Image and Video Compression and Copyright Protection

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Title:
Image and Video Compression and Copyright Protection
Physical Description:
1 online resource (3 p.)
Language:
english
Creator:
Yang,Lei
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Wu, Dapeng
Committee Members:
Li, Xiaolin
Sun, Yijun
Chen, Shigang

Subjects

Subjects / Keywords:
authentication -- clustering -- companding -- compression -- copyright -- hash -- image -- lossless -- optimization -- quantization -- ratedistortion -- registration -- reversible -- scaling -- transform -- video -- watermarking
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre:
Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
An upsoaring number of digital images and videos demand efficient compression to facilitate storage and transmission of images and videos over Internet, and effective security and copyright protection techniques against malicious fabrication and illegal copy of digital contents. First, we focus on transform and quantizer design for compression. We design integer reversible transforms with the stabilized and optimized PLUS factorization for unified lossy/lossless compression. The proposed integer DCTs and integer Lapped Biorthogonal Transform have better lossy/lossless image coding performance than some existing integer DCT and the integer Lapped Transform in JPEG-XR. Moreover, we propose an adaptive quantizer using piecewise companding and scaling for gaussian mixture with three modes. The proposed quantizers approximate MSE performance of Lloyd-Max quantizers but only with similar complexity of uniform quantizers, and achieve higher perceptual quality in High Dynamic Range(HDR) images compression. Furthermore, we propose an optimal vector quantizer approximator by using transforms plus scalar quantizers with small complexity. The system is built on the tri-axis coordinate frame, works for both circular and elliptical distributions, and almost always outperforms restricted/unrestricted polar quantizers, and rectangular quantizers. Second, we study copyright protection of digital contents. The proposed image hashes by using companding and gray code have a small collision rate, strong discriminability and are difficult to analyze by attackers. We also propose an image authentication technique by feature point clustering and matching. Query images are authenticated with anchor images. The query images are registered, and the possible tampered areas are detected. Moreover, a robust track-and-trace video watermarking system is developed with watermarking embedder and detector. In the embedder, we insert a watermark pattern into video frames according to a watermark payload weighted by the human perceptual model, and transform videos with geometric anti-collusion codes. In the detector, Kanade-Lucas-Tomasi feature tracker is used to register the candidate videos, and the cross-correlation sequence is binarized, ECC decoded and decrypted. This system is very robust to both geometric attacks and collusion attacks, and watermarks are perceptually invisible to human vision system.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Lei Yang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Wu, Dapeng.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-02-29

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UFRGP
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Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043097:00001


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IMAGEANDVIDEOCOMPRESSIONANDCOPYRIGHTPROTECTIONByLEIYANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011LeiYang 2

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Tomybelovedparents,eldersisterandZhe 3

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ACKNOWLEDGMENTS Firstandtheforemost,Iamheartilythankfultomyadvisor,ProfessorDapengOliverWu,forhisgreatinspirationandexcellentguidancethroughoutmydissertationandmyPh.D.educationatUniversityofFlorida.Hisextensiveknowledge,valuablesuggestions,researchtrustiness,andkindconsiderationhelpmenishmydissertationandPh.D.study.IwouldalsoliketothankProf.ShigangChen,Prof.XiaolinAndyLi,andProf.YijunSunforservingonmydissertationcommitteeandprovidingvaluablesuggestionsonthisdissertation.Iamindebtedtomymaster'sthesisadvisorProf.PengweiHaotobringmetheresearchworldofsignalandimageprocessing.Hisdeepknowledge,researchinnovation,responsibleattitudeandimpressivekindnesshavehelpedmetodevelopthefundamentalandessentialacademiccompetence.IamfortunatetobeastudentofProf.JianboGao.Ihavelearntalotfromhissignalprocessingclassesandelaboratelydesignedcourseprojects.Iamthankfultomylab-matesinMultimediaCommunicationsandNetworkingLaboratoryatUF.Iamfortunatetojointhisbigfriendlyfamily.Iwouldliketothankseniorlab-matesDr.JunXu,Dr.ZhifengChenandDr.BingHanforresearchdiscussion;Icherisheverycooperationthediscussionwithmycurrentlab-matesQianChen,HuanghuangLi,ZhengYuan,YuejiaHe,ShijieLi;IwouldalsolikethankZongruiDing,Dr.XihuaDong,WenxingYe,Dr.XiaochengLi,YakunHu,Dr.TaoranLv,JiangpingWang,SunbaoHua,QinChen,QingWang,Dr.YounghoJoandChrisPaulson.Wishyouallhavesuccessinyourstudiesandwork.IamgratefultoDr.DebarghaMukhurjee,myinternmentorinGoogleInc.,forhiskindguidance,supportandhelp.IamalsothankfultomymanageMr.Shastra,Dr.YaowuXu,Dr.JimBankoski,Dr.PaulWilkinsfortheirkindinstructionsandsupport.Iaminavideocodingworldwithyou. 4

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Finally,Iowemydeepestgratitudetomyparents,eldersisterandZhefortheirendlessloveandconstantsupport.Withoutmyparents,IwouldhaveneverbeenabletoaccomplishwhatIhadtoday.Tothem,Idedicatethisdissertation. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 11 LISTOFFIGURES ..................................... 12 ABSTRACT ......................................... 15 CHAPTER 1INTRODUCTION ................................... 17 1.1ProblemStatement ............................... 17 1.1.1ResearchBackground ......................... 17 1.1.1.1Compression ......................... 17 1.1.1.2Copyrightprotection ..................... 22 1.1.2ResearchChallenges ......................... 24 1.1.2.1Transformdesign ....................... 24 1.1.2.2Quantizerdesign ....................... 25 1.1.2.3Imagehashingandauthentication ............. 25 1.1.2.4Track-and-tracevideowatermarking ............ 26 1.2ContributionsoftheDissertation ....................... 27 1.3OutlineoftheDissertation ........................... 29 2INTEGERREVERSIBLETRANSFORMSFORLOSSLESSCOMPRESSIONOFIMAGESANDVIDEOS ............................. 32 2.1ResearchBackground ............................. 32 2.1.1TransformDesign ............................ 32 2.1.2IntroductionofPLUSFactorization .................. 35 2.2StablePLUSFactorizationAlgorithms .................... 37 2.3PLUSFactorizationOptimization ....................... 38 2.3.1TransformErrorAnalysis ........................ 39 2.3.2StatementofOptimizationProblem .................. 40 2.3.3OptimizationAlgorithmwithTabuSearch ............... 41 2.4LosslessTransformforLossy/LosslessImageCompression ........ 43 2.5ExperimentalResults ............................. 45 2.5.1ExamplesoftheStablePLUSFactorizationAlgorithms ....... 45 2.5.2ExperimentsforPLUSFactorizationOptimization .......... 46 2.5.3ExperimentsonApplicationsinImageCoding ............ 47 2.5.3.1IntegerDCTwithoptimalPLUSfactorization ....... 47 2.5.3.2IntegerlappedbiothogonaltransformwithoptimalPLUSfactorization ......................... 48 2.6Summary .................................... 51 6

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3ADAPTIVEQUANTIZATIONUSINGPIECEWISECOMPANDINGANDSCALINGFORGAUSSIANMIXTURES ............................ 61 3.1ResearchBackground ............................. 61 3.2Preliminaries .................................. 63 3.2.1MMSEQuantizer ............................ 63 3.2.2GaussianMixtureModelandAfneLaw ............... 64 3.2.3MMSECompander ........................... 65 3.3AdaptiveQuantizerforGaussianMixtureModels .............. 66 3.3.1DesignMethodology .......................... 66 3.3.2ThreeModes .............................. 66 3.3.3ParameterDetermination ....................... 67 3.3.4PiecewiseCompandingofModeII .................. 68 3.3.5AdaptiveQuantizerforAGeneralGMM ............... 69 3.3.5.1GMMestimationbyEM ................... 69 3.3.5.2GeneralizationtoGMM ................... 69 3.4RecongurableA/DconverterwithAdaptiveQuantizer ........... 70 3.5HighDynamicRangeImageCompressionwithJointAdaptiveQuantizerandMultiscaleTechniques ........................... 71 3.6ExperimentalResultsandDiscussion .................... 73 3.6.1ExampleandJusticationofParameterDetermination ....... 73 3.6.2MSEPerformanceComparison .................... 74 3.6.3AnApplicationinImageQuantization ................. 76 3.6.4ExperimentalResultsonHDRImageToneMapping ........ 77 3.7Summary .................................... 78 4APPROXIMATINGOPTIMALVECTORQUANTIZATIONWITHTRANSFORMATIONANDSCALARQUANTIZATION ........................... 84 4.1ResearchBackground ............................. 84 4.2Preliminaries .................................. 86 4.2.1n-dimensionalMMSEQuantizerandScalingLaw .......... 86 4.2.2CircularandEllipticalDistributions .................. 88 4.2.3IdealUniformDistributionandOptimalTwo-dimensionalHexagonLattice .................................. 89 4.3SystemArchitecture .............................. 90 4.3.1QuantizationforCompression ..................... 90 4.3.2TheoremandSystemFramework ................... 90 4.4PreprocessingwithTransforms ........................ 91 4.4.1UnitaryTransforms ........................... 91 4.4.2ScalingTransforms ........................... 92 4.4.3OptimalTransformforArbitraryDistributions ............. 93 4.5OptimalScalarQuantizersinTri-axisCoordinateFrame .......... 93 4.5.1Tri-axisCoordinateFrame ....................... 94 4.5.2Tri-AxisSystemforUniformDistribution ............... 94 4.5.3Tri-AxisCoordinateFrameforCircularandEllipticalDistributions 95 7

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4.5.3.1Elasticquantizationlattices ................. 95 4.5.3.2Designmethodology ..................... 96 4.5.3.3Thenumberofquantizationlevelsineachannulus .... 98 4.5.3.4Expansionrule ........................ 99 4.5.4GeneralizationtoGMMorLMM .................... 101 4.5.5GeneralizationtoHighDimension ................... 101 4.6ExperimentalResultsandDiscussions .................... 101 4.6.1BasicOptimalProperties ........................ 102 4.6.2CircularGaussianDistribution ..................... 103 4.6.3EllipticalGaussianDistribution .................... 103 4.6.4CircularLaplaceDistribution ...................... 103 4.6.5EllipticalLaplaceDistribution ..................... 103 4.6.6Bit-rateSaving ............................. 104 4.7Summary .................................... 104 5CONTENTBASEDIMAGEHASHING ....................... 114 5.1ResearchBackground ............................. 114 5.2SystemOverview ................................ 116 5.3RobustDescriptorofImages ......................... 117 5.3.1Preprocessing .............................. 118 5.3.2FeaturePointExtraction ........................ 118 5.3.3FeaturePointDescription ....................... 119 5.4HashGeneration ................................ 120 5.4.1PseudoRandomPermutationofMorletWaveletCoefcients .... 121 5.4.2QuantizationUsingCompanding ................... 121 5.4.3BinaryCodingUsingGrayCode ................... 122 5.5ExperimentalResults ............................. 122 5.5.1TheRobustnessofFeaturePointDetector .............. 122 5.5.2ParameterDeterminationofSingularityDescriptor ......... 123 5.5.3DiscriminabilityandRobustnessofImageHashes ......... 124 5.5.3.1Discriminabilitybetweendifferentimages ......... 124 5.5.3.2Non-predictabilityofimagehashes ............. 124 5.5.3.3Robustnesstocontent-preservingattacks ......... 124 5.5.3.4Robustnesstotampering .................. 125 5.5.3.5Discriminativethresholds .................. 125 5.6Summary .................................... 125 6CONTENTBASEDIMAGEAUTHENTICATION .................. 132 6.1ResearchBackground ............................. 132 6.2SystemOverview ................................ 134 6.3FeaturePointDetection ............................ 135 6.3.1Preprocessing .............................. 136 6.3.2FeaturePointExtraction ........................ 136 6.4FeaturePointClusteringandMatching .................... 137 8

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6.4.1ClusteringbyFuzzyC-Means ..................... 137 6.4.2OutlierRemoval ............................. 138 6.4.3SpatialOrderingandFeaturePointMatching ............ 139 6.4.4AlgorithmSummary .......................... 139 6.5DistanceEvaluation .............................. 139 6.5.1NormalizedEuclideanDistance .................... 140 6.5.2HausdorffDistance ........................... 141 6.5.3HistogramWeightedDistance ..................... 141 6.5.4MajorityVote .............................. 141 6.5.5StrategyforThresholdDetermination ................. 142 6.6PossibleAttackIdentication ......................... 142 6.6.1GeometricAttackEstimationandRegistration ............ 142 6.6.2TamperingAttackIdentication .................... 143 6.7ExperimentalResults ............................. 143 6.7.1FeaturePointDetection ........................ 143 6.7.2FeaturePointMatchingExample ................... 144 6.7.3AuthenticationPerformance ...................... 144 6.7.4DistanceComparison ......................... 146 6.7.5TamperingDetection .......................... 146 6.8Summary .................................... 147 7ROBUSTTRACK-AND-TRACEVIDEOWATERMARKING ............ 155 7.1ResearchBackground ............................. 155 7.2ArchitectureofRobustVideoWatermarkingSystem ............ 157 7.2.1WatermarkingEmbedder ....................... 158 7.2.2WatermarkingDetector ......................... 158 7.3WatermarkEmbeddingTechniques ...................... 160 7.3.1WatermarkPatternGeneration .................... 160 7.3.2WatermarkPayloadGeneration .................... 160 7.3.3PerceptualWeightingModel ...................... 161 7.3.3.1Temporalperceptualmodeling ............... 161 7.3.3.2Spatialperceptualmodeling ................ 162 7.3.4GeometricAnti-collusionCoding ................... 164 7.4WatermarkDetectionTechniques ....................... 164 7.4.1VideoFrameRegistration ....................... 164 7.4.1.1Spatialregistration ...................... 165 7.4.1.2Temporalregistration .................... 166 7.4.2WatermarkExtractionandPayloadRecovery ............ 167 7.5ExperimentalResults ............................. 168 7.6Summary .................................... 169 8CONCLUSIONS ................................... 177 8.1SummaryoftheDissertation ......................... 177 8.2FutureWork ................................... 180 9

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8.2.1OptimalIntegerReversibleTransformsandtheLosslessVideoCompression .............................. 180 8.2.2ANewVideoCodingStrategyandRDCOptimization ........ 181 APPENDIX APERTURBATIONANALYSISOFPLUS ...................... 182 BPROOFS ....................................... 187 B.1ProofofProposition 3.1 ............................ 187 B.2ProofofProposition 3.2 ............................ 187 B.3ProofofProposition 4.1 ............................ 188 B.4ProofofLemma 2 ............................... 188 B.5ProofofLemma 4 ............................... 189 B.6ProofofLemma 5 ............................... 189 REFERENCES ....................................... 190 BIOGRAPHICALSKETCH ................................ 202 10

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LISTOFTABLES Table page 2-1PerformanceofoptimalfactorizationsforDCTmatriceswithexhaustivesearch. 56 2-2E2(LUS),OMSEandOMEofseveralPLUSfactorizationsforDCTmatrices .. 56 2-3SomeoptimalfactorizationsforDCTfoundbyexhaustivesearch ........ 57 2-4TransformerrorE2ofoptimalfactorizationsfoundbyTS ............. 57 2-5EntropycomparisonofintegerDCTs. ....................... 58 2-6Entropycomparisonamongdifferentintegertransforms. ............. 58 2-7Performancecomparisonofintegerlappedbiorthogonaltransforms ...... 59 3-1Proposedquantizervs.Lloyd-Maxquantizer. ................... 78 3-2Comparisonofcomplexityofquantizaters. ..................... 81 3-3Comparisonamonghistogramsofimages ..................... 82 4-1Averagebit-ratesavingoftheproposedquantizersoverotherquantizers. ... 113 5-1HammingdistancebetweenLenaanditstamperedversion. ........... 130 5-2Hammingdistancebetweendifferenttestimages. ................ 130 5-3Hammingdistancebetweenattackedimagesandtestimages. ......... 131 6-1Authenticationperformancecomparisonamongdifferentmethods. ....... 153 6-2Tamperingdetectionandpercentageoftamperingareaestimation. ....... 154 7-1Stepbystepresultsofwatermarkingembedder .................. 175 7-2Crosscorrelationcoefcienterrorratioperformance ............... 175 7-3CapabilityofKLTbasedvideoregistrationforvariousgeometrictransforms .. 176 11

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LISTOFFIGURES Figure page 1-1Thestructureofthedissertation. .......................... 31 2-1Flowchartof4-pointintegerDCTimplementedwithPLUS. ............ 51 2-2E2comparisonbetweenthefactorizationsfoundbythreealgorithms. ...... 52 2-3ConvergencespeedofoptimizationalgorithmusingTS. ............. 52 2-4Averagebppvs.PSNRwithintegertransformsfortestimages. ......... 52 2-5LossyperformancecomparisonforimageBarbara. ................ 53 2-6LossyperformancecomparisonforimageLena. ................. 54 2-7LossyperformancecomparisonforimageBaboon. ................ 55 3-1CDFofGaussianN(0,1)vs.CDFofSGMMwithm=0.5. ............ 78 3-2Transformationfunctionofapiecewisecompressorm=1.5. ........... 79 3-3CDFofthecatenatedGaussianvs.CDFofSGMMwithm=3. ......... 79 3-4RecongurableA/Dconverter. ........................... 79 3-5GMMestimationbyEMalgorithmonhistogramofBarbara. ........... 79 3-6Tonemappingbyusingjointadaptivequantizerandmultiscaletechniques. .. 80 3-7MSEcomparisonamongdifferentquantizers(s=1). ............... 80 3-8MSEcomparisonamongdifferentquantizers(s=2). ............... 80 3-9Performancecomparisonamongdifferentquantizerswhenk=4. ........ 81 3-10Performancecomparisonamongdifferentquantizerswhenk=5. ........ 81 3-11Performancecomparisonbetweendifferenttonemappingalgorithms. ..... 82 3-12Visualperformancecomparisonbetweendifferenttonemappingalgorithms. .. 83 4-1Generalencodinganddecodingpipeline. ..................... 105 4-2Systemarchitecturewithtransformplusscalarquantization. ........... 105 4-3Transformplusscalarquantizationwithcompandingtechnique. ......... 105 4-4Gaussianmixturemodeldecorrelation. ....................... 106 4-5Two-dimensionaltri-axiscoordinatesystem. .................... 106 12

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4-6Twodimensionaloptimaluniformvectorquantizer. ................ 106 4-7Circularlyexpandedhexagonlatticefortwo-dimensionalcirculardistribution. 107 4-8Ellipticallyexpandedhexagonlattices. ....................... 107 4-9Tri-axisframeforageneraltwo-dimensionalellipticaldistribution. ........ 107 4-10Expandedhexagonlatticefortwo-dimensionalcircularGaussiandistribution. 108 4-11Expandedhexagonlatticefortwo-dimensionalellipticalGaussiandistribution. 108 4-12Therstoptimalquantizationscheme. ....................... 109 4-13Thesecondprogressivequantizationscheme. .................. 109 4-14Voxelforthree-dimensionaluniformdistribution. .................. 110 4-15MSEperdimension. ................................. 110 4-16Optimalmagnitudelevels. .............................. 111 4-17Rate-Distortioncomparisonamongdifferentquantizers1. ............ 111 4-18Rate-Distortioncomparisonamongdifferentquantizers2. ............ 112 4-19Rate-Distortioncomparisonamongdifferentquantizers3. ............ 112 4-20Rate-Distortioncomparisonamongdifferentquantizers4. ............ 113 5-1Flowchartofimagehashgeneration. ....................... 126 5-2The7thframeandthe17thframeinthetestvideoBunny. ............ 126 5-3Thestablefeaturepointdetector. .......................... 127 5-4Hashdistancewithdifferentwaveletsatdifferentscales. ............. 128 5-5Sixtestimagesforimagehashing. ......................... 128 5-6Distancebetweenthehashes. ........................... 129 5-7SixtamperedimagesofLena. ........................... 129 6-1Theowchartoftheproposedimageauthenticationsystem. ........... 148 6-2Thestablefeaturepointdetector. .......................... 149 6-4Sixtestimagesforimagehashing. ......................... 150 6-5Distancecomparisonamongdifferentauthenticationmethods1. ........ 151 6-6Distancecomparisonamongdifferentauthenticationmethods2. ........ 151 13

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6-7ThetwoframesintwoshotsinthetestvideoBunny. ............... 152 6-8Diagramofpossibleattackidentication. ...................... 152 7-1Track-and-tracevideowatermarkingembedder. .................. 171 7-2Track-and-tracevideowatermarkingdetector. ................... 172 7-3Perceptualmodelingdiagram ............................ 173 7-4PSNRofwatermarkedvideosequencesinTable 7-1 .............. 173 7-5Geometrictransform/attacksto5thframeofGrandma .............. 174 14

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyIMAGEANDVIDEOCOMPRESSIONANDCOPYRIGHTPROTECTIONByLeiYangAugust2011Chair:DapengWuMajor:ElectricalandComputerEngineeringAnupsoaringnumberofdigitalimagesandvideosdemandefcientcompressiontofacilitatestorageandtransmissionofimagesandvideosoverInternet,andeffectivesecurityandcopyrightprotectiontechniquesagainstmaliciousfabricationandillegalcopyofdigitalcontents.First,wefocusontransformandquantizerdesignforcompression.WedesignintegerreversibletransformswiththestabilizedandoptimizedPLUSfactorizationforuniedlossy/losslesscompression.TheproposedintegerDCTsandintegerLappedBiorthogonalTransformhavebetterlossy/losslessimagecodingperformancethansomeexistingintegerDCTandtheintegerLappedTransforminJPEG-XR.Moreover,weproposeanadaptivequantizerusingpiecewisecompandingandscalingforgaussianmixturewiththreemodes.TheproposedquantizersapproximateMSEperformanceofLloyd-Maxquantizersbutonlywithsimilarcomplexityofuniformquantizers,andachievehigherperceptualqualityinHighDynamicRange(HDR)imagescompression.Furthermore,weproposeanoptimalvectorquantizerapproximatorbyusingtransformsplusscalarquantizerswithsmallcomplexity.Thesystemisbuiltonthetri-axiscoordinateframe,worksforbothcircularandellipticaldistributions,andalmostalwaysoutperformsrestricted/unrestrictedpolarquantizers,andrectangularquantizers.Second,westudycopyrightprotectionofdigitalcontents.Theproposedimagehashesbyusingcompandingandgraycodehaveasmallcollisionrate,strong 15

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discriminabilityandaredifculttoanalyzebyattackers.Wealsoproposeanimageauthenticationtechniquebyfeaturepointclusteringandmatching.Queryimagesareauthenticatedwithanchorimages.Thequeryimagesareregistered,andthepossibletamperedareasaredetected.Moreover,arobusttrack-and-tracevideowatermarkingsystemisdevelopedwithwatermarkingembedderanddetector.Intheembedder,weinsertawatermarkpatternintovideoframesaccordingtoawatermarkpayloadweightedbythehumanperceptualmodel,andtransformvideoswithgeometricanti-collusioncodes.Inthedetector,Kanade-Lucas-Tomasifeaturetrackerisusedtoregisterthecandidatevideos,andthecross-correlationsequenceisbinarized,ECCdecodedanddecrypted.Thissystemisveryrobusttobothgeometricattacksandcollusionattacks,andwatermarksareperceptuallyinvisibletohumanvisionsystem. 16

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CHAPTER1INTRODUCTION 1.1ProblemStatement 1.1.1ResearchBackgroundDigitalimagesandvideosareinanexponentialincreaseduetotheproliferationofInternet,digitalcameras,andimageandvideoapplications.Withtheimmersivedisplayandimagingtechnologies,increasingpenetrationofhigh-speedbroadband,andavailabilityofcomputingpower,imageandvideocompressionandcopyrightprotectiontechnologyhavecomeofage,enablingimagesandvideosasadrivingforceforgrowthandinnovationovertheyears.Inmanyapplicationsandendequipments,imagesandvideosplayakeyroleasamechanismofinformationexchange,transmissionandstorage,andplayanimportantpartinhumanlives. 1.1.1.1CompressionThesoaringnumberofimagesandvideosrequiresefcientcompression.Therefore,thecompressiontechniquesaredeployedinmanyappliancesandapplicationsincludingcameras,digitalcinema,cableandsatellitedigitalvideotransmissionsforentertainment,low-latencyvideotelephone,storage-constrainedsurveillanceapplications,machinevisionandrecognition,small-formandpower-constrainedmobilehandsets,andotherhandhelddevices.Fortheseapplications,lossyvideocompressionalreadysatisesthehumanperceptualrequirement.Whereas,fortheespeciallypreciousdata,suchashighqualitystudioproducts,neartsandantiquedocuments,medicalandsatellitedata,losslessarchivingisneededtopreserveeverypixelexactly.ThegeneralcompressionsystemcomposesofanencoderandadecoderasshowninFigure 4-1 .Encoderusuallyincludestransform,quantizationandentropycoding,whiledecoderperformsasamirrorofencoder.Therearemanyimageandvideocodingstandardsevolvingwithages.JPEGisforstillcolorimagecompression,beganintheInternationalStandardsOrganization 17

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(ISO)in1982,andwasapprovedinSeptember1992asITU-TRecommendationT.81andin1994asISO/IEC10918-1.JBIG1andJBIG2areforbilevelfaximagesorblackandwhiteimages,andJPEG-2000isforstillcolorimages,buttheyarenotaspopularasJPEG.TheH.26xfamilyofvideocodingstandardsweredevelopedbytheITU-TVideoCodingExpertsGroup(VCEG)inthedomainoftheITU-T.H.261isforconversationalservices.MPEG-1,andMPEG-2areforstorageandbroadcastapplications.H.263andH.263+arevideocompressionstandardsoriginallydesignedasalow-bitratecompressedformatforvideoconferencing.MPEG-7isformultimediametadata.H.264/MPEG-4Part10orAVC(AdvancedVideoCoding)iscurrentlyoneofthemostcommonlyusedformatsfortherecording,compression,anddistributionofhighdenitionvideos.Theyarepresentedintheorderofincreasingcomplexityalthoughthisisnotinaccordancewiththechronologicalorderoftheircompletion.Amongthecompressiontechniques,transformsarewidelyusedinsourcecoding,imageprocessingandcomputergraphics.Transformcodingisamajortechniqueinimageandvideocompressionstandard,suchasJPEG,JPEG2000,JPEG-XR[ 56 ],MPEG[ 81 ]andH.264[ 152 ].Lossycompressionisbasedonthetraditionaltransforms,suchasdiscretecosinetransform(DCT)inJPEGandH.264,discretewavelettransform(DWT)inJPEG2000.Forlosslesscompression,JPEG2000usesintegerdiscretewavelettransform(IntDWT),JPEG-XRusesintegerlappedbiorthogonaltransform(IntLBT)torealizeperfectintegerreversibility,whilelosslessJPEGandH.264arebasicallybasedondifferentialpulse-codemodulation(DPCM).Therearemanyintegerreversibletransformsproposedintheliterature.Besidesthedirectlydesignedintegertransforms[ 160 ],mostofintegertransformsarederivedfromthetraditionaltransforms.Factorizationisaneffectivetooltomakethetraditionaltransforms,suchasDiscreteCosineTransform(DCT)[ 7 ]andDiscreteWaveletTransform(DWT)[ 92 134 ],faster,simplerandintegerreversible.PLUSfactorizationisakindofcustomizabletriangularmatrixfactorization,proposedbyHao[ 63 ]asa 18

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newframeworkofmatrixfactorization,andencompassesandgeneralizesquiteafewtriangularmatrixfactorizations[ 64 133 141 ].Intherstpartofthisdissertation,wewillstabilizeandoptimizethePLUSfactorization,anddoperturbationanalysisonit.Weuseittodesignintegerreversibletransforms,tomakeimageandvideocompressionlossless.Quantizationisanothercriticalstepincompressionaswellasindigitization.Thesourcesareinagreatvariety.Theycouldbeuni-variateormulti-variate,couldbeinGaussiandistribution,Laplacedistributionoramixturedistribution.HowtomaketheoptimalquantizationforanarbitrarysourceintermsofMMSE?Theoptimalvectorquantizersaredesignedforthispurpose.TheresultedMMSEquantizersorLloyd-Maxquantizerssatisfytwooptimalconditions:thenearestneighborcondition(NNC)andthecentroidcondition(CC)asshowninEq.( 3 )andEq.( 3 ).However,thecodebooksizelinearlyincreaseswiththenumberofquantizationlevelsN,moreover,codebookdesigntimeexponentiallyincreaseswiththenumberofquantizationlevelsN.Therefore,researchersarefocusingondesigningvarioussubstitutionsoftheoptimalvectorquantizerswithmuchlessquantizerdesigncomplexity.ThisisthegoalofChapter 3 andChapter 4 .Thereareseveralquantizerdesignsavailablewhichprovidevarioustrade-offsbetweensimplicityandperformance.ItalsofallsintothestudyofRate-Distortion-Complexity(RDC)theory.Foruni-variatequantizer,existingquantizationschemescanbeclassiedintotwocategories,namely,uniformquantizationandnonuniformquantization[ 60 61 ].Uniformquantizationissimple,andoptimalforuniformdistributions,butnotoptimalforsignalswithnonuniformdistributionsifmorecomputationsandstorageareavailable.Nonuniformquantizationismuchmorecomplexandinagreatvariety.Minimummeansquarederror(MMSE)quantization(a.k.a,Lloyd-Maxquantization)isamajortypeofnonuniformquantization.Itisoptimalinthesenseofmeansquarederror(MSE),butincurshighcomputationalcomplexity.Companding,whichconsistsof 19

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nonlineartransformationanduniformquantization,isatechniquecapableoftradingoffquantizationperformancewithcomplexityfornonuniformquantization.Especially,forhighratecompression,theperformanceofcompandingcanapproachthatofLloyd-Maxquantizationasymptotically.Lloyd-MaxquantizersandcompandersarealreadywelldevelopedforGaussiandistributionorLaplaciandistribution[ 61 68 109 ]asconvenience,butnotforGaussianmixturemodel(GMM).SinceGMMservesasagoodapproximationofanarbitrarydistribution,itisimportanttodevelopquantizersandcompandersforGMM,whichareexpectedtondwideapplicationsinADCandhighdynamicrange(HDR)imagecompression,aswellasaudio[ 111 ]andvideo[ 152 ]compression.Toaddressthis,weproposesasuccinctadaptivequantizerwithpiecewisecompandingandscalingforGMMinthispaper.WerstconsiderasimpleGMM(SGMM)thatconsistsoftwoGaussiancomponentswithmean)]TJ /F11 11.955 Tf 1 0 .167 1 343 -298.85 Tm[(mandmrespectively,andthesamevariances2.Theproposedquantizershavethreemodes,makingthemcapableofadaptingtheirreconstructedlevelstothevaryingmeansandvariancesoftheGaussiancomponentsinaGMM.Formulti-variatequantization,toreducethecodebookdesignandlookuptime,alotofresearchhasbeenfocusedontwo-dimensionalrandomvariables(r.v.),especiallythoseincircularGaussiandistributions,sinceGaussiandistributions[ 20 39 ]havealotofelegantclose-formtheorems.TheearliestworkcouldrefertoHuangandSchultheiss'smethod[ 66 ],whichquantizeseachdimensionofrandomvariableswithseparateLloyd-Maxquantizers[ 97 ].Itisefcientandeffective,butdenitelycouldbeimproved.Later,Zador[ 163 ]andGersho[ 57 ]studiedquantizationbyusingcompanderswithalargenumberofquantizationlevels.Theyusedacompressortotransformthedataintoauniformdistribution,andthenappliedtheoptimalquantizersfortheuniformdistribution,andthentransformthedatawithanexpander.Butthisschemedoesnotworkwellunderasmallnumberofquantizationlevels.Anothermajormethodfor 20

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designingquantizersforcirculardistributionsusespolarcoordinates.Polarquantizationincludesseparablemagnitudequantizationandphasequantization.TheoptimalratiobetweenthenumberofmagnitudequantizationlevelsandthenumberofphasequantizationlevelsarestudiedbyPearlman[ 108 ]andBucklewetal.[ 21 22 ],andanMMSErestrictedpolarquantizerisimplementedbyusingauniformquantizerforthephaseanglesandascaledLloyd-MaxRayleighquantizerforthemagnitude.Butthisschemedoesnotconsiderthecenterofacirculardistributionasaquantizationlevel,thus,itsMSEperformanceissometimeworsethanrectangularquantizersandotherlatticequantizers,anditdoesnotworkwellforellipticaldistributions.Wilson[ 153 ]proposedaseriesofnon-continuousquantizationlatticeswhichprovidealmosttheoptimalperformanceamongpolarquantization.Itisakindofunrestrictedpolarquantization,butwithoutDirichletboundaries.Peteretal.[ 135 ]improvedWilson'sschemebyreplacingarcboundarieswithDirichletboundaries.HeshowedtheoptimalcircularlysymmetricquantizersforcircularGaussiandistributions.MostofthesepreviousworksconcentrateonGaussiandistributions,andprovidenumericalresultsonlyforGaussiandistributions.AlthoughGaussiansourceisconsideredastheworstcasesourcefordatacompression,whichisinstructivetoconstructarobustquantizer[ 27 ],itisfarfromtheoptimalforquantizingotherdistributions.Theydidnotconsidertheellipticaldistributionsneither,whoseoptimalquantizersaredifferentfromthoseforcirculardistributions.Also,theydidnotprovideauniedframeworkforarbitrarydistributions.Therefore,theoptimalquantizersforotherdistributionssuchasLaplaciandistributions,ellipticalGaussianandLaplaciandistributionsneedinvestigation.Toaddresstheseproblems,weproposeauniedquantizationsystemtoapproachtheoptimalvectorquantizersbyusingtransformsandscalarquantizers.Theeffectoftransformsonsignalentropyandsignaldistortionisdiscussed,especiallyforunitarytransformsandvolume-keepingscalingtransforms.Theoptimaldecorrelationtransform 21

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isillustratedwhichturnsamemorysourceintoamemorylesssourceinanidealcase.Thenwefocusonthequantizerdesignformemorylesscircularandellipticalsources.Thetri-axiscoordinateframeisproposedtodeterminethequantizationlattice,i.e.thepositionsofquantizationlevels,inspiredbythewell-knownoptimalhexagonlatticefortwo-dimensionaluniformlydistributedsignals[ 68 ].Itprovidesauniedframeworkforbothcircularandellipticaldistributions,andencompassesthepolarquantizationasaspecialcase.Theproposedquantizerisalsoakindofadaptivequantizerwithelasticlattice.WewillpresentthesimpledesignmethodologyandutilizetheLloyd-Maxquantizersforthecorrespondingone-dimensionaldistributions.Theoptimalityofthisschemeisveriedonelliptical/circularGaussianandLaplaciandistributions.Themethodologydescriptionandexperimentsarefocusedonthebivariaterandomvariables,andtheextensiontohighdimensionalrandomvariablesisalsodiscussed. 1.1.1.2CopyrightprotectionBesidescompressiontechniques,thesecurityandcopyrightofdigitalimagesandvideosalsorisetothepositionthatcannotbeignored.Digitalimagesandvideosfacilitatemultimediaprocessing,andatthemeantime,makefabricatingandcopyingofdigitalcontentseasy,withtheincreaseofvariousimageandvideoapplications.Computers,interconnectedviatheInternet,makethedistributionofthedigitalmediafastandeasy,andmaketheexactcopiestobeobtainedwithoutefforts.Toprotectthecopyrightofimagesandvideos,efcientandautomatictechniquesareneededtoidentifyandverifythecontentofdigitalmultimedia.Therefore,itposesgreatchallengestocopyrightprotectionfordigitalmedia.Besidesthelong-timeestablishedcopyrightprotectiontoolimagewatermarking[ 112 ],imagehashing[ 105 ]emergesasaneffectivetooltorepresentimagesandautomaticallyidentifywhetherthequeryimageisafabricationoracopyoftheoriginalone.Asanalternativetoimagewatermarking,imagehashingcanbeappliedtomanyapplicationspreviouslyaccomplishedbywatermarking,suchascopyrightprotection, 22

