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

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 R^N, 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.