Estimation of Regularization Parameters in Multiple-Image Deblurring

R. Vio, P. Ma, W. Zhong, J. Nagy, L. Tenorio and W. Wamsteker

We consider the estimation of the regularization parameter for the simultaneous deblurring of multiple noisy images via Tikhonov regularization. We approach the problem in two ways. We first reduce the problem to single-image deblurring for which the regularization parameter can be estimated through a classic generalized cross-validation (${\rm GCV}$) method. A modification of this function is used for correcting the undersmoothing typical of the original technique. With a second method, we minimize an average least-squares fit to the images and define a new ${\rm GCV}$ function.With a reliable estimator for the regularization parameter, one can fully exploit theexcellent computational characteristics typical of direct deblurring methods, which, especially forlarge images, makes them competitive with the more flexible but much slower iterativealgorithms. The performance of the technique is analyzed through numerical experiments. We find that under the independent homoscedastic and Gaussian assumptions made on the noise,the single-image and the multiple-image approaches provide almost identical resultswith the former having the practical advantage that no new software is required and the same image canbe used with other deblurring algorithms.