UC Irvine

In this work, we introduce two methods to handle nearly isometric shape correspondence problem. The first method, Sparsity-Enforced Quadratic Assignment (SEQA), introduces a novel local pairwise descriptor and then develops a simple, effective iterative method to solve the resulting quadratic assignment through sparsity control. The pairwise descriptor is based on the stiffness and mass matrix of finite element approximation of the Laplace-Beltrami differential operator. The key idea of our iterative algorithm is to select pairs with good correspondence as anchor pairs based on a new criterion we introduced called local mapping distortion, and then solve a regularized quadratic assignment problem the neighborhoods of corresponding selected anchor pairs through sparsity control. Various pointwise global features with reference to these anchor pairs can be used to improve the dense shape correspondence further. Inspired by the powerful ability of local mapping distortion on selecting accurate anchor pairs, we further introduce the second method, Dual Iterative Refinement (DIR). It is a simple and efficient algorithm which combines dual features, spatial and spectral, or local and global, in a complementary and optimal way. DIR first uses local spatial feature, local mapping distortion to obtain anchor pairs which are used to determine an appropriate dimension of the spectral space and the corresponding functional map. Then the functional map is used to update a new correspondence. Both methods allow us to deal with open surfaces, partial matching, and topological perturbations robustly. We use various experiments to show the efficiency, quality, and versatility of our methods on large data sets, patches, and point clouds (without global meshes).