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 | References | Similar Articles | Additional Information

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 , was introduced on the space of bimorphisms between two fixed curves and , and it was proposed that one define a ``best non-rigid match'' between and by minimizing such a functional. In this paper we use the Nash Inverse Function Theorem to show that for strongly convex functions , if and are curves whose shapes are not too dissimilar (specifically, are -close for a certain finite ), 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.

**[A]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[FB]**M. Frenkel and R. Basri,*Curve matching using the fast marching method*, Energy Minimization Methods in Computer Vision and Pattern Recognition: Proc. 4th International Workshop, EMMCVPR 2003, A. Rangarajan et al. (eds.), Springer-Verlag, Berlin, 2003, pp. 35-51.**[GT]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****[G]**D. Groisser,*Certain optimal correspondences between plane curves, I: manifolds of shapes and bimorphisms*, Trans. Amer. Math. Soc., this issue.**[H]**Richard S. Hamilton,*The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 1, 65–222. MR**656198**, 10.1090/S0273-0979-1982-15004-2**[T]**H. D. Tagare,*Shape-based nonrigid correspondence with application to heart motion analysis*, IEEE Trans. Med. Imaging**18**(1999), 570-579.**[TOG]**Hemant D. Tagare, Donal O’Shea, and David Groisser,*Non-rigid shape comparison of plane curves in images*, J. Math. Imaging Vision**16**(2002), no. 1, 57–68. MR**1884465**, 10.1023/A:1013938519103

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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

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