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

TL;DR

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.

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.

Figures (5)

• Figure 1: Comparison of invariant and equivariant networks, subplots A and B, respectively. A layer with an invariant pattern recognition cannot distinguish some images, producing identical feature maps for different inputs. In contrast, equivariant pattern recognition allows the discrimination of these images by building feature maps in 6D.
• Figure 2: A -- Schematic representation of the EquiLoPO ResNet-18 architecture, with a sequence of operations in the Initial and Basic blocks. $L$ is the maximum expansion degree of the last operator in the block, and $D$ is the number of the block's features. B -- Correlation between the Sampled Maximum and its SoftMax approximation of the output functions in the rotational space in the trained model with locally adaptive coefficients on the Vessel dataset.
• Figure 3: Effect of the ReLU approximation with constant and adaptive polynomial coefficients in the SO(3) space for two resolution strategies: A,C -- low-resolution setup, $L_{\text{out}} = L_{\text{in}}$; B,D -- high-resolution setup, $L_{\text{out}} = 2L_{\text{in}}$. Plots A and B show results with constant activation coefficients. Plots C and D show results with adaptive activation coefficients. The $x$ and $y$ axes represent the input and output functions, respectively. The histograms above and to the right of the main plot display the distribution of these functions' values. The top histogram plots also compare the input distribution (blue) with the output distribution (red). Opacity of the main plots indicate the density distributions, overlaid with the ReLU functions.
• Figure I1: Left: Coefficients of the polynomial as functions of $\frac{\mu}{3\sigma}$. Right: Approximation error as a function of $\frac{\mu}{3\sigma}$.
• Figure K1: Schematic representation of the EquiLoPO ResNet-18 architecture, with a sequence of operations in the Initial and Basic blocks. $L$ is the maximum expansion degree of the last operator in the block and $D$ is the number of the block's features.