On the affine heat equation for non-convex curves

Angenent, Sigurd; Sapiro, Guillermo; Tannenbaum, Allen

doi:10.1090/S0894-0347-98-00262-8

On the affine heat equation for non-convex curves
HTML articles powered by AMS MathViewer

by Sigurd Angenent, Guillermo Sapiro and Allen Tannenbaum

J. Amer. Math. Soc. 11 (1998), 601-634

DOI: https://doi.org/10.1090/S0894-0347-98-00262-8

PDF | Request permission

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

Luis Álvarez, Frédéric Guichard, Pierre-Louis Lions, and Jean-Michel Morel, Axiomes et équations fondamentales du traitement d’images (analyse multiéchelle et EDP), C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 2, 135–138 (French, with English and French summaries). MR 1197224
Luis Alvarez, Frédéric Guichard, Pierre-Louis Lions, and Jean-Michel Morel, Axiomatisation et nouveaux opérateurs de la morphologie mathématique, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 3, 265–268 (French, with English and French summaries). MR 1179717
Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), no. 2, 207–230. MR 1424425
Sigurd Angenent, Parabolic equations for curves on surfaces. I. Curves with $p$-integrable curvature, Ann. of Math. (2) 132 (1990), no. 3, 451–483. MR 1078266, DOI 10.2307/1971426
Sigurd Angenent, Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math. (2) 133 (1991), no. 1, 171–215. MR 1087347, DOI 10.2307/2944327
Sigurd Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), no. 3, 601–633. MR 1100205
Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
W. Blaschke, Vorlesungen über Differentialgeometrie II, Verlag Von Julius Springer, Berlin, 1923.
Bu Chin Su, Affine differential geometry, Science Press Beijing, Beijing; Gordon & Breach Science Publishers, New York, 1983. MR 724783
Jean A. Dieudonné and James B. Carrell, Invariant theory, old and new, Academic Press, New York-London, 1971. MR 0279102
Xu-Yan Chen and Hiroshi Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), no. 1, 160–190. MR 986159, DOI 10.1016/0022-0396(89)90081-8
C. L. Epstein and Michael Gage, The curve shortening flow, Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986) Math. Sci. Res. Inst. Publ., vol. 7, Springer, New York, 1987, pp. 15–59. MR 920831, DOI 10.1007/978-1-4613-9583-6_{2}
Michael E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225–1229. MR 726325, DOI 10.1215/S0012-7094-83-05052-4
M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), no. 2, 357–364. MR 742856, DOI 10.1007/BF01388602
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401, DOI 10.4310/jdg/1214439902
Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111. MR 979601, DOI 10.2307/1971486
Heinrich W. Guggenheimer, Differential geometry, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0156266
Benjamin B. Kimia, Allen Tannenbaum, and Steven W. Zucker, On the evolution of curves via a function of curvature. I. The classical case, J. Math. Anal. Appl. 163 (1992), no. 2, 438–458. MR 1145840, DOI 10.1016/0022-247X(92)90260-K
B. B. Kimia, A. Tannenbaum, and S. W. Zucker, “Shapes, shocks, and deformations,” Int. J. Computer Vision 15 (1995), 189-224.
Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401–441. MR 672070
Peter J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. MR 1240056, DOI 10.1007/978-1-4612-4350-2
P. J. Olver, “Differential invariants,” to appear in Acta Appl. Math.
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.
Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum, Classification and uniqueness of invariant geometric flows, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 339–344 (English, with English and French summaries). MR 1289308
Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum, Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), no. 1, 176–194. MR 1429382, DOI 10.1137/S0036139994266311
P. Olver, G. Sapiro, and A. Tannenbaum, “Affine invariant edge maps and active contours,” to appear in CVIU.
Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. MR 965860, DOI 10.1016/0021-9991(88)90002-2
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
Guillermo Sapiro and Allen Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), no. 1, 79–120. MR 1255274, DOI 10.1006/jfan.1994.1004
G. Sapiro and A. Tannenbaum, “Affine invariant scale-space,” Int. J. Computer Vision 11, pp. 25-44, 1993.
Guillermo Sapiro and Allen Tannenbaum, On invariant curve evolution and image analysis, Indiana Univ. Math. J. 42 (1993), no. 3, 985–1009. MR 1254129, DOI 10.1512/iumj.1993.42.42046
G. Sapiro and A. Tannenbaum, “Area and length preserving geometric invariant scale-spaces," IEEE Trans. Pattern Analysis and Machine Intelligence 17 (1995), 1066-1070.
Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532830
Brian White, Some recent developments in differential geometry, Math. Intelligencer 11 (1989), no. 4, 41–47. MR 1016106, DOI 10.1007/BF03025885