Multi-directional Geodesic Neural Networks via Equivariant Convolution

TL;DR

This paper introduces multi-directional geodesic neural networks that utilize equivariant convolution on curved surfaces, enabling better propagation of directional information for geometric deep learning tasks.

Contribution

It presents a novel directional convolution method that preserves all rotational information across layers, improving shape analysis and matching tasks.

Findings

Significant performance improvements over baselines in shape classification.

Effective propagation of directional information across surface regions.

Versatile application in shape segmentation and matching.

Abstract

We propose a novel approach for performing convolution of signals on curved surfaces and show its utility in a variety of geometric deep learning applications. Key to our construction is the notion of directional functions defined on the surface, which extend the classic real-valued signals and which can be naturally convolved with with real-valued template functions. As a result, rather than trying to fix a canonical orientation or only keeping the maximal response across all alignments of a 2D template at every point of the surface, as done in previous works, we show how information across all rotations can be kept across different layers of the neural network. Our construction, which we call multi-directional geodesic convolution, or directional convolution for short, allows, in particular, to propagate and relate directional information across layers and thus different regions on the shape. We first define directional convolution in the continuous setting, prove its key properties and then show how it can be implemented in practice, for shapes represented as triangle meshes. We evaluate directional convolution in a wide variety of learning scenarios ranging from classification of signals on surfaces, to shape segmentation and shape matching, where we show a significant improvement over several baselines.