  In this work, we introduce a novel local pairwise descriptor and then develop
a simple, effective iterative method to solve the resulting quadratic
assignment through sparsity control for shape correspondence between two
approximate isometric surfaces. Our pairwise descriptor is based on the
stiffness and mass matrix of finite element approximation of the
Laplace-Beltrami differential operator, which is local in space, sparse to
represent, and extremely easy to compute while containing global information.
It allows us to deal with open surfaces, partial matching, and topological
perturbations robustly. To solve the resulting quadratic assignment problem
efficiently, the two key ideas of our iterative algorithm are: 1) select pairs
with good (approximate) correspondence as anchor points, 2) solve a regularized
quadratic assignment problem only in the neighborhood of selected anchor points
through sparsity control. These two ingredients can improve and increase the
number of anchor points quickly while reducing the computation cost in each
quadratic assignment iteration significantly. With enough high-quality anchor
points, one may use various pointwise global features with reference to these
anchor points to further improve the dense shape correspondence. We use various
experiments to show the efficiency, quality, and versatility of our method on
large data sets, patches, and point clouds (without global meshes).
