基于PDE模型的图像处理问题的快速数值方法
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摘要
图像处理是一门与国计民生紧密相联的应用科学,它已经渗透到人们生活和工作的各个领域,如航空航天、生物医学工程、工业检测、机器人视觉、军事制导、地球物理以及大气环境等领域,已给人类带来了巨大的经济和社会效益;同时图像处理技术还远远不能满足社会需求.因此,对于图像处理的研究具有重要的意义和实用价值.
本文主要研究了图像处理中的两个基本问题:图像去噪、图像分割,运用了偏微分方程方法.图像去噪属于图像复原范畴,它要求对观测到的图像进行去噪,恢复理想图像的原貌.图像分割即是将图像中感兴趣的对象与图像中的其余部分相分离,以便为更高层图像处理服务.在简要介绍图像处理的一些基本概念和研究现状的基础上,本论文针对图像去噪和图像分割进行了深入研究,所做的主要工作如下所述:
提出了一种解LLT模型(各向同性)的非线性多重网格方法.通过对求解LLT模型的Chambolle对偶迭代(CDA)进行局部傅立叶分析,并分析其光滑速度,使我们知道采用其作为光滑迭代的多重网格迭代收敛会很慢;而且,通过对带参数的修正光滑迭代进行局部傅立叶分析,使我们认识到选择适当的参数有助于改进收敛速度.在此基础上,数值求解时,我们提出了采用改进的对偶迭代作为多重网格方法的光滑迭代.由于是对LLT模型的对偶问题进行多重网格迭代求解,从而求得原问题的解,这样也克服了模型不可微性造成的数值求解困难.所提出的多重网格方法用于灰度图像进行实验,效果明显好于CDA.当图像规模变大,即离散化更加精细时,多重网格方法在计算量上较其他方法增加少.
提出了一种解带两个L1正则项的一般化图像去噪模型的非线性多重网格方法.特别地,我们把该方法应用到了解两个特殊的模型:各向异性ROF模型和各向异性LLT模型.通过对解这两个模型的Chambolle对偶迭代和一种修正光滑迭代分别进行局部傅立叶分析,并分析它们的光滑速度,我们证明了所提出的多重网格方法采用改进的对偶迭代作为光滑迭代是非常合理的.为了克服模型不可微性造成的数值求解困难,我们是对由原问题的对偶问题产生的对偶方程采用多重网格算法求解.在推导出原问题相应的对偶方程时,和以往的方法不同,我们采用了增广拉格朗日方法来推导,更为简单.数值实验验证了解这两类各向异性图像去噪模型的多重网格方法的高效性,并表明该方法更适合处理大规模图像.
研究了二相位分片常数Mumford-Shah模型进行图像分割的改进的对偶算法.原问题被转化为三个子问题来求解,其中一个子问题我们提出采用改进的对偶迭代计算.为了证明该方法的合理性,我们对Chambolle对偶迭代和一种带参数的修正的迭代格式分别进行了局部傅立叶分析.通过数值实验验证了所提出的算法保持了基于对偶算法的水平集图像分割方法的快速分割能力,提高了分割的质量.
此博士论文得到了国家自然科学基金(Nos.60872129,60835004)的资助.
此博士论文用LATEX2ε软件打印.
Image processing is an applied science closely related to the national econo-my and people’s livelihood. It has penetrated into all areas of people’s lives andwork, such as aerospace, biomedical engineering, industrial inspection, robot vi-sion, military guidance, geophysical and atmospheric environment. Furthermore,it has brought huge economic and social benefts to mankind. At the same time,image processing technology is still far from being unable to meet the needs of soci-ety. Therefore, image processing research has important signifcance and practicalvalue.
In this dissertation, by using partial diferential equation method, two basicproblems in image processing are studied: image denoising, image segmentation.Image denoising belongs to the context of image restoration. It requires to denois-ing the observed image and restoring the original appearance of the ideal image.Image segmentation is a technology that separates the object of interest in theimage from the remaining portion in the image in order to serve higher-level imageprocessing. After briefy introducing some basic concepts and research status ofimage processing, this dissertation is devoted to studying in-depth image denoisingand image segmentation. The main work is described as follows:
A nonlinear multigrid method for solving the LLT model(isotropic) is pro-posed. By using the LFA of the Chambolle’s dual iterations (called CDA) forsolving the LLT model and analyzing its smoothing rate, we know that the non-linear multigrid method with the Chambolle’s dual iterations as its smoother willconverge slowly. Furthermore, by using the LFA of a modifed smoother, it is seenthat choosing a suitable parameter contributes to much improved rates. Basedon these two observations, we propose to use an improved dual iteration as thesmoother of the multigrid method. Since we use the multigrid method to solvethe dual problem of the LLT model and then obtain the solution of the originalproblem, the numerical difculties caused by the non-diferentiability of the LLTmodel are also overcome. Using the proposed multigrid method to process grayscale image in experiments, the efect is signifcantly better than the CDA. Whenthe image size becomes larger, that is discretization becomes more sophisticated,multigrid method will increase less calculation amount than other methods.
A nonlinear multigrid method for solving a general image denoising modelwith two L1-regularization terms is studied. In particular, we apply this method to solve two special models: the anisotropic ROF model and the anisotropic LLTmodel. By using the LFAs of the Chambolle’s dual iterations and a modifedsmoother for solving these two models respectively and analyzing their smoothingrates, we prove that the proposed multigrid algorithm with the improved dualiteration as its smoother is very reasonable. To overcome the numerical difcultiescaused by the non-diferentiability of the models, we use the multigrid algorithm tosolve the dual equation derived from the dual problem. Diferent from the previousstudies, we give a simpler derivation of the dual formulation of the general modelby augmented Lagrangian method. Numerical experiments verify the efciency ofthe proposed multigrid method for solving these two anisotropic image denoisingmodels and indicate that such a multigrid method is more suitable to deal withlarge-sized images.
