基于稀疏表示的图像修补研究
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摘要
图像修补技术是图像处理的关键技术之一,被广泛地应用于生物视觉系统研究、计算机模式识别和医学等多个领域。目前,基于稀疏表示的图像修补研究是该领域近几年的一个新的研究分支,同时,稀疏表示的图像修补研究具有很重要的现实意义和广阔的研究前景,基于此,本文在以下几方面进行了研究,并有所收获：
1、系统地研究了MCA稀疏模型,实现了基于MCA模型的图像稀疏分解,并数值实现MCA和TV模型相结合的污损图像修补。更进一步指出MCA模型可以和CDD模型相结合修补污损图像,并算法实现。实验证明,MCA&CDD修补算法在一定程度上克服了MCA&TV的弊病,能够较好修补污损图像。
2、研究VO模型与Bregman迭代理论,利用Bregman迭代求解VO模型,实现基于VO模型的图像分解,并将VO模型、曲波变换、局域DCT变换与CDD模型相结合实现基于VO&CDD的稀疏图像修补,实验证明,VO&CDD可以较好地实现稀疏图像修补。
3、引入稀疏表示与全变分相结合的图像分解模型(SAT模型),并利用Bregman迭代方法求解该模型,将其应用于稀疏分解与稀疏图像修补。实验证明,SAT模型能够获得好的重构图像,并具有好的图像稀疏修补能力。
总之,本文重点研究了稀疏分解模型以及基于稀疏表示的图像修补方法,并算法实现了MCA&TV、MCA&CDD和VO&CDD修补算法,更进一步采用SAT模型实现图像稀疏分解与修补,分别获得好的修补结果。
As a key technology of image processing, image inpainting has been widely used in the research of biological vision systems, computer pattern recognition, medicine and other fields. At present, the research of image inpainting based on sparse representation is a new branch of this field in recent years. The image inpainting of sparse representation has very crucial practical significance and broad researching prospects. According to this, this paper mainly tries to do some research from the following aspects and concludes.
1. The paper systematically studies the Morphological Component Analysis (MCA) sparse model, achieves the sparse decomposition of image based on the model of MCA and deals with the dirty image inpainting by combining the model of MCA &TV. Furthermore, the paper points out that the model of MCA could be combined with CDD to achieve the same result. The experiment turned out that the combination of MCA&CDD overcomes the shortcomings of MCA&TV in some extent and has a better performance in scratched image inpainting.
2. The paper studies the VO model and the theory of Bregman iteration. Through bregman iterative, it solves VO model, thereby, achieve image sparse decomposition based on VO model. Combining the VO model, curvelet transform, local DCT transform with CDD model to achieve sparse image inpainting based on VO&CDD algorithm. Finally, the experiment shows that VO&CDD can achieve better sparse image inpainting.
3. The paper introduces a new image decomposition model, named SAT, which combines the sparse representation with total variation. Using bregman iteration method solves this model and then applies to sparse decompose image and inpainting. The test shows that the SAT model can obtain almost perfect reconstructed image and has a better ability to inpainting the sparse image.
In a word, this paper mainly studies the sparse decomposition model and the image inpainting based on sparse representation methods, and achieves the algorithms for the MCA&TV, MCA&CDD and VO&CDD inpainting and furthermore, achieves the image decomposition and sparse inpainting through the SAT model. Each of them carries out good performance.
1、系统地研究了MCA稀疏模型,实现了基于MCA模型的图像稀疏分解,并数值实现MCA和TV模型相结合的污损图像修补。更进一步指出MCA模型可以和CDD模型相结合修补污损图像,并算法实现。实验证明,MCA&CDD修补算法在一定程度上克服了MCA&TV的弊病,能够较好修补污损图像。
2、研究VO模型与Bregman迭代理论,利用Bregman迭代求解VO模型,实现基于VO模型的图像分解,并将VO模型、曲波变换、局域DCT变换与CDD模型相结合实现基于VO&CDD的稀疏图像修补,实验证明,VO&CDD可以较好地实现稀疏图像修补。
3、引入稀疏表示与全变分相结合的图像分解模型(SAT模型),并利用Bregman迭代方法求解该模型,将其应用于稀疏分解与稀疏图像修补。实验证明,SAT模型能够获得好的重构图像,并具有好的图像稀疏修补能力。
总之,本文重点研究了稀疏分解模型以及基于稀疏表示的图像修补方法,并算法实现了MCA&TV、MCA&CDD和VO&CDD修补算法,更进一步采用SAT模型实现图像稀疏分解与修补,分别获得好的修补结果。
As a key technology of image processing, image inpainting has been widely used in the research of biological vision systems, computer pattern recognition, medicine and other fields. At present, the research of image inpainting based on sparse representation is a new branch of this field in recent years. The image inpainting of sparse representation has very crucial practical significance and broad researching prospects. According to this, this paper mainly tries to do some research from the following aspects and concludes.
