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Groupwise Analysis of Neuroimaging Data

Permanent Link: http://ufdc.ufl.edu/UFE0043689/00001

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Title: Groupwise Analysis of Neuroimaging Data
Physical Description: 1 online resource (107 p.)
Language: english
Creator: Chen, Ting
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: analysis -- atlas -- classification -- complex -- epilepsy -- groupwise -- hippocampus -- registration -- segmentation -- shape
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation proposes techniques for automatic brain structure segmentation and brain disease diagnosis within a computer aided diagnosis system framework. We started by presenting novel shape atlas construction techniques for single shape and shape complex which consists of a collection of neighboring shapes. We use the point set representation for the single structure and develop a group wise point set registration algorithm where the atlas is obtained as the emerging mean during the registration. The distance transform representation is used for building the shape complex atlas and it has its advantage in capturing the details of complicated shapes. A novel technique that maps the distance transform to the space of square root density is developed and the atlas is estimated as the geodesic mean on the high dimensional sphere of the square root density space. As an application of the atlas constructed, we proposed a segmentation algorithm which combines a set of weak atlas based segmentation results and improve their individual performance based on the assumption that the locally weighted combination varies w.r.t. both the weak segmenters and the training images. We learn the weighted combination during the training stage using a discriminative spatial regularization which depends on training set labels and use a sparse regularization scheme to avoid overfitting in the testing stage. Finally, a classification algorithm is presented as a last component of the computer aided diagnosis system, which takes the histogram of the deformation field between the atlas and the segmented structure as its feature and classify between the normal subjects and the ones with diseases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ting Chen.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Vemuri, Baba C.
Local: Co-adviser: Rangarajan, Anand.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043689:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043689/00001

Material Information

Title: Groupwise Analysis of Neuroimaging Data
Physical Description: 1 online resource (107 p.)
Language: english
Creator: Chen, Ting
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: analysis -- atlas -- classification -- complex -- epilepsy -- groupwise -- hippocampus -- registration -- segmentation -- shape
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation proposes techniques for automatic brain structure segmentation and brain disease diagnosis within a computer aided diagnosis system framework. We started by presenting novel shape atlas construction techniques for single shape and shape complex which consists of a collection of neighboring shapes. We use the point set representation for the single structure and develop a group wise point set registration algorithm where the atlas is obtained as the emerging mean during the registration. The distance transform representation is used for building the shape complex atlas and it has its advantage in capturing the details of complicated shapes. A novel technique that maps the distance transform to the space of square root density is developed and the atlas is estimated as the geodesic mean on the high dimensional sphere of the square root density space. As an application of the atlas constructed, we proposed a segmentation algorithm which combines a set of weak atlas based segmentation results and improve their individual performance based on the assumption that the locally weighted combination varies w.r.t. both the weak segmenters and the training images. We learn the weighted combination during the training stage using a discriminative spatial regularization which depends on training set labels and use a sparse regularization scheme to avoid overfitting in the testing stage. Finally, a classification algorithm is presented as a last component of the computer aided diagnosis system, which takes the histogram of the deformation field between the atlas and the segmented structure as its feature and classify between the normal subjects and the ones with diseases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ting Chen.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Vemuri, Baba C.
Local: Co-adviser: Rangarajan, Anand.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043689:00001


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GROUPWISEANALYSISOFNEUROIMAGINGDATAByTINGCHENADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011TingChen 2

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ToEvaandthankherforactingasaverywellbehavednewbornbaby TomyhusbandYingandmyparents 3

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ACKNOWLEDGMENTS IwouldliketorstthankmyadvisorsDr.BabaC.VemuriandDr.AnandRangarajan,withoutwhomthethesiswouldn'thaveexisted.Theirstrongpassion,profoundknowledgeandendlessinnovativeideasforresearchhaveenlightenedmethroughoutmyPhDtodate.Theenthusiasmandrigorousattitudeindoingresearch,theproblemsolvingmethodsandthetechnicalwritingandpresentationskillslearnedfromthemarethemostimportantpartsofmyPhDstudy.IalsowouldliketothankDr.ArunavaBanerjee,Dr.JeffreyHoandDr.BrettPresnellfortheirwillingnesstoserveonmycommitteeandprovidingnumerousconstructiveopinionsonmythesis.Inaddition,specialthanksgotoDr.StephanJ.Eisenschenkforsharingthedatasetsandthegroundtruthfortheclinicalstudyinmyresearch.Finally,IthankmyCVGMIgroupmembers,BingJian,SantoshKodipaka,AngelosBarmpoutis,AjitRajwade,AdrianPeter,RitwikKumar,OzlemSubakan,GuangCheng,YuchenXie,MeizhuLiu,DohyungSeo,O'neilSmith,WenxingYe,YanDeng,SileHu,QiDeng,HesamodinSalehian,YuanxiangWang,TheodoreHaforallthehelpIhavereceivedinworkingtowardsthisdissertation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1HumanBrainandNervousSystemDiseases ................ 12 1.2ComputerAidedDiagnosisSystem:TheEpilepsyToolkit ......... 15 1.3OutlineoftheRemainder ........................... 17 2ATLASCONSTRUCTIONVIAGROUPWISEPOINTSETREGISTRATION .. 21 2.1BackgroundandRelatedWork ........................ 21 2.2PointPatternRegistrationModel ....................... 25 2.2.1DenitionofCDF-HCDivergence ................... 26 2.2.2GroupwisePointSetRegistrationModel ............... 27 2.3EmpiricalCDF-HCEstimation ......................... 29 2.3.1EmpiricalCumulativeDistributionFunction .............. 29 2.3.2AnalyticalGradientComputation ................... 30 2.4ComplexityAnalysisandExperimentalResults ............... 35 2.4.1ComputationalComplexityAnalysis .................. 35 2.4.2GroupwiseRegistrationforAtlasConstruction ............ 36 2.4.3GroupwiseRegistrationAssessmentWithoutGroundTruth .... 42 2.5Summary .................................... 45 3SHAPECOMPLEXATLASCONSTRUCTIONFROM3DBRAINMRI ..... 47 3.1BackgroundandRelatedWork ........................ 47 3.2ShapeComplexAtlasConstructionMethodology .............. 50 3.2.1FromDistanceTransformstoSquare-RootDensityFunctions ... 50 3.2.2SpaceofSquare-RootDensities ................... 54 3.2.3FromSquare-RootDensityFunctionstoDistanceTransforms ... 56 3.3ExperimentalResults ............................. 57 3.3.12DShapeComplexAtlas ....................... 57 3.3.23DShapeComplexAtlas ....................... 59 3.3.3ShapeVariationAnalysis ........................ 62 3.4Summary .................................... 65 5

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4MIXTUREOFSEGMENTERS ........................... 68 4.1BackgroundandRelatedWork ........................ 68 4.2MixtureofSegmentersMethodology ..................... 71 4.2.1SegmentationMixtureSetup ...................... 72 4.2.2TrainingStage:DiscriminativeSpatialWeightRegularization .... 74 4.2.3TestingStage:SparseLinearCombination .............. 76 4.3Experiments .................................. 77 4.3.1HippocampusDataSet ......................... 77 4.3.2WeakSegmenters ........................... 77 4.3.3PerformanceMeasure ......................... 78 4.3.4ExperimentalParameterSettings ................... 78 4.3.5ExperimentalResults .......................... 79 4.4Summary .................................... 81 5CAVIAR:CLASSIFICATIONVIAAGGREGATEDREGRESSION ........ 83 5.1BackgroundandRelatedWork ........................ 83 5.2CAVIARAlgorithm ............................... 85 5.3Optimization .................................. 87 5.4Experiments .................................. 90 5.4.1FeatureSelection ............................ 90 5.4.2WeakLearners ............................. 91 5.4.3ModelSelection ............................. 91 5.4.4ExperimentalResults .......................... 92 5.5Summary .................................... 94 6CONCLUSIONSANDFUTUREWORK ...................... 96 REFERENCES ....................................... 99 BIOGRAPHICALSKETCH ................................ 107 6

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LISTOFTABLES Table page 2-1KS-statisticbasedmeasure ............................. 42 2-2Averagenearestneighbordistance ......................... 43 2-3RegistrationassessmentusingKS-statisticbasedmeasure ........... 45 2-4Registrationassessmentusingaveragenearestneighbordistance ....... 45 3-1Shapevariationsforthecontrolsubjectscomparedtotheatlas. ......... 64 3-2ShapevariationsfortheLATLsubjectscomparedtotheatlas. .......... 64 3-3ShapevariationsfortheRATLsubjectscomparedtotheatlas. ......... 64 4-1TheDICEandDIFindicesforSegMix,GWVandLWV. .............. 80 4-2TheDICEandDIFindicesforSegMixcomparedtothebestweaksegmenters 80 5-1TestingresultsofEpilepsydataset ......................... 94 5-2TestingresultsofOASISdatabase ......................... 94 7

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LISTOFFIGURES Figure page 1-1Thehumanbrain ................................... 13 1-2TheworkowoftheETKmodules ......................... 16 1-3Organizationofthethesis .............................. 18 2-1FeaturelandmarkssampledfromtheMRIscansandthepointsetatlasconstructedfromthem. ............................... 22 2-2Unbiasedgroupwisenon-rigidregistrationviaCDF-HConrealCCdatasets 37 2-3ExamplesofdifferentregularizationparametersofTPSthatproducearelativelystableatlas. ..................................... 38 2-4Unbiasedgroupwisenon-rigidregistrationviaPDF-JSandCDF-HConCCdatasets ....................................... 38 2-5Registrationperformedusingdifferentvaluesof. ................ 39 2-6Biasedgroupwisenon-rigidregistrationviaCDF-JS,PDF-JSandCDF-HConOlympicLogoand2Dshdatasets. ........................ 39 2-7Examplemeshesofthe2Dempiricalcumulativedistributionfunction ...... 42 2-8Atlasconstructionfromfour3Dhippocampidatasets. .............. 43 2-9Atlasconstructionfrom3DDuckdatasets.Eachpointsetcontains235points. ............................................. 44 3-1Theowchartofourframework. .......................... 51 3-2ThisguredepictstheEuclideandistancetransformsandtheexactshapecontoursofthe2Dshapecomplex ......................... 58 3-3Signandunsigndistancetransform ........................ 59 3-4Thetwoviewsof3Dshapecomplexof8brainstructures,includingtheleft/righthippocampus,entorhinalcortex,amygdalaandthalamus. ............ 60 3-5Threeviewsofthe3Dshapecomplexatlas .................... 61 3-6ErroroftheKarchermeaniterationfordifferent~values. ............ 62 3-7Theshapevariationsalongtherstandsecondprincipaldirections.Here~=0.6 ........................................... 62 3-8TheaverageVolumeIndexforeachstructureandforthetestdataset. ..... 65 8

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3-9Theoverlapbetweentheatlasand2examplesfromthecontrol,LATLandRATLdatasetrespectively. ............................. 66 4-1Frameworkoftheproposedalgorithm. ....................... 72 4-2NeighborhoodGraphG. ............................... 73 4-3Thenotations. .................................... 74 4-4ParameterSettings .................................. 79 4-5ThegureshowstheSIMandDIFvaluesforeachimage. ............ 80 4-6Segmentationresults ................................ 82 5-1Theclassicationerrorsforbothvalidationandtestdatasetsw.r.t.differentd(indicatedbythebinsonthex-axis)forMiddleagedvs.Old. .......... 92 5-2Theclassicationerrorsoftheweaklearnersandthenalstronghypothesisw.r.t.differentnumberofweaklearnersforADvs.Control. ........... 93 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyGROUPWISEANALYSISOFNEUROIMAGINGDATAByTingChenDecember2011Chair:BabaC.VemuriMajor:ComputerEngineering Thisdissertationproposestechniquesforautomaticbrainstructuresegmentationandbraindiseasediagnosiswithinacomputeraideddiagnosissystemframework.Westartbypresentingnovelshapeatlas(averageoverapopulation)constructiontechniquesforsingleshapeandshapecomplexwhichconsistsofacollectionofneighboringshapes.Weuseaboundarypointsetrepresentationforthesingleshapeanddevelopagroupwisepointsetregistrationalgorithmtoregister/alignapopulationofshapeswheretheatlasemergesasameanshapeduringtheregistration.Adistancetransformrepresentationisusedforbuildingtheshapecomplexatlaswhichhastheadvantageofbeingabletocapturethedetailsofcomplicatedshapes.Anoveltechniquethatmapsthedistancetransformtothespaceofsquarerootdensitiesisdevelopedandtheatlasisestimatedasthegeodesicmeanonthehighdimensionalunitsphererepresentingthespaceofsquarerootdensities. Asanapplicationoftheconstructedatlas,weproposeasegmentationalgorithmwhichcombinesasetofweakatlasbasedsegmentationsandimprovestheirindividualperformancebasedontheassumptionthatthelocallyweightedcombinationvariesw.r.t.boththeweaksegmentersandthetrainingimages.Welearntheweightedcombinationduringthetrainingstageusingadiscriminativespatialregularizationwhichdependsontrainingsetlabelsanduseasparseregularizationschemetoavoidoverttinginthetestingstage.Finally,aclassicationalgorithmispresentedasthelastcomponentof 10

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thecomputeraideddiagnosissystem,whichtakesthehistogramofthedeformationeldbetweentheatlasandthesegmentedstructureasitsfeatureandclassiesnormalsubjectsandtheoneswithdiseases. 11

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CHAPTER1INTRODUCTION Thischapterbeginswithashortintroductiontothekeybrainanatomiesandthecommonneurodegenerativediseasesorassociatedneuropsychiatrydisorders.Thecontentsandmaterialsarecollectedfromwww.wiki.com,[ 11 73 80 87 92 ]andInternet.Next,acomputeraideddiagnosis(CAD)system,namely,EpilepsyToolkit(ETK)willbebrieydescribed.Asaclinicallypracticalgoalofthisdissertation,ETKaimsatsegmentingthesub-corticalstructures,particularlythehippocampi,fromthe3Dbrainmagneticresonance(MR)imagesandautomaticallydiagnosingthediseasesbasedontheclassicationofthefeaturesobtainedfromthesegmentedstructures.Finally,wepresenttheoutlineofthethesisbydepictingtheowofthechaptersandexhibitingtheirtheoreticalcontributionstotheETKsystem. 1.1HumanBrainandNervousSystemDiseases Asthecenterofthehumannervoussystem,thebrainisthemostcomplexorganinthebody,whichispartitionedintothreelayersbytheneuroscientistsaccordingtotheirfunctions:thecentralcore(Fig. 1-1A ),limbicsystem(Fig. 1-1B )andcerebralcortex(Fig. 1-1C ).Thecentralcoretakeschargeofthehuman'sbasiclifeprocesssuchasbreathing,pulse,heartbeating,waking,sleep,movement,balanceandsensoryattheearlystage,whilethelimbicsystemregulatesthebodytemperature,behaviors,bloodpressure,bloodsugarlevel,aswellastheemotionsandmemoryprocesses,etc.Thehigherlevelemotionalandcognitivefunctionsofthehumanarecontrolledbythecerebralcortex,whichcanbedividedintotwohalvessymmetricallylocatingoneachhemisphere.Eachhalfcontainsthefrontallobe,parietallobe,occipitallobe,andtemporallobe. Thereareseveralimportantanatomies(Fig. 1-1D )thatlieintheaforementionedthreelayersofthebrain,andtheircooperationregulatesthehumanbodies'scomplexfunctionalities.Therefore,manyneurologicaldisordersofthehumanbeings'areclosely 12

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AThecentralcore BThelimbicsystem CThecerebralcortex DThebrainanatomies Figure1-1. Thegures A B and C areadaptedfromhttp://www.learner.organdthegure D ismodiedfromhttp://www.howstuffworks.com. relatedtothosestructures.Thehippocampuswhichbelongstothelimbicsystemandislocatedinthemedialtemporallobe,playsanimportantroleinthememoryprocessingandspatialnavigation.Itisoneofthefewstructuresinthebrainwherenewneuronsarecreatedthroughouttheentirelifetime.Theamygdala,whichisadjacenttothehippocampus,isinvolvedintheprocessingofemotionsanddeterminesthestorageofmemory.Situatednexttothehippocampusandamygdala,istheentorhinalcortex,whichservesasahubinthebrainmemoryandnavigationsystem.Anotherrelatedstructureisnamedthalamus,whichisinthecentralcorelayerandconnectstheareasofthebraincontrollingtheperceptionandmovement.Italsocontrolsthestatesof 13

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sleepandwakefulness.Sinceeachhemisphereofthebrainspecializescertaintypesofinformationprocessing,astructureisneededasaconnectorfortheircommunication.Thecorpuscallosumplaysthiscriticalroleofbridgingthetwohemispheres.Additionally,therearetwomorestructuresinthecentralcorethataremainlyassociatedwiththemovementandsensoryfunctionsofthebrain,thebrainstemandthecerebellum.Thebrainstemisasmallbutextremelyimportantstructurelocatedattheposteriorpartofthebrainandconnectedtothespinalcord.Itisthekeychannelforpassingtheinformationfromthebraintotherestofthebody.Nexttothebrainstemisthecerebellum,whichcontributesinthecoordinationofthesensoryinformationandhencethene-tuningofmovement. Damagerelatedtotheaforementionedbrainanatomiescausedduetovariousdiseases/conditions,suchas,thebraintumor,geneticdisorder,injury,orinfectionetc.,willleadtoallsortsofneurodegenerativediseasesorneuropsychiatrydisorders.Herewebrieydiscusssomeofthemostcommondiseasesofthehumanbrain.Asthemostfrequentlyoccurringformofdementia,theAlzheimer'sdisease(AD)hasanearlysymptomofnotbeingabletoacquirenewmemoriesandleadstolanguagebreakdown,long-termmemorylossandbodilyfunctionlossetc.asthediseaseprogresses.AmorphologicalmarkerforADistheenlargementofventriclesandtheshrinkageofentorhinalcortex,amygdalaandhippocampi.Epilepsyisanothertypeofbraindisorderthatischaracterizedbyseizures.Thereareover40epilepsysyndromesclassiedmainlybasedonthelocationandcauseoftheseizures.Oneofthemostcommonepilepsyoccurringintheadultsisthetemporallobeepilepsy(TLE),inwhichtheaffectedregionisfoundinthemidlinetemporalstructures,includingthehippocampus,amygdalaandparahippocampalgyrus.Mania,whichisusuallythoughtofastheoppositeofdepressionandmostoftenassociatedwithbipolardisorder,servesasanotherexample.In[ 80 ]allthebrainstructuresassociatedwiththeneuralpathwayswereexaminedandtheauthorsclaimedthatthepatientswithmaniahaveasignicantoverallvolume 14

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differenceintheregionsincludingthethalamus,hippocampiandtheamygdala.Asatypeofneurodevelopmentaldisorder,Schizophreniahasthesymptomsofhallucinations,delusionsanddisorganizedthinkingorspeech.In[ 73 ],Seidmanetal.concludedthatoneremarkablevulnerabilityofschizophrenia,isthestructuralabnormalitiesinthethalamusandtheamygdala-hippocampusregion.Therefore,advancedmedicalimagingtechnologiessuchasmagneticresonanceimaging(MRI)canbeusedtoexamineandanalyzetherelatedneuroanatomiesandhencefacilitatethediagnosisofthebraindiseases.Thisdissertationmainlyfocusesondesigning/implementingefcientnovelalgorithmsanddevelopingaCADsystemthatcanautomaticallyandnoninvasivelydetectbraindisordersfromMRIscansandinsomecasesmayevenleadtoearlyprediction. 1.2ComputerAidedDiagnosisSystem:TheEpilepsyToolkit ThisthesishasmadecontributionstothreemajormodulesoftheETKsystem:atlasconstruction,segmentationandclassication/diagnosis.Fig. 1-2 demonstratestheworkowofthesystem. AtlasConstruction:Generallyspeaking,anatlasisavolumeoftables,charts,orplatesthatsystematicallyillustratesaparticularsubject1.Specically,ananatomicalatlasisarepresentativeabstractanatomyoranatomygroupthatconsistsofthegeneralinformationoftheparticularstructures,suchasshape,intensity,volumeorneighborhoodconnectivityetc.,whicharestatisticallyevaluatedfromalargepopulationofsamples.Itcanbeasinglelabeledimageorshapethatcapturesthecommoncharacteristicsoverapopulationofanatomies.Itcanalsobeamulti-spectraldatabaseoflabeledimagescapturedbydifferentimagingtechniques(MRI,CT,X-rayetc.),henceprovidesrichclinicallyusefulinformationontheanatomiesbeingstudied.The 1Webster'sIINewCollegeDictionary,HoughtonMifinHarcourt-HoughtonMifin,2008. 15

