高阶图像去噪模型的快速数值算法研究

项目名称： 高阶图像去噪模型的快速数值算法研究

项目编号： No.11526110

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 张俊

作者单位： 南昌工程学院

项目金额： 3万元

中文摘要： 基于高阶PDE 的图像去噪模型源于对ROF 模型的改进，在达到较好的去噪效果时能克服阶梯效应。由于其离散问题的非线性程度很高且涉及高阶偏导数计算，许多求解非线性方程的迭代算法就显得不是很高效。发展快速迭代算法来解这类问题具有重要的理论价值和实际意义。已提出的增广拉格朗日方法在求解基于四阶PDE 的欧拉弹性去噪模型和均值曲率去噪模型时，需要对某些子问题的欧拉拉格朗日方程进行求解，且求解的子问题较多。本项目从算法应该简单、高效的角度出发，研究两类去噪模型的更加快速高效的增广拉格朗日方法。具体研究工作为：拟分别构造线性化增广拉格朗日方法与基于不动点的增广拉格朗日方法来求解欧拉弹性模型和均值曲率模型，避免了解偏微分方程，研究它们的收敛性，并且通过数值实验来检验算法的高效性；拟研究求解均值曲率模型的另外一种更快速高效的增广拉格朗日算法框架，研究算法的收敛性，并通过数值实验验证算法的高效性。

中文关键词： 欧拉弹性模型；均值曲率模型；线性化增广拉格朗日方法；MRI重构；交替方向乘子法

英文摘要： Higher-order PDE-based image de-noising models stem the improvements from the ROF model. They can overcome the ladder effect at the time of achieving better de-noising effect. For their dispersion problems , because of the high degree of nonlinearity and involving higher-order partial derivatives to be calculated, many iterative algorithms for solving the nonlinear equations are not very efficient. Development of fast iterative algorithms to solve this kind of problem has important theoretical and practical significance. While solving the Euler’s elastic de-noising model and the mean curvature de-noising model based on the fourth-order PDEs , the proposed augmented Lagrangian methods need to solve the associated Euler-Lagrange equations derived from some of sub-problems and they involve too many sub-problems to be solved. Considering that the algorithm should be simple and efficient , this item will study more quickly and efficiently augmented Lagrangian method for these two kinds of de-noising models. The concrete research work is as follows: we intend to construct a linearized augmented Lagrangian method and a fixed point-based augmented Lagrangian method for solving the Euler’s elastic model and the mean curvature model, respectively. It therefore avoids solving partial differential equations. Furthermore, w

英文关键词： Euler's elastica model；Mean curvature model；Linearized augmented Lagrangian method；MRI reconstruction；ADMM