阳明大学放射医学科学研究所

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1、HaarWaveletAnalysis吳育德陽明大學放射醫學科學研究所台北榮總整合性腦功能實驗室AFirstCourseinWaveletswithFourierAnalysisAlbertBoggessFrancisJ.NarcowichPrentice-Hall,Inc.,2001OutlinesWhyWaveletHaarWaveletsTheHaarScalingFunctionBasicPropertiesoftheHaarScalingFunctionTheHaarWaveletHaarDecompositionandReconstructionAlg

2、orithmsDecompositionReconstructionFiltersandDiagramsSummary4.1WhyWaveletWaveletswerefirstappliedingeophysicstoanalyzedatafromseismicsurveys.SeismicsurveygeophonesseismictraceSesimictraceDirectwave(alongthesurface)Subsequentwaves(rocklayersbelowground)FourierTransform(FT)isnotagoodtool

3、–givesnodirectinformationaboutwhenanoscillationoccurred.Short-timeFT:equaltimeinterval,high-frequencyburstsoccurarehardtodetect.Waveletscankeeptrackoftimeandfrequencyinformation.Theycanbeusedto“zoomin”ontheshortbursts,orto“zoomout”todetectlong,slowoscillationsfrequencyfrequency+time(e

4、qualtimeintervals)frequency+time4.2HaarWavelets4.2.1TheHaarScalingFunctionWaveletfunctionsScalingfunctionΦ(fatherwavelet)WaveletΨ(motherwavelet)ThesetwofunctionsgenerateafamilyoffunctionsthatcanbeusedtobreakuporreconstructasignalTheHaarScalingFunctionTranslationDilationUsingHaarblock

5、stoapproximateasignalHigh-frequencynoiseshowsupastall,thinblocks.Needsanalgorithmthateliminatesthenoiseandnotdistributetherestofthesignal.DisadvantagesofHarrwavelet:discontinuousanddoesnotapproximatecontinuoussignalsverywell.Figure2Daubechies8Dubieties3Daubechies4Chap64.2.2BasicProper

6、tiesoftheHaarScalingFunctionTheHaarScalingfunctionisdefinedasΦ(x-k):samegraphbuttranslatedbytotheright(ifk>0)bykunitsLetV0bethespaceofallfunctionsoftheformV0consistsofallpiecewiseconstantfunctionswhosediscontinuitiesarecontainedinthesetofintegersV0hascompactsupport.TypicalelementinV0F

7、igure5Figure6hasdiscontinuitiesatx=0,1,3,and4LetV1bethespaceofpiecewiseconstantfunctionsoffinitesupportwithdiscontinuitiesatthehalfintegershasdiscontinuitiesatx=0,1/2,3/2,and2Supposejisanynonnegativeinteger.Thespaceofstepfunctionsatlevelj,denotedbyVj,,isdefinedtobethespacespannedbythe

8、setov