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02_An Introduction to Wavelets

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Introduction to wavelets
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  AnIntroductiontoWavelets  AmaraGraps  ABSTRACT.Waveletsaremathematicalfunctionsthatcutupdataintodierentfrequencycom-ponents,andthenstudyeachcomponentwitharesolutionmatchedtoitsscale.Theyhavead-vantagesovertraditionalFouriermethodsinanalyzingphysicalsituationswherethesignalcontainsdiscontinuitiesandsharpspikes.Waveletsweredevelopedindependentlyintheeldsofmathemat-ics,quantumphysics,electricalengineering,andseismicgeology.Interchangesbetweentheseeldsduringthelasttenyearshaveledtomanynewwaveletapplicationssuchasimagecompression,turbulence,humanvision,radar,andearthquakeprediction.Thispaperintroduceswaveletstotheinterestedtechnicalpersonoutsideofthedigitalsignalprocessingeld.IdescribethehistoryofwaveletsbeginningwithFourier,comparewavelettransformswithFouriertransforms,stateprop-ertiesandotherspecialaspectsofwavelets,andnishwithsomeinterestingapplicationssuchasimagecompression,musicaltones,andde-noisingnoisydata.1.WAVELETSOVERVIEW Thefundamentalideabehindwaveletsistoanalyzeaccordingtoscale.Indeed,someresearchersin thewaveleteldfeelthat,byusingwavelets,oneisadoptingawholenewmindsetorperspectivein processingdata.Waveletsarefunctionsthatsatisfycertainmathematicalrequirementsandareusedinrepresent-ingdataorotherfunctions.Thisideaisnotnew.Approximationusingsuperpositionoffunctionshasexistedsincetheearly1800's,whenJosephFourierdiscoveredthathecouldsuperposesinesand cosinestorepresentotherfunctions.However,inwaveletanalysis,the scale thatweusetolookatdataplaysaspecialrole.Waveletalgorithmsprocessdataatdierent scales or resolutions. Ifwelookatasignalwithalarge\window, wewouldnoticegrossfeatures.Similarly,ifwelookata signalwithasmall\window, wewouldnoticesmallfeatures.Theresultinwaveletanalysisisto seeboththeforest and  thetrees,sotospeak.Thismakeswaveletsinterestinganduseful.Formanydecades,scientistshavewantedmoreappropriatefunctionsthanthesinesandcosineswhichcomprisethebasesofFourieranalysis,to approximatechoppysignals(1).Bytheirdenition,thesefunctionsarenon-local(andstretchouttoinnity).Theythereforedoaverypoorjobinapproximatingsharpspikes.Butwithwaveletanalysis,wecanuseapproximatingfunctionsthatarecontainedneatlyinnitedomains.Waveletsarewell-suitedforapproximatingdatawithsharpdiscontinuities.Thewaveletanalysisprocedureistoadoptawaveletprototypefunction,calledan  analyzing wavelet or motherwavelet. Temporalanalysisisperformedwithacontracted,high-frequencyversion oftheprototypewavelet,whilefrequencyanalysisisperformedwithadilated,low-frequencyversion ofthesamewavelet.Becausetheoriginalsignalorfunctioncanberepresentedintermsofawavelet1 c   1995InstituteofElectricalandElectronicsEngineers,Inc.Personaluseofthismaterialispermitted.TheoriginalversionofthisworkappearsinIEEEComputationalScienceandEngineering,Summer1995,vol.2,num.2,publishedbytheIEEEComputerSociety,10662LosVaquerosCircle,LosAlamitos,CA90720,USA,TEL+1-714-821-8380,FAX+1-714-821-4010.  