert learns particular visual features at a region. By stacking several space-MoEs, we can easily learn to depict thousands of text concepts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The extracted text tokens are input into the U-Net through a cross-attention layer. The text tokens possess a size of $n _ { y } \\times d _ { y }$ , where $n _ { y }$ represents the number of text tokens, and $d _ { y }$ signifies the dimension of a text token (e.g., $d _ { y } = 7 6 8$ in [14]). ",
+        "bbox": [
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+        "type": "text",
+        "text": "The cross-attention layer can be formulated as attention $\\begin{array} { r } { ( \\mathbf { Q } , \\mathbf { K } , \\mathbf { V } ) = \\operatorname { s o f t m a x } \\left( \\frac { \\mathbf { Q } \\mathbf { K } ^ { \\top } } { \\sqrt { d } } \\right) \\mathbf { V } } \\end{array}$ , where $\\mathbf { Q }$ , $\\mathbf { K }$ , and $\\mathbf { V }$ correspond to the query, key, and value matrices, respectively. These matrices are computed as $\\mathbf { Q } = h \\left( \\mathbf { x } _ { t } \\right) \\mathbf { W } _ { x } ^ { \\mathrm { q r y } }$ , $\\mathbf { K } = E _ { \\theta } ( \\mathbf { y } ) \\mathbf { W } _ { y } ^ { \\mathrm { k e y } }$ , and $\\mathbf { V } = E _ { \\theta } ( \\mathbf { y } ) \\mathbf { W } _ { y } ^ { \\mathrm { v a l } }$ , where $\\mathbf { W } _ { x } ^ { \\mathrm { q r y } } \\in \\mathbb { R } ^ { d \\times d }$ and $\\mathbf { W } _ { y } ^ { \\mathrm { k e y } }$ , $\\mathbf { W } _ { y } ^ { \\mathrm { v a l } } \\in \\mathbb { R } ^ { d _ { y } \\times d }$ represent the parametric projection matrices for the image and text, respectively. Additionally, $d$ denotes the dimension of an image token, $h ( \\mathbf { x } _ { t } ) \\in \\mathbb { R } ^ { n _ { x } \\times d }$ indicates the flattened intermediate representation within the U-Net, with $n _ { x }$ being the number of tokens in an image. A cross-attention map between the text and image, $\\begin{array} { r } { \\mathbf { M } = \\mathrm { s o f t m a x } \\left( \\frac { \\mathbf { Q } \\mathbf { K } ^ { \\top } } { \\sqrt { d } } \\right) \\in \\mathbb { R } ^ { n _ { x } \\times n _ { y } } } \\end{array}$ , is defined, which plays a crucial role in the proposed approach, as described in the following sections. ",
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+    {
+        "type": "text",
+        "text": "3 Our Approach ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The overall framework of RAPHAEL is illustrated in Fig.3, with the network configuration details provided in the Appendix 7.1. Employing a U-Net architecture, the framework consists of 16 transformer blocks, each containing four components: a self-attention layer, a cross-attention layer, a space-MoE layer, and a time-MoE layer. The space-MoE is responsible for depicting different text concepts in specific image regions at a given scale, while the time-MoE handles different diffusion timesteps. ",
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+        "type": "text",
+        "text": "3.1 Space-MoE and Time-MoE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Space-MoE. Regarding the space-MoE layer, distinct text tokens correspond to various regions within an image, as previously mentioned. For instance, when provided with the prompt “a furry bear under the sky”, each text token and its corresponding image region (represented by a binary mask) are fed into separate experts, as illustrated in Fig.3b. The space-MoE layer’s output is the mean of all experts, calculated using the following formula: $\\begin{array} { r } { \\frac { 1 } { n _ { y } } \\sum _ { i = 1 } ^ { \\widehat { n _ { y } } } e _ { \\mathrm { r o u t e } ( \\mathbf { y } _ { i } ) } \\left( h ^ { \\prime } ( \\mathbf { x } _ { t } ) \\circ \\widehat { \\mathbf { M } } _ { i } \\right) } \\end{array}$ . In this equation, $\\widehat { \\mathbf { M } } _ { i }$ is a binary two-dimensional matrix, indicating the image region the $i$ -th text token should correspond to, as shown in Fig.3b. Here, $\\circ$ represents hadamard product, and $h ^ { \\prime } ( \\mathbf { x } _ { t } )$ is the features from time-MoE. The gating (routing) function route $\\left( \\mathbf { y } _ { i } \\right)$ returns the index of an expert in the space-MoE, with $\\{ e _ { 1 } , e _ { 2 } , \\ldots , e _ { k } \\}$ being a set of $k$ experts. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Text Gate Network. The Text Gate Network is employed to distribute an image region to a specific expert, as shown in Fig.3b. The function ${ \\mathrm { r o u t e } } ( \\mathbf { y } _ { i } ) = { \\mathrm { a r g m a x } } \\left( { \\mathrm { s o f t m a x } } \\left( { \\mathcal { G } } \\left( E _ { \\theta } ( \\mathbf { y } _ { i } ) \\right) + \\epsilon \\right) \\right.$ is used, where $\\mathcal { G } : \\mathbb { R } ^ { d _ { y } } \\mapsto \\mathbb { R } ^ { k }$ is a feed forward network, which uses a text token representation $E _ { \\theta } ( \\mathbf { y } _ { i } )$ as input and assigns a space expert. To prevent mode collapse, random noise $\\epsilon$ is incorporated. The argmax function ensures that one expert exclusively handles the corresponding image region for each text token, without increasing computational complexity. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/b115ba8d28166e03abba1bc6e5a2915cb181861d4e3db6cee9a04f61ce9f7e43.jpg",
+        "image_caption": [
+            "Figure 4: Left: We visualize the diffusion paths (routes) from the network input to the output, utilizing 16 space-MoE layers, each containing 6 spatial experts. These paths are closely associated with 100 adjectives, such as “scenic”, “peaceful”, and “majestic”, which represent the most frequently occurring adjectives for describing artworks as suggested by GPT-3.5 [15, 16]. Given that GPT-3.5 has been trained on trillions of tokens, we believe that these adjectives reflect a diverse, real-world distribution. Our findings indicate that different paths distinctively represent various adjectives. Right: We depict the diffusion paths for ten categories (i.e., nouns) within the COCO dataset. Our observations reveal that different categories activate distinct paths in a heterogeneous manner. The display colors blend together where the routes overlap. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "From Text to Image Region. Recall that M is the cross-attention map between text and image, where each element, $\\mathbf { M } _ { j , i }$ , represents a correspondence value between the $j$ -th image token and the $i \\cdot$ -th text token. In the space-MoE, each entry in the binary mask $\\widehat { \\mathbf { M } } _ { i }$ equals “1” if $\\mathbf { M } _ { j , i } \\geq \\eta _ { i }$ , otherwise $\\mathbf { \\vec { \\nabla } } _ { 0 } , \\mathbf { \\vec { \\mathbf { \\phi } } } _ { }$ if $\\mathbf { M } _ { j , i } < \\eta _ { i }$ , as illustrated in Fig.3b. A thresholding mechanism is introduced to determine the values in the mask. The threshold value $\\eta _ { i } = \\alpha \\operatorname* { m a x } ( \\mathbf { M } _ { * , i } )$ is defined, where $\\operatorname* { m a x } ( \\mathbf { M } _ { * , i } )$ represents the maximum correspondence between text token $i$ and all image regions. The hyper-parameter $\\alpha$ will be evaluated through an ablation study. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Discussions. The insight behind the space-MoE is to effectively model the intricate relationships between text tokens and their corresponding regions in the image, accurately reflecting concepts in the generated images. As illustrated in Fig.4, the employment of 16 space-MoE layers, each containing 6 experts, results in billions of spatial diffusion paths (i.e., $6 ^ { 1 6 }$ possible routes). It is evident that each diffusion path is closely associated with a specific textual concept. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To investigate this further, we generate 100 prevalent adjectives that are the most frequently occurring adjectives for describing artworks as suggested by GPT-3.5 [15, 16]. Given that GPT-3.5 has been trained on trillions of tokens, we posit that these adjectives reflect a diverse, real-world distribution. We input each adjective into the RAPHAEL model to generate 100 distinct images and collect their corresponding diffusion paths. Consequently, we obtain ten thousand paths for the 100 words. By treating these pathways as features (i.e., each path is a vector of 16 entries), we train a straightforward classifier (e.g., XGBoost [17]) to categorize the words. The classifier after 5-fold cross-validation achieves over $93 \\%$ accuracy for open-world adjectives, demonstrating that different diffusion paths distinctively represent various textual concepts. We observe analogous phenomena within the 80 object categories of the COCO dataset. Further details on verbs and visualization are provided in the Appendix 7.5. ",
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+    {
+        "type": "text",
+        "text": "Time-MoE. We can further enhance the image quality by employing a time-mixture-of-experts (time-MoE) approach, which is inspired by previous works such as [4, 5]. Given that the diffusion process iteratively corrupts an image with Gaussian noise over a series of timesteps $t = 1 , \\dots , T$ , the image generator is trained to denoise the images in reverse order from $t = T$ to $t = 1$ . All timesteps aim to denoise a noisy image, progressively transforming random noise into an artistic image. Intuitively, the difficulty of these denoising steps varies depending on the noise ratio presented in the image. For example, when $t = T$ , the denoising network’s input image $\\mathbf { x } _ { t }$ is highly noisy. When $t = 1$ , the image $\\mathbf { x } _ { t }$ is closer to the original image. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To address this issue, we employ a time-MoE before each space-MoE in each transformer block. In contrast to [4, 5] , which necessitate hand-crafted time expert assignments, we implement an additional gate network to automatically learn to assign different timesteps to various time experts. Further details can be found in the Appendix 7.3. ",
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+    {
+        "type": "text",
+        "text": "3.2 Edge-supervised Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to further enhance the image quality, we propose incorporating an edge-supervised learning strategy to train the transformer block. By implementing an edge detection module, we aim to extract rich boundary information from an image. These intricate boundaries can serve as supervision to guide the model in preserving detailed image features across various styles. ",
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+    {
+        "type": "text",
+        "text": "Consider a neural network module, $P _ { \\theta } ( \\mathbf { M } )$ , with parameters of $N$ convolutional layers (e.g., $N = 5$ ). This module is designed to predict an edge map given an attention map $\\mathbf { M }$ (refer to Fig.7a in the Appendix 7.2). We utilize the edge map of the input image, denoted as $\\mathbf { I } _ { \\mathrm { e d g e } }$ , to supervise the network $P _ { \\theta }$ . $\\mathbf { I } _ { \\mathrm { e d g e } }$ can be obtained by the holistically-nested edge detection algorithm [18] (Fig.7b). Intuitively, the network $P _ { \\theta }$ can be trained by minimizing the loss function, $\\mathcal { L } _ { \\mathrm { e d g e } } = \\mathrm { F o c a l } ( P _ { \\theta } ( \\mathbf { M } ) , \\mathbf { I } _ { \\mathrm { e d g e } } )$ , where $\\operatorname { F o c a l } ( \\cdot , \\cdot )$ denotes the focal loss [19] employed to measure the discrepancy between the predicted and the “ground-truth” edge maps. Moreover, as discussed in [5, 6], the attention map $\\mathbf { M }$ is prone to becoming vague when the timestep $t$ is large. Consequently, it is essential to adopt a timestep threshold value to inactivate (pause) edge-supervised learning when $t$ is large. This timestep threshold value $( T _ { c } )$ is a hyper-parameter that will be evaluated through an ablation study. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Overall, the RAPHAEL model is trained by combining two loss functions, $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { d e n o i s e } } + \\mathcal { L } _ { \\mathrm { e d g e } }$ As demonstrated in Fig.7d in the Appendix 7.2, edge-supervised learning substantially improves the image quality and aesthetic appeal of the generated images. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This section presents the experimental setups, the quantitative results compared to recent state-ofthe-art models, and the ablation study to demonstrate the effectiveness of RAPHAEL. More artistic images generated by RAPHAEL and comparisons between RAPHAEL and other diffusion models can be found in Appendix 7.6 and 7.7. ",
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+        "type": "text",
+        "text": "Dataset. The training dataset consists of a subset of LAION-5B [20] and some internal datasets, including 730M text-images pairs in total. To collect training data from LAION-5B, we filter the images using the aesthetic scorer same as Stable Diffusion [2] and remove the image-text pairs that have scores smaller than 4.7. We remove the images with watermarks either. Since the text descriptions in LAION-5B are noisy, we clean them by removing useless information such as URLs, HTML tags, and email addresses, inspired by [2, 4, 21]. ",
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+        "text": "Multi-scale Training. To improve text-image alignment, instead of cropping images to a fixed scale [2], we resize an image to its nearest size into different buckets, which has 9 different image scales. Additionally, the GPU resources will be automatically allocated to each bucket depending on the number of images it contains, enabling effective use of computational resources\\*. ",
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+        "text": "Implementations. To reduce training and sampling complexity, we use a Variational Autoencoder (VAE) [22, 23] to compress images using Latent Diffusion Model [2]. We first pre-train an image encoder to transform an image from pixel space to a latent space, and an image decoder to convert it back. Unlike previous works, the cross-attention layers in RAPHAEL are augmented with space-MoE and time-MoE layers. The entire model is implemented in PyTorch [24], and is trained by AdamW [25] optimizer with a learning rate of $1 e - 4$ , a weight decay of 0, a batch size of 2, 000, on 1, 000 NVIDIA A100s for two months. More details on the hyper-parameter settings can be found in the Appendix 7.1. ",
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+        "text": "4.1 Comparisons ",
