Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations
Zhong-Zhi Bai 1*, Yu-Mei Huang 2, Michael K. Ng 3, Xi Yang 11 State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific
2 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China.
3 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
Received 10 August 2009; Accepted (in revised version) 11 January 2010
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 iterate, 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 well.AMS subject classifications: 65F10, 65F50, 65W05, CR: G1.3
Key words: Edge-preserving, image restoration, multiplicative half-quadratic regularization, Newton method, preconditioned conjugate gradient method, constraint preconditioner, eigenvalue bounds.
Email: firstname.lastname@example.org (Z.-Z. Bai), email@example.com (Y.-M. Huang), firstname.lastname@example.org (M. K. Ng), email@example.com (X. Yang)