Robust Face Recognition via Sparse Representation(12)

WRIGHTET

AL.:ROBUSTFACERECOGNITIONVIASPARSEREPRESENTATION221

Fig.9.RecognitionratesonARdatabase,forvariousfeaturetransformationsandclassifiers.(a)SRC(our

approach).(b)NN.(c)NS.(d)SVM(linearkernel).

thefeaturedimensionislarge,asinglerandomprojectionperformsthebest(98.1percentrecogni-tionrateonYale,94.7percentonAR).

4.2PartialFaceFeatures

herehavebeenextensivestudiesinboththehumanandcomputervisionliteratureabouttheeffectivenessofpartialfeaturesinrecoveringtheidentityofahumanface,e.g.,see[21]and[41].Asasecondsetofexperiments,wetestouralgorithmonthefollowingthreepartialfacialfeatures:nose,righteye,andmouthandchin.WeusetheExtendedYaleBdatabasefortheexperiment,withthesametrainingandtestsets,asinSection4.1.1.SeeFig.10foratypicalexampleoftheextractedfeatures.

Foreachofthethreefeatures,thedimensiondislargerthanthenumberoftrainingsamplesðn¼1;207Þ,andthelinearsystem(16)tobesolvedbecomesoverdetermined.Nevertheless,sparseapproximatesolutionsxcanstillbe

obtainedbysolvingthe"-relaxed‘1-minimizationproblem(17)(here,again,"¼0:05).TheresultsinFig.10rightagainshowthattheproposedSRCalgorithmachievesbetterrecognitionratesthanNN,NS,andSVM.Theseexperi-mentsalsoshowthescalabilityoftheproposedalgorithminworkingwithmorethan104-dimensionalfeatures.

4.3RecognitionDespiteRandomPixelCorruptionForthisexperiment,wetesttherobustversionofSRC,whichsolvestheextended‘1-minimizationproblem(22)usingtheExtendedYaleBFaceDatabase.WechooseSubsets1and2(717images,normal-to-moderatelightingconditions)fortrainingandSubset3(453images,moreextremelightingconditions)fortesting.Withoutocclusion,thisisarelativelyeasyrecognitionproblem.Thischoiceisdeliberate,inordertoisolatetheeffectofocclusion.Theimagesareresizedto96Â84pixels,19sointhiscase,B¼½A;I isan8,064Â8,761matrix.Forthisdataset,wehaveestimatedthatthepolytopeP¼convðÆBÞisapproximately1,185neighborly(usingthemethodgivenin[37]),suggestingthatperfectreconstructioncanbeachievedupto13.3percent(worstpossible)occlusion.

Wecorruptapercentageofrandomlychosenpixelsfromeachofthetestimages,replacingtheirvalueswithindependentandidenticallydistributedsamplesfromauniformdistribution.20Thecorruptedpixelsarerandomlychosenforeachtestimage,andthelocationsareunknowntothealgorithm.Wevarythepercentageofcorruptedpixelsfrom0percentto90percent.Figs.11a,11b,11c,and11dshowsseveralexampletestimages.Tothehumaneye,beyond50percentcorruption,thecorruptedimages(Fig.11asecondandthirdrows)are

19.TheonlyreasonforresizingtheimagesistobeabletorunalltheexperimentswithinthememorysizeofMatlabonatypicalPC.Thealgorithmreliesonlinearprogrammingandisscalableintheimagesize.20.Uniformover½0;ymax ,whereymaxisthelargestpossiblepixelvalue.

Fig.10.Recognitionwithpartialfacefeatures.(a)Examplefeatures.(b)RecognitionratesofSRC,NN,NS,andSVMontheExtendedYaleBdatabase.

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