数字图像修复算法研究
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摘要
随着数字时代的到来,许多珍贵的档案图片、古文字画、老电影等被扫描并保存在计算机中。然而,在保存过程中,由于一些自然或非自然因素的影响,图像往往容易出现一些缺损；另外有时还需要对照片中一些不感兴趣的部分进行选择性移除。在这种情况下,数字图像修复技术应运而生,并得到了广泛关注。
数字图像修复是从人的视觉角度出发,在一定的原理或算法的引导下,实现对图像中未知破损区域的修复,使观察者察觉不到修改的痕迹,并使修复后图像的视觉效果清晰自然。
现阶段的数字图像修复算法主要分为两个大方向,即：用于弥补小尺度划痕和斑点的修复技术,以及用于填充大目标去除后空白区域的背景补全技术。目前,用于小尺度修复的技术已经比较成熟,本文中不再作具体介绍；论文主要以修复大区域的缺损信息为目的,对几种修复算法的原理、流程、效果进行了深入研究。
论文首先概括了数字图像修复算法的背景及意义,总结了不同修复技术的发展历程和现状；然后对经典的纹理合成算法进行了深入的讨论,并在此基础上对算法进行了改进,提出了一种非均匀纹理的大区域图像修复算法；该算法通过添加方向性优先权系数,对修复的顺序进行了改善,同时限定了最优匹配块的搜索范围,在提高效率的同时减少了误匹配。
对于具有明显结构信息的图像,本文研究了基于结构传播以及区域分解的修复算法。算法首先对人眼较为敏感的结构信息进行连接恢复,然后利用这些曲线作为分水岭将图像进行分区,最后对剩余区域进行分别修复。仿真结果表明,这一算法可以对复杂的结构图像得到较好的修复效果。
本文最后将图像分解技术应用到修复领域中,深入研究了基于模型分解的修复技术；通过将破损图像分解成纹理和结构两幅子图像可以分别对结构和纹理进行修复。此外,本文还利用该技术对结构传播的修复算法进行了改进,最终通过实验效果分析了上述算法的优点及不足。
With the advent of the digital age, many precious photos were scanned and saved on the computer. However, these photos always have some defects due to some natural or unnatural factors when preserving. Besides that, sometimes we need to selectively remove some parts of the picture which are not interested in. Under this condition, the digital image processing technology emergences and gets more and more attention.
Digital image inpainting is to repair the damaged areas conveniently under the guidance of some principles and algorithms, it needs to be finished from the perspective of human vision. Its target is to make the results more clear and natural, and make the viewer cannot aware of any modification.
Digital image restoration algorithm can be divided into two general directions, the repairing techniques for recovering small-scale damages, such as the scratches and spots, and the completion technology for filling the large-scale blank region after removing some objects. Currently, the technology for small-scale restoration is relatively mature, so there is no specific description in this thesis. This thesis mainly focused on how to repair a large-scale region, and discussed the principles, processes, and effects of the following algorithms.
Firstly, we introduced the background and significance of image inpainting technology, and summed up the research status of different inpainting techniques. Then we in-deeply discussed the classic texture synthesis algorithm, and on this basis proposed a non-regular texture image completion algorithm in large region. This algorithm improved the restoration order by adding a directional priority coefficient, speeded up the efficiency and reduced the mismatch by limiting the searching scope of best-matching patch.
Besides that, we proposed a structure-based and the decomposition-based algorithm for the restoration of images with obvious structure information. Since the human eyes are more sensitive to the structural information, the structure was connected and restored firstly, and then the connected structure was utilized as the watershed for the image partition. Finally the remaining texture information in different parts were filled separately. Experimental results show that the algorithm can achieve good results for complex structure images.
In this thesis, we proposed a decomposition-based restoration algorithm by introducing the technique in the image inpainting field. By decomposing an image into two parts of texture and structure, we can inpaint them respectively. More over, the decomposition technique can also be used to improve the structure propagation algorithm. Finally, we analyzed the advantages and disadvantages of the algorithm with the experimental results in this thesis.
