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Modeling distance functions induced by face recognition algorithms

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Modeling distance functions induced by face recognition algorithms
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Chaudhari, Soumee
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affine space
eigen space
principal component analysis
Optimal affine transformation
biometrics
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ABSTRACT: Face recognition algorithms has in the past few years become a very active area of research in the fields of computer vision, image processing, and cognitive psychology. This has spawned various algorithms of different complexities. The concept of principal component analysis(PCA) is a popular mode of face recognition algorithm and has often been used to benchmark other face recognition algorithms for identification and verification scenarios. However in this thesis, we try to analyze different face recognition algorithms at a deeper level. The objective is to model the distances output by any face recognition algorithm as a function of the input images. We achieve this by creating an affine eigen space from the PCA space such that it can approximate the results of the face recognition algorithm under consideration as closely as possible.Holistic template matching algorithms like the Linear Discriminant Analysis algorithm( LDA), the Bayesian Intrapersonal/Extrapersonal classifier(BIC), as well as local feature based algorithms like the Elastic Bunch Graph Matching algorithm(EBGM) and a commercial face recognition algorithm are selected for our experiments. We experiment on two different data sets, the FERET data set and the Notre Dame data set. The FERET data set consists of images of subjects with variation in both time and expression. The Notre Dame data set consists of images of subjects with variation in time. We train our affine approximation algorithm on 25 subjects and test with 300 subjects from the FERET data set and 415 subjects from the Notre Dame data set. We also analyze the effect of different distance metrics used by the face recognition algorithm on the accuracy of the approximation.We study the quality of the approximation in the context of recognition for the identification and verification scenarios, characterized by cumulative match score curves (CMC) and receiver operator curves (ROC), respectively. Our studies indicate that both the holistic template matching algorithms as well as feature based algorithms can be well approximated. We also find the affine approximation training can be generalized across covariates. For the data with time variation, we find that the rank order of approximation performance is BIC, LDA, EBGM, and commercial. For the data with expression variation, the rank order is LDA, BIC, commercial, and EBGM. Experiments to approximate PCA with distance measures other than Euclidean also performed very well. PCA+Euclidean distance is best approximated followed by PCA+MahL1, PCA+MahCosine, and PCA+Covariance.
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Thesis (M.S.C.S.)--University of South Florida, 2004.
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by Soumee Chaudhari.
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ModellingDistanceFunctionsInducedbyFaceRecognitionA lgorithms by SoumeeChaudhari Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofScienceinComputerScience DepartmentofComputerScienceandEngineering CollegeofEngineering UniversityofSouthFlorida MajorProfessor:SudeepSarkar,Ph.D. RangacharKasturi,Ph.D. RaviSankar,Ph.D. DateofApproval: November9,2004 Keywords:anespace,eigenspace,principalcomponentana lysis,optimalane transformation,biometrics c r Copyright2004,SoumeeChaudhari

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DEDICATION Tomyfamilyandancewhohavebeenaconstantsourceoflovea ndsupport

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ACKNOWLEDGEMENTS IwouldliketothankDr.SudeepSarkar,mymajorprofessor,f orhisguidanceand supportthroughoutmyMaster'sdegreeprogram,andforcare fullyreviewingmythesis writeup.Thisthesishasbeenagreatlearningexperiencefo rme.IsincerelythankDr. SudeepSarkarforgivingmetheopportunitytoworkonthispr oject.Iwouldalsoliketo thankDr.KasturiandDr.Sankarforbeingapartofmythesisc ommittee.Iwouldalso liketothankDr.RossBeveridgeandhisstudentsfromtheCol oradoStateUniversityfor theirpatienceandhelp.Iwouldalsoliketothankmyfellows tudentsLakshmiReguna, PranabMohantyandHimanshuVajariafortheirvaluableandt imelyhelp.

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TABLEOFCONTENTS LISTOFFIGURES iii ABSTRACT v CHAPTER1INTRODUCTION 1 1.1Motivation 3 CHAPTER2RELATEDWORK 5 2.1FaceRecognitionEvaluationProtocols52.2DimensionalityReductionTechniques102.3DistanceMetricLearningTechniques 13 CHAPTER3THEFACERECOGNITIONALGORITHMSEVALUATED15 3.1PrincipalComponentAnalysis 16 3.2LinearDiscriminantAnalysis 16 3.3BayesianIntrapersonal/ExtrapersonalClassier173.4ElasticBunchGraphMatching 18 3.5CommercialFaceRecognitionSoftware19 CHAPTER4DISTANCESINAFFINEAPPROXIMATIONSPACES21 4.1Multi-DimensionalScaling 21 4.2AneTransformationBasedModellingofDistances22 4.2.1SimilaritiesandDissimilarities25 4.3AnAlternativeMethodtoDeriveA 25 CHAPTER5EXPERIMENTSANDANALYSISOFRESULTS27 5.1IssuesAddressed 27 5.2DistanceMeasuresUsedbyFaceRecognitionAlgorithms2 7 5.3DataDescription 29 5.4TrainingandTestSetup 31 5.5AneMatrix 34 5.5.1AneSpaceDimensions 34 5.6PerformanceonDatawithTimeVariation40 5.6.1IdenticationandVericationPerformance40 5.7PerformanceonDataWithExpressionVariation42 5.7.1IdenticationandVericationPerformance42 5.8ApproximatingPCAwithDierentDistanceMeasures445.9EectsofNormalizationontheDistanceMatrix45 i

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CHAPTER6SUMMARYANDCONCLUSION50 6.1Future 51 REFERENCES 52 APPENDICES 55 AppendixAComparisonofResultswithReguna56 ii

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LISTOFFIGURES Figure2.1EvaluationProtocols 5 Figure2.2DimensionalityReductionTechniques10Figure2.3DistanceMetricLearningTechniques13Figure3.1FaceRecognitionAlgorithm 15 Figure4.1FindMatrixAsuchthattheEuclideanDistancesBe tweenTransformed imagesareEqualtotheGivenDistances21 Figure5.1TrainingSet:100Imageswith4ImagesofEachSubj ectoftheFERET DataSet 30 Figure5.2TestSet(FERET):600Imageswith2ImagesofEachS ubject31 Figure5.3TestSet(NotreDame):830Imageswith2ImagesofE achSubject31 Figure5.4HistogramoftheNotreDameImagesAcquiredOverT ime32 Figure5.5SampleImages 32 Figure5.6TrainingSetupfortheAneApproximationalgori thm33 Figure5.7TestingSetupfortheAneApproximationalgorit hm33 Figure5.8Visualizationof A Tnr A nr forLDAAlgorithm35 Figure5.9Visualizationof A Tnr A nr forBICAlgorithm35 Figure5.10Visualizationof A Tnr A nr forEBGMAlgorithm36 Figure5.11Visualizationof A Tnr A nr forCommercialAlgorithm36 Figure5.12VisualizationofDiagonalValuesof A Tnr A nr forAllAlgorithms37 Figure5.13TopDimensionsoftheAneApproximationtotheD ierentAlgorithms.LastRowShowstheCorrespondingPCADimensionforC omparison 38 Figure5.14PCAEigenDimensionsAlongWhichWeNeedtoStret chandShear theMosttoMatchDistances 39 iii

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Figure5.15ROConNotreDameImagesShowingthePerformance Comparisonof theDierentFaceRecognitionAlgorithmalongwiththeAne ApproximationAlgorithm(a)LDA(b)BIC(c)EBGM(d)Commercia l40 Figure5.16CMConNotreDameImagesShowingthePerformance Comparison oftheDierentFaceRecognitionAlgorithmalongwiththeA neApproximationAlgorithm(a)LDA(b)BIC(c)EBGM(d)Commercia l41 Figure5.17ROConFERETDataSetShowingthePerformanceCom parisonofthe DierentFaceRecognitionAlgorithmalongwiththeAneApp roximationAlgorithm.(a)LDA(b)BIC(c)EBGM(d)Commercial42 Figure5.18CMConFERETDataSetShowingthePerformanceCom parisonof theDierentFaceRecognitionAlgorithmalongwiththeAne ApproximationAlgorithm(a)LDA(b)BIC(c)EBGM(d)Commercia l43 Figure5.19ROCCurvesofPCAAlgorithmwithDierentDistan ceMeasures(a) Euclidean(b)Covariance(c)Maha-Cosine(d)MahL144 Figure5.20ROCofPCAAlgorithmwithDierentDistanceMeas ures(a)Euclidean(b)Covariance(c)Maha-Cosine(d)MahL145 Figure5.21Visualizationof A Tnr A nr forPCAAlgorithm(EuclideanDistance)for theFERETDataSet 46 Figure5.22Visualizationof A Tnr A nr forPCAAlgorithm(CovarianceDistance)for theFERETDataSet 46 Figure5.23Visualizationof A Tnr A nr forPCAAlgorithm(MahCosineDistance)for theFERETDataSet 47 Figure5.24Visualizationof A Tnr A nr forPCAAlgorithm(MahL1Distance)forthe FERETDataSet 47 Figure5.25EectofNormalizationontheROCCurvesfortheC ommercialFace RecognitionAlgorithmontheFERETDataSet49 FigureA.1ComparisonofCMCUsingFERETDataonDierentAlg orithms60 FigureA.2ComparisonofROCUsingFERETDataonDierentAlg orithms61 iv

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ModellingDistanceFunctionsInducedbyFaceRecognitionAlgorith ms SoumeeChaudhari ABSTRACT Facerecognitionalgorithmshasinthepastfewyearsbecome averyactiveareaof researchintheeldsofcomputervision,imageprocessing, andcognitivepsychology.This hasspawnedvariousalgorithmsofdierentcomplexities.T heconceptofprincipalcomponentanalysis(PCA)isapopularmodeoffacerecognitiona lgorithmandhasoftenbeen usedtobenchmarkotherfacerecognitionalgorithmsforide nticationandvericationscenarios.Howeverinthisthesis,wetrytoanalyzedierentfa cerecognitionalgorithmsat adeeperlevel.Theobjectiveistomodelthedistancesoutpu tbyanyfacerecognition algorithmasafunctionoftheinputimages.Weachievethisb ycreatingan ane eigen spacefromthePCAspacesuchthatitcanapproximatetheresu ltsofthefacerecognition algorithmunderconsiderationascloselyaspossible. HolistictemplatematchingalgorithmsliketheLinearDisc riminantAnalysisalgorithm(LDA),theBayesianIntrapersonal/Extrapersonalcl assier(BIC),aswellaslocal featurebasedalgorithmsliketheElasticBunchGraphMatch ingalgorithm(EBGM)anda commercialfacerecognitionalgorithmareselectedforour experiments.Weexperimenton twodierentdatasets,theFERETdatasetandtheNotreDamed ataset.TheFERET datasetconsistsofimagesofsubjectswithvariationinbot htimeandexpression.The NotreDamedatasetconsistsofimagesofsubjectswithvaria tionintime.Wetrainour aneapproximationalgorithm on25subjectsandtestwith300subjectsfromtheFERET datasetand415subjectsfromtheNotreDamedataset.Wealso analyzetheeectof dierentdistancemetricsusedbythefacerecognitionalgo rithmontheaccuracyofthe approximation.Westudythequalityoftheapproximationin thecontextofrecognition v

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fortheidenticationandvericationscenarios,characte rizedbycumulativematchscore curves(CMC)andreceiveroperatorcurves(ROC),respectiv ely. Ourstudiesindicatethatboththeholistictemplatematchi ngalgorithmsaswellas featurebasedalgorithmscanbewellapproximated.Wealso ndtheaneapproximation trainingcanbegeneralizedacrosscovariates.Forthedata withtimevariation,wend thattherankorderofapproximationperformanceisBIC,LDA ,EBGM,andcommercial.Forthedatawithexpressionvariation,therankorder isLDA,BIC,commercial, andEBGM.ExperimentstoapproximatePCAwithdistancemeas uresotherthanEuclideanalsoperformedverywell.PCA+Euclideandistancei sbestapproximatedfollowed byPCA+MahL1,PCA+MahCosine,andPCA+Covariance. vi

