The deterministic properties of weighted median (WM) filters are analyzed. Threshold decomposition and the stacking property together establish a unique relationship between integer and binary domain filtering. The authors present a method to find the weighted median filter which is equivalent to a stack filter defined by a positive Boolean function. Because the cascade of WM filters can always be expressed as a single stack filter this allows expression of the cascade of WM filters as a single WM filter. A direct application is the computation of the output distribution of a cascade of WM filters. The same method is used to find a nonrecursive expansion of a recursive WM filter. As applications of theoretical results, several interesting deterministic and statistical properties of WM filters are derived.
Yli-Harja, O., Neuvo, Y., & Astola, J. (1991). Analysis of the properties of median and weighted median filters using threshold logic and stack filter representation. IEEE Transactions on Signal Processing, 39(2), 395 - 410. https://doi.org/10.1109/78.80823