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Sonar Image Modeling for Texture Discrimination and Classification

Permanent Link: http://ufdc.ufl.edu/UFE0043611/00001

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Title: Sonar Image Modeling for Texture Discrimination and Classification
Physical Description: 1 online resource (148 p.)
Language: english
Creator: Cobb, James T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: autocorrelation -- expectation-maximization -- k-distribution -- segmentation -- sonar
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and 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: High-resolution synthetic aperture sonar (SAS) systems yield finely detailedimages of sea bed environments. SAS image texture models must be capable ofrepresenting a wide variety of sea bottom environments including sand ripples, coral or rock formations, and flat hard pack. In this dissertation a parameterized texture modelbased on the autocorrelation functions (ACF) of the SAS imaging point spread functionand the ACF of the seabed texture sonar cross section (SCS) are derived from realisticscattering assumptions. The proposed texture mixture model is analytically tractableand parameterized by component mixing parameters, mixture component correlation lengths, means, the single-point intensity image statistical shape parameter, and therotation of the ACF mixture components in the 2-D imaging plane. These parametersprovide an intuitive, low-dimension representation of the image texture in terms of its contrast, period, orientation, and shape. To estimate the various ACF mixture model parameters, an iterative algorithm based on the Expectation Maximization algorithm for truncated data is presented and tested against various synthetic and real SAS image textures. The accuracy of the parameter estimation algorithm is compared and discussed for synthetically generated data across various image sizes and texture characteristics. The use of the Bayesian information criteria (BIC) as an effective model selection metric is demonstrated and discussed. ACF model parameters are also estimated for a small set of real SAS survey images and are shown to accurately fit the imaging point spread function and seabed SCS ACF for these textures of interest. An unsupervised multi-class k-means segmentation algorithm that uses the features derived from the ACF model is employed to label sand, rock, and ripple textures from a set of real textured SAS images. First, results are compared between increasingly complex intensity ACF models, with the most effective being a four-component model capable of extracting the period of the ripple textures. Later, the results of the four-component ACF segmentation are compared against the performance of the segmentation approach using bi-orthogonal wavelets and Haralick features. In the described experiments, the ACF model features are shown to produce better segmentations than the features based on wavelet coefficients and Haralick features for classifiers of low complexity.
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 James T Cobb.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Principe, Jose C.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043611:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043611/00001

Material Information

Title: Sonar Image Modeling for Texture Discrimination and Classification
Physical Description: 1 online resource (148 p.)
Language: english
Creator: Cobb, James T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: autocorrelation -- expectation-maximization -- k-distribution -- segmentation -- sonar
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and 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: High-resolution synthetic aperture sonar (SAS) systems yield finely detailedimages of sea bed environments. SAS image texture models must be capable ofrepresenting a wide variety of sea bottom environments including sand ripples, coral or rock formations, and flat hard pack. In this dissertation a parameterized texture modelbased on the autocorrelation functions (ACF) of the SAS imaging point spread functionand the ACF of the seabed texture sonar cross section (SCS) are derived from realisticscattering assumptions. The proposed texture mixture model is analytically tractableand parameterized by component mixing parameters, mixture component correlation lengths, means, the single-point intensity image statistical shape parameter, and therotation of the ACF mixture components in the 2-D imaging plane. These parametersprovide an intuitive, low-dimension representation of the image texture in terms of its contrast, period, orientation, and shape. To estimate the various ACF mixture model parameters, an iterative algorithm based on the Expectation Maximization algorithm for truncated data is presented and tested against various synthetic and real SAS image textures. The accuracy of the parameter estimation algorithm is compared and discussed for synthetically generated data across various image sizes and texture characteristics. The use of the Bayesian information criteria (BIC) as an effective model selection metric is demonstrated and discussed. ACF model parameters are also estimated for a small set of real SAS survey images and are shown to accurately fit the imaging point spread function and seabed SCS ACF for these textures of interest. An unsupervised multi-class k-means segmentation algorithm that uses the features derived from the ACF model is employed to label sand, rock, and ripple textures from a set of real textured SAS images. First, results are compared between increasingly complex intensity ACF models, with the most effective being a four-component model capable of extracting the period of the ripple textures. Later, the results of the four-component ACF segmentation are compared against the performance of the segmentation approach using bi-orthogonal wavelets and Haralick features. In the described experiments, the ACF model features are shown to produce better segmentations than the features based on wavelet coefficients and Haralick features for classifiers of low complexity.
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 James T Cobb.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Principe, Jose C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043611:00001


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SONARIMAGEMODELINGFORTEXTUREDISCRIMINATIONAND CLASSIFICATION By JAMESTORYCOBB ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011JamesToryCobb 2

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ToParmy,Lily,andNate 3

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ACKNOWLEDGMENTS WhenIrstbeganworkingatthenCoastalSystemsStationin2001,laterNaval SurfaceWarfareCenterPanamaCity,Itookdistancelearningclassesfromthe UniversityofFlorida.TheseclassesturnedintothepursuitofmyPh.D.withtheurging ofmycolleagueDr.GerryDobeckandmysoon-to-bePh.D.advisorDr.ClintSlatton.I amindebtedtobothofthemforseeingmypotentialandforcounselingmethroughout mystudies.Sadly,Dr.Slattonpassedtragicallyafterabriefghtwithcancerinthe springof2010.IwasextremelyfortunatetohaveDr.JosePrincipeonmycommittee andhegraciouslyvolunteeredtobemyadvisor.IsincerelyappreciateDr.Principe's soundadviceandtimelydeadlinesthataidedmeinnishingmyresearch.Ialso thankmyfellowgraduatestudentsKittipatBotKampaandErionHasanbelliufor collaboratingwithmeonresearchprojectsandhelpingmemeetallofourpublication andpresentationdeadlines. Myfamilysupportedmewithloveandencouragementthroughoutmytime ingraduateschool.MywifeParmy,aBaylorUniversitygraduatewithaPh.D.in mathematics,understoodthestruggleanddemandsofresearchandpublishing. Sheproddedmetoworkhardandnevercomplainedofthetravelandlatehours. WithouthersupportIwouldnothavenishedduetothedemandsofafull-timejoband thecareoftwoyoungchildren.MydaughterLilyandsonNatewerebornduringmy studies,andthoughtheyweretooyoungtocomprehendmyacademicpursuit,they inspiredmewithunconditionallove.Iwasblessedtobebornintoafamilythatvalued education.MyfatherandmotherhavesharedmysetbacksandsuccessesandIowe anyaccomplishmenttotheirhighexpectationsandencouragementthroughoutmylife.I thanktheothersinmyfamily,especiallymybrotherandmygrandmothers,withcontinual supportandpraise. Finally,andmostimportantly,IthankGodforHiscontinualblessings.ThroughHimI amsustainedandinspiredinallmyendeavors. 4

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ThisresearchwassupportedwithfundingfromtheNavalSurfaceWarfareCenter PanamaCityDivisionIn-HouseLaboratoryIndependentResearchILIRprogramand theOfceofNavalResearchCode321.Allinformationhereinisapprovedforpublic release;distributionisunlimited. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................9 LISTOFFIGURES.....................................10 ABSTRACT.........................................13 CHAPTER 1INTRODUCTION...................................15 1.1High-ResolutionSyntheticApertureStripMapImaging...........15 1.2SASImageStatisticalModels.........................17 1.2.1ResolutionCellScatteringModel...................19 1.2.2Single-pointSASImagePixelModels................21 1.2.2.1Rayleighdistribution.....................21 1.2.2.2Exponentialdistribution...................23 1.2.2.3 K distribution.........................23 1.2.3Single-pointStatisticsforTextureCharacterization.........25 1.2.4CorrelatedSASImagePixelModels.................26 1.3DissertationOutline..............................30 2SASIMAGETEXTUREMODELS.........................32 2.1DerivationoftheParameterizedSASImageAutocorrelationFunction..33 2.1.1MeanIntensity.............................34 2.1.2ComplexBeamformedReturnSxAutocorrelationFunction....34 2.1.3IntensityAutocorrelationFunction...................35 2.1.4IntensityACFforGaussianSCSACFandImagingPSFModels..36 2.1.5Two-DimensionalFormoftheIntensityACF.............36 2.2ExtensionofModelIntensityACFofRippledTextures...........38 2.3Summary....................................41 3GENERATINGSYNTHETICSASTEXTURESWITHTHEINTENSITYACF MODEL........................................43 3.1ImageSynthesisProcedure..........................44 3.1.1GeneratingCorrelatedGammaRandomVariables.........45 3.1.2GeneratingCorrelated K -distributedRandomVariables......46 3.2SynthesizedImageSamples.........................47 3.3EffectsofParameterChoiceonCorrelatedImageSynthesis........53 3.3.1ACFDistortionDuetoSmall Values................55 3.3.2DependenceofExpected ValueonImagingPSFParameters..56 3.4Summary....................................58 6

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4AUTOCORRELATIONFUNCTIONMODELPARAMETERESTIMATION....60 4.1ImagingPointSpreadFunctionParameterEstimation...........61 4.2Single-PointShapeParameterEstimation..................61 4.3NormalizedIntensityAutocorrelationFunctionParameterEstimationvia theEMAlgorithm................................64 4.3.1AlgorithmSimplicationsbyCouplingParameters..........66 4.3.2EffectsofIntensityACFTruncationonEstimationAccuracy....67 4.3.3IntensityACFAlgorithmSummary..................69 4.4InitializationofParametersandImplementationConsiderations......71 4.5Summary....................................73 5ESTIMATIONRESULTSONSYNTHETICANDREALSASIMAGETEXTURES74 5.1ParameterEstimationResultsforSyntheticImageTestCases.......74 5.1.1TestCase1...............................75 5.1.2TestCase2...............................75 5.1.3TestCase3...............................79 5.2High-ResolutionSASDataACFParameterEstimation...........82 5.3Summary....................................90 6SASTEXTURESEGMENTATIONUSINGACFMODELPARAMETERS....94 6.1RelatedResearchinRadarandSonarImageSegmentation........94 6.2SASImageSegmentationUsingSingle-PointStatistics..........95 6.3SASImageSegmentationUsingAutocorrelationFunctionParameters..98 6.4SASImageSegmentationUsingWaveletandHaralickFeatures.....101 6.4.1Bi-orthogonalWaveletFeatures....................103 6.4.2HaralickFeatures............................104 6.4.2.1Haralickfeaturecalculation.................106 6.4.2.2Haralickfeaturevector....................107 6.5ComparativeSegmentationResults.....................108 6.6Summary....................................112 7CONCLUSIONSANDFUTUREWORK......................116 7.1Contributions..................................116 7.2FutureWork...................................118 7.2.1MultiscaleDecompositionFeatureExtraction............119 7.2.2Higher-OrderStatistics.........................120 7.2.2.1Fourth-orderdependencemetric..............123 7.2.2.2Autocorrentropy.......................124 7.2.3SASImageSceneAnalysis......................128 7.3Summary....................................129 APPENDIX ADERIVATIONOFCOMPOUNDREPRESENTATIONOF K DISTRIBUTION..131 7

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BDERIVATIONOFTHEINTENSITYACFMIXTUREMODEL...........134 CEMALGORITHMFORESTIMATINGGAUSSIANMIXTUREPARAMETERS OFATWO-DIMENSIONALACF..........................138 REFERENCES.......................................142 BIOGRAPHICALSKETCH................................148 8

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LISTOFTABLES Table page 1-1 2 goodness-of-tstatisticsbyseabedcomposition................26 3-1SimulationparametervaluesforimagesinFigures3-13-4...........52 3-2Valuesof d 2 k forvaryingvaluesofshapeparameter ...............54 5-1EstimatedtextureparametersforTestCase1...................77 5-2EstimatedtextureparametersforTestCase2...................81 5-3RealSAStexture:accuracyversusmodelcomplexity...............90 6-1FeaturevectorentriesforthethreeACFmodels..................99 6-2SASimagesegmentationresultsinpct.correct..................100 6-3Confusionmatrixforone-componentsegmentationresults............100 6-4Confusionmatrixfortwo-componentsegmentationresults............101 6-5Confusionmatrixforfour-componentmodelsegmentationresults........101 6-6SASimagesegmentationresultsinpct.correct..................109 6-7Confusionmatrixforbi-orthogonalwaveletsegmentationresults.........109 6-8ConfusionmatrixforHaralicksubsetsegmentationresults............110 9

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LISTOFFIGURES Figure page 1-1SASstripmapsampleimage.............................16 1-2Sideandtopviewofsonarstripmapimageformation...............18 1-3Resolutioncellscatteringcartoon..........................20 1-4Complex,amplitude,andintensityimagesofarippledseabedtexture......22 1-5Aplotofthe K distributionpdfdenedbyEquation1forvariousvaluesof .25 1-6FitofSASimagesandtextureempiricalprobabilitydensityfunctionwithRayleigh and K distributionprobabilitydensityfunction....................27 1-7FitofSASimageseagrasstextureempiricalprobabilitydensityfunctionwith Rayleighand K distributionprobabilitydensityfunction..............28 1-8FitofSASimagerippletextureempiricalprobabilitydensityfunctionwithRayleigh and K distributionprobabilitydensityfunction....................29 1-9Comparisonofestimated valueparametersbetweenposidonea,sandripple, andseashelltextures.................................31 2-1GraphicalrepresentationoftheintensityACFmodeldenedbyEquation219.39 2-2AcomparisonofintensityACFmodeltstotheone-dimensionalSASintensity ACFestimatedfromarippledseabed........................40 2-3GraphicalrepresentationoftheintensityACFmodeldenedbyEquation222.42 3-1SimulatedsandtextureSASimagedisplayedincomparisonwitharealsand textureSASimage...................................48 3-2SimulatedseagrasstextureSASimagedisplayedincomparisonwithareal seagrasstextureSASimage.............................49 3-3SimulatedrocktextureSASimagedisplayedincomparisonwitharealrock textureSASimage...................................50 3-4SimulatedsandrippletextureSASimagedisplayedincomparisonwithareal sandrippletextureSASimage............................51 3-5Twosyntheticimageswithinsertedtargets.....................53 3-6Coefcientvaluesforvariousvaluesof ......................54 3-7Plotofthemappingbetween and G determinedbyEquations33..55 10

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3-8Plotofthemean-squareerrorinducedbyviolationoftheconstraintinEquation 3..........................................57 3-9Plotoftheeffectonthesimulatedvalueof forincreasingvaluesof inthe imagingpointspreadfunction h x y ........................58 4-1AnexampleofestimatingtheimagingPSFparametersfromatexturedSAS imagesnippet.....................................62 4-2One-dimensionalplotofanintensityACFestimatedfromanimagegenerated usingatwo-componentACF.............................68 4-3NegativeeffectsofACFestimatetruncation....................69 4-4Estimationafterreplacingtruncateddatawithmissingdata............70 5-1Sample 512 512 256 256 128 128 ,and 64 64 realizationsofthesynthetic testimagesinTestCase1..............................76 5-2BarplotofthemeanparameterestimatesforTestCase1.............77 5-3BarplotofthestandarddeviationofparameterestimatesforTestCase1....78 5-4Errorbarplotofthemean-squareerrorofparameterestimatesforTestCase1.79 5-5Sample 256 256 realizationsofthesynthetictestimagesforTestCase2...80 5-6BarplotofthemeanofparameterestimatesforTestCase2...........82 5-7BarplotofthestandarddeviationofparameterestimatesforTestCase2....83 5-8 256 256 syntheticimageandgeneratingACFofimageusedinTestCase3..84 5-9ErrorbarplotofthemeanoftheBICandlog-likelihoodof100syntheticimage samplesgeneratedusingtheACFdepictedinFigure5-8.............85 5-10 256 256 sandtextureandACFmodelt......................86 5-11 256 256 seagrasstextureandACFmodelt...................87 5-12 256 256 rocktextureandACFmodelt......................88 5-13 256 256 rippletextureandACFmodelt.....................89 5-14Plotoflog-likelihoodandMSEforrealsandtexture................91 5-15Plotoflog-likelihoodandMSEforrealseagrasstexture..............91 5-16Plotoflog-likelihoodandMSEforrealrocktexture.................92 5-17Plotoflog-likelihoodandMSEforrealsandrippletexture.............92 11

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6-1Case1:segmentationresultsusingtheshapeparameter ............96 6-2Case2:segmentationresultsusingtheshapeparameter ............97 6-3Imagesofsegmentationresultsusingthefour-componentACFmodel......102 6-4Plotofthewaveletbior1.3chosenfromtheMATLAB R WaveletToolbox....104 6-5Five-scalewaveletdecompositionsofrock,smallripple,andlargerippletextures usingthebi-orthogonaldecompositionwaveletdepictedinFigure6-4......105 6-6Barplotofthemeanscatterover1000instantiationsofthe k -meansalgorithm with k =3 fortheACF,wavelet,andHaralickfeaturesets.............110 6-7ErrorbarplotsoftextureclassicationratesfortherockSASimagetextures..112 6-8ErrorbarplotsoftextureclassicationratesforthesmallrippleSASimage textures.........................................113 6-9ErrorbarplotsoftextureclassicationratesforthelargerippleSASimage textures.........................................114 6-10Errorbarplotsoftheaveragetextureclassicationforcombinedrock,small ripple,andlargerippleSASimagetextures.....................115 7-1 800 800 syntheticimagesamplesofvariouscorrelationlengthsdownsampled by2,4,and8.....................................121 7-2Correlationlengthestimatesacrossvariousimagescales.............122 7-3WindowusedtocomputeCDFhistogramvalues..................123 7-4Multiscaleimagesofsandripple,rock,seagrass,andsandtextures.......125 7-5Plotofcodependencemetric M for3scalesofsand,seagrass,rock,and sandrippleSASimagetextures...........................126 7-6AutocorrentropyestimatesfortheAsand,Brock,andCsandrippletextures ofFigures3-1A,3-3A,and3-4A..........................127 7-7Imagesofthe k -meanscentroidsforthelargeripple,smallripple,androck texturessegmentedinChapter6usingthefour-componentACFmodel.....129 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SONARIMAGEMODELINGFORTEXTUREDISCRIMINATIONAND CLASSIFICATION By JamesToryCobb December2011 Chair:JosePrincipe Major:ElectricalandComputerEngineering High-resolutionsyntheticaperturesonarSASsystemsyieldnelydetailed imagesofseabedenvironments.SASimagetexturemodelsmustbecapableof representingawidevarietyofseabottomenvironmentsincludingsandripples,coralor rockformations,andathardpack.Inthisdissertationaparameterizedtexturemodel basedontheautocorrelationfunctionsACFoftheSASimagingpointspreadfunction andtheACFoftheseabedtexturesonarcrosssectionSCSarederivedfromrealistic scatteringassumptions.Theproposedtexturemixturemodelisanalyticallytractable andparameterizedbycomponentmixingparameters,mixturecomponentcorrelation lengths,means,thesingle-pointintensityimagestatisticalshapeparameter,andthe rotationoftheACFmixturecomponentsinthe2-Dimagingplane.Theseparameters provideanintuitive,low-dimensionrepresentationoftheimagetextureintermsofits contrast,period,orientation,andshape. ToestimatethevariousACFmixturemodelparameters,aniterativealgorithm basedontheExpectationMaximizationalgorithmfortruncateddataispresentedand testedagainstvarioussyntheticandrealSASimagetextures.Theaccuracyofthe parameterestimationalgorithmiscomparedanddiscussedforsyntheticallygenerated dataacrossvariousimagesizesandtexturecharacteristics.TheuseoftheBayesian informationcriteriaBICasaneffectivemodelselectionmetricisdemonstratedand discussed.ACFmodelparametersarealsoestimatedforasmallsetofrealSASsurvey 13

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imagesandareshowntoaccuratelyttheimagingpointspreadfunctionandseabed SCSACFforthesetexturesofinterest. Anunsupervisedmulti-class k -meanssegmentationalgorithmthatusesthe featuresderivedfromtheACFmodelisemployedtolabelsand,rock,andripple texturesfromasetofrealtexturedSASimages.First,resultsarecomparedbetween increasinglycomplexintensityACFmodels,withthemosteffectivebeingafour-component modelcapableofextractingtheperiodoftherippletextures.Later,theresultsof thefour-componentACFsegmentationarecomparedagainsttheperformanceof thesegmentationapproachusingbi-orthogonalwaveletsandHaralickfeatures.In thedescribedexperiments,theACFmodelfeaturesareshowntoproducebetter segmentationsthanthefeaturesbasedonwaveletcoefcientsandHaralickfeaturesfor classiersoflowcomplexity. 14

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CHAPTER1 INTRODUCTION State-of-the-artsyntheticaperturesonarSASimagingsystemsgeneratenely detailedseabedimages.Sensorcharacteristics,seabottomtopography,vegetation cover,andsedimentcharacteristicsallplayaroleintheheterogeneoustexturesthat appearinthesehigh-resolutionimages.Thecharacteristicsoftheimagetexturesina surveyareamayvarywidely,e.g.onesquarenauticalmileofseabottommaycontain sandripples,coralorrockformations,andatfeaturelesshardpack.Automatically segmentingthesesonarimagesintohomogeneoustexturedregionsrequiressome metricornumericmeasureofdistinctionbeusedbyapatternclassicationscheme. Section1.1describesthegenerationofhigh-resolutionsyntheticaperturestripmap imagesbycurrentsensingsystemsingeneraltermsandintroducestheimagegeometry andnotationusedthroughouttheremainderofthestudy.Section1.2discussesthe strengthsandweaknessesofsomecommonprobabilitymodelsintheliteraturethat describeSASimagepixelstatisticsinthecontextofautomatedtexturediscrimination. 1.1High-ResolutionSyntheticApertureStripMapImaging SASstripmapimagingderivesitsnamefromtheconceptofsynthesizinga variable-lengtharraytomaintainaconstantcross-rangeimagingresolutionalong therange-directionbroadsidefromthetransducer[1],[2,Ch.6].AdvancementsinSAS sensortechnologyhaveledtosystemsthatcanproducehighly-resolvedimagerythat revealsobjectdetailsatscalesofjustafewcentimetersatover100metersinrange[3]. InthesehighlydetailedSASimages,complexseabedtexturesandsmallobjects arenowdiscerniblewhereasinolderrealaperturesonarsystemsthesesmallscale featureswerenotdistinguishableatlongrangesandyieldedrange-dependentstrip mapimagepixelstatisticsduetothebroadsidebeampattern.Figure1-1depictsaSAS imagewithmultipleseabedtexturesandsmallobjectstoillustratetherichnessofthe imagingregime.Inthegure,threedistincttextureclassesareevident:arippledtexture 15

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Figure1-1.SampleofSASstripmapimageillustratingtherichnessofseabedtexture andobjectdetail. fromsandridges,agrainy,raisedtexturefromtheseagrassknownasposidonea, andasmoothdarktexturefromthehard-packedsandyseaoor.Allofthesetextures presentdifferentstatisticalcharacteristicswewillmodelwithSASimagepixelprobability distributions. Stripmapimageformation: Asonarstripmapimageisformedbymovinga sonararrayalongaxedpathandrecordingtheechofromasoundwaveprojected alongtheseaoorandreceivedbroadsidetothearray.Eachping,orcollectionof coherentlysummedpingsinthecaseofSAS,representsasendandreceivecycle 16

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oftheprojectedsoundwaveandproducesonehorizontallineinthestripmapimage. Thehorizontallineisfurtherdividedintorangebinsthatrepresentthereceivedsignal timedelayduetotheroundtripofthesoundwavefromtheprojectortoapointinspace ensoniedbythesignalandbacktothereceiver[2].Figure1-2isacartoonillustrating howthesonararrayoperatestoproducethestripmapimage.AtthetopofFigure1-2is asideviewofthesonar.Inthesideviewthesensorensoniesaportionoftheseaoor atrangebin X o .Theslantrangefromrangebin X o tothesensorandbackdetermines thereceivedecho'smappingintothestripmapimageinthebottomgure.I.e.therange bin X N isatarangefurtherfromthesensorthantherangebin X 1 .Thereadershould notethatthetermrangeinthisresearchreferstoslantrangeasinthetopofFigure1-2 andnotthetruegeospatialdistanceorgroundrangefromthesensortotherangebin. Asthesensorcontinuesalongitsintendedtrackandmorepingsarecollected, animageiseventuallyformedofresolutioncellsindexedbyrangeandcross-range directions.Fortheremainderofthispaper,therangedirectionisdenotedby X andthe cross-rangedirectionisdenotedby Y .Inthisresearchallimagesarecreatedinthis manner.Sincethebeginningofresearchinthispaperstartswithafully-formedstrip mapimage,onlyarudimentary,intuitivereviewofSASimageformationisincludedin thissection.ForacomprehensivereviewofSASstripmapbeamformingthereaderis directedto[1],[2]and[4][6]. 1.2SASImageStatisticalModels Thetexturemodelspresentedhereallresultfrominitialassumptionsaboutthe statisticalpropertiesofthepixelsintheSASstripmapimage.Thetwomainmodels presentedinvolvethestatisticsofimagepixelsexaminedas1identicallydrawn, independently-distributedi.i.d.samples,referredtoassingle-pointstatistics,and 2correlationsbetweenimagepixelsinneighboringresolutioncells.Usingthese twostatisticalmeasures,SASimagetexturemodelsandassociatedsynthesisand parameterestimationtechniqueswillbedevelopedandexplained. 17

