基于小波域系数统计模型的图像去噪研究
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摘要
图像信号在获取、传输和存储等环节,往往会受到各种噪声的污染,严重影响了图像的视觉效果,亦给后续处理带来困难,如边缘检测、图像分割、特征提取、目标跟踪和模式识别等。因此,图像去噪成为图像预处理中一项非常重要的工作。随着近年来学者的广泛关注和研究,一种新的时频分析方法——小波变换,因其具有多尺度、多分辨分析的特点能为信号处理提供一种新的、强有力的分析手段,在图像去噪领域得到了成功的应用。本文主要对基于小波域系数的统计模型去噪算法进行了研究,其具体工作如下:
首先,对图像去噪技术的发展和研究现状进行了综述。从图像去噪的原理出发,较为系统的介绍了图像去噪方法的分类,其中重点阐述了基于小波域的图像去噪方法的发展。同时,针对图像噪声模型和图像质量评估体系进行了详细的阐述,并指出实验所采用的方法。
然后,重点对以双变量去噪模型为代表的基于小波系数统计模型的去噪算法进行了深入的研究。在对传统小波变换理论研究的同时,主要分析了双密度双树复小波变换和轮廓波变换、复方向滤波器(PDTDFB)变换两种新型的多尺度几何分析工具,对其原理、结构及算法实现进行了深入的探讨。在此基础上,结合贝叶斯估计理论,提出基于组合双密度双树复小波、PDTDFB两类信号分析工具在贝叶斯最大后验估计理论框架下双变量模型的图像去噪算法。
本文从原理和结构,以及仿真实验的效果,全面总结和分析改进算法的优缺点,同时与国内较为典型的图像去噪算法进行了比较。实验证明,该算法合理有效,在抑制噪声的同时能较好的保留图像的边缘和纹理信息,具有一定的代表性和创新之处。
最后,对全文内容进行了总结,并对图像去噪方向的进一步研究工作进行了展望。
In the process of the acquisition,, transmission and storage, digital image is often interfered by various noises, which does not only seriously affecting the visual result of the image, but also causing difficulties to the subsequent processings such as edge detection, image segmentation, feature extraction, object tracking and pattern recognition. Therefore, image denoising is a very important job in image pre-processings. With the attentions and researches of scholars in recent years, a new time-frequency analytical method--wavelet transform has been applied successfully in image denoising field. By right of multi-scale, multi-resolution analysis characteristics, the transform can provide a new and powerful method to signal processing. Based on statistical characteristics of wavelet coefficients, the paper proposes a new model-based image denoising methods, and the specific work as follows:
Firstly, the image denoising technology development and research is reviewed. Starting from the principle of image denoising, this paper gives a more systematic introduction to the classification of the image denoising methods, and focuses on the wavelet-based image denoising method development. Meanwhile, the image noise model and image quality evaluation system is intimately elaborated, and the methods used in the experiments are pointed out.
Secondly, taking the bivariate denoising model for the represent, the paper has a further study in wavelet statistic coefficients of denoising algorithm. It not only discusses the traditional wavelet transform theory which takes the double-density dual tree complex wavelet transform for represent, but also discusses the new multi-scale geometric analysis tool which takes contour wave transform and multiple orientation filters (PDTDFB) Transformation for present. On this basis, combining with Bayesian estimation theory, two new bivariate denoising method is presented, they are respectively the method based on double-density dual tree complex wavelet transform and PDTDFB within the framework of Bayesian maximum posterior estimation theory.
There is a comprehensive conclusion to the advantages and disadvantages of new method by analyzing algorithm theory, structure and the simulation results. Comparing to the typical image denoising algorithm of home and abroad, it shows the new algorithm is efficient in suppressing image noise and keeping the image edge and texture information, so the algorithm is reasonable and effective.
Finally, there is a summary of the full text, and it also takes a prospect of image denoising direction.