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+        "text": "Results on COCO. Following previous works [1, 2, 4], we evaluate RAPHAEL on the COCO $2 5 6 \\times 2 5 6$ dataset using zero-shot Frechet Inception Distance (FID), which measures the quality and diversity of images. Similar to [1, 2, 4, 5, 32], 30, 000 images are randomly selected from the validation set for evaluation. Table 1 shows that RAPHAEL achieves a new state-of-the-art performance of text-to-image generation, with 6.61 zero-shot FID-30k on MS-COCO, surpassing prominent image generators such as Stable Diffusion, Imagen, ERNIE-ViLG 2.0, and DALL-E 2. ",
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+            "Figure 5: Comparisons of RAPHAEL with DALL-E 2, Stable Diffusion XL (SD XL), ERNIE-ViLG 2.0, and DeepFloyd in a user study using the ViLG-300 benchmark. We report the user’s preference rates with $9 5 \\%$ confidence intervals. We see that RAPHAEL can generate images with higher quality and better conform to the prompts. "
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+        "text": "Human Evaluations. We employ the ViLG-300 benchmark [5], a bilingual prompt set, which enables to systematically evaluate text-to-image models given various text prompts in Chinese and English. ViLG-300 allows us to convincingly compare RAPHAEL with recent-advanced models including DALL-E 2, Stable Diffusion, ERNIE-ViLG 2.0, and DeepFloyd, in terms of both image quality and text-image alignment. For example, human artists are presented with two sets of images generated by RAPHAEL and a competitor, respectively. They are asked to compare these images from two aspects respectively, including image-text alignment, and image quality and aesthetics. Throughout the entire process, human artists are unaware of which model the image is generated from. Fig.5 shows that RAPHAEL surpasses all other models in both image-text alignment and image quality in the user study, indicating that RAPHAEL can generate high-artistry images that conform to the text. ",
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+        "text": "Extensions to LoRA, ControlNet, and SR-GAN. RAPHAEL can be further extended by incorporating LoRA, ControlNet, and SR-GAN. In Appendix 7.8, we present a comparison between RAPHAEL and Stable Diffusion utilizing LoRA. RAPHAEL demonstrates superior robustness against overfitting compared to Stable Diffusion. We also demonstrate RAPHAEL with a canny-based ControlNet. Furthermore, by employing a tailormade SR-GAN model, we enhance the image resolution to $4 0 9 6 \\times 6 1 4 4$ . ",
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+        "text": "4.2 Ablation Study ",
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+        "text": "Evaluate every module in RAPHAEL. We conduct a comprehensive assessment of each module within the RAPHAEL model, utilizing the CLIP [14] score to measure image-text alignment. Given the significance of classifier-free guidance weight in controlling image quality and text alignment, we present ablation results as trade-off curves between CLIP and FID scores across a range of guidance weights [33], specifically 1.5, 3.0, 4.5, 6.0, 7.5, and 9.0. Fig.6b compares these curves for the complete RAPHAEL model and its variants without space-MoE, edge-supervised learning, and time-MoE, respectively. Our findings indicate that all modules contribute effectively. For example, space-MoE substantially enhances the CLIP score and the optimal guidance weight for the sampler shifts from 3.0 to 4.5. Moreover, at the same guidance weight, space-MoE considerably reduces the FID, resulting in a significant improvement in image quality. ",
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+            "Figure 6: Ablation Study. (a) examines the selection of $\\alpha$ and $T _ { c }$ . (b) presents the trade-off between FID and CLIP scores for the complete RAPHAEL model and its variants without space-MoE, time-MoE, and edge-supervised learning. (c) visualizes the correlation between FID- $^ { 5 \\mathrm { k } }$ and runtime complexity (measured in terms of the number of DDIM [34] steps for an image per second) as a function of the number of experts employed. Notably, the computational complexity is predominantly influenced by the number of spatial experts. "
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+        "type": "text",
+        "text": "Choice of $\\alpha$ and $T _ { c }$ . As depicted in Fig.6a, we observe that $\\alpha = 0 . 2$ delivers the best performance, implying a balance between preserving adequate features and avoiding the use of the entire latent features. An appropriate threshold value for $T _ { c }$ terminates edge-supervised learning when the diffusion timestep is large. Our experiments reveal that a suitable choice for $T _ { c }$ is 500, ensuring the effective learning of texture information. ",
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+        "type": "text",
+        "text": "Performance and Runtime Analysis on Number of Experts. We offer an examination of the number of experts, ranging from 0 to 8, in Fig.6c. For each setting, we employ 100 million training samples. Our results demonstrate that increasing the number of experts improves FID (lower values are preferable). However, adding spatial experts introduces additional computations, with the computational complexity bounded by the total number of experts. Once all available experts have been deployed, the computational complexity ceases to grow. In the right-hand side of Fig.6c, we provide a runtime analysis for 40 input tokens, ensuring the utilization of all space experts. For instance, when the number of experts is 6, the inference speed decreases by $24 \\%$ but yields superior fidelity. This remains faster than previous diffusion models such as Imagen [1] and eDiff-I [4]. ",
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+        "text": "5 Related Work ",
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+        "text": "We review related works from two perspectives, mixture-of-experts and text-to-image generation. More related works can be found in Appendix 7.4. Firstly, the Mixture-of-Experts (MoE) method [7, 8] partitions model parameters into distinct subsets, each termed an “expert”. The MoE paradigm finds applicability beyond language processing tasks, extending to visual models [35] and Mixtureof-Modality-Experts within multi-modal transformers [36]. Additionally, efforts are being made to accelerate the training or inference processes for MoE [37, 38]. Secondly, text-to-image generation is to synthesize images from natural language descriptions. Early approaches relied on generative adversarial networks (GANs) [39, 40, 41, 42] to generate images. More recently, with the transformative success of transformers in generative tasks, models such as DALL-E [43], Cogview [44], and Make-A-Scene [29] have treated text-to-image generation as a sequence-to-sequence problem, utilizing auto-regressive transformers as generators and employing text/image tokens as input/output sequences. Recently, another research direction has focused on diffusion models by integrating textual conditioning within denoising steps, like Stable Diffusion [2], DALL-E 2 [3], eDiff-I [4], ERNIE-ViLG 2.0 [5], and Imagen [1]. ",
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+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This paper introduces RAPHAEL, a novel text-conditional image diffusion model capable of generating highly-artistic images using a large-scale mixture of diffusion paths. We carefully design space-MoE and time-MoE within an edge-supervised learning framework, enabling RAPHAEL to accurately portray text prompts, enhance the alignment between textual concepts and image regions, and produce images with superior aesthetic appeal. Comprehensive experiments demonstrate that RAPHAEL surpasses previous approaches, such as Stable Diffusion, ERNIE-ViLG 2.0, DeepFloyd, and DALL-E 2, in both FID-30k and the human evaluation benchmark ViLG-300. Additionally, RAPHAEL can be extended using LoRA, ControlNet, and SR-GAN. We believe that RAPHAEL has the potential to advance image generation research in both academia and industry. ",
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+        "type": "text",
+        "text": "Limitation and Potential Negative Societal Impact. We acknowledge some limitations in our paper that require attention. One limitation is the direct binarization of the attention map, which may result in the loss of some information. An adaptive module should be proposed to address this issue effectively. Additionally, the performance may be affected by failure cases of the edge detector, leading to potential degradation. We plan to explore solutions for these limitations in our future work. The potential negative social impact is to use the RAPHAEL API to create images containing misleading or false information. This issue potentially presents in all powerful text-toimage generators. We will solve this issue (e.g., by prompt filtering) before releasing the API to the public. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This paper is partially supported by the National Key R&D Program of China No.2022ZD0161000 and the General Research Fund of Hong Kong No.17200622. ",
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+        "type": "text",
+        "text": "References ",
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+# Concurrent 3D super resolution on intensity and segmentation maps improves detection of structural effects in neurodegenerative disease
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We propose a new perceptual super resolution (PSR) method for 3D neuroimaging   
+2 and evaluate its performance in detecting brain changes due to neurodegenerative   
+3 disease. The method, concurrent super resolution and segmentation (CSRS), is   
+4 trained on volumetric brain data to consistently upsample both an image intensity   
+5 channel and associated segmentation labels. The simultaneous nature of the method   
+6 improves not only the resolution of the images but also the resolution of associated   
+7 segmentations thereby making the approach directly applicable to existing labeled   
+8 datasets. One challenge to real world evaluation of SR methods such as CSRS   
+9 is the lack of high resolution ground truth in the target application data: clinical   
+10 neuroimages. We therefore evaluate CSRS effectiveness in an adjacent, clinically   
+11 relevant signal detection problem: quantifying cross-sectional and longitudinal   
+12 change across a set of phenotypically heterogeneous but related disorders that   
+13 exhibit known and differentiable patterns of brain atrophy. We contrast several 3D   
+14 PSR loss functions in this paradigm and show that CSRS consistently increases the   
+15 ability to detect regional atrophy both longitudinally and cross-sectionally in each   
+16 of five related diseases.
+
+# 17 1 Introduction
+
+18 Magnetic resonance image (MRI) datasets capturing in vivo longitudinal change in the human brain   
+19 are currently available at unprecedented scale. These data allow us to quantify the complex etiology   
+20 of neurodegenerative disease during life. A fundamental problem in quantifying brain disorders   
+21 from imaging is that many anatomical structures are small in comparison to image resolution. This   
+22 is caused by not only limited image resolution but also the potentially convoluted shape of the   
+23 targeted anatomy [1]. Thinner, more oblate and/or curved structures undergo more distortion due   
+24 to sampling-related aliasing in comparison to larger, more spherical structures. These distortions   
+25 can limit detection power in the context of either clinical trials and/or at the level of patient specific   
+26 medicine [2, 3]. These results also show that, based on first principles, many disease relevant   
+27 anatomical structures in the brain, in particular cortical regions, mid-brain regions and hippocampal   
+28 subfields, should be quantified at higher resolutions (e.g. $\approx 0 . 5 \mathrm { m m ^ { 3 } }$ or smaller rather than the   
+29 more commonly available $\approx 1 \mathrm { m m ^ { 3 } }$ ). The need for increased resolution is only heightened when   
+30 considering aging and neurodegeneration where some brain structures may lose half or more of their   
+31 pre-disease onset volume or thickness.   
+32 Perceptual super resolution (PSR) for 2D RGB imagery consistently demonstrates the ability to   
+33 estimate more “realistic” looking upsampled data in comparison to traditional linear or nearest   
+34 neighbor interpolants [4]. While many competitive methods are available, the deep back projection   
+35 network (DBPN) [5] performed consistently in several competitions including NTIRE 2018 and 2019
+
+[6], AIM 2019 [7] and PIRM 2018 [8]). These large challenges compared dozens of methods with respect to a variety of both perceptual and reconstruction metrics at different levels of upsampling and noise.
+
+39 Can the 2D RGB performance advantages of methods like the DBPN translate to improvements in   
+40 the 3D quantification of brain regions as seen in MRI? If so, then PSR for 3D neuroimaging promises   
+41 to improve quantification by better resolving the brain’s internal structures and tissue boundaries.   
+42 While traditional evaluations of PSR focus on reconstruction error and perceptual impression, these   
+43 measurements do not provide clinically relevant evidence of PSR’s value in quantification. One barrier   
+44 to evaluating PSR’s impact on clinically relevant outcomes (segmentation volumes) is that ground   
+45 truth segmentations do not exist at the super-resolved scale. To address this concern, [9] simulated low   
+46 resolution magnetic resonance images (MRI) of the brain from high-resolution (HR) images obtained   
+47 from the Human Connectome Project [10, 11]. They then applied a very deep super resolution   
+48 (VDSR) model to the simulated data and the high-resolution data and compared the accuracy of an   
+49 automated cortical segmentation method. This careful evaluation study demonstrated that cortical   
+50 segmentation on the VDSR images closely approximated the HR data. However, relatively few   
+51 details are provided about the training of this model and associated loss functions. Furthermore, it   
+52 remains unclear whether these improvements in reconstruction error would translate to the detection   
+53 of population-level effects in real world data particularly in the aging populations that are the target   
+54 of the majority of interventional trials for the brain.   
+55 A more recent effort in volumetric PSR for medical images [12] proposed SOUP-GAN: Super  
+56 resolution Optimized Using Perceptual-tuned Generative Adversarial Network (GAN). SOUP-GAN   
+57 adopts transfer learning from 2D VGG19 to 3D as proposed in [13] to produce a pseudo-volumetric   
+58 perceptual metric [14]. Shan et al. used this metric to denoise low-dose computed tomography   
+59 (CT) images and showed its effectiveness at preserving small anatomical structures. Similarly, the   
+60 SOUP-GAN effort demonstrates that the pseudo-3D perceptual metric improves both PSNR and   
+61 SSIM as well as shows visually appealing upsampling for a variety of medical imaging modalities.   
