Complete classification of roots to one-dimensional median and rank-order filters

The set of roots to the one-dimensional median filter is completely determined. Let 2N+1 be the filter window width. It has been shown that if a root contains a monotone segment of length N+1, then it must be locally monotone N+2. For roots with no monotone segment of length N+1, it is proved that the set of such roots is finite, and that each such root is periodic. The methods used are constructive, so given N, one can list all possible roots of this type. The results developed for the median filter also apply to rank-order filters