A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring

Abstract

AbstractDiscrete ill‐posed inverse problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods aim at reducing this sensitivity. This article considers an iterative regularization method, based on iterated Tikhonov regularization, that was proposed in M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods for ill‐posed problems, with application to image deblurring, Inverse Problems, 29 (2013), Art. 095008, 16 pages. In this method, the exact operator is approximated by an operator that is easier to work with. However, the convergence theory requires the approximating operator to be spectrally equivalent to the original operator. This condition is rarely satisfied in practice. Nevertheless, this iterative method determines accurate image restorations in many situations. We propose a modification of the iterative method, that relaxes the demand of spectral equivalence to a requirement that is easier to satisfy. We show that, although the modified method is not an iterative regularization method, it maintains one of the most important theoretical properties for this kind of methods, namely monotonic decrease of the reconstruction error. Several computed experiments show the good performances of the proposed method.