基于cartoon和texture的图像处理
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摘要
一般的图像复原问题通常可以表为：f=Au+ε,这里,f∈Rl表示已知的观察到的图像,u∈Rn表示未知的真实图像,ε为方差为σ的高斯白噪声,A∈Rl×n是一个线性算子.针对上述问题,本文做了两个部分的工作,分别为图像去噪和图像修复.
由于上述问题一般是不适定的,因此,我们通常使用正则化技巧将上述不适定问题转化为适定问题,再进行求解.在本文第二章第一节中,首先,我们列出了由正则化技巧得到的求解图像去噪问题的各项同性的ROF模型,此模型一般可以表为求解如下极值问题：这里,▽xu,▽yu表示u的微分算子,‖.‖1表示L1范数.其次,我们用分裂Bregman方法求解上述ROF模型.最后,我们给出数值实验结果,从ROF模型去噪的数值实验结果中,我们可以看出,全变差去噪模型虽然在有效去除噪声的同时也能较好地保持图像的边缘信息,但由于它对纹理和噪声的辨别能力弱,所以在去除噪声的过程中,部分纹理信息也被去除了.
在本文第二章第二节中,首先,我们列出在ROF模型的基础上进一步提取纹理细节的去噪方法,即Xiaoqun Zhang在[20]中提出的基于非局部全变差泛函的去噪方法,此方法利用图像的自相似性来合成纹理.一般可表为求解如下的极小值问题：这里,‖.‖表示L1范数,ω是从Ω×Ω到R的非负对称权函数.其次,我们用分裂Bregman方法求解上述极小值问题.最后,我们给出数值实验结果,从非局部全变差泛函去噪的数值实验结果中,我们可以看出,此方法在合成纹理的同时,也将噪声带回到图像中.
由于有些小波对纹理有较强的表示能力,因此,我们考虑在ROF模型的基础上使用小波更好地提取纹理细节,在本文第二章第三节中,首先,我们将小波方法和基于变分的全变差方法结合起来,提出了基于cartoon和texture的图像去噪模型,此模型可表为求解如下的极小值问题：其中,uC表示图像的cartoon部分,uT表示图像的texture部分,uC+uT表示去噪后得到的图像,w为可以稀疏表示图像的texture部分,但不能稀疏表示图像cartoon部分的框架.其次,同列出的前两个模型一样,我们用分裂Bregman方法求解上述极小值问题.再次,我们证明了此分裂Bregman迭代方法的收敛性.最后,我们给出数值实验结果,实验表明,本文提出的图像去噪方法可以在ROF模型的基础上更好地提取纹理细节,去噪效果显著.
由上面对基于cartoon和teture的图像去噪模型及一般图像复原问题正则化的讨论,在第三章,首先,我们提出本文的基于cartoon和texture的图像修复模型.此模型可表为求解如下问题：使得PA(uC+uT)= PAf.其中,uC表示图像的cartoon部分,UT表示图像的texture部分：uC+uT表示修复后得到的图像,w为可以稀疏表示图像的texture部分,但不能稀疏表示图像.cartoon部分的框架,PA为映射算子.其次,我们用分裂Bregman方法求解上述极小值问题.最后,我们给出数值实验结果,实验表明,本文提出的图像修复方法效果显著.
Image restoration is often formulated as an inverse problem:f=Au+ε, here,f∈Rl is the known observed image, u∈Rn is the unknown true iamge,εis a white Gaussian noise with varianceσ, A∈Rl×n is a linear operator.For the above questions,we done two part of the work in this paper,it is image denoising and image inpainting.
Since the above problems are typically ill posed,it is standard to use a regularization technique to make them well-posed. In Chapter 2 Section 1 of this paper,first,we give the isotropic ROF model got by the regularization technique to solve the problem of image denoising.The model can often be formulated as the following minimization problem: here, (?)χu. (?)yu is the difference operator of u,||.||1 denotes the L1 norm. Second,we give the split Bregman iteration of the ROF model. At last,we provide several numerical simulation,from the simulation result of the ROF denoising model,we can see,the TV model can effectively remove noise and preserve edges,but the model does not recognize well texture from noise.Therefore part of texture information was removed in the process of removing noise.
