An intuitive multi-frequency feature representation for SO(3)-equivariant networks

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.