半二次图像复原中结构化方程组的预处理方法研究

A Review of Preconditioners for Structured Systems Arising from Half-Quadratic Image Restoration

摘要: 半二次正则化最小二乘是实现高质量图像复原的重要模型之一。在利用牛顿迭代方法等优化方法求解该模型的过程中，每一步都涉及结构化方程组的求解。预处理共轭梯度法(PCG)是求解此类方程组的有效方法，而其收敛速度取决于预处理后矩阵的特征值性质。构造合适的预处理矩阵对于提高图像复原的性能具有重要的意义。近年来，结合半二次图像复原中方程组的结构化特点，学者们基于矩阵的Schur补近似等策略构造出了一系列的预处理矩阵，并给出了相应的特征值分析。数值结果表明，这些预处理方法有效地降低了图像复原的计算成本。针对半二次图像复原中的结构化方程组，本文整理了近几年出现的预处理方法，并从不同侧面进行对比分析，旨在为进一步的预处理方法改进和研究提供思路参考。

Abstract: Half-quadratic regularized least square method is one of the important models for achieving high-quality image restoration. In the process of solving this model using optimization methods such as Newton method, each step involves solving a structured system. The Preconditioned Conju-gate Gradient (PCG) method is an efficient method for solving such systems while its convergence rate depends on the nature of the eigenvalues of the preconditioner. Constructing a suitable pre-conditioner is meaningful to improving the performance of image restoration. In recent years, com-bined with the structured characteristics of the system in half-quadratic image restoration, scholars have constructed a series of preconditioners based on the Schur complement approximation of the matrices and other strategies, and the corresponding eigenvalue analyses are given. The numerical results show that these preconditioners effectively reduce the computational cost of image restora-tion. For the structured system in half-quadratic image restoration, this paper organizes the pre-conditioners that appeared in recent years and makes a comparative analysis from different sides, aiming to provide ideas reference for further improvement and research of preconditioners.