WRIGHTET
AL.:ROBUSTFACERECOGNITIONVIASPARSEREPRESENTATION219
unconventionalfeatures:randomfacesanddownsampledimages.Wecompareouralgorithmwiththreeclassicalalgorithms,namely,NN,andNS,discussedintheprevioussection,aswellaslinearSVM.15Inthissection,weusethestableversionofSRCinvariouslowerdimensionalfeaturespaces,solvingthereducedoptimizationproblem(17)withtheerrortolerance"¼0:05.TheMatlabimplementationofthereduced(featurespace)versionofAlgorithm1takesonlyafewsecondspertestimageonatypical3-GHzPC.
Fig.7.Facerecognitionwithocclusion.ThecolumnsofÆB¼Æ½A;I spanahigh-dimensionalpolytopeP¼BðP1ÞinIRm.Eachvertexofthispolytopeiseitheratrainingimageoranimagewithjustasinglepixelilluminated(correspondingtotheidentitysubmatrixI).Givenatestimage,solvingthe‘1-minimizationproblemessentiallylocateswhichfacetofthepolytopethetestimagefallson.The‘1-minimizationfindsthefacetwiththefewestpossiblevertices.Onlyverticesofthatfacetcontributetotherepresentation;allotherverticeshavenocontribution.
willusethistechniquetoestimatetheneighborlinessofallthetrainingdatasetsconsideredinourexperiments.
Empirically,wefoundthatthestableversion(10)isonlynecessarywhenwedonotconsiderocclusionorcorruptione0inthemodel(suchasthecasewithfeatureextractiondiscussedintheprevioussection).WhenweexplicitlyaccountforgrosserrorsbyusingB¼½A;I theextended‘1-minimization(22)withtheexactconstraintBww¼yisalreadystableundermoderatenoise.
^1;^^1¼½xe1 iscomputed,Oncethesparsesolutionw
:
settingyr¼yÀ^e1recoversacleanimageofthesubjectwithocclusionorcorruptioncompensatedfor.Toidentifythesubject,weslightlymodifytheresidualriðyÞinAlgorithm1,computingitagainsttherecoveredimageyr:
^1Þk2¼kyÀ^^1Þk2:riðyÞ¼kyrÀA iðxe1ÀA iðx
ð23Þ
4EXPERIMENTALVERIFICATION
Inthissection,wepresentexperimentsonpubliclyavailable
databasesforfacerecognition,whichservebothtodemon-stratetheefficacyoftheproposedclassificationalgorithmandtovalidatetheclaimsoftheprevioussections.Wewillfirstexaminetheroleoffeatureextractionwithinourframework,comparingperformanceacrossvariousfeaturespacesandfeaturedimensions,andcomparingtoseveralpopularclassifiers.Wewillthendemonstratetherobustnessoftheproposedalgorithmtocorruptionandocclusion.Finally,wedemonstrate(usingROCcurves)theeffective-nessofsparsityasameansofvalidatingtestimagesandexaminehowtochoosetrainingsetstomaximizerobustnesstoocclusion.
4.1.1ExtendedYaleBDatabase
TheExtendedYaleBdatabaseconsistsof2,414frontal-faceimagesof38individuals[58].Thecroppedandnormalized192Â168faceimageswerecapturedundervariouslaboratory-controlledlightingconditions[59].Foreachsubject,werandomlyselecthalfoftheimagesfortraining(i.e.,about32imagespersubject)andtheotherhalffortesting.Randomlychoosingthetrainingsetensuresthatourresultsandconclusionswillnotdependonanyspecialchoiceofthetrainingdata.
Wecomputetherecognitionrateswiththefeaturespacedimensions30,56,120,and504.Thosenumberscorre-spondtodownsamplingratiosof1/32,1/24,1/16,and1/8,respectively.16NoticethatFisherfacesaredifferentfromtheotherfeaturesbecausethemaximalnumberofvalidFisherfacesisonelessthanthenumberofclassesk[24],38inthiscase.Asaresult,therecognitionresultforFisherfacesisonlyavailableatdimension30inourexperiment.
ThesubspacedimensionfortheNSalgorithmis9,whichhasbeenmostlyagreeduponintheliteratureforprocessingfacialimageswithonlyilluminationchange.17Fig.8showstherecognitionperformanceforthevariousfeatures,inconjunctionwithfourdifferentclassifiers:SRC,NN,NS,andSVM.
SRCachievesrecognitionratesbetween92.1percentand95.6percentforall120Dfeaturespacesandamaximumrateof98.1percentwith504Drandomfaces.18ThemaximumrecognitionratesforNN,NS,andSVMare90.7percent,94.1percent,and97.7percent,respectively.Tableswithalltherecognitionratesareavailableinthesupplementaryappendix,whichcanbefoundontheComputerSocietyDigitalLibraryathttp://www.77cn.com.cn/10.1109/TPAMI.2008.79.TherecognitionratesshowninFig.8areconsistentwiththosethathavebeenreportedintheliterature,althoughsomereportedondifferentdatabasesorwithdifferenttrainingsubsets.Forexample,Heetal.[25]reportedthebestrecognitionrateof75percentusingEigenfacesat33D,and89percentusingLaplacianfacesat
15.Duetothesubspacestructureoffaceimages,linearSVMisalreadyappropriateforseparatingfeaturesfromdifferentfaces.Theuseofalinearkernel(asopposedtomorecomplicatednonlineartransformations)alsomakesitpossibletodirectlycomparebetweendifferentalgorithmsworkinginthesamefeaturespace.Nevertheless,betterperformancemightbeachievedbyusingnonlinearkernelsinadditiontofeaturetransformations.16.Wecutoffthedimensionat504asthecomputationofEigenfacesandLaplacianfacesreachesthememorylimitofMatlab.Althoughouralgorithmpersiststoworkfarbeyondonthesamecomputer,504isalreadysufficienttoreachallourconclusions.
17.Subspacedimensionssignificantlygreaterorlessthan9eventuallyledtoadecreaseinperformance.
18.Wealsoexperimentedwithreplacingtheconstrained‘1-minimiza-tionintheSRCalgorithmwiththeLasso.Forappropriatechoiceofregularization ,theresultsaresimilar.Forexample,withdownsampledfacesasfeaturesand ¼1;000,therecognitionratesare73.7percent,86.2percent,91.9percent,97.5percent,atdimensions30,56,120,and504(within1percentoftheresultsinFig.8).
4.1FeatureExtractionandClassificationMethodsWetestourSRCalgorithmusingseveralconventionalholisticfacefeatures,namely,Eigenfaces,Laplacianfaces,andFisherfaces,andcomparetheirperformancewithtwo