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imageauthentication.Itcanalsobeusedforimageindexingandretrievalaswellasvideosignature.Unlikewatermarking,imagehashingneednotchangetheimagebyinsertingwatermarksintotheimage.Imagehashisashortbinarystring,mappedfromanimagebyanimagehashfunction.Theimagehashfunctionhassuchapropertythatperceptuallyidenticalimagesshouldhavethesameorsimilarhashvalueswithhighprobability,whileperceptuallydifferentimagesshouldhavequitedifferenthashvalues.Inaddition,thehashfunctionshouldbesecure,sothatanattackercannotpredictthehashvalueofaknownimage.Imageauthentication[ 117 ]issuchapromisingtechniquetoautomaticallyidentifywhetherthequeryimageisafabricationorasimplecopyoftheoriginalone.Itcanutilizetheestablishedimagehashing.Thevideocontentchangesarelessnoticeablethanimagecontentchanges.Therefore,weapplywatermarkingtechniquetovideocopyrightprotection[ 82 ]ratherthanimagecopyrightprotection.Digitalwatermarkembeddingisaprocessofintegratingtheuserandcopyrightinformationintothecarriermediainawaywhichisinvisibletohumanvisionsystem(HVS).Itspurposeistoprotectthedigitalworksfromtheunauthorizedduplicationordistribution.Videowatermarkingsystemisdesiredtoembedwatermarkinsuchawaythatthewatermarkcanbedetectedlaterforauthentication,copyrightprotection,andtrack-and-traceillegaldistribution.Videos,composedofmultipleframes,canutilizeimagewatermarkingtechniquesinaframe-wisemanner.Althoughthewatermarkingembeddingcapacityofvideosismuchlargerthanthatofimages,theattacksthevideowatermarkingsuffersaremorecomplicatedthanimagewatermarking.Theattacksincludenotonlyspatialattacks,butalsotemporalattacks,hybridspatial-temporalattacksandcollusionattacks.Thesemotivateustoinvestigateonimage/videocompressionandcopyrightprotectionbyfocusingontransformdesign,quantizerdesign,imagehashing,imageauthenticationandvideowatermarking. 23

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1.1.2ResearchChallenges 1.1.2.1TransformdesignIntegerreversibletransformsareone-to-onemappings,andanypermutationisanintegerreversibletransform.Abouttheintegerreversibletransformdesignforlosslesscompression,thebestenergyconcentratingability,i.e.,thebestdecorrelationability,isdesired.Afterintegerreversibletransforms,thejointentropyofsubbandcoefcientswillnotchange,butthesumofthemarginalentropyofsubbandcoefcientswilldecrease.Theleastsumofthemarginalentropyofsubbandcoefcientsisachievedwhensubbandcoefcientsareindependentofeachother,i.e.,theirmutualinformationis0.Foramulti-dimensionalinformationsource,suchasanimageoravideo,theleastentropyintegertransform,i.e.,theoptimalpermutation,couldbederivedbyalgorithmsapproximately.However,thecontentofimagesandvideosvarying,theoptimalintegertransformforasourcemaynotoptimalglobally.Thenwewishtodesignanoptimalintegerreversibletransforminastatisticalsense,sincethejointtwo-dimensionaldistributionsofsourcesarestatisticallysimilar.Itiswellknownthatthetraditionalorthogonaltransformshavenicedecorrelationability,andtheyarerotationsinEuclideanspacegeometrically.Therefore,theintegerreversibletransformsderivingfromormimicingthetraditionaltransformsworkwell,butstillneedimproving.Inadditiontothestrongdecorrelationability,thepiecewiselinearityandtheleasttransformerrorofintegerreversibletransformsaredesired.Thesetwopropertiesguaranteetheintegerreversibletransformscouldbeappliedtolossycompressiondirectly,withthecompetentperformanceasthetraditionaltransforms.Moreover,theleasttransformerrorwillpromisecoefcientsofintegerreversibletransformsapproachingthoseoforiginaltransformsascloseaspossible.Lateron,wewillpresentourmethodsbyusingPLUSfactorizationtoaddresstheseproblems. 24

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1.1.2.2QuantizerdesignQuantizerdesignconsidersthetrade-offsbetweensimplicityandperformance.HowtodesignaquantizerthathasthesimilargoodRate-Distortionperformanceastheoptimalvectorquantizer,butwithmuchlesscomplexityisourconcern.Thegoodperformanceapproximationshouldworkforsourcesinarbitrarydistributions.Thesourcecouldbeuni-variateormulti-variate.Itcouldbememorysourceormemorylesssource.Itcouldbeincirculardistributionorellipticaldistribution.UniformdistributionandGaussiandistributionarewellinvestigated.Howabouttheotherdistributions?CanweusecompandingtotransformarbitrarydistributionintouniformorGaussiandistribution,thenusetheLloyd-MaxquantizersforuniformorGaussiandistributiontoquantize.Furthermore,howtocapturethesourcedistributioninformationtoadaptivelytunethequantizersinauniedframeworkneedsinvestigation.Thenwithcertainperformanceguarantee,howmuchcomplexityofquantizerdesignandquantizationimplementationcouldwereduce?Memorylesssourceusuallycouldhavesimplerquantizerdesignthanmemorysource.Withtransforms,wecanturnmemorysourceintomemorylesssource.Thenwiththetransformdesigntechniqueathand,wecouldusetransformtosimplifyquantizer'sdesignandimplementationaswellasimprovedthequantizationperformance.Whichtransformsareeffective?InChapter 3 andChapter 4 ,wewillconcentratetoaddresstheseproblemsandanswerthesequestions. 1.1.2.3ImagehashingandauthenticationForimagecopyrightprotection,imagehashingshouldberobusttomaliciousimageprocessing.Itconsistsofacompactrepresentationofsomeimagefeatures.Itshouldberobusttoimageltering,butsurrenderstogeometricattacksandmaynotbecollisionfree.TheimagehashbasedonScaleInvariantFeatureTransform(SIFT)algorithm[ 48 ]andcompressivesensingtechnique[ 138 ]couldsolvegeometricattacksincertaindegree,butitiscomputationallyexpensive.LinandChang[ 86 ]createdthemutualrelationshipofpairwiseblockDCTcoefcientstodistinguishJPEGcompressionfrom 25

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maliciousmodications.Buttheblockbasedmethodisunreliabletosomegeometricattacks,sincepossibleshiftingandcroppingoperationsmaychangehashvalues.Venkatesanetal.[ 147 ]proposedanimagehashingtechnique.Theirhashesaregeneratedfromstatisticalfeaturesextractedfromrandomtilingofwaveletcoefcients.However,itonlyallowslimitedresistancetogeometricdistortions,andissusceptibletosomemanipulations,suchasluminancechangeandobjectinsertion.Toaddresstheseproblems,weproposecontentbasedimagehashingusingcompandingandGraycode.Imageauthenticationtechniquesusuallyincludeconventionalcryptography,fragileandsemi-fragilewatermarkinganddigitalsignatureandsoon.Theauthenticationprocesscanbeassistedwiththeoriginalimageorintheabsenceoftheoriginalimage.Imageauthenticationmethods,basedoncryptography,useahashfunction[ 79 130 ]tocomputethemessageauthenticationcode(MAC)fromimages.Thegeneratedhashisfurtherencryptedwithasecretekeyfromthesender,andthenappendedtotheimageasanoverhead,whichiseasytoberemoved.Fragilewatermarkingusuallyreferstoreversibledatahiding[ 23 140 160 162 ].Thesemethodscannotdistinguishtolerablechangesfrommaliciouschanges.Semi-fragilewatermarkinghasattack-resistantabilitybetweenfragileandrobustwatermarking.Andthereisatradeoffbetweenimagequalityandwatermarkrobustness.Digitalsignaturebasedtechniquesareimagecontentdependent,whicharealsocalledimagehashing.Animagehashisarepresentationoftheimage.Besidesimageauthentication,itcanalsobeusedforimageretrievalandotherapplications.Butitisnotintendedtoidentifythelocationsofchanges.Toaddresstheseproblems,weproposecontentbasedimageauthenticationbyfeaturepointclusteringandmatching. 1.1.2.4Track-and-tracevideowatermarkingVideowatermarkingsystemisdesiredtoembedwatermarkinsuchawaythatthewatermarkcanbedetectedlaterforauthentication,copyrightprotection,andtrack-and-traceillegaldistribution.Videos,composedofmultipleframes,can 26

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utilizeimagewatermarkingtechniquesinaframe-wisemanner[ 112 ].Althoughthewatermarkingembeddingcapacityofvideosismuchlargerthanthatofimages,theattacksfromwhichthevideowatermarkingsuffersaremorecomplicatedthanthoseofimagewatermarking.Theattacksincludenotonlyspatialattacks,butalsotemporalattacksandhybridspatial-temporalattacks.Intheliteratureoftrack-and-tracevideowatermarking,thealgebra-basedanti-collusioncodeisinvestigated[ 10 17 24 139 144 151 155 ].Itsabilitytotraceoneormultiplecolludersdependsontheassumptionthatthecodeisalwaysavailableanderror-free,whichmaynotbetrueinpractice.Besides,thelengthofanti-collusioncodedeterminesthesystemuser-groupcapacity.Hence,practicalandmulti-functionalwatermarkingsystemsbasedonalgebraanti-collusioncodeareverylimited.Tothisend,weproposearobusttrack-and-tracewatermarkingsystemfordigitalvideocopyrightprotection[ 158 ]. 1.2ContributionsoftheDissertationThemajorcontributionsofourworkaresummarizedasfollows: 1. WedesignintegerreversibletransformsbasedontheoptimizedPLUSfactorization. Westabilize,optimizethePLUSfactorization,anddoperturbationanalysisonittoproveitsnumericalstability. WeproposetheintegerDCTandtheintegerLappedBiorthogonalTransformbyusingtheoptimizedPLUSfactorization. ExperimentalresultsshowthesuperiorityofouralgorithmsoversomeexistingintegerDCTalgorithms,andtheintegerlappedtransformfactorizationinJPEG-XRforlossy/losslessimagecoding. 2. WedesignadaptivequantizationusingpiecewisecompandingandscalingforGaussianmixtureswiththreemodes. Theexperimentalresultsshowthat1)theproposedquantizerisabletoachieveperformanceclosetoLloyd-MaxquantizerinthesenseofMeanSquaredError(MSE),atmuchlowercomputationalcostthanit;2)theproposedquantizerisabletoachievemuchbetterMSEperformancethanauniformquantizer,atacostsimilartoit. WeproposearecongurablearchitecturetoimplementouradaptivequantizerinanADC. 27

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Weuseittoquantizeimagesanddesignthetonemappingalgorithmforhighdynamicrange(HDR)imagecompression,rewardingimprovedvisualperformance. 3. Wedesigntheoptimalvectorquantizerapproximatorswithtransformsandscalarquantizers,especiallyfortwo-dimensionalrandomvectors. Itprovidesanelegantquantizationlatticeforarbitrarynumberofquantizationlevels,especiallyforprimenumbers. ItalmostalwayshavesmallerMSEthantheotherquantizers,andhassmalldesignandimplementationcomplexity. Itconsidersbothmemorylessandmemorysourcewitharbitrarydistributions,suchascirculardistributions,ellipticaldistributionsandmixeddistributions. Itisunderauniedframeworkoftri-axiscoordinateframe. 4. Weproposearobustimagehashingsystem. Thek-largestlocaltotalvariationsarerobusttocontentpreservingattackssuchasgeometricattacksandluminanceattacks,andindicatestablesimilarfeaturepointsinperceptualidenticalimages. TheMorletwaveletcoefcientsarepseudorandomlypermutedwithasecretekey,whichenhancesthesecurityandreducesthecollisionrateoftheimagehashingsystem. Morletwaveletcoefcientsarequantizedusingcompandingtechniqueaccordingtotheprobabilitydistributionofthecoefcients,whichenablestheimagehashingrobusttocontrastchangingandgammacorrectionofimages. Graycodeisusedtobinarilycodethequantizedcoefcients,whichincreasesdiscriminabilitybetweenimagehashes. 5. Weproposearobustimageauthenticationsystem. Featurepointsarerstgeneratedfromagivenimage,buttheirlocationsmaybechangedduetopossibleimageprocessinganddegradation.Accordingly,weproposetouseFuzzyC-meanclusteringalgorithmtoclusterthefeaturepointsandremovetheoutliersfromthefeaturepoints. Histogramweighteddistanceisproposed,whichisequivalenttoHausdorffdistanceafteroutlierremoval. Theauthenticityofthequeryimageisdeterminedbythemajorityvoteofwhetherthreetypesofdistancebetweenmatchedfeaturepointpairarelargerthantheirrespectivethresholds. 28

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Thegeometrictransformsthroughwhichthequeryimagesarealignedwiththeanchorimagesareestimated,andthequeryimagesareregistered. Thepossibletamperedimageblocksareidentied,andthepercentageofthetamperedareaisestimated. 6. Weproposearobusttrack-and-tracevideowatermarkingsystem.Thesystemprovides: Security:userandproductcopyrightinformation,e.g.astringoflengthLs,isrstencryptedwithAdvancedEncryptionStandard(AES);errorcorrectioncode(ECC)isappliedtothesequencetogenerateabinarysequencewitherror-correctionabilityoflengthL,calledwatermarkpayload;aframe-sizewatermarkpatternarisesfromapseudo-randomnoise(PN)sequence. Perceptualinvisibilityandrobustness:Tomakeatrade-offbetweenvisualqualityandrobustnesswhichisdeterminedbytheembeddingstrength,webuildaperceptualmodeltodeterminethesignalstrengththatcanbeembeddedtoeachpixelbyusingstatisticalsourceinformationandJust-Notice-Difference(JND)model. Track-and-trace:geometricanti-collusioncodingisusedfortrackingandtracingcolluders. AniterativeKLTschemeisusedforregistrationforwatermarkingextraction. 1.3OutlineoftheDissertationThestructureandinnerconnectionofthedissertationisshowninFigure 1-1 .Thersttheoreticalcompressionpartontransformandquantizationprovidestoolsforthethesecondsystembasedcopyrightprotectionpart.Intherstpart,transformandscalarquantizationarecombinedtogethertoapproximatetheoptimalvectorquantization.Inthesecondpart,imagehashingisincorporatedintoimageauthenticationsystem.Theoutlineofeachchapterispresentedasfollows.Chapter 2 studiestheintegerreversibletransformforlosslessimage/videocompressionbyusingPLUSfactorization.WeproposestabilizedPLUSfactorizationinSection 2.2 ,anddoperturbationanalysisinAppendix A ,whichprovesthenumericalstabilityofPLUSfactorization.Furthermore,weoptimizePLUSfactorizationtoachievetheleasttransformerrorbyusingTabuSearchalgorithminSection 2.3 .ThenwestudiesthelosslessTransformforlossy/losslessimagecompressioninSection 29

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2.4 .WeproposetheintegerDCTandtheintegerLappedTransformbyusingtheoptimizedPLUSfactorization,andtestthelossy/losslessimagecodingperformance.ExperimentalresultsshowthesuperiorityofouralgorithmsoversomeexistingintegerDCTalgorithms,andthelappedtransformfactorizationinJPEG-XRinSection 2.5 ,Finally,weconcludethischapterinSection 2.6 .Chapter 3 designsadaptivequantizationusingpiecewisecompandingandscalingforGaussianmixtures.Section 3.2 presentsthepreliminariesofoptimaladaptivequantizers.Section 3.3 describestheproposedadaptivequantizerforGMM.InSection 3.4 ,weproposearecongurablearchitecturetoimplementouradaptivequantizerinanADC.Insection 3.5 ,theproposedquantizerisappliedintohighdynamicrangeimagecompression.ExperimentalresultsareexhibitedinSection 3.6 .Section 3.7 concludesthechapter.Chapter 4 designstheoptimalvectorquantizerapproximatorswithtransformsandscalarquantizers.Section 4.2 presentsthepreliminariesofourproposedquantizer.Section 4.3 describesthesystemarchitectureoftransformplusscalarquantizationtoapproachtheoptimalvectorquantizer.ThepreprocessingwithtransformsisdiscussedinSection 4.4 todecorrelatesignals.InSection 4.5 ,wepresentthetri-axiscoordinateframe,andthemethodologytodesigntheoptimalscalarquantizerforbothcircularandellipticaldistributionsindetail.ExperimentalresultsareshowninSection 4.6 .Finally,Section 4.7 concludesthechapter.Chapter 5 proposesarobustimagehashingsystem.InSection 5.2 ,wepresentanoverviewoftheproposedimagehashingsystem.InSection 5.3 ,wedescribehowtoextracttherobustfeatureofimages,whichistheMorletwaveletcoefcientsatfeaturepointswiththek-largestlocaltotalvariations.ThentheMorletwaveletcoefcientsarequantizedandbinarilycodedwithGraycodeasshowninSection 5.4 .Section 5.5 showstheexperimentalresultsthatdemonstratetheeffectivenessandrobustnessoftheproposedimagehashingsystem.Finally,weconcludethischapterinSection 5.6 30

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Chapter 6 investigatesonarobustimageauthenticationsystem.Section 6.2 presentsanoverviewoftheproposedimageauthenticationsystem.Section 6.3 describeshowtodetectfeaturepointsinimages.InSection 6.4 ,weproposeanefcientandeffectivealgorithmtoremoveoutliersoffeaturepoints,andtheremainingfeaturepointsareorderedandmatchedintopairs.HistogramweighteddistanceisproposedandnormalizedEuclideandistanceandHausdorffdistanceareusedinSection 6.5 .Majorityvotingstrategyisusedtodeterminetheauthenticityofimages.InSection 6.6 ,possibleattacksareidentied,thequeryimagesareregistered,thetamperedimageblocksarelocated,andthepercentageoftamperedareaisestimated.ExperimentalresultsareshowninSection 6.7 .Finally,Section 6.8 concludesthechapter.Chapter 7 researchesontrack-and-tracevideowatermarking.Section 7.2 describestheoverallarchitectureoftheproposedtrack-and-tracevideowatermarkingsystem.ThewatermarkingembeddertechniquesarediscussedinSection 7.3 .Section 7.4 introduceswatermarkingdetectortechniques.TheexperimentalresultspresentedinSection 7.5 verifyrobustnessoftheproposedvideowatermarking.Finally,theconclusionandfutureworkaregiveninSection 7.6 Figure1-1. Thestructureofthedissertation. 31

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CHAPTER2INTEGERREVERSIBLETRANSFORMSFORLOSSLESSCOMPRESSIONOFIMAGESANDVIDEOS 2.1ResearchBackground 2.1.1TransformDesignTransformsarewidelyusedinsourcecoding,imageprocessingandcomputergraphics.Transformcodingisamajortechniqueinimageandvideocompressionstandard,suchasJPEG,JPEG2000,JPEG-XR[ 56 ],MPEG[ 81 ]andH.264[ 152 ].Besidesthedirectlydesignedintegertransforms[ 160 ],mostofintegertransformsarederivedfromthetraditionaltransforms.Factorizationisaneffectivetooltomakethetraditionaltransforms,suchasDiscreteCosineTransform(DCT)[ 7 ]andDiscreteWaveletTransform(DWT)[ 92 134 ],aswellasnewemergingripplettransform[ 156 ]faster,simplerandintegerreversible.Theselineartransforms,suchasFouriertransforms,discretecosinetransformsandwavelettransforms,alsohavewideapplicationsingeneralsignalprocessing,withtheabilityofenergycompactfrequencydecomposition.Forprevalentdigitalimagesandvideos,inputsignalsareavailableasintegerdatasequences,ormoregenerallyasxed-pointdatasequences.Insomespecialapplicationssuchasmilitary,medicalandremotesensingimaging[ 113 ],lossofinformationisnottoleratedduringprocessing.Therefore,integerreversibletransformsarehighlydesirable.However,theselineartransformscannotachieveperfectreversibilitydirectlyduetotheprecisionlimitationofcomputers.Intheliteratureoftransforms,theindirectlyimplementedtransformsusuallyrefertointegerdiscretewavelettransforms(IntDWT),integerdiscretecosinetransforms(IntDCT),integerdiscreteFouriertransforms(IntDFT)andsoforth.Generallyspeaking,theintegerreversibletransformsarethemarriageofcorrespondinglineartransformsandanappropriatereversibletransformframeworkwithroundingoperationsinanattempttoapproximatethecorrespondinglineartransforms.Sweldensetal.[ 44 136 137 ]rstproposedfactoringwaveletsintoliftingstepstorealizeinteger 32

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transformation.Withtheliftingscheme,anewgenerationofwaveletswasconstructed,calculatedin-placeandwithfurtherreducedcomputationalcomplexity.Chen[ 30 ]andLiangetal.[ 85 ]combinedtheliftingframeworkwithdiscretecosinetransformstoconstructfastintegerDCT,andOraintaraetal.[ 104 ]proposedfastintegerFouriertransforms(IntFFT)whicharealsobasedontheliftingscheme.Besides,anotherintegertransformframeworkoverlappingroundingtransform(ORT)wasdevelopedbyJungandProst[ 72 ],whichis,laterprovedbyAdams[ 6 ],equivalenttoaspecialcaseofliftingwithonlytrivialextensions.Forgenericlineartransforms,theelementaryreversiblematriceswithGaussianintegerunitsasthediagonalentrieswereproposedbyHaoandShi[ 64 ]fortriangularmatrixfactorizationtorealizeinteger-to-integertransforms.AlineartransformwithunitarydeterminantwasfurtherprovedtobeintegerreversibleifaPLUSfactorizationwasappliedtothetransformmatrix[ 63 ].Differently,Plonka[ 110 ]choseexpansionfactorsfortransformstoexpandtherangesofthetransformslargerthantheirinputdomains,andthistransformdomainredundancywasutilizedforreversibility.Modulotransforms(MT),analternativetolifting,wasproposedbySrinivasan[ 131 ]recently,whichemployedcertainPythagoreantriplesthatcouldbecriticallyquantizedtoproduceareversible,normalized,scale-freetransform,andtheladderstructurewasemployedtoapproximateorthonormaltransformsby2-pointrotations.Forthelineartransforms,thebitwidthofthelow-frequentcoefcientsisgenerallygreaterthanthatoftheoriginaldata,sothedynamicrangesofthetransformsareexpanded[ 78 ].Forexample,thediscretecosinetransformmatrixoftypeIIofordernis: CIIn:=r 2 nen(j)cosj(2k+1)p 2n(2)whereen(0)=p 2=2anden(j)=1forj,k21,,n)]TJ /F8 11.955 Tf 10.95 0 Td[(1.ItiseasytoknowkCIInk=p n.Itindicatesthatm-bitinputsresultinm+log2p nbitoutputs,i.e.16-bitmemoryspaceisgenerallyneededinprogramstostorea9-bitoutputforeach8-bitinputafter4-point 33

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DCTtransformationwithgeneralpersonalcomputers.Evenmore,thecorrespondingintegerreversibletransforms,IntDCT,usuallyneedgreaterexpandedrangestokeepreversibility.However,ifweencounteracomputationalenvironmentwithonlylimitedbuffer,xed-pointarithmeticunitsandxed-widthchannels,orwithxedrepresentingwordlength,thetraditionalintegertransformswillfail,andthedynamic-range-preservingtransformsaredesired.Theyareappealingforsavingmemoryandcomputationalresourcesinbothcompressionanddecompression,andattractiveforspeedingupthecomputationalprocessofcoding,sinceonlyaretheprocessingunitsforxed-bitwidthoperandsneeded.Thesolutiontointegerreversibleanddynamic-range-preservingtransformationisachallengingresearchtopic,sincetheconstantdynamicrange,thecompactcoefcientrepresentationandreversibilityareconictinggoals.Notmuchhasbeendonetondthebestofallpossiblesolutions,exceptthreemethods.Chaoetal.[ 26 ]utilizedthecomplementarycodeandmodulararithmeticautomaticallytopreservethedynamicrangesofintegerwavelettransforms.However,theapplicationofthisapproachislimitedtolosslesscompression,inwhichthelargepositivetransformcoefcientsbecomenegativeandviceversaduetomodulararithmetic.Thus,therecoveredimagessuffersevereblockingartifactsandsalt-and-peppernoisewhentransformcoefcientsarelossilycompressed.TheothertwoapproachestoavoidthedynamicrangeexpansionisTable-LookupHaar-liketransform(TLHaar)andPiecewiseLinearHaar-liketransform(PLHaar)proposedbySenecaletal.[ 123 124 ].TLHaarandPLHaararetwo-pointtransforms,bothevolvingfromtheSTransform,theintegerrealizationofHaarwavelettransform.TLHaarneedsadynamiclook-uptablebuiltbyoptimizedpermutationandPLHaarisaspecial2-Dcaseofourproposedinnity-normrotation.In[ 160 ],weconstructgeneralinteger-reversibleanddynamic-range-preservinginnity-normrotationtransformsbyanalogywithrotationsinthe2-normspace(orthogonaltransforms,e.g.,DCT).AlthoughPLHaarisaspecialcase 34

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ofourproposedinnity-normrotationtransforms,itisnotstraightforwardtoobtainourproposedtransformsfromPLHaar. 2.1.2IntroductionofPLUSFactorizationPLUSfactorizationisakindofcustomizabletriangularmatrixfactorization,proposedbyHao[ 63 ]asanewframeworkofmatrixfactorization,andencompassesandgeneralizesquiteafewtriangularmatrixfactorizations[ 64 133 141 ].ThePLUSfactorizationofageneralnonsingularmatrixAisformulatedas: A=PLUS(2)wherematricesP,LandUare,almostthesameasinLUfactorization,permutation,unitloweranduppertriangularmatrices,respectively,whileSisaveryspecialmatrix,whichisunit,lowerandtriangular,andonlywithnomorethanN)]TJ /F8 11.955 Tf 11.71 0 Td[(1nonzeros.DifferentfromLUfactorization,allthediagonalelementsofUinPLUSfactorizationarecustomizable,i.e.thediagonalelementscanbeassignedalmostfreely,aslongasthedeterminantisequaltothatofAuptoapossiblesignadjustment.WithPLUSfactorization,anonsingularmatrixAiseasilyfactorizedfurtherintoaseriesofspecialmatricessimilartoS.Furthermore,thepermutationmatrixcanalsobesubstitutedwithapseudo-permutationmatrix,whichisasimpleunituppertriangularmatrixwith0,1and)]TJ /F8 11.955 Tf 9.29 0 Td[(1asitsoff-diagonalelements.BesidesPLUS,acustomizablefactorizationalsohasotheralternatives,LUSP,PSULorSULPwithlowerS,andPULS,ULSP,PSLUorSLUPwithupperS,whicharealltakenasvarietiesofPLUSfactorization.Currently,theliftingfactorization[ 44 ]ismostlyusedtofactorizethetransformsintoliftingstepstosimplifythetransformsaswellastomakethetransformintegerreversible,suchasthefactorizationinJPEG-XR[ 51 ].Butthesefactorizationsaremostlybasedonexperiencesandexperiments.However,thePLUSfactorizationprovidesageneralanduniversalwaytofactorizetransformmatricesofanyorderintoproducts 35

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ofElementaryReversibleMatrices[ 64 ].WiththeElementaryReversibleMatricesasfactormatrices,PLUSfactorizationisapowerfultoolforrealizingtheintegerreversibletransform[ 5 ],whenassistedbytheladderstructureandtheroundingoperations[ 64 ].Therefore,ithaspromisingapplicationsinlossless/lossycoding[ 64 67 128 ]andreversibleimageprocessing[ 38 159 ].Meanwhile,anelementaryreversiblematrixisalsoatriangularshearmatrix.Thus,italsohasfoundapplicationsincomputergraphics,suchastransformationaccelerationbyshears[ 28 ],fastimageregistration[ 29 ],inwhichmatrix-basedtransformationsdominate.However,theexistingPLUSfactorizationsuffersfromtwolimitations:instabilityandsub-optimality.Byinstability,wemeanthatthePLUSfactorizationmaystopbecauseofzeroornearzeropivotingduringtheprocessofGaussianElimination.Bysub-optimality,wemeanthatthePLUSfactorizationoftransformmatricesfoundbyexistingfactorizationalgorithmmayleadtolargetransformerror,whichmaydeprivethegoodpropertiesoforiginaltransformmatrices,suchasorthogonalityandhighenergy-compactingability,fromtheproductsofthefactormatricesofPLUSfactorization.Thischapterisproposedtoaddresstheseproblems(otherproblemsofPLUSfactorization,likeblockingfactorizationandparallellingcomputing,arisingfromsolvinglargelinearsystems,couldbefoundin[ 125 127 ]).Ourmaincontributionsinclude: 1. WeproposethreestablePLUSalgorithmsinMatlabpseudocode,andthemethodologytostabilizethefactorization. 2. WeprovethestabletheoremanddidperturbationanalysisofPLUSfactorization,toguaranteethestabilityofouralgorithmstheoretically. 3. Weobtainaclosed-formformulaoftransformerrorofPLUSfactorization,andproposetheoptimizationalgorithmbasedonTabuSearchtoquicklyndtheoptimalfactorizationandtorealizeintegertransformwiththeleasttransformerror. 4. WeapplyouralgorithmstorealizeintegerDCTandintegerLappedBiorthogonalTransform,andtestthelossy/losslessimagecodingperformance.ExperimentalresultsshowthesuperiorityofouralgorithmsoversomeexistingintegerDCTalgorithms,andthelappedtransformfactorizationinJPEG-XR. 36

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Stabilization,optimizationandperturbationanalysisofPLUSfactorizationaredocumentedin[ 161 ].WemainlyfocusedonthenewestworkofPLUSfactorizationoptimizationanditsapplicationsinimageandvideocompression. 2.2StablePLUSFactorizationAlgorithmsWeexhibitaPLUSfactorizationalgorithmwithpartialpivotinginAlgorithm 2.2 ,withthetwopivotsofthelargestmagnitudesinA(i:n,n)andA(i:n,i). Algorithm1PLUSfactorizationwithpartialpivoting Foranonsingularn-by-nmatrixA,therstn)]TJ /F8 11.955 Tf 12.09 0 Td[(1diagonalentriesofUaregiveninvectoru,thenthePLUSfactorizationalgorithmwithpartialpivotingisgivenasfollows: P=1:n fori=1:(n)]TJ /F8 11.955 Tf 10.95 0 Td[(1)do Determinem1withim1n,sothatjA(m1,n)j=kA(i:n,n)k P(i)$P(m1) A(i,1:n)$A(m1,1:n) s(i)=(A(i,i))]TJ /F3 11.955 Tf 10.95 0 Td[(u(i))=A(i,n) A(1:n,i)=A(1:n,i))]TJ /F3 11.955 Tf 10.95 0 Td[(s(i)A(1:n,n) Determinem2withim2n,sothatjA(m2,i)j=kA(i:n,i)k P(i)$P(m2) A(i,1:n)$A(m2,1:n) k=(i+1):n A(k,i)=A(k,i)=A(i,i) A(k,k)=A(k,k))]TJ /F3 11.955 Tf 10.95 0 Td[(kron(A(k,i),A(i,k)) endfor L=I+strict lower tri(A) U=upper tri(A) S=I+[zeros(n)]TJ /F8 11.955 Tf 10.95 0 Td[(1,1);1][s,0]2 PLUSfactorizationwithcompletepivotingisequivalenttoapplyingbothleftandrightpermutationmatricestoAineachiteration.Notonlyisthefollowingalgorithmstablewithout0divisors,butalsoavoidssubtractionbetweentwoveryclosenumberswhencalculatings(pass)inline9.ThecolumnpivotsmustavoidtheelementsinthelastcolumntoguaranteethatSiPRi+1=PRi+1Si.Therefore,PLAPR1S1PR2S2PRn)]TJ /F28 6.974 Tf 5.42 0 Td[(1Sn)]TJ /F5 8.966 Tf 6.97 0 Td[(1=APLAPRS,wherePR=PR1PR2PRn)]TJ /F28 6.974 Tf 5.42 0 Td[(1andS=S1S2Sn)]TJ /F5 8.966 Tf 6.97 0 Td[(1,wherePLisleftpermutationmatrix,PRisrightpermutationmatrixandPRiistherightpermutationmatrixinthei-thiteration. 37

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Algorithm2PLUSfactorizationwithcompletepivoting. PL=1:n PR=1:n fori=1:(n)]TJ /F8 11.955 Tf 10.95 0 Td[(1)do Determinemwithimn,sothatjA(m,n)j=kA(i:n,n)k Determinelwithiln)]TJ /F8 11.955 Tf 11.23 0 Td[(1,sothatjA(m,l)j=maxfjA(m,m))]TJ /F3 11.955 Tf 11.23 0 Td[(d(i)j,m=i:(n)]TJ /F8 11.955 Tf -446.76 -14.45 Td[(1)g PL(i)$PL(m) PR(i)$PR(l) A(i,1:n)$A(m,1:n) A(1:n,i)$A(1:n,l) ifjA(i,n)j>ethen s(i)=(A(i,i))]TJ /F3 11.955 Tf 10.95 0 Td[(u(i))=A(i,n) A(1:n,i)=A(1:n,i))]TJ /F3 11.955 Tf 10.95 0 Td[(s(i)A(1:n,n) Determinenwithinn,sothatjA(n,i)j=kA(i:n,i)k PL(i)$PL(n) A(i,1:n)$A(n,1:n) k=(i+1):n A(k,i)=A(k,i)=A(i,i) A(k,k)=A(k,k))]TJ /F3 11.955 Tf 10.95 0 Td[(kron(A(k,i),A(i,k)) endif endfor L=I+strict lower tri(A) U=upper tri(A) S=I+[zeros(n)]TJ /F8 11.955 Tf 10.95 0 Td[(1,1);1][s,0]2 2.3PLUSFactorizationOptimizationForthebetterperformanceinapplications,optimizationofPLUSfactorizationaimstominimizethreetypesoftransformerror,whichhasthreemainorigins.Therstoneisduetotheprecisionlimitationofcomputers.Thesecondoneresultsfromtheroundingoperationsforintegerreversibletransformations.Thethirdonecomesfromthepossiblequantizationofcompression.Theerrorwillbefurtherpropagatedandampliedaftermultiplicationsbyfactormatrices.ThetransformerrorcausesthedifferencesbetweenthecoefcientsaftertransformwithfactormatricesofPLUSfactorizationandthoseaftertheoriginaltransform.ItiswellknownthatthetraditionallineartransformssuchasDCTandDWTholdmanygoodpropertieslikeorthogonality,highde-correlationandenergy-concentrationabilityforeffectiveimagecoding.Therefore,tokeepthe 38

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samemeritsastheoriginaltransformmatricesisanimportantconcern,whenusingPLUSfactorizationastheeffectivetoolforrealizingintegerreversiblereversionofthetraditionaltransforms.Theleasttransformerrorisdesired.ThisoptimizationproblemofPLUSfactorizationisdenotedasPLUSFOPinthefollowingdiscussion.PLUSfactorizationisreallydiversied,evenwithagivenpatternofSand1asdiagonalentriesofU.Forann-by-nmatrixA,A=PLLUSPR,thereareupton!possibleleftpermutationmatricesPL,n!possiblerightpermutationmatricesPRand2n)]TJ /F5 8.966 Tf 6.97 0 Td[(1possiblecombinationsofrstn)]TJ /F8 11.955 Tf 11.23 0 Td[(1diagonalentriesofU.Itmeansthattherearen!n!2n)]TJ /F5 8.966 Tf 6.97 0 Td[(1possiblePLUSfactorizations.ItisanN-PhardproblemtondtheoptimalPLUSfactorization.Thus,enumeratingallthepossiblesolutionstondthebestisoutofthequestionforthehigh-ordermatrices,butitsresultsforlow-ordermatricescanbeusedasagroundtruthforcomparisonbetweenalgorithms.TosolvePLUSFOP,wepresentourerroranalysisofPLUSfactorization,proposeE2errormetric,anddesigntheoptimizationalgorithmwithTabuSearchtondthefactorizationswiththeleasttransformerror.TheoptimalPLUSfactorizationwiththeleasttransformerrorcanhelpimprovetheperformanceofvarioussystems,e.g.,lossless/lossyimagecoding. 2.3.1TransformErrorAnalysisThetransformerrorafterthetransformationstepswiththefactormatricesofPLUSisactuallyamixtureofdirectround-offerrorandindirecterrorpropagatedandaccumulatedfromtheround-offerrorintheprevioussteps.ForA=PLLUSPR,thetransformerrorcanbeformulatedas: e=PL(eL+L(eU+U(eS+SPRe0)))(2)whereeL,eUandeSareround-offerrorvectorsdirectlybroughtinafterthetransformationwithL,UandS,respectively,ande0isthesystemerrorvectorbeforetransformation.ComparedwiththetransformerrorusingtheoriginalmatrixA,e=eA+Ae0,e0canbe 39

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disregarded,andthefollowingerrormodelisconsideredforPLUSfactorization: e=PL(eL+L(eU+UeS))(2)whereeL=[0,1,1,,1]T,eU=[1,1,,1,0]TandeS=[0,0,,0,1]Tastheupperboundofthemagnitudesoferrorvectors,whentheoor()operatorisused.ecanbeevaluatedbyitsnormasfollows: kek=kPL(eL+L(eU+UeS))k=keL+LeU+LUeSkkeLk+kLeUk+kLUeSk(2)Inourrandomnumericalexperiments,suchuppererrorboundcansometimesbereached.Therefore,intheintegertransformdomain,anerrormetricofPLUSfactorizationcanbedenedbytheerrorboundinEquation( 2 ): Ep(LUS)=keLkp+kLeUkp+kLUeSkp(2)wherekkpisthep-normoperator. 2.3.2StatementofOptimizationProblemTaketheerrormetricdenedinEquation( 2 )astheobjectivefunction,theoptimizationofPLUSfactorizationforanynonsingularmatrixAisformulatedas: minPL,PR,uEp(LUS)s.t.A=PLLUSPR(2)whereuisthevectorcomposedbytherstn)]TJ /F8 11.955 Tf 10.95 0 Td[(1diagonalelementsofU.ForthevectornorminEquation( 2 ),wetestL1,L2andLinourexperiments,anduseE1,E2andEtorepresentthecorrespondingerrormetric,respectively.OurexperimentalresultsinSection 2.5 showthatL2,whichisacontinuousfunctionof 40