2AmaraGraps expansion(usingcoecientsinalinearcombinationofthewaveletfunctions),dataoperationscan beperformedusingjustthecorrespondingwaveletcoecients.Andifyoufurtherchoosethebestwaveletsadaptedtoyourdata,ortruncatethecoecientsbelowathreshold,yourdataissparsely represented.Thissparsecodingmakeswaveletsanexcellenttoolintheeldofdatacompression.Otherappliedeldsthataremakinguseofwaveletsincludeastronomy,acoustics,nuclearengi-neering,sub-bandcoding,signalandimageprocessing,neurophysiology,music,magneticresonanceimaging,speechdiscrimination,optics,fractals,turbulence,earthquake-prediction,radar,human vision,andpuremathematicsapplicationssuchassolvingpartialdierentialequations.2.HISTORICALPERSPECTIVE Inthehistoryofmathematics,waveletanalysisshowsmanydierentorigins(2).Muchofthework wasperformedinthe1930s,and,atthetime,theseparateeortsdidnotappeartobepartsofa coherenttheory. 2.1.PRE-1930  Before1930,themainbranchofmathematicsleadingtowaveletsbeganwithJosephFourier(1807)withhistheoriesoffrequencyanalysis,nowoftenreferredtoasFouriersynthesis.Heassertedthatany2   -periodicfunction  f  ( x  )isthesum  a  0 +  1  X  k =1 ( a  k cos kx  +  b k sin  kx  )(1)ofitsFourierseries.Thecoecients a  0 , a  k and  b k arecalculatedby  a  0 = 12   2  Z  0 f  ( x  ) dxa  k = 1   2  Z  0 f  ( x  )cos( kx  ) dxb k = 1   2  Z  0 f  ( x  )sin( kx  ) dx  Fourier'sassertionplayedanessentialroleintheevolutionoftheideasmathematicianshad aboutthefunctions.Heopenedupthedoortoanewfunctionaluniverse.After1807,byexploringthemeaningoffunctions,Fourierseriesconvergence,andorthogonalsystems,mathematiciansgraduallywereledfromtheirpreviousnotionof frequencyanalysis tothenotionof scaleanalysis. Thatis,analyzing  f  ( x  )bycreatingmathematicalstructuresthatvary inscale.How?Constructafunction,shiftitbysomeamount,andchangeitsscale.Applythatstructureinapproximatingasignal.Nowrepeattheprocedure.Takethatbasicstructure,shiftit,andscaleitagain.Applyittothesamesignaltogetanewapproximation.Andsoon.Itturnsoutthatthissortofscaleanalysisislesssensitivetonoisebecauseitmeasurestheaverageuctuationsofthesignalatdierentscales.TherstmentionofwaveletsappearedinanappendixtothethesisofA.Haar(1909).OnepropertyoftheHaarwaveletisthatithas compactsupport, whichmeansthatitvanishesoutsideofaniteinterval.Unfortunately,Haarwaveletsarenotcontinuouslydierentiablewhichsomewhatlimitstheirapplications.  AnIntroductiontoWavelets3 2.2.THE1930S  Inthe1930s,severalgroupsworkingindependentlyresearchedtherepresentationoffunctionsusing  scale-varyingbasisfunctions. Understandingtheconceptsofbasisfunctionsandscale-varyingbasisfunctionsiskeytounderstandingwaveletsthesidebarbelowprovidesashortdetourlessonforthoseinterested.Byusingascale-varyingbasisfunctioncalledtheHaarbasisfunction(moreonthislater)PaulLevy,a1930sphysicist,investigatedBrownianmotion,atypeofrandomsignal(2).HefoundtheHaarbasisfunctionsuperiortotheFourierbasisfunctionsforstudyingsmallcomplicateddetailsin theBrownianmotion.Another1930sresearcheortbyLittlewood,Paley,andSteininvolvedcomputingtheenergyofafunction  f  ( x  ):energy= 12 2  Z  0 j f  ( x  ) j 2 dx  (2)Thecomputationproduceddierentresultsiftheenergywasconcentratedaroundafewpointsordistributedoveralargerinterval.Thisresultdisturbedthescientistsbecauseitindicatedthatenergymightnotbeconserved.Theresearchersdiscoveredafunctionthatcanvaryinscale and  canconserveenergywhencomputingthefunctionalenergy.TheirworkprovidedDavidMarrwith aneectivealgorithmfornumericalimageprocessingusingwaveletsintheearly1980s.|||||||||||-SIDEBAR. WhatareBasisFunctions?  