Context-Dependent Detection in Hyperspectral Imagery

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Context-Dependent Detection in Hyperspectral Imagery
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english
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Glenn, Taylor C
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University of Florida
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Gainesville, Fla.
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
GADER,PAUL D
Committee Co-Chair:
WILSON,JOSEPH N
Committee Members:
RANGARAJAN,ANAND
BANERJEE,ARUNAVA
LEE,WON SUK

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bayesian -- context-dependent -- detection -- fuzzy -- hyperspectral
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
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Computer Engineering thesis, Ph.D.
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Abstract:
Significant context information often exists in hyperspectral images, but the existing known-signature target detection techniques do not explicitly account for this fact and do not take advantage of the information. This dissertation explores the idea that using context information improves detection algorithms for hyperspectral imagery. In support of this, new context-dependent detection techniques are developed to improve the performance of detection algorithms through the use of context information. These algorithms include a sequential unsupervised context learning and detection method using a piecewise convex model based unmixing approach. The sequential method is then followed by new algorithms using joint unsupervised context learning and detection. In service of developing the joint context learning and detection approaches, a Bayesian formulation for fuzzy clustering is derived, which is a significant advance to that area. Additionally a new approach to fusion called Alarm-Set Fusion is derived, and both supervised and unsupervised learning algorithms are presented for that purpose. Experimental results on both synthetic and real data are given that show the developed methods are an advancement to detection in hyperspectral imagery. Finally, a discussion of these research efforts and future research directions is presented.
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In the series University of Florida Digital Collections.
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by Taylor C Glenn.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: GADER,PAUL D.
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Co-adviser: WILSON,JOSEPH N.

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IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,FEBR UARY2012331Multi-ModalChangeDetection,Applicationto theDetectionofFloodedAreas:Outcomeofthe 2009–2010DataFusionContestNathanLongbotham ,StudentMember,IEEE ,FabioPaci ci ,Member,IEEE ,TaylorGlenn ,StudentMember,IEEE AlinaZare ,Member,IEEE ,MicheleVolpi ,StudentMember,IEEE DevisTuia ,Member,IEEE EmmanuelChristophe ,Member,IEEE ,JulienMichel ,AssociateMember,IEEE ,JordiInglada ,Member,IEEE JocelynChanussot ,Fellow,IEEE ,andQianDu ,SeniorMember,IEEEAbstract— The2009–2010DataFusionContestorganizedby theDataFusionTechnicalCommi tteeoftheIEEEGeoscienceand RemoteSensingSocietywasf ocusedonthedetectionof ooded areasusingmulti-temporalandmulti-modalimages.Bothhigh spatialresolutionopticalandsyn theticapertureradardatawere provided.Thegoalwasnotonlytoidentifythebestalgorithms(in termsofaccuracy),butalsotoinvestigatethefurtherimprovementderivedfromdecisionfusion. Thispaperpresentsthefouraw ardedalgorithmsandtheconclusionsofthecontest,investigatingbothsupervisedandunsupervisedmethodsandtheuseofmulti-modaldatafor ooddetection. Interestingly,asimpleunsupervi sedchangedetectionmethodprovidedsimilaraccuracyassuperv isedapproaches,andadigitalelevationmodel-basedpredictivemethodyieldedacomparableprojectedchangedetectionmapwithoutusingpost-eventdata. IndexTerms— Changedetection,datafusion,decisionfusion, ooddetection,highspatialresolut ion,optical,syntheticaperture radar.ManuscriptreceivedSeptember12,201 1;revisedOctober24,2011;accepted November28,2011.DateofpublicationJanuary31,2012;dateofcurrentversionFebruary29,2012.ThisworkwassupportedinpartbytheSwissNational ScienceFoundationunderGrant200021-126505andGrantPZ00P2-136827. N.LongbothamiswiththeAerospaceEn gineeringSciencesDepartment, UniversityofColorado,Boulder,CO80302USA. F.Paci ciiswithDigitalGlobeInc.,Longmont,CO80504USA. T.GlenniswiththeDepartmentofC omputerandInformationScience andEngineering,UniversityofFlorida,Gainesville,FL32601USA(e-mail: tcg@cise.u .edu). A.ZareiswithDepartmentofElectricalandComputerEngineering,UniversityofMissouri,Columbia,MO65211USA(e-mail:zarea@missouri.edu). M.VolpiiswithInstituteofGeomaticsandAnalysisofRisk,Universityof Lausanne,1015Lausanne,Switzerland(e-mail:michele.volpi@unil.ch). D.TuiaiswiththeLASIGLaboratory,EPFL,1015Lausanne,Switzerland (e-mail:devis.tuia@ep .ch). E.ChristophewaswiththeCentreforRemoteImaging,SensingandProcessing(CRISP),NationalUniversityofSingapore,Singapore119260,andis nowwithGoogleInc.,MountainView,CA94043USA(e-mail:emmanuel. christophe@gmail.com). J.MichelwaswithCommunicationsetSystmes,31506Toulouse,France, andisnowwiththeCentreNationald’tudesSpatiales,ToulouseCedex09, 34401France(e-mail:julien.michel@cnes.fr). J.IngladaiswiththeCentreNationald’tudesSpatiales,ToulouseCedex09, 34401France(e-mail:jordi. inglada@cesbio.cnes.fr). J.ChanussotiswithGIPSA-Lab,GrenobleInstituteofTechnology,Grenoble, 38402France(correspondingauthor,e-mail:jocelyn.chanussot@gipsa-lab. grenoble-inp.fr). Q.DuiswiththeDepartmentofElect ricalandComputerEngineering,MississippiStateUniversity,MississippiState,MS39762USA(e-mail:du@ece. msstate.edu). Colorversionsofoneormoreofthe guresinthispaperareavailableonline athttp://ieeexplore.ieee.org. Dig italObjectIdenti er10.1109/JSTARS.2011.2179638I.INTRODUCTIONTHEDataFusionTechnical Committee(DFTC)ofthe IEEEGeoscienceandRem oteSensingSocietyserves asaglobal,multi-d isciplinarynetworkforgeospatialdata fusion,connectin gpeopleandresources.Itaimsateducating studentsandprofe ssionals,andpromotingthebestpractices indatafusionapp lications[1].TheDataFusionContesthas beenannuallyor ganizedbytheDFTCsince2006[2]–[4].It isopennotonly toIEEEmembers,buttoeveryone,withthe aimofevaluat ingexistingmethodologiesattheresearchor operational leveltosolveremotesensingproblemsusingdata fromdiffer entsources. In2009–2010 ,theaimoftheContestwastoperformchange detectionu singmulti-temporalandmu lti-modaldata.Twopairs ofdataset swereavailableoverGloucester,UK,beforeandafter a oodeve ntoccurredinNovember2000.Theclass“change” wasther iverandclass“nochange”wastheareasthatstayed dry.Th eopticalandsyntheticapertureradar(SAR)imageswere provid edbytheCentreNationald’tudesSpatiales(CNES). Thepa rticipantswereallowedtouseasupervisedoranunsuperv isedmethodwithallthedata,theopticaldataonly,orthe SARd ataonly.Accordingly,six categorieswereconsideredto acc ountfordifferentsubmissions: Sup ervised—Alldata Supe rvised—Opticaldata Super vised—SARdata Unsup ervised—Alldata Unsupe rvised—Opticaldata Unsuper vised—SARdata Asforth epreviouseditionsoftheContest,thegroundtruth usedto assesstheresultswasnotpr ovidedtotheparticipants. Howeve r,about60,000samplesweremadeavailablefor train ingthesupervisedmethods. Thesin gleresultsamongallcateg orieswerevalidatedand ranke dusingtheestimatedCohen’sKappastatistic(or K coef cie nt).Then,thebest5individualalgorithmsamongallsubmiss ionswereusedtoperformdecisionfusionwithmajority vot ing. Thef ourwinningalgorithmsarebrie ydescribedasfollows: 1)Supe rvisedmethodwithalldata:asupervisedneuralnetwork approachwasproposedtoexploitboththeoptical and SARimagestocreateachangedetectionmap;the1939-1404/$31.002012IEEE

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332IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,F EBRUARY2012 oodedandnon oodedmapregionsweresuccessively homogenizedusingaminimummappingunitmorphologicaloperator. 2)Supervisedmethodwithopticaldataonly:itusedspatial contextualfeaturesextracted fromthenear-infrared(NIR) bandofthepost-eventimage(wherethe oodedareas weremoredistinguishable);t hemorphologicalfeaturesextractedwerestackedtoformat en-dimensionalmulti-temporaldatasetandfedasinputstoasupportvectormachine (SVM)classi er. 3)Unsupervisedmethodwithopticaldataonly:thistechniquealsoexploitedtherelativelyhighabsorptionof waterintheNIRbandtoguideanunsupervisedclustering algorithm. 4)Apredictivemodel:apredictivemodelwasusedtoproject post oodchangebyexploitingad igitalelevationmodel (DEM).Thismodelmaybehelpfultorescueplanning duringthevery rsthoursaftera oodoccurswhenthe post-eventdatamaystillbeunavailable. Thispaperisorganizedasfollow s.First,ashortliteraturereviewisillustratedinSectionII, whereasthedatasetusedforthe ContestisdescribedinSectionI II.Thefourwinningalgorithms arepresentedinSectionsIVtoVII.Finally,decisionfusionis discussedinSectionVIII,aswellasconclusionsandperspectivesdrawnbythisContest. II.RELATEDWORKInthepastfewyears,therehasbeenagrowinginterestinthe developmentofchangedetectiontechniquesfortheanalysisof multi-temporalimagery.Thisintereststemsfromthewiderange ofapplicationsinwhichchangedetectionmethodscanbeused, suchasurbanandenvironmentalm onitoring,agriculturaland forestsurveys,and,aswiththisyear’sContest,disastermanagement.Supervisedandunsupervisedapproachesproposedin theliteratureareherebrie yreviewed. A.SupervisedChangeDetection Supervisedchangedetectionaimsatde ningclassi cation rulesbymodelinglabeledexamplesprovidedbytheuser, accountingforthedifferentclassesofland-covertransitions. Whendealingwithhightoveryhighspatialresolution(VHR) imagery(eitheropticalorSARdatasets),thecomprehensive labeledsetcanbeextractedby photo-interpretationofthe multi-temporalimages. Twomainsupervisedapproachesarefoundintheliterature: postclassi cationcomparisons[5],whereclassi cationisperformedindependentlyateachtimeinstant,andasuccessive comparisonde nesachangemap;andmulti-dateclassi cation [6],wheremulti-temporalinformationisconsideredsimultaneouslyforclassi cation. Inthe rstcase,alogicalcomparisonismostoftenperformed ontheclassi cationoutcomesateachtimeinstant[7].ThisapproachmaynotbeoptimalforVHRimagery,becauseitmay accumulatesingleimageclassi cationerrors.Asaresult,postclassi cationcomparisongenerally providesnoisymaps,and changesinviewingacquisitioncanmakethe nalmapdif cult tobeinterpreted.Toavoidthe seerrors,anapproachbasedon post-classi cationmaskinghasbeenproposedin[8],wherea binaryneuralnetworkisusedtoremovespuriousdetections,or in[9],wherechangevectoranalysis[10],[11]isusedforthe samepurpose. Onthecontrary,multi-dateclassi cationconvertschange detectionintoamulti-temporalclassi cationproblem.The twoimagesarecombinedintoamulti-temporalrepresentation (throughvectorstackingormulti -variatedifference)before analysisandamodelisde nedusingthismulti-temporal featurespace.Forexample,sup ervisedmulti-temporalclassi cationwasimplementedusingSVMin[12],whilein[13] Camps-Valls etal. de nedasetofspeci ckernelfunctions fortheproblemofmulti-temporalclassi cation.Finally,recentworksstudiedtherelationshipbetweentheef ciencyof changedetectorsandscarcenes soflabeledinformationusing semi-supervisedmethods[14],[15]. B.UnsupervisedChangeDetection Muchoftherecentworkonunsupervisedmulti-andhyperspectralimageanalysishasattemptedtointegratespatialinformationtoimprovealgorith mperformance[16].In[17],a fuzzyclusteringapproachthatin corporatesspatialinformation forsegmentationofremotesensingdatasetswasinvestigated. Ahierarchicalclusteringalgorithmthatexploitsspatialinformationforsegmentationwasdes cribedin[18].In[19],spatial-spectralclusteringu singGaussianMarkovrandom elds forscenesegmentationandanomalydetectionwaspresented. Similarly,integrationofspatialinformationintospectralunmixingofmulti-andhyper-spectraldatawasinvestigatedin [20]–[22]. Theintegrationofdomainspeci cknowledgehasbeenused inanumberofsystemsfordetectionofvariousmaterialsor objects.Thisknowledgemaybebasedonthesensingplatform,subjectsbeingsensed,orphysicalphenomenaaffecting thesystem.Suchtechniquesarediscussedin[23]wherethe expectedspectralshapeofvege tationinthelongwaveinfrared wasusedtoguideanunsupervisedalgorithmforthedetection ofvegetationinremotelysenseddata.In[24],bridgesare detectedusingruleswhichleveragedtheknownspatialarrangementsofbridgepixelswithrespecttowaterandconcrete inconjunctionwithclassi cationalgorithms. InthecontextofSARimageanalysis,theproblemofunsupervisedchangedetectionhasbeenaddressedwithlessemphasiswithrespecttoopticalim agery.Recentstudieshaveinvestigateddifferentaspects,in cludingimagede-specklingand optimalthresholdselection[25], [26],whereasstatisticaland fuzzyapproacheshavebeendiscussedin[27]. III.DATASETSThedatausedintheContestincludedtwoobservations,one beforeandoneafterthe oodevent,fromtwoseparatesatellite instruments:theopticalSystmeProbatoired’Observationde laTerre(SPOT)andtheSARinstrumentaboardtheEuropean RemoteSensing1(ERS-1). TheSPOTsatellite,operatedbytheSpotImagecompany,is afourbandopticalplatform.T hreeofthechannelscoverthe spectralrange0.50 to0.89 andthefourthisapanchromaticbandwithaspectralrangefrom0.50 to0.73 .The

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LONGBOTHAM etal. :MULTI-MODALCHANGEDETECTION,APPLICATIONTOTHEDETECTIONOFFLOODEDAREAS333TABLEI SPECTRALBANDSOFTHEOPTICALSPOTSATELLITE. ONLYTHEMULTI-SPECTRALBANDSWEREUSEDASINPUTFORTHECHANGEDETECTIONPROBLEM TABLEII ERSSARSATELLITESPECIFICATIONS spatialresolutionisnominally20matnadirforthemulti-spectralbandsand10matnadirforthepanchromatic.Thesatellite isinasun-synchronouspolarorbitandhasasteerablestrip-selectionmirrorthatcancollectimageryupto27 off-nadir.This pointingcapabilitygivesthesatellitearevisitrateof4–5days. AsummaryoftheSPOTspeci cationsisprovidedinTableI. TheERSSARsatellite,launchedbytheEuropeanSpace Agency,isaC-bandplatform.Thesatelliteisinasun-synchronouspolarorbitwitharevisittimeof35days.TheSAR instrumentcanoperateintwomodes:imageandwave.The wavemodeprovidesaspatialresolutionof10mwithaswath widthof5km.Theimagemodeprovidesalargerareaobservationwithaswathwidthof100kmbutreducesthespatial resolutionto30m.TheimageryusedinthisContestwas collectedinwavemode.Itsspeci cationsaresummarizedin TableII. TheSPOTimagesdistributedduringtheContestcontained onlythemulti-spectralbands(i.e.,nopanchromaticinformation wasavailabletotheparticipants ).Theseimageswereacquired onSeptember1999andNovember2000,whiletheERS-1imageswereacquiredonOctober1999andNovember2000.The before-andafter oodimagesfrombothplatformsareshown inFig.1,whereasthegroundsurveyusedtovalidatethechange detectionresultsuploadedbyth eparticipantsisshowninFig.2. IV.SUPERVISEDCHANGEDETECTION—ALLDATAThemethodologydiscussedinthissectionwasverysimilartothewinningapproachesin previousyears’Contests.The methodsfrom2007[3]and2008[4]sharethesameneuralnetworktopologyastheonediscussedhere.However,theinputand outputofthenetworkwerediffer entasrequiredbythedataand changedetectionproblem,butth einternalarchitecturewassimilar.Also,boththelearningandp runingmethodswereidentical. Fig.1.ThreebandopticalandsingleamplitudeSARdata,collectedbefore (left)andafter(right)the oodevent,providedasinputtothechangedetection problemfromtheSPOT(a)and(b),andERS(c)and(d)satellites. Fig..2.GroundsurveyusedfortheContest.Neuralnetworksarenonlinearstatisticalmodelscapableof modelingcomplexsystems[28].Theyarecomposedofasetof

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334IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,F EBRUARY2012interconnectednodes,calledneurons,eachconnectedbyanetworkofweightedinputs.Eachne uronreceivesinputfromeither externalsourcesorneighborin gnodes.Itthenusesaninternal mechanism(networkandactivationfunctions)tocomputean outputvaluefromtheweightedinputsthatisthenpropagatedon tothenodesitisconnectedto[2 9].Thenodes,inturn,calculate anoutputvaluefromtheirweightedinputsandinternalmechanisms.Afterthisprocessiscompletethroughoutthenetwork, theweightingsateachneuronareoptimizedbyminimizingan errorfunctionmeasuringthel earnedaccuracyofthenetwork againstknowninput/outputvalues. Thisarrangementprovidesneuralnetworksthecapabilityto modelhighlynonlinearsystemsinsituationswherethestatisticaldistributionofthedataispoorlyunderstood[30].Thisis oftenthecaseinchangedetectionwherealargerangeofdata frommultiplesensingplatforms,aswellasdataderivedfrom thesesources,canbeavailable. Neuralnetworkshavebeenusedforland-coverchangedetectionfordecades.Earlyattemp tsfocusedonthecapabilityof basicneuralnetworkstoaccura telypredictchangeinmultiple remotesensingapplications[31] –[33].Morerecentstudieshave focusedonalternativenetworktypes[34],[35]andonneural networksaspartofchangedetectionsystems[36]. Sinceneuralnetworksoperateontheprincipleofweighted inputs,itisadvantageoustonor malizetheinputdataspace.This shouldresultinafastertrainingprocessaswellasincreasesaccuracyfortheresultingnetwork[37].Inthepresentedmethod, thedatawasnormalizedintherange Asdiscussedabove,theinputdataconsistedofabeforeand afterimagefrombothathree-bandopticalandsingle-bandSAR satellite.Fromthisinformation,andthedesiredchangecases, theinputandoutputnodesofthenetworkwere xed:eightinput nodescorrespondingtotheeightbandsofbeforeandafterobservationdata,andtwooutput nodescorrespondingtothetwo classi cationcasesofinterests( oodedandnon ooded). Thedesignofthehiddenlayeristhesubjectofalargediscussionincurrentliterature;howev er,itisgenerallyacknowledged thatnomorethantwolayersareneededandthatlayerdepth canbeprunedtooptimizethenetworkcomplexity[38].Forthis Contest,computationalpowerwasnotaconstraint.Therefore, thenetworkwasdesignedatthemorecomplexendofliterature recommendationsandprunedtoa noptimizedtopology.Theinternalstructure,beforepruning,consistedoftwointernallayers of40nodeseach.Thenetworkwastrainedwiththescaledconjugategradientmethodandusedmagnitude-basedpruningto optimizethetopology[38]. Thepixel-levelclassi cationproducesmapsthatdirectlyre ectthespectral-spatialvariabi lityoftheimage,producinga sortof“saltandpepper”noisedrivenbyvalidspectralinformation.Thiscanbealimitationtoclassi cationaccuracywhen comparedagainstgroundsurveyr egionsthatareoftenconsidereduniformbyhumanexperts.Inthecurrentstudy,thiscanbe seeninsmallpatchesofearthexposedinthemiddleofa ooded eldorregionsofstandingwaterinanotherwisenon ooded area. Thiseffectwasaddressedthroughtheuseofmorphological postprocessing[39]toimplementaminimummappingunit andhomogenizestrandedpixelsinthe oodedandnon oodedTABLEIII CONFUSIONMATRIXANDKCOEFFICIENTFORTHESUPERVISEDCHANGEDETECTION—ALLDATA regions.Theimplementationusedasieveandclumpprocess to lteroutgroupsofpixelsbelowaspeci edsize.Clusters ofpixelsbelowagiven ltersizewereremovedusingablob process.Theremovedpixelswerethenre lledusingmorphologicaldilationfromthesurroundingclasses[40]. Inthecaseofchangedetection,removingregionsofagiven classbelowacertainpixelnumberthresholdand llingthearea withtheotherclasswasarelativelysimpleprocess.Forthe Contest,the ltersizechosenwas51pixels.Thisprovidedwell segmentedchangeregionswithoutoversimplifyingthespatial structureofthe oodedregion.Thisstepgeneratedthe nalclassi cationmapsubmittedforthecompetition,whichproduceda coef cientof0.703.TheconfusionmatrixinTableIIIfurther detailsthedistributionofaccura telypredictedchangedetection pixels. V.SUPERVISEDCHANGEDETECTION—OPTICALDATAThemethodologydiscussedinthissectionconsidersthe useofcascadecontextualfeatur esinconjunctionwithsupport vectormachines.Contrarilytothemodelpresentedinthe previoussection,spatialregularizationisintrinsictothemodel, sinceitisintroducedbytheuseofcontextualinformationin theinputinformation.Theuseofcontextualinformationisnot fullyexploitedinthesupervised changedetectionliterature, althoughthebene tsofincludingsuchasv ariablesareclearly demonstratedforclassi cationtasks[41]–[43].In[44],theadvantagesofdeployingreconstructionmorphologicaloperators inthechangevectoranalysis frameworkhavebeendemonstrated.In[45]acontextualp arcel-basedmulti -scaleapproach tounsupervisedchangedetect ionispresented.Finally,in [46]texturalandmorphological ltersarestudiedextensively andtheirimportanceforsuccessfulVHRsupervisedchange detectionisanalyzed. Mathematicalmorphology[40] isacollectionofcontextual ltersbasedonsettheory.Theye xtractlocalcharacteristics ofgray-levelimages,and,atdiff erentscales,provideinformationaboutshapeandstructureoftheobjectsinthescene[47], [48].Morphologicalreconstruction ltersareconsideredinthis studytomaintainthespatialinformationnecessarytodetectthe oodinglimitsprecisely.These lters—openingandclosingby reconstruction—preserveobject s’shapes,whilereturningrespectivelylocalminimaormaxima,andsmoothingthehigh localvariancesthataffectVHRimages. Fourfeatureshavebeenused.The rsttwohavebeenextractedusingopeningbyreconstructionontheNIRbandat timeinstant ,wherethe oodedareawasthemostvisible. Twosizesofthestructuringelements(themovingwindowsdetectinglocalvalleys/peaks)havebeenconsideredtoaccountfor

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LONGBOTHAM etal. :MULTI-MODALCHANGEDETECTION,APPLICATIONTOTHEDETECTIONOFFLOODEDAREAS335 Fig.3.Contextualfeaturesextracted fromthenearinfraredbandattime (a)Openingbyreconstructionwithcircularstructuringelementofradius40 pixels.(b)Openingbyreconstructionwithcircularstructuringelementofradius 100pixels.(c)Closingbyreconstructionwithcircularstructuringelementof radius60pixelsusingerosionofpanel(a)asstartingmarker.(d)Closingby reconstructionwithcircularstructuringelementofradius90pixelsusingerosion ofpanel(a)asstartingmarker.differentscalesofsmoothing:circularelementsof40and100 pixelsinradius.Thechoiceofwideelementsresultedfromthe largeextentofthe oodedarea.Resultsofthis ltering, ,are reportedinFigs.3(a)and3(b),respectively.Inbothcases,light areassaturate,clearlyde ningtheshapeofthe oodedarea. However,theopeningbyreconstructionoperatorsdonot lteroutsmallandmediumsi zedcropsareasshowinglow NIRvalues(resultingindarko bjects).Inordertoeliminate theseundesiredlow-valuedpat ches,weappliedclosingby reconstruction(i.e.theinverseprocess)totheopeningby reconstructionfeatureofFig.3(a):bydoingso,theopening imageisdilatedandthesmallpatchesareabsorbedbythe high-valuedneighboringpatches.Thankstotheshape-preservingcharacterof thereconstruction lter,thecentral ooded arearemainsgeometricallyunchanged,althoughtheaverage valueofitssegmentsincreases(tothemaximalvalueencounteredinthe oodedareareconstructedinFig.3(a)).TheTABLEIV CONFUSIONMATRIXANDKCOEFFICIENTFORTHESUPERVISEDCHANGEDETECTION—OPTICALDATA resultsofthismo rphologicalcascade lter arereportedin Figs.3(d)and3(e)forcircularstructuringelementsofradius 60and90pixelsrespectively. Successively,theoriginalmult i-temporalbandsandtheextractedcontextualfeatureswe restackedina10-dimensional vector Eachfeaturehasbeenconvertedtostandardscorespriorto stacking. Fromthegroundtruthprovided,4000pixels(2000for changesrelatedtowaterand2000correspondingtoun ooded areas)havebeenextractedforthe trainingphase.Theassociated multi-temporalpixelwasthenusedtotrainaone-against-all SVMimplementedusingtheTorch3library[49].Aradial basisfunctionkernelhasbeenused.Modelselectionhasbeen performedbygridsearchto ndSVMoptimalparameters and The nalsubmissionproduceda coef cientof0.650.The confusionmatrixinTableIVfurtherdetailsthedistributionof accuratelypredictedchangedetectionpixels. VI.UNSUPERVISEDCHANGEDETECTION—OPTICALDATAClusteringtechniqueshavebeenwidelyusedinthemultispectralandhyper-spectrallitera ture[50].Forexample,in[51], afuzzyclusteringapproachwasusedtoidentifyland-cover typesinLandsat,QuickBird ,andMODISdata.Theapproach describedinthissectionusedclusteringtechniquestofusethe informationfromthethreespectralbandsandproducelabelsfor eachpixelbycontentgrouping.Su ccessively,contextualinformationwasusedtoproduceconsistentlabelsoverlargerareas. Finally,physics-basedruleswereusedtoselectlabeledgroups containing oodwater. The rststepwastopre-processthedatatoremovesome anomalies.Pixelvaluesforeachchannelwereintegersinthe range[0,127].Afewanomalouspixelswerefoundwitha 128 valueacrossbands,andsuchpixelsweresimplyclippedinto standardboundsbysettingtheirvalueto0. Theopticaldatawasthenclusteredintotengroupsusingthe fuzzyC-meanclusteringalgori thmwithaEuclideandistance measure.Clusteringwasperformedusingonlythethreeopticalbandvaluesforeachpixel(i.e.,nospatialinformationwas used).Pixelswerethenassignedthelabeloftheclustertowhich theyhadhighestmembership.T heclusteringwasperformed usingalloftheapproximately9.8millionpixelsintheimage, andgeneratedrobustresultstoaqualitativevisualinspection. UponcloseinspectionoftheclusteringresultinFig.4,itis possibletonotethatthelargecentral ooded/riversectionwas assignedtocluster5(cyan).Manyareasoflandcorresponding

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336IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,F EBRUARY2012 Fig.4.Pixelsassignedtocolor/cl assbyhighestFuzzyC-Meanscluster membership. Fig.5.(a)Non-smoothedclusterlabels(detailviewofFig.4).(b)Clusterlabelsaftersmoothingbyvotingin5 5window.toagricultural eldswereassignedtoclusters6and7(yellow andlightorange),whereasar easinshadowfromcloudcover wereassignedtocluster1(darkblue). Thenextstepintegratedspatialinformationtothecluster labelsbyapplyinganeighborhood-votingprocedure.Thisstep wasperformedtoremoveanysmall,discontinuous,ornoisy clusters.Clusterlabelsforeachpixelweresmoothedusing voting.Thevotingm ethodreassignedeachpixelthevalue ofthemostfrequentlyoccurringlabelwithina5 5pixel neighborhoodcenteredonthepixelofinterest.Fig.5showsa comparisonoftheoriginalclusteringresultwiththesmoothed labelsonasubsectionofthefullimage.Itispossibletonote thatmanyofthesmall(butirrelevantforthetaskathand) detailshavebeenremoved. The nalstepofthealgorithmaimedatconvertingthe unsupervisedclusteringresultsintoausable“ ooddetection Fig.6.Resultofapplyingdistancetransformtothe“water”pixels. TABLEV CONFUSIONMATRIXANDKCOEFFICIENTORTHEUNSUPERVISEDCHANGEDETECTION—OPTICALDATA map”throughtheuseofdomain-knowledgeofthesensorand problem.Speci cally,the rststeptowards ooddetection wastoapplyclusteringto ndpixelswhichwerelikelytobe water.Thisclusterwasidenti edusingthefactthatwateris moreabsorptiveintheNIRwavelengths.Therefore,thecluster havingthelowestaverageNIRintensitywasthenconsidered astheoneinwhichpixelswer emostlikelytobewater. ThelowNIRspectralruleidenti edthemajorityofthewater oodedregions,butityieldedmanyfalsepositivesthatwerefar awayfromthecentral oodedregion.Toremovethesefalsepositives,onlypixelsclosetothelargestcentralwaterregionswere retained.Theseregionsweremostlyconnected,butrequired somefurtherprocessingto ll-intheholes.Tothisend,adistancetransformwasappliedtotheimage.Thetransformlabeled eachnon-waterpixelbyitsdistancetotheclosestwatersample. Then,clusterswithinadistanceoflessthan15pixelswereretainedalongwiththeoriginalwatersamples.Fig.6showsthe resultsofthedistancetransform.Toremovethefalsepositives, onlythetwolargestconnectedco mponentswereretained.This rulemaintainedthe oodedareasanddisregardedthewaterand falsepositiveareasfarfromtheriver. The nalsubmissionproduceda coef cientof0.688.The confusionmatrixinTableVfurtherdetailsthedistributionof accuratelypredictedchangedetectionpixels.

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LONGBOTHAM etal. :MULTI-MODALCHANGEDETECTION,APPLICATIONTOTHEDETECTIONOFFLOODEDAREAS337VII.APREDICTIVEMETHODFORPROJECTEDPOST-FLOODCHANGEDETECTIONWhenadisasterstrikes,the rstfewhoursarecriticaltocoordinatetheresponse.Import antdecisionshavetobemaderegardingwheretosendthe rstavailablerescueteams,howthe transportandcommunicationn etworkshavebeenimpacted,and whethersomepopulationarethreatenbydisasterdevelopments, suchasnew ooding,tsunami,orearthquakereplicas.Unfortunately,inthe rstfewhoursofthedisastermitigationperiod, rescueresourcesarescarce.Giventhatlivesareatstake,theimportanceofallocatingtheseresourcesef cientlyisparamount. Thesedecisionsrequiretheclear estviewofthesituation,with themostup-to-dateinformationavailable.Yet,localmonitoring oftheplacewherethedisasterst ruckisoftenaffectedand,dependingonthetopographyofthearea,itmaybedif c ulttobear newmonitoringteamsanddevicesfromneighboringplaces. Inthissituation,satellitesareinvaluabletools:theiroperation isnotaffectedbythelocalcrisis,theyprovideglobalcoverage andareabletoacquireseverallargeareasatatime.However, satellitesarenotabletoproviderealtimeimagery.Asanexample,weatherisafactorthatmaydelayth eacquisitionofopticalimagery.Thisisparticularlytrueinthecaseof ooding, wherecloudcoverisindicativeo ftheevent.Further,manyareas intheworldhaveanaveragecloudcoveragehigherthan80%, whichincreasestheaveragede laybeforehigh-qualityspaceborneopticalimagesareavaila ble.InthecaseoftheInternationalCharter,thedelaybetwee nthedisasterandthe rstimagesisatleast1or2days,andsometimesevenlonger[52], whereastheCOSMO-SkyMedconstellationshas12hourrevisit timeunderemergencyconditions.Therefore,itmaybeimportanttostartevaluatingtheareaoftheeventevenbeforethe rst post-eventimagecomes.Atthisstage,anyinformationisuseful. Manysourcesareavailable concerningpre-disasterimagery, formostplaces;GoogleEarthor BingMapsenabletoexamine placesforabetterideaaboutwh eretherisksmaybeconcentrated(usuallyinthesettlements),Landsatimageryisavailableglobally,andarchivedimagesoftheareaarecommonly availableindataprovider’scatalogs.For oodevents,theseimagesandmapsourcesc anbecomplementedwithanothersource ofinformation:theelevationmap.InitiativesliketheShuttle RadarTopographyMission(S RTM)ortheAdvancedSpaceborneThermalEmissionandRe ectionRadiometer(ASTER) provideareliableelevationmapforanyplacebetweenlatitude 60Northandlatitude60SouthforSRTMandbetweenlatitude 83Northandlati tude83SouthforASTER.Thisprovidescoverageformostoftheworldpopulation. Ontheotherhand,thedataava ilablealsovariessigni cantly innatureandquality.Soitiscriticaltohave exibletoolstodeal withunexpectedsituations.Forthisstudy,theOrfeoToolbox (OTB)[53]wasused.OTBisanopen-sourceimageprocessing librarywhic hhandlesimagegeometryandmapprojection,accessavarietyofdata,suchasdigitalelevationmodels,andstandardimageprocessingtechniquesina exibleandintegrated way.Thistoolprovidesadvancedalgorithmsaswellaslow levelaccesstodata. InthecaseoftheContest,severalfactsmadeiteasiertouse theinfor mationderivedfromtheDEM.Infact,theterrainwas relatively at,withoutsharpmountains,andtheobjectivewas tomeasurethe oodextensionafewhoursafterithappened. Fig.7.SRTMDEMofthe oodedarea. TABLEVI CONFUSIONMATRIXANDKCOEFFICIENTFORTHEPREDICTIVEMETHOD Themetadataincludedintheim agesiscriticaltounderstand thecontext.However,thebefore oodSPOTimagehadthe wrongmetadataprovidingawrong geolocation.Unfortunately, thiskindofsituationiscommoninreal-casescenariosand shouldbeaccountedfor.Itisimportanttobecautiousofthis informationatanytimeanditisanotherreasonwhy exible toolsarecritical. Fig.7showstheSRTMDEMextractedfromthemetadataof thepost-eventSPOTimage.Usinga oodlevelof13mabove sealevel,the ooddetectionaccuracywas0.627,whichwas areasonablevaluecomparedtootherresultsinthesedif cult conditions.Theconfusionmatri xinTableVIfurtherdetailsthe distributionofaccuratelypred ictedchangedetectionpixels. VIII.DISCUSSIONANDCONCLUSIONAttheendoftheContest,morethan200usersdownloaded thedatasetandmorethan350di fferentchangedetectionmaps havebeenuploadedtothesystemtoberanked.Ingeneral,the Contestturnedouttobeverytoughforthemethodsinallthe categories,asevidencedbyamaximumaccuracyof0.703.The relativelylowaccuracymaybeduetothepresenceofpixellevel regionmixingissue,inwhichanisolatedpatchofdryareasurroundedbywatermaybeconsideredas oodbyahumanoperatorthatde nestheextensionofthe oodedarea.Further,the oodedareachangedsigni cantlybetweenthetwosensorobservations(about4daysapart ).Thismayhavecreatedfundamentallydifferentinformation,increasingthedif cultytodeal withthetemporalshiftbetweentheopticalandSARdatasets. Thetop-rankedchangedetectionmapssubmittedforeach categoryareshowninFig.8,whe reastherelativebestaccuraciesarereportedinTableVI I.Asshown,themethodsthat

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338IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,F EBRUARY2012 Fig.8.Top-rankedmapsthatweresubmittedforeachcategory:(a)Supervised—Alldata,(b)Supervised—Optic aldata,(c)Supervised—SARdata, (d)Unsupervised—Alldata,(e)Unsupervised—Opticaldata,(f)Unsupervised—SARdata.exploitedSARimageryalonedidnotprovideaccuratechange detectionmaps.Forthe“Super vised—SARdata”category,theTABLEVII COEFFICIENTOFTHEBESTRESULTFOREACHCATEGORY Fig.9.Decisionfusionresultproducedusingmajorityvotingbetweenthebest 5individualresultsamongallsubmissions. TABLEVIII CONFUSIONMATRIXANDKCOEFFICIENTFORTHEDECISIONFUSIONRESULT methodin[54]wasexploited.Forthe“Unsupervised—SAR data”categoryapatch-basedsimilaritybetweenthetwoSAR imageswascomputedateachpixelfordifferentpatchsizes. Thedecisionfusionofthebest5individualresultsamong allsubmissionswasachievedusingmajorityvoting(shownin Fig.9).TableVIIIpresentsthecorresponding nalconfusion matrix.The coef cientwas0.706.Eventhoughthe nal scoreislessthan1%higherthanthebestalgorithm,majority votingstilloutperformedalltheotherresultssubmittedonthis dataset.Further,thestatisticalsigni canceofthechangedetectionmapswasevaluatedwiththeMcNemartestandalltheresultswerestatisticallysigni cant(tothe95%con dencelevel). Failureorsuccessofachangede tectionalgorithmcannotbe analyzedonlywithaconfusionmatrix,asitisimportanttounderstandthecontextoftheapplication.Forexample,missed alarmsmaybemoreimportantth anafalsealarmincatastrophic scenariosasitisbettertocheckanon-destroyedbuildingthan nottovisitadestroyedone.Forthe ooddetectionapplication discussedinthispaper,thenumberoftruepositivesandtrue negativesoftheconfusionmatri cesreportedinTablesIII–VI

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LONGBOTHAM etal. :MULTI-MODALCHANGEDETECTION,APPLICATIONTOTHEDETECTIONOFFLOODEDAREAS339wereapproximatelythesameforallthewinningmethods.The maindifferencescanbefoundintheoff-diagonalterms.The “Supervised—Opticaldata”and“Unsupervised—Opticaldata” entriesshowedtheoppositebehavior,the rsthavingabout twicefalsenegativesthanfalsepositives,andthesecondhaving abouttwicefalsepositivescomparedtothefalsenegatives.The submissions“Supervised—Alld ata”and“Unsupervised—All data”showedmorebalancevaluesalongtheoff-diagonalterms. Thiswasalsothecaseofthedecisionfusionconfusionmatrix. Alltheproposedmethodsreliedondifferentmodelingsolutionsandshoweddifferentwaystoapproachthechangedetectionproblem.However,itisimportanttoemphasizethatphysicalproperties(suchastheadmis sibledistancefromwaterand theelevationinformationextractedfromtheDEM)havebeen successfullyusedinadditiontotr aditionalapproachesbasedon patternrecognition,imageproce ssingtechniques,andcontextualinformation.Theseappro achesopennewresearchpossibilitiesfordatafusion,whereintegrationofthephysicsofthe problemmayplayanimportantroleforthesuccessofthemodelingtaskinvolvingcomplexphenomena. Further,itisimportanttohighlightthesmalldifferenceinaccuracyachievedbetweensuperv isedandunsupervisedmethods. Thisisremarkableconsideringthequickresponsethatmaybe necessaryindisasterscenariosw ithouttrainingsamplesavailable.Forthecaseofa oodevent,thisstudyalsoshowedthat evenwithoutup-to-datedata,reasonableresultscanbeobtained withinminutesofthe oodcrisis(andevenbeforethe ood eventitself)withaDEM-based predictivemodel.Ofcourse, combiningthisearlyinformationwithpost-eventimagesmay furtherimprovetheresults.Additionally,theelevationmapcan be,forinstance,usedasafurtherinputfortheclassi cation phase. Toconclude,inarelativelyr ecentpaper[55],Wilkinson showedsatelliteimageclassi cationresultshavenotbeen improvingoverthepast20years.Thisiscorroboratedbythe resultsofthisContest,wherethe variousmachinelearningtechniqueshaveshownlittlediffer encetothegeneralclassi cation problem.Speci cally,oldermethods,suchasthesupervised neuralnetworkandtheunsupervisedC-mean,providedthe samelevelofaccuracyasdidnewermethods.Asamatterof fact,theexactsameneuralnetworkapproachprovidedthebest individualperformanceamongallsubmissionsintheprevious 2007and2008Contests[3],[4].Theseconclusionssuggest thatresearchshouldbedirected toinvestigatingnewandmore powerfulinputfeatures(asanexample,multi-temporalaswell asmulti-angularinformationcanbeexploited)tobefedintothe variousmachinelearningschemes,ortoabetterunderstanding ofthephysicalbehavioroftheEar thsurfacebeinginvestigated. REFERENCES [1]2011,IEEEGRSSDataFusionTechnicalCommittee.[Online].Available:http://www.grss-ieee .org/community/technical-committees/datafusion/ [2]L.Alparone,L.Wald,J.Chanussot,C.Thomas,P.Gamba,andL. M.Bruce,“Comparisonofpansharpeningalgorithms:Outcomeofthe 2006GRS-Sdatafusioncontest,” IEEETrans.Geosci.RemoteSens. vol.45,no.10,pp.3012–3021,Oct.2007. [3]F.Paci ci,F.DelFrate,W.Emery,P.Gamba,andJ.Chanussot, “UrbanmappingusingcoarseSARandopticaldata:Outcomeofthe 2007GRSSdatafusioncontest,” IEEEGeosci.RemoteSens.Lett. vol.5,no.3,pp.331–335,Jul.2008. [4]G.Licciardi,F.Paci ci,D.Tuia,S.Prasad,T.West,F.Giacco,C. Thiel,J.Inglada,E.Christophe,J. Chanussot,andP.Gamba,“Decision fusionfortheclassi cationofhyperspectraldata:Outcomeofthe2008 GRS-Sdatafusioncontest,” IEEETrans.Geosci.RemoteSens. ,vol. 47,no.11,pp.3857–3865,Nov.2009. [5]P.Coppin,I.Jonckheere,K.Nackaerts,andM.B.,“Digitalchange detectionmethodsinecosystemmonitoring:Areview,” Int.J.Remote Sens. ,vol.25,no.9,pp.1565–1596,2004. [6]A.Singh,“Digitalchangedetectiontechniquesusingremotely-sensed data,” Int.J.RemoteSens. ,vol.10,no.6,pp.989–1003,1989. [7]F.D.Frate,F.Paci ci,andD.Solimini,“Monitoringurbanlandcover inRome,Italyanditschangesbysingle-polarizationmulti-temporal SARimages,” IEEEJ.Sel.TopicsAppl.EarthObserv.RemoteSens. vol.1,no.2,pp.87–97,Jun.2008. [8]F.Paci ci,F.D.Frate,C.Solimini,andW.Emery,“Aninnovative neural-netmethodtodetecttemporalchangesinhigh-resolutionoptical satelliteimagery,” IEEETrans.Geosci.RemoteSens. ,vol. 45,no.9,pp. 2940–2952,2007. [9]J.Chen,X.Chen,X.Cui,andJ.Ch en,“Changevectoranalysisinposteriorprobabilityspace:Anewmethodforlandcoverchangedetection,” IEEEGeosci.RemoteSens.Lett. ,vol.PP,no.99,pp.317–321, 2010. [10]W.A.Malila,“Changevectoranalysis:Anapproachfo rdetecting forestchangewithLandsat,”in IEEEProc.AnnualSymp.Machine ProcessingofRemotelySensingData ,1980,pp.326–336. [11]F.BovoloandL.Bruzzone,“Asplit-basedapproachtounsupervised changedetectioninlargesizemult itemporalimages:Applicationto Tsunami-damageassessment,” IEEETrans.Geosci.RemoteSens. ,vol. 45,no.6,pp.1658–1671,2007. [12]H.NemmourandY.Chibani,“Multiplesupportvectormachinesfor landcoverchangedetection:Anapplicationformappingurbanextensions,” J.Photogr.RemoteSens. ,vol.61,pp.125–133,2006. [13]G.Camps-Valls,L.Gmez-Chova,J.Muoz-Mar,J.L.Rojo-lvarez, andM.Martnez-Ramn,“Kernel-basedframe workformulti-temporal andmulti-sourceremotesensingdataclassi cationandchangedetection,” IEEETrans.Geosci.RemoteSens. ,vol.46,no.6,pp.1822–1835, 2008. [14]F.Bovolo,L.Bruzzone,andM.Marconcini,“Anovelapproachto unsupervisedch angedetectionbasedonasemisupervisedSVMand asimilaritymeasure,” IEEETrans.Geo sci.RemoteSens. ,vol.46,no. 7,pp.2070–2082,Jul.2008. [15]J.Muoz-Mar,F.Bovolo,L.Gmez-Chova,L.Bruzzone,andG. Camps-Valls,“Semisupervisedone -classsupportvectormachinesfor classi cationofremotesensingdata,”IEEETrans.Geosci.Remote Sens. ,vol.48,no.8,pp.3188–3197,2010. [16]T.N.Tran,R.Wehrens,andL.M.C.Buyde ns,“Sparef:Aclustering algorithmformultispectralimages,” AnalyticaChimicaActa vol.490, no.1–2,pp.303–312,2003[Online].Available:http://www.sciencedirect.com/science/article/pii/S0003267003007207,paperspresented atthe8thInternationalConferenceonChemometricsandAnalytical Chemistry [17]M.HasanzadehandS.Kasaei,“Amul tispectralimagesegmentation methodusingsize-weightedfuzzyclusteringandmembershipconnectedness,” IEEEGeosci.RemoteSens.Lett. ,vol.7,no.3,pp.520–524, Jul.2010. [18]A.MarcalandL.Castro,“Hierarchicalclusteringofmultispectralimagesusingcombinedspectralandspatialcriteria,” IEEEGeosci.RemoteSens.Lett. ,vol.2,no.1 ,pp.59–63,Jan.2005. [19]G.Hazel,“MultivariateGaussianMRFformultispectralscenesegmentationandanomalydetection,” IEEETrans.Geosci.RemoteSens. vol.38,no.3,pp.1199–1211,May2000. [20]G.MartinandA.Plaza,“Regionbasedspatialpreprocessingforendmemberextractionandspectralunmixing,” IEEEGeosci.RemoteSens. Lett. ,vol.8,no.4,pp.745–7 49,Jul.2011. [21]A.Zare,O.Bchir,H.Frigui,andP.Gader,“Spatially-smoothpiecewiseconvexendmemberdetection,”in 20102ndWorkshoponHyperspectralImageandSignalProcessing:EvolutioninRemoteSensing (WHISPERS) ,Jun.2010,pp.1–4. [22]M.ZorteaandA.Plaza,“Spatialpreprocessingforendmemberextraction,” IEEETrans.Geosci. RemoteSens. ,vol.47,no.8,pp.2679–2693, Aug.2009. [23]A.Zare,J.Bolton,P.Gader,an dM.Schatten,“Vegetationmappingforlandminedetectionusinglong-wavehyperspectralimagery,” IEEETrans.Geosci.RemoteSens. ,vol.46,no.1,pp.172–178,Jan. 2008.

