基于双树复小波变换的图像去噪
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摘要
图像在采集、获取、编码和传输的过程中会受到各种噪声的干扰,从而使图像的质量下降,对图像信息的处理、传输和存储造成极大的影响。为了抑制噪声,以求改善图像质量所进行的图像处理叫做图像去噪。寻求一种既能有效地减小噪声,又能很好地保留图像边缘信息的方法,是人们一直追求的目标。近年来,随着小波理论的不断完善,凭借其良好的时频局部化特征、尺度变化特征和方向性特征,在图像去噪领域取得了广泛的应用。离散小波变换虽然在图像去噪领域得到了广泛应用,但是存在两个局限性,主要体现在两个方面：
1、缺乏平移不变性。这意味着输入信号有很小的移位都可能导致在各个尺度上的小波系数能量分布有明显的变化。
2、缺乏方向选择性。在普通的小波分解中,每一个尺度空间只能分解成三个方向(水平、垂直、对角),对于方向的选择性相当有限。
利用双树复小波变换进行图像去噪,可以克服一般离散小波变换的上述不足。本论文主要围绕对树复小波变换及其在图像去噪中的应用来进行研究。主要工作包括：
(1)讨论了小波变换去噪的一般原理,介绍了小波去噪的几种方法,重点学习了四种经典的基于离散小波变换的去噪算法：小波变换模极大值去噪,小波变换尺度间相关性的去噪,小波阈值收缩法去噪以及平移不变量小波阈值去噪法。并对四种去噪算法进行了综合比较。
(2)介绍了对偶树复小波变换的原理和特性。对偶树复小波变换具有近似的平移不变性、良好的方向选择性,与此同时,它还具有完全重构特性。对偶树复小波变换在每一层产生六个具有方向选择性的子带,分别指向±15°,+45°,±75°。将对偶树复小波变换应用于图像去噪,可以更好地表示图像的边缘和纹理特征,从而得到较小波更好的去噪效果。
(3)研究了小波系数尺度间相关性模型,利用BKF函数刻画小波系数外尺度之间的相关性。并提出一种结合小波系数外尺度和内尺度相结合的多元BKF模型。在该模型下应用双树复小波变换进行小波去噪。先利用双树复小波进行图像分解,应用最大后验估计(MAP)的得到新的阈值函数,并与同样考虑内外尺度关系的Bishrink,Probshrink和BLS-GSM算法进行实验比较,取得了更好的去噪效果。
Image signal if often influenced by many kinds of noise when collected, acquisited, encoded and transmitted; and the image degradation maybe impair the quality of the image information processing, transmission and storage. Therefore it is a very important work to reduce image noise. The processing in order to reduce noise, and to improve the quality of image is called Image denoising. We are pursuing an effective method not only can it reduce the noise, but also retain the image edge information. In recent years, with the development of the wavelet theory, diserete wavelet transform(DWT) has been widely used in image denoising with its good time-frequency localization characteristics, scale characteristics and direction characteristics.Although the discrete wavelet transform denoising has been widely applied, it has two limitations, mainly in two aspects. Firs,the lack of translation invariance:this means that the input signal with a very small shift can lead to various scales in the wavelet coefficients so that energy distribution will have significant changes. Second, lack of direction selectivity:diserete wavelet transoform coeffieients reveal only threes spatial orientations. (horizontal, vertical, diagonal), so the direction selectivity is limited. In order to overeome the shortcoming of the commonly—used denoisng methods, the image denoising method based on dual tree complex wavelet trnasoform(DT-CWT) is proposed.The paper uses Dual Tree Complex Wavelet Transform for image denoising to overcome the shortage of DWT above-mentioned. The main work can be summarized as ofllows:
(1) We discussed the general principles of wavelet denoising, wavelet denoising introduced several methods, focusing on learning the four types of classical discrete wavelet transform based denoising algorithm:wavelet transform modulus maxima denoising, wavelet transform correlation between scale denoising, wavelet shrinkage threshold denoising method and the translation invariant wavelet thresholding denoising method. And the four denoising algorithms were compared.
(2) We introduced the dual tree complex wavelet transform principle and character. Dual tree complex wavelet transform provides approximate shift invariance, good directional selectivity, at the same time, it also has perfect reconstruction feature. Dual tree complex wavelet transform in the direction of each layer with the selective production of six sub-band, respectively. Dual tree complex wavelet transform will be applied to image denoising, and it can better represent the image edge and texture features, which has smaller waves better denoising effect.
