图像分割的变分正则化模型—非凸、稀疏理论与算法
|
摘要
近年来，变分正则化方法在图像处理中的应用取得了很多研究成果，建立了许多经典模型和方法。在此基础上，本文针对图像分割问题，提出了一些新的概念、模型和方法。本文的主要工作有：
1.图像分割的变分模型包含数据项和正则项，通常，数据项利用L2范数，正则项由先验假设确定。在最大后验概率框架下，给出了一个新的离散模型。数据项利用迭代重加权L2范数，同时具备L2范数计算简单和混合L21范数对野点鲁棒的优点；在正则项中，给出了一个新的边界罚函数，增强了模型去除野点的能力，并能保弱边。提出一种改进的graph cuts算法求解模型。数值实验表明，新模型具有很好的图像分割效果，尤其体现在去除野点和保弱边两方面。
2.研究了一类向量值极小化问题的凸松弛方法，给出了适用于split-Bregman快速算法的一般性等价模型。Vese-Chan多相分割方法和基于分片常数水平集函数的Mumford-Shah方法是新模型的特例。数值实验表明，在Vese-Chan方法和Mumford-Shah方法中应用split-Bregman算法，具有较快的运算速度和较好的分割效果，且对初始条件是鲁棒的。
3.将图像超像素分割看作子空间聚类问题。给出一个约束条件，等价于以干净数据为字典。利用系数矩阵的非凸迫近p范数作为稀疏约束，利用系数矩阵奇异值的非凸迫近p范数作为低秩约束，建立非凸极小化模型。运用增广Lagrangian方法和交替极小化(AM, alternating minimization)方法给出数值计算方法。数值实验表明，新约束条件下的分割效果优于原始数据作为字典；非凸迫近p范数的分割效果优于凸的核范数和l1范数。
4.给出总变差(TV, total variation)正则化的两种改进模型。第一种是加权总广义变差(TGV, total generalized variation)正则化的Mumford-Shah模型。给出了加权TGV的定义。新模型利用图像的2阶加权TGV半范作为正则项，利用水平集函数的2阶加权TGV半范近似边界长度。对未知函数分别利用交替split-Bregman方法、Fenchel对偶方法及FISTA(fast iterative shrinkage-thresholding algorithm)给出数值计算模型。仿真实验结果表明，利用图像的2阶加权TGV半范的去噪效果优于常用的梯度模2范数和加权TV半范正则化；利用水平集函数的2阶加权TGV半范近似边界长度的边缘检测效果优于传统的TV半范和加权TV半范约束。第二种是非局部总变差正则化的活动轮廓模型。在活动轮廓模型的全局连续极小化方法基础上，利用非局部总变差作为边界长度正则项。数值实验表明，新模型能将图像中的主体结构和有用的精细结构很好地分割出来。
5.提出了基于拓扑优化的非线性复扩散方法。针对线性扩散会使图像边缘模糊，基于拓扑优化思想，对每个像素点的线性复扩散系数扰动，使得拓扑导数最小的扩散系数是最优的，选择拓扑导数足够小的像素点，对这些像素点用最优扩散系数进行扩散。给出了使算法迭代终止的判据。这里选取的扩散系数具有各向异性特性，有利于去除边缘上的噪声，并能很好地保留边缘。实验表明，利用新方法对原始噪声图像处理后，实部图像具有很好的去噪效果，并能有效减少阶梯效应；虚部图像则很好地保留了图像边缘。
Recently, many researches are focused on the applications of variationalregularization method in image processing. Many classical models and methods werebuilt up. Some new notations, models and methods are proposed in this thesis. Imagesegmentation is mainly concernd. The main work of this thesis includes the followingaspects:
Variational model for image segmentation consists of data term and regularizationterm. Usually data term is chosen as L2norm, and regularization term is determined bythe prior assumption. We present a novel model in the maximum a posterior probabilityframework. A new reweighted L2norm is used in the data term, which shares theadvantages of L2and mixed L21norm. An edge weighting function is addressed in theregularization term, which enforces the ability to reduce the outlier effects and preserveweak edges. An improved region-based graph cuts algorithm is proposed to solve thismodel efficiently. Numerical experiments show our method can get better segmentationresults, especially in terms of removing outliers and preserving weak edges.
A general equivalent model is introduced based on the convex relaxation model ofa class of vector-valued minimization problems. The presented model can be solvedwith the split-Bregman algorithm. Therefore the computational efficiency is greatlyimproved. The method is applied to the Vese-Chan multi-phase segmentation model andMumford-Shah model. Numerical experiments show our method has fast computingspeed and good segmentation results, and is robust to the initial condition.
Image superpixels segmentation is considered as the subspace clustering problem.A new constraint condition is presented to be equivalent to use the clean data asdictionary. The nonconvex proximal p-norm of the coefficients matrix is used forsparse constraint, and the nonconvex proximal p-norm of the singular values of thecoefficients matrix is used for low-rank constraint. Then a nonconvex minimizationmodel is proposed. The augmented Lagrangian method and the AM (alternatingminimization) method are applied for solving the unknown matrices. The numericalexperiments show that the presented constraint condition is better than using the originaldata as dictionary, and the nonconvex proximal p-norm has better segmentation resultthan the convex nuclear norm andl1norm.
Two improved models of the TV (total variation) regularization are presented. Oneis the Mumford-Shah model based on weighted TGV (total generalized variation). Theweighted TGV is defined. The second order weighted TGV semi-norm of images isused as the regularization term. Besides, the second order weighted TGV semi-norm ofthe level set function is used for approximating length of the boundary. The numericalcalculation model for solving the unknown functions is presented by using alternatingsplit-Bregman method、 Fenchel dual method and FISTA (fast iterativeshrinkage-thresholding algorithm) separately. Simulation results show that the use ofsecond order weighted TGV semi-norm of images has better denoising effect than thecommon L2norm of gradient norm and the weighted TV semi-norm. And the result ofedge detection is better than the traditional TV semi-norm and weighted TV semi-normby using second order weighted TGV semi-norm of the level set function toapproximate length of the boundary. Another is the active contour model based on thenonlocal TV regularization. Based on the continuous global minimization approach forthe active contour model, the nonlocal TV is used for the regularization term ofboundary length. Numerical experiments show the new model can segment the mainstructures and the useful fine structures well.
The nonlinear complex diffusion method based on topological optimization ispresented. Based on the topological optimization idea, the linear complex diffusioncoefficients at each pixel are perturbed. The diffusion coefficients corresponding to thesamllest topological derivative are optimal. Then the pixels having the enough smalltopological derivatives are chosen, and diffusion is applied to them using the optimaldiffusion coefficients. Stop criteria of the algorithm is introduced. Diffusion coefficientschosen here have the anisotropic property thus can remove noise along edges andpreserve edges well. Experiments show the real part shows better denoising effect andedges are well preserved in the imaginary part after the original noisy image isprocessed using our method. Besides, our method can reduce staircase effect effectively.
1.图像分割的变分模型包含数据项和正则项，通常，数据项利用L2范数，正则项由先验假设确定。在最大后验概率框架下，给出了一个新的离散模型。数据项利用迭代重加权L2范数，同时具备L2范数计算简单和混合L21范数对野点鲁棒的优点；在正则项中，给出了一个新的边界罚函数，增强了模型去除野点的能力，并能保弱边。提出一种改进的graph cuts算法求解模型。数值实验表明，新模型具有很好的图像分割效果，尤其体现在去除野点和保弱边两方面。
2.研究了一类向量值极小化问题的凸松弛方法，给出了适用于split-Bregman快速算法的一般性等价模型。Vese-Chan多相分割方法和基于分片常数水平集函数的Mumford-Shah方法是新模型的特例。数值实验表明，在Vese-Chan方法和Mumford-Shah方法中应用split-Bregman算法，具有较快的运算速度和较好的分割效果，且对初始条件是鲁棒的。
3.将图像超像素分割看作子空间聚类问题。给出一个约束条件，等价于以干净数据为字典。利用系数矩阵的非凸迫近p范数作为稀疏约束，利用系数矩阵奇异值的非凸迫近p范数作为低秩约束，建立非凸极小化模型。运用增广Lagrangian方法和交替极小化(AM, alternating minimization)方法给出数值计算方法。数值实验表明，新约束条件下的分割效果优于原始数据作为字典；非凸迫近p范数的分割效果优于凸的核范数和l1范数。
4.给出总变差(TV, total variation)正则化的两种改进模型。第一种是加权总广义变差(TGV, total generalized variation)正则化的Mumford-Shah模型。给出了加权TGV的定义。新模型利用图像的2阶加权TGV半范作为正则项，利用水平集函数的2阶加权TGV半范近似边界长度。对未知函数分别利用交替split-Bregman方法、Fenchel对偶方法及FISTA(fast iterative shrinkage-thresholding algorithm)给出数值计算模型。仿真实验结果表明，利用图像的2阶加权TGV半范的去噪效果优于常用的梯度模2范数和加权TV半范正则化；利用水平集函数的2阶加权TGV半范近似边界长度的边缘检测效果优于传统的TV半范和加权TV半范约束。第二种是非局部总变差正则化的活动轮廓模型。在活动轮廓模型的全局连续极小化方法基础上，利用非局部总变差作为边界长度正则项。数值实验表明，新模型能将图像中的主体结构和有用的精细结构很好地分割出来。
5.提出了基于拓扑优化的非线性复扩散方法。针对线性扩散会使图像边缘模糊，基于拓扑优化思想，对每个像素点的线性复扩散系数扰动，使得拓扑导数最小的扩散系数是最优的，选择拓扑导数足够小的像素点，对这些像素点用最优扩散系数进行扩散。给出了使算法迭代终止的判据。这里选取的扩散系数具有各向异性特性，有利于去除边缘上的噪声，并能很好地保留边缘。实验表明，利用新方法对原始噪声图像处理后，实部图像具有很好的去噪效果，并能有效减少阶梯效应；虚部图像则很好地保留了图像边缘。
Recently, many researches are focused on the applications of variationalregularization method in image processing. Many classical models and methods werebuilt up. Some new notations, models and methods are proposed in this thesis. Imagesegmentation is mainly concernd. The main work of this thesis includes the followingaspects:
Variational model for image segmentation consists of data term and regularizationterm. Usually data term is chosen as L2norm, and regularization term is determined bythe prior assumption. We present a novel model in the maximum a posterior probabilityframework. A new reweighted L2norm is used in the data term, which shares theadvantages of L2and mixed L21norm. An edge weighting function is addressed in theregularization term, which enforces the ability to reduce the outlier effects and preserveweak edges. An improved region-based graph cuts algorithm is proposed to solve thismodel efficiently. Numerical experiments show our method can get better segmentationresults, especially in terms of removing outliers and preserving weak edges.
