多分辨率隐马尔可夫模型图像去噪研究
|
摘要
图像去噪是图像分割、编码、模式识别等图像处理应用研究的基础，去噪效果的好坏直接影响这些应用研究的结果，因此去噪算法的研究一直受到了广泛关注。图像去噪的主要方法分为空间域和变换域两类：在空间域处理方法中，主要是利用相邻像素之间的相似性，采用均值滤波、中值滤波、维纳滤波和马尔科夫随机场模型等方法；在变换域处理法方法中，主要是利用傅立叶变换、小波变换和滤波器组分解等工具将图像投影到变换域，然后对变换域的系数进行处理以达到去噪的目的。小波变换和滤波器组分解因为其具有多分辨率分析及去高阶相关的特性，已成为图像去噪算法研究的主流方向。
本文的主要工作和贡献如下：
第一，小波系数的边缘分布为对称非高斯分布且具有尺度内聚簇性和尺度间继承性，因此对小波系数的边缘分布和条件分布的分析是建立图像去噪统计模型的基础。传统HMT模型较少考虑到尺度内系数之间的相关性，本文在定性分析了小波系数的边缘分布及条件分布的基础上，利用基于模糊理论的随机场方法，在隐马尔可夫树模型训练过程中，为该模型引入了聚簇约束，从而有效地提高了模型近似的精度。
第二，不可分离方向性滤波器组能够将图像分解为多分辨率形式，并能更有效地提取图像的方向性信息。本文使用直方图方法，定性分析了分解系数的联合分布及条件分布，在此基础上利用方向性滤波器组的多尺度结构，为其系数建立了隐马尔可夫树模型。在实验中通过与DB4小波变换隐马尔可夫树模型的去噪效果对比，表明该模型去噪性能优异，尤其是在处理纹理信息丰富的图像时，该模型的去噪效果更加明显。
第三，M带小波具有比2带小波更精细的频带划分，当前基于M带小波变换的图像去噪算法多为阈值方法。考虑到其尺度间的系数构成了十六叉树结构，条件分布和联合分布的定性分析也表明尺度内系数之间具有很强的相关性，适合采用多尺度随机场模型来处理。传统多尺度随机场模型多为各向同性，不能表达M带小波系数在同一尺度下不同子带的相关性，且随机场模型参数的精确估计需要消耗大量的计算时间和存储。本文借鉴基于模糊逻辑的随机场参数估计方法，为M带小波系数构建了一种自适应的多尺度隐马尔可夫随机场模型，采用伪似然方法计算似然函数，模型参数估计采用了EM-MAP算法。实验结果表明，与基于M带的阈值方法去噪效果相比，该模型的去噪效果在大噪声场合的表现非常优异。
Image denoising is the foundation of high level image processing,such as imagesegmentation,coding,pattern recognition,etc.The research for image denoising algorithm hasreceived extensive attention because the applications of these high level image processingtechniques are directly influnced by the denoising results.There are two different ways of imagedenoising: spatial domain algorithm and transform domain algorithm.The main methods in spatialdomain include mean filter,median filter,wiener filter,and Markov Random Field Model.Allthese methods are mainly based on the similarity of neighboring pixcels.The methods intransform domain first transform the image using Fourier transform,wavelet transform and filterbanks decomposition,then images are denoised by.adjusting these coefficients.The researchs forimage denoising are mainly concerntrated on wavelet transform and filter bank decomposition inresent years,because the coefficients have the characteristics of multiresolution analysis and highorder decorrelation,.
The main work and contribution in this thesis include:
The wavelet coefficients have a non-gaussian symmetric marginal distribution,and haveproperties of intra-scale clustering and inter-scale persistence.So it is the foundation for theconstruction of statistical image processing models to analysis the marginal and conditionaldistribution of wavelet coefficients.In traditional Hidden Markov Tree model,the clusteringproperty of coefficients is less take into account.In this thesis,based on the qualitative analysis ofthe marginal and conditional distribution of wavelet coefficients,and using random field that isbased on the fuzzy logic,the clustering constraints are take account into the training process ofhidden markov tree model to efficiently enhance the precision of parameter estimation.
Non seperable directional filter bank can decompose an image into multiresolution structureand can efficiently extract directional information.In this thesis,using histogram technique,aqualitative analysis of the marginal and conditional distribution is implemented.Then weconstructed a hidden markov tree model for the coefficients,and used this model for imagedenoising.Compared with the denoising results based on DB4 wavelet transform,it can be seenthat the model based on directional filter bank has very excellent denoising performance,especially for the images having rich textual information.
Compared with 2-band wavelet transform,M-band wavelet transform can partition thefrequency domain with finer structure and its interscale coefficients have a structure ofhexadecimal tree.Currentely,the M-Band based image denoising algorithms are mainlythreadsholding methods,Through qualitative analysis of conditional and joint distribution of thecoefficients,it can be seen that multiscale random field model is suitable for the modeling of thecoefficients.Traditional markov random field model has an isotropic structure and is not suitablefor the correlation structure of the M-band coefficients.The estimation of field model parametersis also time consuming and need a large amount of memory.In the reference of the random fieldmodel based on fuzzy logic,in this thesis,a spatially adatptive multiscale hidden markov randomfield is constructed for the M-band coefficients.In the model,the likelihood function is computedusing psudo likelihood and the ML-MAP algorithm is used for parameter estimation.Experimentsshow that compared with the performence of threadtholding algorithms,the denoising results is very excellent,especially in high level noise situation.