+62 That is, the surprising utility (in 2D) of VGG weights as a feature space [15] appears to at least   
+63 partially transfer to PSR in 3D medical imaging.   
+64 The current research provides perhaps the first broadly scoped, real world evaluation of MRI PSR for   
+65 quantification of neurodegenerative disease. Moreover, we demonstrate that a regression network   
+66 (ResNet) that predicts T1w image quality can yield a directly useful perceptual feature space that   
+67 performs competitively with pseudo-3D VGG19 features. We build these contributions upon the   
+68 backbone of a set of methods that we call concurrent super resolution and segmentation (CSRS) that   
+69 extends the proven 2D DBPN to 3D and also includes extra output channel(s) enabling segmentation   
+70 maps to be upsampled concurrently. We use this framework to test the impact of different loss   
+71 functions on a set of domain-specific, clinically relevant segmentation measurements related to   
+72 brain atrophy. Specifically, we evaluate CSRS on the quantification of frontotemporal disorders [3]   
+73 from publicly available longitudinal T1-weighted (T1w) neuroimaging (i.e. MRI). Of the several   
+74 combinations of losses that we evaluate, the best model improves not only segmentation performance   
+75 (when ground truth is available) but also detection power across all our related disorders: structural   
+76 changes in behavioral variant frontotemporal dementia (bvFTD), semantic variant primary progressive   
+77 aphasia (svPPA), nonfluent/agrammatic PPA (naPPA), progressive supranuclear palsy (PSP) and   
+78 corticobasal syndrome (CBS) each of which impacts known networks in the brain. CSRS with a new   
+79 perceptual loss based on a shallow ResNet layer performs as well or better than VGG-based models   
+80 in this test of the practical usefulness of PSR.
+
+81 The primary contributions of this work include:
+
+• new PSR that upsamples multi-label segmentations at the same time as intensity; • a new real world evaluation paradigm for PSR in neuroimaging; • comparison of three perceptual loss functions for PSR, two of which are new; • demonstration that loss choice impacts detection power in natural history studies of neurodegenerative disease. Standard intensity similarity and segmentation overlap metrics, on the other hand, do not discriminate performance between the candidate CSRS options.
+
+88 Model weights, sample data, and training code will be made publicly available after anonymous   
+89 review.   
+91 Software platform: We employ the ANTsX platform [16] version 2.3.5 for anatomical labeling,   
+92 data augmentation/sampling during model training and to form the tabular data for the statistical   
+93 evaluation. All MRI processing details follow [16]. Tensorflow 2.6.2 is used for deep learning   
+94 including a ResNet implementation and the CSRS architecture. R version 4.1 is used for statistical   
+95 analysis with packages lmer and ggplot2. All MRI processing was done on Amazon Web Services   
+96 parallel cluster with 24 cores and 32GB RAM per process (Intel(R) Xeon(R) Platinum 8259CL   
+97 CPU $\ @ \ 2 . 5 0 \mathrm { G H z }$ ).   
+98 Data: Human Connectome Project (HCP): We downloaded 1,113 high-resolution $0 . 7 \mathrm { m m ^ { 3 } }$ T1-   
+99 weighted images from the HCP on which to train CSRS. These T1w data were acquired using a   
+100 magnetization-prepared rapid gradient-echo (MPRAGE) sequence on a customized 3T Siemens   
+101 Skyra; see [10] for all details of acquisition. As such, these images provide both high resolution and   
+102 high quality in comparison to the majority of publicly available T1w MRI. Critically, they provide   
+103 superior resolution for the thin convoluted cortical layer that is critical to the measurement of brain   
+104 atrophy in frontotemporal disorders. We transformed these data into numpy blocks with randomly   
+105 selected high-resolution $6 4 ^ { 3 }$ patches and paired low-resolution $3 2 ^ { 3 }$ patches. For each patch pair, we   
+106 also provide a high-resolution binary segmentation and a low-resolution downsampled version of   
+107 that binary segmentation. Each patch segmentation was gained by 2-class $\mathbf { k }$ -means performed on   
+108 the patch where the center voxel’s label determines which class (1 or 2) is used as foreground. This   
+109 collection of 16,640 patches is then divided randomly into train $\scriptstyle \mathrm { n = 1 6 } , 3 8 4 ,$ ) and test sets.   
+110 Data: Parkinson’s Progression Markers Initiative (PPMI): PPMI is a longitudinal multi-center   
+111 clinical study of PD patients and age-matched healthy controls http://www.ppmi-info.org.   
+112 PPMI employed(s) over 20 data collection sites with scanners that span the primary manufacturers   
+113 (Siemens, GE, Phillips), a variety of head coils and also magnet strengths (1.5T, 3T). This heterogene  
+114 ity of data collection provides a rich set of T1w images with highly variable image contrast, resolution   
+115 and quality. We manually reviewed and labelled 1,431 raw T1w from PPMI to capture the range of   
+116 quality in an ordinal scale. This resulted in a ground truth dataset with 456 images given grade “A”   
+117 (superior), 568 given grade “B”, 350 given grade “C” and 57 given grade “F” which represents images   
+118 that are of little to no use for quantitative studies of brain structure. We then employed a standard   
+119 3D ResNet (antspynet.create_resnet_model_3d with parameters lowest_resolution $^ { \mathtt { = 3 2 } }$ ,   
+120 number_of_classification_label $\mathtt { s } { = } 4$ , cardinality ${ \tt = } 1$ , 39,424,004 parameters, 53 3D convo  
+121 lutional layers) to learn to predict this scale automatically and reliably from the input T1w. We denote   
+122 this network as a T1w Quality Rating Resnet (T1wQRResNet). Details of training T1wQRResNet   
+123 are in Supplementary Information.   
+124 Data: Frontotemporal Lobar Degeneration Neuroimaging Initiative (NIFD) & 4-Repeat   
+125 Tauopathy Neuroimaging Initiative (4RTNI): These inter-related multi-site studies share the goal   
+126 of improving the quantification of frontotemporal spectrum disorders with both imaging and clinical   
+127 scores. Like PPMI and HCP, these studies provide longitudinal T1w images that enable measurement   
+128 of not only the baseline brain structure differences between controls (individuals without a disease   
+129 i.e. normal aging) and disease groups but also differences in rates of change due to neurodegeneration.   
+130 We downloaded and curated 4RTNI and NIFD T1w data and merged these images into a common   
+131 database. These images were collected at three different sites using protocols consistent with ADNI   
+132 3T guidelines [2]. The images overall have a median spacing that is isotropically $1 \mathrm { m m }$ with a minority   
+133 of subjects with out-of-plane spacing up to $1 . 2 \mathrm { m m }$ . As such, these data suit the goals of testing PSR   
+134 for benefits to the quantification of neurodegenerative disease. After filtering data for very low quality   
+135 images and the presence of longitudinal data collected within 2 years of baseline, we obtained 128   
+136 baseline/171 followup images for controls, 60/112 for bvFTD, 38/72 for naPPA, 37/71 for svPPA,   
+137 55/70 for CBS and 75/102 for PSP. Further cohort details (age, education, sex, etc) are available in   
+138 supplementary information. We processed all images consistently and automatically with default   
+139 ANTsX pipelines to gain cortical, medial temporal lobe and deep brain structure segmentations for   
+140 every subject as described in [16]. By consensus, co-authors selected a priori regions for testing   
+141 within each of four groups CBS/PSP [17], bvFTD, svPPA and naPPA [18–24]. Details of the regions   
+142 and rationale for their selection are available in the Supplementary Information. See Figure 1 for an   
+143 overview of processing, the CSRS method and a visualization of the regions (1.C).
+
+![](images/4b5c9141c17579e01bca0f86e4338eef20afa5551c25144ea159f8bda44a630f.jpg)  
+Figure 1: (A) Image processing begins with raw MRI, extracts the brain, labels cortical regions, labels medial temporal lobe regions and labels deep brain regions. (B) The CSRS method is used, here, to upsample data by a factor of 2 isotropically; the sketch of the algorithm provides an example of how two nearby regions would flow through the method and be stitched back together at high resolution. (C) The impact of SR on quantifying neurodegeneration is assessed on a priori regions that are specific to each clinical diagnostic group; all regions are bilateral except for svPPA which uses only left hemisphere cortical and medial temporal labels.
+
+# 144 2.1 Concurrent super resolution and segmentation methods
+
+145 CSRS uses, as a sub-algorithm, a three-dimensional and multi-output version of the neural network   
+146 architecture defined by the 2D deep back projection network (DBPN) [5]. The DBPN is uniquely   
+147 relevant to medical imaging in that it is perhaps the first published SR method that integrates the   
+148 downsampling-upsampling error (i.e. residual layers) as a feature map. This novel architecture may   
+149 prevent feature hallucination and constrain the high-resolution image to maintain features that are   
+150 consistent with the low-resolution input. We extend the 2D DBPN to 3D MRI data by, first, translating   
+151 2D convolutions, padding, striding and other relevant parameters to 3D. To generalize the architecture   
+152 further, we allow options for not only convolutional upsampling (transposed convolution) but also   
+153 nearest neighbor (or linear) interpolation layers at the user’s choice. Lastly, we implement flexible   
+154 choices of input channels, the number of residual layers (backprojection) layers and the number of   
+155 outputs. This 3D DBPN network implementation is available within R and python. All parameters   
+156 were the same as the published work [25] (though in translation to 3D) with the exception of the   
+157 number of back projection layers which has a large impact on the number of parameters. We reduced   
+158 the number of backprojection layers to 5 (16,264,322 parameters) due to the memory limitations   
+159 caused by working with large 3D images and limited GPU resources (all GPU computations in this   
+160 work were implemented with Nvidia V100s locally).   
+161 Efficient computational strategy is essential for CSRS to be applied to large 3D images (a brain   
+162 image may contain 10 million voxels) when CPUs and RAM are limited. As such, a local patch-work   
+163 strategy is necessary. Sampling, upsampling, mapping and unification (SUMU) are the common steps   
+164 needed for not only training but also inference. “Sampling” decides the form of the input data: full   
+165 images (not used here), image patches (used here in training) or anatomical image regions (used here   
+166 in inference). Upscaling determines the core approach to transferring the low-resolution data to a   
+167 higher-resolution output. Mapping compensates for shape or intensity distortion. Finally, “unification”   
+168 is an ensembling or merging step that brings together several sub-estimates of an SR image into a   
+169 single joined (final/full) SR image. We detail each of the 4 components below.   
+170 Sampling: We choose a patch-based model for training as these can easily be applied to input data   
+171 with different resolutions and fields of view. A second reason for patch-based modeling is that a   
+172 candidate network does not need to learn the full scope of image variation. This results in shallower   
+173 and faster to train networks that fit more easily onto readily available GPUs. The choices made   
+174 during sampling step define the feature basis set. Because prior super-resolution competitions suggest   
+175 larger patches lead to better performance, we choose the largest patches that would permit efficient   
+176 batch sizes of 4 (64x64x64). An additional ad hoc support for this choice is that cortical features   
+177 are relatively well-resolved in sub- $1 \mathrm { m m }$ training images when voxel cubes of this sized are used.   
+178 However, there is no direct evidence that this size of patch domain is optimal for this problem.   
+179 Upscaling: is done with the CSRS’s DBPN architecture using nearest neighbor interpolation for   
+180 the upsampling layers. The software interface to CSRS also allows the user to optionally employ   
+181 standard linear (tri-linear) interpolation. We use the linear option as a reference in evaluation studies   
+182 below.   
+183 Mapping: may be used to compensate for distortions in the image shape or intensity space. Because   
+184 each patch is scaled independently on training data (to have an intensity range of -127.5 to 127.5),   
+185 the output of the PSR upsampled image intensities must be mapped back to the original quantitative   
+186 space. This is performed by directly comparing the output of the PSR upsampled patch/region to the   
+187 original data upsampled by nearest neighbor or linear interpolation. As such, we can accurately retain   
+188 quantitative intensity data at the original scale/units with minimal distortion and/or stitching artifacts.   
+189 Unification: this is a general term that, here, refers to the algorithm that is used to derive a single   
+190 CSRS image and multi-label segmentation from multiple CSRS sub-images (not necessarily isotropic   
+191 patches as in training). In 2D, multiple input images are typically generated from a single input by   
+192 “augmentation” e.g. random flipping, translation, etc thus allowing a practitioner to gain multiple   
+193 “votes” about how the SR image should appear at any given voxel. Such a step is used in most PSR   
+194 competitions to reduce aliasing or artifacts and may involve averaging, sharpening or more complex   
+195 modeling such as joint intensity fusion, multi-channel deep learning or other ensemble methods.   
+196 Due to the high memory and computation cost of running CSRS on 3D images, we instead apply   
+197 CSRS to either sub-regions of interest or, when a full T1w brain image is desired, each hemisphere.   
+198 The unification step then maps each local patch intensity range back to the original MRI range and   
+199 then joins the sub-regions back together to complete the SR reconstruction. Augmentation can be   
+200 employed beyond this but at substantial increase in computation time (e.g. 10x to see meaningful   
+201 gains due to augmentation).   