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vectors,consistswiththedenitionofMeanSquareErrortoachieveLeastMeanSquareError.Thus,thetransformerrormetricisdenedasE2.TheglobaloptimalfactorizationresultsfortheDCTmatricesfoundbyexhaustivesearcharegiveninTable 2-3 ,whenmatrixordern=2,4,8.Itiseasytoobtaintheoptimawhenn=2or4.Butwhenn=8,aPCwith0.7GCPUand128MMemorytakesnearly3weekstotryallthepossibilities,nottomensionthematricesofhigherorders.Thefactsfoundedbyexhaustivesearchare:(i)Duetosymmetryofsolutionspace,theoptimalsolutionsarenotunique:4optimalresultsfor22DCT,4for44DCT,and32for88DCTarefoundinourexperiments.(ii)Therearealotoffactorizationresults,whichapproximatetheoptimal,scatteringinthefeasiblesolutionspace.Inourexperiments,thereare8suboptimalfactorizationsneartheoptimalonefor44DCT,with0.08moretransformerrorthantheleasttransformerroroftheoptimalfactorization.Therefore,attemptstosolvethisNP-hardnonlinearcombinatorialoptimizationproblemcanbeachievedbyheuristicmethods,whichyieldtheapproximatelyoptimalsolutionsinapolynomialtime.SuchmethodsincludeNeuralNetwork(NN),SimulatedAnnealing(SA),GeneticAlgorithm(GA),TabuSearch(TS)andsoon[ 149 ].Inthischapter,weuseTabuSearchtosolvethePLUSFOP,sinceitisacombinatorialoptimizationproblem,andcanbewellmodelledintheTSframeworkandwellsolvedasshownintheexperiments. 2.3.3OptimizationAlgorithmwithTabuSearchTabusearch(TS)isameta-heuristictechniqueproposedbyGlover[ 59 ]tosolvecombinatorialoptimizationproblems,suchasvehicleroutingandshopschedulingproblems[ 149 ].TSavoidsbeingtrappedatlocalminimabyallowingthetemporalacceptanceoftheworsesolutions,andavoidscyclicallyrevisitingsolutionsbykeepingtrackoftherecentmigrationsofthesolutionsintabulist.ItconsistentlyoutperformsthealgorithminSection 2.2 andprovidesamazingoptimizationperformance.BasedontheessentialproceduresoftheTSalgorithmandthecharacteristicsofpossible 41

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solutionsmentionedinSection 2.3.2 ,weseethatPLUSFOPisatypicalproblemintheTSframeworkandTSworkswellforourPLUSFOP,whichisalsoveriedbytheexperiments.SomekeytermsintheTSalgorithmare: 1. ObjectivefunctionTheobjectivefunctionEisdenedinEquation( 2 )withp=2. 2. PossiblesolutionsetApossiblesolutioncanberepresentedasatriplet(PL,PR,u).Thepossiblesolutionsetis=f(PL,PR,u)g,andjj=n!n!2n)]TJ /F5 8.966 Tf 6.97 0 Td[(1. 3. Neighbors8X2,X=(PL,PR,u),theneighborsofXisdenedas:B(X)=f(P0L,P0R,u0)jd((PL,PR,u),(P0L,P0R,u0))=1g,whered((PL,PR,u),(P0L,P0R,u0))=d0(PL,P0L)+d0(PR,P0R)+d00(u,u0),d0(P,P0)=(i,jdP(i,j),P0(i,j))=2,d00(u,u0)=idu(i),u0(i),anddi,j=0wheni=j,di,j=1wheni6=j. 4. Candidatelist8X2,C(X)B(X),iscomposedofthetopkbettercandidatesamongtheneighborsofthecurrentpointwithlessE.ThesizeofthecandidatelistjC(X)j=kisatunableparameter. 5. Tabutenets,tabulistandtabutenureTabutenetsareimplementedusingtabulistandtabutenure.Tabulistrecordstherecentmigrationsofcurrentpointandforbidsrevisitingthesepointsintabutenure. 6. AspirationcriteriaCriterion1.IfE(Xnext)
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Algorithm3TheoptimizationofPLUSfactorization. Initialization:RandomlyChooseaninitialpointXin. SetXmin=X,Emin=E(X)andi=0. whilenoterminationcriteriaaremeetdo i=i+1 FindB(X)and8X02B(X)calculateE(X0). ConstructC(X)andndXnext. ifXnextsatisesaspirationcriteriathen X=Xnext. ifE(X)
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Forlosslesscompression,weevaluatetheperformanceoftheintegertransformsintermsofentropyH(C)ofcoefcients,whichissimilartocodingdatarateinunitofbitsperpixel(bpp).(Notethatthedistortioncausedbylosslesscodingiszero.)H(C)istheaverageentropyoftransformcoefcientsofallsubbands,givenasbelow: H(C)=1 ssi=1H(Ci)(2)whereH(Ci)istheentropyofthetransformcoefcientsofthei-thsubband,andsisthetotalnumberofthesubbands.Forn-point2dimensionalDCT,s=nn.Hereweuseentropyoftransformcoefcientsinsteadofimplementinganentropycodingscheme(e.g.,Huffmancodeorarithmeticcode)tocodetheresultingtransformcoefcientsfortworeasons.First,thischapterfocusesonmatrixfactorizationforintegertransformimplementation;hence,weshouldcomparethedecorrelationperformanceoftheresultingintegertransforms,whichisusuallycharacterizedbytheaverageentropydenedinEq.( 2 ).Second,entropycodingforspecicintegertransformsisoutofthescopeofthischapter;wewillleavethisforfuturestudy.Forlossycompression,tocomparetheperformanceoftheintegertransforms,weusebppvs.PeakSignal-to-NoiseRatio(PSNR),andStructuralSIMilarity(SSIM)indexandthequalitymapproposedbyWang[ 150 ].Inthelossyencoderthatweimplement,aninputimageisrsttransformedbyanintegertransform;thentheintegertransformcoefcientsareuniformlyquantized;thequantizedcoefcientsarecodedwithxedlengthcodinginsteadofvariablelengthcoding(orentropycoding),sinceentropycodingisoutofthescopeofthischapter.Finally,thedecoderreconstructstheimage. 44

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2.5ExperimentalResults 2.5.1ExamplesoftheStablePLUSFactorizationAlgorithmsExample:A=0BBBBBBB@43203432234312341CCCCCCCATheoriginalgeneralPLUSfactorizationalgorithmstopsintherstiterationduetoa14=0.Withpartialpivoting,theresultofAlgorithm 2.2 is:A=0BBBBBBB@01000010000110001CCCCCCCA0BBBBBBB@100041003)]TJ /F8 11.955 Tf 9.28 0 Td[(0.5102)]TJ /F8 11.955 Tf 9.29 0 Td[(0.251.511CCCCCCCA0BBBBBBB@110.3340)]TJ /F8 11.955 Tf 9.29 0 Td[(10.67)]TJ /F8 11.955 Tf 9.29 0 Td[(16001)]TJ /F8 11.955 Tf 9.29 0 Td[(18000181CCCCCCCA0BBBBBBB@10000100001000.250.6711CCCCCCCAWithcompletepivoting,theresultofAlgorithm 2.2 is:A=0BBBBBBB@01000001001010001CCCCCCCA0BBBBBBB@100021002.53.131021.251.2211CCCCCCCA0BBBBBBB@12.51.9740)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.28 0 Td[(0.94)]TJ /F8 11.955 Tf 9.29 0 Td[(800118000)]TJ /F8 11.955 Tf 9.29 0 Td[(181CCCCCCCA 45

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0BBBBBBB@1000010000100.5)]TJ /F8 11.955 Tf 9.29 0 Td[(0.380.00711CCCCCCCA0BBBBBBB@00011000010000011CCCCCCCARemark:Thedifferencesbetweenthetwoabovefactorizationsarethatthelattergiveslowerintegertransformerror,sincecompletepivotingresultsinelementswithsmallermagnitudesinfactormatrices.E2fortherstfactorizationis17.03,andE2forthesecondfactorizationis15.68. 2.5.2ExperimentsforPLUSFactorizationOptimizationDCTmatricesareusedintheexperimentstoexemplifytheeffectivenessofPLUSfactorizationoptimization,duetothepopularityofDCTinimageandvideocoding.ForeachoptimalPLUSfactorization,10000randomlygeneratedmatrices[ 103 ].Therangeofelementsinrandomlygeneratedmatricesis[0,255],whichsimulateblocksingrayimages.TheyaretestedfortheaveragetransformerrorwithOverallMeanSquareError(OMSE)andOverallMeanError(OME): OMSE=ni=1nj=110000k=1e2k(i,j) nn10000(2) OME=ni=1nj=110000k=1jek(i,j)j nn10000(2)whereek(i,j)=xk(i,j))]TJ /F3 11.955 Tf 11.15 0 Td[(xk(i,j),xk(i,j)arethecoefcientsoftherandomlygeneratedmatricesafterDCTtransform,andxk(i,j)arethecoefcientsafterintegertransformwiththePLUSfactormatricesoftheDCTmatrices.TheresultsinTable 2-1 andTable 2-2 revealthat: OMSEandOMEoftheglobaloptimalPLUSfactorizationsfoundwitherrormetricsdenedusingL2arelessthanthosedenedusingL1orL. ForoptimalPLUSfactorizationswithdifferentn,smallE(LUS)isrelatedtosmallOMSEandOME,andE(LUS)increasesslowlywiththeordersofmatrices. 46

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FordifferentPLUSfactorizationswiththesamen,smallE2(LUS)isrelatedtosmallOMSEandOME.BasedontheresultsinTable 2-4 ,Figure 2-2 andFigure 2-3 ,someremarksaregivenasfollows: Forn=4,whenthesolutionspaceofoptimizationisrelativelysmall,ouralgorithmcanalwaysndtheoptima. Forn=8,whenthesolutionspaceofoptimizationisexpanded,ouralgorithmcanndtheglobaloptimaandothersub-optimalsolutions.Thetransformerrorsofsub-optimalsolutionsareveryclosetothatoftheglobaloptima. Forn=16,whenthesolutionspaceofoptimizationisverylarge,ouralgorithmcanndthesub-optimalsolutionswithlittleuctuation.Thesub-optimalsolutionsfoundwithourfastTSmethodareallmuchbetterthanrandomlyfoundonesandthosefoundbyanyPLUSfactorizationalgorithm. TheconvergencespeedofouroptimizationalgorithmusingTabuSearchisveryfast.Incontrasttoexhaustivesearchforweeks,eachiterationforfactorizationofthe88DCTmatrixonlycosts0.2msonaPCwith0.7GCPUand128Mmemory,andthesub-optimalsolutionscanbefoundinonlyafewiterations. Whennissmall,thePLUSfactorizationsobtainedfromalgorithmsinSection2arewithlittlelargertransformerror.Thus,thesealgorithmsarepracticalintheapplicationswhennissmall. 2.5.3ExperimentsonApplicationsinImageCoding 2.5.3.1IntegerDCTwithoptimalPLUSfactorizationWeapplytheoptimalPLUSfactorizationofDCTfoundbyourproposedalgorithmtolosslessimagecoding.Thetransformmatricesare2-point,3-pointand4-pointDCTs.TheirintegerreversibleimplementationbyoptimalPLUSfactorizationaredenotedas`Opt2',`Opt3'and`Opt4'respectively.Theirintegerreversibleimplementationbyexpansionfactors[ 110 ]aredenotedas`IntDCT2',`IntDCT3'and`IntDCT4'.Table 2-5 showstheentropyoftransformcoefcientsobtainedbyourproposedoptimalPLUSfactorizationschemesandintegerDCTwithexpansionfactors.TheentropyobtainedbyouralgorithmislessthanthatobtainedbyintegerDCTwithexpansionfactorsforalltest 47

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images.ItindicatesthattheintegerDCTwithoptimalPLUSfactorizationhasstrongerabilitytoreduceredundancyinimagesthanintegerDCTwithexpansionfactors.ThePLUSfactorization`Opt2'for2-pointDCTCII2is: CII2=0B@01101CA0B@100.414211CA0B@1)]TJ /F8 11.955 Tf 9.29 0 Td[(0.7071011CA0B@100.414211CAThePLUSfactorization`Opt3'for3-pointDCTCII3is: CII3=0BBBB@0101000011CCCCA0BBBB@1000.3382100.2391)]TJ /F8 11.955 Tf 9.29 0 Td[(0.517611CCCCA0BBBB@1)]TJ /F8 11.955 Tf 9.29 0 Td[(0.3660)]TJ /F8 11.955 Tf 9.28 0 Td[(0.7071010.81650011CCCCA0BBBB@1000100.4142)]TJ /F8 11.955 Tf 9.29 0 Td[(0.517611CCCCAThePLUSfactorization`Opt4'for4-pointDCTCII4is: CII4=0BBBBBBB@00010010100001001CCCCCCCA0BBBBBBB@10000.3827100)]TJ /F8 11.955 Tf 9.29 0 Td[(0.9239)]TJ /F8 11.955 Tf 9.29 0 Td[(0.66821000.33180.693411CCCCCCCA0BBBBBBB@1)]TJ /F8 11.955 Tf 9.28 0 Td[(0.33180.3318)]TJ /F8 11.955 Tf 9.28 0 Td[(0.501)]TJ /F8 11.955 Tf 9.29 0 Td[(0.0761)]TJ /F8 11.955 Tf 9.29 0 Td[(0.4619001)]TJ /F8 11.955 Tf 9.28 0 Td[(0.500011CCCCCCCA0BBBBBBB@10000100001010.3364)]TJ /F8 11.955 Tf 9.29 0 Td[(0.336411CCCCCCCA0BBBBBBB@00100001010010001CCCCCCCA 2.5.3.2IntegerlappedbiothogonaltransformwithoptimalPLUSfactorizationWealsoapplyoptimalPLUSfactorizationtomakeLappedTransform[ 93 95 ]integerreversible.ThePhotoCoreTransform(PCT)andPhotoOverlapTransform 48

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(POT)aredenedin[ 145 ].WeobtaintheoptimalPLUSfactorizationfor4-pointPOTand4-pointDCT,andapplyittolossy/losslessimagecoding,whichisdenotedas`PLUS 1'.WealsoapplyoptimalPLUSfactorizationto4-pointPOTand4-pointPCTwhichisanapproximationofDCT.Thiscodingschemeisdenotedas`PLUS 2'.TheoptimalPLUSfactorizationofPOTis: 0BBBBBBB@)]TJ /F8 11.955 Tf 9.29 0 Td[(0.14480.2313)]TJ /F8 11.955 Tf 9.29 0 Td[(0.23130.97200.23130.9720)]TJ /F8 11.955 Tf 9.29 0 Td[(0.1448)]TJ /F8 11.955 Tf 9.29 0 Td[(0.2313)]TJ /F8 11.955 Tf 9.29 0 Td[(0.2313)]TJ /F8 11.955 Tf 9.29 0 Td[(0.14480.97200.23130.9720)]TJ /F8 11.955 Tf 9.29 0 Td[(0.23130.2313)]TJ /F8 11.955 Tf 9.29 0 Td[(0.14481CCCCCCCA=0BBBBBBB@00010100001010001CCCCCCCA0BBBBBBB@10000.276100)]TJ /F8 11.955 Tf 9.29 0 Td[(0.276)]TJ /F8 11.955 Tf 9.29 0 Td[(0.17310)]TJ /F8 11.955 Tf 9.29 0 Td[(0.3330.328)]TJ /F8 11.955 Tf 9.29 0 Td[(0.08511CCCCCCCA0BBBBBBB@1)]TJ /F8 11.955 Tf 9.28 0 Td[(0.25850.2311)]TJ /F8 11.955 Tf 9.29 0 Td[(0.144801)]TJ /F8 11.955 Tf 9.29 0 Td[(0.2089)]TJ /F8 11.955 Tf 9.29 0 Td[(0.19130010.158300011CCCCCCCA0BBBBBBB@10000100001010.3364)]TJ /F8 11.955 Tf 9.29 0 Td[(0.336411CCCCCCCATheoptimalPLUSfactorizationofDCTis: 0BBBBBBB@0.50.50.50.50.65330.2706)]TJ /F8 11.955 Tf 9.29 0 Td[(0.2706)]TJ /F8 11.955 Tf 9.29 0 Td[(0.65330.5)]TJ /F8 11.955 Tf 9.29 0 Td[(0.5)]TJ /F8 11.955 Tf 9.29 0 Td[(0.50.50.2706)]TJ /F8 11.955 Tf 9.29 0 Td[(0.65330.6533)]TJ /F8 11.955 Tf 9.29 0 Td[(0.27061CCCCCCCA=0BBBBBBB@01001000000100101CCCCCCCA0BBBBBBB@10000.23461000.4142)]TJ /F8 11.955 Tf 9.29 0 Td[(0.7654100.23460)]TJ /F8 11.955 Tf 9.29 0 Td[(0.693411CCCCCCCA0BBBBBBB@1)]TJ /F8 11.955 Tf 9.29 0 Td[(0.2929)]TJ /F8 11.955 Tf 9.29 0 Td[(0.0137)]TJ /F8 11.955 Tf 9.29 0 Td[(0.6533010.30660.65330010.500011CCCCCCCA0BBBBBBB@1000010000100.5307)]TJ /F8 11.955 Tf 9.29 0 Td[(0.86260.393311CCCCCCCA 49

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TheoptimalPLUSfactorizationoftheapproximationofDCTis: 0BBBBBBB@0.50.50.50.50.707100)]TJ /F8 11.955 Tf 9.29 0 Td[(0.70710.5)]TJ /F8 11.955 Tf 9.29 0 Td[(0.5)]TJ /F8 11.955 Tf 9.29 0 Td[(0.50.500.7071)]TJ /F8 11.955 Tf 9.29 0 Td[(0.707101CCCCCCCA=0BBBBBBB@01001000001000011CCCCCCCA0BBBBBBB@10000.29291000.292901000.7071)]TJ /F8 11.955 Tf 9.29 0 Td[(2.121311CCCCCCCA0BBBBBBB@1)]TJ /F8 11.955 Tf 9.29 0 Td[(0.5)]TJ /F8 11.955 Tf 9.29 0 Td[(1.5)]TJ /F8 11.955 Tf 9.29 0 Td[(0.70710120.70710010.707100011CCCCCCCA0BBBBBBB@1000010000100.4142)]TJ /F8 11.955 Tf 9.28 0 Td[(0.7071)]TJ /F8 11.955 Tf 9.29 0 Td[(2.121311CCCCCCCATheliftingfactorizationschemeinJPEG-XR[ 51 ]isdenotedas`JPEG-XR'.Theentropyperformance,aswellaslossyperformanceof`PLUS 1'and`PLUS 2'isbetterthanthatof`JPEG-XR',asshownintheTable 2-6 ,Table 2-7 ,Figure 2-4 ,Figure 2-5 ,Figure 2-6 andFigure 2-7 .InTable 2-6 ,theentropyobtainedby`PLUS 1'issmallerthanthatobtainedby`PLUS 2'and`JPEG-XR'foralltestimages.InTable 2-7 ,thePSNRandSSIMindexareshownforalltestimagesatthebitratesof4,2,1,0.5,0.25,0.125bpp.TheblockforSSIMcalulationis4.`PLUS 1'hasbetterperformance,i.e.,largerPSNRandSSIM,than`PLUS 2'and`JPEG-XR'.Forexample,forimageLena,atthebitrateof0.25bpp,`PLUS 1'achieves2.5dBgaininPSNRthan`JPEG-XR'.Inaddition,theperformanceof`PLUS 1'isbetterthanthatof`PLUS 2',whichmeansthattheDCTimplementedin`PLUS 1'hashigherdecorrelationabilitythanitsapproximationin`PLUS 2'.Wealsocomparethesubjectiveperformanceof`JPEG-XR',`PLUS 1'and`PLUS 2'intermsofthevisualqualityofthereconstructedimages`Barbara'inFigure 2-5 ,`Lena'inFigure 2-6 and`Baboon'inFigure 2-7 .The`Barbara',`Lena',`Baboon'reconstructedby`PLUS 1'hasthebestvisualquality,andthenthe`PLUS 2',andtheworstis`JPEG-XR'.Thequalitymapsforthesereconstructedimagesarealsoshown.More 50

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contoursinqualitymapsimplieshigherqualitywithlargerSSIMindexs,whichcouldbefoundaroundtableclothof`Barbara',hatowersof`Lena'andnoseof`Baboon'.Itindicatesthatourproposed`PLUS 1'and`PLUS 2'hasbetterobjectiveandsubjectiveperformanceinlossyimagecompressionthan`JPEG-XR'. 2.6SummaryInthischapter,westudiedtheintegerreversibletransformforlosslessimage/videocompressionbyusingPLUSfactorization.WeproposedstabilizedPLUSfactorization,anddoperturbationanalysis,whichprovesthenumericalstabilityofPLUSfactorization.Furthermore,weoptimizedPLUSfactorizationtoachievetheleasttransformerrorbyusingTabuSearchalgorithm.ThenwestudiedthelosslessTransformforlossy/losslessimagecompression.WeproposedtheintegerDCTandtheintegerLappedTransformbyusingtheoptimizedPLUSfactorization,andcomparethelossy/losslessimagecodingperformancewiththestandards.ExperimentalresultsshowthesuperiorityofouralgorithmsoversomeexistingintegerDCTalgorithms,andthelappedtransformfactorizationinJPEG-XR.Theoptimalintegerreversibletransformswiththeleastentropyneedtobeinvestigatedfurther. Figure2-1. Flowchartof4-pointintegerDCTimplementedwithPLUS.([]denotesaround-offoperation,anedgewithanumberdenotesmultiplication,Ldenotesaddition.) 51

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Figure2-2. E2comparisonbetweenthefactorizationsfoundbythreealgorithms. Figure2-3. ConvergencespeedofoptimizationalgorithmusingTS. Figure2-4. Averagebppvs.PSNRwithintegertransformsfortestimages. 52

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A B C D E F G Figure2-5. Lossyperformancecomparisonof`JPEG-XR',`PLUS 1'and`PLUS 2'at0.25bppforimageBarbara.(A)Originalimage(B)`PLUS 1'PSNR32.51dB(C)`PLUS 2'PSNR31.78dB(D)`JPEG-XR'PSNR30.33dB(E)Qualitymapwith`PLUS 1'SSIM0.6556(F)Qualitymapwith`PLUS 2'SSIM0.6427(G)Qualitymapwith`JPEG-XR'SSIM0.5835. 53

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A B C D E F G Figure2-6. Lossyperformancecomparisonof`JPEG-XR',`PLUS 1'and`PLUS 2'at0.25bppforimageLena.(A)Originalimage(B)`PLUS 1'PSNR33.94dB(C)`PLUS 2'PSNR31.78dB(D)`JPEG-XR'PSNR33.52dB(E)Qualitymapwith`PLUS 1'SSIM0.4831(F)Qualitymapwith`PLUS 2'SSIM0.4831(G)Qualitymapwith`JPEG-XR'SSIM0.4734. 54

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A B C D E F G Figure2-7. Lossyperformancecomparisonof`JPEG-XR',`PLUS 1'and`PLUS 2'at0.25bppforimageBaboon.(A)Originalimage(B)`PLUS 1'PSNR29.52dB(C)`PLUS 2'PSNR29.30dB(D)`JPEG-XR'PSNR28.85dB(E)Qualitymapwith`PLUS 1'SSIM0.7765(F)Qualitymapwith`PLUS 2'SSIM0.7669(G)Qualitymapwith`JPEG-XR'SSIM0.7599 55

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Table2-1. E(LUS),OMSEandOMEofoptimalfactorizationsforDCTmatriceswithexhaustivesearch nerrormetricE(LUS)OMSEOME E12.19070.38210.000152E21.78090.12720.00011E1.02050.30000.0003 E13.21340.38550.000154E22.88930.14850.00011E1.32050.30000.0003 E117.58340.40110.000168E24.67660.15490.00011E4.56550.31230.00049 nistheorderoftheDCTmatrices. Table2-2. E2(LUS),OMSEandOMEofseveralPLUSfactorizationsforDCTmatrices nE2(LUS)OMSEOME 8.54371.42130.0201523.01190.47280.001031.78090.12720.00011 16.74216.51220.0798246.81021.32010.019712.88930.14850.00011 nistheorderoftheDCTmatrices. 56

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Table2-3. SomeoptimalfactorizationsforDCTfoundbyexhaustivesearch nPLPRuE2 212121)]TJ /F8 11.955 Tf 9.29 0 Td[(11.78092112111.7809 4204312341112.889343212413)]TJ /F8 11.955 Tf 9.28 0 Td[(1112.8893 4718653232547186)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(11)]TJ /F8 11.955 Tf 9.29 0 Td[(111114.676684718653232547816)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(11)]TJ /F8 11.955 Tf 9.29 0 Td[(1111)]TJ /F8 11.955 Tf 9.29 0 Td[(14.676623148576146352781)]TJ /F8 11.955 Tf 9.29 0 Td[(11)]TJ /F8 11.955 Tf 9.29 0 Td[(11)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(14.676623148576146357281)]TJ /F8 11.955 Tf 9.29 0 Td[(11)]TJ /F8 11.955 Tf 9.29 0 Td[(11)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(114.6766 Table2-4. TransformerrorE2ofoptimalfactorizationsfoundbyTS t nk5101520EaeEmse 42.892.892.892.89482.892.892.892.8900112.892.892.892.89 44.684.814.824.78864.764.764.764.760.080.001684.764.764.714.76 58.358.148.317.941688.198.238.128.230.011118.288.288.288.28 kisthesizeofcandidatelist;tistabutenure;Eae=1 NNi=1E(i))]TJ /F3 11.955 Tf 10.95 0 Td[(Emin;EminisE2inTable 2-3 ;Emse=1 NNi=1(E(i))]TJ ET q .478 w 279.39 -569.71 m 288.27 -569.71 l S Q BT /F3 11.955 Tf 279.39 -579.69 Td[(E)2; Eistheaverageerror. 57

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Table2-5. EntropycomparisonofintegerDCTs ImageIntDCT2Opt2IntDCT3Opt3IntDCT4Opt4 Barbara6.945.958.026.656.925.57Lena6.375.387.435.976.365.03Boat6.415.427.466.066.395.12Jet5.895.116.895.725.874.95Mandrill7.596.598.667.007.576.62Goldhill6.695.707.786.236.685.43Average6.655.697.716.276.635.45 Table2-6. EntropycomparisonamongintegerlappedbiorthogonaltransformsimplementedbyJPEG-XR,PLUS 1andPLUS 2 ImageJPEG-XRPLUS 1PLUS 2 Lena4.934.484.61Baboon6.346.116.22Barbara5.644.985.29Boat5.104.544.71Goldhill5.274.925.01Peppers5.094.764.83Average5.404.975.11 58

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Table2-7. Bppvs.PSNRcomparisonofintegerlappedbiorthogonaltransforms ImagebppJPEG-XRPLUS 1PLUS 2 PSNRSSIMPSNRSSIMPSNRSSIM 444.140.910345.210.928645.030.9261242.350.876942.970.893942.930.8930Lena138.900.784039.820.805239.690.80350.535.160.586636.950.620636.690.61500.2531.450.422633.940.483133.520.47340.12528.460.286530.100.356829.780.3484 444.130.937045.220.950245.060.9482242.360.913542.990.925642.930.9482Barbara138.890.848939.960.869439.730.86670.534.680.716736.540.763536.020.75500.2530.340.583532.510.655831.780.64270.12526.430.447528.140.529127.420.5086 444.140.986845.440.990145.300.9897242.390.980743.130.983743.060.9834Baboon138.850.959739.280.963639.210.96330.533.890.895734.310.904134.240.90110.2528.850.759929.520.776529.300.76690.12524.460.557425.390.607925.060.5916 59

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Table 2-7 .Continued ImagebppJPEG-XRPLUS 1PLUS 2 PSNRSSIMPSNRSSIMPSNRSSIM 444.090.883445.170.906244.970.9027242.310.844942.920.866842.910.8655Boat139.020.749040.220.766240.050.76410.535.260.580737.010.604736.670.60000.2530.910.448833.140.499532.740.48750.12527.400.303329.120.374428.850.3614 444.130.949545.290.960445.090.9587242.370.930543.010.939942.960.9397Goldhill138.880.873839.550.884039.410.88240.534.420.739935.640.767335.450.76140.2530.380.541332.110.603931.860.58820.12527.510.328228.790.415328.630.4017 444.150.933645.360.949045.170.9473242.350.905143.050.919343.000.9193Peppers138.870.824239.490.842339.330.83880.534.590.625135.670.646735.570.63270.2531.170.390933.270.424933.090.41820.12528.570.261930.190.316029.990.3017 60

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CHAPTER3ADAPTIVEQUANTIZATIONUSINGPIECEWISECOMPANDINGANDSCALINGFORGAUSSIANMIXTURES 3.1ResearchBackgroundQuantizationisacriticaltechniqueforanalog-to-digitalconversionandsignalcompression.Ononehand,manyinputsignalsarecontinuousanalogsignals,therefore,quantizationisindispensableforanalog-to-digitalconverters(ADC)[ 70 ],whichareimportantcomponentsofmanydigitalproducts.Ontheotherhand,withtheexponentialgrowthofusageofcomputersandInternet,countlessdigitalcontents,especiallydigitalimagesandvideos,demandsignalcompressionforefcientstorageandtransmission.Accordingly,quantizationprovidesameanstorepresentsignalsefcientlywithacceptabledelityforsignalcompression.Existingquantizationschemescanbeclassiedintotwocategories,namely,uniformquantizationandnonuniformquantization[ 60 61 ].Uniformquantizationissimple,butnotoptimalforsignalswithnonuniformdistributionintermsofMMSEifmorecomputationsandstorageareavailable.Whilenonuniformquantizationismuchmorecomplexandinagreatvariety.Minimummeansquarederror(MMSE)quantization(a.k.a,Lloyd-Maxquantization)isamajortypeofnonuniformquantization.Itisoptimalinthesenseofmeansquarederror(MSE),butincurshighcomputationalcomplexity.Companding,whichconsistsofnonlineartransformationanduniformquantization,isatechniquecapableoftradingoffquantizationperformancewithcomplexityfornonuniformquantization.Especially,forhighratecompression,theperformanceofcompandingcanapproachthatofLloyd-Maxquantizationasymptotically.Lloyd-MaxquantizersandcompandersarealreadywelldevelopedforGaussiandistributionorLaplaciandistribution[ 61 68 109 ]asconvenience,butnotforGaussianmixturemodel(GMM).SinceGMMservesasagoodapproximationofanarbitrarydistribution,itisimportanttodevelopquantizersandcompandersforGMM,which 61

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areexpectedtondwideapplicationsinADCandhighdynamicrange(HDR)imagecompression,aswellasaudio[ 111 ]andvideo[ 152 ]compression.Toaddressthis,weproposesasuccinctadaptivequantizerwithpiecewisecompandingandscalingforGMMinthischapter.WerstconsiderasimpleGMM(SGMM)thatconsistsoftwoGaussiancomponentswithmean)]TJ /F11 11.955 Tf 1 0 .167 1 343 -107.59 Tm[(mandmrespectively,andthesamevariances2.Theproposedquantizershavethreemodes,makingthemcapableofadaptingtheirreconstructedlevelstothevaryingmeansandvariancesoftheGaussiancomponentsinaGMM.Specically,forSGMMs,ifmissmall,ourquantizeroperatesinModeI,andtreatstheinputasifitwerefromtwooverlappingGaussianrandomvariables(r.v.)ratherthanaGMMr.v..ForModeI,ourquantizercanbeimplementedbyacompanderorascaledLloyd-Maxquantizerofaunit-varianceGaussian.Ifmislarge,ourquantizeroperatesinModeIII,i.e.,iftheinputisnegative,treattheinputasifitwereaGaussianr.v.withmean)]TJ /F11 11.955 Tf 1 0 .167 1 42.81 -322.76 Tm[(m;iftheinputispositive,treattheinputasifitwereaGaussianr.v.withmeanm.ForModeIII,ourquantizercanbeimplementedbytwocompandersortwoscaledLloyd-Maxquantizers,eachofwhichcorrespondstooneofthetwoGaussianr.v.s.Ifmisofmediumvalue,ourquantizeroperatesinModeII,i.e.,withpiecewisecompanding.Moreover,weproposearecongurablearchitecturetoimplementouradaptivequantizerinanADC.Theproposedadaptivequantizeristunedbytheinformationfromasignalhistogramestimatortooptimallyquantizesignalswithavailablespeedandpowerfromdevices.Furthermore,theproposedquantizerisappliedintoimagequantizationandhighdynamicrangeimagecompression.WedesignHDRtonemappingalgorithmbyjointlyusingadaptivequantizersandmultiscaletechniques.Therefore,theproposedalgorithmcouldmitigatethehaloartifactsintheresultedlowdynamicrangeimage,aswellaskeepthecontrastofimagedetailscrossingthelargestgamut.Theexperimentalresultsshowthat1)ourproposedquantizerisabletoachieveMSEperformanceclosetoLloyd-MaxquantizerforGMM,atmuchlowercostthan 62

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Lloyd-MaxquantizerforGMM;2)ourproposedquantizerisabletoachievemuchbetterMSEperformancethanauniformquantizer,atacostsimilartotheuniformquantizer.TheexperimentalresultsalsoshowthattheproposedadaptivequantizerholdsgreatpotentialofrevolutionizingtheexistingADCandHDRimagecompression.Itworkswellwithbothhighrateandlowratequantization.Therestofthechapterisorganizedasbelow.Section 3.2 presentsthepreliminariesofoptimaladaptivequantizers.Section 3.3 describestheproposedadaptivequantizerforGMM.InSection 3.4 ,weproposearecongurablearchitecturetoimplementouradaptivequantizerinanADC.Insection 3.5 ,theproposedquantizerisappliedintohighdynamicrangeimagecompression.ExperimentalresultsareexhibitedinSection 3.6 .Section 3.7 concludesthechapter. 3.2Preliminaries 3.2.1MMSEQuantizerTheperformanceofaquantizercanbeevaluatedbymeansquareerror(MSE)betweeninputsignalXandthereconstructedsignalX,i.e., MSE=E[(X)]TJ /F8 11.955 Tf 13.79 2.92 Td[(X)2](3)Lloyd-Maxquantizer[ 58 ]isanMMSEquantizer.Lettk(k=0,,N)denoteboundarypointsofquantizationintervals,andletrk(k=0,,N)]TJ /F8 11.955 Tf 10.3 0 Td[(1)denotequantizationlevels.ThenLloyd-Maxquantizerischaracterizedby: ftk,rkg=argminftk,rkgMSE=argminftk,rkgN)]TJ /F5 8.966 Tf 6.97 0 Td[(1k=0Ztk+1tk(x)]TJ /F3 11.955 Tf 10.95 0 Td[(rk)2fX(x)dx(3)wherefX(x)istheprobabilitydensityfunction(pdf)ofX,Nisthenumberofquantizationlevels.DerivingrespecttotkandrKinEq. 4 ,wehavethecentroidandthenearest 63

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neighborconditionsasfollowing: tk=rk)]TJ /F5 8.966 Tf 6.97 0 Td[(1+rk 2,k=1,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1,(3)and rk=Rtk+1tkxp(x)dx Rtk+1tkp(x)dx,k=0,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1,(3)where[t0,tN]istherangeofthequantizerinput.TheLloyd-MaxquantizerforGaussiandistributionwithzeromeanandunitvariancehasbeenwellstudied.GiventhenumberofquantizationlevelsN,theLloyd-Maxquantizerforzeromean,unitvarianceGaussiancouldbeobtainedfromtablesin[ 68 ].GiventheLloyd-Maxquantizerforzeromean,unitvarianceGaussian,wecanusetheafnelawinProposition 3.1 toobtaintheLloyd-MaxquantizerforGaussiandistributionwitharbitrarymeanmandarbitraryvariances2. 3.2.2GaussianMixtureModelandAfneLawGaussiandistributioniswildlyusedinsignalmodelingbecauseofitssimplicity,ubiquity,andtheCentralLimitTheorem.However,signalsintherealworld,suchaspixelintensityofnaturalimages,mayhaveanarbitrarydistribution,whichcanbebetterapproximatedbyaGMMthanbyaGaussiandistribution.ThepdfofaGMMr.v.Xisgivenasbelow: fX(x)=Ngi=1pigi(x)(3)whereNgisthenumberofGaussiancomponentsintheGMM;gi(x)istheGaussianpdfforcomponenti(i=1,,Ng);pidenotestheprobabilityofcomponenti(i=1,,Ng);andNgi=1pi=1.Inthischapter,werstlyconsideraSimpleGMM(SGMM)givenasbelow: fX(x)=1 2p 2p(e)]TJ /F28 6.974 Tf 8.17 3.53 Td[(1 2(x)]TJ /F50 8.966 Tf 1 0 .167 1 250.88 -610.95 Tm[(m)2+e)]TJ /F28 6.974 Tf 8.16 3.53 Td[(1 2(x+m)2)(3) 64

