小波技术在信号滤波中的应用研究
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摘要
小波分析是在傅立叶分析基础上发展而来的新的时频分析工具,具备良好的时频局部化性质和多分辨率特性,在信号处理领域中得到广泛的应用。本论文对小波变换在信号滤波中的应用进行了研究。
信号在采集、转换和传输过程中,由于受到设备、环境及人为因素的影响,使信号不可避免地受到噪声干扰。因此,如何去除信号中的噪声,得到感兴趣的信息是信号处理过程中的一项关键技术。根据小波变换的性质和噪声的统计特性,Donoho提出了小波阈值滤波方法,通过选择合适的阈值和阈值函数,对含噪信号的小波系数进行阈值化处理,可有效地去除噪声。
本文研究了基于小波阈值的信号滤波问题,在阈值选取上,依据信号与噪声的小波系数在多尺度上的变化规律的不同,提出一种改进的阈值的选取方法,数值实验结果表明了该算法的有效性和可行性。针对传统的硬阈值函数和软阈值函数存在的缺陷,即硬阈值函数的不连续性,软阈值函数存在恒定偏差,提出了一种新的阈值函数,有效地克服了硬阈值函数和软阈值函数的缺点。实验结果表明,改进后的方法可获得更好的滤波效果。
总之,本文所提出的新的阈值滤波方法获得了很好的结果,具有很好的稳定性和可靠性。
Wavelet analysis developed from Fourier analysis is a new time-frequency analysis tool which has favorable time-frequency localized and multi-resolution properties. Wavelet analysis has been widely applied in signal processing field and other fields. In this paper, a fast signal reconstruction algorithm and a signal de-noising algorithm from its wavelet transform modulus maxima are presented.
In the process of signal collection, transform and transmission, the signal often mix noise ineluctably because the equipments, environments and even human errors. De-noising with the purpose of extracting desired information has been a crucial technique in signal processing. Based on the properties of wavelet transform and the statistical characteristics of noise, Donoho presented the threshold de-noising in wavelet transform domain. By selecting appropriate threshold and threshold function, the noise can be suppressed.
According to the different characters of Wavelet coefficients of signal and noise, a new threshold selection method is put forward. The experimental results show that this method is efficient and practical. But there are discontinuity of hard-threshold function and biased estimation of soft-threshold function. In this paper, a new threshold function which overcomes the inherent disadvantages of hard-threshold and soft-threshold is proposed. Experiments show that the improved method has better performance of de-noising than the traditional methods.
In short, the new thresholding algorithm can reach the expected outcome and has good stability and reliability.
信号在采集、转换和传输过程中,由于受到设备、环境及人为因素的影响,使信号不可避免地受到噪声干扰。因此,如何去除信号中的噪声,得到感兴趣的信息是信号处理过程中的一项关键技术。根据小波变换的性质和噪声的统计特性,Donoho提出了小波阈值滤波方法,通过选择合适的阈值和阈值函数,对含噪信号的小波系数进行阈值化处理,可有效地去除噪声。
本文研究了基于小波阈值的信号滤波问题,在阈值选取上,依据信号与噪声的小波系数在多尺度上的变化规律的不同,提出一种改进的阈值的选取方法,数值实验结果表明了该算法的有效性和可行性。针对传统的硬阈值函数和软阈值函数存在的缺陷,即硬阈值函数的不连续性,软阈值函数存在恒定偏差,提出了一种新的阈值函数,有效地克服了硬阈值函数和软阈值函数的缺点。实验结果表明,改进后的方法可获得更好的滤波效果。
总之,本文所提出的新的阈值滤波方法获得了很好的结果,具有很好的稳定性和可靠性。
Wavelet analysis developed from Fourier analysis is a new time-frequency analysis tool which has favorable time-frequency localized and multi-resolution properties. Wavelet analysis has been widely applied in signal processing field and other fields. In this paper, a fast signal reconstruction algorithm and a signal de-noising algorithm from its wavelet transform modulus maxima are presented.
In the process of signal collection, transform and transmission, the signal often mix noise ineluctably because the equipments, environments and even human errors. De-noising with the purpose of extracting desired information has been a crucial technique in signal processing. Based on the properties of wavelet transform and the statistical characteristics of noise, Donoho presented the threshold de-noising in wavelet transform domain. By selecting appropriate threshold and threshold function, the noise can be suppressed.
According to the different characters of Wavelet coefficients of signal and noise, a new threshold selection method is put forward. The experimental results show that this method is efficient and practical. But there are discontinuity of hard-threshold function and biased estimation of soft-threshold function. In this paper, a new threshold function which overcomes the inherent disadvantages of hard-threshold and soft-threshold is proposed. Experiments show that the improved method has better performance of de-noising than the traditional methods.
In short, the new thresholding algorithm can reach the expected outcome and has good stability and reliability.
引文
[1]潘泉,张磊,孟晋丽,et al.小波滤波方法及应用,北京:清华大学出版社,2005.