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CHAPTER1 INTRODUCTION Recentlybiometricbasedauthenticationandidenticatio nsystemshavereceiveda greatimpetus.Theyhavefoundnumerousapplicationsinsur veillance,securesiteaccess, transactionsecurityandremoteaccesstoresources.Butun likenonbiometricsystems theyhaveseveralinherentadvantages.Unlikeapinorapass worditcannotbeusedbyan unauthorizeduser,itdoesnotneedtobecarried,andprovid esforpositiveidentication. Howeverbiometricbasedngerprint,irisorDNArecognitio nareintrusiveandrequirethe cooperationoftheuser.Thisisonethemajorfactorswhyfac erecognitionhasbecome highlypopularandanactiveareaofresearch.Ithastheadva ntageofbeingnoninvasive andcanbeperformedevenatadistance,sometimesevenwitho uttheknowledgeofthe userasisnecessaryinmanysecurityandsurveillanceappli cations. Facerecognitiontechnologycanbedonewith2Dimagesaswel las,astherecenttrend hasbeen,with3Dimages.Algorithmstoperformfrontal,pro leandviewtolerantrecognitionhavebeendeveloped.Beingachallengingyetinteresti ngproblem,variousalgorithms andtechniqueshavebeendevelopedforfacerecognitionpro blems.Itcanbebroadlyclassiedintothreetechniques.Holisticmethods,feature-bas edmethodsandhybridmethods. Holisticmethodsincludemethodssuchasprincipalcompone ntanalysis[1],lineardiscriminantanalysis[2],andsupportvectormachines[3].Featur e-basedmethodsincludepure geometrymethods,dynamiclinkarchitecture[4],andhidde nMarkovmodels[5].Hybrid methodsincludemodulareigenfaces,hybridLFA,andcompon entbasedmethods[6].The processmayincludeautomaticallydetectingorlocatingth efaceintheimage,extraction oftheorientationoftheface,handlingofconditionssucha silluminationandshadowsand generationofnewviewsfromexistingviews. 1

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Inthewakeoftherecentterrorismincidents,especiallyaf ter9/11therehasbeen renewedinterestinfacerecognitiontechnology.Howeveri thascomeinforalotofcriticism foritsfailuretoperformadequatelyinvariousairportsan dothersensitivelocations,like bordercheckpointsthatithasbeeninstalled.Severalothe rgovernmentagencieshave abandonedfacerecognitionsystemsafterndingthattheir performancewasnotcloseto whatwasadvertised.Sincemostfacerecognitionsystemsar eeasilythrownobychangesin hairstyle,facialhair,aging,weightgainorlossandminor disguisesweneedmoreresearch andeectivealgorithmstodealwiththeseissues.Conseque ntlywealsoneedeective protocolstoevaluatethesealgorithmssoastomaintainast andardofperformance. TwomajorevaluationmethodologiesusedrecentlyweretheF ERET[7,8,9,10,11] andtheFRVT[12,13,14]evaluationprotocols.TheFERETtes tingmodelevaluated thealgorithmsbasedondierentscenarios,dierentcateg oriesofimagesandversionsof algorithms.Performancewascomputedundertwoscenarios. Theidenticationscenario andthevericationscenario.Inanidenticationapplicat ion,thealgorithmispresented withanunknownfacewhichneedstobeidentiedfromasetofi mages.Intheverication application,thealgorithmneedstodeterminethesubjecti ntheimageisindeedwhoit claimstobefromasetofimages.TheFERETevaluationstookp lacein1994,1995 and1996.FRVT2000and2002evaluationsmeasuredthecapabi litiesofsystemsonreal lifelargescaledatabasesandintroducednewexperimentst ounderstandfacerecognition experimentsbetter.Thesize,complexityanddicultyleve lwasincreasedwitheach successiveevaluationinordertorerecttheincreasingmat urationofthefacerecognition technologyaswellasevaluationtheory. Thetechnologicalevaluationofthesefacerecognitionalg orithmsiscarriedoutbygenerationofempiricalstatisticsoftheirperformanceforth eidenticationandverication scenarios.Howeverthesestatisticsaredependentonthesi zeandcharacteristicsofthe databaseandoeronlyaglobal(orgross)characterization ofthealgorithms.Ouraimin thisthesishasbeentoevaluatethefacerecognitionalgori thmsatadeeperlevel.Wetry toapproximatetheperformanceofanyalgorithmbyanapproa chthatjustperformsane transformation(rotation,shear,stretch)oftheimagespa ce. 2

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1.1Motivation Atthecoreofanyfacerecognitionalgorithmisamodulethat computesdistance(or similarity)betweentwofaceimages.Justaslinearsystems theoryallowsustocharacterize asystembasedoninputsandoutputs,weseektocharacterize afacerecognitionalgorithm basedonthedistances(the\outputs")computedbetweentwo faces(the\inputs").Canwe modelthedistances, d ij ,computedbyanygivenfacerecognitionalgorithm,asafunc tion ofthegivenfaceimages, ~x i ( ~x i )and ~x j ( ~x j ),where ~x i and ~x j aretherowscanned vectorrepresentationsofthegivenimagearray?Mathemati cally,whatisthefunction suchthat d ij jj ( ~x i ) ( ~x j ) jj isminimized?Thebenetsofsuchascharacterizationis twofold.First,itwillallowustocompareanytworecogniti onalgorithmsatadeeperlevel thanispresentlypossible,usingjustoverallperformance scores,thataredependentonthe sizeandcharacteristicsofthedatabase.Forinstance,if isanorthogonaloperatorthen itwouldsuggestthattheunderlyingfacerecognitionalgor ithmsisessentiallyperforminga rigidrotationandtranslationtothefacerepresentations ,i.e.likeinprincipalcomponent analysis(PCA).If isageneralaneoperator,thenitwouldsuggesttheunderly ingalgorithmscanbeapproximatedfairlywellbyalineartransform ation(rotation,shear,stretch) ofthefacerepresentations.Thesecondbenetisthatthepr oposedcharacterizationwill facilitatefutureanalyticalmodellingofcomplete,possi blymulti-modal,biometricsystems, wheretherecognitionmodulewouldbeapartofnetworkofdi erentmodules.Afunction representationofthefacerecognitionmodulewillallowus toanalyticallyexpressvariations intheoutput(thedistances)intermsofthevariationsinth einputimages. Inthisthesiswespecicallyconsiderane 'sandderiveaclosedformsolutionthat usethestatisticalmethodofmulti-dimensionalscaling(MD S)[15].Givenadistance matrixbetweenasetofdata,MDScanarriveatanembeddingof thedatainamultidimensionalspace,wherethedistancesareclosetothegive ndistancematrix.However, unlikeMDS,whichprovidesanembeddingjustforthegivenda taset,welearna function thatcanbeusedfornewdatapoints.Weanalyzesomeofthepop ularfacerecognition algorithms,namelyEigenfaces(PCA+distancemetrics)[1] ,LinearDiscriminantAnalysis (LDA)[2],BayesianIntrapersonal/ExtrapersonalClassi er(BIC)[16],andElasticBunch 3

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GraphMatching(EBGM)facerecognitionalgorithm[4]andac ommercialfacerecognition algorithm.Thechoiceofthefacerecognitionalgorithmssp antemplatebasedapproaches tofeaturebasedones,suchastheEGBMandcommercialalgori thm.Wetestthequality oftheapproximationonthewellknownFERET[17]datasetand ontherecentNotre Damedata[18].Wemeasuretheapproximationqualitybyther ecognitionratesusingthe learntanemodels.Surprisingly,theresults,whichisbas edonacompleteseparationof trainandtestdataandonlargetestdatasets,indicatestha tthealgorithmscanbefairly wellapproximatedbyanetransformationmodels. ThisstudyisafollowupontheworkdonebyLakshmiRegunainh erMastersthesis[19]. ThecurrentthesisisdierentfromReguna'sinthefollowin gways, 1.Wehaveanewapproximationstrategywhichissignicantl yfasterthanReguna's. 2.Weexperimentandtestonmorealgorithmsincludingtempl atematchingalgorithms aswellasfeaturebasedgraphalgorithms,thanReguna.Cert ainexperimentswhich tookupmorethanadaytoexecutenowtakejustfewminuteswit hourapproach. 3.Wealsotestwithtwodierentdatasetswithvariationine xpressionandtime. 4.Wealsohavecompleteseparationoftrainandtestsetinte rmsofcollectionsite. Thefollowingchaptersofthisthesishasbeenorganizedint hefollowingway.Inthe nextchapter,wetakealookatsomeoftherelatedworkthatha sbeendone.Next,wetake abrieflookatthetestfacerecognitionalgorithmsthatweh aveexperimentedwith.Chapter4describesthemathematicalformulationbehindachiev ingtheanetransformation. Finally,wehavedescribedthestudiesandtheresultsobtai nedfollowedbyconclusions. 4

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CHAPTER2 RELATEDWORK Thischapterisdividedintothreemainsections.Giventhec losesimilarityofgoals withourapproach,wediscussthedominantevaluationproto colsandmethodologiesthat havebeenusedtoevaluatetheperformanceoffacerecogniti onalgorithms.Wealsodiscuss variousmethodsandtechniquesthatarebeingusedfordimen sionalityreductionincases ofhighdimensionaldata.Thenwetakealookatworksonlearn ingdistancemetricfrom thecontextofmachinelearning.2.1FaceRecognitionEvaluationProtocols Face Recognition Evaluation Protocols Comparison With Human Vision FERET FRVT Statistical Evaluations Figure2.1.EvaluationProtocols TheFERETevaluations[7,8,9,10,11]aresomeofthemostsig nicantresearchthat hasbeendonetoevaluatetheperformanceoffacerecognitio nalgorithms.Ithadthreepri5

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marypurposes.Thegoalwasdevelopmentandadvancementoff acerecognitiontechnology. Thesecondgoalwastocreatealargeimagedatabasethatcoul dbeusedasastandardfor algorithmdevelopment,testandevaluation.Thethirdgoal wastoprovidestandardized testsandprotocolsforfacerecognitionalgorithm.Thetes tsmeasuredtheperformanceof thealgorithmsonvariousaspects,likeitsabilitytohandl elargedatabases,variationin illumination,scale,poseandchangesinbackground.Perfo rmancewasmeasuredinterms ofprobabilityoffalsealarmandfalsenegatives.Afalseal armisthescenariowhereaface isfalselyrecognizedtoexistinthegalleryandafalsenega tiveisthescenariowhenaface isnotrecognizedeventhoughitdoesexistinthegallery.Th eFERETprogramranin phasesfromtheyears1993to1996.TherstFERETevaluation tookplaceonAugust94. Theaimofthesetestsweretoevaluatetheabilityofalgorit hmstolocate,normalizeand identifyfacesfromadatabase.Thetestsincludedabilityt orecognizefromagallery,rate offalsealarmsandresiliencetoposevariation.Thesecond evaluationtookplaceonMarch 1995andwasdesignedtomeasuretheprogressofthealgorith mssincetherstevaluation. Theywerealsotestedonlargerdatabasesandwithduplicate imagestakenoveraperiod oftime.ThethirdandnaltestwasdoneonSeptember96where theperformancewas testedonover3000imagesandagainstmultipleprobeandgal lerysets.Thegalleryand probesetswereselectedbasedonvariouscombinationsofva riationovertimeandlightingconditions.ThealgorithmstestedwerethePCA[1],Fish erdiscriminantanalysis[2], Greyscaleprojection[20],DynamicLinkArchitecture[4], andBayesianclassication[16] basedapproaches. TheFERETevaluationsexaminedtheperformanceforboththe closeduniverseas wellastheopenuniversescenarios.Theopenuniversescena rioimpliesthatnotevery probeexistsinthegallery.Inthecloseduniversescenario everyprobeexistsinthegallery. Thecloseduniversescenarioisalsocalledtheidenticati onscenariobecauseitallowsus toknowiftheprobeimageisthetopidentifyingmatchorhowm anyimagesneedto beexaminedtoknowthegetthedesiredlevelofperformance. Theyarecharacterized byCumulativeMatchscoreCurves(CMC)wherethex-axisdenot estherankandthe y-axisdenotesthepercentageofcorrectmatches.Thehigher thepercentageofcorrect matchesatrankonethebettertherecognitionrate.Intheve ricationscenario,the 6

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systemeitherconrmsordeniesifaprobeexistsinthegalle ry.Itischaracterizedby theReceiverOperatorcharacteristicCurve(ROC).Thex-axi srepresentsthefalsealarm rateandthey-axisrepresentstheprobabilityofvericatio n.Thehighertheprobability ofvericationatlowfalsealarmsthehigheristheaccuracy ofthesystem.Identication scenarioapplicationsincludeidentifyingafacefromaset ofmugshotsorsurveillance imagesatairports.Vericationscenarioapplicationsinc ludeautomatedconrmationof identitiesatATMmachines,accesscontroltosecurebuildi ngsandsites. SomeoftheconclusionsdrawnfromtheFERETevaluationswer e. 1.Algorithmperformanceincreasedwithincreaseinsizeof database. 2.Algorithmperformancealsodependedontheselectionofp robeandgallerysets. 3.Sincesomealgorithmsperformedbetterforidenticatio nthanverication,performanceononetaskisnotpredictiveofperformanceonanother 4.Identicationperformanceonduplicatescores(matchfr ontalstakenondierentdates) waslowerthanperformanceonfrontalstakenonsamedate.Ve ricationresultsalso hadsimilarconclusionsregardingthis. 5.Somealgorithmswereinsensitivetochangeinilluminati onandsomeshowedperformancedegradationafter40%changeinillumination. 6.Mostalgorithmswereinsensitivetochangeinimagesize. DuetotheFERETevaluations,itwaspossibleforresearcher stodevelopandtesttheir algorithmacrossacommondatabaseusingstandardevaluati onprotocols.Itenabledthe facerecognitioncommunitytoassessthestrengthsandweak nessesintheeldandlaythe foundationsforfuturedirectionofresearch. TheFaceRecognitionVendorTest(FRVT)[12,13,14]conduct edin2000and2002 followedtheFERETevaluationsexceptthattheywereconduc tedoncommerciallyavailableandmaturecommercialfacerecognitionsystems.FRVT2 000followedthethreestep evaluationprotocoloftechnologyevaluation,scenarioev aluation,andoperationalevaluation.TheRecognitionPerformanceTest(TechnologyEvalu ation)showedhowwellthe 7

