Algorithms for Tracking on the Manifold of Symmetric Positive Definite Matrices

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Algorithms for Tracking on the Manifold of Symmetric Positive Definite Matrices
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Cheng, Guang
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Doctorate ( Ph.D.)
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University of Florida
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Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Vemuri, Baba C
Committee Members:
Rangarajan, Anand
Banerjee, Arunava
Ho, Jeffrey
Presnell, Brett D

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kalman -- manifold -- matrix -- spd -- tracking -- tractography
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Computer Engineering thesis, Ph.D.
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The problem of tracking on the manifold of $n\times n$ symmetric positive definite (SPD) matrices is an important problem and has many applications in several areas such as computer vision and medical imaging. The aim of this dissertation is to develop novel tracking algorithms on Pn for several different applications. One of the basic tracking problems on $P_n$ is to recursively estimate the Karcher expectation -- an generalization of the expectation to the Riemannian manifold, which can be viewed as tracking a static system. In this dissertation, we proposed a novel recursive Karcher expectation estimator (RKEE), and we further proved its unbiasedness and L2-convergence to the Karcher expectation under symmetric distribution on $P_n$. Synthetic experiments showed RKEE the similar accuracy as the Karcher mean but more efficient for sequential data. We then developed a fast DTI (diffusion tensor imaging) segmentation algorithm based RKEE. The experiments on the real data of rat spinal cord and rat brain with comparison to Karcher mean and other type of centres based algorithms demonstrated the accuracy and efficiency of RKEE. To further tackle the dynamic system tracking on $P_n$, we studied and discovered several properties of the generalized Gaussian distribution on $P_n$, based on which a novel probabilistic dynamic model is proposed in conjunction with an intrinsic recursive filter for tracking a time sequence of SPD matrix measurements in a Bayesian framework. This newly developed filtering method can then be used for the covariance descriptor updating problem in covariance tracking, leading to new efficient video tracking algorithms. To show the the accuracy and efficiency of our covariance tracker in comparison to the state-of-the-art, we present synthetic experiments on Pn, and real data experiments for tracking in video sequences. To handle the non-$P_n$ inputs and the non-linear observation model, a novel intrinsic unscented Kalman filter tracking points on Pn is presented. With the combination of the stream line tracking strategy, an efficient fiber tracking method is proposed to track white matter fibers from diffusion weighted (DW) MR images of mammalian brains specifically, human and rats. Different from the first method, the input of filter could be the diffusion weighted MR signal, which makes it possible to track fibers directly without the pre-process step commonly required by existing methods. Real data experiments on data sets of human brain and rat spinal cords are presented and depicted the accuracy and efficiency of the method. For group-wise analysis of the white matter fiber bundles from our tracking algorithm, a novel group-wise registration and atlas construction algorithm for the DW MR datasets represented by Gaussian mixture fields is proposed and applied to the spinal cord dataset. The group-wise analysis result of the spinal cord fiber bundle in this dissertation showed the significant difference between injured and healthy rats.
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by Guang Cheng.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Vemuri, Baba C.
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ALGORITHMSFORTRACKINGONTHEMANIFOLDOFSYMMETRICPOSITIVEDEFINITEMATRICESByGUANGCHENGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFTHEPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012GuangCheng 2

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TomywifeYandi,mydaughterLerong,andmyparents 3

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ACKNOWLEDGMENTS IwouldliketogratefullythankDr.BabaVemurimymajoradvisor,forhisinsightfulguidance,uninchingpatienceandencouragementthroughoutmyPhDstudy.Thisdissertationwouldnothavebeenwrittenwithouthisguidanceandsupport.Iwouldalsoliketothankmycommitteemembers,Dr.JeffreyHo,Dr.AnandRangarajan,Dr.ArunavaBanerjeeandDr.BrettPresnell,notonlyforagreeingtobemycommittee,butalsoforbeingalwayssupportiveduringtheentireacademicprogramandthebroadexposureIgainedthroughtheircourseofferings.IthankthegenerousresearchsupportprovidedbytheNIHgrantsNS066340andEB007082tomyadvisor,Dr.Vemuri,thatmadeitpossibleformetohaveanuninterruptedRAshipduringthecourseofmyPhDIalsohavereceivedtravelgrantsfromtheCISEdepartmentattheUniversityofFlorida.IthankDr.DenaHowland,Dr.JohnForder,Dr.Min-SigHwangandDr.SarahE.Mondelloforprovidingthedataandsharingtheknowledge.IthankDr.BingJian,Dr.AngelosBarmpoutisandDr.SanthoshKodipakafortheproductivediscussionsandtheirhelpontheproject.Ithankallmylab-matesYuchenXie,TingChen,MeizhuLiu,WenxingYe,DohyungSeo,SileHu,YuanxiangWang,YanDeng,HesamodinSalehian,QiDengandTheodoreHa,forallthehelpstheygaveme.SpecialthanksareextendedtoHesamodinSalehianforhisgreatworkinourcollaboration,especiallytheexperimentsinthechapterofrecursiveKarcherexpectationestimator.IwouldliketothankmywifeYandi,forherpatiencesupportandlove.Ialsothankmyparents,KexinandQixin,forthefaithinmeandallowingmetobewhatIwanttobe. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFABBREVIATIONS ................................ 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1Motivation .................................... 12 1.2MainContributions ............................... 13 1.2.1RecursiveKarcherExpectationEstimator .............. 13 1.2.2IntrinsicRecursiveFilter ........................ 14 1.2.3IntrinsicUnscentedKalmanFilter ................... 15 1.3Outline ...................................... 15 2RIEMANNIANGEOMETRYONPn ......................... 17 2.1GL-invariantmetricvs.EuclideanmetriconPn ............... 19 2.2Log-EuclideanvsGL-invariance ....................... 19 2.3AlgorithmsontheFieldofSPDMatrices ................... 20 3RECURSIVEKARCHEREXPECTATIONESTIMATION ............. 22 3.1BackgroundandPreviousWork ........................ 22 3.2Methods ..................................... 23 3.2.1TheRecursiveKarcherExpectationEstimator ............ 23 3.2.2RecursiveformofthesymmetrizedKL-divergencemean ...... 26 3.2.3RecursivemeanfortheLog-EuclideanMetric ............ 27 3.3Experiments .................................. 28 3.3.1PerformanceoftheRecursiveEstimators .............. 28 3.3.2ApplicationtoDTISegmentation ................... 29 4INTRINSICRECURSIVEFILTERONPn ...................... 34 4.1BackgroundandPreviousWork ........................ 34 4.2IRF:ANewDynamicTrackingModelonPn ................. 36 4.2.1GeneralizationoftheNormalDistributiontoPn ........... 36 4.2.1.1Themeanandthevarianceofthegeneralizednormaldistribution .......................... 40 4.2.2TheProbabilisticDynamicModelonPn ............... 46 5

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4.3IRF-basedTrackingAlgorithmonPn ..................... 46 4.3.1TheBayesianTrackingFramework .................. 46 4.3.2TheTrackingAlgorithm ......................... 48 4.4Experiments .................................. 50 4.4.1TheSyntheticDataExperiment .................... 50 4.4.2TheRealDataExperiment ....................... 52 5INTRINSICUNSCENTEDKALMANFILTER ................... 56 5.1BackgroundandPreviousWork ........................ 56 5.2IntrinsicUnscentedKalmanFilterforDiffusionTensors ........... 57 5.2.1TheStateTransitionandObservationModels ............ 57 5.2.2TheIntrinsicUnscentedKalmanFilter ................ 59 5.3Experiments .................................. 61 6ATLASCONSTRUCTIONFORHARDIDATASETREPRESENTEDBYGAUSSIANMIXTUREFIELDS .................................. 64 6.1BackgroundandPreviousWork ........................ 64 6.2Methods ..................................... 65 6.2.1ImageAtlasConstructionFramework ................. 65 6.2.2L2DistanceandRe-orientationforGMs ............... 67 6.2.3MeanGMFComputation ........................ 68 6.3Experiments .................................. 69 6.3.1SyntheticDataExperiments ...................... 69 6.3.2RealDataExperiments ........................ 71 7DISCUSSIONANDCONCLUSIONS ........................ 73 REFERENCES ....................................... 76 BIOGRAPHICALSKETCH ................................ 83 6

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LISTOFTABLES Table page 3-1Time(inseconds)formeancomputationintheDTIsegmentationonsyntheticdataset ........................................ 31 3-2Timinginsecondsforsegmentationofgreymatterinaratspinalcord ..... 32 4-1Trackingresultfortherealdataexperiment. .................... 53 7

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LISTOFFIGURES Figure page 3-1Accuracyandspeedcomparisonsoftherecursiveversusnon-recursivemeancomputationalgorithmsfordataonP3 ....................... 29 3-2ResultsfortheDTIsegmentationexperimentsonthesyntheticdataset ..... 31 3-3Segmentationresultsofgreymatterinaratspinalcordfor6differentmethods. 32 3-4Segmentationresultsofthemolecularlayerinarathippocampusfor3differentmethods. ....................................... 33 4-1Meanestimationerrorfrom20trialsforthesyntheticdataexperiment. ..... 52 4-2Headtrackingresultforvideosequenceswithmovingcamera. ......... 55 5-1Fibertrackingresultsonrealdatasetsfromratspinalcords.c[2012]IEEE .. 62 5-2Biomarkerscapturedbycomputingdensitymapforeachberbundle.c[2012]IEEE .......................................... 63 6-1Imageregistrationresultsonsyntheticdataset.c[2011]IEEE ......... 70 6-2Registrationresultsforrealdatasetfromratspinalcord.c[2011]IEEE .... 72 8

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LISTOFABBREVIATIONS DTIdiffusiontensorimagingDWMRIdiffusionweightedmagneticresonanceimagingGMFGaussianmixtureeldHARDIhighangularresolutiondiffusionimagingRKEErecursiveKarcherexpectationestimatorIRFintrinsicrecursivelterIUKFintrinsicunscentedKalmanlterMRImagneticresonanceimagingSPDsymmetricpositivedenite 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofthePhilosophyALGORITHMSFORTRACKINGONTHEMANIFOLDOFSYMMETRICPOSITIVEDEFINITEMATRICESByGuangChengMay2012Chair:BabaC.VemuriMajor:ComputerEngineeringTheproblemoftrackingonthemanifoldofnnsymmetricpositivedenite(SPD)matricesisanimportantproblemandhasmanyapplicationsinseveralareassuchascomputervisionandmedicalimaging.TheaimofthisdissertationistodevelopnoveltrackingalgorithmsonPnforseveraldifferentapplications.OneofthebasictrackingproblemsonPnistorecursivelyestimatetheKarcherexpectationangeneralizationoftheexpectationtotheRiemannianmanifold,whichcanbeviewedastrackingastaticsystem.Inthisdissertation,weproposedanovelrecursiveKarcherexpectationestimator(RKEE),andwefurtherproveditsunbiasednessandL2-convergencetotheKarcherexpectationundersymmetricdistributiononPn.SyntheticexperimentsshowedRKEEthesimilaraccuracyastheKarchermeanbutmoreefcientforsequentialdata.WethendevelopedafastDTI(diffusiontensorimaging)segmentationalgorithmbasedRKEE.TheexperimentsontherealdataofratspinalcordandratbrainwithcomparisontoKarchermeanandothertypeofcentresbasedalgorithmsdemonstratedtheaccuracyandefciencyofRKEE.TofurthertacklethedynamicsystemtrackingonPn,westudiedanddiscoveredseveralpropertiesofthegeneralizedGaussiandistributiononPn,basedonwhichanovelprobabilisticdynamicmodelisproposedinconjunctionwithanintrinsicrecursivelterfortrackingatimesequenceofSPDmatrixmeasurementsinaBayesianframework.Thisnewlydevelopedlteringmethodcanthenbeusedforthecovariance 10

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descriptorupdatingproblemincovariancetracking,leadingtonewefcientvideotrackingalgorithms.Toshowthetheaccuracyandefciencyofourcovariancetrackerincomparisontothestate-of-the-art,wepresentsyntheticexperimentsonPn,andrealdataexperimentsfortrackinginvideosequences.Tohandlethenon-Pninputsandthenon-linearobservationmodel,anovelintrinsicunscentedKalmanltertrackingpointsonPnispresented.Withthecombinationofthestreamlinetrackingstrategy,anefcientbertrackingmethodisproposedtotrackwhitematterbersfromdiffusionweighted(DW)MRimagesofmammalianbrainsspecically,humanandrats.Differentfromtherstmethod,theinputofltercouldbethediffusionweightedMRsignal,whichmakesitpossibletotrackbersdirectlywithoutthepre-processstepcommonlyrequiredbyexistingmethods.Realdataexperimentsondatasetsofhumanbrainandratspinalcordsarepresentedanddepictedtheaccuracyandefciencyofthemethod.Forgroup-wiseanalysisofthewhitematterberbundlesfromourtrackingalgorithm,anovelgroup-wiseregistrationandatlasconstructionalgorithmfortheDWMRdatasetsrepresentedbyGaussianmixtureeldsisproposedandappliedtothespinalcorddataset.Thegroup-wiseanalysisresultofthespinalcordberbundleinthisdissertationshowedthesignicantdifferencebetweeninjuredandhealthyrats. 11

