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On the affine heat equation for non-convex curves
Author(s):
Sigurd
Angenent;
Guillermo
Sapiro;
Allen
Tannenbaum
Journal:
J. Amer. Math. Soc.
11
(1998),
601-634.
MSC (1991):
Primary 35K22, 53A15, 58G11
MathSciNet review:
1491538
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Abstract:
In this paper, we extend to the non-convex case the affine invariant geometric heat equation studied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the well-known Euclidean geometric heat equation.
References:
- 1.
- L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, ``Axiomes et equations fondamentales du traitement d`images,'' C. R. Acad. Sci. Paris 315, pp. 135-138, 1992. MR 94d:47066
- 2.
- L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, ``Axiomatisation et nouveaux operateurs de la morphologie mathematique,'' C. R. Acad. Sci. Paris 315, pp. 265-268, 1992. MR 94d:47067
- 3.
- B. Andrews, ``Contraction of convex hypersurfaces by their affine normal,'' J. Differential Geometry 43, pp. 207-230, 1996. MR 97m:58045
- 4.
- S. Angenent, ``Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature,'' Annals of Mathematics 132, pp. 451-483, 1990. MR 91k:35102
- 5.
- S. Angenent, ``Parabolic equations for curves on surfaces, Part II. Intersections, blow-up, and generalized solutions,'' Annals of Mathematics 133, pp. 171-215, 1991. MR 92b:58039
- 6.
- S. Angenent, ``On the formation of singularities in the curve shortening flow,'' J. Differential Geometry 33, pp. 601-633, 1991. MR 92c:58016
- 7.
- S. Angenent, ``The zero set of a solution of a parabolic equation,'' J. für die reine and angewandte Mathematik 390, pp. 79-96, 1988. MR 89j:35015
- 8.
- W. Blaschke, Vorlesungen über Differentialgeometrie II, Verlag Von Julius Springer, Berlin, 1923.
- 9.
- S. Buchin, Affine Differential Geometry, Gordon and Breach, Science Publishers, Inc., New York, 1983. MR 85g:53010
- 10.
- J. Dieudonné and J. Carrell, Invariant Theory: Old and New, Academic Press, London, 1970. MR 43:4828
- 11.
- X.-Y. Chen and H. Matano, ``Convergence, asymptotic periodicity and one point blow-up in one dimensional semilinear heat equations,'' Journal of Differential Equations 33, pp. 160-190 (1989). MR 90e:35018
- 12.
- C. L. Epstein and M. Gage, ``The curve shortening flow,'' in Wave Motion: Theory, Modeling, and Computation, A. Chorin and A. Majda, Editors, Springer-Verlag, New York, 1987. MR 89f:58128
- 13.
- M. Gage, ``An isoperimetric inequality with applications to curve shortening,'' Duke Mathematical Journal 50, pp. 1225-1229, 1983. MR 85d:52007
- 14.
- M. Gage, ``Curve shortening makes convex curves circular,'' Invent. Math. 76, pp. 357-364, 1984. MR 85i:52004
- 15.
- M. Gage and R. S. Hamilton, ``The heat equation shrinking convex plane curves,'' J. Differential Geometry 23, pp. 69-96, 1986. MR 87m:53003
- 16.
- M. Grayson, ``The heat equation shrinks embedded plane curves to round points,'' J. Differential Geometry 26, pp. 285-314, 1987. MR 89b:53005
- 17.
- M. Grayson, ``Shortening embedded curves,'' Annals of Mathematics 129, pp. 71-111, 1989. MR 90a:53050
- 18.
- H. W. Guggenheimer, Differential Geometry, McGraw-Hill Book Company, New York, 1963. MR 27:6194; MR 58:12737
- 19.
- B. B. Kimia, A. Tannenbaum, and S. W. Zucker, ``On the evolution of curves via a function of curvature, I: the classical case,'' J. of Math. Analysis and Applications 163, pp. 438-458, 1992. MR 93a:58037
- 20.