首先,对图像去噪技术的发展和研究现状进行了综述。从图像去噪的原理出发,较为系统的介绍了图像去噪方法的分类,其中重点阐述了基于小波域的图像去噪方法的发展。同时,针对图像噪声模型和图像质量评估体系进行了详细的阐述,并指出实验所采用的方法。
然后,重点对以双变量去噪模型为代表的基于小波系数统计模型的去噪算法进行了深入的研究。在对传统小波变换理论研究的同时,主要分析了双密度双树复小波变换和轮廓波变换、复方向滤波器(PDTDFB)变换两种新型的多尺度几何分析工具,对其原理、结构及算法实现进行了深入的探讨。在此基础上,结合贝叶斯估计理论,提出基于组合双密度双树复小波、PDTDFB两类信号分析工具在贝叶斯最大后验估计理论框架下双变量模型的图像去噪算法。
本文从原理和结构,以及仿真实验的效果,全面总结和分析改进算法的优缺点,同时与国内较为典型的图像去噪算法进行了比较。实验证明,该算法合理有效,在抑制噪声的同时能较好的保留图像的边缘和纹理信息,具有一定的代表性和创新之处。
最后,对全文内容进行了总结,并对图像去噪方向的进一步研究工作进行了展望。
In the process of the acquisition,, transmission and storage, digital image is often interfered by various noises, which does not only seriously affecting the visual result of the image, but also causing difficulties to the subsequent processings such as edge detection, image segmentation, feature extraction, object tracking and pattern recognition. Therefore, image denoising is a very important job in image pre-processings. With the attentions and researches of scholars in recent years, a new time-frequency analytical method--wavelet transform has been applied successfully in image denoising field. By right of multi-scale, multi-resolution analysis characteristics, the transform can provide a new and powerful method to signal processing. Based on statistical characteristics of wavelet coefficients, the paper proposes a new model-based image denoising methods, and the specific work as follows:
Firstly, the image denoising technology development and research is reviewed. Starting from the principle of image denoising, this paper gives a more systematic introduction to the classification of the image denoising methods, and focuses on the wavelet-based image denoising method development. Meanwhile, the image noise model and image quality evaluation system is intimately elaborated, and the methods used in the experiments are pointed out.
Secondly, taking the bivariate denoising model for the represent, the paper has a further study in wavelet statistic coefficients of denoising algorithm. It not only discusses the traditional wavelet transform theory which takes the double-density dual tree complex wavelet transform for represent, but also discusses the new multi-scale geometric analysis tool which takes contour wave transform and multiple orientation filters (PDTDFB) Transformation for present. On this basis, combining with Bayesian estimation theory, two new bivariate denoising method is presented, they are respectively the method based on double-density dual tree complex wavelet transform and PDTDFB within the framework of Bayesian maximum posterior estimation theory.
There is a comprehensive conclusion to the advantages and disadvantages of new method by analyzing algorithm theory, structure and the simulation results. Comparing to the typical image denoising algorithm of home and abroad, it shows the new algorithm is efficient in suppressing image noise and keeping the image edge and texture information, so the algorithm is reasonable and effective.
Finally, there is a summary of the full text, and it also takes a prospect of image denoising direction.
引文
[1] R. C. Gonzalez and R. Woods.数字图像处理(第二版)[M].阮秋琦,阮宇智.北京:电子工业出版社, 2003.
[2] M. C. Motwani, M. C. Gadiya and R. C. Motwani. Survey of image denoising techniques[C]. In: proceedings of Global signal Processing Conference, 2004.
[3]章毓晋.图像处理和分析[M].北京:清华大学出版社, 1999.
[4] R. Yang, L. Yin and M. Gabbouj. Optimal weighted median filtering under structural constraints[J]. IEEE Trans. Signal Processing, 1995, 43(3): 591-604.
[5]李军,丁明跃.一种改进的B超图像自适应加权中值滤波[J].华中理工大学学报. 2000, 28(6): 71-73.
[6] S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation[J]. IEEE Trans. Pattern Analysis and Machine Intelligence, 1989, 11(7): 674-693.
[7] S. Mallat and W. L.Hwang. Singularity detection and processing with wavelets[J]. IEEE Trans. Inform. Inform. Theory, 1992, 38(2): 617-643.
[8] Yuan Yan Tang, Lihua Yang, Jiming Liu. Wavelet Theory and its Application to Pattern Recognition[M], The World Scientific Publishing Co. Pte, Ltd. , Singapore, 2000.
[9] S. Mallat and S. Zhong. Characterization of signal from multiscale edges[J]. IEEE Trans. PAMI, 1992, 14(7): 710-732.
[10] Xu Y, J.B. Weaver, Healy D M, et al. Wavelet transform domain filters: a spatially selective noise filtration technique[J]. IEEE Trans. Image Processing, 1994, 3(6): 747-758.
[11] D.L. Donoho and I.M. Johnstone. Ideal spatial adaptation via wavelet shrinkage[J]. Biometrika, 1994, 81(3): 425-455.
