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+# 2D Amodal Instance Segmentation Guided by 3D Shape Prior
+
+Zhixuan Li $^{1,2}$ , Weining Ye $^{2}$ , Tingting Jiang $^{\text{串}1,2}$ , and Tiejun Huang $^{2}$
+
+<sup>1</sup> Advanced Institute of Information Technology, Peking University, Hangzhou, China
+
+$^{2}$  National Engineering Research Center of Visual Technology, School of Computer Science, Peking University, Beijing, China {zhixuanli,ywning,ttjiang,tjhuang}@pku.edu.cn
+
+Abstract. A modal instance segmentation aims to predict the complete mask of the occluded instance, including both visible and invisible regions. Existing 2D AIS methods learn and predict the complete silhouettes of target instances in 2D space. However, masks in 2D space are only some observations and samples from the 3D model in different viewpoints and thus can not represent the real complete physical shape of the instances. With the 2D masks learned, 2D amodal methods are hard to generalize to new viewpoints not included in the training dataset. To tackle these problems, we are motivated by observations that (1) a 2D amodal mask is the projection of a 3D complete model, and (2) the 3D complete model can be recovered and reconstructed from the occluded 2D object instances. This paper builds a bridge to link the 2D occluded instances with the 3D complete models by 3D reconstruction and utilizes 3D shape prior for 2D AIS. To deal with the diversity of 3D shapes, our method is pretrained on large 3D reconstruction datasets for high-quality results. And we adopt the unsupervised 3D reconstruction method to avoid relying on 3D annotations. In this approach, our method can reconstruct 3D models from occluded 2D object instances and generalize to new unseen 2D viewpoints of the 3D object. Experiments demonstrate that our method outperforms all existing 2D AIS methods.
+
+Keywords: Amodal, occlusion, instance segmentation
+
+# 1 Introduction
+
+Different from visible instance segmentation (VIS) [9,1,14,31] which only predicts the visible region of each instance, amodal instance segmentation (AIS) [19, 37, 30] task poses a harder challenge that demands to predict both the visible and occluded parts. AIS has many potential applications, including auto-driving [27], automatic checkout in the market [7] and image editing [36].
+
+The concept of AIS was proposed in 2016 [19], and several datasets [38, 5, 27, 13, 7] have been provided. Most of the existing amodal methods [19, 38, 5, 7, 13, 27, 34, 17] are developed based on visible instance segmentation methods [18, 9] that directly minimize the discrepancy between amodal prediction and ground-truth masks. Recently, some methods consider the characteristics of the amodal
+
+![](images/65d11ac77ee73fa04e5e0ff41bf6c5fd50d7018743fb54136c9293bcfa6e8807.jpg)  
+(a)  
+(b)  
+Fig. 1. Overview and comparison of the 2D shape prior dictionary (SPD) based method ShapeDict [30] and our proposed 3D shape prior generation-based method. For the left input RGB images, the first stage of both methods conduct instance segmentation to obtain amodal bounding boxes and visible masks for each instance. (a) ShapeDict regards each visible mask as query for SPD and retrieves the matched amodal shape prior masks. Due to the limited diversity of prestored shape prior, the retrieved shape prior is more appropriate for samples that have been seen (green box) in the dictionary rather than the unseen ones (red box). (b) Our proposed method adaptively generates 3D shape prior models from visible masks and performs projection for 2D amodal masks without needing for prestoring.
+
+problem itself and propose new solutions. For example, relative depth order of instances is used to help comprehend the scene [37, 36]. Weakly supervised methods are proposed [23, 36, 25] without needing ground-truth amodal mask while taking ground-truth visible mask as input and supervision.
+
+Besides, a natural solution is to use the shape-prior knowledge for handling the occluded instance, which lacks the shape and appearance information of the invisible region. In 2020, ShapeDict [30] proposes to utilize 2D shape prior knowledge to deal with the amodal segmentation problem. As shown in Fig. 1(a), ShapeDict first establishes a 2D shape prior dictionary (SPD) by applying the K-means algorithm on the ground-truth 2D amodal masks in the training set, and takes the cluster centers as shape priors. During inference, the closet shape prior to the predicted visible mask is retrieved from the SPD and used for amodal segmentation. However, this nearest neighbor search approach can only work for amodal masks having been seen during training (as shown in the green box of Fig. 1(a)). Otherwise, it will fetch inappropriate shape prior and lead to wrong amodal segmentation. For example, an occluded airplane photographed from a new viewpoint, whose amodal mask is not stored in the SPD, cannot be correctly matched. Therefore, ShapeDict is limited to the number and variety of shape prior masks stored in the dictionary, making it hard to generalize on unseen viewpoints. In this paper, we consider if it is possible to adaptively generate the
+
+shape prior masks rather than prestoring the shape prior masks in a dictionary, and tackle the challenges of viewpoint changes?
+
+With these problems in mind, we hope to learn the shape prior in the 3D space, which is a unified representation of 2D masks from all viewpoints and can generalize to new perspectives. Meanwhile, to avoid the shortcoming of the SPD method in ShapeDict, we hope to generate the 3D shape prior with learned shape knowledge adaptively and need no requirement for a prestored shape dictionary. To achieve these two purposes, we propose to reconstruct the complete 3D shape prior from the 2D occluded instance, as shown in Fig. 1(b). To accomplish 3D reconstruction, either multi-view images as input or 3D models as supervision signals are usually needed. Unfortunately, both are not available in any existing 2D AIS datasets. However, the good news is that, in recent years, single-view unsupervised 3D reconstruction methods [24, 20, 12] can avoid the requirements of multiple views and 3D model for training, which makes 2D amodal datasets usable for 3D reconstruction. For single-view unsupervised 3D reconstruction methods, the 3D model is first reconstructed from the single-view input RGB image and then projected along the estimated viewpoints to 2D masks. During reconstruction, only the 2D visible mask is provided as the supervision signal and the 3D reconstruction model is supervised indirectly by the consistency between 2D projection of the 3D model and 2D visible mask. In our method, the 2D amodal mask is used as the supervision signal for the single-view unsupervised 3D reconstruction.
+
+In this work, we propose Amodal 3D Network (A3D), a novel coarse-to-fine architecture that combines category-specific 3D shape prior with 2D AIS. As shown in Fig. 2, for an input RGB image, we first apply the visible instance segmentation method to obtain visible masks of each instance. Next, we use a two-branch structure, in which the upper branch utilizes an Encoder Decoder Network for Category-specific 3D shape prior reconstruction, and the lower branch predicts camera viewpoint parameters by the Viewpoint Estimator. Then the Differentiable Render projects the 3D shape prior model according to the predicted viewpoint to the 2D coarse amodal mask. Finally, the Region-specific Edge Refine module refines the edges with the guidance of the visible mask and predicts the final amodal mask. With this coarse-to-fine pipeline, A3D can benefit from the power of 3D shape prior modelling and 2D edge refinement at the same time.
+
+It is worth noting that our 3D shape prior reconstruction method only requires 2D amodal masks as ground truth, without supervision signals like 3D models, which are expensive to obtain. Because the 3D reconstruction module plays a crucial role in our method, we need to ensure that the reconstruction module can generate high-quality 3D shape prior models when facing 2D occluded instances. We design a pretrain-and-finetune pipeline that the single-view 3D reconstruction module is first pretrained on a large 3D reconstruction dataset like ShapeNet [2] for common shape in unsupervised approach without using 3D annotations. Then we conduct finetuning on the 2D AIS dataset for specific shape knowledge learning.
+
+The effectiveness of our proposed method A3D is evaluated on several challenging datasets, including D2SA for market goods, KINS for person and vehicle, and COCOA-cls for life scene. We achieve state-of-the-art on all datasets.
+
+We summarize our final contributions as follows:
+
+1. A new method A3D is proposed for AIS, which utilizes the single-view unsupervised 3D reconstruction for 3D shape prior learning, to tackle the problem that 2D amodal segmentation methods are hard to generalize on new viewpoints. To our best knowledge, it is the first time the 3D shape prior is used for 2D AIS.
+
+2. A coarse-to-fine pipeline is designed, which learns the 3D coarse shape prior and then refines edges with region-specific loss. It is end-to-end trainable and profits from both 3D and 2D information.
+
+# 2 Related Work
+
+# 2.1 2D Instance Segmentation
+
+Amodal Instance Segmentation Comparing to visible instance segmentation, due to the shape of both visible and occluded regions needing to be predicted, the Amodal Instance Segmentation (AIS) task has fewer clues to infer the complete silhouette of instance and more ambiguity because of the occlusion. Existing methods mainly solve the task in 4 ways, including (1) directly minimizing between the prediction [19,38,5,27] and the target, (2) using relative depth order to comprehend the relationship between different objects [37, 36], (3) mutual helping from visible and amodal masks [7,17] and (4) using prestored shape prior knowledge [30,23]. In the meanwhile, several datasets have been proposed including realistic ones [38,27,7] and synthetic ones [5,13]. In addition, some papers [36,23] are working on a relevant task amodal completion, which aims to predict the complete amodal shape based on the given visible mask, while there are no visible masks given in the AIS task.
+
+All of the existing AIS algorithms are working in 2D space, which lacks comprehension of the real shape in 3D space. In this paper, our method learns the shape knowledge in 3D space to overcome the drawbacks of 2D AIS methods.
+
+# 2.2 Unsupervised Single View 3D Reconstruction
+
+Based on deep learning, supervised 3D reconstruction methods [4,32,33] are relying on high-quality 3D models for supervision, which are expensive to build. And because 2D supervision signals like segmentation masks are easier to obtain, unsupervised 3D model reconstruction methods are more popular. The pipeline of unsupervised 3D reconstruction consists of two steps, including 3D model reconstruction from the 2D image and rendering the 3D model into the 2D space, which is called rasterization. Whether the rasterization is differentiable decides whether the second rendering step can be included in the deep learning model with end-to-end training.
+
+Before 2018, most methods [35, 15] take rasterization by discrete assignment and cannot be trained end-to-end for the whole network. To make the rasterization differentiable, in 2018, NMR [16] proposes an approximate gradient approach to make the backward gradient progress in rasterization differentiable. SoftRas [24] and DIB-R [3] methods make both of the forward and backward steps in rasterization differentiable and can be trained in an end-to-end manner. Based on the differentiable rasterization technique, UMR [20] utilizes the semantic parts consistency between 2D and 3D spaces as supervision. SMR [12] proposes landmark and interpolation consistency for self-supervision.
+
+Because our method aims to represent and learn the complete shape in 3D space, we utilize the unsupervised single-view algorithm to gain a deeper understanding of the real shape in 3D space and help with 2D AIS with the reconstructed complete 3D shape.
+
+# 3 Amodal 3D Network (A3D)
+
+In this section, we develop a novel coarse-to-fine structure by combining the strength of 3D shape prior reconstruction and 2D edge refinement. We will first show the overall architecture of A3D and introduce each stage of A3D in detail.
+
+![](images/1bd6b30855eaaf0bab2e99dd6fc13625c9a64f9a597a9068b6b4833d99f2704c.jpg)  
+Fig. 2. The pipeline of our proposed Amodal 3D Network (A3D).
+
+# 3.1 Overall Architecture
+
+Given an input image  $I \in R^{H \times W \times 3}$  containing  $N$  instances, for the  $i$ -th instance, 2D AIS algorithms aim to predict the class ID  $c_i \in \{1, 2, \dots, K\}$  and 2D amodal masks  $M_i^a$ .
+
+The overall architecture is illustrated in Fig. 2. We take Amodal BBox Detection and Visible Instance Segmentation as the first stage to predict amodal
+
+bounding box  $B_{i}^{a}$  and the visible mask  $M_{i}^{v}$ . Then the RGB image  $I_{i}^{a}$  is cropped by using  $B_{i}^{a}$ . For each instance, we concatenate  $M_{i}^{v}$  and  $I_{i}^{a}$  as input to the second proposed 3D Amodal Shape Modeling (3D-ASM) stage for predicting the final amodal segmentation mask. 3D-ASM contains three important modules including Category-specific 3D Modeling, 2D Mask Generation and Region-specific Edge Refine. (1) The Category-specific 3D Modeling module reconstructs the 3D complete model as shape prior. (2) In the 2D Mask Generation module, the Viewpoint Estimator first predicts the camera parameters and apply the transformation to the reconstructed 3D shape prior model to the appropriate observation viewpoint. Then a Differentiable Render projects the 3D shape prior along the predicted viewpoint to obtain a coarse amodal segmentation mask. (3) Finally, the Region-specific Edge Refine module utilizes the visible mask, whose edge is accurate because the appearance information of the visible region is available to modify edges of the amodal mask. The whole network is end-to-end trainable.
+
+We will introduce the details of our network in the following sections.
+
+# 3.2 Amodal BBox Detection and Visible Instance Segmentation
+
+In this stage, we aim to predict the amodal bounding boxes and the visible masks for each instance. Following [7,37,30] we choose the popular Mask-RCNN [9] method for this stage. In Mask-RCNN, the first detection stage is set to predict the amodal bounding box (BBox) and the second segmentation stage is set to predict the visible mask. For Mask-RCNN, we choose Faster-RCNN [28] for bounding box detection and ResNet-50 [11] combining Feature Pyramid Network [21] as the backbone network. We use the ground truth of amodal bounding box, category  $ID$  and visible mask as supervision signals. From this stage, we can obtain the predicted amodal bounding box  $B_{i}^{a}$ , class  $IDc_{i}$ , and visible foreground binary mask  $M_{i}^{v}$  in the region of amodal bounding box for the  $i$ -th instance. Then we crop the input RGB image  $I$  with amodal bounding box  $B_{i}^{a}$  to get the image  $I_{i}^{a}$  in the region of the  $i$ -th instance.
+
+# 3.3 Category-specific 3D Modeling
+
+In this module, we aim to reconstruct the complete 3D model based on the occluded instance in 2D space. To achieve the 3D reconstruction of the  $i$ -th occluded instance, traditional 3D reconstruction methods [4, 32, 33] requires either multiple-view inputs or 3D supervision signals, which are not available in any existing 2D amodal segmentation datasets [7, 27, 38]. Therefore we choose to use the single-view unsupervised 3D reconstruction methods [24, 3, 20, 12] considering the dataset limitation.
+
+In the single-view unsupervised 3D reconstruction framework, the 3D model is first reconstructed from 2D inputs  $(2\mathrm{D}\rightarrow 3\mathrm{D})$  and then projected to 2D space  $(3\mathrm{D}\rightarrow 2\mathrm{D})$  for 2D amodal mask predictions. The 3D reconstruction network is indirectly supervised by the ground-truth of the 2D amodal masks. There are not any 3D models used as supervision signals.
+
+For the  $i$ -th instance, the input for 3D reconstruction is  $A_{i} = [I_{i}^{a}, M_{i}^{v}]$ , which is the concatenation of the image region  $I_{i}^{a}$  and visible mask  $M_{i}^{v}$ . We use the simple and classic Encoder Decoder structure following [16,24] for 3D shape modeling, leaving room for improvement by using more complex models. The Encoder contains five conv-bn-relu blocks for visible feature extraction, and three fully connected (fc) layers for linearly feature mapping. There is also a classification branch taking the feature from Encoder output and predicts the category of each instance, making the 3D shape reconstruction in a class-specific manner. It is worthy to notice that we use one model to handle all categories.
+
+We take a sphere as the initial object model  $O_{i}^{0}(V_{i}^{0})$ , in which  $V_{i}^{0}$  is the initial vertices. The Decoder consists of two fc-relu blocks to predict the offset  $\triangle V_{i}$  between reconstructed 3D model and initial 3D model. Finally we can obtain the vertices  $V_{i}^{r} = V_{i}^{0} + \triangle V_{i}$  and the reconstructed 3D object  $O_{i}^{r}(V_{i}^{r})$ . The detailed network architecture is described in the supplementary.
+
+# 3.4 2D Mask Generation
+
+For the  $i$ -th reconstructed 3D shape prior model  $O_{i}^{r}$ , if we want to obtain the 2D amodal mask, it is necessary to transform the 3D model with the correct parameters of the camera and project the 3D model to the camera plane. In this module, we take a Viewpoint Estimator to predict the camera parameters and utilize a Differentiable Render for projection.
+
+Following SMR [12], we use an Encoder Network to construct the Viewpoint Estimator, which consists of five conv-bn-relu blocks and three fully-connected layers. The Viewpoint Estimator predicts the camera parameters  $[e_i,d_i,(a_i^x,a_i^y)]$  representing elevation  $e_i$ , distance  $d_i$  and azimuth  $(a_i^x,a_i^y)$  in Cartesian coordinates, in which azimuth  $a_i = \arctan 2(a_i^x,a_i^y)$ . With the predicted viewpoint  $[e_i,d_i,(a_i^x,a_i^y)]$ , the 3D model is transformed appropriately. The Viewpoint Estimator is supervised indirectly by the ground-truth of 2D amodal masks, and no ground truth of viewpoints is used for supervision.
+
+Finally to project the transformed 3D shape prior model  $O_{i}^{r}$  for the coarse 2D amodal mask  $\widetilde{M_i^a}$ , we utilize the Differentiable Render SoftRas [24], which can maintain the gradient flow for end-to-end training.
+
+# 3.5 Region-specific Edge Refine
+
+In previous modules, we have obtained the 3D reconstructed shape prior model  $O_{i}^{r}$  and the projected 2D coarse amodal mask  $\widetilde{M}_i^a$ . However, the quality of the existing 3D reconstruction method is affected by the number of vertices in the initial sphere  $O_{i}^{0}$  and topological changes like holes which are hard to be learned in the mesh format. Therefore we design the Region-specific Edge Refine module to use the 2D amodal image region  $I_{i}^{a}$  and visible mask  $M_{i}^{v}$  to improve the edge of the amodal mask  $\widetilde{M}_i^a$ , because the appearance textures are only available in the visible region.
+
+We take 3 repeated conv-bn-relu layers as the module architecture with kernel size=3. This module takes the concatenation of coarse 2D amodal mask  $\widetilde{M}_i^a$ , the
+
+2D amodal image region  $I_{i}^{a}$  and visible mask  $M_{i}^{v}$  as input, and uses visible mask  $M_{i}^{v}$  to help with loss function, which is designed to punish more on the visible edge and less on the occluded edge. Visualization example of Edge Refine are shown in Fig. 3.
+
+![](images/bf355f4c67c2528ffe194ce0478440159bee549d9140be5181d4750b86438d89.jpg)  
+Fig. 3. Visualization of Region-specific Edge Refine.  $\oplus$  means concatenation.
+
+![](images/c37b23ab64b483d2f16daa6003078d60ed0a785aeea265b5ec5db51f195bbaab.jpg)
+
+![](images/f382dd09b0205b53094656efacdfd87851672b2f763b4a6444c8ca9de6175560.jpg)
+
+![](images/9e5183962ce338e6d1decabbb2ac95cedcc89437df9660b99a2e2a2d4683404d.jpg)
+
+# 3.6 Loss Functions
+
+In this section, we will introduce loss functions for our 3D modelling and the edge refinement modules.
+
+3D Modeling To get rid of dependence on the expensive 3D model annotation, we choose to use the unsupervised 3D reconstruction method without needing 3D models as supervision signals. We train both the Category-specific 3D Modeling module and Region-specific Edge Refine module simultaneously because only the ground-truth 2D amodal masks  $\overline{M}_i^a$  are available for supervision. Therefore the loss function is designed to encourage the predicted coarse amodal mask  $\widetilde{M}_i^a$  being close to  $\overline{M}_i^a$ , which indirectly supervises the quality of reconstructed 3D shape prior model  $O_i^r$ . The loss function for unsupervised 3D reconstruction is:
+
+$$
+\mathcal {L} _ {\mathcal {R}} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left(1 - I o U \left(\widetilde {M} _ {i} ^ {a}, \overline {{M}} _ {i} ^ {a}\right)\right) \tag {1}
+$$
+
+where  $IoU$  computes the intersection over union between the predicted coarse amodal mask  $\widetilde{M}_i^a$  and the ground-truth amodal mask  $\overline{M}_i^a$ .
+
+Edge Refine In the Region-specific Edge Refine module, the loss function of the Region-specific Amodal Edge Refine module is designed as following:
+
+$$
+\mathcal {L} _ {E} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left(\sum_ {p \in S _ {i}} \mathcal {L} _ {B} \left(\widehat {M} _ {i, p} ^ {a}, \overline {{M}} _ {i, p} ^ {a}\right) + \lambda \sum_ {p \notin S _ {i}} \mathcal {L} _ {B} \left(\widehat {M} _ {i, p} ^ {a}, \overline {{M}} _ {i, p} ^ {a}\right)\right) \tag {2}
+$$
+
+where  $N$  is the instance number in the input image. For the  $i$ -th instance,  $S_{i}$  represents the visible region indicated by the ground-truth visible mask, and  $p$  denotes the pixel  $p$ .  $\mathcal{L}_B$  is the Binary Cross Entropy loss function, computing the difference between the predicted values of pixel  $p$  from refined prediction  $\widehat{M}_{i,p}^{a}$  and ground-truth mask  $\overline{M}_{i,p}^{a}$ . We set the loss weight  $\lambda = 0.5$  to rely more on the visible region for edge refinement.
+
+# 3.7 Pretrain and Finetune
+
+In previous subsections, the whole framework of A3D and loss functions are introduced. In this subsection, a carefully designed pretrain and finetune strategy is introduced to improve the performance of 3D shape reconstruction for better 2D amodal segmentation results. It is worth noting that the ground truth of 3D models and viewpoints of the pretrain and finetune datasets are never used, to make our method applicable in real applications.
+
+We first pretrain our A3D network on the 3D reconstruction dataset by unsupervised approach (as described in Sec 3.3), and then finetune on the train set of 2D AIS dataset, finally conduct inference on its test set.
+
+To handle the problem that the categories of pretrain and finetune datasets are different, including overlapped and non-overlapped categories, we deal with them in different approaches. For overlapped categories, we reuse the network weights after pretraining as the initialization for training on the finetune dataset. With the weights being reused, the category-specific knowledge can be transferred for the overlapped categories between pretraining and finetune datasets. For non-overlapped categories, the weight parameters are randomly initialized [10]. The performance of overlapped and non-overlapped categories are shown in Tab. 3.
+
+Besides, to improve the performance of 3D reconstruction supervised by single view, in the pretrain process, we use the cross-view technique [24], which requires the 3D model reconstructed from two viewpoints for the same object to be similar. The cross-view technique only additionally uses the correspondence information that two images from different viewpoints corresponds to the same object, and never takes the ground truth of viewpoints and 3D models as supervisions. The details are shown in the supplementary. The cross-view technique is optional.
+
+# 4 Experiments
+
+In order to evaluate our proposed method, extensive experiments have been conducted on three public amodal segmentation datasets, including COCOA-cls, KINS and D2SA, as well as a 3D dataset ShapeNet. Our method is compared with several SOTA amodal segmentation methods, and results show the advantage of our approach.
+
+# 4.1 Datasets
+
+2D AIS Datasets For 2D AIS task, we conduct experiments on a large-scale synthetic dataset ShapeNet [2] and three real 2D AIS datasets including COCOA-cls [7], KINS [27] and D2SA [7] for scenes of outdoor, street and indoor supermarkets. ShapeNet dataset contains 735,432 instances for training and 210,288 instances for testing. There are 13 categories, including various objects, and each object is rendered in 24 different viewpoints. We randomly occlude the RGB image and mask to simulate the occluded inputs. We do not
+
+use any 3D shape and viewpoint supervision signals in all experiments even they are available. COCOA-cls dataset, which annotates a subpart of COCO [22] dataset with amodal masks, has 3,501 images and 10,592 instances in 80 categories. KINS dataset is the biggest street scene amodal dataset, which can be applied on tasks like auto driving, built on KITTI dataset [8] with re-annotated amodal masks. KINS has two super-classes, including person and vehicle, and seven sub-classes with 7,474 and 7,517 images for training and testing. D2SA dataset is built upon D2S [6] dataset with amodal mask re-annotated, including plenty kinds of goods placed in different postures and occlusion approaches on a rotatable supermarket platform with varying light conditions. D2SA contains 5,600 images totally and 28,720 instances in 60 classes.
+
+Pretraining Datasets For using the pretrain and finetune strategy claimed in Sec. 3.7, both ShapeNet and PASCAL3D+ [29] are used. PASCAL3D+ dataset contains 55,867 3D models in 12 categories, which is more than 39,405 3D models in ShapeNet dataset, providing richer shape knowledge for 3D reconstruction.
+
+# 4.2 Implementation Details and Evaluation Metric
+
+Our method is implemented based on the Pytorch [26] framework. For all of the 2D amodal segmentation methods used in our experiments, we use the same configuration following ShapeDict [30]. For our proposed A3D Network, the learning rate is set to 0.0001, and we use the Adam algorithm for gradient descent with 64 batches. All experiments are conducted on a single 2080Ti GPU card. All categories including rigid and non-rigid are used in all experiments. Faster-RCNN with ResNet-50 is used for all methods for object detection.
+
+We choose the mean Intersection over Union (mIoU) and mean Average Precision (mAP) as metrics for performance evaluation. It is worthy to notice that the commonly chosen metric mean Average Precision (mAP) measures the performance of two sub-tasks in Amodal Instance Segmentation simultaneously, including Object Detection and Semantic Segmentation. Therefore for ShapeDict dataset we only report mIoU because there are only one object in each image and mAP which measures the performance of object detection is not reported.
+
+# 4.3 2D Amodal Instance Segmentation
+
+This section evaluates 2D AIS methods on the challenging ShapeNet dataset and three amodal datasets, including COCOA-cls, D2SA and KINS for different scenes. Following state-of-the-art methods are used for comparison.
+
+(1) Mask-RCNN [9] is trained using ground-truth amodal bounding boxes and masks to show the transferability of the visible instance segmentation method on the amodal problem. (2) ORCNN [7] is a two-branch approach that predicts and supervises the visible, amodal and occluded region at the same time. (3) BCNet [17] decouples the occluding and occluded instances combining graph
+
+convolution. (4) ShapeDict [30] establishes a 2D shape prior dictionary by clustering the ground-truth amodal masks and uses the query-and-retrieve approach to provide prior knowledge. Besides, we also take Deocclusion [36], a weakly-supervised amodal completion method, for comparison to show the performance gap between weakly and fully supervised AIS methods.
+
+Table 1. Results (mIoU) on the ShapeNet dataset. For each category, bold performance is the best, and the second-best is underlined. The subscript numbers are the subtraction results between ours and the second-best methods. SU means the Supervision signal type. W and F mean weakly and fully supervised.  
+
+<table><tr><td>Methods</td><td>SU</td><td>Airplane</td><td>Bench</td><td>Dresser</td><td>Car</td><td>Chair</td><td>Display</td><td>Lamp</td><td>Speaker</td><td>Rifle</td><td>Sofa</td><td>Table</td><td>Phone</td><td>Vessel</td><td>mIoU</td></tr><tr><td>Deocclusion [36]CVPR20</td><td>W</td><td>24.9</td><td>67.4</td><td>45.3</td><td>58.8</td><td>83.7</td><td>78.4</td><td>77.9</td><td>15.2</td><td>48.7</td><td>48.1</td><td>39.5</td><td>23.8</td><td>71.9</td><td>52.2</td></tr><tr><td>Mask-RCNN [9]ICCV17</td><td>F</td><td>73.4</td><td>66.0</td><td>92.4</td><td>93.5</td><td>89.3</td><td>90.0</td><td>77.4</td><td>88.5</td><td>30.0</td><td>86.0</td><td>73.1</td><td>89.8</td><td>80.5</td><td>79.2</td></tr><tr><td>ORCNN [7]WACV19</td><td>F</td><td>71.5</td><td>61.1</td><td>92.0</td><td>92.7</td><td>88.8</td><td>88.8</td><td>79.5</td><td>88.7</td><td>32.8</td><td>85.6</td><td>72.5</td><td>89.0</td><td>80.0</td><td>78.7</td></tr><tr><td>BCNet [17]CVPR21</td><td>F</td><td>73.0</td><td>75.1</td><td>93.8</td><td>89.4</td><td>86.6</td><td>88.7</td><td>81.6</td><td>90.2</td><td>32.8</td><td>83.4</td><td>77.5</td><td>88.7</td><td>74.8</td><td>78.2</td></tr><tr><td>ShapeDict [30]AAAI21</td><td>F</td><td>75.2</td><td>68.5</td><td>93.7</td><td>93.6</td><td>88.4</td><td>89.3</td><td>78.1</td><td>88.6</td><td>34.4</td><td>87.3</td><td>74.8</td><td>90.7</td><td>80.9</td><td>80.3</td></tr><tr><td>Ours (no pretrain)</td><td>F</td><td>77.9</td><td>80.8</td><td>94.2</td><td>92.8</td><td>79.7</td><td>87.5</td><td>67.8</td><td>90.5</td><td>69.9</td><td>90.3</td><td>86.2</td><td>92.1</td><td>81.3</td><td>83.9</td></tr></table>
+
+Table 2. Results (mIoU and mAP) on the 2D AIS datasets. SU means the Supervision signal type. W and F mean weakly and fully supervised. FLOPs and Params measure the computational efficiency and model size.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">SU</td><td colspan="3">mIoU ↑</td><td colspan="3">mAP ↑</td><td rowspan="2">FLOPs(G) ↓</td><td rowspan="2">Params(M) ↓</td></tr><tr><td>D2SA</td><td>KINS</td><td>COCOA-cls</td><td>D2SA</td><td>KINS</td><td>COCOA-cls</td></tr><tr><td>Deocclusion [36]CVPR&#x27;20</td><td>W</td><td>73.8</td><td>59.2</td><td>39.2</td><td>61.7</td><td>27.5</td><td>19.9</td><td>160.4</td><td>44.1</td></tr><tr><td>Mask-RCNN [9]ICCV&#x27;17</td><td>F</td><td>74.6</td><td>60.1</td><td>63.8</td><td>63.6</td><td>30.0</td><td>33.7</td><td>160.4</td><td>44.1</td></tr><tr><td>ORCNN [7]WACV&#x27;19</td><td>F</td><td>74.1</td><td>55.1</td><td>57.6</td><td>64.2</td><td>30.6</td><td>28.0</td><td>229.4</td><td>46.8</td></tr><tr><td>BCNet [17]CVPR&#x27;21</td><td>F</td><td>74.9</td><td>44.0</td><td>15.1</td><td>50.9</td><td>22.1</td><td>16.2</td><td>263.5</td><td>63.2</td></tr><tr><td>ShapeDict [30]AAAI&#x27;21</td><td>F</td><td>75.0</td><td>63.7</td><td>64.5</td><td>70.3</td><td>32.1</td><td>35.4</td><td>271.3</td><td>48.0</td></tr><tr><td>Ours (w/o pretrain)</td><td>F</td><td>74.7</td><td>61.4</td><td>64.2</td><td>68.5</td><td>31.4</td><td>34.9</td><td>229.8</td><td>57.2</td></tr><tr><td>Ours (w/ pretrain)</td><td>F</td><td>78.4</td><td>65.5</td><td>67.4</td><td>73.5</td><td>36.2</td><td>40.6</td><td>229.8</td><td>57.2</td></tr></table>
+
+The comparison results of AIS methods are shown in Tab. 1 for ShapeNet dataset and Tab. 2 for the three 2D AIS datasets, including D2SA, KINS and COCOA-cls. In Tab. 1, all methods are trained on the train set of ShapeNet and there are no extra data used for pretraining in our method. Our method achieves the best performance on nine categories. Compared with the methods of the second-best performance, A3D gains significant IoU improvement on Bench, Rifle and Table with  $5.7\%$ ,  $21.6\%$  and  $8.7\%$  respectively, which shows the effectiveness of our proposed A3D method. Fig. 4 shows the qualitative results of our A3D method on the ShapeNet dataset.
+
+In Tab. 2, all methods except our methods are trained on the train split of respective datasets, and our method additionally uses ShapeNet and PASCAL3D+ for pretraining to make our method applicable in real applications. Our A3D network outperforms all methods with certain advantages for mIoU and mAP. In terms of the number of FLOPs and Parameters, our method is com
+
+parable to previous work. More visualizations for D2SA, KINS and COCOA-cls datasets are given in the supplementary.
+
+![](images/e075099442be76a289641420950d7d8fa1239cbc6131cfb228555f4c8b5991b4.jpg)  
+Fig. 4. Visualization result of our method on the ShapeNet dataset. The reconstructed 3D shape prior models are shown from two viewpoints.
+
+# 4.4 Effectiveness of Pretraining
+
+Table 3. Ablation study results (mAP) of pretraining. N, S, P and S+P means no pretraining, pretraining with ShapeNet, with PASCAL3D+, and with both ShapeNet & PASCAL3D+. #Overlapped means the number of categories overlapped between pretrain and finetune datasets. SU means supervision signal (W and F for weakly and fully supervised). Category number is noted in the brackets after each dataset name.  
+
+<table><tr><td rowspan="2">Index</td><td rowspan="2">Methods</td><td rowspan="2">SU</td><td colspan="4">D2SA (60)</td><td colspan="4">KINS (7)</td><td colspan="4">COCOA-cls (80)</td></tr><tr><td>N</td><td>S</td><td>P</td><td>S+P</td><td>N</td><td>S</td><td>P</td><td>S+P</td><td>N</td><td>S</td><td>P</td><td>S+P</td></tr><tr><td colspan="3">#Overlapped</td><td>-</td><td>2</td><td>2</td><td>4</td><td>-</td><td>1</td><td>4</td><td>4</td><td>-</td><td>13</td><td>12</td><td>19</td></tr><tr><td>1</td><td>Deocclusion</td><td>W</td><td>61.7</td><td>61.9</td><td>62.1</td><td>62.3</td><td>27.5</td><td>27.9</td><td>28.2</td><td>28.8</td><td>19.9</td><td>20.4</td><td>20.9</td><td>21.3</td></tr><tr><td>2</td><td>Mask-RCNN</td><td>F</td><td>63.6</td><td>63.9</td><td>64.2</td><td>64.8</td><td>30.0</td><td>30.5</td><td>30.8</td><td>31.1</td><td>33.7</td><td>33.9</td><td>34.2</td><td>34.6</td></tr><tr><td>3</td><td>ORCNN</td><td>F</td><td>64.2</td><td>64.8</td><td>65.1</td><td>65.5</td><td>30.6</td><td>30.9</td><td>31.2</td><td>31.8</td><td>28.0</td><td>28.3</td><td>28.7</td><td>29.1</td></tr><tr><td>4</td><td>BCNet</td><td>F</td><td>50.9</td><td>51.2</td><td>51.5</td><td>51.9</td><td>22.1</td><td>22.6</td><td>22.8</td><td>23.0</td><td>16.2</td><td>16.8</td><td>17.1</td><td>17.4</td></tr><tr><td>5</td><td>ShapeDict</td><td>F</td><td>70.3</td><td>70.5</td><td>70.9</td><td>71.2</td><td>32.1</td><td>32.2</td><td>32.4</td><td>32.5</td><td>35.4</td><td>35.7</td><td>36.1</td><td>36.4</td></tr><tr><td>6</td><td>Ours</td><td>F</td><td>68.5</td><td>71.4</td><td>72.6</td><td>73.5</td><td>31.4</td><td>33.7</td><td>35.2</td><td>36.2</td><td>34.9</td><td>37.4</td><td>39.2</td><td>40.6</td></tr></table>
+
+In Tab. 2, all the previous methods do not use the pretrain dataset while our method does. To make a fair comparison, we design an experiment such that each previous method can also take advantage of the pretrain dataset. Specifically, in pretrain process, each previous method can take all images in the pretrain dataset, each of which contains one non-occluded object with white background
+
+and the corresponding 2D amodal mask, as training data. With this pretrain strategy, all the previous methods can also make use of the pretrain dataset and thus the comparison between previous methods and ours is fair. We compare the performance of all methods with & without pretraining on three 2D AIS datasets. Results are shown in Tab. 3.
+
+Comparing methods in different lines, we can conclude that by directly adding the 3D representation without pretraining, our method outperforms the baseline method Mask-RCNN but cannot beat ShapeDict. This is because the 2D AIS datasets provide not enough supervision for training 3D reconstruction in our method. In the last line of our method, for each finetune dataset, the performance increases with more pretrain data used. Meanwhile the performance of 2D AIS methods (Tab. 3, Line #1 to #5) do not increase much with pretraining. This is because for both pretrain datasets, each input RGB image contains only one non-occluded object with white background, which is easy to be segmented and not very helpful for amodal segmentation. However the pretrain datasets are very helpful for 3D reconstruction, making our method gains much improvements. With limited number of overlapped categories between pretrain and finetune datasets, our method can still achieve good performances.
+
+# 4.5 Ablation Study
+
+In this section, we conduct ablation experiments to validate the effectiveness of our proposed modules and pretraining for 3D reconstruction.
+
+Effectiveness of 3D Modeling and Edge Refine In our proposed A3D network, we design a Category-specific 3D Modeling module for 3D shape prior generation and a Region-specific Edge Refine module for 2D edge refinement. In this section, we validate the effectiveness of the two proposed modules on the ShapeNet dataset. (1) The Mask-RCNN method, which directly predicts the 2D amodal mask from the input image, is the baseline method. (2) If the Category-specific 3D Modeling module is added,  $4\%$  mIoU improvement will be obtained, which shows the effectiveness of 3D modeling. (3) Then after further combining the Region-specific Edge Refine module, the performance can boost  $0.7\%$  mIoU, which improves the quality of some details of the edge and not drastically modifies the predicted mask. Visualizations of the effectiveness are shown in Fig. 3, and Edge Refine can improve the quality of boundary to some extent.
+
+Effectiveness of cross-view technique Cross-view technique supervises the reconstructed 3D model from two viewpoints to be consistent. It does not use any additional supervision signals like viewpoint information or 3D model, only use the correspondence of two images from the same object of different viewpoints. We evaluate the performance without and with cross-view technique on ShapeNet dataset, and the mIoU results are  $82.5\%$  and  $83.9\%$  respectively. Cross-view technique brings  $1.4\%$  improvement with the reconstruction consistency between different viewpoints.
+
+# 4.6 Methodology Limitation
+
+![](images/0963dc07220224fdaf74ed29ef56247216e1bff065d3902e1980dd02f07a8dcb.jpg)  
+Fig. 5. Examples of chairs and lamps. In each four-tuple, images from left to right are input RGB images, ground-truth amodal masks, reconstructed 3D shape prior models, and predicted amodal masks.
+
+![](images/33d5a7cda52344f96008c6bacd14b4777a7cda00f4e0a061539920df237c28b4.jpg)
+
+![](images/3a319b2661ae4e7f7171c23f075714453ff0f3f7f99f29a86428bfe374ce2087.jpg)
+
+![](images/36e8823fe529d5bb620d75587ab7ba0b7a0379cc6526a12855b19ea61905eba4.jpg)
+
+As shown in Tab. 1, for the categories of Chair and Lamp, ours A3D method fails to perform well with large margin, dropping for  $9.6\%$  and  $11.7\%$  IoU compared with the best performance. As illustrated in Fig. 5, there are plenty of holes in the Chair category and complicated structures in the Lamp category. However, in our A3D method, the 3D shape prior model is reconstructed by predicting vertices offset from the initial sphere, remaining the topology unchanged. Complicated structures in both Chair and Lamp categories require the topology changes, where our A3D network is incapable at present. We leave this problem to future work.
+
+# 5 Conclusion
+
+In this paper, we propose a novel coarse-to-fine Amodal 3D (A3D) network. A3D is a brand new framework which for the first time tackles the 2D AIS problem by reconstructing the 3D complete shape prior model. With the benefits of 3D modelling, A3D can alleviate the shortcoming that 2D AIS methods are difficult to generalize on untrained new viewpoints of the occluded 3D object. Our A3D achieves state-of-the-art performance on multiple AIS datasets.
+
+# Acknowledgments
+
+This work was partially supported by the Natural Science Foundation of China under contracts 62088102. This work was also partially supported by Qualcomm. We also acknowledge High-Performance Computing Platform of Peking University for providing computational resources.
+
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+# 2D GANs Meet Unsupervised Single-view 3D Reconstruction
+
+Feng Liu, Xiaoming Liu
+
+Michigan State University, Computer Science & Engineering {liufeng6, liuxm}@msu.edu
+
+Abstract. Recent research has shown that controllable image generation based on pre-trained GANs can benefit a wide range of computer vision tasks. However, less attention has been devoted to 3D vision tasks. In light of this, we propose a novel image-conditioned neural implicit field, which can leverage 2D supervisions from GAN-generated multiview images and perform the single-view reconstruction of generic objects. Firstly, a novel offline StyleGAN-based generator is presented to generate plausible pseudo images with full control over the viewpoint. Then, we propose to utilize a neural implicit function, along with a differentiable renderer to learn 3D geometry from pseudo images with object masks and rough pose initializations. To further detect the unreliable supervisions, we introduce a novel uncertainty module to predict uncertainty maps, which remedy the negative effect of uncertain regions in pseudo images, leading to a better reconstruction performance. The effectiveness of our approach is demonstrated through superior single-view 3D reconstruction results of generic objects. Code is available at http://cvlab.cse.msu.edu/project-gansvr.html.
+
+Keywords: 2D GANs, Multi-view Pseudo Images, Unsupervised, Single-view 3D Reconstruction, Generic objects, Uncertainty
+
+# 1 Introduction
+
+Realistic image synthesis is an important research area of computer vision. There has been remarkable progress in this field with the advent of 2D Generative Adversarial Networks (GANs) [9], such as StyleGAN [23] and its variations [21,22,24], which can generate high-fidelity images of diverse object categories with a wide variety of attributes (e.g., pose, identity). Such superior capabilities of modeling the semantic image manifold enable the generated photorealistic images to be leveraged for many vision tasks, such as image editing [20,55], domain translation [73], face recognition [30], and video generation [60]. However, it remains much less explored in 3D vision tasks, e.g., 3D reconstruction [31,32].
+
+The 2D GAN manifolds appear to learn 3D geometrical properties implicitly, where recent GAN interpretation methods [14, 51] have shown that manipulating the latent code of the pre-trained GAN models can produce images of the same object under different viewpoints. Our work aims to answer the following
+
+![](images/64cc9a4cbdb44fcb42be7f2aaf06428cd4875f14fbdaa02e53d4f5cab26ffd95.jpg)  
+Fig.1: Our approach leverages StyleGAN-generated multi-view pseudo images to learn a 3D model without 3D supervision, which can perform single-view 3D reconstruction for a variety of generic objects, e.g., airplanes, birds, cars, horses, motorbikes, potted plants, etc. In addition, our framework produces uncertainty maps, indicating the unreliable local areas in the pseudo images.
+
+question. Using the GAN-generated multi-view images, can we learn a category-specific multi-view stereo system without 3D supervision that can reconstruct 3D shapes from a single image?