数字图像修复是从人的视觉角度出发,在一定的原理或算法的引导下,实现对图像中未知破损区域的修复,使观察者察觉不到修改的痕迹,并使修复后图像的视觉效果清晰自然。
现阶段的数字图像修复算法主要分为两个大方向,即：用于弥补小尺度划痕和斑点的修复技术,以及用于填充大目标去除后空白区域的背景补全技术。目前,用于小尺度修复的技术已经比较成熟,本文中不再作具体介绍；论文主要以修复大区域的缺损信息为目的,对几种修复算法的原理、流程、效果进行了深入研究。
论文首先概括了数字图像修复算法的背景及意义,总结了不同修复技术的发展历程和现状；然后对经典的纹理合成算法进行了深入的讨论,并在此基础上对算法进行了改进,提出了一种非均匀纹理的大区域图像修复算法；该算法通过添加方向性优先权系数,对修复的顺序进行了改善,同时限定了最优匹配块的搜索范围,在提高效率的同时减少了误匹配。
对于具有明显结构信息的图像,本文研究了基于结构传播以及区域分解的修复算法。算法首先对人眼较为敏感的结构信息进行连接恢复,然后利用这些曲线作为分水岭将图像进行分区,最后对剩余区域进行分别修复。仿真结果表明,这一算法可以对复杂的结构图像得到较好的修复效果。
本文最后将图像分解技术应用到修复领域中,深入研究了基于模型分解的修复技术；通过将破损图像分解成纹理和结构两幅子图像可以分别对结构和纹理进行修复。此外,本文还利用该技术对结构传播的修复算法进行了改进,最终通过实验效果分析了上述算法的优点及不足。
With the advent of the digital age, many precious photos were scanned and saved on the computer. However, these photos always have some defects due to some natural or unnatural factors when preserving. Besides that, sometimes we need to selectively remove some parts of the picture which are not interested in. Under this condition, the digital image processing technology emergences and gets more and more attention.
Digital image inpainting is to repair the damaged areas conveniently under the guidance of some principles and algorithms, it needs to be finished from the perspective of human vision. Its target is to make the results more clear and natural, and make the viewer cannot aware of any modification.
Digital image restoration algorithm can be divided into two general directions, the repairing techniques for recovering small-scale damages, such as the scratches and spots, and the completion technology for filling the large-scale blank region after removing some objects. Currently, the technology for small-scale restoration is relatively mature, so there is no specific description in this thesis. This thesis mainly focused on how to repair a large-scale region, and discussed the principles, processes, and effects of the following algorithms.
Firstly, we introduced the background and significance of image inpainting technology, and summed up the research status of different inpainting techniques. Then we in-deeply discussed the classic texture synthesis algorithm, and on this basis proposed a non-regular texture image completion algorithm in large region. This algorithm improved the restoration order by adding a directional priority coefficient, speeded up the efficiency and reduced the mismatch by limiting the searching scope of best-matching patch.
Besides that, we proposed a structure-based and the decomposition-based algorithm for the restoration of images with obvious structure information. Since the human eyes are more sensitive to the structural information, the structure was connected and restored firstly, and then the connected structure was utilized as the watershed for the image partition. Finally the remaining texture information in different parts were filled separately. Experimental results show that the algorithm can achieve good results for complex structure images.
In this thesis, we proposed a decomposition-based restoration algorithm by introducing the technique in the image inpainting field. By decomposing an image into two parts of texture and structure, we can inpaint them respectively. More over, the decomposition technique can also be used to improve the structure propagation algorithm. Finally, we analyzed the advantages and disadvantages of the algorithm with the experimental results in this thesis.
引文
[1]M. Bertalmio, G. Sapiro, V. Caselles, et al. Image inpainting. Proceedings of ACM SIGGRAPH 2000. New York:ACM Press,2000:417-424.
[2]T. F. Chan, J. H. Shen. Mathematical models for local non-texture inpaintings. SIAM J. APPL. MATH,2002,62:1019-1043.
[3]T. F. Chan, J. H. Shen. Non-texture inpainting by curvature driven diffusions(CDD). Vision Comm. Image Rep,2001,4:436-449.
[4]M. Oliveira, B. Bowen, R. McKenna, et al. Fast digital image inpainting. Proceedings of the Visualization, Imaging, and Image Processing IASTED Conference, Marbella, Spain,2001:261-266.
