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1、ImageCompressionUsingtheDiscreteCosineTransformAbstractThediscretecosinetransform(DCT)isatechniqueforconvertingasignalintoelementaryfrequencycomponents.Itiswidelyusedinimagecompression.HerewedevelopsomesimplefunctionstocomputetheDCTandtocompressimages.ThesefunctionsillustratethepowerofMathema
2、ticaintheprototypingofimageprocessingalgorithms.Therapidgrowthofdigitalimagingapplications,includingdesktoppublishing,multimedia,teleconferencing,andhigh-definitiontelevision(HDTV)hasincreasedtheneedforeffectiveandstandardizedimagecompressiontechniques.AmongtheemergingstandardsareJPEG,forcomp
3、ressionofstillimages[Wallace1991];MPEG,forcompressionofmotionvideo[Puri1992];andCCITTH.261(alsoknownasPx64),forcompressionofvideotelephonyandteleconferencing.Allthreeofthesestandardsemployabasictechniqueknownasthediscretecosinetransform(DCT).DevelopedbyAhmed,Natarajan,andRao[1974],theDCTisacl
4、oserelativeofthediscreteFouriertransform(DFT).ItsapplicationtoimagecompressionwaspioneeredbyChenandPratt[1984].Inthisarticle,IwilldevelopsomesimplefunctionstocomputetheDCTandshowhowitisusedforimagecompression.Wehaveusedthesefunctionsinourlaboratorytoexploremethodsofoptimizingimagecompressionf
5、orthehumanviewer,usinginformationaboutthehumanvisualsystem[Watson1993].ThegoalofthispaperistoillustratetheuseofMathematicainimageprocessingandtoprovidethereaderwiththebasictoolsforfurtherexplorationofthissubject.TheOne-DimensionalDiscreteCosineTransformThediscretecosinetransformofalistofnreal
6、numberss(x),x=0,...,n-1,isthelistoflengthngivenby:s(u)=C(u)u=0,…nwhereforu=0=1otherwiseEachelementofthetransformedlistS(u)istheinner(dot)productoftheinputlists(x)andbasisvector.Theconstantfactorsarechosensothatthebasisvectorsareorthogonalandnormalized.Theeightbasisvectorsforn=8areshowninFigur
7、e1.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},{j,0,7}]//N;Wecancheckthatthematr