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varioussystemsrespondedtochangesinpose,lighting,and imagecompressionlevel.The ProductUsabilityTest(Limitedscenarioevaluation)demo nstratedtheabilityofaface recognitionsystemtooperateunderaliveenvironment.Par ticipantsoftheFRVT2000 includeBanque-TecInternationalPtyLimited,C-VisCompute rVisionandAutomation, eTrue(formerlyMiros),LauTechnologies,andVisionicsCo rporation. FRVT2002attemptedtoassesstheperformanceofthealgorit hmstomeetrealworld requirements.Itconsistedoftwosub-tests,thehighcomput ationalintensity(HCInt)test andmediumcomputationalintensity(MCInt)test.TheHCInt testwasusedtotestthe performanceofthesystemsonverychallengingreal-worldpr oblems.Ontheotherhand, MCIntwasdesignedtoevaluatetheperformanceofsystemson varioustypesofimages (stillaswellasvideo)undervariousconditions.Therewer emanymoreparticipantsin theFRVT2002includingAcSysBiometrics,C-VISComputerVis ionandAutomation, CognitecSystems,DreamMirhCo.,Ltd,EyematicInterfaces Inc.,Iconquest,Identix, ImagisTechnologiesInc.,ViisageTechnology,andVisionS phereTechnologies. SomeoftheconclusionsdrawnfromtheFRVTtestsinclude. 1.Performancedecreaseslinearlywithincreaseinelapsed timebetweendatabaseand newimages.Performancedroppedroughlyatarateof0.05poi ntsperyearfor identicationscenario.Howevertheperformancedropforv ericationscenariowas slowerthanforidentication. 2.Goodfacerecognitionsystemsareresilienttoindoorlig htingchange. 3.Nonfrontalfacesarebetterrecognizedbyre-mappingthem intofrontalonesusing 3Dmorphablemodels[21]. 4.Videoimagesdonotnecessarilyresultinbetterperforma ncethanstillimage. 5.Oldermaturefaceareeasiertorecognizethanyoungerfac es. 6.Malefacesareeasiertorecognizethanfemalefaces.7.Outdoorimagesdonotperformaswellasindoorimages. 8

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Therehavebeenseveralothereortstobenchmarkandevalua tefacerecognitionalgorithms.Guttaetal.[22]conductedbenchmarkstudiesonsev eralsimplebutwellknown algorithmsaswellasnovelrecognitionschemes.Someofthe conclusionsthattheydrewof theirstudies,wasthatthefutureoffacerecognitionalgor ithmslaywithhybridrecognition systemsandthatholisticalgorithmsoutperformfeaturean dcorrelationmethods. Variouseortswerealsomadetodevelopevaluationframewo rksusingstatisticalevaluationmethodologiesofrecognitionalgorithms.In[23]th eauthorshavesuggesteda newframeworkforevaluatingrecognitionsystems.Often,j ustblindassumptionofi.i.d datacanreducetheaccuracyofasystem.Theirworkallowedt hesystemtoobtaintight condenceintervalsofevaluationestimates.Italsosimul taneouslyreducedtheamountof dataandcomputationrequiredtoreachthoseconclusions.T heyhaveachievedthisusing stratiedsamplingmethodsandtheapplicationofareplica testatisticaltechniquecalled balancedrepeatedsampling(BRR).Someparametricandnonp arametricmethodsforthe statisticalevaluationofrecognitionalgorithmswereexp loredin[24].Oneofthemethods proposedwasaparametricmethodthatequatedsuccessorfai lureofalgorithmsonprobe imagestoBernoullitrials.Thismethodtriestoprovideapr obabilityofsuccessofthe algorithmarisingduetothesizeofthesample.Thesecondno nparametricmethodbased ontheMonteCarlobasedsamplingtechniquecapturesthepro babilitydistributionofthe recognitionratebysamplingthespaceofallpossiblegalle ryandprobesets. AninterestingpieceofworkdonebyPhilipsetal.[25]evalu atedvariousfacerecognitionalgorithmtoassessthequalitativeaccordbetweenf acerecognitionalgorithmsand humanperceivers.Theaimofthisresearchwastoanswertheq uestionifbothhumansand algorithmsfoundthesamefacessimilar.UnliketheFERETev aluationsitconcentrated onthequalitativeratherthanthequantitativeaspectsofp erformancebyhumansand models.Italsotriedtousefacesthatwereconfusedwitheac hother.Itconcludedthat mostfacerecognitionalgorithmsperformedsimilartohuma ns.Bycomparingsimilarity scoresbetweenpairsoffacesandmeasuregeneratedbyalgor ithmicmodelsandhumans.It wasalsoobservedthatalgorithmswithdierentrepresenta tionscouldclustertogetherin termsofdistancecomputed.Mostalgorithmswithsamerepre sentationsclustertogether 9

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althoughitwasnottheonlyfactor.Anotherimportantobser vationwasthatalgorithms havingthesamerepresentationsbutusingdierentdistanc emetricsclustereddierently. 2.2DimensionalityReductionTechniques Interestinvisualizationofhighdimensionaldatafacethe problemofdimensionality reduction.Beingabletondmeaningfullowerdimensionsto representthedatahidden intheirhighdimensionalobservationshasbeenachallenge facedbymanyeldsofinterest.Thegoalistoestimatelowdimensionalplacementofagi vensetofpointssoasto approximatedistancesinahigherdimensionalspace.Inthi saspectourgoalissimilarto MultidimensionalScaling(MDS)[15].Howeverwealsoseekt olearnamappingsothatwe canmapnewdatapointsintothisspace.Someofthedimension alityreducingtechniques areshowninFig.2.2. Dimensionality Reduction Techniques PCA MDS Isomap LLE Kernel Matrices Figure2.2.DimensionalityReductionTechniques Embeddingonespaceintoalowerdimensionalspacehasgener allyemployedlineartechniqueslikePCAornonlineartechniqueslikeMDS.Embedding impliesthemappingofone spaceintoanother.InPCAtheoptimal\m"dimensionsareret ained.Thechosencoordinateaxescoincidewiththeeigenvectorswiththelargeeige nvalues.Euclideandistancesin thisreducedsubspaceapproximatetheEuclideandistances intheoriginalspace.InMDS, 10

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alowdimensionalspaceiscreatedinwhichthedissimilarit yorsimilaritybetweenanytwo objectsispreserved.Howeverboththesemethodsbecomecom putationallyexpensivefor highdimensionaldataorifthenumberofpointsislarge.Als oPCAistypicallysuitable fordatathathassimplelinearcorrelations.MDSontheothe rhand,isaniterativeprocess exceptforthesimplestcaseanddoesnotguaranteeoptimali tyoruniquenessinoutput. Newermethodslikemanifoldlearningmethodswhichinclude theIsomap[26]andLLE[27] algorithmsuseacollectionoflocalneighborhoodsorexplo itthespectralpropertiesofadjacencygraphsfromwhichtheglobalgeometryofthemanifol dcanbereconstructed.A distributionalscalingmethodby[28]describesanewmetho dforembeddingmetricas wellasnonmetricspacesinlowdimensionalEuclideanspace s.Thismethodworkswith metricaswellasnonmetricdatasets.Themethodcombinesbo thpairwisedistortionas wellasgeometricdistortion.Ittriestopreservetheorigi nalstructureorgeometryofthe data.Itisalsoresilienttothepresenceofnoise.Itusescl usteringalgorithmstodirectthe embeddingprocessandimproveitsconvergenceproperties. Theyalsosuggestmethodsto estimatetherightdimensionalityofdata.Thesemethodsar ebasedonalocalgeometric approachandaglobalheuristicapproach. Thetheoryofmultidimensionalscaling[15](MDS)hasbeenu sedinthepastmostlyin relationtofacerecognitionasavisualizationtoolfordis tancesbetweenfaces.MDScanbe denedasasearchforalowdimensionalEuclideanspaceinwh ichthedistancebetween thepointsinspacematchaswellaspossibletotheoriginald issimilarities.Inourstudy multidimensionalscalingisthetheorybehindtheAneAppr oximationAlgorithm.From asetofknownsquareddistances d rs intheoriginalspace,wecalculatetheinnerproduct matrix B andfrom B thecoordinatesinthereducedspace.Theobjectiveistond coordinatessuchthatthedistancebetweenthepointsinthereduce dspace"matches"aswellas possiblethedistancesintheoriginalspace.Thedissimila rities/distancestakenasinputin practicalcasesmayormaynotbeEuclidean.Ifthesedissimi laritiesarenotEuclideanthe B matrixwillnotbepositivesemi-denitewhichisindicatedb ysomenegativeeigenvalues. Tomakeitpositivesemi-denite,aconstantcanbeaddedtoal lthedissimilaritiesexcept theselfdissimilarities[15]. 0 rs = rs + c (1 rs ) 11

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where c = 2 n isaconstantand rs isKroneckerdelta. n isthesmallesteigenvalue. Anonlineardimensionalityreductionmethoddescribedin[ 26]usesthealgorithmic benetsofPCAandmultidimensionalscaling(MDS)butlearn sabroadclassofnonlinear manifolds.DatawhichliesonatwodimensionalSwissrollha vepointslyingfarapart asmeasuredbygeodesicdistancesonanunderlyingmanifold mayappeartobecloseto eachotheriftheyaremeasuredbyEuclideandistances.Theg eodesicdistancesbetween farawaypointsgiventheinputspacedistancescanbemeasur edinEuclideanorany otherdomainspecicmetric.Fornearorneighboringpoints theinputspacedistances givesagoodestimateofthegeodesicdistances.Geodesicdi stancesforfarawaypoints areapproximatedbyndingtheshortestpathbetweenthem,f oundbyconstructinga graphconnectingneighboringpoints.Therststepinthism ethod,calledIsomap,entails estimatingneighboringpointsonamanifoldbasedonaxedr adiusorknearestneighbor methods.Thesecondstepestimatesthegeodesicdistancesb etweenallpairsofinput pointsbytakingtheshortestpathbetweenthem.Thenalste pistoapplyclassicalMDS tothematrixofgraphdistances.Ad-dimensionalEuclideans paceisobtainedwhichbest preservesthemanifoldsintrinsicgeometry.Itiscapableo fdiscoveringnonlineardegrees offreedomunliketheclassicaltechniquesofPCAandMDS. Othertechniquesofnonlineardimensionalityreductionin cludethelocallinearmethod (LLE)asdescribedin[27].Itisanunsupervisedalgorithmt hatcomputeslowdimensional, neighborhoodpreserving,embeddingofhighdimensionalin puts.Unlikeotherclustering methodsforlocaldimensionalityreduction,LLEmapsitsin puttoalowdimensionalglobal coordinatesystemanddoesnotgetstuckinalocalminimapro blem.Thismethodinvolves takingasamplesetofdatapointsrepresentingtheunderlyi ngmanifold.Theunderlying geometryofthesepatchesisreconstructedfromthelinearc oecientsthatreconstructeach datapointfromitsneighbors.Acostfunctionrepresentsth ereconstructionerrorwhichis thenminimizedtogettheoptimalreconstruction.Thismeth odscaleswellwithmanifold dimensionality.Itavoidstheproblemofsolvinglargedyna micprogrammingproblems andusessparsematricesthatcanbeexploitedtosavecomput ationtimeandspace.These methodsarepowerful,non-iterative,andguaranteeglobalo ptimality.Howevertheydonot performwellifthenumberofdatapointsislessorifthedata isintrinsicallynon-metric. 12