1. The paper systematically studies the Morphological Component Analysis (MCA) sparse model, achieves the sparse decomposition of image based on the model of MCA and deals with the dirty image inpainting by combining the model of MCA &TV. Furthermore, the paper points out that the model of MCA could be combined with CDD to achieve the same result. The experiment turned out that the combination of MCA&CDD overcomes the shortcomings of MCA&TV in some extent and has a better performance in scratched image inpainting.
2. The paper studies the VO model and the theory of Bregman iteration. Through bregman iterative, it solves VO model, thereby, achieve image sparse decomposition based on VO model. Combining the VO model, curvelet transform, local DCT transform with CDD model to achieve sparse image inpainting based on VO&CDD algorithm. Finally, the experiment shows that VO&CDD can achieve better sparse image inpainting.
3. The paper introduces a new image decomposition model, named SAT, which combines the sparse representation with total variation. Using bregman iteration method solves this model and then applies to sparse decompose image and inpainting. The test shows that the SAT model can obtain almost perfect reconstructed image and has a better ability to inpainting the sparse image.
In a word, this paper mainly studies the sparse decomposition model and the image inpainting based on sparse representation methods, and achieves the algorithms for the MCA&TV, MCA&CDD and VO&CDD inpainting and furthermore, achieves the image decomposition and sparse inpainting through the SAT model. Each of them carries out good performance.
引文
[1]乔雅丽.基于稀疏表示的图像去噪算法研究[D].北京：北京交通大学,2009
[2]章毓晋.图像处理和分析[M].北京：清华大学出版社,1999
[3]祝轩,周明全,耿国华等.曲率驱动与边缘停止相结合的非纹理图像修复[J].计算机科学,2008,35(12)：212-213
[4]邓承志.图像稀疏表示理论及其应用研究[D].武汉：华中科技大学,2008
[5]庄永文.基于稀疏编码理论的自然图像处理研究[D].厦门：厦门大学,2008
[6]Olshausen B.A., Field D.J. Emergence of simple—cell receptive field properties by learning a Sparse code for natural images [J]. Nature,1996,381:607-609
[7]张跃飞.基于稀疏分解的图像压缩[D].成都：西南交通大学,2003
[8]严德志.匹配追逐算法的研究及在信号稀疏表示中的应用[D].无锡：江南大学,2006
[9]Maloney L.T. Evaluation of linear models of surface spectral reflectance with small numbers of parameters [J]. J. Opt. Soc. Am. A,1986,3(10):1673-1683
[10]Bergeau E., Mallat S. Matching pursuit of images[C]. International Conference on Image Processing (ICIP'95).1995,1:53-56
[11]冯象初,甘小冰,宋国乡.数值泛函与小波理论[M].西安：西安电子科技大学出版社,2003
[12]Vinge WE., Gallant JL. Sparse coding and decorrelation in primary visual cortex during natural vision [J]. Science,2000,287:1273-1276
[13]Alliney S., A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing,1997,45(4): 913-917
[14]Wotao Yin, D. Goldfarb, S. Osher. A comparison of three total variation based texture extraction models [J]. J. Vis. Commun. Image R.,2007,18:240-252
[15]Starck J.L., M. Elad, and D. Donoho. Image decomposition via the combination of sparse representations and variational approach[J]. IEEE Trans. Image Process,2005, vol.14, no. 10:1570-1582
[16]Fadili M.J., Starck J.L., Bobin J., et al. Decomposition and Separation Using Sparse Representations:An Overview [J]. IEEE Trans. Image Process,2010,98(6):983-994
[17]李映,张艳宁,许星.基于信号稀疏表示的形态成分分析进展和展望[J].电子学报,2009,37(1)：987：990
[18]张红英,彭启琮.数字图像修复技术综述[J].中国图象图形学报,2007,12(1)：450-455
[19]J Bobin, Y Moudden, J L Starek, et al. Morphological diversity and source sepamtion [J]. IEEE Signal Processing Letters.2006,13(7):409-412
[20]M. Bertalmio, G. Sapiro, V. Caselles, et al. Image inpainting[J]. Computer Graphics Proceedings Annual Conference Series, ACM SIGGRAPH, New Orleans,2000,417-424