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GivenasuboptimalquantizerforSGMM,wecanusetheafnelawinProposition 3.1 toobtainasuboptimalquantizerforaGMMthatconsistsoftwoGaussiancomponentswitharbitrarymean)]TJ /F11 11.955 Tf 1 0 .167 1 114.79 -59.77 Tm[(mandm(m>0),respectivelyandthesamevariances2(s2>0).ItcanalsobeusedtoobtainthesuboptimalquantizerforaGMMwitharbitrarynumberofcomponents. Proposition3.1. (AfneLaw)Forar.v.Xwithzeromeanandunitvariance,assumethatitsN-levelLloyd-Maxquantizerisspeciedbytk(k=0,,N)andrk(k=0,,N)]TJ /F8 11.955 Tf -458.71 -23.91 Td[(1).Thenforr.v.Y=sX+m,withmeanmandvariances,itsLloyd-Maxquantizerisspeciedbytk=stk+m(k=0,,N)andrk=srk+m(k=0,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1).ForaproofofProposition 3.1 ,seeAppendix B.1 3.2.3MMSECompanderAcompanderconsistsofacompressor,auniformquantizer,andanexpandor;thecompressorperformsnonlineartransformationandtheexpandorisaninverseofthecompressor.Thecompressorisintendedtoconverttheinputr.v.ofarbitrarydistributionintoauniformly-distributedr.v.,sothatwecanuseasimpleuniformquantizer,whichistheoptimalquantizerfortheone-dimensionaluniformdistributioninthesenseofMMSE.Proposition 3.2 givesanonlineartransformationforan(suboptimal)MMSEcompanderforanydistribution. Proposition3.2. Assumethatar.v.XhasCumulativeDistributionFunction(CDF)FX(x)(x2R).Thenr.v.Y=FX(X)isuniformlydistributedin[0,1];andthecompanderwithcompressorY=FX(X)isanoptimal/suboptimalMMSEquantizerofX,especiallywhenXisquantizedwithhighrate.ForaproofofProposition 3.2 ,seeAppendix B.2 .ForGaussiandistributionwithzeromeanandunitvariance,aMMSEcompressorperformstransformationby1)]TJ /F3 11.955 Tf 10.95 0 Td[(Q(X),where Q(X)=1 p 2pZXexp()]TJ /F3 11.955 Tf 10.48 8.1 Td[(u2 2)du.(3) 65

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SincetheintegralinQ(X)hashighcomputationalcomplexity,inthischapter,weproposeasimplecompressor,whichonlyneedscomputationofpiecewisemonomials(seeSection 3.3.4 ). 3.3AdaptiveQuantizerforGaussianMixtureModelsInthissection,werstpresentouradaptivequantizerforSGMMinEq.( 3 )andthenextendittoamorecomplicatedGMMwitharbitrarymands2,andarbitrarynumberofcomponents,byusingProposition 3.1 3.3.1DesignMethodologyBecauseProposition 3.2 statesthatthecompanderwithcompressorY=FX(X)isaMMSEquantizerofinputX,ourdesignmethodologyistondacompressorwhosetransformationfunctionissimple,butcanachieveagoodapproximationofCDFFX(X).Therobustquantizer[ 75 ]willbeprovidedthroughthedeterminationoftherequiredparameters.Figure 3-1 showstheCDFofGaussianN(0,1)vs.thatofSGMMwithm=0.5.Wecanobservethattheyaresimilar.Figure 3-2 showsthetransformationfunctionofapiecewisecompressorspeciedbyEq.( 3 )vs.CDFofSGMMwithm=1.5.FromFigure 3-2 ,wecouldobservethatthetransformationfunctionofapiecewisecompressorspeciedbyEq.( 3 )issimilartotheCDFofSGMMwithm=1.5.Figure 3-3 showstwocatenatedCDFsoftwoGaussiansvs.theCDFofSGMMwithm=3,wherethecatenatedCDFoftwoGaussiansisgivenbyEq.( 3 ).FromFigure 3-3 ,itisobservedthattheCDFofthecatenatedGaussianissimilartotheCDFofSGMMwithm=3.Forthisreason,ourproposedadaptivequantizeroperatesunderthreemodes,whichcorrespondtosmallm,medium-valuedm,andlargem,respectively. 3.3.2ThreeModesLetqg(X)denotetheLloyd-MaxquantizationfunctionforaGaussianr.v.XN(0,1). 66

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Ourproposedadaptivequantizeroperatesinoneofthefollowingthreemodes,dependingonthevalueofm. 1. If0m
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thedataofSGMMasModeI.Inconclusion,fortheproposedquantizermS=1andmL=3. 3.3.4PiecewiseCompandingofModeIIForModeII,wechoosethemonomialf(x)=axbtoapproximatetheidealcompressorofSGMM,i.e.theCDFofSGMM,piecewisely.Therearemanymoreaccurateandmorecomplicatedapproximativefunctions,likethesumofmonomialsf(x)=iaixbi,i>1,sigmoidfunctionf(x)=1 1+e)]TJ /F26 6.974 Tf 5.41 0 Td[(x,andf(x)=arctan(x).Buttheircorrespondingexpandors,i.e.theinversesofcompressors,arehardtoobtainorcomputationallyexpensive.However,f(x)=axbhassimpleinverseandisagoodapproximationtothesegmentsoftheCDFofSGMM.ThepiecewisecompressorsymmetricaltotheorigincanbedescribedbyEq.( 3 ). f(x)=8>>>>>>><>>>>>>>:a(x+m)b+0.25,x)]TJ /F11 11.955 Tf 1 0 .167 1 301.53 -299.68 Tm[(m (3a) a0(x+m)b0+0.25,)]TJ /F11 11.955 Tf 1 0 .167 1 285.66 -326.58 Tm[(mm (3d) with fa,a0,b,b0g=argfminfa,a0,b,b0gZ31(Z)]TJ /F4 8.966 Tf 6.97 0 Td[((FSGMM(x,m))]TJ /F3 11.955 Tf 10.95 0 Td[(f(x,m))2dx)dmg(3)Bythesteepestdescentmethod,weobtainb=1 3,b0=1 2,a=0.15anda0=0.125(whichcanberealizedbyrightshifting3bits)forsimplicityandfastcomputation.ThecompressorisshowninFigure 3-2 whenm=1.5.Whenx<)]TJ /F11 11.955 Tf 1 0 .167 1 376.47 -524.49 Tm[(mandx>m,thePDFdecayingfaster,weusef(x)=ax1 3.Whenx>)]TJ /F11 11.955 Tf 1 0 .167 1 284.89 -548.4 Tm[(mandx
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AlthoughtherearemoreaccuratecompressorstoapproximatetheCDFwithcertainm,theymaynothavegoodapproximationstotheCDFwithotherm2[1,3)inaverage.Theproposedcompressorisagoodtradeoffbetweenaccuracyandgeneralizability.Itprovidesastablegoodperformancewhenm2[1,3)asshowninexperimentsinSection 3.6 .Itisrobust.Therefore,theproposedcompanderhasthreeadvantages. 1. ItiseasytodesigncompanderbyEq.( 3 ); 2. Itisfasttoquantizedatawiththiscompander; 3. IthasgoodaverageMSEperformancewhenm2[1,3). 3.3.5AdaptiveQuantizerforAGeneralGMMInthissection,wedesigntheadaptivequantizerforageneralGMMbasedontheadaptivequantizerforSGMM. 3.3.5.1GMMestimationbyEMTheGMM(GaussianMixtureModel)isaprobabilitydistributionmodelconsistingnitenumberofGaussiancomponentsasshowninEq.( 4 ).TheExpectation-Maximum(EM)algorithm[ 15 ]isageneralmethodtondthemaximumlikelihoodestimationofGMM.EMalgorithmcanefcientlyestimatethecomponentsofGMM[ 157 ]asshowninFigure 3-5 .ThenumberofcomponentsofGMMshouldbeassignedtotheEMalgorithmbyexperienceandrestrictedbytheavailablecomputationalresourcesandN,thenumberofthereconstructionlevelsofquantizers.NgcouldbeN=5orsmaller.mi,siandpi(i=1,,Ng)ofeachGaussiancomponentinEq.( 4 )aredeterminedbytheEMalgorithm.TheGMMestimationofsignalsisobtainedforlaterquantizationonceforall. 3.3.5.2GeneralizationtoGMMFortheGeneralGMMasshowninEq.( 4 ),withthescalinglawinProposition 3.1 ,thefollowinggeneralizationsaremadefromSGMMbyprocessingneighboringpairwiseGaussiancomponentsofaGMM. 69

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AssumingtheGaussiancomponentsaresortedbytheirmeansmi,fortheneighboringGaussiancomponentsCiandCi+1,weconsidersupport(mi,mi+1),wheni6=1,Ng,elseconsider()]TJ /F7 11.955 Tf 9.29 0 Td[(,m1)or(mNg,+). 1. AllocatethenumberNifromthetotalreconstructionlevelsNforeachGaussiancomponentaccordingtoitspercentagepi.Ni=[Npi]where[]isroundoffoperator.ForeachNi,itissymmetricallylocatedwithrespecttothemeanofthecorrespondingGaussiancomponent. 2. OriginShift:ForanytwoadjacentGaussiancomponentswithmeansandvariancesof(mi,s2i)and(mi+1,s2i+1),theirpdfsequalaroundxo=simi+si+1mi+1 si+si+1(theeffectofpiisomitted).Thenweshifttheorigintoxo. 3. Thethree-modeboundariesmSandmLarescaledby(si+si+1). 4. Scalethereconstructionlevelsaccordingtothevariance:FortheGaussiancomponentiwith(mi,si),scalethereconstructionlevelsobtainedfromSGMMbysi. 5. TunemodeII:Sincehalfsupport(mi,mi+1)ofGaussiancomponentsisconsideredeachtime,thecompressorinEq.( 3b )( 3c )areneeded,andshouldbescaledbypias: f(x)=(pi(a0(x+m)b0+0.25),)]TJ /F11 11.955 Tf 1 0 .167 1 307.64 -455.4 Tm[(m
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witharbitrarydistribution,wequicklysampleanddiscretizeitwithuniformquantizertoestimatethedistributionofthesignal.Thisinformationissentbacktotheproposedadaptivequantizertodomodeselection.Thentheadaptivequantizercouldgiveamoreaccuratediscretesignalbycapturingthesignalcharacteristicsasmuchaspossiblewithappropriatemodes.Theresidualsignalcouldalsobeiterativelysentbacktotheadaptivequantizertominimizethequantizationerror.TheFPGAimplementationoftheadaptivequantizercouldbereconguredinTqmilliseconds,whereTq<10.ThenthesystemcanbeupdatedatthebeginningofeverycycleofTqmilliseconds,accordingtothedistributionoftheinputsignal.Thenumberofquantizationlevelscouldbeadjustedaccordingtothespeed,resolutionandpowerconsumptionofthedevices.OurschemebasedonhistogramestimationandtheGMMmodelingmayoutstandpreviousiterativeDPCMschemes[ 36 ].TherecongurableADCarchitectureinFigure 3-4 candynamicallyadjustthequantizationspeed,resolutionandpowerconsumptiontomatchinputdatacharacteristics.Therefore,itwillhavewideapplicationsinmanyADCsandsensorapplications. 3.5HighDynamicRangeImageCompressionwithJointAdaptiveQuantizerandMultiscaleTechniquesHighdynamicrangeimaging(HDRIorjustHDR)isoneofthefrontiertechniquesinimageprocessing,computergraphicsandphotography[ 46 47 49 ],whereimagepixelstakeoatingvaluesintherangeof[0,1]ratherthanthetraditional8bitsperpixelforgrayimagesand24bitsperpixelforRGBimages.HDRItrytocapturethedynamicrangeofnaturalscenes,whichcanexceedthreeordersofmagnitudesofdisplaydevices.Thedynamicrangeofnaturalscenescanbecapturedbyhumaneyes,manylms,andnewcamerasensors.Whereas,displaydevices,suchasCRTs,LCDs,andprintmaterials,arerestrictedtolowdynamicrange.Therefore,compressingthehighdynamicrangeofHDRItoadapttothelowdynamicrangeofdisplaydevicesandkeepingthevividcolors 71

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andtherichdetailsoftheoriginalimagesasmuchaspossible,isgettingmoreandmoreattention.Itiscalledtonemapping,whichisanimportantcomponentintheHDRimagingpipeline,andwidelyusedinvirtualreality,videoadvertising,visualsimulation,remotesensingimages,aerospace,medicalandmanyotherelds[ 45 ].Thetonemappingtechniquescanbedividedintotwocategories:tonereproductioncurves(TRCs)andtonereproductionoperators(TROs).Theycouldbeappliedtoimagesbothgloballyandlocally.TRCsusecompressivepointnonlinearitymapping,suchasapowerfunctionf(),toshrinkthehighdynamicrangeimagesintothelowdynamicrangeimages.K.Chiuetal.proposedspatiallynonuniformscalingfunctionsforhighcontrastimages[ 35 ].F.Dragoetal.usedanadaptivelogrithmicmappingfordisplayinghighcontrastscenes[ 50 ].I.R.Khanetal.[ 77 ]andA.Boschettietal.[ 19 ]proposedatonemappingalgorithmbasedonhistogramequalization.Thesealgorithmsaresimpleandefcient,butthecontrastofimagedetailsmaybelostobviously.TheTROsadjustpixelintensitybyusingspacialcontexttopreservelocalimagecontrast,whichusuallyusemultiscaletechniques.Stockham[ 132 ]separatedanHDRimageH(x,y)intoaproductofanilluminationimageI(x,y)andareectanceimageR(x,y)inanearlyliterature.Lateron,Jobsonetal.[ 69 ],Pattanaiketal.[ 107 ]improvedthemultiscaletechniquesbyintroducingmechanismofthehumanvisualsystem.Thesemultiscalemethodshavehaloartifacts,whichhappenaroundthesharpedgesandarecausedbytheblurringeffectoflters.ThemostrecentmultiscaletechniqueproposedbyYuanzhenLi[ 84 ]properlyusedasymmetricanalysis-synthesislterbank,andlocalgaincontrolofeachsubbandtomitigatethehaloartifacts.Buttheluminanceoftheresultedlowdynamicrangeimagesseemslow,andtheboundaryofthedynamicrangeisclipped,whichcouldbeseenfromtheirhistograms.Toaddresstheseproblems,weproposedajointTRCandTROmethodsforhighdynamicrangeimagetonemappingbasedonLi'smethod[ 84 ]andourproposedadaptivequantizer. 72

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TheproposedtonemappingalgorithmbyusingjointadaptivequantizerandmultiscaletechniquesisshowninFigure 3-6 withprocessinginbothwaveletdomainandimagedomain.TheHDRimageisrstdecomposedintoseveralsubbandswithwaveletanalysis.Forthesignalineachsubband,webuildagainmapfromit,andapplythegainmaptothesubbandsignaltoreleaseitfromthecompressedlogdomain.Thensubbandsignalsaresynthesizedbacktoimagedomain.Furthermore,theproposedadaptivequantizerisappliedonallthegamutofthereconstructedimage,tunetheoatingpixelvaluesintouserassignedshrinkeddynamicrange,suchasintegersin[0,255]forordinarydigitalimages.OursystemenjoysthebenetsfrommultiscalemethodtokeepthecontrastofHDRasmuchaspossible.ByusingLi'slocalgainmap[ 84 ],oursystemcouldmitigatehaloartifacts.Theproposedadaptivequantizerkeepsthemostgamutinformationinthelargestavailablelowdynamicrange.Therefore,oursystemoutperformsLi'salgorithmandhistogramequalizationbasedsystem. 3.6ExperimentalResultsandDiscussionWecomparetheproposedquantizerwiththeactualLloyd-MaxquantizersforSGMM,bycomparingthecorrespondingapproximateCDFsoftheproposedquantizerwiththeactualCDFsofSGMM.WealsocomparetheproposedquantizerwiththeLloyd-MaxquantizersforSGMMandtheuniformquantizerintermsofMSEperformance.TheproposedadaptivequantizerisdescribedindetailinSection 3.3 .TheLloyd-MaxquantizerforSGMMisfoundbytheLBGalgorithmnumerically[ 61 ].Theuniformquantizerwecomparewithistheoptimaluniformquantizerwhichisapplieduniformlytotheniteregioncontaining99.8%ofthedataoftheGMMdistribution. 3.6.1ExampleandJusticationofParameterDeterminationThereproductionvaluesof2-bitLloyd-MaxquantizerforN(0,1)are[)]TJ /F8 11.955 Tf 9.29 0 Td[(1.5104,)]TJ /F8 11.955 Tf -437.33 -23.91 Td[(0.4528,0.4528,1.5104].Whenm3,forthe3-bitquantizerforSGMM,the8reproductionvaluesare[)]TJ /F8 11.955 Tf 9.29 0 Td[(1.5104)]TJ /F11 11.955 Tf 1 0 .167 1 117.14 -634.81 Tm[(m,)]TJ /F8 11.955 Tf 10.99 0 Td[(0.4528)]TJ /F11 11.955 Tf 1 0 .167 1 192.21 -634.81 Tm[(m,0.4528)]TJ /F11 11.955 Tf 1 0 .167 1 254.59 -634.81 Tm[(m,1.5104)]TJ /F11 11.955 Tf 1 0 .167 1 316.98 -634.81 Tm[(m,)]TJ /F8 11.955 Tf 10.99 0 Td[(1.5104+m,)]TJ /F8 11.955 Tf 10.99 0 Td[(0.4528+ 73

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m,0.4528+m,1.5104+m]asinModeIII.Whenm<1,i.e.inmodeI,forthe2-bitquantizerforSGMM,thereproductionvaluesare[)]TJ /F8 11.955 Tf 9.29 0 Td[(1.5104,)]TJ /F8 11.955 Tf 11.18 0 Td[(0.4528,0.4528,1.5104].When1m<3,i.e.inmodeII,thecompanderischosenasshowninEq.( 3 ).ThedifferencesbetweenreproductionvaluesoftheproposedquantizerandthoseoftheLloyd-MaxquantizerforSGMMareevaluatedbyaverageabsolutedifference(AAD)asfollowing: AAD=Zdc1 NN)]TJ /F5 8.966 Tf 6.97 0 Td[(1k=0jrpk(m))]TJ /F3 11.955 Tf 10.95 0 Td[(rlk(m)jdm(3)whererpkandrlkarethereproductionvaluesoftheproposedquantizerandtheLloyd-MaxquantizerforSGMM,misthemeaninSGMM,(c,d)isthesupportforaveraging,i.e.theregionofmforeachmode.TheapproximationerrorbetweentheCDFapproximatorsintheproposedquantizerandthoseofSGMMisevaluatedby: Zdc(Z)]TJ /F4 8.966 Tf 6.97 0 Td[((FSGMM(x,m))]TJ /F3 11.955 Tf 10.95 0 Td[(FA(x,m))2dx)dm(3)whereFSGMMistheCDFofSGMM,FAistheCDFapproximatorsintheproposedquantizer.ForModeI,c=0,d=1,FA(x)=Q(x)whereQ(x)isdenedinEq.( 3 );forModeII,c=1,d=3,FA(x)isinEq.( 3 );forModeIII, FA(x)=8><>:(1+Q(x+m))=2,x<0(1+Q(x)]TJ /F11 11.955 Tf 1 0 .167 1 233.26 -462.15 Tm[(m))=2+1=2,x0(3)ThenumericalexperimentsshowthattheAADin10)]TJ /F16 8.966 Tf 6.96 0 Td[(norderandtheapproximationerroroftheproposedquantizerissmallaslistedinTable 3-1 .Table 3-1 ,Figure 3-1 ,Figure 3-2 ,andFigure 3-3 indicatetheclosenessoftheproposedquantizertotheLloyd-Maxquantizeraswellastherobustness[ 65 ]oftheproposedquantizer. 3.6.2MSEPerformanceComparisonWerandomlygenerate10000datafromthedistributionofSGMMinEq.( 3 ).Thentheproposedadaptivequantizer,Lloyd-Maxquantizeranduniformquantizerare 74

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usedtoquantizethedatainto8quantizationlevels.Wereconstructthedatafromthequantizedvalues,andcomparethemwiththeoriginaldataintermsofMSEwithrespecttodifferentmasshowninFigure 3-7 .TheMSEperformanceoftheproposedquantizerisveryclosetothatofLloyd-Maxquantizerandmuchbetterthanthatoftheuniformquantizer.InmodeI,sincewetaketheoptimaluniformquantizationinnitehighprobabilityregion,theMSEgapbetweenuniformquantizerandLloyd-Maxquantizerissmall.Buttheproposedquantizerstillhasperformancegainthanuniformquantizer.InmodeII,theproposedpiecewisecompanderprovidesagoodstableMSEperformancewithasimpledesign.InmodeIII,theMSEoftheuniformquantizerincreasesdramaticallywithm,sincedistributionisfarawayfromuniformdistributionwhenmislarge,andtheuniformquantizerwasteslotsofbitsforvalueswithsmallprobabilityaroundorigin.ButtheproposedquantizerisstillwithMSEperformanceveryclosetothatofLloyd-Maxquantizer.Again,weapplyourmethodto: G2(x)=2i=11 (2p)1=2se)]TJ /F29 6.974 Tf 8.16 5.22 Td[((x)]TJ /F54 6.974 Tf 1 0 .167 1 293.43 -346.99 Tm[(mi)2 2s2(3)Whenm1=)]TJ /F11 11.955 Tf 1 0 .167 1 71.92 -394.48 Tm[(m2=mands=2,wedrawMSEresultsoftheproposedquantizer,theLloyd-MaxquantizerandtheuniformquantizerinFigure 3-8 .FromFigure 3-8 ,wecouldseethatinModeIandIII,thequantizationerroroftheproposedadaptivequantizerisveryclosetothatoftheLloyd-Maxquantizer,andthequantizationerrorisalittlehigherinModeII.mSandmLfors=2arealmostthetwiceofthosefors=1.TheproposedadaptivequantizerhasMSEperformanceclosetothatoftheLloyd-Maxquantizer,withsimilarcomputationsastheuniformquantizer.ItveriestheafnelawofquantizersinProposition 3.1 .Bytheway,duetothegoodMSEperformanceoftheproposedquantizer,thereproductionvaluesoftheproposedquantizerareeffectiveinitialsofLloyd-MaxalgorithmforquicklyndingtheLloyd-MaxquantizersforGMM. 75

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Thetimecomplexityandspacecomplexityoftheuniformquantizer,theLloyd-MaxquantizerandtheproposedquantizerforNquantizationlevelsareshowninTable 3-2 .Quantizerdesigningtime,quantizationrunningtimepersampleandmemorycostofthequantizersarecompared.TheuniformquantizerdesignneedstheinverseofCDFtoobtaintheoptimalquantizationrange.Theuniformquantizationfunction[x=N]+t0needs3operationspersample,i.e.amultiplication,aroundingoperationandanaddition.InmodeIandmodeIII,thecomputationoftheproposedadaptivequantizerusingqIandqIIIisjustatable-lookup.WhenthenumberofquantizationlevelsNissmall,therunningtimeoftheproposedquantizerpersamplelog(N)orlog(N=2)issimilartothatofuniformquantization.InmodeII,theadaptivequantizerusescompandingtechnique.Itscomputationisapproximative4operationspersample,i.e.amultiplication,anexponentiation,aroundingoperationandanaddition.InmodeIandmodeIII,ifcompandingisused,thecomplexityisthesameasModeII.ThecomputationofLloyd-Maxalgorithmincludesanaddition,adivisioninEq.( 3 )andtwointegralsinEq.( 3 )foreachreconstructionlevelinoneiteration.Thememorycostsarealsocompared.ForuniformquantizerandproposedquantizerinmodeIIandthecompandinginmodeIandIII,O(1)spaceisneededforcomputation.OthersneedO(N)spacefortablelookup.Inshort,theproposedquantizerismuchmorecomputationallyefcientthantheLloyd-Maxquantizerandclosetotheuniformquantizer. 3.6.3AnApplicationinImageQuantizationWeapplytheproposedadaptivequantizeringrayimagequantization.Assumethatweonlyhavealowdynamicrangedisplayer,suchasprintedpaper,withmbitsperpixel,wherem<8,i.e.weshouldhaveambitquantizationschemeforproperdisplay.Thenwhatisthebestimagequalitywecanobtainfromoriginalgrayimageswithquantization?Thequantizershouldutilizetheinformationofimagepixeldistribution. 76

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Again,wecomparetheproposedadaptivequantizerwithuniformquantizerandLloyd-Maxquantizer.Weshowthecaseswhenm=4,5inFigure 4-8 andFigure 3-10 respectivelyonimageBarbara,whosehistogramandGMMestimationareshowninFigure 3-5 before.FromFigure 4-8 andFigure 3-10 ,wecouldseethattheproposedadaptivequantizerresultslessblockartifactsthantheuniformquantizerandsimilartoLloyd-Maxquantizer.Theproposedquantizerhasbetterperformancethantheuniformquantizer,andapproximatetotheoptimalLloyd-MaxquantizerintermsofperceptualqualityandPSNR. 3.6.4ExperimentalResultsonHDRImageToneMappingWeshowtheexperimentalresultsoftheproposedtonemappingalgorithmofHDRimagesbyusingjointadaptivequantizerandmultiscaletechniquesinthissection.Wecompareouralgorithmwiththerecentalgorithms:aTRCbasedmethod[ 84 ]andaTRObasedmethod[ 77 ].FromFigure 3-11 ,itisobservedthatLi'sresult[ 84 ]isalittledarkduetotheconcentratedhistograms,andthehistogrambasedalgorithm[ 77 ]losessomedetailsbetweentreesandbackground,whileourresultlooksbetter.WealsocomparetheresultedLDRimagesandtheirhistogramsofRGBcomponentsontheHDRimagechairsinFigure 3-12 andTable 3-3 .FromFigure 3-12 ,wecanseethattheboardonthewallinourresultisclearerthanthatinLi'sresult,andtheilluminationinformationinouralgorithmisricher.InTable 3-3 ,therstrowshowsthehistogramsofresultsfromLi'salgorithm[ 84 ]andthesecondrowshowsthehistogramsofresultsfromourproposedalgorithm.ThehistogramsofRGBcomponentsfromourresultsaremorespreadoutthanLi'salgorithm.Inaddition,Li'salgorithmhasclippingonbothhighandlowendofdynamicranges,whichwilllossinformationandmaycausefalsecolorartifacts. 77

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3.7SummaryInthischapter,weproposedanoveladaptivequantizerforGaussianMixtureModel.TheproposedquantizerisadaptivetothevaryingmeansandvariancesofthecomponentsofGaussianMixture.TheadaptivequantizerhaslessMeanSquareErrorthanuniformquantizer,andveryclosetoLloyd-Maxquantizer,onlywithsimilarcomputationsasuniformquantizer.WealsoproposedarecongurableA/Dconverterwithouradaptivequantizer.TheproposedquantizercanalsohaveapplicationsinimagequantizationandHighDynamicRangeImagecompression.ThequantizedgrayimageswithourquantizerhavebettervisualqualityandhigherPNSRthanthosewiththeuniformquantizer,andaresimilartothosewithLloyd-Maxquantizer.ForHDRimagecompression,weproposedthetonemappingalgorithmbyusingouradaptivequantizerandmultiscaletechniques.TheexperimentalresultsshowthattheproposedadaptivequantizerholdsgreatpotentialofrevolutionizingtheexistingADCandHDRimagecompression.Ourfutureworkwillfocusonextendingone-dimensionalquantizersofGaussianMixtureModeltohighdimensionalspace.Thepotentialapplicationswillincludehighdimensionalsignalprocessingandclustering. Figure3-1. CDFofGaussianN(0,1)vs.CDFofSGMMwithm=0.5. Table3-1. Proposedquantizervs.Lloyd-Maxquantizer. ModeIModeIIModeIII AAD10)]TJ /F5 8.966 Tf 6.97 0 Td[(210)]TJ /F5 8.966 Tf 6.96 0 Td[(110)]TJ /F5 8.966 Tf 6.96 0 Td[(4ApproximationError2.5116.690.03 78

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Figure3-2. Transformationfunctionofapiecewisecompressorvs.CDFofSGMMwithm=1.5. Figure3-3. CDFofthecatenatedGaussianvs.CDFofSGMMwithm=3. Figure3-4. RecongurableA/Dconverter. Figure3-5. GMMestimationbyEMalgorithmonhistogramofBarbara. 79

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Figure3-6. Tonemappingbyusingjointadaptivequantizerandmultiscaletechniques. Figure3-7. MSEcomparisonamongtheproposedadaptivequantizer,Lloyd-MaxquantizerandUniformquantizerforSGMM(s=1). Figure3-8. MSEcomparisonamongtheproposedadaptivequantizer,Lloyd-MaxquantizerandUniformquantizerforthedatainEq.( 3 )(s=2). 80

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A B C Figure3-9. Performancecomparisonamongdifferentquantizerswhenk=4.(A)UniformQuantizer(34.76dB).(B)ProposedQuantizer(36.21dB).(C)Lloyd-MaxQuantizer(36.84dB). A B C Figure3-10. Performancecomparisonamongdifferentquantizerswhenk=5.(A)UniformQuantizer(40.72dB).(B)ProposedQuantizer(41.85dB).(C)Lloyd-MaxQuantizer(42.45dB). Table3-2. Comparisonofcomplexityofquantizaters. QuantizersDesignTimeRunningTimeperSampleMemory UniformQuantizersInv3O(1) ProposedModeIqI(x)NlogNO(N)CompandingN4O(1)AdaptiveModeIICompandingN4O(1)ModeIIIqIII(x)N=2log(N=2)O(N)QuantizerCompandingN4O(1)theLloyd-MaxQuantizerforGMM2kN(n+1)logNO(N) (Nisthenumberofquantizationlevels;InvdenotesthecomplexityofcomputingtheinverseofCDF;kisthenumberofiterationsinLloyd-Maxalgorithm;nisthenumberoftrainingsamplesinLloyd-Maxalgorithm.) 81

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A B C Figure3-11. PerformancecomparisonbetweendifferenttonemappingalgorithmsonHDRimagempi atrium(copyrightbyRafalMantiuk).(A)Li'salgorithm[ 84 ].(B)Histogrambasedalgorithm[ 77 ].(C)Ourproposedalgorithm. Table3-3. HistogramsofimagesobtainedbyLi'salgorithm[ 84 ](therstrow)andouralgorithm(thesecondrow) RComponentHistogramGComponentHistogramBComponentHistogram Thehorizontalaxesarenormalizedintotherange[01]. 82

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A B Figure3-12. VisualperformancecomparisonbetweendifferenttonemappingalgorithmsonHDRimagechairs.(A)Li'salgorithm[ 84 ].(B)Theproposedalgorithm. 83

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CHAPTER4APPROXIMATINGOPTIMALVECTORQUANTIZATIONWITHTRANSFORMATIONANDSCALARQUANTIZATION 4.1ResearchBackgroundQuantization[ 61 ]isacriticaltechniqueforanalog-to-digitalconversionandsignalcompression.Scalarquantizationissimpleandfast,whilevectorquantization[ 58 ]inhighdimensioncouldachievesmallermeansquareerror(MSE)andbetterrate-distortionperformance[ 39 ],byjointlyconsideringallthedimensions,butatthecostofexponentialincreasingquantizerdesigntimeandmorequantizationcomputations,i.e.atthecostofmorecodebookdesignandlookuptime.Toreducethecodebookdesignandlookuptime,alotofresearchhasbeenfocusedontwo-dimensionalrandomvariables(r.v.),especiallythoseincircularGaussiandistributions,sinceGaussiandistributions[ 20 39 ]havealotofelegantclose-formtheorems.TheearliestworkcouldrefertoHuangandSchultheiss'smethod[ 66 ],whichquantizeseachdimensionofrandomvariableswithseparateLloyd-Maxquantizers[ 97 ].Itisefcientandeffective,butdenitelycouldbeimproved.Later,Zador[ 163 ]andGersho[ 57 ]studiedquantizationbyusingcompanderswithalargenumberofquantizationlevels.Theyusedacompressortotransformthedataintoauniformdistribution,andthenappliedtheoptimalquantizersfortheuniformdistribution,andthentransformthedatawithanexpander.Butthisschemedoesnotworkwellunderasmallnumberofquantizationlevels.Anothermajormethodfordesigningquantizersforcirculardistributionsusespolarcoordinates.Polarquantizationincludesseparablemagnitudequantizationandphasequantization.TheoptimalratiobetweenthenumberofmagnitudequantizationlevelsandthenumberofphasequantizationlevelsarestudiedbyPearlman[ 108 ]andBucklewetal.[ 21 22 ],andanMMSErestrictedpolarquantizerisimplementedbyusingauniformquantizerforthephaseanglesandascaledLloyd-MaxRayleighquantizerforthemagnitude.Butthisschemedoesnotconsiderthecenterofacirculardistributionasaquantizationlevel,thus,itsMSE 84

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performanceissometimeworsethanrectangularquantizersandotherlatticequantizers,anditdoesnotworkwellforellipticaldistributions.Wilson[ 153 ]proposedaseriesofnon-continuousquantizationlatticeswhichprovidealmosttheoptimalperformanceamongpolarquantization.Itisakindofunrestrictedpolarquantization,butwithoutDirichletboundaries.Peteretal.[ 135 ]improvedWilson'sschemebyreplacingarcboundarieswithDirichletboundaries.HeshowedtheoptimalcircularlysymmetricquantizersforcircularGaussiandistributions.MostofthesepreviousworksconcentrateonGaussiandistributions,andprovidenumericalresultsonlyforGaussiandistributions.AlthoughGaussiansourceisconsideredastheworstcasesourcefordatacompression,whichisinstructivetoconstructarobustquantizer[ 27 ],itisfarfromtheoptimalforquantizingotherdistributions.Theydidnotconsidertheellipticaldistributionsneither,whoseoptimalquantizersaredifferentfromthoseforcirculardistributions.Also,theydidnotprovideauniedframeworkforarbitrarydistributions.Therefore,theoptimalquantizersforotherdistributionssuchasLaplaciandistributions,ellipticalGaussianandLaplaciandistributionsneedinvestigation.Toaddresstheseproblems,weproposeauniedquantizationsystemtoapproachtheoptimalvectorquantizersbyusingtransformsandscalarquantizers.Theeffectoftransformsonsignalentropyandsignaldistortionisdiscussed,especiallyforunitarytransformsandvolume-keepingscalingtransforms.Theoptimaldecorrelationtransformisillustratedwhichturnsamemorysourceintoamemorylesssourceinanidealcase.Thenwefocusonthequantizerdesignformemorylesscircularandellipticalsources.Thetri-axiscoordinateframeisproposedtodeterminethequantizationlattice,i.e.thepositionsofquantizationlevels,inspiredbythewell-knownoptimalhexagonlatticefortwo-dimensionaluniformlydistributedsignals[ 68 ].Itprovidesauniedframeworkforbothcircularandellipticaldistributions,andencompassesthepolarquantizationasaspecialcase.Theproposedquantizerisalsoakindofadaptiveelasticlatticequantizer. 85

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WewillpresentthesimpledesignmethodologyandutilizetheLloyd-Maxquantizersforthecorrespondingone-dimensionaldistributions.Theoptimalityofthisschemeisveriedonelliptical/circularGaussianandLaplaciandistributions.Themethodologydescriptionandexperimentsarefocusedonthebivariaterandomvariables,andtheextensiontohighdimensionalrandomvariablesisalsodiscussed.Theadvantagesofourschemeincludethefollowing: 1. Itprovidesanelegantquantizationlatticeforarbitrarynumberofquantizationlevels,especiallyforprimenumbers. 2. ItalmostalwayshavesmallerMSEthantheotherquantizers. 3. Itconsidersbothmemorylessandmemorysourcewitharbitrarydistributions,whichincludecirculardistributions,ellipticaldistributionsandmixeddistributions. 4. Itisunderauniedframeworkoftri-axiscoordinateframe. 5. Ithassmalldesignandimplementationcomplexity.Therestofthechapterisorganizedasbelow.Section 4.2 presentsthepreliminariesofourproposedquantizer.Section 4.3 describesthesystemarchitectureoftransformplusscalarquantizationtoapproachtheoptimalvectorquantizer.ThepreprocessingwithtransformsisdiscussedinSection 4.4 todecorrelatesignals.InSection 4.5 ,wepresentthetri-axiscoordinateframe,andthemethodologytodesigntheoptimalscalarquantizerforbothcircularandellipticaldistributionsindetail.ExperimentalresultsareshowninSection 4.6 .Finally,Section 4.7 concludesthechapter. 4.2Preliminaries 4.2.1n-dimensionalMMSEQuantizerandScalingLawUsually,therearethreetoolstoevaluatetheperformanceofaquantizer.Firstly,meansquareerror(MSE)betweeninputsignalXandthereconstructedsignalX,whereX,X2n,isconsideredasfollowing, MSE=E[(X)]TJ /F8 11.955 Tf 13.79 2.92 Td[(X)2](4) 86