本文的主要工作和贡献如下：
第一，小波系数的边缘分布为对称非高斯分布且具有尺度内聚簇性和尺度间继承性，因此对小波系数的边缘分布和条件分布的分析是建立图像去噪统计模型的基础。传统HMT模型较少考虑到尺度内系数之间的相关性，本文在定性分析了小波系数的边缘分布及条件分布的基础上，利用基于模糊理论的随机场方法，在隐马尔可夫树模型训练过程中，为该模型引入了聚簇约束，从而有效地提高了模型近似的精度。
第二，不可分离方向性滤波器组能够将图像分解为多分辨率形式，并能更有效地提取图像的方向性信息。本文使用直方图方法，定性分析了分解系数的联合分布及条件分布，在此基础上利用方向性滤波器组的多尺度结构，为其系数建立了隐马尔可夫树模型。在实验中通过与DB4小波变换隐马尔可夫树模型的去噪效果对比，表明该模型去噪性能优异，尤其是在处理纹理信息丰富的图像时，该模型的去噪效果更加明显。
第三，M带小波具有比2带小波更精细的频带划分，当前基于M带小波变换的图像去噪算法多为阈值方法。考虑到其尺度间的系数构成了十六叉树结构，条件分布和联合分布的定性分析也表明尺度内系数之间具有很强的相关性，适合采用多尺度随机场模型来处理。传统多尺度随机场模型多为各向同性，不能表达M带小波系数在同一尺度下不同子带的相关性，且随机场模型参数的精确估计需要消耗大量的计算时间和存储。本文借鉴基于模糊逻辑的随机场参数估计方法，为M带小波系数构建了一种自适应的多尺度隐马尔可夫随机场模型，采用伪似然方法计算似然函数，模型参数估计采用了EM-MAP算法。实验结果表明，与基于M带的阈值方法去噪效果相比，该模型的去噪效果在大噪声场合的表现非常优异。
Image denoising is the foundation of high level image processing,such as imagesegmentation,coding,pattern recognition,etc.The research for image denoising algorithm hasreceived extensive attention because the applications of these high level image processingtechniques are directly influnced by the denoising results.There are two different ways of imagedenoising: spatial domain algorithm and transform domain algorithm.The main methods in spatialdomain include mean filter,median filter,wiener filter,and Markov Random Field Model.Allthese methods are mainly based on the similarity of neighboring pixcels.The methods intransform domain first transform the image using Fourier transform,wavelet transform and filterbanks decomposition,then images are denoised by.adjusting these coefficients.The researchs forimage denoising are mainly concerntrated on wavelet transform and filter bank decomposition inresent years,because the coefficients have the characteristics of multiresolution analysis and highorder decorrelation,.
The main work and contribution in this thesis include:
The wavelet coefficients have a non-gaussian symmetric marginal distribution,and haveproperties of intra-scale clustering and inter-scale persistence.So it is the foundation for theconstruction of statistical image processing models to analysis the marginal and conditionaldistribution of wavelet coefficients.In traditional Hidden Markov Tree model,the clusteringproperty of coefficients is less take into account.In this thesis,based on the qualitative analysis ofthe marginal and conditional distribution of wavelet coefficients,and using random field that isbased on the fuzzy logic,the clustering constraints are take account into the training process ofhidden markov tree model to efficiently enhance the precision of parameter estimation.
Non seperable directional filter bank can decompose an image into multiresolution structureand can efficiently extract directional information.In this thesis,using histogram technique,aqualitative analysis of the marginal and conditional distribution is implemented.Then weconstructed a hidden markov tree model for the coefficients,and used this model for imagedenoising.Compared with the denoising results based on DB4 wavelet transform,it can be seenthat the model based on directional filter bank has very excellent denoising performance,especially for the images having rich textual information.
Compared with 2-band wavelet transform,M-band wavelet transform can partition thefrequency domain with finer structure and its interscale coefficients have a structure ofhexadecimal tree.Currentely,the M-Band based image denoising algorithms are mainlythreadsholding methods,Through qualitative analysis of conditional and joint distribution of thecoefficients,it can be seen that multiscale random field model is suitable for the modeling of thecoefficients.Traditional markov random field model has an isotropic structure and is not suitablefor the correlation structure of the M-band coefficients.The estimation of field model parametersis also time consuming and need a large amount of memory.In the reference of the random fieldmodel based on fuzzy logic,in this thesis,a spatially adatptive multiscale hidden markov randomfield is constructed for the M-band coefficients.In the model,the likelihood function is computedusing psudo likelihood and the ML-MAP algorithm is used for parameter estimation.Experimentsshow that compared with the performence of threadtholding algorithms,the denoising results is very excellent,especially in high level noise situation.
引文
[1] R. C. Gonzalez, R. E. Woods. Digital image Processing [M]. Prentice Hall, 2004.
[2] J. Besag. On the statistical analysis of dirty pictures [J].Journal of royal statistical society, 1986, serial B (48): 259-302.
[3] S. G Chang, B. Yu, M. Vetterli. Adaptive wavelet thresholding for image denoising and compression [J]. IEEE Transactions on image processing, 2000, 9(9): 1532-1546.
[4] Park Sil, M. J. T. Smith, J. J. Lee. Fingerprint enhancement based on the directional filter bank [C]. In ICIP00, Vancouver, BC, Canada, 2000, 3: 793-796.
[5] Sang Keun Oh, J. Jae Lee. New fingerprint image enhancement using directional filter bank [C]. Journal of WSCG, Plzen-Bory, Czech Republic, 2003,11(1).
[6] Guoliang Fan. Wavelet Domain Statistical Image Modeling and Processing [D]. PhD thesis, University of Delaware, 2001.
[7] S. Geman, D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images [J]. IEEE Transaction on Pattern Anallysis and Machine Intellegence [J]. 1984,6: 721-741.
[8] G Strang. Wavelet transforms versus Fourier transforms [J]. Bull. Amer. Math. Soc. (N.S.), 1993,28(2):288-305.
[9] B. Ripley. Statistical inference for spatial processes [M]. Cambridge University Press, 1988.
[10] D. Wang, V. Haese-coat, A. Bruno. Some Statistical Properties of Mathematical Morphology [J].IEEE Transactions on signal processing, 1995,43(8): 1955-1965.
[11] R. Kinderman, J. Snell. Markov random fields and their applications [J].American Mathematical Society, Providence, Rhode Island, 1980, 253-258,.
[12] M. Hassner, J. Sklansky. The use of Markov random fields as models of texture [J]. Computer Graphics and Image Processing, 1980, 12:357-370.
[13] J. Besag. Spatial interaction and the statistical analysis of lattice systems (with discussions) [J]. Journal of the Royal Statistical Society, Series B, 1974, 36:192-236.
[14] V. Kolmogorov, R. Zabih. What energy functions can be minimized via graph cuts? [J]. IEEE Trans Pattern Anal Mach Intell, 2004, 26:147-159.
[15] Y. Weiss, W. Freeman. Correctness of belief propagation in Gaussian graphical models of arbitrary topology [J]. Neural Computation, 2001,13:2173-2200.
[16] L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition [J]. Proceedings of the IEEE, 1989,77(2):257-286.
[17] A. Kundu, Y. He, P. Bahl. Recognition of handwritten word: First and second order hidden Markov model based approach [J]. Pattern Recognition, 1989,22(3):283-297.
[18] J. ZHANG The mean field theory in EM procedures for Markov random fields [J]. IEEE Transactions on signal processing, 1992,40(10):2570-2583.
[19] D. Tran, M. Wagner. Generalised Fuzzy Hidden Markov Models for Speech Recognition [C]. In Proceedings of the International Conference on Fuzzy Systems,Calcutta, India, 2002, 30(2):68-82.
[20] J. Sun, D. Gu, S. Zhang. Hidden markov bayesian texture segmentation using complex wavelet transform [J]. IEEE Proceedings Vision, Image and Signal Processing, 2004, 151(3):215-223.
[21] F. Samaria. Face Recognition Using Hidden Markov Models [D]. PhD Thesis, University of Cambridge, 1994.