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Figure1-2. TheworkowoftheETKmodules constructionofneuroanatomicalatlasesofthehumanbrainisofparticularinterestanditsimportancehasbeenemphasizedinanumberofrecentstudies([ 1 67 75 97 ]).Inbrief,anatlasprovidesareferenceforapopulationofshapes/imageswhichisusefulinnumerousapplications:(i)statisticalanalysisofvolumetricchangesincontrolandpatientpopulations,(ii)atlas-guidedsegmentationofstructuresofinterestwhichisneededinfurtherdiagnosticprocedures,and(iii)automateddetectionofdiseaseregionsbasedonshapevariationsbetweentheatlasandindividualsubjects. TheatlasconstructionmoduleinoursysteminvolvesconstructinganatlasfromMRIimagesofasingleanatomyormultipleanatomies.InChapter 2 ,weproposeanovelgroup-wisepointsetregistrationalgorithmforshapeatlasconstructionfromagroupofsamplesusingpointsetrepresentationandthisisfollowedbyashapecomplex 16

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atlasconstructionalgorithmforbuildinganatlasofmultipleanatomiesthatrespectstheinter-structurerelationshipsinChapter 3 Segmentation:Asdiscussedintheprevioussection,manyneurologicaldiseasesareassociatedwithparticularbrainstructures,hence,itisofgreatclinicalimportanceifwecanaccuratelysegmenttheseanatomiesfromthebrainMRscansandfocusfurtherstudiesordiagnosisonthem.IntheETKsystem,thesegmentationmodulewillautomaticallysegmentthehippocampusregionsfromagivenbrainMRscanbycombiningtheresultsofseveralweaksegmentationalgorithmsandproducingastrongandimprovedsegmentationoutput.Theatlasbasedsegmentationschemesareusedasweaksegmentersinoursystem.InChapter 4 ,wewillpresentthisnovelsegmentationalgorithm. ClassicationandDiagnosis:Theclassicationanddiagnosismoduleisthekernelpartfromaclinicalviewpoint.Thismoduletakesthesegmentedstructuresasinputandproducesthediagnosisresultofclassifyingitasbelongingtooneoftheservaldenedclasses.Inparticular,ETKwillautomaticallydetermineifthepatientbeingexaminedhasacertaintypeofepilepsyornot.Werstextractfeaturesfromthestructuressegmentedandthenperformaclassicationalgorithmtodistinguishthepatientsfromthenormalsubjects.Forinstance,agoodfeatureforepilepsydiagnosisistheasymmetrybetweentheleftandrighthippocampus[ 46 ].Withthehippocampicorrectlysegmentedintheprevioussegmentationmodule,theelasticdeformationeldrequiredtonon-rigidlymorphtheleft(right)hippocampustotheright(left)canbecomputeddirectlyandusedasthefeaturefortheclassication.ThedetailsofthisalgorithmwillbepresentedinChapter 5 1.3OutlineoftheRemainder Inthischapter,wehavediscussedthemajorbrainanatomiesandtheneurologicaldiseasesrelatedtothem,whichserveastheclinicalbackgroundofthisdissertation.Wealsohaveprovidedabriefoverviewofthecomputeraideddiagnosissystem.The 17

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Figure1-3. Organizationofthethesis remainderofthethesiswillfocusonthealgorithmsandtechniquesappliedwithinETKandtheorganizationoftheremainingchaptersisillustratedinFig. 1-3 Chapter 2 buildstheshapeatlasforstructuresrepresentedbylandmarks/pointsets.Anovelandrobusttechniqueforgroup-wiseregistrationofpointsetswithunknowncorrespondenceispresented.WebeginbydeningaHavrda-Charvat(HC)entropyvalidforcumulativedistributionfunctions(CDFs).Basedonthisdenition,weproposeanewmeasurecalledtheCDF-HCdivergencewhichisusedtoquantifythedis-similaritybetweenCDFsestimatedfromeachpoint-setinthegivenpopulationofpointsets.Aclosed-formformulafortheanalyticgradientofthecostfunctionwithrespecttothenon-rigidregistrationparametershasbeenderived,whichisconduciveforefcientquasi-Newtonoptimization.OurCDF-HCalgorithmisespeciallyusefulforunbiasedpoint-setatlasconstructionandcandosowithouttheneedtoestablishcorrespondences. 18

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Chapter 3 ,anovelshapecomplexatlasconstructiontechniqueispresentedwhichisbasedonaninformationgeometryframeworkandisappliedtobrainanatomicalstructures.Ashapecomplexisacollectionofneighboringshapesthatmayexhibitchangesinshapeacrossmultiplestructuresduringaknowndiseaseprocess,forexample,thethalamus,amygdalaandhippocampuscircuitwhichareprovetochangesinepilepsy.WerepresenttheboundaryoftheentireshapecomplexusingthezerolevelsetofadistancefunctionandthenderivearelationshipbetweenthestationarystatewavefunctionoftheSchrodingerequationandtheeikonalequationsatisedbyanydistancefunction.Thisrelationshipisfurtherexploitedbymappingthestaticwavefunctiontoaunithypersphere.AshapecomplexatlasisconstructedbycomputingtheKarchermeanonthehypersphere. InChapter 4 ,anovelsegmentationalgorithmformedicalimagesisproposedandappliedtothesegmentationofthehippocampi.Thealgorithmautomaticallylearnsthecombinationofweaksegmentersandbuildsastrongonebasedontheassumptionthatthelocallyweightedcombinationvariesw.r.t.boththeweaksegmentersandthetrainingimages.Welearntheweightedcombinationduringthetrainingstageusingadiscriminativespatialregularizationwhichdependsontrainingsetlabels.Inthetestingstage,asparseregularizationschemeisimposedtoavoidovertting. Chapter 5 presentsaclassicationalgorithmbasedonaggregatedregressionanditisusedforthediagnosisofthesegmentedstructures.Thephilosophybehindthisalgorithmissimilartothesegmentationtechniquethatweproposed.Thealgorithmcombinesasetofweaklearnersbasedontheassumptionthattheweightcombinationforthenalstronghypothesisdependsonboththeweaklearnersandthetrainingdata.Toavoidovertting,aregularizationschemeusingthenearestneighbormethodisappliedinthetestingstage.Anovelfeature,thehistogramofthedeformationeldbetweenthepatientMRIbrainscanandtheatlas,isadoptedintheclassication,whichcapturesthestructuralchangesinthescanwithrespecttotheatlasbrain. 19

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WeconcludethethesisinChapter 6 bysummarizingthealgorithmsthatweproposedandpresentingpossibledirectionsforfutureresearch. 20

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CHAPTER2ATLASCONSTRUCTIONVIAGROUPWISEPOINTSETREGISTRATION 2.1BackgroundandRelatedWork Inmedicalimagebasedcomputeraideddiagnosis,thefeaturelandmarksoftheanatomicalstructureareusuallymanuallyannotatedbyanexpertneurologistorautomaticallydetectedbyafeaturedetectionalgorithmalongtheboundaryoftheanatomy1.Thispointsetrepresentationhasitsmeritineffectivelycapturingthedescriptiveshapeinformationforeachstructureofinterest.Bymatchingthepointsetssampledfromapopulationofvolume(MRI,CTetc.)scans,theatlasofthatparticularanatomycanbeconstructedasameanshape.WeshowanillustrativeexampleinFig. 2-1 .Sincethesignicanceofanatlashasbeenfullydiscussedinthepreviouschapter,wewillmainlyfocusonproposinganovelatlasconstructionalgorithmbasedongroupwisepointsetregistrationinthischapter. Thekeyprobleminpointsetregistrationregardlessofthedimensioninwhichthepointsareembeddedinistoestimatethetransformationbetweenthecoordinatesusedtorepresentthepointsineachset.Thetransformationmaybecharacterizedasbeinglinearornonlinear,parameterizedornon-parameterized.Theliteraturehasmanytechniquestosolveforthelinearandnonlineartransformationsrequiredtoregisterthepointsets.Below,webrieydiscusssomeoftheprominentmethodsandthenestablishmotivationfortheworkreportedhere. ThepriorworkinpointsetregistrationcanbetracedbacktoBaird'seffortin1985[ 4 ],whereinatechniquewasproposedforpair-wiseshaperegistrationunderasimilaritytransformation.Heconstructedalinearprogrammingmodeltosolvefortheregistration 1ThecontentofthischapterhasbeentakenwithpermissionfromGroupwisePointSetRegistrationUsingaNovelCDF-basedHavrda-CharvatDivergence,TingChen,BabaC.Vemuri,AnandRangarajanandStephanJ.Eisenschenk,InternationalJournalofComputerVision,Volume86,2010cSpringer 21

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Figure2-1. ThisgureshowsthefeaturelandmarkpointsetssampledfromapopulationofMRIscansandtheatlas(averageshape)constructedfromthem. parametersandmadeeffectiveuseofthefeasibility-testingalgorithms,suchastheSimplexalgorithmandtheSovietEllipsoidalgorithm,toallowforasystematicsearchforthecorrespondences.Theoreticalanalysisindicatedthattheruntimeofthealgorithmisasymptoticallyquadraticinthenumberofpointsn,butinpracticeitislinearinnforn<100.However,rigidregistrationistoorestrictivearequirement.Sincethen,abundantresearchonpairwisenon-rigidpointsetregistrationcanbefoundinliterature.Forinstance,Belongieetal.[ 6 ]aimedatnon-rigidlyregisteringtwoshapesrepresentedbypointsetsbyrstsolvingforthecorrespondences.Theirmethodismoretunedtoshapeindexingthanregistration.ChuiandRangarajan[ 17 ]proposedamethodthatjointlyrecoversthecorrespondencesandthenon-rigidregistrationbetweentwopointsetsviadeterministicannealingandsoftassign.Theirworkrequiresoutlierrejectionparameterstobespeciedandtheuseofdeterministicannealingfrequentlyresultsinaslowtoconvergealgorithminpractice.Notethatthesetwopreviouslydiscussednon-rigidregistrationmethodsemploynon-rigidspatialmappings,particularlythinplatesplines(TPSs)[ 8 66 ]asthedeformationmodel.Inrecentwork,Glaunesetal.[ 30 ]attemptsolvingthepointsetmatchingprobleminadiffeomorphismsetting.Thissuccessfullyovercomesthedrawbackssuchaslocalfoldsandreectionsinducedbysplinebasedmodelsasin[ 88 ].However,itrequiresalargeamountofcomputationin 22

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3Dduetotheneedofcomputingaspatialintegral.Furthermore,themethodhasnotbeenextendedtothegroupwisesetting. Alternatively,thereexistsaclassofmethodsthatachievesmorerobustnesswithunknowncorrespondencesinthepresenceofoutliers.Thegeneralideaforthesemethodsistorepresenteachpointsetbyaprobabilitydensityfunctionandcomputethedis-similaritybetweenthemusinganinformationtheoreticmeasure.Thisclassofmethodsiscloselyrelatedtoourworkreportedhere.Themostillustrativeexampleisthemethodproposedin[ 94 ]byWangetal.,whoattemptedminimizingtherelativeentropybetweentwodistributionsestimatedfromthepointsetsw.r.t.theregistrationparameters,soastoregisterthetwopointsets.Themaindrawbackofthisapproachisthatonlyrigidpair-wiseregistrationproblemisaddressed.Besidesthis,TsinandKanade[ 84 ]proposedaKernelCorrelation(KC)basedpointsetregistrationalgorithmbymaximizingthekernelcorrelationcostfunctionoverthetransformationparameters,wherethecostfunctionisproportionaltothecorrelationofthetwokerneldensitiesestimated.JianandVemuriin[ 42 ]modeledeachofthepointsetsasaGaussianmixturemodel(GMM)andminimizedtheL2distanceoverthespaceoftransformationparameters,yieldingthedesiredtransformation.Whilethemethodhasattractivespeedandrobustnessproperties,theL2distanceisnotadivergencemeasureandtheoverlayofthepointsetsisnotmodeled,makingitdifculttoextendthistotheunbiasedregistrationofmultiplepointsets. Tosummarize,inallthetechniquesdiscussedthusfar,oneofthetwogivenpointsetsisxedasareferencewhichdenitelyleadstoabiasinthedeformationtowardthechosendataset.Moreover,allthepointsetregistrationmethodsmentionedabovearedesignedtoachievepair-wisepointsetregistration,andarenoteasilygeneralizabletoachievegroupwiseregistrationofmultiplepointsets.Consideringgroupwisealignmentalgorithms,mostoftheeffortswerededicatedtogroupwiseimageregistrations,i.e.constructingimageatlas.Forinstance,in[ 51 65 68 85 ],severalnon-rigidgroupwise 23

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imageregistrationmethodswereproposed.However,itisanontrivialtasktoextendtheseeleganttechniquestogroupwisepointsetregistration.Therefore,wewillnotfurtherdiscusstheseimagebasedmethodshereandonlyfocusonpointsetatlasconstructioninstead.Beforemovingtogroupwisepointsetregistration,weneedtobrieymentionthatratherthanthefeaturepointrepresentationscheme,shaperepresentationusingcurvesorsurfaces[ 45 72 ]hasalsoreceivedattentionintheliterature.Sincestatisticalshapeanalysisincurve/surfacespaceisverydifcult,methodsusingthisrepresentationhaveusuallyresortedtocomputingmeansetc.ofsplineparameters(usedforcurve/surfacerepresentation)[ 7 ]whichisanextrinsicapproach. A2DaverageshapemodelingtechniquewithautomaticshapeclusteringandoutlierdetectionwasproposedbyDutaetal.in[ 24 ].Theirmatchingmethodtookthepointsetsextractedfromtheshapecontoursasinputandperformedpairwiseregistrationoftwopointsetswithoutanyrequirementsofsettingtheinitialposition/scaleofthetwoobjectsorneedinganymanuallytunedparameters.However,theProcrustesanalysisprocedureimposedintheirmodelrequirestheknowledgeofcorrespondences.In[ 18 ],agroupwisepointsetregistrationalgorithmwasproposedasageneralizationof[ 17 ],butthismethodhasthesameshortcomingsas[ 17 ]inthatanexplicitcorrespondenceproblemneedstobesolved.Furthermore,themethodisslowduetotheuseofdeterministicannealing.Intherecentpast,severalresearcharticlesongroupwisepointsetregistrationhavebeenpublishedbyWangetal.[ 89 90 ].Themainstrengthoftheirworkisinsimultaneouslygroupwiseregisteringthedataandcomputingthemeanatlasshapeformultipleunlabeledpointsetswithoutchoosinganyspecicdatasetasareferenceorsolvingforcorrespondences,thusyieldinganunbiasedgroupwiseregistrationaswellasanatlas.TheirapproachistominimizetheJensen-Shannon(JS)divergenceamongtheprobabilitydensityfunctions(PDF)in[ 90 ](orcumulativedistributionfunctions(CDF)in[ 89 ]),estimatedfromthegivenpopulation 24

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ofpointsets,withTPSsadoptedasthedeformationmodel.Themainclaimin[ 89 ]isthattheCDF-JSismoreimmunetonoisethanPDF-JSbecauseoftherobustnesspropertyoftheCDF(beinganintegralmeasure).However,themainbottleneckintheirCDF-basedworkisthecomputationalcostandthecomplexityinimplementation,sincetheyusecubicsplineParzenwindowstoestimatethePDFandCDF.Hence,anumericalapproximationisinvolvedinCDFestimationandextensivecomputationisinvolvedinCDFentropyestimationsincetheentropiesandtheirderivativesarenotavailableinclosedform.Unliketheirwork,wegeneralizetheCDF-JSanddevelopanalgorithm,namelyCDFbasedHavrda-Charvat(CDF-HC)divergence,thatiscomputationallymuchfasterandmoresimpleandaccuratefromanimplementationperspective(thantheworkin[ 89 ]),withoutlosingtheinherentstatisticalrobustnessoraccuracyinCDFbasedmodels.WewillcomparethecomputationalcomplexityofCDF-JSandCDF-HCinSection 2.4.1 andshowthatCDF-HCtremendouslyreducesthecomplexitycomparedtoCDF-JS.WealsodemonstratetherobustnessandaccuracyoftheCDF-HCmethodbyshowingasetofexperimentalresultsonCDF-HC,CDF-JSandPDF-JSmethods. Therestofthechapterisorganizedasfollows:InSection 2.2 ,wepresentthedenitionofCDF-HCdivergenceandintroduceourpointsetregistrationmodel.Section 2.3 containsofthedescriptionofanoveltechniqueforestimatingtheempiricalCDF-HCwhichisusedfortheimplementationofouralgorithm.ThealgorithmisthenanalyzedandvalidatedexperimentallyinSection 2.4 andwepresentconcludingremarksinSection 2.5 2.2PointPatternRegistrationModel Inthissection,wepresentthedetailsofourproposednon-rigidpointsetregistrationmodel.Thebasicideaistomodeleachpointsetbyasurvivalfunction(complementofacumulativedistributionfunctionabbreviatedasCDF),andthenquantifythedistancebetweentheseprobabilitydistributionsviaaninformation-theoreticmeasure.This 25

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dis-similaritymeasureisthenoptimizedoverthespaceofcoordinatetransformationparameters.Wewillbeginbyderivingthisnewdis-similaritymeasure,namelytheCDF-HCdivergence. 2.2.1DenitionofCDF-HCDivergence TheCDF-HCdivergenceinourworkparallelsthedenitionofCDF-JS[ 89 ]andisdenedovertheHavrda-Charvat(HC)CumulativeResidualEntropy(HC-CRE),thedenitionofwhichwillbegivenlater.Forconvenience,wereproducethedenitionofCRE,CDF-JSandHavrdaCharvatdifferentialentropy(HC)here, Denition1(CRE,[ 91 ]):LetXbearandomvectorinRd,theCREofXisdenedby E(X)=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(ZRd+(P(jXj>)logP(jXj>))d(2) whereX=fx1,x2,...,xdg,=f1,2,...,dg,andjXj>meansjxij>i,Rd+=fxi2Rd;xi0;i2f1,2,...,dgg. Denition2(CDF-JS,[ 89 ]):GivenNcumulativeprobabilitydistributionsPk,k2f1,...,Ng,theCDF-JSdivergenceofthesetfPkgisdenedas JS(P1,P2,...,PN)=E(XkkPk))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XkkE(Pk)(2) where=f1,2,...,Nj0k1,Pkk=1garetheweightsoftheprobabilities,andEistheCumulativeResidualEntropy(CRE)asdenedin[ 91 ]. Denition3(HC,[ 39 ]):TheHavrdaCharvatentropyisdenedas H(X)=)]TJ /F12 7.97 Tf 17.64 14.94 Td[(nXi=1()]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F11 7.97 Tf 6.59 0 Td[(1(p(xi))]TJ /F6 11.955 Tf 11.95 0 Td[(p(xi))(2) wherex1,...,xnarepossiblevaluesfortherandomvariableX,pdenotestheprobabilitymassfunctionofXandisitsinherentparameter. NowwedeneHC-CREbyreplacingthedensityfunctioninEq. 2 withthesurvivalfunction.ThisdenitionparallelstheCumulativeResidualEntropyEwhichisbasedonCDFs. 26

