Fully constrained least squares linear mixture anal

IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.39,NO.3,MARCH2001529FullyConstrainedLeastSquaresLinearSpectralMixtureAnaly,StudentMember,IEEE,andChein-IChang,SeniorMember,IEEEAbstract—Linearspectralmixtureanalysis(LSMA)isawidelyusedtechniqueinremotesensrforanLSMA-basedestimatortoproduceaccurateamountsofmaterialabundance,itgenerallyrequirestwoconstraintsimposedonthelinearmixturemodelusedinLSMA,whicharetheabundstcon-straintrequiresthesumoftheabundancefractionsofmaterialspresentinanimagepixeltobeoneandthesehefirstconstraintiseasytodealwith,thesecondconstraintisdiffi-culttoimplementsinceituently,mostLSMA-basedmethodsareunconstrainedandproducesolutionstcase,theycanonlybeusedforthepurposesofmaterialdetection,discrimination,andclassification,paper,wepresentafullyconstrainedleastsquares(FCLS)oclosedformcanbederivedforthismethod,rtofurtherapplythedesignedalgorithmtounknownimagescenes,anunsupervisedleastsquareserror(LSE)-basedsofcomputersimulationsandrealhyperspectraldataex-perimentswereconductedtodemonstratetheperms—Fullyconstrainedleastsquares(FCLS),linearspectralmixtureanalysis(LSMA),nonnegativelyconstrainedleastsquares(NCLS),sum-to-oneconstrainedleastsquares(SCLS),unsupervisedFCLS(UFCLS).NNCLSNSCLSOSPSCLSUFCLSnormalizedNCLS;normalizedSCLS;orthogonalsubspaceprojection;sum-to-oneconstrainedleastsquares;UCTIONNOMENCLATUREANCASCAVIRISFCLSHYDICELSELSMALSNCLSabundancenonnegativityconstraint;abundancesum-to-oneconstraint;airbornevisible/infraredimagingspectrometer;fullyconstrainedleastsquares;hyperspectraldigitalimagerycollectionexperi-ment;leastsquareserror;linearspectralmixtureanalysis;leastsquares;nonnegativelyconstrainedleastsquares;ManuscriptreceivedDecember14,1999;revisedMay30,rkwassupportedbytheBechtelNevadaCorporationunderContractDE-AC08-96NV11718throughtheDepartmentofEnergy,Washington,horsarewiththeRemoteSensingSignalandImageProcessingLaboratory,DepartmentofComputerScienceandElectricalEngineering,UniversityofMaryland-BaltimoreCounty,Baltimore,MD21250USA(e-mail:cchang@).PublisherItemIdentifierS0196-2892(01)Lmixtureanalysishasbeenwidelyusedinre-motesensingforversatileapplicationssuchasmaterialdis-crimination,detection,hofspectralmixtureanalysistechniqueshavebeenreportedinthelitera-ture[1]–[12].Inmultispectral/hyperspectralimagery,hemacroscopicmixture[11],whichmodelsamixedmodelsuggestedbyHapkein[12],calledtheintimatespectralmixture,heless,Hapke’smodelcanbelinearizedbyamethodproposedbyJohnsonetal.[13].Consequently,onlylinearspectralmixtureanalysis(LSMA)ngadvantageofthelinearmixturemodel,r,thereisaprincipaldifferencebetweenpureandmixedpixelprocessing,wheretheformerisasmple,mixedpixelclassificationattemptstoestimatetheabundancefractionsofmaterialsofinterestinapixelandclassifiesthesematerialsinaccordancewiththeirestimatedabundancefractionsasopposedtoult,themixedpixelclassificationgenerallygeneratesagrayscaleimagewhosegraylevelvaluesarede-terminedbytheestimatedabundancefractionsofthematerialsresidentintheimagepixelsincontrast-basedmethodsrequireaprioriknowledgeofthesignaturesofmaterialspresentintheimagescene,hiscircumstance,selectionofanappropriatesetofmaterialsdealcase,thesesignatureswounately,thiscaseisrarelytrueinpracticalsituationssinceallthesignaturestobeusedaregenerallyobtaineddirectlyfromtheimagescene,ethatS0196–2892/01$10.00©2001IEEE

530IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.39,NO.3,MARCH2001materialsignaturesForinstance,whensomesignature,pliesthatanimagescenemaycontainasmanyas0forall1equationswiththenumberofbandsandis

HEINZANDCHANG:MIXTUREMODELLinearspectralmixtureanalysisisawidelyuseverypixelisacquiredbyspectralbandsatdifferentwave-lengths,theycanberepresentedascolumnvectors,ethatbean1columnpixelvectorinamuan,wherethmaterialresidentintheimagescene,and,icalapproachtosolvingmixedpixelclassificationproblemislinearunmixing,whichassumesthatthereareislinearlymixedbythesecanberepresentedbyalinearregressionmodelasfollows:(1)tlossofgenerality,thisdesiredmaterialsignatureisas-sume(1)canberewrittenasistheabundancevectorassociatedwithfrom(3)quares(LS)ProjectionClassifierIntheOSPclassifiergivenby(3),theabundancevectorisgenerallynotknown,rtoestimate,formodel(1)isgivenby,isgivenbyclassifierspecifiedby(6)imposednoconstraintsontheabundancevector,wefirstconsiderapartiallyconstrainedleastsquareslinearmixingproblemthatimposesonlytheASConsubjectto

532IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.39,NO.3,softheLSandFCLSmethodsindetectionandquantificationofcreosoteleaves,whereMconsistedofthreematerials:drygrass,redsoil,utionto(7)canbeobtainedby0foreach10foreachmixingproblemswhilediscardingtheASC.)onlinearconsistingofallindicesgivencorrespondingtopositivecomponentsintheestimateby(5)ativityConstrainedLeastSquares(NCLS)MethodUnliketheSCLSmethodwhichproducesaclosed-formso-lution,theNCLSmethoddoesnothaveananalyticalsolutionsincetheANCisformedbyasetofmustsatisfythefollowingKuhn-Tuckerconditions:subjecttofromthe0forall10for1,wecanformaLagrangianandisinmple,

HEINZANDCHANG:sofLS,SCLS,NSCLS,NCLS,NNCLS,andFCLSmethodsindetectionandquantificationofcreosoteleaves,whereMconsistedoffivematerials:drygrass,redsoil,creosoteleaves,sagebrush,THODANDUNSUPERVISEDFCLSMETHODSOnesimpleapproachtosolvingforfullyconstrainedlineaCLSsolution,wesimplythrowoutthemate-rialsignatureswithnegativeabundancefractionsandnormalultingsolutioniscalledanormalizedSCLS(NSCLS)CLSsolution,wecannormalizeittounity,whichresultsinthenormalizedNCLS(NNCLS)unately,aswillbeshownintheexperiments,nei-thertheNSCLSnortheNNCLSmethodwillyieldoptimalso-lutionsdthatsimultaneouslyimple-mentedANCandASCwasrecentlyproposedin[31].However,itstillproducedonlyanearlyoptimalsolutionbecauseitdidnotsatisfy(15).Inthissection,wepresentanFCLSalgorithmthatwillgenerateanoptimalsolutionbymakinguseoftheNCaneouslyrequiringboththeASC2002200pixelsubsectionofanAVIRISimagescene(LunarCraterVolcanicField).Smethodisthesameoneconsideredin[32]thatextendedthenonnegativeleastsquaresalgorithmin[33]byincludingtheASC.

534IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.39,NO.3,sofLS,SCLC,NSCLS,NCLS,NNCLS,andFCLSmethods,wherefigureslabeledby(a),(b),(c),(d),and(e)aredetectionandquantificationresultsofcinders,playa,rhyolite,vegetation,andshade,thodInordertotakecareoftheASC,weincludetheASCinthesignaturematrix,definedby(17)rforittoapplytoasituationwherenoaprioriinformationisavailable,weneedanunsu-pervisedprocthenearestneighborrule,fromwhichanunsupervisedvectorquantizationmethodwasderivedtofindunknownmaterialorsignalsinanimagescene[24].Anotheristhetargetgenerationprocessproposedin[34],[35],section,weproposeanLSE-basedcriterionasanalternativebe-causethcriterionminimizesthegood-acanbedescribedasfollows.

HEINZANDCHANG:sofUFCLSmethodfor(a)cinders,(b)playa,(c)rhyolite,(d)vegetation,and(e)shadewithpartialmaterialknowledge,lly,wemakeanattempttoselectapurepixelvectorofamaterialpresenmaximumLSEvalues:erscriptindicatesthenumberofiterationscur-and,ulatetheleastsquareserror(LSE)thatyieldsthelargestLSEwillbeselectedtobethethirdmaterialpixelisrepeateduntcedureoutlinedasaboveiscalledUnsupervisedFCLS(UFCLS)Algorithm,atethematrixandisincludedforeachpixelvector

536IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.39,NO.3,,thetobeaprescribederrorthreshold,andlet.2),,toestimatetheabundancefractionsof,,and.4)FindtheleastsquareserrordefinedbyLSEalgorithmstops,otherwisecontinue.5)FindLSE,thus,onally,ananal-ogousapproachtoUFCLScanalsobeusedtoextendLS,SCLS,NSCLS,NCLS,andNNCLSmethodstoUnsupervisedLS,Un-supervisedSCLS,UnsupervisedNSCLS,UnsupervisedNCLS,andUnsupervisedNNCLS[36].ERSIMULATIONSANDEXPERIMENTSThissectioncontainsaseriesofcomputersimulationsandexperimentstoevaluatecomparativeperformanceoftheLS,SCLS,NSCLS,NCLS,NNCLS,,weconductedcom-putersimulationstodemonstrateadvantagesoftheFCLSand

HEINZANDCHANG:sofLS,SCLS,NSCLS,NCLS,NNCLS,alhyperspectralimagedatawereusedtoshowthesuperiementingtheFCLSandUFCLSmethods,thevalueof10,exceptforHYDICEexperimentswherevariouswasformedfromthedrygrass,redsoil,andcreosoteleavessignatures,withtheirassociatedabundancefractionsgivenby

538IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.39,NO.3,sofLS,SCLS,NSCLS,NCLS,NNCLS,andFCLSmethods,wherefigureslabeledby(a),(b),(c),(d),and(e)aredetectionandquantificationresultsofP1,P2,P3,P4,andP5,thodwere2.58810for10forNCLS,SCLS,-1.303viously,theFCLSmethodproducedthebestquantr,e2:EffectsofTwoAdditional,LessSpectrallyDistinct,Materials:Thematerialsignaturematrixwasassumedtoconsistofthesefivespectralsignatureswithabundancefractionsgivenby10fortheLS,1.94510fortheNSCLSweresignificantlyworsethanthoseproducedbytheotherthreemethods,4.82310fortheNNCLS,and2.806,butabsentinapixel,r,twoadditionalsigna-tures,blackbrushandsagebrush,wereaddedtothesignaturematrixusedinthismethodnulledtheundesiredsignatures,blaformanceoftheSCLSandNSCLSmethodswasreducedbecausetheyassumedtherewerefivesignaturesandtheirestimatedabundancefractionsmustbe

HEINZANDCHANG:sofFCLSmethods.(a)