Root images of median filters: Semi-topological approach

Median filters are frequently used in signal analysis because they are robust edge-preserving smoothing filters. Since median filters are nonlinear filters, the tools of linear theory are not applicable to them. One approach to deal with nonlinear filters consists in investigating their root images (fixed elements or signals transparent to the filter). Whereas for one-dimensional median filters the set of all root signals can be completely characterized, this is not true for higher dimensional filters. Tyan (1981) and Dohler (1989) proposed a method for construction of small root images for two-dimensional median filters. Although the Tyan-Dohler construction is valid for a wide class of median filters, their arguments were not correct and their assertions do not hold universally. In this paper we give a rigorous treatment for the construction of Tyan and Dohler. Moreover, the approach is generalized to the d-dimensional case.