数值计算与计算机应用
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数值计算与计算机应用  2016, Vol. 37 Issue (4): 273-286    DOI:
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数字图像修复的变分方法与实现过程
邱俊1, 胡晓2,3, 王汉权4
1. 云南财经大学统计与数学学院, 昆明 650221;
2. 云南财经大学统计与数学学院, 昆明 650221;
3. 四川省攀枝花市第七高级中学, 四川攀枝花 617005;
4. 云南财经大学统计与数学学院, 昆明 650221
VARIATIONAL METHOD FOR IMAGE INPAINTING AND ITS IMPLEMENTATION
Qiu Jun1, Hu Xiao2,3, Wang Hanquan4
1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China;
2. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China;
3. The Seventh High School of Panzhihua City, Panzhihua 617005, Sichuan, China;
4. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
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摘要 图像修复是数字图像处理过程的一个很重要的方面.图像修复目的是将图像中污损或破损的部分运用相关的方法将其恢复.本文主要讨论数字图像恢复的变分方法及其实现过程,重点讨论变分方法之中的偏微分方程模型建立的基本过程和求解方法.图像恢复的变分方法的核心思想是将恢复过程归结为求解一个含约束条件的泛函极小值问题.为得到此泛函极小值问题的解,先根据拉格朗日乘子法,将含约束条件的泛函极小值问题化为无约束的泛函极小值问题.由于无约束的泛函极小值问题的解满足一偏微分方程,于是可构造一梯度流并通过它找出该偏微分方程的解.最终用偏微分方程数值方法-有限差分法来离散得到此梯度流的稳态解的近似,并将此近似解作为图像修复之后的结果表示.
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关键词数字图像修复   有限差分法   能量泛函   梯度下降流   欧拉-拉格朗日方程   变分方法     
Abstract: Image inpainting is one of the most important part in digital image processing. It aims to fill in missing parts of damaged or degraded images based on the information around. In this paper, we discuss how to do image inpainting with variational methods and how to implement it in computers. We mainly investigate the general way on how to build partial differential equations based on variational methods and how to solve subsequent models. The key idea of variational methods for image inpainting is turning the inpainting problem into a functional minimization problem with constraints. To solve this minimization problem, we first apply the Lagrangian multiplier and solve a free functional minimization problem instead. The solution of the latter one satisfies Euler-Lagrangian equation. We next construct a continuous gradient flow and discretize it with finite difference method in order to find the steady state solution of the gradient flow, which is in fact the solution of related Euler-Lagrangian equation. We finally use the numerical solution as the approximation of inpainted image.
Key wordsDigital image inpainting   finite difference method   energy functional   gradient flow   Euler-Lagarangian equation   variational methods   
收稿日期: 2015-09-30;
基金资助:

国家自然科学基金(11261065、91430103)和教育部新世纪优秀人才基金(NCET-13-0995)资助.

通讯作者: 王汉权,E-mail:hanquan.wang@gmail.com     E-mail: hanquan.wang@gmail.com
引用本文:   
. 数字图像修复的变分方法与实现过程[J]. 数值计算与计算机应用, 2016, 37(4): 273-286.
. VARIATIONAL METHOD FOR IMAGE INPAINTING AND ITS IMPLEMENTATION[J]. Journal of Numerical Methods and Computer Applicat, 2016, 37(4): 273-286.
 
[1] Aubert G, Kornprobst P. Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations[M]. Springer Verlag, 2001.
[2] Bertalmio M, Sapiro G, Caselles V, Ballester C. Image inpainting[C]. Proceedings of SIGGRAPH, ACM, 2000, 417-424.
[3] Chan T, Esedoglu S. Aspects of total variation regularized L1 function approximation[J]. SIAM J. Appl. Math., 2004, 65: 1817-1837.
[4] Chan T, Shen J. Image processing and analysis: variational, pde, wavelet, and stochastic methods[M]. SIAM, 2009.
[5] Le T, Chartrand R, Asaki T. A variational approach to constructing images corrupted by poisson noise[J]. Journal of Mathematical Imaging and Vision, 2007, 27: 257-263.
[6] M Bertalmo, A L Bertozzi, G Sapiro. Navier-Stokes, fluid dynamics, and image and video imaging[C]. Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2001, 1: 355-362.
[7] Schonlieb C B, Bertozzi A. Unconditionally stable schemes for higher order inpainting[J]. Communications in Mathematical Sciences, 2011, 9: 413-457.
[8] Bertalmio M, Sapiro G, Caselles V, Ballester C Image Inpaiting[C]. Proceedings of ACM SIGGRAPH 2000 New York:ACM Press, 2000, 417-424
[9] Burger M, He L, Schonlieb C. Cahn-Hilliard inpainting and a generalization for grayvalue images[J]. SIAM J. Imaging Sci., 2009, 2: 1129-1167.
[10] Rudin L, Osher S, and Fatemi E. Nonlinear total variation based noise removal algorithms[J]. Physica D, 1992, 60: 259-268.
[11] Vese L. A study in the BV space of a denoising-deblurring variational problem[J]. Appl. Math. Optim., 2001, 44: 131-161.
[12] Geman S, Geman D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[J]. IEEE Trans. Pattern Anal. Machine Intell., 1984, 6: 721-741.
[13] Daubechies I. Ten Lectures on Wavelets[M]. SIAM, 1992.
[14] Pennec E L, Mallat S. Image compression with geometrical wavelets[C]. Proc. of Int'l Conf. Image Processing, 2000, 1: 661-664.
[15] Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems[J]. Comm. Pure Appl. Math., 1989, 42: 577-685.
[16] Likas A C, Galatsanos N P. A variational approach for Bayesian blind image deconvolution[J]. IEEE Trans. Image Process, 2004, 52: 2222-2233.
[17] Acar R, Vogel R. Analysis of total variation penalty methods for ill-posed problems[J]. Inverse Problems, 1997, 10: 1217-1229.
[18] Chan T, Shen J. Mathematical models for local nontexture inpaintings[J]. SIAM J. Appl. Math, 2015, 62: 1019-1043.
[19] Chan T, Shen J. Luminita Vese L. Variational PDE models in image processing[J]. Notices of the AMS, 2002, 50: 14-26.
[20] Chan T, Shen J. Variational image processing[J]. Commun. Pure Appl. Math., 2005, 58: 579-619.
[21] Schonlieb B. Modern PDE techniques for image inpainting, PhD Thesis, University of Cambridge, 2009.
[22] Wang Y, Yang J, Yin W, Zhang Y. A new alternating minimization algorithm for total variation image reconstruction[J]. SIAM J. Appl. Math., 2015, 1: 248-272.
[23] Wang X, Deng L. Blind image inpainting based on TV model and edge detection[C]. Proceedings of the 2015 International Conference on Applied Mechanics, Mechatronics and Intellligent Systems, 2015, 385-392.
[24] Dong B, Ji H, Li J. Wavelet frame based blind image inpainting[J]. Appli. Comput. Harmonic Analysis, 2012, 32: 268-279.
[25] Shen J, Kang S H, Chan T. Euler's elastica and curvature-based inpainting[J]. SIAM J. Appl. Math., 2010, 63: 564-592.
[26] Dahl J, Hansen P, Jensen S, Jensen T. Algorithms and software for total variation image reconstruction via first-order methods[J]. Numerical algorithms, 2010, 53: 67-92.
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