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1、Thisisanexpandedversionofthe`QuickStudy'inPhysicsTodaymagazine,October,2007.WaveletTransforms
2、AQuickStudyIvanW.SelesnickPolytechnicUniversityBrooklyn,NYSeptember27,2007Thewavelettransformhasbecomeausefulcomputationaltoolforavarietyofsignalandimageprocessingapplications.Forexam
3、ple,thewavelettransformisusefulforthecompressionofdigitalimage les;smaller lesareimportantforstoringimagesusinglessmemoryandfortransmittingimagesfasterandmorereliably.TheFBIuseswavelettransformsforcompressingdigitallyscanned ngerprintimages.NASA'sMarsRoversusedwavelettransform
4、sforcompressingimagesacquiredbytheir18cameras.Thewavelet-basedalgorithmimplementedinsoftwareonboardtheMarsRoversisdesignedtomeetthespecialrequirementsofdeep-spacecommunication.Inaddition,JPEG2K(thenewerJPEGimage leformat)isbasedonwavelettransforms.Wavelettransformsarealsousefu
5、lfor`cleaning'signalsandimages(reducingunwantednoiseandblurring).Somealgorithmsforprocessingastronomicalimages,forexample,arebasedonwaveletandwavelet-liketransforms.ThisQuickStudydescribesthewavelettransform,illustrateswhyitise ectivefornoisere-duction,andbrie
ydescribessevera
6、limprovementsofthebasicwavelettransformandbasicnoisereductionmethodusedintheillustration.Wedescribewhatthewavelettransformis,andwedescribealgorithmsforprocessingasignalafteritswavelettransformhasbeencomputed.Firstweshouldpointoutthattherearetwobasictypesofwavelettransform.Onet
7、ypeofwavelettransformisdesignedtobeeasilyreversible(invertible);thatmeanstheoriginalsignalcanbeeasilyrecoveredafterithasbeentransformed.Thiskindofwavelettransformisusedforimagecompressionandcleaning(noiseandblurreduction).Typically,thewavelettransformoftheimageis rstcom-puted,
8、thewaveletrepresentationisthenmodi edappropriately,andthenthewavelettransformisreversed(inverted)toobtainanewimage.Thesecondtypeofwavelettransformisdesignedforsignalanalysis;forexample,todetectfaultsinmachineryfromsensormeasurements,tostudyEEGorotherbiomedicalsignals,todetermi
9、nehowthefrequencycontentofasignalevolvesovertime.Inthesecases
