MAP estimation of finite gray-scale digital images corrupted by supremum/infimum noise

Finite gray-scale digital images are modeled as realizations of discrete random functions (DRF), and then the estimation of realizations of DRF corrupted by a supremum/infimum noise model is considered. It is proved that morphological operators such as openings, closings, supremum of openings and infimum of closings are optimal maximum a posteriori (MAP) estimators under an appropriate and minimal set of assumptions relating to the structural and statistical constraints on image DRF and noise DRF. These results are obtained for independent, identically distributed (i.i.d.) noise for single and multiframe observation scenarios. Next, the assumption of i.i.d. noise is relaxed and the MAP optimality and strong consistency of morphological filters for filtering image DRF degraded by morphologically smooth noise (i.e., colored noise) is proved. Simulations on actual image data are carried out in support of the validity of theoretical results presented.