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DigitalImageProcessingWaveletandMultiresolutionProcessing
MultiresolutionAnalysisManysignalsorimagescontainfeaturesatvariouslevelsofdetail(i.e.,scales). Smallsizeobjectsshouldbeexaminedatahigh
resolution.Largesizeobjectsshouldbeexaminedatalow
resolution.MultiresolutionAnalysis(cont’d)Localimagestatisticsarequitedifferentfromglobalimagestatistics.Modelingentireimageisdifficultorimpossible.Needtoanalyzeimagesatmultiplelevelsofdetail.Transform:AmathematicaloperationthattakesafunctionorsequenceandmapsitintoanotheroneTransformsaregoodthingsbecause…Thetransformofafunctionmaygiveadditional/hiddeninformationabouttheoriginalfunction,whichmaynotbeavailable/obviousotherwiseThetransformofanequationmaybeeasiertosolvethantheoriginalequationThetransformofafunction/sequencemayrequirelessstorage,henceprovidedatacompression/reductionAnoperationmaybeeasiertoapplyonthetransformedfunction,ratherthantheoriginalfunction(recallconvolution)Introduction(RobiPolikar,RowanUniversity)
WhatisaTransformandWhydoWeNeedOne?Mostusefultransformsare:Linear:wecanpulloutconstants,andapplysuperpositionOne-to-one:differentfunctionshavedifferenttransformsInvertible:foreachtransformT,thereisaninversetransformT-1usingwhichtheoriginalfunctionfcanberecovered(kindof–sortoftheundobutton…)Continuoustransform:mapfunctionstofunctionsDiscretetransform:mapsequencestosequencesTfFT-1fIntroduction
PropertiesofTransformsComplexfunctionrepresentationthroughsimplebuildingblocksCompressedrepresentationthroughusingonlyafewblocks(calledbasisfunctions/kernels)Sinusoidsasbuildingblocks:FouriertransformFrequencydomainrepresentationofthefunctionIntroduction
WhatDoesaTransformLookLike?FourierseriesContinuousFouriertransformLaplacetransformDiscreteFouriertransformZ-transformIntroduction
WhatTransformsareAvailable?JeanB.JosephFourier(1768-1830)“Anarbitraryfunction,continuousorwithdiscontinuities,definedinafiniteintervalbyanarbitrarilycapriciousgraphcanalwaysbeexpressedasasumofsinusoids” J.B.J.FourierDecember,21,1807Introduction
FourierWho…?RecallthatFTusescomplexexponentials(sinusoids)asbuildingblocks.Foreachfrequencyofcomplexexponential,thesinusoidatthatfrequencyiscomparedtothesignal.Ifthesignalconsistsofthatfrequency,thecorrelationishighlargeFTcoefficients.Ifthesignaldoesnothaveanyspectralcomponentatafrequency,thecorrelationatthatfrequencyislow/zero,small/zeroFTcoefficient.Introduction
HowDoesFTWorkAnyway?Introduction
FTatWorkFFFIntroduction
FTatWorkFIntroduction
FTatWorkComplexexponentials(sinusoids)asbasisfunctions:FAnultrasonicA-scanusing1.5MHztransducer,sampledat10MHzIntroduction
FTatWorkFTidentifiesallspectralcomponentspresentinthesignal,howeveritdoesnotprovideanyinformationregardingthetemporal(time)localizationofthesecomponents.Why?StationarysignalsconsistofspectralcomponentsthatdonotchangeintimeallspectralcomponentsexistatalltimesnoneedtoknowanytimeinformationFTworkswellforstationarysignalsHowever,non-stationarysignalsconsistsoftimevaryingspectralcomponentsHowdowefindoutwhichspectralcomponentappearswhen?FTonlyprovideswhatspectralcomponentsexist
,notwhereintimetheyarelocated.NeedsomeotherwaystodeterminetimelocalizationofspectralcomponentsIntroduction
StationaryandNon-stationarySignalsStationarysignals’spectralcharacteristicsdonotchangewithtimeNon-stationarysignalshavetimevaryingspectraConcatenationIntroduction
StationaryandNon-stationarySignals5Hz25Hz50HzPerfectknowledgeofwhatfrequenciesexist,butnoinformationaboutwherethesefrequenciesarelocatedintimeIntroduction
Non-stationarySignalsComplexexponentialsstretchouttoinfinityintimeTheyanalyzethesignalglobally,notlocallyHence,FTcanonlytellwhatfrequenciesexistintheentiresignal,butcannottell,atwhattimeinstancesthesefrequenciesoccurInordertoobtaintimelocalization
ofthespectralcomponents,thesignalneedtobeanalyzedlocally,BUTHOW?Introduction
FTShortcomingsChooseawindowfunctionoffinitelengthPutthewindowontopofthesignalatt=0TruncatethesignalusingthiswindowComputetheFTofthetruncatedsignal,save.SlidethewindowtotherightbyasmallamountGotostep3,untilwindowreachestheendofthesignalForeachtimelocationwherethewindowiscentered,weobtainadifferentFTHence,eachFTprovidesthespectralinformationofaseparatetime-sliceofthesignal,providingsimultaneoustimeandfrequencyinformationIntroduction
ShortTimeFourierTransform(STFT)Introduction
ShortTimeFourierTransform(STFT)STFTofsignalx(t):Computedforeachwindowcenteredatt=t’TimeparameterFrequencyparameterSignaltobeanalyzedWindowingfunctionWindowingfunctioncenteredatt=t’FTKernel(basisfunction)Introduction
