A theoretical analysis of the properties of median filters

Necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived. These conditions state that a signal must be locally monotone to pass through a median filter unchanged. It is proven that the form of successive median filtering of a signal (i.e., the filtered output is itself again filtered) eventually reduces the original signal to an invariant signal called a root signal. For a signal of length L samples, a maximum of
\frac{1}{2}(L - 2)
repeated filterings produces a root signal.