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Hyperspectral Endmember Detection and Band Selection Using Bayesian Methods

Permanent Link: http://ufdc.ufl.edu/UFE0022888/00001

Material Information

Title: Hyperspectral Endmember Detection and Band Selection Using Bayesian Methods
Physical Description: 1 online resource (140 p.)
Language: english
Creator: Zare, Alina
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bayesian, dimensionality, dirichlet, endmember, hyperspectral, sparsity, spectra, spectral
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Four methods for endmember detection and spectral unmixing which estimate endmember spectra and proportion values for each pixel are described. The first method simultaneously determines the number of endmembers in addition to estimating endmember spectra and proportion values. The second method treats endmembers as distributions and estimates each endmember distribution while simultaneously learning proportion values. The third endmember detection method autonomously partitions the input data set into convex regions for which endmember distributions and proportion values are simultaneously estimated. The fourth method which performs hyperspectral band selection in addition to endmember detection, spectral unmixing, and determinating of the number of endmembers is also described. Few endmember detection algorithms estimate the number of endmembers in addition to determining their spectral shape. Also, methods which treat endmembers as distributions or treat hyperspectral images as piece-wise convex data sets have not been previously developed. A hyperspectral image is a three-dimensional data cube containing radiance values collected over an area (or scene) in a range of wavelengths. Endmember detection and spectral unmixing attempt to decompose a hyperspectral image into the pure - separate and individual - spectral signatures of the materials in a scene, and the proportions of each material at every pixel location. Each spectral pixel in the image can then be approximated by a convex combination of proportions and endmember spectra. The first method described is the Sparsity Promoting Iterated Constrained Endmembers (SPICE) algorithm, which incorporates sparsity-promoting priors to estimate the number of endmembers. The algorithm is initialized with a large number of endmembers. The sparsity promotion process drives all proportions of some endmembers to zero. These endmembers can be removed by SPICE with no effect on the error incurred by representing the image with endmembers. The second method, the Endmember Distributions detection (ED) algorithm, models each endmember as a distribution rather than a single spectrum. This view can incorporate an endmember's spectral variation which may occur due to varying environmental conditions as well as inherent variability in a material. The third method is the Piece-wise Convex Endmember (PCE) detection algorithm which partitions the input hyperspectral data set into convex regions and determines endmembers for each of these regions. The number of convex regions are determined autonomously using the Dirichlet process while simultaneously estimating endmember distributions and proportion values for each pixel in the input data set. The SPICE, ED and PCE algorithms are effective at handling highly-mixed hyperspectral images where all of the pixels in the scene contain mixtures of multiple endmembers. These methods are capable of extracting endmember spectra from a scene that does not contain pure pixels composed of only a single endmember's material. Furthermore, the methods conform to the Convex Geometry Model for hyperspectral imagery. This model requires that the proportions associated with an image pixel be non-negative and sum to one. The fourth method is known as the Band Selecting Sparsity Promoting Iterated Constrained Endmember (B-SPICE) algorithm and is an extension of SPICE that performs hyperspectral band selection in addition to all of SPICE's endmember detection and spectral unmixing features. This method applies sparsity promoting priors to discard those hyperspectral bands which do not aid in distinguishing between endmembers in a data set. Results indicate that SPICE and B-SPICE consistently produce the correct number of endmembers and the correct spectral shape for each endmember. The B-SPICE algorithm is shown to significantly decrease the number of hyperspectral bands while maintaining competitive classification accuracy for a data set. The ED algorithm results indicate that the algorithm produces accurate endmembers and can incorporate spectral variation into the endmember representation. The PCE algorithm results on hyperspectral data indicate that PCE produces endmember distributions which represent the true ground truth classes of the input data set.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alina Zare.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Gader, Paul D.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022888:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022888/00001

Material Information

Title: Hyperspectral Endmember Detection and Band Selection Using Bayesian Methods
Physical Description: 1 online resource (140 p.)
Language: english
Creator: Zare, Alina
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bayesian, dimensionality, dirichlet, endmember, hyperspectral, sparsity, spectra, spectral
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Four methods for endmember detection and spectral unmixing which estimate endmember spectra and proportion values for each pixel are described. The first method simultaneously determines the number of endmembers in addition to estimating endmember spectra and proportion values. The second method treats endmembers as distributions and estimates each endmember distribution while simultaneously learning proportion values. The third endmember detection method autonomously partitions the input data set into convex regions for which endmember distributions and proportion values are simultaneously estimated. The fourth method which performs hyperspectral band selection in addition to endmember detection, spectral unmixing, and determinating of the number of endmembers is also described. Few endmember detection algorithms estimate the number of endmembers in addition to determining their spectral shape. Also, methods which treat endmembers as distributions or treat hyperspectral images as piece-wise convex data sets have not been previously developed. A hyperspectral image is a three-dimensional data cube containing radiance values collected over an area (or scene) in a range of wavelengths. Endmember detection and spectral unmixing attempt to decompose a hyperspectral image into the pure - separate and individual - spectral signatures of the materials in a scene, and the proportions of each material at every pixel location. Each spectral pixel in the image can then be approximated by a convex combination of proportions and endmember spectra. The first method described is the Sparsity Promoting Iterated Constrained Endmembers (SPICE) algorithm, which incorporates sparsity-promoting priors to estimate the number of endmembers. The algorithm is initialized with a large number of endmembers. The sparsity promotion process drives all proportions of some endmembers to zero. These endmembers can be removed by SPICE with no effect on the error incurred by representing the image with endmembers. The second method, the Endmember Distributions detection (ED) algorithm, models each endmember as a distribution rather than a single spectrum. This view can incorporate an endmember's spectral variation which may occur due to varying environmental conditions as well as inherent variability in a material. The third method is the Piece-wise Convex Endmember (PCE) detection algorithm which partitions the input hyperspectral data set into convex regions and determines endmembers for each of these regions. The number of convex regions are determined autonomously using the Dirichlet process while simultaneously estimating endmember distributions and proportion values for each pixel in the input data set. The SPICE, ED and PCE algorithms are effective at handling highly-mixed hyperspectral images where all of the pixels in the scene contain mixtures of multiple endmembers. These methods are capable of extracting endmember spectra from a scene that does not contain pure pixels composed of only a single endmember's material. Furthermore, the methods conform to the Convex Geometry Model for hyperspectral imagery. This model requires that the proportions associated with an image pixel be non-negative and sum to one. The fourth method is known as the Band Selecting Sparsity Promoting Iterated Constrained Endmember (B-SPICE) algorithm and is an extension of SPICE that performs hyperspectral band selection in addition to all of SPICE's endmember detection and spectral unmixing features. This method applies sparsity promoting priors to discard those hyperspectral bands which do not aid in distinguishing between endmembers in a data set. Results indicate that SPICE and B-SPICE consistently produce the correct number of endmembers and the correct spectral shape for each endmember. The B-SPICE algorithm is shown to significantly decrease the number of hyperspectral bands while maintaining competitive classification accuracy for a data set. The ED algorithm results indicate that the algorithm produces accurate endmembers and can incorporate spectral variation into the endmember representation. The PCE algorithm results on hyperspectral data indicate that PCE produces endmember distributions which represent the true ground truth classes of the input data set.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alina Zare.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Gader, Paul D.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022888:00001


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Iwouldliketothankmyadvisor,Dr.PaulGader,forallofhisguidance,supportandthenumerousopportunitiesheprovidedmethroughoutmystudiesandresearch.Iwouldalsoliketothankmycommitteemembers,Dr.JeeryHo,Dr.GerhardRitter,Dr.ClintSlatton,andDr.JosephWilson,foralloftheirhelpandvaluablesuggestions.ThankyoutoMirandaSchattenSilviousofNVESD,Dr.RussellHarmonofARL/ARO,Dr.WilliamClarkofARO,andDr.MichaelCathcartofGTRIfortheirsupportthroughoutthisresearch.Additionally,thankyoutomymanyformerandcurrentlabmates.IamparticularlygratefultoJeremyBoltonandXupingZhangforthecountlessnumberoftimestheyprovidedinsightanddiscussionduringmystudies.Thankyoutomyparents,RoobikandEmikZare,mysister,AnahitaZare,myboyfriend,MichaelBlack,andallofmyfamilyfortheirnever-endinglove,supportandunderstanding.Finally,manythankstomygodfather,ArtooshAvanessian,whoseinuencerstledmetocomputerscience. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 LISTOFSYMBOLSANDABBREVIATIONS ..................... 10 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 14 1.1HyperspectralImageDataandAnalysis ................... 14 1.1.1EndmemberDetection ......................... 15 1.1.2HyperspectralBandSelection ..................... 17 1.2StatementofProblem ............................. 17 1.3OverviewofResearch .............................. 18 2LITERATUREREVIEW .............................. 20 2.1ExistingEndmemberDetectionAlgorithms .................. 20 2.1.1PixelPurity ............................... 20 2.1.2ConvexHull ............................... 23 2.1.3NonnegativeMatrixFactorization ................... 25 2.1.4MorphologicalAssociativeMemories .................. 30 2.1.5EvolutionarySearch ........................... 35 2.1.6IndependentComponentsAnalysis ................... 36 2.1.7EstimatingtheNumberofEndmembers ................ 38 2.2ExistingHyperspectralBandSelectionAlgorithms .............. 44 2.3SummaryofLiteratureReview ........................ 47 3TECHNICALAPPROACH ............................. 56 3.1ReviewofSparsityPromotionTechniques .................. 57 3.2ReviewoftheIteratedConstrainedEndmembersDetectionAlgorithm ... 58 3.3NewEndmemberDetectionAlgorithmUsingSparsityPromotingPriors .. 60 3.4NewBandSelectionAlgorithmUsingSparsityPromotingPriors ...... 62 3.5NewEndmemberDistributionDetectionAlgorithm ............. 65 3.6ReviewofMarkovChainMonteCarloSamplingAlgorithms ........ 69 3.7ReviewoftheDirichletDistributionandtheDirichletProcess ....... 71 3.7.1DirichletProcessMixtureModel .................... 73 3.7.2GibbsSamplingfortheDirichletProcessMixtureModel ...... 74 5

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...................................... 76 4RESULTS ....................................... 82 4.1SparsityPromotingIteratedConstrainedEndmember(SPICE)AlgorithmResults ...................................... 82 4.1.1TheSPICETwo-DimensionalExampleResults ............ 82 4.1.2TheSPICEAVIRISCupriteDataResults .............. 83 4.1.3TheSPICEAVIRISIndianPinesResults ............... 85 4.1.4TheSPICEAHIVegetationDetectionResults ............ 86 4.2BandSelectingSPICE(B-SPICE)AlgorithmResults ............ 91 4.2.1TheB-SPICEAVIRISCupriteDataResults ............. 91 4.2.2TheB-SPICEAVIRISIndianPinesResults .............. 92 4.2.3TheB-SPICEAVIRISIndianPinesResultsusingSampledParameterValues .................................. 94 4.3EndmemberDistribution(ED)DetectionResults .............. 95 4.3.1ResultsonTwo-DimensionalDatausingED ............. 95 4.3.2ResultsonAVIRISCupritedatausingED .............. 96 4.4Piece-wiseConvexEndmember(PCE)DetectionResults .......... 96 4.4.1DetectionResultsonTwo-DimensionalDatausingPCE ....... 96 4.4.2DetectionResultsontheAVIRISIndianPinesDatausingPCE .. 97 5CONCLUSION .................................... 130 REFERENCES ....................................... 132 BIOGRAPHICALSKETCH ................................ 140 6

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Table page 4-1NumberofendmembersfoundbySPICEandICEonAVIRISCupritedata ... 102 4-2MinedistributionsinoverlapregionsofAHIandLynximagery ......... 102 4-3FalsealarmratereductionusingblackbodymaskinAHIimage1 ........ 102 4-4FalsealarmratereductionusingblackbodymaskinAHIimage2 ........ 102 4-5FalsealarmratereductionusingblackbodymaskinAHIimage3 ........ 103 4-6MeanandstandarddeviationofthenumberofendmembersandbandsretainedusingSPICEandB-SPICEonthesimulatedAVIRISCupritedataset ..... 103 4-7StatisticsoftheaveragedsquarederrorperabundancevalueusingSPICEandB-SPICE ....................................... 103 4-8IndianPinesdatasetclassicationresultsusingSPICEandB-SPICE ...... 104 4-9TheAVIRISIndianPinesdataB-SPICEresultsusingsampledparametervalues 104 4-10ParametervaluesusedtogenerateEDresultsontwo-dimensionaldatasets ... 104 4-11ParametervaluesusedtogenerateEDresultsonhyperspectraldatasets .... 104 4-12ParametervaluesusedtogeneratePCEresults .................. 105 7

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Figure page 2-1Three-dimensionaldatapointsandendmemberresultsusingconvexconeanalysis 49 2-2SubsetofAVIRISCupritedatascene ........................ 49 2-3TheCCAendmemberresultsonAVIRISCupritedata .............. 50 2-4Morphologicalassociativememoriesendmemberdetectionresultsusingtheminmemoryontwo-dimensionaldata .......................... 50 2-5Morphologicalassociativememoriesendmemberdetectionresultsusingbothmemoriesontwo-dimensionaldata ......................... 51 2-6ThreedimensionaldatasetgeneratedfromtwoendmemberswithGaussiannoise. 52 2-7NormalizedAVIRISCupritespectra ........................ 53 2-8DatasetgeneratedfromAVIRISCupriteendmemberswithasmallamountofGaussiannoise .................................... 54 2-9DatasetgeneratedfromAVIRISCupriteendmemberswithalargeamountofGaussiannoise .................................... 55 3-1Endmemberdistributionalgorithm'sabundanceprior ............... 80 3-2Endmemberdistributionalgorithm'sabundancepriorasafunctionofc 80 3-3Datapointsgeneratedfromtwoendmemberdistributions ............. 81 4-1Two-dimensionalexampledataset ......................... 105 4-2ComparisonofICEandSPICEalgorithmresultsontwo-dimensionaldata ... 105 4-3TheSPICEresultsontwo-dimensionaldata .................... 106 4-4EndmembersfoundusingSPICEonAVIRISCupritedata ............ 106 4-5ComparisonofSPICEendmemberandUSGSAlunitespectrum ......... 107 4-6SelectedendmembersfromAVIRISCupritedata ................. 107 4-7TestpixelsfromAVIRISCupritedata ....................... 108 4-8EndmemberSPICEresultsonAVIRISCupritescene ............... 108 4-9TheAVIRISIndianPinesdatasetandgroundtruth ............... 109 4-10AbundancemapsonAVIRISIndianPinesdatageneratedbySPICE ...... 110 4-11AbundancemapsonlabeledAVIRISIndianPinesdatageneratedbySPICE .. 111 8

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. 112 4-13SubsetofAHIhyperspectralimage ......................... 113 4-14VegetationmasksforAHIhyperspectralimagesgeneratedusingSPICE ..... 114 4-15SelectedAVIRISCupriteendmembers ....................... 115 4-16TheBSPICEAVIRISCupritesimulateddataendmemberresults ........ 115 4-17HistogramsofthenumberofendmembersandbandsfoundusingB-SPICE ... 116 4-18Resultsontwo-dimensionaltriangledatafoundusingED ............. 116 4-19Datapointsgeneratedfromthreeendmemberdistributions ............ 117 4-20ResultsontwodimensionaldatausingED ..................... 117 4-21ResultsontwodimensionaldatageneratedfromendmemberdistributionsusingSPICE ......................................... 118 4-22ResultsonsimulatedAVIRISCupritedatafoundusingED ............ 118 4-23ResultsonsubsetofAVIRISCupritedatafoundusingED ............ 119 4-24ResultsonsubsetofAVIRISCupritedatafoundusingED ............ 120 4-25Two-dimensionaldatageneratedfromthreesetsofendmembers ......... 120 4-26Two-dimensionaldataresultsfoundusingPCE .................. 121 4-27AbundancemapsfoundusingPCEonlabeledPCA-reducedAVIRISIndianPinesdata .......................................... 122 4-28HistogramsofPCEendmemberresultsonlabeledPCA-reducedAVIRISIndianPinesdata ....................................... 123 4-29AbundancemapsfoundusingSPICEonlabeledPCA-reducedAVIRISIndianPinesdata ....................................... 124 4-30HistogramsofSPICEendmemberresultsonlabeledPCA-reducedAVIRISIndianPinesdata ....................................... 125 4-31AbundancemapsfoundusingPCEonlabeledAVIRISIndianPinesdatawithhierarchicaldimensionalityreduction ........................ 126 4-32HistogramsofPCEendmemberresultsonlabeledAVIRISIndianPinesdatawithhierarchicaldimensionalityreduction ..................... 127 4-33AbundancemapsfoundusingPCEonlabeledAVIRISIndianPinesdata .... 128 4-34HistogramsofPCEendmemberresultsonlabeledAVIRISIndianPinesdata .. 129 9

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AMEE Automatedmorphologicalendmemberextractionalgorithm AVIRIS Airbornevisible/infraredimagingspectrometer BSPICE Bandselectingsparsitypromotingiteratedconstrainedendmemberdetectionalgorithm CCA Convexconeanalysis DECA Dependentcomponentsanalysis DPEM Dirichletprocessendmemberdetectionalgorithm DWT Discretewavelettransform Endmemberdistributiondetectionalgorithm EMD Euclideanminimumdistance FCM Fuzzyc-meansclusteringalgorithm HFC Harsanyi-Farrand-Changmethod ICA Independentcomponentsanalysis ICE Iteratedconstrainedendmembersalgorithm iid Independentlyandidenticallydistributed MCMC Markovchainmontecarlo MEI Morphologicaleccentricityindex MNF Maximumnoisefractiontransform MVC-NMF Minimumvolumeconstrainednon-negativematrixfactorizationalgorithm MVT Minimumvolumetransform NATGD Noise-adjustedtransformedGerschogorindisk NMF Non-negativematrixfactorization 10

