  We propose a novel framework for computing descriptors for characterizing
points on three-dimensional surfaces. First, we present a new non-learned
feature that uses graph wavelets to decompose the Dirichlet energy on a
surface. We call this new feature wavelet energy decomposition signature
(WEDS). Second, we propose a new multiscale graph convolutional network (MGCN)
to transform a non-learned feature to a more discriminative descriptor. Our
results show that the new descriptor WEDS is more discriminative than the
current state-of-the-art non-learned descriptors and that the combination of
WEDS and MGCN is better than the state-of-the-art learned descriptors. An
important design criterion for our descriptor is the robustness to different
surface discretizations including triangulations with varying numbers of
vertices. Our results demonstrate that previous graph convolutional networks
significantly overfit to a particular resolution or even a particular
triangulation, but MGCN generalizes well to different surface discretizations.
In addition, MGCN is compatible with previous descriptors and it can also be
used to improve the performance of other descriptors, such as the heat kernel
signature, the wave kernel signature, or the local point signature.