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340IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,F EBRUARY2012[24]D.ChaudhuriandA.Samal,“Anautomaticbridgedetectiontechnique formultispectralimages,” IEEETrans.Geosci.RemoteSens. ,vol.46, no.9,pp.2720–2727,Sep.2008. [25]Y.Bazi,L.Bruzzone,andF.M.Me lgani,“Anunsup ervisedapproach basedonthegeneralizedGaussianm odeltoautomaticchangedetection inmultitemporalSARimages,” IEEETrans.Geosci.RemoteSens. vol.43,no.4,pp.874–887,April2005. [26]G.MoserandS.B.Serpico,“Generalizedminimum-errorthresholding forunsupervisedchangedetectionfromSARamplitudeimagery,” IEEETrans.Geosci.RemoteSens. ,vol.44,no.10,pp.2972–2982, Oct.2006. [27]C.Carincotte,S.Derrode,andS.Bourennane,“Unsupervisedchange detectiononSARimagesusingfuzzyhiddenMarkovchains,” IEEE Trans.Geosci.RemoteSens. ,vol.44,no.2,pp.432–441,Feb.2006. [28]T.Hastie,R.Tibshirani,andJ.Friedman ,TheElementsOfStatistical Learning:DataMining,Inference,andPrediction ,2nded.New York:Springer,2009. [29]J.F.MasandJ.J.Flores,“Theapplicationofarti cialneuralnetworks totheanalysisofremotelysenseddata,” Int.J.RemoteSens. ,vol.29, no.3,pp.617–663,Feb.2008. [30]J.Benediktsson,P.Swain,andO.Ersoy,“Neuralnetworkappro aches versusstatisticalmethodsinclassi cationofmultisourceremote sensingdata,” IEEETrans.Geosci.RemoteSens. ,vol.28,no.4,pp. 540–552,Jul.1990. [31]S.GopalandC.Woodcock,“Remotesensingofforestchangeusing arti cialneuralnetworks,” IEEETrans.Geosci.RemoteSens. ,vol.34, no.2,pp.398–404,Mar.1996. [32]G.Carpenter,S.Gopal,B.Shock,andC.Woodcock,Aneuralnetwork methodforlandusechangeclassi cation,withapplicationtotheNile riverdeltaBostonUniv.,Ctr.AdaptiveSystems,Dept.Cognitiveand NeuralSystems,Boston,MA,Tech.Rep.,2001[Online].Available: http://techlab.bu.edu/ les/resources/articles_cns/CarpenterGopalShockWoodcock2003.pdf [33]I.Olthof,“MappingdeciduousforesticestormdamageusingLandsat andenvironmentaldata,” RemoteSens.Environ. ,vol.89,no.4,pp. 484–496,Feb.2004. [34]S.Ghosh,L.Bruzzone,S.Patra,F.Bovolo,andA.Ghosh,“Acontext-sensitivetechniqueforunsupervisedchangedetectionbasedon hop eld-typeneuralnetworks,” IEEETrans.G eosci.RemoteSens. vol.45,no.3,pp.778–789,Mar.2007. [35]F.Paci ciandW.J.Emery ,PulseCoupledNeuralNetworksforAutomaticUrbanChangeDetectionatVeryHighSpatialResolution ,ser. LectureNotesinComputerScien ce.Berlin,Heidelberg,Germany: Springer,2009,vol.5856,pp.929–942. [36]M.Chini,F.Paci ci,W.Emery,N.Pierdic ca,andF.DelFrate,“Comparingstatisticalandneuralnetworkmethodsappliedtoveryhighresolutionsatelliteimagesshowingchangesinman-madestructuresat rocky ats,” IEEETrans.Geosci.RemoteSens. ,vol.46,no.6,pp. 1812–1821,Jun.2008. [37]J.SolaandJ.Sevilla,“Importanceofinputdatanormalizationforthe applicationofneuralnetworkstocompl exindustrialproblems,” IEEE Trans.Nucl.Sci. ,vol.44,no.3,pp.1464–1468,Jun.1997. [38]F.Paci ci,M.Chini,andW.J.Emery,“Aneuralnetworkapproachusingmulti-scaletexturalmetricsfromveryhighresolutionpanchromaticimageryforurbanland-useclassi cation,” RemoteSens.Environ. ,vol.113,no.6,pp.1276–1292,Jun.2009. [39]J.Serra ,ImageAnalysisandMathema ticalMorphology .NewYork: AcademicPress,1982. [40]P.Soille ,MorphologicalImageAnalysis ,2nded.Berlin,Germany: Springer,2004. [41]M.Fauvel,J.A.Benediktsson, J.Chanussot,andJ.R.Sveinsson, “Spectralandspatialclassi cationofhyperspectraldatausingSVMs andmorphologicalpro les,” IEE ETrans.Geosci.RemoteSens. ,vol. 46,no.11,pp.3804–3814,2008. [42]D.Tuia,F.Paci ci,M.Kanevski,andW.Emery,“Classi cationof veryhighspatialresolutionimageryusingmathematicalmorphology andsupportvectormachines,” IEEETrans.Geosci.RemoteSens. ,vol. 47,no.11,pp.3866–3879,2009. [43]D.Tuia,G.Camps-Valls,G.Matas ci,andM.Kanevski,“Learningrelevantimagefeatureswithmultiplekernelclassi cation,” IEEETrans. Geosci.RemoteSens. ,vol.48,no.10,pp.3780–3791,2010. [44]M.DallaMura,J.A.Benediktsson,F.Bovolo,andL.Bruzzone,“An unsupervisedtechniquebasedonmorphological ltersforchangedetectioninveryhighresolutionimages,” IEEETrans.Geosci.Remote Sens. ,vol.5,no.3,pp.433–4 37,2008. [45]F.Bovolo,“Amultilevelparcel-basedapproachtochangedetection inveryhighresolutionmultitemporalimages,” IEEEGeosci.Remote Sens.Lett. ,vol.6,no.1,pp.33–38,2009. [46]M.Volpi,D.Tuia,F.Bovolo,M.Kanevski,andL.Bruzzone,“SupervisedchangedetectioninVHRimagesusingcontextualinformation andsupportvectormachines,” Int.J.Appl.EarthObserv.Geoinform. 2011,inpress. [47]M.PesaresiandJ.Benediktsso n,“Anewapproachforthemorphologicalsegmentationofhigh-resolutionsatelliteimages,” IEEETrans. Geosci.RemoteSens. ,vol.39,no.2,pp.309–320,2001. [48]J.A.Benediktsson,M.Pesaresi,andK.Arnason,“Classi cationand featureextractionforremotesens ingimagesfromurbanareasbasedon morphologicaltransformations,” IEEETrans.Geosci.RemoteSens. vol.41,no.9,pp.1940–1949,2003. [49]R.Collobert,S.Bengio,andJ.Marithoz,“Torch:AModularMachine LearningSoftwareLibrary,”IDIAP,Tech.Rep.RR02-46,2002. [50]T.N.Tran,R.Wehrens,andL.M.Buydens,“Clusteringmultispectral images:Atutorial,” ChemometricsandIntelligentLaboratorySystem s vol.77,no.1–2,pp.3–17,2005. [51]C.Yang,L.Bruzzone,F.Sun,L.Lu,R.Guan,andY.Liang,“Afuzzystatistics-basedaf nitypropagationtechniqueforclusteringinmultispectralimages,” IEEETrans.Geosci.RemoteSens. ,vol.48,no.6,pp. 2647–2659,Jun.2010. [52]Disastercharter.[Online].Available:http://www.disaste rscharter.org/ home [53]TheOrfeoTeam.[Online].Available:http://www.orfeo-toolbox.org [54]G.Mercier,G.Moser,andS.Serpico,“Conditionalcopulaforchange detectiononheterogeneousSARdata,” IEEETrans.Geosci.RemoteSens. ,vol.45,no.5,pp.1428–1441,May2008. [55]G.Wilkinson,“Resultsandimplicationsofastudyof fte enyearsof satelliteimageclassi cationexperiments,” IEEETrans.Geosci.RemoteSens. ,vol.43,no.3,pp.433–440,Mar.2005. NathanLongbotham (S’11)iscurrentlyaPh.D.studentstudyingremotesensingintheDepartmentof AerospaceEngineeringSciencesattheUniversityof ColoradoatBoulder.HereceivedanM.S.degreein opticalscienceandengin eeringfromtheUniversity ofNewMexicoandaB.S.degreeinphysics( magna cumlaude ;universityscholar;presidentialscholar) fromAbileneChristianUniversity. WhilepursuingtheM.S.degree,heheldagraduateinternshippositionatSandiaNationalLaboratories,whereheconductedresearchintotheproperties ofQ-switchedmicrolasersandtheirapplicabilitytoLIDARsystems.Priorto hiscurrentstudies,heheldanOpticalEngineerpositionwiththeholographic datastoragestartup,InPhaseTechnologi es,developingaspecialized,tunable ultravioletlasersystem.HeiscurrentlyaResearchAssistantattheUniversity ofColoradoatBouldercollaboratingwiththeR&DdepartmentatDigitalGlobe inLongmont,CO,developingurbanremotesensingtechniquesthatleverage multi-angleopticalimagery.Hisresearchinterestsincludeimageanalysis,data fusion,multi-temporaldataanalysis,andfeatureextraction. Mr.Longbothamearned rstplaceinthe2009IEEEGeoscienceandRemote SensingDataFusionContest.HeservesasareviewerfortheIEEEJOURNAL OFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONANDREMOTESENSING. FabioPaci ci (S’03–M’10)receivedthePh.D. degreeinGeoInformationfromTorVergataUniversity,Rome,Italy,in2010.Healsoreceivedthe LaureaSpecialistica(M.S.; cumlaude )andLaurea (B.S.; cumlaude )degreesintelecommunication engineeringfromthesameUniversity,in2003and 2006,respectively. Since2009,heisworkingatDigitalGlobeasR&D Scientist.Between2005and2009,hecollaboratedas VisitorScientistwiththeDepartmentofAerospace EngineeringSciences,UniversityofColorado, Boulder.Hehasbeeninvolvedinvariousremotesensingprojectssupported bytheEuropeanSpaceAgency.Hisresear chactivitiesincludeprocessingof remotesensingimages,datafusion,featureextraction,activelearning,and analysisofmultitemporaldata.Inparticular,hisresearchinterestsarerelated tothedevelopmentofclassi cationandchangedetectiontechniquesforurban remotesensingapplicationsusingveryhighspatialresolutionopticaland/or syntheticapertureradarimagery,wit hspecialemphasisonmachinelearning. Heisauthor(orco-author)of13scienti cpublicationsinreferredinternational Journals,2bookchapters,andmorethan40contributionsininternational conferences.

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LONGBOTHAM etal. :MULTI-MODALCHANGEDETECTION,APPLICATIONTOTHEDETECTIONOFFLOODEDAREAS341Dr.Paci ciisthecurrentChairoftheIEEEGeoscienceandRemote SensingDataFusionTechnicalCommitteeandservesasAssociateEditorfor theIEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONS ANDREMOTESENSING(JSTARS).Hewastherecipientofthe2009Joint UrbanRemoteSensingEventstudentpapercompetition.Hereceivedthe rst prizeinthe2007,2008andthe2009IEEEGeoscienceandRemoteSensing DataFusionContest.Heservedasam emberofthe2011JointUrbanEvent TechnicalCommitteeandSessionChairattheInternationalGeoscienceand RemoteSensingSymposium.HehasbeentheGuestEditorofaspecialissue oftheJSTARSonmultiangularremotesensing. TaylorGlenn (S’10)receivedtheB.S.andM.E. degreesincomputerengineeringfromtheUniversity ofFlorida,Gainesville,in2003and2004.Heis currentlyaPh.D.candidateandGraduateFellow intheDepartmentofComputerandInformation ScienceandEngineeringattheUniversityofFlorida. From2004to2009hewasapartnerandleadengineerat2GEngineering,LLC.Hiscurrentresearch interestsareinthe eldsofmachinelearning,computervision,andremotesensing. AlinaZare (M’08)receivedthePh.D .degreeincomputerengineeringfromtheUniver sityofFloridain December2008. SheisanAssistantProfessorinthe Department ofElectricalandComputerEngine eringattheUniversityofMissouri,Columbia.He rresearchinterests includeremotesensing,spars itypromotion,machine learning,imageanalysis,an dpatternrecognition.She hasbeeninvolvedinlandmine ,explosiveobjectand traceexplosivesdetection researchusinghyperspectralimagersinavarietyofm odalitiessuchairborne orground-basedforward-l ookingsensors.Shehasalsoconductedresearchon targetdetectionusingwid ebandelectromagneticinductionsensorsandinvestigatedagent-basedmodeli ngtechniquesforhumangeographyapplications. MicheleVolpi (S’08) wasborninLugano,Switzerland,in1985.Herec eivedtheB.S.degreeinphysicalgeographyand theM.S.degreeinenvironmental sciencesfromthe UniversityofLausanne,Lausanne, Switzerland,in2 007andin2009,respectively.He iscurrentlypurs uingthePh.D.degreeattheInstituteofGeomatic sandAnalysisofRisk,University ofLausanne,und eraSwissNationalScienceFoundationgrant. Hisresearchact ivitiesareintheareaofremote sensingimagepr ocessingandmultitemporalimage analysis.Inp articular,hisinterestsincludethedevelopmentandapplicationof machinelearn ingalgorithms(speci callykernel-basedmethods)forchangedetection,mul titemporalimageclassi cation,featureextraction,andclassi cation formultispe ctralveryhighresolutiondata. Mr.Volpiwaso neofthewinnersoftheIEEEGeosciencesandRemote SensingData FusionContest,in2009.HeisarefereeofIEEETRANSACTIONS ONGEOSCIENC EANDREMOTESENSINGand IEEEGeoscienceandRemote SensingLett ers DevisTuia (S’07–M’09)wasborninMendrisio, Switzerland,in1980.Hereceivedthediplomain GeographyattheUniversityofLausannein2004, theMasterofAdvancedStudiesinEnvironmental EngineeringattheFederalInstituteofTechnology ofLausanne(EPFL)in2005,andthePh.D.inenvironmentalsciencesattheUniversityofLausannein 2009. HewasapostdocresearcheratboththeUniversity ofValncia,SpainandtheUniveristyofColoradoat BoulderunderaSwissNationalFoundationprogram. HeisnowaSeniorResearchAssociateatt heLaboratoiredes Systmesd’InformationGographiques,EPFL.Hisresearchinterestsincludethedevelopment ofalgorithmsforinformationextractionandclassi cationofveryhighresolutionremotesensingimagesandsocioeconomicdatausingmachinelearning algorithms.Hiswebsiteishttp://devis.tuia.googlepages.com/. EmmanuelChristophe (M’07)receivedtheEngineeringdegreeinTelecommunicationsfromcole NationaleSuprieuredesTlcommunicationsde Bretagne,Brest,France,andtheDEAinTelecommunicationsandimageprocessingfromUniversity ofRennes1in2003.InOctober2006,hereceived thePh.D.degreefromSupaeroandUniversityof Toulouseinhyperspectralimagecompressionand imagequality. Hehasbeenavisitingscholarin2006atRensselaerPolytechnicInstitute,Troy,NY,USA.From2006 to2008,hewasaresearchengineeratCNES,theFrenchSpaceAgency,focusingoninformationextractionforhighresolutionopticalimages.Between 2008and2010,hemovedtoSingaporeat CRISP,NationalUniversityofSingapore,wherehewastacklingnewchallenges forremotesensingintropicalareas. HeisnowwithGoogleInc.inCalifornia. JulienMichel (A’10)receivedt heTelecommunicationsEngineerdegreefromt hecoleNationale SuprieuredesTlcommunica tionsdeBretagne, Brest,France,in2006. From2006to2010,hehasbeenwi thCommunicationsetSystmes,Toulo use,France,wherehe hasbeenworkingonstudiesan ddevelopmentsin the eldofremotesensingim ageprocessing.He isnowwiththeCentreNation ald’tudesSpatiales (FrenchSpaceAgency),To ulouse,France,wherehe isinchargeofthedevelo pmentofimageprocessing algorithmsfortheexplo itationofEarthobservationimages,mainlyinthe eld ofveryhighresolution imageanalysis. JordiInglada (M’09)receivedtheTelecommunicationsEngineerdegreefromboththeUniversitat PolitcnicadeCatalunya,Barcelona,Spain,andthe coleNationaleSuprieu redesTlcommunications deBretagne,Brest,France,in1997andthePh.D. degreeinsignalprocessingandtelecommunications in2000fromUniversitdeRennes1,Rennes, France. HeiscurrentlywiththeCentreNationald’tudes Spatiales(FrenchSpaceAgency),Toulouse,France, workinginthe eldofremotesensingimageprocessingattheCESBIOlaboratory.Heisinchargeofthedevelopmentofimage processingalgorithmsfortheoperationalexploitationofEarthobservationimages,mainlyinthe eldofmultitemporalimageanalysisforlanduseandcover change. Dr.IngladaisanAssociateEditoroftheIEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING.

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342IEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING,VOL.5,NO.1,F EBRUARY2012 JocelynChanussot (M’04–SM’04–F’12)received theM.Sc.degreeinelectricalengineeringfromthe GrenobleInstituteofTechnology(GrenobleINP), Grenoble,France,in1995,andthePh.D.degree fromSavoieUniversity,Annecy,France,in1998. In1999,hewaswiththeGeographyImageryPerceptionLaboratoryforth eDelegationGeneralede l’Armement(DGA—FrenchNationalDefenseDepartment).Since1999,hehasbeenwithGrenoble INP,wherehewasanAssistantProfessorfrom1999 to2005,anAssociateProfessorfrom2005to2007, andiscurrentlyaProfessorofsignalandimageprocessing.Heiscurrently conductinghisresearchattheGrenobleImagesSpeechSignalsandAutomatics Laboratory(GIPSA-Lab).Hisresearchi nterestsincludeimageanalysis,multicomponentimageprocessing,nonlinear ltering,anddatafusioninremote sensing. Dr.ChanussotisthefoundingPresidentofIEEEGeoscienceandRemote SensingFrenchchapter (2007–2010)whichrecei vedthe2010IEEEGRS-S ChapterExcellenceAward“forexcellenceasaGeoscienceandRemote SensingSocietychapterdemonstratedbyexemplaryactivitiesduring2009.” Hewastherecipientofthe2011IEEEGRSSSymposiumBestPaperAward. HewasamemberoftheIEEEGeoscienceandRemoteSensingSociety AdCom(2009–2010),inchargeofmembershipdevelopment.Hewasthe GeneralChairofthe rstIEEEGRSSWorkshoponHyperspectralImageand SignalProcessing,EvolutioninRemotesensing(WHISPERS).Hewasthe Chair(2009–2011)andtheCo-Chair(2005–2008)oftheGRSDataFusion TechnicalCommittee.Hewasamember oftheMachineLearningforSignal ProcessingTechnicalCommitteeoftheIEEESignalProcessingSociety (2006–2008)andtheProgramChairoftheIEEEInternationalWorkshopon MachineLearningforSignalProcessing ,(2009).HewasanAssociateEditor fortheIEEEGeoscienceAndRemoteSensingLetters(2005–2007)andfor PatternRecognition(2006–2008).Since2007,hehasbeenanAssociateEditor fortheIEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING.Since 2011,hehasbeentheEditor-in-ChiefoftheIEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING. QianDu (S’98–M’00–SM’05)receivedthePh.D. degreeinelectricalengin eeringfromtheUniversity ofMarylandBaltimoreCountyin2000. ShewaswiththeDepartmentofElectricalEngineeringandComputerScience,TexasA&MUniversity,Kingsville,from2000to2004.Shejoinedthe DepartmentofElectricalandComputerEngineering atMississippiStateUniversityinFall2004,where sheiscurrentlyanAssociateProfessor.Herresearch interestsincludehyperspectralremotesensingimage analysis,patternclassi cation,datacompression,and neuralnetworks. Dr.DucurrentlyservesasCo-Chair fortheDataFusionTechnicalCommitteeofIEEEGeoscienceandRemoteSensingSociety.Shealsoservesas AssociateEditorforIEEEJOURNALOFSELECTEDTOPICSINAPPLIEDEARTHOBSERVATIONSANDREMOTESENSING(J-STARS).Shereceivedthe2010Best ReviewerawardfromIEEEGeoscienc eandRemoteSensingSociety.Sheis theGeneralChairforthefourthIEEEGRSSWorkshoponHyperspectralImage andSignalProcessing,EvolutioninR emoteSensing(WHISPERS).Dr.Duis amemberofSPIE,ASPRS,andASEE.



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CONTEXT-DEPENDENTDETECTIONINHYPERSPECTRALIMAGERY By TAYLORC.GLENN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c 2013TaylorC.Glenn 2

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ACKNOWLEDGMENTS Iwouldliketoacknowledgeandgivemythankstoeveryonewhohashelpedmealong thewaytocompletingthePhD.Thoughtherearetoomanytomention,ifyouknowme andarereadingthis,Iamtalkingaboutyou.InparticularIwouldliketoextendmy gratitudetomylabmatesinthelandmineslab,myresearchadvisors,andmyparents andfamily.MostespeciallyIwanttothankDr.AlinaZare,whosecontributionstothis work{andmylife{areenormous. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................3 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 LISTOFABBREVIATIONS...............................11 ABSTRACT........................................14 CHAPTER 1INTRODUCTION..................................15 2LITERATUREREVIEW..............................19 2.1Introduction...................................19 2.2DetectionMethods...............................19 2.2.1CategorizationofSignatureBasedDetectionMethods........19 2.2.2SpectralAngleMapper.........................21 2.2.3SpectralMatchedFilter.........................22 2.2.4ConstrainedEnergyMinimization...................23 2.2.5TargetConstrainedInterferenceMinimizedFilter..........24 2.2.6AdaptiveCosine/CoherenceEstimator................26 2.2.7SubpixelSpectralMatchedFilter...................27 2.2.8SpectralMatchedFilterswithBackgroundClustering........28 2.2.8.1UsingGaussianmixturemodels...............28 2.2.8.2UsingK-means........................29 2.2.9OrthogonalSubspaceProjection....................30 2.2.10TargetSubspaceDetectors.......................32 2.2.11AdaptiveMatchedSubspaceDetector.................34 2.2.12ConstrainedLeastSquaresAbundanceEstimation..........34 2.2.13HybridSubpixelDetector........................37 2.2.14ConstrainedSignalDetector......................39 2.2.15QuadraticMatchedFilter........................40 2.2.16WaveletBasedDetectors........................41 2.2.17KernelBasedMethods.........................42 2.2.18SparseReconstructionBasedDetectors................43 2.2.19SupportVectorDataDescription...................44 2.2.20KalmanFilter..............................44 2.3DetectionRelatedTechniquesintheLiterature...............45 2.3.1FalseAlarmMitigationStatistics...................45 2.3.1.1Mixturetunedmatchedlter................45 2.3.1.2Subpixelreplacementmodel.................46 4

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2.3.1.3Finitetargetmatchedlter.................47 2.3.1.4Leastangleregression....................48 2.3.2DeterminingTargetSignatures.....................49 2.3.3ImprovingDetectorPerformance....................50 2.4UnmixingandEndmemberExtractionMethods...............51 2.4.1Sample-PCE...............................52 2.4.2FuzzPop.................................56 2.5Context-DependentMethods..........................58 2.5.1UsesofContextinLiterature......................58 2.5.2ContextDependenceinClassication.................60 2.5.3ContextDependenceinFusion.....................62 2.6FuzzyClustering................................63 2.7Summary....................................65 3TECHNICALAPPROACH.............................70 3.1SequentialUnsupervisedContextLearningandDetection..........71 3.2JointUnsupervisedContextLearningandDetection.............72 3.2.1FCM+Detector+CEM.........................72 3.2.2FuzzyConstrainedEnergyMinimization...............74 3.2.3PossibilisticFuzzyandSpatialConstrainedEnergyMinimization..77 3.3BayesianFuzzyClustering...........................79 3.3.1BayesianFuzzyClusteringModel...................80 3.3.2InferenceintheBayesianFuzzyClusteringModel..........84 3.3.3BayesianModelforEstimatingtheNumberofClusters.......87 3.3.4InferenceintheInniteBayesianFuzzyClusteringModel......91 3.3.5DerivationoftheFuzzyDataLikelihoodNormalizationConstant..95 3.4BayesianJointUnsupervisedContextLearningandDetection.......96 3.4.1InniteBayesianFuzzyConstrainedEnergyMinimization......96 3.4.2BayesianFuzzyACE..........................98 3.5Alarm-SetFusion................................104 3.5.1FARE-ASF................................105 3.5.2RunPacking...............................107 3.5.3GreedyPackingStrategy........................109 3.5.4DynamicProgrammingStrategy....................111 3.5.5ROCInterpretation...........................112 4EXPERIMENTSANDRESULTS..........................115 4.1Datasets.....................................115 4.1.1SyntheticGaussianContextData...................115 4.1.2SyntheticEndmemberContextData.................115 4.1.3MUUFLGulfportCollection......................117 4.2SequentialUnsupervisedContextLearningandDetection..........118 4.2.1ComparingContextEstimationMethods...............118 4.2.2ComparingUseofLiDARInformation................119 5

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4.3JointUnsupervisedContextLearningandDetection.............120 4.3.1FCM+ACE+CEM...........................120 4.3.2FCEM..................................121 4.3.2.1Syntheticdata........................121 4.3.2.2Gulfportbatchtest......................122 4.3.2.3Gulfportindividual......................122 4.3.3PFSCEM.................................123 4.3.3.1Syntheticdata........................123 4.3.3.2Gulfportbatchtest......................124 4.3.3.3Gulfportindividual......................126 4.3.4IBFCEM.................................126 4.3.4.1Syntheticdata........................126 4.3.4.2Realdata...........................127 4.3.5BFACE..................................129 4.3.5.1Syntheticdata........................129 4.3.5.2Realdata...........................130 4.4BayesianFuzzyClustering...........................131 4.4.1BFC...................................131 4.4.2IBFC...................................132 4.4.2.1IBFCversusCompetitiveAgglomeration..........133 4.5Alarm-SetFusion................................135 4.5.1Rule-BasedSegmentation........................136 4.5.2UnsupervisedAlarm-SetFusion....................138 4.5.3SupervisedAlarm-SetFusion......................140 5DISCUSSIONANDFUTUREWORK.......................187 5.1Discussion....................................187 5.2FutureWork...................................189 5.2.1DetectionMethods...........................189 5.2.2BayesianFuzzyClustering.......................191 5.2.3Alarm-SetFusion............................194 5.3ConcludingRemarks..............................194 REFERENCES.......................................196 BIOGRAPHICALSKETCH................................208 6

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LISTOFTABLES Table page 2-1Detectorcategorizations...............................69 4-1FACEMmaximumAUCandnumberofclustersoncampusdataforvarious settingsof and m ..................................181 4-2AverageAUCforFCEMonbrowntargetsupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 FA/ m 2 ..........181 4-3AverageAUCforFCEMondarkgreentargetsupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FA/ m 2 ........181 4-4AverageAUCforFCEMonfauxvineyardgreentargetsupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FA/ m 2 ...181 4-5AverageAUCforFCEMonpeagreentargetsupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 FA/ m 2 ........182 4-6PFSCEMAUConsyntheticendmembercontextdataforvarioussettingsof possibilisticweightandspatialsmoothing.....................182 4-7Maximummean-AUCparametercongurationforPFSCEMonGulfportcampusdata........................................183 4-8AUCvaluesupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 forIBFCEM,FCEM,ACE,andSMFonvecampus images.........................................183 4-9IBFCEMobjectivevaluesforIBFCEMandFCEMresults............183 4-10AUCvaluesupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 forIBFCEMandFCEMonvecampusimageswith targetsexcludedfromtraining............................183 4-11AverageRandindexforCompetitiveAgglomerationandIBFC..........184 4-12AverageEarthMover'sDistanceforCompetitiveAgglomerationandIBFC...184 4-13Fractionoftrialswithsamenumberofclustersastruth..............185 4-14AveragenumberofclustersfoundbyIBFCandCA................185 4-15Detectorperformancebysegment..........................186 4-16Learnedthresholdsinalarm-setfusionversussegmentindependentthresholds.186 7

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LISTOFFIGURES Figure page 2-1Detectionblockdiagram...............................67 2-2SchematicrepresentationofSample-PCEprobabilisticmodel...........67 2-3Detectormodelprogression.............................67 2-4Progressiontocontext-dependentdetection.....................68 3-1Context-DependentDetection............................113 3-2Beforeandafteroneiterationofrunpackingwiththegreedystrategy......113 3-3ROCcurveinterpretationofrunpacking......................114 3-4ResultingROCcurveafterrunpacking.......................114 4-1Scatterplotoftwo-componentGaussiansyntheticcontextdata.........142 4-2False-colorRGBimageoftwo-componentGaussiansyntheticcontextdata...143 4-3Scatterplotoftwo-componentGaussiansyntheticcontextdatawithtargets..143 4-4Proportionmapoftargetmixturefortwo-dimensionaltwo-componentGaussian syntheticcontextdata................................144 4-5Materialspectraforthefour-contextsyntheticendmemberdataset.......145 4-6Materialspectraforthefour-contextsyntheticendmemberdataset.......146 4-7Materialspectraforthefour-contextsyntheticendmemberdataset.......147 4-8Pea-greentargetspectrum..............................148 4-9Targetemplacementgridforthefour-contextsyntheticendmemberdata....148 4-10Scatterplotsoffour-contextsyntheticendmemberdatainrstthreeprincipal componentdimensions................................149 4-11RGBandLiDARDEMofGulfportcampusdataset................150 4-12ExampleemplacedtargetsintheGulfportcampusdatasetin4colorsand3sizes151 4-13ROCcurvescomparingglobalACEandHSDtoFuzzyversionsusingFuzzPop clustering.......................................152 4-14ROCsforFuzzPop-ACEandFuzzPop-HSDonGulfportcampusdatabytarget type..........................................153 8

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4-15ROCsofFuzzPop-HSDusingFuzzPopandFuzzPop-LiDARcomparedtoglobal HSD..........................................154 4-16ROCsforFACEMvsACE,SMF,GMM-CCMFonGaussiansyntheticdata..155 4-17FACEMvsGMM-CCMFcondencescatterplotsonGaussiansyntheticdata..155 4-18ROCsforFACEMvsACE,SMF,GMM-CCMFonGulfportcampusdata...156 4-19ROCsforFCEMvsACE,SMF,GMM-CCMF,FCM-MFonGaussiansynthetic data..........................................156 4-20FCEMvsFCM-MFcondencescatterplotsonGaussiansyntheticdata.....157 4-21ROCsforFCEMonGulfportcampusdata.....................158 4-22Pixellabelingsbyspatialwindowsize........................159 4-23PFSCEMAUChistogramsonbrowntargetsvaryingspatialandpossibilistic usage..........................................159 4-24PFSCEMAUChistogramsondarkgreentargetsvaryingspatialandpossibilisticusage........................................160 4-25PFSCEMAUChistogramsondarkgreentargetsvaryingspatialandpossibilisticusage........................................161 4-26PFSCEMAUChistogramsonpeagreentargetsvaryingspatialandpossibilisticusage........................................162 4-27ROCsforPFSCEMonGulfportcampusdata...................163 4-28HistogramsofROCAUConsyntheticGaussiancontextdataover50Trials..164 4-29MedianROCcurvesforIBFCEM,FCEM,ACE,andSMFonsyntheticGaussiancontextdataover50trials...........................165 4-30BFACEROCsonsmallsyntheticGaussiancontextdatasetcomparedtoFCEM, ACE,andSMF....................................166 4-31ScatterplotofsmallsyntheticGaussiancontextdatasetcoloredbyBFACEcondence.........................................167 4-32BFACEROCandscatterplotonfullGaussiansyntheticdataset.........168 4-33BFACEROCandscatterplottrainedonsmalldatasetthentestedonfullsyntheticdataset.....................................169 4-34ROCsforBFACEonGulfportcampusdata....................170 4-35HistogramofBFACEcondencesforbrowntargets.................171 9

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4-36CondencemapsofBFACEdetectorforbrowntargets..............172 4-37BFCandFCMoutputsfor m =2..........................173 4-38BFCresultsfor m =1and m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10........................174 4-39BFCresultsforcombinationsof m and .....................174 4-40IBFCandFCMresultsonfourcomponentdata..................175 4-41IBFCparticlelteralternativesolutions......................176 4-42Segmentationmap..................................177 4-43SegmentspecicROCcurves............................178 4-44ROCcurvesforSMF,ACE,HSD,andCC-ACEandFARE-ASFresult.....179 4-45ACEalgorithmROCs................................180 10

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LISTOFABBREVIATIONS ACE AdaptiveCosine/CoherenceEstimator ANC AbundanceNonnegativityConstraint AMSD AdaptiveMatchedSubspaceDetector ASC AbundanceSumConstraint ASF Alarm-SetFusion AUC AreaUndertheCurve AVIRIS AirborneVisible/InfraredImagingSpectrometer BFACE BayesianFuzzyACE BFC BayesianFuzzyClustering CCMF ClassConditionalSpectralMatchedFilter CDF ContextDependentFusion CELF ContextExtractionforLocalFusion CEM ConstrainedEnergyMinimization CFAR ConstantFalse-AlarmRate CMPP conditionalmappedposteriorprobability CODE context-dependent CPD ConstantProbabilityofDetection DEM DigitalElevationMap DP DirichletProcess FACEM FCM+ACE+CEM FAR FalseAlarmRate FARE-ASF FalseAlarmRateEstimationforAlarm-SetFusion FCEM FuzzyConstrainedEnergyMinimization FCLS FullyConstrainedLeast-Squares FCM FuzzyC-Means FLICM FuzzyLocalInformationC-Means 11

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GLRT GeneralizedLikelihoodRatioTest GMM GaussianMixtureModel HME HierarchicalMixturesofExperts HSD HybridSubpixelDetector HSI hyperspectralimagery HUD HybridUnstructuredDetector IBFC InniteBayesianFuzzyClustering IBFCEM InniteBayesianFuzzyConstrainedEnergyMinimization JSD JointSubspaceDetector LiDAR LightDetectionandRanging LARS Least-AngleRegressionStatistic LMM linearmixingmodel MACE MinimumAverageCorrelationEnergy MAP Maximum aposteriori MCMC MarkovChainMonteCarlo MIL MultipleInstanceLearning MRF MarkovRandomField MTMF MixtureTunedMatchedFilter NCLS NonnegativelyConstrainedLeast-Squares OSP OrthogonalSubspaceProjection PALM PairwiseAdaptiveLinearMatchedFilter PCA PrincipalComponentAnalysis PCE Piece-wiseConvexEndmember PCOMMEND Piece-wiseConvexMultipleModelEndmemberDetection PD ProbabilityofDetection PDF ProbabilityDensityFunction PFA ProbabilityofFalseAlarm 12

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PFSCEM PossibilisticFuzzyandSpatialConstrainedEnergyMinimization RGB RedGreenBlue ROC ReceiverOperatingCharacteristics RP RunPacking RX Reed-Xiaoli SAM SpectralAngleMapper SCAD SimultaneousClusteringandAttributeDiscrimination SCLS Sum-to-OneConstrainedLeast-Squares SDIA Signal-DecomposedInterference-Annihilated SEM StochasticExpectationMaximization SMF SpectralMatchedFilter SNR Signal-to-NoiseRatio SPICE SparsityPromotingIteratedConstrainedEndmember SPSMF SubpixelSpectralMatchedFilter SSD SubpixelSubspaceDetector SVDD SupportVectorDataDescription TCIMF TargetConstrainedInterferenceMinimizedFilter 13

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CONTEXT-DEPENDENTDETECTIONINHYPERSPECTRALIMAGERY By TaylorC.Glenn December2013 Chair:PaulGader Cochair:JosephWilson Major:ComputerEngineering Signicantcontextinformationoftenexistsinhyperspectralimages,buttheexisting known-signaturetargetdetectiontechniquesdonotexplicitlyaccountforthisfactand donottakeadvantageoftheinformation.Thisdissertationexplorestheideathatusing contextinformationimprovesdetectionalgorithmsforhyperspectralimagery.Insupport ofthis,newcontext-dependentdetectiontechniquesaredevelopedtoimprovetheperformanceofdetectionalgorithmsthroughtheuseofcontextinformation.Thesealgorithms includeasequentialunsupervisedcontextlearninganddetectionmethodusingapiecewise convexmodelbasedunmixingapproach.Thesequentialmethodisthenfollowedbynew algorithmsusingjointunsupervisedcontextlearninganddetection.Inserviceofdevelopingthejointcontextlearninganddetectionapproaches,aBayesianformulationforfuzzy clusteringisderived,whichisasignicantadvancetothatarea.AdditionallyanewapproachtofusioncalledAlarm-SetFusionisderived,andbothsupervisedandunsupervised learningalgorithmsarepresentedforthatpurpose.Experimentalresultsonbothsynthetic andrealdataaregiventhatshowthedevelopedmethodsareanadvancementtodetection inhyperspectralimagery.Finally,adiscussionoftheseresearcheortsandfutureresearch directionsispresented. 14

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CHAPTER1 INTRODUCTION Hyperspectralimagerscaptureincominglightatalargenumberofwavelengthsat eachpixel.Whereasatraditionalcolorimageisformedusinglightfromonlythered, green,andblueRGBwavelengthsbecausethesearepreciselythewavelengthsto whichthehumanvisualsystemresponds,ahyperspectralimage,incontrast,measures lightattenstohundredsofwavelengths.Thesewavelengthsarespacedthroughoutthe electro-magneticspectrumfromultraviolet,throughthevisible,andbeyondtoinfrared wavelengths.Hyperspectralimagesthuscaptureaspectrumateachpixellocation. Distinctmaterials,ingeneral,interactwitheachwavelengthoflightinaunique manner,andthusideallyproducedistinctspectralsignatures.Hyperspectralimages canallowformaterialidenticationatthesinglepixelscale,ataskwhichwouldbevery dicultgivenjustanRGBimage.Additionally,whenmultiplematerialsarepresent withintheimagedareaofahyperspectralpixel,theirspectralresponsesarecombinedto formanewspectrum.Thisspectrumhasthepotentialtobeinvertedintoitsconstituent components,providinghyperspectralimageswithaveryuniquecapability,thattheymay beusedtorevealthepresenceofobjectswhicharesmallerthanasinglepixel. Hyperspectralimagesareusedformanytasks.Oneoftheseistheuseofcomputationalalgorithmsonhyperspectralimagedatatoautomaticallydetectthepresenceof targetsubstancesinthescene.Thistechnologyhasbeenappliedtotaskssuchasdisease detectionincitrusplants[1],airbornesearchandrescue[2],anddefenserelatedproblems. Theaimofthisresearchistoimproveuponthestateoftheartindetectionalgorithms.Insurveyingtheacademicliteratureonthissubject,itisevidentthatthereare fourmainthemesformakingsuchimprovements.Theseare: Improvethemodel.Modelingthehyperspectralsignalencapsulatesourknowledgeof itsexpectedformmathematically.Withbettermodelsthesensedsignalorportionsof itcanbeattributedtotargetornon-targetbackgroundsourceswithhigheraccuracy andprecision. 15

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Improvetheestimation.Givenamodelofthesensedsignal,thenextstepisoftento estimatetheparametersofthemodel.Performanceindetectioncanbeaectedby theaccuracyoftheparameterestimates. Preprocessing.Preprocessingcanremovenoisefromtheimagemakingestimation morerobust,oritcanconvertthedataintoadomainwhichismoreappropriatefor thegivenmodel. Fusion.Datafusiontechniquescombinemultiplesourcesofinformationtocreatean outputthatisbetterthananyindividualsource. Theresearchandmethodspresentedhereargueforasimplecentralthesis:Using contextinformationimprovesdetectionalgorithmsforhyperspectralimagery.Insupport ofthisthesis,thisresearchpresentsnewmethodsthatcreatecontext-dependentdetectors. Additionally,toenablethesemethods,anewBayesianapproachtofuzzyclustering isdevelopedwhichismoreexibleandpowerfulthantheexistingfuzzyclustering techniques.Finally,newfusionmethodsarepresentedwhichworkwithcontext-dependent detectors. Ageneraldenitionofcontext[3]isthecircumstancesthatformthesettingforan event,statement,oridea,andintermsofwhichitcanbefullyunderstoodandassessed." Itisinthesenseofthisdenitionthatcontextwillbeusedthroughoutthiswork,though itmaybegivenotherformaldenitionsformathematicaloralgorithmicpurposes.A context-dependentdetectionalgorithmisonewhichprovidesdierentdetectionbehavior contingentuponthecontextassociatedwiththedataundertest. Contextualinformationandcontextdependenceisexhibitedinseveralformsin hyperspectralimageryHSI.Afewcategoriesarelistedhere,buttherearecertainly others. ThersttypeofcontextualinformationencounteredinHSIparticularlyinremote sensingapplicationscomesfromthesolarandatmosphericpropertiesoftheimagecollection.Imagersarepassivedevices;theymeasuretheinteractionoflightemittedbythe sunwiththematerialsofinterest.Theintensityandgeometryofthesolarillumination canaecttheimage,andonEarthitisdependentuponthetimeofday,thedaywithin 16

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theyear,andthelatitudewheretheimagewascollected.Thecharacteristicsoftheatmospherebetweentheobjectsandthesensorcanalsoaecttheimage.Forinstancethe amountofwatervaporintheatmospherecanvaryacrosslocationsandthisaectsthe absorbedandscatteredenergyatvariousbandsthroughoutthespectrum. Reectancecalibrationisacommonlyusedmethodtoaccountforthesesolarand atmosphericcontextualdependences.Accuratereectancecalibrationmaynotalways beavailable,andmethodswhichareapplicableintheradiancedomainareoftendesired. Forsucharadiancedomainmethodtobebroadlyapplicable,itshouldeitherberobust tothesechangesoritshouldbeabletodetectanddenethecontextandperformina context-dependentmanner. Anothertypeofcontextinformationistheobjects,regions,orgroundcoverclassesin theimage.Thespectrumatanyindividualpixelmaybeambiguousfordeterminingthe truecontentofthepixel{especiallyifspectralresolutionistradedforspatialresolution. Howeverifcontextualknowledgeofthegeneralsurroundingobjectofthepixelcanbe determined,thenacontext-dependentdetectorcouldusethistomakeanappropriate decision. Athirdformofcontextualinformationcomesfromlocalchangesinilluminationand shading.Allbuttheleastcomplicatednaturalscenescontainthreedimensionalobjects whichmayhavevariationsinilluminationthroughouttheobjectandwhichcancast shadowsuponotherobjects.Generalreectancecalibrationmethodsdonotaccountfor theselocalchangesinilluminationthroughoutthescene.Somedetectionmethodshave invariancetochangesinthescaleofthespectrumduetoshadow.Ingeneralhowever, itcanbeambiguousifthechangeisduetoilluminationortochangeinthematerial.A betterapproachmaybetodetecttheregionsofshadowandapplyappropriatecontextdependentmethods. ExistingdetectionalgorithmsinHSIdonotexplicitlymodelcontextdependence. Becauseofthistheyaskthewrongquestionsabouthowtoadvanceperformance.Some 17

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existingdetectionalgorithms,discussedinsection2.2.8,useclusteringtopartitionthe spectralfeature-space,thenuseadetectorspecictoeachcluster.Thoughnotinherently abadstrategy,theissuehereisoneofmotivation.Lookingatthosemethods,onemay askthequestions,HowdoIdoabetterjobofclustering?,"orHowdoIndtheright numberofclusters?"Butclusteringisanambiguousproblemwithasubjectiveevaluation ofgoodness.Furthermore,thereisnoguaranteethattheclusteringwillimprovedetection results. Iftheguidingprincipleofcontext-dependentdetectionisused,thenbetterquestions canbeasked.Suchquestionsare,Whatcontextualinformationcanbeusedtoimprove detectionperformance?,"Howcantherelevantcontextualinformationoftheimagebe found?,"andWhatmethodscanbeusedtoexploitthisinformationtomakebetter detectors?"Inthismindset,clusteringmethodsmayormaynotbetheappropriatetool toautomaticallyndthecontextinformation,andimprovingdetectionperformanceisthe overridinggoal. Inthefollowingchapters,thisdocumentwillreviewtheexistingliteraturerelatedto thethesis,presentnewmethodsforcontext-dependentdetection,evaluatetheperformance ofthesemethods,anddiscussfutureworkinthisdirection.InChapter2,acomprehensive reviewofdetectionalgorithmsispresented,aswellasareviewofcontext-dependent methodsbothingeneralandasapplicabletoHSI.Additionallyareviewofunmixing methodsfocusingontherelevantpiecewiseconvexapproachesispresented.Chapter3 presentsnewapproachestodetectionandfusioninHSIusingcontextdependence,aswell asaBayesianapproachtofuzzyclustering.Chapter4showstheresultsobtainedinthe investigationofthenewmethods.Chapter5closeswithadiscussionofthepresentedwork andavenuesforfutureinvestigation. 18

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CHAPTER2 LITERATUREREVIEW 2.1Introduction AlargebodyofliteratureexistsrelatedtohyperspectralimageryHSIanalysis. CommontasksinHSIanalysisincludedetection,classicationandsegmentation,and unmixing.Thisresearchfocusesonimprovingsignature-matchedtargetdetection,and assuchpresentsacomprehensivereviewoftheliteratureonthissubject.Background literatureonunmixingmethodswillalsobecoveredtotheextentthattheyareused inthisworkintheserviceoftargetdetection.Additionally,theliteratureoncontextdependentmethods,ingeneralandasappliedtoHSI,iscoveredaswell.Finally,areview oftherelevantliteratureonfuzzyclusteringisgiven,asthesemethodsareextendedby theBayesianFuzzyClusteringmodeldevelopedinthisdocument. 2.2DetectionMethods ThreebroadtypesofdetectiontasksarecommonlystudiedforuseinHSI.Theseare anomalydetection,signature-matcheddetection,andchangedetection.Signature-matched detectorslookforpixelswhichmaycontainsomeamountofagivenmaterialwitha knownspectralsignature.Ontheotherhand,anomalydetectorsidentifypixelswhichare somehowdierentfromtheirsurroundingbackgroundintheimage,butwhosestructure andappearanceisnotknown.Finally,changedetectionalgorithmsndcorresponding pixelsfrommultipleimageswhichhaveundergonesomemeaningfulchangeinappearance. Thisliteraturereviewisfocusedonsignature-matcheddetectionalgorithms.Muchwork hasalsobeendoneinbothanomalyandchangedetection,butasitisnotthefocusofthis reviewitwillnotbecoveredhereindepth.Eismanncoversthesemethodswellinsections 14.2anomalydetectionand14.6changedetectionofhisbook[4]. 2.2.1CategorizationofSignatureBasedDetectionMethods Thoughitcansometimesbeambiguousintheliterature,signature-matchedtarget detectioniscommonlyreferredtoasjustdetection,"andaspecicsignature-matched 19

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detectionalgorithmisfrequentlysimplycalledadetector."Ingeneraladetectorisa devicewhichtakesasinputahyperspectralimageandthespectraofoneormoretarget signaturesandthenoutputsadetectionscoreforeachpixeloftheinputimage.Alarger detectionscorerepresentsmorebeliefbythedetectorthatthetargetmaterialispresentin thatpixel.Figure2-1showsablockdiagramforthecomponentsinadetectionsystem. Detectionalgorithmscanoftenbespeciedbythefunctiontheyusetoproducethe detectionscoreateachpixel.Inthisdocumentdetectorswillbegivenintheformofa functiondenitionsuchas r name x = f x ; ^ y; ^ z ; {1 where r name x denotesthedetectionstatisticasafunctionofasinglepixel x ,andthe functionmayincludethevaluesofotherestimatedparameterssuchas^ y and^ z Manydetectionalgorithmshavebeenproposedforndingtargetsinhyperspectral images.Thougheachdetectorhassomeuniqueunderlyingidea,somebroadclasses ofdetectorscanbeidentied.Thefollowingsectionsdetailmanydetectorsandthe followingnon-exclusivecategorizationsaregiventohelpguidethereader'sintuitionof thedetector'sbehavior.Table2-1givesalistingofsomecategoriesandplacesdetectors amongthem. Onedistinctionamongdetectorsisinhowthedetectorhandlesvariationinthe signals.Onemethodistomodelthetargetorbackgroundsignalsasbeingrandomly generatedaccordingtoanunderlyingprobabilitydistribution.Thismaybedonein additiontomodelingthesensornoiseprobabilistically.Suchmethodsarecategorizedas statistical methods. Incontrast,anothermethodistondasubspacewhichspanstherangeofpossible valuesandvariationofthetargetorbackgroundsignals,whileideallynotincludingthe otherclassofsignals.Detectorsusingsuchtechniquescanbelabeledas subspace methods. Thephysicsofimaginggivesthatlightreectedfrommaterialswithinapixelis mixedapproximatelyaccordingtothelinearmixingmodelLMM.Thisisespecially 20

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commonwhenpixelshavelargearealextentasiscommoninairborneorsatelliteremote sensingapplications.AcommontechniqueinHSIanalysisisunmixing,orsolvingforthe proportionsandendmembersinthelinearmixingmodel.Somedetectorsuseunmixingin theiralgorithm,andmightbelabeledas unmixing baseddetectors. Anotheraxisofdistinctionamongdetectorsisiftheyaredesignedforndingtargets at sub-pixel proportionsoriftheyaredesignedtodetectonly purepixels ofthetarget signature.Unmixingbaseddetectorsareinherentlylookingatsubpixelproportions,yet otherdetectorsallowforthesubpixeltargetproportionswithoutfullyunmixingthesignal. Eventhoughadetectordoesnotuseasubpixelformulationexplicitly,itmaystillobtain adequateperformancewhenthetargetispresentatsubpixelproportions. Thefollowingsectionsshowthecommondetectorsforhyperspectralimagerythat arepresentintheliterature.Becauseofthemanyaxesofclassicationtheirpresentation orderisnotnecessarilygivenbytheirprimaryclassication,butconceptuallyrelated classiersarelooselypresentedtogether. 2.2.2SpectralAngleMapper OneoftheearliestdetectionalgorithmsdevelopedspecicallyforHyperspectral ImageryistheSpectralAngleMapper[5].Intheir1993paperdescribingtheirSpectral ImageProcessingSystemsuiteoftoolsforanalyzingAirborneVisible/InfraredImaging SpectrometerAVIRISimagery,Kruseetal.proposetheSpectralAngleMapperSAM asamethodforcomparingthespectralsimilarityofapixeltoareferencespectra.The SAMismotivedassimplyndingtheanglebetweentwospectratakenas N -dimensional vectors.Givenatestspectrum x andareferencespectrum s ,theSAMoutputis cos )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x T s jj x jjjj s jj : {2 Largeranglesindicatemoredierencebetweenthetwospectra.Thismeasureisinvariant toilluminationscalefactordierencesbetweenthetestandreferencespectra.Thiswas 21