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CHAPTER1INTRODUCTION 1.1MotivationTrackingingeneralisatasktorecursivelyestimatethecurrentsystemstatebasedonasequentialdataset.Itisveryimportanttaskbothcomputervisionandmedicalimaging.Incomputervision,trackingiscrucialforvideosurveillance,augmentedreality,human-computerinteraction,etc.Itisalsoanecessarypreprocessingstepforhighlevelcomputervisiontaskssuchasvisualsceneanalysis.Inmedicalimaging,trackingisveryusefulinnotonlyanalysingtimesequencessuchascardiaccycleinmedicalimagingbutalsoinneuralbertractographyfromdiffusionweight(DW)MRIdataset.Trackingistraditionallyatimeseriesanalysisproblem,itiscloselyrelatedtoprediction.Predictionhasawideclinicalapplications,suchasdiseaseprediction.Bothtrackingandpredictiontaketimesequencesasdatainput,andareusuallybasedondynamicmodels.Manytrackingmethods,suchasthewellknownKalmanlter,arebasedonthepredict-updateframework,wherepredictionisacrucialpartinthetrackingalgorithm.Themaindifferencebetweentrackingandpredictionisthat,intrackingweestimatethecurrentstateofacertainprocessbasedonthecurrentandpreviousobservations,whileinpredictiontheestimationisforfuturestatewherenodirectobservationisavailable.MostclassicaltrackingtechniquesareinEuclideanspace.However,incertainapplications,theproblemsmightnotnaturallybeintheEuclideanspace.Instead,theyusuallylieonaRiemannianmanifold,butnotinavectorspace.Also,theinputdatadimensioninmodernproblemsisusuallyhuge.Manylinearandnon-lineardimensionalreductiontechniquesarehenceusedtondthemeaningfullowerdimensionalrepresentationsofthedata,andtheserepresentationsmightnotbeintheEuclideanspace.Therefore,trackingandpredictionalgorithmsontheRiemannianmanifoldcouldbeappliedinmanypracticalproblems. 12

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Thisdissertationfocusesthetrackingprobleminthespaceofnnsymmetricpositivedenite(SPD)matricesrepresentedasPn.Manyfeaturedescriptorssuchascovariancematrices,Cauchydeformationtensors,diffusiontensors,metrictensors,etc,canberepresentedinPn.ThusalgorithmsonPncouldbewidelyappliedtopracticalproblemsindifferentareassuchascomputervision,medicalimaging,etc.PnisknowntobeaRiemannianspacewithnon-positivecurvature.ManyresearcheshavereportedondifferentproblemsonPnsuchasthecomputationofintrinsic/extrinsicmean,linear/non-lineardimensionalreduction,statistics,etcwithapplicationsinmanydifferentareas.ThisdissertationisprimarilymotivatedbypracticaltrackingproblemssuchasvideotrackingandotherproblemswheretrackingalgorithmscanbeappliedincludingsegmentationofDTI(diffusiontensorimaging)datasetandbertractography. 1.2MainContributions 1.2.1RecursiveKarcherExpectationEstimatorFindingthemeanofapopulationofSPDmatrices/tensorsisanoftenencounteredprobleminmedicalimageanalysisandcomputervision,specicallyindiffusionMRIprocessing,tensor-basedmorphometry,textureanalysisusingthestructuretensoretc.Themeantensorcanbeusedtorepresentapopulationofstructuretensorsintextureanalysis,diffusiontensorsindiffusiontensorimage(DTI)segmentationorforinterpolationofdiffusiontensorsorinclusteringapplications.Ameanisusuallyusedasagoodestimatoroftheexpectation.Ifthedatasamplesaregivensequentially,themeanndingproblemcanalsobeviewedasatrackingproblemforastaticprocess.Itiswellknownthatcomputationofthemeancanbeposedasaminimizationprobleminwhichoneminimizesthesumofsquareddistancesbetweentheunknownmeanandthemembersofthesetwhosemeanisbeingsought.Mathematicallyspeaking,wewanttond,=minPnid2(xi,),where,disthechosendistance,xiarethedatasampleswhosemeanisbeingsoughtandisthemean.Dependingonthedenitionofdistanced,onegetsdifferentkindsofmeans.Forexample,ifwechoose 13

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theEuclideandistanceford,wegetthearithmeticmean,whereasifwechosetheL1-norminsteadoftheL2normintheaboveformula,wegetthemedian.Ifwechoosethegeodesicdistanceinthedomainofxi,wegettheKarchermean[ 38 ].Currently,therearenoexistingclosedformsolutionforKarchermeancomputationonPnformorethantwosamplepoints[ 54 ].Agradientbasedoptimizationalgorithm[ 54 ]isusedinpracticewhichisknowntobeinefcient.Inthisdissertation,weproposeanovelrecursiveKarcherexpectationestimator(RKEE),whichisanalgorithmtorecursivelyestimatetheKarcherexpectation.TheproofoftheunbiasednessandL2-convergenceofRKEEunderanysymmetricdistributionsonPnisalsopresented.SyntheticdataexperimentsshowedthesimilaraccuracyofRKEEandKarchermeanasanestimatorofKarcherexpectation,butRKEEismoreefcientespeciallyforsequentialdataset.FurtherweappliedRKEEtoDTIsegmentationproblemandcomparedwithKarchermeanandothercentresonrealdatasetofratspinalcordsandratbrains. 1.2.2IntrinsicRecursiveFilterInrecentyears,thecovarianceregiondescriptorwhichisthecovariancematrixofthefeaturevectorsateachpixelintheregion,areshowntoberobustandefcientinvideotrackinganddetections[ 61 ].Severalworks[ 43 75 76 83 83 84 84 ]werereportedtoaddresstheproblemofupdatingcovariancedescriptorinvideotracking.Here,anovelprobabilisticdynamicmodelonPnbasedongeometryandprobabilitytheoryispresented.Thenoisystateandobservationsaredescribedbymatrix-variaterandomvariableswhosedistributionisageneralizednormaldistributiontobasedontheGL-invariantmeasure.Thenannovelintrinsicrecursivelter(IRF)onPnisdevelopedbasedonthedynamicmodel,andappliedtocovariancetracking,whichformsanrealtimevideotrackingalgorithm.Syntheticandrealdataexperimentsarepresentedtosupporttheeffectivenessandtheefciencyoftheproposedalgorithm. 14

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1.2.3IntrinsicUnscentedKalmanFilterDiffusion-WeightedMRImaging(DW-MRI)isauniquenon-invasivetechniquethatcanlocallyinfertheimagedtissuestructureinvivobyMRsignalthatissensitivetothewatermoleculediffusion.TheDW-MRIdatasetisthena3Dimagethatcontainstissue(suchasbrainwhitematter)directionalinformationateachofitsvoxel.Thisdirectionalinformationateachvoxelforsinglebercasecanbemodelledbya2ndorderpositivedenitetensor(SPDmatrix)whichistheclassicaldiffusiontensorimage(DTI).Highordermodelssuchasmulti-tensor,highordertensor,etchavebereportedinordertohandlemorecomplexcases,e.g.bercrossing.However,the33SPDmatrixwhichcanbeviewedasapointinP3isstillaveryusefuldescriptorinrepresentingthelocalberinformation.Tofurthervisualizeandanalysethetissuestructure,thebertrackingtechniqueisneededtoreconstructtheimagedtissuewhichisveryimportantinbothresearchandclinicalapplicationsinneuroscience.ThebertractographyisformulatedasatrackingprobleminPn.HerewepresentanovelintrinsicunscentedKalmanlter(IUKF)onPn,whichtothebestofourknowledgeistherstextensionoftheunscentedKalmanltertoPn.Weapplythisltertobothestimateandtrackthetensorsinmulti-tensormodelusingtheintrinsicformulationtoachieveasdemonstratedthroughexperiments.Weperformrealdemonstratetheaccuracyandefciencyofourmethod.Also,agroup-wiseregistrationandatlasconstructionmethoddevelopedtoregisterDW-MRdatasetsrepresentedbyGaussianMixtureFieldsisproposedforgroupberanalysis. 1.3OutlineTheremainingchaptersareorganizedasfollows:ThebasicpropertiesofPnandcommonRiemannianmanifoldscanbefoundinChapter2.TheRKEEandapplicationtoDTIsegmentationisintroducedinChapter 3 ,followedbytheIRFonthespaceofSPDmatricesanditsapplicationstocovariancetrackingisdiscussedinChapter 4 ;TheFibertrackingwithintrinsicunscentedKalmanlterispresentedinChapter 5 ;This 15

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isfollowedbytheatlasconstructionmethoddevelopedforthegroupberanalysisinChapter 6 .AndnallytheconclusioncanbefoundinChapter 7 .Alargepartofthisthesishasbeenpublishedinseveralpapers[ 18 20 ]. 16

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CHAPTER2RIEMANNIANGEOMETRYONPNInthischapterweintroducethebasicconceptsofRiemanniangeometryonPn,andreferthereaderto[ 34 53 70 ]fordetails.Pnisthespaceofnnsymmetricpositivedenite(SPD)matrices,whichisaRiemannianmanifold.ItcanbeidentiedwiththequotientspaceO(n)nGL(n)[ 70 ],whereGL(n)denotestheGeneralLineargroupthegroupof(nn)non-singularmatrices,andO(n)istheorthogonalgroupthegroupof(nn)orthogonalmatrices.ThismakesPntobeahomogeneousspacewithGL(n)asthegroupthatactsonitandthegroupactiondenedforanyX2PnbyX[g]=gXgt.OnecannowdeneGL-invariantquantitiessuchastheGL-invariantinnerproductbasedonthegroupactiondenedabove.WewillnowbeginwithinnerproductinthetangentspaceofPn.FortangentvectorsUandV2TXPn(thetangentspaceatpointX,whichisthespaceofsymmetricmatricesofdimension(n+1)n=2andavectorspace)theGLinvariantinnerproductisdenedas8g2GL(n),X=gXgt.OnPnthisGLinvariantinnerproducttakestheform, X=tr(X)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2UX)]TJ /F5 7.97 Tf 6.59 0 Td[(1VX)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2).(2)Withmetric/innerproductdenedonthemanifold,thelengthofanycurveinPn,:[0,1]!Pnisdenedaslength()2=R10<_,_>(t)dt.Thedistancebetween8X,Y2PnisdenedasthelengthoftheshortestcurvebetweenXandY(Geodesicdistance).WiththeGL-invariantmetric,thedistancebetweenX,Y2Pnis dist(X,Y)2=tr(log2(X)]TJ /F5 7.97 Tf 6.59 0 Td[(1Y))(2)wherelogisthematrixlogoperator.SincethisdistanceisinducedfromtheGL-invariantmetricinEquation 2 ,thisdistanceisnaturallyGL-invarianti.e. dist2(X,Y)=dist2(gXgt,gYgt)(2) 17

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WithGL-invariantmetricdenedonPn,theintrinsicorKarchermeanofasetofelementsXi2Pncanbecomputedbyperformingthefollowingminimization: =argminXidist2(Xi,)(2)usingagradientbasedtechnique,wheretheupdateequationineachiterationis new=Expold(PiLogold(Xi) N)(2)whereisthestepsize,andExpold()andLogold()aretheLogandExpentialmapsatpointold2Pn.TheLogandExponentialmaps[ 34 ]areveryusefultoolsontheRiemannianmanifold.TheExponentialmapdenotedasExpX(),whereX2Pn,mapsavectorrootedattheoriginofthetangentspaceTXPntoageodesicemanatingfromX.TheLogmap(LogX())istheinverseoftheExponentialmap.TheExponentialandLogmaponPnaregivenby: ExpX(V)=X1=2exp(X)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2VX)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2)X1=2LogX(Y)=X1=2log(X)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2YX)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2)X1=2(2)whereX,Y2Pn,V2TXPn,andlogandexpdenotethematrixexpandlogoperators.TheKarchermeancanbeviewedasanextensionofthearithmeticmeanfromtheEuclideanspacetotheRiemannianmanifold.Similarly,theexpectationandthevariancecanalsobeextended.GivenarandomvariableM2PnwithaprobabilitydensityP(M) E(M)=argminZPndist(,X)2P(X)[dX] (2) Var(M)=ZPndist(E(M),X)2P(X)[dX] (2) 18