[21]T.Chan, J.Shen. Mathematical Models for Local Non-Texture Inpaintings [R]. UCLACAM TR00-11,2000
[22]T.Chan, J.Shen. Non-texture inpainting by curvature-driven diffusions (CDD)[J]. J. Visual Comm,2001,4 (12):436-449
[23]刘旭.数字图像修复技术的研究[D].上海：上海交通大学,2009
[24]Elad M, Starck J L, Querre P, et al. Simultaneous Cartoon and Texture Image inpainting Using Morphological Component Analysis(MCA) [J]. Applied and Computational Harmonic Analysis,2005,19(3):340-358
[25]Chan T F, Shen J H. Mathematical models for local non-texture inpainting [J]. SIAM Journal of Applied Mathematics,2001,62(3):1019-1043
[26]Criminisi A, P'erez P, Toyama K. Region filling and object removal by exemplar-based image inpainting [J]. IEEE Transactions on Image Processing.2004,13(9):1200-1212
[27]Kedar S, Qin c, Wangs z. Layered image inpainting based on image decomposition [J]. Journal of Shanghai University (English Edition),2007,11(6):580-584
[28]Y. Meyer. Oscillating patterns in image processing and nonlinear evolution equations [R]. Vol.22 of University Lecture Series, AMS,2002.
[29]Vese L., Osher S.. Modeling textures with total variation minimization and oscillating patterns in image processing [J]. Journal of Scientific Computing 2003,19(1-3):553-572
[30]Alliney S..A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing 1997,45(4): 913-917
[31]Wotao Yin, D Goldfarb, S Osher. A comparison of three total variation based texture extraction models [J]. J. Vis. Commun. Image R,2007,18:240-252
[32]Starck J.L., Elad M., Donoho D.. Image decomposition via the combination of sparse representations and variational approach[J]. IEEE Trans. Image Process,2005,14(10): 1570-1582
[33]Bruckstein A.M., Elad M., A generalized uncertainty principle and sparse representation in pairs of rn bases [J]. IEEE Trans. Image Process,2002,48:2558-2567
[34]Chen, Donolio, Saunders M.. Atomic decomposition by basis pursuit[J]. SIAM Journal On Scientific Computing,1999,20(1):33-61
[35]W. Pan, T. D. Bui and C. Y. Suen. Rotation invariant texture classification by ridgelet transform and frequency-orientation space decomposition[J]. Signal Processing,2008, 88(1):189-199
[36]Chan T., Shen J.. Non-Texture Inpalnfing by Culvature-Driv-on Diffusions(CDD)[R]. Technical Report CAM 00-35, Image processing Research Group, UCLA,2000
[37]Demanet L, Ying Lexing. Wave Atoms and Sparsity of Oscillatory Patterns[J]. Applied and Computational Harmonic Analysis,2007,23(3):368-387
[38]Fadili M J, Starck J L. EM algorithm for sparse representation based image inpainting [J]. IEEE Trans. Image Process,2005,2:61-64
[39]Vese L, Osher S, Modeling textures with total variation minimization and oscillating patterns in image processing [J]. Journal of Scientific Computing.2003,19(1-3): 553-572
[40]Shen B, Hu W, Zhang Y M, et al. Image inpainting via sparse representation[J]. IEEE International Conference on Acoustics,2009,43:697-700
[41]Bobin J, Starck J L, Fadili J, et al. Mor-phological component analysis:all adaptive thresholding strategy[J]. IEEE Transactiom on Image Processing,2007,16(11):2675-2681
[42]赖炎连,贺国平.最优化方法[M].北京：清华大学出版社,2008
[43]Cai J-F, Osher S, Shen Z.. Split Bregman Method And Frame Based Image Restoration[J]. IEEE Transactions on Signal Processing,2009,78(4):945-949
[44]J-F. Cai, S. Osher, Z. Shen. Convergence of the Linearized Bregman Iteration for (?)l-norm Minimization[J]. Math. Comp.78,2009:2127-2136
[45]T. Goldstein, S. Osher, The split Bregman algorithm for l1 regularized problems[J]. SIAM J. Imaging Sci.2009,2:323-343