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Noisesubspaceprojectionmethod NWHFC NoisewhitenedHarsanyi-Farrand-Changmethod Principalcomponentstransform PCE Piece-wiseconvexendmemberdetectionalgorithm PNACP Partitionednoise-adjustedprincipalcomponentsalgorithm PPI Pixelpurityindex RVM Relevancevectormachine SAM Spectralanglemapper SIE Singleindividualevolutionarystrategy SPICE Sparsitypromotingiteratedconstrainedendmembersalgorithm SSEE Spatial-spectralendmemberextractionalgorithm SVD Singularvaluedecomposition TGD TransformedGerschogorindisk VD Virtualdimensionality 11

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KeshavaandMustard 2002 ; Manolakis,Marden,andShaw 2003 ).Radiancemeasuredbyahyperspectralsensorisacombinationofradiationthatisreectedand/oremittedbymaterialsontheground( Manolakisetal. 2003 ).Inpassivesystems,thereectedportionofthesignalistheamountofradiationreectedfromsunlightshiningongroundmaterials( KeshavaandMustard 2002 ).Theatmospherebetweenthesensorandmaterialsonthegroundaectstheradiancemeasurements.Watervaporandoxygenintheatmospherecausethelargesteect.Incertainwavelengths,thoseknownasabsorptionbands,watervaporandoxygenabsorbalargeportionofthesignal,causingpoorsignal-to-noiseratios( Manolakisetal. 2003 ).Inadditiontoabsorptioncharacteristics,thedierentwavelengthsacrosswhichradiancecanbemeasuredhavevaryingproperties.Forexample,inthe0.4to2.5mrange,sunlightoranotheractiveilluminationsourceisneededsincereectedradiancedominatesthisportionofthespectrum.Incontrast,thethermalinfraredregionfrom8to14misdominatedbyemittedradianceandcan,therefore,bemeasuredduringthenightwithoutanactiveilluminationsource( Manolakisetal. 2003 ).Themainappealforhyperspectralimagingistheconceptthatdierentmaterialsreectandemitvaryingamountsofradianceacrosstheelectromagneticspectrum.Inotherwords,dierentmaterialsgenerallyhaveuniquespectralsignatures.Itisforthis 14

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Manolakisetal. 2003 ).Twoimportantcharacteristicsofahyperspectralsensorareitsspectralandspatialresolution.Spectralresolutionofasensorcorrespondstotherangeofwavelengthsoverwhichradiancevaluesaremeasuredandcombinedtobecomeasinglebandinahyperspectralimage.Spatialresolutioncorrespondstothesizeofthephysicalareaonthegroundfromwhichradiancemeasurementsaretakenforasingleimagepixel.Astheareacorrespondingtoapixelincreases,thespatialresolutionoftheimagedecreases( KeshavaandMustard 2002 ; Manolakisetal. 2003 ).Forairbornesystems,spatialresolutionisgenerallyconstantacrossanimage.However,formanyforward-lookingground-basedsystems,thespatialresolutionmayvarywithinanimage.Thevaryingspatialresolutionisaresultoftheanglefromwhichahyperspectralsensorimagesaregion.Pixelsclosertothesensorhavehigherspatialresolutionthanthosefartheraway.Spatialresolutionisoneofthecausesofmixedpixelsinahyperspectraldataset( KeshavaandMustard 2002 ; Manolakisetal. 2003 ).Amixedpixelisapixelwhichcombinestheradiancevaluesofmultiplematerials.Apurepixelcorrespondstoasinglematerial'sradiancevalues.Mixedpixelscanoccurfromlowspatialresolutionsince,asapixel'scorrespondingareaonthegroundincreases,neighboringmaterialsarelikelytobecombinedintotheimagepixel.Mixedpixelsalsooccurwhenthedierentmaterialsaremixedontheground.Beachsandisacommonexampleforthistypeofmixedpixelsincegrainsofdierentmaterialsareintermingled( KeshavaandMustard 2002 ). KeshavaandMustard 2002 ).Duetothepresenceofmixedpixelsinahyperspectralimage,spectralunmixingisoftenperformedtodecomposemixedpixelsintotheirrespectiveendmembersandabundances.Abundancesaretheproportionsof 15

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KeshavaandMustard 2002 ; NascimentoandBioucas-Dias 2005a ).Theconvexgeometrymodelassumesthateverypixelisaconvexcombinationofendmembersinthescene.ThismodelcanbewrittenasshowninEquation 1{1 ( KeshavaandMustard 2002 ; Manolakisetal. 2003 ; NascimentoandBioucas-Dias 2005a ), 1{2 KeshavaandMustard 2002 ).Theendmemberdetectionproblemisthetaskofdeterminingthepurespectralsignaturesinagivenhyperspectralscene.Endmemberdetectionalgorithmsoftenassumetheconvexgeometrymodelandperformspectralunmixingtoreturntheendmembersandabundancesinanimage( KeshavaandMustard 2002 ). 16

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Manolakisetal. 2003 ).Althoughtheresolutionprovidedallowsfortheextractionofmaterialspectra,thevolumeofdataposesmanychallengingproblemssuchasdatastorage,computationaleciency,andthecurseofdimensionality( Chang,Du,Sun,andAlthouse 1999 ; HuangandHe 2005 ).Onemethodtoovercomethesechallengesistheuseofdatareductiontechniques( HuangandHe 2005 ).Hyperspectralbandselectionisonemethodofdatareductionthatalsoretainsthephysicalmeaningofthedataset( Guo,Gunn,Damper,andNelson 2006 ).Hyperspectralbandselectionselectsasetofbandsfromtheinputhyperspectraldatasetwhichretaintheinformationneededforsubsequenthyperspectralimagespectroscopy. 1{2 .Existingendmemberdetectionalgorithmsrepresentendmembersassinglespectralpoints,whichdoesnotincorporatethespectralvariabilitythatoccursduetodieringenvironmentalconditions.Furthermore,existingendmemberdetectionalgorithmsgenerallyassumethatthehyperspectraldatapointslieinasingleconvexregionwithonesetofendmembers.However,itmaybethecasethatmultiplesetsofendmembers,deningseveraloverlappingconvexregions,canbetterdescribethehyperspectralimage.Existinghyperspectralbandselectionalgorithmsoftenrequireaninputofthenumberofhyperspectralbandstoberetained.Furthermore,manyhyperspectraldata 17

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1{1 andthusconstraintheproportionvaluestobenon-negativeandsumtoone.Thegeneralapproachinvolvesintegrating 18

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1{1 ( KeshavaandMustard 2002 ).Mostexistingalgorithmsrequireadvanceknowledgeofthenumberofendmembersinagivenscene.However,thisvalueisoftenunknownforagivendataset.Severalmethodsmakethepixelpurityassumptionandassumethatpurepixelsexistintheinputdatasetforeveryendmemberinthescene.Thisassumptioncausesalgorithmstobeinaccurateforhighly-mixeddatasetswherepurepixelsforeachmaterialcannotbefoundintheimagery.Additionally,somemethodsdonotencompassallofthedatapointsand,therefore,eitherpreventspectralunmixingwithabundancevaluesthatconformtotheconstraintsinEquation 1{2 orhavelargereconstructionerrorsusingtheestimatedendmemberandabundancematrices.Theexistingmethodsgenerallyrepresenteachendmemberasasinglespectrumwhichdoesnotaccountforthespectralvariationthatmayoccurduetovaryingenvironmentalconditions.Themajorityofthesemethodsalsoassumethatthehyperspectraldatapointslieinasingleconvexregionandcanbedescribedbyasinglesetofendmemberswhichencompassthedataset.Inthischapter,asummaryofmanyoftheseexistingendmemberdetectionalgorithmsisprovided. 20

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Winter 1999 )andthePixelPurityIndexalgorithm( Boardman,Kruse,andGreen 1995 )bothofwhicharedescribedindetailbelow.Additionally,theAutomatedMorphologicalEndmemberExtraction(AMEE)algorithmdenesmultispectraldilationanderosionoperatorsusedtocomputethemor-phologicaleccentricityindex(MEI)( Plaza,Martinez,Perez,andPlazas 2002 ).TheMEIisusedtoidentifyspectrallypurepixelsintheimagewhicharereturnedasendmembers( Plazaetal. 2002 ).TheSpatial-SpectralEndmemberExtraction(SSEE)algorithmprojectstheimageontoeigenvectorscomputedfromtheSingularValueDecomposition(SVD)ofsubsetsintheinputdataset( Rogge,Rivard,Zhang,Sanchez,Harris,andFeng 2007 ).SSEEidentiescandidateendmembersasthosethatfallontheextremeendsoftheprojectionandreturnseitherthepixelsorthemeanofpixelsthatarespatiallycloseandspectrallysimilar.ThemethodbasedonMorphologicalAssociativeMemoriesdescribedby Grana,Sussner,andRitter ( 2003 )alsodependsonthepixelpurityassumptionforendmemberextractionasdescribedinSection 2.1.4 .VertexComponentAnalysisaddsendmemberssequentiallybyselectingpixelswhichprojectfarthestinadirectionorthonormaltothespacespannedbythecurrentendmemberset( NascimentoandBioucas-Dias 2005b ).Thus,VertexComponentAnalysisalsoreliesonthepixelpurityassumption( NascimentoandBioucas-Dias 2005b ). Winter 1999 ).NFindrseeksthesetofinputpixelsthatencompassthelargestvolume( Winter 1999 ).Thealgorithmbeginsbyrandomlyselectingasetofpixelsfromtheimagetobetheinitialendmemberset.Then,eachendmemberisreplaced,insuccession,byallother 21

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Winter 1999 ).ThevolumeenclosedbyeachsetofpotentialendmembersiscomputedusingEquation 2{1 (M1)!abs(jEj)(2{1)where Green,Berman,Switzer,andCraig 1988 ; Lee,Woodyatt,andBerman 1990 ).Thedatadimensionalitymustbeonelessthanthedesirednumberofendmemberssincethedeterminantofanon-squarematrixisnotdened( Winter 1999 ).Thisalgorithmworksbymaximizingthevolumebytheendmembersinscribedwithinthehyperspectraldatacloud.Sincetheendmembersarefoundwithinthedatacloud,theendmembersmaynotencloseallthedatapoints.Inadditiontoassumingpurepixelscanbefoundintheimage,thisalgorithmrequiresknowledgeofthenumberofendmembersinadvance( Winter 1999 ). Boardmanetal. 1995 ).ThePPIalgorithmranksimagepixelsbasedontheirpixelpurityindices.Then,theMpixelswiththehighestpixelpurityvaluesarereturnedaspotentialendmembers.Thenumberofendmembers,M,isnotdeterminedbythisalgorithm.PPIisoftenusedforgeneratingcandidateendmemberswhicharethenusedasinputstootherendmemberextractionalgorithms( Berman,Kiiveri,Lagerstrom,Ernst,Donne,andHuntington 2004 )orloaded 22

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Roggeetal. 2007 ).ThePPIalgorithmassignseachpixelapixelpurityvaluebyrepeatedlyprojectingallofthepixelsontorandomlydirectedvectors.Thealgorithmisinitializedbyassigningallpixelsapixelpurityvalueofzero.Thepixelpurityvaluesareupdatedfollowingeachrandomprojectionbyaddingonetothevaluesofthepixelsthatfallneareitherendofeveryprojection.SincePPIvaluesaregeneratedusingrandomvectors,theresultsaredependentonthenumberofrandomprojectionsandthethresholdfordeterminingifapixel'sprojectionisconsiderednearanend-point( Boardmanetal. 1995 ). Craig 1994 )ndsthesmallestsimplexthatcircumscribesthehyperspectraldatapoints.ThisisincontrasttotheNFindrmethodwhichobtainsthelargestsimplexinscribedwithintheinputdataset( Winter 1999 ).MVTsearchesforhyperplanesthatminimizetheirenclosedvolumewhileencompassingallofthedata.Thealgorithmtheniterativelyvariesthehyperplanesusinglinearprogrammingmethodstoprovideaprogressivelytightertaroundthedata.Afterminimizingthevolumeenclosedbythehyperplaneswhileencompassingalloftheinputdata,theintersectionsoftheplanesarereturnedasendmembers.AlthoughMVTdoesnotrequirepurepixelstobeinthedataset,itdoesrequirethenumberofendmembersinadvance.ThemethodperformstheMaximumNoiseFractiontransform( Greenetal. 1988 )toreducethedatadimensionalityto(M1)whereMisthenumberofendmembers. 23

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IfarraguerriandChang 1999 )ofendmemberextractionalsosearchesfortheboundariesofaconvex,non-negativeregionthatenclosetheinputdatapoints.Thismethodreliesonthefactthatradiancevaluesarenon-negativeand,therefore,canrestricttheendmemberstobenon-negativepoints.Themethodrequiresaninputofthenumberofdesiredendmembers,M.GivenM,theeigenvectorsofthesamplecorrelationmatrixthatcorrespondtotheMlargesteigenvaluesarecomputed( IfarraguerriandChang 1999 ). 2{4 IfarraguerriandChang 1999 ).Thersteigenvectorofthesamplecorrelationmatrix,u1,willpointtowardsthedataset.Equation 2{4 canbeinterpretedasperturbingthersteigenvectorbyalinearcombinationoftheotherorthogonaleigenvectorswhileconstrainingtheendmemberstobenon-negative.SinceeacheigenvectorisofdimensionD,solvingforthe(M1)ajcoecientsinEquation 2{4 isanover-determinedproblem.Becauseofthis,theCCAmethoditeratesthrougheachsubsetofbandsofsize(M1)andsolvesasetof(M1)linearequationsfortheajcoecients, 24

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2{4 .ThevectorscomputedbypluggingbackintoEquation 2{4 willcontain(M1)zerovalues.Theremainingbandvaluesarecheckedtoensurethattheyarenon-negative.Ifapotentialendmemberisfoundtobenon-negative,thenitiskeptasanendmember,otherwise,thatvectorisdiscarded( IfarraguerriandChang 1999 ).TheCCAmethodsearchesthroughDM1potentialendmemberswhichcanbeprohibitivefordatasetswithalargenumberofhyperspectralbands.Furthermore,sinceDM1maybegreaterthanM,moreendmembersthanspeciedmaybefound. IfarraguerriandChang ( 1999 )listpotentialmethodsforremovingtheadditionalendmemberssuchasremovingendmembersthatarecollinearwithotherendmemberspectra.Thisalgorithmdoesnotprovideendmemberswhichtightlysurroundthedatapoints.Thisisanartifactfromthe(M1)zerosineachendmemberspectraascanbeseeninFigure 2-1 .Figure 2-1 showsthethree-dimensionaldatasetandthethreeendmembersfoundusingCCA.Sinceeachendmemberhastwozerosintheirspectra,theendmemberspectraliealongthex-,y-andz-axisratherthantightlysurroundingthedataset.ThisisfurtherillustratedinFigures 2-2 and 2-3 .ThenormalizeddatapointsinFigure 2-2 arethersttwenty-vebands(approximately1978to2228nm)fromasubsetofpixelsintheAirborneVisible/InfraredImagingSpectrometer(AVIRIS)Cuprite\Scene4"dataset( AVIRIS ).Figure 2-3 showsthenineendmembersdeterminedusingCCAwithMsettothree.Thenormalizeddatasetvaluesrangefrom0.138to0.244whiletherangeofendmembervaluesis0to0.904. 25

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LeeandSeung 1999 ; MiaoandQi 2007 ), MiaoandQi 2007 ; Pauca,Piper,andPlemmons 2005 ).OneNMFalgorithmproposedby LeeandSeung ( 2000 )minimizestheobjectivefunctioninEquation 2{7 ( Paucaetal. 2005 ), 2kXEPk2F=DXi=1NXj=1(Xij(EP)ij)2:(2{7)TheNMFupdatedevelopedby LeeandSeung ( 2000 )usesthemultiplicativeupdaterulesinEquations 2{8 and 2{9 wherekindicatestheiteration.TheelementsofthePandEmatricesareupdatedsimultaneouslybyiteratingbetweentheelementsofthetwomatrices( Paucaetal. 2005 ). LeeandSeung ( 2000 )provethatthedistanceinEquation 2{7 doesnotincreasewhenusingtheupdatesinEquations 2{8 and 2{9 .Asshownby LeeandSeung ( 2000 ),themultiplicativeupdaterulesareequivalenttostandardgradientdescentupdateswhenthestepsizeparameterissettopkij 26

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MiaoandQi 2007 )attemptstominimizetheobjectivefunctioninEquation 2{10 solvingforendmembersandtheabundancevaluesforeachpixel( MiaoandQi 2007 ). minf(E;P)=1 2DXi=1NXj=1(XijEPij)2+1 2(M1)!0B@1TM~E1CA2(2{10)where 2{11 inordertobeabletocomputethedeterminantinEquation 2{10 ( MiaoandQi 2007 ).Thersttermoftheobjectivefunctionisasquarederrorterm.Byminimizingtherstterm,theerrorbetweentheinputdatasetandtheestimatedpixelscomputedfromtheabundancevaluesandendmembersareminimized.Thesecondtermoftheobjectivefunctionisthevolumeofthespacedenedbytheendmembers.Byminimizingthesecondterm,theendmembersprovideatighttaroundthedata.Thesetwotermscanbeseenasan\internalforce"andan\externalforce"( MiaoandQi 2007 ).Thersttermcanbeinterpretedasanoutwardforcethatprefersendmemberswhichcompletelyencompassthedataandthesecondtermisaninwardforcethatwantstominimizethevolumeenclosedbytheendmembers( MiaoandQi 2007 ).InMVC-NMF,theobjectivefunctioninEquation 2{10 isminimizedusinggradientdescentwithclipping.Thevaluesfortheendmembersandtheirproportionsareupdatedinanalternatingfashion.Inotherwords,ineachiterationofthealgorithm,eithertheendmembersortheproportionsareupdatedwhiletheotherisheldconstant( MiaoandQi 2007 ). 27