(3) We presents an algorithm in image denoising based on multi-BKF using Dual-Tree Complex Wavelet.This method incorporates both interscale and intrascale wavelet coefficients into the model of the multi-BKF and it sufficiently takes into accout the relevance of interscale and intrascale coefficients. New corresponding nonlinear threshold functions are derived from the models using MAP estimation theory. Ultimately, we obtain the denoised image by computing the inverse transform using the modified coefficients. At the same time, the Dual-Tree Complex Wavelet is discussed(DT-CWT); compared with the traditional discrete Wavelete transform, DT-CWT has the properties of approximate shift invariance and more directionality. The experimental results show that the algorithm performs better than other traditional de-noising algorithms.
1、缺乏平移不变性。这意味着输入信号有很小的移位都可能导致在各个尺度上的小波系数能量分布有明显的变化。
2、缺乏方向选择性。在普通的小波分解中,每一个尺度空间只能分解成三个方向(水平、垂直、对角),对于方向的选择性相当有限。
利用双树复小波变换进行图像去噪,可以克服一般离散小波变换的上述不足。本论文主要围绕对树复小波变换及其在图像去噪中的应用来进行研究。主要工作包括：
(1)讨论了小波变换去噪的一般原理,介绍了小波去噪的几种方法,重点学习了四种经典的基于离散小波变换的去噪算法：小波变换模极大值去噪,小波变换尺度间相关性的去噪,小波阈值收缩法去噪以及平移不变量小波阈值去噪法。并对四种去噪算法进行了综合比较。
(2)介绍了对偶树复小波变换的原理和特性。对偶树复小波变换具有近似的平移不变性、良好的方向选择性,与此同时,它还具有完全重构特性。对偶树复小波变换在每一层产生六个具有方向选择性的子带,分别指向±15°,+45°,±75°。将对偶树复小波变换应用于图像去噪,可以更好地表示图像的边缘和纹理特征,从而得到较小波更好的去噪效果。
(3)研究了小波系数尺度间相关性模型,利用BKF函数刻画小波系数外尺度之间的相关性。并提出一种结合小波系数外尺度和内尺度相结合的多元BKF模型。在该模型下应用双树复小波变换进行小波去噪。先利用双树复小波进行图像分解,应用最大后验估计(MAP)的得到新的阈值函数,并与同样考虑内外尺度关系的Bishrink,Probshrink和BLS-GSM算法进行实验比较,取得了更好的去噪效果。
Image signal if often influenced by many kinds of noise when collected, acquisited, encoded and transmitted; and the image degradation maybe impair the quality of the image information processing, transmission and storage. Therefore it is a very important work to reduce image noise. The processing in order to reduce noise, and to improve the quality of image is called Image denoising. We are pursuing an effective method not only can it reduce the noise, but also retain the image edge information. In recent years, with the development of the wavelet theory, diserete wavelet transform(DWT) has been widely used in image denoising with its good time-frequency localization characteristics, scale characteristics and direction characteristics.Although the discrete wavelet transform denoising has been widely applied, it has two limitations, mainly in two aspects. Firs,the lack of translation invariance:this means that the input signal with a very small shift can lead to various scales in the wavelet coefficients so that energy distribution will have significant changes. Second, lack of direction selectivity:diserete wavelet transoform coeffieients reveal only threes spatial orientations. (horizontal, vertical, diagonal), so the direction selectivity is limited. In order to overeome the shortcoming of the commonly—used denoisng methods, the image denoising method based on dual tree complex wavelet trnasoform(DT-CWT) is proposed.The paper uses Dual Tree Complex Wavelet Transform for image denoising to overcome the shortage of DWT above-mentioned. The main work can be summarized as ofllows:
(1) We discussed the general principles of wavelet denoising, wavelet denoising introduced several methods, focusing on learning the four types of classical discrete wavelet transform based denoising algorithm:wavelet transform modulus maxima denoising, wavelet transform correlation between scale denoising, wavelet shrinkage threshold denoising method and the translation invariant wavelet thresholding denoising method. And the four denoising algorithms were compared.
(2) We introduced the dual tree complex wavelet transform principle and character. Dual tree complex wavelet transform provides approximate shift invariance, good directional selectivity, at the same time, it also has perfect reconstruction feature. Dual tree complex wavelet transform in the direction of each layer with the selective production of six sub-band, respectively. Dual tree complex wavelet transform will be applied to image denoising, and it can better represent the image edge and texture features, which has smaller waves better denoising effect.
