Robust Face Recognition via Sparse Representation(6)

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AL.:ROBUSTFACERECOGNITIONVIASPARSEREPRESENTATION215

Fig.4.Nonsparsityofthe‘2-minimizer.(a)Coefficientsfrom‘2-minimizationusingthesametestimageasFig.3.Therecoveredsolutionisnotsparseand,hence,lessinformativeforrecognition(largecoefficientsdonotcorrespondtotrainingimagesofthistestsubject).(b)Theresidualsof

^Þofthecoefficientsobtainedby‘2-minimization.Theratiobetweenthetwosmallestthetestimagefromsubject1withrespecttotheprojection iðx

residualsisabout1:1.3.Thesmallestresidualisnotassociatedwithsubject1.

Example1(‘1-minimizationversus‘2-minimization).ToillustratehowAlgorithm1works,werandomlyselecthalfofthe2,414imagesintheExtendedYaleBdatabaseasthetrainingsetandtherestfortesting.Inthisexample,wesubsampletheimagesfromtheoriginal192Â168tosize12Â10.Thepixelvaluesofthedownsampledimageareusedas120-Dfeature-s—stackedascolumnsofthematrixAinthealgorithm.Hence,matrixAhassize120Â1,207,andthesystemy¼Axxisunderdetermined.Fig.3aillustratesthesparsecoefficientsrecoveredbyAlgorithm1foratestimagefromthefirstsubject.Thefigurealsoshowsthefeaturesandtheoriginalimagesthatcorrespondtothetwolargestcoefficients.Thetwolargestcoefficientsarebothassociatedwithtrainingsamplesfromsubject1.Fig.3bshowstheresidualswithrespecttothe38projected

^1Þ,i¼1;2;...;38.With12Â10down-coefficients iðx

sampledimagesasfeatures,Algorithm1achievesanoverallrecognitionrateof92.1percentacrosstheExtendedYaleBdatabase.(SeeSection4fordetailsandperformancewithotherfeaturessuchasEigenfacesandFisherfaces,aswellascomparisonwithothermethods.)Whereasthemoreconventionalminimum

x‘2-normsolutiontotheunderdeterminedsystemy¼Ax

1

istypicallyquitedense,minimizingthe‘-normfavorssparsesolutionsandprovablyrecoversthesparsestsolutionwhenthissolutionissufficientlysparse.Toillustratethiscontrast,Fig.4ashowsthecoefficientsofthesametestimagegivenbytheconventional‘2-minimization(4),andFig.4bshowsthecorrespondingresidualswithrespecttothe38subjects.Thecoefficientsaremuchlesssparsethanthosegivenby‘1-minimization(inFig.3),andthedominantcoefficientsarenotassociatedwithsubject1.Asaresult,thesmallestresidualinFig.4doesnotcorrespondtothecorrectsubject(subject1).

testsamplebasedonhowsmallthesmallestresidualis.However,eachresidualriðyÞiscomputedwithoutanyknowledgeofimagesofotherobjectclassesinthetrainingdatasetandonlymeasuressimilaritybetweenthetestsampleandeachindividualclass.

^1Inthesparserepresentationparadigm,thecoefficientsx

arecomputedglobally,intermsofimagesofallclasses.Inasense,itcanharnessthejointdistributionofallclassesfor

^arebettervalidation.Wecontendthatthecoefficientsx

statisticsforvalidationthantheresiduals.Letusfirstseethisthroughanexample.

Example2(concentrationofsparsecoefficients).WerandomlyselectanirrelevantimagefromGoogleanddownsampleitto12Â10.WethencomputethesparserepresentationoftheimageagainstthesameExtendedYaleBtrainingdata,asinExample1.Fig.5aplotstheobtainedcoefficients,http://www.77cn.com.cnparedtothecoefficientsofavalidtest

^herearenotimageinFig.3,noticethatthecoefficientsx

concentratedonanyonesubjectandinsteadspreadwidelyacrosstheentiretrainingset.Thus,thedistributionofthe

^containsimportantinfor-estimatedsparsecoefficientsx

mationaboutthevalidityofthetestimage:Avalidtestimageshouldhaveasparserepresentationwhosenonzeroentriesconcentratemostlyononesubject,whereasaninvalidimagehassparsecoefficientsspreadwidelyamongmultiplesubjects.Toquantifythisobservation,wedefinethefollowingmeasureofhowconcentratedthecoefficientsareonasingleclassinthedataset:

Definition1(sparsityconcentrationindex(SCI)).TheSCIofacoefficientvectorx2IRnisdefinedas

:kÁmaxik iðxÞk1=kxk1À1

SCIðxÞ¼

kÀ1

2½0;1 :

ð14Þ

2.4ValidationBasedonSparseRepresentationBeforeclassifyingagiventestsample,wemustfirstdecideifitisavalidsamplefromoneoftheclassesinthedataset.Theabilitytodetectandthenrejectinvalidtestsamples,or“outliers,”iscrucialforrecognitionsystemstoworkinreal-worldsituations.Afacerecognitionsystem,forexample,couldbegivenafaceimageofasubjectthatisnotinthedatabaseoranimagethatisnotafaceatall.

SystemsbasedonconventionalclassifierssuchasNNorNS,oftenusetheresidualsriðyÞforvalidation,inadditiontoidentification.Thatis,thealgorithmacceptsorrejectsa

^foundbyAlgorithm1,ifSCIðx^Þ¼1,theForasolutionx

testimageisrepresentedusingonlyimagesfromasingle

^Þ¼0,thesparsecoefficientsarespreadobject,andifSCIðx

evenlyoverallclasses.9Wechooseathreshold 2ð0;1Þandacceptatestimageasvalidif

9.DirectlychoosingxtominimizetheSCImightproducemoreconcentratedcoefficients;however,theSCIishighlynonconvexanddifficulttooptimize.Forvalidtestimages,minimizingthe‘1-normalreadyproducesrepresentationsthatarewell-concentratedonthecorrectsubjectclass.

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