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Denition4(HC-CRE):LetXbearandomvectorinRd:WedenetheHC-CREofXby EH(X)=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(ZRd+()]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F11 7.97 Tf 6.58 0 Td[(1(P(jXj>))]TJ /F6 11.955 Tf 11.95 0 Td[(P(jXj>))d(2) whereX,,andRd+aredenedasinDenition1. TherelationshipbetweenHC-CREandCREisstraightforwardandwehaveshownthattheHC-CREapproachesCREastendsto1in[ 16 ].BasedonHC-CRE,wedenetheCDF-HCdivergencebetweenNprobabilitydistributionsPk.Therefore,wewillconsiderXtobeintheR+domainandwriteXinsteadofjXj. Denition5(CDF-HCdivergence):TheCDF-HCdivergenceisdenedas HC(P1,P2,...,PN)=EH(XkkPk))]TJ /F13 11.955 Tf 11.96 11.36 Td[(XkkEH(Pk)(2) Here,hasthesamedenitionasinDenition2.WerewriteHC(P1,P2,...,PN)bysubstitutingEq. 2 intoEq. 2 togetasimpliedversionofDenition5. Property(Simplication):LetPbetheconvexcombinationoffPkg:P=PkkPk.WesimplifyHCtobe HC(P1,P2,...,PN)=)]TJ /F4 11.955 Tf 9.3 0 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F11 7.97 Tf 6.59 0 Td[(1(ZRd+P(X>)d)]TJ /F13 11.955 Tf 9.96 11.36 Td[(XkkZRd+Pk(Xk>)d) ThissimpliedCDF-HCdivergenceformulawillbeusedinallourcomputationsandimplementations. 2.2.2GroupwisePointSetRegistrationModel DenotetheNpointsetstoberegisteredasXk,k2f1,2,,Ng.EachpointsetXkconsistsofpointsxik2Rd,i2f1,2,,Dkg,DkbeingthenumberofpointsinthepointsetXk.AssumeeachpointsetXkisrelatedtothenallyregistereddataXkviaanunknowntransformationfunctionfk,andletk2RDkd,k2f1,2,,Ngbethesetoftransformationparametersassociatedwitheachfunctionfk,i.e.Xk=fk(Xk;k)andeachXkconsistsofpointsxik2Rd,i2f1,2,,Dkg. 27

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TogroupwiselyregisterallthegivenNpointsets,weneedtorecoverthetransformationparameterskassociatedwitheachfk.ThisproblemcanbemodeledasanoptimizationproblemwiththeobjectivefunctionbeingtheCDF-HCdivergencebetweentheNsurvivalfunctionscomputedfromthedeformedpointsets,representedasPk,k2f1,2,,Ng. Thegroupwiseregistrationproblemasknownastheatlasconstructionproblemcannowbeformulatedas, minkHC(P1,P2,...,PN)+NXk=1jjLfkjj2=mink(EH(XkkPk))]TJ /F13 11.955 Tf 9.96 11.35 Td[(XkkEH(Pk))+NXk=1jjLfkjj2 (2) InEq. 2 ,astandardregularizationofthetransformationfunctionsfkisused.Theparameterisapositiveconstant,whichactsasthetradeoffbetweenthetwoenergies.Itpreventsthedatasetfromcollapsingintoasinglepoint.Bytuning,wecancontrolthedegreeofdeformation,thedemonstrationofwhichisshownintheexperiment. LetLdenotetheregularizationoperator.Forexample,Lcouldbeadifferentialoperatorsuchasasecondorderlineardifferentialoperatorcorrespondingtothethin-platespline(TPS).Inourimplementation,wechooseTPSasthenon-rigiddeformation.GivenasetofcontrolpointsinRd,wewriteTPSasageneralnon-rigidmappingf:Rd!Rd,suchthatf(x)=WU(x)+A[x;I],whereA[x;I]istheafnepartoftheTPStransformationandthenon-rigidpartisdeterminedbythetransformationparametersstoredinthednmatrixW.HereU(x)isann1vectorconsistingofnbasisfunctionsUi(x)=U(x,xi)=U(jjx)]TJ /F8 11.955 Tf 12.58 0 Td[(xijj).Letr=jjx)]TJ /F8 11.955 Tf 12.58 0 Td[(xijj.ThenU(r)isthereproducingkernelofthethin-platespline.NotethatthereexistsaboundaryconditionPWT=0[ 66 ]forTPS,wherePisa(d+1)nmatrixwiththerstcolumnbeingonesandtherestofthecolumnsbeingthecoordinatesofthepointsinthepointset.Thisconditionensuresthatthenon-rigidpartofthetransformationiszeroatinnity. 28

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UsingtheconstraintthatWliesinthenull-spaceofP,thedimensionofWisreducedto(d+1)d.Hence,theTPStransformationparameterk=[A,W]isadnmatrix.Therefore,theobjectivefunctionfornon-rigidregistrationcanbeformulatedasanenergyfunctionalinaregularizationframework,wherethebendingenergyoftheTPSwarpingisexplicitlygivenbytrace(WKWT),withK=(Kij),andKij=U(xi,xj)dependingonthesplinekernelandthecontrolpointsets. Havingintroducedtheobjectivefunctionandthetransformationmodel,thetasknowistodesignanefcientwaytoestimatetheempiricalCDF-HCdivergenceandderivetheanalyticgradientoftheestimateddivergenceinordertoefcientlyachieveagood(albeitsuboptimal)solution. 2.3EmpiricalCDF-HCEstimation Inthissection,weproposeatechniqueforestimatingtheempiricalCDF-HC.Asmentionedpreviously,in[ 89 ]theParzenwindowtechniqueisused.SpecicallyacubicsplineParzenwindowisusedtoestimatethesmoothedprobabilitydensityfunctionpofagivenpointset.Thecumulativeresidualdistributionfunctioniscomputedbyintegratingoverp.Thisisaconstructivemethodandrequiresnumericalintegrationwhichcanaffectperformance(aswewillseeintheexperiments).However,herewepresentatechniquetoconstructtheCDFsurfaceusingtheHeavisidefunctionwhichiscomputationallyfasterandsimplerfromanimplementationperspective.WethenderivetheanalyticgradientofCDF-HC,whentheparameterforHCequalsto2.Withoutlossofgenerality,weonlydiscussthederivationforthe2Dcase,sincethederivationcanbeeasilyextendedtothe3Dcase. 2.3.1EmpiricalCumulativeDistributionFunction Mixturemodels[ 54 ]havebeenaneffectivetoolformodelingshapes[ 20 42 90 ],especiallywhentheshapesarerepresentedbyfeaturepointsorlandmarks.HerewerstresorttocomputingtheempiricalcumulativedistributionfunctionforagivepointsetXk,k2f1,2,...,Ng. 29

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Webeginwiththegeneraldenitionoftheempiricaldistributionfunction.AsdiscussedbyvanderVaartin[ 86 ],theempiricaldistributionfunctionPnforni.i.d.observations(x1,...,xn)isdenedasthefollowing,Pn(t)=1 nnXi=11xit where1xitistheindicatorfunction,equalto1ifxitandequalto0otherwise. LetDkbethenumberofpointswithinthekthpointset,withoutpriorinformation,weassumethateachpointisthei.i.d.samplefromthedataandhasthesameweight1 Dk.ByusingtheHeavisideStepfunctionHtorepresenttheindicatorfunction,theempiricalsurvivalfunctionPkforthekthpointsetisachievedbysummingupalltheHfunctionsevaluatedateverypointxik2Rd,i2f1,2,,Dkg. Pk(x)=1 DkDkXi=1H(x,xik)(2) wherethedenitionofH(t,t0)inone-dimensionisgivenby H(t,t0)=8><>:1tt00t>t0(2) TheseparabilityoftheHeavisidefunctioninhigherdimensionsallowsustoturnthemulti-dimensionalHeavisidefunctionintothemultiplicationofone-dimensionalHs.AlsonotethattheintegralofH(t,t0),from0to1,ist0.Thissimplebutimportantpropertywillbeusedingradientcomputationinthenextsubsection. 2.3.2AnalyticalGradientComputation NowwewillderivetheanalyticgradientofCDF-HCwhen=2.NotethatitistheonlycaseforwhichCDF-HChasaconciseexpressionforestimatingtheCDFandaclosedformsolutionforthegradientofthecostfunction.ThiswillhaveramicationsintheoptimizationstrategysinceCDF-HCisnotavailableinclosedformfor6=2.Therefore,wewillfocuson=2inthiswork. 30

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Let=2inEq. 2 .Weget HC(P1,P2,...,PN)=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(ZRd+P2d+XkkZRd+P2kd Webeginbyderivingthedetailedformulaforeachtermofthisequation.Takethe2Dcaseasanexampleinourdiscussion,whered=2andthetwocoordinatesofxikaredenotedas[xi,yi].First,wecomputethesecondtermofEq. 2 .SincetheHeavisideFunctionisseparable,foragivenpointsetXk,wehave Pk(x,y)=1 DkDkXi=1H(x,y,xi,yi)=1 DkDkXi=1H(x,xi)H(y,yi) Hence, P2k(x,y)=1 Dk2DkXi=1H(x,y,xi,yi)DkXj=1H(x,y,xj,yj)=1 Dk2Xi,jH(x,min(xi,xj))H(y,min(yi,yj)) (2) Now,wecomputetherstterm,i.e.theconvexcombinationterm,ofEq. 2 .Forsimplicity,weabbreviateP(x,y)asP.Recallthat P=NXk=1kPk(2) Wehave P2=(1P1++NPN)2=Xk(kPk)2+Xl6=slsPlPs TherstpartofEq. 2 coincideswithEq. 2 .Forthesecondpart,wehaveasimilarexpression,i.e. 31

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PlPs=1 DlDlXi=1H(x,y,xi,yi)1 DsDsXj=1H(x,y,xj,yj)=1 DlDsXi,jH(x,min(xi,xj))H(y,min(yi,yj)). Inordertogetananalyticalgradientforourcostfunction,weemployananalyticalformoftheminoperator(namedanlmin)inourexpressionsanddene anlmin(x,y)=)]TJ /F4 11.955 Tf 10.99 8.09 Td[(1 log(e)]TJ /F16 7.97 Tf 6.59 0 Td[(x+e)]TJ /F16 7.97 Tf 6.59 0 Td[(y).(2) Therefore, @anlmin(x,y) @x=fe)]TJ /F20 5.978 Tf 5.76 0 Td[(x e)]TJ /F20 5.978 Tf 5.76 0 Td[(x+e)]TJ /F20 5.978 Tf 5.75 0 Td[(yx6=y1x=y(2) whichislaterused.Itisobviousthatthetwoequations,Eq. 2 andEq. 2 ,shareauniformexpressionexceptforascalingconstantc.Wenallysolveforthefollowinguniformtermbasedontheanalyticalminimumfunction g(x,y)=cXi,jH(x,anlmin(xi,xj))H(y,anlmin(yi,yj))(2) wherecisastand-inforthetwodifferentconstants1 Dk2or1 DlDs.Sinceweareworkingin2DEuclideanspace,theintegralinEq. 2 isreplacedwiththe2DintegralintheR2+domain. HC(P1,P2,...,PN)=)]TJ /F13 11.955 Tf 11.3 16.27 Td[(ZZR2+P2dxdy+XkkZZR2+P2kdxdy SincetheHeavisidefunctionhasastraightforwardintegralexpression,wecomputethe2Dintegraloftheuniformexpressiong(x,y)inthefollowing. 32

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G=Z10Z10g(x,y)dxdy (2) =Z10Z10cXi,jH(x,min(xi,xj))H(y,min(yi,yj))dxdy=cXi,jZH(x,min(xi,xj))dxZH(y,min(yi,yj))dy=cXi,jmin(xi,xj)min(yi,yj) Therefore,thekeyissueremainingistoderivetheanalyticgradientforG,sincethecostfunctionisalinearcombinationofG.Weusethechainruletoget @G @k=@G @Xk@Xk @k (2) where@G @Xk=[@G @x1k,...,@G @xDkk],@G @xik=[@G @xik,@G @yik] and@G @xik=cXi,janlmin(yi,yj)@anlmin(xi,xj) @xi. NotethatinEq. 2 ,xiandxjrefertothex-coordinatesofpointsfromthesamepointset,whereasinEq. 2 ,theyrefertothex-coordinatesfromdifferentpointsets. LetBkbetheTPSbasismatrixcomputedinadvancefromthegivenpointsetXk.TherstthreecolumnsofBkspantheafnebasisandtheremainingcolumnsspanthenonlinearwarpingbasis.Thetransformationfk(Xk;k)thereforecanbeformulatedasXk=Bkk.Hence, @Xk @k=BTk(2) 33

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Finally,weproducethegradientoftheobjectivefunctionasfollows, @HC @k=@HC @Xk@Xk @k=[@HC @x1k,@HC @x2k,...,@HC @xDkk]BTk where @HC @xik=@()]TJ /F13 11.955 Tf 11.29 11.35 Td[(X2kZZP2kdxdy (2) )]TJ /F13 11.955 Tf 11.3 11.36 Td[(Xl6=slsZZPlPsdxdy+XkkZZP2dxdy)=@xik=)]TJ /F13 11.955 Tf 11.3 11.36 Td[(X2k1 Dk2Xi,janlmin(yi,yj)@anlmin(xi,xj) @xi)]TJ /F13 11.955 Tf 11.3 11.36 Td[(Xl6=sls1 DlDsXi,janlmin(yi,yj)@anlmin(xi,xj) @xi+Xkk1 Dk2Xi,janlmin(yi,yj)@anlmin(xi,xj) @xi Theexpressionisalsosimilarwith@HC @yik.TheabovederivationisdirectlyextensibletohigherdimensionsasaresultoftheseparabilityoftheHeavisideFunction. WecanseethattheequationfortheobjectivefunctionEq. 2 andgradientEq. 2 aresimpletoimplementandcomputationallyfast.Withtheanalyticalgradientbeingexplicitlyderived,wecanusethegradient-basednumericaloptimizationmethodssuchasquasi-Newton[ 57 ]toyieldagoodsolution.Meanwhile,andfromtheoverallperspective,robustnessisachievedbyusingaCDFbasedobjectivefunction.Notethatouralgorithmcanalsobeappliedtoyieldabiasedregistrationifwexoneofthedatasetsasthemodelandestimatethetransformationfromthescenedatasetstothemodel.Inthenextsection,wewillusethisbyproductabiasedgroupwiseregistrationtoproposeaseriesofcomparisonexperiments,bytakingthexedmodeldatasetasgroundtruth. 34

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2.4ComplexityAnalysisandExperimentalResults WerstbrieyanalyzethecomputationalcomplexityofbothCDF-JSandCDF-HCalgorithmsandshowthatourmodelgreatlyreducesthecomputationalcomplexityintermsofCDFestimationandthusincreasestheefciencyoftheoverallapproach.Next,weshowademonstrativeexampleof2Datlasconstructionandillustratetheeffectoftheregularizationparameterbyshowingthegroupwiseregistrationresultsforavarietyof.TodemonstratetheaccuracyandrobustnessofourCDFbasedmethodoverthecorrespondingPDFbasedapproach,asetofcomparisonexperimentswerecarriedoutonbothsyntheticandreal2Ddatasets.Finally,weshowtheresultsofgroupwise3Dregistrationandanon-rigidgroupwiseregistrationassessmentmethodisproposedtoevaluatetheregistrationwithoutknowingthegroundtruth. 2.4.1ComputationalComplexityAnalysis Here,wecomparethecomputationalcomplexityfortheobjectivefunctionsofCDF-JSandCDF-HC.Withoutlossofgeneralityinthemathematicalanalysis,weassumethereexistNpointsetswithdimensiond,eachconsistingofnpoints.Alsoweassumetheweights(i.e.kinDenition2&5)forallthepointsetstobeequalto1 N.SincebothmethodsuseTPSasthenon-rigidtransformationmodel,weonlycomparethecomplexityoftheinformationtheoreticmeasurepartoftheircostfunctions,sincetheTPSregularizationparthasthesamecomputationalcomplexity.Takingthed=2caseasanexample,wereproducethecostfunctionsforCDF-JS(Eq.7in[ 89 ]butusingournotation)andCDF-HChere: CDF)]TJ /F21 11.955 Tf 11.95 0 Td[(JS:C(P1,P2,...,PN)=)]TJ /F12 7.97 Tf 14.93 14.94 Td[(NxXNyXPlogP+1 NNXk=1NxXNyXPklogPk wherePk=Pni=1(xki)(yki),P=1 NPNk=1Pkand()isthecumulativeresidualfunctionofthecubicsplinekernelusedtocomputeCDF-JS. CDF)]TJ /F21 11.955 Tf 11.96 0 Td[(HC:HC(P1,P2,...,PN)=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(ZZR2+P2dxdy+1 NNXk=1ZZR2+P2kdxdy 35

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whereP=1 NPNk=1PkandRRR2+Pk1Pk2dxdy=Pni=1Pnj=1min(xi,xj)min(yi,yj). Sincethecomputationalcomplexityforthefunctions(),min()andlog()donotdependonthenumberoftheinputs,itisvalidtoassumethatthosefunctionshaveaxedO(1)cost.Therefore,itisclearthatthecomplexityfortheCDF-JSalgorithmisO(NxNyNn)andthecomplexityfortheCDF-HCalgorithmisO(Nn2).SinceintheParzenwindowestimation,NxandNyarethenumberofdiscretecoordinatevaluesinthexandyaxisrespectively,wethushaveNxNynand,consequently,weconcludethatthecomputationalcomplexityforCDF-JSisasymptoticallymuchlargerthanthatforCDF-HC.(Asabriefaside,wewouldalsoliketomentionthatourexperiencewithbothalgorithmsCDF-JSandCDF-HChasbeenoverwhelminglyinfavorofthelatter,butthisperspectiveisdrivenbyourchoiceofquasi-Newton-basedoptimizationalgorithmsandthefactthattheobjectivefunctionandgradientareavailableinclosedformfor=2forCDF-HC.) 2.4.2GroupwiseRegistrationforAtlasConstruction Inthissection,werstshowademonstrativeexampleofourCDF-HCalgorithmforunbiased2DatlasconstructiononarealCorpusCallosum(CC)dataset.Inthisexperiment,wemanuallyextracted63pointsontheoutercontouroftheCCfromsevennormalsubjects.OuralgorithmcansimultaneouslyalignmultipleshapesintoameanshapeasshowninFig. 2-2 .Next,weperturbedtheseventhdatasetandaddedoutlierstoit(asshownintherstgureofFig. 2-4 ,denotedby'+').TheregistrationresultsofbothPDF-JSandCDF-HCareshownintheFig. 2-4 .WefoundthattheCDF-HCmethodcanbetterregistertheoutliershapetotheemergingmeanshape.Intheseexperiments,theinitializationofthenon-rigidregistrationparametersaresimply[I;0]foralltheafnepartsand0forallthenon-rigidones.AlltheexperimentsinthisworkwereimplementedinMATLABRusingtheBroyden-Fletcher-Goldfarb-Shanno(BFGS)quasi-Newtonmethodofoptimizationwithamixedquadraticandcubiclinesearchprocedure.Thisquasi-NewtonmethodusestheBFGSformulaforupdatingtheapproximationofthe 36

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Figure2-2. Therstandsecondrowsandtheleftmostimageinthethirdrowshowthedeformationofeachpointsettotheatlas.Theinitialpointsetisdenotedwith'+'andthedeformedone''.Themiddleimageinthethirdrowshowsthesuperimposedpointsetsbeforeregistration.Therightmostimageinthethirdrowshowsthesuperimposedpointsetsafterregistration. Hessianmatrix.Whenanalyticgradientsareused,(andthisisthebasecaseforourapproachwith=2)cubiclinesearchesarepreferred.Whennumericalapproximationofthegradientsisused,thequadraticlinesearchmethodispreferredsinceitrequiresfewergradientevaluations. Ouralgorithmrequiresustochooseagoodvalueoftheregularizationparameter.Toillustratetheeffectofregistrationviaaltering,wenallyconstructasetofatlasesfordifferent.ThesameCorpusCallosumdatasetswereusedinthisexperiment.Fig. 2-3 showstheatlaswithdifferent.Experimentalresultsindicatethatthethat 37

