摘要
多尺度几何分析和极小化能量泛函方法是当前数学图像处理和计算机视觉等领域最具代表性的两种研究范式,这方面的研究已引起广大学者的普遍关注。一方面新的成果不断涌现,另一方面,已有的成果仍存在许多值得进一步研究的问题。本文基于这两种范式,主要提出并解决了以下问题:
1、针对图像的曲线状奇异性表征问题和Candes单尺度脊波框架的冗余问题,利用局部化原理和Donoho构造的正交脊波,提出了一种单尺度正交脊波紧框架。该框架不仅保留了Candes单尺度脊波的方向性,而且具有正交性。另外,针对边缘具有一定光滑性的图像,讨论了该框架对其的非线性逼近性。实验结果表明,所提单尺度正交脊波在图像压缩、图像恢复及图像去噪中均得到较好地应用。
2、针对Starck分解模型在算法复杂度、纹理表征和噪声约束诸方面所存在的一些不足,提出一种基于基追踪的极小化能量泛函模型。该改进模型利用第二代曲波和波原子,分别表征含噪图像中的结构分量和纹理分量,并采用全变差半范约束分片光滑部分的结构性,同时利用Meyer所建议的广义齐型Besov范数对噪声分量进行约束,最后利用基追踪去噪算法,对新模型进行迭代求解。实验结果表明,所提能量泛函模型不但对噪声具有较强的鲁棒性,而且能使边缘和细小纹理信息保持稳定。
3、为了解决多尺度几何分析在图像抑噪应用中出现的“虚假”效应问题,提出一种带变换域约束条件的极小化全变差能量泛函。该能量泛函将多尺度几何分析和极小化能量泛函方法有机地结合起来。首先对降质图像利用多尺度几何分析进行相应变换和非线性阈值,然后根据保留的变换系数确定可行域,从而建立所提能量泛函模型,最后利用投影梯度算法对其进行求解,并以有限脊波变换和第二代曲波变换为例,进行了图像抑噪仿真实验。实验结果表明,所提能量泛函在有效抑噪和保持边缘的同时,能够有效地抑制伪吉布斯振荡、“卷绕”伪直线和“曲波状”伪曲线等“虚假”效应,取得了较为理想的视觉效果。
4、针对经典极小化能量泛函中平衡参数对图像振荡分量先验信息的过分依赖性,首先讨论了一类更为一般的Meyer分解模型,并证明了解的存在性和唯一性。然后将平衡参数视为尺度参数,提出一种分级多尺度极小化能量泛函,并导出一种图像的多尺度表示方法。同时,对这一多尺度表示方法的收敛性进行了理论分析。最后,利用BV近似W 1,1导出一种新的近似求解算法。数值实验结果表明,所提能量泛函在各类图像处理中均有较好的应用。
5、针对图像恢复应用中,经典全变差正则化方法的阶梯效应和细小纹理信息丢失问题,从两种不同视角提出如下解决方案:
(1)提出一种自适应正则化的极小化能量泛函,将图像分解为结构分量和振荡分量,其中对结构分量的正则化是通过TV光滑化和各向同性光滑化之间的插值得到,即依据图像局部特征进行一种自适应正则化;振荡分量被置于div(BMO)空间中加以讨论。此外,我们对所提能量泛函解的存在唯一性进行了理论证明,并导出其相应的Euler-Lagrange方程。实验结果表明,所提能量泛函在实际图像分解应用中,不但能够较好地保持边缘和细小纹理,而且有效地抑制了阶梯效应。
(2)利用Meyer的振荡模式分解理论,提出了一种磨光流场的全变差正则化抑噪方法。该方法首先引入负指数Hilbert-Sobolev范数来度量逼近项,对图像水平曲线的法向量场进行全变差正则化磨光,然后构造一个曲面拟合能量泛函,对磨光后的流场进行拟合。最后,导出各能量泛函所对应的Euler-Lagrange方程,并利用有限差分法进行数值求解。实验结果表明,该方法在有效去噪的同时,能够较好地保持边缘和纹理信息,并且使阶梯效应也得到有效地抑制。
In mathematical imaging and visual of computer, multi-scale geometric analysis and minimization energy functional are the most representative two paradigms at current. It has attracted attention of researchers widely. A lot of new productions are presented about the two paradigms. On the other hand, the productions proposed still exist some problems which are worthy of researching further. In this dissertation, based on the two paradigms, the following problems are discussed:
1.To represent curve singularity of image and deal with redundancy of Candes’s monoscale ridgelets, a new ridgelets frame, monoscale orthonormal ridgelets frame (MORF), is presented, which use the localization principle and the orthonormal ridgelet constructed by Donoho. The MORF not only remains directionality, but also bears orthonormality. We discuss the nonlinear approximation of the MORF for image which is smooth away from discontinuities across curves. The experiments demonstrate that the new frame have preferable application for image compression, image reconstruction and image denoising.
