Robust Face Recognition via Sparse Representation(7)

216IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE,VOL.31,NO.2,FEBRUARY

2009

Fig.5.Exampleofaninvalidtestimage.(a)SparsecoefficientsfortheinvalidtestimagewithrespecttothesametrainingdatasetfromExample1.Thetestimageisarandomlyselectedirrelevantimage.(b)Theresidualsoftheinvalidtestimagewithrespecttotheprojection iðx^Þof

1

thesparserepresentationcomputedby‘-minimization.Theratioofthetwosmallestresidualsisabout1:1.2.

^Þ! ;SCIðxð15Þ

andotherwiserejectasinvalid.Instep5ofAlgorithm1,one

maychoosetooutputtheidentityofyonlyifitpassesthiscriterion.

UnlikeNNorNS,thisnewruleavoidstheuseoftheresidualsriðyÞforvalidation.NoticethatinFig.5,evenforanonfaceimage,withalargetrainingset,thesmallestresidualoftheinvalidtestimageisnotsolarge.Ratherthanrelyingonasinglestatisticforbothvalidationandidentification,ourapproachseparatestheinformationrequiredforthesetasks:theresidualsforidentificationandthesparsecoefficientsforvalidation.10Inasense,theresidualmeasureshowwelltherepresentationapprox-imatesthetestimage;andthesparsityconcentrationindexmeasureshowgoodtherepresentationitselfis,intermsoflocalization.

Onebenefittothisapproachtovalidationisimprovedperformanceagainstgenericobjectsthataresimilartomultipleobjectclasses.Forexample,infacerecognition,http://www.77cn.com.cningresidualsforvalidationmorelikelyleadstoafalsepositive.However,agenericfaceisunlikelytopassthenewvalidationruleasagoodrepresentationofittypicallyrequirescontribu-tionfromimagesofmultiplesubjectsinthedataset.Thus,thenewrulecanbetterjudgewhetherthetestimageisagenericfaceorthefaceofoneparticularsubjectinthedataset.InSection4.7,wewilldemonstratethatthenewvalidationruleoutperformstheNNandNSmethods,withasmuchas10-20percentimprovementinverificationrateforagivenfalseacceptrate(seeFig.14inSection4orFig.18inthesupplementaryappendix,whichcanbefoundontheComputerSocietyDigitalLibraryathttp://www.77cn.com.cn/10.1109/TPAMI.2008.79).

3.1TheRoleofFeatureExtraction

Inthecomputervisionliterature,numerousfeatureextrac-tionschemeshavebeeninvestigatedforfindingprojectionsthatbetterseparatetheclassesinlowerdimensionalspaces,whichareoftenreferredtoasfeaturespaces.OneclassofmethodsextractsholisticfacefeaturessuchasEigen-faces[23],Fisherfaces[24],andLaplacianfaces[25].Anotherclassofmethodstriestoextractmeaningfulpartialfacialfeatures(e.g.,patchesaroundeyesornose)[21],[41](seeFig.6forsomeexamples).Traditionally,whenfeatureextractionisusedinconjunctionwithsimpleclassifierssuchasNNandNS,thechoiceoffeaturetransformationisconsideredcriticaltothesuccessofthealgorithm.Thishasledtothedevelopmentofawidevarietyofincreasinglycomplexfeatureextractionmethods,includingnonlinearandkernelfeatures[42],[43].Inthissection,wereexaminetheroleoffeatureextractionwithinthenewsparserepresentationframeworkforfacerecognition.

Onebenefitoffeatureextraction,whichcarriesovertotheproposedsparserepresentationframework,isreduceddatadimensionandcomputationalcost.Forrawfaceimages,thecorrespondinglinearsystemy¼Axxisverylarge.Forinstance,ifthefaceimagesaregivenatthetypicalresolution,640Â480pixels,thedimensionmisintheorderof105.AlthoughAlgorithm1reliesonscalablemethodssuchaslinearprogramming,directlyapplyingittosuchhigh-resolutionimagesisstillbeyondthecapabilityofregularcomputers.

Sincemostfeaturetransformationsinvolveonlylinearoperations(orapproximatelyso),theprojectionfromtheimagespacetothefeaturespacecanberepresentedasamatrixR2IRdÂmwithd(m.ApplyingRtobothsidesof(3)yields

:~¼Ryyy¼RAxx0

2IRd:

ð16Þ

3

TWOFUNDAMENTALISSUES

RECOGNITION

IN

FACE

Inthissection,westudytheimplicationsoftheabovegeneralclassificationframeworkfortwocriticalissuesinfacerecognition:1)thechoiceoffeaturetransformation,and2)robustnesstocorruption,occlusion,anddisguise.

10.Wefindempiricallythatthisseparationworkswellenoughinourexperimentswithfaceimages.However,itispossiblethatbettervalidationandidentificationrulescanbecontrivedfromusingtheresidualandthesparsitytogether.

Inpractice,thedimensiondofthefeaturespaceistypicallychosentobemuchsmallerthann.Inthiscase,thesystemof

~¼RAxequationsyx2IRdisunderdeterminedintheun-knownx2IRn.Nevertheless,asthedesiredsolutionx0is

sparse,wecanhopetorecoveritbysolvingthefollowingreduced‘1-minimizationproblem:ð‘1rÞ:

^1¼argminkxk1x

subjectto

~k2 ";kRAxxÀy

ð17Þ

foragivenerrortolerance">0.Thus,inAlgorithm1,thematrixAoftrainingimagesisnowreplacedbythematrix

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