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.

图像恢复是一个病态问题，因此经常使用正则化方法来稳定解。在本文中，我们考虑了加法半二次正则化图像恢复问题，并使用牛顿法对其进行了求解。在每个牛顿迭代步骤中，都需要求解具有对称正定系数矩阵的线性方程组的结构化系统。通过在系数矩阵中采用近似Schur补码，我们构造了一个限制性的预处理器，并将其组合到共轭梯度法中，以求解线性系统。还分析了预处理基质的光谱性质。数值实验证明了该方法在图像复原中的有效性。