Certain optimal correspondences between plane curves, II: Existence, local uniqueness, regularity, and other properties

Abstract:

This paper is a companion to the author’s paper (this volume), in which several theorems were proven concerning the nature, as infinite-dimensional manifolds, of the shape-space of plane curves and of spaces of certain curve-correspondences called bimorphisms. In Tagare, O’Shea, Groisser, 2002, a class of objective functionals, depending on a choice of cost-function $\Gamma$, was introduced on the space of bimorphisms between two fixed curves $C_1$ and $C_2$, and it was proposed that one define a “best non-rigid match” between $C_1$ and $C_2$ by minimizing such a functional. In this paper we use the Nash Inverse Function Theorem to show that for strongly convex functions $\Gamma$, if $C_1$ and $C_2$ are $C^\infty$ curves whose shapes are not too dissimilar (specifically, are $C^j$-close for a certain finite $j$), and neither is a perfect circle, then the minimum of a certain regularized objective functional exists and is locally unique. We also study certain properties of the Euler-Lagrange equation for the objective functional, and obtain regularity results for “exact matches” (bimorphisms for which the objective functional achieves its absolute minimum value of 0) that satisfy a genericity condition.