+202 Loss functions for CSRS: We employ a loss function that seeks to balance reconstruction error   
+203 (intensity difference, abbreviated here as R), edge preserving denoising (total variation, abbreviated   
+204 as TV), perceptual quality (based on VGG or ResNet) and segmentation overlap (Dice, abbreviated   
+205 as D). Each of these terms can be up or down weighted to control the network’s performance where   
+206 mean squared error (L2 intensity error) leads to smoother results, L1 (or total variation) provides   
+207 denoising and the perceptual loss yields more natural appearing output textures and shapes. The   
+208 Dice loss term seeks to minimize distortions in the shape of segmentation objects on the output   
+209 of CSRS. The Dice loss is only applied to the second output channel of the network which uses a   
+210 sigmoid activation function appropriate for probabilistic/binary data. We refer to CSRS trained with   
+211 specific combinations of these losses by concatenation of the abbreviations above. For example,   
+212 CSRS.R.TV.D.Res6 refers to a network trained with reconstruction loss, TV regularization, Dice loss   
+213 and the 6th layer of the T1wQRResNet for perceptual loss.
+
+Recent research demonstrates that deep learning models trained on large-scale object detection reference datasets (e.g. imagenet) encode a feature space that may mimic human perception [26]. Such perceptual spaces typically arise from the activations that occur within the layers of convolutional networks trained on massive classification datasets. Here, however, we compare a standard VGG based perceptual space (block2_conv2) (mapped to 3D as described before) to those defined by the T1wQRResNet. From T1wQRResNet, we choose two different deep layers that have similar numbers of parameters to the 3D version of the VGG19 block2_conv2 network: res_conv_block_6 (the 2nd convolutional block) and res_conv_block_21 (the 7th convolutional block). This allows us to compare perceptual metrics based on either pseudo-3D VGG19 or our intrinsically 3D res_conv_block choices.
+
+24 Quantification of medical images requires a high degree of faithfulness to the input data. "Halluci  
+25 nated" features are undesirable. As such, our baseline loss function focuses on reconstruction error   
+226 and TV for both intensity and segmentation images. We then add perceptual and Dice losses for   
+227 further comparison. If we denote $I$ as the estimated super-resolution, $I _ { s }$ as the estimated segmentation   
+228 from the sigmoid output channel, $J$ as the real high resolution image, $J _ { s }$ as the real high resolution   
+229 segmentation, then the final loss function that we optimize is:
+
+![](images/c1c932135400cbcabfbf50ed1c493f50e263d8d369711c3faf4f78969e9debd1.jpg)  
+Figure 2: Best model CSRS applied to three categories of anatomy where row (A) is the original resolution (OR) and segmentation and row (B) is the output of CSRS.R.TV.D.Res6.
+
+$$
+\begin{array} { r } { I - J \| ^ { 2 } w _ { r } ^ { i } + \| I _ { s } - J _ { s } \| ^ { 2 } w _ { r } ^ { s } + T V ( I , J ) w _ { t } ^ { i } + T V ( I _ { s } , J _ { s } ) w _ { t } ^ { s } + \| f _ { n } ( I ) - f _ { n } ( J ) \| ^ { 2 } w _ { f } + D i c e ( I _ { s } , J _ { s } ) w _ { d } } \end{array}
+$$
+
+230 where the term $\| \cdot \|$ indicates the euclidean norm, $T V ( \cdot , \cdot )$ indicates the total variation norm (which   
+231 provides denoising), $w _ { r , t , f , d }$ (superscripts for intensity or segmentation) indicates a term-specific   
+232 scalar weight and $f _ { n } ( . )$ indicates a perceptual feature map. The weight terms can be tuned for   
+233 performance and application area given an objective and quantitative evaluation metric. We initially   
+234 manually tuned the training of a DBPN model with only the reconstruction metrics $( \| I - J \| ^ { 2 } w _ { r } ^ { I } \dot { + }$   
+235 $\lVert I _ { s } - \dot { J _ { s } } \rVert ^ { 2 } w _ { r } ^ { s }$ with $w _ { r } ^ { i } = 5 e - 4$ and $w _ { r } ^ { s } = 1$ ) using adam optimizer and learning rate 5e-5. We   
+236 then set weights relative to the value of the reconstruction error after convergence such that: the TV   
+237 loss is roughly $2 / 3$ the reconstruction term (R); the perceptual loss is roughly $3 \mathrm { x } \ \mathrm { R }$ ; the Dice loss is   
+238 roughly equivalent to the perceptual loss. This strategy, based on our task-specific goals, enables us   
+239 to compare models consistently and add/subtract terms without extensive weight optimization.   
+40 Computation and inference: All models were implemented with tensorflow. The computation to   
+241 double magnification – for a single T1w – takes (generally on a modern computational platform)   
+42 between 10 and 40 minutes. Results are computed region-wise over the set of segmentation labels   
+243 where CSRS is run on each cropped label and its associated intensity. When multiple regions are   
+244 used (as is done here), then results are stitched back together while using a linear mapping back   
+245 to the original intensity space and a arg_max operation to define the hard segmentation labels at   
+46 every voxel in the stitched, joint intensity/probability double magnification space. See Figure 2 for   
+47 an example result of CSRS as applied to the variety of brain regions in this study. Figure 3 shows a   
+48 zoomed visual comparison of the impact on intensity and the lack of stitching artifacts.
+
+# 2.2 Quantification of CSRS impact on segmentation and intensity in ground truth data
+
+Evaluation of PSR results on simulated downsampled-upsampled data does not constitute real world conditions. However, for reference, we include evaluation results based on an independent set of labeled brain images [27]. For these images, we downsample with nearest neighbor interpolation and upsample with linear interpolation (for the intensity) and a “generic label” interpolation that is designed for multi-label images [28] thereby allowing us to report standard metrics of Dice overlap, PSNR and SSIM to complement our study of brain atrophy detection. Figure 4 demonstrates example results illustrating this component of our evaluation.
+
+![](images/e5ad374190f490e8c5abd0d89955a3089fdac304feb8b087fbb3b132df729771.jpg)  
+Figure 3: Comparison of CSRS with different loss functions to original resolution and linear upsampling. The bold (panel E) is the best performing model according to quantitative criteria. However, visual differences between the perceptual models (D,E,F) are not easy to discern.
+
+![](images/f3b1e61e60f74e47e18322e106c4ced735805a3652dae28f93cd6626678bbe98.jpg)  
+Figure 4: Panel (A) shows the original $\mathrm { 1 m m ^ { 3 } }$ resolution ground truth image and its segmentation. Panel (B) shows the impact of linear/generic label upsampling of ground truth data artifically downsampled to $2 \mathrm { m m ^ { 3 } }$ . Panel (C) shows a CSRS result where other models are visually similar to this. Panel (D) demonstrates that all regions improve with CSRS (all differences $> 0$ ) and that regions with lower Dice overlap under the linear/generic label model improve more when upsampled with CSRS.
+
+# 257 2.3 Quantification of effect sizes in frontotemporal disorder atrophy
+
+The frontotemporal disorders produce a profound and debilitating effect on patients with concomitant, symptom-related atrophy. Measuring this atrophy is critical to detecting the effects, for instance, of disease modifying therapies that may slow atrophy. Such measurements are challenged by low resolution and this challenge is compounded by the degeneration process itself.
+
+262 We use statistical modeling to determine if CSRS can mitigate the known limitations of resolution on   
+263 atrophy measurement. We adopt an interpretable mixed effects modeling approach (lmer)[29] to   
+264 estimate effect sizes per brain region, per diagnostic category and per resolution/CSRS model. The   
+265 baseline performance is determined by the effect sizes estimated on the original resolution (OR) data.   
+266 We estimate effect sizes following [30, 31]. Better methods, under this design, should more reliably   
+267 identify disease-related atrophy which will be reflected in increased effect sizes for a given set of $a$   
+268 priori diagnosis-specific regions. The model for the region of interest $i$ $( R O I _ { i } )$ ) is:
+
+$$
+R O I _ { i } \approx A g e _ { b } + S e x + B V _ { b } + D X + \Delta T * D X + ( 1 | I D ) ,
+$$
+
+269 with $( 1 | I D )$ representing a subject-specific random effect, $A g e _ { b }$ is the subject’s age at the first visit,   
+270 $B V _ { b }$ is the first visit brain volume, $D X$ is the diagnosis for the subject, $\Delta T$ is the change in time   
+271 since baseline and the $\Delta T * D X$ represents an interaction between time and diagnosis. The $R O I _ { i }$   
+272 represents the volume for all regions. However, for cortical regions, we also use the region’s thickness   
+273 measurement as a second outcome (as this is a standard measurement in morphometry of the human   
+274 cortex). We estimate effect sizes for cross-sectional effects via the model’s parameter fit for the   
+275 diagnosis $( D X )$ term; we estimate longitudinal effect sizes via the parameter on the interaction term.
+
+Table 1: Summary of results where the comparison of the model impact on effect size is computed by bootstrapped $\scriptstyle ( \mathrm { n = 1 0 0 0 } )$ ) paired t.test. The number of pairs is 274 (see Table 2 for further breakdown by category). CSRS losses are abbreviated as $\mathbf { R } =$ reconstruction, $\mathrm { T V } { = }$ total variation, $\scriptstyle \mathbf { D = }$ dice, VGG $\circeq$ VGG19 pseudo 3D features, Res6 is from the 6th layer of T1wQRResNet and Res21 is the 21st layer of T1wQRResNet. srmeanES indicates the mean effect size for the model averaged over all a priori regions; boot.95ci is the 95 percent confidence interval for the improvement in effect size due to the model. t represents the $t$ -statistic and boot.p represents the bootstrapped p-value for the significance of the improvement in effect size. Columns psnr and ssim show the standard PSNR and SSIM values for an image for which we have ground truth high-resolution intensity and segmentation. The dice columns show the mean and standard deviation of the Dice overlap between ground truth and the upsampled simulated data with each model, estimated over all regions. Best $=$ bold.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>srmeanES</td><td rowspan=1 colspan=1>boot.95ci</td><td rowspan=1 colspan=1>t</td><td rowspan=1 colspan=1>boot.p</td><td rowspan=1 colspan=1>psnr</td><td rowspan=1 colspan=1>ssim</td><td rowspan=1 colspan=1>dice.mean</td><td rowspan=1 colspan=1>dice.sd</td></tr><tr><td rowspan=1 colspan=1>OR</td><td rowspan=1 colspan=1>0.559</td><td rowspan=1 colspan=1>0/0</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>0.468</td><td rowspan=1 colspan=1>-0.1006/-0.08163</td><td rowspan=1 colspan=1>-18.61</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>40.6</td><td rowspan=1 colspan=1>0.996</td><td rowspan=1 colspan=1>0.769</td><td rowspan=1 colspan=1>0.047</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV</td><td rowspan=1 colspan=1>0.574</td><td rowspan=1 colspan=1>0.01124/0.01854</td><td rowspan=1 colspan=1>7.96</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.0</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.884</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D</td><td rowspan=1 colspan=1>0.582</td><td rowspan=1 colspan=1>0.01868/0.0273</td><td rowspan=1 colspan=1>10.38</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>41.8</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.883</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>0.581</td><td rowspan=1 colspan=1>0.01775/0.02559</td><td rowspan=1 colspan=1>10.76</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.0</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.885</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.VGG</td><td rowspan=1 colspan=1>0.577</td><td rowspan=1 colspan=1>0.01403/0.02209</td><td rowspan=1 colspan=1>8.86</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>41.9</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.884</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.Res6</td><td rowspan=1 colspan=1>0.572</td><td rowspan=1 colspan=1>0.009741/0.01665</td><td rowspan=1 colspan=1>7.49</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.4</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.886</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>0.588</td><td rowspan=1 colspan=1>0.02497/0.0331</td><td rowspan=1 colspan=1>14.02</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.4</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.885</td><td rowspan=1 colspan=1>0.032</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.Res21</td><td rowspan=1 colspan=1>0.577</td><td rowspan=1 colspan=1>0.01462/0.02143</td><td rowspan=1 colspan=1>10.40</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.3</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.884</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res21</td><td rowspan=1 colspan=1>0.581</td><td rowspan=1 colspan=1>0.01814/0.02627</td><td rowspan=1 colspan=1>10.59</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.3</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.887</td><td rowspan=1 colspan=1>0.03</td></tr></table>
+
+![](images/2b07a81d2d8a589a8b1bb114132c9aa7f9a79ccb050cea1dd7e921d7dc6686f0.jpg)  
+Figure 5: Bland-Altman plots for model CSRS.R.TV.D.Res6 demonstrate variability in the performance by type of anatomy and by diagnostic grouping with some individual points generating substantially greater $\%$ improvement than suggested by the overall trend. Similarly, a few points show decreased performance relative to OR.
+
+# 276 3 Results
+
+Table 1 summarizes overall results where we show original resolution and results from linear upsampling, as baseline, and compare to eight variants of CSRS. Two of these do not use perceptual metrics. The remaining six add or subtract Dice loss and each of our candidate perceptual losses. Table 1 shows both the aggregate impact of model on effect size estimates in the neurodegeneration data as well as intensity similarity (reconstruction) and Dice overlap in the ground truth data. Dice overlap (a measure that varies between zero and one) improves by a margin of 0.11 to 0.123 $9 5 \%$ CI bootstrapped percentile confidence interval, $p < 1 e - 1 6$ . See Figure 4.