+
+Early attempts [44,52,63] are made to mine 3D geometric cues from the pretrained 2D GAN models in an unsupervised manner. However, without modeling objects in the 3D space, these methods only recover 2.5D representations (depth or normals). Recently, StyleGANRender [71] integrates StyleGAN to generate multi-view images, which may be used to train an inverse graphics network for 3D reconstruction. However, the method focuses more on performing independent manipulation of 3D properties in GAN's latent space by fine-tuning the GAN models. In addition, the unreliable texture existing in GAN-generated multi-view images is a common issue, and has not been investigated.
+
+It remains a challenge to leverage GAN-generated multi-view supervision for single-view 3D reconstruction. First of all, the pre-trained 2D GAN models lack explicit and precise camera pose to control over generated images, which is a necessity for classic multi-view stereo. Second, the GAN-generated multi-view images often suffer from local distortion and low perceptual quality, which severely breaks the consistency of either object shape or texture across views, and thereby ruins the cornerstone of multi-view stereo.
+
+To address these challenges, we propose a novel framework to leverage GAN-generated multi-view images (termed 'pseudo images') in learning generic object shape models, for the purpose of 3D reconstruction from a single image (Fig. 1). To first generate multi-view imagery by a pre-trained GAN, e.g., StyleGAN, we carefully study the latent space of StyleGAN and devise a simple but effective technique, which generates plausible images with an azimuth range of  $0 - 360^{\circ}$ . Consequently, during training, given a realistic image generated by StyleGAN, we can produce a set of pseudo images of the same object under different view-
+
+points. We then introduce a neural implicit network to simultaneously learn the unknown geometry, texture, and camera parameters for the objective of reconstructing the pseudo images, by incorporating a differentiable renderer.
+
+A key component is that we introduce a learning framework that enables the neural implicit network to be conditioned on a single image. Specifically, we adopt an image encoder as a hypernetwork to predict the network parameters of the implicit function. This image conditioning allows the framework to be trained on multi-view images, where it learns object geometry priors within the category to perform single-view reconstruction. Moreover, to address the unreliable texture supervision issue in pseudo images, we devise an uncertainty prediction module, together with an uncertainty-aware photometric loss to estimate uncertainty maps, which can effectively filter out the unreliable supervision signals/inconsistencies within/across multi-view pseudo images, leading to a more precise reconstruction. Comprehensive experiments show the superiority of our method over existing methods in unsupervised single-view 3D reconstruction.
+
+In summary, the contributions of this work include:
+
+$\diamond$  We propose a novel image-conditioned neural implicit network, which can exploit 2D supervision from GAN-generated multi-view pseudo images and performs single-view 3D reconstruction of generic objects.  
+We introduce a multi-view image generation mechanism based on the pretrained StyleGAN models, which can produce plausible images with full control over viewpoints.  
+$\diamond$  We propose an uncertainty prediction module to ignore unreliable texture supervision in pseudo images, enabling a reliable self-supervised learning.  
+$\diamond$  Our method shows superior single-view 3D reconstruction for rigid and non-rigid generic objects in the wild.
+
+# 2 Prior Work
+
+Application of Pre-trained 2D GANs While research on GANs is rapidly growing, our review mainly focuses on the pre-trained unconditional 2D GAN models. The capability to produce high-quality images makes 2D GANs applicable to many vision tasks, e.g., image restoration [61,67], image editing ( inpainting, super-resolution, semantic manipulation) [12, 45], segmentation [72], and DeepFake attack and defense [2,6,7,48]. Further, the pre-trained 2D GAN models have been applied to data augmentation to reduce overfitting and bias in deep models [47,54]. To expand to 3D vision applications, prior works [36,56,62,63,74] adopt GANs to learn 3D shapes from images but rely on either 3D supervision or a 3D generator, which suffers from heavy memory consumption or extra training difficulties. Recently, LiftedGAN [52] lifts a pre-trained StyleGAN and distill it into a 3D aware generator, producing depth maps as a by-product. Similarly, GAN2Shape [44] produces an unsupervised decomposition by using a GAN model as supervision. However, those methods require inefficient online image generation during training and infer 2.5D representations only. StyleGAN-Render [71] exploits StyleGAN as a multi-view generator to learn an mesh-based
+
+Table 1: Comparison of unsupervised shape learning methods. [Keys: Cam. = camera poses per training sample, Requ. or Cons. = requirement or constraint, Real data= whether can train on real-world images,  $\bullet$  = camera poses for a set of reference images]  
+
+<table><tr><td>Method</td><td>Output Representation</td><td>Required Template</td><td>Required Cam.</td><td>Additional Requ. or Cons.</td><td>Real data</td></tr><tr><td>LiftedGAN [52]</td><td>2.5D, depth</td><td>X</td><td>X</td><td>GAN models, pre-trained</td><td>✓</td></tr><tr><td>GAN2Shape [44]</td><td>2.5D, depth</td><td>X</td><td>X</td><td>GAN models, pre-trained</td><td>✓</td></tr><tr><td>StyleGANRender [71]</td><td>3D, mesh</td><td>X</td><td>X</td><td>GAN models, fine-tuning</td><td>✓</td></tr><tr><td>DVR [42]</td><td>3D, implicit</td><td>X</td><td>✓</td><td>multi-view</td><td>X</td></tr><tr><td>DIST [34]</td><td>3D, implicit</td><td>X</td><td>✓</td><td>multi-view</td><td>X</td></tr><tr><td>SDFDiff [17]</td><td>3D, implicit</td><td>X</td><td>✓</td><td>multi-view</td><td>X</td></tr><tr><td>CSDM [58]</td><td>3D, mesh</td><td>✓</td><td>✓</td><td>2D semantic</td><td>✓</td></tr><tr><td>CMR [18]</td><td>3D, mesh</td><td>✓</td><td>X</td><td>2D semantic</td><td>✓</td></tr><tr><td>U-CMR [8]</td><td>3D, mesh</td><td>✓</td><td>X</td><td>viewpoint distribution</td><td>✓</td></tr><tr><td>UMR [28]</td><td>3D, mesh</td><td>X</td><td>X</td><td>3D semantic</td><td>✓</td></tr><tr><td>CSM [27]</td><td>3D, mesh</td><td>✓</td><td>X</td><td>-</td><td>✓</td></tr><tr><td>A-CSM [26]</td><td>3D, mesh</td><td>✓</td><td>X</td><td>-</td><td>✓</td></tr><tr><td>DRC [59]</td><td>3D, voxel</td><td>X</td><td>✓</td><td>multi-view</td><td>✓</td></tr><tr><td>SRN [53]</td><td>3D, implicit</td><td>X</td><td>✓</td><td>multi-view</td><td>X</td></tr><tr><td>NeRF [38]</td><td>3D, implicit</td><td>X</td><td>✓</td><td>multi-view</td><td>✓</td></tr><tr><td>SDF-SRN [29]</td><td>3D, implicit</td><td>X</td><td>✓</td><td>-</td><td>✓</td></tr><tr><td>ShSMesh [69]</td><td>3D, volumetric</td><td>X</td><td>X</td><td>-</td><td>✓</td></tr><tr><td>Proposed</td><td>3D, implicit</td><td>X</td><td>◎</td><td>GAN models, pre-trained</td><td>✓</td></tr></table>
+
+inverse graphics network to turn the StyleGAN into a controllable render. However, they require a fine-tuning step for the entire StyleGAN model, which is not desirable in this work. Moreover, they do not tackle the unreliability in pseudo multi-view images. In contrast, our method focuses on leveraging pretrained 2D GANs for single-view 3D reconstruction. Despite both methods utilizing GAN-generated pseudo images for 3D modeling, we step forward in more plausible multi-view generation, robust shape and texture representation, and uncertainty-aware photometric supervision mechanism.
+
+3D-aware Generative Models Understanding the latent representation of GANs has resulted in a body of works disentangling various factors of generated objects in a 3D-controllable manner, e.g., viewpoint. These approaches can be classified into two groups. One adds additional modules or losses in training to explicitly disentangle 3D factors [37]. For example, HoloGAN [39] controls the object pose by rigid-body transformations via a 3D feature module. StyleFlow [1] learns non-linear paths in the latent space by normalizing flows conditioned on the attribute. Recently there has been a trend combining of the neural radiance fields (NeRF) [5,11,40,41,49] with GANs to devise 3D-aware generators. Another line of works, such as InterFaceGAN [50], SeFa [51], GANSpace [14], discover the latent semantic directions of a pre-trained GAN model that can manipulate object rotation unaware of its underlying 3D model. It is preferable to exploit the knowledge contained in a pre-trained GAN image manifold for the goal of recovering 3D object shapes without retraining the GAN models.
+
+Shape Learning without 3D Supervision While 3D reconstruction, especially for faces [3,33,57], is a long-standing topic, we focus our review on shape learning from real-world images of generic objects without 3D supervision. Recent neural networks tackle this ill-posed problem via a differentiable renderer
+
+![](images/33d40091cde891e84ed58964dbf8b1304e6bc2e159cf172a929f2f5629495fb2.jpg)  
+Fig. 2: Overview. The proposed framework is composed of two key modules: an offline StyleGAN-based multi-view generation and an image-conditioned neural implicit network. During training, the neural implicit network learns unknown geometry, texture, and camera poses for the objective of approximating the multi-view pseudo images. At inference time, the learned neural implicit function performs 3D reconstruction for the object from a single image.
+
+along with a choice of 3D shape representation [17, 34, 35, 42, 53, 59]. However, in these works, multiple views of the same object with known cameras are required, which limits their learning from real-world images. Another branch of works show promising reconstruction from real-world images [8, 26-28, 64]. SDF-SRN [29] mines more supervision from 2D silhouette for superior reconstruction, yet still requires camera pose. ShSMesh [69] further discards the need for former constraints, but their reconstructions are of lower quality. NeRF [38] and its variations are scene- or object-specific models, which limit their applications for 3D reconstruction from unseen objects or scenes. In contrast, our models are category-specific, and can perform single-view 3D reconstruction for novel instances. Tab. 1 summarizes the differences between our method and prior work.
+
+# 3 Proposed Method
+
+We start with an overview of the proposed framework (Fig. 2) and then present the individual modules in detail. We first introduce an offline and effective multiview generator based on the pre-trained StyleGAN models, which produce plausible multi-view images with full control over viewpoints. Then, we detail the proposed image-conditioned neural implicit field learning framework, including a neural implicit network, differentiable rendering procedure, and an uncertainty prediction module. These three modules work jointly for the objective of exploiting the pseudo images to learn generic object shape priors and perform 3D reconstruction from a single input image.
+
+# 3.1 StyleGAN based Multi-view Generation
+
+We briefly review the embedding space of the StyleGAN [23,24]. Typically, a generator  $G(\cdot)$  samples a latent code  $\mathbf{z}$  from a pre-defined distribution  $\mathcal{Z}$  such as the normal distribution, and produces an output image  $\mathbf{I}$ . The code  $\mathbf{z}$  is first mapped to an intermediate latent space  $\mathcal{W}$  via a Multi-layer Perceptron (MLP), and then  $\mathcal{W}$  is transformed to  $\mathcal{W}^+$  space by 16 learned affine transformations. The generator  $G(\cdot)$  projects  $\mathbf{W}$  to the final image:  $G(\mathbf{W}) = \mathbf{I}$ . Such latent codes have been shown to learn various disentangled semantics [14,51]. For instance, StyleGAN-Render [71] finds that the latent codes  $\mathbf{W}_v := (\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4) \in \mathbb{R}^{4 \times 512}$  in the first 4 layers control camera viewpoints. That is, given a source and reference generated image pair  $(\mathbf{I}^S, \mathbf{I}^R)$  with their latent codes  $(\mathbf{W}^S, \mathbf{W}^R)$ , we can generate an image of the source object with the reference viewpoint by swapping  $(\mathbf{W}_v^S, \mathbf{W}_v^R)$  and keeping the rest dimensions of  $\mathbf{W}^S$ . We denote this multi-view generation strategy as **Baseline**. However, while  $\mathbf{W}_v$  indeed alters the object viewpoint, it still perceives the shape of the reference object, as shown in Fig. 3a.
+
+It is difficult to develop a multi-view stereo system from noisy multi-view images with inconsistent shapes. To tackle this issue, inspired by SeFa [51], we propose a novel offline multi-view generation mechanism that generates plausible images with enhanced cross-view consistency. SeFa suggests that the weight parameters in early affine transformations contain essential knowledge of image variations. One can obtain interpretable directions in the latent space by computing eigenvectors of their weight matrices and selecting eigenvectors with the largest eigenvalues. With this observation, we propose to filter the viewpoint-irrelevant features in  $\mathbf{W}_v^R$  guided by the eigenvectors of the  $k$  largest eigenvalues, which are computed from the weight parameters in the first 4 transformations. It can be formulated as:
+
+$$
+\underset {\alpha} {\arg \min } | | \hat {\mathbf {W}} _ {v} - \mathbf {W} _ {v} ^ {R} | | ^ {2}, \quad \hat {\mathbf {W}} _ {v} = \alpha \mathbf {V} + \mathbf {W} _ {v} ^ {S}, \tag {1}
+$$
+
+where  $\hat{\mathbf{W}}_v$  is the enhanced viewpoint latent code.  $\alpha$  is a  $k$ -dim viewpoint coefficient.  $\mathbf{V} \in \mathbb{R}^{k \times 512}$  denotes the eigenvectors of  $\mathbf{A}^T\mathbf{A}$  associated with the  $k$  largest eigenvalues.  $\mathbf{A} \in \mathbb{R}^{m \times 512}$  are the weights of transformations.  $\alpha\mathbf{V}$  is duplicated into four rows in Eqn. 1 and  $\alpha$  can be solved by gradient descent. Finally, we generate a new image by combining  $\hat{\mathbf{W}}_v$  and remaining the dimensions of  $\mathbf{W}^S$ . Training Pseudo Images Given a pre-trained StyleGAN model, we first synthesize training images and filter out images that have more than one instance or an unrealistic instance, resulting in  $N$  training samples  $\{\mathbf{I}^j\}_{j=1}^N$ . Then, we manually select  $n$  reference view samples, which roughly cover the common object viewpoints ranging from  $0 - 360^\circ$  in azimuth. Finally, for each training sample  $\mathbf{I}^j$ , we produce  $n$  multi-view images of the same object with fixed camera poses as the pseudo images  $\{\mathbf{I}_i^{*j}\}_{i=1}^n \in \mathbb{R}^{W \times H \times 3}$ . We further apply the work [46] to obtain corresponding instance segmentation  $\{\mathbf{M}_i^j\}_{i=1}^n$  of pseudo images.
+
+Pose Sampling Space We assume a pinhole camera model  $\mathcal{C} = (\mathbf{t},\mathbf{K})$ , where  $\mathbf{K}\in \mathbb{R}^{3\times 3}$  is the intrinsic parameter. We represent the camera pose/extrinsic parameters  $\mathbf{t} = (\mathbf{c}_0,\mathbf{q})$  based on its 3D position  $\mathbf{c}_0\in \mathbb{R}^3$  and its rotation from a
+
+![](images/d0e4ef39d872420c58e9a7e23227923c5288859dfad9f54b824c677993f9d0a5.jpg)  
+(a)
+
+![](images/2c18b8f6b2297133ccfbeb4ac6ab31a1af7cb18e6624e700ac7c5a2b13361da9.jpg)  
+(b)  
+Fig. 3: (a) Multi-view generator comparisons of Baseline [71] and our approach (Ours). As can be observed, besides the viewpoint, the baseline perceives shape cues from the references (yellow circle, best view in zoom in). While our generated images show more consistency in object shape across views, which benefits shape learning. Please refer to Supp for more categories. (b) The architecture of our neural implicit fields and differentiable renderer.
+
+canonical view.  $\mathbf{q} \in \mathbb{R}^4$  is the quaternion vector representing the camera rotation. We assume the observed object is approximately inside the unit sphere. We further assume a known intrinsic and the principal point at the image center, as commonly assumed in stereo systems [19]. For the pose initialization, we manually annotate the camera poses  $\{\widetilde{\mathbf{t}}_i\}_{i=1}^n$  for the  $n$  reference images, which initializes the pseudo images' poses,  $\mathbf{t}_i^j = \widetilde{\mathbf{t}}_i, j \in [1,N]$ . Since we only need to annotate  $n$  reference samples per object category, it is far more practical than prior works that require camera pose label per training sample (see Tab. 1).
+
+# 3.2 Image-conditioned Neural Implicit Field
+
+3D Geometry Representation As illustrated in Fig. 3b, the object geometry to be reconstructed is represented by the function [68]:  $\mathcal{F}:\gamma (\mathbf{x})\to (s,\mathbf{f})$  that maps a point  $\mathbf{x}\in \mathbb{R}^3$  to its signed distance value  $s$  to the object surface and a local geometry feature  $\mathbf{f}\in \mathbb{R}^{d_f}$ .  $\gamma (\cdot)$  denotes a positional encoding operator on  $\mathbf{x}$  with 6 exponentially increasing frequencies introduced in NeRF [38]. The surface  $S$  is represented as the zero level set of MLP  $\mathcal{F}$  with learnable parameters  $\theta$ :
+
+$$
+\mathcal {S} = \{\mathbf {x} \in \mathbb {R} ^ {3} | \mathcal {F} _ {\theta} ^ {(s)} (\gamma (\mathbf {x})) = 0 \}. \tag {2}
+$$
+
+Neural Renderer Given a pixel  $p$  of a masked input image, we march a ray  $\mathbf{r} = \{\mathbf{c}_0 + t\mathbf{v}|t\geqslant 0\}$ , where  $\mathbf{c}_0$  is the camera position and  $\mathbf{v}$  the viewing direction.  $\hat{\mathbf{x}}$  denotes the first intersection between the ray  $\mathbf{r}$  and the surface, which can be efficiently detected via the sphere tracing [15] and implemented in a differentiable manner. As in Fig. 3b, the rendered color of the pixel  $p$  is encoded as [68]
+
+$$
+\mathcal {G}: (\hat {\mathbf {x}}, \hat {\mathbf {n}}, \mathbf {f}, \mathbf {v}) \rightarrow \mathbf {c}, \tag {3}
+$$
+
+which is a function of the surface properties at  $\hat{\mathbf{x}}_p$ , including the surface point  $\hat{\mathbf{x}}_p$ , the surface normal  $\hat{\mathbf{n}}_p \in \mathbb{R}^3$ , the local geometry feature  $\mathbf{f}_p$ , and a viewing direction  $\mathbf{v}_p \in \mathbb{R}^3$ . Similarly, the function  $\mathcal{G}$  is implemented as an MLP with learnable parameters  $\phi$ . The surface normal  $\hat{\mathbf{n}}_p$  can be computed by the spatial derivative  $\frac{\delta\mathcal{F}^{(s)}}{\delta\mathbf{x}_p}$  via back-propagation through the network  $\mathcal{F}$ . Incorporating the surface normal and view direction enables  $\mathcal{G}$  to represent the light reflected from a surface point  $\mathbf{x}$  across different viewpoints [38, 68]. Also, according to [68], introducing the local geometry feature vector  $\mathbf{f}$  allows the renderer to handle more complex appearances, which might appear in GAN-generated images. The neural implicit field can be learned through back-propagation without any 3D supervision by comparing the rendered images with the multi-view pseudo images. Minimizing this error encourages multi-view photo consistency because only when the point is on the actual surface, the MLPs  $(\mathcal{F},\mathcal{G})$  will predict accurate color with fewer multi-view variations.
+
+Image Encoder We propose to utilize architecture to condition on a single image, such that the learned neural implicit field can generalize to a new object instance without re-training. Specifically, as shown in Fig. 2, we use an image encoder  $\mathcal{E}$  as a hyper network [13,29] to predict  $\theta$  (the parameters of  $\mathcal{F}$ ), written as  $\theta = \mathcal{E}_{\Phi}(\mathbf{I})$ , where  $\varPhi$  is the neural network weights.
+
+# 3.3 Model Learning
+
+Given  $N$  sets of masked pseudo images  $\{\mathbf{I}^j,\{\mathbf{I}_i^{*j}\}_{i = 1}^n\}_{j = 1}^N$ , along with their initial camera poses  $\{\{\mathbf{t}_i^j\}_{i = 1}^n\}_{j = 1}^N$ , we optimize the encoder parameters  $\varPhi$ , texture MLP parameters  $\phi$  and camera poses  $\{\{\mathbf{t}_i^j\}_{i = 1}^n\}_{j = 1}^N$  by minimizing the loss:
+
+$$
+\mathcal {L} = \sum_ {j = 1} ^ {N} \left(\mathcal {L} _ {R G B} + \lambda_ {\text {m a s k}} \mathcal {L} _ {\text {M a s k}} + \lambda_ {\text {e i k}} \mathcal {L} _ {\text {E i k}}\right), \tag {4}
+$$
+
+where  $\mathcal{L}_{RGB}$  is photometric loss,  $\mathcal{L}_{Mask}$  is silhouette loss,  $\mathcal{L}_{Eik}$  is Eikonal regularization, and  $\lambda_{*}$  are loss weights.
+
+Photometric Loss Let  $\mathbf{I}_p^*$ ,  $\mathbf{M}_p \in \{0,1\}$  be the RGB and silhouette values of pixel  $p$  in an image sample  $\mathbf{I}_i^*$  taken at the view direction  $\mathbf{v}_p$  associated with camera  $\mathcal{C}_i$ .  $p \in P$  indexes all pixels in the input image set  $\{\mathbf{I}_i^*\}_{i=1}^n$ . The photometric loss is defined on mini-batches of pixels in  $P$ :
+
+$$
+\mathcal {L} _ {R G B} = \frac {1}{| P |} \sum_ {p \in P ^ {i n}} \left| \mathbf {I} _ {p} ^ {*} - \mathcal {G} \left(\hat {\mathbf {x}} _ {p}, \hat {\mathbf {n}} _ {p}, \mathbf {f} _ {p}, \mathbf {v} _ {p}\right) \right|, \tag {5}
+$$
+
+where  $\mathbf{f}_p, \hat{\mathbf{x}}_p, \hat{\mathbf{n}}_p$  are defined in Eqn. 3.  $P^{in} \subset P$  represents the subset of pixels  $P$  where intersection has been found and  $\mathbf{M}_p = 1$ .  $|\cdot|$  denotes the  $L_{1}$  loss.
+
+Silhouette Loss We define the silhouette loss as
+
+$$
+\mathcal {L} _ {\text {M a s k}} = \frac {1}{| P |} \sum_ {p \in P ^ {\text {o u t}}} C E \left(\mathbf {M} _ {p}, \hat {\mathbf {M}} _ {p}\right), \tag {6}
+$$
+
+where  $\hat{\mathbf{M}}$  is the masked rendering.  $P^{out} = P - P^{in}$  represents the indices in the mini-batch for which there is no ray-geometry intersection or  $\mathbf{M}_p = 0$ .  $CE(\cdot, \cdot)$  denotes the cross-entropy loss. Conventionally, given the ray  $\mathbf{r}_p = \{\mathbf{c}_0 + t\mathbf{v}_p | t \geqslant 0\}$  of pixel  $p$ ,  $\hat{\mathbf{M}}_p$  is defined as:
+
+$$
+\hat {\mathbf {M}} _ {p} = \left\{ \begin{array}{l l} 1 & \mathbf {r} _ {p} \cap \mathcal {S} \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {7}
+$$
+
+To make this differentiable, we follow [68] and compute
+
+$$
+\hat {\mathbf {M}} _ {p} = \operatorname {s i g m o i d} \left(- \beta \min  _ {t \geq 0} \mathcal {F} ^ {(s)} \left(\mathbf {c} _ {0} + t \mathbf {v} _ {p}\right)\right). \tag {8}
+$$
+
+When  $\beta \to \infty$ ,  $\mathcal{F}^{(s)} < 0$  means inside the surface and  $\mathcal{F}^{(s)} > 0$  outside.
+
+Eikonal Regularization A special property of signed distance functions is their differentiability with a gradient of unit norm, satisfying the Eikonal equation  $||\nabla \mathcal{F}||_2 = 1$  [10,43]. We thus encourage our implicit geometry representation to satisfy the Eikonal property:
+
+$$
+\mathcal {L} _ {E i k} = \sum_ {\tilde {\mathbf {x}}} \left\| \left\| \nabla_ {\tilde {\mathbf {x}}} \mathcal {F} ^ {(s)} (\tilde {\mathbf {x}}) \right\| _ {2} - 1 \right\| _ {2} ^ {2}, \tag {9}
+$$
+
+where  $\tilde{\mathbf{x}}$  is uniformly sampled at the 3D region of interest.
+
+# 3.4 Uncertainty Prediction Module
+
+The pseudo images are inherently aleatoric, i.e., there might be areas with either notable artifacts in one image or with inconsistent shape/texture across images. To adaptively treat these problematic areas in learning, we propose to use Bayesian learning [25, 66] to model this aleatoric uncertainty. Specifically, we exploit the feature space of the image encoder  $\mathcal{E}$  and train a shallow decoder to estimate an uncertainty map  $\mathbf{U}$ , which has the same size as pseudo images. Formally, we model the observed color  $\mathbf{I}_p^*$  at pixel  $p$  with a likelihood function  $\mathcal{P}(\mathbf{I}_p^*)$  that follows the Laplacian distribution with ray-dependent variance  $\sigma_p$ :
+
+$$
+\mathcal {P} \left(\mathbf {I} _ {p} ^ {*}\right) = \frac {1}{2 \sigma_ {p}} \exp \left(- \frac {\left| \mathbf {I} _ {p} ^ {*} - \mathcal {G} \left(\hat {\mathbf {x}} _ {p} , \hat {\mathbf {n}} _ {p} , \mathbf {f} _ {p} , \mathbf {v} _ {p}\right) \right|}{\sigma_ {p}}\right), \tag {10}
+$$
+
+where  $\sigma_{p}$  denotes the uncertainty. Since  $L_{1}$  distance is less sensitive to outliers, which is more suitable for optimizing the rendered appearance against the pseudo RGB values. Thus, we adopt Laplacian likelihood to model the inconsistent uncertainty distribution. To find the parameters best explaining the model, we maximize the likelihood function, i.e., minimizing the negative log-likelihood:
+
+$$
+- \log (\mathcal {P} \left(\mathbf {I} _ {p} ^ {*}\right)) = \frac {\left| \mathbf {I} _ {p} ^ {*} - \mathcal {G} \left(\hat {\mathbf {x}} _ {p} , \hat {\mathbf {n}} _ {p} , \mathbf {f} _ {p} , \mathbf {v} _ {p}\right) \right|}{\sigma_ {p}} + \log \sigma_ {p} + \log 2. \tag {11}
+$$
+
+Therefore, we update the photometric loss as:
+
+$$
+\mathcal {L} _ {R G B} = \frac {1}{| P |} \sum_ {p \in P _ {i n}} \left(e ^ {- \mathbf {U} _ {p}} \left| \mathbf {I} _ {p} ^ {*} - \mathcal {G} \left(\hat {\mathbf {x}} _ {p}, \hat {\mathbf {n}} _ {p}, \mathbf {f} _ {p}, \mathbf {v} _ {p}\right) \right| + \mathbf {U} _ {p}\right). \tag {12}
+$$
+
+![](images/9a903aa88a5785a0ed178ca06ca95b83037c04c80ceb9c56b3a72132e58052e7.jpg)  
+Fig. 4: The uncertainty maps produced by our proposed method and humans.
+
+We train this loss on mini-batches of pixels in  $P$ . Here  $P$  indexes all pixels in the input multi-view image set  $\{\mathbf{I}_{i=1}^{*}\}_{i=1}^{n}$ , which contribute to the same object's depth, texture and uncertainty learning. During training, for the surface point  $\hat{\mathbf{x}}_p$ , the first term  $e^{-\mathbf{U}_p}$  can be seen as a weighted distance which assigns larger weights to less uncertain pixels. The second term  $\mathbf{U}_p$  is a penalty term.
+
+$\mathcal{L}_{RGB}$  encourages the neural implicit representation to bring together the texture information of all cross-view corresponding pixels in all images. Consequently, the pixel-wise uncertainty value is able to mine the multi-view inconsistencies among those pixels which contribute to the same 3D surface point. In practice, we train a 2-layer convolutional network to predict the log variance  $\mathbf{U}_p \coloneqq \log \sigma_p$  (please refer to Supp for more details).
+
+# 3.5 Implementation Details
+
+The encoder  $\mathcal{E}$  is implemented as a ResNet-18 [16] followed by fully-connected layers. Both  $\mathcal{F}$  and  $\mathcal{G}$  have 4 fully-connected layers. For the main experiment, we set  $N = 2,000$ ,  $n = 40$ ,  $k = 5$ ,  $W = H = 256$ ,  $d_{f} = 256$ ,  $\beta = 50$ ,  $\lambda_{mask} = 0.01$ ,  $\lambda_{eik} = 0.1$ . We implement our model in Pytorch and use the Adam optimizer with a learning rate of  $1e - 4$  for both network and camera pose parameters.
+
+# 4 Experimental Results
+
+We evaluate our approach on six category-specific StyleGAN models, including rigid objects such as airplanes, cars, motorbikes, as well as non-rigid objects such as birds, horses, and potted plants. We use the official car and horse models from StyleGAN2 repo  $^{1}$ , trained on the LSUN dataset [70]. For other 4 categories, we train the StyleGAN models with StyleGAN2-ADA-Pytorch library  $^{2}$  on LSUN airplanes, birds, motorbikes, and potted plants with  $200K$  images respectively.
+
+# 4.1 Human Study vs Our Method on the Uncertainty Prediction
+
+Although GAN interpretation methods have shown that manipulating the latent code of StyleGAN produces multi-view images of the same object [14, 51], no studies have quantitatively evaluated the unreliable/inconsistent object shape
+
+or texture in/across the multi-view pseudo images. Thanks to our multi-view-stereo-like neural implicit network, our uncertainty map can serve as a means to detect the unreliable/inconsistent areas in the GAN-generated multi-view images. On the other hand, volunteers were asked to label the potentially problematic areas in the pseudo images. As shown in Fig. 4, given an image, the multiview generator is able to generate pseudo images with varying viewpoints. We believe humans are able to reason unreliable/inconsistent regions in/across the pseudo images. Specifically, this is accomplished using a random set of 100 images from the PASCAL3D+ car category. For each image, we generate a pseudo image with a different viewpoint (Sec. 3.1). Then, the volunteers manually labels polygon-based regions of interest (uncertainty region) on the pseudo images, using the Matlab Image Labeler app. As can be observed in Fig. 4, human labels mainly focus on the global object shape inconsistency across views, which might not have the granularity to evaluate the pixel-level inconsistency. Nevertheless, we quantify the detection ability of our uncertainty maps by using the human labels as the ground-truth. We achieve  $34.6\%$  Intersection over Union (IoU), which shows our capability in detecting the inconsistent object shape in GAN-generated multi-view pseudo images. To our knowledge, this is the first method that tries to investigate the unreliable supervision across the GAN-generated pseudo multi-view images in 3D object modeling.
+
+# 4.2 Quantitative 3D Reconstruction Evaluation
+
+We quantitatively evaluate on the PASCAL3D+ dataset [65], a 3D reconstruction benchmark of real-world images with (approximate) CAD model annotations. Similar to prior work [29, 59], we use annotations of airplane and car categories on the test set for evaluation.
+
+Evaluation Metrics. We adopt standard 3D reconstruction metrics: IoU and Chamfer- $L_{1}$  Distance (CD). Following [59], we compute 3D IoU between ground truth and prediction with the resolution of  $32^{3}$ . Following [29], we uniformly sample 3D points from the ground truth and prediction to compute CD.
+
+Baselines We compare against SoTA unsupervised single-view 3D reconstruction baselines: CSDM [58], DRC [59], CMR [18], U-CMR [8] and SDF-SRN [29]. As detailed in Tabs. 1 and 2a, some baselines require additional unsupervised constraints, e.g., DRC (implicit) and SDF-SRN (implicit) both require ground-truth camera pose for each training sample. The mesh-based methods such as CSDM, CMR, and U-CMR need expert object-specific templates as additional constraints. Here, we do not compare with ShSMesh [69] as it neither quantitatively evaluates on PASCAL3D+, nor trains on real-world car/airplane images. Also, we do not quantitatively compare with StyleGANRender [71] as the code or trained model is not publicly available, and StyleGANRender does not report full 3D shape reconstruction errors as our baselines did.
+
+Results We present the comparisons of our approach in Tab. 2a and visualize sample predictions in Fig. 5. It can be observed that the Proposed model is significantly better than baselines in both CD (10.1% relative over SDF-SRN) and IoU (7.8% relative over DRC). It is worth noting that our models trained with
+
+Table 2: (a) Quantitative 3D reconstruction results on PASCAL3D+. During training, CSDM, DRC and SDF-SRN require ground-truth camera pose per training sample, CMR uses 2D keypoints and object-specific templates as additional constraints, and U-CMR only relies on object-specific templates. [Keys: Requ. or Cons. = requirement or constraint in training, T = category-specific templates, C = poses per training sample,  $\mathrm{C}^{*} =$  poses for reference images, K = 2D keypoints, S = semantic information, GANs = pre-trained GAN models]. All CD values are scaled by 10 following [29]. (b) Ablation of uncertainty prediction and the number of pseudo images, using the PASCAL3D+ car category.
+
+(a)  
+
+<table><tr><td rowspan="2">Category</td><td rowspan="2">Req. or Cons.</td><td colspan="2">Airplane</td><td colspan="2">Car</td></tr><tr><td>CD (↓)</td><td>IoU (↑)</td><td>CD (↓)</td><td>IoU (↑)</td></tr><tr><td>CSDM [58]</td><td>T, C</td><td>-</td><td>0.400</td><td>-</td><td>0.600</td></tr><tr><td>DRC [59]</td><td>C</td><td>-</td><td>0.420</td><td>-</td><td>0.670</td></tr><tr><td>CMR [18]</td><td>T, K</td><td>0.625</td><td>-</td><td>0.474</td><td>0.640</td></tr><tr><td>U-CMR [8]</td><td>T</td><td>-</td><td>-</td><td>-</td><td>0.646</td></tr><tr><td>UMR [28]</td><td>S</td><td>-</td><td>-</td><td>-</td><td>0.620</td></tr><tr><td>SDF-SRN [29]</td><td>C</td><td>0.303</td><td>0.405</td><td>0.233</td><td>0.653</td></tr><tr><td>SDF-SRN* [29]</td><td>C</td><td>0.297</td><td>0.412</td><td>0.230</td><td>0.661</td></tr><tr><td>Proposed</td><td>GANs, C*</td><td>0.286</td><td>0.473</td><td>0.195</td><td>0.702</td></tr></table>
+
+(b)  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Proposed w/o Uncertainty (n=40)</td><td colspan="4">Proposed (n =)</td></tr><tr><td>10</td><td>20</td><td>40</td><td>50</td></tr><tr><td>CD (↓)</td><td>0.208</td><td>0.336</td><td>0.243</td><td>0.195</td><td>0.191</td></tr><tr><td>IoU (↑)</td><td>0.681</td><td>0.612</td><td>0.643</td><td>0.702</td><td>0.714</td></tr></table>
+
+Proposed setting only require ground-truth pose annotations for the reference images. It is more practical for real-world scenarios than DRC and SDF-SRN, which require ground-truth camera pose per training sample. Further, we retrain SDF-SRN with our GAN-generated training data and report the results in Tab. 2a (SDF-SRN*). It can be observed, despite the minor improvement over the original SDF-SRN due to our pseudo images, the new model still performs worse than ours. Fig. 5 shows visual comparisons to SDF-SRN [29] and StyleGANRender [71] results. As can be observed, our approach suffers slightly from shape ambiguity, e.g., windows of cars tend to be concave. Nonetheless, our predictions more closely resemble the ground truth.
+
+# 4.3 Ablation Study
+
+All ablations use the models trained on the car category.
+
+Effect on Uncertainty Prediction We evaluate single-view 3D reconstruction on a model trained without uncertainty prediction, i.e. using Eqn. 5 instead of Eqn. 12. As shown in Tab. 2b, our uncertainty prediction module can remedy the negative impact of uncertain texture in GAN-generated multi-view images, leading to improved 3D reconstruction (CD:  $0.208 \rightarrow 0.195$ ).
+
+Effect on  $n$  The key assumption, as well as motivation of our work, is that GAN-generated multi-view images can be leveraged to learn a multi-view stereo system. To validate the impact of the amount of pseudo images, we train models with different numbers of image viewpoints,  $n = 10,20,40,50$ . Tab. 2b shows that the model trained with  $n = 40$  images significantly outperforms the ones with  $n = 10,20$ , and saturates when  $n = 40 \rightarrow 50$ . Considering the tradeoff between reconstruction accuracy and training cost, we use  $n = 40$  for all experiments.
+
+![](images/1a78445560a7a0a1ff7a4f10e4b618ab9a7008544d1599580659c17076647791.jpg)  
+Fig. 5: Qualitative comparisons with SDF-SRN [29] (SOTA baseline) and StyleGANRender [71] on PASCAL3D+ car or airplane categories. Our approach recovers significantly more accurate 3D shapes and topologies from the images.
+
+Effect on Multi-view Generation Following the same setting, we re-train a model with pseudo imaged produced by Baseline method (Sec. 3.1). Quantitatively, such a model only obtain the IoU of 0.679, much worse than ours (0.702). The comparisons show the superior quality of our multi-view generation method.
+
+# 4.4 Qualitative Evaluation
+
+Comparison with U-CMR [8] and DRC [59] We show qualitative comparisons with U-CMR and DRC on the PASCAL3D+ cars, motorbikes or airplanes, in Fig. 6. Our approach achieves more faithful reconstructions than U-CMR. Note that U-CMR requires expert templates while our approach does not.
+
+Results on More Categories While our quantitative evaluation is on the PASCAL3D+ airplane and car, we provide more qualitative results for the birds, horses, motorbikes and potted plants in Fig. 7. As can be seen, our approach can effectively capture the thin structure present in 3D shapes from single-view images, e.g. horses' legs and motorbikes' hand clutch. All testing images are from the LSUN dataset, which never appear in the training set of our GAN models.
+
+# 5 Conclusions
+
+To leverage pre-trained GAN models for 3D vision tasks, we propose an image-conditioned neural implicit network that can learn the shape priors from GAN-generated multi-view images and perform single-view 3D reconstruction. Moreover, we naturally introduce a novel uncertainty prediction module, which can
+
+![](images/3a89a0911831e992a305994d181e82cc17fc248a741d2e3006d6b1b50c3d2aac.jpg)  
+Fig. 6: Additional qualitative comparisons with U-CMR [8] and DRC [59] on cars and motorcycles or airplanes.
+
+![](images/7ddd0d424d152107674559843825e295903c043e4bcd8fc1d4f6b1b2faaf489b.jpg)  
+Fig. 7: Qualitative results of birds, horses, motorbikes, and potted plants.
+
+avoid invalid supervisions for better single-view 3D reconstruction. Experimentally, our approach significantly outperforms the SoTA unsupervised single-view 3D reconstruction methods, while requiring less supervision during training. We believe this work opens up a path for improving the ability to semantically control GAN generation and facilitates 2D GAN priors for 3D vision tasks.
+
+Limitations For some categories (e.g., chair), StyleGAN is unable to converge to satisfying results, partially due to chairs' large topology variations. We believe the rapid development of GANs will extend to these challenging categories, and thus our method can leverage them for 3D reconstruction of more categories. Also, similar to most prior works, our model is category-specific. One future direction is to develop a single model for multiple categories based on universal GAN models, e.g., BigGAN [4], improving generalization to unseen categories.
+
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+# 2DPASS: 2D Priors Assisted Semantic Segmentation on LiDAR Point Clouds
+
+Xu Yan $^{1\dagger}$ , Jiantao Gao $^{2\dagger}$ , Chaoda Zheng $^{1\dagger}$ , Chao Zheng $^{3}$ , Ruimao Zhang $^{1}$ , Shuguang Cui $^{1}$ , Zhen Li $^{1\star}$
+
+<sup>1</sup>The Chinese University of Hong Kong (Shenzhen), The Future Network of Intelligence Institute, Shenzhen Research Institute of Big Data, <sup>2</sup>Shanghai University, <sup>3</sup>Tencent Map, T Lab {xuyan1@link., lizhen}@cuhk.edu.cn
+
+Abstract. As camera and LiDAR sensors capture complementary information in autonomous driving, great efforts have been made to conduct semantic segmentation through multi-modality data fusion. However, fusion-based approaches require paired data, i.e., LiDAR point clouds and camera images with strict point-to-pixel mappings, as the inputs in both training and inference stages. It seriously hinders their application in practical scenarios. Thus, in this work, we propose the 2D Priors Assisted Semantic Segmentation (2DPASS) method, a general training scheme, to boost the representation learning on point clouds. The proposed 2DPASS method fully takes advantage of 2D images with rich appearance during training, and then conduct semantic segmentation without strict paired data constraints. In practice, by leveraging an auxiliary modal fusion and multi-scale fusion-to-single knowledge distillation (MSFSKD), 2DPASS acquires richer semantic and structural information from the multi-modal data, which are then distilled to the pure 3D network. As a result, our baseline model shows significant improvement with only point cloud inputs once equipped with the 2DPASS. Specifically, it achieves the state-of-the-arts on two large-scale recognized benchmarks (i.e., SemanticKITTI and NuScenes), i.e., ranking the top-1 in both single and multiple scan(s) competitions of SemanticKITTI.
+
+Keywords: Semantic Segmentation, Multi-Modal, Knowledge Distillation, LiDAR Point Clouds
+
+# 1 Introduction
+
+Semantic segmentation plays a crucial role in large-scale outdoor scene understanding, which has broad applications in autonomous driving and robotics [1-3]. In the past few years, the research community has devoted significant effort to understanding natural scenes using either camera images [4-7] or LiDAR point clouds [2, 8-12] as the input. However, these single-modal methods inevitably face challenges in complex environments due to the inherent limitations of the
+
+![](images/23fdc84be108ce7ba027f85d7837e5b1b425216c0487ce08e95a890bcfb40968.jpg)  
+Fig. 1. Limitation of fusion-based methods. When the self-driving car only has front-cameras with limited perspective such as SemanticKITTI [16] dataset while the 360-degree LiDAR has a much larger sensing range, fusion-based methods that require strict alignment between camera and LiDAR can only identify a small proportion of the point cloud (see the red region).
+
+![](images/27cccaf5cb21393a83377d5cfa1e9f7a19287b433b0368e8b1471b173b2f2aa3.jpg)
+
+![](images/33c5ba078cd40732e38a400c331f5ee008d7c29b3f995a8a067862f80429fd94.jpg)
+
+input sensors. Concretely, cameras provide dense color information and fine-grained texture, but they are ambiguous in depth sensing and unreliable in low light conditions. In contrast, LiDARs robustly offer accurate and wide-ranging depth information regardless of lighting variances but only capture sparse and textureless data. Since cameras and LiDARs complement each other, it is better to perceive the surrounding with both sensors.