In Chapter 2 Section 2 of this paper,first,we give the denoising method which extract the detail of texture from the image got by the ROF model,it is the denoising method based on nonlocal total variation functional given by Xiaoqun Zhang in [20]. This method used similarities in images to synthesize textures and can often be formulated as the following minimization problem: here,||.|| stands for the L1 norm,ωis the nonnegative symmetric weight function fromΩ×Ωto R,second,we use the split Bregman method to solve it.At last,we provide several numerical simulation,from the simulation result of the nonlocal total variation functional denoising model,we can see,when this method synthesizes textures, the noise is brought back to the image.
Since some wavelet have been proved or verified to represent the texture part well,we consider using wavelet to extract the detail of texture from the image got by the ROF model.In Chapter 2 Section 3 of this paper, first,we combine the denoising method based on wavelet shrinkage with the method based on total variation,give the denoising model based on cartoon and texture. This model can be formulated as the following minimization problem: here, uc is the cartoon part of image, uT is the texture part of image, uC+uT is the denoised image, w is the framelet which can represent the texture part sparsely but not represent the cartoon part sparsely.Second,similar to the above two model,we use the split Bregman method to solve it.Third,we prove the convergence of the model.At last,we provide several numerical simulation.from the simulation result of the denoising model based on cartoon and texture,we can see,the denoising method provided in this paper can extract the detail of texture from the image got by the ROF model and get the better denoisng effect.
Due to the above discussion about the image denoising model based on cartoon and texture and the regularization of image restoration problem,in chapter 3,first,we provide the image inpainting model based on cartoon and texture,this model can be formulated as the following minimization problem: subject to PΛ(uC+uT)=PΛf. here, uc is the cartoon part of image, uT is the texture part of image, uC+UT is the restored image, w is the framelet which can represent the texture part sparsely but not represent the cartoon part sparsely, PΛis projection.Second,similar to the denoising model,we use the split Bregman method to solve it.At last,we provide several numerical simulation,from the simulation result of the image inpainting model based on cartoon and texture,we can see,the image inpainting method provided in this paper get better effect.
由于上述问题一般是不适定的,因此,我们通常使用正则化技巧将上述不适定问题转化为适定问题,再进行求解.在本文第二章第一节中,首先,我们列出了由正则化技巧得到的求解图像去噪问题的各项同性的ROF模型,此模型一般可以表为求解如下极值问题：这里,▽xu,▽yu表示u的微分算子,‖.‖1表示L1范数.其次,我们用分裂Bregman方法求解上述ROF模型.最后,我们给出数值实验结果,从ROF模型去噪的数值实验结果中,我们可以看出,全变差去噪模型虽然在有效去除噪声的同时也能较好地保持图像的边缘信息,但由于它对纹理和噪声的辨别能力弱,所以在去除噪声的过程中,部分纹理信息也被去除了.
在本文第二章第二节中,首先,我们列出在ROF模型的基础上进一步提取纹理细节的去噪方法,即Xiaoqun Zhang在[20]中提出的基于非局部全变差泛函的去噪方法,此方法利用图像的自相似性来合成纹理.一般可表为求解如下的极小值问题：这里,‖.‖表示L1范数,ω是从Ω×Ω到R的非负对称权函数.其次,我们用分裂Bregman方法求解上述极小值问题.最后,我们给出数值实验结果,从非局部全变差泛函去噪的数值实验结果中,我们可以看出,此方法在合成纹理的同时,也将噪声带回到图像中.