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Figure2-3. ExamplesofdifferentregularizationparametersofTPSthatproducearelativelystableatlas. Figure2-4. Theseventhpointsetistheoutlier.Wedenotetheinlierswith''andtheoutlierwith'+'. producerelativelystableatlasisintherangeof[510)]TJ /F11 7.97 Tf 6.58 0 Td[(6,510)]TJ /F11 7.97 Tf 6.59 0 Td[(5],anorderofmagnituderange. Inthenextsetofexperiments,weexaminedthe(anecdotal)variabilityw.r.t..Intheseexperiments,weusedthenumericalgradientintheBFGSquasi-Newtonmethodevenfor=2forthesakeofafaircomparison.Whileitisdifculttoreachaconclusion,weobservedthattheoptimizationprocesswasmuchlongerespeciallysinceeachresulthadtobeobtainedforthebestsettingoftheregularizationparameter.Theincreaseddifcultyoftheoptimizationduetotheabsenceoftheanalyticalcostfunctionandgradientappearedtonarrowtherange(ofregularizationparametervalues)overwhichweobtainedgoodregistrations.Weobservedadeteriorationofthequalityoftheregistrationresultsasisincreasedbeyond3. TodemonstratetheaccuracyandrobustnesstonoiseofouralgorithmoverCDF-JSandPDF-JS,wedesignedthefollowingproceduretoperformabiased2Datlasconstructiononsyntheticdatasetswithandwithoutoutliersusingbothmethods.Werstmanuallyextract113pointsontheoutercontouroftheBeijing2008Olympic 38

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Figure2-5. Registrationperformedusingdifferentvaluesof.Inalltheseexperiments,thenumericalgradientwasused. Figure2-6. Theinitialpointsetsaredenotedwith'+'andthedeformedones''.Fromlefttoright:Therstcolumnshowsthesuperimposedpointsetsbeforeregistration.ThesecondcolumnshowsthesuperimposedpointsetsafterregistrationusingtheCDF-JSmethod,thethirdshowsthesameexceptwithPDF-JSandthelastcolumnshowstheresultsafterCDF-HCregistration. 39

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Logo2,namelypointsetB,andrandomlygenerate6setsofTPStransformationparameters.WethenappliedthesetransformationstoBtoget6randomlytransformeddatasets(asshownintherstgureinRow#1ofFig. 2-6 ,indicatedby0+0).ThesameprocedurewasappliedtoashshapetakenfromGatorBait1003,namelypointsetF,soastogetanother6non-rigidtransformeddatasets(therstgureinRow#2ofFig. 2-6 ,indicatedby0+0),eachcontaining150points.Wealsodisturbthedatasetswith10randomlygeneratedpointjitterforeachoftheshdatasets.TheCDF-HC,CDF-JSandPDF-JSalgorithmsareusedforgroupwiseregistrationofthe6newlycreateddatasetstotheoriginalBorF.Forallthreemethods,weusethesameinitializationfortheoptimizationparameters,thatisweinitializetheafneportionwith[I;0]and0forthenon-rigidparameters.Byimplementingtheaboveprocedure,weestablishedthegroundtruthatlasesforbothOlympicLogoandshdatasets,i.e.BandF,andhenceweareabletocomparethethreemethodswiththesamexedgroundtruth. TheKolmogorov-Smirnov(KS)statistic[ 25 ]wascomputedtomeasurethedifferencebetweentheCDFsofthegroundtruthpointsetandthenewlyregisteredpointsets.Anaturalquestiontoaskatthisjunctureiswhyweareusingadifferentstatistictogaugeregistrationaccuracy.Thereasonissimple.Wedonotwanttousethesameregistrationmeasure-theHCdivergence-toalsogaugetheregistrationaccuracy.Furthermore,theHCdivergenceisnotanestablishedmeasure(thoughwe'retoilingashardaswecantochangethat)whereastheKSstatisticisextremelywellknownandwidelyused(albeitusuallyforsignicancetesting).NotethatherewearenotdoingstatisticalsignicancetestsbutinsteadusingtheKS-statisticasameasureofthedis-similaritybetweentwounderlyingprobabilitydistributions.Whiletheone-dimensional 2Thisdatasetisavailableathttp://en.beijing2008.cn/en index.shtml3ThisdatasetisavailableunderthetermsoftheGNUGeneralPublicLicense(GPLversion2)athttp://www.cise.u.edu/anand/GatorBait 100.tgz 40

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KS-statisticdenotedasK(1)isindependentofthespecialformofthedistributionthisisnottrueinhigherdimensions.Apopular2DextensionK(2)wasproposedbyPeacockin[ 59 ],whereinheshowedthatformostcases,Peacock'sversionofthe2DKSstatisticwassufcientlydistribution-free.Laterin[ 34 ],GossetgeneralizedK(2)andproposeda3DKSstatisticK(3).Notethatin2D,thereare3independentdirections(4directionsintotal)toperformthecumulativeintegrationandin3Dthereareatotalof8suchdirections.TheproceduresbyPeacockandGossetweretoadoptthelargestdifferencesbetweenthetwoempiricalcumulativedistributionfunctions(eCDF),amongallthe4or8cumulativedirectionsin2Dand3Drespectively. Inthisexperiment,weconstructedthe2D=3DeCDFsfromthepointsetsfollowingtheformulaeproposedin[ 34 ].Anexampleofthe2DeCDFcomputedfromoneofthe4directionsisshowninFig. 2-7 .WecomputedtheKS-statisticbetweeneCDFsfromthegroundtruthandtheregisteredpointsets.LetFgbetheeCDFofthegroundtruthandFkbetheeCDFestimatedfromthekthregisteredpointset.TheaveragedifferenceoftheeCDFbetweenthegroundtruthpointsetandregisteredpointsetsareevaluatedusing1 NPNk=1K(2)(Fg,Fk).TheKS-statisticsforbothOlympicLogoandshdatasetsarepresentedinTable 2-1 andtheregistrationresultsareshowninFig. 2-6 .TheyclearlyindicatethattheCDF-HCmethodyieldsasmallerKS-statistic,hencebetterregistrationandmoreimmunetonoise. WealsopresentthecomparisonusingtheaveragenearestneighbordistanceinTable 2-2 ,whichalsofavorstheCDF-HCmethod.Forapairofgivenpointsets,theaveragenearestneighbordistanceisdenedbyndingthenearestneighborfromthesecondpointsetforeachpointintherstpointsetandviceversa,andthencomputingtheaveragedistanceoverallthepoints.Here,wecomputethedistancebetweeneachpointsetandthegroundtruthpointsetandthentaketheaverage. Finally,wepresenttheresultsfor3Datlasconstruction.Theinitializationintheoptimizationhereissimilartothepreviousexperiments,i.e.[I;0]forafneand0for 41

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Figure2-7. Examplemeshesofthe2DempiricalcumulativedistributionfunctionestimatedfromtheOlympicLogoandtheshdatausedforcomputingtheKS-statisticswiththeintegralperformedfromoneofthefourdirections. Table2-1. KS-statisticbasedmeasure KSCDF-JSPDF-JSCDF-HC OlympicLogo0.11030.10180.0324Fish0.13140.12670.0722 non-rigidparameters.Theexperimentswerecarriedoutonthehippocampusandduckdatasets,thelatterofwhichisextractedfromaweb-based3Ddataset4.Eachdatasetcontains4pointsets.TheunbiasedgroupwiseregistrationresultsforthehippocampusandduckdatasetsusingtheCDF-HCalgorithmareshowninFig. 2-8 andFig. 2-9 ,respectively.Theseexperimentsclearlydemonstratethatourpointsetregistrationalgorithmcansimultaneouslyregistermultiplepointsets,whichcanbeusedtocomputeameaningfulmeanshape/atlas. 2.4.3GroupwiseRegistrationAssessmentWithoutGroundTruth Intheprevioussection,asetofatlasconstructionexperimentsforrealdatawerepresented.However,thereisnostandardvalidationmethodtoevaluatethegoodnessofacomputedatlasshape(orinourcaseanatlasprobabilitydistribution). 4http:==www.3dxtras.com 42

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Table2-2. Averagenearestneighbordistance ANNCDF-JSPDF-JSCDF-HC OlympicLogo0.03670.03070.0019Fish0.09700.06100.0446 Figure2-8. Atlasconstructionfromfour3Dhippocampidatasets.Eachpointsetcontains400,429,554,310pointsrespectively.Therstrowandtheleftmostimageinthesecondrowshowthedeformationofeachpointsettotheatlas.Theinitialpointsetisdenotedwith''andthedeformedone'o'.Themiddleimageinthesecondrowshowsthesuperimposedpointsetsbeforeregistration.Therightmostimageinthesecondrowshowsthesuperimposedpointsetsafterregistration. Therefore,wepresentagroupwiseregistrationassessmentcriterion,withthegroundtruthunknown,inordertovalidateourregistration. SincetheKS-statisticisastandardmeasureofdis-similaritybetweentwoCDFs,wedecidedtogeneralizetheKS-statistictomeasurethequalityofunbiasedgroupwiseregistration.Aprocedurethatissimilartotheonepresentedinprevioussubsection 2.4.2 isusedforcomputingtheempiricalcumulativedistributionfunctioninthe 43

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Figure2-9. Atlasconstructionfrom3DDuckdatasets.Eachpointsetcontains235points.Therstrowandtheleftmostimageinthesecondrowshowthedeformationofeachpointsettotheatlas.Theinitialpointsetisdenotedwith''andthedeformedone'o'.Themiddleimageinthesecondrowshowsthesuperimposedpointsetsbeforeregistration.Therightmostimageinthesecondrowshowsthesuperimposedpointsetsafterregistration. estimationoftheKS-statistic.Intuitivelyspeaking,ifthepointsetsarebetterregistered,theestimated2D/3DeCDFshouldbemoresimilartoeachother,andhence,weshouldobtainasmallerKS-statisticbetweeneacheCDFestimatedfromtheregisteredpointsetspair-wiselythantheeCDFestimatedfromtheinitialpointsetpairs.WeusethefollowingmeasuretoevaluatetheCDF-HCatlasconstruction: M=1 N2NXk,s=1,k6=sK(Fk,Fs)(2) 44

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whereFkistheeCDFfromtheonepointsetoutoftheNpointsets.KisK(2)orK(3)in2Dor3Dcaserespectively,thatis,theKS-statisticbetweentwo2D/3DeCDFs.KthusevaluatestheaveragepairwiseKS-statisticamongthesepointsets. TheresultsforCDF-HCgroupwiseregistrationassessmentusingMforthe2Dand3DunbiasedatlasconstructionexperimentsarelistedinTable 2-3 .OurassessmentmeasureMiscomputedforeachdatasetbeforeandafterregistration.Similarly,theaveragenearestneighbordistanceisalsocomputedhereinTable 2-4 forreference. Table2-3. Non-rigidgroupwiseregistrationassessmentwithoutgroundtruthusingKS-statisticbasedmeasure BeforeRegistrationAfterRegistration CorpusCallosum0.32260.0635CorpusCallosumwithoutlier0.31800.0742OlympicLogo0.15590.0308Fishwithoutlier0.11020.0544Hippocampus0.26200.0770Duck0.22870.0160 Table2-4. Non-rigidgroupwiseregistrationassessmentwithoutgroundtruthusingaveragenearestneighbordistance BeforeRegistrationAfterRegistration CorpusCallosum0.02910.0029CorpusCallosumwithoutlier0.02880.0092OlympicLogo0.08250.0022Fishwithoutlier0.14610.0601Hippocampus13.76793.1779Duck15.47250.3280 Obviously,afterCDF-HCregistration,thepointsetsachieveamuchlowervalueforbothMaswellastheaveragenearestneighbordistance,ascomparedtothemeasuresbeforeregistration.Thisindicatesthatthenewlyregisteredpointsetsmoreresembleeachotherthanthepointsetsbeforeregistration. 2.5Summary Inthischapter,wepresentedanovelandrobustalgorithmthatutilizesaninformationtheoreticmeasuretheCDF-basedHavrdaCharvat(CDF-HC) 45

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divergencetosimultaneouslyregistermultipleunlabeledpointsetswithunknowncorrespondence.InspiredbytheseparabilityoftheHeavisideFunction,weemployitastheindicatorfunctionintheempiricalCDFestimation,whichgreatlysimpliesthecomputation.Wealsodiscoveredthatthecostfunctionhasaclosed-formsolutionforitsderivativeiftheparameterfortheCDF-HCdivergenceissetto=2.Thisenabledustoreachsimplebutelegantformulaeforboththecostfunctionanditsderivatives.TheadvantagesofouralgorithmoverexistingtechniquesarethatCDF-HCcanbeusedforunbiasedgroupwisepointsetregistrationanditisrobust,computationallyfasterandmuchsimplerfromanimplementationperspective.WecomparedthecomputationalcomplexityofobjectivefunctioncomputationforCDF-HCandCDF-JS,showingthatCDF-HCismuchmoreefcient.WealsocomparedtheperformanceofCDF-HC,CDF-JSandPDF-JSmethodsandshowed2DexperimentalresultsonavarietyofdatasetstodemonstratetheadvantageofcorrectnessandrobustnessofourCDF-HCalgorithmoverthecorrespondingCDF-andPDF-basedapproaches.Finally,wedenedaKS-statisticbasedmeasuretoevaluatethequalityofthegroupwiseregistrationforrealdatasetsforthecasewhenthegroundtruthatlasisunknown.NotethatweusetheTPSasourtransformationmodelsincethismodelisgoodenoughforthedeformationsofthepointsetsinthisatlasconstructionproblemandwedidnotobserveanylocalfolding.Apromisingimmediateavenueforfutureresearchistheincorporationofadiffeomorphismmodelofdeformation[ 35 ].Asthiseffortexpandstolargerdatasets,wemayhavetoconsidersimultaneouslylearningmultipleexemplaratlases. 46

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CHAPTER3SHAPECOMPLEXATLASCONSTRUCTIONFROM3DBRAINMRI 3.1BackgroundandRelatedWork HumanbrainMRIanalysisbasedonbrainatlasisanimportanttaskduetoitsapplicationinthediagnosisandtreatmentofneurologicaldiseasesanddisorders,forexample,theatlasguidedsegmentationandtheabnormalityanalysisofbrainstructures,etcetera.1However,mostexistingshapeatlasconstructiontechniquesarebasedonanisolated,singleanatomicalshape([ 26 50 92 ])whichdoesnotcontainanyinter-structureinformation.Forexample,thespatialrelationshipsamongdifferentneighboringstructuresmaychangeduetotheeffectofnon-uniformvolumeshrinkageorexpansionofneighborhoodstructures.Ratherthanasinglestructure,manyneurologicaldisordersarediagnosedbythestructuralabnormalities(e.g.volumechange)ascribedtoseveralbrainstructuresasdiscussedinChapter 1 .Therefore,aneuroanatomicalshapecomplexatlaswhichcapturestheanatomicalconductivitiesaswellastheinter-structurerelationshipswillbeofprimaryclinicalimportance. Inthecontextofatlasconstructionformultiplebrainstructures,mostoftheeffortsinthepastwerefocusedonbuildingthefullbrainimageprobabilisticatlases.Forinstance,in[ 3 43 77 95 ],severalimageatlasconstructionmethodsfortheentirebrainwereproposedbasedontheacquisitionof3DbrainMRscans.Thetraditionaltechniquesforimageatlasconstructionusuallyfocusondevelopingeffectiveimagedeformationmethodstoregisterapopulationofbrainimages.Subsequently(orintandem),theatlasimageisestimatedasanaverageovertheregisteredimagepopulation.Morerecentworksarebasedondevelopingspecictechniquesformeancomputation. 1ThecontentofthischapterhasbeentakenwithpermissionfromConstructionofNeuroanatomicalShapeComplexAtlasfrom3DBrainMRI,TingChen,AnandRangarajan,StephanJ.Eisenschenk,andBabaC.Vemuri,MedicalImageComputingandComputer-AssistedIntervention,Volume6363,2010cSpringer 47

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Forexample,inthemulti-regionalatlas([ 77 ]),theregionspecicmeanisestimatedwhereasin[ 95 ],thegeodesicmeanofapopulationofbrainimagesiscomputedviaanintrinsicaveragingmethod.Thebrainimageatlashasitsadvantageingeneralbrainanalysis.Thevariationsoftheentirebrainduetoagingcanbestudied([ 67 ])andthesegmentationofbrainstructures(viaregistrationoftheatlas)isachieved([ 43 ])withtheaidofthewholebrainimageatlas.However,imageregistration(andhencetheanalysisbasedonit)maynotbeaccurateforparticularstructuresofinterestduetothemisalignmentcausedbytheoveralldeformationoftheconvolutedcortexwithitsgyrencephalicdetails.Furthermore,itisanon-trivialtasktoextendthesetechniquestoshapeatlasconstruction.Consequently,wewillforgofurtherdiscussionofimagebasedatlaseshereandrestrictourfocustoshapebasedatlasconstruction.Ashapeatlasisofgreatimportancewhentheanalysisisfocusedonacertainstructureoraneuralpathwaycontainingseveralrelatedstructuresinthebrain:examplesarethediseasesassociatedwithhippocampiandamygdalas. AspresentedinChapter 2 ,featurepoint-sets(orlandmarkswhenspecicidentitiesareascribedtothefeatures)areoneofthemostcommonshaperepresentationsintheliterature.Unbiasedatlasconstructionofhippocampiviagroupwisepoint-setregistrationofmixturemodelprobabilitydensityfunctionsisdescribedin[ 16 18 90 ].Whileexplicitpointtopointcorrespondencesarerecoveredin[ 18 ],information-theoreticmethodologiesareadoptedin[ 16 90 ]resultinginimplicitcorrespondence.In[ 21 ],astatisticalshapemodelisdirectlyconstructedondiffeomorphicdeformationelds.Othermethodsthatrepresentshapesin2Dusingparametriccurvesandin3Dusingparametricsurfaceshavealsoreceivedconsiderableattentionintheliterature([ 45 72 ]).In[ 81 ],acharacteristic3DshapemodeldubbedtheM-repwasproposed,andbasedonthisrepresentation,amathematicalcharacterizationofthespaceofM-repswasdeveloped.AnatlaswasthenconstructedinthisspaceviacomputationofthegeodesicmeanofapopulationofshapesrepresentedbyM-reps([ 26 ]).Recentworkin[ 50 ] 48

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describesaninterestingmodelusingcontinuoussphericalshapestoanalyzetheanatomicalshapedifferencesinthehippocampusofacontrolgroupandblindsubjects. Tosummarize,inallthetechniquesdiscussedthusfar,theshapeatlasisdevelopedonlyforanisolatedanatomicalstructureanditisdifculttogeneralizethesemethodstomultipleconnectedanatomicalstructuresinaneighborhood.Ashapecomplexanalysisalgorithmwasproposedin[ 13 ],wheretheshapesarerepresentedbypointsetsandthecorrespondencesacrosstheshapecomplexesareoptimizedviaminimizinganentropybasedcostfunction.Althoughthismodelleadstostraightforwardstatisticalshapeanalyses,ithastoresorttoagradientdescentstrategyfortheoptimization.In[ 33 63 ]multi-objectshapeanalysisframeworkswerepresentedwhereeachshapeofthemulti-objecthadanindependentrepresentation,andhenceextrainformationonthestructuralrelationshipsbetweendifferentshapesneededtobemaintained.In[ 49 ],amulti-objectshapedistributionwasusedasapriorfor2Dimagesegmentation,whereinthedistributionofasetofshapesisdenedastheaverageofthedistributioncorrespondingtotheindividualshapesinthegroup.Thismethoddoesextractfeaturesfromashapecomplexbutthisshapeinformationislostafteraveraging. Inthiswork,weproposeanoveltechniqueforconstructingtheatlasofaneuroanatomicalshapecomplexconsistingofmultipleneuroanatomicalstructureswheretheinter-structuralrelationshipsarecapturedimplicitlywithoutlossofinformationofanyoftheconstituentstructures.Inourframework,werstusethezerolevelsetofthedistancetransformfunctiontorepresenttheboundariesoftheentireshapecomplexandbasedonthemathematicalrelationshipderivedinSection 3.2 ,wethenmapthedistancetransformfunctionstothespaceofsquare-rootdensitieswhereageodesicmean(atlas)iscomputed.Finally,theactualshapecomplexatlasisrealizedviatheinversemapbacktothespaceofdistancetransforms. Thekeycontributionsofthisworkareasfollows:(i)Wederiveanovelrelationshipbetweenthestationarystatewavefunction (x)oftheSchrodingerequationand 49