+
+277   
+278   
+279   
+280   
+281   
+282   
+283   
+284   
+285   
+286   
+287   
+288
+
+Table 2 focuses on the two perceptual models with the greatest improvement from original resolution as assessed by pairwise $t$ -test. It breaks down the effect size results in relation to which type of effect size is being analyzed (cross-sectional or longitudinal) and by brain region / diagnostic grouping. Relatedly, Figure 5 shows a Bland-Altman style plot that demonstrates, for the CSRS.R.TV.D.Res6 model, the range of effect size changes due to CSRS across all 274 measurement points.
+
+Table 2: Summary of results for the two best perceptual models broken down by anatomical class, type of predictor (longitudinal or cross-sectional) and diagnostic groups. The n column indicates the number of samples used in the statistical testing. The codes in the AnatClass column are: CtxV - cortical volume; CtxT - cortical thickness; MB - deep brain (for CBS/PSP); MTL - medial temporal lobe (for svPPA). The columns that have non-NA DX2 means that both DX and DX2 groups were aggregated in the computation of the bootstrapped paired $t$ -test for the given group of anatomy.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>AnatClass</td><td rowspan=1 colspan=1>isLong</td><td rowspan=1 colspan=1>DX</td><td rowspan=1 colspan=1>DX2</td><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=1>srmeanES</td><td rowspan=1 colspan=1>boot.95ci</td><td rowspan=1 colspan=1>t</td><td rowspan=1 colspan=1>boot.p</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>Both</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>274</td><td rowspan=1 colspan=1>0.581</td><td rowspan=1 colspan=1>0.01775/0.02559</td><td rowspan=1 colspan=1>10.763</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.826</td><td rowspan=1 colspan=1>0.007153/0.01626</td><td rowspan=1 colspan=1>4.936</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>1.016</td><td rowspan=1 colspan=1>-0.009547/0.01006</td><td rowspan=1 colspan=1>0.033</td><td rowspan=1 colspan=1>0.9769</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.600</td><td rowspan=1 colspan=1>0.03683/0.1208</td><td rowspan=1 colspan=1>3.458</td><td rowspan=1 colspan=1>0.0246</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>svPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>1.003</td><td rowspan=1 colspan=1>0.02801/0.06981</td><td rowspan=1 colspan=1>4.190</td><td rowspan=1 colspan=1>0.0112</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.645</td><td rowspan=1 colspan=1>0.01848/0.03313</td><td rowspan=1 colspan=1>6.719</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.527</td><td rowspan=1 colspan=1>0.03521/0.05297</td><td rowspan=1 colspan=1>9.445</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.201</td><td rowspan=1 colspan=1>0.01409/0.03941</td><td rowspan=1 colspan=1>3.924</td><td rowspan=1 colspan=1>0.0110</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>svPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0.701</td><td rowspan=1 colspan=1>-0.0159/0.002513</td><td rowspan=1 colspan=1>-1.309</td><td rowspan=1 colspan=1>0.2652</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>Both</td><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>274</td><td rowspan=1 colspan=1>0.588</td><td rowspan=1 colspan=1>0.02497/0.0331</td><td rowspan=1 colspan=1>14.021</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.831</td><td rowspan=1 colspan=1>0.01199/0.02182</td><td rowspan=1 colspan=1>6.573</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>1.023</td><td rowspan=1 colspan=1>-0.002896/0.01832</td><td rowspan=1 colspan=1>1.394</td><td rowspan=1 colspan=1>0.1604</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.589</td><td rowspan=1 colspan=1>0.02166/0.1131</td><td rowspan=1 colspan=1>2.718</td><td rowspan=1 colspan=1>0.0454</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>SvPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>1.004</td><td rowspan=1 colspan=1>0.02782/0.07253</td><td rowspan=1 colspan=1>4.021</td><td rowspan=1 colspan=1>0.0102</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.658</td><td rowspan=1 colspan=1>0.03203/0.04756</td><td rowspan=1 colspan=1>9.751</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.545</td><td rowspan=1 colspan=1>0.05421/0.06995</td><td rowspan=1 colspan=1>15.151</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.198</td><td rowspan=1 colspan=1>0.006017/0.04464</td><td rowspan=1 colspan=1>2.233</td><td rowspan=1 colspan=1>0.0166</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>svPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0.704</td><td rowspan=1 colspan=1>-0.0177/0.01214</td><td rowspan=1 colspan=1>-0.369</td><td rowspan=1 colspan=1>0.7511</td></tr></table>
+
+# 289 4 Discussion
+
+The PSNR and SSIM improve similarly across all CSRS models and do not substantively differentiate performance. Dice overlap is consistently superior than linear upsampling across all models but shows little difference between models with perhaps a small advantage for the ResNet features. Greater stratification may be seen when looking at results that relate to quantifying the phenotypic heterogeneity of brain atrophy in frontotemporal spectrum diagnostic groups. Model CSRS.R.TV.D.Res6 stands out under this criteria with Table 2 suggesting that the majority of the improvement arises for cortical measurements, particularly longitudinally. Performance improvements are not, however, perfectly consistent. Figure 5 shows that CSRS augments effect size in the large majority of regions (some greatly so) but a few regions are subtly better at OR. Additional discussion of performance implications with respect to individual regions and diagnoses is in supplementary information.
+
+300 The extension of PSR to 3D raises opportunities as well as challenges. Parameter exploration is   
+301 fundamentally limited because training a model on our patch dataset for 1 epoch takes over 12 hours   
+302 (we trained each model for 2 epochs or until convergence). Other architectures than DBPN may   
+303 perform better with CSRS such as ESRGAN [32] or, potentially, methods with stronger modality   
+304 specific priors on the convolutional kernels [33]. Specifically, fast-training, fewer parameter models   
+305 may ease some of the computational burden and facilitate more parameter exploration.   
+306 CSRS performance is fundamentally limited by the quality of its segmentation inputs. It may be   
+307 more beneficial to develop new methods that operate at high resolution (HR) – adding substantial   
+308 computational cost if the goal is to take advantage of HR features – or that take advantage of   
+309 intrinsically HR ground truth data. The primary barrier to such an effort is the current lack of HR   
+310 ground truth labels for neuroimaging and in particular for neurodegenerative disease. Moreover,   
+311 most methods embed resolution assumptions in their own processing choices and optimize for these   
+312 choices. As such, CSRS bridges a performance gap with a practical solution readily available today.   
+313 Retooling existing methods and segmentation labels for HR (e.g. 7T MRI) is costly both computation  
+314 ally and in terms of the effort of human experts due to the already high volume of 3D neuroimaging.   
+315 We demonstrated that CSRS, in most of its variants, leads to significant performance improvements   
+316 over our reference of original resolution $\mathrm { { ( l m m ^ { 3 } ) } }$ ) image processing and ground truth labels. Because   
+317 CSRS operates on existing images and labels, new HR method and segmentation development is   
+318 not required. Thus, CSRS may be used to improve existing ground truth datasets and existing   
+319 processed data, today. However, comparison to other and/or larger real world datasets is needed to   
+320 help determine the extent to which our results may be deployed to new data without concern.   
+321 References   
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+329 3. Boxer AL, Gold M, Feldman H, Boeve BF, Dickinson SL-J, Fillit H, et al. New directions in   
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+332 4. Blau Y, Michaeli T. The perception-distortion tradeoff. Proceedings of the IEEE Conference   
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We provide the data, in supplementary material, that is needed to generate the tables shown in the paper. The raw data is publicly available but we do not have permission to redistribute these data. However, we do provide the ability to reproduce key results in the Tables of the main manuscript via supplementary information.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See beginning of Methods section and CSRS methods section.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
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+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] in supplemental information.
+
+5. If you used crowdsourcing or conducted research with human subjects...
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+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
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+        "text": "Concurrent 3D super resolution on intensity and segmentation maps improves detection of structural effects in neurodegenerative disease ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 We propose a new perceptual super resolution (PSR) method for 3D neuroimaging   \n2 and evaluate its performance in detecting brain changes due to neurodegenerative   \n3 disease. The method, concurrent super resolution and segmentation (CSRS), is   \n4 trained on volumetric brain data to consistently upsample both an image intensity   \n5 channel and associated segmentation labels. The simultaneous nature of the method   \n6 improves not only the resolution of the images but also the resolution of associated   \n7 segmentations thereby making the approach directly applicable to existing labeled   \n8 datasets. One challenge to real world evaluation of SR methods such as CSRS   \n9 is the lack of high resolution ground truth in the target application data: clinical   \n10 neuroimages. We therefore evaluate CSRS effectiveness in an adjacent, clinically   \n11 relevant signal detection problem: quantifying cross-sectional and longitudinal   \n12 change across a set of phenotypically heterogeneous but related disorders that   \n13 exhibit known and differentiable patterns of brain atrophy. We contrast several 3D   \n14 PSR loss functions in this paradigm and show that CSRS consistently increases the   \n15 ability to detect regional atrophy both longitudinally and cross-sectionally in each   \n16 of five related diseases. ",
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+        "text": "17 1 Introduction ",
+        "text_level": 1,
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+        "text": "18 Magnetic resonance image (MRI) datasets capturing in vivo longitudinal change in the human brain   \n19 are currently available at unprecedented scale. These data allow us to quantify the complex etiology   \n20 of neurodegenerative disease during life. A fundamental problem in quantifying brain disorders   \n21 from imaging is that many anatomical structures are small in comparison to image resolution. This   \n22 is caused by not only limited image resolution but also the potentially convoluted shape of the   \n23 targeted anatomy [1]. Thinner, more oblate and/or curved structures undergo more distortion due   \n24 to sampling-related aliasing in comparison to larger, more spherical structures. These distortions   \n25 can limit detection power in the context of either clinical trials and/or at the level of patient specific   \n26 medicine [2, 3]. These results also show that, based on first principles, many disease relevant   \n27 anatomical structures in the brain, in particular cortical regions, mid-brain regions and hippocampal   \n28 subfields, should be quantified at higher resolutions (e.g. $\\approx 0 . 5 \\mathrm { m m ^ { 3 } }$ or smaller rather than the   \n29 more commonly available $\\approx 1 \\mathrm { m m ^ { 3 } }$ ). The need for increased resolution is only heightened when   \n30 considering aging and neurodegeneration where some brain structures may lose half or more of their   \n31 pre-disease onset volume or thickness.   \n32 Perceptual super resolution (PSR) for 2D RGB imagery consistently demonstrates the ability to   \n33 estimate more “realistic” looking upsampled data in comparison to traditional linear or nearest   \n34 neighbor interpolants [4]. While many competitive methods are available, the deep back projection   \n35 network (DBPN) [5] performed consistently in several competitions including NTIRE 2018 and 2019 ",
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+        "text": "[6], AIM 2019 [7] and PIRM 2018 [8]). These large challenges compared dozens of methods with respect to a variety of both perceptual and reconstruction metrics at different levels of upsampling and noise. ",