由于有些小波对纹理有较强的表示能力,因此,我们考虑在ROF模型的基础上使用小波更好地提取纹理细节,在本文第二章第三节中,首先,我们将小波方法和基于变分的全变差方法结合起来,提出了基于cartoon和texture的图像去噪模型,此模型可表为求解如下的极小值问题：其中,uC表示图像的cartoon部分,uT表示图像的texture部分,uC+uT表示去噪后得到的图像,w为可以稀疏表示图像的texture部分,但不能稀疏表示图像cartoon部分的框架.其次,同列出的前两个模型一样,我们用分裂Bregman方法求解上述极小值问题.再次,我们证明了此分裂Bregman迭代方法的收敛性.最后,我们给出数值实验结果,实验表明,本文提出的图像去噪方法可以在ROF模型的基础上更好地提取纹理细节,去噪效果显著.
由上面对基于cartoon和teture的图像去噪模型及一般图像复原问题正则化的讨论,在第三章,首先,我们提出本文的基于cartoon和texture的图像修复模型.此模型可表为求解如下问题：使得PA(uC+uT)= PAf.其中,uC表示图像的cartoon部分,UT表示图像的texture部分：uC+uT表示修复后得到的图像,w为可以稀疏表示图像的texture部分,但不能稀疏表示图像.cartoon部分的框架,PA为映射算子.其次,我们用分裂Bregman方法求解上述极小值问题.最后,我们给出数值实验结果,实验表明,本文提出的图像修复方法效果显著.
Image restoration is often formulated as an inverse problem:f=Au+ε, here,f∈Rl is the known observed image, u∈Rn is the unknown true iamge,εis a white Gaussian noise with varianceσ, A∈Rl×n is a linear operator.For the above questions,we done two part of the work in this paper,it is image denoising and image inpainting.
Since the above problems are typically ill posed,it is standard to use a regularization technique to make them well-posed. In Chapter 2 Section 1 of this paper,first,we give the isotropic ROF model got by the regularization technique to solve the problem of image denoising.The model can often be formulated as the following minimization problem: here, (?)χu. (?)yu is the difference operator of u,||.||1 denotes the L1 norm. Second,we give the split Bregman iteration of the ROF model. At last,we provide several numerical simulation,from the simulation result of the ROF denoising model,we can see,the TV model can effectively remove noise and preserve edges,but the model does not recognize well texture from noise.Therefore part of texture information was removed in the process of removing noise.
In Chapter 2 Section 2 of this paper,first,we give the denoising method which extract the detail of texture from the image got by the ROF model,it is the denoising method based on nonlocal total variation functional given by Xiaoqun Zhang in [20]. This method used similarities in images to synthesize textures and can often be formulated as the following minimization problem: here,||.|| stands for the L1 norm,ωis the nonnegative symmetric weight function fromΩ×Ωto R,second,we use the split Bregman method to solve it.At last,we provide several numerical simulation,from the simulation result of the nonlocal total variation functional denoising model,we can see,when this method synthesizes textures, the noise is brought back to the image.
Since some wavelet have been proved or verified to represent the texture part well,we consider using wavelet to extract the detail of texture from the image got by the ROF model.In Chapter 2 Section 3 of this paper, first,we combine the denoising method based on wavelet shrinkage with the method based on total variation,give the denoising model based on cartoon and texture. This model can be formulated as the following minimization problem: here, uc is the cartoon part of image, uT is the texture part of image, uC+uT is the denoised image, w is the framelet which can represent the texture part sparsely but not represent the cartoon part sparsely.Second,similar to the above two model,we use the split Bregman method to solve it.Third,we prove the convergence of the model.At last,we provide several numerical simulation.from the simulation result of the denoising model based on cartoon and texture,we can see,the denoising method provided in this paper can extract the detail of texture from the image got by the ROF model and get the better denoisng effect.
