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Surface comparison and matching is a challenging problem in computer vision.While reparametrization-invariant Sobolev metrics provide meaningful elasticdistances and point correspondences via the geodesic boundary value problem,solving this problem numerically tends to be difficult. Square root normalfields (SRNF) considerably simplify the computation of certain elasticdistances between parametrized surfaces. Yet they leave open the issue offinding optimal reparametrizations, which induce elastic distances betweenunparametrized surfaces. This issue has concentrated much effort in recentyears and led to the development of several numerical frameworks. In thispaper, we take an alternative approach which bypasses the direct estimation ofreparametrizations: we relax the geodesic boundary constraint using anauxiliary parametrization-blind varifold fidelity metric. This reformulationhas several notable benefits. By avoiding altogether the need forreparametrizations, it provides the flexibility to deal with simplicial meshesof arbitrary topologies and sampling patterns. Moreover, the problem lendsitself to a coarse-to-fine multi-resolution implementation, which makes thealgorithm scalable to large meshes. Furthermore, this approach extends readilyto higher-order feature maps such as square root curvature fields and is alsoable to include surface textures in the matching problem. We demonstrate theseadvantages on several examples, synthetic and real.

A numerical framework for elastic surface matching, comparison, and interpolation