一种新的振荡纹理的TV模型
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摘要
本文主要分为两部分,分别为对已有图像去噪方法的讨论和一种新的针对振荡纹理的TV模型的提出。第一部分中对几个典型的去噪方法进行讨论,包括小波去噪领域中的双树复小波变换及其与隐马尔科夫模型相结合的去噪算法；Curvelet变换阈值去噪算法；Wave atoms变换阈值去噪算法；PDE模型去噪领域中的分裂Bregman迭代求解ROF模型去噪算法。对上述问题从理论上进行分析,包括小波基和模型的构造与算法的细节的讨论,并通过数值实践的结果对各类方法的做了比较,最后,基于对去噪问题的理解,提出了对去噪方法的一些展望。第二部分中提出一种新的针对振荡纹理图像的TV去噪模型,利用分裂Bregman迭代给出的求解方法,并给出了收敛性证明,较之传统TV模型,新模型对振荡纹理图像有较好的去噪和保持纹理的效果。
Image processing is widely used in astronomy, medicine, computer vision and many other fields. But the image will receive the noise pollution frequently. If people directly carry on other processing to the image with noise, such as edge detection, image segmentation, feature extraction, pattern recognition and so on, then the result is clearly unsatisfactory. For further processing, carring on the de-noising to the image is reliable guarantee. The image de-noising belonging to the technical category of the image restores is an ancient and challenging task.
With the variety of large-scale digital images and digital TV appearance, people need one kind of image restoration method which the condition necessary to be simple, real-time stronger, and effect better. However, most algorithms have not yet attained a desirable level of applicability.
For a long time, according to the actual characteristics of the image, the statistical characteristics of noise, and the frequency spectrum distributed rule, people has developed many de-noising methods. Generally speaking, the de-nosing method can be used to remove the noise, but, it will often lose the high-frequency characteristic of the target image, causing the edge and texture fuzziness. In other words, it has contradiction between noise reduction and edge、texture protection in the denoising process. Moreover, the algorithm complexity direct impact on the effectiveness of its practical applications, and is also the factor which we need to consider.
This article mainly divides into two parts, respectively be the discussion of the image denoising method and the statement of a new TV model for the oscillation texture. In the first part, this article mainly carries on the discussion to several typical de-noising methods, including the dual-tree complex wavelet combine with the hidden Markov model denoising, Curvelet transform threshold denoising, wave stoms transform threshold denoising, PDE model combine with split Bregman method denoising. This article carries on the theoretical analysis to the above question, including the construction of the basic elements and the discussion of the details of the algorithms. Then we achieve the results through numerical implementation and sum up the advantages and disadvantages and associated research tendency of each kind of the method. At the end, the future trend of image de-noising is pointed out, though, in personal opinion. In the second part, we propose a new TV model for the oscillation texture. The solution method is given by using split Bregman iteration. Then we give the proof of the astringency. Compared with the traditional TV model, the new model has a good denoising effect and can maintain the texture better.
In the chapter one of this paper, we outline the significance of image de-noising, as well as domestic and foreign scholars for research to image denoising. We classify and discuss all kinds of denoising method, including linear filtering and nonlinear filtering algorithm, and wavelet based denoising algorithm, as well as PDE based image denosing algorithm.
In the chapter two, we first summarize the mathematical essence of wavelet transform denoising. The wavelet-based denoising approach can be sum up as follows:
1) uses the appropriate wavelet function space, in order to achieve the best image representation.
2) formulates the appropriate metrics, namely, carries on corresponding processing to the wavelet coefficient, in order to achieve the best denoising results.
Then we respectively discusse the dual-tree complex wavelet transform denoising, Curvelet transform denoising and wave atoms transform denoising, form the construction of one-dimensional wavelet bases to the construction of two-dimensional wavelet bases, then, to the discretization, the fast algorithm and the numerical implementation. Separable wavelet, dual-tree complex wavelet, curvelet and wave atoms have basically reflected the various stages of the development of the wavelet basis. DT-CWT has multi-scale, approximate shift-invariant and multi-directional nature, which makes them easily maintain the edge features in image denoising, compared with may separate wavelet transform. But there is limitation that the direction is insufficient. Curvelet has the directional parameters, achieve the heterosexual relationship between scales, and its approaching rates superior to the traditional wavelet transform. However, for more complex images, such as the texture image, curvelet transform can not reach optimal. The wave atoms transform has given the vibration function most optimal sparse representation, therefore achieves good results in maintaining the texture. Although the wavelet transform develops gradually in the image characteristic representation, there is big redundancy in the de-noising applications, and the computing time is long. Wavelet transform denoising is difficult to achieve optimal. If we use wavelet threshold denoising, the denoisied image will has seriously pseudo-Gibbs phenomena. Because of the multi-directional of the basis, the processed image appears scratches, and this phenomena greatly reduce the de-noising effect. If we use the complex model to carry on coefficient processing, the operation complexity will be increased, but it can maintain the texture better, moreover the computer operating speed enhanced unceasingly has provided the feasibility for this reason.
