图像恢复中的数值计算方法
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摘要
图像作为人类感知世界的视觉基础,是人类获取信息、表达信息和传递信息的重要手段。在过去的几十年里,我们见证了一个影像科学的时代。从卫星图像、X光成像,到现代医学中的电脑断层扫描(CT)、核磁共振成像(MRI)和正电子发射断层显像(PET),数字图像越来越普遍和重要,同时也得到了越来越多的研究和成果。总的来说,图像处理包括图像增强、图像恢复(或图像复原)、图像修补、图像分割等方面。
本文主要研究基于基本数学模型上的图像恢复问题的数值计算方法。首先,简单介绍数字图像的基本知识和涉及到的两个基本数学模型,并且简单介绍图像处理中的反问题和求解反问题的正则化方法。
其次,研究预处理方法在半二次图像恢复问题中的应用。图像恢复是图像处理中的基本问题,除了采用恢复图像的各种滤波方法之外,常常应用正则化方法将问题等价于一个优化问题,目标函数包含一个数据保真项和正则项。特别地,半二次正则项能够有效恢复图像的基本特征,因此我们的目标函数采用半二次正则项。我们利用牛顿法求解该优化问题,在每步牛顿迭代中,都有一个结构化的、系数矩阵是对称正定的线性方程组,从而可以利用预处理共轭梯度法求解这个方程组。我们提出了两个有效的预处理算子,并且给出一些理论结果。数值试验表明预处理共轭梯度法求解半二次图像恢复问题是有效的、快速的。
同时,本文还研究乘性噪声下图像恢复的快速算法。乘性噪声下图像恢复问题是应用数学中的一个富有挑战性的问题,近年来吸引了很多的关注和研究。受到前人对脉冲噪声下的图像恢复的方法的启示,我们提出一个快速的两步方法解决乘性噪声下的图像恢复问题。该方法的第一步,利用非局部滤波来降低乘性噪声,然后利用正则化方法将问题转换为一个优化问题,目标函数由混合保真项和一个总变差正则项组成。我们利用著名的交替迭代法求解这个优化问题,并证明算法至少线性收敛。数值试验表明两步法求解乘性噪声下图像恢复问题是非常有效的,在恢复图像的质量和速度上都优于目前其它方法。
Images, as an important bridge of the human and the world, play an significant role in human's obtaining information, expressing information and transmitting information. In the last few decades, we have witnessed an era of imaging sciences. From satellite imaging, X-ray imaging to modern medical computer tomography (CT), magnetic reso-nance imaging (MRI) and positron emission tomography (PET), digital images became more and more common and important. In general, image processing includes image enhancement, image restoration, image inpainting, image segmentation, etc.
In this paper, we focus on the numerical methods in image restoration problems. First of all, we simply introduce some basic concepts of digital image and two basic mathematical models, then study the inverse problem and regularization method.
Then, we apply the preconditioned conjugate gradient method to solve the half-quadratic regularization image restoration problem. Image restoration is the fundamental problem in image processing area. Except for many different filters applied to obtain an restored image, a degraded image can often be recovered efficiently by solving a mini-mization function which consists of a data-fidelity term and a regularization term. In spe-cific, half-quadratic regularization can effectively preserve image edges in the recovered images and a fixed-point iteration method is usually employed to solve the minimization problem. In this paper, Newton method is applied to solve the half-quadratic regular-ization image restoration problem. And at each step of the Newton method, a structured linear system of a symmetric positive definite coefficient matrix arises and preconditioned conjugate gradient method is applied to solve it with a product preconditioner.The exper-imental results demonstrate that the product preconditioned conjugate gradient method is efficient for the half-quadratic regularization image restoration in terms of the numerical performance and image recovering quality.
