Discrete Cosine Transform, Second Edition

389 Pages·2019·17.397 MB·English

by  Humberto Ochoa-Dominguez, K R Rao

Discrete Cosine
Transform
Second Edition

Discrete Cosine
Transform
Second Edition
Humberto Ochoa-Dom´ınguez
K. R. Rao
CRCPress
Taylor&FrancisGroup
6000BrokenSoundParkwayNW,Suite300
BocaRaton,FL33487-2742
(cid:2)c 2019byTaylor&FrancisGroup,LLC
CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness
NoclaimtooriginalU.S.Governmentworks
Printedonacid-freepaper
InternationalStandardBookNumber-13:978-1-138-30472-7(Hardback)
Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Reasonableefforts
havebeenmadetopublishreliabledataandinformation,buttheauthorandpublishercannotassume
responsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse.Theauthorsandpublishers
haveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublicationandapologize
tocopyrightholdersifpermissiontopublishinthisformhasnotbeenobtained.Ifanycopyrightmaterial
hasnotbeenacknowledgedpleasewriteandletusknowsowemayrectifyinanyfuturereprint.
ExceptaspermittedunderU.S.CopyrightLaw,nopartofthisbookmaybereprinted,reproduced,trans-
mitted,orutilizedinanyformbyanyelectronic,mechanical,orothermeans,nowknownorhereafter
invented,includingphotocopying,microfilming,andrecording,orinanyinformationstorageorretrieval
system,withoutwrittenpermissionfromthepublishers.
For permission to photocopy or use material electronically from this work, please access
www.copyright.com(http://www.copyright.com/)orcontacttheCopyrightClearanceCenter,Inc.(CCC),
222RosewoodDrive,Danvers,MA01923,978-750-8400.CCCisanot-for-profitorganizationthatpro-
videslicensesandregistrationforavarietyofusers.Fororganizationsthathavebeengrantedaphotocopy
licensebytheCCC,aseparatesystemofpaymenthasbeenarranged.
TrademarkNotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andare
usedonlyforidentificationandexplanationwithoutintenttoinfringe.
LibraryofCongressCataloging-in-PublicationData
Names:Rao,K.Ramamohan(KamisettyRamamohan),author.|Ochoa-Dom´ınguez,
Humberto,author.
Title:Discretecosinetransform/HumbertoOchoa-Dom´ınguezandK.R.Rao.
Description:Secondedition.|BocaRaton,FL:Taylor&FrancisGroup,CRCPress,
2019.|Revisededitionof:Discretecosinetransform:algorithms,advntages,
applications/K.R.Rao,P.Yip.1990.|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019000195|ISBN9781138304727(hardback:alk.paper)
Subjects:LCSH:Signalprocessing--Mathematics.|Discretecosinetransforms.
Classification:LCCTK5102.5.R3352019|DDC621.382/23--dc23
LCrecordavailableathttps://lccn.loc.gov/2019000195
VisittheTaylor&FrancisWebsiteat
http://www.taylorandfrancis.com
andtheCRCPressWebsiteat
http://www.crcpress.com
Acknowledgments
Thisbookistheresultofthelongtermassociationoftheauthors,
HumbertoOchoa-Dom´ınguez andK.R.Rao.Dr.Ochoa’ssignificant
andsustainedcontributionsmadethisbookpossible.Specialthanks
gototheirrespectivefamiliesfortheirsupport,perseveranceand
understanding.Dr.Ochoawouldliketoexpresshisgratitudeto
ClaudiaValenzuela,forhercontinuoussupportduringthewritingof
thisbook,andtotheUniversidadAuto´nomadeCiudadJua´rez
(UACJ).Dr.Raowouldliketoacknowledgethesupportprovidedin
variousformsbyDr.Crouch,DeanoftheCollegeofEngineering,and
Dr.J.W.Bredow,Chair,DepartmentofElectricalEngineering,both
atUTA.Dr.G.J.Sullivan, Microsoft,Dr.H.Kalva,FloridaAtlantic
UniversityandDr.W.-K.Cham,ChineseUniversityofHongKong
providedvariousresourcesinthisregard.ShibaKuanar,Harsha
Nagathihalli Jagadish,VenkataSivaReddyAnushaandSwaroop
KrishnaRaoatUTAcontributed countlesshoursintyingupallloose
ends(references,copyrightreleases,proofreadingandamillionother
details).ThegraduatestudentsandalumnaeintheMultimedia
ProcessingLab(MPL)atUTAinvariouswayshavemade
constructivecomments.

