Relaxations for Minimizing Metric Distortion and Elastic Energies for 3D Shape Matching

Abstract

We present two methods for non-rigid shape matching. Both methods formulate shape matching as an energy minimization problem, where the energy measures distortion of the metric defined on the shapes in one case, or directly describes the physical deformation relating the two shapes in the other case. The first method considers a parametrized relaxation of the widely adopted quadratic assignment problem (QAP) formulation for minimum distortion correspondence between deformable shapes. In order to control the accuracy/sparsity trade-off a weighting parameter is introduced to combine two existing relaxations, namely spectral and game-theoretic. This leads to an approach for deformable shape matching with controllable sparsity. The second method focuses on computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points it matches infinitesimal surface patches while preserving the geometric structures. In this spirit, matchings are considered as diffeomorphisms between the objects' surfaces which are by definition geometrically consistent. Based on the observation that such diffeomorphisms can be represented as closed and continuous surfaces in the product space of the two shapes, this leads to a minimal surface problem in this product space. The proposed discrete formulation describes the search space with linear constraints. Computationally, the approach results in a binary linear program whose relaxed version can be solved efficiently in a globally optimal manner.