(3) We presents an algorithm in image denoising based on multi-BKF using Dual-Tree Complex Wavelet.This method incorporates both interscale and intrascale wavelet coefficients into the model of the multi-BKF and it sufficiently takes into accout the relevance of interscale and intrascale coefficients. New corresponding nonlinear threshold functions are derived from the models using MAP estimation theory. Ultimately, we obtain the denoised image by computing the inverse transform using the modified coefficients. At the same time, the Dual-Tree Complex Wavelet is discussed(DT-CWT); compared with the traditional discrete Wavelete transform, DT-CWT has the properties of approximate shift invariance and more directionality. The experimental results show that the algorithm performs better than other traditional de-noising algorithms.
引文
[1]孙延奎.小波分析及其应用[M].北京：清华大学出版社,2005.219-243
[2]王香菊.基于中值滤波和小波变换的图像去噪研究[D].西安：西安科技大学,2008
[3]冷建华.傅立叶变换[M].北京：清华大学出版社,2004
[4]Yong R. An introduction to nonlinear Fourier seriers [M]. New York:Academix,1980
[5]崔丽鸿,程正兴.M带紧支撑正交对称复尺度函数的构造[J].高等学校计算数学学报,2003,25(2)：160-166.
[6]程正兴.小波分析算法及应用[M].北京：世界图书出版社,1996
[7]Chui C K,Wang J Z. A general framework of compactly supported splines and wavelets [J].Journal of Approximation Theory,1992,71(3):263-304
[8]Mallat S. Multiresolution approximation and wavelet orthonormal bases of L2(R)[J]. Trans Amer Math Soc,1989,315:69-87
[9]Mallat S. Multifrequency channel decompositions of image and wavelet model [J]. IEEE Trans. ASSP,1989,37(12):2091-2110
[10]Mallat S.A theory for multiresolution signal decomposition:the wavelet representation.IEEE Trans.on PAMI,1989,11(7):674-693
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[38]Cui L H, Wang W X, Li W G Biorthogonal balanced multiwavelets with armlet order and its application in imagine denoising[J]. Mathematics and Computers Simulation.己修改
[39]崔丽鸿,李文国.使用贝塞尔K形式密度的多比特倍乘水印的最优解码算法[J].工程数学学报.已投稿
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[2]王香菊.基于中值滤波和小波变换的图像去噪研究[D].西安：西安科技大学,2008
[3]冷建华.傅立叶变换[M].北京：清华大学出版社,2004
[4]Yong R. An introduction to nonlinear Fourier seriers [M]. New York:Academix,1980
[5]崔丽鸿,程正兴.M带紧支撑正交对称复尺度函数的构造[J].高等学校计算数学学报,2003,25(2)：160-166.
[6]程正兴.小波分析算法及应用[M].北京：世界图书出版社,1996
[7]Chui C K,Wang J Z. A general framework of compactly supported splines and wavelets [J].Journal of Approximation Theory,1992,71(3):263-304
[8]Mallat S. Multiresolution approximation and wavelet orthonormal bases of L2(R)[J]. Trans Amer Math Soc,1989,315:69-87
[9]Mallat S. Multifrequency channel decompositions of image and wavelet model [J]. IEEE Trans. ASSP,1989,37(12):2091-2110
[10]Mallat S.A theory for multiresolution signal decomposition:the wavelet representation.IEEE Trans.on PAMI,1989,11(7):674-693
[11]Grossman A, Morlet J, Morrn K. Decomposition of hardy functions into square integrable wavelets of constant shape [J]. SIAM J Math. Anal.,1984,1(5):723-736
[12]Daubechies I. Ten lectures on wavelets [M]. Philadelphia, PA:SIAM,1992.
