基于薄壳层能量的隐式曲面匹配的对称性和尺度极限
Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies
摘要：
在Iglesias、Rumpf和Scherzer最近发表的一篇论文中，Math.18（4），2018）提出了用于匹配作为水平集函数给出的一对形状的变形的变分模型。它的主要特征是各向异性能量的存在，仅在超曲面周围的窄带中起作用，类似于弹性壳的行为。在这项工作中，我们考虑一些扩展和进一步分析该模型。首先，我们提出了一个对称的能量泛函，使给定的两个特定的形状，它分配相同的能量给任何给定的变形，其逆形状的角色互换时，并引入足够的参数缩放恢复的表面问题时，窄带的宽度消失。然后，在宽度足够小的情况下，得到了对称能量的极小化变形的存在性，并证明了相应的非对称能量在宽度趋于零时的$Γ$ -收敛结果.最后，数值模拟结果的实际形状匹配的应用程序演示的对称能量的效果。
英文摘要：
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 $Γ$-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.
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