- B. B. Kimia, A. Tannenbaum, and S. W. Zucker, ``Shapes, shocks, and deformations,'' Int. J. Computer Vision 15 (1995), 189-224.
- 21.
- H. Matano, ``Non-increase of the lapnumber of a solution for a one dimensional semilinear parabolic equation,'' J. Fac. Sci. Univ. Tokyo Math. 29 (1982), 401-441. MR 84m:35060
- 22.
- P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer-Verlag, New York, 1993. MR 94g:58260
- 23.
- P. J. Olver, ``Differential invariants,'' to appear in Acta Appl. Math.
- 24.
- P. Olver, G. Sapiro, and A. Tannenbaum, ``Differential invariant signatures and flows in computer vision: A symmetry group approach,'' Geometric Driven Diffusion, edited by Bart ter har Romeny, Kluwer, 1994.
- 25.
- P. Olver, G. Sapiro, and A. Tannenbaum, ``Classification and uniqueness of invariant geometric flows,'' Comptes Rendus Acad. Sci. (Paris) 319 (1994), 339-344. MR 95j:58180
- 26.
- P. Olver, G. Sapiro, and A. Tannenbaum, ``Invariant geometric evolutions of surfaces and volumetric smoothing,'' SIAM J. Applied Math. 57 (1997), 176-194. MR 97j:53014
- 27.
- P. Olver, G. Sapiro, and A. Tannenbaum, ``Affine invariant edge maps and active contours,'' to appear in CVIU.
- 28.
- S. J. Osher and J. A. Sethian, ``Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,'' Journal of Computational Physics 79, pp. 12-49, 1988. MR 89h:80012
- 29.
- M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New York, 1967. MR 36:2935
- 30.
- G. Sapiro and A. Tannenbaum, ``On affine plane curve evolution,'' Journal of Functional Analysis 119, pp. 79-120, 1994. MR 94m:58049
- 31.
- G. Sapiro and A. Tannenbaum, ``Affine invariant scale-space,'' Int. J. Computer Vision 11, pp. 25-44, 1993.
- 32.
- G. Sapiro and A. Tannenbaum, ``Invariant curve evolution and image processing,'' Indiana Univ. Journal of Math. 42, pp. 985-1009, 1993. MR 94m:58048
- 33.
- G. Sapiro and A. Tannenbaum, ``Area and length preserving geometric invariant scale-spaces," IEEE Trans. Pattern Analysis and Machine Intelligence 17 (1995), 1066-1070.
- 34.
- M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc, Berkeley, California, 1979. MR 82g:53003a; MR 82g:53003b; MR 82g:53003c; MR 82g:53003d; MR 82g:53003e
- 35.
- B. White, ``Some recent developments in differential geometry,'' Mathematical Intelligencer 11, pp. 41-47, 1989. MR 90k:53003
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Additional Information:
Sigurd
Angenent
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Guillermo
Sapiro
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455
Allen
Tannenbaum
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455
Email:
tannenba@ece.umn.edu
DOI:
10.1090/S0894-0347-98-00262-8
PII:
S 0894-0347(98)00262-8
Received by editor(s):
April 24, 1997
Received by editor(s) in revised form:
January 20, 1998
Additional Notes:
This work was supported in part by grants from the National Science Foundation DMS-9058492, ECS-9122106, ECS-99700588, NSF-LIS, by the Air Force Office of Scientific Research AF/F49620-94-1-00S8DEF, AF/F49620-94-1-0461, AF/F49620-98-1-0168, by the Army Research Office DAAL03-92-G-0115, DAAH04-94-G-0054, DAAH04-93-G-0332, MURI Grant, Office of Naval Research ONR-N00014-97-1-0509, and by the Rothschild Foundation-Yad Hanadiv.
Copyright of article:
Copyright
1998,
American Mathematical Society
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