+
+Recently, many commercial cars have been equipped with both cameras and LiDARs. This excites the research community to improve the semantic segmentation by fusing the information from two complementary sensors [13-15]. These approaches first establish the mapping between 3D points and 2D pixels by projecting the point clouds onto the image planes using the sensor calibrations. Based on the point-to-pixel mapping, the models fuse the corresponding image features into the point features, which are further processed to obtain the final semantic scores. Despite the improvements, fusion-based methods have the following unavoidable limitations: 1) Due to the difference of FOVs (field of views) between cameras and LiDARs, the point-to-pixel mapping cannot be established for points that are out of the image planes. Typically, the FOVs of LiDAR and cameras only overlap in a small portion (see Fig. 1), which significantly limits the application of fusion-based methods. 2) Fusion-based methods consume more computational resources since they process both images and point clouds (through multitask or cascade manners) at runtime, which introduces a great burden on real-time applications.
+
+To address the above two issues, we focus on improving semantic segmentation by leveraging both images and point clouds through an effective design in this work. Considering the sensors are moving in the scenes, the non-overlap part of the 360-degree LiDAR point clouds corresponding to image in the same timestamp (see the gray region of the right part in Fig. 1) can be covered by images from other time-stamp. Besides, the dense and structural information of images provides useful regularization for both seen and unseen point cloud regions. Based on these observations, we propose a "model-independent" training scheme, namely 2D Priors Assisted Semantic Segmentation (2DPASS), to enhance the representation learning of any 3D semantic segmentation networks with minor
+
+structure modification. In practice, on the one hand, for above-mentioned non-overlap regions, 2DPASS takes pure point clouds as the inputs to train the segmentation model. On the other hand, for subregions with well-aligned point-to-pixel mappings, 2DPASS adopts an auxiliary multi-modal fusion to aggregate image and point features in each scale, and then aligns the 3D predictions with the fusion predictions. Unlike previous cross-modal alignment [17] apt to contaminate the modal-specific information, we design a multi-scale fusion-to-single knowledge distillation (MSFSKD) strategy to transfer extra knowledge to the 3D model as well as retaining its modal-specific ability. Compared with fusion-based methods, our solution has the following preferable properties: 1) Generality: It can be easily integrated with any 3D segmentation model with minor structural modification; 2) Flexibility: The fusion module is only used during the training to enhance the 3D network. After training, the enhanced 3D model can be deployed without image inputs. 3) Effectively: Even with only a small section of overlapped multi-modality data, our method can significantly boost the performance. As a result, we evaluate 2DPASS with a simple yet strong baseline implemented with sparse convolutions [3]. The experiments show 2DPASS brings noticeable improvements even over this strong baseline. Equipped with 2DPASS using multi-modal data, our model achieves the top-1 results on the single and multiple-scan leaderboards of SemanticKITTI [16]. The state-of-the-art results on the NuScenes [18] dataset further confirm the generality of our method.
+
+In general, the main contributions are summarized as follows.
+
+- We propose 2D Priors Assisted Semantic Segmentation (2DPASS) that assists 3D LiDAR semantic segmentation with 2D priors from cameras. To the best of our knowledge, 2DPASS is the first method that distills multi-modal knowledge to single point cloud modality for semantic segmentation.  
+- Equipped with the proposed multi-scale fusion-to-single knowledge distillation (MSFSKS) strategy, 2DPASS achieves the significant performance gains on SemanticKITTI and NuScenes benchmarks, ranking the 1st on single and multiple tracks of SemanticKITTI.
+
+# 2 Related Work
+
+# 2.1 Single-Sensor Methods
+
+Camera-Based Methods. Camera-based semantic segmentation aims to predict the pixel-wise labels for input 2D images. FCN [19] is the pioneer in semantic segmentation, which proposes an end-to-end fully convolutional architecture based on image classification networks. Recent works have achieved significant improvements via exploring multi-scale features learning [4,20,21], dilated convolution [5,22], and attention mechanisms [7,23]. However, camera-only methods are ambiguous in depth sensing and not robust in low light conditions.
+
+LiDAR-Based Methods. The LiDAR data is generally represented as point clouds. There are several mainstreams to process point clouds with different representations. 1) Point-based methods approximate a permutation-invariant
+
+set function using a per-point Multi-Layer Perceptron (MLP). PointNet [24] is the pioneer in this field. Later on, many studies design point-wise MLP [25, 26], adaptive weight [27, 28] and pseudo grid [29, 30] based methods to extract local features of point clouds or exploit nonlocal operators [31-33] to learn long distance dependency. However, point-based methods are not efficient in the LiDAR scenario since their sampling and grouping algorithms are generally time-consuming. 2) Projection-based methods are very efficient approaches for LiDAR point clouds. They project point clouds onto 2D pixels so that traditional CNN can play a normal role. Previous works project all points scanned by the rotating LiDAR onto 2D images by plane projection [34-36], spherical projection [37, 38] or both [39]. However, the projection inevitably causes information loss. And the projection-based methods currently meet the bottleneck of the segmentation accuracy. 3) Most recent works adopt voxel-based frameworks since they balance the efficiency and effectiveness, where sparse convolution (SparseConv) [3] are most commonly utilized. Compared to traditional voxel-based methods (i.e., 3DCNN) directly transforming all points into the 3D voxel grids, SparseConv only stores non-empty voxels in a Hash table and conducts convolution operations only on these non-empty voxels in a more efficient way. Recently, many studies have used SparseConv to design more powerful network architectures. Cylinder3D [40] changes original grid voxels to cylinder ones and designs an asymmetrical network to boost the performance. AF $^2$ -S3Net [41] applies multiple branches with different kernel sizes, aggregating multi-scale features via an attention mechanism. 4) Very recently, there is a trend of exploiting multi-representation fusion methods. These methods combine multiple representations above (i.e., points, projection images, and voxels) and design feature fusion among different branches. Tang et.al. [10] combines point-wise MLPs in each sparse convolution block to learn a point-voxel representation and uses NAS to search for a more efficient architecture. RPVNet [42] proposes range-point-voxel fusion network to utilizes information from three representations. Nevertheless, these methods only take sparse and textureless LiDAR point clouds as inputs, thus appearance and texture in the camera images have not been fully utilized.
+
+# 2.2 Multi-Sensor Methods
+
+Multi-sensor methods attempt to fuse information from two complementary sensors and leverage the benefits of both camera and LiDAR [14, 15, 43, 44]. RGBAL [14] converts RGB images to a polar-grid mapping representation and designs early and mid-level fusion strategies. PointPainting [15] exploits the segmentation logits of images and projects them to the LiDAR space by bird's-eye projection [23] or spherical projection [45] for LiDAR network performance improvement. Recently, PMF [13] exploits a collaborative fusion of two modalities in camera coordinates. However, these methods require multi-sensor inputs in both training and inference phases. Moreover, the paired multi-modality data is usually computation-intensive and unavailable in practical application.
+
+![](images/eb009fa4cbf474cd5d6f4b659baca3641c3e9520ff7702976ebe6b16d422233c.jpg)  
+Fig. 2. 2D Priors Assisted Semantic Segmentation (2DPASS). It first crops a small patch from the original camera image as the 2D input. Then the cropped image patch and LiDAR point cloud independently pass through the 2D and 3D encoders to generate multi-scale features in parallel. Afterwards, for each scale, complementary 2D knowledge is effectively transferred to the 3D network via the multi-scale fusion-to-single knowledge distillation (MSFSKD). The feature maps (in the form of either pixel grid or point set) are used to generate the final semantic scores using modal-specific decoders, which are supervised by pure 3D labels.
+
+# 2.3 Cross-modal Knowledge Transfer
+
+Knowledge distillation was initially proposed for compressing the large teacher network to a small student one [46]. Over the past few years, several subsequent studies enhanced knowledge transferring through matching feature representations in different manners [47-50]. For instance, aligning attention maps [49] and Jacobean matrixes [50] were independently applied. With the development of multi-modal computer vision, recent research apply knowledge distillation to transfer priors across different modalities, e.g., exploiting extra 2D images in the training phase and improving the performance in the inference [51-55]. Specifically, [56] introduces the 2D-assisted pre-training, [57] inflates the kernels of 2D convolution to the 3D ones, and [58] applies well-designed teacher-student framework. Inspired but different from the above, we transfer 2D knowledge through a multi-scale fusion-to-single manner, which additionally takes care of the modal-specific knowledge.
+
+# 3 Method
+
+# 3.1 Framework Overview
+
+This paper focuses on improving the LiDAR point cloud semantic segmentation, which aims to assign the semantic label to each point. To handle difficulties in large-scale outdoor LiDAR point clouds, i.e., sparsity, varying density, and lack of texture, we introduce the strong regularization and priors from 2D camera images through a fusion-to-single knowledge transferring.
+
+The workflow of our 2D Priors Assisted Semantic Segmentation (2DPASS) is shown in Fig. 2. Since the camera images are pretty large (e.g.,  $1242 \times 512$ ), sending the original ones to our multi-modal pipeline is intractable. Therefore,
+
+we randomly sample a small patch  $(480 \times 320)$  from the original camera image as the 2D input [17], accelerating the training processing without performance drop. Then the cropped image patch and LiDAR point cloud independently pass through independent 2D and 3D encoders, where multi-scale features from the two backbones are extracted in parallel. Afterwards, multi-scale fusion-to-single knowledge distillation (MSFSKD) is conducted to enhance the 3D network using multi-modal features, i.e., fully utilizing texture and color-aware 2D priors as well as retaining the original 3D-specific knowledge. Finally, all the 2D and 3D features at each scale are used to generate semantic segmentation predictions, which are supervised by pure 3D labels. During inference, the 2D-related branch can be discarded, which effectively prevents extra computational burden in real application compared with fusion-based approaches.
+
+# 3.2 Modal-Specific Architectures
+
+Multi-Scale Feature Encoders. As shown in Fig. 2, we use two different networks to independently encode multi-scale features from 2D image and 3D point cloud. We apply ResNet34 [59] encoder with 2D convolution as the 2D network. For the 3D network, we adopt sparse convolution [3] to construct the 3D network. One merit of sparse convolution lies in the sparsity, with which the convolution operation only considers the non-empty voxels. Specifically, we design a hierarchical encoder as SPVCNN [10], and we adopt the ResNet bottleneck [59] design in each scale while replacing the ReLU activation with Leaky ReLU activation [60]. In both network, we extract  $L$  feature maps from different scales, obtaining the 2D and 3D features, i.e.,  $\{F_l^{2D}\}_{l=1}^L$  and  $\{F_l^{3D}\}_{l=1}^L$ .
+
+Prediction Decoders. After processing the features from images and point clouds at each scale, two modal-specific prediction decoders are independently applied to restore the down-sampled feature maps to their original sizes.
+
+For the 2D network, we adopt FCN [19] decoder to up-sample the features from each encoder layer. Specifically, the feature map  $D_{l}^{2D}$  from the  $l$ -th decoder layer can be gained by up-sampling the feature map from the  $(L - l + 1)$ -th encoder layer, where all the up-sampled feature maps will be merged through element-wise addition. Finally, the semantic segmentation of the 2D network is obtained by passing the fused feature map through a linear classifier.
+
+For the 3D network, we do not adopt the U-Net decoder used in previous methods [10, 40, 41]. In contrast, we up-sample the features from different scales to the original size and concatenate them together before feeding them into the classifier. We find out that such a structure can better learn hierarchical information while gaining the prediction in a more efficient way.
+
+# 3.3 Point-to-Pixel Correspondence
+
+Since the 2D features and 3D features are generally represented as pixels and points, respectively, it is difficult to directly transfer information between two modalities. In this section, we aim to generate paired features of two modalities
+
+![](images/7af580495bab3f7c17b80e2f35f0013ccdcf14adda9f81946559227dd32aa616.jpg)  
+(a) 2D Feature Generation
+
+![](images/e6192c50167c054bd4b387b0aad029995a88ed3944c692fbe9c4a48c7fdd734c.jpg)  
+(b) 3D Feature Generation  
+Fig. 3. 2D and 3D feature generation. Part (a) demonstrates the 2D feature generation, where the point cloud will first be projected onto the image patch and generate the point-to-pixel (P2P) mapping. After that, it transfers the 2D feature map to the point-wise 2D features according to P2P mapping. Part (b) shows the 3D feature generation. The point-to-voxel (P2V) mapping is easy to obtain, and the voxel features will be interpolated onto the point cloud.
+
+for further knowledge distillation, using the point-to-pixel correspondence. The details of paired feature generation in two modalities are demonstrated in Fig. 3. 2D Features. The process of 2D feature generation is illustrated in Fig. 3 (a). By cropping a small patch  $I \in \mathbb{R}^{H \times W \times 3}$  from the original image and passing it through a 2D network, multi-scale features can be extracted in the hidden layers with different resolution. Taking the feature map  $F_{l}^{2D} \in \mathbb{R}^{H_{l} \times W_{l} \times D_{l}}$  from  $l$ -th layer as an example, we first conduct a decovolution operation to upscale its resolution to the original one  $\tilde{F}_{l}^{2D}$ . Similar to the recent multi-sensor method [13], we adopt perspective projection and calculate a point-to-pixel mapping between point clouds and images. Specifically, given a LiDAR point cloud  $P = \{p_{i}\}_{i=1}^{N} \in \mathbb{R}^{N \times 3}$ , the projection of each 3D point  $p_{i} = (x_{i}, y_{i}, z_{i}) \in \mathbb{R}^{3}$  to a point  $\hat{p}_{i} = (u_{i}, v_{i}) \in \mathbb{R}^{2}$  in the image plane is given as:
+
+$$
+\left[ u _ {i}, v _ {i}, 1 \right] ^ {T} = \frac {1}{z _ {i}} \times K \times T \times \left[ x _ {i}, y _ {i}, z _ {i}, 1 \right] ^ {T}, \tag {1}
+$$
+
+where  $K \in \mathbb{R}^{3 \times 4}$  and  $T \in \mathbb{R}^{4 \times 4}$  are the camera intrinsic and extrinsic matrices respectively.  $K$  and  $T$  are directly provided in KITTI [61]. Since the lidar and cameras operate at different frequencies in NuScenes [18], we need to transform the LiDAR frame at timestamp  $t_l$  to camera frame at timestamp  $t_c$  via the global coordinate system. The extrinsic matrix  $T$  in NuScenes dataset [18] is given as:
+
+$$
+T = T _ {\mathrm {c a m e r a} \leftarrow \mathrm {e g o} _ {\mathrm {t} _ {c}}} \times T _ {\mathrm {e g o} _ {\mathrm {t} _ {c}} \leftarrow \mathrm {g l o b a l}} \times T _ {\mathrm {g l o b a l} \leftarrow \mathrm {e g o} _ {\mathrm {t} _ {l}}} \times T _ {\mathrm {e g o} _ {\mathrm {t} _ {l}} \leftarrow \mathrm {l i d a r}} \tag {2}
+$$
+
+After the projection, the point-to-pixel mapping is represented as
+
+$$
+M ^ {i m g} = \left\{\left(\left\lfloor v _ {i} \right\rfloor , \left\lfloor u _ {i} \right\rfloor\right) \right\} _ {i = 1} ^ {N} \in \mathbb {R} ^ {N \times 2}, \tag {3}
+$$
+
+where  $\lfloor \cdot \rfloor$  is the floor operation. According to the point-to-pixel mapping, we extract a point-wise 2D feature  $\hat{F}^{2D}\in \mathbb{R}^{N^{img}\times D_l}$  from the original feature map
+
+![](images/76da6a8bee6e0ccd44a97a361c701fa5a6ed7fe211ab7ebec29779ca11e99710.jpg)  
+Fig. 4. Internal structure of Multi-Scale Fusion-to-Single Knowledge Distillation (MSFSKD), which consists of the modality fusion and Modality-Preserving KD. For each scale, modality fusion is first utilized to achieve an enhanced multi-modality feature  $\hat{F}_l^{2D3D_e}$ . Afterwards, the enhanced feature  $\hat{F}_l^{2D3D_e}$  promotes the 3D representation  $\hat{F}_l^{3D_e}$  through the uni-directional Modality-Preserving KD.
+
+$F^{2D}$  if any pixel on the feature map is included in  $M^{img}$ . Here  $N^{img} < N$  represents the number of points that are included in  $M^{img}$ .
+
+3D Features. The process of 3D features is relatively straightforward (as shown in Fig. 3 (b)). Specifically, for the point cloud  $P = \{(x_i,y_i,z_i)\}_{i = 1}^N$ , we obtain a point-to-voxel mapping in the  $l$ -th layer through
+
+$$
+M _ {l} ^ {v o x e l} = \left\{\left(\left\lfloor x _ {i} / r _ {l} \right\rfloor , \left\lfloor y _ {i} / r _ {l} \right\rfloor , \left\lfloor z _ {i} / r _ {l} \right\rfloor\right) \right\} _ {i = 1} ^ {N} \in \mathbb {R} ^ {N \times 3}, \tag {4}
+$$
+
+where  $r_l$  is the voxelization resolution in the  $l$ -th layer. After that, given the 3D feature  $F_l^{3D} \in \mathbb{R}^{N_l' \times D_l}$  from a sparse convolution layer, we gain a point-wise 3D feature  $\tilde{F}_l^{3D} \in \mathbb{R}^{N \times D_l}$  through nearest interpolation on the original feature map  $F_l^{3D}$  according to  $M_l^{voxel}$ . Finally, we filter the points by discarding points outside the image FOV:
+
+$$
+\hat {F} _ {l} ^ {3 D} = \{f _ {i} | f _ {i} \in \tilde {F} _ {l} ^ {3 D}, M _ {i, 1} ^ {i m g} \leq H, M _ {i, 2} ^ {i m g} \leq W \} _ {i = 1} ^ {N} \in \mathbb {R} ^ {N ^ {i m g} \times D _ {l}}, \tag {5}
+$$
+
+2D Ground Truths. Considering only 2D images is provided, the 2D ground-truths are obtained by projecting the 3D point labels to the corresponding image plane using above point-to-pixel mapping. Afterwards, the projected 2D ground truths can work as the supervision for the 2D branch.
+
+Features Correspondence. Since both 2D and 3D feature use the same point-to-pixel mapping, 2D features  $\hat{F}_l^{2D}$  and 3D features  $\hat{F}_l^{3D}$  in arbitrary  $l$ -th layer have the same number of point  $N^{img}$  and point-to-pixel correspondence.
+
+# 3.4 Multi-Scale Fusion-to-Single Knowledge Distillation (MSFSKD)
+
+As the key of 2DPASS, MSFSKD aims at improving the 3D representation in each scale using auxiliary 2D priors through a fusion-then-distillation manner. The knowledge distillation (KD) design of MSFSKD is partially inspired by [17]. However, [17] conducts KD in a naive cross-modal manner, i.e., simply
+
+aligning the outputs from two sets of single modal features (i.e. either 2D or 3D), which inevitably pushes the features from two modals to their overlapped space. Therefore, such a manner actually discards the modal-specific information, which is crucial in multi-sensor segmentation. Although this issue can be relieved by introducing extra segmentation heads [17], it is inherent for the cross-modal distillation, resulting in biased predictions. To this end, we propose multi-scale fusion-to-single knowledge distillation (MSFSKD) module as shown in Fig. 4, which first fuses features of both images and point clouds and then conducts unidirectional alignment between the fused and the point cloud features. In our fusion-then-distillation manner, the fusion well retains the complete information from multi-modal data. Besides, the unidirectional alignment ensures boosted point cloud features from fusion without losing modal-specific information.
+
+Modality Fusion. For each scale, considering the 2D and 3D feature gaps owing to different backbones, it is ineffective to directly fuse the raw 3D features  $\hat{F}_l^{3D}$  into their 2D counterparts  $\hat{F}_l^{2D}$ . Thus, we firstly transform  $\hat{F}_l^{3D}$  to  $\hat{F}_l^{\mathrm{learner}}$  through a "2D learner" MLP, which struggles to narrow the feature gap. Afterwards, the  $\hat{F}_l^{\mathrm{learner}}$  not only flows into the subsequent concatenation with 2D features  $\hat{F}_l^{2D}$  to gain the fused features  $\hat{F}_l^{2D3D}$  through another MLP, but also goes back into the original 3D features via a skip connection to yield enhanced 3D features  $\hat{F}_l^{3D_e}$ . Besides, similar to attention mechanism, the final enhanced fused features  $\hat{F}_l^{2D3D_e}$  is obtained by:
+
+$$
+\hat {F} _ {l} ^ {2 D 3 D _ {e}} = \hat {F} _ {l} ^ {2 D} + \sigma (\mathsf {M L P} (\hat {F} _ {l} ^ {2 D 3 D})) \odot \hat {F} _ {l} ^ {2 D 3 D}, \tag {6}
+$$
+
+where  $\sigma$  denotes Sigmoid activation function.
+
+Modality-Preserving KD. Although the  $\hat{F}_l^{\mathrm{learner}}$  is generated from pure 3D features, it is influenced by the segmentation loss of the 2D decoder as well, which takes enhanced fused feature  $\hat{F}_l^{2D3D_e}$  as inputs. Acting like a residual between fused and point features, the 2D learner feature  $\hat{F}_l^{\mathrm{learner}}$  well prevents the distillation from contaminating the modal-specific information in  $\hat{F}_l^{3D}$ , achieving a Modality-Preserving KD. Finally, two independent classifiers (fully-connected layers) are respectively applied on top of  $\hat{F}_l^{2D3D_e}$  and  $\hat{F}_l^{3D_e}$  to obtain the semantic scores  $S_l^{2D3D}$  and  $S_l^{3D}$ . We choose KL divergence as the distillation loss  $L_{xM}$  as follows:
+
+$$
+L _ {x M} = D _ {K L} \left(S _ {l} ^ {2 D 3 D} \| S _ {l} ^ {3 D}\right), \tag {7}
+$$
+
+In our implementation, we detach  $S_{l}^{2D3D}$  from the computational graph when computing  $L_{xM}$ , enforcing the uni-directional distillation by only pushing  $S_{l}^{3D}$  closer to  $S_{l}^{2D3D}$ .
+
+By taking such a knowledge distillation scheme, there are several advantages in our framework: 1) The 2D learner and the fusion-to-single distillation provides rich texture information and structural regularization to enhance the 3D feature learning without losing any modal-specific information in 3D. 2) The fusion branch is only adopted in the training phase. Therefore, the enhanced model can almost run without extra computational cost during the inference.
+
+Table 1. Semantic segmentation results on the SemanticKITTI test benchmark. Only approaches published before 03/08/2022 are compared.  
+
+<table><tr><td>Method</td><td>mIoU</td><td>road</td><td>sidewalk</td><td>parking</td><td>other-ground</td><td>building</td><td>car</td><td>truck</td><td>bicycle</td><td>motorcycle</td><td>other-vehicle</td><td>vegetation</td><td>trunk</td><td>terrain</td><td>person</td><td>bicyclist</td><td>motorcyclist</td><td>fence</td><td>pole</td><td>traffic sign</td><td>speed (ms)</td></tr><tr><td>SqueezeSegV2 [38]</td><td>39.7</td><td>88.6</td><td>67.6</td><td>45.8</td><td>17.7</td><td>73.7</td><td>81.8</td><td>13.4</td><td>18.5</td><td>17.9</td><td>14.0</td><td>71.8</td><td>35.8</td><td>60.2</td><td>20.1</td><td>25.1</td><td>3.9</td><td>41.1</td><td>20.2</td><td>26.3</td><td>-</td></tr><tr><td>DarkNet53Seg [16]</td><td>49.9</td><td>91.8</td><td>74.6</td><td>64.8</td><td>27.9</td><td>84.1</td><td>86.4</td><td>25.5</td><td>34.7</td><td>22.6</td><td>22.6</td><td>78.3</td><td>50.1</td><td>64.0</td><td>36.2</td><td>33.6</td><td>4.7</td><td>55.0</td><td>38.9</td><td>52.2</td><td>-</td></tr><tr><td>RangeNet3++ [45]</td><td>52.2</td><td>91.8</td><td>75.2</td><td>65.0</td><td>27.8</td><td>87.4</td><td>91.4</td><td>25.7</td><td>25.7</td><td>34.4</td><td>23.0</td><td>80.5</td><td>55.1</td><td>64.6</td><td>38.3</td><td>38.8</td><td>4.8</td><td>58.6</td><td>47.9</td><td>55.9</td><td>83.3</td></tr><tr><td>3D-MiniNet [62]</td><td>55.8</td><td>91.6</td><td>74.5</td><td>64.2</td><td>25.4</td><td>89.4</td><td>90.5</td><td>28.5</td><td>42.3</td><td>42.1</td><td>29.4</td><td>82.8</td><td>60.8</td><td>66.7</td><td>47.8</td><td>44.1</td><td>14.5</td><td>60.8</td><td>48.0</td><td>56.6</td><td>-</td></tr><tr><td>SqueezeSegV3 [8]</td><td>55.9</td><td>91.7</td><td>74.8</td><td>63.4</td><td>26.4</td><td>89.0</td><td>92.5</td><td>29.6</td><td>38.7</td><td>36.5</td><td>33.0</td><td>82.0</td><td>58.7</td><td>65.4</td><td>45.6</td><td>46.2</td><td>20.1</td><td>59.4</td><td>49.6</td><td>58.9</td><td>238</td></tr><tr><td>PointNet++ [25]</td><td>20.1</td><td>72.0</td><td>41.8</td><td>18.7</td><td>5.6</td><td>62.3</td><td>53.7</td><td>0.9</td><td>1.9</td><td>0.2</td><td>0.2</td><td>46.5</td><td>13.8</td><td>30.0</td><td>0.9</td><td>1.0</td><td>0.0</td><td>16.9</td><td>6.0</td><td>8.9</td><td>5900</td></tr><tr><td>TangentConv [36]</td><td>40.9</td><td>83.9</td><td>63.9</td><td>33.4</td><td>15.4</td><td>83.4</td><td>90.8</td><td>15.2</td><td>2.7</td><td>16.5</td><td>12.1</td><td>79.5</td><td>49.3</td><td>58.1</td><td>23.0</td><td>28.4</td><td>8.1</td><td>49.0</td><td>35.8</td><td>28.5</td><td>3000</td></tr><tr><td>PointASNL [31]</td><td>46.8</td><td>87.4</td><td>74.3</td><td>24.3</td><td>1.8</td><td>83.1</td><td>87.9</td><td>39.0</td><td>0.0</td><td>25.1</td><td>29.2</td><td>84.1</td><td>52.2</td><td>70.6</td><td>34.2</td><td>57.6</td><td>0.0</td><td>43.9</td><td>57.8</td><td>36.9</td><td>-</td></tr><tr><td>RandLA-Net [1]</td><td>55.9</td><td>90.5</td><td>74.0</td><td>61.8</td><td>24.5</td><td>89.7</td><td>94.2</td><td>43.9</td><td>29.8</td><td>32.2</td><td>39.1</td><td>83.8</td><td>63.6</td><td>68.6</td><td>48.4</td><td>47.4</td><td>9.4</td><td>60.4</td><td>51.0</td><td>50.7</td><td>880</td></tr><tr><td>KPConv [29]</td><td>58.8</td><td>90.3</td><td>72.7</td><td>61.3</td><td>31.5</td><td>90.5</td><td>95.0</td><td>33.4</td><td>30.2</td><td>42.5</td><td>44.3</td><td>84.8</td><td>69.2</td><td>69.1</td><td>61.5</td><td>61.6</td><td>11.8</td><td>64.2</td><td>56.4</td><td>47.4</td><td>-</td></tr><tr><td>PolarNet [63]</td><td>54.3</td><td>90.8</td><td>74.4</td><td>61.7</td><td>21.7</td><td>90.0</td><td>93.8</td><td>22.9</td><td>40.3</td><td>30.1</td><td>28.5</td><td>84.0</td><td>65.5</td><td>67.8</td><td>43.2</td><td>40.2</td><td>5.6</td><td>61.3</td><td>51.8</td><td>57.5</td><td>62</td></tr><tr><td>JS3C-Net [2]</td><td>66.0</td><td>88.9</td><td>72.1</td><td>61.9</td><td>31.9</td><td>92.5</td><td>95.8</td><td>54.3</td><td>59.3</td><td>52.9</td><td>46.0</td><td>84.5</td><td>69.8</td><td>67.9</td><td>69.5</td><td>65.4</td><td>39.9</td><td>70.8</td><td>60.7</td><td>68.7</td><td>471</td></tr><tr><td>SPVNAS [10]</td><td>67.0</td><td>90.2</td><td>75.4</td><td>67.6</td><td>21.8</td><td>91.6</td><td>97.2</td><td>56.6</td><td>50.6</td><td>50.4</td><td>58.0</td><td>86.1</td><td>73.4</td><td>71.0</td><td>67.4</td><td>67.1</td><td>50.3</td><td>66.9</td><td>64.3</td><td>67.3</td><td>259</td></tr><tr><td>Cylinder3D [40]</td><td>68.9</td><td>92.2</td><td>77.0</td><td>65.0</td><td>32.3</td><td>90.7</td><td>97.1</td><td>50.8</td><td>67.6</td><td>63.8</td><td>58.5</td><td>85.6</td><td>72.5</td><td>69.8</td><td>73.7</td><td>69.2</td><td>48.0</td><td>66.5</td><td>62.4</td><td>66.2</td><td>131</td></tr><tr><td>RPVNet [42]</td><td>70.3</td><td>93.4</td><td>80.7</td><td>70.3</td><td>33.3</td><td>93.5</td><td>97.6</td><td>44.2</td><td>68.4</td><td>68.7</td><td>61.1</td><td>86.5</td><td>75.1</td><td>71.7</td><td>75.9</td><td>74.4</td><td>43.4</td><td>72.1</td><td>64.8</td><td>61.4</td><td>168</td></tr><tr><td>\( (\mathrm{AF})^{2} \)-S3Net [41]</td><td>70.8</td><td>92.0</td><td>76.2</td><td>66.8</td><td>45.8</td><td>92.5</td><td>94.3</td><td>40.2</td><td>63.0</td><td>81.4</td><td>40.0</td><td>78.6</td><td>68.0</td><td>63.1</td><td>76.4</td><td>81.7</td><td>77.7</td><td>69.6</td><td>64.0</td><td>73.3</td><td>-</td></tr><tr><td>Baseline</td><td>67.4</td><td>89.8</td><td>73.8</td><td>62.1</td><td>33.5</td><td>91.9</td><td>96.3</td><td>54.9</td><td>51.1</td><td>55.8</td><td>51.6</td><td>86.5</td><td>72.3</td><td>71.3</td><td>76.8</td><td>79.8</td><td>30.3</td><td>68.7</td><td>63.7</td><td>70.2</td><td>62</td></tr><tr><td>2DPASS(Ours)</td><td>72.9</td><td>89.7</td><td>74.7</td><td>67.4</td><td>40.0</td><td>93.5</td><td>97.0</td><td>61.1</td><td>63.6</td><td>63.4</td><td>61.5</td><td>86.2</td><td>73.9</td><td>71.0</td><td>77.9</td><td>81.3</td><td>74.1</td><td>72.9</td><td>65.0</td><td>70.4</td><td>62</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Experiment Setups
+
+Datasets. We extensively evaluate 2DPASS on two large-scale outdoor benchmarks: SemanticKITTI [16] and Nuscene [18]. SemanticKITTI provides dense semantic annotations for each individual scan of sequences 00-10 in KITTI dataset [61]. According to the official setting, sequence 08 is the validation split, while the remaining are the train split. SemanticKITTI uses sequences 11-21 in KITTI as the test set, whose labels are held on for blind online testing<sup>1</sup>. NuScenes contains 1000 scenes which show a great diversity in inner cities traffic and weather conditions. It officially divides the data into 700/150/150 scenes for train/val/test. Similar to SemanticKITTI, the test set of NuScenes is used for online benchmarking<sup>2</sup>. For 2D sensors, KITTI has only two front-view cameras, while NuScenes has six cameras covering the full  $360^{\circ}$  fields of view.
+
+Evaluation Metrics. We evaluate methods mainly using mean intersection over union (mIoU), which is defined as the average IoU over all classes. Additionally, we report the overall accuracy (Acc)/ frequency-weighted IOU (FwIOU) provided by the online leaderboard of two benchmarks. FwIOU is similar to mIoU except that each IoU is weighted by the point-level frequency of its class.
+
+Network Setup. We apply ResNet34 [59] encoder with 2D convolution as the 2D network, where features after each down-sampling layers are extracted to generate 2D features. The 3D encoder is a modified SPVCNN [10] (voxel size 0.1) with fewer parameters, whose hidden dimensions are 64 for SemanticKITTI and 128 for NuScenes to speed up the network. The number of layers  $L$  for MSFSKD
+
+Table 2. Comparison to the state-of-the-art methods on the test set of SemanticKITTI multiple scans challenge. -s indicates static and -m stands for moving.  
+
+<table><tr><td>Method</td><td>mIoU</td><td>Acc</td><td>car-s</td><td>car-m</td><td>truck-s</td><td>truck-m</td><td>other-s</td><td>other-m</td><td>person-s</td><td>person-m</td><td>bicyclist-s</td><td>bicyclist-m</td><td>motorcyclist-s</td><td>motorcyclist-m</td></tr><tr><td>LatticeNet [64]</td><td>45.2</td><td>89.3</td><td>91.1</td><td>54.8</td><td>29.7</td><td>3.5</td><td>23.1</td><td>0.6</td><td>6.8</td><td>49.9</td><td>0.0</td><td>44.6</td><td>0.0</td><td>64.3</td></tr><tr><td>TemporalLidarSeg [65]</td><td>47.0</td><td>89.6</td><td>92.1</td><td>68.2</td><td>39.2</td><td>2.1</td><td>35.0</td><td>12.4</td><td>14.4</td><td>40.4</td><td>0.0</td><td>42.8</td><td>0.0</td><td>12.9</td></tr><tr><td>KPConv [29]</td><td>51.2</td><td>89.3</td><td>93.7</td><td>69.4</td><td>42.5</td><td>5.8</td><td>38.6</td><td>4.7</td><td>21.6</td><td>67.5</td><td>0.0</td><td>67.4</td><td>0.0</td><td>47.2</td></tr><tr><td>Cylinder3D [40]</td><td>52.5</td><td>91.0</td><td>94.6</td><td>74.9</td><td>41.3</td><td>0.0</td><td>38.8</td><td>0.1</td><td>12.5</td><td>65.7</td><td>1.7</td><td>68.3</td><td>0.2</td><td>11.9</td></tr><tr><td>(AF)2-S3Net [41]</td><td>56.9</td><td>88.1</td><td>91.8</td><td>65.3</td><td>15.7</td><td>5.6</td><td>27.5</td><td>3.9</td><td>16.4</td><td>67.6</td><td>15.1</td><td>66.4</td><td>67.1</td><td>59.6</td></tr><tr><td>2DPASS(Ours)</td><td>62.4</td><td>91.4</td><td>96.2</td><td>82.1</td><td>48.2</td><td>16.1</td><td>52.7</td><td>3.8</td><td>35.4</td><td>80.3</td><td>7.9</td><td>71.2</td><td>62.0</td><td>73.1</td></tr></table>
+
+![](images/adaa7f71b168b597ed7c7c279d5abedf770b9b04acba51e623295dc026b96b52.jpg)  
+Fig. 5. Qualitative results of 2DPASS on the validation set of SemanticKITTI. Our baseline has a higher error recognizing small objects and region boundaries, while 2DPASS recognizes small objects better thanks to the prior of 2D modality.
+
+is set to 4 and 6 for SemanticKITTI and NuScenes, respectively. In each scale of knowledge distillation, 2D and 3D features are reduced to 64 dimensions through deconvolution or MLPs. Similarly, the hidden size of MLPs and 2D learner in MSFSKD are identically 64.
+
+Training and Inference Details. We employ the cross-entropy and Lovasz losses as [40] for semantic segmentation. For the knowledge distillation, we set the proportion of segmentation loss and KL divergence as  $1:0.05$ . Test-time augmentation [40] is applied during the inference. Training details will be introduced in supplementary material.
+
+# 4.2 Benchmark Results
+
+SemanticKITTI. SemanticKITTI evaluates segmentation performance using two settings: single scan and multiple scans. For methods using a single scan as input, moving and non-moving are mapped to a single class. While methods using multiple scans as inputs should distinguish between moving and non-moving
+
+Table 3. Semantic segmentation results on the Nuscenes test benchmark. Only approaches published before 03/08/2022 are compared.  $L$  and  $C$  stand for LiDAR and camera, respectively. (*) The speed reported in PMF [13] is accelerated by TensorRT, and we test their model without such technique in the same environment.  
+
+<table><tr><td>Method</td><td>Input</td><td>mIoU</td><td>fW mIoU</td><td>barrier</td><td>bicycle</td><td>bus</td><td>car</td><td>construction</td><td>motorcycle</td><td>pedestrian</td><td>traffic cone</td><td>trailer</td><td>truck</td><td>drivable</td><td>other flat</td><td>sidewalk</td><td>terrain</td><td>mannade</td><td>vegetation</td><td>speed (ms)</td></tr><tr><td>PolarNet [63]</td><td>L</td><td>69.4</td><td>87.4</td><td>72.2</td><td>16.8</td><td>77.0</td><td>86.5</td><td>51.1</td><td>69.7</td><td>64.8</td><td>54.1</td><td>69.7</td><td>63.5</td><td>96.6</td><td>67.1</td><td>77.7</td><td>72.1</td><td>87.1</td><td>84.5</td><td>-</td></tr><tr><td>JS3C-Net [2]</td><td>L</td><td>73.6</td><td>88.1</td><td>80.1</td><td>26.2</td><td>87.8</td><td>84.5</td><td>55.2</td><td>72.6</td><td>71.3</td><td>66.3</td><td>76.8</td><td>71.2</td><td>96.8</td><td>64.5</td><td>76.9</td><td>74.1</td><td>87.5</td><td>86.1</td><td>-</td></tr><tr><td>Cylinder3D [40]</td><td>L</td><td>77.2</td><td>89.9</td><td>82.8</td><td>29.8</td><td>84.3</td><td>89.4</td><td>63.0</td><td>79.3</td><td>77.2</td><td>73.4</td><td>84.6</td><td>69.1</td><td>97.7</td><td>70.2</td><td>80.3</td><td>75.5</td><td>90.4</td><td>87.6</td><td>63</td></tr><tr><td>AMVNet [39]</td><td>L</td><td>77.3</td><td>90.1</td><td>80.6</td><td>32.0</td><td>81.7</td><td>88.9</td><td>67.1</td><td>84.3</td><td>76.1</td><td>73.5</td><td>84.9</td><td>67.3</td><td>97.5</td><td>67.4</td><td>79.4</td><td>75.5</td><td>91.5</td><td>88.7</td><td>85</td></tr><tr><td>SPVCNN [10]</td><td>L</td><td>77.4</td><td>89.7</td><td>80.0</td><td>30.0</td><td>91.9</td><td>90.8</td><td>64.7</td><td>79.0</td><td>75.6</td><td>70.9</td><td>81.0</td><td>74.6</td><td>97.4</td><td>69.2</td><td>80.0</td><td>76.1</td><td>89.3</td><td>87.1</td><td>63</td></tr><tr><td>\( (AF)^2-S3Net[41] \)</td><td>L</td><td>78.3</td><td>88.5</td><td>78.9</td><td>52.2</td><td>89.9</td><td>84.2</td><td>77.4</td><td>74.3</td><td>77.3</td><td>72.0</td><td>83.9</td><td>73.8</td><td>97.1</td><td>66.5</td><td>77.5</td><td>74.0</td><td>87.7</td><td>86.8</td><td>270</td></tr><tr><td>PMF [13]</td><td>L+C</td><td>77.0</td><td>89.0</td><td>82.0</td><td>40.0</td><td>81.0</td><td>88.0</td><td>64.0</td><td>79.0</td><td>80.0</td><td>76.0</td><td>81.0</td><td>67.0</td><td>97.0</td><td>68.0</td><td>78.0</td><td>74.0</td><td>90.0</td><td>88.0</td><td>125*</td></tr><tr><td>2D3DNet [66]</td><td>L+C</td><td>80.0</td><td>90.1</td><td>83.0</td><td>59.4</td><td>88.0</td><td>85.1</td><td>63.7</td><td>84.4</td><td>82.0</td><td>76.0</td><td>84.8</td><td>71.9</td><td>96.9</td><td>67.4</td><td>79.8</td><td>76.0</td><td>92.1</td><td>89.2</td><td>-</td></tr><tr><td>Baseline</td><td>L</td><td>77.6</td><td>88.5</td><td>80.8</td><td>37.9</td><td>92.7</td><td>90.5</td><td>65.4</td><td>77.6</td><td>71.5</td><td>70.9</td><td>83.1</td><td>75.3</td><td>97.0</td><td>69.3</td><td>78.1</td><td>75.6</td><td>89.1</td><td>86.8</td><td>44</td></tr><tr><td>2DPASS(Ours)</td><td>L</td><td>80.8</td><td>90.1</td><td>81.7</td><td>55.3</td><td>92.0</td><td>91.8</td><td>73.3</td><td>86.5</td><td>78.5</td><td>72.5</td><td>84.7</td><td>75.5</td><td>97.6</td><td>69.1</td><td>79.9</td><td>75.5</td><td>90.2</td><td>88.0</td><td>44</td></tr></table>
+
+objects, which is more challenging. All the reported results are from the official blind test competition website of SemanticKITTI.
+
+Tab. 1 shows our performance under the single scan setting. Our baseline without 2DPASS already performs on par with a strong model Cylinder3D [40] while runs at a faster speed. Even so, the application of 2DPASS still brings a significant improvement over the baseline. Thanks to the auxiliary knowledge distillation, 2DPASS does not put any extra burden on the original model and thus does not sacrifice the running speed of the baseline. Overall, 2DPASS achieves the best result in terms of mIoU and running speed, outperforming the state-of-the-art (i.e.,  $(\mathrm{AF})^{2}$ -S3Net [41]) by  $2.1\%$ . The visualization results on SemanticKITTI single scan are shown in Fig. 5.
+
+Tab. 2 reports the results under the multiple scans setting. The mIoU and overall accuracy are calculated over all 25 classes. Due to the limited space, we only report the per-class IOUs for dynamic objects with non-moving/moving properties. Under this challenge setting, 2DPASS surprisingly surpasses previous approaches with even larger margins, i.e., achieving better mIoU (5.5% improvement over  $(\mathrm{AF})^{2}$ -S3Net [41]) and overall accuracy.
+
+NuScenes. The results on NuScenes are reported in Tab. 3, where 2DPASS achieves the 1st place as well. Note that we only include published works in Tab. 3 and the results are directly taken from the official leaderboard of NuScenes, where our model also ranks the 3rd place with slight disadvantage when considering unpublished works. Besides surpassing all single-modal methods, 2DPASS surprisingly outperforms those fusion-based approaches (the last two rows in Tab. 3). Note that NuScenes provides images covering the whole FOV of the LiDAR, and fusion-based approaches achieve such results by using both point clouds and image features during the inference. In contrast, our method only takes point clouds as input.
