基于小波变换的图像去噪方法研究
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摘要
图像是人类传递信息的主要媒介。然而,图像在生成和传输的过程中会受到各种噪声的干扰,对信息的处理、传输和存储造成极大的影响。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。小波分析是局部化时频分析,它用时域和频域联合表示信号的特征,是分析非平稳信号的有力工具。它通过伸缩、平移等运算功能对信号进行多尺度细化分析,能有效地从信号中提取信息。随着小波变换理论的完善,小波在图像去噪中得到了广泛的应用,与传统的去噪方法相比小波分析有着很大的优势,它能在去噪的同时保留图像细节,得到原图像的最佳恢复。
本文对基于小波变换的图像去噪方法进行了深入的研究分析,首先详细介绍了几种经典的小波变换去噪方法。对于小波变换模极大值去噪法,详细介绍了其去噪原理和算法,分析了去噪过程中参数的选取问题,并给出了一些选取依据;详细介绍了小波系数相关性去噪方法的原理和算法;对小波变换阈值去噪方法的原理和几个关键问题进行了详细讨论。最后对这些方法进行了分析比较,讨论了它们各自的优缺点和适用条件,并给出了仿真实验结果。
在众多基于小波变换的图像去噪方法中,运用最多的是小波阈值萎缩去噪法。传统的硬阈值函数和软阈值函数去噪方法在实际中得到了广泛的应用,而且取得了较好的效果。但是硬阈值函数的不连续性导致重构信号容易出现伪吉布斯现象;而软阈值函数虽然整体连续性好,但估计值与实际值之间总存在恒定的偏差,具有一定的局限性。鉴于此,本文提出了一种基于小波多分辨率分析和最小均方误差准则的自适应阈值去噪算法。该方法利用小波阈值去噪基本原理,在基于最小均方误差算法LMS和Stein无偏估计的前提下,引出了一个具有多阶连续导数的阈值函数,利用其对阈值进行迭代运算,得到最优阈值,从而得到更好的图像去噪效果。最后,通过仿真实验结果可以看到,该方法去噪效果显著,与硬阈值、软阈值方法相比,信噪比提高较多,同时去噪后仍能较好地保留图像细节,是一种有效的图像去噪方法。
Image is an important information source for human beings. However, in the course of its acquisition and transmission, noise is often introduced, which makes great influence to the processing, delivering and saving of information. Therefore, hunting for a method of denoising effectively and keeping the edge information simultaneously is a goal people have been pursuing all the time. Wavelet analysis is local analysis in the time domain and frequency domain, which represents the signal property using combination of the time domain and frequency domain. It is a useful tool to analyze the unstationary signal that implements multi-scale analysis to the signal by the translation and dilation of the mother wavelet, so it can effectively extract information from signal. Recently, with the improvement of wavelet theory, wavelet analysis has applied to image denoising successfully. Compared with traditional methods, wavelet has incomparable advantage in image denoising. It can not only wipe off noise but also retain the image details.
Based on the profound analysis on wavelet image denoising, several classical wavelet denoising methods are introduced in detail. The principles and algorithm of wavelet transform modulus maxima denoising method are introduced in detail and an analysis of the choice of some parameters in the process of denoising is made in detail. The principles and the algorithm of the relativity of the wavelet coefficient denoising method are introduced. Some key problems on denoising method based on wavelet threshold are discussed in detail. The advantages and disadvantages of these methods and their applicable condition are discussed at last and the simulation experiments show the results of image denoising.
