Approximate Schur Complement Preconditioners for Half-Quadratic Image Restoration With Zero Boundary Conditions

Abstract

In this paper, the additive half-quadratic image restoration problem with zero boundary conditions is investigated. The Newton method is used to solve this problem and a structured linear system needs solving at each step. In order to accelerate this process, we have proposed a preconditioning method based on approximations of the Schur complement and the blurring matrix. The block Toeplitz matrix is approximated as a block circulant matrix and the fast Fourier transform is used to implement matrix–vector multiplications. We give an analysis of the eigenvalue property of the preconditioned Hessian matrix. Numerical results demonstrate the effectiveness of the proposed preconditioning method.