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Affine invariant scale-space

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Abstract

A newaffine invariant scale-space for planar curves is presented in this work. The scale-space is obtained from the solution of a novel nonlinear curve evolution equation which admits affine invariant solutions. This flow was proved to be the affine analogue of the well knownEuclidean shortening flow. The evolution also satisfies properties such ascausality, which makes it useful in defining a scale-space. Using an efficient numerical algorithm for curve evolution, this continuous affine flow is implemented, and examples are presented. The affine-invariant progressive smoothing property of the evolution equation is demonstrated as well.

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References

  1. U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions,J. Diff. Geom. 23: 175–196, 1986.

    Google Scholar 

  2. 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: 135–138, 1992.

    Google Scholar 

  3. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, Axiom-atisation et nouveaux operateurs de la morphologie mathematique,C.R. Acad. Sci. Paris 315: 265–268, 1992.

    Google Scholar 

  4. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, Axioms and fundamental equations of image processing, to appear,Arch. for Rational Mechanics.

  5. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, Équations fondamentales de l'analyse multiechélle des filmes,C. R. Acad. Sci. Paris 315: 1145–1148, 1992.

    Google Scholar 

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

    Google Scholar 

  7. S. Angenent, Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature,Ann. Math. 132: 451–483, 1990.

    Google Scholar 

  8. S. Angenent, Parabolic equations for curves on surfaces, Part II. Intersections, blow-up, and generalized solutions,Ann. Math. 133: 171–215, 1991.

    Google Scholar 

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

    Google Scholar 

  10. J. Babaud, A.P. Witkin, M. Baudin, and R.O. Duda, Uniqueness of the Gaussian kernel for scale-space filtering,IEEE Trans. Patt. Anal. Mach. Intell. 8: 26–33, 1986.

    Google Scholar 

  11. G. Barles, Remarks on a flame propagation model,Rapports de Recherce 464:INRIA, Sophia Antipolis, 1985.

  12. W. Blaschke,Vorlesungen über Differentialgeometrie II, Verlag Von Julius Springer: Berlin, 1923.

    Google Scholar 

  13. H. Blum, Biological shape and visual science,J. Theor. Biol. 38: 205–287, 1973.

    Google Scholar 

  14. A.M. Bruckstein and A.N. Netravali, On differential invariants of planar curves and recognizing partially occluded planar shapes,Proc. Visual Form Workshop, Capri, May 1991, Plenum Press: New York.

    Google Scholar 

  15. S. Buchin,Affine Differential Geometry, Gordon and Breach, Science Publishers: New York, 1983.

    Google Scholar 

  16. M.P. DoCarmo,Differential Geometry of Curves and Surfaces, Prentice-Hall: Englewood Cliffs, 1976.

    Google Scholar 

  17. M.H. Chen and P.F. Yan, A multiscale approach based on morphological filtering,IEEE Trans. Patt. Anal. Mach. Intell. 11: 694–700, 1989.

    Google Scholar 

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

    Google Scholar 

  19. D.L. Chopp, Computing minimal surfaces via level set curvature flow, Ph.D. Dissertation, Mathematics Department, Lawrence Berkeley Laboratory, 1991.

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

    Google Scholar 

  21. M.G. Crandall, H. Ishii, and P.L. Lions, User's guide to viscosity solutions of second order partial linear differential equations,Bull. Amer. Math. Soc. 27: 1–67, 1992.

    Google Scholar 

  22. J. Dieudonné and J. Carrell,Invariant Theory: Old and New, Academic Press: London, 1970.

    Google Scholar 

  23. G. Dudek and J.K. Tsotsos, Shape representation and recognition from curvature,Proc. IEEE Conf. Comput. Vis. Patt. Recog., Hawaii, 1991.

  24. C.L. Epstein and M. Gage, The curve shortening flow. InWave Motion: Theory, Modeling, and Computation, A. Chorin and A. Majda, eds., Springer-Verlag: New York, 1987.

    Google Scholar 

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

    Google Scholar 

  26. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, Scale and the differential structure of images,Image and Vis. Comput. 10: 376–388, 1992.

    Google Scholar 

  27. D. Forsyth, J.L. Mundy, A. Zisserman, C. Coelho, A. Heller, and C. Rothwell, Invariant descriptors for 3-D object recognition and pose,IEEE Trans. Patt. Anal. Mach. Intell. 13: 971–991, 1991.

    Google Scholar 

  28. M. Gage, An isoperimetric inequality with applications to curve shortening,Duke Math. J. 50: 1225–1229, 1983.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  33. H.W. Guggenheimer,Differential Geometry, McGraw-Hill: New York, 1963.

    Google Scholar 

  34. A. Hummel, Representations based on zero-crossings in scale-space,Proc. IEEE Conf. Comput. Vis. Patt. Recog., pp. 204–209, Miami Beach, 1986.

  35. B.B. Kimia,Toward a Computational Theory of Shape, Ph.D. Dissertation, Department of Electrical Engineering, McGill University, Montreal, Canada, August 1990.

    Google Scholar 

  36. B.B. Kimia, A. Tannenbaum, and S.W. Zucker, Toward a computational theory of shape: An overview,Lect. Notes Comput. Sci. 427: 402–407, Springer-Verlag: New York, 1990.

    Google Scholar 

  37. B.B. Kimia, A. Tannenbaum, and S.W. Zucker, Shapes, shocks, and deformations, I, to appear,International Journal of Computer Vision.