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Anewmethodfordimensionalityreductionorlearningunder lyingmanifoldswasproposedin[29],basedonsemideniteprogramming.Itcombine stheconceptsofsemideniteprogrammingforlearningkernelmatriceswithspec tralmethodsofnonlinear dimensionalityreduction.LiketheIsomapandtheLLEalgor ithmsitisnotaectedby localminimaproblems.Therststepofthisalgorithminclu desndingtheknearest neighborofeachinputandcreateagraphlinkingeachneighb oraswellaseachneighborto otherneighborsofthesameinput.TheGrammatrixofthemaxi mumvarianceembedding iscomputedwhichiscenteredattheoriginandpreservesthe distancesofalledgesin theneighborhoodgraph.Thenalstepextractsthelowdimen sionalembeddingfromthe dominanteigenvectorsoftheGrammatrixlearnedbysemi-de niteprogramming. 2.3DistanceMetricLearningTechniques Severaleldslikearticialintelligence,patternclassi cation,machinelearning,statisticsdataanalysisrequireustobeabletolearnimportantin formationhiddeninmultivariate data.Ofparticularinterestareworksthatseektolearndis tancesforexamplesofsimilar andnonsimilarclasses. Distance Metric Learning Techniques Convex Optimization Problem Non Parametric Kernel Adaptation Relative Comparisons Distributional Scaling Figure2.3.DistanceMetricLearningTechniques 13

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In[30]adistancemetriclearningmethodisproposedthatle arnsadistancemetric thatseekstopreservethesimilarity/dissimilarity(bina ry0or1)relationshipbetweena setofpoints.Themethodisbasedonposingdistancemetricl earningproblemasaconvex optimizationproblem.ThismethodscoredoverMDSandother distancemetriclearning algorithmsinthatitcanlearnthemetricoverthewholeinpu tspaceandnotjustthe trainingpoints.Henceitperformsbetterwithpreviouslyu nseendata.Thismethodisalso ecientandlocaloptimafree.Howeverthismethodinvolves aniterativeprocedureand eigendecompositionandcanbecomeexpensivewithlargenum berofdimensions. Adistancemetriclearningalgorithmwithkernelswaspropo sedin[31].Itdescribes afeatureweightingmethodthatworkedintheinputspaceasw ellasthekernelspace.It basicallyperformsanonparametrickerneladaptation.Man yclusteringorclassication algorithmsmakeuseofadistancemeasurebetweenpatterns. Oneofthemostpopular methodbeingtheEuclideandistancemetric.HowevertheEuc lideandistanceassumes equalweightagetoalldimensions.Inrealworldapplicatio nsthisisrarelytrue.Hence featureweightingtechniquesareused.Butthelimitationi sthatthenumberofparameters orweightsincreaseswiththeincreaseinnumberoffeatures .Hencetheycannotbeeasily kernelizedandcantypicallyselectfeaturesonlyintheinp utspaceandnotinthefeature space. Thedistancelearningmethoddescribedin[32]learnsbyrel ativecomparisons,which isarexiblewayfordescribingqualitativetrainingdataas asetofconstraints.Theseconstraintsleadtoaconvexquadraticprogrammingproblemtha tcanbesolvedbyadapting standardmethodsforSVMtraining.Itcanlearnadistanceme tricfromqualitativeand relativeexamples.Thealgorithmsearchesaparameterized familyofdistancemetricsand discriminatelysearchesfortheparametersthatfulllthe trainingexamples.Atraining problemisthenformulatedasaconvexquadraticprogramfor learningtheweightsassociatedwiththesedimensions.Theadvantageofthisparti cularalgorithmisthatthe qualitativenatureoftheconstraintenablesittobeusedfo rawiderangeofapplications. 14

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CHAPTER3 THEFACERECOGNITIONALGORITHMSEVALUATED Forourexperimentsvefacerecognitionalgorithmswerese lected.TheColoradoState Universitysourceimplementation[33]wasusedtogettheou tputdistancematricesforthe PCA,LDA[34],BIC[35]andEBGM[36]algorithms.Atopperfor mingcommercialface recognitionsoftwarewasthefthalgorithmtested.Asshow ninFig.3wehaveselected algorithmbasedonbothholistictemplatematchingtechniq uesaswellaslocalfeature basedgraphalgorithms.ThePCA,LDAandBICalgorithmsaret hetemplatematching algorithmsandlocalfeaturebasedgraphalgorithmsinclud etheEBGMandthecommercial facerecognitionalgorithm. Face Recognition Algorithms Local Feature based graph methods Holistic Template Matching Methods PCA + Different Distance Measures LDA BIC EBGM Commercial Algorithm Figure3.1.FaceRecognitionAlgorithm 15

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3.1PrincipalComponentAnalysis Thistheoryismotivatedbybothphysiologyandinformation theory.Itrecognizesfaces basedonthefactthatfacescanberecognizedbasedoncertai nsetofimagefeatureswhich approximatetheimageface.Howeverthesefeaturesdonotne cessarilycorrespondtothe intuitivenotionoffacialfeatures.Inmathematicalterms itcanbesaidthatweneedtond theprincipalcomponentsofthedistributionoftheinputda tasetwhichcanbeshowntobe theprincipaleigenvectorsofthecovariancematrixofthei nputimages.Theseeigenvalues andcorrespondingeigenvectorsarethenordered.Eacheige nvaluecorrespondstoacertain amountofvariationalongeachdimension.Wecanchoosetoke epthetopMeigenvectors thatbestdescribethevarianceintheimages.Eachimagecan nowberepresentedina linearcombinationoftheprojectionsonthesetopeigenvec tors.Recognitionofnewfaces isperformedbyprojectingthisfaceinaneweigenspaceandt hencomparingthedistance ofthisfacefromthefacesinthetrainingset.Ifthelowestd istanceiswithinacertain amount,itisconsideredtobethatindividualelseitisclas siedasunknown.Inareal worldapplication,thetradeobetweenaccuracyandspeedn eedstobeassessed.Accuracy increaseswiththenumberoffacesinthedatabase,howevert hisalsodecreasesthespeed ofrecognition.Afewissuesneedtobetakencareofbeforeas ystemcansuccessfully work.Backgroundinformationcanhaveasignicanteecton theperformancesince theeigenvectorsarecalculatedonthewholeimageandnotju stthefaceimage.Hence eliminationofthebackgroundisanimportantprerequisite forgoodperformance.Asecond considerationisthattheperformancedecreasesifthescal e/sizeofthefacesisnotclose tothetrainedimages.Theorientationoftheheadalsohasan eectonperformance. Orientationoftheheadalongthelinesoftheeigenfacesben etstherecognitionrate greatly.Recognitionratesfalloastheheadorientationd iersfromthatinthetraining setofimages.3.2LinearDiscriminantAnalysis ThisalgorithmisacombinationofPCAandLDA.PCAisusedast hepreliminary steptocreateafacesubspacetoapplyLDAsoastoobtaintheb estlinearclassier.The 16

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combinationalsohelpsinclassicationwhentherearefewn umberofsamplesfromeach classavailable.ThisisbecauseapureLDAalgorithmisnel ytunedtothetrainingdata anddoesnotgeneralizewell.Hencebygivingonlytheprinci palcomponentsastheinput totheclassier,bettergeneralizationisachieved.Thego alofthisclassieristoreduce thewithinclassscatterandincreasethebetweenclassscat terandformasubspacethat linearlyseparatesbetweenclasses.Theprocedureinclude smappinganimage x ontothe facesubspacetoget y .Thisinturnisprojectedonthediscriminantspace W y toget z ~y =( ~x ~m ) ~z = ~ W T y ~y x isthemeanimage.Finally,classicationisperformedbase donsomedistancemetriclike theEuclideandistance.Performanceusingthishybridmeth odwasshowtohaveimproved overapureLDAclassier.3.3BayesianIntrapersonal/ExtrapersonalClassier ThisalgorithmproposedbyMoghaddamandPentland[35]uses thedierencebetween twoimagestoprobabilisticallydeterminewhethertheybel ongtothesamesubjectornot. Dierenceimagesarisingfromtheimagesofthesamesubject arecalledintra-personal imagesanddierenceimagesarisingfromtheimagesoftwodi erentsubjectsiscalledas extra-personalimages.Eachofthesedierenceimagesiscon sideredtobeapointina highdimensionalspace.Thehighdimensionalspaceishowev erverysparselypopulatedas majorityofthevacantspacescorrespondtodierenceimage sthatneveroccurinpractice. Thesedierenceimageswilltendtoformclusters.Moghadda mandPentlandassume thateachdierenceimagebelongstooneofthetwointerpers onalandextrapersonal clusters.AlsothattheyaredistinctandlocalizedGaussia ndistributionswithinthespace ofallpossibleimages.Howevertheparameterstothesedist ributionsareunknown.These parameterscanbeestimatedbyusingthemaximumaposterior imethodorthemaximum likelihoodmethod.Forthisthesiswewillrestrictourselv estoonlytheMaximumLikelihood 17

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methodsinceithasbeenfoundtohaveequallygoodresultsas theMaximumaposteriori methodandthesametimeismuchlesscomputationallyintens ive. Inthetrainingphase,PCAisperformedtoestimatethestati sticalpropertiesofthe twosubspacesietheinterpersonalclasscalledn I andextrapersonalclassn E .Ifis adierenceimageofunknownmembershipthenthesimilarity scoreforthemaximum likelihoodisgivenas S ML = P ( j n I ) P ( j n I )= e 0 : 5 P Ti =1 y 2 i i (2) T= 2 Q Ti =1 1 = 2 i whereT-nooftruncateddimensionsfromtheoriginaldimens ionsofthedata(in ordertoreducecomputationalcomplexity) -Teigenvalues y =[ y 1 ;y 2 ; ;y M ] T ofeachdierenceimageembeddedintothe PCAsubspace. Duringthetestingphasetheclassiertakesadierenceima geofunknownmembership anduses P ( = n I )asameansofidentication.Themaximumlikelihoodestima teignores theextra-personalclassinformation.Whencomparinganove limageto n knowngallery images,thegalleryimageyieldingthehighestsimilaritys coreistakenastobetheperson intheprobeimage.3.4ElasticBunchGraphMatching Thisalgorithmdiersfromtheotheralgorithmsbecauseitr ecognizesfacesbycomparingparts,insteadofperformingmatchingthewholeimag e.Thefeaturesoftheimages arerepresentedbyGaborjetsalsocalledasmodeljets.This isobtainedbyconvolvingan imagewithGaborlters.Themodeljetsarecollectivelycal ledbunchgraphs.Eachnode inthisgraphisacollectionofmodeljetsofaparticularlan dmark.Thesejetshavebeen extractedfrommanuallyselectedlandmarklocationsfromt hemodelimagesandaddingto theappropriatebunch.Thesebunchgraphsarethenusedasre ferencedataforlandmark descriptionswhilelocatinglandmarksinnovelimages. 18

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Locatingalandmarkisbasedontwosteps.Thelocationisrs testimatedbytheknown locationofotherlandmarksintheimage.Thisestimatedloc ationisthenfurtherrened byextractingaGaborjetatthatpointandcomparingitagain stasetofmodels.Themost similarjetisselectedfromabunchgraphandthisthenserve sasamodel.Thealgorithm beginsbyestimatingtheeyecoordinatesrstbecausethese estimatesareveryreliable. Thealgorithmworksiterativelytolocatetherestofthelan dmarkstillithasreachedthe edgeofthehead. Facegraphsarecreatedforeachimagebyextractingjetsfro mthelandmarklocations likeeyes,nosetip,andcorneroflips.Thesegraphscontain thephysicallocationofthe landmarksaswellasthevalueofthejets.Jetsarealsoextra ctedfromlocationsatthe midpointbetweentwolandmarks.Sinceanimageisrepresent edonlybyitsfacegraph now,theoriginalimagedatacanbediscarded. Similaritybetweentwoimagesiscalculatedasafunctionof thelandmarklocationsand theirjetvalues.Jetsimilaritycanbecomputedusingvario usmethodsofmagnitudeonly, phaseordisplacementcompensatedGaborjetsimilarity.An othermethodtocomputethe similarityisbasedonthepositionofthelandmarkpoints.A simplewayistocomputethe Euclideandistancebetweentheselocations.Thepresumpti onbeingthatimagesbelonging tothesamesubjectwilldierverylittleinthelandmarkloc ations. 3.5CommercialFaceRecognitionSoftware Acommercialfacerecognitionsystemwaschosenforourexpe riments.Thisparticular systemwasamongstthetopperformingalgorithminalltheFR VTevaluations.Itis capableofcapturingimagesatadistanceandinmotionusing CCTVandcanperformreal timeidenticationofsubjects.ItusesLocalFeatureAnaly sis(LFA)torepresenttheface. ThemathematicaltechniqueofLFAassumesthatafacialimag ecanbesynthesizedfrom anirreduciblesetofbuildingelements.Theseelementscan bederivedfromasetofmodel faceimagesusingstatisticaltechniques.Foridenticati onpurposestherelativepositionsof theseelementsareasimportantasthecharacteristicsofth eelementsthemselves.Although severalelementsarepossible,onlyafewareneededtodescr ibeafacecompletely.However 19

