State description for the root-signal set of median filters

Median filtering is a simple digital technique for smoothing signals. One main characteristic of the filter is that it maps the input signal space into a root signal space, where signals invariant to median filters are called roots of the signal. In this paper, we develop the theory for the root signal set of median filters. A tree structure for the root signal set is obtained for binary signals. The number of roots R (n) for a signal of length "n" and window size filter "2s- 1" is exactly represented by the difference equation R(n) = R(n - 1) + R(n - s). A general solution is obtained in a Z domain approach. Finally, a method for faster one dimensional median filter operation is introduced.