Robust Face Recognition via Sparse Representation(13)

222IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE,VOL.31,NO.2,FEBRUARY2009

Fig.11.Recognitionunderrandomcorruption.(a)TestimagesyfromExtendedYaleB,withrandomcorruption.Toprow:30percentofpixels

^1.arecorrupted.Middlerow:50percentcorrupted.Bottomrow:70percentcorrupted.(b)Estimatederrors^e1.(c)Estimatedsparsecoefficientsx

(d)Reconstructedimagesyr.SRCcorrectlyidentifiesallthreecorruptedfaceimages.(e)Therecognitionrateacrosstheentirerangeofcorruptionforvariousalgorithms.SRC(redcurve)significantlyoutperformsothers,performingalmostperfectlyupto60percentrandomcorruption(seetablebelow).

barelyrecognizableasfaceimages;determiningtheiridentityseemsoutofthequestion.Nevertheless,eveninthisextremecircumstance,SRCcorrectlyrecoverstheidentityofthesubjects.

Wequantitativelycompareourmethodtofourpopulartechniquesforfacerecognitioninthevisionliterature.ThePrincipalComponentAnalysis(PCA)approachin[23]isnotrobusttoocclusion.TherearemanyvariationstomakePCArobusttocorruptionorincompletedata,andsomehavebeenappliedtorobustfacerecognition,e.g.,[29].Wewilllaterdiscusstheirperformanceagainstoursonmorerealisticconditions.Here,weusethebasicPCAtoprovideastandardbaselineforcomparison.21Theremainingthreetechniquesaredesignedtobemorerobusttoocclusion.IndependentComponentAnalysis(ICA)architectureI[53]attemptstoexpressthetrainingsetasalinearcombinationofstatisticallyindependentbasisimages.LocalNonnega-tiveMatrixFactorization(LNMF)[54]approximatesthetrainingsetasanadditivecombinationofbasisimages,computedwithabiastowardsparsebases.22Finally,todemonstratethattheimprovedrobustnessisreallyduetotheuseofthe‘1-norm,wecomparetoaleast-squarestechniquethatfirstprojectsthetestimageontothesubspacespannedbyallfaceimagesandthenperformsNS.

Fig.11eplotstherecognitionperformanceofSRCanditsfivecompetitors,asafunctionofthelevelofcorruption.Weseethatthealgorithmdramaticallyoutperformsothers.From0percentupto50percentocclusion,SRCcorrectlyclassifiesallsubjects.At50percentcorruption,noneoftheothersachieveshigherthan73percentrecognitionrate,whiletheproposedalgorithmachieves100percent.Evenat70percentocclusion,therecognitionrateisstill90.7percent.

21.Following[58],wenormalizetheimagepixelstohavezeromeanandunitvariancebeforeapplyingPCA.

22.ForPCA,ICA,andLNMF,thenumberofbasiscomponentsischosentogivetheoptimaltestperformanceovertherangef100;200;300;400;500;600g

.

Thisgreatlysurpassesthetheoreticalboundoftheworst-casecorruption(13.3percent)thatthealgorithmisensuredtotolerate.Clearly,theworst-caseanalysisistooconserva-tiveforrandomcorruption.

4.4RecognitionDespiteRandomBlockOcclusionWenextsimulatevariouslevelsofcontiguousocclusion,from0percentto50percent,byreplacingarandomlylocatedsquareblockofeachtestimagewithanunrelatedimage,asinFig.12a.Again,thelocationofocclusionisrandomlychosenforeachimageandisunknowntothecomputer.Methodsthatselectfixedfacialfeaturesorblocksoftheimage(e.g.,[16]and[57])arelesslikelytosucceedhereduetotheunpredictablelocationoftheocclusion.ThetoptworowsinFigs.12a,12b,12c,and12dshowsthetworepresentativeresultsofAlgorithm1with30percentocclusion.Fig.12aistheoccludedimage.Inthesecondrow,theentirecenterofthefaceisoccluded;thisisadifficultrecognitiontaskevenforhumans.Fig.12bshows

e1themagnitudeoftheestimatederror^e1.Noticethat^

compensatesnotonlyforocclusionduetothebaboonbutalsofortheviolationofthelinearsubspacemodelcausedbytheshadowunderthenose.Fig.12cplotstheestimatedcoefficientvectorx^1.Theredentriesarecoefficientscorrespondingtotestimage’strueclass.Inbothexamples,theestimatedcoefficientsareindeedsparseandhavelargemagnitudeonlyfortrainingimagesofthesameperson.Inbothcases,theSRCalgorithmcorrectlyclassifiestheoccludedimage.Forthisdataset,ourMatlabimplementa-tionrequires90secondspertestimageonaPowerMacG5.ThegraphinFig.12eshowstherecognitionratesofallsixalgorithms.SRCagainsignificantlyoutperformstheotherfivemethodsforalllevelsofocclusion.Upto30percentocclusion,Algorithm1performsalmostperfectly,correctlyidentifyingover98percentoftestsubjects.Evenat40percentocclusion,http://www.77cn.com.cnparedtotherandompixelcorruption,contiguous

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