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	"content": "vision tasks, e.g., object tracking [36, 37], 3D reconstruction [8, 17], and SLAM [20, 41]. Frame-based feature tracking methods [5, 24, 25, 33, 35] have been extensively investigated in the past decades. However, all existing methods focus on tracking 2D feature trajectories in the image plane."
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	"content": "For the 3D feature tracking of high-speed moving objects, the main challenges lie in three folds. (1) With the limited frame rate of traditional frame-based cameras, the motion of high-speed moving objects may not be consistently captured due to the blind time between consecutive frames. Therefore, how to continually record valid motion information of high-speed moving objects is the first chal"
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	"content": "This CVPR paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore."
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	"content": "lenge. (2) The second challenge lies in establishing the correlation between the 3D position of the feature and the 2D visual data acquired by cameras to generate a continuous and smooth 3D feature trajectory. (3) To the best of our knowledge, there are currently no existing high-speed 3D feature tracking datasets. This is primarily due to the difficulty in capturing ground truth 3D feature trajectories of high-speed moving objects, which is constrained by the insufficient capture frequency of existing 3D vision sensors. Thus, the lack of high-speed 3D feature tracking dataset is the third challenge, which is also a principal impediment to the advancement of research within this domain."
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	"content": "To address the aforementioned technical challenge, we propose a high-speed 3D feature tracking method based on stereo event cameras, predicting the long-term 3D trajectories of target features from stereo event streams and template patches. To achieve 3D feature tracking, our proposed method leverages a joint framework to predict the 2D feature motion offsets and the feature spatial position at each timestamp simultaneously. A motion compensation module is leveraged to adapt to the feature deformation, and a patch matching module based on bi-polarity hypergraph modeling is proposed to accurately estimate the feature spatial position. In addition, we introduce a stereo motion consistency mechanism that establishes the constraint between the feature motion offsets and the spatial position to achieve smooth 3D trajectory estimation."
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	"content": "scanned by FARO Quantum ScanArm, resulting in the generation of the ground truth 3D trajectories of each feature at a rate of 250 FPS. To the best of our knowledge, our dataset is the first event-based feature tracking dataset containing high-speed moving objects and providing 3D ground truth feature trajectories."
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	"content": "- We collect the first real-world 3D feature tracking dataset containing multiple high-speed moving objects, named E-3DTrack. Our dataset contains stereo event streams and 250 FPS ground truth 3D feature trajectories, which could be used as the 3D feature tracking benchmark."
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	"content": "Trajectory Prediction via Event Camera. Event-based feature tracking methods have been developed rapidly within the last decade. Earlier works [18, 38] treat the events as a point set and used ICP [6] to estimate feature motion trajectories. Then, EKLT [10] is proposed to obtain feature patch from the reference frame as template, and use the event stream to track the template and predict the trajectory. Meanwhile, some event-by-event trackers [2, 3] are proposed to exploit the asynchronicity of event camera, e.g., eCDT [16] employs a clustering method to cluster adjacent events, and uses cluster descriptors to find continual feature tracks. Recently, DeepEvT [26] is proposed as the first data-driven event-based feature tracking method, which achieves state-of-the-art 2D feature tracking performance."
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	"content": "An alternative approach for trajectory prediction is optical flow estimation, wherein the pixel-level motion field is predicted using the input event stream. Compared with feature tracking, these methods [1, 4, 12, 29, 30] focus more on estimating the motion field between adjacent moments and lack modeling of long-term trajectory consistency."
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	"content": "However, all these existing trajectory prediction methods could only predict 2D feature trajectories in the image plane while the real objects are moving in 3D space, i.e., the predicted feature motion trajectories are information-deficient."
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	"content": "Event-based 3D Position Estimation. As the 2D feature trajectories can be predicted, a simple and straightforward solution is to use a monocular or stereo depth estimation method to predict the depth of the feature and calculate the 3D position. In recent years, several event-based"
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	"content": ". Meanwhile, a patch matching module based on bi-polarity hypergraph modeling is leveraged to predict the disparity. Finally, a projection operation is performed to update the 3D trajectory."
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	"content": "However, we will show that the simple combination of these two types of methods could not achieve satisfying long-term 3D feature trajectories prediction performance in Sec. 5.2. Therefore, high-speed 3D feature tracking is still a challenging open problem."
