On the Fourier analysis in the SO(3) space : EquiLoPO Network
This work addresses the challenge of rotational equivariance in deep learning for volumetric data, offering a flexible framework with potential applications across domains, though it appears incremental by building on prior group and steerable convolutional networks.
The paper tackles the problem of analyzing volumetric data with rotational invariance or equivariance by proposing the EquiLoPO Network, a novel equivariant neural network architecture that achieves analytical equivariance to local pattern orientation on the continuous SO(3) group with unconstrained trainable filters, and it consistently outperforms state-of-the-art methods on diverse 3D medical imaging datasets from MedMNIST3D.
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.