[22] C.A. Bouman, M. Shapiro. A Multiscale Random Field Model for Bayesian Image Segmentation [J]. IEEE Trans. Image Processing, 1994, 3:162-176.
[23] D. L. Donoho. De-Noising by Soft-Thresholding [J]. IEEE Transactions on information theory, 1995,41(3): 613-627.
[24] D. L. Donoho, I. M. Johnstone. Adapting to unknown smoothness via wavelet shrinkage [J]. Amer. Statist. Assoc, 1995, 90(432):1200-1224.
[25] Sendur Levent, Selesnick Iva W. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency [J]. IEEE Transactions on signal processing, 2002, 50(11): 2744-2756.
[26] F. Abramovich, T. Sapatinas, B.W. Silverman. Wavelet thresholding via a Bayesian approach [J]. J. R. Statist. Soc. ser. B, 1998,1(2): 725-749.
[27] H. Chipman, E. Kolaczyk, R. McCulloch. Adaptive Bayesian wavelet shrinkage [J]. Journal of American statistical association, 1997, 92(440):1413-1421.
[28] J. Portilla, V. Strela, M. Wainwright, E. P. Simoncelli. Image denoising using scale mixtures of gaussians in the wavelet donlain [J]. IEEE Transactions on Image Processing, 2004,12(11):1338-1351.
[29] A. P. Dempster, N. M. Laird, D. B. Bubin. Maximum likelihood from imcomplete data via EM algorithm [J]. Journal of the Royal Statistical Society, Series B, 1977, 39:1-38.
[30] M. S. Crouse, R. D. Nowak, R. G Baraniuk. Wavelet-based statistical signal processing using hidden Markov models [J]. IEEE Trans. Signal Processing, 1998, 46(4): 886-902.
[31] Guo-Liang FAN, Xiang-Gen XIA. A joint multicontext and multiscale approach to Bayesian image segmentation [J]. IEEE Transactions on Geoscience and Remote Sensing, 2001, 39(12):2680-2688
[32] F. Desbouvries, J. Lecomte, W. Pieczynski. Kalman filtering in pairwise Markov trees [J]. Signal Processing (Elsevier), 2006, 86(5):1049-1054.
[33] F. X. Yan, S. L. Peng, L. Z. Cheng. Dual-tree complex wavelet hidden Markov tree model for image denoising [J]. Electronics Letters, 2007,43:973-975.
[34] J. K. Romberg, H. Choi, R. G Baraniuk, N.G Kingsbury. Hidden Markov tree models for complex wavelet transforms [J]. Proceedings of the IEEE, 2002, 8: 1396-1458.
[35] D. D. Y. Po, M. N. Do. Directional multiscale modeling of images using the contourlet transform [J]. IEEE Trans. Image Processing, 2006,15(6): 1610-1620.
[36] M. Malfait, D. Roose. Wavelet-based denoising using a Markov random field a prior model [J]. IEEE Transactions on image processing, 1997, 6(4):549-565.
[37] M. Jansen, A. Bultheel. Empirical Bayes approach to improve wavelet thresholding for image noise reduction [J]. Journal of the American statistical association, 2001,96(454): 629-639.
[38] A. Pizurica, W. Philips, L. Lemahieu, et al. A joint inter- and intrascale statistical model for Bayesian wavelet based image denoising [J]. IEEE Transactions on image processing, 2002, 11(5):545-557.
[39] S. Mallat, S. Zhong. Characterization of signals from multiscale edges [J]. IEEE Transaction on Pattern analysis and Machine Intelligence, 1992,14:710-732.
[40] M.T. Orchard, K. Ramchandran. An investigation of wavelet-based image coding using an entropy constrained quantization framework [C]. In Data Compression Conference, Snowbird, Utah, 1994,341-350.
[41] J. M. Hammersley, P. Clifford. Markov field on finite graphs and lattices. unpublished 1971.
[42] S. Mallat. A Wavelet Tour of Signal Processing [M]. SanDiego, CA: Academic Press, 1997.
[43] I. Daubechies, Ten Lectures on Wavelets [M]. New York: SIAM, 1992.
[44] X. Lin, M. T. Orchard. Rate-distortion optimized image coding via least square esimation quantization [C]. In Proc. 337-d Asilomar Conf. on Signal, System and Computers, Pacific Grove, CA, Oct. 1999,12:1183-1187.
[45] H. Sun, BV. Kwok. Concealment of damaged block transform coded images using projections onto convex sets [J]. IEEE Transactions on image Processing, Apr.1995,4(4):470-477.
[46] Gelman. Bridges between deterministic and probabilistic classification models. Manuscript submitted for publication -Leenen, 2003
[47] J. R. Cross, A. K. Jain. Markov random field texture models [J]. IEEE Transaction on Pattern Anallysis and Machine Intellegence. 1983, 5(3):25-39.
[48] R. Paget, I. D. Longstaff. Texture synthesis via a noncausal nonparametric multiscale Markovrandom field [J]. IEEE Transactions on Image Processing, 1998,7(6): 925-931.
[49] S. Z. Li. Markov Random Field Modeling in Image Analysis [M]. Springer, 1st edition in 1995 and 2nd edition in 2001
[50] 陈木法.随机场概论 [J]. 数学进展, 1989,18(3):294-322.
[51] R. Chellapqsa, S. Chateerjee. Classificatiol of textures using gaussian markov random fields [J]. IEEE Transactions on Acoustics Speech and Signal Processing, 1985,33:959-963.
[52] B. S. Everitt. A finite mixture model for the clustering of mixed-mode data [J]. Statistical Probability Letter, 6 (5):305-309.
[53] G W. Milligan, M. C. Cooper. An examination of procedures for determining the number of clusters in a data set [J]. Psychometrika, 1985,50:159-179.
[54] L. E. Baum, T. Petrie. Statistical inference for probabilistic functions of finite state Markov chains [J]. Ann. Math. Statist, 1966, 37:1554-1563.
[55] G Fan, X.G Xia. Image denoising using local contextual hidden markov model in the wavelet domain [J]. IEEE Signal Processing Letters, 2001, 8(5):125-128.
[56] J. M. Laferte, F. Heitz, P. Perez. A multiresolution EM algorithm for unsupervised image classification [C]. Proc. Int. Conf. of Pattern Recognition, Vienna, Austria, 1996, 849-853.
[57] M. Vetterli, J. Kovacevic. Wavelet and Subband Coding [M]. Prentice Hall, 1995.
[58] C. Morimoto. Y. Yacoob, L. Davis. Recognition of head gestures using hidden markov models [C]. Proc. of Internatzonal Conference on Pattern Recognition, Vienna, Arastria, Aug. 1996,461-465.