In many kinds of wavelet denoising methods, wavelet threshold shrinkage is widely used. In practice, the hard-thresholding and soft-thresholding algorithm is frequently used to denoising and has obtained a good effect. Discontinuity of the hard-thresholding function results in Pseudo-Gibbs phenomenon of the reconstructed signal. Soft-thresholding function has good continuity but a constant deviation of the estimated value from the actual value confines its application. To improve the performance of wavelet threshold denoising algorithm, a new adaptive threshold denoising method based on wavelet multi-resolution analysis and LMS algorithm is proposed. Based on the wavelet threshold denoising theory, and also based on the least mean square error algorithm and Stein's unbiased risk estimate optimizing algorithm, the method chooses a new threshold function with multi-order continuous derivatives, which makes it possible to construct an optimal threshold through the method of iteration operation. At last, the results of the simulation experiments show that the new method usually obtains better performance and improves image's PSNR than classical threshold image denoising methods. The new method can also preserve more image details, so it is an effective method on image denoising
本文对基于小波变换的图像去噪方法进行了深入的研究分析,首先详细介绍了几种经典的小波变换去噪方法。对于小波变换模极大值去噪法,详细介绍了其去噪原理和算法,分析了去噪过程中参数的选取问题,并给出了一些选取依据;详细介绍了小波系数相关性去噪方法的原理和算法;对小波变换阈值去噪方法的原理和几个关键问题进行了详细讨论。最后对这些方法进行了分析比较,讨论了它们各自的优缺点和适用条件,并给出了仿真实验结果。
在众多基于小波变换的图像去噪方法中,运用最多的是小波阈值萎缩去噪法。传统的硬阈值函数和软阈值函数去噪方法在实际中得到了广泛的应用,而且取得了较好的效果。但是硬阈值函数的不连续性导致重构信号容易出现伪吉布斯现象;而软阈值函数虽然整体连续性好,但估计值与实际值之间总存在恒定的偏差,具有一定的局限性。鉴于此,本文提出了一种基于小波多分辨率分析和最小均方误差准则的自适应阈值去噪算法。该方法利用小波阈值去噪基本原理,在基于最小均方误差算法LMS和Stein无偏估计的前提下,引出了一个具有多阶连续导数的阈值函数,利用其对阈值进行迭代运算,得到最优阈值,从而得到更好的图像去噪效果。最后,通过仿真实验结果可以看到,该方法去噪效果显著,与硬阈值、软阈值方法相比,信噪比提高较多,同时去噪后仍能较好地保留图像细节,是一种有效的图像去噪方法。
Image is an important information source for human beings. However, in the course of its acquisition and transmission, noise is often introduced, which makes great influence to the processing, delivering and saving of information. Therefore, hunting for a method of denoising effectively and keeping the edge information simultaneously is a goal people have been pursuing all the time. Wavelet analysis is local analysis in the time domain and frequency domain, which represents the signal property using combination of the time domain and frequency domain. It is a useful tool to analyze the unstationary signal that implements multi-scale analysis to the signal by the translation and dilation of the mother wavelet, so it can effectively extract information from signal. Recently, with the improvement of wavelet theory, wavelet analysis has applied to image denoising successfully. Compared with traditional methods, wavelet has incomparable advantage in image denoising. It can not only wipe off noise but also retain the image details.
Based on the profound analysis on wavelet image denoising, several classical wavelet denoising methods are introduced in detail. The principles and algorithm of wavelet transform modulus maxima denoising method are introduced in detail and an analysis of the choice of some parameters in the process of denoising is made in detail. The principles and the algorithm of the relativity of the wavelet coefficient denoising method are introduced. Some key problems on denoising method based on wavelet threshold are discussed in detail. The advantages and disadvantages of these methods and their applicable condition are discussed at last and the simulation experiments show the results of image denoising.