2.A new minimization energy functional based on the basis pursuit is presented to improve on Starck’s decomposition model. In this new functional, the second generation curvelets and wave atoms are used to represent structure and texture respectively, and the total variational semi-norm is added for restricting structure parts. In addition, the generalized homogeneous Besov norm proposed by Meyer is used to constrain noisy components. Finally, the basis pursuit denoisiing algorithm is used to solve the new model. Experiments show that the new model is very robust for noise, and that can keep edges and textures stably.
3. To reduce the“aliasing effects”resulted from using multi-scale geometric analysis for image denoising, a minimization total variational energy functional with constraint condition on transform domain is presented. Firstly, the nonlinear thresholding strategy associated with certain multi-scale geometric analysis is applied to the transform coefficients of noisy image. And then, the feasible domain of the proposed model is determined by the coefficients remained. Finally, the projected gradient algorithm is used to solve the proposed model. Experiments show that the presented model can remove noisy and remain edges, while the“aliasing effects”such as psudo-Gibbs effects,“wrap around”effects and the“curvelet like”aliased curves are suppressed efficiently, when the finite ridgelet transform and the curvelet transform are applied respectively.
4.In the classical minimization energy functional, the balance parameter often depends on the priori information of the oscillatory component. To reduce this dependence, Firstly, a more general decomposition model is discussed, while the existence and uniqueness of solution about this model are proved. And then, the balance parameter is taken as scale. Instead of L2 space, we discuss the oscillatory component in distribution space W ?1 ,∞. To this end, a multiscale hierarchical decomposition model is established, which still obtains a multiscale representation of image, at the same time, its convergence is analyzed. Finally, a new algorithm is proposed by applying BV to approximate W 1,1. The numerical experiments show that the multiscale hierarchical decomposition model has well application in many image processing.
5.In the two different viewpoints, we investigate staircasing effect and losing of small scale texture which are caused by TV regularization in the classical minimization energy functional, and present two solutions as follows:
(1) A new minimization energy functional with adaptive regularization is proposed, which decomposes image into structure component and oscillatory component, where the regularization for the structure component is obtained by interpolation of TV regularization and isotropic smoothing, namely, take adaptive regularization in terms of the local feature of image; the oscillatory component is investigated in div(BMO) space. In addition, we proof the existence and uniqueness of the presented model. Finally, the corresponding Euler-Lagrange equations are derived to numerically implement. Experimental results and comparisons demonstrate that the proposed model has an advantage of improving visual effect of image decomposition, namely, both edge and texture are well remained, while the staircasing effect is avoided efficiently.
(2) Using oscillating patterns theory in image processing proposed by Meyer, a total variation image denoising method is presented, which based on smoothing flow field. Firstly, through applying Hilbert-Sobolev norm to measure fidelity term, a total variation filter is used to smooth the normal vectors of the level curves in noise image. And then, a model is constructed to find a surface which fit smoothed normal vectors. Finally, finite difference schemes are used to solve the Euler-Lagrange functions derived from above models. The experiments show that the approach not only can remove noisy efficiently, but also can retain edges and texture and the staircasing effect also is avoided.