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notedtobeusefulforcomparinglaboratoryspectratoremotesensingspectrawhichhave unknownilluminationcharacteristics. Laterauthors[4, x 14.3.1]havere-derivedtheSAMmeasureinatraditionalstatistical signalprocessingGeneralizedLikelihoodRatioTestGLRTframework.Underthemodel assumptions H 0 : x = b H 1 : x = s + b {3 where b N 0 ; 2 I istherandombackground, s isthetargetreferencespectrawitha scalefactor toaccountforilluminationormixingeects,and x isthespectraundertest. AConstantFalse-AlarmRateCFARdetectorcanbeshowntoresultfromthemaximum likelihoodestimatesof and 2 .Thedetector r SAM x = s T x 2 s T s x T x {4 ismonotonicallyrelatedtothenegativeoftheinversecosineexpressiongivenin2{2. 2.2.3SpectralMatchedFilter TheMatchedFilterisacentralresultofstatisticalsignalprocessingtheoryandis usedfordetectingaknowndeterministicsignalinadditiveGaussiandistributedrandom noise[6].Whenappliedtohyperspectralimageryfordetectionatthepixel/spectralevel, theformulationofthematchedlterisoftenreferredtoasaSpectralMatchedFilter SMF.Thisisincontrasttoamatchedlterinthetraditionalimageprocessingsenseof matchingthepatternofaresolvedtargettemplate[7].Eismanngivesagoodoverviewand derivationoftheSMF[4, x 14.3.2],andnumerousarticleshavebeenpublishedwhichuse theSMForavariationupontheconcept[2,8{12]. TheSMFcanbederivedbyfollowingtheGLRTframeworkunderthemodelgiven in2{3.UnliketheSAM,wewilluseafullcovariancematrixandapotentiallynonzero meanvectorforthebackground b N ; .Theresultofmaximizingthelikelihoodof 22

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theunknownparameters and isthedetectionstatistic r SMF x = [ s T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ ] 2 s T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s {5 where ^ and ^ arethesamplebackgroundmeanandcovarianceestimatedfromthe image. ThetraditionalSMFhasalimitationinhyperspectralimageryinthatitassumesthat thetargetsignalisaddedtoanexistingbackgroundsignalvsreplacingthebackground signalwithtargetinformation.Toalleviatesomeofthiseect[4, x 14.3.2],thetarget signature s canbeadjustedto s )]TJ/F40 11.9552 Tf 12.66 0 Td [( .Secondly,theSMF'ssquaredstatisticcanresult inpositiveoutputsforspectrathataresimilartothetargetbutorientedintheopposite direction.Assuch,forhyperspectralimageryapplications,takingthepositivesquare rootoftheSMFresultsinamorerobustdetectionstatistic.Finally,thedenominatorof theSMFfractionisaconstantvalueandcanbesafelyignoredtogetthemonotonically relatedone-sidedSMFstatistic, r SMF x = s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ : {6 2.2.4ConstrainedEnergyMinimization TheConstrainedEnergyMinimizationCEMlterisalinearlteroriginally developedbyHarsanyi[13].Ithassincebecomeastandardtopiccoveredinthedetection literature[4, x 14.3.3],[14, x 4.2.1],andithasbeenusedsuccessfullyinapplications[15,16]. ItcanalsobenotedthatCEMiseectivelyanapplicationoftheMinimumAverage CorrelationEnergyMACElterbyMahalanobisetal.[17]tospectralcorrelation insteadofspatialcorrelation. TheCEMlterisformedbylookingforalinearoperator w ,appliedas r x = w T x whichisconstrainedsuchthattheoutputonpuretargetspectraisone, r s = w T s =1. Amongsuchoperators,theonewithminimumenergyoverallimagepixelsisfound,giving theCEMoperator. 23

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Formally,denetheaggregateenergyofthelteroverallpixelsinanimage x i tobe E = 1 N N X i =1 r 2 x i = 1 N N X i =1 x T i w T x T i w = w T 1 N N X i =1 x i x T i w = w T Rw {7 where R isthesamplecorrelationmatrixoftheimage. Theminimizationproblembecomesthen, min w w T Rw subjectto w T s =1 : {8 Thesolutiontothisproblemisgivenby, w = R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 s s T R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s ; {9 whichyieldsthedetector r CEM x = s T R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x s T R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s : {10 ThoughitisalinearlterliketheSMF,theCEMlterhasadierentconceptual underpinningandderivationandcanperformdierentlyfromtheSMF[4, x 14.3.3].Ifthe samplemeanoftheimagedataissubtractedbeforeapplyingCEMhowever,theCEMand theone-sidedSMFresultinthesameoutputtowithinascalefactor. ShiandYang[18]proposeaconstrainedlinearlterwhichminimizedtheexpected valueofahigher-orderstatisticsuchasKurtosis.Thoughconceivedofasaformofrobust matchedlter,initsderivationitismoreconceptuallysimilartotheCEMlter. 2.2.5TargetConstrainedInterferenceMinimizedFilter TheconceptsoftheCEMlterwereextendedbyRenandChangtocreatethe TargetConstrainedInterferenceMinimizedFilterTCIMF[19,20].TheTCIMFaddsto 24

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thedetectortheconceptofusingknownsignaturesofundesiredtargets.Ifsuchknown undesiredsignatureswereprovided,theCEMlterwouldtreatthemnodierently frombackgrounddatareferredtoasinterferersinthepublicationsfromChang'sgroup, aggregatedin[14],whereastheTCIMFattemptstospecicallysettheresponseofthe ltertoknownundesiredsignaturestobe0eventhoughtheymaybequitesimilartothe knowndesiredsignatures. TheTCIMFisderivedbyextendingthesingletargetconstraintofCEMtobe constraintvector c whichwillconsistofentriesof1forknowndesiredtargets,andentries of0forknownundesiredtargets.Formally,let D =[ d 1 d 2 ::: d p ]and U =[ u 1 u 2 ::: u q ] bethematricesofdesiredandundesiredtargetsignatures,respectively.Thendenethe targetsignaturematrix M =[ DU ]andtheconstraintvector c =[ 1 1 p 0 1 q ] T ,sothatthe constrainedobjectivebecomes min w w T Rw subjectto M T w = c : {11 Theoptimalsolutiontothisobjectiveisthen w TCIMF = R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 M M T R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c ; {12 whichyieldsthedetector r TCIMF x = w TCIMF T x : {13 WhiletheCEMlterisanobvioussubcaseoftheTCIMF,lessobviousconnections betweentheTCIMFandOrthogonalSubspaceProjectionOSPformulationsare mentionedin[20]andexploredin[14, x 4.3].DuandChangfollowedtheTCIMFidea in[21]withtheSignal-DecomposedInterference-AnnihilatedSDIAapproach.TheSDIA methodsusetheideasof apriori knownundesiredtargets,andaddinestimationof a posteriori discoveredbackgroundsignatures.Severaldetectorscanthenformulatedinthis frameworkbasedonGLRT,OSP,andTCIMFderivations. 25

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2.2.6AdaptiveCosine/CoherenceEstimator TheAdaptiveCosine/CoherenceEstimatorACEcanbederivedfromtheGLRT withthefollowingbackgroundreplacementmodels H 0 : x = 0 b H 1 : x = s + 1 b ; {14 where and areunknownparameters,andthebackgroundisassumedtobedistributed accordingtoazeromeanGaussianwithunknowncovariance, b N 0 ; .Duetothe zeromeanbackgroundassumption,thesamplemeanoftheimageshouldberemovedfrom allimagepixelsandtargetspectrabeforeoperationofthisdetector. ThesignalreplacementmodelmakestheACEdetectormoresuitablethanthe purelyadditivemodelseenintheSMFforthesubpixeltargetresponsesoftenseenin hyperspectralremotesensingimagery. Thedetectorundermaximumlikelihoodestimatesoftheunknownparametersis showntobe r ACE x = s T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x 2 s T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s x T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x : {15 AmoreaccuratedescriptionoftheACEdetectorasitisoftenusedinapplicationsis however r ACE x = s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ 2 s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ : {16 Ifweviewthisdetectorasrstadaptivelywhiteningboththetargetsignature andpixelundertestbythebackgroundstatistics,forming ~ s = ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = 2 s )]TJ/F15 11.9552 Tf 14.566 0.166 Td [(^ and ~ x = ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = 2 x )]TJ/F15 11.9552 Tf 13.301 0.166 Td [(^ ,then2{15istheequivalentoftheSpectralAngleMapper2{4inthe adaptivelywhitenedspace[22].HencethenameAdaptiveCosineEstimatorisappropriate forthisdetectoreventhoughitisperhapsnotobviousfromitsderivation. TheformalderivationbyKrautandScharofthisdetectorasaGLRTwasrst givenin[23]whichwascloselyrelatedtoKelly'searlierworkondetectioninradar 26

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applications[24].Earlierheuristicformulationsofthedetectorcanalsobefoundinthe literature[25,26].KrautandScharpublishedseveralpapersaboutACEandrelated detectors[23,27,28].Ithasalsoenteredthestandardcanonofhyperspectralimaging detectionliteratureandiscoveredin[22,29],[4, x 14.3.4].TheuseofCosine/Coherencein thenameseemstohavebecomestandardinthemodernliterature,whereasearlypapers referredtoitastheAdaptiveCoherenceEstimatormostfrequently,andtheCosinename followsfromtheearliermentionedheuristicconnection. 2.2.7SubpixelSpectralMatchedFilter TheSubpixelSpectralMatchedFilterSPSMFproposedin[4, x 14.3.5]isaGLRT derivedalgorithmunderahypothesismodelthatshouldbemoresuitableforhyperspectral imagerythaneithertheSMForACEmodels.TakingtheACEmodelasastartingpoint, theSPSMFremovesthescaleterm underthenullhypothesis,andallowsforanonzero backgroundmean.Thusthemodelhypothesesare H 0 : x = b N ; H 1 : x = s + b : {17 Theparameters and areestimatedbythesamplemeanandcovarianceofthe trainingbackgrounddata,whiletheunknowns and areestimatedviathemaximum likelihoodmethodonperpixelbasis.Thisestimationyieldsthefollowingequationsforthe detector: r SPSMF x = x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ )]TJ/F15 11.9552 Tf 13.15 8.088 Td [( x )]TJ/F15 11.9552 Tf 13.115 0 Td [(^ s )]TJ/F15 11.9552 Tf 13.639 3.155 Td [(^ ^ T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.115 0 Td [(^ s )]TJ/F15 11.9552 Tf 13.64 3.155 Td [(^ ^ ^ 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 K ln ^ {18 where K isthenumberofbandsinthespectra,and ^ = s T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F15 11.9552 Tf 13.64 3.155 Td [(^ s T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 s ^ = )]TJ/F21 11.9552 Tf 9.299 0 Td [(a 1 p a 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 a 2 a 0 2 a 2 27

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a 0 = x T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x s T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 s )]TJ/F15 11.9552 Tf 11.956 0 Td [( s T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s 2 a 1 = s T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x s T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ^ )]TJ/F15 11.9552 Tf 11.955 0 Td [( s T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s ^ T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x a 2 = )]TJ/F21 11.9552 Tf 9.298 0 Td [(K s T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 s : Thesignalreplacementmodelgivenby2{17shouldbemoresuitablethanthe additivemodel2{3underwhichtheSMF2{6isderived.Itshouldbenotedhowever that and arenotconstrainedinthismodeltobeinphysicallymeaningfulranges 0 ; 1with + =1,anditisunclearwhethervaluesoutsideoftheserangesoccur inapplication.Thoughinterestinginitsderivation,itwasobservedin[4, x 14.3.5]thatthe SPSMFdoesnotshowaperformanceadvantageoversimplerdetectorssuchasACEor eventhenon-subpixelSMF. 2.2.8SpectralMatchedFilterswithBackgroundClustering Severalresearchershavenoticedthattheperformanceofdetectionalgorithmsmaybe improvedbyrstusingaclusteringalgorithmtopartitionthespectralfeature-spaceofan imageintoregions,andthenusingaseparatedetectorforeachfeature-spaceregion.Two primaryapproachestothishavebeentakentoidentifythefeature-spaceregions,therst usingGaussianmixturemodels,andthesecondusingK-meansclustering.Thepublished literatureseemstofavorfollowingtheclusteringstepwithmatchedlterbaseddetectors, thoughwenotethatanydetectionstatisticcouldbeused. 2.2.8.1UsingGaussianmixturemodels ThepreviouslydiscussedmodelshaveusedaunimodalmultivariateGaussian distributiontocharacterizetheimagebackground.Oneextensionofthesemodelsis touseaGaussianmixturemodelinstead.Thisintheory,thoughnotnecessarilyin theseapplicationsallowsforaprincipledextensionofthemodel-basedGLRTdetection approachesthatcapturesthemulti-modalnatureoftrueimagery.Eismann[4, x 14.3.6] givestwoexampledetectorsusingthisidea. 28

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TherstistheClassConditionalSpectralMatchedFilterCCMF.IntheCCMF, theimagepixelsarerstdividedintomultipleclassesbyaGaussianmixturemodel.For eachclass q ,thesamplemean ^ q andcovariance ^ q arecomputed.Thedetectoroutputis thencomputedasaSMF,wheretheclassassignmentforthepixelistreatedasknown a priori information,yielding r CCSMF x = s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ q T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 q x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ q : {19 AseconddetectorleveragingtheGaussianmixturemodelisknownasthePairwise AdaptiveLinearMatchedFilterPALMlter.InthePALMdetector,themostsimilar backgroundclasstothepixelundertestisused,yielding r PALM =min q s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ q T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 q x )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ q : {20 OneotherreferencetoPALMwasfoundonline[30],buttheactualpublicationfor thisreferencewasnotavailable.Asitisassumedthatthebackgroundcomesfromthe sameunderlyingclassinboththe H 0 and H 1 hypothesis,itseemsreasonabletoassume thatthepairwise"portionofthisdetector'snamereferstothepairsofbackground classescreatedbythetwohypotheses.Withinaselectedclassthenwehaveastandard adaptivelinearmatchedlteri.e.aSMF. Thoughagoodideaintheory,ithasbeennotedinEismann[4, x 14.3.6]thatthese detectorsdonotalwaysworkwellinpractice.Eismannnotesthatthisisperhapsbecause oftheinuenceofoutlierortargetpixelsonthestatisticsofthesmallerfeature-space regions.Wenotethatthiscouldalsobeduetodistributiondierencesbetweentheclass outputs,whichmaybehandledbytheAlarm-SetFusionmethodsdevelopedinsection3.5. 2.2.8.2UsingK-means Funketal.[10]proposeusingavariationofK-meanstoclustertheimagewhich usesarandomsubsetoftheimagepixelsateachiteration,presumablyforperformance reasons.Theirdetectorisaclassconditionalmatchedlterasin2{19.Theypropose 29

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aninitializationstrategybaseduponextremepixelvaluesineachprinciplecomponent axisthatpromotesamorerapidandstableconvergence.Mostinterestinglytheyshowremarkableimprovementsindetectionperformanceoftracegasversusaglobalbackground SMFasthenumberofclustersincreasesuptowhatcouldbeconsideredalargenumber of22,thoughsubjectiveperformancediddegradewithevenhighernumbersofclasses. BajorskipresentsamethodofmakingdetectorsfrommultipleclusterscalledGeneralizedDetectionFusion[31],andevaluatesoneimplementationofthiscalledaMax-Type Detectorin[32].Herstpresentsamethodforclusteringthespectraofahyperspectral imagebaseduponK-meansthathecallsFlexibleSpectralClustering.Hethendenes fuseddetectorsoverthesegmentsdenedbytheclustering.TheMaxMintypedetectoris denedas r MaxMin x =max 1 2 1 min 0 2 0 D 0 : 1 x ; {21 where D 0 : 1 x = p x ; 1 p x ; 0 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 0 ; 1 {22 isalikelihoodratioforthetwoclasshypothesistestdenedby2{3, 0 ; 1 isafunctionforthedecisionthresholdwhichcanbechosentogiveCFAR,ConstantProbability ofDetectionCPD,orstandardgeneralizedlikelihoodratiobychoosingasingleconstantforallvaluesof 0 ; 1 performance[31].Thesets 0 and 1 denotethemeanand covarianceparametersforeachofthesegmentsfoundthroughtheinitialFCSclustering algorithm.SimilarlytotheMaxMindetector,aMinMaxdetectorisdenedbyreversing theorderoftheminandmaxin2{21. 2.2.9OrthogonalSubspaceProjection TheOrthogonalSubspaceProjectionOSPdetectorwasrstproposedbyChangand Harsanyi[33].Theprimaryideabehindthisdetectoristhedevelopmentofaprojection operator P B ? whichprojectsanyvectorintotheorthogonalcomplementofthesubspace spannedbythematrixofbackgroundundesired"inChang'snotationresponses B .The followingdescriptionroughlyfollowsEismann'ssummaryofOSPin[4, x 14.3.7]. 30

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OSPassumesthatHSIsignalsaregovernedbythefollowingmodel: H 0 : x = B + n H 1 : x = s + B + n : {23 Inthismodelthebackgroundbasis B spansasubspacethespaceofitscolumnsandis notassumedtobestatisticallydistributed.Thenoisevector n isassumedtoberandom, butnodistributionassumptionsneedtobemadetoderivethedetector. Thebackgroundbasis B isestimatedfromthedata,usuallybytakingsomenumber M
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whichhasamaximumat h = s .ThisyieldstheOSPdetector r OSP x = s T P B ? x : {28 EismannfurthergivesaderivationofOSPundertheGLRTframeworkfollowing[34] whichyieldsnoiseadaptiveversionsofthedetector. Changandhisstudents/collaboratorscontinuedtodeveloptheOSPideasoverthe yearssincetheirintroduction.IntheinitialversionsofOSP,themixingproportionsof thetargetsignaturewereassumedknown apriori ,thiswasextendedtouseleastsquares estimationoftheproportionsin[35].TheLeastSquaresOSPwasfurtherextendedin[36] tocreatetherelatedSignatureSubspaceProjection,TargetSubspaceProjection,and ObliqueSubspaceProjectiondetectors.TheUnconstrainedGaussianMaximumLikelihood detectorof[37]isshownin[38]tobeaspecialcaseoftheOSPdetectorswhenthenoise distributionisassumedtobeGaussian.Chang'sbook[14, x 3.3]hasacomprehensive overviewofOSPanditsfollow-ons. Matteolietal.[39]proposedanextensiontotheOSPideawhichestimatesalocal backgroundsubspaceperpixel.Thelocallyestimatedsubspaceisthenremovedbyan orthogonalprojectionfromboththepixelundertestandthetargetsignatureandthen astandardspectralmatchedlter2{6isapplied.Matteolietal.showanorderof magnitudeormoreperformancegainoverusingasingleglobalbackgroundsubspace. 2.2.10TargetSubspaceDetectors TheACEdetectorcanbeformulatedusingasubspaceoftargets, S ,insteadofsimply asingletargetsignaturevector s asin[4, x 14.3.4].Infact,theACEdetectorismore oftenseenwiththetargetsubspaceformulation[22,29],[4, x 14.5.2].Whenusingatarget subspace,theACEdetectorbecomesameasureoftheanglebetweenthepixelunder testandthehyperplanespanningthecolumnsofthetargetsubspacematrix S inthe 32

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adaptivelywhiteneddomain[29].ThesubspaceACEdetectorisgivenas r SS )]TJ/F20 7.9701 Tf 6.587 0 Td [(ACE x = x T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S S T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x x T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x : {29 Ifboththebackgroundandtargetareassumedtoberepresentablebysubspaces andwithzeromeanGaussiannoiseandknowncovarianceaddedbythesensor,thenwe caneasilyformulateadetectorknownastheJointSubspaceDetectorJSDfromthe likelihoodratio[4, x 14.5.3].Giventhemodel H 0 : x = B + nn N 0 ; n H 1 : x = S + n ; {30 thisyieldsthedetectionstatistic r JSD x = x T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n S S T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n S )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n x )]TJ/F37 11.9552 Tf 11.955 0 Td [(x T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 n B B T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 n B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 B T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 n x : {31 Formingtheprojectionoperators P S = S S T S )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S T ,and P B = B B T B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 B T ,and assuminganisotropicnoisecovariance N = 2 n I ,yieldsthesimplerformulationof r JSD x = 1 2 n x T P S )]TJ/F37 11.9552 Tf 11.955 0 Td [(P B x : {32 TheJSDformulationisnotasubpixelmodel.ExtendingtheJSDmodeltoallowfor potentialsubpixeltargetsyieldsthehypotheses H 0 : x = B + nn N 0 ; n H 1 : x = S + B + n : {33 Thismodelthenyieldsthedetector,whichEismanncallstheSubpixelSubspaceDetector SSDandwhichManolakiscallsaClairvoyantDetectorcalledsobecauseitknowsthe noiselevel 2 [40], r SSD x = 1 2 n x T P SB )]TJ/F37 11.9552 Tf 11.955 0 Td [(P B x = 1 2 n x T P B ? )]TJ/F37 11.9552 Tf 11.955 0 Td [(P SB ? x {34 33

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where SB =[ SB ]istheconcatenationofthetargetandbackgroundsubspacesand P SB isthecorrespondingprojectionoperator P SB =[ SB ][ SB ] T [ SB ] )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 [ SB ] T : {35 Thesecondrighthandsidetermin2{34showstheequivalentdetectorwith orthogonalprojections P B ? = I )]TJ/F37 11.9552 Tf 12.661 0 Td [(P B and P SB ? = I )]TJ/F37 11.9552 Tf 12.662 0 Td [(P SB inordertodemonstrate thecommonalityofthisdetectortoformulationsoftheAdaptiveMatchedSubspace detector2.2.11andtheGLRTderivationsoftheOrthogonalSubspaceProjectiondetector in[4, x 14.3.7]. 2.2.11AdaptiveMatchedSubspaceDetector InthecasewherethevarianceofthenoisedistributionintheSSDmodel2{33is unknownandevenpotentiallynon-stationary,wecanfallbackontheGLRTframework. DoingsoyieldstheAdaptiveMatchedSubspaceDetectorAMSD,alsocalledthe AdaptiveSubspaceDetectorin[4, x 14.5.4] r AMSD x = x T P B ? )]TJ/F37 11.9552 Tf 11.955 0 Td [(P SB ? x x T P SB ? x : {36 Thederivationandtheoreticalanalysisofthisdetectorforhyperspectralimagery isgivenin[40],whichitselfwastakenfromearlierwork[25,34].Manolakisetal.have publishedextensivelyabouttheAMSD[22,29,41].ThaiandHealyalsoderivethis detectorandcallitanInvariantSubpixelDetector[42].Intheirworkthetargetsubspace comesfrommappingasingletargetreectancesignatureintotheradiancedomainovera rangeofatmosphericandilluminationparametersettingsandthenndingbasisvectors forthegeneratedradiancesignatures. 2.2.12ConstrainedLeastSquaresAbundanceEstimation HyperspectraldataisoftenrepresentedbytheLMM,whichgivesthatthelight receivedforeachpixelisacombinationofthelightreectedfromeachmaterialwithin thepixel'sareainproportiontoitsabundancewithinthatarea.Analysisofthespectral 34

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mixtureisanimportantresearchtoolandalgorithmsforunmixingareanactiveareasof research. Formally,thelinearmixingmodelgivesthatahyperspectralpixel x istheweighted sumofsomenumberofendmembersubstances e i for i =1 :::M x = M X i =1 e i a i {37 wheretheabundancesoftheendmembersaresubjecttotheconstraintthattheysumto onetheAbundanceSumConstraintASC M X i =1 a i =1 ; {38 andalsoaresubjecttotheconstraintthatallabundancesarenon-negativethe AbundanceNonnegativityConstraintANC a i 0 ; 8 i =1 :::M: {39 Intheory,ifthehyperspectraldatafollowalloftheassumptionsoftheLMMand couldbeunmixedusingalloftheconstraints,thentheabundanceoftheendmember correspondingtoatargetofinterestwouldmakeaperfectdetectionstatistic.Eventhough theLMMandunmixingarenotperfectinreality,usefuldetectorscanstillbederived throughabundanceestimationfromunmixingmethods. IfonlytheASCisemployed,thenanecientleastsquaresmethodofestimating abundancesknownasSum-to-OneConstrainedLeast-SquaresSCLScanbeused[43,44], [14, x 3.4].Posedasanoptimizationproblem,SCLSminimizestheresidualmagnitude betweenthesensedpixel x anditsapproximationfromendmembers E =[ e 1 ::: e M ], min a x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea T x )]TJ/F37 11.9552 Tf 11.956 0 Td [(Ea subjectto M X i =1 a i =1 : {40 35

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Thesolutiontothiscanbederivedas[14, x 3.4], a SCLS = P ? M; 1 a LS x + E T E )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 T E T E )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 {41 where a LS x = E T E )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 E T x isthestandardleastsquaressolution,theprojection operatorisdenedas P ? M; 1 = I LxL )]TJ/F15 11.9552 Tf 11.955 0 Td [( E T E )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 T E T E )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; {42 andthevector 1 isan M dimensionalcolumnvectorofallones. IfonlytheANCconstraintisemployed,thenaNonnegativelyConstrainedLeastSquaresNCLSapproachcansolvethecorrespondingoptimizationproblem.The nonnegativelyconstrainedleastsquaresproblemingeneraliscoveredin[45,46],while somespecicapplicationstoHSIareexploredin[47],[14, x 3.5].ThoughtheNCLS methodsmaynotbeasaccurateasafullyconstrainedapproachtounmixing,ithas beenshown[48]thatNCLSmethodsmaymakeabetterdetectorthanFCLSevenifthe abundanceestimatesarelessaccurate. TheNCLSmethodattemptstooptimizethefollowingproblem: min a x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea T x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea subjectto a i 0 ; 8 i =1 :::M: {43 UsingLagrangemultipliers,theequationsforthesolutioncanbefoundas a NCLS = E T E )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 E T x )]TJ/F15 11.9552 Tf 11.955 0 Td [( E T E )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = a LS x )]TJ/F15 11.9552 Tf 11.955 0 Td [( E T E )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; {44 where = E T x )]TJ/F37 11.9552 Tf 11.956 0 Td [(Ea NCLS {45 andtheoptimalsolutionisthenfoundonlybyiteratingbetweenequations2{44and 2{45.AfastalgorithmforsolvingNCLSproblemswasgivenin[46]andrecounted by[14, x 3.5]inapplicationtohyperspectralimagery. 36

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IfboththeASCandtheANCareapplied,thenaFullyConstrainedLeast-Squares FCLSalgorithmcanbeusedtosolveforendmemberabundances[48],[14, x 10.2].The equationsfromtheNCLSsolution2{44,2{45canbeusedtoapproximatelycomputean FCLSsolutionbyreplacing E by ~ E and x by ~ x giveninthefollowingequations: ~ E =[ E1 ] T {46 ~ x =[ x 1] T {47 where isasmallnumbersay1 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 whichcontrolsthetrade-obetweennonnegativityandsum-to-oneconstraints[49]. HeinzandChang[48]giveaniterativealgorithmforsolvingtheFCLSproblemthat shouldbecomputationallyecientforHSIapplicationsandproduceanswerswithin acceptableboundsofaccuracytotheoptimalanswers. Itshouldbenotedthat,givenknownendmembers,theabundancescanalsobesolved bygeneralquadraticprogramming[43,50,51],whichhasbecomecomputationallytractable withmodernsolverimplementations.Theunmixingandendmemberestimationproblem isoftenencounteredinhyperspectralimageanalysis,andmanycommonunmixing approacheswereexaminedin[52]. 2.2.13HybridSubpixelDetector Theuseofsimpleabundanceestimatesfromunmixedhyperspectralimagerydoes notalwaysprovidethebestdetectionalgorithm.Broadwateretal.[49,53]haveproposed detectionstatisticsthatperformbetterthantraditionalnon-unmixingbaseddetectorsand thatmaybemorerobustinthefaceofinaccurateendmemberandabundancecalculations thansimplyusingtheabundanceasadetectionstatistic. TheHybridStructuredDetectorHSDhasaderivationthatisrootedinManolakis etal.'sAMSD[40]andHeinzandChang'sFCLS[48]algorithm.Startingwiththe 37

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hypotheses H 0 : x N Ba b; 0 ; 2 0 \051 H 1 : x N Sa s + Ba b; 1 = Ea ; 2 1 \051 ; {48 theHSDisderivedusingtheGLRTframework.Thelocalnonstationarityscalingparametersofthenoisecanbeestimatedas ^ 2 0 = 1 L x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ba b; 0 T )]TJ/F24 7.9701 Tf 7.314 4.937 Td [()]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ba b; 0 {49 ^ 2 1 = 1 L x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea T )]TJ/F24 7.9701 Tf 7.314 4.937 Td [()]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea : {50 Theabundancesarethenestimatedusingtwoinstancesofanoiseadjustedvariantof FCLSwhichminimizestheobjective min a x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea T )]TJ/F24 7.9701 Tf 7.314 4.936 Td [()]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F37 11.9552 Tf 11.955 0 Td [(Ea subjectto a i 0 8 i; X i a i =1 : {51 Giventheestimatesoftheparameters,thenaldetectionstatisticisfoundas r HSD x = x )]TJ/F37 11.9552 Tf 11.956 0 Td [(B ^ a b T )]TJ/F24 7.9701 Tf 7.315 4.338 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F37 11.9552 Tf 11.955 0 Td [(B ^ a b x )]TJ/F37 11.9552 Tf 11.955 0 Td [(E ^ a T )]TJ/F24 7.9701 Tf 7.314 3.454 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F37 11.9552 Tf 11.955 0 Td [(E ^ a : {52 TheoriginalHybridSubpixelDetectorHSD[53]assumedthatthesensornoisewas scaledwhitenoiseequalto 2 I ,whichthencanbeomittedfromtheequation2{52. HowevertheHybridStructuredDetectorcanbeconsidertosupplantthisoriginalHybrid SubpixelDetectorderivation,andisnowwhatisactuallyreferredtowhenusingthename HybridSubpixelDetectorortheacronymHSD. Anotherhybriddetectorgivenin[49]istheHybridUnstructuredDetectorHUD. ThisdetectorisderivedfromtheACEdetector[22]which,whenstatedintheterminology of[49],hasthezeromeanbackgroundmodel H 0 : x N 0 ; 2 0 \051 H 1 : x N Sa s ; 2 1 \051 : {53 38

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BroadwatergivestheHUDdetectoras r HUD x = x T )]TJ/F24 7.9701 Tf 7.314 4.338 Td [()]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S ^ a s x T )]TJ/F24 7.9701 Tf 7.314 3.454 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x ; {54 wheretheestimateoftheabundancevectorforthetargetsignaturesisgivenbytheFCLS unmixingof2{51. Thoughnotstatedin[49]itseemsappropriategiventhezeromeanbackground assumptionof2{53thatthebackgroundestimateberemovedbeforeapplyingtheHUD asisdonebeforeapplyingACEin[4, x 14.3.4].Givenamodiedpixel z = x )]TJ/F37 11.9552 Tf 12.238 0 Td [(Ba b; 1 ,the modiedHUDwouldbe r MHUD z = z T )]TJ/F24 7.9701 Tf 7.314 4.338 Td [()]TJ/F20 7.9701 Tf 6.587 0 Td [(1 S ^ a s z T )]TJ/F24 7.9701 Tf 7.315 3.454 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 z : {55 In[54],Zhangetal.demonstrateimprovedperformanceoftheHSDandHUD detectorsbyusinganalternativeendmemberselectionalgorithminsteadofFCLSto generateabundancevalues.Theirmethodusesagreedyheuristicmethodsimilarto matchingpursuits[55]whichselectsthenextendmemberbaseduponitscross-correlation withtheunapproximatedresidual. 2.2.14ConstrainedSignalDetector TheConstrainedSignalDetector,originallyderivedin[56]andthenextendedbetoa familyofdetectorsin[57]whichisanimpressivepaperwithover125equationsgivenin just15pages,isaGLRTbaseddetectorwhichrespectssomeoftheconstraintsofthe linearmixingmodel.Theuseofthelinearmixingmodelconstraintseectivelymakesthis detectorasubpixelversionoftheclassicalGLRTdetectorsliketheSMForACE,butthe formofthedetectorisalsorelatedtothesubspaceprojectionbaseddetectorslikeOSP andAMSD.Theintuitionhereisthatthedetectorworksliketheunconstrainedsubspace detectorswhichsubtracttheprojectionintothebackgroundsubspace,butthenapplies correctionfactorswhichconstrainthebackgroundabundancecoecientstosumtoone. Followingthisintuition,theCSDdetectorshouldreducetotheHybridSubpixelDetector iftheunmixingisperformedasaseparateoperationbeforeapplyingthedetector. 39

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TheCSDfamilyofdetectorsassumesalinearmixingmodelwithadditiveGaussian noise.Thisgivesthemodelhypothesesof H 0 : x = Ba b; 0 + n ; n N 0 ; 2 0 I H 1 : x = Sa s + Ba b; 1 + n ; n N 0 ; 2 1 I ; {56 subjecttotheconstraints a T b; 0 1 = a T b; 1 1 + a T s 1 =1 ; {57 where B isa D dimensionby M elementmatrixofthebackgroundendmembersascolumn vectors. Perhapsthemostcommonlyapplicabledetectorderivedin[57]istheconstrained CFARdetectorforarankonetargetmatrix,thoughthepaperalsogivesformsofthe detectorsforthecaseofmultipletargetvectorsanddetectorsforthecasewhenthenoise varianceisknown. TheCFARconstrainedsignaldetectoris r CSD x = D )]TJ/F21 11.9552 Tf 11.956 0 Td [(M c B T c B c SB T c SB s T P B ? s 1 = 2 c B T c B s T P B ? x + )]TJ/F37 11.9552 Tf 11.955 0 Td [(c B T s )]TJ/F37 11.9552 Tf 11.955 0 Td [(c B T x x T P SB ? x + )]TJ/F38 7.9701 Tf 6.587 0 Td [(c SB T x 2 c SB T c SB 1 = 2 ; {58 where c B = B B T B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 {59 c SB = c B + P B ? S S T P B ? S )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 1 )]TJ/F37 11.9552 Tf 11.956 0 Td [(S T c B ; {60 andtheorthogonalprojectionoperators P B ? and P SB ? arethesameasusedinother subspacedetectors2{24,2{25,2{35. 2.2.15QuadraticMatchedFilter EismanngivesaformulationwhichhecallstheQuadraticSpectralFilterfora substitutionmodelofhyperspectraldetectionwhichtakesintoaccountdistributionsfor 40

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thetargets,background,andsensornoise[4, x 14.4.1].Thisgivesthehypotheses H 0 : x = b + n H 1 : x = s + n ; {61 whereweassumemultivariatenormaldistributionsforthebackground b N b ; b targetsignature s N s ; s ,andsensornoise n N 0 ; n .Assumingthatthemeans andcovariancesareknownorareestimatedfromthedata,thedetectionstatisticis r QSF x = x )]TJ/F40 11.9552 Tf 11.955 0 Td [( b T b + n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F40 11.9552 Tf 11.956 0 Td [( b )]TJ/F15 11.9552 Tf 11.955 0 Td [( x )]TJ/F40 11.9552 Tf 11.956 0 Td [( s T s + n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F40 11.9552 Tf 11.956 0 Td [( s +log j b + n j j s + n j : {62 ThedetectionproblemundertheseassumptionsGaussiandistributionsandno subpixelmixingandassumingequalclasspriorprobabilitybecomesthewellknowntwo classBayesianclassier[58]whichresultsinaquadraticdecisionsurface. 2.2.16WaveletBasedDetectors In[59,60],Bruceetal.proposeamethodofsubpixeltargetdetectionusingwavelet analysis.Theirmethodisquitedierentfromstandardhyperspectralimagedetection techniquesinthatitlooksonlyforcharacteristicabsorptionfeaturesduetothepresence oftargetmaterials.Inthiswayitismoresimilartotraditionalspectroscopytechniques, anditisclaimedtobeusefulfortracegasdetectioninremotesensinghyperspectral imageryfromtheHYDICEsensor.Firstadatasetofspectrabothwithandwithout thecharacteristicabsorptioniscreated.Thenthesespectraaredecomposedalonga waveletbasis,andasinglemostdiscriminativefeatureisselectedusinglineardiscriminant analysis.Finallyaclassieristrainedtodiscriminatespectrawithandwithoutthe absorptionanomaly. Salvadoretal.morerecentlyproposedadetectionmethodbasedupondecomposition ofspectraintoawaveletpacketsubspace[61].Asetofbestbasiselementsforboth thetargetspectraandthebackgroundspectraareindependentlyselectedviasome 41

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objectiveL1norminthiscase.Thenanyoverlappingbasiselementsfromthetargetand backgroundarethrownout.Thespectralanglebetweenpuretargetandthebackground pixelsisthencomputedusingtheselectedwaveletpackettransformbasesinsteadofthe originalfullspectra. Thoughalsoproposedfortracegasdetection,themethodin[61]couldmorereadily beappliedtogeneraltargetdetectionthanthewaveletbasedcharacteristicabsorption detectorsgivenin[59,60].Forexampleoncethewaveletbasishasbeenselected,mostof thedetectionalgorithmsreviewedhereshouldbeapplicablebysimplysubstitutingthe waveletbasiscoecientsfortheoriginalspectra.Thewaveletpacketdecompositioncould beviewedinthislightasapreprocessingstepakintoPCA. 2.2.17KernelBasedMethods KwonandNasrabadihavederivedkernelspaceversionsofmanysignaturedetection methodsusedinhyperspectralimagery[12,62{64]aswellastheReed-XiaoliRX anomalydetector[65]. Theideahereisthatthehyperspectraldataismappedusingapotentiallynonlinear mappingintoaevenhigherdimensionalfeaturespace.Theoriginaldetectionalgorithms arethenperformedinthishigherdimensionalspace.Thiseectivelyperformsacomplicatednonlinearoperationintheoriginalfeaturespacebyusingasimplelinearoperation inthehighdimensionalspace.Workinginthehighdimensionalspacedirectlyhowever maynotbecomputationallyfeasible.So,usingwhatisknownastheKernelTrick,"the operationcanbemodeledintermsofinnerproductsinthekernelspacewhichhavesome easilycomputedrepresentationintheoriginalspace. AkernelizedSMFcalledKMFDKernelMatchedFilterDetectorwasdeveloped in[12].AkernelizedversionoftheAMSDcalledtheKMSDKernelMatchedSubspace Detectorwasdevelopedin[63].TheydevelopaversionoftheACEdetectorcalledthe KASDKernelAdaptiveSubspaceDetectorin[64],andakernelizedOSPcalledKOSP 42

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in[62].Capobiancoetal.in[66]extendtheKOSPalgorithmtouseasemi-supervised learningframework. 2.2.18SparseReconstructionBasedDetectors Chen,Nasrabadi,andChanproposetargetdetectionalgorithmsin[67,68]which modelpixelsinthehyperspectralimageasasparsecombinationofelementsfroma dictionaryoftrainingsamples.In[68]ajointsparsitymodelisproposedwhichselects onlythoseelementsfromthedictionarywhichbestjointlytboththepixelundertest anditsimmediateneighborpixels.In[67]asimilareectisachievedwithasmoothness constraint.Inbothcasesagreedyorthogonalmatchingpursuitsstylealgorithmis usedtosolveforthesparseapproximation.Thejointsparsitymodelhasalsobeen independentlydiscoveredandsuccessfullyappliedtolandminedetectionapplications withelectromagneticinductionsensors[69],andwasdiscussedbeforeeitherapplication in[70,71]. Chenetal.'ssparsereconstructiondetectorusesthefollowingfunction: r SRD x = jj X )]TJ/F37 11.9552 Tf 11.955 0 Td [(A b ^ S b jj F )-222(jj X )]TJ/F37 11.9552 Tf 11.955 0 Td [(A t ^ S t jj F : {63 Where X isthematrixconsistingofthepixelundertestaswellasitsneighboring pixels X =[ xx n ::: x n M ],thedictionariesareseparatedbetweenbackground A b and target A t ,and ^ S b and ^ S t arethecorrespondingsparsecoecientmatrices. Thesparsecoecientmatricesarefoundbyminimizing{usingtheSOMPalgorithm presentedin[70]{theoptimizationproblem minimize jj AS )]TJ/F37 11.9552 Tf 11.955 0 Td [(X jj F subjectto jj S jj row ; 0 K 0 ; {64 where A =[ A b A t ], S =[ S b S t ],and K 0 isthedesiredsparsitylevelnumberofdictionary elementsusedintheapproximation. 43

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Thisdetectorwasshownbyitsauthorstoperformwellfortargetsconsistingof multipleneighboringpurepixels.Itappearstobelesssuitablehoweverfordetecting individualsub-pixeltargets. 2.2.19SupportVectorDataDescription AnapproachtotargetdetectionusingaSupportVectorDataDescriptionSVDD methodwasgivenbySaklaetal.in[72].TheSVDDmethodissupervisedsingleclass classier,whichattemptstoenclosethetrainingsetbyaminimumvolumehypersphere whiletradingovolumeofthesphereversusoutlierpoints.Tocreateatrainingset fortheSVDD,theauthorsgeneratedrandomsamplesfromamultivariateGaussian distributionwiththetargetsignatureasthemean.Thecovarianceofthedistributionused aninterestingToeplitzmatrixcongurationwhichattemptstomodelcovariancebetween adjacentspectralbands.Thedegreeofadjacentbandcorrelationisestimatedfromthe backgroundhyperspectralimage.Thoughtrainedwithonlypurepixels,theSVDD methodwasshowntooutperformamatchedlteratlowfalsealarmratesonasynthetic datasetcontainingsomemixedpixels.OthershaveappliedSVDDtohyperspectralimage classication[73],anomalydetection[74],andfeatureselection[75]. 2.2.20KalmanFilter In[76],ChangandBrumbleyproposeusingaKalmanltertodetermineunknown abundancesofknowntargetsignaturesinalinearmixingmodel.Thepixelsareviewed asathemeasurementtimeseriesscanningdownanimageline,andabundancesare theunobservedstatevariablesateachtimestep.Optimalgiventhemodelassumptions unconstrainedabundancesaresolvedforbytheKalmanlter,whichtakesintoaccount knownnoiselevelsandpixeltopixeltransitiondynamics,makingthisadetectorby abundanceestimation. 44

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2.3DetectionRelatedTechniquesintheLiterature 2.3.1FalseAlarmMitigationStatistics Thedetectionalgorithmsstudiedtodateintheliteratureareoftencapableof detectingmosttargets,butdosoattheexpenseofahighfalsealarmrate.Therefore, manyfalsealarmmitigationstatisticshavebeenpresentedintheliterature. Onecouldthinkofthesestatisticsasprovidinganadditionaldecisionboundarythus augmentingthesimplelineardetectorstocreatenonlineardecisioncontours. 2.3.1.1Mixturetunedmatchedlter TheMixtureTunedMatchedFilterMTMFisaprocessforremovingfalsealarms fromaSMFdetectoroutput.TheMTMFwasoriginallyin[77],whichdoesnotseem tobeavailableonline,andwascoveredrecentlyingreatdetailin[78]andsummarized in[4, x 14.4.3].TheMTMFprocessgivesastatisticusefulforrejectingfalsealarms, thoughperhapsafteranalysisofthestatisticbyanexpertoperator. TheMTMFdenestheideaofafeasiblemixturebetweenthetargetsignatureand backgroundsignatures.Itthendenesapseudo-simplexbetweenthetargetandthe backgrounddistribution.Pixelswhichlieoutsideofthissimplexarelikelytobefalse alarmsastheyareunlikelytohaveresultedfromamixtureoftargetsignatureand backgrounddata.Theinfeasibilitymeasure,whichdependsupontheSMFoutputforthe pixelundertest,is r INF x = z T [ ^ ] )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 z ; {65 where z isthewhiteneddatapoint x ^ =^ 2 P s ? D )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 b P s ? + )]TJ/F15 11.9552 Tf 13.115 0 Td [(^ 2 P s ? + I {66 ^ = r SMF z = s T w z ; {67 and isasmallpositivenumbermultipliedbytheidentitymatrixpreventthematrix ^ frombecomingsingular. D b isthediagonalizedbackgroundcovariancematrix, P s is theprojectionoperatorontothetargetsubspace,and s w isthewhitenedtargetsignature. 45

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2.3.1.2Subpixelreplacementmodel TheSubpixelReplacementModelisthebasisforafalsealarmmitigationtechnique givenin[79],alsocoveredin[4, x 14.4.2].Underthismodel,apixelisassumedtocome fromatwoendmembermixturebetweenthetargetdistributionandthebackground distribution.Giventhatthetargetabundancecanbeestimated,thismodelprovidesa measureofthelikelihoodofthepixelunderthemixturedistribution,andcanbeusedto rejectoutlierpointswhichwouldotherwisebefalsealarmsunderasimplerdetectorlike theSMF. Giventhehypotheses H 0 : x = b H 1 : x = s + )]TJ/F21 11.9552 Tf 11.956 0 Td [( b ; {68 where b N b ; b and s N s ; s .Themixturedistributionstatisticsaregivenby = s + )]TJ/F21 11.9552 Tf 11.955 0 Td [( b {69 = 2 s + )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 b : {70 Takingamonotonicfunctionofthelog H 1 hypothesisyieldsthestatistic r FAM x = x )]TJ/F40 11.9552 Tf 11.955 0 Td [( T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F40 11.9552 Tf 11.955 0 Td [( +log j j ; {71 whichisanindicatorofhowlikelythepixelundertestistobeatargetatthegiven mixtureproportion. Inapplication,thetargetdistributionisunknownandnoteasilyestimated.Sothe targetsignatureistakentobethemeanofthetargetdistribution s = s ,andthe covarianceofthetargetistakentobethesameasthebackgroundcovariance s = b Undertheseassumptionsthefalsealarmmitigationstatisticbecomes r FAM x = x )]TJ/F40 11.9552 Tf 11.955 0 Td [( b )]TJ/F21 11.9552 Tf 11.955 0 Td [( s )]TJ/F40 11.9552 Tf 11.955 0 Td [( b T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 b x )]TJ/F40 11.9552 Tf 11.955 0 Td [( b )]TJ/F21 11.9552 Tf 11.955 0 Td [( s )]TJ/F40 11.9552 Tf 11.955 0 Td [( b : {72 46

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Finally,thetargetabundance isunknownandmustbeestimated.Thespectral matchedlteroutputisoftentakenasanestimateforsimplicity ^ = r SMF x = s T x s T s ; {73 whereitisassumedthatthebackgroundmeanhasbeensubtractedbeforeapplyingthe lter. 2.3.1.3Finitetargetmatchedlter TheFiniteTargetMatchedFilterisanalgorithmthatcombinesafalsealarm mitigationstatisticwiththeoutputofamatchedlterinordertoprovideamoreusable singledetectionstatistic.ItisattributedoriginallytoStockerandSchaum[80,81]andis alsocoveredin[79]aswellas[4, x 14.4.4]. Usingthegeneralizedlikelihoodratiotestforthehypotheses2{68,thentakingthe log,multiplyingby2anddiscardingconstants,thefollowingdetectorcanbederived: r FTMF x = x )]TJ/F40 11.9552 Tf 11.956 0 Td [( b T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 b x )]TJ/F40 11.9552 Tf 11.955 0 Td [( b )]TJ/F15 11.9552 Tf 11.955 0 Td [( x )]TJ/F40 11.9552 Tf 11.955 0 Td [( ^ T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F40 11.9552 Tf 11.955 0 Td [( ^ )]TJ/F15 11.9552 Tf 11.955 0 Td [(log j ^ j ; {74 where and aregivenby2{70. Themaximumlikelihoodestimatefor mustbefoundhoweverbysolvingthe minimizationproblem ^ =argmax f x j ; =argmin x )]TJ/F40 11.9552 Tf 11.955 0 Td [( T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F40 11.9552 Tf 11.955 0 Td [( +log j j : {75 Itisshownin[79]thatifthetargetcovariancehasaknownproportiontothebackground covariance,thenthisoptimizationcanbeperformedanalyticallybysolvingacubic function.Alternativelymultiplevaluesof cansimplybetested,incrementsof0 : 05inthe interval[0 ; 1]aresuggested. 47

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2.3.1.4Leastangleregression Anotherfalsealarmmitigationtechniquepresentedin[4, x 14.4.5]isknownas regression or materialidentication .Theideaisthatadetectionalgorithmsuchas theSMForACEisrstperformed,andthenasecondthresholdisappliedtosome metricwhichmeasuresnumericalttothetarget.Eismann[4, x 14.4.5]proposedthe Euclideandistanceasonesuchmetric,butnotesthatitsperformancecanbeaected bynormalizationandatmosphericcorrectionissuesaswellassubpixelmixingwiththe background. Villeneuveetal.[82]proposeamethodtheylabeltheLeast-AngleRegression StatisticLARS.Inthismethod,theyconstrainthepixelundertesttothelinearmixing modelwiththeconstraintthattheL1normoftheproportionsbelessthanorequalto1. Thealgorithmsolvesaconstrainedoptimizationproblemforboththebackgroundcase ^ =argmin jj x )]TJ/F37 11.9552 Tf 11.955 0 Td [(B jj 2 : M X m =1 j m j 1 ; {76 where B isthematrixofthe M backgroundendmembersascolumnvectors,andthe targetcase ^ =argmin jj x )]TJ/F37 11.9552 Tf 11.955 0 Td [(S jj 2 : M +1 X m =1 j m j 1 ; {77 where S isthematrixwiththebackgroundendmembersconcatenatedwiththetarget signature. Thetworegressionsarethencomparedusingtheratiooftheerrorsnormalizedby theirdegreesoffreedomtheerrortermsareviewedas 2 statistics r LARS x = M +1 M jj x )]TJ/F37 11.9552 Tf 11.955 0 Td [(B ^ jj 2 jj x )]TJ/F37 11.9552 Tf 11.955 0 Td [(S ^ jj 2 : {78 ItshouldbenotedthattheLeastAngleRegressiontechniqueisessentiallytheHybrid SubpixelDetectorfromsection2.2.13underadierentmixingconstraint,anditisalso connectedthroughthe L 1normconstrainedminimizationtothesparseapproximation techniquesdiscussedinsection2.2.18. 48

