Stack filters and selection probabilities

Based on the facts that the output of a given stack filter can be determined if the ranks of the input samples are known and that this output always equals one of the samples in the input window, rank- and sample-selection probabilities are defined. The output distribution function of a stack filter of size N with continuous, independently identically distributed (IID) inputs can be expressed as a weighted sum of the distribution functions of the ith (i=1,2,. . .,N) order statistics, where the rank-selection probabilities are the weights. The sample-selection probabilities equal the impulse response coefficients of the FIR (finite-impulse-response) filter whose output spectrum is closest, of all linear filters, to that of the stack filter for IID Gaussian inputs. Some statistical properties of stack filters are derived. A method of computing the selection probabilities from the positive Boolean function of the stack filter is also given.<>