外文翻译利用离散余弦变换进行图像压缩.docx

外文翻译利用离散余弦变换进行图像压缩.docx

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外文翻译利用离散余弦变换进行图像压缩

中文4500字

ImageCompressionUsingtheDiscreteCosineTransform

Abstract

Thediscretecosinetransform（DCT）isatechniqueforconvertingasignalintoelementaryfrequencycomponents.Itiswidelyusedinimagecompression.HerewedevelopsomesimplefunctionstocomputetheDCTandtocompressimages.

ThesefunctionsillustratethepowerofMathematicaintheprototypingofimageprocessingalgorithms.

Therapidgrowthofdigitalimagingapplications,includingdesktoppublishing,multimedia,teleconferencing,andhigh-definitiontelevision（HDTV）hasincreasedtheneedforeffectiveandstandardizedimagecompressiontechniques.AmongtheemergingstandardsareJPEG,forcompressionofstillimages[Wallace1991];MPEG,forcompressionofmotionvideo[Puri1992];andCCITTH.261（alsoknownasPx64）,forcompressionofvideotelephonyandteleconferencing.

Allthreeofthesestandardsemployabasictechniqueknownasthediscretecosinetransform（DCT）.DevelopedbyAhmed,Natarajan,andRao[1974],theDCTisacloserelativeofthediscreteFouriertransform（DFT）.ItsapplicationtoimagecompressionwaspioneeredbyChenandPratt[1984].Inthisarticle,IwilldevelopsomesimplefunctionstocomputetheDCTandshowhowitisusedforimagecompression.Wehaveusedthesefunctionsinourlaboratorytoexploremethodsofoptimizingimagecompressionforthehumanviewer,usinginformationaboutthehumanvisualsystem[Watson1993].ThegoalofthispaperistoillustratetheuseofMathematicainimageprocessingandtoprovidethereaderwiththebasictoolsforfurtherexplorationofthissubject.

TheOne-DimensionalDiscreteCosineTransform

Thediscretecosinetransformofalistofnrealnumberss（x）,x=0,...,n-1,isthelistoflengthngivenby:

s（u）=

C（u）

u=0,…n

where

foru=0

=1otherwise

EachelementofthetransformedlistS（u）istheinner（dot）productoftheinputlists（x）andbasisvector.Theconstantfactorsarechosensothatthebasisvectorsareorthogonalandnormalized.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},{j,0,7}]//N;

Wecancheckthatthematrixisorthogonal:

DCTMatrix.Transpose[DCTMatrix]//Chop//MatrixForm

1.0000000

01.000000

001.00000

0001.0000

00001.000

000001.00

0000001.0

00000001.

Eachbasisvectorcorrespondstoasinusoidofacertainfrequency:

Show[GraphicsArray[Partition[

ListPlot[#,PlotRange->{-.5,.5},PlotJoined->True,

DisplayFunction->Identity]&

/@DCTMatrix,2]]];

Figure1.Theeightbasisvectorsforthediscretecosinetransformoflengtheight.

Thelists（x）canberecoveredfromitstransformS（u）byapplyingtheinversecosinetransform

（IDCT）:

=

x=0,…,n

where

foru=0

=1otherwise

Thisequationexpressessasalinearcombinationofthebasisvectors.ThecoefficientsaretheelementsofthetransformS,whichmayberegardedasreflectingtheamountofeachfrequencypresentintheinputs.

Wegeneratealistofrandomnumberstoserveasatestinput:

input1=Table[Random[Real,{-1,1}],{8}]

{0.203056,0.980407,0.35312,-0.106651,0.0399382,0.871475,-0.648355,0.501067}

TheDCTiscomputedbymatrixmultiplication:

output1=DCTMatrix.input1

{0.775716,0.3727,0.185299,0.0121461,-0.325,-0.993021,0.559794,-0.625127}

Asnotedabove,theDCTiscloselyrelatedtothediscreteFouriertransform（DFT）.Infact,itispossibletocomputetheDCTviatheDFT（see[Jain1989,p.152]）:

Firstcreateanewlistbyextractingtheevenelements,followedbythereversedoddelements.ThenmultiplytheDFTofthisre-orderedlistbyso-called"twiddlefactors"andtaketherealpart.Wecancarryoutthisprocessforn=8usingMathematica'sDFTfunction.

DCTTwiddleFactors=N@Join[{1},

Table[Sqrt[2]Exp[-IPik/16],{k,7}]]

{1.,1.38704-0.275899I,1.30656-0.541196I,1.17588-0.785695I,1.-1.I,0.785695-1.17588I,0.541196-1.30656I,0.275899-1.38704I}

ThefunctiontocomputetheDCTofalistoflengthn=8isthen:

DCT[list_]:

=Re[DCTTwiddleFactors*

InverseFourier[N[list[[{1,3,5,7,8,6,4,2}]]]]]

NotethatweusethefunctionInverseFouriertoimplementwhatisusuallyinengineeringcalledtheforwardDFT.Likewise,weuseFouriertoimplementwhatisusuallycalledtheinverseDFT.ThefunctionNisusedtoconvertintegerstorealsbecause（inVersion2.2）FourierandInverseFourierarenotevaluatednumericallywhentheirargumentsareallintegers.Thespecialcaseofalistofzerosneedstobehandledseparatelybyoverloadingthefunctions,sinceNoftheinteger0