Statistical Inference, Filtering, and Modeling of Discrete Random Sets

In the second part, we consider the problem of estimating realizations of uniformly bounded discrete random sets, distorted by a degradation process which can be described by a union/intersection noise model. Two different optimal filtering approaches are considered. The first involves a class of filters which arises quite naturally from the set-theoretic analysis of optimal filters. We call this the class of mask filters. The second approach deals with optimal Morphological filters. First, we provide some fresh statistical insight into certain "folk theorems" of Morphological filtering. We do so by exploiting the uniformly bounded discrete random set formulation of the filtering problem. Then we show that, by using an appropriate (under a given degrataion model) expansion of the optimal filter, we can obtain "universal" characterizations of optimality, in terms of the fundamental functionals of random set theory, namely the generating functionals of the signal and the noise.