+
+Table 4. Comparison with different knowledge distillation.  
+
+<table><tr><td>Method</td><td>SemanticKITTI</td></tr><tr><td>Hinton et.al. [46]</td><td>66.34</td></tr><tr><td>Huang et.al. [67]</td><td>66.46</td></tr><tr><td>Yang et.al. [68]</td><td>66.75</td></tr><tr><td>xMUDA [17]</td><td>67.88</td></tr><tr><td>2DPASS</td><td>69.32</td></tr></table>
+
+Table 5. Ablation study on the SemanticKITTI validation set.  
+
+<table><tr><td>baseline</td><td colspan="3">MSFSKD
+KL Div Modality Fusion 2D Learner</td><td>SemanticKITTI</td></tr><tr><td>✓</td><td></td><td></td><td></td><td>65.58</td></tr><tr><td>✓</td><td>✓</td><td></td><td></td><td>66.34</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td></td><td>69.13</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>69.32</td></tr></table>
+
+# 4.3 Comprehensive Analysis
+
+Comparing with Other Knowledge Distillation. To further verify the effectiveness of our fusion-to-single knowledge distillation paradigm upon common teach-student architecture and other cross-modal manners, we compare 2DPASS with typical approaches of knowledge transfer in Tab. 4, where we utilize these methods in each scale for fair comparison. Among all the methods, Hinton et.al. [46], Huang et.al. [67] and Yang et.al. [68] are pure knowledge distillation designs, where the former is the pioneer for the research field and the latter is newly proposed. As shown in the Tab. 4, pure knowledge distillation manners cannot be directly adopted on the LiDAR semantic segmentation, and their improvement upon the baseline model is limited. Recently, [17] adopts cross-modal feature alignment technique in the task of domain adaptation on semantic segmentation. However, their improvement is still marginal. To the end, in the Tab. 4, 2DPASS significantly performs better, which illustrates the effectiveness of our multi-scale fusion-to-single knowledge distillation (MSFSKD).
+
+Design Analysis of MSFSKD. Tab. 5 demonstrates the ablation study on SemanticKITTI validation set. As shown in the table, our baseline only achieves a lower result of  $65.58\mathrm{mIoU}$ . Note that simply using feature alignment between two modalities cannot effectively improve the result, where the metric of mIoU will be only increased to 66.34. After using 2D-3D fusion in each knowledge distillation scale, there is a significant improvement to 69.13. This improvement mainly comes from the knowledge provided by the stronger fusion prediction. Finally, we find out that 2D learner design can slightly improve the performance by about  $0.2\%$ . Note that the results on SemanticKITTI validation set is lower than that on benchmark since small object category (i.e., motocyclist) only occupies a small proportion.
+
+Distance-based Evaluation. We investigate how segmentation is affected by distance of the points to the ego-vehicle, and compare 2DPASS, current state-of-the-art and the baseline on the SemanticKITTI validation set. Fig. 6 (a) illustrates the mIoU of 2DPASS as opposed to the baseline and  $(\mathrm{AF})^2$ -S3Net. The results of all the methods get worse by increasing the distance since points are relatively sparse in the long distance. 2DPASS improves the performance greatly within  $10m$ , i.e., from 61.2 to 89.1, which is the best distance for the camera to capture objects' color and texture. There is also a significant improvement upon  $(\mathrm{AF})^2$ -S3Net within this distance, i.e., 84.4 v.s. 89.1.
+
+![](images/f6773aa2ce54dbdeb52ddd503eed0c9e1ce5cf0283cfb8268c48606c1005a030.jpg)  
+(a)
+
+![](images/480a8259b797a0e638dfc8e70ed3577cdb4445f031f1d3799656d303159e1d67.jpg)  
+(b)  
+Fig.6. Extensive experiment results. The part (a) shows the results on SemanticKITTI validation set with different distance-range. Part (b) demonstrates the results before and after exploiting 2DPASS on MinkowskiNet [10] and SPVCNN [10].
+
+Generality. We show our 2DPASS can be a "model-independent" training scheme that boosts the performance of other networks. We additionally trained two open-sourced baselines, i.e., MinkowskiNet and SPVCNN implemented in [10] with 2DPASS. During the experiment, we keep all the setups the same except for the 2D-related components. As shown in Fig. 6 (b), 2DPASS improves the former one from 63.1 to 66.2 and the latter from 63.8 to 66.9. These results sufficiently demonstrate the effectiveness and generality of 2DPASS.
+
+# 5 Conclusion
+
+This work proposes the 2D Priors Assisted Semantic Segmentation (2DPASS), a general training scheme, to boost the performance of LiDAR point cloud semantic segmentation via 2D prior-related knowledge distillation. By leveraging an auxiliary modal fusion and knowledge distillation in a multi-scale manner, 2DPASS acquires richer semantic and structural information from the multimodal data, effectively enhancing the performance of a pure 3D network. Eventually, it achieves the state-of-the-arts on two large-scale benchmarks (i.e., SemanticKITTI and NuScenes). We believe that our work can be applied to a wider range of other scenarios in the future, such as 3D detection and tracking.
+
+Acknowledgment. This work was supported in part by NSFC-Youth 61902335, by Key Area R&D Program of Guangdong Province with grant No.2018B030338 001, by the National Key R&D Program of China with grant No.2018YFB1800 800, by the Basic Research Project No. HZQB-KCZYZ-2021067 of Hetao Shenzhen HK S&T Cooperation Zone, by Shenzhen Outstanding Talents Training Fund, by Guangdong Research Project No.2017ZT07X152, by Guangdong Regional Joint Fund-Key Projects 2019B1515120039, by the NSFC 61931024&8192 2046, by zelixir biotechnology company Fund, by Tencent Open Fund, and by High-Performance Computing Portal under the administration of the Information Technology Services Office (ITSO) at CUHKSZ.
+
+# References
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+# 3D Clothed Human Reconstruction in the Wild
+
+Gyeongsik Moon $^{1*}$ , Hyeongjin Nam $^{2*}$ , Takaaki Shiratori $^{1}$ , and Kyoung Mu Lee $^{2,3}$
+
+Meta Reality Labs Research
+
+$^{2}$  Dept. of ECE & ASRI, Seoul National University, Korea
+
+$^{3}$  IPAI, Seoul National University, Korea
+
+{mks0601,tshiratori}@fb.com,{namhjsnu28,kyoungmu}@snu.ac.kr
+
+Abstract. Although much progress has been made in 3D clothed human reconstruction, most of the existing methods fail to produce robust results from in-the-wild images, which contain diverse human poses and appearances. This is mainly due to the large domain gap between training datasets and in-the-wild datasets. The training datasets are usually synthetic ones, which contain rendered images from GT 3D scans. However, such datasets contain simple human poses and less natural image appearances compared to those of real in-the-wild datasets, which makes generalization of it to in-the-wild images extremely challenging. To resolve this issue, in this work, we propose ClothWild, a 3D clothed human reconstruction framework that firstly addresses the robustness on in-the-wild images. First, for the robustness to the domain gap, we propose a weakly supervised pipeline that is trainable with 2D supervision targets of in-the-wild datasets. Second, we design a DensePose-based loss function to reduce ambiguities of the weak supervision. Extensive empirical tests on several public in-the-wild datasets demonstrate that our proposed ClothWild produces much more accurate and robust results than the state-of-the-art methods. The codes are available in here.
+
+# 1 Introduction
+
+3D clothed human reconstruction aims to reconstruct humans with various poses, body shapes, and clothes in 3D space from a single image. It is an essential task for various applications, such as 3D avatars and virtual try-on. Most of the recent 3D clothed human reconstruction methods [33,37,2,34,15,13,14,6,17,7,3] require 3D scans for the training; hence, they are trained on synthetic datasets, which consist of 3D scans [31,4] and rendered images from the scans. Although significant progress has been made by utilizing such synthetic datasets, all of them fail to produce robust results on in-the-wild images.
+
+The in-the-wild images are taken in our daily environments, such as cluttered offices and concert halls, and have diverse human poses, appearances, and severe human occlusions. On the other hand, the 3D scans are captured from restricted
+
+![](images/9c5ca17fa931132fc17cddf041c67fcd81f07344e84b7ff4896c85c33e6efb8d.jpg)  
+(a) Domain gap (pose, appearance, occlusion, etc)
+
+![](images/add192f5d134d573d2f515c255b8f5a9f4ba9ac5eafbf6b9ac931e255f2d25e7.jpg)  
+(b) Qualitative comparison  
+Fig. 1. (a) Domain gap between rendered images and in-the-wild images. (b) Due to the large domain gap, existing methods (e.g. PIFuHD [34] and BCNet [17]) fail on in-the-wild images, while our ClothWild successfully reconstructs. Colors of reconstructed 3D clothes are manually assigned to represent cloth types.
+
+environments, such as motion capture studios. To ease 3D scan capturing, subjects are often instructed to take simple poses. Therefore, rendered images from the 3D scans have artificial appearances and simple human poses compared to those of in-the-wild images, as shown in Fig. 1(a). This large domain gap between in-the-wild and rendered images makes the methods trained on synthetic datasets fail to generalize to in-the-wild images, as shown in Fig. 1(b).
+
+For the robustness to the domain gap, we present ClothWild, a 3D clothed human reconstruction framework that leverages a weak-supervision strategy to learn 3D clothed humans from in-the-wild datasets. Here, the weak supervision means that the supervision target is not full 3D data, but 2D data defined only in the single-camera viewpoint (i.e., 2D cloth segmentations [9]). For the weak-supervision strategy, our reconstruction pipeline is divided into two networks: a cloth generative model and a regressor, ClothNet. The cloth generative model generates 3D cloth geometry around the T-posed 3D body of SMPL [23] from the cloth latent codes, where the 3D cloth geometry is animatable using a skinning function of SMPL. It is fully supervised on the synthetic datasets beforehand, and we freeze it during the weakly supervised training. Although the cloth generative model covers various cloth types and styles, it does not cover complicated human poses and diverse image appearances of in-the-wild datasets. For the robustness to such in-the-wild datasets, we design a regressor, called ClothNet, and weakly supervise it on in-the-wild datasets to predict latent codes of the cloth generative model. The final output can be obtained by passing the output of the ClothNet to the cloth generative model. As the cloth generative model is fixed during the weakly supervised training, we can protect the generative model from being hurt by imperfect 2D supervision targets of in-the-wild datasets, while making our ClothNet robust to diverse human poses and image appearances of in-the-wild images. Our weak-supervision strategy can recover full 3D geometry only from the evidence of the single-camera viewpoint by regularizing the predicted cloth latent codes to be in the cloth latent space.
+
+Note that such regularization is not possible for previous approaches [33,34,36,3] as they do not model cloth latent space.
+
+For the weak supervision, we can enforce the projection of 3D reconstruction results to be close to 2D supervision targets (i.e., cloth segmentations [9]) of in-the-wild datasets. However, naively adopting such a strategy has three difficulties in learning 3D clothed humans. First, the 2D supervision targets provide information for only a single camera viewpoint. Therefore, there is no guarantee that the weak supervision can recover 3D geometry of other viewpoints. Second, severe depth ambiguity of the cloth segmentations hampers learning accurate 3D clothes. Since each pixel in cloth segmentations corresponds to innumerable points in 3D space, it is difficult to specify 3D points corresponding to clothes. Third, pixel-level misalignment can occur between projected 3D reconstruction results and 2D cloth segmentation. As camera parameters are not available for in-the-wild images, we should additionally predict the camera parameters to project the 3D clothed humans. This is a highly ill-posed problem as various combinations of camera parameters and 3D clothed humans correspond to the same 2D cloth segmentation. Due to the ill-posedness, camera prediction can often fail, which results in the wrong projection to the 2D image space.
+
+In this regard, we design a DensePose-based loss function to resolve the issues which can occur when weakly supervising 3D clothed human reconstructions based on their projections. DensePose [10] informs where each human pixel corresponds to the 3D human body surface of SMPL. Using DensePose, our DensePose-based loss function obtains 3D points that correspond to the cloth segmentations around the SMPL surface. Then, it enforces such 3D points to be a part of the 3D cloth surface. As the DensePose effectively limits possible positions of 3D points around the SMPL surface, it significantly reduces the depth ambiguity. In addition, DensePose is already aligned with cloth segmentations in the image space; therefore, we do not have to predict camera parameters and project 3D reconstructions to the 2D image space. Hence, our DensePose-based loss function does not suffer from pixel-level misalignment.
+
+We show that the proposed ClothWild produces far more robust and better results than the previous 3D clothed human reconstruction methods from in-the-wild images. As robustness on in-the-wild images has not been extensively studied in the 3D human reconstruction community, we believe ours can give useful insights to the following research.
+
+Our contributions can be summarized as follows.
+
+- We present ClothWild, which reconstructs robust 3D clothed humans from a single in-the-wild image. To the best of our knowledge, it is the first to explicitly address robustness on in-the-wild images.  
+- For the robustness to the domain gap between synthesized and in-the-wild datasets, we propose a weakly supervised pipeline that is trainable with 2D supervision targets of in-the-wild datasets.  
+- Our DensePose-based loss function resolves the ambiguities of weak supervision by effectively limiting possible positions of 3D points using DensePose.  
+- ClothWild largely outperforms previous methods on in-the-wild images.
+
+# 2 Related works
+
+3D clothed human reconstruction. Varol et al. [35] and Jackson et al. [16] proposed volumetric regression networks that directly predict a voxel representation of a 3D clothed human from a single image. Saito et al. [33,34] and He et al. [13] presented a pixel-aligned implicit function that predicts 3D occupancy fields of clothed humans. Alldieck et al. [1] utilized displacement vectors between a human body surface and clothes for reconstruction. Alldieck et al. [2] presented a method estimating normal and displacement maps on top of a human body model, such as SMPL [23]. Bhatnagar et al. [6] and Jiang et al. [17] presented 3D clothed human reconstruction systems that predict PCA coefficients of cloth generative model space. Huang et al. [15] and He et al. [14] handle implicit function representation of 3D clothed humans in arbitrary poses and produce animatable reconstruction results. Xiu et al. [36] presented ICON, which utilizes local features for robust 3D clothed human reconstruction. Alldieck et al. [3] proposed PHORHUM to photorealistically reconstruct the 3D geometry and appearance of a dressed person. As all of the above methods require 3D scans for training, they are trained on synthetic datasets, which consist of 3D scans and the rendered images from the scans. As the synthetic datasets mainly contain simple human poses with artificial appearances, the methods trained on such datasets fail to generalize to in-the-wild datasets. In contrast, our ClothWild is the first work that can be weakly supervised with 2D supervision targets of in-the-wild datasets, which results in robust outputs on in-the-wild images.
+
+Recently, Corona et al. [7] presented a fitting framework that fits their cloth generative model to 2D cloth segmentations. Although their fitting framework handles reconstruction on in-the-wild images, there are two major differences from ours. First, their fitting framework is not a learning-based system; hence, it cannot utilize image features. As it only relies on 2D cloth segmentations for fitting, it produces wrong results when cloth segmentations are not available due to truncations or occlusions. On the other hand, ClothWild is a learning-based system that utilizes image features; hence, it is much more robust to occlusions and truncations by considering contextual information of image features, as shown in Fig. 5. Second, their fitting framework suffers from the depth ambiguity and pixel-misalignment as described in Section 1, since it projects 3D reconstruction results to the 2D image space and compares the projected one with 2D cloth segmentations. On the other hand, ClothWild leverages our newly proposed DensePose-based loss function. Hence, ClothWild suffers much less from the depth ambiguity and is free from the pixel-misalignment issue. The detailed description of it is provided in Section 6.4.
+
+3D cloth generative model. 3D cloth generative models parameterize 3D clothes by embedding acquired 3D cloth scans in a latent space. Bhatnagar et al. [6] and Jiang et al. [17] used the PCA algorithm to represent clothes in a latent space. Ma et al. [25] used a graph convolutional neural network-based generative model to embed the clothes in a latent space. Bertiche et al. [5] embedded displacements from the human body surface to registered cloth meshes and cloth types as latent codes. Patel et al. [29] decomposed 3D clothes into low-
+
+and high-frequency components and represented them as a function of human poses, shapes and cloth styles. Corona et al. [7] presented a cloth generative model, SMPLicit, which embeds 3D clothes as latent codes that represent cloth styles and cloth cuts. SMPLicit covers a wide variety of clothes, which differ in their geometric properties, such as sleeve length and looseness. In our framework, we employ SMPLicit as a cloth generative model.
+
+3D human body reconstruction. 3D human body reconstruction methods predict parameters of the SMPL [23] body model from a single image. They perform well on in-the-wild images via weakly supervision with 2D GTs of in-the-wild images. Kanazawa et al. [18] proposed an end-to-end framework that utilizes adversarial loss to reconstruct plausible human bodies. Kolotouros et al. [20] combined a regressor and iterative fitting framework. Moon et al. [27] presented Pose2Pose utilizing local and global image features. We use Pose2Pose as an off-the-shelf 3D human body reconstruction method of our ClothWild due to its superior accuracy compared to other works.
+
+# 3 ClothWild
+
+Fig. 2 shows the overall pipeline of our ClothWild, which consists of ClothNet and BodyNet. We provide a detailed description of each module below.
+
+# 3.1 ClothNet
+
+Given an input image  $\mathbf{I}$ , ClothNet predicts cloth existence scores  $\mathbf{c} = (c_{1},\dots ,c_{N_{c}})$  a set of cloth latent codes  $\{\mathbf{z}_i\}_{i = 1}^{N_c}$ , and a gender  $\mathbf{g}$ .  $N_{c} = 5$  denotes the number of clothes we consider, which include upper cloth, coat, pants, skirt, and shoes. The  $i$ th cloth existence score  $c_{i}$  represents the probability that a human is wearing  $i$ th cloth. The  $i$ th cloth latent code  $\mathbf{z}_i\in \mathbb{R}^{d_i}$  represents a low-dimensional code of  $i$ th 3D cloth, embedded in latent space of the 3D cloth model, SMPLicit [7]. We set  $d_{i} = 4$  for shoes and  $d_{i} = 18$  for other clothes following SMPLicit [7]. The gender  $\mathbf{g}\in \mathbb{R}^2$  is a one-hot encoded vector that represents a label for the appropriate human body model (i.e., male and female). We use ResNet-50 [12] to extract an image feature vector  $\mathbf{f}\in \mathbb{R}^{2048}$  from the input image after removing the fully-connected layer of the last part of the original ResNet. The image feature vector  $\mathbf{f}$  is passed into a fully-connected layer, followed by a sigmoid activation function, to predict the cloth existence scores  $\mathbf{c}$ . In addition, we use  $N_{c}$  fully-connected layers to predict a set of cloth latent codes  $\{\mathbf{z}_i\}_{i = 1}^{N_c}$  from  $\mathbf{f}$ , where  $i$ th fully-connected layer predicts the latent codes of  $i$ th cloth,  $\mathbf{z}_i$ . Finally, another fully-connected layer, followed by a softmax activation function, predicts the gender from  $\mathbf{f}$ . The predicted set of cloth latent codes  $\{\mathbf{z}_i\}_{i = 1}^{N_c}$  and gender  $\mathbf{g}$  are passed to SMPLicit.
+
+# 3.2 BodyNet
+
+The BodyNet takes the input image  $\mathbf{I}$  and predicts a shape parameter  $\beta \in \mathbb{R}^{10}$  and a pose parameter  $\theta \in \mathbb{R}^{72}$  of the SMPL human body model [23]. The shape
+
+![](images/476c956787c7f5960d74845bec381734573d407667c18a68bdbdb1f334ef5fca.jpg)  
+Fig. 2. The overall pipeline of ClothWild. ClothNet predicts cloth latent codes of the cloth generative model, SMPLicit, and produces an unsigned distance field by passing the codes to the SMPLicit. BodyNet predicts SMPL pose and shape parameters. At the training stage, the unsigned distance field is supervised with a combination of Dense-Pose and cloth segmentations with our DensePose-based loss function. At the inference stage, the final 3D clothed human reconstruction is obtained through Marching Cubes and pose deformation steps.
+
+parameter  $\beta$  represents PCA coefficients of T-posed human body shape space, and the pose parameter  $\theta$  represents 3D rotations of human body joints. The shape parameter  $\beta$  is forwarded to SMPLicit that is described in Section 3.3. The pose parameter  $\theta$  is used in the inference stage, of which detailed descriptions are in Section 3.4. We use Pose2Pose [27] as the BodyNet, which achieves state-of-the-art performance on in-the-wild benchmarks.
+
+# 3.3 Cloth generative model
+
+The cloth generative model embeds 3D clothes in the latent space. We use SMPLicit [7] as our cloth generative model, which produces a continuous unsigned distance field of a 3D cloth from a cloth latent code, gender, and human shape. We use the pre-trained SMPLicit and fix it while training ClothWild. Given the cloth latent codes  $\{\mathbf{z}_i\}_{i=1}^{N_c}$ , gender  $\mathbf{g}$ , and human shape  $\beta$ , SMPLicit outputs the unsigned distance field for the  $i$ th cloth, as follows:
+
+$$
+C (\mathbf {x}, \mathbf {z} _ {i}, \mathbf {g}, \beta) \longrightarrow \mathbb {R} ^ {+}, \quad i = 1, 2, \dots , N _ {c}, \tag {1}
+$$
+
+where  $\mathbf{x} \in \mathbb{R}^3$  is a 3D query point in a canonical 3D space where the human is in a T-pose. In the canonical 3D space, a T-posed naked human body mesh is founded by deriving from the gender  $\mathbf{g}$  and human shape  $\beta$ . Around the T-posed naked human body, a set of 3D query points indicate the closest distance to the 3D cloth surface as the unsigned distance field. The above step only proceeds for the  $i$ th cloth when its cloth existence score  $c_i$  is larger than a threshold, which we empirically set to 0.25.
+
+# 3.4 Integration of clothes and body
+
+Marching Cubes. In the inference stage, we obtain cloth meshes using Marching Cubes [24], which extracts a mesh of an isosurface from a 3D discrete scalar field, such as the unsigned distance field. We calculate the unsigned distance field by densely sampling 3D query points in the 3D space and forwarding them to the SMPlicit model. Then, we extract cloth meshes on the T-posed from the unsigned distance field by Marching Cubes. At the end of this step, we obtain a T-posed clothed human mesh that comprises a T-posed naked human body mesh and cloth meshes.
+
+Pose deformation. Finally, we apply the pose deformation to the naked human body mesh and cloth meshes of the T-posed clothed human mesh, respectively, following Corona et al. [7]. To deform the naked human body, we use the skinning deformation of SMPL with the pose parameter  $\theta$  predicted by the BodyNet. To deform cloth meshes, we allocate each cloth vertex of the cloth meshes to its closest naked human body vertex and apply the same pose deformation of the human body vertex. Such the SMPL-driven pose deformation has the strength that the reconstruction results can be deformed with an arbitrary pose instead of the predicted pose, which enables an animation.
+
+# 4 Learning from in-the-wild datasets
+
+In this section, we describe loss functions to train our framework on in-the-wild datasets without any 3D scan GTs. ClothNet is the only trainable module in our framework, and all other modules, including BodyNet and SMPLicit, are fixed during the training. The overall loss function is
+
+$$
+L _ {\text {t o t a l}} = \lambda_ {\mathrm {d p}} L _ {\mathrm {d p}} + \lambda_ {\mathrm {r e g}} L _ {\mathrm {r e g}} + \lambda_ {\text {e x i s t}} L _ {\text {e x i s t}} + \lambda_ {\text {g e n d e r}} L _ {\text {g e n d e r}}, \tag {2}
+$$
+
+where  $\lambda_{\mathrm{dp}} = 1$ ,  $\lambda_{\mathrm{reg}} = 0.1$ ,  $\lambda_{\mathrm{exist}} = 0.01$ , and  $\lambda_{\mathrm{gender}} = 0.01$ .
+
+# 4.1 DensePose-based loss function
+
+The DensePose-based loss function  $L_{\mathrm{dp}}$ , which uses a combination of cloth segmentations and DensePose, makes reconstructed clothes close to cloth segmentations. The loss is computed in three steps: cloth-to-body mapping, query point selection, and loss calculation.
+
+Cloth-to-body mapping. In the first step, we place pixels of 2D cloth segmentations in 3D space. Fig. 3 shows the procedure of the cloth-to-body mapping step. In the area where DensePose is defined, we sample 2D cloth points  $\{\mathbf{p}_k^{2\mathrm{D}}\}_{k = 1}^{N_p}$  on cloth segmentations, where  $N_{p} = 196$  is the number of the sampled 2D points. We utilize DensePose that informs where each human pixel corresponds to the 3D human body surface of SMPL to map the 2D cloth points  $\{\mathbf{p}_k^{2\mathrm{D}}\}_{k = 1}^{N_p}$  to 3D cloth points  $\{\mathbf{p}_k^{3\mathrm{D}}\}_{k = 1}^{N_p}$  on the T-posed human body surface. Each of the mapped 3D cloth points represents a cloth label (e.g., upper cloth, pants, or non-cloth) elicited from the cloth segmentations.
+
+![](images/d50afecff2ae6b872684a0ac9d8fd4f2cd72755ae54ec12b72439f40013ae9d1.jpg)  
+(a) image
+
+![](images/877a1dc420c3ad9acb8c91061813914799d86bf47f8a139c17ade6aff3b7440d.jpg)  
+(b) DensePose
+
+![](images/a98f08023b2511f2e9f765e505cfa9360cfa7999631506122bbb83bb67566501.jpg)  
+(c) cloth segm.
+
+![](images/0968072ce3f80a847cebe6ee2677bf9a48548cbafeabc8992f4816e3c3ea2dd4.jpg)  
+(d) T-posed human body  
+Fig. 3. The procedure of the cloth-to-body mapping step. From (a) an in-the-wild image, we exploit 2D supervision targets: (b) DensePose and (c) cloth segmentations. (d) We sample a 2D cloth point  $\mathbf{p}^{2\mathrm{D}}$  of the cloth segmentations, and map the 2D cloth point  $\mathbf{p}^{2\mathrm{D}}$  to the 3D cloth point  $\mathbf{p}^{3\mathrm{D}}$  on the T-posed human body surface. We select the 3D query point  $\mathbf{x}$  within constant distance from the 3D cloth points, and the DensePose-based loss function supervises the selected 3D query point.
+
+Query point selection. In this step, we select 3D query points around the T-posed human body for loss calculation. For each cloth type, we uniformly sample 3D query points at a resolution of  $21 \times 21 \times 21$  from a 3D bounding box, where the sampling strategy is described in the supplementary material. We select 3D query points  $\{\mathbf{x}_j\}_{j=1}^{N_q}$  within a distance threshold  $\tau$  from the 3D cloth points  $\{\mathbf{p}_k^{3\mathrm{D}}\}_{k=1}^{N_p}$  on the 3D human body surface, where  $N_q$  is the number of the sampled 3D query points. The distance threshold  $\tau$  is set  $10\mathrm{cm}$  for the coat and  $3\mathrm{cm}$  for others.
+
+Loss calculation. To the formal description of the loss function, we define a cloth binary function  $S_{i}(\cdot)$ . If the closest 3D cloth point from a 3D query point  $\mathbf{x}_j$  belongs to the  $i$ th cloth,  $S_{i}(\mathbf{x}_{j})$  becomes 1 and 0 else. With the cloth binary function  $S_{i}(\cdot)$ , the DensePose-based loss function follows:
+
+$$
+\begin{array}{l} L _ {\mathrm {d p}} = \frac {1}{N _ {c} N _ {q}} \sum_ {i = 1} ^ {N _ {c}} \sum_ {j = 1} ^ {N _ {q}} \left(S _ {i} \left(\mathbf {x} _ {j}\right) \mid C \left(\mathbf {x} _ {j}, \mathbf {z} _ {i}, \mathbf {g}, \beta\right) \right| \tag {3} \\ + (1 - S _ {i} (\mathbf {x} _ {j})) | C (\mathbf {x} _ {j}, \mathbf {z} _ {i}, \mathbf {g}, \beta) - d _ {\max} |), \\ \end{array}
+$$
+
+where  $d_{\mathrm{max}}$  is a maximum cut-off distance of the unsigned distance fields, set to 0.01 for shoes and 0.1 for others. The first loss term forces the 3D query point  $\mathbf{x}$  to belong to  $i$ th cloth by making its unsigned distance close to zero when the 3D point matches cloth label  $i$ . The second loss term forces the 3D query point  $\mathbf{x}$  to not belong to  $i$ th cloth when the 3D point does not match cloth label  $i$ . As a result, the DensePose-based loss function supervises the 3D query points established by the above steps without the depth ambiguity of the cloth segmentations.
+
+# 4.2 Other loss functions
+
+Regularization loss. The regularization loss function  $L_{\mathrm{reg}}$  makes predicted latent codes close to the mean of the cloth latent space, which results in plau
+
+sible 3D clothes. The regularization loss function is defined as follows:  $L_{\mathrm{reg}} = \sum_{i=1}^{N_c} \alpha_i \| \mathbf{z}_i \|_2$ , where  $\alpha_i$  is set 0.1 for shoes and 1.0 for others. As the DensePose-based loss supervises 3D points that belong to partial areas of clothes, there is a possibility to learn implausible clothes (e.g., overly thick cloth or torn cloth) that only rely on the areas. The regularization loss prevents it by constraining output clothes to be close to the mean.
+
+- Cloth existence loss. For the cloth existence prediction, we calculate a binary cross-entropy as follows:  $L_{\mathrm{exist}} = -\frac{1}{N_c}\sum_{i = 1}^{N_c}(c_i^*\log c_i + (1 - c_i^*)(1 - \log c_i))$ , where the asterisk denotes the groundtruth.
+
+Gender classification loss. For the gender classification, we calculate a cross-entropy as follows:  $L_{\mathrm{gender}} = -\left(g_m^*\log g_m + g_f^*\log g_f\right)$ , where  $g_{m}$  and  $g_{f} = 1 - g_{m}$  is the probability of being male and female of the input human, respectively. The asterisk denotes the groundtruth.
+
+# 5 Implementation details
+
+PyTorch [28] is used for implementation. The backbone part is initialized with the publicly released ResNet50 [12] pre-trained on ImageNet [32]. The weights are updated by Adam optimizer [19] with a mini-batch size of 8. The human body region is cropped using a GT box in both training and testing stages following previous works [33,34]. The cropped image is resized to  $256 \times 192$ . Data augmentations, including scaling, rotation, random horizontal flip, and color jittering, are performed in training. The initial learning rate is set to  $10^{-4}$  and reduced by a factor of 10 after the 5th epoch. We train the model for 8 epochs with an NVIDIA GTX 2080 Ti GPU.
+
+# 6 Experiment
+
+# 6.1 Datasets
+
+MSCOCO. MSCOCO [22] is a large-scale in-the-wild dataset, which provides in-the-wild images with diverse human poses and appearances. We train ours on the training split and evaluate on the validation split. We use outputs of the DensePose regression model [10] for the DensePose-based loss function as GT. DensePose provides only sparse annotations. For the cloth segmentations, we use LIP [9], which contains cloth segmentation annotations of MSCOCO images. We acquire gender annotations by running Homogenus [30] on MSCOCO and use its predictions as supervision targets.
+
+DeepFashion2. DeepFashion2 [8] is a comprehensive dataset that focuses on capturing clothed humans with a wide variety of cloth styles, and we use it as the additional train set along with the MSCOCO. We obtain DensePose and gender annotations by the same procedure as MSCOCO, described above. For cloth segmentation annotations, we use SCHP [21] to obtain cloth segmentations of the dataset.
+
+![](images/c3def66fe7aa5beee36ccf93a23f13b61ff28329ee9b3ad13616c64ec8088a58.jpg)  
+(a) Chamfer distance (CD)
+
+![](images/3a614817ab631f027c650ae12ac364ee38954fe39d336bf6540a185743573824.jpg)  
+(b) Body-cloth correspondence (BCC)  
+Fig. 4. Descriptions of our evaluation metrics. (a) Chamfer distance (CD) is a 3D distance between reconstruction and GT surface. It is measured after aligning them based on each 2D projection. (b) BCC is a proportion of 3D points that have correctly matched cloth types. The purple and red points on the right body surface are 3D points that are mapped from cloth segmentations using DensePose. The purple points represent 3D points that are covered with a correct type of reconstructed 3D clothes, while the red points represent others. It is measured after normalizing 3D poses.
+
+3DPW. 3DPW [26] is an in-the-wild dataset, and we use it only for the evaluation purpose after sampling every 25th frame of test set videos. We use two items of 3DPW: T-posed clothed human meshes registered to each subject's 3D scan and SMPL pose parameters of humans in images. As the registered 3D clothed human meshes have the same mesh topology as that of SMPL body mesh, we deform the registered ones with SMPL pose parameters following the same skinning algorithm of SMPL. We use the posed registered meshes as GT 3D clothed humans of 3DPW.
+
+# 6.2 Evaluation metrics
+
+Fig. 4 shows our two evaluation metrics. Since there has been no work dealing with 3D clothed human reconstruction on the in-the-wild dataset, we propose two evaluation metrics in the following.
+
+Chamfer distance (CD). Chamfer Distance (CD) is a 3D distance between the 3D clothed human reconstruction and the GT surface, widely used in previous works [33,34,17]. Before measurement, we rigidly align global rotation, scale, and translation of the reconstruction to the GT surface. The alignment is not trivial as there are no semantically matching pairs between the reconstruction and GT surface. For the alignment, we assume that the same human parts (e.g., shoulders and knees) of both the reconstruction and the GT surface are projected to the same pixel in the image. We rasterize both the reconstruction and GT surface and pair two vertices (i.e., one from the reconstruction and the other from the GT surface) that correspond to the same pixel. Based on the matching pairs, we align the reconstruction and measure the Chamfer distance (CD) in millimeter. More details of CD metric is described in the supplementary material.
+
+Body-cloth correspondence (BCC). Body-cloth correspondence (BCC) is a proportion of 3D points that have correct cloth types on the T-posed naked human body surface. It only considers 3D cloth predictions, excluding 3D human
+
+![](images/327823a875f3ae7c595a37501499138515cc05348952284f4c2911873c7ccd9f.jpg)  
+Fig. 5. Qualitative result comparisons on MSCOCO validation set. PIFu additionally uses human segmentation obtained from Mask R-CNN [11] for reconstruction. SMPLicit fits use SMPL parameter and cloth segmentations obtained from our BodyNet and SCHP [21], respectively. Colors of reconstructed 3D clothes are manually assigned to represent cloth types.
+
+pose predictions. With cloth-to-body mapping described in Section 4.1, we map 2D points in a GT cloth segmentation for one cloth label to 3D points on the T-posed human body surface with the cloth-to-body mapping. We consider 3D points are correctly covered ones if the distance between them and reconstructed cloth is shorter than  $3\mathrm{cm}$ . We calculate the proportion of the correctly covered points and average the proportions for all cloth labels (e.g., upper cloth, pants, or non-cloth). This metric is used for methods that support cloth-to-body mapping based on the T-posed naked body of SMPL. For example, as PIFu [33], PIFuHD [34], and ICON [36] do not tell us cloth type which points belong, we could not evaluate them using the BCC metric.
+
+# 6.3 Comparison with state-of-the-art methods
+
+We compare our ClothWild with recent 3D clothed human reconstruction methods: PIFu [33], PIFuHD [34], ICON [36], BCNet [17], and the fitting framework of SMPLicit [7]. For the inference, PIFu requires human segmentation, and the fitting framework of SMPLicit requires SMPL parameter and cloth segmentations. We provide human segmentation, obtained by Mask R-CNN [11], to PIFu. The same SMPL parameters of our BodyNet and cloth segmentations from SCHP [21] are provided to the SMPLicit fitting framework. All of their results are obtained by using their officially released codes and pre-trained weights. Please note that
+
+Table 1. CD comparison with state-of-the-art methods on 3DPW. Methods with * and † additionally use human segmentation and cloth segmentations as an input for the inference, respectively.  
+
+<table><tr><td>Methods</td><td>CD ↓</td></tr><tr><td>PIFuHD [34]</td><td>137.50</td></tr><tr><td>BCNet [17]</td><td>118.75</td></tr><tr><td>ICON [36]</td><td>75.00</td></tr><tr><td>PIFu [33]*</td><td>67.25</td></tr><tr><td>SMPLicit fits [7]†</td><td>45.66</td></tr><tr><td>ClothWild (Ours)</td><td>40.34</td></tr></table>
+
+Table 2. BCC comparison with state-of-the-art methods on MSCOCO. Methods with † additionally use cloth segmentations as an input for the inference.  
+
+<table><tr><td>Methods</td><td>upper body</td><td>lower body</td><td>non-cloth</td><td>average</td></tr><tr><td>BCNet [17]</td><td>0.415</td><td>0.729</td><td>0.800</td><td>0.648</td></tr><tr><td>SMPLicit fits [7]†</td><td>0.645</td><td>0.493</td><td>0.961</td><td>0.700</td></tr><tr><td>ClothWild (Ours)</td><td>0.830</td><td>0.820</td><td>0.887</td><td>0.846</td></tr></table>
+
+the fitting framework of SMPLlicit fits their cloth latent codes to the cloth segmentations. We call the fitting results of their framework as SMPLlicit fits.
+
+Qualitative results. Fig. 5 shows that our ClothWild produces much better reconstruction results than previous state-of-the-art 3D clothed human reconstruction methods on MSCOCO [22]. PIFu [33], PIFuHD [34], and ICON [36] suffer from undesirable results on in-the-wild images with diverse human poses and appearances, especially in the occluded human scene. ICON produces better results than PIFu and PIFuHD; however, it still suffers from missing human parts. On the other hand, our ClothWild successfully reconstructs invisible parts of clothed humans even when the input human is occluded or truncated. Like PIFu, PIFuHD, and ICON, BCNet [17] also requires 3D scans for the training. Accordingly, it is trained on synthetic datasets without in-the-wild images, the reason for weak results on in-the-wild images. SMPLcit fits [7] does not utilize image features, which provide contextual information of occluded and truncated human parts. Hence, it cannot reconstruct clothes at occluded or truncated human parts. In addition, the depth ambiguity and pixel-level misalignment of their silhouette-based loss function make the fitting attain the incorrect cloth styles. Our ClothWild resolves such issues by utilizing image features and the DensePose-based loss function.
+
+Quantitative results. Table 1 shows that ClothWild achieves the best CD on 3DPW [22]. Also, Table 2 shows that ClothWild produces much better BCC than previous methods on MSCOCO [22]. Unlike ours, BCNet [17] handles only three cloth types (i.e., upper cloth, pants, and skirt). Hence, we categorize upper
+
+Table 3. Running time (seconds per image) comparisons.  
+
+<table><tr><td>PIFuHD [34] 
+(CVPR 20)</td><td>SMPLicit fits [7] 
+(CVPR 21)</td><td>ICON [36] 
+(CVPR 22)</td><td>ClothWild 
+(Ours)</td></tr><tr><td>20.43</td><td>105.43</td><td>87.88</td><td>10.21</td></tr></table>
+
+cloth and coat into the upper body and pants and skirts into the lower body for a fair comparison. Then, we calculate BCC only for such clothes and exclude shoes. As the BCC evaluates only predicted 3D clothes without considering 3D human poses, the comparisons show that our ClothWild predicts the cloth styles of the image accurately, such as sleeveless and length of upper cloth and pants.
+
+Running time. Table 3 shows that ours takes the shortest computational time to process a single image than recently presented works. The running times are measured in the same environment with Intel Xeon Gold 6248R CPU and NVIDIA RTX 2080 Ti GPU. We exclude pre-processing stages, such as 3D body pose estimation, human segmentation, and cloth segmentations. The fitting framework of SMPLicit takes a much longer time, although it is based on the same cloth generative model [7] as ours. This is because the fitting framework of SMPLicit forwards the cloth generative model about 200 iterations to fit its cloth latent codes to cloth segmentations. ICON suffers from a similar problem as it iteratively fits initial results based on normal maps and silhouettes. In contrast, our ClothWild performs the feed-forward only a single time. The above methods, including ours, require Marching Cubes to obtain final 3D geometry, one of the main bottlenecks of the running time. The running time of ClothWild's each component will be reported in the supplementary material.
+
+# 6.4 Ablation study
+
+Effectiveness of DensePose-based loss. The top block of Table 4 shows that the proposed DensePose-based loss has a vital role in properly learning 3D clothed humans. Fig. 6 (a) and (b) additionally shows the effectiveness of our DensePose-based loss compared to previous silhouette loss [7]. The silhouette loss enforces 3D cloth reconstruction results to be projected onto the 2D cloth segmentations. The top block of the table shows that using the DensePose-based loss achieves better BCC than using the silhouette loss. The silhouette loss suffers from depth ambiguity and pixel-misalignment, as described in Section 1. On the other hand, the proposed DensePose-based suffers much less from the depth ambiguity and is free from the misalignment issue as it designates accurate 3D points around the human body surface. Hence, it has significant benefits to learning precise 3D clothes.
+
+Effectiveness of regularization loss. The bottom block of Table 4 shows the effectiveness of the regularization loss. Fig. 6 (b) and (c) additionally demonstrates the effectiveness. The regularization loss makes predicted clothes be in the latent space of the 3D cloth model. It is necessary because 2D supervision
+
+Table 4. BCC comparison among different loss configurations.  
+
+<table><tr><td>DensePose</td><td>Silhouette</td><td>Regularization</td><td>BCC ↑</td></tr><tr><td colspan="4">* Effectiveness of DensePose-based loss</td></tr><tr><td>X</td><td>✓</td><td>✓</td><td>0.644</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>0.684</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>0.689 (Ours)</td></tr><tr><td colspan="4">* Effectiveness of regularization loss</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>0.381</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>0.689 (Ours)</td></tr></table>
+
+![](images/3306190b2bbafef600db7e814eb18c7f54d70bd7dff450e386cef2254f737390.jpg)  
+(a) DP. + Reg. (Ours)
+
+![](images/6548c184764e9aa7f129012dc6c19fadd7ef207803155098145e53c50274bb8f.jpg)  
+(b) Sil.  $+\mathrm{Reg}$
+
+![](images/e9ddd4d620bbabce94d6ce9c716081f643c600a64c888cb031499c2d82d2ecfd.jpg)  
+(c) Ours-Reg.
+
+![](images/bd0f708e913392f7a5d781be2c972e6951252c01c179cb6c2f1aca9c5c84a090.jpg)
+
+![](images/de714b1944fb030ef16775c8cab18b526eb6cb9c0fa8f40e77f64a5259fff9f7.jpg)  
+Fig.6. Comparison of three different loss configurations: (a) our proposed loss configuration, (b) replacing our DensePose-based loss with the silhouette loss, and (c) removing the regularization loss.
