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

We model digital binary image data as realizations of a uniformly bounded discrete random set (or discrete random set, for short), which is a mathematical object that can be directly defined on a finite lattice. We consider the problem of estimating realizations of discrete random sets distorted by a degradation process that can be described by a union/intersection noise model. Two distinct optimal filtering approaches are pursued. The first involves a class of "mask" filters, which arises quite naturally from the set-theoretic analysis of optimal filters. The second approach involves a class of morphological filters. We prove that under i.i.d noise morphological openings, closings, unions of openings, and intersections of closings can be viewed as MAP estimators of morphologically smooth signals. Then, we show that by using an appropriate (under a given degradation model) expansion of the optimal filter, we can obtain universal characterizations of optimality that do not rely on strong assumptions regarding the spatial interaction of geometrical primitives of the signal and the noise. The results generalize to gray-level images in a fairly straightforward manner.