基于Directionlet变换的图像去噪和融合
|
摘要
图像去噪和融合是图像处理的两个重要组成部分。图像在获取和传输过程中,常常受到各种噪声的干扰,这将对后续的图像处理,如分割、压缩等产生不利影响。图像去噪的目的是尽可能恢复图像的原貌,改善图像的质量。图像融合是将多个图像数据所含的优势信息进行互相补充,并有机结合起来产生新的、信息含量更大的图像,来提高图像的信息可用程度,弥补单一信息源的不足。
近年来,小波变换在图像处理中得到广泛应用。小波分析对含点状奇异性的目标函数具有最优的非线性逼近性能,但在高维情况下,小波并不是最优的或最稀疏的表示方法,小波系数不再稀疏,因此不能很好地挖掘图像中的方向信息。新提出的多尺度几何分析工具,致力于构建最优逼近意义下高维函数的表示方法,在图像去噪和融合中有着良好的应用前景。
本文主要研究了多尺度几何分析工具中的Directionlet变换,并将其用于图像的去噪和融合中。本文的主要工作如下:
(1)提出了一种改进的多尺度Directionlet变换的方法。当Directionlet基函数的方向与图像中各向异性目标匹配时,对图像的逼近效果较好,不匹配时则退化为小波,逼近效果很差。本文通过自适应地寻找图像的主要方向,构造Directionlet变换的采样矩阵,使Directionlet的变换方向和队列方向尽量与图像的主要方向一致,能够自适应地捕捉图像中的各向异性特征。
(2)提出了基于改进Directionlet变换的图像去噪方法。针对图像改进Directionlet系数的高尖峰、重拖尾特点,用广义高斯分布模型对图像Directionlet变换系数的各高频子带系数进行建模。在去噪时,结合广义高斯分布形状参数的大小,将形状参数分为两个区间,采用不同的无噪系数估计方法,利用局部标准差估计,对图像进行去噪处理。
(3)将改进的Directionlet变换用于图像融合中。改进的Directionlet变换能自适应地捕捉图像沿不同方向的奇异信息,因而具有更好的方向性和各向异性。在融合时,采用基于区域特性量测的融合规则,改善了融合图像的主观视觉效果和客观评价指标。
Image denoising and fusion are two important components of image processing. During acquisition and transmission, images are often corrupted by diversified noises, which could have a negative impact on the following process such as image segmentation and compression. Denoising is to obtain clearer object and increase the recognition rate through filtering noises in the images. Fusion is to combine advantageous information from multiple images of the same scene to acquire more exact and comprehensive description of the image.
Wavelet transform has found wide applications in image processing in recent years. Wavelet is the optimal bases for functions with point singularity. But in the case of high dimensional, wavelet analysis can not take advantages of the data geometrical features. Then it is not the optimal or the sparsest representation of the functions, so it can not make good use of direction information in images. To solve this problem, a series of new multiscale geometric analysis emerges to building the optimal representation of high dimensional functions, which have a wonderful prospect in the research of image denoising and fusion.
This paper studies one of the multiscale geometric analysis tools, i.e. Directionlet, and its application on image denoising and fusion. The main innovative points are as follows:
(1) We proposed an improved multiscale Directionlet transformation method. When the direction of the Directionlet bases matches that of the anisotropic object in images, Directionlet can represent images well, otherwise bases of Directionlet will degenerate into wavelets and have poor approximation power. In this paper, we find main directions of an image and construct the sample matrix adaptively, which can adaptively catch the anisotropic features in images.
(2) An image denoising algorithm based on improved Directionlet is presented. For the high peaks and heavy tail character of Directionlet coefficients, we model each subband with a generalized Gaussian distribution, then denoise images combining the values of shape parameter and local variance estimate.
(3) We also study the application of the improved Directionlet transform. As the improved Directionlet can catch image information from different directions, so it has better directionality and anisotropy. Image fusion with improved Directionlet transform and regional measures outperforms wavelet fusion method.
This research is supported by National Nature Science Foundation (No.60672126, 60702062,60971128), National High Technology Research and Development Program (863) (No.2007AA12Z136), Major State Basic Research Development Program of China (973) (No.2006CB705707).
