Root Images of Median Filters

Abstract

Median filters are frequently used in signal analysis because of their smoothing properties and their insensitivity with respect to outliers in the data. 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. In 1989, Dohler stated a result on certain root images for two-dimensional median filters. Although Dohlers results are true for a wide class of median filters, his arguments were not correct and his assertions do not hold universally. In this paper we give a rigorous proof of Dohlers results. Moreover, his approach is generalized to the d-dimensional case.