Statistical threshold decomposition for recursive and nonrecursive median filters

The statistical analysis of recursive nonlinear filters is generally difficult. The analysis of recursive median filters has been limited to the trivial cases of signals with a small number of quantization levels and to small window sizes. A block state description of recursively filtered signals is developed, and by applying this description to threshold decomposition, closed-form expressions for the statistics of recursive median filters are obtained. In this case, the number of quantization levels and the window size do not increase the analysis complexity since the output statistics depend on the distribution of a single-threshold filtered binary signal. The statistical decomposition is also developed for nonrecursive median filter operations yielding a connection from classical order statistics to the threshold decomposition approach. Finally, some statistical properties are derived for recursively median-filtered signals.