At last, we propose two step method to handle the multiplicative noise restoration problem. Restoration of images corrupted by blur and multiplicative noise is a challeng-ing problem in applied mathematics and it has attracted much attention in recent years. Inspired by previous works on image recovery under impulse noise with blur, we pro-pose a two-step approach to handle the multiplicative noise restoration problem. In the proposed method, the multiplicative noise is first reduced by nonlocal filters, and then a convex variational model is further adopted on the result of the first step. The variational model is composed of an L1-L2data-fidelity term and a total variation (TV) regulariza-tion term. The alternating direction method (ADM) is utilized to solve this variational problem, and we also demonstrate that the ADM algorithm converges at least linearly. Experimental results are given to show that the proposed two-step approach performs bet-ter than the existing methods for multiplicative noise image restoration both in the quality of the restored images and the convergence speed of the algorithms.
本文主要研究基于基本数学模型上的图像恢复问题的数值计算方法。首先,简单介绍数字图像的基本知识和涉及到的两个基本数学模型,并且简单介绍图像处理中的反问题和求解反问题的正则化方法。
其次,研究预处理方法在半二次图像恢复问题中的应用。图像恢复是图像处理中的基本问题,除了采用恢复图像的各种滤波方法之外,常常应用正则化方法将问题等价于一个优化问题,目标函数包含一个数据保真项和正则项。特别地,半二次正则项能够有效恢复图像的基本特征,因此我们的目标函数采用半二次正则项。我们利用牛顿法求解该优化问题,在每步牛顿迭代中,都有一个结构化的、系数矩阵是对称正定的线性方程组,从而可以利用预处理共轭梯度法求解这个方程组。我们提出了两个有效的预处理算子,并且给出一些理论结果。数值试验表明预处理共轭梯度法求解半二次图像恢复问题是有效的、快速的。
同时,本文还研究乘性噪声下图像恢复的快速算法。乘性噪声下图像恢复问题是应用数学中的一个富有挑战性的问题,近年来吸引了很多的关注和研究。受到前人对脉冲噪声下的图像恢复的方法的启示,我们提出一个快速的两步方法解决乘性噪声下的图像恢复问题。该方法的第一步,利用非局部滤波来降低乘性噪声,然后利用正则化方法将问题转换为一个优化问题,目标函数由混合保真项和一个总变差正则项组成。我们利用著名的交替迭代法求解这个优化问题,并证明算法至少线性收敛。数值试验表明两步法求解乘性噪声下图像恢复问题是非常有效的,在恢复图像的质量和速度上都优于目前其它方法。
Images, as an important bridge of the human and the world, play an significant role in human's obtaining information, expressing information and transmitting information. In the last few decades, we have witnessed an era of imaging sciences. From satellite imaging, X-ray imaging to modern medical computer tomography (CT), magnetic reso-nance imaging (MRI) and positron emission tomography (PET), digital images became more and more common and important. In general, image processing includes image enhancement, image restoration, image inpainting, image segmentation, etc.
In this paper, we focus on the numerical methods in image restoration problems. First of all, we simply introduce some basic concepts of digital image and two basic mathematical models, then study the inverse problem and regularization method.
Then, we apply the preconditioned conjugate gradient method to solve the half-quadratic regularization image restoration problem. Image restoration is the fundamental problem in image processing area. Except for many different filters applied to obtain an restored image, a degraded image can often be recovered efficiently by solving a mini-mization function which consists of a data-fidelity term and a regularization term. In spe-cific, half-quadratic regularization can effectively preserve image edges in the recovered images and a fixed-point iteration method is usually employed to solve the minimization problem. In this paper, Newton method is applied to solve the half-quadratic regular-ization image restoration problem. And at each step of the Newton method, a structured linear system of a symmetric positive definite coefficient matrix arises and preconditioned conjugate gradient method is applied to solve it with a product preconditioner.The exper-imental results demonstrate that the product preconditioned conjugate gradient method is efficient for the half-quadratic regularization image restoration in terms of the numerical performance and image recovering quality.
At last, we propose two step method to handle the multiplicative noise restoration problem. Restoration of images corrupted by blur and multiplicative noise is a challeng-ing problem in applied mathematics and it has attracted much attention in recent years. Inspired by previous works on image recovery under impulse noise with blur, we pro-pose a two-step approach to handle the multiplicative noise restoration problem. In the proposed method, the multiplicative noise is first reduced by nonlocal filters, and then a convex variational model is further adopted on the result of the first step. The variational model is composed of an L1-L2data-fidelity term and a total variation (TV) regulariza-tion term. The alternating direction method (ADM) is utilized to solve this variational problem, and we also demonstrate that the ADM algorithm converges at least linearly. Experimental results are given to show that the proposed two-step approach performs bet-ter than the existing methods for multiplicative noise image restoration both in the quality of the restored images and the convergence speed of the algorithms.
