On two-dimensional root structures of separable and cross median filters

Deterministic properties of separable and cross median filters are presented. It is shown that the roots of a separable median filter without the presence of vertical binary oscillations are also the roots of the corresponding cross median filter. The sufficient and necessary condition for a point to be invariant to cross median filtering is given. Finally, it is indicated that there exist three different types of regions in the root structures of cross median filters based on the one-dimensional features of rows and columns