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2.1GL-invariantmetricvs.EuclideanmetriconPnTherearetwoprimarytheoreticalreasonsforthechoiceofaGL-invariantmetricovertheconventionalEuclideanmetricwhendoingoperationsonPnandsinceourdynamicmodelisonPn,thisishighlyrelevant.Firstly,PnisanopensubsetofthecorrespondingEuclideanspaceR(n+1)n=2,whichimpliesthatPnwouldbeincompletewithaEuclideanmetricsinceitspossibletondaCauchysequencewhichmightnotconvergeforthiscase.ThisimpliesthatforsomeoftheoptimizationproblemssetinPn,theoptimumcannotbeachievedinsidePn.Thisinturnmeansthatthecovarianceupdatescouldleadtomatricesthatarenotcovariancematrices,anunacceptablesituationinpractice.ThisproblemwillnotarisewhenusingtheGL-invariantmetric,sincethespaceofPnisgeodesicallycompletewithaGL-invariantmetric[ 70 ].Secondly,ingeneral,thefeaturevectorsmightcontainvariablesfromdifferentsources,e.g.objectposition,objectcoloretc.Inthiscase,anormalizationofthe(ingeneral)unknownscalesofdifferentvariableswouldbenecessarywhenusingtheEuclideandistance,whichisnontrivialandmayleadtouseofadhocmethods.However,withaGLinvariantmetric,thisscalingissuedoesnotarisesince,thepresenceofdifferentscalesfortheelementsofafeaturevectorfromwhichthecovariancematrixisconstructed,isequivalenttomultiplicationofthecovariancematrixwithapositivedenitediagonalmatrix.ThisoperationisaGLgroupoperationandsinceGL-invarianceimpliesinvariancetoGLgroupoperations,thescalingissueisanonissuewhenusingaGL-invariantmetric. 2.2Log-EuclideanvsGL-invarianceLog-Euclideandenedin[ 3 ]isaframeworkthatinducesthemetricfromtheEuclideanspacetotheRiemannianmanifold(calledLog-Euclideanmetric)throughtheLogmapatanarbitrarilychosenpointonthemanifold.Fromthisdenitionwecansaythat,ingeneralforPn(n>1),theLog-Euclideanmetricisnotintrinsic.Moreover, 19

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itisnotGL-invariantandisdependentontheaforementionedarbitrarilychosenpoint.Hence,itisnotanaturalmetric.AtypicalLog-Euclideanoperationisathreestepprocedure.Intherststep,allthedatapointsonthemanifoldareprojectedtothetangentspaceatanarbitrarilychosenpoint,usuallytheidentity,throughtheLogmap.Then,standardvectorspaceoperationscanbeappliedinthetangentspace.Inthelaststep,theresultofvectorspaceoperationswhichlieinthetangentspaceareprojectedbacktothemanifoldviatheExponentialmap.IfoneweretousetheLog-Euclideanoperationstocomputetheintrinsic/KarchermeanofagivenpopulationofdataonPn,theresultwillnotbethetrueKarchermean.Log-Euclideanoperationshavebeenusedincovariancetrackingin[ 75 ],wherein,thebasepointisarbitrarilychosenintherstframeanditerativelyupdatedforsubsequentframesusingapredenedandconstantstate-transitionmatrix.Hence,thebasepointwillneverconvergetoanymeaningfulstatisticofthedataset.BecausetheLog-EuclideanframeworkisusedtoapproximatetheGL-invariantmetricusingtheLog-Euclideanmetric,theapproximationerrorwillaffectthetrackingresult,asshownintheSection 4.4.1 2.3AlgorithmsontheFieldofSPDMatricesAeldofSPDmatricesisamapfrom2Dor3DEuclideanspacetoPn.Inthediscretecase,itcanbeviewedasanimage(volume)wheretheimagevalueateachpixel(voxel)isaSPDmatrix.ThiseldofSPDmatricesisalsoreferredasatensoreld.TheGL-invariancepropertyisusuallyrequiredforalgorithmsontensorelds.Thisisbecauseinmanyapplications,e.g.metrictensoreld,deformationtensoreld,etc,thetensorvaluesateachpixel(voxel)isdirectlyrelatedtothelocalcoordinatesystemoftheimagelatticeinsuchawaythatwhenevertheimageisdeformed,thetensorvaluesshouldchangedlinearlyaccordingtotheJacobianofthetransformation.AssumingthetransformationisTontheimagelattice,andthetensorvalueisI(x)=Datpointx 20

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beforethetransformation.Afterthetransformation,thetensorvaluewouldbe I(T(x))=JT(x)DJT(x)t(2)whereJT(x)istheJacobianofTatpointx.(NotethatinDTI(diffusiontensorimage)registrationEquation 2 iscalledre-transformation[ 78 ]whichisoneofthere-orientationstrategiesinDTIregistration[ 19 ]).SooperationssuchasinterpolationanddissimilaritymeasurementsonthesetensorarerequiredtohavetheGL-invariantpropertysuchthattheycanbepreservedbeforeandafterthedeformation.OneexampleistheDTIsegmentation.ThesegmentationresultcouldnotguaranteedtobethesamebeforeandafterafnetransformationifthedistanceusedisnotGL-invariant[ 78 ]. 21

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CHAPTER3RECURSIVEKARCHEREXPECTATIONESTIMATION 3.1BackgroundandPreviousWorkTomesofresearchhasbeenpublishedonndingthemeantensorusingdifferentkindsofdistances/divergencesandhasbeenappliedtoDTIaswellasstructuretensoreldsegmentation,interpolationandclustering.In[ 79 ],authorsgeneralizedthegeometricactivecontourmodel-basedpiece-wiseconstantsegmentation[ 17 72 ]tosegmentationofDTIsusingtheEuclideandistancetomeasurethedistancebetweentwoSPDtensors.Authorsin[ 27 ],presentageometricactivecontour[ 16 49 ]basedapproachfortensoreldsegmentationthatusedinformationfromthediffusiontensorstoconstructthesocalledstructuretensorwhichisasumofstructuretensorsformedfromeachcomponentofthediffusiontensor.ARiemannianmetriconthemanifoldofSPDmatriceswasusedin[ 32 40 77 ]and[ 11 54 59 ]forDTIsegmentationandforcomputingthemeaninterpolantofdiffusiontensorsrespectively.In[ 78 80 91 ]and[ 54 ]thesymmetrizedKL-divergencewasusedforDTIsegmentationandinterpolationrespectively.TheLog-EuclideandistancewasintroducedtosimplifythecomputationsonthemanifoldofSPDmatricesandthiswasachievedbyusingtheprincipalLog-mapfromthemanifoldtoitstangentspaceandthenusingtheEuclideanmetricontheLog-mappedmatricesinthetangentspaceattheidentity[ 4 ].Morerecently,in[ 77 ],astatisticallyrobustmeasurecalledthetotalBregmandivergence(tBD)familywasintroducedandusedforinterpolationaswellasDTIsegmentation.NoneoftheabovemethodsforcomputingthemeanofSPDmatriceswhichareusedwithinthesegmentationalgorithmsorintheirownrightforinterpolationpurposesareinrecursiveform.Arecursiveformulationwouldbemoredesirableasitwouldyieldacomputationallyefcientalgorithmforcomputingthemeansofregionsinthesegmentationapplication.Also,inmanyapplicationssuchasDTIsegmentation,clusteringandatlasconstruction,dataareincrementallysuppliedtothealgorithmfor 22

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classicationorassimilationforupdationofthemeanandanalgorithmthatrecursivelyupdatesthemeanratherthanonethatrecomputesthemeaninabatchmodewouldbemuchmoreefcientanddesirable.Inthisdissertation,wepursuethisverytaskofrecursivemeancomputation.Thekeycontributionsare:(i)rst,wepresentnoveltheoreticalresultsprovingtheL2-convergenceoftherecursiveintrinsicKarcherexpectationcomputationofasetofSPDmatricestothetrueKarcherexpectation.(ii)Additionally,wepresentrecursiveformulationsforcomputingthemeanusingcommonlyuseddistance/divergencemeasuresmentionedaboveandpresentexperimentsthatdepictsignicantgainsincomputetimeovertheirnon-recursivecounterparts.(iii)WepresentsyntheticandrealdataexperimentsdepictinggainsincomputetimeforDTIsegmentationusingtheserecursivealgorithms.Therestofthesectionisorganizedasfollows:inSection 3.2 wepresentnoveltheoreticalresultsleadingtotheintrinsicKarcherexpectationcomputationalgorithm.Inaddition,wepresenttherecursiveformulationsforthecommonlyusedsymmetrizedKL-divergencebasedmeancomputationaswellastheLog-Euclideandistancebasedmean.Section 3.3 containssyntheticandrealdataexperimentsdepictingtheimprovementsincomputationtimeoftheDTIsegmentationtask. 3.2Methods 3.2.1TheRecursiveKarcherExpectationEstimatorWenowdevelopanestimatorfortheintrinsic(Karcher)expectationthatcanbeusedtorepresentasetofdatapointsinPn(thespaceofdiffusiontensors)andcanbecomputedrecursively.Thisrecursivecomputationpropertyisaveryimportantpropertyespeciallyforonlineproblemswherethedatapointsareprovidedsequentially.ThisisverypertinenttoapplicationssuchasDTIandstructuretensoreldsegmentation,diffusion/structuretensorclusteringetc. 23

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LetXk2Pn,k=1,2,...beiidsamplesinPnfromprobabilitymeasureP(M).TherecursiveKarcherexpectationestimatorcanbedenedas: M1=X1 (3) Mk+1(wk+1)=M1 2k(M)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2kXk+1M)]TJ /F15 5.978 Tf 7.79 3.26 Td[(1 2k)wk+1M1 2k (3) Herewesetwk+1=1 k+1.WenowprovethefollowingpropertiesoftherecursiveKarcherexpectationestimatorpresentedhereintheformoftheoremswiththeirproofs.Theorem1.:Leti.i.d.samplesXkbegeneratedfromadensityP(X;)thatissymmetricw.r.t.toitsexpectation,thenMkisaunbiasedestimator.Bysymmetrywemeanthat8X2PnP(X;)=P(X)]TJ /F5 7.97 Tf 6.58 0 Td[(1;),notethatX,,X)]TJ /F5 7.97 Tf 6.59 0 Td[(1areonthesamegeodesicanddist(,X)=dist(,X)]TJ /F5 7.97 Tf 6.59 0 Td[(1) Proof. Withoutlossgeneralityweassumethat=I,whereIistheidentitymatrix.Nowwecanprovethetheorembyinduction.Fork=1,E(M1)=E(X1)=IwhereEdenotestheKarcherexpectation.Pm1(M1;I)isobviouslysymmetric.AssumingE(Mk)=I,andthetheposteriorPmk(Mk)issymmetric.Then,Pmk+1(Mk+1)=ZPnPx(Xk+1)Pmk(X1 2k+1(X)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2k+1Mk+1X)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2k+1)wk+1)]TJ /F5 7.97 Tf 6.59 0 Td[(1X1 2k+1)[dXk+1]=Pmk+1(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1k+1)sincePx,Pmkaresymmetricand(X1 2k+1(X)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2k+1Mk+1X)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2k+1)wk+1)]TJ /F5 7.97 Tf 6.59 0 Td[(1X1 2k+1))]TJ /F5 7.97 Tf 6.59 0 Td[(1=X)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2k+1(X1 2k+1M)]TJ /F5 7.97 Tf 6.58 0 Td[(1k+1X1 2k+1)wk+1)]TJ /F5 7.97 Tf 6.59 0 Td[(1X)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2k+1Thus,Pmk+1issymmetricwithrespecttoI,andE(Mk+1)=I=sinceZPnLog(M)Pmk+1(M)[dM]=ZPnLog(N)]TJ /F5 7.97 Tf 6.59 0 Td[(1)Pmk+1(N)]TJ /F5 7.97 Tf 6.58 0 Td[(1)[dN)]TJ /F5 7.97 Tf 6.59 0 Td[(1]=0 24

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Theorem2:8A,B2Pn,andw2[0,1] jLog(A1 2(A)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2BA)]TJ /F15 5.978 Tf 7.79 3.26 Td[(1 2)wA1 2)jj2tr(((1)]TJ /F3 11.955 Tf 11.96 0 Td[(w)Log(A)+wLog(B))2)(3)NotethattheleftsideoftheinequalityisthesquaredistancebetweentheidentitymatrixandageodesicinterpolationbetweenAandB,andtherightsideoftheinequalityisthesquaredistancebetweentheidentityandtheloglinearinterpolation.ThisinequalityistruebasedonthefactthatPnisaspacewithnon-positivesectionalcurvature. Proof. Let(w)=A1 2(A)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2BA)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2)wA1 2.Then(w)isageodesicbetweenA,BSincePnisaHadamardspace,basedonLurie'snotesonHadamardspace[ 46 ],weknowthat lhs=dist(I,(w))2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(w)dist(I,A)2+wdist(I,B)2)]TJ /F3 11.955 Tf 11.96 0 Td[(w(1)]TJ /F3 11.955 Tf 11.95 0 Td[(w)dist(A,B)2(3)Also rhs)]TJ /F8 11.955 Tf 11.95 0 Td[(((1)]TJ /F3 11.955 Tf 11.95 0 Td[(w)dist(I,A)2+wdist(I,B)2)]TJ /F3 11.955 Tf 11.95 0 Td[(w(1)]TJ /F3 11.955 Tf 11.96 0 Td[(w)dist(A,B)2)=w(1)]TJ /F3 11.955 Tf 11.96 0 Td[(w)(dist(A,B)2)]TJ /F3 11.955 Tf 11.96 0 Td[(tr(LogA)]TJ /F3 11.955 Tf 11.95 0 Td[(LogB)2)0(3)wherethelastinequalityisbasedontheCosineinequalityin[ 6 ].Thus,wehaveprovedthatlhsrhs. Theorem3.Letwk=1 k,wethenhave,Var(Mk)1 ku2,whereu2=Var(Xi),i=1,2.... Proof. :Wecanprovethistheoremalsobyinduction.WestillassumethatE(Xk)=I.Whenk=1,Var(M1)=Var(X1)=u2. 25