A general equivalent model is introduced based on the convex relaxation model ofa class of vector-valued minimization problems. The presented model can be solvedwith the split-Bregman algorithm. Therefore the computational efficiency is greatlyimproved. The method is applied to the Vese-Chan multi-phase segmentation model andMumford-Shah model. Numerical experiments show our method has fast computingspeed and good segmentation results, and is robust to the initial condition.
Image superpixels segmentation is considered as the subspace clustering problem.A new constraint condition is presented to be equivalent to use the clean data asdictionary. The nonconvex proximal p-norm of the coefficients matrix is used forsparse constraint, and the nonconvex proximal p-norm of the singular values of thecoefficients matrix is used for low-rank constraint. Then a nonconvex minimizationmodel is proposed. The augmented Lagrangian method and the AM (alternatingminimization) method are applied for solving the unknown matrices. The numericalexperiments show that the presented constraint condition is better than using the originaldata as dictionary, and the nonconvex proximal p-norm has better segmentation resultthan the convex nuclear norm andl1norm.
Two improved models of the TV (total variation) regularization are presented. Oneis the Mumford-Shah model based on weighted TGV (total generalized variation). Theweighted TGV is defined. The second order weighted TGV semi-norm of images isused as the regularization term. Besides, the second order weighted TGV semi-norm ofthe level set function is used for approximating length of the boundary. The numericalcalculation model for solving the unknown functions is presented by using alternatingsplit-Bregman method、 Fenchel dual method and FISTA (fast iterativeshrinkage-thresholding algorithm) separately. Simulation results show that the use ofsecond order weighted TGV semi-norm of images has better denoising effect than thecommon L2norm of gradient norm and the weighted TV semi-norm. And the result ofedge detection is better than the traditional TV semi-norm and weighted TV semi-normby using second order weighted TGV semi-norm of the level set function toapproximate length of the boundary. Another is the active contour model based on thenonlocal TV regularization. Based on the continuous global minimization approach forthe active contour model, the nonlocal TV is used for the regularization term ofboundary length. Numerical experiments show the new model can segment the mainstructures and the useful fine structures well.
The nonlinear complex diffusion method based on topological optimization ispresented. Based on the topological optimization idea, the linear complex diffusioncoefficients at each pixel are perturbed. The diffusion coefficients corresponding to thesamllest topological derivative are optimal. Then the pixels having the enough smalltopological derivatives are chosen, and diffusion is applied to them using the optimaldiffusion coefficients. Stop criteria of the algorithm is introduced. Diffusion coefficientschosen here have the anisotropic property thus can remove noise along edges andpreserve edges well. Experiments show the real part shows better denoising effect andedges are well preserved in the imaginary part after the original noisy image isprocessed using our method. Besides, our method can reduce staircase effect effectively.
引文
[1] Averintsev M B. The description of Markov random fields by Gibbs distribution.Teor. Verojatrost. Primen,1972,17(1):21-35.
[2] Besag J E. Spatial interaction and the statistical analysis of lattice systems (withdisscussion). J. Roy. Statist. Soc. B.,1974,36(2):192-236.
[3] Besag J E. On the statistic analysis of dirty pictures. J. Roy. Statist. Soc. B.,1986,48(3):259-302.
[4] Zhu S C, Mumford D. Prior learning and Gibbs reaction-diffusion. IEEE Trans. onPAMI,1997,19:1236-1250.
[5] Giordana N, Pieczynaki W. Estimation of generalized multisensor Hidden MarkovChains and unsupervised image segmentation. IEEE Trans. on PAMI,1997,19(5):465-475.
[6] Huber P J. Robust Statistics. New Yoke: Wiley,1981.
[7] Geman S, Geman D. Stochastic relaxation, Gibbs distributions, and the Bayesianrestoration of images. IEEE Trans. PAMI,1984,6:721-741.
[8] Meyer Y. Wavelets Algorithms and Applications. SIAM, Philadelphia,1993.
[9] Calderon, Albert P. An atomic decomposition of distributions in parabolic HPspaces.Advances in Math.,1977,25(3):216-225.
[10] Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. OnPure and Appl. Math.,1988,41(7):909-996.
[11] Mallat S. Multifrequency channel decomposition of images and wavelet models.IEEE Transaction in Acoustic Speech and Signal Processing,1989,37(12):2091-2110.
[12] Mallat S. A Wavelet Tour of Signal Processing2nd. Academic Press:1999.
[13] Mallat S著,扬力华等译.信号处理的小波导引.北京:机械工业出版社,2002.
[14] Perona P, Malik J. Scale space and edge detection using anisotropic diffusion.IEEE Trans. on Pattern Analysis and Machine Intelligence,1990,12(7):629-639.
[15] Meyer Y. Oscillating Patterns in Image Processing and Nonlinear EvolutionEquations. AMS, Providence, RI,2001.
[16] Catte F, Lions P L, Morel J M, et al. Image selective smoothing and edge detectionby nonlinear diffusion. SIAM Journal of Numerical Analysis,1992,29(1):182-193.
[17] Osher S, Rudin L. Feature-oriented image enhancement using shock filters. SIAMJournal of Numerical Analysis,1990,27(4):919-940.
[18] Weickert J. Theoretical foundation of anisotropic diffusion in image processing.Computing Supplement,1996,11:221-236.
[19] Weickert J. Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart,1998.
[20] Weickert J. Coherence-enhancing diffusion filtering. International Journal ofComputer Vision,1999,31(2):111-127.
[21] Alvarez L, Mazorra L. Signal and image restoration using shock filter andanisotropic diffusion. SIAM Journal of Numerical Analysis,1994,31(2):590-605.
[22] Aubert G, Kornprobst P. Mathematical Problems in Image Processing: PartialDifferential Equations and the Calculus of Variations. Applied MathematicalSciences147, Springer-Verlag,2006.
[23] Mumford D, Shah J. Optimal Approximation by piecewise smooth functions andassociated variational problems. Comm. Pure. Appl. Math.1989,42(5).577-685
[24] Rudin L I, Osher S J, Fatemi E. Nonlinear total variation based noise removalalgorithms. Phys. D,1992,60:259–268.
[25] Guirgu De, Carriero M, Leac A. Existence theorem for a minimum problem withfree discontinuity set. Archive for Rational Mechanics and Analysis,1989,108(3):195-218.
[26] Geman S, Geman D. Stochastic relaxation, Gibbs distributions, and the bayesianrestoration of images. IEEE Transaction on Pattern Analysis and MachineIntelligence,1984,6:721-741.
[27] Chan T, Jianhong Shen. Image processing and analysis: Variational, PDE, Wavelet,and Stochastic methods. Society for Industrial and Applied Mathematics (SIAM),Philadelphia,2005.
[28] Kass M, Witkin A, Terzopoulos D. Snakes: Active contour models. InternationalJournal of Computer Vision,1987,1(4):321-331.
[29] Osher S, Sethian J A. Fronts propagating with curvature-dependent speed:Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys.,79:12-49.
[30] Tikhonov A N, Arsenin V Y. Solutions of Ill-posed Problems. Winston and Sons,Washington, D.C.,1977.
[31] Meyer Y. Oscillating patterns in image processing and nonlinear evolutionequations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, UniversityLecture Series, No.22, Oxford: American Mathematical Society, Providence, RI,2001.
[32] Vese L A, Osher S J. Modeling textures with total variation minimization andoscillating patterns in image processing. Journal of Scientific Computing,2003,December,19(1-3).