In the chapter three, we first summarize the PDE based denoising methods. We divide it into axiomatic approach based on partial differential equation, geometric partial differential equation method and minimization of energy function method based on variation method. Then we focused on the variational-based methods, including the model proposed and development, as well as solving methods. We have theoretically analyzed the advantages and disadvantages of such methods.
After again, we introduces splits Bregman method, and solve the ROE denoising model with it. Finally, adoption of numerical experimentation, we prove that RDE model denoising can maintain image edge well, but for the image texture features, it falls short. Using split Bregman method to solve the ROE denoising model demonstrated efficient.
In the chapter four, we propose a novel model for the oscillation texture image——TV model for oscillatory patterns.
The fifth chapter is a summary of the above three chapters and some future prospects: Whichμis the weight determined to the experience,Du is the local direction variation, namely, the partial difference direction is the texture derivative direction. Then we give the solution method by using the split Bregman iteration.
And eventually solution procedure translates into the following iteration step:
Then we give the proof of the astringency.
Theorem 4.1 Let "u be the unique solution of the minimization problem(4.1). For
The split Bregman iteration greatly increased the speed of the solution. The effectiveness is guaranteed, moreover, the nature of the model itself had decided that the model is simple and effective in image denoising application of oscillation texture image.
The fourth chapter is a summary of the above three chapters and some future prospects. Considering the advantages and disadvantages of the total variation-based model and the wavelet transform denoising, the TV-synthesized wavelet transform will be one of focus of the future research. If split the Bregman iteration and the wavelet fast algorithm unifies, that will definitely raise the operating speed greatly.
In the near future, with the increase in computer performance, high-precision denoising model and the algorithm will have better prospects.
Image processing is widely used in astronomy, medicine, computer vision and many other fields. But the image will receive the noise pollution frequently. If people directly carry on other processing to the image with noise, such as edge detection, image segmentation, feature extraction, pattern recognition and so on, then the result is clearly unsatisfactory. For further processing, carring on the de-noising to the image is reliable guarantee. The image de-noising belonging to the technical category of the image restores is an ancient and challenging task.
With the variety of large-scale digital images and digital TV appearance, people need one kind of image restoration method which the condition necessary to be simple, real-time stronger, and effect better. However, most algorithms have not yet attained a desirable level of applicability.
For a long time, according to the actual characteristics of the image, the statistical characteristics of noise, and the frequency spectrum distributed rule, people has developed many de-noising methods. Generally speaking, the de-nosing method can be used to remove the noise, but, it will often lose the high-frequency characteristic of the target image, causing the edge and texture fuzziness. In other words, it has contradiction between noise reduction and edge、texture protection in the denoising process. Moreover, the algorithm complexity direct impact on the effectiveness of its practical applications, and is also the factor which we need to consider.
This article mainly divides into two parts, respectively be the discussion of the image denoising method and the statement of a new TV model for the oscillation texture. In the first part, this article mainly carries on the discussion to several typical de-noising methods, including the dual-tree complex wavelet combine with the hidden Markov model denoising, Curvelet transform threshold denoising, wave stoms transform threshold denoising, PDE model combine with split Bregman method denoising. This article carries on the theoretical analysis to the above question, including the construction of the basic elements and the discussion of the details of the algorithms. Then we achieve the results through numerical implementation and sum up the advantages and disadvantages and associated research tendency of each kind of the method. At the end, the future trend of image de-noising is pointed out, though, in personal opinion. In the second part, we propose a new TV model for the oscillation texture. The solution method is given by using split Bregman iteration. Then we give the proof of the astringency. Compared with the traditional TV model, the new model has a good denoising effect and can maintain the texture better.
In the chapter one of this paper, we outline the significance of image de-noising, as well as domestic and foreign scholars for research to image denoising. We classify and discuss all kinds of denoising method, including linear filtering and nonlinear filtering algorithm, and wavelet based denoising algorithm, as well as PDE based image denosing algorithm.
