小波分析及其工程应用讲义

WAVELTTRANSFORMANDITSAPPLICATIONS小波变换与工程运用WAVELTTRANSFORMANDITSAPPLICATIONS李艳讲师自动化科学与工程学院TheDiscreteWaveletTransform(1)Inwaveletanalysis,weoftenspeakofapproximationsanddetails.Theapproximationsarethehigh-scale,low-frequencycomponents.Thedetailsarethelow-scale,high-frequencycomponents.TheDiscreteWaveletTransform(2)ThesesignalsAandDareinteresting,butweget2000valuesinsteadofthe1000wehad.Bylookingcarefullyatthecomputation,wemaykeeponlyonepointoutoftwoineachofthetwo2000-lengthsamplestogetthecompleteinformation.Thisisthenotionofdown-sampling.WeproducetwosequencescalledcAandcD.TheDiscreteWaveletTransform(3)ExampleoftheDiscreteWaveletTransformMultipleLeveldecompositionThedecompositionprocesscanbeiterated,withsuccessiveapproximationsbeingdecomposedinturn,sothatonesignalisbrokendownintomanylowerresolutioncomponents.ThisiscalledthewaveletSincetheanalysisprocessisiterative,intheoryitcanbecontinuedindefinitely.Inreality,thedecompositioncanproceedonlyuntiltheindividualdetailsconsistofasinglesampleorpixel.Inpractice,you'llselectasuitablenumberoflevelsbasedonthenatureofthesignal,oronasuitablecriterionsuchasentropyWaveletReconstructionTheprocessofAssemblingthesecomponentsbackintotheoriginalsignalwithoutlossofinformationiscalledreconstruction,orsynthesis.Themathematicalmanipulationthateffectssynthesisiscalledtheinversediscretewavelettransform(IDWT)ReconstructionFiltersThedownsamplingofthesignalcomponentsperformedduringthedecompositionphaseintroducesadistortioncalledaliasing.Itturnsoutthatbycarefullychoosingfiltersforthedecompositionandreconstructionphasesthatarecloselyrelated(butnotidentical),wecan"cancelout"theeffectsofaliasing.ReconstructionofApproximationsandDetails(1)Reconstructouroriginalsignalfromthecoefficientsoftheapproximationsanddetails.Reconstructtheapproximationsanddetailsthemselvesfromtheircoefficientvectors.Reconstructthefirst-levelapproximationA1fromthecoefficientvectorcA1Reconstructthefirst-leveldetailD1fromthecoefficientvectorcD1S=A1+D1ReconstructionofApproximationsandDetails(2)thecoefficientvectorscA1andcD1--halfthelengthoftheoriginalsignal--cannotdirectlybecombinedtoreproducethesignal.Itisnecessarytoreconstructtheapproximationsanddetailsbeforecombiningthem.Extendingthistechniquetothecomponentsofamultilevelanalysis,wefindthatsimilarrelationshipsholdforallthereconstructedsignalconstituents.Thatis,thereareseveralwaystoreassembletheoriginalsignal:RelationshipofFilterstoWaveletshapesThewaveletfunctionisdeterminedbythehigh-passfilter,whichalsoproducesthedetailsofthewaveletdecomposition.Thereisanadditionalfunctionassociatedwithsome,butnotall,wavelets.Thisistheso-calledscalingfunction,.Thescalingfunctionisverysimilartothewaveletfunction.Itisdeterminedbythelow-passquadraturemirrorfilters,andthusisassociatedwiththeapproximationsofthewaveletdecomposition.Inthesamewaythatiterativelyupsamplingandconvolvingthehigh-passfilterproducesashapeapproximatingthewaveletfunction,iterativelyupsamplingandconvolvingthelow-passfilterproducesashapeapproximatingthescalingfunction.Multi-stepDecompositionandReconstrctionThisprocessinvolvestwoaspects:breakingupasignaltoobtainthewaveletcoefficients,reassemblingthesignalfromthecoefficients.WaveletPackageAanlysisInwaveletpacketanalysis,thedetailsaswellastheapproximationscanbesplit.Thisyieldsmorethandifferentwaystoencodethesignal.Thisisthewaveletpacketdecompositiontree.Forinstance,waveletpacketanalysisallowsthesignalStoberepresentedasA1+AAD3+DAD3+DD2.小波包函数除了尺度和平移两个参数外，添加了一个频率参数，抑制了小波时间分辨率高时频率分辨率低的缺陷。IntroduceofWaveletFunction〔1〕IntroduceofWaveletFunction〔2〕根据不同的规范，小波函数具有不同的类型小波函数和尺度函数及其傅立叶变换的支撑长度。即当时间或频率趋向无穷大时，函数从一个有限值收敛到0的速度；对称性。在图像处置中用于防止移相；消逝矩阶数。有利于数据紧缩；正那么性。有利于信号或图像的重构获得较好的平滑效果。在MATLAB命令行输入：waveinfo(‘’)命令可以查看函数简要阐明例如：waveinfo(‘db’)在MATLAB命令行输入：wavemenu，翻开小波工具箱GUI可以查看详细协助参考文献：缺点信号检测的小波基选择方法.PDF小波函数的性质及其运用研讨.PDFApplications一维小波分析用于信号奇特性检测一维小波分析用于用于信号消噪处置一维小波分析用于识别含噪信号的有用信号开展趋势二维小波分析用于图像紧缩二维小波分析用于图像消噪二维小波分析用于图像加强二维小波分析用于图像交融利用小波包进展特征提取利用小波包进展信号消噪处置利用小波包进展图像紧缩一维小波分析用于信号奇特性检测(1)信号中的奇特点及不规那么突变部分经常带有比较重要的信息，例如在缺点诊断中，缺点通常表现为输出信号发生突变。在这些奇特信号中,信号的奇特程度是不同的,根据研讨的需要,常将其分为剧变奇特信号和缓变奇特信号。剧变奇特信号是指信号本身具有突变,缓变奇特信号那么指信号本身是延续的,但其某阶导数具有延续或奇变。对信号进展多尺度分析，在信号出现突变时，小波变换后的系数具有模值极大值，可以经过对极大值点的检测确定缺点发生的时间。小波的选择，需求留意具有良好的正那么性。例程：test_1_01.m test_1_02.m一维小波分析用于信号奇特性检测(2)Test_1_01.m第一类延续点一维小波分析用于信号奇特性检测(3)Test_1_02.m一维小波分析用于用于信号消噪处置(1)一维小波分析用于用于信号消噪处置(2)对平稳信号消噪一维小波分析用于用于信号消噪处置(3)对非平稳信号消噪一维小波分析用于识别含噪信号的有用信号开展趋势〔1〕一维小波分析用于识别含噪信号的有用信号开展趋势〔2