+
+![](images/6b536f944556f5c2740db1ca3b33cb9f5307e7fa21a10ef8eac93c254366cc2f.jpg)  
+(a) DP. + Reg. (Ours)
+
+![](images/9fea6368455fdaec36e0a2506b4f3bdaf78ad788c8501bb230ff236cfac64a46.jpg)  
+(b) Sil. + Reg.
+
+![](images/a20c968481dd90a2cd40f326ca406c5ea4756db7089753affa70ced1de4a8235.jpg)  
+(c) Ours-Reg.
+
+targets, such as cloth segmentations, only provide information of partial areas of clothes. Such limited information can lead our ClothWild to produce implausible clothes, such as overly thick cloth and torn cloth. The regularization loss prevents such improper learning and encourages reconstructing reasonable 3D clothes, despite only information of partial areas of clothes.
+
+# 7 Conclusion
+
+We propose ClothWild, a 3D clothed human reconstruction framework that produces significantly robust results from in-the-wild images. For the robustness to the domain gap between synthesized and in-the-wild datasets, we propose a weakly supervised pipeline and a DensePose-based loss function. As a result, our ClothWild outperforms previous 3D clothed human reconstruction methods on in-the-wild images.
+
+Acknowledgements. This work was supported in part by IITP grant funded by the Korea government (MSIT) [No. 2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University), No.2022-0-00156], and in part by the Bio & Medical Technology Development Program of NRF funded by the Korean government (MSIT) [No. 2021M3A9E4080782].
+
+# References
+
+1. Alldieck, T., Magnor, M., Bhatnagar, B.L., Theobalt, C., Pons-Moll, G.: Learning to reconstruct people in clothing from a single RGB camera. In: CVPR (2019)  
+2. Alldieck, T., Pons-Moll, G., Theobalt, C., Magnor, M.: Tex2shape: Detailed full human body geometry from a single image. In: ICCV (2019)  
+3. Alldieck, T., Zanfir, M., Sminchisescu, C.: Photorealistic monocular 3D reconstruction of humans wearing clothing. In: CVPR (2022)  
+4. aXYZ: (2018), https://secure.axyz-design.com  
+5. Bertiche, H., Madadi, M., Escalera, S.: CLOTH3D: Clothed 3D humans. In: ECCV (2020)  
+6. Bhatnagar, B.L., Tiwari, G., Theobalt, C., Pons-Moll, G.: Multi-Garment Net: Learning to dress 3D people from images. In: ICCV (2019)  
+7. Corona, E., Pumarola, A., Alenya, G., Pons-Moll, G., Moreno-Noguer, F.: SMPLicit: Topology-aware generative model for clothed people. In: CVPR (2021)  
+8. Ge, Y., Zhang, R., Wang, X., Tang, X., Luo, P.: Deepfashion2: A versatile benchmark for detection, pose estimation, segmentation and re-identification of clothing images. In: CVPR (2019)  
+9. Gong, K., Liang, X., Zhang, D., Shen, X., Lin, L.: Look into person: Self-supervised structure-sensitive learning and a new benchmark for human parsing. In: CVPR (2017)  
+10. Güler, R.A., Neverova, N., Kokkinos, I.: DensePose: Dense human pose estimation in the wild. In: CVPR (2018)  
+1. He, K., Gkioxari, G., Dolar, P., Girshick, R.: Mask R-CNN. In: ICCV (2017)  
+2. He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: CVPR (2016)  
+3. He, T., Collomosse, J., Jin, H., Soatto, S.: Geo-PIFu geometry and pixel aligned implicit functions for single-view human reconstruction. In: NeurIPS (2020)  
+4. He, T., Xu, Y., Saito, S., Soatto, S., Tung, T.: ARCH++: Animation-ready clothed human reconstruction revisited. In: ICCV (2021)  
+5. Huang, Z., Xu, Y., Lassner, C., Li, H., Tung, T.: ARCH: Animatable reconstruction of clothed humans. In: CVPR (2020)  
+6. Jackson, A.S., Manafas, C., Tzimiropoulos, G.: 3D human body reconstruction from a single image via volumetric regression. In: ECCV (2018)  
+7. Jiang, B., Zhang, J., Hong, Y., Luo, J., Liu, L., Bao, H.: Bcnet: Learning body and cloth shape from a single image. In: ECCV (2020)  
+8. Kanazawa, A., Black, M.J., Jacobs, D.W., Malik, J.: End-to-end recovery of human shape and pose. In: CVPR (2018)  
+9. Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: ICLR (2014)  
+20. Kolotouros, N., Pavlakos, G., Black, M.J., Daniilidis, K.: Learning to reconstruct 3D human pose and shape via model-fitting in the loop. In: ICCV (2019)  
+21. Li, P., Xu, Y., Wei, Y., Yang, Y.: Self-correction for human parsing. IEEE TPAMI (2020)  
+22. Lin, T.Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Dóllar, P., Zitnick, C.L.: Microsoft COCO: Common objects in context. In: ECCV (2014)  
+23. Loper, M., Mahmood, N., Romero, J., Pons-Moll, G., Black, M.J.: SMPL: A skinned multi-person linear model. ACM TOG (2015)  
+24. Lorensen, W.E., Cline, H.E.: Marching Cubes: A high resolution 3D surface construction algorithm. ACM siggraph computer graphics (1987)
+
+25. Ma, Q., Yang, J., Ranjan, A., Pujades, S., Pons-Moll, G., Tang, S., Black, M.J.: Learning to dress 3D people in generative clothing. In: CVPR (2020)  
+26. von Marcard, T., Henschel, R., Black, M.J., Rosenhahn, B., Pons-Moll, G.: Recovering accurate 3D human pose in the wild using imus and a moving camera. In: ECCV (2018)  
+27. Moon, G., Choi, H., Lee, K.M.: Accurate 3d hand pose estimation for whole-body 3d human mesh estimation. In: CVPRW (2022)  
+28. Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., Lerer, A.: Automatic differentiation in pytorch (2017)  
+29. Patel, C., Liao, Z., Pons-Moll, G.: TailorNet: Predicting clothing in 3D as a function of human pose, shape and garment style. In: CVPR (2020)  
+30. Pavlakos, G., Choutas, V., Ghorbani, N., Bolkart, T., Osman, A.A., Tzionas, D., Black, M.J.: Expressive body capture: 3D hands, face, and body from a single image. In: CVPR (2019)  
+31. Renderpeople: (2018), https://renderpeople.com/3d-people  
+32. Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., et al.: ImageNet large scale visual recognition challenge (2015)  
+33. Saito, S., Huang, Z., Natsume, R., Morishima, S., Kanazawa, A., Li, H.: PIFu: Pixel-aligned implicit function for high-resolution clothed human digitization. In: ICCV (2019)  
+34. Saito, S., Simon, T., Saragih, J., Joo, H.: PIFuHD: Multi-level pixel-aligned implicit function for high-resolution 3D human digitization. In: CVPR (2020)  
+35. Varol, G., Ceylan, D., Russell, B., Yang, J., Yumer, E., Laptev, I., Schmid, C.: Bodynet: Volumetric inference of 3D human body shapes. In: ECCV (2018)  
+36. Xiu, Y., Yang, J., Tzionas, D., Black, M.J.: ICON: Implicit Clothed humans Obtained from Normals. In: CVPR (2022)  
+37. Zheng, Z., Yu, T., Wei, Y., Dai, Q., Liu, Y.: DeepHuman: 3D human reconstruction from a single image. In: ICCV (2019)

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+# 3D CoMPaT: Composition of Materials on Parts of 3D Things
+
+Yuchen Li $^{1,\ast}$ , Ujjwal Upadhyay $^{1,\star}$ , Habib Slim $^{1,\ast}$ , Ahmed Abdelreheem $^{1}$ , Arpit Prajapati $^{2}$ , Suhail Pothigara $^{2}$ , Peter Wonka $^{1}$ , Mohamed Elhoseiny $^{1}$
+
+1 KAUST, Thuwal, Saudi Arabia: {firstname_lastname}@kaust.edu.sa  
+$^{2}$  Poly9 Inc., San Francisco, California: {firstname}@polynine.com
+
+Abstract. We present 3D CoMPaT, a richly annotated large-scale dataset of more than 7.19 million rendered compositions of Materials on Parts of 7262 unique 3D Models; 990 compositions per model on average. 3D CoMPaT covers 43 shape categories, 235 unique part names, and 167 unique material classes that can be applied to parts of 3D objects. Each object with the applied part-material compositions is rendered from four equally spaced views as well as four randomized views, leading to a total of 58 million renderings (7.19 million compositions  $\times 8$  views). This dataset primarily focuses on stylizing 3D shapes at part-level with compatible materials. We introduce a new task, called Grounded CoMPaT Recognition (GCR), to collectively recognize and ground compositions of materials on parts of 3D objects. We present two variations of this task and adapt state-of-art 2D/3D deep learning methods to solve the problem as baselines for future research. We hope our work will help ease future research on compositional 3D Vision. The dataset and code are publicly available at https://www.3dcompat-dataset.org/
+
+# 1 Introduction
+
+Various datasets have been proposed to facilitate 3D visual understanding including ShapeNet [4], ModelNet [33], and PartNet [26]. Recently, 3D-FUTURE [12] was proposed, which contains 9,992 industrial 3D CAD shapes of furniture with textures developed by professional designers. Despite these significant efforts to create 3D datasets, current 3D object datasets (e.g., [4,33,26]) and 3D Scene datasets (e.g., [8]) lack part-level material information. The availability of material information has multiple benefits. First, material information provides additional semantic information about an object. Second, material information enables more realistic renderings making the models more suitable for synthetic to real transfer. Third, the same geometric 3D shape can be rendered with different material assignments leading to more variability during training (see Fig. 1).
+
+We introduce a richly annotated large-scale dataset, dubbed as 3D CoMPaT, Compositions of Materials on Parts of 3D Things. The dataset contains more
+
+![](images/bce37e798cef9c9c4fcd60f105da6a2e25a86b9bcde093209c9352d5f82ba123.jpg)  
+Fig. 1: Stylized models in the 3D CoMPaT dataset. We show several compositions of 8 selected models in the dataset, stylized with different materials.
+
+than 7.19 million rendered model styles from 8 views, covers 43 shape categories, 235 unique and distinguishable part names, and 167 unique and distinguishable materials from 10 material classes that can be applied to parts of 3D objects. Each object with the applied part-material compositions is rendered from four equally spaced views, leading to 58 million (7.19 million compositions  $\times 8$  views) images in total. Examples of some rendered compositions and views can be seen in Fig. 1 and 2 respectively.
+
+We start with 7262 unique shapes with a total of 37198 segmented parts (i.e., 5.12 segmented parts per shape on average), and we annotate the list of compatible/applicable materials for each part. Then, we sample a model by enumerating randomly over the compatible materials for each part with a limit of 1000 compositions per shape, leading to 7.19 million compositions of 3D objects. Connection and differences to existing datasets. The proposed dataset is different from the currently available datasets in the literature in the following ways. First, the dataset contains a diverse set of high-quality materials beyond mere texture maps. Second, for each part found in every 3D model, the dataset defines a set of materials that may be applied to this part in that model, allowing us to generate multiple material combinations for a single model (we call each combination a style). The models in 3D-FUTURE [12] and ShapeNet [4] do not have multiple styles, and also, in the ShapeNet dataset, only a small portion of 3D shapes are stylized. The following four key aspects can characterize our 3D CoMPaT dataset in contrast to existing datasets.
+
+$-(a)$  human-generated vs. 3D scanned geometry. For example, ScanNet [8] and Matterport3D [3] datasets are scanned 3D geometry. Conversely, ShapeNet [4] and our 3D CoMPaT dataset are human-generated.  
+$-(b)$  part segmentation information. For some datasets, none or only a subset of the shapes have segmented part information, which is an important aspect of datasets like PartNet [26] and is also a characteristic of our dataset.
+
+![](images/cb245a7fe214b6f0ded6298cd58993f01e0ef40b3ee48b7308f1ec45daaa9070.jpg)  
+Fig. 2: 3D CoMPaT Dataset. Left: Examples of a stylized cabinet. The cabinet has five parts, shelves, drawers, handles, back and side panel, indicated as highlighted. The box below contains the material names for the indicated parts in different stylized cabinets. Middle: Each 3D object is rendered in four canonical and four randomized views. Right: Part segmentation masks for randomly selected shapes from our dataset.
+
+$-(c)$  texture coordinates, textures, and materials. Since stylizing the composition of 3D model parts is at the heart of our work, our models have texture coordinates and material compatibility information to enable high-quality rendering of hundreds of compositions of materials on each shape. This is the most important distinguishing characteristic of our 3D CoMPaT dataset compared to existing datasets. There was some earlier effort to augment a subset of ShapeNet with material information [24]. This dataset has fewer shapes (3080 vs 7262), parts, and materials compared to our work.  
+$-(d)$  automatically generated vs. human-generated information. 3D CoMPaT part names are consistent and come from a list of allowable part names per model category. All models are manually segmented at a part level rather than being segmented with deep learning models like OpenRooms [23]. Furthermore, in 3D CoMPaT all texture coordinates are developed and verified by humans (refer to Sec. 3 for more details).
+
+We validate our dataset with experiments covering main 3D recognition tasks, including 3D object classification, 3D part recognition, and material tagging.
+
+Grounded CoMPaT Recognition (GCR) Task. Finally, we introduce a novel task, dubbed as CoMPaT recognition. It aims at recognizing and grounding the shape category collectively with the part-material pairs associated with the shape, e.g., recognizing that the example in Fig. 2 is a "Cabinet", with a handle made of "shiny nickel (metal)" and a back made of "maple coffee wood".
+
+# Contributions.
+
+- We propose a new dataset of 7.19 million stylized models to study composition of Materials on Parts of 3D Things. Our dataset contains (a) a diverse
+
+set of 167 materials for 3D shapes. (b) The material assignment is done at part-level. (c) Segmentation masks in 2D and 3D are provided, alongside (d) human-verified texture coordinates. We hope this dataset may also facilitate future research on retrieving objects in 3D scenes (e.g., localizing a specific "chair" or "table" from Fig. 2).
+
+- We validate our dataset by a set of experiments involving 2D/3D shape classification, part recognition (detection and segmentation), and material tagging.
+
+- We also propose Grounded CoMPaT Recognition, a novel task of collectively recognizing and grounding compositions of materials on parts of 3D objects. We introduce two variants of this task, and adapt 2D/3D state-of-the-art methods as baselines for this problem.
+
+# 2 Related Work
+
+Datasets of 3D shapes and scenes. ModelNet [33] is a large-scale 3D CAD model dataset covering 40 categories. ShapeNet [4] is a richly-annotated, large-scale repository of shapes with semantic categories and organizes them under the WordNet taxonomy. PartNet [26] assigned rich fine-grained segmentation labels on the part level. Recently, 3D-FUTURE [12] was proposed, which contains 9,992 unique industrial 3D CAD shapes of furniture with high-resolution informative textures developed by professional designers. In contrast to 3D-FUTURE, PartNet, and ShapeNet, where only a small portion of 3D shapes can be assigned materials, our dataset contains 7262 models, all of which can be stylized with different textures. PhotoShape [27] is a dataset similar to ours. More specifically, it uses a technique to automatically apply materials to existing models, mainly from ShapeNet [4]. Some rendered models are not that realistic. The texture coordinates are generated automatically, while ours are human-generated and human-verified. PhotoShape only has a single shape category, i.e., chair, and has only five material classes (leather, fabric, wood, metal, plastic). In comparison, our dataset has 43 shape categories and thirteen material classes (wood, metal, fabric, marble, ceramic, glass, leather, paint, paper, plastic, rubber, granite, wax). OpenRooms [23] is a large dataset containing indoor scenes. The authors automatically segment CAD models of the scenes into parts based on a segmentation model trained on the PartNet dataset. Hence, OpenRooms parts are restricted by the part classes present in PartNet. This also may introduce some segmentation errors in part localization and naming since learned predictions are not as accurate as human annotations. In contrast, our models are manually annotated and verified. Furthermore, our dataset contains models of objects and not indoor scenes, which gives users more flexibility. The major difference between OpenRooms and 3D CoMPaT is that OpenRooms uses scanned geometry captured by sensors while our dataset is manually constructed. We also note that Lin et.al. [24] introduced 3080 stylized models for three categories with five material classes and part information. However, the scale of our dataset is much larger.
+
+Texture Generation. TM-Net [13] is a novel deep generative model that generates meshes with detailed textures and synthesizes plausible textures for a given shape. Their work is inspired by SDM-NET [14]. Their method produces texture maps for each part, which means it works in a part-aware fashion. Each part is represented as a deformed box. They encode geometry and texture separately and learn the texture probability distribution conditioned on the geometry. This allows their method to be a generic framework for different application scenarios.
+
+High-Level 3D Vision. Encouraging progress in 3D scene understanding, ScanNet [8] introduced a large-scale dataset of 1513 real-world scenes. More recently and at the intersection of 3D vision and natural language, ScanRefer [5] and Referit3D [2] datasets were recently introduced on top of ScanNet to study 3D object identification based on free-form natural language descriptions. The detailed composition of the shape category and part-material pairs provided in 3D CoMPaT can serve as a rich semantic description of shapes, and hence may facilitate more fine-grained visual grounding of language referring to 3D objects and scenes.
+
+# 3 3D CoMPaT: Data Collection, Benchmark, and Validation
+
+The 3D CoMPaT dataset collection pipeline comprises three main processes: 3D CAD models collection, materials collection, material assignment, and rendering.
+
+# 3.1 3D CAD Models Collection
+
+3D CoMPaT is based on a collection of 3D CAD models managed by Poly9 Inc.. The initial data has high-quality 3D models, but the part names, segmentation information, class information, and material information is often missing or faulty. Repetitions of the same CAD model may be present, and some CAD models contain a set of similar parts. The team for building 3D CoMPaT consisted of professional CAD modelers, researchers, and crowd-sourced workers from AMT. The process for creating 3D CoMPaT consists of frequent review meetings between researchers and professional modelers discussing issues with part names, shape categories, materials, and shape segmentation continuing for over one year. Based on these reviews, the professional modelers would adapt their processes, such as labeling instructions, or the allowable list of part names. While professional modelers did all the labeling and modeling work, researchers focused on automatic and manual quality control. While ultimately most of the class names, part names, and material assignments had to be changed during our effort, we only selected shapes that already had high geometric quality and texture coordinates so that little effort was needed to fix problems in the geometry. Due to our multi-stage verification process, each 3D shape was manually inspected more than once. Models that failed a stage of the quality control pipeline were sent back to the team of professional modelers.
+
+- (A) Shape and Part Category Labeling: Each 3D CAD model is assigned a shape category label (e.g., chair, desk, table). All models are consistently segmented into parts and every part in every model is assigned a part name (e.g., "seat, back, or legs"). Each part in each 3D CAD model is also assigned a list of compatible material types, e.g., for one particular chair a modeler could assign that the legs of the chair can be made of either metal or wood. Designing a consistent list of allowable part names for each shape category is a considerable effort. We sourced information from online retailers, other datasets such as PartNet [26], names used in 3D CAD models, crowdsourcing services and our own experience. In particular, we started with a smaller subset of shapes and some initial labeling of part names to verify these annotations using Amazon Mechanical Turk (AMT). Even though our goal to fix the list of allowable part names early in the process, we had to adapt the list over time in the review meetings as new shapes were being processed.  
+- (B) Duplicates and Near-duplicates Removal: Some 3D CAD models are repeated more than once or contain multiple instances of the same model (e.g., a 3D CAD model representing a set of vases with different sizes). we implemented an automatic procedure to detect duplicates and near-duplicates to remove them from the dataset.  
+- (C) Part Segmentation: Every CAD model should be part-segmented; i.e., every CAD model consists of a set of separated part meshes. We manually check the segmentation of each shape in a 3D viewer and correct them if they are not consistent with the defined part categories.  
+- (D) Texture Coordinates Quality Check. For a proper material assignment, the quality of texture coordinates was verified qualitatively. We followed two strategies. First, we overlay different materials over each part and check how it renders in different settings (we used an increasing level of light bounces to see how textures look). Second, we applied checkered textures to visually inspect the texture coordinates; as illustrated in Fig. 4 (left). For the evaluation of the 3D geometry, we checked that the models are watertight and that they have outward-pointing normals.
+
+![](images/574e9b9a5a383ea61710d51eb7149e470d95b867f6155420b089a37043bca9c2.jpg)  
+Fig. 4: Left: Examples of texture coordinate checks. Right: Blender rendering environment. The environment contains three light sources (a directional light source and three area lights) and a plane at the bottom. The 3D CAD model is normalized, centered on the origin, and placed on the plane.
+
+![](images/661c8d9ffed2ee32aae1c5e6fe9f48933f2725257806a7478dc823b654178285.jpg)  
+Fig.3: Examples of materials found in the dataset. We show examples of wood, metal, and fabric materials in the first, second, and third rows respectively.
+
+Crowdsourcing the Verification of Shape and Category labels. To verify annotated class names of a given 3D model, we asked five MTurk participants to choose from the following four options: (1) "yes, I would name the model the same," "(2) yes, but I would have given the model a different name," (3) "no, this is a wrong name (please specify a name)" or (4) "no, the model cannot be given a specific name." We used a similar interface to verify the part class names; the part-annotation and model-annotation verification interface used in our AMT experiments is shown in the supplementary material [22].
+
+# 3.2 Materials Collection.
+
+For materials, we use the free and open-source Nvidia vMaterials library. The Nvidia vMaterials library has over 2150 real-world materials and continues to grow. These materials are defined by the Nvidia MDL specification, allowing PBR materials with higher visual quality than basic materials based on diffuse textures. Materials from this library have the infinite tiling feature, allowing textures to be spread across large areas without a clear repeating pattern. The library provides class labels for every material organized in a hierarchical tree (e.g., an antique oxidized aluminum is a rusted aluminum metal). Tab. 1 presents the count of distinct subtypes for each material class (10). In total, the number of materials in our dataset is 167.
+
+We manually inspected these materials, ignoring materials that may not be realistic when applied to specific parts. For example, a fabric material that has a mesh-like appearance is not suitable to be used for chair cushions (see Fig. 3).
+
+Table 1: 3D CoMPaT material classes and number of materials per class.
+
+<table><tr><td>Wood</td><td>Metal</td><td>Fabric</td><td>Marble</td><td>Ceramic</td><td>Glass</td></tr><tr><td>40</td><td>32</td><td>35</td><td>14</td><td>8</td><td>6</td></tr><tr><td>Leather</td><td>Plastic</td><td>Rubber</td><td>Granite</td><td>Wax</td><td>Total</td></tr><tr><td>13</td><td>10</td><td>5</td><td>3</td><td>1</td><td>167</td></tr></table>
+
+# 3.3 Part-Material Assignment
+
+One of the novel aspects of 3D CoMPaT is the presence of material compatibility information for parts present in each 3D model. Human workers conduct the process of material assignment. This process was realized at the instance level, i.e., the shape and parts of each 3D model were considered to compose them with appropriate materials. For example, the legs of one particular table could be assigned either metal or wood and the legs of another table could be assigned wood or plastic. The assignment space for this process is  $7262 \times 5.12 \times 10$  (where 10 is the number of material categories). We only assign material classes. For example, all 32 metals can be assigned to a shape part in the material sampling stage if a metal is a possible assignment. We also control compatibility to some extent through grouping information. For example, all table legs have to be assigned the same material. However, we do not explicitly control complex material combinations, as this is hard to integrate into the currently used 3D modeler. This has advantages and disadvantages. An advantage of the current solution is that we allow a greater variety of models which can be a good source
+
+of data augmentation. As a disadvantage, several sampled stylized models may not be aesthetically beautiful. Overall, we believe that the consistency of the material assignments is better controlled at sampling process that assigns materials. For example, the sampler could select the same material for chair back and legs significantly more often than different materials. We believe that is more efficient and compatible with our current approach while an explicit control of material combinations suffers from an exponential explosion of possible combinations that need to be controlled. To analyze this issue further will require a significant effort in synthetic to real transfer which we leave to future work.
+
+# 3.4 Rendering Composition of Materials on Parts of the Collected CAD Models
+
+Once material assignments for each part are available, this information is used to sample materials from the assigned categories. For example, if a tabletop is made of wood, we can sample one of the 40 wood types, like teak, oak, hazelnut, etc. In what follows, we refer to the combination of these materials assigned to parts of a given CAD model as a composition. The application of one such composition to the CAD model is called a style of the model.
+
+For each 3D CAD model, we randomly select a material for each part from the list of its compatible materials. We sample at most 1000 styles per 3D model.
+
+- (A) Rendering. We use Blender [1] to render each CAD model into RGB images from 8 different views, with the camera placed far enough for the entire object to be visible. For the lighting setup, we use three light sources; see Fig. 4 (right). We render each stylized model in 4 standard views (front and back with default model orientation and front left and back right with the model rotated with 30 degrees around the z (up) axis). Further, we also render each model from 4 random views. The camera for random views is parameterized with elevation angle  $\theta_{cam}$  (in degree)  $\in$  [0, 90], while keeping the  $x, y$  same. The model is rotated parameterized with random rotation angle  $\theta_{model}$  (in degree)  $\in$  [0, 360].  
+- (B) Segmentation and Depth Maps. The rendered images in the 3D CoMPaT dataset will be accompanied by corresponding segmentation maps and depth maps. These maps will be rendered with the same four fixed views and four random views.  
+- (C) Stylized 3D models. We plan to release the stylized 3D models, which will enable their use in many 3D computer vision applications, including retrieval, reconstruction, and 3D generation.
+
+# 3.5 Dataset Statistics
+
+3D CoMPaT contains 7262 unique 3D shapes covering 43 shape categories. The top 6 classes are (table, tray, bowl, chair, desk, cabinet). The dataset contains 37198 part instances covering 235 part classes, and 167 different materials from 10 material classes. The top 5 material classes are (metal, wood, fabric, paint, marble); please see the supplementary for more details about shape classes, the number of object instances per shape class, part, and material classes [22]. In
+
+Table 2: Comparison of 3D CoMPaT with other datasets in the literature. To our knowledge, our dataset is the first one to have many different materials applied to different parts in the same 3D model.  $\checkmark^{*}$ : only a subset of shapes are textured and the remaining shapes are with unidentified textures. ?: unknown. HVT stands for Human Verified Textures.  
+
+<table><tr><td>Benchmarks</td><td>Shapes No.</td><td>Categories</td><td>Material Classes</td><td>Materials</td><td>Shape Source</td><td>Stylized Models</td><td>HVT</td><td>Parts per Shape</td></tr><tr><td>3D-Future[12]</td><td>9992</td><td>34</td><td>14(+1)</td><td>?</td><td>industry</td><td>102972</td><td>×</td><td>10.3</td></tr><tr><td>3D Front[11]</td><td>13151</td><td>50</td><td>23</td><td>?</td><td>3D-Future</td><td>13151</td><td>×</td><td>6.5</td></tr><tr><td>PhotoShape[27]</td><td>5830</td><td>1</td><td>5</td><td>363</td><td>ShapeNet, industry</td><td>11000</td><td>×</td><td>1-3</td></tr><tr><td>ShapeNetCore[4]</td><td>51300</td><td>55</td><td>×</td><td>✓*</td><td>online, crowdsourced</td><td>×</td><td>×</td><td>×</td></tr><tr><td>ShapeNetSem[4]</td><td>~12000</td><td>270</td><td>×</td><td>✓*</td><td>online</td><td>×</td><td>×</td><td>×</td></tr><tr><td>ShapeNetPart[35]</td><td>31963</td><td>16</td><td>×</td><td>✓*</td><td>online</td><td>×</td><td>×</td><td>2.92</td></tr><tr><td>ModelNet[33]</td><td>151128</td><td>660</td><td>×</td><td>×</td><td>online</td><td>×</td><td>×</td><td>×</td></tr><tr><td>ObjectScans[6]</td><td>1,900</td><td>44</td><td>×</td><td>×</td><td>Scans</td><td>×</td><td>×</td><td>×</td></tr><tr><td>PartNet[26]</td><td>26671</td><td>24</td><td>×</td><td>×</td><td>ShapeNet</td><td>×</td><td>×</td><td>21.5</td></tr><tr><td>Lin et.al.,[24]</td><td>3080+115</td><td>3</td><td>5</td><td>×</td><td>online, ShapeNet</td><td>3080+115?</td><td>×</td><td>5.16</td></tr><tr><td>3D CoMPaT</td><td>7262</td><td>43</td><td>10</td><td>167</td><td>industry</td><td>7.19 million</td><td>✓</td><td>5.12</td></tr></table>
+
+Table 2, we show how our proposed dataset has more variety in the number of materials and the number of materials assignments (styles) than the currently available datasets.
+
+Fig. 5a and Fig. 5b show the frequency with which different shape classes and parts occur in the dataset. We observe that the dataset has an uneven distribution as some parts, model classes, and materials are more frequent than others. Fig. 5c shows the distribution of subsets of parts in different model classes. The size of the bubble represents the occurrences of a part in a certain model class, which we further categorized as very frequent, frequent, or less frequent. It can be inferred from Fig. 5c that some parts are centered around a single "model class"; for example, the "top" is mostly centered around the "table" class, indicating the very frequent tabletop part in the dataset. Some parts have high variability across different models. From Fig. 5c, we can see that "leg" is a part that frequently occurs in "table" models but also in several other models like "chair", "cabinet", and "desk".
+
+Table 3 shows the statistics of 3D CoMPaT in different aspects including parts, model classes, material and styles. The scale of material compositions on model parts is a key difference of 3D CoMPaT when compared to existing datasets, enabling styling of all existing model classes with different part-material combinations that are compatible. As we pointed out earlier, some materials cannot be applied to some parts (e.g., wood in exchange for glass).
+
+Table 3: Dataset statistics.  
+
+<table><tr><td>Total Number of Models</td><td>7262</td></tr><tr><td>Total Number of Model Classes</td><td>43</td></tr><tr><td>Total Number of Parts</td><td>37198</td></tr><tr><td>Total Number of Parts Classes</td><td>235</td></tr><tr><td>Minimum Number of Parts Per Model</td><td>1</td></tr><tr><td>Maximum Number of Parts Per Model</td><td>17</td></tr><tr><td>Average Number of Parts Per Model</td><td>5.12</td></tr><tr><td>Average Compositions Per Model</td><td>990</td></tr><tr><td>Total Number of Materials</td><td>167</td></tr><tr><td>Total Number of Stylized Models</td><td>7.19 million</td></tr></table>
+
+# 3.6 Dataset Split and Non-Compositional Validation Experiments
+
+We create training, validation, and test splits for the renderings. All renderings and stylized versions of a 3D shape have to be assigned together to either training, validation, or test. Therefore, the splits are defined on shapes to prevent data leak. The training set has 5597 shapes, the test set 924 shapes and the validation set 477 shapes. Despite compositional recognition being the focus of our work, a variety of standard tasks can benefit from our proposed dataset, including 3D object classification, 3D semantic segmentation, shape classification, image shape retrieval, shape reconstruction from single/multiple images. We conducted experiments on some of these tasks to validate the properties of our proposed 3D CoMPaT dataset.
+
+3D Shape and Part Classification. Our dataset has an uneven distribution, so some models and parts have more examples and hence help in better generalization. Some parts and models resemble their more frequently occurring counterparts (e.g. jar and container), making classification challenging. Note that there is some intersection between model class and part names, namely
+
+![](images/2c9d5299b3818b5ad59a06e71a3d7248999d4c828532051acf1a2e8124f48113.jpg)
+
+![](images/855cf10d1a2d5f7b2a15ef3b48c628e6927c054257f3abdf849c36f119725e93.jpg)
+
+![](images/8c911039c8b91f48f9dcf897d4f54169d0b254eef5cc5c3be5b0e6df15be4ddf.jpg)  
+(a)  
+(c)  
+Fig.5: 3D CoMPaT. (a) Number of samples per shape category. (b) Number of samples per part class. (c) Frequency of occurrence of part and model pair. Note that both visualizations do not cover all shape and part labels because of space constraints; more details are provided in the supplementary [22].
+
+basket, bowl, candle holder, glass, shelf, table, tray, vase. This is because some models are parts of other larger models (e.g. shelf as part of shelf structure or cabinet). The presence of various
+
+materials to style the 3D object gives our dataset an edge over other existing datasets. We conduct shape classification experiments for the 3D models and the 3D parts. Results are reported in Table 4, where we benchmark Point Cloud Transformer (PCT) [16], DGCNN [32], and Pointnet++ [29] on shape classification and Pointnet++ and PCT
+
+Table 4: 3D CoMPaT non-compositional validation experiments.  
+
+<table><tr><td>Architecture</td><td>Task</td><td>Test Performance</td></tr><tr><td>Pointnet++[29]</td><td>3D Shape Classification</td><td>57.95% Accuracy</td></tr><tr><td>DGCNN[32]</td><td></td><td>68.32% Accuracy</td></tr><tr><td>PCT[15]</td><td></td><td>69.09% Accuracy</td></tr><tr><td>Pointnet++[29]</td><td>3D Part Classification</td><td>24.18% Accuracy</td></tr><tr><td>PCT[15]</td><td></td><td>37.37% Accuracy</td></tr><tr><td>BPNet 2D [19]</td><td>2D Material Segmentation</td><td>35.75% mIOU</td></tr><tr><td>BPNet 3D [19]</td><td>3D Material Segmentation</td><td>17.03% mIOU</td></tr><tr><td>ResNet50 [18]</td><td>2D Material Tagging</td><td>0.53 F1, 0.67 AP</td></tr><tr><td>ResNet50 [18]</td><td>2D Shape Classification</td><td>76.82% Accuracy</td></tr></table>
+
+on part classification; see results in Table 4.
+
+2D and 3D Material Segmentation. We benchmark BPNet [19], a 2D and 3D joint UNet, for our 2D and 3D Material Segmentation in Table 4.
+
+2D Material Tagging/ Shape Classification. We use a ResNet50 [18] backbone for encoding the rendered images, to train a multi-label classifier over the 167 materials in the rendered images. The F1 score and average precision were 0.53 and 0.67 respectively. We also report a 2D shape classification performance of  $76.82\%$  using ResNet50, on 50 canonical compositions; see Table 4.
+
+Sim2Real 3D shape recognition. We trained a PointMLP [25] model on ModelNet40 and 3D CoM-PaT (with only one sampled composition/shape) and evaluated on the hardest variant of ScanOb-
+
+jectNN [31] (without finetuning). Table 5 shows 3D shape classification results for 9 classes. Results show that pretraining on 3DCoMPaT shapes leads to better generalization to real-world data than pretraining on ModelNet40.
+
+Table 5: Sim2Real transfer: Accuracy results for PointMLP [25] trained on ModelNet40 and 3DCoM-PaT (1 random composition), on ScanObjectNN's hardest variant.  
+
+<table><tr><td>Dataset</td><td>Acc. (%)</td></tr><tr><td>ModelNet40</td><td>24.33</td></tr><tr><td>3DCoMPaT</td><td>29.21</td></tr></table>
+
+# 4 2D/3D Grounded CoMPaT Recognition (GCR) Task, Baselines, and Results
+
+The goal of compositional modeling on 3D CoMPaT is to recognize the entire composition of materials on parts of a given 3D model. More specifically, we aim at correctly predicting the object category, part categories and the associated material for every part in the 3D model. Fig. 6 visualizes some ground truth and prediction examples for this task. This task is challenging because  $96.31\%$  of the compositional frames at test time are unseen. We define a 3D CoMPaT compositional frame as a shape category and a set of part-material categorical pairs. Two compositional frames are different if they differ in a single part or material assignment. In standard recognition settings the model only has to select
+
+![](images/36a419e56be64a92857a7f6b7df679abd60e3d786d31d15bbf33bdd90693b0dc.jpg)  
+Fig. 6: Example realized materials over parts of certain model classes. Below each image is a table where the first row is the model class, the left column is part names, and the right column is material name for those parts. On the left outlined in gold are ground truth. The output from our material recognition, part recognition, and model recognition model is on the right. Incorrect part names and material names are highlighted in red, whereas correct ones are green.
+
+the correct shape class and examples of all shape classes have been seen before. By contrast, several proposed metrics require the correct recognition of compositional frames. The number of compositional frames is much higher than the number of shape classes and for most compositional frames in the test set there are no examples in the training set. This can be related to zero-shot recognition, which aims at recognizing unseen categories that are defined by unseen compositions of visual attributes (e.g., [21,10,9,17]). This is also connected to existing yet different compositional 2D computer vision tasks, including situation recognition [28,34], which aims at identifying an activity like "surfing" in an image, the engaged entities with their roles (e.g., "agent: woman", "tool: surfboard", and "place: ocean"), and bounding-box groundings of entities.
+
+Metrics. Inspired by the metrics proposed in [34,28] for compositional situation recognition of activities in images, we define the compositional metrics of the 2D/3D Grounded CoMPaT Recognition (GCR) task as follows:
+
+(a) Shape Accuracy: accuracy of the predicted shape category. (b) Value: accuracy of predicting both part category and the material of a given part correctly. (c) Value-all: accuracy of predicting all the (part, material) pairs of a shape correctly. We similarly define grounding metrics to check segmentation masks. A grounding is correct if the IoU of a predicted part and ground truth part is more than 0.5; it can be IoU on segmentation masks. (d) Grounded-
+
+![](images/2f3a094cb6af529793d980e6f291c070c6a8b236dbc72089c601c8e958f67163.jpg)  
+Fig. 7: 2D/3D GCR-SEG: Modified 2D/3D BPNet segmentation [19] architecture with 2D UNet [30] on the left and 3D MinkowskiUNet [7] on the right with same number of pyramid levels.
+
+value: accuracy of predicting both part category and the material of a given part as well as correctly grounding it. (e) Grounded-value-all: accuracy of predicting all the (part, material) pairs of a given shape correctly and grounding all of them correctly. All these metrics are calculated for each shape and then averaged across them to avoid bias toward shapes with more parts.
+
+Given the shape dependence of metrics, we define three settings: (a) Ground Truth Shape: the ground truth shape is assumed to be correct. (b) Top-1 Shape: Shape category is predicted correctly. (C) Top-5 Shape: Shape category is in the top-5 predictions. For (b) and (c), part-material pairs and their groundings are considered incorrect if shape is not in top-1 or top-5 predictions, respectively. We investigate two variants of the GCR task:
+
+(A) Joint 2D/3D GCR-SEG Setting and Baseline: 2D/3D GCR-SEG aims at solving the GCR Task in 2D or 3D, where grounding of parts is measured at the pixel precision for 2D and mesh triangle precision for 3D. Hence, we adopted a segmentation approach to solve it, specifically the joint 2D/3D BPNet segmentation model [19]; see Fig. 7. We adapted [19] to jointly predict shape, part, and material recognition in the Grounded CoMPat recognition task. As shown in Fig. 7, our adapted network consists of two branches: according to their functional domains, we denote the left one as the 2D UNet branch and the right as the 3D MinkowskiUNet branch. A Bidirectional Projection Module (BPM) bidirectionally fuses the multi-view 2D and 3D features between two branches. Features from the encoder of the U-Net are fed to a fully connected layer for shape classification. Images and voxels can be aggregated in a coarse-to-fine manner. BPNet can collect low-level and advanced complementary information; see more details in the supplementary [22].
+
+- (B) 3D GCR-SEG Setting and Baseline: Similar to (A), but in 3D only, where part label and material labels are predicted at the point precision for grounding. [20]. We built on PointGroup [20], a point cloud based method for 3D segmentation; see PointGroup adaptation details in the supplementary [22].
+
+Results. Table 6 shows the results for 2D GCR-SEG using BPNet, 3D GCR-SEG using our joint 2D/3D BPNet-based baseline. We report the "Standard"
+
+Table 6: 2D/3D Grounded CoMPaT recognition (GCR) Results.  
+
+<table><tr><td rowspan="2">Exp</td><td colspan="5">Top-1 predicted shape</td><td colspan="5">Top-5 predicted shape</td><td colspan="4">Ground Truth Shape</td></tr><tr><td>Shape Acc.</td><td>Value</td><td>Value-all</td><td>Value-grnd</td><td>Value-all grnd</td><td>Shape Acc.</td><td>Value</td><td>Value-all</td><td>Value-grnd</td><td>Value-all grnd</td><td>Value</td><td>Value-all</td><td>Value-grnd</td><td>Value-all grnd</td></tr><tr><td colspan="15">2D GCR-SEG (BPNet)</td></tr><tr><td>Standard</td><td>36.91</td><td>6.81</td><td>3.54</td><td>3.29</td><td>0.07</td><td>39.07</td><td>7.29</td><td>3.74</td><td>3.48</td><td>0.07</td><td>65.07</td><td>27.72</td><td>36.15</td><td>2.89</td></tr><tr><td>GT Material</td><td>36.91</td><td>7.10</td><td>3.54</td><td>4.30</td><td>0.60</td><td>39.07</td><td>7.59</td><td>3.74</td><td>4.51</td><td>0.60</td><td>69.54</td><td>27.72</td><td>40.46</td><td>4.80</td></tr><tr><td>GT Part</td><td>36.91</td><td>7.52</td><td>5.77</td><td>4.94</td><td>1.20</td><td>39.07</td><td>8.07</td><td>6.28</td><td>5.37</td><td>1.49</td><td>86.54</td><td>71.73</td><td>71.83</td><td>44.80</td></tr><tr><td colspan="15">2D GCR-SEG (BPNet) + Separate 2D Shape Classifier</td></tr><tr><td>Standard</td><td>67.86</td><td>40.39</td><td>16.57</td><td>23.23</td><td>1.78</td><td>77.60</td><td>49.28</td><td>20.67</td><td>27.50</td><td>1.89</td><td>65.07</td><td>27.72</td><td>36.15</td><td>2.89</td></tr><tr><td>GT Material</td><td>67.86</td><td>42.98</td><td>16.57</td><td>26.40</td><td>3.46</td><td>77.60</td><td>52.33</td><td>20.67</td><td>31.18</td><td>3.64</td><td>69.54</td><td>27.72</td><td>40.46</td><td>4.80</td></tr><tr><td>GT Part</td><td>67.86</td><td>55.12</td><td>46.58</td><td>46.02</td><td>30.08</td><td>77.60</td><td>66.89</td><td>57.36</td><td>55.81</td><td>37.30</td><td>86.54</td><td>71.73</td><td>71.83</td><td>44.80</td></tr><tr><td colspan="15">3D GCR-SEG (BPNet)</td></tr><tr><td>Standard</td><td>36.91</td><td>5.39</td><td>2.02</td><td>0.65</td><td>0.03</td><td>39.07</td><td>5.68</td><td>2.07</td><td>0.70</td><td>0.03</td><td>41.35</td><td>11.48</td><td>4.62</td><td>0.03</td></tr><tr><td>GT Material</td><td>36.91</td><td>6.10</td><td>2.02</td><td>1.34</td><td>0.15</td><td>39.07</td><td>6.43</td><td>2.07</td><td>1.40</td><td>0.15</td><td>46.39</td><td>11.48</td><td>7.80</td><td>0.27</td></tr><tr><td>GT Part</td><td>36.91</td><td>6.30</td><td>4.13</td><td>2.05</td><td>0.82</td><td>39.07</td><td>6.72</td><td>4.40</td><td>2.29</td><td>0.92</td><td>77.40</td><td>66.47</td><td>46.73</td><td>39.03</td></tr><tr><td colspan="15">3D GCR-SEG (BPNet) + Separate 3D Shape Classifier</td></tr><tr><td>Standard</td><td>67.53</td><td>29.83</td><td>8.12</td><td>3.25</td><td>0.46</td><td>87.23</td><td>38.34</td><td>11.07</td><td>4.30</td><td>0.47</td><td>44.59</td><td>12.52</td><td>4.92</td><td>0.47</td></tr><tr><td>GT Material</td><td>67.53</td><td>33.72</td><td>8.12</td><td>6.07</td><td>0.63</td><td>87.23</td><td>43.15</td><td>11.07</td><td>7.38</td><td>0.66</td><td>50.71</td><td>12.52</td><td>8.23</td><td>0.66</td></tr><tr><td>GT Part</td><td>67.53</td><td>50.08</td><td>42.42</td><td>27.86</td><td>22.65</td><td>87.23</td><td>63.53</td><td>54.06</td><td>36.44</td><td>29.31</td><td>77.27</td><td>65.30</td><td>44.83</td><td>36.57</td></tr></table>
+
+compositional metrics described earlier in this section, on ten compositions. To demonstrate how perfect prediction of either material or part influence the performance, we also report results where we use ground truth materials as the predicted material labels ("GT Material"), and ground truth parts as predicted parts ("GT Part"). It is not surprising that compared to "Standard", some metrics improve under "GT Material" and "GT Part" evaluation, especially for the value and value-all metrics that depend on the predicted part and the material labels. Note that all these baselines are composed of one model that jointly predicts shape and part material pairs in 2D or 3D. These models have a shape recognition performance ranging between  $15.64\%$  and  $38.29\%$  top-1 accuracy., and between  $62.32\%$  and  $85.2\%$  top-5 accuracy.; see Table 6. This limits the compositional performance, especially as we showed earlier in Table 4 that separate 2D and 3D Shape classifiers can reach  $76.8\%$  and  $69.1\%$  Top-1 Acc respectively. Hence, we also evaluated our BPNet-based 3D GCR-SEG approach where shape classes are predicted with a separate 3D PCT [15] classifier, leading to improved compositional performance. We observe similar behavior with PointGroup-based 3D GCR-SEG solution; see supplementary for materials for details [22]. The results suggest that designing a single model capable of performing well on GCR metrics is a challenge, and we hope that our 3D CoMPaT dataset and GCR baselines help ease future research.