引文
[1]G. Aubert and J. Aujol, A variational approach to remove multiplicative noise, SIAM J. Appl. Math., Vol.68 (2008), pp.925-946.
[2]Z.-Z. Bai, A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations, Adv. Comput. Math., Vol.10 (1999), pp.169-186.
[3]Z.-Z. Bai, Construction and analysis of structured preconditioners for block two-by-two matrices, J. Shanghai Univ. (English Edition), Vol.8 (2004), pp.397-405.
[4]Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., Vol.75 (2006), pp.791-815.
[5]Z.-Z. Bai, Y.-M. Huang and M. K. Ng, Block-triangular preconditioners for systems arising from edge-preserving image restoration, J. Comput. Math., Vol.6 (2010), pp. 848-863.
[6]J. Bioucas-Dias and M. Figueiredo, Multiplicative noise removal using variable split-ting and constrained optimization, IEEE Trans. Image Process., Vol.19 (2010), pp. 1720-1730.
[7]S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found, and Trends Mach. Learning, Vol.3 (2010), pp.1-122.
[8]A. Buades, B. Coll and J.-M. Morel, A non-local algorithm for image denoising, In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol. 2 (2005), pp.60-65.
[9]M. J. Buckley, Fast computation of a discretized thin-plate smoothing spline for image data, Biometrika, Vol.81 (1994), pp.247-258.
[10]J. Cai, R. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Problems and Imaging, Vol.2 (2008), pp.187-204.
[11]A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., Vol.76 (1997), pp.167-188.
[12]R. H. Chan, T. F. Chan and C. K. Wong, Cosine transform based preconditioners for total variation deblurring, IEEE Trans. Image Processing, Vol.8 (1999), pp.1472-1478.
[13]R. Chan, C. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and edge-preserving regularization, IEEE Transactions on Image Processing, Vol.14 (2005), pp.1479-1485.
[14]R. Chan, C. Hu and M. Nikolova, An iterative procedure for removing random-valued impulse noise, IEEE Signal Process. Lett, Vol.11 (2004), pp.921-924.
[15]T. Chan and J. Shen, Image Processing And Analysis:Variational, PDE, Wavelet, And Stochastic Methods, SI AM Publisher,2005.
[16]R. H. Chan, J. F. Yang and X. M. Yuan, Alternating direction method for image in-painting in wavelet domain, SIAM J. Imaging Sci, Vol.4 (2011), pp.807-826.
[17]S. Chen, X. Yang and G. Cao, Impulsive noise suppression with an augmentation of ordered difference noise detector and an adaptive variational method, Pattern Recog-nition Lett., Vol.30 (2009), pp.460-467.
[18]C.-A. Deledalle, L. Denis and F. Tupin, Iterative weighted maximum likelihood de-noising with probabilistic patch-based weights, IEEE Trans. Image Process,18(12) (2009), pp.2661-2672.
[19]C.-A. Deledalle, L. Denis and F. Tupin, How to compare noisy patches? Patch similar-ity beyond gaussian noise, International Journal of Computer Vision, Vol.99 (2012), pp.86-102.
[20]Y. Dong, T. Zeng, A Convex Variational Model for Restoring Blurred Images with Multiplicative Noise, UCLA CAM Report, pp.12-23,2012. ftp://ftp.math.ucla.edu/pub/camreport/cam12-23.pdf
[21]S. Durand, J. Fadili and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, J. Math. Imaging Vis., Vol.36 (2010), pp.201-226.
[22]J. Eckstein, and D. Bertsekas, On the Douglas-Rachford splitting method and the prox-imal point algorithm for maximal monotone operators, Mathematical Programming, Vol.55 (1992), pp.293-318.
[23]M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process, Vol.19 (2010), pp.3133-3145.