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+        "text": "39 Can the 2D RGB performance advantages of methods like the DBPN translate to improvements in   \n40 the 3D quantification of brain regions as seen in MRI? If so, then PSR for 3D neuroimaging promises   \n41 to improve quantification by better resolving the brain’s internal structures and tissue boundaries.   \n42 While traditional evaluations of PSR focus on reconstruction error and perceptual impression, these   \n43 measurements do not provide clinically relevant evidence of PSR’s value in quantification. One barrier   \n44 to evaluating PSR’s impact on clinically relevant outcomes (segmentation volumes) is that ground   \n45 truth segmentations do not exist at the super-resolved scale. To address this concern, [9] simulated low   \n46 resolution magnetic resonance images (MRI) of the brain from high-resolution (HR) images obtained   \n47 from the Human Connectome Project [10, 11]. They then applied a very deep super resolution   \n48 (VDSR) model to the simulated data and the high-resolution data and compared the accuracy of an   \n49 automated cortical segmentation method. This careful evaluation study demonstrated that cortical   \n50 segmentation on the VDSR images closely approximated the HR data. However, relatively few   \n51 details are provided about the training of this model and associated loss functions. Furthermore, it   \n52 remains unclear whether these improvements in reconstruction error would translate to the detection   \n53 of population-level effects in real world data particularly in the aging populations that are the target   \n54 of the majority of interventional trials for the brain.   \n55 A more recent effort in volumetric PSR for medical images [12] proposed SOUP-GAN: Super  \n56 resolution Optimized Using Perceptual-tuned Generative Adversarial Network (GAN). SOUP-GAN   \n57 adopts transfer learning from 2D VGG19 to 3D as proposed in [13] to produce a pseudo-volumetric   \n58 perceptual metric [14]. Shan et al. used this metric to denoise low-dose computed tomography   \n59 (CT) images and showed its effectiveness at preserving small anatomical structures. Similarly, the   \n60 SOUP-GAN effort demonstrates that the pseudo-3D perceptual metric improves both PSNR and   \n61 SSIM as well as shows visually appealing upsampling for a variety of medical imaging modalities.   \n62 That is, the surprising utility (in 2D) of VGG weights as a feature space [15] appears to at least   \n63 partially transfer to PSR in 3D medical imaging.   \n64 The current research provides perhaps the first broadly scoped, real world evaluation of MRI PSR for   \n65 quantification of neurodegenerative disease. Moreover, we demonstrate that a regression network   \n66 (ResNet) that predicts T1w image quality can yield a directly useful perceptual feature space that   \n67 performs competitively with pseudo-3D VGG19 features. We build these contributions upon the   \n68 backbone of a set of methods that we call concurrent super resolution and segmentation (CSRS) that   \n69 extends the proven 2D DBPN to 3D and also includes extra output channel(s) enabling segmentation   \n70 maps to be upsampled concurrently. We use this framework to test the impact of different loss   \n71 functions on a set of domain-specific, clinically relevant segmentation measurements related to   \n72 brain atrophy. Specifically, we evaluate CSRS on the quantification of frontotemporal disorders [3]   \n73 from publicly available longitudinal T1-weighted (T1w) neuroimaging (i.e. MRI). Of the several   \n74 combinations of losses that we evaluate, the best model improves not only segmentation performance   \n75 (when ground truth is available) but also detection power across all our related disorders: structural   \n76 changes in behavioral variant frontotemporal dementia (bvFTD), semantic variant primary progressive   \n77 aphasia (svPPA), nonfluent/agrammatic PPA (naPPA), progressive supranuclear palsy (PSP) and   \n78 corticobasal syndrome (CBS) each of which impacts known networks in the brain. CSRS with a new   \n79 perceptual loss based on a shallow ResNet layer performs as well or better than VGG-based models   \n80 in this test of the practical usefulness of PSR. ",
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+        "text": "81 The primary contributions of this work include: ",
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+        "text": "• new PSR that upsamples multi-label segmentations at the same time as intensity; • a new real world evaluation paradigm for PSR in neuroimaging; • comparison of three perceptual loss functions for PSR, two of which are new; • demonstration that loss choice impacts detection power in natural history studies of neurodegenerative disease. Standard intensity similarity and segmentation overlap metrics, on the other hand, do not discriminate performance between the candidate CSRS options. ",
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+        "text": "88 Model weights, sample data, and training code will be made publicly available after anonymous   \n89 review.   \n91 Software platform: We employ the ANTsX platform [16] version 2.3.5 for anatomical labeling,   \n92 data augmentation/sampling during model training and to form the tabular data for the statistical   \n93 evaluation. All MRI processing details follow [16]. Tensorflow 2.6.2 is used for deep learning   \n94 including a ResNet implementation and the CSRS architecture. R version 4.1 is used for statistical   \n95 analysis with packages lmer and ggplot2. All MRI processing was done on Amazon Web Services   \n96 parallel cluster with 24 cores and 32GB RAM per process (Intel(R) Xeon(R) Platinum 8259CL   \n97 CPU $\\ @ \\ 2 . 5 0 \\mathrm { G H z }$ ).   \n98 Data: Human Connectome Project (HCP): We downloaded 1,113 high-resolution $0 . 7 \\mathrm { m m ^ { 3 } }$ T1-   \n99 weighted images from the HCP on which to train CSRS. These T1w data were acquired using a   \n100 magnetization-prepared rapid gradient-echo (MPRAGE) sequence on a customized 3T Siemens   \n101 Skyra; see [10] for all details of acquisition. As such, these images provide both high resolution and   \n102 high quality in comparison to the majority of publicly available T1w MRI. Critically, they provide   \n103 superior resolution for the thin convoluted cortical layer that is critical to the measurement of brain   \n104 atrophy in frontotemporal disorders. We transformed these data into numpy blocks with randomly   \n105 selected high-resolution $6 4 ^ { 3 }$ patches and paired low-resolution $3 2 ^ { 3 }$ patches. For each patch pair, we   \n106 also provide a high-resolution binary segmentation and a low-resolution downsampled version of   \n107 that binary segmentation. Each patch segmentation was gained by 2-class $\\mathbf { k }$ -means performed on   \n108 the patch where the center voxel’s label determines which class (1 or 2) is used as foreground. This   \n109 collection of 16,640 patches is then divided randomly into train $\\scriptstyle \\mathrm { n = 1 6 } , 3 8 4 ,$ ) and test sets.   \n110 Data: Parkinson’s Progression Markers Initiative (PPMI): PPMI is a longitudinal multi-center   \n111 clinical study of PD patients and age-matched healthy controls http://www.ppmi-info.org.   \n112 PPMI employed(s) over 20 data collection sites with scanners that span the primary manufacturers   \n113 (Siemens, GE, Phillips), a variety of head coils and also magnet strengths (1.5T, 3T). This heterogene  \n114 ity of data collection provides a rich set of T1w images with highly variable image contrast, resolution   \n115 and quality. We manually reviewed and labelled 1,431 raw T1w from PPMI to capture the range of   \n116 quality in an ordinal scale. This resulted in a ground truth dataset with 456 images given grade “A”   \n117 (superior), 568 given grade “B”, 350 given grade “C” and 57 given grade “F” which represents images   \n118 that are of little to no use for quantitative studies of brain structure. We then employed a standard   \n119 3D ResNet (antspynet.create_resnet_model_3d with parameters lowest_resolution $^ { \\mathtt { = 3 2 } }$ ,   \n120 number_of_classification_label $\\mathtt { s } { = } 4$ , cardinality ${ \\tt = } 1$ , 39,424,004 parameters, 53 3D convo  \n121 lutional layers) to learn to predict this scale automatically and reliably from the input T1w. We denote   \n122 this network as a T1w Quality Rating Resnet (T1wQRResNet). Details of training T1wQRResNet   \n123 are in Supplementary Information.   \n124 Data: Frontotemporal Lobar Degeneration Neuroimaging Initiative (NIFD) & 4-Repeat   \n125 Tauopathy Neuroimaging Initiative (4RTNI): These inter-related multi-site studies share the goal   \n126 of improving the quantification of frontotemporal spectrum disorders with both imaging and clinical   \n127 scores. Like PPMI and HCP, these studies provide longitudinal T1w images that enable measurement   \n128 of not only the baseline brain structure differences between controls (individuals without a disease   \n129 i.e. normal aging) and disease groups but also differences in rates of change due to neurodegeneration.   \n130 We downloaded and curated 4RTNI and NIFD T1w data and merged these images into a common   \n131 database. These images were collected at three different sites using protocols consistent with ADNI   \n132 3T guidelines [2]. The images overall have a median spacing that is isotropically $1 \\mathrm { m m }$ with a minority   \n133 of subjects with out-of-plane spacing up to $1 . 2 \\mathrm { m m }$ . As such, these data suit the goals of testing PSR   \n134 for benefits to the quantification of neurodegenerative disease. After filtering data for very low quality   \n135 images and the presence of longitudinal data collected within 2 years of baseline, we obtained 128   \n136 baseline/171 followup images for controls, 60/112 for bvFTD, 38/72 for naPPA, 37/71 for svPPA,   \n137 55/70 for CBS and 75/102 for PSP. Further cohort details (age, education, sex, etc) are available in   \n138 supplementary information. We processed all images consistently and automatically with default   \n139 ANTsX pipelines to gain cortical, medial temporal lobe and deep brain structure segmentations for   \n140 every subject as described in [16]. By consensus, co-authors selected a priori regions for testing   \n141 within each of four groups CBS/PSP [17], bvFTD, svPPA and naPPA [18–24]. Details of the regions   \n142 and rationale for their selection are available in the Supplementary Information. See Figure 1 for an   \n143 overview of processing, the CSRS method and a visualization of the regions (1.C). ",
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+        "image_caption": [
+            "Figure 1: (A) Image processing begins with raw MRI, extracts the brain, labels cortical regions, labels medial temporal lobe regions and labels deep brain regions. (B) The CSRS method is used, here, to upsample data by a factor of 2 isotropically; the sketch of the algorithm provides an example of how two nearby regions would flow through the method and be stitched back together at high resolution. (C) The impact of SR on quantifying neurodegeneration is assessed on a priori regions that are specific to each clinical diagnostic group; all regions are bilateral except for svPPA which uses only left hemisphere cortical and medial temporal labels. "
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+        "type": "text",
+        "text": "144 2.1 Concurrent super resolution and segmentation methods ",
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+        "text": "145 CSRS uses, as a sub-algorithm, a three-dimensional and multi-output version of the neural network   \n146 architecture defined by the 2D deep back projection network (DBPN) [5]. The DBPN is uniquely   \n147 relevant to medical imaging in that it is perhaps the first published SR method that integrates the   \n148 downsampling-upsampling error (i.e. residual layers) as a feature map. This novel architecture may   \n149 prevent feature hallucination and constrain the high-resolution image to maintain features that are   \n150 consistent with the low-resolution input. We extend the 2D DBPN to 3D MRI data by, first, translating   \n151 2D convolutions, padding, striding and other relevant parameters to 3D. To generalize the architecture   \n152 further, we allow options for not only convolutional upsampling (transposed convolution) but also   \n153 nearest neighbor (or linear) interpolation layers at the user’s choice. Lastly, we implement flexible   \n154 choices of input channels, the number of residual layers (backprojection) layers and the number of   \n155 outputs. This 3D DBPN network implementation is available within R and python. All parameters   \n156 were the same as the published work [25] (though in translation to 3D) with the exception of the   \n157 number of back projection layers which has a large impact on the number of parameters. We reduced   \n158 the number of backprojection layers to 5 (16,264,322 parameters) due to the memory limitations   \n159 caused by working with large 3D images and limited GPU resources (all GPU computations in this   \n160 work were implemented with Nvidia V100s locally).   \n161 Efficient computational strategy is essential for CSRS to be applied to large 3D images (a brain   \n162 image may contain 10 million voxels) when CPUs and RAM are limited. As such, a local patch-work   \n163 strategy is necessary. Sampling, upsampling, mapping and unification (SUMU) are the common steps   \n164 needed for not only training but also inference. “Sampling” decides the form of the input data: full   \n165 images (not used here), image patches (used here in training) or anatomical image regions (used here   \n166 in inference). Upscaling determines the core approach to transferring the low-resolution data to a   \n167 higher-resolution output. Mapping compensates for shape or intensity distortion. Finally, “unification”   \n168 is an ensembling or merging step that brings together several sub-estimates of an SR image into a   \n169 single joined (final/full) SR image. We detail each of the 4 components below.   \n170 Sampling: We choose a patch-based model for training as these can easily be applied to input data   \n171 with different resolutions and fields of view. A second reason for patch-based modeling is that a   \n172 candidate network does not need to learn the full scope of image variation. This results in shallower   \n173 and faster to train networks that fit more easily onto readily available GPUs. The choices made   \n174 during sampling step define the feature basis set. Because prior super-resolution competitions suggest   \n175 larger patches lead to better performance, we choose the largest patches that would permit efficient   \n176 batch sizes of 4 (64x64x64). An additional ad hoc support for this choice is that cortical features   \n177 are relatively well-resolved in sub- $1 \\mathrm { m m }$ training images when voxel cubes of this sized are used.   \n178 However, there is no direct evidence that this size of patch domain is optimal for this problem.   \n179 Upscaling: is done with the CSRS’s DBPN architecture using nearest neighbor interpolation for   \n180 the upsampling layers. The software interface to CSRS also allows the user to optionally employ   \n181 standard linear (tri-linear) interpolation. We use the linear option as a reference in evaluation studies   \n182 below.   \n183 Mapping: may be used to compensate for distortions in the image shape or intensity space. Because   \n184 each patch is scaled independently on training data (to have an intensity range of -127.5 to 127.5),   \n185 the output of the PSR upsampled image intensities must be mapped back to the original quantitative   \n186 space. This is performed by directly comparing the output of the PSR upsampled patch/region to the   \n187 original data upsampled by nearest neighbor or linear interpolation. As such, we can accurately retain   \n188 quantitative intensity data at the original scale/units with minimal distortion and/or stitching artifacts.   \n189 Unification: this is a general term that, here, refers to the algorithm that is used to derive a single   \n190 CSRS image and multi-label segmentation from multiple CSRS sub-images (not necessarily isotropic   \n191 patches as in training). In 2D, multiple input images are typically generated from a single input by   \n192 “augmentation” e.g. random flipping, translation, etc thus allowing a practitioner to gain multiple   \n193 “votes” about how the SR image should appear at any given voxel. Such a step is used in most PSR   \n194 competitions to reduce aliasing or artifacts and may involve averaging, sharpening or more complex   \n195 modeling such as joint intensity fusion, multi-channel deep learning or other ensemble methods.   \n196 Due to the high memory and computation cost of running CSRS on 3D images, we instead apply   \n197 CSRS to either sub-regions of interest or, when a full T1w brain image is desired, each hemisphere.   \n198 The unification step then maps each local patch intensity range back to the original MRI range and   \n199 then joins the sub-regions back together to complete the SR reconstruction. Augmentation can be   \n200 employed beyond this but at substantial increase in computation time (e.g. 10x to see meaningful   \n201 gains due to augmentation).   \n202 Loss functions for CSRS: We employ a loss function that seeks to balance reconstruction error   \n203 (intensity difference, abbreviated here as R), edge preserving denoising (total variation, abbreviated   \n204 as TV), perceptual quality (based on VGG or ResNet) and segmentation overlap (Dice, abbreviated   \n205 as D). Each of these terms can be up or down weighted to control the network’s performance where   \n206 mean squared error (L2 intensity error) leads to smoother results, L1 (or total variation) provides   \n207 denoising and the perceptual loss yields more natural appearing output textures and shapes. The   \n208 Dice loss term seeks to minimize distortions in the shape of segmentation objects on the output   \n209 of CSRS. The Dice loss is only applied to the second output channel of the network which uses a   \n210 sigmoid activation function appropriate for probabilistic/binary data. We refer to CSRS trained with   \n211 specific combinations of these losses by concatenation of the abbreviations above. For example,   \n212 CSRS.R.TV.D.Res6 refers to a network trained with reconstruction loss, TV regularization, Dice loss   \n213 and the 6th layer of the T1wQRResNet for perceptual loss. ",