+
+# 5 Conclusion
+
+We introduce 3D CoMPaT, a large-scale dataset of Compositions of Materials on Parts of 3D Things. It contains 7.19 million styled models stemming from 7262 CAD models from 43 object categories. The unique aspect of 3D CoMPaT is that it contains 3D shapes, part segmentation information, texture coordinates, and material compatibility information, so that multiple high-quality PBR materials can be assigned to the same shape part. We also propose a new task, dubbed as 2D/3D Grounded CoMPaT Recognition (GCR), that the dataset enables and introduce baseline methods to solve them.
+
+Acknowledgments. The authors wish to thank Poly9 Inc. participants for all the hard work, without whom this work would not be possible. This research is supported by King Abdullah University of Science and Technology (KAUST).
+
+# References
+
+1. Blender foundation, blender.org - home of the blender project - free and open 3d creation software (2021) 8  
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+# 3D Compositional Zero-shot Learning with DeCompositional Consensus
+
+Muhammad Ferjad Naeem $^{1}$ , Evin Pinar Örnek $^{2}$ , Yongqin Xian $^{1}$ , Luc Van Gool $^{1}$ , and Federico Tombari $^{2,3}$
+
+$^{1}$  ETH Zürich,  $^{2}$  TUM,  $^{3}$  Google
+
+Abstract. Parts represent a basic unit of geometric and semantic similarity across different objects. We argue that part knowledge should be composable beyond the observed object classes. Towards this, we present 3D Compositional Zero-shot Learning as a problem of part generalization from seen to unseen object classes for semantic segmentation. We provide a structured study through benchmarking the task with the proposed Compositional-PartNet dataset. This dataset is created by processing the original PartNet to maximize part overlap across different objects. The existing point cloud part segmentation methods fail to generalize to unseen object classes in this setting. As a solution, we propose De-Compositional Consensus, which combines a part segmentation network with a part scoring network. The key intuition to our approach is that a segmentation mask over some parts should have a consensus with its part scores when each part is taken apart. The two networks reason over different part combinations defined in a per-object part prior to generate the most suitable segmentation mask. We demonstrate that our method allows compositional zero-shot segmentation and generalized zero-shot classification, and establishes the state of the art on both tasks.
+
+Keywords: 3D Compositional Zero-shot Learning, Compositionality.
+
+# 1 Introduction
+
+A centaur is a mythological creature with the upper body of a human and the bottom body of a horse. This creature was never observed in our world, yet even a child can label its body parts from the human head to the horse legs. We humans can dissect the knowledge of basic concepts as primitives, like parts from human head to horse legs, to generalize to unseen objects. Cognitive studies have shown that humans learn part-whole relations in hippocampal memory to achieve object understanding through compositionality [18,47]. Compositionality has evolved as a survival need since every combination of every primitive cannot be observed.
+
+Parts represent a basic primitive of geometric and semantic similarity across objects. Recently, PartNet dataset has been introduced to study fine-grained semantic segmentation of parts [35]. This has inspired several architectural works
+
+![](images/70e3bfee22efc24ae546868a5377f46b2937e2e80b41bb13301a6a0cb2112627.jpg)  
+Fig.1: We aim to compose parts (e.g., display screen, key, horizontal surface) from seen (e.g., Display, Keyboard) to unseen object classes (e.g., Laptop) for semantic segmentation and classification in 3D point clouds.
+
+towards improving supervised fine-grained segmentation in 3D models [54,5,57]. A parallel line of work uses the concept of parts to improve tasks like 3D reconstruction with hierarchical decomposition [39], unsupervised segmentation by finding repeated structural patterns [29], and instance segmentation in unseen objects [9]. However, these works do not predict semantic part classes.
+
+With the availability of RGB-D sensors and the ease of acquiring 3D data in domains from augmented reality to robotic perception, the need for object understanding beyond seen object classes has emerged [38,2,51]. A model is unlikely to be trained for all possible existing objects [10,12], however, man-made environments consist of objects that share similarities through their parts. In this scenario, reasoning over learned parts can present an avenue for generalization to unseen objects classes. Zero-shot learning with 3D data has received far less attention compared to 2D domain. In this work, we introduce a new task, namely 3D Compositional Zero-Shot Learning (3D-CZSL), aiming at jointly segmenting and classifying 3D point clouds of both seen and unseen object classes (see Figure 1). 3D-CZSL is a challenging task as it requires generalizing parts from seen object classes to unseen classes that can be composed entirely of these parts.
+
+Our contributions are as follows: (1) We formalize zero-shot compositionality for 3D object understanding with semantic parts and introduce the 3D-CZSL task. To the best of our knowledge, we present the first work for joint classification and semantic part labeling for compositional zero-shot learning in 3D. (2) We establish a novel benchmark through Compositional PartNet (CPartNet), which enables research in 3D-CZSL through 16 seen and 8 unseen object classes. (3) We show that existing point cloud models fail to generalize beyond the seen object classes, whereas the performance of existing 2D zero-shot methods is severely limited in the 3D domain. (4) We propose a novel method, DeCompositional Consensus, which maximizes agreement between a segmentation hypothesis and its decomposed parts. Our method sets the state of the art for 3D-CZSL.
+
+# 2 Related Work
+
+Our work lies at the intersection of compositionality, zero-shot learning, and 3D point cloud part segmentation and discovery.
+
+Compositionality is the notion of describing a whole through its parts, studied thoroughly in many disciplines such as mathematics, physics, and linguistics. Hoffman [19] and Biederman [4] suggested that human object recognition is based on compositionality. They heavily influenced both traditional and modern computer vision research, such as describing objects by their primitives in Deformable Part Models [15], images as a hierarchy of features in Convolutional Neural Networks [61,26], understanding a scene through its components as in Scene Graphs [22], events as a set of actions as in Space-Time Region Graphs [53]. Parts have been used as semantic and geometric object primitives, which were seen to be captured within CNN kernels implicitly [17,16].
+
+Zero-shot learning (ZSL) addresses the task of recognizing object classes whose instances have not been seen during training [25,49,56]. This is attained through auxiliary information in the form of attributes (ALE [1]), word embeddings (SPNet [55]), or text descriptions [46]. Compositional zero-shot learning (CZSL) focuses on detecting unseen compositions of already observed primitives. The current literature on the topic focuses on state-object compositionality. Towards this, one line of research aims to learn a transformation between objects and states [34,37,28]. Another line proposes a joint compatibility function with respect to the image, state, and object [42,58,31]. Graph methods are also recently used in this direction including learning a causal graph of state object transformations [3] and using the dependency structure of state object compositions to learn graph embeddings [36,30]. There have been some preliminary works exploring zero-shot learning in 3D as an extension of 2D methods including projecting on word embeddings [12], using transductive approaches [10], along with some unlabeled data [11], and using generative models to learn the label distribution of unseen classes [32].
+
+3D part segmentation and discovery aims at parsing 3D objects into semantically and geometrically significant parts. The PartNet dataset [35] enabled studying fine-grained 3D semantic segmentation, hierarchical segmentation, and instance segmentation. The existing point cloud processing methods accomplish the task through conditioning the model over the known object class. PointNet models [43,44] provide multi-layer-perceptron (MLP) based solutions, DGCNN [54] uses graph convolutions for point clouds, ConvPoint [5] pre-processes points to define neighborhoods for convolutions, GDANet [57] uses attention in addition to MLP and currently holds state of the art for part semantic segmentation. Capsule Networks [48,62] propose architectural changes that implicitly model parts for tasks like object classification and segmentation.
+
+An alternate line of work uses the idea of parts in objects for downstream tasks like instance segmentation and point cloud reconstruction. This includes discovering geometrically similar part prototypes, similar to superpixels [29], predicting category-agnostic segmentation through a clustering approach [52], finding repetitive structural patterns in instances of an object [9], modeling 3D objects as compositions of cuboids [50], superquadrics [21,40,39], convex functions [13], and binary space partitioning planes [8] through deep learning.
+
+![](images/ff140648259ef83ec9f2df197f82a4222a879a25c7ea5d867c4a61c36c08f188.jpg)  
+Fig. 2: DeCompositional Consensus(DCC) combines our compositional part segmentation function  $\mathcal{F}$  with our part scoring function  $\mathcal{G}$ . We use the Part Prior  $\mathcal{P}_o$  of which parts can exist in each object class to populate the Hypothesis Bank with multiple segmentation masks. These hypotheses are used in a Hypothesis Driven Part Pooling to get a part descriptor of each part as an input to the part scoring function  $\mathcal{G}$  to calculate the DCC Score. This score measures the agreement of the segmentation mask with its part scores when each part is taken apart like lego blocks. Hypothesis with the maximum DCC score is selected for compositional zero-shot segmentation and zero-shot classification.
+
+Our work lies at the intersection of these three areas. Similar to CZSL works [34,37,28,36,3], we study the compositionality of learned primitives, however, we are interested in parts of objects rather than state-object relations. Similar to ZSL works [25,49,56,1,55,46], we learn classification scores of unseen object classes, however, our method only uses parts as side information and does not rely on any pretrained models like word embeddings. Similar to part discovery in objects [29,9,39,23,62], we rely on parts as a basic unit of understanding an object. However, instead of geometric primitives, we use human-defined semantic parts, which tightly couple geometry, semantics, and affordances [14]. Our method further has parallels to ensemble learning, where a combination of learners solves the same downstream task [7], however, we use an agreement between different tasks to improve generalization.
+
+# 3 Proposed Approach
+
+In the following, we formalize the problem and explain the proposed solution.
+
+Problem formulation. Let  $\mathcal{T}$  define the training set with instances  $(x,o,z)$ , where  $x$  is an input object point cloud described as a set of points in  $\mathbb{R}^3$ ,  $o$  is the object class label from the set of seen object classes  $\mathcal{O}_s$  and  $z$  is the part segmentation mask labelled with parts  $p$  from the set of all possible parts  $\mathcal{P}$ . We task a model to generalize to a set of unseen object classes  $\mathcal{O}_u$ , i.e.,  $\mathcal{O}_s \cap \mathcal{O}_u = \emptyset$ . We assume that  $\mathcal{O}_u$  is labelled with the same part set  $P$  for part segmentation that was completely observed in seen object classes  $\mathcal{O}_s$ . This makes
+
+the part segmentation task as a compositional zero-shot problem and the object classification task as a generalized zero-shot problem, i.e., we predict over the full object set  $\mathcal{O} = \mathcal{O}_s \cup \mathcal{O}_u$  at inference for object classification. We further assume that the model has access to a part prior for all object classes. For an object class  $o$ , this prior is defined as the set of parts  $\mathcal{P}_o = \{p_1, \dots, p_l\}$  that it can be labelled with for part segmentation.
+
+Method overview. Part segmentation is a challenging task, as it requires one model to adapt to parts of varying scale, orientation, and geometry for all objects. Existing point cloud part segmentation methods simplify this by learning an object class conditioned model, either by training separate models specialized for each object class [35], or by feeding the object class label as an input to the model (one-hot class vector [43,54,57,62]). However, this requires an object class input at test time which is not available for unseen object classes. In this work, we refer to this case as object prior, i.e., the model has access to the ground truth object class. The first step of our approach removes the object prior assumption and proposes Compositional Part Segmentation. In the second step, we propose our model DeCompositional Consensus which predicts the object class using the part segmentation from the previous step. It learns an agreement over a segmentation hypothesis and its part-based object classification score based on the idea of an object being taken apart like Lego blocks. The full model is depicted in Figure 2.
+
+# 3.1 Compositional Part Segmentation
+
+We reformulate part segmentation to allow compositional reasoning by encoding the part prior into the optimization criterion and the model inference. Formally, given an input point cloud  $x$ , we define  $\mathcal{F}(x,p)$  as the part segmentation function, with learnable parameters  $W$ , which returns a part score for each part  $p$  in the full part set  $\mathcal{P}$ . At training time, we compute the segmentation loss  $L_{Seg}$  from [43,44] as a cross entropy over parts in the part prior  $\mathcal{P}_o$  of the ground truth object class  $o$  rather than the full part set  $\mathcal{P}$ . At inference, the predicted part segmentation mask  $\hat{z}(o)$  of an object class  $o$  is computed over the scores of parts in its part prior:
+
+$$
+\hat {z} (o) = \underset {p \in \mathcal {P} _ {o}} {\arg \max } (\mathcal {F} (x, p)) \tag {1}
+$$
+
+With the proposed changes, a part segmentation model such as PointNet[43] can now compositionally generate a part segmentation mask for any object class we have a part prior for. Furthermore, the proposed improvements also prevent unintended biases against similar parts in different object classes as shown experimentally later in Table 3b. Notably, when an object prior is available as ground truth class, it defines the upper bound of the part segmentation performance for a model (see Table 1a "Object Prior"). In the absence of an object prior (GT object class), we can predict  $|\mathcal{O}|$  segmentation masks (hypotheses) for each object class we have a part prior for, generating the Hypothesis Bank (HB). Next, we introduce our novel method which allows for selecting the most suitable segmentation hypothesis for an input point cloud.
+
+# 3.2 DeCompositional Consensus
+
+We propose a novel method, DeCompositional Consensus (DCC), which learns an agreement (Consensus) over a segmentation hypothesis and its part-based object classification score when the object is taken apart (DeComposed) into parts like lego blocks as segmented in the hypothesis. DCC is based on the idea that we can learn what a valid part descriptor is from seen object classes to generalize to unseen object classes.
+
+Hypothesis driven part pooling. We extract a part descriptor for each part in a segmentation hypothesis from the point-wise features of the segmentation backbone as shown in Figure 2. Part segmentation models like PointNet[43] generate this feature representation in the penultimate layer of the model, i.e., before the final per-part segmentation scoring layer. We use the segmentation hypothesis as the pooling mask to pool over the point dimensions of the feature map for each part. This results in a permutation invariant part feature vector for each part, i.e., part descriptor representing the features responsible for that part segmentation in this hypothesis. We choose maxpool as the pooling operation due to its wide adoption in point cloud literature [43,44]. For the segmentation hypothesis  $\hat{z}(o)$  of an object class  $o$ , this operation returns a set  $\mathcal{D}(o)$  with part descriptors  $d$  for each part  $p$  found in this hypothesis. Note that  $|\mathcal{D}(o)|$  is not always equal to  $|\mathcal{P}_o|$  as a segmentation hypothesis might not contain all parts defined in the  $\mathcal{P}_o$ , e.g., an instance of a chair might or might not contain sidearms.
+
+Learning DeCompositional Consensus. Our DCC model learns a part scoring function  $\mathcal{G}$  with weights  $\Theta$ . For a part descriptor  $d$ , the function returns a score  $\mathcal{G}(d,p)$  which measures the likelihood of this part descriptor to belong to the part  $p$ . We define DeCompositional Consensus score as the agreement between the segmentation hypothesis and the part scores. For an object hypothesis  $\hat{z}(o)$ , the DCC score is defined as:
+
+$$
+s (x, o) = \frac {1}{| \mathcal {D} (o) |} \sum_ {n = 1} ^ {| \mathcal {D} (o) |} \mathcal {G} \left(d _ {n}, p _ {n}\right) \tag {2}
+$$
+
+Our novel DCC score measures the individual consensus of each part descriptor with the full segmentation mask to define an object classification score. We optimize DCC score for classification with a cross entropy loss over  $\mathcal{O}_s$  as:
+
+$$
+L _ {D e C o m p} = - \log \left(\frac {\exp s (x , o)}{\sum_ {o ^ {\prime} \in \mathcal {O} _ {s}} \exp s (x , o ^ {\prime})}\right) \tag {3}
+$$
+
+Since  $L_{DeComp}$  is computed over the Hypothesis Bank generated by  $\mathcal{F}$ , an additional part classification loss  $L_{Part}$  is computed using the ground truth segmentation mask  $z$  of each input to prevent bias against parts that are hard to segment.  $L_{Part}$  uses the ground truth segmentation mask to extract part descriptor set  $\mathcal{D}_{gt}$  and optimizes them for part classification over  $\mathcal{P}$ .
+
+$$
+L _ {P a r t} = \sum_ {n = 1} ^ {| \mathcal {D} _ {g t} |} - l o g \left(\frac {\exp \mathcal {G} \left(d _ {n} , p _ {n}\right)}{\sum_ {p ^ {\prime} \in \mathcal {P}} \exp \mathcal {G} \left(d _ {n} , p ^ {\prime}\right)}\right) \tag {4}
+$$
+
+![](images/4defda7222680404c0b2d6c96729e41374b8f3d08f7cf3f12e1230707f9f1b7d.jpg)  
+Fig. 3: Compositional PartNet refines the labels of PartNet dataset to maximize shared parts across different object classes, and enables studying 3D-CZSL task. The available 24 object classes are divided into 16 seen classes for training and 8 unseen classes for inference in zero-shot. We depict the shared labels between seen and unseen object classes in same colors.
+
+Inference. For generalized zero-shot inference, the HB is populated over all object classes  $\mathcal{O} = \mathcal{O}_s + \mathcal{O}_u$ . The object class prediction  $\hat{o}$  for an input point cloud  $x$  is retrieved by selecting the object class with the highest DeCompositional Consensus score:
+
+$$
+\hat {o} = \underset {o ^ {\prime} \in \mathcal {O}} {\arg \max } \left(s \left(x, o ^ {\prime}\right)\right) \tag {5}
+$$
+
+The corresponding hypothesis of the predicted class  $\hat{o}$  becomes the final part segmentation output, i.e.,  $\hat{z} (\hat{o})$ . Our technical novelty lies in defining part descriptors as features responsible for part segmentation in a hypothesis; and using their likelihood to define an object class level consensus score to achieve zero-shot compositionality. In contrast to several zero-shot baselines [55,36], our method does not require any supervised calibration step over the unseen classes.
+
+# 4 Compositional PartNet Benchmark
+
+Zero-shot compositionality in machine learning algorithms has mainly been studied for state-object relations in image datasets like MIT-States [20], UT-Zappos [60], AO-CLEVr [3], and more recent C-GQA [36]. These datasets have several limitations such as including label noise [20,3], lacking visual cues [60,36], being too simple [3], or missing multilabel information [36].
+
+We believe that 3D part object relations provide an ideal avenue to study zero-shot compositionality, as they tend to be more well-defined albeit challenging. There have been several attempts in a part-based benchmark [6,59] for 3D object understanding. Recently, ShapeNet has been extended with fine-grained part labels to form the new dataset PartNet [35]. PartNet provides 24 distinct object classes, annotated with fine-grained, instance-level, and hierarchical 3D part information, consisting of around 26K 3D models with over 500K part instances and 128 part classes. However, these part class labels are not unified
+
+across different object categories, preventing a study into zero-shot compositionality. We refine PartNet into Compositional PartNet (C-PartNet) with a new labeling scheme that relates the compositional knowledge between objects by merging and renaming the repeated labels as shown in Figure 3.
+
+Unifying part labels. While PartNet provides three levels of hierarchical part labels, not all objects are labeled at the deepest level. We take the deepest level available for each object. We find similar parts within and across different objects by training a supervised segmentation model and compute pairwise similarities between parts across PartNet. Parts that share a high similarity and have the same semantic meaning (e.g., bed horizontal surface in object Bed and horizontal surface in Storage Furniture) are merged into a single general part label (horizontal surface). Furthermore, parts with a similar function but different name (e.g., screen side of Laptop and display screen of Display) are merged together. The relabeled C-PartNet consists of 96 parts compared to 128 distinct part labels in the original PartNet. Details in the supplementary.
+
+Selecting test time unseen object classes. Objects that share a similar function tend to have similar parts [4]. We divide PartNet objects into several functional categories. Details of this categorization and the dataset statistics can be found in the supplementary. We identify three easy to compose unseen object classes (i.e., Mug, Bowl and TrashCan), that share large similarities with seen object classes (Bottle and Vase). Furthermore, we choose three object classes of medium difficulty that require generalizing parts beyond the context they were observed in (i.e., Dishwasher, Refrigerator, and Laptop). Finally, Scissors and Door present two hard-to-compose object classes that require generalizing beyond scale, context, and number instances of parts compared to seen object classes. The validation set contains all seen and 2 unseen object classes (Bowl and Dishwasher). The test set consists of 16 seen  $O_{s}$  and 8 unseen classes  $O_{u}$ .
+
+# 5 Experiments
+
+Since our proposed benchmark lies at the intersection of point cloud processing, attribute learning, zero-shot learning, and its specialized sub-domain compositional zero-shot learning, we adapt baselines representing these lines of works.
+
+Baselines. Object Prior uses a point cloud part segmentation model trained with our framework and evaluates the segmentation performance on the ground truth object. This is the oracle upper bound for the zero-shot models. Direct Seg trains a point cloud part segmentation model  $\mathcal{F}$  without a part prior to predict over all parts  $\mathcal{P}$  in the dataset. PartPred is inspired from a classic zero-shot baseline DAP [24] and trains a part prediction network from the global feature of each point cloud. The predicted parts are used as  $P_{o}$  for equation 1 to condition the compositional part segmentation network [24]. For zero-shot classification baselines, we use the predicted class to select the corresponding segmentation mask from the Hypothesis Bank. Among these, SPNet [55] learns classification by projecting the global feature of an input on a pretrained distribution where both seen and unseen objects lie e.g., word embeddings. CGE
+
+[36] proposes to model compositional relations using a graph consisting of parts connected to objects they occur in. We reformulate CGE to a multitask setup and use part nodes for segmentation and object nodes for classification. PartPred DCC uses the part prediction network's scores for parts found in each segmentation hypothesis to calculate the consensus score from Equation 2. Finally, 3D Capsule Networks [62] aim to discover part prototypes through unsupervised reconstruction. Segmentation is subsequently learned by a linear mapper from capsules to part labels. We give additional details about these baselines in the supplementary and also compare with the current SOTA for part class agnostic segmentation method, Learning to Group [29], on unseen object classes.
+
+Metrics. The proposed benchmark consists of two jointly learned tasks. For the compositional zero-shot segmentation, we report the mean object class-wise Intersection-over-Union (mIoU) for part labels over seen and unseen object classes. We also report the harmonic mean over seen and unseen object classes to study the best generalized zero-shot performance. In addition, we report a per-object mIoU to study model performance on each unseen object across the three difficulty levels. For generalized zero-shot classification, we report for the per-object class top-1 classification accuracy over unseen classes, mean accuracy over seen classes, unseen classes and their harmonic mean. For models that apply joint classification and segmentation, we choose the checkpoint with the best segmentation performance to encourage compositional part understanding. Part based classification baselines can give the same scores across two objects if an instance does not have all parts, e.g., an empty Vase has the same parts in Vase Hypothesis and Bowl. This is counted as an accurate classification, since the ground truth object still receives the highest score and achieves compositional segmentation.
+
+Training details. For its simplicity and competitive performance in our ablations (see Table 3), we choose PointNet [43] as the backbone model for  $\mathcal{F}$  in our baseline comparisons in Table 1a, 1b. We also report further results on DGCNN [54], ConvPoint [5] and GDANet [57] in Table 3. All backbones are pretrained with the author's implementations extended by our framework. The pretrained models are then used as initialization for the zero-shot models and are finetuned. For our model DCC, we use a 2-layer MLP with 512 hidden dimensions, ReLU, and dropout followed by a linear layer as function  $\mathcal{G}$ . We use a step size learning rate scheduler between  $1e^{-3}$  and  $1e^{-5}$  with Adam optimizer. We use cross entropy as segmentation loss  $L_{Seg}$  for part segmentation similar to [43,44,54,57].  $L_{Seg}$ ,  $L_{DeComp}$  and  $L_{Part}$  are equally weighted and the network is trained until convergence on the validation set. We use Word2Vec [33] for models that rely on word embeddings [55,36]. For CGE, we choose the graph configuration that achieved the best result on the validation set at 2 layers of GCN with a hidden dimension of 1024. Our framework is implemented in PyTorch [41] and all experiments are conducted using Nvidia A100 GPUs. The dataset and experimental framework will be released upon acceptance.
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">Unseen object classes</td></tr><tr><td>HM</td><td>S</td><td>U</td><td>Bowl</td><td>Dish</td><td>Door</td><td>Lap</td><td>Mug</td><td>Refr</td><td>Scis</td></tr><tr><td>Object Prior [43]</td><td>47.9</td><td>52.8</td><td>43.8</td><td>77.0</td><td>40.2</td><td>25.1</td><td>72.4</td><td>47.1</td><td>31.9</td><td>22.5</td></tr><tr><td>Direct Seg [43]</td><td>28.5</td><td>48.7</td><td>20.1</td><td>62.9</td><td>4.0</td><td>1.6</td><td>19.9</td><td>35.7</td><td>0.9</td><td>0.0</td></tr><tr><td>SPNet* [55]</td><td>8.5</td><td>28.5</td><td>5.0</td><td>12.6</td><td>2.5</td><td>0.5</td><td>0.0</td><td>2.6</td><td>2.6</td><td>0.0</td></tr><tr><td>CGE* [36]</td><td>30.8</td><td>37.0</td><td>26.4</td><td>67.0</td><td>19.5</td><td>0.3</td><td>35.1</td><td>39.6</td><td>11.2</td><td>0.0</td></tr><tr><td>3D-PointCapsNet [62]</td><td>4.4</td><td>9.4</td><td>2.9</td><td>4.3</td><td>0.0</td><td>0.2</td><td>1.2</td><td>11.2</td><td>0.1</td><td>0.1</td></tr><tr><td>PartPred [24]</td><td>26.3</td><td>33.6</td><td>21.6</td><td>66.2</td><td>2.3</td><td>7.2</td><td>19.4</td><td>43.1</td><td>0.5</td><td>0.0</td></tr><tr><td>PartPred DCC [24]</td><td>20.9</td><td>41.3</td><td>14.0</td><td>35.5</td><td>2.1</td><td>7.2</td><td>17.2</td><td>29.2</td><td>0.7</td><td>0.0</td></tr><tr><td>DCC (ours)</td><td>35.2</td><td>38.0</td><td>32.7</td><td>66.1</td><td>30.9</td><td>5.3</td><td>56.3</td><td>40.4</td><td>28.4</td><td>0.0</td></tr></table>
+
+(a) Compositional Zero-shot Segmentation  
+
+<table><tr><td rowspan="2">Method</td><td colspan="11">Unseen object classes</td></tr><tr><td>HM</td><td>S</td><td>U</td><td>Bowl</td><td>Dish</td><td>Door</td><td>Lap</td><td>Mug</td><td>Refr</td><td>Scis</td><td>Trash</td></tr><tr><td>SPNet* [55]</td><td>3.8</td><td>46.7</td><td>2.0</td><td>12.0</td><td>0.0</td><td>3.1</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>CGE* [36]</td><td>33.1</td><td>54.3</td><td>23.8</td><td>31.9</td><td>0.0</td><td>0.0</td><td>52.0</td><td>1.0</td><td>33.7</td><td>0.0</td><td>71.9</td></tr><tr><td>PartPred DCC [24]</td><td>19.9</td><td>74.0</td><td>11.5</td><td>4.3</td><td>3.3</td><td>25.8</td><td>0.0</td><td>13.5</td><td>0.0</td><td>0.0</td><td>45.2</td></tr><tr><td>DCC(ours)</td><td>55.9</td><td>73.2</td><td>45.2</td><td>79.8</td><td>57.1</td><td>5.3</td><td>55.4</td><td>71.9</td><td>55.6</td><td>0.0</td><td>36.8</td></tr></table>
+
+(b) Generalized Zero-shot Classification
+
+Table 1: Baseline comparison. We compare our proposed method, DeCompositional Consensus (DCC), against baseline and report results for the two tasks. * marks baselines that require supervised calibration. For (a), we report mIoU % over part labels per object class over seen objects, unseen objects, and their harmonic mean. We also report the mIoU over each unseen object class. For (b), we report the top-1 classification accuracy. DCC achieves SOTA on both tasks.
+
+# 5.1 Comparing with State of the Art
+
+We compare our method with baselines on compositional zero-shot segmentation in Table 1a and generalized zero-shot classification in Table 1b. Our method outperforms all baselines on almost all metrics and establishes state of the art on both tasks.
+
+Compositional zero-shot segmentation performance. Our method demonstrates remarkable performance gains on all unseen classes and achieves the best harmonic mean on compositional zero-shot segmentation in Table 1a. We achieve a  $50\%$  improvement over the direct segmentation demonstrating that the introduction of object class conditioned inference with DCC can improve compositional zero-shot segmentation in point cloud models. This improvement is observed most in unseen object classes that have large variations in parts from
+
+![](images/1ffded7bcdbcfa21630c45a611b608abb4702169e12a5ad06eb8f96299949397.jpg)  
+Fig. 4: Qualitative results. Direct segmentation tends to segment an input point cloud to parts from seen objects with large geometric similarities. While this works for the objects from Container category, it fails in more complex objects that share similarity with Furniture while being composed of parts from other categories. In contrast, DeCompositional Consensus builds an implicit understanding of what parts can occur together in different categories, and achieves meaningful segmentations for all object classes but Door and Scissors.
+
+the seen object classes like Dishwasher  $(7.5\times)$ , Laptop  $(2.5\times)$  and Refrigerator  $(28\times)$  as shown in Fig. 4. Unseen object classes that share large geometric and semantic similarities with respect to parts to seen object classes also have significant improvements. This includes improvements in Bowl  $(4\%)$ , Mug  $(14\%)$ , and TrashCan  $(1.4\%)$  that have very similar parts with seen Bottle and Vase.
+
+Comparing with zero-shot learning baselines, we observe that our method achieves the best performance in 6 out of 8 classes and establishes a state of the art in overall harmonic mean and unseen mIoU while achieving competitive seen IoU. PartPred [24] learns to dynamically predict parts and generalizes to unseen objects that share part and geometric similarities with seen objects in the Container category but fails in other objects. As SPNet [55] does not use any part information, it fails to generalize to unseen objects by projecting on word embeddings alone. Compared to SPNet, CGE [36] performs much better as it uses the part prior and refines the word embeddings by using the dependency structure defined in the graph. Although being competitive on Bowl, Dishwasher, Mug and TrashCan, it performs much poorer on other unseen objects. 3D-PointCapsNet [62], while conceptually engineered for part-whole relations, fails to generalize to unseen objects, likewise having very low performance on seen objects. We relate this performance to the capsules' inability to generalize without object prior as further shown in the supplementary material. Finally, PartPred DCC, achieves impressive performance on seen objects but fails on the unseen objects showing the importance of learning consensus over the features responsible for a hypothesis. We observe that while some methods have almost zero classification
+
+accuracy, they can still achieve some segmentation performance due to confusion with objects that share some parts with the ground truth object. All methods fail to generalize to challenging object classes Door and Scissors. We discuss that in qualitative analysis in Section 5.3.
+
+Zero-shot classification performance. Our method also achieves significant gains on generalized zero-shot classification as seen in Table 1b. DCC attains the best harmonic mean and unseen classification accuracy while maintaining a competitive seen performance. In fact, the best seen performance is achieved by PartPred DCC which extends our DCC score to a simple attribute (part) prediction model. This shows the power of enforcing consensus in different decisions of a model. Specifically, DCC is able to classify 6 out of the 8 unseen object classes with an outstanding accuracy. SPNet is only able to classify Bowl with a low accuracy of  $12\%$ . CGE is again a competitive baseline here. However, it is only able to receive reasonable classification scores on 4 of the 8 unseen object classes while maintaining a competitive seen class performance.
+
+# 5.2 Ablations
+
+We ablate our design choices and compare performance against different point cloud backbones.
+
+**Optimization criteria.** We ablate over the two optimization criteria for DCC in Table 2. As seen from row a) that only training for  $L_{DeComp}$  is unable to attain high performance as it can introduce bias against hard to predict parts to increase classification performance. Similarly, only training for  $L_{Part}$  in row b) achieves low performance as the model is not optimized for the downstream classification task of predicting the consensus score. Row c) and d) combine both of these losses and see a big performance gain. In row c) we replace the predicted segmentation mask corresponding to the ground truth object class in HB with the ground truth segmentation mask. Comparing row c) and d) in Table 2, we see that when we learn DCC score exclusively on the model's predicted segmentation instead of using ground truth segmentation mask, we see a large improvement in seen and unseen performance. We conjecture that the part scoring function  $\mathcal{G}$  learns the segmentation network's limitations in this setting, i.e., if a part is not predicted well by  $\mathcal{F}$ ,  $\mathcal{G}$  can look for cues from other parts.
+
+Comparing point cloud backbones. We compare point cloud backbones under Direct Segmentation and DCC in Table 3a. We see that all models are unable to achieve competitive performance over unseen object classes with direct segmentation. In fact, ConvPoint [5] completely fails under this setting. The introduction of DCC to every backbone leads to a major increase in performance on the unseen object while being competitive over seen classes. This shows that our model can be readily extended to various families of point cloud backbones. In Table 3b, we compare the oracle segmentation performance over the ground truth object class when trained with and without our part prior optimization criterion ( $L_{Seg}$  over  $\mathcal{P}_o$  or over  $\mathcal{P}$ ). In absence of our criterion, we observe a large difference between the performance on seen and unseen object classes. We
+
+<table><tr><td></td><td colspan="3">Hyperparameters</td><td colspan="3">Classification</td><td colspan="3">Segmentation</td></tr><tr><td></td><td>LDeComp</td><td>LPart</td><td>Segonly</td><td>HM</td><td>S</td><td>U</td><td>HM</td><td>S</td><td>U</td></tr><tr><td>a)</td><td>✓</td><td></td><td>✓</td><td>29.0</td><td>38.1</td><td>23.4</td><td>23.3</td><td>24.2</td><td>22.5</td></tr><tr><td>b)</td><td></td><td>✓</td><td></td><td>14.9</td><td>34.8</td><td>9.4</td><td>24.8</td><td>22.1</td><td>28.3</td></tr><tr><td>c)</td><td>✓</td><td>✓</td><td></td><td>52.6</td><td>54.4</td><td>50.9</td><td>39.1</td><td>35.9</td><td>42.9</td></tr><tr><td>d)</td><td>✓</td><td>✓</td><td>✓</td><td>72.8</td><td>76.6</td><td>69.3</td><td>45.1</td><td>40.9</td><td>50.2</td></tr></table>
+
+Table 2: Ablating over  $L_{DeComp}$  and  $L_{Part}$ , we see that both the criterion complement each other to achieve the best performance.  
+
+<table><tr><td rowspan="2">Backbone</td><td colspan="3">Direct Seg</td><td colspan="3">DCC</td><td rowspan="2">Backbone</td><td colspan="3">LSeg over P</td><td colspan="3">LSeg over Po</td></tr><tr><td>HM</td><td>S</td><td>U</td><td>HM</td><td>S</td><td>U</td><td>HM</td><td>S</td><td>U</td><td>HM</td><td>S</td><td>U</td></tr><tr><td>PointNet [43]</td><td>28.5</td><td>48.7</td><td>20.1</td><td>35.2</td><td>38.0</td><td>32.7</td><td>PointNet [43]</td><td>43.2</td><td>51.7</td><td>37.1</td><td>47.9</td><td>52.8</td><td>43.8</td></tr><tr><td>DGCNN [54]</td><td>29.5</td><td>50.0</td><td>20.9</td><td>36.2</td><td>45.1</td><td>30.2</td><td>DGCNN [54]</td><td>44.6</td><td>52.4</td><td>38.8</td><td>50.0</td><td>55.0</td><td>46.3</td></tr><tr><td>ConvPoint [5]</td><td>2.9</td><td>5.2</td><td>2.0</td><td>29.5</td><td>35.0</td><td>25.5</td><td>ConvPoint [5]</td><td>29.1</td><td>28.7</td><td>29.5</td><td>43.5</td><td>42.4</td><td>43.0</td></tr><tr><td>GDANet [57]</td><td>28.7</td><td>47.7</td><td>20.5</td><td>33.5</td><td>46.2</td><td>26.4</td><td>GDANet [57]</td><td>43.4</td><td>53.1</td><td>36.8</td><td>48.0</td><td>53.7</td><td>43.4</td></tr><tr><td colspan="7">(a) CZSL Segmentation</td><td colspan="7">(b) Oracle Performance</td></tr></table>
+
+Table 3: Backbone ablation. (a) We see DCC results in a large improvement compared to direct segmentation across all ablated point cloud models (b) We further see that our Part Prior optimization criterion greatly benefits all backbones under oracle evaluation especially on unseen objects.
+
+conjecture that the model overfits to seen object classes, limiting compositionality to unseen object classes. With our criterion, e.g., PointNet segmentation network improves up to  $19\%$  on unseen and  $1\%$  on seen classes.
+
+# 5.3 Qualitative and Model Limitation
+
+In Figure 4, we show some qualitative results for direct segmentation versus top-3 results of our model across unseen objects. We further validate our results from Table 1a, and see that for the easy object Mug, the direct segmentation can give a meaningful result. However, it fails for other relatively harder objects, which can be attributed to the lack of affordance, i.e., how an agent interacts with an object. Our method, despite not having access to affordances, builds an implicit understanding of what parts occur in each object category and is thus able to learn a reasonable consensus score. This brings about an increased generalization and meaningful results for all objects. We see a correct prediction for even Dishwasher and Refrigerator, which are closer geometrically to Furniture than Microwave, their closest functional seen object. However, although less, DCC also suffers from lack of affordances. For example, among the top-3 result for a Laptop in Figure 4 is a Chair which shares geometrical similarity to an open Laptop. This indicates an upper bound to the performance that can be
+
+![](images/ef2e9ecef9e12aa10e84256d9f3606507b5e09b3a098329dee609dfd1bd482ec.jpg)  
+Fig. 5: Error plots. We find that PointNet[43] is comparable in mIoU to a much newer model, GDANet[57], across unseen objects.
+
+achieved from visual data alone [27,45]. An affordance prior can help address this limitation for part-object relations.
+
+Another aspect that limits our model performance is the generalization limitations of point cloud backbones. In Figure 5, we compare the object prior per part performance on the unseen objects between PointNet [43] and GDANet [57], which were released five years apart. A surprising insight we observe is that years of progress in point cloud processing, while making a significant advance on seen object classes, does not translate to improvement on unseen object classes. We see that there is no clear consensus on which model is better for unseen object class generalization. Even using the right part prior, some parts are unlikely to be segmented in unseen classes. An example of this is handle, which is unable to be reasonably segmented for Mug, Dishwasher, and Refrigerator. A more extreme case of this is observed in Door and Scissors, where the segmentation fails completely as shown in last two columns of Figure 4. These objects have a large variation with respect to parts from the seen objects in scale, the number of instances (two blades in Scissors vs one in Knife), and orientation.
+
+# 6 Conclusion and Future work
+
+We introduce 3D-CZSL as a joint compositional zero-shot segmentation and generalized zero-shot classification task. We provide a structured study into zero-shot compositionality through a novel benchmark on the proposed C-PartNet dataset and show that previous models do not generalize beyond the training object classes. Towards this, our novel approach, DeCompositional Consensus, maximizes the agreement between a segmentation hypothesis and its parts when taken apart, and sets a new SOTA. We also show that while there has been a lot of progress in part segmentation in a supervised setting, simple models like PointNet are still competitive in unseen object classes, arguably because the current research has not focused on this task. There are several future directions that can stem from this work, including introducing affordance priors and extension of the capsules paradigm for part reasoning on unseen object classes. We also hope to inspire future research into compositional point cloud models.
+
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+# 3D Equivariant Graph Implicit Functions
+
+Yunlu Chen<sup>1</sup>, Basura Fernando<sup>2</sup>, Hakan Bilen<sup>3</sup>, Matthias Nießner<sup>4</sup>, and Efstratios Gavves<sup>1</sup>
+
+<sup>1</sup> University of Amsterdam  
+2 CFAR, IHPC, A*STAR  
+<sup>3</sup> University of Edinburgh  
+4 Technical University of Munich
+
+ychen9201@gmail.com, fernandopbc@ihpc.a-star.edu.sg, hbilen@ed.ac.uk, niessner@tum.de, e.gavves@uva.nl
+
+Abstract. In recent years, neural implicit representations have made remarkable progress in modeling of 3D shapes with arbitrary topology. In this work, we address two key limitations of such representations, in failing to capture local 3D geometric fine details, and to learn from and generalize to shapes with unseen 3D transformations. To this end, we introduce a novel family of graph implicit functions with equivariant layers that facilitates modeling fine local details and guaranteed robustness to various groups of geometric transformations, through local  $k$ -NN graph embeddings with sparse point set observations at multiple resolutions. Our method improves over the existing rotation-equivariant implicit function from 0.69 to 0.89 (IoU) on the ShapeNet reconstruction task. We also show that our equivariant implicit function can be extended to other types of similarity transformations and generalizes to unseen translations and scaling.
