Conformal curvature flows: From phase transitions to active vision

Abstract

In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.

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References

  1. 1.

    S. M. Allen & J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27, 1085–1095, 1979.

    Google Scholar 

  2. 2.

    L. Alvarez, F. Guichard, P. L. Lions & J. M. Morel, Axiomes et equations fondamentales du traitement d'images, C. R. Acad. Sci. Paris 315, 135–138, 1992.

    Google Scholar 

  3. 3.

    L. Alvarez, F. Guichard, P. L. Lions & J. M. Morel, Axioms and fundamental equations of image processing, Report #9216, CEREMADE, Université Paris Dauphine, 1992; Arch. Rational Mech. Anal. 123, 200–257, 1993.

  4. 4.

    L. Alvarez, F. Guichard, P. L. Lions & J. M. Morel, Axiomatisation et nouveaux operateurs de la morphologie mathematique, C. R. Acad. Sci. Paris 315, 265–268, 1992.

    Google Scholar 

  5. 5.

    L. Alvarez, P. L. Lions & J. M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Num. Anal. 29, 845–866, 1992.

    Google Scholar 

  6. 6.

    L. Alvarez & J. M. Morel, Formalization and computational aspects of image analysis, Report #0493, Department of Information and Systems, Universidad de las Palmas de Gran Canaria, 1993.

  7. 7.

    B. Andrews, Contraction of convex hypersurfaces by their affine normal, submitted for publication, 1994.

  8. 8.

    S. Angenent, On the formation of singularities in the curve shortening flow, J. Diff. Geom. 33, 601–633, 1991.

    Google Scholar 

  9. 9.

    S. Angenent & M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2: Evolution of an isothermal surface, Arch. Rational Mech. Anal. 108, 323–391, 1989.

    Google Scholar 

  10. 10.

    S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly 87, 359–370, 1980.

    Google Scholar 

  11. 11.

    A. Blake & A. Yuille, Active Vision, MIT Press, 1992.

  12. 12.

    K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, 1978.

  13. 13.

    L. Bronsard & R. V. Kohn, Motion by mean curvature as a singular limit of Ginzburg-Landau dynamics, J. Diff. Eqs. 90, 211–237, 1991.

    Google Scholar 

  14. 14.

    L. Bronsard & F. Reitich, On three phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal. 124, 355–379, 1993.

    Google Scholar 

  15. 15.

    G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of phase field equations, Phys. Rev. A 39, 887–896, 1989.

    Google Scholar 

  16. 16.

    V. Caselles, F. Catte, T. Coll & F. Dibos, A geometric model for active contours in image processing, Technical Report #9210, CEREMADE, Université Paris Dauphine, 1992.

  17. 17.

    V. Caselles & C. Sbert, What is the best causal scale-space for 3D images?, Technical Report, Department of Math, and Comp. Sciences, University of Illes Balears, 07071 Palma de Mallorca, Spain, March 1994.

    Google Scholar 

  18. 18.

    Y.-G. Chen, Y. Giga & S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom. 33, 749–786, 1991.

    Google Scholar 

  19. 19.

    B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Diff. Geom. 22, 117–138, 1985.

    Google Scholar 

  20. 20.

    M. G. Crandall & H. Ishii, The maximum principle for semicontinuous functions, Diff. Integral Eqs. 3, 1001–1014, 1990.

    Google Scholar 

  21. 21.

    M. G. Crandall, H. Ishii & P.-L. Lions, Users guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27, 1–67, 1992.

    Google Scholar 

  22. 22.

    M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.

  23. 23.

    M. P. Do Carmo, Riemannian Geometry, Prentice-Hall, 1992.

  24. 24.

    C. L. Epstein & M. Gage, The curve shortening flow, in Wave Motion: Theory, Modeling, and Computation, A. Chorin & A. Majda, Editors, Springer-Verlag, 1987.

  25. 25.

    L. C. Evans & J. Spruck, Motion of level sets by mean curvature, I, J. Diff. Geom. 33, 635–681, 1991.

    Google Scholar 

  26. 26.

