Statistical Dependence Measure Based Multi-Model Image Registration and Registration Assisted Non-Parametric Image Segme...

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Title:
Statistical Dependence Measure Based Multi-Model Image Registration and Registration Assisted Non-Parametric Image Segmentation
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1 online resource (95 p.)
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english
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Shi, Jiangli
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University of Florida
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Gainesville, Fla.
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Chen, Yunmei
Committee Members:
Rao, Murali
Shen, Li C
Mccullough, Scott A
Gilland, David R

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Subjects / Keywords:
dependence -- image -- measure -- registration -- segmentation -- statistical
Mathematics -- Dissertations, Academic -- UF
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Mathematics thesis, Ph.D.
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

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Abstract:
A series of mathematical models for image registrations are investigated in this work to provide a variety of techniques especially for medical image alignment. Analysis of robustness, efficiency and accuracy for each model is included in order to give comparisons and therefore to show advantages and limitations for each model. As shown in this thesis matching similar structures between several image modalities poses extreme difficulties for images with complex structures. Those difficulties primarily stem from the nonlinear relation and unknown quantization effects of intensity values. However, potential of a large variety of industrial applications makes it inevitable to register similar structures between several image modalities, for which direct intensity comparison is not possible. In particular, in the applications of medical imaging, images with different modalities often need to be registered for an accurate fusion of complementary information. To tackle with those difficulties, we introduced a statistical dependence measure into multi-model image registration. A novel variational model for deformable multi-modal image registration is then proposed. As an alternative to the models based on maximizing mutual information, the Renyi’s statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent samples. Moreover, the theory of reproducing kernel Hilbert spaces is used to simplify the computation. Experimental results with comparisons are provided to show the effectiveness of the proposed model. Finally we introduced a registration assisted non-parametric segmentation model as an application of Renyi’s statistical dependence measure. This model generates a label image , which represents the location of the edges separating the homogeneous regions in the given image, by assigning different constants on each of the sub-regions. Registration is then applied to the image to be segmented and the generated label image. This is accomplished by maximizing the similarity between those two images. As the label image is deformed from the process of registration, the location of the edges in the label image gradually moves to the same location of the edges  in the segmenting image. Finally, the best alignment of those two images provides a best segmentation result, rendering a suitable choice of similarity measure extremely important for this problem. Since those two images are no longer of the same modalities, the direct comparison of their intensities fails. By using R\'{e}nyi's statistical dependence measure the proposed model deals directly with independent samples and does not need to estimate the continuous joint probability density function, which is time consuming. Furthermore, the computation is greatly simplified by using the theory of reproducing kernel Hilbert spaces. Experimental results for given medical and real images are provided to demonstrate the effectiveness and efficiency of the proposed method.
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In the series University of Florida Digital Collections.
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Includes vita.
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Description based on online resource; title from PDF title page.
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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 Jiangli Shi.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Chen, Yunmei.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-02-28

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STATISTICALDEPENDENCEMEASUREBASEDMULTI-MODELIMAGEREGISTRATIONANDREGISTRATIONASSISTEDNON-PARAMETRICIMAGESEGMENTATIONByJIANGLISHIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JiangliShi 2

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Tomyfamily 3

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ACKNOWLEDGMENTS Firstofall,Iwouldliketosincerelythankmyadvisor,YunmeiChen,forherencouragement,inspirationandinvaluableadvicethroughoutmyPh.D.program.I'mverygratefulforhertremendoussupportandpatientguidance,notonlyinthetheoreticalinstructionsbutalsointhepracticalimplementations,duringmytimeofacademicresearch.ThanksgoouttomycommitteemembersDavidGilland,ScottMcCullough,MuraliRaoandLi-ChienShenfortheirvaluablesuggestionsandcommentsregardingmydissertation.I'mespeciallygratefultoMuraliRaoforhisadviceduringdiscussionandhisguidanceonpaperrevision.IwouldliketothankJianGeandhisstudentsRuiLiandJiWangfromAstronomyDepartmentforcollaborationandhelpfulsuggestions.ThanksalsogoouttoXiaojingYeforallhelpfuldiscussiononseveraltopicsinoptimizationtechniques.IwouldacknowledgeallthosewhohaveworkedwithmeduringmyPh.D.program,includingFuhuaChen,Jin-SeopLee,MengLiu,YuyuanOuyangandHailiZhang.Finally,Iwouldliketothankmyfamilyfortheirencouragement,endlessloveandcontinuoussupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 1.1ImageRegistration:anOverview ....................... 11 1.1.1BackgroundandMotivation ...................... 11 1.1.2ProblemDenition ........................... 12 1.1.3Data ................................... 12 1.2MethodsandClassication .......................... 13 1.2.1LinearandNonlinearTransformations ................ 13 1.2.2SpatialandFrequencyDomainMethods ............... 13 1.2.3SingleandMulti-ModalityMethods .................. 14 1.2.4IntensityBasedandFeatureBased .................. 15 1.2.5AutomaticandInteractiveMethods .................. 15 1.3SimilarityCriteria ................................ 16 1.3.1MeasuresforLinearRelation ..................... 16 1.3.2MeasuresforNonlinearRelation ................... 17 1.4Applications ................................... 17 1.4.1ApplicationsinOtherFields ...................... 18 1.4.2AssistancetoOtherProceduresinImagingScience ........ 18 1.5OrganizationofThisThesis .......................... 18 2PARAMETRICIMAGEREGISTRATIONS ..................... 20 2.1ParametricGeometricTransformations .................... 20 2.1.1RigidTransformations ......................... 20 2.1.2AfneTransformations ......................... 21 2.2ImageRegistrationbyUsingPhaseCorrelation ............... 22 2.2.1Rationale ................................ 23 2.2.2ExperimentsandApplications ..................... 27 3DEFORMABLEIMAGEREGISTRATION ..................... 28 3.1EstimationofProbabilityDensityFunction .................. 28 3.2CrossCorrelationBasedImageRegistration ................ 29 3.3CorrelationRatioBasedImageRegistration ................. 31 3.4InformationTheoreticApproaches ...................... 33 5

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3.4.1Shanon'sEntropy ............................ 33 3.4.2MutualInformation ........................... 35 4DEFORMABLEMULTI-MODALIMAGEREGISTRATIONBYMAXIMIZINGRENYI'SSTATISTICALDEPENDENCEMEASURE ............... 39 4.1Introduction ................................... 39 4.2ReproducingKernelHilbertSpacesandDensenessProperties ...... 46 4.2.1ReproducingKernelHilbertSpace .................. 46 4.2.2DensenessProperties ......................... 47 4.3ProposedModel ................................ 47 4.4TheIterativeAlgorithm ............................. 50 4.5ExtensiontoLocalMaximumCorrelationCoefcient ............ 52 4.6ExperimentalResults ............................. 53 4.6.1Parameters ............................... 53 4.6.2TestsonSyntheticImagesandSimulatedDeformationField .... 54 4.6.3TestonCTandMRImages ...................... 57 4.6.4ComparisonswithOtherDeformableModels ............ 60 4.7ConclusionandDiscussion .......................... 63 5REGISTRATIONASSISTEDNON-PARAMETRICIMAGESEGMENTATION .. 65 5.1Introduction ................................... 65 5.2ProblemStatementandRelatedWorks ................... 68 5.3Renyi'sStatisticalMeasure .......................... 70 5.4ProposedModelandNumericalMethod ................... 71 5.4.1LevelSetFormulationandNumericalMethod ............ 72 5.4.2ASoftFormulationandNumericalMethod .............. 74 5.5ExperimentalResults ............................. 76 5.6Conclusion ................................... 80 APPENDIX ......................................... 82 APROOFOFTHEDENSETHEOREMINSECTION4.2.1 ............ 82 BE-LEQUATIONOFFORTHEGLOBALVERSIONOFMCCMODEL ...... 85 CE-LEQUATIONOFFORTHELOCALVERSIONOFMCCMODEL ...... 87 REFERENCES ....................................... 89 BIOGRAPHICALSKETCH ................................ 95 6

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LISTOFTABLES Table page 4-1ResultsofRegistrationModelswithSuitableParameters ............ 61 4-2FinalMIValuesforMIandMCCModels ...................... 62 4-3FinalMCCValuesforMIandMCCModels .................... 62 7

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LISTOFFIGURES Figure page 2-1MotionCorrectionbyUsingPhaseCorrelation .................. 27 3-1VennDiagramforShannonEntropyandMutualInformation ........... 36 3-2VennDiagramwhenMutualInformationMI(X,Y)ReachesitsMaximumH(X) 38 4-1DemoofImageDecomposition ........................... 45 4-3RegistrationResultsforSyntheticImages ..................... 54 4-4RegistrationResultforSyntheticImageswithRegionalInhomogeneities .... 56 4-5RegistrationResultsforT2andPDBrainImages ................. 56 4-6RegistrationResultsforCTandMRImages .................... 57 4-7RegistrationResultsforCTandMRImageswithGaussianNoise ........ 58 4-8RegistrationResultsforT1andT2Images .................... 59 4-9RegistrationResultsforT1andT2ImageswithGaussianNoise ........ 60 4-10ComparisonswithOtherRegistrationModels ................... 62 5-1SegmentationResultsofaSyntheticImage .................... 77 5-2SegmentationResultsofaBrainImage ...................... 79 5-3SegmentationResultsofaCameramanImage .................. 80 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySTATISTICALDEPENDENCEMEASUREBASEDMULTI-MODELIMAGEREGISTRATIONANDREGISTRATIONASSISTEDNON-PARAMETRICIMAGESEGMENTATIONByJiangliShiAugust2013Chair:YunmeiChenMajor:Mathematics Aseriesofmathematicalmodelsforimageregistrationsareinvestigatedinthisworktoprovideavarietyoftechniquesespeciallyformedicalimagealignment.Analysisofrobustness,efciencyandaccuracyforeachmodelisincludedinordertogivecomparisonsandthereforetoshowadvantagesandlimitationsforeachmodel.Asshowninthisthesismatchingsimilarstructuresbetweenseveralimagemodalitiesposesextremedifcultiesforimageswithcomplexstructures.Thosedifcultiesprimarilystemfromthenonlinearrelationandunknownquantizationeffectsofintensityvalues.However,potentialofalargevarietyofindustrialapplicationsmakesitinevitabletoregistersimilarstructuresbetweenseveralimagemodalities,forwhichdirectintensitycomparisonisnotpossible.Inparticular,intheapplicationsofmedicalimaging,imageswithdifferentmodalitiesoftenneedtoberegisteredforanaccuratefusionofcomplementaryinformation. Totacklewiththosedifculties,weintroducedastatisticaldependencemeasureintomulti-modelimageregistration.Anovelvariationalmodelfordeformablemulti-modalimageregistrationisthenproposed.Asanalternativetothemodelsbasedonmaximizingmutualinformation,theRenyisstatisticaldependencemeasureoftworandomvariablesisproposedasameasureofthegoodnessofmatchinginourobjectivefunctional.Theproposedmodeldoesnotrequireanestimationof 9

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thecontinuousjointprobabilitydensityfunction.Instead,itonlyneedsobservedindependentsamples.Moreover,thetheoryofreproducingkernelHilbertspacesisusedtosimplifythecomputation.Experimentalresultswithcomparisonsareprovidedtoshowtheeffectivenessoftheproposedmodel. Finally,weintroducedaregistrationassistednon-parametricsegmentationmodelasanapplicationofRenyisstatisticaldependencemeasure.Thismodelgeneratesalabelimage,whichrepresentsthelocationoftheedgesseparatingthehomogeneousregionsinthegivenimage,byassigningdifferentconstantsoneachofthesub-regions.Registrationisthenappliedtotheimagetobesegmentedandthegeneratedlabelimage.Thisisaccomplishedbymaximizingthesimilaritybetweenthosetwoimages.Asthelabelimageisdeformedfromtheprocessofregistration,thelocationoftheedgesinthelabelimagegraduallymovestothesamelocationoftheedgesinthesegmentingimage.Finally,thebestalignmentofthosetwoimagesprovidesabestsegmentationresult,renderingasuitablechoiceofsimilaritymeasureextremelyimportantforthisproblem.Sincethosetwoimagesarenolongerofthesamemodalities,thedirectcomparisonoftheirintensitiesfails.ByusingRenyi'sstatisticaldependencemeasure,theproposedmodeldealsdirectlywithindependentsamplesanddoesnotneedtoestimatethecontinuousjointprobabilitydensityfunction,whichistimeconsuming.Furthermore,thecomputationisgreatlysimpliedbyusingthetheoryofreproducingkernelHilbertspaces.Experimentalresultsforgivenmedicalandrealimagesareprovidedtodemonstratetheeffectivenessandefciencyoftheproposedmethod. 10

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CHAPTER1INTRODUCTION 1.1ImageRegistration:anOverview 1.1.1BackgroundandMotivation Imageregistrationhasbeenwidelyusedinmedicalimaging,computervision,astrophysics,militarytargetrecognition,remotesensingandsatellitesdataanalyzing.Itisnecessarytoalignimagesinordertobeabletocompareorintegratethedataobtainedfromdifferentmeasurementsoratdifferenttimes. Especiallyinmedicalimaging,twoormoreimagesareusuallyneededtoberegisteredforanaccuratefusionofcomplementaryinformation.Forinstance,magneticresonance(MR)imagesarerealignedwithCTimagesofthesamesubjecttopreciselylocalizethetumortoassistsurgeryplanning.Also,functionalMRbrainimagesareoftenrealignedtohighresolutionT1-weightedanatomicalimagestocorrectdistortionsforaccuratelocalizationofbrainactivationmaps.Thetransformationsthatproducedfromregistrationprocesscanbeappliedinsomeclinicallymeaningfulway,makingtheregistrationbenecialinmedicaldiagnosisortreatment.Amedicalsystemwillcombinethetworegisteredimagesbyproducingareorientedversionofoneimagewhichcanbefusedwiththeother.Thisfusingoftwoimagesintoonemaybeaccomplishedbysimplysummingintensityvaluesintwoimages,byimposingoutlinesfromoneimageoverthegraylevelsoftheother,orbyencodingoneimageinhueandtheotherinbrightnessinacolorimage.Imageregistrationisanecessarystepbeforefusioncanbesuccessful. Duringrecentdecades,abroadrangeofmethods,techniquesandalgorithmshasbeendevelopedforregistrationproblemswithdifferenttypesofdata.Twocomprehensivesurveysofimageregistrationmethodswerepublishedin1992byBrown[ 13 ]andin2003byBarbaraZitovaandJanFlusser[ 77 ].However,growingdemandofhighaccuracyandfastalignmenthasinvokedtheresearchonimageregistrationwithmoreadvancedtechniques. 11

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1.1.2ProblemDenition Imageregistrationistheprocessofestimatinganoptimalgeometrictransformationbetweentwoimagessuchthatoneimageafterthetransformationisbestalignedwiththeother.Theoriginalimagesmaybetakenfromdifferentsensors,atdifferenttimes,orfromdifferentperspectives,butaretakenofthesamesceneorthesameobject. Moreprecisely,giventwoimagesI(x)andJ(x),imageregistrationaimstodetermineaspatialtransformationu(x)insomeadmissiblespaceVsuchthatthefollowingenergyfunctionalisminimized:J(I(x),Jh(x))+R(u), orequivalentlytosolvethefollowingoptimizationproblem:minu(x)2VJ(I(x),Jh(x))+R(u), whereJisdesignedtomeasurethedissimilaritybetweenthereferenceimageI(x)andtheimageJh(x),whichisobtainedbyapplyingtransformationh(x)onthesourceimageJ(x).isaparameterbalancestheregularizationtermandthedissimilarityterm.Thelargeris,themoreweightwillbeonR(u)andthemoresmoothnessthedeformationu(x)willhave. Ifthetransformationispresumablytoberigidorafne,thetermR(u)canremovedfromtheenergyfunctionalastheregularizationisalreadyindicatedbytheparameterswhichdeterminetherigidorafnetransformations. 1.1.3Data Inmedicalimaging,theimagedataforregistrationscanbetwo-dimensional(2D)x-rayprojectionscapturedonlm,adigitalradiograph,aphotographcapturedbyprojectionsofvisiblelights,oravideoframe.Forregistrationofvolumeimages,three-dimensional(3D)images,suchascomputedtomography(CT),magneticresonance(MR)imaging,single-photonemissioncomputedtomography(SPECT), 12

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andpositronemissiontomography(PET),areacquiredbytomographicmodalities.Ineachofthesemodalities,acontiguoussetoftwo-dimensionalslicesprovidesathree-dimensionalarrayofimageintensityvalues. Duringimageprocessing,thoseimagescanbemathematicallyregardedasfunctions,vectors,matricesorsamplesofrandomvariables.Theintensityvaluesmaybesmoothedbysomekernelfunctiontobeusedinopticalowmethods,ormaybebinnedintodifferentintensityintervalstogetprobabilitydensityfunctionestimated. 1.2MethodsandClassication Dependingonthegivendataandmethodsusedforregistration,thereareseveraldifferentwaystoclassifyregistrationmodels. 1.2.1LinearandNonlinearTransformations Accordingtothetransformationusedtorelateoneimagetotheother,imageregistrationalgorithmscanalsobeclassiedintothosedealingwithlineartransformationandthosedealingwithnonlineartransformation.Thelineartransformationincludestranslation,rotation,scaling,andotherafnetransformssuchasshearing.Lineartransformationscannotmodellocalgeometricdifferencesbetweenimagesastheyareglobalinnature. Thenonlineartransformationsallow'elastic'or'nonrigid'transformationsandtheyarecapableoflocallywarpingthetargetimagetoalignwiththereferenceimage.Nonrigidtransformationsincluderadialbasisfunctions(thin-plateorsurfacesplines,compactly-supportedtransformations),physicalcontinuummodels(viscousuids),andlargedeformationmodels(diffeomorphisms). 1.2.2SpatialandFrequencyDomainMethods Spatialmethodsoperateintheimagedomain,matchingintensitypatternsorfeaturesbetweenthereferenceimageandthesourceimage.Similaritymeasures,suchascrosscorrelation,correlationratioandmutualinformation,areusuallyusedinthespatialdomainforimageregistration. 13

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Frequency-domainmethodsrsttransformtheoriginalimagedataintoK-spacedatausingfastFouriertransform.TheregistrationworkisdoneontheK-spacedomainanditaimstondthetransformationparametersbetweentheimagesforregistration.Suchmethodsworkforsimpletransformations,suchastranslation,rotationandscaling.Thephasecorrelationmethodisresilienttonoiseandocclusions.TheuseoffastFouriertransformgreatlyincreasedtheperformanceandreducedthecomputationalcomplexity. 1.2.3SingleandMulti-ModalityMethods Dependingonthegivenimagedatatypeforregistrationandhowthosedatadiffersfromeachother,theregistrationmethodscanalsobeclassiedassingle-modalityandmulti-modalitymethods. Singe-modalityimagesareacquiredusingthesamesensorsbutatdifferenttimesorindifferentviewpoints.Theycanbeassumedeitheridenticalorinlinearrelationafterregistration.Intheformercasedirectintensitycomparisonispossibleandcriteriasuchas(I(x))]TJ /F5 11.955 Tf 12.09 0 Td[(J(x))2candothefairlywelljob.AswewillexplainindetailsinChapterthree,thecorrelationcoefcientcanmeasuresimilarityuptoalinearrelation,thelattercasecanbesolvebythemeasureofcorrelationcoefcientorit'svariants. Generally,imagesofthesamescenebutacquiredbydifferentsensorscannotbecompareddirectlyusingintensityvalues.Andthoseimageshavemuchmorecomplexrelationfortheintensitiesvalues.Theycanbeundernonlinearfunctionalrelationorevenlocallyfunctionalrelation.Multi-modalityregistrationmethodstendedtoregisterimagesacquiredbydifferentsensors.Multi-modalityregistrationmethodsareoftenusedinmedicalimagingasimagesofasubjectarefrequentlyobtainedfromdifferentscanners.ExamplesincluderegistrationofbrainCTandMRIimagesorwholebodyPETandCTimagesfortumorlocalization,registrationofcontrast-enhancedCTimagesagainstnon-contrast-enhancedCTimagesforsegmentationofspecicpartsofthe 14

