S变换轮廓术中消除条纹非线性影响的方法
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摘要
S变换是短时傅里叶变换和小波变换的延伸和推广,是一种无损可逆的非平稳信号时频分析方法。它不仅具有线性、多分辨性和逆变换唯一性等特点,而且其反变换与傅里叶变换保持着直接的联系。在S变换中,以简谐波作为基波,以可以同时进行伸缩和平移的高斯函数作为窗函数。同短时傅里叶变换相比,S变换的时频分辨率可以同时达到最佳,同小波变换相比,S变换谱同傅里叶变换谱保持着直接联系。讨论了S变换在受到非线性因素影响下的变形条纹解相中的应用,推导了相应的条纹图的S变换公式;分别研究了S变换"脊"分析方法和S变换滤波分析方法在非线性因素影响下的变形条纹解相中的应用,完成了相应的计算机模拟和实验验证,并将S变换三维重建效果与傅里叶变换和小波变换的结果进行对比,由此表明了S变换具有更好的恢复效果。
S-Transform,a hybrid and extension of the short-time(or windowed) Fourier transform and the wavelet transform,is one of lossless and reversible time-frequency analysis methods,which is suitable to analyze non-stationary signals.It not only has advantages of linearity,multi-resolution and uniqueness of inverse,but also its inverse transform directly keeps in contact with the Fourier transform.In S-Transform,the harmonic wave is used as a basic element function,and the window function is a Gaussian function with the ability of both dilation and translation,which is controlled by a frequency parameter.Compared with the short-time(or windowed) Fourier transform,it has optimized the time-resolution and frequency resolution simultraneously.Compared with wavelet transform,it keeps in contact with the Fourier transform.S transform in the application of demodulation of fringe pattern with nonlinear parts has been deeply discussed,S-transform expression of the deformed fringe pattern considering by nonlinear effects is deduced and two ways,including S transform filtering method and S transform ridge method are proposed,which are used to eliminate the nonlinear error in three-dimensional optical measurement based on the structured light projection.Computer simulations and experiments have verified the proposed two methods.Compared with Fourier transform and wavelet transform,the proposed methods based on S transform have better reconstruction results.
S-Transform,a hybrid and extension of the short-time(or windowed) Fourier transform and the wavelet transform,is one of lossless and reversible time-frequency analysis methods,which is suitable to analyze non-stationary signals.It not only has advantages of linearity,multi-resolution and uniqueness of inverse,but also its inverse transform directly keeps in contact with the Fourier transform.In S-Transform,the harmonic wave is used as a basic element function,and the window function is a Gaussian function with the ability of both dilation and translation,which is controlled by a frequency parameter.Compared with the short-time(or windowed) Fourier transform,it has optimized the time-resolution and frequency resolution simultraneously.Compared with wavelet transform,it keeps in contact with the Fourier transform.S transform in the application of demodulation of fringe pattern with nonlinear parts has been deeply discussed,S-transform expression of the deformed fringe pattern considering by nonlinear effects is deduced and two ways,including S transform filtering method and S transform ridge method are proposed,which are used to eliminate the nonlinear error in three-dimensional optical measurement based on the structured light projection.Computer simulations and experiments have verified the proposed two methods.Compared with Fourier transform and wavelet transform,the proposed methods based on S transform have better reconstruction results.
引文
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11Liu Zhenming,Gu Hanming,Zhan Fenglin.Application of S-transform time-frequency analysis in seismic prospection for karstcaves under tunnel[J].Chinese Journal of EngineeringGeophysicals,2010,7(1):50～54刘振明,顾汉明,詹凤林.S变换时频分析在隧道岩溶地震勘探解释中的应用[J].工程地球物理学报,2010,7(1):50～54
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16Serhat zder,zlem Kocahan,Emre Coskun et al..Opticalphase distribution evaluation by using an S-transform[J].Opt.Lett.,2007,32(6):591～593
17Jiang Mohua,Chen Wenjing,Zheng Zhiping.Research of phasedemodulation technique based on S-transform[J].Acta OpticaSinica,2011,31(4):0412001蒋模华,陈文静,郑志平.基于S变换的解相技术研究[J].光学学报,2011,31(4):0412001
18M.Takeda,K.Mutoh.Fourier transform profilometry for theautomatic measurement of 3-D object shapes[J].Appl.Opt.,1983,22:3977～3982