[12] D.L. Donoho and I. M. Johnstone. Adapting to unknown smoothness via wavelet shrinkage[J]. Journal of the American Statistical Assoc, 1995, 90(432): 1200-1224.
[13] D.L. Donoho. De-noising by soft-thresholding[J]. IEEE Trans. Inform. Theory, 1995, 41(3): 613-627.
[14] S.G. Chang, B. Yu and M. Vetterli. Adaptive wavelet thresholding for image denoising and compression[J]. IEEE Trans. Image Processing, 2000, 9(9): 1532-1546.
[15] S.G. Chang, B. Yu and M. Vetterli. Spatially adaptive wavelet thresholding with context modeling for image denoising[J]. TEEE Trans. Image Processing, 2000, 9(9): 1522-1531.
[16] P. Moulin and Liu J. Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors[J]. IEEE Trans. Inform. Theory, 1999, 45(3): 909-919.
[17] T.T. Cai and B.W. Silverman. Incorporating information on neighboring coefficients into wavelet estimation[J]. Sankhya: The Indian Journal of Statistics, 2001, 63, Series B, Pt. 2: 127-148.
[18] M. Hansen and B. Yu. Wavelet thresholding via MDL for natural images[J]. IEEE Trans. Inform. Theory, 2000, 46(5): 1778-1788.
[19] A. Pizurica and W. Philips. Estimating the probability of the presence of a signal of interest in multiresolution single- and multiband image denoising[J]. IEEE Trans. Image Processing, 2006, 15(3): 654-665.
[20] M.S. Crouse, R.D. Nowak and R.G. Baraniuk. Wavelet-based statistical signal processing using hidden Markov models[J]. IEEE Trans. Signal Processing, 1998, 46(4): 886-902.
[21] L. Sendur and I.W. Selesnick. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency[J]. IEEE Trans. Signal Processing, 2002, 50(11): 2744-2756.
[22] L. Sendur and I.W. Selesnick. Bivariate shrinkage with local variance estimation[J]. IEEE Signal Processing Letters, 2002, 9(12): 438-441.
[23] Z. Cai, T.H. Cheng, C. Lu, et al. Efficient wavelet-based image denoising algorithm[J]. IEEE Electronics Letters, 2001, 37(11): 683-685.
[24] M. Malfait and D. Roose. Wavelet-based image denoising using a Markov random field a priori model[J]. IEEE Trans. Image Processing, 1997, 6(4): 549-565.
[25] A. Pizurica, W. Philips, I. Lemahieu, et al. A Joint Inter-and Intrascale Statistical Model for Bayesian Wavelet Based Image Denoising[J]. IEEE Trans. Image Processing, 2002, 11(5): 545-557.
[26] J. Liu and P. Moulin. Image denoising based on scale-space mixture modeling of wavelet coefficients[C]. In: Proceedings of IEEE International Conference on Image Processing, ICIP 1999, Vol. 1: 386-390.
[27] I.K. Eom and Y.S. Kim. Spatially adaptive denoising based on mixture modeling and interscale dependence of wavelet coefficients[C]. In: Proceedings of IEEE International Conference on Neural Networks & Signal Processing, 2003, Vol. 2: 1070-1073.
[28] R.J. Claudio and S. Jacob. Adaptive image denoising and edge enhancement in scale-space using wavelet transform[J]. Pattern Recognition Letters, 2003, 24(7): 965-971.
[29] D. Cho, T.D. Bui and G.Y. Chen. Multiwavelet statistical modeling for image denoising using wavelet transforms[J]. Signal Processing: Image Communication, 2005, 20(1): 77-89.
[30] G. Strang. Wavelets and dilation equations: A brief introduction[J]. SIAM Review, 1989, 31(4): 614-627.
[31] David L. Donoho, Ana Georgina Flesia. Can recent innovations in harmonic analysis explainkey findings in natural image statistics[M]. Network: Computation in Neural Systems, 2001, 12(3): 371~393.
[32] E.J. Candès, D.L. Donoho. Ridgelets: the Key to Higher-dimensional Intermittency[J]. Phil. Trans. R. Soc. Lond. A, 1999, 357(1760): 2495–2509.
[33] E.J. Candès. On the Representation of Mutilated Sobolev Functions[R]. Stanford Technical Report, Department of Statistics, Stanford University, 1999.
[34] D.L. Donoho. Wedgelets: nearly minimax estimation of edges[J]. Ann Statist, 1999, 27: 859-897.
