Flows under min/max curvature flow and mean curvature: Applications in image processing

  • R. Malladi
  • J. A. Sethian
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1064)


We present a class of PDE-based algorithms suitable for a wide range of image processing applications. The techniques are applicable to both salt- and-pepper grey-scale noise and full-image continuous noise present in black and white images, grey-scale images, texture images and color images. At the core, the techniques rely on a level set formulation of evolving curves and surfaces and the viscosity in profile evolution. Essentially, the method consists of moving the isointensity contours in a image under curvature dependent speed laws to achieve enhancement. Compared to existing techniques, our approach has several distinct advantages. First, it contains only one enhancement parameter, which in most cases is automatically chosen. Second, the scheme automatically stops smoothing at some optimal point; continued application of the scheme produces no further change. Third, the method is one of the fastest possible schemes based on a curvature-controlled approach.


Curvature Flow Multiplicative Noise White Image Image Smoothing Initial Hypersurface 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Alvarez, P. L. Lions, and J. M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion. II,” SIAM Journal on Numerical Analysis, Vol. 29(3), pp. 845–866, 1992.Google Scholar
  2. 2.
    J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAMI-8, pp. 679–698, 1986.Google Scholar
  3. 3.
    M. Gage, “Curve shortening makes convex curves circular,” Inventiones Mathematica, Vol. 76, pp. 357, 1984.Google Scholar
  4. 4.
    M. Grayson, “The heat equation shrinks embedded plane curves to round points,” J. Diff. Geom., Vol. 26, pp. 285–314, 1987.Google Scholar
  5. 5.
    R. Malladi and J. A. Sethian, “Image processing via level set curvature flow,” Proc. Natl. Acad. of Sci., USA, Vol. 92, pp. 7046–7050, July 1995.Google Scholar
  6. 6.
    R. Malladi and J. A. Sethian, “Image processing: Flows under min/max curvature and mean curvature,” to appear in Graphical Models and Image Processing, March 1996.Google Scholar
  7. 7.
    R. Malladi and J. A. Sethian, “A unified approach for shape segmentation, representation, and recognition,” Report LBL-36069, Lawrence Berkeley Laboratory, University of California, Berkeley, August 1994.Google Scholar
  8. 8.
    R. Malladi, D. Adalsteinsson, and J. A. Sethian, “Fast method for 3D shape recovery using level sets,” submitted.Google Scholar
  9. 9.
    R. Malladi, J. A. Sethian, and B. C. Vemuri, “Evolutionary fronts for topology-independent shape modeling and recovery,” in Proceedings of Third European Conference on Computer Vision, LNCS Vol. 800, pp. 3–13, Stockholm, Sweden, May 1994.Google Scholar
  10. 10.
    R. Malladi, J. A. Sethian, and B. C. Vemuri, “Shape modeling with front propagation: A level set approach,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 17(2), pp. 158–175, Feb. 1995.Google Scholar
  11. 11.
    D. Marr and E. Hildreth, “A theory of edge detection,” Proc. of Royal Soc. (London), Vol. B207, pp. 187–217, 1980.Google Scholar
  12. 12.
    S. Osher and J. A. Sethian, “Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulation,” Journal of Computational Physics, Vol. 79, pp. 12–49, 1988.Google Scholar
  13. 13.
    P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 12(7), pp. 629–639, July 1990.Google Scholar
  14. 14.
    L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Modelisations Matematiques pour le traitement d'images, INRIA, pp. 149–179, 1992.Google Scholar
  15. 15.
    G. Sapiro and A. Tannenbaum, “Image smoothing based on affine invariant flow,” Proc. of the Conference on Information Sciences and Systems, Johns Hopkins University, March 1993.Google Scholar
  16. 16.
    J. A. Sethian, “Curvature and the evolution of fronts,” Commun. in Mathematical Physics, Vol. 101, pp. 487–499, 1985.Google Scholar
  17. 17.
    J. A. Sethian, “Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws,” Journal of Differential Geometry, Vol. 31, pp. 131–161, 1990.Google Scholar
  18. 18.
    J. A. Sethian, “Curvature flow and entropy conditions applied to grid generation,” Journal of Computational Physics, Vol. 115, No. 2, pp. 440–454, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • R. Malladi
    • 1
  • J. A. Sethian
    • 1
  1. 1.Lawrence Berkeley National LaboratoryUniversity of CaliforniaBerkeleyUSA

Personalised recommendations