A Total Fractional-Order Variation Model for Image Restoration with Nonhomogeneous Boundary Conditions and Its Numerical Solution
A Total Fractional-Order Variation Model for Image Restoration with Nonhomogeneous Boundary Conditions and Its Numerical Solution
复制标题
非齐次边界条件图像恢复的全分数阶变分模型及其数值解
DOI：
10.1137/14097121x
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发表时间：
2015-01
影响因子：
2.1
通讯作者：
Chen, Ke
中科院分区：
文献类型：
--
作者：
Zhang, Jianping;Chen, Ke
为了克服基于全变分的图像恢复模型的弱点，最近提出并研究了各种高阶（通常是二阶）正则化模型。在本文中，我们分析和测试了基于分数阶导数的总$\alpha$阶变分模型，该模型可以优于当前流行的高阶正则化模型。之前存在一些使用总 $\alpha$ 阶变化进行图像恢复的工作；然而，首先尚未进行分析，其次所有测试的公式彼此不同，都利用了不现实的零狄利克雷边界条件（而非零边界条件违反了分数阶导数的定义）。本文首先回顾了分数阶导数的一些结果，然后严格分析了所提出的总$\alpha$阶变分模型的理论性质。然后，它开发了四种解决变分问题的算法，一种基于变分 Split-Bregman 思想，三种基于离散优化问题的直接解决方案。数值实验表明，在恢复质量和求解效率方面，对于平滑图像，该模型能够产生与已建立的两个高阶模型：平均曲率和总广义变分相比极具竞争力的结果。
To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.
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DOI：
10.1109/icwapr.2007.4420804
发表时间：
2007-11
期刊：
2007 International Conference on Wavelet Analysis and Pattern Recognition
影响因子：
--
作者：
Wei Guo;L. Qiao
通讯作者：
Wei Guo;L. Qiao
影响因子：
2
作者：
A. Chambolle
通讯作者：
A. Chambolle
DOI：
10.1137/s0036139901390088
发表时间：
2003
期刊：
SIAM J. Appl. Math.
影响因子：
--
作者：
Jianhong Shen;S. Kang;T. Chan
通讯作者：
Jianhong Shen;S. Kang;T. Chan
DOI：
10.1093/acprof:oso/9780198528418.001.0001
发表时间：
2004-02
期刊：
--
影响因子：
--
作者：
J. Modersitzki
通讯作者：
J. Modersitzki
DOI：
10.1017/s1004897900001239
发表时间：
2013-02
影响因子：
1.3
作者：
R. Chan;A. Lanza;S. Morigi;F. Sgallari
通讯作者：
R. Chan;A. Lanza;S. Morigi;F. Sgallari