近年来,小波变换在图像处理中得到广泛应用。小波分析对含点状奇异性的目标函数具有最优的非线性逼近性能,但在高维情况下,小波并不是最优的或最稀疏的表示方法,小波系数不再稀疏,因此不能很好地挖掘图像中的方向信息。新提出的多尺度几何分析工具,致力于构建最优逼近意义下高维函数的表示方法,在图像去噪和融合中有着良好的应用前景。
本文主要研究了多尺度几何分析工具中的Directionlet变换,并将其用于图像的去噪和融合中。本文的主要工作如下:
(1)提出了一种改进的多尺度Directionlet变换的方法。当Directionlet基函数的方向与图像中各向异性目标匹配时,对图像的逼近效果较好,不匹配时则退化为小波,逼近效果很差。本文通过自适应地寻找图像的主要方向,构造Directionlet变换的采样矩阵,使Directionlet的变换方向和队列方向尽量与图像的主要方向一致,能够自适应地捕捉图像中的各向异性特征。
(2)提出了基于改进Directionlet变换的图像去噪方法。针对图像改进Directionlet系数的高尖峰、重拖尾特点,用广义高斯分布模型对图像Directionlet变换系数的各高频子带系数进行建模。在去噪时,结合广义高斯分布形状参数的大小,将形状参数分为两个区间,采用不同的无噪系数估计方法,利用局部标准差估计,对图像进行去噪处理。
(3)将改进的Directionlet变换用于图像融合中。改进的Directionlet变换能自适应地捕捉图像沿不同方向的奇异信息,因而具有更好的方向性和各向异性。在融合时,采用基于区域特性量测的融合规则,改善了融合图像的主观视觉效果和客观评价指标。
Image denoising and fusion are two important components of image processing. During acquisition and transmission, images are often corrupted by diversified noises, which could have a negative impact on the following process such as image segmentation and compression. Denoising is to obtain clearer object and increase the recognition rate through filtering noises in the images. Fusion is to combine advantageous information from multiple images of the same scene to acquire more exact and comprehensive description of the image.
Wavelet transform has found wide applications in image processing in recent years. Wavelet is the optimal bases for functions with point singularity. But in the case of high dimensional, wavelet analysis can not take advantages of the data geometrical features. Then it is not the optimal or the sparsest representation of the functions, so it can not make good use of direction information in images. To solve this problem, a series of new multiscale geometric analysis emerges to building the optimal representation of high dimensional functions, which have a wonderful prospect in the research of image denoising and fusion.
This paper studies one of the multiscale geometric analysis tools, i.e. Directionlet, and its application on image denoising and fusion. The main innovative points are as follows:
(1) We proposed an improved multiscale Directionlet transformation method. When the direction of the Directionlet bases matches that of the anisotropic object in images, Directionlet can represent images well, otherwise bases of Directionlet will degenerate into wavelets and have poor approximation power. In this paper, we find main directions of an image and construct the sample matrix adaptively, which can adaptively catch the anisotropic features in images.
(2) An image denoising algorithm based on improved Directionlet is presented. For the high peaks and heavy tail character of Directionlet coefficients, we model each subband with a generalized Gaussian distribution, then denoise images combining the values of shape parameter and local variance estimate.
(3) We also study the application of the improved Directionlet transform. As the improved Directionlet can catch image information from different directions, so it has better directionality and anisotropy. Image fusion with improved Directionlet transform and regional measures outperforms wavelet fusion method.
This research is supported by National Nature Science Foundation (No.60672126, 60702062,60971128), National High Technology Research and Development Program (863) (No.2007AA12Z136), Major State Basic Research Development Program of China (973) (No.2006CB705707).
引文
[1]冈萨雷斯等著,阮秋琦等译.数字图像处理.第二版.北京:电子工业出版社. 2003.
[2] Berger C.S.. Self-tuning control of offset using a moving average filter. IEEEproceedings Part D. Control theory and applications, 1985, 133(44): 184-188.
[3] Lopes A., Nezry E., Touzi R, et al. Maximum a posteriori speckle filtering andfirst order texture models in SAR images. Proceedings of IGARSS, 1990:2409-2412.
[4] Baraldi A., Parmigiani E. A refined Gamma MAP SAR speckle filter withimproved geometrical adaptivity. IEEE Transactions on Geoscience and RemoteSensing. 1995, 33(5): 1245-1257.
[5] Lee J. S.. Digital image enhancement and noise filtering by use of localstatistics. IEEE Transactions on Pattern Analysis Machine Intelligence. 1980,PAMI-2: 165-168.