[33] Osher S, Sole A, Vese L. Image decomposition and restoration using totalvariation minimization and theH1norm. SIAM Journal on MultiscaleModeling and Simulation,2003,1-3:349-370.
[34] Aujol J F, Aubert G, Feraud B L, et.al. Image decomposition: application totextured images and SAR images, Technical Report ISRN I3S/RR-2003-01-FR,INRIA—Project ARIANA, Sophia Antipolis,2003.
[35] Aujol J F, Chambolle A. Dual norms and image decomposition models. Intl. J.Comp. Vis.,2005,63(1):85-104.
[36] Daubechies I, Teschke G. Wavelet based image decomposition by variationalfunctionals. Proc. SPIE, Wavelet Applications in Industrial Processing,2004,5266:94-105.
[37] Daubechies I, Teschke G. Variational image restoration by means of wavelets:simultaneous decomposition, deblurring and denoising. Applied andComputational Harmonic Analysis,2005,19(1):1-16.
[38] Linh Lieu, Luminita Vese. Image Restoration and Decompostion via BoundedTotal Variation and Negative Hilbert-Sobolev Spaces. Applied Mathematics andoptimization,2008,58(2):167-193.
[39] Le T, Vese L. Image decomposition using the total variation and div (BMO).Multiscale Modeling and Simulation,2005,4(2):390-423.
[40] Starck J L, Elad M, Donoho D L. Image decomposition via the combination ofsparse representations and a variational approach. IEEE Trans. Image Processing2005,14(10):1570-1582.
[41] Aubert G, Aujol J F. Modeling very oscillating signals. Application to imageprocessing, AMO2005,51(2):163-182.
[42] Kindermann S, Osher S, Xu J. Denoising by BV-Duality. Journal of ScientificComputing,2006,28(2-3):411–444.
[43] Vese L. A study in the BV space of a denoising-deblurring variational problem.App. Math. Optm,2001,44:131-161.
[44] Burger M, Frick K, Osher S, et al. Inverse total variation flow. UCLA CAMReport06-24,2006.
[45] Burger M, Gilboa G, Osher S, et al. Nonlinear inverse scale space methods. Comm.Math. Sci.,2006,4(1):175-208.
[46] Burger M, Osher S, Jinjun Xu, et al. Nonlinear inverse scale space methods forimage restoration. Lecture Notes in Computer Science,2005,3752:25-36.
[47] Chan T F, Esedoglu S. Aspects of total variation regularizedL1functionapproximation. SIAM.Appl. Math.,2005,65(5):1817-1837.
[48] Chan T F, Esedoglu S, Park F. A fourth order dual method for staircase reductionin texture extraction and image restoration problems. UCLA CAM Report05-28,2005.
[49] Chambolle A. An algorithm for total variation regularization and denoising.Journal of Mathematical Imaging and Vision,2004,20:89-97.
[50] Chan T F, Shen J. Mathematical models for local nontexture inpaintings. SIAMJournal on Applied Mathematics,2002,62:1019-1043.
[51] Shen J. Inapainting and the fundamental problem of image processing. SIAMNews,2003,36(5):1-4.
[52] Chan T F, Shen J, Vese L. Variational pde models in image processing. Notice ofthe AMS,2003,50(1):14-26.
[53] Bertalmio M, Sapiro G, Caselles V, et al. Image inpainting. Proceedings of ACMSIGGRAPH, also ACM Transactions on Graphics,2000,19(3):417-424.
[54] Sapiro G. Image inpainting. SIAM News,2002,35(4):1-2.
[55] Rane S D, Sapiro G, Bertalmio M. Structure and texture filling-in of missingimage blocks in wireless transmission and compression applications. IEEETransactions on Image Processing,2003,12(3):296-303.
[56] Bertalmio M, Vese L, Sapiro G, et al. Simultaneous structure and texture imageinpainting. IEEE Trans. Image. Process,2003,12:882–889.
[57] Chan T F, Shen J. Non-texture inpainting by curvature-driven diffusion. Journal ofVisual Communication and Image Representation,2001,12(4):436-449.
[58] Chan T F, Shen J. Image Processing and Analysis. SIAM, Philadelphia,2005.
[59] Chandler D. Introduction to Modern Statistical Mechanics. Oxford UniversityPress, New York, Oxford,1987.
[60] Gibbs W. Elementary Principles of Statistical Mechanics. Yale University Press,1902.
[61] Aubert G, Kornprobst P. Mathematical Problems in Image Processing.Springer-Verlag, New York,2001.
[62] Hadamard J, Sur Les. Problems and Derivees Partielle et Leur SignificationPhysique. Bull University Princeton,1902:49-52.
[63]肖庭延,于慎根,王彦飞.反问题的数值解法.科学出版社,2003.
[64]刘继军.不适定问题的正则化方法及应用.科学出版社，2005.
[65]马兴瑞,陶良,黄文虎.弹性波反演方法及其应用.科学出版社,1999.
[66]黄光远,刘小军.数学物理反问题.山东科学技术出版社,1993.
[67] You Y, Kaveh M. Fourth-order partial differential equations for noise removal.IEEE Trans. Image Process.,2000,9(10):1723-1730.
[68] Lysaker M, Lundervold A, Tai X C. Noise removal using fourth order partialdifferential equation with applications to medical magnetic resonance images inspace and time. IEEE Trans. Image Process.,2003,12(12):1579-1590.
[69] Mohammad R H. An anisotropic fourth-order diffusion filter for image noiseremoval. Int. J. Comput. Vis.,2011,92(2):177-191.
[70] Guidotti P, Longo K. Two enhanced fourth order diffusion models for imagedenoising. J. Math. Imaging and Vis.,2011,40(2):188-198.
[71] Lysaker M, Tai X C. Iterative image restoration combining total variationminimization and a second order functional. Int. J. Comput. Vis.,2006,66(1):5-18.
[72] Lysaker M, Osher S, Tai X C. Noise removal using smoothed normals and surfacefitting. IEEE Trans. Image Process.,2004,13(10):1345-1357.
[73] Rahman T, Tai X C, Osher S. A TV-stokes denoising algorithm. SSVM,2007,473-483.
[74] Tony Chan, Selim Esedoglu, Mila Nikolova. Algorithms for finding globalminimizers of image segmentation and denoising models. SIAM Journal onApplied Mathematics,2006,66:1932-1648.
[75] Chan T, Vese L. Active contours without edges. IEEE Transactions on ImageProcessing,2001,10:266-277.
[76] Caselles V, Kimmel R, Sapiro G. Geodesic active contours. Computer Vision,IEEE International Conference on,1995,0:694.
[77] 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.
[78] Burger M, Hackl B, Ring W. Incorporating topological derivatives into level setmethods. Journal of Computational Physics,2004,194(1):344–362.
[79] Bae E, Tai X C. Efficient global optimization for the multiphase Chan-Vese modelof image segmentation by graph cuts. In EMMCVPR, Lecture Notes in ComputerSciences.2009,5681:28-41.
[80] Zhu S C, Yuille A. Region Competition: Unifying Snakes, Region Growing, andBayes/mdl for Multiband image Segmentation. IEEE Trans. Pattern Anal.Machine Intell,1996,18(9):884-900.
[81] Chen S F, Cao L L, Wang Y M, et al. Image Segmentation by MAP-MLEstimations. IEEE Trans. on Image Processing,2010,19(9):2254-2264.
[82] Nikolova M. Minimizers of Cost-Functions Involving Nonsmooth Data-FidelityTerms: Application to the Processing of Outliers. SIAM Journal on NumericalAnalysis,2003,40(3):965-994.
[83] Daubechies I, DeVore R, Fornasier M, et al. Iteratively Re-weighted LeastSquares Minimization for Sparse Recovery. Communications on Pure and AppliedMathematics,2010,63(1):1-38.
[84] Boykov Y, Veksler O, Zabih R. Fast Approximate Energy Minimization viaGraph Cuts. IEEE Trans. Pattern Anal. Mach. Intell.,2001,23(11):1222–1239.
[85] Tu Z, Zhu S C. Image Segmentation by Data-Driven Markov Chain Monte Carlo.IEEE Trans. Pattern Anal. Mach. Intell,2002,24(5):657–673.
[86] Tu Z. An Integrated Framework for Image Segmentation and PerceptualGrouping. Proc.10thInt. Conf. on Computer Vision, Beijing, China, Oct.17-21,2005,670–677.
[87] Kolmogorov V, Zabih R. What Energy Functions Can Be Minimized via GraphCuts. IEEE Transactions on Pattern Analysis and Machine Intelligence,2004,26(2):147–159.
[88] Komodakis N, Tziritas G, Paragios N. Fast, Approximately Optimal Solutions forSingle and Dynamic MRFs. Proc. IEEE Conference on Computer Vision andPattern Recognition, Minneapolis, MN, June17-22,2007,1–8.
[89] Boykov Y, Kolmogorov V. Computing Geodesics and Minimal Surfaces viaGraph Cuts. Proc.9thIEEE International Conf. on Computer Vision, Nice, France,Oct.13-16.2003,26-33.
