Cross-validation And Maximum Likelihood Space Varying Regularized Image Restoration
In a number of image processing problems, such as image restoration problems, ill-conditioned equations must be solved [l]. Because of the presence of singularities and noise in the data, regularization is an effective method for obtaining a satisfactory solutions to this problem [6]. The application of regularizing necessitates a choice of the regularization parameter. This parameter trades fidelity to the data with smoothness of the solution. For most problems of interest the choice of the regularizing parameter is not known a priori and has to be computed. For an M x M image and linear shift invariant degradation with white additive noise, the imaging equation is given by g = H f + n , (1) where the vectors g, f, and n are M 2 x 1 lexicographic of the degraded observed image, the original image and the noise, respectively. The M 2 x M 2 degradation matrix H is Block Toeplitz and ill-conditioned [l]. The regularized estimate f h a t for this problem is given by f̂ = ( H t H + XQtQ)-lHtg = A(X)g, (2) where Q is the regularizing functional and X the regularizing parameter. A number of approaches have been suggested to compute the regularizing parameter. These approaches can be classified into two broad categories ones requiring a priori knowledge of the noise variance such as the constrained least squares method [l] and the set theoretic approach [4] and ones that do not such as the cross-validation (CV) [5] and the maximum likelihood (ML) approach [3]. For the CV approach the regularizing parameter is obtained by minimizing the cross-validation function given by and for the ML approach by minimizing (3)