[24]D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinu-ities, IEEE Trans. Pattern Anal. Machine Intelligence, Vol.14 (1992), pp.367-383.
[25]D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Processing, Vol.4 (1995), pp.932-946.
[26]P. C. Hansen, J. G. Nagy and D. P. O'Leary, Deblurring Images:Matrices, Spectra, and Eiltering, SIAM, Philadelphia,2006.
[27]Y.-M. Huang and D.-Y. Lu, On decomposition-based block preconditioned iterative methods for half-quadratic image restoration, J. Comput. Appl. Math., Vol.237 (2013), pp.162-170.
[28]Y.-M. Huang and D.-Y. Lu, A preconditioned conjugate gradient method for multi-plicative half-quadratic image restoration, Appl. Math. Comput., Vol.219 (2013), pp. 6556-6564.
[29]Y.-M. Huang and M. K. Ng, Product-type block preconditioners for electromagnetic cavity problems, Appl. Math. Comput., Vol.216 (2010), pp.1741-1751.
[30]Y. Huang, M. Ng and Y. Wen, A fast total variation minimization method for image restoration, SIAM Multiscale Modeling and Simulation Vol.7(2) (2008), pp.774-795.
[31]Y. Huang, M. Ng and Y. Wen, A new total variation method for multiplicative noise removal, SIAM Journal on Imaging Sciences, Vol.2(1) (2009), pp.20-40.
[32]Y. Huang, M. Ng and Y. Wen, Fast image restoration methods for impulse and Gaussian noises removal, IEEE Signal Processing Lett., Vol.16 (2009), pp.457-460.
[33]Y. Huang, M. Ng and T. Zeng, The Convex Relaxation Method on Deconvolution Model with Multiplicative Noise, CICP,2012.
[34]J. Idier, Convex half-quadratic criteria and auxiliary interacting variables for image restoration, IEEE Trans. Image Processing, Vol.10(2001), pp.1001-1009.
[35]C. Kervrann, J. Boulanger and P. Coupe, Bayesian non-local means filter, image redun-dancy and adaptive dictionaries for noise removal, in:Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol.4485, pp.520-532. Springer, Heidelberg (2007).
[36]M. K. Ng, Iterative Methods for Toeplitz Systems, Oxford University Press, New York, 2004.
[37]M. K. Ng, R. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., Vol.21 (1999), pp.851-866.
[38]M. P. Nikolova and M. K. Ng, Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning, Intern. Conference on Image Processing Proceedings, Vol.1 (2001), pp.277-280.
[39]M. P. Nikolova and M. K. Ng, Analysis of half-quadratic minimization methods for signal and image recovery, SIAM J. Sci. Comput., Vol.27 (2005), pp.937-966.
[40]J. Nocedal and S. Wright, "Numerical Optimization", Springer,2006.
[41]L. Rudin, P. Lions and S. Osher, Multiplicative denoising and deblurring:theory and algorithms, In S. Osher and N. Paragios, editors, Geometric Level Sets in Imaging, Vision, and Graphics, Springer,2003, pp.103-119.
[42]L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algo-rithms, Physica D, Vol.60 (1992), pp.259-268.
[43]J. Shi and S. Osher, A nonlinear inverse scale method for a convex multiplicative noise model, SIAM Journal on Imaging Sciences, Vol.1 (2008), pp.294-321.
[44]G. Steidl and T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis., vol.36 (2010), pp.168-184.
[45]W. Sun and Y. Yuan, "Optimization Theory and Methods:Nonlinear Programming". Springer,2006.
[46]T. Teuber and A. Lang, Nonlocal filters for removing multiplicative noise, In A. M. Bruckstein, B. ter Haar Romeny, A. M. Bronstein, and M. M. Bronstein, editors, Scale Space and Variational Methods in Computer Vision, Third International Conference, Vol.6667 (2011) of LNCS Springer, pp.50-61.
[47]T. Teuber and A. Lang, A new similarity measure for nonlocal filtering in the presence of multiplicative noise, Computational Statistics and Data Analysis, Vol.56(2012), pp.3821-3842.
