Calculating the output distribution of stack filters that are erosion-dilation cascades, in particular LULU-filters

Abstract

Two procedures to compute the output distribution ϕs of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields ϕs for some important LULU-operators S. Properties of ϕs can be used to characterize smoothing properties of S. Also, in the same way as our polynomials ϕs are computed one could compute the reliability polynomial of a connected graph, or more generally the reliability polynomial w.r.t. any positive Boolean function.

Keywords: Stack filter, output distribution, LULU filter, reliability polynomial, inclusion-exclusion.