On the Fourier analysis in the SO(3) space : EquiLoPO Network

TL;DR

This paper introduces EquiLoPO Network, a novel neural network architecture that achieves continuous rotational equivariance on SO(3) for volumetric data, using irreducible representations and local activation functions, outperforming existing methods.

Contribution

The work presents a new equivariant neural network with unconstrained trainable filters and analytical SO(3) equivariance, overcoming limitations of prior group convolutional approaches.

Findings

Outperforms state-of-the-art on MedMNIST3D datasets

Demonstrates effective continuous SO(3) equivariance

Provides a flexible framework for volumetric data analysis

Abstract

Analyzing volumetric data with rotational invariance or equivariance is an active topic in current research. Existing deep-learning approaches utilize either group convolutional networks limited to discrete rotations or steerable convolutional networks with constrained filter structures. This work proposes a novel equivariant neural network architecture that achieves analytical Equivariance to Local Pattern Orientation on the continuous SO(3) group while allowing unconstrained trainable filters - EquiLoPO Network. Our key innovations are a group convolutional operation leveraging irreducible representations as the Fourier basis and a local activation function in the SO(3) space that provides a well-defined mapping from input to output functions, preserving equivariance. By integrating these operations into a ResNet-style architecture, we propose a model that overcomes the limitations of prior methods. A comprehensive evaluation on diverse 3D medical imaging datasets from MedMNIST3D demonstrates the effectiveness of our approach, which consistently outperforms state of the art. This work suggests the benefits of true rotational equivariance on SO(3) and flexible unconstrained filters enabled by the local activation function, providing a flexible framework for equivariant deep learning on volumetric data with potential applications across domains. Our code is publicly available at https://gricad-gitlab.univ-grenoble-alpes.fr/GruLab/ILPO/-/tree/main/EquiLoPO.