[2]Mallat S G,Hwang W L.Singularity detection and processing with wavelets.Information Theory,IEEE Transactions on,1992,38(2):617-643.
[3]Lu J,et al.Contrast enhancemeent of medical images using multiscale edge representations.Optical Engineering,1994,33(7):2151-2161.
[4]Witkin A.Scale space filtering.In:Proc.8th Int.Joint Conf Artificial Intell,Karlsruhe,West Germany,1983,1019-1021.
[5]Mallat S G,Zhong S.Characterization of Signals From Multiscales Edges.IEEE Trans.Pattern Anal.Machine Intell,1992,14(7):710-732.
[6]Rosenfeld A.A nonlinear edge detection technique.In:Proce.IEEE,1970,58(5):814-816.
[7]Xu Yansu,et al.Wavelet transform domain filters:a spatially selective noisefiltration technique.IEEE Trans.Image Processing,1994,3(6):747-758.
[8]Donoho D L.Wavelet shrinkage and W.V.D - a ten-minute tour.http://www.standord.edu/toulous.gs,.
[9]Donoho D L.De-noising by soft-thresholding.IEEE Trans.Information Theory,1995,41(3):613-627.
[10]Donoho D L.Ideal spatial adaptation by wavelet shrinkage:Johnstone I M.1994:81,424-455.
[11]Donoho D L.Adapting to unknown smoothness via wavelet shrinkage.Journal of the American Statistical Association,1995,90:1200-1224.
[12]Donoho D L,Johnstone I M.Minimax estimation via wavelet shrinkage.Annals of Statistis,1998,26(3):879-921.
[13]Donoho D L,Johnstone I M.Ideal denoising in an orthonormal basis chosen from a library of bases.Compt.Rend.Acad.Sci.Paris ser A,1994,:1317-1322.
[14]彭玉华.小波变换与工程应用,北京:科学出版社,2000.
[15]秦前清,杨宗凯.实用小波分析,西安:西安交通大学出版社,1994.
[16]Chui C K.An introduction to wavelets,New York:Academic Press,1992.
[17]Daubechies I.Ten lectures on wavelets,Philadephia:SIAM,1992.1-351.
[18]Wei Dong,Tian Jun,Raymond O W.A new class of Biorthogonal wavelet systems for image transform coding.IEEE Transaction on image processing,1987,7(7):1000-1013.
[19]Daubechies I.Orthonormal bases of compactly supported wavelets.Comm.Pure and Appl.Math,1988,41:909-996.
[20]Zhou D X.Multivariate orthogonal bases of wavelet.Journal of Mathematical Research & Exposition,1992,12(2).
[21]Cohen A,Dabuechies I,Feauveau J C.Biorthogonal bases of compactly supported.Commun.On Pure and Appl.Math,1992,45:485-560.
[22]Mallat S G.A theory for multiresolution signal decomposition:The wavelet representation.IEEE Trans.Pattern Anal.Machine Intell,1989,11(7):674-693.
[23]Porat B.Digital Processing of Random Signals:Theory and Method,Prentice-Hall,Englewood Cliffs.NJ,1978.
[24]Gao Hong-Ye.WaveShrink and semisoft shrinkage.StaSci Research Report,1995,.
[25]Bruce A G.Waveshrink with firm shrinkage.Statistica Sinica,1997,7(4):855-874.
[26]曲天书,戴逸松,王树勋.基于SURE无偏估计的自适应小波阈值去噪.电子学报,2002,30(2):266-268.
[27]赵瑞珍.小波理论及其在图像、信号处理中的算法研究:[博士学位论文].西安:西安电子科技大学,2001.
[28]Bruce A G.Understanding waveshrink:variance and bias estimation.Biometrika,1996,83(4):727-745.
[29]Cetin A E,Ansari R.Signal recovery from Wavelet transform maxima.IEEE Trans.Signal Processing,1994,42(1):194-196.
[30]Coifman R R,Donoho D L.Translation-Invariant De-Noising,New York:Springer-Verlag,1994.125-150.
[31]Lang M,et al.Noise Reduction Using an Undecimated Discrete Wavelet Transform.IEEE signal Processing Letters,1996,3(1):10-12.
[32]Mallat S G.Wavelets for a vision.Proceedings of the IEEE,1996,84(4):604-614.
[33]刘贵忠,张志明,冯牧.由小波变换的模极大值快速重构信.自然科学进展,2000,10(7):660-664.
[34]张维强,宋国乡.基于一种新的阈值函数的小波域信号去噪.西安电子科技大学学报,2004,31(2):296-299.
[35]朱高中,王艳红.一种改进的小波阈值法在信号消噪中的研究.继电器,2007,35(18):41-45.
[36]何焰兰,苏勇,高永媚.一种自适应小波去噪算法.电子学报,2000,28(10):138-140.
[37]张磊,潘泉,张洪才,et al.小波域滤波阈值参数c的选取.电子学报,2001,29(3).
