LP Relaxation for Elastic Matching

According to [2], in this paper we discuss a new method to match two 3D shapes in a geometrically consistent way. We understand a matching as a diffemorphism and want to find the matching that minimizes the physical energy that is necessary to deform one shape into the other. Because of the fact that diffeomorphisms can be completely characterized by their graph, we propose an equivalent formulation of the problem that minimizes the surface in the product space of the two shapes. So in the two-dimensional case, we get a shortest path problem that can easily be solved with Dijkstra’s algorithm. The runtime is less than cubic in the number of discretized shape points. The discretization of the 3D model leads to an integer linear program that matches infinitesimal surface patches while preserving the geometric structures rather than mapping points to points. The relaxation of the integer constraint computes high-quality matchings even compared to existing works.