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Signal-to-Noiseratioisanotherevaluationtool. SNR=jj MSE(4)whereisthecovariancematrixofX,andjjisthematrixdeterminantoperator.Therate-distortioncurveisthethirdone.Lloyd-Maxquantizer[ 58 97 ]isanMMSEquantizer.Forone-dimensionalsignals,lettk(k=0,,N)denoteboundarypointsofquantizationintervals,andletrk(k=0,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1)denotequantizationlevels.ThenLloyd-Maxquantizerischaracterizedby: ftk,rkg=argminftk,rkgMSE=argminftk,rkgN)]TJ /F5 8.966 Tf 6.97 0 Td[(1k=0Ztk+1tk(x)]TJ /F3 11.955 Tf 10.95 0 Td[(rk)2fX(x)dx(4)wherefX(x)istheprobabilitydensityfunction(pdf)ofX,Nisthenumberofquantizationlevels.From( 4 ),wehavethecentroid,andthenearestneighborconditions: tk=rk)]TJ /F5 8.966 Tf 6.97 0 Td[(1+rk 2,k=1,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1,(4)and rk=Rtk+1tkxp(x)dx Rtk+1tkp(x)dx,k=0,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1,(4)[t0,tN]istherangeoftheinputofquantizers.TheLloyd-Maxquantizerforone-dimensionalGaussiandistributionwithzeromeanandunitvariancehasbeenwellstudied.GiventhenumberofquantizationlevelsN,theLloyd-Maxquantizerforzeromean,unitvarianceGaussianisshowninthetablesin[ 68 ].GiventheLloyd-Maxquantizerforzeromean,unitvarianceGaussian,wecanusetheafnelawinProposition 4.1 toobtaintheLloyd-MaxquantizerforGaussiandistributionwitharbitrarymeanmandarbitraryvariances2. 87

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Proposition4.1. (AfneLaw)Forar.v.Xwithzeromean,assumethatitsN-levelLloyd-Maxquantizerisspeciedbytk(k=0,,N)andrk(k=0,,N)]TJ /F8 11.955 Tf 11.13 0 Td[(1).Thenforr.v.Y=)]TJ /F28 6.974 Tf 8.16 3.53 Td[(1 2X+m,withmeanmandcovariancematrix,where=cI,c>0,itsLloyd-Maxquantizerisspeciedbytk=)]TJ /F28 6.974 Tf 8.16 3.53 Td[(1 2tk+m(k=0,,N)andrk=)]TJ /F28 6.974 Tf 8.16 3.53 Td[(1 2rk+m(k=0,,N)]TJ /F8 11.955 Tf 10.95 0 Td[(1).ForaproofofProposition 4.1 ,seeAppendix B.3 .Itisindicatedthatforarandomvariableobtainedfromanotherrandomvariablebyanafnetransform,theoptimalquantizercouldbeobtainedfromtheoriginalquantizerbythesameafnetransform.Later,wefocusourinvestigationonrandomvariableswithzeromeansanddiagonalcovariancematrices. 4.2.2CircularandEllipticalDistributionsThesourceinacirculardistributionismemorylesssource.Eachdimensionofdataisinexactlythesameone-dimensionaldistribution.Forexample,atwo-dimensionalcircularGaussiandistributionwithunitaryvarianceisshownasfollowing: f(x1,x2)=1 2pe)]TJ /F26 6.974 Tf 8.16 6.03 Td[(x21+x22 2(4)Itcouldalsoberepresentedinpolarcoordinateframeasfollowing: f(r,f)=1 2pre)]TJ /F26 6.974 Tf 8.16 3.53 Td[(r2 2(4)ThatiswhyLloyd-MaxquantizerforRayleighdistributionispreferredtoquantizesignalmagnitudesinpolarquantization.Thesourceinanellipticaldistributioncouldbememorylesssourcewitherectprincipleaxes,andmemorysourcewithskewedprincipleaxes.Memorysourcecouldbedecorrelatedwithunitarytransformsintomemorylesssourcewhosecomponentineachdimensionisinthesamedistributionbutwithpossibledifferentvariances.The 88

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memorylessellipticalGaussiansourcecouldberepresentedasfollowing: f(x1,x2)=1 2ps1s2e)]TJ /F29 6.974 Tf 8.16 5.05 Td[((x1)]TJ /F54 6.974 Tf 1 0 .167 1 262.48 -34.76 Tm[(m1)2 2s21e)]TJ /F29 6.974 Tf 8.16 5.05 Td[((x2)]TJ /F54 6.974 Tf 1 0 .167 1 315.7 -34.76 Tm[(m2)2 2s22(4)Uniformdistributionisbothanordinarycirculardistributionandanordinaryellipticaldistribution. 4.2.3IdealUniformDistributionandOptimalTwo-dimensionalHexagonLatticeVectorquantizationaimstoMMSE,butitscomputationalcomplexityisreallyhighandincreasesexponentiallywiththenumberofquantizationlevels.Foruniformdistribution,wegenerallyacceptthattheoptimalVQsareregularvoxelsinthesourcedomain.Whereas,duetonitedomainconstrain,regularvoxelscannotjustpadthespacebyexactlyintegernumberofvoxels.Therefore,theVQsofuniformdistributionsfoundbyvariousalgorithmscomposeofirregularvoxels,whicharedegeneratedfromtheregularones.Webuildourtheoremontheinnitedomain,i.e.innitedynamicrangetoavoidboundarydilemma.Thus,wedeneidealuniformdistributionintheinnitedynamicrangeasfollows. Denition1. IdealUniformDistributionGivendomainofdistributionwhosevolumeisV,theuniformdistributionis f(x)=8><>:1 V,x2V,0,o=w.(4)WhenV!,thenf(x)isinidealuniformdistribution.ThereisnoboundaryconstrainforoptimalVQdesigninginidealuniformdistributioncase. Lemma1. TheoptimalVQforthetwo-dimensionalidealuniformdistributionisregularhoneycombasshowninFigure 4-5 89

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4.3SystemArchitecture 4.3.1QuantizationforCompressionThegeneralcodingsystemusuallyincludestransform,quantizationandentropycodingasshowninFigure 4-1 .Theoptimaltransformcouldsimplifyvectorquantizationschemeintoscalarquantization,evenreplaceentropycodingfromthecodingsystembyxedlengthcoding.Rate-DistortioncodeisoptimalcodeproposedbyShannon[ 39 ].Itisanoptimalvectorcodewhenblocklengthn!.Onlyisitsexistenceknown,butnotitsdesigninageneralcase.VectorquantizationhastheabilitytoapproachRate-DistortionboundwhennumberofquantizationlevelsN!,butisoverwhelmedbytheexponentiallyincreasingcomplexity.Therefore,vectorquantizationisdesiredtobereplacedbytransformfollowedbyscalarquantizationwiththesameRate-DistortionperformancebutmuchlessdesignandimplementationcomplexityasadoptedbyageneraltransformcompressionsystemshowninFigure 4-1 .Therefore,anoptimaltransformplusanoptimalscalarquantizergivesusanewpromisingguidelinetoachieveRate-DistortionboundasstudiedinthenextSections. 4.3.2TheoremandSystemFrameworkTherefore,weclaimastatementasfollowing: Theorem4.1. TheMMSEvectorquantizationcouldbeachievedbyatransformfollowedbyscalarquantization.ItwillbeprovedinSection 4.5 .FollowingTheorem 4.1 ,weproposesystemarchitectureasshowninFigure 4-2 .Avectorquantizerisimplementedbytransformsandscalarquantization.Thetransformswefocusonarelineartransformswithhighdecorrelationability.Wewilldiscusstheunitarytransforms,volume-keepingscalingtransformsandtheoptimaldecorrelationtransformsinSection 4.4 .Thescalarquantizerisimplementedinthetri-axiscoordinateframewhichwillbedescribedindetailinSection 4.5 .ThetransformplusscalarquantizationhastheadvantageofpossiblesmallcomplexityandgoodR-Dperformance,butstillhastradeoffbetweencomplexity,rate 90

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anddistortion.Therefore,thesystemdesignshouldconsistwiththeC-R-Dtheory.Crepresentscomplexity,Rrepresentsrate,andDrepresentsdistortion.BestR-Dperformancewithleastcomplexityisdesired.Thissystemisexible,inwhichthecompandingtechniquecouldalsobepluggedasshowninFigure 4-3 .Aswewillpointoutlater,thecompandingtechniqueisasymptoticallyoptimal,butmaynotworkwellinlowratesituation.Butourtri-axiscoordinateframeworksalmostuniversally. 4.4PreprocessingwithTransformsTransformishelpfulforquantization.Anymappingisatransform.Nonlineartransformswillintroducenonlinearerrorafterquantization.Therefore,lineartransformisconsideredinthissection.Topreserveconstantenergy,lineartransforms,representedbymatriceswithunitarydeterminant,arefocusedon,suchasunitarytransformsandvolume-keepingscalingtransforms. 4.4.1UnitaryTransformsUnitaryTransformsarerotationsinEuclideanspace,aimingathighdecorrelationability.WeknowthatKarhunen-Loevetransform(KLT)dependentonsignalsisoptimalinthesenseofthehighestdecorrelationabilityforniteblocklengthsignals.WhileDCTisaxedtransformandagoodsubstitutionofKLT. Lemma2. MeanSquareErrorisinvariantunderunitarytransforms.ForaproofofLemma 2 ,seeAppendix B.4 Lemma3. TheMMSEvectorquantizersofrandomvectorsafterunitarytransformationaretheMMSEvectorquantizersoftherandomvectorafterthesameunitarytransforma-tion.Lemma 3 iseasytoobtainfromLemma 2 ,andisarotation-invariantpropertyofMMSEvectorquantizers. 91

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Lemma4. ThesumofentropyofeachcomponentofrandomvectorXwilldecreaseafterdecorrelationbyunitarytransforms.ForaproofofLemma 4 ,seeAppendix B.5 4.4.2ScalingTransformsFromAfneLawinProposition 4.1 ,weknowthattheMMSEquantizerswillundergothesimilarexpansionorshrinkageasinputsignals,whosedimensionsarescaledbyasamefactor.Thesescalingtransformswillincreaseordecreasesignalvolumn.Thus,westeertoVolume-keepinguniformscalingtransform. Denition2. Volume-keepinguniformscalingtransformcouldberepresentedbyadiagonalmatrixwithunitarydeterminant,i.e.,theproductofdiagonalelementsofthediagonalmatrixisoneasfollowing:0BBBBBBB@a1000a20.........00an1CCCCCCCAwhereni=1ai=1.UsuallytheMSEandenergyofsignalswillchangeaftervolume-keepingscaling,sinceni=1x2i6=ni=1a2ix2i.Therate-distortiontheoryrequirestheMSEuniformlydistributedamongeachcomponentofrandomvector,iftheMSEdoesnotexceedthevarianceofthatcomponent.Therefore,theMMSEvectorquantizerforaellipticaldistributioncouldnotbeobtainedfromtheMMSEvectorquantizerforacirculardistributionbyasimplescaling.Theyshouldbeconsideredseparately. Lemma5. ThesumofentropyofeverycomponentofarandomvectorinGaussianorLaplacedistributionskeepsconstantaftervolume-keepingscalingtransform.ForaproofofLemma 5 ,seeAppendix B.6 92

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4.4.3OptimalTransformforArbitraryDistributionsSincethevolume-keepingscalingtransformsperturbtheMSEofeachcomponent,weonlyconsiderunitarytransformsbeforequantization.Ifanarbitrarydistributionisconsidered,themostpowerfuldecorrelationcannotbeachievedbyasimpleunitarytransform,sinceunitarytransformdecreasetheintra-component/blockcorrelation,butitdidnotconsidertheinter-component/blockcorrelation.AnarbitrarydistributioncanbeapproximatedbyaGaussianMixtureModel(GMM)foundbyexpectationmaximization(EM)algorithm.ThepdfofaGMMrandomvariableXisgivenasbelow: fX(x)=Ngi=1pigi(x)(4)whereNgisthenumberofGaussiancomponentsintheGMM;gi(x)isaGaussianpdfforcomponenti(i=1,,Ng)shownasellipsesinFigure 4-4 ;pidenotestheprobabilityofcomponenti(i=1,,Ng);andNgi=1pi=1.Assumethedatacomefromthepixelpairsinagrayimage.ThenGaussiancomponentsofdataalmostfallalongdiagonalduetothecorrelationbetweenpixelpairs.Therststepisintra-componentdecorrelation,thesecondstepisinter-componentdecorrelation.Thetwostepscouldchangeorder.LaterthescalarquantizerscouldbeappliedtoeachdecorrelatedGaussiancomponent. 4.5OptimalScalarQuantizersinTri-axisCoordinateFrameAftertransforms,weobtainrandomvectorswithindependentcomponents.ForTheorem 4.1 ,one-dimensionalvectorquantizationisscalarquantization.Notransformisneeded.Itisatrivialcase.Fortwo-dimensionalvectorquantization,wewillprovethistheoreminTri-axiscoordinatesystem.Forhighdimensionalvectorquantization,multi-axiscoordinatesystemisneeded. 93

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4.5.1Tri-axisCoordinateFrame Denition3. Thetri-axiscoordinateframeintwo-dimensionalspacehasthreeaxesX,YandZ,andthereare120anglesbetweenthethreeaxesX,YandZ.Thetri-axiscoordinateframeisshowninFigure 4-5 .Everypointinthetwo-dimensionplanecouldberepresentedinthistri-axiscoordinateframe.Forsomesymmetricaldistribution,twoaxesoroneaxisissufcient.Forexample,theoptimaltwo-dimensionalvectorquantizationforuniformdistributioncouldbedeterminedbytwoaxesX,Yhaving120inbetweenspanningahexagonallattice.Theoptimaltwo-dimensionalvectorquantizationforcirculardistributionalsoneedstwoaxes,oneofwhichdeterminesthemagnitudequantization,anotheronedeterminesthephasequantization.Wewillshowtheminnextsubsections. 4.5.2Tri-AxisSystemforUniformDistributionItiswellknownthattheoptimalvectorquantizerforuniformdistributionsintwo-dimensionalspaceisregularhoneycomb[ 68 ],whichisfromthegeometryofnumbers,alsofromdiscretegeometryintheEuclideanplane.Wewillimplementitwithscalarquantizationintri-axissystemasshowninFigure 4-6 Proposition4.2. Hexagonlatticeintri-axissystemisstillRate-Distortionoptimalforuniformdistribution. Proof. 1. Vectorquantizationlevelsarethecentroidsofthehexagons.Thecentroidofeachhexagonoftheoptimalquantizercouldberepresentedbyaxedlengthcode.R=log2N,whereNisthenumberofquantizationlevels. 2. Everypointintwo-dimensionalspacecouldberepresentedbythevector~r=c1~r1+c2~r2,asshowninFigure 4-6 ,where~r1and~r2arethebasisvectorsofVQ.c1andc2areintegersanduniformlydistributedifthecentroidisuniformlydistributed. 3. Scalarquantizerscomposeoftwoindependentscalarquantizersalongtwoaxes~r01and~r02.Everypointcouldberepresentedbytwoindicesinthecodebook.Theindicesareobtainedbyprojectingthepointtothenearestcodeontheaxes.Forexample,apointrepresentationis~r=x~r01+y~r02whichisquantizedto~r=xm~r01+yn~r02. 94

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4. Let~r1=~r01and~r2=~r02,thenwehavec1=xmandc2=ym.TheMSEdistortionofbothSQandVQaretheinnerproductsofvectors((x)]TJ /F3 11.955 Tf 11.15 0 Td[(xm)~r01and(y)]TJ /F3 11.955 Tf 11.15 0 Td[(ym)~r2)0,((x)]TJ /F3 11.955 Tf 10.95 0 Td[(c1)~r1and(y)]TJ /F3 11.955 Tf 10.95 0 Td[(c2)~r2).Thus,VQandSQhavethesamedistortion. 5. Foruniformdistribution,onlyonecodebookisneededfortwoaxes~r01and~r02.Theindicescouldbexedlengthcoded.Ri=log2Ni,whereNiisthenumberofquantizationlevelsalongaxisi(i=1,2). 6. N1N2=Nasymptotically,i.e.R1+R2=log2N1+log2N2=log2N=R,VQandSQhavethesamerate. 7. Therefore,SQandVQarewiththesameR-Dperformance.Therefore,theoptimalMMSEvectorquantizationcouldbeachievedbyatransform(anidenticaltransform)followedbyuniformscalarquantizationfortwo-dimensionalidealuniformdistribution. Inthisway,SQandVQhavethesameR-Dperformance,whilethecodebooksizeofSQisaroundthesquarerootofthatofVQ.BecauseofthereducedcomplexitybroughtbySQ,wetrytouseSQtoreplaceVQinthischapter.Anotherthingneededtomentionisthatthenumbersofquantizationlevelsforhexagonlatticeareprimenumbers1,7,19,37,,growingalongcircleswithlargerandlargerradius,whichcouldbefoundinFigure 4-7 4.5.3Tri-AxisCoordinateFrameforCircularandEllipticalDistributionsHowaboutthedistributionisnotuniform,whatwilltheshapeofoptimallatticebe?Thisistheinverseproblemofndingthetransformfromnon-uniformdistributiontouniformdistribution. 4.5.3.1ElasticquantizationlatticesWewillshowtheelasticquantizationlatticesforcircularandellipticaldistributions.Fromtheoptimalhexagonlatticesforauniformdistribution,westatethattheoptimalvectorquantizerforacirculardistributionformsanexpandedhexagonallattice,asshowninFigure 4-7 .Theexpansionratiobetweentheoptimallatticeforatwo-dimensionalcirculardistributionandthatofatwo-dimensionaluniformdistribution 95

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alongtheradiusdirectionmayapproximatelyfollowtheexpansionratiobetweentheLloyd-Maxquantizerforthecorrespondingone-dimensionaldistributionandthatofaone-dimensionaluniformdistribution,i.e.theLloyd-Maxquantizerforthecorrespondingone-dimensionaldistribution. 4.5.3.2DesignmethodologyWerstlyfocusonthepositionsofquantizationlevelsofatwo-dimensionalvectorquantizer.Thelatticepatternsoftheproposedquantizer,wherequantizationlevelsfallon,aredeterminedbeforehand,asshowninFigure 4-10 andFigure 4-11 .Thequantizationlevelsapproximatelyfallonthecentroidsofthelattice,whichareuniformlydistributedineachannulus.Werestrictthemtobeinthesamecircleforsimplicity,andeachquantizationregionoflatticedonotneedtobehexagon.Indifferentannuluses,quantizationlevelsarestaggeredarrangedsimilartotherotatedpolarquantization[ 135 ].Theoptimaldistancebetweenthequantizationlevelsandtheorigin,i.e.,themagnitudequantization,isdeterminedbyweightingtheLloyd-Maxquantizerofthecorrespondingone-dimensionaldistributionwiththeunitary-variance.FormorepreciselocationsofMMSEmagnitudequantizationlevels,theyarefurthersearchedoutwardalongradialdirectionintermsofMMSE.Tobespecic,foracirculardistribution,itspdfcouldberepresentedinpolarcoordinateframeasf(r,q)=f1(r)f2(q).Foranarbitraryellipticaldistribution,thedatacouldbetransformedbyunitarytransformsintoadistributionwhoseprincipleaxesareparalleltothecoordinateframes,andthenshiftedtotheorigin.Thenthedistributionscouldbeuniformlyrepresentedbythefollowingequationincartesiancoordinatesystem: ni=1x2i b2i=1(4)b1=b2==bn=bforcirculardistributions;bisarenotallequalforellipticaldistributions.Forcirculardistributions,f1(r),f2(q)areseparable,whileitisnotforellipticaldistributions.Theweightingeffectfromb1andb2forellipticaldistributionsis 96

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important.Ifthequantizersofcirculardistributionsisusedforellipticaldistributions,theresultedMSEpereachdimensionhasratioofb21 b22.Whereas,fromShannonRate-Distortiontheory(i.e.reversewaterling),weknowthattheiftheMSEislessthanthevarianceofeachcomponent,bitrateshouldbeallocatedsuchthattheMSEperdimensionisnearlyequal.Therefore,weshoulduseb1andb2toweightquantizationlevelstowardsthiswayforellipticaldistributions.Themagnitudequantizationisnon-uniform.Forbothcircularandellipticaldistributions,thetwo-dimensionalquantizationlevelsfalloneachcircleorovalcouldberepresentedbythecoordinates(cb1cosq,cb2sinq)shownasstarsinFigure 4-9 .cincreasesnon-uniformlyalongtheradialdirection.ccouldbedeterminedbysearchingoutwardstartingfromLloyd-MaxquantizationforGaussiandistributionalongradialdirection.Theuniformphasequantizationisoptimalforcirculardistributions,butmaynotforellipticaldistributions.Wetakeuniformphasequantizationforbothkindsofdistributions,sincetheoptimalphasequantizationforellipticaldistributionsisalittleperturbationfromtheuniformquantization.Wewillshowitssub-optimalityforellipticaldistributioninexperiments.AsshowninFigure 4-10 ,thenumberofquantizationlevelsineachannulusis1,6,12,18,similartothatoftheregularhexagonlattice.Withineachmagnitudeannulus,thekphaseregionsallhaveequalsize,whoseboundariesarerepresentedasfollowing: (j)]TJ /F8 11.955 Tf 10.95 0 Td[(1)2p kq
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4.5.3.3ThenumberofquantizationlevelsineachannulusHowmanyquantizationlevelsshouldweassigntoeachannulus?Previously,fortherestrictedpolarquantization[ 108 ],quantizationlevelsNisfactorizedintoN=NqNr,whereNqisthenumberofquantizationlevelsineachannulus,andNristhenumberofannuluses.AlthoughtheoptimalratiobetweenNqandNrisstudied,somenumberNcannotbeperfectlyfactorized,nottomentiontheprimenumber.Thisdifcultyalsoliesintheunrestrictedpolarquantization[ 153 ].Thenon-continuityofquantizationpatternsexistsinallthepreviousworks.Itisalsoanimperfectioninourschemes.Wehavetwoschemestoarrangemagnitudequantizationlevelsvs.phasequantizationlevels.Therstoneistheoptimalarrangementaccordingtoquantizationlevels,similartoWilson's[ 153 ].Thesecondoneisprogressivesemi-continuousscheme.Ourquantizerdesignandoptimizationmethodologyismuchsimplerthanthatoftheunrestrictedpolarquantization.Therstschemeallowsfreedominthenumberofphasesassignedateachmagnitudelevel.Theoptimalpatternsarederivedfromexperiments,whicharecoincidentwithWilson'sscheme[ 153 ]butwithbetterperformanceandDirichletboundaries,asshowninFigure 4-12 .Anotheroneistheprogressivequantizationscheme[ 115 ]asshowninFigure 4-13 .ThenumberofannulusesLincreaseswhenthenumberofquantizationlevelsN=1,7,19,,1+6(1+2+3+).DenesetNL=f1,7,19,,1+6(1+2+3+)gandNL(l)isthel-thelementinsetNL.ThatisthenumberofannulusesLisdeterminedby: L=8><>:inffl:N)]TJ /F8 11.955 Tf 10.95 0 Td[(3lNL(l)g,N4inffl:N+7)]TJ /F8 11.955 Tf 10.95 0 Td[(6lNL(l)g,o=w(4)whereinfistheinmum.Therefore,thequantizercouldbeimplementedprogressivelyalongtheincreaseofN.Thepreviouslylocatedquantizationlevelsneednotchangetheirrelativepositions,onlytheirmagnitudesshouldbeshrinkedalittleassuggested 98

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bytheLloyd-Maxquantizerofone-dimensionalGaussian.Orhierarchically,wecouldfurtherquantizeeachexistingquantizationregionwithourscheme.Comparingthetwoschemesofquantizationlatticepatterns,thequantitativedescriptionsoftherstoptimalschemearedifculttoprovide.ForsmallN,thesecondschemehasperformanceclosetotherstscheme,althoughwithpossibledifferentlatticepatterns;forlargeN,theirperformancedifferencedecreases,andthequantizationpatternsofthesecondschemeisasymptoticallyapproachesthoseoftherstscheme.Schemeoneandschemetwohavealotofcommonquantizationlatticepatterns. 4.5.3.4ExpansionruleHowfarawayistheexpansionrulealongradialdirectionfortwo-dimensionaldistributionsfromtheLloyd-Maxquantizerforthecorrespondingone-dimensionaldistributions?Takegaussiandistributionasanexample.Fortwo-dimensionalGaussianwithjointdensitygivenby: PX(x1,x2)=1 2pexpf)]TJ /F3 11.955 Tf 16.47 8.23 Td[(x21+x22 2g)(4)where)]TJ /F7 11.955 Tf 9.29 0 Td[(
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Gaussiandistribution. N1=8><>:2L,L=12L)]TJ /F8 11.955 Tf 10.95 0 Td[(1,L2(4)ThentheexpansionruleforrinEq.( 4 )withLannulusesisfoundintableoftheLloyd-Maxquantizerforone-dimensionalGaussianwithN1quantizationlevels.Forexample,N=7,L=2caseasshowninFigure 4-12 .ItcorrespondstoN1=3ofthequantizerforone-dimensionalGaussian,i.e.,r1=0,r2=1.2240.Thenhowfarawayofr1=0,r2=1.2240fromtheoptimalr1,r2?Consideranupperboundofthedifferencebetweenr1andr1whenL=1.TheradiusexpansionisfollowingtheruleforRayleighdistribution.ButthereisnoquantizationlevelatoriginforRayleighdistribution,sowehavetoutilizethequantizerforGaussiandistributionforourquantizer.R0Rq1q0r2 2pe)]TJ /F26 6.974 Tf 8.16 3.53 Td[(r2 2dqdr R0Rq1q0r 2pe)]TJ /F26 6.974 Tf 8.16 3.53 Td[(r2 2dqdr=r p 2TheLloyd-Maxquantizerforone-dimensionalGaussiandistributionasfollowing:R0x p 2pe)]TJ /F26 6.974 Tf 8.16 3.53 Td[(x2 2dx R01 p 2pe)]TJ /F26 6.974 Tf 8.16 3.53 Td[(x2 2dx=r 2 pThentheupperboundofdifferenceisaround0.46forunitary-variancedistributions.AsNbecomeslarger,thesedifferencesbecomesmallers.Thesearealsothemaximalsearchingrangestondtheoptimalmagnitudequantizers.TheLloyd-Maxquantizerforone-dimensionGaussiancaseisgoodinitialforndingtheoptimalmagnitudequantizationforone-dimensionaldistributions.Themagnitudequantizationisalmostindependentofthephasequantization.Itmeansthatwhenphasequantizationchanges,themagnitudequantizationsuffersalittleperturbationatmost. 100

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4.5.4GeneralizationtoGMMorLMMWecanuseEMalgorithmtoidentifythecomponentsinGaussianMixtureModel(GMM)orLaplacianMixtureModel(LMM).Thenforeachcomponent,theproposedtransformsplusscalarquantizercouldbeusedtoreplacevectorquantizerstoapproximatetheoptimalquantizationperformance. 4.5.5GeneralizationtoHighDimensionThedodecahedronistheoptimalvoxelofvectorquantizationforsignalsinthree-dimensionaluniformdistribution[ 9 ].Thethree-dimensionalsix-axiscoordinatesystem(sixaxesforsixparallelsurfacepairsofdodecahedron)canbebuiltupsimilarlyasshowninFigure 4-14 ,anditscoordinates(a,b,c,d,e,f)havethreeindependentcomponents.Analogously,vectorquantizationcouldbeapproximatedbytransformsplusscalarquantizationforthree-dimensionaldistributions.Forevenhigherdimensionalspace,oncetheoptimalvoxelofvectorquantizationforsignalsinuniformdistributionisobtained,wecanobtaintheoptimaltransformsplusscalarquantizationtoreplacevectorquantizationaccordingly. 4.6ExperimentalResultsandDiscussionsInthissection,wewillrstshowthebasicpropertiesoftheproposedscalarquantizer.LateronwewillshowexperimentalresultsonmemorylesssourceinunitarycircularGaussianandLaplacedistribution,anderectellipticalGaussianandLaplacedistributionwithb1=2,b2=1.WewillcompareMSEandtheRate-Distortionperformanceofourproposedquantizersbasedontherstschemewithoptimalquantizationlattices,theunrestrictedpolarquantizers[ 153 ](indicatedby`UPQ'),therestrictedpolarquantizers[ 108 ](indicatedby`PQ'),therectangularquantizers[ 66 ](indicatedby`Rectangular').Theratehereisdenedaslog2N 2.DistortionisMSEperdimension.Thebenchmarkis 101

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Rate-DistortionfunctionfortheGaussianmemorylesssource: R(D)=1 2log2(1 D)(4)where0
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N,andgraduallyslowdown.Thisgivesusaguidanceonhowtotunetheoptimalmagnitudequantizationlevels. 4.6.2CircularGaussianDistributionWeshowtheR-Dperformanceofdifferentquantizersonauni-variancecircularGaussiandistributioninFigure 4-17 .FromFigure 4-17 ,wecanseethattheR-DperformanceofourproposedquantizersisalwaysalittlebetterthanthatofUPQs,andmuchbetterthanthatofPQsandthatofRectangularquantizers.TheyhavethesameR-DperformancewhenN=4,duetothesamequantizationlevels.RectangularquantizersmayhavebetterperformancethanPQswithsomen2quantizationlevels. 4.6.3EllipticalGaussianDistributionWeshowtheR-DperformanceofdifferentquantizersonanellipticalGaussiandistributioninFigure 4-18 .FromFigure 4-18 ,wecanseethatUPQsdoesnotconsiderthedifferentvariancesamongdifferentrandomvectorcomponents,thusdonotperformwell.Ourproposedquantizersalmostalwaysperformbetterthanrectangularquantizers,exceptwhenN=8.SinceN=8=42isthebestfactorizationfortherectangularquantizeronellipticaldistributionswhenratioofdatacomponentvariancesequals2.Whereas,forotherNnon-factorable,rectangularquantizersperformmuchworseasexpected,althoughwedonotplotitinthegure. 4.6.4CircularLaplaceDistributionWeshowtheR-Dperformanceofdifferentquantizersonauni-variancecircularLaplacedistributioninFigure 4-19 .ItindicatesinFigure 4-19 thatourproposedquantizersalwaysperformalittlebetterthanUPQs,andmuchbetterthanPQsandrectangularquantizers. 4.6.5EllipticalLaplaceDistributionWeshowtheR-DperformanceofdifferentquantizersonanellipticalLaplacedistributioninFigure 4-20 .ItindicatesinFigure 4-20 thatourproposedquantizersalwaysperformbetterthanUPQs,andbetterthanrectangularquantizersexcept 103

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whenN=8=42.OurproposedquantizershavepredominantadvantageouswhenN=7,19,37,. 4.6.6Bit-rateSavingWealsoevaluatetheaveragebit-ratesavingofourquantizerscomparingtootherquantizers.Averagebit-rateiscalculatedbyusingBjontegaard'smethod[ 16 106 ]withttingpolynomialsofdegree3.Bit-ratesavingisevaluatedbasedonrelativeaveragebit-rateinpercentageasshowninthefollowingequation. Rc)]TJ /F3 11.955 Tf 10.95 0 Td[(Rp Rp100%(4)whereRcistheaveragebit-rateofthecomparedquantizers,andRpistheaveragebit-rateoftheproposedquantizers.FromTable 4-1 ,wecanseethattheproposedquantizersaves0.4%-24.5%bit-rateonaverage,comparedtounrestrictedpolarquantizers,restrictedpolarquantizersandrectangularqantizers.Wedidnotlisttheaveragebit-rategainoverrestrictedpolarquantizersforellipticaldistributions,whichisevenhigherthanthatoverunrestrictedpolarquantizers. 4.7SummaryInthispaper,weproposedaschemetousetransformationplusscalarquantizationtoreplacetheoptimalvectorquantization.Theunitarytransformsratherthanscalingtransformswereneededfortheoptimalvectorquantizerapproximation.Aftertransformation,scalarquantizationforbothcircularandellipticaldistributionswasstudiedintheproposedtri-axiscoordinatesystem.Theoptimalquantizationlevelswerefoundintheelastichexagonallattices,whichincludedtheoptimalandtheprogressivequantizerlatticepatterns.TheexperimentalresultsshowedthatourproposedquantizersalmostalwayshadbetterperformancethanUPQs,PQsandrectangularquantizersonbothGaussianandLaplacedistributions,especiallywithprimenumberofquantizationlevels.WeachievedO(N2)designcomplexityand0.4%-24.5%bitratesaving,whereNisthe 104

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numberofquantizationlevelsperdimension.Therefore,weclaimedthattransformsplusscalarquantizerscouldapproximatetheoptimalvectorquantizersintermsofR-Dperformancebutwithmuchlesscomputationalcomplexity.Ourfutureworkwillfocusontheoptimalvectorquantizerapproximationinhighdimensionalspaces,andtheapplicationsinimageandvideocoding. Figure4-1. Generalencodinganddecodingpipelinewithtransformsandquantization. Figure4-2. Systemarchitecturewithtransformplusscalarquantization. Figure4-3. Transformplusscalarquantizationwithcompandingtechnique. 105

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Figure4-4. Gaussianmixturemodeldecorrelation. Figure4-5. Two-dimensionaltri-axiscoordinatesystem. Figure4-6. Twodimensionaloptimaluniformvectorquantizer. 106

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Figure4-7. Circularlyexpandedhexagonlatticefortwo-dimensionalcirculardistribution. A B Figure4-8. Ellipticallyexpandedhexagonlatticesfortwo-dimensionalellipticalGaussiandistribution.(A)Horizontalellipticalhexagonallattice.(B)Verticalellipticalhexagonallattice. Figure4-9. Tri-axisframeforageneraltwo-dimensionalellipticaldistribution. 107

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Figure4-10. Expandedhexagonlatticefortwo-dimensionalcircularGaussiandistribution. Figure4-11. Expandedhexagonlatticefortwo-dimensionalellipticalGaussiandistribution. 108

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Figure4-12. Therstoptimalquantizationscheme. Figure4-13. Thesecondprogressivequantizationscheme. 109

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Figure4-14. Voxelforthree-dimensionaluniformdistribution. Figure4-15. MSEperdimensionofquantizationfor10000samplesfromaunitary-variancecircularGaussiandistribution. 110

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Figure4-16. OptimalmagnitudelevelsfordifferentnumberofquantizationlevelsNforunitary-variancecircularGaussiandistribution. Figure4-17. Rate-DistortioncomparisonamongdifferentquantizersforcircularGaussiandistribution. 111

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Figure4-18. Rate-DistortioncomparisonamongdifferentquantizersforellipticalGaussiandistribution. Figure4-19. Rate-DistortioncomparisonamongdifferentquantizersforcircularLaplacedistribution. 112

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Figure4-20. Rate-DistortioncomparisonamongdifferentquantizersforellipticalLaplacedistribution. Table4-1. Averagebit-ratesavingoftheproposedquantizersoverotherquantizers. UPQ PQ Rectangular CircularGaussian 0.36% 6.78% 3.22% EllipticalGaussian 22.4% 16.9% CircularLaplace 0.94% 24.5% 5.62% EllipticalLaplace 19.8% 6.32% 113

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CHAPTER5CONTENTBASEDIMAGEHASHING 5.1ResearchBackgroundDigitalimagesareinanexponentialincreaseduetoproliferationofdigitalcamerasandimageapplications.Thehugenumberofdigitalimagesrequiresefcientclassicationandretrievalofimages[ 43 ].Digitalimagesfacilitatemultimediaprocessing,andatthemeantime,makefabricatingandcopyingofdigitalcontentseasy.Toprotectthecopyrightoftheimages,efcientandautomatictechniquesareneededtoidentifyandverifythecontentofdigitalmultimedia.Besidesthelong-timeestablishedcopyrightprotectiontoolimagewatermarking[ 154 ],imagehashingemergesasaneffectivetooltorepresentimagesandautomaticallyidentifywhetherthequeryimageisafabricationoracopyoftheoriginalone.Asanalternativetoimagewatermarking,imagehashingcanbeappliedtomanyapplicationspreviouslyaccomplishedbywatermarking,suchascopyrightprotection,imageauthentication.Itcanalsobeusedforimageindexingandretrievalaswellasvideosignature.Unlikewatermarking,imagehashingneednotchangetheimagebyinsertingwatermarksintotheimage.Imagehashisashortbinarystring,mappedfromanimagebyanimagehashfunction.Theimagehashfunctionhassuchapropertythatperceptuallyidenticalimagesshouldhavethesameorsimilarhashvalueswithhighprobability,whileperceptuallydifferentimagesshouldhavequitedifferenthashvalues.Inaddition,thehashfunctionshouldbesecure,sothatanattackercannotpredictthehashvalueofaknownimage.Manytechniquesforimagehashinghavebeenproposedintheliterature.Thesealgorithmsaretypicallybasedonstatistics[ 147 ],relations[ 86 ],low-levelimagefeatures[ 13 99 ],non-negativematrixfactorizations[ 100 ],andsoon.FridrichandGoljan[ 55 ]proposedarobustvisualhashingmethod.TheirhashdigestsofdigitalimagesarecreatedbyprojectionsofDCTcoefcientstozero-meanrandomsmoothpatterns, 114

