Dept. of Electr. Eng., Indian Inst. of Technol., Kanpur.
IEEE Trans Image Process. 1997;6(8):1077-88. doi: 10.1109/83.605405.
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.
有限灰度数字图像被建模为离散随机函数（DRF）的实现，然后考虑由上确界/下确界噪声模型污染的 DRF 的实现的估计。证明了形态算子（例如开运算、闭运算、开运算的上确界和闭运算的下确界）在与图像 DRF 和噪声 DRF 的结构和统计约束相关的适当且最小的假设集下是最优后验最大似然（MAP）估计器。这些结果是针对单帧和多帧观测场景下独立同分布（i.i.d.）噪声的情况得到的。接下来，放宽了 i.i.d.噪声的假设，并证明了形态平滑噪声（即彩色噪声）退化的图像 DRF 的形态滤波器的 MAP 最优性和强一致性。实际图像数据的模拟支持了所提出的理论结果的有效性。
