A preconditioned conjugate gradient method for multiplicative half-quadratic image restoration

In mathematics, the image restoration problem is often settled by minimizing a cost function. This cost function usually consists of a data-fidelity term and a regularization term. In this paper, we study the image restoration where the regularization term is a half-quadratic term of multiplicative form. The Newton method is applied to solve the half-quadratic regularization image restoration problem. A structured linear system arises at each step of the Newton iteration. Preconditioned conjugate gradient method is applied to solve it with some product-type preconditioners of block triangular and block diagonal matrices. The spectral analysis is presented and bounds are given for the eigenvalues of the preconditioned matrices. The experimental results show that the preconditioned conjugate gradient method is efficient for solving the half-quadratic regularization image restoration with multiplicative form in both numerical performance and image recovering quality.