Regularized supervised Bayesian approach for image deconvolution with regularization parameter estimation

Image deconvolution consists in restoring a blurred and noisy image knowing its point spread function (PSF). This inverse problem is ill-posed and needs prior information to obtain a satisfactory solution. Bayesian inference approach with appropriate prior on the image, in particular with a Gaussian prior, has been used successfully. Supervised Bayesian approach with maximum a posteriori (MAP) estimation, a method that has been considered recently, is unstable and suffers from serious ringing artifacts in many applications. To overcome these drawbacks, we propose a regularized version where we minimize an energy functional combined by the mean square error with H-1 regularization term, and we consider the generalized cross validation (GCV) method, a widely used and very successful predictive approach, for choosing the smoothing parameter. Theoretically, we study the convergence behavior of the method and we give numerical tests to show its effectiveness.