基于空间分数阶偏微分方程图像去噪的隐式差分方法
Implicit Difference Numerical Method of Image Denoising Based on Space Fractional PDE
DOI: 10.12677/AAM.2016.51012, PDF,    国家自然科学基金支持
作者: 杨泽凡, 杨晓忠*：华北电力大学数理学院信息与计算研究所，北京
关键词: 图像去噪；空间分数阶偏微分方程；隐式差分方法；数值试验；Image Denoising； Space Fractional Partial Differential Equations； Implicit Difference Method； Numerical Experiment
摘要: 图像去噪的空间分数阶偏微分方程方法是图像去噪领域中的一个重要方向，对于它的数值方法研究有重要的理论意义和实用价值。本文研究基于空间分数阶偏微分方程的图像去噪方法，对空间分数阶图像去噪模型构造隐式差分格式，分析格式解的存在唯一性和格式的稳定性、收敛性，并给出精度分析。理论分析和数值试验证实：隐式差分格式对求解空间分数阶偏微分方程是可行的，且去噪效果优良。
Abstract: Image denoising numerical methods based on space fractional Partial Differential Equations is an important direction of image denoising field, and its study of numerical methods has important theoretical significance and practical value. This paper constructs implicit difference scheme for solving the space fractional partial differential equation. Through theoretical analysis and numer-ical experiments, we found that implicit difference scheme for solving space fractional partial dif-ferential equations is feasible, it can ensure good denoising effect.
文章引用：杨泽凡, 杨晓忠. 基于空间分数阶偏微分方程图像去噪的隐式差分方法[J]. 应用数学进展, 2016, 5(1): 79-86. https://dx.doi.org/10.12677/AAM.2016.51012

参考文献

[1] Bai, J. and Feng, X.-C. (2007) Fractional-Order Anisotropic Diffusion for Image Denoising. IEEE Transactions on Image Processing, 16, 2492-2502. http://dx.doi.org/10.1109/TIP.2007.904971 [Google Scholar] [CrossRef] 
[2] Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P. (2006) A Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213, 205-213. http://dx.doi.org/10.1016/j.jcp.2005.08.008 [Google Scholar] [CrossRef] 
[3] 张军, 韦志辉. SAR图像去噪的分数阶多尺度变分PDE模型及自适应算法[J]. 电子与信息学报, 2010, 32(7): 1654-1659.
[4] 吴亚东, 孙世新, 张红英, 韩永国, 陈波. 一种基于图割的全变差图像去噪算法[J]. 电子学报, 2007, 35(2): 265- 266.
[5] Wu, Y.D. and Zhang, H.Y. (2006) Move Space Based Total Variation Image Denosing. Proceedings of the International Conference on Sensing, Compu-ting and Automation, 10, 1039-1043.
[6] 王磊, 莫玉龙. 精确复原退化图象的连续Hopfield网络研究[J]. 上海交通大学学报, 1997, 31(12): 43-46.
[7] 王彪, 田杰, 黄海宁, 李淑秋. 基于偏微分方程和小波图像去噪研究[J]. 计算机工程与应用, 2005, 41(4): 72-74.
[8] 朱立新, 王平安, 夏德深. 引入耦合梯度保真项的非线性扩散图像去噪方法[J]. 计算机研究与发, 2007, 44(8): 1390-1398.
[9] 黄果, 许黎, 蒲亦非. 分数阶微积分在图像处理中的研究综述[J]. 计算机应用研究, 2012, 29(2): 414-420.
[10] 黄果, 蒲亦非, 陈庆利. 基于分数阶积分的图像去噪[J]. 系统工程与电子技术, 2011, 33(4): 925-932.
[11] Pu, Y.-F., Zhou, J.-L. and Yuan, X. (2010) Fractional Differential Mask: A Fractional Differential-Based Approach for Multiscale Texture Enhancement. IEEE Transactions on Image Processing, 19, 491-511. http://dx.doi.org/10.1109/TIP.2009.2035980 [Google Scholar] [CrossRef] 
[12] Zhang, Y. and Yang, X.Z. (2010) Accelerated Additive Operator Splitting Schemes for Nonlinear Diffusion Filtering. International Conference on Biomedical Engineering and Computer Science, Wuhan, 23-25 April 2010, 1-4. http://dx.doi.org/10.1109/icbecs.2010.5462498 [Google Scholar] [CrossRef] 
[13] Weilbee, M. (2005) Efficient Numerical Methods for Fractional Differential Equations and Their Analytical Background. Technical University of Braunschweig, Braun-schweig.
[14] 郭柏灵. 分数阶偏微分方程及其数值解[M]. 北京: 科学出版社, 2011.

为你推荐