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1{2 ,aftersolvingforeithertheendmemberortheproportions,anynegativevaluesaresettozero. whereandarethegradientdescentlearningrates( MiaoandQi 2007 ).Topromotethesum-to-oneconstraintoftheproportions,whenupdatingproportionvalues,theendmemberanddatamatricesareaugmentedbyarowofconstantpositivevalues.Thelargertheconstant,themoreemphasisisplacedonthesum-to-oneconstraint( MiaoandQi 2007 ).Thealgorithmseeksendmembersthatminimizethesquaredreconstructionerror.Thealgorithmalsoallowsforsomeresiliencetonoiseandselectsendmembersthatprovideatighttaroundthedata.Still,theMVC-NMFalgorithmdoeshavesomedrawbacks.Thealgorithmrequiresknowledgeofthenumberofendmembersinadvanceanddoesnotstrictlyenforcethesum-to-oneconstraint. Pauca,Piper,andPlemmons ( 2005 )alsodevelopamethodofendmemberextractionbasedontheNMFalgorithm.TheirconstrainedNMFalgorithmincorporatessmoothnessconstraintsintotheNMFobjectionfunctiondescribedinEquation 2{7 .TheresultingobjectivefunctionisshowninEquation 2{14 minE;PfkXEPk2F+akEk2F+kPk2Fg(2{14)whereaandbareregularizationparametersbalancingtheerrorandsmoothnessterms( Paucaetal. 2005 ).ThesmoothnesstermsencouragesparsitywithinthematricesandareequivalenttoapplyingaGaussianpriorontheendmembersandabundances.Thisobjectiveisminimizedusinggradientdescent.Following LeeandSeung ( 2000 ),thestepsizeparametersaresettopkij Paucaetal. 2005 ). 28

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( 2005 ),aftercomputingendmembersusingtheConstrainedNMFmethod,retainendmembersbasedontheirsimilaritytolaboratorymeasuredspectra.EndmembersarecomparedtolaboratoryspectrausingthesymmetricKullback-LeiblerDivergence, Paucaetal. 2005 ).LikethepreviousNMF-basedalgorithms,thismethodrequiredanestimateofthenumberofendmemberspriortorunningtheNMFalgorithm.However,thisnumbermaychangebasedonthenalpruningstepwhichrequiresaccesstoaspectrallibrarycontainingsignaturesthatcanbefoundintheimage. LiouandYang ( 2005 )alsodevelopedanendmemberextractionmethodbasedonNMF.TheirmethodreliesonthebasicNMFmultiplicativeupdaterule( LeeandSeung 1999 )butprovidesamethodofinitializingthetwonon-negativematrices.TheEandPmatricesareinitializedusingtheFuzzyC-Means(FCM)clusteringmethod.TheFCMclusteringalgorithmclustersthedataintoMclusterswitheachinputpointhavingvaryingdegreesofmembershipineachcluster.TheobjectivefunctionforFCMisdenedinEquation 2{17 29

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LiouandYang 2005 ; TheodoridisandKoutroumbas 2003 ).FCMminimizestheobjectivefunctionbyupdatingmembershipvaluesandclustercenters.SinceFCMprovidesbothclustercentersandmembershipvaluesforeachdatapoint,thematricesfortheNMFalgorithmareinitializedusingtheclustercentersastheinitialendmembersandthemembershipvaluesastheinitialabundancevalues.SinceNMFisdependentoninitialization,wellchoseninitialmatricescanimproveperformance( LiouandYang 2005 ).ThismethodalsorequiresadvanceknowledgeofthenumberofendmemberstoperformboththeFCMandtheNMFalgorithms. LiouandYang ( 2005 )utilizethePartitionedNoise-AdjustedPrincipalComponentAnalysismethod( Tu,Huang,andChen 2001 )totrytoestimatethenumberofendmemberspriortoapplyingtheendmemberdetectionalgorithm.ThismethodofestimatingthenumberofendmembersisdescribedinSection 2.1.7 RitterandGader 2006 ).Therearetwotypesofmorphologicalmemoriesthatcanbecomputed,theminmemoryandthemaxmemory.Givenasetofinputvectors,X=fx1;:::;xNgandassociateddesiredoutputs,Y=fy1;:::;yNg.Theminandmaxmorphologicalassociativememories,WXYandMXY,arecomputedusing 30

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2{18 and 2{19 (2{18) (2{19) wherexisthelatticeconjugatetransposeofxwhichisdenedtobex=(x)T.Auto-associativemorphologicalmemoriesarethemorphologicalassociativememorieswhichassociateasetXtoitself.Theminandmaxauto-associativememoriesarerelatedtoeachotherusingtheconjugatetransposeoperator,WXX=MXX.Patternsarerecalledusingassociativemorphologicalassociativememoriesthrougheitherthemaxproductortheminproduct,y=WXY 2{20 Grana,Sussner,andRitter 2003 ; Myers 2005 ).TheconvexhullofendmembersthatfollowtheconvexgeometrymodelinEquation 1{1 denesavolumeinD-dimensionalspacewhichsurroundthehyperspectraldatapointsinanimage.TheconvexhullofD+1anelyindependentpointsdenesasimplexinD-dimensionalspace( Myers 2005 ; RitterandUrcid 2008 ).Therefore,themotivationtondanelyindependentpointsusingmorphologicalassociativememoriesisthatthesimplexdenedbythesepointsboundsavolumeisD-dimensionalspacewhichcanbeusedtotrytosurroundthehyperspectralimagepoints( Myers 2005 ).Morphologicalassociativememoriescanalsobeusedtodeterminewhetherpointsaremorphologicallyindependent( Granaetal. 2003 ).Someendmemberdetectionalgorithms 31

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Granaetal. 2003 ). Grana,Sussner,andRitter ( 2003 )developedamethodusingMorphologicalAssociativeMemoriestoextractendmembers.Theimagepixelsareallshiftedbytheirmean,X0=fx0ijx0i=xig.Then,thealgorithmbeginsbyrandomlyselectingasingleinputpixeltobetheinitialendmember.Usingthisinitialendmember'sbinaryrepresentation,minandmaxauto-associativememoriesarecreated.Thebinaryrepresentationofpixelxisdenedtobesgn(x)( GranaandGallego 2003 ; Granaetal. 2003 ).Afterselectingtheinitialpixel,allotherimagepixelsaresequentiallyconsideredtobeendmembers.Whenbeingconsidered,apixelisshiftedandtestedformorphologicalindependenceagainstallofthecurrentendmembers'binaryrepresentations( Granaetal. 2003 ). (2{21) (2{22) whereaisaconstantvalue,Bmatrixofthecurrentendmembers'binaryrepresentations,andisthevectorofvariancesofeachbandoftheinputimage.Apixelisdeterminedtobemorphologicallyindependentifx+i=2Bandxi=2B.Ifashiftedpixelisfoundtobemorphologicallyindependent,itisaddedtothesetofendmembersandnewauto-associativememoriesarecomputed.Ifitisnotmorphologicallyindependent,thenthepixeliscomparedagainstexistingendmemberstoseeifitismoreextremethanthecurrentendmember.Ifthepixelismoreextreme,thenitisreplacesthatendmember;otherwise,thepixelisdiscarded( Granaetal. 2003 ).UsingD-lengthvectors,thereare2Dpossiblebinaryvectors.Thisalgorithmwillreturnoneendmemberforeachsetofshiftedinputdatapointswiththesamebinaryrepresentations.Therefore,upto2Dendmembersmaybereturned. 32

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Grana,Hernandez,andd'Anjou ( 2005 )developedanalgorithmthatcombinesanevolutionarysearchandendmemberdetectionusingMorphologicalAssociativeMemories.ThisalgorithmusesanevolutionarysearchtondasetofmorphologicallyindependentendmembersthatminimizethetnessfunctioninEquation 2{25 ( Granaetal. 2005 ).Thealgorithmproceedsbyevolvingasetofbinaryvectorsusingamutationoperatorandroulettewheelselectionbasedonthetnessfunction( Granaetal. 2005 ).Everymutationistestedformorphologicalindependence.Ifthemutatedsetofbinaryvectorsisnotmorphologicallyindependent,itisrejected.Givenasetofbinaryvectors,thecorrespondingendmembersaretheextremepixelsinthedirectionidentiedbythebinaryvectors( Granaetal. 2005 ).Thisalgorithmprovidesbothendmemberspectraandabundancevalues.However,thisalgorithmrequirespriorknowledgeofthedesirednumberofendmembersanddoesnotstrictlyenforcethenon-negativityandsum-to-oneconstraintsontheabundancevalues. RitterandUrcid ( 2008 )developedamethodofextractingendmembersthatusesthecolumnsoftheminauto-associativememoriesinEquation 2{18 .Thismethodissimilartotheonepresentedby Myers ( 2005 )whichalsoreturnsthestronglatticeindependentcolumnsoftheminormaxauto-associativememories.Theauto-associativememoriesinthismethodarecreatedusingthepointsoftheinputhyperspectralimage.Aftercomputingtheauto-associativememories,anyduplicatecolumnsoftheminmemoryareremovedensuringthattheremainingcolumnarelinearlyindependentas 33

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RitterandUrcid ( 2008 ).Linearlyindependentsetsarecreatedsincelinearindependenceimpliesaneindependence.Themembersofthelinearlyindependentsetsarethenshiftedbytheelementsofthebrightpoint(thecomponent-wisemaximumofallinputpoints) RitterandUrcid 2008 ).Theelementsalongthediagonalofthememoryareequaltozero.Therefore,aftershifting,thesevaluesaresettothemaximumvalueofthedatapointsinthecorrespondinghyperspectralband.Thisprovidesphysicalmeaningbetweentheendmembersandtheinputdataset( RitterandUrcid 2008 )Aftershifting,theuniquelinearlyindependentvectors,W=fw1;:::;w2g,arereturnedasendmembers.Additionally,theshadepoint(thecomponent-wiseminimumofallinputpoints)isreturnedasanendmember( Myers 2005 ; RitterandUrcid 2008 ).Usingthemin-memoryandtheshadepointprovidesuptoD+1endmembers.Thismethodisveryecient;itrequiresonlyasinglepassthroughalloftheinputpixels( Myers 2005 ; RitterandUrcid 2008 ).However,thisalgorithmdoesnotcomputeabundancevaluesanditdoesnotguaranteethatallpixelswillbeencompassedbytheselectedendmembers.Figure 2-4 displaystheendmembersdeterminedusingthismethodontwo-dimensionaldata.Thedatasetwasgeneratedfromfourendmembers([10,30],[13,24],[15,31],[22,25]).Also,Gaussianrandomnoisewasaddedtoeachcoordinateofthedataset.Theminmemory,priortoshifting,foundendmembers(0,1.38)and(-21.42,0).Thesewereshiftedby(the1stand2ndcoordinatesofthebrightpoint,thecomponent-wisemaxofthedata)22.64and32.97,respectively,toobtain(22.64,24.02)and(11.55,32.97).Asshowninthegure,allofthepixelsarenotencompassedbytheendmembers.Usingthebrightpoint,theshadepointandtheuniquecolumnsofboththemin-andmax-memoriesasendmembersguaranteesthatallinputdatapointswillbeencompassed 34

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RitterandUrcid 2008 ).ThisisshowninFigure 2-5 .However,thisdoesnotprovideatighttaroundthedatapointsanditwillreturnupto2D+2endmembers.Withhighdimensionaldatasets,themethodwouldreturnaverylargenumberofendmembers.Methodstoreducethenumberofendmemberswhenusingeitherbothmemoriesoronlythemin-memoryarediscussedby RitterandUrcid ( 2008 ).Forexample,everyothercolumnofthememorymaybediscardedsincecontiguouscolumnsareoftenhighlycorrelated.Anothermethodpresentedistocomputelinearcorrelationcoecientsbetweeneachoftheendmembersandretainasubsetofendmemberswhosecorrelationcoecientsfallbelowasetthreshold( RitterandUrcid 2008 ). Grana,Hernandez,andGallego 2004 ).TheSIEalgorithmbeginsbysamplingMendmembersfromaGaussiandistributioncenteredatthemeanofthehyperspectraldataset 35

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Granaetal. 2004 ; Whitley 2001 ).Oncechosen,anendmemberismutatedbyaddingarandomGaussianperturbation. (2{27) (2{28) (2{29) 2{25 isrecomputed.Ifthetnessfunctionimproves,thenthemutatedendmemberreplacestheoriginalendmember.Anendmembercanbemutateduptotimes.Thisalgorithmsearchesforendmembersthatminimizethesquarederrorbetweenthepixelsandtheirestimationandminimizetheamountabundancevaluesarenegativeordonotprescribetothesum-to-oneconstraint.Likemanyofotherofexistingendmemberdetectionalgorithms,thisalgorithmrequirestheinputofthenumberofdesiredendmembers. Chiang,Chang,andGinsberg 2000 ; Tu 2000 ; Tu,Huang,andChen 2001 ; WangandChang 2006 ).IndependentComponentAnalysisperformsunsupervisedseparationofstatisticallyindependentsourcesinadataset( NascimentoandBioucas-Dias 2005a ).The 36

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HyvarinenandOja 2000 ). HyvarinenandOja 2000 ).Equivalently,ICAcanbeperformedbysearchingforthecomponentswhichmatchthedataandare\non-Gaussian"( HyvarinenandOja 2000 ).InICA,thenumberofsignalsourcesfoundisthesameasthedimensionalityofthedata.Therefore,eitherDsignalsourcesarefoundwhereDisthedatadimensionality,ordimensionalityreductionisusedtondMDsignals( HyvarinenandOja 2000 ; Tuetal. 2001 ). Chiangetal. ( 2000 )directlyapplyICAtotheproblemofdeterminingtheendmembersforahyperspectralimagewheretheabundancevaluesareassumedtobestatisticallyindependent\randomsignalsources". Tu ( 2000 )alsoappliedtheICAalgorithmforendmemberextraction.PriortorunningICA,TuestimatesthenumberofendmembersandwhitensthedatatoreducethedatadimensionalityusingtheNoise-AdjustedTransformedGerschgorindisk(NATGD)( Tu 2000 ).Similarly,theSpectralDataExploreralgorithm(SDE)usesthePartitionedNoise-AdjustedPrincipalComponentsAlgorithm(PNAPCA)towhitenthedataanddeterminethenumberofendmemberafterwhichICAisperformed( Tuetal. 2001 ).NATGDandPNAPCAmethodsforestimatingthenumberofendmembersaredescribedinSection 2.1.7 WangandChang ( 2006 )applyICAforendmemberextractionwhere,priortorunningICA,thenumberofendmembers,M,isestimatedusingtheVirtualDimensionality(describedinSection 2.1.7 )ofthedataset.Giventheestimatednumberofendmembers,theindependentcomponentsdeterminedusingtheICAalgorithmareprioritizedusingthe3rdand4thorderstatisticsofthecomponent( WangandChang 2006 ).ForeachoftheMhighestprioritycomponents,imagepixelswiththelargestabsolutevalueoftheabundancearereturnedasendmembers( WangandChang 2006 ). 37

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( 2005a )arguethattheuseofIndependentComponentsAnalysisforendmemberdetectionisnotanaccuratemethodsincethesum-to-oneconstraintontheabundancevaluescausethesourcesinthecorrespondingICAproblemtobedependent.ThisdependencyviolatesthebasicICAassumptionofstatisticallyindependentsources( NascimentoandBioucas-Dias 2005a ). NascimentoandBioucas-Dias ( 2005a )provideresultstoarguethatthesomeendmembersareincorrectlyunmixedusingICAmethods.Asanalternative,theDependentComponentAnalysismethod(DECA)wasdevelopedwhichassumestheabundancevaluesaredrawnfromaDirichletdistribution( NascimentoandBioucas-Dias 2007a b ).TheDirichletenforcesthenon-negativityandsum-to-oneconstraintsontheabundancevalues.DECAdeterminesabundancesandendmembervaluesusingtheExpectation-Maximization(EM)method( NascimentoandBioucas-Dias 2007a b ).However,likeICA-basedmethods,DECAalsorequiresthenumberofendmemberstobeknowninadvance. ChangandDu 2004 ; WangandChang 2006 ).Also, Tu ( 2000 )reliesontheTransformedGerschgorinDisk(TGD)andtheNoise-AdjustedTGDmethodofestimatingthenumberofendmembers.ThePartitionedNoise-AdjustedPrincipalComponentsAnalysis(PNAPCA)methodofcomputingthenumberofendmembersisbasedonpartitioningandtransformingthenoise-adjustedcovariancematrix( LiouandYang 2005 ; Tuetal. 2001 ). ChangandDu 2004 ).VDiscomputedusingtheeigenvaluesofthecovarianceandcorrelationmatricesoftheinputdataset.Letn^1^2^dobetheeigenvaluesfromthesamplecorrelationmatrixandletf12dgbetheeigenvaluesfromthesample 38