An improved dual algorithm for solving two-phase piecewise constant Mumford-Shah model for image segmentation is studied. The original problem is dividedinto three subproblems to solve. We propose to use an improved dual iterative tosolve one of these subproblems. In order to prove the rationality of the proposedmethod, we use the LFA to analyze the convergence rate of the Chambolle’s dualiterations and a modifed iterative scheme for solving the subproblem, respectively.The numerical experiments show that the proposed algorithm has maintained fastsegmentation ability of the level-set image segmentation method based on the dualiterations, and improves the segmentation quality.
This dissertation is supported by the National Natural Science Foundation ofChina(Nos.60872129,60835004).
This dissertation is typeset by software LATEX2ε.
本文主要研究了图像处理中的两个基本问题:图像去噪、图像分割,运用了偏微分方程方法.图像去噪属于图像复原范畴,它要求对观测到的图像进行去噪,恢复理想图像的原貌.图像分割即是将图像中感兴趣的对象与图像中的其余部分相分离,以便为更高层图像处理服务.在简要介绍图像处理的一些基本概念和研究现状的基础上,本论文针对图像去噪和图像分割进行了深入研究,所做的主要工作如下所述:
提出了一种解LLT模型(各向同性)的非线性多重网格方法.通过对求解LLT模型的Chambolle对偶迭代(CDA)进行局部傅立叶分析,并分析其光滑速度,使我们知道采用其作为光滑迭代的多重网格迭代收敛会很慢;而且,通过对带参数的修正光滑迭代进行局部傅立叶分析,使我们认识到选择适当的参数有助于改进收敛速度.在此基础上,数值求解时,我们提出了采用改进的对偶迭代作为多重网格方法的光滑迭代.由于是对LLT模型的对偶问题进行多重网格迭代求解,从而求得原问题的解,这样也克服了模型不可微性造成的数值求解困难.所提出的多重网格方法用于灰度图像进行实验,效果明显好于CDA.当图像规模变大,即离散化更加精细时,多重网格方法在计算量上较其他方法增加少.
提出了一种解带两个L1正则项的一般化图像去噪模型的非线性多重网格方法.特别地,我们把该方法应用到了解两个特殊的模型:各向异性ROF模型和各向异性LLT模型.通过对解这两个模型的Chambolle对偶迭代和一种修正光滑迭代分别进行局部傅立叶分析,并分析它们的光滑速度,我们证明了所提出的多重网格方法采用改进的对偶迭代作为光滑迭代是非常合理的.为了克服模型不可微性造成的数值求解困难,我们是对由原问题的对偶问题产生的对偶方程采用多重网格算法求解.在推导出原问题相应的对偶方程时,和以往的方法不同,我们采用了增广拉格朗日方法来推导,更为简单.数值实验验证了解这两类各向异性图像去噪模型的多重网格方法的高效性,并表明该方法更适合处理大规模图像.
研究了二相位分片常数Mumford-Shah模型进行图像分割的改进的对偶算法.原问题被转化为三个子问题来求解,其中一个子问题我们提出采用改进的对偶迭代计算.为了证明该方法的合理性,我们对Chambolle对偶迭代和一种带参数的修正的迭代格式分别进行了局部傅立叶分析.通过数值实验验证了所提出的算法保持了基于对偶算法的水平集图像分割方法的快速分割能力,提高了分割的质量.
此博士论文得到了国家自然科学基金(Nos.60872129,60835004)的资助.
此博士论文用LATEX2ε软件打印.
Image processing is an applied science closely related to the national econo-my and people’s livelihood. It has penetrated into all areas of people’s lives andwork, such as aerospace, biomedical engineering, industrial inspection, robot vi-sion, military guidance, geophysical and atmospheric environment. Furthermore,it has brought huge economic and social benefts to mankind. At the same time,image processing technology is still far from being unable to meet the needs of soci-ety. Therefore, image processing research has important signifcance and practicalvalue.
In this dissertation, by using partial diferential equation method, two basicproblems in image processing are studied: image denoising, image segmentation.Image denoising belongs to the context of image restoration. It requires to denois-ing the observed image and restoring the original appearance of the ideal image.Image segmentation is a technology that separates the object of interest in theimage from the remaining portion in the image in order to serve higher-level imageprocessing. After briefy introducing some basic concepts and research status ofimage processing, this dissertation is devoted to studying in-depth image denoisingand image segmentation. The main work is described as follows:
A nonlinear multigrid method for solving the LLT model(isotropic) is pro-posed. By using the LFA of the Chambolle’s dual iterations (called CDA) forsolving the LLT model and analyzing its smoothing rate, we know that the non-linear multigrid method with the Chambolle’s dual iterations as its smoother willconverge slowly. Furthermore, by using the LFA of a modifed smoother, it is seenthat choosing a suitable parameter contributes to much improved rates. Basedon these two observations, we propose to use an improved dual iteration as thesmoother of the multigrid method. Since we use the multigrid method to solvethe dual problem of the LLT model and then obtain the solution of the originalproblem, the numerical difculties caused by the non-diferentiability of the LLTmodel are also overcome. Using the proposed multigrid method to process grayscale image in experiments, the efect is signifcantly better than the CDA. Whenthe image size becomes larger, that is discretization becomes more sophisticated,multigrid method will increase less calculation amount than other methods.