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anatomy,andregistrationofultrasoundandCTimagesforprostatelocalizationinradiotherapy. Registrationmethodsworkingformulti-modalimagescangenerallybeappliedtosingle-modalimages.However,theconverseisgenerallynottrue.Registrationofmulti-modalimagesisachallengingtaskduetotheunknownrelationofintensityvalues.Butitisoftennecessarytosolve,especiallyinmedicalimaging.Withtheassistanceofmulti-modalimageregistration,thecomparisonofanatomicalandfunctionalimagesofthepatientsbodybecomespossibleandcanleadtoamoreaccuratediagnosis. 1.2.4IntensityBasedandFeatureBased Imageregistrationalgorithmscanbeclassiedintotwomajorcategories:intensity-basedandfeature-based.Intensity-basedmethodsuseintensitiesofpredenedwindowsorevenentireimagesforthecorrespondenceestimationduringregistrationprocess.Thosemethodscompareintensitypatternsinimagesviacorrelationmeasures. Inthemeanwhile,feature-basedmethodsaimstoestablishpairwisecorrespondencebetweenimagefeaturessuchaspoints,lines,andcontoursusingtheirspatialrelationsorvariousdescriptorsoffeatures.Knowingthecorrespondenceofanumberofpointsbetweentwoimages,thecomputerperformstransformationsononeimagetomakemajorfeaturesalignwithasecondimageandatransformationisthenestimatedtoestablishingpoint-by-pointcorrespondencebetweenthosetwoimage.Featuredetectionisalsorequiredbeforeregistrationbythenatureofthistypeofmethods. Intensity-basedmethodsandfeature-basedmethodsmaybeusedcollaboratively.Forinstance,ifsub-imagesareregisteredusingintensity-basedmethods,centersofcorrespondingsub-imagescanbetreatedascorrespondingfeaturepoints. 1.2.5AutomaticandInteractiveMethods Registrationmethodsmaybeclassiedbasedonthelevelofautomationtheyprovide.Manual,interactive,semi-automatic,andautomaticmethodshavebeen 15

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developed.Manualmethodsprovidetoolstoaligntheimagesmanually.Interactivemethodsreduceuserbiasbyperformingcertainkeyoperationsautomaticallywhilestillrelyingontheusertoguidetheregistration.Semi-automaticmethodsperformmoreoftheregistrationstepsautomaticallybutdependontheusertoverifythecorrectnessofaregistration.Automaticmethodsdonotallowanyuserinteractionandperformallregistrationstepsautomatically. Someofthefeaturematchingalgorithmsareoutgrowthsoftraditionaltechniquesforperformingmanualimageregistration,inwhichanoperatorchoosescorrespondingcontrolpoints(CPs)inimages.Whenthenumberofcontrolpointsexceedstheminimumrequiredtodenetheappropriatetransformationmodel,iterativealgorithmslikeRANSACcanbeusedtorobustlyestimatetheparametersofaparticulartransformationtype(e.g.afne)forregistrationoftheimages. 1.3SimilarityCriteria Imagesimilaritymeasuresarebroadlyusedinmedicalimaging.Animagesimilaritymeasurequantiesthedegreeofsimilaritybetweenintensitypatternsintwoimages.Thechoiceofanimagesimilaritymeasuregreatlydependsonthemodalityoftheimagestoberegistered.Amongmanypossiblecriteria,thesumofsquaredifference,thecrosscorrelation,thecorrelationratioandthemutualinformationprovideuswithaconvenienthierarchyaccordingtotherelationtheycanenforcebetweentheintensitiesofthetwoimages. 1.3.1MeasuresforLinearRelation Whenimagesforregistrationarelinearrelated,simplecorrelationmeasuressuchascrosscorrelationwerethenproposedinordertocopewithimageregistration.measuressuchascrosscorrelation(CC)canbeused.Sumofsquareddifferencesiscommonlyusedforregistrationofimagesinthesamemodalityandwhoseintensitiescanbecompareddirectly. 16

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SumofSquaredDifference:Thissimilaritymeasurecanbeappliedonimageswhoseintensitiescanbecompareddirectly.Itisthemostconstrainedoftheallcriteria.Tokeepthiswork,thereferenceimageandthesourceimageshouldhavethesamerangeofintensityvalues. CrossCorrelation:Ithasbeenwidelyusedasarobustcomparisonfunctionforimagematching.Beingameasureofthelineardependencybetweenimageintensities,itsdenitionreliesonthemeans,variancesandcovariancesgivenimages.WerefertotheworksdonebyFaugerasandKeriven(1998)[ 28 ],CachierandPennec(2000)[ 16 ],Netschetal.(2001)[ 51 ],andG.Hermosilloetal.(2002)[ 34 ] 1.3.2MeasuresforNonlinearRelation Althoughsimilaritymeasuressuchassumofsquareddifferenceandcrosscorrelationhavebeenusedextensivelyinmedicalimaging,theybasicallyassumealinearrelationshipbetweentheimageintensities.Suchahypothesisisgenerallytoocrudeinmulti-modalregistrations.Measureswhichcanreectsnonlinearrelationofimageintensitiesneedtobeemployedformulti-modalimageregistrations. CorrelationRatio:Correlationratioisameasureoftherelationshipbetweenthestatisticaldispersionwithinindividualcategoriesandthedispersionacrossthewholesamples.Themeasureisdenedastheratiooftwostandarddeviationsrepresentingthesetypesofvariation.RelatedworksaredonebyAlexisRocheetal.(1998)[ 58 ] MutualInformation:Mutualinformationisaconceptborrowedfrominformationtheory.Itispositiveandsymmetric,andmeasurestheamountofinformationthatthetwoimagesimagesshareorhowtheintensitydistributionsoftwoimagesfailtobeindependent.GiventworandomvariablesXandY,theirmutualinformationisdenedasMI(X,Y)=H(X)+H(Y))]TJ /F5 11.955 Tf 11.95 0 Td[(H(X,Y) whereHisShannonEntropy.ItcanalsobedenedintermsofthejointpdfpX,Y(x,y)anditsmarginalspdfpX(x)andpY(x).MI(X,Y)=ZpX,Y(x,y)logpX,Y(x,y) pX(x)pY(x) 1.4Applications Duetothevastapplicationstowhichimageregistrationcanbeapplied,itisimpossibletodevelopageneralmethodthatisoptimizedforalluses.Hence,the 17

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registrationmodelalongwithitstunedparametersisusuallysetupforaspecicpurposeofuse. 1.4.1ApplicationsinOtherFields Typically,registrationisrequiredinremotesensingforenvironmentalmonitoring,changedetection,imagemosaicing,weatherforecasting,creatingsuper-resolutionimagesandintegratinginformationintogeographicinformationsystems(GIS).Inmedicineitisusedforcombiningcomputertomography(CT)andMRdatatoobtainmorecompleteinformationaboutthepatient,monitoringtumorgrowth,treatmentverication,comparisonofthepatientsdatawithanatomicalatlases,incartographyitisusedformapupdating,andincomputervisionitisusedfortargetlocalization,automaticqualitycontrol. 1.4.2AssistancetoOtherProceduresinImagingScience Realimageshavemorecomplexstructuresthansyntheticones,whichareonlyusedformodeltestingandparametertuning.Toachievebetterresults,allpossiblescenariosintherealimagesneedtobeconsideredinordertofurtherimprovetheperformanceofaproposedmodel.Hybridmodelsareproposedduringrecentdecadesandtheyhavebeenextensivelyappliedtomedicalimagingandremotesensing.Forexample,imageregistrationisemployedasassistancetoatlas-basedsegmentation[ 65 ].Theatlasisagivenground-truthsegmentationandcanbeconstructedinvariousways,oneofwhichisthatexpertsincorporatepriorinformationabouttheanatomicalstructureofinterestintotheatlasanduseitasatemplate.Thetemplateisthenregisterednonrigidlytotheimagebeingsegmentedtoachievethedesiredsegmentation. 1.5OrganizationofThisThesis InChaptertwo,werstintroducedregistrationmodelsforrigidtransformationsandafnetransformations.Then,abriefdescriptionofthesimilaritymeasuresisprovided,followedbydetailsonmeasureswhichcanbeusedforrigidandafneregistration. 18

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Wederivemathematicalmodelsinboth2Dand3Dcases.Thenthemethodofphasecorrelationisintroducedasatypicalmethodbasedonfrequencydomain. AcomprehensiveinvestigationofsimilaritymeasuresisgiveninChapterthreeandtypicaldeformablemodelsareintroducedafterwards.ChapterfourisdedicatedtointroducinganewregistrationmodelusingRenyi'sstatisticaldependencemeasure.Experimentalresultsarepresentedtoshowtheefciencyandeffectivenessofthismodel.Finally,aregistrationassistednon-parametricsegmentationmodelisintroducedinChapterve,asanexampleofhowregistrationassistsinotherproceduresinmedicalimaging. 19

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CHAPTER2PARAMETRICIMAGEREGISTRATIONS Generally,thegeometrictransformationsinimageregistrationcanbeclassiedastheparametricandthenonparametric.Theaccordingparametricregistrationmodelaimstondalltheoptimalparameterswhichdenethetransformation,whilethenonparametricmodeltriestodeterminedensedeformationeldwhichisinanonparametricfashion.Therstapproachemploysasmallnumberofparameterstodenethewarp,whereasthesecondoneusesamotiondisplacementateachpixellocation. 2.1ParametricGeometricTransformations Parametricgeometrictransformationsincludetranslations,rotations,rigidtransformation,afnetransformation,projectivetransformations,justtonameafew.Thosetransformationsarerelativelyeasiertodeterminecomparedtothoseinnonparametricapproaches,whereadensedeformationeldisinvolved.Inthissection,threetypicaltransformationswillbediscussed:rigid,afneandprojective. 2.1.1RigidTransformations Rigidtransformationsaredenedasgeometricaltransformationsthatpreservedistancesforeverypairpoints.Directlyfollowedbydenition,rigidtransformationsalsopreservethestraightnessoflinesandallnonzeroanglesbetweenstraightlines.Itincluderotations,translations,reections,ortheircombinations.SometimesreectionsareexcludedfromthedenitionofarigidtransformationbyimposingthatthetransformationalsopreservethehandednessofguresintheEuclideanspace(areectionwouldnotpreservehandedness). Rigidregistrationproblemsareregistrationproblemslimitedtorigidtransformations.Sinceanyrigidtransformationcanbedecomposedasarotationfollowedbyatranslation,rigidregistrationcanbespeciedastwocomponents:atranslationandarotation.Letusconsiderrigidtransformationsinthethreedimensionalcaseasthose 20

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intwodimensionalcaseareeasierandcanfollowdirectlyfromthethreedimensionalcase.LetRbetherotationmatrixandlettbethetranslationvector,thentherigidtransformationcanberepresentedbyy=Rx+t. TherotationRcanbefurtherparameterizedintermsofthreeanglesofrotationabouttherespectiveCartesianaxes:x,y,z.Andwiththisnotation,Risaproductofthreerotationsaboutthex,y,andzaxes:R=0BBBB@cosz)]TJ /F3 11.955 Tf 11.29 0 Td[(sinz0sinzcosz00011CCCCA0BBBB@cosy0siny010)]TJ /F3 11.955 Tf 11.29 0 Td[(siny0cosy1CCCCA0BBBB@1000cosx)]TJ /F3 11.955 Tf 11.29 0 Td[(sinx0sinxcosx1CCCCA=0BBBB@cosycosz)]TJ /F3 11.955 Tf 11.29 0 Td[(cosxsinz+sinxsinycoszsinxsinz+cosxsinycoszcosysinzcoszcosz+sinzsinysinz)]TJ /F3 11.955 Tf 11.29 0 Td[(sinxcosz+cosxsinysinz)]TJ /F3 11.955 Tf 11.29 0 Td[(sinysinxcosycosxcosy1CCCCA Thetranslationtcanberepresentedeasilybythreecomponentstx,tyandtz:t=0BBBB@txtytz1CCCCA ThenregistrationworkcanbedonebysolvingthefollowingoptimizationproblemwithasuitablechoiceofdissimilaritymeasureJ:minx,y,z,tx,ty,tzJ(I(x),J(Rx+t)) 2.1.2AfneTransformations Afnetransformationsaretransformationspreservingstraightlinesandratiosofdistancesbetweenpointslyingonastraightline.Andhence,itpreservestheplanarityofsurfacesandparallelism,butitallowsanglesbetweenlinestochange. 21

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Afnetransformationisanappropriatetransformationclasswhentheimagemayhavebeenskewedduringacquisition.Forexample,whentheCTgantryangleisincorrectlyrecorded. Allrigidtransformationsareafnetransformationsandalllineartransformationsareafnetransformations.Rigidtransformationswhichinvolveatranslationarenotlineartransformations. Afnetransformationsconsistsofatranslationandalinearmap.IfthelinearmapisrepresentedasamultiplicationbyamatrixAandthetranslationastheadditionofavectorb,anafnetransformationactingonavectorxcanberepresentedbyy=Ax+b, inwhichthereisnorestrictionontheelementsaijofthematrixA. Usinganaugmentedmatrixandanaugmentedvector,itispossibletorepresentboththetranslationandthelinearmapusingasinglematrixmultiplication:0B@y11CA=0B@Ab0,,011CA0B@x11CA. TherelatedoptimizationproblemforregistrationwithasuitablechoiceofdissimilaritymeasureJ:minaij,b1,b2,b3J(I(x),J(Ax+b)) ismoredifculttosolvethantheoneintherigidcase.TheincreaseoffreedomsinmatrixAsignicantlyincreasethedifcultiesinsolvingtheproblem. 2.2ImageRegistrationbyUsingPhaseCorrelation Inthissection,wewillinvestigatethephasecorrelationapproach,aregistrationmethodbasedonfrequencydomain.Byapplyingthephasecorrelationmethodtoapairofimages,weobtainathirdimagewhichcontainsasinglepeak.Thelocationofthispeakcorrespondstotherelativetranslationbetweenthosetwoimages.Unlike 22

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manyspatial-domainalgorithms,thephasecorrelationmethodisresilienttonoise,occlusions,andotherdefectstypicalofmedicalorsatelliteimages.Additionally,thephasecorrelationusesthefastFouriertransformtocomputethecross-correlationbetweenthetwoimages,generallyresultinginlargeperformancegains. Themethodisoriginallydesignedtodeterminetranslations,butcanbeextendedtodeterminerotationandscalingdifferencesbetweentwoimagesbyrstconvertingtheimagestolog-polarcoordinates.Furthermore,therotationandscalingparameterscanbedeterminedinamannerinvarianttotranslation,duetopropertiesoftheFouriertransform. 2.2.1Rationale ForapairofgivenimagesI0(x)andJ0(x)ofsizeMN,repeatedlyextendthemintoimagesI(x)andJ(x)onthedomainZZ.SupposeJ(x)isobtainedbyapplyingtranslation(x,y)onimageI(x):J(x,y)=I((x)]TJ /F3 11.955 Tf 11.95 0 Td[(x)modM,(y)]TJ /F3 11.955 Tf 11.96 0 Td[(y)modN) AndsupposeI(x,y)andJ(x,y)arestillofsizeMN.ThenthediscreteFouriertransformFgivesFI(u,v)=X1xM1yNI(x,y)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(2i((x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1) M+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1) N) andFJ(u,v)=X1xM1yNI(x)]TJ /F3 11.955 Tf 11.96 0 Td[(x,y)]TJ /F3 11.955 Tf 11.96 0 Td[(y)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(2i((x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1) M+(y)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.96 0 Td[(1) N). AfurthercomputationofFJ(u,v)givethefollowingrelationbetweenFI(u,v)andFJ(u,v): 23

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FJ(u,v)=X1xM1yNI(x)]TJ /F3 11.955 Tf 11.95 0 Td[(x,y)]TJ /F3 11.955 Tf 11.96 0 Td[(y)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(2i((x)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1) M+(y)]TJ /F3 11.955 Tf 11.96 0 Td[(y)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.96 0 Td[(1) N)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(2i(x(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1) M+y(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1) N)=FI(u,v)exp)]TJ /F3 11.955 Tf 9.3 0 Td[(2i(x(u)]TJ /F3 11.955 Tf 11.96 0 Td[(1) M+y(v)]TJ /F3 11.955 Tf 11.96 0 Td[(1) N). NowletR(u,v)=FI(u,v)FJ(u,v) jFI(u,v)FJ(u,v)j=FI(u,v)FI(u,v)exp2i(x(u)]TJ /F9 7.97 Tf 6.59 0 Td[(1) M+y(v)]TJ /F9 7.97 Tf 6.58 0 Td[(1) N) jFI(u,v)FJ(u,v)exp2i(x(u)]TJ /F9 7.97 Tf 6.58 0 Td[(1) M+y(v)]TJ /F9 7.97 Tf 6.59 0 Td[(1) N)j=exp2i(x(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1) M+y(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1) N). WehaveF)]TJ /F9 7.97 Tf 6.59 0 Td[(1R(x,y)=1 MNX1uM1vNexp2i(x(u)]TJ /F3 11.955 Tf 11.96 0 Td[(1) M+y(v)]TJ /F3 11.955 Tf 11.96 0 Td[(1) N)exp2i((x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1) M+(y)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.96 0 Td[(1) N)=1 MNX1uM1vNexp2i((x+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.96 0 Td[(1) M+(y+y)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1) N)=1 MNX1uMexp2i(x+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.96 0 Td[(1) MX1vMexp2i(y+y)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1) N Notethatifx+x)]TJ /F3 11.955 Tf 11.96 0 Td[(1=0,thenX1uMexp2i(x+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(u)]TJ /F3 11.955 Tf 11.96 0 Td[(1) M=M. 24

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Nowif1><>>:Mifx+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1=00ifx+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1isanintegerand1x+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1M)]TJ /F3 11.955 Tf 11.95 0 Td[(1. 25