引文
[1] Candμes E J. Donoho D L. New Tight Frames of Curvelets and Optimal Representations of Objects with C2 Singularities[J]. Communications on Pure and Applied Mathematics,2004,57(2):219-266.
[2] Ali Haddad and Y. Meyer. Variational methods in image processing. UCLA CAM Report 04-52, 2004.
[3] T.Chan and Jianhong Shen. Image processing and analysis: Variational, PDE, Wavelet, and Stochastic methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2005.
[4] D.Mumford, B.Gidas. Stochastic Models for Generic Images. Quart. Appl.Math.2001, 59(3):85-111.
[5] R.J.Adams, J.J.F.Fournier. Sobolev Spaces. Academic Press, Amsterdam, second edition, 2003.
[6] G. B. Folland. Real Analysis——Modern Techniques and Their Application. John Wiley & Sons, New York, second edition, 1999.
[7] E,H,Lieb, M.Loss. Analysis. AMS, Providence, RI, second edition,2001
[8] Y. Meyer. Oscillating patterns in image processing and nonlinear evolution equations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, University Lecture Series, No.22, Oxford: American Mathematical Society,Providence, RI, 2001.
[9] L. I. Rudin, S. J. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Phys. D, 1992, 60. 259–268.
[10] Mumford D. and Shah J. Optimal Approximation by piecewise smooth functions and associated variational problems. Comm. Pure. Appl. Math. 1989, 42(5).577-685.
[11] Y. Meyer. Wavelets and Operators. Cambridge University Press, Cambridge, UK, 1992.
[12] I. Daubechies. Ten Lectures on Wavelets. CBMS Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.
[13] T. Cai, J. Shen. Boundedness is Redundant in a Theorem of Daubechies. Appl. Comput. Harmon. Anal. 2003,6:400-404.
[14] Donoho D L and Ana Georgina Flesia. Can recent innovations in harmonicalanalysis‘explain’key findings in nature image statistics. Network: Computation in Neural System, 12(3): 371-393, 2001.
[15] Chui C K. An introduction to wavelets. Academic Press, Inc., 1992.
[16] Y. Meyer. Wavelets-Algorithms and Application. SIAM. 1993
[17] I.Daubechies. Orthonormal bases of compactly supported wavelets. Comm. on Pure and Appl. Math., 1988, 41(7). 909-996.
[18] Mallat S. A wavelet tour of signal processing. Boston: Academic Press, 1998.
[19] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. PAMI, 11(7): 674-693,1989.
[20] Mallat S. Multiresolution approximations and wavelet orthonormal bases of . Trans. Amer. Math. Soc.,9(315): 69-87, 1989.
[21] Mallat S. Multi-frequency channel decomposition of images and wavelet. IEEE Trans. ASSP., 12(37): 2091-2110, 1989.
[22] Coifman R R. and Wickerhauser M V. Entropy based algorithms for best basis selection. IEEE Trans. on Inform. Threory, 1992 , 38(2).713-718.
[23] Donoho D.L. Interpolating wavelet transforms. Technical Report, Dept. of Statistics Stanford Univ., 1992.10
[24] Dubuc S. Interpolating through an iterative scheme. J. Math. Anal. and Appl., 1986, 114. 185-204
[25] Cohen A., Daubechies I. and Vial P. Wavelets on the interval and fast wavelet transform. Appl. Comp. Harm. Anal., 1993, 1. 54-80
[26] Goodman T.N.T. and Lee S.L. Wavelets of multiplicity. Trans. Amer. Math. Soc., March 1994, 342(1). 307-324.
[27] Sweldens W. The lifting scheme: A construction of second generation wavelets. SIAM J.Math.Anal., 1997, 29(2). 511-546.
[28] Devore R.A. Nonlinear approximation, Acta Numerica, Cambridge University Press. 1998. 51-150.