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theeikonalequationkrSk=1fortheEuclideandistancetransformproblem,whichservesasabridgethatconnectsthedistancetransformrepresentationoftheshapetothespaceofsquare-root-densities.(ii)Theinter-structuralrelationshipsarewellcapturedinourdistancetransformrepresentationoftheshapecomplex,whichisofgreatclinicalimportanceforstudyingtheshapevariationsacrossmultiplestructuresinbothontogenesisandinvariousneurologicaldiseases.(iii)Werepresentshapecomplexesusingsquare-rootdensities.Sincethemanifoldofsquare-rootdensityfunctionsisaunitHilbertiansphereanditsgeometryiswellunderstood,itallowsustouseintrinsicgeometrytocompareshapecomplexesandcarryoutastatisticalanalysisofthem. Therestofthechapterisorganizedasfollows:InSection 3.2 ,wepresentthedetailsofourshapecomplexatlasconstructionmethodology.WedemonstrateourtechniqueinSection 3.3 ona2Dshapecomplexdatasetcomprisingthecorpuscallosum,brainstemandthecerebellum(takenfromthemidsagittalplane)and3Dbrainstructuresincludingleft/righthippocampus,entorhinalcortex,amygdalaandthalamus.Thedataarefromapopulationof463DbrainMRscanswithalltheneuroanatomicalstructureslabeledbyanexpertneurologist. 3.2ShapeComplexAtlasConstructionMethodology Inthissection,wederivetherelationshipbetweendistancetransformfunctionandthesquare-rootdensityrepresentation,whichallowsustomodeltheshapecomplexinthesquare-rootdensityspace,performthestatisticalanalysisoftheshapesandrecoverthemeanshapebackinthedistancetransformfunctionspace. 3.2.1FromDistanceTransformstoSquare-RootDensityFunctions Thedistancetransformisawellestablishedtechniqueforshaperepresentationandhasitsadvantageincapturingthedetailsofcomplicatedshapes.Inourmodel,eachshapecomplexdatasampleisrepresentedbyadistancetransformfunction,thezerolevelsetofwhichgivestheindividualboundariesofthevariousshapesconstitutingthe 50

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Figure3-1. Theowchartofourframework.Here,wevisualizethedistancetransformandsquare-rootdensityin2Dcase.EachsampledataisrepresentedbyabluepointonthehighdimensionalsphereandtheredpointistheKarchermean. shapecomplex.Atleasttwodecadesofefforthavegoneintolevelsetanddistancefunctionrepresentationsofshapes([ 52 58 70 74 ])theprincipaladvantagebeingtheabilitytocombinedifferentshapesintoasinglescalareldrepresentation.However,sincevariationalandpartialdifferentialequationmethodsareatthefoundationoflevelsets,itisanon-trivialtasktoemploystatisticalmethodsonscalarelddistancefunctionrepresentations.Alternatively,thereexistsaclassofmethodsthatperformshapeanalysisbyrepresentingsingleshapesusingprobabilitydensityfunctions([ 16 49 78 ]),andobtaininginterestingandpracticalresults.Forinstance,despitesacricingtheabilitytorepresentasetofshapesorashapecomplex,inthisframework,themean,varianceandprincipalmodesoftheshapepopulationarealleasilycomputed.Oneofthemaincontributionsofthisworkistosuccessfullybridgethetwodisparatedomains-variationalandlevelsetmethodsontheonehandandprobabilisticmethodsontheother-anddirectlyobtainthedensityfunctionofashapecomplexfromadistancetransformfunctionrepresentation. 51

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Morerecently,Pohletal.in[ 62 ]embedsigneddistancefunctionsintothelinearspaceofLogOdds,whereadditionandscalarmultiplicationareinclosedform.ThismanipulationallowstheaveragingandotheroperationsofSDFs,howeveramonotonicfunctionisneededtoimposetheSDFintotheprobabilityspaceof[0,1].Inthispaper,wepresentatechniquethatisdifferentfromtheaforementionedframeworkandprovidesanexplicitfunctionthatmappingtheSDFstothespaceofsquare-rootdensities.In[ 36 ],theauthorsbeginbyexpressingtheEuclideandistancefunctionprobleminaSchrodingerwaveequationrepresentation.TheysolvetheSchrodingerwaveequationinsteadofthecorrespondingstaticHamilton-Jacobiequationtoobtainthedistancetransform.WhiletheyemphasizethatthemainadvantageoftheirapproachisthelinearityoftheSchrodingerequation(asopposedtothenon-linearityoftheHamilton-Jacobiequation),wewishtodrawupontheobvious,historicalprecedentinquantummechanicsofmotivatingtheSchrodingerwavefunctionasasquare-rootdensity([ 9 ]).Inspiredbythisvoluminouspreviouswork,weadopttheinterpretationofthestationarystateSchrodingerwavefunctionfortheEuclideandistancetransformasasquare-rootdensity. Generallyspeaking,ourshapecomplexatlasconstructiontechniqueconstitutesthestagesdepictedinFig. 3-1 .WerstrepresenttheshapecomplexusingadistancetransformS(x).Then,wederiveaone-to-onemap(uptoscale)betweenS(x)andthesquare-rootrepresentation (x)asinEq. 3 andconverttheshapeintothesquare-rootdensityspace,wherestatisticalanalysiscanbeeasilyperformed.Finally,theatlasiscomputedbymappingthemeansquare-rootdensity (x)backintothedistancetransformspaceS(x)viaEq. 3 followedbytheextractionofthezerolevelset.Inthisprocedure,therelationshipsinEq. 3 andEq. 3 playakeyrole. WeprovidethederivationoftherelationshipbetweenS(x)and (x)below.Let (x)bethestationarystatewavefunction(whichisinterpretedassquare-rootdensity)andlet~Planck'sconstantbeafreeparameterinthismodel.Thestaticwave 52

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equationfortheEuclideandistancefunctionproblemis ~2r2 (x)= (x).(3) Pleasesee[ 36 ]foramoredetailedderivation. When (x)=exp()]TJ /F12 7.97 Tf 6.59 0 Td[(S(x) ~)andsatisesEq. 3 ,S(x)asymptoticallysatisestheeikonalequationkrSk=1as~!0.Hereisanormalizationconstantsuchthat (x)isasquare-rootdensity,Rj (x)j2dx=1whereisaboundeddomaininR2orR3. Proof. Fromthedenitionofasquare-rootdensity,2=1 Rexp()]TJ /F15 5.978 Tf 5.76 0 Td[(2S(x) ~)dx,whichisaconstantforeachS(x).Takingthe2Dcaseasanexample,when (x1,x2)=exp()]TJ /F12 7.97 Tf 6.58 0 Td[(S(x1,x2) ~),wehavefortherstpartialsof (x1,x2): @ @x1=)]TJ /F6 11.955 Tf 10.68 8.09 Td[(a ~exp()]TJ /F6 11.955 Tf 9.29 0 Td[(S ~)@S @x1,(3) @ @x2=)]TJ /F6 11.955 Tf 10.68 8.09 Td[(a ~exp()]TJ /F6 11.955 Tf 9.29 0 Td[(S ~)@S @x2,(3) andthesecondpartials(requiredfortheLaplacianinEq. 3 ): @2 @x21= ~2exp()]TJ /F6 11.955 Tf 9.3 0 Td[(S ~)(@S @x1)2)]TJ /F3 11.955 Tf 13.15 8.09 Td[( ~exp()]TJ /F6 11.955 Tf 9.3 0 Td[(S ~)@2S @x21,(3) @2 @x22= ~2exp()]TJ /F6 11.955 Tf 9.3 0 Td[(S ~)(@S @x2)2)]TJ /F3 11.955 Tf 13.15 8.09 Td[( ~exp()]TJ /F6 11.955 Tf 9.3 0 Td[(S ~)@2S @x22.(3) FromEq. 3 ,wehave(@S @x1)2+(@S @x2)2)]TJ /F5 11.955 Tf 11.95 0 Td[(~(@2S @x21+@2S @x22)=1whichimplies krSk2)]TJ /F5 11.955 Tf 11.95 0 Td[(~r2S=1.(3) Sincer2Sisbounded,weobtainkrSk=1as~!0. 53

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Thepropositionaboveallowsustorecoverthedistancetransformfunctionfromthesquare-rootdensityrepresentationbycomputingtheinversemapof (x)=exp()]TJ /F6 11.955 Tf 9.3 0 Td[(S(x) ~),(3) whichis S(x)=~log())]TJ /F5 11.955 Tf 11.95 0 Td[(~log( (x)).(3) NotethatEq. 3 Fisasurjectivefunction,whereatleasttwodistancetransformsaremappedtothesamesquare-rootdensityfunction.However,thesedistancetransformfunctionsonlydifferbyashift.Simplealgebrawillleadtothefollowing:S0(x)=S(x)+c0,wherec0isaconstantandS0,Saremappedtothesame .Thisshiftingeffectisremovedwhenweafneregisteralltheshapecomplexesintheinitializationprocess. Thisimportantrelationshipbuildsadirectconnectionbetweenthetworealms,i.e.thelevelsetframeworkandprobabilitydensityfunctions.Hence,ashapecomplexofanycomplextopologycanberepresentedusingasingledistancetransformfunctionandfurtherstatisticalanalysisoftheshapepopulationcanbeaccomplishedinthespaceofsquare-rootdensitiesnamely,theunithypersphereasaresultofthetransformationfromthedistancefunctiontothesquare-rootdensityrepresentation. 3.2.2SpaceofSquare-RootDensities Thesquare-rootdensityhasbeenwidelyusedintheareasofcomputervisionandmedicalimageanalysis,seeforexample[ 31 60 79 ].ThisisduetothefactthattheresultingmanifoldisaunitsphereinHilbertspace,whereinboththeinnerproductsofthetangentvectorsandtheelementsinthespacearewelldened.Hence,avarietyofRiemannianoperationsareinclosed-formsincethespaceisaconvexsubsetofasphereinL2. NotethatEq. 3 doesnotyieldasolutionthatisasquare-rootdensity.However,itdoesbuildtherelationshipbetweenexponentiateddistancefunctionsandthe 54

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Schrodingerequation.SincetheSchrodingerwavefunctionhasthepropertyofbeingasquare-rootdensity([ 9 ]),wefurtherrestrictthesolutiontobeinthesquare-rootdensityspace.Thereasonsforustofocusonthesquare-rootdensityspaceratherthantheexponentiatedfunctionspaceareasfollows.Probabilitydensityfunctionsareveryusefulshaperepresentationsasshownbyseveralresearchersintheliterature.Forinstance,onecancomputemomentsofthedensityandgetglobal/localshapedescriptors([ 40 ])whichcannotbeachievedwithanun-normalizedexponentiateddistancefunction.Onecanalsomatcheitherthedensitiesortheirmomentsforthepurposeofregistration.Also,probabilitydensityfunctionsallowustorelateourunknownsmoothingparameter(Planck'sconstant)~touncertainty.Furthermore,computingaveragesofun-normalizedexponentiatedfunctionsistheoreticallyadifcultproblem.Thespaceofexponentiatedfunctionsispositivesemidenitewhereasthesquare-rootdensityspaceisthehyperspherewhichleadstoaclosed-formmetric(andgeodesic)thatisefcienttocompute. Forconvenience,wereproducethefollowingwell-knownoperationsinthespaceofthehighdimensionalsphere.Let i,i=1,...,nbeasetofsquare-rootdensitiesinthespaceofsuchfunctions.Denev2T 1 asavectorinthetangentspaceof 1. GeodesicDistance:d( 1, 2)=cos)]TJ /F11 7.97 Tf 6.59 0 Td[(1h 1, 2i ExponentialMap: 2=exp 1(v)=cos(jvj) 1+sin(jvj)v jvj LogMap:v=log 1( 2)=ucos)]TJ /F11 7.97 Tf 6.58 0 Td[(1h 1, 2i=p hu,ui,whereu= 2)-222(h 2, 1i 1 KarcherMean: =argmin 2Pni=1d2( i) Equippedwiththesebasictools,wearenowabletoconstructanatlasfortheshapecomplexbycomputingtheKarchermeanofthegivenshapecomplexpopulationinthespaceoftheHilbertsphere.WeillustratetheideaofourframeworkonasimpleexampleinFig. 3-1 .NotethatthenotionofatlasherecorrespondstothemeancomputedusingtheL2norm.However,anynormisapplicableinourframework.Forexample,wecan 55

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envisageestimatingthemedianofthepopulationviatheL1norm.Finally,withthesquare-rootdensityrepresentation,wecanperformcompletestatisticalanalysiswhenrequired. 3.2.3FromSquare-RootDensityFunctionstoDistanceTransforms InSection 3.2.1 ,wediscussedthetechniqueformappingtheshapecomplexrepresentedbydistancetransformtothespaceofsquare-rootdensityfunctions.Thegeodesicmean (x)onthesphereofsquare-rootdensitiesisthereforecomputedfromapopulationofshapecomplexes.Sinceeachshapecomplexisnallyrepresentedby i(x)=iexp()]TJ /F12 7.97 Tf 6.58 0 Td[(Si(x) ~),i=1,...,n,itisvalidtoassumethattheirmeanshape (x)hasthesamerepresentationaseachshapecomplexdatasample,i.e. (x)=exp()]TJ /F11 7.97 Tf 7.18 1.77 Td[(S(x) ~).Therefore,thedistancetransformfunctionofthemeanshape(atlas)isgivenby S(x)=~log())]TJ /F5 11.955 Tf 11.95 0 Td[(~log( (x)).(3) SincePlanck'sconstant~asmoothingparameterinourframeworkissettoaxedvalueduringthewholeprocedure,istheonlyparameterweneedtoestimateinordertorecoverS(x). Foreachshapecomplexsamplerepresentedby i,iisthenormalizationparameterthatimposesthesquare-rootdensitypropertyon i.Henceiisalwaysgreaterthan0,i.e.i2R+.WedenotethehighdimensionalHilbertsphereby.Now,letP=R+betheproductmanifold,andEq. 3 maptheelementsinPtothespaceofdistancetransformfunctions.Weevaluate bycomputingtheKarchermeanoff igontheunitHilbertsphereandestimateviatheKarchermeanoffiginR+.Thegeodesicmeanoff ighasbeeninvestigatedintheprevioussectionandherewediscusshowtocomputeinR+.ItisknownthatthecorrespondingRiemanniandistancebetweentwoelementsx,y2R+isjlogx)]TJ /F4 11.955 Tf 12.81 0 Td[(logyj([ 5 55 ]).Therefore,the 56

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geodesicmeanoffigissimplytheexponentialoftheaverageoflog1,...,logn,i.e. =exp(1 nnXi=1logi).(3) 3.3ExperimentalResults Inthissection,wepresentseveralexperimentsdemonstratingtheperformanceofouralgorithmonapopulationofreal(shapecomplex)data.TheimagesareaprioriafneregisteredusinganITK-basedmutualinformationregistrationalgorithm([ 82 ]).Notethatforvisualizationpurposes,wetransferthelabelsofeachstructureintheshapecomplexbymappingthelabelimagetoouratlas.Thetransformationparametersofthemappingarecomputedbyanon-rigidwarpingfromabinaryimageoftheshapecomplextemplatetothebinaryimageestimatedfromtheshapecomplexatlas. 3.3.12DShapeComplexAtlas Toillustrateourtechnique,webeginwitha2DshapecomplexdatasetwiththestepsofourapproachfollowingtheowchartinFig. 3-1 .The2DMRimagesaretakenfromthemidsagittalplaneofthe3DbrainMRIwith3structures(corpuscallosum,brainstemandcerebellum)labeledbyanexpertneurologist.Theowchartcontainsthefollowingmajorsteps. 1. Weestimatethedistancetransformfunctionfromeachbinarylabeledimageoftheshapecomplexsegmentedbytheexpertneurologist(Dr.Eisenschenk).Thezerolevelsetofthedistancetransformcapturestheboundarycontoursoftheshapecomplex. 2. Fromthedistancetransformrepresentationoftheshapecomplex,wecomputeitssquare-rootdensityrepresentationviaEq. 3 3. Eachsquare-rootdensityrepresentationoftheshapecomplexcorrespondstoasinglepointonthehighdimensionalsphere.InthisHilbertiansphere,theKarchermeanisdirectlycomputed.Asabyproduct,statisticalanalysissuchastheprincipalgeodesicanalysis(PGA)([ 26 ]),isperformed. 4. Thedistancetransformrepresentationoftheatlasisrecoveredfromthegeodesicmean(whichisasquare-rootdensityfunctiononthesphere)viaEq. 3 57

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Figure3-2. ThisguredepictstheEuclideandistancetransformsandtheexactshapecontoursofthe2Dshapecomplex(corpuscallosum,brainstemandcerebellum)atlascorrespondingtodifferent~values.As~increases,theatlasbecomesmoresmooth. 5. Finally,theshapecontoursarerecoveredfromthea-levelsetofthedistancetransformfunction,whereaisestimatedfromEq. 3 andEq. 3 .Theneedforthisstepisexplainedbelow. InFig. 3-2 ,wedemonstratethe2Dshapecomplexatlasconstructedfromouralgorithmwith8givensamples.Therstrowshowsthedistancetransforms(step4)oftheatlascomputedatdifferentvaluesof~.Theresultsofstep5areillustratedinthesecondrow.Wecanseethatas~isincreased,thesmoothnessoftherecoveredatlasshapecontoursincreases. Notethatthesigneddistancetransformwasusedinthisexperiment.Ouralgorithmisvalidforbothsignedandunsigneddistancetransformfunctions.AsshowninFig. 3-3 ,whereasthesquare-rootdensityoftheunsigneddistancetransformhaspeaksontheshapeboundary,itssigneddistancecounterpartcapturestheskeletonoftheshapecomplex.However,thesigneddistancetransformrepresentationismorerobustinpracticesincewetakethea-levelsetofthedistancetransformtorecovertheatlasshapecontours.Duetonumericalissues,wecannotguaranteethatthedistancetransformfunctiongeneratedfromtheinversemappinghasallthelocalminimaonthesamelevelset.Hence,theboundariesweextractfromthea-levelsetmightbenoisy.Wehavenotobservedthistobeaprobleminthecaseofthesigneddistancetransform. 58

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Figure3-3. Therstrowofthegureshowstheunsigneddistancetransformofthemeanshapecomplex(atlas)ontheleftanditssquare-rootdensityrepresentationontheright.Thesecondrowshowsthesigneddistancetransformanditscorrespondingsquare-rootdensityforthesamedata. 3.3.23DShapeComplexAtlas Next,weapplyourframeworktoa3Dshapecomplexdataset(Fig. 3-4 ).Thisdatasetcontainsthe3DbrainMRIfrom32controlsand14patientswithepilepsy(7rightand7leftanteriormedialtemporallobeepilepsycasesrespectively).Epilepsyreferstoagroupofrelatedneurologicaldisorderscharacterizedbyrecurrentseizures.Intemporallobeepilepsy,thehippocampus,amygdala,andparahippocampalregionsareconsideredtobetheepileptogenicfocusedstructures.Inthisdataset,wehavethefollowing8relatedstructureslabeledbytheexpert:left/righthippocampus,entorhinalcortex,amygdalaandthalamus.Therstexperimentistoconstructtheshapecomplexatlasusingourtechnique.WeshowtheatlasconstructedfromthecontrolsasthemeanshapeinthreedifferentanglesofviewinFig. 3-5 .Thefreeparameter~actsasasmoothing/regularizationtermforatlasconstructionandisexpectedtoactasanuncertaintycontrolsimilartotheroleplayedbyPlanck'sconstantinquantum 59

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Figure3-4. Thetwoviewsof3Dshapecomplexof8brainstructures,includingtheleft/righthippocampus,entorhinalcortex,amygdalaandthalamus. mechanics.Wedemonstratethevariationoftheatlaswhendifferent~valuesareused.As~increases,theatlasbecomesmoresmooth. SincethereisnoanalyticalsolutionforcomputingtheKarchermean,weuseagradient-basedapproach([ 79 ])toiterativelycomputethegeodesicmeanofthesquare-rootdensityfunctions.TodemonstratetheconvergenceofKarchermeancomputation,weestimatetheerrorateachiterationastheL2normofthedifferencebetweenthecurrentmeanvalueandtheoneevaluatedatthepreviousiteration.Weshowtheerrorsw.r.t.theiterationnumberfordifferent~valuesinFig. 3-6 .Thealgorithmconvergeswithin50integrationsforeachsetting,whichisveryefcient. Astheshapecomplexisrepresentedusingasquare-rootdensity,wearecapableofperformingasetofstatisticalanalysisoftheshape.Applyingprincipalgeodesicanalysis(PGA)([ 26 ])toourdataset,werecoverthemodesofdeformationandtheshapevariationalongtherstandsecondprincipaldirectionsasshowninFig. 3-7 60