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Similarly,wehave,X1vNexp2i(y+y)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1) N=8>><>>:Nify+y)]TJ /F3 11.955 Tf 11.95 0 Td[(1=00ify+y)]TJ /F3 11.955 Tf 11.95 0 Td[(1isanintegerand1y+y)]TJ /F3 11.955 Tf 11.96 0 Td[(1N)]TJ /F3 11.955 Tf 11.95 0 Td[(1. Hence,F)]TJ /F9 7.97 Tf 6.59 0 Td[(1R(x,y)=8>>>>>><>>>>>>:1ifx+x)]TJ /F3 11.955 Tf 11.96 0 Td[(1=0andy+y)]TJ /F3 11.955 Tf 11.95 0 Td[(1=0,0ifx+x)]TJ /F3 11.955 Tf 11.96 0 Td[(1isanintegersatisfying1x+x)]TJ /F3 11.955 Tf 11.95 0 Td[(1M)]TJ /F3 11.955 Tf 11.96 0 Td[(1ory+y)]TJ /F3 11.955 Tf 11.96 0 Td[(1isanintegersatisfying1y+y)]TJ /F3 11.955 Tf 11.95 0 Td[(1N)]TJ /F3 11.955 Tf 11.95 0 Td[(1. NowwecanviewF)]TJ /F9 7.97 Tf 6.58 0 Td[(1R(x,y)asafunctionofxandyandregardxandyasparameterstobedetermined.SinceF)]TJ /F9 7.97 Tf 6.59 0 Td[(1R(x,y)achievesitsmaximumvalueatx+x)]TJ /F3 11.955 Tf 11.96 0 Td[(1=0andy+y)]TJ /F3 11.955 Tf 11.95 0 Td[(1=0, wecanrstdeterminethelocationwhereF)]TJ /F9 7.97 Tf 6.59 0 Td[(1R(x,y)achievesitsmaximumvalueandthenplugintotheaboveequationstogetxandy. Inpractice,itismorelikelythattheimageJ(x)willbeasimplelineartranslationofI(x)ratherthanacircularshiftasstatedatthebeginningofthissection.Insuchcases,awindowfunctionsuchasGaussianwindowshouldbeemployedtoreducetheedgeeffects.Ortheimagesshouldbezeropaddedsothattheedgeeffectscanbeignored.Iftheimagesconsistofaatbackground,withalldetailsituatedawayfromtheedges,thenalinearshiftwillbeequivalenttoacircularshift,andtheabovederivationwillholdexactly.Thepeakcanbesharpenedbyusingedgeorvectorcorrelation. 26

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Onedifferencebetweenphasecorrelationmethodsandthevariationalregistrationmethodsisthatafterapplyingphasecorrelation,thefunctionalforvariationalmethodsbecomesafunction,whichiseasiertodetermineitsmaximum. 2.2.2ExperimentsandApplications Thefollowingexperimentexaminesthephasecorrelationapproachforregistrationoftwoimageswithtranslationsonly.Theoriginalrelativetranslationis(45.3562,30.5610)forpixelswheretherstentryisforhorizontalmotionandthesecondoneisforverticalmotion.Therecoveredmotionis(45.3482,30.5827)with(0.0080,)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0217)pixelerror. (a)(b)(c) Figure2-1. MotionCorrectionbyUsingPhaseCorrelation:(a)and(b).imageclownwithrelativemotion(45.3562,30.5610).(c).imageforF)]TJ /F9 7.97 Tf 6.59 0 Td[(1R(x,y) 27

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CHAPTER3DEFORMABLEIMAGEREGISTRATION Inanonparametricapproacheachimagepixelistransformedindependently.Onepopulartechniquetoimposesomeregularizationonthisformulationemploysaglobalobjectivefunctionthatconsistsoftwoterms:thealignmentmeasureandanexternalregularizationtermthatreectsourexpectationsbypenalizingunlikelytransformations. Medicalimageregistration(fordataofthesamepatienttakenatdifferentpointsintimesuchaschangedetectionortumormonitoring)oftenadditionallyinvolveselastic(alsoknownasnonrigid)registrationtocopewithdeformationofthesubject(duetobreathing,anatomicalchanges,andsoforth).Nonrigidregistrationofmedicalimagescanalsobeusedtoregisterapatient'sdatatoananatomicalatlas,suchastheTalairachatlasforneuroimaging. Deformableimageregistrationallowsmoreexibilityinthetypesofimagesandapplicationsinwhichitcanbeusedthanrigidorafneregistrations.Manycurrentworksinmedicalimageregistrationsarefocusedondeformableregistrationinwhichthetransformationbetweentheimagescontainslocalizednon-rigidwarps.However,deformableregistrationtechniqueshasmorecomputationalcomplexityandrequiresignicantlymoretimethanrigidorafneregistrationtechniques.Besides,averylargenumberofparametersmayneedtobedetermined. 3.1EstimationofProbabilityDensityFunction Asstatisticalsimilaritymeasuresorinformation-theoreticapproachesforimageregistrationinvolveprobabilitydensityfunctions(PDFs).Theestimationofprobabilitydensityfunctionsforapairofgivenimagesisusuallytherststep,wheretheimagesforregistrationsareregardedassamplesofrandomvariables. OnewaytoestimatethejointprobabilitydensityfunctionsistouseGaussiankernel:G(x,y)=1 22e)]TJ /F9 7.97 Tf 6.59 0 Td[((x2+y2)=2 28

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whereisthevariance.ThenthethejointprobabilitydensityfunctionspX,Y(i,j)isestimatedas:pX,Y(x,y)=1 jjZG(X(x))]TJ /F5 11.955 Tf 11.95 0 Td[(i,Y(x))]TJ /F5 11.955 Tf 11.96 0 Td[(j)dx wherexisthepixelpositionforimagesXandY. ThecorrespondingmarginalprobabilitydensityfunctionspX(i)andpY(j)canbeestimatedbypX(i)=ZRpX,Y(i,j)dj andpY(j)=ZRpX,Y(i,j)di. Asintherealmedicalimages,especiallyinthebrainimages,relationbetweenintensitiesbecomesmuchmorecomplexthanexpected,andabetterwaytoestimatethejointprobabilitydensityfunctionsistogeneralizeoftheaboveestimatortothelocalcase.ThiscanbedonebyestablishedanestimatepX,Y(i,j,x0)inthetheneighborhoodofeachpointx0in. pX,Y(i,j,x0)=1 W(x0)ZG(X(x))]TJ /F5 11.955 Tf 11.96 0 Td[(i,Y(x))]TJ /F5 11.955 Tf 11.96 0 Td[(j)G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)dx whereW(x0)=ZG(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)dx 3.2CrossCorrelationBasedImageRegistration GiventheestimatedjointprobabilitydensityfunctionpX,Y(i,j)andmarginalprobabilitydensityfunctionspX(i)andpY(j),thecrosscorrelationarereadytobedenedandthedenitionreliesonthemeansX=ZRipX(i)di,Y=ZRjpY(j)dj, 29

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variancesX=ZRi2pX(i)di)]TJ /F7 11.955 Tf 11.95 0 Td[(X2,Y=ZRj2pY(j)dj)]TJ /F7 11.955 Tf 11.95 0 Td[(Y2, andcovarianceofXandYX,Y=ZR2ijpX,Y(i,j)didj)]TJ /F7 11.955 Tf 11.96 0 Td[(XY. ThefollowinggivestheformaldenitionofcrosscorrelationCC(X,Y)ofXandYisdenedasCC(X,Y)=1,2 p 12 NowifweviewtheintensityvaluesofimageI(x)andimageJ(x)asidenticalindependentdistributed(i.i.d.)samplesofrandomvariableXandrandomvariableY,respectively.Thenaccordingthemeans,variancesandcovarianceofXandYcanbeestimatedbyX=1 jjXxI(x),Y=1 jjXxJ(x), X=1 jjXx(I(x))]TJ /F7 11.955 Tf 11.95 0 Td[(X)2,Y=1 jjXx(J(x))]TJ /F7 11.955 Tf 11.95 0 Td[(Y)2, andX,Y=1 jjXx(I(x))]TJ /F7 11.955 Tf 11.96 0 Td[(X)(J(x))]TJ /F7 11.955 Tf 11.96 0 Td[(Y), whereisthedomainforimageI(x)andJ(x)andjjrepresentsthetotalpixelnumberinthedomain.ThenthecrosscorrelationCC(X,Y)canbeestimatedbyCC(X,Y)=Px(I(x))]TJ /F7 11.955 Tf 11.95 0 Td[(X)(J(x))]TJ /F7 11.955 Tf 11.95 0 Td[(Y) p Px(I(x))]TJ /F7 11.955 Tf 11.95 0 Td[(X)2Px(J(x))]TJ /F7 11.955 Tf 11.95 0 Td[(Y)2. Ontheotherhand,wecanviewI(x))]TJ /F7 11.955 Tf 12.44 0 Td[(XandJ(x))]TJ /F7 11.955 Tf 12.44 0 Td[(YasvectorsinthespaceRjj.ThentheaboveCC(X,Y)denetheanglebetweenthevectorI(x))]TJ /F7 11.955 Tf 12.3 0 Td[(XandthevectorJ(x))]TJ /F7 11.955 Tf 11.95 0 Td[(Y.BytheCauchyCSchwarzinequality,)]TJ /F3 11.955 Tf 9.3 0 Td[(1CC(X,Y)1 30

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andtheequalityholdsifandonlyifthereexistsaconstantk2RsuchthatI(x))]TJ /F7 11.955 Tf 11.96 0 Td[(X=k(J(x))]TJ /F7 11.955 Tf 11.96 0 Td[(Y)orI(x)=kJ(x)+(X)]TJ /F5 11.955 Tf 11.96 0 Td[(kY) orJ(x))]TJ /F7 11.955 Tf 11.95 0 Td[(Y=k(I(x))]TJ /F7 11.955 Tf 11.96 0 Td[(X)orJ(x)=kI(x)+(Y)]TJ /F5 11.955 Tf 11.96 0 Td[(kX), bothofwhichindicatesalinearrelationofI(x)andJ(x). Hence,ifweapplythedeformationeldu(x)onimageJ(x),namely,Ju(x)andviewJu(x)assamplesofrandomvariableY,thenmaximizingCC(X,Y)to1orminimizingCC(X,Y)to-1canenforcethelinearrelationonimageI(x)andJu(x).Asaresult,thedissimilaritymeasureusuallychoose)]TJ /F5 11.955 Tf 9.29 0 Td[(CC(X,Y)2asapartofenergyfunctiontobeminimized. Crosscorrelationhasbeenwidelyusedasarobustsimilaritymeasureforimagematching.However,aswehaveseenabove,registrationbyusingcrosscorrelationcanonlyworkonimageswhoseintensityvaluesarelinearlyrelated.Whenitcomestomulti-modalimages,crosscorrelationfails. 3.3CorrelationRatioBasedImageRegistration ForrandomvariablesXandY,theconditionalexpectationE[YjX]andconditionalvarianceVar[YjX]arebothrandomvariables.ForE[YjX]:P(E[YjX]=E[YjX=x])=P(X=x) whereE[YjX=x]=XyyP(Y=yjX=x). Thefollowingequationcalledlawoftotalexpectation,whichwillbeusedfortheproofoflawoftotalvariance.E[E[YjX]]=E[Y] 31

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Thelawoftotalexpectationcanbesimplyprovedasfollowing:E[E[YjX]]=E[XyyP(Y=yjX)]=Xx XyyP(Y=yjX=x)!P(X=x)=XyyXxP(Y=yjX=x)P(X=x)=XyXxP(X=x,Y=y)=XyyP(Y=y)=E[Y]. Byusingthelawoftotalexpectation,lawoftotalvarianceVar(Y)=E[Var(YjX)]+Var(E[YjX]) canbeprovedasVar(Y)=E[Y2])]TJ /F5 11.955 Tf 11.96 0 Td[(E2[Y]=E[E[Y2jX]])]TJ /F5 11.955 Tf 11.96 0 Td[(E2[E[YjX]]=E[Var(YjX)+E2[YjX]])]TJ /F5 11.955 Tf 11.95 0 Td[(E2[E[YjX]]=E[Var(YjX)]+E[E2[YjX]])]TJ /F5 11.955 Tf 11.96 0 Td[(E2[E[YjX]]=E[Var(YjX)]+Var(E[YjX]) ThecorrelationratioforrandomvariablesXandYisdenedasCR(X,Y)=Var(E[YjX]) Var(Y) BythelawoftotalvarianceVar(Y)=E[Var(YjX)]+Var(E[YjX]), 32

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WecanrewritethecorrelationratioasCR(X,Y)=Var(E[YjX]) Var(Y)=1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(E[Var(YjX)] Var(Y). ThefollowinggiveshowrandomvariableXisrelatedtoYinthetwoextremecases:CR(X,Y)=1andCR(X,Y)=0. IfCR(X,Y)=1,E[Var(YjX)]=0,thenVar(YjX=x)=0.ItfollowsthatthereisnodispersionforYwithgivenvalueofX,whichmeansYcanbepredictedforsureifXisknown,or,YisafunctionofXandYisfunctionaldependentonX. IfCR(X,Y)=0,Var(E[YjX])=0,whichmeansE[YjX=x]isaconstantandthereisnodispersionamongmeansE[YjX=x]foreachgivenx.Hence,CR(X,Y)=0doesnotindicateanyindependence. SinceCR(X,Y)=1indicatesthatYisfunctionalrelatedtoX:Y=(X)forsomemeasurablefunction, Wecanregisterimageswithintensityvalueswhichisnotlinearlyrelatedbutfunctionalrelated.Hence,bymaximizingthecorrelationratiowecandoregistrationformulti-modalimages. 3.4InformationTheoreticApproaches 3.4.1Shanon'sEntropy ShannonstatesthatameasureoftheamountofinformationH(X)foradiscreterandomvariableXwithpossiblevaluesx1,x2,,xNandprobabilitymassfunctionp(x1),p(x2),,p(xN)shouldsatisfythefollowingthreeconditions[ 61 ]: 1).Hshouldbecontinuousineachp(xi) 2).Ifallthep(xi)areequal,p(xi)=1 N,theHshouldbeamonotonicincreasingfunctionofN. 3).Hshouldbeadditive.Theamountofentropyshouldbeindependentofhowtheprocessisregardedasbeingdividedintoparts.Thislastfunctionalrelationshipcharacterizestheentropyofasystemwithsub-systems.Itdemandsthattheentropy 33

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ofasystemcanbecalculatedfromtheentropiesofitssub-systemsiftheinteractionsbetweenthesub-systemsareknown. Givenanensembleofnuniformlydistributedelementsthataredividedintokboxes(sub-systems)withb1,...,bkelementseach,theentropyofthewholeensembleshouldbeequaltothesumoftheentropyofthesystemofboxesandtheindividualentropiesoftheboxes,eachweightedwiththeprobabilityofbeinginthatparticularbox. Hn1 n,,1 n=Hk(b1 n,,bk n+kXi=1Hbi1 bi,,1 bi Hisafunctionalofprobabilitymassfunction,notafunctionofarandomvariable.ButwecansayHisameasureoftherandomvariable. Anydenitionofentropysatisfyingalltheaboveassumptionshastheform )]TJ /F5 11.955 Tf 9.3 0 Td[(KXip(xi)logp(xi) whereK>0.Inthefollowingcontext,Ktakesvalue1andShannon'sentropyforrandomvariableXwithprobabilitymassfunctionpX(xi)isdenedasH(X)=)]TJ /F11 11.955 Tf 11.29 11.36 Td[(XipX(xi)logpX(xi). AsaresultthejointentropyforrandomvariableXandYwithjointprobabilitymassfunctionpX,Y(xi,yj),respectively,isthendenedasH(X,Y)=)]TJ /F11 11.955 Tf 11.29 11.36 Td[(Xi,jpX,Y(xi,yj)logpX,Y(xi,yj). Consequently,theconditionalentropyofXgiveYcanbedenedas:H(XjY)=H(X,Y))]TJ /F5 11.955 Tf 11.95 0 Td[(H(Y) whichmeasurestheamountofinformationleftinXifsubtractedbytheamountofinformationXandYshares.FromthedenitionofH(X,Y)andH(Y),wecaneasily 34

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gettheexpressionofH(XjY)giventhejointprobabilitymassfunctionpX,Y(xi,yj)andtheprobabilitymassfunctionpY(yj)ofY:H(XjY)=)]TJ /F11 11.955 Tf 11.3 11.36 Td[(Xi,jpX,Y(xi,yj)logpX,Y(xi,yj)+XipY(yi)logpY(yi)=)]TJ /F11 11.955 Tf 11.3 11.36 Td[(Xi,jpX,Y(xi,yj)logpX,Y(xi,yj)+Xi,jpX,Y(xi,yi)logpY(yi)=Xi,jpX,Y(xi,yj)logpY(yi) pX,Y(xi,yj) Similarly,wecangetH(YjX)=)]TJ /F11 11.955 Tf 11.29 11.36 Td[(Xi,jpX,Y(xi,yj)logpX,Y(xi,yj)+XipX(xi)logpX(xi)=)]TJ /F11 11.955 Tf 11.29 11.35 Td[(Xi,jpX,Y(xi,yj)logpX,Y(xi,yj)+Xi,jpX,Y(xi,yi)logpX(xi)=Xi,jpX,Y(xi,yj)logpX(xi) pX,Y(xi,yj) 3.4.2MutualInformation Mutualinformationisaninformation-theoreticapproachtoimageregistrationthatwasproposedindependentlybyViolaandWells[ 68 ]andCollignonetal.[ 23 ]in1995.Foranoverviewofmutualinformation-basedimageregistration,see[ 54 ]. ThemutualinformationMI(X,Y)oftworandomvariablesXandYmeasuresthecommoninformationthatXandYshareormeasuresthequantityofthemutualdependenceofthetworandomvariables.Itcanbedenedas:MI(X,Y)=H(X)+H(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(H(X,Y)=H(X))]TJ /F5 11.955 Tf 11.95 0 Td[(H(XjY)=H(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(H(YjX)=H(X,Y))]TJ /F5 11.955 Tf 11.96 0 Td[(H(XjY))]TJ /F5 11.955 Tf 11.96 0 Td[(H(YjX) 35

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whereH(X)andH(Y)areentropiesofXandY,respectively,andH(X,Y)isthejointentropyofXandY.ThemutualinformationMI(X,Y)canberewrittenasMI(X,Y)=Xi,jpX,Y(xi,yj)logpX,Y(xi,yj) pX(xi)pY(yi) TheVenndiagramFig 3.4.2 depictstherelationofShannonentropyandmutualinformation.TheleftdiscrepresentstheentropyofXandtherightdiscrepresentstheentropyofY.Themutualinformationistheoverlappedareaofthetwodiscs. Figure3-1. VenndiagramforShannonentropyandmutualinformation.TheleftdiscrepresentstheentropyofXandtherightdiscrepresentstheentropyofY.Themutualinformationistheoverlappedareaofthetwodiscs. PropertiesofMutualInformation(notethatentropymaynotbepositiveinthecontinuouscasebutmutualinformationisalwayspositive):FirstwenotethatMI(X,Y)0because)]TJ /F3 11.955 Tf 11.29 0 Td[(logisaconvexfunctionandMI(X,Y)=Xi,jpX,Y(xi,yj)logpX,Y(xi,yj) pX(xi)pY(yj)=Xi,jpX,Y(xi,yj))]TJ /F3 11.955 Tf 11.29 0 Td[(logpX(xi)pY(yj) pX,Y(xi,yj))]TJ /F3 11.955 Tf 30.55 0 Td[(log Xi,jpX,Y(xi,yj)pX(xi)pY(yj) pX,Y(xi,yj)!=)]TJ /F3 11.955 Tf 11.29 0 Td[(log Xi,jpX(xi)pY(yj)!=0 36

