  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.