[59] L. Meng, T. Q. Nguyen, D.A. Castanon. An Image-Based Bayesian Framework for Face Detection [C]. Proc. IEEE Conf. on Computer Vision and Pattern Recognition, Hilton Head, South Carolina, 2000, 1:302-307.
[60] L. E. Baum, J. A. Eagon. An inequality with applications to statistical estimation for probabilistic functions of markov processes and to a model for ecology [J]. Bulletin of American Mathematical Statistics, 1967,37:360-363.
[61] L. E. Baum, T. Petrie. Statistical inference for probabilistic funceions of finite state markov chains [J]. Annals of Mathematical Statistics, 1966,37:1554-1563.
[62] H. Elliott, H. Derin, R. Cristi, D. Geman. Application of the Gibbs distribution to image segmentation [C]. Proceedings of the International Conference on Acoustic, Speech and Signal Processing, San Diego, 1984,678- 681.
[63] H. Derin, H. Elliott. Modeling and segmentation of noisy and textured images using Gibbs random fields [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1987,9(1):39-55.
[64] R. Bellman. Dynamic Programming [M]. Princeton, NJ: Princeton Univ. Press,1957.
[65] A. J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm [J]. IEEE Transactions on Information Theory, 1967, IT-13:260-269.
[66] Y. He and A. Kundu. 2-D shape classification wing hmms [J]. IEEE Transactions on Pattern -Analysis and Machine Intelligence, 1991,13(11):1172-1184.
[67] B. H. Juang, L. R. Rabiner. The segmental k-means algorithm for estimating the parameters of hidden markov models [J]. IEEE Transactions on Accowstics, Speech, Signal Processing, 1990,38(9): 1639-1641.
[68] L. R. Rabiner, B. H. Juang. An introduction to hidden markov models [J]. IEEE Acoustics, Speech and Signal Processing Magazine, 1986, 3:4-16.
[69] D. Murray, B. Buxton. Scene segmentation from visual motion using global optimization [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1987, 8:220-228.
[70] W. E. L. Grimson. From Images to Surfaces: A Computational Study of the Human Early Visual System [D]. MIT Press, Cambridge, MA. W. E. L. 1981.
[71] J. Besag. Statistical analysis of non-lattice data [J]. The Statistician, 1975, 24(3):179-195.
[72] J. Besag. Towards Bayesian image analysis [J]. Journal of Applied Statistics, 1989,16(3):395-406
[73] D. Chandler. Introduction to modern statistical mechanics [M]. Oxford University, Press, 1987.
[74] N. Metropolis. Equations of state calculations by fast computational machine [J]. Journal of Chemical Physics, 1953, 21:1087-1091
[75] A.Gersho, B.Ramaurthi. Image coding using vector quantization [C]. In Proc. IEEE Int, Conf. Acoust, speech signal Processing, 1982, 428-431.
[76] D. Gabor. Theory of Communication [J]. IEEE Trans. On IT, 1964, 9(3):76-87.
[77] J.O. Stromberg. A modified Franklin system and higher order spline systems on R' as unconditional bases for Hardy spaces [C]. Conf. on Harmonic Analysis in Honor of A.Zygmund,1982,Ⅱ:475-494．
[78]李建平．小波分析与信号处理—理论、应用及软件实现[M]．重庆：重庆出版社，1997．
[79] P.Goupillaud,A.Grossman,J.Morlet.Cycle-Octave and Related Transforms in Seismic Signal Analysis [J].Geoexploration,1984,23:85-102.
[80] Y.Meyer.Wavelets and Operators [J].Cambridge Studies in Advanced Math.,1992,3(2):45-64.
[81] S.Mallat.Multiresolution approximations and wavelet orthonormal bases of L'(R) [J].Trans.Amer.Math.Soc.1989,9:69-87.
[82]S.Mallat.A theory of multiresolution signal decomposition:The wavelet transform [J].IEEE Trans.PAMI,1989,11 (7):674-693.
[83] S.Mallat.Multifrequency channel decompositions of images and wavelet models [J].IEEE Transactions on Acoustics,Speech,and Signal Processing,1989 37(12):2091-2110.
[84] L.Daubechies.Orthonormal bases of compactly supported wavelets comm [J].Pure & Appl.Math.,1988,4(1):909-996.
[85] M.Wckerhauser.Adapted wavelet analysis from theory to software [M].A.K.Peters Wellesley,MA.
[86] R.Coifman,M.Wickerhauser.Entropy based algorithms for best basis selection [J].IEEE Trans.on IT.1992,38(2):713-718.
[87] T.N.T.Goodman,S.L.Lee.Wavelets of multiplicity [J].Trans.Amer.Math.Soc.,1994,342:307-324.
[88] Peter Schr(?)der,Wim Sweldens.The Lifting Scheme:A New Philosophy in Biorthogonal Wavelet Constructions [J].SIGGRAPH 1995:161-172.
[89] G.C.Donovan,G.S.Geronimo,D.P.Hardin.Construction of Orthogonal Wavelets Using Fractal Interpolation Function [J].SIAM.J Math Anal,1996,27:1158-1192.
[90] E.J.D.Candes,L.Donoho.Ridgelets:Theory and applications [D].Ph.D.Thesis,Technical Report,Department of Statistics,Stanford University,1998.
[91] Donoho,Duncan.Digital curvelet transform:Strategy,implementation and experiments [C].Proc SPIE,2000,4056:12-29.
[92] J.L.Starck,E.Candes,D.L.Donoho.The Curvelet Transform for Image Denoising [J].IEEE Transactions on Image Processing,2002,670-684
[93] M.N.Do.Directional Multiresolution Image Representations [D].Ph D Thesis.Department of Communication Systems,Swiss Federal Institute of Technology Lausanne,2001.
[94] M.N.Do,M.Vetterli.The contourlet transform:an efficient directional multiresolution image representation [J].IEEE Transactions on Image Processing,2005,14(12):2091-2106.
[95]飞思科技产品研发中心．小波分析理论与MATLAB 7实现[M]．电子工业出版社．2005．
[96]杨福生．小波变换的工程分析与应用[M]．中国科学出版社，1999．
[97] J.Huang,D.Mumford.Statistics of natural images and models [C].Proceedings of CVPR,1999,541-547.
[98] E.Simoncelli.Modeling the joint statistics of images in the wavelet domain [C].Proceedings of the SPIE 44th Annual Meeting,1999,188-195.
[99] S.Sanjay-Gopal,T.Hebert.Bayesian pixel classification using spatially variant finite mixtures and the generalized EM algorithm [J].IEEE Transactions on Image Processing,1998,7(7):1014-1028.
[100] D.Andrews,C.Mallows.Scale mixtures of normal distributions [J].J.Royal Stat.Soc.I974,36:99-102.