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IfMI(X,Y)=0,theaboveequalitymusthold.Since)]TJ /F3 11.955 Tf 11.29 0 Td[(logisastrictconvexfunction,pX(xi)pY(yj) pX,Y(xi,yj)=C,forsomeconstantC. Hence,pX(xi)pY(yj)=CpX,Y(xi,yj),foranyiandj. Sumonbothsidesoveralliandj,wehaveXi,jpX(xi)pY(yj)=CXi,jpX,Y(xi,yj), whichleadstoC=1. Hence,pX,Y(xi,yj)=pX(xi)pY(yj),foranyiandj, whichmeanstherandomvariableXandYareindependent. SincepY(yj)pX,Y(xi,yj),wehaveH(XjY)=Xi,jpX,Y(xi,yj)logpY(yj) pX,Y(xi,yj)0 ItfollowsthatMI(X,Y)=H(X))]TJ /F5 11.955 Tf 11.96 0 Td[(H(XjY)H(X). Similarly,wehaveMI(X,Y)=H(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(H(YjX)H(Y). Hence,MI(X,Y)minfH(X),H(Y)g. Now,wewillseehowthetworandomvariablesrelatedifthemutualinformationMI(X,Y)reachesitsmaximumvalue,eitherH(x)orH(Y).TakingH(x)forexample,we 37

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haveH(XjY)=Xi,jpX,Y(xi,yj)logpY(yj) pX,Y(xi,yj)=0 Sinceeachtermintheabovesummationisnonnegative,pX,Y(xi,yj)logpY(yj) pX,Y(xi,yj)=0foranyxediandj. ThismeaneitherpX,Y(xi,yj)=0 orpY(yj)=pX,Y(xi,yj) TherstequationindicatesatrivialcasewherethejointeventX=xi,Y=yjwillneverhappen.ThesecondequationisequivalenttoP(X=xijY=yj)=pX,Y(xi,yj) pY(yj)=1. whichmeanstherandomvariableXistotallypredictablegiventherandomvariableY.ThiscanalsobeseenfromtheVenndiagram 3.4.2 Figure3-2. VennDiagramwhenMutualInformationMI(X,Y)ReachesitsMaximumLeft:MI(X,Y)=H(X).Right:MI(X,Y)=H(Y) 38

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CHAPTER4DEFORMABLEMULTI-MODALIMAGEREGISTRATIONBYMAXIMIZINGRENYI'SSTATISTICALDEPENDENCEMEASUREOutline TheworkpresentedinthischapteriscompletedasaresultofcollaborationwithYunmeiChen,MuraliRaoandJin-SeopLee. Anovelvariationalmodelfordeformablemulti-modalimageregistrationispresentedinthischapter.Asanalternativetothemodelsbasedonmaximizingmutualinformation,theRenyi'sstatisticaldependencemeasureoftworandomvariablesisproposedasameasureofthegoodnessofmatchinginourobjectivefunctional.Theproposedmodeldoesnotrequireanestimationofthecontinuousjointprobabilitydensityfunction.Instead,itonlyneedsobservedindependentsamples.Moreover,thetheoryofreproducingkernelHilbertspacesisusedtosimplifythecomputation.Experimentalresultswithcomparisonsareprovidedtoshowtheeffectivenessofthemodel. 4.1Introduction Imageregistrationisafundamentalproblemincomputervisionanddataanalysis.Ithasbeenincreasinglyappliedtomedicalimageanalysistoassistindiagnosisandtreatment.Whentheintensitiesoftwoimagesarelinearlyrelated,theycanberealignedbydirectcomparisonofthedata,suchasminimizingtheirdifferenceunderanoptimizedlineartransform,ormaximizingtheircrosscorrelation(CC).However,inmanyapplicationsweencountertheproblemofmatchingsimilarstructuresbetweenseveralimagemodalities,forwhichintensitycomparisonisnotpossible.Inparticular,intheapplicationsofmedicalimagingimageswithdifferentmodalitiesoftenneedtoberegisteredforanaccuratefusionofcomplementaryinformation.Forinstance,magneticresonance(MR)imagesarerealignedwithCTimagesofthesamesubjecttopreciselylocalizethetumortoassistsurgeryplanning.Also,functionalMRbrainimagesareoftenrealignedtohighresolutionT1-weightedanatomicalimagestocorrectdistortionsforaccuratelocalizationofbrainactivationmaps. 39

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Thechallengeinmatchingmulti-modalimagescomesfromthefactthatthereisnodirectcomparisonofintensities.Tocopewiththisdifcultyanumberofmethodsbasedoninformationtheoreticalapproachorstatisticaldependencemeasureshavebeenproposed.In[ 23 36 63 ]jointentropywassuggestedasasimilaritymeasureformulti-modalimageregistration.TheShannonentropyforajointdistributionoftworandomvariables/vectorsXandYisdenedas H(X,Y)=)]TJ /F11 11.955 Tf 11.29 16.28 Td[(ZRNZRNpX,Y(x,y)logpX,Y(x,y)dxdy.(4) In[ 23 36 63 ]theregistrationwasndingthetransformation/deformationeldthatminimizesthejointentropyofapairofimages,sinceentropymeasuresthedispersionofaprobabilitydistribution. However,ithasbeenfoundthatinpracticeminimizingjointentropymaynotwork.Largeoverlapareaofbackgroundcouldleadtosmallentropy,buttheoverlapareadoesnotcontaininformation.After,mutualinformation(MI)wassoonintroducedfortheregistrationofmulti-modalityimages.ItwaspioneeredbyCollignonetal.[ 22 ]andbyViolaandWells[ 70 ]touseMIasasimilaritymeasuretorigidlyalignimagesofdifferentmodalities.SincethenMIhasbeenbecomethemostinvestigatedmeasureformulti-modalimageregistration,andvariousmodelsbasedonmaximizingMIhavebeendevelopedforrigidandnon-rigidregistration,andshowedpromisingresults(e.g.[ 33 34 40 42 46 48 54 62 64 65 70 73 78 ]). TheMIbetweentworandomvariablesXandYisdenedas MI(X,Y)=H(X)+H(Y))]TJ /F1 11.955 Tf 11.96 0 Td[(H(X,Y)(4) whereH(X)andH(Y)aretheentropiesofXandY,respectively,denedinasimilarmannerasthejointentropyH(X,Y)withthejointpdfpX,Y(x,y)in( 4 )replacedbytheirmarginalpdfspX(x)andpY(y),respectively.RegistrationbasedonmaximizingMIcanbeinterpretedasthatwhentheimagesbeingalignedtheamountofinformation 40

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theycontainabouteachotherismaximal.TheMI(X,Y)canalsobewrittenas MI(X,Y)=ZZpX,Y(x,y)logpX,Y(x,y) pX(x)pY(y)dxdy,(4) Therefore,itcanbeinterpretedastheKLdivergencebetweenthejointpdfpX,Y(x,y)andtheproductofthemarginalpdfspX(x)pY(y),andisconsideredameasureofthestatisticaldependence.MI(X,Y)=0,ifandonlyifXandYareindependent.Ontheotherhand,ifXandYarefunctionstoeachother,MI(X,Y)=H(X)=H(Y)ismaximized.BesidesMIthecorrelationratio(CR)[ 34 58 ],Kullback-Leibler(KL)divergence[ 24 25 30 32 54 ],andlocalCC[ 16 ]havealsobeenproposedassimilarity/dissimilaritymeasuresformultimodalimageregistration.TheCCisameasureofthelineardependencybetweentheintensities,henceitislimitedtomatchingimagesthatarelinearlyrelated.UsinglocalCCasasimilaritymeasure,itisimplicitlyassumedthatthepairofimagestobealignedhaveafnedependencylocally. MIhasbeenconsideredasthestate-ofthe-artsimilaritymeasureformulti-modalimageregistrationalthoughtherearecertainissuesinvolvedinimplementation.ThemaindifcultyinusingMIasasimilaritymeasureisintheneedofestimatingthejointpdf.Methodsforconstructingdiscretepdfsarehistogrambasedandeasytoimplement,buttheiraccuracyislimitedbythequantizationoftheintensities.Inmanyefcientoptimizationschemesthederivativeofthejointpdfandmarginalpdfsareusedtoestimatethedeformationeld,andhence,theconstructionofcontinuousjointpdfisnecessary.Parzen-windowdensityestimatorisoneofthekernelbasedmethodstoestimatecontinuouspdf.ThismethodestimatesthejointpdfoftworandomvariablesXandYby^pX,Y(x,y)=1 mPmi=1G(x)]TJ /F5 11.955 Tf 12.1 0 Td[(xi,y)]TJ /F5 11.955 Tf 12.1 0 Td[(yi),whereGistheGaussianfunctionwithmeanzeroandvariance2((xi,yi)(i=1,,m)aresamples).InimageregistrationthecontinuousjointpdfestimatebyusingParzen-windowisaffectedbythekernelwidthandbinningstrategy. 41

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TheaimofthisworkistointroduceanalternativedependencemeasuretoMI,whichiscapableofhandlingmulti-modalimagematching,butdoesnotneedestimationofcontinuousjointpdf.In1959Renyi[ 56 ]proposedasetofpostulatesforanappropriatemeasureQ(X,Y)ofdependencefortworandomvariablesXandYonaprobabilityspace[,F,P].Thisworkhasdrawnmuchattentioneversince.Thesepostulatesinclude: 1.Q(X,Y)isdenedforanypairofrandomvariablesXandY,neitherofthembeingconstantwithprobabilityone; 2.Q(X,Y)=Q(Y,X); 3.0Q(X,Y)1; 4.Q(X,Y)=0ifandonlyifX,Yareindependent; 5.Q(X,Y)=1ifY=f(X)orX=g(Y),wherefandgareBorelmeasurablefunctions; 6.IftheBorelmeasurablefunctionsfandgmaptherealaxisinonetoonewayontoitself,Q(f(X),g(Y))=Q(X,Y). Then,RenyishowedinthisworkthatonemeasuresatisfyingtheseconditionsisthemaximumcorrelationcoefcientMCCdenedasfollows. MCC(X,Y)=supf,g2VCC(f(X),g(Y)),(4) whereVisthespaceofallBorelmeasurablefunctionswithnitepositivevariance.CC(f(X),g(Y))isthecorrelationcoefcientoff(X)andg(Y),i.e., CC(f(X),g(Y))=Cov(f(X),g(Y)) p Var(f(X))p Var(g(Y)), whereCovandVarstandforcovarianceandvariance,respectively.Cov(f(X),g(Y))=E[(f(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E[f(X)])(g(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(E[g(Y)])]=E[f(X)g(Y)])]TJ /F5 11.955 Tf 11.96 0 Td[(E[f(X)]E[g(Y)]. 42

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Var(f(X))=E[(f(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E[f(X)])2]=E[f(X)2])]TJ /F5 11.955 Tf 11.95 0 Td[(E[f(X)]2, andVar(g(Y))isdenedinthesameway. InthisworkweproposetouseRenyi'sstatisticaldependencemeasureMCCasanalternativetoMIasasimilaritymeasureinmulti-modalimageregistration.Sincethereisnoconstructivedenitionofdependenceoftworandomvariables,statisticaldependenciesaremostoftendenedmerelyastheabsenceofindependence.Considerthe4thpostulate,weimplicitlyassumethatthesmaller1)]TJ /F1 11.955 Tf 12.69 0 Td[(MCC(X,Y)is,themoredependentXandYare.ThisissimilartothecaseswhereMIisusedasadependencemeasure.ThelargerMIis,themoredependentthetworandomvariablesare.TheadvantageofusingRenyi'sstatisticaldependencemeasureisthatinpractice,tocomputeMCC(X,Y)in( 4 )wedonotdealwiththemeasurePX,Yitself,butinsteadobservedsamplesdrawnindependentlyaccordingtoit.ThisavoidsthedifcultiesofcontinuousjointpdfestimationencounteredinthevariationalmodelsbasedonmaximizingMI. Theframeworkofvariationalmodelsforimageregistrationisestimatingadisplacementvectoreldh(x)=x+u(x),whereu(x)isthecorrespondingdeformationeld,byminimizinganenergyfunctionalinasetofadmissiblefunctions.Theenergyfunctionalenforcestheregularityofthedeformationeldandstatisticaldependencyofthedeformedimageandtargetimage.Therehavebeenmanyregularizationmethodsdevelopedintheliterature,suchasminimizingafunctionofthederivativesofuinasuitablespace[ 75 ],linearelasticregularization[ 8 9 11 ],andtensorbasedsmoothing[ 3 ],wherethetensorisdesignedtopreventthetransformationeldsfrombeingsmoothedacrosstheboundariesoffeatures.Sincethefocusofthisworkisonanalternativestatisticalmeasureofdependenceforimageregistration,theregularitytermisjustdenedasR(u)=RjDuj2dx.Inourvariationalmodelthestatisticaldissimilaritymeasureisdenedas1)]TJ /F1 11.955 Tf 10.83 0 Td[(MCC(S(x+u(x)),T(x)),whereS(x)andT(x)arethesourceandtargetimages,respectively. 43

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MCCisanextensionofCC.CC(X,Y)measuresthelineardependencyofXandY,whileMCC(X,Y)measuresthelineardependencyforf(X)andg(Y),wherefandgaresearchedfromallboundedBorelmeasurablefunctionswithnitepositivevariance.Therefore,evenwhenXandYarenotlineallyrelated,MCC(X,Y)maystillapply. Thecorrelationratio(CR)isdenedas CR(X,Y)=Var(E(YjX)) Var(Y)(4) alsoprovidesafunctionaldependenceofXandY.WhenCR(XY)ismaximized,i.e.,CR(X,Y)=1,thereexistsafunctionsuchthatY=(X).WhileMCC(X,Y)=1indicatesthatthereexiststwofunctionsfandgsuchthatf(X)andg(Y)arelinearlyrelated.However,forgisnotnecessarilyinvertible.Hence,YmaynotbeafunctionofX.Fromthispointofview,MCCmaybeamoresuitablesimilaritymeasurethanCR,sinceinmanycasesapairofimagestobealignedmaynothavearelationY=(X). InthedenitionofMCC( 4 ),ndingthefunctionsfandgamongallboundedBorelmeasurablefunctionsisdifcult.Wewillshowinthenextsectionthatthesearchforfandgcanberestrictedtoamuchsmallerspaceinwhichthesupremumispreserved,i.e.theMCCisthesame.ThissmallerspacecanbeareproducingkernelHilbertspace(RKHS)associatedwithareproducingkernelthatiscontinuous,symmetric,positivedeniteandvanishingrapidlyoutsidethekernelwindow.InthisworkwechoosetousethefollowingGaussiankernel K(x,y)=1 p 2expf)]TJ 16.47 8.09 Td[(jx)]TJ /F5 11.955 Tf 11.96 0 Td[(yj2 22g.(4) BythetheoryofRKHS(seesection2below),anyfunctionfandgintheRKHSassociatedwiththekernelin( 4 )canbeapproximatedbythesumofnitelymany 44

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functionsfK(,)2RgandfK(,)2Rg,respectively,i.e.,f(x)mXi=1i p 2expf)]TJ 16.47 8.09 Td[(jx)]TJ /F7 11.955 Tf 11.95 0 Td[(ij2 22g,g(x)nXi=1i p 2expf)]TJ 16.47 8.09 Td[(jx)]TJ /F7 11.955 Tf 11.95 0 Td[(ij2 22g, Therefore,thecomputationofndingfandgcanbereducedtoestimatingthecoefcientsiandjinfandg,wheni,iandiarechosen. (a)(b)(c)(d) Figure4-1. Theeffectoff(x)appliedonanimageS(x).(a)ImageS(x).(b)Imagesforeachterminf(S(x))wherem=6,=0.4andisoptimizedtomaximizeCC(f(S(x)),T(x)).(c)Imagef(S(x))asthesumoftheprevioussiximages.(d)ImageT(x). Withoutlossofgeneralitywecantakem=n.Then,theMCCoftwoimagesI1andI2canbeestimatedbyMCC(I1,I2)=supi,iCC(Xi p 2expf)]TJ 16.47 8.09 Td[(jI1)]TJ /F7 11.955 Tf 11.95 0 Td[(ij2 22g,Xi p 2expf)]TJ 16.47 8.09 Td[(jI2)]TJ /F7 11.955 Tf 11.95 0 Td[(ij2 22g). Hence,minimizing1)]TJ /F1 11.955 Tf 12.85 0 Td[(MCC(I1,I2)aimstondalloptimaliandimakingtheCCofPi p 2expf)]TJ /F10 7.97 Tf 16.48 5.7 Td[(jI1)]TJ /F12 7.97 Tf 6.59 0 Td[(ij2 22gandPi p 2expf)]TJ /F10 7.97 Tf 16.47 5.7 Td[(jI2)]TJ /F12 7.97 Tf 6.58 0 Td[(ij2 22g,i.e.,CCoff(I1)andg(I2),asclosetooneaspossible.TheideaofusingMCCasasimilaritymeasureformulti-modelimageregistrationisthatalthoughtheintensitiesofI1andI2arefarawayfromalinearrelationship,wecanndtheoptimaliandisuchthatf(I1)andg(I2)havearelationclosetolineardependence. Inthenextsection,forconvenienceofreaders,webrieyrecallthedenitionandsomepropertiesofRKHSusedinthiswork,andshowthatthesupremumin( 4 )ispreservedinRKHS.Inthethirdsectionwedescribeourmodel.Then,weprovideexperimentalresultswithcomparisonsofMIbasedalgorithmsinthefourthsection. 45

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4.2ReproducingKernelHilbertSpacesandDensenessProperties Inthissectionweshowthatthesupremumin( 4 ),infact,canbetakenoveraRKHSassociatedwithasymmetricpositivedenitekernel,suchasaGaussiankernelin( 4 ). 4.2.1ReproducingKernelHilbertSpace Denition1. ARKHSisaHilbertspaceinwhichallthepointevaluationsareboundedlinearfunctionals([ 2 ]). LetH(E)beareproducingkernelHilbertspaceoffunctionsonadomainE.BytheRieszrepresentationtheorem,foreveryx2E,thereexistsanelementhx2H(E),suchthatthepointevaluationex(f)satises ex(f)=f(x)=hf,hxiH,for8f2H(E).(4) LetK(x,y)=hhx,hyiH.ThenK(x,y)issymmetricandpositivedeniteonEE.Thismeansthatforeveryn,anyx1,x2,...,xn2Eandany1,2,...,n2R, nXi,jijK(xi,xj)0,(4) andequalityholdsifandonlyifi'sareallzero.Thisisequivalent,underverygeneralconditions,to ZZK(x,y)m(dx)m(dy)0,(4) forallboundedsignedmeasuresmwithequalityonlyifm=0.ThefunctionKiscalledareproducingkernelforH(E). FromthedenitionofKandthefactthathx2H(E),wehavehx(y)=hhx,hyiH=K(x,y).Therefore,onehas (a).Forallx2E,K(x,)2H(E), (b).Forallx2E,8f2H(E),hf,K(x,)i=f(x)2H(E)(Thisconditioniscalledreproducingproperty. (c).K(x,y)=hK(x,),K(y,)iH. 46

