#### Document Type

Journal Article

#### Department/Unit

Department of Mathematics

#### Title

Preconditioned iterative methods for algebraic systems from multiplicative half-quadratic regularization image restorations

#### Language

English

#### 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 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. © 2010 Global-Science Press.

#### Keywords

Constraint preconditioner, Edge-preserving, Eigenvalue bounds, Image restoration, Multiplicative half-quadratic regularization, Newton method, Preconditioned conjugate gradient method

#### Publication Date

2010

#### Source Publication Title

Numerical Mathematics -English Series-

#### Volume

3

#### Issue

4

#### Start Page

461

#### End Page

474

#### Publisher

Nanjing University Press

#### DOI

10.4208/nmtma.2010.m9014

#### Link to Publisher's Edition

#### ISSN (print)

10048979

#### ISSN (electronic)

20797338

#### Recommended Citation

Bai, Zhong-Zhi,
Yu-Mei Huang,
Michael K. Ng,
and
Xi Yang.
"Preconditioned iterative methods for algebraic systems from multiplicative half-quadratic regularization image restorations."
*Numerical Mathematics -English Series-*
3.4
(2010): 461-474.