【作者】 李艳红；

【导师】 肖运海；

【作者基本信息】 河南大学 ， 运筹学与控制论， 2012， 硕士

【摘要】 数字图像处理是利用计算机对图像信息进行加工以满足人的视觉心理或应用需求的行为.图像重构是为了提高图像的质量,建立模型,采用某种数值方法,重构或重建原始图像.本文提出求解基于全变分正则化优化模型压缩传感图像重构的交替极小化算法和交替方向算法,分析算法的收敛性,通过数值仿真试验验证算法的有效性.第一章,简单介绍问题的研究背景和研究意义;介绍压缩传感和全变分模型的基本理论;回顾求解l1-范数优化问题和全变分模型的已有算法;给出求解可分离凸优化问题交替方向法的基本知识;列出本文所用到的一些符号.第二章,提出基于全变分模型压缩传感图像重构的两种算法.所提算法分别是交替极小化原始问题罚函数和增广拉格朗日函数.利用全变分模型特征,线性化拟合项并加临界点项确保子问题具有显式解.所提两种算法迭代形式简单,每步迭代仅需计算两次矩阵向量内积和一次收缩算子.在一定条件下,分析算法全局收敛性.数值试验验证所提算法的有效性.第三章,结合Xiao-Yang-Yuan提出的算法和第二章所提算法,提出交替线性化的杂交算法.在每步迭代中,本章所提算法交替线性化增广拉格朗日函数的二次拟合项.数值仿真试验表明,本章所提算法提高了效率.第四章,给出本文的总结,并提出一些值得继续探讨的方向.更多还原

【Abstract】 In order to improve the quality of the image, the task of image reconstruction isto construct model, use some numerical methods, restore or recovery the original image.In this thesis, we propose alternating minimization algorithm and alternating directionalgorithm for recovering the compressive sensing image based on total variation regular-ization optimization model. We analyze the algorithms’ convergence properties and dosome numerical experiments to show the efciency of each algorithm.In chapter1, we introduce the background and signifcance of the thesis. We briefyreview the theory of compressive sensing and total variation, and recall some recent algo-rithm for solving1-norm and total variation regularized minimization problems. More-over, we give the preliminaries of alternating direction algorithm for convex separableminimization problems, and list some notations which used in this thesis.In chapter2, based on the total variation model, we proposed double algorithms forthe compressive sensing image restoration. Both algorithms minimize the penalty functionand augmented Lagrangian function respectively. We linearize the ftting term and adda proximal point term to ensure that both subproblems have closed-form solutions. Theiterative form of both algorithms is very simple and requires a shrinkage and two matrix-vector produce at each iteration. With some mild conditions, we show that both algorithmconverges globally. Moreover, we also do some numerical experiments which show thatboth proposed algorithm are promising.In Chapter3, combining the Xiao-Yang-Yuan’s algorithm with the proposed algo-rithm in the previous chapter, we develop a hybrid linearized algorithm. At each iteration,the proposed algorithm alternatively linearized the quadratic ftting term of the aug-mented Lagrangian function. The performance comparisons illustrate that the proposedalgorithm is superior to the algorithm in the previous section.In Chapter4, we give a summary of this thesis and list some further research topics.更多还原