CNNs on Surfaces using Rotation-Equivariant Features
This addresses a fundamental issue in geometric deep learning for applications like 3D shape analysis, though it is an incremental improvement over existing surface-based CNNs.
The paper tackles the problem of rotational ambiguity in convolutional neural networks on curved surfaces by proposing a network architecture using rotation-equivariant features, enabling local alignment of features in convolution layers without dependence on coordinate systems. The approach is implemented on triangle meshes and evaluated on shape correspondence and classification tasks, showing competitive performance compared to other methods.
This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution layer. The resulting network is agnostic to the choices of coordinate systems for the tangent spaces on the surface. We implement our approach for triangle meshes. Based on circular harmonic functions, we introduce convolution filters for meshes that are rotation-equivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.