  M\"obius 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, M\"obius-equivariant spherical convolution operator which we call
M\"obius convolution, and with it, develop the foundations for
M\"obius-equivariant spherical CNNs. Our approach is based on a simple
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 M\"obius 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
both flexible and descriptive, and we demonstrate its utility by achieving
promising results in both shape classification and image segmentation tasks.