[101] M.S.Crouse,G.Richard,Baraniuk.Contextual Hidden Markov Models for Wavelet-domain Signal Processing [C].In Proc.31th Asilomar Conf.Signals,Systems,and Computers,Pacific Grove,CA,1997,95-100.
[102] K.Justin,Romberg,G.Richard,Baraniuk.Bayesian Tree-Structured Image Modeling Using Wavelet-Domain Hidden Markov Models [J].IEEE Trans.Image Processing,2001,10(7):1056-1068.
[103] Z.Azimifar.Image Models for Wavelet Domain Statistics [D].PhD Thesis,Systems Design Engineering,University of Waterloo,2005.
[104] Din-Chang Tseng,Ming-Yu Shih.Wavelet-based image denoising using contextual hidden Markov tree model [J].Journal of Electronic Imaging,2005,14(3):1-12.
[105] R.H.Bamberger,M.J.T.Smith.A filter bank for the directional decomposition of images:theory and design [J].IEEE Transactions on Signal Processing,1992,40(7):882-893.
[106] T.T.Nguyen,S.Oraintara.A directional decomposition:theory,design and implementation [C].International Symposium on Circuits and Systems,Vancouver,Canada,2004,281-284.
[107] T.T.Nguyen,S.Oraintara.Multiresolution direction filter banks:Theory,design and application [J].IEEE Trans.Signal Process,2005,53(10):3895-3905.
[108] J.McClellan.The design of two-dimensional filters by transformations [C].Seventh Ann.Princeton Conf.on ISS,Princeton,NJ,1973,247-251.
[109] E.Dubois.The sampling and reconstruction of time-varying imagery with application in video systems [J].Proc.IEEE,1985,73(4):502-522.
[110] P.Steffen,P.N.Heller,R.A.Gopinath,C.S.Burrus.Theory of regular M-band wavelet bases [J].IEEE Trans.Signal Process,1993,41 (12):3497-3511.
[111] H.Zou,A.H.Tewfik.Discrete Orthogonal M-band Wavelet Decompositions [C].In proceeding of ICASSP,San Francisco,CA,1992,4:605-608.
[112]唐向宏．图像多通道多分辨分析的实现[J]．信号处理，1997，31(4)：290-297．
[113]唐向宏，龚宇．M带小波变换在图像中的应用[J]．电子科学学刊．1998，20(1)：7-12．
[114] P.E Vaidyanathan.Theory and design of M-Channel maximally decimated quadrature mirror filters with arbitrary M,having the perfect-reconstruction property [J].IEEE Transactions on Signal Processing,1987,ASSP-35(4):467-492.
[115] A.K.Somna,P.P.Vaidyanathan,T.Q.Nguyen.Linear phase paraunitary filter banks:theory,factorizations and designs [J].IEEE Transactions on Signal Processing,1993,41(12):3480-3496.
[116] C.K.Chui,J.Lian.Construction of compactly supported symmetric and anti-symmetric orthonormal wavelets with scale=3 [J].Applied and Computational Harmonic Analysis,1995,2:21-51.
[117] B.Han.Symmetric orthonormal scaling functions and wavelets with dilation factor d=4 [J].Unpublished,1998.
[118] S.Qraintara,T.D.Tran,P.N.Heller,T.Q.Nguyen.Lattice structure for regular paraunitary linear-phase filter banks and M-band orthogonal symmetric wavelets [J].IEEE Transactions on Signal Processing,2001,49(11):2659-2672.
[119] Nikou,C.Galatsanos,N.P.Likas,A.C.A Class-Adaptive Spatially Variant Mixture Model for Image Segmentation [J].IEEE Transactions on Image Processing,2007,16(4):1121-1130.
[120] K.Blekas,A.Likas,N.P.Galatsanos,I.E.Lagaris.A spatially constrained mixture model for image segmentation [J].IEEE Transactions on Neural Networks,2005,16(2):494-498.
[121] M.Chans,E.Levitan,G.T.Herman.Image-modeling Gibbs distributions for Bayesian restoration [J].IEEE Transactions on image analysis and interpretation,1994,21(4):7-12.
[122]张红梅，袁泽剑，蔡忠闽等．基于层次MRF的MR图像分割[J]．软件学报，2002，13(9)：1779-1786．
[123]黄宁，朱敏慧，张守融．一种采用高斯隐马尔可夫随机场模型的遥感图像分类算法[J]．电子与信息学报，2003，25(1)：50-53．
[124] H.S.Choi,D.R Haynor,Y Kim.Partial volume Tissue classification of multichannel magnetic resonance images-a mixel Model [J].IEEE Transactions on medical image,1991,10(3):395-407.
[125] Jun Zhang.The mean field theory in EM procedures for blind markov random field image restoration [J].IEEE Transactions on image processing,1993,2(1):27-40.
[126] Wenhui WANG,Qianjin FENG,Lei Liu.Segmentation of Brain MR Images Through Class-adaptive Gauss-Markov Random Field Model and the EM Algorithm [J].Journal of Image and Graphics,2008,13(3):488-493.
[127] M.J.Beal.Variational Algorithms for Approximate Bayesian Inference [D].PhD Thesis,University College London,2003.
[2] J. Besag. On the statistical analysis of dirty pictures [J].Journal of royal statistical society, 1986, serial B (48): 259-302.
[3] S. G Chang, B. Yu, M. Vetterli. Adaptive wavelet thresholding for image denoising and compression [J]. IEEE Transactions on image processing, 2000, 9(9): 1532-1546.
[4] Park Sil, M. J. T. Smith, J. J. Lee. Fingerprint enhancement based on the directional filter bank [C]. In ICIP00, Vancouver, BC, Canada, 2000, 3: 793-796.
[5] Sang Keun Oh, J. Jae Lee. New fingerprint image enhancement using directional filter bank [C]. Journal of WSCG, Plzen-Bory, Czech Republic, 2003,11(1).
[6] Guoliang Fan. Wavelet Domain Statistical Image Modeling and Processing [D]. PhD thesis, University of Delaware, 2001.
[7] S. Geman, D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images [J]. IEEE Transaction on Pattern Anallysis and Machine Intellegence [J]. 1984,6: 721-741.
[8] G Strang. Wavelet transforms versus Fourier transforms [J]. Bull. Amer. Math. Soc. (N.S.), 1993,28(2):288-305.
[9] B. Ripley. Statistical inference for spatial processes [M]. Cambridge University Press, 1988.
[10] D. Wang, V. Haese-coat, A. Bruno. Some Statistical Properties of Mathematical Morphology [J].IEEE Transactions on signal processing, 1995,43(8): 1955-1965.
[11] R. Kinderman, J. Snell. Markov random fields and their applications [J].American Mathematical Society, Providence, Rhode Island, 1980, 253-258,.