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ChangandDu 2004 ). ^rr;forr=1;:::;VD ^r=r;forr=VD+1:::;d Theeigenvaluescomputedfromthesamplecovariancematrixequalthevariancesofthetransformeddata( TheodoridisandKoutroumbas 2003 )andtheeigenvaluesofthesamplecorrelationmatrixarerelatedtothevarianceofthedatafromtheorigin.Assumingthatthenoisehaszero-meanandunit-varianceandsignalsinthedatahavenon-zerovalues,theeigenvaluesofthesamplecorrelationmatrixcorrespondingtosignalsinthedatawillhavealargervaluethanthecorrespondingeigenvaluesfromthesamplecovariancematrix.Theeigenvaluesofthesamplecorrelationandcovariancematrixcorrespondingtonoisewillbeequal( ChangandDu 2004 ), ^rr2nlforr=1;:::;VD ^r=r=2nlforr=VD+1:::;d where2nlisthenoisevariance.AnexampleofthisconceptisshownusingthedatainFigure 2-6 .Thethree-dimensionaldatasetwasgeneratedusingtwoendmembers,[2,5,0]and[3,6,1].Zero-meanGaussiannoisewithavarianceof0.03wasaddedtoeachcoordinateofthedata.Theeigenvaluesofthecovarianceandcorrelationmatriceswerecomputed.Theeigenvaluesfromthecovariancematrixwerefoundtobe0.1814,0.0010,and0.0008.Theeigenvaluesforthecorrelationmatrixwerefoundtobe36.4991,0.0647,and0.0008.Therefore,thevirtualdimensionalitycorrectlydeterminesthenumberofendmemberstobetwo,sincethethirdeigenvaluefromboththecovarianceandcorrelationmatricesareequal.Inordertodeterminethattheeigenvaluesdier,theirdierencesarethresholded.ChangandDu( ChangandDu 2004 )describethreethresholdingmethodsfordeterminingthevirtualdimensionality.ThesemethodsincludetheHarsanyi-Farrand-Chang(HFC) 39

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Solvingforlgivesthethresholdfortheeigenvaluedierences.Thisthresholdingmethodrequiresanestimateofthevarianceofthedierencebetweentheeigenvaluesateachband,zl.TheHFCmethoduses2zl2^2l ChangandDu 2004 ).TheNoiseSubspaceProjectionmethodissimilartotheNWHFC.Anestimateofthenoisecovarianceforthedataisusedtowhitenthecovariancematrix.However,insteadofcomputingdierences.Thismethodrecognizesthattheeigenvaluescorrespondingtonoiseshouldbeequaltoone.Therefore,thecomputedthresholdisapplieddirectlytotheeigenvaluestodetermineiftheirdierencefromoneissignicant( ChangandDu 2004 ).Thismethodissensitivetothevarianceandcovarianceestimatesusedindeterminingthethresholdsforeacheigenvalue.Therefore,theVDestimateofthenumberofendmembersissensitivetonoiseintheinputdataset.TheVDwasrunontwodatasetsgeneratedfromthethreeAVIRISCupritespectrashowninFigure 2-7 .Theasmall 40

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2-8 .Theseconddataset,showninFigure 2-9 ,hasmoreaddednoise.Thethreethresholdingmethodswereappliedtobothsetsofdata.Ontherstdataset,theNSPmethodcorrectlydeterminedthenumberofendmembersbyestimating3signals.TheHFCandNWHFCmethodsincorrectlyestimatedthenumberofendmemberswith2and5,respectively.Ontheseconddatasetwithlargeramountsofnoise,noneofthethresholdingmethodscorrectlyestimatedthreeendmembers.TheHFCmethodestimated2,theNWHFCmethodestimated7,andtheNSPmethodestimated2endmembers. Tuetal. 1999 2001 )ofestimatingthenumberofendmembersisbasedontheMaximumNoiseFraction(MNF)(alsoknownastheNoiseAdjustedPrincipalComponentsAnalysismethod)( Greenetal. 1988 ; Leeetal. 1990 ).TheMNFtransformusesanestimateofthenoisecovariancematrixtotransformthedataintocomponentswhicharesortedbasedontheirsignal-to-noiseratio.UsingtheconvexgeometrymodelinEquation 1{1 andassumingthenoiseandsignalcomponentsofthedataareuncorrelated,thecovariancematrixofthedatacanbewrittenas( Leeetal. 1990 ; Tuetal. 1999 ) Leeetal. 1990 ; Tuetal. 1999 ), WTNW=argmaxWWTXW WTNW1:(2{38)Thesignal-to-noiseratioismaximizedbyassigningW=N1 2NAwhereNistheeigenvectormatrixofthenoisecovariancematrix,Nisthediagonaleigenvaluematrixof 41

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2NTXN1 2N( Leeetal. 1990 ; Tuetal. 1999 ).UsingthematrixW,theMNFtransformsimultaneouslydiagonalizesthedatacovariancematrixandwhitensthenoisecovariancematrix.TheMNFtransformationrequiresanestimateofthenoisecovariancematrix,N( Tuetal. 1999 ).Asdescribedby Tuetal. ( 1999 2001 ),PNAPCAmethodpartitionsthenoiseadjustedcovariancematrixfoundbyMNFanddiagonalizesthetwopartitions. Tuetal. ( 1999 2001 )claimthatbyexaminingtheeigenvaluesofthetwopartitionssimultaneously,theeectsofincorrectlyestimatingthenoisecovariancematrixarelessened. Wuetal. ( 1995 )and Tu ( 2000 )developedmethodsofestimatingthenumberofsignalsinadatasetbasedonGerschgorin'sdisktheorem( HornandJohnson 1985 ).Gerschgorin'sdisktheoremprovidesamethodofestimatingthelocationsofeigenvaluesofamatrix.Thetheoremstatesthattheeigenvaluesofamatrix,A,arelocatedwithintheunionofthedisksdenedby HornandJohnson 1985 ).ThetheoremalsostatesthatiftheunionofkoftheDdisksformaconnectedregionandiftheconnectedregionisdisjointfromalloftheremainingdisks,thenkeigenvaluesarelocatedwithintheregiondenedbytheunionofthekdisks( HornandJohnson 1985 ).ThetransformedGerschgorinDiskmethoddevelopedby Wuetal. ( 1995 )denesatransformationonthecovariancematrixofaninputdatasetsothattheGerschgorindisks 42

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Tu ( 2000 )appliesthetransformedGerschgorindiskmethodtothenoise-adjustedcovariancematrix.ThetransformationmatrixusedintheTransformedGerschgorinDiskandtheNoise-AdjustedTransformedGerschgorinDiskmethodsisdeterminedbydiagonalizingtheD1D1leadingsub-matrixoftheinputcovariancematrix, 2{39 ,theGerschgorindisksofthetransformedinputcovariancematrixinEquation 2{43 haveradiiequaltoj1j;j2j;:::;jD1jandcentersat1;2;:::;D1.Assumingthatthenoiseinthe 43

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Greenetal. 1988 ; Leeetal. 1990 ; TheodoridisandKoutroumbas 2003 )havebeenusedtoprojectthedataintoalowerdimensionalspaceandthusreducethedimensionalityofthedata.Althoughthesemethodsareeectiveatdatareduction,theydonotretainphysicallymeaningfulbandsthatcorrespondtowavelengthsintheoriginaldataset. HarsanyiandChang ( 1994 )provideanorthogonalsubspaceprojectionsapproachthatalsotransformsthedata. Bruceetal. ( 2002 )conductsdimensionalityreductionbyextractingfeaturesthatdistinguishbetweenlabeledclassesinatrainingsetusingtheDiscreteWaveletTransform(DWT).Thismethodextractsfeaturesthatincorporatebothfrequencyinformationanddetailedlocalizedfeaturesoftheinputhyperspectralsignal.ThisfeaturesetisfurtherreducedusingFisher'sLinearDiscriminantAnalysis.SincethismethodusingtheDWTandFisher'sLinearDiscriminant,extractedfeaturesdonotcorrespondtowavelengthsinthe 44

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DeBackeretal. ( 2005 )and Kumaretal. ( 2001 )bothpresentmethodsthatmergemanyadjacenthyperspectralbands. DeBackeretal. ( 2005 )mergesbandsintogroupswhichoptimizetheBhattacharyadistancebetweenlabeledclassesinatrainingset.Thesupervisedbandselectionmethodpresentedby RiedmannandMilton ( 2003 )mergesneighboringbandstoimproveaccuracyinaclassicationtask.ThehyperspectraldimensionalityreductionmethodbasedonLocalizedDiscriminantBases( Venkataramanetal. 2005 )alsomergesadjacentbandsforfeatureextraction. Martinez-Usoetal. ( 2007 )presentabandmergingalgorithmusinginformationmeasuresandhierarchicalclustering.Adivergencemeasurebetweeneverypairofbandsiscomputedandusedtoperformhierarchicalclusteringofthebands.Abandrepresentativeisthencomputedforeachcluster. LinandBruce ( 2004 )useaProjectionPursuitsmethodstoreducethedimensionalityofahyperspectraldatasetbydeterminingaprojectionmatrixthataidsindistinguishingbetweenclassesinthedataset.Insteadofmergingbandsortransformingthedata,thismethodmaintainsonlythosebandsthatareusefulforthehyperspectralimageanalysistask.Theadvantageofphysicallymeaningfulbandsistoidentifyusefulwavelengthsforaparticularclassicationtask.Identifyingimportantwavelengthscanalsobeusedinthedesignofhyperspectralsensors.Byreducingthenumberofwavelengthsthatneedtobecollected,datacollectionwillbeperformedfasterandwithlessrequiredstoragespace.Additionally,mostofthepreviouslymentionedbandselectionalgorithms( DeBackeretal. 2005 ; Greenetal. 1988 ; HarsanyiandChang 1994 ; Leeetal. 1990 ; Martinez-Usoetal. 2007 )requiretheknowledgeofthedesirednumberofbands.Serpicoandcolleagues'searchmethodforbandselection,Duandcolleagues'methodofbandprioritizationbasedontheIndependentComponentAnalysis'weightmatrix,Hanandcolleagues'eigenvalueweightedbandprioritizationmethod,andGuoandcolleagues'( Duetal. 2003 ; Guoetal. 2006 ; Hanetal. 2004 ; SerpicoandBruzzone 2001 )mutualinformationbasedbandselectionmethodrequirethedesirednumberofbands. Petrieetal. ( 1998 )outlines 45

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Petrieetal. ( 1998 )requirethedesirednumberofbands.ThebandselectionmethodbasedontheNFindralgorithm( Wangetal. 2006 )retainsthebandswhichmaximizethevolumebetweentheendmembersfoundusingtheNFindralgorithm( Winter 1999 ).Thismethodattemptstondbandswhichaidinspectralunmixing,however,thenumberofbandstoretainmustbeknowninadvance.Often,thenumberofrequiredbandsisnotknown.KeshavapresentedamethodbasedontheSpectralAngleMapper(SAM)distanceortheEuclideanMinimumDistance(EMD)measures( Keshava 2001 2004 ).ThealgorithmincrementallyaddsbandsthatincreasetheSAMorEMDmeasurebetweentwolabeledclassesuntilsomestoppingcriterionisreached.Althoughthismethoddoesnotrequirethenumberofbandsinadvance,themethodislimitedtodistinguishingbetweentwolabeledclassesinthedataset.Similarly,theSparseLinearFiltersalgorithm( TheilerandGlocer 2006 )developssparselinearlterstodistinguishbetweentwolabeledclassesinthedata.TheltersuseasparsesetofthehyperspectralbandsbyutilizinganL1-penaltytermtoselectthebandsandthenumberofbands( Tibshirani 1996 ).Althoughthenumberofbandsisestimated,themethodrequirestwolabeledclasses( TheilerandGlocer 2006 ). Changetal. ( 1999 )ranksallbandsbasedonloadingfactorsconstructedusingmaximum-variancePCA,MNF,orthogonalsubspaceprojectionandminimummisclassicationcanonicalanalysismethods.Followingranking, Changetal. ( 1999 )removecorrelatedbandsusingadivergencemeasure. ChangandWang ( 2006 )employamethodbasedonconstrainedenergyminimization.Themethodselectsbandsthathavetheminimalcorrelationwitheachotherandusestheconceptofvirtualdimensionalitytodeterminethenumberofchosenspectralbands. 46

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1{2 .Thesealgorithmalsodoaccountforspectralvariabilityintheirendmemberrepresentations.Furthermore,theexistingalgorithmsdonotconsidercasesinwhichmultipleconvexregionsandsetsofendmembersmaymoreaccuratelydescribethedata.Hyperspectralbandselectionalgorithmsoftenrequirethenumberofneededhyperspectralbandspriortorunningthebandselectionalgorithm.Datareductiontechniqueswhichperformprojectionsareoftenusedtoreducethedimensionalityofahyperspectralimage.However,theprojectionmethodslosethephysicalmeaningassociatedwiththehyperspectralbands.Hyperspectralbandselectionalgorithmsarealsooftentiedtoclassicationproblems.Inthesecases,labeledtrainingdataisneededtodeterminethebandswhichdistinguishbetweentheclasses.Bandselectionandendmemberdetectionmethodspresentedautonomouslydeterminethenumberofendmemberandhyperspectralbandsneededforanimage.Methodsarepresentedwhichaccountforanendmember'sspectralvariabilityandcanautonomouslydeterminethenumberofconvexregionsneededtodescribeaninputdataset.Furthermore,allpresentedmethodsprovideabundancevalueswhichconformtotheconstraintsinEquation 1{2 .Thenewalgorithmsarealsocapableofdeterminingendmembersforhighlymixeddatasetssincethepixelpurityassumptionisnotemployed.Thepresentedhyperspectralbandselectionalgorithmalsoretainsphysicallymeaningful 47

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48

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Three-dimensionaldatapointsandendmemberresultsusingconvexconeanalysis. First25bands(1978to2228nm)ofasubsetofnormalizedpixelsfromtheAVIRIScuprite\scene4"dataset 49

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CCAendmemberresultsonasubsetofAVIRISCupritedata Morphologicalassociativememoriesendmemberresultsusingtheminmemoryontwo-dimensionaldata.Endmembersfoundfromthecolumnsoftheminmemoryareshowninblue.Theshadepointisgreen.Datapointswithintheareadenedbytheendmembersareinblack.Datapointsoutsideoftheareadenedbytheendmembersareinred. 50

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Morphologicalassociativememoriesendmemberresultsusingbothmemoriesontwo-dimensionaldata.Endmembersfoundfromthecolumnsofthemaxmemoryareshowninred.Endmembersfromthecolumnsoftheminmemoryareshowninblue.Thebrightpointandshadepointaregreen. 51

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ThreedimensionaldatasetgeneratedfromtwoendmemberswithGaussiannoise. 52

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NormalizedAVIRISCupritespectra 53

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DatasetgeneratedfromAVIRISCupriteendmemberswithasmallamountofGaussiannoise 54

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DatasetgeneratedfromAVIRISCupriteendmemberswithalargeamountofGaussiannoise 55

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56

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Williams 1995 ).Weightdecaytermshasbeenpreviouslyappliedinneuralnetworkapplicationstopromoteregularization( Haykin 1999 ; Williams 1995 ).Consideraleastsquaresobjectivewithaweightmatrix,P.AweightdecaytermappliedtothePparametersattemptstopreventthemfrombecominglarge.LSWD=lnexp8<:1 2NXi=1XiMXk=1pikEk!2NXi=1MXk=1p2ik9=; 2NXi=1XiMXk=1pikEk!29=;exp(NXi=1MXk=1p2ik) 3{2 canbeinterpretedinaprobabilisticmannerwherethesecondexponentialcanbeseenasazero-meanGaussian( Williams 1995 ).Therefore,Equation 3{2 canbeviewedasthelogoftheproductinEquation 3{3 ( Williams 1995 ), Williams 1995 ).InsteadofusingaGaussiandistributionfortheparameters'prior,azero-meanLaplaciandistributioncanbeusedwhichismoreeectiveatsparsitypromotion( Figueiredo 2003 ; Williams 57

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), 2NXi=1XiMXk=1pikEk!2MXk=1kNXi=1jpikj:(3{4)TheapplicationofaLaplaciandistributionisrelatedthetoleastabsoluteshrinkageandselectionoperator(LASSO)( Tibshirani 1996 ).Asdescribedby Tibshirani ( 1996 ),theLASSOappliestheconstraintthatthesumoftheabsolutevaluesoftheweights,PMk=1PNi=1jpikjmustbelessthanathreshold.ThisisequivalenttoapplyingaLaplacianprior( Tibshirani 1996 ).Oftenduringoptimization,thederivativeofanobjectivefunctionmustbecomputed.InthecasewheretheLaplacianpriorhasbeenusedtopenalizelargeparametervalues,thederivativeisnotdenedatzeroduetotheabsolutevaluefunction( Williams 1995 ).Inthesecases,theLaplacianpriorontheparameterscanbedenedinahierarchicalfashionwheretheparametersaredistributedaccordingtoaGaussiandistributionwhosevariancehasanexponentialhyper-prior( Figueiredo 2003 ).Byintegratingoverallpossiblevaluesforthevariancegiventhehyper-prior,thehierarchicalexpressionisequivalentthetotheLaplaciandistribution( Figueiredo 2003 ).Sparsitypromotionhasbeenappliedtoanumberofapplicationsincludingneuralnetworks( Williams 1995 ),classicationandregressionusingexpectation-maximization(EM)( Figueiredo 2003 ),featureselectionandclassication( Krishnapurametal. 2004 ),classicationandregressionusingtheChoquetintegral( Mendez-VazquezandGader 2007 )andothers. Bermanetal. 2004 )performsaminimizationofaresidualsumofsquares(RSS)termbasedontheconvexgeometrymodelinEquation 1{1 .TheerrorbetweenthepixelspectraandthepixelestimatefoundbytheICEalgorithmusingtheendmembersandtheirproportionsisminimizedwhentheresidualsumofsquares(RSS) 58