[2]Mallat S G,Hwang W L.Singularity detection and processing with wavelets.Information Theory,IEEE Transactions on,1992,38(2):617-643.
[3]Lu J,et al.Contrast enhancemeent of medical images using multiscale edge representations.Optical Engineering,1994,33(7):2151-2161.
[4]Witkin A.Scale space filtering.In:Proc.8th Int.Joint Conf Artificial Intell,Karlsruhe,West Germany,1983,1019-1021.
[5]Mallat S G,Zhong S.Characterization of Signals From Multiscales Edges.IEEE Trans.Pattern Anal.Machine Intell,1992,14(7):710-732.
[6]Rosenfeld A.A nonlinear edge detection technique.In:Proce.IEEE,1970,58(5):814-816.
[7]Xu Yansu,et al.Wavelet transform domain filters:a spatially selective noisefiltration technique.IEEE Trans.Image Processing,1994,3(6):747-758.
[8]Donoho D L.Wavelet shrinkage and W.V.D - a ten-minute tour.http://www.standord.edu/toulous.gs,.
[9]Donoho D L.De-noising by soft-thresholding.IEEE Trans.Information Theory,1995,41(3):613-627.
[10]Donoho D L.Ideal spatial adaptation by wavelet shrinkage:Johnstone I M.1994:81,424-455.
[11]Donoho D L.Adapting to unknown smoothness via wavelet shrinkage.Journal of the American Statistical Association,1995,90:1200-1224.
[12]Donoho D L,Johnstone I M.Minimax estimation via wavelet shrinkage.Annals of Statistis,1998,26(3):879-921.
[13]Donoho D L,Johnstone I M.Ideal denoising in an orthonormal basis chosen from a library of bases.Compt.Rend.Acad.Sci.Paris ser A,1994,:1317-1322.
[14]彭玉华.小波变换与工程应用,北京:科学出版社,2000.
[15]秦前清,杨宗凯.实用小波分析,西安:西安交通大学出版社,1994.
[16]Chui C K.An introduction to wavelets,New York:Academic Press,1992.
[17]Daubechies I.Ten lectures on wavelets,Philadephia:SIAM,1992.1-351.
[18]Wei Dong,Tian Jun,Raymond O W.A new class of Biorthogonal wavelet systems for image transform coding.IEEE Transaction on image processing,1987,7(7):1000-1013.
[19]Daubechies I.Orthonormal bases of compactly supported wavelets.Comm.Pure and Appl.Math,1988,41:909-996.
[20]Zhou D X.Multivariate orthogonal bases of wavelet.Journal of Mathematical Research & Exposition,1992,12(2).
[21]Cohen A,Dabuechies I,Feauveau J C.Biorthogonal bases of compactly supported.Commun.On Pure and Appl.Math,1992,45:485-560.
[22]Mallat S G.A theory for multiresolution signal decomposition:The wavelet representation.IEEE Trans.Pattern Anal.Machine Intell,1989,11(7):674-693.
[23]Porat B.Digital Processing of Random Signals:Theory and Method,Prentice-Hall,Englewood Cliffs.NJ,1978.
[24]Gao Hong-Ye.WaveShrink and semisoft shrinkage.StaSci Research Report,1995,.
[25]Bruce A G.Waveshrink with firm shrinkage.Statistica Sinica,1997,7(4):855-874.
[26]曲天书,戴逸松,王树勋.基于SURE无偏估计的自适应小波阈值去噪.电子学报,2002,30(2):266-268.
[27]赵瑞珍.小波理论及其在图像、信号处理中的算法研究:[博士学位论文].西安:西安电子科技大学,2001.
[28]Bruce A G.Understanding waveshrink:variance and bias estimation.Biometrika,1996,83(4):727-745.
[29]Cetin A E,Ansari R.Signal recovery from Wavelet transform maxima.IEEE Trans.Signal Processing,1994,42(1):194-196.
[30]Coifman R R,Donoho D L.Translation-Invariant De-Noising,New York:Springer-Verlag,1994.125-150.
[31]Lang M,et al.Noise Reduction Using an Undecimated Discrete Wavelet Transform.IEEE signal Processing Letters,1996,3(1):10-12.
[32]Mallat S G.Wavelets for a vision.Proceedings of the IEEE,1996,84(4):604-614.
[33]刘贵忠,张志明,冯牧.由小波变换的模极大值快速重构信.自然科学进展,2000,10(7):660-664.
[34]张维强,宋国乡.基于一种新的阈值函数的小波域信号去噪.西安电子科技大学学报,2004,31(2):296-299.
[35]朱高中,王艳红.一种改进的小波阈值法在信号消噪中的研究.继电器,2007,35(18):41-45.
[36]何焰兰,苏勇,高永媚.一种自适应小波去噪算法.电子学报,2000,28(10):138-140.
[37]张磊,潘泉,张洪才,et al.小波域滤波阈值参数c的选取.电子学报,2001,29(3).