Itissimplertoexplainabasisfunctionifwemoveoutoftherealmofanalog(functions)andinto therealmofdigital(vectors)(*).Everytwo-dimensionalvector( xy  )isacombinationofthevector(1  0)and(0  1) : Thesetwo vectorsarethebasisvectorsfor( xy  ) : Why?Noticethat x  multipliedby(1  0)isthevector( x 0)  and  y  multipliedby(0  1)isthevector(0 y  ) : Thesumis( xy  ) : Thebestbasisvectorshavethevaluableextrapropertythatthevectorsareperpendicular,ororthogonaltoeachother.Forthebasis(1  0)and(0  1)  thiscriteriaissatised.Nowlet'sgobacktotheanalogworld,andseehowtorelatetheseconceptstobasisfunctions.Insteadofthevector( xy  )  wehaveafunction  f  ( x  ) : Imaginethat f  ( x  )isamusicaltone,saythenote A  inaparticularoctave.Wecanconstruct A  byaddingsinesandcosinesusingcombinationsofamplitudesandfrequencies.Thesinesandcosinesarethebasisfunctionsinthisexample,andtheelementsofFouriersynthesis.Forthesinesandcosineschosen,wecansettheadditionalrequirementthattheybeorthogonal.How?Bychoosingtheappropriatecombinationofsineandcosinefunction termswhoseinnerproductadduptozero.Theparticularsetoffunctionsthatareorthogonal and  thatconstruct f  ( x  )areourorthogonalbasisfunctionsforthisproblem.  4AmaraGraps WhatareScale-varyingBasisFunctions?  Abasisfunctionvariesinscalebychoppingupthesamefunctionordataspaceusingdierentscalesizes.Forexample,imaginewehaveasignaloverthedomainfrom0to1.Wecandividethesignalwithtwostepfunctionsthatrangefrom0to1/2and1/2to1.Thenwecandividetheoriginalsignalagainusingfourstepfunctionsfrom0to1/4,1/4to1/2,1/2to3/4,and3/4to1.Andso on.Eachsetofrepresentationscodetheoriginalsignalwithaparticularresolutionorscale. Reference  (  )G.Strang,\Wavelets, AmericanScientist, Vol.82,1992,pp.250-255.|||||||||||- 2.3.1960-1980  Between1960and1980,themathematiciansGuidoWeissandRonaldR.Coifmanstudiedthesimplestelementsofafunctionspace,called  atoms, withthegoalofndingtheatomsforacommon functionandndingthe\assemblyrules thatallowthereconstructionofalltheelementsofthefunctionspaceusingtheseatoms.In1980,GrossmanandMorlet,aphysicistandanengineer,broadlydenedwaveletsinthecontextofquantumphysics.Thesetworesearchersprovidedaway ofthinkingforwaveletsbasedonphysicalintuition. 2.4.POST-1980  In1985,StephaneMallatgavewaveletsanadditionaljump-startthroughhisworkindigitalsignalprocessing.Hediscoveredsomerelationshipsbetweenquadraturemirrorlters,pyramidalgorithms,andorthonormalwaveletbases(moreontheselater).Inspiredinpartbytheseresults,Y.Meyerconstructedtherstnon-trivialwavelets.UnliketheHaarwavelets,theMeyerwaveletsarecontin-uouslydierentiablehowevertheydonothavecompactsupport.Acoupleofyearslater,Ingrid DaubechiesusedMallat'sworktoconstructasetofwaveletorthonormalbasisfunctionsthatareperhapsthemostelegant,andhavebecomethecornerstoneofwaveletapplicationstoday.3.FOURIERANALYSIS Fourier'srepresentationoffunctionsasasuperpositionofsinesandcosineshasbecomeubiquitousforboththeanalyticandnumericalsolutionofdierentialequationsandfortheanalysisandtreatmentofcommunicationsignals.Fourierandwaveletanalysishavesomeverystronglinks. 3.1.FOURIERTRANSFORMS  TheFouriertransform'sutilityliesinitsabilitytoanalyzeasignalinthetimedomainforitsfrequencycontent.Thetransformworksbyrsttranslatingafunctioninthetimedomainintoa functioninthefrequencydomain.ThesignalcanthenbeanalyzedforitsfrequencycontentbecausetheFouriercoecientsofthetransformedfunctionrepresentthecontributionofeachsineandcosinefunctionateachfrequency.AninverseFouriertransformdoesjustwhatyou'dexpect,transform datafromthefrequencydomainintothetimedomain.
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