Optimal Morphological Filters for Discrete Random Sets under a Union or Intersection Noise

We consider the problem of optimal binary image restoration under a union or intersection noise model. Union noise is well suited to model random clutter (obscuration), whereas intersection noise is a good model for random sampling. Our approach is random set-theoretic, i.e. digital images are viewed as realizations of a uniformly bounded discrete random set. First we provide statistical proofs of some "folk theorems” of Morphological filtering. In particular, we prove that, under some reasonable worst-case statistical scenarios, Morphological openings, closings, unions of openings, and intersections of closings, can be viewed as MAP estimators of the signal based on the noisy observation. Then we propose a "generic” procedure for the design of optimal Morphological filters for independent union or intersection noise.