非局部总变差正则化图像恢复模型的快速子空间校正算法

项目名称： 非局部总变差正则化图像恢复模型的快速子空间校正算法

项目编号： No.11426165

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 常慧宾

作者单位： 天津师范大学

项目金额： 3万元

中文摘要： 非局部总变差正则化模型被用于恢复退化的图像，恢复结果不仅具有锐利的边界，而且保持了较好的纹理结构，同时有效消除了传统有界总变差正则化模型带来的阶梯效应，因而成为了近年来的研究热点之一。现有的模型求解算法具有较高的算法复杂度，不能完全满足实际高精度、高维图像以及视频处理的需要。本项目将探讨采用以区域分解算法、多重网格算法为代表的子空间校正算法求解此类问题，实现并行计算，有效处理此类大规模、高复杂度的问题。此模型的目标泛函具有不可微、不可加性以及导数运算的全局相关性质，因而为算法设计带来巨大困难。在前期研究中，我们设计了单水平区域分解算法，巧妙借助算子分裂技巧，克服了上述困难。数值试验表明该算法是收敛，且具有较好的加速效率，在此基础上，将进一步分析算法的收敛性；其次，设计并分析两水平和多水平子空间校正算法，使之具有对于正则化参数更好的稳定性和更快的收敛速度。

中文关键词： 非局部；总变正则化模型；对偶；区域分解算法；子空间校正算法

英文摘要： Nonlocal Total Variation (NLTV) regularized image restoration model can be adopted to restore the degraded images, and the restored results own sharper edges and better textures. It also removes the staircase artifacts, which exists in the restored images by the traditional total variation regularized model. Therefore, the study of this model has recently become more and more popular. In this project, we will explore the subspace correction methods in order to solve the NLTV model, which mainly consists of the domain decomposition methods (DDMs) and multigrid methods. They can solve such problems with large scales and high computational complexities by parallel computing. However, the objective functional of the NLTV model is non-differentiable, non-additive and non-local, that causes the great difficulty for designing the efficient algorithm. In our preliminary studies, we propose the one level DDM for the NLTV model, where the operator splitting method is used in a clever way to overcome the special difficulty of the objective functional. Numerical examples demonstrate the convergence of the proposed algorithm, and the satisfactory speed-up efficiency. The convergence will be proved furthermore. Moreover, we shall develop and analysis the two-level DDMs and multilevel subspace correction methods with better co

英文关键词： Subspace correction method；Nonlocal；Dual；Domain decomposition method；Total variation