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generatedusingasecretekey.Royetal.[ 120 ]proposedcontent-hashing,whichconsistsconsistsofacompactrepresentationofsomeimagefeatures.Theresultingimagehashisrobusttoimageltering,butsurrenderstogeometricattacksandmaynotbecollisionfree.TheimagehashbasedonScaleInvariantFeatureTransform(SIFT)algorithm[ 48 ]andcompressivesensingtechnique[ 138 ]couldsolvegeometricattacksincertaindegree,butitiscomputationallyexpensive.LinandChang[ 86 ]createdthemutualrelationshipofpairwiseblockDCTcoefcientstodistinguishJPEGcompressionfrommaliciousmodications.Buttheblockbasedmethodisunreliable,sincepossibleshiftingandcroppingoperationsmaychangehashvalues.Venkatesanetal.[ 147 ]proposedanimagehashingtechnique.Theirhashesaregeneratedfromstatisticalfeaturesextractedfromrandomtilingofwaveletcoefcients.However,itonlyallowslimitedresistancetogeometricdistortions,andissusceptibletosomemanipulations,suchasluminancechangeandobjectinsertion.Toaddresstheseproblems,weproposecontentbasedimagehashingusingcompandingandGraycode.Contentbasedimagehashingismorerobusttocontentpreservingimageprocessinglikegeometricandilluminanceattacks,andmoresensitivetomaliciouscontenttamperingattacksthanstatisticsbasedimagehashing.Ourmethodcombinesrobustfeaturepointdetectorandrobustcontentsingularitydescriptoratthesefeaturepoints.Thefeaturepointsarechosenfromcrosspointsoflines,withthek-largestlocaltotalvariations[ 8 ]inimages.Thelocaltotalvariationsaredeterminedbythestructureofimages,androbusttogeometricattacksandluminanceattacks.Therefore,itwillobtainstablesimilarfeaturepointsinperceptualidenticalimagesandisresistanttocontent-preservingattacks.Morletwavelet[ 14 ]isgoodatdescribingthesingularityofsignals.TheMorletwaveletcoefcientsatfeaturepointsareobtainedtorepresentimages.TheMorletwaveletcoefcientsarepseudorandomlypermutedwithasecretekey,whichenhancesthesecurityoftheimagehashingsystem.Morletwaveletcoefcientsarecomputationallyefcientlyquantizedusingcompandingtechnique 115

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accordingtotheprobabilitydistributionofthecoefcients.Thus,theproposedimagehashingisrobusttocontrastchangingandgammacorrectionofimages.Graycodeisusedtobinarilycodethequantizedcoefcients,whichincreasesdiscriminabilityofimagehashes.Therestofthechapterisorganizedasfollowing.InSection 5.2 ,wepresentanoverviewoftheproposedimagehashingsystem.InSection 5.3 ,wedescribehowtoextracttherobustfeatureofimages,whichistheMorletwaveletcoefcientsatfeaturepointswiththek-largestlocaltotalvariations.ThentheMorletwaveletcoefcientsarequantizedandbinarilycodedwithGraycodeasshowninSection 5.4 .Section 5.5 showstheexperimentalresultsthatdemonstratetheeffectivenessandrobustnessoftheproposedimagehashingsystem.Finally,weconcludethischapterinSection 5.6 5.2SystemOverviewImagehashesshouldhavesmallcollisionprobability,andhighdiscriminability.FromtwoinputimagesIandI0,aimagehashingsystemfextractstwocorrespondingbinaryhasheshandh0usingasecretekeyKasshowninEquation( 5 ).ThedistancefunctionsuchasnormalizedHammingdistanceisdenotedbyd(,),anddiscriminativethresholdsaredenotedbyz1andz2. 8><>:h=f(I,K)h0=f(I0,K)(5)Forthedesignofimagehashingsystem,threeobjectivesshouldbeconsidered. 1. 8I,I0,ifI6=I0thend(h,h0)z1; 2. 8I,I0,ifI=I0thend(h,h0)
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Thesecondobjectiveimpliesthatthedistancesbetweensimilarimagesshouldbesmallerthanathresholdz2,wherez2z1,whichensurestherobustnessofimagehashingunderintentionalorunintentionalattacks.Forsimilarimages,itisexpectedthattheimagehashisabletodiscriminateimagesunderintentionalandunintentionalattacksusingathresholdforimageauthenticationpurpose.Thethirdobjectiveprovidestheunpredictabilityofimagehasheswhosebinaryvaluesaredistributedwithequalprobability.OurproposedimagehashingsystemisshowninFigure 5-1 .First,featurepointsinimagesareextracted.Thefeaturepointsareexpectedtobesimilarforsimilarimages,suchthatdistancesbetweenhashesofsimilarimagesaresmallandthatimagehashesarerobustagainstperceptuallypreservingattacks.Weextractfeaturepointswiththek-largestlocaltotalvariations,whichcapturethestructureofimages.Second,weobtaintheMorletwaveletcoefcientsatfeaturepointstodescribethedegreeofsingularityatfeaturepoints.Third,pseudorandompermutationoftheMorletwaveletcoefcientswithasecretekeyincreasesthesecurityandreducesthecollisionprobabilityofimagehashes.Forth,thecoefcientsarequantizedwithcompandingtechniqueandbinarilycodedwithGraycodetoformthenalimagehashes.InverseErrorCorrectionCodingtocompressimagehashesisoptionalinoursystem. 5.3RobustDescriptorofImagesMostinformationofsignalsisconveyedbyirregularstructuresandtransientphenomenaofsignals.Featurepointssuchascornersaresalientcontentdescriptorsofimages.Therearethreestagestoextracttherobustdescriptorofimagesinourproposedmethod. Imagepreprocessing; Findingthelocationsoffeaturepoints; Evaluatingthesingularityofimagesignalsatfeaturepointsbycontinuouswavelettransform. 117

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5.3.1PreprocessingPreprocessingwillchangeimagepixelsandmayinuencethedetectionanddescriptionoffeaturepoints.Wetrytoavoidanychangestoimagesandextracttheoriginalinformationfromimages.Therefore,theonlypreprocessinginourmethodisresizingimagesintothesamesizetofacilitatelateralgorithmsteps. 5.3.2FeaturePointExtractionFordifferentapplicationsandcorrespondingperformancerequirements,differenttechniquestoextractfeaturepointsareexploredintheliterature[ 146 ].Sinceimagehashesshouldbeinvarianttocontent-preservingprocessing,robustrepeatablefeaturepointdetectorswithsmallcomputationsaredesired.Jaroslavetal.[ 74 ]proposedafeaturepointdetectorinblurredimages,whichwecallBFPinthechapter.BFPcanyieldhighrepetitionrateondifferentlydistortedimages.WewillproposeamorerobustfeaturepointdetectorbasedonBFP.BFPistoefcientlydetectpointswhichbelongtotwoedgesregardlesstheirorientations.Itselectspointswiththek-largestlocalvariances.Thelocalvariance(LV)isdenedontheimageblockinEquation( 6 ). LV=X2(I(X))]TJ /F8 11.955 Tf 11.35 2.92 Td[(I)2(5)whereistheimageblockcenteredatafeaturepoint,Xisavectorrepresentingthepixelcoordinates,I(X)isthepixelvalue,Iisthemeanofthepixelvaluesintheblock.LVdependsonpixelvalues,thus,iseasilychangedbyanyimageprocessing.Therefore,weproposetoselectfeaturepointswiththek-largestlocaltotalvariations(LTV)[ 8 ].LTVisdenedas: LTV=X2jI0(X)j2(5) 118

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whereistheimageblockcenteredatthecurrentfeaturepoint,I0(X)isthegradientofpixelvaluesatcoordinateX=(x1,x2) jI0(X)j=s (I(X) x1)2+(I(X) x2)2(5)LTVdependsonlocalstructureinimages.Itisrobustagainstcontent-preservingimageprocessing.Therefore,ourmodiedfeaturepointextractionalgorithmismorerobustthanBFP.Weusethismethodtodeterminethecoordinatesofthemostsalientfeaturepointswiththek-largestlocaltotalvariationsinimages,asshowninFigure 5-3 .Featurepointsareextractedinthehighrepetitionrate.Coordinatesoffeaturepointsarenotinvarianttothegeometrictransformsofimages,butthesingularityoffeaturepointsisinvarianttothegeometrictransformsofimages.Therefore,afterlocatingthemostsalientfeaturepointsinimages,weuseMorletwavelettoevaluatethedegreeofsingularityatthefeaturepoints. 5.3.3FeaturePointDescriptionHarmonicanalysis[ 73 ]ofsignalscandetectandlocatethesingularityofsignals.Waveletbaseshavegoodlocalizationabilityinbothtimeandfrequencydomain.Therefore,theycanlocateandcharacterizethesingularityofsignalsverywell.ThelocalsingularityoffunctionsismeasuredbyLipschitzexponentmathematically.Afunctionf(x)iswithsingularityofLipschitza,atpointx0,ifandonlyifthereexistsaconstantAsuchthatallthepointsxinaneighborhoodofx0satisfyjf(x))]TJ /F3 11.955 Tf 11.26 0 Td[(f(x0)jAjx)]TJ /F3 11.955 Tf 11.26 0 Td[(x0ja.ThewaveletcoefcientsWf(s,x0)off(x)atx0andscaleshasrelationwithLipschitzexponentsashowninEquation( 5 ). jWf(s,x0)jAesa(5)whereAeisaconstant. 119

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Continuouswavelettransform[ 91 ]isdesignedtodetectthesingularityofsignalsbetterthandiscretewavelettransform.Thelocationsofsingularityfoundbycontinuouswavelettransform[ 14 ]maybeinuencedbythenoiseinimages.Falsepositivemayhappenatpointswhicharenotatcornersbutclosetostraightlines.Falsenegativemayhappenatpointswhichareatcornersbutwithsmallvariationofgraylevels.Butthedegreeofsingularityislessinuenced.Therefore,basedonrobustlyextractedfeaturepoints,wecalculatethecontinuouswaveletcoefcientsrow-by-rowandcolumn-by-column,andusethemagnitudesofthecoefcientstorepresentfeaturepoints.MorletwaveletisacontinuouswaveletwithsinglefrequencyandSinemodulatedGaussianfunction.Morletwaveletisusedtodetectlinearstructuresperpendiculartotheorientationofthewavelet.2DMorletwaveletisdenedas jM(X)=(eiK0X)]TJ /F3 11.955 Tf 10.95 0 Td[(e)]TJ /F5 8.966 Tf 6.97 0 Td[(1=2jK0j2)e)]TJ /F5 8.966 Tf 6.97 0 Td[(1=2jXj2(5)whereX=(x1,x2)isthe2Dspatialcoordinates,andK0=(k1,k2)isthewave-vectorofthemotherwavelet,whichdeterminesthescale-resolvingpowerandangularresolvingpowerofthewavelet.BecausethedirectionsofthestrongestresponsesofMorletwaveletlteratfeaturepointsmaybeperturbedbynoise.OnlyhorizontalandverticaldirectionsofMorletwaveletareconsidered.AlthoughthemagnitudesofMorletwaveletcoefcientsinhorizontalandverticaldirectionswillbeslightlyperturbedbyasmalldegreeofrotationofimages,itwillbenormalizedinthelaterquantizationstep. 5.4HashGenerationAfterextractingsalientfeatureinimages,wewillfurthergeneratethebinaryhashsequencesfromtheobtainedMorletwaveletcoefcientsinthissection. 120

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5.4.1PseudoRandomPermutationofMorletWaveletCoefcientsToenhancethesecurityofimagehashes,i.e.,toavoidtheforgeryinputsdesignedbyanadversaryresultinginthesamehashes,weuseasecretekeyKtopseudorandomlypermutetheMorletwaveletcoefcients.Therandompermutationcanalsodecreasethecollisionprobabilityfordifferentinputsbyusingdifferentsecretekeys. 5.4.2QuantizationUsingCompandingQuantizationusingcompandingisefcientandunbiasedforcoefcientswithdifferentprobability.Quantizationcanobtaindiscreterepresentationofimagehash,normalizetherangeofoutputhash,aswellasweightdifferentpartsofhasheswithdifferentvalues.VectorquantizationwithLloyd-Maxalgorithmisclassical,butisdependentontheinitialcongurationandcomputationallyexpensive.Therefore,weproposetousecompandingtechnique[ 61 ]toquantizetheoat-pointMorletwaveletcoefcientstonitelevelbinaries.Thealgorithmofcompandingfordiscretevaluesissimilartothealgorithmofhistogramequalization.ThecomputationalcomplexityofthealgorithmisO(n),wherenisthenumberofcoefcients.QuantizationusingcompandingtechniqueassumesthattheshapeofdistributionofMorletwaveletcoefcientsofsimilarimagesaresimilarwhichisinlinewiththefact.Itisakindofprobabilisticquantization.Ittriestobefairtoeverycoefcients,i.e.,coefcientvalueswithlargeprobabilitywillbequantizedwithsmallstepsizes,whilecoefcientvalueswithsmallprobabilitywillbequantizedwithlargestepsizes.Acompandorconsistsofacompressor,auniformquantizer,andanexpandor.Thecompressorisanonlineartransformationanddesignedtoconvertthedistributionintotheuniformdistribution.Theexpandorisaninverseofthecompressor,whichisusedforrecoveryoforiginalcoefcientsandthusdisregardedinourimagehashingsystem.Usingthecompandingtechnique,wequantizethedataintoLlevels.Thecoefcientprobabilityofeachlevelisthesame,i.e.,1 L.Lshouldbe2m(m2Z+)foreasybinarizationofcoefcients. 121

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5.4.3BinaryCodingUsingGrayCodeWeproposetouseGraycode[ 122 ]tocodethequantizedcoefcients.Graycode,alsoknownasthereectionbinarycode,isabinarycode,inwhichtwosuccessivevaluesdifferinonlyonebit.Thusthehammingdistancesbetweensuccessivevaluesare1s,andhammingdistancesbetweennonsuccessivevaluesareproportionaltotheirdifferences.However,itdoesnotholdforordinarybinarycode.Inthisway,thedistancesbetweensimilarimagesdecreaseandthosebetweendifferentimagesincrease.Thishelpsincreasethediscriminabilityofthesystem.Sincethelengthofahashis5inourexperiments,a32bytearrayisusedasalookuptableforconstructinghashwithGrayCode.Forarbitrarylength,Graycodemaybeconstructedrecursively. 5.5ExperimentalResultsInourexperiments,theimageblockis15x15tocalculateLTV;40pointswithlargestLTVsarechosenasfeaturepoints;thequantizationlevelLis32,i.e.25;thelengthofimagehashNis200.ThedistancesbetweendifferenthashesareevaluatedbythenormalizedHammingdistance. d(h,h0)=1 Nni=1d(h(i),h0(i))(5)where d(h(i),h0(i))=8><>:1,h(i)=h0(i)0,h(i)6=h0(i)(5)handh0aretwohashvectors,theirtheithvaluesaredenotedash(i)andh0(i),andNisthelengthofanimagehash.Theseparameterscouldbetunedforbetterperformanceinspecicapplications. 5.5.1TheRobustnessofFeaturePointDetectorTherobustnessofproposedfeaturepointdetectorisshowninFigure 5-3 .Theimage`Lena'istampered,rotated,contrastenhancedwithhistogramequalization 122

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asshowninFigure 5-3 (b)(c)(d),respectively.InFigure 5-3 ,theextractedfeaturepointsaredenotedbyred`o'.Thefeaturepointsextractedfromtheoriginalimage,thetamperedimage,therotatedimageandtheimageafterhistogramequalizationarealmostthesame.Itindicateshighcorrectdetectionrateoftheproposedfeaturepointdetector.Hence,theproposedfeaturepointdetectorisrobustagainsttampering,rotationandhistogramequalization. 5.5.2ParameterDeterminationofSingularityDescriptorBesidestheeleganceofMorletwavelet,thereasonwhyweuseMorletwaveletisduetoitsstrongdiscriminabilityintheproposedimagehashingsystem,whichwewillshowinthissubsection.TheoptimalscaleofMorletwaveletisdeterminedbyusing24framesoftwoshotsinthevideosequencebig buck bunny 480p h264.mov[ 3 ].Eachshothas12frames.Thereferenceimagerandomlyselectedisthe7thframeintherstshot.The7thframeandthe17thframeamong24framesareshowninFigure 6-7 .TheframesintherstshotaresimilartoFigure 6-7 (a),andtheframesinthesecondshotaresimilartoFigure 6-7 (b).FromFigure 5-4 ,wecanseethatthedistancebetweenthe7thframeandthereferenceimage,i.e.,itself,is0.Thedistancesbetweensimilarframesinthesameshotaremuchsmallerthanthedistancesbetweendifferentframesindifferentshots.Figure 5-4A comparesthediscriminabilityofimagehasheswithMorletwaveletatdifferentscales.Thegapsbetweentheaveragedistancesoftherstshotandtheaveragedistancesofthesecondshotare0.2342,0.2704,0.2628,0.2645atscale6,8,10,12,respectively.Thelargestgap0.2704atscale8indicatesthatthestrongestdiscriminabilityofMorletwaveletatscale8forvideosignals.Figure 5-4B comparesthediscriminabilityofimagehasheswithdifferentwaveletsatscale8.Thegapsbetweentheaveragedistancesoftherstshotandtheaveragedistancesofthesecondshot 123

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are0.2704,0.2335,0.2083,0.2582forMorletwavelet,Splinewavelet,Haarwavelet,Symmetricwaveletrespectively.Morletwavelethasthestrongestdiscriminability. 5.5.3DiscriminabilityandRobustnessofImageHashesWealsotesttherobustnessofproposedimagehashingsystemonnaturalimagesundervariousattacks.Thesixtestimagesare512x512grayimagesshowninFigure 6-4 5.5.3.1DiscriminabilitybetweendifferentimagesThenormalizedHammingdistancesofsixtestimagesareshowninTable 5-2 .Theyarerelativelylarge.Itindicatesthatthediscriminabilityoftheproposedimagehashingsystemisgood. 5.5.3.2Non-predictabilityofimagehashesAdesirablepropertyofanimagehashfunctionisthatthedistancebetweenthehashofanimageandanyrandomsequencewiththesamelengthislarge.Ifthispropertyisachieved,itisunlikelyforanattackertogenerateanimposterofthehashofanimagebyusingarandomsequence.Weevaluatethedistancebetweenanimagehashandarandombinarysequence.Wegenerate50differentrandomsequences;eachrandomsequenceisabinaryrandomvectorwithprobabilityp(0)=p(1)=0.5andhasthesamelengthastheimagehash.ThenormalizedHammingdistancesbetweenthehashofLenaandthe50differentrandomsequencesareillustratedinFigure 5-6 .Thedistancesarerelativelylargeandconstant,whichimpliesthenon-predictabilityofimagehashes. 5.5.3.3Robustnesstocontent-preservingattacksWemakethefollowingtypesofattacksontheseimages:scalingimagesto0.5and1.5oftheirsizes,compressingimagesusingJPEGwithqualityfactor50[ 116 ],rotatingimagesby5degrees,cropping20%ofimages,addingwhiteGaussiannoise(variances2=20)toimages,lteringimageswithGaussianandMedianlters.ThenormalizedHammingdistancesbetweentheattackedimagesandtheoriginaltestimagesare 124

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listedinTable 5-3 .Thehashdistancesaresmallunderthesecontent-preservingimageprocessing,showingtherobustnessoftheproposedimagehashingsystem. 5.5.3.4RobustnesstotamperingWemakeseveraltamperedversionsofimage`Lena'asshowninFigure 5-7 .Thetamperedareasareindicatedbyredrectangles.ThenormalizedHammingdistancesbetweenthehashofLenaandthoseofthetamperedLenaareshowninTable 5-1 .ThenormalizedHammingdistancesbetweenthehashofLenaandthoseofthetamperedLenaarelargerthanthenormalizedHammingdistancesbetweenthehashofLenaandthoseofthecontent-preservingprocessedLena,andaresmallerthanthenormalizedHammingdistancesbetweenthehashesofdifferentimages.Thus,theproposedimagehashingsystemhastheabilitytoidentifytampering. 5.5.3.5DiscriminativethresholdsBasedontheexperimentsabove,thediscriminativethresholdsinoursystemaredeterminedasz1=z2=0.26.Withthethresholds,thefalsepositiveandfalsenegativerateonthetestimagesareboth0. 5.6SummaryInthischapter,weproposedanewmethodtogeneraterobustimagehashes.Thefeaturepointsareextractedfromimageswiththek-largestlocaltotalvariations.Morletwaveletcoefcientsarecalculatedatthefeaturepoints.Theyarepseudorandompermutated,quantizedwithcompandingtechniqueandbinarilycodedwithGraycode.Thegeneratedimagehashesarerobusttocontent-preservingimageprocessing.ThenormalizedHammingdistancesbetweenthehashesofLenaandthoseofthetamperedLenaarelargerthanthenormalizedHammingdistancesbetweenthehashofLenaandthoseofthecontent-preservingprocessedLena,andaresmallerthanthenormalizedHammingdistancesbetweenthehashesofdifferentimages.Ourfutureresearchwillbeexploringitsapplicationsonimageauthenticationandvideosignature,sinceour 125

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proposedimagehashhasgooddiscriminabilitybetweendifferentimagesanddifferentvideoshots,andstrongabilitytorecognizesimilarimages. Figure5-1. Flowchartofimagehashgeneration. A B Figure5-2. The7thframeandthe17thframeinthetestvideoBunny.(A)The7thframe.(B)The17thframe. 126

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A B C D Figure5-3. Thestablefeaturepointdetector.(A)Theoriginalimageandextractedfeaturepoints.(B)Thehat-tamperedimageandextractedfeaturepoints.(C)Therotatedimageandextractedfeaturepoints.(D)Theimageafterhistogramequalizationandextractedfeaturepoints. 127

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A B Figure5-4. Hashdistancewithdifferentwaveletsatdifferentscales.(A)HashdistancewithMorletwaveletatdifferentscales.(B)Hashdistancewithdifferentwaveletsatscale8. A B C D E F Figure5-5. Sixtestimagesforimagehashing.(A)Lena(B)Barbara(C)Boat(D)Mandrill(E)Jet(F)Pepper 128

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Figure5-6. DistancebetweenthehashofLenaandthen-thrandomsequence.(n=1,2,,50) ATamper1 BTamper2 CTamper3 DTamper4 ETamper5 FTamper6 Figure5-7. SixtamperedimagesofLena. 129

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Table5-1. HammingdistancebetweenLenaanditstamperedversion. (Lena,Tamper1)(Lena,Tamper2)(Lena,Tamper3) Distance0.280.320.33 (Lena,Tamper4)(Lena,Tamper5)(Lena,Tamper6) Distance0.260.350.29 Table5-2. Hammingdistancebetweendifferenttestimages. (Lena,Barbara)(Lena,Boat)(Lena,Mandrill) Distance0.440.390.42 (Barbara,Boat)(Barbara,Mandrill)(Barbara,Jet) Distance0.390.480.41 (Boat,Jet)(Boat,Pepper)(Mandrill,Jet) Distance0.370.370.41 (Lena,Jet)(Lena,Pepper)(Barbara,Pepper) Distance0.350.390.39 (Boat,Mandrill)(Mandrill,Pepper)(Jet,Pepper) Distance0.410.400.36 130

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Table5-3. Hammingdistancebetweenattackedimagesandtestimages. AttacksLenaBarbaraBoatMandrillJetPepper Scale0.50.050.030.250.080.250.03Scale1.50.030.030.050.080.050.03JPEG500.050.300.150.030.030.03Rotate5o0.030.340.250.030.230.17Crop20%0.030.330.250.080.250.25AWGNs2=200.030.120.030.030.040.03GaussianFiltering0.120.030.030.120.060.06MedianFiltering0.060.120.250.120.030.15 131

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CHAPTER6CONTENTBASEDIMAGEAUTHENTICATION 6.1ResearchBackgroundDigitalimagesbecomeanimportantpartofourdailylivesduetotherapidgrowthofInternetandtheincreasingdemandofmultimediacontentsfrompeople.Theupsoaringnumberofimageapplicationsfacilitateimageprocessing,andatthemeantime,makefabricatingandcopyingofdigitalcontentseasy,andleadusdoubtfulwhendigitalimagesareusedasevidencesincourt.Therefore,efcientandautomatictechniquesaredesiredtoidentifyandverifythecontentsofdigitalimages.Imageauthenticationissuchapromisingtechniquetoautomaticallyidentifywhetheraqueryimageisadifferentone,orafabrication,orasimplecopyofananchorimage.Here,theanchorimageisthegroundtruthimageortheoriginalimageasanauthenticationreference,andthequeryimageistheoneundersuspicion.Imageauthenticationtechniquesusuallyincludeconventionalcryptography,fragileandsemi-fragilewatermarkinganddigitalsignatureandsoon.Theauthenticationprocesscanbeassistedwiththeoriginalimageorintheabsenceoftheoriginalimage.Imageauthenticationmethods,basedoncryptography,useahashfunction[ 79 130 ]tocomputethemessageauthenticationcode(MAC)fromimages.Thegeneratedhashisfurtherencryptedwithasecretekeyfromthesender,andthenappendedtotheimageasanoverhead,whichiseasytoberemoved.Fragilewatermarkingusuallyreferstoreversibledatahiding[ 23 140 160 162 ].Awatermarkisembeddedintoanimageinareversibleandunnoticeableway.Iftheoriginalimageisreconstructedandtheembeddedmessageisrecoveredexactly,thentheimageisdeclaredasauthentic.Theconventionalcryptographyandreversiblewatermarkingcanguaranteetheintegrityofimages,buttheyarevulnerabletoanychanges.Aone-bitdifferentversionoftheimagewillbetreatedasatotallydifferentimage.Thesemethodscannotdistinguishtolerablechangesfrommaliciouschanges.Semi-fragilewatermarkinghas 132

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attack-resistantabilitybetweenfragileandrobustwatermarking.Ithastheabilityoftamperingidentication.Fridrich[ 53 54 ]proposedblockDiscreteCosineTransform(DCT)basedmethodstoidentifythetamperedareas.Buttheblockbasedmethodissusceptibletotranslationandcroppingattacks.Besides,semi-fragilewatermarkingtechniqueswillchangethepixelvalues,anddegradetheimagequalityoncethewatermarksareembedded,whichisundesirable.Andthereisatradeoffbetweenimagequalityandwatermarkrobustness.Digitalsignaturebasedtechniquesareimagecontentdependent,whicharealsocalledimagehashing.Animagehashisarepresentationoftheimage.Besidesimageauthentication,itcanalsobeusedforimageretrievalandotherapplications.Kozatetal.[ 80 ]proposedanimagehashtechniquebasedonSingularValueDecomposition(SVD).Itisassumedthatthesingularvaluesarerobusttogeneralimageprocessing,butnottomaliciousimagetampering.Itachieveshighprobabilityofdetectingatamperedimageatthecostofhighfalsealarmprobability.Venkatesanetal.[ 147 ]developedanimagehashbasedonastatisticalpropertyofwaveletcoefcients,whichisinvarianttocontent-preservingmodicationsofimages.Butitisnotintendedtoidentifythelocationsofchanges.TheimageauthenticationsystemproposedbyMongaetal.[ 101 ]isbasedonfeaturepointsofimages.Thesystemisnotsufcientlyrobustduetotheoutlierfeaturepointsproducedbyimageprocessing,althoughHausdorffdistanceisusedtoevaluatethedistancesbetweenfeaturepoints.Mongaetal.[ 99 ]alsoproposedaperceptualimagehashing.TheextractedfeaturesarethequantizedmagnitudesoftheMorletwaveletcoefcientsatfeaturepoints.AlthoughthedistributionofthemagnitudesoftheMorletwaveletcoefcientsmaybepreservedunderperceptuallyinsignicantdistortions,thelocationinformationislost.Inthischapter,weproposeaperceptualimageauthenticationtechniquebasedonclusteringandmatchingoffeaturepointsofimagestoaddressthelimitationsoftheaforementionedschemes.Featurepointsarerstgeneratedfromagivenimage,but 133

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theirlocationsmaybechangedduetopossibleimageprocessinganddegradation.Accordingly,weproposetouseFuzzyC-meanclusteringalgorithmtoclusterthefeaturepointsandremovetheoutliersfromthefeaturepoints.Inthemeanwhile,thefeaturepointsinthequeryimageandtheanchorimagearematchedintopairsinzigzagorderingalongdiagonalsoftheimagesclusterbycluster.Threetypesofdistanceareusedtomeasurethedistancesbetweenthematchedfeaturepointpairs.Histogramweighteddistanceisproposed,whichisequivalenttoHausdorffdistanceafteroutlierremoval.Theauthenticityofthequeryimageisdeterminedbythemajorityvoteofwhetherthreetypesofdistancebetweenmatchedfeaturepointpairsarelargerthantheirrespectivethresholds.Thegeometrictransformsthroughwhichthequeryimagesarealignedwiththeanchorimagesareestimated,andthequeryimagesareregisteredaccordingly.Moreover,thepossibletamperedimageblocksareidentied,andthepercentageofthetamperedareaisestimated.Therestofthechapterisorganizedasfollows.Section 6.2 presentsanoverviewoftheproposedimageauthenticationsystem.Section 6.3 describeshowtodetectfeaturepointsinimages.InSection 6.4 ,weproposeanefcientandeffectivealgorithmtoremoveoutliersoffeaturepoints,andtheremainingfeaturepointsareorderedandmatchedintopairs.HistogramweighteddistanceisproposedandnormalizedEuclideandistanceandHausdorffdistanceareusedinSection 6.5 .Majorityvotingstrategyisusedtodeterminetheauthenticityofimages.InSection 6.6 ,possibleattacksareidentied,thequeryimagesareregistered,thetamperedimageblocksarelocated,andthepercentageoftamperedareaisestimated.ExperimentalresultsareshowninSection 6.7 .Finally,Section 6.8 concludesthechapter. 6.2SystemOverviewTheservicesprovidedbytheproposedimageauthenticationsysteminclude: Identifyaqueryimageasasimilarimage,oratamperedimage,oradifferentimage,withregardtoananchorimage; 134

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Evaluatesimilarityoftwoimagesbydistancebetweenthem; Identifyandlocatethreetypesoftamperedarea,i.e.,addedarea,removedarea,changedarea; Estimatethepercentageoftamperedarea.TheowchartoftheproposedimageauthenticationsystemisshowninFigure 6-1 .First,featurepointsareextractedfromtheanchorimageandthequeryimagewiththek-largestlocaltotalvariations.Second,thefeaturepointsareclustered,thenoutliersoffeaturepointsareremoved,andcorrespondingfeaturepointpairsintheanchorandqueryimagesarezigzagalignedalongthediagonalsofimages.Third,histogramweighteddistanceisproposed.Threetypesofdistancesbetweentwoimagesareevaluatedandcomparedtothresholds.Thelowmissingrateofauthenticationisdesiredinoursystem.Thus,majorityvotingstrategyisusedtomakeauthenticationdecisionsofimages.Ifatleasttwodistancesaregreaterthantheirthresholds,thetwoimagesaredeclaredasdifferent.Otherwise,thetwoimagesaredeclaredassimilarforfurtherexamination.Forth,ifthetwoimagesareconsideredtobesimilar,thepossibleattacksonthequeryimage,i.e.,geometricattacksandtamperingattacks,aresubjecttodetection.Thequeryimageisfurtherregistered.Thelocationsandpercentageoftamperedareaareestimated. 6.3FeaturePointDetectionFeaturepointsaregeometricdescriptorsofthecontentsofimages.Mostinformationofsignalsisconveyedbyirregularstructuresandtransientphenomenaofsignals.Featurepointssuchascornerscanbeusedtocharacterizethesaliencyofimages.Featurepointbaseddescriptorismorerobusttogeometricattacksthanstatistics-baseddescriptors.Featurepointsarealsousefulforregistrationandidenticationofpossibleunderlyingattacks(geometricornon-geometric),onqueryimages. 135

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6.3.1PreprocessingPreprocessingwillchangeimagepixelsandmayinuencethedetectionanddescriptionoffeaturepoints.Toextracttheoriginalinformationfromthequeryimage,wekeepthequeryimageintactexceptadaptingitssizetothesizeoftheanchorimage. 6.3.2FeaturePointExtractionFordifferentapplications,differenttechniquestoextractfeaturepointsareexploredintheliterature[ 146 ].Sinceimageauthenticationneedstobeinvarianttocontent-preservingprocessing,hence,robustandrepeatablefeaturepointdetectorswithsmallcomputationoverheadaredesired.Jaroslavetal.[ 74 ]proposedafeaturepointdetectorforblurredimages,whichwecallBFPinthechapter.Inourchapter,amorerobustfeaturepointdetectorisproposedbasedonBFP.BFPisintendedtoefcientlydetectpointswhichbelongtotwoedgesregardlesstheirorientations.Itselectspointswiththek-largestlocalvariances.Thelocalvariance(LV)isdenedontheimageblockinEquation( 6 ). LV=X2(I(X))]TJ /F8 11.955 Tf 11.35 2.92 Td[(I)2(6)whereistheimageblockcenteredatthecurrentfeaturepoint,Xisavectorrepresentingthepixelcoordinates,I(X)isthepixelvalueatX,Iisthemeanofthepixelvaluesintheblock.LVdependsonpixelvalues,thus,iseasilychangedbyanyimageprocessing.Therefore,weproposetoselectfeaturepointswiththek-largestlocaltotalvariations(LTV)[ 8 ].LTVisdenedas: LTV=X2jI0(X)j2(6) 136

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whereistheimageblockcenteredatthecurrentfeaturepoint,I0(X)isthegradientofimageatcoordinateX=(x1,x2) jI0(X)j=s (I(X) x1)2+(I(X) x2)2(6)LTVdependsonlocalstructureofimages.Itismorerobustagainstcontent-preservingimageprocessingthanLV.Therefore,ourproposedfeaturepointextractionalgorithmismorerobustthanBFP.Weusethismethodtodeterminethecoordinatesofthemostsalientfeaturepointswiththek-largestlocaltotalvariationsinimages,asshowninFigure 6-2 6.4FeaturePointClusteringandMatchingDuetopossiblechangesappliedtothequeryimage,suchasluminancechangesandgeometrictransforms,theextractedfeaturepointsofthequeryimagearedifferentfromthoseoftheanchorimage,nomatterthequeryimageandtheanchorimagearesimilarornot.Thepossiblymissing,emergingandmovingfeaturepointsmaydefeattheimageauthentication.Iftwoimagesaresimilar,thepossiblemissing,emergingandmovingfeaturepointsofthequeryimage,mayenlargetheirdistance,andaffectthesimilaritymeasure.Ifthequeryimageandtheanchorimagearetotallydifferentimages,thepossiblechangesoffeaturepointsinthequeryimagemaydecreasethedistancebetweenthetwodifferentimages,anddegradethediscriminabilityofthesystem.Besides,fordistanceevaluation,thefeaturepointmatchingisneededbetweentheanchorimageandthequeryimage.Therefore,toimprovetheperformanceofthesystem,thefollowingclusteringprocessiscriticaltoremoveoutliersandmatchfeaturepointsintopairsincertainspatialordering.WeproposetouseFuzzyC-meanclusteringtoimplementoutlierremovalandfeaturepointmatchinginonepass. 6.4.1ClusteringbyFuzzyC-MeansFuzzyC-meansclusteringalgorithmisusedtoclusterthefeaturepoints.FuzzyC-meansclusteringmethod,developedbyDunn[ 52 ]in1973andimprovedbyBezdek 137

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[ 12 ]in1981,isbasedonminimizationofthefollowingobjectivefunction: Jm=Ni=1Cj=1umijkxi)]TJ /F3 11.955 Tf 10.95 0 Td[(cjk2(6)where1m<,uijisthedegreeofmembershipofxibelongingtotheclusterj,xiistheithfeaturepoint,cjisthecenteroftheclusterj,kkisanynormevaluatingthedistancebetweenanyfeaturepointandthecenter,Nisthenumberofsamples,andCisthenumberofclusters.Themembershipdegreeuijandtheclustercenterscjareupdatedby: uij=1 Ck=1(kxi)]TJ /F16 8.966 Tf 6.97 0 Td[(cjk kxi)]TJ /F16 8.966 Tf 6.97 0 Td[(ckk)1 m)]TJ /F28 6.974 Tf 5.42 0 Td[(1(6) cj=Ni=1umijxi Ni=1umij(6) 6.4.2OutlierRemovalTheoutliersaredenedasextrapointsunmatchedincorrespondingclustersinthequeryimageandtheanchorimage.Forexample,therearenfeaturepointsinclusterjintheanchorimage,andn+1featurepointsinthecorrespondingclusterj0inthequeryimage,thentheoneextraemergingfeaturepointinclusterjinthequeryimagewithleastdegreeofmembershipisregardedasoutlier,andviceversa.Likenoise,thesepointsshouldnotbeconsideredinthemeasurementofdistancebetweentheanchorimageandthequeryimage,andtheregistrationofthequeryimage.Ifthenumberofoutliersinaclusterisgreaterthanathreshold,thisclusterisdeclaredas`tampered'.Ifanimagehasatleastonetamperedcluster,thisimageisdeclaredas`tampered'.Thelocationsoftheoutliersareusedtodeterminethelocationsoftamperedarea. 138