[46]Cai J.F, Osher S, Shen Z.W. Split Bregman Methods And Frame Based Image Restorartioin[R]. UCLA CAM Report (09-28),2009
[47]L. Vese, S. Osher. Modeling textures with total variation minimization and oscillating patterns in image processing[J]. Journal of Scientific Computing2003,19 (1-3):553-572
[48]H.H. Bauschke, J. M. Borwein, P. L. Combettes. Bregman monotone optimization algorithms [J]. SIAM J. Control Optim.,2003,20:596-636
[49]S. Alliney. A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing,1997,45(4): 913-917
[50]李长洋.基于稀疏性的图像分层修复[D].成都：西南交通大学,2010
[2]章毓晋.图像处理和分析[M].北京：清华大学出版社,1999
[3]祝轩,周明全,耿国华等.曲率驱动与边缘停止相结合的非纹理图像修复[J].计算机科学,2008,35(12)：212-213
[4]邓承志.图像稀疏表示理论及其应用研究[D].武汉：华中科技大学,2008
[5]庄永文.基于稀疏编码理论的自然图像处理研究[D].厦门：厦门大学,2008
[6]Olshausen B.A., Field D.J. Emergence of simple—cell receptive field properties by learning a Sparse code for natural images [J]. Nature,1996,381:607-609
[7]张跃飞.基于稀疏分解的图像压缩[D].成都：西南交通大学,2003
[8]严德志.匹配追逐算法的研究及在信号稀疏表示中的应用[D].无锡：江南大学,2006
[9]Maloney L.T. Evaluation of linear models of surface spectral reflectance with small numbers of parameters [J]. J. Opt. Soc. Am. A,1986,3(10):1673-1683
[10]Bergeau E., Mallat S. Matching pursuit of images[C]. International Conference on Image Processing (ICIP'95).1995,1:53-56
[11]冯象初,甘小冰,宋国乡.数值泛函与小波理论[M].西安：西安电子科技大学出版社,2003
[12]Vinge WE., Gallant JL. Sparse coding and decorrelation in primary visual cortex during natural vision [J]. Science,2000,287:1273-1276
[13]Alliney S., A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing,1997,45(4): 913-917
[14]Wotao Yin, D. Goldfarb, S. Osher. A comparison of three total variation based texture extraction models [J]. J. Vis. Commun. Image R.,2007,18:240-252
[15]Starck J.L., M. Elad, and D. Donoho. Image decomposition via the combination of sparse representations and variational approach[J]. IEEE Trans. Image Process,2005, vol.14, no. 10:1570-1582
[16]Fadili M.J., Starck J.L., Bobin J., et al. Decomposition and Separation Using Sparse Representations:An Overview [J]. IEEE Trans. Image Process,2010,98(6):983-994
[17]李映,张艳宁,许星.基于信号稀疏表示的形态成分分析进展和展望[J].电子学报,2009,37(1)：987：990
[18]张红英,彭启琮.数字图像修复技术综述[J].中国图象图形学报,2007,12(1)：450-455
[19]J Bobin, Y Moudden, J L Starek, et al. Morphological diversity and source sepamtion [J]. IEEE Signal Processing Letters.2006,13(7):409-412
[20]M. Bertalmio, G. Sapiro, V. Caselles, et al. Image inpainting[J]. Computer Graphics Proceedings Annual Conference Series, ACM SIGGRAPH, New Orleans,2000,417-424
[21]T.Chan, J.Shen. Mathematical Models for Local Non-Texture Inpaintings [R]. UCLACAM TR00-11,2000
[22]T.Chan, J.Shen. Non-texture inpainting by curvature-driven diffusions (CDD)[J]. J. Visual Comm,2001,4 (12):436-449
[23]刘旭.数字图像修复技术的研究[D].上海：上海交通大学,2009
[24]Elad M, Starck J L, Querre P, et al. Simultaneous Cartoon and Texture Image inpainting Using Morphological Component Analysis(MCA) [J]. Applied and Computational Harmonic Analysis,2005,19(3):340-358
[25]Chan T F, Shen J H. Mathematical models for local non-texture inpainting [J]. SIAM Journal of Applied Mathematics,2001,62(3):1019-1043
[26]Criminisi A, P'erez P, Toyama K. Region filling and object removal by exemplar-based image inpainting [J]. IEEE Transactions on Image Processing.2004,13(9):1200-1212
[27]Kedar S, Qin c, Wangs z. Layered image inpainting based on image decomposition [J]. Journal of Shanghai University (English Edition),2007,11(6):580-584
[28]Y. Meyer. Oscillating patterns in image processing and nonlinear evolution equations [R]. Vol.22 of University Lecture Series, AMS,2002.
