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

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.

Image comparison and scaling via nonlinear elasticity

“…Our work is closest in spirit to that of Droske & Rumpf [14], Rumpf [27] and Rumpf & Wirth [26], who like us make use of the existence theory for polyconvex energies in [4], as also do Burger, Modersitski & Ruthotto [9], Debroux et al [12], Iglesias, Rumpf & Scherzer [16] and Iglesias [15]. Other nonlinear elasticity approaches are due to Lin, Dinov, Toga & Vese [19], Ozeré, Gout & Le Guyader [22], Ozeré & Le Guyader [23], Simon, Sheorey, Jacobs & Basri [28] and Debroux & Le Guyader [13].…”

“…Such a symmetry condition was introduced by Cachier & Rey [10] and subsequently used by Kolouri, Slepčev & Rohde [18] and Iglesias [15]. It is also implicit in the work of Iwaniec & Onninen [17].…”