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Ontheotherhand,theMoore-Aronszajntheorem[ 4 ]statesthatforeverysymmetric,positivedenitefunctionK:EE!C,thereisauniqueHilbertspaceH(E)offunctionsonEforwhichKisareproducingkernel.Infact,denehx=K(x,).LetH0(E)bethelinearspanofthefunctionsfK(x,)x2Eg,anddeneaninnerproductonitbyhnX1aiK(xi,),mX1bjK(yj,)iH=nX1mX1aibjK(xi,yj). LetalsoH(E)bethecompletionofH0(E)withrespecttothisinnerproduct.Then,itisnotdifculttocheckthatKisareproducingkernelforH(E),andH(E)istheuniqueRKHSassociatedwiththiskernel. 4.2.2DensenessProperties LetEbeatopologicalspace(inourapplicationE=R).DenotebyC0(E)thespaceofallthecontinuousfunctionsonEvanishingatinnitywiththesupremenorm.SupposethatKonEEisafunctionsatisfying (a).Kissymmetric,positivedenitefunction(see( 4 and 4 )). (b).K(x,)2C0(E)forallx2E. Then,asstatedabove,thereisauniqueHilbertspace,denotedbyH(E),associatedwiththisK.ThefollowingtheoremisprovedbycombiningthreelemmasinAppendixA. Theorem4.1. supf,g2H0(E)CC(f(X),g(Y))=supf,g2VCC(f(X),g(Y)), whereVisthespaceofallBorelmeasurablefunctionsonRwithnitepositivevariance. 4.3ProposedModel Ourproposedvariationalmodelestimatesadeformationeldbyminimizinganenergyfunctionalconsistingoftwoterms.Oneofthemregularizesthedeformationeld.Theothermeasuresthegoodnessofmatching,whichisformulatedbyusingtheRenyi'sstatisticaldependencemeasureMCCbetweenthedeformedimageandthe 47

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targetimage.Moreover,WeintroducereproducingkernelHilbertspacesassociatedwithGaussiankernelstosimplifythecomputationinndingthesupremuminthedenitionofMCC. LetS(x)andT(x)bethesourceimageandtargetimageontheimagedomain,respectively,andu(x)bethedeformationeldthatdeformsStoT.FirstweshowhowtoestimatetheMCCbetweenthedeformedimageSu(x)=S(x+u(x))andtargetimageT(x)usingthetheoryofRKHS.TakeK(x,y)=1 p 2expf)]TJ /F3 11.955 Tf 16.47 8.09 Td[((x)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2 22g. BytheTheorem 4.1 ,MCC(Sfu(x),Tg(x))canbeapproximatedbysup,CC(Sfu(x),Tg(x)),where=(1,,m)T,=(1,,n),and Sfu(x)=mXi=1iK(Su(x),i),Tg(x)=nXi=1iK(T(x),i). (4) Here,m,n,iiareparameters.ThemeanofSfu(x)andTg(x)is1(u,)=1 jjZmXi=1iK(Su(x),i)dx,2()=1 jjZnXi=1iK(T(x),i)dx, respectively.ThevarianceandthecovarianceofSfu(x)andTg(x)arethenestimatedas: 1(u,)=1 jjZfmXi=1iK(Su(x),i)g2dx)]TJ /F7 11.955 Tf 11.96 0 Td[(1(u,)2,2()=1 jjZfnXi=1iK(T(x),i)g2dx)]TJ /F7 11.955 Tf 11.95 0 Td[(2()2, (4) 12(u,,)=1 jjZfmXi=1iK(Su(x),i))]TJ /F7 11.955 Tf 11.96 0 Td[(1(u,)gfnXi=1iK(T(x),i))]TJ /F7 11.955 Tf 11.96 0 Td[(2()gdx, 48

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respectively. Therefore CC(Sfu,Tg)=12(u,,) p 1(u,)p 2(), (4) andMCC(Su,T)=supf,gCC(Sfu,Tg)=max,12(u,,) p 1(u,)p 2(). NowweregistertheimageS(x)toT(x)bysolvingthefollowingminimizationproblemwithrespecttothedeformationeldu(x): minu,,Zjru(x)j2dx+jj(1)]TJ /F1 11.955 Tf 11.96 0 Td[(MCC(Su,T))p, (4) whereparameter>0balancestheregularizationofthedeformationeldandgoodnessofthematching,andtheparameterpinthesecondtermof( 4 )inuencesthespeedofMCC(Su,T)increasingtowardsone.Fig. 4.3 depictsthegraphof(x)=(1)]TJ /F5 11.955 Tf 12.28 0 Td[(x)p,0x1fordifferentvaluesofp.Itcanbeseenthatwhenxapproaches1,thederivativeof(x)atxisincreasingif0
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Byminimizingthisenergywecangetaregularizeddeformationeld,whichforcesthedeformedimageandthetargetimagetohavemaximumdependenceintermsofRenyi'sstatisticaldependencemeasureMCC.Intheextremecasethatthedeformedimageisafunctionofthetargetimageorviseversa,MCC(S,T)=1,andhence,thesecondtermiszero. Usingnotationabove,theminimizationproblem( 4 )canberewrittenas: minufR(u)+jjmin,J(u,,)g. (4) where R(u)=Zjru(x)j2dx,J(u,,)=(1)]TJ /F7 11.955 Tf 34.7 8.08 Td[(12(u,,) p 1(u,)p 2())p. (4) 4.4TheIterativeAlgorithm Anexplicitalternatingalgorithmisusedheretondtheminimizeru(x),,.Startingfromaninitialu0(x),forinstance,u0(x)=0,thismethodcomputesasequenceof:(1),(1),u(1),(2),(2),u(2),,(k),(k),u(k), suchthat(k),(k)=argmin,J(u(k)]TJ /F9 7.97 Tf 6.58 0 Td[(1),,),u(k)=argminuR(u)+J(u,(k),(k)). NotethatsearchingforandwhichminimizeJ(u,,)isequivalenttosearchingforandwhichmaximize( 4 )Inthediscreteversion,arrangeSfu(x),Tg(x),pi=K(Su(x),i)andqj=K(T(x),j)column-wisein( 4 )and( 4 )asvectorsoflengthL,whichisequaltotheimagesizeofS(x)andT(x).ThenSfu=PandTg=Q,where=(1,...,m)T,=(1,...,n)T,P=(pij)Lm=(p1,p2,,pm) 50

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andQ=(qij)Ln=(q1,q2,,qn).Nowitisreadilycheckedthat12(u,,) p 1(u,)p 2()= jP0jjQ0j, whereP0=(pij0)Lm=(pij)]TJ /F9 7.97 Tf 12.13 4.71 Td[(1 LPLi=1pij)LmandQ0=(qij0)Ln=(qij)]TJ /F9 7.97 Tf 12.14 4.71 Td[(1 LPLi=1qij)Ln.Let^PbeamatrixwhosecolumnsformanorthonormalbasisforthevectorspacespannedbythecolumnsofP0and^QbeamatrixobtainedsimilarlyfromQ0.SincemaximizingCC(Sfu,Tg)isequivalenttomaximizingwithrespecttoandundertheconditionjP0j=jQ0j=1,itisalsoequivalenttomaximizing<^P^,^Q^>withrespectto^and^undertheconditionj^j=j^j=1.Since<^P^,^Q^>=^T^PT^Q^,wecarryoutasingularvaluedecompositionfor^PT^Q=UVTwhereUandVareunitarymatricesandisadiagonalmatrixwhosediagonalentriesarenonnegativeandarelistedinadescendingorderfromtheupperlefttothelowerright.Hence,wecanchoose^and^suchthat^TU=(1,0,0,,0)and^TV=(1,0,0,,0),whichimplies^=U(1,0,0,,0)Tand^=V(1,0,0,,0)T.Finally,wecanjustsetSfu=P0=^PU(1,0,0,,0)TandTg=Q0=^QV(1,0,0,,0)Twhichwillnotaffectthesubsequentresults.Ashaveshownherethatwedidnotexplicitlygetthesolutionfortheminimizerand,butitimpliestheexistenceofthoseminimizersandwegetSfu(x)andTg(x)asweneed. Next,letandbetheminimizersfromthepreviousstep,andlet1,2,1,2and12in( 4 )be1(u)=1(u,),2=2(),1(u)=1(u,),2=2(),12(u)=12(u,,),andJ(u),(1)]TJ /F7 11.955 Tf 31.56 8.09 Td[(12(u) p 1(u)p 2)p. BycomputingtherstvariationofR(u)+jjJ(u)withrespecttou(x)(Appendix??),wegetthefollowingEuler-Lagrange(EL)equation:)]TJ /F3 11.955 Tf 9.3 0 Td[(24u(x))]TJ /F5 11.955 Tf 11.95 0 Td[(F(x,u)=0 51

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whereF(x,u)=p 1)]TJ /F7 11.955 Tf 30.03 8.09 Td[(12(u) 1=21(u)1=22!p)]TJ /F9 7.97 Tf 6.58 0 Td[(1(Tg(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2 1=21(u)1=22)]TJ /F5 11.955 Tf 13.15 8.09 Td[(Sfu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(1(u) 3=21(u)1=2212(u))rSfu(x). TheevolutionequationcorrespondingtotheELequationofu(x,t)is @u(x,t) @t=24u(x,t)+F(x,u). (4) 4.5ExtensiontoLocalMaximumCorrelationCoefcient Inthissection,weextendourmodeltothelocalversion,whichprovidesustheexibilityneededtocopewithnon-stationarityinintensitydistributionsoftheimagestoberegistered.Tothisend,weestimatethesimilaritymeasuresbetweentheneighborhoodsofeachpointx02intwoimages.Thisisachievedbyweightingourpreviousestimatesofthemeans,variancesandcovarianceswithanormalizedGaussiankernelofvariance. Moreprecisely,themeanofSfu(x)andTg(x)intheneighborhoodofzcanbeestimatedby:u1(z)=ZSfzu(x)G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx,2(z)=ZTgz(x)G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)dx. ThevarianceofSfu(x),Tg(x)andthecovarianceofthosetwointheneighborhoodofzarethencomputedby:u1(z)=ZSfzu(x)2G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx)]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z)2,2(z)=ZTgz(x)2G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx)]TJ /F7 11.955 Tf 11.96 0 Td[(2(z)2,u12(z)=ZSfzu(x)Tgz(x)G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)dx)]TJ /F7 11.955 Tf 11.95 0 Td[(u1(z)2(z), 52

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whichleadstothelocalversionofthemodel( 4 )basedonRenyi'sstatisticalmeasure: minu(x),(x),(x)Zjru(x)j2dx+jjZ(1)]TJ /F7 11.955 Tf 41.3 8.09 Td[(u12(x) p u1(x)p 2(x))2dx, (4) wherexandxaretobeoptimized:[x,x]=argminx,x(1)]TJ /F7 11.955 Tf 41.3 8.09 Td[(u12(x) p u1(x)p 2(x))2 Hence,theELequationforthelocalversionoftheproposedmodelis(Appendix??))]TJ /F3 11.955 Tf 9.3 0 Td[(24u(x))]TJ /F5 11.955 Tf 11.96 0 Td[(p1)]TJ /F11 11.955 Tf 11.95 16.27 Td[(Zu1,2(z) [u1(z)]1=2[2(z)]1=2dzp)]TJ /F9 7.97 Tf 6.58 0 Td[(1ZG(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)F(x,z,u)dz=0 whereF(x,z,u)=Tgz(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2(z) [u1(z)]1=2[2(z)]1=2)]TJ /F5 11.955 Tf 28.28 8.08 Td[(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z) [u1(z)]3=2[2(z)]1=2u1,2(z)rSfzu(x). 4.6ExperimentalResults InthissectionweexaminetheefciencyandrobustnessofthealgorithmdescribedinSection4throughtheexperimentsbelowonavarietyofimages.Weuseamulti-resolutionapproach.Thiscoarse-to-nestrategyhelpsavoidthedeformationeldfromgettingstuckatlocalminimaoftheobjectivefunctional.Tospeedupthecomputation,theadditiveoperatorsplitting(AOS)algorithm[ 45 71 ]isappliedinoursemi-implicitnitedifferenceschemetosolvetheequation 4 4.6.1Parameters Wediscussherehowtheparametersareselectedinthealgorithmandtrytomaketheparameterselectionadaptivetothegivenimages. :Thisistheparameterwhichbalancestheregularizationtermandthedissimilarityterm.Wechoose=0.06forthemodel( 4 ),and=0.2foritslocalversion. m,n,i,i:TheK-meanalgorithmisusedtodeterminetheseparameters.Werepeatthek-meanalgorithmaswegraduallyincreasethenumbersofclusterstopartitionthesourceandtargetimagesuntilthevarianceofeachclusteris 53

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smallerthanaprexedvariance0=0.02.Themandnarethesmallestnumbersofclusters,whosevariancesarelessthan0,inthesourceandtargetimages,respectively.Thenwetakethei'sandi'sasthemeansoftheseclusters. :Weselectasthesmallestdifferenceofi'sori's,whicheverissmaller. :Thevalueof,whichwillappearinSection 4.5 ,describesthelocalwindowsizeinourlocalversionof( 4 ).Wefound=(M+N)=64workedwellforimagesofsizeMN. dt:Thestepsizedtinourexperimentsis8forthemodel( 4 ),and0.9foritslocalversion. 4.6.2TestsonSyntheticImagesandSimulatedDeformationField Tovalidatetheproposedmodelwerstappliedmodel( 4 )toapairofsyntheticimagesS(x)andT(x),shownastherstandsecondimagesinFig. 4-3 ,respectively.ThesourceimageS(x)consistsoftwoobjectswiththeirintensities1.00and0.70,andabackgroundwithintensity0.15.ThetargetimageT(x)isgeneratedinasimilarmanner.ButthetwocorrespondingobjectsinT(x)withtheirintensities0.08and1.00,respectively,havedifferentshapesfromthoseinS(x).ThebackgroundintensityofT(x)is0.65.TheintensitiesofS(x)andT(x)arefarfromlinearlyrelated. (a)S(x)(b)T(x)(c)Su(x)(d)u(x) (e)Sfu(x)(f)Tg(x)(g)jSfu(x))]TJ /F5 11.955 Tf 11.95 0 Td[(Tg(x)j(h)MCCandMI Figure4-3. Registrationresultsforsyntheticimages.(a)SourceimageS(x).(b)TargetimageT(x).(c)DeformedimageSu(x).(d)Deformationeldu(x)appliedtoaregulargrid.(e)ImageSfu(x)withoptimizedf.(f)ImageTg(x)withoptimizedg.(g)ImagejSfu(x))]TJ /F19 10.909 Tf 10.91 0 Td[(Tg(x)j.(h)MCC(Su(x),T(x))andMI(Su(x),T(x))vs.iterations. 54

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Weappliedtheequation( 4 )toS(x)andT(x)showninFig. 4-3 (a)and(b).AsaresultwegetthedeformedimageSu(x)showninFig. 4-3 (c),andtheoptimizeddeformationeldu(x)showninFig. 4-3 (d).TheparametersdeterminedbySection 4.6.1 forthistestare:m=n=3,1=0.15,2=0.70,3=1.00,1=0.08,2=0.65,3=1.00,=0.30. AtthesametimeweobtaintheoptimalSfu(x),Tg(x)andSfu(x))]TJ /F5 11.955 Tf 13.23 0 Td[(Tg(x)associatedwiththeoptimized,andu(x)using( 4 )showninFig. 4-3 (e)-(g),respectively.OnecanseethattheimagesSfu(x)andTg(x)canbeconsideredassamemodalityimages,althoughthedeformedimageSuandtargetimageTarenot.ThedifferencebetweenSfu(x)andTg(x)showsthattheirintensitiesareveryclosetoeachother.Thisisanindicationofagoodmatching.WealsoplotthegraphofMCCbetweenSu(x)andT(x)(orequivalently,CCbetweenSfu(x)andTg(x))vs.iterations,whichshowsthattheMCCquicklyincreasesto0.999. Furthermorewestudiedthequestion:istheincreaseinMCCconsistentwiththeincreaseoftheMIbetweenSu(x)andT(x)?ToanswerthisquestionwealsoplotthegraphofMI(Su(x),T(x))vs.iterationsinthesamegureasthegraphofMCC(Su(x),T(x)).Comparingthesetwographs,onecanseethatwhenMCC(Su(x),T(x))increases,theMI(Su(x),T(x))alsoconsistentlyincreaseviatheproposedmodel. Next,weapplyourmodeltoSyntheticimageswithinhomogeneousintensitiesshownasthersttwoimagesinFig. 4-4 .TheresultingdeformedimageSu(x)anddeformationeldu(x)areshowninFig. 4-4 (c)and(d),respectively.Fig. 4-4 showsalmostperfectmatchingfortheobjectsinimages.Thisindicatestherobustnessoftheproposedmodeltotheinhomogeneityandbiasofimageintensities.TheparametersdeterminedinSection 4.6.1 forthistestare:m=n=4,1=0.14,2=0.37,3=0.66,4=0.91,1=0.04,2=0.33,3=0.59,4=0.89,=0.23. Finally,toexaminetheefciencyofouralgorithmweappliedouralgorithmtoapairofPDandT2brainimagesshowninFig. 4-5 (b)and(c),respectively.TheT2image 55

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(a)S(x)(b)T(x)(c)Su(x)(d)u(x) Figure4-4. Registrationresultforsyntheticimageswithregionalinhomogeneities.(a)SourceimageS(x).(b)TargetimageT(x).(c)DeformedimageSu(x).(d)Deformationeldu(x)appliedtoaregulargrid. O(x)andPDimageT(x)wereobtainedsimultaneouslybyonescansession.Thereforethestructuresinthesetwoimagesshouldbealignedperfectlyalthoughthemodalitiesaredifferent.Weappliedapredetermineddeformationeldu0(x)toO(x)togeneratethesourceimageS(x)showninFig. 4-5 (b).TheparametersdeterminedbySection 4.6.1 forthistestare:m=n=6,1=0.05,2=0.24,3=0.41,4=0.67,5=0.87,6=0.95,1=0.04,2=0.21,3=0.39,4=0.51,5=0.68,6=0.77,=0.12. (a)O(x)(b)S(x)(c)T(x)(d) (e)Su(x)(f)jO(x))]TJ /F5 11.955 Tf 11.95 0 Td[(S(x)j(g)jO(x))]TJ /F5 11.955 Tf 11.96 0 Td[(Su(x)j(h) Figure4-5. RegistrationresultsforT2andPDbrainimages(a)OriginalT2imageO(x).(b)SourceimageS(x)obtainedbyapplyingapredetermineddeformationeldu0(x)onO(x).(c)PDTargetimageT(x).(d)CompositeviewofS(x)andT(x)beforeregistration.(e)DeformedimageSu(x).(f)ImagejS(x))]TJ /F19 10.909 Tf 10.91 0 Td[(O(x)j.(g)ImagejSu(x))]TJ /F19 10.909 Tf 10.91 0 Td[(O(x)j.(h)CompositeviewofSu(x)andT(x)afterregistration 56

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WeappliedourmodeltoS(x)andT(x)togetthedeformedimageSu(x)showninFig. 4-5 (e).Sincetheintensitiesoftwoimagesaremorecomplexthanthesyntheticimagesabove,wetakem=n=11ratherthansixinthisexperiment.NotethatO(x)canbeperfectlyalignedtoT(x)byidentitymap.Ifouralgorithmcanprovideanaccurateregistrationresult,Su(x)shouldbeveryclosetoO(x).TheresultsshowninFig. 4-5 indicatethisbyshowingthedifferenceimagejS(x))]TJ /F5 11.955 Tf 12.71 0 Td[(O(x)jin(g)andthecompositeviewofSu(x)andT(x)inFig. 4-5 (h). 4.6.3TestonCTandMRImages Thisexperimentaimstovalidatetheeffectivenessoftheproposedmodelinrealmedicalimages.WeapplyourmodeltoapairofCTandMRlungimagesshowninFig. 4-6 (a)and 4-6 (b). (a)S(x)(b)T(x)(c)Su(x)(d)u(x) (e)Sfu(x)(f)Tg(x)(g)jSfu(x))]TJ /F5 11.955 Tf 11.96 0 Td[(Tg(x)j(h)MCCandMI Figure4-6. RegistrationresultsforCTandMRimages.(a)CTsourceimageS(x).(b)MRtargetimageT(x).(c)DeformedimageSu(x).(d)Deformationeldu(x)appliedtoaregulargrid.(e)ImageSfu(x)withoptimizedf.(f)ImageTg(x)withoptimizedg.(g)ImagejSfu(x))]TJ /F19 10.909 Tf 10.91 0 Td[(Tg(x)j.(h)MCC(Su(x),T(x))andMI(Su(x),T(x))vs.iterations. InapatternsimilartoFig. 4-3 thedifferencebetweenSfu(x)andTg(x)inFig. 4-6 showsthattheirintensitiesareclosetoeachotherandthestructureofS(x)ispreservedinSfu(x).ThegraphsofMCCandMIin( 4 )betweenSu(x)andT(x)areshownin(h).FromthisgureonecanseethattheMCC(Su(x),T(x))risesto0.98,andhence,thealignmentisadequate.Moreover,theincreasingtrendinMCCisconsistentwiththatinMIasobservedintherstexperiment.TheparametersdeterminedinSection 4.6.1 for 57