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Figure1-2.Sideandtopviewofsonarstripmapimageformation. 18

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SonarbackscatterfromtheseaoorhasbeenwidelymodeledasRayleighdistributedsincethedetectedenvelopeofthesonarreturnatanygiventimeisassumed tobethesquarerootofthesumoftheenergyfrommanyindependentcomplex Gaussian-distributedscatterers[7,Ch.9,pp.259-260].Thisstatisticalmodelholds providedthenumberofscattererscontributingtothereturnatagivenrangecellis largeandthephaseofeachreturnisuniformlydistributedsuchthattheCentralLimit Theoremcanbeinvoked.However,ithasbeenshowninvariousradarandsonar clutterenvironmentswheretheassumptionsrequiredfortheRayleighdistribution approximationdonothold.Thenumberofscatterersperrangecellmaybediminished duetoocclusioncausedbythenatureoftheclutterandthedepressionangleofthe sensoratincreasingrangesorreturnsmaybehighlycorrelatedinphasedueto naturaltextures,resultinginaprobabilitymodelforpixelintensityknownasthe K distribution[8,9].HerewereviewRayleighand K distributionSASimagepixelmodels andcomparetheireffectivenessinmodelinguncorrelatedandcorrelatedortextured SASimagesviatheirstatisticalparameters. 1.2.1ResolutionCellScatteringModel Thescatteringmodelfortheimagepixelstatisticsassumesthattheechofroman ensoniedvolumeatagivenrangeandcross-rangeresolutioncellisasummationof energyfromthecomponentscattererscontainedinthecell.Coherentbeamforming requirestheechofromagivenresolutioncelltocontainbothin-phaseandquadrature componentsandisthusacomplexsignal.Withtheseassumptions,the totalscattering a x y ismodeledas a x y = N X i =1 a i x y = N X i =1 r i x y e j i where r i x y istheamplitudeand i thephaseofthecomponentscatter a i x y .The cartooninFigure1-3depictsthetotalscatteringmodelforasingleresolutioncell.Two 19

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Figure1-3.Resolutioncellscatteringcartoon. furtherassumptionsaremadethattheamplitudesofcomponentscatterersarei.i.d.and thecomponentscatterphasesaredistributeduniformlyfrom )]TJ/F25 11.9552 Tf 9.298 0 Td [( to Anindividualstripmapimagepixeliscreatedwithintheframeworkdescribedin Section1.1bytheconvolutionofthetotalscattering a x y withanimagingpointspread functionPSF h x y toformthecompleximagepixel s x y or s x y = a x y h x y TheimagingPSFisthemodelforthecoherentbeamformingprocessandallsystem andmediumeffectsthatdistortthescatteredenergyfromthetotalscatteringatagiven resolutioncell.Inthischapter h x y isassumedtobeidealoradeltafunction x y forthepurposesofillustratingthevariousimagestatisticalmodels.Thepropertiesand effectsofaGaussianimagingPSFwillbeexploredinChapter2. Thestripmapcompleximagepixel s x y isnotaveryusefulmediumforvisually analyzingsonarimagedataduetotherandomphasecomponentandisusually 20

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transformedintoan amplitude imagepixel A x y or intensity imagepixel I x y by theoperations A x y = j s x y j or I x y = j s x y j 2 respectively.Examplesofthe differentrepresentationsofacomplexsonarimageareillustratedinFigure1-4.Inthis gure,arippledseabottomtextureisdepictedastherealandimaginarycomponentsof thecomplexsonarimage,theamplitudeimage,andtheintensityimage.Thecomplex imagedoesnotrevealmanyvisualcluesaboutthenatureoftheimage,whereasin theamplitudeandintensityimages,thetexturepatterniseasilydiscernible.Asimilar comparisonbetweencomplex,amplitude,andintensityimagesismadewithstripmap syntheticapertureradarSARimageryin[10,Ch.4,pp.86]. 1.2.2Single-pointSASImagePixelModels InSections1.2.2.11.2.2.3allpixelmodelstatisticalmeasuresassumethepixels aredrawni.i.d.fromaSASstripmapimage.Thei.i.d.orsingle-pointimagestatistics assumenocorrelationbetweenneighboringpixels.Additionally,anassumptionis madethatthestatisticsofthepixelsinquestionarestationaryordonotvaryspatially. CorrelatedpixelmodelswillbeexaminedinChapter2. 1.2.2.1Rayleighdistribution Basedonthetotalscatteringmodelthusfar,itcanbeeasilyshownthatfor a i 's withnitevariance, a x y isdistributedcomplexGaussianforlarge N byinvoking theCentralLimitTheorem[11,Ch.4,pp.225-228]inEquation1.Theimage pixelstatisticsforanamplitudeimageformedwiththisassumptionaredistributed Rayleigh[11,Ch.3,pp.150-151].Thisisacommonassumptioninsonarecho statisticalanalysis[7,Ch.9,pp.259-261]andwillbeshowninlatersectionstot single-pointstatisticswellinamplitudeimagepixelregionswithlowcontrast.The Rayleighprobabilitydensityfunction[12,Ch.5,pp.148] p R x = x 2 e )]TJ/F40 5.9776 Tf 9.947 3.259 Td [(x 2 2 2 x > 0, 21

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A Re f s x y g B Im f s x y g C j s x y j D j s x y j 2 Figure1-4.Complex,amplitude,andintensityimagesofarippledseabedtexture.Inthe ArealandBimaginarypartsofthecompleximage s x y theseabed textureisdifculttodistinguish,whereasintheCamplitudeandDintensity imagestherippledtextureisclearlyrevealed. 22

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hasasingleparameter relatedtoboththemean R = p 2 andvariance 2 R = )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F26 7.9701 Tf 13.15 4.707 Td [( 2 2 ofthepixelstatisticsunderexamination.Thesinglevariable giveslittle freedomforthedistributiontomodelheavy-tailedpixeldistributions. 1.2.2.2Exponentialdistribution UsingthesameassumptionspresentedinSection1.2.2.1leadstoanexponential pixeldistributionfortheintensityimage I x y [11,Ch.4,pp.149-150].Similartothe Rayleighprobabilitydensityfunction,theexponentialprobabilitydensityfunction p E x = 1 e )]TJ/F40 5.9776 Tf 8.236 3.258 Td [(x x > 0, hasasingleparameter thatdenesboththemean E = andvariance 2 E = 2 ofthepixelstatistics.BoththeRayleighandtheexponentialdistributionswillplaya roleindevelopingthe K probabilitydensityfunctionforamplitudeandintensityimage single-pointstatistics,respectively. 1.2.2.3 K distribution AsmentionedinSection1.2,assumptionsforRayleigh-distributedbackscattering inamplitudedataorexponentially-distributedscatteringinintensitydatadonotholdin highlyresolvedandtexturedseabottomregions.Tobetterdescribethestatisticalnature ofsonarscattering,researchershaveusedthetwo-parameter K distributiontomodel resolutioncellamplitudeandintensityvalues[8,9]. Inseveralpapersthe K distributionforresolutioncellscatteringisderivedusing ageneralizedmodelofEquation1wherethenumberofresolutioncellscattersis distributednegativebinomial[13],thecomponentscattererswithinaresolutioncellhave anexponentially-distributedsize[9],orinthecaseoflow-grazingangle,high-resolution stripmapsonarsystems,thecomponentscatterershavenon-uniformphase[14].The K distributionalsoarisesusingalternateassumptionsabouttheunderlyingnoiseprocess. Alternatively,Wardproposedacompoundrepresentationformodelinghighresolution syntheticapertureradarSARclutterbasedonaproductofRayleighandgamma 23

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densities[15].Theintuitionbehindtheformulationwasthathigh-resolutionechoes createdbySARsystemsappeartobethemodulationofuncorrelatednoisebyanother backgroundnoiseprocesswithlongercorrelationlengths.Thismodelwasshowntot thesingle-pointstatisticsofSARseasurfaceclutter. Inthecompoundrepresentationofthe K distribution,thestatisticsofthecomplex signalamplitudearemodeledbyletting x denoteaRayleighrandomvariablewith parameter y ,where y isdistributedsquare-rootgammawithparameters and b Throughmarginalizationover y ,theunconditionaldensityof x isshowntobe p A x = Z 1 p x j y p y dy = 2 \050 x 2 K )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 x u x > 0 where K )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 x isthemodiedBesselfunctionofthesecondkind, isapositive constantcalledthescaleparameterand = p 2 b isapositiveconstantcalledthe shapeparameter,and u x istheunitstepfunction.Theprobabilitydensityfunction p A x isthedensityfunctionforthe K distribution.Usingasimilarderivation,the statisticsofthecomplexsignalintensityorenergyis p I x = \050 2 x )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 2 K )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( p x u x > 0. InbothpdfformsEquations1and1thescaleparameter isdirectly proportionaltothemagnitudeoftheenergyinthepixelvalue x .Theparameter denestheshapeofthepdfof x withlow valuesindicatingheavy-tailedpixel distributionsandhigh valuesindicatingRayleigh-likedistributions.Infact,as !1 the K distributionapproachestheRayleighdistributioninthederivationofJakeman [13].Thusthe K distributionofthesignalamplitudecanbethoughtofasafamilyofpdf curvesthattendtowardtheRayleighpdfas becomeslarge.Toillustratethisconcept, Equation1isplottedinFigure1-5forvariousvaluesof alongwiththepdfcurvefor theRayleighdistribution. 24

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Figure1-5.AplotoftheKdistributionpdfdenedbyEquation1forvariousvaluesof .Forsmallvaluesof thepdfisheavytailed.As becomeslarge,thepdf approachestheshapeoftheRayleighpdf. Theproductmodelderivationofthe K distributionisaconvenientformulation thatlendsmoreinsightintochoosingparametersforcorrelatedtwo-dimensional K distributionsynthesisandwillberelieduponinfurtherdiscussion.Thereaderisdirected toAppendixAfordetailedderivationsofEquations1and1.Furtherdiscussion oftheproductmodelanditscorrelationpropertieswillclearlyidentifywhichform, amplitudeorintensity,isassumedtoavoidconfusion. 1.2.3Single-pointStatisticsforTextureCharacterization Ingeneral,thesingle-pointorone-dimensionalstatisticsofsonarimageswith non-uniformtexturesarebetterrepresentedviathe K distributionthantheRayleigh 25

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distribution.IntheseriesofFigures1-61-8theempiricalprobabilitydensityfunctions ofamplitudepixelvaluesareplottedversustheestimated K andRayleighdistribution functionforatuniformsand,seagrass,andsandripple. Qualitativelytheplotssuggestthatthe K distributionisabetterstatisticaltthanthe Rayleighdistributionforboththesandrippleandseagrasstextureswhiletheuniform sandseabedtsthe K distributionslightlybetter.Toquantitativelycomparethemodel ttothedata,achi-squaredgoodnessoftstatisticiscalculatedforeach 256 256 imagetextureblockversushypothetical K andRayleighdistributionswithestimated parameters.Thechi-squarestatisticsarelistedinTable1-1foreachtexture.Inallthree casesthe K distributionisabetterttotheamplitudepixelvaluesofthetexturesin question.The K distributiontisamarkedlybettertfortheseagrassandsandripple textureswhoseempiricaldistributionshaveheaviertailsthantheuniformsandtexture. Table1-1. 2 goodness-of-tstatisticsforsand,seagrass,andsandrippleamplitude pixelsfor K andRayleighdistributions. K DistributionRayleighDistribution Texture 2 Statisticdof* 2 Statisticdof* UniformSand 1.254 10 2 1.846 10 3 Seagrass 1.444 10 3 4.513 10 4 SandRipple 3.384 10 3 5.214 10 4 *Note:theabbreviationdofmeansdegreesoffreedom. 1.2.4CorrelatedSASImagePixelModels Dunlopsuggestedcertainbottomtypessand,rock,mud,etc.maybeidentiedby theirrespective K distributionparameters[16].Cobbusedthe K distributionparameters asinputstoadynamictreesegmentationschemetodistinguishcomplexversusuniform seabeds[17].Inbothcasestheauthorsciteddifferencesinparametervaluesfor single-pointstatisticsgatheredfromsonarreturnsasthemeasureofdiscernability betweenobjectclasses.However,therearecaseswhenanalgorithmcanbefooledby single-pointparametermeasuresthatdonottaketwo-dimensionalpixelcorrelationinto 26

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Figure1-6.FitofSASimagesandtextureempiricalprobabilitydensityfunctionwith Rayleighand K distributionprobabilitydensityfunction. 27

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Figure1-7.FitofSASimageseagrasstextureempiricalprobabilitydensityfunctionwith Rayleighand K distributionprobabilitydensityfunction. 28

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Figure1-8.FitofSASimagerippletextureempiricalprobabilitydensityfunctionwith Rayleighand K distributionprobabilitydensityfunction. 29

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account.InFigure1-9the parametersasafunctionofrangeforthetexturesatthetop ofthegureareestimatedviaRaghavan'smethod[18]usingaslidingwindow10pixels widecenteredateachrangebin.Theplotshowsthatsingle-pointestimatedparameter valuesfor arevirtuallyidentical,yetitiseasilyvisuallyveriedtheposidoneaatype ofseagrass,sandripple,andseashelltexturesarenotsimilartexturesduetotheir differentpixelcorrelationlengthsandorientations.Thisobservationsuggeststhat ameasureoftwo-dimensionalspatialcorrelationisnecessaryforaccuratetexture discriminationusing K distributionparameters.Chapters26willdescribestatistical modelsthatincorporatecorrelationparametersinanaimtoaccuratelydescribecomplex SAStexturesandprovideusefulfeaturesforsegmentation. 1.3DissertationOutline HereweintroducedthebasicconceptsofSASstripmapimagingandmotivatedthe useofRayleigh,exponential,and K single-pointprobabilitydistributionstomodelSAS amplitudeandintensityimagepixels.Theremainderofourstudyisoutlinedasfollows. Chapter2willintroduceandderiveaparameterizedstatisticaltexturemodelbased ontheintensityimageautocorrelationfunctionACF.Chapter3willdetailamethod tosimulateSAStextureswithpredenedparametersusingtheintensityACFmodel. Chapter4willdescribeanestimationmethodfortheintensityACFmodel.Chapter5 willdetailresultsforsimulatedandrealtextureACFparameterestimates.Chapter6will explainapplicationsoftheACFmodeltounsupervisedimagesegmentationandpresent segmentationresultsusingACFmodelfeaturescomparedtosegmentationresultsusing otherrecentlyproposedseabedtexturefeatures.Chapter7willsummarizethendings oftheresearchanddescribesomeavenuesforfuturework. 30

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APosidonea BSandripple CSeashell Figure1-9.Comparisonofestimated valueparametersbetweenAposidonea,B sandripple,andCseashelltextures. valuesbetweenthethreetextures arevirtuallyindistinguishable,necessitatingtheneedforcorrelation measurestoproperlysegmentsonarimages. 31

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CHAPTER2 SASIMAGETEXTUREMODELS Theseriesofpapers[1923]describeatheoreticalapproachtotexturemodeling ofimagescollectedfromacoherentsensor.Intheestablishedtheory,thestatisticsof theintensityorpowervalueoftheimagepixelscreatedbyaSARimagingsystem aregovernedbytheACFoftheunderlyingradarcross-sectionandthePSFof theimagingsystem[10,Ch.5,pp.138-139].Recentworkdescribedin[24]uses similartheoreticalprinciplesrstproposedbyOlivertoestimatemultivariateGaussian autocovarianceparametersinanonstationarytexturedSARimage,howeverthe imagingPSFisassumedtobeadeltafunctionandisthusignoredintheestimation oftheautocovarianceparameters.Theassumptionofadeltafunctionimagingpoint spreadfunctiongreatlysimpliesthederivationofcoherentimagetexturemodels, yetisnotrealisticformostreal-worldsystems.Additionally,althoughimagesmay beresampledtoyielddeltafunctionimagepointspreadfunctions,theseschemes usuallyresultinasignicantlossofdataandthusareductionintheaccuracyofspatial correlationlengthparameterestimates[25].Textureresolutioncanberetainedand parametersestimatedmoreaccuratelybyincorporatingarealisticimagingpointspread functionintotheintensityACFmodel. ThederivationspresentedhereafterseektoextendtheworkofSARimagetexture analysispresentedin[1924,26]tothemodelingoftexturesincoherentSASimages byincludingmoreaccuraterepresentationsoftheimagingPSFinmodelderivations andexplicitlyincludingtheeffectsoftheimagingPSFonthemeasuredtexture parameters.AnewGaussianmixturetexturemodelispresentedandincorporated intotheintensityACFformulationyieldingmixturecomponentparametersthatmeasure spatialcorrelationlengthsalongtwoorthogonalaxes,mixturecomponentrotation,and K distributionshapeparameteroftheimagepixelsingle-pointstatistics[27]. 32

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2.1DerivationoftheParameterizedSASImageAutocorrelationFunction Aone-dimensionalintensityACFisderivedinthissectionusingtwoassumptions: 1acomplexGaussianscatteringfunctionislocatedatarangecelllabeled x ,and2in SASprocessingtheimagingPSF h x israngeindependentusingstate-of-the-artstrip mapimagereconstructionalgorithms[4,Ch.4,pp.274-299].InSection2.1.5weextend theone-dimensionalderivationtothetwo-dimensionalcasefortheimagingdomain. Thefollowingdenitionsandidentitieswillbeusedtoderiveaparameterizedformof theintensityACFthatincludesboththescatteringACFandimagingPSFcontributions forcertaincasesofparameterizedGaussianfunctions.Someofthesederivationswill mirrorworkrstpresentedin[1923,26]andwillbenotedwhenpertinent.Forclarity andcompleteness,thederivationsarereproducedwithnotationconsistentwiththenew ndingsandresultspresentedintheremainderoftheproposal.Thefollowingdenitions andidentitieswillbeusedthroughoutthisdissertation: Denition1: a x isthe totalscattering emittedfromaresolutioncellatposition x [26].Writtenincomplexexponentialform, a x = r x e j x ,wherethemagnitude component r x andphasecomponent e j x areassumedtobeindependently distributed.Thephasecomponentisassumedtobedistributeduniformlybetween )]TJ/F25 11.9552 Tf 9.298 0 Td [( and ,thus E f a x g =0 .For a x and a y x 6 = y ,thephasecomponents e j x and e j y areassumedtobeindependent. Denition2: h x istherange-independentSAS imagingpointspreadfunction PSF. Denition3: x a x a x andiscalledthe sonarcrosssection SCSat position x .Theterm x isanalogoustotheenergyemittedbyagivenresolutioncellat position x .Inthispaper, x isdistributedgammawithshapeparameter [20]. Denition4: S x = a x h x ,where istheconvolutionoperator,iscalledthe complexbeamformedreturn fromtheresolutioncellatspatiallocation x Denition5: I x j S x j 2 ,andiscalledthe intensity atrange x 33

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Denition6: R f x x + X E f f x f x + X g andiscalledthe autocorrelation function ACFof f x Identity1: E f a x a y g = E f x g ,if y = x ,otherwise =0 if y 6 = x Thisidentityfollowsfromtheindependenceofthephasecomponentsof a x E f x g willhereafterbedenoted x .Thenotation impliesthatthemeanis stationarywithrespecttospatiallocation. Identity2: E f a x a y a u a v g E f x u g + E f y v g Thisidentityisprovenbytakinganensembleaverageacrossalargenumberof realizations[26],[10,Ch.5,pp.138-140]andnotingthatthelargestcontributionstothe expectedvaluearepairingswiththephasecancellationswhere y = x v = u and x = v u = y .Thecontributionofthephasecancellationfor x = y = u = v isnegligibleoveran ensembleaveragewithmanyrealizations. 2.1.1MeanIntensity Themeanintensityoftheimage E f I x g ,denoted I x ,isnowshowntobea functionofboththeimagingPSFandthemeanSCS[10,Ch.5,pp.138-139] I x = E 8 < : ZZ h h a x )]TJ/F25 11.9552 Tf 11.955 0 Td [( a x )]TJ/F25 11.9552 Tf 11.955 0 Td [( d d 9 = ; ByinvokingIdentity1,theexpressionisreducedto I x = x Z j h j 2 d InthespecialcaseofanimagingPSFwithunitenergy, I x = x 2.1.2ComplexBeamformedReturnSxAutocorrelationFunction InvokingIdentity1againandassumingastationarymeanSCS, ,theACFofthe complexbeamformedreturnisshowntobetheproductofthetheACFoftheimaging 34

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PSFand [20], R s x x + X = ZZ E f a a g h x )]TJ/F25 11.9552 Tf 11.955 0 Td [( h x + X )]TJ/F25 11.9552 Tf 11.956 0 Td [( d d = Z h x )]TJ/F25 11.9552 Tf 11.955 0 Td [( h x + X )]TJ/F25 11.9552 Tf 11.955 0 Td [( d = R h x x + X WhennormalizedbythemeanSCS,theACFof S x istheimagingpointspread functionACF.ThissimplicationisexploitedinChapter4toestimatetheSASimaging functionforaGaussianimagingPSFmodel. 2.1.3IntensityAutocorrelationFunction ThefunctionalrelationshipbetweentheimagingPSFandtheSCSACFwillnow beshownthroughthederivationoftheintensityACF[10,22].Usingtheresultsfrom Identity2andpairing 2 = 1 4 = 3 intherstsummandand 1 = 4 3 = 2 inthe secondsummandtheintensityACFisfoundtobe R I x x + X = ZZ 1 3 E f 1 3 gj h x )]TJ/F25 11.9552 Tf 11.956 0 Td [( 1 j 2 j h x + X )]TJ/F25 11.9552 Tf 11.956 0 Td [( 3 j 2 d 1 d 3 + ZZ 2 4 E f 2 4 g h x )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 h x + X )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 h x )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 h x + X )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 d 2 d 4 R I x x + X = ZZ 1 3 R 1 3 j h x )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 j 2 j h x + X )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 j 2 d 1 d 3 | {z } 1 + ZZ 2 4 R 2 4 h x )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 h x + X )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 h x )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 h x + X )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 d 2 d 4 | {z } 2 TheintensityACFiscomposedoftwoparts:1theleftsummandofEquation2isthe convolutionofthesquareoftheimagingPSFandasquaredimagingPSFshiftedby X withtheACFoftheSCS x and2therightsummandisaslightlymorecomplicated 35

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integralinvolvingsimilarcomponentsbutwhichisonlynon-zeroforsmallvaluesof X assumingtypicaldelta-likeimagingPSFs. 2.1.4IntensityACFforGaussianSCSACFandImagingPSFModels InAppendixBtheclosedformanalyticalsolutionofEquation2isderivedfora GaussianSCSACFmixturemodelpairedwithaGaussianimagingPSF.Thisresult leadstosimplerparameterestimationtechniqueswhencarryingouttheinverseproblem ofrecoveringACFparametersfrommeasureddata.Theaforementionedderivation leadstotheclosedformsolutionofEquation2 R I x x + X = 2 1+ [ R h x x + X ] 2 + 1 X i i l x i p 2 2 + l 2 x i e )]TJ/F40 5.9776 Tf 21.825 3.259 Td [(X 2 2 2 2 + l 2 x i + e )]TJ/F40 5.9776 Tf 9.137 3.258 Td [(X 2 4 2 #! 2.1.5Two-DimensionalFormoftheIntensityACF ForSASintensityimage I x y ,thederivationinAppendixBisalsoapplicableto thetwo-dimensionalcasewithrange X andcross-range Y lagvariablesduetothe functionalseparabilityofthetwo-dimensionalGaussiansintheimagingPSFandSCS ACFi.e., h X Y = 1 p 2 x y e )]TJ/F40 5.9776 Tf 9.137 3.259 Td [(X 2 4 2 x )]TJ/F40 5.9776 Tf 9.008 3.259 Td [(Y 2 4 2 y = 1 p 2 1 p x e )]TJ/F40 5.9776 Tf 9.137 3.258 Td [(X 2 4 2 x 1 p y e )]TJ/F40 5.9776 Tf 9.008 3.259 Td [(Y 2 4 2 y and R X Y = 2 1+ 1 X i i e )]TJ/F40 5.9776 Tf 8.658 3.258 Td [(X 2 2 l 2 x i )]TJ/F40 5.9776 Tf 8.529 3.258 Td [(Y 2 2 l 2 y i = 2 1+ 1 X i i e )]TJ/F40 5.9776 Tf 8.659 3.258 Td [(X 2 2 l 2 x i e )]TJ/F40 5.9776 Tf 8.53 3.258 Td [(Y 2 2 l 2 y i Allintegraloperationsperformedintheprevioussubsectionsfortheone-dimensional casecanbecarriedoutontheconstituentcomponentsofEquations2and2 36