[29] Devore R.A. and Lucier B J. Classifying the smoothness of images: theory and application to wavelet image peocessing, in Proc.1994 IEEE Int. Conf. Image Processing, Austin, TX, IEEE Press. 1994, 2. 6-10
[30] J. Ma, M. Fenn. Combined complex ridgelet shrinkage and total variation minimization. SIAM Journal on Scientific Computing, 2006, 28 (3). 984-1000.
[31] Candes E J. Ridgelets: Theory and Applications [D]. USA: Department of Statistics, Stanford University,1998. 1-166.
[32] Candès E. J, Donoho D. L. Ridgelets: the Key to Higher-dimensionalIntermittency? Phil. Trans. R. Soc. Lond. A, 1999, 357(1760): 2495–2509.
[33] Candès E. J. On the Representation of Mutilated Sobolev Functions, Stanford Technical Report, Department of Statistics, Stanford University, 1999.
[34] Donoho D L. Orthonormal Ridgelet and Linear Singularities[J]. SIAM Journal on Mathematical Analysis,2000,31(5): 1062-1099.
[35] Candes E J. Monoscale Ridgelets for the Representation of Images with Edges. USA:Department of Statistics,Stanford University,1999. 1-26.
[36] Candes E J,Donoho D L. Curvelets: a Surprisingly Effective Nonadaptive Representation for Objects with Edges[A]Curves and Surfaces[M]. Nashville :Vanderbilt University Press,1999.1-10.
[37] Do M N and Vetterli M. The finite ridgelet transform for image representation. IEEE Trans on Image Processing, 2003, 12(1). 16-28.
[38] Donoho D L, Flesia A G. Digital ridgelet transform based on true ridge functions. Stanford: Stanford Univ, 2001.
[39] Donoho D L, Duncan M R. Digital curvelet transform: Strategy, implementation and experiments. Proc. SPIE, Wavelet Application VII[C]. Orlando, USA, 2000, 4056. 12-29.
[40] Starck J.L., Candes E.J., Donoho D.L.: The Curvelet transform for image denoising .IEEE Trans. Image Proc, 2002, 11(6). 670-684.
[41] Starck J L, Murtagh F, Candes E J and Donoho DL. Gray and color image contrast enhancement by the curvelet transform. IEEE Trans. Image Processing, 2003, 12(6). 706-717.
[42] Candes E. J., Donoho D. L.. New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities [J]. Comm. Pure Appl. Math, 2004, 57: 219–266.
[43] Demanet L,Ying L X. Wave Atoms and Sparsity of Oscillatory Patterns[J]. Appl. Comput. Harmon. Anal.,2007, 23(3):368-387.
[44] Do M N, Vetterli M. The Contourlet Transform: an Efficiet Directional Multiresolution Image Representation[J]. IEEE Transactions Image on Processing,2005,14(12):2091-2106.
[45] Lu Y and Do M N. CRISP-Contourlet: A critically sampled directional multiresolution image representation. Proc. SPIE Conf on Wavelets X. San Diego, Aug. 2003.
[46] Donoho D L. Wedgelets: nearly minimax estimation of edges. Ann Statist, 1999, 27. 859-897.
[47] Pennec E Le, Mallat S. Image compression with geometrical wavelets. Proceeding of CIP 2000. Vancouver, 2000.661-664.
[48] Donoho D L, Huo Xiao-ming. Beamlets and multiscale image analysis. BARTH T J, CHAN T, HAMSER Multiscale and multiresolution methods Berliin: Springer, 2001.
[49] Sapiro G, Ringach D. Anisotropic diffusion of multivalued images with pplication to color filtering. IEEE Trans. Image Processing , 1996, 5(11): 1582–1585.
[50] P. V. Blomgren, Total Variation Methods for Restoration of Vector Valued Images, Ph.D. dissertation, Los Angeles: Univ. California, 1998.
[51] Kimmel R. Demosaicing: image reconstruction from color CCD samples [J]. IEEE Transactions on Image Processing, 1999,8(9): 1221~1228.