[90] Boykov Y, Kolmogorov V, Cremers D, et al. An Integral Solution to SurfaceEvolution PDEs via Geo-Cuts. Proc.3rdEuropean conference on Computer Vision(3), Vol.3953,2006,409–422.
[91] Darbon J, Sigelle M. Image Restoration with Discrete Constrained TotalVariation Part I: Fast and Exact Optimization. J. Math. Imaging Vis.,2006,26(3):261–276.
[92] Carson C, Belongie S, Greenspan H, et al. Blobworld: Image SegmentationUsing Expectation-Maximization and Its Application to Image Querying. IEEETrans. Pattern Anal. Mach. Intell.,2002,24(8):1026–1038.
[93] F rstner W. A Framework for Low Level Feature Extraction. Proc.3rd Europeanconf. on Computer Vision,1994,801:383–394.
[94] G rding J, Lindeberg T. Direct Computation of Shape Cues Using Scale-AdaptedSpatial Derivative Operators. Int. J. Comput. Vis.,1996,17(2):163–191.
[95] Malik J, Perona P. Preattentive Texture Discrimination with Early VisionMechanisms. J. Opt. Soc. Amer. A, Opt. Image Sci.,1990,7(5):923–932.
[96] Osborne M R. Finite Algorithms in Optimization and Data Analysis. John Wiley&Sons Ltd., Chichester,1985.
[97] Bae E, Tai X C. Graph Cut Optimization for the Piecewise Constant Level SetMethod Applied to Multiphase Image Segmentation. Proc.2nd Int. Conf. on ScaleSpace and Variational Methods in Computer Vision, Voss, Norway, Jun.1-5,2009,1-13.
[98] Felzenszwalb P, Huttenlocher D. Efficient Graph-Based Image Segmentation. Int.J. Comput. Vis.,2004,59(2):167–181.
[99] Unnikrishnan R, Pantofaru C, Hebert M. Toward Objective Evaluation of ImageSegmentation Algorithms. IEEE Trans. Pattern Anal. Mach. Intell.,2007,29(6):929–944.
[100] Yin W, Osher S, Goldfarb D. Bregman iterative algorithms for l1-minimizationwith applications to compressed sensing. SIAM Journal on Imaging Sciences,2008,1(1):142-168.
[101] Goldstein T, Osher S. The split Bregman algorithm for L1regularized problems.SIAM J on Imaging Sciences,2(2):323-343.
[102] Yin W, Wang Y, Yang J, et al. A new alternating minimization algorithm fortotal variation image reconstruction. SIAM Journal on Imaging Sciences, Toappear.
[103] Osher S, Burger M, Goldfarb D, et al. An iterative regularization method for totalvariation-based image restoration. MMS,2005,4:460-489.
[104] Goldstein T, Bresson X. Geometric application of the split Bregman method:segmentation and surface reconstruction. Journal of Scientific Computing,2009,45(1-3):272-293.
[105] Cai J F, Osher S, Shen Z W. Split Bregman Methods and Frame Based ImageRestoration. Multiscale Model. Simul.,8(2):337–369.
[106] Plonka G, Ma J W. Curvelet-wavelet regularized split bregman iteration forcompressed sensing. Int. J. Wavelets Multiresolut Inf. Process,2011,09:79.
[107] Bresson X, Esedoglu S, Vandergheynst P, et al. Fast global minimization of theactive contour/snake model. Journal of Mathematical Imagingand Vision,2007,28:151-167.
[108] Blomgren P, Chan T F. Color TV: total variation methods for restoration ofvector-valued images. IEEE Trans. Image Process.,1998,7(3):304-309.
[109] Almgren F, Taylor J E, Wang L. Curvature-driven flows: a variational approach.SIAM J. Control Optim.,1993,31(2):387-438.
[110] Chambolle A. An algorithm for mean curvature motion. Interfaces Free Bound,2004,6(2):195-218.
[111] Xavier Bresson, Tony Chan. Active contours based on chambolle's meancurvature motion. IEEE International Conference on Image Processing,2007,33-36.
[112] Brown E S, Chan T, Bresson X. A convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Sha-h Segmentation. Technical report, UCLA,2010. ftp://ftp.math.ucla.edu/pub/camreport/cam10-43.pdf.
[113] Arrow K J, Hurwicz L, Uzawa H. Studies in linear and non-linear programming.Vol. II, Stanford, Calif: Stanford University Press,1958:124-158.
[114] Popov L D. A modification of the Arrow-Hurwitz method of search for saddlepoints. Mathematics Zametki,1980,28(5):777–784.
[115] Potts R B, Domb C. Some generalized order-disorder transformations.Mathematical Proceedings of the Cambridge Philosophical Society,1952,48:106–109.
[116] Boykov Y, Veksler O, Zabih R. A new algorithm for energy minimization withdiscontinuities. In EMMCVPR’99: Proceedings of the Second InternationalWorkshop on Energy Minimization Methods in Computer Vision and PatternRecognition, pages205–220, London, UK,1999. Springer-Verlag.
[117] Lie J, Lysaker M, Tai X C. A variant of the level set method and applications toimage segmentation. Mathematics of Computation,2006,75(255):1155–1174.
[118]汪云飞,毕笃彦,黄飞.一种用于图像分割的等周算法改进.西安电子科技大学学报,2012,39(2):87-94.
[119]刘建磊,冯大政.一种全局最优的非匀质图像分割算法.西安电子科技大学学报,2011,38(2):66-77.
[120]刘汉强,焦李成,赵风.基于局部相似性测度的SAR图像多层分割.西安电子科技大学学报,2010,37(6),1071-1076.
[121] Elhamifar E, Vidal R. Sparse subspace clustering. IEEE Conference onComputer Vision and Pattern Recognition, Miami, FL, June20-252009,2790–2797.
[122] Cheng B, Liu G C, Wang G D et al. Multi-task low-rank affinity pursuit forimage segmentation. International Conference on Computer Vision, Barcelona,Nov.6-132011. Washington, DC, USA: IEEE Computer Society,2439-2446.
[123] Liu G C, Lin Z C, Yan S C et al. Robust recovery of subspace structures bylow-rank representation. IEEE Transactions on Pattern Analysis and MachineIntelligence,2013,35(1):171-184.
[124] Zhuang L S, Gao H Y, Ma Y et al. Non-nenagtive low rank and sparse graph forsemi-supervised learning. IEEE Conference on Computer Vision and PatternRecognition, Providence, RI, June16-212012,2328-2335.
[125] Chartrand R, Yin W T. Iteratively reweighted algorithms for compressivesensing. IEEE International Conference on Acoustics, Speech, and SignalProcessing, March31-April42008,3869-3872.
[126] Chartrand R, Staneva V. Restricted isometry properties and nonconvexcompressive sensing. Inverse Problems,2008,24(035020):1–14.
[127] Saab R, Chartrand R, Yilmaz O. Stable sparse approximations via nonconvexoptimization. IEEE International Conference on Acoustics, Speech, and SignalProcessing, Las Vegas, NV, March31-April4,2008,3885-3888.
[128] Chartrand R. Nonconvex splitting for regularized low-rank+sparsedecomposition. IEEE Transactions on Signal Processing,2012,60(11):5810–5819.
[129] Mori G, Ren X F, Alexei A E et al. Recovering human body configurations:combining segmentation and recognition. Proceedings of the2004IEEEComputer Society Conference on Computer Vision and Pattern Recognition,27June-2July2004,2:II-326-333.
[130] Wei S M, Lin Z C. Analysis and improvement of low rank representation forsubspace segmentation.[2011.7]. http://arxiv.org/abs/1107.1561v1.
[131] William M R. Objective criteria for the evaluation of clustering methods. Journalof the American Statistical Association,1971,66(336):846-850.
[132] Marina M. Comparing clustering: An axiomatic view. Proceedings of the22ndinternational conference on Machine learning. New York, NY, USA: ACM,2005:577-584.
[133] Arbelaez P, Maire M, Fowlkes C et al. From contour to regions: An empiricalevaluation. IEEE conference on Computer Vision and Pattern Recognition, Miami,FL, June20-252009,2294-2301.
[134]李小毛朱琳琳唐延东.基于Mumford-Shah模型和开样条曲线的边界检测.自动化学报,2009,35(2):132-136.
[135] Wang L L, Shi Y Y, Tai X C. Robust edge detection using Mumford-Shahmodel and binary level set method. In: Proceedings of the Third InternationalConference on Scale Space and Variational Methods in Computer Vision. VerlagBerlin, Heidelberg: Springer,2011.291-301.
[136] Rudin L I, Osher S, Fatemi E. Nonlinear total variation based noise removalalgorithms. Physica D,1992,60(1-4):259-268.
[137]何进荣.基于加权范数迭代算法的总变差正则化图像复原方法[硕士学位论文],武汉理工大学,2009.
[138] Chambolle A, Lions P L. Image recovery via total variation minimization andrelated problems. Numeric Mathematic,1997,76(2):167-188.
[139] Chan T F, Esedoglu S, Park F E. A fourth order dual method for staircasereduction in texture extraction and image restoration Problems, In: Proceedings ofthe17thIEEE International Conference on Image Processing. Hong Kong: IEEE,2010.4137-4140.