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2.3.2DeterminingTargetSignatures Animportantpartoftargetdetectionthatisoftenleftoutofthediscussionisthe determinationofthetarget'ssignature. Onecommonapproachisusingknownreectancesignaturesofthetarget.In thiscasetheimagerymustbetransformedintoapparentreectance,whichcanbea dicultproceduretoperformaccuratelyforremotesensingdata.Alternatively,the knownreectancesignaturescanbetranslatedintoradiancedatausingthesameknown atmosphericparameters,althoughthisassumesthattheatmosphericparametersare constantthroughoutthescene. Anotherapproachistousepixelsfromtheimagethatcorrespondtoknowntarget locations.Thisisofcoursedicultincaseswheregroundtruthisunknown,anditalsois dicultiftherearenopurepixelrepresentationsofthetarget. AnewapproachforndingtargetsignaturesofsubpixeltargetscalledFUMIfor FUnctionsofMultipleInstanceswaspresentedbyZareet.al.[83].Inthisapproachone ormorepixelwhichcontainthetargetsignatureinsomenonzeroproportionisselected, andinaddition,severalpixelsknowntocontainnotargetinformationareselected.From theseselectedpixels,usingmultipleinstancelearningandunmixingtechniques,atarget signaturecanthenbededuced. Anapproachmentionedin[42,49,84,85]and[4, x 14.5.1]istotranslateknownreectancesignaturesintoarangeofmultipleradiancesignatures.Thespaceofparameters fortheradiancetransformisdiscretized,andmanypotentialtargetsignaturesaredeveloped.Thevariationofthetargetsignaturecanthenbecharacterized,perhapsby statisticalorsubspacemethods,andadetectorwhichacceptsthisvariationcanthenbe used. AninterestinglineofresearchbyStefanouandKerekes[86,87]assessesthesuitability orutility"ofagivenimageforsubpixeldetectionofagiventargetsignature.Thisis donebyarticiallymixinginthetargetsignaturetoeachpixeloftheimageatsome 49

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fractionalabundanceandthenevaluatingdetectorperformance.Dierentchoicesof detectionalgorithm,signature,image,andoperatingpointcanbecomparedwiththis methodonarelativescale. 2.3.3ImprovingDetectorPerformance Manydetectionmethodsrequireanestimateofthecovarianceofthebackground interference"matrix.LiandMichels[88]giveamethodforestimatingthiscovariance usingparametricauto-regressivemodels.Thesemodelscanbeestimatedwithless computationalcostandlesstrainingdatathanthewholeimagecovariance,andthey canpotentiallyprovideimproveddetectionperformanceoverthestandardtechniques.Li andMichelsadditionallyshowthatleavingsomeoutlier"pointsoftheestimationofthe covariancematrixcanimprovethedetector'sperformance. Alsoin[88],theauthorsshowthatlowpasslteringalongthespectraldimension canpotentiallyimproveperformance.RenardandBourennane[89]andBourennaneet al.[90]presentcombinedspatial-spectralltersthatalsoimprovetheperformanceofsome detectorssuchastheSMFandACE. Itwasshownin[91]thatdetectorssuchastheSMFcanbeimprovedbyusingthe localbackgroundestimationtechniquesmostoftenassociatedwiththeRXalgorithm. Inthismethodaregionaroundthepixelundertestisusedtoestimatethebackground statistics,andaguardregionisexcludedfromtheinteriorofthisregiontoaccountforthe potentialmulti-pixelextentoftargets.Nasrabadifurthershowedin[9]thatregularization ofthecovariancematrixdiagonalloadingwithasmallcoecientaround0 : 01improves performanceoftheSMFwhendoinglocalbackgroundestimation. Similarlytothelocalbackgroundestimationin[91],Matteolietal.[39]recommend alocalbackgroundremovalprocess,usingittoshowimprovementinmatchedlter performance.Thisprocessusesamovingbackgroundwindowcenteredaroundthepixel undertest,andthenitestimatesthebackgroundsubspacedimensionalityusingthenoisewhitenedHarsanyi-Farrand-Changprocedure[92].Nextitdeterminesthesubspacebasis 50

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vectorusingtheleadingsingularvectorsfromanSVDdecompositionofthebackground data.FinallythebackgroundissuppressedusinganorthogonalcomplementasinOSP 2.2.9. 2.4UnmixingandEndmemberExtractionMethods UnmixingandEndmemberExtractionarecommonlyusedmethodsinHSIanalysis andanactiveareaofresearch.Suchmethodsattempttoinvertthemixingmodelto obtainthepurematerialsendmembersandmixingproportionsforeachpixel.Endmemberextractionalsocalledendmemberdetectionorendmemberestimationrefers totheautomatedprocessofdeterminingendmembersfromahyperspectralimage,while unmixingcanrefertoeithertheprocessofdeterminingmixingproportiongivenknown endmembers,ortothecombinedprocessofndingendmemberandproportionvalues. Surveysofunmixingandendmemberextractionmethodsarepresentedin[52]and[93]. Similarlytoclusteringtechniques,mostendmemberextractionmethodsrequire thenumberofendmembersdesiredtobefoundtobespeciedasaparameter.Zare andGaderproposedtheSparsityPromotingIteratedConstrainedEndmemberSPICE method[94]whichcansimultaneouslyextractendmembersandestimatethenumberof endmembersinthedata.SPICEusessparsitypromotioninitsobjectivefunctionand pruningofunusedendmembersinordertondanappropriatenumber.Thusaninitial numberhigherthanthebelievedtruenumberischosen,andthealgorithmprunesaway unusedendmembersasititerates. Traditionally,endmemberextractionalgorithmshaveoperatedundertheassumptions ofthelinearmixingmodel,alsoknownastheconvexgeometrymodel.Inthismodel,a pixelisassumedtobeaconvexcombinationofendmembervectors,writtenas: x i = M X k =1 p ik e k + i ;i =1 :::N; {79 where N isthenumberofpixelsintheimage, M isthenumberofendmembersinthe scene, i isanerrorterm,and p ik istheproportionofendmember e k inpixel x i .The 51

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proportionsaresubjecttotheconstraints p ik 0 8 i;k; M X k =1 p ik =1 8 i: {80 Theconvexgeometrymodelmaybeareasonableassumptionforanygivenpixelina hyperspectralimage,butincommonremotesensingscenarioshowever,itwillnotapply tothesceneasawhole.Commonly,thedatasetoftheentireimageisnon-convexdueto separateimageregionscontainingdierentmaterials.Inthissituationtheendmembers whichlieinsideanoutermostconvexshelloftheimagedatasetcannotbefoundusing theconvexgeometrymodel.Thisobservationleadstoanimportantlineofresearch,the Piece-wiseConvexEndmemberPCEmodelproposedbyZareandGaderin[95].The PCEmodelbasedalgorithmsndmultipleconvexregionswithinanimage,andthen estimateendmembersandproportionswithineachoftheconvexregions.Thisisachieved usingaGibbssamplingimplementationofaDirichletProcessDPtoclustertheimage intoconvexregionswhilesimultaneouslytheendmembersandproportionsarefoundin eachregionusingtheSPICEalgorithm. In[96]avariantofthePCEmethodispresentedasaformofcontext-basedendmemberdetection.ThisillustratesthatthePCEideasarerelatedtothecontext-dependent themespresentedinthiswork. AnalternativeapproachtothepiecewiseconvexmodelscalledPiece-wiseConvex MultipleModelEndmemberDetectionPCOMMENDisdevelopedin[97].Thisapproach usesfuzzyclusteringtosimultaneouslyndtheconvexregionsandendmembersand proportionswithinthoseregions.Anextensionofthismethodtoincorporateaspatial smoothnessconstraintisproposedin[98]. 2.4.1Sample-PCE TheDPbasedPCEmodelrstproposedin[95]wasimproveduponin[99]to createtheS-PCUEalgorithm{alsoknowntoitsauthorsandhereafterreferredtoas Sample-PCE.ThismethodusesafullyBayesiansamplingbasedinferencestrategyto 52

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estimateendmemberdistributionsandproportions,whereastheoriginalPCEuseda hybridstrategyofsamplingandoptimizationwhichwasnotguaranteedtoconverge toaglobaloptimum.TheSample-PCEmethodusesaMetropolis-within-Gibbs[100] methodandsatisesMarkovChainMonteCarloMCMCconvergencepropertiesand hasthecorrespondingconvergenceguarantees.AdditionallytheSample-PCEmodeluses endmemberdistributionsinsteadofxedvalueendmemberestimates.Theendmember distributionshelpaccountforspectralvariabilityintheendmembersand,togetherwith thepiecewiseconvexapproach,maketheSample-PCEmodelacompellingadvancein accuratephysicalmodelingofhyperspectralimages. Aswithanygoodsamplingmethod,westartbyspecifyingaprobabilisticmodelof thedata.IntheSample-PCEmodel,eachpixelintheimageisamixtureofendmembers fromoneandonlyonesetofendmemberdistributions.Thustheimageispartitionedinto adisjointunionofconvexsets X = [ R r =1 )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(r ,eachsetsatisfyinglinearmixingmodel. Theendmemberdistributionsaremodeledasnormaldistributions, N e m;r ; S m;r where e m;r isthemeanofthedistribution m inconvexset r ,and S m;r isthecovariance matrixforthatsamedistribution.IntheSample-PCEimplementationhowever,all endmemberdistributionsusethesameisotropicdiagonalcovariancematrix S .Thedistributioncreatedbyalinearmixtureofnormaldistributionsisalsoanormaldistribution, andintheHSIliteraturethisisknownastheNormalCompositionalModel[101,102].Underthenormalcompositionalmodelthen,eachpixelcanbemodeledasasamplefromthe resultantcomposednormaldistributionfortheconvexregionandthemixingproportions ofthepixel.Thisisstatedmathematicallyas x j j E r ; p j f x j j E r ; p j = N p j E r ; M X m =1 p 2 jm S m;r ; {81 where x j isthe j th pixel, r istheindexoftheconvexsettowhichthepixelhasbeen assigned, E r isthematrixofendmemberdistributionmeansforconvexset r p j isthe 53

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vectorofproportionsfortheendmembersintheconvexset,and p jm isthe m th proportion forpixel j inthevectorofproportionsforthe M endmembersintheconvexset. Thejointconditionallikelihoodforallpixelsintheimageunderamodelwith theconvexsets,mixingproportions,andconvexsetassignmentsspeciedisthensimply Q R r =1 Q j 2 I r f x j j E r ; p j ,where I r isthesetofindicesforallpixelsgeneratedbyconvexset r ofthe R totalsets. EachendmemberdistributionwithinaconvexsethasaGaussianprioronitsmean e m;r N r ; C r ;m =1 :::M: {82 Eachofthesepriordistributionsinturnhashyperpriordistributionsforthehyperparameters r ,and C r .Thecovariancesoftheendmemberdistributionsaresettoaxedvalue S m;r = S Thehyperpriorforthehyperparameter r isalsoGaussiandistributed r N 1 N N X j =1 x j ; C ; {83 wherethemeanofthehyperprioristakentobethemeanoftheentireimage.The covarianceofthehyperpriorissetto C = I ,where istakentobesomexedlarge numbercomparedtothespreadoftheentiredataset. Thehyperpriorforcovariancehyperparameter C r isgivenanInverse-Wishart distribution.Thisischosenbecauseitistheconjugatepriorofthecovarianceofa Gaussian,whichsimpliesthelatersamplingmodel. Theproportionsforeachendmemberwithintheconvexsetareassumedtobe generatedfromaDirichletdistribution p j j z j = r D M 1 ;r ;:::; M;r ; {84 54

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where D M isthe M -factorDirichletdistributionwithdensity D m p j j z j = r = \050 P M m =1 m;r Q M m =1 \050 m;r M Y m =1 p m;r )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 jm : {85 Interestingly,tomaintainlikelihoodvaluesformembershipineachconvexsetneededfor theDirichletProcesssteps,asetofproportionvaluesforeachconvexsetiskept,even thoughapixelisonlyassignedtooneofthesetsatatime. AschematicrepresentationoftheSample-PCEprobabilisticmodelisgivenin Figure2-2.Thisschematicnotationfollowsalongthestandardgraphicalmodelplate notation[103],butaddsbusesandmultiplexerswhicharecommonelementsinelectrical engineeringschematics.Thepurposehereofthebus,shownbyathicklinewitha forward-slashgivingthebuswidth,istodenotegroupsofsimilarrandomvariables.Small diamondshapesontheedgeofaplateindicatewheremultipleinputsareaggregatedinto ordisaggregatedfromabus.Themultiplexers,shownbyagraytrapezoid,selectoneof theinputsfromthelargerbusside,andoutputasingleinstanceontheoppositeoutput side.Thesmallsidehasaninputwithanarrow,thisistheswitchinput,whichselectsthe inputthemultiplexerwillpasstotheoutput. SamplingfortheSample-PCEmodelisdoneusingaDirichletProcessbasedGibbs sampler.Becausetheposteriordistributionsofeachofthelatentvariablesinthemodel arediculttosamplefrom,aMetropolis-Hastingsstepisusedtogeneratethesamples foreach.Metropolis-Hastingsusesproposaldistributionsandacceptsaproposedvalueif itismorelikelythanthepreviousvalue,oracceptstheproposedsampleonlywithsome probabilityifitislesslikelythatthepreviousvalue. TheDirichletProcessclusteringstep,inshort,rstproposes K newpotentialconvex setsineachineachsamplinground.Eachdatapointisthenevaluatedforitslikelihoodin thenewproposedconvexsetsandeachexistingconvexset.Thedatapointisthenplaced randomlyintooneofthesetswithprobabilityinproportiontoitslikelihoodineachset. 55

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Aninnovationparameter helpstocontroltherateatwhichpointsareplacedintothe newlygeneratedpartitions. Whilethesamplerisrunning,themodelwithlargestlikelihoodvalueforeveryunique numberofconvexsetsisretained.Thetotalcountofsamplesusingeachnumberofconvex setsisalsotallied.Thenalnumberofconvexsetsisthendecidedtobethenumber whichwasusedinthemostsamples.Withthisnumberofconvexsetsdecided,themost likelymodelfoundwiththisnumberofconvexsetsisthenusedasthenalresult. 2.4.2FuzzPop Recently,thelineofresearchintofuzzyclusteringforpiecewiseconvexmodelsthat startedwithPCOMMEND[97]hasbeenextendedintothePFCM-FLICM-PCEalgorithm [104].ThealgorithmnamecanbebrokendownapproximatelyasPossibilisticand FuzzyClusteringMethodwithFuzzyLocalInformationC-MeansforPiecewiseConvex Endmembers,butitisknowncolloquiallytoitsauthorsandwillhereafterbereferred toasFuzzPop.ThisalgorithmusesPossibilisticandFuzzyclustering[105]toestimate theendmembersundertheassumptionofmultipleconvexregionswithintheimage. Thefuzzyclusteringisusedtondtheconvexregions,whilethepossibilistictypicality weightingaddsrobustnesstooutliersintheestimationoftheregions'endmembersand proportions.Thealgorithmalsoincludesaspatiallyconstrainedunmixingusingthe FLICMmodel[98,106],whichencouragesspatiallyneighboringpixelstohavemembership inthesameconvexregion. TheFuzzPopalgorithmisperformedbyusinganalternatingoptimizationofthe objectivefunction J = )]TJ/F21 11.9552 Tf 11.955 0 Td [( C X i =1 N X j =1 au m ij )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(p ij E i x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(p ij E i T + G ij + bt n ij x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(p ij E i x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(p ij E i T + C X i =1 M trace E i E T i )]TJ/F37 11.9552 Tf 11.955 0 Td [(1 1 M E i E T i 1 M 1 + C X i =1 i N X j =1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(t ij n ; {86 56

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where a b aretuningparametersofthealgorithm, m isthefuzzierforthefuzzy clustering, n istheparametersimilartoafuzzierusedforthetypicalities, C isthe speciednumberofconvexregions, N isthenumberofpixelsintheimage,and M isthe numberofendmemberperconvexregion.Theindex i =1 :::;C runsoverconvexregions, and j =1 ;:::;N runsoverpixelsintheimage.Thelocalspatialinformationtermisgiven as G ij = X k 2 N j ;k 6 = j 1 d jk +1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(u ik m jj x k )]TJ/F37 11.9552 Tf 11.956 0 Td [(p ik E i jj 2 2 ; {87 where d jk isthedistanceingroundcoordinatesbetweenpixel j anditsneighborpixel k withinthelocalneighborhood N j .Thetypicalityscaleterm, i ,issetateachiterationto betheresidualerrorofpointstotheconvexregion i = 1 N N X j =1 jj x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(p ij E i jj 2 2 : {88 Theupdateequationsfortheendmembersmatrices E i ,proportionvalues p ij memberships u ij ,andtypicalities t ij aregiveninthefollowingequations: E i = X j au m ij + bt n ij p T ij p ij +2 M I M M )]TJ/F37 11.9552 Tf 11.955 0 Td [(1 M M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 X j au m ij + bt n ij p T ij x j ; {89 p T ij = E i E T i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 E i x j )]TJ/F37 11.9552 Tf 13.151 9.168 Td [(1 1 M E i E T i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 E i x T j )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 1 1 M E i E T i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 M 1 1 M 1 {90 u ij = 1 P C q =1 x j )]TJ/F38 7.9701 Tf 6.587 0 Td [(p ij E i x j )]TJ/F38 7.9701 Tf 6.587 0 Td [(p ij E i T + G ij x j )]TJ/F38 7.9701 Tf 6.587 0 Td [(p qj E q x j )]TJ/F38 7.9701 Tf 6.587 0 Td [(p qj E q T + G qj 1 m )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 ; {91 t ij = 1 1+ b i jj x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(p ij E i jj 2 2 1 = n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 : {92 FuzzPophastheadditionalabilitytointegrateelevationmeasurementsfromaLiDAR sensor.Todoso,weaugmentthedistancecomputation d ij usedinthelocalinformation term2{87tobeathreedimensionalEuclideandistance.Thersttwoofthethree dimensionscomefromtheassignedUTMcoordinatesoftheimagepixelsandthethird fromtheelevationoftheLiDARrstreturn.Thisweightingthusdecreasestheenforced 57

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similarityofpixelsiftheyhavedieringelevation,whichwouldbeanindicationthatthey arefromdierentconvexregionsoftheimage. 2.5Context-DependentMethods 2.5.1UsesofContextinLiterature Severalauthorshavediscussedtheideaofcontextintheirworkinremotesensing imageryanalysis.Thissectiondiscussestheseaspectsofcontextandothersthatappearin asurveyoftheliterature. In[107],theauthorsattempttodetectthepresenceofinvasive Lepidiumlatifolium perennialpepperweedatthreesitesintheSanFranciscoBay/Sacremento-SanJoaquin RiverDeltausingairbornehyperspectralimageryfromtheHyMapsensor.UsingaMTMF fordetectionandClassicationandRegressionTreesasatrainedclassier,theyfound thatperformanceoftheiralgorithmsvariedacrosssites,andhypothesizethatenvironmentalhabitatcharacteristicssuchastheidentityofco-occurringspecies,phenologiesof targetandco-occurringspecies,anddegreeofmixingandtypeofbackgroundmaterial allcontributedtothesuccessoftheirmethods,withthemostcomplexandvariablesites exhibitingtheleastsuccess. In[108],theauthorsrelyoncontexttohelpdistinguishotherwiseindistinguishable materialsingroundcoverclassicationofremotesensingimages.Materialssuchasroong bitumen,roongtar-paper,andasphaltroadsarenormallydiculttodistinguish.Using hyperspectraldatafromtheHyMapsensor,theauthorsrstidentifystreetsandbuildings usingknowledgesuchasshadowinformationfortheheightofbuildingsandidentifying streetsaslongstraightregionswithoutshadow.Then,giventhecontextofapixel,the actualcovermaterialcanbemoreeasilydistinguished. Duetotechnicalconstraintsintheirdesign,imagingspectroscopysensorstendto tradespectralresolutionforspatialresolutionandviceversa.In[109],theauthorsare workingwithhighresolutionmultispectralremotesensingimagery-BandQuickbird andIkonos.Heretheyaugmentthespectralinformationforeachpixelwithcontext 58

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informationfromregionpropertiesofthepixel'scontainingsegmentatmultiplehierarchicallevelsofresolution.Thespectralandcontextfeaturevectorsarethenclassiedinto ground-coverclasseswithanSVM.Theargumenthereisthatwhenthespectralresolution isgenerallyinsucientforclassicationsuchasinRGB+NIRmultispectralimagery, contextualinformationcanbeusedhelpidentifyanygivenpixel. Anotherformofcontextoftenusedinimagedataistheclassicationvalueof neighboringpixels.Therangeofvaluesforsomeclassesmayoverlapinspectralfeature space,suchthatthevalueofanygivenpixelmaybeambiguous.However,ifwecan makeassumptionsthatneighboringpixelsshouldcomefromthesameclass,perhaps becauseobjectsarelargerthanthespatialresolutionofthesensor,thenmethodscanbe developedtousemorecertainlabelsonsomepixelstoresolveambiguouslabelsonothers. MarkovRandomFieldsMRFsareoneoftenusedapproach[110{112]toexploitingsuch contextualinformationintheclassicationofremotesensingimagery. Contextdependencehasbeendiscussed[113,114]asbeingrelatedtotheideaof conceptdrift[115{117].Theideaofconceptdriftexistsinthemachinelearningliterature atleastsincethe1980's,whentheeldwasverynewandabitmoreinfusedwiththe symboliclearningandrepresentationconceptsnowmoreassociatedwiththearticial intelligencearea.Theideain[115]isthattherepresentationofthetargetconceptcould changeovertimedrift,andthatalearningsystemshouldbeabletohandlethischange. Inthe1990'stheideacameupagaininthecontextofon-linelearningsystems[116].Here theideacomesupofanunderlyingcontextthatcouldaecthowthetargetisexpressed, andasthecontextchangesthetargetconceptcoulddrift.Inthe2000'sthisideawas studiedagainin[117]inamorecomprehensivereviewthatgenerallyequatesconcept drifttochangeovertime.Thedriftmaybecyclicalornot,andmaychangegraduallyor abruptly,andmaybereal"duetoanactualchangeinthetargetconceptorvirtual" duetoachangeintheobservationorsamplingoftheconcept.Theideaofconceptdrift iscertainlyanexpressionofcontextdependence,andtheunderlyingcontextmaybeas 59

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simpleastheprogressoftime.Inthiswork,wetakeamoregeneralviewtowardscontext dependence,andnotethatthelearnedconceptsmayneedtoreectthetargetunder multiplecontextssimultaneouslywithinasingleimage. 2.5.2ContextDependenceinClassication Theideaofacontext-dependentclassierhasonlyappearedrelativelyrecentlyin theliterature,mostlyconcurrentwiththeworkoncontext-dependentfusion.Earlier methods,suchasMixturesofExperts,certainlyexistwhichembodysomeoftheseideas withoutexplicitlylabelingthemascontext-dependentclassication.Thissectiondiscusses context-dependentclassicationmethods. In[114],theauthorsapplycontext-dependentdetectiontogroundpenetratingradar GPRbasedsystemforlandminedetection.TheGPRdatahasbeenshowntoexhibit strongcontextdependenceintermsofsurfaceroughness,subsurfaceheterogeneity,and soilmoisture.Theauthorsfoundthatbyrstclassifyingthecontextofthedata,then usingdiscriminationalgorithmstrainedforeachcontext,overallsystemperformance couldbeimproved.Inthiscaseboththecontextclassierandtargetclassierscouldbe addressedwithsupervisedlearningtechniques,andspecializedfeaturesweredevelopedto helpidentifythecontext. Torrioneetal.[118]integrateMultipleInstanceLearningMILandcontextdependentclassicationinHSI.TheMILhelpstoaddresstheuncertaintyinpixel labelingfortargetclassicationtasks,whichisveryusefulforconvertingthetraditionallyunsupervisedlearningbaseddetectionproblemstosomethingwhichismoreakinto supervisedlearningtechniques.Contextfeaturesweretakenfromthesurroundingpixels ofanalarmlocation.TheauthorsthenuseaBayesianclassierforthecontextposterior distribution,andintegratethecontext-dependenttargetlikelihoodoverallcontextsto yieldatotalposteriorprobabilityoftarget. Mortonetal.[119]extended[118]byusingunsupervisedlearningofthecontextusing aDPmixturemodel.Twovariantsofthisapproachwerederivedusingeithergenerative 60

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ordiscriminativecontext-dependentclassiers.Thediscriminativeclassierbasedversion achievedthehighestperformance,butthecontextslearnedinthedatanolongermatched theknowntime-of-daycontextusedinthesupervisedcontextlearningcases.Thiswork isinterestingbecauseitappearsthatthoughitisnotthoroughlyexplainedduetothis beingaconferencepapertheunsupervisedcontextlearningandsupervisedcontextdependentclassicationarejointlyoptimized. BoltonandGader[113,120]addresstheproblemofcontext-basedclassicationin hyperspectralimageryviaarandomsetframework.Theauthorsmakeseveralgoodpoints abouttheroleofdisguisingversusnon-disguisingcontextbasedtransformations.The authorsalsopointoutthatcontextcanbeduetoglobalexternalenvironmentalfactors suchasthesun'sintensityduetothetimeofday,andcanalsoexhibitlocalcontext variationsuchaslocalshadingorsoilpropertychanges.Additionallytheyshowthatoften especiallywithdisguisingtransformationssimplefeaturespaceanalysiscannotbeused tocorrectlyinfercontext,andthatapopulationofsamplesmustbeused.Inthiscase thecontextsandcontext-dependentclassiersweretrainedwithsupervisedlearning.In testing,thecontextisrstestimatedusingthefeaturespacedistributionofthepopulation ofpixelsinthelocalwindowaroundanalarmpixel,thenthealarmisclassiedvia weightedaggregationofcontext-dependentclassiers.Randommeasureswerealsousedby theauthorsforcontextestimationin[121]. MixturesofExperts[122]areamethodwhichlearnsregionsoffeaturespacein anunsupervisedmannerandlearnsclassiersforthoseregionssimultaneously.Jordan etal.extendthisworkintoHierarchicalMixturesofExpertsHME[123],andthis techniquewasappliedtocontext-dependentlandminedetectionproblemsin[124].Bolton pointsout[113]thatsuchfeature-spaceonlymethodsofcontext-dependentclassication areunabletohandletheeectsofdisguisingtransforms.Theauthorsof[123]mention however,thattheiruseofsoftcontextdecisionboundariessomewhatreduceseectsof incorrectcontextestimation,andsoHMElikemethodsmightgeneralizebettertothe 61

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ambiguouscasesthanamethodwhichmakesahardcontextdecision.Thoughitshouldbe recognizedthatambiguouscontextisrelatedto,butnotnecessarilythesameas,context whichcanberesolvedthroughpopulationanalysis. 2.5.3ContextDependenceinFusion Muchoftheexistingworkincontext-dependentmethodshasfocusedonmethodsof classierfusionandothersystemsforusingmultipleclassiers.Someoftheearliestwork incontext-dependentmethodscamebeforethephrasecontext-dependentwascoinedand beforetheuseofthatmodeofthinkingshapedthedirectionofresearch.Earlierwork includestheideaofclassierselection[125,126],andusinglocalaccuracyinfusion[127]. TheContextDependentFusionCDFframeworkproposedin[128],usesaunsupervisedclusteringmethodofthefeaturespacefollowedbycontext-dependentdecisionlevel fusionofclassieroutputs.TheauthorsproposeusingtheSimultaneousClusteringand AttributeDiscriminationSCAD[129]fuzzyclusteringtechniquewhichperformsacluster specicfeatureselectionbylearningweightsfor,inthiscase,thefeaturevectorscorrespondingtoeachclassierbeingfused.Thentheauthorsproposealinearfusionmethod withineachcontextwhichgivesweightproportionaltohowwelleachclassierseparates itstrueandfalsealarms.Duetothedynamicrangedierencesbetweenclassication algorithms,amethodtonormalizeeachtotherange[0 ; 1]basedontheircumulative posteriorprobabilitywithinthecontextisproposed,thisnormalizedcontext-dependent outputiscalledtheconditionalmappedposteriorprobabilityCMPP.TheCDFmethod wasappliedtoHSIin[130]. TheContextExtractionforLocalFusionCELFmethod[131]combinestheproblemsofunsupervisedcontextlearningandsupervisedlearninginfusionintoajoint optimizationproblem.ThismethodcombinestheFuzzyC-means[132]objectivefunction withacontext-locallinearcombinationfusionmethod.TheCELFobjectivefunctionis J U ; W ; C = N X j =1 C X i =1 u m ij jj x j )]TJ/F37 11.9552 Tf 11.955 0 Td [(c i jj 2 + N X j =1 C X i =1 u m ij K X k =1 w ik y kj )]TJ/F21 11.9552 Tf 11.955 0 Td [(t j 2 {93 62

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subjecttotheconstraints: u ij 2 [0 ; 1] 8 i;j; C X i =1 u i;j =1 8 j and K X k =1 w ik =1 8 i where N isthenumberofsamples, C isthenumberofclusters, u ij isthemembershipof sample j incluster i m isthefuzziercontrollingthecrispnessofthepartitioning, x j is theconcatenationofallrelevantfeaturevectorsintoacolumnvectorforsample j c i is theclustercenterforcluster i isascalarthatbalancesimportanceofclassicationerror tothesumofintraclusterdistances, K isthenumberofalgorithmsbeingfused, w ik isthe weightofalgorithm k incluster i y kj istheoutputofalgorithm k forsample j ,and t j is thetruelabelofsample j ThelearningalgorithmforCELFfollowsastandardalternatingoptimizationstrategy where U and W arerstinitializedrandomly,andthennewclustercenters,memberships, andfusionweightsarecomputedbyclosedformupdateequations.Iterationproceeds untiltheparametersaredeemedtohaveconverged.Analpost-processingstepwhich recomputes U and W withslightlychangedupdateequationsisthenincludedtohelp ensurecompactclusters.Intesting,theclustermembershipsforthenewtestpoint, x l arerstcomputed,andthenthecontext-dependentfusionoperationsareaggregatedby clustermembership: t l = C X i =1 u il K X k =1 w ik y kl {94 TheCELFjointoptimizationframeworkhasbeenextendedtoimproveperformance byaddingfeatureselectionasinSCADin[133],toimproverobustnessbyadding regularizationin[134],andtoreplacethelinearcombinationfusionwithafuzzyintegral methodin[135]. 2.6FuzzyClustering Fuzzyclusteringcanbeviewedastheproblemofndingapartitioningofthedata intofuzzysets.Eachdatapoint, x n for n =1 :::N ,hasnonnegativemembership u nc ineachofthefuzzysets,indexedby c =1 :::C ,andatotalmembershipofoneshared 63

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betweenallofthesets,i.e. P C c =1 u nc =1.Forexample,datapointsrepresentingtheheight valuesofpeoplecouldbedividedintotwofuzzysets.First,thesetofalltall"people andsecond,thesetofallshort"people.Everydatapointsimultaneouslybelongsto bothfuzzysetswithavaryingdegreeofmembership.Atallbasketballplayermayhavea membershipvaluenear1inthesetoftallpeople,andbydenitionthevalueofoneminus thatmembershipinthesetofshortpeople.Incontrasttoprobabilisticmixturemodels however,neitherofthesesetsisconsideredtobeanunderlyinggeneratingdistributionof thedatapoint. Thegoaloffuzzyclusteringistondanoptimalgroupofsetstoexplainthedata points,wheretheoptimalityisdenedasaminimumtotalmembershipweighteddistance ofeachdatapoint x n toaprototypeofeachset c .Thusthefuzzyclusteringproblemhas twogroupsofunknownvariables,thesetprototypes c andthesetmemberships u nc for eachdatapoint. ThefuzzyclusteringproblemhasclassicallybeentackledwiththeFuzzyC-Means algorithm[132].TheFuzzyC-Meansalgorithmisderivedbyminimizationoftheobjective function J X ; U ; = N X n =1 C X c =1 u m nc d x n ; c 2 ; {95 where m isthefuzzierparameterwhichaectsthedegreetowhichmembershipsare mixed,andwheretheformofthedistancefunction, d x n ; c canbeusedtocontrolthe typeofclustersfound.ThisandisoftentakentobetheEuclideandistance d x n ; y c = k x n )]TJ/F37 11.9552 Tf 12.379 0 Td [(y c k .Valuesofthemembershipsandtheclusterprototypescanbefoundusingan alternatingoptimizationstrategy.Closedformalternatingupdateequationsarederived throughdierentiationoftheobjectivefunctionwithLagrangemultiplierconstraintsto yield u nc = =d x n ; c 2 1 = m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P C k =1 =d x n ; k 2 1 = m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 {96 64

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y c = 1 P N n =1 u m nc N X n =1 u m nc x n : {97 Wenotethatthisalgorithmisguaranteedtoconvergeasnoupdatecanincreasethe objectivevaluebutonlytoalocalminimumorsaddlepointoftheobjectivefunction,and itisthussubjecttotheinitializationconditionsastowhichlocalminimumwillbefound. Thecommonlycitedproblemwithtraditionalclusteringapproachesisthatthe numberofclustersmustbespeciedasaparametertothealgorithm,thoughitsvalue isoftenunknowntotheuserofthealgorithm.IntheFuzzyClusteringliterature,one approachoftentakenistorunthealgorithmwiththenumberofclusterssettoeachvalue withinanexpectedreasonablerange,andtocomparetheresultsusingaclustervalidity index.SuchanapproachwastakeninaninuentialearlyworkbyGathandGeva[136]. AfuzzyclusteringalgorithmthatattemptstolearnthenumberofclustersisCompetitiveAgglomeration[137].Thisalgorithmstartswithahigherthanreasonablyexpected numberofclusters,andposesthemincompetitionagainsteachothertoreceivethemembershipofthedatapoints.Asthealgorithmrunsitprogressivelyprunesclusterswhich havelowtotalmembership. AnotherfuzzyclusteringapproachpresentedbyLietal.[138]alsostartswithalarge numberofclusters,butthenallowsclusterprototypestobecomeco-located.Thecolocatedclustersarethenmergedinapostprocessingstepafterthealgorithmconverges. 2.7Summary Fromthesurveyofdetectionmethods,itcanbenotedthatthereisaprogression ofmodelswithincreasingaccuracyintheirphysicalrepresentationofthesignalsinHSI. ThemostbasicmodelisthatpresentedbytheSAM.TheSAMusesanadditivesignal modelandanormaldistributionforthebackgroundwithzeromeanandasimpleisotropic diagonalcovariance. 65

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ThenextmostaccuratemodelisusedbytheSMFdetector.TheSMFstillusesan additivetargetmodel,thoughthebackgroundgainsanunknownmeanandfullcovariance matrix. FollowingtheSMFistheACEdetector.IntheACEdetectorweseetheuseofa signalreplacementmodel,wherethemagnitudeofthebackgroundvariancecannowbe scaled.Thescalingallowsforthelevelofthebackgroundtobereducedwhenastronger targetsignalisobserved,andthusitmorecloselyapproximatestruemixingbehavior. ThenalnotabledetectorinthisprogressionistheHSD.TheHSDfullyintegrates thelinearmixingmodel,whichisthenextmostphysicallyaccuraterepresentationofHSI signals. Thoughmanyotherdirectionshavebeentakenindetectorresearch,nodetectorshave beenpresentedwithcompellingimprovementstothemodelalongthisline,asillustrated inFigure2-3. TherecentresearchintopiecewiseconvexmodelsforHSIunmixingprovidesa promisingcandidateforthenextmodelinthedetectorprogression.Piecewiseconvex modelsdividethelinearmixingmodelupintomultipleconvexmixtureswhichcanprovide forcontextswithintheimage. Theresearchintocontext-dependentfusionnextprovidessometechniquesthatwill beusefulforcombiningsuchenhancedmodelswithdetectors.TheCELFmethodgivesan importantideaforjointcontextestimationanddetectoroptimization. Takentogether,theprogressionofresearchindetectors,unmixingmethods,and context-dependentmethodsgivesafoundationforanextstepinadvancingdetection: context-dependentdetection.ThisisillustratedinFigure2-4. 66

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Figure2-1.Detectionblockdiagram Figure2-2.SchematicrepresentationofSample-PCEprobabilisticmodel Figure2-3.Detectormodelprogression 67

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Figure2-4.Progressiontocontext-dependentdetection 68

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Table2-1.Detectorcategorizations StatisticalSubspaceUnmixingPurePixelSub-pixel SAMx SMFxx CEMx TCIMFx ACExx SPSMFxxx CCSMFxx PALMxx OSPxx SS-ACExxx JSDxx AMSDx SCLSxx NCLSxx FCLSxx HSDxx HUDxx CSDxx QMFxx Waveletsx KernelMethodsxxxx SparseReconstructionx SVDDx 69

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CHAPTER3 TECHNICALAPPROACH Theoriginalcontributionsofthisworkarepresentedinveconceptualclasses.The rstisasequentialcontextlearninganddetectionstrategyyieldingtheFuzzPop-ACEand FuzzPop-HSDdetectors.Nextisajointcontextlearninganddetectionstrategyyielding theFCM+ACE+CEMFACEM,FuzzyConstrainedEnergyMinimizationFCEM, andPossibilisticFuzzyandSpatialConstrainedEnergyMinimizationPFSCEMdetectors.Followingthat,Bayesiantechniquesforfuzzyclusteringaredeveloped,yielding theBayesianFuzzyClusteringBFCandInniteBayesianFuzzyClusteringIBFC clusteringalgorithms.Fourthly,theBayesianfuzzyclusteringtechniquesareappliedtodetection,yieldingtheInniteBayesianFuzzyConstrainedEnergyMinimizationIBFCEM andBayesianFuzzyACEBFACEdetectors.Finally,thenewconceptofAlarm-Set FusionASFispresentedthroughtheFalseAlarmRateEstimationforAlarm-Set FusionFARE-ASFandRunPackingRPalgorithms. Thesecontext-dependentdetectionmethodsaugmentthetraditionaldetection processwithacontextestimationstep.Figure3-1showsablockdiagramforthecontextdependentdetectors.Thecontextestimatorblockwithinthelargercontext-dependent detectionblockisresponsibleforidentifyingtherelevantcontextualinformationforeach pixeloftheinputimage.Thecontextestimatorcanutilizeadditionalinputinformation suchasLiDARelevationmaps,localenvironmentalmeasurements,orotherinformation likespectrallibrariestoperformthistask. InthesequentialFuzzy-ACEandFuzzy-HSDmethods,thecontextisestimated rstandthensimplypassedontothecontext-dependentdetector.InthejointFACEM, FCEM,andPFSCEMmethods,thereisatwo-wayowofinformationbetweenthetwo blocks.Forthesemethods,thecontextinformationisavectorofclustermembershipsfor eachpixel.Thecontext-dependentdetectorthenperformsparameterestimationforeach clusterandformsdetectionoutputsbasedonthecluster-specicparameters. 70

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Additionally,thecontextinformationandcontext-dependentdetectionoutputcanbe usedasaninputforASF.TheASFmethodsrecombinethecontext-dependentoutputs, whichmayhavedierentdistributions,inamannerthattriestooptimizesystemlevel performance. 3.1SequentialUnsupervisedContextLearningandDetection Arstapproachtocontext-dependentdetection,onesharedbyexistingclusterbased detectors,isasequentialprocesswhichrstestimatesthecontextsthenusesaconditional detectorbasedonapixel'smembershipineachcontext.Thisbasicideacanbeextended tofuzzyclusteringoutputs.Thissectionpresentsageneraltargetdetectionalgorithm strategywhichusestheFuzzPopalgorithmsee x 2.4.2torstestimatebackground contexts,andthenintegratesthisoutputwithcontextdependentdetectors.Underthis strategytheFuzzPop-HSDandFuzzPop-ACEalgorithmsaredeveloped. InthissectionaslightchangeinnotationoverthepreviousFuzzPopdescription isused.Theconvexregionsareindexedby c =1 :::C ,thedatapointsareindexedby n =1 :::N ,andthetypicalityfuzzicationexponentisdenotedby TheFuzzPop-HSDdetectoriscreatedthroughthefollowingsteps: 1.UseFuzzPoptondbackgroundendmembersets E c ,convexregionmemberships u nc ,andtypicalities t nc 2.Addthetargetsignaturetoeachendmembersetcreating S c =[ s E c ],thenndnew proportions p s nc ,memberships u s nc ,andtypicalities t s nc usingtheFuzzPopupdate equations2{90,2{91,and2{92. 3.Computefuzzyandpossibilisticweightedmeans ^ c andcovariances ^ c foreachclass usingequations3{1,3{2. ^ c = 1 P N n =1 u m nc t nc N X n =1 u m nc t nc x n {1 ^ c = 1 P N n =1 u m nc t nc N X n =1 u m nc t nc x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T {2 71

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4.Computethedetectionstatisticforeachpixelusingequation3{3. r FuzzPop )]TJ/F20 7.9701 Tf 6.587 0 Td [(HSD x n = C X c =1 u nc x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(p T nc E c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(p T nc E c x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(p s T nc S c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(p s T nc S c {3 Thepatternoutlinedabovecanbeusedtomakeafuzzyweightedorcontextconditionalversionofanyexistingdetectionalgorithm.Forexampleamembershipweighted FuzzPop-ACEdetectorwouldusethefollowingdetectionstatistic: r FuzzPop )]TJ/F20 7.9701 Tf 6.586 0 Td [(ACE x n = C X c =1 u nc s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c 2 s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c : {4 Thisapproachhasalsobeenextendedtointegrateelevationmeasurementsfrom aLightDetectionandRangingLiDARsensor.Todoso,weaugmentthedistance computation d nc usedinthemembershipupdateequation2{91tobeathreedimensional Euclideandistance.ThersttwoofthethreedimensionscomefromtheassignedUTM coordinatesoftheimagepixelsandthethirdfromtheelevationoftheLiDARrstreturn. Thisweightingthusdecreasestheenforcedsimilarityofpixelsiftheyhavediering elevation,whichwouldbeanindicationthattheyarefromdierentconvexregionsofthe image. 3.2JointUnsupervisedContextLearningandDetection Afurtheradvancementtotheframeworkofcontext-dependentdetectioncomesfrom methodswhichlearncontextsthatoptimizedetectionperformance.Threealgorithmsfor suchajointunsupervisedcontextlearninganddetectionapproacharepresentedhere. ThesearetheFACEM,FCEM,andPFSCEMalgorithms. 3.2.1FCM+Detector+CEM Onecontext-dependentdetectorcanbecreatedusingacombinationofFuzzyCMeansFCM,adetectorsuchasACE,andCEM.Aspeciccaseofthisapproachisthe FACEMdetector.ThisdetectorwouldusetheFCMobjectivefunctioncombinedwitha membership-weightedcontext-dependentACEdetectionstatistic,yieldingtheobjective 72

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function J = C X c =1 N X n =1 u m nc jj x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c jj 2 + N X n =1 C X c =1 u m nc s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c 2 s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c ; {5 where ^ c = 1 P N n =1 u m nc N X n =1 u m nc x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T : {6 Theoptimizationproblemisthen min u nc ; ^ c 8 n;c J subjectto C X c =1 u nc =1 ;u nc 0 : {7 Inordertomakeoptimizationoftheobjectivefunctiontractable,asimplifying assumptionismadethatthedetectortermisnotdependentuponthecurrentvaluesof theparameters,anditthussimplyaddsaconstantoset.Minimizationoftheobjective functionwiththisassumptionthenfollowsanalternatingoptimizationstrategybasedon closedformupdateequationsasintraditionalFCMandotherFuzzyClusteringalgorithms baseduponthatobjective.Theupdateequationscanbefoundas: ^ c = 1 P N n =1 u m nc N X n =1 u m nc x n {8 u nc = 1 P C k =1 jj x n )]TJ/F20 7.9701 Tf 7.463 0.111 Td [(^ c jj 2 + ACE x n ; ^ c ; ^ c jj x n )]TJ/F20 7.9701 Tf 7.463 0.111 Td [(^ k jj 2 + ACE x n ; ^ k ; ^ k 1 = m )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 : {9 Inoperation,thenaldetectionstatisticisanaggregationoverthemembership weighteddetectionoutputfromeachcluster r FACEM x n = C X c =1 u m nc s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c 2 s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c s )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c T ^ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c x n )]TJ/F15 11.9552 Tf 13.26 0.166 Td [(^ c : {10 73

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PsuedocodeforthisalgorithmisgiveninAlgorithm1.Convergenceismeasured bychangeintheobjectivefunctionvalue3{5beinglessthanauserdenedthreshold, whereavalueontheorderof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 isoftenanappropriatechoice. Algorithm1: FACEMDetector Data :inputdata X size D N bandsbypixels,targetspectralsignature s numberofclusters C ,maximumiterations N iter ,fuzzier m Result :detectioncondences z n ,memberships u nc ,clustercenters ^ c ,cluster covariances ^ c ,for n =1 :::N c =1 :::C initializecenters y c for c =1 :::C withrandomlyselectedpixels for iter= 1 :::N iter do updatememberships u nc using3{9forall n =1 :::N c =1 :::C updateclustercenters c using3{8forall c =1 :::C updateclustercovariances ^ c ,using3{6for c =1 :::C stopifconverged computedetectioncondences z n using3{10forall n =1 :::N 3.2.2FuzzyConstrainedEnergyMinimization TestingoftheFACEMdetectionalgorithmwasfoundtoprovidelittleadvantageover sequentiallyperformingFCMclusteringandthenusingaclusterweightedFuzzyACE detector.Thisisperhapsunsurprising,becauserstlythemulti-objectiveisatradeo betweencompactsphericalclustersanddetectionoutputs,andsecondlybecausethe optimizationinFACEMisbiasedtowardstheFCMtermofthemulti-termobjective. Continuingtheideahoweverthatcontextestimationshouldndcontextsthatareuseful fordetection,weinvestigatedifdetectorfunctionscouldinsteadbeclustereddirectly.This yieldsanewalgorithmcalledFCEM.InFCEM,thegoalistominimizethefollowing genericobjective J = C X c =1 N X n =1 u m nc d x n ; c {11 subjecttothetraditionalfuzzyclusteringconstraintsandtotheadditionalconstraintthat thedetectoroutputbe1whenpresentedwiththetargetsignature. 74

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Whenusingalinearlterdetectionoperatorasthedistancefunctionforclusteringwe havethefollowingobjectivefunction: J = C X c =1 N X n =1 u m nc )]TJ/F37 11.9552 Tf 5.479 -9.684 Td [(w T c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c 2 ; {12 subjecttotheconstraints w T c s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c =1{13 0 u nc 1{14 C X c =1 u nc =1{15 forall c =1 :::C and n =1 :::N Ifmultipletargetsareused,theconstraintsbecome: M T c w c = 1 K ; {16 where K isthenumberoftargetsignatures, M isthe d k matrixofthetargetsignatures with c subtracted,and 1 k isa k -dimensionalvectorofones. TheobjectivecanthenbewrittenastheLagrangian L = C X c =1 w T c C c w c + C X c =1 c )]TJ/F37 11.9552 Tf 5.479 -9.683 Td [(w T c s )]TJ/F40 11.9552 Tf 11.956 0 Td [( c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + N X n =1 n C X c =1 u nc )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; {17 orwithmultipletargets L = C X c =1 w T c C c w c + C X c =1 c K X k =1 )]TJ/F37 11.9552 Tf 5.479 -9.684 Td [(w T c s k )]TJ/F40 11.9552 Tf 11.955 0 Td [( c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + N X n =1 n C X c =1 u nc )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : {18 TheclosedformupdateequationscanbefoundfromthisLagrangianas u nc = 1 P C k =1 w T c x n )]TJ/F41 7.9701 Tf 6.587 0 Td [( c 2 w T k x n )]TJ/F41 7.9701 Tf 6.587 0 Td [( k 2 1 = m )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 {19 w c = C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c M c M T c C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c M c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 k {20 c = x c + w c w T c w c w T c 1 K K X k =1 s k )]TJETq1 0 0 1 381.857 100.501 cm[]0 d 0 J 0.478 w 0 0 m 7.098 0 l SQBT/F37 11.9552 Tf 381.857 93.514 Td [(x c )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 {21 75

