Further Results on MAP Optimality and Strong Consistency of Certain Classes of Morphological Filters

In two recent papers [1], [2], Sidiropoulos et al. have obtained statistical proofs of Maximum A Posteriori} (MAP) optimality and strong consistency of certain popular classes of Morphological filters, namely, Morphological Openings, Closings, unions of Openings, and intersections of Closings, under i.i.d. (both pixel-wise, and sequence-wide) assumptions on the noise model. In this paper we revisit this classic filtering problem, and prove MAP optimality and strong consistency under a different, and, in a sense, more appealing set of assumptions, which allows the explicit incorporation of geometric and Morphological constraints into the noise model, i.e., the noise may now exhibit structure; Surprisingly, it turns out that this affects neither the optimality nor the consistency of these field-proven filters.