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theseelementsdonotnecessarilycorrespondtofacialfeat ureseventhoughtheyspanjust afewpixels.ComparedtomethodssuchasthePCA,LFAismuchm oreresilientto changesinexpressionandhencemuchmorerobust.Itisinsen sitivetohairstylechanges andgrowthoffacialhair.Itisalsoposeinvariantupto10-15 degrees.Howeverpose anglecanbeestimatedandcompensatedfor,therebyimprovi ngperformance.Italso workssuccessfullywithpeoplewearingeyeglasses.Itisin varianttogenderandraceof individuals.Someofthetestsinwhichitwasatopperformer atFRVTinclude,high vericationaccuracy,one-to-manysearchonalargedatabase ,highcorrectalarmratefor watchlistapplications,minimalsensitivitytolightinga ndtemporalvariation.Thisstateof-the-artfacialrecognitionsystemhasbeendeployedinair ports,towncentersandborder crossingsworldwide. 20

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CHAPTER4 DISTANCESINAFFINEAPPROXIMATIONSPACES 4.1Multi-DimensionalScaling xj x1 x2 y1 y2Input Image Space xiyiLinearTransform xkyj Euclidean Distances in this Space Should“match”DistancesComputed by the AlgorithmUnder Consideration Figure4.1.FindMatrixAsuchthattheEuclideanDistancesBetwe enTransformedimagesareEqualtotheGivenDistances Theaneapproximationalgorithmutilizesthetechniqueof multidimensionalscaling tondtheeigenspacethatcanapproximatethedistancescom putedbyanygiventestface recognitionalgorithm.Therearedierenttypesofscaling techniqueslikeclassicalscaling, metricleastsquaresscaling,uni-dimensionalscalingandn onmetricscaling. 21

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4.2AneTransformationBasedModellingofDistances Ourgoalistondananetransformationofthegivenimagess othattheEuclidean distancebetweenthetransformedimagesmatchthegivendis tanceset.Webuildthesolutionbasedonthestatisticalmethodofmultidimensional scaling[15],Let n faceshave dissimilarities f rs g (maybenon-Euclidean)computedbetweenthem.Thegoalofmul tidimensionalscaling(MDS)istondacongurationofpoints ,representingthesefaces,in a p dimensionalspacesuchthatthedistancebetweentwopoints r and s bedenotedby d rs \matches"thecorrespondingcomputeddissimilarities.An importantdistinctionofthe currentworkfromMDS,isthatthatunliketraditionalMDSwh ichjustseekstoembeda given setofdistances,wealsoseeka mapping fromtheinputimagestoMDScoordinates soastobeabletomapany new image.Butrst,afewnotationaldenitionsareinorder. Let, 1. ~x i bethe N 2 1sizedcolumnvectorformedbyrowscanningthe N Ni -thimage. 2. X =[ ~x 1 ; ;~x K ]bethematrixcomposedoutoftheimagevectors. 3. ij bethedistancebetweentwoimages, ~x i and ~x j ,thatagivenalgorithmcomputes. Thesedistancescanbearrangedasa K K matrix D =[ 2 ij ],where K isthegiven numberofimages.Notethatthematrixisconstructedoutoft he squared distances. 4. A ( M N 2 )matrixisusedtolinearlytransformtheinputimagevector ~y i = A ~x i (4.1) 5.The squared Euclideandistancebetween ~y i and ~y j isgivenby, d 2ij =( ~x i ~x j ) T ( A T A )( ~x i ~x j ) (4.2) Thedistancesarestoredina K K matrix =[ d 2ij ] Let, = D (4.3) 22

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Thematrix A ,whichistheanetransform,hastobedeterminedsuchthat ( AX ) T ( AX )= 1 2 HH (4.4) where H =( I 1 N ~ 1 ~ 1 T )(4.5) where I istheidentitymatrix, ~ 1isthevectorofones.Thisoperator H isreferredtoas thecenteringoperator.Applyingthisoperatortobothside sofEq.4.3,wehave ( AX ) T ( AX )= 1 2 HDH = B (4.6) Thematrix B isreferredtoasthe\centered"versionof D .If D isEuclideanthenit canbeshownthatthematrix B istheinnerproductmatrixofthecoordinates[15].We willrefertothetransformedcoordinatesas X MDS .Thus, ( AX ) T ( AX )=( X MDS ) T ( X MDS )(4.7) ThesecoordinatescanbearrivedatbydierentMDSembeddin gschemes,suchas classical,leastsquares,orISOMAP.However,wechosethes implestpossiblescheme,the classicalscheme[37,38,15]thatarrivesatthesolutionba sedonthesingularvaluedecompositionof B = V MDS MDS V T MDS where V MDS MDS aretheeigenvectorsand eigenvaluesrespectively.Assumingthat B representstheinnerproductdistancesofan Euclideandistancematrix,thecoordinateswhicharegiven by X MDS =( V MDS 1 2 MDS ) T (4.8) Thisdecompositionispossibleiftheunderlyingdistancem atrix D isEuclidean.To handlenonmetricornon-Euclideandissimilaritiesandalso tohandle similarities werst transformthemintoEuclideandistance.Forthis,werelyon GowerandLegendre[39,15], whohaveshownhowdissimilaritiescanbetestedformetrica ndEuclideanpropertiesand canbetransformedtopossessthesepropertiesiftheyareab sent. 23

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If D isnonmetricthenthematrixconstructedfromelements rc + c (forevery r 6 = c ) ismetric,where c max i;j;k j ij + jk jk j D isEuclideanifandonlyifthematrix B ispositivesemi-denite.If B ispositive semi-deniteofrank p ,thenacongurationin p dimensionalEuclideanspacecanbe found If D isadissimilaritymatrix,thenthereexistsaconstant h suchthatthematrix withelements( 2 rs + h ) 1 2 isEuclidean,where h 2 n ,thesmallesteigenvalueof B .Inourapplicationcontextoffacerecognition,additivec onstantstothecomputed dissimilaritiesdonotalterperformance. If S isapositivesemi-denitesimilaritymatrixwithelements0 s rs 1and s rr =1, thenthedissimilaritymatrixwithelements 0 rs = rs + c (1 rs )isEuclidean,where c = 2 n isaconstantand rs isKroneckerdelta. n isthesmallesteigenvalue. ToarriveatasolutiontoEq.4.7,wend A suchthat, X MDS = AX (4.9) A canbeconsideredtohavetotwoparts,thenonrigidpart( A nr )andtherigidpart ( A r ).Hence, A canalsobeexpressedas A = A nr A r .Eq.4.9cannowbewrittenas, X MDS = A nr A r X (4.10) Therigidpart A r canbearrivedatbyPCA.LetthePCAcoordinatesbedenotedby X PCA = A r X ,where X PCA aretheoriginalcoordinatesprojectedontothePCAspace. Thuswehave, X MDS = A nr X PCA (4.11) Substitutingeq.4.11ineq.4.8weget, A nr X PCA =( V MDS 1 2 MDS ) T (4.12) 24

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Nowitcanbeshownthat X PCA X TPCA = PCA where PCA isthediagnonalmatrix withthePCAeigenvalues. X PCA X TPCA =( A r X )( A r X ) T X PCA X TPCA = A r ( XX T ) A Tr X PCA X TPCA = A r ( A Tr PCA A r ) A Tr However, A r A Tr = I where I isanidentitymatrix.Thus, X PCA X TPCA = PCA (4.13) Multiplyingbothsidesofeq.4.12by X TPCA A nr X PCA X TPCA =( V MDS 1 2 MDS ) T X TPCA (4.14) Finally,fromeq.4.14andeq.4.15weget, A nr =( V MDS 1 2 MDS ) T X TPCA 1 PCA (4.15) 4.2.1SimilaritiesandDissimilarities Facerecognitionalgorithmssometimescomputesimilariti esinsteadofdistances.Similaritycoecientscanbeconvertedintodissimilaritieso rdistancesusing[15]. 1. rs =1 s rs 2. rs = c s rs forsomeconstantc 3. rs =2(1 s rs ) 1 = 2 4.3AnAlternativeMethodtoDeriveA Analternativemethodtondtheanetransformationmatrix wasproposedbyLakshmiRegunaandcanbereferredtoinhermastersthesis[19]. Itrequiredthecomputation 25

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oftheeigenvectorsofaverylargematrixandhencewasverys low.Ourapproachisalso computationallylessexpensive.Wealsoachievethesamepe rformanceandinthecase ofROCbetterperformancewithourapproachonthesameexper iments.Theapproach describedbyRegunahasbeenreproducedfromherthesis[19] inAppendixAforreference. SomecomparisonofresultsarealsopresentedinAppendixA. 26

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CHAPTER5 EXPERIMENTSANDANALYSISOFRESULTS 5.1IssuesAddressed Inthissectionwepresentresultsonasetofstudiesdesigne dtoaddressthefollowing. Canweapproximatenotonlytemplatebasedalgorithms,such asPCA,LDAbutalso featurebasedalgorithmslikeEBGMandthecommercialfacer ecognitionalgorithm?. Doestheaneapproximationtraininggeneralizeacrossdat asetscollectedatdierentsites? Doesthetraininggeneralizeacrossdierentcovariateae ctingfacerecognition,such as,expressionandtime? Howclosedothe recognition performanceoftheaneapproximatedalgorithmcome totheoriginalfacerecognitionalgorithm? Whateect,ifany,doesthedistancemetricusedintheorigi nalalgorithm,haveon theperformanceofAneApproximation? 5.2DistanceMeasuresUsedbyFaceRecognitionAlgorithms Dierentalgorithmscomewithdierentdistancemeasuresc omputedbetweentwoimages.Herewesummarizetheonesthatweconsidered.Ifanalg orithmcomputesasimilarity measureinsteadofadistancemeasure,wediscusshowweconv ertittoadistancemeasure. ExperimentswiththePCAalgorithmwereperformedwithdie rentdistancemetrics aredescribedbelow.Let ~u and ~v betwoimagesrepresentedasvectors. CityBlock(L1) : D CityBlock ( ~u;~v )= X i jj ~u i ~v i jj 27

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Euclidean(L2) : D Euclidean ( ~u;~v )= s X i ( ~u i ~v i ) 2 Covariance : S Covariance ( ~u;~v )= P i ~u i ~v i q P i ~u 2i q P i ~v 2 i D Covariance ( ~u;~v )= S Covariance ( ~u;~v )+ c where c isasuitableconstantaddedtoconvertanynegativevaluest opositiveones. MahalanobisSpace :TheMahalanobisspaceisthespacewherethevariancealong each dimensionisone.Itcanbeobtainedfromtheimagespacebydi vidingeachcoecientof thevectorbyitscorrespondingstandarddeviation.Let u and v bevectorsintheimage spaceand m and n bevectorsintheMahalanobisspace.Let i bePCAeigenvaluesand i bethestandarddeviation,then i = 2 i .Thevectors u,v arerelatedto m,n inthe followingmanner. m i = u i i n i = v i i MahaL1 : D MahaL 1 ( u;v )= X i jj m i n i jj MahaL2 : D MahaL 2 ( u;v )= s X i ( m i n i ) 2 MahaCosine : S MahCosine ( u;v )= m:n j m jj n j D MahCosine ( u;v )= S MahCosine + c where c isasuitableconstantaddedtoconvertanynegativevaluest opositiveones. ExperimentsperformedwithontheLDAalgorithmwereperfor medusingtheL2norm. 28

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TheBayesianalgorithmhastwovariants,the MaximumLikelihood andthe Maximum aposteriori classier.Forthepurposesofthisthesiswehaveusedonlyt he Maximum Likelihood classier. MLSimilarityMeasure : S ML = P ( j n I ) P ( j n I )= e 0 : 5 P Ti =1 y 2 i i (2) T= 2 Q Ti =1 1 = 2 i whereTisthenumberoftruncateddimensionsfromtheorigin aldimensionsofthe data(inordertoreducecomputationalcomplexity) aretheTeigenvaluesthatspanthedierenceimages. Theelasticbunchgraphingalgorithmprovidesvariousfeat uresbased,aswellasgeometrybasedmethodstondthesimilaritybetweenfaces.In ourexperimentswehave usedafeaturebasedmethodcalledtheFGNarrowingLocalSea rch.Thismeasureisbased onaveragesimilarityofallthefacegraphjetsbasedonthe S D graphGaborjetsimilarity. D = log ( S ( J;J 0 ; ~ d )) 5.3DataDescription Imagesfromtwodierentdatasets,theFERETdatasetandthe NotreDamedata set,wereusedtotesttheaneapproximationdistances.Par toftheFERETdatasetwas usedfortrainingandpartwasusedfortesting.TheNotreDam edatasetwasusedfor testing.FromtheFERETdatasetweusedimagesoftypefa(reg ularfacialexpressions) andfb(alternatefacialexpressionofthesubjecttakenwit hthesamelightingconditions). Theyareimagesofthesubjectstakenonthesamedaywiththes amelightingconditions. TrainingsetasseeninFig.5.3consistsof100imagesof25su bjectswith4imagesper subjectofbothfa,fbtypeimages.Itconsistedof2fa,fbima gestakenonthesameday and2fa,fbimagestakenafteratimeintervalrangingfromaf ewdaystoafewyears.Test setconstructedoutoftheFERETdatasetasseeninFig.5.3co nsistedof600imagesof 300subjectswithtwoimagespersubjectofbothfa,fbtypeim ages.Thistestsetfrom 29