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Figure3-5. Thisgureshowsthethreeviewsofthe3Dshapecomplexatlascorrespondingtodifferent~valuesineachrow.As~increases,theatlasbecomesmoresmooth. 61

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Figure3-6. ErroroftheKarchermeaniterationfordifferent~values. Figure3-7. Theshapevariationsalongtherstandsecondprincipaldirections.Here~=0.6. 3.3.3ShapeVariationAnalysis Inthestatisticalanalysisoftheshapevariations,suchasthestructuraldeformationsoccurredintheepilepsyprocess,thevolumebasedanalysishasbeenpopularfordecades([ 22 29 ]).Beside,ithasbeenshowninthestudythatbrainstructurevolumeshrinkageexistedintheepilepsypatients([ 53 ]).Therefore,toinvestigatethebrainstructureshapevariationsintheepilepsypatients,wedesignedthefollowingexperiments.Werandomlytake7ofthecontrolsubjectsandusethemfor 62

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testing.Theremaining25samples(data)areusedtoconstructtheatlas.Thetestdatathencontains7control,7LATLand7RATLsamples.Wecomputetheshapevariationsbetweenthetestingdataandtheatlasasfollows: VolumeIndex=Volume(subject) Volume(atlas),(3) SimilarityIndex=2Volume(subject\atlas) Volume(subject)+Volume(atlas),and(3) DierenceIndex=2jVolume(subject))]TJ /F4 11.955 Tf 11.96 0 Td[(Volume(atlas)j Volume(subject)+Volume(atlas),(3) wheretheVolumeIndexdenotestheentirevolumechangebetweenthetestsubjectw.r.t.theatlas,theSimilarityIndexindicatestheregionofoverlapbetweenthesubjectandtheatlas,andtheDifferenceIndexshowsthedifferenceofthetwointermsofvolumesize.TheresultsarelistedinTables 3-1 3-2 and 3-3 .Notethatlargebrainvolumeshrinkageisobservedfortheepilepsypatientscomparedtothecontrols.Tobetterillustratetheseresults,weplottheoverlapsofthetestingshapecomplexes(ingray)andtheatlas(incoloredmesh)inFig. 3-9 .StudieshaveshownthattoautomaticallydistinguishbetweenLATLandRATLinepilepsyisahardproblemandweneedtodesignsophisticatedfeaturesinordertobetterclassifythem([ 46 ]).Tables 3-2 and 3-3 indicatethattheindicesforbothLATLandRATLaresimilarhencetheleftandrightanteriormedialtemporallobefocusesareundistinguishablew.r.tvolumebasedanalysis.Apromisingimmediateavenueforfutureresearchistoincorporatethehistogramofthedeformationeldbetweentheshapecomplexofthesubjectandtheatlas([ 46 ])andconductfurtheranalysisonthat. Inourapproach,asingledistancetransformisusedtorepresentashapecomplex.Consequently,despitegoingagainstthegrainofourphilosophy,wecanbuildatlasesfortheindividualbrainstructuresintheshapecomplex.Todemonstratethisby-product 63

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Table3-1. Shapevariationsforthecontrolsubjectscomparedtotheatlas. ShapeComplexControl subjectIDVolumeIndexSimilarityIndexDifferenceIndex 11.08820.80550.084521.04350.83480.042630.91300.83850.091040.91780.83490.085750.94790.81860.053560.92620.80870.076770.88530.79550.1217mean0.96030.81950.0794std.0.07560.01690.0259 Table3-2. ShapevariationsfortheLATLsubjectscomparedtotheatlas. ShapeComplexLATL subjectIDVolumeIndexSimilarityIndexDifferenceIndex 10.93370.74780.068620.84650.72800.166230.89830.74030.107240.82500.72540.191850.88540.79750.121660.57160.64650.545170.89850.78240.1070mean0.83700.73830.1868std.0.12240.04870.1632 Table3-3. ShapevariationsfortheRATLsubjectscomparedtotheatlas. ShapeComplexRATL subjectIDVolumeIndexSimilarityIndexDifferenceIndex 10.86140.78190.148920.75590.71540.278130.84340.75010.169940.88160.78990.125850.90620.76800.098460.88650.80570.120370.76280.75820.2691mean0.84260.76960.1729std.0.06020.03050.0724 64

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Figure3-8. TheaverageVolumeIndexforeachstructureandforthetestdataset. ofouralgorithmandtheeffectofepilepsyoneachbrainstructure,webuildtheatlasofeachstructureseparatelyandcomputetheVolumeIndexbetweentheatlasandthetestdataforeachstructure.TheresultsareshowninFig. 3-8 .ForLATL,weobservelargervolumeshrinkageforthestructuresinleftbrainthanintherightbrainwhileforRATL,rightbrainstructuresexperiencelargeratrophy.Thisisobviouslyanecdotalbutindicatestheneedformoredetailedempiricalanalyses. OuratlasconstructionalgorithmwasimplementedinMatlabrona2.80GHZIntelCore(TM)i7CPUPC.Ittakeslessthan2secondstoconstructanatlasfrom25labeledbrainMRIwiththedimensionsoftheROIbeing716579.Thisservestolooselyillustratethecomputationaltimeinvolved. 3.4Summary Theraisond'^etreforournewapproachisthepremisethatatlasesofnearby3DMRIbrainstructuresshoulduseanintegratedrepresentationinwhichindividualstructuresarenotcompromised.Tothisend,wedesignedaneuroanatomicalatlasconstructionframework(andalgorithm)forshapecomplexdata.Wederivedandutilized 65

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Figure3-9. Theoverlapbetweentheatlasand2examplesfromthecontrol,LATLandRATLdatasetrespectively.Theatlasisdepictedusingacoloredmeshandthetestdataisshowningray. 66

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therelationshipsbetweentheEuclideandistancetransformandthesquare-rootdensitySchrodingerwavefunctionrepresentationandthissuccessfullybuildsonaconnectionbetweentherealmsofthelevelsetframeworkandprobabilitydensityfunctions.Ourmodelisnotonlycapableofpreservingthespatialrelationshipsamongthedifferentstructuresintheshapecomplexbutalsoofcarryingoutavarietyofstatisticalanalysesoftheshapecomplexpopulation.WeexperimentallydemonstratetheshapecomplexatlascomputationalgorithmapopulationofbrainMRIscans.WealsopresentmodesofvariationfromthecomputedatlasforthecontrolpopulationaswellastheshapecomplexandsinglestructurevariationanalysesforLATLandRATLpatients. Mostexistingatlasconstructionmethodsarebasedonregisteringshapes/imagestoacommonspaceandestimatingthemean.Topologypreservingdeformationtechniques([ 38 93 96 ])canbeusedtoregisterthedata,however,themeanshape/imageevaluationcannotguaranteethenaltopology.Thatis,theboundariesofadjacentstructureswillbeblurredandmergedduringtheaveraging.Sincetheatlasconstructionprocessinouralgorithmisalsobasedonmeancomputation,theatlastopologyisthereforenotguaranteed.Recently,severaltechniqueshavebeenproposedtoconstructatopologicallymeaningfulatlasfromnearbydatasamples([ 28 37 95 ]).Whilethesemethodsobtaingoodempiricalresultsw.r.t.shapetopology,thereisnoguaranteethattopologyispreserved.Thisisclearlyagoodavenueforfutureresearch. 67

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CHAPTER4MIXTUREOFSEGMENTERS 4.1BackgroundandRelatedWork BrainMRimageanalysisanditsassociatedapplicationinthediagnosisandtreatmentofbrain-baseddiseaseshasattractedimmenseattentioninthepasttwodecades1.Aswementionedbefore,thesegmentationofbrainneuroanatomyisoneofthekeystepsinmedicalimageanalysisandcomputeraideddiagnosissystem.Forexample,researchersareinterestedinthestudyofhippocampalstructuresduetothecriticalroletheyplayinmanyneuro-disordersincludingdementia,epilepsyandschizophrenia,etc.Specically,Alzheimer'sdiseaseisidentiedpartiallybytheshrinkageofthehippocampi[ 19 ]andthestudyofthedeformationofhippocampalshapesalsoassistsinthediagnosisofepilepsy[ 46 ].Therefore,inordertoavoidtheintensiveandtediouslaborinvolvedinmanualsegmentation,atechniquethatisabletoautomaticallysegmentthestructureofinterestfrom3DbrainMRscansisofgreatclinicalinterest.Notethatthischapterwillmainlyfocusonthesegmentationofhippocampifortheshowcaseofthealgorithm,however,thetechniqueproposedisapplicableforsegmentingallkindsofanatomicalstructures. Severaltechniqueshavebeenproposedintheliteraturetosegmentthehippocampus.Onedirectapproachin[ 12 ]seekstobuildanatlasfromthetrainingimageswithmanuallabelsanddeformittothetestimageusingadeformableregistrationandhencefacilitatetheautomatictransferofthesegmentationlabels.Thismethodallowstheautomaticsegmentationofnotonlythehippocampusbutofallbrainstructures.However,ithaslimitationsinaccuracyduetotheintensity-guided 1ThecontentofthischapterhasbeentakenwithpermissionfromMixtureofSeg-menterswithDiscriminativeSpatialRegularizationandSparseWeightSelection,TingChen,BabaC.Vemuri,AnandRangarajanandStephanJ.Eisenschenk,MedicalImageComputingandComputer-AssistedIntervention,Volume6893,2011cSpringer 68

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deformationinvolvedwhichmakesitdifculttosegmentthehippocampusduetobothitssmallsize(whencomparedtothewholebrain)andduetolowcontrastintensityboundaries.In[ 76 ],amorecomplicatedmodelthatintegratesgeometric,statistical,anduser-denedinformationisproposedforhippocampalsegmentation,whereintheshapepriorforthehippocampusisdeformedtothetestimage.Thismethodrequiresmanuallandmarkstobelocatedalongtheimpreciselydenedhippocampalboundarieswhichmakesitdependonuserinteractiontosomeextent.Asimilarsemi-automaticapproachwasproposedbyYushkevichetal.[ 98 ],theperformanceofwhichalsodependsontheuser'sinitialization. Alternatively,thereexistsanotherclassofmethodsthatachievemorerobustnessandfullautomationbyextractingfeaturesfromtheimageateachvoxelandresortingtomachinelearningtechniquestodistinguishtheanatomicalstructuresfromthebackground.In[ 32 ],Gollandetal.usethesupportvectormachine(SVM)toclassifythefeatureschosenbyprincipalcomponentanalysis(PCA)fromalargefeatureset.In[ 56 ],Morraetal.adoptAdaboosttoselectthefeaturesandinvestigatethecombinationofthisautomaticfeatureselectionmethodologywiththeSVMclassicationmethod.ThisAda-SVMframeworkobtainsarelativelygoodapproximationtotheboundaryofthehippocampusandthefeatureselectionmethodsimpliestheexperts'effortinchoosinginformativefeaturesfromalargefeatureset.However,onestillneedstodevelopaverylargefeaturepoolthatcontainspotentiallyusefulfeaturesandthisisannon-trivialtaskingeneral. Recently,itwasshownthatcombiningmultipleatlasbasedsegmentationsimprovesthesegmentationaccuracy[ 2 44 69 71 ].Asoneofthemostpopularcombinationstrategies,majorityvotingwasshowntoimprovetheaccuracyandrobustnessofweakhypotheses.In[ 2 ],Artaechevarriaetal.proposeanimagesegmentationalgorithmthatcombinesmultipleatlas-basedsegmentersbasedonweightedvotingwiththeweightsestimatedfromthelocalsimilaritybetweeneachatlasandthetestimage. 69

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Duetothisspecicdesign,theweaksegmentersforthisalgorithmarelimitedtoatlas-basedsegmentation.Besides,inorderforthevotingbasedtechniquetoworkwell,asufcientlylargenumberofatlasesareneededandarobustregistrationalgorithmisrequiredsoastohavearelativelyaccuratesegmentationforeachatlas.AdifferentcombinationstrategycalledSuperDynwasproposedbyKhanetal.[ 44 ],whereinsupervisedlearningwasusedforcomputingtheweightedcombination.Subsequently,dynamicinformationbasedonregistrationaccuracybetweentheatlasandthetestimagewasemployedfortheweightselection.SuperDynindependentlyestimatestheweightsforeachweaksegmenterateachvoxel.However,thisisclearlyinadequatesinceithasbeenknownforlongthatstrongspatialdependenciesexistinnaturalimages.Meanwhile,thedynamicselectionmechanismalsorestrictsthistechniquetoatlas-basedweaksegmenters. Inthiswork,weproposeanovelsegmentationalgorithmdubbedSegMixwhichisdifferentfromalloftheaforementionedframeworks.Spatialsmoothnessandboundarydiscontinuitiesintheanatomicalstructuresareexplicitlyincorporatedintoadiscriminativeregularizerinthetrainingstage,resultinginageneraltechniquecapableofutilizingavastarrayofweaksegmenters.SegMixassumesthatthecombinationweightsdependnotonlyontheweaklearnersbutalsoonthetrainingdata.Thisisanalogoustoamedicalconsultationcarriedoutbyagroupofdoctorsonanumberofpatients.Itiswell-justiedtoassumethateachpatient'spersonalconditionhasadifferenteffectontheexperts'naldecision.Wecarefullytreattheproblemofovertting(whichcanoccurfromhavingtoomanyweights)byutilizingthepreviouslymentionedspatialweightregularizationandviaanovelnon-parametrictestingsieve.ThismakesSegMixsubstantiallyandthematicallydifferentfromtheothercombinationstrategies(e.g.SuperDynandVoting),whichbasicallyassumethattheweightsonlyvaryw.r.t.theweaksegmenters.Duetothisnovelaspectofourframework,combiningaverysmall 70

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numberofweaksegmenterscanleadtodramaticallysignicantimprovementswhichisvalidatedinourexperiments. Morespecically,welearntheweightsinthetrainingstageforeachtrainingimageandtheweightsvaryoverboththeweaksegmentersandovereachlocalregioninthetrainingimages.Hence,itispronetooverttingwhichispartiallyalleviatedbyspatial,discriminativeregularization.Furthermore,anovelschemeisimposedinthetestingphase,whereinastrongsegmenterisconstructedbyapproximatingthetestimagewithasparsecombinationofthetrainingdataandonlychoosingthelearnedweightscorrespondingtothosesparselyselectedtrainingimages.Intuitivelyspeaking,inthetrainingstage,weuseacooperationmechanismontheweaksegmenterssoastoachievethebestsegmentationforeachtrainingimage,whileinthetestingphase,weuseacompetitionmechanismtoselectonlytherelevantimagesfromthetrainingsetforaparticulartestimage.Theresultisanalgorithmdrivenbyco-opetitionwhichusesthepreviouslylearnedcooperationskillsofthecompetitivelyselectedtrainingdatatolettheweaksegmenterscollaborateandobtainastrongsegmentation.Notethatasmoreexpertdrivenmanualdelineationsbecomeavailable,theycanbeusedasweaksegmentersinourframework. Therestofthechapterisorganizedasfollows:InSection 4.2 ,wepresentthedetailsofthesegmentationmethodology.ThealgorithmisvalidatedexperimentallyinSection 5.4 andwemakeconcludingremarksinSection 5.5 4.2MixtureofSegmentersMethodology Inthissection,werstintroducethesigneddistancefunctionandneighborhoodgraphrepresentationswhichwillbeusedinouralgorithm.Thenweproposethedetailsofthetrainingandtestingphasesofthetechniqueandderiveaclosed-formsolutiontotheminimizationproblemofourcostfunction. 71

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Figure4-1. Frameworkoftheproposedalgorithm. 4.2.1SegmentationMixtureSetup ThebasicframeworkoftheproposedtechniqueisillustratedinFig. 4-1 .Inthetrainingstage,theoptimallocallyweightedcombinationoftheweaksegmentationsareestimatedtobestapproximatethegroundtruthlabelforeachtrainingimage.Asshowninthegure,theweightmatrixWntisassociatedwiththenthtrainingimageandtthweaksegmenter.Sincewecomputelocalweights,thespatialinteractionsamongvoxelsaremodeledcloselyfollowingtheDiscriminativeRandomField(DRF)[ 47 ].Inthetestingstage,wecomputethesparsecombinationofthetrainingimagestoapproximatethetestimageandonlythelearnedweightsassociatedwiththoseselectedtrainingimagesareusedtoconstructthestrongsegmenterforthatparticulartestimage.ThegureshowsanexampleofpickingI2andINasthesparserepresentationofthetestimageandonlytheweightsassociatedwiththem,W21,...W2TandWN1,...WNT,areusedtoconstructthenalstrongsegmenterfortestimageY. Beforegettingintothedetailsofthealgorithm,werstintroducetwokeytermsthatwillbeusedintherestofthischapter. SignedDistanceFunctions(SDF)areusedtorepresenttheshapeofthestructureinthesegmentation.Eachweaksegmentationoutputisrepresentedbyasigneddistancetransformandthelocallyweightedcombinationofthemcorresponds 72

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Figure4-2. NeighborhoodGraphG. toastrongsegmenter.ThereareseveraltechniquesexistingintheliteratureforlinearcombinationofSDFs.In[ 61 ],Pohletal.embedSDFintothelinearspaceofLogOdds,whereadditionandscalarmultiplicationareinclosedform,whilein[ 15 ]aSDFismappedtothesquare-rootdensityspaceviaaSchroingerwavefunction,whereavarietyofRiemannianoperationsareeasilycomputed.However,wewillresorttoasimplerideapointedoutbyLeventonetal.in[ 48 ].Theyclaimthatusingthesigneddistancetransformshaperepresentationistoleranttoslightmisalignmentandhencearoughalignmentofthedataduringpre-processingwillavoidsolvingforthegeneralcorrespondenceprobleminSDFcombination.Inthiswork,weadoptthisaforementionedphilosophyandobservehighaccuracyoftheresultsintheexperiments. ANeighborhoodGraphGiscomputedforeachtrainingdatatostorethespatialinteractionsanddependenciesbetweenvoxels.Givenatrainingsample,letL(x)bethesigneddistancetransformofthelabelimageandMbethenumberofvoxels.TheMM-dimensionalneighborhoodgraphmatrixGiscomputedfromthefollowingDRFformulation:G(i,j)=exp(jj(L(xi))]TJ /F6 11.955 Tf 11.96 0 Td[(L(xj))jj22), wherej2N(i).In3D,theneighborhoodregionoftheithvoxelN(i)iscomputedthrough6,18or26-connectivity(Fig. 4-2 ).Thisneighborhoodgraphwillbeusedtogatethedistancebetweentheweightscorrespondingtotheithandjthvoxels. 73

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Figure4-3. Thenotations. 4.2.2TrainingStage:DiscriminativeSpatialWeightRegularization Oursegmentationalgorithmtakesasinputasetofweaksegmentationresultsandcombinesthemviaaregressionmodel.AssumethereareTweaksegmentersandtheoutputsofthemarebinaryimagesbt(x),t=1,2,...,Tdistinguishingthestructuresfromthebackground.ForagiventrainingimageI(x),t(x)isthesigneddistancetransformcomputedfromthetthweaksegmentationoutputbt(x)andL(x)isthesigneddistancetransformofthegroundtruth,representingthetruesegmentationofI(x).ThealgorithmassumesthatL(x)isthelocallyweightedcombinationoft(x). Sincetheparameterstobeoptimizeddependonboththevoxellocationsandtheweaksegmenters,wesolvethisminimizationproblemvoxel-wisebyre-arrangingeacht(x)intoacolumnvectorandstackingthemtogethercolumnbycolumnintheMTmatrix,whereMisthenumberofvoxelsintheimage.Letthecolumnvector~lbethere-arrangementofL(x)anddenotecolumnvector~iastakenfromtheithrowof,~liastheithentryof~landcolumnvector~wiastheweightsassociatedwith~i.Fig. 4-3 illustratestheaforementionednotations,whereistheHadamard-Schurmatrixmultiplication[ 41 ],whichispreciselytheentry-wiseproductandR()isthere-arrangement. Itiswell-justiedtoassumethattheweights~wiand~wjareexpectedtobesimilarifthejthvoxelisintheneighborhoodoftheithvoxel.Wethereforeadoptaregularizationtermbasedonthepre-computedmatrixG,whichcapturesthesimilarityofthelabels 74