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leadingtothetwo-dimensionalintensityACF R I X Y = 2 1+ [ R h X Y ] 2 + 1 X i i l x i p 2 2 x + l 2 x i l y i p 2 2 y + l 2 y i e )]TJ/F40 5.9776 Tf 21.825 3.259 Td [(X 2 2 2 2 x + l 2 x i )]TJ/F40 5.9776 Tf 21.695 3.259 Td [(Y 2 2 2 2 y + l 2 y i + [ R h X Y ] 2 #! where [ R h X Y ] 2 = e )]TJ/F40 5.9776 Tf 9.137 3.259 Td [(X 2 4 2 x )]TJ/F40 5.9776 Tf 9.008 3.259 Td [(Y 2 4 2 y Interestedreadersaredirectedto[22]forashortsummaryofthetwo-dimensional derivationofEquation2. ThenalformoftheintensityACFisnowpresentedtosimplifynotationand toincludearotationparameterinthemixturecomponentsofEquation2.The two-dimensionalGaussianimagingPSFisdenedas h x y 1 p 2 x y e )]TJ/F23 5.9776 Tf 7.782 3.259 Td [(1 4 [ xy ] B )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 [ xy ] T where B = 2 6 4 2 x 0 0 2 y 3 7 5 andEquation2hasthefollowingsquaredautocorrelationfunctionoverlagvariables X and Y [ R h X Y ] 2 = e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 4 [ XY ] B )]TJ/F23 5.9776 Tf 5.757 0 Td [(1 [ XY ] T Wedenethetwo-dimensionalSCSACFas R X Y 2 1+ 1 X i i e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 [ XY ] S )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 i [ XY ] T 37

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where and i aredenedasinEquationsBandB, S i = i i T i ,where i isa diagonalmatrixofmixturecomponentcorrelationlengthparameters[24] i = 2 6 4 l 2 x i 0 0 l 2 y i 3 7 5 and i isamixturecomponentrotationmatrix[24] i = 2 6 4 cos i sin i )]TJ/F22 11.9552 Tf 11.291 0 Td [(sin i cos i 3 7 5 where i isthecounterclockwiserotationoftheSCSACFmixturecomponentfromthe X -axis.CombiningEquations2and2,andsetting i = i i +2 B T i yields theclosedformoftheintensityACF R I X Y = 2 1+ [ R h X Y ] 2 + 1 X i i l x i l y i j i j )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 h e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 [ XY ] )]TJ/F23 5.9776 Tf 5.757 0 Td [(1 i [ XY ] T + [ R h X Y ] 2 i Atwo-dimensionalpictorialrepresentationofasingle-componentintensityACFwithits associatedparametersisdepictedinFigure2-1. 2.2ExtensionofModelIntensityACFofRippledTextures TheintensityACFmodelpresentedthusfardoesnothaveacapacitytomodel periodicACFsthatareproducedbysandrippletexturedepictedinFigure1-8. Oliverproposedsuchaone-dimensionalperiodicACFbymodulatingtheexponential componentofthecorrelatedgammaSCSACFinEquationBwithacosineofangular frequency X [25] R x x + X = 2 1+ 1 2 e )]TJ/F40 5.9776 Tf 7.897 3.258 Td [(X 2 2 l 2 x 1+cos X X 38

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Figure2-1.GraphicalrepresentationoftheintensityACFmodeldenedbyEquation 2.Theparameters l x and l y denethemajorandminoraxesofthe ellipse,respectively,and denesacounterclockwiserotationoftheellipse fromthespatiallag X axis.Theparameters x and y denetheshapeof theimagingPSF. Followingthesamederivationstepsfortheone-dimensionalintensityACFpresentedin AppendixB,theclosedformsolutionfortheperiodicACFis R I X = 2 1+ [ R h X ] 2 + 1 2 l x p 2 2 + l 2 x e )]TJ/F40 5.9776 Tf 21.064 3.258 Td [(X 2 2 2 2 + l 2 x + e )]TJ/F40 5.9776 Tf 9.137 3.259 Td [(X 2 4 2 + e )]TJ/F27 5.9776 Tf 7.782 5.335 Td [(! 2 X l 2 x 2 2 2 + l 2 x e )]TJ/F40 5.9776 Tf 21.063 3.259 Td [(X 2 2 2 2 + l 2 x cos X l 2 x X 2 2 + l 2 x + e )]TJ/F40 5.9776 Tf 9.137 3.259 Td [(X 2 4 2 !#! AtoftheperiodicACFmodelinEquation2using X =0.159 =0.01 ,and l x =52 toaone-dimensionalrippledtextureestimatedfromhigh-resolutionSASdata normalizedbythemeanintensityisplottedinFigure2-2.Itisclearintheplotthatthe periodicACFmodelofEquation2denotedbythegreencirclesdoesnottthemain lobeoftheestimatedintensityACFcenteredaround X =0 verywell.Additionallythet 39

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aroundtherstsetoflobesat X = 38 areoverestimatedaswell.Toreducethiserror, anextensionofEquation2isproposedwithnonzero-meancomponents. Figure2-2.AcomparisonofintensityACFmodeltstotheone-dimensionalSAS intensityACFestimatedfromarippledseabed,denotedbythesolidredline. TheintensityACFbasedonamodulationbyacosinedenedbyEquation 2anddenotedwithgreencirclesdoesnotttheestimatedintensityACF aswellastheintensityACFmixturemodeldenedbyEquation2and denotedbybluex's.Theparametersforthetwomodelsarelistedin Section2.2. Makingthesubstitution [ XY ]=[ X Y ]+[ X i Y i ] inEquation2,where X i is themeanofthe i th mixturecomponentinthe X spatiallagdirectionand Y i isthemean ofthe i th mixturecomponentinthe Y spatiallagdirectionleadstotheintensityACF 40

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equation R I X Y = 2 1+ [ R h X Y ] 2 + 1 X i i l x i l y i j i j )]TJ/F23 5.9776 Tf 7.782 3.259 Td [(1 2 h e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 [ X Y ] )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 i [ X Y ] T + [ R h X Y ] 2 i Byvisualinspectionitisclearthata6-componentintensityACFmodelofEquation2 with =0.15 ,twozero-meancomponents 1 =0.345 2 =0.081 1 = 2 =0 l x 1 =1.25 l x 2 =5.5 ,andfournonzero-meancomponents 3 = 4 =0.64 5 = 6 =0.64 3 =38.5= )]TJ/F25 11.9552 Tf 9.299 0 Td [( 4 5 =77= )]TJ/F25 11.9552 Tf 9.299 0 Td [( 6 l x 3 = l x 4 =5.5 and l x 5 = l x 6 =7 denotedby bluex'sintheplotinFigure2-2tstheestimatedintensityACFoftherippletexture betterthanthemodeldescribedbyEquation2.Inrippledtexturesthemeans 3 and 5 correspondtotheperiodoftheunderlyingtimeseries.Figure2-3graphically depictsathree-componentintensityACFmodeldescribedbyEquation2withone zero-meancomponentandtwononzero-meancomponents.Itisclearbycomparing thenumberofparametersbetweenFigures2-1and2-3thatthetradeoffofaccurately modelingcomplextexturessuchassandrippleisanincreaseinthenumberofmixture componentsorparameters. 2.3Summary HerewehavederivedaparameterizedmixturemodelforSASintensityimage texturefromrealisticscatteringassumptions.ThemodicationoftheoriginalACF texturemodelintroducedbyOliver[20]describedheremoreaccuratelymodelsrippled texturesthroughadditionaldegreesoffreedom.InChapter3wewillintroduceamethod forgeneratingsyntheticSASimagetextureswithknownparametersfromthismodel.In theremainderofourstudywewillintroduceamethodforestimatingtheintensityACF parametersusingtheEMalgorithm,presentestimationaccuracyndings,anddescribe thesuitabilityoftheACFmodelforextractingSASimagetexturesegmentationfeatures. 41

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Figure2-3.GraphicalrepresentationoftheintensityACFmodeldenedbyEquation 2.Withadditionalparametersdescribingthecorrelationlength,mean, androtationofthethreemixturecomponents,thismodelisconsiderably morecomplexthanthemodelinFigure2-1,yetmaybenecessarytomodel complexseabedtexturessuchassandripple. 42

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CHAPTER3 GENERATINGSYNTHETICSASTEXTURESWITHTHEINTENSITYACFMODEL Inthischapter,theintensityACFmodelderivedintheChapter2willbeusedto generateSASimagetextureswithknownparameters.Syntheticdataisusefulfortwo mainreasons:1SASdatafromeldtestsisbothexpensiveanddifculttocollect duetothecomplexityandscarcityofthesesensingsystems,and2itisimpossibleto constructadatabasewithpreciselydescribedimagestatisticsoverawidevarietyof operatingenvironmentsobtainedfromreal-worldsurveys.SimulatedSASimagesfulll thisneedbybeingbothinexpensivetogenerateandallowingthedesignertoprecisely setthesingle-pointandtexturestatisticsofagivenimage. Somemethodstogenerateside-lookingsonarimageryarebasedonphysical modelsoftheprojector,receivearray,andtheoceanandseabedenvironment. Thesephysics-basedsimulationsrelyuponspecicsensinggeometryandscattering assumptionsfromtargetandseabedmodelstogeneratesyntheticimageryfrom SASandrealaperturearraydesigns.Typically,theresultingsimulationsrelyupon ray-tracingandothernumericaltechniquestocalculatetheenergyreectedfrom pointsinthesimulatedsonarimage[28,29].Physics-basedmethodsareexcellent simulationchoiceswhenthesimulatedacousticalscatteringfromtargetsandthesea oorbathymetrymustbeveryaccurate. Whentheseabedtexturecanbedescribedviaspatialstatisticalmodels,simulation techniquesbasedonphysicalmodelsmaynotbenecessary.Thesimulationmethod describedherereliesonapurelystatisticalmodeloftheimagebasedonthesingle-point parametersandautocorrelationfunctionACFofthe K -distributedintensityimage denedinChapter2.Severalreferencesdescribeaprocesstogeneratecorrelated K -distributedrandomvariableswithhalf-integershapeparametervaluesthatonly requiressumsofsquaredcorrelatedGaussianrandomvariablestoproducethe correlatedgammacomponentoftheproductmodel[20,24,30].Althoughthe 43

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procedureiscomputationallysimplewiththisapproach,therestrictionplacedonthe resolutionoftheshapeparameterpreventsdetailedestimationanalysisforshape parametervaluesontheinterval ,2 whichoccurintexturedimageswithhighpixel contrast.WefollowtheapproachdescribedinworkbyLiu,Tough,andCobbtoproduce imagerywithanarbitrary K distributionshapeparameterviamemorylessnonlinear transformationofcorrelatedGaussianrandomvariables[3133]. Thechapterorganizationfollows.Section3.1presentsamethodforsimulatingSAS imageswithprescribedACFandsingle-pointstatisticalparameters.Section3.2depicts somegeneratedtexturesamplesalongsidetheirrealSAScounterpartsanddescribes anapplicationofimagesynthesistocreatingrealisticdatasetsforautomatictarget recognitionalgorithmdevelopment.Section3.3describesthethedistortionsinthe simulatedintensityimageACFsforchoicesofverysmallshapeparametervaluesand expectedchangestoestimatedshapeparametervaluesbasedontheselectedwidthof theimagingPSF,controlledby x and y 3.1ImageSynthesisProcedure ThesimulationmethoddescribedinthissectionreliesontheSASimagepixel statisticalmodeldescribedinChapter2thatwasbasedontheproductofacomplex Gaussianspecklecomponentwithacorrelatedgammasonarcross-sectionSCS component[10].Recallthesingle-pointpdfequationforintensityimagepixels p I x = Z 1 p x j y p y dy = \050 2 x )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 2 K )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( p x u x fromChapter1.AsdemonstratedinAppendixA,theterm p x j y p y isformedfrom theproductofanegativeexponentialrandomvariable p x j y thesquared-magnitude ofaGaussianrandomvariableandagammarandomvariable p y .Thisequation servesasthebasisforgeneratingtheSASimagesimulationsdescribedinthischapter. TheresultingsimulationswillproducesyntheticSASintensityimageswithprescribed single-pointstatisticsdenedbyEquation1andACFsdenedbyEquation2[33]. 44

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Duetothealgorithmiccomplexity,theprocessofgeneratingcorrelatedgammavariables isrstdescribedthentheprocedureforproducingsyntheticimagesisdetailedlater. 3.1.1GeneratingCorrelatedGammaRandomVariables Thegenerationofacorrelatedgammarandomvariablerequiresamapping ofcorrelationcoefcientsbetweenazeromeanunitvariancebivariateGaussian randomvariableandaunitvariancebivariategammarandomvariable.Usingthe proceduredenedbyLiuandMunsonandextendingitfromone-dimensiontothe two-dimensionalspatialimagedomain,acorrelatedgammarandomvariable \050 x y iscreatedinfoursteps[31].Therststepistodeneatwo-dimensionalgamma distributionautocovariancefunction X Y ,where X Y = R X Y )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 R X Y isthegammaACF,and isthemeanofthegammarandomvariable.The correlatedgammaRVisalsoconstrainedtounitvarianceor 0,0 =1 .Thesecond stepistomapthevaluesof ateach X )]TJ/F39 11.9552 Tf 11.904 0 Td [(Y lagtoazero-meanunitvarianceGaussian autocovariancefunction G X Y viathesolutionof X Y = 1 X k =1 d 2 k [ G X Y ] k where 1 X k =1 d 2 k =1 followsfrom 0,0 = G 0,0 =1 .Thecoefcients d 2 k ofEquation3arefoundby solvingtheequation d 2 k = 1 k 1 Z g x H k x p G x dx 45

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where p G x isthezero-meanunitvariancenormaldensityfunction, H k x isthe k th Hermitepolynomial[34,Ch.22,pp.771-802],and g x isthenonlineartransform g x F )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 F G y j 1 p InEquation3 F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 P y j isthegammainversecumulativedistributionfunction withshapeparameter andscaleparameter fortheprobability P y and F G y isthe zeromean,unitvarianceGaussiancumulativedistributionfunctionevaluatedat y .The thirdstepofthetransformationistolterazeromeanunitvarianceGaussianprocess usingtheautocovariancefunction G X Y foundbythesolutionofEquation3.The fourthandnalstepistomapthevaluesfromthelteredGaussianintoacorrelated gammarandomvariablewiththepropersingle-pointstatisticsusing g x 3.1.2GeneratingCorrelated K -distributedRandomVariables Incorporatingtheaboveprocedureforgeneratingacorrelatedgammarandom variable,theproceduretogeneratean N N syntheticSASintensityimagewithmean I ,shapeparameter andanACFdenedbyEquation2follows: 1.Generatean N N zero-meanunitvariancewhiteGaussiannoiseimage G x y 2.UsingEquations2and3withtheassumption =1 andnormalizingby to guaranteeunitvariance,assignthegammaRVautocovariancefunction X Y = X i i e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 [ X Y ] i i T i )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 [ X Y ] T 3.SolveEquation3tondthezero-meanunitvarianceautocovariancefunction G X Y 4.CreatecorrelatedGaussiannoiseimage CG x y byltering G x y with G X Y CG x y = IDFT n DFT f G x y g p DFT f G X Y g o where IDFT istheinversediscreteFouriertransform, DFT isthediscreteFourier transform,and istheelement-wisearraymultiplicationoperator. 46

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5.Create N N correlatedgammanoiseimage \050 x y bymappingeachpixelfrom CG x y into \050 x y via g x inEquation3 \050 x y )]TJ/F39 11.9552 Tf 22.582 0 Td [(F )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 F G CG x y j 1 p 6.Createcomplexsonarimage S x y byformingaproductofthesquare-root ofthecorrelatedgammanoisewithan N N zero-meanunitvariancewhite complexGaussiannoiseimage C x y andlteringtheresultwiththeimaging PSF h x y [25] S x y = p \050 x y C x y h x y where h x y 1 p 2 x y e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 4 [ xy ] B )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 [ xy ] T aspreviouslydenedinChapter2,and istheconvolutionoperator.Atthisstep itiseasytodiscernthatwhen h x y isdelta-likethesquaredmagnitudeof theRHSofEquation3istheproductformoftheintensitypdfdescribedby Equation3. 7.Theintensityimage I x y withmean I andACFdenedbyEquation2isthe scaledsquaredmagnitudeof S x y I x y = j S x y j 2 I p 3.2SynthesizedImageSamples Asademonstrationofthealgorithmdescribedabove,fourreal 256 256 SAS imagetexturesaredepictedalongsidetheirsimulatedcounterpartsinFigures3-13-4. Forthesandtexture,threezero-meancomponentswereused.Fortheseagrass,rock, andsandrippletexturesatotalof7componentswereused,twopairednonzero-mean, andthreezero-mean.Theparametersusedtogeneratethefourtexturesarelisted inTable3-1.Therstandsecondcomponentsoftheseagrass,rock,andsandripple texturesarethepairedcomponentsthusonlyoneofthecomponents'parametersare listedsincetheothermemberofthepairisthemirrorimage.Thereadershouldalso notethatbecauseofthenonzero-meancomponentpairingthe valuesinTable3-1for theseagrass,rock,andsandrippletextureswillnotsumtoonesince 1 and 2 values arehalved. 47

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ASASimagesandtexture BSASimagesandACF CSimulatedSASsandtexture DSimulatedsandACF Figure3-1.SimulatedsandtextureSASimagedisplayedincomparisonwitharealsand textureSASimage:Arealsandtexture,BestimatedintensityACFofA,C simulatedsandtextureusingparameterslistedinTable3-1,andDthe simulatedintensityACFusingtheparametersinTable3-1. 48

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ASASimageseagrasstexture BSASimageseagrassACF CSimulatedSASseagrasstexture DSimulatedseagrassACF Figure3-2.SimulatedseagrasstextureSASimagedisplayedincomparisonwithareal seagrasstextureSASimage:Arealseagrasstexture,Bestimatedintensity ACFofA,CsimulatedseagrasstextureusingparameterslistedinTable 3-1,andDthesimulatedintensityACFusingtheparametersinTable3-1. 49

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ASASimagerocktexture BSASimagerockACF CSimulatedrocktexture DSimulatedrockACF Figure3-3.SimulatedrocktextureSASimagedisplayedincomparisonwitharealrock textureSASimage:Arealrocktexture,BestimatedintensityACFofA,C simulatedrocktextureusingparameterslistedinTable3-1,andDthe simulatedintensityACFusingtheparametersinTable3-1. 50

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ASASimagesandrippletexture BSASimagesandrippleACF CSimulatedSASsandrippletexture DSimulatedsandrippleACF Figure3-4.SimulatedsandrippletextureSASimagedisplayedincomparisonwitha realsandrippletextureSASimage:Arealsandrippletexture,Bestimated intensityACFofA,Csimulatedsandrippletextureusingparameterslisted inTable3-1,andDthesimulatedintensityACFusingtheparametersin Table3-1. 51

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Table3-1.SimulationparametervaluesforimagesinFigures3-13-4. TextureTextureType ParametersSandSeagrassRockSandRipple 3 51 50 50 5 0 50 50 50 5 l X 1 0 315 06 04 0 l X 2 1 010 05 04 0 l X 3 0 59 050 025 0 l X 4 4 55 52 8 l X 5 2 82 12 5 l Y 1 2 855 038 246 5 l Y 2 4 535 043 062 0 l Y 3 1 785 0105 0100 0 l Y 4 23 090 026 5 l Y 5 7 510 046 0 1 0 40000 03640 04950 1485 2 0 35000 04560 03250 0735 3 0 25000 04370 11140 0010 4 0 12130 07300 4700 5 0 67100 65160 0850 1 0 0 o 30 8 o )]TJ/F22 11.9552 Tf 9.299 0 Td [(25 7 o 44 2 o 2 0 0 o 21 8 o )]TJ/F22 11.9552 Tf 9.298 0 Td [(7 9 o 45 0 o 3 0 0 o 15 7 o )]TJ/F22 11.9552 Tf 9.299 0 Td [(43 8 o 45 0 o 4 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 8 o )]TJ/F22 11.9552 Tf 9.299 0 Td [(26 2 o 43 6 o 5 20 9 o )]TJ/F22 11.9552 Tf 9.298 0 Td [(0 7 o 36 7 o [ X 1 Y 1 ][ 0,0 ][ 63,52 ][ 50, )]TJ/F22 11.9552 Tf 9.298 0 Td [(60 ][ 32, )]TJ/F22 11.9552 Tf 9.298 0 Td [(18 ] [ X 2 Y 2 ][ 0,0 ][ 72, )]TJ/F22 11.9552 Tf 9.299 0 Td [(46 ][ 85, )]TJ/F22 11.9552 Tf 9.298 0 Td [(45 ][ 60,33 ] [ X 3 Y 3 ][ 0,0 ][ 0,0 ][ 0,0 ][ 0,0 ] [ X 4 Y 4 ] [ 0,0 ][ 0,0 ][ 0,0 ] [ X 5 Y 5 ] [ 0,0 ][ 0,0 ][ 0,0 ] Targetinsertionapplication: Asmentionedintheintroductiontothechapter, eldsurveydataisdifcultandexpensivetocollectforawidevarietyofseabed environments.Additionally,collectingdataforsonarimagetargetdetectionalgorithm developmentaddsevenmorecomplexconstraints.Inadditiontotheneedfortarget trainingsamplesindifferentseabedcontexts,targetposeandrangemustalsobe 52

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ASynthetichomogeneousbackground BSyntheticheterogeneousbackground Figure3-5.Twosyntheticimageswithinsertedtargets.AsyntheticSASimagewitha homogeneousbackgroundand3syntheticcone-shapedtargetsinserted.B syntheticSASimagewithaheterogeneousbackgroundcomposedoftwo homogeneousbackgroundsandthreesyntheticcylindricaltargetsinserted. varied,addingmanymultiplesofdatasurveystoensurealgorithmrobustness.Robust targetdetectionalgorithmdesigncanbedonemorecheaplyandinlesstimewith syntheticsonarimagescontainingsynthetictargetsbecausetheymitigatetheneedfor large,expensivedatasurveys. Figure3-5depictstwoexampleimagesofsynthetictargetsinsertedintosynthetically generatedseabedenvironments.InFigure3-5A,syntheticcone-shapedtargetsare insertedintoasimplesyntheticsandenvironmentatvariousranges.InFigure3-5 B,syntheticcylindricaltargetsatvariousorientationsandrangesareinsertedinto aheterogeneoussyntheticseabedenvironmentcomprisingtwodifferenttextures separatedbyadistinctboundary. 3.3EffectsofParameterChoiceonCorrelatedImageSynthesis Itisimportanttounderstandtheramicationsofparameterchoiceinthesimulation processandhowthesechoicesaffectthenalsimulatedimagery.Thechoiceinthe 53

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Figure3-6.Coefcientvaluesforvariousvaluesof .Asthevalueof decreases,more coefcientsareneededtoaccuratelymapcorrelationcoefcientsbetween thegammaandGaussianrandomvariables. Table3-2.Valuesof d 2 k forvaryingvaluesofshapeparameter d 2 1 d 2 2 d 2 3 d 2 4 d 2 5 d 2 6 d 2 7 0.10.3480.4262.00 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1.99 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 2.27 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 3.35 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 1.93 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 0.30.5830.3546.09 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 8.04 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 1.39 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 4.64 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(6 1.43 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 0.70.7570.2291.39 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 1.74 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 9.63 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(5 1.80 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 2.05 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(6 1.50.8690.1282.80 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 6.50 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 1.38 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(6 1.47 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(6 4.13 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(8 3.00.9470.0533.35 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 4.97 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(6 1.36 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(9 8.22 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(9 8.95 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(10 numberofsummationtermsinEquation3denestheaccuracyofthecovariance mappingbetweengammaandGaussiancorrelationcoefcientsanddependsheavily onthechoiceoftheshapeparameter .Figure3-6depictsthecoefcientvalues d 2 k forvariousvaluesof .ThesevaluesarealsopresentedinTable3-2.Itisclearas becomesverysmall,moretermsareneededinEquation3toaccuratelymap correlationcoefcientsbetweenthetworandomvariables. 54