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+        "text": "Recent research demonstrates that deep learning models trained on large-scale object detection reference datasets (e.g. imagenet) encode a feature space that may mimic human perception [26]. Such perceptual spaces typically arise from the activations that occur within the layers of convolutional networks trained on massive classification datasets. Here, however, we compare a standard VGG based perceptual space (block2_conv2) (mapped to 3D as described before) to those defined by the T1wQRResNet. From T1wQRResNet, we choose two different deep layers that have similar numbers of parameters to the 3D version of the VGG19 block2_conv2 network: res_conv_block_6 (the 2nd convolutional block) and res_conv_block_21 (the 7th convolutional block). This allows us to compare perceptual metrics based on either pseudo-3D VGG19 or our intrinsically 3D res_conv_block choices. ",
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+        "text": "24 Quantification of medical images requires a high degree of faithfulness to the input data. \"Halluci  \n25 nated\" features are undesirable. As such, our baseline loss function focuses on reconstruction error   \n226 and TV for both intensity and segmentation images. We then add perceptual and Dice losses for   \n227 further comparison. If we denote $I$ as the estimated super-resolution, $I _ { s }$ as the estimated segmentation   \n228 from the sigmoid output channel, $J$ as the real high resolution image, $J _ { s }$ as the real high resolution   \n229 segmentation, then the final loss function that we optimize is: ",
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+        "image_caption": [
+            "Figure 2: Best model CSRS applied to three categories of anatomy where row (A) is the original resolution (OR) and segmentation and row (B) is the output of CSRS.R.TV.D.Res6. "
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+        "type": "equation",
+        "img_path": "images/59e0068b476efa1151e61e5f664050baca9ca632b1d6a56116751df774fac640.jpg",
+        "text": "$$\n\\begin{array} { r } { I - J \\| ^ { 2 } w _ { r } ^ { i } + \\| I _ { s } - J _ { s } \\| ^ { 2 } w _ { r } ^ { s } + T V ( I , J ) w _ { t } ^ { i } + T V ( I _ { s } , J _ { s } ) w _ { t } ^ { s } + \\| f _ { n } ( I ) - f _ { n } ( J ) \\| ^ { 2 } w _ { f } + D i c e ( I _ { s } , J _ { s } ) w _ { d } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "230 where the term $\\| \\cdot \\|$ indicates the euclidean norm, $T V ( \\cdot , \\cdot )$ indicates the total variation norm (which   \n231 provides denoising), $w _ { r , t , f , d }$ (superscripts for intensity or segmentation) indicates a term-specific   \n232 scalar weight and $f _ { n } ( . )$ indicates a perceptual feature map. The weight terms can be tuned for   \n233 performance and application area given an objective and quantitative evaluation metric. We initially   \n234 manually tuned the training of a DBPN model with only the reconstruction metrics $( \\| I - J \\| ^ { 2 } w _ { r } ^ { I } \\dot { + }$   \n235 $\\lVert I _ { s } - \\dot { J _ { s } } \\rVert ^ { 2 } w _ { r } ^ { s }$ with $w _ { r } ^ { i } = 5 e - 4$ and $w _ { r } ^ { s } = 1$ ) using adam optimizer and learning rate 5e-5. We   \n236 then set weights relative to the value of the reconstruction error after convergence such that: the TV   \n237 loss is roughly $2 / 3$ the reconstruction term (R); the perceptual loss is roughly $3 \\mathrm { x } \\ \\mathrm { R }$ ; the Dice loss is   \n238 roughly equivalent to the perceptual loss. This strategy, based on our task-specific goals, enables us   \n239 to compare models consistently and add/subtract terms without extensive weight optimization.   \n40 Computation and inference: All models were implemented with tensorflow. The computation to   \n241 double magnification – for a single T1w – takes (generally on a modern computational platform)   \n42 between 10 and 40 minutes. Results are computed region-wise over the set of segmentation labels   \n243 where CSRS is run on each cropped label and its associated intensity. When multiple regions are   \n244 used (as is done here), then results are stitched back together while using a linear mapping back   \n245 to the original intensity space and a arg_max operation to define the hard segmentation labels at   \n46 every voxel in the stitched, joint intensity/probability double magnification space. See Figure 2 for   \n47 an example result of CSRS as applied to the variety of brain regions in this study. Figure 3 shows a   \n48 zoomed visual comparison of the impact on intensity and the lack of stitching artifacts. ",
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+        "type": "text",
+        "text": "2.2 Quantification of CSRS impact on segmentation and intensity in ground truth data ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Evaluation of PSR results on simulated downsampled-upsampled data does not constitute real world conditions. However, for reference, we include evaluation results based on an independent set of labeled brain images [27]. For these images, we downsample with nearest neighbor interpolation and upsample with linear interpolation (for the intensity) and a “generic label” interpolation that is designed for multi-label images [28] thereby allowing us to report standard metrics of Dice overlap, PSNR and SSIM to complement our study of brain atrophy detection. Figure 4 demonstrates example results illustrating this component of our evaluation. ",
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+        "img_path": "images/e5ad374190f490e8c5abd0d89955a3089fdac304feb8b087fbb3b132df729771.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of CSRS with different loss functions to original resolution and linear upsampling. The bold (panel E) is the best performing model according to quantitative criteria. However, visual differences between the perceptual models (D,E,F) are not easy to discern. "
+        ],
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+        "img_path": "images/f3b1e61e60f74e47e18322e106c4ced735805a3652dae28f93cd6626678bbe98.jpg",
+        "image_caption": [
+            "Figure 4: Panel (A) shows the original $\\mathrm { 1 m m ^ { 3 } }$ resolution ground truth image and its segmentation. Panel (B) shows the impact of linear/generic label upsampling of ground truth data artifically downsampled to $2 \\mathrm { m m ^ { 3 } }$ . Panel (C) shows a CSRS result where other models are visually similar to this. Panel (D) demonstrates that all regions improve with CSRS (all differences $> 0$ ) and that regions with lower Dice overlap under the linear/generic label model improve more when upsampled with CSRS. "
+        ],
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+        "text": "257 2.3 Quantification of effect sizes in frontotemporal disorder atrophy ",
+        "text_level": 1,
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+        "text": "The frontotemporal disorders produce a profound and debilitating effect on patients with concomitant, symptom-related atrophy. Measuring this atrophy is critical to detecting the effects, for instance, of disease modifying therapies that may slow atrophy. Such measurements are challenged by low resolution and this challenge is compounded by the degeneration process itself. ",
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+        "text": "262 We use statistical modeling to determine if CSRS can mitigate the known limitations of resolution on   \n263 atrophy measurement. We adopt an interpretable mixed effects modeling approach (lmer)[29] to   \n264 estimate effect sizes per brain region, per diagnostic category and per resolution/CSRS model. The   \n265 baseline performance is determined by the effect sizes estimated on the original resolution (OR) data.   \n266 We estimate effect sizes following [30, 31]. Better methods, under this design, should more reliably   \n267 identify disease-related atrophy which will be reflected in increased effect sizes for a given set of $a$   \n268 priori diagnosis-specific regions. The model for the region of interest $i$ $( R O I _ { i } )$ ) is: ",
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+    {
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+        "img_path": "images/85481eb268e4a6aafc8cc4feecdead2221cf94106ee4cf1b4070db59ca054451.jpg",
+        "text": "$$\nR O I _ { i } \\approx A g e _ { b } + S e x + B V _ { b } + D X + \\Delta T * D X + ( 1 | I D ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "269 with $( 1 | I D )$ representing a subject-specific random effect, $A g e _ { b }$ is the subject’s age at the first visit,   \n270 $B V _ { b }$ is the first visit brain volume, $D X$ is the diagnosis for the subject, $\\Delta T$ is the change in time   \n271 since baseline and the $\\Delta T * D X$ represents an interaction between time and diagnosis. The $R O I _ { i }$   \n272 represents the volume for all regions. However, for cortical regions, we also use the region’s thickness   \n273 measurement as a second outcome (as this is a standard measurement in morphometry of the human   \n274 cortex). We estimate effect sizes for cross-sectional effects via the model’s parameter fit for the   \n275 diagnosis $( D X )$ term; we estimate longitudinal effect sizes via the parameter on the interaction term. ",
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+        "text": "Table 1: Summary of results where the comparison of the model impact on effect size is computed by bootstrapped $\\scriptstyle ( \\mathrm { n = 1 0 0 0 } )$ ) paired t.test. The number of pairs is 274 (see Table 2 for further breakdown by category). CSRS losses are abbreviated as $\\mathbf { R } =$ reconstruction, $\\mathrm { T V } { = }$ total variation, $\\scriptstyle \\mathbf { D = }$ dice, VGG $\\circeq$ VGG19 pseudo 3D features, Res6 is from the 6th layer of T1wQRResNet and Res21 is the 21st layer of T1wQRResNet. srmeanES indicates the mean effect size for the model averaged over all a priori regions; boot.95ci is the 95 percent confidence interval for the improvement in effect size due to the model. t represents the $t$ -statistic and boot.p represents the bootstrapped p-value for the significance of the improvement in effect size. Columns psnr and ssim show the standard PSNR and SSIM values for an image for which we have ground truth high-resolution intensity and segmentation. The dice columns show the mean and standard deviation of the Dice overlap between ground truth and the upsampled simulated data with each model, estimated over all regions. Best $=$ bold. ",
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+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>srmeanES</td><td rowspan=1 colspan=1>boot.95ci</td><td rowspan=1 colspan=1>t</td><td rowspan=1 colspan=1>boot.p</td><td rowspan=1 colspan=1>psnr</td><td rowspan=1 colspan=1>ssim</td><td rowspan=1 colspan=1>dice.mean</td><td rowspan=1 colspan=1>dice.sd</td></tr><tr><td rowspan=1 colspan=1>OR</td><td rowspan=1 colspan=1>0.559</td><td rowspan=1 colspan=1>0/0</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>0.468</td><td rowspan=1 colspan=1>-0.1006/-0.08163</td><td rowspan=1 colspan=1>-18.61</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>40.6</td><td rowspan=1 colspan=1>0.996</td><td rowspan=1 colspan=1>0.769</td><td rowspan=1 colspan=1>0.047</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV</td><td rowspan=1 colspan=1>0.574</td><td rowspan=1 colspan=1>0.01124/0.01854</td><td rowspan=1 colspan=1>7.96</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.0</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.884</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D</td><td rowspan=1 colspan=1>0.582</td><td rowspan=1 colspan=1>0.01868/0.0273</td><td rowspan=1 colspan=1>10.38</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>41.8</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.883</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>0.581</td><td rowspan=1 colspan=1>0.01775/0.02559</td><td rowspan=1 colspan=1>10.76</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.0</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.885</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.VGG</td><td rowspan=1 colspan=1>0.577</td><td rowspan=1 colspan=1>0.01403/0.02209</td><td rowspan=1 colspan=1>8.86</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>41.9</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.884</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.Res6</td><td rowspan=1 colspan=1>0.572</td><td rowspan=1 colspan=1>0.009741/0.01665</td><td rowspan=1 colspan=1>7.49</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.4</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.886</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>0.588</td><td rowspan=1 colspan=1>0.02497/0.0331</td><td rowspan=1 colspan=1>14.02</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.4</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.885</td><td rowspan=1 colspan=1>0.032</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.Res21</td><td rowspan=1 colspan=1>0.577</td><td rowspan=1 colspan=1>0.01462/0.02143</td><td rowspan=1 colspan=1>10.40</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.3</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.884</td><td rowspan=1 colspan=1>0.031</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res21</td><td rowspan=1 colspan=1>0.581</td><td rowspan=1 colspan=1>0.01814/0.02627</td><td rowspan=1 colspan=1>10.59</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>42.3</td><td rowspan=1 colspan=1>0.997</td><td rowspan=1 colspan=1>0.887</td><td rowspan=1 colspan=1>0.03</td></tr></table>",
