The output distribution of important LULU-operators

TL;DR

This paper presents two methods for computing the output distribution of certain stack filters, enabling analysis of their smoothing properties and applications to reliability polynomials of Boolean functions.

Contribution

It introduces novel procedures for calculating output distributions of LULU-operators, including a method for reliability polynomial computation of positive Boolean functions.

Findings

Methods effectively compute output distributions for important LULU-operators.

Properties of the output distribution characterize the smoothing behavior of stack filters.

The inclusion-exclusion approach can determine the reliability polynomial of Boolean functions.

Abstract

Two procedures to compute the output distribution phi_S 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 phi_S for some important LULU-operators S. Properties of phi_S can be used to characterize smoothing properties of S. One of the methods discussed also allows for the calculation of the reliability polynomial of any positive Boolean function (e.g. one derived from a connected graph).