Preconditioned Iterative Methods For The Linear Systems Arising From Regularized Image Restoration

The recorded images are often degraded by noise and blur in image applications.The goal of image restoration is to process the degraded image,and get an estimate which is as close as possible to the original image.Image restoration is the basis of image processing,and is an important content on image research.With the popularity and development of information visualization,imaging applications become more and more important,the fields involved in image processing are becoming increasingly broad.Due to the importance of image restoration in image processing,the study of mathematical methods in image restoration problem is of greatly theoretical and practical significance.Image restoration is a highly ill-posed inverse problem,so the regularization methods are the key methods in image restoration.In the regularized image restoration methods,the recovered image can usually be obtained by minimizing a cost function which consists of a data fidelity term and a regularization term.In this paper,we consider the half-quadratic regularization and total variation regularization image restoration,and apply the Newton method and the lagged fixed point method to solve the regularized models,respectively.The linear systems are generated during the process of solving the regularized models,and the preconditioned iterative methods with the preconditioned matrices we constructed are applied to solve the linear systems,the corresponding theoretical analyses and numerical experiments are given.We first consider the multiplicative half-quadratic regularization model in image restoration.The Newton method is applied to solve the Euler-Lagrange equation.In each Newton iteration step,a symmetric positive definite linear system is needed to be solved.We construct a modified block Symmetric Successive Overrelaxation(SSOR)preconditioned matrix for the coefficient matrix,and the preconditioned conjugate gradient method is applied to solve the linear system.We estimate the condition number of the preconditioned matrix,its upper bound and the optimal relaxation parameter are given.We also construct an improved block product preconditioner for the symmetric positive definite coefficient matrix,and the preconditioned conjugate gradient method is employed to solve the linear system.The spectral properties of the preconditioned matrix are also analyzed.Both theoretical analyses and numerical results demonstrate that the modified block SSOR preconditioned matrix and the improved block product preconditioned matrix we proposed improve the computing efficiency in image restoration.Then,we consider the additive half-quadratic regularization model in image restoration.The Newton method is applied to solve the Euler-Lagrange equation.A linear system of equations with symmetric positive definite coefficient matrix arises in each Newton iteration step.The approximation of Schur complement can be obtained by replacing its diagonal matrix with a scalar matrix,we construct a restrictive preconditioner,and the preconditioned conjugate gradient method is employed to solve the symmetric positive definite linear system.Both theoretical analysis and numerical results illustrate that the preconditioned matrix we proposed further accelerates the computing speed of the additive half-quadratic image restoration.Finally,the total variation regularization model in image restoration is also considered.The lagged fixed point iteration method is employed to solve the EulerLagrange equation.At each iteration,we need to solve a linear system,which can be transformed into solving a block 2 × 2 nonsymmetric positive definite linear system by variable substitution.We construct a restrictive preconditioned matrix of the coefficient matrix by approximating its Schur complement,based on the block triangular factorization form of the coefficient matrix,and the restrictive preconditioned conjugate gradient method is employed to solve the nonsymmetric positive definite linear system.A splitting preconditioner is proposed for the 2 × 2 block nonsymmetric positive definite linear system,in terms of the iteration scheme of Hermitian and skew-Hermitian splitting method,and use the preconditioned Generalized Minimal Residual method to solve the nonsymmetric positive definite linear system.Both theoretical analyses and numerical results demonstrate the proposed preconditioned iterative methods are feasible and effective for solving the linear system arising from the total variation regularization image restoration.