[5]T. F. Chan, J. Shen. Morphologically invariant PDE inpaintings. UCLA CAM Report,2001-15 at:www.math.ucla.edu/-imagers. IEEE Trans. Image process, 2002.
[6]Tsai A, Yezzi A J, Willsky A S. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation and magnification. IEEE Transactions on Image Processing,2001,10(8): 1169-1186.
[7]T. F. Chan, J. H. Shen. Mathematical models for local non-texture inpaintings. SIAM J. APPL. MATH,2002,62:1019-1043.
[8]Esedoglu S, Shen J H, Digital inpainting based on the Mumford-Shah-Euler image model. Euro, Journal of Applied Mathematics,2002(13):353-370.
[9]邵肖伟,刘政凯,宋璧.一种基于TV模型的自适应图像修复方法.电路与系统学报,2004,9：113-117.
[10]T. K. Shih, R. C. Chang. Digital inpainting-survey and multilayer image inpainting algorithms. Information Technology and Applications,2005,7(1): 15-24.
[11]Y. H. Fung, Y. H. Chan. A POCS-based restoration algorithm for restoring halftoned color-quantized images. Image Processing,2006,7(15):1985-1992.
[12]R. H. Chan, Y. W. Wen, and A. M. Yip. A fast optimization transfer algorithm for image inpainting in wavelet domains. IEEE Transactions on Image Processing, 2009,7(18):1467-1476.
[13]周廷方,汤峰,王进,等.基于径向基函数的图像修复技术.中国图像图形学报,2004,9：1190-1197.
[14]Efros, T. Leung. Texture synthesis by non-parametric sampling. In International Conference on Computer Vision,1999,1033-1038.
[15]S. Zelinka, M. Garland. Towards real-time texture synthesis with the jump map. Proceedings of the Thirteenth Eurographics Workshop on Rendering Techniques, 2002:99-104.
[16]Efros, W. T. Freeman. Image quilting for texture synthesis and transfer. Proceedings of SIGGRAPH 2001,2001:341-346.
[17]V. Kwatra, A. Schoedl, I. Essa, G. Turk, and A. Bobick. Graphcut textures: Image and video synthesis using graph cuts. Proceedings of SIGGRAPH 2003, 2003:277-286.
[18]Nealen, M. Alexa, Hybrid texture synthesis. Proceedings of the 14th Eurographics workshop on Rendering,2003:97-105.
[19]Drori, D. Cohen-Or, and Yeshurun, H. Fragment-based image completion. Proceedings of ACM SIGGRAPH 2003,2003:303-312.
[20]Criminisi, P. Perez, and K. Toyama, Object removal by exemplar-based inpainting. Proceedings of 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognitiont. Madison, Wisconsin,2003,2: 721-728.
[21]张岩,李文辉,孟宇,庞云阶.应用PSO的快速纹理合成算法.计算机研究与发展,2005,42(3)：424-430.
[22]Jiying Wu, Qiuqi Ruan. Object Removal By Cross Isophotes Exemplar-based Inpainting, IEEE, the 18th international conference on pattern recognition,2006.
[23]汪忠庆,檀结庆,何蕾.方向性纹理图像的修复算法.计算机工程与应用,2006,42(34)：29-32.
[24]Dongdong Nie, Lizhuang Ma, Shuangjiu Xiao. Similarity Based Image Inpainting Method. IEEE Transaction,2006,1-4244-0028-7/06.
[25]Y. B. Xue, H. Zhang, F. Y. Wang, et al. Exemplar-Based Image Completion Using Color Ratio Gradient. Image and Signal Processing,2008,5(3):569-572.
[26]M. Kuo, N. C. Yang, W. H. Chang, et al. Image Recovery Based on Effective Image Completion. Intelligent Information Hiding and Multimedia Signal Processing,2008,8:393-396.
[27]Yedidia, J. S., Freeman, W. T., and Weiss, Y.2002. Understanding belief propagation and its generalizations. Technical Report, Mitsubishi Electric Research Laboratories, MERL-TR-2001-22.
[28]C. Barcelos, M. Batista, A. Martins, and A. Nogueira. Level Lines Continuation based Digital Inpainting. IEEE Computer Graphics and Image Prcessing, October 2004,10:50-57.
