Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed to solve the nonlinear system of equations resulted from the optimization problem for image restoration. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. The preconditioned conjugate gradient method with the proposed modified block SSOR (symmetric successive over-relaxation) preconditioner is applied to solve this linear system of equations. The condition number of the preconditioned matrix is estimated and numerical experiments are also implemented for image restoration. Both theoretical and numerical results show that the modified block SSOR preconditioned PCG methods can greatly improve the computation efficiency when solving the multiplicative half-quadratic regularized image restoration problem.

中文翻译：

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共轭梯度法，采用改进的块SSOR迭代进行预处理，以实现乘法半二次图像复原

通常通过最小化由数据保真度项和正则化项组成的成本函数来解决图像恢复问题。半二次正则化的优点是可以很好地保留恢复的图像中的图像细节。在本文中，我们考虑解决涉及乘法半二次正则项的图像恢复模型。牛顿法被用来解决由图像恢复的优化问题导致的非线性方程组。在每个牛顿迭代步骤中，都会出现具有对称正定系数矩阵的线性方程组。采用带有改进的块SSOR（对称连续超松弛）预处理器的预处理共轭梯度方法来求解该线性方程组。估计预处理矩阵的条件数，并且还进行了数值实验以进行图像恢复。理论和数值结果均表明，改进的块SSOR预处理PCG方法在解决乘法半二次正则化图像恢复问题时可以大大提高计算效率。