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

Author: David Groisser
Journal: Trans. Amer. Math. Soc. 361 (2009), 2959-3000
MSC (2000): Primary 53A04, 49K15
Published electronically: December 23, 2008
MathSciNet review: 2485414
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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$. A class of objective functionals 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 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\leq j<\infty$, the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for $ 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^\infty$ curves and bimorphisms to show that if $ \Gamma$ is strongly convex, if $ C_1$ and $ C_2$ are $ C^\infty$ curves whose shapes are not too dissimilar ($ C^j$-close for a certain finite $ j$) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.

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Additional Information

David Groisser
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Keywords: Shape analysis, shape space, non-rigid correspondence, plane curve, bimorphism
Received by editor(s): April 5, 2004
Received by editor(s) in revised form: February 11, 2007
Published electronically: December 23, 2008
Article copyright: © Copyright 2008 American Mathematical Society