In many kinds of wavelet denoising methods, wavelet threshold shrinkage is widely used. In practice, the hard-thresholding and soft-thresholding algorithm is frequently used to denoising and has obtained a good effect. Discontinuity of the hard-thresholding function results in Pseudo-Gibbs phenomenon of the reconstructed signal. Soft-thresholding function has good continuity but a constant deviation of the estimated value from the actual value confines its application. To improve the performance of wavelet threshold denoising algorithm, a new adaptive threshold denoising method based on wavelet multi-resolution analysis and LMS algorithm is proposed. Based on the wavelet threshold denoising theory, and also based on the least mean square error algorithm and Stein's unbiased risk estimate optimizing algorithm, the method chooses a new threshold function with multi-order continuous derivatives, which makes it possible to construct an optimal threshold through the method of iteration operation. At last, the results of the simulation experiments show that the new method usually obtains better performance and improves image's PSNR than classical threshold image denoising methods. The new method can also preserve more image details, so it is an effective method on image denoising
引文
[1]阮秋琦.数字图像处理学.第二版.北京.电子工业出版社.2007.1-2
[2]Donoho D L.Ideal spatial adaptation via wavelet shrinkage.Biometrika.1994.81.425-455
[3]A Gersho,B Ramaurthi.Image coding using vector quantization in Proc.IEEE Int.Conf.Acost.speech sinai processing.1982.5.430-432
[4]M V Wickerhauser.Adapted wavelet analysis from theory to software.Wellesley.MA.A K Peters.1995
[5]A Mertins.Signal analysis wavelet,filter banks,time-frequency transforms and applications.NJ.John Wiley and Sons.1999
[6]秦前清.实用小波分析.西安.西安电子科技大学出版社.1994
[7]S Mallat.A wavelet tour of signal processing.San Diego.Academic Press.1998
[8]A Grossman and J Morlet.Decomposition of hardy functions into square integrable wavelets of constant shape.SIAM J of Math.Anal.1984.15(4).723-736
[9]迈耶.小波算子.第一卷.北京.世界图书出版公司.1992
[10]迈耶.小波算子.第二卷.北京.世界图书出版公司.1994
[11]Meyer Y.Wavelets and operators.Cambridge.UK.Combridge Univerity Press.1992
[12]S Mallat.Multifrequency channel decomposition of images and wavelet models.IEEE Trans.Acoust.Speech.Signal and process.1989(37)2091-2110
[13]I Daubechies.Orthonormal bases of compactly supported wavelets.Comm.on Pure and Appl.Math.1988.41.909-996
[14]S Mallat.Multiresolution approximations and wavelet orthonormal bases of L~2(R).Trans.Amer.Math.Soc.1989.315.70-88
[15]S Mallat.Zero-crossings of a wavelet transform.IEEE Trans.IT.1991(37).1019-1033
[16]R Coifman.Wavelet analysis and signal processing.Wavelets and their applications,Bost-on.Jones an Bartlett.1992.153-178
[17]R Coifman,et al.Signal processing and compression with wave packets.In processing's of the conference on wavelets.Marseilles.1989
[18]C K Chui,J Z Wang.On compacting supported spline wavelets and a duality principle.Tans.Amer.Math.Soc.1992.330(2).903-915
[19]A Cohen,et al.Biorthogonal bases of compactly supported wavelets.Comm.on Pure and Appl.Math.1992.45.485-560
[20]I Daubechies.Orthonormal bases of compactly supported wavelets Ⅱ:Variations on a theme.SIAM.J.Math.Anal.1993124.499-519
[21]Goodman,et al.Construction of orthogonal wavelets using fractal interpolation function.SIAM.J.Math.Anal.1996.27.1158-1192
[22]W Swelden.Ten lift scheme:A construction of second generation wavelets.SIAM.J.Math.Anal.1998.29(2).511-546
[23]S Mallat.A theory for multiresolution aignal decomposition:the wavelet trpresentation.IEEE Trans.PAMI.1989.11.673-693
[24]S Mallat,Hwang W L.Singularity detection and processing with wavelets.IEEE Trans.on IT.1992(38).617-643
[25]S Mallat,et al.Characterization of signals from multiscale edges.IEEE Trans.PAMI.1992.14(7).710-732
[26]Y Xu,et al.Wavelet transform domain filters,A spatially selective noise filtration teeh-nique.IEEE Trans.on IP.1994(3).217-237
[27]Q Pan,et al.Two denoising methods by wavelet transform.IEEE Trans.on SP.1999(47).3401-3406
[28]Donoho D L.De-noising by soft-thresholding.IEEE Trans.on IT.1995.41(3).612-626.