[29]Vese L., Osher S.. Modeling textures with total variation minimization and oscillating patterns in image processing [J]. Journal of Scientific Computing 2003,19(1-3):553-572
[30]Alliney S..A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing 1997,45(4): 913-917
[31]Wotao Yin, D Goldfarb, S Osher. A comparison of three total variation based texture extraction models [J]. J. Vis. Commun. Image R,2007,18:240-252
[32]Starck J.L., Elad M., Donoho D.. Image decomposition via the combination of sparse representations and variational approach[J]. IEEE Trans. Image Process,2005,14(10): 1570-1582
[33]Bruckstein A.M., Elad M., A generalized uncertainty principle and sparse representation in pairs of rn bases [J]. IEEE Trans. Image Process,2002,48:2558-2567
[34]Chen, Donolio, Saunders M.. Atomic decomposition by basis pursuit[J]. SIAM Journal On Scientific Computing,1999,20(1):33-61
[35]W. Pan, T. D. Bui and C. Y. Suen. Rotation invariant texture classification by ridgelet transform and frequency-orientation space decomposition[J]. Signal Processing,2008, 88(1):189-199
[36]Chan T., Shen J.. Non-Texture Inpalnfing by Culvature-Driv-on Diffusions(CDD)[R]. Technical Report CAM 00-35, Image processing Research Group, UCLA,2000
[37]Demanet L, Ying Lexing. Wave Atoms and Sparsity of Oscillatory Patterns[J]. Applied and Computational Harmonic Analysis,2007,23(3):368-387
[38]Fadili M J, Starck J L. EM algorithm for sparse representation based image inpainting [J]. IEEE Trans. Image Process,2005,2:61-64
[39]Vese L, Osher S, Modeling textures with total variation minimization and oscillating patterns in image processing [J]. Journal of Scientific Computing.2003,19(1-3): 553-572
[40]Shen B, Hu W, Zhang Y M, et al. Image inpainting via sparse representation[J]. IEEE International Conference on Acoustics,2009,43:697-700
[41]Bobin J, Starck J L, Fadili J, et al. Mor-phological component analysis:all adaptive thresholding strategy[J]. IEEE Transactiom on Image Processing,2007,16(11):2675-2681
[42]赖炎连,贺国平.最优化方法[M].北京：清华大学出版社,2008
[43]Cai J-F, Osher S, Shen Z.. Split Bregman Method And Frame Based Image Restoration[J]. IEEE Transactions on Signal Processing,2009,78(4):945-949
[44]J-F. Cai, S. Osher, Z. Shen. Convergence of the Linearized Bregman Iteration for (?)l-norm Minimization[J]. Math. Comp.78,2009:2127-2136
[45]T. Goldstein, S. Osher, The split Bregman algorithm for l1 regularized problems[J]. SIAM J. Imaging Sci.2009,2:323-343
[46]Cai J.F, Osher S, Shen Z.W. Split Bregman Methods And Frame Based Image Restorartioin[R]. UCLA CAM Report (09-28),2009
[47]L. Vese, S. Osher. Modeling textures with total variation minimization and oscillating patterns in image processing[J]. Journal of Scientific Computing2003,19 (1-3):553-572
[48]H.H. Bauschke, J. M. Borwein, P. L. Combettes. Bregman monotone optimization algorithms [J]. SIAM J. Control Optim.,2003,20:596-636
[49]S. Alliney. A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing,1997,45(4): 913-917
[50]李长洋.基于稀疏性的图像分层修复[D].成都：西南交通大学,2010