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Figure3-7.Plotofthemappingbetween and G determinedbyEquations33. Forsmall thenegativedomainof G becomesrestricted. 3.3.1ACFDistortionDuetoSmall Values Thereareconstraintsonthevaluesthat G maytakesuchthattheautocovariance isalegitimatenonnegativedenitefunction[31].Figure3-7depictsvariousplotsofthe mappingbetween and G determinedbyEquations33fordifferent values. As becomesverysmall,thenegativedomainof G becomesquiterestricted.For non-symmetricinversecumulativedistributionfunctionssuchasthegammadistribution, asolutionexistsforeach G inthemappingdenedbyEquation3if 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 d 2 1 55

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DuetotheresultsdepictedinTable3-2andFigure3-6,thisconditionwilllikelynotbe metformanyautocovariancefunctionswhentheshapeparameter isverysmall.When theconstraintinEquation3isviolatedthereisdistortionintheresultingsimulated image'sspectrumandACF.UsingtheperiodicintensityACFdenedinEquation2 wemeasuredmean-squareerrorbetweenthetrue R I andtheestimate ^ R I forvarious valuesof ^ R I wasestimatedby ^ R I X Y = 1 NM F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 f F f I x y g conj F f I x y g g where F fg isthe2-DdiscreteFouriertransformoperator, conj istheconjugation operator, I x y isthe N M intensityimage,and istheelement-wisearray multiplicationoperator. SincethecorrelationcoefcientsproducedbyEquation2areallpositive numbers,onecanassumedistortioncausedbythemappingbetweengammaand Gaussiancorrelationcoefcientswilloccurmainlywhen valuesarelessthan 0.5365 whichisthevalueforwhichEquation3issatisedforallpositivevaluesof .Figure 3-8depictsthemeansquareerrorover100 512 512 synthesizedimageswithvarying parametersof andxedparameters 1 = 2 =0.5 1 = 2 =0 l X 1 = l Y 1 =4 l X 1 = l Y 1 =20 ,fortheACFdenedbyEquation2.Theshapeparameter was estimatedineachsimulatedimageusingamethoddescribedbyRaghavan[18].The errorbarsintheplotsignifyonestandarddeviationfromthemeanvalueandthe -axis isplottedinlogarithmicscaleforclarity.Figure3-8conrmsthatspectraldistortionis highestforsmallvaluesof thensteadilydecreasestoarelativelyconstantvaluefor > 0.9 3.3.2DependenceofExpected ValueonImagingPSFParameters Finally,thesmearingofthetrueSCSresolutionbytheimagingpointspread functionPSF h x y willchangethe valueinthesimulatedimagedependingon theaveragingeffectof h x y .ExaminingEquation3,largevaluesof x and y will 56

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Figure3-8.Plotofmean-squareerrorinducedbyviolationoftheconstraintinEquation 3.Distortioninthesimulatedimageryisgreatestforsmallvaluesof The -axisisplottedinlogarithmicscaleforclarity. causetheimagingPSFtosmoothlargeneighborhoodsofpixelvalues.Abrahamand Lyonsdiscussthiseffectintermsofsummingsonarreturnsfromhorizontalelementsin anearlierpaper[9].Figure3-9depictsthechangein causedbydifferentvaluesof whereweset x = y = inEquation3.Itisclearfromtheplotthatasthesizeof theneighborhoodincreases,i.e., becomeslarge,thevalueof inthesimulatedimage increasesfromthedesired choseninthesimulationprocess.EachpointinFigure3-9 isthemeanvalueof estimatedfrom1000imagesimulationsusingatwocomponent intensityACFfromEquation2withinitialparameters =1.0 l x 1 = l y 1 =4 l x 2 = l y 2 =20 1 = 2 =0.5 ,and 1 = 2 =0 o .Errorbarsintheplotrepresentone standarddeviationfromthemean. 57

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Figure3-9.Plotoftheeffectonthesimulatedvalueof forincreasingvaluesof inthe imagingpointspreadfunction h x y .As increases,thesimulatedvalueof increasesfromtheoriginaldesiredvalueof =1.0 3.4Summary Herewepresentedaprobability-basedmethodforsimulatingSASimageswith desiredstatisticalandtexturalfeatures.Varioussimulatedimagesaredepicted alongsidetheirrealcounterpartsforseveralcommontexturesofinterest.Effects ofparameterchoiceonthesimulatedimagesisexaminedforthenumberofterms inthecovariancemappingfunction,thedistortionproducedinnon-positivedenite correlationfunctions,andthedegradationof causedbysmoothingeffectsofthe imagingpointspreadfunction.Whilethistypeofsimulationdoesnotmodelcause andeffectrelationshipscausedbybathymetry,e.g.occlusion,itcanbeveryusefulfor experimentswherethestatisticalandtexturalfeaturesmustbepreciselyknown apriori ThesimulationtechniquespresentedthusfarwillbeusedtogeneratesyntheticSAS 58

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imageswithknownACFparameters.InChapter5theseimageswillthenbeusedtotest theperformanceoftheparameterestimationmethodpresentedinChapter4. 59

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CHAPTER4 AUTOCORRELATIONFUNCTIONMODELPARAMETERESTIMATION HerewepresentaniterativeExpectationMaximizationEMalgorithmtoestimate theparametersoftheintensityACFdescribedbyEquation2basedonavariationof themaximumlikelihoodformulationforgroupedandtruncateddata[35][38].Equation 2denesatwo-dimensionalACFwiththeparameter i describingtherotationof theSCSACFmixturecomponentscounterclockwisefromthe X -axisviamatrix i parameters x y deningtheimagingPSFwidth, l x i and l y i deningtheSCSspatial correlationlengthsofthemixturecomponentsintherotated X )]TJ/F20 11.9552 Tf 12.622 0 Td [(and Y )]TJ/F20 11.9552 Tf 12.622 0 Td [(directions, x i and y i deningthemeanofthemixturecomponents, i deningthemixingparameters ofthemixturecomponents,andthesingle-pointstatisticalparameter deningthe probabilitydensityfunctionshapeparameterofthecorrelatedgammadistribution. Theautocorrelationfunctionestimatesoftheimagingpointspreadfunctionand theintensityautocorrelationfunctionaredenoted ^ R h X Y and ^ R I X Y respectively. Tondtheseestimates,theFourierdomainimplementationofthe2-Dautocorrelation functionisused.Assumingan N M complexbeamformedimage S x y ^ R h X Y isfoundusingEquation2,themeanestimate ,andtheestimatedcomplex beamformedimageautocorrelationfunction R S X Y ^ R h X Y = 1 ^ R S X Y where ^ = 1 NM X x i X y i j S x i y i j 2 and ^ R S X Y iscalculatedas ^ R S X Y = 1 NM F )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 f F f S x y g conj F f S x y g g where F fg isthe2-DdiscreteFouriertransformoperator, conj istheconjugation operator,and istheelement-wisearraymultiplicationoperator. ^ R I X Y isfoundwith 60

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thesameoperationasEquation4,substituting S x y withtheintensityimage I x y andwhere I x y j S x y j 2 ^ R I X Y = 1 NM F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 f F f I x y g conj F f I x y g g 4.1ImagingPointSpreadFunctionParameterEstimation UsinganestimatedACFofthecomplex N N beamformedimage S x y ,the simpleformofthecomplexACFnormalizedbythemeanintensityinEquations2 and2allowsforadirectestimationofparameters x and y inisolationfromthe parametersthatdene R X Y .Notingthat R h X Y isanon-negativefunction,we constructanobjectivefunction J x y = X X X Y ln ^ R h X Y )]TJ/F22 11.9552 Tf 11.955 0 Td [(ln R h X Y x y 2 where ^ R h X Y isestimatedusingEquation4and R h X Y x y = e )]TJ/F40 5.9776 Tf 9.137 3.259 Td [(X 2 8 2 x )]TJ/F40 5.9776 Tf 9.008 3.259 Td [(Y 2 8 2 y J x y maynowbeminimizedbytheleast-squaresolutionofthematrixequation XY 2 6 4 1 2 x 1 2 y 3 7 5 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(8 2 6 6 6 6 4 ln ^ R h X 1 Y 1 ln ^ R h X N Y N 3 7 7 7 7 5 where [ XY ] isa N 2 2 matrixofsquaredspatiallagindicesforimage S x y .The estimatedparameters ^ x and ^ y arethenusedtocalculateamodelimagingPSF ~ R h X Y ^ x ^ y thatremainsxedforthesolutionoftheremainingparametersof Equation2. 4.2Single-PointShapeParameterEstimation Theparameter ,whichisthesingle-pointpdfshapeparameteroftheintensity imagedata[10,Ch.5,pp.131-134],isestimatedviaamethoddevelopedbyRaghavan in[18].Notingthattheshapeparameterofthegammadistributioncloselyapproximates 61

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A B C D Figure4-1.AnexampleofestimatingtheimagingPSFparametersfromatexturedSAS imagesnippet.ThecompleximageoftherippledtextureAisusedtond thecomplexACFB,where x and y arefoundvialeast-squarest.The estimatedimagingPSF ^ R h X Y isplottedforspatiallag X Candspatial lag Y Ddimensionsandoverlaidwith ~ R h X Y ^ x ^ y forestimated parameters ^ x =0.6136 and ^ y =0.5636 62

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the K distributionshapeparameter,theestimateiscalculatedviaratioofgeometricto arithmeticsamplemeansandasimplefunctionalmapping. Raghavandevelopedthemethodforthe K distributionofamplitudepixelsand wheretheshapeparameterhasthesupport 2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1, 1 .Thustheamplitudeimage pixel A x y isdenedtobethemagnitudeofthecompleximagepixelor A x y = j S x y j andaunitconstantisaddedtotheestimationoutputfromRaghavan'smethod tomatchtheformulationthroughoutthispaper.Tondtheshapeparameterestimate ^ usingRaghavan'smethod,anintermediatevariable ^ n isrstestimatedby ^ n = 1 n P n i =1 A i Q n i =1 A i 1 n = m a m g where A i i 2f 1,2,..., n g ,are n independentlyidenticallydrawnamplitudeimagepixel samples, m a isthearithmeticsamplemean,and m g isthegeometricsamplemean.The estimate ^ isthenfoundbygraphicallyinvertingthefunction ^ n = )]TJ/F30 11.9552 Tf 8.767 9.684 Td [()]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(1.5+ )]TJ/F22 11.9552 Tf 11.956 0 Td [(0.5 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F30 11.9552 Tf 8.767 9.684 Td [()]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(1.5 )]TJ/F22 11.9552 Tf 11.955 0 Td [(0.5 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ 8.767 -0.166 Td [( +1 )]TJ/F30 11.9552 Tf 8.767 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(0.5 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F30 11.9552 Tf 8.767 9.684 Td [()]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(0.5 n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ 8.767 -0.166 Td [( +1 # n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 for n > [ 2 +1 ] )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 WhilenotasaccurateastheMLestimationprocedure[39]for > 2.0 ,thismethod issufcientforthepresentsensingdomainwherethedataofinteresttypicallyhave low values.Ininstanceswhere > 2.0 ,anotherfastestimationtechniqueusinga ratioofrstandsecondordersamplemomentscanbeused,howeverthevariance oftheestimate ^ willlikelybemuchgreaterthantheMLestimateorthemethod describedabove[18].Thereareothertechniquesforestimatingthe K distributionshape parameter,e.g.[40][42],andthesemaybeusedinterchangeablyintheparameter estimationalgorithmdescribedinthenextsection. 63

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4.3NormalizedIntensityAutocorrelationFunctionParameterEstimationviathe EMAlgorithm GiventheestimatedintensityACF,or ^ R I X Y ,themodeledautocorrelation functionoftheimagingPSF ~ R h X Y x y foundinSection4.1fromacoherent SASimage,andtheshapeparameterestimate ^ foundinSection4.2,weestimate theremainingparametersofEquation2viatheExpectation-MaximizationEM algorithm[35,43].Specically,weuseaformulationoftheEMalgorithmfortruncated dataduetothepracticalproblemsassociatedwithcalculatingtheACFestimateusing anautocorrelationblockoperator,namelytruncatedandnoisyACFtailestimates[36]. First,thenormalizedformoftheintensityACFisdenedas[22] R I N X Y R I X Y 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 = [ R h X Y ] 2 1+ 1 f i l x i l y i + 1 f i l x i l y i e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 [ X Y ] i )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 [ X Y ] T where f i l x i l y i X i i l x i l y i j i j )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 Droppingnotation X Y forbrevityandassumingthat ~ R h X Y and ^ arexed throughouttheestimationalgorithm,theestimate ^ R I N X Y issolvedatstep k +1 by manipulatingEquation4into ^ R k +1 I N )]TJ/F22 11.9552 Tf 13.315 2.656 Td [(~ R 2 h 1+ 1 ^ f k i l k x i l k y i = ^ R I ^ 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 )]TJ/F22 11.9552 Tf 13.314 2.656 Td [(~ R 2 h 1+ 1 ^ f k i l k x i l k y i = 1 ^ f k +1 i l k +1 x i l k +1 y i e )]TJ/F23 5.9776 Tf 7.782 3.259 Td [(1 2 [ X Y ] k +1 i )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 [ X Y ] T andestimatingtheparametersinEquation4,where ^ R I istheestimatedintensity ACFfromthesonarimagedata, ^ istheestimatedintensityimagemean,and wherethenotation x k denotestheestimateofvariable x atstep k .Inthisestimation 64

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schemethereisanacceptedtradeoffofinaccuracyfromusingtheoldervaluefor 1 ^ f k i l k x i l k y i inEquation4tomaintainacleanerfunctionalformforEquation4. Theright-handsideofEquation4isrecognizableasan M -componentGaussian mixturemodeloftheform C M X i =1 i G i X Y j i where C =2 P j j l x j l y j ^ and i = i l x i l y i P j j l x j l y j sothat P i i =1 i = f X i Y i i g ,and G i X Y j i = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 j i j )]TJ/F23 5.9776 Tf 7.782 3.259 Td [(1 2 e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 [ X Y ] )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 i [ X Y ] T UsingtheEMalgorithmfortruncatedtwo-dimensionalGaussianmixturemodels describedinAppendixC,theparameters k = n k 1 ,..., k M o atstep k arefoundby maximizingthelikelihoodfunction L k = X X X Y g k X Y ln C k M X i =1 k i G k i X Y j k i where g k X Y = ^ R I ^ 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 )]TJ/F22 11.9552 Tf 13.315 2.657 Td [(~ R 2 h 1+ 1 ^ f k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 i l k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 x i l k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 y i # Afterconvergenceatstep k ,thenewvaluesfor l k x i l k y i ,and k i ,arecalculatedby 2 6 4 l k x i 2 0 0 l k y i 2 3 7 5 = ^ k i T k i ^ k i )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 ^ B 65

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and k i = k i l x k i l y k i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 P j k j l x k j l y k j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 undertheconstraint P j j =1 ,andwhere ^ B isthediagonalmatrixofestimatedimaging PSFcorrelationlengthsand ^ k i isfoundviasingularvaluedecompositionof k i Equations4and4arethenupdatedwithanewvaluefor f i l x i l y i k X i Y i ,and k i .Theparameterset k +1 isupdatedviatheEMalgorithminthenext iteration.Thealgorithmiteratesbetweenthesetwoupdatestepsuntilconvergence. Uponconvergence,theestimates ^ l x i ^ l y i ,and ^ i i =1,..., M ,arerecoveredvia Equations44.Thenonzeromeans ^ X i ^ Y i arefoundviaEquationC. UsingEquation2thevariable ^ i isrecoveredby ^ i =tan )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ^ i ,2 ^ i ,1 where ^ i m n istheelementonthe m -throwand n -thcolumnofmatrix ^ i 4.3.1AlgorithmSimplicationsbyCouplingParameters SincethedatastructureswithintheintensityACFaresymmetricaboutspatiallag X =0, Y =0 somesimplicationstoparameterestimationstepscanbemadein thecaseofcouplednonzero-meanmixturecomponents.Theparameterestimation equationsforthevariablesinEquation2canbemodiedslightlytoreectthe constraintsofthesymmetryofthemixturecomponents,namelythattwocoupled componentswillhavethetwo-dimensionalmeans [ X 1 Y 1 ]= )]TJ/F22 11.9552 Tf 9.299 0 Td [([ X 2 Y 2 ] andthesame variance.Thiscouplingsimpliestheestimationofthenonzero-meancomponentssince twoofthecomponentparameterscanbeestimatedwithonestep.Thissimplication isbestillustratedwithanexample.Weassume j measurementsdenoted x j aredrawn fromathree-componentone-dimensionalGaussianmixturemodelGMM p x j j 2 withonecomponentmean 1 =[00] ,twononzerocomponentmeans 2 = )]TJ/F25 11.9552 Tf 9.298 0 Td [( 3 ,three 66

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componentvariances 2 1 and 2 2 = 2 3 ,andmixingparameters 1 and 2 = 3 .The updateequationsfor 1 and 1 aretheGMMupdateequationsin[43,Ch.2,pp.63-64]. Forparameters 2 3 2 3 2 2 ,and 2 3 updateequationscanbewrittenas z k ij = k i p i x j j i 2 i 3 P l =1 k l p l x j j l 2 l i =2,3, k +1 2 = P j h z k 2 j x j )]TJ/F39 11.9552 Tf 11.955 0 Td [(z k 3 j x j i P j h z k 2 j + z k 3 j i k +1 3 = )]TJ/F25 11.9552 Tf 9.299 0 Td [( k +1 2 2 3 k +1 = 2 2 k +1 = P j h z k 2 j x j )]TJ/F25 11.9552 Tf 11.955 0 Td [( k +1 2 2 + z k 3 j x j + k +1 2 2 i P j h z k 2 j + z k 3 j i and k +1 3 = k +1 2 = 1 2 X j h z k 2 j + z k 3 j i ThissimplicationisappliedtotheintensityACFparameterestimationbyextending theapproachtotwodimensions, X and Y ,andmodifyingtheappropriateparameter estimationstepsinAppendixC. 4.3.2EffectsofIntensityACFTruncationonEstimationAccuracy AnynoiseandoutliersoutsidethemainlobeoftheestimatedintensityACF determinedbyEquation4candegradeparameterestimationaccuracy.Figure 4-2isaone-dimensionalexampleoftheeffectsofnoiseonparameterestimationfor atwo-componentintensityACF.InthegureripplesintheestimatedintensityACFat edgesofthespatiallagdomaincausethecorrelationlength l x tobeoverestimated. Inthisexample,anintensityimageisgeneratedusingatwo-componentACFwith parameters: l X 1 =5 l X 2 =20 ,and 1 = 2 =0.5 .Theestimatedvalueof ^ l X 1 =17.8 is muchlargerthanthetruevalue. 67

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Figure4-2.One-dimensionalplotofanintensityACFdenotedbytheredlineestimated fromanimagegeneratedusingatwo-componentACFdenotedbytheblack x'swithparameters l X 1 =5 l X 2 =20 ,and 1 = 2 =0.5 .Therippleatthe edgeofthespatialdomaincausesalargeinaccuracyinthemodelACF denotedbythebluedotsintheestimateof ^ l X 1 =17.8 Tomitigatetheseinaccuracies,thedataoutsidethisregionistruncatedordiscarded inthecalculationoftheACFparameters.IftheEMalgorithmisthenemployedto estimateparameterswiththissmallerregionofsupportinthespatiallagdomain, parameterestimatesoflargecorrelationlengthswillbeunderestimated.Thiseffectis showninFigure4-3,where ^ R I istruncatedat 20 andtheestimatedmodelcorrelation length l X 2 =10.2 Tomitigatetheerrorscausedbyestimatenoisewithoutcausingalossofaccuracy duetotruncation,thedatainthetruncatedregionisreplacedwithmissingdatarather thandiscarded[36].ThismissingdataisestimatedalongwiththeACFmixture componentparametersintheEMalgorithmpresentedinAppendixC.InFigure4-4, ^ R I isagaintruncatedat 20 ,butmissingvaluesareestimatedforthedomain 100 68

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Figure4-3.NegativeeffectsofACFestimatetruncation.Ifnoiseoutsidethemainlobeis mitigatedbysimplydiscardingthedata,longcorrelationlengthsare underestimated.ForthemodelACFinthisgure ^ l X 1 =9.93 and ^ l X 2 =10.2 versustrueACFparameters l X 1 =5 and l X 2 =20 producingmoreaccurateestimatesoftheACFcorrelationlengths ^ l X 1 =4.98 and ^ l X 2 =15.2 versustrueACFparameters l X 1 =5 and l X 2 =20 4.3.3IntensityACFAlgorithmSummary Thealgorithmicstepstoestimatetheparametersof R I X Y aresummarized below.Inputs:Errorthresholds 1 and 2 N N complexbeamformedSASimage S x y ,numberofGaussianmixturecomponents M 1.Estimatemeanintensity ^ : ^ = 1 N 2 X x i X y i j S x i y i j 2 2.EstimateImagingPSFACF ^ R h X Y viadiscrete2-DFouriertransform: ^ R h X Y = ^ )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 N 2 F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 f F f S x y g conj F f S x y g g 69

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Figure4-4.Estimationafterreplacingtruncateddatawithmissingdata.Thetruncated regionbeyond 20 intheestimate ^ R I isreplacedwithmissingdataandthe domainextendedto 100 intheEMalgorithm.Thisadditionalstepyields moreaccuratecorrelationlengthparameters, ^ l X 1 =4.98 ^ l X 2 =15.2 where F fg isthediscreteFouriertransformoperator, conj istheconjugation operator,and istheelement-wisearraymultiplicationoperator. 3.EstimateIntensityACF ^ R I X Y : ^ R I X Y = 1 N 2 F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 F j S x y j 2 conj F j S x y j 2 4.Estimate ^ x and ^ y vialeast-squaresmatrixcomputationusing ^ R h X Y and ^ 5.Estimate ^ viaRaghavan'smethod. 6.Initializeparameterset 0 i = 0 i 0 X i 0 Y i 0 i for i =1,..., M 7.Set k =0 8.Initializeerrorvariables E 1 >> 1 E 2 >> 2 9.Initialize L 0 viaEquations44. 70

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10.WHILE E 1 > 1 DO a k = k +1 bSet p =0 cWHILE E 2 > 2 DO i. p = p +1 ii.Calculate E j p +1 i [ x j y j ] j p i viaEquationC iii.Calculate p +1 i viaEquationC iv.Calculate p +1 X i p +1 Y i viaEquationC v.Calculate p +1 i viaEquationC vi. E 2 = j L p +1 )]TJ/F39 11.9552 Tf 11.956 0 Td [(L p j vii.ENDDO dRETURN k i l k x i l k y i k X i k Y i eUsing ^ R I X Y ^ ~ R h X Y k i l k x i l k y i k X i k Y i calculate ^ R k +1 I N X Y via Equations4and4 f E 1 = j ^ R k +1 I N X Y )]TJ/F22 11.9552 Tf 13.314 2.657 Td [(^ R k I N X Y j gENDDO 11.RETURN ^ l x i ^ l y i ^ i ^ X i ^ Y i ^ i for i =1,..., M 4.4InitializationofParametersandImplementationConsiderations DuetothenonlinearityoftheobjectivefunctioninEquation417theglobal minimumsolutionismorelikelytobereachedgivenagoodinitialparameterestimate asaninputtotheiterativeparameterestimationalgorithm.Inallestimationattempts describedinthiswork,thespatialcorrelationlengthswereinitializedwitharandom valuebetween1and3.Itwasfoundinmanycasesthealgorithmwasslowtoconverge orfailedtoconvergetosmallspatialcorrelationlengthvalueswhentheinitialspatial correlationlengthswereinitializedtolargevalues.Conversely,initializingwithsmall valuesdidnotadverselyaffecttheconvergencerateoraccuracyoflargecorrelation lengthestimates.Themixingparameterswereinitializeduniformlyasrandom 71

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numbersbetween0.1and0.9.IftheACFcomponentshadanonzero-mean,the meanparameterswereinitializedathalfthetruncationvalue. Thereareothermethodsthatcouldbeusedtoinitializetheseparametersmore accurately,especiallythemeanparametersofthenonzero-meanmixturecomponents. Spectralestimationtechniques,thespectrumbeingthedualoftheautocorrelationinthe frequencydomain,arepowerfulmethodstodeterminemodesorpeaksinthefrequency domain.AsmentionedinSection2.2thesandrippletextureACFmodelassumesthat nonzero-meanmixturecomponentsarespacedattheperiodoftheripple.Ofcourse thesemixturecomponentmeansaredirectlyanalogoustopeaksinthespectrumof thetexture.Preliminaryresultsusingtechniquesdescribedin[44]and[45]toestimate thefrequenciesinherentinsandrippletexturesindicatethatspectralpeakestimation couldalsobeusedtoinitializethemeansofthemixturecomponents.Otherclassical spectrumestimationmethods,e.g.theconventionalperiodogram[46,ch.2pp.37-39] ortheFouriertransformoftheACF,maybesuitableformeanparameterinitializationas well. Forablockprocessingapproachtotextureparameterestimation,asimpleintuitive approachtoparameterinitializationistondanestimatefortherstblockthroughthe meansdescribedabovethensettheinitialestimatesforthenextsequentialblockwith theestimatedparametersofthepreviousblock.Assumingthattextureparametersare somewhatstationarybetweenblocks,subsequentblocksshouldbeinitializedcloseto theparameterestimationsolution. Numberofmixturecomponentparameters: Wheneveramixturemodelisused totadatamodelcarefulconsiderationmustbegiventothe apriori choiceforthe numberofmixturecomponentparameters.Abalancebetweentoomanyparameters, whichleadstooverttingthedata,andtoofewparameters,wherethemodeldoesnot haveenoughdegreesoffreedomtoaccuratelydescribethedata,mustbestruck.Inthe 72