[48]F. Wang and M. K. Ng, A fast minimization method for blur and multiplicative noise removal, International Journal of Computer Mathematics,2012,1-14, iFirst
[49]Y. Wang, J. Yang, W. Yin, and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imag. Sci., Vol.1 (2008), pp.248-272.
[2]Z.-Z. Bai, A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations, Adv. Comput. Math., Vol.10 (1999), pp.169-186.
[3]Z.-Z. Bai, Construction and analysis of structured preconditioners for block two-by-two matrices, J. Shanghai Univ. (English Edition), Vol.8 (2004), pp.397-405.
[4]Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., Vol.75 (2006), pp.791-815.
[5]Z.-Z. Bai, Y.-M. Huang and M. K. Ng, Block-triangular preconditioners for systems arising from edge-preserving image restoration, J. Comput. Math., Vol.6 (2010), pp. 848-863.
[6]J. Bioucas-Dias and M. Figueiredo, Multiplicative noise removal using variable split-ting and constrained optimization, IEEE Trans. Image Process., Vol.19 (2010), pp. 1720-1730.
[7]S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found, and Trends Mach. Learning, Vol.3 (2010), pp.1-122.
[8]A. Buades, B. Coll and J.-M. Morel, A non-local algorithm for image denoising, In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol. 2 (2005), pp.60-65.
[9]M. J. Buckley, Fast computation of a discretized thin-plate smoothing spline for image data, Biometrika, Vol.81 (1994), pp.247-258.
[10]J. Cai, R. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Problems and Imaging, Vol.2 (2008), pp.187-204.
[11]A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., Vol.76 (1997), pp.167-188.
[12]R. H. Chan, T. F. Chan and C. K. Wong, Cosine transform based preconditioners for total variation deblurring, IEEE Trans. Image Processing, Vol.8 (1999), pp.1472-1478.
[13]R. Chan, C. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and edge-preserving regularization, IEEE Transactions on Image Processing, Vol.14 (2005), pp.1479-1485.
[14]R. Chan, C. Hu and M. Nikolova, An iterative procedure for removing random-valued impulse noise, IEEE Signal Process. Lett, Vol.11 (2004), pp.921-924.
[15]T. Chan and J. Shen, Image Processing And Analysis:Variational, PDE, Wavelet, And Stochastic Methods, SI AM Publisher,2005.
[16]R. H. Chan, J. F. Yang and X. M. Yuan, Alternating direction method for image in-painting in wavelet domain, SIAM J. Imaging Sci, Vol.4 (2011), pp.807-826.
[17]S. Chen, X. Yang and G. Cao, Impulsive noise suppression with an augmentation of ordered difference noise detector and an adaptive variational method, Pattern Recog-nition Lett., Vol.30 (2009), pp.460-467.
[18]C.-A. Deledalle, L. Denis and F. Tupin, Iterative weighted maximum likelihood de-noising with probabilistic patch-based weights, IEEE Trans. Image Process,18(12) (2009), pp.2661-2672.
[19]C.-A. Deledalle, L. Denis and F. Tupin, How to compare noisy patches? Patch similar-ity beyond gaussian noise, International Journal of Computer Vision, Vol.99 (2012), pp.86-102.
[20]Y. Dong, T. Zeng, A Convex Variational Model for Restoring Blurred Images with Multiplicative Noise, UCLA CAM Report, pp.12-23,2012. ftp://ftp.math.ucla.edu/pub/camreport/cam12-23.pdf
[21]S. Durand, J. Fadili and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, J. Math. Imaging Vis., Vol.36 (2010), pp.201-226.
[22]J. Eckstein, and D. Bertsekas, On the Douglas-Rachford splitting method and the prox-imal point algorithm for maximal monotone operators, Mathematical Programming, Vol.55 (1992), pp.293-318.
[23]M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process, Vol.19 (2010), pp.3133-3145.
[24]D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinu-ities, IEEE Trans. Pattern Anal. Machine Intelligence, Vol.14 (1992), pp.367-383.
[25]D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Processing, Vol.4 (1995), pp.932-946.