A nonlinear multigrid method for solving a general image denoising modelwith two L1-regularization terms is studied. In particular, we apply this method to solve two special models: the anisotropic ROF model and the anisotropic LLTmodel. By using the LFAs of the Chambolle’s dual iterations and a modifedsmoother for solving these two models respectively and analyzing their smoothingrates, we prove that the proposed multigrid algorithm with the improved dualiteration as its smoother is very reasonable. To overcome the numerical difcultiescaused by the non-diferentiability of the models, we use the multigrid algorithm tosolve the dual equation derived from the dual problem. Diferent from the previousstudies, we give a simpler derivation of the dual formulation of the general modelby augmented Lagrangian method. Numerical experiments verify the efciency ofthe proposed multigrid method for solving these two anisotropic image denoisingmodels and indicate that such a multigrid method is more suitable to deal withlarge-sized images.
An improved dual algorithm for solving two-phase piecewise constant Mumford-Shah model for image segmentation is studied. The original problem is dividedinto three subproblems to solve. We propose to use an improved dual iterative tosolve one of these subproblems. In order to prove the rationality of the proposedmethod, we use the LFA to analyze the convergence rate of the Chambolle’s dualiterations and a modifed iterative scheme for solving the subproblem, respectively.The numerical experiments show that the proposed algorithm has maintained fastsegmentation ability of the level-set image segmentation method based on the dualiterations, and improves the segmentation quality.
This dissertation is supported by the National Natural Science Foundation ofChina(Nos.60872129,60835004).
This dissertation is typeset by software LATEX2ε.
引文
[1]冈萨雷斯,伍兹著.数字图像处理(第二版).阮秋琦,阮宇智等译.北京:电子工业出版社,2003
[2]冯象初,王卫卫.图像处理的变分和偏微分方程方法.北京:科学出版社,2009
[3]王大凯,侯榆青,彭进业.图像处理的偏微分方程方法.北京:科学出版社,2008
[4] Tai X C, Lie K A, Chan T F, et al. Image Processing Based on Partial DiferentialEquations. Heidelberg: Springer,2007
[5] Aubert G, Kornprobst P. Mathematical problems in image processing: partialdiferential equations and the calculus of variations. New York: Springer,2002
[6]吴斌,吴亚东,张红英.基于变分偏微分方程的图像复原技术.北京:北京大学出版社,2008
[7] Chan T F, Shen J. Theory and Computation of Variational Image Deblurring.2005
[8] Guichard F, Morel J M. Partial diferential equations and image iterative fltering.New York: Oxford University Press,1997
[9] Alvarez L, Guichard F, Lions P L, et al. Axioms and fundamental equations ofimage processing. Archive for Rational Mechanics and Analysis,1993,123(3):199-257
[10] Witkin A. Scale-space fltering. Proceedings of the International Joint Conferenceon Artifcial Intelligence,1983,2:1019-1021
[11] Charbonnier P, Blanc-Feraud L, Aubert G, et al. Two deterministic half-quadraticregularization algorithms for computed imaging. IEEE International Conferenceon Image Processing,1994,2:168-172
[12] Perona P, Malik J. Scale-space and edge detection using anisotropic difusion. IEEETransactions on Pattern Analysis and Machine Intelligence,1990,12(7):629-639
[13] Weickert J. Theoretical foundations of anisotropic difusion in image processing.Computing-Wien-Supplements,1996,11:221-236
[14] Weickert J. A review of nonlinear difusion fltering. Scale-Space Theory in Com-puter Vision,1997,1252:1-28
[15] Weickert J. Coherence-Enhancing Difusion Filtering. International Journal ofComputer Vision,1999,31(2-3):111-127
[16] Rudin L I, Osher S, Fatemi E. Nonlinear total variation based noise removal algo-rithms. Physica D: Nonlinear Phenomena,1992,60(1-4):259-268
[17] Ambrosio L, Fusco N, Pallara D. Functions of bounded variation and free discon-tinuity problems. Oxford: Clarendon Press,2000
[18] Chan T, Shen J. Image Processing and Analysis: Variational, PDE, Wavelet, andStochastic Methods. Philadelphia: SIAM Publisher,2005
[19] Paragios N, Chen Y, Faugeras O D. Handbook of Mathematical Models in Com-puter Vision. Heidelberg: Springer,2010
[20] You Y L, Kaveh M, Xu W Y, et al. Analysis and design of anisotropic difusion forimage processing. Proceedings of the1st IEEE International Conference on ImageProcessing,1994,2:497-501
[21] You Y L, Xu W Y, Tannenbaum A, et al. Behavioral analysis of anisotropic dif-fusion in image processing. IEEE Transactions on Image Processing,1996,5(11):1539-1553
[22] You Y L, Kaveh M. Fourth-order partial diferential equations for noise removal.IEEE Transactions on Image Processing,2000,9(10):1723-1730
[23] Lysaker M, Lundervold A, Tai X C. Noise removal using fourth-order partial dif-ferential equation with applications to medical magnetic resonance images in spaceand time. IEEE Transactions on Image Processing,2003,12(12):1579-1590
[24] Zhu W, Chan T. Image denoising using mean curvature of image surface. SIAMJournal on Imaging Sciences,2012,5(1):1-32