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where C c = N X n =1 u m nc x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c T {22 x c = 1 P N n =1 u m nc N X n =1 u m nc x n : {23 Inoperation,thenaldetectoroutputwouldthenbe r FCEM x n = C X c =1 u nc w T c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c : {24 Fordetectionstatisticsotherthanthelineardetectorhowever,closedformupdates maybediculttoderive.Generalnonlinearoptimizationstrategiescanbeusedtond goodparametervalues,howeverthesetendtobeveryslow,anditcanbedicultto imposeconstraintssuchaspositivedenitenessofacovariancematrix. PsuedocodefortheFCEMdetectorisgiveninAlgorithm2.Convergenceismeasured bychangeintheobjectivefunctionvalue3{12beinglessthanauserdenedthreshold, whereagain,avalueontheorderof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 isoftenanappropriatechoice.Asanalternative initialization,theFCMalgorithmcanberunforafewiterationstoobtaininitialcenters andmemberships. Algorithm2: FCEMDetector Data :inputdata X size D N bandsbypixels,targetspectralsignature s numberofclusters C ,maximumiterations N iter ,fuzzier m Result :detectioncondences z n ,memberships u nc ,detectorcenters c ,detector weights w c ,for n =1 :::N c =1 :::C initializecenters c for c =1 :::C withrandomlyselectedpixels initializememberships u nc =1 =C forall n =1 :::N c =1 :::C initializedetectorweights w c using3{20forall c =1 :::C for iter= 1 :::N iter do updatememberships u nc using3{19forall n =1 :::N c =1 :::C updatedetectorcenters c using3{21forall c =1 :::C updatedetectorweights w c using3{20forall c =1 :::C stopifconverged computedetectioncondences z n using3{24forall n =1 :::N 76

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3.2.3PossibilisticFuzzyandSpatialConstrainedEnergyMinimization TheFuzzPopalgorithm x 2.4.2hasdemonstratedthepotentialeectivenessof incorporatingpossibilisticclusteringandspatialmembershipsmoothingtermstothefuzzy clusteringofhyperspectralimages.Thepossibilistictermsaddrobustnesstotheclustering byreducingtheeectsofoutliers,andtheFuzzyLocalInformationC-MeansFLICM spatialsmoothingtermhelpstondmeaningfulcontextsthatarespatiallycoherent. BecausetheFCEMjointclusteringanddetectionalgorithmalsousesafuzzyclustering basedobjective,thesesamepossibilisticandFLICMtermscanbeintegratedwithFCEM tocreateanewcontext-dependentdetectionalgorithm. Thenewapproach,calledPossibilisticFuzzyandSpatialConstrainedEnergyMinimizationPFSCEM,createsagenericframeworkfordetectorsparameterizedbythe detectionstatistic.Inamannersimilartofuzzyclusteringalgorithms,theparametersof thisdetectionstatisticcanbeestimatedthroughtheminimizationoftheenergyobjective function J = C X c =1 N X n =1 [ au m nc d x n ; c + G nc + bt n nc d x n ; c ]+ C X c =1 c N X n =1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(t nc n ; {25 where a and b aretuningparametersofthealgorithm, m isthefuzzierforthefuzzy clusteringwithmembership u nc n istheparametersimilartoafuzzierusedforthe typicalities t nc C isthespeciednumberofconvexregions,and N isthenumberofpixels intheimage.Theindex c =1 :::C runsoverconvexregions,and n =1 :::N runsover pixelsintheimage.Thelocalspatialinformationtermisgivenas G nc = X k 2 N n ;k 6 = n 1 d s nk +1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(u ck m d x n ; c ; {26 where d s nk isthespatialdistanceingroundcoordinatesbetweenpixel n anditsneighbor pixel k withinthelocalneighborhood N n .Thetypicalityscaleterm, c = 1 P N n =1 u nc N X n =1 u nc d x n ; c ; {27 77

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issetateachiterationtobeaveragemembershipweighteddetectoroutputwithinthe cluster. Underalinearmatchedlterstyledetector,wecansetthedetectionstatistictobe asquaredlinearoperationwithaweightvectorandaclustercenter d x n ; c = d x n ; w c ; c = w T c x n )]TJ/F21 11.9552 Tf 11.955 0 Td [( c 2 = x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c T w c w T c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c : {28 Usingthisdetectionstatisticwecanobtainclosedformupdateequationsforthejoint clusteringanddetectorestimationalgorithm w c = m T c C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c m T c C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c m c {29 c = x c + w c w T c w c )]TJ/F37 11.9552 Tf 5.48 -9.684 Td [(w T c s )]TJETq1 0 0 1 341.298 440.765 cm[]0 d 0 J 0.478 w 0 0 m 7.098 0 l SQBT/F37 11.9552 Tf 341.298 433.777 Td [(x c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 {30 m c = s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c {31 C c = 1 P N n =1 nc N X n =1 nc x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c T {32 nc = au m nc + bt n nc {33 x = 1 P N n =1 nc N X n =1 nc x n : {34 Additionallythestandardtypicalityupdateequationsofpossibilisticclusteringis foundas t nc = 1 b c d x n ; c 1 = n )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 +1 ; {35 andthemembershipupdateisnearlythesameastradtionalFCMbutnowincludesthe FLICMterm u nc = 1 P C k =1 d x n ; c + G nc d x n ; k + G kn 1 = m )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 : {36 78

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PsuedocodeforthePFSCEMdetectorisgiveninAlgorithm3.LiketheFCEM algorithm,convergenceismeasuredbychangeoftheobjectivefunction3{25being lessthanauserdenedthreshold,whereagain,avalueontheorderof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 isoftenan appropriatechoice.Asanalternativeinitialization,theFCMalgorithmcanberunfora fewiterationstoobtaininitialcentersandmemberships.Thenaldetectionoutputofthe PFSCEMdetectoristhesameasfortheFCEMdetectorusingequation3{24. Algorithm3: PFSCEMDetector Data :inputdata X size D N bandsbypixels,targetspectralsignature s numberofclusters C ,maximumiterations N iter ,fuzzytermweight a ,typicality termweight b ,fuzzier m ,typicalityexponent n ,spatialwindowsize S win Result :detectioncondences z n ,memberships u nc ,typicalities t nc ,detectorcenters c ,detectorweights w c ,for n =1 :::N c =1 :::C initializecenters c for c =1 :::C withrandomlyselectedpixels initializememberships u nc =1 =C forall n =1 :::N c =1 :::C initializetypicalities t nc =1 initializedetectorweights w c using3{29forall c =1 :::C for iter= 1 :::N iter do updatetypicalityscale c using3{27forall c =1 :::C updateFLICMterms G nc using3{26forall n =1 :::N c =1 :::C updatememberships u nc using3{36forall n =1 :::N c =1 :::C updatetypicalities t nc using3{35forall n =1 :::N c =1 :::C updatedetectorcenters c using3{30forall c =1 :::C updatedetectorweights w c using3{29forall c =1 :::C stopifconverged computedetectioncondences z n using3{24forall n =1 :::N 3.3BayesianFuzzyClustering FuzzyclusteringisoftencomparedtotheprobabilistictechniqueofGaussianmixture modelestimation,asthetwomethodsdohavemanysimilaritiesonthesurface.The supercialsimilarityviewiswheremostcomparisonsstophowever,leavingthetrue dierencesmuddled.Atleastpartlybecausethetwomethodsareseeminglysimilaryet somehowdierent,twocampsarise,oneofprobabilisticmethodsforsolvingproblemsand anotheroffuzzymethods.Theworkinthissectioncrossesbetweencampsbypresenting aprobabilisticmodelforfuzzyclustering.Then,throughtheuseofBayesianprobabilistic 79

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inference,wedevelopnewalgorithmsforfuzzyclusteringwithexpandedcapabilitiesand improvedresults.Furthermore,itcanbeshownbythismodelthatthefuzzyclustering problemisindeedadierentproblemfrommixturemodelestimation,asthetwoproblems havedierentprobabilisticmodels. FortheBayesianFuzzyClusteringBFCapproach,werepresenttheunknownset memberships u nc andsetprototypes c offuzzyclusteringsee x 2.6asrandomvariables, andwerepresentourknowledgeoftheformof,anduncertaintyaboutthevalueofthese randomvariableswithprobabilitydistributions.Wetheninferthemostlikelyvalue ofthesevariablesgiventhedatawehaveobserved.TheBFCapproachcanthenbe extendedtorepresentthenumberofclusters C asarandomvariableaswell.Thisyields anapproachtoestimatingthenumberofclustersthatisperhapssimplerthanthefuzzy methodsandprobabilisticmethodscommonlyusedtodate. Themotivationforthisapproachcameaboutbecausemanyofthealgorithms developedinthisdissertationhaverelieduponfuzzyclusteringtolearncontextual information.However,thealgorithmictoolscommonlyappliedtotraditionalfuzzy clusteringwerenotsucientforthemethodswewantedtodevelop,astheyaredicult tooptimize.ABayesianprobabilisticinterpretationoffuzzyclusteringwasdesiredfor abilitytoapplyMCMCinferencetechniques.Thisapproachtofuzzyclusteringisa signicantcontributionbyitself,andthemodelsandinferencealgorithmsdevelopedfor thispurposearedetailedinthefollowingsections.FirsttheBayesianfuzzyclustering modelispresented,andthenanMCMCinferencealgorithmisgivenforthismodel.Next thismodelisextendedtoavariablenumberofclusters,andthenwepresentaparticle lterinferencealgorithmforthismodel. 3.3.1BayesianFuzzyClusteringModel TheBayesianFuzzyClusteringBFCmodeliscomposedofadatalikelihood distributioncalledtheFuzzyDataLikelihoodFDL,apriordistributionforthecluster membershipscalledtheFuzzyClusterPriorFCPandaGaussianpriordistributionon 80

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clusterprototypes p X j U ; Y = N Y n =1 FDL x n j u n ; Y = N Y n =1 1 Z u n ;m; Y C Y c =1 N x n j = y c ; = u m nc I D D {37 ~ p U j Y = N Y n =1 FCP u n j Y = N Y n =1 Z u n ;m; Y C Y c =1 u )]TJ/F22 7.9701 Tf 6.587 0 Td [(mD= 2 nc Dirichlet u n j # {38 p Y = C Y c =1 N y c j y ; y ; {39 where N isthenumberofdatapoints, C isthenumberofclusters, D isthedimensionalityofthedata, u nc isthemembershipofdatapoint x n incluster c m isthefuzzier, y c aretheclusterprototypes, I D D isthe D dimensionalidentitymatrix,and Z u n ;m; Y isanormalizationconstantgivenin3{40.Theargumentstotheprobabilitydensity functionsaregroupedintothe D N matrixofdatapoints X =[ x 1 j ::: j x N ],memberships U withdimensions C N wheretheelementatrow c ,column n is u nc ,andprototypes Y =[ y 1 j ::: j y C ]withdimensions D C IntheFuzzyDataLikelihood3{37,thedataaregivenalikelihoodproportionalto theproductof C normallikelihoodfunctions,eachwithadierentprecisionparameter. Thismightbeviewedasthelikelihoodofthe C normaldistributionssimultaneouslygeneratingthedatapoint.Theprecisioninversevarianceofeachofthenormalcomponents = u m nc isspecictoeachdatapointandisequaltoitsclustermembershipraisedto apower.Thus,farawaycomponentsofthelikelihoodarestretched-outbyusingalow membership/precisionhighvariancevalue.Notethatinthismodel,eachdatapointhas auniquevectorofmembershipparametersforthedatalikelihood,andthereforeeachdata pointcanbeconsideredtohaveitsowngeneratingprobabilitydistribution.Acrossall 81

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datapointshowever,thenormallikelihoodsaregroupedsuchthattheysharemeanvalues = y c whicharetheclusterprototypes. WenotethattheproductofGaussianlikelihoodfunctionsoverthesamevariable x n isnolongeranormalizeddistribution,thoughthenewnormalizationconstantis computableas Z u n ;m; Y = )]TJ/F22 7.9701 Tf 6.587 0 Td [(D C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = 2 C Y c =1 u m nc D= 2 C X c =1 u m nc )]TJ/F22 7.9701 Tf 6.586 0 Td [(D= 2 exp 8 > < > : )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 0 B @ C X c =1 u m nc y T c y c )]TJ/F26 11.9552 Tf 13.151 25.704 Td [( P C c =1 u m nc y c T P C c =1 u m nc y c P C c =1 u m nc 1 C A 9 > = > ; ; {40 wherethederivationofthisfunctionisgivenasanappendix.Intheinferencealgorithms discussedherehowever,thenormalizationconstantiscanceledbytheFCPandneednot becomputed.WenotealsothattheproductofnormalsintheFDLcanbere-written asasinglenormaldistributionforwhichthemeanandcovariancecanbecomputed byexpandingtermsandcompletingthesquare.Thissinglenormalversionshowshow toderivetheFDLnormalizationconstant.Weprefertheproductofnormalsnotation howeverasitmoreclearlyexpressestheintentofthemodelandmoredirectlysetsupthe inferencealgorithms. Themembershipvariablesareassumedtohaveapriordistribution3{38derived forthispurposewhichwecalltheFuzzyClusterPrior.Thispriorconsistsofthreefactors whichwewillcall F 1 = Z u n ;m; Y F 2 = Q C c =1 u )]TJ/F22 7.9701 Tf 6.587 0 Td [(mD= 2 nc ,and F 3 =Dirichlet u n j .The rsttwofactorsareacounter-balancetothedatalikelihood,with F 1 exactlycancelingthe normalizationconstantoftheFDL. Thefactor F 2 occursbecausetheGaussiancomponentsof3{37willessentially promotehighmembershipvalues,havingthefactorof u mD= 2 nc whichislargeforlargememberships.Theindividualfactorsof F 2 u )]TJ/F22 7.9701 Tf 6.586 0 Td [(mD= 2 nc howeverarelargestforsmallmembership values.Takentogetherthefactorsexactlycancel,causingthejointdistributiontobe moreagnosticaboutmembershipvalues.Becauseofthenegativevalueoftheexponent 82

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in F 2 however,theFCPcannotbenormalizedovertheinterval[0 ; 1]atleastformany relevantvaluesof m and D .Thusitbelongstothecommonlyusedclassofimproper priors.Thoughwhereasimproperpriorsaremoretypicallyusedtoprovideadistribution whichisuninformativeaboutpriorbelief,theFCPisusedasamathematicalconvenience toexactlyreplicatethebehaviorofFCMbytheBayesianmodel.Wenotethataproper priorsuchasaproductofInverse-Gammadistributionscouldbeusedinplaceof F 2 in theFCP.IftheshapeparameteroftheInverse-Gammaissetto = mD= 2 )]TJ/F15 11.9552 Tf 12.165 0 Td [(1andscale parameter issmall,thentheperformanceofthemodelwouldbeunchanged. The F 3 factoroftheFCPisaDirichletlikelihoodfunctionparameterizedbythe vector ,denedas Dirichlet x j = \050 P K k =1 k Q K k =1 \050 k K Y k =1 x k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 k ; {41 onthedomainofthestandardsimplex, x k 0forall k =1 :::K P K k =1 x k =1.When takentobeasymmetricDirichletdistributionwithparameter = 1 C ,thatisa C 1 columnvectorofallones,thisfactorvanishesandhasnoeectontheclusteringoutput. Thus,thisfactorisnotstrictlynecessarytoreplicatethebehavioroftheFuzzyC-Means algorithm.TheadditionoftheDirichletfactorhowevernicelyexpressesthepositivity andsum-to-oneconstraintsonthememberships,anditprovidesadditionalexibilityand capabilitytotheclusteringalgorithmwhilecontainingFCMasasub-caseofthemore generalmethod. FinallytheBFCmodelplacesaGaussianprior3{39oneachoftheclusterprototypeparameters.Thehyper-parametersofthisdistributioncanbesetintheempirical Bayesfashiontousethemeanofthedataset y = 1 N N X n =1 x n ; {42 andawidecovarianceintheshapeofthedatacovariance y = N N X n =1 x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( y x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( y T ; {43 83

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where isausersetparameteraectingthestrengthoftheprior,weused =3inour applications.Consideringthelog-likelihoodofthefullBFCmodelasafuzzyclustering objectivefunction,theclusterprototypeprioractsasaregularizationterm. Whenmultipliedtogether,theFDL3{37,FCP3{38,andprototypeprior3{39 formthejointlikelihoodofthedataandparameters p X ; U ; Y = p X j U ; Y ~ p U j Y p Y / exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 N X n =1 C X c =1 u m nc k x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c k 2 N Y n =1 C Y c =1 u c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 nc exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 C X c =1 y c )]TJ/F21 11.9552 Tf 11.955 0 Td [( y T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 y y c )]TJ/F21 11.9552 Tf 11.956 0 Td [( y ; {44 whichisproportionaltotheposteriordistributionoftheparametersgiventhedata, p U ; Y j X / p X ; U ; Y Theequivalentobjectivefunctionformofthejointlikelihoodisthenegativeofits logarithm, J X ; U ; Y = N X n =1 C X c =1 u m nc k x n )]TJ/F37 11.9552 Tf 10.739 0 Td [(y c k 2 )]TJ/F15 11.9552 Tf 10.738 0 Td [(2 N X n =1 C X c =1 c )]TJ/F15 11.9552 Tf 10.738 0 Td [(1log u nc + C X c =1 y c )]TJ/F21 11.9552 Tf 10.739 0 Td [( y T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 y y c )]TJ/F21 11.9552 Tf 10.738 0 Td [( y ; {45 whereafactorof2hasbeenmultipliedthroughtosimplify. 3.3.2InferenceintheBayesianFuzzyClusteringModel TheprimaryinferencetaskweareconcernedwithisndingtheMaximum aposteriori MAPvaluesoftheparametersintheBFCmodelgiventhedata.Thisisequivalent tondingthegloballyoptimalvaluesoftheparametersforaregularizedFCMobjective function.ToperformtheMAPinferenceweuseanMCMCtechniquetoleverageits optimalityguarantees. AMetropoliswithinGibbs[100]samplercanbeusedtogeneratesamplesfromthe BFCmodel'sposteriordistributiongiventhedata.Asthesesamplesaregeneratedthey canbeevaluatedfortheirposteriorlikelihoodandthecurrentbestsampleretained. Algorithm4givestheprocedureforthismethod. 84

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Algorithm4: BayesianFuzzyClusteringMAPSearch Data :Datamatrix X ,fuzzier m ,numberofclusters C ,numberofsampling iterations N iter Result :MAPestimatesformembership U andclusterprototypes Y initializehyperparameters y ; y from3{42and3{43 sampleinitial u n Dirichlet = 1 C forall n =1 :::N sampleinitial y c N y ; y forall c =1 :::C setMAPsamplestocurrent u n u n ; y c y c forall n =1 :::N;c =1 :::C for iter=1 ;:::;N iter do /*sample U p U j X ; Y / p X ; U ; Y */ for n =1 :::N do sampleproposednewmembershipvector u y n from3{46 acceptproposal u n u y n withprobability a u from3{48 if p x n ; u y n j Y >p x n ; u n j Y ,using 3{47 then u n u y n /*sample Y p Y j X ; U / p X ; U ; Y */ for c =1 :::C do sampleproposednewclusterprototype y y c from3{49 acceptproposal y c y y c withprobability a y from3{51 if p X ; y y c j U >p X ; y c j U ,using 3{50 then y c y y c /*checkfullsamplefornewmaximumlikelihood*/ if p X ; U ; Y >p X ; U ; Y ,using 3{44 then U U Y Y Samplingfromtheconditionaldistributionofthemembershipsgiventhedataand clusterprototypes p U j X ; Y isaccomplishedwithaMetropolis-Hastingssamplingstep using,forsimplicity,auniformsymmetricDirichletproposaldistribution u y n Dirichlet = 1 C : {46 Theconditionaldistributionofthememberships, p U j X ; Y ,isproportionaltothejoint distributionofdata,memberships,andprototypes, p X ; U ; Y ,givenxedvaluesofthe clusterprototypes.Additionally,foranyproposedmembershipvector u y n foradatapoint index n ,thetermsrelatedtotheothermembershipvectorsandclusterprototypesare 85

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unchanged.Thereforeweonlyneedtoevaluatethefollowingquantity: ~ p x n ; u n j Y = p x n j u n ; Y ~ p u n j Y = C Y c =1 N x n j y c ;u m nc u )]TJ/F22 7.9701 Tf 6.586 0 Td [(mD= 2 nc Dirichlet u n j / C Y c =1 exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 u m nc k x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c k 2 u c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 nc : {47 Aproposednewmembershipsample u y n isthusacceptedtoreplace u n withprobability equaltotheratio a u =min 1 ; ~ p x n ; u y n j Y ~ p x n ; u n j Y ; {48 where,becausetheproposaldistributionisnotdependentuponthecurrentsample,a Hastingscorrectionisnotneeded. ThenextGibbssamplingstepistosamplenewclusterprototypesfromtheconditionaldistributionoftheprototypesgiventhedataandmemberships p Y j X ; U .Again thisdistributionisproportionaltothejointdistribution p X ; U ; Y forxedvaluesofthe dataandmemberships.Proposednewvaluesfortheclusterprototypesaresampledfrom aGaussiandistributionwithsmallvarianceintheshapeofthepriorandcenteredonthe currentMarkov-chainstate y y c N y c ; 1 y ; {49 where isausersetparameterthatcontrolstightnessoftheproposalaroundthecurrent stateandrelatestothesampleacceptancerate,inapplicationweset =10.Forasingle proposedclusterprototype, y y c ,thetermsdependingupontheotherclusterprototypesand membershipswillbeunchanged,leavingonlythefollowingquantitytobeevaluated: p X ; y c j U = p X j U ; y c p y c / exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 N X n =1 u m nc k x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c k 2 exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 y c )]TJ/F21 11.9552 Tf 11.955 0 Td [( y T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 y y c )]TJ/F21 11.9552 Tf 11.955 0 Td [( y : {50 Thisgivesthattheproposedsamplecanbeacceptedaccordingtotheratio a y =min 1 ; p X ; y y c j U p X ; y c j U ; {51 86

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wherebecausetheGaussianproposalissymmetric,aHastingscorrectionisagainnot needed. Thesearchcompareseachproposedvectorofmembershipvalues u y n whetherornot itisacceptedasthenewMarkov-chainstate u n forimprovementover u n inthecurrent MAPsample f U ; Y g using3{47.If u y n increasesthelikelihoodoftheMAPsample, thenitbecomesthenew u n .Similarlyeachproposedclusterprototype y y c istestedfor improvementusing3{50,andretainedasthenew y c ifitisanimprovement.After eachfullsample f U ; Y g isgenerated,itslikelihoodiscomparedagainstthecurrentMAP sample f U ; Y g using p X ; U ; Y ,andisretainedasthenewMAPsampleifithas higherlikelihood.WerefertothecomparisonofproposalsamplestoMAPsamplesas aratchetingsearch,andndthatitspeedsthesearchforagoodanswerearlyon,while leavingopenthepossibilitytondanewMAPfullsampleasthesamplerprogresses. Wealsonotethatinthisinferencemethod,becausewearenotusingtheclosedform FCMupdateequations2{96and2{97,wearefreetosetthevalueofthefuzzier parameter m to1toobtainacrispclustering,oreventovalueslessthanoneornegative toobtainsomeinterestingresults. 3.3.3BayesianModelforEstimatingtheNumberofClusters Comparingtheobjectiveorlikelihoodvalueoftwomodelsofdierentsizesisnot generallyastraightforwardtaskasintheobjectivefunctioncasetheobjectivevalue isdependentuponthemodelsize.Thisisbecausetheadditionaltermsinalarger modeltendtoincreasetheobjectivevalueregardlessofifthelargermodelisabetter solutiontotheproblem.Forourapproachtolearningthesizeofthemodel,wemake theassumptionthatsimplytakingtheaverageoftheobjectivevalueoverthenumberof modelcomponentswillremovethisdependenceuponmodelsize.Havingremovedthe dependenceoftheobjectivevalueonmodelsize,andassumingwearecomparingtwo modelswhichworkequallywellonaverage,wethenfavormodelsusingfewercomponents. Forexample,ifanobjectivefunctionisstructuredas J X ; = P C c =1 f X ; c ,we 87

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restructurethisastheaveragedobjectiveplusapenalty, J 0 X ; ;C = 1 C P C c =1 f X ; c + Penalty C ApplyingthisphilosophynowtotheBFCobjective3{45,wecanderiveanew objectiveaveragedovermodelsize J X ; U ; Y ; C = 1 C N X n =1 C X c =1 u m nc k x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c k 2 )]TJ/F15 11.9552 Tf 14.841 8.087 Td [(2 C N X n =1 C X c =1 c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1log u nc + 1 C C X c =1 y c )]TJ/F40 11.9552 Tf 11.955 0 Td [( y T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 y y c )]TJ/F40 11.9552 Tf 11.955 0 Td [( y +Penalty C : {52 Treatingthisbasicformofthisobjectiveastheprototypeforthenegativelog-likelihood, wederiveanewprobabilisticmodelwecalltheInniteBayesianFuzzyClusteringIBFC model.ThisisnamedinthestyleofmanyBayesiannonparametricmodelssuchasinnite GaussianmixturemodelDirichletProcess[139]ortheinnitelatentfeaturemodel IndianBuetProcess[140],thoughthesemodelsaretakenastheinnitelimitoverall modelsizes,whereastheIBFCmodelsimplyaveragesovertheunboundedmodelsize. TheIBFCmodelconsistsofthefollowinglikelihoods: p X j U ; Y ;C = N Y n =1 1 Z X C Y c =1 N x n j = y c ; = 1 C u m nc I D D {53 ~ p U j Y ;C = N Y n =1 Z X C Y n =1 u m nc C D= 2 Dirichlet u n j 1 =C # )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(C )]TJ/F22 7.9701 Tf 6.587 0 Td [(CD= 2 = 2 N=C exp N C {54 p Y j C = C Y c =1 N y c j y ;C y {55 P C =Poisson C j = C C exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( g : {56 TherststepincreatingthisIBFCmodelisincludingthenumberofclusters C asa randomvariable.AsthisisapositiveintegervaluedparameteritisgivenaPoissonprior 3{56.Themeanhyper-parameterofthePoissoncanthenbesetinanempiricalfashion 88

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tobeincreasingwiththenumberofdatapoints,forexampleinourapplicationswechoose = log N butnotethatotherchoicesmaybereasonabledependingonthedataset. ThedatalikelihoodoftheIBFCmodel3{53issimilartopreviousBFClikelihood, buttheaveragingoverthemodelsizenowappearsintheprecisionparameter.Thechange intheprecisionvalueaectsthenormalizationconstant,andthisiscancelledagainbythe rsttwofactorsoftheIBFCfuzzyclusterpriorIFCP3{54. Thearithmaticmeaninthelog-likelihoodcanbeinterpretedasageometricmean intheregularlikelihoodspace.WeusethisfactfortheDirichletfactoroftheIFCP.The IFCPalsohasthreeadditionalmultiplicativefactorsbeyondtheFCPfrom3{38.The fourthmultiplicativefactor F 4 = )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(C )]TJ/F22 7.9701 Tf 6.586 0 Td [(CD= 2 isusedtocancelthenormalizationtermsof theclustercenterprior3{55inamannersimilartothatusedforthedatalikelihoodby F 2 ThefthandsixthfactorsintheIFCPareusedtopromotemodelsparsity.These comefromplacinganadditionalLaplacedistributionfactoronthememberships.The Laplacedistribution,withmeanparameter andscaleparameter ,hastheprobability density p x = = 2exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( j x )]TJ/F21 11.9552 Tf 11.956 0 Td [( jg : TheideabehindtheLaplaceprioristhatamodelwithtoomanyclusterswillhavelow membershipvaluesonaverage.Tocounteractthis,theLaplacepriorwith =1rewards membershipsbeinghighandthuspromotesfewerclusters. CombiningtherstfourfactorsoftheIFCPwithaLaplacedistributionwith =1 foreachmembership,andtakingthegeometricmeanoverthemodelsize,resultsin ~ p U j Y;C = N Y n =1 Z X C Y n =1 u m nc C )]TJ/F22 7.9701 Tf 6.586 0 Td [(D= 2 Dirichlet u n j 1 =C # )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(C CD= 2 N Y n =1 C Y c =1 = 2exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( j u nc )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 jg 1 =C : 89

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FocusingontheLaplaceportiononly,becausethemembershipswillalwaysbelessthanor equalto1wecanrewritethisas, = 2exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( j u nc )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 jg = = 2exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( )]TJ/F21 11.9552 Tf 11.956 0 Td [(u nc g : Additionally,becausethemembershipssumto1overallclusters, C Y c =1 = 2exp f)]TJ/F21 11.9552 Tf 15.277 0 Td [( )]TJ/F21 11.9552 Tf 11.955 0 Td [(u nc g 1 =C = = 2 1 =C exp )]TJ/F21 11.9552 Tf 11.475 8.088 Td [( C C X c =1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(u nc = = 2 1 =C exp )]TJ/F21 11.9552 Tf 9.299 0 Td [( + C C X c =1 u nc = = 2 1 =C exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( g exp C : Consideringnowtheproductoveralldatapointsgives N Y n =1 = 2 1 =C exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( g exp C = = 2 N=C exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [(N g exp N C : Finally,removingtheunneededconstanttermexp f)]TJ/F21 11.9552 Tf 15.276 0 Td [(N g ,thisyieldsthelasttwofactors, = 2 N=C exp N C ,oftheIFCPprior3{54. Thejointlog-likelihoodoftheIBFCmodel,formedfromtheproductof3{53, 3{54,3{55,and3{56canthenbefoundtobe log p X ; U ; Y ;C =log p X j ; U ; Y ;C ~ p U j Y ;C p Y j C P C /)]TJ/F15 11.9552 Tf 29.723 8.088 Td [(1 2 C N X n =1 C X c =1 u m nc k x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c k 2 + 1 C N X n =1 logDirichlet u n j + N C log 2 + N C )]TJ/F15 11.9552 Tf 17.767 8.088 Td [(1 2 C C X c =1 y c )]TJ/F40 11.9552 Tf 11.955 0 Td [( y T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 y y c )]TJ/F40 11.9552 Tf 11.955 0 Td [( y + C )]TJ/F22 7.9701 Tf 17.29 14.944 Td [(C X i =1 log i : {57 Takingthenegativeofthisequationgivesanobjectivefunctionformulationwhichmatches theproposedobjectivein3{52,wheretheprioron C andthemodelsparsityterms combinetobecometheproposedpenaltyfunction. 90

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3.3.4InferenceintheInniteBayesianFuzzyClusteringModel WepresentaparticlelterinferencealgorithmcalledtheInniteBayesianFuzzy ClusteringParticleFilterIBFC-PFforndinglocalMAPparametervaluesintheIBFC model.OneofthemostwellknowninstancesofaparticlelterisintheCondensation algorithm[141]designedforobjecttracking,andassuchparticleltersaremostoften associatedwithdynamicmodelshavingatimecomponent.Theparticleltercanalsobe appliedforapproximateinferenceoftheparametersofastaticmodelanditisusedinthat modehere.KollerandFriedman[103]provideachapteronparticle-basedapproximate inferenceinprobabilisticmodels. Aparticlelterrepresentsaprobabilitydensityfunctionusingadiscreteapproximation.Thisapproximationconsistsofasetofsamplevaluesontheinputdomainofthe function,knownasparticles,andtheoutputvalueofthedensityfunctionforeachsample, knownastheparticle'sweight.The lter partofthenamecomesfromitssimilaritytothe Kalmanlterintrackingapplications.Aparticlelterconsistsofaniterationoffourbasic steps: 1.DeterministicDrift:theparticlestateisupdatedaccordingtosystem'stimedynamicse.g.equationsofmotion. 2.RandomDiusion:theparticlestateisupdatedaccordingtosystem'srandom dynamicse.g.randomposition/directionchange. 3.Weighting:theparticlesareweightedaccordingtotheirlikelihoodgiventhecurrent systemobservatione.g.measurementnoise. 4.Resampling:theparticlepopulationisresampledwithreplacementbyprobabilities equaltothenormalizedparticleweights. ToapplythelterprocesstoMAPsearchinastaticmodel,thedriftstepbecomes aclosedformconditionalmaximizationofasubsetofthestatevariables.Thediusion stepbecomesarandomsamplingofthevariablesforwhichnoclosedformmaximization isavailable.Theweightingstepevaluatesthelikelihoodoftheparticles,andthenthe resamplingsteprepopulatestheparticleswithavarietyofgoodanswers.Theparticle 91

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ltersearchcanalsobethoughtofasaparallelversionofastochasticexpectationmaximizationapproach. TheIBFC-PFalgorithmshowninAlgorithm5,usesamixofstochasticandclosed formupdates.Inordertousetheclosedformupdates,thisalgorithmrequiresthatthe Dirichletconcentrationparameterbesetto =1andthefuzzierbe m> 1. Afterinitialization,thesearchloopsuntilamaximumnumberofiterationsor convergencehasbeenreached.Ineachiteration,theparticleltergoesthroughthe fourphasesofrandomdiusion,deterministicdrift,weighting,andthenresampling. Performingtherandomstepbeforethedeterministicstepherespeedsconvergenceand simpliestheimplementation. Intherstphasecorrespondingtotherandomdiusion,arandomnewmodelsize C isdrawnfromaPoissondistributionwithameanvalueattheparticle'scurrentsize, andthentheclusterprototypesarechangedaccordingly.Ifthenewsizeisgreaterthan thecurrent,newclusterprototypesaresampledfromtheirprior.Ifthenewsizeissmaller thanthecurrent,arandomsubsetoftheclusterprototypesareretained. Nextthemembershipsandclusterprototypesareupdated,thisisthedeterministic driftstep.Themembershipupdateusesthesameclosedformupdateastraditional FCM2{96.Fortheclusterprototypeshowever,theadditionofthepriordistribution regularizestheobjectivefunction.Thusaclosedformupdatefornewprototypesmust takethisintoaccount.Takingthelog-likelihoodoftheIBFCmodel,thentakingthe partialderivativeofthelog-likelihoodwithrespecttotheprototypeandsettingitequalto zeroyieldsthenewupdateequation y c = N X n =1 u m nc I D D + )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 y )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 N X n =1 u m nc x n + y : {58 Inthethirdphaseeachparticleisrstweightedbytheexponentiationofitslog-jointlikelihood3{57,asthisisproportionaltothelog-posterior-likelihoodoftheparameters 92

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Algorithm5: ParticleFilterIBFCMAPSearch Data :Datamatrix X ,fuzzier m ,numberofparticles P ,numberofsampling iterations N iter Result :MAPestimatesfornumberofclusters C ,memberships U andcluster prototypes Y initializehyperparameters y ; y createinitialsample s ,where s:C 1, s:u n 1 1forall n =1 :::N s: y 1 N y ; y evaluatesamplelog-likelihood s: llfrom3{53 setinitialMAPestimateformodelsize1,MAP[1] s initializeparticlesPCL[ i ] s forall i =1 :::P for iter=1 :::N iter do for i =1 :::P do samplenewmodelsize C PoissonPCL[ i ] :C if C< PCL[ i ] :C then inds randomsubsetlength C ofrange1 ::: PCL[ i ] :C PCL[ i ] : Y PCL[ i ] : Y [inds] elseif C> PCL[ i ] :C then N new C )]TJ/F15 11.9552 Tf 11.956 0 Td [(PCL[ i ] :C samplenewprototypes y new k N y ; y for k =1 :::N new PCL[ i ] : Y [PCL[ i ] : Y ; Y new ] PCL[ i ] :C C updatemembershipsPCL[ i ] : U U-update X; PCL[ i ] : Y from2{96 updateclusterprototypesPCL[ i ] : Y Y-update X; PCL[ i ] : U from3{58 evaluateparticlelog-likelihoodPCL[ i ] : llfrom3{57 if PCL[ i ] : ll > MAP[ C ] : ll then MAP[ C ] PCL[ i ] constructlistofparticlesforresamplingPCLS [PCL ; MAP] for i =1 :::P do ind sampleexp f [PCLS : ll] g = P [exp f PCLS : ll g ] PCL[ i ] PCLS[ind] stopifconverged pickMAPmodelsize, C argmax[MAP : ll] C C; U MAP[ C ] : U ; Y MAP[ C ] : Y giventhedata.Beforeresampling,eachparticleischeckedtoseeifitimprovestheMAP estimateforthatmodelsize.IfsoitisretainedasacandidateforthenalMAPsolution. TheparticlesarethensampledwithreplacementfromthelistofparticlesconcatenatedwiththelistofpotentialMAPsolutions.Asthisisasearchalgorithmandnota 93

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trackinglter,theadditionoftheMAPsolutionsintheresamplingspeedsconvergence byencouragingthesearchtolooknearthecurrentbestanswers. Convergenceoftheparticlelterisdeterminedbythechangeinthelikelihood valueofthemostlikelyMAPsamplebeingbelowathresholdformultipleiterationsin arow.Becauseoftherandomnatureofthesearch,thesamplesmayconvergetoalocal maximum,butthennotbeimproveduponforanindeterminatenumberofiterations. Eventually,throughtherandomaddingandremovingofclusterprototypeshowever,a newmorelikelyareaofparameterspaceisfoundandimprovementsbeginagain.Thus assessmentofconvergenceofthealgorithmisdicult.Wehavefoundthatforcaseswith fewclusterinlowdimensions,asmallnumberofiterationsnear10afterconvergenceis sucient,whileforhighdimensionalproblemswithmanyclusters,alargernumbernear 100isoftenneeded. Becauseoftherandomselectionofmodelsizeandnewclusterinitializations,the IBFC-PFalgorithmasgivenismorelikelytondagoodmaximumvaluethanan algorithmwhichusesnorandomness.However,itisnotguaranteedtobeabletoexplore theentiresolutionstatespace,andthereforecannotguaranteeaglobaloptimumanswer. If,intherandomportionoftheparticleupdate,theclusterprototypeswererandomly re-initializedwithsomelowprobability,thiswouldbeenoughtoguaranteeaccessibility toallofthestatespace.Becausetheexplorationofthespacewouldbeexceedinglyslow however,thisstepwasdeemedunnecessary.Thusconvergenceisonlyguaranteedtoa localoptima. Wealsonotethat,insteadofthehybriddeterministic/stochasticparticlelter,a fullMCMCsamplingapproachcouldbederivedfortheIBFCmodelaswell.Suchan algorithmwouldallowtheDirichletconcentrationtobevariedandthefuzziersetto exoticvalues,aswellasguaranteeconvergencetoaglobaloptimum.Howeversuchan algorithmwouldbebemuchslowerthantheIBFC-PFinoperation. 94

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3.3.5DerivationoftheFuzzyDataLikelihoodNormalizationConstant Thenormalization Z u n ;m; Y fortheFuzzyDataLikelihood3{37isderivedby rstexpontiatingtheFCMobjective2{95termsforagivendatapoint x n multipliedby afactorof )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 2 .AssumingEuclideandistancethisyields exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 C X c =1 u m nc x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c T x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c : AssumingthatthisexponentialexpressionispartofaGaussianPDF f x n ,completethe squarewithintheexponentialtondtheparametersofthecorrespondingnormal f x n / exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 C X c =1 u m nc x T n x n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 x T n y c + y T c y c / exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 C X c =1 u m nc x T n x n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 x T n C X c =1 u m nc y c + C X c =1 u m nc y T c y c !# ; setting = 1 P C c =1 u m nc P C c =1 u m nc y c and = P C c =1 u m nc I f x n / exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 x T n x n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 x T n + T exp )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 C X c =1 u m nc y T c y c )]TJ/F40 11.9552 Tf 11.955 0 Td [( T # {59 f x n = N x n j ; : Next,thenormalization Z u n ;m; Y factorcanbesolvedforbyequating N x n j ; = 1 Z u n ;m; Y C Y c =1 N x n j = y c ; = u m nc I D D Z u n ;m; Y = Q C c =1 N x n j = y c ; = u m nc I D D N x n j ; = Q C c =1 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(D= 2 u mD= 2 nc exp )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 2 u m nc x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c T x n )]TJ/F37 11.9552 Tf 11.955 0 Td [(y c 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(D= 2 P C c =1 u m nc D= 2 exp )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 2 x T n )]TJ/F40 11.9552 Tf 11.955 0 Td [( T x T n )]TJ/F40 11.9552 Tf 11.956 0 Td [( ; 95

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which,usingtheexpansionfortheoriginalFCMobjectiveobtainedin3{59,canbe simpliedto Z u n ;m; Y = )]TJ/F22 7.9701 Tf 6.587 0 Td [(D C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = 2 C Y c =1 u m nc D= 2 C X c =1 u m nc )]TJ/F22 7.9701 Tf 6.587 0 Td [(D= 2 exp 8 > < > : )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 0 B @ C X c =1 u m nc y T c y c )]TJ/F26 11.9552 Tf 13.151 25.704 Td [( P C c =1 u m nc y c T P C c =1 u m nc y c P C c =1 u m nc 1 C A 9 > = > ; : 3.4BayesianJointUnsupervisedContextLearningandDetection TheBayesianfuzzyclusteringtechniquesdevelopedintheprevioussection,though generalinnature,cameaboutthroughadesiretooptimizethenonlineardetectorobjectivefunctionssuchasACE.ThissectionshowstheapplicationoftheBayesianfuzzy clusteringtechniquestocontext-dependentdetectionintheformoftheIBFCEMand BFACEdetectors. 3.4.1InniteBayesianFuzzyConstrainedEnergyMinimization UsingtheIBFCmodeldevelopedinsection3.3.3asastartingpoint,ajointclustering anddetectionalgorithmcanbedeveloped.Thismethod,calledIBFCEMplacesthe detectionstatisticasthedistancetermwithintheIBFCdatalikelihood,andthenplaces priordistributionsontheparametersofthedetector.Thus,thismethodcanbeseenasa versionofFCEMwhichestimatesthebestnumberofclustercomponents. TheIBFCEMmodelforagenericdetectorfunctionis p X j U ; ;C = N Y n =1 1 Z X C Y c =1 N d x n ; c j =0 ; = u m nc 1 =C {60 ~ p U j C =exp N C N Y n =1 Z X C Y c =1 u )]TJ/F22 7.9701 Tf 6.587 0 Td [(m= C nc Dirichlet u n j 1 =C # {61 p j C = C Y c =1 1 Z p c 1 =C {62 P C j =Poisson C j = C C exp f)]TJ/F21 11.9552 Tf 15.276 0 Td [( g : {63 96

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Whenusingthelinearlteroperatorasthedetector, d x n ; c becomesthelinear operator w T c x n )]TJ/F40 11.9552 Tf 12.281 0 Td [( c .Gaussianpriortermsareusedfortheparametersofthedetector. Thusthelikelihoodandpriorparametersinthismodelbecome p X j U ; W ; M ;C = N Y n =1 1 Z X C Y c =1 N w T c x n )]TJ/F21 11.9552 Tf 11.956 0 Td [( c j =0 ; = u m nc 1 =C {64 p W ; M j C = C Y c =1 1 Z WM N w c j =0 ; w 1 =C N c j m ; m 1 =C I w T c s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c =1 ; {65 where s isthetargetsignature,andthehyperparametersoftheprioraresetinan EmpiricalBayesfashion.Fortheweightvectors,adiagonalisotropiccovarianceproportionaltothemagnitudeofthedatavariancecanbechosen,i.e. w = I ,where =max f diag X g .Forthemeanvectors,thedatasetmeanandabroadcovariancein thesameshapecanbeused,i.e. m = X ,and m =5 X ThesameparticlelterinferencealgorithmdevelopedforIBFCisapplicabletothe IBFCEMmodel.Theparticlelterallowsclosedformupdatesforsomeparameters,thus makingitwellsuitedforthelineardetectorobjectivefunction.Toderivetheupdate equations,wemaximizethelogofthejointdistributionasitisproportionaltothe posteriordistributiontondlocallymaximalvaluesofeachsetofparametersintermsof theothers. WhentheDirichletpriorconcentrationissetto =1,thenthelog-joint-likelihoodof thelinearIBFCEMbecomes log p X ; U ; W ; M ;C = )]TJ/F15 11.9552 Tf 17.768 8.088 Td [(1 2 C N X n =1 C X c =1 u q nc )]TJ/F37 11.9552 Tf 5.479 -9.684 Td [(w T c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c 2 )]TJ/F15 11.9552 Tf 17.768 8.088 Td [(1 2 C C X c =1 w T c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 w w c )]TJ/F15 11.9552 Tf 17.768 8.088 Td [(1 2 C C X c =1 c )]TJ/F40 11.9552 Tf 11.955 0 Td [( m T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 m c )]TJ/F40 11.9552 Tf 11.955 0 Td [( m + C )]TJ/F22 7.9701 Tf 17.29 14.944 Td [(C X i =1 log i + K; {66 97

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where K isaconstanttermrepresentingallxedconstantsintermsofthehyperparameters.Thisexpressionisvalidsubjecttotheconstraintthat w T c s )]TJ/F40 11.9552 Tf 12.16 0 Td [( c =1andthefuzzy clusteringmembershipconstraints. Maximizingtheexpressionin3{66fortheparameters w c and c individuallywhile keepingtheremainingtermsxedandusingLagrangemultipliersfortheconstraintsyields closedformupdateequationsforeach.Fortheweightvectors, w c = )]TJ/F37 11.9552 Tf 5.479 -9.683 Td [(V + )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 w )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c T )]TJ/F37 11.9552 Tf 5.48 -9.684 Td [(V + )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 w )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c ; {67 where V = P N n =1 u q nc x n )]TJ/F40 11.9552 Tf 12.091 0 Td [( c x n )]TJ/F40 11.9552 Tf 12.091 0 Td [( c T .Thisiseectivelyaregularizedversionofathe FCEMupdate3{20. Fortheclustercenterstheupdateequationis c = A )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c z c + A )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c w c w T c A )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c w c )]TJ/F37 11.9552 Tf 5.479 -9.684 Td [(w T c )]TJ/F37 11.9552 Tf 5.48 -9.684 Td [(s )]TJ/F37 11.9552 Tf 11.955 0 Td [(A )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c z c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; {68 where A c = w c w T c + 1 P N n =1 u q nc )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 m z c = w c w T c x c + 1 P N n =1 u q nc )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 m m ; and x c = 1 P N n =1 u q nc N X n =1 u q nc x n : Thisseriesofequationscanbeseenagainasaregularizedversionofthesimplerformin 3{21fromFCEM. 3.4.2BayesianFuzzyACE TheBFCmodelwasderivedinpartsothatclusteringobjectiveswithnonlinear distancefunctionscouldbeoptimizedusingprobabilisticinferencetechniquesMCMCin particular.TherstapplicationoftheBFCmethodtosuchanobjectivefunctioniswith theACEdetector,wecallthistheBFACEmodel. 98

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LiketheIBFCEMdetector,BFACEmodelplacesthedetectionstatisticintothe normaltermsoftheBFCdatalikelihood.Priordistributionsarethenplacedonthe meanandcovarianceparametersoftheACEdetector.UnlikeintheCEMapproaches however,BFACEdoesnotneedtoconstrainthedetectoroutput,astheformofthe detectorisnaturallyconstrainedandwillreturnthevalue1whenpresentedwiththe targetsignature. WeuseanormalpriorontheACEdetectorcenters,andaWishartprioronthe detectorinverse-covariances.ThefullBFACEmodelis: p X j U ; M ; 1 ;:::; C = N Y n =1 1 Z u n ;q C Y c =1 N d x n ; c ; c j =0 ; = u q nc {69 ~ p U = N Y n =1 Z u n ;q C Y c =1 u )]TJ/F22 7.9701 Tf 6.587 0 Td [(mD= 2 nc Dirichlet u n j # {70 p M = C Y c =1 N c j m ; m {71 p 1 ;:::; C = C Y c =1 Wishart c j V l ; l ; {72 where d x ; ; isthesquarerootoftheACEdetectorgiventargetsignature s ,and d x ; ; = s )]TJ/F40 11.9552 Tf 11.955 0 Td [( T x )]TJ/F40 11.9552 Tf 11.955 0 Td [( [ s )]TJ/F40 11.9552 Tf 11.955 0 Td [( T s )]TJ/F40 11.9552 Tf 11.955 0 Td [( x )]TJ/F40 11.9552 Tf 11.955 0 Td [( T x )]TJ/F40 11.9552 Tf 11.955 0 Td [( ] 1 = 2 : {73 ThehyperparametersofthepriorsaresetintheEmpiricalBayesfashion.Forthedetector centers,themeanisthedatasetmean m = X ,andthecovarianceisawidecovariance intheshapeofthedataset m =5 X Forthedetectorinversecovariances 1 ;:::; C ,aWishartpriorisused.TheWishart distributionisaprobabilitydistributionoverpositivedenitematrices,parameterizedbya scalematrix V andascalarnumberofdegreesoffreedom ,andgivenbytheProbability DensityFunctionPDF Wishart X j V ; = 1 2 D= 2 j V j = 2 )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(D 2 j X j )]TJ/F22 7.9701 Tf 6.587 0 Td [(D )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = 2 exp f)]TJ/F15 11.9552 Tf 16.472 8.087 Td [(1 2 trace V )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 X g ; {74 99