[29]R Coifman and Donoho D L.Translation-invariant de-noising.Wavelets and Statistics.1994.125-151
[30]Gao H.Wavelet shrinkage denoising using the non-negative Garrote.Journal of Computational and GraPhical Statistics.1998.7(4).469-490
[31]Johnstonel M and Silverman B.Wavelet threshold estimators for data with Correlated noise.J.R.Statist.Soc.1997.59(23).345-349
[32]Jansen,Malfait M and Bultheel A.Generalized cross validation for wavelet thresholding.Signal Processing.1997.56.33-44
[33]Nowak R D.Optimal signal estimation using cross validation.IEEE Signal Processing Letters.1997.4(1).23-25
[34]Hsung T C,Lun D P-K and Siu W C.Denoising by singularity detection.IEEE Trans.on SP.1999.47(11).3139-3144
[35]Chang S G,Yu B and Veterli M.Spatially adaptive wavelet thresholding with context modeling for image denoising.IEEE Trans.on IP.2000.9(9).1522-1531
[36]Ingrid Daubechies著.李建平,杨万年 译.小波十讲.北京.国防工业出版社.2004.5
[37]Lei Zhang,Paul Bao.Denoising by spatial correlation thresholding.IEEE Trans.on Circuits and System for Video Technology.2003.13.535-538
[38]程正兴,张玲玲.多小波分析与应用.工程数学学报.2001.18(1).99-107
[39]杨福生.小波变换工程分析与应用.北京.科学出版社.2003
[40]孙延奎.小波分析及其应用.北京.机械工业出版社.2005
[41]杨宗凯.小波去噪及其在信号检测中的应用.华中理工大学学报.1997.25(2).1-4
[42]程正兴.小波分析算法与应用.西安.西安交通大学出版社.1998
[43]I Daubechies.The wavelet transform,time-frequency localization and signal analysis.IEEE Trans.on IT.1990.36(5).960-1005
[44]I Daubechies.Ten lectures on wavelets.Philadelphia.Society for Industrial and Applied Mathematics.1992
[45]Burrus C S.Introduction to wavelets and wavelet transforms.北京.机械工业出版社.2005
[46]刘贵忠.小波分析及其应用.西安.西安电子科技大学出版社.1992
[47]Werich N and Warhola G T.Wavelet shrinkage and generalized cross validation for image denoising.IEEE Trans.on IP.1998.7(1).82-90
[48]Tu Dan and Shen Jianjun.The design of wavelet domain wiener filter and its application in image denoising.Systems Engineering and Electronics.2001.23.4-7
[49]Shubhankar R,Bani K.A Bayesian transformation model for wavelet shrinkage.IEEE Trans.on IP.2003.12(12).1512-1520
[50]Bruce A and Gao H.Waveshrink:Shrinkage function and thresholds.SPIE.1996.2569.270-281
[51]Gao H and Bruce A.Waveshrink with firm shrinkage.Statistic Sinica1997.7.855-875
[52]Donoho D L,Johnstone I.Adapting to unknown smoothness via wavelet shrinkage.Journal of Amercan Stat.Assoc.1995.90.1201-1225
[53]胡昌华等.基于Matlab的系统分析与设计—小波分析.西安.西安电子科技大学出版社.1999
[54]Femandes,et al.Multidimensional mapping-based complex wavelet-transform.IEEE Trans.on SP.2002.50(3).487-500
[55]J Tian,et al.Vanishing moments and wavelet approximation.Tech.Rep.CMLTR95-01.Compact.Math.Lab.Riee Univ.Houston.1995
[56]黄继武等.DCT域图像水印:嵌入对策和算法.电子学报.2006.28(4):57-60
[57]Glentis G O,Berberidis K,Theodoridis S.Efficient least squares adaptive algorithms for FIR transversal filtering.IEEE Trans.on SP.1999.16(4).13-41
[58]Stein C.Estimation of the mean of a multivariate normal distribution.The Annals of Statistics.1981.9(6).1135-1151