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Bermanetal. 2004 ), Bermanetal. ( 2004 ),theminimizerfortheRSStermisnotunique.Therefore,theICEalgorithmaddsasumofsquareddistances(SSD)termtotheobjectivefunction. Bermanetal. ( 2004 )showthattheSSDisequivalenttoEquation 3{7 Bermanetal. 2004 ).TheobjectivefunctionusedintheICEalgorithmisshowninEquation 3{8 N+V(3{8)whereisaregularizationparameterthatbalancestheRSSandSSDtermsoftheobjectivefunction.TheICEalgorithmminimizesthisobjectivefunctioniteratively.First,givenendmemberestimates,theproportionsforeachpixelareestimated.Fortherstiterationofthealgorithm,endmemberestimatesmaybesettorandomlychosenpixelsfromtheimage.EstimatingtheproportionsrequiresaleastsquaresminimizationofeachterminEquation 3{5 .SinceeachofthesetermsisquadraticandsubjectedtothelinearconstraintsinEquation 1{2 ,theminimizationisdoneusingquadraticprogramming.Aftersolvingfortheproportions,theendmembersareupdatedusingthecurrentproportion 59

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Bermanetal. ( 2004 )describedinSection 3.2 .TheSPICEalgorithm,whichisasparsitypromotingextensionofICE,isdevelopedinthissection.TheRSStermoftheICEobjectivefunctionisaleastsquarestermwhoseminimizationisequivalenttothemaximizationofEquation 3{10 ( Williams 1995 ) 2NXi=1XiMXk=1pikEk!2=lnexp8<:1 2NXi=1XiMXk=1pikEk!29=;:(3{10)WhenexaminingtheexponentialinEquation 3{10 ,itcanbeseenthatthisisproportionaltotheGaussiandensitywithmeanPMk=1pikEkandvariance1, 2NXi=1XiMXk=1pikEk!29=;/exp8<:1 2NXi=1XiMXk=1pikEk!29=;:(3{11) 60

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3{4 ,thesparsitypromotingtermtobeaddedtotheICEobjectivefunctionshouldbeoftheformshowninEquation 3{12 1{2 .Forthiswork,kissetasshowninEquation 3{13 ZareandGader 2007a ),RSSreg=(1)RSS N+V+SPT NNXi=1XiMXk=1pikEk!TXiMXk=1pikEk!+V+MXk=1kNXi=1pik NNXi=124XiMXk=1pikEk!TXiMXk=1pikEk!+N 3{9 sincetheSPTtermdoesnotdependontheendmembers.Whensolvingfortheproportionvaluesgiventhe 61

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3{17 needtobeminimizedgiventheconstraintsinEquation 1{2 usingquadraticprogramming. NNXi=124XiMXk=1pikEk!TXiMXk=1pikEk!+MXk=1kpik35(3{17)where 1(3{18)Duringtheiterativeminimizationprocess,endmemberscanberemovedastheirproportionvaluesdropbelowathreshold.Aftereveryiterationoftheminimizationprocess,themaximumproportionvaluesforeveryendmembercanbecalculated, 3{14 ( ZareandGader 2008 ).IncorporatingthebandweightsandthebandsparsitypromotingtermyieldsEquation 3{20 62

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3{24 .Thebandsparsitypromotingterm(BST)isdenedasaweightedsumofbandweightswithonetermforeachband, PMk=1(ekj0)2+1(3{25)isatunableparametercontrollingthedegreeofsparsityamongthebandweights,0istheglobaldatamean,xijisthejthbandoftheithpixel,andekjisthejthbandofthekthendmember.ThebandweightsaresubjecttotheconstraintsinEquation 3{26 3{26 allowsforthesecondequalityinEquation 3{24 .ThejvaluesarerelatedtothemethodofrankingbandsaccordingtotheMinimumMisclassicationCanonicalAnalysis(MMCA)usedby Changetal. ( 1999 ).Chang:1999rankbandsaccordingtotheMMCAvaluewhichisderivedfromFisher'sdiscriminantfunction.AlthoughtheproposedmethodusesaweightthatisrelatedtotheFisher'sdiscriminantvalue,thisalgorithmdiersfromthemethodusedby Changetal. ( 1999 )by 63

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3{20 ,theiterativeprocedureusedinSPICEcanstillbeapplied.Theminimizationprocessiteratesbetweensolvingfortheproportions,endmembersandbandweights.TheendmemberscanbesolvedfordirectlyaswasdoneinEquation 3{9 .Whensolvingfortheproportionvaluesgivenendmemberandbandweightestimates,Nquadraticprogrammingsteps,oneforeachdatapoint,canbeemployedtominimizeEquation 3{20 withrespecttotheconstraintsinEquation 1{2 .Similarly,whensolvingforthebandweightsgiventheproportionandendmemberestimates,Equation 3{20 canbeminimizedusingasinglequadraticprogrammingstepgiventheconstraintsinEquation 3{26 .Afterupdatingbandweights,bandsareremovedfromdatapointsandendmemberswhenthecorrespondingbandweightdropsbelowaprescribedthreshold.Sincethebandweightsandendmembervaluesdependoneachother,anoptimizationscheduleneedstobeemployed.Anestimateoftheendmembersisneededbeforedeterminingwhichbandsareusefulindistinguishingbetweentheendmembers.Therefore,anupdatescheduleallowstheendmembersandproportionstosettlebeforedetermining 64

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3.5NewEndmemberDistributionDetectionAlgorithmThenewEndmemberDistribution(ED)detectionalgorithmhastheuniquepropertyofrepresentingendmembersasrandomvectors,therebycalculatingendmembersdistributionsratherthansinglespectra.Endmemberdistributionsarefoundbyassumingamodelforeachendmemberanditerativelyupdatingendmemberdistributionsandproportionvectorsforeachpixel.EDwasdevelopedforusewithinthePiece-wiseConvexEndmemberdetectionalgorithminSection 3.8 .However,sinceEDincorporatesspectralvariabilitywhenperformingspectralunmixingandendmemberdetermination,applicationsforEDmayextendbeyondusewithinthePCEalgorithm.AssumingtheconvexgeometrymodelinEquation 1{1 ,eachinputhyperspectralpixelisalinearcombinationoftheendmembers.Inthefollowing,allendmemberdistributions 65

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2xjMXk=1pjkek!TMXk=1p2jkVk!1xjMXk=1pjkek!9=;(3{27)whereekandVkarethemeanspectrumandcovariancematrixforthekthendmemberdistribution,Misthenumberofendmemberdistributionsbeingdetermined,andpjkisthejthdatapoint'sproportionvalueforthekthendmember( Wackerlyetal. 1996 ).Thejointlikelihoodforallthehyperspectralpixelsisassumedtobetheproductoftheindividuallikelihoods, 2xjMXk=1pjkek!TMXk=1p2jkVk!1xjMXk=1pjkek!9=;:(3{28)Eachhyperspectraldatapointhasauniqueabundancevector.Althoughallthedatapointssharethesamesetofendmemberdistributions,theiruniqueabundancevectorsresultineachdatapointhavingauniqueGaussiandistribution.InEquation 3{28 ,themaximumlikelihoodvalueofthedatapointxjispjE.Inordertoprovideatighttaroundtheinputhyperspectraldataset,thepriorontheendmembersisdenedusingthesumofsquareddistancesbetweenthemeansoftheendmemberdistributions.ThisissimilartothepriorontheendmembersusedbySPICEalgorithm. (2)D 2exp(1 4MXk=1MXl=1(ekel)TS1(ekel))(3{29)Initially,theDirichletdistributionwasconsideredforthepriorontheabundancevalues.However,sincetheDirichletdistributionisnotaconjugatepriortoP(XjE;P),asimpleupdateformulacannotbeused.Instead,constrainednon-linearoptimizationisrequiredwhenupdatingabundancevalues.Asabundancesapproachzero(which 66

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3{30 wasdevelopedforthepriorontheabundancevectors. (M+1)M2!:Thepandcvectorsareconstrainedtobenon-negativeandsum-to-one,pjk08k=1;:::;M;MXk=1pjk=1;ck08k=1;:::;M;MXk=1ck=1:Thevectorcisthemaximumlikelihoodvalueforp.Thebktermscontrolthesteepnessoftheprior.Thisabundancepriorprefersabundancevectorswhicharebinary;thatis,vectorswithasingleabundancewithvalue1andtherestwithvalue0.Thisisaresultofthenormalizationconstant,Z.Thenumeratoroftheabundancepriorismaximizedwhencisequaltop.Thenormalizationconstantinthedenominatorisminimizedwhencisbinary.Thus,whenboththepandcvectorsarebinary,theabundancepriorismaximized.Thispropertyintroducessparsitywithinabundancevectorswhich,whencombinedwiththeexibilityachievedbyrepresentingendmembersbydistributions,representsamajoradvanceinautomateddeterminationofmeaningfulendmembersandabundances.Ifseveralendmembersadequatelydescribeadatapoint,theabundancepriorwillplaceallweightononeendmemberratherthanspreadingtheabundanceacrossendmembersencouragingthemethodtousetheminimumnumberofendmembersneeded.Furthermore,manydierentpointscanbeassignedabundancevaluesofonewithrespect 67

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3-1 .Also,plotsshowingtheabundancepriorasafunctionofcareshowninFigure 3-2 .ThealgorithmproceedsbyiterativelymaximizingP(XjE;P)P(E)P(P)whereP(P)isthejointlikelihoodofalltheabundancevectors.Giveninitialestimatesoftheendmemberdistributionsandcfromtheabundanceprior,abundancevectorsareupdatedbymaximizingthelogoftheproductofEquations 3{28 and 3{30 withrespecttoP.Thisisaconstrainednon-linearoptimizationproblem.InthecurrentMatlabimplementation,thisismaximizedusingMatlab'sfminconfunctionintheoptimizationtoolbox.Followinganupdateoftheabundancevectors,theproductofEquations 3{28 and 3{29 aremaximizedwithrespecttomeansoftheendmemberdistributions,ekfork=1;:::;M.Thismaximizationisperformeddirectlybytakingthederivativeofthelogoftheproductandsettingitequaltozero. 68

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3-3 weregeneratedusingtwoendmemberdistributions.Thestandardmodelusingconvexcombinationsofsingleendmemberspectrawouldrequirethreeendmemberstorepresentthedatawhilemaintainingasmallreconstructionerror. ChibandGreenberg 1995 ; MacKay 2003 ).Samplesareproducedinasequencewhereeachnewsampleisgeneratedbasedonthepreviousoneusingatransitionkernel.Thetransitionkerneldenestheconditionalprobabilityofmovingtoaparticularsample(oranysubsetofsamples)giventhecurrentsamplevalue( ChibandGreenberg 1995 ).Sinceeachsampleinthesequenceisproducedbasedonthepreviousone,consecutivesamplesgeneratedusingMCMCmethodsarenotindependent( MacKay 2003 ).OneMCMCsamplingmethodistheMetropolis-Hastingsalgorithmwhichusesanormalizedcandidate-generatingdensitytoprovidepotentialsamples( ChibandGreenberg 1995 ).Thesecandidatesamplesarethenevaluatedusinganacceptanceratiowhichdenestheprobabilityofretainingorrejectingthecandidatesample( ChibandGreenberg 1995 ).TheMetropolis-Hastingsalgorithmcanbeusedwhenitisdiculttogeneratesamplesdirectlyfromthetargetdistributionbutsamplescanbeeasilyevaluatedinthedistribution( MacKay 2003 ). 69

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3{32 ( ChibandGreenberg 1995 ), whereqisthecandidate-generatingdistributionwhichcanrelyonaprevioussampleandfisthetargetdensityfromwhichthesamplesaredesired( ChibandGreenberg 1995 ).Thecandidatesampleisacceptedwithprobabilitya(si1;ci).Ifthesampleisrejected,thens1=s0otherwises1=c1.Samplesaregeneratedinthissequentialmannerforalargenumberofiterations.InMetropolis-HastingsandallMCMCmethods,samplesgeneratedduringaninitialperiodofrunningthealgorithmarediscarded.Thesesamplesgeneratedduringtheburn-inperiodarediscardedsinceconvergencetothedesiredtargetdistributionhasnotyetbeenreachedandabiasbasedonthearbitrarystartingpointispresent( CasellaandGeorge 1992 ; ChibandGreenberg 1995 ).Thenumberofsamplesthatneedtobediscardedisdiculttodetermine.Onetechniquetodeterminethelengthoftheburn-inperiodisdescribedby ChibandGreenberg ( 1995 ).ThistechniqueusesseveralMetropolis-Hastingsgeneratedsamplesequenceswithvaryinginitializationpoints.Assamplesarecollectedineachsequence,variancesacrosssamplesarecomparedbetweenthechains.Thetechniqueofusingseveralchainsisalsodescribedby CasellaandGeorge ( 1992 )togenerateindependentsamplesfromanMCMCmethod.Sequenceswithdistinctstartingpointsaregeneratedforlargenumberofiterations.Thenalsamplesfromeachchainarethenusedasindependentandidenticallydistributed(iid)samplesfromthetargetdistribution( CasellaandGeorge 1992 ).Anothertechniqueusesasinglesequenceandreturnseverykthpositioninthe 70

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CasellaandGeorge 1992 ).ThepopularGibbssamplerisasignicantspecialcaseoftheMetropolis-Hastingsalgorithm( CasellaandGeorge 1992 ; ChibandGreenberg 1995 ).TheGibbssamplerproducessamplesfromamulti-variatedistributionbyiterativelysamplingfromtheconditionaldistributionofeachvariablegivenalltheothers( CasellaandGeorge 1992 ).Inthiscase,thecandidate-generatingdistributionistheconditionaldistributionforthevariablebeingsampled( ChibandGreenberg 1995 ; MacKay 2003 ).Itcanbeshownthatusingtheconditionaldistributionsforproducingcandidatesamplescausestheacceptanceratioforeverytransitiontobe1( ChibandGreenberg 1995 ).Considerthemulti-variatejointdensityoftherandomvariablesR,SandT,f(r;s;t).TheGibbssequence,r0;s0;t0;r1;s1;t1;:::isgeneratedbyiteratingbetweentheconditionalsinEquation 3{33 giveninitialvaluesforr0ands0( CasellaandGeorge 1992 ), Neal 1991 ; Tehetal. 2006 ; Xingetal. 2007 ).AreviewoftheDirichletProcessMixtureModelanditsapplicationtoadatasetforclusteringisdescribedinthissection.First,thedenitionsfortheDirichletdistribution,theDirichletProcess,andtheDirichletProcess 71

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Devroye 1986 ) D(;m)=() 3{36 3{37 and 3{38 1+ ByexaminingEquation 3{36 ,itcanbeseenthatas,theconcentrationparameter,isvaried,themeanoftheDirichletdistributiondoesnotchange.Incontrast,asisincreased,thecovariancedecreases.GiventhedenitionofaDirichletDistribution,theDirichletProcesscanbedened. Antoniak 1974 ; Ferguson 1973 ; Tehetal. 2004 ).

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Neal 1991 ).FinitemixturemodelscanbeexpressedusingEquation 3{39 JainandNeal 2000 ).ConsiderNdatapoints,fx1;:::;xNgeachofwhichareassumedtohavebeenindependentlygeneratedbysomedistributionfi(;i)whereiisthevectorofparametersthatdenestheprocessgeneratingobservationxi.UndertheDirichletProcessMixtureModel,iisgeneratedbysomeunknowndistributionG( Westetal. 1994 ).Then,GisdistributedaccordingtotheDirichletprocess,D(G0)whereG0isthebasedistributionandistheconcentrationparameter( JainandNeal 2000 ).Therefore,thecompletemodelcanbewrittenas( JainandNeal 2000 ; Neal 1998 ) 73

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Westetal. 1994 ).Tosimplifythemodel,Gcanbeintegratedouttoexpressthepriorofeachiintermsofthebasedistribution,G0,andallotherparametersets( JainandNeal 2000 ; Neal 1998 ; Rasmussen 2000 ; Westetal. 1994 ), +N1G0(3{41)whereiisthesetofcomponentdistributionsforalldatapointsotherthani,Nisthenumberofdatapoints,(i)isthedistributionoverparameterswithallweightconcentratedatparameterseti,andG0isthepriordistributionforthecomponentparameters( Neal 1998 ; Ranganathan 2006 ). Neal ( 1998 ),thelikelihoodofadatapointgivencomponentparameterscanbecombinedwiththeprobabilityofaclasslabelgivenallotherlabelsinEquation 3{41 .Then,theGibbssamplercanbeusedtosampleindicatorvariablevaluesandcomponentparametervalues.TheconditionalprobabilitiesforanindicatorvariablearedenedinEquation 3{42 +N1Zf(xij)G0()d(3{42)whereCisanormalizingconstantcomputedbyEquation 3{43 1 +N1Rf(xij)G0():(3{43)TheMarkovChainfortheGibbssamplerusingtheconditionalsinEquation 3{42 consistsofalltheindicatorvariablescandallcomponentdistributionparameters( Neal 1998 ).IfG0isaconjugatepriortothelikelihooddistributions(componentdistributions)f(j),thentheintegralinEquation 3{42 canbeanalyticallycomputed( Neal 1998 ).Assuming 74