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thistestare:m=7,n=9,1=0.03,2=0.21,3=0.35,4=0.57,5=0.63,6=0.84,7=0.91,1=0.05,2=0.17,3=0.24,4=0.32,5=0.53,6=0.61,7=0.69,8=0.76,9=0.92,=0.07. Theaimofthenextexperimentistotesttherobustnessoftheproposedmodeltonoise.Forthispurposeweappliedmodel( 4 )withthesameparametersasinthepreviousexperimenttoapairofnoisyCTandMRlungimagesshownasthersttwoimagesinFig. 4-7 .ThesetwoimagesaregeneratedbyaddingaGaussiannoiseofzeromeanandvariance0.1totheimagesusedinthepreviousexperimentwiththeintensityrangein[0,1].Model( 4 )stillworkswellinthiscase.TheMCCbetweenSu(x)andT(x)isupto0.97,andthedifferencebetweenSu(x)andT(x)issmall.TheconsistencyoftheincreasingtrendforMCC(Su(x),T(x))andMI(Su(x),T(x))isstillpreserved.TheparametersdeterminedbySection 4.6.1 forthistestare:m=7,n=10,1=0.03,2=0.22,3=0.35,4=0.56,5=0.63,6=0.85,7=0.90,1=0.05,2=0.19,3=0.24,4=0.31,5=0.52,6=0.61,7=0.69,8=0.77,9=0.87,10=0.95,=0.05. (a)S(x)(b)T(x)(c)Su(x)(d)u(x) (e)Sfu(x)(f)Tg(x)(g)jSfu(x))]TJ /F5 11.955 Tf 11.96 0 Td[(Tg(x)j(h)MCCandMI Figure4-7. RegistrationresultsforCTandMRimageswithGaussiannoise.(a)CTsourceimageS(x).(b)MRtargetimageT(x).(c)DeformedimageSu(x).(d)Deformationeldu(x)appliedtoaregulargrid.(e)ImageSfu(x)withoptimizedf.(f)ImageTg(x)withoptimizedg.(g)ImagejSfu(x))]TJ /F19 10.909 Tf 10.91 0 Td[(Tg(x)j.(h)MCC(Su(x),T(x))andMI(Su(x),T(x))vs.iterations. Fig. 4-8 showstheresultofregisteringT1andT2imagesusingtheabovelocalversion( 4 ).Theaimofthisexperimentistotestifthelocalversioncanresultinagoodregistrationforimageswithmorecomplexintensities.Weappliedthelocal 58

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versiontoalignapairofT1andT2brainimages,forwhicharelativelylargedeformationisrequired.TheimagesshownasthersttwoimagesinFig. 4-8 havecomplexandnonuniformintensities,andthedeformationfromonetotheothercannotbetoosmall.Theglobalversionofmodel( 4 )can'tprovideadesirableregistrationresultinthiscase.Thelocalversionof( 4 ),which,roughlyspeaking,applies( 4 )withineachofthelocalwindowsoftheimage,providesreasonablygoodresults. (a)S(x)(b)T(x)(c)Su(x)(d)u(x) (e)Sfu(x)(f)Tg(x)(g)jSfu(x))]TJ /F5 11.955 Tf 11.95 0 Td[(Tg(x)j(h)MCCandMI Figure4-8. RegistrationresultsforT1andT2images.(a)T2sourceimageS(x).(b)T1targetimageT(x).(c)DeformedimageSu(x).(d)Deformationeldu(x)appliedtoaregulargrid.(e)ImageSfu(x)withoptimizedf.(f)ImageTg(x)withoptimizedg.(g)ImagejSfu(x))]TJ /F19 10.909 Tf 10.91 0 Td[(Tg(x)j.(h)MCC(Su(x),T(x))andMI(Su(x),T(x))vs.iterations. AsshowninFig.4thedifferencebetweenSfu(x)andTg(x)issmall,andMCCreachesupto0.98.Thedeformationisrelativelylarge.TheincreasingofMCCvs.iterationsisconsistentwiththatofMI.Alltheseindicatetheeffectivenessoftheproposedmodelevenforimageswithcomplexintensities. Toexaminetherobustnessofthelocalversion( 4 )tonoise,wetestedthismodelonapairofnoisyimages,whicharegeneratedbyaddingGaussiannoiseofmean0andvariance0.1tothepairofimagespresentedinFig.( 4-8 )withtheintensityrange 59

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in[0,1].Thesamesetofparametersofthepreviousexperimentareused.TheresultsareshowninFig.( 4-9 ).Model( 4 )stillworkswellinthiscase.TheMCCbetweenSu(x)andT(x)isupto0.975,andthedifferencebetweenSu(x)andT(x)issmall.TheconsistencyoftheincreasingtrendsforMCC(Su(x),T(x))andMI(Su(x),T(x))arestillpreserved. (a)S(x)(b)T(x)(c)Su(x)(d)u(x) (e)Sfu(x)(f)Tg(x)(g)jSfu(x))]TJ /F5 11.955 Tf 11.95 0 Td[(Tg(x)j(h)MCCandMI Figure4-9. RegistrationresultsforT1andT2imageswithGaussiannoise.(a)T2sourceimageS(x).(b)T1targetimageT(x).(c)DeformedimageSu(x).(d)Deformationeldu(x)appliedtoaregulargrid.(e)ImageSfu(x)withoptimizedf.(f)ImageTg(x)withoptimizedg.(g)ImagejSfu(x))]TJ /F19 10.909 Tf 10.91 0 Td[(Tg(x)j.(h)MCC(Su(x),T(x))andMI(Su(x),T(x))vs.iterations. 4.6.4ComparisonswithOtherDeformableModels Inthissection,wemakecomparisonsoftheproposedmodel( 4 )withotherthreevariationalmodels.Theenergyfunctionalsofthesethreemodels,similartotheproposedmodel,consistofaregularizationtermforthedeformationeld,andadissimilaritymeasuretoforceagoodmatching.Theregularizationterminthesethreemodelsarethesameasintheproposedmodel( 4 ),whilethesimilaritymeasuresarethecrosscorrelation(CC),correlationratio(CR),andmutualinformationofSuandT,respectively. 60

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Table4-1. Parametersandresultsfordifferentmodels. method dt iter. CC CR MI MCC CPUtime(s) 4 0.15 6 0.2 1.1 300 0.8431 0.9291 1.2364 0.9573 638 4 0.15 6 0.2 1.1 100 0.8682 0.9333 1.4864 0.9763 472 4 0.15 6 0.1 0.6 100 0.8753 0.9597 1.5193 0.9898 1413 4 0.15 6 0.2 0.9 100 0.8734 0.9663 1.5122 0.9950 1381 Theyareasfollows: minuZjru(x)j2dx)]TJ /F1 11.955 Tf 11.95 0 Td[(CC(Su,T), (4) minuZjru(x)j2dx)]TJ /F1 11.955 Tf 11.95 0 Td[(CR(Su,T), (4) minuZjru(x)j2dx)]TJ /F1 11.955 Tf 11.95 0 Td[(MI(Su,T), (4) whereCR(Su,T)isdenedin( 4 )andMI(Su,T)isdenedin( 5 ). WeimplementedallthosefourmodelsintheirlocalversionsonapairofT1andT2brainimagesasshowninFig. 4-10 .WegetSufromeachmodel,thenuseCC,CR,MIandMCCbetweeneachresultingSuandTascriteriatomeasurethegoodnessofmatching.Table. 4-1 givestheparametersusedineachmodel,andtheCC,CR,MIandMCCofSuandTforeachmodel.Sinceinthelocalversionofourmodeltheparametersmayvaryfordifferentpatchesoftheimage,wedon'tlisttheparametervalueshere.However,Section 4.6.1 givethegeneralruleforparameterselection FromtheTable 4-1 ,weseethatthemodels( 4 )and( 4 )yieldrelativelybetteralignmentthantheothertwomethods.TheyprovidedhighterCC(Su,T),CR(Su,T),MI(Su,T)andMCC(Su,T). 61

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Figure4-10. Comparisonswithothermodels.Firstcolumn:SourceimageS(x)andtargetimageT(x).From2nd-5thcolumnsresultingfromCC,CR,MIandMCCrespectively:Top:deformedimages;Bottom:deformationeld. Table4-2. MI(Su,T) MI(Su,T) mean variance PPPPPPPPPmodels 0.11 0.13 0.15 0.17 0.19 0.15 0.0001 localversionof 4 0.6923 1.2343 1.5193 1.2623 0.7232 1.0862 0.1317 localversionof 4 1.4358 1.4492 1.5122 1.4438 1.4358 1.4550 0.0010 Table4-3. MCC(Su,T) MCC(Su,T) mean variance PPPPPPPPPmodels 0.11 0.13 0.15 0.17 0.19 0.15 0.0001 localversionof 4 0.7928 0.8783 0.9898 0.9138 0.8283 0.8805 0.0058 localversionof 4 0.9583 0.9738 0.9950 0.9727 0.9568 0.9709 0.0002 62

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Tocomparethesensitivityofthelocalversionofthemodels( 4 )and( 4 )totheselectionof,weappliedbothmodelstothepairofimagesshowninFigure 4-8 toestimatethedeformationelduandhence,thedeformedimageSuwithvedifferentvaluesof.ThenwecomputedMI(Su,T)withtheresultingSufromeachoftheve'sandTable 4-3 isdesignedinthesamemannerexceptMI(Su,T)beingreplacedbyMCC(Su,T).Theresultsinthesetwotablesindicatethelocalversionof( 4 )islesssensitivetothechoiceof. 4.7ConclusionandDiscussion Renyi'sstatisticaldependencemeasureMCCisproposedasanalternativesimilaritymeasuretoMIformulti-modalimageregistration. Indoingthis,wearemotivatedbythefollowingtwoconsiderations:First,itisdesirabletohaveanalgorithmwhichdoesnotneedtheestimationofcontinuousjointpdf.Inpractice,toosmallwindowsizehasundesirableconsequenceswhereastoolargewindowsizesmoothsoutfeatures.MCCnotonlydependsonquantization(theparametersinK(x,i)andK(x,i)),butalsoonthecoefcientsiandiinthelinearcombination(whichapproximatesthefunctionintakingsupremum)(see( 4 )).Therefore,evenwhenthequantizationisgiven,westillgetagoodregistrationbyoptimizingi's,i'susingdata.Secondly,itisnotcleartheoreticallythattheestimatedMIcomputedfromestimatedjointpdfusingParzenwindowconvergestothetrueMIwhenthenumberofthesamplesgoestoinnity,whilethisconvergencepropertycanbeprovedforMCC,sincetheestimationofMCCusesonlythecovariancesandvariances.Ourexperimentalresultsalsoindicatethattheproposedmodelislesssensitivetothechoiceofthewindowsize,henceismoremanageablethantheMImethodalthoughtheybothdemonstratedcomparablepowerforregistration.ThetheoryofreproducingkernelHilbertspacesisusedtolessenthecomputationaldifcultyinndingthesupremum.However,iftheintensitiesoftheimagestobealignedarecomplex,alargenumberofkernels(mandnin(3.1))areneededtoapproximatethefunctions 63

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tobeoptimized.Thiscanleadtoheavycomputation.Alocalversionoftheproposedmodelmightbeasolution,sincelocallytheimagestobealignedcanberepresentedbyasmallnumberofkernels.Theselectionoftheparametersintheproposedmodel,andcomparisonswiththemodelusingMIasasimilaritymeasurewillbefurtherstudiedinthefuture. ItisprovedusingthetheoryofreproducingkernelHilbertspacesthatthesupremeinthecomputationofMCCcanbetakeninthespace,wherethefunctionscanbeapproximatedbyalinearcombinationsofthekernelfunctions.Therefore,thecomputationcanbemuchsimplied.Alocalversionoftheproposedmodelisalsopresentedtocopewiththedifcultiesinregisteringimageswithnon-stationaryintensitiesornoise.ComparisonsofthelocalversionoftheproposedmodelwithlocalversionofMI,CRorCCbasedmodelsarepresented.AlltheexperimentalresultsindicatethatMCCisagoodalternativetoMIformulti-modalimageregistration. However,parameterselectionisstillanissuethatneedsmoreinvestigationtomaketheimplementationmoreefcientandpractical.Iftheintensitiesoftheimagestobealignedarecomplex,alargenumberofkernels(mandnin( 4 ))areneededtoapproximatethefunctionsfandginMCC.Thiscanleadtoheavycomputation.Weconsiderlocalversionoftheproposedmodelasasolution,sincelocallytheimagestobealignedcanberepresentedbyasmallnumberofkernels.Thechoiceofthelocalwindow(thewidth)dependsontheimagestobealigned.Itisnotaeasyproblem,willbefurtherstudiedinthefuture. 64

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CHAPTER5REGISTRATIONASSISTEDNON-PARAMETRICIMAGESEGMENTATIONOutline TheworkpresentedinthischapteriscompletedasaresultofcollaborationwithHailiZhangandYunmeiChen. Inthischapter,wepresentanovelnonparametricactiveregionmodelforimagesegmentation.Thismodelpartitionsanimagebymaximizingthesimilaritybetweenthatimageandalabelimage,whichisgeneratedbysettingdifferentconstantsastheintensitiesofpartitionedsubregions.Theintensitiesofthesetwoimagescannotbecompareddirectlyastheyareofdifferentmodalities.InthisworkweuseRenyi'sstatisticaldependencemeasure,maximumcrosscorrelation,asacriteriontomeasuretheirsimilarity.Byusingthismeasuretheproposedmodeldealsdirectlywithindependentsamplesanddoesnotneedtoestimatethecontinuousjointprobabilitydensityfunction.Moreover,thecomputationisfurthersimpliedbyusingthetheoryofreproducingkernelHilbertspaces.Experimentalresultsbasedonmedicalandrealimagesareprovidedtodemonstratetheeffectivenessoftheproposedmethod. 5.1Introduction Imagesegmentationistheprocessofpartitioningadigitalimageintomultipleregionsorsetofhomogeneouspixels,whicharemoremeaningfulandeasiertoanalyze.Moreprecisely,imagesegmentationistheprocessofassigningalabeltoeverypixelinanimagesuchthatpixelswiththesamelabelsharecertainvisualcharacteristicsorcertainmeaningfulfeatures.Imagesegmentationorpatternclassicationisoffundamentalimportanceintheeldofmedicalimageprocessing.Duringthelastfewdecades,aconsiderableamountofapproacheshaveemergedtotacklethisissue.However,thedifcultiescausedbyintensityinhomogeneity,higherlevelofnoiseandunevenlydistributedilluminationstillneedtobeaddressed. 65

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LetbeaboundedopensubsetofRn,andI0:!Rbeanobservednoisyimage.ThecelebratedworkfromMumfordandShah[ 50 ]providesthefollowingmodelforsimultaneoussmoothingandsegmentation: minI,)]TJ /F11 11.955 Tf 13.23 18.88 Td[(XiZi(Ii)]TJ /F5 11.955 Tf 11.96 0 Td[(I0)2dx+Zn)]TJ /F2 11.955 Tf 7.31 10.8 Td[(jrIj2dx+j)]TJ /F2 11.955 Tf 6.77 0 Td[(j,(5) where)]TJ /F1 11.955 Tf 10.09 0 Td[(separatesintotworegionsi(i=1,2),j)]TJ /F2 11.955 Tf 6.77 0 Td[(jrepresentsthelengthof)]TJ /F1 11.955 Tf 6.77 0 Td[(,IisapiecewisesmoothapproximationofI0,andIiistherestrictionofItoi.WhenIisaconstantciineachi,model( 5 )reducestothefollowingform, minCi,s,)]TJ /F11 11.955 Tf 9.28 18.89 Td[(XiZ(ci)]TJ /F5 11.955 Tf 11.95 0 Td[(I0)2dx+j)]TJ /F2 11.955 Tf 6.78 0 Td[(j.(5) ThispiecewiseconstantMumford-ShahmodelhasbeenwellstudiedbyChanetal.in[ 18 ][ 67 ].Themajoradvantageofthismodelisthatitcanseparatetworelativelyhomogeneousregionswithoutusinganyedgeinformation.However,thehomogeneityassumptionlimitsitsapplications. Amoregeneralapproachisparametricregionbasedactivecontourmethod.Thismethodisbasedontheassumptionthatateachx2i,theimageintensityI0(x)isanindependentrandomvariabledrawnfromtheprobabilitydensityfunction(p.d.f.)P(I0i(x)ji),whereI0iistherestrictionofI0toiandiisaparametervectorwhichneedstobeestimated.Theframeworkofthismethodminimizesthenegativelog-likelihoodfunctionaltogetherwiththelengthterm,i.e., min,i)]TJ /F11 11.955 Tf 11.3 11.36 Td[(XiZilogP(I0i(x)ji)dx+jCj.(5) IntheregioncompetitionmodelbyZhuetal.[ 76 ]andgeodesicactiveregionmodelsbyRoussonetal.[ 59 ]andParagiosetal.[ 53 ],P(I0i(x)ji)ischosentobeaGaussiandistribution: P(I0i(x)jci,i)=1 p 2iexp )]TJ /F3 11.955 Tf 11.3 -.17 Td[((I0i(x))]TJ /F5 11.955 Tf 11.95 0 Td[(ci)2 22i!.(5) 66

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Ifallthei'sarethesameandprexed,model( 5 )( 5 )reducestomodel( 5 ).ThisisaglobalGaussianmodelasitassumesallrandomvariablesI0i(x)intheregionisharethesamemeanciandvariancei.Model( 5 )providesdesirablesegmentationresultswhentheparametricformoftheintensitydistributionisknown.However,aspecicassumptionoftheintensitydistributioncanbeasignicantrestrictioninrealapplications,especiallywhentheimagehasheavynoiseortheimageisofmulti-modalintensitydistribution. Toovercomethisproblem,nonparametricmodels[ 26 ]havebeendevelopedtoincreasetherobustnessandsuccessfullyappliedtoimagesegmentationandregistration.Thesemethodsarefeaturedbyusingnonparametricdensityestimationtoreplacetheparametricdensityestimation.Forinstance,thenonparametricactivecontourmodelin[ 1 ]isdrivenbythedisparityoftheforegroundandbackgroundp.d.f.'s,whichareapproximatedbyParzenwindowdensitymethod.In[ 5 ]thedynamicsegmentationofvideoimagesequencesisobtainedbyminimizingthedisparityofthep.d.f.ofthecurrentframewiththepreviousoneandthep.d.f.'sarealsoestimatedusingParzenwindowmethod.Thevariationalsegmentationmodelin[ 37 ]incorporatesboundaryinformationwithregioninformation,wheretheboundaryinformationisobtainedfromtheedgemapimageandtheinteriorregioninformationisrepresentedbytheintensityp.d.f.capturedusingParzenwindowdensityestimation.Moryetal.[ 38 ]regardtheforegroundandbackgroundp.d.f.'stobeunknown,whichareintegratedintheregioncomputationmodel.Theproposedmodelcouldsimultaneouslyperformsegmentationandnonparametricdensityestimation,whichareupdatedusingtheParzenwindowdensitymethod.[ 14 ]regardstheforegroundandbackgroundcumulativedistributionfunction(c.d.f.)asunknownsandutilizestheWassersteindistancetomeasurethedisparityoflocalc.d.f.withtheestimatedc.d.f.'s.Aworkcloselyrelatedtothispaperis[ 41 ],inwhichKimetal.segmentimagesthroughmaximizingmutualinformationbetweentheimagetobesegmentedanditscorrespondinglabelimage 67

