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.

中文翻译：

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基于薄壳能量的隐式曲面匹配的对称性和标度限制

在 Iglesias、Rumpf 和 Scherzer（Found. Comput. Math. 18(4), 2018）最近的一篇论文中，提出了一种变形变分模型，该模型与作为水平集函数给出的一对形状相匹配。它的主要特征是存在仅在超曲面周围的窄带中活跃的各向异性能量，类似于弹性壳的行为。在这项工作中，我们考虑对该模型进行一些扩展和进一步分析。首先，我们提出了一个对称的能量泛函，这样给定两个特定的形状，当形状的角色互换时，它为任何给定的变形分配相同的能量，并引入足够的参数缩放来恢复表面问题。窄带的宽度消失。然后，我们获得了对于足够小宽度的双索博列夫同胚类中对称能量的最小变形的存在，并证明了当宽度趋于零时对应的非对称能量的 $\Gamma$-收敛结果。最后，给出了实际形状匹配应用的数值结果，证明了对称能量的影响。