Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies

TL;DR

This paper extends a variational model for shape matching using thin shell energies, introducing symmetry, analyzing limits, proving existence of minimizers, and demonstrating applications with numerical results.

Contribution

It introduces a symmetric energy functional for shape matching, analyzes its scaling limits, proves existence of minimizers, and demonstrates practical applications.

Findings

Symmetric energy functional ensures consistent deformation energies.

Gamma-convergence of non-symmetric energies as band width vanishes.

Numerical results show effectiveness in realistic shape matching.

Abstract

In a recent paper by Iglesias, Rumpf and Scherzer (Found. Comput. Math. 18(4), 2018) a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic energies active only in a narrow band around the hypersurfaces that resemble the behavior of elastic shells. In this work we consider some extensions and further analysis of that model. First, we present a symmetric energy functional such that given two particular shapes, it assigns the same energy to any given deformation as to its inverse when the roles of the shapes are interchanged, and introduce the adequate parameter scaling to recover a surface problem when the width of the narrow band vanishes. Then, we obtain existence of minimizing deformations for the symmetric energy in classes of bi-Sobolev homeomorphisms for small enough widths, and prove a $\Gamma$-convergence result for the corresponding non-symmetric energies as the width tends to zero. Finally, numerical results on realistic shape matching applications demonstrating the effect of the symmetric energy are presented.