## Optimal Filtering of Digital Binary Images Corrupted by Union/Intersection Noise

**Title**: Optimal Filtering of Digital Binary Images Corrupted by Union/Intersection Noise

**Authors**:

Berenstein, Carlos A.

Sidiropoulos, N D

**Journal**: IEEE Transactions on Image Processing Vol. 3, No. 4, pp. 382-403, July 1994

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 which can be described by a union/intersection noise model. Two distinct optimal filtering approaches are pursued. The first involved a class of “mask” filters, which arises quite naturally from the set-theoretic analysis of optimal 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 given degradtion model) expansion of the optimal filter, we can obtain universal characterizations of optimality that do not rely on strong assummptions 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.

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