[6] Lee J. S.. Refined filtering of image noise using local statistics. ComputerGraphics and Image Processing. 1981, 15(3): 380-389.
[7] V. S. Frost, J. A. Stiles, K. S. Shanmugan, et al. A model for radar images and itsapplication to adaptive digital filtering of multiplicative noise. IEEE Trans.Pattern Anal. Machine Intell. 1982, PAMI-4: 157-166.
[8] D. T. Kuan, A. A. Sawchuk, T. C. Strand, et al. Adaptive noise smoothing filterfor images with signal-dependent noise. IEEE Trans. Pattern Anal. MachineIntell. 1985, PAMI-2: 165-177.
[9] Sthephane G Mallat. A Theory for Multiresolution Signal Decomposition: TheWavelet Representation. IEEE Transactions on Patterns Analysis and MachineIntelligence. 1989, 11(7): 674-693.
[10] A. Grossman, J. Morlet. Decomposition of Hardy Functions into SquareIntegrable Wavelets of Constant Shape, SIAM J. Appl. 1984, 15: 723-737.
[11] C. K. Chui. An Introduction to Wavelets. San Diego, Academic Press. 1992.
[12] Donoho D. L.. Denoising by soft-thresholding. IEEE Transactions onInformation Theory. 1995, 41(3): 613-627.
[13] Donoho D. L., Johhstone I. M.. Adapting to unknown smoothness via waveletshrinkage. Journal of the American Statistical Assoc. 1995, 90(12): 1200-1224.
[14] Jansen M, Bultheel A.. Multiple wavelet threshold estimation by generalized cross validation for images with correlated noise. IEEE Transactions on Image Processing, 1999, 8(7): 947-953.
[15]胡海平,莫玉龙.基于贝叶斯估计的小波阈值图像降噪方法.红外与毫米波学报. 2002, 21(1):74-76.
[16] Huang X., Woolsey G. A.. Image denoising using wiener filtering and waveletthresholding. IEEE International Conference on Image Processing. 2000, 3:1759-1762.
[17] Candes E. J. Ridgelets: theory and applications. USA: Department of Statistics,Stanford University. 1998.
[18] Donoho D. L., Mark R. Duncan. Digital Curvelet Transform: Strategy,Implementation and Experiments. SPIE proceedings series. 1999: 12-29.
[19] Emmanuel J. Candes, Donoho D. L.. New Tight Frames of Curvelets andOptimal Representations of Objects with C2 Singularities. Commun Pure Appl.2004, 57(2): 219-266.
[20] Do M. N., Vetterli M. The Contourlet transform: an efficient directional multiresolution image representation. IEEE Transactions on Image Processing.2005, 14(12): 2091-2106.
[21] Velisavljevic V., Beferull-Lozano B., Vetterli M., Dragotti P. L. Directionlets: anisotropic multi-directional representation with separable filtering. IEEE Transactions on Image Processing. 2006, 15(7): 1916-1933.
[22] J-L. Starck, E. J. Candes, Donoho D.L.. The Curvelet transform for image denoising. IEEE Transactions on Image Processing. 2002, 6(11): 670-684.
[23] HongJun Li, ZhiMin Zhao. Image Denoising Algorithm Based on Improved Filter in Contourlet Domain. Computer Science and Information Engineering. 2009 WRI World Congress .2009, 6: 499-503.
[24]邵芸.合成孔径雷达.北京:科学出版社. 2001.
[25]张澄波.综合孔径雷达原理系统分析与应用.北京:科学出版社. 1989.
[26] Keydel W.. Perspectives and visions for future SAR systems. IEEE processingon Radar Sonar Navig. 2003, 3: 7-103.
[27] Arsenault H. H., April G.. Properties of speckle integrated with a finite apertureand logarithmically transformed. Journal of the optical society of America. 1976,66: 1160-1163.
[28] S. Fukuda, H. Hirosawa. Suppression of speckle in synthetic aperture radarimages using wavelet. International Journal of Remote Sensing. 1998, 19(3):507-519.
[29] Yocky D. A. Image Merging and Data Fusion by Means of the Discrete TwoDimensional Wavelet Transform. J Opt Soc Am A. 1995, 2(9): 34-1841.
[30] Meek T.R.. Multiresolution Image Fusion of The Maticmapper Imagery withSynthetic Aperture Radar Imagery. USA: tah State University, 1999.