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+            "Figure 5: Bland-Altman plots for model CSRS.R.TV.D.Res6 demonstrate variability in the performance by type of anatomy and by diagnostic grouping with some individual points generating substantially greater $\\%$ improvement than suggested by the overall trend. Similarly, a few points show decreased performance relative to OR. "
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+        "type": "text",
+        "text": "276 3 Results ",
+        "text_level": 1,
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+        "text": "Table 1 summarizes overall results where we show original resolution and results from linear upsampling, as baseline, and compare to eight variants of CSRS. Two of these do not use perceptual metrics. The remaining six add or subtract Dice loss and each of our candidate perceptual losses. Table 1 shows both the aggregate impact of model on effect size estimates in the neurodegeneration data as well as intensity similarity (reconstruction) and Dice overlap in the ground truth data. Dice overlap (a measure that varies between zero and one) improves by a margin of 0.11 to 0.123 $9 5 \\%$ CI bootstrapped percentile confidence interval, $p < 1 e - 1 6$ . See Figure 4. ",
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+        "text": "277   \n278   \n279   \n280   \n281   \n282   \n283   \n284   \n285   \n286   \n287   \n288 ",
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+        "text": "Table 2 focuses on the two perceptual models with the greatest improvement from original resolution as assessed by pairwise $t$ -test. It breaks down the effect size results in relation to which type of effect size is being analyzed (cross-sectional or longitudinal) and by brain region / diagnostic grouping. Relatedly, Figure 5 shows a Bland-Altman style plot that demonstrates, for the CSRS.R.TV.D.Res6 model, the range of effect size changes due to CSRS across all 274 measurement points. ",
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+            "Table 2: Summary of results for the two best perceptual models broken down by anatomical class, type of predictor (longitudinal or cross-sectional) and diagnostic groups. The n column indicates the number of samples used in the statistical testing. The codes in the AnatClass column are: CtxV - cortical volume; CtxT - cortical thickness; MB - deep brain (for CBS/PSP); MTL - medial temporal lobe (for svPPA). The columns that have non-NA DX2 means that both DX and DX2 groups were aggregated in the computation of the bootstrapped paired $t$ -test for the given group of anatomy. "
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+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>AnatClass</td><td rowspan=1 colspan=1>isLong</td><td rowspan=1 colspan=1>DX</td><td rowspan=1 colspan=1>DX2</td><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=1>srmeanES</td><td rowspan=1 colspan=1>boot.95ci</td><td rowspan=1 colspan=1>t</td><td rowspan=1 colspan=1>boot.p</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>Both</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>274</td><td rowspan=1 colspan=1>0.581</td><td rowspan=1 colspan=1>0.01775/0.02559</td><td rowspan=1 colspan=1>10.763</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.826</td><td rowspan=1 colspan=1>0.007153/0.01626</td><td rowspan=1 colspan=1>4.936</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>1.016</td><td rowspan=1 colspan=1>-0.009547/0.01006</td><td rowspan=1 colspan=1>0.033</td><td rowspan=1 colspan=1>0.9769</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.600</td><td rowspan=1 colspan=1>0.03683/0.1208</td><td rowspan=1 colspan=1>3.458</td><td rowspan=1 colspan=1>0.0246</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>svPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>1.003</td><td rowspan=1 colspan=1>0.02801/0.06981</td><td rowspan=1 colspan=1>4.190</td><td rowspan=1 colspan=1>0.0112</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.645</td><td rowspan=1 colspan=1>0.01848/0.03313</td><td rowspan=1 colspan=1>6.719</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.527</td><td rowspan=1 colspan=1>0.03521/0.05297</td><td rowspan=1 colspan=1>9.445</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.201</td><td rowspan=1 colspan=1>0.01409/0.03941</td><td rowspan=1 colspan=1>3.924</td><td rowspan=1 colspan=1>0.0110</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.VGG</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>svPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0.701</td><td rowspan=1 colspan=1>-0.0159/0.002513</td><td rowspan=1 colspan=1>-1.309</td><td rowspan=1 colspan=1>0.2652</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>Both</td><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>274</td><td rowspan=1 colspan=1>0.588</td><td rowspan=1 colspan=1>0.02497/0.0331</td><td rowspan=1 colspan=1>14.021</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.831</td><td rowspan=1 colspan=1>0.01199/0.02182</td><td rowspan=1 colspan=1>6.573</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>1.023</td><td rowspan=1 colspan=1>-0.002896/0.01832</td><td rowspan=1 colspan=1>1.394</td><td rowspan=1 colspan=1>0.1604</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.589</td><td rowspan=1 colspan=1>0.02166/0.1131</td><td rowspan=1 colspan=1>2.718</td><td rowspan=1 colspan=1>0.0454</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Cross</td><td rowspan=1 colspan=1>SvPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>1.004</td><td rowspan=1 colspan=1>0.02782/0.07253</td><td rowspan=1 colspan=1>4.021</td><td rowspan=1 colspan=1>0.0102</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxV</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.658</td><td rowspan=1 colspan=1>0.03203/0.04756</td><td rowspan=1 colspan=1>9.751</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>CtxT</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>bvFTD</td><td rowspan=1 colspan=1>naPPA</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0.545</td><td rowspan=1 colspan=1>0.05421/0.06995</td><td rowspan=1 colspan=1>15.151</td><td rowspan=1 colspan=1>0.0000</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MB</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>CBS/PSP</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.198</td><td rowspan=1 colspan=1>0.006017/0.04464</td><td rowspan=1 colspan=1>2.233</td><td rowspan=1 colspan=1>0.0166</td></tr><tr><td rowspan=1 colspan=1>CSRS.R.TV.D.Res6</td><td rowspan=1 colspan=1>MTL</td><td rowspan=1 colspan=1>Long</td><td rowspan=1 colspan=1>svPPA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0.704</td><td rowspan=1 colspan=1>-0.0177/0.01214</td><td rowspan=1 colspan=1>-0.369</td><td rowspan=1 colspan=1>0.7511</td></tr></table>",
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+        "text": "289 4 Discussion ",
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+        "text": "The PSNR and SSIM improve similarly across all CSRS models and do not substantively differentiate performance. Dice overlap is consistently superior than linear upsampling across all models but shows little difference between models with perhaps a small advantage for the ResNet features. Greater stratification may be seen when looking at results that relate to quantifying the phenotypic heterogeneity of brain atrophy in frontotemporal spectrum diagnostic groups. Model CSRS.R.TV.D.Res6 stands out under this criteria with Table 2 suggesting that the majority of the improvement arises for cortical measurements, particularly longitudinally. Performance improvements are not, however, perfectly consistent. Figure 5 shows that CSRS augments effect size in the large majority of regions (some greatly so) but a few regions are subtly better at OR. Additional discussion of performance implications with respect to individual regions and diagnoses is in supplementary information. ",
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+        "text": "300 The extension of PSR to 3D raises opportunities as well as challenges. Parameter exploration is   \n301 fundamentally limited because training a model on our patch dataset for 1 epoch takes over 12 hours   \n302 (we trained each model for 2 epochs or until convergence). Other architectures than DBPN may   \n303 perform better with CSRS such as ESRGAN [32] or, potentially, methods with stronger modality   \n304 specific priors on the convolutional kernels [33]. Specifically, fast-training, fewer parameter models   \n305 may ease some of the computational burden and facilitate more parameter exploration.   \n306 CSRS performance is fundamentally limited by the quality of its segmentation inputs. It may be   \n307 more beneficial to develop new methods that operate at high resolution (HR) – adding substantial   \n308 computational cost if the goal is to take advantage of HR features – or that take advantage of   \n309 intrinsically HR ground truth data. The primary barrier to such an effort is the current lack of HR   \n310 ground truth labels for neuroimaging and in particular for neurodegenerative disease. Moreover,   \n311 most methods embed resolution assumptions in their own processing choices and optimize for these   \n312 choices. As such, CSRS bridges a performance gap with a practical solution readily available today.   \n313 Retooling existing methods and segmentation labels for HR (e.g. 7T MRI) is costly both computation  \n314 ally and in terms of the effort of human experts due to the already high volume of 3D neuroimaging.   \n315 We demonstrated that CSRS, in most of its variants, leads to significant performance improvements   \n316 over our reference of original resolution $\\mathrm { { ( l m m ^ { 3 } ) } }$ ) image processing and ground truth labels. Because   \n317 CSRS operates on existing images and labels, new HR method and segmentation development is   \n318 not required. Thus, CSRS may be used to improve existing ground truth datasets and existing   \n319 processed data, today. However, comparison to other and/or larger real world datasets is needed to   \n320 help determine the extent to which our results may be deployed to new data without concern.   \n321 References   \n322 1. Mulder MJ, Keuken MC, Bazin PL, Alkemade A, Forstmann BU. Size and shape matter: The   \n323 impact of voxel geometry on the identification of small nuclei. PLoS ONE. 2019. https://doi.   \n324 org/10.1371/journal.pone.0215382.   \n325 2. Veitch DP, Weiner MW, Aisen PS, Beckett LA, Cairns NJ, Green RC, et al. Understanding disease   \n326 progression and improving Alzheimer’s disease clinical trials: Recent highlights from the Alzheimer’s   \n327 disease neuroimaging initiative. Alzheimer’s & Dementia : the Journal of the Alzheimer’s Association.   \n328 2019;15:106–52.   \n329 3. Boxer AL, Gold M, Feldman H, Boeve BF, Dickinson SL-J, Fillit H, et al. New directions in   \n330 clinical trials for frontotemporal lobar degeneration: Methods and outcome measures. Alzheimer’s &   \n331 dementia : the journal of the Alzheimer’s Association. 2020;16:131–43.   \n332 4. Blau Y, Michaeli T. The perception-distortion tradeoff. Proceedings of the IEEE Conference   \n333 on Computer Vision and Pattern Recognition, pp 6228-6237, 2018. 2017. https://doi.org/10.   \n334 1109/CVPR.2018.00652.   \n335 5. Haris M, Shakhnarovich G, Ukita N. Deep back-projection networks for single image super  \n336 resolution. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2020; 43(12):4323-   \n337 4337.   \n338 6. Agustsson E, Timofte R. NTIRE 2017 challenge on single image super-resolution: Dataset and   \n339 study. In: Proceedings of the IEEE conference on computer vision and pattern recognition workshops   \n340 2017. pp. 126-135.   \n341 7. Gu S, Danelljan M, Timofte R, Haris M, Akita K, Shakhnarovic G, et al. AIM 2019 challenge on   \n342 image extreme super-resolution: Methods and results. In: 2019 IEEE/CVF International Conference   \n343 on Computer Vision Workshop (ICCVW). Seoul, Korea (South): IEEE. pp. 3556–64.   \n344 8. Blau Y, Mechrez R, Timofte R, Michaeli T, Zelnik-Manor L. The 2018 PIRM challenge on   \n345 perceptual image super-resolution. In: 2018 Proceedings of the European Conference on Computer   \n346 Vision (ECCV) Workshops. pp. 334–55.   \n347 9. Tian Q, Bilgic B, Fan Q, Ngamsombat C, Zaretskaya N, Fultz NE, et al. Improving in vivo human   \n348 cerebral cortical surface reconstruction using data-driven super-resolution. Cerebral cortex (New   \n349 York, NY : 1991). 2021;31:463–82.   \n350 10. Glasser MF, Smith SM, Marcus DS, Andersson JLR, Auerbach EJ, Behrens TEJ, et al. The human   \n351 connectome project’s neuroimaging approach. Nature Neuroscience. 2016; 19(9): pp. 1175-1187.   \n352 11. 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Dementia and Geriatric Cognitive Disorders.   \n386 2009. https://doi.org/10.1159/000194658.   \n387 25. Haris M, Shakhnarovich G, Ukita N. Deep back-projection networks for super-resolution. In:   \n388 Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition.   \n389 2018.   \n390 26. Zhang R, Isola P, Efros AA, Shechtman E, Wang O. The unreasonable effectiveness of deep   \n391 features as a perceptual metric. In: Proceedings of the IEEE Computer Society Conference on   \n392 Computer Vision and Pattern Recognition. 2018.   \n393 27. Klein A, Tourville J. 101 labeled brain images and a consistent human cortical labeling protocol.   \n394 Frontiers in neuroscience. 2012;6:171.   \n395 28. Schaerer J, Roche F, Belaroussi B. A generic interpolator for multi-label images. 2014. https:   \n396 //doi.org/10.54294/nr6iii.   \n397 29. Bates D, Mächler M, Bolker B, Walker S. Fitting linear mixed-effects models using lme4 | bates |   \n398 journal of statistical software. Journal of Statistical Software. 2015;67.   \n399 30. Brysbaert M, Stevens M. Power analysis and effect size in mixed effects models: A tutorial.   \n400 Journal of Cognition. 2018;1.   \n401 31. Ben-Shachar M, Lüdecke D, Makowski D. Effectsize: Estimation of effect size indices and   \n402 standardized parameters. Journal of Open Source Software. 2020;5.   \n403 32. Wang X, Yu K, Wu S, Gu J, Liu Y, Dong C, et al. ESRGAN: Enhanced super-resolution generative   \n404 adversarial networks. arXiv e-prints. 2018;arXiv:1809.00219.   \n405 33. Bell-Kligler S, Shocher A, Irani M. Blind super-resolution kernel estimation using an internal-gan.   \n406 In: Advances in Neural Information Processing Systems. 2019. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We provide the data, in supplementary material, that is needed to generate the tables shown in the paper. The raw data is publicly available but we do not have permission to redistribute these data. However, we do provide the ability to reproduce key results in the Tables of the main manuscript via supplementary information.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See beginning of Methods section and CSRS methods section. ",
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