[59]杨兴明等.基于小波多分辨率分析和新的阈值自适应的信号去噪.合肥工业大学学报.2007.30(12).1580-1583
[60]Donoho D L.Denoising by soft-thresholding.IEEE Trans on IT.1995.3.613-627
[2]Donoho D L.Ideal spatial adaptation via wavelet shrinkage.Biometrika.1994.81.425-455
[3]A Gersho,B Ramaurthi.Image coding using vector quantization in Proc.IEEE Int.Conf.Acost.speech sinai processing.1982.5.430-432
[4]M V Wickerhauser.Adapted wavelet analysis from theory to software.Wellesley.MA.A K Peters.1995
[5]A Mertins.Signal analysis wavelet,filter banks,time-frequency transforms and applications.NJ.John Wiley and Sons.1999
[6]秦前清.实用小波分析.西安.西安电子科技大学出版社.1994
[7]S Mallat.A wavelet tour of signal processing.San Diego.Academic Press.1998
[8]A Grossman and J Morlet.Decomposition of hardy functions into square integrable wavelets of constant shape.SIAM J of Math.Anal.1984.15(4).723-736
[9]迈耶.小波算子.第一卷.北京.世界图书出版公司.1992
[10]迈耶.小波算子.第二卷.北京.世界图书出版公司.1994
[11]Meyer Y.Wavelets and operators.Cambridge.UK.Combridge Univerity Press.1992
[12]S Mallat.Multifrequency channel decomposition of images and wavelet models.IEEE Trans.Acoust.Speech.Signal and process.1989(37)2091-2110
[13]I Daubechies.Orthonormal bases of compactly supported wavelets.Comm.on Pure and Appl.Math.1988.41.909-996
[14]S Mallat.Multiresolution approximations and wavelet orthonormal bases of L~2(R).Trans.Amer.Math.Soc.1989.315.70-88
[15]S Mallat.Zero-crossings of a wavelet transform.IEEE Trans.IT.1991(37).1019-1033
[16]R Coifman.Wavelet analysis and signal processing.Wavelets and their applications,Bost-on.Jones an Bartlett.1992.153-178
[17]R Coifman,et al.Signal processing and compression with wave packets.In processing's of the conference on wavelets.Marseilles.1989
[18]C K Chui,J Z Wang.On compacting supported spline wavelets and a duality principle.Tans.Amer.Math.Soc.1992.330(2).903-915
[19]A Cohen,et al.Biorthogonal bases of compactly supported wavelets.Comm.on Pure and Appl.Math.1992.45.485-560
[20]I Daubechies.Orthonormal bases of compactly supported wavelets Ⅱ:Variations on a theme.SIAM.J.Math.Anal.1993124.499-519
[21]Goodman,et al.Construction of orthogonal wavelets using fractal interpolation function.SIAM.J.Math.Anal.1996.27.1158-1192
[22]W Swelden.Ten lift scheme:A construction of second generation wavelets.SIAM.J.Math.Anal.1998.29(2).511-546
[23]S Mallat.A theory for multiresolution aignal decomposition:the wavelet trpresentation.IEEE Trans.PAMI.1989.11.673-693
[24]S Mallat,Hwang W L.Singularity detection and processing with wavelets.IEEE Trans.on IT.1992(38).617-643
[25]S Mallat,et al.Characterization of signals from multiscale edges.IEEE Trans.PAMI.1992.14(7).710-732
[26]Y Xu,et al.Wavelet transform domain filters,A spatially selective noise filtration teeh-nique.IEEE Trans.on IP.1994(3).217-237
[27]Q Pan,et al.Two denoising methods by wavelet transform.IEEE Trans.on SP.1999(47).3401-3406
[28]Donoho D L.De-noising by soft-thresholding.IEEE Trans.on IT.1995.41(3).612-626.
