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,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,WAVELTTRANSFORMANDITSAPPLICATIONS,小波,变,变换,与,与工,程,程应,用,用,WAVELTTRANSFORMANDITSAPPLICATIONS,李艳,讲,讲,师,师,自动,化,化科,学,学与,工,工程,学,学院,TheDiscreteWaveletTransform(1),In wavelet analysis,weoften speakof approximations and details.,Theapproximations are the high-scale,low-frequencycomponents.,Thedetailsarethelow-scale,high-frequency components.,TheDiscreteWaveletTransform(2),These signals AandD are interesting,but we get 2000 values instead ofthe1000 wehad.Bylookingcarefully at the computation,we may keep onlyonepoint out oftwoineachofthetwo2000-lengthsamplesto get the completeinformation.This isthenotionof down-sampling.Weproducetwosequences called cAandcD.,TheDiscreteWaveletTransform(3),Exampleof the DiscreteWaveletTransform,MultipleLevel decomposition,Thedecomposition process can beiterated,withsuccessive approximations beingdecomposed in turn,so thatonesignal is brokendown into manylower resolutioncomponents.This iscalledthewavelet,Since the analysis process is iterative,intheory it can becontinued indefinitely.Inreality,thedecomposition can proceed onlyuntil the individualdetailsconsistofa singlesampleor pixel.Inpractice,youll selecta suitablenumber of levelsbased on the natureofthesignal,or on asuitablecriterion suchas entropy,WaveletReconstruction,Theprocessof Assembling thesecomponents backintotheoriginal signalwithoutloss ofinformationiscalled reconstruction,or synthesis.Themathematical manipulation thateffectssynthesis iscalledtheinversediscretewavelettransform(IDWT),Reconstruction Filters,Thedownsamplingofthesignal components performedduring the decompositionphase introduces adistortion called aliasing.It turnsoutthat bycarefully choosingfiltersforthedecomposition and reconstructionphasesthatarecloselyrelated(but not identical),wecancancelout the effects ofaliasing.,Reconstruction of ApproximationsandDetails(1),Reconstructouroriginalsignalfromthecoefficients oftheapproximationsanddetails.,Reconstructtheapproximations and details themselves from theircoefficientvectors.,Reconstructthefirst-levelapproximation A1from the coefficient vectorcA1,Reconstruct thefirst-level detailD1 from the coefficient vectorcD1,S=A1+D1,Reconstructionof Approximations and Details(2),the coefficientvectors cA1 and cD1,-half the length of theoriginalsignal,-cannotdirectly be combinedto reproduce the signal.,It isnecessaryto reconstructtheapproximationsand details before combining them.,Extendingthistechniqueto the components ofa multilevel analysis,we find that similarrelationships hold for all thereconstructedsignal constituents.Thatis,thereareseveral ways toreassemble theoriginalsignal:,Relationship ofFilters to Waveletshapes,The wavelet functionisdeterminedby the high-pass filter,which also producesthedetails ofthewavelet decomposition.,Thereis an additional function associated with some,but not all,wavelets.This isthe so-called scaling function,.,The scaling functionis very similar tothe wavelet function.Itis determined by thelow-passquadraturemirror filters,andthusis associatedwiththe approximations of thewavelet decomposition.,In the same waythatiteratively upsampling and convolving the high-pass filterproduces ashape approximatingthewavelet function,iterativelyupsamplingandconvolvingthelow-pass filterproducesa shape approximating thescaling function.,Multi-stepDecompositionand Reconstrction,Thisprocess involves twoaspects:,breaking up a signalto obtainthewavelet coefficients,reassembling the signal from the coefficients.,Wavelet PackageAanlysis,In waveletpacket analysis,the detailsas well astheapproximationscan be split.This yieldsmorethandifferent ways toencode thesignal.This is thewavelet packetdecompositiontree.,For instance,wavelet packet analysis allowsthe signalS tobe representedas A1+AAD3+DAD3+DD2.,小波包函数,除,除了尺度和,平,平移两个参,数,数外，增加,了,了一个频率,参,参数，,克服了小波,时,时间分辨率,高,高时频率分,辨,辨率低的缺,陷,陷。,Introduceof WaveletFunction（1）,Introduceof WaveletFunction（2）,根据不同的,标,标准，小波,函,函数具有不,同,同的类型,小波函数和,尺,尺度函数及,其,其傅立叶变,换,换的支撑长,度,度。即当时,间,间或频率趋,向,向无穷大时,，,，函数从一,个,个有限值收,敛,敛到0的速,度,度；,对称性。在,图,图像处理中,用,用于避免移,相,相；,消失矩阶数,。,。有利于数,据,据压缩；,正则性。有,利,利于信号或,图,图像的重构,获,获得较好的,平,平滑效果。,在MATLAB命令行,输,输入：waveinfo(),命,命令可以查,看,看函数简要,说,说明,例如：waveinfo(db,),在MATLAB命令行,输,输入：wavemenu，打开小,波,波工具箱GUI,可以查看详,细,细帮助,参考文献：,故障信号检,测,测的小波基,选,选择方法.PDF,小波函数的,性,性质及其应,用,用研究.PDF,Applications,一维小波分,析,析用于信号,奇,奇异性检测,一维小波分,析,析用于用于,信,信号消噪处,理,理,一维小波分,析,析用于识别,含,含噪信号的,有,有用信号发,展,展趋势,二维小波分,析,析用于图像,压,压缩,二维小波分,析,析用于图像,消,消噪,二维小波分,析,析用于图像,增,增强,二维小波分,析,析用于图像,融,融合,利用小波包,进,进行特征提,取,取,利用小波包,进,进行信号消,噪,噪处理,利用小波包,进,进行图像压,缩,缩,一维小波分,析,析用于信号,奇,奇异性检测(1),信号中的奇,异,异点及不规,则,则突变部分,经,经常带有比,较,较重要的信,息,息，例如在,故,故障诊断中,，,，故障通常,表,表现为输出,信,信号发生突,变,变。,在这些奇异,信,信号中,信,号,号的奇异程,度,度是不同的,根据研究,的,的需,要,常将其,分,分为剧变奇,异,异信号和缓,变,变奇异信号,。,。剧变奇异,信,信号是指信,号,号本身具有,突,突变,缓变,奇,奇异信号则,指,指信号本身,是,是连续的,但,但其某阶导,数,数具有间断,或,或奇变。,对信号进行,多,多尺度分析,，,，在信号出,现,现突变时，,小,小波变换后,的,的系数具有,模,模值极大值,，,，,可以通过对,极,极大值点的,检,检测确定故,障,障发生的时,间,间。,小波的选择,，,，需要注意,具,具有良好的,正,正则性。,例程：test_1_01.mtest_1_02.m,一维小波分,析,析用于信号,奇,奇异性检测(2),Test_1_01.m 第一类,间,间断点,一维小波分,析,析用于
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