In the chapter two, we first summarize the mathematical essence of wavelet transform denoising. The wavelet-based denoising approach can be sum up as follows:
1) uses the appropriate wavelet function space, in order to achieve the best image representation.
2) formulates the appropriate metrics, namely, carries on corresponding processing to the wavelet coefficient, in order to achieve the best denoising results.
Then we respectively discusse the dual-tree complex wavelet transform denoising, Curvelet transform denoising and wave atoms transform denoising, form the construction of one-dimensional wavelet bases to the construction of two-dimensional wavelet bases, then, to the discretization, the fast algorithm and the numerical implementation. Separable wavelet, dual-tree complex wavelet, curvelet and wave atoms have basically reflected the various stages of the development of the wavelet basis. DT-CWT has multi-scale, approximate shift-invariant and multi-directional nature, which makes them easily maintain the edge features in image denoising, compared with may separate wavelet transform. But there is limitation that the direction is insufficient. Curvelet has the directional parameters, achieve the heterosexual relationship between scales, and its approaching rates superior to the traditional wavelet transform. However, for more complex images, such as the texture image, curvelet transform can not reach optimal. The wave atoms transform has given the vibration function most optimal sparse representation, therefore achieves good results in maintaining the texture. Although the wavelet transform develops gradually in the image characteristic representation, there is big redundancy in the de-noising applications, and the computing time is long. Wavelet transform denoising is difficult to achieve optimal. If we use wavelet threshold denoising, the denoisied image will has seriously pseudo-Gibbs phenomena. Because of the multi-directional of the basis, the processed image appears scratches, and this phenomena greatly reduce the de-noising effect. If we use the complex model to carry on coefficient processing, the operation complexity will be increased, but it can maintain the texture better, moreover the computer operating speed enhanced unceasingly has provided the feasibility for this reason.
In the chapter three, we first summarize the PDE based denoising methods. We divide it into axiomatic approach based on partial differential equation, geometric partial differential equation method and minimization of energy function method based on variation method. Then we focused on the variational-based methods, including the model proposed and development, as well as solving methods. We have theoretically analyzed the advantages and disadvantages of such methods.
After again, we introduces splits Bregman method, and solve the ROE denoising model with it. Finally, adoption of numerical experimentation, we prove that RDE model denoising can maintain image edge well, but for the image texture features, it falls short. Using split Bregman method to solve the ROE denoising model demonstrated efficient.
In the chapter four, we propose a novel model for the oscillation texture image——TV model for oscillatory patterns.
The fifth chapter is a summary of the above three chapters and some future prospects: Whichμis the weight determined to the experience,Du is the local direction variation, namely, the partial difference direction is the texture derivative direction. Then we give the solution method by using the split Bregman iteration.
And eventually solution procedure translates into the following iteration step:
Then we give the proof of the astringency.
Theorem 4.1 Let "u be the unique solution of the minimization problem(4.1). For
The split Bregman iteration greatly increased the speed of the solution. The effectiveness is guaranteed, moreover, the nature of the model itself had decided that the model is simple and effective in image denoising application of oscillation texture image.
The fourth chapter is a summary of the above three chapters and some future prospects. Considering the advantages and disadvantages of the total variation-based model and the wavelet transform denoising, the TV-synthesized wavelet transform will be one of focus of the future research. If split the Bregman iteration and the wavelet fast algorithm unifies, that will definitely raise the operating speed greatly.
In the near future, with the increase in computer performance, high-precision denoising model and the algorithm will have better prospects.
引文
[1]孙即祥编著.图像分析[M].北京：科学出版社,2005
[2]L Rudin, S Osher, E Fatemi. Nonlinear total variation based noise removal algorithms[M]. Physica D,1992,60:259-268.
[3]T Goldstein, S Osher. The split Bregman method for L1-regularized problem[R]. UCLA CAM Report,2008.math.ucla.edu.
[4]焦李成,侯彪,王爽,刘芳.图像多尺度几何分析[M].西安：西安电子科技大学出版社.2008
[5]S Mallat, W L Hwang. Singularity detection and processing with wavelets[J]. IEEE Transactions on Information Theory,1992,38(2):617-643.
[6]D L Donoho. De-noising by soft thresholding[J]. IEEE Trans. Inform Theory,vol.41, pp.613-627, May 1995.
[7]D L Donoho, I M Johnston. Ideal spatial adaptation by wavelet shrindage[J]. Biometrika,vol.81, no.3,pp,425-455,1994.