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	"content": " using our proposed patch matching module based on bipolarity hypergraph modeling. In addition, inspired by the fact that the 2D motion offsets in both camera planes have a constraint with the disparity change, we use the event patch "
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	"content": "\\mathbf{M}_{t_i}^1 \\in \\mathbb{R}^{d \\times d \\times c}"
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	"content": "\\mathbf{M}_{t_i}^2 \\in \\mathbb{R}^{d \\times W \\times c}"
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	"content": " from "
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	"content": " and "
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	"content": ", respectively, where "
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	"content": " is the feature channel and "
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	"content": " is the image width, i.e., the number of candidate matching positions. We further calculate the cost volume "
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	"content": "\\mathbf{C}_{t_i} \\in \\mathbb{R}^{W \\times c}"
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	"content": " composed of the feature similarity between "
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	"content": " and "
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	"content": " at each matching position, which represents the pair-wise similarity between "
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	"content": " and the sub-patch of "
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	"content": " at each matching position. Then, the "
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	"type": "text",
	"content": " matching positions are used as vertices to construct bi-polarity hypergraphs. Compared to the pair-wise correlation contained in the cost volume, each hyperedge of a hypergraph could connect multiple vertices, i.e., high-order correlations among multiple vertices could be constructed. In practice, we use the Euclidean distance of the vertex feature as metric and calculate the "
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	"content": " nearest neighbors of each vertex. For each vertex, we use a hyperedge to connect the vertices in its "
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	"content": " neighbor vertices with spatial distance smaller than a certain threshold "
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	"content": "\\delta"
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	"content": ". Therefore, a positive hypergraph "
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	"content": " with the adjacency matrix "
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	"content": " could be constructed. Besides, for each vertex, vertices with spatial distance larger than "
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	"content": " in its "
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	"content": " neighbor vertices are connected by another hyperedge. Thus, a negative hypergraph "
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	"type": "text",
	"content": " with the adjacency matrix "
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	"content": " could be constructed. Each hyperedge of "
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	"content": " connects matching patches that are semantic similar and spatially close to "
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	"content": ". These connections are expected to be enhanced. In contrast, each hyperedge of "
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	"content": " connects matching patches that are semantic similar but spatially distant from "
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	"content": "P_{t_i}^1"
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	"type": "text",
	"content": ", which are interference and needs to be suppressed. Then, inspired by [9], we propose a feature aggregation method based on bi-polarity hypergraphs:"
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	"content": "\\begin{array}{l} \\hat {\\mathbf {C}} _ {t _ {i}} = \\mathbf {C} _ {t _ {i}} + \\sigma \\left(\\left(\\mathbf {D} _ {v} ^ {+}\\right) ^ {- 1} \\mathbf {H} ^ {+} \\left(\\mathbf {D} _ {e} ^ {+}\\right) ^ {- 1} \\left(\\mathbf {H} ^ {+}\\right) ^ {\\top} \\mathbf {C} _ {t _ {i}} \\boldsymbol {\\Theta} ^ {+}\\right), \\tag {2} \\\\ - \\sigma \\left(\\left(\\mathbf {D} _ {v} ^ {-}\\right) ^ {- 1} \\mathbf {H} ^ {-} \\left(\\mathbf {D} _ {e} ^ {-}\\right) ^ {- 1} \\left(\\mathbf {H} ^ {-}\\right) ^ {\\top} \\mathbf {C} _ {t _ {i}} \\boldsymbol {\\Theta} ^ {-}\\right) \\\\ \\end{array}",
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	"content": " are the diagonal matrices of hyperedge degree and vertex degree, respectively. "
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	"content": "\\Theta^{*}"
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	"content": " is the learnable parameter, and "
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	"content": "\\sigma (\\cdot)"
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	"type": "text",
	"content": " is the non-linear activation function. Using Eq. (2), features are aggregated to enhance vertices with similar features and spatial close and suppress vertices with similar features but spatially distant. Finally, "
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	"content": " is forwarded into a 1D convolutional layer with the kernel size of"
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	"content": "3 to regress the matching result. Using the patch matching module, the disparity "
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	"content": " could be computed using projection."
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	"content": "Stereo Motion Consistency. For objects moving in 3D space captured by stereo cameras, the 2D motion offsets are strongly constrained with the disparity. Meanwhile, our offset estimation module is also deeply coupled with the patch matching module. Therefore, inspired by [21], we leverage a stereo motion consistency constraint to reinforce this correlation. Consider a point "
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	"content": ", it's 2D coordinates in the camera plane could be calculated by "
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	"content": " is the camera focal length and "
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	"content": " the coordinate convert factor. For calibrated stereo cameras with the baseline distance of "
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	"content": ", the disparity of "
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	"content": ". By taking the time derivative, we could obtain that "
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	"content": "\\frac{\\Delta d}{\\Delta t} = -\\frac{f}{s}\\frac{b}{z^2}\\frac{\\Delta z}{\\Delta t}"
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	"content": "d _ {t _ {i}} - d _ {t _ {i - 1}} = \\Delta d = - \\frac {f}{s} \\frac {b}{z _ {t _ {i}} ^ {2}} \\left(z _ {t _ {i}} - z _ {t _ {i - 1}}\\right). \\tag {3}",
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	"content": "\\frac{\\Delta\\mathbf{u}}{\\Delta t} = \\frac{f}{zs} (\\frac{\\Delta x}{\\Delta t},\\frac{\\Delta y}{\\Delta t}) - \\frac{f}{z^2s}\\frac{\\Delta z}{\\Delta t} (x,y)"
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	"content": ", i.e., we have "
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	"content": "\\Delta \\mathbf{u} = (\\Delta u,\\Delta v) = \\frac{f}{zs} (\\Delta x,\\Delta y) - \\frac{f\\Delta z}{z^2s} (x,y)"
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	"content": "In addressing the deficiency of high-speed 3D feature tracking datasets, we establish a hybrid vision system containing stereo event cameras and Optitrack, as shown in Fig. 3 (a), and curate the first event-based 3D feature tracking dataset, named E-3DTrack. Compared to existing event-based feature tracking datasets that contain only static scenes and 2D trajectories, our dataset is the first to contain high-speed moving objects and ground truth 3D feature trajectories."