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texturemodelingproblem,itispossibletousetheBayesianInformationCriterionBIC tostrikesuchabalance[47,ch.4,pp.215-217]. TheBICattemptstobalancethemodelaccuracyexpressedthroughthemaximum likelihoodfunctiongiventheestimatedparameterswithapenaltyplacedonthemodel complexityexpressedasthenumberofmodelparametersusingtheformula BIC D = lnp D j )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(1 2 M N where aretheparametersofthemixturemodel, M isthenumberofcomponents in lnp D j isthemaximumlikelihoodfunctionofthedatagiventheestimated parameters,and N isthenumberofdatapoints.Fortextureestimationusingthe intensityACFmodel,theBICiscalculatedascomponentsareaddedforagiven texturedimageblock.Theoptimalchoiceof isselectedwhentheBICismaximized. InChapter5theuseoftheBICtoautomaticallydeterminemodelcomplexityis demonstratedforatexturemodelingscenario. 4.5Summary HerewedescribeanalgorithmforestimatingtheintensityACFparametersfrom SASimageswaspresented.StartingwiththemodelderivedinChapter2,parameters thatdenetheimagingpointspreadfunction,componentmean,componentorientation, componentcorrelationlength,andcomponentweightwereestimatedusinganEM algorithmfortruncateddata.InChapter5theestimationalgorithmwillbetestedagainst syntheticandrealSASimagetexturestoevaluatetheperformanceofthetechniquein termsofitsaccuracyandvariance. 73

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CHAPTER5 ESTIMATIONRESULTSONSYNTHETICANDREALSASIMAGETEXTURES TheSASimagetextureestimationmethoddescribedinChapter4wastestedon bothsyntheticdataandrealdatacollectedineldtests.Therstthreeexperiments weredesignedtotesttheperformanceoftheACFmodelestimationmethodagainst syntheticSAStexturesofvarioussizes,composition,andshapeparameterstatistics. SynthetictexturesgeneratedusingthetechniquedescribedinChapter3wereusedto measureestimationperformanceagainstknownACFparametersandtoproducealarge datasetofconsistenttextures.Inthelastexperiment,parameterestimationresultsfor4 SAStexturesfromdatacollectedinaeldsurveyarethendetailedanddescribed.The realSAStextureswerechosenasarepresentativesampleofthevarietyofseabottoms onemightencounterinatypicalseabottomsurvey. 5.1ParameterEstimationResultsforSyntheticImageTestCases SyntheticimageswithknownACFparametersweretestedtomeasurethe performanceoftheparameterestimationmethodandBayesianinformationcriterion BICmodelselectionagainstvariousimagesamplesizesandsingle-pointstatistical shapeparametervalues.InTestCase1,theparameterestimationmethodperformance ismeasuredforatwo-componentmixtureofACFestimatestakenfromvariousimage blocksizes.InTestCase2,theestimationmethodperformanceismeasuredfora three-componentmixturefora 256 256 imagewiththesameACFparametersbut different parameters.InTestCase3theBICwastestedforutilityinACFmodel selectionagainstaperiodictextureoftypicalcomplexity.Inthersttwotestcases,the ACFestimatesweretruncatedto 1 2 thesizeofthesyntheticinputimageblockstoreduce theeffectsofnoiseintheACFestimatetailregion.Followingtruncationthemixture modelsupportregionwasextendedtothegreaterof 120 spatiallagsorthesizeof thetruncation 2 inboth X and Y directions.E.g.,a 128 128 imageblockisrst truncatedto 32 followingestimationoftheintensityACF,thenthesupportextended 74

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to 120 spatiallagspriortomixturemodelestimation.InTestCase3thetruncationof theACFissetat 40 spatiallagsandthesupportextendedto 150 spatiallagspriorto estimation. 5.1.1TestCase1 Inthistestcasesyntheticimageblocksofvarioussizesweregeneratedviathe procedureexplainedinChapter4.Tomeasuretheeffectofparameterestimation performanceagainstthevariousblocksizes,allACFparameterswerethesamein eachimagerealization.Inthistestcaseimagesweregeneratedwiththefollowing parameters: =0.5 1 = 2 =0.5 l x 1 = l y 1 =4 l x 2 = l y 2 =20 1 = 2 =0 o ,and x = y =0.5 .Tocreatetherealizations 512 512 imageswererstsynthesizedthen croppedintonon-overlappingsmallerblocksizesof 64 64 128 128 and 256 256 Figure5-1depictsthetexturesforthevariousblocksizesorscales.Table5-1lists themeanandstandarddeviationinparenthesesoftheparameterestimationresults over100realizationsforeachimageblocksize.Table5-1resultsarealsoreproduced inthebarplotsinFigures5-2and5-3andtheerrorbarplotinFigure5-4foreasier visualization.Astheimageblocksizeincreases,themeanparameterestimates ^ ^ l x 1 ^ l y 1 ,and ^ l y 2 tendtowardthetruevalueoftheparameterswithalowervariance.The parametervalueforbecomeslightlylessaccurateastheblocksizeincreasesbut theparameterstandarddeviationdoesdecreasesignicantlyforthe ^ l x 2 forthelarger blocksizes.Notably,themean-squareerroranderrorvariancebetweenthetruncated estimatedACFandthemodelACFdecreasessignicantlyastheblocksizeincreases asdepictedinFigure5-4. 5.1.2TestCase2 Inthistestcase 256 256 syntheticimagesofthesameACFtextureparameters butvariousvaluesoftheshapeparameter weregeneratedtomeasuretheeffect ofparameterestimationperformanceagainstvariousshapesofthesingle-pointpixel histogramortheshapeofthesingle-pointdistributionascontrolledby .Imageswere 75

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A B C D Figure5-1.Sample 512 512 A, 256 256 B, 128 128 C,and 64 64 D realizationsofthesynthetictestimagesinTestCase1.InTestCase1 imagesweregeneratedwiththefollowingparameters: =0.5 1 = 2 =0.5 l x 1 = l y 1 =4 l x 2 = l y 2 =20 1 = 2 =0 o ,and x = y =0.5 76

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Table5-1.EstimatedtextureparametersforTestCase1. =0.5 x =0.5 y =0.5 1 =0.5 l x 1 =4 l y 1 =4 2 =0.5 l x 2 =20 l y 2 =20 Block ^ ^ x ^ y ^ 1 ^ l x 1 ^ l y 1 ^ 2 ^ l x 2 ^ l y 2 MSE Size ^ ^ x ^ y ^ 1 ^ l x 1 ^ l y 1 ^ 2 ^ l x 2 ^ l y 2 MSE 640 6700 5650 5610 6715 0264 7870 33019 34917 3212.24 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 257 034 040 241 814 273 241 564 364.24 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 1280 5820 5240 5300 6604 2684 1860 34019 07216 4347.32 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 098 037 030 123 570 635 123 888 754 5.23 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 2560 5410 5010 5070 6513 9813 9920 34918 18318 5752.61 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 060 028 027 059 593 668 059 871 233 1.18 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 5120 5410 5000 4970 6453 9833 9680 35518 31018 1921.17 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 023 017 016 037 453 432 037 005 751 3.87 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 Figure5-2.BarplotofthemeanparameterestimatesforTestCase1.Estimates calculatedforasyntheticSASimagetexturewithinputparametersof =0.5 1 = 2 =0.5 l x 1 = l y 1 =4 l x 2 = l y 2 =20 1 = 2 =0 o ,and x = y =0.5 .100trialswereconductedforimageblocksizesof 64 64 128 128 256 256 ,and 512 512 77

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Figure5-3.BarplotofthestandarddeviationofparameterestimatesforTestCase1. EstimatescalculatedforasyntheticSASimagetexturewithinput parametersof =0.5 1 = 2 =0.5 l x 1 = l y 1 =4 l x 2 = l y 2 =20 1 = 2 =0 o and x = y =0.5 .100trialswereconductedforimageblocksizesof 64 64 128 128 256 256 ,and 512 512 generatedwith parameters0.3,1.1,1.9,and2.9whileholdingtheremainingtexture parametersconstant: 1 = 2 = 3 = 1 3 1 = 3 =45 o 2 =45 o l x 1 2 l y 1 =15 l x 2 =15 l y 2 =2 l x 3 = l y 3 =20 ,and x = y =0.5 .Asampleimagefromeach valueis depictedinFigure5-5.Table5-2liststhemeanandstandarddeviationinparentheses oftheparameterestimationresultsover100realizationsforeachvalueof .Overall, theestimationtechniqueappearstobesomewhatinvarianttotheshapeparameterof theimagetexture.Meanparameterestimatesarefairlysimilarforthefourcasessave fortheslightinaccuracyofestimating l y 1 and l x 2 when =0.3 .Thereappeartobefairly largestandarddeviationsfor l y 1 when =1.1 andfor l x 3 and l y 3 when =0.3 .Inthe 78

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Figure5-4.Errorbarplotofthemean-squareerrorofparameterestimatesforTestCase 1.MSEcalculatedforasyntheticSASimagetexturewithinputparameters of =0.5 1 = 2 =0.5 l x 1 = l y 1 =4 l x 2 = l y 2 =20 1 = 2 =0 o ,and x = y =0.5 .100trialswereconductedforimageblocksizesof 64 64 128 128 256 256 ,and 512 512 caseof =1.1 theremaybeseveraloutliersinthesetofestimates.Inthecaseof =0.3 ,thesmall valuemaybecausingdistortioninthegeneratingACFasmentioned inSection3.3.1.Ofnotehereisthattheestimateoftheorientationofthetextureis highlyaccurate,withonlyasmallstandarddeviationforall values. 5.1.3TestCase3 InthistestcasetheuseoftheBayesianinformationcriterionBICwasvalidatedas ausefulmetricfordeterminingthenumberofcomponentsneededtoadequatelymodel agiventexture.A 256 256 SASimagetexturewith4components,twozero-mean componentsandacoupledpairofnon-zeromeancomponents,wassynthetically generated.Parametervalues =0.5 1 = 2 =0.125 l x 1 = l x 2 =3 l y 1 = l y 2 =60 l x 3 =7 79

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A B C D Figure5-5.Sample 256 256 realizationsofthesynthetictestimagesforTestCase2. InTestCase2imagesweregeneratedwiththesametextureparametersbut differingshapeparameters.ImagesADweregeneratedwiththe parametersA =0.3 ,B =1.1 ,C =1.9 ,andD =2.9 whilethe remainingtextureparameterswereheldconstant: 1 = 2 = 3 = 1 3 1 = 3 =0 o 2 =45 o l x 1 = l y 2 =2 l x 2 = l y 1 =15 l x 3 = l y 3 =20 ,and x = y =0.5 80

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Table5-2.EstimatedtextureparametersforTestCase2. = 2 6 6 6 4 0.33 0.33 0.34 3 7 7 7 5 l x = 2 6 6 6 4 2 15 20 3 7 7 7 5 l y = 2 6 6 6 4 15 2 20 3 7 7 7 5 2 =45 o ^ )]TJ/F84 10.9091 Tf 5 -8.837 Td [( ^ ^ l x ^ l x ^ l y ^ l y ^ 2 ^ 2 0.3 0 314 1561 996 73313 038 52145 748 o 612 o 0 287 14012 329 9122 410 317 0 399 09717 231 16323 187 811 1.1 0 311 0892 290 10915 582 94945 468 o 258 o 0 312 08615 760 5542 784 415 0 378 08417 543 10822 848 274 1.9 0 311 0722 056 65915 130 35044 963 o 064 o 0 317 07015 947 9432 407 957 0 372 07118 048 28321 415 559 2.9 0 316 0662 292 62216 020 64844 983 o 777 o 0 330 06516 627 7382 515 007 0 354 06217 607 89921 742 190 l y 3 =45 l x 4 =7 l y 4 =45 x 1 =30= )]TJ/F25 11.9552 Tf 9.299 0 Td [( x 2 y 1 = y 2 = y 3 = y 4 = x 3 = x 4 =0 1 = 2 = 3 = 4 =0 o ,and x = y =0.5 werechosentogeneratethesyntheticimages. Figure5-8depictsandimageofthesyntheticimagealongsidethegeneratingACF. Thetexturewasprogressivelyestimatedusingmodelswith1zero-meancomponent, 2zero-meancomponents,4zero-meanandacoupledpairofnon-zeromean components,and6zero-meananda2coupledpairsofnon-zeromeancomponents with3,7,12,and17freeparametersrespectively.TheBICforeachestimationwas recordedfor100samplesforeachACFmodelconguration.AnerrorbarplotoftheBIC andlog-likelihoodversusthenumberofmodelparametersisdepictedinFigure5-9with thewidthoftheerrorbarrepresenting 1 2 ofonestandarddeviation.TheBIClevelsoffat 12parametersandonlyslightlyincreasesfor17parametersindicatingthe12parameter modelisanadequaterepresentationoftheACFdata. 81

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Figure5-6.BarplotofthemeanofparameterestimatesforTestCase2.Estimates calculatedforasyntheticSASimagetexturewithinputparametersof 1 = 2 = 3 = 1 3 1 = 3 =0 o 2 =45 o l x 1 = l y 2 =2 l x 2 = l y 1 =15 l x 3 = l y 3 =20 ,and x = y =0.5 .100trialswereconductedfor valuesof 0.3,1.1,1.9,and2.9. 5.2High-ResolutionSASDataACFParameterEstimation Four 256 256 imagetextureswereselectedtoillustratetheabilityofthe intensityACFmixturemodeltotseabottomtexturesinhigh-resolutionSASimages. ThetexturesandtheirassociatedestimatedACFsaredepictedinthetoprowof Figures5-105-13.ThetextureinFigure5-10Aisfromarelativelyhomogeneoussea bottommadeupofhardpacksandandcoveredwithsmallpoint-likescatteringobjects. Figure5-11Aisatexturecausedbyreectionsfromseagrassandisacommon biologicalclutterelementinmanysonarimages.Figure5-12Aisatexturefromarocky orcoralenvironment.ThenaltextureinFigure5-13Aisfromasandrippleeldthat 82

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Figure5-7.BarplotofthestandarddeviationofparameterestimatesforTestCase2. StandarddeviationcalculatedforasyntheticSASimagetexturewithinput parametersof 1 = 2 = 3 = 1 3 1 = 3 =0 o 2 =45 o l x 1 = l y 2 =2 l x 2 = l y 1 =15 l x 3 = l y 3 =20 ,and x = y =0.5 .100trialswereconductedfor valuesof0.3,1.1,1.9,and2.9. hasalongcorrelationlengthinonespatiallagdirection,andaperiodiccorrelationalong theperpendicularspatiallagdirection. Realdataresults: AsetoffourprogressivelycomplexACFmodelswereestimated foreachtextureinFigures5-105-13andthereconstructedACFsbasedonthemodel tsdepictedinthebottomfourimagesofeachofthegures.Thetruncationofthe estimatedACFwaschosentobe 40 spatiallagsforallthetexturesandtheregion ofsupportwas 150 spatiallags.Eachmodelasinitializedwithmixturecomponents withrandomcorrelationlengthschosendistributeduniformlybetween1and4.The numberofmodelparametersforeachtwerethesameasinTestCase3,i.e.3,7, 12,and17freeparametersrespectively.ThenaturallogoftheintensityACFsare 83

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ASyntheticimage BACF Figure5-8. 256 256 syntheticimageAandgeneratingACFBofimageusedinTest Case3.Imagesweregeneratedusingthefollowingparameters: =0.5 1 = 2 =0.125 l x 1 = l x 2 =3 l y 1 = l y 2 =60 l x 3 =7 l y 3 =45 l x 4 =5 l y 4 =45 x 1 =30= )]TJ/F25 11.9552 Tf 9.299 0 Td [( x 2 y 1 = y 2 = y 3 = y 4 = x 3 = x 4 =0 1 = 2 = 3 = 4 =0 o ,and x = y =0.5 displayedforeasiervisualcomparison.Thelog-likelihood,BIC,andmean-square errorbetweentheestimatedACFandthemodelACFwithinthetruncationregionare collectedinTable5-3.Intheseagrass,rock,andsandripplegures,theACFmodelt isqualitativelymoreaccurateasmorecomponentsareaddedintothemixturemodel. Theimprovementinaccuracyisalsoborneoutinthediminishingmean-squareerror betweenthethedataandthemodelACFs.Dependingonthecomplexityofthetexture, itisalsoclearthatthereissomeacceptablemodelrepresentationthataccurately capturesthegrosscharacteristicsyetmaynotrequireall17parameters.Inthesand texture,theadditionalnonzero-meancomponentsdoimprovethemodelingaccuracy intermsofmean-squareerrorandlog-likelihoodbutthenonzero-componentsovert theregionsoutsidethezero-meanlobe.Inthiscase,thetruncationregionistoo largefortheestimatedACFbecausethemodelisttingthenoisyregionawayfrom 84

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Figure5-9.ErrorbarplotofthemeanoftheBICandlog-likelihoodof100synthetic imagesamplesgeneratedusingtheACFdepictedinFigure5-8.Thewidth oftheerrorbarrepresents 1 2 ofonestandarddeviation.TheBICbeginsto leveloffwhenthetruenumberofparametersisusedtomodeltheestimated ACF. thezero-meanmainlobe.ThemodelACFsrepresentthedatawellintermsofgross orientationandperiodicityforthenon-sandACFs. TheseriesofplotsinFigures5-145-17depictthethelog-likelihoodand mean-squareerrorfromTable5-3toillustratethegaininaccuracyasthenumber ofmodelparametersincreases.Forthesand,seagrass,andsandrippletextures, thechangeinaccuracyfromgoingto17parametersfrom12isnotsignicant, indicatingthatfairlycomplextexturessuchasthesecanbemodeledwithrelatively fewparameters.Therocktextureisvisuallycomplex,withmultipletexturalorientations 85

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ASandtexture BSandestimatedACF C3parameters D7parameters E12parameters F17parameters Figure5-10.A 256 256 realsandtexture,B 81 81 estimatedACF,CmodeledACF with3freeparametersonezero-meancomponent,DmodeledACFwith 7freeparameterstwozero-meancomponents,EmodeledACFwith12 freeparameterstwozero-meancomponentsandonecoupled nonzero-meancomponent,andFmodeledACFwith17freeparameters twozero-meancomponentsandtwocouplednonzero-meancomponents. 86

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ASeagrasstexture BSeagrassestimatedACF C3parameters D7parameters E12parameters F17parameters Figure5-11.A 256 256 realseagrasstexture,B 81 81 estimatedACF,Cmodeled ACFwith3freeparametersonezero-meancomponent,DmodeledACF with7freeparameterstwozero-meancomponents,EmodeledACFwith 12freeparameterstwozero-meancomponentsandonecoupled nonzero-meancomponent,andFmodeledACFwith17freeparameters twozero-meancomponentsandtwocouplednonzero-meancomponents. 87

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ARocktexture BRockestimatedACF C3parameters D7parameters E12parameters F17parameters Figure5-12.A 256 256 realrocktexture,B 81 81 estimatedACF,CmodeledACF with3freeparametersonezero-meancomponent,DmodeledACFwith 7freeparameterstwozero-meancomponents,EmodeledACFwith12 freeparameterstwozero-meancomponentsandonecoupled nonzero-meancomponent,andmodeledFACFwith17freeparameters twozero-meancomponentsandtwocouplednonzero-meancomponents. 88

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ASandrippletexture BSandrippleestimatedACF C3parameters D7parameters E12parameters F17parameters Figure5-13.A 256 256 realsandrippletexture,B 81 81 estimatedACF,Cmodeled ACFwith3freeparametersonezero-meancomponent,DmodeledACF with7freeparameterstwozero-meancomponents,EmodeledACFwith 12freeparameterstwozero-meancomponentsandonecoupled nonzero-meancomponent,andFmodeledACFwith17freeparameters twozero-meancomponentsandtwocouplednonzero-meancomponents. 89

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Table5-3.RealSAStexture:accuracyversusmodelcomplexity. 3Parameters7Parameters12Parameters17Parameters L L L L BIC BIC BIC BIC SASTexture MSEMSEMSEMSE Sand )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.114 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.053 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.820 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.821 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.246 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.361 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.347 10 2 )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.568 10 2 5.564 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 6.077 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(5 5.443 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 5.369 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(5 Seagrass )]TJ/F22 10.9091 Tf 8.485 0 Td [(2.054 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(2.060 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(1.970 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(1.974 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(2.067 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(2.091 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(2.022 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(2.048 10 3 1.256 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 1.553 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 6.279 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 7.561 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4 Rock )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.072 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.925 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.790 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.709 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(4.086 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.956 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.843 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(3.784 10 3 4.824 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 1.077 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 2.846 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 2.067 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 SandRipple )]TJ/F22 10.9091 Tf 8.485 0 Td [(8.738 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(8.011 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(7.340 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(7.265 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(8.751 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(8.041 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(7.393 10 3 )]TJ/F22 10.9091 Tf 8.485 0 Td [(7.340 10 3 8.011 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 3.146 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 4.262 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 3.559 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(3 andcorrelationsizes,thustheaccuracygainissignicantasmoremodelparameters areadded. 5.3Summary Heretheempiricalperformanceoftheestimationtechniqueagainstsyntheticand realSAStextureswasmeasuredintermsofestimateaccuracyandvariance.Inthe rsttwotests,synthetictextureswereusedtotestparameterestimateaccuracyagainst variousdesignparameterssuchasACFestimateblocksizeandimagecontrastas measuredbytheshapeparameter .ThesmallblocksizesforcalculatingtheACF estimatewasshowntodegradedparameterestimationresults,buttheparameter estimateaccuracywasfairlyrobusttochangesintheshapeparameter.Inthethird test,syntheticdatawasusedtoshowtheutilityoftheBICinforACFmodelselection usingafairlyrealisticandcomplexperiodictexture.FourrealSAStextures,namely 90

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Figure5-14.Plotoflog-likelihoodandMSEforsandtextureagainstincreasingmodel complexity. Figure5-15.Plotoflog-likelihoodandMSEforseagrasstextureagainstincreasing modelcomplexity. 91

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Figure5-16.Plotoflog-likelihoodandMSEforrocktextureagainstincreasingmodel complexity. Figure5-17.Plotoflog-likelihoodandMSEforsandrippletextureagainstincreasing modelcomplexity. 92

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sand,seagrass,rock,andsandrippleweretusingprogressivelycomplexACFtexture modelsandthelog-likelihood,BIC,andmean-squareerrorwasexaminedforeachcase. IntheChapter6,ACFmodelparameterestimateswillbeusedasfeaturesin anunsupervisedtexturesegmentationalgorithm.Thusthequestionofintra-class parametervarianceexaminedinthissectionisonlyoneimportantfactorwhenadditional parametersthatdistinguishtexturesareconsidered.Thequestionnowconcerns discernability,orwhethertheestimatedACFparametersofdifferenttextureshave differentmeansandlowintra-classvariabilitysuchthattheyclusterintodistinguishable modesinfeaturespace. 93

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CHAPTER6 SASTEXTURESEGMENTATIONUSINGACFMODELPARAMETERS InChapters2and4aSASimagetexturemodelandassociatedparameter estimationalgorithmthatincorporatesbothaGaussianimagingpointspreadfunction PSFandaseabedtextureGaussianmixturemodelweredeveloped.Resultsfromtting themodeltosyntheticallygeneratedimagesofknownparametersandhigh-frequency SASimagescollectedinaeldsurveywerepresented.Themodelwasshownto adequatelytthesetexturesandproduceestimatedparametervaluesthatintuitively describethetextureintermsofcorrelationlengths,rotation,andthesingle-point statisticalshapeparameter. HereweemploytheACFtextureparametersexplainedthusfarinSASimage scenesegmentation.Priorworkonradarandsonarimagesegmentationisrstbriey explored.Anunsupervised k -meanssegmentationisthenintroducedforSASimage segmentationusingtheshapeparameter andACFtextureparametersaselements inafeaturevector.Acomparisonofsegmentationresultsusing3increasinglycomplex ACFmodelsisrstpresented,thenresultsarecomparedagainsttwodifferentsonar imagesegmentationfeaturesetsrecentlypublished. 6.1RelatedResearchinRadarandSonarImageSegmentation Variousmethodshavebeenappliedtosegmentradarandsonarimagescenes. Fosgate,etal.presentamultiscaleapproachtosyntheticapertureradarSAR supervisedimagesegmentationwheretwotextureclasses,grassandforest,are separatedusingfeaturesfromvariousimagescales[48].Inthisworkresidualsfrom autoregressivemodelsformedfromasupervisedtrainingsetfromvariousimage scaleswerecomparedinabinaryhypothesistest.VariousauthorshaveusedMarkov randomeldsbasedontheauto-binomialmodeltoassignclasslabelstodifferentimage scenecomponents[4951].Mignotteetal.usedabinaryunsupervisedsegmentation algorithmforseabedandobjectshadowpixelsinside-scansonar[49].Amultiscale 94