[31] Chavez P. S., etal. Comparison of Three Different Methods to MergeMultire solution and Multispectral Data, Landsat TM and SPOT Panchromatic.PE&RS.1991, 57(3): 265-303.
[32] Hayden R., Dalke G. W., Henkel J., Bare J. E. Application of the HIS ColorTransform to the Processing of Multisensor Data and Image Enhancement.Proceedings of the International Symposium on Remote Sensing of Arid andSemi-arid Lands, Cairo, Egypt. 1982, 599-616.
[33]王文武.应用主成分分解(PCA)法的图像融合技术.微计算机信息, 2007, 23 (4): 285-286.
[34] Mallat S., Zhong S. Characterization of Signals from Multiscale Edges. IEEETran on Pattern Analysis and Machine Intelligence. 1990, 12: 629-639.
[35] Li H., Manjunath B. S., Mitra S. K.. Multisensor Image Fusion Using theWavelet Transform. Graphical Models and Image Processing. 1995, 27(3):235-244.
[36] David L. H. An Introduction to Multisensor Data Fusion. Proceedings of theIEEE. 1997, 85(l):6-23.
[37] Meyer EG, Coifman R. R. Brushlets: A Tool for Directional Image Analysisand Image Compression. Journal of Application and Computer, HarmonicAnalysis. 1997,5:147-187.
[38] Candes E. J. Monoscale Ridgelets for the Representation of Images with Edges.Tech Report, Department of Statistics, Stanford University, 1999.
[39] Pennec E.L., Mallat S. Image Compression with Geometrical Wavelets. InProcessing of ICIP'2000. Vancouver, Canada, September, 2000: 661-664.
[40] Donoho D. L., Huo Xiaoming.. Beamlets and multiscale image analysis.Multiscale and Multire solution Methods, Berlin, Springer Presss.2001: 149-196.
[41] J.D. Foley, A.V. Dam, S.K. Feiner, J.F. Hughes. Computer graphics: Principlesand practice. Reading. MA: Adilson-Wesley Publishing Company, 1990.
[42] Pesquet J.-C, Krim H., Carfantan H.. Time-invariant orthonormal waveletrepresentations, IEEE Transactions on Signal Processing, 1996, 44(8):1964-1970.
[43] S. Mallat. A theory for multiresolution decomposition: the wavelet represention.IEEE Trans on Pattern Anal Mach Intell. 1989,11(7): 674-693.
[44] S Mallat, S Zhong. Characterization of signals from multiscale edges [J]. IEEETransactions on Pattern Analysis Machine Intelligence. 1992, 14(7): 709-732.
[45] D. L. Donoho, I. M. John stone, Ideal spatial adaptation via wavelet shrinkage.Biometrika.1994, 81: 425-455.
[46] Chang S. G., Yu B., Vetterli M.. Adaptive wavelet thresholding for imagedenoising and compression. IEEE Transactions on Image Processing. 2000, 9(9):1532-1546.
[47] Pierre Moulin, Juan Liu. Analysis of Multiresolution Image Denoising SchemesUsing Generalized Gaussian and Complexity Priors. IEEE Transactions onInformation Theory. APRIL 1999, 45(3).
[48]李军侠,水鹏朗.基于广义高斯最大似然估计的小波域类LMMSE滤波算法.电子与信息学报. 2007, 29(12): 2853-2857.
[49] Mortazavi S. H., Shahrtash S. M. Comparing denoising performance ofDWT,WPT, SWT and DT-CWT for Partial Discharge signals, UniversitiesPower Engineering Conference, Padova, Italy. Sept. 1-4, 2008: 1-6.
[50] Kang Li, Jinghuai Gao, Wei Wang. Adaptive Shrinkage for Image DenoisingBased on Contourlet Transform. Second International Symposium on IntelligentInformation Technology Application, Shanghai, China, Dec.20-22. 2008, 2:995-999.
[51] D. L. Donoho, I. M. John stone, Ideal spatial adaptation via wavelet shrinkage.Biometrika.1994, 81: 425-455.
[52] Po D.D.-Y., Do M. N.. Directional Multiscale Modeling of Images Using theContourlet Transform. IEEE Transactions on Image Processing. 2006, 15(6):1610-1620.
[53] J. R. Sveinsson, J. A. Benediktsson. Speckle reduction and enhancement ofSAR images in the Wavelet domain. Proceedings of IEEE InternationalGeoscience and Remote Sensing Symp 1996. 1996: 63-66.