[12] M. Hassner, J. Sklansky. The use of Markov random fields as models of texture [J]. Computer Graphics and Image Processing, 1980, 12:357-370.
[13] J. Besag. Spatial interaction and the statistical analysis of lattice systems (with discussions) [J]. Journal of the Royal Statistical Society, Series B, 1974, 36:192-236.
[14] V. Kolmogorov, R. Zabih. What energy functions can be minimized via graph cuts? [J]. IEEE Trans Pattern Anal Mach Intell, 2004, 26:147-159.
[15] Y. Weiss, W. Freeman. Correctness of belief propagation in Gaussian graphical models of arbitrary topology [J]. Neural Computation, 2001,13:2173-2200.
[16] L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition [J]. Proceedings of the IEEE, 1989,77(2):257-286.
[17] A. Kundu, Y. He, P. Bahl. Recognition of handwritten word: First and second order hidden Markov model based approach [J]. Pattern Recognition, 1989,22(3):283-297.
[18] J. ZHANG The mean field theory in EM procedures for Markov random fields [J]. IEEE Transactions on signal processing, 1992,40(10):2570-2583.
[19] D. Tran, M. Wagner. Generalised Fuzzy Hidden Markov Models for Speech Recognition [C]. In Proceedings of the International Conference on Fuzzy Systems,Calcutta, India, 2002, 30(2):68-82.
[20] J. Sun, D. Gu, S. Zhang. Hidden markov bayesian texture segmentation using complex wavelet transform [J]. IEEE Proceedings Vision, Image and Signal Processing, 2004, 151(3):215-223.
[21] F. Samaria. Face Recognition Using Hidden Markov Models [D]. PhD Thesis, University of Cambridge, 1994.
[22] C.A. Bouman, M. Shapiro. A Multiscale Random Field Model for Bayesian Image Segmentation [J]. IEEE Trans. Image Processing, 1994, 3:162-176.
[23] D. L. Donoho. De-Noising by Soft-Thresholding [J]. IEEE Transactions on information theory, 1995,41(3): 613-627.
[24] D. L. Donoho, I. M. Johnstone. Adapting to unknown smoothness via wavelet shrinkage [J]. Amer. Statist. Assoc, 1995, 90(432):1200-1224.
[25] Sendur Levent, Selesnick Iva W. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency [J]. IEEE Transactions on signal processing, 2002, 50(11): 2744-2756.
[26] F. Abramovich, T. Sapatinas, B.W. Silverman. Wavelet thresholding via a Bayesian approach [J]. J. R. Statist. Soc. ser. B, 1998,1(2): 725-749.
[27] H. Chipman, E. Kolaczyk, R. McCulloch. Adaptive Bayesian wavelet shrinkage [J]. Journal of American statistical association, 1997, 92(440):1413-1421.
[28] J. Portilla, V. Strela, M. Wainwright, E. P. Simoncelli. Image denoising using scale mixtures of gaussians in the wavelet donlain [J]. IEEE Transactions on Image Processing, 2004,12(11):1338-1351.
[29] A. P. Dempster, N. M. Laird, D. B. Bubin. Maximum likelihood from imcomplete data via EM algorithm [J]. Journal of the Royal Statistical Society, Series B, 1977, 39:1-38.
[30] M. S. Crouse, R. D. Nowak, R. G Baraniuk. Wavelet-based statistical signal processing using hidden Markov models [J]. IEEE Trans. Signal Processing, 1998, 46(4): 886-902.
[31] Guo-Liang FAN, Xiang-Gen XIA. A joint multicontext and multiscale approach to Bayesian image segmentation [J]. IEEE Transactions on Geoscience and Remote Sensing, 2001, 39(12):2680-2688
[32] F. Desbouvries, J. Lecomte, W. Pieczynski. Kalman filtering in pairwise Markov trees [J]. Signal Processing (Elsevier), 2006, 86(5):1049-1054.
[33] F. X. Yan, S. L. Peng, L. Z. Cheng. Dual-tree complex wavelet hidden Markov tree model for image denoising [J]. Electronics Letters, 2007,43:973-975.
[34] J. K. Romberg, H. Choi, R. G Baraniuk, N.G Kingsbury. Hidden Markov tree models for complex wavelet transforms [J]. Proceedings of the IEEE, 2002, 8: 1396-1458.
[35] D. D. Y. Po, M. N. Do. Directional multiscale modeling of images using the contourlet transform [J]. IEEE Trans. Image Processing, 2006,15(6): 1610-1620.
[36] M. Malfait, D. Roose. Wavelet-based denoising using a Markov random field a prior model [J]. IEEE Transactions on image processing, 1997, 6(4):549-565.
[37] M. Jansen, A. Bultheel. Empirical Bayes approach to improve wavelet thresholding for image noise reduction [J]. Journal of the American statistical association, 2001,96(454): 629-639.
[38] A. Pizurica, W. Philips, L. Lemahieu, et al. A joint inter- and intrascale statistical model for Bayesian wavelet based image denoising [J]. IEEE Transactions on image processing, 2002, 11(5):545-557.
[39] S. Mallat, S. Zhong. Characterization of signals from multiscale edges [J]. IEEE Transaction on Pattern analysis and Machine Intelligence, 1992,14:710-732.
[40] M.T. Orchard, K. Ramchandran. An investigation of wavelet-based image coding using an entropy constrained quantization framework [C]. In Data Compression Conference, Snowbird, Utah, 1994,341-350.
[41] J. M. Hammersley, P. Clifford. Markov field on finite graphs and lattices. unpublished 1971.
[42] S. Mallat. A Wavelet Tour of Signal Processing [M]. SanDiego, CA: Academic Press, 1997.
[43] I. Daubechies, Ten Lectures on Wavelets [M]. New York: SIAM, 1992.
[44] X. Lin, M. T. Orchard. Rate-distortion optimized image coding via least square esimation quantization [C]. In Proc. 337-d Asilomar Conf. on Signal, System and Computers, Pacific Grove, CA, Oct. 1999,12:1183-1187.
[45] H. Sun, BV. Kwok. Concealment of damaged block transform coded images using projections onto convex sets [J]. IEEE Transactions on image Processing, Apr.1995,4(4):470-477.
[46] Gelman. Bridges between deterministic and probabilistic classification models. Manuscript submitted for publication -Leenen, 2003
[47] J. R. Cross, A. K. Jain. Markov random field texture models [J]. IEEE Transaction on Pattern Anallysis and Machine Intellegence. 1983, 5(3):25-39.
[48] R. Paget, I. D. Longstaff. Texture synthesis via a noncausal nonparametric multiscale Markovrandom field [J]. IEEE Transactions on Image Processing, 1998,7(6): 925-931.