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withintheneighborhood.Weeventuallyformalizeourcostfunctioninthefollowing:~w=argmin~wMXi=1jj~wi~i)]TJ /F3 11.955 Tf 10.03 3.15 Td[(~lijj22+MXi,j2N(i)G(i,j)jj~wi)]TJ /F3 11.955 Tf 12.93 .5 Td[(~wjjj22. Aclosedformsolutioncanbederivedforthisobjectivefunction.WestartbyexpandingthecostfunctionandgetE(f~wig)=MXi=1~wti~i~ti~wi+MXi=1~l2i)]TJ /F4 11.955 Tf 11.96 0 Td[(2MXi=1~li~wti~i+MXi,j2N(i)G(i,j)(~wti~wi+~wtj~wj)]TJ /F4 11.955 Tf 11.96 0 Td[(2~wti~wj). Withthefollowingnotations: Hi=~i~ti, Wt=[~wt1,...~wtM], Bk=Hk+2(Pi=k,j2N(k)G(i,j)+Pi2N(k),j=kG(i,j))ITT, pt=[~l1~t1,...~lM~tM] andaftersomealgebra,wenallyre-arrangethecostfunctionintoamatrixformgivenby,E=Wt0BBBBBBB@B1)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(1,2)ITT...)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(1,N)ITT)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(2,1)ITTB2...)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(2,N)ITT............)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(N,1)ITT)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(N,2)ITT...BN1CCCCCCCAW)]TJ /F4 11.955 Tf 19.27 0 Td[(2ptW+NXn=1~l2n. WetakethederivativeofEw.r.t.Wandsettheresultto0inordertosolvefortheweights.Wethenhave@E @W=(Dt+D)W)]TJ /F4 11.955 Tf 12.73 0 Td[(2p=0,withDbeingthematrixintheequationabovethatcontainsBkasdiagonal.Theproblemisnallyreducedtosolving 75

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thefollowinglinearsystem(Dt+D)W=2p.NotethatsinceDt+Disasparsematrix,wenallysolveasparseleast-squaresproblemwhichcanbeefcientlyperformed. 4.2.3TestingStage:SparseLinearCombination AssumethereareNtrainingimagesandforeachimage,wesolveforthelocalweightsWtocombinetheweaksegmenters.WedenotebyWnt,theweightmatrixforthenthtrainingimageandthetthweaksegmenter,whichisbasicallyasinglematrixinvolvingthere-arrangementofWthatwesolvedforinthetrainingstage.Toavoidovertting,inthetestingstage,notallthetrainingresultsareused.Thisissimilartothesituationwhenanewpatientcomesinformedicalconsultation,weexpectthatagoodstrategyfortheexpertsinvolvessearchingforusefulrelevantcasestudiesfromtheoldpatientsinordertoarriveataconsensusdiagnosis.Therefore,onlyasubsetofthetrainedparametersarehelpfulintesting.Severaltechniquescanbeusedtoachievethisgoal,forinstancetheK-NearestNeighbor(kNN)andtheSparseRepresentationmethods.ThekNNbasedsearchforthemostsimilarcasestorepresentthetestingdatawillpotentiallyfailwhenallthetrainingimagesdifferfromthetestsample.Besides,onehastoresorttoarelativelycomplicateddatastructuresuchasthekd-tree[ 14 ]forfastkNNimplementationwhenthefeaturedimensionishigh.Recently,sparsityhasbeeninvestigatedforfeatureselection[ 99 ].Followingthisphilosophy,wecomputetheoptimalsparsecombinationofthetrainingimagestoapproximatethetestdata. Formally,letAbeamatrixwithNcolumnswheretheithcolumncontainsthefeaturesoftheithtrainingimage.LetYbethefeaturesforthegiventestimage.Toobtainasparsecombinationoftrainingsetimages,weuseanL1normregularizer.Theproblemisformalizedasfollows:=argminjjA)]TJ /F6 11.955 Tf 11.95 0 Td[(Yjj22+jjjj1 76

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whichcanbesolvedusingexistingtechniques,suchasLASSO[ 83 ].ThenalstrongsegmenteristhengivenbyS(x)=NXn=1TXt=1nWnt(x)t(x). 4.3Experiments Inthissection,weempiricallyvalidateourSegMixalgorithmandcompareitwiththewidelyusedvotingbasedmethod.Ouralgorithmsignicantlyimprovestheweaksegmentationresultsgivenasmallnumberoflow-accuracyweaksegmenters. 4.3.1HippocampusDataSet Thisexperimentisperformedonahippocampaldatasetcontaining60brainMRI(T1)images,withtherighthippocampimanuallysegmentedbyanexpertneurologist.Wedividethedatasetinto2groups.Therstgroupcontains20imagesusedinbuildingthemultipleatlases.10-foldcrossvalidationisappliedtotheremaining40images.Theoriginalbrainimagesarerstcorrectedforintensityinhomogeneityandnormalized,thenregisteredtothesamecoordinatesystemusingasimilaritytransformation.Sincethehippocampiarewithinacertainregionofthebrain,wethereforedeneaboundingboxthatapproximatelyencloseseachhippocampusandonlytaketheseROIsastheinputtooursegmentationalgorithm.Intheexperiment,thesizeofROIis563930.WeextracttheROIforeachtestimagebyrstdeformingthebrainMRIscantoalabeledbraintemplateandndingtheROIbasedonthetemplateinformation. 4.3.2WeakSegmenters Notethatanysegmentationmethodisapplicableasaweaksegmenterwithinourframework.However,inordertodemonstratetherobustnessandperformanceofouralgorithmandcompareittotheexistingmulti-atlassegmentationmethods,weuseatlas-basedsegmentationastheweaksegmenter.Werstclusterthe20imagesintoasetofgroupsbasedonthehippocampalshapeinformationfromthelabels.Thesigneddistancetransformrepresentationofeachshapeismappedtothesquare-rootdensity 77

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spaceviaaSchrodingerwavefunction(Chapter 3 fordetaileddiscussion).Hence,eachshapecorrespondstoasinglepointonthehighdimensionalsphereandthesimilaritybetweentheshapesiscomputedintrinsicallyusingthegeodesicdistanceontheunitsphere.Armedwiththisintrinsicsimilaritymeasure,anyclusteringmethodmaybeusedhere.Weemployafnitypropagationsinceitdoesnotrequirethenumberofclusterstobespecied.Weget5clustersandtheatlases/centersforeachgroupareusedfortheweaksegmentations. 4.3.3PerformanceMeasure Theevaluationmetricsformeasuringtheperformanceofthealgorithmusedinthischapterincludethe similarityindex(knownasDicecoefcient)SIM=2V(A\B) V(A)+V(B),whichcomputestheoverlapoftwovolumes, andthedifferenceindexDIF=2jV(A)V(B)j V(A)+V(B),whichmeasuresthesizedifferenceofthetwovolumes. HereV()computesthevolume.Agoodsegmentationhaslargeroverlapwiththegroundtruthhencehighersimilarityindex,butlowerdifferenceindex. 4.3.4ExperimentalParameterSettings ThefreeparametersinvolvedinourSegMixalgorithmincludeandfortheregularizationinthetrainingandtestingstagesrespectively.Werandomlyselect20imagesfromthe40testingimagesetandusethemforvalidation.InFig. 4-4A ,weshowtheaverageSIMandDIFvaluesduringtrainingstagefordifferentsettingswithin[0.00001,0.001,0.1,10,1000,100000]whilex=0.5.Theresultsindicatedthatsmallleadstodramaticovertting,thatisSIM=1andDIF=0.However,largeresultsinbadperformance.ThesimilaritymeasureintestingstageisshowninFig. 4-4B ,whereinthesametrendisobserved.Therefore,weset=0.1soastoobtainbestperformance.Ontheotherhand,Fig. 4-4C demonstratestheparameterselectionfor.Alargerthannecessarymeanshighsparsityandthealgorithmwillreducetothe 78

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ASimilaritymeasurewithdifferentduringtrain-ingstage BSimilaritymeasurewithdifferentduringtestingstage CSimilaritymeasurewithdifferent Figure4-4. ParameterSettings nearestneighborselectionmethod.Smallercorrespondstodenserdataselectionandstableperformance.Todecreasethedatasizewhileensuringgoodperformance,wechoose=1.An18-connectedneighborhoodisusedforcomputingthegraphG.Thenonrigidregistrationalgorithmusedforatlas-basedsegmentationisDemons,whereweuseallthedefaultparametersettings,i.e.alltheregistrationsareperformedwiththesameparameters.Duetotheuseofstandarddefaults,atlas-basedsegmentationleadstolow-accuracyforeachweaksegmenter.However,theexperimentalresultsindicatethatSegMixsignicantlyimprovesonthoseweaksegmentations. 4.3.5ExperimentalResults Tovalidateourproposedalgorithm,theexperimentsareperformedonthe3DhippocampalMRIimagesusing(1)SegMix,(2)GlobalWeightedVoting(GWV)and(3) 79

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Figure4-5. ThegureshowstheSIMandDIFvaluesforeachimage. Table4-1. TheDICEandDIFindicesforSegMix,GWVandLWV. ave.weakDICEnalDICEIncreased SegMix0.680.040.800.0317.65%GWV0.680.040.730.047.35%LWV0.680.040.740.048.82% ave.weakDIFnalDIFDecreased SegMix0.420.100.090.0678.57%GWV0.420.100.370.1111.90%LWV0.420.100.390.107.14% LocalWeightedVoting(LWV)[ 2 ]with5weaksegmenters.InTable. 4-1 and 4-2 ,welisttheaverageperformanceevaluationforthe10-foldcrossvalidationofthe40imagesandtheperformancecomparisonbetweenthebestweaksegmenterandSegMix.Fig. 4-6 showssamplesegmentationresultsfromouralgorithmandtheweaksegmentersaswellasthemanuallabelingfromtheexpertneurologist. SincebettersegmentationcorrespondstolargerSIMbutsmallerDIF,weshowtheincreasedSIMvalueanddecreasedDIFvaluew.r.t.theweaksegmentations.Duetothelowaccuracyandlimitednumberoftheweaksegmenters,theperformanceofthevoting-basedmethodispoorasexpected.WealsopresenttheSIMandtheDIF Table4-2. TheDICEandDIFindicesforSegMixcomparedtothebestweaksegmenters bestweakDICEnalDICEIncreased SegMix0.760.040.800.035.26% bestweakDIFnalDIFDecreased SegMix0.290.110.090.0668.97% 80

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valuesforboththeweaksegmentationsandthenalstrongsegmentationandshowtheimprovementsofourtechniquesw.r.t.theweaksegmentersforeachtestimageinFig. 4-5 4.4Summary Inthischapter,weintroducedanovelweaksegmentationcombinationstrategybasedontheassumptionthatthelocallyweightedcombinationvariesw.r.t.boththeweaksegmentersandthetrainingimages.Welearnedtheweightedcombinationduringthetrainingstageusingadiscriminativespatialregularizationwhichdependsontrainingsetlabels.Inthetestingstage,asparseregularizationschemewasimposedtoavoidovertting.WepresentedvalidationexperimentsandshowedresultsofcomparisonwithGlobalWeightedVoting.Theexperimentalresultsindicatedthatouralgorithmnotonlyoutperformsthevotingbasedmethodbutalsosignicantlyimprovestheperformancesoftheweaksegmenters. 81

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Figure4-6. Thegureshowssamplesegmentationresults.Firstthreerowsaresegmentationsfromtheweaksegmenters,followedbytheoutputsofourMixSegalgorithmandthegroundtruthsegmentationsfor4subjects. 82

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CHAPTER5CAVIAR:CLASSIFICATIONVIAAGGREGATEDREGRESSION 5.1BackgroundandRelatedWork Asoneofthemajorstepsinthecomputeraideddiagnosissystem,theautomaticdiagnosisofthebrain-baseddiseaseshasbeenintensivelystudiedinthepasttwodecades1.Thereareallkindsofwell-knownneurologicaldisorders,someofwhichhavebeendiscussedintheintroductionchapter.Herewereinvestigatethediseasesofdementiaandepilepsytoillustratethemethodologyproposedinthischapteranddevelopanovelclassicationalgorithmaswellasafeaturedescriptorthatbestdistinguishthediseasesaforementioned.Dementiaisanillnesssyndromethatmayoccuratanystageofadulthoodandleadtolong-termdeclineincognitivefunction,whileepilepsyisanothertypeofneurologicaldisorderthatinvolvesspontaneousseizures.Neuroanatomicalchangessuchasenlargingoftheventricleorshrinkageofthehippocampusareobservedinthediseaseprocesses.Therefore,atechniquethatisabletodetectthechangesinbrainstructuresduetotheonsetofdementiaorepilepsyandusethisinformationtoclassifythesubjectsautomaticallyisofgreatvalue. Oneofthemainchallengesinclassicationisthatitisnoteasytondasingleclassierthatachievesalowerrorrate.DuetothedifcultyinobtaininggoodfeaturesfromMRI,theclassiersweobtainareactuallyweakclassiers.However,canasetofweakclassierscreateasinglestronglearner?Numerousvariantsofalgorithmsforclassierensembleshavebeenproposedinliterature,forinstance,boostingandbagging.Generallyspeaking,theboostingmethoditerativelyreneseachweaklearnerandre-adjuststheweightedcombinationofthetrainingdataaftereachiteration. 1ThecontentofthischapterhasbeentakenwithpermissionfromCAVIAR:ClassicationviaAggregatedRegressionandItsApplicationinClassifyingtheOASISBrainDatabase,TingChen,AnandRangarajanandBabaC.Vemuri,IEEEInternationalSymposiumonBiomedicalImaging,2010.cIEEE 83

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Anothertypeofalgorithmiscalledbootstrapaggregation(bagging)proposedbyBreiman[ 10 ].Herebootstrapsamplesaretrainedusingweaklearnersandtheoutputoftheweakpredictorsarecombinedbyaveragingorvoting.InAdaboost[ 27 ],itisrequiredthattheperformanceofweaklearnersbeslightlybetterthanmererandomassignment.Therefore,averyweakweakclassierisnotacceptable.Besides,thisalgorithmalsoneedsalargenumberofweaklearnersinordertoconverge.Meanwhile,duetotheequallyweightedvotingscheme,baggingalsohasitslimitationintermsofimprovingonlinearmodels. Inthischapter,weproposeanovelensembleclassier-calledCAVIAR-whichisdifferentfromthebaggingandboostingmentionedaboveandoffersasignicantlybetterperformance.ThephilosophybehindCAVIARissimilartothesegmentationtechniquethatweproposedinChapter 4 ,however,ourmethoddiffersfromitintermsofbothmathematics(L1regularizervs.L2)andobjectives(segmentationvs.classication).CAVIARisaregressionbasedclassicationalgorithm,whichassumesthattheweightsforcombiningtheweaklearnersdependnotonlyontheweaklearnersbutalsoonthetrainingdata.Itisanalogoustoamedicalconsultationcarriedbyagroupofdoctorsonanumberofpatients.Wethinkthatit'swell-justiedtoassumethateachpatient'spersonalconditionhasadifferentaffectontheexperts'consensusnaldecisions.SinceinCAVIARtheweightsvaryoverboththeweaklearnersandthetrainingset,itispronetoovertting.Weimposearegularizationschemewhereintheweightscorrespondingtoclosedtrainingpatternsareforcedtobesimilar.Furthermore,inthetestingphase,astrongclassierisconstructedbychoosingtheweightscorrespondingtothenearestneighborsofthetestpatternamongthetrainingdataset.Intuitivelyspeaking,asanewpatientcomesinformedicalconsultation,weexpectthatagoodstrategyfortheexpertsinvolvessearchingthroughsimilarcasestudiesinordertoarriveataconsensusdiagnosis.Tothebestofourknowledge,thiskindofdataadaptivetestingtechniquehasneverbeenreportedinliterature. 84

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Therestofthechapterisorganizedasfollows:InSection 5.2 ,wepresentthedetailsoftheCAVIARalgorithm.WederivetheclosedformsolutionsforthecostfunctionsinSection 5.3 .ThetechniqueisvalidatedexperimentallyinSection 5.4 andwemakeconcludingremarksinSection 5.5 5.2CAVIARAlgorithm Formallyspeaking,letXbethesetoftrainingfeaturevectorsandYbethesetoflabels,where(xn,yn)aredrawnrandomlyfromXYbasedonsomeunknowndistribution.Inthecaseoftwo-classclassication,Yisabinarysetcontainingtwolabelsf+1,)]TJ /F4 11.955 Tf 9.3 0 Td[(1g.AssumehistheweakhypothesisappliedtotheinstancetakenfromthesamplesetX,withitsmagnitudejhjbeingthecondenceofthepredictionanditssigndistinguishingtheclasstowhichitbelongs.LetW=fwntg,withn=1,2,...,Nandt=1,2,...,T,betheweightmatrixthatcorrespondstothetrainingsamplesandweaklearners.ThegoalintheclassicationproblemistondapropercombinationC(xn)=Ptwntht(xn)=wnh(xn)foreachdatathatminimizesagivenclassicationerrordiscriminantfunctionPnDist(C(xn),yn)forthewholetrainingdataset.Tosimplifythenotation,weusewntodenotethevector(wn1,wn2,...,wnT)tandlethbethevector(h1(xn),h2(xn),...,hT(xn))t. Sofar,thecuriousreadermayhavetwoconcerns.First,sincethedimensionoftheweightstobeoptimizedinCAVIARisNTwhichishugeforalargedataset,theoptimizationmightbedifcultandtimeconsuming.Second,thistrainingtechniqueisverylikelytoinduceover-tting.Wenowrelegatethediscussionoftheoptimizationissuetothenextsectionandaddressindetailourmethodforsolvingtheover-ttingproblemhere. Itisjustiedtoassumethattheweightswnandwmareexpectedtobesimilarifthetrainingsamplesxnandxmareclosetoeachotherinthefeaturespace.Wethereforeadoptanaggregatedregularizationterminordertopreventover-tting.AnearestneighborgraphGispre-computed(onceandforallforthetrainingdata)andstored, 85

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withG(n,m)=1ifxnandxmareneighbors(appropriatelysymmetrized)andG(n,m)=0,otherwise.WeregularizeourobjectivefunctionusingPNn,m=1G(n,m)Dist(wn,wm). Assumeweobtainasetofweightsforcombiningtheweaklearnersinthetrainingstage.Inthetestingphase,alteringprocedureisimposedinordertoconstructastronglearnerbasedonthetestdataandtoovercomeover-tting.Insteadofusingallthetrainingresults,weonlyusethelearnedweightsthatareassociatedwiththetrainingpatternswhichareinthenearestneighborhoodofthetestpattern.Formally,letxbeanincomingtestinstance.WethentakeintoaccountasetofnearestneighborsofxfromthetrainingdatasetX.Moreover,eachneighborisassignedadifferentweightaccordingtoitsdistancetotheinstancex.Wechoosetheweighttobeinverselyproportionaltothedistancewhichaccordswiththeassumptionthatthetrainingdatabeingmoresimilartothetestsamplexshouldhavemorecontributiontothenalstrongclassierforthatparticularx.Hereweresorttothekd-tree[ 14 ]forfastkNNimplementation. ThenalhypothesisHofCAVIARforatestsamplexistheweightedcombinationoftheTweakhypothesesweightedbytheKnearestneighbors'contribution,thatis,H(x)=sign(PKk=1skPTt=1wsktht(x)).Herewsktistheoptimalweightobtainedfromthetraining.Thepseudocodeof2-classCAVIARalgorithmislistedinAlgorithm1.Notethatanydistancemeasurecanbeusedtodenetheobjectivefunctioninstep3ofthetrainingstage.ThemostpopularoneistheL2distancewhichleadstoaclosedformsolutionintheoptimization. Next,weshowhowtogeneralizethisalgorithmtothemulti-classclassicationcase,whereeachtrainingsamplecorrespondstoaparticularclasslabelfromthesetofintegersY=f1,2,...,Cg,andCisthenumberofclasses.Inourapproach,wedeneaC-dimensionallabelvectorLnforeachtrainingsamplexn,withthecthentrybeing1onlyifthelabelofxniscandtheremainingentriesbeing)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Meanwhile,theweaklearnerhtinthiscaseisavectorfunction,whichmapsaninstanceintoa 86