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Assumethattheclaimistruefork=i,thatisVar(Mi)1 iu2.Whenk=i+1,usingthelemmaabove,weknowthat, Var(Mi+1(w))ZPnZPnjj(1)]TJ /F16 10.909 Tf 10.91 0 Td[(w)Log(Mi)+wLog(Xi+1)jj2P(Mi)P(Xi+1)[dMi][dXi+1]=(1)]TJ /F16 10.909 Tf 10.9 0 Td[(w)2Var(Mi)+w2Var(Xi+1)(1)]TJ /F16 10.909 Tf 10.9 0 Td[(w)21 iu2+w2u2(3)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(w)21 iu2+w2u2=1 i+1u2whenw=1 i+1. Fromthetheoremsabove,wecanndthattherecursiveKarcherexpectationestimatorisanunbiasedestimatorfortheKarcherexpectationwhenthesamplesaredrawnfromasymmetricdistributiononPn.AnditconvergesintheL2sensetotheexpectation.Ofcourse,thisrecursiveKarcherexpectationestimatorcanbeviewedasanapproximationoftheKarchersamplemean.However,inourexperiments,wendthatitactuallyhassimilaraccuracyastheKarchersamplemean.Also,becauseitisarecursiveestimator,itwouldbefarmorecomputationallyefcienttouseourestimatorthanthestandardnon-recursiveKarchermeanalgorithmwhenthediffusiontensorsareinputsequentiallytotheestimationalgorithmasinalltheaforementionedapplications. 3.2.2RecursiveformofthesymmetrizedKL-divergencemeanWenowpresentarecursiveformulationforcomputingthesymmetrizedKL-divergencebasedmean.Let'srecall,thesymmetrizedKLdivergencealsocalledtheJ-divergence,isdenedbyJ(p,q)=1 2(KL(pjjq)+KL(qjjp))UsingthesquarerootofJ,onecandeneadivergencebetweentwogivenpositivedenitetensors.ThesymmetrizedKL(KLs)divergencebasedmeanofasetofSPDtensorsistheminimizerofthesumofsquaredKLdivergences.ThisminimizationproblemhasaclosedformsolutionasshowninWangetal.[ 78 ]andrepeatedhereforconvenience, MKL=p B)]TJ /F5 7.97 Tf 6.58 0 Td[(1[q p BAp B]p B)]TJ /F5 7.97 Tf 6.58 0 Td[(1(3) 26

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whereA=1 NPiTiisthearithmeticmean,B=1 NPiT)]TJ /F5 7.97 Tf 6.58 0 Td[(1iistheharmonicmean,T=fTigisthegiventensoreldandNisthetotalnumberoftensors.TheclosedformEquation 3 canbecomputedinanrecursivemannerasfollows.Letthearithmeticandharmonicmeansatiterationnbedenotedby(An)and(Bn),respectively.Whenanew(n+1)sttensor,Tn+1augmentsthedataset,thequantitiesAnandBnarerecursivelyupdatedviathefollowingsimpleequations, An+1=n n+1An+1 n+1Tn+1and (3) Bn+1=n n+1Bn+1 n+1T)]TJ /F5 7.97 Tf 6.59 0 Td[(1n+1. (3) Usingtheaboverecursiveformofthearithmeticandharmonicmeansofasetoftensorsandtheclosedformexpression(Equation 3 ),wecanrecursivelycomputetheKLsmeanofasetofSPDtensors. 3.2.3RecursivemeanfortheLog-EuclideanMetricWenowformulatetherecursiveformoftheLog-Euclidean(LE)basedmean.ItiswellknownthatPncanbediffeomorphicallymappedtotheEuclideanspaceusingthematrixLogfunction,whichmakesitpossibletodirectlyinducetheEuclideanmetriconPncalledtheLog-Euclideanmetric[ 4 ].Then,theLog-Euclideandistancecanbedenedas, DLE(T1,T2)=jjLog(T1))]TJ /F3 11.955 Tf 11.95 0 Td[(Log(T2)jj(3)wherejj.jjistheEuclideannorm.TheLE-meanonasetofSPDmatrices,isobtainedbyminimizingthesumofthesquaredLEdistanceswhichleadstoaclosedformsolution MLE=Exp(nXi=1Log(Ti))(3)Thisclosedformexpressioncanberewritteninarecursiveformformoreefcientcomputation.LetMnbetheLog-Euclideanmeaninthenthiteration.Whenthe(n+1)sttensor,sayTn+1isadded,thecurrentmeancanberecursivelyupdatedusingthe 27

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followingequation, Mn+1=Exp(n n+1Log(Mn)+1 n+1Log(Tn+1)).(3) 3.3Experiments 3.3.1PerformanceoftheRecursiveEstimatorsTojustifytheperformanceoftherecursiveestimators,werstgeneratei.i.d.samplesfromtheLog-normaldistribution[ 64 ]onP3withtheexpectationattheidentitymatrix.Then,weinputthe100randomsamplessequentiallytoallestimatorsincludingtherecursiveKarcherexpectationestimator(RKEE),Karchermean(KM),recursiveKLsmean(RKLS),non-recursiveKLs(KLS)mean,recursiveLog-Euclideanmean(RLEM)andthenon-recursiveLog-Eucildeanmean(LEM)respectively.TocomparetheaccuracyofRKEEandKM,weevaluatetheerroroftheestimatorusingthesquareddistanceinEquation 2 betweenthegroundtruth(theidentitymatrix)andthecomputedestimate.TheaccuracytestoftheremainingalgorithmsisnotincludedbecauseforKLsandLog-Euclideanmetrics,therecursiveandnon-recursivealgo-rithmwillgeneratetheexactsameresults.Also,thecomputationtimeforeachstep(eachsample)isrecorded.Forcomparison,wehavethesamesettingsforallthemeancomputationalgorithms.Weruntheexperiment20timesandplottheaverageerrorandtheaveragecomputationtimeateachstepinFigure 4-1 .InFigure 4-1 (a),weseethattheaccuracyofcomputedmeanisnearlythesameforboththenon-recursiveKarchermeanandtherecursiveKarcherexpectationestimatorsaftertheyaregiven10samples.Thecomputationtime(inCPUsecondsonanI-7,2.8GHZprocessor)fortheKarchermeanhoweverincreaseslinearlywiththenumberofsteps,whilethatfortherecursiveKarcherexpectationestimatorisnearlyaconstantandfarlessthanthenon-recursivecase.ThismeansthattherecursiveKarcherexpectationestimatoriscomputationallyfarsuperiorespeciallyforlargesizeproblemswheredataisinputincrementally,forexampleinalgorithmsforsegmentation,clustering,classicationandatlasconstruction.Similar 28

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Figure3-1. Accuracyandspeedcomparisonsoftherecursiveversusnon-recursivemeancomputationalgorithmsfordataonP3.Figure(a)isthemeanerroroftheKarchermean(reddashedline)andtherecursiveKarcherexpectationestimator(bluesolidline)foreachstep.Figure(b)(c)(d)arethecomparisonsofcomputationtime(inseconds)betweentherecursive(reddashedline)andnon-recursive(bluesolidline)meancomputationalgorithmsfordifferentmetrics.ResultfortheRiemannianmetric(Karchermean)isinFigure(b),KLsinFigure(c),Log-EuclideaninFigure(d). conclusionscanalsobedrawninFigure 4-1 (c)and(d),whereforsequentiallyinputdatatherecursivemeanalgorithmforKLsdivergenceandLog-Euclideanmeanaremuchmoreefcientthantheirownbatchversions. 3.3.2ApplicationtoDTISegmentationInthissection,wepresentresultsofapplyingourrecursivealgorithmstotheDTIsegmentationproblem.In[ 78 ],theclassicallevelsetbasedsegmentationalgorithm[ 17 ]wasextendedtotheeldofdiffusiontensors.Inthisalgorithm,basedonapiecewiseconstantmodel,thesegmentationprocedurebecameaEMlikealgorithm,whereateachiteration,themeantensoriscomputedovereachregionandtheregionboundaryisthenevolvedbasedonthemeantensor.Inthissection,weusethisalgorithmtosegmentDTIs,andplugindifferenttensoreldmeancomputationtechniquesforcomparison.Firstly,experimentsonDTIsegmentationofsyntheticdatasetsispresentedhere.Wemanuallygeneratedanimageregionofsize(6464)whichcontainstwodifferentkindoftensors(differinginorientation,oneverticalandanotherhorizontal).ThenDWMRsignalisgeneratedbasedon[ 66 ]with5differentlevelofRicciannoisedaddedto 29

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theDWMRsignal=0.1,0.15,0.2,0.25,0.3,wheresigma2isvarianceoftheGaussiannoiseaddedtotherealandimagepartoftheDWMRsignal.DTIsareconstructedbyusingthetechniquein[ 9 ].Exactsamedatasetandsamesettingareusedforallsixmethods.TheinitializationcurveoverlayedonanoisydatasetisdepictedinFigure 3-2 (a).Toevaluatethesegmentationresult,thedicecoefcientbetweenthegroundtruthsegmentationandtheestimatedsegmentationarecomputed.TheseresultsareshowninFigure 3-2 withgure(b)depictingthedicecoefcientsandgure(c)showingthecomparisonoftherunningtimes.Fromthegure(b)wecanseethatthesegmentationaccuraciesareverysimilarfortherecursiveandnon-recursivemethodsforthesamedistancemetric.Fordifferentdistancemetrics,resultoftheRiemannian(GLinvariant)metricisthemostaccurate,sincetheGLinvariantmetricisthenaturalmetriconPn.InFigure 3-2 ,wecanndthatsegmentationusingtheKMtakessignicantlylongertimethanothermethods,thisisbecausethereisnoclosedformcomputationformulaforKarchermeanonPn,andhencetheKarchermeancomputationisverytimeconsumingwhichcanalsobeseeninTable 3-1 .TherecursiveKarcherexpectationestimatorisabout2timesfasterandhasthesimilaraccuracy.ForKLsandLog-Euclideanmetrics,thetimesavedbytherecursivemethodisnotsosignicantasfortheGL-invariantmetric.Thisisbecause,althoughthemeancomputationtimefortherecursivemethodisatleastonetenthofthenon-recursivemethod(0.01versus0.1inTable 3-1 ),thetimeusedforcurveevolutionisabout14secondswhichmakesthesavingsintotalsegmentationtimenotsignicant.FromtheseresultswecanndthattherecursiveKarcherexpectationestimatoristhemostattractivefromanaccuracyandefciencyviewpoint.Fortherealdataexperiment,theDTIareestimated[ 78 ]fromaDW-MRscanofaratspinalcord.TheDWMRdatawereacquiredusingaPGSEwithTR=1.5s,TE=28.3ms,bandwidth=35Khz,21diffusionweightedimageswithab-valueof1250s=mm2werecollected.Theimagesizeis12812810.Weusedthesameinitializationforeachsegmentation.Weappliedallof 30

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Figure3-2. ResultsfortheDTIsegmentationexperimentsonthesyntheticdataset.Figure(a)istheinitializationcurveoverlayedonthesyntheticdatasetatoneofthenoiselevelsusedintheexperiments.Figure(b)isthesegmentationaccuracyevaluatedbythedicecoefcientofthesegmentationsfromallmethodsatallnoiselevels.Figure(c)isthetotalsegmentationtime(inseconds)forallsegmentationmethodsatallnoiselevels. NoiseLevelRKEEKMRKLSKLSRLEMLEM 0.10.010.750.010.120.010.130.150.0071.020.0060.230.010.150.20.021.550.010.420.0050.220.250.012.220.010.630.0060.280.30.014.520.0080.550.010.45 Table3-1. Time(inseconds)formeancomputationintheDTIsegmentationonsyntheticdataset thesixmethods(recursiveandnon-recursiveforeachofthethreedistancemeasures)toperformthisexperiment.Inordertocomparethetimeefciency,wereportthewholesegmentationrunningtime,includingthetotaltimerequiredtocomputethemeans.Table 3-2 showstheresultofthiscomparison,fromwhichwecanndthatitismuchmoreefcienttousetherecursivemeanestimatorinthesegmentationthanusingthebatchmeanestimator.Especially,inthecaseoftheKarchermean,whichhasnoclosedformformulaandtakesnearlyhalfofthetotalreportedsegmentationtime,whereas,usingtherecursiveKarcherexpectationestimatormakesthecomputationmuchfaster,andalsosignicantlyreducesthetotalsegmentationtime.ThesegmentationresultsaredepictedinFigure 3-3 foreachmethod.Each(3,3)diffusiontensorintheDTIdataareillustratedasanellipsoidwhoseaxisdirectionsandlengthscorrespondto 31