19Su Ke,Chen Wenjing.Analyzing wavelet transform profilometryin the restraining CCD nonlinear characteristic[J].OpticalTechnique,2009,35(1):37～40苏轲,陈文静.小波变换轮廓术抑制CCD非线性的分析[J].光学技术,2009,35(1):37～40
20Zhao Huandong,Zhu Bijin,Li Zhineng.Effect on the Fouriertransform profilometry due to the CCD nonlinearity[J].OpticalInstruments,2000,22(5):7～13赵焕东,祝必进,李志能.CCD的非线性效应对改进型傅里叶变换轮廓术的影响[J].光学仪器,2000,22(5):7～13
21Ervin Sejdic,Igor Djurovic,Jin Jiang.A window width optimizedS-transform[J].EURASIP Journal on Advances in SignalProcessing,2008.1～13
2Sai Siva Gorthi,Pramod Rastogi.Fringe projection techniques:whither we are?[J].Opt.&Lasers in Engng.,2010,48(2):133～140
3Xianyu Su,Wenjing Chen.Fourier transform profilometry:a review[J].Opt.&Lasers in Engng.,2001,35(5):263～284
4Qian Kemao,Haixia Wang,Wenjing Gao.Windowed Fouriertransform for fringe pattern analysis:theoretical analyses[J].Appl.Opt.,2008,47(29):5408～5419
5Sun Juan,Chen Wenjing,Su Xianyu et al..Study themeasurement range of wavelet transform profilometry[J].ActaOptica Sinica,2007,27(4):647～653孙娟,陈文静,苏显渝等.小波变换轮廓术的测量范围研究[J].光学学报,2007,27(4):647～653
6J.Zhong,J.Weng.Phase retrieval of optical fringe patternsfrom the ridge of a wavelet transform[J].Opt.Lett.,2005,30(19):2560～2562
7Lei Huang,Qian Kemao,Bing Panet al..Comparison of Fouriertransform,windowed Fourier transform,and wavelet transformmethods for phase extraction from a single fringe pattern in fringeprojection profilometry[J].Opt.&Lasers in Engng.,2010,48(2):141～148
8Rafael C.Gonzalez,Richard E.Woods.Digital Image Processing(Second Edtion)[M].Ruan Qiuqi,Ruan Yuzhi Transl.Beijing:Publishing House of Electranics Industry,2007.63～66拉斐尔.C.冈萨雷斯,理德.E.伍兹.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,2007.63～66
9R.G.Stockwell,L.Mansinha,R.P.Lowe.Localization of thecomplex spectrum:the S-transform[J].IEEE Transactions onSignal Processing,1996,44(4):998～1001
10Wang Limei,Sun Fengrong,Zhang Mingqiang et al..Application of S transform in myocardial contrast echocardiographimage de-noising[J].Computer Engineering,2009,35(13):216～218王丽梅,孙丰荣,张明强等.S变换在心肌声学造影图像降噪中的应用[J].计算机工程,2009,35(13):216～218
11Liu Zhenming,Gu Hanming,Zhan Fenglin.Application of S-transform time-frequency analysis in seismic prospection for karstcaves under tunnel[J].Chinese Journal of EngineeringGeophysicals,2010,7(1):50～54刘振明,顾汉明,詹凤林.S变换时频分析在隧道岩溶地震勘探解释中的应用[J].工程地球物理学报,2010,7(1):50～54
12Ma Jianqing,Li Qingchun,Fan Jinsheng et al..Rayleigh wavedispersion analysis based on S transform[J].Journal of EarthSciences and Environment,2010,32(3):319～323马见青,李庆春,樊金生等.基于S变换的瑞利面波频散分析[J].地球科学与环境学报,2010,32(3):319～323
13Wang Tiewang,Yu Xianhai,Liu Mingqianet al..The attenuation ofpower interference based on S-transform[J].Geophysical&Geochemical Exploration,2010,34(4):532～535汪铁望,于相海,刘明乾等.基于S变换的工频干扰压制[J].物探与化探,2010,34(4):532～535
14Liu Haipeng,Li Yanjun,Zhang Ke.Time-frequency filtering ofradar signal based on generalized S transform[J].AeronauticalComputing Technique,2010,40(5):23～25刘海鹏,李言俊,张科.基于广义S变换进行雷达信号时频滤波去噪[J].航空计算技术,2010,40(5):23～25
15Mansinha L.,Stoekwell R.G.,Lowe R.P..Patten analysis withtwo dimensional spectral localization:application of two dimensional Stransform[J].Physica A,1997,239(1-3):286～295
16Serhat zder,zlem Kocahan,Emre Coskun et al..Opticalphase distribution evaluation by using an S-transform[J].Opt.Lett.,2007,32(6):591～593
17Jiang Mohua,Chen Wenjing,Zheng Zhiping.Research of phasedemodulation technique based on S-transform[J].Acta OpticaSinica,2011,31(4):0412001蒋模华,陈文静,郑志平.基于S变换的解相技术研究[J].光学学报,2011,31(4):0412001
18M.Takeda,K.Mutoh.Fourier transform profilometry for theautomatic measurement of 3-D object shapes[J].Appl.Opt.,1983,22:3977～3982
19Su Ke,Chen Wenjing.Analyzing wavelet transform profilometryin the restraining CCD nonlinear characteristic[J].OpticalTechnique,2009,35(1):37～40苏轲,陈文静.小波变换轮廓术抑制CCD非线性的分析[J].光学技术,2009,35(1):37～40
20Zhao Huandong,Zhu Bijin,Li Zhineng.Effect on the Fouriertransform profilometry due to the CCD nonlinearity[J].OpticalInstruments,2000,22(5):7～13赵焕东,祝必进,李志能.CCD的非线性效应对改进型傅里叶变换轮廓术的影响[J].光学仪器,2000,22(5):7～13
21Ervin Sejdic,Igor Djurovic,Jin Jiang.A window width optimizedS-transform[J].EURASIP Journal on Advances in SignalProcessing,2008.1～13
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