On the root structures of weighted median filters

A class of weighted median filters (WMFs) is considered whose weights are symmetric about and nondecreasing upon going to the window center. It is shown that the structure of root signals of these WMFs can be sharply different from those of the standard median filter. The root signal may contain three parts: edges that are defined in the same way as that of the median filter; constant neighborhoods which, compared to that of the median filter, can have a shorter minimum-length; and a third part that exhibits an oscillatory behavior