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Algorithm1CAVIAR:for2classes Trainingstage: 1: InputNlabeledtrainingsamplesh(x1,y1),...,(xN,yN)i,whereyi2f)]TJ /F32 10.909 Tf 39.31 0 Td[(1,1gandTweaklearnersh1,...,hT,ht:X![)]TJ /F32 10.909 Tf 8.49 0 Td[(1,1] 2: Initializetheweightmatrix:W=fwntg,forn=1,2,...,Nandt=1,2,...,T 3: Minimizethefollowingobjectivefunction: w=argminwNXn=1Dist(wnh(xn),yn)+NXn,m=1G(n,m)Dist(wn,wm) (5) Testingstage: 1: Inputthetestsamplex 2: Computethenearestneighborsofx:xs1,xs2,...xsK2X,attainedfromXwithinthedistancethresholdd 3: Assignweightstothechosentrainingsamplesusing: sk=exp(dsk) PKk=1exp(dsk)(5) wheredsk=Dist(x,xsk) 4: Outputthestronghypothesis H(x)=sign(KXk=1skTXt=1wsktht(x))(5) C-dimensionalvectorspacewithapositiveentryindicatingthatthedatabelongtothatclass,negativeotherwiseandthemagnitudebeingthecondenceoftheprediction.ThenalhypothesiswillassignclassctothetestdataifthecthentryhasthemaximumpositivevalueinthefollowingvectorPKk=1skPTt=1wsktht(x).Therefore,theobjectivefunctionisasfollows, w=argminw(NXn=1Dist(wnh(xn),Ln))+NXn,m=1G(n,m)Dist(wn,wm). Wedonotpresentthedetailedpseudocodeheresincethewholestructureofthealgorithmissimilartothe2-classcase. 5.3Optimization Inthissection,webrieydiscusstheoptimizationtechniqueandderivetheclosedformsolutionsforboth2-classandmulti-classalgorithmswhenusingtheL2distance. 87

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ByusingtheL2distanceasthedistancemeasureinthe2-classobjectivefunction,weobtainEfwng=NXn=1jjwnh(xn))]TJ /F6 11.955 Tf 11.96 0 Td[(ynjj2+NXn,m=1G(n,m)jjwn)]TJ /F8 11.955 Tf 11.95 0 Td[(wmjj2. Expandingtherighthandsideoftheequation,wegetE=NXn=1wtnh(xn)ht(xn)wn+NXn=1y2n)]TJ /F4 11.955 Tf 11.96 0 Td[(2NXn=1ynwtnh(xn)+NXn,m=1G(n,m)wtnwn+NXn,m=1G(n,m)wtmwm)]TJ /F4 11.955 Tf 11.96 0 Td[(2NXn,m=1G(n,m)wtnwm. LetHn=h(xn)ht(xn)anddenotethecolumnvectorofWasWt=[wt1,wt2,...,wtN],thematrixBkasBk=Hk+2(Pn=k,m6=1G(n,m)+Pn6=1,m=kG(n,m))ITTandthecolumnvectorbasbt=[y1ht(x1),y2ht(x2),...,yNht(xN)].Thecostfunctioncanbere-arrangedintoamatrixformasfollows:E=Wt0BBBBBBB@B1...)]TJ /F4 11.955 Tf 9.29 0 Td[(2G(1,N)ITT)]TJ /F4 11.955 Tf 9.29 0 Td[(2G(2,1)ITT...)]TJ /F4 11.955 Tf 9.29 0 Td[(2G(2,N)ITT.........)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(N,1)ITT...BN1CCCCCCCAW)]TJ /F4 11.955 Tf 19.26 0 Td[(2btW+NXn=1y2n. TakingthederivativeofEw.r.t.Wandsettingtheequationto0,wehave,@E @W=(Dt+D)W)]TJ /F4 11.955 Tf 12.83 0 Td[(2b=0,withDbeingthematrixintheequationabovethatcontainsBkasdiagonal.Theproblemisnallyreducedtosolvingthefollowinglinearsystem(Dt+D)W=2b. Forthemulti-classcase,assumethereareCclasses,foreachweaklearnerht,theoutputisaCdimensionalvectordenotedas[ht1(xn),...,htC(xn)].Were-arrangethoseTC-dimensionalvectorsbytakingthecorrespondingcthentryofeachandstackingthemintoonevector,andgethc(xn)=[h1c(xn),...,hTc(xn)].Recallthatthetruelabel 88

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foreachdataisaCdimensionalvector.Withtheabovenotation,theobjectivefunctionbecomesE=NXn=1jj[wnh1(xn),wnh2(xn),...,wnhC(xn)])]TJ /F4 11.955 Tf 7.97 0 Td[([yn1,yn2,...,ynC]jj2+NXn,m=1G(n,m)jjwn)]TJ /F8 11.955 Tf 7.97 0 Td[(wmjj2=NXn=1jjwnh1(xn))]TJ /F6 11.955 Tf 7.97 0 Td[(yn1jj2+NXn=1jjwnh2(xn))]TJ /F6 11.955 Tf 7.97 0 Td[(yn2jj2++NXn=1jjwnhC(xn))]TJ /F6 11.955 Tf 7.97 0 Td[(ynCjj2+NXn,m=1G(n,m)jjwn)]TJ /F8 11.955 Tf 7.97 0 Td[(wmjj2. Similartothe2-classcase,wechangesomenotationsanddeneHn=h1(xn)ht1(xn)+h2(xn)ht2(xn)++hC(xn)htC(xn),whichisaTTmatrix,Bk=Hk+2(Pn=k,m6=1G(n,m)+Pn6=1,m=kG(n,m))ITTandbtc=[y1htc(x1),y2htc(x2),...,yNhtc(xN)].Thus,wehaveE=Wt0BBBBBBB@B1)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(1,2)ITT...)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(1,N)ITT)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(2,1)ITTB2...)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(2,N)ITT............)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(N,1)ITT)]TJ /F4 11.955 Tf 9.3 0 Td[(2G(N,2)ITT...BN1CCCCCCCAW)]TJ /F4 11.955 Tf 19.27 0 Td[(2bt1W)]TJ /F4 11.955 Tf 11.96 0 Td[(2bt2W)-221()]TJ /F4 11.955 Tf 40.51 0 Td[(2btCW+NXn=1(y2n1+y2n2++y2nC). TakethederivativeofEw.r.t.Wandsettheequationto0.Weget@E @W=(Dt+D)W)]TJ /F4 11.955 Tf 12.7 0 Td[(2(b1+b2++bC)=0,thatistosolvethelinearsystem(Dt+D)W=2(b1+b2++bC).NotethatDhereisthecorrespondingmatrixintheaboveequationwithBkinitsdiagonal. Notethatinbothcases,Dt+Daresparsematrices.Finally,theoveralloptimizationreducestoonebasicproblem:solvingasparselinearsystemAx=b[ 23 ]withAbeingDt+Dinthepreviousequations.SparseCholeskyfactorizationisusedwhenAisa 89

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positivedenitematrix.Sincethepositivedenitepropertyisnotguaranteedforsmall,weresorttotheLDLtfactorizationwhenAisindenite. 5.4Experiments Inthissection,wedemonstratethestabilityandeffectivenessoftheclassicationmodel,alongwithitsnoveltestingtechniquebyclassifyingtheOASISMRIdatabaseintotheconstituentclassesnamely,young(Y),old(O),middleaged(M),controlsandverymildtomoderateAlzheimerdisease(AD)andtheEpilepsydatasetintocontrolandpatientswithepilepsy.Wealsoindicatetheepilepticfocusinthepatientsbydistinguishingbetweenrightanteriormedialtemporallobe(RATL)andleftanteriormedialtemporallobe(LATL). 5.4.1FeatureSelection Inadditiontothebehavioralassessmentsandcognitivetests,amorphologicalmarkerfortheAlzheimer'sdiseaseistheenlargementofventriclesandtheshrinkageofcortexandhippocampi.In[ 67 ],Mertetal.revealedthestructuralchangesinthebrainacrossdifferentagegroupsandbetweenthehealthyandpatientswithdementia.Similarly,thehippocampusisthemajorstructurewithlesionsforepilepsyandisespeciallyvulnerabletoepilepticseizure.Thisledustothehypothesisthatagoodfeaturemightbeonethatcapturesthestructuraldifferences.ToidentifythechangesintheanatomicalstructuresrelatedtoAlzheimer'sdisease,wecomputethedeformationeld(vectorsinthe3Dspace)ofeachbrainMRscanbyregisteringittoanemergingatlasachievedbyperformingagroupwiseregistration[ 43 ]oftheMRimageswithintheOASISdataset.FortheEpilepsydataset,weregisterthelefthippocampustotherightonewithineachsampleandobtainthedeformationeld.The3Dhistogramsofthedeformationeldsareevaluatedasourfeaturesets,wherethenumberofbinsineachdirectionissetto(666)forconstructingthehistogramsofthevectors. 90

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5.4.2WeakLearners InordertodemonstratethepowerofCAVIAR,wechoosethemostsimpleweaklearnersbyrandomlyselectingacomponentfromthefeaturevectorandpickingarandomthreshold.Thesampleswereassignedtoparticularclassesbasedontheirvaluesinthatchosencomponentofthefeaturevectorbycomparingtothethreshold.Forinstance,assumethedataxhasad-dimensionalfeature.Theweaklearnerh(x)assignsthedatatoclass1ifthekthdimension(randomlychosen)ofthefeaturevectorislargerthanarandomvaluet,andassignstoclass2otherwise. 5.4.3ModelSelection ThefreeparametersinvolvedinourCAVIARalgorithmincludetheregularizationparameter,thenumberofnearestneighborsKinsettingupthenearestneighborgraphG,theweightingparameterandthedistancethresholdd.TheexperimentsindicatethatonlydiscrucialforCAVIAR'sperformance,andrequirestuningwiththeremainingparametersxedapriori,withoutsacricingperformance.Theparameterisusedtotunetheamountofsimilarityoftheweightswnandwmcorrespondingtothenearestneighborsxnandxm.CAVIARworkswellintheregion2(0,1)andwesetittobe0.01intheexperiments.Empirically,Kischosentobeapproximately5%ofthenumberofthetrainingdata.isusedtoadjusttheweightsofthechosentrainingdataasinEq. 5 .Asettingof=0impliesthatthedataareequallyweightedanda=1correspondstoasinglenearestneighborchoice.Stableperformanceisachievedwhenisinverselyproportionaltotheaveragedistanceamongthesamples. Themostcrucialparameterforthisalgorithmisthedistancethresholdd.NotethatEuclidiandistanceisusedasourdistancemeasurehere.Alargerthannecessarydmeansmoretrainingdataareinvolvedinthetestingwhichresultsinover-tting.Meanwhileasmallerthannecessarydleadstoasmallportionoftrainingresultsbeingusedwhichpushesthealgorithmclosertobeinganearestneighbormethod.Therefore,aproperchoiceofdisofgreatimportance. 91

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Figure5-1. Theclassicationerrorsforbothvalidationandtestdatasetsw.r.t.differentd(indicatedbythebinsonthex-axis)forMiddleagedvs.Old. Tothisend,wediscretizethesearchspacefordintolbinsandthisbinnumberonlyaffectstheresolutionofthesearch.Bysearchingovertheldifferentdvaluesandcomputetheerror,wendthebestdforeachdataset.Theoptimizationcurvesoftheclassicationerrorsw.r.t.d(indicatedbylbins)forboththevalidationandtestdataareshowninFig. 5-1 .Weobtainthebestdaccordingtothevalidationerrorcurveandusethisdinthetestingstage. 5.4.4ExperimentalResults TheOASISdatasetcontainscross-sectionalcollectionof416subjectsaged18to96.Wedividedthesubjectsintothreegroups,withagesbelow40designatedasyoung,above60asoldandtheonesinbetweenasmiddleaged.Amongtheoldpeople,wetook70subjects,and35ofthemwerediagnosedwithverymildtomoderateADwhiletherestwerecontrols.Wehave39controlsand44epilepsypatientsintheEpilepsydataset,whereinthepatientsaredividedinto19LATLsand25RATLs. 92

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Figure5-2. Theclassicationerrorsoftheweaklearnersandthenalstronghypothesisw.r.t.differentnumberofweaklearnersforADvs.Control. Foreachexperiment,werandomlydividedthedatasetinto5portionsandtook4ofthemfortrainingand1fortesting,with1ofthetrainingdataselectedasavalidationset.Fordifferentnumbersofweaklearners,theexperimentswererepeated20timesandtheaveragesweretakenastheresulttobereported. WerstclassiedthesamplesbetweencontrolsandepilepsyintheEpilepsydatasetandanytwoagegroupsintheOASISdatabaseusingthe2-classversionofCAVIARandthendemonstratedtheperformanceofourmulti-classclassicationalgorithmbyclassifyingthethreeagegroupssimultaneously.Finally,wepresentedmoreseriouschallengestoouralgorithmbyclassifyingtheLATLandRATLfortheepilepsypatientsanddistinguishingthehealthyfromtheverymildtomildADpatientsinOASISdataset.Forcomparison,theresultsoftheAdaboostalgorithmwiththesameweaklearnersettings(20weaklearners)arealsoreported. 93

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InTable 5-1 and 5-2 ,weshowtheaverageerrorfortheweaklearnersinthesecondcolumnandthetesterrorofthenalstronghypothesisinthethirdcolumn,followedbytheimprovementoftheperformanceofouralgorithmw.r.t.theweaklearners,andreportedthetesterrorforAdaboostinthelastcolumn.Theerrorismeasuredbythemis-classicationrate.Toillustratethechangeofperformancew.r.t.theincreasingnumberofweaklearners,weshowtheexperimentalresultsofADvs.ControlinFigure. 5-2 Table5-1. TestingresultsofEpilepsydataset Ave.WLErrTestErrImproveAda.Err Controlvs.Epilepsy0.49370.286042.07%0.4250LATLvs.RATL0.52940.269849.03%0.4889 Table5-2. TestingresultsofOASISdatabase Ave.WLErrTestErrImproveAda.Err Yvs.M0.32200.023392.76%0.0400Mvs.O0.67460.016497.57%0.0320Ovs.Y0.47700.008698.19%0.0125Yvs.Mvs.O0.79250.037595.27%0.0912ADvs.Control0.68760.041793.94%0.0975 TheexperimentalresultsindicatethatthedeformationeldcapturesthestructuralchangesacrossthebrainsverywellandtheCAVIARalgorithmsignicantlyimprovestheperformancesoftheweakclassiers,whenpresentedwithasmallnumberofsimpleweaklearnersinboth2-classandmulti-classcases. 5.5Summary Inthischapter,weintroducedanovelclassicationalgorithmthatcombinesweakclassiersbasedontheassumptionthatthelocallyweightedcombinationvariesw.r.t.boththeweakclassiersandthetrainingdatasamples.Aregularizationschemeusingthenearestneighbormethodwasimposedinthetestingstagetoavoidovertting.WeempiricallyshowedthatCAVIARsignicantlyincreasestheperformanceoftheweak 94

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classiersbyshowcasingtheperformanceofourtechniqueonOASISbraindatabaseandEpilepsydataset. 95

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CHAPTER6CONCLUSIONSANDFUTUREWORK InthisthesiswehavedevelopedacomputeraideddiagnosispipelinenamedEpilepsyToolkit(ETK),whichaimsatautomaticallysegmentingstructuresofinterestfromagivenMRIscanandclassifyingthescanintothenormalordiseasesaccordingtothefeaturevectorsestimatedfromtheanatomicalstructuressegmented.ETKconsistsofthreemajormodules:atlasconstruction,segmentationandclassication.Ourcontributionstoeachmodulecanbesummarizedasfollows. InChapter 2 ,wedevelopedanovelandrobustgroupwisepointsetregistrationalgorithmwhichutilizedtheCDF-basedHavrdaCharvat(CDF-HC)divergenceasthesimilaritymeasuretoconstructashapeatlasofasinglestructurerepresentedbyanatomicallandmarks.Sincemanyneurologicaldisordersarediagnosedbythestructuralabnormalitiesascribedtoseveralbrainstructuresratherthanasingleone,wethereforeproposedanatlasconstructionalgorithmforashapecomplexinChapter 3 .Thismethodbuildsarelationshipbetweenthespaceofdistancetransformrepresentationoftheshapecomplexanditssquarerootdensityrepresentation.Theatlaswasestimatedfromthegeodesicmeanofthesquarerootdensitiesandmappedbacktothedistancetransformspace.Anovelmulti-atlassegmentationtechniquewasproposedinChapter 4 ,whereinastrongsegmenterwascomposedbyoptimallycombiningasetofweakatlasbasedsegmentations.Chapter 5 demonstratedthediagnosticresultsbyclassifyingdatasetsintocontrolsandepilepticsbasedonthehistogramofthedeformationelds(asafeature)computedfromthepreviouslysegmentedstructures. Bypresentingsolidtheoreticaldiscussionsandprovidingabundantexperimentalresultsforeachproposedalgorithm,thisthesisdepictedapromisingworkowoftheETKsystem.Herewewillpresentthepossibledirectionsforfutureresearchesbasedoncurrentwork. 96

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AnimmediatenextstepforthegroupwisepointsetregistrationalgorithmproposedinChapter 2 istogeneralizeittoimageregistration.DuetotherobustnessandefciencyweobservedbyemployingtheCDF-HCdivergenceinthepointsetregistration,webelievethattheapplicationofCDF-HCwillalsoimprovetheexistinggroupwiseimageregistrationframework.Oneofthemajorchallengesingroupwiseimageregistrationisthescalingprobleminthejointdensityestimationofmultipleimages.Forexample,10imagesofsize256256256with64intensitybinsgivesahistogramofsize6410=260,butonly2563=224samples.Anoveldensityestimationtechniqueproposedby[ 64 ]successfullyavoidsthisaforementionedproblembyusingageometricapproachwhichregardstheprobabilitydensityasbeingproportionaltotheoverlappingareaamongtheisocontoursofthegivenNimagefunctions.Equippedwiththistechnique,thederivationoftheCDF-HCbasedgroupwiseimageregistrationbecomesapossibleresearchdirection. Intheshapecomplexatlasconstructionalgorithm,thedistancetransform(DT)functionhasitsmeritinrepresentinganycomplicatedstructuresandtheSchrodingerwavefunctionconnectstheDTfunctionspacetosquarerootdensityspacewhereallkindsofstatisticalanalysisareperformed.However,amajorconcernrelatedtothisalgorithmisthatthetopologicalconstraintsamongthestructureswithintheshapecomplexwerenotretainedsincewerepresentedeachshapecomplexusingasingledistancetransform.Asaresult,thetopologyofthenalatlasmaynotbeguaranteed.Thisissueneedstobefurtheraddressedinthefuturework. Inthisthesis,thesegmentationalgorithmdevelopedthusfarwasforsingleanatomicalstructure,forinstance,thehippocampus.Basedontheshapecomplexatlasconstructionmethod,itispossibletogeneralizeittomulti-structuraltechnique.Morespecically,thesegmentationresultforoneparticularstructurecanbeimprovedbyutilizingtheinformationfromitsneighboringstructures.Thehippocampusandamygdalaregionservesasagoodexamplewhereasingleamygdalaisverydifculttosegment 97

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evenbyexpertneurologist,however,duetoitsadjacencytothehippocampusthesegmentationaccuracywouldbeimprovedbyattainingthecorrectpositioninformationofthehippocampusandhenceroughlyobtainthepriorforamygdalasegmentation.Ontheotherhand,inthesegmentationalgorithmweproposedforsinglestructure,thediscriminativespatialregularizationwasdenedonthegroundtruthlabelsaroundtheneighborhoodofeachvoxel.Whengeneralizingittoshapecomplexsegmentation,weneedtodenethenotionofneighborhoodamongstructures.Thesepreviouslymentionedproblemsalsoprovideinterestingresearchperspectives. 98

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BIOGRAPHICALSKETCH TingChenwasborninLinhai,Zhejiang,P.R.China.ShereceivedherBachelorofSciencedegreefromtheUniversityofScienceandTechnologyofChinain2006.SherecievedherPh.D.fromtheDepartmentofComputerandInformationScienceandEngineeringatUniversityofFloridainthefallof2011.Herresearchinterestsaremedicalimaging,machinelearningandcomputervision. 107