[35] E.J. Candes, D.L. Donoho. Curvelets[R]. USA: Department of Statistics, Stanford University, 1999.
[36] Pennec. E. Le, S. Mallat. Image compression with geometrical wavelets[J]. Proceeding of CIP 2000. Vancouver, 2000: 661-664.
[37] Do. M. N, M. Vetterli. The Contourlet Transform: an Efficiet Directional Multiresolution Image Representation[J]. IEEE Transactions Image on Processing, 2005, 14(12): 2091-2106.
[38] Truong T. Nguyen, Soontorn Oraintara, The Shiftable Complex Directional Pyramid, PartI: Theoretical Aspects[J]. IEEE Transactions on Signal Processing, 2008, 56(10): 4651–4660.
[39] Truong T. Nguyen, Soontorn Oraintara, The ShiftableComplex Directional Pyramid PartII: Implementation and Applications. IEEE Transactions on Signal Processing[J], 2008, 56(10): 4661–4672.
[40] S.M. Kay.统计信号处理基础——估计与检测理论[M].罗鹏飞,张文明,刘忠等译.北京,电子工业出版社, 2003.
[41]张贤达.现代信号处理第二版[M].北京,清华大学出版社, 2002.
[42] N.G. Kingsbury. Image Processing with the complex wavelet[D]. Phil. Trans Royal Society London A September 1999.
[43] N.G. Kingsbury. Shift invariant prosperities of the Dual-Tree Complex Wavelet Transform[J]. Proc. IEEE Icassp’99, March 1999, paper SPTM3. 6.
[44] F.A. Fernandes, R.L.C van Spaendonck. A New Framework for Complex Wavelet Transform[J]. IEEE Trans. Signal Processing, 2003, 51(7): 1825-1835.
[45] R.L.C. van Spaendonck, F.M. Hindriks, F. C. A. Fernandes, et al. Three- dimensional attribute for seismic interpretation[J]. Ann. Mtg. Abstracts Soc. Exploration Geophys[J]. 2000, 2059-2062.
[46] I.W. Selesnick. The double-density dual-tree DWT. IEEE Trans[J]. on Acoustics, Speech, and Signal Processing, 2004, 52(5): 1304-1314.
[47] Guy Gilboa, Nir Sochen, Yehoshua Y Zeevi. Texture Preserving Variational Denoising Usingan Adaptive Fidelity Term[C]. Proc. VLSM 2003, Nice, France, Oct. 2003.
[48] G.Y. Chen, T.D. Bui, A. Krzy˙zak, Image denoising with neighbour dependency and customized wavelet and threshold[J]. Pattern Recognition, 2005, 38: 115– 124.
[49] Zhou Dengwen, Shen Xiaoliu. Image denoising using block thresholding[J]. Image and Signal Processing, 2008, CISP '08, 27-30 May 2008 , 3: 335– 338.
[50]贾建,项海林.基于剪切不变的递归Contourlet变换图像去噪[J].计算机科学, 2009, 36(5): 254-256
[51]夏君君.多分辨率多方向性变换在图像处理中的应用.中国科技大学硕士学位论文[D]. 2006. 9.
[52] Zhu Beibei, Shang Zhaowei, Zhang Feng, Yuan Bo. Chinese handwriting-based writer identiication with PDTDFB transform[C], Wavelet Analysis and Pattern Recognition, 2009. ICWAPR 2009. International Conference on 12-15 July 2009 Page(s): 205–210.
[53] P.J. Burt, E.H. Adelson. The Laplacian pyramid as acompact image code[J]. IEEE Trans. Communication, 1983, 31(4): 532-540.
[54] R.H. Bamberger, M.J. Smith. A filter bank for the directional decomposition of images[J]. IEEE Trans, Theory and Design. Signal Processing, 1992, 40(4): 882-893.
[55]杨帆,赵瑞珍,胡绍海.基于Contourlet系数相关特性的自适应图像去噪算法[J].光学学报, 2009, 29(2): 357–361.
[2] M. C. Motwani, M. C. Gadiya and R. C. Motwani. Survey of image denoising techniques[C]. In: proceedings of Global signal Processing Conference, 2004.
[3]章毓晋.图像处理和分析[M].北京:清华大学出版社, 1999.
[4] R. Yang, L. Yin and M. Gabbouj. Optimal weighted median filtering under structural constraints[J]. IEEE Trans. Signal Processing, 1995, 43(3): 591-604.