[140] Chan T F, Marquina A, Mulet P. Higher order total variation-based imagerestoration. SIAM Journal on Scientific Computing,2000,22(2):503-516.
[141] P schl C, Scherzer O. Characterization of minimizers of convex regularizationfunctional. Contemporary Mathematics,2008,451:219-248.
[142] Bredies K, Kunisch K, Pock T. Total generalized variation. SIAM Journal onImaging Sciences,2010,3(3):492–526.
[143] Carter J. Dual Methods for Total Variation Based Image Restoration [Ph. D.dissertation], University of California, Los Angeles,2001.
[144] Beck A, Teboulle M. A fast iterative shrinkage-thresholding algorithm for linearinverse problems. SIAM Journal on Imaging Sciences,2009,2(1):183-202.
[145]王旭东,冯象初,霍雷刚.去除乘性噪声的重加权各向异性全变差模型.自动化学报,2012,38(3),444-451.
[146]韩雨,王卫卫,冯象初.基于迭代重加权的非刚性图像配准.自动化学报,2011,37(9):1059-1066.
[147] Chambolle A. An algorithm for total variation minimization and applications.Journal of Mathematical Imaging and Vision,2004,20(1-2):89–97.
[148] Canny J F. A computational approach to edge detection. IEEE Transactions onPattern Analysis and Machine Intelligence,1986,8(6):679-698.
[149] Smith S M, Brady J M. SUSAN-a new approach to low level image processing.International Journal of Computer Vision,1997,23(1):45-78.
[150] Budes A, Coll B, Morel J M. A review of image denoising algorithms, with anew one. SIAM Multiscale Modeling and Simulation (MMS),2005,4(2):490–530.
[151] Gilboa G, Osher S. Nonlocal Linear Image Regularization and SupervisedSegmentation. SIAM Multiscale Modeling and Simulation (MMS),2007,6(2):595630.
[152] Zhou D, Scholkopf B. A Regularization Framework for Learning from GraphData. In Work shop on Statistical Relational Learning and Its Connections toOther Fields,2004.
[153] Gilboa G, Osher S. Nonlocal Operators with Applications to Image Processing.SIAM Multiscale Modeling and Simulation (MMS),2008,7(3):1005-1028.
[154] Gilboa G, Darbon J, Osher S, et al. Nonlocal Convex Functionals for ImageRegularization. UCLA CAM Report,2006.
[155] Kindermann S, Osher S, Jones P W. Deblurring and Denoising of Images byNonlocal Functionals. SIAM Multiscale Modeling and Simulation (MMS),2005,4(4):1091-1115.
[156] Kichenassamy S, Kumar A, Olver P, et al. Conformal Curvature Flows: FromPhase Transitions to Active Vision. In Archive for Rational Mechanics andAnalysis,1996,134:275–301.
[157] Merriman B, Bence J K, Osher S J. Motion of multiple junctions: a level setapproach. Journal of Computational Physics,1994,112(2):334–363.
[158] Bresson X, Chan T F. Nonlocal Unsupervised Variational Image SegmentationModels. UCLA CAM Report,2008.
[159] Céa J, Garreau S, Guillaume P, et al. The shape and topological optimizationsconnection. Comput Method Appl Mech Eng,2000,188(4):713-726.
[160] Eschenauer H A, Kobelev V V, Schumacher A. Bubble method for topology andshape optimization of structures. Struct Optim,1994,8:42-51.
[161] Novotny A A, Feijóo R A, Taroco E, Padra C. Topological sensitivity analysis.Comput Method Appl Mech Eng,2003,192:803-29.
[162] Belaid L J, Jaoua M, Masmoudi M, et al. Application of the topological gradientto image restoration and edge detection. Engineering Analysis with BoundaryElement,2008,32(11):891-899.
[163] Frangakis A S, Hegerl R. Noise reduction in electron tomographicreconstructions using nonlinear anisotropic diffusion. J Struct Biol,2001,1(135):239-250.
[164] Giboa G, Sochen N, Zeevi Y Y. Image enhancement and denoising by complexdiffusion processes. IEEE Transactions on Pattern Analysis and MachineIntelligence,2004,26(8):1020–1036.
[165] Gonzalez C R, Woods R E. Digital image processing. Seconded. Prentice Hall,2001.
[166] Sethian J A. Level Set Methods and Fast Marching Methods. CambridgeUniversity Press,1999.
[167] Malladi R, Sethian J A. Level set methods for curvature flow, imageenhancement, and shape recovery in medical images. In: Proceedings of theconference on visualization and mathematics, Springer-Verlag,1997,329–45.
[168] Xu C, Pham D, Prince J. Medical image segmentation using deformable models.SPIE handbook on medical imaging, vol. III: medical image analysis;2000. p.129–74.
[169] Xu C, Pham D, Rettmann M E, et al. Reconstruction of the human cerebralcortex from magnetic resonance images. IEEE Trans Med Imag1999,18(6):467–80.
[170] Li H, Deklerck R, DeCuyper B, et al. Object recognition in brain ct-scans:knowledge-based fusion of data from multiple feature extractors. IEEE Trans MedImag,1995,14:212–29.
[171] Boscolo R, Brown M, McNitt G M. Medical image segmentation withknowledge-guided robust active contours. Radiographics,2002,22(2):437–448.
[172] Held K, Kops E R, Krause B J, et al. Markov random field segmentation of brainmr images. IEEE Trans Med Imag,1997,16(6):878–86.
[173] Zhang Y, Brady M, Smith S. Segmentation of brain MRI images through ahidden markov random field model and the expectationmaximization algorithm.IEEE Trans Med Imag,2001,20(1):45–57.
[174] Larrabide I, Feijóo R A, Novotny A A, et al. Topological derivative: A tool forimage processing. Computers and Structures,2008,86:1386-1403.
[175] Larrabide I, Novotny A A, Feijóo R A, et al. Topological derivative as a tool forimage processing: image segmentation.Technical report15/2006, LNCC,2006.
[176] Banga C, Ghorbel F. Optimal bootstrap sampling for fast image segmentation:application to retina images. In IEEE ICASP93, Minneapolis USA,1993,5:638–641.
[177] Koch I, Marshall G. Bootstrap coverage plots for image segmentation. In IEEEProceedings of ICPR1996,447–451.
[178] Dunn J C. A fuzzy relative of the isodata process and its use in detecting compact,well separated clusters. Journal of Cybernetics,1974,3:32–57.
[179] Bezdek J. Pattern Recognition with fuzzy Objective Function Algorithms. NewYork: Plenum Press,1981.
[180] Gray R M, Linde Y. Vector quantization and predictive quantizer forgauss-markov sources. IEEE Transaction on Communications,1982,30:381–389,.
[181] Pappas T N. An adaptive clustering algorithm for image segmentation. IEEETransactions on Signal Processing,1992,40:901–914.
[2] Besag J E. Spatial interaction and the statistical analysis of lattice systems (withdisscussion). J. Roy. Statist. Soc. B.,1974,36(2):192-236.
[3] Besag J E. On the statistic analysis of dirty pictures. J. Roy. Statist. Soc. B.,1986,48(3):259-302.
[4] Zhu S C, Mumford D. Prior learning and Gibbs reaction-diffusion. IEEE Trans. onPAMI,1997,19:1236-1250.
[5] Giordana N, Pieczynaki W. Estimation of generalized multisensor Hidden MarkovChains and unsupervised image segmentation. IEEE Trans. on PAMI,1997,19(5):465-475.
[6] Huber P J. Robust Statistics. New Yoke: Wiley,1981.
[7] Geman S, Geman D. Stochastic relaxation, Gibbs distributions, and the Bayesianrestoration of images. IEEE Trans. PAMI,1984,6:721-741.
[8] Meyer Y. Wavelets Algorithms and Applications. SIAM, Philadelphia,1993.
[9] Calderon, Albert P. An atomic decomposition of distributions in parabolic HPspaces.Advances in Math.,1977,25(3):216-225.
[10] Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. OnPure and Appl. Math.,1988,41(7):909-996.
[11] Mallat S. Multifrequency channel decomposition of images and wavelet models.IEEE Transaction in Acoustic Speech and Signal Processing,1989,37(12):2091-2110.
[12] Mallat S. A Wavelet Tour of Signal Processing2nd. Academic Press:1999.
[13] Mallat S著,扬力华等译.信号处理的小波导引.北京:机械工业出版社,2002.
[14] Perona P, Malik J. Scale space and edge detection using anisotropic diffusion.IEEE Trans. on Pattern Analysis and Machine Intelligence,1990,12(7):629-639.
[15] Meyer Y. Oscillating Patterns in Image Processing and Nonlinear EvolutionEquations. AMS, Providence, RI,2001.
[16] Catte F, Lions P L, Morel J M, et al. Image selective smoothing and edge detectionby nonlinear diffusion. SIAM Journal of Numerical Analysis,1992,29(1):182-193.
[17] Osher S, Rudin L. Feature-oriented image enhancement using shock filters. SIAMJournal of Numerical Analysis,1990,27(4):919-940.