Due to the above discussion about the image denoising model based on cartoon and texture and the regularization of image restoration problem,in chapter 3,first,we provide the image inpainting model based on cartoon and texture,this model can be formulated as the following minimization problem: subject to PΛ(uC+uT)=PΛf. here, uc is the cartoon part of image, uT is the texture part of image, uC+UT is the restored image, w is the framelet which can represent the texture part sparsely but not represent the cartoon part sparsely, PΛis projection.Second,similar to the denoising model,we use the split Bregman method to solve it.At last,we provide several numerical simulation,from the simulation result of the image inpainting model based on cartoon and texture,we can see,the image inpainting method provided in this paper get better effect.
引文
[1]姚敏等.数字图像处理.机械工业出版社.2006.
[2]Jianfeng Cai, Stanley Osher, Zuowei Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation:A SIAM Interdisciplinary Journal 8(2), (2009),337-369.
[3]Rong-Qing Jia, Hanqing Zhao, Wei Zhao, Convergence analysis of the Bregman method for the variational model of image denoising, Appl. Comput. Harmon. Anal. (2009), doi:10.1016/j.acha.2009.05.002
[4]L. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization, USSR Comput. Math. Math. Phys.7 (1967) 200-217.
[5]M. Elad, J.-L. Starck, P. Querre, and D. L. Donoho,Simultaneous cartoon and texture im-age inpainting using morphological component analysis (MCA),Appl. Comput. Harmon. Anal.,19 (2005), pp.340-358
[6]J. Starck. M. Elad, D. Donoho, Image decomposition via the combination of sparse rep-resentation and a variational approach, IEEE Trans. Image Process.,14 (10),1570-1582
[7]L.Rudin,S.Osher,E.Fatemi,Nonlinear total variation based noise removal algo-rithms,Phys.D 60 (1992)259-268.
[8]Jianfeng Cai, Raymond Chan, and Zuowei Shen, Simultaneous cartoon and texture in-painting, Inverse Problem and Imaging 4 (3), (2010),379-395.
[9]I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math.,57 (2004), pp. 1413-1457.
[10]p.l.Combettes and V.R.Wajs,Signal recovery by proximal forward-backward split-ting,Multiscale Model.Simul.,4(2005),pp.1168-1200(electronic).
[11]W.Yin,S.Osher,D.Goldfarb,and J.Darbon,Bregman iterative algorithms for l1 minimizatio with applications to compressed sensing,SIAM J.Imaging Sci.,1(2008),pp.143-168.
[12]A.Buades,B.Coll,and J.M.Morel. A review of image denoising algorithms,with a new one.Multiscale Model.simul.,4(2),2005.6
[13]A.A.Efros and T.K.Leung. Texture synthesis by non-parametric sampling.In Proc.Int.Conf.Computer Vision,volume 2,pages 1022-1038,1999.6
[14]T. Chan, H. Zhou, Total variation improved wavelet thresholding in image compression, in:Proc. of ICIP'2000 16 (5),1289-1302 (2007).
[15]S. Durand, J. Froment, Reconstruction of wavelet coefficients using total variation mini-mization, SIAM J. Sci. Comput.,24,1754-1767 (2003)
[16]J. Starck, M. Elad, D. Donoho, Image decomposition via the combination of sparse rep-resentation and a variational approach, IEEE Trans. Image Process.,14 (10),1570-1582 (2005).
[17]E. Cand'es, F. Guo, New multiscale transforms, minimum total variation synthesis:Ap-plications to edge-preserving image reconstruction, Signal Process.,82 (11),1519-1543 (2002).
[18]R. Coifman, A. Sowa, Combining the calculus of variations and wavelets for image en-hancement, Appl. Comput. Harmon. Anal.,9,1-18 (2000).
[19]G. Easley, D. Labate, F. Colonna, Shearlet based total variation for denoising, IEEE Trans. Image Process.,18 (2),260-268 (2009).
[20]Zhang, X., Burger, M., Bresson, X., Osher, S., Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, UCLA CAM Report [09-03] 2009.