[52] S. Osher, M. Burger, D. Goldfarb, and et.al. An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul., 2005, 4(2). 460-489.
[53] Lin He, M.Burger and S.J.Osher. Iterative total variation regularization with non-quadratic fidelity. J.Math. Imaging Vis., 2006, 26. 167-184.
[54] Jinjun Xu. Iterative regularization and nonlinear inverse scale space methods in image restoration. Ph. D. thesis, UCLA-CAM-Report 06-33, Los Angeles, CA, USA, 2006.
[55] J.Xu, S.Osher. Iterative regularization and nonlinear inverse scale space applied to wavelet based denoising. IEEE Trans. Image Proc., 2007, 16 (2). 534-544.
[56] T.F.Chan and S. Esedoglu. Aspects of total variation regularized L1 function approximation. SIAM.Appl. Math., 2005, 65 (5). 1817-1837.
[57] S. Esedoglu and S. Osher. Decomposition of Images by the Anisotropic Rudin-Osher-Fatemi Model. Comm. Pure Appl. Math.. 2004, 57:1609–162.
[58] A.Chambolle and P.Lions. Image recovery via total variation minimization and related problems. Numer. Math., 1997, 66(2. 167-188.
[59] P. Blomgren, T. F. Chan, P. Mulet, and C. Wong, Total variation image restoration: Numerical methods and extensions, Proceedings of the 1997 IEEE International Conference on Image Processing, III . 1997, 384-387.
[60] Y. Chen, S. Levine, and M. Rao. Variable Exponent, Linear Growth Functionals in Image Processing [J]. SIAM J. Appl. Math. 2006, 66(4): 1383-1406.
[61] R. DeVore and B. Lucier, High order regularity for conservation laws, Indiana Univ.Math.J, 39 (1990), 413-430.
[62] L. A. Vese and S. J. Osher. Modeling textures with total variation minimizationand oscillating patterns in image processing. Journal of Scientific Computing, 2003, December, 19(1-3).
[63] S. Osher, A. Sole. And L. Vese. Image decomposition and restoration using total variation minimization and the H ?1 norm. SIAM Journal on Multiscale Modeling and Simulation, 2003, 1-3. 349-370.
[64] I. Daubechies and G. Teschke. Wavelet based image decomposition by variational functionals. Proc. SPIE, Wavelet Applications in Industrial Processing, 2004, 5266. 94-105.
[65] I. Daubechies and G.Teschke. Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising, Applied and Computational Harmonic Analysis, 2005,19(1).1-16
[66] E. Tadmor, S. Nezzar, and L. Vese. A multiscale image representation using hierarchical (BV, L2) decompositions. Multiscale Model. Simul., 2004, 2. 554-579.
[67] J. Garnett, P. Jones and T. Le et al. Modeling oscillatory components with the homogeneous space BMO ?αand W ?α,p[R]. Technical Report. UCLA CAM, USA, 2007.
[68] J.F.Aujol, G.Aubert, L.Blanc-Feraud, and et.al. Image decomposition: application to textured images and SAR images, Technical Report ISRN I3S/RR-2003-01-FR, INRIA—Project ARIANA, Sophia Antipolis, 2003.
[69] J.-F.Aujol, A.Chambolle. Dual norms and image decomposition models. Intl. J. Comp. Vis., 2005, 63(1).85-104.
[70] Linh Lieu and Luminita Vese. Image Restoration and Decompostion via Bounded Total Variation and Negative Hilbert-Sobolev Spaces. Applied Mathematics and optimization (in press).
[71] T.Le and L.Vese. Image decomposition using the total variation and div ( BMO). Multiscale Modeling and Simulation, 2005, 4(2). 390-423.
[72] J.-L. Starck, M. Elad and D.L.Donoho. Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Processing 2005, 14(10).1570-1582.
[73] G.Aubert and J.F. Aujol. Modeling very oscillating signals. Application to image processing, AMO, 2005, 51 (2). 163-182.