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3{42 requiresthatonlytheindicatorvariablesoftheobservationsneedtobesampled.Inthiscase,theconditionaldistributionsareexpressedasinEquation 3{44 ( JainandNeal 2000 ), +N1Zf(xij)G0()d(3{44)whereCisanormalizingconstantandHi;cjistheposteriordistributionofthecomponentparametersgivenpriorG0andcurrentindicatorvaluesci( JainandNeal 2000 ).TheseintegralsremovetheneedtoincludecomponentparametersintheMarkovChainwhichsignicantlyreducesthesearchspacefortheGibbssampler( JainandNeal 2000 ).ForcasesinwhichG0isnotaconjugatepriortothelikelihoodfunctions,techniqueshavebeendevelopedtoeitherestimatetheintegralvaluesorusesamplingtechniquestoavoidtheneedtocomputetheintegralvalues.Someofthesetechniquesarediscussedby Neal ( 1998 ).OnemethodusestheMetropolis-Hastingsalgorithmwherethecandidatedistributionforaparameterset,i,is1 N1+G0andtheacceptanceprobabilityisa(i;thetai)=minh1;f(xi;i) Neal 1998 ).SpeciccasesofthisGibbsalgorithmfortheDirichletProcessMixtureModelarederivedintheliterature. Rasmussen ( 2000 )derivesthealgorithmwherethecomponentdistributionsandpriorsareallGaussian. Neal ( 1991 )derivesthemethodforcategoricaldatausingaBernoullidistributionsforthecomponentdistributionsandaBetadistributionsforthepriors. 75

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Rasmussen ( 2000 )and Westetal. ( 1994 ),thismodelcanalsobeextendedbyaddinghyper-priorsfortheparameterandtheparametersofthepriordistributionG0. 3{27 ,eachdatapointisarandomvariablewithauniquedistribution.EachdatapointhavingauniquedistributioncontrastswiththeDPMMapproachwheredatapointsfromeachclusterareassumedtobeidenticallydistributed.ThePCEalgorithmperformsGibbssamplingwithDirichletprocesspriorstosamplethepartitiontowhicheachdatapointbelongs.Theprobabilityofsamplingapartitioniscomputedusingthelikelihoodofadatapointbelongingtoaconvexcombinationofthe 76

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+N1Zf(xijE)G0(E)dE=NV0(V0+V)1xi+V(V0+V)10;V10+V11+V(3{45)whereriistheindicatorvariableforthecurrentdatapoint,xi,Cisanormalizationconstant,ni;jisthenumberofdatapointsexcludingxiinpartitionrj,Nisthetotalnumberofdatapoints,andistheinnovationparameterfortheDirichletprocess.Thematrices,TandS,correspondtoPkc2kVkandPkp2ikVk,respectively.ThematricesVandVrjareallthecovariancematricesassociatedwithnewandexistingendmemberdistributions.Inthecurrentimplementationofthisalgorithm,allcovariancematricesforendmemberdistributionsaresettothesameconstantmatrixvalue.Thepriordistribution,G0,isGaussianwherethemean,0,issettothemeanoftheinputdatasetandthecovariance,V0,isconstant, 3{27 .Thevector,prji,containstheproportionvaluesforthecurrentdatapointinpartitionrj.TheseproportionvaluesaredeterminedbymaximizingtheproductofEquations 3{30 and 3{27 giventheendmembersofthepartition,Erj.Thelikelihood 77

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3{30 givenPrandVr.Byincorporatingthisupdatedprior,thelikelihooddependsnotonlyonthedistancetopEbutalsotocE.Whenthecovariancematricesforendmemberdistributionsareequal,theupdatedpriordependsonthedistancetoapointonthelinesegmentconnectingpEandcE,namely,w1pE+w2cEwherew1=PMk=1c2k 3{45 .Theunitintervalisthendividedintoregionswhoselengthsareequaltoeachpartition'snormalizedlikelihoodvalue.Arandomvaluefromtheunitintervalisthengenerated.Thecorrespondingpartitionwhoseregionincludesthegeneratedrandomvalueisthepartitionwhichissampledforthecurrentdatapoint.Afterapartitionissampled,theparametersofthesampledpartitionareupdated.Thisisdonebyupdatingthepriorontheabundances,Equation 3{30 ,withrespecttocforthegivenpartition.AfteroneormoreiterationsofthepartitionsamplingschemeusingtheDirichletprocess,theendmemberdistributionsandallproportionvaluesareupdatedusingadesignatednumberofiterationsoftheEDalgorithm.SeveralitemsinthefollowingPCEpseudo-codedierfromthestandardDPMMmethod.Asstatedinlines10and13ofthepseudo-code,inPCE,apartition'sparameters 78

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3{45

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PlotsofED'sabundancepriorforM=2andvariouscandbvalues.Thex-axisisthe1stabundancevalue.They-axisisthepriorprobabilityvaluefortheabundancevector.A)c=[:5;:5]B)c=[.75,.25]. PlotsofED'sabundancepriorforM=2andvariouspandbvalues.Thex-axisisthe1stcvalue.They-axisisthepriorprobabilityvalueforc.A)p=[:45;:55]B)p=[.5,.5]. 80

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Datapointsgeneratedfromlinearcombinationsof2endmemberdistributions.Theendmemberdistributioncenteredat(5,5)hasadiagonalcovariancewhoseelementsareallequalto0.005.Theendmemberdistributioncenteredat(1,1)hasadiagonalcovariancewhoseelementsareallequalto0.5.Datapointsareshowninblue.Meanspectraandstandarddeviationcurvesfortheendmemberdistributionsareshowninred. 81

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4.1.1 to 4.1.4 4-1 showsthedatasetandtheendmembersfromwhichthedataweregenerated.Thesedatapointsweregeneratedinthesamefashionasthetwo-dimensionalexampleshownby Bermanetal. ( 2004 ).Theendmembersthatwereusedtogeneratethe100datapointswere(10p Bermanetal. ( 2004 )assignthenumberofendmembersforthisexampletothree.InSPICE,thenumberofendmembersdoesnotneedtobeknowninadvance.Therefore,thealgorithmcanbeinitializedwithalargenumberofendmembers.TheresultsofthreeexperimentscomparingtheICEandSPICEalgorithmsareshowninFigure 4-2 .Theparametersforeachalgorithm,otherthanthesparsity-promotingterm,weresettobethesameduringtheexperiments.Theinitialnumberofendmembersforallthreerunswas20,andwassetto0.001.TheparameterfortheSPICEalgorithmwassetto10,20,and5forthethreeruns,respectively.ICEandSPICEwereinitializedtothesameendmembersforeachexperiment.Theseinitialendmemberswerechosenrandomlyfromthedataset. 82

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4-2D ,twooftheendmembersthatarefoundbyICEwere(-3.62,7.94)and(-3.68,7.94);theyappearasoneendmemberinthegure.SPICEwasalsoappliedtothetwo-dimensionaldatausedtothetesttheMorphologicalAssociativeMemorymethodinSection 2.1.4 .TheSPICEresultsonthisdatasetareshowninFigure 4-3 .Fortheresultsshown,wassetto20andwassetto0.01. AVIRIS ).ThisdatasetwaschosentobeabletocomparetheresultswiththeNFINDRresultspresentedby Winter ( 1999 ).Following Bermanetal. ( 2004 ),SPICEwasrunonasubsetofpixelsfromtheimagetoreducecomputationaltime.Thesecandidatepointswereselectedusingthe 83

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Bermanetal. 2004 ; Boardmanetal. 1995 ).Intheseexperiments,thecandidatepointswerechosenfrom10,000randomprojections.Pointswithinadistanceoftwofromtheboundaryoftheprojectionreceivedincreasedpurityindices.The1011pixelswiththehighestPPIwereusedasthecandidatepoints.1000pixelswereusedby Bermanetal. ( 2004 )duringtheexperimentsontherealimagesets.APPIthresholdthatproducedascloseto1000pixelsaspossible(manypixelshadthesamePPI)waschosen.Also,fastimplementationsforthealgorithmcanbecreatedaswasdoneby Bermanetal. ( 2004 )toavoidtheneedtoselectasubsetofthepixels.ThespectralprolesofthenineendmembersthatwerefoundbySPICEtorepresentthisimageareshowninFigure 4-4 .ThethreeendmembersintheFigures 4-4C 4-4G ,and 4-4I comparewelltothreeendmembersthatarefoundandidentiedaskaolinite,alunite,andcalciteby Winter ( 1999 ),respectively.Figure 4-5 showsacomparisonof 4-4I totheU.S.GeologicalSurvey(USGS)spectrallibrarydataonalunite( Clarketal. 2004 ).AlthoughitisclearthatSPICEwasabletondsomeofthesameendmembersthatareidentiedby Winter ( 1999 ),itisnotclearifthecorrectnumberofendmemberswasfound.Thedicultyofusingrealimagedataisthatthecorrectnumberofendmembersinthesceneisunknown.Toovercomethisproblem,asubsetoftheCupritedatawasusedforfurthertestingofthemethod.Threeendmembers,showninFigure 4-6 ,wereselectedfromthehyperspectralimagebyhand.ThesquaredEuclideandistancewascalculatedfromeverypixelintheimagetothesethreeendmembers.Thepixelswithin500,000squaredEuclideandistancefromthesethreehand-selectedendmemberswerecollectedandusedasatestsetforSPICE.ThetestsetwasnormalizedandisshowninFigure 4-7 .Table 4-1 showsthenumberofendmembersthatarefoundusingSPICEforarangeof'sandinitialnumberofendmembers.Asshown,SPICEconsistentlyndsthreeendmembersforthisdataset.TheresultsinTable 4-1 andinFigure 4-2 showthattheSPICEalgorithmisfairlystablewithrespectto.SPICEisalsoverystablewithrespect 84

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4-8 showstheendmembersthatarefoundusingSPICEintheseexperiments.Theseendmembersareclearlyverysimilartothethreehand-selectedendmembersusedforthisexperiment.+ GranaandGallego 2003 ; SerpicoandBruzzone 2001 ).Theremainingeldareaissoilcoveredwithresiduefromthepreviouscrop.Thenotill,mintill,andcleantilllabelsindicatetheamountofpreviouscropresidueremaining.Notillcorrespondstoalargeamountofresidue,mintillhasamoderateamount,andcleantillhasaminimalamountofresidue( SerpicoandBruzzone 2001 ).Figure 4-9 showsband10(approximately0.49m)andthegroundtruthofthedataset.Only49%ofthepixelsintheimagehavegroundtruthinformation( SerpicoandBruzzone 2001 ).SPICEwasrunonasubsetoftheimagepixels.1100pixelswererandomlyselectedfromtheimage.BeforerunningSPICE,thesepixelswerenormalized.Theinitialnumberofendmembers,,andweresetto60,0.1,and1,respectively.TenendmemberswerefoundforthisdatasetusingSPICE.TheresultingabundancemapsareshowninFigure 4-10 .SPICEprunedunnecessaryendmembersandprovidedinterpretableresultsthatcomparewelltopreviouslypublishedresultsonthisdataset( GranaandGallego 2003 ; Granaetal. 2003 ; Miaoetal. 2006 ).InFigure 4-10 ,theimageswerefoundtoroughlycorrespondtothefollowing:(A)and(I)arewoodsandtreecanopies;(B),(C),and(J)areamixtureofsoybeanandcorn 85

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4-11 .Theendmembersroughlycorrespondtothefollowingclasses:(A)grass/pastureandwoods,(B)hay-windrowed,alfalfaandgrass/pasture-mowed,(C)and(E)correspondtocornandsoybean,(D)stone-steeltowers,and(F)grass/trees,wheat,woods.NormalizedhistogramsshowingthedistributionofabundancesvaluesamongendmembersineachgroundtruthclassareshowninFigure 4-12 .Thesehistogramswerecomputedbysummingalltheabundancevaluesassociatedwithanendmemberineachgroundtruthclass.Eachhistogramwasnormalizedbydividingbythenumberofpointsinthecorrespondinggroundtruthclass, Luceyetal. 1998 ; ZareandGader 2007b ).AHIcollects256spectralbandsofdatafromthelongwaveinfraredregionintherangeof7.88to11.49microns( Luceyetal. 1998 ).TheAHIdatasetcollectedistrimmedandbinneddownto70bandsoverthesamewavelengths( Luceyetal. 1998 ).Thedatasetusedfortheseresultswascollectedfrom 86

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ZareandGader 2007b ).ThreeAHIimageswereusedandscoringwasperformedtodeterminethereductioninfalsealarmrate.Scoringfortheresultsinthispaperwascarriedoutoverregionsofinterestintheimagery.TheregionsofinterestforthisstudyweredenedastheareaswherecollectedLynxSyntheticApertureRadarandAHIimageryintersect( LYNXSAR ).Fourminetypesweredistributedintheintersectingregions.Twooftheminetypeswereplasticcased(PC)andtwoweremetalcased(MC).ThedistributionofminestypesintheintersectingregionsoftheAHIandLynximagesaredisplayedinTable 4-2 ( ZareandGader 2007b ). Frenchetal. 2000 ).Additionally,skewnessofemissivityacrossspectralbandshasbeenseentobehelpfulindistinguishingvegetation( Zareetal. 2008 ).Toexploitthisinformation,theSPICEalgorithmcanberunontheemissivityspectracalculatedfromLWIRhyperspectraldata.Forthisstudy,theemissivityspectrumofeachpixelintheimageiscalculatedusingtheEmissivityNormalizationMethod( KealyandHook 2000 ).AfterapplyingtheSPICEalgorithmtotheemissivityspectra,theendmembersdeterminedbythealgorithmareexamined.Theendmemberwiththehighestmeanandtheloweststandarddeviationisdeterminedtobetheblackbodyendmember, (4{3) 87

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(4{4) wherejcorrespondstothejthpixelintheimage.AmaskV,isdenedusingtheblackbodymapbyinvertingthevaluesandenhancingthemapusingalocal3minimumlter, (4{6) wheretisthethresholddeterminedusingOtsu'sthresholdingmethod( Otsu 1979 ).Thepartialthresholdisappliedsothattheonlyvaluesmodiedbythemaskarethosethatareassociatedwithpixelsthatbehavelikeablackbody( ZareandGader 2007b ).Following Bermanetal. ( 2004 ),SPICEwasrunonasubsetofpixelsfromtheimage.ThesubsetwasselectedusingthePixelPurityIndex(PPI)algorithm( Boardmanetal. 1995 ).Thesubsetwaschosenusing30,000randomprojections.Pointswithinadistanceofthreefromtheboundaryoftheprojectionreceivedincreasedpixelpurityvalues.Athresholdwasselectedtoallowascloseto1000pixelsaspossible(manypixelshavethesamePPI).Thenumberofpointsselectedwas1095,767and1103forAHIImages1,2, 88

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Zareetal. ( 2008 ).Since Zareetal. ( 2008 )usedonlythestatisticsofemissivity(mean,standarddeviationandskewnessacrossspectralbands)insteadofthefullemissivitycurve,theresultsdisplayedarethosegeneratedbyrunningSPICEononlythestatisticsofemissivityinsteadofthefullemissivityspectra.Thismethodwasusedtobeabletocompareperformanceoftheclusteringmethodby Zareetal. ( 2008 )andtheSPICEmethoddirectlywithoutaddingconfusionoverwhetherthedierenceinperformanceresultedfromthemethodsortheinputdata.Furthermore,apartialthresholdasdenedinEquation 4{6 wasalsoappliedtothemaskgeneratedbytheclusteringmethodby Zareetal. ( 2008 ).IncontrasttoSPICE,whichndsthedesirednumberofendmembersforadataset,themethodby Zareetal. ( 2008 )requiresthenumberofclusterstobesuppliedtothealgorithm.Themethodwasrunonthisdatawiththenumberofclustersrangingfromthreetosix.Theresultsdisplayedarethebestresultsobtainedoverthisrangeofnumberofclustervalues.Figure 4-14 showstheblackbodymaskgeneratedusingSPICEandtheclusteringmethodforfourandveclusters.Whencomparingthetwomasksgeneratedbytheclusteringmethod,itcanbeseenthatwhenveclustersischoseninsteadoffour,manyofthevegetationpixelsarebeingsplitbetweenmultipleclustersand,thus,arefartherfromtheselectedvegetationclustercenter.SinceSPICEautomaticallyselectsthedesirednumberofendmembers,thisdicultyiseliminated.WhenexaminingtheSPICEmaskandtheclusteringmaskgeneratedwith4clusters,itcanbeseenthattheSPICEmaskprovidesmoresolidvegetationregionsthusprovidingabettermappingofthevegetationpixelsthantheclusteringmethod.PointsofInterest,POIs,intheoverlapregionsoftheimagerywerefoundusingtheRXdetectoralgorithm( Yuetal. 1993 ).TheRXdetectorappliedwasanimplementation 89

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Winter ( 2004 ).TheRXalgorithmwasappliedtodetectburiedminesintheLWIRhyperspectralimagery.TheblackbodymaskisincorporatedbymultiplyingtheRXcondenceofeveryPOIwiththeircorrespondingblackbodymaskvalue.Thisdiersfromthedetectionalgorithmsusedby Zareetal. ( 2008 )wheretheblackbodymaskisappliedtotheoutputofaChoquetfusionsystemincorporatingseveraldetectionalgorithms.Intheseresults,onlythecomparativeperformanceofthetwoblackbodymasksarebeingexamined.TheresultsineachofthethreeoverlapregionsareshowninTables 4-3 4-4 and 4-5 .Theprobabilityofdetection,PD,isdenedasthenumberofmineswithacondenceabovethethresholddividedbythetotalnumberofmines.Thefalsealarmrate,FAR,isdenedasthenumberofnon-minesabovethethresholddividedbythenumberofsquaremetersintheoverlapregion.AlthoughRXwasappliedtodetectburiedmines,theresultsareshownoverallminetypesintheoverlapregions.Ifdetected,ducialmarkersinthesceneareconsideredfalsealarms.TherstlineineachtabledisplaysthefalsealarmrateswithoutanyblackbodymaskbeingusedontheRXvalues.ThesecondlinedisplaystheFARsafterapplyingtheblackbodymaskgeneratedusingtheclusteringmethod.ThethirdlineshowsthereductionintheFARafterusingtheblackbodymaskfromtheclusteringmethod.ThefourthlinedisplaystheFARafterapplyingtheblackbodymaskgeneratedusingSPICE.Finally,thefthlineshowsthereductioninFARafterusingtheblackbodymaskfromSPICEwhencomparedtotheresultswithoutusingablackbodymask( ZareandGader 2007b ).TheblackbodymaskgeneratedusingtheSPICEalgorithmcanprovidefalsealarmreductionduringlandminedetection.Incomparisontotheclusteringmethodby Zareetal. ( 2008 ),theSPICEmethodprovidesimprovedvegetationdetectionandeliminatestheneedtosetthenumberofclustersorendmembersneeded. 90