[26]P. C. Hansen, J. G. Nagy and D. P. O'Leary, Deblurring Images:Matrices, Spectra, and Eiltering, SIAM, Philadelphia,2006.
[27]Y.-M. Huang and D.-Y. Lu, On decomposition-based block preconditioned iterative methods for half-quadratic image restoration, J. Comput. Appl. Math., Vol.237 (2013), pp.162-170.
[28]Y.-M. Huang and D.-Y. Lu, A preconditioned conjugate gradient method for multi-plicative half-quadratic image restoration, Appl. Math. Comput., Vol.219 (2013), pp. 6556-6564.
[29]Y.-M. Huang and M. K. Ng, Product-type block preconditioners for electromagnetic cavity problems, Appl. Math. Comput., Vol.216 (2010), pp.1741-1751.
[30]Y. Huang, M. Ng and Y. Wen, A fast total variation minimization method for image restoration, SIAM Multiscale Modeling and Simulation Vol.7(2) (2008), pp.774-795.
[31]Y. Huang, M. Ng and Y. Wen, A new total variation method for multiplicative noise removal, SIAM Journal on Imaging Sciences, Vol.2(1) (2009), pp.20-40.
[32]Y. Huang, M. Ng and Y. Wen, Fast image restoration methods for impulse and Gaussian noises removal, IEEE Signal Processing Lett., Vol.16 (2009), pp.457-460.
[33]Y. Huang, M. Ng and T. Zeng, The Convex Relaxation Method on Deconvolution Model with Multiplicative Noise, CICP,2012.
[34]J. Idier, Convex half-quadratic criteria and auxiliary interacting variables for image restoration, IEEE Trans. Image Processing, Vol.10(2001), pp.1001-1009.
[35]C. Kervrann, J. Boulanger and P. Coupe, Bayesian non-local means filter, image redun-dancy and adaptive dictionaries for noise removal, in:Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol.4485, pp.520-532. Springer, Heidelberg (2007).
[36]M. K. Ng, Iterative Methods for Toeplitz Systems, Oxford University Press, New York, 2004.
[37]M. K. Ng, R. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., Vol.21 (1999), pp.851-866.
[38]M. P. Nikolova and M. K. Ng, Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning, Intern. Conference on Image Processing Proceedings, Vol.1 (2001), pp.277-280.
[39]M. P. Nikolova and M. K. Ng, Analysis of half-quadratic minimization methods for signal and image recovery, SIAM J. Sci. Comput., Vol.27 (2005), pp.937-966.
[40]J. Nocedal and S. Wright, "Numerical Optimization", Springer,2006.
[41]L. Rudin, P. Lions and S. Osher, Multiplicative denoising and deblurring:theory and algorithms, In S. Osher and N. Paragios, editors, Geometric Level Sets in Imaging, Vision, and Graphics, Springer,2003, pp.103-119.
[42]L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algo-rithms, Physica D, Vol.60 (1992), pp.259-268.
[43]J. Shi and S. Osher, A nonlinear inverse scale method for a convex multiplicative noise model, SIAM Journal on Imaging Sciences, Vol.1 (2008), pp.294-321.
[44]G. Steidl and T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis., vol.36 (2010), pp.168-184.
[45]W. Sun and Y. Yuan, "Optimization Theory and Methods:Nonlinear Programming". Springer,2006.
[46]T. Teuber and A. Lang, Nonlocal filters for removing multiplicative noise, In A. M. Bruckstein, B. ter Haar Romeny, A. M. Bronstein, and M. M. Bronstein, editors, Scale Space and Variational Methods in Computer Vision, Third International Conference, Vol.6667 (2011) of LNCS Springer, pp.50-61.
[47]T. Teuber and A. Lang, A new similarity measure for nonlocal filtering in the presence of multiplicative noise, Computational Statistics and Data Analysis, Vol.56(2012), pp.3821-3842.
[48]F. Wang and M. K. Ng, A fast minimization method for blur and multiplicative noise removal, International Journal of Computer Mathematics,2012,1-14, iFirst
[49]Y. Wang, J. Yang, W. Yin, and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imag. Sci., Vol.1 (2008), pp.248-272.