    L. C. Evans, H. M. Soner & P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45, 1097–1123, 1992.

    Google Scholar 

  27. 27.

    M. Gage, Curve shortening makes convex curves circular, Invent. Math. 76, 357–364, 1984.

    Google Scholar 

  28. 28.

    M. Gage & R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom. 23, 69–96, 1986.

    Google Scholar 

  29. 29.

    I. M. Gelfand & S. V. Fomin, Calculus of Variations, Prentice-Hall, 1963.

  30. 30.

    C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Diff. Geom. 32, 299–314, 1990.

    Google Scholar 

  31. 31.

    M. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58, 555–558, 1989.

    Google Scholar 

  32. 32.

    M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom. 26, 285–314, 1987.

    Google Scholar 

  33. 33.

    M. Grayson, Shortening embedded curves, Ann. Math. 129, 71–111, 1989.

    Google Scholar 

  34. 34.

    A. Gupta, L. Von Kurowski, A. Singh, D. Geiger, C.-C. Liang, M.-Y. Chiu, L. P. Adler, M. Haacke & D. L. Wilson, Cardiac MRI analysis: segmentation of myocardial boundaries using deformable models preprint.

  35. 35.

    M. E. Gurtin, Toward a non-equilibrium thermodynamics of two-phase materials, Arch. Rational Mech. Anal. 100, 275–312, 1988.

    Google Scholar 

  36. 36.

    M. Gurtin, Multiphase thermomechanics with interfacial structure, 1: Heat conduction and the capillary balance law, Arch. Rational Mech. Anal, 104, 185–221, 1988.

    Google Scholar 

  37. 37.

    G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom. 20, 237–266, 1984.

    Google Scholar 

  38. 38.

    T. Ilmanen, Generalized flow of sets by mean curvature on a manifold, Indiana Univ. Math. J. 41, 671–705, 1992.

    Google Scholar 

  39. 39.

    M. Kass, A. Witkin & D. Terzopoulos, Snakes: active contour models, Int. J. Computer Vision 1, 321–331. 1987.

    Google Scholar 

  40. 40.

    S. Kichenassamy, A. Kumar, P. J. Olver, A. Tannenbaum & A. Yezzi, Gradient flows and geometric active contours, Proceedings of the Fifth International Conference on Computer Vision, 810–816, 1995.

  41. 41.

    B. B. Kimia, A. Tannenbaum & S. W. Zucker, Toward a computational theory of shape: An overview, Lecture Notes in Computer Science 427, 402–407, Springer-Verlag, 1990

  42. 42.

    B. B. Kimia, A. Tannenbaum & S. W. Zucker, Shapes, shocks and deformations, I, to appear in Int. J. Computer Vision.

  43. 43.

    B. B. Kimia, A. Tannenbaum & S. W. Zucker, On the evolution of curves via a function of curvature, I: the classical case, J. Math. Anal. Appls. 163, 438–458, 1992.

    Google Scholar 

  44. 44.

    A. Kumar, Visual Information in a Feedback Loop, Ph.D. thesis, University of Minnesota, 1995.

  45. 45.

    R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992.

  46. 46.

    P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, 1982.

  47. 47.

    R. Malladi, J. Sethian & B. Vemuri, Shape modeling with front propagation: a level set approach, IEEE Trans. Pattern Anal. Machine Intell. 17, 158–175, 1995.

    Google Scholar 

  48. 48.

    F. Mokhatarian & A. Mackworth, A theory of multiscale, curvature-based shape representation for planar curves, IEEE Trans. Pattern Anal. Machine Intell. 14, 789–805, 1992.

    Google Scholar 

  49. 49.

    F. Morgan, Riemannian Geometry, John and Bartlett, 1993.

  50. 50.

    P. De Mottoni & M. Schatzman, Evolution géométrique d'interfaces, C. R. Acad. Sci. Paris, sér. I, Math. 309, 453–58, 1989.

    Google Scholar 

  51. 51.

