Certain optimal correspondences between plane curves, I: Manifolds of shapes and bimorphisms

Abstract

In previous joint work, a theory introduced earlier by Tagare was developed for establishing certain kinds of correspondences, termed bimorphisms, between simple closed regular plane curves of differentiability class at least C 2 C^2 . A class of objective functionals was introduced on the space of bimorphisms between two fixed curves C 1 C_1 and C 2 C_2 , and it was proposed that one define a “best non-rigid match” between C 1 C_1 and C 2 C_2 by minimizing such a functional. In this paper we prove several theorems concerning the nature of the shape-space of plane curves and of spaces of bimorphisms as infinite-dimensional manifolds. In particular, for 2 ≤ j > ∞ 2\leq j>\infty , the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for C ∞ C^\infty curves is the space of bimorphisms an infinite-dimensional manifold, and then only a Fréchet manifold, not a Banach manifold. This paper lays the groundwork for a companion paper in which we use the Nash Inverse Function Theorem and our results on C ∞ C^\infty curves and bimorphisms to show that if Γ \Gamma is strongly convex, if C 1 C_1 and C 2 C_2 are C ∞ C^\infty curves whose shapes are not too dissimilar ( C j C^j -close for a certain finite j j ) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.