[13]Daubechies I. Orthonormal bases of compactly supported wavelets [J]. Communications on Pure and Applied Math,1988,41:909-996
[14]Chui C K,Wang J Z. A general framework of compactly supported splines and wavelets [J].Journal of Approximation Theory,1992,71(3):263-304
[15]Goodman T N T, Lee S L. Wavelets of multiplicity r [J]. Trans. Amer. Math. Soc.,1994, 342(1):307-324
[16]Sweldens W. Wavelets and the Lifting Scheme:a 5 minute tour [J]. Zeitschrift FAur Angewandte Mathematic and Mechanic,1996,76(2):41-44
[17]Kingsbury N G. Image processing with complex wavelets [J]. Phil. Trans. Royal Society London Ser.A,1999,357:2543-2560
[18]Selesnick I W, Baraniuk R G, and Kingsbury N G, The dualtree complex wavelet transform—A coherent framework for multiscale signal and image processing[J]. IEEE Signal Process.Mag,2005,22(6):123-151
[19]Mallat S, Hwang W L. Singularity detection and processing with wavelets[J].IEEE Trans on information theory,1992,38(2):617-643
[20]Xu yansun,et al.Wavelet transform domain filters:a spatially selective noise filtration technique[J].IEEE Trans. Image Processing,1994,3(6):747-758
[21]Donovan G, Geronimo J S, Hardin D P, Massopust P. Construction of orthogonal wavelets using fractal interpolation functions [J]. SIAM J. Math. Anal.,1996 27(4):1158-1192
[22]Donoho D L. De-noising by sotf-thresholding[J].IEEE Trans.Inofmr.Theory 1995, 5,41(3):613-627
[23]Donho D L, Johnstone I M.AdaPting to unknown Smoothness Via wavelet Shrinkage[J]. Ame.rStat. Assoe.1995.12,0:1200-1224
[24]Donho D L, Johnstone I M.Ideal spatial adaptation via wavelet shrinkage[J].Biometrika,1994,81:425-455
[25]Chnag S G,Yu B,Vetterli M. Spatially adaptive wavelet thresholding with Context modeling for image denoising[J].EIEETarns.on IP,2000,9,9(9):1522—1531
[26]Chnag S G,Yu B,VetterliM.Adpative wavelet thresholding for image denoising ande compression[J].IEEETrans.Image Proeessing,2000,9(9):1532—1546
[27]Levent Sendur, W. Selesnick., Bivariate Shrinkage with local variance estimation[J]. Functions for wavelet-based denoising exploiting interscale dependency[J]. IEEE Trans. Signal Processing, Lett.,2002,9(12):438-441.
[28]Levent Sendur, SelesnickW. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency[J]. IEEE Trans. Image Processing,2002, 50(11):2744-2756.
[29]Pizurica A, Philips W.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.
[30]Portilla J, Strela V, Wainwright M J, Simoncelli E P. Image denoising using a scale mixture of gaussians in the wavelete domain[J]. IEEE Trans. Image Processing,2003, 12(11):1338-1351.
[31]阮秋琦,阮宇智.数字图像处理[M].北京：电子工业出版社,2007,176-180
[32]张凌霜.非均匀噪声图像的小波去噪研究[D].西安电子科技大学,2005
[33]章毓晋.图像工程上册—图像处理和分析[M].北京：清华大学出版社,2001.9
[34]楼顺天,李博菡.基于MATLAB的系统分析与设计——信号信号处理[M].西安：西安电子科技大学出版社,1999,1-150
[35]Cui Lihong. Some properties and construction of multiwavelets related to different symmetric centers[J]. Mathematics and Computers in Simulation,2005,70(2):69-89.
[36]Cui L H, Li W G Adaptive Multiwavelet-Based Watermarking Through JPW masking.IEEE Trans on Image Processing.已接受
[37]Cui L H, Li W G Image Denoising Using Multivariate SURE-LET Approach Exploiting Interscale and Intrascale Dependencies[J]. Image and Vision Computing.已修改
[38]Cui L H, Wang W X, Li W G Biorthogonal balanced multiwavelets with armlet order and its application in imagine denoising[J]. Mathematics and Computers Simulation.己修改
[39]崔丽鸿,李文国.使用贝塞尔K形式密度的多比特倍乘水印的最优解码算法[J].工程数学学报.已投稿
[40]Cohen A, Daubechies I, Plonka G Regularity of refinable function vectors[J]. J Fourier Anal Appl,1997,3:295-324.
[41]Shen Z. Refinable function vectors[J]. SIAM Journal on Mathematical Analysis,1998, 29(1):235-250.
[42]杨守志,程正兴.紧支撑正交小波的构造[J].工程数学学报,1998,15(2)：23-28
[43]Donoho D L, Duncan M R. Digital curvelet transform:strategy, implementation and experiments[J]. Proc. SPIE,2000,4056:12-29
[44]Shen Z. Refinable function vectors [J]. SIAM J Math. Anal.,1998,29(1):235-250
[45]Lian Jian-ao. Orthogonal criteria for multiscaling functions[J]. Appl.Comp.Harm.Anal. 1998,5(3):277-311.
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