ShortTimeFourierTransform(STFT)0100200300-1.5-1-0.500.510100200300-1.5-1-0.500.510100200300-1.5-1-0.500.510100200300-1.5-1-0.500.51WindowedsinusoidallowsFTtobecomputedonlythroughthesupportofthewindowingfunctionIntroduction
STFTatWorkIntroduction
STFT300Hz200Hz100Hz50HzSTFTprovidesthetimeinformationbycomputingadifferentFTsforconsecutivetimeintervals,andthenputtingthemtogetherTime-FrequencyRepresentation(TFR)Maps1-Dtimedomainsignalsto2-Dtime-frequencysignalsConsecutivetimeintervalsofthesignalareobtainedbytruncatingthesignalusingaslidingwindowingfunctionHowtochoosethewindowingfunction?Whatshape?Rectangular,Gaussian,Elliptic…?Howwide?Introduction
STFTTwoextremecases:W(t)infinitelylong:
STFTturnsintoFT,providingexcellentfrequencyinformation(goodfrequencyresolution),butnotimeinformationW(t)infinitelyshort:
STFTthengivesthetimesignalback,withaphasefactor.Excellenttimeinformation(goodtimeresolution),butnofrequencyinformationIntroduction
SelectionofSTFTWindowWideanalysiswindowpoortimeresolution,goodfrequencyresolutionNarrowanalysiswindowgoodtimeresolution,poorfrequencyresolutionOncethewindowischosen,theresolutionissetforbothtimeandfrequency.Timeresolution:HowwelltwospikesintimecanbeseparatedfromeachotherinthetransformdomainFrequencyresolution:HowwelltwospectralcomponentscanbeseparatedfromeachotherinthetransformdomainBothtimeandfrequencyresolutionscannotbearbitrarilyhigh!!!
Wecannotpreciselyknowatwhattimeinstanceafrequencycomponentislocated.WecanonlyknowwhatintervaloffrequenciesarepresentinwhichtimeintervalsIntroduction
HeisenbergUncertaintyPrincipleIntroduction
STFTGaussianwindowfunction:a=0.01a=0.0001a=0.00001OvercomesthepresetresolutionproblemoftheSTFTbyusingavariablelengthwindowAnalysiswindowsofdifferentlengthsareusedfordifferentfrequencies:AnalysisofhighfrequenciesUsenarrowerwindowsforbettertimeresolutionAnalysisoflowfrequenciesUsewiderwindowsforbetterfrequencyresolutionThisworkswell,ifthesignaltobeanalyzedmainlyconsistsofslowlyvaryingcharacteristicswithoccasionalshorthighfrequencybursts.Heisenbergprinciplestillholds!!!Thefunctionusedtowindowthesignaliscalledthewavelet
Introduction
WaveletTransformContinuouswavelettransformofthesignalx(t)usingtheanalysiswavelet(.)Translationparameter,measureoftimeScaleparameter,measureoffrequencyThemotherwavelet.Allkernelsareobtainedbytranslating(shifting)and/orscalingthemotherwaveletAnormalizationconstantSignaltobeanalyzedScale=1/frequencyIntroduction
WaveletTransformHighfrequency(smallscale)Lowfrequency(largescale)Introduction
WTatWorkIntroduction
WTatWorkIntroduction
WTatWorkIntroduction
WTatWorkIntroduction
TimeandFrequencyResolutionBackgroundImagePyramidsComputeareduced-resolutionapproximationoftheinputimageFiltering(Averaging,Gaussian)Down-samplingUp-sampletheoutputofthepreviousbyafactor2Computethedifferencebetweenthepredictionofstep2andtheinputto
Step1.ImagePyramidsImagePyramidsInMulti-resolutionAnalysis(MRA),aScalingFunctionisusedtocreateaseriesofapproximationsofafunctionorimage,eachdifferingbyafactor2fromitsnearestneighboringapproximations.Additionalfunctions,calledWavelet,areusedtoencodethedifferenceininformationbetweenadjacentapproximationMulti-ResolutionExpansionMulti-ResolutionExpansion
SeriesExpansionReal-valuedexpansioncoefficientsReal-valuedexpansionfunctionsIftheexpansionisUNIQUE-thatis,thereisonlyonesetofforanygiven-thearecalledbasisfunctions,andtheexpansionset,,iscalledaBASISfortheclassoffunctionsthatcanbesoexpressed.TheexpressiblefunctionsformafunctionspacethatisreferredtoastheclosespanoftheexpansionsetMulti-ResolutionExpansion
SeriesExpansionDualFunctionsMulti-ResolutionExpansion
SeriesExpansionCASE1:ExpansionfunctionsformanorthonormalbasisCASE2:Expansionfunctionsarenotorthonormal,butareanorthogonalbasis(biorthogonalbasis)CASE3:ExpansionsetisnotabasisMulti-ResolutionExpansion
ScalingFunctionsMulti-ResolutionExpansion
ScalingFunctionsThescalingfunctionsisORTHOGONALtoitsintegertranslations.Thesubspacespannedbythescalingfunctionatlowscalesarenestedwithinthosespannedathigherscales.TheonlyfunctionthatiscommontoallVj
is
f(x)=0AnyfunctioncanberepresentedwitharbitraryprecisionMulti-ResolutionExpansion
MRARequirementsMulti-ResolutionExpansion
MRARequirementsScalingVectorMulti-ResolutionExpansion
WaveletFunctionsUnionofSpacesMulti-ResolutionExpansion
Wavele
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