Projected Landweber method and preconditioning

The projected Landweber method is an iterative method for solving constrained least-squares problems when the constraints are expressed in terms of a convex and closed set C. The convergence properties of the method have been recently investigated. Moreover, it has important applications to many problems of signal processing and image restoration. The practical difficulty is that the convergence is too slow. In this paper we apply to this method the so-called preconditioning which is frequently used for increasing the efficiency of the conjugate gradient method. We discuss the significance of preconditioning in this case and we show that it implies a modification of the original constrained least-squares problem. However, when the original problem is ill-posed, the approximate solutions provided by the preconditioned method are similar to those provided by the standard method if the preconditioning is suitably chosen. Moreover, the number of iterations can be reduced by a factor of 10 and even more. A few applications to problems of image restoration are also discussed.