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approachwasusedbeginningatthelargestscaletoregularizethesegmentation solutionatnerscales.Dengetal.presentedamulti-classunsupervisedsegmentation algorithmforseaiceinSARimageryusinggamma-distributedpixelintensityand gray-levelco-occurrenceprobabilitycontrastandentropyfeatures[50]. Inrecentworkonseabedtexturesegmentation,variousauthorshavebuiltuponthe unsupervisedtexturesegmentationschemeofJainandFarrokhnia,wherecoefcients fromGaborlterbanksareusedasfeaturesfora k -meanssegmentationalgorithm[52]. Williamsusedcoefcientsfromabi-orthogonalwaveletdecompositiontosegment rippledandrockyseabedtextures.Thecoefcientswerethencomparativelysegmented usingspectralclusteringand k -meansclustering[53].LianantonakisandPetillotused HaralickfeaturesandChan-Veseactivecontourstosegmentsandrippleandrocky regionsfromhard-packedsandandothertextures[54].SamieeandRadcombined theapproachesofJain,Farrokhnia,Lianantonakis,andPetillotusingGaborlterbank coefcientsandChan-Veseactivecontourstosegmentseabedtextures[55]. Inthenextsections,aSASimagesegmentationalgorithmisrstdescribed.Next, theSASimagesegmentationresultsusingtexturefeaturesextractedfromparameters oftheintensityACFmodelarecomparedwiththeresultsusingthefeaturesextracted fromthetwoapproaches,[53]and[54],describedabove.Segmentationresultsusing thevariousfeaturesetsarecomparedbasedonaveragecorrectclassicationrates.An analysisofthethemultimodalityofthetexturefeaturespaceisalsoconductedtoinfer supervisedclassicationperformanceamongstthethreefeaturesets. 6.2SASImageSegmentationUsingSingle-PointStatistics Severalauthorshaveproposedusingtheestimated valueasafeatureforimage segmentation[16,17].Thisapproachcanbeeffectivewhensegmentingregionsofhigh contrastwithregionsoflowcontrastinsonarimages,e.g.,whensandrippleorrock regionsareisolatedinregionsofhard-packedsand.Figure6-1depictssegmentation resultsintwoexampleswherearockyregionandasandrippleregionareisolatedina 95

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AOriginal BOriginal CSegmented DSegmented Figure6-1.Segmentationresultsusingtheshapeparameter .Segmentationresults aregoodwhenthereisalargedifferenceinimagepixelcontrastbetween differenttexturedregionssuchasatsandandsandripple. atsandybottom.Inthisexample,the parameterisestimatedinasliding 20 20 pixel windowusingRaghavan'smethod[18].Theestimated valuesarethensegmented usinga k -meansalgorithmwith k =2 forimagesinFigures6-1a-band6-2Aand k =3 fortheimageinFigure6-2B. 96

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AOriginal BOriginal CSegmented DSegmented Figure6-2.Segmentationresultsusingtheshapeparameter .Segmentationresults arepoorwhentexturedregionshavesimilarcontrastbutdifferenttextures. 97

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Thisapproachissufcientforsimplesegmentationtaskssuchasseparatingsand frommorecomplextexturesasevidencedinFigure6-1.However,thisapproachis noteffectivewhensegmentingtwotexturedregionswithsimilar values.Figure6-2 depictsasituationwherethe valuealonedoesnotsegmentanimagewithtwodened regionsofsandrippleandseagrassinFigure6-2AoramixtureoftexturesFigure 6-2B.Thissecondexperimentillustratestheneedformoredescriptivesegmentation parameterswhentexturedregionsaremorecomplicatedthanmerelyseparating regionsofhighcontrastfromlowcontrast.Inthenextsectionweuseparametersfrom theintensityACFmodeldescribedinChapter2forsegmentationfeatures.Whilethe modelandparameterestimationmethodismorecomplexthanusingthe value,the higherorderstatisticalinformationprovidedbytheACFmodelismoreeffectivewhen segmentingtexturesofsimilarcontrast. 6.3SASImageSegmentationUsingAutocorrelationFunctionParameters SegmentationfeaturesforSASimagescanbeextractedusingtheparameters oftheintensityACFmodelasdemonstratedinChapter5[56].However,thenumber ofcomponentsinthemodelmustbedenedpriortoestimatingtheparameters,and alsowiththeparticularformofEquation2,whethernonzero-meancomponentsare allowed.Ratherthanspecifyonesetofresultsforaparticularnumberofcomponents, resultsherearegivenforthreedifferentmodels:1asingle-componentmodelwith azero-meanparameter,2atwo-componentmodelwithbothcomponentshavinga zero-mean,and3afour-componentmodelwithpairednonzero-meancomponentsand twozero-meancomponents. Asetof40high-resolution 1000 1000 labeledSASimagescontainingdistinct texturesofsand,rock,smallrippleperiod 20-25spatialsamples,andlarge rippleperiod 45-60spatialsampleswereusedtotestthesegmentationalgorithm. Featureswerecalculatedfora 200 200 pixelwindowandshiftedin25-pixelincrements pixeloverlapovertheentireimage.These 40 40 featureimageswerethen 98

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medianlteredwitha 3 3 windowtoeliminateoutliersintheparameterestimationstep. Thelteredfeaturevector f wasrecordedforeach 200 200 pixelwindowpopulated withentriesdenedbytheTable6-1.Thereadershouldnotethatthefour-component modelonlyliststhreeparameterssincethenonzero-meancomponentparameterpairs areconstrainedtothesamevaluesforadetaileddiscussionseeChapter2. Table6-1.FeaturevectorentriesforthethreeACFmodels. ModelTypeFeatureVector One-Component f = f l x 1 l y 1 g Two-Component f = f l x 1 l x 2 l y 1 l y 2 g Four-Component f = l x 1 l x 2 l x 3 l y 1 l y 2 l y 3 p 2 x 1 + 2 y 1 Thegoalofthesegmentationtaskistoautomaticallylabelthedominanttexture classesinaseriesofsonarimages,oramulti-imagesegmentationtask.Thusthe featurevectorsforthe40testimageswerecombinedintoonelargetestsetand segmentedtogether.Atexturelabelwasassignedtoeach 25 25 pixelblockinthe originalimageresultinginatotalof64000labeledtexturesamplesforall40images. Thebreakdownofnumbersoftexturesamplesbyclassareasfollows:37553sand samples,8184rocksamples,2190smallripplesamples,and16073largeripple samples. Segmentationwascarriedoutintwostages.Intherststage,theshapeparameter wasusedtodifferentiatebetweensandlowcontrastandnonsandhighcontrast texturesina k -meansalgorithmwith k =2 [57],[58,ch.10pp.526-528].The well-known k -meansalgorithmassignsfeaturevectorsto k clusters,suchthatthe within-clusterscatterorEuclideandistancefrommemberfeaturevectorstothecluster meanisminimized.Inthesecondstage,nonsandtextureswerefurthersegmented using k -means, k =3 ,intorock,smallsandripple,andlargesandrippleclasses usingtheremainingfeaturesinthevectorsofTable6-1.Table6-2summarizesthe averagecorrectsegmentationresultsfor1000randominitializationsofthe k -means 99

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segmentationalgorithmfortheone-,two-,andfour-componentfeaturesetsplusor minusonestandarddeviation.Tables6-36-5depictthemeanconfusionmatrices foreachofthefeaturevectorcongurations.Thefour-componentmodelfeaturevector yieldssuperiorsegmentationresultsovertheone-andtwo-componentmodels.The one-componentmodeldiscriminateslargeripplefairlywellbutgroupstherockcategory featureswiththesmallripplefeaturesveryconsistently.Thetwo-componentmodel performsbetteratdiscriminatingtherockyseabedbuthashighvariabilityintherock andsmallripplelabelingindicatingthatthefeaturesarenotisolatedintodistinctclusters inEuclideanfeaturespace.Thefour-componentmodelhasgoodsegmentationdelity betweensmallandlargerippleswithfairlylowvariability.Therocksegmentationforthe four-componentmodelisconsiderablybetterthantheone-andtwo-componentACF models. Table6-2.SASimagesegmentationresultsinpct.correct. ModelTypeSandRockSmallRippleLargeRipple One-Component 92.9 0.0 % 4.2 0.0 % 86.8 0.0 % 62.8 0.0 % Two-Component 92.9 0.0 % 41.4 39.4 % 53.4 41.0 % 43.3 5.0 % Four-Component 92.9 0.0 % 72.0 12.4 % 54.8 13.0 % 66.6 6.6 % Table6-3.Confusionmatrixforone-componentsegmentationresults. SegmentationResults TextureLabelSandRockSmallRippleLargeRipple Sand 92.9 % 1.6 0.0 % 4.6 0.0 % 0.8 0.0 % Rock 13.7 % 4.2 0.0 % 82.2 0.0 % 0.0 0.0 % SmallRipple 4.5 % 7.9 0.0 % 86.8 0.0 % 0.8 0.0 % LargeRipple 17.0 % 18.9 0.0 % 1.4 0.0 % 62.8 0.0 % Aseriesof6ofthe40texturedSASimagespairedwiththeirsegmentationresults aredepictedinFigure6-3.Inthesegmentedimages,thecolorblackisassignedto 100

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Table6-4.Confusionmatrixfortwo-componentsegmentationresults. SegmentationResults TextureLabelSandRockSmallRippleLargeRipple Sand 92.9 % 3.0 1.5 % 3.0 1.8 % 1.0 0.3 % Rock 13.7 % 41.4 39.4 % 43.8 39.8 % 1.2 0.6 % SmallRipple 4.5 % 40.7 42.3 % 53.4 41.0 % 1.5 1.4 % LargeRipple 17.0 % 23.0 16.8 % 16.8 12.1 % 43.3 5.0 % Table6-5.Confusionmatrixforfour-componentmodelsegmentationresults. SegmentationResults TextureLabelSandRockSmallRippleLargeRipple Sand 92.9 % 3.0 0.5 % 1.8 0.6 % 2.2 0.1 % Rock 13.7 % 72.0 12.4 % 11.5 12.0 % 2.8 0.5 % SmallRipple 4.5 % 40.3 13.3 % 54.8 13.0 % 0.4 0.2 % LargeRipple 17.0 % 3.7 0.1 % 16.1 6.6 % 66.6 6.6 % thesandclass,thecolordarkgrayisassignedtotherockclass,thecolorlightgrayis assignedtothesmallrippleclass,andthecolorwhiteisassignedtothelargeripple class.InFigure6-3A,thesegmentationalgorithmassignsanincorrectsmallripple labeltotheleftsideoftherockytexture,butperformswellintheremainingimages.Of particularinterestisthesegmentationofFigure6-3C.InFigure6-3Cthealgorithm doeswellseparatingthesmallrippleinterspersedwiththerocktexture. 6.4SASImageSegmentationUsingWaveletandHaralickFeatures Twoothersonarimagetexturefeaturesetsbasedonbi-orthogonalwavelets[53] andHaralickfeatures[59]respectivelywerecomparedagainstthefour-component ACFtexturefeaturesusingthepreviouslydescribedunsupervised k -meansSASimage segmentationalgorithm[60].AsintheACFfeaturesegmentationtaskintheprevious section,thefeaturevectorswerecombinedintoonelargetestsetforthe40imagesand weresegmentedtogether,i.e.amulti-imagesegmentationtask. 101

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AOriginal#1 BOriginal#2 COriginal#3 DSegmented#1 ESegmented#2 FSegmented#3 GOriginal#4 HOriginal#5 IOriginal#6 JSegmented#4 KSegmented#5 LSegmented#6 Figure6-3.Imagesofsegmentationresultsusingthefour-componentACFmodel.SAS imagetexturesofArock,Bsmallrippleeld,Csmallripplesmixedwith rock,sandripplesG-I,andtheirrespectivesegmentedimagesD-Fand J-L. 102

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Asintheprevioussection,segmentationwascarriedoutintwostages.Inthe rststage,theestimatedshapeparameter foundusingRaghavan'smethod[18]was usedtodifferentiatebetweensandandnonsandtexturesina k -meansalgorithmwith k =2 .Inthesecondstage,segmentednonsandtextureswerefurthersegmented using k -means, k =3 ,intorock,smallsandripple,andlargesandrippleclassesusing theACF,wavelet,andHaralickfeaturesrespectively.AsintheACFsegmentationin theprevioussection,featurevectorswerenormalizedtozero-meanandunitvariance toadjusttherelativescaleofthefeatureswithineachfeatureextractionalgorithm.A descriptionofthewaveletandHaralickfeaturesetsaredetailedinSections6.4.1and 6.4.2. 6.4.1Bi-orthogonalWaveletFeatures Arecentapproachtomulti-classunsupervisedseabedsegmentationuses bi-orthogonalwavelettransformcoefcients[61,Ch.7,pp.266-272]toseparate ripple,rock,andsandclassesinSASimagery[53].Usingtheapproachof[53]awavelet featurevectorof16elementswasformedforoverlappingwindowsofsize 200 200 byperformingavescalewaveletdecompositionofthewindowusingtheMATLAB R functiondwt2forthebior1.3waveletfamily.Theroot-meansquarevalueofeach waveletscaleanddirectionwasstoredasanelementinthefeaturevector.Figure6-4 isaplotofthedecompositionwaveletofbior1.3.Ingeneralthewaveletselectionfor texturefeatureextractionisspecictothesegmentationtaskathand.Thisparticular waveletwaschosenbecauseitcancapturetheperiodicityofsandripplesthatare inherentinthisdataset.Otherwaveletsmaybebettersuitedfordifferentseabed segmentationtasks.Windowsizewaschosensothatatleastoneperiodofthelargest ripplewouldbecapturedinthedecomposition. WaveletfeaturedecompositionsamplesaredepictedinFigure6-5forthethree textureclasses:rock,smallripple,andlargeripple.Thewaveletcoefcientintensities areplottedtocoincidewiththefeaturevectorcompositiondescribedabove.The 103

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Figure6-4.Plotofthewaveletbior1.3chosenfromtheMATLAB R WaveletToolbox. Thewaveletwaschosenbecauseitcancapturetheperiodicityofsand ripplesthatareinherentinthisdataset. strongverticallinestructureofthelargerippleisrepresentedinthelargeresponse inverticallteringcoefcientsatallscalesverticallteringblocksareontheleftside oftheimage.Thesmallrippleandrocktextureshavestrongercoefcientsatthe smallerscalesthandoesthelargerippleasonewouldexpect.Onedrawbackofusing waveletsisapparentinthesegures.Thetwo-dimensionalwaveletdecompositionis inherentlydirectional.Thismeansthattheorientationofthetextureisadiscriminating factorinsegmentation.Inthecaseofmulti-imagesegmentationaspresentedhere, rotation-invariancemaybedesirabletogroupsandripplesorotherdirectionaltexturesin thesameclassacrossalargesampleset.ThefeaturevectorofACFparametersused inthesegmentationtasksdeliberatelyignoresthetextureorientationforthisreason. 6.4.2HaralickFeatures Awell-knownsetoftexturaldescriptorsderivedfromthegray-levelco-occurrence GLCMmatrixistheHaralickfeatureset[59].Thesefeaturesarebasedonspatial relationshipsbetweenadiscretenumberofgraylevelpixelvaluesinanimageblock. 104

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ARocktexture BRockwaveletcoefcients CSmallrippletexture DSmallripplewaveletcoefcients ELargerippletexture FLargeripplewaveletcoefcients Figure6-5.Five-scalewaveletdecompositionsofrock,smallripple,andlargeripple texturesusingthebi-orthogonaldecompositionwaveletdepictedinFigure 6-4.Theintensityofthewaveletcoefcientsareplottedateachscale. 105

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Foragivenimageblockorwindow,thesizeandcontentofaGLCMisdenedby threeparameters:graylevelquantization,orientationin 45 o degreeincrements,and interpixeldistance.TheGLCMisasquarematrixwithrowandcolumnsizeequalto thenumberofgraylevels.Anentry G i j inaGLCMoforientation anddistance d iscomputedbycountingthenumberoftimesapixelofvalue i wasatadistance d and orientation fromapixelvalue j .SeveraldetailedexamplesofcalculatingaGLCMare in[59]and[62,Ch.11,pp.666-669].AftercreatingaparameterizedGLCMvarious texturalfeaturesarecalculatedfromthematrixentries.In[59],14GLCMfeaturesare described.Inthiscomparison, energy contrast correlation entropy homogeneity from[59]and clustershade ,and clusterprominence from[63]arechosentomatchthe featuresetin[54]. 6.4.2.1Haralickfeaturecalculation If G i j isaGLCMofagivenorientation,gray-levelquantization,andinterpixel distance,denethenormalizedGLCM G N i j as G N i j G i j P i j G i j ThefeatureslistedinSection6.4.2aredescribedandcalculatedbelow. 1. Energy: thesumofthesquaredentriesinthenormalizedGLCM.Energyis calculatedas X i j G N i j 2 2. Contrast: ameasureoftheintensitydifferencebetweenneighboringelementsin thenormalizedGLCM.Contrastiscalculatedas X i j j i )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j 2 G N i j 2 3. Correlation: themeancorrelationcoefcientbetweenneighboringelementsinthe GLCM.Correlationiscalculatedas P i j ij G N i j )]TJ/F25 11.9552 Tf 11.956 0 Td [( i j i j 106

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4. Entropy: thisinformationmetricisameasureofuniformityintheGLCMentries. Uniformvaluesforentriesresultsinalargeentropywhileafewnonzerovalues resultsinasmallentropy.Entropyiscalculatedas )]TJ/F30 11.9552 Tf 11.956 11.358 Td [(X i j G N i j log G N i j 5. Homogeneity: ameasureofthehowcloselytheelementsinthenormalizedare distributedtotheGLCMdiagonal.Avaluecloseto1indicatesthatsimilarpixel valuesaregroupedtogetherandpixelvariationissmoothovertheimageblockat theparticularorientationandinterpixeldistance.Homogeneityiscalculatedas X i j G N i j 1+ j i )]TJ/F39 11.9552 Tf 11.956 0 Td [(j j 6. ClusterShade: ameasureofhowsymmetricthedistributionofnormalizedGLCM valuesarearoundthemeanGLCMvalue.Thismeasureisanaverageskewness oftheGLCMrowsandcolumnsaboutthemeanrowandcolumnGLCMvalue. Clustershadeiscalculatedas X i j i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i + j )]TJ/F25 11.9552 Tf 11.955 0 Td [( j 3 G N i j 7. ClusterProminence: ameasureofhowmanyoutliersareinthedistributionof normalizedGLCMvalues,orhowpeakedthedistributionisaroundthemean GLCMvalue.ThismeasureisanaveragekurtosisoftheGLCMrowsandcolumns aboutthemeanrowandcolumnGLCMvalue.Clusterprominenceiscalculatedas X i j i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i + j )]TJ/F25 11.9552 Tf 11.956 0 Td [( j 4 G N i j 6.4.2.2Haralickfeaturevector Inthisfeatureextractionapproach,parameterizedGLCMsarecalculatedfor quantizedwindowsofimagetexturesandstoredaselementsinafeaturevector.In[54], Haralickfeaturemapswereusedasinputstoalevel-setsegmentationalgorithmfor vector-valuedimages.UsingthesameGLCMparametersasin[54],afeaturevector wascreatedforoverlapping 200 200 windowsbycalculatingtheGLCMfororientations of 0 o )]TJ/F22 11.9552 Tf 9.299 0 Td [(90 o ,45 o )]TJ/F22 11.9552 Tf 9.299 0 Td [(45 o ,interpixeldistance 2 ,and32-bitquantizationoftheoriginal gray-scaleimage.Eachfeaturevectorcontained28elementscorrespondingtofour orientationsoftheenergy,contrast,correlation,entropy,homogeneity,clustershade, 107

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andclusterprominencevaluescalculatedfromtheGLCMsasdetailedinSection6.4.2.1 above.Thesefeaturevectorswerethenusedasinputsintothe k -meanssegmentation algorithm. 6.5ComparativeSegmentationResults Table6-6summarizestheaveragecorrectsegmentationresultsfor1000random initializationsofthe k -meanssegmentationalgorithmforthethreefeaturesetsplusor minusonestandarddeviation.Featurevectorswerenormalizedtozeromeanandunit variancepriortosegmentationusingthethreemethods.Tables6-5,6-7,and6-8depict themeanconfusionmatricesforeachofthefeaturevectorcongurations.Usingthis particularsegmentationapproach,thefour-componentACFmodelfeaturevectoryields superiorsegmentationresultsoversegmentationusingwaveletandHaralickfeatures fortherockandlargerippletextures.ACFmodelsegmentationresultsareslightly poorerforthesmallrippleclassagainstthewaveletfeatureset.TheHaralickfeature setincorrectlygroupsthesmallrippleandrockclassestogetherwithhighvariability, implyingthatthefeaturevectorsarenotverydiscriminatorybetweenthesetwoclasses. AnimportantobservationoftheseresultsisthattheACFmodelperformsratherwellin thissegmentationtaskbetweencomplextexturesusingonlyasevenelementfeature vector.Thisindicatesthatthesalientinformationintheimagepixelsiscapturedbythe model. AsfurtherevidenceoftheabilityoftheACFmodeltocapturethekeyelementsof sonarimagetexturethewithin-classvarianceofeachclassiedfeaturesetisexamined. Recallthatthe k -meansalgorithmminimizesthewithin-clusterscatter.Within-cluster scatterisdenedas S below S M X i =1 X x 2 C i jj x )]TJ/F25 11.9552 Tf 11.955 0 Td [( i jj 2 where M isthenumberofclusters, x isafeaturevector, C i arethemembersofcluster i ,and i isthecentroidof C i .Thususing k -meansasanunsupervisedsegmentation 108

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methodmakesanimplicitassumptionthatthethreetextureclassesareunimodalin aEuclideanfeaturespace,andthelargertheinterclusterdistanceandthelowerthe intraclustervariancethebetterthesegmentationresultsinthenallabelassignment. ThebarplotinFigure6-6depictstheaveragewithin-clustervarianceforeachthethree featuresetsover1000instantiationsofthe k -meansalgorithm.Theplotshowsthat theACFfeatureshavethelowestmeanscatteramongthethreefeaturessets,with thewaveletfeatureshavingonlyaslightlyhighermeanscatter.TheHaralickfeatures havethelargestmeanscatterimplyingthatthefeaturevectorclustersareeitherhighly multimodalorareverydiffuse.ThisisanindicationthatHaralickfeaturesmaynotbeas suitableastheotherfeaturesetsfordiscriminatingthetexturedataintheexperiment. Table6-6.SASimagesegmentationresultsinpct.correct. ModelTypeSandRockSmallRippleLargeRipple ACF 92.9 % 72.0 12.4 % 54.8 13.0 % 66.6 6.6 % Wavelet 92.9 % 37.1 5.7 % 65.5 3.5 % 51.2 5.6 % Haralick 92.9 % 30.9 19.7 % 75.8 21.8 % 45.4 6.2 % Table6-7.Confusionmatrixforbi-orthogonalwaveletsegmentationresults. SegmentationResults TextureLabelSandRockSmallRippleLargeRipple Sand 92.9 % 0.7 0.0 % 4.8 0.7 % 1.7 0.7 % Rock 13.7 % 37.1 5.7 % 40.0 3.9 % 9.3 1.8 % SmallRipple 4.5 % 17.5 6.0 % 65.5 3.5 % 12.5 2.5 % LargeRipple 17.0 % 6.7 1.3 % 25.2 6.9 % 51.2 5.6 % Tobetterunderstandthediscriminatoryabilityandmodalityofeachfeaturesetin asupervisedclassicationsetting,anotherexperimentwasconductedwhereclass labelswhereassignedbyamajorityvotewithinafeaturecluster.Inthissecond experimenta k -meansalgorithmforvaluesof k = f 3,6,10,15,20,30,50,100 g ran 109

