Eigenface方法详细过程

EigenfacesforFaceDetection/Recognition(M.TurkandA.Pentland,EigenfacesforRecognition,JournalofCognitiveNeuroscience,vol.3,no.1,pp.71-86,1991,hardcopy)•FaceRecognition-Thesimplestapproachistothinkofitasatemplatematchingproblem:-Problemsarisewhenperformingrecognitioninahigh-dimensionalspace.-Significantimprovementscanbeachievedbyfirstmappingthedataintoalower-dimensionalityspace.-Howtofindthislower-dimensionalspace?•Mainideabehindeigenfaces-SupposeΓisanN2x1vector,correspondingtoanNxNfaceimageI.-TheideaistorepresentΓ(Φ=Γ-meanface)intoalow-dimensionalspace:ˆΦ-mean=w1u1+w2u2+...wKuK(KN2)-2-ComputationoftheeigenfacesStep1:obtainfaceimagesI1,I2,...,IM(trainingfaces)(veryimportant:thefaceimagesmustbecenteredandofthesamesize)Step2:representeveryimageIiasavectorΓiStep3:computetheaveragefacevectorΨ:Ψ=1MMi=1ΣΓiStep4:subtractthemeanface:Φi=Γi-ΨStep5:computethecovariancematrixC:C=1MMn=1ΣΦnΦTn=AAT(N2xN2matrix)whereA=[Φ1Φ2...ΦM](N2xMmatrix)-3-Step6:computetheeigenvectorsuiofAATThematrixAATisverylarge--notpractical!!Step6.1:considerthematrixATA(MxMmatrix)Step6.2:computetheeigenvectorsviofATAATAvi=miviWhatistherelationshipbetweenusiandvi?ATAvi=mivi=AATAvi=miAvi=CAvi=miAviorCui=miuiwhereui=AviThus,AATandATAhavethesameeigenvaluesandtheireigenvec-torsarerelatedasfollows:ui=Avi!!Note1:AATcanhaveuptoN2eigenvaluesandeigenvectors.Note2:ATAcanhaveuptoMeigenvaluesandeigenvectors.Note3:TheMeigenvaluesofATA(alongwiththeircorrespondingeigenvectors)correspondtotheMlargesteigenvaluesofAAT(alongwiththeircorrespondingeigenvectors).Step6.3:computetheMbesteigenvectorsofAAT:ui=Avi(important:normalizeuisuchthat||ui||=1)Step7:keeponlyKeigenvectors(correspondingtotheKlargesteigenvalues)-4-Representingfacesontothisbasis-Eachface(minusthemean)ΦiinthetrainingsetcanberepresentedasalinearcombinationofthebestKeigenvectors:ˆΦi-mean=Kj=1Σwjuj,(wj=uTjΦi)(wecalltheuj’seigenfaces)-EachnormalizedtrainingfaceΦiisrepresentedinthisbasisbyavector:Ωi=⎡⎢⎢⎢⎣wi1wi2...wiK⎤⎥⎥⎥⎦,i=1,2,...,M-5-FaceRecognitionUsingEigenfaces-GivenanunknownfaceimageΓ(centeredandofthesamesizelikethetrainingfaces)followthesesteps:Step1:normalizeΓ:Φ=Γ-ΨStep2:projectontheeigenspaceˆΦ=Ki=1Σwiui(wi=uTiΦ)Step3:representΦas:Ω=⎡⎢⎢⎢⎣w1w2...wK⎤⎥⎥⎥⎦Step4:finder=minl||Ω-Ωl||Step5:iferTr,thenΓisrecognizedasfacelfromthetrainingset.-Thedistanceeriscalleddistancewithinthefacespace(difs)Comment:wecanusethecommonEuclideandistancetocomputeer,however,ithasbeenreportedthattheMahalanobisdistanceperformsbetter:||Ω-Ωk||=Ki=1Σ1li(wi-wki)2(variationsalongallaxesaretreatedasequallysignificant)-6-FaceDetectionUsingEigenfaces-GivenanunknownimageΓStep1:computeΦ=Γ-ΨStep2:computeˆΦ=Ki=1Σwiui(wi=uTiΦ)Step3:computeed=||Φ-ˆΦ||Step4:ifedTd,thenΓisaface.-Thedistanceediscalleddistancefromfacespace(dffs)-7--Reconstructionoffacesandnon-faces-8-•Timerequirements-About400msec(Lisp,Sun4,128x128images)•Applications-Facedetection,tracking,andrecognition•Problems-Background(deemphasizetheoutsideoftheface,e.g.,bymultiplyingtheinputimagebya2DGaussianwindowcenteredontheface)-Lightingconditions(performancedegradeswithlightchanges)-Scale(performancedecreasesquicklywithchangestotheheadsize)*multiscaleeigenspaces*scaleinputimagetomultiplesizes)-Orientation(perfomancedecreasesbutnotasfastaswithscalechanges)*planerotationscanbehandled*out-of-planerotationsmoredifficulttohandle-9-•Experiments-16subjects,3orientations,3sizes-3lightingconditions,6resolutions(512x512...16x16)-Totalnumberofimages:2,592-10-Experiment1*Usedvarioussetsof16imagesfortraining*Oneimage/person,takenunderthesameconditions*Eigenfaceswerecomputedoffline(7eigenfaceswereused)*Classifytherestimagesasoneofthe16individuals*Norejections(i.e.,nothresholdfordifs)-Performedalargenumberofexperimentsandaveragedtheresults:96%correctaveragedoverlightvariation85%correctaveragedoverorientationvariation64%correctaveragedoversizevariation-11-Experiment2-Theyconsideredrejections(i.e.,bythresholdingdifs)-Thereisatradeoffbetweencorrectrecognitionandrejections.-Adjustingthethresholdtoachieve100%recognitionacurracyresultedin:*19%rejectionswhilevaryinglighting*39%rejectionswhilevaryingorientation*60%rejectionswhilevaryingsizeExperiment3-Reconstructionusingpartialinformation