Emerging from low-level vision theory, steerable filters found their counterpart in prior work on steerable convolutional neural networks equivariant to rigid transformations. In our work, we propose a steerable feed-forward learning-based approach that consists of neurons with spherical decision surfaces and operates on point clouds. Such spherical neurons are obtained by conformal embedding of Euclidean space and have recently been revisited in the context of learning representations of point sets. Focusing on 3D geometry, we exploit the isometry property of spherical neurons and derive a 3D steerability constraint. After training spherical neurons to classify point clouds in a canonical orientation, we use a tetrahedron basis to quadruplicate the neurons and construct rotation-equivariant spherical filter banks. We then apply the derived constraint to interpolate the filter bank outputs and, thus, obtain a rotation-invariant network. Finally, we use a synthetic point set and real-world 3D skeleton data to verify our theoretical findings. The code is available at https://github.com/pavlo-melnyk/steerable-3d-neurons.

翻译：从低水平视觉理论中产生的低水平视觉理论中,可移植过滤器在先前关于可控进动神经网络的工作中找到了对应的可控进化神经网络,这种神经网络的变异性与僵硬的变异性。在我们的工作中,我们建议了一种可控进化进化学习法,由具有球状决定表面的神经元组成,在点云上运行。这种球状神经元是通过对欧几里得空间的同步嵌入获得的,最近又在点集学习演示中重新审视。聚焦于3D几何,我们利用球状神经元的异性属性,并产生3D可导力制约。在对球状神经元进行对点云进行分类培训后,我们使用四面基元基元基础来对神经元进行四重修补,并构建旋转-异性透析库。然后,我们运用衍生的制约来对过滤库输出进行内置,从而获得一个旋转-变异性网络。最后,我们利用一个合成点组和真实世界的3D骨质数据来验证我们的理论发现。代码可在 http://gnistrus-drobormus-stal/dormus-sturvaxmuslon//dormvvpormus/s/slentrvp