[54]白静,侯彪,王爽,焦李成.基于提升Directionlet域高斯混合尺度模型的SAR图像噪声抑制.计算机学报. 2008, 7(31): 1234-1241.
[55] Durand J.M. SAR data filtering for classification. IEEE Transactions onGeoscience and Remote Sensing. 1996, 25(5): 629-637.
[56]刘贵喜,杨万海[56] .基于多尺度对比度塔的图像融合方法及性能评价.光学学报. 2001, 21(11): 1336-1342.
[57]王海晖,彭嘉雄,吴巍.基于小波包变换的遥感图像融合.中国图像图形学报. 2002. Geoscience and Remote Sensing. 1996, 25(5): 629-637.
[58] Burt P. J., Kolczynski, R. J.. Enhanced image capture through fusion.Proceedings of Fourth International Conference on Computer Vision. 1993, 5:173-182.
[59] Nunez J., Otazu X., Fors O., Prades A., Pala V, rbiol R. Multiresoution-BasedImage Fusion with Additive Wavelet Decomposition. IEEE Transactions onGeoscience and Remote Sensing, 1999, 37(3).
[60] Li Hui, Manjunath B.S., Sanjit Mitra. Multi-Sensor Image Fusion UsingWavelet Transform. Proceedings of the 1994 IEEE International Conferenceon Image Processing. 1994, 170: 51-55.
[61] Wang Hai-hui, Peng Jia-xiong, Wu Wei. Remote Sensing Image Fusion UsingWavelet Packet Transform, Journal of Image and Graphics. 2002, 9(7): 922-937.
[62]刘贵喜,杨万海.基于小波分解的图像融合方法及性能评价.自动化学报.2002, 11(28): 927-934.
[2] Berger C.S.. Self-tuning control of offset using a moving average filter. IEEEproceedings Part D. Control theory and applications, 1985, 133(44): 184-188.
[3] Lopes A., Nezry E., Touzi R, et al. Maximum a posteriori speckle filtering andfirst order texture models in SAR images. Proceedings of IGARSS, 1990:2409-2412.
[4] Baraldi A., Parmigiani E. A refined Gamma MAP SAR speckle filter withimproved geometrical adaptivity. IEEE Transactions on Geoscience and RemoteSensing. 1995, 33(5): 1245-1257.
[5] Lee J. S.. Digital image enhancement and noise filtering by use of localstatistics. IEEE Transactions on Pattern Analysis Machine Intelligence. 1980,PAMI-2: 165-168.
[6] Lee J. S.. Refined filtering of image noise using local statistics. ComputerGraphics and Image Processing. 1981, 15(3): 380-389.
[7] V. S. Frost, J. A. Stiles, K. S. Shanmugan, et al. A model for radar images and itsapplication to adaptive digital filtering of multiplicative noise. IEEE Trans.Pattern Anal. Machine Intell. 1982, PAMI-4: 157-166.
[8] D. T. Kuan, A. A. Sawchuk, T. C. Strand, et al. Adaptive noise smoothing filterfor images with signal-dependent noise. IEEE Trans. Pattern Anal. MachineIntell. 1985, PAMI-2: 165-177.
[9] Sthephane G Mallat. A Theory for Multiresolution Signal Decomposition: TheWavelet Representation. IEEE Transactions on Patterns Analysis and MachineIntelligence. 1989, 11(7): 674-693.
[10] A. Grossman, J. Morlet. Decomposition of Hardy Functions into SquareIntegrable Wavelets of Constant Shape, SIAM J. Appl. 1984, 15: 723-737.
[11] C. K. Chui. An Introduction to Wavelets. San Diego, Academic Press. 1992.
[12] Donoho D. L.. Denoising by soft-thresholding. IEEE Transactions onInformation Theory. 1995, 41(3): 613-627.
[13] Donoho D. L., Johhstone I. M.. Adapting to unknown smoothness via waveletshrinkage. Journal of the American Statistical Assoc. 1995, 90(12): 1200-1224.
[14] Jansen M, Bultheel A.. Multiple wavelet threshold estimation by generalized cross validation for images with correlated noise. IEEE Transactions on Image Processing, 1999, 8(7): 947-953.
[15]胡海平,莫玉龙.基于贝叶斯估计的小波阈值图像降噪方法.红外与毫米波学报. 2002, 21(1):74-76.