+
+Keywords: Implicit neural representations; equivariance; graph neural networks; 3D reconstruction; transformation.
+
+# 1 Introduction
+
+Neural implicit representations are effective at encoding 3D shapes of arbitrary topology [32,30,10]. Their key idea is to represent a shape by a given latent code in the learned manifold and for each point in space, the neural implicit function checks whether a given coordinate location is occupied within the shape or not. In contrast to traditional discrete 3D representations such as triangle meshes or point clouds, this new paradigm of implicit neural representations has gained significant popularity due to the advantages such as being continuous, grid-free, and the ability to handle various topologies.
+
+Despite their success, latent-code-conditioned implicit representations have two key limitations. First, the latent code of the shape captures coarse high-level shape details (i.e., the global structure) without any explicit local spatial information, hence it is not possible to learn correlations between the latent code and local 3D structural details of the shape. As a result, the surface reconstruction from latent-code-conditioned implicit functions tends to be over-smoothed and they are not good at capturing local
+
+![](images/14b959d10c2ae4be3c807b673124cdd1ad7cddb9ac67af6303baeb8412c98f81.jpg)  
+Fig. 1: Our equivariant graph implicit function infers the implicit field  $F(\cdot|\mathbf{X})$  for a 3D shape, given a sparse point cloud observation  $\mathbf{X}$ . When a transformation  $T_{g}$  (rotation, translation, or/and scaling) is applied to the observation  $\mathbf{X}$ , the resulting implicit field  $F(\cdot|T_{g}(\mathbf{X}))$  is guaranteed to be the same as applying a corresponding transformation  $T_{g}^{*}$  to the inferred implicit field from the untransformed input (middle). The property of equivariance enables generalization to unseen transformations, under which existing models such as ConvONet [34] often struggle (right).
+
+surface detail [34,11,25,21,18]. Second, implicit representations are sensitive to various geometric transformations, in particular to rotations [16]. The performance of the implicit representations heavily relies on the assumption that shape instances in the same category are required to be in the same canonical orientation such that shape structures of planes and edges are in line with the coordinate axes. While data augmentation loosely addresses this second issue to some degree, a principled approach is to enable the representations to be inherently aware of common geometric operations such as rotations, translations, and scaling, which are found commonly in real-world 3D objects.
+
+To address the first challenge of modeling local spatial information, recent methods [34,11] first discretize 3D space into local 2D or 3D grids and then store implicit codes locally in the respective grid cells. However, these methods are still sensitive to transformations as the grid structure is constructed in line with the chosen coordinate axes. This results in deteriorated performance under transformations as shown in Fig. 1. In addition, the grid discretization often has to trade fine details of the shape, hence the quality of shape reconstruction, for better computational efficiency through a low resolution grid. Deng et al. [17] propose VN-ONet to tackle the second challenge of pose-sensitivity with a novel vector neuron formulation, which enables the network architecture to have a rotation equivariant representation. Nevertheless, similar to implicit representations, the VN-ONet encodes each shape with a global representation and hence fails to capture local details. As grid discretization is not robust to transformations, this solution is not compatible with the grid-based local implicit methods. Thus, integrating VN-ONet with grid-approach is not a feasible solution.
+
+In this work, our goal is to simultaneously address both challenges of encoding local details in latent representations and dealing with the sensitivity to geometric transfor-
+
+mations such as rotations, translations, and scaling. To this end, we propose a novel equivariant graph-based local implicit function that unifies both of these properties. In particular, we use graph convolutions to capture local 3D information in a non-Euclidean manner, with a multi-scale sampling design in the architecture to aggregate global and local context at different sampling levels. We further integrate equivariant layers to facilitate generalization to unseen geometric transformations. Unlike the grid-based methods [34,11] that requires discretization of 3D space into local grids, our graph-based implicit function uses point features from the input point cloud observation directly without interpolation from grid features. Our graph mechanism allows the model to attend detailed information from fine areas of the shape surface points, while the regularly-spanned grid frame may place computations to less important areas. In addition, our graph structure is not biased towards the canonical axis directions of the given Cartesian coordinate frame, hence less sensitive than the grid-local representations [34,11]. Therefore, our graph representation is maximally capable of realizing an equivariant architecture for 3D shape representation. In summary, our contributions are as follows:
+
+- We propose a novel graph-based implicit representation network that enables effective encoding of local 3D information in a multi-scale sampling architecture, and thus modeling of high-fidelity local 3D geometric detail. Our model features a non-Euclidean graph representation that naturally adapts with geometric transformations.  
+- We incorporate equivariant graph layers in order to facilitate inherent robustness against geometric transformations. Together with the graph embedding, our equivariant implicit model significantly improves the reconstruction quality from the existing rotation equivariant implicit method [17].  
+- We extend our implicit method to achieve a stronger equivariant model that handles more types of similarity transformations simultaneously with guaranteed perfect generalization, including rotation, translation and scaling.
+
+# 2 Related Work
+
+Implicit 3D representations. Neural implicits have been shown to be highly effective for encoding continuous 3D signals of varying topology [32,30,10]. Its variants have been used in order to reconstruct shapes from a single image [38,52,53], or use weaker supervision for raw point clouds [1,2,3] and 2D views [29,31,26].
+
+Local latent implicit embeddings. ConvONet [34] and IF-Net [11] concurrently propose to learn multi-scale local grid features with convolution layers to improve upon global latent implicit representations. Other variants of grid methods [25,5] takes no global cues, hence restricted by requiring additional priors during inference such as normals [25] or the partial implicit field [5]. The paradigm is extended to adaptive grids or octrees [44,45,47]. Inspired by non-grid local embeddings in point cloud networks [36,48,28,20], we aim for non-grid local implicit methods, which are less explored. The existing non-grid local implicits [22,18] are limited without multi-scale hierarchical designs, and thus restricted to single objects. The pose-sensitivity problem is not addressed in all these local methods. In contrast, we use a hierarchical graph embedding that effectively encodes multi-scale context, as the first local implicit method to address the pose-sensitivity problem.
+
+Pose-sensitivity in implicit functions. As generalization becomes a concern for 3D vision [42,4,9,8], Davies et al. [16] first point out that the implicit 3D representations are biased towards canonical orientations. Deng et al. [17] introduce a rotation equivariant implicit network VN-ONet that generalises to random unseen rotations, but yet with the restrictions from the global latent. Concurrent to our work, [41,7,54] extend equivariance of implicit reconstruction to SE(3) group for reconstruction and robotic tasks, while our method further handles scaling transformation, and recovers significantly better local details with the graph local embedding design. As grid embeddings are sensitive to rotation, seeking a compatible local latent embedding is a non-trivial problem.
+
+Rotation equivariance with 3D vector features. Equivariance has drawn attention in deep learning models with inductive priors of physical symmetries, e.g., the success of ConvNets are attributed to translation equivariance. Advanced techniques are developed for equivariance to rotation [14,51,46], scale [43,56] and permutation [55,35]. Recently, a new paradigm for rotation equivariance uses 3D vectors as neural features [17,40] with improved effectiveness and efficiency upon methods based on spherical harmonics [51,46,50,49,19]. Shen et al. [40] first introduced pure quaternion features that are equivalent to 3D vectors. Deng et al. [17] proposed a similar design with improved nonlinear layers. Satorras et al. [39] proposed to aggregate vector inputs in graph message passing, but without vector nonlinearities involved. Leveraging on existing work [40,17,39], we introduce hybrid vector and scalar neural features for better performance and efficiency. We also adapt the paradigm for scale equivariance, for the first time in literature.
+
+# 3 A Definition of Equivariance for Implicit Representations
+
+While equivariance to common geometric transformations is widely studied for explicit representations of 2D and 3D data [13,51,46], the property for implicit representations that encode signals in a function space is more challenging since continuous queries are involved, yet an important problem for 3D reconstruction. We discuss standard 3D implicit functions and then define equivariance for the representation.
+
+3D implicit representations. We build our model on neural 3D occupancy field functions [30], widely used as a shared implicit representation for a collection of 3D shapes. Given an observation  $\mathbf{X} \in \mathcal{X}$  of a 3D shape, the conditional implicit representation of the shape,  $F(\cdot|\mathbf{X}) : \mathbb{R}^3 \to [0,1]$ , is a 3D scalar field that maps the 3D Euclidean domain to occupancy probabilities, indicating whether there is a surface point at the coordinate. In this work, we consider  $\mathbf{X}$  as a sparse 3D point cloud, such that the information from the observation is indifferent under an arbitrary global transformation, as required for equivariance. For each 3D query coordinate  $\vec{p} \in \mathbb{R}^3$ , the conditioned implicit representation is in the form of
+
+$$
+F (\vec {p} | \mathbf {X}) = \Psi (\vec {p}, \vec {z}) = \Psi (\vec {p}, \Phi (\mathbf {X})), \tag {1}
+$$
+
+where  $\varPhi(\mathbf{X})=\vec{z}\in\mathbb{R}^{C_{\vec{z}}}$  is the latent code in the form of a  $C_{\vec{z}}$ -dimensional vector from the observation  $\mathbf{X}$ , and  $\varPhi$  is the latent feature extractor that encodes the observation data  $\mathbf{X}$ .  $\varPsi$  is the implicit decoder implemented using a multi-layered perceptron with
+
+ReLU activations. Occupancy probabilities are obtained by the final sigmoid activation function of the implicit decoder. Following [30,34], the model is trained with binary cross-entropy loss supervised by ground truth occupancy. The underlying shape surface is the 2-manifold  $\{\vec{p^{\prime}} |F(\vec{p^{\prime}} |\mathbf{X}) = \tau \}$ , where  $\tau \in (0,1)$  is the surface decision boundary.
+
+Preliminary of equivariance. Consider a set of transformations  $T_{g} : \mathcal{X} \to \mathcal{X}$  on a vector space  $\mathcal{X}$  for  $g \in G$ , where  $G$  is an abstract group. Formally,  $T_{g} = T(g)$  where  $T$  is a representation of group  $G$ , such that  $\forall g, g' \in G, T(gg') = T(g)T(g')$ . In the case that  $G$  is the 3D rotation group SO(3),  $T_{g}$  instantiates a 3D rotation matrix for a rotation denoted by  $g$ . We say a function  $\Xi : \mathcal{X} \to \mathcal{Y}$  is equivariant with regard to group  $G$  if there exists  $T_{g}^{*} : \mathcal{Y} \to \mathcal{Y}$  such that for all  $g \in G$ :  $T_{g}^{*} \circ \Xi = \Xi \circ T_{g}$ . We refer to [13,46] for more detailed background theory. Next, we define equivariance for implicit representations as follows:
+
+Definition 1 (Equivariant 3D implicit functions). Given a group  $G$  and the 3D transformations  $T_{g}$  with  $g \in G$ , the conditioned implicit function  $F(\cdot|\mathbf{X})$  is equivariant with regard to  $G$ , if
+
+$$
+F \left(\cdot \mid T _ {g} (\mathbf {X})\right) = T _ {g} ^ {*} \left(F (\cdot | \mathbf {X})\right), \quad \text {f o r a l l} g \in G, \mathbf {X} \in \mathcal {X}. \tag {2}
+$$
+
+where the transformation  $T_{g}^{*}$  applied on the implicit function associated to  $T_{g}$  is applying the inverse coordinate transform on query coordinates  $T_{g}^{*}(F(\cdot |\mathbf{X}))\equiv F(T_{g}^{-1}(\cdot)|\mathbf{X})$ .
+
+Remark 1 Eq. (2) can be reformulated more intuitively as:
+
+$$
+F (\cdot | \mathbf {X}) = F \left(T _ {g} (\cdot) \mid T _ {g} (\mathbf {X})\right), \quad \text {f o r a l l} g \in G, \mathbf {X} \in \mathcal {X}. \tag {3}
+$$
+
+Eq. (3) indicates that the equivariance is satisfied if for any observation  $\mathbf{X}$  and query  $\vec{p}$ , the implicit output of  $F(\vec{p}|\mathbf{X})$  is locally invariant to any  $T_g$  applied jointly to  $\mathbf{X}$  and  $\vec{p}$  in the implicit model.
+
+# 4 Transformation-robust Graph Local Implicit Representations
+
+Our goal is to design an equivariant implicit function model using local feature embeddings to capture fine details of the 3D geometry. However, existing grid-based local implicit functions are sensitive to geometric transformations such as rotations, thus not suitable for equivariant implicit representations. To address this limitation, we propose a graph-based local embedding which is robust to geometric transformations.
+
+Background: grid local implicit representations. To overcome the limitation of a global latent feature, recent methods, such as ConvONet [34] and IF-Net [11], propose to learn spatially-varying latent features  $\mathbf{z}_{\mathbf{p}} = \varPhi_{\mathrm{grid}}(\mathbf{p};\mathbf{X})$ . The main idea is to partition the 3D space into a grid and compute latent codes locally. Specifically, these methods formulate the local latent implicit function on the grid as  $F_{\mathrm{grid}}(\mathbf{p}|\mathbf{X}) = \varPsi(\mathbf{p},\mathbf{z}_{\mathbf{p}}) = \varPsi(\mathbf{p},\varPhi_{\mathrm{grid}}(\mathbf{p};\mathbf{X}))$ , where the grid local latent extractor  $\varPhi_{\mathrm{grid}}$  is further decomposed as  $\varPhi_{\mathrm{grid}}(\mathbf{p};\mathbf{X}) = \psi_{\mathrm{grid}}(\mathbf{p},\phi_{\mathrm{grid}}(\mathbf{X}))$ . The function  $\phi_{\mathrm{grid}}$  is the grid feature encoder that learns to generate a 2D or 3D grid-based feature tensor  $\mathbf{M}$  from the entirety of point observations  $\mathbf{X}$ . For each grid location, point features are aggregated for all  $\mathbf{x}_i \in \mathbf{X}$  in
+
+![](images/ceff0c5b5cc6314f74551c13c71743338c7dd3aa2c77bee821a36ea98284aea0.jpg)  
+Fig. 2: Graph (left) vs. grid (right) local implicit feature embeddings under rotation.  $\varPhi_{\mathrm{graph}}$  extracts  $k$ -NN graphs and applies graph convolutions;  $\varPhi_{\mathrm{grid}}$  partitions points into regular grids and applies regular convolutions. Left: given a point cloud observation  $\mathbf{X}$  (navy) (i), our method aggregates the local latent feature at any query coordinate  $\mathbf{p}$  (orange) from a local  $k$ -NN graph connecting its neighbours in  $\mathbf{X}$  (ii). Moreover, when a transformation  $T_{g}$ , e.g. rotation, is applied to the shape, (iv) the constructed local graph is in the same structure as (v) applying  $T_{g}$  to the graph from untransformed data. Right: in contrast, (vi) visualizes discretized grid features. (vii) The off-the-grid query location  $\mathbf{p}$  interpolates the neighboring on-grid features. However, (viii) with  $T_{g}$  applied to the raw observation, often (ix) the sub-grid point patterns for the local features are different from (x) applying  $T_{g}$  to the untransformed local grids. This makes the grid local implicit models sensitive to transformations such as rotation.
+
+![](images/9478f766032d09012ee83cbed968e9a7e87475da547590cc9abfa4193012f985.jpg)
+
+the corresponding bin. Convolutional layers are applied to the grid-based  $\mathbf{M}$  to capture multi-scale information and maintain translation equivariance. Given the local 3D feature tensor  $\mathbf{M}$ , the local latent aggregator  $\psi_{\mathrm{grid}}$  computes local latent feature  $\mathbf{z}_{\mathbf{p}}$  on any off-the-grid query coordinate  $\mathbf{p}$  using a simple trilinear interpolation. We refer to [25,5] for other variants of grid-based representations using purely local information without global cues, while restricted by requiring additional priors during inference such as the normals. Overall, grid partitioning is not robust to general transformations such as rotations, especially when the sub-grid structure is considered for the resolution-free implicit reconstruction. Fig. 2 (right) shows an illustration of this limitation, which we will address in our method.
+
+# 4.1 Graph-structured local implicit feature embeddings
+
+We propose to use graphs as a non-regular representation, such that our local latent feature function is robust to these transformations and free from feature grid resolutions. The graph-local implicit function extends the standard form of Eq. (1) to
+
+$$
+F _ {\text {g r a p h}} (\mathbf {p} | \mathbf {X}) = \Psi (\mathbf {p}, \mathbf {z} _ {\mathbf {p}}) = \Psi (\mathbf {p}, \Phi_ {\text {g r a p h}} (\mathbf {p}; \mathbf {X})) = \Psi (\mathbf {p}, \psi (\mathbf {p}, \phi (\mathbf {X})). \tag {4}
+$$
+
+$\varPhi_{\mathrm{graph}}$  is a deep network that extracts local latent features  $\mathbf{z}_{\mathbf{p}}$  on the graph, composed of two sub-networks  $\phi$  and  $\psi$ . The point feature encoder  $\phi$  maps the point set  $\mathbf{X} =$
+
+![](images/41bc47af413e5327a3d86ee3b65dd11da130362ed9619453f1c617f710dd62fa.jpg)  
+Fig. 3: Multi-scale design, with enlarged receptive fields of local  $k$ -NN graphs for the graph point encoder  $\phi$  (orange) and the graph local latent aggregator  $\psi$  (violet).
+
+$\{\mathbf{x}_i\}$  to the associated features  $\{\mathbf{h}_i\}$ , and is invariant to the sampled query location  $\mathbf{p}$ . The graph local latent feature aggregator  $\psi$  propagates the input point feature to the query coordinate  $\mathbf{p}$ . Unlike grid-based methods, we directly aggregate local information from the point cloud feature  $\{(\mathbf{x}_i,\mathbf{h}_i)\}$  without an intermediate grid feature tensor. In particular, we construct a local  $k$ -nearest neighbor ( $k$ -NN) graph  $(\mathcal{V},\mathcal{E})$  for every query point  $\mathbf{p}$ , where the vertices  $\mathcal{V} = \mathbf{X} \cup \{\mathbf{p}\}$  include the point set elements and the query coordinate. The edges  $\mathcal{E} = \{(\mathbf{p},\mathbf{x}_i)\}$  are between the query point  $\mathbf{p}$  and its  $k$ -NN points from the observation point set  $\mathbf{x}_{i'} \in \mathcal{N}_k(\mathbf{p},\mathbf{X})$ , with  $\mathcal{N}_k(\mathbf{p},\mathbf{X})$  denoting the set of the  $k$ -NN points of  $\mathbf{p}$  from  $\mathbf{X}$ . Last, with graph convolutions, we aggregate into the local feature vector  $\mathbf{z}_{\mathbf{p}}$  the point features of the neighbors of  $\mathbf{p}$ . We adopt a simple spatial graph convolution design in the style of Message Passing Neural Network (MPNN) [23], which is widely used for 3D shape analysis [48,24]. For each neighboring point  $\mathbf{x}_{i'}$  from the query  $\mathbf{p}$ , messages are passed through a function  $\eta$  as a shared two-layer ReLU-MLP, where the inputs are the point features  $\mathbf{h}_{i'}$  and the query coordinate  $\mathbf{p}$  as node feature as well as the displacement vector  $\mathbf{x}_{i'} - \mathbf{p}$  as the edge feature, followed by a permutation-invariant aggregation AGGRE over all neighboring nodes, e.g., max- or mean-pooling:
+
+$$
+\mathbf {z} _ {\mathbf {p}} = \underset {i ^ {\prime}} {\operatorname {A G G R E}} \eta (\mathbf {p}, \mathbf {h} _ {i ^ {\prime}}, \mathbf {x} _ {i ^ {\prime}} - \mathbf {p}). \tag {5}
+$$
+
+For each neighboring point as a graph node, the edge function  $\eta$  take as inputs, the query coordinate  $\mathbf{p}$ , the node point feature  $\mathbf{h}_{i'}$ , and the relative position  $\mathbf{x}_{i'} - \mathbf{p}$ .
+
+As all the graph connections are relative between vertices, the local latent feature aggregation is robust to transformations like rotations, as illustrated in Fig. 2 (left).
+
+# 4.2 Learning multi-scale local graph latent features
+
+To capture the context of the 3D geometry at multiple scales, both ConvONet [34] and IF-Net [11] rely on a convolutional U-Net [37], with progressively downsampled and then upsampled feature grid resolutions to share neighboring information at different scales. Our graph model enables learning at multiple scales by farthest point sampling (FPS). That is, we downsample the point set  $\mathbf{X}$  to  $\mathbf{X}^{(l)}$  at sampling levels  $l = 1,\dots ,L$  with a progressively smaller cardinality  $|\mathbf{X}^{(l)}| < |\mathbf{X}^{(l - 1)}|$ , where  $\mathbf{X}^{(0)} = \mathbf{X}$  is the original set.
+
+Moreover, we use a graph encoder for the point encoder  $\phi$ , instead of PointNet [35]. This way, without involving regular grid convolutions, we can still model local features and facilitate a translation-equivariant encoder, which is beneficial in many scenarios,
+
+especially for learning scene-level implicit surfaces [34]. Next, we sketch the multi-scale graph point encoder and latent feature aggregator, with Fig. 3 as a conceptual illustration.
+
+The graph point encoder  $\phi$  learns point features  $\{\mathbf{h}_i^{(l)}\}$  for the corresponding points  $x_{i} \in \mathbf{X}^{(l)}$  at each sampling level  $l = 0, \dots, L$ . The encoder starts from the initial sampling level  $l = 0$  where the input features are the raw coordinates. At each sampling level, a graph convolution is applied to each point to aggregate message from its local  $k$ -nearest neighbor point features, followed by an FPS operation to the downsampled level  $l + 1$ . The graph convolution is similar to that in Eq. (5), with the point features from both sides of the edge and the relative position as inputs. The graph convolutions and FPS downsampling are applied until the coarsest sampling level  $l = L$ . Then the point features are sequentially upsampled back from  $l = L$  to  $l = L - 1$ , until  $l = 0$ . At each sampling level  $l$ , the upsampling layer is simply one linear layer followed by ReLU activation. For each point, the input of the upsampling layer is the nearest point feature from the last sampling level  $l + 1$ , and the skip-connected feature of the same point at the same sampling level from the downsampling stage. Thus far, we obtain multi-scale point features  $\{(\mathbf{x}_i, \mathbf{h}_i^{(l)})\}$  for  $\mathbf{x}_i \in \mathbf{X}^{(l)}$  at sampling levels  $l = 0, \dots, L$ , as the output from the graph point encoder  $\phi$ .
+
+For the graph latent feature aggregator  $\psi$ , at each query coordinate  $\mathbf{p}$ , we use graph convolutions to aggregate the  $k$ -neighboring features  $\{(\mathbf{x}_i, \mathbf{h}_i^{(l)})\}$  at different sampling levels  $l$ , as described in Sec 4.1. The aggregated features from all sampling levels  $l$  are concatenated to yield the local latent vector  $\mathbf{z}_{\mathbf{p}}$  as output. The detailed formulations of  $\phi$  and  $\psi$  are provided in the supplementary material.
+
+# 5 Equivariant Graph Implicit Functions
+
+The local graph structure of the proposed implicit function, with  $\mathbf{X} \cup \{\mathbf{p}\}$  as the set of vertices, is in line with the requirement of equivariance in Sec. 3 and Eq. (3). As a result, the local graph implicit embedding can be used for an equivariant model to achieve theoretically guaranteed generalization to unseen transformations. To do this, we further require all the graph layers in the latent extractor  $\varPhi_{\mathrm{graph}}$  to be equivariant in order to obtain equivariant local latent feature. We present the equivariant layers for 3D rotation group  $G = SO(3)$ , a difficult case for implicit functions [16].
+
+To build the equivariant model from equivariant layers, we additionally remove the query coordinate input  $\mathbf{p}$  to ensure the implicit decoder  $\varPsi$  spatially invariant in Eq. (4), as the local spatial information is already included in the latent  $\mathbf{z}_{\mathbf{p}}$ . See Appendix A.1 for details. We also extend the method to other similarity transformations, such as translation and scaling. See Appendix A.2.
+
+# 5.1 Hybrid feature equivariant layers
+
+Our equivariant layers for the graph convolution operations is inspired by recent methods [17,40] that lift from a regular neuron feature  $h \in \mathbb{R}$  to a vector  $\mathbf{v} \in \mathbb{R}^3$  to encode rotation, and the list of 3D vector features  $\mathbf{V} = [\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_{C_{\mathbf{v}}}]^\top \in \mathbb{R}^{C_{\mathbf{v}} \times 3}$  that substitutes the regular features  $\mathbf{h} \in \mathbb{R}^{C_h}$ . However, using only vector features in the
+
+![](images/fb4b48c22102ac0a846bffc712badb3e6a6f7ba2717ee64b7b7833ca6b23c80c.jpg)  
+Fig. 4: Hybrid feature equivariant layers. Visualization of how vector and scalar features share information in linear and nonlinear layers. Vector features go through an invariant function  $\Omega$  that is added to the scalar part. Scalar features are transformed with normalizing  $\frac{\cdot}{\|\cdot\|}$  to scale the vector feature channels.
+
+network is non-optimal for both effectiveness and efficiency, with highly regularized linear layers and computation-demanding nonlinearity projections.
+
+To this end, we extend the method and propose hybrid features  $\{\mathbf{s},\mathbf{V}\}$  to replace the regular feature  $\mathbf{h}$ , where vector features  $\mathbf{V}$  encode rotation equivariance, and scalar features  $\mathbf{s} \in \mathbb{R}^{C_s}$  are rotation-invariant. In practice, hybrid features show improved performance and computation efficiency by transferring some learning responsibility to the scalar features through more powerful and efficient standard neural layers.
+
+Linear layers. For input hybrid hidden feature  $\{\mathbf{s},\mathbf{V}\}$  with  $\mathbf{s}\in \mathbb{R}^{C_s}$ , and  $\mathbf{V} = [\mathbf{v}_1,\mathbf{v}_2,\dots ,\mathbf{v}_{C_{\mathbf{v}}}]^\top \in \mathbb{R}^{C_{\mathbf{v}}\times 3}$ , we define a set of weight matrices,  $\mathbf{W}_s\in \mathbb{R}^{C_s'\times C_s}$ ,  $\mathbf{W}_{\mathbf{v}}\in \mathbb{R}^{C_{\mathbf{v}}'\times C_{\mathbf{v}}}$ ,  $\mathbf{W}_{s\mathbf{v}}\in \mathbb{R}^{C_{s}^{\prime}\times C_{s}}$ , and  $\mathbf{W}_{\mathbf{v}s}\in \mathbb{R}^{C_s'\times C_v}$  for the linear transformation, with information shared between scalar and vector features in the inputs. The resulting output features  $\{\mathbf{s}',\mathbf{V}'\}$  become:
+
+$$
+\mathbf {s} ^ {\prime} = \mathbf {W} _ {s} \mathbf {s} + \mathbf {W} _ {\mathbf {v} s} \Omega (\mathbf {V}) \tag {6}
+$$
+
+$$
+\mathbf {V} ^ {\prime} = \mathbf {W} _ {\mathbf {v}} \mathbf {V} \odot \left(\mathbf {W} _ {s \mathbf {v}} \mathbf {s} / \| \mathbf {W} _ {s \mathbf {v}} \mathbf {s} \|\right) \tag {7}
+$$
+
+where  $\odot$  is channel-wise multiplication between  $\mathbf{W}_{\mathbf{v}}\mathbf{V} \in \mathbb{R}^{C_{\mathbf{s}}^{\prime} \times 3}$  and  $\mathbf{W}_{s\mathbf{v}}\mathbf{h} \in \mathbb{R}^{C_{\mathbf{v}}^{\prime} \times 1}$ , and the normalized transformed scalar feature  $\mathbf{W}_{s\mathbf{v}}\mathbf{s} / \| \mathbf{W}_{s\mathbf{v}}\mathbf{s}\|$  learns to scale the output vectors in each channel;  $\varOmega(\cdot)$  is the invariance function that maps the rotation equivariant vector feature  $\mathbf{V} \in \mathbb{R}^{C_{\mathbf{v}} \times 3}$  to the rotation-invariant scalar feature  $\varOmega(\mathbf{V}) \in \mathbb{R}^{C_{\mathbf{v}}}$ , to be added on the output scalar feature. The design of the invariance function  $\varOmega(\cdot)$  is introduced later in Eq. (9).
+
+Nonlinearities. Nonlinearities apply separately to scalar and vector features. For scalar features, it is simply a  $\mathrm{ReLU}(\cdot)$ . For vector features, the nonlinearity  $\mathbf{v}$ - $\mathrm{ReLU}(\cdot)$  adopts the design from Vector Neurons [17]: the vector feature  $\mathbf{v}_c$  at each channel  $c$  takes an inner-product with a learnt direction  $\mathbf{q} = [\mathbf{W_qV}]^\top \in \mathbb{R}^3$  from a linear layer  $\mathbf{W_q} \in \mathbb{R}^{1 \times C_v}$ . If the inner-product is negative,  $\mathbf{v}_c$  is projected to the plane perpendicular to  $\mathbf{q}$ .
+
+$$
+[ \mathbf {v} - \operatorname {R e L U} (\mathbf {V}) ] _ {c} = \left\{ \begin{array}{l l} \mathbf {v} _ {c} & \text {i f} \left\langle \mathbf {v} _ {c}, \frac {\mathbf {q}}{\| \mathbf {q} \|} \right\rangle \geq 0, \\ \mathbf {v} _ {c} - \left\langle \mathbf {v} _ {c}, \frac {\mathbf {q}}{\| \mathbf {q} \|} \right\rangle \frac {\mathbf {q}}{\| \mathbf {q} \|}. & \text {o t h e r w i s e .} \end{array} \right. \tag {8}
+$$
+
+Invariance layer. The invariance function  $\Omega(\cdot)$  maps the rotation equivariant vector feature  $\mathbf{V} \in \mathbb{R}^{C_{\mathbf{v}} \times 3}$  to a rotation-invariant scalar feature  $\Omega(\mathbf{V}) \in \mathbb{R}^{C_{\mathbf{v}}}$  with the same
+
+Table 1: ShapeNet implicit reconstruction from sparse noisy point clouds. The grid resolutions are marked for ConvONet and IF-Net.  
+
+<table><tr><td></td><td colspan="4">SO(3) equiv. Mean IoU ↑ Chamfer-ℓ1 ↓ Normal consist. ↑</td></tr><tr><td>ONet</td><td>×</td><td>0.736</td><td>0.098</td><td>0.878</td></tr><tr><td>ConvONet-2D (3×642)</td><td>×</td><td>0.884</td><td>0.044</td><td>0.938</td></tr><tr><td>ConvONet-3D (323)</td><td>×</td><td>0.870</td><td>0.048</td><td>0.937</td></tr><tr><td>IF-Net (1283)</td><td>×</td><td>0.887</td><td>0.042</td><td>0.941</td></tr><tr><td>GraphONet (ours)</td><td>×</td><td>0.904</td><td>0.038</td><td>0.946</td></tr><tr><td>VN-ONet</td><td>✓</td><td>0.694</td><td>0.125</td><td>0.866</td></tr><tr><td>E-GraphONet-SO(3) (ours)</td><td>✓</td><td>0.890</td><td>0.041</td><td>0.936</td></tr></table>
+
+![](images/9e6e4751e1eed50e1f458295cdc43f7d339971200cdf0ab6e18d4b2cfdeecd00.jpg)  
+Fig.5: ShapeNet object reconstruction in canonical space. GraphONet is among the best performing non-equivariant implicit representation methods; E-GraphONet significantly improves on the existing equivariant method VN-ONet [17].
+
+channel dimension  $C_{\mathbf{v}}$ . At each layer  $c = 1, \dots, C_{\mathbf{v}}$ ,  $[\Omega(\mathbf{V})]_c \in \mathbb{R}$  takes the inner product of  $\mathbf{v}_c$  with the channel-averaged direction:
+
+$$
+[ \varOmega (\mathbf {V}) ] _ {c} = \left\langle \mathbf {v} _ {c}, \frac {\overline {{\mathbf {v}}}}{\| \overline {{\mathbf {v}}} \|} \right\rangle , \tag {9}
+$$
+
+where  $\overline{\mathbf{v}} = \frac{1}{C_{\mathbf{v}}} \sum_{c'=1}^{C_{\mathbf{v}}} \mathbf{v}_{c'}$  is the channel-averaged vector feature. One can verify that applying any rotation on the vector feature does not change the inner product. Our  $\Omega$  adopts from [17] with modification. Ours is parameter-free using averaged direction, while [17] learns this vector.
+
+The invariance function is applied in each linear layer in Eq. (7) to share the information between vector and scalar parts of the feature. In addition, at the end of equivariant graph feature network  $\varPhi_{\mathrm{graph}}$ ,  $\varOmega(\mathbf{V})$  is concatenated with the scalar feature as the final invariant local latent feature  $\mathbf{z}_{\mathbf{p}} = \varOmega(\mathbf{V})\|\mathbf{s}$ , as to be locally invariant with transformed  $\mathbf{p}$  and  $\mathbf{X}$  in line with Eq. (3).
+
+# 6 Experiments
+
+We experiment on implicit surface reconstruction from sparse and noisy point observations. In addition, we evaluate the implicit reconstruction performance under random
+
+Table 2: Implicit surface reconstruction with random rotation, IoU↑. The left table evaluates non-equivariant methods, and the right one compares the best performing model with equivariant methods.  $I$  denotes canonical pose and  $SO(3)$  random rotation.  $I / SO(3)$  denotes training with canonical pose and test with random rotation, and so on. *:  
+models not re-trained with augmentation due to guaranteed equivariance.  
+
+<table><tr><td>training / test</td><td>1/1</td><td>I/SO(3)</td><td>SO(3)/SO(3)</td></tr><tr><td>ONet</td><td>0.742</td><td>0.271 [-0.471]</td><td>0.592 [-0.150]</td></tr><tr><td>ConvONet-2D</td><td>0.884</td><td>0.568 [-0.316]</td><td>0.791 [-0.093]</td></tr><tr><td>ConvONet-3D</td><td>0.870</td><td>0.761 [-0.109]</td><td>0.838 [-0.032]</td></tr><tr><td>GraphONet (ours)</td><td>0.904</td><td>0.846 [-0.058]</td><td>0.887 [-0.017]</td></tr><tr><td>VN-ONet</td><td colspan="2">SO(3)</td><td>0.694 0.694 [-0.000]</td></tr><tr><td>E-GraphONet (ours)</td><td colspan="2">SO(3)</td><td>0.890 0.890 [-0.000]</td></tr></table>
+
+transformations of rotation, translation and scaling. Our method is referred to as Graph Occupancy networks, or GraphONet, while E-GraphONet is the equivariance model with 3 variants: SO(3), SE(3) and similarity transformations (Sim.).
+
+Implementation details. We implement our method using PyTorch [33]. The number of neighbours in  $k$ -NN graph is set as 20. For the multi-scale graph implicit encoder, we take  $L = 2$ , i.e., the point set is downsampled twice with farthest point sampling (FPS) to  $20\%$  and  $5\%$  of the original cardinality respectively. The permutation invariant function AGGRE adopts mean-pooling for vector features and max-pooling for scalar features. We use an Adam optimizer [27] with  $\gamma = 10^{-3}$ ,  $\beta_{1} = 0.9$  and  $\beta_{2} = 0.999$ . Main experiments are conducted on the ShapeNet [6] dataset with human designed objects, where the train/val/test splits follow prior work [12,34] with 13 categories. Following [34], we sample 3000 surface points per shape and apply Gaussian noise with 0.005 standard deviation. More details are provided in the Appendix. Code is available at https://github.com/yunlu-chen/equivariant-graph-implicit.
+
+# 6.1 Canonical-posed object reconstruction
+
+We experiment on ShapeNet object reconstruction following the setups in [34]. For quantitative evaluation in Table 1, we evaluate the IoU, Chamfer- $\ell_{1}$  distance and normal consistency, following [30,34]. The qualitative results are shown in Fig. 5. Our GraphONet outperforms the state-of-the-arts methods ConvONet [34] and IF-Net [11], as our graph-based method aggregates local feature free of spatial grid resolution and captures better local details. IF-Net is better than ConvONet but at large memory cost per batch with  $128^{3}$  resolution grid feature. For rotation equivariant models, our E-GraphONet significantly outperforms VN-ONet [17], benefiting from the graph local features.
+
+# 6.2 Evaluation under geometric transformations
+
+Rotation. First, we investigate how implicit models perform under random rotations, which is challenging for neural implicits [16]. In Table 2 left, GraphONet shows the smallest performance drop among all non-equivariant methods under rotations, either with  $(SO(3) / SO(3))$  or without augmentation  $(I / SO(3))$  during training, since the graph structure is more robust to rotations. ConvONet [34]-2D is more sensitive than the 3D
+
+Table 3: Implicit surface reconstruction performance under various types of seen/unseen transformations, mIoU↑.  
+
+<table><tr><td>Transformation(s) Training augmentation</td><td>equiv.</td><td>- -</td><td colspan="2">translation × √</td><td colspan="2">scale × √</td><td colspan="2">rot. &amp; transl. × √</td><td colspan="2">rot. &amp; scale × √</td><td colspan="2">transl. &amp; scale × √</td><td>all × √</td><td></td></tr><tr><td>ONet</td><td>×</td><td>0.738</td><td>0.221</td><td>0.716</td><td>0.423</td><td>0.685</td><td>0.154</td><td>0.585</td><td>0.235</td><td>0.591</td><td>0.202</td><td>0.713</td><td>0.121</td><td>0.573</td></tr><tr><td>ConvONet-2D (3×1282)</td><td>ׇ</td><td>0.882</td><td>0.791</td><td>0.878</td><td>0.812</td><td>0.850</td><td>0.532</td><td>0.771</td><td>0.542</td><td>0.789</td><td>0.723</td><td>0.838</td><td>0.481</td><td>0.728</td></tr><tr><td>ConvONet-3D (643)</td><td>ׇ</td><td>0.861</td><td>0.849</td><td>0.856</td><td>0.797</td><td>0.837</td><td>0.759</td><td>0.836</td><td>0.742</td><td>0.832</td><td>0.771</td><td>0.835</td><td>0.721</td><td>0.803</td></tr><tr><td>GraphONet (ours)</td><td>ׇ</td><td>0.901</td><td>0.884</td><td>0.898</td><td>0.857</td><td>0.893</td><td>0.837</td><td>0.881</td><td>0.798</td><td>0.880</td><td>0.852</td><td>0.888</td><td>0.798</td><td>0.874</td></tr><tr><td>VN-ONet</td><td>SO(3)</td><td>0.682</td><td>0.354</td><td>0.667</td><td>0.516</td><td>0.662</td><td>0.357</td><td>0.658</td><td>0.511</td><td>0.666</td><td>0.360</td><td>0.638</td><td>0.309</td><td>0.615</td></tr><tr><td>E-GraphONet (ours)</td><td>SO(3)‡</td><td>0.887</td><td>0.823</td><td>0.876</td><td>0.824</td><td>0.880</td><td>0.825</td><td>0.877</td><td>0.825</td><td>0.882</td><td>0.726</td><td>0.872</td><td>0.729</td><td>0.870</td></tr><tr><td>E-GraphONet (ours)</td><td>SE(3)</td><td>0.884</td><td>0.884</td><td>0.884*</td><td>0.840</td><td>0.880</td><td>0.884</td><td>0.884*</td><td>0.838</td><td>0.880</td><td>0.841</td><td>0.878</td><td>0.840</td><td>0.878</td></tr><tr><td>E-GraphONet (ours)</td><td>Sim.†</td><td>0.882</td><td>0.882</td><td>0.882*</td><td>0.882</td><td>0.882*</td><td>0.882</td><td>0.882*</td><td>0.882</td><td>0.882*</td><td>0.882</td><td>0.882*</td><td>0.882</td><td>0.882*</td></tr></table>
+
+$^{\dagger}$ Similarity transformation group.  $\ddagger$  The (graph) convolution subnetwork is translation equivariant. * with no augmentation due to guaranteed equivariance.
+
+![](images/52c128cff34d5746df144daac4692d6ce6949eb8d3271272544bbce9a4ff0e88.jpg)  
+Fig. 6: Implicit surface reconstruction under unseen transforms. Visualizing back-transformed shapes. Shapes are scaled by a factor of 0.25, and translation is from the unit cube center to the corner. E-GraphONet-Sim is robust to all similarity transformations. See the appendix video for reconstructions under different poses and scales.
+
+version, as 3D rotation would lead to highly distinct 2D projections. In Table 2 right, E-GraphONet, equipped with equivariant layers, achieves better performance under random rotations, even when the non-equivariant methods are trained with augmentation. It also outperforms the previous equivariant method VN-ONet [17] by a large margin.
+
+Scale, translation and combinations. We evaluate how implicit methods perform under various similarity transformations besides SO(3) rotation, including scale, translation and combinations. We apply random scales and rotations in a bounded unit cube  $[-0.5, 0.5]^3$ , as assumed by grid methods, and set the canonical scale to be half of the cube. ConvONet resolutions are doubled to keep the effective resolution. Random scaling and translation are added under the constraint of the unit bound, with the minimum scaling factor of 0.2.
+
+From the results in Table 3, GraphONet is more robust to transformations than other non-equivariant models. For equivariant models, VN-ONet and the SO(3) E-GraphONet models perform poorly on other types of transformations, as they are optimized towards the rotation around origin only. Similarly, the SE(3) E-GraphONet does not generalize to scaling. Our model with full equivariance performs well on all similarity transformations
+
+Table 4: Parameter- and data-efficiency. Evaluated on the full test set with 5 runs of 130 randomly sampled training examples.  
+
+<table><tr><td></td><td>#param.</td><td>IoU↑</td></tr><tr><td>ConvONet-2D</td><td>2.0 × 106</td><td>0.727 ±0.009</td></tr><tr><td>ConvONet-3D</td><td>1.1 × 106</td><td>0.722 ±0.008</td></tr><tr><td>GraphONet (ours)</td><td>1.9 × 105</td><td>0.867 ±0.004</td></tr><tr><td>E-GraphONet (ours)</td><td>6.5 × 104</td><td>0.873 ±0.002</td></tr></table>
+
+![](images/acce6418b4e1b2593b334c16c594374a2d3159a8105ac3607e1c5a3d07f456bd.jpg)  
+Fig. 7: Ablation on hybrid feature channels. Mixed vectors and scalars are more effective and efficient than pure vectors.