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Figure6-6.Barplotofthemeanscatterover1000instantiationsofthe k -means algorithmwith k =3 fortheACF,wavelet,andHaralickfeaturesets.The k -meansclusteringalgorithmminimizeswithin-clusterscatterandismost accuratewhenclassesareunimodalandhavesmallwithin-classvariance. Table6-8.ConfusionmatrixforHaralicksubsetsegmentationresults. SegmentationResults TextureLabelSandRockSmallRippleLargeRipple Sand 92.9 % 1.5 0.3 % 4.1 0.4 % 1.5 0.1 % Rock 13.7 % 30.9 19.7 % 37.9 20.6 % 17.6 2.0 % SmallRipple 4.5 % 18.7 22.5 % 75.8 21.8 % 1.0 0.7 % LargeRipple 17.0 % 28.9 8.8 % 8.8 3.6 % 45.4 6.2 % 110

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oneachfeatureset.Featureswerenotnormalizedbytheirmeanandstandarddeviation priortosegmentationasinpreviousexperimentsinordertobetterdifferentiatethe performancebyfeaturespacecomplexity.Thenalrock,smallripple,orlargeripple labelforeachtexturesamplewasbasedonthemajorityclassthatoccupiedthesame cluster,excludinganymisclassiedsandsamples.Voteswerenormalizedforpriorclass probabilitysothattextureclasseswithfewersampleshavethesamevoteastexture classeswithlargenumbersofsamples.Thissecondexperimentinfershowwelleach featuresetcoulddiscriminatebetweenclassesforagivenclassiercomplexity,where classiercomplexityisindicatedbythenumber k inthe k -meansalgorithm. Inthissecondexperiment,the k -meansalgorithmwasrandomlyinitialized100 timesforeach k andthecorrectclassicationraterecorded.Figures6-76-9depict themeanresultsforeachfeaturesetfortherock,smallripple,andlargerippletextures respectively.Figure6-10depictsthemeanresultsforallthetexturescombined.The errorbarsintheplotsrepresentplusorminusonestandarddeviation.Forsmall k theACFtexturemodelfeaturesprovidegooddiscriminatoryinformationfortheripple textures,butperformspoorlyonrockytextures.TheHaralickfeaturesunderperform insegmentingthelargerippletexturesbutsegmenttherockandsmallrippletextures betterthantheACFandwaveletfeaturesforsmallvaluesof k .Thewaveletfeatures segmentallclassesfairlywellfor k > 20 ,suggestingthatalltexturesdescribedbythese featuresarehighlymulti-modal.Themulti-modalityofwaveletfeaturesislikelyexplained bythemultipleorientationsandscalesofthetextureclassesbeinggroupedunderthe samelabelinthismulti-imagesegmentationtask.Onaverage,theACFmodelfeatures performwellfor k < 30 suggestingthatthisapproachtofeatureextractionleadsto simplerexpressionsofthedatainfeaturespacefewerclustersormodesandrequires lessclassiercomplexity.TheHaralickfeaturesunderperformintherockandlarge rippleclassicationtasksforallvaluesof k .Thereadershouldnotethattheresultsof 111

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Figure6-7.Errorbarplotsoftextureclassicationratesusingthethreefeaturesetson 100instantiationsofthe k -meansalgorithmatvariousvaluesof k forthe rockSASimagetextures.Theextremesofeacherrorbarrepresentone standarddeviation. thesegmentationfor k =3 willnotmatchtheresultsoftherstexperimentbecausethe featurevectorswerenotnormalizedpriortoclassication. 6.6Summary Hereweusedanunsupervisedmulti-classsegmentationapproachusingthe k -meansalgorithmtosegmentrock,smallripple,andlargerippleseabedtextures fromasetof40SASimagesusingthefeaturesderivedfromtheACFmodel.Three parameterizedACFtexturemodelsofvariouscomplexitywereusedtoextractfeatures 112

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Figure6-8.Errorbarplotsoftextureclassicationratesusingthethreefeaturesetson 100instantiationsofthe k -meansalgorithmatvariousvaluesof k forthe smallrippleSASimagetextures.Theextremesofeacherrorbarrepresent onestandarddeviation. fromtexturedSASimages.Tabulatedresultsofthesand,rock,andrippletexture segmentationwerepresentedonasetof40realSASimages.TheACFmodelwiththe mostdegreesoffreedomwasshowntobestdiscriminatebetweenthesand,rock,small sandripple,andlargesandrippletextures. Thesesegmentationresultswerethencomparedtotwootherrecentlyproposed seabedtexturefeaturesetsbasedonwaveletcoefcientsandHaralickfeatures, respectively.Intherstcomparativeexperiment,theACFmodelfeaturesproduced 113

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Figure6-9.Errorbarplotsoftextureclassicationratesusingthethreefeaturesetson 100instantiationsofthe k -meansalgorithmatvariousvaluesof k forthe largerippleSASimagetextures.Theextremesofeacherrorbarrepresent onestandarddeviation. bettersegmentationresultsintermsofcorrectclassicationrateintherockandlarge rippleclassesthantheotherfeaturessetswhenthe k -meansalgorithmassumedthe numberofmodesorclusterswasequaltothenumberofclasses.Asecondexperiment wasconductedtocalculatethecomplexityofthefeaturespacescreatedbythethree featureextractionmethods.Forclassiersoflowcomplexity,theACFmodelfeatures producedbettersegmentationsthanthefeaturesbasedonwaveletcoefcientsand Haralickfeatures.Asclassiercomplexityincreased,thewaveletcoefcientandHaralick 114

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Figure6-10.Errorbarplotsoftheaveragetextureclassicationratesusingthethree featuresetson100instantiationsofthe k -meansalgorithmatvarious valuesof k forcombinedrock,smallripple,andlargerippleSASimage textures.Theextremesofeacherrorbarrepresentonestandarddeviation. featuresproducedslightlybetterorequalsegmentationresults,inferringthatthefeature spacesfromthesecompetingfeatureextractiontechniquesarehighlymultimodal. 115

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CHAPTER7 CONCLUSIONSANDFUTUREWORK Chapters16detailanapproachtothemodeling,simulation,andsegmentation ofSASstripmapimagetextureswithaparameterizedintensityautocorrelationfunction ACF.Theclosed-formparameterizedACFisrstderivedfromthethescattering cross-sectionandimagingpointspreadfunctionofthebeamformingprocess.The ACFmodelisthenusedgenerativelytosynthesizeSASimagetextureswithknown ACFparameters.AnExpectation-Maximizationalgorithmfortruncateddataisused toestimatedmodelparametersfromtheestimatedACFofaSASimagetextureand performanceoftheestimationtechniqueiscompiledforacollectionofsyntheticand realSAStextures.Finally,theextractedACFmodelparametersareemployedin anunsupervisedtexturesegmentationalgorithmandresultsarecomparedacross increasingACFmodelcomplexityandagainsttwocompetingsonarimagetexture featuressets,bi-orthogonalwaveletsandtheHaralickfeatureset. Theworkhereinpresentsathoroughmodeldesignfortheextractionofrstand second-orderinformationcontainedinSASimagesandthedetailedcontributionsare listedinthenextsection.Thesecondsectionoutlinessomenewideasforfutureareas ofresearchinSASimagetexturemodelingandsegmentation,specicallythemultiscale modelingofSAStextureandtheuseofhigherordermomentinformationmoments > 2 fortexturediscrimination.Anoverallsummaryconcludesthechapter. 7.1Contributions ThisdissertationdetailedseveralcontributionstothestatisticalmodelingofSAS imagetextures.Thecontributionsareenumeratedandsummarizedbelow: 1. ExaminedthestatisticalmodelingofSASimagetexturesusingtheRayleigh, exponential,and K distributions: TheRayleighandexponentialdistributions wereshowntotthesingle-pointorimagepixelhistogramstatisticsoflow-contrast amplitudeandintensitytextures,respectively.Howeverfortextureswithhigh contrastsuchasrockandsandripples,the K distributionwasshowntobeabetter t.Acasewasalsomadetoincludehigher-ordermomenttexturaldescriptorsto 116

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betterdiscriminatebetweenhigh-contrasttexturesinSASimagesandjustiedthe developmentoftheACFmodeldescribedinChapter2. 2. AppliedoriginalSARintensityimagetexturemodeltoSAStextures: TheACF modeloriginallyproposedbyOliver[20,22,23,25,26]yieldsvaluableinformation aboutthedirectionandgrossspatialcorrelationlengthfornaturalscenetextures inSARimagery.Thismodelhasutilityintheclosely-linkedworldofSAS,where beamformingandimageacquisitiontechniquesareverysimilar.However,textures commontoSASmaybelesscommoninSARandtheACFmodelwasmodied toaccommodatefrequentlyencounteredseabedtextures.Asanexample,seabed texturepatternsaregeneratedbydifferentnaturalandbiologicalprocesses,e.g. sandripplesandseagrasselds.Intheauthor'sexperience,theperiodicityor quasi-periodicityandthedirectionofripplesandotherphenomenaaresalient featurescapturedwellbytheproposedmodel. 3. DerivedparameterizedintensityACFmodelfromrealisticscatteringassumptions: Aclosed-formoftheparameterizedintensityACFtexturemodelwas derivedusinglinearshift-invariantsystemstheory.Someattractiveproperties oftheACFmodelaretheinclusionofthevariable thatdenesshapeofthe single-point K pdfandtheimagingPSFthatdenesthetwo-dimensionalspatial lteringcausedbythebeamformingprocess. 4. ExtendedthemultivariateGaussianACFmodelinSARliteraturetoamixture modelwithnonzero-meanandrotationparameters: Oliver'soriginalSAR texturemodellackstheplasticitynecessarytoproperlymodelmanySAStextures ofinterest.Amixturemodelwithnonzero-meancomponentswasproposedand showntotavarietyofSAStexturesincludingavarietyofperiodictextures commontoSAStextures. 5. Developedamethodtosimulatecomplextexturedseabedsusingintensity ACFmixturemodel: Usingacompoundmodelforthe K distribution,correlation textureswerecreatedwithknownintensityACFmixturemodelparameters.Using thisgenerativeprocess,performanceofACFmodelestimatescanbeempirically calculatedfromsynthetictextures.Thesimulationmethodcanalsobeusedto createsyntheticyetrealisticSASimagedatasetswithtexturedseabedregions. ThissimulateddataisveryusefulsincegatheringrealSASimagedatafroma varietyofseabedenvironmentsisexpensiveandtime-consuming. 6. DevelopedmethodtoestimateACFmodelparametersusingtheEMalgorithmfortruncateddata: AftermanipulatingtheintensityACFequation,theEM algorithmwasemployedtoestimatetheACFmodelparametersfromanintensity ACFestimate.SincetheestimatedACFwasnoisyawayfromthemainlobe,the estimatewasrsttruncatedorremovedandtreatedasmissingdatatoproduce smootherACFestimates. 117

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7. TestedACFestimationmethodperformanceagainsttextureblocksizeand statisticalshapeparameterandusedtheBICtoquantifymodelcomplexity vs.goodnessoft: TheperformanceoftheACFestimationmethodwastested againstvarioussyntheticandrealSAStextures.Thesynthetictextureswere chosentomeasurehowtheaccuracyofthemodeltdegradesagainstchangesin textureestimateblocksizeandcontrast.Sincethenumberofmodelparameters isgenerallynotknown apriori ,theBICwasemployedtoestimatethenumber ofmixturecomponentsandshowntobeausefulmodelselectionmetricfora synthesizedperiodictexture.ACFmodelsforsand,rock,seagrass,andsand rippletextureswerealsoestimatedandtheaccuracyandcomplexityforeach textureclasswascompared. 8. UsedACFmodeltoextractparametersfortexturesegmentationandcomparedACFfeatureperformanceagainsttwootherrecentfeaturesetsproposedforunsupervisedseabedsegmentation: TheACFmodelparameters wereextractedforasetofSASimagesthatcontainedsand,rock,smallripple,and largerippletextures.Theseextractedparameterswereusedinanunsupervised multi-imagesegmentationalgorithmbasedon k -meansclusteringandaccuracy versusACFmodelcomplexitywascompared.Thesegmentationresultsusingthe ACFmodelparameterswasalsocomparedagainstwaveletandHaralicktexture featuresandshowntobesuperiorforthegivensegmentationtask. 7.2FutureWork Asthisworkfocusedmainlyonthemodeling,simulation,featureextraction,and segmentationofSASimagetexturesaspresentedbyahigh-frequencySASstripmap imagingsystem,otherapproachestotexturemodeling,texturesegmentation,andscene analysiswerenotexamined.Multiscalerepresentationsoftexturesusingwavelets [52,6466],Markovrandomelds[49,6770],andrecentlydynamictrees[7173]all haveutilityinsupervisedandunsupervisedclassicationschemes.Inmultiresolution ormultiscalemodeling,thetextureofinterestisrepresentedbybasiscoefcientsor modelparameters,andmathematicalrelationshipsbetweentexturedescriptorsat differentscalesareexploitedasadditionalinformationtoaidindiscrimination.These approachesareveryeffectiveatregularizingorsmoothingsegmentationresultsat verysmallscalesasthemultiresolutionmodelsusuallyassumethattexturedregions ofthesameclassareclosedwithdistinctboundaries.Aswillbeseeninthenext section,theACFmodelrepresentationyieldsconsistentandpredictableresultsat 118

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multiplescales,implyingthatthemodelparameterscouldbeincorporatedintoa tree-structuredmultiscaleimagemodel.Inthecaseofahierarchicalautomatedscene analysisalgorithm,segmentationmaybetherststepindeterminingtheunderlying characteristicsorconstructionofdifferentscenecomponents.E.g.,asonarimage maycontainsandripples,darksand,lightsand,androcks.Afterinitialsegmentation, additionalsensorsorphysics-basedalgorithmsmaybeusedtodeterminesedimenttype ofthesandregions,rippleheight,andaveragesizeoftherockregions.TheACFmodel usedherehastheaddedadvantagethatsomehigherlevelinformationaboutthetexture iscarriedforwardsuchasrippleperiodandrelativetexturecomponentsize. Sections7.2.17.2.3describesomeavenuesoffutureresearchforSASimage texturemodelingandclassicationusingextensionsofthecurrentACFmodel, examiningtheSAStexturehigher-ordermoments,andusingtheACFmodelparameters toqueueanautomatedphysics-basedrippleheightestimationalgorithm. 7.2.1MultiscaleDecompositionFeatureExtraction Textureinformationisfrequentlyexaminedatmultipleimagescaleseithertomatch thewindowsizeofabasisdecompositiontechniquesuchasawaveletdecomposition ortocheckforsomefeatureresiliencetoimagedownsamplinganindicationoftexture size.InthecaseoftheintensityACFmodelparameters,textureswithlargecorrelation lengthswillretainconsistentparametervaluesacrossscale,takingintoaccountthe downsamplingfactor.Thisparametervaluescaleinvarianceprovidesadditional featuresfortextureclassication.Conversely,textureswithsmallcorrelationlengths willdecorrelateasanimageisdownsampledandtheestimatedcorrelationlengths willbecomedominatedbynoise.ThisisdemonstratedinFigure7-2wheretheACF parametersofthreetextures,onewithashortcorrelationlength l x = l y =4 ,one withamediumcorrelationlength, l x = l y =8 ,andonewithalongcorrelationlength l x = l y =24 areestimatedforscalesof 800 800 400 400 200 200 ,and 100 100 for100simulations.Thescalesaredenoted0,1,2,and3with0beingthelargest 119

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scale 800 800 and3beingthesmallscale 100 100 .Theparameterestimates for l x and l y arecombinedtogivea200totalindependentsimulationsinthemeanand standarddeviationcalculations.Themeanparameterestimateforeachscaleisplotted alongwithanerrorbarrepresentingtwicethestandarddeviationnormalizedbythe meanoftheparameterestimate.Thecorrelationlengthsof8and24areshowntobe resilienttodownsamplingwhilethesmallercorrelationlengthbecomesdecorrelated atsmallscaleasevidencedbytheparameterestimatesamplemeanandstandard deviation.Usingamultiscaleapproach,afeaturevectorofACFparameterestimates atvariousscalescouldprovidemoreinformationthananestimateatasinglescale. Additionallytheconsistencyofaparameterestimateacrossscalecouldalsogive additionaldiscriminatoryinformation. 7.2.2Higher-OrderStatistics Inadditiontotherstorderstatisticalparametersuchasthe K distributionshape parameterandsecondorderinformationsuchastheintensityACFparameterestimates, higherorderstatisticalmomentinformationmaybeusefulindiscriminatingbetween differentSASseabedtexturesofdifferentscale.Thereferences[74,75]describe researchintotheuseofstatisticaldependencemeasuresforclassierfeatureselection. Themetricsdescribedusethedata'scumulativedistributionfunctionandthusall momentsofthedataandareinvarianttomonotonictransformsofthedata.Thislatter propertyisusefulwhenestimatingpropertiesofnonnegativeimagedatasuchasSAS imagerysincetheintensity,amplitude,andlog-intensitypixelvaluesandanyother multiplicativelyampliedorattenuatedimageswillhavethesamedependencevalue. Anotherhigherordermeasurementfortimeseriesinformationknownascorrentropy hasalsorecentlybeendeveloped[76,77].Thisnonlinearfunctionoperatessimilarlyto autocorrelationbutratherthanperformingtheinnerproductinthedatadomain R 2 in thecaseofaSASimage,theinnerproductiscarriedoutinaninnitedimensional reproducingkernelHilbertspaceviaaGaussiankernelfunction.Aninteresting 120

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A l x =4,800 800 B l x =8,800 800 C l x =24,800 800 D l x =4,400 400 E l x =8,400 400 F l x =24,400 400 G l x =4,200 200 H l x =8,200 200 I l x =24,200 200 J l x =4,100 100 K l x =8,100 100 L l x =24,100 100 Figure7-1. 800 800 syntheticimagesamplesofvariouscorrelationlengths downsampledby2,4,and8.ThetopleftsampleAwasgeneratedwitha singlecomponentACFwith l x = l y =4 .ThetopcentersampleBwas generatedwith l x = l y =8 .ThetoprightsampleCwasgeneratedwith l x = l y =24 .Imagepixelsinleftandcentercolumnofimagesdecorrelateas theyaredownsampled,yettherightcolumnimagesareresilientto downsamplingduetothelongcorrelationlength. 121

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Figure7-2.Correlationlengthestimatesacrossvariousimagescales.Thecorrelation lengthestimatesof l x = l y =24 and l x = l y =8 areresilienttodownsampling byratesof2,4,and8whilethesmallestcorrelationlength l x = l y =4 decorrelatesatadownsamplingrateof8asevidencedbythemeanand normalizedstandarddeviationvalues. propertyofcorrentropyisthatthemeasuredvaluecontainsinformationfromallthe evenmomentsofthedataandthusmoreinformationbeyondthatmeasuredintheACF mayobtained.Sections7.2.2.1and7.2.2.2brieydescribesomeinitialresultsfrom applyingco-dependenceandcorrentropymeasurementstoSAStexturesandhintatuse oftheseapproachesindiscrimination. 122

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7.2.2.1Fourth-orderdependencemetric Afourth-orderdependencemetricwasusedtocalculatethedependencebetween4 neighboringpixelsinaSASimageandisdenedas M = X i X j X k X l j ^ F X 1 X 2 X 3 X 4 i j k l )]TJ/F22 11.9552 Tf 13.174 2.657 Td [(^ F X 1 i ^ F X 2 j ^ F X 3 k ^ F X 4 l j where ^ F X 1 X 2 X 3 X 4 istheempiricaljointcumulativedistributionfunctionCDFofthe variables X 1 X 2 X 3 X 4 and ^ F X i istheempiricalmarginalCDFofvariable X i .The empiricaljointCDFforvariables X 1 X 2 X 3 X 4 iscalculatedbycomputingthejoint histogramofpixelvaluesthatllthe 2 2 windowinFigure7-3translatedovereach scaleoftheimageinquestion.Thisdependencemetrichassmallvaluesfortextures thatarecomposedofindependentpixelvaluesand M increasesastexturepixels takeonsimilarneighboringvalues.Inthefollowingseriesofguresthefourthorder Figure7-3.WindowusedtocomputeCDFhistogramvalues. dependencedenedbythewindowinFigure7-3wasestimatedforscalesof 400 400 200 200 ,and 100 100 foreachtexture. M forfourtextures,sand,ripple,seagrass, androckwereplottedagainsttheirrespectivescalesforcomparisonanddepictedin Figure7-4.TheplotinFigure7-5showsthattherippleandrocktextureshavethe highestcodependencevaluesacrossthescales.Seagrassishigherthanthesand 123

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whichhasthesmallestcodependenceasexpected.Asthescalegetslarger,the dependenceforalltexturesdecreasesastheimagesdecorrelateduetodownsampling, butofnoteisthatthesandrippletexturedecreasestheleastamount.Thisisveriedin Figure7-4bythevisiblestructuraldetailsinthesandridges. 7.2.2.2Autocorrentropy Thecorrentropyfunction V beappliedtoarandomsequence X t inamanner similartoautocorrelationandisthuscalled autocorrentropy .Theautocorrentropyof X t isdenedas[77] V X t X t + E [ X t )]TJ/F39 11.9552 Tf 11.955 0 Td [(X t + ] where X t )]TJ/F39 11.9552 Tf 11.955 0 Td [(X t + istheGaussiankernel X t )]TJ/F39 11.9552 Tf 11.955 0 Td [(X t + = 1 p 2 e )]TJ/F23 5.9776 Tf 5.757 0 Td [( X t )]TJ/F40 5.9776 Tf 5.756 0 Td [(X t + 2 2 2 Thekernelbandwidth playsanimportantroleintheautocorrentropyoutputasit determinesthesensitivityofthefunctionoutputtothedifferencesin X t and X t + Atoneextreme,averysmall valuemeanstheautocorrentropyvaluewillbezero unlessthetwosequences X t and X t + areidentical.Attheotherextreme,a large valuecausessecond-ordermomentstodominatethecalculation,thusthe autocorrentropyvalueapproachesthatofautocorrelation. Anexperimentwasconductedtouseautocorrentropytocalculateacontinuum ofoutputsusingvariouskernelbandwidthsinordergeneratehigherordermoment informationforafewSASimagetextures.Usingthesand,rock,andrippletextures ofFigures3-1A,3-3A,and3-4Arespectively.Thesonarimagelogintensity inputwasde-meanedandnormalizedbythestandarddeviation.Theautocorrentropy estimationwascarriedoutrow-wisewithasinglerowofthesonarimagebeingusedas 124

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ARipple400 400 BRipple200 200 CRipple100 100 DRock400 400 ERock200 200 FRock100 100 GSeagrass400 400 HSeagrass200 200 ISeagrass100 100 JSand400 400 KSand200 200 LSand100 100 Figure7-4.Multiscaleimagesofsandripple,rock,seagrass,andsandtextures. 125

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Figure7-5.Plotofcodependencemetric M for3scalesofsand,seagrass,rock,and sandrippleSASimagetextures. theinputtotheautocorrentropysampleestimator ^ V X t X t + = 1 N N X i =1 X t i )]TJ/F39 11.9552 Tf 11.955 0 Td [(X t i + Themaximumlag fortheautocorrentropyestimatewas100andthesequence length N was250forallthreetextures.Thekernelparameter wasvariedfrom e )]TJ/F23 7.9701 Tf 6.587 0 Td [(2.5 to e 2.5 .Theautocorrentropyestimatesforasinglerowofeachtextureareplotted againstthe X -spatiallagvariable andkernelbandwidthinFigure7-6AC.Each autocorrentropyestimateisnormalizedrelativetothemaximumandminimumofACFof thesametimeseries.Thesandrippletexturewasrotatedcounterclockwiseby 37 o so thedirectionoftherippleeldwasperpendiculartothe X -spatiallagdirection. 126

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ASandACE:singlerow BRockACE:singlerow CRippleACE:singlerow Figure7-6.AutocorrentropyestimatesfortheAsand,Brock,andCsandripple texturesofFigures3-1A,3-3A,and3-4A. 127

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ExaminingtheACEwaterfallsforthesinglerowofeachtexture,itappearsthat eachtexturehassomedifferentiatinginformationduetothedifferencesintheplots.At rstglance,theautocorrentropyestimatesforsmallkernelwidthsmayprovidesome interestingdiscriminatinginformation.However,inanydiscriminationorclassication schemethisfeaturemustbeprovenreliable,i.e.haveasmallintra-classvariance andlargeinter-classmeandifference.Inordertousetheautocorrentropyfunctionfor discriminationabetterideaofintra-andinter-classfeaturevariabilityisneededanda consolidatingmeasureornumericalvaluesuchasentropyshouldbeassignedtothe waterfallmeasure,amultirowestimate,oramultiscaledecomposition. 7.2.3SASImageSceneAnalysis Typicallyarststepinanimagesceneanalysistaskistoautomaticallysegment regionsofinterest.TheintensityACFmodelhasshownutilitywithregardtoSASimage texturesegmentation.Anadditionaladvantageofthemodelisthattheparameters arerelatedtosomeintrinsicpropertyofthetexture.Asanexample,arippletexture isparameterizedbyseveralkeyproperties:contrast,period,orientation,etc.After segmentation,thesepropertiescouldbeusedinafollow-onsceneanalysistasktobuild moredetailedknowledgeaboutthesegmentedtextures. InFigure7-7,the k -meanscentroidsofthefour-componentACFmodelareplotted toillustratethedescriptorsthatcanbederivedfromtheACFmodelparameters.In Figure7-7AtheregeneratedACFofthelargerippledepictsacommonperiodof approximately50spatialsamplesandnearidenticalspatialcorrelationinallofthe mixturecomponents.ThesmallrippleandrockACFsareverysimilar,aswasclearfrom thesegmentationresults,butthesmallrippleACFhasawell-denednonzero-mean mixturecomponentconsistentwithaperiodofapproximately35spatialsamples.The nonzero-meanmixturecomponentandzero-meanmixturecomponentsfortherock texturearelessdened.Perhaps,theseparametersmaybepairedwithsomerecent physics-basedapproachestoroughnessmodelingandrippleheightdeterminationin 128