[25] Kim S, Lim H. Fourth-order partial diferential equations for efective image denois-ing. UAB Conference on Diferential Equations and Computational Simulations,2009,17:107-121
[26] Li F, Shen C, Fan J, et al. Image restoration combining a total variational flter anda fourth-order flter. Journal of Visual Communication and Image Representation,2007,18(4):322-330
[27] Didas S, Weickert J, Burgeth B. Properties of Higher Order Nonlinear DifusionFiltering. Journal of Mathematical Imaging and Vision,2009,35(3):208-226
[28] Chan T F, Golub G H, Mulet P. A nonlinear primal-dual method for totalvariation-based image restoration. SIAM Journal on Scientifc Computing,1999,20(6):1964-1977
[29] Carter J L. Dual methods for total variation-based image restoration. Universityof California Los Angeles,2001
[30] Chambolle A. An algorithm for total variation minimization and applications.Journal of Mathematical Imaging and Vision,2004,20:89-97
[31] Chambolle A. Total variation minimization and a class of binary MRF models.Energy Minimization Methods in Computer Vision and Pattern Recognition,2005,136-152
[32] Zhu M. Fast numerical algorithms for total variation based image restoration.Ph.D. thesis, UCLA,2008
[33] Zhu M, Chan T. An efcient primal-dual hybrid gradient algorithm for total vari-ation image restoration. UCLA CAM Report,2008,08-34
[34] Zhu M, Wright S J, Chan T F. Duality-based algorithms for total-variation-regularized image restoration. Computational Optimization and Applications,2010,47(3):377-400
[35] Chen H, Song J, Tai X C. A dual algorithm for minimization of the LLT model.Advances in Computational Mathematics,2009,31(1-3):115-130
[36] Chan T F, Chen K, Carter J L. Iterative methods for solving the dual formulationarising from image restoration. Electronic Transactions on Numerical Analysis,2007,26:299-311
[37] Brito-Loeza C, Chen K. Multigrid algorithm for high order denoising. SIAM Jour-nal on Imaging Sciences,2010,3(3):363-389
[38] Savage J, Chen K. An improved and accelerated non-linear multigrid method fortotal-variation denoising. International Journal of Computer Mathematics,2005,82(8):1001-1015
[39] Chan T F, Chen K. On a nonlinear multigrid algorithm with primal relaxationfor the image total variation minimisation. Numerical Algorithms,2006,41(4):387-411
[40] Chan T F, Chen K. An optimization-based multilevel algorithm for total variationimage denoising. Multiscale Modeling and Simulation,2006,5(2):615-645
[41] Chan R H, Chen K. Multilevel algorithm for a Poisson noise removal model withtotal-variation regularization. International Journal of Computer Mathematics,2007,84(8):1183-1198
[42] Chan R H, Chen K. A multilevel algorithm for simultaneously denoising and de-blurring images. SIAM Journal on Scientifc Computing,2010,32(2):1043-1063
[43] Chen K, Savage J. An accelerated algebraic multigrid algorithm for total-variationdenoising. BIT Numerical Mathematics,2007,47(2):277-296
[44] Brito-Loeza C, Chen K. On high-order denoising models and fast algorithms forvector-valued images. IEEE Transactions on Image Processing,2010,19(6):1518-1527
[45] Chen K, Tai X C. A nonlinear multigrid method for total variation minimizationfrom image restoration. Journal of Scientifc Computing,2007,33(2):115-138
[46] Badshah N, Chen K. Multigrid method for the Chan-Vese model in variationalsegmentation. Communications in Computational Physics,2008,4(2):294-316
[47] Badshah N, Chen K. On two multigrid algorithms for modeling variational multi-phase image segmentation. IEEE Transactions on Image Processing,2009,18(5):1097-1106
[48] Zhang J, Chen K, Yu B. A3D multi-grid algorithm for the Chan–Vese model ofvariational image segmentation. International Journal of Computer Mathematics,2012,89(2):160-189
[49] Ma W Y, Manjunath B S. Edge fow: a framework of boundary detection and imagesegmentation. Computer Vision and Pattern Recognition,1997. Proceedings.,1997IEEE Computer Society Conference on. IEEE,1997:744-749
[50] Castleman K R. Digital image processing. New Jersey: Prentice Hall Press,1996
[51] Kass M, Witkin A, Terzopoulos D. Snakes: Active contour models. InternationalJournal of Computer Vision,1988,1(4):321-331
[52] Caselles V, Kimmel R, Sapiro G. Geodesic active contours. International Journalof Computer Vision,1997,22(1):61-79
[53] Mumford D, Shah J. Optimal approximations by piecewise smooth functions andassociated variational problems. Communications on Pure and Applied Mathe-matics,1989,42(5):577-685
[54] Chan T F, Vese L A. Active contours without edges. IEEE Transactions on ImageProcessing,2001,10(2):266-277
[55] Chan T F, Sandberg B Y, Vese L A. Active Contours without Edges for Vector-Valued Images. Journal of Visual Communication and Image Representation,2000,11(2):130-141
[56] Vese L A, Chan T F. A multiphase level set framework for image segmentationusing the Mumford and Shah model. International Journal of Computer Vision,2002,50(3):271-293