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SegmentationMethodRKEEKMRKLSKLSRLEMLEM Meancomputationtime0.023.560.010.560.010.4Totalsegmentationtime5.098.133.414.415.455.82 Table3-2. Timinginsecondsforsegmentationofgreymatterinaratspinalcord Figure3-3. Segmentationresultsofgreymatterinaratspinalcordfor6differentmethods.Figure(a)isRKEEbasedsegmentation.Figure(b)issegmentationusingtheKarchermeanKM.Figure(c)and(d)areresultsfortherecursiveandnon-recursiveKLsmeanestimatorsrespectively.Figure(e)and(f)areresultsfortherecursiveandnon-recursiveLog-Euclideanmeanrespectively. theeigen-vectorsandeigen-valuesrespectively.Fromthegurewecanseethatthesegmentationresultsarevisuallysimilartoeachother,whileourrecursiveKarcherexpectationbasedmethodtakesmuchlesstimewhichwouldbeveryusefulinpractice. 32

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Figure3-4. Segmentationresultsofthemolecularlayerinarathippocampusfor3differentmethods.Figure(a)RKEEbasedsegmentation.(b)RecursiveKLsbasedsegmentation(c)RecursiveLog-Euclideanbasedsegmentation. Asecondrealdatasetfromanisolatedrat-hippocampuswasusedtotestthesegmentationalgorithms.Figure 3-4 depictsthesegmentationofthemolecularlayerintherathippocampus.Forthesakeofspace,wepresentonlythesegmentationresultsfromtherecursivealgorithmspresentedandnotthenon-recursivecounterpartsastheirresultsarevisuallysimilarandthekeydifferenceisinthetimesavings. 33

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CHAPTER4INTRINSICRECURSIVEFILTERONPN 4.1BackgroundandPreviousWorkSincePnisaRiemannianmanifold,butnotavectorspace,manyoperationsandalgorithmsinEuclideanspacecannotbeapplieddirectlytoPnandthishasleadtoaurryofresearchactivityintherecentpast.SeveraloperationsinEuclideanspacehavebeenextendedtoRiemannianmanifold.Forexample,theextensionofarithmeticmeantoRiemannianmanifoldisKarcherMean[ 38 ];TheextensionofPrincipalComponentAnalysis(PCA)isthePrincipleGeodesicAnalysis[ 28 29 ];Meanshift[ 23 ]hasalsobeenextendedtoRiemannianmanifolds[ 69 ].However,forlteringoperationsindynamicscenessuchasthepopularKalmanlter[ 67 ],anintrinsicextensiondoesnotexistinliteraturetodate.Recursivelteringisatechniquetoreducethenoiseinthemeasurementsbyusingtheoryofrecursionappliedtoltering.Itisoftenusedintimesequencedataanalysisespeciallyinthetrackingproblemwherethemodelofthetargetneedstobeupdatedbasedonthemeasurementandprevioustrackingresults.ManyrecursivelteringtechniqueshavebeendevelopedintheEuclideanspace,suchasKalmanlter,ExtendedKalmanlteretc,wheretheinputsandoutputsofthelterareallvectors[ 37 67 ].However,severaltrackingproblemsarenaturallysetinPn,aRiemanniansymmetricspace[ 34 ].Recentworkreportedin[ 61 ]oncovariancetrackingusesacovariancematrix(constructedfrompixel-wisefeaturesinsidetheobjectregion)thatbelongstoPninordertodescribetheappearanceofthetargetbeingtracked.Thiscovariancedescriptorhasprovedtoberobustinbothvideodetection[ 71 74 ]andtracking[ 21 39 43 44 60 61 75 83 ].Thecovariancedescriptorisacompactfeaturerepresentationoftheobjectwithrelativelylowdimensioncomparedtootherappearancemodelssuchassthehistogrammodelin[ 24 ].In[ 73 ]anefcientalgorithmforgeneratingcovariancedescriptorsfromfeaturevectorsisreportedbasedonthe 34

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integralimagetechnique,whichmakesitpossibletousecovariancedescriptorsinrealtimevideotrackingandsurveillance.Onemajorchallengeincovariancetrackingishowtorecursivelyestimatethecovariancetemplate(acovariancedescriptorthatservesasthetargetappearancetemplate)basedontheinputvideoframes.In[ 61 ]andalsoin[ 44 60 61 ]theKarchermeanofsamplecovariancedescriptorsfromaxednumberofvideoframesisusedasthecovariancetemplate.ThismethodisbasedonthenaturalRiemanniandistancetheGL-invariantdistanceinPn.Currently,thisKarchermeancannotbecomputedinclosedform,andthecomputationisachievedusingagradientbasedoptimizationtechniquewhichisinefcientespeciallywhentheinputcontainsalargenumberofsamples.Tosolvethisproblem,aLog-Euclideanmetricisusedin[ 39 43 ],anarithmeticmeanlikemethodisusedin[ 83 ],andanextensionoftheoptimalltertoPnwasdevelopedin[ 75 ].However,noneoftheseareintrinsicduetotheuseofapproachesthatareextrinsictoPn.Recently,somemethodswerereportedaddressingtherecursivelteringproblemonRiemannianmanifoldsotherthanPn.Forexample,thegeometricparticlelterinhandling2Dafnemotions(2-by-2non-singularmatrix)wasreportedin[ 39 42 ],andanextensiontotheRiemannianmanifoldwasdevelopedin[ 65 ].However,sincethecovariancedescriptorisusuallyahighdimensionaldescriptor,e.g.thedegreesoffreedomofa55covariancematrixare15,thenumberofsamplesrequiredfortheparticlelterwouldbequitelargeinthiscase.Additionally,computingtheintrinsic(Karcher)meanonPniscomputationallyexpensiveforlargesamplesizes.Thus,usinganintrinsicparticleltertoupdatecovariancedescriptorwouldbecomputationallyexpensiveforthetrackingproblem.TherearealsoexistingtrackingmethodsonGrassmannmanifolds[ 22 68 ]however,itisnon-trivialtoextendthesetoPn,sincetheGrassmannmanifoldsandPnhaveverydifferentgeometricproperties,e.g.Grassmannmanifoldsarecompactandhaveanon-negativesectionalcurvature 35

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whenusinganinvariantRiemannianmetric[ 81 ],whilePnisnon-compactandhasnon-positivesectionalcurvaturewhenusinganinvariant(tothegenerallineargroup(GL))Riemannianmetric[ 34 ].Inthisdissertation,wefocusontheproblemofdevelopinganintrinsicrecursivelterabbreviatedIRFfortherestofthischapteronPn.AnovelprobabilisticdynamicmodelonPnbasedonRiemanniangeometryandprobabilitytheoryispresented.Here,thenoisystateandobservationsaredescribedbymatrix-variaterandomvariableswhosedistributionisageneralizednormaldistributiontoPnbasedontheGL-invariantmeasure.In[ 41 58 ]authorsprovidealinearapproximationofthisdistributionforcaseswhenthevarianceofthedistributionisverysmall.Incontrast,inthisdissertation,weexploredseveralpropertiesofthisdistributionforthearbitraryvariancecase.WethendeveloptheIRFbasedonthisdynamicmodelandtheBayesianframeworkdescribedin[ 22 ].Byapplyingthisrecursiveltertoachievecovariancetrackinginconjunctionwithaparticlepositiontracker[ 5 ],weobtainanewefcientrealtimevideotrackingalgorithmdescribedinSection 4.3.2 .Wepresentexperimentswithcomparisonstoexistingstate-of-the-artmethodsandquantitativeanalysisthatsupporttheeffectivenessandefciencyoftheproposedalgorithm.Theremainderofthischapterisorganizedasfollows:InSection 4.2 weintroducetheprobabilisticdynamicmodelonPn.ThentheIRFandthetrackingalgorithmsarepresentedinSection 4.3 ,followedbytheexperimentsinSection 4.4 4.2IRF:ANewDynamicTrackingModelonPn 4.2.1GeneralizationoftheNormalDistributiontoPnTodeneaprobabilitydistributiononamanifold,rstweneedtodeneameasureonthemanifold.HereweusetheGL-invariantmeasure[dX]onPn.GLinvariancehereimplies8g2GL(n)and8X2Pn,[dgXgt]=[dX].From[ 70 ],weknow[dX]=jXj)]TJ /F5 7.97 Tf 6.59 0 Td[((n+1)=2Q1ijndxij,wherexijistheelementinthei-throwandj-thcolumnofthe 36

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SPDmatrixX.Also,thismeasureisconsistentwiththeGLinvariantmetricdenedonPndenedearlierandalsopresentedin[ 58 ].SimilartotheKarchermean,theKarcherExpectationfortherandomvariableXinanyRiemannianManifoldMcanbedenedastheresultofthefollowingminimizationproblem, E(X)=argminY2MZMdist2(X,Y)d(X)(4)where(X)istheprobabilitymeasuredenedinM.Similarly,thevariancecanbedenedbasedonthisexpectationby, Var(X)=ZMdist2(X,E(X))d(X)(4)Notethat,inEuclideanspaceRmwhichisalsoaRiemannianmanifold,theKarcherExpectationisequivalenttothetraditionaldenitionofexpectation,andthevarianceinEquation 4 isthetraceofthecovariancematrix.InPn,bytakinggradientoftheenergyfunctioninEquation 4 andsettingittozero,wendthattheexpectationoftherandomvariablewillsatisfythefollowingequation. ZPnlog(E(X))]TJ /F5 7.97 Tf 6.58 0 Td[(1=2XE(X))]TJ /F5 7.97 Tf 6.58 0 Td[(1=2)p(X)[dX]=0(4)ThegeneralizationofthenormaldistributiontoPnusedherecanbedenedasfollows: dP(X;M,!2)=p(X;M,!2)[dX]=1 Zexp()]TJ /F3 11.955 Tf 10.49 8.08 Td[(dist(X,M)2 2!2)[dX](4)whereP()andp()aretheprobabilitydistributionanddensityrespectivelyoftherandomvariableX2Pn,withtwoparametersM2Pnand!22R+,andZisthescalarnormalizationfactor.dist()isdenedinEquation 2 .Asshownin[ 58 ],thisdistributionhasminimuminformationgiventheKarchermeanandvariance.Thatis,intheabsenceofanyinformationthisdistributionwouldbethebestpossibleassumptionfromaninformationtheoreticviewpoint.Also,thisdistributionisdifferentfromtheLog-normaldistributionwhichwasusedin[ 64 75 ].Actually,thetwodistributionshave 37

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verysimilardensities,butthedensityusedhereisbasedonGLinvariantmeasurewhileLog-normaldensityisbasedontheLebesguemeasureintheEuclideanspace.Averyimportantpropertyoftheabovegeneralizednormaldistributionissummarizedinthefollowingtheorem. Theorem4.1. ThenormalizationfactorZinEquation 4 isaniteconstantwithrespecttoparameterM2PnTheconsequenceofTheorem 4.1 isthat,ifthepriorandthelikelihoodarebothbasedonthegeneralizednormaldistributiondenedusingtheGL-invariantmeasure,computingthemodeoftheposteriordensitycanbeachievedbyminimizingthesumofsquaredGL-invariantdistancesfromtheunknownexpectationtothegivensamples.ToprovetheTheorem 4.1 ,weneedtorstprovethefollowinglemma: Lemma1. W=ZPnexp()]TJ /F3 11.955 Tf 10.49 8.09 Td[(Tr(logXlogXt) 2!2)[dX]<1 Proof. ThislemmaindicatesthatthenormalizationfactorZisconstant,andhencep(X;M,!2)isaprobabilitydensityfunctiononPn.Toprovethislemma,werstrepresentXinpolarcoordinatesfig,Rbasedontheeigendecomposition,X=RRt,where=diag(1,...n),RRt=Inn.From[ 70 ]weknowthat [dX]=cnnYj=1)]TJ /F5 7.97 Tf 6.59 0 Td[((n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=2jY1i
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changeofvariables,yi=log(i),weget W=cnZRnexp()]TJ /F4 7.97 Tf 17.65 14.94 Td[(nXi=1(1 2!2y2i+n)]TJ /F8 11.955 Tf 11.96 0 Td[(1 2yi))Y1i
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Proof. )]TJ /F3 11.955 Tf 9.3 0 Td[(log(p(X1,X2,....Xm;M,!2))=)]TJ /F10 11.955 Tf 11.29 11.36 Td[(Xilog(p(Xi;M,!2))=nlogZ+Pidist2(Xi,M) 2!2SinceZisconstantwithrespecttoMasprovedtheTheorem 4.1 ,wehaveargmaxMp(X1,X2,....Xm;M,!2)=argminMXidist2(Xi,M)Thus,MLEoftheparameterMofthedistributiondP(X;M)equalstotheKarchermeanofsamples. FromTheorem 4.1 weknowthatthenormalizationfactorZinEquation 4 isafunctionof!.TheintegralinEquation 4 isnon-trivial,andcurrentlynoexactsolutionisavailableforarbitraryn.Forn=2wecanhave, Z2(!)=2c2Z1Zy1exp()]TJ /F5 7.97 Tf 17.7 14.95 Td[(2Xi=1(1 2!2y2i+1 2yi))(exp(y1))]TJ /F3 11.955 Tf 11.96 0 Td[(exp(y2))dy2dy1=p 2c2!exp(1 4!2)Z1(exp()]TJ /F8 11.955 Tf 10.5 8.09 Td[((y1)]TJ /F8 11.955 Tf 11.95 0 Td[(0.5!2)2 2!2)(1+erf(y1+0.5!2 p 2!2)))]TJ /F3 11.955 Tf 9.3 0 Td[(exp()]TJ /F8 11.955 Tf 10.5 8.09 Td[((y1+0.5!2)2 2!2)(1+erf(y1)]TJ /F8 11.955 Tf 11.96 0 Td[(0.5!2 p 2!2))))dy1=4c2!2exp(1 4!2)erf(! 2)(4)whereerf(x)=2 p Rx0exp()]TJ /F3 11.955 Tf 9.3 0 Td[(t2)dtistheerrorfunction. 4.2.1.1ThemeanandthevarianceofthegeneralizednormaldistributionSimilartothenormaldistributioninEuclideanspace,themeanandthevarianceofthegeneralizednormaldistributiononPninEquation 4 arecontrolledbytheparametersMand!2respectively.TherelationbetweenManddP(X;M,!2)isgivenbythefollowingtheorem. Theorem4.2. MistheKarcherExpectationofthegeneralizednormaldistributiondP(X;M,!2). 40