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where)]TJ/F22 7.9701 Tf 41.131 -1.793 Td [(D isthe D dimensionalmultivariategammafunction.TheWishartdistribution isdenedfor X positivedeniteand >D )]TJ/F15 11.9552 Tf 12.64 0 Td [(1,where D isthedimensionalityofthe matrix X .ThemeanofaWishartisequalto V ,sowesetthescalematrixoftheprior to V l = 1 l 1 C X )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ,i.e.proportionaltotheinverseofascaleddownversionofthedataset covariance.Thedegreesoffreedomarethensettoproduceawidedistributionaroundthis mean, l =10+ D )]TJ/F15 11.9552 Tf 11.956 0 Td [(1. MAPinferencefortheBFACEmodelcanbeperformedusingaStochasticExpectationMaximizationSEMapproach,outlinedinAlgorithm6.StochasticEMusesa samplingheuristictogeneratevaluesoftheunknowndetectorcenterandinversecovarianceparameters,butthenusesclosedformupdatestondtheconditionalmaximum valuesoftheclustermemberships.Thisalgorithmthusallowsforaconstrainedrandom searchoverthemeanandinverse-covarianceparametersforwhichclosedformupdates cannotbederived,butafastclosedformupdateforthememberships.Thusthespeedof thealgorithminndinggoodsolutionsisimprovedoverafullsamplerapproach,though attheexpenseofglobaloptimality.TheSEMapproachaspresentedcanbeconsidereda modiedMetropolis-within-Gibbssamplingapproach.Eachsetofrandomvariablesexcept forthosewithclosedformupdatesaresampledfromtheirconditionaldistributionsusinga Metropolis-Hastingsstrategy.Forthosewithclosed-formupdates,theGibbssamplingstep isreplacedwiththegenerationofthemaximumconditionallikelihoodsample. TheclosedformupdateformembershipsusestheACEdetector2{15inplaceofthe distanceinthestandardfuzzyclusteringupdates,yielding u nc = 1 ACE x n ; c ; c 1 = q )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P C k =1 1 ACE x n ; k ; k 1 = q )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 : {75 Togeneratenewvaluesforthedetectorcenters,wetreatthevariablesasifwewere performingaMetropolis-within-Gibbssamplingiteration.Thuswegeneratearandom samplefromaproposaldistribution,andthenacceptthisnewlygeneratedvaluewith 100

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Algorithm6: BayesianFuzzyACEStochasticEMSearch Data :Datamatrix X ,fuzzier m ,numberofclusters C ,numberofsampling iterations N iter Result :MAPestimatesformembership U anddetectormeans M andinverse covariances 1 ;:::; C initializehyperparameters m mean X m 5 cov X l 10+ D )]TJ/F15 11.9552 Tf 11.955 0 Td [(1, V l 1 l 1 C X )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 initializedetectorcenters M withsamplesfromprior3{71 initializedetectorinverse-covariances c withsamplesfromprior3{72 M M c c for c =1 :::C /*StochasticEM*/ for iter=1 :::N iter do U updatecurrentsamplemembershipsusing3{75 U updatemaxsamplemembershipsusing3{75 /*sample M p M j X ; U ; / p X ; U ; M ; */ for c =1 :::C do sampleproposednewdetectormean y c from3{76 acceptproposal c y c withprobabilityfrom3{77 if p X ; U ; [ y c M f)]TJ/F22 7.9701 Tf 10.821 0 Td [(c g ] ; 1 ;:::; C >p X ; U ; M ; 1 ;:::; C then c y c /*sample p j X ; U ; M / p X ; U ; M ; */ for c =1 :::C do sampleproposednewdetectorinversecovariance y c from3{78 acceptproposal c y c withprobabilityfrom3{79 if p X ; U ; M ; f)]TJ/F22 7.9701 Tf 10.821 0 Td [(c g ; y c >p X ; U ; M ; 1 ;:::; C then c y c /*checkfullsamplefornewmaximumlikelihood*/ if p X ; U ; M ; 1 ;:::; C >p X ; U ; M ; 1 ;:::; C then U U M M c c for c =1 :::C probabilityequaltothecomputedacceptanceratio.Forthedetectorcenters,weusea Gaussianproposaldistributioncenteredonthecurrentsamplewithanarrowcovariancein thesameshapeasthedataset,thatis y c N j c ; 0 : 1 X : {76 101

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AstheGaussianproposaldistributionissymmetric,noHastingscorrectionisneededin theacceptanceratio.Thustheacceptanceratioissimplytheratioofthelikelihoodinthe jointdistributionusingtheproposedsampletothelikelihoodinthejointdistributionof theprevioussample a =min 1 ; p X j U ; M ; 1 ;:::; c p [ p M f)]TJ/F22 7.9701 Tf 10.821 0 Td [(p g ] p X j U ; M ; 1 ;:::; c p M : {77 Thedetectorinverse-covariancesareproposedbyanarrowWishartdistribution centeredatthecurrentsample y c Wishart j V = 1 p c ; = p : {78 Thedegreesoffreedomparameter p ischangedtoaecttheacceptancerateofthe samples,withlargervaluesgeneratingatighterdistributionandthusahigheracceptance. BecausetheWishartdistributionisnotsymmetric,weapplytheHastingscorrectionto theacceptanceratioforthesesamples,yielding a =min 1 ; p X j U ; M ; 1 ;:::; c p [ p f)]TJ/F22 7.9701 Tf 10.821 0 Td [(p g ]Wishart c j 1 p p ; p p X j U ; M ; 1 ;:::; c p 1 ;:::; c Wishart p j 1 p c ; p : {79 Iterationproceedsbyrstupdatingthemembershipsforboththecurrentsampleand themaximumlikelihoodsample.Next,newsamplesforeachdetectorcenterareproposed andpotentiallyaccepted.Regardlessifthesamplewasaccepted,itischeckedtoseeif itimprovesthemaximumlikelihoodanswer,andifsoitisretainedforthatpurpose. Thirdly,newinverse-covariancesaresampledandcheckedforamaximumlikelihood increase.Finallythejointsampleofmemberships,centers,andcovariancesiscompared againstthemaximumlikelihoodsamplevaluesandreplacethemifmorelikely.Attheend ofthespeciednumberofsamplingiterations,themaximumlikelihoodsamplevaluesare returned. 102

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Therandomupdatesofthedetectormeanscanbeveryslowmixinginimplementation.Analternativeimplementationofthismodelmaychoosetoinsteaduseanoptimizationsteptondnew,morelikely,valuesofthemeanparametersateachiteration.The formulationoftheACEdetectorprecludesderivingclosed-formupdatesforconditional maxima.However,theconditionalgradientcanbefollowedtowardslocalmaximausing eitherasimplelinesearchorperhapsanoptimizationpackage.FormulatingtheBFACE modelasanoptimizationobjectiveintermsofthedetectormeansyields J = C X c =1 N X n =1 u m nc )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c T )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c )]TJ/F15 11.9552 Tf 12.952 -9.684 Td [( x n )]TJ/F40 11.9552 Tf 11.956 0 Td [( c T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c + C X c =1 c )]TJ/F40 11.9552 Tf 11.955 0 Td [( m T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 m c )]TJ/F40 11.9552 Tf 11.955 0 Td [( m : {80 Tosimplifythemath,denetheterms m c s )]TJ/F40 11.9552 Tf 11.955 0 Td [( c z nc x n )]TJ/F40 11.9552 Tf 11.955 0 Td [( c A nc m T c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c z nc B c m T c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c m c C nc z T nc )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c z nc p c c )]TJ/F40 11.9552 Tf 11.955 0 Td [( m P c p T c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 m p c whichgivestherewrittenobjective J = C X c =1 N X n =1 u m nc A 2 nc B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 nc + C X c =1 P c : {81 103

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Then,giventhefollowingidentityfor = y T Ax ,when y and x arefunctionsof z but A isnot, @ @ z = x T A T @ y @ z + y T A @ x @ z ; wecanndthederivatives @A nc @ c = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 m c + z nc T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c @B c @ c = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 m T c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c @C nc @ c = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 z T nc )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c @P c @ c = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p T c )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 m : Theseyieldanequationforthegradientwithrespecttothedetectormeansof @ @ c J =2 N X j = n u m nc A 2 nc B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 nc f)]TJ/F21 11.9552 Tf 15.276 0 Td [(A )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 nc m c + z nc T + B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 c m T i + C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 nc z T nc g )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 c )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 p T c )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 m : {82 3.5Alarm-SetFusion Oneapproachthatcanbetakentocontext-dependentdetectionistorstusean imagesegmentationalgorithmasthecontextestimationstep.Suchsegmentsmayconsist ofsinglelandcoverclassesliketrees,elds,roads,buildings,etc.Afterthesegmentshave beenfound,adetectionalgorithmcanberunonthepixelsfromeachsegmentindependently.Indoingsohowever,itcanbeobservedthatthecondencedistributionvaries betweensegments,andthatcorrespondingly,ifthecondencesaresimplyconcatenated andasingleglobaldetectionthresholdused,thensystemperformanceissuboptimal.A betterstrategymaybetosetalowdetectionthresholdindicultsegmentse.g.trees, shadowsandahighdetectionthresholdineasysegmentse.g.openelds.Whatis neededthenisanautomatedtechniqueforndingsuchcombinationsofthresholds.This istheproblemaddressedbythetheoryofAlarm-SetFusion. 104

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Alarm-SetFusionASFisanewconceptinfusionthatiswellsuitedtocontextdependentdetectors.TheobjectiveofASFistomaptheindependentoutputsofmultiple detectorstoasharedrange.ThisrangeallowsasingleROCtobecreatedthatshows globalsystemperformancetrade-osacrossallalarmsources.Thresholdsproducing desiredsystemlevelPD/FARperformancecanthenbetracedtocorrespondingthresholds fortheindividualdetectors.ThissectionpresentstwoalgorithmsforASF,therstcalled FARE-ASFthatcanaccommodateunlabeledalarm-sets,andthesecond,RPthatworks withlabeledalarm-sets. 3.5.1FARE-ASF FARE-ASFisanewunsupervisedlearningmethodforfusingalarm-sets.Thisis performedbyestimatingthresholdsformultipleFARsacrossalarm-sets.TheFARE-ASF algorithmattemptstobalancethefalsealarmdistributionsofallalarm-setsbeingfusedby assigningthesameoutputcondencetoalarmswhichwouldoccuratapproximatelythe samerate.Thiscanbeachievedinanunsupervisedlearningmanner,thatiswithoutany knowledgeofthetrueorfalselabelofindividualalarms,undertheassumptionthatan insignicantnumberoftruealarmsarepresentinthealarm-sets. PseudocodefortrainingtheFARE-ASFalgorithmisgiveninalgorithm7.Training datashouldeitherbefreeoftruealarmsorthenumberoftruealarmsshouldbenegligible comparedtothenumberoffalsealarmssothattheydonotaecttheestimationofthe FalseAlarmRateFAR. Algorithm7: FARE-ASFtraining Data :Alarm-sets A 1 ;:::;A M from M alarmsources, N falsealarmrates f 1 ;:::;f N atwhichtondthresholds Result :Thresholds t ij 8 i =1 :::M;j =1 :::N s sortfalsealarmrates f intodescendingorder for i =1: M do A 0 i sortalarmsof A i bydescendingcondence r i areaencompassedbyalarm-set i for j =1: N do t ij condenceofalarmatindex d s j r i e in A 0 i 105

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ThelearneddatainFARE-ASFarethethresholdvaluesineachalarm-setthat correspondtoagivensetoffalsealarmrates.Thesearelearnedintrainingbyrst ensuringthefalsealarmratesaresortedindescendingorder.Then,thresholdsare estimatedforeachalarm-setseparately. ToestimatethethresholdforaFAR,thealarm-setmustrstbesortedinorderof decreasingcondence.Nexttheareaencompassedbythealarm-setmustbeestimated. InHSIdata,thiscanbedeterminedbymultiplyingtheimagedareagroundsample distanceofapixelbythenumberofpixelsinthealarm-set.Inothercasesthearea encompassedbythealarm-setwillbeanexternalquantityassociatedwiththealarm-set andgivenbytheparametersofthedatacollection.Withaknownareaforthealarm-set, thenumberoffalsealarmscorrespondingtothegivenrateisequaltothedesiredFAR timesthealarm-setarea. FinallytheestimateofthethresholdforaFARisgivenbythelowestvalueofthe highestcondence M alarms.Where M isthenumberoffalsealarmsneededtoachieve thespeciedFARinthesegment.Becausethealarmsaresortedindescendingorder,this isjustthecondencevalueatindex M PseudocodefortestingphaseofFARE-ASFisgiveninalgorithm8.Inoperation,a testalarm, c new i ,isgivenanoutputcondence, c out ,proportionaltothefalsealarmrateof thatcondenceforthatalarmsource, i ,inthetrainingset.Alarmswithcondencehigher thanorequaltothehighestlearnedthresholdlowestFARforthisalarmsourcearegiven condence1,andsimilarlyalarmswithcondencelowerthanthelowestlearnedthreshold highestFARaregivencondence0. Todeterminetheoutputcondencevaluewhenthealarmiswithintherangeofthe learnedthresholdsforthealarmsource,werstndthehighestthresholdlessthanor equaltothisnewvalue.Wethenlinearlyinterpolatethenewalarmbetweenthelower thresholdandnexthigherthresholdtogiveavalue, intherange[0 ; 1.Thisvalue is thenaddedtothebasecondencelevelforthisfalsealarmrate.Thebasecondencefora 106

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Algorithm8: FARE-ASFtesting Data :newalarmcondence c new i fromalarmsourceindex i learnedthresholds t kj forallalarmsources k =1 :::M andindices j =1 :::N Result :outputcondenceofalarm c out if c new i t i 1 then c out 1 elseif c new i
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Algorithm9: RunPackingtraining Data :Alarmsets A 1 ;:::;A M from M detectors Result :RunSets R i = f i;k j k =1 :::K i g8 i =1 ;:::;M ,andruntooutput mappings )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i;k 8 i;k for i =1: M do A 0 i sortalarmsof A i bydescendingcondence R i determinerunsfrom A 0 i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 applypackingstrategyto R 1 ;:::;R M Algorithm10: RunPackingtesting Data :newalarmcondence c new i ,detectorindex i ,runset R i ,runtooutput mapping )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i;k 8 i;k ,andtotalnumberofruns K Result :outputcondenceofalarm c out k indexinrunset R i with c min i;k c new i
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where c max i;k = c i;r k )]TJ/F22 7.9701 Tf 6.586 0 Td [(c i;r k )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 2 and c min i;k = c i;r k +1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(c i;r k +1 )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 2 .Ifnopreviousrunexists,thenthe condenceleveloftherstalarmisusedfor c max i; 1 .Ifnonextrunexists,thecondenceof thelastalarmisusedfor c min i;K i Finally,therunsaregloballyorderedinajointsequence ;:::; K ,where K = P M i =1 K i and j mapstheindexintheglobalordertoasensorandrun-indexto abusenotation, j : j i;k .Theorderingisdeterminedbythepackingstrategyand ranksrunsindecreasingorderofcondenceinthenaloutputscale.Theglobalordering oftherunsandtheruntuplesarethelearnedinformationinRP. Anoutputcondence, c out isassignedtoanewalarmfromdetector i ,byrstnding therunfromthetrainingsetwhichwouldcontainthisnewalarm'scondence, c new i Restated,nd i;k suchthat c min i;k c new i
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Wepresentthefollowinggreedypackingheuristic:Ateachiteration,packtherun withthemosttruealarms.Ifnorunswithtruealarmsareavailable,packtherunwith thefewestfalsealarms.Pseudocodeforthegreedypackingstrategyisgiveninalgorithm 11. Algorithm11: GreedyPackingStrategy Data :Runsets R i = f i; 1 ;:::; i;K i g for i =1 :::M ,totalnumberofruns K Result :runtooutputmapping )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i;k 8 i;k /*Initialize*/ for i =1 :::M do for k =1 :::K i do i;k t i;k )]TJ/F21 11.9552 Tf 11.955 0 Td [(f i;k u i 1 /*PackRuns*/ for j =1 :::K do i argmax m 2f 1 :::M g m;u m )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i;u i j u i u i +1 Formally,thegreedymethodisasfollows.First,denethesignedlengthofrun k fromalarm-set i tobe i;k = t i;k )]TJ/F21 11.9552 Tf 11.955 0 Td [(f i;k ; {87 where,becausetheruncontainseitheralltrueorallfalsealarms,thisissimplyeither t i;k or )]TJ/F21 11.9552 Tf 9.299 0 Td [(f i;k Thepackingstartsbykeepinganindex, u i ,totherstunpackedrunfromeach sensor,setto1initially.Theniterationproceedsuntilallrunshavebeenpacked.At eachiteration,selectfromthealarm-setthathastheunpackedrunwiththelargest i;u i Tiesarebrokenarbitrarily.Packtheselectedrunintotheglobalorder,thenadvancethe unpackedindexforthissensortothenextrun.Figure3-2givesagraphicalillustrationof twoiterationsofthegreedystrategyinoperation. 110

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3.5.4DynamicProgrammingStrategy Adynamicprogrammingbasedpackingstrategyisderivedherefortwodetectors. Pseudocodeisgiveninalgorithm12.Thedynamicprogrammingmethodcanbeextended tomoredetectorsbymergingtheinitialtwodetectors,anditerativelymergingtheresult ofthelastmergewiththenextdetector.ThisstrategycanbeshowntooptimizeAUC. Algorithm12: RunPackingbyDynamicProgramming Data :Runsets R 1 = f 1 ; 1 ;:::; 1 ;K 1 g and R 2 Result :runtooutputmapping )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i;k 8 i;k W zeros K 1 ;K 2 /*AUCtable*/ B zeros K 1 ;K 2 /*Backpointers*/ for u =0 ;:::;K 1 do for v =0 ;:::;K 2 do if u ==0&& v ==0 then B u;v 0 elseif u ==0 then W u;v maxAUCfromequation3{92 B u;v 2 elseif v ==0 then W u;v maxAUCfromequation3{91 B u;v 1 else W u;v maxAUCfromequation3{89 B u;v argumentindexofmaxAUC /*Backwardpass*/ k K 1 + K 2 ; u K 1 ; v K 2 while k> 0 do if B u;v ==1 then )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;u k ; u u )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 else )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;v k ; v v )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 k k )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 WerstshowthatAUCforagloballyorderedsequenceofrunscanbecomputed recursively: AUC k = k )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 X j =1 t j f k + AUC k )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 {88 111

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Next,denotetheAUCofapartialmergeofthetwosetsas W u;v ,consistingofallruns fromdetector1fromrun1uptorun u andallrunsfromdetector2fromrun1uptorun v .TheoptimalAUCforthecompletemergeofthetwodetectors'runsisgivenby W K 1 ;K 2 TherecurrencerelationforDPis W u;v =max W u )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;v + f 1 ;u T u )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;v ;W u;v )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 + f 2 ;v T u;v )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; {89 where T u;v = u X g =1 t 1 ;g + v X h =1 t 2 ;h : {90 Aback-pointertabletrackswhichdetectorcontributedtheoptimalAUCrunforallpairs ofrunindices u and v Thebasecasesfortherecursionare: W u; 0 = W u )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; 0 + f 1 ;u T u )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; 0 {91 W 0 ;v = W 0 ;v )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 + f 2 ;v T 0 ;v )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 {92 W 0 ; 0 =0 : {93 AftercomputingAUCsandback-pointersforallindexpairs,theoptimalmergeis determinedbyfollowingback-pointersfromindex K 1 ;K 2 backwardstoindex ; 0. 3.5.5ROCInterpretation TherunsinRPcorrespondtosegmentsofROCcurves.Runsoftruealarmsare verticalsegmentsofthecurve,andrunsoffalsealarmsarehorizontalsegmentsofthe curve.Inthislight,RPcanbethoughtofasconstructingajointROCcurvepieceby piecefromthesegmentsoftheindividualsensorROCcurves. Figure3-3givesavisualizationoftheRPalgorithmandthegreedypackingstrategy inthecontextofROCcurves,wheretherunlengthsarethesameasingure3-2.Figure 3-4thenshowstheresultingROCcurveafterallrunshavebeenpackedintotheoutput list.Inbothguresthenumbersbesidelinesegmentsaresignedrunlength i;k values. 112

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Figure3-1.Context-DependentDetection ABefore BAfter Figure3-2.Beforeandafteroneiterationofrunpackingwiththegreedystrategy.The lefthandsideshowsthegloballyorderedoutputsetofruns.Therighthand sideshowstheremainingunpackedrunsfromeachalarm-set.Numbersinthe boxesare i;k values. 113

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Figure3-3.ROCcurveinterpretationofrunpacking Figure3-4.ResultingROCcurveafterrunpacking 114

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CHAPTER4 EXPERIMENTSANDRESULTS 4.1Datasets 4.1.1SyntheticGaussianContextData Totestandillustratethebasicideaofimageswithcontextinformation,alowdimensional,easilyinterpretableimagewasdesired.Asyntheticexamplewasgeneratedby samplingtwodisjointcloudsoftwodimensionaldatapointsfromGaussiandistributions. TherstGaussianusedtheparameters 1 =[5 ; 5] T ,and 1 =[1 : 5; : 51].Thesecond usedtheparameters 2 =[15 ; 5] T ,and 1 =[1 )]TJ/F21 11.9552 Tf 10.939 0 Td [(: 5; )]TJ/F21 11.9552 Tf 9.298 0 Td [(: 51].Thedatapointsweresampled andarrayedintoanimagewith100rowsand100columns,wheretherst50columns weregeneratedbycomponent1,andthesecond50weregeneratedbycomponent2.This createdatotalof10 ; 000samples,5 ; 000fromeachcomponent.Ascatterplotofthis datasetisshowninFigure4-1.Figure4-2showstheimagedatamappedintoafalsecolor RedGreenBlueRGBimagewiththedimension1valuedividedby20mappedtogreen, dimension2dividedby20mappedtoblue,andtheredcomponentsettozero. Atargetsignatureatlocation s =[103] T wasthenselectedandmixedinwiththe backgrounddata.Arandomsubsetof100pixelsfromeachcomponentwereselectedand linearlymixedwiththetargetsignaturewitharandomproportionuniformontheinterval [0 : 251 : 0].AscatterplotofthedatasetwithtargetmixedpixelsisshowninFigure4-3, herepixelswithtargetcontentareshownaslledredcircles.Figure4-4showsamapof theproportionoftargetsignaturepresentineachpixeloftheimage. 4.1.2SyntheticEndmemberContextData Asecondsyntheticdatasetwascreatedwiththegoalofmakingamorerealisticthan Gaussianblobssimulationofhyperspectralimagedataasitwouldbemeasuredina lowdensityurbanenvironment,butwithknowngroundtruthoftargetandbackground content.Todoso,hand-heldspectrometermeasurementsfrommaterialspresentatthe MUUFLGulfportcollectionwereselectedandmixedinfourdistinctimageregions. 115

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Withineachregion,agridofpixelswasselectedandatargetsignaturewasmixedinat variousproportions.Finally,whiteGaussiannoisewasaddedtotheimages. Thebackgroundofthesyntheticimageconsistsoffoursub-imagesorcontexts,each sized107rowsby107columns.Thisoddseemingsizewasselectedsothataregularly spacedgridoftargetpixelscanlaterbemixedinwithasucientmarginaroundallof thetargets.Foreachcontextregion,twotofourmaterialswereselectedtobemixed.For eachmaterial,multiplespectralmeasurementsarepresent,sooneofthemeasurements isselectedatrandomineachpixel.Thiswasdoneinordertoincreasethespectral variationofthesyntheticdata,asselectingasinglematerialrepresentativesuchasthe meanspectrumcreatedadatasetthatwastooeasyofadetectiontask.Thespectrawere reducedto97bands,spacedapproximatelyevery7nmfrom325nmto997nm. Therstcontextcontainsamixtureofthematerialslabeledbyaroughtranslation oftheirlenames,whichcontainsomeadditionalidentifyinginformation,Grassby Building",Dirt",GrassClumpinSun".Thesecondcontextcontainsamixtureofthe materials,GrassByBuilding",BeachSand",FriendshipOak".Thethirdcontained, Bark31-39",Live-oakLeaves",Dirt",FriendshipOak".Thefourth,Asphaltby Hardy1-10",SidewalkinSun",andSidewalkinShade".Thereectancespectrafor thesematerialsareshowninFigures4-5,4-6,and4-7. Tocreatethemixedbackgrounddataforeachpixel,rstaspectrumfromeach materialischosenrandomly.Thenmixingproportionsforthesematerialsaredrawnwith 5%chancefromasymmetricDirichletdistributionwith =0 : 5,orwith95%chance fromasymmetricDirichletdistributionwith =2.Then,with25%chance,thepixel isselectedtocontainshadow.Ifselected,thenarandomshadowamountuniformon [0 : 010 : 99]isselected,andthemixingproportionsofthematerialendmembersarescaled downbythisamount. Forthisdatasetthepeagreenclothcolor,showninFigure4-8isusedasthetarget signature.Thetargetsignatureismixedintoeachcontextregioninaregulargrid 116

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pattern.Thepatternconsistsofvegroupsof100pixels,arrayedinfourrowsof25pixels. Thepixelsarespacedsothat3pixelsseparatealltargetswithinagroup,and5rows separateeachgroup.Thetargetpixelswithineachgrouphavearandomlyselectedtarget proportionwithinthatgroupsrange.Therangesare[5 ; 12],[12 ; 25],[25 ; 50],[50 ; 75],and [75 ; 100].Theresultingtargetproportionmapfortheasinglecontextsyntheticsegmentis giveninFigure4-9. Finally,whiteGaussiannoiseisaddedtothedataset.ASignal-to-NoiseRatioSNR of25dBwaschosen,wherethemeanpowerofthespectrumoverpixelsisusedasthe signalmeanlevel.TheSNRdenedasSNR=10log 10 = thenyieldsthatthestandard deviationoftheGaussiannoiseis = 10 SNR = 10 Togiveasenseoftheshapeofthisdataset,thePrincipalComponentAnalysisPCA transformwasapplied,andtherstthreeprincipaldimensionsofthedataarescatter plottedinFigure4-10.Pointsarecolorcodedaccordingtotheircontextasshowninthe legend.ThetargetsignaturelocationisgivenbytheT",andtheshadowpointisgiven bythe0. 4.1.3MUUFLGulfportCollection Datawerecollectedbyanairborneimagingspectrometerproducingseventy-two9.5 nmbandsfrom367to1043nm,geo-referencedtoa1 : 0m 1 : 0mgridandconverted toreectance.Aco-locatedLiDARsensormeasuredelevationandintensity.LiDAR measurementswereregisteredtothesamegrid,producingDigitalElevationMapsDEMs oftherstandlastreturn. ImageswereacquiredovertheUniversityofSouthernMississippicampusnear Gulfport,Mississippi,USA.AnRGBimageandDEMareshowninFigure4-11.The scenecontainsbuildings,roads,automobiles,grass,sidewalks,trees,bushes,abeach,etc. Fourcoloredcalibrationclothsarevisibleinthecenterofthescene. Clothswatchesstretchedoverwoodenframeswereplacedaroundthecampusas detectiontargets.FourcolorsnamedBrown,DarkGreen,VineyardGreen,andPea 117

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Greenand3sizes : 5m 0 : 5m,1 : 0m 1 : 0m,and3 : 0m 3 : 0mwereused.Examples ofthesetargetsareshowninFigure4-12.Atotalof57targetswereplacedontheground inopenareas,inshadow,orpartiallyorfullyoccludedbytrees.Manyofthelarger targetsspanmultipleregionse.g.bothundertreecoverandintheopen.Someare veryoccludedandprobablynotdetectable.The3sizescreatetargetsthataredenitely sub-pixel,probablysub-pixel,orcontainatleastonepurepixelifun-occluded. Approximategroundtruthlocationsforeachemplacedtargetwererecordedwith ahandheldGPSdevice.Afterthecollectioneachtargetwasmanuallyre-locatedand categorizedineachimage.Targetswerecategorizedbytheirocclusionpropertiesas unoccluded,partiallyorfullyobscuredbyshadowbutnottrees,orpartiallyorfully occludedbytreecover.Condenceinthetargetpresenceatthestatedlocationswasgiven foreachtargetasvisible,probablyvisible,possiblyvisible,ornotvisible. Imageswerecollectedonvepassesoverthecampusattwoaltitudes.Threelow altitudeimageswerecollectedat3500feet,andtwohighaltitudeimageswerecollectedat 6700feet.Thelowaltitudeimageshaveanativepixelsizeof0 : 6m 1 : 0m,whilethehigh altitudeimageshaveanativeresolutionof1 : 0m 1 : 0m. 4.2SequentialUnsupervisedContextLearningandDetection 4.2.1ComparingContextEstimationMethods Totesttheideaofperformingunsupervisedcontextlearningfollowedbycontextdependentdetection,theFuzzPopalgorithmwasusedtosegmenttheGulfportcampus imageintothreeconvexregions.Figure4-13showsROCcurvesoftheperformanceof theFuzzPop-ACEandFuzzPop-HSDdetectorsfromsection3.1ontheGulfportdataset. TheACEROCcurvescomparetheFuzzPop-ACE,toaGaussianMixtureModelGMM class-conditionalACE,toaglobalimagestatisticsbasedACE.Theseresultsshowthatthe FuzzPopclusteringndscontextswhichbenetdetectorperformance,whiletheclusters learnedbytheGMMareaslessusefulascontextualinformationfordetection.Similarly, 118

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theHSDROCsshowalargeimprovementusingthecontext-dependentFuzzPop-HSDover theglobalHSDwithendmembersandstatisticsfoundfromtheentireimageusingSPICE. ThesameFuzzPopoutputcanalsobeusedtoevaluateeachtargetindependently. TheresultsofthisexperimentaregiveninFigure4-14. 4.2.2ComparingUseofLiDARInformation Forthisexperiment,threedetectionresultswereevaluated.Asaperformance baseline,theHSDdetectorwasusedwiththeglobalendmembersfoundbySPICE. NexttheFuzzPopalgorithmwasusedtondfuzzypartitionsandendmembersfor eachpartition.TheFuzzPop-HSDdetectorgivenin3{3wasusedontheresultsof theFuzzPoprun.FinallyFuzzPopwiththeLiDARDEMusedinthespatialtermwas runandtheFuzzPop-HSDdetectorwasusedonthisoutput.ForboththeFuzzPop andFuzzPop-LiDARrunstheparametersweresetto C =3convexregions, M =3 endmembersperregion,membershipfuzzier m =2andtypicalityfuzzier n =1 : 5, a3 3spatialneighborhood, a =20, b =0 : 1,and =0 : 001.Bothmethodsshared initializationvaluesfoundbyusingFuzzyC-means[132]clusteringoftheimagespectraas initialpartitionsandthenusingVCA[142]tondinitialendmembersineachregion. ResultsofthedetectionexperimentareshowninFigure4-15.Theresultsaregiven asROCcurves,whichshowthetradeoofProbabilityofDetectionPDversusFAR asthedecisionstatisticthresholdisvariedoverallcondencelevelsintheoutput.The rangeoffalsealarmratesusedshowsperformanceuptoareasonableupperlimitforan operationalsystemof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 or1in1000pixels.InpartatheROCsshowthatusing FuzzPopmethodndsregionsofdistinctmixturesthatareusefulcontextualinformation fortargetdetection,withtheMagentacurveexceedingtheRedatnearlyallfalsealarm rates.UsingtheLiDARelevationinformationBluecurvefurtherimprovesdetection performance,ndinganadditionaltargetatthefalsealarmratecuto.Partbgivesa zoomedinviewofthesameROCcurveswith95%condenceintervalbandsonthefalse alarmrates.Condenceintervalswereestimatedfromabinomialttothenumberof 119

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observedfalsealarmsateachPDoutofthetotalnumberofopportunitiespixelsinthe image.Theseintervalsshowthattheproposedmethodsachievedetectionratesof0.49 atsignicantlylowerfalse-alarmratesthantheglobalmethod.Theyalsoshowthatthe methodusingLiDARachievesitshighestdetectionlevelsatsignicantlybetterfalsealarm ratesthanwithoutLiDAR. 4.3JointUnsupervisedContextLearningandDetection 4.3.1FCM+ACE+CEM AsaninitialtestFACEMalgorithmwasrunontheGaussiansyntheticcontextdata withthesettings m =2, C =2,and =1.Theresultswerescoredandcomparedagainst theglobalACEandSMFalgorithms,aswellasaGMMbasedCCMF.Theseresultsare showninFigure4-16.Itcanbeseenthatforthisdataset,theclusterbasedFACEMdoes poorlyatlowfalsealarmprobabilities,indeedonlyexceedingtheglobalACEalgorithm aboveaProbabilityofFalseAlarmPFAofaround0 : 1. Thepoorabsoluteperformanceonthisdatasetcanbeunderstoodhoweverbylooking atthescatterplotsofFACEMandtheGMM-CCMFpresentedinFigure4-17.Itcanbe seenthatFACEMidentiesthebackgroundclustersinthedata,butbecauseofitsnature asaanglebasedtwo-sideddetector,manyhighcondencefalsealarmsaregeneratedaway fromthetruetargetdirection.BycomparisonthematchedlterbasedGMM-CCMFonly increasescondenceinthetargetdirection.Itisanartifactofthedatasethoweverthat fewerofthekindoffalsealarmsthatwouldaectthematchedlters.Truehyperspectral datatendstohaveacharactermoresuitedtoACEstylealgorithms. AlargescaletestontheMUUFLGulfportcampusdatawasconstructedtodetermine theperformanceoftheFCM+ACE+CEMalgorithmoverarangeof and m parameter settings.Resultsofthetestbyperformanceofthealgorithmindetectingallfourtarget classessimultaneouslywasmeasuredbyAUCupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 PFA.The parameterwas variedoveralogarithmicallyspacedrangefrom0to100usingthevaluesintheset f 0 ; 0 : 001 ; 0 : 01 ; 0 : 1 ; 1 ; 10 ; 100 g .Thefuzzierwastestedatthreesettings, m =1 : 75, 120

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m =2 : 0,and m =2 : 25.TheseresultsaresummarizedinTable4-1.Theseresultsshow thatthebestperformancewasachievedwiththefuzziersetto m =2 : 25,butalsoshow thatthebestresultssetthe parameterto0.InotherwordsusingACEasaweighting terminthejointoptimizationwasnobetterandpotentiallyworsethansimplynding theclustersviaFCMandthenrunningtheACEalgorithmwiththeclustercentersand fuzzyweightedcovariances.Thismaybebecausetheobjectiveatbestndsatrade-o betweencompactsphericalclustersandcontextswhicharegoodfordetection,andthe approximateoptimizationisbiasedtowardsndingthecompactclusters.Thusother approacheswhichcanndcontextswhicharebestfordetectionarestilldesired. IncomparisontotheresultspresentedinTable4-1,aglobalACEdetectorhasan AUCof0 : 458,SMFtakingmaxcondenceovertargettypes0 : 397,andtheGMMCCMF : 491.TheseresultscanbeseenintheROCcurvesshowninFigure4-18,wherethe plottedFACEMcurveisfromasinglerunwithparameters m =2, =0 : 1, C =5,and hasanAUCupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 of0 : 527. 4.3.2FCEM 4.3.2.1Syntheticdata TotestifthebasicconceptbehindtheFCEMalgorithmwasworking,itswasrst evaluatedonthesyntheticGaussiandatasetfromsection4.1.1.Forthistest,theFCEM andotheralgorithmsbeingcomparedtoitweretrainedusingthedatawithoutany targetpixelspresent.Thelearnedparametersfromthisdatasetwerethenappliedby thedetectorswiththetargetdatapresent.TheFCEMfuzzierwassetto m =2and numberofclusters C =2,andrunfor100iterationsoruntilconvergence.Figure4-19 comparesFCEMtotheACE,SMF,aGMMbasedCCMF,andaFCMbasedMatched FilterdetectorFCM-MF.TheFCM-MFusesasequentialclusteringthendetector strategyasinsection3.1,butusesamatchedlterforeachcluster.Theseresultsshow thatFCEMandtheGMM-CCMFarecapableoflearningthecontextstructurepresent 121

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inthesyntheticdata,andperformverywellcomparedtotheotheralgorithmsandnearly identicallytoeachother. TheFCM-MFapproachisabletoout-performthenoncontextdependentdetectors, butbecauseitdoesnotlearnacovariance,thedetectioncondencedistributionsdonot workaswellasthosefromFCEM.Figure4-20showsthesyntheticdatascatterplotscolor codedbythedetectoroutputfortheFCEMandFCM-MFalgorithms.Whencompared toFigure4-3,theFCM-MFiso-linesarenotquiteangledproperlytomaximizethe detectionsatlowfalsealarmrates. 4.3.2.2Gulfportbatchtest TheFCEMalgorithmhastwoprimaryparameters,thenumberofclusters C ,and thefuzzier m .TodeterminegoodsettingsoftheparametersontheGulfportdataset, theseparametersweretriedatavarietyofsettingsforeachtargettype.Thefuzzierwas triedatthevaluesintheset f 1 : 75 ; 2 : 0 ; 2 : 25 g ,whilethenumberofclusterswastriedfor allintegersintherange2through10.Tenrepetitionswithrandominitializationwererun ateachparametersettingcombination,andtheFCEMalgorithmwasrunforamaximum of100iterationsoruntilconvergence.Thedetectorresultswerescored,andtheAUCof theROCuptoaFARof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FA = m 2 wascomputedforeachiteration.Themeanand standarddeviationoftheAUCoverthe10repetitionsaregivenintheTables4-2,4-3, 4-4,and4-5.Table4-2showsthemeanAUCvalueandstandarddeviationonthebrown targets.Table4-3showsthesameforthedarkgreentarget,Table4-4thefauxvineyard greentargets,andTable4-5thepeagreentargets.Theboldvaluesineachtableshow locationswheremaximalvalueswereachievedinthetests. 4.3.2.3Gulfportindividual UsingtheparametervaluesdeterminedbythemaxAUCinpreviousbatchexperiment,theFCEMdetectorwasrunagainforeachtargettypeandcomparedtoexisting detectionalgorithms.Figure4-21showsROCcurvesforeachofthesetests.Baseline ROCsforACEandtheSMFareshownforcomparison,aswellasotherclusterbased 122

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matchedltertypealgorithms,theGaussian-Mixture-ModelClass-ConditionalMatched FilterGMM-CCMF,andtheFuzzyC-MeansMatchedFilterFCM-MF.TheFCEM algorithmgreenlinedetectsonetargetfewerofthebrowntargetsthantheotheralgorithms,buthasalowfalsealarmrateperformanceequaltoACEandbetterthanthe otherclusterbasedalgorithms.Forthedarkgreentargets,FCEMisrsttothemax detectionrate,thoughalltheclusterbasedalgorithmsandACEallhaveperformancethat iswithinjustafewfalsealarmsofeachother.FCEMoutperformstheothersontheFaux VineyardGreenandPeaGreentargets. 4.3.3PFSCEM 4.3.3.1Syntheticdata ThePFSCEMalgorithmhastheabilitytoaddaspatialconstraintonthecluster membershipsaswellasatypicalityweightforeachdatapointtoeachcluster.The syntheticendmembercontextdatawasusedtodemonstratetheeectoftheseparameters ofthePFSCEMalgorithm.ToinvestigatetheeectoftheFLICMspatialsmoothing term,thePFSCEMalgorithmwasrunwiththefuzzier m =2,numberofclasses C =4, thepossibilistictermsdisabled b =0,andthespatialwindowsetatavarietyofsizes,0 nospatialwindow,sameasFCEMalgorithm,3 3pixels,and5 5pixels.Alltestwere conductedbytrainingondatawithnotargetspresent,andthentestingonthedatawith targetsmixedin.Examplesforeachwindowsizeofpixellabelingsbymaximumcluster membershipareshowninFigure4-22. Itisinterestingtonotethatwhilethemembershipareindeedmademorespatially contiguousbytheFLICMterm,theydonotendupmatchingthegeneratingcontexts ofthedataset.Thisindicatesthatperhapsduetothesomewhatoverlappingnature ofthecontextsegmentsthePFSCEMalgorithmwasmoreeectiveatminimizingthe backgroundenergybychoosingsegmentsotherthanthegeneratingcontexts. Furthertestswerecarriedoutvaryingthepossibilisticweightwhilenotperforming spatialsmoothingandthenrepeatedwithpossibilisticweightandspatialsmoothing.For 123

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thesetests,evenlyvaluedpossibilisticandfuzzyweights a;b = ; 1,lowpossibilistic weights a;b = ; 1,andhighpossibilisticweights a;b = ; 10weretested.The spatialtermwassetto0or3 3pixels,membershipfuzzier m =2,typicalityfuzzier n =1 : 8,andnumberofclusters C =4.Thesetestswererepeated10timeswithrandom initializations,andtheresultsweremeasuredbytheAUCoftheROCupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 FAR. TheseresultsareshowninTable4-6,wheretheentriesarethemeanAUCplusorminus onestandarddeviation. ItcanbeseenfromtheresultsinTable4-6thatthebestresultswithspatialsmoothingbutnopossibilisticweightwereachievedwitha3 3window.Slightlybetterresults wereobtainedusingthepossibilisticweightbutwithnospatialsmoothing.Interestingly though,whencombiningthepossibilisticweightandspatialsmoothing,theresultswere worsethaneitheronitsown.Thisisperhapsindicativethatthisdatasetwasnotdesigned withoutliersinmind,andsothepossibilistictermssimplyallowmoredegreesoffreedom inttingthedataset.Perhapswhenimposingthespatialmembershipconstraintsthese typicalitiesovertanddecreaseperformanceintesting. 4.3.3.2Gulfportbatchtest ThePFSCEMalgorithmhasseveralparameterstoset.Alargetestwasperformedat manydierentparametersettingsinordertodeterminethebestperformingcombinations ontheGulfportdata.Thenumberofcomponents C wastestedatthevalues4 ; 6 ; 10 ; 15, thefuzzier m atthevalues2 : 0 ; 2 : 25 ; 2 : 5,andthespatialwindowatthevalues0 ; 3 3 ; 5 5.Thepossibilistictermsencompassthreevariables,andweretestedwiththepossibilistic weightingdisabled b =0,andwhileenabledtheparameterssetovertheranges:typicality fuzzier n in1 : 75 ; 2 : 0 ; 2 : 25,fuzzytermweight a in1 ; 2 ; 3,possibilistictermweight b in 1 ; 2 ; 3.Intotal,1008parametercongurationsweretestedforeachtargettype,andeach congurationwastested5timeswithdierentrandominitializations.TheAUCvalueof theROCuptoaFARof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 wasrecordedforeachtest.Inordertoreducetheoverall runtimeofthetest,amaximumof30iterationswasusedforeachconguration.Thus 124

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theresultsheremaynotbedirectlycomparabletotheFCEMbatchtestwhereupto 100updateiterationswereused.Evenwiththeseeortstoreducethesearchspaceand runtime,andtestingeachtargettypeinparallelonaquad-coremachine,theoverall runtimeofthistestwasanextraordinarilylong9 : 7days! Table4-7showsthemaximummeanoverrepetitionsAUCvalueandparameter congurationforeachtargettype.Interestingly,thepossibilistictermwasnotshowntobe ofanybenet,atleastattheparametersettingstested.Thismaybebecauseoutlierswere generallynotsucientinnumbertoskewthebackgroundstatisticsandlayinadierent directionfromthetargetsignatures,sotheydidnotaecttheresultsmuch.Thespatial membershipsmoothingwasfoundtobeusefulforalltargetsexceptthedarkgreen.Also, ahighernumberofclustersascomparedtoFCEMwasgenerallyselected. Themaximummeanvalueoverthetestisofcoursenottheentirestory.Thelack ofbenetofthepossibilisticterm,andtheselectionofnospatialsmoothingforthedark greentargetsraisesthequestionofwhatthemarginaldistributionofperformancelooks likeisolatingfortheseparameters.Toanalyzethesedistributions,histogramswerecreated overtheAUCvaluesofalloftherepetitionswithallparametersvariedexceptforthe spatialon/oandpossibilisticon/osetting.Figure4-23showsthesehistogramsfor thebrowntargets,Figure4-24thedarkgreen,Figure4-25thefauxvineyardgreen,and Figure4-26thepeagreentargets. ThehistogramsinFigure4-24indicatethatthenon-selectionofthespatialterm forthedarkgreentargetswaslikelyananomaly,asthewithspatial,nopossibilistic" histogramhasafattertailinhighestAUCs,whilethenospatial,nopossibilistic" histogramhasonlysmallnumberoftestsatthehigherAUCs.Ageneralpatterncanbe observedhoweverthatusingthepossibilistictermsproducedadistributionwithlower AUCsonaveragethannotusingit.Clearevidenceofusingthespatialsmoothingismore diculttodiscern,asthedistributionsareoftenquitesimilar.Thoughthefauxvineyard greentargetsinFigure4-25clearlyshowbenetfromthespatialterm.Certainlythetests 125

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indicatenoharminusingthespatialterm,andthebalanceofevidenceincludingthemax meanresultsfromTable4-7supportstheuseofthisterm. 4.3.3.3Gulfportindividual ThePFSCEMalgorithmwasnexttestedforeachofthefourtargettypeswiththe maxAUCparameterslearnedfromthebatchtestwiththeexceptionofthedarkgreen wherethespatialtermisusedinthistest.Foralltests,thefuzzierwassetto m =2, thepossibilistictermwasturnedo b =0,andtheFLICMspatialtermwassettousea 3 3window.Thenumberofcomponentswassetforthebrowntargetsto C =15,forthe darkgreento C =6,thefauxvineyardgreento C =15,andthepeagreento C =10. ThePFSCEMalgorithmwasrunfor200iterationsoruntilconvergencetoachangein theobjectiveoflessthan10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 .Figure4-27showsROCcurvessummarizingtheresultsof thesetests. ComparingtheresultsinFigure4-27tothoseinFigure4-21showstheresultsof addingthespatialsmoothingterminPFSCEMversustheresultswithoutitforFCEM. PFSCEMndstheonebrowntargetthatwasmissedbyFCEM,andforalltargets PFSCEMachievesitsmaximalPDatalowerFARthanFCEM. 4.3.4IBFCEM 4.3.4.1Syntheticdata TheIBFCEMalgorithmwasrsttestedonthesyntheticGaussiancontextdatato testthatcanndanappropriatenumberofclustersandsimilarperformancetotheFCEM algorithm. BoththeIBFCEMandFCEMalgorithmswererun50timesonthesyntheticGaussiancontextdataset.Thefuzzierwassetto m =2forboth.TheFCEMalgorithm wassetto C =2,whileIBFCEMsetthesparsityweight =0 : 05.Therangeof =[0 : 03 ; 0 : 08]wasfoundtoproducemaximalsolutionsat C =2forIBFCEM.The priortermsontheweightvectorsandclustercentersforIBFCEMwerenotincludedin theupdatesorlikelihoodcomputationastheywerefoundtostronglyaectthesolution. 126

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Thisisnumericallyequivalenttousingverybroadpriorsfortheseterms.FCEMwasset toinitializewithrandompointsfromthedatasetasclustercentersandequalmemberships toeachcluster.Bothalgorithmsweretrainedondatawithnotargetspresent,thentested ondatawiththetargetsmixedin. IBFCEMfoundasolutionhaving2clustersin49ofthe50tests,andused3clusters intheremainingtest.Thenumberofclustersisnothoweveraindicatorofdetection performance,assometimesbettersolutionscanbefoundwithmoreclustersthoughthis datasetdoespreferasolutionalignedwiththetwocontexts.TheAUCoverthefullrange ofPFAofthesolutionsfoundbyIBFCEMandFCEMwasmeasured,andhistogramsof thesevaluesareshowninFigure4-28.Theseresultsshowapproximatelyequalspreadsto thedistributionsofvalues,butIBFCEMproducedthehighestAUCsolutionsfourtimes asoften. TheROCcurvescorrespondingtothemedianAUCsolutionsfoundinthe50testsare showninFigure4-29.ThevaluesinthelegendaretheAUCovertherangeoftheROCs. WhiletheFCEMresultsintheseROCsappearquitepoor,thatagoodsolutionequalto IBFCEMwasachievedin20%ofthetrials.Additionally,ifFCMisusedtoinitializethe FCEMalgorithm,theresultsareimprovedonthisdataset,thoughthisismostlikelydue tothestructureofthisparticulardatasethavingwellseparatedcompactclusters. 4.3.4.2Realdata Totesttheperformanceandabilitytoconsistentlyndausefulnumberofclusters, theIBFCEMalgorithmwastestedontheGulfportCampusimagesfromvedierent ights.Foraperformancecomparison,theFCEM,ACE,andSMFalgorithmswererunon theseimagesaswell.TheAUCupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FARwascomputedforeachalgorithmoneach image,andtheresultsareshowninTable4-8.Usingidenticalparametersettingsforeach test,theIBFCEMalgorithmfound11clustersineachcase. Forthistest,theparametersoftheIBFCEMalgorithmweresetto m =2,and =10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 .Thepriorsontheclustercentersandweightswerenotused.Thesparsity 127