[21]T.Goldstein,S.Osher, The split Bregman method for L1 regularized problems,UCLA CAM Report (08-29).
[22]S.osher,Y.Mao,B.Dong and W.Yin,Fast linearized bregman iteration for compressed sens-ing and sparse denoising,2008.ULCA CAM Report(08-37).
[2]Jianfeng Cai, Stanley Osher, Zuowei Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation:A SIAM Interdisciplinary Journal 8(2), (2009),337-369.
[3]Rong-Qing Jia, Hanqing Zhao, Wei Zhao, Convergence analysis of the Bregman method for the variational model of image denoising, Appl. Comput. Harmon. Anal. (2009), doi:10.1016/j.acha.2009.05.002
[4]L. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization, USSR Comput. Math. Math. Phys.7 (1967) 200-217.
[5]M. Elad, J.-L. Starck, P. Querre, and D. L. Donoho,Simultaneous cartoon and texture im-age inpainting using morphological component analysis (MCA),Appl. Comput. Harmon. Anal.,19 (2005), pp.340-358
[6]J. Starck. M. Elad, D. Donoho, Image decomposition via the combination of sparse rep-resentation and a variational approach, IEEE Trans. Image Process.,14 (10),1570-1582
[7]L.Rudin,S.Osher,E.Fatemi,Nonlinear total variation based noise removal algo-rithms,Phys.D 60 (1992)259-268.
[8]Jianfeng Cai, Raymond Chan, and Zuowei Shen, Simultaneous cartoon and texture in-painting, Inverse Problem and Imaging 4 (3), (2010),379-395.
[9]I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math.,57 (2004), pp. 1413-1457.
[10]p.l.Combettes and V.R.Wajs,Signal recovery by proximal forward-backward split-ting,Multiscale Model.Simul.,4(2005),pp.1168-1200(electronic).
[11]W.Yin,S.Osher,D.Goldfarb,and J.Darbon,Bregman iterative algorithms for l1 minimizatio with applications to compressed sensing,SIAM J.Imaging Sci.,1(2008),pp.143-168.
[12]A.Buades,B.Coll,and J.M.Morel. A review of image denoising algorithms,with a new one.Multiscale Model.simul.,4(2),2005.6
[13]A.A.Efros and T.K.Leung. Texture synthesis by non-parametric sampling.In Proc.Int.Conf.Computer Vision,volume 2,pages 1022-1038,1999.6
[14]T. Chan, H. Zhou, Total variation improved wavelet thresholding in image compression, in:Proc. of ICIP'2000 16 (5),1289-1302 (2007).
[15]S. Durand, J. Froment, Reconstruction of wavelet coefficients using total variation mini-mization, SIAM J. Sci. Comput.,24,1754-1767 (2003)
[16]J. Starck, M. Elad, D. Donoho, Image decomposition via the combination of sparse rep-resentation and a variational approach, IEEE Trans. Image Process.,14 (10),1570-1582 (2005).
[17]E. Cand'es, F. Guo, New multiscale transforms, minimum total variation synthesis:Ap-plications to edge-preserving image reconstruction, Signal Process.,82 (11),1519-1543 (2002).
[18]R. Coifman, A. Sowa, Combining the calculus of variations and wavelets for image en-hancement, Appl. Comput. Harmon. Anal.,9,1-18 (2000).
[19]G. Easley, D. Labate, F. Colonna, Shearlet based total variation for denoising, IEEE Trans. Image Process.,18 (2),260-268 (2009).
[20]Zhang, X., Burger, M., Bresson, X., Osher, S., Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, UCLA CAM Report [09-03] 2009.
[21]T.Goldstein,S.Osher, The split Bregman method for L1 regularized problems,UCLA CAM Report (08-29).
[22]S.osher,Y.Mao,B.Dong and W.Yin,Fast linearized bregman iteration for compressed sens-ing and sparse denoising,2008.ULCA CAM Report(08-37).