Journal of Mathematical Imaging and Vision

, Volume 12, Issue 2, pp 109–119

Vector Median Filters, Inf-Sup Operations, and Coupled PDE's: Theoretical Connections

  • Vicent Caselles
  • Guillermo Sapiro
  • Do Hyun Chung
Article

Abstract

In this paper, we formally connect between vector median filters, inf-sup morphological operations, and geometric partial differential equations. Considering a lexicographic order, which permits to define an order between vectors in RN, we first show that the vector median filter of a vector-valued image is equivalent to a collection of infimum-supremum morphological operations. We then proceed and study the asymptotic behavior of this filter. We also provide an interpretation of the infinitesimal iteration of this vectorial median filter in terms of systems of coupled geometric partial differential equations. The main component of the vector evolves according to curvature motion, while, intuitively, the others regularly deform their level-sets toward those of this main component. These results extend to the vector case classical connections between scalar median filters, mathematical morphology, and mean curvature motion.

vector median filtering inf-sup operations asymptotic behavior anisotropic diffusion curvature motion coupled PDE's 

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References

  1. 1.
    L. Alvarez, P.L. Lions, and J.M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal., Vol. 29, pp. 845-866, 1992.Google Scholar
  2. 2.
    V. Caselles, J.M. Morel, G. Sapiro, and A. Tannenbaum (Eds.), “Special issue on partial differential equations and geometrydriven diffusion in image processing and analysis,” IEEE Trans. Image Processing, Vol. 7, pp. 269-273, 1998.Google Scholar
  3. 3.
    L.C. Evans and J. Spruck, “Motion of level-sets by mean curvature II,” in Trans. American Mathematical Society, Vol. 30, No. 1, pp. 321-332, 1992.Google Scholar
  4. 4.
    F. Guichard and J.M. Morel, Introduction to Partial Differential Equations in Image Processing. Tutorial Notes, IEEE Int. Conf. Image Proc., Washington, DC, Oct. 1995.Google Scholar
  5. 5.
    H.J.A.M. Heijmans and J.B.T.M. Roerdink (Eds.), Mathematical Morphology and Its Applications to Image and Signal Processing, Kluwer: Dordrecht, The Netherlands, 1998.Google Scholar
  6. 6.
    D.G. Karakos and P.E. Trahanias, “Generalized multichannel image-filtering structures,” IEEE Trans. Image Processing, Vol. 6, pp. 1038-1045, 1997.Google Scholar
  7. 7.
    R. Kimmel, R. Malladi, and N. Sochen, “Image processing via the Beltrami operator,” in Proc. of 3rd Asian Conf. on Computer Vision, Hong Kong, Jan. 8-11, 1998.Google Scholar
  8. 8.
    R. Kimmel, Personal communication.Google Scholar
  9. 9.
    P. Maragos and R.W. Schafer, “Morphological systems for multidimensional image processing,” Proc. IEEE, Vol. 78, pp. 690-710, 1990.Google Scholar
  10. 10.
    P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern. Anal. Machine Intell., Vol. 12, pp. 629-639, 1990.Google Scholar
  11. 11.
    G. Sapiro and D. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Processing Vol. 5, pp. 1582-1586, 1996.Google Scholar
  12. 12.
    B. Tang, G. Sapiro, and V. Caselles, “Direction diffusion,” ECE Department Technical Report, University of Minnesota, Feb. 1999.Google Scholar
  13. 13.
    B. Tang, G. Sapiro, and V. Caselles, “Direction diffusion,” in Proc. Int. Conference Comp. Vision, Greece, Sept. 1999.Google Scholar
  14. 14.
    B. Tang, G. Sapiro, and V. Caselles, “Color image enhancement via chromaticity diffusion,” ECE Department Technical Report, University of Minnesota, March 1999.Google Scholar
  15. 15.
    P.E. Trahanias and A.N. Venetsanopoulos, “Vector directional filters-A new class of multichannel image processing filters,” IEEE Trans. Image Processing, Vol. 2, pp. 528-534, 1993.Google Scholar
  16. 16.
    P.E. Trahanias, D. Karakos, and A.N. Venetsanopoulos, “Directional processing of color images: Theory and experimental results,” IEEE Trans. Image Processing, Vol. 5, pp. 868-880, 1996.Google Scholar
  17. 17.
    R.T. Whitaker and G. Gerig, “Vector-valued diffusion,” in Geometry Driven Diffusion in Computer Vision, B. ter Haar Romeny (Ed.), Kluwer: Boston, MA, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Vicent Caselles
    • 1
  • Guillermo Sapiro
    • 2
  • Do Hyun Chung
    • 3
  1. 1.Department of Informatics and MathematicsUniversity of the Illes BalearsPalma de MallorcaSpain
  2. 2.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA
  3. 3.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA

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