  38. B.B. Kimia, A. Tannenbaum, and S.W. Zucker, Entropy scale-space,Proc. Visual Form Workshop, Capri, May 1991, Plenum Press: New York.

    Google Scholar 

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

    Google Scholar 

  40. R. Kimmel, K. Siddiqi, B.B. Kimia, and A.M. Bruckstein, Shape from shading via level sets, CIS Report, Computer Science Department, Technion, I.I.T., Haifa 32000, Israel, June 1992, submitted.

  41. J.J. Koenderink, The structure of images,Biological Cybernetics 50: 363–370, 1984.

    Google Scholar 

  42. J.J. Koenderink and A.J. van Doorn, Dynamic shape,Biological Cybernetics 53: 383–396, 1986.

    Google Scholar 

  43. J.J. Koenderink and A.J. van Doorn, Representation of local geometry in the visual system,Biological Cybernetics 55: 367–375, 1987.

    Google Scholar 

  44. J.J. Koenderink,Solid Shape, MIT Press: Cambridge, MA, 1990.

    Google Scholar 

  45. E.P. Lane,A Treatise on Projective Differential Geometry, University of Chicago Press, 1941.

  46. E. Lutwak, On the Blaschke-Santalo inequality,Ann. N.Y. Acad. Sci, Discrete Geometry and Convexity 440: 106–112, 1985.

    Google Scholar 

  47. E. Lutwak, On some affine isoperimetric inequalities,J. Diff. Geom. 23: 1–13, 1986.

    Google Scholar 

  48. F. Mokhatarian and A. Mackworth, Scale-based description of planar curves and two-dimensional shapes,IEEE Trans. Patt. Anal. Mach. Intell. 8: 34–43, 1986.

    Google Scholar 

  49. F. Mokhatarian and A. Mackworth, A theory of multiscale, curvature-based shape representation for planar curves,IEEE Trans. Patt. Anal. Mach. Intell. 14: 789–805, 1992.

    Google Scholar 

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

    Google Scholar 

  51. S. Osher and L.I. Rudin, Feature-oriented image enhancement using shock filters,SIAM J. Numer. Anal. 27: 919–940, 1990.

    Google Scholar 

  52. S. Osher and C.W. Shu, High-order essentially nonoscialltory schemes for Hamilton-Jacobi equations,SIAM J. Numer. Anal. 28: 907–922, 1991.

    Google Scholar 

  53. S. Osher, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations,UCLA Report, Los Angeles, September 1992.

  54. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,IEEE Trans. Patt. Anal. Mach. Intell. 12: 629–639, 1990.

    Google Scholar 

  55. C.M. Petty, Affine isoperimetric problems,Ann. N.Y. Acad. Sci., Discrete Geometry and Convexity 440: 113–127, 1985.

    Google Scholar 

  56. M.H. Protter and H.F. Weinberger,Maximum Principles in Differential Equations, Springer-Verlag: New York, 1984.

    Google Scholar 

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

    Google Scholar 

  58. L.A. Santaló,Introduction to Integral Geometry, Hermann & Company: Paris, 1953.

    Google Scholar 

  59. G. Sapiro, R. Kimmel, D. Shaked, B.B. Kimia, and A.M. Bruckstein, Implementing continuous-scale morphology via curve evolution, to appear,Pattern Recognition.

  60. G. Sapiro and A. Tannenbaum, On affine plane curve evolution, to appear,Journal of Functional Analysis.

  61. G. Sapiro and A. Tannenbaum, Affine shortening of non-convex plane curves,EE Publication 845, Department of Electrical Engineering, Technion, I.I.T., Haifa 32000, Israel, July 1992, submitted.

    Google Scholar 

  62. G. Sapiro and A. Tannenbaum, On invariant curve evolution and image analysis, to appear, “Indiana University Journal of Mathematics.

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

    Google Scholar 

  64. J.A. Sethian, A review of recent numerical algorithms for hyper-surfaces moving with curvature dependent speed,J. Diff. Geom. 31: 131–161, 1989.

    Google Scholar 

  65. J.A. Sethian and J. Strain, Crystal growth and dendritic solidification,J. Computat. Phys. 98, 1992.

  66. J. Smoller,Shock Waves and Reaction-diffusion Equations, Springer-Verlag: New York, 1983.

    Google Scholar 

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

    Google Scholar 

  68. M. Spivak,A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc: Berkeley, CA, 1979.

    Google Scholar 

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

    Google Scholar 

  70. E.J. Wilczynski,Projective Differential Geometry of Curves and Ruled Surfaces, Leipzig: Teubner, 1906.

    Google Scholar 

  71. A.P. Witkin, Scale-space filtering,8th Intern. Joint Conf. Artif. Intell., Karlsruhe, pp. 1019–1021, 1983.

  72. A.L. Yuille and T.A. Poggio, Scaling theorems for zero crossings,IEEE Trans. Patt. Anal. Mach. Intell. 8: 15–25, 1986.

    Google Scholar 

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Author notes

  1. Guillermo Sapiro

    Present address: Laboratory for Information and Decision Systems, M.I.T., 02139, Cambridge, MA

Authors and Affiliations

  1. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Guillermo Sapiro

  2. Department of Electrical Engineering, University of Minnesota, 55455, Minneapolis, MN

    Allen Tannenbaum

  3. Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Allen Tannenbaum

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  1. Guillermo Sapiro
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Sapiro, G., Tannenbaum, A. Affine invariant scale-space. Int J Comput Vision 11, 25–44 (1993). https://doi.org/10.1007/BF01420591

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