Möbius Convolutions for Spherical CNNs

Möbius transformations play an important role in both geometry and spherical image processing – they are the group of conformal automorphisms of 2D surfaces and the spherical equivalent of homographies. Here we present a novel, Möbius-equivariant spherical convolution operator which we call Möbius convolution; with it, we develop the foundations for Möbius-equivariant spherical CNNs. Our approach is based on the following observation: to achieve equivariance, we only need to consider the lower-dimensional subgroup which transforms the positions of points as seen in the frames of their neighbors. To efficiently compute Möbius convolutions at scale we derive an approximation of the action of the transformations on spherical filters, allowing us to compute our convolutions in the spectral domain with the fast Spherical Harmonic Transform. The resulting framework is flexible and descriptive, and we demonstrate its utility by achieving promising results in both shape classification and image segmentation tasks.