[29]R Coifman and Donoho D L.Translation-invariant de-noising.Wavelets and Statistics.1994.125-151
[30]Gao H.Wavelet shrinkage denoising using the non-negative Garrote.Journal of Computational and GraPhical Statistics.1998.7(4).469-490
[31]Johnstonel M and Silverman B.Wavelet threshold estimators for data with Correlated noise.J.R.Statist.Soc.1997.59(23).345-349
[32]Jansen,Malfait M and Bultheel A.Generalized cross validation for wavelet thresholding.Signal Processing.1997.56.33-44
[33]Nowak R D.Optimal signal estimation using cross validation.IEEE Signal Processing Letters.1997.4(1).23-25
[34]Hsung T C,Lun D P-K and Siu W C.Denoising by singularity detection.IEEE Trans.on SP.1999.47(11).3139-3144
[35]Chang S G,Yu B and Veterli M.Spatially adaptive wavelet thresholding with context modeling for image denoising.IEEE Trans.on IP.2000.9(9).1522-1531
[36]Ingrid Daubechies著.李建平,杨万年 译.小波十讲.北京.国防工业出版社.2004.5
[37]Lei Zhang,Paul Bao.Denoising by spatial correlation thresholding.IEEE Trans.on Circuits and System for Video Technology.2003.13.535-538
[38]程正兴,张玲玲.多小波分析与应用.工程数学学报.2001.18(1).99-107
[39]杨福生.小波变换工程分析与应用.北京.科学出版社.2003
[40]孙延奎.小波分析及其应用.北京.机械工业出版社.2005
[41]杨宗凯.小波去噪及其在信号检测中的应用.华中理工大学学报.1997.25(2).1-4
[42]程正兴.小波分析算法与应用.西安.西安交通大学出版社.1998
[43]I Daubechies.The wavelet transform,time-frequency localization and signal analysis.IEEE Trans.on IT.1990.36(5).960-1005
[44]I Daubechies.Ten lectures on wavelets.Philadelphia.Society for Industrial and Applied Mathematics.1992
[45]Burrus C S.Introduction to wavelets and wavelet transforms.北京.机械工业出版社.2005
[46]刘贵忠.小波分析及其应用.西安.西安电子科技大学出版社.1992
[47]Werich N and Warhola G T.Wavelet shrinkage and generalized cross validation for image denoising.IEEE Trans.on IP.1998.7(1).82-90
[48]Tu Dan and Shen Jianjun.The design of wavelet domain wiener filter and its application in image denoising.Systems Engineering and Electronics.2001.23.4-7
[49]Shubhankar R,Bani K.A Bayesian transformation model for wavelet shrinkage.IEEE Trans.on IP.2003.12(12).1512-1520
[50]Bruce A and Gao H.Waveshrink:Shrinkage function and thresholds.SPIE.1996.2569.270-281
[51]Gao H and Bruce A.Waveshrink with firm shrinkage.Statistic Sinica1997.7.855-875
[52]Donoho D L,Johnstone I.Adapting to unknown smoothness via wavelet shrinkage.Journal of Amercan Stat.Assoc.1995.90.1201-1225
[53]胡昌华等.基于Matlab的系统分析与设计—小波分析.西安.西安电子科技大学出版社.1999
[54]Femandes,et al.Multidimensional mapping-based complex wavelet-transform.IEEE Trans.on SP.2002.50(3).487-500
[55]J Tian,et al.Vanishing moments and wavelet approximation.Tech.Rep.CMLTR95-01.Compact.Math.Lab.Riee Univ.Houston.1995
[56]黄继武等.DCT域图像水印:嵌入对策和算法.电子学报.2006.28(4):57-60
[57]Glentis G O,Berberidis K,Theodoridis S.Efficient least squares adaptive algorithms for FIR transversal filtering.IEEE Trans.on SP.1999.16(4).13-41
[58]Stein C.Estimation of the mean of a multivariate normal distribution.The Annals of Statistics.1981.9(6).1135-1151
[59]杨兴明等.基于小波多分辨率分析和新的阈值自适应的信号去噪.合肥工业大学学报.2007.30(12).1580-1583
[60]Donoho D L.Denoising by soft-thresholding.IEEE Trans on IT.1995.3.613-627