  We introduce pointwise map smoothness via the Dirichlet energy into the
functional map pipeline, and propose an algorithm for optimizing it
efficiently, which leads to high-quality results in challenging settings.
Specifically, we first formulate the Dirichlet energy of the pulled-back shape
coordinates, as a way to evaluate smoothness of a pointwise map across discrete
surfaces. We then extend the recently proposed discrete solver and show how a
strategy based on auxiliary variable reformulation allows us to optimize
pointwise map smoothness alongside desirable functional map properties such as
bijectivity. This leads to an efficient map refinement strategy that
simultaneously improves functional and point-to-point correspondences,
obtaining smooth maps even on non-isometric shape pairs. Moreover, we
demonstrate that several previously proposed methods for computing smooth maps
can be reformulated as variants of our approach, which allows us to compare
different formulations in a consistent framework. Finally, we compare these
methods both on existing benchmarks and on a new rich dataset that we
introduce, which contains non-rigid, non-isometric shape pairs with
inter-category and cross-category correspondences. Our work leads to a general
framework for optimizing and analyzing map smoothness both conceptually and in
challenging practical settings.
