Seventh MDSP Workshop 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 prob-lems 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 f ̂ = ( 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 minimizin