Field Convolutions for Surface CNNs

TL;DR

This paper introduces a new surface convolution operator called field convolution that operates on vector fields, combining intrinsic spatial convolution with parallel transport, leading to improved surface learning tasks.

Contribution

The paper proposes a novel field convolution method that is invariant to isometries and robust to noise, enhancing CNNs on surface data with increased descriptive power.

Findings

Achieved state-of-the-art results in shape classification.

Improved performance in surface segmentation and correspondence tasks.

Demonstrated robustness to noise and nuisance factors.

Abstract

We present a novel surface convolution operator acting on vector fields that is based on a simple observation: instead of combining neighboring features with respect to a single coordinate parameterization defined at a given point, we have every neighbor describe the position of the point within its own coordinate frame. This formulation combines intrinsic spatial convolution with parallel transport in a scattering operation while placing no constraints on the filters themselves, providing a definition of convolution that commutes with the action of isometries, has increased descriptive potential, and is robust to noise and other nuisance factors. The result is a rich notion of convolution which we call field convolution, well-suited for CNNs on surfaces. Field convolutions are flexible, straight-forward to incorporate into surface learning frameworks, and their highly discriminating nature has cascading effects throughout the learning pipeline. Using simple networks constructed from residual field convolution blocks, we achieve state-of-the-art results on standard benchmarks in fundamental geometry processing tasks, such as shape classification, segmentation, correspondence, and sparse matching.