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theFERETdatabasewasusedtoconductexperimentsinvolvin gvariationinexpression. ThePCA,LDA,Bayesianalgorithmsweretrainedonthisset.T herewerespecialtraining setsfortheEBGMandthecommercialfacerecognitionalgori thms.Weusedthetrained dataprovidedbytheCSUimplementationbecausethealgorit hmrequiredaspecialtool forgroundtruthingtheimageswhichwasnotavailabletous. TheEBGMwasdtrainedon 68imagesof68subjects.Similarly,thecommercialfacerec ognitionalgorithm,wastrained onthetrainingsetthatwasmadeavailablewiththesoftware ThetestsetfromtheNotreDamedatabaseasseeninFig.5.3co nsistedof830images from415subjects.Theimageswereonlyoftypefaandwithsim ilarlightingastheFERET dataset.Foreachsubject,twoimagesinthedatasetwiththe maximumtimedierence betweenthemwaschosenforeachsubject.Fig.5.3showsahis togramofthetimevariation intheimages.Ascanbeobservedfromthehistogramthevaria tionisconcentratedovera periodof100days.Theseimageswereusedtoconductexperim entsinvolvingvariationin time.SampleimagesfromtheFERETaswellasNotreDamedatas etcanbeobservedin Fig.5.3 Figure5.1.TrainingSet:100Imageswith4ImagesofEachSubjectoftheFERETDataSet Thefollowingpreprocessingandnormalizationcodedevelo pedatNIST/CSUwasused onalltheimages. 1.Integertoroatconversion-Convert256graylevelsintor oatingpointequivalents. 2.Geometricnormalization-Linesuphumanchoseneyecoordi nates 3.Masking-Cropstheimageusinganellipticalmaskandimag eborderssuchthatonly thefacefromforeheadtochinandcheektocheekisvisible. 30

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Figure5.2.TestSet(FERET):600Imageswith2ImagesofEachSubject Figure5.3.TestSet(NotreDame):830Imageswith2ImagesofEachSubjec t 4.Histogramequalization-Equalizesthehistogramoftheu nmaskedpartoftheimage. 5.Pixelnormalization-scalesthepixelvaluestohaveamea nofzeroandastandard deviationofone 5.4TrainingandTestSetup Fig.5.4showsthestepsofthetrainingphase.First,thefac erecognitionalgorithm underconsiderationisgivenasetoftrainingimagesasinpu tandweobtainadistance matrixasoutput.ThebaselinePCAalgorithmisalsogiventh esamesetoftrainimages asinput.Weobtaintheprojectedcoordinatesoftheseimage sinthePCAspaceandthe A r asoutput.ThesePCAcoordinatesareobtainedbyretaininga llthecomputedPCA dimensionswithnon-zeroeigenvalues.TheAneApproximati onalgorithmisgiventhe distancematrixasinputalongwiththeprojectedimagecoor dinates.Withtheseasinput, theAneApproximationalgorithmcomputesthenon-rigidpar toftheanetransfor31

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0 50 100 150 200 250 0 20 40 60 80 100 120 140 160 Time in terms of daysNo of imagesHistogram of Images acquired over Time Figure5.4.HistogramoftheNotreDameImagesAcquiredOverTime FERETSampleImagesNotreDameSampleImages Figure5.5.SampleImages mation A nr .The A r fromthePCAalgorithmand A nr fromtheAneApproximation algorithmcombinedformtheanetransformationmatrix A ThetestsetupisshowninFig.5.4.Thefacerecognitionalgo rithmunderconsideration isgivenasetoftestimagesasinputandwegetadistancematr ixasoutput.Thesame setoftestimagesarealsoprojectedintotheAnespaceobta inedbythetrainingprocess. 32

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PCA Affine Approximation MDS Distance Matrix Between Train Images Train ImagesFERET(100 images from 25 subjects) Face Recognition Algorithm A r A nr A affine = A nr A r Convert To EuclideanDistances X PCA X MDS Figure5.6.TrainingSetupfortheAneApproximationalgorithm A r A nr Affine Approximation Distance Matrix Between Test Images Test Images FERET (300 subjects) Notre Dame (415 subjects) Face Recognition Algorithm Distance Matrix Between Test Images Compare Recognition Performance (CMC + ROC) Figure5.7.TestingSetupfortheAneApproximationalgorithm 33

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WecomputeEuclideandistancesinthisprojectedspacewhic histhencomparedwiththe actualdistancematrixintermsofbiometricperformance. Onecouldcomputeerrormeasuresbasedonthedistancescomp uted,however,since theultimategoalortaskisrecognition,weperformevaluat ionintermsofhowwellthe aneapproximateddistancescanbeusedinrecognition.Ino urexperimentswecompare performanceusingbothCMCsandROCs.5.5AneMatrix Itisworthwhiletovisualizetheanetransformationmatri x.Figs.5.5,5.5,5.5,5.5 showthe A Tnr A nr foreachofthealgorithmswhere A nr representsthenonrigidpartof theAnetransformationmatrix. A Tnr A nr = I ifthePCAspacedoesnotneedtobe modied.Fromthevaryingvaluesalongthediagonalwecanse ethatinBICandEBGM algorithmsarewellapproximatedbythePCAdimensionwiths hearsandstretchalong thesedimensions.However,thesignicantnon-zeroo-diago nalvaluesfortheLDAand thecommercialalgorithmdenotesthatweneedtoshearandst retchthePCAspacealong dimensionsthatarenotalignedalongthePCAdimensions.Fi g.5.5showstheplotof thediagonalvaluesofthe A Tnr A nr foreachalgorithmversus 1 p PCAi .Wenotethatthere issteadyincreaseintheamountofstretchorshear,inthea netransformationmatrix, alongthedimensionswithlowerPCAeigenvalues.Thisimpli esthattheleastdominant PCAdimensionsareundergoingthemaximumamountoftransfo rmation. 5.5.1AneSpaceDimensions Wecanalsolookattopthreeanespacedimensions,visualiz edasfaces,forallthe facerecognitionalgorithms.Thedimensionscapturethese tofdominantfeaturesusedto characterizeaface.Eachoftheseimageshighlightthevari ationinaparticularfeaturefor thegivensetofimages.Bothverydarkandverybrightregion signifyimportance.The topdimensionswiththemaximumeigenvaluewillthussignif yfeatureswithmaximum variationacrosssubjects.InFig.5.5.1wetakealookatthe PCAeigenvectorsbefore andaftertheyhavebeentransformedbytheanetransformat ion.Weseethatthemost 34

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0 5 10 15 20 25 0 5 10 15 20 25 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Dimension Dimension Value Figure5.8.VisualizationofA Tnr A nr forLDAAlgorithm 0 20 40 60 80 100 0 20 40 60 80 100 -0.01 0 0.01 0.02 0.03 0.04 0.05 Dimension Dimension Value Figure5.9.VisualizationofA Tnr A nr forBICAlgorithm importantfeaturetothePCAisthelightingascanbeobserve dfromtheintensecontrast ofbrightanddarkregionsalongthetwohalvesoftheface.Ot hertemplatematchingalgorithmsliketheLDAandtheBICalsocapturesimilarfeatures astheprincipalcomponent ofvariation.However,theEBGMandthecommercialfacereco gnitionalgorithm,which 35

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0 10 20 30 40 50 60 70 0 20 40 60 80 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10-4 Dimension Dimension Value Figure5.10.VisualizationofA Tnr A nr forEBGMAlgorithm 0 20 40 60 80 100 0 20 40 60 80 100 -0.01 -0.005 0 0.005 0.01 0.015 0.02 Dimension Dimension Value Figure5.11.VisualizationofA Tnr A nr forCommercialAlgorithm arefeaturebasedalgorithms,usemorelocalfacialfeature sasincapturedintheisolated brightanddarkpatchesintheeigen-dimensions. Fig5.5.1showthePCAeigenfacesalongwhichthereismaximu mneedforshearand stretchbytheAneApproximationalgorithm.Weseethatloc alfeaturesarebeingreemphasized. 36

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0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 DimensionsValuesDiagonal Values of Affine Matrix PCA LDA BIC EBGM Commercial 1/SQRT(PCA Eigen Values) Figure5.12.VisualizationofDiagonalValuesofA Tnr A nr forAllAlgorithms 37

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20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 LDA 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 BIC 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 EBGM 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 Commercial 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 PCA Figure5.13.TopDimensionsoftheAneApproximationtotheDierentAl gorithms.LastRowShowstheCorrespondingPCADimensionforCompari son 38

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20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 LDA 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 BIC 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 EBGM 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 140 50 100 150 200 250 Commercial Figure5.14.PCAEigenDimensionsAlongWhichWeNeedtoStretchandSheartheMosttoMatchDistances 39

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5.6PerformanceonDatawithTimeVariation Inthissection,welookattheresultsofexperimentswithal lthedierentalgorithms performedontheNotreDamedataset.Intheseexperiments,t heAneApproximation hasbeentrainedonthe25subjectsinFERETdataset.Thegall eryandprobesetconsisted of415imageseachof415subjectsintheNotreDamedataset.T hegallerysetcontained theimagesofthese415subjectswhentheyrsttaken.Thepro besetcontainingtheimages ofthesame415subjectswiththemaximumtimegapavailablef romthesubsequenttimes whentheimageswerere-acquired.5.6.1IdenticationandVericationPerformance 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve LDA face recognition algorithm Affine face recognition algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve Bayesian face recognition algorithm Affine face recognition algorithm (a) (b) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve EBGM face recognition algorithm Affine face recognition algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve Commercial face recognition algorithm Affine Approximation algorithm (c) (d) Figure5.15.ROConNotreDameImagesShowingthePerformanceCompari-sonoftheDierentFaceRecognitionAlgorithmalongwiththeAneAppr oximationAlgorithm(a)LDA(b)BIC(c)EBGM(d)Commercial 40

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ThevericationperformanceisshownasROCcurvesinFig.5. 6.1forboththeAne Approximationalgorithmandthefacerecognitionalgorith m.TheidenticationperformanceareshowninFig.5.6.1Wenotethat, TheCMCandROCperformanceofallthealgorithmsarewellapp roximated. IncaseofBICandEBGMalgorithms,the recognition performanceishigherthan theoriginalalgorithms. 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve LDA face recognition algorithm Affine face recognition algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve Bayesian face recognition algorithm Affine face recognition algorithm (a) (b) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve EBGM face recognition algorithm Affine face recognition algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve Commercial face recognition algorithm Affine Approximation algorithm (c) (d) Figure5.16.CMConNotreDameImagesShowingthePerformanceCom-parisonoftheDierentFaceRecognitionAlgorithmalongwiththeAneApproximationAlgorithm(a)LDA(b)BIC(c)EBGM(d)Commercial 41

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5.7PerformanceonDataWithExpressionVariation Inthissection,wepresenttheresultsoftestingonthethed atathatinvolveexpression change.Asbeforethetrainsetconsistsof100imagesof25su bjectswith4imagesper subject.Thetestsetconsistsof600imagesof300subjects. Thegallerycontainedof300 "fa"imagesandtheprobeconsistedof300"fb"imagesintheF ERETdataset.Thereis nooverlapbetweentheidentityofsubjectsinthetrainseta ndthetestsets. 5.7.1IdenticationandVericationPerformance 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve LDA Face Recognition Algorithm Affine Face Recognition Algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve Bayesian face recognition algorithm Affine Approximation algorithm (a) (b) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve EBGM face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve Commercial face recognition algorithm Affine Approximation algorithm (c) (d) Figure5.17.ROConFERETDataSetShowingthePerformanceComparisonoftheDierentFaceRecognitionAlgorithmalongwiththeAneApproxi mation Algorithm.(a)LDA(b)BIC(c)EBGM(d)Commercial FromthevericationperformancepresentedinFig.5.7.1we seethattheAneApproximationalgorithmperformsaswellastheLDAalgorithm andincaseofBIC,hasa 42