[74] S. Kindermann, S.Osher, and J. Xu. Denoising by BV-Duality. Journal of Scientific Computing, 2006, 28(2-3). 411– 444.
[75] Gilles Aubert, Pierre Kornprobst. Mathematical problems in image processing:partial differential equations and the calculus of variations. Applied mathematical sciences. Springer, 2001.
[76] S. Chen, D. Donoho, and M. Saunder, Atomic Decomposition by Basis Pursuit, SIAM Journal on Scientific Computing, 1998, 20 (1): 33-61.
[77] J. L. Starck, E. Candes, and D. Donoho, Astronomical Image Representation by the Curvelet Transform, Astronomy and Astrophysics, 2003, 398 (2): 785-800.
[78] E.J.Candes, L.Demanet, and D. L. Donoho, Fast Discrete Curvelet Transforms, Applied and Computational Mathematics, California Institute of Technology, Pasadena, California, USA, 2005, 1–43.
[79] BAI Jian, FENG Xiangchu, Image Decomposition by Curvelet Transform, Acta Electronica Sinica, 2007, 35 (1): 123-126.
[80] G. Steidl, J.Weickert and T. Brox et al, On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs, SIAM Numerical Analysis, 2004, 42(2): 686-713.
[81] J.F. Aujol, G. Aubert and L. Blanc-Féraud et al, Image Decomposition Into a Bounded Variation Component and an Oscillating Component [J], Journal of Mathematical Imaging and Vision. 2005, 22 (1): 71-88.
[82] L.Villemoes, Wavelet packets with uniform time-frequency localization, Comptes-Rendus Mathematique, 2002, 335(10): 793-796.
[83] A. Chambolle, R. A. Devore, N. Lee, et al. Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. on Image Processing, 1998, 7(3): 319-335.
[84] S. Durand, J. Froment. Reconstruction of wavelet coefficients using total variation minimization. SIAM.J.SCI. COMPUT, 2003, 24(4): 1754–1767.
[85] F. Alter, S. Durand, J. Froment. Adapted total variation for artifact free decompression of JPEG images [J]. Journal of Mathematical Imaging and Vision, 2005, 23 (2): 199–211.
[86] F. Malgouyres. Minimizing the total variation under a general convex constraint for image restoration [J]. IEEE Trans. on Image Processing, 2002, 11(12): 1450-1456.
[87] You Yuli, Xu Wenyuan, Tannenbaum Allen, et al. Behavioral analysis of anisotropic diffusion in image processing [J]. IEEE Trans. on Image Processing. 1996, 5(11): 1539–1553.
[88] L. Vese. A study in the BV space of a denoising-deblurring variational problem. App. Math. Optm, 2001, 44. 131-161.
[89] M. Burger, K. Frick, S. Osher, and et.al. Inverse total variation flow. UCLA CAM Report 06-24, 2006.
[90] M. Burger, G. Gilboa, S. Osher, and et.al. Nonlinear inverse scale space methods. Comm. Math. Sci., 2006, 4(1).175-208.
[91] M. Burger, S. Osher, Jinjun Xu, and et.al. Nonlinear inverse scale space methods for image restoration. Lecture Notes in Computer Science, 2005, 3752. 25-36.
[92] M. Bertalmio, L. Vese, G. Sapiro, and et.al. Simultaneous structure and texture image inpainting. IEEE Trans. Image. Process, 2003, 12. 882–889.
[93] Chan T. F. and Shen J. Non-texture inpainting by curvature-driven diffusion. Journal of Visual Communication and Image Representation, 2001, 12(4).436-449.
[94] Chan T. F. and Shen J. and Kang S. Euler’s elastica and curvature-based image inpainting. SIAM Journal on Applied Mathematics, 2002, 63(2).564-592.
[95] Donoho D L. and Johnstone I M. Ideal spatial adaptation via wavelet shrinkage. Biometrika, 1994, 81(3). 425-455.
[96] Dirk A. Lorenz. Wavelet shrinkage in signal and image processing-An investigation of relations and equivalences. Ph.D. thesis, 2004.