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weightparameter wasselectedsuchthataround10clusterswerefoundinatrialrun oftherstGulfportCampusimage.FortheFCEMalgorithm,theparameterswereset to m =2,and C =10,andveiterationsofFCMrandomlyinitializedwereusedto initializetheclustercentersandmemberships.BothIBFCEMandFCEMwererununtil aconvergenceof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 changeintheobjectivevalue,oramaximumof500iterations.The Campusimageswerepreprocessedbyahierarchicaldimensionalityreductionstrategy[143] to20bandsinordertoreduceruntime. TheperformanceofthesolutionfoundbyIBFCEMasshowninTable4-8seems inconsistentfromimagetoimage.Inimages2and5,thealgorithmperformedquite well,whileinimages3and4itswasoutperformedbytheglobalstatisticsbasedACE andSMF.Thisraisesthequestionofwhetherthepoorperformancecasesaredueto convergencetolocalmaximathatarenotasgoodastheFCEMsolution,orifthereisa deciencyintheprobabilitymodelthatcausespoorsolutionstohaveahigherlikelihood. Thelog-likelihoodofIBFCEMsolutionandFCEMsolutionintheIBFCEMmodel areshowninTable4-9.Fromthisinformationitcanbeseenthatthetwoalgorithmsnd solutionswithsimilarlikelihoods,buttheIBFCEMsolutionhasthehigherlikelihood valueineachcase.Onereasonforthismaybethatthetestsareperformedwithtargets presentindata,undertheassumptionthattheyaresonegligibleinquantitythatthe learnedsolutionswilleectivelyignorethem.Thisassumptionmaynotbevalid. Toprovideevidencefororagainstthisassumption,thetestswereperformedagain withthetargetregionsexcluded.Whileunrealisticfortruealgorithmoperation,thiswill allowustotestiftheCEMparadigmistrulyvalid,butthattheIBFCEMalgorithmwas inasensetooeectiveatminimizingitsobjective.Trainingthealgorithmswiththetest regionsexcluded,thentestingwithouttheregionsexcludedgivestheresultsinTable 4-10.ItcanbeseenfromtheseresultsthattheIBFCEMalgorithmperformsequallywith FCEM.TheresultsalsoindicatethattheIBFCEMalgorithmcanminimizemanyofthe partialtargetsignaturesiftheyarepresentinthescene,andassuch,careshouldbetaken 128

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thattargetsarenotpresentinthetrainingdata.Alsolistedinthetablearethenumberof componentsfoundforeachimage. 4.3.5BFACE 4.3.5.1Syntheticdata TheBFACEalgorithmhasproventobeveryslowtondgoodanswersonlarge datasets.InitialtestsusingasmallversionoftheGaussiansyntheticdatasetthough showthatBFACEiscapableofachievinghighperformance.AninitialtestusingBFACE withagradientsearchbaseddetectormeanupdatestep,thefuzziersetto m =2, C =2,andrunningfor10000samplingiterationsyieldstheresultsshowninFigure 4-30.ForcomparisontheFCEM,ACE,andSMFalgorithmswerealsorunonthesame smalldataset.Thedatasetconsistsof50samplesfromeachoftwoGaussiancomponents havingthesamecongurationasthedatasetdescribedinsection4.1.1.Ascatterplotof thedatasetwithdetectormeanlocationsmarkedandpointscolorcodedbycondenceis showninFigure4-31. RunningthesametestonthestandardGaussiansyntheticdatasethowevergives apoorresult.ThebestsolutionfoundbyBFACEwiththelargerdatasetisapparently stuckinalocalmaximum.TheequivalentROCcurvesandscatterplotforthislarger datasetareshowninFigure4-32.Additionally,whiletheFCEMalgorithmrunsto convergenceonthisdatasetin0 : 18seconds,theBFACEsamplerneeded601 : 30seconds timesaslongtoreach10000samplingiterations. However,thesolutionfoundonthesmalldatasetforBFACEworkswellwhenapplied tothelargerdatasetthathasthesamestatistics,thissolutioncanbeseeninFigure4-33. HeretheBFACE,ACE,SMF,andFCEMalgorithmswerealltrainedonthesmalldataset withnotargetspresentthentestedonthelargedatasetwithtargetsmixedin.These resultsshowalargeimprovementbyBFACEoverthesolutionfoundbytrainingonthe largesyntheticdataset.Itshouldbenotedthatthisdatasetwasgeneratedonthemixing 129

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assumptionsoftheSMFdetector'ssignalmodel,andassuch,theFCEMalgorithmdoes haveaperformanceadvantage. 4.3.5.2Realdata ForaninitialtestoftheBFACEalgorithm,itwasrunonadownsampledversion oftheGulfportCampusdatawhichhadbeenreducedto20bandsusinganinformation theoretichierarchicaldimensionalityreductionstrategy[143].Thedownsamplingprocedureusesthemeanspectralvalueofeach10 10pixelnon-overlappingwindowinthe originalimagetocreatethedownsampledpixels.Targetregionsandinvalidpixelsare maskedoutoftheoriginalimagebeforedownsampling,andmaskedpixelswereexcluded fromthewindowmeans.Thedownsamplingcreated33 34 20pixelimagesuitablefor trainingBFACEwithareasonableruntime.TheBFACEalgorithmwastrainedonthe downsampledimagewiththeparameters m =2, C =4,andrunfor20 ; 000sampling iterations.Thelearnedparameterswerethentestedonthefullresolutiondimensionality reducedcampusimage.ForcomparisonpurposestheFCEM,ACE,andSMFalgorithms wererunwiththesametrainingandtestingsetup.FCEMusedtheparameters m =2, and C =10.TheresultingROCcurvesfromthistestareshowninFigure4-34. TheresultsinFigure4-34showtheBFACEalgorithmmostlyunder-performingthe FCEMandeventhenon-cluster-basedACEalgorithm.Perhapsinterestinglyhowever, BFACEperformsslightlybetterthoughcertainlyofnostatisticalsignicancethanthe otheralgorithmsonthePeaGreentarget. ItappearsasiftheBFACEalgorithmistooeectiveatminimizingthebackground energy,whilestillreturningamaximalvaluewhenpresentedthetargetsignature.Figure 4-35showsahistogramofoutputcondencesbyBFACEforthebrowntargets.Itcanbe seenthatonetargetpixelisgiventhemaximumcondenceof1,whilethevastmajorityof therestaregivenverylowcondences.Thelowestvaluedbinofthishistogramactually continuesabovethey-axis,andcontains106 ; 851oftheimages106 ; 923validpixels. 130

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ThespatialdistributionoftheseoutputcondencesisshowninFigure4-36.Inpart Athecondencesareshownoverfourbrowntargetsindicatedbywhitesquareswith thecolorscalesettoanupperlimitat0 : 001condence.Hereitcanbeseenthatatleast twoofthetargetshaverelativelyhighercondencethanthesurroundingpixels.However inpartB,thesameregionisshownusingacolorscalefrom0to1.Thisshowshigh condenceonlyon8pixelsfromasinglebrowntarget.Thehighestcondencepixelat thecenterofthetargethascondenceexactly1 : 0,asthiswasthepixelselectedfromthe imageasthetargetsignature.Itisclearfromtheseresultsthat,althoughtheBFACE algorithmdoesfunctionproperlyandgiveahighcondencetothetargetsignature,ithas overtrainedandonlyrespondstopixelsverynearlythesameasthetarget.Thisiseven moreimpressiveconsideringtheverylimitedtrainingsetusedinthisexperiment. FutureexperimentsmaytrymodifyingtheBFACEalgorithmtouseatargetsubspace formulation,withmanytargetpixelsusedtobroadenthedetector'sresponse.Additionally,furtherconstraintsmayneedtobeplacedonthecovariancematrixtoprevent overtting. 4.4BayesianFuzzyClustering 4.4.1BFC TodemonstratetheBFCsampler,atwodimensionalsyntheticdatasetwithtwo Gaussiancomponents,generating250datapointseach,wassampledfrom 1 =2 T 1 = I ,and 2 =4 T 2 = I .Inthefollowingexperiments,FCMwasrunto convergenceandtheBFCsearchwasrunfor1000sampleiterations.Weshowresultson thissimpledatasettogivesomeintuitionofthebehavioroftheBFCmodelatvarious parametersettings.WealsodemonstratethatbecausetheBFCsearchdoesnotuse theclosedformupdateequations,itcanberunforsettingsofparametersnotnormally feasibleforFCM. Figure4-37showstheresultsofbothtraditionalFCMandtheBFCMAPsearch forthefuzziersetat m =2and m =10,respectively.Theseresultsdemonstratethat 131

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theratchetingBFCsearch,inrelativelyfewsamplingiterations,canndacomparable solutiontoFCM. WheretherstexperimentsshowthatBFCandFCMproducesimilarresults,the experimentsshowninFigure4-38givetheBFCoutputforfuzziersettingsnotnormally consideredvalidforFCMimplementations.Intheseexperimentsthefuzzierwasset to m =1and m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10,andtheDirichletconcentrationparameterto = 1 .For m =1,theBFCproducesacrisppartitioningsimilartothatofK-meansclustering. Whileforthenegativevalue m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10theclusterprototypesareinsimilarlocationsbut highmembershipvaluesarefoundoppositeoftheprototypelocations.Theseopposite membershipsarenotthesameasoneminusthemembershipinaclass,becauseinthe caseofmorethantwoclusterstheyarestillconstrainedtosumto1overallcomponents. ChangingtheDirichletconcentrationparameterprovidessomeclusteringbehaviors notpossiblewithtraditionalFCMclustering.Figure4-39ashowsresultsusing m =2 andastrongDirichletconcentrationat =3.Heretheresultsappearspatiallysimilar toalinearclassieroutput.Themembershipsincreasetotheirmaximumaround0.6 asdistancefromtheclusterborderincreases,whereastraditionalFCMmemberships tendtofall-oradiallyawayfromtheclusterprototype.Contrastingthecharacterofthis distributioninthelineplottothelineplotsofFigure4-37candd,thisdistributionis morebinarizedwithintheenvelopeofthe0 : 6to0 : 4membershiprange. Figure4-39bshowsanotherinterestingbehaviorwhenthevalueofthefuzzieris relativelyhigh,at m =4,whiletheDirichletconcentrationislessthanone,at =0 : 95. Thisparametercombinationshowsaninnercorenearesttheclusterprototypeswhere clustermembershipiscrisp,whilebeyondthisregionthemembershipsaremoremixed. 4.4.2IBFC TodemonstratetheabilityoftheIBFCmodelandparticlelterinference,we constructedatwodimensionalsyntheticdataset200randomsamplesfromeachoffour 132

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Gaussiancomponentswiththefollowingmeansandcovariances: 1 =2 T 2 =9 T 3 =2 T 4 =9 T 1 =2 I 2 2 2 =0 : 25 I 2 2 3 =0 : 75 I 2 2 4 =2 I 2 2 Figure4-40ashowstheresultsobtainedbyrunningtheIBFCParticleFilteron thefourcomponentdataset.TheIBFC-PFfoundfourcomponentstobethemostlikely solution.PartbshowstheFCMresultsforthisdatasetwiththecorrectnumberof clustersspecied.Figure4-41showsthemostlikelysolutionsfoundbytheparticlelter atalternativemodelsizes.Allofthesesolutionshavealowerlikelihoodthanthecorrectly identiedfourclustersolution. 4.4.2.1IBFCversusCompetitiveAgglomeration ToevaluatetheeectivenessoftheIBFC-PFalgorithminidenticationofthenumber ofclustersinthedata,weranasyntheticdataexperimentwithrandomlygenerated clusters.Wevariedthenumberofclustersfrom2to10,andthedimensionalityofthe datafrom2to10dimensions.Foreachcluster,arandommeanwasdrawnfromthe hypercubeontheinterval[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 ; 10]foreachdimension.Thenarandomnumberof sampleswasselecteduniformlyfrom30to100,andthatnumberofsampleswasdrawn fromaGaussianwiththeidentityascovariance.Forthesedatasets,especiallyinlower dimensionaldata,someoftheclusterswilloverlap,andthealgorithmsarenotexpectedto alwaysndthesamenumberofclustersaswasusedtogeneratethedataset. Foreachnumberofclustersanddimensionality,30trialswereconducted.Ineach trialanewdatasetwasgenerated,thenbothIBFCandCompetitiveAgglomerationCA wererunonthedataset.ThealgorithmswerethenscoredusingtheRandclustervalidity index[144],theEarthMover'sDistanceclustervaliditymetricproposedbyAndersonet al.[145],andthefractionoftrialsinwhichthealgorithmreturnedthesamenumberof clustersaswasusedtogeneratethedataset. 133

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TheIBFC-PFusedtheparametersettings m =2forthefuzzier,sparsityweight equaltothedimensionalityofthedata = D ,numberofparticles P =10,maximum numberofiterations N iter =500,andimplicitlyfortheIBFC-PFtheDirichletconcentrationsetto =1.Thestoppingconditionwasnochangegreaterthan10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 for10 iterationswithdimensionality2through7,orfor100iterationswithdimensionality8or higher. Thecompetitiveagglomerationalgorithmusedthesettings,fuzzier m =2,the initialaveragedistancetoclustercardinalityratioweightingof 0 =5,anexponential weightingdecaytimeconstant =10,maximumof50iterations,initialnumberofclusters C init =20.Theseparametersettingsarelikelysuboptimalatsomepointsintherangeof testeddatasets,buttheywerechosensuchthattheyhadsubjectivelygoodperformance ininitialtesting,andbecausetheyarebasedonthemethodsdiscussedintheoriginal paper[137]likelytobetherstsettingsattemptedinanyuseofthealgorithm. Table4-11showstheaverageRandindexbetweenboththeCAoutputandthetrue generatingcomponents,andbetweentheIBFC-PFandthetruenumberofcomponents. Thenumberofdimensions D ofthedatasetincreasesalongtherowsofthetable,and thenumberofcomponents C inthegeneratingdatasetincreasesalongthecolumns.To computethisindex,rstthefuzzymembershipscomputedbyIBFC-PFandCAwere convertedtocrispclusterassignmentsbymaximummembership.ThentheRandindex computesthesumofthenumberofpairsofpointsplacedinthesameclusterinboth clusteringsandthenumberofpairsplacedindierentclustersinbothclusterings,then dividesbythetotalnumberofpairsofpoints.Theresultshereshowthat,onaverage,for everytesteddimensionalityandnumberofclusters,theIBFC-PFalgorithmproduceda clusteringthatwasassimilartoormoresimilartothetruegeneratingcomponentsthan didCA. TheRandindexisjustoneofmanymethodsofmeasuringclusteringsimilarity,and itdoesnottakeintoaccountfuzzymembershipsorthelocationofclusterprototypes. 134

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Totakethemembershipsandprototypesintoaccount,weusedaversionofAnderson etal.'sEarthMover'sDistancemeasureoffuzzyclusteringsimilarity[145].Wechoseto usetheEuclideandistancebetweenclusterprototypesandthegeneratingcomponent's meanasthegrounddistance.Thiswasdoneinordertoexplicitlytaketheaccuracyofthe clusterprototypesintoaccount.Smallernumbersofthismeasureindicatelessdissimilarity betweentheclusteringoutputandthetruegeneratingcomponents. Table4-12showstheaverageEMDmeasurebetweentheclusteringoutputand generatingcomponentsoverthe30trialsateach D and C size.Theresultsinboldface showwhereCAperformedbetterinthismeasureonaveragethantheIBFC-PF.Intotal theIBFC-PFperformedaswellorbetterthanCAin77of81tests%. Table4-13showsthefractionoftestsinwhichtheclusteringalgorithmsfoundthe samenumberofclustersandthegeneratingdata,andTable4-14showstheaverage numberofclustersfoundbyeachalgorithm.TheIBFC-PFperformedaswellorbetter thanCAallofthe81tests.Bothalgorithmsdidnotoftencorrectlyidentifythenumber ofclustersinlowdimensionaldatawithalargenumberofclusters,butthisismost likelyduetothegenerateddataoverlappingandbeingindistinguishableasseparate clusters.Lookingattheresultsfromanycolumn,theIBFC-PFshowsanencouraging trendofgenerallygettingthenumberofclusterscorrectmoreoftenwithincreasing dimensionality.WhileCAontheotherhandgetsbetterwithdimensionalitytoapoint, butthenperformancegenerallyfallsoatthehighestnumbersofdimensions. 4.5Alarm-SetFusion TotesttheASFtechniques,rstarule-basedsegmentationoftheGulfportdataset wascreated.Thissegmentationgivescontexts,andthencontext-dependentdetectorswere runineachcontext.ExperimentsusingboththeunsupervisedFARE-ASFmethodand thesupervisedRunPackingmethodwerethenperformed. 135

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4.5.1Rule-BasedSegmentation Arule-basedsegmentationoftheimagewascreatedinordertoexploredetection performancewithineachregion.Thesesegmentsandbriefdescriptionsareasfollows: 1.Groundlevel,imperviouscoverprimarilyroads-Notargetswereemplacedinthis region. 2.Groundlevel,perviouscovergrass,lowbushes,non-beachsand-Targetsemplaced inthisregionareintheopen. 3.Groundlevel,shadows-Targetsemplacedinthisregionareobscuredbytheshadow. 4.Trees-Targetsplacedunderthetreecanopyarepotentiallyoccluded. 5.Buildings-Notargetswereplacedontopofthebuildings. 6.Beach-Notargetswereplacedonthebeach,anditsbrightreectancemakesitan outlierfromtheothersegments. 7.CalibrationCloths-Theclothscoverasignicantnumberofpixelsandhavevery distinctspectralcharacteristicsthatwouldskewthebackgrounddistributions.As suchtheyaretreatedasaseparatesegmentandremovedfromtheimage. Tocreatethesegmentation,rstanNDVImapfortheimagewascomputed.Next height-basedsegmentationwasperformedtondthelowelevationground,highelevationbuildingsandtreetops,andmixedelevationtreecanopysections.The segmentationfollowsthroughaseriesofrulesandlteringstepsalongthelinesofthe pseudocodeinAlgorithm13,withsomesmalldetailsleftoutforclarity.Theltering functionconsistsofoperationssuchasamorphologicalclosingtoconnectnearlyadjacent components,llingofconnectedcomponents,andremovalofconnectedcomponentsless thantheminimumspeciedsize. Thebeachsegmentisfoundbythresholdsonpixelintensityandelevation.The calibrationclothsaredetectedbymatchedlterresponsetohandselectedprototype spectrafromcoordinatesselectedeasilyfromtheRGBimage. TheshadowmapiscomputedusingtheLiDARrstreturnDEMandthesun illuminationangleobtainedfromtheknownlocation,time-of-day,andday-of-yearofthe 136

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Algorithm13: Rule-basedsegmentation Data :HSIImage hsi data ,NDVImap ndvi ,LiDARrstandlastreturnDEMs lidar rst lidar last Result :Segmentmaps ground impv seg ground perv seg ground shadow seg tree seg bldg seg beach seg ,and cloth seg high veg ndvi thresh =0 : 4 gnd veg ndvi thresh =0 : 2 low elev = lidar rst 10m high elev =! low elev mixed elev = lidar rst )]TJ/F50 11.9552 Tf 11.955 0 Td [(lidar last > 1m bldg base = ndvi high elev < high veg ndvi thresh bldg seg = filter bldg base ; min size =20 tree base = ndvi high elev jj mixed elev > high veg ndvi thresh tree seg = filter tree base ; min size =5 ground seg =! tree seg jj bldg seg ground veg base = ndvi ground seg > gnd veg ndvi thresh ground perv seg = filter ground veg base ; min size =10 ground impv base = ndvi ground seg < gnd veg ndvi thresh ground impv seg = filter ground impv base ; min size =10 intensity map = jj hsi data jj 2 = n bands beach base = and intensity map > 0 : 06 ; lidar last < 3m beach seg = filter beach base ; min size =100 cloth base = detect cloths hsi data cloth seg = filter cloth base ; min size =50 shadow map = compute shadow map lidar rst ground shadow seg = ground seg shadow map ground perv seg = remove intersection ground perv seg ; union ground seg ; beach seg ; cloth seg ground impv seg = remove intersection ground impv seg ; union ground shadow seg ; beach seg ; cloth seg collection.Foreachpixel,arayisprojectedoutattheilluminationangle,ifanyother pixelsintheDEMarefoundthatintersectthisray,thentheoriginalpixelisconsideredto beinshadow. 137

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Theresultingimagewitheachsegmentofthesevensegmentslabeledbycoloris showninFigure4-42.Theareainthelowerrightoftheimagecontainsnodataandis labeledasclass0. 4.5.2UnsupervisedAlarm-SetFusion Onemotivationfortheseexperimentsistoidentifyacontext-dependentmethodthat boostsdetectionperformanceontheoccludedandobscuredtargetsinourdataset.The datasetwassegmentedusingtherule-basedsegmentationthatspecicallyidentiesthe segmentswheretargetsmaybeobscuredbyshadowandwheretargetsmaybeoccluded bytreecover. Threeoftheimagesegments,theGround/Pervious,Ground/Shadow,andTreesegments,containalloftheemplacedtargets.Figure4-43showsROCcurvesfordetection algorithmsrunonlyontheimagepixelsfromeachofthesesegmentsindividually.ProbabilitiesofdetectionfortheseROCsarerelativetotheentirepopulationof57emplaced targetsintheimage.Becausesometargetshaveamultiplepixelextentandareonthe boundarybetweensegments,theyareobservableinmultiplesegmentsandthetotalPDof theaggregateddetectorswillnotnecessarilybethesumofthePDacrossallsegments. TheseresultinFigure4-43showthatintheopenunoccludedandunobscured Ground/Pervioussegment,theHSDoutperformstheotherdetectorsatthefalsealarm ratesofinterestthoseat10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 FA = m 2 andbelow.IntheobscuredGround/Shadow segment,theHSDalsoshowsthebestperformance.IntheTreesegmenthowever,the SMFdetectorndsonemoreofthetargetsatthefalsealarmratesofinterest. GiventheresultsinFigure4-43itwouldseemlogicaltopartitiontheimageintosegmentswithknownproperties,andthenapplyindividualdetectorsinacontext-dependent mannerforeachofthosesegments.Thatwaythedetectorwiththebestperformancein eachcontextcanbechosen.This,however,makesevaluationandcomparisontoother 138

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nonsegmentbaseddetectorsdicult.TheFARE-ASFalgorithmcanbeusedtorecombinethecontext-dependentsegmentoutputsthatleveragethebestperformingdetectors withineachsegment. Anexperimentwasperformedtodetermineiffusingthecontext-dependentsegmentbaseddetectorsusingFARE-ASFwouldprovidebetteroverallperformancethan runningthedetectionalgorithmsgloballyontheentireimage.TheFARE-ASFalgorithmfusedthealarm-setsofasegmentspecicHSDdetectorintheGround/Pervious segment,asegmentspecicHSDdetectorintheGround/Shadowsegment,andthe SMFdetectorintheTreesegment.Forallothersegments,theoutputofaglobal ACEdetectorwasused.Thresholdswereestimatedforthefalsealarmratesintheset f 0 ;: 00002 ;: 0001 ;: 0002 ;: 0003 ;: 0004 ;: 0005 ;: 0006 ;: 0007 ;: 0008 ;: 0009 ;: 001 ; 1 g .Figure4-44 showsROCsforthisFARE-ASFcombinedoutputaswellastheACE,SMF,andHSDdetectors.Additionally,theCC-ACEalgorithmwasusedwith6clustersforacomparison tomethodswhichaggregatemultipledetectorsfromclusteringbasedsegments. TheresultsinFigure4-44showthattheFARE-ASFcombinationofcontextdependentdetectorsinthisexperimentismostbenecialatthelow )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 FA = m 2 FARs,whilethedetectorstestedalldetectapproximatelythesamenumberoftargetsat thehigher10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FA = m 2 FAR. AninterestingfeatureoftheseROCcurvesisthattheFARE-ASFcurvegoes diagonallytozerofalsealarmsafterthelowestdetection.ThisisbecausetheFARE-ASF algorithmplacesthehighestcondencefalsealarminasegmentwithnotruealarmsat thesamecondenceasthehighestcondencedetectionsintheothersegments.Intrue operationthesefewfalsealarmswillcontributeproportionallylesstothefalsealarmrate astheareaofthesegmentsgrows. AsecondexperimentwasconductedtoexamineiftheFARE-ASFalgorithmhelps performancewhengivenasingledetectionalgorithmruninmultiplesegments.Figure 4-45showstheresultswhentheACEalgorithmisrunwithgloballyestimatedparameters 139

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shownasACE,wheneachsegmenthasacontext-dependentdetectorandthesegment resultsconcatenatedshownasSegACE,andwhentheFARE-ASFalgorithmisused tobalancethecondencesofthesegmentsshownasFARE-ASF.Thisresultshowsa substantialincreaseofdetectionperformanceatlowfalsealarmrateswhenthesegments arebalancedwiththeFARE-ASFalgorithm.InthesimpleconcatenationcaseSegACE thebenetsofthecontext-dependentdetectorsarelostbecauseofthedierencesinthe distributionsofcondenceamongsegments. 4.5.3SupervisedAlarm-SetFusion Thesupervisedalarm-setfusionexperimentstesttheperformanceoftheRun PackingalgorithmwiththeDynamicProgrammingpackingstrategyinfusingcontextdependentdetectorsoncontextsdenedbytherule-basedsegmentation.Asonlythe Ground/Pervious,Ground/Shadow,andTreesegmentscontainalltargets,theseexperimentalresultsusedatafromtheaforementionedsegmentsonly.Aspectrumforeachcloth colorwashand-selectedfromtheimagefromanun-occluded3 : 0mtargetpixel.Because theHSDrequiresunmixing,SPICE[94]wasusedtondendmembersandabundancesfor eachsegment. TheACEandHSDwererunoneachofthetargetsegmentsindependently.A summaryofthedetectionresultsisshowninTable4-15.TheHSDperformsbetterthan ACEintheGround/PerviousandGround/Shadowsegments,whilethereverseistruein theTreesegment.Thisprovidesmotivationforusingthebest"detectorforeachsegment andthencombiningtheirrespectiveoutputswithASF. FortrainingdatatoRPAUC,wecreatedamixeddetectorcompositeoutputfrom theHSDresponseontheGround/PerviousandGround/Shadowsegments,andtheACE responseontheTreesegment.ToprovideafairestimateoftheperformanceofRPAUC, wesplitthemixeddetectorimageintotwohalvesalongcolumn168ofthe325 337 imageandperformedatwo-foldcrossvalidation.Theresultofconcatenatingthetwo test-imageoutputswasthenscored.Table4-15givesthedetectionperformanceofthe 140

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RPAUCalarm-setfusionMixedASFaswellasACEandHSDrungloballyacrossthe threesegments.NotethatsometargetsspanmultiplesegmentssotheAll"resultsdetect fewertargetsthanthesumoftheindividualsegmentresults. Tocomparewithanothercontext-dependentmethod,aslidingwindowRXlike versionofACEwasalsotested,showninTable4-15asACE-RX.Thisisanextremecase ofacontext-dependentdetector,usingadierentcontextforeverypixeloftheimage.The crossvalidatedASFshowscomparabledetectionperformancetoACE-RX,whileavoiding therun-timepenaltycausedbyinvertingacovariancematrixforeverypixel.Additionally, atest-on-trainRPAUCmixeddetectorASFresultispresentedtogiveanupperboundon performance. Oneinterpretationofalarm-setfusionisthejointselectionofthresholdswithineach alarm-setyieldingasinglethresholdlevelinthejointROC.Table4-16showsthelearned thresholdsforeachcomponentinthemixeddetectorASFtest-on-trainresultaswellas thethresholdsfortheindependentdetectorstoachievethecorrespondingsegmentlocal FAR.Thelearnedthresholdsshowthat,inmovingfroma10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 to10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 globalFAR,itis besttokeeptheACEthresholdintheTreesegmentxedatalowlocalFAR,andthen lowerthethresholdintheGround/Shadowsegmentyieldingahigherthan10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 local FARinthatsegmenttogetmoredetectionsoverall. 141

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Figure4-1.Scatterplotoftwo-componentGaussiansyntheticcontextdata 142

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Figure4-2.False-colorRGBimageoftwo-componentGaussiansyntheticcontextdata Figure4-3.Scatterplotoftwo-componentGaussiansyntheticcontextdatawithtargets 143

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Figure4-4.Proportionmapoftargetmixturefortwo-componentGaussiansynthetic contextdata 144

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AGrassByBuilding" BDirt" CGrassClumpinSun" Figure4-5.Materialspectraforthefour-contextsyntheticendmemberdataset. 145

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ABeachSand" BFriendshipOak" CBark31-39" Figure4-6.Materialspectraforthefour-contextsyntheticendmemberdataset. 146

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ALive-oakLeaves" BAsphaltbyHardy1-10" CSidewalkinSun" DSidewalkinShade" Figure4-7.Materialspectraforthefour-contextsyntheticendmemberdataset. 147

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Figure4-8.Pea-greentargetspectrum Figure4-9.Targetemplacementgridforthefour-contextsyntheticendmemberdata.This patternisrepeatedineachofthefourcontexts. 148

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Figure4-10.Scatterplotsoffour-contextsyntheticendmemberdatainrstthreeprincipal componentdimensions.TheletterT"indicatesthetargetsignatureandthe 0indicatestheshadowpoint. 149

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ARGB BLiDARFirstReturnDEM Figure4-11.RGBandLiDARDEMofGulfportcampusdataset.ARGBimage.B LiDARdigitalelevationmap 150

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Figure4-12.ExampleemplacedtargetsintheGulfportcampusdatasetin4colorsand3 sizes.Thetwobrowntargetsarethesamematerial,whilethethreegreen targetsarealldierentmaterials. 151

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AACE BHSD Figure4-13.ROCcurvescomparingglobalACEandHSDtoFuzzyversionsusing FuzzPop.AClass-conditionalFuzzPop-ACEvsglobalACEandaclass conditionalACEfromGaussianmixturemodel.BFuzzPop-HSDvsglobal HSDusingSPICE. 152

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ABrown BDarkGreen CFauxVineyardGreen DPeaGreen Figure4-14.ROCsforFuzzPop-ACEandFuzzPop-HSDonGulfportcampusdataby targettype 153

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AWithouterrorbars BWitherrorbars Figure4-15.ROCsofFuzzPop-HSDusingFuzzPopandFuzzPopwithLiDARcompared toglobalHSD.AWithouterrorbars.BWith95%condenceinterval bandsonFAR 154

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AFullRange BUpto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FAR Figure4-16.ROCsforFACEMvsACE,SMF,GMM-CCMFonGaussiansyntheticdata. AUCsoverthex-axisrangeareshowninthelegend.AResultsoverthefull PFArange.BResultsoverPFArangeupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 AFACEM BGMM-CCMF Figure4-17.FACEMvsGMM-CCMFscatterplotsofGaussiansyntheticdatacoloredby condence.AFACEMshowsthetwosidednatureofACEdetectors.B GMM-CCMFusesaonesideddetectorandincreasescondencetowards target. 155

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Figure4-18.ROCsforFACEMvsACE,SMF,GMM-CCMFonGulfportcampusdata Figure4-19.ROCsforFCEMvsACE,SMF,GMM-CCMF,FCM-MFonGaussian syntheticdata.AUCvaluesareshowninthelegend. 156

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AFCEM BFCM-MF Figure4-20.FCEMvsFCM-MFcondencescatterplotsonGaussiansyntheticdataA FCEMcondencedistributionisangledtominimizefalsealarms.B FCM-MFangleofthecondencedistributiondoesnotminimizefalsealarms 157

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ABrown BDarkGreen CFauxVineyardGreen DPeaGreen Figure4-21.ROCsforFCEMonGulfportcampusdata 158

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ANoSpatialSmoothing B3 3SpatialWindow C5 5SpatialWindow Figure4-22.Pixellabelingsbyspatialwindowsize Figure4-23.PFSCEMAUChistogramsonbrowntargetsvaryingspatialandpossibilistic usage 159

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Figure4-24.PFSCEMAUChistogramsondarkgreentargetsvaryingspatialand possibilisticusage 160

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Figure4-25.PFSCEMAUChistogramsonfauxvineyardgreentargetsvaryingspatialand possibilisticusage 161

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Figure4-26.PFSCEMAUChistogramsonpeagreentargetsvaryingspatialand possibilisticusage 162

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ABrown BDarkGreen CFauxVineyardGreen DPeaGreen Figure4-27.ROCsforPFSCEMonGulfportcampusdata.ABrowntargets.BDark greentargets.CFauxvineyardgreentargets.DPeagreentargets. 163

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AIBFCEM BFCEM Figure4-28.HistogramsofROCAUConsyntheticGaussiancontextdataover50Trials 164

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AFullrange BUpto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 PFA Figure4-29.MedianROCcurvesforIBFCEM,FCEM,ACE,andSMFonsynthetic Gaussiancontextdataover50trials.AResultsoverfullPFArage.AUC valuesoverthisrangeareshowninthelegend.BResultsupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 PFA. AUCvaluesoverthereducedrangeshowninthelegend. 165

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Figure4-30.BFACEROCsonsmallsyntheticGaussiancontextdatasetcomparedto FCEM,ACE,andSMF.AUCvaluesareshowninthelegend. 166

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Figure4-31.ScatterplotofsmallsyntheticGaussiancontextdatasetcoloredbyBFACE condence 167

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AROCCurves BScatterPlot Figure4-32.BFACEROCandscatterplotonfullGaussiansyntheticdataset.AROC curvesshowingBFACEperformancerelativetoFCEM,ACEandSMF.AUC valuesareshowninthelegend.BScatterplotofdatasetcoloredbyBFACE condence. 168

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AROCCurves BScatterPlot Figure4-33.BFACEROCandscatterplottrainedonsmalldatasetthentestedonfull syntheticdataset.AROCcurvesshowingBFACEperformancerelativeto FCEM,ACEandSMF.AUCvaluesareshowninthelegend.BScatterplot ofdatasetcoloredbyBFACEcondence. 169

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ABrown BDarkGreen CFauxVineyardGreen DPeaGreen Figure4-34.ROCsforBFACEonGulfportcampusdata.Algorithmsweretrainedon downsampledimagesthentestedonfullresolutionimages. 170

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Figure4-35.HistogramofBFACEcondencesforbrowntargets.Thelowestbincontinues abovey-axisandcontains106 ; 851pixels. 171

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ARange[0 ; 0 : 001] BRange[0 ; 1] Figure4-36.CondencemapsofBFACEdetectorforbrowntargets.AResultsonacolor rangefrom0to0.001,valuesabovemaxareclipped.BSameresultsona colorrangeof0to1,showingasingletargetdominatingthecondence range. 172

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ABFC, m =2, = 1 BFCM, m =2 CBFC, m =10, = 1 DFCM, m =10 Figure4-37.BFCandFCMoutputsfor m =2.Scatterplotsarecolorcodedby membershipincluster1redishighmembership,blueislowmembership. Lineplotsdenotemembershipincluster1bydatapointindex.Indices1-250 generatedby N 1 ; 1 ,indices251-500generatedby N 2 ; 2 .aBFC MAPsearchresults m =2, = 1 .bFCMresults m =2cBFCresults for m =10, = 1 .dFCMresultsfor m =10. 173

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ABFC m =1 ; =1 BBFC m = )]TJ/F17 10.9091 Tf 8.485 0 Td [(10 ; =1 Figure4-38.BFCresultsfor m =1and m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10.aThefuzzier m =1resultsina crisppartitioningallmemberships0or1.bThefuzzier m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10places clustermembershipawayfromtheprototypelocation. ABFCMAPSearch m =2 ; =3 BBFCMAPSearch m =4 ; =0 : 95 Figure4-39.BFCresultsforcombinationsof m and .aResultsforalowfuzzier m =2,andhighDirichletconcentration =4,membershipsincreasewith distancefromclusterboundary.bResultsshowingabinaryinnercorewith highfuzzier m =4andlowDirichletconcentration =0 : 95. 174

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AIBFC-PF m =2 ; =1 ; =2 BFCM m =2 ;C =4 Figure4-40.IBFCandFCMresultsonfourcomponentdata,colorcodedbymaximum membershipcluster 175

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ABest2ClusterSolution BBest3ClusterSolution CBest5ClusterSolution DBest6ClusterSolution Figure4-41.IBFCparticlelteralternativesolutionswithlowerlikelihood.Allsolutions generatedbyasingleIBFC-PFrunwithparameters m =2 ; =1 ; =2 176

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Figure4-42.Segmentationmap 177

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AGround/Pervious BGround/Shadow CTree Figure4-43.SegmentspecicROCcurves.AResultsontargetsintheGround/Pervious segment.BResultsontargetsintheGround/Shadowsegment.CResults ontargetsintheTreesegment 178

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AAlltargets BOccludedandobscuredtargetsonly Figure4-44.ROCcurvesforSMF,ACE,HSD,andCC-ACEandFARE-ASFresult.A Resultsonalltargets.BResultsonoccludedandobscuredtargetsonly. 179

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Figure4-45.ACEalgorithmROCs 180

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Table4-1.FACEMmaximumAUCandnumberofclustersoncampusdataforvarious settingsof and m # m 1.752.02.25 00.516@ C =5 0.532 @ C =5 0.536 @ C =9 0.0010.516@ C =50.532@ C =50.535@ C =9 0.010.516@ C =50.532@ C =50.533@ C =9 0.10.518@ C =50.532@ C =50.532@ C =9 1 0.521 @ C =40.526@ C =40.519@ C =5 100.498@ C =50.485@ C =90.474@ C =8 1000.478@ C =90.465@ C =80.459@ C =5 Table4-2.AverageAUCforFCEMonbrowntargetsupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 FA/ m 2 C # m 1.7522.25 20.68 0 : 0000.68 0 : 0000.68 0 : 000 30.69 0 : 0000.69 0 : 0000.69 0 : 000 40.56 0 : 000 0.75 0 : 000 0.75 0 : 000 50.49 0 : 0760.70 0 : 0610.74 0 : 012 60.63 0 : 0230.69 0 : 0020.71 0 : 000 70.70 0 : 0020.69 0 : 0000.71 0 : 000 80.70 0 : 0000.65 0 : 0050.71 0 : 000 90.74 0 : 0300.73 0 : 0260.64 0 : 051 100.63 0 : 0570.72 0 : 0730.61 0 : 023 Table4-3.AverageAUCforFCEMondarkgreentargetsupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FA/ m 2 C # m 1.7522.25 20.46 0 : 0000.46 0 : 0000.46 0 : 000 30.35 0 : 0000.38 0 : 0000.46 0 : 000 40.36 0 : 0000.41 0 : 000 0.47 0 : 000 50.38 0 : 0080.42 0 : 0270.43 0 : 002 60.34 0 : 0140.37 0 : 0090.39 0 : 021 70.44 0 : 0050.43 0 : 0030.46 0 : 001 80.44 0 : 0010.41 0 : 0010.42 0 : 002 90.41 0 : 0310.40 0 : 0390.39 0 : 009 100.40 0 : 0160.44 0 : 0220.42 0 : 050 Table4-4.AverageAUCforFCEMonfauxvineyardgreentargetsupto10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FA/ m 2 C # m 1.7522.25 20.57 0 : 0000.57 0 : 0000.57 0 : 000 30.56 0 : 0000.57 0 : 0000.57 0 : 000 40.57 0 : 0000.56 0 : 0000.56 0 : 000 50.58 0 : 0000.58 0 : 0050.58 0 : 003 60.58 0 : 0000.58 0 : 0000.58 0 : 013 70.58 0 : 0000.60 0 : 0000.65 0 : 001 80.58 0 : 000 0.71 0 : 000 0.71 0 : 000 90.65 0 : 0150.63 0 : 0070.64 0 : 025 100.67 0 : 0250.63 0 : 0110.64 0 : 014 181

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Table4-5.AverageAUCforFCEMonpeagreentargetsupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 FA/ m 2 C # m 1.7522.25 20.44 0 : 0000.44 0 : 0000.44 0 : 000 30.46 0 : 0000.46 0 : 0000.46 0 : 000 40.46 0 : 0000.46 0 : 0000.46 0 : 000 50.46 0 : 0010.46 0 : 0010.46 0 : 000 60.47 0 : 0060.46 0 : 0000.46 0 : 000 70.45 0 : 0000.46 0 : 0000.46 0 : 000 80.31 0 : 0020.46 0 : 0000.47 0 : 000 90.46 0 : 0010.47 0 : 0100.46 0 : 000 100.46 0 : 032 0.48 0 : 011 0.46 0 : 000 Table4-6.PFSCEMAUConsyntheticendmembercontextdataforvarioussettingsof possibilisticweightandspatialsmoothing Window a;b = ; 0 ; 1 ; 1 ; 10 00.9147 0 : 00500.9274 0 : 00010.9279 0 : 00520.9260 0 : 0007 3 30.9268 0 : 00060.9248 0 : 00000.9253 0 : 00010.9242 0 : 0003 5 50.9244 0 : 0001--182

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Table4-7.Maximummean-AUCparametercongurationforPFSCEMonGulfport campusdata AUC Cmnab spatial brown0.771522103 darkgreen0.51622100 f.v.green0.681522103 peagreen0.461022103 Table4-8.AUCvaluesupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 forIBFCEM,FCEM,ACE,andSMFonvecampus images Image1Image2Image3Image4Image5 IBFCEM0.3540.3540.2650.2440.315 FCEM0.4300.3330.3170.2710.341 ACE0.3810.3330.2670.2650.267 SMF0.3520.3210.2670.2610.264 Table4-9.IBFCEMobjectivevaluesforIBFCEMandFCEMresults Image1Image2Image3Image4Image5 IBFCEM-1965114.64-1932216.73-1848299.29-2111446.36-2037104.26 FCEM-1965114.66-1932216.77-1848299.30-2111446.40-2037104.27 Table4-10.AUCvaluesupto10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 forIBFCEMandFCEMonvecampusimageswith targetsexcludedfromtraining Image1Image2Image3Image4Image5 IBFCEM0.400.330.310.270.32 FCEM0.400.330.320.270.32 183

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Table4-11.AverageRandindexforCompetitiveAgglomerationvstrueandIBFCvstrue largerisbetter D # C 2345678910 2 CA0.930.930.930.930.940.930.930.910.92 IBFC0.990.960.950.950.960.940.950.950.95 3 CA0.980.980.990.980.980.960.950.940.92 IBFC1.000.990.990.990.990.970.980.980.98 4 CA0.991.000.990.990.990.960.980.960.90 IBFC1.001.001.001.000.990.990.990.990.98 5 CA0.991.000.980.970.930.990.960.950.93 IBFC1.001.001.001.000.991.001.001.000.99 6 CA0.980.980.950.890.970.970.960.970.97 IBFC1.001.001.001.001.001.001.001.001.00 7 CA0.950.950.951.000.860.940.970.910.98 IBFC1.001.001.001.001.001.001.001.001.00 8 CA0.890.950.850.890.920.970.940.990.92 IBFC1.001.001.001.001.001.001.001.001.00 9 CA0.850.950.800.790.921.000.910.970.99 IBFC1.001.001.001.001.001.001.001.001.00 10 CA0.800.820.750.890.891.000.880.910.88 IBFC1.001.001.001.001.001.001.001.001.00 Table4-12.AverageEarthMover'sDistanceforCompetitiveAgglomerationvstrueand IBFCvstruesmallerisbetter D # C 2345678910 2 CA0.590.790.931.111.221.471.521.782.01 IBFC0.350.600.810.971.051.281.381.481.66 3 CA0.520.790.951.211.421.632.042.403.04 IBFC0.440.710.891.071.331.511.671.872.09 4 CA0.550.801.021.251.522.062.162.623.72 IBFC0.510.760.991.211.481.691.942.122.46 5 CA0.57 0.84 1.151.652.261.982.463.093.73 IBFC0.550.851.091.321.621.822.072.342.59 6 CA0.881.301.742.852.052.442.993.143.67 IBFC0.620.921.181.451.741.992.232.532.77 7 CA1.611.602.07 1.52 3.602.872.893.863.57 IBFC0.660.991.271.531.832.052.432.612.94 8 CA2.681.873.943.403.172.583.512.994.77 IBFC0.691.031.331.671.902.202.522.763.02 9 CA3.831.775.015.263.33 2.30 4.053.413.56 IBFC0.771.081.421.742.092.322.682.963.23 10 CA5.494.576.203.684.13 2.47 4.544.646.09 IBFC0.751.131.461.822.152.482.793.163.39 184

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Table4-13.Fractionoftrialswithsamenumberofclustersastruthlargerisbetter D # C 2345678910 2 CA0.430.330.470.600.400.100.130.030.00 IBFC1.000.900.800.700.630.630.270.070.10 3 CA0.700.630.830.670.570.330.200.130.00 IBFC1.000.970.970.930.800.800.430.400.20 4 CA0.830.800.900.900.770.730.470.300.03 IBFC1.001.001.000.970.830.830.630.530.33 5 CA0.870.970.730.930.730.730.670.470.37 IBFC1.001.000.971.000.870.870.800.770.57 6 CA0.930.830.830.830.870.730.700.600.43 IBFC1.001.001.001.000.930.930.900.730.73 7 CA0.900.900.900.970.730.830.730.570.60 IBFC1.001.001.001.001.001.000.930.930.80 8 CA0.630.770.670.770.700.900.830.830.50 IBFC1.001.001.001.001.001.000.970.900.93 9 CA0.600.730.630.600.700.970.770.930.67 IBFC1.001.001.001.001.001.000.970.970.93 10 CA0.570.630.630.830.730.870.770.800.47 IBFC1.001.001.001.001.001.001.000.931.00 Table4-14.AveragenumberofclustersfoundbyIBFCandCA D # C 2345678910 2 CA2.93.94.45.15.55.96.96.37.3 IBFC2.02.93.84.75.66.06.87.47.9 3 CA2.33.34.15.25.76.56.87.37.0 IBFC2.03.04.04.95.86.57.48.28.8 4 CA2.23.24.05.05.96.87.37.87.6 IBFC2.03.04.05.05.86.87.68.59.1 5 CA2.13.04.15.06.36.87.98.38.6 IBFC2.03.04.05.05.96.97.88.89.5 6 CA2.33.54.65.76.17.27.88.28.8 IBFC2.03.04.05.05.96.97.98.79.7 7 CA2.53.44.85.06.87.38.18.79.3 IBFC2.03.04.05.06.07.07.98.99.8 8 CA3.03.55.05.66.57.18.28.88.9 IBFC2.03.04.05.06.07.08.08.99.9 9 CA3.73.55.66.26.57.08.49.09.6 IBFC2.03.04.05.06.07.08.09.09.9 10 CA4.34.45.85.76.77.18.58.99.2 IBFC2.03.04.05.06.07.08.08.910.0 185

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Table4-15.Detectorperformancebysegment.Column#@..."givesnumberoftargets detectedatindicatedFAR.ColumnFAR@Max#"giveslowestFARunder 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 wheredetectorndsitsmaximumnumberoftargets. DetectorSegment #@10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 /#@10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 FAR@Max# 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 ACE-segGnd/Pervious11/127.9 HSD-segGnd/Pervious14/156.4 ACE-segGnd/Shadow3/68.1 HSD-segGnd/Shadow6/61.2 ACE-segTree16/160.8 HSD-segTree14/151.3 ACE-globalAll18/306.6 HSD-globalAll27/304.7 ACE-RXAll28/336.9 MixedASF CrossvalidatedAll30/311.4 MixedASF Test-on-TrainAll31/332.6 Table4-16.Learnedthresholdsinalarm-setfusionversussegmentindependentthresholds DetectorSegment10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 thresh.10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 thresh. MixedASFAll0.860.82 HSD-ASFGnd/Pervious1.221.18 HSD-segGnd/Pervious1.241.16 HSD-ASFGnd/Shadow1.361.10 HSD-segGnd/Shadow1.321.16 ACE-ASFTree0.490.49 ACE-segTree0.490.33 186

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CHAPTER5 DISCUSSIONANDFUTUREWORK 5.1Discussion Itiswellknownthattheenergyminimizationstyleobjectivefunctionoptimization andlikelihoodmaximizationproblemsunderaprobabilisticframeworkareinherentlythe samething.Itisperhapslesswellknown,andcertainlyunrealizedtomewhenIstarted thiswork,theconsequencethenthatconditionallikelihoodmaximizationandcoordinatewiseoptimizationarealsothesameproblem.ThishasadeeperconnectionrelatingGibbs samplingandcoordinate-ascentstrategiesforoptimization,whichhasbeenusedinthe inferencealgorithmsdevelopedhere. Forexample,anyMAPsearchproblemcanbeaddressedbyusinganMCMCsampler togeneratesamplesfromthedesiredposteriordistribution,andthenbykeepingtrack ofthegeneratedsamplewiththehighestlikelihoodvalue.TheeasiestMCMCapproach isthenGibbssampling,generallywithaMetropolis-within-Gibbsapproach.Anyofthe Gibbssamplingstepscanbereplacedhoweverwithaconditionalorcoordinate-wise maximizationstepinstead,andthealgorithmbecomeaformofgeneralizedstochastic expectationmaximization.IfalloftheGibbsstepsarereplacedwithcoordinate-wisemaximizations,thentheprobabilistictechniqueisknownastheIteratedConditionalModes algorithm.ThustheFCMalgorithmissimplytheapplicationofIteratedConditional ModesinferencetotheBayesianFuzzyClusteringprobabilisticmodel. UsingthemixofstochasticandoptimizationbasedGibbssteps,asisdoneinsome algorithmsinthisdissertation,mayoftenbedesirablebecauseitimprovesspeedin ndinggoodsolutionwheretheMCMCchainmaybeslowtomix.Whilethestochastic stepseasilyhandleconstraints,discontinuities,andnonlinearitiesthatmaybeotherwise diculttohandlewithclassicaloptimization.Itdoeshoweverlosetheglobaloptimality guarantees,buttheseguaranteesareoftenmootbecausetheyonlytrulyapplyinthelimit ofinnitesamples. 187