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Proof. Toprovethis,weneedtoshowthatdP(X;M,!2)satisesEquation 4 .Let=ZPnlog(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2XM)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2)dP(X;M,!2)thenintheintegral,usingachangeofvariable,XtoY=MX)]TJ /F5 7.97 Tf 6.59 0 Td[(1M(X=MY)]TJ /F5 7.97 Tf 6.59 0 Td[(1M).SincePnisasymmetricspaceandthemetric/measureisGL-invariantweknowthat[dX]=[dY],anddist(X,M)=dist(Y,M).Thuswehave,=ZPnlog(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2XM)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2)1 Zexp()]TJ /F3 11.955 Tf 10.5 8.09 Td[(dist2(X,M) 2!2)[dX]=ZPnlog(M1=2Y)]TJ /F5 7.97 Tf 6.59 0 Td[(1M1=2)1 Zexp()]TJ /F3 11.955 Tf 10.49 8.09 Td[(dist2(Y,M) 2!2)[dY]=)]TJ /F8 11.955 Tf 9.3 0 Td[(=0.SincePnhasnon-positivecurvature,thesolutionofEquation 4 isunique[ 38 ].ThusMistheKarcherExpectationofdP(X;M,!2). ThevarianceofdP(X;M,!2)iscontrolledbytheparameter!2.Unlikethemulti-variatenormaldistributionintheEuclideanspace,wheretheKarchervariance(Equation 4 )isequalton!2,therelationbetweenthevarianceand!2ofthegeneralizednormaldistributionismuchmorecomplex.WithoutlossofgeneralityweassumeX2Pnisamatrix-valuedrandomvariablefromdP(X;I,!2).ThevarianceVar(X)=1 ZRPnjjlog(X)jj2exp()]TJ /F14 7.97 Tf 10.49 5.7 Td[(jjlog(X)jj2 2!2)[dX].AsinEquation 4 ,byusingthePolarcoordinatesandtakinglogoftheeigenvalues,wecanget Var(X)=!2Varq(y)(4)whereyisarandomvariableinRnhavingthedistributionwithdensityfunction, q(y)=1 z(!)exp()]TJ /F8 11.955 Tf 10.49 8.08 Td[(1 2Xiy2i)Y1i
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asinEquation 4 Varq(y)=! p exp(1 4!2)erf(! 2)+2(1+!2 4)(4)FromEquation 4 wecanndthat,inP2when!isclosedtozero,Var(X)3!2,andwhen!islargeVar(X)!4 2.ThisisbecausePncanbelocallyapproximatedbyanEuclideanspace.When!isclosedtozero,themajorportionofthedistributionwouldbeinasmallregioninPn,wheretheEuclideanapproximationisrelativelyaccurate.Hence,Var(X)isproportionalto!2whichissimilartothenormaldistributionintheEuclideanspace.When!2isnotclosedtozero,thetheEuclideanapproximationisnolongeraccurate,andtheVar(X)becomesacomplicatedfunctionof!2.Thispropertyhasbeenusedtogettheapproximationofthegeneralizednormaldistributionwithsmall!2in[ 41 58 ].Thefollowingtwotheoremsshowthattheabovestatedapproximationswillstillbesatisedforn>2. Theorem4.3. lim!!0Var(X) !2=n(n+1) 2(4) Proof. Let v(y,!)=X2Snsgn()exp()]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2nXi=1(y2i+!(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(2(i))yi))(4)where,Snandsgn()arerelatedtothepermutationof0,1,...,n)]TJ /F8 11.955 Tf 12.33 0 Td[(1whichisdenedtobethesameasinEquation 4 .Alsowecanndthatq(y)=jv(y,!)j z(!)andz(!)=RRnjv(y,!)jdy.TheTaylorexpansionofv(y,!)upton(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1) 2-thorderwithrespectto!aroundzerois v(y,!)=X2Snsgn()n(n)]TJ /F15 5.978 Tf 5.76 0 Td[(1) 2Xk=0()]TJ /F22 10.909 Tf 8.48 0 Td[(!)k k!exp()]TJ /F20 10.909 Tf 9.68 15.76 Td[(Pni=1y2i 2)(nXi=1(n)]TJ /F17 10.909 Tf 10.91 0 Td[(1 2)]TJ /F22 10.909 Tf 10.91 0 Td[((i))yi)k+O(!n2)]TJ /F13 5.978 Tf 5.76 0 Td[(n+2 2)=C()]TJ /F22 10.909 Tf 8.48 0 Td[(!)n(n)]TJ /F15 5.978 Tf 5.76 0 Td[(1) 2exp()]TJ /F20 10.909 Tf 9.68 15.76 Td[(Pni=1y2i 2)Y1i
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whereCisaconstant.TheEquation 4 aboveusedthefactthatgivennnon-negativeintegersi,andPni=1in(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1) 2, Xsgn()nYi=1(i)i=0iffig=2Sn(4)So,intheTaylorexpansionallthetermswithdegreelessthann(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1) 2arezeros.Inthen(n)]TJ /F5 7.97 Tf 6.58 0 Td[(1) 2-thorderterms,onlytermswithpowersinSnwillbenon-zero.Letthedensity^q(y)=1 ^zexp()]TJ /F24 7.97 Tf 10.49 12.46 Td[(Pni=1y2i 2)Q1i
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isthenormalizedlogoftheeigenvaluesofX,wecanseethatwhen!isclosetozerothegeneralizednormaldistributioncanbeapproximatedbyaLog-Normaldistribution. Theorem4.4. lim!!1Var(x) !4=(n3)]TJ /F3 11.955 Tf 11.95 0 Td[(n) 12(4) Proof. Werstdenetheupperboundandlowerboundonq(y). qu(y)=1 zu(!)Xexp()]TJ /F8 11.955 Tf 10.49 8.09 Td[(1 2nXi=1(y2i+!(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(2(i))yi))q(y)=1 z(!)exp()]TJ /F8 11.955 Tf 10.49 8.08 Td[(1 2Xiy2i)Y1i
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Fromthedenitionwecancomputethenormalizationconstantsandthevariancesofqandquinaclosedform. zu=(2)n=2Xexp(!2Pni=1(i)2)]TJ /F7 11.955 Tf 11.96 0 Td[(!2n(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1)2=4 2)=(2)n=2n!exp(!2n(n2)]TJ /F8 11.955 Tf 11.95 0 Td[(1) 24)z=(2)n=2X2Bnexp(!2Pni=1(i)2)]TJ /F7 11.955 Tf 11.96 0 Td[(!2n(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)2=4 2)=zu+(2)n=2X2BnnSnexp(!2Pni=1(i)2)]TJ /F7 11.955 Tf 11.95 0 Td[(!2n(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)2=4 2)Varqu(y)=n+n3)]TJ /F3 11.955 Tf 11.95 0 Td[(n 12!2Varq(y)=n)]TJ /F3 11.955 Tf 13.15 8.08 Td[(n(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)2 4!2+(2)n=2 z!2X2Bnexp(!2Pni=1(i)2)]TJ /F7 11.955 Tf 11.95 0 Td[(!2n(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)2=4 2)(nXi=1(i)2)(4)Since0cosh(x))]TJ /F8 11.955 Tf 13 0 Td[(1jsinh(x)jandjPixijPijxij,wehave8y2Rn,zq(y)zq(y)
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4.2.2TheProbabilisticDynamicModelonPnToperformtrackingonPn,obviouslytheobservationYkandthestateXkattimek2Pnrespectively.Thestatetransitionmodelandtheobservationmodelcanthenbedenedasp(XkjXk)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1 Zsexp()]TJ /F3 11.955 Tf 10.5 8.09 Td[(dist2(Xk,gXk)]TJ /F5 7.97 Tf 6.58 0 Td[(1gt) 2!2)p(YkjXk)=1 Zoexp()]TJ /F3 11.955 Tf 10.5 8.09 Td[(dist2(Yk,hXkht) 22)whereg,h2GL(n).!2,2>0aretheparametersthatcontrolthevarianceofthestatetransitionandtheobservationnoise.TheabovetwodensitiesarebothbasedontheGLinvariantmeasureonPn,unlikein[ 64 75 ]wheretheyarebasedontheLebesguemeasure.Whatdoesthisimply?ThekeyimplicationofthisisthatthenormalizationfactorinthedensitiesisaconstantfortheGLinvariantmeasureandnotsofortheLebesguemeasurecase.Ifthenormalizationfactorwasnotaconstant,onedoesnothaveavaliddensity. 4.3IRF-basedTrackingAlgorithmonPn 4.3.1TheBayesianTrackingFrameworkForsimplicityweusetheBayesiantrackingframeworkdescribedin[ 22 ].Thetrackingproblemcanbeviewedas,givenatimesequenceofobservationsY s=fY1,Y2,......,Ysgfromtime1totimes,howcanonecomputethestateXsattimes?Tosolvethisproblem,rstwemaketwoassumptions:(1)ThestatetransitionisMarkovian,i.e.,thestateXsdependsonlyonXs)]TJ /F5 7.97 Tf 6.58 0 Td[(1,orsay p(XsjX s)]TJ /F5 7.97 Tf 6.58 0 Td[(1,Y s)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=p(XsjXs)]TJ /F5 7.97 Tf 6.58 0 Td[(1)(4)(2)TheobservationYsisdependentonlyonthestateXsatthecurrenttimepoints,inotherwords, p(YsjX s,Y s)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=p(YsjXs)(4) 46

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Andhence,p(XsjXs)]TJ /F5 7.97 Tf 6.58 0 Td[(1)iscalledthestatetransitionmodelandp(YsjXs)iscalledtheobservationmodel.Thegoaloftrackingcanthusbeviewedascomputingtheposteriorp(X sjY s).Firstwehave p(X sjY s)=p(X s,Y s)=p(Y s)/p(X s,Y s)(4)Andalso,p(X s,Y s)=p(YsjXs)p(XsjXs)]TJ /F5 7.97 Tf 6.59 0 Td[(1)p(X s)]TJ /F5 7.97 Tf 6.59 0 Td[(1,Y s)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=sYk=1p(YkjXk)p(XkjXk)]TJ /F5 7.97 Tf 6.58 0 Td[(1).So,ifXk)]TJ /F5 7.97 Tf 6.58 0 Td[(2,Xk)]TJ /F5 7.97 Tf 6.58 0 Td[(3,...,X0havealreadybeencomputed,wecanthencompute^Xk,^Xk)]TJ /F5 7.97 Tf 6.59 0 Td[(1bysolvingthefollowingoptimizationproblem:^Xk,^Xk)]TJ /F5 7.97 Tf 6.58 0 Td[(1=argmaxXk,Xk)]TJ /F15 5.978 Tf 5.75 0 Td[(1kYj=k)]TJ /F5 7.97 Tf 6.58 0 Td[(1p(YjjXj)p(XjjXj)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=argminXk,Xk)]TJ /F15 5.978 Tf 5.76 0 Td[(1Ek(Xk,Xk)]TJ /F5 7.97 Tf 6.58 0 Td[(1)whereEk(Xk,Xk)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=)]TJ /F5 7.97 Tf 6.58 0 Td[(2dist2(h)]TJ /F5 7.97 Tf 6.59 0 Td[(1Ykh)]TJ /F5 7.97 Tf 6.58 0 Td[(t,Xk)+!)]TJ /F5 7.97 Tf 6.58 0 Td[(2dist2(gXk)]TJ /F5 7.97 Tf 6.59 0 Td[(1gt,Xk)+)]TJ /F5 7.97 Tf 6.58 0 Td[(2dist2(h)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yk)]TJ /F5 7.97 Tf 6.59 0 Td[(1h)]TJ /F4 7.97 Tf 6.58 0 Td[(t,Xk)]TJ /F5 7.97 Tf 6.59 0 Td[(1)+!)]TJ /F5 7.97 Tf 6.59 0 Td[(2dist2(Xk)]TJ /F5 7.97 Tf 6.59 0 Td[(1,gXk)]TJ /F5 7.97 Tf 6.59 0 Td[(2gt)ThisproblemcanbesolvedbyusingthegradientdescentmethodonPn,whereateachstepwecomputethegradientwhichliesinthetangentspace,andgetthenewstatebymovingalongthegeodesicinthecorrespondingdirection.Atthei-thiterationstep,@Ek @X(i)k2TPX(i)k,and@Ek @X(i)k)]TJ /F15 5.978 Tf 5.75 0 Td[(12TPX(i)k)]TJ /F15 5.978 Tf 5.75 0 Td[(1.Therefore,X(i+1)k=ExpX(i)k(@Ek @X(i)k)and,X(i+1)k)]TJ /F5 7.97 Tf 6.59 0 Td[(1=ExpX(i)k)]TJ /F15 5.978 Tf 5.76 0 Td[(1(@Ek @X(i)k)]TJ /F15 5.978 Tf 5.76 0 Td[(1),whereisthestepsizeandExp(.)istheExponentialmapas 47