+
+with numerically the same performance. Fig 6 shows qualitative examples, where our E-GraphONet-Sim handles all types of unseen similarity transformations.
+
+# 6.3 Analysis
+
+We show some ablation experiments while more results are provided in the Appendix.
+
+Learning from very few training examples. We show that our graph method is both parameter- and data-efficient, and the transform-robust modeling inherently benefits generalization. As reported in Table 4, we use less than  $10\%$  of the parameters of ConvONet as the graph conv kernel shares parameters for all directions. We evaluate the test set performance when training on only 130 examples - 10 per class - instead of the full training set size of 30661. While ConvONets fail to achieve good performance, GraphONets does not drop by far from the many-shot results in Table 1. The E-GraphONet demonstrates even better performance, with more parameter-sharing from the equivariance modeling, indicating better power of generalization.
+
+Ablation on vector and scalar feature channels. We validate our design of hybrid features by experimenting different ratio of vector and scalar channels. We constrain in total 48 effective channels, with one vector channel counted as three scalars. In Fig. 7, using both vectors and scalars with a close-to-equal ratio of effective channels obtains higher performance with less memory cost than using pure vectors, i.e., in the Vector Neurons [17]. This indicates the expressive power of the scalar neuron functions. We provide additional results for non-graph-based equivariant models in the Appendix.
+
+# 6.4 Scene-level reconstructions
+
+In addition to the ability of handling object shape modeling under transformations, our graph implicit functions also scale to scene-level reconstruction. We experiment on two datasets: (i) Synthetic Rooms [34], a dataset provided by [34], with rooms constructed with walls, floors, and ShapeNet objects from five classes: chair, sofa, lamp, cabinet and table.
+
+Table 5: Indoor scene reconstructions.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">Synthetic room</td><td>ScanNet</td></tr><tr><td>IoU↑</td><td>Chamfer↓</td><td>Normal↑</td><td>Chamfer↓</td></tr><tr><td>ONet</td><td>0.514</td><td>0.135</td><td>0.856</td><td>0.546</td></tr><tr><td>ConvONet-2D (3×1282)</td><td>0.802</td><td>0.038</td><td>0.934</td><td>0.162</td></tr><tr><td>ConvONet-3D (643)</td><td>0.847</td><td>0.035</td><td>0.943</td><td>0.067</td></tr><tr><td>GraphONet (ours)</td><td>0.883</td><td>0.032</td><td>0.944</td><td>0.061</td></tr><tr><td>E-GraphONet (ours)</td><td>0.851</td><td>0.035</td><td>0.934</td><td>0.069</td></tr></table>
+
+![](images/782bfc035fecdb1544f8b9c681be7cd338619b026d125b0e6c96a22b3168ca17.jpg)  
+Fig. 8: Indoor scene reconstructions on Synthetic rooms (left) and ScanNet (right). Our GraphONet produces shaper edges and better finer details.
+
+(ii) ScanNet [15], a dataset of RGB-D scans of real-world rooms for testing synthetic-to-real transfer performance.
+
+We train and evaluate our model on Synthetic room dataset [34] using 10,000 sampled points as input. The quantitative results are shown in Table 5 and qualitative results in Fig. 8 (left). Our GraphONet performs better than ConvONets [34] at recovering detailed structures. We evaluate the SE(3) variant of our equivariance model, and it performs generally well, but less smooth at flat regions. In addition, we evaluate the model transfer ability of our method on ScanNet [15] with the model trained on synthetic data, for which we report the Chamfer measure in Table 5. Our GraphONet ourperforms other methods. Fig. 8 (right) shows a qualitative example of our reconstructions.
+
+# 7 Conclusion
+
+In this paper, we introduce graph implicit functions, which learn local latent features from  $k$ -NN graph on sparse point set observations, enabling reconstruction of 3D shapes and scenes in fine detail. By nature of graphs and in contrast to regular grid representations, the proposed graph representations are robust to geometric transformations. What is more, we extend the proposed graph implicit functions with hybrid feature equivariant layers, thus guarantee theoretical equivariance under various similarity transformations, including rotations, translations and scales, and obtain models that generalize to arbitrary and unseen transformations.
+
+Acknowledgement This research was supported in part by SAVI/MediFor project, ERC Starting Grant Scan2CAD (804724), EPSRC programme grant Visual AI EP/T028572/1, and National Research Foundation Singapore and DSO National Laboratories under its AI Singapore Programme (AISG Award No: AISG2-RP-2020-016). We thank Angela Dai for the video voice over.
+
+# References
+
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+2. Atzmon, M., Lipman, Y.: Sal: Sign agnostic learning of shapes from raw data. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 2565-2574 (2020) 3  
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+# 3D Face Reconstruction with Dense Landmarks
+
+Erroll Wood, Tadas Baltrusaitis, Charlie Hewitt, Matthew Johnson, Jingjing Shen, Nikola Milosavljevic, Daniel Wilde, Stephan Garbin, Toby Sharp, Ivan Stojiljkovic, Tom Cashman, and Julien Valentin
+
+Microsoft
+
+Abstract. Landmarks often play a key role in face analysis, but many aspects of identity or expression cannot be represented by sparse landmarks alone. Thus, in order to reconstruct faces more accurately, landmarks are often combined with additional signals like depth images or techniques like differentiable rendering. Can we keep things simple by just using more landmarks? In answer, we present the first method that accurately predicts  $10 \times$  as many landmarks as usual, covering the whole head, including the eyes and teeth. This is accomplished using synthetic training data, which guarantees perfect landmark annotations. By fitting a morphable model to these dense landmarks, we achieve state-of-the-art results for monocular 3D face reconstruction in the wild. We show that dense landmarks are an ideal signal for integrating face shape information across frames by demonstrating accurate and expressive facial performance capture in both monocular and multi-view scenarios. Finally, our method is highly efficient: we can predict dense landmarks and fit our 3D face model at over 150FPS on a single CPU thread. Please see our website: https://microsoft.github.io/DenseLandmarks/.
+
+Keywords: Dense correspondences, 3D morphable model, face alignment, landmarks, synthetic data
+
+# 1 Introduction
+
+Landmarks are points in correspondence across all faces, like the tip of the nose or the corner of the eye. They often play a role in face-related computer vision, e.g., being used to extract facial regions of interest [34], or helping to constrain 3D model fitting [26, 79]. Unfortunately, many aspects of facial identity or expression cannot be encoded by a typical sparse set of 68 landmarks alone. For example, without landmarks on the cheeks, we cannot tell whether or not someone has high cheek-bones. Likewise, without landmarks around the outer eye region, we cannot tell if someone is softly closing their eyes, or scrunching up their face.
+
+In order to reconstruct faces more accurately, previous work has therefore used additional signals beyond color images, such as depth images [64] or optical flow [13]. However, these signals may not be available or reliable to compute. Instead, given color images alone, others have approached the problem using analysis-by-synthesis: minimizing a photometric error [26] between a generative 3D face model and an observed image using differentiable rendering [18, 27].
+
+![](images/102d67f0224a99371918f3dd47e7f7f943214081483a2102ff407ef19e9163dc.jpg)  
+Fig. 1. Given a single image (top), we first robustly and accurately predict 703 landmarks (middle). To aid visualization, we draw lines between landmarks. We then fit our 3D morphable face model to these landmarks to reconstruct faces in 3D (bottom).
+
+Unfortunately, these approaches are limited by the approximations that must be made in order for differentiable rendering to be computationally feasible. In reality, faces are not purely Lambertian [23], and many important illumination effects are not explained using spherical harmonics alone [18], e.g., ambient occlusion or shadows cast by the nose.
+
+Faced with this complexity, wouldn't it be great if we could just use more landmarks? We present the first method that predicts over 700 landmarks both accurately and robustly. Instead of only the frontal "hockey-mask" portion of the face, our landmarks cover the entire head, including the ears, eyeballs, and teeth. As shown in Figure 1, these landmarks provide a rich signal for both facial identity and expression. Even with as few as 68, it is hard for humans to precisely annotate landmarks that are not aligned with a salient image feature. That is why we use synthetic training data which guarantees consistent annotations. Furthermore, instead of representing each landmark as just a 2D coordinate, we predict each one as a random variable: a 2D circular Gaussian with position and uncertainty [38]. This allows our predictor to express uncertainty about certain landmarks, e.g., occluded landmarks on the back of the head.
+
+Since our dense landmarks represent points of correspondence across all faces, we can perform 3D face reconstruction by fitting a morphable face model [6] to them. Although previous approaches have fit models to landmarks in a similar way [77], we are the first to show that landmarks are the only signal required to achieve state-of-the-art results for monocular face reconstruction in the wild.
+
+The probabilistic nature of our predictions also makes them ideal for fitting a 3D model over a temporal sequence, or across multiple views. An optimizer can discount uncertain landmarks and rely on more certain ones. We demonstrate this with accurate and expressive results for both multi-view and monocular facial performance capture. Finally, we show that predicting dense landmarks
+
+![](images/3a5f74b674b522b859fce987cc8d1d84571d054520cbba8f0660a5aa4edc3e1c.jpg)  
+a)
+
+![](images/ed0459cc302e2a07ebc48f52aaabb186612e4b6fa846a37a7c4dbd1da757aea3.jpg)  
+b)  
+Fig. 2. Compared to a typical sparse set of 68 facial landmarks (a), our dense landmarks (b) cover the entire head in great detail, including ears, eyes, and teeth. These dense landmarks are better at encoding facial identity and subtle expressions.
+
+![](images/3ade194cc42d64df914ed1448fafabb7167d8dc2bb3105ae9e2a1c48489add5e.jpg)
+
+![](images/7678baf509e28a66885cb671e70e74cca432695460b1a936330a0832a58b0601.jpg)
+
+and then fitting a model can be highly efficient by demonstrating real-time facial performance capture at over 150FPS on a single CPU thread.
+
+In summary, our main contribution is to show that you can achieve more with less. You don't need parametric appearance models, illumination models, or differentiable rendering for accurate 3D face reconstruction. All you need is a sufficiently large quantity of accurate 2D landmarks and a 3D model to fit to them. In addition, we show that combining probabilistic landmarks and model fitting lets us intelligently aggregate face shape information across multiple images by demonstrating robust and expressive results for both multi-view and monocular facial performance capture.
+
+# 2 Related work
+
+Reconstructing faces in 3D from images is a mature field at the intersection of vision and graphics. We focus our literature review on methods that are closer to our own, and refer the reader to Morales et al. [47] for an extensive survey.
+
+Regression-based 3D face reconstruction DNN-based regression has been extensively used as a tool for 3D face reconstruction. Techniques fall into two broad categories: supervised, and self-supervised. Approaches either use 3D Morphable Models (3DMMs) [7, 28, 41], or eschew linear models and instead learn a non-linear one as part of the training process [66].
+
+Fully supervised techniques either use parameter values from a 3DMM that is fit to the data via optimization as labels [14, 68, 73], or known face geometry is posed by sampling from a 3DMM and rendered to create synthetic datasets [21, 27, 51, 57]. Self-supervised approaches commonly use landmark reprojection error and/or perceptual loss via differentiable rendering [17, 23, 27, 31, 32, 45, 52, 55, 62, 63, 66, 67]. Other techniques augment this with 3D or multiview constraints [20, 44, 58, 61, 74, 75]. While this is similar to our technique, we only use a DNN to regress landmark positions which are then used to optimize 3DMM parameters, as in the large body of hybrid model-fitting methods [8, 33].
+
+Optimization-based 3D face reconstruction Traditionally, markerless reconstruction of face geometry is achieved with multi-view stereo [4, 56], followed by optical flow based alignment, and then optimisation using geometric and temporal priors [5, 9, 50]. While such methods produce detailed results, each step takes hours to complete. They also suffer from drift and other issues due to their reliance on optical flow and multi-view stereo [15]. While our method cannot reconstruct faces in such fine detail, it accurately recovers the low-frequency shape of the face, and aligns it with a common topology. This enriches the raw data with semantics, making it useful for other tasks.
+
+If only a single image is available, dense photometric [18, 65], depth [64], or optical flow [13] constraints are commonly used to recover face shape and motion. However, these methods still rely on sparse landmarks for initializing the optimization close to the dense constraint's basin of convergence, and coping with fast head motion [79]. In contrast, we argue that dense landmarks alone are sufficient for accurately recovering the overall shape of the face.
+
+Dense landmark prediction While sparse landmark prediction is a mainstay of the field [12], few methods directly predict dense landmarks or correspondences. This is because annotating a face with dense landmarks is a highly ambiguous task, so either synthetic data [71], pseudo-labels made with model-fitting [16, 25, 78], or semi-automatic refinement of training data [36, 37] are used. Another issue with predicting dense landmarks is that heatmaps, the de facto technique for predicting landmarks [11, 12], rise in computational complexity with the number of landmarks. While a few previous methods have predicted dense frontal-face landmarks via cascade regression [36] or direct regression [16, 29, 37], we are the first to accurately and robustly predict over 700 landmarks covering the whole head, including eyes and teeth.
+
+Some methods choose to predict dense correspondences as an image instead, where each pixel corresponds to a fixed point in a UV-unwrapping of the face [1, 25] or body [30, 60]. Such parameterization suffers from several drawbacks. How does one handle self-occluded portions of the face, e.g., the back of the head? Furthermore, what occurs at UV-island boundaries? If a pixel is half-nose and half-cheek, to which does it correspond? Instead, we choose to discretize the face into dense landmarks. This lets us predict parts of the face that are self-occluded, or lie outside image bounds. Having a fixed set of correspondences also benefits the model-fitter, making it more amenable to running in real-time.
+
+# 3 Method
+
+In recent years, methods for 3D face reconstruction have become more and more complicated, involving differentiable rendering and complex neural network training strategies. We show instead that success can be found by keeping things simple. Our approach consists of two stages: First we predict probabilistic dense 2D landmarks  $L$  using a traditional convolutional neural network (CNN). Then, we fit a 3D face model, parameterized by  $\Phi$ , to the 2D landmarks by
+
+![](images/f933d69e6bef927bd0fd8eceb0110aa933651f1d58ac70b33341457c4af5f97e.jpg)  
+1) Dense landmark prediction
+
+![](images/844eb12879a57a654013442e34898e3ddc45d3e9340f0723858325eddb90e48d.jpg)  
+2) 3D model fitting  
+Fig. 3. Given an image, we first predict probabilistic dense landmarks  $L$ , each with position  $\mu$  and certainty  $\sigma$ . Then, we fit our 3D face model to  $L$ , minimizing an energy  $E$  by optimizing model parameters  $\Phi$ .
+
+![](images/32446bcd8c2915d0ad53927a1e68b4dba394b900750f66f375355ccfd9df66bd.jpg)  
+Fig. 4. Examples of our synthetic training data. Without the perfectly consistent annotations provided by synthetic data, dense landmark prediction would not be possible.
+
+minimizing an energy function  $E(\Phi; L)$ . Images themselves are not part of this optimization; the only data used are 2D landmarks.
+
+The main difference between our work and previous approaches is the number and quality of landmarks. No one before has predicted so many 2D landmarks, so accurately. This lets us achieve accurate 3D face reconstruction results by fitting a 3D model to these landmarks alone.
+
+# 3.1 Landmark prediction
+
+Synthetic training data. Our results are only possible because we use synthetic training data. While a human can consistently label face images with e.g., 68 landmarks, it would be almost impossible for them to annotate an image with dense landmarks. How would it be possible to consistently annotate occluded landmarks on the back of the head, or multiple landmarks over a largely featureless patch of skin e.g., the forehead? In previous work, pseudo-labelled real images with dense correspondences are obtained by fitting a 3DMM to images [1], but the resulting label consistency heavily depends on the quality of the 3D fitting. Using synthetic data has the advantage of guaranteeing perfectly consistent labels. We rendered a training dataset of  $100\mathrm{k}$  images using the method of Wood et al. [71] with some minor modifications: we include expression-dependent
+
+![](images/d5aaf8fd8692a24c29e73093ef4da388c04d15a6a3cf11ef025a0dc909bef85c.jpg)  
+Fig. 5. When parts of the face are occluded by e.g. hair or clothing, the corresponding landmarks are predicted with high uncertainty (red), compared to those visible (green).
+
+wrinkle texture maps for more realistic skin appearance, and additional clothing, accessory, and hair assets. See Figure 4 for some examples.
+
+Probabilistic landmark regression. We predict each landmark as a random variable with the probability density function of a circular 2D Gaussian. So  $L_{i} = \{\pmb{\mu}_{i},\sigma_{i}\}$ , where  $\pmb{\mu}_i = [x_i,y_i]$  is the expected position of that landmark, and  $\sigma_{i}$  (the standard deviation) is a measure of uncertainty. Our training data includes labels for landmark positions  $\pmb{\mu}_i^{\prime} = [x_i^{\prime},y_i^{\prime}]$ , but not for  $\sigma$ . The network learns to output  $\sigma$  in an unsupervised fashion to show that it is certain about some landmarks, e.g., visible landmarks on the front of the face, and uncertain about others, e.g., landmarks hidden behind hair (see Figure 5). This is achieved by training the network with a Gaussian negative log likelihood (GNLL) loss [38]:
+
+$$
+\operatorname {L o s s} (L) = \sum_ {i = 1} ^ {| L |} \lambda_ {i} \left(\underbrace {\log \left(\sigma_ {i} ^ {2}\right)} _ {\text {L o s s} _ {\sigma}} + \underbrace {\frac {\left\| \boldsymbol {\mu} _ {i} - \boldsymbol {\mu} _ {i} ^ {\prime} \right\| ^ {2}}{2 \sigma_ {i} ^ {2}}} _ {\text {L o s s} _ {\mu}}\right) \tag {1}
+$$
+
+$\mathrm{Loss}_{\sigma}$  penalizes the network for being too uncertain, and  $\mathrm{Loss}_{\mu}$  penalizes the network for being inaccurate.  $\lambda_{i}$  is a per-landmark weight that focuses the loss on certain parts of the face. This is the only loss used during training.
+
+The probabilistic nature of our landmark predictions is important for accuracy. A network trained with the GNLL loss is more accurate than a network trained with L2 loss on positions only. Perhaps this is the result of the CNN being able to discount challenging landmarks (e.g., fully occluded ones), and spend more capacity on making precise predictions about visible landmarks.
+
+Landmarks are commonly predicted via heatmaps [11]. However, generating heatmaps is computationally expensive [42]; it would not be feasible to output over 700 heatmaps in real-time. Heatmaps also prevent us predicting landmarks outside image bounds. Instead, we keep things simple, and directly regress position and uncertainty using a traditional CNN. We are able to take any off-the-shelf architecture, and alter the final fully-connected layer to output three values per-landmark: two for position and one for uncertainty. Since this final layer represents a small percentage of total CNN compute, our method scales well with landmark quantity.
+
+Training details. Landmark coordinates are normalized from  $[0, S]$  to  $[-1, 1]$ , for a square image of size  $S \times S$ . Rather than directly outputting  $\sigma$ , we predict
+
+![](images/319e5654b848e4ad5f7919fcb2381cdc20e2a8d6715c3870d80b88dd242b5b43.jpg)  
+Fig. 6. We implemented two versions of our approach: one for processing multi-view recordings offline (a), and one for real-time facial performance capture (b).
+
+![](images/6ad9b0b0ba5f1930f6aa29db296d8d9c3ea1f5715531a4a8ca56a0ecd37dea6e.jpg)
+
+$\log \sigma$ , and take its exponential to ensure  $\sigma$  is positive. Using PyTorch [48], we train ResNet [35] and MobileNet V2 [54] models from the timm [70] library using AdamW [46] with automatically determined learning rate [22]. We use data augmentation to help our synthetic data cross the domain gap [71].
+
+# 3.2 3D model fitting
+
+Given probabilistic dense 2D landmarks  $L$ , our goal is to find optimal model parameters  $\Phi^{*}$  that minimize the following energy:
+
+$$
+E (\Phi ; L) = \underbrace {E _ {\mathrm {l a n d m a r k s}}} _ {\mathrm {D a t a t e r m}} + \underbrace {E _ {\mathrm {i d e n t i t y}} + E _ {\mathrm {e x p r e s s i o n}} + E _ {\mathrm {j o i n t s}} + E _ {\mathrm {t e m p o r a l}} + E _ {\mathrm {i n t e r s e c t}}} _ {\mathrm {R e g u l a r i z e r s}}
+$$
+
+$E_{\text{landmarks}}$  is the only term that encourages the 3D model to explain the observed 2D landmarks. The other terms use prior knowledge to regularize the fit.
+
+Part of the beauty of our approach is how naturally it scales to multiple images and cameras. In this section we present the general form of our method, suitable for  $F$  frames over  $C$  cameras, i.e., multi-view performance capture.
+
+3D face model. We use the face model described in [71], comprising  $N = 7,667$  vertices and  $K = 4$  skeletal joints (the head, neck, and two eyes). Vertex positions are determined by the mesh-generating function  $\mathcal{M}(\pmb {\beta},\pmb {\psi},\pmb {\theta}): \mathbb{R}^{|\pmb {\beta}| + |\pmb {\psi}| + |\pmb {\theta}|}\rightarrow \mathbb{R}^{3N}$  which takes parameters  $\pmb {\beta}\in \mathbb{R}^{|\pmb{\beta}|}$  for identity,  $\pmb {\psi}\in \mathbb{R}^{|\pmb{\psi}|}$  for expression, and  $\pmb {\theta}\in \mathbb{R}^{3K + 3}$  for skeletal pose (including root joint translation).
+
+$$
+\mathcal {M} (\boldsymbol {\beta}, \psi , \boldsymbol {\theta}) = \mathcal {L} (\mathcal {T} (\boldsymbol {\beta}, \psi), \boldsymbol {\theta}, \mathcal {J} (\boldsymbol {\beta}); \mathbf {W})
+$$
+
+where  $\mathcal{L}(\mathbf{V},\pmb {\theta},\mathbf{J};\mathbf{W})$  is a standard linear blend skinning (LBS) function [40] that rotates vertex positions  $\mathbf{V}\in \mathbb{R}^{3N}$  about joint locations  $\mathbf{J}\in \mathbb{R}^{3K}$  by local joint rotations in  $\pmb{\theta}$ , with per-vertex weights  $\mathbf{W}\in \mathbb{R}^{K\times N}$ . The face mesh and joint locations in the bind pose are determined by  $\mathcal{T}(\beta ,\psi):\mathbb{R}^{|\beta | + |\psi |}\to \mathbb{R}^{3N}$  and  $\mathcal{J}(\beta):\mathbb{R}^{|\beta |}\to \mathbb{R}^{3K}$  respectively. See Wood et al. [71] for more details.
+
+Cameras are described by a world-to-camera rigid transform  $\mathbf{X} \in \mathbb{R}^{3 \times 4} = [\mathbf{R}|\mathbf{T}]$  comprising rotation and translation, and a pinhole camera projection matrix  $\Pi \in \mathbb{R}^{3 \times 3}$ . Thus, the image-space projection of the  $j^{\text{th}}$  landmark in the  $i^{\text{th}}$  camera is  $\mathbf{x}_{i,j} = \Pi_i \mathbf{X}_i \mathcal{M}_j$ . In the monocular case,  $\mathbf{X}$  can be ignored.
+
+![](images/91c0c3a945b0c4c0011af48bde4f0fcfca5575cd7b86c42663ae26edd1792a0a.jpg)  
+$E_{\mathrm{intersect}}$  encourages these skin vertices  
+to remain outside these convex shapes.
+
+![](images/f0731428ee725fe41ea31fef88e56733fb71e10662e7196b2f3005340354a2ac.jpg)  
+Without  $E_{\text{intersect}}$  
+Fig. 7. We encourage the optimizer to avoid face mesh self-intersections by penalizing skin vertices that enter the convex hulls of the eyeballs or teeth parts.
+
+![](images/2eaa5331f64e4782ce08342cf5e1aa53d367465797656a3bffbb7921e0c011cf.jpg)  
+With  $E_{\mathrm{intersect}}$
+
+Parameters  $\Phi$  are optimized to minimize  $E$ . The main parameters of interest control the face, but we also optimize camera parameters if they are unknown.
+
+$$
+\boldsymbol {\Phi} = \{\underbrace {\boldsymbol {\beta} , \boldsymbol {\Psi} _ {F \times | \boldsymbol {\psi} |} , \boldsymbol {\Theta} _ {F \times | \boldsymbol {\theta} |}} _ {\text {F a c e}}; \underbrace {\mathbf {R} _ {C \times 3} , \mathbf {T} _ {C \times 3} , \mathbf {f} _ {C}} _ {\text {C a m e r a (s)}} \}
+$$
+
+Facial identity  $\beta$  is shared over a sequence of  $F$  frames, but expression  $\Psi$  and pose  $\Theta$  vary per frame. For each of our  $C$  cameras we have six degrees of freedom for rotation  $\mathbf{R}$  and translation  $\mathbf{T}$ , and a single focal length parameter  $f$ . In the monocular case, we only optimize focal length.
+
+$E_{\mathrm{landmarks}}$  encourages the 3D model to explain the predicted 2D landmarks:
+
+$$
+E _ {\text {l a n d m a r k s}} = \sum_ {i, j, k} ^ {F, C, | L |} \frac {\left\| \mathbf {x} _ {i j k} - \boldsymbol {\mu} _ {i j k} \right\| ^ {2}}{2 \sigma_ {i j k} ^ {2}} \tag {2}
+$$
+
+where, for the  $k^{\mathrm{th}}$  landmark seen by the  $j^{\mathrm{th}}$  camera in the  $i^{\mathrm{th}}$  frame,  $[\pmb{\mu}_{ijk},\sigma_{ijk}]$  is the 2D location and uncertainty predicted by our dense landmark CNN, and  $\mathbf{x}_{ijk} = \pmb{\Pi}_j\mathbf{X}_j\mathcal{M}(\pmb {\beta},\pmb {\psi}_i,\pmb {\theta}_i)_k$  is the 2D projection of that landmark on our 3D model. The similarity of Equation 2 to  $\mathrm{Loss}_{\mu}$  in Equation 1 is no accident: treating landmarks as 2D random variables during both prediction and modeling allows our approach to elegantly handle uncertainty, taking advantage of landmarks the CNN is confident in, and discounting those it is uncertain about.
+
+$E_{\mathrm{identity}}$  penalizes unlikely face shape by maximizing the relative log-likelihood of shape parameters  $\beta$  under a multivariate Gaussian Mixture Model (GMM) of  $G$  components fit to a library of 3D head scans [71].  $E_{\mathrm{identity}} = -\log(p(\beta))$  where  $p(\beta) = \sum_{i=1}^{G} \gamma_i \mathcal{N}(\beta | \nu_i, \Sigma_i)$ .  $\nu_i$  and  $\Sigma_i$  are the mean and covariance matrix of the  $i^{\text{th}}$  component, and  $\gamma_i$  is the weight of that component.
+
+$E_{\mathrm{expression}} = \| \psi \| ^2$  and  $E_{\mathrm{joint}} = \| \pmb{\theta}_{i:i\in [2,K]}\| ^2$  encourage the optimizer to explain the data with as little expression and joint rotation as possible. We do not penalize global translation or rotation by ignoring the root joint  $\pmb{\theta}_1$ .
+
+$E_{\mathrm{temporal}} = \sum_{i=2,j,k}^{F,C,|L|}\|\mathbf{x}_{i,j,k} - \mathbf{x}_{i-1,j,k}\|^2$  reduces jitter by encouraging face mesh vertices  $\mathbf{x}$  to remain still between neighboring frames  $i-1$  and  $i$ .
+
+$E_{\text{intersect}}$  encourages the optimizer to find solutions without intersections between the skin and eyeballs or teeth (Figure 7). Please refer to the supplementary material for further details.
+
+# 3.3 Implementation
+
+We implemented two versions of our system: one for processing multi-camera recordings offline, and one for real-time facial performance capture.
+
+Our offline system produces the best quality results without constraints on compute. We predict 703 landmarks with a ResNet 101 [35]. To extract a facial Region-of-Interest (ROI) from an image we run a full-head probabilistic landmark CNN on multi-scale sliding windows, and select the window with the lowest uncertainty. When fitting our 3DMM, we use PyTorch [48] to minimize  $E(\Phi)$  with L-BFGS [43], optimizing all parameters across all frames simultaneously.
+
+For our real-time system, we trained a lightweight dense landmark model with MobileNet V2 architecture [54]. To compensate for a reduction in network capacity, we predict 320 landmarks rather than 703, and modify the ROI strategy: aligning the face so it appears upright with the eyes a fixed distance apart. This makes the CNN's job easier for frontal faces at the expense of profile ones.
+
+Real-time model fitting. We use the Levenberg-Marquardt algorithm to optimize our model-fitting energy. Camera and identity parameters are only fit occasionally. For the majority of frames we fit pose and expression parameters only. We rewrite the energy  $E$  in terms of the vector of residuals,  $\mathbf{r}$ , as  $E(\Phi) = \| \mathbf{r}(\Phi)\| ^2 = \sum_i r_i(\Phi)^2$ . Then at each iteration  $k$  of our optimization, we can compute  $\mathbf{r}(\Phi_k)$  and the Jacobian,  $J(\Phi_k) = \frac{\partial\mathbf{r}(\Phi)}{\partial\Phi}\big|_{\Phi = \Phi_k}^{\Phi = \Phi_k}$ , and use these to solve the symmetric, positive-semi-definite linear system,  $(J^{T}J + \lambda \mathrm{diag}(J^{T}J))\pmb{\delta}_{k} = -J^{T}\mathbf{r}$  via Cholesky decomposition. We then apply the update rule,  $\pmb{\Phi}_{k + 1} = \pmb{\Phi}_k + \pmb{\delta}_k$ .
+
+In practice we do not actually form the residual vector  $\mathbf{r}$  nor the Jacobian matrix  $J$ . Instead, for performance reasons, we directly compute the quantities  $J^{T}J$  and  $J^{T}\mathbf{r}$  as we visit each term  $r_i(\Phi_k)$  of the energy. Most of the computational cost is incurred in evaluating these products for the landmark data term, as expected. However, the Jacobian of landmark term residuals is not fully dense. Each individual landmark depends on its own subset of expression parameters, and is invariant to other expression parameters. We performed a static analysis of the sparsity of each landmark term with respect to parameters,  $\partial r_i / \partial \varPhi_j$ , and we use this set of  $i,j$  indices to reduce the cost of our outer products from  $O(|\Phi|^2)$  to  $O(m_i^2)$ , where  $m_i$  is the sparsified dimensionality of  $\partial r_i / \partial \Phi$ . We further enhance the sparsity by ignoring any components of the Jacobian with an absolute value below a certain empirically-determined threshold.
+
+By exploiting sparsity in this way, the landmark term residuals and their derivatives become very cheap to evaluate. This formulation avoids the correspondence problem usually seen with depth images [59], which requires a more expensive optimization. In addition, adding more landmarks does not significantly increase the cost of optimization. It therefore becomes possible to implement a very detailed and well-regularized fitter with a relatively small compute
+
+<table><tr><td>Method</td><td>Common NME</td><td>Challenging NME</td><td>Private FR10%</td></tr><tr><td>LAB [72]</td><td>2.98</td><td>5.19</td><td>0.83</td></tr><tr><td>AWING [69]</td><td>2.72</td><td>4.52</td><td>0.33</td></tr><tr><td>ODN [76]</td><td>3.56</td><td>6.67</td><td>-</td></tr><tr><td>3FabRec [10]</td><td>3.36</td><td>5.74</td><td>0.17</td></tr><tr><td>Wood et al. [71]</td><td>3.09</td><td>4.86</td><td>0.50</td></tr><tr><td>LUVLi [39]</td><td>2.76</td><td>5.16</td><td>-</td></tr><tr><td>ours (L2)</td><td>3.30</td><td>5.12</td><td>0.33</td></tr><tr><td>ours (GNLL)</td><td>3.03</td><td>4.80</td><td>0.17</td></tr></table>
+
+![](images/4e46617e52c40e89712f326bfe2067043837f283cb1faee7e84633366d771eb8.jpg)  
+Fig. 8. Left: results on 300W dataset, lower is better. Note competitive performance of our model (despite being evaluated across-dataset) and importance of GNLL loss. Right: sample predictions (top row) with label-translated results (bottom row).
+
+burden, simply by adding a sufficient number of landmarks. The cost of the Cholesky solve for the update  $\pmb{\delta}_k$  is independent of the number of landmarks.
+
+# 4 Evaluation
+
+# 4.1 Landmark accuracy
+
+We measure the accuracy of a ResNet 101 dense landmark model on the 300W [53] dataset. For benchmark purposes only, we employ label translation [71] to deal with systematic inconsistencies between our 703 predicted dense landmarks and the 68 sparse landmarks labelled as ground truth (see Figure 8). While previous work [71] used label translation to evaluate a synthetically-trained sparse landmark predictor, we use it to evaluate a dense landmark predictor.
+
+We use the standard normalized mean error (NME) and failure rate  $(\mathrm{FR}_{10\%})$  error metrics [53]. Our model's results in Figure 8 are competitive with the state of the art, despite being trained with synthetic data alone. Note: these results provide a conservative estimate of our method's accuracy as the translation network may introduce error, especially for rarely seen expressions.
+
+Ablation study We measured the importance of predicting each landmark as a random variable rather than as a 2D coordinate. We trained two landmark prediction models, one with our proposed GNLL loss (Equation 1), and one with a simpler L2 loss on landmark coordinate only. Results in Figure 8 confirm that including uncertainty in landmark regression results in better accuracy.
+
+Qualitative comparisons are shown in Figure 9 between our real-time dense landmark model (MobileNet V2) and MediaPipe Attention Mesh [29], a publicly available dense landmark method designed for mobile devices. Our method is more robust, perhaps due to the consistency and diversity of our synthetic training data. See the supplementary material for additional qualitative results, including landmark predictions on the Challenging subset of 300W.
+
+![](images/c459d1f533623494ecbe76d422dd7aa1c3a0bfa2f3d04e4b3b81518b6b9c78dc.jpg)  
+Fig. 9. We compare our real-time landmark CNN (MobileNet V2) with MediaPipe Attention Mesh [29], a publicly available method for dense landmark prediction. Our approach is more robust to challenging expressions and illumination.
+
+# 4.2 3D face reconstruction
+
+Quantitatively, we compare our offline approach with recent methods on two benchmarks: the NoW Challenge [55] and the MICC dataset [2].
+
+The NoW Challenge [55] provides a standard evaluation protocol for measuring the accuracy and robustness of 3D face reconstruction in the wild. It consists of 2054 face images of 100 subjects along with a 3D head scan for each subject which serves as ground truth. We undertake the challenge in two ways: single view, where we fit our face model to each image separately, and multi-view, where we fit a per-subject face model to all image of a particular subject. As shown in Figure 10, we achieve state of the art results.
+
+The MICC dataset [2] consists of 3D face scans and videos of 53 subjects. The videos were recorded in three environments: a "cooperative" laboratory environment, an indoor environment, and an outdoor environment. We follow Deng et al. [17], and evaluate our method in two ways: single view, where we estimate one face shape per frame in a video, and average the resulting face meshes, and multi-view, where we fit a single face model to all frames in a video jointly. As shown in Table 1, we achieve state of the art results.
+
+Note that many previous methods are incapable of aggregating face shape information across multiple views. The fact ours can benefit from multiple views highlights the flexibility of our hybrid model-fitting approach.
+
+Ablation studies We conducted an experiment to measure the importance of landmark quantity for 3D face reconstruction. We trained three landmark CNNs,
+
+![](images/45184e092845f4d505529e67486e448e251ecbe80709bc1ed2700f7efc81a610.jpg)  
+Fig. 10. Results for the NoW Challenge [55]. We outperform the state of the art on both single- and multi-view 3D face reconstruction.
+
+<table><tr><td>Method</td><td colspan="3">Error (mm)</td></tr><tr><td>Single view</td><td>Median</td><td>Mean</td><td>Std</td></tr><tr><td>Deng et al. [17]</td><td>1.11</td><td>1.41</td><td>1.21</td></tr><tr><td>RingNet [55]</td><td>1.21</td><td>1.53</td><td>1.31</td></tr><tr><td>3DFAv2 [31]</td><td>1.23</td><td>1.57</td><td>1.39</td></tr><tr><td>DECA [24]</td><td>1.09</td><td>1.38</td><td>1.18</td></tr><tr><td>Dib et al. [19]</td><td>1.26</td><td>1.57</td><td>1.31</td></tr><tr><td>ours</td><td>1.02</td><td>1.28</td><td>1.08</td></tr><tr><td>Multi-view</td><td></td><td></td><td></td></tr><tr><td>Bai et al. [3]</td><td>1.08</td><td>1.35</td><td>1.15</td></tr><tr><td>ours</td><td>0.81</td><td>1.01</td><td>0.84</td></tr></table>
+
+Table 1. Results on the MICC dataset [2], following the single and multi-frame evaluation protocol of Deng et al. [17]. We achieve state-of-the-art results.  
+
+<table><tr><td>Method</td><td colspan="3">Error (mm), mean</td><td>Method</td><td colspan="3">Error (mm), mean</td></tr><tr><td>Single view</td><td>Coop.</td><td>Indoor</td><td>Outdoor</td><td>Multi-view</td><td>Coop.</td><td>Indoor</td><td>Outdoor</td></tr><tr><td>Tran et al. [68]</td><td>1.97</td><td>2.03</td><td>1.93</td><td>Piotraschke and Blanz [49]</td><td>1.68</td><td>1.67</td><td>1.72</td></tr><tr><td>Genova et al. [27]</td><td>1.78</td><td>1.78</td><td>1.76</td><td>Deng et al. [17]</td><td>1.60</td><td>1.61</td><td>1.63</td></tr><tr><td>Deng et al. [17]</td><td>1.66</td><td>1.66</td><td>1.69</td><td>ours</td><td>1.43</td><td>1.42</td><td>1.42</td></tr><tr><td>ours</td><td>1.64</td><td>1.62</td><td>1.61</td><td></td><td></td><td></td><td></td></tr></table>
+
+predicting 703, 320, and 68 landmarks respectively, and used these on the NoW Challenge (validation set). As shown in Figure 11, fitting with more landmarks results in more accurate 3D face reconstruction.
+
+In addition, we investigated the importance of using landmark uncertainty  $\sigma$  in model fitting. We fit our model to 703 landmark predictions on the NoW validation set, but using fixed rather than predicted  $\sigma$ . Figure 11 (bottom row of table) shows that fitting without  $\sigma$  leads to worse results.
+
+Qualitative comparisons between our work and several publicly available methods [17, 24, 31, 55, 58] can be found in Figure 13.
+
+# 4.3 Facial performance capture
+
+Multi-view Good synthetic training data requires a database of facial expression parameters from which to sample. We acquired such a database by conducting markerless facial performance capture for 108 subjects. We recorded each subject in our 17-camera studio, and processed each recording with our offline multi-view model fitter. For a 520 frame sequence it takes 3 minutes to predict dense landmarks for all images, and a further 9 minutes to optimize face model parameters. See Figure 12 for some of the 125,000 frames of expression data captured with our system. As the system which is used to create the database
+
+![](images/25a4484866b7f2155bb9d69b3257c8026a7ee0b51a8ecb88bf59007136ca7d01.jpg)  
+Fit with: 68 ldmks. 320 ldmks. 703 ldmks.
+
+Number of Error (mm)  
+
+<table><tr><td>Landmarks</td><td>Median</td><td>Mean</td><td>Std</td></tr><tr><td>68</td><td>1.10</td><td>1.38</td><td>1.16</td></tr><tr><td>320</td><td>1.00</td><td>1.24</td><td>1.02</td></tr><tr><td>703</td><td>0.95</td><td>1.17</td><td>0.97</td></tr><tr><td>703 (without σ)</td><td>1.02</td><td>1.26</td><td>1.03</td></tr></table>
+
+![](images/bee2edc609d614197c763d711fc9ae05237dd925be14acec57216e47bab09a75.jpg)  
+Fig. 11. Ablation studies on the NoW [55] validation set confirm that denser is better: model fitting with more landmarks leads to more accurate results. In addition, we see that fitting without using  $\sigma$  leads to worse results.  
+Fig. 12. We demonstrate the robustness and reliability of our method by using it to collect a massive database of 125,000 facial expressions, fully automatically.
+
+is then subsequently re-trained with it, we produced several databases in this manner until no further improvement was seen. We do not reconstruct faces in fine detail like previous multi-view stereo approaches [5, 9, 50]. However, while previous work can track a detailed 3D mesh over a performance, our approach reconstructs the performance with richer semantics: identity and expression parameters for our generative model. In many cases it is sufficient to reconstruct the low-frequency shape of the face accurately, without fine details.
+
+Real-time monocular See the last two columns of Figure 13 for a comparison between our offline and real-time systems for monocular 3D model-fitting. While our offline system produces the best possible results by using a large CNN and optimizing over all frames simultaneously, our real-time system can still produce accurate and expressive results fitting frame-to-frame. Please refer to the supplementary material for more results. Running on a single CPU thread (i5-11600K), our real-time system spends 6.5ms processing a frame (150FPS), of which 4.1ms is spent predicting dense landmarks and 2.3ms is spent fitting our face model.
+
+# 5 Limitations and future work
+
+Our method depends entirely on accurate landmarks. As shown in Figure 14, if landmarks are poorly predicted, the resulting model fit suffers. We plan to address this by improving our synthetic training data. Additionally, since our model does not include tongue articulation we cannot recover tongue movement.
+
+![](images/3f47a33b00016057ba509aa483ef6f9061c718f15cc4407fe50d9c252e1ce61b.jpg)  
+Fig. 13. Compared to previous recent monocular 3D face reconstruction methods, ours better captures gaze, expressions like winks and sneers, and subtleties of facial identity. In addition, our method can run in real time with only a minor loss of fidelity.
+
+![](images/df87b885d9b1c769a3ff3028915fe26fe156960f809e33816dea7995168e6bb4.jpg)  
+Fig. 14. Bad landmarks result in bad fits, and we are incapable of tracking the tongue.
+
+Heatmaps have dominated landmark prediction for some time [11, 12]. We were pleasantly surprised to find that directly regressing 2D landmark coordinates with unspecialized architectures works well and eliminates the need for computationally-costly heatmap generation. In addition, we were surprised that predicting  $\sigma$  helps accuracy. We look forward to further investigating direct probabilistic landmark regression as an alternative to heatmaps in future work.
+
+In conclusion, we have demonstrated that dense landmarks are an ideal signal for 3D face reconstruction. Quantitative and qualitative evaluations have shown that our approach outperforms those previous by a significant margin, and excels at multi-view and monocular facial performance capture. Finally, our approach is highly efficient, and runs at over 150FPS on a single CPU thread.
+
+Acknowledgements Thanks to Chirag Raman and Jamie Shotton for their contributions, and Jiaolong Yang and Timo Bolkart for help with evaluation.
+
+# Bibliography
+
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