[18] Weickert J. Theoretical foundation of anisotropic diffusion in image processing.Computing Supplement,1996,11:221-236.
[19] Weickert J. Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart,1998.
[20] Weickert J. Coherence-enhancing diffusion filtering. International Journal ofComputer Vision,1999,31(2):111-127.
[21] Alvarez L, Mazorra L. Signal and image restoration using shock filter andanisotropic diffusion. SIAM Journal of Numerical Analysis,1994,31(2):590-605.
[22] Aubert G, Kornprobst P. Mathematical Problems in Image Processing: PartialDifferential Equations and the Calculus of Variations. Applied MathematicalSciences147, Springer-Verlag,2006.
[23] Mumford D, Shah J. Optimal Approximation by piecewise smooth functions andassociated variational problems. Comm. Pure. Appl. Math.1989,42(5).577-685
[24] Rudin L I, Osher S J, Fatemi E. Nonlinear total variation based noise removalalgorithms. Phys. D,1992,60:259–268.
[25] Guirgu De, Carriero M, Leac A. Existence theorem for a minimum problem withfree discontinuity set. Archive for Rational Mechanics and Analysis,1989,108(3):195-218.
[26] Geman S, Geman D. Stochastic relaxation, Gibbs distributions, and the bayesianrestoration of images. IEEE Transaction on Pattern Analysis and MachineIntelligence,1984,6:721-741.
[27] Chan T, Jianhong Shen. Image processing and analysis: Variational, PDE, Wavelet,and Stochastic methods. Society for Industrial and Applied Mathematics (SIAM),Philadelphia,2005.
[28] Kass M, Witkin A, Terzopoulos D. Snakes: Active contour models. InternationalJournal of Computer Vision,1987,1(4):321-331.
[29] Osher S, Sethian J A. Fronts propagating with curvature-dependent speed:Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys.,79:12-49.
[30] Tikhonov A N, Arsenin V Y. Solutions of Ill-posed Problems. Winston and Sons,Washington, D.C.,1977.
[31] Meyer Y. Oscillating patterns in image processing and nonlinear evolutionequations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, UniversityLecture Series, No.22, Oxford: American Mathematical Society, Providence, RI,2001.
[32] Vese L A, Osher S J. Modeling textures with total variation minimization andoscillating patterns in image processing. Journal of Scientific Computing,2003,December,19(1-3).
[33] Osher S, Sole A, Vese L. Image decomposition and restoration using totalvariation minimization and theH1norm. SIAM Journal on MultiscaleModeling and Simulation,2003,1-3:349-370.
[34] Aujol J F, Aubert G, Feraud B L, et.al. Image decomposition: application totextured images and SAR images, Technical Report ISRN I3S/RR-2003-01-FR,INRIA—Project ARIANA, Sophia Antipolis,2003.
[35] Aujol J F, Chambolle A. Dual norms and image decomposition models. Intl. J.Comp. Vis.,2005,63(1):85-104.
[36] Daubechies I, Teschke G. Wavelet based image decomposition by variationalfunctionals. Proc. SPIE, Wavelet Applications in Industrial Processing,2004,5266:94-105.
[37] Daubechies I, Teschke G. Variational image restoration by means of wavelets:simultaneous decomposition, deblurring and denoising. Applied andComputational Harmonic Analysis,2005,19(1):1-16.
[38] Linh Lieu, Luminita Vese. Image Restoration and Decompostion via BoundedTotal Variation and Negative Hilbert-Sobolev Spaces. Applied Mathematics andoptimization,2008,58(2):167-193.
[39] Le T, Vese L. Image decomposition using the total variation and div (BMO).Multiscale Modeling and Simulation,2005,4(2):390-423.
[40] Starck J L, Elad M, Donoho D L. Image decomposition via the combination ofsparse representations and a variational approach. IEEE Trans. Image Processing2005,14(10):1570-1582.
[41] Aubert G, Aujol J F. Modeling very oscillating signals. Application to imageprocessing, AMO2005,51(2):163-182.
[42] Kindermann S, Osher S, Xu J. Denoising by BV-Duality. Journal of ScientificComputing,2006,28(2-3):411–444.
[43] Vese L. A study in the BV space of a denoising-deblurring variational problem.App. Math. Optm,2001,44:131-161.
[44] Burger M, Frick K, Osher S, et al. Inverse total variation flow. UCLA CAMReport06-24,2006.
[45] Burger M, Gilboa G, Osher S, et al. Nonlinear inverse scale space methods. Comm.Math. Sci.,2006,4(1):175-208.
[46] Burger M, Osher S, Jinjun Xu, et al. Nonlinear inverse scale space methods forimage restoration. Lecture Notes in Computer Science,2005,3752:25-36.
[47] Chan T F, Esedoglu S. Aspects of total variation regularizedL1functionapproximation. SIAM.Appl. Math.,2005,65(5):1817-1837.
[48] Chan T F, Esedoglu S, Park F. A fourth order dual method for staircase reductionin texture extraction and image restoration problems. UCLA CAM Report05-28,2005.
[49] Chambolle A. An algorithm for total variation regularization and denoising.Journal of Mathematical Imaging and Vision,2004,20:89-97.
[50] Chan T F, Shen J. Mathematical models for local nontexture inpaintings. SIAMJournal on Applied Mathematics,2002,62:1019-1043.
[51] Shen J. Inapainting and the fundamental problem of image processing. SIAMNews,2003,36(5):1-4.
[52] Chan T F, Shen J, Vese L. Variational pde models in image processing. Notice ofthe AMS,2003,50(1):14-26.
[53] Bertalmio M, Sapiro G, Caselles V, et al. Image inpainting. Proceedings of ACMSIGGRAPH, also ACM Transactions on Graphics,2000,19(3):417-424.
[54] Sapiro G. Image inpainting. SIAM News,2002,35(4):1-2.
[55] Rane S D, Sapiro G, Bertalmio M. Structure and texture filling-in of missingimage blocks in wireless transmission and compression applications. IEEETransactions on Image Processing,2003,12(3):296-303.
[56] Bertalmio M, Vese L, Sapiro G, et al. Simultaneous structure and texture imageinpainting. IEEE Trans. Image. Process,2003,12:882–889.
[57] Chan T F, Shen J. Non-texture inpainting by curvature-driven diffusion. Journal ofVisual Communication and Image Representation,2001,12(4):436-449.
[58] Chan T F, Shen J. Image Processing and Analysis. SIAM, Philadelphia,2005.
[59] Chandler D. Introduction to Modern Statistical Mechanics. Oxford UniversityPress, New York, Oxford,1987.
[60] Gibbs W. Elementary Principles of Statistical Mechanics. Yale University Press,1902.
[61] Aubert G, Kornprobst P. Mathematical Problems in Image Processing.Springer-Verlag, New York,2001.
[62] Hadamard J, Sur Les. Problems and Derivees Partielle et Leur SignificationPhysique. Bull University Princeton,1902:49-52.
[63]肖庭延,于慎根,王彦飞.反问题的数值解法.科学出版社,2003.
[64]刘继军.不适定问题的正则化方法及应用.科学出版社，2005.
[65]马兴瑞,陶良,黄文虎.弹性波反演方法及其应用.科学出版社,1999.
[66]黄光远,刘小军.数学物理反问题.山东科学技术出版社,1993.
[67] You Y, Kaveh M. Fourth-order partial differential equations for noise removal.IEEE Trans. Image Process.,2000,9(10):1723-1730.
[68] Lysaker M, Lundervold A, Tai X C. Noise removal using fourth order partialdifferential equation with applications to medical magnetic resonance images inspace and time. IEEE Trans. Image Process.,2003,12(12):1579-1590.
[69] Mohammad R H. An anisotropic fourth-order diffusion filter for image noiseremoval. Int. J. Comput. Vis.,2011,92(2):177-191.
[70] Guidotti P, Longo K. Two enhanced fourth order diffusion models for imagedenoising. J. Math. Imaging and Vis.,2011,40(2):188-198.
[71] Lysaker M, Tai X C. Iterative image restoration combining total variationminimization and a second order functional. Int. J. Comput. Vis.,2006,66(1):5-18.
[72] Lysaker M, Osher S, Tai X C. Noise removal using smoothed normals and surfacefitting. IEEE Trans. Image Process.,2004,13(10):1345-1357.
[73] Rahman T, Tai X C, Osher S. A TV-stokes denoising algorithm. SSVM,2007,473-483.
[74] Tony Chan, Selim Esedoglu, Mila Nikolova. Algorithms for finding globalminimizers of image segmentation and denoising models. SIAM Journal onApplied Mathematics,2006,66:1932-1648.
[75] Chan T, Vese L. Active contours without edges. IEEE Transactions on ImageProcessing,2001,10:266-277.
[76] Caselles V, Kimmel R, Sapiro G. Geodesic active contours. Computer Vision,IEEE International Conference on,1995,0:694.
[77] 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.
[78] Burger M, Hackl B, Ring W. Incorporating topological derivatives into level setmethods. Journal of Computational Physics,2004,194(1):344–362.
[79] Bae E, Tai X C. Efficient global optimization for the multiphase Chan-Vese modelof image segmentation by graph cuts. In EMMCVPR, Lecture Notes in ComputerSciences.2009,5681:28-41.
