## Vector median filters, inf-sup operations, and coupled PDE's: Theoretical connections

Publication ,  Journal Article
Caselles, V; Sapiro, G; Chung, DH
Published in: Journal of Mathematical Imaging and Vision
January 1, 2000

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 IRN, 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.

## Published In

Journal of Mathematical Imaging and Vision

0924-9907

January 1, 2000

12

2

## Start / End Page

109 / 119

• Artificial Intelligence & Image Processing
• 4901 Applied mathematics
• 4606 Distributed computing and systems software
• 4603 Computer vision and multimedia computation
• 0802 Computation Theory and Mathematics
• 0801 Artificial Intelligence and Image Processing
• 0102 Applied Mathematics

### Citation

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Caselles, V., Sapiro, G., & Chung, D. H. (2000). Vector median filters, inf-sup operations, and coupled PDE's: Theoretical connections. Journal of Mathematical Imaging and Vision, 12(2), 109–119. https://doi.org/10.1023/A:1008310305351
Caselles, V., G. Sapiro, and D. H. Chung. “Vector median filters, inf-sup operations, and coupled PDE's: Theoretical connections.” Journal of Mathematical Imaging and Vision 12, no. 2 (January 1, 2000): 109–19. https://doi.org/10.1023/A:1008310305351.
Caselles V, Sapiro G, Chung DH. Vector median filters, inf-sup operations, and coupled PDE's: Theoretical connections. Journal of Mathematical Imaging and Vision. 2000 Jan 1;12(2):109–19.
Caselles, V., et al. “Vector median filters, inf-sup operations, and coupled PDE's: Theoretical connections.” Journal of Mathematical Imaging and Vision, vol. 12, no. 2, Jan. 2000, pp. 109–19. Scopus, doi:10.1023/A:1008310305351.
Caselles V, Sapiro G, Chung DH. Vector median filters, inf-sup operations, and coupled PDE's: Theoretical connections. Journal of Mathematical Imaging and Vision. 2000 Jan 1;12(2):109–119.

## Published In

Journal of Mathematical Imaging and Vision

0924-9907

January 1, 2000

12

2

109 / 119