[29]J. Sun, L. Yuan, J. Y. Jia, et al. Image Completion with Structure Propagation. ACM Transactions on Graphics,2005,24(3):861-868.
[30]A. Rares, M.J.T. Reinders, J. Biemond. Edge-Based Image Restoration[J]. IEEE Transactions on Image Processing,2005,14(10):1454-1468.
[31]Shen Minfen, Li Bin. Structure and texture image inpainting based on region segmentation//Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. Hawaii,2007:701-704.
[32]G. Hao, O. Nobutaka, and S. Shiqeki. A Structure-Synthesis Image Inpainting Algorithm Based on Morphological Erosion Operation. Image and Signal Processing,2008,5(3):530-535.
[33]Wang Minqin, Han Guoqiang, Tu Yongqiu. Edge-based image completing guided by region segmentation.//Proceedings of ISECS International Colloquium on Computing, Communication, Control, and Management. Guangzhou, China, 2008:152-156.
[34]H. F. Shen, L. P. Zhang. A MAP-Based Algorithm for Destriping and Inpainting of Remotely Sensed Images. IEEE Trans. Geosci. Remote Sens., May 2009, 5(47):1492-1502.
[35]L. Ruain, S. Osher, E. Fatemi. Nonlinear total variation based noise removal algorithm. Physica D,1992,60:259-268.
[36]Mumford, J. Shah. Optimal approximation by Piece2wise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics,1989,42(5):577-685.
[37]P. Perona, J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE Tram on PatternAnal Machine Intell,1990:12(7):629-639.
[38]Y. Meyer. Oscillating patterns in image processing and nonlinear evolution equatious. University Lecture Series,2001,22:1047-3998.
[39]Luminita A.Vese and Stanley J.Osher. Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing, Journal of Scientific Computing,2002,19(1-3):553-572.
[40]WRAY I. Integrating EAM and PdM formaintenance productivity. Plant Engineering,2003,57(10):65-66.
[41]S Levine. An adaptive variational model for image decomposition, in LNCS, 3757,2005:382-397.
[42]J F Aujol, A Chambolle. Dual norms and image decomposition models. Int. J. Comput. Vis.,2005,63(1):85-104.
[43]W. YIN, D. GOLDFARB, AND S. OSHER. Total variation-based image cartoon-texture decomposition. Columbia University CORC Report TR-2005-01, UCLA CAM Report,2005:05-27.
[44]Tadmor, S.Nezzar, L.Vese. A multiscale image representation using heirarchical (BV,L2) decompositions. Multiscale Model. Simul.,2004,2:554-579.
[45]J. B. Garnett, T. M. Le, Y. Meyer, and L. A. Vese. Image decompositions using bounded variation and generalized homogeneous Besov spaces. Appl. Comput. Harmon. Anal.,2007,23(1):25-56.
[46]Buades, T. Le, J.-M. Morel, and L. Vese. Fast cartoon+ texture image filters. IEEE Transactions on Image Processing,2010,19(8):1978-1986.
[47]Vese, L.A., Osher, S.J.. Color texture modeling and color image decomposition in a variational-PDE approach. IEEE Proceedings of the Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC'06), New York,2006:103-110.
[48]S. Dubois, M. Lugiez, R. P'eteri, and M. M'enard. Adding a noise component to a color decomposition model for improving color texture extraction. Proceedings CGIV 2008 and MCS08,2008:394-398.
[49]E. Tadmor, S. Nezzar and L. Vese. Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation. Communications in Mathematical Sciences,2008,6:281-307.
[50]V. Duval, J.-F. Aujol, and L. A. Vese. Mathematical modeling of textures: Application to color image decomposition with a projected gradient algorithm. J Math Imaging Vis.,2010,37:232-248.
[51]S. RANE, G. Sapiro, and M. Bertalmio. Structure and texture filling-in of missing image blocks in wireless transmission and compression applications. IEEE Tran Image Processing, Feb.2003,12(2):34-41.
[52]M. Bertalmio, L. VESE, G. Sapiro, et al. Simultaneous structure and texture image inpainting. IEEE Trans. Image Process., Aug.2003,12(8):882-889.
[53]S.Osher, A.Sole, L.Vese. Image decomposition and restoration using total variation minimization and the H-1 norm. Multiscale Model. Simul.,2003,1: 349-370.
