Volume 3, Issue 4
Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 461-474.

Published online: 2010-03

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• Abstract

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, i.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iteration, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably good.

• Keywords

Edge-preserving, image restoration, multiplicative half-quadratic regularization, Newton method, preconditioned conjugate gradient method, constraint preconditioner, eigenvalue bounds.

65F10, 65F50, 65W05, CR: G1.3

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@Article{NMTMA-3-461, author = {Zhong-Zhi Bai , and Yu-Mei Huang , and Michael K. Ng , and Xi Yang , }, title = {Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2010}, volume = {3}, number = {4}, pages = {461--474}, abstract = {

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, i.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iteration, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably good.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2010.m9014}, url = {http://global-sci.org/intro/article_detail/nmtma/6009.html} }
TY - JOUR T1 - Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations AU - Zhong-Zhi Bai , AU - Yu-Mei Huang , AU - Michael K. Ng , AU - Xi Yang , JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 461 EP - 474 PY - 2010 DA - 2010/03 SN - 3 DO - http://doi.org/10.4208/nmtma.2010.m9014 UR - https://global-sci.org/intro/article_detail/nmtma/6009.html KW - Edge-preserving, image restoration, multiplicative half-quadratic regularization, Newton method, preconditioned conjugate gradient method, constraint preconditioner, eigenvalue bounds. AB -

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, i.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iteration, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably good.

Zhong-Zhi Bai, Yu-Mei Huang, Michael K. Ng & Xi Yang. (2020). Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations. Numerical Mathematics: Theory, Methods and Applications. 3 (4). 461-474. doi:10.4208/nmtma.2010.m9014
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