Some Statistical Properties of Weighted Median Filters

In this paper, based on the fact that the output of a weighted median (WM) filter is always one of the samples in the input window, rank and sample selection probabilities are defined. The former is the probability that a certain ranked sample will appear as the output and latter is the probability that the output equals one of the time-indexed samples. Using the rank selection probabilities, it is shown here that the output distribution of the WM filter of size N with independent identically distributed (i.i.d.) inputs is a weighted sum of the distributions of the 1th, , 2, . . . , N order statistics. The weights are given by the rank selection probabilities. The sample selection probabilities are the coefficients of the finite impulse response (FIR) filter whose output, of all linear filters, is closest to that of the WM filter. Several statistical properties of WM fiters using selection probabilities are then derived. A method to compute the selection probabilities from the weights of the WM filter is also given.