Image restoration using recursive estimators

In this paper, edge preserving recursive estimators are proposed For restoring images corrupted by noise. Edge detection using a 5×5 Graeco-Latin squares (GLS) mask is carried out as the first step for preserving edges. The GLS mask preprocessor determines the orientation of edges in horizontal, vertical, 45° diagonal, or 135° diagonal directions. The actual removal of noise is done in the second step. If the noise is Gaussian, the center pixel in the 5×5 mask is estimated using a multiple linear regression model fitted to the noisy image on the same side of the edge. The parameters of the regression model are estimated using the least squares estimator. The least squares estimator is made recursive using the Robbins-Monro stochastic approximation (RMSA) algorithm. The RMSA guarantees convergence of the estimate in the mean square sense and with probability one. If the Gaussian noise is contaminated by a small percentage of heavy tailed (impulsive) noise (salt and pepper noise), the recursive least square estimator is robustized using a symmetrical version of Wilcoxon signed rank statistic. The GLS mask for edge detection uses an F-ratio test which is robust for small deviations from normality assumption of the noise. The mathematical properties and various forms of convergence of the robustized algorithm are shown in the appendix. The efficacy of the proposed restoration procedures are demonstrated on two types of images (“girl” and “house”)