Optimal filtering of digital binary images corrupted by union/intersection noise

Digital binary image data are modeled as realizations of a uniformly bounded discrete random set, a mathematical object which can be directly defined on a finite lattice. The problem of estimating realizations of discrete random sets distorted by a degradation process that can be described by a union/interaction noise model is considered. Some theoretical justification of the popularity of certain morphological filters, namely morphological openings, closings, unions of openings, and intersections of closings is provided. The authors prove that if the signal is `smooth´, then these filters are optimal under reasonable worst-case statistical scenarios. A class of filters that arises quite naturally from the set-theoretic analysis of optimal filters is considered. It is called the class of mask filters. Both fixed and adaptive mask filters are considered, and explicit formulas for the optimal mask filter under quite general assumptions on the signal and the degradation process are derived