[49] S. Z. Li. Markov Random Field Modeling in Image Analysis [M]. Springer, 1st edition in 1995 and 2nd edition in 2001
[50] 陈木法.随机场概论 [J]. 数学进展, 1989,18(3):294-322.
[51] R. Chellapqsa, S. Chateerjee. Classificatiol of textures using gaussian markov random fields [J]. IEEE Transactions on Acoustics Speech and Signal Processing, 1985,33:959-963.
[52] B. S. Everitt. A finite mixture model for the clustering of mixed-mode data [J]. Statistical Probability Letter, 6 (5):305-309.
[53] G W. Milligan, M. C. Cooper. An examination of procedures for determining the number of clusters in a data set [J]. Psychometrika, 1985,50:159-179.
[54] L. E. Baum, T. Petrie. Statistical inference for probabilistic functions of finite state Markov chains [J]. Ann. Math. Statist, 1966, 37:1554-1563.
[55] G Fan, X.G Xia. Image denoising using local contextual hidden markov model in the wavelet domain [J]. IEEE Signal Processing Letters, 2001, 8(5):125-128.
[56] J. M. Laferte, F. Heitz, P. Perez. A multiresolution EM algorithm for unsupervised image classification [C]. Proc. Int. Conf. of Pattern Recognition, Vienna, Austria, 1996, 849-853.
[57] M. Vetterli, J. Kovacevic. Wavelet and Subband Coding [M]. Prentice Hall, 1995.
[58] C. Morimoto. Y. Yacoob, L. Davis. Recognition of head gestures using hidden markov models [C]. Proc. of Internatzonal Conference on Pattern Recognition, Vienna, Arastria, Aug. 1996,461-465.
[59] L. Meng, T. Q. Nguyen, D.A. Castanon. An Image-Based Bayesian Framework for Face Detection [C]. Proc. IEEE Conf. on Computer Vision and Pattern Recognition, Hilton Head, South Carolina, 2000, 1:302-307.
[60] L. E. Baum, J. A. Eagon. An inequality with applications to statistical estimation for probabilistic functions of markov processes and to a model for ecology [J]. Bulletin of American Mathematical Statistics, 1967,37:360-363.
[61] L. E. Baum, T. Petrie. Statistical inference for probabilistic funceions of finite state markov chains [J]. Annals of Mathematical Statistics, 1966,37:1554-1563.
[62] H. Elliott, H. Derin, R. Cristi, D. Geman. Application of the Gibbs distribution to image segmentation [C]. Proceedings of the International Conference on Acoustic, Speech and Signal Processing, San Diego, 1984,678- 681.
[63] H. Derin, H. Elliott. Modeling and segmentation of noisy and textured images using Gibbs random fields [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1987,9(1):39-55.
[64] R. Bellman. Dynamic Programming [M]. Princeton, NJ: Princeton Univ. Press,1957.
[65] A. J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm [J]. IEEE Transactions on Information Theory, 1967, IT-13:260-269.
[66] Y. He and A. Kundu. 2-D shape classification wing hmms [J]. IEEE Transactions on Pattern -Analysis and Machine Intelligence, 1991,13(11):1172-1184.
[67] B. H. Juang, L. R. Rabiner. The segmental k-means algorithm for estimating the parameters of hidden markov models [J]. IEEE Transactions on Accowstics, Speech, Signal Processing, 1990,38(9): 1639-1641.
[68] L. R. Rabiner, B. H. Juang. An introduction to hidden markov models [J]. IEEE Acoustics, Speech and Signal Processing Magazine, 1986, 3:4-16.
[69] D. Murray, B. Buxton. Scene segmentation from visual motion using global optimization [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1987, 8:220-228.
[70] W. E. L. Grimson. From Images to Surfaces: A Computational Study of the Human Early Visual System [D]. MIT Press, Cambridge, MA. W. E. L. 1981.
[71] J. Besag. Statistical analysis of non-lattice data [J]. The Statistician, 1975, 24(3):179-195.
[72] J. Besag. Towards Bayesian image analysis [J]. Journal of Applied Statistics, 1989,16(3):395-406
[73] D. Chandler. Introduction to modern statistical mechanics [M]. Oxford University, Press, 1987.
[74] N. Metropolis. Equations of state calculations by fast computational machine [J]. Journal of Chemical Physics, 1953, 21:1087-1091
[75] A.Gersho, B.Ramaurthi. Image coding using vector quantization [C]. In Proc. IEEE Int, Conf. Acoust, speech signal Processing, 1982, 428-431.
[76] D. Gabor. Theory of Communication [J]. IEEE Trans. On IT, 1964, 9(3):76-87.
[77] J.O. Stromberg. A modified Franklin system and higher order spline systems on R' as unconditional bases for Hardy spaces [C]. Conf. on Harmonic Analysis in Honor of A.Zygmund,1982,Ⅱ:475-494．
[78]李建平．小波分析与信号处理—理论、应用及软件实现[M]．重庆：重庆出版社，1997．
[79] P.Goupillaud,A.Grossman,J.Morlet.Cycle-Octave and Related Transforms in Seismic Signal Analysis [J].Geoexploration,1984,23:85-102.
[80] Y.Meyer.Wavelets and Operators [J].Cambridge Studies in Advanced Math.,1992,3(2):45-64.
[81] S.Mallat.Multiresolution approximations and wavelet orthonormal bases of L'(R) [J].Trans.Amer.Math.Soc.1989,9:69-87.
[82]S.Mallat.A theory of multiresolution signal decomposition:The wavelet transform [J].IEEE Trans.PAMI,1989,11 (7):674-693.
[83] S.Mallat.Multifrequency channel decompositions of images and wavelet models [J].IEEE Transactions on Acoustics,Speech,and Signal Processing,1989 37(12):2091-2110.
[84] L.Daubechies.Orthonormal bases of compactly supported wavelets comm [J].Pure & Appl.Math.,1988,4(1):909-996.
[85] M.Wckerhauser.Adapted wavelet analysis from theory to software [M].A.K.Peters Wellesley,MA.
[86] R.Coifman,M.Wickerhauser.Entropy based algorithms for best basis selection [J].IEEE Trans.on IT.1992,38(2):713-718.
[87] T.N.T.Goodman,S.L.Lee.Wavelets of multiplicity [J].Trans.Amer.Math.Soc.,1994,342:307-324.
[88] Peter Schr(?)der,Wim Sweldens.The Lifting Scheme:A New Philosophy in Biorthogonal Wavelet Constructions [J].SIGGRAPH 1995:161-172.
[89] G.C.Donovan,G.S.Geronimo,D.P.Hardin.Construction of Orthogonal Wavelets Using Fractal Interpolation Function [J].SIAM.J Math Anal,1996,27:1158-1192.
[90] E.J.D.Candes,L.Donoho.Ridgelets:Theory and applications [D].Ph.D.Thesis,Technical Report,Department of Statistics,Stanford University,1998.
