The usage of 3D vision algorithms, such as shape reconstruction, remains limited because they require inputs to be at a fixed canonical rotation. Recently, a simple equivariant network, Vector Neuron (VN) (Deng et al., 2021) has been proposed that can be easily used with the state-of-the-art 3D neural network (NN) architectures. However, its performance is limited because it is designed to use only three-dimensional features, which is insufficient to capture the details present in 3D data. In this paper, we introduce an equivariant feature representation for mapping a 3D point to a high-dimensional feature space. Our feature can discern multiple frequencies present in 3D data, which, as shown by Tancik et al. (2020), is the key to designing an expressive feature for 3D vision tasks. Our representation can be used as an input to VNs, and the results demonstrate that with our feature representation, VN captures more details, overcoming the limitation raised in its original paper. Figure 1: EGAD (Morrison et al., 2020) meshes constructed from the embeddings given by different models based on OccNet (Mescheder et al., 2019) at canonical poses. As already noted in their original paper, VN-OccNet (3rd column), the VN version of OccNet, fails to capture the details present in the ground-truth shapes and does worse than OccNet (2nd column). Using our feature representation, VN-OccNet qualitatively performs better than OccNet (4th column). Note that each of these shapes consists of multiple frequencies - in some parts of the object, the shape changes abruptly, while in some parts, it changes very smoothly. SO(3) equivariant neural networks (NN) change the output accordingly when the point cloud input is rotated without additional training.

Our goal is to develop an efficient contact detection algorithm for large-scale GPU-based simulation of non-convex objects. Current GPU-based simulators such as IsaacGym and Brax must trade-off speed with fidelity, generality, or both when simulating non-convex objects. Their main issue lies in contact detection (CD): existing CD algorithms, such as Gilbert-Johnson-Keerthi (GJK), must trade off their computational speed with accuracy which becomes expensive as the number of collisions among non-convex objects increases. We propose a data-driven approach for CD, whose accuracy depends only on the quality and quantity of offline dataset rather than online computation time. Unlike GJK, our method inherently has a uniform computational flow, which facilitates efficient GPU usage based on advanced compilers such as XLA (Accelerated Linear Algebra). Further, we offer a data-efficient solution by learning the patterns of colliding local crop object shapes, rather than global object shapes which are harder to learn. We demonstrate our approach improves the efficiency of existing CD methods by a factor of 5-10 for non-convex objects with comparable accuracy. Using the previous work on contact resolution for a neural-network-based contact detector, we integrate our CD algorithm into the open-source GPU-based simulator, Brax, and show that we can improve the efficiency over IsaacGym and generality over standard Brax. We highly recommend the videos of our simulator included in the supplementary materials.