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4.2.1 to 4.2.3 AVIRIS ).ThechosenendmembersareshowninFigure 4-15 .ThedatasetwasgeneratedfromtheendmembersfollowingtheconvexgeometrymodelinEquation 1{1 .Asimulateddatasetwasusedtoverifythatthemethodcanrecovertheendmembers,performeectivebandselection,andproduceaccurateabundancevaluesforeachpixel.Thesecanbetestedusingsimulateddatasincethetrueendmembersandabundancesareknown.B-SPICEandSPICEwererunonthisdatasetforarangeofvalues.Allparameters,otherthanthoseinvolvedwithbandselection,wereheldconstantforeachrunofthealgorithm.Theparameterwassetto2000,was0.3,was0.3fortherst150iterationsandthensetto0,theinitialnumberofendmemberswassetto20,andtheendmemberpruningthresholdwas1108.Theinitialendmemberswereselectedrandomlyfromthedataset.WhenrunningB-SPICE,bandselectionwasnotstarteduntilthe100thiteration,afterwhich,thebandweightswereupdatedeveryfthiteration.Thebandpruningthresholdwassetto1105,andthebandweightchangethresholdwassetto1105.wassetto0(forSPICE),0.25,0.5,0.75,and1(forB-SPICE).B-SPICEandSPICEwererunonthedata50timesforeachparameterset.AnexampleoftheendmembersfoundusingeachvalueisshowninFigure 4-16 .Table 4-6 showsthemeanandstandarddeviationofthenumberofendmembersandthenumberofbandsretainedforeachparametersetoverthe50runsofthealgorithm.Ascanbeseen,bothSPICEandB-SPICEareabletoconsistentlydeterminethecorrectnumberofendmembers. 91

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4-7 .Themedianaveragesquarederrorperabundancevaluewascomputedbytakingthemedianover50runsofthealgorithmoftheaveragesquarederrorbetweeneachpixelstrueandcomputedabundancevalues.Asshown,themedianaveragesquarederrorperabundancevalueisfairlystableacrossthevalues,indicatingthatB-SPICEcanbeaseectiveatdeterminingthetrueabundancevaluesastheSPICEalgorithm.Therefore,byusingB-SPICE,thenumberofbandscanbereducedwhilemaintainingtheabilitytodetermineabundances.However,whenexaminingthestandarddeviationoftheaveragesquarederrorperabundancevalue,itisseenthatSPICEismoreconsistentthanB-SPICE.ThereisanorderofmagnitudedierencebetweenthestandarddeviationsofSPICEandB-SPICE. 4.1.3 .SPICEandB-SPICEwereruntwiceforvedierentvalues.Allparameters,otherthantheparameter,wereheldconstantforeachrunofthealgorithm.Toreduceruntime,SPICEandB-SPICEwererunon1000pixelsrandomlychosenfromthedataset.Afterdeterminingtheendmembersandselectedbandsusingthesubset,unmixingwasperformedontheentiredatasettondabundancevaluesforeverypixel.Theparameterswassetto5000,to0.3,andto0.2fortherst100iterationsandthento0.Theinitialnumberofendmemberswassetto20andtheendmemberpruningthresholdwas1108.Initialendmemberswereselectedrandomlyfromthedataset.WhenrunningB-SPICE,bandselectionwasnotstarteduntilthe100thiteration,afterwhich,thebandweightswereupdatedeveryfthiteration.Thebandpruningthresholdwassetto1105,andthebandweightchangethresholdwassetto1105.wassetto0(forSPICE),0.5,1,5,and10(forB-SPICE).ThenumberofendmembersandthenumberofbandsfoundareshowninTable 4-8 92

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Guoetal. ( 2006 ),supervisedclassicationwasperformed.Thefeaturesusedforsupervisedclassicationweretheabundancevaluescomputedforeachpixelinthe16classesofthedataset.Theunlabeledpixelswerenotincludedintheseexperiments.Sincetheabundancevalueswerethefeaturesusedforclassication,thedimensionalityofthefeaturevectorsisequaltothenumberofendmembersfoundforthedataset.Two-foldcross-validationwasperformedonthedatasetusinga1-versus-1RelevanceVectorMachine(RVM)classicationmethod( Tipping 2001 ).Thetrainingandtestingsetsweredenedbyrandomlysplittingeachofthe16classesinhalf.AnRVMwastrainedforeachpairofclasses.Sincethereare16classes,120RVMsweretrainedforeachtestset.TestpixelswereclassiedbycountingthenumberofRVMsthatassignedthepixeltoeachclass. 4-8 .Sincetheclassicationaccuraciesdependontherandomsplittingofthedataintotrainingandtestingsets,classicationwasperformedthreetimesforeachrunoftheB-SPICEalgorithm. Wangetal. ( 2006 )providesupervisedclassicationresultswithbandselectionontheIndianPinesdataset.Theresultsshownby Wangetal. ( 2006 )showverygoodclassicationaccuraciesrangingfrom90%to94.5%withlessthan50bands;however, 93

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ArchibaldandFann ( 2007 )and HuangandHe ( 2005 ),buttheresultsareprovidedononlyasubsetofthelabeledclasses.Table 4-8 alsoshowsresultsfrom Martinez-Usoetal. ( 2006 );onlytheresultswithlessthan50bandswereprovided. 4-17 showsthehistogramsofthenumberofendmembers,thenumberofbands,andthenumberoftimeseachbandwasretained.ModesofthehistogramsinFigure 4-17 are7and114,respectively.Themostfrequentlyretainedbandsoverthe240runswere1-57,61-76,81-100,and118-138.Byusingthesemodesandthemostfrequentlyretainedbands,ICEcanberuntondendmembersandabundancevalues.Inotherwords,thenumberofendmembersandthebandstoretainweredeterminedusingthehistogramsfoundbyrunningB-SPICEoversampledparametervalues.ThesevalueswerethenusedtosetthenumberofendmembersandthebandstousefortheICEalgorithm.TheclassicationaccuraciesusingthesampledparametervaluesweredeterminedusingthesameclassicationmethoddoneinthepreviousIndianPinesexperimentinSection 4.2.2 .Table 4-9 showstworunsoftheICEalgorithmandwiththreerunsof1-versus-1classication. 94

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4.3.1 4-1 .TheresultsfoundonthisdatasetusingtheEDalgorithmwiththeparametersvalueslistedinTable 4-10 areshowninFigure 4-18 .Afterrunningthealgorithm,thenalcvectorfoundfortheabundancepriorwas[.47.27.26]wherethevaluescorrespondtotheendmemberdistributionscenteredat(-11.9,1.9),(-0.1,18.5)and(7.5,6.9),respectively.Ascanbeseen,EDperformedasexpected.TheendmemberdistributionssurroundthedatapointsandcomparewelltotheendmemberresultsfoundbySPICEinFigure 4-2 .EDwasalsorunonthetwo-dimensionaldatashowninFigure 4-19 .Thisdatawasgeneratedbysamplingendmembersfromthreeendmemberdistributionsandcomputingthedatapointsasconvexcombinationsofthesampledendmembersusingrandomlygeneratedabundancevalues.TheEDresultsonthisdataareshowninFigure 4-20 .ParametersusedtogeneratetheseresultsareshowninTable 4-10 .Again,EDgeneratedtheexpectedresults.Theendmemberdistributionsthatwerefoundareverysimilartothoseusedtogeneratethedata.Forcomparison,SPICEwasalsorunonthetwo-dimensionaldatainFigure 4-19 .TheSPICEwasrunonthedatasetwith=0:01,=1and10initialendmembers.TheresultingendmembersareshowninFigure 4-21A .Asshowninthegure,SPICEneededfourendmemberstorepresentthedataset.SPICEresultsfoundusing=0:001,=1and10initialendmembersareshowninFigure 4-21B .Bydecreasing,lessemphasisisplacedonthesum-of-squareddistancestermintheSPICEobjectivefunctionwhichmayresultinSPICErequiringasmaller 95

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4-15 .ThedatasetwasgeneratedbasedontheconvexgeometrymodelinEquation 1{1 .TheresultsusingEDonthisdatasetareshowninFigure 4-22 .Parametervaluesusedtogeneratetheseresultsareshownin 4-11 .AscanbeseeninFigure 4-22 ,themeansoftheendmemberdistributionsmatchthetrueendmemberswell.EDwasalsorunonthesubsetofAVIRISCupritedatashowninFigure 4-7 .ThisdatasetisacompilationofthepixelsspectrallysimilartothreeendmembersselectedfromtheAVIRISCupritedata.TheendmembersareshowninFigure 4-6 .ResultsonthisdatasetfoundusingtheEDalgorithmareshowninFigure 4-23 .Ascanbeseeninthegure,themeansoftheendmemberdistributionscloselymatchthetrueendmembersandthedataset.TheseresultssuperimposedontheinputdatasetareshowninFigure 4-24 4-25 .Thisdatasetwasgeneratedfromthreesetsofendmembers,eachsetformingatriangleofdatapoints. 96

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4-26 .Priortorunningthealgorithm,partitionswereinitializedusingtheKernelGlobalFuzzyC-Means(KG-FCM)algorithmandtheDirichletProcessMixtureModelalgorithmresultingin8partitions( HeoandGader 2008 ).EndmembersandabundancesforeachpartitionweretheninitializedbyrunningtheEDalgorithmoneachcluster.TheparametersusedtogeneratetheseresultsareshowninTable 4-12 .Ascanbeseen,PCEpartitionedthedatasetintothecorrectnumberofconvexregions.Furthermore,PCEwasabletoidentifyanappropriatesetofendmembersforeachconvexregion. 4.1.3 .PriortorunningthePCEalgorithm,thedatadimensionalitywasreducedfrom220bandsto6dimensionsusingprincipalcomponentsanalysis.Atotalof1037pixels(every10thlabeledpixel)wereselectedfromthedatasetandusedinthePCEalgorithm.PartitionsonthisdatawereinitializedusingtheKG-FCMalgorithmandtheDPMMalgorithmresultingin3partitions.Afterinitialpartitionswerefound,endmembersforeachpartitionwereinitializedusingtheEDalgorithm.Eachpartitionwasrestrictedto3endmembers.TheparametersusedtogenerateresultsshownarelistedinTable 4-12 .Inordertocomputeabundancemapsfortheentireimage,everydatapointwasunmixedusingeachpartitions'setofendmembersandthelikelihoodundereachpartitionwascomputed.Everydatapointwasthenassignedtopartitionwiththelargestlikelihoodvalue.Also,allendmemberswhosemaximumproportionvaluewaslessthan0.05wereremoved.Followingthesesteps,13clusterswerefoundwithatotalof14endmembers.Figure 4-27 showstheabundancesmapsassociatedwitheachendmember.ForcomparisonwiththeSPICEresultsinFigure 4-12 ,normalizedhistogramsshowingthedistributionofabundancevaluesacrosseachendmemberwerecomputedusingEquation 4{2 .ThehistogramsfoundareshowninFigure 4-28 .Whencomparing 97

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Bishop 2006 ).Asmallerentropyvalueindicatesthatafewernumberofendmembersarebeingusedtodescribeeachgroundtruthclassandthattheendmembersarebetterrepresentativesofthegroundtruthclasses.ThesumoftheShannonentropiesfortheSPICEhistogramscomesto19.0.Incontrast,thesumoftheShannonentropiesforthePCEhistogramsissignicantlylowerat9.4.ThisindicatesthatPCEproducesendmemberswhichbetterrepresentthegroundtruthclasses.ThehistogramsandabundancemapsassociatedwithseveralofthegroundtruthclassesverifythatPCEisproducingendmemberswhichprovideabetterrepresentationofthedatathantheendmembersproducedbySPICE.Someofthesegroundtruthclassesarewheat,stone-steeltowers,hay-windrowed.ConsiderthewheatgroundtruthclassintheSPICEandPCEresults.TheSPICEabundancemapassociatedwiththemostamountofwheatisshowninFigure 4-11F andthecorrespondinghistogramisfoundinFigure 4-12M .Byexaminingtheabundancemap,itcanbeseenthatmanypixelsotherthanwheathavenon-zeroabundancevaluesassociatedwithwheat'sSPICEendmember.Incontrast,veryfewpixelsoutsideofthewheatgroundtruthclasssharewheat'sendmember.ThisisshowninthePCEabundancemapinFigure 4-27J .Furthermore,byexaminingtheSPICEhistogramforwheat,onlyabout60%ofthewheatpixels'abundancevaluesareassociatedwiththatendmemberwhereas100%ofwheat'sabundancevaluesareplacedwiththeassociatedendmemberfoundusingPCE.Forthestone-steeltowersgroundtruthclass,morethan70%ofthepixelsassignedtoasingleendmemberusingPCEandthatendmemberisnotassociatedwithanyothergroundtruthclasses.TheSPICEendmembermostassociatedwiththestone-steeltowersgroundtruthclassisalsousedbyeveryothergroundtruthclass. 98

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4-28H ),grass/pasture-mowed(Figure 4-28G )andalfalfa(Figure 4-28A )PCEhistogramsshowthattheyareassociatedwiththesameendmember.ThiscanalsobeseenintheabundancemapinFigure 4-27I .ThecorrespondingSPICEhistogramsforhay-windrowed,grass/pasture-mowedandalfalfainFigures 4-12H 4-12G ,and 4-12A showthatthethreeground-truthclasseshavesimilarhistogramshapesandsharethesameendmembers.However,theabundancesfoundbySPICEarespreadamongthreeendmemberswhereasPCEplacedtheirfullweightwithoneendmember.SoybeanandcorngroundtruthclassesconstitutealargemajorityoftheIndianPinesscene.IntheSPICEresults,abundancevaluesassociatedwiththesoybeanandcornclassesarespreadoverallofthesixendmemberfound.Incontrast,thePCEendmemberresultsplacesnearlyallsoybeanandcornabundanceswiththe2nd,6th,and10thendmembers.AnotherindicationthatPCEisproducingrepresentativeendmembersisfoundwiththeBuilding/Grass/Trees/Drivegroundtruthclass.Thisclassiscomposedofavarietyofmaterialtypes.Interestingly,thisisclearlyshownintheclass'PCEhistogram(Figure 4-28O ).Theabundancevaluesfortheclassarespreadacrossmanyendmembers.InordertoverifythatthedierenceintheresultsbetweenPCEandSPICEarenotduetodierentdatadimensionalityandadierentnumberofendmembers,theICEalgorithmwasrunonthesameAVIRISPCA-reducedIndianPinesdatasetdiscussedinthissection.TheICEalgorithmwasemployedratherthanSPICEsincethenumberofendmemberscanbesettothesamenumberfoundbyPCE.TheICEalgorithmwasrestrictedto14endmembersandtheparameterwassetto0.01.TheresultingabundancemapsareshowninFigure 4-29 andthecorrespondinghistogramsareinFigure 4-30 .ThesumoftheICEhistogramentropiesfromtheseresultsis29.2.Incomparison,PCE'svaluewas9.4.Therefore,althoughICEwasrestrictedtothesamenumberofendmembersfoundusingPCE,ICEdidnotproduceendmembersthatrepresentthe 99

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4-30I 4-30J 4-30K ,and 4-30L andthePCEhistogramsinFigures 4-28I 4-28J 4-28K ,and 4-28L areindicativeofthis.Thesehistogramscorrespondtotheoatsandsoybeanclasses.IntheSPICEhistograms,theabundancevaluesarespreadacrossalloftheendmembers.Incontrast,thePCEhistogramsforthesegroundtruthclassesconcentratetheabundancevaluestoafewendmembers.ThePCEresultsinthissectionstronglyindicatethatthealgorithmproducesendmemberswhichcorrespondverywelltothetruegroundtruthclasses. Martinez-Usoetal. 2006 ).Thedatadimensionalitywasreducedfrom220to3dimensions.Thehierarchicaldimensionalityreductioncomputedthepair-wiseKL-divergencesbetweenthebands'normalizedhistogramsofintensityvalues.TheKL-divergenceswerethenusedtohierarchicallygroupsimilarbands.Theaveragevalueacrosseachgroupofbandswasusedtoformthereduceddimensionalitydataset.PartitionswereinitializedusingtheKG-FCMalgorithmandtheDPMMalgorithmresultingin3clusters.InitialendmemberswerefoundforeachpartitionusingtheEDalgorithm.Atotalof1037pixels(every10thlabeledpixel)wereselectedfromthedatasetandusedinthePCEalgorithm.ParametervaluesusedtogeneratetheresultsshownonthisdatasetarelistedinTable 4-12 .Inordertocomputeabundancemapsfortheentireimage,afterndingendmembersonthesubsetofpixelsusingPCE,everydatapointwasunmixedusingeachclusters'setofendmembersandthelikelihoodundereachclusteriscomputed.Everydatapointwasassignedtoclusterwiththelargestlikelihoodvalue.Furthermore,partitionswithlessthan5assignedpixelswerepruned.Followingthesesteps,2clusterswerefoundwithatotalof6endmembers.Figure 4-31 showstheabundancesmapsassociatedwitheachendmember. 100