[57] Gu Y, Wang L L, Tai X C. A Direct Approach Toward Global Minimization forMultiphase Labeling and Segmentation Problems. IEEE Transactions on ImageProcessing,2012,21(5):2399-2411
[58] Bresson X, Esedoglu S, Vandergheynst P, et al. Global minimizers of the activecontour/snake model. UCLA CAM Report,2005:05-04
[59] Nikolova M, Esedoglu S, Chan T F. Algorithms for fnding global minimizers ofimage segmentation and denoising models. SIAM Journal on Applied Mathematics,2006,66(5):1632-1648
[60] Bresson X, Esedoglu S, Vandergheynst P, et al. Fast global minimization of theactive contour/snake model. Journal of Mathematical Imaging and vision,2007,28(2):151-167
[61] Wang L L, Gu Y. Efcient Dual Algorithms for Image Segmentation Using TV-Allen-Cahn Type Models. Communications in Computational Physics,2011,9(4):859-877
[62]张恭庆,林源渠.泛函分析讲义(上册).北京:北京大学出版社,1987
[63]袁亚湘,孙文瑜.最优化理论与方法.北京:科学出版社,1997
[64]李董辉,童小娇,万中.数值最优化.北京:科学出版社,2005
[65] Rockafellar R T. Convex analysis. New Jersey: Princeton university press,1996
[66]冯德兴.凸分析基础.北京:科学出版社,1995
[67]韩渭敏,程晓良.变分不等式简介—基本理论、数值分析及应用.北京:高等教育出版社,2007
[68] Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods.Heidelberg: Springer-Verlag,1994
[69]王烈衡,许学军.有限元方法的数学基础.北京:科学出版社,2004
[70] Blomgren P, Chan T F, Mulet P, et al. Variational PDE models and methods forimage processing. In Research Notes in Mathematics,2000,420:43-68
[71] Chan T F, Vese L A. Image segmentation using level sets and the piecewise-constant Mumford-Shah model.2000
[72] Kenigsberg A, Kimmel R, Yavneh I. A multigrid approach for fast geodesic activecontours. CIS report,2004,6:2004
[73] Chan T F, Tai X C. Identifcation of discontinuous coefcients in elliptic problemsusing total variation regularization. SIAM Journal on Scientifc Computing,2003,25(3):881-904
[74] Vogel C R, Oman M E. Iterative methods for total variation denoising. SIAMJournal on Scientifc Computing,1996,17(1):227-238
[75] Vogel C R, Oman M E. Fast, robust total variation-based reconstruction of noisy,blurred images. IEEE Transactions on Image Processing,1998,7(6):813-824
[76] Chan T F, Golub G H, Mulet P. A nonlinear primal-dual method for totalvariation-based image restoration. SIAM Journal on Scientifc Computing,1999,20(6):1964-1977
[77] Chan T F, Mulet P. Iterative methods for total variation image restoration. De-partment of Mathematics, University of California, Los Angeles,1996
[78] Savage J, Chen K. An improved and accelerated non-linear multigrid method fortotal-variation denoising. International Journal of Computer Mathematics,2005,82(8):1001-1015
[79] Donatelli M, Serra-Capizzano S. On the regularizing power of multigrid-type al-gorithms. SIAM Journal on Scientifc Computing,2006,27(6):2053-2076
[80] W.哈克布思.多重网格方法.北京:科学出版社,1988
[81] Brandt A. Multi-level adaptive solutions to boundary-value problems. Mathemat-ics of Computation,1977,31:333-390
[82] Chambolle A, Lions P L. Image recovery via total variation minimization andrelated problems. Numerische Mathematik,1997,76(2):167-188
[83] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in imagerestoration. SIAM Journal on Applied Mathematics,2006,66(4):1383-1406
[84] Chan T, Marquina A, Mulet P. High-order total variation-based image restoration.SIAM Journal on Scientifc Computing,2000,22(2):503-516
[85] Osher S, Sol′e A, Vese L. Image decomposition and restoration using total variationminimization and the H1norm. Multiscale Modeling and Simulation,2003,1(3):349-370
[86] Liu Q, Yao Z, Ke Y. Entropy solutions for a fourth-order nonlinear degenerateproblem for noise removal. Nonlinear Analysis: Theory, Methods Applications,2007,67(6):1908-1918
[87] Liu Q, Yao Z, Ke Y. Solutions of fourth-order partial diferential equations in anoise removal model. Electronic Journal of Diferential Equations,2007,2007(120):1-11
[88] Bramble J H. Multigrid methods. New York: Chapman and Hall/CRC,1993
[89] Hackbusch W. Multi-grid methods and applications. Berlin: Springer-Verlag,1985
[90] Chen K, Dong Y, Hintermu¨ller M. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variationbased image restoration. Inverse Problems and Imaging,2011,5(2):323-339
[91] Briggs W L, Henson V E, McCormick S F. A Multigrid Tutorial,2nd edn. Philadel-phia: SIAM,2000
[92] Chen K. Matrix Preconditioning Techniques and Applications. London: Cam-bridge University Press,2005
[93] Trottenberg U, Oosterlee C W, Schuller A. Multigrid. Massachusetts: AcademicPress,2001
[94] Pang Z F, Wang L L, Yang Y F. Fast algorithms for the anisotropic LLT modelin image denoising. East Asian Journal of Applied Mathematics,2011,1:264-283
[95] Steidl G. A note on the dual treatment of higher-order regularization functionals.Computing,2006,76:135-148
[96] Goldstein T, Osher S. The split Bregman method for L1-regularized problems.SIAM Journal on Imaging Sciences,2009,2(2):323-343
[97] Jia R Q, Zhao H. A fast algorithm for the total variation model of image denoising.Advances in Computational Mathematics,2010,33:231-241