Contents
Preface.....................................................................................................................xiii
Acronyms...............................................................................................................xvii
Contributors..........................................................................................................xxiii
ListofFigures........................................................................................................xxv
ListofTables.........................................................................................................xxix
Chapter1 Introduction.....................................................................................1
Applications.....................................................................................1
Chapter2 TheKarhunen–Loe`veTransform....................................................5
Introduction......................................................................................5
TheKarhunen–Loe`veTransformfora1DRandomField...............5
EigenvaluesandEigenvectorsofR.........................................7
ApplicationofKLTtoDataCompression.......................................8
KLTfora2DRandomField..........................................................11
Applications...................................................................................17
Summary........................................................................................18
Problems........................................................................................19
Projects...........................................................................................20
Chapter3 TheDiscreteCosineTransform.....................................................23
Introduction....................................................................................23
TheFourierCosineTransform.......................................................24
PropertiesoftheFCT............................................................24
Definitions......................................................................................27
TheUnitarityofDCTs...................................................................29
ScalinginTime..............................................................................33
ShiftinTime..................................................................................34
TheDifferenceProperty.................................................................38
TheConvolutionProperty..............................................................39
RelationtotheKLT........................................................................42
DCT-I....................................................................................43
FastAlgorithms..............................................................................46
DCTviaFFT.........................................................................47
DirectComputationbySparseMatrixFactorizations..........53
vii
viii Contents
Decimation-in-Time(DIT)andDecimation-in-Frequency
(DIF)Algorithms.....................................................57
TheDITAlgorithm..................................................57
TheDIFAlgorithm..................................................63
TheDCTviatheWalsh–HadamardTransform(WHT).......69
TheDCTviatheDiscreteHartleyTransform(DHT)..........71
OtherFastAlgorithms..........................................................73
PrimeFactorAlgorithm(PFA)................................74
AFastRecursiveAlgorithm.....................................79
DCT-IIRealizationviaPlanarRotations.................83
Applications...................................................................................88
Summary........................................................................................88
Problems........................................................................................89
Projects...........................................................................................91
Chapter4 TheDiscreteSineTransform.........................................................95
Introduction....................................................................................95
TheFourierSineTransform...........................................................95
PropertiesoftheFST............................................................96
TheDiscreteSineTransform.........................................................99
TheDiscreteFourierTransformofaSkewedSequence....100
TheDiscreteSineTransformviatheDFT..........................101
FastAlgorithms............................................................................102
TheFastDST-IviatheSSTAlgorithm..............................105
SparseMatrixFactorizationoftheDST-I...........................107
TheSplit-RadixAlgorithmforDST-I................................108
Applications.................................................................................110
Summary......................................................................................110
Problems......................................................................................110
Projects.........................................................................................111
Chapter5 TheModifiedDiscreteCosineTransform...................................113
Introduction..................................................................................113
Definitions....................................................................................113
MDCTastheShiftedDiscreteFourierTransform......................117
PropertiesoftheMDCT...............................................................118
EvenlyStackedMDCT................................................................120
MDCTBlockTransform....................................................121
SymmetryPropertiesofMDCTBlockTransform.............122
MatrixRepresentationoftheMDCT..................................123
OddlyStackedMDCT.................................................................125
MDCTBlockTransform....................................................125
SymmetryPropertiesofMDCTBlockTransform.............126
MatrixRepresentationoftheMDCT..................................128
Contents ix
Applications.................................................................................129
Summary......................................................................................130
Problems......................................................................................131
Projects.........................................................................................132
Chapter6 TheIntegerDiscreteCosineTransform......................................133
Introduction..................................................................................133
IntegerDCTMatrix.....................................................................134
TheICTbyDirectScaling..................................................135
H.2644x4and8x8ICTMatrices..........................135
WMV94x4and8x8ICTMatrices........................137
AVS1China8x8ICTMatrix.................................138
H.2654x4,8x8,16x16and32x32ICTMatrices.....139
AVS2 China 4x4, 8x8, 16x16 and 32x32 ICT
Matrices.................................................141
VersatileVideoCoding(VVC)..............................143
TheICTbyKernelComponentReplacement....................145
FastAlgorithm.......................................................148
TheLiftingMatrixandtheLiftingStep.............................149
TheIntegerDCT-IIbyLiftingMatrix...................155
Low-ComplexityICT(LCICT)..........................................156
SquareWaveTransform......................................................159
Round-offApproximationofIntDCT.................................162
TheTchebichefPolynomials.......................................................163
Orthogonality......................................................................164
DiscreteTchebichefPolynomials.......................................164
Orthogonality.........................................................165
EvenSymmetry......................................................165
TheDiscreteTchebichefTransform...................................166
4x4and8x8DiscreteTchebichefTransforms....................167
IntegerDiscreteTchebichefTransform..............................168
MetricstoEvaluateaTransform..................................................172
Mean-SquareError.............................................................172
CodingGain........................................................................172
TransformEfficiency..........................................................172
VarianceDistribution..........................................................173
EnergyPackingEfficiency(EPE).......................................173
ResidualCorrelation...........................................................173
Implementations...........................................................................174
Applications.................................................................................185
Summary......................................................................................186
Problems......................................................................................187
Projects.........................................................................................188