[8]M S Crouse, R Baraniuk. Wavelet-based Statistical Signal Processing Using Hidden Markov Models[J]. IEEE Trans. on Signal Processing,1998,46(4):886-902
[9]Li Xuchao, Zhu Shanan. Image Denoising Based on Wavelet Domain Spatial Context Modeling [A]//The 6th World Congress on Intelligent Control and Automation[C]. Dalian,2006:9 504-9 508.
[10]N G Kingsbuty. The dual-tree complex wavelet transform:a new technique for shift invariance and directional filters [J]. IEEE Digital Signal Processing Workshop, DSP98, Bryce Canyon,1998:86.
[11]N G Kingsbuty. Shift invariant properties of the Dual-Tree complex wavelet transform[J]. In Proc. IEEE Conf. on Acoustics, Speech and Signal Processing, Phoecix,AZ,1999.
[12]I W Selesnick. The Double-Density Dual-Tree DWT [J]. IEEE Transactions on Signal Processing, 2004,52(5):1304-1314.
[13]E J Candes. Ridgelets:theory and applications[D]. Stanford University,1998.
[14]E J Candes. Monoscale ridgelets for the representation of images with edges [R]. Stanford University,1999.
[15]D L Donoho. Orthonormal ridgelet and linear singularities [J]. SIAM Journal on Mathematical Analysis,2000,31(5):1062-1099.
[16]E J Candes, D L Donoho. Curvelets-A surprisingly effective nonadaptive representation for objects with edges [C]. In Curves and Surfaces Fitting:Saint-Malo1999.
[17]E J Candes, D L Donoho. New tight frames of curvelets and optimal representation of objects with C2 singularities [J]. Commun. on Pure and Appl-Math,2004,57(2)-219-266.
[18]M Fadili, J Starck. Curvelets and ridgelets, in:Encyclopedia of Complexity and Systems Science, Meyers, Robert(Ed.), Springer New York, in press.
[19]M N Do, M Vetterli. Contourlets. J. Stoeckler and G. V. Welland, Eds. Beyond Wavelets, Academic Press, New York,2002, to appear.
[20]M N Do, M Vetterli. Framing pyramids[J]. IEEE Trans,2003, Signal Proc-51(9):2329-2342
[21]Y Lu, M N Do. CRISP-contourlets:a critically sampled directional multiresolution image representation. Proc. SPIE Conf. on Wavelets X, San Diego, Aug.2003.
[22]E L Pennec, S Mallat. Sparse geometric image representation with bandelets [J]. IEEE Transactions on Image Processing,2005,14(4):423-438.
[23]K Guo, D Labate. Optimally sparse multidimensional representation using Shearlets [J]. SIAM Journal on Mathematical Analysis,2007,39(1):298-318.
[24]E J Candes, L Demanet, D L Donoho, et al. Fast Discrete Curvelet Transforms [R]. Applied and Computational Mathematics. California Institute of Technology,2005,1-43.
[25]L Demanet, L X Ying. Wave atoms and sparsity of oscillatory patterns[J]. Applied and Computational Harmonic Analysis,2007,23(3):368-387
[26]L Villemoes. Wavelet packets with uniform time-frequency location[J]. Comptes-Rendus Mathematique,2002,335(10):793-796.
[27]P Pernoa, J Malik. Scale space and edge detection using anisotropic diffusion[J]. IEEE Transactions on Pattern Analysis and machine Intelligence,1990,12(7):619-639.
[28]F Catte, T Coll, P Lions, J Morel. Image selective smoothing and edge detection by nonlinear diffusion[J]. SIAM Journal on Numerical Analysis,1992,29:182-193.
[29]R Malladi, J Sethian. Image processing:flows under min/max curvature and mean curvature. Graphical models and image processing,1996,58(2):127-141.
[30]N Sochen, R Kimmel, R Malladi. A general framework for low level vision[J]. IEEE Transactions on Image Processing,1998,7(3):310-318.
[31]S H Lee, J Seo. Noise removal with gauss curvature-driven diffusion[J]. IEEE Transactions on Image Processing,2005,14(7):940-909.
[32]A N Tikhonov, V Y Arsenin. Solutions of ill-posed Problems[D]. Winston & Sons, Washington, 1977.
[33]A Chambolle, P Lions. Image recovery via total variation minimization and related problems[J]. Numerische Mathematik,1997,76(2):167-188.
[34]G Aubert, L Vese. A variational method in image recovery[J]. SIAM Journal on Numerical Analysis,1997,34(5):1948-1979
[35]T Chan, A Marquina, P Mulet. High-order total variation-based image restoration[J]. SIAM Journal on Scientific Computing,2000,22(2):503-516.
