Estimation of regularization parameters in multiple-image deblurring

We consider the estimation of the regularization parameter for the simultaneous deblurring of multiple noisy images via Tikhonov regularization. We approach the problem in three ways. We first reduce the problem to a 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. In the last approach, we use the classical ${\rm GCV}$on a single higher-dimensional image obtained by concatenating all the images into a single vector. With a reliable estimator of the regularization parameter, one can fully exploit the excellent computational characteristics typical of direct deblurring methods, which, especially for large images, makes them competitive with the more flexible but much slower iterative algorithms. The performance of the techniques is analyzed through numerical experiments. We find that under the independent homoscedastic and Gaussian assumptions made on the noise, the three approaches provide almost identical results with the first single image providing the practical advantage that no new software is required and the same image can be used with other deblurring algorithms.