Steerable 3D Spherical Neurons

TL;DR

This paper introduces a novel 3D spherical neuron model for point cloud classification that achieves rotation invariance by leveraging steerability constraints and spherical filter banks.

Contribution

It proposes a new steerable 3D spherical neuron approach with a theoretical framework for rotation-equivariance and invariance in point cloud processing.

Findings

Successfully classifies point clouds with rotation invariance

Derives a 3D steerability constraint for spherical neurons

Validates approach on synthetic and real-world 3D data

Abstract

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.