[5]李军,丁明跃.一种改进的B超图像自适应加权中值滤波[J].华中理工大学学报. 2000, 28(6): 71-73.
[6] S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation[J]. IEEE Trans. Pattern Analysis and Machine Intelligence, 1989, 11(7): 674-693.
[7] S. Mallat and W. L.Hwang. Singularity detection and processing with wavelets[J]. IEEE Trans. Inform. Inform. Theory, 1992, 38(2): 617-643.
[8] Yuan Yan Tang, Lihua Yang, Jiming Liu. Wavelet Theory and its Application to Pattern Recognition[M], The World Scientific Publishing Co. Pte, Ltd. , Singapore, 2000.
[9] S. Mallat and S. Zhong. Characterization of signal from multiscale edges[J]. IEEE Trans. PAMI, 1992, 14(7): 710-732.
[10] Xu Y, J.B. Weaver, Healy D M, et al. Wavelet transform domain filters: a spatially selective noise filtration technique[J]. IEEE Trans. Image Processing, 1994, 3(6): 747-758.
[11] D.L. Donoho and I.M. Johnstone. Ideal spatial adaptation via wavelet shrinkage[J]. Biometrika, 1994, 81(3): 425-455.
[12] D.L. Donoho and I. M. Johnstone. Adapting to unknown smoothness via wavelet shrinkage[J]. Journal of the American Statistical Assoc, 1995, 90(432): 1200-1224.
[13] D.L. Donoho. De-noising by soft-thresholding[J]. IEEE Trans. Inform. Theory, 1995, 41(3): 613-627.
[14] S.G. Chang, B. Yu and M. Vetterli. Adaptive wavelet thresholding for image denoising and compression[J]. IEEE Trans. Image Processing, 2000, 9(9): 1532-1546.
[15] S.G. Chang, B. Yu and M. Vetterli. Spatially adaptive wavelet thresholding with context modeling for image denoising[J]. TEEE Trans. Image Processing, 2000, 9(9): 1522-1531.
[16] P. Moulin and Liu J. Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors[J]. IEEE Trans. Inform. Theory, 1999, 45(3): 909-919.
[17] T.T. Cai and B.W. Silverman. Incorporating information on neighboring coefficients into wavelet estimation[J]. Sankhya: The Indian Journal of Statistics, 2001, 63, Series B, Pt. 2: 127-148.
[18] M. Hansen and B. Yu. Wavelet thresholding via MDL for natural images[J]. IEEE Trans. Inform. Theory, 2000, 46(5): 1778-1788.
[19] A. Pizurica and W. Philips. Estimating the probability of the presence of a signal of interest in multiresolution single- and multiband image denoising[J]. IEEE Trans. Image Processing, 2006, 15(3): 654-665.
[20] M.S. Crouse, R.D. Nowak and R.G. Baraniuk. Wavelet-based statistical signal processing using hidden Markov models[J]. IEEE Trans. Signal Processing, 1998, 46(4): 886-902.
[21] L. Sendur and I.W. Selesnick. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency[J]. IEEE Trans. Signal Processing, 2002, 50(11): 2744-2756.
[22] L. Sendur and I.W. Selesnick. Bivariate shrinkage with local variance estimation[J]. IEEE Signal Processing Letters, 2002, 9(12): 438-441.
[23] Z. Cai, T.H. Cheng, C. Lu, et al. Efficient wavelet-based image denoising algorithm[J]. IEEE Electronics Letters, 2001, 37(11): 683-685.
[24] M. Malfait and D. Roose. Wavelet-based image denoising using a Markov random field a priori model[J]. IEEE Trans. Image Processing, 1997, 6(4): 549-565.
[25] A. Pizurica, W. Philips, I. Lemahieu, et al. A Joint Inter-and Intrascale Statistical Model for Bayesian Wavelet Based Image Denoising[J]. IEEE Trans. Image Processing, 2002, 11(5): 545-557.
[26] J. Liu and P. Moulin. Image denoising based on scale-space mixture modeling of wavelet coefficients[C]. In: Proceedings of IEEE International Conference on Image Processing, ICIP 1999, Vol. 1: 386-390.
[27] I.K. Eom and Y.S. Kim. Spatially adaptive denoising based on mixture modeling and interscale dependence of wavelet coefficients[C]. In: Proceedings of IEEE International Conference on Neural Networks & Signal Processing, 2003, Vol. 2: 1070-1073.
