Rotation Equivariant Graph Convolutional Network for Spherical Image Classification

Convolutional neural networks (CNNs) designed for low-dimensional regular grids will unfortunately lead to non-optimal solutions for analyzing spherical images, due to their different geometrical properties from planar images. In this paper, we generalize the grid-based CNNs to a non-Euclidean space by taking into account the geometry of spherical surfaces and propose a Spherical Graph Convolutional Network (SGCN) to encode rotation equivariant representations. Specifically, we propose a spherical graph construction criterion showing that a graph needs to be regular by evenly covering the spherical surfaces in order to design a rotation equivariant graph convolutional layer. For the practical case where the perfectly regular graph does not exist, we design two quantitative measures to evaluate the degree of irregularity for a spherical graph. The Geodesic ICOsahedral Pixelation (GICOPix) is adopted to construct spherical graphs with the minimum degree of irregularity compared to the current popular pixelation schemes. In addition, we design a hierarchical pooling layer to keep the rotation-equivariance, followed by a transition layer to enforce the invariance to the rotations for spherical image classification. We evaluate the proposed graph convolutional layers with different pixelations schemes in terms of equivariance errors. We also assess the effectiveness of the proposed SGCN in fulfilling rotation-invariance by the invariance error of the transition layers and recognizing the spherical images and 3D objects.