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6.4.3SpatialOrderingandFeaturePointMatchingAfteroutlierremoval,thenumbersofremainingfeaturepointsincorrespondingclustersinthequeryimageandtheanchorimagearethesame.Thefeaturepointmatchingalgorithmprocessesfeaturepointsclusterbycluster.Ineachcluster,thefeaturepointsintwoimagesareorderedzigzagalongdiagonalsofimages.Theproposedfeaturepointmatchingalgorithmmaynotresultinexactpairsbetweenfeaturepoints,butitissub-optimalandveryfast.GivenNfeaturepointsinthequeryimage,ndingthecorrespondingNfeaturepointsintheanchorimage,incursacomputationalcomplexityofN!.Whereas,thecomputationalcomplexityofourproposedfeaturepointmatchingalgorithmisO(Nlogn),wherenistheaveragenumberoffeaturepointspercluster.Assumetherearenfeaturepointsperclusteronaverage.Thus,thereareN nclusters.Foreachcluster,thecomputationoforderingisO(nlogn).Afterclusteringandoutlierremoval,thecomputationalcomplexityoffeaturepointmatchingreducestoO(Nlogn)byclusterorderingandspatialordering.Thespatialmatchingbydiagonalorderingisoptimaltorasterorderingintermsofcorrectmatchingrateundertheperturbationofpossibleattacks.Theproposedmatchingalgorithmisrobusttooutliers,andthecasewherefeaturepointsareremoved,emergeorchangetheirlocationsduetopossiblenoiseorattacks.Itincreasesthesimilaritymeasureofsimilarimages,andincreasesthedistancebetweentwodifferentimages. 6.4.4AlgorithmSummary 6.5DistanceEvaluationThreetypesofdistanceareusedtoevaluatethedistancesbetweenimages,amongwhichhistogramweighteddistanceisproposed.Ifatleasttwotypesofdistancearelargerthantheircorrespondingthreshold,thetwoimagesareconsidereddifferent,otherwisesimilar.Thethresholdsareobtainedbystatisticalexperiments. 139

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Algorithm4FeaturePointClusteringandMatching ForfeaturepointsetXAintheanchorimageandfeaturepointsetXQinthequeryimage. 1. PerformfuzzyC-meansclusteringonXAandXQ,whichareclusteredintoclustersXAjandXQj(j=1,,C),Cisthenumberofclusters. 2. Forclusterj(j=1,,C)do:OrderingfeaturepointsinXAjandXQjaccordingtotheircoordinates(x1,x2)inzigzagorderingalongdiagonalsoftheimages,i.e.,orderingfeaturepointswithrespectto(x1+x2). (a) iflength(XAj)=length(XQj),match(X(i)Aj,X(i)Qj)intopairs,whereX(i)AjistheithfeaturepointinthejthclusteroftheanchorimageandX(i)Qjistheithfeaturepointinthejthclusterofthequeryimage. (b) iflength(XAj)>length(XQj),foreachfeaturepointX(i)QjinXQj,sequentiallyndtheclosestunmatchedfeaturepointsX(i0)AjinXAj.Forpairs(X(i01)Aj,X(i1)Qj)and(X(i02)Aj,X(i2)Qj),ifi1>i2,theni01>i02.OtherunmatchedfeaturepointsinXAjareconsideredasoutliersofXAj. (c) iflength(XAj)i2,theni01>i02.OtherunmatchedfeaturepointsinXQjareconsideredasoutliersofXQj. 6.5.1NormalizedEuclideanDistanceThersttypeofdistanceisnormalizedEuclideandistancebetweenthematchedfeaturepointpairs,whichisgivenby: E(XA,XQ)=1 NNi=1kX(i)A)]TJ /F3 11.955 Tf 10.95 0 Td[(X(i)QkE(6)whereNisthenumberoffeaturepointpairs,X(i)Aisthecoordinateofthecorrespondingithfeaturepointintheanchorimage,X(i)Qisthecoordinateoftheithfeaturepointinthequeryimage,kkEisEuclideannorm. 140

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6.5.2HausdorffDistanceTheHausdorffdistance[ 37 ]isdenedby: H(XA,XQ)=max(h(XA,XQ),h(XQ,XA))(6)where h(XA,XQ)=maxx2XAminy2XQkx)]TJ /F3 11.955 Tf 10.95 0 Td[(yk(6)Sinceitisminimaxbaseddistance,itisrobusttooutliersoffeaturepoints.Itisalsousedinanimagehashingsysteminchapter[ 99 ]. 6.5.3HistogramWeightedDistanceWeproposethethirdtypeofdistance,i.e.,histogramweighteddistance,whichisaperceptualbaseddistance.Thesignicanceofafeaturepointisweightedbypercentageofpixelvaluesatthatposition.Ifthepixelvaluesoffeaturepointshavehigherpercentageinthehistogramofpixels,thedistancesbetweenthesepairsoffeaturepointsshouldbetrustedmorethanothers.Thehistogramweighteddistanceisgivenby: W(XA,XQ)=max(1 NNi=1w(i)AkX(i)A)]TJ /F3 11.955 Tf 10.95 0 Td[(X(i)QkE,1 NNi=1w(i)QkX(i)A)]TJ /F3 11.955 Tf 10.95 0 Td[(X(i)QkE)(6)whereNisthenumberoffeaturepointpairs,X(i)Aisthecoordinateoftheithfeaturepointintheanchorimage,X(i)Qisthecoordinateoftheithfeaturepointinthequeryimage,w(i)Aistheluminancepercentageoftheithfeaturepointsintheanchorimage,w(i)Qistheluminancepercentageoftheithfeaturepointsinthequeryimage,andkkEistheEuclideannorm. 6.5.4MajorityVoteThenaldecisionismadefrommajorityvoteamongwhetherthreetypesofdistancearelargerthantherespectivethresholdsornotasshowninFigure 6-1 .Itisduetotheabilityandlimitationofthreetypesofdistance.NormalizedEuclideandistanceismostlyused,butiseasilyperturbedbyoutlierfeaturepoints;Hausdorff 141

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distanceisakindofminimaxdistance,repelstheoutliers,butmaylosesomegeometricinformationofimages;histogramweighteddistanceconsiderspixel/colorinformation,makesdecisionmorerobust,althoughitisinuencedbyoutlierstoo.Therefore,majorityvoteisnecessarytotakeadvantageousofthesetypesofdistance.Threetypesofdistanceareequalimportantandaretreatedwiththesameweightintheproposedsystem.Theyarediverseenoughinourexperimentstolowerauthenticationerrorrate.Moredistancemeasuresmayrepeattheperformanceofexistingdistanceordilutetheirfunctions,andwillincreasethesystemcomplexity. 6.5.5StrategyforThresholdDeterminationThethresholdsofdistancetodifferentiatesimilarimagesanddifferentimagesaredeterminedbasedonthestatisticalexperiments.Anovelstrategywetakeistocalculatedistanceamongtwovideoshots.Aframeinonevideoshotistakenastheanchorimage.Theotherframesarequeryimages.Thenthemiddlevaluebetweentheaveragedistanceinthesamevideoshotandtheaveragedistanceinthedifferentvideoshotsistakenasthethreshold.MoreresultscouldbefoundinexperimentsinSection 6.7 ,especiallyasshowninFigure 6-6 6.6PossibleAttackIdenticationAfterdistanceevaluation,ifthetwoimagesareconsideredsimilar,thepossiblegeometricattacksandtampering,whichthequeryimagemayexperiencearesubjecttofurtherdetection. 6.6.1GeometricAttackEstimationandRegistrationRegistrationalgorithms,suchasiterativeclosepoint(ICP)algorithm[ 164 ]andKanade-Lucas-TomasiFeatureTracker(KLT)[ 90 ]estimatethetranslationandrotationtransformsbetweenfeaturepointpairs,butdonotconsiderscalingtransform.Scale-invariantfeaturetransform(SIFT)algorithm[ 88 89 ]considersthescalingtransform,butrequireshighcomputationoverhead.Inthischapter,weproposetoestimateandrecoverimagesfrompossiblegeometricattacksintwostages.First, 142

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iterativeclosepoint(ICP)algorithm[ 164 ]isusedtoestimatetherotationandtranslationbasedonthematchedfeaturepointpairs.Thenthequeryimageisrecoveredfromtherotationandtranslationtransforms.Second,thescalingtransformsareestimated.Weproposetousetheratioofthestandarddeviation(STD)offeaturepointsofthequeryimagetothestandarddeviationoffeaturepointsoftheanchorimagetoestimatethepossiblescalingtransformsafterrotationandtranslationregistration. 6.6.2TamperingAttackIdenticationThepossibletamperedimageblocksaredetectedandthepercentageofthetamperedareaisestimated.Thetamperedimageblocksaredeterminedbythedistancesbetweenlocalhistogramsofimageblocksaroundthefeaturepointsintwoimages.Thedistanceweuseisearthmoverdistance(EMD)[ 83 121 ].Wedividethetamperingintothreecategories:addingnewfeatures,removingexistingfeatures,andchangingexistingfeatures.Feature-addedareasareidentiedaroundtheoutlierfeaturepointsinthequeryimage,whichdonotappearintheanchorimage.Feature-removedareasareidentiedaroundtheoutlierfeaturepointsintheanchorimage,whichdonotappearinthequeryimage.Feature-changedareasaretheareaswithmatchedfeaturepoints,whichhavelargelocalhistogramdistancesfromthecorrespondingareaintheanchorimage.IftheEMDbetweenlocalhistogramsofimageblocksaroundfeaturepointsintheanchorimageandthelocalhistogramsofthecorrespondingblocksinthequeryimageislargerthanthethreshold,theblocksinthequeryimagearedeclaredastamperedareas.Afterdetectionpossibletamperedareas,wesumuptheareaofthesetamperedblocks,andusetheratioofthesumofthetamperedareatotheareaofthewholeimageasthepercentageofthetamperedarea. 6.7ExperimentalResults 6.7.1FeaturePointDetectionWewillshowtherobustnessoftheproposedfeaturepointdetectorinthissubsection.Wecreatedifferentlydistortedversionsofimage`Lena'bytamperingLena's 143

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hat,rotatingtheimageby3degrees,andhistogramequalization.InFigure 6-2 ,featurepointsaredenotedbyred`o'.Thefeaturepointsextractedfromtheoriginalimage,thetamperedimage,therotatedimageandtheimageafterhistogramequalizationarealmostthesame.Itindicatestherobustnessofourproposedfeaturepointdetectoragainstattacks. 6.7.2FeaturePointMatchingExampleFigure 6-3 showstheresultoftheproposedfeaturepointmatchingalgorithm.Specically,Figure 6-3 (A)showstheoriginalimage;Figure 6-3 (B)showsthetamperedandcompressedimageandFigure 6-3 (C)showsextracted,clusteredandmatchedfeaturepoints.TheaxesinFigure 6-3 (C)denotethepixelcoordinatesinimages.Eachclusterconcentratesinoneellipse.Featurepointsofdifferentclustersareillustratedwithdifferentcolors,`+'and`'denotethematchedandoutlierfeaturepointsintheoriginalimage,and`o'and`'denotethematchedandoutlierfeaturepointsinthequeryimagerespectively.TamperingthecornerofthehatofLenainFigure 6-3 (B)willaddandremovefeaturepoints.Byusingtheproposedfeaturepointmatchingalgorithm,theoutliersoffeaturepointscanbeefcientlyandcorrectlydetected,andcorrespondingfeaturepointsbetweenimagesinFigure 6-3 (A)andFigure 6-3 (B)arematchedintopairsinlinewiththefact.Itshowstheeffectivenessofourfeaturepointmatchingalgorithm.Andouralgorithmrunsfast. 6.7.3AuthenticationPerformanceWecompareauthenticationperformanceoffourimageauthenticationsystems:theproposedimageauthenticationsystem,imagehashingbasedonfeaturepoints[ 99 ],imagehashingbasedonSingularValueDecomposition(SVD)[ 80 ]andimagehashingbasedonWavelet[ 147 ]inimagehashingtoolbox[ 2 ].Wetestimageauthenticationsystemon6testimages.Theyare512x512grayimagesshowninFigure 6-4 144

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Severaltypesofattacksaremadeontheseimages:scalingimagesto0.5and1.5oftheirsizes,compressingimagesusingJPEGwithqualityfactor50[ 116 ],rotatingimagesby5degrees,cropping20%ofimages,addingwhiteGaussiannoise(s2=20)toimages,lteringimageswithGaussianandMedianlters.Thethresholdstodistinguishsimilarimagesanddifferentimagesare1.5,0.2and0.2fornormalizedEuclideandistance,histogramweighteddistanceandHausdorffdistancerespectivelyintheproposeimageauthenticationsystem.Theproposedimageauthenticationsystemcanmakecorrectauthenticationdecisionsinthecaseswherefeaturepointbased,SVDbasedandWaveletbasedimageauthenticationinimagehashingtoolbox[ 2 ]maynot.SomeexperimentalresultsareshowninTable 6-1 .Featurepointbased,SVDbasedandWaveletbasedimageauthenticationaredenotedbyFP,SVDandWaveletinTable 6-1 respectively.Thedecisionsoftheimageauthenticationsystemsarerepresentedby`S'forsimilarimagesand`D'fordifferentimages.FPfailstoauthenticatetheLenaanditstamperedversion,Lenaanditscompressedandenhancedversion,sincetheirfeaturepointextractionisnotthatrobust,andtheydonothaveoutlierexclusion.SVDunderestimatesthedistancesandconsidersLenaandMandrillaresimilar.Waveletfailstorecognizesimilaritybetweentheimage`Goldhill'anditsenhancedversion.ItindicatesthattheSVDbasedmethodandwaveletbasedmethodarenotrobustasfeaturepointbasedmethodsinsomecases.Itfailsinluminanceadjustedcases.Ourproposedauthenticationsystemmakescorrectdecisionsinthesecases.Wealsocreate84attackedimagesfromsixtestimagesinFigure 6-4 .Eachtestimagehas14attackedversions,whichsuffersfromscaling0.5,scaling1.5,JPEGcompressionwithquality50,5degreerotation,cropping20%,whitegaussiannoiseaddition,lteringwithGaussianlterandMedianlter,and6tamperingattacks.Wetestsimilarityanddifferenceamong3486imagepairs.Thecorrectprobabilityoftheproposedsystem,FP,SVDandWaveletare84.5%,81.9%,83.2%,79.1%respectively. 145

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6.7.4DistanceComparisonWecomparethethreetypesofdistanceintheproposedimageauthenticationsystemwiththedistanceofimagehashbasedonfeaturepoints[ 99 ],thedistanceofimagehashbasedonSingularValueDecomposition(SVD)[ 80 ]andthedistanceofimagehashbasedonWavelet[ 147 ]inimagehashingtoolbox[ 2 ].TheyaredenotedbyEuclidean,Histogramweighted,Hausdorff,FP,SVDandWaveletinFigure 6-5 andFigure 6-6 .Ourexperimentsusetheframesoftwoshotsinvideosequencebig buck bunny 480p h264.mov[ 3 ].Eachshothas30frames.The20thframesinthetwoshotsareshowninFigure 6-7 .Thedistancesbetweenthe20thframeandtheotherframesintherstshotareshowninFigure 6-5 .Thedistancesareallverysmall.Thedistancesbetweenthe20thframeintherstshotandtheotherframesintwoshotsareshowninFigure 6-6 .Themethodscandistinguishtwoshots.DiscriminabilityofSVDisthelowest,whilediscriminabilityofFPisthehighest.Thedistancesusedinourauthenticationsystemhavebothrobustnessanddiscriminability,andthenon-constantdistancesreectthesimilaritybetweenframesbetterthanothermethodssinceperceptuallysimilarimageshavesmalldistancebetweenthem.AndFP,SVDandWaveletbasedmethodsdonotprovidetamperinglocationidentication. 6.7.5TamperingDetectionWedetecttamperingsuchasadding,removingandchangingfeaturesasshowninTable 6-2 .Threetypesoftampering,i.e.,adding,changing,andremovingfeatures,areshownineachrowoftheTable 6-2 .IntheimagesintherstcolumnofTable 6-2 ,the`'sinimagesarebasicimageblocksusedinlocalhistogramdistanceevaluation.The`'sindicatethedetectedtamperedblocksaroundsomefeaturepoints.Thetamperedversionsof`Lena'areshownintherstcolumnoftheTable 6-2 .ThepercentageoftamperingareaisalsoestimatedinthesecondcolumnoftheTable 6-2 ,butis 146

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under-estimated.Ifweincreasesizeofthe`',missingratewillbehighforsmall-areatamperedimages,anditwillincreaseEMDcomputationoverhead.Thuswejustchoose11x11asthesizeofimageblocks.Thesizeofimageblocksshouldbehierarchicalandadaptiveinourfuturework. 6.8SummaryWeproposedanefcientrobustimageauthenticationsystem.Thefeaturepointswiththek-largestlocaltotalvariationsareextracted.FeaturepointsareclusteredbyFuzzyC-meansalgorithm.Thentheoutliersoffeaturepointsareremovedandfeaturepointspairsbetweenthequeryimageandtheanchorimagearematchedinzigzagorderclusterbyclusteratthesametime,whichincreasestherobustnessoftheproposedimageauthenticationsystem.Furthermore,thenormalizedEuclideandistance,theHausdorffdistance,thehistogramweighteddistancebetweenthequeryimageandtheanchorimageareevaluated.Basedonthedistances,whethertheimagessimilarornotaredeterminedbymajorityvoting.Forsimilarimages,possiblegeometricattacksaresubjecttodetectionandimageregistrationisperformed.Possibletamperedareasaredeterminedandclassied,andthepercentageoftamperedareaisestimated.Theproposedimageauthenticationsystemcouldserveasabuildingblockinmanyapplicationssuchascopyrightprotection,imageretrievalandvideosignature. 147

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Figure6-1. Theowchartoftheproposedimageauthenticationsystem. 148

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A B C D Figure6-2. Thestablefeaturepointdetector.(A)Theoriginalimageandextractedfeaturepoints.(B)Thehat-tamperedimageandextractedfeaturepoints.(C)Therotatedimageandextractedfeaturepoints.(D)Theimageafterhistogramequalizationandextractedfeaturepoints. 149

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A B C Figure6-3. Featurepointclusteringandmatching.(A)Theoriginalimageandextractedfeaturepoints.(B)Thehat-tamperedimageandextractedfeaturepoints.(C)Featurepointclusteringandmatching. A B C D E F Figure6-4. Sixtestimagesforimagehashing.(A)Lena(B)Barbara(C)Boat(D)Mandrill(E)Jet(F)Pepper 150

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Figure6-5. Distancecomparisonamongdifferentauthenticationmethodsinonevideoshot. Figure6-6. Distancecomparisonamongdifferentauthenticationmethodsintwovideoshots. 151

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A B Figure6-7. ThetwoframesintwoshotsinthetestvideoBunny.(A)The20thframeintherstshot.(B)The20thframeinthesecondshot. Figure6-8. Diagramofpossibleattackidentication. 152

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Table6-1. Authenticationperformancecomparisonamongdifferentmethods. (showsomelimitationsofmethodsfrom[ 80 99 147 ]) 153

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Table6-2. Tamperingdetectionandpercentageoftamperingareaestimation. DetectionresultsPercentageoftamperingarea 1.53% 0.84% 1.02% 154

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CHAPTER7ROBUSTTRACK-AND-TRACEVIDEOWATERMARKING 7.1ResearchBackgroundNowadays,computers,interconnectedviatheInternet,makethedistributionofthedigitalmediafastandeasy.However,italsorequireslessefforttoobtaintheexactcopies.Therefore,itposesgreatchallengestocopyrightprotectionfordigitalmedia.Digitalwatermarkembeddingisaprocessofintegratingtheuserandcopyrightinformationintothecarriermediainawayinvisibletohumanvisionsystem(HVS).Itspurposeistoprotectthedigitalworksfromtheunauthorizedduplicationordistribution.Videowatermarkingsystemisdesiredtoembedwatermarkinsuchawaythatthewatermarkcanbedetectedlaterforauthentication,copyrightprotection,andtrack-and-traceillegaldistribution.Videos,composedofmultipleframes,canutilizeimagewatermarkingtechniquesinaframe-wisemanner[ 112 ].Althoughthewatermarkingembeddingcapacityofvideoismuchlargerthanthatofimage,theattacksthevideowatermarkingsuffersaremorecomplicatedthanimagewatermarking.Theattacksincludenotonlyspatialattacks,butalsotemporalattacksandhybridspatial-temporalattacks.Intheliteratureoftrack-and-tracevideowatermarking,thealgebra-basedanti-collusioncodeisinvestigated[ 10 17 24 139 144 151 155 ].Itsabilitytotraceoneormultiplecolludersdependsontheassumptionthatthecodeisalwaysavailableanderror-free,whichmaynotbetrueinpractice.Besides,thelengthofanti-collusioncodehinderthesystemusercapacity.Hence,practicalandmulti-functionalwatermarkingsystemsbasedonalgebra-basedanti-collusioncodeareverylimited.Tothisend,weproposearobusttrack-and-tracewatermarkingsystemfordigitalvideocopyrightprotection[ 158 ].Itconsistsoftwoindependentbodies,watermarkingembedderandwatermarkingdetector.Atembedder,userandproductcopyrightinformation,e.g.astringoflengthLs,isrstencryptedwithAdvancedEncryption 155

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Standard(AES)[ 42 ]toformabinarysequence.Wethenapplyerrorcorrectioncode(ECC)[ 87 ]tothesequencetogenerateabinarysequencewitherror-correctionabilityoflengthL,calledwatermarkpayload.Meanwhile,aframe-sizewatermarkpatternarisesfromapseudo-randomnoise(PN)sequence[ 41 118 ].Eachbinarybitinwatermarkpayloadisassociatedwithonevideoframeanddetermineshowwatermarkisembeddedtothisframe.Bit0indicatessubtractingthewatermarkpatternfromthecurrentframe,whilebit1indicatesaddingthewatermarkpatterntothecurrentframe.WewillrepeatedlyembedLbitsifthevideoislongerthanLframes,andsyncthewatermarkpayloadatthebeginningofdramaticvideoscenechangestoresisttemporalattacks.Furthermore,inordertomeettheperceptualquality,webuildaperceptualmodeltodeterminethesignalstrengththatcanbeembeddedtoeachframepixel.Notethatthestrongertheembeddedsignal,andhencetheeasierthewatermarkcanbecorrectlydetected.However,watermarkpatternislikerandomnoise,andtoostrongofthenoisesignalcancausenoticeabledistortiontothepicture.TherandomnessofPNsequencesalsomaketheembeddedwatermarkinformationblindtotheattackers.Tomakeatrade-offbetweencapacityandvisualquality,webuildaperceptualmodeltodeterminethesignalstrengththatcanbeembeddedtoeachpixel.Finally,sincedistributedvideosarepronetocollusionattacks,weproposetoapplygeometrictransformstothewatermarkedvideos.Thisiscalledgeometricanti-collusioncodinginthischapter.Thesetransformsincluderotation,resizingandtranslation,andshouldbemoderateenoughtocausenodefecttoHVS,butalsoenhancethecapabilitytoresistcollusionattacks.Thewatermarkingdetectorjustcarriesoutthereverseprocessoftheembedder.Inthissystem,weassumethedetectorcanalwayshaveaccesstotheoriginalvideoastheprototypeofthecandidatevideo.Becauseofthegeometricanti-collusioncodingatembedder,watermarkusuallycannotbecorrectlyextractedwithoutanypre-processingeventhecandidatevideoisanerror-freecopy.Additionally,spatial 156

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attackssuchasfurthergeometricmanipulationsandtemporalattacksmayoccurtodistributedvideos.Inthischapter,weproposetoregisterthecandidatevideototheoriginalvideospatiallyandtemporally.AniterativeKLT[ 142 ]basedschemeisappliedforspatialregistration,whereastemporalregistrationistomatchframesthatminimizethemean-square-error(MSE).Wethencomputecross-correlationcoefcientsbetweenre-generatedwatermarkpatternandframedifferenceoftheregisteredframeanditscorrespondingoriginalframe,demodulatethecoefcientsequencetorecoverthewatermarkpayload.ItisthenECCdecoded(convolutionalcodingforspecic)andAESdecryptedtoderivetheoriginaluserorcopyrightinformation.Successfuldetectionindicatestheuserorcopyrightinformationiscorrectlyextracted,otherwisewesaythedetectorfailstodetectthewatermark.Thechapterisorganizedasfollows:Section 7.2 describestheoverallarchitectureoftheproposedtrack-and-tracevideowatermarkingsystem.ThewatermarkingembeddertechniquesarediscussedinSection 7.3 .Section 7.4 introduceswatermarkingdetectortechniques.TheexperimentalresultspresentedinSection 7.5 verifyrobustnessoftheproposedvideowatermarking.Finally,theconclusionandfutureworkaregiveninSection 7.6 7.2ArchitectureofRobustVideoWatermarkingSystemThearchitectureofthetrack-and-tracevideowatermarkingsystemincludestwoindependentcomponents,i.e.,watermarkingembedder(Figure 7-1 )andwatermarkingdetector(Figure 7-2 ).Itisanadditivewatermarkingsystem.Watermarkingembedderconsistsoftwofunctionalcomponents,watermarkgeneratortogeneratewatermarkpayload(Figure 7-1A ),andwatermarkembeddertoembedthepayloadtovideoframes(Figure 7-1B ).Watermarkingdetectorextractspayloadfromcandidatevideo(Figure 7-2A ),andthenrecoveruserorcopyrightinformationfromthepayload(Figure 7-2B ). 157

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7.2.1WatermarkingEmbedderTheproposedvideowatermarkingsystemisanadditivesystem,i.e.addingwatermarksignaltotheoriginalvideo.Theinputsofembedderaretheoriginalvideo,userIDandcopyrightinformation.Thekeycongureparametersaretheframesize(widthxheight)oftheinputvideo,AESencryptionkey(Key1inFigure 7-1B ),patternID(Key2inFigure 7-1B )togeneratewatermarkpatternandKey3togenerategeo-transformparametersforlmdistributors.String-typeuser/copyrightinformationarebinarized,encrypted,andECCcodedintowatermarkpaylaod.Ifconvolutioncoderate=1/2isused,theinformationofLscharactersistransformedintoL=16Lsbits.WatermarkpatternbyusingorthogonalPNsequencescanresistframe-averagingcollusion.Thepseudorandomwatermarkpatternsweightedbyperceptualmodelingofeachframeareembeddedwiththelargeststrengthunderimperceivableconstraint.ThelengthofPNsequenceNisframesize(widthxheight).ThenumberoftheorthogonalsequencesoflengthNisexactlyN.Forgeometrytransform,thebicubicinterpolationisusedtokeeporiginalvideoinformationasmuchaspossible.Thereare5orotation,5pixeltranslationand5%resizing.Thus,theproposedsystemcouldaccommodate1000Ndistributorsideally.Afterembedding,thewatermarkedvideosaredistributed.Theymaysufferfromintentionalmanipulationsorunintentionaldegradationslater.Theseattacksincludebutnotrestricttogeometricmanipulations,erroneoustransmission,andcollusion. 7.2.2WatermarkingDetectorTheinputsofdetectorarethecandidatevideoanditsoriginalcopy.Thegoalistoextractwatermarkpayloadfromthecandidatevideoandrecovertheuser/copyrightinformationwithreferencetotheoriginalcopy.Someofthekeycongureparametersarethesize(widthandheight)ofthetwoinputvideos,AESdecryptionkey(Key10inFigure 7-2A )andpatternID(Key20inFigure 7-2A )tore-generatewatermarkpattern. 158

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Usually,wesetKey10=Key1,Key20=Key2forconsistencyofsymmetricAESandPN-sequencegenerationatbothends.Thedistributedvideomaybeenlargedorcroppedinsize,referringtoasresizingattack.Hence,thecandidatevideomaydifferinframesizewiththeoriginalvideo.Thedetectoremploysaresizealgorithmtothecandidatevideotomatchtheminsizewherevernecessary.Thealgorithmisbicubicinterpolation(expanding)ordecimation(shrinking).Notethatweapplygeometricanti-collusioncodingatembedder.Also,maliciousattacksmayimposespatialandtemporaltransformsattemptingtoremovethewatermarkinformation.Ontheotherhand,thedetectorisverysensitivetothesetransformsandoftenfailsindetectionwithoutanypre-processingtothecandidatevideo.Accordingly,werstregisterthecandidatevideotothereferencevideo,bothspatiallyandtemporally.Normalizedcross-correlationcoefcientsarecomputedbetweeneachpairoftheregisteredcandidateframeandthereferenceframe.Theanti-collusiongeometrictransforminformationbroughtbyKey30isusedtotracepossibleillegaldistributors.Thenwedobinaryharddecisiontoget+1/-1sequencefromthecoefcientsbasedonathreshold,anddemodulateittoabinary0/1sequence,whichistheextractedwatermarkpayload.Finally,thepayloadisECCdecodedandAESdecryptedtorecovertheuser/copyrightinformationofstringtype,asillustratedinFigure 7-2B .HereViterbialgorithmisusedforECCdecodingregardingconvolutionalcodeforECCencodingatembedder.Theproposedvideowatermarkingsystemisintegratedwithvarioustechniques,includespectrumspreading,AESencryptionanddecryption,ECCencodinganddecoding,perceptualweightingmodel,geometricanti-collusioncodingandframeregistration.Thefollowingsectionwillintroducethesetechniquesindetailrespectively. 159

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7.3WatermarkEmbeddingTechniques 7.3.1WatermarkPatternGenerationAseeddenotedasKey2inFigure 7-1A isrequiredtogenerateaPN-sequenceaswatermarkpatternusingspectrumspreading.Itshouldbeofthesamesizewiththevideoframeinordertodomatrixaddition.ThePN-sequencecanbem-sequence,WalshcodesorKasamisequencewithoptimalcross-correlationvalues.TheorthogonalPN-sequencesaredesiredbetweendifferentvideostoresistaveragingcollusion,anddesiredbetweendifferentwatermarkpayload(+1/-1)toresisttemporalattacks.Orthogonalm-sequenceisusedinoursystem.Forframe-sizeN(widthxheight),thelengthofm-sequencesisN,hence,thereareNorthogonalm-sequences. 7.3.2WatermarkPayloadGenerationProductcopyrightanduserinformationrequireencryptiontokeepitfromattackerswhowanttodetectortamperthecontent.Afterencryption,theinformationappearsasnoisetotheattackers.TheencryptiontechniquecouldbeRivest-Shamir-Adlemancryptography(RSA),DataEncryptionStandard(DES),AdvanceEncryptionStandard(AES)andsoon.TheencryptionkeydenotedasKey1inFigure 7-1A couldbethechoicefromthewatermarkcreatorfollowingcertainrules.Inoursystem,wechooseAESforencryptionandsetthelengthofstandardkeytobe128bits.Key1couldbebothuserandvideorelated.Weassumethatitisacommonkeytobothembedderanddetector,knownasasymmetricalencryptionsystem.Ifunsymmetricalencryptionsystemisused,theembedderhasprivatekeyKey1,andthedetectorhaspublickeyKey10.Moreover,videodistributionprocesscanbeviewedastransmissioninchannel,andtheattackstothemediaisregardedaschannelnoise.Therefore,1/2convolutioncodeisadoptedinoursystemforerrorcorrectioncoding(ECC).Afterencryptionandencoding,abinarysequenceoflengthLisgeneratedaswatermarkpayload.Thebinarypayloadisfurthermodulatedinto+1/-1sequencesas: X0=2X)]TJ /F8 11.955 Tf 10.94 0 Td[(1,(7) 160

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wherefXg2f0,1gisthebinarypayload,X0isthemodulatedsequence. 7.3.3PerceptualWeightingModelAsmentionedinsection 7.1 ,thereisatrade-offbetweenwatermarkcapacityandvisualqualityindeterminingthesignalstrengththatcanbeembeddedintovideoframes.Webuildaperceptualmodelinbothtemporaldomainandspatialdomain.Theobjectiveistomaximizethewatermarkcapacitywithoutcausingnoticeabledegradationtovisualqualityofvideos.ThemodeldiagramisshowninFigure 7-3 .Embeddingstrengthisdeterminedinapixel-wisemannerforeveryframe.Hence,itisformulatedasaheightwidthmaskmatrixM,witheachentrydescribingtheweightforthecollocatedwatermarkpatternsignal.ThenforlengthLwatermarkpayload,thewatermarkcorrespondingtotheithpayloadbitinframekL+i,k2Z+is: W=sign(X0(i))MP(7)whereiselement-wiseproduct,Piswatermarkpattern,X0(i)istheithwatermarkpayload.ThepixelvaluesinthewatermarkedframeWshouldbeclippedto[0,255]. 7.3.3.1TemporalperceptualmodelingTheperceptualmodelintemporaldomainisbasedonthefactthathumaneyesaresensitivetochangeswithslowmotion,butnottofastmovingchanges.Generally,thelargerthedifferencebetweenthecurrentframeandthepreviousframe,thestrongertheembeddedsignalstrengthcouldbe.Butthesimpledifferencebetweenadjacentframesisnotgoodenoughtodescribeobjectmoving.Forexample,ifanobjectismovingleavingasmoothbackground,wecannotembedhighstrengthwatermarkintothesmoothbackground.Therefore,weproposeablockmotionmatchingalgorithmtondthedifferencebetweenblocksincurrentframeandpreviousframewiththeleastsumofabsolutedifferences(SAD),whichisdenedas: SAD(,0)=(i,j)2,(i0,j0)20jIc(i,j))]TJ /F3 11.955 Tf 10.95 0 Td[(Ip(i0,j0)j(7) 161

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whereistheblockinthecurrentframe,0istheblockinthepreviousframe,(i,j),(i0,j0)arethepixelcoordinates,Icisthecurrentframe,Ipisthepreviousframe.Thealgorithmforperceptualmodelintemporaldomainissummarizedasfollows: Algorithm5PerceptualModelinginTemporalDomain foreachblockinthecurrentframedo foreachblock0inNB()inthepreviousframedo ifSAD(,0)
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possible.Map1iscalculatedbyaconvolutionoftheoriginalframeandhighpasslterH,whichisdenedbelow: H=24)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(18)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(1)]TJ /F8 11.955 Tf 9.29 0 Td[(135(7) ThesecondmetricMap2isthevarianceoftheblockwhichiscenteredatthecurrentpixel.Thelargerthevariance,thehighertheembeddedsignalstrengthcanbe. ThethirdmetricMap3istheentropyoftheblockwhichiscenteredatthecurrentpixel.Thehighertheentropyis,therichertexturetheareahas,andthehigherembeddedsignalstrengthcouldbe.Eachofthethreemetricscandescribehowrichthetextureisofthelocalareaaroundpixel,butnoneofthemissufcientbyitsown.Therefore,wedenethespatialastheproductofthethreemetrics: SpatialMask1=bMap1Map2Map3(7)wherebisascalingfactor.Model2isbasedonsaliencymap[ 4 ]andJustNoticeableDifference(JND)model[ 18 40 ]ofvideoframes.Thesaliencymaphighlightssalienttextureareasinaimage,wherecouldbeimperceivableembeddinglocations.Toobtainsaliencymap,theframeisdown-sampledandlowpasslteredinthefouriertransformdomain,andthenup-sampledtotheoriginalframesize.Themagnitudesofthesaliencymapdescribethefrequencyofframeinformation.Visualjustnoticabledifferencereectsnonlinearresponseofhumaneyestospatialfrequency.BasedonJNDhumanperceptualmodel,thesaliencymapisfurthermappedintospatialmaskby: SpatialMask2=h (SaliencyMap+d)(7)Toguaranteegoodvisualqualityofvideos,wechoosetheminimalvaluebetweenthespatialmaskandtemporalmaskforeachpixel.Andthenalembeddedsignal 163

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strengthisalsoboundedbyaminimumandamaximumvalue.Theperceptualweightingmap(PWM)isdenedas: PWM=min(maxStrength,max(minStrength,min(TemporalMask,SpatialMask)))(7) 7.3.4GeometricAnti-collusionCodingAfterembeddingthewatermarkpayloadintothecarriervideo,weapplygeometricaltransformstoeachcopyofthevideo.Thetransformisacombinationofshifting,resizingandrotation,andvariesamongdifferentvideodistributors.Foreachvideocopy,itsspecictransformindexisarandomvariablegeneratedbyKey3,relatedtouserandcopyrightinformation.TheextentofthetransformshouldbemoderateenoughinordernottobeawaredbyHVS,butcanstillbedetectedbycomputers.Ifcolluderstrytolinearlyornonlinearlycombinethemultiplevideocopiestoeliminatetheembeddedwatermark,theresultedvideowillusuallybeblurredandbecomeunacceptedbyhumaneyes.Thus,thegeometricaltransformprotectsvideowatermarkfrominter-videocollusionattacks.Thisprocessiscalledgeometricalanti-collusioncoding.Topreserveasmuchinformationaspossible,bi-cubicsplineinterpolation[ 76 ]isusedtolltheblankareaaftertransform. 7.4WatermarkDetectionTechniques 7.4.1VideoFrameRegistrationApartfromthegeometricanti-collusioncoding,theinputcandidatevideoatdetectormaygothroughmanychanges,eitheraccidentalmanipulationsormaliciousattacks.Twomajorcategoriesamongthechangesareafnetransforminspatialdomain,andframeadd/dropintemporaldomain.Sincedetectorhasaccesstooriginalvideo,wecanuseoriginalcopyasreferenceandregistercandidatevideotothereferenceinbothspatialdomainandtemporaldomain. 164