[28] R.J. Claudio and S. Jacob. Adaptive image denoising and edge enhancement in scale-space using wavelet transform[J]. Pattern Recognition Letters, 2003, 24(7): 965-971.
[29] D. Cho, T.D. Bui and G.Y. Chen. Multiwavelet statistical modeling for image denoising using wavelet transforms[J]. Signal Processing: Image Communication, 2005, 20(1): 77-89.
[30] G. Strang. Wavelets and dilation equations: A brief introduction[J]. SIAM Review, 1989, 31(4): 614-627.
[31] David L. Donoho, Ana Georgina Flesia. Can recent innovations in harmonic analysis explainkey findings in natural image statistics[M]. Network: Computation in Neural Systems, 2001, 12(3): 371~393.
[32] E.J. Candès, D.L. Donoho. Ridgelets: the Key to Higher-dimensional Intermittency[J]. Phil. Trans. R. Soc. Lond. A, 1999, 357(1760): 2495–2509.
[33] E.J. Candès. On the Representation of Mutilated Sobolev Functions[R]. Stanford Technical Report, Department of Statistics, Stanford University, 1999.
[34] D.L. Donoho. Wedgelets: nearly minimax estimation of edges[J]. Ann Statist, 1999, 27: 859-897.
[35] E.J. Candes, D.L. Donoho. Curvelets[R]. USA: Department of Statistics, Stanford University, 1999.
[36] Pennec. E. Le, S. Mallat. Image compression with geometrical wavelets[J]. Proceeding of CIP 2000. Vancouver, 2000: 661-664.
[37] Do. M. N, M. Vetterli. The Contourlet Transform: an Efficiet Directional Multiresolution Image Representation[J]. IEEE Transactions Image on Processing, 2005, 14(12): 2091-2106.
[38] Truong T. Nguyen, Soontorn Oraintara, The Shiftable Complex Directional Pyramid, PartI: Theoretical Aspects[J]. IEEE Transactions on Signal Processing, 2008, 56(10): 4651–4660.
[39] Truong T. Nguyen, Soontorn Oraintara, The ShiftableComplex Directional Pyramid PartII: Implementation and Applications. IEEE Transactions on Signal Processing[J], 2008, 56(10): 4661–4672.
[40] S.M. Kay.统计信号处理基础——估计与检测理论[M].罗鹏飞,张文明,刘忠等译.北京,电子工业出版社, 2003.
[41]张贤达.现代信号处理第二版[M].北京,清华大学出版社, 2002.
[42] N.G. Kingsbury. Image Processing with the complex wavelet[D]. Phil. Trans Royal Society London A September 1999.
[43] N.G. Kingsbury. Shift invariant prosperities of the Dual-Tree Complex Wavelet Transform[J]. Proc. IEEE Icassp’99, March 1999, paper SPTM3. 6.
[44] F.A. Fernandes, R.L.C van Spaendonck. A New Framework for Complex Wavelet Transform[J]. IEEE Trans. Signal Processing, 2003, 51(7): 1825-1835.
[45] R.L.C. van Spaendonck, F.M. Hindriks, F. C. A. Fernandes, et al. Three- dimensional attribute for seismic interpretation[J]. Ann. Mtg. Abstracts Soc. Exploration Geophys[J]. 2000, 2059-2062.
[46] I.W. Selesnick. The double-density dual-tree DWT. IEEE Trans[J]. on Acoustics, Speech, and Signal Processing, 2004, 52(5): 1304-1314.
[47] Guy Gilboa, Nir Sochen, Yehoshua Y Zeevi. Texture Preserving Variational Denoising Usingan Adaptive Fidelity Term[C]. Proc. VLSM 2003, Nice, France, Oct. 2003.
[48] G.Y. Chen, T.D. Bui, A. Krzy˙zak, Image denoising with neighbour dependency and customized wavelet and threshold[J]. Pattern Recognition, 2005, 38: 115– 124.
[49] Zhou Dengwen, Shen Xiaoliu. Image denoising using block thresholding[J]. Image and Signal Processing, 2008, CISP '08, 27-30 May 2008 , 3: 335– 338.
[50]贾建,项海林.基于剪切不变的递归Contourlet变换图像去噪[J].计算机科学, 2009, 36(5): 254-256
[51]夏君君.多分辨率多方向性变换在图像处理中的应用.中国科技大学硕士学位论文[D]. 2006. 9.
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