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denedbysettingdifferentconstantsasimageintensitiesofpartitionedsubregions(ref.Section2),whichturnsouttobeminimizingthedisplacementoftheLogarithmicoftheforegroundandbackgroundp.d.f.'sandthep.d.f.'sareagainestimatedusingParzenwindowmethod. Borrowingtheideafrom[ 41 ],inthispaperweproposeanewapproachofnonparametricimagesegmentationthatusesRenyi'sstatisticaldependencemeasure,maximumcorrelationcoefcient,asasimilaritymeasureoftwoimagesindifferentmodalities.Byusingthismeasureasanalternativechoiceofdependencemeasuretomutualinformation,wedon'tneedtoestimatethecontinuousjointprobabilitydensityfunctionoftwoimages,whichissensitivetoimagequantizationandmaketheoptimizationprocesscomplicated.Moreover,thecomputationisfurthersimpliedbyapplyingthetheoryofreproducingkernelHilbertspaces. Theremainderofthispaperisorganizedasfollows:InSection2,weintroduceourproblemandgiveabriefreviewofthemutualinformationbasednonparametricimagesegmentationapproach;Section3containssomebackgroundinformationaboutRenyi'sstatisticaldependencemeasureandreproducingkernelHilbertspaceassociatedwithGaussiankernels;WeproposeourapproachandnumericalschemesinSection4;thenumericalexperimentalresultsarepresentedinSection5andweconcludethispaperinSection6. 5.2ProblemStatementandRelatedWorks LetbeaboundedLipschitzdomain,I:!Rbeagivenimageand~cbeanarbitrarycurveinthedomain.Thesegmentationproblemistomove~csuchthatitseparatestheforegroundfromthebackground.Orequivalently,tondacurve~csuchthatthelabelimageLcorrespondingtothecurve~cmatchesthebestwiththeoriginalimageI,wherethelabelimageL:!fF,Bgisdenedas L(x)=8><>:Fifx2R;Bifx2Rc.(5) 68

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RandRcdenotetheregioninsideandoutside~crespectively. Thechallengeinmatchingthesetwoimagesliesinthefactthattheyareofdifferentmodalities,soitdoesnotmakesensetodirectlycomparetheirintensities.Tocopewiththisdifculty,anumberofsimilaritymeasuresbasedonstatisticaldependencehavebeenproposed.Forinstance,[ 41 ]choosestomaximizethemutualinformationbetweenoriginalimageIanditslabelimageL,togetherwithaconstraintofthelengthterm,i.e., E(~c)=)]TJ /F5 11.955 Tf 9.3 0 Td[(MI(I(X),L(X))+I~cds.(5) In( 5 ),Xisviewedasarandomvariableuniformlydistributedovertheimagedomain,L(X)isarandomvariabletakingtwovaluesandI(X)isalsoregardedasarandomvariableviaX.ThemutualinformationMI(I(X),L(X))isdenedasfollows: MI(I(X),L(X))=h(I(X)))]TJ /F5 11.955 Tf 11.95 0 Td[(h(I(X)jL(X))=h(I(X)))]TJ /F5 11.955 Tf 11.95 0 Td[(Pr(L(X)=F)h(I(X)jL(X)=F))]TJ /F5 11.955 Tf 11.96 0 Td[(Pr(L(X)=B)h(I(X)jL(X)=B),(5) wheretheentropyofacontinuousrandomvariableZis h(Z)=)]TJ /F11 11.955 Tf 11.3 16.27 Td[(ZRNpZ(z)logpZ(z)dz.(5) Sinceh(I(X))isindependentofthecurve~c,soweonlyneedtoestimateh(I(X)jL(X)=F)andh(I(X)jL(X)=B),whichareestimatedbyusingthenonparametricParzenwindowdensitystrategy,i.e., h(I(X)jL(X)=F))]TJ /F3 11.955 Tf 27.8 8.08 Td[(1 jRjZRlog1 jRjZRK(I(x))]TJ /F5 11.955 Tf 11.96 0 Td[(I(^x))dx,(5) h(I(X)jL(X)=B))]TJ /F3 11.955 Tf 30.36 8.09 Td[(1 jRcjZRclog1 jRcjZRcK(I(x))]TJ /F5 11.955 Tf 11.95 0 Td[(I(^x))dx.(5) Themutualinformationcouldbeeffectivelyusedasasimilaritymeasuretomatchtheimagetobesegmentedanditslabelimage.However,itrequirestoestimatethejointp.d.f.ofIandL,whichissensitivetoimagequantizationandincreasesthecomplexity 69

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ofcomputation.Inthiswork,wechoosetouseRenyi'sstatisticalmeasure,maximumcrosscorrelation,asasimilaritymeasure.Thismeasuredealsdirectlywithsamplesanddoesnotneedtoestimatethecontinuousjointp.d.f..Abriefreviewofthismeasureisprovidedinthenextsection. 5.3Renyi'sStatisticalMeasure In[ 56 ]RenyiproposedasetofpostulatesforasuitabledependencemeasureQoftworandomvariables/vectorsXandY,whichhasdrawnmuchattentioneversince.Thesepostulatesinclude 1. Q(PX,Y)iswell-dened; 2. 0Q(PX,Y)1; 3. Q(PX,Y)=0ifandonlyifX,Yareindependent; 4. Q(PX,Y)=1ifY=f(X)orX=g(Y),wherefandgareBorelmeasurablefunctions. ThenRenyishowedthatonemeasuresatisfyingtheseconditionsis Q(PX,Y)=supf,g2VCC(f(X),g(Y)),(5) whereVisthespaceofallBorelmeasurablefunctionswithnitepositivevariance,andCC(f(X),g(Y))isthecorrelationcoefcientoff(X)andg(Y),i.e., CC(f(X),g(Y))=Cov(f(X),g(Y)) p Var(f(X))p Var(g(Y)).(5) ThedifcultyofusingRenyi'smeasureliesinthefactthatweneedtondtheoptimalfandginthespaceV,whichisthesetofallBorelmeasurablefunctionswithnitepositivevariance.Itisextremelydifculttosearchfandginsuchahugespace.Fortunately,wehaveshownin[ 74 ]thatthesupremuminVcouldbeattainedinamuchsmallerspace,whichisareproducingkernelHilbertspace(RKHS)associatedwithareproducingkernelthatiscontinuous,symmetric,positivedeniteandvanishingat 70

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innity.Inthiswork,wechoosetheGaussianfunctiontobethereproducingkernel,i.e., K(x,y)=1 p 2exp()]TJ /F3 11.955 Tf 10.49 7.92 Td[((x)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2 22).(5) AccordingtothetheoryofRKHS(see[ 74 ]),anytwofunctionsfandgintheRKHSassociatedwiththeGaussiankernelcanbeapproximatedbyfunctionspandqoftheform, p(x)=nXi=1i p 2exp()]TJ /F3 11.955 Tf 10.5 7.92 Td[((x)]TJ /F5 11.955 Tf 11.95 0 Td[(yi)2 22),(5) and q(x)=mXj=1j p 2exp()]TJ /F3 11.955 Tf 10.49 7.92 Td[((x)]TJ /F5 11.955 Tf 11.95 0 Td[(zj)2 22),(5) forsomeparameters,yi,i,zj,j,i=1,2,,n,j=1,2,,m.Inpractice,wecanchoosefandgtobetheaboveformandx,yi,zj.Thereforeweonlyneedtoestimatethecoefcientsiandj,whichcouldsignicantlyreducethecomputation. 5.4ProposedModelandNumericalMethod Inthissection,weproposeourmodelandcorrespondingnumericalschemes.Ouraimistondacurve~csuchthattheresultedlabelimageLdenedin(5)matchesbestwiththeoriginalimageI.InthisworkweusetheRenyi'sstatisticaldependencemeasure,maximumcorrelationcoefcient,asasimilaritymeasureforaligningIandL.Byusingthismeasurewedon'tneedtoestimatethecontinuousjointp.d.f.oftwoimagesasinthemodelsbasedonmutualinformation. Asisshowninthepreviouspostulates,whentheimageIanditslabelimageLarefunctionsofeachother,i.e.,L=f(I)orI=g(L)forsomefunctionforg,themaximumcrosscorrelationattainsitsmaximumvalue1.NotethatLispiecewiseconstant,soistheresultingimageg(L).ItdoesnotmakeabigdifferencebymaximizingthecrosscorrelationbetweenIandLorg(L),sointhefollowingwechoosetomaximizethecrosscorrelationbetweenf(I)andthelabelimageL.Theobjectiveenergyfunctionalis 71

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obtainedbycombiningthecrosscorrelationoff(I)andLandthelengthof~c,i.e., E(~c,a1,,an)=I~cds+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))2,(5) where f(I(x))=nXiaiK(I,yi),(5) andthecorrespondinglabelimageL(x)isdenedby L(x)=8><>:c1ifx2R;c2ifx2Rc.(5) 5.4.1LevelSetFormulationandNumericalMethod Energyfunctional( 5 )canbeminimizedusingthelevelsetapproach[ 52 ][ 20 ][ 21 ].Thecurve~cisrepresentedbythezerolevelofaLipschitzfunction:!Randtheresultingenergyfunctionalbecomes E(,a1,,an)=ZjrH((x))jdx+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))2,(5) whereHistheHeavisidefunctionand L(x)=c1H((x))+c2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(H((x))).(5) Thealternateminimization(AM)approach[ 19 ]isemployedtosolvethisproblem.First,wekeepxedandminimize( 5 )orequivalently,maximizeCC(f(I),L)withrespecttoai'sasthelengthtermisindependentofai's. Forsimplicity,weintroducesomenotations.TheimageIandLareviewedasvectorsoflengthN,whereNisthenumberofpixels.Foreachi=1,2,,n,setpi=K(I)]TJ /F5 11.955 Tf 12.76 0 Td[(yi)beavectoroflengthN.LetPtobetheNnmatrixwhosecolumnvectorsarep1,p2,,pnandabeacolumnvectorwhoseithentryisai.Thus f(I)=nXiaiK(I,yi)=nXi=1aipi=Pa,(5) 72

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and CC(f(I),L)=hPa)]TJ ET q .478 w 229.1 -17.8 m 242.71 -17.8 l S Q BT /F5 11.955 Tf 229.1 -27.78 Td[(Pa,L)]TJ ET q .478 w 269.58 -17.8 m 276.35 -17.8 l S Q BT /F5 11.955 Tf 269.58 -27.78 Td[(Li jPa)]TJ ET q .478 w 228.52 -35.27 m 242.13 -35.27 l S Q BT /F5 11.955 Tf 228.52 -45.24 Td[(PajjL)]TJ ET q .478 w 270.16 -35.27 m 276.93 -35.27 l S Q BT /F5 11.955 Tf 270.16 -45.24 Td[(Lj=hP0a,L0i jP0ajjL0j,(5) whereP0ij=Pij)]TJ /F9 7.97 Tf 14.23 4.71 Td[(1 NPLk=1Pk,j,L0=L)]TJ ET q .478 w 213.92 -56.27 m 220.7 -56.27 l S Q BT /F5 11.955 Tf 213.92 -66.25 Td[(L. SincenN,rank(P0)=n,i.e.,thecolumnvectorsofP0areindependent.ApplytheGram-SchmidtProcesstothecolumnvectorsofP0,wegetasequenceoforthogonormalvectorse1,e2,,en.AssumeP0a=Pni=1bieiforsomebi2R,i=1,2,,n.LetEtobethematrixwhosecolumnvectorsaree1,e2,,enandbbeacolumnvectorwhoseithentryisbi,thenP0a=Pni=1biei=Eb.Foreachi=1,2,,n,denoteci=hei,L0i.Setc=[c1,c2,,cn]T,weget CC(f(I),L)=hP0a,L0i jP0ajjL0j=hPnieibi,L0i jEbjjL0j=Pnibici jbjjL0jjbjjcj jbjjL0j=jcj jL0j.(5) Theaboveequalityholdsifandonlyifbandcaredependent,i.e.,b=rcforsomer2R.Notethatthevaluerdoesnotaffectthenalresult.Withoutlossofgenerality,letr=1,i.e.,b=c.Therefore,P0a=Ecanda=(PTP))]TJ /F9 7.97 Tf 6.58 0 Td[(1PTEc. Nextwekeepa1,,anxedandsolveforusingthegradientdescentapproach,i.e., @ @t=()divr jrj+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))F,(5) whereistheregularizedDiracfunctionand F=(f(I))]TJ ET q .478 w 148.87 -448.48 m 169.88 -448.48 l S Q BT /F5 11.955 Tf 148.87 -459.12 Td[(f(I))Var(L))]TJ /F5 11.955 Tf 11.95 0 Td[(Cov(f(I),L)(L)]TJ ET q .478 w 315.12 -449.14 m 321.89 -449.14 l S Q BT /F5 11.955 Tf 315.12 -459.12 Td[(L) Var(f(I))1 2Var(L)3 2(c1)]TJ /F5 11.955 Tf 11.96 0 Td[(c2).(5) Theabovetwoprocessesarealternativelyupdateduntilwereachasatisfactoryresult.Toincreasetherateofconvergence,thesemi-implicitdifferenceschemeisappliedin( 5 ),i.e., n+1)]TJ /F7 11.955 Tf 11.95 0 Td[(n t=(n)divrn+1 jrnj+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(CC(f(I)n,Ln))Fn.(5) 73

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Thisequationcouldbeeffectivelysolvedbyusingtheadditiveoperatorsplitting(AOS)method.Notethatthecurvaturetermin( 5 )isapproximatedby @ @x x p 2x+2y+2!+@ @y y p 2x+2y+2!,(5) whereisasmallpositivenumberincasethatthedenominatorsbecomezero.However,itmaystillcausestabilityissuesandlimittheconvergencerate. 5.4.2ASoftFormulationandNumericalMethod Toavoidlocalminimumproblemweproposeasoftformulationoftheenergyfunctionalin( 5 ),anduseChambolle'sdualmethod[ 17 ][ 15 ]tosolveit. Letu:![0,1]beafuzzymembershipfunctionandrewritetheenergyfunctionalas E(u,a1,,an)=Zjru(x)jdx+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))2,(5) where L(x)=c1u(x)+c2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(u(x)).(5) Inrealapplications,wewantLbeabinaryimagesothatitcouldgiveareasonablesegmentation.Soduringeachiteration,wereset L(x)=c1(u>.5)+c2(1)]TJ /F7 11.955 Tf 11.95 0 Td[((u>.5)),(5) andreferstothecharacteristicfunction. Followingthestrategyin[ 17 ][ 15 ],weintroduceanauxiliaryvariablev:![0,1]andconsiderthefollowingapproximatedenergyfunctional E(u,v,a1,,an)=Zjru(x)jdx+1 2ku)]TJ /F5 11.955 Tf 11.96 0 Td[(vk2+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))2,(5) where L(x)=c1v(x)+c2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(v(x)),(5) 74

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andischosentobesmallenoughsuchthattheminimizersu?andv?areclosetoeachother. Westillemploythealternateminimization(AM)approachtosolvethisminimizationproblem,i.e.,wegoontoalternativelysolvethefollowingtwoproblems: minuZjru(x)jdx+1 2ku)]TJ /F5 11.955 Tf 11.96 0 Td[(vk2(5) and min0v1,ai1 2ku)]TJ /F5 11.955 Tf 11.95 0 Td[(vk2+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))2.(5) Theminimizationproblem( 5 )couldbeeffectivelysolvedbyapplyingChambolle'smethod[ 17 ]andthesolutionis u(x)=v(x))]TJ /F7 11.955 Tf 11.95 0 Td[(divp(x),(5) wherep=(p1,p2)isgivenby )-222(r(divp)]TJ /F5 11.955 Tf 11.96 0 Td[(v)+jr(divp)]TJ /F5 11.955 Tf 11.96 0 Td[(v)jp=0.(5) Equation( 5 )couldbesolvedbyaxedpointmethod,i.e., pn+1=pn+r(divpn)]TJ /F5 11.955 Tf 11.95 0 Td[(v=) 1+rjdivpn)]TJ /F5 11.955 Tf 11.95 0 Td[(v=j.(5) Followingthesamestrategyin[ 15 ],thesolutionvof( 5 )isgivenby v=min(max(u+G,0),1),(5) where G=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(CC(f(I),L))(f(I))]TJ ET q .478 w 177.7 -537.64 m 198.71 -537.64 l S Q BT /F5 11.955 Tf 177.7 -548.28 Td[(f(I))Var(L))]TJ /F5 11.955 Tf 11.95 0 Td[(Cov(f(I),L)(L)]TJ ET q .478 w 343.95 -538.31 m 350.72 -538.31 l S Q BT /F5 11.955 Tf 343.95 -548.28 Td[(L) Var(f(I))1 2Var(L)3 2(c1)]TJ /F5 11.955 Tf 11.96 0 Td[(c2).(5)ai'sareonlyincludedinthecrosscorrelationterm,sotheoptimizationschemeisexactlythesameasthelevelsetapproach. 75

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Experimentalexperiencetellsusbothofthesetwomethods(levelsetmethodandChambolle'smethod)arequiteeffective.However,itispossiblethatwemayresultinalocalminimum.Thisisduetothenon-convexityofthecrosscorrelationtermintheenergyfunctional( 5 ).Therefore,( 5 )mayhavemorethanoneminimizerandwecannotguaranteethatthisapproachconvergestoaglobalminimizer. 5.5ExperimentalResults Inthissectionweshowourexperimentalresultsonvariousimagestodemonstratetheeffectivenessoftheproposedmodelforsegmentation.Forthelevelsetapproach,theinitialissettobethesigneddistancefunctionoftheinitialcirclesintheimage.Inpracticewetrytoplaceasmanycirclesaspossibleandletthemcovertheentireregion.However,Chambolle'sapproachismorerobusttotheinitialization.Theinitialuandvcanbegeneratedasrandomeldsintherange[0,1].Allofthetestimagesarerescaledtotheinterval[0,1]andtheparametersc1,c2inthelabelimageareprexedtobe1or2.Ineachgure,weincludethetestimageI,thenaltransformedimagef(I),thenallabelimageLandthesegmentationresult,i.e.,thecontours(=0oru=0.5)issuperposedontheoriginalimage. ThepurposeoftheExperimentI(Figure1)istoshowtheperformanceoftheproposedmodelontwosyntheticimagesthatarefeaturedbyobjectsintensitiesoverlappingwiththatofthebackground(seeimage(a)),andintensityinhomogeneitycausedbyseriousunevenlydistributedillumination(seeimage(f)).ThepiecewiseconstantMumford-Shahmodelcouldnotworkasindicatedintheresults(e)(j).Thereasonisthattheintensityoverlappingeffect(a)andtheunevenlydistributedillumination(f)makethepiecewiseconstantassumptionfail.However,theproposednonparametricmodelsuccessfullyseparatestheobjectfromthebackgroundshownin(d)and(i)respectively.Thisisbecausethebackgroundofthesetwoimagesarecomparativelyhomogeneous,wecouldselectyi'sfromtheintervalswheretheintensitiesofthebackgroundlie.Thusafterapplyingthefunctionf,linercomposition 76