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	"content": " FPS for LiDAR), it is difficult to accurately record the 3D feature trajectories of high-speed moving objects at a high frame rate. To tackle this problem, we use the motion capture system, i.e., Optitrack, to record the trajectory of each object attached with fixed markers. To explicitly obtain feature-level 3D trajectories, we use a scanner, i.e., FARO Quantum ScanArm, to capture the high precision point cloud of each object. Then, the 3D affine transform, incorporating a homogeneous scale, is calculated from the object coordinate system to the Optitrack coordinate system based on the markers' coordinates. This leads to the acquisition of the time-series point cloud sequence of the moving objects under the Optitrack coordinate system. Finally, the feature trajectories can be derived from the time-series point cloud sequence based on the feature point index. Hence, our dataset comprises ground truth 3D feature trajectories of high-speed moving objects at 250 FPS, surpassing the capturing frequency of most existing 3D vision sensors."
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	"content": "isons, respectively. Finally, we conduct ablation studies to demonstrate the effectiveness of each proposed module."
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	"content": "Table 2. Quantitative results on our E-3DTrack dataset. Feature age (FA), tracked feature ratio (TFR), and root mean square error (RMSE) are selected as the metrics. Bold numbers represent the best scores, and underlined numbers represent the second-best scores."
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	"content": "with slight depth variation, the comparison method achieves 3D feature tracking with slight oscillations. However, for the red star with large depth variation and rotation, it could not be tracked accurately by the comparison method. The last two rows show two extreme scenarios, i.e., the 3D motions of the objects are with large depth variation and rotation, which will cause significant feature shape deformation. Under such scenarios, our comparison method tracks the features with fatal errors. In contrast, our proposed method tracks the 3D trajectories of the high-speed moving features robustly and continuously due to our motion compensation module and patch matching module."
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	"content": "Figure 6 further shows the tracking error (RMSE) and the tracked feature ratio (TFR) over time on our E-3DTrack dataset. The threshold is selected as "
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	"content": " to calculate TFR. From the figure, we could observe that our proposed method can continuously track 3D trajectories of target features, i.e., our method maintains a high TFR consistently. From the figure, we could also observe that the TFR of E-RAFT + SDE is comparable with DeepEvT + SDE in initial stage, but gradually decreases over time. This is because the optical flow estimation method is lack of long-term consistent modeling. In contrast, our proposed method maintains a high TFR and a low tracking error over all time."
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	"content": "To demonstrate the effectiveness of each proposed module, we validate the performance of our model with and without the motion compensation module (denoted as MC), stereo motion consistency mechanism (denoted as "
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	"content": "), and the bi-polarity hypergraph-based high-order correlation modeling mechanism (denoted as BiHCM), respectively. The ablation experimental results are shown in Tab. 4. See supplementary material for detailed settings."
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	"content": "Bi-Polarity Hypergraph Modeling. From Tab. 4 we could observe that compared with our base model (row (1)), the addition of BiHCM will increase TFR from 0.5586 to 0.6082. Compared with our full model, the removal of the BiHCM will lead to an RMSE increase of "
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	"content": ". This is due to the fact that our proposed BiHCM could enhance the connection between patches with similar features that"
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	"content": "are spatially close, and suppress patches with similar features but spatially distant, which could eliminate mismatching and further improve 3D feature tracking performance."
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	"content": "Stereo Motion Consistency. As shown in Tab. 4, compared with the base model, the addition of stereo motion consistency constraint will reduce RMSE from 0.1807 to 0.1512. Compared with the full model, the removal of the stereo motion consistency constraint will increase RMSE by "
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	"content": ". This is due to the fact that the stereo motion consistency could effectively constrain the correlation between the 2D trajectory and 3D spatial position of the objects, making our method predict more accurate and smooth 3D trajectory."
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	"content": " decrease and increase in RMSE, respectively. With the addition of the motion compensation module, our proposed method could better deal with feature deformation caused by depth changes and rotations of moving objects, and achieve more robust 3D feature tracking."
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	"content": "In this paper, we propose the first high-speed 3D feature tracking method that takes stereo event streams as input to estimate 3D feature trajectories. Our proposed method leverages a joint framework to obtain 3D feature trajectories by estimating the feature motion offsets and spatial position simultaneously. A motion compensation module and a patch matching module based on bi-polarity hypergraphs are proposed to achieve robust feature tracking. Meanwhile, the first 3D feature tracking dataset containing high-speed moving objects and ground truth 3D feature trajectories at 250 FPS is constructed, named E-3DTrack, which can be used as the first 3D feature tracking benchmark."
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	"content": "This work was supported by National Natural Science Funds of China (No. 62021002 and No. 62088102), Beijing Natural Science Foundation (No. 4222025)."
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