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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
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.
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.
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.
D.G. Karakos and P.E. Trahanias, “Generalized multichannel image-filtering structures,” IEEE Trans. Image Processing, Vol. 6, pp. 1038-1045, 1997.
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.
R. Kimmel, Personal communication.
P. Maragos and R.W. Schafer, “Morphological systems for multidimensional image processing,” Proc. IEEE, Vol. 78, pp. 690-710, 1990.
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.
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.
B. Tang, G. Sapiro, and V. Caselles, “Direction diffusion,” ECE Department Technical Report, University of Minnesota, Feb. 1999.
B. Tang, G. Sapiro, and V. Caselles, “Direction diffusion,” in Proc. Int. Conference Comp. Vision, Greece, Sept. 1999.
B. Tang, G. Sapiro, and V. Caselles, “Color image enhancement via chromaticity diffusion,” ECE Department Technical Report, University of Minnesota, March 1999.
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.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Caselles, V., Sapiro, G. & Chung, D.H. Vector Median Filters, Inf-Sup Operations, and Coupled PDE's: Theoretical Connections. Journal of Mathematical Imaging and Vision 12, 109–119 (2000). https://doi.org/10.1023/A:1008310305351
Issue Date:
DOI: https://doi.org/10.1023/A:1008310305351