Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

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 $ \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.

References [Enhancements On Off] (What's this?)

  • [A] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [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] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition. Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [G] D. Groisser, Certain optimal correspondences between plane curves, I: manifolds of shapes and bimorphisms, Trans. Amer. Math. Soc., this issue.
  • [H] R. S. Hamilton, The Inverse Function Theorem of Nash and Moser, Bull. (New Ser.) Amer. Math. Soc. 7 (1982), 65-222. MR 656198 (83j:58014)
  • [T] H. D. Tagare, Shape-based nonrigid correspondence with application to heart motion analysis, IEEE Trans. Med. Imaging 18 (1999), 570-579.
  • [TOG] H. D. Tagare, D. O'Shea, and D. Groisser, Non-rigid shape comparison of plane curves in images, J. Math. Imaging and Vision 16 (2002), 57-68. MR 1884465 (2002m:68120)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53A04, 49K15

Retrieve articles in all journals with MSC (2000): 53A04, 49K15

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

American Mathematical Society