Improved parameterized preconditioner for linear system from multiplicative half-quadratic regularized image restoration

Image restoration is a fundamental problem in image processing, recovering a clear image from its degraded observation is an ill-posed problem, which can be addressed effectively by regularized techniques. Image restoration can usually be solved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the multiplicative half-quadratic regularized image restoration problem and the Newton method is employed to solve the regularized model. At each step of the Newton iteration, a linear system of equations with symmetric positive definite coefficient matrix needs to be solved. Based on the block triangular decomposition of the coefficient matrix, by introducing a parameter, and taking the truncated Taylor expansion form of the Schur complement inverse matrix, we design a parameterized preconditioner of the coefficient matrix and combine it into the conjugate gradient method to solve the linear system efficiently. The spectral properties of the preconditioned matrix are further analyzed. Numerical experiments demonstrate that the proposed preconditioner for solving the multiplicative half-quadratic regularization image restoration problem is more robust and effective, in reducing both the number of iterations and the overall computation time compared with the existing preconditioners.