Vector Median Filters, Morphology, and PDE's: Theoretical Connections

In this paper, we formally connect between vector median lters, morphological operators, and geometric partial diierential equations. Considering a lexicographic order, which permits to deene an order between vectors in IR N , we rst show that the vector median lter of a vector-valued image is equivalent to a collection of innmum-supremum morphological operations. We then proceed and study the asymptotic behavior of this lter. We also provide an interpretation of the innnites-imal iteration of this vectorial median lter in terms of systems of coupled geometric partial diierential 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 l-ters, mathematical morphology, and mean curvature motion.