[97] G. Aubert, J. Bect, L. Blanc Féraud, and et.al. A Unified Variational Framework for Image Restoration. ECCV (4), 2004, 3024: 1-13.
[98] Tony F.Chan, Chiu-Kwong Wong. Total variation blind deconvolution. IEEE Transaction on Image Processing, 1998, 7(3). 370-395.
[99] Y You, M Kaveh. A regularization approach to joint blur identification and image restoration. IEEE Trans Image Process, 1996, 5 (3). 416 - 428.
[100] G. Gilboa, N. Sochen, Y. Y. Zeevi. Texture Preserving Variational Denoising Using an Adaptive Fidelity Term. Proc. VLSM 2003 [C] France, Nice, 2003, 137-144.
[101] J. Weickert. A review of nonlinear diffusion filtering. B. ter Haar romeny, L. Florack, J. Koenderink, M. Viergever (Eds.), Scale-Space Theory in Computer Vision, London. UK: Springer-Verlag, 1997, Vol. 1252: 3-28.
[102] W. Ring. Structural properties of solutions of total variation regularization problem. Math. Model. Numer. Anal., 2000, 34:799-810.
[103] Guy Chavent and Karl Kunisch. Regularization of linear least squares problems by total bounded variation. ESAIM Control Optim. Calc. Var., 1997,2:359-376.
[104] Lin He. Application and generalizations of the iterative refinement method. Ph.D thesis, University of Calfornia, 2006.
[105] G. Gilboa, N. Sochen, Y. Zeevi. Variational denoising of partly-textured images by spatially varying constraints. IEEE Transactions on Image Processing, 2006, 15(8): 2281-2289.
[106] David Strong and Tony Chan. Edge-preserving and scale-dependent properties of total variation regularization. Inverse Problems, 2003, 19: 165-187.
[107] W. Yin, D. Goldfarb and S. Osher. Total variation based image cartoon-texture decomposition, UCLA CAM Report 05-27, 2005.
[108] J. B. Garnett, T. M. Le, and L. A. Vese. Image decomposition using bounded variation and homogenous Besov spaces, UCLA CAM Report 05-57, 2005.
[109] Peter W. Jones and Triet M. Le. Local scales and multiscale image decomposition. UCLA CAM Report 07-11, 2007.
[110] S. Nezzar. A multiscale image representation using hierarchical (BV, L2) decompositions, Ph. D. Thesis, University of California, Los Angeles, CA, 2003.
[111] Lorenz D A. Solving Variational Problems in Image Processing via Projections——A Common View on TV Denoising and Wavelet Shrinkage. Zeitschrift für angewandte Mathematik und Mechanik. 2007, 81(1):247-256.
[112] Acar. R, Vogel. C. Analysis of Total Variation Penalty Methods, Inverse Problems, 1994, 10(6):1217-1229.
[113] Hans Triebel. Theory of function spaces II. Monographs in Mathematics. Borkhauser, 1992.
[114] L. Ambrosio and V. Tortorelli. On the approximation of functionals depending on jumps by elliptic functionals via gama-convergence. Comm. Pure Appl. Math., 1990, 43: 999-1036.
[115] A. Cohen, R. Devore, P. Petrushev and H. Xu. Nonlinear approximation and the space BV(R2). Amer. J. Math, 1999, 121: 587-628.
[116] R. Devore and G. Lorentz. Constructive approximation, Grundlehren Math. Wiss. 303, Springer-Verlag, Berlin, 1993.
[117] A. Chambolle. An algorithm for total variation regularization and denoising. Journal of Mathematical Imaging and Vision, 2004, 20. 89-97.
[118] T.F.Chan, S. Esedoglu and F. Park. A fourth order dual method for staircase reduction in texture extraction and image restoration problems. UCLA CAM Report 05-28, 2005.
[119] Luis Alvarez, Jean-Michel Morel. Formalization and computational aspects of image analysis. United States of America: Cambridge University Press, 1994, 1-59.
