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1、ImageCompressionUsingtheDiscreteCosineTransformAbstractThediscretecosinetransform(DCT)isatechniqueforconvertingasignalintoelementaryfrequencycomponents.Itiswidelyusedinimagecompression.HerewedevelopsomesimplefunctionstocomputetheDCTandtocompressimages.Thesefunctionsillustratethepo
2、werofMathematicaintheprototypingofimageprocessingalgorithms.Therapidgrowthofdigitalimagingapplications,includingdesktoppublishing,multimedia,teleconferencing,andhigh-definitiontelevision(HDTV)hasincreasedtheneedforeffectiveandstandardizedimagecompressiontechniques.Amongtheemerging
3、standardsareJPEG,forcompressionofstillimages[Wallace1991];MPEG,forcompressionofmotionvideo[Puri1992];andCCITTH.261(alsoknownasPx64),forcompressionofvideotelephonyandteleconferencing.Allthreeofthesestandardsemployabasictechniqueknownasthediscretecosinetransform(DCT).DevelopedbyAhme
4、d,Natarajan,andRao[1974],theDCTisacloserelativeofthediscreteFouriertransform(DFT).ItsapplicationtoimagecompressionwaspioneeredbyChenandPratt[1984].Inthisarticle,IwilldevelopsomesimplefunctionstocomputetheDCTandshowhowitisusedforimagecompression.Wehaveusedthesefunctionsinourlaborat
5、orytoexploremethodsofoptimizingimagecompressionforthehumanviewer,usinginformationaboutthehumanvisualsystem[Watson1993].ThegoalofthispaperistoillustratetheuseofMathematicainimageprocessingandtoprovidethereaderwiththebasictoolsforfurtherexplorationofthissubject.TheOne-DimensionalDis
6、creteCosineTransformThediscretecosinetransformofalistofnrealnumberss(x),x=0,...,n-1,isthelistoflengthngivenby:s(u)=C(u)u=0,…nwhereforu=0=1otherwiseEachelementofthetransformedlistS(u)istheinner(dot)productoftheinputlists(x)andbasisvector.Theconstantfactorsarechosensothatthebasisvec
7、torsareorthogonalandnormalized.Theeightbasisvectorsforn=8areshowninFigure1.TheDCTcanbewrittheproductofavector(theinputlist)andthenxnorthogonalmatrixwhoserowsarethevectors.Thismatrix,forn=8,canbecomputedasfollows:DCTMatrix=Table[Ifk==0,Sqrt[1/8],Sqrt[2/8]Cos[Pi(2j+1)k/16]],{k,0,7},
8、{j,0,7}]//N;Wecancheckthatthematr
