On MAP optimality of gray-scale morphological filters

The estimation of finite-gray-scale digital images corrupted by supremum/infimum noise is posed within the statistical framework of discrete random functions (DRF). It is shown that morphological operators such as openings, closings, supremum of openings and infimum of closings are optimal Maximum-A-Posteriori (MAP) estimators under the assumption that noise is independent and identically distributed (lid) for single and multiframe observation scenarios. Simulation results supporting the validity of theoretical results are also included.