A parameterized Schur complement preconditioner for linear system arising from additive HQ image restoration

Image restoration is to estimate the clean image from the recorded image, it is a highly ill-posed inverse problem. Regularization method is an important approach for solving such problem, which can usually be achieved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the additive half-quadratic (HQ) regularized method for image restoration problem, and utilize the Newton method to solve the resulting minimization problem. At each Newton iteration step, a system of linear equations with symmetric positive definite coefficient matrix arises. In order to solve the linear system efficiently, we design a parameterized approximation matrix of the Schur complement inverse matrix, and construct a block preconditioner with parameter correspondingly, according to the block triangular factorization of coefficient matrix and the form of its Schur complement, then the preconditioned conjugate gradient (PCG) method is applied to solve the linear system of equations. Spectral analyses of the preconditioned matrix are also given, numerical experimental results demonstrate the effectiveness of the proposed parameterized preconditioner for solving linear system arising from additive HQ image restoration problem.