[54]H. Yamauchi, J. Haber, H.-P. Seidel. Image Restoration using Multiresolution Texture Synthesis and Image Inpainting. IEEE Proceedings of the Computer Graphics International(CGI'03),2003:120-125.
[55]L. LIEU, L. VESE. Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim.,2008,58: 167-193.
[56]Andreu, C. Ball ester, V. Caselles, J. M. Mazon. Minimizing total variation flow. CRAS I—Mathematique,2000,331(11):867-872.
[57]L. vese. A Study in the BV space of a denosing-deblurring variationl problem. Applied Mathematics and Optimization,2001,44(2):131-161.
[2]T. F. Chan, J. H. Shen. Mathematical models for local non-texture inpaintings. SIAM J. APPL. MATH,2002,62:1019-1043.
[3]T. F. Chan, J. H. Shen. Non-texture inpainting by curvature driven diffusions(CDD). Vision Comm. Image Rep,2001,4:436-449.
[4]M. Oliveira, B. Bowen, R. McKenna, et al. Fast digital image inpainting. Proceedings of the Visualization, Imaging, and Image Processing IASTED Conference, Marbella, Spain,2001:261-266.
[5]T. F. Chan, J. Shen. Morphologically invariant PDE inpaintings. UCLA CAM Report,2001-15 at:www.math.ucla.edu/-imagers. IEEE Trans. Image process, 2002.
[6]Tsai A, Yezzi A J, Willsky A S. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation and magnification. IEEE Transactions on Image Processing,2001,10(8): 1169-1186.
[7]T. F. Chan, J. H. Shen. Mathematical models for local non-texture inpaintings. SIAM J. APPL. MATH,2002,62:1019-1043.
[8]Esedoglu S, Shen J H, Digital inpainting based on the Mumford-Shah-Euler image model. Euro, Journal of Applied Mathematics,2002(13):353-370.
[9]邵肖伟,刘政凯,宋璧.一种基于TV模型的自适应图像修复方法.电路与系统学报,2004,9：113-117.
[10]T. K. Shih, R. C. Chang. Digital inpainting-survey and multilayer image inpainting algorithms. Information Technology and Applications,2005,7(1): 15-24.
[11]Y. H. Fung, Y. H. Chan. A POCS-based restoration algorithm for restoring halftoned color-quantized images. Image Processing,2006,7(15):1985-1992.
[12]R. H. Chan, Y. W. Wen, and A. M. Yip. A fast optimization transfer algorithm for image inpainting in wavelet domains. IEEE Transactions on Image Processing, 2009,7(18):1467-1476.
[13]周廷方,汤峰,王进,等.基于径向基函数的图像修复技术.中国图像图形学报,2004,9：1190-1197.
[14]Efros, T. Leung. Texture synthesis by non-parametric sampling. In International Conference on Computer Vision,1999,1033-1038.
[15]S. Zelinka, M. Garland. Towards real-time texture synthesis with the jump map. Proceedings of the Thirteenth Eurographics Workshop on Rendering Techniques, 2002:99-104.
[16]Efros, W. T. Freeman. Image quilting for texture synthesis and transfer. Proceedings of SIGGRAPH 2001,2001:341-346.
[17]V. Kwatra, A. Schoedl, I. Essa, G. Turk, and A. Bobick. Graphcut textures: Image and video synthesis using graph cuts. Proceedings of SIGGRAPH 2003, 2003:277-286.
[18]Nealen, M. Alexa, Hybrid texture synthesis. Proceedings of the 14th Eurographics workshop on Rendering,2003:97-105.
[19]Drori, D. Cohen-Or, and Yeshurun, H. Fragment-based image completion. Proceedings of ACM SIGGRAPH 2003,2003:303-312.
[20]Criminisi, P. Perez, and K. Toyama, Object removal by exemplar-based inpainting. Proceedings of 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognitiont. Madison, Wisconsin,2003,2: 721-728.
[21]张岩,李文辉,孟宇,庞云阶.应用PSO的快速纹理合成算法.计算机研究与发展,2005,42(3)：424-430.
[22]Jiying Wu, Qiuqi Ruan. Object Removal By Cross Isophotes Exemplar-based Inpainting, IEEE, the 18th international conference on pattern recognition,2006.
