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

Abstract

Discrete 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.

An Arnoldi-based preconditioner for iterated Tikhonov regularization

“…Interestingly, the AIT method also required fewer iterations and displayed lower RRE values for solutions with x 0 = 0 as its starting vector for all noise levels. This is in contrast with the examples considered in [6] and [4]. We comment on this further below.…”

“…Standard images from MATLAB's image processing toolbox as well as constructed PSFs were used. For proper comparison between the IAT and AIT methods, we set ρ = 10 − 3 and q = 0.7 in our examples, as was done in [6] and [4].…”

“…This section reviews the AIT method and the theoretical results shown by Donatelli and Hanke [6]. A modification of this method is discussed in [4]; we will not consider this modification in the present paper. As mentioned above, the AIT method is a regularization method in a well-defined sense under the assumption that A and C are spectrally equivalent; this condition is stated as follows:…”

“…The algorithm in [6], termed the approximated iterated Tikhonov (AIT) method, was demonstrated to give accurate image reconstructions for a reasonable computational cost when A is a matrix that has dominant block-Toeplitz with Toeplitzblocks (BTTB) substructure and C is a suitably chosen block-circulant with circulant-blocks (BCCB) matrix. Analyses of the AIT method are provided in [4,6]. Under the assumption that A and C are spectrally equivalent (see below), it is shown in [6] that the AIT method is a regularization method in a well-defined sense.…”

“…Under the assumption that A and C are spectrally equivalent (see below), it is shown in [6] that the AIT method is a regularization method in a well-defined sense. However, oftentimes in applications, the spectral equivalence assumption is not satisfied; this issue is addressed in [4]. Irrespective of this, the AIT method provides fast and reliable image reconstructions when the available image, represented by the vector b δ , is contaminated by sufficiently much noise.…”