    W. W. Mullins, Theory of thermal grooving, J. Appl. Phys. 28, 333–339, 1957.

    Google Scholar 

  52. 52.

    P. Neskovic & B. Kimia, Three-dimensional shape representation from curvature-dependent deformations, Technical Report #128, LEMS, Brown University, 1994.

  53. 53.

    R. H. Nochetto, M. Paolini & C. Verdi, A dynamic mesh method algorithm for curvature dependent evolving interfaces, Technical Report, University of Maryland, 1994.

  54. 54.

    P. J. Olver, G. Sapiro & A. Tannenbaum, Geometric invariant evolution of surfaces and volumetric smoothing, to appear in SIAM J. Math. Anal., 1994.

  55. 55.

    S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21, 217–235, 1984.

    Google Scholar 

  56. 56.

    S. J. Osher & J. A. Sethian, Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys. 79, 12–49, 1988.

    Google Scholar 

  57. 57.

    S. Osher & L. I. Rudin, Feature-oriented image enhancement using shock filters, SIAM J. Num. Anal. 27, 919–940, 1990.

    Google Scholar 

  58. 58.

    P. Perona & J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639, 1990.

    Google Scholar 

  59. 59.

    M. H. Protter & H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984.

  60. 60.

    J. Rubinstein, P. Sternberg & J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49, 116–133, 1989.

    Google Scholar 

  61. 61.

    G. Sapiro & A. Tannenbaum, Affine invariant scale-space, Int. J. Computer Vision 11, 25–44, 1993.

    Google Scholar 

  62. 62.

    G. Sapiro & A. Tannenbaum, On invariant curve evolution and image analysis, Indiana Univ. Math. J. 42, 985–1009, 1993.

    Google Scholar 

  63. 63.

    J. A. Sethian, An Analysis of Flame Propagation, Ph.D. Dissertation, University of California, 1982.

  64. 64.

    J. A. Sethian, Curvature and the evolution of fronts, Comm. Math. Phys. 101, 487–499, 1985.

    Google Scholar 

  65. 65.

    J. A. Sethian, A review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed, J. Diff. Geom. 31, 131–161, 1989.

    Google Scholar 

  66. 66.

    J. A. Sethian & J. Strain, Crystal growth and dendritic solidification, J. Comp. Phys. 98, 1992.

  67. 67.

    J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer-Verlag, 1983.

  68. 68.

    G. A. Sod, Numerical Methods in Fluid Dynamics, Cambridge University Press, 1985.

  69. 69.

    M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, 1979.

  70. 70.

    H. Tek & B. Kimia, Deformable bubbles in the reaction-diffusion space, Technical Report #138, LEMS, Brown University, 1994.

  71. 71.

    D. Terzopoulos & A. Witkin, Constraints on deformable models: recovering shape and non-rigid motion, Artificial Intelligence 36, 91–123, 1988.

    Google Scholar 

  72. 72.

    D. Terzopoulos & R. Szelski, Tracking with Kalman snakes, in Active Vision edited by A. Blake & A. Zisserman, MIT Press, 1992.

  73. 73.

    B. White, Some recent developments in differential geometry, Math. Intelligencer 11, 41–47, 1989.

    Google Scholar 

  74. 74.

    A. Yezzi, S. Kichenassamy, A. Kumar, P. Olver & A. Tannenbaum, Geometric active contours for segmentation of medical imagery, to appear in IEEE Trans. Medical Imaging.

  75. 75.

    A. Yezzi, S. Kichenassamy, P. Olver & A. Tannenbaum, A gradient surface approach to 3D segmentation, to appear in Proceedings of IS&T 49th Annual Conference.

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Communicated by J. Serrin

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Kichenassamy, S., Kumar, A., Olver, P. et al. Conformal curvature flows: From phase transitions to active vision. Arch. Rational Mech. Anal. 134, 275–301 (1996). https://doi.org/10.1007/BF00379537

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Keywords

  • Phase Transition
  • Evolution Equation
  • Active Surface
  • Surface Model
  • Electromagnetism