Statistical optimization of gray-scale morphological filters

The problem of optimal morphological filtering is posed within the statistical framework of discrete random function theory. The optimal morphological estimator of realizations of the discrete random function (DRF) (modeling finite-gray scale digital images) corrupted by supremum/infimum noise is obtained. The mean absolute error (MAE) is used as a distance measure between the original and the estimated image. The optimal filter (which yields minimum MAE) is characterized in terms of the discrete generating functional of the image DRF and the noise DRF under the assumption of statistical independence of image and noise. The MAE of estimation by the morphological filter is represented in terms of MAE incurred by each of the basis filter elements. Recursive expressions can be used to compute the MAE for an n element basis filter in terms of an (n-1) element basis filter. Simulation results on actual image data are included to corroborate the claims of the paper