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A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring
Numerical Linear Algebra with Applications ( IF 4.3 ) Pub Date : 2020-12-03 , DOI: 10.1002/nla.2353
Alessandro Buccini 1 , Marco Donatelli 2 , Lothar Reichel 3 , Wei‐Hong Zhang 4
Affiliation  

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.

中文翻译:

线性离散不适定问题的一种新的非平稳预处理迭代方法及其在图像去模糊中的应用

离散不适定的逆问题出现在科学和工程的许多领域。他们的解决方案对数据的扰动非常敏感。正则化方法旨在降低这种敏感性。本文考虑了一种基于迭代Tikhonov正则化的迭代正则化方法,该方法在M. Donatelli和M. Hanke中提出,不适定问题的快速非平稳预处理迭代方法,适用于图像去模糊,反问题,29(2013),Art。095008,16页。在此方法中,精确运算符由易于使用的运算符近似。但是,收敛理论要求近似算子在频谱上等同于原始算子。在实践中很少满足此条件。但是,这种迭代方法在许多情况下都可以确定准确的图像恢复。我们提出了一种迭代方法的修改方案,该方案将频谱等效的需求放宽到更容易满足的需求。我们表明,尽管改进的方法不是迭代正则化方法,但它保持了此类方法最重要的理论特性之一,即重构误差的单调减小。若干计算实验证明了该方法的良好性能。
更新日期:2021-02-03
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