Optimal Filtering of Digital Binary Images Corrupted by Union/Intersection

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## Abstract

We model digital binary image data as realizations of a bounded discrete random set, a mathematical object which can be directly defined on a finite lattice. We consider the problem of estimating realizations of discrete random sets distorted by a degradation process which can be described by a union/intersection model. First we present an important structural result concerning the probabilistic specification of discrete random sets defined on a finite lattice. Then we formulate the optimal filtering problem for the case of discrete random sets. Two distinct filtering approaches are pursued. For images which feature strong spatial statistical variations we propose a simple family of spatially varying filters, which we call mask filters, and, for each degradation model, derive explicit formulas for the optimal Mask filter. We also consider adaptive mask filters, which can be effective in a more general setting. For images which exhibit a stationary behavior, we consider the class of Morphological filters. First we provide some theoretical justification for the popularity of certain Morphological filtering schemes. In particular, we show that if the signal is smooth, then these schemes are optimal (in the sense of providing the MAP estimate of the signal) under a reasonable worst-case statistical scenario. 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 which do not rely on strong assumptions regarding the spatial interaction of geometrical primitives of the signal and the noise. This approach corresponds to a somewhat counter- intuitive use of fundamental morphological operators; however it is exactly this mode of the use that enables us to arrive at characterizations of optimality in terms of the fundamental functionals of random set theory, namely the generating functionals of the signal and the noise.