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denedinChapter 2 .Thegradientisgivenby,@Ek @X(i)k=)]TJ /F5 7.97 Tf 6.59 0 Td[(2LogXk(h)]TJ /F5 7.97 Tf 6.59 0 Td[(1Ykh)]TJ /F5 7.97 Tf 6.59 0 Td[(t)+!)]TJ /F5 7.97 Tf 6.59 0 Td[(2LogXk(gXk)]TJ /F5 7.97 Tf 6.59 0 Td[(1gt)@Ek @X(i)k)]TJ /F5 7.97 Tf 6.58 0 Td[(1=)]TJ /F5 7.97 Tf 6.58 0 Td[(2LogXk)]TJ /F15 5.978 Tf 5.76 0 Td[(1(h)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yk)]TJ /F5 7.97 Tf 6.59 0 Td[(1h)]TJ /F4 7.97 Tf 6.59 0 Td[(t)+!)]TJ /F5 7.97 Tf 6.59 0 Td[(2LogXk)]TJ /F15 5.978 Tf 5.76 0 Td[(1(g)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xkg)]TJ /F4 7.97 Tf 6.59 0 Td[(t)+!)]TJ /F5 7.97 Tf 6.59 0 Td[(2LogXk)]TJ /F15 5.978 Tf 5.75 0 Td[(1(gXk)]TJ /F5 7.97 Tf 6.59 0 Td[(2gt)Itiseasytoshowthat,thestateupdatehereisanestimationofthemodeoftheposteriorp(X sjY s),whichisdifferentfromtheusualKalmanlterandparticleltermethods,wherethestateupdateisthemeanoftheposteriorp(XsjY s).Intheproposedupdateprocess,thecovarianceoftheposteriorisnotnecessaryforupdatingthestate.Wedonotprovideanupdateofthecovariancehere,partlybecausethecovarianceupdateishardtocomputeforthisdistributiononPn.Actually,there'snoexistingclosedformsolutionforthecovariancematricesevenforthedistributionp(XkjXk)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1 Zsexp()]TJ /F4 7.97 Tf 10.49 6.37 Td[(dist2(Xk,gXk)]TJ /F15 5.978 Tf 5.76 0 Td[(1gt) 2!2). 4.3.2TheTrackingAlgorithmTherecursivelterforcovariancematrices(descriptors)onPnpresentedabovecanbeusedincombinationwithmanyexistingtrackingtechniques.Manyalgorithmsbasedoncovariancedescriptorslikethosein[ 61 75 ]canuseourIRFasthemodelupdatingmethodforcovariancedescriptors.HerewecombinetheIRFwithaparticlepositiontrackerandgetareal-timevideotrackingalgorithm.FeatureExtraction:AssumewehaveanrectangularregionRwithwidthWandheightHwhichrepresentsthetargetobjectinacertainimageIinthevideosequence.Thefeaturevectorf(x,y),where(x,y)2R,canbeextractedtoincludetheinformationofappearance,positionandetctodescribeinformationatthepoint(x,y).In[ 61 ],thefeaturevectorwaschosentobef=[x,y,I(x,y),jIx(x,y)j,jIy(x,y)j]whereIxandIyarethecomponentsofthegradientrI.Forcolorimages,I(x,y)=[R,G,B]isavector.Withthefeaturevectorsateachpointintheregionoftheobject,thecovariance 48

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matrixcanbecomputedasCR=1 WHPk2R(fk)]TJ /F7 11.955 Tf 12.36 0 Td[(R)(fk)]TJ /F7 11.955 Tf 12.36 0 Td[(R)t.ThiscovariancematrixcanbecomputedinconstanttimewithrespecttothesizeoftheregionRbyusingthetechniquecalledtheintegralimageaswasdonein[ 73 ].WecanalsoaddthemeanRintothecovariancedescriptorandstillobtainasymmetricpositivedenitematrixinthefollowingmanner, ^CR=264CR+2tt1375(4)whereisaparameterusedtobalancetheaffectofthemeanandvarianceinthedescriptor(intheexperiments=0.001).Asin[ 73 ]weuseseveralcovariancedescriptorsforeachobjectinthescene.Verybriey,eachregionenclosinganobjectisdividedinto5regionsandineachofthese,acovariancedescriptoriscomputedandtrackedindividually.Amatchingscore(likelihood)iscomputedusing4ofthemwithrelativelysmalldistancetothecorrespondingtemplateinthetemplatematchingstagedescribedbelow.Thisapproachisusedinordertoincreasetherobustnessofouralgorithm.TrackingandTemplateMatching:Weuseasamplingimportancere-sampling(SIR)particlelter[ 5 ]asapositionandvelocitytracker.Thestatevectoroftheparticlelterisnowgivenby,(x,y,vx,vy,log(s))t,wherex,y,vx.vydenotethepositionandvelocityoftheobjectinthe2Dimage,andlog(s)isthelogofthescale.ThestatetransitionmatrixisdenedbasedonNewton'srstlaw F=266666666664100.01000100.010001000001000001377777777775(4)Thevarianceofthestatetransitionisadiagonalmatrixandinourworkreportedhere,theyweresetto,(42,42,202,202,0.0152).Thesestatetransitionparametersare 49

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dependentonthevideosbeingtracked.Theycouldbelearnedfromthemanuallylabeledtrainingsets.ThelikelihoodfortheparticlelterisbasedonthegeneralizednormaldistributiongiveninEquation 4 .Atstepk,werstcomputethepredictionoftheobjectcovariancetemplateusing^Yk=hgXk)]TJ /F5 7.97 Tf 6.58 0 Td[(1gtht,andthepredictionoftheposition&scaleoftheobjectrepresentedinthesetofparticlesbasedonthestatetransitionmatrix(Equation 4 ).Covariancedescriptorsthenarecomputedforeachofthepredictedparticlestateatthecorrespondingobjectregions.Thelikelihoodforeachcovariancedescriptoriscomputedbasedonthegeneralizednormaldistributioncenteredatthepredictedcorrespondingcovariancetemplate.Andthelikelihoodforeachparticle'sstateiscomputedasmultiplicationofthelikelihoodsofcovariancedescriptorsthatareclosertotheircorrespondingtemplateasmentionedabove.Aftermultiplyingthelikelihoodwiththeweightofeachparticle,themeanofthesamplesetiscomputed.Thisisfollowedbycomputationofcovariancedescriptorsatthelocationofthemeanoftheparticleset.ThiscovariancedescriptorthenformstheinputtoourIRF.Inourexperiments,weuse300particlesfortheparticleset.Ourtrackingalgorithmrunsinaround15Hzforvideoswithaframesizeof352288,onadesktopwitha2.8GHzCPU. 4.4ExperimentsInthissection,wepresenttheresultsofapplyingourintrinsicrecursiveltertobothsyntheticandrealdatasets.Therealdatasetsweretakenfromstandardvideosequencesusedinthecomputervisioncommunityfortestingtrackingalgorithms.Firstwepresentthesyntheticdataexperimentsandthentherealdata. 4.4.1TheSyntheticDataExperimentTovalidatetheproposedlteringtechnique,werstperformedsyntheticdataexperimentsonP3,thespaceof33SPDmatrices.Atimesequenceofi.i.dsamplesofSPDmatriceswererandomlydrawnfromtheLog-normaldistribution[ 64 ]centeredattheidentitymatrix.ThiswasdonebyrstdrawingsamplesfviginR6(isomorphicto 50

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Sym(3))fromtheNormaldistributionN(0,2I6).Then,thesesamplesareprojectedtoP3(denotedbyfXig)usingtheexponentialmapatthepointI3(identitymatrix).Thus,fXigcanbeviewedasatimesequenceofrandommeasurementsoftheidentitymatrix.Ourrecursiveltercanthenbeusedasanestimatorofthisrandomprocess.TheestimationerrorattimepointkcanbecomputedastheRiemanniandistancebyEquation 2 betweentheestimate^Xkandthegroundtruth(theidentitymatrix).Wetestedourrecursivelter(IRF)andevaluatedbycomparingitsperformancewiththeoptimalrecursivelterforlinearsystemsonPn(denotedbyORF)reportedin[ 75 ].TheparametersofORFaresettobeexactlythesameasin[ 75 ]exceptfortheinitialbasepointXb,whereallthesamplesareprojectedtothetangentspaceTXbPnandthenprocessedwithORF.WesetXbtobetheobservationintherststep.Inthisproblem,settingXbtobethegroundtruthwouldgivethebestresultforORF,becauseinthiscaseORFwouldreducetotheKalmanlteronthetangentspace.Sinceinthepracticalcase,thegroundtruthisunknown,herewesetXbastheobservationattherststepwhichisthebestinformationweknowaboutthedatasequencebeforetracking.WealsotriedtorandomlysetXb,andthisdidnotleadtoanyobservablebigdifferences.Fortheproposedmethod,theGLactionsg,hwerebothsettobetheidentity,and2=!2=200.Weperformedexperimentswiththreedifferentnoiselevels,2=0.1,1and2.Ateachnoiselevelweexecutethewholeprocess20timesandcomputedmeanerrorforthecorrespondingtimestep.TheresultsaresummarizedinFigure 4-1 .Fromthegure,wecanseethattheORFperformsbetterwhen2=0.1,andourmethod(IRF)performsbetterwhenthedataismorenoisy(2=1,2).ThereasonisthatORFusesseveralLog-Euclideanoperations,whichisinfactanapproximation.Forlownoiseleveldata,thedatapointsareinarelativelysmallregionaroundthegroundtruth(Identity),inwhichcase,theLog-Euclideanapproximationisquiteaccurate.Butforhighernoiselevelsinthedata,theregionbecomeslargerandtheapproximationbecomesinaccuratewhichleadsto 51

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Figure4-1. Meanestimationerrorfrom20trialsforthesyntheticdataexperiment.Thex-axisdenotesthetimestep.They-axisdenotestheestimationerrormeasuredusingtheRiemanniandistancebetweentheestimatesandthegroundtruth.Inallthreesub-gurestheredcurvesdenotetheestimationerrorforourIRF,thebluecurvesforORFwithXbsetastheobservationintherststep. largeestimationerror.Incontrast,ourlteringmethodisfullybasedontheRiemanniangeometrywithoutanyLog-Euclideanapproximation,soitperformsconsistently,andcorrectlyconvergesforallthethreenoiselevels.Inconclusion,althoughourrecursiveltermightconvergealittlebitslowerthantheORF,itismorerobusttolargeramountsnoisewhichiscommoninrealtrackingsituations. 4.4.2TheRealDataExperimentFortherealdataexperiment,weappliedourIRFtomorethan3000framesindifferentvideosequences.Twoothercovariancedescriptorupdatingmethodswerealsoappliedtothesesequencesforcomparisonnamely,(1)theORFreportedin[ 75 ];(2)TheupdatingmethodusingKarchermeanoftrackingresultsinpreviousframes(KM)reportedin[ 61 ].Theimagefeaturevectorsforthetargetregionwerecomputedasreportedin[ 61 ].ThebuffersizeTintheKMmethodweresettobe20whichmeanstheKarchermeanofcovariancedescriptorsin20previousframeswereusedforthepredictionofthecovariancedescriptorincurrentframe.TheparametersfortheORFweresettovaluesgiveninthepaper[ 75 ].TheparametercontrollingthestatetransitionandobservationnoiseinourIRFaresetto!2=0.0001and2=0.01.SinceourIRFis 52

