  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, straight-forward to incorporate into surface
learning frameworks, 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.