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ofaseriesGaussianfunctionscenteredatyi's,totheimageI,thevaluesoff(I)intheinhomogeousregionbecomeextremelysmallsuchthatthetransformedimagesf(I)aremorehomogeous.Therefore,bycarefullychoosingyi'sinthefunctionfandmaximizingthecorrelationcoefcientofthetransformedimagesf(I)in(b)(g)withthelabelimageL,wegetdesirableresults.ThecomparisonwiththepiecewiseconstantMumford-Shahmodelindicatesthattheparametricmodelsfailswhentheimageintensitydistributionismorecomplicated.Thatiswhyweneedtoexploreforthemoregeneralnonparametricmodels. (a)(b)(c)(d)(e) Figure5-1. Segmentationresultsofasyntheticimage.(a)TestimagesI;(b)Finalf(I);(c)LabelimagesL;(d)Segmentationresultoftheproposedmodel;(e)SegmentationresultsobtainedfrompiecewiseconstantMumford-Shahmodel. ThesecondexperimentII(Figure2)aimstovalidatetheeffectivenessoftheproposedmodelinrealmedicalimages.Inthisexperimentweappllyourmodeltoacleanbrainimageanditsnoisyversion.Thesegementationresultsarecomparedwiththoseobtainedfromtheparametricmodel( 5 )( 5 ).Wersttestourmodelonacleanbrainimage(rstrow)andthenweaddsomeGaussianwhitenoisetotestitsrobustness(secondrow).Thesegmentationresultsoftheparametricmodel( 5 )( 5 )areplacedattheendofeachrow(e)(j).Theparametricmodel( 5 )( 5 )separatesthebackground,cerebrospinaluid(csf)fromthewhitematterandgraymatter.However,thesearequitedifferentfromtheresults(d)(i)obtainedfromtheproposedmodel.From(d)(i),wecanseethatthegraymatterisseparatedfromtherestandthewholeimageisactuallysegmentedintothreepartsevenifweonlydothetwophasesegmentation. 77

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Thisisreasonablebecauseweselectyi'sfromaparticularinterval[0.4,0.5](ref.(k)),wherethegraymatterlies.Therefore,afterweapplythefunctionftothetestimagesI,intensitiesofthewhitematter,backgroundandcsf,whicharenotinthisinterval,becomealmostzerowhileintensitiesofthegraymatterareenlarged.Thatiswhythewhitematter,csfandbackgroundlookdarkwhilethegraymatterlooksbrightinthetransformedimagesf(I)(b)(g).Inthiscase,thewhitematter,csfandbackgroundareregardedasawhole,whicharethenseparatedfromthegraymatter.Thelastimage(l)showsthatthecrosscorrelationbetweentheimagef(I)andthelabelimageL,i.e.,CC(f(I),L),keepsincreasingastheiterationprocessgoes.AnotherpointworthmentioningisthatthistrendcoincideswiththemutualinformationbetweenthetestimageIandthelabelimageL.Sowecanconcludethattheproposedmethodisconsistentwiththemutualinformationbasednonparametricimagesegmentationmethod[ 41 ]. Theabovetwoexperimentsindicatethatbychoosingaseriesofspecicyi's,theproposedmodelworkswellforimageswithinhomogeneity,unevenlydistributedilluminationanditcangetmultiphasesegmentationresultswhileonlyusingtwophases.Inthefollowing,wedonotpaytoomuchattentionontheselectionofyi'sandletthemtobeequallyspacedintheinterval[0,0.5]. ExperimentIII(Figure3)aimstotestwhethertheproposedmodelworksforimageswithnestructures.Herewechoosethetestimagetobealungimagewithlotsofnedetails.AsExperimentII,werstsegmentthecleanimageandthenweaddsomenoisetomakeitmoreinhomogeneous.AfterapplyingthefunctionftotheoriginalimagesI(a)(e),theresultedimagesf(I)(b)(f)havemorestrongcontrastbetweendifferentfeatureswhilestillpreservingthedetailedstructures.Thesameparametersareappliedforthesetwotests.Fromthenalresults(d)(h),wecanseethatmostofthenestructuresarecapturedandthenoiseinhomogeneitydoesnotexertabigdifference. 78

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(a)(b)(c)(d)(e) (f)(g)(h)(i)(j) Figure5-2. Segmentationresultsofacleanbrainimage(rstrow)anditsnoisyversion(secondrow).Thersttworows(fromlefttoright):(a)(f)TestimageI;(b)(g)Imagesf(I);(c)(h)thelabelimagesL;(d)(i)Segmentationresultsoftheproposedmodel;(e)(j)Segmentationresultsobtainedfromtheparametricmodel( 5 )( 5 ).Thethirdrow:(k)Histogramofthetestimage(a);(l)theinformationofCC(f(I),L)andMI(I,L)duringeachiteration. InExperimentIV(Figure4),wetestourmodelonthecameramanimageanditsnoisyversion.Thetestimagesarenotpiecewiseconstantandcannotbeverywellapproximatedbyimagestakingonlytwovalues,whichseemschallengingforthetwophasesegmentationwork.Wecanseethatthetransformedimages(b)(f)lookspiecewiseconstantandcouldbemoreeasilysegmentedthantheoriginalimages(a)(e).Thecontoursgotfromthelabelimages(c)(g)twellwiththeboundariesinthetestimagesIandgivethecorrectsegmentation. 79

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Figure5-3. Segmentationresultsofacleancameramanimage(rstrow)andanoisyone(secondrow).Fromlefttoright:TestimagesI,transformedimagesf(I),labelimagesLandthenalcontour(u=0.5)superimposedonthetestimages. 5.6Conclusion Inthispaper,weproposedanovelimagesegmentationframeworkbasedonRenyi'sstatisticaldependencemeasure.Thismodelexploitsthebenetofmulti-modalimageregistrationbyextractingtheedgeinformationfromalabelimagebeingdeformedandbyapplyingittotheimagetobesegmented.Suchaprocedureintegratesmorefeatureinformationintotheprocessofedgeevolving,comparedtothosewithoutregistrationassisted.Furthermore,theedgeswereoptimizedandnalizedbycompetingnearbyfeaturesduringtheregistrationprocess,yieldingbettersegmentationresult.Atthemeantime,themodelissimpliedbyapplyingthetheoryofreproducingkernelHilbertspace.SinceRenyisstatisticaldependencemeasurecanbeestimateddirectlywithindependentsamples,themodelbecomemoreefciencyasthereisnoneedtoestimatethecontinuousjointprobabilitydensityfunction,whichistimeconsuming. 80

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Twonumericalapproaches,thelevelsetmethodandChambolle'sdualapproach,areemployedduringtheimplementation. Imagesegmentationisoneofthemostimportanttaskswhenextractingandanalyzingfeatureinformationfromvarioussources.Althoughalotofworkhasbeendone,automaticimagesegmentationwithhigherrobustnessandaccuracystillremainsanopenproblem.Segmentationofimageswithcomplexinhomogeneousregionsandlocaldistortions,andsegmentationofhighdimensionalimagesbelongtothemostchallengingtasksatthismoment.Themajordifcultyofhighdimensionalimagesegmentationresidesinitscomputationalcomplexity.Althoughthespeedofcomputershasbeenincreasing,thevolumeofdataforprocessingisgrowingandtheneedtodecreasethecomputationaltimeofmethodspersists. 81

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APPENDIXAPROOFOFTHEDENSETHEOREMINSECTION4.2.1 Lemma1. H0(E)isdenseinC0(E). Proof:IfH0(E)isnotdenseinC0(E),byHahn-Banachtheoremthereisafunctiong2C0(E)nH0(E),andaboundedsignedmeasurem2C0(E),thedualspaceofC0(E),suchthat ZEgdm6=0,(A) andforallf2H0(E),REfdm=0.Inparticular,foranyx2E,REK(x,y)dmy=0.Hence,ZK(x,y)dmxdmy=0. SinceKispositivedenite,thisimpliesm=0,whichcontradicts( A ). Lemma2. LetVB(E)bethespaceofboundedmeasurablefunctionsonE.LetalsoXandYbetworandomvariableswithjointpdfPXY.Then,supf,g2VB(E)CC(f(X),g(Y))=supf,g2C0(E)CC(f(X),g(Y)). Proof:Foranyfunctionsf,g2VB(E)therearetwosequencesoffunctionsfn,gn2C0(E),suchthatforanyq>0,asn!1, E[jfn(X))]TJ /F5 11.955 Tf 11.95 0 Td[(f(X)jq]!0,E[jgn(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(g(Y)jq]!0, (A) Bytakingq=1,2onecangetthatE[jfn(X)j]isuniformlyboundedinn,thenfrom( A )withq=2 E[fn(X)gn(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(f(X)g(Y)]E[jfn(X))]TJ /F5 11.955 Tf 11.96 0 Td[(f(X)jjg(Y)j]+E[gn(Y))]TJ /F5 11.955 Tf 11.95 0 Td[(g(Y)jjfn(X)j]C(E[jfn(X))]TJ /F5 11.955 Tf 11.95 0 Td[(f(X)j2]1=2+E[gn(Y))]TJ /F5 11.955 Tf 11.95 0 Td[(g(Y)j2]1=2)!0, (A) 82

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asn!1.Furthermore,from( A )withq=1and( A )E[(fn(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E[fn(X)])(gn(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(E[gn(Y)])]=E[fn(X)gn(Y)])]TJ /F5 11.955 Tf 11.95 0 Td[(E[fn(X)]E[gn(Y))]!E[f(X)g(Y)])]TJ /F5 11.955 Tf 11.95 0 Td[(E[f(X)]E[g(Y))]=E[(f(X))]TJ /F5 11.955 Tf 11.95 0 Td[(E[f(X)])(g(Y))]TJ /F5 11.955 Tf 11.95 0 Td[(E[g(Y)])]. Similarly,wecanshowthatasn!1,E[(fn(X))]TJ /F5 11.955 Tf 11.95 0 Td[(E[fn(X)])2]=E[fn(X)2])]TJ /F5 11.955 Tf 11.95 0 Td[(E[fn(X)]2!E[f(X)2])]TJ /F5 11.955 Tf 11.95 0 Td[(E[f(X)]2=E[(f(X))]TJ /F5 11.955 Tf 11.95 0 Td[(E[f(X)])2], andE[(gn(Y))]TJ /F5 11.955 Tf 11.95 0 Td[(E[gn(Y)])2]!E[(g(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(E[g(Y)])2]. Then,bythedenitionofCC,wegetCC(fn(X),gn(Y))!CC(f(X),g(Y)). Thus,supf,g2VB(E)CC(f(X),g(Y))=supf,g2C0(E)CC(f(X),g(Y)). Lemma3. LetV(E)bethespaceofallBorelmeasurablefunctionsonEwithnitepositivevariance,andXandYbethesameasinLemma2.Thensupf,g2VB(E)CC(f(X),g(Y))=supf,g2V(E)CC(f(X),g(Y)). Proof:Foranyfunctionf2V(E),denefnasfn=f,ifjfjn,andfn=nsign(f),ifjfj>n. Then,fn2VB(E),fnconvergestofpointwiseandasn!1, E[fn(X)]!E[f(X)].(A) 83

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Moreover,f2nf2,and E[f2n]E[f2].(A) From( A ),E[(fn(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E[fn(X)])2]=E[fn(X)2])]TJ /F5 11.955 Tf 11.96 0 Td[(E[fn(X)]2E[fn(X)2]E[f(X)2]. Nowbyusingthedominatedconvergencetheoremwith( A )weconcludethatasn!1,E[(fn(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E(fn(X)))2]!E[(f(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E(f(X)))2]. Similarly,wecanhaveE[(gn(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(E(gn(Y)))2]!E[(g(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(E(g(Y)))2].E[(fn(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E(fn(X)))(gn(Y))]TJ /F5 11.955 Tf 11.96 0 Td[(E(gn(Y)))]!E[(f(X))]TJ /F5 11.955 Tf 11.96 0 Td[(E(f(X)))(g(Y))]TJ /F5 11.955 Tf 11.95 0 Td[(E(g(Y)))]. ThenbythedenitionofCC,wegetCC(fn(X),gn(Y))!CC(f(X),g(Y)). Thelemmafollowsfromthisimmediately. 84

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APPENDIXBE-LEQUATIONOFFORTHEGLOBALVERSIONOFMCCMODEL Byadoptingthenotationsinthepaper,wehavethefollowingestimatesforthemeans,variancesandcovariances:Sfu+v(x)=f(S(x+u(x)+v(x))),Tg(x)=g(T(x)),u+v1=1 jjZSfu+v(x)dx,u1=1 jjZSfu(x)dx,2=1 jjZTg(x)dx.u+v1=1 jjZ(Sfu+v(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u+v1)2dx,u1=1 jjZ(Sfu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1)2dx,2=1 jjZ(Tg(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2)2dx,u+v1,2=1 jjZ(Sfu+v(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u+v1)(Tg(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2)dx,u1,2=1 jjZ(Sfu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1)(Tg(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2)dx. Itiseasytoseethatd d=0Sfu+v(x)=rSfu(x)v(x),1 jjZu+v1(Tg(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2)dx=0. Hence, d d=0u+v1,2=d d=01 jjZSfu+v(x)(Tg(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2)dx=1 jjZ(Tg(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2)d d=0Sfu+v(x)dx=1 jjZ(Tg(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2)rSfu(x)v(x)dx (B) Bynotingthat2 jjZ(Sfu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1)d d=0u+v1dx=0, 85

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wehave d d=0u+v1=2 jjZ(Sfu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1)d d=0(Sfu+v(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u+v1)dx=2 jjZ(Sfu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1)d d=0Sfu+v(x)dx=2 jjZ(Sfu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1)rSfu(x)v(x)dx (B) Byusing B and B ,wehaved d=0CC(Sfu+v(x),Tg(x))=d d=0u+v1,2(u+v1))]TJ /F9 7.97 Tf 6.59 0 Td[(1=2(2))]TJ /F9 7.97 Tf 6.59 0 Td[(1=2=(u1d d=0u+v1,2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 2u1,2d d=0u+v1)(u1))]TJ /F9 7.97 Tf 6.59 0 Td[(3=2(2))]TJ /F9 7.97 Tf 6.58 0 Td[(1=2=1 jjZTg(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2 (u1)1=2(2)1=2)]TJ /F5 11.955 Tf 23.91 8.09 Td[(Sfu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1 (u1)3=2(2)1=2u1,2rSfu(x)v(x)dx Hence,theE-LequationforCC(Sfu(x),Tg(x))withrespecttou(x)is1 jjTg(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2 (u1)1=2(2)1=2)]TJ /F5 11.955 Tf 23.9 8.09 Td[(Sfu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1 (u1)3=2(2)1=2u1,2rSfu(x)=0. 86

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APPENDIXCE-LEQUATIONOFFORTHELOCALVERSIONOFMCCMODEL Byadoptingthenotationsinthepaper,wehavethefollowingestimatesforthemeans,variancesandcovariances:Sfzu+v(x)=fz(S(x+u(x)+v(x))),Tgz(x)=gz(T(x)),u+v1(z)=ZSfzu+v(x)G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx,u1(z)=ZSfzu(x)G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)dx,2(z)=ZTgz(x)G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx,u1(z)=Z(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z))2G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx,u+v1(z)=Z(Sfzu+v(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u+v1(z))2G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx, 2(z)=Z(Tgz(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2(z))2G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)dx,u+v1,2(z)=Z(Sfzu+v(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u+v1(z))(Tgz(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2(z))G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)dx,u1,2(z)=Z(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z))(Tgz(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2(z))G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx. Itiseasytoseethatd d=0Sfzu+v(x)=rSfzu(x)v(x),Zu+v1(z)(Tgz(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2(z))G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)dx=0. Hence, d d=0u+v1,2(z)=d d=0ZSfzu+v(x)(Tgz(x))]TJ /F7 11.955 Tf 11.96 0 Td[(2(z))G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)dx=Z(Tgz(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2(z))G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)d d=0Sfzu+v(x)dx=Z(Tgz(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2(z))G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)rSfu(x)v(x)dx (C) Bynotingthat2Z(Sfzu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1(z))G(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)d d=0u+v1(z)dx=0 87

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wehave d d=0u+v1(z)=2Z(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z))G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)d d=0(Sfzu+v(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u+v1(z))dx=2Z(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z))G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)d d=0Sfzu+v(x)dx=2Z(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z))G(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)rSfzu(x)v(x)dx (C) Byusing C and C ,wehaved d=0CC(Sfu+v(x),Tg(x))=d d=0Zu+v1,2(z)[u+v1(z)])]TJ /F9 7.97 Tf 6.59 0 Td[(1=2[2(z)])]TJ /F9 7.97 Tf 6.59 0 Td[(1=2dz=Zd d=0u+v1,2(z)[u+v1(z)])]TJ /F9 7.97 Tf 6.58 0 Td[(1=2[2(z)])]TJ /F9 7.97 Tf 6.58 0 Td[(1=2dz=Z(u1(z)d d=0u+v1,2(z))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2u1,2(z)d d=0u+v1(z))[u1(z)])]TJ /F9 7.97 Tf 6.59 0 Td[(3=2[2(z)])]TJ /F9 7.97 Tf 6.59 0 Td[(1=2dz=ZZG(x)]TJ /F5 11.955 Tf 11.95 0 Td[(z)Tgz(x))]TJ /F7 11.955 Tf 11.95 0 Td[(2(z) [u1(z)]1=2[2(z)]1=2)]TJ /F5 11.955 Tf 28.28 8.09 Td[(Sfzu(x))]TJ /F7 11.955 Tf 11.96 0 Td[(u1(z) [u1(z)]3=2[2(z)]1=2u1,2(z)rSfzu(x)v(x)dxdz Hence,theE-LequationforCC(Sfu(x),Tg(x))withrespecttou(x)isZG(x)]TJ /F5 11.955 Tf 11.96 0 Td[(z)Tgz(x))]TJ /F7 11.955 Tf 11.96 -.01 Td[(2(z) [u1(z)]1=2[2(z)]1=2)]TJ /F5 11.955 Tf 28.27 8.09 Td[(Sfzu(x))]TJ /F7 11.955 Tf 11.95 0 Td[(u1(z) [u1(z)]3=2[2(z)]1=2u1,2(z)rSfzu(x)dz=0. 88

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BIOGRAPHICALSKETCH JiangliShiwasborninZhejiang,China.Hereceivedhisbachelor'sdegreeinJuly2004andmaster'sdegreeinMarch2007inpureandappliedmathematicsfromtheDepartmentofMathematicsatShanghaiJiaotongUniversity,Shanghai,China.HereceivedtheDoctorofPhilosophyinmathematicsinAugust2013fromtheDepartmentofMathematicsatUniversityofFlorida 95