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Animportantconsiderationwhendevelopingalgorithmsforhyperspectralimage analysisisspeed.Forevenmoderatelysizedimages,afewhundredpixelsonaside, hundredsofthousandsofpixelsarepresent,eachofwhichmaybeone-hundredormore dimensionalvector.Foroptimizationproblemsthatusesomefunctionofeachpixel, thespeedoftheimplementationofthisfunctionistheprimarybottleneckformost algorithms.Oncethefunctionhasbeenappliedandsingledimensionalmapisgenerated, thealgorithmscanoperatequitequickly.Fortheiterativealgorithmspresentedinthis work,thebottleneckistheapplicationofthedetectionstatistictoeachpixelgivenanew setofparametervalues.Anyimplementationofthesealgorithmsshouldtakeparticular caretooptimizethesedetectionfunctions,withmassivelyparallelGPUimplementations probablybeingideal. Anotheraspectofprobabilisticmodelsistheiruseinrandomdatageneration, i.e.thegenerativemodel.FortheBayesianFuzzyClusteringmodel,theFuzzyData Likelihoodgivesthatthedistributionofrandomdatagiventhesetprototypes Y anda vectoroffuzzy-setmemberships u willbeaGaussiandistribution.Thissuggeststhat randomdatasetcouldbegeneratedbythemodel Y N ; u Dirichlet X N = f u ; Y ; = g u ; Y .ItisinterestingtothinkthatrunningFuzzyClustering onadatasetgeneratedinsuchafashionmaycorrectlyndtheclusterprototypes,but seemsunlikelytondthecorrectmembershipvalues.Thisisbecause,asisshownby theBFCmodel,eachdatapointessentiallyusesamaximum-likelihoodestimateofits ownmembershipvalues,whilethegeneratingmodelhoweverhasaddedvariance g u ; Y aroundthemeanlocation f u ; Y .Statedanotherway,ifalargegroupofsampleswere assumedtoallsharethesamemembershipvector,thenthatmembershipvectorcouldbe estimatedfromthedata.Howeverthemaximumlikelihoodestimateforthatmembership vectorfromasingledatapointwillalmostsurelybeincorrect. BayesianFuzzyClusteringwasanunexpectedoutcomeofthisresearch.Itwas initiallyformulatedasawaytoapplysamplingtechniquesintheoptimizationoffuzzy 188

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clusteringoftheACEdetectorfunction.Thestraightforwardfuzzyclusteringmodelwith Euclideandistanceswasthendevelopedrsttoprovetheideas,andthenitwasfollowed byapplicationstodetectors. AftertheBFCmodelwasrealized,theIBFCmethodofestimatingthenumberof clusterswasnotanimmediatenextstep.Withthegoalinmindofdevelopingafuzzy clusteringthatlearnedthenumberofclusters,muchtimewasspentdevelopingamodel forPossibilisticclusteringwhichusedtheIndianBuetProcess.Aworkablemodelwas developedthere,andmaybecontinuedinfuturework,buttheinferencemechanismswere tooslowtovalidatethemodelevenforsmallproblems. TheIBFCapproachgrewfromfrustrationswiththatwork,withthethoughtthatthe fancysamplersfortheIndianBuetProcesswerenotneeded,whynotsimplyaveragethe likelihoodovermodelcomponentsandhaveapenaltyforlargermodels.Thiswastriedas acrazyidea,butturnedoutasthekernelofaworkablesolution. 5.2FutureWork 5.2.1DetectionMethods Mostoftheresultspresentedinthisdissertationaddressthefourrelevanttarget typefromtheGulfportdataseparately.Anothercontributionofthisworkhoweveristhe Alarm-SetFusionmethodofcombiningmultipledetectors.TheFARE-ASFalgorithm inparticularcouldbeusedtocombinethemultipledetectoroutputsintoacomplete detectionsystem.Investigationintothisideastillremainsforfuturework. TheBFACEmethodsareveryslowtondananswer,andthislimitstheirusability inrealapplication.Ideally,theestimationofthedetectormeansandcovariancescould beachievedthroughoptimizationapproachesinsteadofstochastically.Howeverdoing soforthecovariancematricesrequirespositivedeniteconstraints.Methodsknownas semideniteprogrammingaddresstheproblemforlinearobjectives[146],butoptimizing ACEwillrequireanonlineartechniquewithsemideniteconstraints. 189

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Thendingofcontextualsegmentsofimagesideallyincludessomespatialinformation aswellastheper-pixelspectralinformation.ThePFSCEMalgorithmattemptedto addressthisinformationbytheinclusionoftheFLICMterm.TheFLICMspatialterm worksverywellinalternatingoptimizationstyleapproachescommonforFCM,however itisnotusedproperlyintheoptimizationasitistreatedasaconstantwhentaking derivativeswithrespecttothemembershipparameters.ThevalueoftheFLICMconstant ishoweverre-computedateachoptimizationiteration,andsoitisnottrulyaconstantat all.InordertoapplythisterminmodelsderivingfromtheIBFCapproach,theFLICM termwillneedtobewellbehavedacrossdieringmodelsizes,andnotre-computed betweeniterations.Thus,correctlyoptimizingforthistermneedstobeinvestigated,the workbyCelikandLee[147]maybeagoodstartingpoint. ThepossibilisticclusteringmethodsusedinthePFSCEMalgorithmandtheFuzzpop [104]algorithmwhichinspireditneedmoreinvestigation.Inclusteringforbackground estimationasinthesequentialcluster-then-detectapproaches3.1,thepossibilistic weightsimprovethecontextestimationbyrejectingoutliers.Howeverwhenclustering detectoroutputsdirectly,itisnotasclearwhattheimpactsofsuchtermsare.Theywere foundtobebenecialforsyntheticdata,butnotbenecialinrealdata.Thisneedsmore investigation. TheDirichletconcentrationparametersaddedoftheBayesianFuzzyClusteringmodel bearinvestigationwhenappliedtodetectors.ForexampletheFCEMalgorithmcouldbe restatedintheBFCmanner,andthenrunforvarioussettingsoftheconcentrationparameters.OfparticularinterestarethecaseswithahighfuzzierbutDirichletconcentration lessthanone.Inthesecasestheclusteringoutputiscrispneartheclusterprototypes,but thenbecomesfuzzyafteracertainradius.Whenappliedtoadetectorthismayuseaparticulardetectorclosetotheprototypewhereitismorelikelytobecorrect,butthenuse afuzzycombinationofdetectorswhenfurtherawayfromtheprototypeandthecorrect choiceofdetectorsislesscertain. 190

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Acommontechniqueusedindetectionalgorithmsisthesliding-windowapproach. Mostanydetectionalgorithm,canbeappliedinthismanner,andtheinherentlocalization ofthewindowproducesastyleofcontextdependentdetector.However,thisapproach onlycorrectlyestimatesthecontextualeectsiftheguessofwindowsizeandshape happenstomatchtheunderlyingformofthecontextualinformation.Thismaywork wellwithlocalilluminationoratmosphericvariation,butisunlikelytoworkforthe multiplelargescaleobjectssuchastreesorbuildingswithinthescene.Allofthedetection algorithmspresentedherecouldbeappliedinaslidingwindowfashion,howeverreestimationoftheentiremodelforthewindowcentereduponeachpixelseemsimpractical. Somemethodofchoosinglocalizedsubsetsoftheimagemaybeabetterapproach,and mayhelpthesealgorithmsscaletoverylargeimagesizes.Thisisoneavenueforfuture investigation. Thedetectionmethodspresentedherehavemostlyaddressedcontextestimationwith thelinearMatchedFilterandACEdetectionstatistics.Futureworkcouldinvestigate thejointcontextanddetectoroptimizationofotherdetectionstatisticssuchastheHybrid SubpixelDetector. 5.2.2BayesianFuzzyClustering IndesigningtheIBFCmodel,wendthatwehavetradedoneparameter,the speciednumberofclusters,foranotherlessinterpretableone,themodelsparsityweight. Wemustbecarefulthatthemodelsparsityweightparameterisindeedamoreusefulone thansimplyspecifyingthenumberofclusters.Ourexperimentsshowthatasinglesetting forthesparsityweightthedimensionalityofthedata,wassucientforaccurately learningthenumberofclustersoveramoderaterange.Futureworkwillneedtofocuson thesetting,sensitivity,andinterpretationofthisparameter,orindeedevenexplorationof othermodelsparsitypromotingterms. AlsovariousmethodscanbeimplementedtoimprovetheperformanceIBFC-PF algorithm.Individualparticlescanbemarkedwhentheyconverge,sothatiterations 191

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thatdonotchangethemodelsizeofthatparticlecanbebypassed.Also,ifaparticleis representedmultipletimesintheparticleset,anditssizeisunchangedaftertherandom samplingstep,thenthosecopieswillduplicatethecomputationofthemembership update.Thisextraworkcanbeeliminated. Anotherstepthatcanbetakeninordertoreducememoryconsumptiononlarge datasetsittolimitthelistofMAPestimatestoretainonlythetop K mostlikelysizes. Additionally,ifanyofthecurrentparticlesispresentinthelistofMAPsample,then thatsampleisover-representedinthere-samplingstep,whichmayslowtheexplorationof statespaceabit.Thiscouldbeavoidedwithadditionalchecks.Finally,theparticlescan triviallybeupdatedinparalleltoeachother. Anaspectwhichmayimproveconvergenceoftheparticlelterisremovalofclusters inproportiontotheircardinality.Thuswhenalowernumberofclustersisselectedfor aparticle,theclusterscouldberandomlyselectedforremovalinproportiontotheir cardinality.Thismightalsorestrictexplorationofthesolutionspaceduetothekeepingof largebutbadclusters.Thusanalysisofsuchtechniquesisleftforfuturework. TheBFCmodelhassofaronlyaddressedcompactsphericalclusters,thatisthose modeledbyanormaldistributionwithunitcovariance.Theextensionofthemodelto ellipticalclustersisstraightforwardbyaddingthecovarianceparametersandaWishart orInverse-Wishartpriordistribution.Thepriordistributiononthecovarianceswill actasregularization,allowingclosedformupdatesofthecovariancesinaIBFC-PF likealgorithmwithouttheuseofcorrectionfactorslikethemodiedGustafson-Kessel approachescommonlyseen[148].Otherclustershapesshouldbeimplementablebyusinga singledimensionalzeromeanGaussianaroundthesquareofthedistancefunctionforthat clustershape.Futureworkshouldexploretheseapproaches. Oneaspectofprobabilisticmodelsistheiruseinrandomdatageneration,i.e.the generativemodel.FortheBayesianFuzzyClusteringmodel,theFuzzyDataLikelihood givesthatthedistributionofrandomdata,giventhesetprototypes Y andavectorof 192

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fuzzy-setmemberships u ,willbeaGaussiandistribution.Thissuggeststhatrandom datasetcouldbegeneratedbythemodel y c N ; u Dirichlet X N = f u ; Y ; = g u ; Y .ItisinterestingtothinkthatrunningFuzzyClustering onadatasetgeneratedinsuchafashionmaycorrectlyndtheclusterprototypes,but seemsunlikelytondthecorrectmembershipvalues.Thisisbecause,asisshownby theBFCmodel,eachdatapointessentiallyusesamaximum-likelihoodestimateofits ownmembershipvalues,whilethegeneratingmodelhoweverhasaddedvariance g u ; Y aroundthemeanlocation f u ; Y .Statedanotherway,ifalargegroupofsampleswere assumedtoallsharethesamemembershipvector,thenthatmembershipvectorcouldbe estimatedfromthedata.Howeverthemaximumlikelihoodestimateforthatmembership vectorfromasingledatapointwillalmostsurelybeincorrect. AnextensionofBayesianFuzzyClusteringmaybeinestimatingappropriatevalues ofthefuzzierparameter m .Traditionallythelikelihood/objectivefunctionwillimprove rapidlyasthevalueofthefuzzierisincreased,makingitdiculttocomparethevalidity oftwomodelswithdierentfuzziersbytheirlikelihoodalone.Apriordistributionfor m couldconceivablybeconstructedsuchthatthistradeoisbalanced,thoughthistermwill needtobeproportionaltothesizeofthedatasetorthedatasetlikelihoodgeometrically averagedoverthesamples,asthefuzzieraectsthelikelihoodofeachdatapoint. Anotheralternativemaybetodropthefuzziercompletelyfromthemodelandinstead controlthisdistributionofmembershipvaluesthroughtheDirichletpriorconcentration parameters. TheBayesianfuzzyclusteringtechniquesdevelopedherecancontinuetobeapplied tootherareaswhichusefuzzyclustering.Forexample,thePCOMMENDalgorithmfor endmemberestimation[97]isalikelycandidatefortheIBFCtechniques.TheIBFCEM andBFACEalgorithmsaredirectapplicationsofthethisworktowardsdetectionin hyperspectralimagery,andcontinuedworkcanbedonetoapplysuchtechniqueswith 193

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otherdetectionfunctions,toanomalydetection,ortootherareasoftargetdetection outsideofhyperspectralimagery. 5.2.3Alarm-SetFusion TheAlarm-SetFusionapproachesbearmuchcontinuedinvestigationandimprovement.ThesupervisedlearningRun-Packingalgorithmhasbeencontinuedtobeadvanced byBrandonSmock,andtheseimprovementswillbedetailedinotherco-authoredworks. Ontheunsupervisedlearningfront,theFARE-ASFalgorithmhasaquirkthathighvaluesnotfoundinthetrainingsetaresettothemaximumvalueoftheoutputintesting. Futureworkonthisalgorithmshouldndawaytosmoothlyincorporatehighvalue condencesintotheoutputsetintesting.PerhapsthelargestfuturegoalforAlarm-Set Fusionisinthepopularizationoftheconcept,sothatitmaybeadoptedbyothertarget detectioncommunitiesandseecontinuedadvancement. 5.3ConcludingRemarks Thisworkhasyieldedthreesignicantadvances:theconceptandalgorithmsfor Alarm-SetFusion,aBayesianapproachtothefuzzyclusteringproblem,andimproved algorithmsfordetectioninhyperspectralimagery. TheideaofAlarm-SetFusionisanewconceptthataddressaproblemthatfrequently occursintargetdetectionalgorithms,butthathasuntilnownotbeenrecognizedassuch. Bygivingtheproblemanameanddescribingtheconcept,algorithmscanbedevelopedto addressitandthenthosealgorithmsimprovedupon.TheRunPackingandFARE-ASF algorithmsaretherstfewtinystepsinthisdirection. TheBayesianrestatementoffuzzyclusteringhassignicantimpactonthiswork, asallofthedetectionmethodspresentedusefuzzyclustering.Fuzzyclusteringiswidely usedelsewhere,andtheBayesianmodelscouldimpactthoseworksaswell.Twoofthe mainadvantagesincludetheabilitytousestochasticapproachestolearnparameters whicharetoodiculttoupdateviastandardoptimization,andtheabilitytoestimatean appropriatenumberofclusters. 194

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Finally,theultimategoalofthisresearchwastondmethodstoimprovedetection inhyperspectralimagery.Thecontext-dependentdetectionideawasputforthasthe nextstepinaprogressiontowardsmorephysicallyaccuratemodelsofthesignalsin hyperspectralimages.Thecontextdependentdetectionmethodsdevelopedinthiswork havemetthegoalofimproveddetectorperformance,andtheywillhopefullybeastepping stoneforfurtherimprovementstocomeinthisarea. 195

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REFERENCES [1]J.Gomez,J.Blasco,E.Molto,andG.Camps-Vails,Hyperspectraldetectionof citrusdamagewithMahalanobiskernelclassier," ElectronicsLetters ,vol.43, no.20,pp.1082{1084,27-Sep.2007. [2]M.Eismann,A.Stocker,andN.Nasrabadi,Automatedhyperspectralcueingfor civiliansearchandrescue," Proc.IEEE ,vol.97,no.6,pp.1031{1055,Jun.2009. [3]denecontext,"Googlesearch,2013.[Online].Available:https: //www.google.com/search?rls=en&q=dene+context [4]M.T.Eismann, HyperspectralRemoteSensing .SPIEPress,2012. [5]F.Kruse,A.Lefko,J.Boardman,K.Heidebrecht,A.Shapiro,P.Barloon,and A.Goetz,ThespectralimageprocessingsystemSIPS{interactivevisualization andanalysisofimagingspectrometerdata," RemoteSens.Environ. ,vol.44,no.23,pp.145{163,1993. [6]S.M.Kay, FundamentalofStatisticalSignalProcessing:VolumeII{Detection Theory .Prentice-Hall,1993. [7]G.X.RitterandJ.N.Wilson, HandbookofComputerVisionAlgorithmsinImage Algebra ,2nded.CRCPress,2001. [8]P.Villeneuve,H.Fry,J.Theiler,W.Clodius,B.Smith,andA.Stocker,Improved matched-lterdetectiontechniques,"in ProceedingsofSPIE ,vol.3753,1999,p.278. [9]N.Nasrabadi,Regularizedspectralmatchedlterfortargetrecognitionin hyperspectralimagery," IEEESignalProcess.Lett. ,vol.15,pp.317{320,2008. [10]C.Funk,J.Theiler,D.Roberts,andC.Borel,Clusteringtoimprovematchedlter detectionofweakgasplumesinhyperspectralthermalimagery," IEEETrans.Geosci. RemoteSens. ,vol.39,no.7,pp.1410{1420,Jul.2001. [11]D.Manolakis,R.Lockwood,T.Cooley,andJ.Jacobson,Robustmatchedltersfor targetdetectioninhyperspectralimagingdata,"in Proc.IEEEInt.Conf.onAcoust., SpeechandSignalProcess.,2007 ,vol.1,Apr.2007,pp.I{529{I{532. [12]H.KwonandN.Nasrabadi,Kernelspectralmatchedlterforhyperspectral imagery," Int.J.Comput.Vision ,vol.71,pp.127{141,2007. [13]J.C.Harsanyi,Detectionandclassicationofsubpixelspectralsignaturesin hyperspectralimagesequences,"Ph.D.dissertation,UniversityofMarylandBaltimore County,1993. [14]C.-I.Chang, HyperspectralImaging:TechniquesforSpectralDetectionandClassication .KluwerAcademic/PlenumPublishers,2003. 196

PAGE 197

[15]W.H.FarrandandJ.C.Harsanyi,Mappingthedistributionofminetailings intheCoeurd'Alenerivervalley,Idaho,throughtheuseofaconstrainedenergy minimizationtechnique," RemoteSens.Environ. ,vol.59,no.1,pp.64{76,1997. [16]R.G.Resmini,M.E.Kappus,W.S.Aldrich,J.C.Harsanyi,andM.Anderson, MineralmappingwithHYperspectralDigitalImageryCollectionExperiment HYDICEsensordataatCuprite,Nevada,U.S.A." Int.J.ofRemoteSens. ,vol.18, no.7,pp.1553{1570,1997. [17]A.Mahalanobis,B.V.K.V.Kumar,andD.Casasent,Minimumaveragecorrelation energylters," AppliedOptics ,vol.26,no.17,pp.3633{3640,Sep.1987.[Online]. Available:http://ao.osa.org/abstract.cfm?URI=ao-26-17-3633 [18]Z.ShiandS.Yang,Robusthigh-ordermatchedlterforhyperspectraltarget detection," ElectronicsLetters ,vol.46,no.15,pp.1065{1066,22-Jul.2010. [19]H.RenandC.-I.Chang,Target-constrainedinterference-minimizedapproachto subpixeltargetdetectionforhyperspectralimages," OpticalEngineering ,vol.39, no.12,pp.3138{3145,2000. [20]C.-I.Chang,H.Ren,andS.-S.Chiang,Real-timeprocessingalgorithmsfortarget detectionandclassicationinhyperspectralimagery," IEEETrans.Geosci.Remote Sens. ,vol.39,no.4,pp.760{768,Apr.2001. [21]Q.DuandC.-I.Chang,Asignal-decomposedandinterference-annihilatedapproach tohyperspectraltargetdetection," IEEETrans.Geosci.RemoteSens. ,vol.42,no.4, pp.892{906,Apr.2004. [22]D.Manolakis,Taxonomyofdetectionalgorithmsforhyperspectralimaging applications," OpticalEngineering ,vol.44,no.6,pp.066403{066403{11,2005. [23]S.KrautandL.Scharf,TheCFARadaptivesubspacedetectorisascale-invariant GLRT," IEEETrans.SignalProcess. ,vol.47,no.9,pp.2538{2541,Sep.1999. [24]E.Kelly,Anadaptivedetectionalgorithm," IEEETrans.Aerosp.Electron.Syst. vol.AES-22,no.2,pp.115{127,1986. [25]L.ScharfandL.McWhorter,Adaptivematchedsubspacedetectorsandadaptive coherenceestimators,"in Conf.Recordofthe30thAsilomarConf.onSignals, SystemsandComputers,1996. ,Nov.1996,pp.1114{1117vol.2. [26]L.McWhorter,L.Scharf,andL.Griths,Adaptivecoherenceestimationforradar signalprocessing,"in Conf.Recordofthe30thAsilomarConf.onSignals,Systems andComputers,1996. ,vol.1,Nov.1996,pp.536{540vol.1. [27]S.Kraut,L.Scharf,andL.McWhorter,Adaptivesubspacedetectors," IEEETrans. SignalProcess. ,vol.49,no.1,pp.1{16,Jan.2001. 197

PAGE 198

[28]S.Kraut,L.Scharf,andR.Butler,Theadaptivecoherenceestimator:auniformly most-powerful-invariantadaptivedetectionstatistic," IEEETrans.SignalProcess. vol.53,no.2,pp.427{438,Feb.2005. [29]D.ManolakisandG.Shaw,Detectionalgorithmsforhyperspectralimaging applications," IEEESignalProcess.Mag. ,vol.19,no.1,pp.29{43,Jan.2002. [30]E.Crist,C.Schwartz,andA.Stocker,Pairwiseadaptivelinearmatched-lter algorithm,"in Proc.DARPAAdaptiveSpectralReconnaissanceAlgorithmWorkshop 1999. [31]P.Bajorski,Generalizeddetectionfusionforhyperspectralimages," IEEETrans. Geosci.RemoteSens. ,vol.50,no.4,pp.1199{1205,Apr.2012. [32]||,Practicalevaluationofmax-typedetectorsforhyperspectralimages," IEEEJ. Sel.TopicsAppl.EarthObserv.RemoteSens. ,vol.5,no.2,pp.462{469,Apr.2012. [33]J.HarsanyiandC.-I.Chang,Hyperspectralimageclassicationanddimensionality reduction:anorthogonalsubspaceprojectionapproach," IEEETrans.Geosci. RemoteSens. ,vol.32,no.4,pp.779{785,Jul.1994. [34]L.ScharfandB.Friedlander,Matchedsubspacedetectors," IEEETrans.Signal Process. ,vol.42,no.8,pp.2146{2157,Aug.1994. [35]T.-M.Tu,C.-H.Chen,andC.-I.Chang,Aposteriorileastsquaresorthogonal subspaceprojectionapproachtodesiredsignatureextractionanddetection," IEEE Trans.Geosci.RemoteSens. ,vol.35,no.1,pp.127{139,Jan.1997. [36]C.-I.Chang,X.-L.Zhao,M.Althouse,andJ.J.Pan,Leastsquaressubspace projectionapproachtomixedpixelclassicationforhyperspectralimages," IEEE Trans.Geosci.RemoteSens. ,vol.36,no.3,pp.898{912,May1998. [37]J.Settle,Ontherelationshipbetweenspectralunmixingandsubspaceprojection," IEEETrans.Geosci.RemoteSens. ,vol.34,no.4,pp.1045{1046,Jul.1996. [38]C.-I.Chang,Furtherresultsonrelationshipbetweenspectralunmixingandsubspace projection," IEEETrans.Geosci.RemoteSens. ,vol.36,no.3,pp.1030{1032,May 1998. [39]S.Matteoli,N.Acito,M.Diani,andG.Corsini,Anautomaticapproachtoadaptive localbackgroundestimationandsuppressioninhyperspectraltargetdetection," IEEE Trans.Geosci.RemoteSens. ,vol.49,no.2,pp.790{800,Feb.2011. [40]D.Manolakis,C.Siracusa,andG.Shaw,Hyperspectralsubpixeltargetdetection usingthelinearmixingmodel," IEEETrans.Geosci.RemoteSens. ,vol.39,no.7, pp.1392{1409,Jul.2001. 198

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[41]||,Adaptivematchedsubspacedetectorsforhyperspectralimagingapplications," in Proc.IEEEInt.Conf.onAcoust.,SpeechandSignalProcess.,2001 ,vol.5,2001, pp.3153{3156vol.5. [42]B.ThaiandG.Healey,Invariantsubpixelmaterialdetectioninhyperspectral imagery," IEEETrans.Geosci.RemoteSens. ,vol.40,no.3,pp.599{608,Mar.2002. [43]J.J.SettleandN.A.Drake,Linearmixingandtheestimationofgroundcover proportions," Int.J.RemoteSens. ,vol.14,no.6,pp.1159{1177,1993. [44]E.AshtonandA.Schaum,Algorithmsforthedetectionofsub-pixeltargetsin multispectralimagery," PhotogrammetricEng.&RemoteSens. ,vol.64,no.7,pp. 723{731,1998. [45]C.LawsonandR.Hanson, Solvingleastsquaresproblems .SIAM,1995,vol.15. [46]R.BroandS.DeJong,Afastnon-negativity-constrainedleastsquaresalgorithm," JournalofChemometrics ,vol.11,no.5,pp.393{401,1997. [47]C.-I.ChangandD.Heinz,Constrainedsubpixeltargetdetectionforremotelysensed imagery," IEEETrans.Geosci.RemoteSens. ,vol.38,no.3,pp.1144{1159,May 2000. [48]D.HeinzandC.-I.Chang,Fullyconstrainedleastsquareslinearspectralmixture analysismethodformaterialquanticationinhyperspectralimagery," IEEETrans. Geosci.RemoteSens. ,vol.39,no.3,pp.529{545,Mar.2001. [49]J.BroadwaterandR.Chellappa,Hybriddetectorsforsubpixeltargets," IEEE Trans.PatternAnal.Mach.Intell. ,vol.29,no.11,pp.1891{1903,Nov.2007. [50]J.W.Boardman,Inversionofhighspectralresolutiondata," Proc.SPIE ,vol.1298, pp.222{233,1990. [51]Y.Shimabakuro,Shadeimagesderivedfromlinearmixingmodelsofmultispectral measurementsofforestedareas,"Ph.D.dissertation,DepartmentofForestandWood Science,ColoradoStateUniversity,FortCollins,1987. [52]A.Plaza,P.Martinez,R.Perez,andJ.Plaza,Aquantitativeandcomparative analysisofendmemberextractionalgorithmsfromhyperspectraldata," IEEETrans. Geosci.RemoteSens. ,vol.42,no.3,pp.650{663,Mar.2004. [53]J.Broadwater,R.Meth,andR.Chellappa,Ahybridalgorithmforsubpixeldetection inhyperspectralimagery,"in Proc.IEEEInt.GeoscienceandRemoteSens.Symp., 2004 ,vol.3,Sep.2004,pp.1601{1604vol.3. [54]L.Zhang,B.Du,andY.Zhong,Hybriddetectorsbasedonselectiveendmembers," IEEETrans.Geosci.RemoteSens. ,vol.48,no.6,pp.2633{2646,Jun.2010. 199

PAGE 200

[55]S.MallatandZ.Zhang,Matchingpursuitswithtime-frequencydictionaries," IEEE Trans.SignalProcess. ,vol.41,no.12,pp.3397{3415,Dec.1993. [56]S.Johnson,Theconstrainedsignaldetector," IEEETrans.Geosci.RemoteSens. vol.40,no.6,pp.1326{1337,Jun.2002. [57]||,Familyofconstrainedsignaldetectorsforhyperspectralimagery," IEEETrans. Aerosp.Electron.Syst. ,vol.41,no.1,pp.34{49,Jan.2005. [58]C.M.Bishop, PatternRecognitionandMachineLearning ,1sted.Springer,Oct. 2006. [59]L.Bruce,C.Morgan,andS.Larsen,Automateddetectionofsubpixelhyperspectral targetswithcontinuousanddiscretewavelettransforms," IEEETrans.Geosci. RemoteSens. ,vol.39,no.10,pp.2217{2226,Oct.2001. [60]L.Bruce,J.Li,andY.Huang,Automateddetectionofsubpixelhyperspectral targetswithadaptivemultichanneldiscretewavelettransform," IEEETrans.Geosci. RemoteSens. ,vol.40,no.4,pp.977{980,Apr.2002. [61]M.Salvador,R.Resmini,andR.Gomez,DetectionofsulfurdioxideinAIRSdata withthewaveletpacketsubspace," IEEEGeosci.RemoteSens.Lett. ,vol.6,no.1, pp.137{141,Jan.2009. [62]H.KwonandN.Nasrabadi,Kernelorthogonalsubspaceprojectionforhyperspectral signalclassication," IEEETrans.Geosci.RemoteSens. ,vol.43,no.12,pp.2952{ 2962,Dec.2005. [63]||,Kernelmatchedsubspacedetectorsforhyperspectraltargetdetection," IEEE Trans.PatternAnal.Mach.Intell. ,vol.28,no.2,pp.178{194,Feb.2006. [64]||,Kerneladaptivesubspacedetectorforhyperspectralimagery," IEEEGeosci. RemoteSens.Lett. ,vol.3,no.2,pp.271{275,Apr.2006. [65]||,KernelRX-algorithm:anonlinearanomalydetectorforhyperspectral imagery," IEEETrans.Geosci.RemoteSens. ,vol.43,no.2,pp.388{397,Feb.2005. [66]L.Capobianco,A.Garzelli,andG.Camps-Valls,Targetdetectionwithsemisupervisedkernelorthogonalsubspaceprojection," IEEETrans.Geosci.RemoteSens. vol.47,no.11,pp.3822{3833,Nov.2009. [67]Y.Chen,N.Nasrabadi,andT.Tran,Sparserepresentationfortargetdetectionin hyperspectralimagery," IEEEJ.Sel.TopicsSignalProcess. ,vol.5,no.3,pp.629 {640,Jun.2011. [68]||,Simultaneousjointsparsitymodelfortargetdetectioninhyperspectral imagery," IEEEGeosci.RemoteSens.Lett. ,vol.8,no.4,pp.676{680,Jul.2011. 200

PAGE 201

[69]S.Goldberg,T.Glenn,J.Wilson,andP.Gader,Landminedetectionusing two-tappedjointorthogonalmatchingpursuits,"in Proc.SPIE ,vol.8357,2012,p. 83570B. [70]J.A.Tropp,A.C.Gilbert,andM.J.Strauss,Algorithmsforsimultaneoussparse approximation.PartI:Greedypursuit," SignalProcessing ,vol.86,no.3,pp.572{ 588,2006. [71]S.Cotter,B.Rao,K.Engan,andK.Kreutz-Delgado,Sparsesolutionstolinear inverseproblemswithmultiplemeasurementvectors," IEEETrans.SignalProcess. vol.53,no.7,pp.2477{2488,Jul.2005. [72]W.Sakla,A.Chan,J.Ji,andA.Sakla,AnSVDD-basedalgorithmfortarget detectioninhyperspectralimagery," IEEEGeosci.RemoteSens.Lett. ,vol.8,no.2, pp.384{388,Mar.2011. [73]J.Munoz-Marf,L.Bruzzone,andG.Camps-Vails,Asupportvectordomain descriptionapproachtosupervisedclassicationofremotesensingimages," IEEE Trans.Geosci.RemoteSens. ,vol.45,no.8,pp.2683{2692,Aug.2007. [74]A.Banerjee,P.Burlina,andR.Meth,Fasthyperspectralanomalydetectionvia SVDD,"in Proc.IEEEInt.Conf.onImageProcess. ,vol.4,Oct.2007,pp.IV{101{ IV{104. [75]A.Banerjee,R.Juang,J.Broadwater,andP.Burlina,Sparsefeatureextractionfor supportvectordatadescriptionapplications,"Jul.2010,pp.4236{4239. [76]C.-I.ChangandC.Brumbley,AKalmanlteringapproachtomultispectralimage classicationanddetectionofchangesinsignatureabundance," IEEETrans.Geosci. RemoteSens. ,vol.37,no.1,pp.257{268,Jan.1999. [77]J.W.Boardman,LeveragingthehighdimensionalityofAVIRISdataforimproved sub-pixeltargetunmixingandrejectionoffalsepositives:mixture-tunedmatched ltering," Sum.7thAnn.JPLAirborneGeosci.,JPLPublication97-21 ,vol.1,p.55, 1998. [78]J.BoardmanandF.Kruse,AnalysisofimagingspectrometerdatausingNdimensionalgeometryandamixture-tunedmatchedlteringapproach," IEEETrans. Geosci.RemoteSens. ,vol.49,no.11,pp.4138{4152,Nov.2011. [79]R.S.DiPietro,D.Manolakis,R.Lockwood,T.Cooley,andJ.Jacobson,Performance evaluationofhyperspectraldetectionalgorithmsforsubpixelobjects,"pp.76951W{ 76951W{11,2010. [80]A.SchaumandA.Stocker,Spectrally-selectivetargetdetection," Proc.Int.Symp. SpectralSensingResearch ,1997. [81]A.D.StockerandA.P.Schaum,Applicationofstochasticmixingmodelsto hyperspectraldetectionproblems," Proc.SPIE ,vol.3071,pp.47{60,1997. 201

PAGE 202

[82]P.V.Villeneuve,A.R.Boisvert,andA.D.Stocker,Hyperspectralsub-pixel targetidenticationusingleast-angleregression," Proc.SPIE ,vol.7695,pp. 76951V{76951V{11,2010. [83]A.Zare,P.Gader,J.Bolton,S.Yuksel,T.Dubroca,R.Close,andR.Hummel, Sub-pixeltargetspectraestimationanddetectionusingfunctionsofmultiple instances,"in 20113rdWorkshoponHyperspectralImageandSignalProcessing: EvolutioninRemoteSensingWHISPERS ,Jun.2011,pp.1{4. [84]G.HealeyandD.Slater,Modelsandmethodsforautomatedmaterialidentication inhyperspectralimageryacquiredunderunknownilluminationandatmospheric conditions," IEEETrans.Geosci.RemoteSens. ,vol.37,no.6,pp.2706{2717,Nov. 1999. [85]A.Schaum,Methodsofhyperspectraldetectionbasedonasinglesignaturesample," vol.10,no.3,pp.518{523,Mar.2010. [86]M.StefanouandJ.Kerekes,Amethodforassessingspectralimageutility," IEEE Trans.Geosci.RemoteSens. ,vol.47,no.6,pp.1698{1706,Jun.2009. [87]||,Image-derivedpredictionofspectralimageutilityfortargetdetection applications," IEEETrans.Geosci.RemoteSens. ,vol.48,no.4,pp.1827{1833,Apr. 2010. [88]H.LiandJ.Michels,Parametricadaptivesignaldetectionforhyperspectral imaging," IEEETrans.SignalProcess. ,vol.54,no.7,pp.2704{2715,Jul.2006. [89]N.RenardandS.Bourennane,Improvementoftargetdetectionmethodsby multiwayltering," IEEETrans.Geosci.RemoteSens. ,vol.46,no.8,pp.2407 {2417,Aug.2008. [90]S.Bourennane,C.Fossati,andA.Cailly,Improvementoftarget-detectionalgorithms basedonadaptivethree-dimensionalltering," IEEETrans.Geosci.RemoteSens. vol.49,no.4,pp.1383{1395,Apr.2011. [91]N.Acito,G.Corsini,andM.Diani,Adaptivedetectionalgorithmforfullpixel targetsinhyperspectralimages," IEEProc.-Vision,ImageandSignalProcess. ,vol. 152,no.6,pp.731{740,Dec.2005. [92]C.-I.ChangandQ.Du,Estimationofnumberofspectrallydistinctsignalsourcesin hyperspectralimagery," IEEETrans.Geosci.RemoteSens. ,vol.42,no.3,pp.608{ 619,Mar.2004. [93]J.Bioucas-Dias,A.Plaza,N.Dobigeon,M.Parente,Q.Du,P.Gader,and J.Chanussot,Hyperspectralunmixingoverview:Geometrical,statistical,andsparse regression-basedapproaches," IEEEJ.Sel.TopicsAppl.EarthObserv.Remote Sens. ,vol.5,no.2,pp.354{379,2012. 202

PAGE 203

[94]A.ZareandP.Gader,Sparsitypromotingiteratedconstrainedendmemberdetection inhyperspectralimagery," IEEEGeosci.RemoteSens.Lett. ,vol.4,no.3,pp.446 {450,Jul.2007. [95]||,PCE:Piecewiseconvexendmemberdetection," IEEETrans.Geosci.Remote Sens. ,vol.48,no.6,pp.2620{2632,Jun.2010. [96]||,Context-basedendmemberdetectionforhyperspectralimagery,"in 2009 1stWorkshoponHyperspectralImageandSignalProcessing:EvolutioninRemote SensingWHISPERS ,2009,pp.1{4. [97]A.Zare,P.Gader,O.Bchir,andH.Frigui,Piecewiseconvexmultiple-model endmemberdetectionandspectralunmixing," IEEETrans.Geosci.RemoteSens. vol.PP,no.99,pp.1{10,2012. [98]A.Zare,O.Bchir,H.Frigui,andP.Gader,Spatially-smoothpiece-wiseconvex endmemberdetection,"in 20102ndWorkshoponHyperspectralImageandSignal Processing:EvolutioninRemoteSensingWHISPERS ,2010,pp.1{4. [99]A.Zare,P.Gader,andG.Casella,Samplingpiecewiseconvexunmixingand endmemberextraction," IEEETrans.Geosci.RemoteSens. ,vol.PP,no.99,pp.1 {11,2012. [100]C.RobertandG.Casella, MonteCarloStatisticalMethods ,2nded.Springer,2005. [101]M.T.EismannandD.Stein,Stochasticmixturemodeling," Hyperspectraldata exploitation:theoryandapplications ,vol.148,2007. [102]O.Eches,N.Dobigeon,C.Mailhes,andJ.Y.Tourneret,Bayesianestimationof linearmixturesusingthenormalcompositionalmodel.Applicationtohyperspectral imagery," IEEETrans.ImageProcess. ,vol.19,no.6,pp.1403{1413,2010. [103]D.KollerandN.Friedman, ProbabilisticGraphicalModels:Principlesand Techniques .TheMITPress,2009. [104]A.ZareandP.Gader,Piece-wiseconvexspatial-spectralunmixingofhyperspectral imageryusingpossibilisticandfuzzyclustering,"in Proc.IEEEConf.FuzzySystems, 2011 ,Jun.2011,pp.741{746. [105]N.Pal,K.Pal,J.Keller,andJ.Bezdek,Apossibilisticfuzzyc-meansclustering algorithm," IEEETrans.FuzzySyst. ,vol.13,no.4,pp.517{530,2005. [106]S.KrinidisandV.Chatzis,Arobustfuzzylocalinformationc-meansclustering algorithm," IEEETrans.ImageProcess. ,vol.19,no.5,pp.1328{1337,2010. [107]M.E.AndrewandS.L.Ustin,Theroleofenvironmentalcontextinmapping invasiveplantswithhyperspectralimagedata," RemoteSens.Environ. ,vol.112, no.12,pp.4301{4317,2008. 203

PAGE 204

[108]M.Mueller,K.Segl,andH.Kaufmann,Discriminationbetweenroongmaterialsand streetswithinurbanareasbasedonhyperspectral,shape,andcontextinformation,"in 2ndGRSS/ISPRSJointWorkshoponRemoteSensingandDataFusionoverUrban Areas,2003. ,May2003,pp.196{200. [109]L.BruzzoneandL.Carlin,Amultilevelcontext-basedsystemforclassicationof veryhighspatialresolutionimages," IEEETrans.Geosci.RemoteSens. ,vol.44, no.9,pp.2587{2600,Sep.2006. [110]Q.JacksonandD.Landgrebe,AdaptiveBayesiancontextualclassicationbasedon Markovrandomelds," IEEETrans.Geosci.RemoteSens. ,vol.40,no.11,pp.2454 {2463,Nov.2002. [111]X.JiaandJ.Richards,Managingthespectral-spatialmixincontextclassication usingMarkovrandomelds," IEEEGeosci.RemoteSens.Lett. ,vol.5,no.2,pp.311 {314,Apr.2008. [112]F.MelganiandS.Serpico,AMarkovrandomeldapproachtospatio-temporal contextualimageclassication," IEEETrans.Geosci.RemoteSens. ,vol.41,no.11, pp.2478{2487,Nov.2003. [113]J.BoltonandP.Gader,Randomsetframeworkforcontext-basedclassicationwith hyperspectralimagery," IEEETrans.Geosci.RemoteSens. ,vol.47,no.11,pp.3810 {3821,Nov.2009. [114]C.Ratto,P.Torrione,andL.Collins,Exploitingground-penetratingradar phenomenologyinacontext-dependentframeworkforlandminedetectionand discrimination," IEEETrans.Geosci.RemoteSens. ,vol.49,no.5,pp.1689{1700, May2011. [115]J.C.SchlimmerandR.H.Granger,Incrementallearningfromnoisydata," Machine learning ,vol.1,no.3,pp.317{354,1986. [116]G.WidmerandM.Kubat,Learninginthepresenceofconceptdriftandhidden contexts," Machinelearning ,vol.23,no.1,pp.69{101,1996. [117]A.Tsymbal,Theproblemofconceptdrift:denitionsandrelatedwork," Computer ScienceDepartment,TrinityCollegeDublin ,2004. [118]P.Torrione,C.Ratto,andL.Collins,Multipleinstanceandcontextdependent learninginhyperspectraldata,"in 20091stWorkshoponHyperspectralImageand SignalProcessing:EvolutioninRemoteSensingWHISPERS ,Aug.2009,pp.1{4. [119]K.Morton,P.Torrione,andL.Collins,Dirichletprocessbasedcontextlearningfor minedetectioninhyperspectralimagery,"in 20102ndWorkshoponHyperspectral ImageandSignalProcessing:EvolutioninRemoteSensingWHISPERS ,Jun.2010, pp.1{4. 204

PAGE 205

[120]J.BoltonandP.Gader,Thebenetsofcontextestimationfortargetspectra detectioninhyperspectralimagery,"in Proc.IEEEInt.GeoscienceandRemote Sens.Symp.,2008 ,vol.2,Jul.2008,pp.II{363{II{366. [121]||,Arandommeasureapproachforcontextestimationinhyperspectralimagery," in 20091stWorkshoponHyperspectralImageandSignalProcessing:Evolutionin RemoteSensingWHISPERS ,2009,pp.1{4. [122]R.A.Jacobs,M.I.Jordan,S.J.Nowlan,andG.E.Hinton,Adaptivemixturesof localexperts," Neuralcomputation ,vol.3,no.1,pp.79{87,1991. [123]M.I.JordanandR.A.Jacobs,HierarchicalmixturesofexpertsandtheEM algorithm," Neuralcomputation ,vol.6,no.2,pp.181{214,1994. [124]S.Yuksel,G.Ramachandran,P.Gader,J.Wilson,D.Ho,andG.Heo,Hierarchical methodsforlandminedetectionwithwidebandelectro-magneticinductionandground penetratingradarmulti-sensorsystems,"in Proc.IEEEInt.GeoscienceandRemote Sens.Symp.,2008 ,vol.2,2008,pp.II{177{II{180. [125]L.I.Kuncheva,'Change-glasses'approachinpatternrecognition," Pattern RecognitionLetters ,vol.14,no.8,pp.619{623,1993. [126]L.Kuncheva,Switchingbetweenselectionandfusionincombiningclassiers:an experiment," IEEETrans.Syst.,Man,Cybern.B ,vol.32,no.2,pp.146{156,Apr. 2002. [127]K.Woods,J.Kegelmeyer,W.P.,andK.Bowyer,Combinationofmultipleclassiers usinglocalaccuracyestimates," IEEETrans.PatternAnal.Mach.Intell. ,vol.19, no.4,pp.405{410,Apr.1997. [128]H.Frigui,L.Zhang,andP.Gader,Context-dependentmultisensorfusionandits applicationtolandminedetection," IEEETrans.Geosci.RemoteSens. ,vol.48, no.6,pp.2528{2543,Jun.2010. [129]H.FriguiandS.Salem,Fuzzyclusteringandsubsetfeatureweighting,"in Proc. IEEEConf.FuzzySystems,2003 ,vol.2.IEEE,2003,pp.857{862. [130]L.Zhang,H.Frigui,P.Gader,andJ.Bolton,Context-dependentfusionfor minedetectionusingairbornehyperspectralimagery,"in 20091stWorkshop onHyperspectralImageandSignalProcessing:EvolutioninRemoteSensing WHISPERS ,2009,pp.1{4. [131]H.Frigui,P.Gader,andA.B.Abdallah,Agenericframeworkforcontext-dependent fusionwithapplicationtolandminedetection,"in SPIEDefenseandSecurity Symposium ,2008,pp.490{495. [132]J.C.Bezdek, Patternrecognitionwithfuzzyobjectivefunctionalgorithms .Kluwer AcademicPublishers,1981. 205

PAGE 206

[133]A.Abdallah,H.Frigui,andP.Gader,Contextextractionforlocalfusionusingfuzzy clusteringandfeaturediscrimination,"in Proc.IEEEConf.FuzzySystems,2009 2009,pp.490{495. [134]G.Heo,P.Gader,andH.Frigui,Anoiserobustvariantofcontextextractionfor localfusion,"in Proc.IEEEConf.FuzzySystems,2010 ,Jul.2010,pp.1{6. [135]A.Abdallah,H.Frigui,andP.Gader,Adaptivelocalfusionwithfuzzyintegrals," IEEETrans.FuzzySyst. ,vol.20,no.5,pp.849{864,Oct.2012. [136]I.GathandA.Geva,Unsupervisedoptimalfuzzyclustering," IEEETrans.Pattern Anal.Mach.Intell. ,vol.11,no.7,pp.773{780,1989. [137]H.FriguiandR.Krishnapuram,Clusteringbycompetitiveagglomeration," Pattern Recognition ,vol.30,no.7,pp.1109{1119,1997. [138]M.Li,M.Ng,Y.-M.Cheung,andJ.Huang,AgglomerativefuzzyK-meansclustering algorithmwithselectionofnumberofclusters," IEEETrans.Knowl.DataEng. vol.20,no.11,pp.1519{1534,2008. [139]C.E.Rasmussen,Theinnitegaussianmixturemodel."in NIPS ,vol.12,1999,pp. 554{560. [140]T.GrithsandZ.Ghahramani,Innitelatentfeaturemodelsandtheindianbuet process,"GatsbyUnit,Tech.Rep.TR2005001,2005. [141]M.IsardandA.Blake,CONDENSATION|conditionaldensitypropagationforvisualtracking," Int.J.ComputerVision ,vol.29,pp.5{28,1998, 10.1023/A:1008078328650. [142]J.NascimentoandJ.BioucasDias,Vertexcomponentanalysis:afastalgorithmto unmixhyperspectraldata," IEEETrans.Geosci.RemoteSens. ,vol.43,no.4,pp. 898{910,2005. [143]A.Martinez-Uso,F.Pla,J.Sotoca,andP.Garcia-Sevilla,Clustering-based hyperspectralbandselectionusinginformationmeasures," IEEETrans.Geosci. RemoteSens. ,vol.45,no.12,pp.4158{4171,Dec.2007. [144]S.TheodoridisandK.Konstantinos, PatternRecognition ,2nded.ElsevierAcademic Press,2003. [145]D.Anderson,A.Zare,andS.Price,Comparingfuzzy,probabilistic,andpossibilistic partitionsusingtheearthmover'sdistance," IEEETrans.FuzzySyst. ,vol.21,no.4, pp.766{775,2013. [146]S.BoydandL.Vandenberghe, ConvexOptimization .CambridgeUniversityPress, 2004. 206

PAGE 207

[147]T.CelikandH.K.Lee,CommentsonArobustfuzzylocalinformationC-means clusteringalgorithm"," IEEETrans.ImageProcess. ,vol.22,no.3,pp.1258{1261, 2013. [148]R.Babuska,P.J.VanderVeen,andU.Kaymak,Improvedcovarianceestimation forGustafson-Kesselclustering,"in Proc.IEEEConf.FuzzySystems,2002 ,vol.2, 2002,pp.1081{1085. 207

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BIOGRAPHICALSKETCH TaylorGlennreceivedaBachelorofScienceinComputerEngineeringinMay2003, andaMasterofEngineeringdegreeincomputerengineeringinDecember2004,bothfrom theUniversityofFlorida.Afterhismasters'degreeheco-founded2GEngineeringLLC, acompanyspecializinginembeddedsystemsdesignandquickturnmanufacturing.In 2009hereturnedtotheUniversityofFloridaforhisPhDandgraduatedwithaDoctorof PhilosophyincomputerengineeringinDecember2013.Hisresearchworkhasspecialized inpatternrecognitionandmachinelearningandtheapplicationofthesemethodsto sensordata,inparticulargroundpenetratingradarandelectromagneticinductiondatafor buriedlandminedetection,andhyperspectralimageanalysis. 208