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4-31 andhistogramsareshowninFigure 4-32 .TherstpartitionfoundusingPCEonthethree-dimensionalitydatacorrespondedtothemajorityofthecornandsoybeangroundtruthclasses.Hayandalfalfawerealsoassociatedwiththerstpartition.Thesecondpartitionincludedthemajorityofthegrass,treesandwoods.ThesumoftheShannonentropyvaluesoverthehistogramsfromthePCEresultscameto16.3comparedtoSPICE's19.0value.Again,PCEprovidedmorecompacthistogramsandSPICEindicatingthattheendmembersarebetterrepresentativesofthetruegroundtruthclasses. 4-12 .AfterndingendmembersonthesubsetofpixelsusingPCE,everydatapointintheimagewasunmixedusingeachclusters'setofendmembersand,foreverydatapoint,thelikelihoodundereachclusterwascomputed.Eachdatapointwasthenassignedtothepartitionwiththemaximumlikelihoodvalue.Partitionswithlessthan3pointswereremoved.Followingthesesteps,twopartitionswerefoundwithatotalofsixendmembers.Figure 4-33 showstheabundancesmapsassociatedwitheachendmember.Figure 4-34 containsthenormalizedhistogramsforthissetofresults.Therstpartitionroughlycorrespondstothevarioustendedeldsintheimagerywhereasthesecondpartitionhasmanyoftheabundancesassociatedwithtrees,grassandwoods.Thesumoftheentropiesofthehistogramsfromthisresultscameto15.4.ThisisvaluesmallerthantheSPICEresultsof19.0indicatingthattheendmembersarebetterrepresentativesofthegroundtruthclassesthantheendmembersfoundbySPICE. 101

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NumberofendmembersfoundbySPICEandICEontestpixelsfromAVIRISCupritedataoverarangeofvaluesandinitialnumberofendmembers.EachexperimenthadthesameinitializationforICEandSPICE.Theparameterwas0.1forallexperiments.Thepruningthresholdwassetto1109. NumberNumberInitialnumberGammaconstantofendmembersofendmembersExperimentofendmembersforSPICEfound,SPICEfound,ICE 151.0352100.5393100.53841010.03951010.0386151.03127301.03128401.03139501.0311 Table4-2. MinedistributionsinoverlapregionsofAHIandLynximagery AHIimage1AHIimage2AHIimage3MinetypeDepthquantityquantityquantity PC110cm441717MC110cm574826MC1Flush343420MC1Surface161616MC2Surface14140PC2Surface500Total17012979 FalsealarmratereductionusingblackbodymaskinAHIimage1 RXwithoutBBmask2:31033:31035:81036:71039:0103ClusteringBBmask2:11033:01035:31036:11038:3103FARreduction8:7%9:1%8:6%9:0%7:78%SPICEBBmask1:01031:21032:31032:81034:2103FARreduction56:5%63:6%60:3%58:2%53:3% Table4-4. FalsealarmratereductionusingblackbodymaskinAHIimage2 RXwithoutBBmask1:71032:61033:81036:11038:5103ClusteringBBmask1:41032:21032:81034:21036:1103FARreduction17:6%15:4%26:3%31:1%28:2%SPICEBBmask1:21031:91032:51033:81035:8103FARreduction29:4%26:9%34:2%37:7%31:8% 102

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FalsealarmratereductionusingblackbodymaskinAHIimage3 RXwithoutBBmask3:71035:21039:21031:21031:6103ClusteringBBmask3:71035:21039:21031:21031:6103FARreduction0%0%0%0%0%SPICEBBmask3:31034:41036:81031:01021:4102FARreduction10:8%15:4%26:1%16:7%12:5% Table4-6. Meannumberandstandarddeviationofendmembersandbandsfoundover50runsofSPICEorB-SPICEonthesimulateddataset.Thetruenumberofendmembersforthisdatasetis4. StandardMeanStandarddeviationMeandeviationnumberofnumbernumberofofnumberofbandsofbandsendmembersofendmembersretainedretained 0(SPICE)4051.00.00.254034.61.10.504025.01.40.754020.91.21.004016.23.7 Table4-7. Statisticsoftheaveragedsquarederrorperabundancevaluebetweencalculatedandtrueabundancevalues StandarddeviationMedianaverageMeanaverageofaveragesquarederrorsquarederrorsquarederrorperabundanceperabundanceperabundance 0(SPICE)0.0050.0050.00050.250.0040.0080.00660.500.0040.0070.00570.750.0040.0070.00501.000.0060.0100.0069 103

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IndianPinesDataSetResultsandComparison.ComparisonValuesEstimatedfromGraphsin( Guoetal. 2006 )and( Martinez-Usoetal. 2006 ) Num.ofNum.ofClassicationaccuracyComparisonExp.endmembersbandsinpercentageresultsinpercentage Ref.Ref.Run1Run2Run3Guo,etal.M.-Uso,etal. 10.0822093.693.993.990-20.0722093.193.192.990-30.5712493.393.793.790-40.5712293.092.993.290-51.078993.493.393.690-61.0710393.393.393.590-75.073486.486.486.3888085.083486.586.086.48880910.071983.483.982.582811010.081877.880.078.38282 Table4-9. IndianPinesDataSetresultsusingsampledparametervaluesandcomparisonwith( Guoetal. 2006 ) ExperimentNumberofNumberofClassicationaccuracyComparisonresultsnumberendmembersbandskeptinpercentageinpercentage Run1Run2Run3Ref.Guo,etal. 1711492.192.192.2902711492.692.492.590 Table4-10. ParametervaluesusedtogenerateEDresultsontwo-dimensionaldatasets.Allcovariancematricesarediagonalwithelementsequaltothevaluesshowninthetable. VarianceLikelihoodSSDDataSetofdatavariancevariancebk 4-1 )55.90.10.50.01DatafromDists.(Fig. 4-19 )5.20.51.00.01 Table4-11. ParametervaluesusedtogenerateEDresultsonhyperspectraldatasets.Allcovariancematricesarediagonalwithelementsequaltothevaluesshowninthetable. DimensionalityVarianceLikelihoodSSDDataSetofdataofdatavariancevariancebk 104

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ParametervaluesusedtogeneratePCEresults.Allcovariancematricesarediagonalwithelementsequaltothevaluesshowninthetable. DataVarianceLikelihoodEDlikelihoodEDSSDDataSetdimen.ofdatavariancevariancevariancebk Two-dimensionalSPICEexampledataset.100datapointsgeneratedfromthecornersofthesimplexshown. ComparisonofSPICE(top)andICEwithpruning(bottom).Inthesethreeexperiments,=0:001andthepruningthresholdwassetto0:0005.Initialnumberofendmemberswas20. 105

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TheSPICEresultsontwo-dimensionaldata EndmembersfoundusingSPICEonAVIRISCupritehyperspectraldata.was0.1forallexperiments.Thepruningthresholdwassetto1109.Thelimitsofthex-axisare1978to2477nmandthelimitsofthey-axisare1000to7000inunitsof10,000timesthereectancefactor( Clarketal. 2004 ). 106

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ComparisonofoneendmemberfoundbySPICEandaUSGSAlunitespectrum(\AluniteSUSTDA-20W1R1BaAREF")fromthe2005USGSspectrallibrary. EndmembersselectedfromAVIRISCupritedataimagebyhand. 107

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NormalizedtestpixelsselectedfromAVIRISCupritedata. SPICEendmemberresultsfoundonnormalizedtestdataselectedfromtheAVIRISCupritescene 108

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Band10(s0.5m)oftheAVIRISIndianPinesdatasetandgroundtruth. 109

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AbundancemapsgeneratedbySPICEontheIndianPinesdataset. 110

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AbundancemapsgeneratedbySPICEonthelabeledAVIRISIndianPinesdataset.Pixelsinwhitecorrespondtounlabeledpixels.Remainingpixelsrangefromblack(abundancevalueofzero)tored(abundanceofone). 111

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HistogramofSPICEendmemberresultsonlabeledAVIRISIndianPinesdata.Histogramsshowdistributionofabundancesvaluesamongendmembersineachgroundtruthclass.HistogramswerecomputedaccordingtoEquation 4{2 .Thesumofthesehistograms'Shannon'sentropyvaluesis19.0.Thehistogramscorrespondtothefollowinggroundtruthclasses:(A)alfalfa,(B)corn-notill,(C)corn-min,(D)corn,(E)grass/pasture,(F)grass/trees,(G)grass/pasture-mowed,(H)hay-windrowed,(I)oats,(J)soybeans-notill,(K)soybeans-min,(L)soybean-clean,(M)wheat,(N)woods,(O)building-grass-trees-drive,and(P)stone-steeltowers. 112

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Subsetat9.19micronsofAHIhyperspectralimage2includingtheoverlapregion 113

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BlackbodymaskscreatedusingSPICEandtheclusteringmethod.A)istheblackbodymaskgeneratedusingSPICEandB)isthethresholdedSPICEmask.C)isthemaskgeneratedusing4clustersintheclusteringmethod;D)isthethresholdedversionofthismask.E)isthemaskgeneratedusing5clustersintheclusteringmethodandF)isthethresholdedversionofthismask. 114

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EndmembersusedtogeneratesimulateddatasetselectedbyhandfromtheAVIRISCupritedataset. EndmembersdeterminedusingSPICEandB-SPICEwithparameters=0,0.25,0.5,0.75,and1onthesimulateddataset. 115

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Histogramsof(a)thenumberofendmembers(b)bandsfoundand(c)thenumberoftimeseachbandisretainedover240runsofB-SPICEusingsampledparametervalues. Resultsontwo-dimensionaltriangledatafoundusingED.Bluepointsshowtheinputdataset.RedpointsarethemeanendmembersoftheendmemberdistributionsfoundbyED.Redcurvescorrespondtothe1stand2ndstandarddeviationsineachendmemberdistribution. 116

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Datapointsgeneratedfromthreeendmemberdistributions.Bluepointsshowthegenerateddataset.Redpointsarethemeanendmembersoftheendmemberdistributionsusedtogeneratethedatapoints.Redcurvescorrespondtothe1stand2ndstandarddeviationsineachendmemberdistributionusedtogeneratedthedatapoints. ResultsontwodimensionaldatausingED.Bluepointsshowthegenerateddataset.RedpointsarethemeanendmembersoftheendmemberdistributionsfoundbyED.Redcurvescorrespondtothe1stand2ndstandarddeviationsineachendmemberdistribution. 117

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ResultsontwodimensionaldatausingSPICE.Bluepointsshowthegenerateddataset.RedpointsaretheendmembersfoundbySPICE. ResultsonsimulatedAVIRISCupritedatausingED.Solidbluecurvesshowthetrueendmembersfromwhichthedatawasgenerated.SolidredcurvesarethemeanendmembersoftheendmemberdistributionsfoundbyED.Dashedredcurvescorrespondtothe1ststandarddeviationineachendmemberdistribution. 118

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ResultsonasubsetofAVIRISCupritedatafoundusingED.SolidredcurvesarethemeanendmembersoftheendmemberdistributionsfoundbyED.Dashedredcurvescorrespondtothe1ststandarddeviationineachendmemberdistribution. 119

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ResultsonasubsetofAVIRISCupritedatafoundusingED.Bluecurvesshowtheinputdataset.SolidredcurvesarethemeanendmembersoftheendmemberdistributionsfoundbyED.Dashedredcurvescorrespondtothe1ststandarddeviationineachendmemberdistribution. Two-dimensionaldatageneratedfromthreesetsofendmembers.Bluepointscorrespondtotheinputdataset.Redpointscorrespondtotheendmembersfromwhichthedatawasgenerated.Eachtriangleofdatapointswasgeneratedfromthreeoftheendmembers. 120

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Two-dimensionaldataresultsfoundusingPCE.Smallbluepointscorrespondtotheinputdataset.Largepointscorrespondtothemeanendmembersforeachendmemberdistribution.Thincurvescorrespondtothe1stand2ndstandarddeviationcurvesfromeachendmemberdistribution.Thecolorofeachendmemberdistributioncorrespondstotheconvexregiontowhichitbelongs. 121

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AbundancemapsfoundusingPCEonlabeledPCA-reducedAVIRISIndianPinesdata.Pixelsinwhiteareunlabeled.Pixelsingrayindicatepixelsfromanotherconvexpartition.Remainingpixelsrangefromblue(abundancevalueofzero)tored(abundancevalueofone). 122

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HistogramofPCEendmemberresultsonlabeledPCA-reducedAVIRISIndianPinesdata.Histogramsshowdistributionofabundancesvaluesamongendmembersineachgroundtruthclass.HistogramswerecomputedaccordingtoEquation 4{2 .Thesumofthehistograms'Shannon'sentropyvaluesis9.4.Thehistogramscorrespondtothefollowinggroundtruthclasses:(A)alfalfa,(B)corn-notill,(C)corn-min,(D)corn,(E)grass/pasture,(F)grass/trees,(G)grass/pasture-mowed,(H)hay-windrowed,(I)oats,(J)soybeans-notill,(K)soybeans-min,(L)soybean-clean,(M)wheat,(N)woods,(O)building-grass-trees-drive,and(P)stone-steeltowers. 123

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AbundancemapsfoundusingSPICEonlabeledPCA-reducedAVIRISIndianPinesdata.Pixelsinwhiteareunlabeled.Remainingpixelsrangefromblue(abundancevalueofzero)tored(abundancevalueofone). 124

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HistogramofSPICEendmemberresultsonlabeledPCA-reducedAVIRISIndianPinesdata.Histogramsshowdistributionofabundancesvaluesamongendmembersineachgroundtruthclass.HistogramswerecomputedaccordingtoEquation 4{2 .Thesumofthehistograms'Shannon'sentropyvaluesis29.2.Thehistogramscorrespondtothefollowinggroundtruthclasses:(A)alfalfa,(B)corn-notill,(C)corn-min,(D)corn,(E)grass/pasture,(F)grass/trees,(G)grass/pasture-mowed,(H)hay-windrowed,(I)oats,(J)soybeans-notill,(K)soybeans-min,(L)soybean-clean,(M)wheat,(N)woods,(O)building-grass-trees-drive,and(P)stone-steeltowers. 125

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AbundancemapsfoundusingPCEonlabeledAVIRISIndianPinesdatawithhierarchicaldimensionalityreduction.Pixelsinwhiteareunlabeled.Pixelsingrayindicatepixelsfromanotherconvexpartition.Remainingpixelsrangefromblue(abundancevalueofzero)tored(abundancevalueofone). 126

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HistogramofPCEendmemberresultsonlabeledAVIRISIndianPinesdatawithhierarchicaldimensionalityreduction.Histogramsshowdistributionofabundancesvaluesamongendmembersineachgroundtruthclass.Thesumofthehistogram'sShannon'sentropyvaluesis16.3.HistogramswerecomputedaccordingtoEquation 4{2 .Thehistogramscorrespondtothefollowinggroundtruthclasses:(A)alfalfa,(B)corn-notill,(C)corn-min,(D)corn,(E)grass/pasture,(F)grass/trees,(G)grass/pasture-mowed,(H)hay-windrowed,(I)oats,(J)soybeans-notill,(K)soybeans-min,(L)soybean-clean,(M)wheat,(N)woods,(O)building-grass-trees-drive,and(P)stone-steeltowers. 127

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AbundancemapsfoundusingPCEonlabeledAVIRISIndianPinesdata.Pixelsinwhiteareunlabeled.Pixelsingrayindicatepixelsfromanotherconvexpartition.Remainingpixelsrangefromblue(abundancevalueofzero)tored(abundancevalueofone). 128

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HistogramofPCEendmemberresultsonlabeledAVIRISIndianPinesdata.Histogramsshowdistributionofabundancesvaluesamongendmembersineachgroundtruthclass.HistogramswerecomputedaccordingtoEquation 4{2 .Thesumofthehistogram'sShannon'sentropyvaluesis15.4.Thehistogramscorrespondtothefollowinggroundtruthclasses:(A)alfalfa,(B)corn-notill,(C)corn-min,(D)corn,(E)grass/pasture,(F)grass/trees,(G)grass/pasture-mowed,(H)hay-windrowed,(I)oats,(J)soybeans-notill,(K)soybeans-min,(L)soybean-clean,(M)wheat,(N)woods,(O)building-grass-trees-drive,and(P)stone-steeltowers. 129

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A.ZareandP.Gader.Sparsitypromotingiteratedconstrainedendmemberdetectionforhyperspectralimagery.IEEEGeoscienceandRemoteSensingLetters,4(3):446{450,July2007a. A.ZareandP.Gader.SPICE:Asparsitypromotingiteratedconstrainedendmemberextractionalgorithmwithapplicationstolandminedetectionfromhyperspectralimagery.InProceedingsoftheSPIE,pageCID:655319,Orlando,Fl,2007b. A.ZareandP.Gader.Hyperspectralbandselectionandendmemberdetectionusingsparsitypromotingpriors.IEEEGeoscienceandRemoteSensingLetters,5(2):256{260,Apr.2008. A.Zare,J.Bolton,P.Gader,andM.Schatten.Vegetationmappingforlandminedetectionusinglongwavehyperspectralimagery.IEEETransactionsonGeoscienceandRemoteSensing,46(1):172{178,January2008. 139

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AlinaZarereceivedherBachelorofSciencedegreeincomputerengineeringfromtheUniversityofFloridain2003.ShecontinuedherstudiesattheUniversityofFloridatograduatewithherMasterofScienceandDoctorofPhilosophydegreesfromthedepartmentofComputerandInformationScienceandEngineeringin2008.Herresearchinterestsincludesparsitypromotion,machinelearning,Bayesianmethods,imageanalysisandpatternrecognition. 140