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ALargeripplecentroid BSmallripplecentroid CRockcentroid Figure7-7.Imagesofthe k -meanscentroidsforthelargerippleA,smallrippleB,and rockCtexturessegmentedinChapter6usingthefour-componentACF model.Thenonzero-meanperiodicmixturecomponentsinthecentroids revealcommoninformationinthesegmentedlargerippleandsmallripple texturesthatcouldbeexploitedwithfollow-onsceneanalysisalgorithms. anautomatedapproachtogaindetailedknowledgeoftheseabedwithinasurveyed area[78,79]. 7.3Summary Thebodyofworkpresentedherethoroughlydescribestheuseofrstand second-ordermomentsinsyntheticaperturesonartexturediscriminationalgorithms throughtheemploymentofanintensityACFmodel.Thederivationofthemodel frominitialscatteringassumptionsofindividualresolutioncells,utilityofthemodel insynthesizingrealisticparameterizedseabedenvironmentsforclassicationtasks, detaileddescriptionoftheestimationalgorithmfortexturefeatureextraction,and comparativetestingofthemodel'sdiscriminatorypoweragainstotherfeatureextraction methodspresentasystematic,comprehensiveapproachtoSASimagetexturemodeling anddiscrimination.Thequestionforthereaderisnow,howcanthemodelbeextended orimprovedtoovercomeunaddressedchallengesofSASimagetextureclassicationin automatedsystems? Section7.2hintsatsomepossibleapproachesthatmayimproveSAStexture discriminationperformance.Inthecontextoftheproblemofunsupervisedmulti-scene 129

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seabedsegmentation,itisclearthatanyfeatureextractionandclassicationtechnique mustberobusttovariationsincommontexturesbetweenimages.Orientation,variable intensityduetoimagingbeampatternsandsensinganglesareafewsuchpossible variations.Additionally,thetextureextractiontechniquemustaddresstheissueof heterogeneousandill-denedtextureboundariesthatareendemictoremotelysensed images.In[48]thisproblemisaddressedthroughamultiscaleimplementationwhich maybedirectlyapplicabletoSAStexturesegmentationaswell.Itisimportantto noteonlyafewdominanttextureclassesweredescribedhere,namelysand,rock, seagrass,andsandripple;variationsandmixturesofthesepresentacontinuumof possibletextureclasses.Certainly,higher-ordermomentinformationispresentinSAS imagesandcouldbeusedtoaidsegmentationinsubtleclassdistinctionsaswell.The challengingtechnicalproblemswithSAStexturemodelingandsegmentationcoupled withnewpowerfulinformationmodelingandextractiontechniquesshouldyieldarich eldoffutureresearchformanyyears. 130

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APPENDIXA DERIVATIONOFCOMPOUNDREPRESENTATIONOF K DISTRIBUTION Herewederivethe K distributionprobabilitydensityfunctionforsingle-point amplitudeandintensitypixelstatistics.Theamplitudepdfisderivedrstandthe intensitypdfderivationissummarizedwithappropriatelynotedsubstitutions. Themodelforthestatisticsofanamplitudepixelassumesthepixelisgeneratedby theproductofaRayleighandsquare-rootgammarandomvariable.Assign p X j Y x j y the Rayleighprobabilitydensityfunctionpdfwithparameter Y p X j Y x j y = x y 2 e )]TJ/F40 5.9776 Tf 5.756 0 Td [(x 2 2 y 2 u x A where u x istheunitstepfunction. Assume p Z z isthegammapdfwithparameters and b p Z z = b \050 z )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e )]TJ/F40 7.9701 Tf 6.587 0 Td [(bz u z A Thesquare-rootgammapdf p Y y isformedbythefunctionalrelationship Y = p Z A sothat p Y y =2 yp Z y 2 A Thusthefunctionalformofthesquare-rootgammapdfintermsof y is p Y y = 2 yb \050 y 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 e )]TJ/F40 7.9701 Tf 6.586 0 Td [(by 2 u y A Let p x representthepdfofthesignalenvelope.Thecompoundrepresentation followsfromtheuseofBayesruleandmarginalizationovervariable y : p X x = 1 Z p X Y x y dy = 1 Z p X j Y x j y p Y y dy A 131

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substitutingfromEquationsAandA A p X x = 1 Z 0 x y 2 e )]TJ/F40 5.9776 Tf 5.756 0 Td [(x 2 2 y 2 2 yb \050 y 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 e )]TJ/F40 7.9701 Tf 6.586 0 Td [(by 2 dy A p X x = xb \050 1 Z 0 y )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 y 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 e )]TJ/F40 5.9776 Tf 9.5 3.259 Td [(x 2 2 y 2 )]TJ/F40 7.9701 Tf 6.587 0 Td [(by 2 2 ydy A let C 1 = xb \050 p X x = C 1 1 Z 0 y 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 e )]TJ/F40 5.9776 Tf 9.5 3.259 Td [(x 2 2 y 2 )]TJ/F40 7.9701 Tf 6.587 0 Td [(by 2 2 ydy A let u = y 2 du =2 ydy p X x = C 1 1 Z 0 u )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 e )]TJ/F40 5.9776 Tf 5.756 0 Td [(x 2 2 u )]TJ/F40 7.9701 Tf 6.587 0 Td [(bu du A let = )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 p X x = C 1 1 Z 0 u )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e )]TJ/F40 5.9776 Tf 7.782 3.259 Td [(x 2 2 u )]TJ/F40 7.9701 Tf 6.587 0 Td [(bu du A from[80] 1 R 0 u )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e )]TJ/F27 5.9776 Tf 7.782 3.693 Td [( u )]TJ/F26 7.9701 Tf 6.587 0 Td [( u du =2 2 K )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(2 p ,setting = x 2 2 and = b p X x =2 C 1 x 2 2 b 2 K 2 x r b 2 A 132

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substituting )]TJ/F22 11.9552 Tf 11.956 0 Td [(1= and C 1 = xb \050 p X x = 2 xb \050 x 2 2 b 2 )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(1 2 K )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 x p 2 b A p X x = 2 xb \050 p 2 b x x 2 2 b 2 K )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 x p 2 b A p X x = 2 p 2 b \050 x r b 2 K )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 x p 2 b A let a = p 2 b p X x = 2 a \050 ax 2 K )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ax u x A TheinterestedreadercanconrmthatEquationAisthefamiliarpdfofthe K distributionpresentedin[9,13,15]withshapeparameter andscaleparameter a Forthecaseofthesingle-pointintensitypixelstatisticstheproductmodelassumes theintensitypixelistheproductofanegativeexponentialandgammarandom variable[10,Ch.5pp.130-131].Followingtheseinitialassumptionsandmaking similarchangesinvariablesasoutlineaboveleadstothe K distribution p I i = \050 2 i )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 2 K )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 p i A Theequationisalsoeasilyderivednotingthefunctionalrelationshipbetweenamplitude andintensity I = A 2 andsubstitutingvariablesappropriatelyusingtherelation p I i = 1 2 p i p A p i A 133

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APPENDIXB DERIVATIONOFTHEINTENSITYACFMIXTUREMODEL HerewendaclosedformanalyticalsolutionforEquation2usingaGaussian SCSACFmixturemodelpairedwithaGaussianimagingPSF.Tobeginweassumea simplebutusefulwide-sensestationarycorrelatedgammaSCSACFwithparameters and l x fortheone-dimensionalgammadistributionshapeandspatialcorrelationlength respectively[20] R x x + X = 2 1+ 1 e )]TJ/F40 5.9776 Tf 7.898 3.259 Td [(X 2 2 l 2 x B ThereareseveralformsfortheSCSACFthatcanbechosenbasedoninitialscattering assumptionsforthesensedenvironment[20,22].ThemodelinEquationBassumes anarrowbandsourceilluminatingaresolutioncellcontaining scatterers.Modelsthat includemodulationoftheSCSACFbyasinusoidmighttrippledseabedtexturesmore accuratelythantheACFinEquationB. IntheanalysisinthissectiontheimagingPSFisassumedtobeunitenergy R j h j 2 d =1 andhastheform h = 1 2 1 4 1 2 e )]TJ/F27 5.9776 Tf 9.484 3.258 Td [( 2 4 2 B WewillexamineeachsummandofEquation2separately.Webeginbyinserting EquationsBandBintoSummand1ofEquation2,substituting = x and = x + X ,andsimplifying 2 Z 1 j h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 j 2 d 1 Z 3 j h )]TJ/F25 11.9552 Tf 11.956 0 Td [( 3 j 2 d 3 + ZZ 1 3 1 e )]TJ/F23 5.9776 Tf 7.782 4.395 Td [( 1 )]TJ/F27 5.9776 Tf 5.757 0 Td [( 3 2 2 l 2 x j h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 j 2 j h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 j 2 d 1 d 3 = 2 0 @ 1+ 1 Z 3 e )]TJ/F23 5.9776 Tf 7.782 4.394 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 3 2 2 l 2 x 1 p 2 e )]TJ/F27 5.9776 Tf 9.484 3.259 Td [( 2 2 2 j h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 j 2 d 3 1 A B 134

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where istheconvolutionoperator.TheconvolutioninEquationBiscalculatedvia theFouriertransform.TheFouriertransformpairofaGaussianfunctionis e )]TJ/F40 5.9776 Tf 9.947 3.258 Td [(x 2 2 2 FT p 2 e )]TJ/F27 5.9776 Tf 7.782 3.259 Td [( 2 2 2 UsingthistransformpairandtheshiftpropertyoftheFouriertransform,theconvolution yields F )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 F e )]TJ/F23 5.9776 Tf 7.782 4.394 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 3 2 2 l 2 x 1 p 2 e )]TJ/F27 5.9776 Tf 9.485 3.258 Td [( 2 2 2 = l x p 2 + l 2 x e )]TJ/F23 5.9776 Tf 10.633 4.394 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 3 2 2 2 + l 2 x B InsertingEquationBbackintoEquationBandmakingthesubstitution = 3 )]TJ/F25 11.9552 Tf 12.214 0 Td [( wendanotherconvolution 2 1+ 1 l x p 2 + l 2 x e )]TJ/F23 5.9776 Tf 12.492 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 2 + l 2 x 1 p 2 e )]TJ/F23 5.9776 Tf 7.782 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 2 #! B TheconvolutionviaFouriertransformiscarriedoutagainasinEquationBand substituting = )]TJ/F25 11.9552 Tf 11.955 0 Td [( yields F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 F e )]TJ/F27 5.9776 Tf 20.038 3.693 Td [( 2 2 2 + l 2 x 1 p 2 e )]TJ/F27 5.9776 Tf 9.775 3.693 Td [( 2 2 2 = s 2 + l 2 x 2 2 + l 2 x e )]TJ/F27 5.9776 Tf 21.701 3.693 Td [( 2 2 2 2 + l 2 x B WeinserttheresultofEquationBbackintoEquationBforthenalformof Summand1 2 1+ 1 l x p 2 2 + l 2 x e )]TJ/F23 5.9776 Tf 14.155 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 2 2 + l 2 x B OperationsonSummand2ofEquation2willrequireaslightlydifferentapproach.We rstsubstitutefor R andrearrangetermstosimplifytheexpression 2 [ R h ] 2 + ZZ 2 4 1 e )]TJ/F23 5.9776 Tf 7.782 4.394 Td [( 2 )]TJ/F27 5.9776 Tf 5.756 0 Td [( 4 2 2 l 2 x h )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 h )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 h )]TJ/F25 11.9552 Tf 11.956 0 Td [( 4 d 2 d 4 B TocompletetheanalysisofEquation2,wewillsolvefortheclosedformofEquationB. Tosimplifynotation,welet c = 1 e )]TJ/F23 5.9776 Tf 7.782 3.859 Td [( )]TJ/F27 5.9776 Tf 5.757 0 Td [( 2 2 l 2 x andassignEquationBtoa 135

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function g g = ZZ 2 4 c 2 4 h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 h )]TJ/F25 11.9552 Tf 11.955 0 Td [( 4 d 2 d 4 B Weinsertthefunction h = 1 p p 2 e )]TJ/F27 5.9776 Tf 9.485 3.258 Td [( 2 4 2 andcombinenormalizingconstants g = 1 2 2 ZZ 2 4 c 2 4 e )]TJ/F23 7.9701 Tf 7.782 4.302 Td [( )]TJ/F27 5.9776 Tf 5.757 0 Td [( 2 2 4 2 e )]TJ/F23 7.9701 Tf 7.782 4.302 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 4 2 e )]TJ/F23 7.9701 Tf 7.782 4.302 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 4 2 4 2 e )]TJ/F23 7.9701 Tf 7.782 4.302 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 4 2 4 2 d 2 d 4 B g = 1 2 2 ZZ 2 4 c 2 4 e )]TJ/F23 5.9776 Tf 11.933 3.259 Td [(1 2 2 )]TJ/F27 5.9776 Tf 7.782 4.395 Td [( 2 + 4 2 2 e )]TJ/F23 5.9776 Tf 11.933 3.259 Td [(1 2 2 )]TJ/F27 5.9776 Tf 7.782 4.395 Td [( 2 + 4 2 2 e )]TJ/F23 5.9776 Tf 11.933 3.259 Td [(1 4 2 2 )]TJ/F26 7.9701 Tf 6.587 0 Td [( 4 2 d 2 d 4 B NowwetaketheFouriertransformof g withrespectto and G j j = 1 2 2 ZZ 2 4 c 2 4 e )]TJ/F23 5.9776 Tf 11.933 3.259 Td [(1 4 2 2 )]TJ/F26 7.9701 Tf 6.586 0 Td [( 4 2 Z e )]TJ/F23 5.9776 Tf 11.934 3.258 Td [(1 2 2 )]TJ/F27 5.9776 Tf 7.782 4.394 Td [( 2 + 4 2 2 e )]TJ/F40 7.9701 Tf 6.587 0 Td [(j d Z e )]TJ/F23 5.9776 Tf 11.934 3.258 Td [(1 2 2 )]TJ/F27 5.9776 Tf 7.782 4.394 Td [( 2 + 4 2 2 e )]TJ/F40 7.9701 Tf 6.587 0 Td [(j d d 2 d 4 B rearrangeterms, G j j = ZZ 2 4 c 2 4 e )]TJ/F23 5.9776 Tf 11.933 3.258 Td [(1 4 2 2 )]TJ/F26 7.9701 Tf 6.587 0 Td [( 4 2 e )]TJ/F40 7.9701 Tf 6.587 0 Td [(j 2 + 2 e )]TJ/F40 7.9701 Tf 6.586 0 Td [(j 4 + 2 d 2 d 4 e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 2 + 2 2 B andsubstitute 1 e )]TJ/F23 5.9776 Tf 7.782 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 l 2 x for c G j j = 1 Z 4 2 4 Z 2 e )]TJ/F23 5.9776 Tf 7.782 4.523 Td [(2 2 + l 2 x 4 2 l 2 x 2 )]TJ/F26 7.9701 Tf 6.586 0 Td [( 4 2 e )]TJ/F40 7.9701 Tf 6.586 0 Td [(j 2 + 2 d 2 3 5 e )]TJ/F40 7.9701 Tf 6.587 0 Td [(j 4 + 2 d 4 e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 + 2 2 B 136

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ThenextfewstepsfollowtheidenticationoftheFouriertransformintegralsinsidethe bracketsofEquationBwithrespectto 2 and 4 : G j j = 1 Z 4 p 2 p 2 l x p 2 2 + l 2 x e )]TJ/F23 5.9776 Tf 10.252 4.523 Td [(4 2 l 2 x 2 2 + l 2 x h + 2 i 2 e )]TJ/F40 7.9701 Tf 6.587 0 Td [(j 4 + d 4 e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 + 2 2 B G j j = 1 2 + p 2 p 2 l x p 2 2 + l 2 x e )]TJ/F23 5.9776 Tf 10.252 4.523 Td [(4 2 l 2 x 2 2 + l 2 x h + 2 i 2 e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 2 + 2 2 B g = 1 l x p 2 2 + l 2 x e )]TJ/F23 5.9776 Tf 7.782 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 4 2 B ThecombinationofEquationsB,B,andByieldstheclosedformsolutionofthe intensityACF[25] R I = 2 1+ [ R h ] 2 + 1 l x p 2 2 + l 2 x e )]TJ/F23 5.9776 Tf 14.155 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 2 2 + l 2 x + e )]TJ/F23 5.9776 Tf 7.782 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 4 2 #! B ItisasimpleextensionoftheabovederivationtocalculatetheintensityACF forthecasewhentheSCSmodelisaGaussianmixturewithparameters l 2 x i and i i = f 1,..., M g R = 2 1+ 1 X i i e )]TJ/F23 5.9776 Tf 7.782 3.858 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 l 2 x i B where P i i =1 .Sincethesummationcanbecarriedthroughalltheintegral relationshipstheintensityACFinthiscaseissimply R I = 2 1+ [ R h ] 2 + 1 X i i l x i p 2 2 + l 2 x i e )]TJ/F23 5.9776 Tf 14.917 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 2 2 2 + l 2 x i + e )]TJ/F23 5.9776 Tf 7.782 3.859 Td [( )]TJ/F27 5.9776 Tf 5.756 0 Td [( 2 4 2 #! B 137

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APPENDIXC EMALGORITHMFORESTIMATINGGAUSSIANMIXTUREPARAMETERSOFA TWO-DIMENSIONALACF In[35]anExpectationMaximizationmethodisproposedtoestimateprobability densityparametersoffunctionmeasurementsthataregroupedorbinnedintodiscrete intervals.Insomeintervalsthereisalsoassumedtobemissingdataduetotruncation ofthemeasurementsupport.ThisEMmethodisappliedtoGaussianmixturemodeling ofredbloodcellpopulationsthathavebinnedandtruncatedhistogramcountsin[36] and[43,Ch.2,pp.66-73].In[37]and[38]theauthorsalsoapplythemethodtomodel andtracktargetsandspectrummeasurementsinsonarapplications.Inthispaper theEMalgorithmforgroupedandtruncateddataisusedtoestimateparametersof anon-negativetwo-dimensionalfunctionthatrelatetotheintensityACFmodelas describedinSection4.3.ThisappendixbrieydescribestheEMalgorithmusedinthis workanddirectsthereadertoapplicablereferencesforamoredetaileddescription. Theparametersofadiscretepositivefunction g x y thathassupportovera two-dimensionalgridof v binsisestimatedviathefollowingmethod.Assume r measurementsof g x y viathefunction g T x y arereceived,where r < v .Asin [37]and[38],quantize g T x y n j = b N Q g T x j y j c i = f 1,..., r g C where y = b x c isafunctionthatassigns y thegreatestintegerlessthanorequalto x and N Q isalargeintegerthatcontrolstheaccuracyandscaleofthequantization. Nowassume n j isthebincountofatwo-dimensionalhistogramthathasamultinomial distributionconsistingof n drawson r categories,where n = P r j =1 n j .Proceedingas in[35,36]and[43]thebinprobability P j / P isdenedbyintegratingaGaussian mixturemodelprobabilitydensityfunctionoverthebinareaof g T x j y j or P j = ZZ x j y j f x y j dxdy C 138

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where f x y j = M X i =1 i G i x y j x i y i i C Theparameters i x i y i ,and i arethemixingparameter, x mean, y mean,and covarianceparameterofthe i -thzero-meanGaussianmixturecomponent G i ,and P = r X j =1 P j C Giventheaboveassumptionstheparameters = f i x i y i i g i = f 1,..., M g ,of g x y areestimatedviatheEMalgorithm. TheEMalgorithmrstdetailedin[35]providesamethodtoestimateparameters viastandardmaximumlikelihoodmathematicaloperationsbytheinclusionofhidden variablesrepresentingmissinginformationintheobservedmeasurements.Inthis case g x y hasthreepiecesofmissinginformation:missingmeasurementsdueto truncationin g T x y ,themixturecomponentmembershipofagivenbinmeasurement representedby j ,andunknownmeasurementassignmenttoaspecic x )]TJ/F39 11.9552 Tf 11.389 0 Td [(y coordinate of f x y j duetothequantizationof g x y intodiscreteintervals.Introducingrandom variables n r +1 ,..., n v torepresentmissingbincountsand [ x jk y jk ] j = f 1,..., v g k = f 1,..., n j g ,tobethenumberofindependentobservationsof g x y withdensity f x y j / P j leadstothecomplete-datalog-likelihoodfunction[36] M X i =1 v X j =1 n j X k =1 z ijk [ ln G i [ x jk y jk ] j x i y i i +ln i ] C where z ijk isarandomvariableintroducedtoassignmembershipofobservation [ x jk y jk ] tomixturecomponent i withprobability P z ijk =1 j [ x jk y jk ] i [ x jk y jk ] j i = i G i [ x jk y jk ] j x i y i i P M i =1 i G i [ x jk y jk ] j x i y i i C TheE-stepoftheEMalgorithmrequirestakingtheexpectedvalueofEquation Cconditionedonthecurrentsetofparametersandtheobserveddata g T x y .This 139

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conditionalexpectationorQ-functionforstep p +1 isformedbyreplacingthehiddenand missingvariablesinEquationCbytheirexpectedvaluesandtheparametersatstep p or[36] Q p +1 j p = M X i =1 v X j =1 m j p E p j f i [ x j y j ] j p i g ln G i [ x jk y jk ] j p x i p y i p i +ln p i C where m j p = 8 > < > : n j j =1,..., r n P j p j P p j = r +1,... v C andthevector [ x j y j ] meansthatobservations [ x jk y jk ] aretakentobeatthecoordinates ofthebinareacenteron g x j y j [43,Ch.2,pp.70]. TheM-stepofEMalgorithmndstheparameterestimatesatstep p +1 by maximizingEquationCwithrespecttotheobserveddataandcurrentestimates atstep p .Theupdateequationsfor i [ x j y j ] j i i x i y i ,and i areasfollows: h p i i j E j f p i [ x j y j ] j p i g = p i G i )]TJ/F22 11.9552 Tf 5.479 -9.85 Td [([ x j y j ] j p x i p y i p i M P i =1 p i G i [ x j y j ] j p x i p y i p i C p +1 i = v P j =1 m j p h p i i j v P j =1 m j p C p +1 x i p +1 y i = v P j =1 m j p h p i i j [ x j y j ] v P j =1 m j p h p i i j C 140

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and p +1 i = v P j =1 m j p h p i i j x j )]TJ/F25 11.9552 Tf 11.955 0 Td [( p +1 x i y j )]TJ/F25 11.9552 Tf 11.955 0 Td [( p +1 y i T x j )]TJ/F25 11.9552 Tf 11.955 0 Td [( p +1 x i y j )]TJ/F25 11.9552 Tf 11.955 0 Td [( p +1 y i v P j =1 m j p h p i i j C 141

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BIOGRAPHICALSKETCH J.ToryCobb,anativeofAnniston,AL,receivedtheB.S.degreeinelectrical engineeringfromtheUnitedStatesCoastGuardAcademyin1994andtheM.S.degree inelectricalengineeringfromAuburnUniversityin2001. From1994to1999hewasanactivedutyofcerintheCoastGuardandservedas adeckwatchofcerandexecutiveofceraboardthecutters Courageous and Wrangell Since2001hehasbeenemployedasaresearchengineerattheNavalSurfaceWarfare Center,PanamaCity,Florida.AssignedtotheAdvancedSignalProcessingandATR Branch,hehasservedasPrincipalInvestigatororCo-PrincipalInvestigatorforvarious automatictargetrecognitionandsensorfusionprojectsfundedbytheOfceofNaval Research.Hiscurrentresearchinterestsincludestatisticalmodelingofsonarsignals withapplicationstoautomatictargetrecognitionandautomatedseabedsegmentation. TorylivesinPanamaCityBeach,FLwithhiswifeParmjeet,daughterLily,andson Nate.Hisinterestsareoutdoorsportsandshing. 148