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betterperformance.IncaseoftheEBGM,whenwecomparethep erformanceingraph (c)ofFig.5.6.1andgraph(c)inFig.5.7.1algorithmwenoti cethereisadropinthe performance.ThisisprobablyduetothefactthattheEBGMal gorithmanditsAne Approximationwasdierent.EBGMwastrainedonjust"fa"im agesandtheAneApproximationalgorithmtrainingincluded"fa"and"fb"imag es.FortheNotreDamedata set,whichconsistsofonly"fa"images(variationintime), wegetabetterperformance approximation.Wealsoobservefromgraph(d)inFig.5.7.1t hattheperformanceof Commercialfacerecognitionalgorithmisbetteratlowfals ealarm,buttheperformance oftheAneApproximationalgorithmismuchbetterathigher valuesoffalsealarm.The identicationperformancewhichispresentedinFig.5.7.1 alsofollowsasimilarpattern. 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve LDA Affine Face Recognition Algorithm Affine Approximation Algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve Bayesian face recognition algorithm Affine Approximation algorithm (a) (b) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve EBGM face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve Commercial face recognition algorithm Affine Approximation algorithm (c) (d) Figure5.18.CMConFERETDataSetShowingthePerformanceComparisonoftheDierentFaceRecognitionAlgorithmalongwiththeAneApprox imationAlgorithm(a)LDA(b)BIC(c)EBGM(d)Commercial 43

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5.8ApproximatingPCAwithDierentDistanceMeasures 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve PCA face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve PCA-Covariance Affine Approximation algorithm (a) (b) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve PCA-MahaCosine Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve PCA-MahL1 Affine Approximation algorithm (c) (d) Figure5.19.ROCCurvesofPCAAlgorithmwithDierentDistanceMeasur es (a)Euclidean(b)Covariance(c)Maha-Cosine(d)MahL1 Fig.5.8,5.8showtheCMCandROCcurvesrespectivelywhenth ePCAalgorithmis usedasthetestfacerecognitionalgorithm.ForthePCA,the non-rigidpart, A nr ,shouldbe anidentitymatrix.Weshowresultsonthe600imagesfrom300 subjectsfromtheFERET dataset.Asbefore,thegalleryconsistsofall"fa"imagesa ndtheprobeconsistsofall"fb" images.Weobservefromgraph(a)inFig.5.8andgraph(a)in5 .8thatboththeROCand theCMCcurvesvalidatethisfact.InFig.5.8weobservethat theAneTransformation matrix A nr ,visualizedas A Tnr A nr ,isanidentitymatrix. FromthegraphsinFig.5.8,5.8wecanalsoobservetheeects ontheAneApproximationalgorithmofusingdierentdistancealongwit hthePCA.Weobservefrom 44

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0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve PCA-Euclidean Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve PCA-Covariance Affine Approximation Algorithm (a) (b) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve PCA-MahaCosine Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve PCA-MahL1 Affine Approximation algorithm (c) (d) Figure5.20.ROCofPCAAlgorithmwithDierentDistanceMeasures(a)Euclidean(b)Covariance(c)Maha-Cosine(d)MahL1graphs(c)and(d)inFig.5.8andFig.5.8thatPCAwithMahana lobiscosinedistance andMahanalobisL1distancealsoarewellapproximated.The performanceoftheane approximationisbetterthanthatofthePCA+Covariancedis tancemeasureasseenin Fig.5.8(b)andFig.5.8(b).5.9EectsofNormalizationontheDistanceMatrix FromFig.5.7.1and5.7.1weobservethatthereisasignican tdropintheperformance oftheCommercialalgorithmascomparedtotheothertestalg orithmsatlowfalsealarm rates.Althoughthereasonforthisisnotclearatpresentan dmoreresearchneedsto bedonetoinvestigatethereasonsforthegapintheperforma nce,oneofthefactors aectingtheapproximationcouldbetheunknownnormalizat ionperformedonthedistance 45

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0 20 40 60 80 100 0 20 40 60 80 100 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Dimension Dimension Value Figure5.21.VisualizationofA Tnr A nr forPCAAlgorithm(EuclideanDistance) fortheFERETDataSet 0 20 40 60 80 100 0 20 40 60 80 100 -0.5 0 0.5 1 1.5 2 2.5 3 x 10-3 Dimension Dimension Value Figure5.22.VisualizationofA Tnr A nr forPCAAlgorithm(CovarianceDistance) fortheFERETDataSet 46

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0 20 40 60 80 100 0 20 40 60 80 100 -5 0 5 10 15 20 x 10 -4 Dimension Dimension Value Figure5.23.VisualizationofA Tnr A nr forPCAAlgorithm(MahCosineDistance) fortheFERETDataSet 0 20 40 60 80 100 0 20 40 60 80 100 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Dimension Dimension Value Figure5.24.VisualizationofA Tnr A nr forPCAAlgorithm(MahL1Distance)for theFERETDataSet 47

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matrixoutputbytheCommercialFacerecognitionalgorithm .InFig.5.9weseethe eectofdierentnormalizationproceduresonthedistance scoresoutputbytheAne ApproximationalgorithmontheROCcurves.Weobservethata ftertheZnormalization, thereisadeniteshiftintheROCcurvetowardstheoriginal ROCcurveoutputbythe facerecognitionalgorithm. G-theimagesinthegallery. P-theimagesintheprobe. S pg -matchscoreofaprobe'p'withgallerytemplate'g'. S pG -vectorofmatchscoresobtainedwhenaprobe'p'withentire galleryG. Inmin-maxnormalization,theminimumandmaximumscoresare usedtonormalize thescorevectoraccordingtoequation5.1 S 0 pG = S pG min ( S pG ) max ( S pG min ( S pG )) (5.1) InZ-normalization,usingthemeanandstandarddeviationof thescorevector S pG wegetthenormalizedscoreasinequation5.2.Theresulting S 0 pG haveameanofzeroand astandarddeviationofone. S 0 pG = S pG mean ( S pG ) std ( S pG ) (5.2) 48

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-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1true positive ratefalse positive rate ROC curve Affine + Z normalizationAffine + Min-Max normalizationAffineCommercial algorithm Figure5.25.EectofNormalizationontheROCCurvesfortheCommercialFaceRecognitionAlgorithmontheFERETDataSet 49

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CHAPTER6 SUMMARYANDCONCLUSION Inthisthesis,wehaveproposedanapproachtomodeldistanc efunctionsoffacerecognitionalgorithmssoastobeabletoevaluatethealgorithms atadeeperlevel.Givenany distancematrixcomputedbyafacerecognitionalgorithm,w e"learn"ananetransformationofthePCAeigenspacesuchthatitcanmatchthedistan ces.Wehavetestedthis anetransformationontwodierentsetsoftestdata,theFE RETdatabasehavingface imageswithvariationinexpressionandtimeandNotreDamed atabasehavingimageswith variationintime.Wehavealsoperformedourexperimentson bothtemplatematching algorithmsliketheBICandLDAaswellasgraphbasedalgorit hmslikeEBGMandthe commercialfacerecognitionalgorithm. Thecoretechniqueusedtocomputetheanetransformationm atrixisclassicalmultidimensionalscaling.Multidimensionalscalingallowsdataw ithahighnumberofdimensions tobeembeddedinalowerdimensionalspace.Toaugmentthisp rocess,wealso"learn"a mappingfunctionsoastobeabletocomputedistancebetween anytwoimages.Thisis unlikeaplainMDSthatjustprovidesanembeddingofthe given data. Wehaveusedtwodierentdatabaseswithtwodierentcovari atessuchasexpression andtimetotesttheperformanceoftheaneapproximation.S incetheapproximationis quiteclosebothsetsoftestdata,wecanconcludethatthea neapproximationtraining canbegeneralizedacrosscovariates. Sincetheultimategoal,isfacerecognitionitisimportant toalsodiscusshowthe recognitionperformanceoftheaneapproximationalgorit hmandtheoriginalfacerecognitioncompare.Aswehaveseeninthepreviouschapter,from theROCsandCMCs,the recognitionperformanceoftheaneapproximationalgorit hmisquiteclosetotheoriginal facerecognitionalgorithmandinsomecasesliketheBIC,ev enperformsbetter.Wend 50

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thattheAneApproximationalgorithmapproximatesthefac erecognitionalgorithmsin thefollowingrankorder.LDAisbestapproximatedfollowed byBIC,commercial,and EBGMforthedatawithexpressionvariation.Forthedatawit htimevariationwend thattheorderoftherankisBIC,LDA,EBGM,commercial.From theexperimentsof thePCAwithdierentdistancemeasures,wendthatPCA+Euc lideandistanceisbest approximatedfollowedbyPCA+MahL1,PCA+MahCosine,andPC A+Covariance. Theaneapproximationalgorithm,canserveasanimportant tooltoevaluateother facerecognitionalgorithm.Forfacerecognitionalgorith mstomatureintomoreecient systems,itisnotonlyimportanttodevelopgoodbaselineal gorithmsbutalsoessentialto developgoodevaluationprotocols.6.1Future Moreresearchandinvestigationisneededtoseehowwellthe anetransformations performwithdatasetswithcovariateswhichhavenotbeenex ploredinthisthesis.Wealso needtostudytheeectofdistancemeasuresandnormalizati ontechniquesonthequality ofanetransformations. 51

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REFERENCES [1]M.TurkandA.Pentland,\Eigenfacesforrecognition," JournalofCognitiveNeuroscience ,vol.3,no.1,pp.71{86,1991. [2]W.Zhao,R.Chellapa,A.Krishnaswamy,D.Swets,andJ.We ng,\Discriminant analaysisofpricipalcomponentsforfacerecognition,"in InternationalConferenceon AutomaticFaceandGestureRecognition ,pp.336{341,1998. [3]B.Heisele,P.Ho,andT.Poggio,\Facerecognitionwiths upportvectormachines: Globalversuscomponent-basedapproach,"in InternationalConferenceonComputer Vision ,vol.2,pp.688{694,2001. [4]L.Wiskott,J.Fellous,N.Kruger,andC.Malsburg,\Face recognitionbyelasticbunch graphmatching," IEEETransactionsonPatternAnalysisandMachineIntellig ence vol.2,no.7,pp.1160{1169,1997. [5]A.NeanandM.Hayes,\Facedetectionandrecognitionus inghiddenmarkovmodels,"in InternationalConferenceofImageProcessing [6]Y.Ivanov,B.Heisele,andT.Serre,\Usingcomponentfea turesforfacerecognition," in InternationalConferenceonAutomaticFaceandGestureRec ognition [7]P.Philips,H.Moon,S.Rizvi,andR.P.J.,\TheFERETeval uationmethodologyfor face-recognitionalgorithms," IEEETransactionsonPatternAnalysisandMachine Intelligence ,vol.22,no.10,pp.1090{1104,2000. [8]S.Rizvi,P.Phillips,andH.Moon,\TheFERETvericatio ntestingprotocolforface recognitionalgorithms,"in InternationalConferenceonAutomaticFaceandGesture Recognition ,pp.48{53,1998. [9]P.Philips,H.Wechsler,J.Huang,andP.Rauss,\TheFERE Tdatabaseandevaluationprocedureforfacerecognitionalgorithms," ImageandVisionComputingJournal vol.16,pp.295{306,1998. [10]\TheFERETdatabase."http://www.itl.nist.gov/iad/ humanid/feret/,June2004. [11]P.Philips,A.Martin,C.Wilson,andM.Przybocki,\Ani ntroductiontoevaluating biometricsystems," IEEEComputer ,vol.33,no.2,pp.56{63,1998. [12]\Facerecognitionvendortest."www.frvt.org,Octobe r2004. [13]D.Blackburn,B.J.M.,andP.Phillips,\FRVT2000:Eval uationreport,"tech.rep., http://www.frvt.org/FRVT2000/documents.htm,May2000. 52