[120] Perona P. and Malik J. Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence,1990, 2(7).629-639.
[121] G. Aubert and L. Vese. A Variational Method in Image Recovery. SIAM Journal of Numerical Analysis. 1997, 34(5): 1948-1979.
[122] CattéF, Lions P L. and Morel J M. et al. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numerical Analysis, 1992, 29(1).182-193.
[123] D. C. Dobson and F. Santosa. Recovery of Blocky Images from Noisy and Blurred Data [J]. SIAM J. Appl. Math. 1996, 56: 1181-1198.
[124] M. Nikolova. Weakly Constrained Minimization: Application to the Estimation of Images and Signals Involving Constant Regions. J. Math. Imaging and Vision. 2004, 21(2): 155-175.
[125] Y. Gousseau and J.-M. Morel. Are Natural Images of Bounded Variation?. SIAM J. Math. Anal. 2001, 33(3): 634-648.
[126] J. M. Morel and S. Solimini. Variational methods in image segmentation: with seven image processing experiments, (Progress in Nonlinear Differential Equations and Their Applications), Birkhauser Boston 1994.
[127] J. Weickert, Bar M. ter Haar Romeny. Efficient and Reliable Schemes for nonlinear Diffusion Filtering. IEEE Trans. Image Processing, 1998, 7(3).398-410.
[128] J. Weickert. Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart, 1998.
[129] S. Levine, An Adaptive Variational Model for Image Decomposition, to appear in EnergyMinimization Methods in Computer Vision and Pattern Recognition (2005)
[130] Lysaker M, Osher S, and Tai X C. Noise removal using smoothed normal and surface fitting. IEEE Trans. on Image Processing. 2004, 13 (10): 1345-1357.
[131] Tasdizen T, Whitaker R, and Burchard P. Geometric surface processing via normal maps [J]. ACM Trans. Graph.. 2003, 22(4): 1012–1033.
[132] Besl P, Jain R. Segmentation through variable-order surface fitting [J]. IEEE Trans. on Pattern Anal. Machine Intell.. 1988, 10(2): 167–192.
[133] Chan T, Marquina A, and Mulet P. High-order total variation-based image restoration [J], SIAM J. Sci.Compu. 2000, 22(2): 503–516.
[134] Vese L A, Osher S. Numerical methods for p-harmonic flows and applications toimage processing [J].SIAM J. Numer. Anal.. 2002, 40 (6): 2085–2104.
[135] I.Ekeland and R.Temam. Convex analysis and variational problems. Translated from the French, Studies in Mathematics and it’s Applications, vol.1, North-Holland Publishing Co., Amsterdam, 1976.
[136]章毓晋.图像处理和分析.北京:清华大学出版社, 1999.
[137]阮秋琦著.数字图像处理学.电子工业出版社,2001.
[138]李朝晖主编.数字图像处理及应用.北京:机械工业出版社, 2004.
[139]张贤达,保铮.非平稳信号分析与处理.北京:国防工业出版社, 1998
[140]冯象初,甘小冰,宋国乡.数值泛函与小波理论.西安电子科技大学出版社,2003.
[141]崔锦泰(著),程正兴(译).小波分析导论.西安:西安交通大学出版社,1995.
[142] 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.
[143]邹谋炎.反卷积和信号复原.北京:国防出版社, 2001, 3.
[144]白键,冯象初,基于对偶框架的数字脊波重构,中国科学(E),2005,35 (10): 1072-1082.
[145]焦李成,侯彪,刘芳.脊函数网络逼近,进展与展望.工程数学学报, 2002, 19(1): 21-36.
[146]姜东焕,冯象初,宋国乡.依赖Besov权的平移不变小波阈值方法,工程数学学报. 2006, 23:3443-449.
[147]白键.脊波、曲线波和偏微分方程在图像处理中的算法研究.[西安电子科技大学博士学位论文],2007,9.
[148]李敏.基于小波和变分PDE的图像建模理论、算法及应用. [西安电子科技大学博士学位论文],2008,3.