Approach to regularization preconditioners for image processing

Preconditioning techniques for linear systems are widely used in order to speed up the convergence of iterative methods. Unfortunately, linear systems arising in image processing are highly ill-conditioned and preconditioners often give bad results, since the noise components on the data are strongly amplified already at the early iterations. In this work, we propose filtering strategies which allow to obtain preconditioners with regularization features for Toeplitz systems of image deblurring. Regularization preconditioners are able to speed up the convergence in the space less sensitive to the noise and, simultaneously, they slow down the restoration from components mainly corrupted by noise. A 2-d numerical simulation concerning astronomical image deblurring confirms the effectiveness of the arguments.