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7.4.1.1SpatialregistrationThespatialregistrationisbasedonKanade-Lucas-Thomasi(KLT)featuretracker.Afnemodelisusedinspatialregistration[ 90 129 ].Theafnetransformmodeltoanypixel(x,y)is: 264x0y0375=264abcd375264xy375+264ef375(7)Theobjectiveofspatialregistrationistondthe6afnetransformparametersa)]TJ /F3 11.955 Tf 11.33 0 Td[(finthemodel,soastodoinverseafnetransformbeforedetection.KLTachievesthisbymatchingthecorrespondingfeaturepointsinthecandidateframeandtheoriginalframe,andgetthesolutiontotheparameterset..WecalltherectiedframeF(1).Foreachpixel(x,y)ofF(1),wecomputeitspixelposition(x0,y0)incandidateframeF(0).TakeF(0)(x0,y0)asthematchinF(1)(x,y)ifx0,y0areintegers;otherwise,weinterpolateF(0)(x0,y0)at(x0,y0).However,duetothecomplexityofthetransformandtheimperfectnessofKLTalgorithm,therectiedframeafterone-timeinverseafnetransformisoftennotgoodenoughtoextractwatermarkfrom.Therefore,weproposetorenetheestimateF(1)byapplyingKLTiteratively.Specically,wehaveafnetransformdisplacementexpressedas: 264dxdy375=264x0)]TJ /F3 11.955 Tf 10.95 0 Td[(xy0)]TJ /F3 11.955 Tf 10.95 0 Td[(y375=264a)]TJ /F8 11.955 Tf 10.94 0 Td[(1bcd)]TJ /F8 11.955 Tf 10.95 0 Td[(1375264xy375+264ef375(7)WhentheithKLTiterationgetsF(i)andthecorrespondingafneparametersetai)]TJ /F3 11.955 Tf 11.12 0 Td[(fi,wecomputethedisplacementofeachpixel.Wekeepdoingthisuntiltheconvergenceconditionissatisedorwereachthemaximumnumberofiteration.Inthissystem,wecheckthemaximumpixeldisplacementbetweentwoconsecutiverectications: maxxi,yi2F(i)fjdxij,jdyijg
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where dxi=xi)]TJ /F3 11.955 Tf 10.95 0 Td[(xi)]TJ /F5 8.966 Tf 6.97 0 Td[(1 (7) dyi=yi)]TJ /F3 11.955 Tf 10.95 0 Td[(yi)]TJ /F5 8.966 Tf 6.97 0 Td[(1 (7) andeisapre-denedthreshold.Inmostcases,weexpectspatialregistrationbasedonKLTtoimprovethedetectionperformanceifthecandidatevideoactuallyexperiencescertainafnetransform.However,thedetectorisunawareoftheexactmanipulationtothecandidateframe.Ifitisnotafnetransform,KLTgiveswrongparameters,andtheperformanceafterspatialregistrationcanbeworsethanthatwithoutit.Hence,spatialregistrationissetoptionalinourdetector.Typically,detectorcancontroltoswitchon/offthespatialregistrationifithasmanipulationinformation.Otherwise,wecanalwaystrybothandchoosetheonewithbetterdetectionresult. 7.4.1.2TemporalregistrationIntemporalregistration,weusethesimpleruleofminimizingthemean-square-error(MSE)toregistercandidateframetotheoriginalframe.WescanoriginalsequencetondthebestmatchforcurrentcandidateframethatminimizeMSE[ 33 ].Onecausalconstraintisputsothatnoframedisplayedinthepastcanbecapturedinthefuture.Thatis,iftwoframesi,jincandidatevideowithi
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7.4.2WatermarkExtractionandPayloadRecoveryAfterregistration,itisassumedeachframeincandidatevideohasfounditsmatchframeintheoriginalvideo.Notethewatermarkisaadditivesystemthataddswatermarkpatternintooriginalframe.Hence,wecandetecttheexistenceofwatermarksignalbycomputingthecross-correlationbetweenthewatermarkpatternandthetrueframedifference.WeuseakeyexactlycorrespondingtoKey3atembeddertore-generatethewatermarkpattern,aframe-sizePNsequenceatthedetector.Thenormalizedcross-correlationisdenedas: NC(P,P)=hP kPkF,P kPkFi(7)wherePisthewatermarkpattern,Pisthetruedifferencebetweencandidateframeanditsregisteredframe;<,>denotesinnerproduct,andkkFisFrobeniusnorm.Therangeofthenormalizedcross-correlationis[)]TJ /F8 11.955 Tf 9.29 0 Td[(1,1].Thelargertheabsolutevalueofthecoefcient,thebetterchancethecandidateframecontainstheregeneratedpattern,i.e.ithasthewatermarkinformationembedded.Eachcandidateframecorrespondstoonecoefcientvalue.Aharddecisionthresholdof0isusedtomakethecoefcientsequencetoabinary-1/+1sequence.Ifthecoefcientislargerthan0,wedenoteitas,otherwiseitis-1.Theextracted-1/+1sequencesfX0gisthendemodulatedto0/1sequencefXgas: X=(X0+1)=2(7)Thewatermarkpayloadrecoveryisthereverseprocessofpayloadgeneration.ThebinarypayloadsequenceX0needstobedecodedanddecryptedtoderivethestring.ForECCdecoding,weuseViterbialgorithmtodecodetheconvolutionalcode[ 102 ].AndAESdecryptionmethodisdescribedinstandard[ 42 ].The128-bitAESkeyusedindecryptionisdenotedasKey10,usuallysettobeidenticaltoKey1. 167

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7.5ExperimentalResultsThestepbystepresultsofwatermarkembeddingarelistedinTable 7-1 .Thetestvideosequencesaredownloadedfrom[ 1 ]forwatermarkingembedding.TheyareYUVsequencesofCIFformatincludingForeman,Mobile,NewsandStefan.Column1showstheYcomponentsofthe90thframesintheoriginalsequences.Column2representstheircorrespondingwatermarkpatternsafterperceptualweightingmodel.Column3showsthewatermarkedsequencesfromwhichthewatermarkisimperceptible.AndColumn4isthewatermarkedframesaftergeometrictransformwithanti-collusionability.Theyallundergoup-rightshifting3pixels,clockwiserotate2o,andresizing101%.WecanhardlydistinguishthewatermarkedframesinColumn3withtheoriginalframesinColumn1,whichmeetsourperceptualrequirementforwatermarkingsystem.Furthermore,geometricanti-collusioncodedoesnotcausemuchdistortionneither,asframesinColumn4andColumn3lookexactlyalike.ThePSNRofthewatermarkedsequencesareshowninFigure 7-4 ,whichfallsintherangeof34and47dB.Atdetectorside,wetestthecapabilityofdetectortocorrectlyextractwatermarkinformationundervariousattacks.Amongthem,themostimportanttaskistoverifythecapabilitytoresistafnetransforms,notonlybecausetheyareusedforanti-collusioncodingatembedderforsecuritypurpose,butthedistributedvideoscanencountermaliciousgeometricmanipulationsaswell.Figure 7-5 liststhewatermarkedframeundervariousafnetransforms.Thetestsequenceis300framesQCIFGrandma,andthe5thframeisselectedtoillustratetheeffectofmultiplegeometrictransforms,including25pixelrotation(around8orotation)( 7-5C ),10pixelexpanding(around105.7%)( 7-5D ),10pixelshrinking(around94.3%)( 7-5E )and40pixelshifting 7-5F .Notehowsignicantlythelastthreetransformschangetheframestructure.Thegeometrictransformstosuchextenthavebeeneasilydetectedbyhumaneyes,hencetheymaybeoutoftherangetheanti-collusioncodingcancarryoutonwatermarkedvideo,andverylikelytheresultofthirdpartyattacks.Therefore,theperformancethe 168

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detectorachievesonthesevideoscanfullyjustifyitunderafnetransformattacks.Table 7-2 showstheerrorrateofcross-correlationcoefcientsthedetectorobtainsundertheabovementionedtransformscenarios.Itisdenedas: Re=Me M(7)whereMeisthenumberoferroneousdemodulatedbinarybits,andMisthenumberofframesinthesequence.NotethiserrorrateisobtainedbeforeECCdecoding,whichcanfurthercorrectthebiterror.Therstrowshowstheerrorratewithoutframeregistration.TheerrorrateistoohighforECCtocorrect.Anditturnsoutwecannotgetthecorrectwatermarkinformationatdetector.OnetimeKLTregistrationhassignicantlyreducedtheerrorrate.TheiterativeKLTregistrationcanfurtherimprovetheperformance(thethirdrow)butnotsosignicantaswhatonetimeregistrationtonoregistrationatall.Wenoticefor25pixelrotation,iterativeKLTisactuallyidenticaltoonetimeKLTasitonlyoperatestheregistrationonce.Thisisbecausetheseareallsinglekindtransforms,eitherrotation,resizingorshifting.AndonetimeKLTisgoodenoughtotrackthecorrecttransformparameters.WhilecombinationaltransformsposegreaterchallengeforKLTbasedspatialregistration.AsshowninTable 7-3 ,complexafnetransformsandsingletransformofhighermagnituderequireiterativeKLTtoenablethedetectortoextractthecorrectwatermark.Notethatforresizing,positivevaluemeansshrinking( 7-5E ),andnegativevalueindicatesexpanding( 7-5D ).TherearesomecombinatorialtransformsinwhichKLTregistrationfails(indicatedbyN/A),i.e.iterativeKLTregistrationwillnotconvergeaftermaximumiterationtimesandfailstoestimatethecorrectparameters. 7.6SummaryInthischapter,weproposearobusttrack-and-traceanti-collusionwatermarkingsystem.Attheembedder,theuserandcopyrightinformationissecurelymappedtobinarysequencesusingAES,ECC,whichresultsinwatermarkpayload.Orthogonal 169

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framesizePN-sequenceisgeneratedwithsecretekeyaswatermarkpattern.Thepatternisthenperceptuallyweightedandintegratedwiththeoriginalvideosequenceframebyframeaccordingtowatermarkpayload.Foranti-collusionpurpose,thewatermarkedvideowillbegeometricallytransformedbeforedistribution.Atthedetector,candidatevideowillbespatiallyandtemporallyregisteredtotheoriginalvideoifneeded.Wecomputethecrosscorrelationbetweenthere-generatedwatermarkpatternandframedifferencetoextractthepayload.ThenthepayloadisECCdecodedandAESdecryptedtogetthenalwatermarkinformation.Experimentalresultsshowthattheproposedsystemisrobustagainstgeometricattacksandcollusionattacks,andmeetstherequirementofinvisibilitytoHVS.Meantime,italsoshowsthatiterativeKLTregistrationhaslimitations.Ourfutureworkincludesfurtherinvestigationintothetransformattacksandenhancingthedetectorcapabilitytocopewithcomplicatedcombinatorialafnetransformsandnon-afnetransforms.Moreover,wewilltestthedetectorunderothertypesofvideo-relatedattackssuchascompression,erroneoustransmission,andreverseorderdisplay. 170

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A B Figure7-1. Track-and-tracevideowatermarkingembedder.(A)Watermarkgenerator(B)Watermarkembedder 171

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A B Figure7-2. Track-and-tracevideowatermarkingdetector.(A)Watermarkextractor(B)Payloadrecovery 172

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Figure7-3. Perceptualmodelingdiagram A B C D Figure7-4. PSNRofwatermarkedvideosequencesinTableI.(A)Foreman.(B)Mobile.(C)News.(D)Stefan. 173

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A B C D E F Figure7-5. Geometrictransform/attacksto5thframeofGrandma.(A)Originalframe.(B)Watermarkframe.(C)Watermarkframewith25pixelrotation.(D)Watermarkframewith10pixelexpanding.(E)Watermarkframewith10pixelshrinkingandtruncatedtooriginalsize.(F)Watermarkframewith40pixelshifting. 174

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Table7-1. Stepbystepresultsofwatermarkingembedder SequencesPWMWatermarkedGeo-transformed Table7-2. Crosscorrelationcoefcienterrorratio(%)withframeregistrationinGrandmasequence Attack1Attack2Attack3Attack4 Noregistration41.67348.6747.671-KLTRegistration02.330.672.33IterativeKLTRegistration0000.33 (Attack1:25PixelRotation;Attack2:10PixelExpanding;Attack3:10PixelResizing;Attack4:40PixelShifting) 175

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Table7-3. CapabilityofKLTbasedvideoregistrationforvariousgeometrictransforms shiftresizerotateKLTiterationtimeCapability 0-1502Y0-200N/AN01502Y0200N/AN00402Y1010104Y201020N/AN205204Y20-5203Y30510N/AN (N/A:notapplicable) 176

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CHAPTER8CONCLUSIONS 8.1SummaryoftheDissertationInthissection,wesummarizetheresearchpresentedinthisdissertation.Thisdissertationexploredalgorithmsandtheoriesinimageandvideocompressionandcopyrightprotection.Forcompression,integerreversibletransformsandlowcomplexityquantizationwerestudied.Forcopyrightprotection,imagehashing,imageauthenticationandvideowatermarkingtechniquesweredeveloped.Chapter 2 studiedtheintegerreversibletransformdesignproblemforlosslesssignalcompression.ForthePLUSfactorization,whichfactorizesarbitrarytransformmatrixwithunitarydeterminantintoaproductofpermutationmatrix,alowertriangularelementaryreversiblematrix,anuppertriangularelementaryreversiblematrixandasinglerowelementaryreversiblematrix,westabilizedandoptimizedthefactorizationanddidperturbationanalysisonit.StabilizationbyusingpivotingmadePLUSfactorizationstable;optimizationbyusingTabusearchmadeithavethesmallesttransformerror;perturbationanalysisproveditnumericallystable.BasedontheoptimizedPLUSfactorization,webuiltupintegerreversibletransformsfromthetraditionaltransforms,suchasDCTandLBT.Applyingtheproposedintegerreversibletransformsintolosslessimagecompression,theexperimentalresultsshowedthatourintegerDCTwiththeoptimalPLUSfactorizationoutperformedtheintegerDCTwithexpansionfactors;ourintegerlappedbiothogonaltransformswiththeoptimalPLUSfactorizationoutperformedthatinJPEG-XR.Chapter 3 studiedtheone-dimensionalquantizerdesignandproposedanadaptivequantizationusingpiecewisecompandingandscalingforGaussianmixture.OuradaptivequantizerhadthreemodescorrespondingtothreetypesofGaussianMixture.Ourexperimentalresultsshowedthat1)theproposedquantizerwasabletoachieveperformanceclosetotheoptimalquantizer(i.e.,Lloyd-MaxquantizerforGMM)in 177

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thesenseofMeanSquaredError(MSE),atmuchlowercomputationalcostthanit;2)theproposedquantizerwasabletoachievemuchbetterMSEperformancethanauniformquantizer,atacostsimilartotheuniformquantizer.Furthermore,weproposedarecongurablearchitecturetoimplementouradaptivequantizerinanADC.Wealsousedittoquantizeimagesanddesignthetonemappingalgorithmforhighdynamicrange(HDR)imagecompression,rewardingimprovedvisualperformance.Chapter 4 studiedthehighdimensionalquantizerdesignandproposedanoptimalvectorquantizerapproximatorswithtransformsandscalarquantizers.VectorquantizersaimattheoptimalRate-Distortionperformance,buttheirdesigncomplexityincreasesexponentiallywiththenumberofquantizationlevels.Toreducequantizerdesignandimplementationcomplexity,weproposedtocombinethetransformandscalarquantization.Transformwasusedtodecorrelatedataandfacilitatequantization.Unitarytransformsandvolume-keepingscalingtransformswerediscussed,butonlyunitarytransformswereused.Aftertransform,thememorylessdatawerepluggedintoatri-axiscoordinateframe,andthenaneffectivescalarquantizationwasappliedonthedata.Theproposedquantizationframeworkwerealmostsuitableforarbitrarydistribution.Thetri-axiscoordinateframeworkedformemorylesssourcesinbothcircularandellipticaldistributions.WetestedourproposedquantizeronbothGaussianandLaplacedistributions.Theresultedperformancewasalmostalwaysbetterthanthatofrestricted/unrestrictedpolarquantizersandrectangularquantizers.TheresultedRate-Distortionperformanceoftheproposedquantizationapproachedtheperformanceoftheoptimalvectorquantization.Chapter 5 proposedarobustimagehashingsystem.Thesystemwasbuiltupby:arobustdescriptorofimagesthek-largestlocaltotalvariations,whichwasthenquantizedwithcompandingandbinarilycodedwithGraycode.Thek-largestlocaltotalvariationswererobusttocontentpreservingattackssuchasgeometricattacksandluminanceattacks,andindicatedstablesimilarfeaturepointsinperceptualidentical 178

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images.TheMorletwaveletcoefcientswerepseudorandomlypermutedwithasecretekey,whichenhancedthesecurityandreducedthecollisionrateoftheimagehashingsystem.Morletwaveletcoefcientswerecomputationallyefcientlyquantizedusingcompandingtechniqueaccordingtotheprobabilitydistributionofthecoefcients.Thus,theproposedimagehashingwasrobusttocontrastchangingandgammacorrectionofimages.Graycodewasusedtobinarilycodethequantizedcoefcients,whichincreaseddiscriminabilityofimagehashes.Chapter 6 proposedarobustimageauthenticationsystem.Itwasacontentbasedimageauthenticationsystembyfeaturepointclusteringandmatching.Featurepointsweredetectedfromimagesrstly.Thenthefeaturepointsfromtheanchorimagesandthequeryimageswereclusteredandmatched.Thedistanceandmatchinginformationbetweenpointpairswereusedtoidentifytheauthenticityofthequeryimagesandpossibleattackitsuffers.Featurepointswererstlygeneratedfromagivenimage,buttheirlocationsmaybechangedduetopossibleimageprocessinganddegradation.Accordingly,weproposedtouseFuzzyC-meanclusteringalgorithmtoclusterthefeaturepointsandremovetheoutliersfromthefeaturepoints.Histogramweighteddistancewasproposed,whichwasequivalenttoHausdorffdistanceafteroutlierremoval.Theauthenticityofthequeryimagewasdeterminedbythemajorityvoteofwhetherthreetypesofdistancebetweenmatchedfeaturepointpairwerelargerthantheirrespectivethresholds.Thegeometrictransformsthroughwhichthequeryimageswerealignedwiththeanchorimageswereestimated,andthequeryimageswereregisteredaccordingly.Thepossibletamperedimageblockswereidentied,andthepercentageofthetamperedareawasestimated.Chapter 7 proposedarobusttrack-and-tracevideowatermarkingsystem.Thesystemincludesawatermarkembedderandawatermarkdetector.Atembedder,wepseudorandomlygeneratedwatermarkpatternsandtranslatedtheuserinformation,videoinformationintowatermarkpayload.Thentheywereembeddedintovideoframes 179

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accordingtoperceptualweightingmodels.Wealsoembeddedgeometricanti-collusioncodetoresistcollusionattacks.Atdetector,weusedKLTtoregistervideoframesandthenextractedwatermarkandrecoverpayload.Thevideocopyrightinformationandthepossiblemaliciousattackswereidentied.Thesystemprovided:Security:userandproductcopyrightinformation,e.g.astringoflengthLs,wasrstencryptedwithAdvancedEncryptionStandard(AES);errorcorrectioncode(ECC)wasappliedtothesequencetogenerateabinarysequencewitherror-correctionabilityoflengthL,calledwatermarkpayload;aframe-sizewatermarkpatternarisedfromapseudo-randomnoise(PN)sequence.Perceptualinvisibilityandrobustness:Tomakeatrade-offbetweenvisualqualityandrobustnesswhichwasdeterminedbytheembeddingstrength,webuiltaperceptualmodeltodeterminethesignalstrengththatcouldbeembeddedtoeachpixelbyusingstatisticalsourceinformationandJust-Notice-Difference(JND)model.Track-and-trace:geometricanti-collusioncodingwasusedfortrackingandtracingcolluders.AniterativeKLTbasedschemewasappliedforspatialregistrationforwatermarkingextraction. 8.2FutureWorkInthissection,wepointoutfutureresearchdirections. 8.2.1OptimalIntegerReversibleTransformsandtheLosslessVideoCompres-sionTheoptimalintegerreversibletransformswiththeleastentropyneedtobeinvestigatedfurther.Nowadays,capturing,creating,editingandprocessingmovingimagesemployawiderangeoftechniquesforreducingtheamountofdatatobestoredandtransmitted.Signicantadvancetechniquesareoccurringinthereductionofbit-ratesforend-usedistributionandconsumerapplicationssuchasInternetvideostreaming,theDCI(DigitalCinemaInitiative),Blu-RayDVD,andhigh-denitionTV.Videoarchiving,especiallylosslessvideoarchiving,isimportantforhighqualitystudioproducts,medicalvideos, 180

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neartsandantiquedocuments,andsatellitedata.ThereislosslesscompressionmodeinH.264AdvancedVideoCompression/MPEG-4Part10whichisanindustrystandardforvideocoding[ 119 ].Itsimplementationinx264isbasicallyDPCMwithouttransform.WewilltrytoimplementintegerDCTintovideocodeVP8,testitsperformance,andcompareitwithlosslesscompressionmodeofvideocodecx264.Besides,theRDCoptimaltransformsarestillunknown,andhowtoeffectivelyusedirectionaltransformsstillneedinvestigation. 8.2.2ANewVideoCodingStrategyandRDCOptimizationVideosbecomeanimportantpartofhumanlifeinthedigitalage.Thesoaringnumberofvideosdemandsefcientvideocompression,whichisstandarnizedinH.264/MPEG-4Part10orAVC(AdvancedVideoCoding)[ 119 152 ]andtheemergingH.265/HEVC[ 71 96 148 ].Thesestandardsaimtoencodevideoswiththeleastbitrate,theleastdistortionandtheleastcomputationalcomplexitywithcertainconstraint.TheexistingH.264videocodecsincludemanyencodingstrategies,suchasone-passandmulti-passAverageBitrateEncoding(ABR),ConstantBitrateEncoding(CBR),ConstantQuantizerEncoding(CQP)andConstantRateFactorEncoding(CRF)[ 32 98 ].Theseencodingstrategiesgenerallyhavesingleobjectiveandthefollowingproperties.Basedontheexistingrate-distortion-complexityframework,wecanincorporateRDCoptimaltransformandRDCoptimalquantizationintovideocoding.TheRDCtheorycouldbestudiedandbuiltup,andtestedonpracticalvideocodingsystem. 181

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APPENDIXAPERTURBATIONANALYSISOFPLUSAssumethatanonsingularmatrixA2RnnhasuniquePLUSfactorization,A=LUS.LetA2RnnbeaperturbationsuchthatA+AalsohasuniquePLUSfactorization: A+A=(L+L)(U+U)(S+S)(A)NotethatPisnotconsidered,becausetheanalysisofthesensitivityofgeneralPLUSfactorizationalgorithmissimplerandwithoutmuchlossofgenerality.ThemeasureofL,UandSisdeducedasfollowing.Weusethematrix-vectorequationapproach[ 25 ],topresentperturbationanalysisforPLUSfactorization.ComparedwithChang'sperturbationanalysisforLUfactorization, 1. ThepatternofUinPLUSfactorizationisdifferentfromthatofLUfactorization. 2. Sincreasestheanalysiscomplexity.Weusethefollowingnotationsfortheperturbationanalysis.A(t)representsmatrixfunction,andA(t)representsitsderivative.ForanymatrixA2Rnn,A=[a1,,an],aiistheithcolumnvectorofA,andvec(A)2Rn21isthevectorformofAwithaiinsuccession.eL2Rn(n)]TJ /F5 8.966 Tf 6.97 0 Td[(1)=21iscomposedofelementsinnonzerolocationsofL(0),eU2R(n2)]TJ /F16 8.966 Tf 6.97 0 Td[(n+2)=21iscomposedofelementsinnonzerolocationsofU(0),andeS2R(n)]TJ /F5 8.966 Tf 6.97 0 Td[(1)1iscomposedofelementsinnonzerolocationsofS(0).TheuniquenessconditionofPLUSfactorizationis:UniquenessCondition.det([an,a1,a2,,ak)]TJ /F5 8.966 Tf 6.97 0 Td[(1](k))6=0whenk>1,anda1n6=0whenk=1,whereA(k)denotesthek-thleadingsub-matrixofA.Lemma1.AssumethatanonsingularmatrixA2RnnhasuniquePLUSfactor-ization,A=LUS.LetA2Rnn,A=eE,whereeissmallenough,suchthatA+tEsatisesUniquenessConditionforalljtje,thenA+tEhasuniquePLUSfactorization: A+tE=L(t)U(t)S(t),jtje,(A) 182

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whichleadsto L(0)US+LU(0)S+LUS(0)=E.(A)Fort=e,weobtainA+AwiththeuniquePLUSfactorization A+A=(L+L)(U+U)(S+S),(A)whereL,UandSsatisfyL=eL(0)+O(e2) (Aa)U=eU(0)+O(e2) (Ab)S=eS(0)+O(e2) (Ac)ItcanbeeasilyprovedusingTaylorexpansioninthesimilarwayasin[ 25 ].NotethatL(0)=L,L(e)=L+L,U(0)=U,U(e)=U+U,S(0)=S,andS(e)=S+S.FromEquation( A ),weobtain ei=L(0)Usi+LU(0)si+LUs(0)i,i=1,,n.(A)Byrearrangingequation( A ),weobtainamatrix-vectorequation: WLWUWS0BBBB@eLeUeS1CCCCA=vec(E)(A)whereWL2Rn2n(n)]TJ /F5 8.966 Tf 6.97 0 Td[(1)=2iscomposedofn(n)]TJ /F8 11.955 Tf 10.95 0 Td[(1)sub-matriceswithfollowingpattern:0BBBBBBBBBB@WL1,1WL1,2WL1,n)]TJ /F69 5.978 Tf 4.65 0 Td[(2WL1,n)]TJ /F69 5.978 Tf 4.64 0 Td[(1WL2,1WL2,2WL2,n)]TJ /F69 5.978 Tf 4.65 0 Td[(2WL2,n)]TJ /F69 5.978 Tf 4.64 0 Td[(1...............WLn)]TJ /F69 5.978 Tf 4.65 0 Td[(1,1WLn)]TJ /F69 5.978 Tf 4.64 0 Td[(1,2WLn)]TJ /F69 5.978 Tf 4.65 0 Td[(1,n)]TJ /F69 5.978 Tf 4.64 0 Td[(2WLn)]TJ /F69 5.978 Tf 4.64 0 Td[(1,n)]TJ /F69 5.978 Tf 4.65 0 Td[(1WLn,1WLn,2WLn,n)]TJ /F69 5.978 Tf 4.65 0 Td[(2WLn,n)]TJ /F69 5.978 Tf 4.64 0 Td[(11CCCCCCCCCCA 183

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WLij2Rn(n)]TJ /F16 8.966 Tf 6.97 0 Td[(j),1i,jn, WLij=8>>>>>>>><>>>>>>>>:(siujn+uji)0B@0In)]TJ /F16 8.966 Tf 6.96 0 Td[(j1CA,i
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1,,1| {z }n)asthediagonalentries.Therefore,W=[WL,WU,WS]isinvertible.LetW)]TJ /F5 8.966 Tf 6.96 0 Td[(1=0BBBB@YLYUYS1CCCCA,thenweobtain 0BBBB@eLeUeS1CCCCA=W)]TJ /F5 8.966 Tf 6.97 0 Td[(1vec(E)=0BBBB@YLYUYS1CCCCAvec(E)(A)kL(0)kF=keLk2kYLkFkvec(E)k2=kYLkFkEkF (Aa)kU(0)kF=keUk2kYUkFkvec(E)k2=kYUkFkEkF (Ab)kS(0)kF=keSk2kYSkFkvec(E)k2=kYSkFkEkF (Ac)kLkF kLkFkYLkFkEkF kLkFe+O(e2)=kL(A)kAkF kAkF+O(e2) (Aa)kUkF kUkFkYUkFkEkF kUkFe+O(e2)=kU(A)kAkF kAkF+O(e2) (Ab)kSkF kSkFkYSkFkEkF kSkFe+O(e2)=kS(A)kAkF kAkF+O(e2) (Ac)wherekL(A)=kYLkFkAkF=kLkFkU(A)=kYUkFkAkF=kUkFkS(A)=kYSkFkAkF=kSkFTheorem3.PerturbationanalysisII.AssumethatAandA+AarebothnonsingularandtheirPLUSfactorizationsexist:A=LUSandA+A=(L+L)(U+U)(S+S),thenEquations( A )( Aa )( Ab )( Ac )( Aa )( Ab )( Ac )hold.Perturbationanalysisexample: 185

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TheoriginalPLUSfactorizationis: A=LUS=0B@0.89130.45650.76210.01851CA=0B@100.766511CA0B@10.45650)]TJ /F8 11.955 Tf 9.29 0 Td[(0.33141CA0B@10)]TJ /F8 11.955 Tf 9.29 0 Td[(0.238111CAWithaperturbationA,thePLUSfactorizationis: A+A=(L+L)(U+U)(S+S)=0B@0.89130.45650.76210.01851CA+0B@0.0090.0010.0070.0041CA=0B@100.774611CA0B@10.45750)]TJ /F8 11.955 Tf 9.29 0 Td[(0.33161CA0B@10)]TJ /F8 11.955 Tf 9.29 0 Td[(0.217911CATherefore,theperturbationerrorofPLUSfactorizationislimited. 186

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APPENDIXBPROOFS B.1ProofofProposition 3.1 Proof. TheLloyd-Maxquantizeris: minr0k,t0kNk=1Zt0kt0k)]TJ /F28 6.974 Tf 5.42 0 Td[(1(x)]TJ /F3 11.955 Tf 10.95 0 Td[(r0k)2fX(x)dx(B)withftkgNk=0andfrkgNk=1asthesolution.ForY=sX+m, minr00k,t00kNk=1Zt00kt00k)]TJ /F28 6.974 Tf 5.42 0 Td[(1(y)]TJ /F3 11.955 Tf 10.95 0 Td[(r00k)2fY(y)dy=Nk=1Zt00k)]TJ /F54 6.974 Tf 1 0 .167 1 200.22 -231.95 Tm[(m st00k)]TJ /F28 6.974 Tf 5.41 0 Td[(1)]TJ /F54 6.974 Tf 1 0 .167 1 204.21 -252.13 Tm[(m s(sx+m)]TJ /F3 11.955 Tf 10.95 0 Td[(r00k)2fX(x)dx=s2Nk=1Zt00k)]TJ /F54 6.974 Tf 1 0 .167 1 213.7 -273.79 Tm[(m st00k)]TJ /F28 6.974 Tf 5.41 0 Td[(1)]TJ /F54 6.974 Tf 1 0 .167 1 217.69 -293.97 Tm[(m s(x)]TJ /F3 11.955 Tf 12.15 8.46 Td[(r00k)]TJ /F11 11.955 Tf 1 0 .167 1 272.92 -283.01 Tm[(m s)2fX(x)dx(B)ifandonlyift00k)]TJ /F28 6.974 Tf 5.42 0 Td[(1)]TJ /F50 8.966 Tf 1 0 .167 1 92.62 -325.32 Tm[(m s=tkandr00k)]TJ /F50 8.966 Tf 1 0 .167 1 169.52 -325.9 Tm[(m s=rk,Eq.( B )isminimal,i.e.fstk+mgNk=0andfsrk+mgNk=1isthesolutionforEq.( B ). B.2ProofofProposition 3.2 Proof. TheF)]TJ /F5 8.966 Tf 6.97 0 Td[(1X(y)shouldbewelldenedasF)]TJ /F5 8.966 Tf 6.96 0 Td[(1X(y)=inffx:FX(x)y,0
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AssumethatXhasanitesupportoratruncatedsupport(a,b),andfX(x)>0onthesupport.ByusingBennettt'sBennett[ 11 ]approximateexpressionforthemeansquaredistortionforverylargenumberNofquantizeroutputlevels,wehave: E[(X)]TJ /F8 11.955 Tf 13.79 2.92 Td[(X)2]=1 12N2Zba1 fX(x)dx(B)Itisboundedby1 12N2(b)]TJ /F3 11.955 Tf 10.95 0 Td[(a)max(a,b)f1 fX(x)gTherefore,Eq.( B )istowards0asN!. B.3ProofofProposition 4.1 Proof. Forone-dimensionalr.v.s,=s)]TJ /F5 8.966 Tf 6.97 0 Td[(2,theLloyd-Maxquantizeris: minr0k,t0kNk=1Zt0kt0k)]TJ /F28 6.974 Tf 5.42 0 Td[(1(x)]TJ /F3 11.955 Tf 10.95 0 Td[(r0k)2fX(x)dx(B)withftkgNk=0andfrkgNk=1asthesolution.ForY=sX+m, minr00k,t00kNk=1Zt00kt00k)]TJ /F28 6.974 Tf 5.42 0 Td[(1(y)]TJ /F3 11.955 Tf 10.95 0 Td[(r00k)2fY(y)dy=Nk=1Zt00k)]TJ /F54 6.974 Tf 1 0 .167 1 200.22 -397.32 Tm[(m st00k)]TJ /F28 6.974 Tf 5.41 0 Td[(1)]TJ /F54 6.974 Tf 1 0 .167 1 204.21 -417.49 Tm[(m s(sx+m)]TJ /F3 11.955 Tf 10.95 0 Td[(r00k)2fX(x)dx=s2Nk=1Zt00k)]TJ /F54 6.974 Tf 1 0 .167 1 213.7 -439.16 Tm[(m st00k)]TJ /F28 6.974 Tf 5.41 0 Td[(1)]TJ /F54 6.974 Tf 1 0 .167 1 217.69 -459.33 Tm[(m s(x)]TJ /F3 11.955 Tf 12.15 8.46 Td[(r00k)]TJ /F11 11.955 Tf 1 0 .167 1 272.92 -448.38 Tm[(m s)2fX(x)dx(B)ifandonlyift00k)]TJ /F28 6.974 Tf 5.42 0 Td[(1)]TJ /F50 8.966 Tf 1 0 .167 1 92.62 -490.68 Tm[(m s=tkandr00k)]TJ /F50 8.966 Tf 1 0 .167 1 169.52 -491.26 Tm[(m s=rk,Eq.( B )isminimal,i.e.fstk+mgNk=0andfsrk+mgNk=1isthesolutionforEq.( B ).Similarly,itholdsforn-dimensionalr.v.s. B.4ProofofLemma 2 Proof. AunitarytransformUsatisesUTU=UUT=I.Forvectorsxandy,theEuclideandistancebetweenthemiskx)]TJ /F3 11.955 Tf 11.53 0 Td[(yk.AftertransformbyU,theEuclidean 188

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distancebetweenUxandUyiskUx)]TJ /F3 11.955 Tf 10.94 0 Td[(Uyk=kUkkx)]TJ /F3 11.955 Tf 10.94 0 Td[(yk=kx)]TJ /F3 11.955 Tf 10.94 0 Td[(yk.Thus,MeanSquareErrorisinvariantunderunitarytransforms. B.5ProofofLemma 4 Proof. DenotethecomponentsofrandomvectorXas(X1,X2)andaunitarytransformasU.Aftertransform,therandomvectorbecomesX0=(X01,X02).Neglecttheprecisionlossbroughtbythecomputer,Uisaone-to-onemapping.Therefore,H(X1,X2)=H(X01,X02).IfX01andX02areindependent,thenH(X01)+H(X02)=H(X01,X02)=H(X1,X2)H(X1)+H(X2) B.6ProofofLemma 5 Proof. Denotethevarianceofeverycomponentofann-dimensionalGaussianvectorX=(X1,X2,,Xn)as(s21,s22,,s2n).Aftervolume-keepingscalingtransforms,theresultedGaussianvectorX0hasvariancesof(s210,s220,,s2n0)foreachcomponentandni=1s2i=ni=1s2i0.Therefore,ni=1H(Xi)=ni=11 2ln(2pes2i)=1 2ln((2pe)nni=1s2i)=1 2ln((2pe)nni=1s2i0)=ni=1H(X0i)Similarly,itholdsforrandomvectorsinLaplaciandistribution. 189

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BIOGRAPHICALSKETCH LeiYangreceivedherB.S.degreeincomputersciencefromBeijingInformationandTechnologyInstitute,andM.E.inelectricalandcomputerengineering,PekingUniversity,Beijing,China,in2008.ShereceivedherPh.D.inelectricalandcomputerengineeringfromtheUniversityofFloridainthesummerof2011.FromMay2010toDec.2010,shewasaninternatGoogleInc.,MountainView,CA,wheresheworkedonrate-distortion-complexityoptimizationofvideotranscodingforYouTube,andopensourcevideocodecVP8.ShewilljoinGoogleInc.afterAugust2011.Herresearchinterestsincludeadvancedimageandvideocoding,imageprocessing,imageandvideosecurity,transformdesignandoptimization,computervisionandmachinelearning. 202