[23]汪忠庆,檀结庆,何蕾.方向性纹理图像的修复算法.计算机工程与应用,2006,42(34)：29-32.
[24]Dongdong Nie, Lizhuang Ma, Shuangjiu Xiao. Similarity Based Image Inpainting Method. IEEE Transaction,2006,1-4244-0028-7/06.
[25]Y. B. Xue, H. Zhang, F. Y. Wang, et al. Exemplar-Based Image Completion Using Color Ratio Gradient. Image and Signal Processing,2008,5(3):569-572.
[26]M. Kuo, N. C. Yang, W. H. Chang, et al. Image Recovery Based on Effective Image Completion. Intelligent Information Hiding and Multimedia Signal Processing,2008,8:393-396.
[27]Yedidia, J. S., Freeman, W. T., and Weiss, Y.2002. Understanding belief propagation and its generalizations. Technical Report, Mitsubishi Electric Research Laboratories, MERL-TR-2001-22.
[28]C. Barcelos, M. Batista, A. Martins, and A. Nogueira. Level Lines Continuation based Digital Inpainting. IEEE Computer Graphics and Image Prcessing, October 2004,10:50-57.
[29]J. Sun, L. Yuan, J. Y. Jia, et al. Image Completion with Structure Propagation. ACM Transactions on Graphics,2005,24(3):861-868.
[30]A. Rares, M.J.T. Reinders, J. Biemond. Edge-Based Image Restoration[J]. IEEE Transactions on Image Processing,2005,14(10):1454-1468.
[31]Shen Minfen, Li Bin. Structure and texture image inpainting based on region segmentation//Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. Hawaii,2007:701-704.
[32]G. Hao, O. Nobutaka, and S. Shiqeki. A Structure-Synthesis Image Inpainting Algorithm Based on Morphological Erosion Operation. Image and Signal Processing,2008,5(3):530-535.
[33]Wang Minqin, Han Guoqiang, Tu Yongqiu. Edge-based image completing guided by region segmentation.//Proceedings of ISECS International Colloquium on Computing, Communication, Control, and Management. Guangzhou, China, 2008:152-156.
[34]H. F. Shen, L. P. Zhang. A MAP-Based Algorithm for Destriping and Inpainting of Remotely Sensed Images. IEEE Trans. Geosci. Remote Sens., May 2009, 5(47):1492-1502.
[35]L. Ruain, S. Osher, E. Fatemi. Nonlinear total variation based noise removal algorithm. Physica D,1992,60:259-268.
[36]Mumford, J. Shah. Optimal approximation by Piece2wise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics,1989,42(5):577-685.
[37]P. Perona, J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE Tram on PatternAnal Machine Intell,1990:12(7):629-639.
[38]Y. Meyer. Oscillating patterns in image processing and nonlinear evolution equatious. University Lecture Series,2001,22:1047-3998.
[39]Luminita A.Vese and Stanley J.Osher. Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing, Journal of Scientific Computing,2002,19(1-3):553-572.
[40]WRAY I. Integrating EAM and PdM formaintenance productivity. Plant Engineering,2003,57(10):65-66.
[41]S Levine. An adaptive variational model for image decomposition, in LNCS, 3757,2005:382-397.
[42]J F Aujol, A Chambolle. Dual norms and image decomposition models. Int. J. Comput. Vis.,2005,63(1):85-104.
[43]W. YIN, D. GOLDFARB, AND S. OSHER. Total variation-based image cartoon-texture decomposition. Columbia University CORC Report TR-2005-01, UCLA CAM Report,2005:05-27.
[44]Tadmor, S.Nezzar, L.Vese. A multiscale image representation using heirarchical (BV,L2) decompositions. Multiscale Model. Simul.,2004,2:554-579.
[45]J. B. Garnett, T. M. Le, Y. Meyer, and L. A. Vese. Image decompositions using bounded variation and generalized homogeneous Besov spaces. Appl. Comput. Harmon. Anal.,2007,23(1):25-56.
[46]Buades, T. Le, J.-M. Morel, and L. Vese. Fast cartoon+ texture image filters. IEEE Transactions on Image Processing,2010,19(8):1978-1986.
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