[80] Zhu S C, Yuille A. Region Competition: Unifying Snakes, Region Growing, andBayes/mdl for Multiband image Segmentation. IEEE Trans. Pattern Anal.Machine Intell,1996,18(9):884-900.
[81] Chen S F, Cao L L, Wang Y M, et al. Image Segmentation by MAP-MLEstimations. IEEE Trans. on Image Processing,2010,19(9):2254-2264.
[82] Nikolova M. Minimizers of Cost-Functions Involving Nonsmooth Data-FidelityTerms: Application to the Processing of Outliers. SIAM Journal on NumericalAnalysis,2003,40(3):965-994.
[83] Daubechies I, DeVore R, Fornasier M, et al. Iteratively Re-weighted LeastSquares Minimization for Sparse Recovery. Communications on Pure and AppliedMathematics,2010,63(1):1-38.
[84] Boykov Y, Veksler O, Zabih R. Fast Approximate Energy Minimization viaGraph Cuts. IEEE Trans. Pattern Anal. Mach. Intell.,2001,23(11):1222–1239.
[85] Tu Z, Zhu S C. Image Segmentation by Data-Driven Markov Chain Monte Carlo.IEEE Trans. Pattern Anal. Mach. Intell,2002,24(5):657–673.
[86] Tu Z. An Integrated Framework for Image Segmentation and PerceptualGrouping. Proc.10thInt. Conf. on Computer Vision, Beijing, China, Oct.17-21,2005,670–677.
[87] Kolmogorov V, Zabih R. What Energy Functions Can Be Minimized via GraphCuts. IEEE Transactions on Pattern Analysis and Machine Intelligence,2004,26(2):147–159.
[88] Komodakis N, Tziritas G, Paragios N. Fast, Approximately Optimal Solutions forSingle and Dynamic MRFs. Proc. IEEE Conference on Computer Vision andPattern Recognition, Minneapolis, MN, June17-22,2007,1–8.
[89] Boykov Y, Kolmogorov V. Computing Geodesics and Minimal Surfaces viaGraph Cuts. Proc.9thIEEE International Conf. on Computer Vision, Nice, France,Oct.13-16.2003,26-33.
[90] Boykov Y, Kolmogorov V, Cremers D, et al. An Integral Solution to SurfaceEvolution PDEs via Geo-Cuts. Proc.3rdEuropean conference on Computer Vision(3), Vol.3953,2006,409–422.
[91] Darbon J, Sigelle M. Image Restoration with Discrete Constrained TotalVariation Part I: Fast and Exact Optimization. J. Math. Imaging Vis.,2006,26(3):261–276.
[92] Carson C, Belongie S, Greenspan H, et al. Blobworld: Image SegmentationUsing Expectation-Maximization and Its Application to Image Querying. IEEETrans. Pattern Anal. Mach. Intell.,2002,24(8):1026–1038.
[93] F rstner W. A Framework for Low Level Feature Extraction. Proc.3rd Europeanconf. on Computer Vision,1994,801:383–394.
[94] G rding J, Lindeberg T. Direct Computation of Shape Cues Using Scale-AdaptedSpatial Derivative Operators. Int. J. Comput. Vis.,1996,17(2):163–191.
[95] Malik J, Perona P. Preattentive Texture Discrimination with Early VisionMechanisms. J. Opt. Soc. Amer. A, Opt. Image Sci.,1990,7(5):923–932.
[96] Osborne M R. Finite Algorithms in Optimization and Data Analysis. John Wiley&Sons Ltd., Chichester,1985.
[97] Bae E, Tai X C. Graph Cut Optimization for the Piecewise Constant Level SetMethod Applied to Multiphase Image Segmentation. Proc.2nd Int. Conf. on ScaleSpace and Variational Methods in Computer Vision, Voss, Norway, Jun.1-5,2009,1-13.
[98] Felzenszwalb P, Huttenlocher D. Efficient Graph-Based Image Segmentation. Int.J. Comput. Vis.,2004,59(2):167–181.
[99] Unnikrishnan R, Pantofaru C, Hebert M. Toward Objective Evaluation of ImageSegmentation Algorithms. IEEE Trans. Pattern Anal. Mach. Intell.,2007,29(6):929–944.
[100] Yin W, Osher S, Goldfarb D. Bregman iterative algorithms for l1-minimizationwith applications to compressed sensing. SIAM Journal on Imaging Sciences,2008,1(1):142-168.
[101] Goldstein T, Osher S. The split Bregman algorithm for L1regularized problems.SIAM J on Imaging Sciences,2(2):323-343.
[102] Yin W, Wang Y, Yang J, et al. A new alternating minimization algorithm fortotal variation image reconstruction. SIAM Journal on Imaging Sciences, Toappear.
[103] Osher S, Burger M, Goldfarb D, et al. An iterative regularization method for totalvariation-based image restoration. MMS,2005,4:460-489.
[104] Goldstein T, Bresson X. Geometric application of the split Bregman method:segmentation and surface reconstruction. Journal of Scientific Computing,2009,45(1-3):272-293.
[105] Cai J F, Osher S, Shen Z W. Split Bregman Methods and Frame Based ImageRestoration. Multiscale Model. Simul.,8(2):337–369.
[106] Plonka G, Ma J W. Curvelet-wavelet regularized split bregman iteration forcompressed sensing. Int. J. Wavelets Multiresolut Inf. Process,2011,09:79.
[107] Bresson X, Esedoglu S, Vandergheynst P, et al. Fast global minimization of theactive contour/snake model. Journal of Mathematical Imagingand Vision,2007,28:151-167.
[108] Blomgren P, Chan T F. Color TV: total variation methods for restoration ofvector-valued images. IEEE Trans. Image Process.,1998,7(3):304-309.
[109] Almgren F, Taylor J E, Wang L. Curvature-driven flows: a variational approach.SIAM J. Control Optim.,1993,31(2):387-438.
[110] Chambolle A. An algorithm for mean curvature motion. Interfaces Free Bound,2004,6(2):195-218.
[111] Xavier Bresson, Tony Chan. Active contours based on chambolle's meancurvature motion. IEEE International Conference on Image Processing,2007,33-36.
[112] Brown E S, Chan T, Bresson X. A convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Sha-h Segmentation. Technical report, UCLA,2010. ftp://ftp.math.ucla.edu/pub/camreport/cam10-43.pdf.
[113] Arrow K J, Hurwicz L, Uzawa H. Studies in linear and non-linear programming.Vol. II, Stanford, Calif: Stanford University Press,1958:124-158.
[114] Popov L D. A modification of the Arrow-Hurwitz method of search for saddlepoints. Mathematics Zametki,1980,28(5):777–784.
[115] Potts R B, Domb C. Some generalized order-disorder transformations.Mathematical Proceedings of the Cambridge Philosophical Society,1952,48:106–109.
[116] Boykov Y, Veksler O, Zabih R. A new algorithm for energy minimization withdiscontinuities. In EMMCVPR’99: Proceedings of the Second InternationalWorkshop on Energy Minimization Methods in Computer Vision and PatternRecognition, pages205–220, London, UK,1999. Springer-Verlag.
[117] Lie J, Lysaker M, Tai X C. A variant of the level set method and applications toimage segmentation. Mathematics of Computation,2006,75(255):1155–1174.
[118]汪云飞,毕笃彦,黄飞.一种用于图像分割的等周算法改进.西安电子科技大学学报,2012,39(2):87-94.
[119]刘建磊,冯大政.一种全局最优的非匀质图像分割算法.西安电子科技大学学报,2011,38(2):66-77.
[120]刘汉强,焦李成,赵风.基于局部相似性测度的SAR图像多层分割.西安电子科技大学学报,2010,37(6),1071-1076.
[121] Elhamifar E, Vidal R. Sparse subspace clustering. IEEE Conference onComputer Vision and Pattern Recognition, Miami, FL, June20-252009,2790–2797.
[122] Cheng B, Liu G C, Wang G D et al. Multi-task low-rank affinity pursuit forimage segmentation. International Conference on Computer Vision, Barcelona,Nov.6-132011. Washington, DC, USA: IEEE Computer Society,2439-2446.
[123] Liu G C, Lin Z C, Yan S C et al. Robust recovery of subspace structures bylow-rank representation. IEEE Transactions on Pattern Analysis and MachineIntelligence,2013,35(1):171-184.
[124] Zhuang L S, Gao H Y, Ma Y et al. Non-nenagtive low rank and sparse graph forsemi-supervised learning. IEEE Conference on Computer Vision and PatternRecognition, Providence, RI, June16-212012,2328-2335.
[125] Chartrand R, Yin W T. Iteratively reweighted algorithms for compressivesensing. IEEE International Conference on Acoustics, Speech, and SignalProcessing, March31-April42008,3869-3872.
[126] Chartrand R, Staneva V. Restricted isometry properties and nonconvexcompressive sensing. Inverse Problems,2008,24(035020):1–14.
