Tikhonov regularization with conjugate gradient least squares method for large-scale discrete ill-posed problem in image restoration

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2024-06-13 DOI:10.1016/j.apnum.2024.06.010

Wenli Wang , Gangrong Qu , Caiqin Song , Youran Ge , Yuhan Liu

引用次数: 0

Abstract

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests.

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用共轭梯度最小二乘法对图像复原中的大规模离散失当问题进行提霍诺夫正则化处理

图像复原是一个大规模离散问题，它可以转化为一个可以逼近原始图像的 Tikhonov 正则化问题。在 Tikhonov 正则化问题中引入了 Kronecker 积近似，从而产生了求解广义 Sylvester 矩阵方程的替代问题，缩小了图像复原问题的规模。本文考虑采用共轭梯度最小二乘法（CGLS）来解决这一替代问题，该方法已被证明高效简洁。本文对 CGLS 方法的收敛性进行了分析，结果表明 CGLS 方法能在有限的迭代步数内收敛到最小二乘法解。通过数值试验验证了 CGLS 方法的有效性和优越性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.