Robust Face Recognition via Sparse Representation(15)

224IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE,VOL.31,NO.2,FEBRUARY

2009

WeverifytheefficacyofthisschemeontheARdatabaseforfacesdisguisedwithsunglassesorscarves.Wepartitiontheimagesintoeight(4Â2)blocksofsize20Â30pixels.Partitioningincreasestherecognitionrateonscarvesfrom59.5percentto93.5percentandalsoimprovestherecogni-tionrateonsunglassesfrom87.0percentto97.5percent.ThisperformanceexceedsthebestknownresultsontheARdataset[29]todate.Thatworkobtains84percentonthesunglassesand93percentonthescarfs,ononly50subjects,usingmoresophisticatedrandomsamplingtechniques.Alsonoteworthyis[16],whichaimstorecognizeoccludedfacesfromonlyasingletrainingsamplepersubject.OntheARdatabase,thatmethodachievesalowercombinedrecogni-tionrateof80percent.24

Fig.13.(a)-(d)Partitionschemetotacklecontiguousdisguise.ThetoprowvisualizesanexampleforwhichSRCfailedwiththewholeimage(holistic).Thetwolargestcoefficientscorrespondtoabeardedmanandascreamingwoman,twoimageswhosemouthregionresemblestheoccludingscarf.Iftheocclusionisknowntobecontiguous,onecanpartitiontheimageintomultiplesmallerblocks,applytheSRCalgorithmtoeachoftheblocksandthenaggregatetheresultsbyvoting.Thesecondrowvisualizeshowthispartition-basedschemeworksonthesametestimagebutleadstoacorrectidentification.(a)Thetestimage,occludedbyscarf.(b)Estimated

^1.(d)Reconstructedsparseerror^e1.(c)Estimatedsparsecoefficientsx

image.(e)TableofrecognitionratesontheARdatabase.Thetableshowstheperformanceofallthealgorithmsforbothtypesofocclusion.SRC,itsholisticversion(righttop)andpartitionedversion(rightbottom),achievesthehighestrecognitionrate.

rejectionruleintroducedinSection2.4).Themajordifficultywiththisapproachisthattheocclusioncannotbeexpectedtorespectanyfixedpartitionoftheimage;whileonlyafewblocksareassumedtobecompletelyoccluded,someoralloftheremainingblocksmaybepartiallyoccluded.Thus,insuchascheme,thereisstillaneedforrobusttechniqueswithineachblock.

WepartitioneachofthetrainingimagesintoLblocksofsizeaÂb,producingasetofmatricesAð1Þ;...;AðLÞ2IRpÂn,

:

wherep¼ab.Wesimilarlypartitionthetestimageyintoyð1Þ;...;yðLÞ2IRp.WewritethelthblockofthetestimageasasparselinearcombinationAðlÞxðlÞoflthblocksofthetrainingimages,plusasparseerroreðlÞ2IRp:yðlÞ¼AðlÞxðlÞþeðlÞ.WecanrecovercanagainrecoverasparsewðlÞ¼½xðlÞeðlÞ 2IRnþpby‘1minimization:

hi

ðlÞ:ðlÞðlÞ^1¼argminkwksubjecttoAIw¼y:ð24Þw1

w2IRnþpWeapplytheclassifierfromAlgorithm1withineachblock23

andthenaggregatetheresultsbyvoting.Fig.13illustratesthisscheme.

23.Occludedblockscanalsoberejectedvia(15).Wefindthatthisdoesnotsignificantlyincreasetherecognitionrate.

4.7RejectingInvalidTestImages

Wenextdemonstratetherelevanceofsparsityforrejectinginvalidtestimages,withorwithoutocclusion.Wetesttheoutlierrejectionrule(15)basedontheSparsityConcentra-tionIndex(14)ontheExtendedYaleBdatabase,usingSubsets1and2fortrainingandSubset3fortestingasbefore.Weagainsimulatevaryinglevelsofocclusion(10percent,30percent,and50percent)byreplacingarandomlychosenblockofeachtestimagewithanunrelatedimage.However,inthisexperiment,weincludeonlyhalfofthesubjectsinthetrainingset.Thus,halfofthesubjectsinthetestingsetarenewtothealgorithm.Wetestthesystem’sabilitytodeterminewhetheragiventestsubjectisinthetrainingdatabaseornotbysweepingthethreshold througharangeofvaluesin[0,1],generatingthereceiveroperatorcharacteristic(ROC)curvesinFig.14.Forcomparison,wealsoconsideredoutlierrejectionbythresholdingtheeuclideandistancebetween(featuresof)thetestimageand(featuresof)thenearesttrainingimageswithinthePCA,ICA,andLNMFfeaturespaces.ThesecurvesarealsodisplayedinFig.14.Noticethatthesimplerejectionrule(15)performsnearlyperfectlyat10percentand30percentocclusion.At50percentocclusion,itstillsignificantlyoutperformstheotherthreealgorithmsandistheonlyoneofthefouralgorithmsthatperformssignificantlybetterthanchance.Thesupplementaryappen-dix,whichcanbefoundontheComputerSocietyDigitalLibraryathttp://www.77cn.com.cn/10.1109/TPAMI.2008.79,containsmorevalidationresultsontheARdatabaseusingEigenfaces,againdemonstratingsig-nificantimprovementintheROC.

4.8DesigningtheTrainingSetforRobustness

Animportantconsiderationindesigningrecognitionsys-temsisselectingthenumberoftrainingimages,aswellastheconditions(lighting,expression,viewpoint,etc.)underwhichtheyaretobetaken.Thetrainingimagesshouldbeextensiveenoughtospantheconditionsthatmightoccurinthetestset:theyshouldbe“sufficient”fromapatternrecognitionstandpoint.Forinstance,Leeetal.[59]showshowtochoosethefewestrepresentativeimagestowellapproximatetheilluminationconeofeachface.ThenotionofneighborlinessdiscussedinSection2providesadifferentquantitativemeasureforhow“robust”thetrainingsetis:theamountofworstcaseocclusionthealgorithmcantolerateisdirectlydeterminedbyhowneighborlytheassociatedpolytopeis.Theworstcaseisrelevantinvisualrecognition,

24.Fromourownimplementationandexperiments,wefindtheirmethoddoesnotgeneralizewelltomoreextremeilluminations.

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