[16] Huang X., Woolsey G. A.. Image denoising using wiener filtering and waveletthresholding. IEEE International Conference on Image Processing. 2000, 3:1759-1762.
[17] Candes E. J. Ridgelets: theory and applications. USA: Department of Statistics,Stanford University. 1998.
[18] Donoho D. L., Mark R. Duncan. Digital Curvelet Transform: Strategy,Implementation and Experiments. SPIE proceedings series. 1999: 12-29.
[19] Emmanuel J. Candes, Donoho D. L.. New Tight Frames of Curvelets andOptimal Representations of Objects with C2 Singularities. Commun Pure Appl.2004, 57(2): 219-266.
[20] Do M. N., Vetterli M. The Contourlet transform: an efficient directional multiresolution image representation. IEEE Transactions on Image Processing.2005, 14(12): 2091-2106.
[21] Velisavljevic V., Beferull-Lozano B., Vetterli M., Dragotti P. L. Directionlets: anisotropic multi-directional representation with separable filtering. IEEE Transactions on Image Processing. 2006, 15(7): 1916-1933.
[22] J-L. Starck, E. J. Candes, Donoho D.L.. The Curvelet transform for image denoising. IEEE Transactions on Image Processing. 2002, 6(11): 670-684.
[23] HongJun Li, ZhiMin Zhao. Image Denoising Algorithm Based on Improved Filter in Contourlet Domain. Computer Science and Information Engineering. 2009 WRI World Congress .2009, 6: 499-503.
[24]邵芸.合成孔径雷达.北京:科学出版社. 2001.
[25]张澄波.综合孔径雷达原理系统分析与应用.北京:科学出版社. 1989.
[26] Keydel W.. Perspectives and visions for future SAR systems. IEEE processingon Radar Sonar Navig. 2003, 3: 7-103.
[27] Arsenault H. H., April G.. Properties of speckle integrated with a finite apertureand logarithmically transformed. Journal of the optical society of America. 1976,66: 1160-1163.
[28] S. Fukuda, H. Hirosawa. Suppression of speckle in synthetic aperture radarimages using wavelet. International Journal of Remote Sensing. 1998, 19(3):507-519.
[29] Yocky D. A. Image Merging and Data Fusion by Means of the Discrete TwoDimensional Wavelet Transform. J Opt Soc Am A. 1995, 2(9): 34-1841.
[30] Meek T.R.. Multiresolution Image Fusion of The Maticmapper Imagery withSynthetic Aperture Radar Imagery. USA: tah State University, 1999.
[31] Chavez P. S., etal. Comparison of Three Different Methods to MergeMultire solution and Multispectral Data, Landsat TM and SPOT Panchromatic.PE&RS.1991, 57(3): 265-303.
[32] Hayden R., Dalke G. W., Henkel J., Bare J. E. Application of the HIS ColorTransform to the Processing of Multisensor Data and Image Enhancement.Proceedings of the International Symposium on Remote Sensing of Arid andSemi-arid Lands, Cairo, Egypt. 1982, 599-616.
[33]王文武.应用主成分分解(PCA)法的图像融合技术.微计算机信息, 2007, 23 (4): 285-286.
[34] Mallat S., Zhong S. Characterization of Signals from Multiscale Edges. IEEETran on Pattern Analysis and Machine Intelligence. 1990, 12: 629-639.
[35] Li H., Manjunath B. S., Mitra S. K.. Multisensor Image Fusion Using theWavelet Transform. Graphical Models and Image Processing. 1995, 27(3):235-244.
[36] David L. H. An Introduction to Multisensor Data Fusion. Proceedings of theIEEE. 1997, 85(l):6-23.
[37] Meyer EG, Coifman R. R. Brushlets: A Tool for Directional Image Analysisand Image Compression. Journal of Application and Computer, HarmonicAnalysis. 1997,5:147-187.
[38] Candes E. J. Monoscale Ridgelets for the Representation of Images with Edges.Tech Report, Department of Statistics, Stanford University, 1999.
[39] Pennec E.L., Mallat S. Image Compression with Geometrical Wavelets. InProcessing of ICIP'2000. Vancouver, Canada, September, 2000: 661-664.
[40] Donoho D. L., Huo Xiaoming.. Beamlets and multiscale image analysis.Multiscale and Multire solution Methods, Berlin, Springer Presss.2001: 149-196.
