Vector median filters, morphology, and PDE´s: theoretical connections

We formally connect between vector median filters, morphological operators, and geometric partial differential equations. Considering a lexicographic order, which permits us to define an order between vectors in IR^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