[98] Li Y, Santosa F. A computational algorithm for minimizing total variation inimage restoration. IEEE Transactions on Image Processing,1996,5(6):987-995
[99] Tai X C, Wu C. Augmented Lagrangian method, dual methods and split Breg-man iteration for ROF model. Scale Space and Variational Methods in ComputerVision, Lecture Notes in Computer Science,2009,5567:502-513
[100] Micchelli C A, Shen L, Xu Y. Proximity algorithms for image models: denoising.Inverse Problems,2011,27(4):045009,30pp
[101] Chung G, Vese L A. Image segmentation using a multilayer level-set approach.Computing and Visualization in Science,2009,12(6):267-285
[102] Lie J, Lysaker M, Tai X C. A binary level set model and some applicationsto Mumford-Shah image segmentation. IEEE Transactions on Image Processing,2006,15(5):1171-1181
[103] Lie J, Lysaker M, Tai X C. A variant of the level set method and applications toimage segmentation. Mathematics of Computation,2006,75:1155-1174
[2]冯象初,王卫卫.图像处理的变分和偏微分方程方法.北京:科学出版社,2009
[3]王大凯,侯榆青,彭进业.图像处理的偏微分方程方法.北京:科学出版社,2008
[4] Tai X C, Lie K A, Chan T F, et al. Image Processing Based on Partial DiferentialEquations. Heidelberg: Springer,2007
[5] Aubert G, Kornprobst P. Mathematical problems in image processing: partialdiferential equations and the calculus of variations. New York: Springer,2002
[6]吴斌,吴亚东,张红英.基于变分偏微分方程的图像复原技术.北京:北京大学出版社,2008
[7] Chan T F, Shen J. Theory and Computation of Variational Image Deblurring.2005
[8] Guichard F, Morel J M. Partial diferential equations and image iterative fltering.New York: Oxford University Press,1997
[9] Alvarez L, Guichard F, Lions P L, et al. Axioms and fundamental equations ofimage processing. Archive for Rational Mechanics and Analysis,1993,123(3):199-257
[10] Witkin A. Scale-space fltering. Proceedings of the International Joint Conferenceon Artifcial Intelligence,1983,2:1019-1021
[11] Charbonnier P, Blanc-Feraud L, Aubert G, et al. Two deterministic half-quadraticregularization algorithms for computed imaging. IEEE International Conferenceon Image Processing,1994,2:168-172
[12] Perona P, Malik J. Scale-space and edge detection using anisotropic difusion. IEEETransactions on Pattern Analysis and Machine Intelligence,1990,12(7):629-639
[13] Weickert J. Theoretical foundations of anisotropic difusion in image processing.Computing-Wien-Supplements,1996,11:221-236
[14] Weickert J. A review of nonlinear difusion fltering. Scale-Space Theory in Com-puter Vision,1997,1252:1-28
[15] Weickert J. Coherence-Enhancing Difusion Filtering. International Journal ofComputer Vision,1999,31(2-3):111-127
[16] Rudin L I, Osher S, Fatemi E. Nonlinear total variation based noise removal algo-rithms. Physica D: Nonlinear Phenomena,1992,60(1-4):259-268
[17] Ambrosio L, Fusco N, Pallara D. Functions of bounded variation and free discon-tinuity problems. Oxford: Clarendon Press,2000
[18] Chan T, Shen J. Image Processing and Analysis: Variational, PDE, Wavelet, andStochastic Methods. Philadelphia: SIAM Publisher,2005
[19] Paragios N, Chen Y, Faugeras O D. Handbook of Mathematical Models in Com-puter Vision. Heidelberg: Springer,2010
[20] You Y L, Kaveh M, Xu W Y, et al. Analysis and design of anisotropic difusion forimage processing. Proceedings of the1st IEEE International Conference on ImageProcessing,1994,2:497-501
[21] You Y L, Xu W Y, Tannenbaum A, et al. Behavioral analysis of anisotropic dif-fusion in image processing. IEEE Transactions on Image Processing,1996,5(11):1539-1553
[22] You Y L, Kaveh M. Fourth-order partial diferential equations for noise removal.IEEE Transactions on Image Processing,2000,9(10):1723-1730
[23] Lysaker M, Lundervold A, Tai X C. Noise removal using fourth-order partial dif-ferential equation with applications to medical magnetic resonance images in spaceand time. IEEE Transactions on Image Processing,2003,12(12):1579-1590
[24] Zhu W, Chan T. Image denoising using mean curvature of image surface. SIAMJournal on Imaging Sciences,2012,5(1):1-32
[25] Kim S, Lim H. Fourth-order partial diferential equations for efective image denois-ing. UAB Conference on Diferential Equations and Computational Simulations,2009,17:107-121
[26] Li F, Shen C, Fan J, et al. Image restoration combining a total variational flter anda fourth-order flter. Journal of Visual Communication and Image Representation,2007,18(4):322-330
[27] Didas S, Weickert J, Burgeth B. Properties of Higher Order Nonlinear DifusionFiltering. Journal of Mathematical Imaging and Vision,2009,35(3):208-226
[28] Chan T F, Golub G H, Mulet P. A nonlinear primal-dual method for totalvariation-based image restoration. SIAM Journal on Scientifc Computing,1999,20(6):1964-1977
[29] Carter J L. Dual methods for total variation-based image restoration. Universityof California Los Angeles,2001
[30] Chambolle A. An algorithm for total variation minimization and applications.Journal of Mathematical Imaging and Vision,2004,20:89-97
