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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: David Groisser
Journal: Trans. Amer. Math. Soc. 361 (2009), 3001-3030
MSC (2000): Primary 53A04, 49K15
Published electronically: December 23, 2008
MathSciNet review: 2485415
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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.


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Additional Information

David Groisser
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
Email: groisser@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04497-8
PII: S 0002-9947(08)04497-8
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