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阳明大学放射医学科学研究所(2)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.,2001OutlinesWhyWaveletHaarWaveletsTheHaarScalingFunctionBasicPropertiesoftheHaarScalingFunctionTheHaarWaveletHaarDecompos...

阳明大学放射医学科学研究所(2)
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.,2001OutlinesWhyWaveletHaarWaveletsTheHaarScalingFunctionBasicPropertiesoftheHaarScalingFunctionTheHaarWaveletHaarDecompositionandReconstructionAlgorithmsDecompositionReconstructionFiltersandDiagramsSummary4.1WhyWaveletWaveletswerefirstappliedingeophysicstoanalyzedatafromseismicsurveys.SeismicsurveygeophonesseismictraceSesimictraceDirectwave(alongthesurface)Subsequentwaves(rocklayersbelowground)FourierTransform(FT)isnotagoodtool–givesnodirectinformationaboutwhenanoscillationoccurred.Short-timeFT:equaltimeinterval,high-frequencyburstsoccurarehardtodetect.Waveletscankeeptrackoftimeandfrequencyinformation.Theycanbeusedto“zoomin”ontheshortbursts,orto“zoomout”todetectlong,slowoscillationsfrequencyfrequency+time(equaltimeintervals)frequency+time4.2HaarWavelets4.2.1TheHaarScalingFunctionWaveletfunctionsScalingfunctionΦ(fatherwavelet)WaveletΨ(motherwavelet)ThesetwofunctionsgenerateafamilyoffunctionsthatcanbeusedtobreakuporreconstructasignalTheHaarScalingFunctionTranslationDilationUsingHaarblockstoapproximateasignalHigh-frequencynoiseshowsupastall,thinblocks.Needsanalgorithmthateliminatesthenoiseandnotdistributetherestofthesignal.DisadvantagesofHarrwavelet:discontinuousanddoesnotapproximatecontinuoussignalsverywell.Figure2Daubechies8Dubieties3Daubechies4Chap64.2.2BasicPropertiesoftheHaarScalingFunctionTheHaarScalingfunctionisdefinedasΦ(x-k):samegraphbuttranslatedbytotheright(ifk>0)bykunitsLetV0bethespaceofallfunctionsoftheformV0consistsofallpiecewiseconstantfunctionswhosediscontinuitiesarecontainedinthesetofintegersV0hascompactsupport.TypicalelementinV0Figure5Figure6hasdiscontinuitiesatx=0,1,3,and4LetV1bethespaceofpiecewiseconstantfunctionsoffinitesupportwithdiscontinuitiesatthehalfintegershasdiscontinuitiesatx=0,1/2,3/2,and2Supposejisanynonnegativeinteger.Thespaceofstepfunctionsatlevelj,denotedbyVj,,isdefinedtobethespacespannedbythesetovertherealnumbers.Vjisthespaceofpiecewiseconstantfunctionsoffinitesupportwhosediscontinuitiesarecontainedinthesetmeansnoinformationislostastheresolutiongetsfiner.Vjcontainsallrelevantinformationuptoaresolutionscaleorder2-jAfunctionf(x)belongstoV0ifff(2jx)belongstoVjAfunctionf(x)belongstoVjifff(2-jx)belongstoV0HowtodecomposeasignalintoitsVj-componentsWhenjislarge,thegraphofΦ(2jx)issimilartooneofthespikesofasignalthatwemaywishtofilterout.OnewayistoconstructanorthonormalbasisforVjusingtheL2innerproductTheorem:4.2.4TheHaarWaveletWewanttoisolatethespikesthatbelongtoVj,butthatarenotmembersofVj-1ThewayistodecomposeVjasanorthonormalsumofVj-1anditscomplement.StartwithV1,assumetheorthonormalcomplementofVoisgeneratedbytranslatesofsomefunctionsΨ,weneed:HarrwaveletTheorem4.8(extendtoVj)DecomposingVjTheorem:4.3HaarDecompositionandReconstructionAlgorithmsImplementationStep1:ApproximatetheoriginalsignalfbyastepfunctionoftheformExample4.11GeneraldecompositionschemeWj-1-componentVj-1-componentTheorem4.12(HaarDecomposition)Example4.13V8-componentV7-componentV6-componentV4-componentW7-component4.3.2ReconstructionGeneralreconstructionschemeGeneralreconstructionschemeTheorem4.14(HaarReconstruction)Example4.1580%compression90%compressionsamplesignal4.3.3FiltersandDiagramsk=-1,0k=-1,0DecompositionalgorithmdownsamplingoperatorReconstructionk=0,1k=0,1upsamplingoperatorSummary6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#pXlUiQfNcK8H5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E2B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUiRfOcG5D2A-x*t$qZnVkShPdMaI7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3B0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPd6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*pXmUjRfOcK9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!q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