[41] J.D. Foley, A.V. Dam, S.K. Feiner, J.F. Hughes. Computer graphics: Principlesand practice. Reading. MA: Adilson-Wesley Publishing Company, 1990.
[42] Pesquet J.-C, Krim H., Carfantan H.. Time-invariant orthonormal waveletrepresentations, IEEE Transactions on Signal Processing, 1996, 44(8):1964-1970.
[43] S. Mallat. A theory for multiresolution decomposition: the wavelet represention.IEEE Trans on Pattern Anal Mach Intell. 1989,11(7): 674-693.
[44] S Mallat, S Zhong. Characterization of signals from multiscale edges [J]. IEEETransactions on Pattern Analysis Machine Intelligence. 1992, 14(7): 709-732.
[45] D. L. Donoho, I. M. John stone, Ideal spatial adaptation via wavelet shrinkage.Biometrika.1994, 81: 425-455.
[46] Chang S. G., Yu B., Vetterli M.. Adaptive wavelet thresholding for imagedenoising and compression. IEEE Transactions on Image Processing. 2000, 9(9):1532-1546.
[47] Pierre Moulin, Juan Liu. Analysis of Multiresolution Image Denoising SchemesUsing Generalized Gaussian and Complexity Priors. IEEE Transactions onInformation Theory. APRIL 1999, 45(3).
[48]李军侠,水鹏朗.基于广义高斯最大似然估计的小波域类LMMSE滤波算法.电子与信息学报. 2007, 29(12): 2853-2857.
[49] Mortazavi S. H., Shahrtash S. M. Comparing denoising performance ofDWT,WPT, SWT and DT-CWT for Partial Discharge signals, UniversitiesPower Engineering Conference, Padova, Italy. Sept. 1-4, 2008: 1-6.
[50] Kang Li, Jinghuai Gao, Wei Wang. Adaptive Shrinkage for Image DenoisingBased on Contourlet Transform. Second International Symposium on IntelligentInformation Technology Application, Shanghai, China, Dec.20-22. 2008, 2:995-999.
[51] D. L. Donoho, I. M. John stone, Ideal spatial adaptation via wavelet shrinkage.Biometrika.1994, 81: 425-455.
[52] Po D.D.-Y., Do M. N.. Directional Multiscale Modeling of Images Using theContourlet Transform. IEEE Transactions on Image Processing. 2006, 15(6):1610-1620.
[53] J. R. Sveinsson, J. A. Benediktsson. Speckle reduction and enhancement ofSAR images in the Wavelet domain. Proceedings of IEEE InternationalGeoscience and Remote Sensing Symp 1996. 1996: 63-66.
[54]白静,侯彪,王爽,焦李成.基于提升Directionlet域高斯混合尺度模型的SAR图像噪声抑制.计算机学报. 2008, 7(31): 1234-1241.
[55] Durand J.M. SAR data filtering for classification. IEEE Transactions onGeoscience and Remote Sensing. 1996, 25(5): 629-637.
[56]刘贵喜,杨万海[56] .基于多尺度对比度塔的图像融合方法及性能评价.光学学报. 2001, 21(11): 1336-1342.
[57]王海晖,彭嘉雄,吴巍.基于小波包变换的遥感图像融合.中国图像图形学报. 2002. Geoscience and Remote Sensing. 1996, 25(5): 629-637.
[58] Burt P. J., Kolczynski, R. J.. Enhanced image capture through fusion.Proceedings of Fourth International Conference on Computer Vision. 1993, 5:173-182.
[59] Nunez J., Otazu X., Fors O., Prades A., Pala V, rbiol R. Multiresoution-BasedImage Fusion with Additive Wavelet Decomposition. IEEE Transactions onGeoscience and Remote Sensing, 1999, 37(3).
[60] Li Hui, Manjunath B.S., Sanjit Mitra. Multi-Sensor Image Fusion UsingWavelet Transform. Proceedings of the 1994 IEEE International Conferenceon Image Processing. 1994, 170: 51-55.
[61] Wang Hai-hui, Peng Jia-xiong, Wu Wei. Remote Sensing Image Fusion UsingWavelet Packet Transform, Journal of Image and Graphics. 2002, 9(7): 922-937.
[62]刘贵喜,杨万海.基于小波分解的图像融合方法及性能评价.自动化学报.2002, 11(28): 927-934.