A restrictive preconditioner for the system arising in half-quadratic regularized image restoration

Abstract

Image restoration is an ill-conditioned problem, so the regularization method is often applied to stabilize the solution. In this paper, we consider an additive half-quadratic (HQ) regularized image restoration problem and use the Newton method to solve it. At each Newton iteration step, a structured system of linear equations with symmetric positive definite coefficient matrix is needed to be solved. By taking an approximate Schur complement in the coefficient matrix, we construct a restrictive preconditioner and combine it into the conjugate gradient method in order to solve the linear system. The spectral properties of the preconditioned matrix are also analyzed. The numerical experiments demonstrate the effectiveness of the proposed method for image restoration.