[127] Saab R, Chartrand R, Yilmaz O. Stable sparse approximations via nonconvexoptimization. IEEE International Conference on Acoustics, Speech, and SignalProcessing, Las Vegas, NV, March31-April4,2008,3885-3888.
[128] Chartrand R. Nonconvex splitting for regularized low-rank+sparsedecomposition. IEEE Transactions on Signal Processing,2012,60(11):5810–5819.
[129] Mori G, Ren X F, Alexei A E et al. Recovering human body configurations:combining segmentation and recognition. Proceedings of the2004IEEEComputer Society Conference on Computer Vision and Pattern Recognition,27June-2July2004,2:II-326-333.
[130] Wei S M, Lin Z C. Analysis and improvement of low rank representation forsubspace segmentation.[2011.7]. http://arxiv.org/abs/1107.1561v1.
[131] William M R. Objective criteria for the evaluation of clustering methods. Journalof the American Statistical Association,1971,66(336):846-850.
[132] Marina M. Comparing clustering: An axiomatic view. Proceedings of the22ndinternational conference on Machine learning. New York, NY, USA: ACM,2005:577-584.
[133] Arbelaez P, Maire M, Fowlkes C et al. From contour to regions: An empiricalevaluation. IEEE conference on Computer Vision and Pattern Recognition, Miami,FL, June20-252009,2294-2301.
[134]李小毛朱琳琳唐延东.基于Mumford-Shah模型和开样条曲线的边界检测.自动化学报,2009,35(2):132-136.
[135] Wang L L, Shi Y Y, Tai X C. Robust edge detection using Mumford-Shahmodel and binary level set method. In: Proceedings of the Third InternationalConference on Scale Space and Variational Methods in Computer Vision. VerlagBerlin, Heidelberg: Springer,2011.291-301.
[136] Rudin L I, Osher S, Fatemi E. Nonlinear total variation based noise removalalgorithms. Physica D,1992,60(1-4):259-268.
[137]何进荣.基于加权范数迭代算法的总变差正则化图像复原方法[硕士学位论文],武汉理工大学,2009.
[138] Chambolle A, Lions P L. Image recovery via total variation minimization andrelated problems. Numeric Mathematic,1997,76(2):167-188.
[139] Chan T F, Esedoglu S, Park F E. A fourth order dual method for staircasereduction in texture extraction and image restoration Problems, In: Proceedings ofthe17thIEEE International Conference on Image Processing. Hong Kong: IEEE,2010.4137-4140.
[140] Chan T F, Marquina A, Mulet P. Higher order total variation-based imagerestoration. SIAM Journal on Scientific Computing,2000,22(2):503-516.
[141] P schl C, Scherzer O. Characterization of minimizers of convex regularizationfunctional. Contemporary Mathematics,2008,451:219-248.
[142] Bredies K, Kunisch K, Pock T. Total generalized variation. SIAM Journal onImaging Sciences,2010,3(3):492–526.
[143] Carter J. Dual Methods for Total Variation Based Image Restoration [Ph. D.dissertation], University of California, Los Angeles,2001.
[144] Beck A, Teboulle M. A fast iterative shrinkage-thresholding algorithm for linearinverse problems. SIAM Journal on Imaging Sciences,2009,2(1):183-202.
[145]王旭东,冯象初,霍雷刚.去除乘性噪声的重加权各向异性全变差模型.自动化学报,2012,38(3),444-451.
[146]韩雨,王卫卫,冯象初.基于迭代重加权的非刚性图像配准.自动化学报,2011,37(9):1059-1066.
[147] Chambolle A. An algorithm for total variation minimization and applications.Journal of Mathematical Imaging and Vision,2004,20(1-2):89–97.
[148] Canny J F. A computational approach to edge detection. IEEE Transactions onPattern Analysis and Machine Intelligence,1986,8(6):679-698.
[149] Smith S M, Brady J M. SUSAN-a new approach to low level image processing.International Journal of Computer Vision,1997,23(1):45-78.
[150] Budes A, Coll B, Morel J M. A review of image denoising algorithms, with anew one. SIAM Multiscale Modeling and Simulation (MMS),2005,4(2):490–530.
[151] Gilboa G, Osher S. Nonlocal Linear Image Regularization and SupervisedSegmentation. SIAM Multiscale Modeling and Simulation (MMS),2007,6(2):595630.
[152] Zhou D, Scholkopf B. A Regularization Framework for Learning from GraphData. In Work shop on Statistical Relational Learning and Its Connections toOther Fields,2004.
[153] Gilboa G, Osher S. Nonlocal Operators with Applications to Image Processing.SIAM Multiscale Modeling and Simulation (MMS),2008,7(3):1005-1028.
[154] Gilboa G, Darbon J, Osher S, et al. Nonlocal Convex Functionals for ImageRegularization. UCLA CAM Report,2006.
[155] Kindermann S, Osher S, Jones P W. Deblurring and Denoising of Images byNonlocal Functionals. SIAM Multiscale Modeling and Simulation (MMS),2005,4(4):1091-1115.
[156] Kichenassamy S, Kumar A, Olver P, et al. Conformal Curvature Flows: FromPhase Transitions to Active Vision. In Archive for Rational Mechanics andAnalysis,1996,134:275–301.
[157] Merriman B, Bence J K, Osher S J. Motion of multiple junctions: a level setapproach. Journal of Computational Physics,1994,112(2):334–363.
[158] Bresson X, Chan T F. Nonlocal Unsupervised Variational Image SegmentationModels. UCLA CAM Report,2008.
[159] Céa J, Garreau S, Guillaume P, et al. The shape and topological optimizationsconnection. Comput Method Appl Mech Eng,2000,188(4):713-726.
[160] Eschenauer H A, Kobelev V V, Schumacher A. Bubble method for topology andshape optimization of structures. Struct Optim,1994,8:42-51.
[161] Novotny A A, Feijóo R A, Taroco E, Padra C. Topological sensitivity analysis.Comput Method Appl Mech Eng,2003,192:803-29.
[162] Belaid L J, Jaoua M, Masmoudi M, et al. Application of the topological gradientto image restoration and edge detection. Engineering Analysis with BoundaryElement,2008,32(11):891-899.
[163] Frangakis A S, Hegerl R. Noise reduction in electron tomographicreconstructions using nonlinear anisotropic diffusion. J Struct Biol,2001,1(135):239-250.
[164] Giboa G, Sochen N, Zeevi Y Y. Image enhancement and denoising by complexdiffusion processes. IEEE Transactions on Pattern Analysis and MachineIntelligence,2004,26(8):1020–1036.
[165] Gonzalez C R, Woods R E. Digital image processing. Seconded. Prentice Hall,2001.
[166] Sethian J A. Level Set Methods and Fast Marching Methods. CambridgeUniversity Press,1999.
[167] Malladi R, Sethian J A. Level set methods for curvature flow, imageenhancement, and shape recovery in medical images. In: Proceedings of theconference on visualization and mathematics, Springer-Verlag,1997,329–45.
[168] Xu C, Pham D, Prince J. Medical image segmentation using deformable models.SPIE handbook on medical imaging, vol. III: medical image analysis;2000. p.129–74.
[169] Xu C, Pham D, Rettmann M E, et al. Reconstruction of the human cerebralcortex from magnetic resonance images. IEEE Trans Med Imag1999,18(6):467–80.
[170] Li H, Deklerck R, DeCuyper B, et al. Object recognition in brain ct-scans:knowledge-based fusion of data from multiple feature extractors. IEEE Trans MedImag,1995,14:212–29.
[171] Boscolo R, Brown M, McNitt G M. Medical image segmentation withknowledge-guided robust active contours. Radiographics,2002,22(2):437–448.
[172] Held K, Kops E R, Krause B J, et al. Markov random field segmentation of brainmr images. IEEE Trans Med Imag,1997,16(6):878–86.
[173] Zhang Y, Brady M, Smith S. Segmentation of brain MRI images through ahidden markov random field model and the expectationmaximization algorithm.IEEE Trans Med Imag,2001,20(1):45–57.
[174] Larrabide I, Feijóo R A, Novotny A A, et al. Topological derivative: A tool forimage processing. Computers and Structures,2008,86:1386-1403.
[175] Larrabide I, Novotny A A, Feijóo R A, et al. Topological derivative as a tool forimage processing: image segmentation.Technical report15/2006, LNCC,2006.
[176] Banga C, Ghorbel F. Optimal bootstrap sampling for fast image segmentation:application to retina images. In IEEE ICASP93, Minneapolis USA,1993,5:638–641.
[177] Koch I, Marshall G. Bootstrap coverage plots for image segmentation. In IEEEProceedings of ICPR1996,447–451.
[178] Dunn J C. A fuzzy relative of the isodata process and its use in detecting compact,well separated clusters. Journal of Cybernetics,1974,3:32–57.
[179] Bezdek J. Pattern Recognition with fuzzy Objective Function Algorithms. NewYork: Plenum Press,1981.
[180] Gray R M, Linde Y. Vector quantization and predictive quantizer forgauss-markov sources. IEEE Transaction on Communications,1982,30:381–389,.
[181] Pappas T N. An adaptive clustering algorithm for image segmentation. IEEETransactions on Signal Processing,1992,40:901–914.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |