Advertisement

Area and length preserving geometric invariant scale-spaces

  • Guillermo Sapiro
  • Allen Tannenbaum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 801)

Abstract

In this paper, area preserving geometric multi-scale representations of planar curves are described. This allows geometric smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via invariant geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. The flows are geometrically intrinsic to the curve, and exactly satisfy all the basic requirements of scale-space representations. In the case of the Euclidean heat flow for example, it is completely local as well. The same approach is used to define length preserving geometric flows. The geometric scalespaces are implemented using an efficient numerical algorithm.

Keywords

Heat Flow Curve Evolution Planar Curf Initial Curve Euclidean Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, Axioms and fundamental equations of image processing, to appear in Arch. for Rational Mechanics.Google Scholar
  2. 2.
    L. Alvarez, P. L. Lions, and J. M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29, pp. 845–866, 1992.Google Scholar
  3. 3.
    S. Angenent, Parabolic equations for curves on surfaces, Part II. Intersections, blow-up, and generalized solutions, Annals of Mathematics 133, pp. 171–215, 1991.Google Scholar
  4. 4.
    S. Angenent, G. Sapiro, and A. Tannenbaum, On the affine heat equation for nonconvex curves, Technical Report MIT — LIDS, January 1994.Google Scholar
  5. 5.
    J. Babaud, A. P. Witkin, M. Baudin, and R. O. Duda, Uniqueness of the Gaussian kernel for scale-space filtering, IEEE-PAMI 8, pp. 26–33, 1986.Google Scholar
  6. 6.
    W. Blaschke, Vorlesungen über Differentialgeometrie II, Verlag Von Julius Springer, Berlin, 1923.Google Scholar
  7. 7.
    C. L. Epstein and M. Gage, The curve shortening flow, in Wave Motion: Theory, Modeling, and Computation, A. Chorin and A. Majda (Ed.), Springer-Verlag, New York, 1987.Google Scholar
  8. 8.
    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 Vision Comp. 10, pp. 376–388, 1992.Google Scholar
  9. 9.
    M. Gage, On an area-preserving evolution equation for plane curves, Contemporary Mathematics 51, pp. 51–62, 1986.Google Scholar
  10. 10.
    M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geometry 23, pp. 69–96, 1986.Google Scholar
  11. 11.
    M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geometry 26, pp. 285–314, 1987.Google Scholar
  12. 12.
    B. K. P. Horn and E. J. Weldon, Jr., Filtering closed curves, IEEE-PAMI 8, pp. 665–668, 1986.Google Scholar
  13. 13.
    B. B. Kimia, A. Tannenbaum, and S. W. Zucker, Shapes, shocks, and deformations, I, to appear in International Journal of Compute Vision.Google Scholar
  14. 14.
    J. J. Koenderink, The structure of images, Biological Cybernetics 50, pp. 363–370, 1984.Google Scholar
  15. 15.
    T. Lindeberg and J. O. Eklundh, On the computation of a scale-space primal sketch, Journal of Visual Comm. and Image Rep. 2, pp. 55–78, 1991.Google Scholar
  16. 16.
    D. G. Lowe, Organization of smooth image curves at multiple scales, International Journal of Computer Vision 3, pp. 119–130, 1989.Google Scholar
  17. 17.
    F. Mokhatarian and A. Mackworth, A theory of multiscale, curvature-based shape representation for planar curves, IEEE-PAMI 14, pp. 789–805, 1992.Google Scholar
  18. 18.
    J. Oliensis, Local reproducible smoothing without shrinkage, IEEE-PAMI 15, pp. 307–312, 1993.Google Scholar
  19. 19.
    P. J. Olver, G. Sapiro, and A. Tannenbaum, Differential invariant signatures and flows in computer vision: A symmetry group approach, Technical Report MIT — LIDS, December 1993. Also in Geometry Driven Diffusion, B. ter har Romeny Ed., 1994.Google Scholar
  20. 20.
    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.Google Scholar
  21. 21.
    P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE-PAMI 12, pp. 629–639, 1990.Google Scholar
  22. 22.
    J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA Journal of Applied Mathematics 48 pp. 249–264, 1992.Google Scholar
  23. 23.
    G. Sapiro, R. Kimmel, D. Shaked, B. B. Kimia, and A. M. Bruckstein, Implementing continuous-scale morphology via curve evolution, Pattern Recognition 26:9, pp. 1363–1372, 1993.Google Scholar
  24. 24.
    G. Sapiro and A. Tannenbaum, On affine plane curve evolution, February 1992, to appear in Journal of Functional Analysis.Google Scholar
  25. 25.
    G. Sapiro and A. Tannenbaum, Affine invariant scale-space, International Journal of Computer Vision 11:1, pp. 25–44, 1993.Google Scholar
  26. 26.
    G. Sapiro and A. Tannenbaum, Image smoothing based on an affine invariant flow, Proc. of Conf.on Information Sciences and Systems, Johns Hopkins University, March 1993.Google Scholar
  27. 27.
    G. Sapiro and A. Tannenbaum, On invariant curve evolution and image analysis, Indiana University Mathematics Journal 42:3, 1993.Google Scholar
  28. 28.
    G. Sapiro and A. Tannenbaum, Area and length preserving geometric invariant scale-spaces, Technical Report MIT — LIDS 2200, September 1993.Google Scholar
  29. 29.
    M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc, Berkeley, California, 1979.Google Scholar
  30. 30.
    A. P. Witkin, Scale-space filtering, Int. Joint. Conf. Artificial Intelligence, pp. 1019–1021, 1983.Google Scholar
  31. 31.
    A. L. Yuille and T. A. Poggio, Scaling theorems for zero crossings, IEEE-PAMI 8, pp. 15–25, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Guillermo Sapiro
    • 1
  • Allen Tannenbaum
    • 2
  1. 1.EE&CS Department-LIDSMITCambridge
  2. 2.EE DepartmentUniversity of MinnesotaMinneapolis

Personalised recommendations