[91] Donoho,Duncan.Digital curvelet transform:Strategy,implementation and experiments [C].Proc SPIE,2000,4056:12-29.
[92] J.L.Starck,E.Candes,D.L.Donoho.The Curvelet Transform for Image Denoising [J].IEEE Transactions on Image Processing,2002,670-684
[93] M.N.Do.Directional Multiresolution Image Representations [D].Ph D Thesis.Department of Communication Systems,Swiss Federal Institute of Technology Lausanne,2001.
[94] M.N.Do,M.Vetterli.The contourlet transform:an efficient directional multiresolution image representation [J].IEEE Transactions on Image Processing,2005,14(12):2091-2106.
[95]飞思科技产品研发中心．小波分析理论与MATLAB 7实现[M]．电子工业出版社．2005．
[96]杨福生．小波变换的工程分析与应用[M]．中国科学出版社，1999．
[97] J.Huang,D.Mumford.Statistics of natural images and models [C].Proceedings of CVPR,1999,541-547.
[98] E.Simoncelli.Modeling the joint statistics of images in the wavelet domain [C].Proceedings of the SPIE 44th Annual Meeting,1999,188-195.
[99] S.Sanjay-Gopal,T.Hebert.Bayesian pixel classification using spatially variant finite mixtures and the generalized EM algorithm [J].IEEE Transactions on Image Processing,1998,7(7):1014-1028.
[100] D.Andrews,C.Mallows.Scale mixtures of normal distributions [J].J.Royal Stat.Soc.I974,36:99-102.
[101] M.S.Crouse,G.Richard,Baraniuk.Contextual Hidden Markov Models for Wavelet-domain Signal Processing [C].In Proc.31th Asilomar Conf.Signals,Systems,and Computers,Pacific Grove,CA,1997,95-100.
[102] K.Justin,Romberg,G.Richard,Baraniuk.Bayesian Tree-Structured Image Modeling Using Wavelet-Domain Hidden Markov Models [J].IEEE Trans.Image Processing,2001,10(7):1056-1068.
[103] Z.Azimifar.Image Models for Wavelet Domain Statistics [D].PhD Thesis,Systems Design Engineering,University of Waterloo,2005.
[104] Din-Chang Tseng,Ming-Yu Shih.Wavelet-based image denoising using contextual hidden Markov tree model [J].Journal of Electronic Imaging,2005,14(3):1-12.
[105] R.H.Bamberger,M.J.T.Smith.A filter bank for the directional decomposition of images:theory and design [J].IEEE Transactions on Signal Processing,1992,40(7):882-893.
[106] T.T.Nguyen,S.Oraintara.A directional decomposition:theory,design and implementation [C].International Symposium on Circuits and Systems,Vancouver,Canada,2004,281-284.
[107] T.T.Nguyen,S.Oraintara.Multiresolution direction filter banks:Theory,design and application [J].IEEE Trans.Signal Process,2005,53(10):3895-3905.
[108] J.McClellan.The design of two-dimensional filters by transformations [C].Seventh Ann.Princeton Conf.on ISS,Princeton,NJ,1973,247-251.
[109] E.Dubois.The sampling and reconstruction of time-varying imagery with application in video systems [J].Proc.IEEE,1985,73(4):502-522.
[110] P.Steffen,P.N.Heller,R.A.Gopinath,C.S.Burrus.Theory of regular M-band wavelet bases [J].IEEE Trans.Signal Process,1993,41 (12):3497-3511.
[111] H.Zou,A.H.Tewfik.Discrete Orthogonal M-band Wavelet Decompositions [C].In proceeding of ICASSP,San Francisco,CA,1992,4:605-608.
[112]唐向宏．图像多通道多分辨分析的实现[J]．信号处理，1997，31(4)：290-297．
[113]唐向宏，龚宇．M带小波变换在图像中的应用[J]．电子科学学刊．1998，20(1)：7-12．
[114] P.E Vaidyanathan.Theory and design of M-Channel maximally decimated quadrature mirror filters with arbitrary M,having the perfect-reconstruction property [J].IEEE Transactions on Signal Processing,1987,ASSP-35(4):467-492.
[115] A.K.Somna,P.P.Vaidyanathan,T.Q.Nguyen.Linear phase paraunitary filter banks:theory,factorizations and designs [J].IEEE Transactions on Signal Processing,1993,41(12):3480-3496.
[116] C.K.Chui,J.Lian.Construction of compactly supported symmetric and anti-symmetric orthonormal wavelets with scale=3 [J].Applied and Computational Harmonic Analysis,1995,2:21-51.
[117] B.Han.Symmetric orthonormal scaling functions and wavelets with dilation factor d=4 [J].Unpublished,1998.
[118] S.Qraintara,T.D.Tran,P.N.Heller,T.Q.Nguyen.Lattice structure for regular paraunitary linear-phase filter banks and M-band orthogonal symmetric wavelets [J].IEEE Transactions on Signal Processing,2001,49(11):2659-2672.
[119] Nikou,C.Galatsanos,N.P.Likas,A.C.A Class-Adaptive Spatially Variant Mixture Model for Image Segmentation [J].IEEE Transactions on Image Processing,2007,16(4):1121-1130.
[120] K.Blekas,A.Likas,N.P.Galatsanos,I.E.Lagaris.A spatially constrained mixture model for image segmentation [J].IEEE Transactions on Neural Networks,2005,16(2):494-498.
[121] M.Chans,E.Levitan,G.T.Herman.Image-modeling Gibbs distributions for Bayesian restoration [J].IEEE Transactions on image analysis and interpretation,1994,21(4):7-12.
[122]张红梅，袁泽剑，蔡忠闽等．基于层次MRF的MR图像分割[J]．软件学报，2002，13(9)：1779-1786．
[123]黄宁，朱敏慧，张守融．一种采用高斯隐马尔可夫随机场模型的遥感图像分类算法[J]．电子与信息学报，2003，25(1)：50-53．
[124] H.S.Choi,D.R Haynor,Y Kim.Partial volume Tissue classification of multichannel magnetic resonance images-a mixel Model [J].IEEE Transactions on medical image,1991,10(3):395-407.
[125] Jun Zhang.The mean field theory in EM procedures for blind markov random field image restoration [J].IEEE Transactions on image processing,1993,2(1):27-40.
[126] Wenhui WANG,Qianjin FENG,Lei Liu.Segmentation of Brain MR Images Through Class-adaptive Gauss-Markov Random Field Model and the EM Algorithm [J].Journal of Image and Graphics,2008,13(3):488-493.
[127] M.J.Beal.Variational Algorithms for Approximate Bayesian Inference [D].PhD Thesis,University College London,2003.
© 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 |