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

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