[36]Tony F. Chan, Gene H. Golub, and Pep Mulet. A nonlinear primal-dual method for total variationbased image restoration[J]. SIAM J. Sci. Comput:1999,20:1964-1977.
[37]C Vogel. A multigrid method for total variation-based image denoising[R]. Computation and Control IV, Progress in Systems and Control Theory,20, Birkhauser,1995。
[38]M Lysaker, S Osher, X C Tai. Noise removal using smoothed normals and surface fitting[J]. IEEE Transection on image processing,2004,13(10):1345-1457.
[39]M Burger, S Osher, J Xu, G Gilboa. Nonlinear inverse scale space methods for image restoration[J]. VLSM, LNCS,2005,3752:85-96.
[40]J Nocedal, S Wright. Numerical Optimization. Springer Verlag,2006.
[41]S Osher, M Burger, D Goldfarb, et. al. An iterative regularization method for total variation-based image restoration[J]. MMS,4:460-489,2005.
[42]S Boyd, L Vandenberghe. Coves Optimization[M]. Cambridge:Cambridege University Press.2004.
[43]Y Wang, W Yin, and Y Zhang. A fast algorithm for image deblurring with total variation regularization[R]. CAAM Technical Reports,2007.
[44]W Yin, S Osher, D Goldfarb, and J Darbon. Bregman iterative algorithms for L1-minimization with applications to compressed sensing. Siam J. Imaging Science,1:142-168,2008.
[45]褚标.小波理论在图像去噪与纹理分析中的应用研究[D].合肥工业大学,2008.
[46]李波.基于PDE的图像去噪、修补及分解研究[D].大连理工大学,2008.
[2]L Rudin, S Osher, E Fatemi. Nonlinear total variation based noise removal algorithms[M]. Physica D,1992,60:259-268.
[3]T Goldstein, S Osher. The split Bregman method for L1-regularized problem[R]. UCLA CAM Report,2008.math.ucla.edu.
[4]焦李成,侯彪,王爽,刘芳.图像多尺度几何分析[M].西安：西安电子科技大学出版社.2008
[5]S Mallat, W L Hwang. Singularity detection and processing with wavelets[J]. IEEE Transactions on Information Theory,1992,38(2):617-643.
[6]D L Donoho. De-noising by soft thresholding[J]. IEEE Trans. Inform Theory,vol.41, pp.613-627, May 1995.
[7]D L Donoho, I M Johnston. Ideal spatial adaptation by wavelet shrindage[J]. Biometrika,vol.81, no.3,pp,425-455,1994.
[8]M S Crouse, R Baraniuk. Wavelet-based Statistical Signal Processing Using Hidden Markov Models[J]. IEEE Trans. on Signal Processing,1998,46(4):886-902
[9]Li Xuchao, Zhu Shanan. Image Denoising Based on Wavelet Domain Spatial Context Modeling [A]//The 6th World Congress on Intelligent Control and Automation[C]. Dalian,2006:9 504-9 508.
[10]N G Kingsbuty. The dual-tree complex wavelet transform:a new technique for shift invariance and directional filters [J]. IEEE Digital Signal Processing Workshop, DSP98, Bryce Canyon,1998:86.
[11]N G Kingsbuty. Shift invariant properties of the Dual-Tree complex wavelet transform[J]. In Proc. IEEE Conf. on Acoustics, Speech and Signal Processing, Phoecix,AZ,1999.
[12]I W Selesnick. The Double-Density Dual-Tree DWT [J]. IEEE Transactions on Signal Processing, 2004,52(5):1304-1314.
[13]E J Candes. Ridgelets:theory and applications[D]. Stanford University,1998.
[14]E J Candes. Monoscale ridgelets for the representation of images with edges [R]. Stanford University,1999.
[15]D L Donoho. Orthonormal ridgelet and linear singularities [J]. SIAM Journal on Mathematical Analysis,2000,31(5):1062-1099.
[16]E J Candes, D L Donoho. Curvelets-A surprisingly effective nonadaptive representation for objects with edges [C]. In Curves and Surfaces Fitting:Saint-Malo1999.
[17]E J Candes, D L Donoho. New tight frames of curvelets and optimal representation of objects with C2 singularities [J]. Commun. on Pure and Appl-Math,2004,57(2)-219-266.