[31] Chambolle A. Total variation minimization and a class of binary MRF models.Energy Minimization Methods in Computer Vision and Pattern Recognition,2005,136-152
[32] Zhu M. Fast numerical algorithms for total variation based image restoration.Ph.D. thesis, UCLA,2008
[33] Zhu M, Chan T. An efcient primal-dual hybrid gradient algorithm for total vari-ation image restoration. UCLA CAM Report,2008,08-34
[34] Zhu M, Wright S J, Chan T F. Duality-based algorithms for total-variation-regularized image restoration. Computational Optimization and Applications,2010,47(3):377-400
[35] Chen H, Song J, Tai X C. A dual algorithm for minimization of the LLT model.Advances in Computational Mathematics,2009,31(1-3):115-130
[36] Chan T F, Chen K, Carter J L. Iterative methods for solving the dual formulationarising from image restoration. Electronic Transactions on Numerical Analysis,2007,26:299-311
[37] Brito-Loeza C, Chen K. Multigrid algorithm for high order denoising. SIAM Jour-nal on Imaging Sciences,2010,3(3):363-389
[38] Savage J, Chen K. An improved and accelerated non-linear multigrid method fortotal-variation denoising. International Journal of Computer Mathematics,2005,82(8):1001-1015
[39] Chan T F, Chen K. On a nonlinear multigrid algorithm with primal relaxationfor the image total variation minimisation. Numerical Algorithms,2006,41(4):387-411
[40] Chan T F, Chen K. An optimization-based multilevel algorithm for total variationimage denoising. Multiscale Modeling and Simulation,2006,5(2):615-645
[41] Chan R H, Chen K. Multilevel algorithm for a Poisson noise removal model withtotal-variation regularization. International Journal of Computer Mathematics,2007,84(8):1183-1198
[42] Chan R H, Chen K. A multilevel algorithm for simultaneously denoising and de-blurring images. SIAM Journal on Scientifc Computing,2010,32(2):1043-1063
[43] Chen K, Savage J. An accelerated algebraic multigrid algorithm for total-variationdenoising. BIT Numerical Mathematics,2007,47(2):277-296
[44] Brito-Loeza C, Chen K. On high-order denoising models and fast algorithms forvector-valued images. IEEE Transactions on Image Processing,2010,19(6):1518-1527
[45] Chen K, Tai X C. A nonlinear multigrid method for total variation minimizationfrom image restoration. Journal of Scientifc Computing,2007,33(2):115-138
[46] Badshah N, Chen K. Multigrid method for the Chan-Vese model in variationalsegmentation. Communications in Computational Physics,2008,4(2):294-316
[47] Badshah N, Chen K. On two multigrid algorithms for modeling variational multi-phase image segmentation. IEEE Transactions on Image Processing,2009,18(5):1097-1106
[48] Zhang J, Chen K, Yu B. A3D multi-grid algorithm for the Chan–Vese model ofvariational image segmentation. International Journal of Computer Mathematics,2012,89(2):160-189
[49] Ma W Y, Manjunath B S. Edge fow: a framework of boundary detection and imagesegmentation. Computer Vision and Pattern Recognition,1997. Proceedings.,1997IEEE Computer Society Conference on. IEEE,1997:744-749
[50] Castleman K R. Digital image processing. New Jersey: Prentice Hall Press,1996
[51] Kass M, Witkin A, Terzopoulos D. Snakes: Active contour models. InternationalJournal of Computer Vision,1988,1(4):321-331
[52] Caselles V, Kimmel R, Sapiro G. Geodesic active contours. International Journalof Computer Vision,1997,22(1):61-79
[53] Mumford D, Shah J. Optimal approximations by piecewise smooth functions andassociated variational problems. Communications on Pure and Applied Mathe-matics,1989,42(5):577-685
[54] Chan T F, Vese L A. Active contours without edges. IEEE Transactions on ImageProcessing,2001,10(2):266-277
[55] Chan T F, Sandberg B Y, Vese L A. Active Contours without Edges for Vector-Valued Images. Journal of Visual Communication and Image Representation,2000,11(2):130-141
[56] Vese L A, Chan T F. A multiphase level set framework for image segmentationusing the Mumford and Shah model. International Journal of Computer Vision,2002,50(3):271-293
[57] Gu Y, Wang L L, Tai X C. A Direct Approach Toward Global Minimization forMultiphase Labeling and Segmentation Problems. IEEE Transactions on ImageProcessing,2012,21(5):2399-2411
[58] Bresson X, Esedoglu S, Vandergheynst P, et al. Global minimizers of the activecontour/snake model. UCLA CAM Report,2005:05-04
[59] Nikolova M, Esedoglu S, Chan T F. Algorithms for fnding global minimizers ofimage segmentation and denoising models. SIAM Journal on Applied Mathematics,2006,66(5):1632-1648
[60] Bresson X, Esedoglu S, Vandergheynst P, et al. Fast global minimization of theactive contour/snake model. Journal of Mathematical Imaging and vision,2007,28(2):151-167
[61] Wang L L, Gu Y. Efcient Dual Algorithms for Image Segmentation Using TV-Allen-Cahn Type Models. Communications in Computational Physics,2011,9(4):859-877
[62]张恭庆,林源渠.泛函分析讲义(上册).北京:北京大学出版社,1987
[63]袁亚湘,孙文瑜.最优化理论与方法.北京:科学出版社,1997
[64]李董辉,童小娇,万中.数值最优化.北京:科学出版社,2005
[65] Rockafellar R T. Convex analysis. New Jersey: Princeton university press,1996
