  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 and straight-forward to implement, 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.