[18]M Fadili, J Starck. Curvelets and ridgelets, in:Encyclopedia of Complexity and Systems Science, Meyers, Robert(Ed.), Springer New York, in press.
[19]M N Do, M Vetterli. Contourlets. J. Stoeckler and G. V. Welland, Eds. Beyond Wavelets, Academic Press, New York,2002, to appear.
[20]M N Do, M Vetterli. Framing pyramids[J]. IEEE Trans,2003, Signal Proc-51(9):2329-2342
[21]Y Lu, M N Do. CRISP-contourlets:a critically sampled directional multiresolution image representation. Proc. SPIE Conf. on Wavelets X, San Diego, Aug.2003.
[22]E L Pennec, S Mallat. Sparse geometric image representation with bandelets [J]. IEEE Transactions on Image Processing,2005,14(4):423-438.
[23]K Guo, D Labate. Optimally sparse multidimensional representation using Shearlets [J]. SIAM Journal on Mathematical Analysis,2007,39(1):298-318.
[24]E J Candes, L Demanet, D L Donoho, et al. Fast Discrete Curvelet Transforms [R]. Applied and Computational Mathematics. California Institute of Technology,2005,1-43.
[25]L Demanet, L X Ying. Wave atoms and sparsity of oscillatory patterns[J]. Applied and Computational Harmonic Analysis,2007,23(3):368-387
[26]L Villemoes. Wavelet packets with uniform time-frequency location[J]. Comptes-Rendus Mathematique,2002,335(10):793-796.
[27]P Pernoa, J Malik. Scale space and edge detection using anisotropic diffusion[J]. IEEE Transactions on Pattern Analysis and machine Intelligence,1990,12(7):619-639.
[28]F Catte, T Coll, P Lions, J Morel. Image selective smoothing and edge detection by nonlinear diffusion[J]. SIAM Journal on Numerical Analysis,1992,29:182-193.
[29]R Malladi, J Sethian. Image processing:flows under min/max curvature and mean curvature. Graphical models and image processing,1996,58(2):127-141.
[30]N Sochen, R Kimmel, R Malladi. A general framework for low level vision[J]. IEEE Transactions on Image Processing,1998,7(3):310-318.
[31]S H Lee, J Seo. Noise removal with gauss curvature-driven diffusion[J]. IEEE Transactions on Image Processing,2005,14(7):940-909.
[32]A N Tikhonov, V Y Arsenin. Solutions of ill-posed Problems[D]. Winston & Sons, Washington, 1977.
[33]A Chambolle, P Lions. Image recovery via total variation minimization and related problems[J]. Numerische Mathematik,1997,76(2):167-188.
[34]G Aubert, L Vese. A variational method in image recovery[J]. SIAM Journal on Numerical Analysis,1997,34(5):1948-1979
[35]T Chan, A Marquina, P Mulet. High-order total variation-based image restoration[J]. SIAM Journal on Scientific Computing,2000,22(2):503-516.
[36]Tony F. Chan, Gene H. Golub, and Pep Mulet. A nonlinear primal-dual method for total variationbased image restoration[J]. SIAM J. Sci. Comput:1999,20:1964-1977.
[37]C Vogel. A multigrid method for total variation-based image denoising[R]. Computation and Control IV, Progress in Systems and Control Theory,20, Birkhauser,1995。
[38]M Lysaker, S Osher, X C Tai. Noise removal using smoothed normals and surface fitting[J]. IEEE Transection on image processing,2004,13(10):1345-1457.
[39]M Burger, S Osher, J Xu, G Gilboa. Nonlinear inverse scale space methods for image restoration[J]. VLSM, LNCS,2005,3752:85-96.
[40]J Nocedal, S Wright. Numerical Optimization. Springer Verlag,2006.
[41]S Osher, M Burger, D Goldfarb, et. al. An iterative regularization method for total variation-based image restoration[J]. MMS,4:460-489,2005.
[42]S Boyd, L Vandenberghe. Coves Optimization[M]. Cambridge:Cambridege University Press.2004.
[43]Y Wang, W Yin, and Y Zhang. A fast algorithm for image deblurring with total variation regularization[R]. CAAM Technical Reports,2007.
[44]W Yin, S Osher, D Goldfarb, and J Darbon. Bregman iterative algorithms for L1-minimization with applications to compressed sensing. Siam J. Imaging Science,1:142-168,2008.
[45]褚标.小波理论在图像去噪与纹理分析中的应用研究[D].合肥工业大学,2008.
[46]李波.基于PDE的图像去噪、修补及分解研究[D].大连理工大学,2008.