[66]冯德兴.凸分析基础.北京:科学出版社,1995
[67]韩渭敏,程晓良.变分不等式简介—基本理论、数值分析及应用.北京:高等教育出版社,2007
[68] Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods.Heidelberg: Springer-Verlag,1994
[69]王烈衡,许学军.有限元方法的数学基础.北京:科学出版社,2004
[70] Blomgren P, Chan T F, Mulet P, et al. Variational PDE models and methods forimage processing. In Research Notes in Mathematics,2000,420:43-68
[71] Chan T F, Vese L A. Image segmentation using level sets and the piecewise-constant Mumford-Shah model.2000
[72] Kenigsberg A, Kimmel R, Yavneh I. A multigrid approach for fast geodesic activecontours. CIS report,2004,6:2004
[73] Chan T F, Tai X C. Identifcation of discontinuous coefcients in elliptic problemsusing total variation regularization. SIAM Journal on Scientifc Computing,2003,25(3):881-904
[74] Vogel C R, Oman M E. Iterative methods for total variation denoising. SIAMJournal on Scientifc Computing,1996,17(1):227-238
[75] Vogel C R, Oman M E. Fast, robust total variation-based reconstruction of noisy,blurred images. IEEE Transactions on Image Processing,1998,7(6):813-824
[76] Chan T F, Golub G H, Mulet P. A nonlinear primal-dual method for totalvariation-based image restoration. SIAM Journal on Scientifc Computing,1999,20(6):1964-1977
[77] Chan T F, Mulet P. Iterative methods for total variation image restoration. De-partment of Mathematics, University of California, Los Angeles,1996
[78] Savage J, Chen K. An improved and accelerated non-linear multigrid method fortotal-variation denoising. International Journal of Computer Mathematics,2005,82(8):1001-1015
[79] Donatelli M, Serra-Capizzano S. On the regularizing power of multigrid-type al-gorithms. SIAM Journal on Scientifc Computing,2006,27(6):2053-2076
[80] W.哈克布思.多重网格方法.北京:科学出版社,1988
[81] Brandt A. Multi-level adaptive solutions to boundary-value problems. Mathemat-ics of Computation,1977,31:333-390
[82] Chambolle A, Lions P L. Image recovery via total variation minimization andrelated problems. Numerische Mathematik,1997,76(2):167-188
[83] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in imagerestoration. SIAM Journal on Applied Mathematics,2006,66(4):1383-1406
[84] Chan T, Marquina A, Mulet P. High-order total variation-based image restoration.SIAM Journal on Scientifc Computing,2000,22(2):503-516
[85] Osher S, Sol′e A, Vese L. Image decomposition and restoration using total variationminimization and the H1norm. Multiscale Modeling and Simulation,2003,1(3):349-370
[86] Liu Q, Yao Z, Ke Y. Entropy solutions for a fourth-order nonlinear degenerateproblem for noise removal. Nonlinear Analysis: Theory, Methods Applications,2007,67(6):1908-1918
[87] Liu Q, Yao Z, Ke Y. Solutions of fourth-order partial diferential equations in anoise removal model. Electronic Journal of Diferential Equations,2007,2007(120):1-11
[88] Bramble J H. Multigrid methods. New York: Chapman and Hall/CRC,1993
[89] Hackbusch W. Multi-grid methods and applications. Berlin: Springer-Verlag,1985
[90] Chen K, Dong Y, Hintermu¨ller M. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variationbased image restoration. Inverse Problems and Imaging,2011,5(2):323-339
[91] Briggs W L, Henson V E, McCormick S F. A Multigrid Tutorial,2nd edn. Philadel-phia: SIAM,2000
[92] Chen K. Matrix Preconditioning Techniques and Applications. London: Cam-bridge University Press,2005
[93] Trottenberg U, Oosterlee C W, Schuller A. Multigrid. Massachusetts: AcademicPress,2001
[94] Pang Z F, Wang L L, Yang Y F. Fast algorithms for the anisotropic LLT modelin image denoising. East Asian Journal of Applied Mathematics,2011,1:264-283
[95] Steidl G. A note on the dual treatment of higher-order regularization functionals.Computing,2006,76:135-148
[96] Goldstein T, Osher S. The split Bregman method for L1-regularized problems.SIAM Journal on Imaging Sciences,2009,2(2):323-343
[97] Jia R Q, Zhao H. A fast algorithm for the total variation model of image denoising.Advances in Computational Mathematics,2010,33:231-241
[98] Li Y, Santosa F. A computational algorithm for minimizing total variation inimage restoration. IEEE Transactions on Image Processing,1996,5(6):987-995
[99] Tai X C, Wu C. Augmented Lagrangian method, dual methods and split Breg-man iteration for ROF model. Scale Space and Variational Methods in ComputerVision, Lecture Notes in Computer Science,2009,5567:502-513
[100] Micchelli C A, Shen L, Xu Y. Proximity algorithms for image models: denoising.Inverse Problems,2011,27(4):045009,30pp
[101] Chung G, Vese L A. Image segmentation using a multilayer level-set approach.Computing and Visualization in Science,2009,12(6):267-285
[102] Lie J, Lysaker M, Tai X C. A binary level set model and some applicationsto Mumford-Shah image segmentation. IEEE Transactions on Image Processing,2006,15(5):1171-1181
[103] Lie J, Lysaker M, Tai X C. A variant of the level set method and applications toimage segmentation. Mathematics of Computation,2006,75:1155-1174
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