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[14]P.Phillips,P.Grother,R.Micheals,D.Blackburn,E.T abassi, andJ.Bone,\FRVT2002:Evaluationreport,"tech.rep.,http://www.frvt.org/FRVT2002/documents.htm,March200 2. [15]T.CoxandM.Cox, MultidemensionalScaling .ChapmanandHall,2nded.,1994. [16]B.MoghaddamandA.Pentland,\Beyondeigenfaces:Prob abilisticmatchingforface recognition,"in InternationalConferenceonAutomaticFaceandGestureRec ognition pp.30{35,1998. [17]http://www.itl.nist.gov/iad/humanid/feret/feret master.html,October2004. [18]J.Min,K.Bowyer,andP.Flyn.http://www.nd.edu/%7Ec vrl/UNDBiometricsDatabase.html, October2004. [19]L.Reguna,\Anindepthanalysisoffacerecognitionalg orithmsusinganeapproximations,"Master'sthesis,UniversityofSouthFlorida,20 03. [20]J.Wilder, FaceRecognitionusingTransformCodingofGrayscaleProje ctionsandthe NeuralTreeNetwork .ChapmanandHall,1994. [21]V.BlanzandT.Vetter,\Amorphablemodelforthesynthe sisof3dfaces,"in Proceedingsofthe26thAnnualConferenceonComputerGraphicsandI nteractiveTechniques pp.187{194,ACMPress/Addison-WesleyPublishingCo.,1999 [22]S.Gutta,J.Huang,D.Singh,I.Shah,B.Takacs,andH.We chsler,\Benchmark studiesonfacerecognition,"in ProceedingsofInternationalWorkshoponAutomatic Face-andGestureRecognition(IWAFGR) ,1995. [23]R.J.MichealsandT.Boult,\Ecientevaluationofclas sicationandrecognition systems.,"in IEEEConferenceonComputerVisionandPatternRecognition ,pp.50{ 57,2001. [24]J.Beveridge,K.She,B.Draper,andG.Givens,\Paramet ricandnonparametric methodsforthestatisticalevaluationofhumanidalgorith ms,"in ThirdWorkshopon theEmpiricalEvaluationofComputerVisionSystems ,2001. [25]Y.Cheng,H.A.Wild,A.J.O'Toole,P.J.Phillips,andB. Ross.NISTIR6348,June 1999. [26]J.Tenenbaum,V.Silva,andL.J.C.,\Aglobalgeometric frameworkfornonlinear dimensionalityreduction," ScienceMagazine ,vol.290,no.5500,pp.2319{2323,2000. [27]S.RoweisandL.Saul,\Nonlineardimensionalityreduc tionbylocallylinearembedding," ScienceMagazine ,vol.290,no.5500,pp.2323{2326,2000. [28]M.QuistandG.Yona,\Distributionalscaling:Analgor ithmforstructure-preserving embeddingofmetricandnonmetricspaces," JournalofMachineLearningResearch vol.5,pp.399{420,2004. [29]K.WeinbergerandL.Saul,\Unsupervisedlearningofim agemanifoldsbysemidenite programming," IEEEConferenceonComputerVisionandPatternRecognition ,vol.2, pp.988{995,2004. 53

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[30]E.Xing,N.A.Y.,M.Jordan,andS.Russell,\Distanceme triclearningwithapplicationtoclusteringwithside-information,"in AdvancesinNeuralInformationProcessingSystems16 (S.T.S.BeckerandK.Obermayer,eds.),pp.505{512,Cambri dge, MA:MITPress,2003. [31]I.TsangandK.J.T.,\Distancemetriclearningwithker nels,"in InternationalConferenceonArtitialNeuralNetworks ,pp.126{129,2003. [32]M.SchultzandT.Joachims,\Learningadistancemetric fromrelativecomparisons," in AdvancesinNeuralInformationProcessingSystems15 (S.Thrun,L.Saul,and B.Scholkopf,eds.),Cambridge,MA:MITPress,2004. [33]\Evaluationoffacerecognitionalgorithms."http:// www.cs.colostate.edu/evalfacerec/, October2004. [34]W.S.Yambor,\AnalysisofPCA-basedandFisherdiscrimi nant-basedimagerecognitionalgorithms,"Master'sthesis,ColoradoStateUnive rsity,2003. [35]M.L.Teixeira,\Thebayesianintrapersonal/extraper sonalclassier,"Master'sthesis, ColoradoStateUniversity,2003. [36]D.S.Bolme,\Elasticbunchgraphmatching,"Master'st hesis,ColoradoStateUniversity,2003. [37]I.Schoenberg,\RemarkstoMauriceFr'echet'sarticle 'surlad'enitionaxiomatique d'uneclassed'espacesdistanci'esvectoriellementappli cablessurl'espacedehilbert'," TheAnnalsofMathematics ,no.36,pp.724{732,1935. [38]G.YoungandA.Householder,\Discussionofasetofpoin tsintermsoftheirmutual distances," Psychometrika ,no.3,pp.19{22,1938. [39]J.C.GowerandP.Legendre,\Metricandeuclideanprope rtiesofdissimilaritycoefcients," JournalofClassication ,no.3,pp.5{48,1986. 54

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APPENDICES 55

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AppendixAComparisonofResultswithReguna Thefollowingnotationwillbeusedtodescribethelineartr ansformstrategyusedby Reguna. 1.Let ~x i bethe N 2 1sizedcolumnvectorformedbyrowscanningthe N Ni -th image. 2.Let K denotethenumberofimages. 3.Let d Aij bethedistancebetween ~x i and ~x j thatagivenalgorithmcomputes.These distancescanbearrangedasa K K matrix D ,where K isthegivennumberof images. 4.Letthematrix A bea M N 2 sizedarraythatisusedtolinearlytransformthe inputimagevector. ~y i = A~x i (A.1) Therowsofthematrix A denotetheaxesofthereduced M dimensionalspace.For aPCAbasedspace,therowsof A willbeorthogonaltoeachother. 5.The(square)Euclideandistancebetween ~y i and ~y j canbedenotedby d E ( ~y i ;~y j )= P Mk =1 ( ~y i ( k ) ~y j ( k )) 2 =( ~y i ~y j ) T ( ~y i ~y j ) (A.2) ProblemDenition:Thematrix A ,whichistheanetransform,hastobedetermined suchthat d E ( ~y i ;~y j )= d Aij (A.3) d E ( ~y i ;~y j )=( ~y i ~y j ) T ( ~y i ~y j ) =( A~x i A~x j ) T ( A~x i A~x j ) =( A ( ~x i ~x j )) T ( A ( ~x i ~x j )) =( ~x i ~x j ) T ( A T A )( ~x i ~x j ) (A.4) 56

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AppendixA(Continued) (Asanaside,itisworthnotingthatiftherowsofAwereortho normal(e.g.inPCA) andsizeof A was N 2 N 2 then A T A = AA T = I ,theidentitymatrix.Orinotherwords, Euclideandistancearepreserved: d E ( ~y i ;~y j )= d E ( ~x i ;~x j )). Let, 1. B = A T A ,where B isa N 2 N 2 sizedmatrix.Notethat B issymmetric,i.e. B T = B 2. ~ ij = ~x i ~x j isa N 2 1sizedcolumnvector. Usingtheabovenotations d E ( ~y i ;~y j )= ~ Tij B~ ij = P N 2 k =1 P N 2 l =1 B ( k;l ) ~ ij ( k ) ~ ij ( l ) (A.5) Theabovedoublesumcanbeexpressedasproductoftwocolumn vectorsasfollows.Let twocolumnsvectorsbedenedasfollows 1. ~ b isa N 2 ( N 2 +1) 2 sizedcolumnvectorbyscanningthelowertriangularentrie s(including thediagonal)ofB.Thus, ~ b ( k ( k +1) 2 + l )= B ( k;l ),for l k;k =1 ; ;N 2 (A.6) 2. ~ ij isa N 2 ( N 2 +1) 2 sizedcolumnvectorsuchthat ~ ij ( k ( k +1) 2 + l )= 8><>: ~ ij ( k ) 2 for k = l 2 ~ ij ( k ) ~ ij ( l )for l k (A.7) Usingtheaboveequation d E ( ~y i ;~y j )= ~ T ij ~ b (A.8) ~ b shouldbedeterminedsuchthat ~ T ij ~ b = d Aij (A.9) 57

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AppendixA(Continued)foreverypairofimages.These K ( K 1) 2 equationscanbecompactlyexpressedinmatrix notionasfollows T ~ b = ~ d A (A.10) where isa K ( K 1) 2 N 2 ( N 2 +1) 2 sizedmatrixformedbyconcatenatingthecolumnvectors ~ ij .And, ~ d A iscolumnvectorofthegivendistances. Intheaboveequation, ~ b isunknownandcanbesolvedusinganystandardlinear equationsolver.Theonlyconstraintisthat K N 2 +1sothatthenumberequationisat leastequaltothenumberofunknowns.Given ~ b ,thematrixBcanbeformed,fromwhich wewouldliketoformAsuchthat B = A T A .Thiscanbedoneusingtheeigenvectors( ~u i ) andeigenvalues( i )of B Thematrix B isfactoredinto U U T ,wherethecolumnsof U aretheeigenvectors, ~u i ,of B andisadiagonalmatrixformedoutoftheeigenvalues.Sinc eBisguaranteed bythefactthat B issymmetric.Infact,wecanalsoclaimthattheeigenvalues wouldreal andpositive.Symmetricmatriceshaverealeigenvalues.Fr omEq.A.5isfollowsthat B is positivesemi-denitebecausedistancesarealwaysaregrea terthanorequaltozero.And, eigenvaluesofpositivesemi-denitematricesaregreatert hanorequaltozero. Giventheeigenvalueandeigenvectordecompositionof B wecanchoose A = 1 2 U T (A.11) orinotherwordstherowsof A arethescaledeigenvectors ~u i 's.Inparticular,the i -throw of A willbe p i ~u i T .Thus,thenon-zerorowsof A wouldbedeterminedbythenumberof non-zeroeigenvaluesof B A istheaneapproximationmatrixwhichwhenwillattempt toduplicatetheresultsofanyinputfacerecognitionalgor ithm. 58

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AppendixA(Continued) Herewehaveshownasimultaneouscomparisonofsomeofthere sultsgotbyusand thosebyRegunausingherapproachonthesameexperiments.T heseexperimentswere conductedontheFERETdatabaseusingatrainingsetof100im ageswith4imagesper subject.Thetestsetconsistedof600imagesof300subjects .Thegallerysetcontained 300imagesoftype"fa"andtheprobesetconsistedof300imag esoftype"fb". WecanobservefromFig.AthattheCMCsfromourapproachandh erapproachis almostidentical.HowevertheROCsshowaremarkableimprov ementbyourmethodas canbeobservedinFig.A. 59

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AppendixA(Continued) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve PCA face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve PCA-Euclidean Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve LDA Affine Face Recognition Algorithm Affine Approximation Algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve LDA face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve Bayesian face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Probability of Correct MatchRank CMC Curve Bayesian face recognition algorithm Affine Approximation algorithm OurApproachReguna'sApproach FigureA.1.ComparisonofCMCUsingFERETDataonDierentAlgorithms 60

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AppendixA(Continued) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve PCA face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve PCA face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve LDA Face Recognition Algorithm Affine Face Recognition Algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve LDA face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve Bayesian face recognition algorithm Affine Approximation algorithm 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Probability of VerificationFalse Alarm Rate ROC curve Bayesian face recognition algorithm Affine Approximation algorithm OurApproachReguna'sApproach FigureA.2.ComparisonofROCUsingFERETDataonDierentAlgorithms 61


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Modeling distance functions induced by face recognition algorithms
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ABSTRACT: Face recognition algorithms has in the past few years become a very active area of research in the fields of computer vision, image processing, and cognitive psychology. This has spawned various algorithms of different complexities. The concept of principal component analysis(PCA) is a popular mode of face recognition algorithm and has often been used to benchmark other face recognition algorithms for identification and verification scenarios. However in this thesis, we try to analyze different face recognition algorithms at a deeper level. The objective is to model the distances output by any face recognition algorithm as a function of the input images. We achieve this by creating an affine eigen space from the PCA space such that it can approximate the results of the face recognition algorithm under consideration as closely as possible.Holistic template matching algorithms like the Linear Discriminant Analysis algorithm( LDA), the Bayesian Intrapersonal/Extrapersonal classifier(BIC), as well as local feature based algorithms like the Elastic Bunch Graph Matching algorithm(EBGM) and a commercial face recognition algorithm are selected for our experiments. We experiment on two different data sets, the FERET data set and the Notre Dame data set. The FERET data set consists of images of subjects with variation in both time and expression. The Notre Dame data set consists of images of subjects with variation in time. We train our affine approximation algorithm on 25 subjects and test with 300 subjects from the FERET data set and 415 subjects from the Notre Dame data set. We also analyze the effect of different distance metrics used by the face recognition algorithm on the accuracy of the approximation.We study the quality of the approximation in the context of recognition for the identification and verification scenarios, characterized by cumulative match score curves (CMC) and receiver operator curves (ROC), respectively. Our studies indicate that both the holistic template matching algorithms as well as feature based algorithms can be well approximated. We also find the affine approximation training can be generalized across covariates. For the data with time variation, we find that the rank order of approximation performance is BIC, LDA, EBGM, and commercial. For the data with expression variation, the rank order is LDA, BIC, commercial, and EBGM. Experiments to approximate PCA with distance measures other than Euclidean also performed very well. PCA+Euclidean distance is best approximated followed by PCA+MahL1, PCA+MahCosine, and PCA+Covariance.
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