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Table4-1. Trackingresultfortherealdataexperiment. Seq.Obj.StartEndErr(IRF)Err(ORF)Err(KM) C3ps112007004.321310.846710.7666C3ps172007004.19987.35244.7929C3ps182007002.62585.58446.2928C3ps275008002.760510.647812.4289C3ps285008003.44517.211311.6432Cosow215009003.76124.6136.0196Cosow235009004.98715.85527.9788 combinedwithaparticlelterasapositiontracker,forthepurposeofcomparisons,theKMandORFarealsocombinedwithexactlythesameparticlelter.Firstly,weusedthreevideosequencesfromthedatasetCAVIAR:1.ThreePastShop1cor(C3ps1);2.ThreePastShop2cor(C3ps2);3.OneShopOneWait2cor(Cosow2).Allthreesequencesarefromaxedcameraandaframesizeof384288.7objectsweretrackedseparately.Givengroundtruthwasusedtoquantitativelyevaluatethetrackingresults.Tomeasuretheerrorforthetrackingresults,weusedthedistancebetweenthecenteroftheestimatedregionandthegroundtruth.Withallthethreemethodshavingthesameinitialization,thetrackingresultsareshownintheTable 4-1 ,wherealltheerrorsshownaretheaverageerrorsoverallthetrackedframes.FromthetablewecanseethatORFismoreaccuratethanKMbasedmethodsinmostoftheresults,andourIRFoutperformsboththesemethods.TheKMdriftsawayfromthetargetbecauseitisbasedonaslidingwindowapproach.Ifthenumberofconsecutivenoisyframesisclosetothewindowsize,thetrackerwilltendtotrackthenoisyfeatures.ForORF,sinceitisanon-intrinsicapproach,theapproximationoftheGL-invariantmetricwouldintroduceerrorsthataccumulateovertimeacrosstheframescausingittodriftaway.SinceIRFisanintrinsicrecursivelter,whichusestheGL-invariantmetric,thereislesserrorintroducedinthecovariancetrackerupdates.Thisinturnleadstohigheraccuracyintheexperimentsabove.Inthesecondexperiment,weperformedheadtrackinginvideosequenceswithamovingcamera.Twovideosequenceswereused:(i)Seq mbsequence(trackingface) 53

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and(ii)Seq sb.Eachofthesequenceshave500frameswithframesize96128.Bothsequencesarechallengingbecauseofcomplexbackground,fastappearancechangesandocclusions.TheresultsaresummarizedinFigure 4-2 .InSeq mb,KMfailsatframe450wheretheocclusionoccurs,whileORFandIRFdonotlosetrack.BothKMandORFproducerelativelylargeerrorsincapturingthepositionofthegirl'sfaceafterthegirlturnsaroundthersttimebetweenframes100to150duetothecompletechangeinappearanceofthetarget(girl'sface).ORFproducesrelativelylargererrorinestimatingthescale(comparedtotheinitialization)ofthefacebetweenframes400to500,whichcanbefoundinthesnapshotsincludedinFigure 4-2 .Theresultofourmethod(IRF)hasrelativelylargererrorataroundframe100and180,becauseatthistime,thecameraistrackingthehairofthegirlwherenofeaturecanbeusedtolocatethepositionoftheface.However,forotherframes,IRFtracksthefacequiteaccurately.InSeq sbbothKMandORFfailatframe200,andIRFhoweversuccessfullytrackedthewholesequencewithrelativelyhighaccuracyevenwithfastappearancechangesandocclusions,asshowninthequantitativeanalysisinFigure 4-2 .Theseexperimentsthusdemonstratetheaccuracyofourmethodinbothmovingcameraandxedcameracases. 54

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Figure4-2. Headtrackingresultforvideosequenceswithmovingcamera.TopandbottomrowsdepictsnapshotsandquantitativeevaluationsoftheresultsfromtheSeq mbandSeq sbrespectively.Thetrackingerrorismeasuredbythedistancebetweentheestimatedobjectcenterandthegroundtruth.Trackingresultsfromthethreemethodsareshownbyusingdifferentcoloredboxessuperposedontheimagesanddifferentcoloredlinesintheplots.Resultsfromourmethod(IRF)areinred,ORFingreenandKMinblue. 55

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CHAPTER5INTRINSICUNSCENTEDKALMANFILTER 5.1BackgroundandPreviousWorkDiffusionWeightedMRImaging(DWMRI)isthetechniquethatcanmeasurethelocalconstrainedwaterdiffusionpropertiesindifferentspatialdirectionsinMRsignalsandthusinfertheunderlyingtissuestructure.Itisauniquenon-invasivetechniquethatcanrevealtheneuralberstructuresin-vivo.Thelocalwaterdiffusionpropertycanbedescribedeitherviaadiffusivityfunctionoradiffusionpropagatorfunction.ThediffusivityfunctioncanbeestimatedfromtheDW-MRsignalsandrepresentedbya2ndordertensorateachimagevoxelyieldingthesocalledDiffusionTensorImaging(DTI)pioneeredin[ 12 ].ItisnowwellknownthatDTIfailstoaccuratelyrepresentlocationsthedatavolumecontainingcomplextissuestructuressuchasbercrossings.Tosolvethisproblem,severalhigherordermodelswereproposedsuchas[ 1 26 51 ].Tofurtherrevealthebrousstructuressuchasbrainwhitematter,bertrackingmethodswereproposedtoanalyzetheconnectivitiesbetweendifferentregionsinthebrain.Existingbertrackingmethodsfallmainlyintwocategories,deterministicandprobabilistic.Onepopulardeterministictrackingmethodisthestreamlinemethod[ 13 50 ],wherethetrackingproblemistackledusingalineintegration.Thedeterministictrackingmethodcanalsobasedonthe(RiemannianorFinsler)geometryimposedbythediffusivityfunction[ 62 ]wherethetrackingproblemisposedasashortestpathproblem.Inprobabilisticbertrackingmethods[ 14 63 89 ],aprobabilisticdynamicmodelisrstbuiltandthenalteringtechniquesuchasparticlelterisapplied.Mostoftheexistingbertrackingmethodsarebasedontwostagesnamely,rstestimatingthetensorsfromDWIandthentrackingusingtheseestimatedtensors.Recently,in[ 48 ]alteredmulti-tensortractographymethodwasproposedinwhichthebertrackingandthemulti-tensorreconstructionwasperformedsimultaneously.Therearemainlytwoadvantagesofthisapproach:(1)Thereconstructionisperformed 56

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onlyatlocationswhereitisnecessary,whichwouldsignicantlyreducethecomputationalcomplexitycomparedtotheapproachesthatrstreconstructwholetensoreldandthenapplytractography,(2)bertrackingisusedasaregularizationinthereconstructioni.e.,thesmoothnessoftheberpathisusedtoregularizethereconstruction.However,in[ 48 ]thelteringisappliedonlytothetensorfeatures(majoreigenvectorsetc.)allofwhichhavestrictmathematicalconstraintsthatoughttobesatisedbutnotalloftheconstraintswereenforced.Forexample,theconstraintoneigenvectorstolieontheunitspherewasnotenforced.Ingeneralitwouldbemorefavorabletotrackthefulltensorandenforcethenecessaryconstraints.Itisknownthatdiffusiontensorsareinthethespaceofsymmetricpositivedenite(SPD)matricesdenotedasPn,whichisnotaEuclideanspacebutaRiemannianmanifold.VectoroperationsarenotavailableonPn.Soalgorithmsthatarebasedonvectoroperationscannotbeapplieddirectlytothesespaces,andnon-trivialextensionsareneeded.Inthisdissertation,weproposeanovelintrinsicunscentedKalmanlteronPn,whichtothebestofourknowledgeistherstextensionoftheunscentedKalmanltertoPn.Weapplythisltertobothestimateandtrackthetensorsinthemulti-tensormodelusingtheintrinsicformulationtoachievebetteraccuracyasdemonstratedthroughexperiments.Weperformrealdataexperimentstodemonstratetheaccuracyandefciencyofourmethod.Therestofthechapterisorganizedasfollows:theintrinsicunscentedKalmanlterisdescribedinSection 5.2 ,whereanoveldynamicmodeldenedforthemulti-tensormodel.WethenpresenttheintrinsicunscentedKalmanlteralgorithmandnallytheexperimentsarepresentedinSection 5.3 5.2IntrinsicUnscentedKalmanFilterforDiffusionTensors 5.2.1TheStateTransitionandObservationModelsThestatetransitionmodelonPnhereisbasedontheGLoperationandtheLogNormaldistribution.Forthebi-tensor(sumoftwoGaussians)model,thestate 57

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transitionmodelatstepkisgivenby, D(1)k+1=ExpFD(1)kFt(v(1)k)D(2)k+1=ExpFD(2)kFt(v(2)k)(5)where,D(1)k,D(2)karethetwotensorstatesatstepk,FisthestatetransitionGLbasedoperation,v(1)kandv(2)karetheGaussiandistributedstatetransitionnoiseforD(1)kandD(2)kinthetangentspaceTD(1)kP3andTD(2)kP3respectively.Hereweassumethatthetwostatetransitionnoisemodelsareindependentfromeachotherandthepreviousstates.ThecovariancematricesofthetwostatetransitionnoisemodelsareQ(1)kandQ(2)krespectively.ThecovariancematrixQ(i)ki=1,2isa66matrixdenedforthetangentvectorsinTD(1)kP3.NotethatQ(i)kisnotinvarianttoGLcoordinatetransformonPn.AssumearandomvariableX=Exp(v)inPn,wherevisarandomvectorfromazeromeanGaussianwithQbeingthecovariancematrix.Then,afteraGLcoordinatetransformg2GL(n),thenewrandomvariableY=gXgt=Expggt(u).Thecovariancematrixofuis Q(g)=(gg))]TJ /F5 7.97 Tf 6.59 0 Td[(1Q(gg))]TJ /F4 7.97 Tf 6.59 0 Td[(t(5)wheredenotestheKroneckerproduct.HerewerstdenethecovariancematrixattheidentityQI33=qI66,whereqisapositivescalar.AndthecovariancematrixatpointXcanbecomputedusingEquation 5 bysettingg=X1 2.Withthisdenitionthestatetransitionnoiseisindependentwithrespecttothesystemstate.Theobservationmodelisbasedonthebi-tensordiffusionmodel. S(n)k=S0(e)]TJ /F4 7.97 Tf 6.59 0 Td[(bngtnD(1)kgtn+e)]TJ /F4 7.97 Tf 6.58 0 Td[(bngtnD(2)kgtn)(5)wheregndenotesthedirectionofn-thmagneticgradient,andbnisthecorrespondingb-value,andS(n)kistheMRsignalforn-thgradientatiterationstepk.Thecovariancematrixoftheobservationmodelforallthemagneticgradientsisadiagonalmatrix 58

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denotedbyR.Thisassumesthatthemeasurementsfromdistinctgradientdirectionsareindependent. 5.2.2TheIntrinsicUnscentedKalmanFilterJustasinthestandardKalmanlter,ateachiterationstepoftheunscentedKalmanlter[ 48 ]therearetwostages,thepredictionandupdatestagesrespectively.Inthepredictionstage,thestateofthelteratthecurrentiterationispredictedbasedontheresultfromthepreviousstepandthestatetransitionmodel.Intheupdatestep,theinformationfromtheobservationatthecurrentiterationisusedintheformofthelikelihoodtocorrecttheprediction.SincethestatesarenowdiffusiontensorswhichareinthespaceofPn,wherenovectoroperationsareavailable,weneedanon-trivialextensionoftheUnscentedKalmanlter,especiallyforthepredictionstagetobevalidonPn.Tobeginwith,wedenetheaugmentedstateforthebi-(diffusion)tensorstateatiterationstepktobe Xk=[u(1),tk,u(2),tk,v(1),tk,v(2),tk]t(5)wherev(i)ki=1,2isthestatetransitionnoisevectorfordiffusiontensorstateD(i)kandu(i)k=LogEK(D(i)k)(D(i)k)whichistherepresentationofthestaterandomvariableinthetangentplaneatitsKarcherexpectation(EK()).XkiszeromeanandwithcovariancematrixdenotedbyPak.Thecovariancematrixforthestate[u(1),tk,u(2),tk]tisdenotedbyPk,DD.NotethatPakisablock-wisediagonalmatrixcomposedfromPk,DD,Q(1)kandQ(2)k.Inthepredictionstage,2L+1weightedsamplesfromthedistributionofXtkarerstcomputedbyadeterministicsamplingschemegivenbelow.Here,L=24anddenotesthedimensionofXtk. Xk,0=0,w0==(L+) (5) Xk,j=(q (L+)Pak)j,wj=1=2(L+) (5) Xk,j+L=)]TJ /F8 11.955 Tf 9.29 0 Td[((q (L+)Pak)j,wj+n=1=2(L+) (5) 59

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wherewjistheweightforthecorrespondingsample,2Risaparametertocontrolthescatterofthesamples,and(p (L+)Pak)jisthej-thcolumnvectorofmatrixp (L+)Pak.SincesamplesXk,j=[u(1),tk,j,u(2),tk,j,v(1),tk,j,v(2),tk,j]taregeneratedfromthejointdistributionofposteriorandstatetransitionatframek,wecangetthesamplesfromthedistributionofpredictioninframek+1basedonXk,jthroughatwo-stepprocedure.Firstwecangetthesamplesfromtheposterior D(i)k,j=Exp^D(i)k(u(i)k,j)(5)where^D(i)kisthestateestimatefromthelastiteration(theestimatorofEK(D(i)k)).AndthenthesamplesfromthepredicteddistributioncanbegeneratedbasedonD(i)k,jandv(i)k,j, D(i)k+1,j=ExpD(i)k,j(v(i)k,j)(5)whereD(i)k+1,jdenotesthej-thsamplefromthedistributionoftheprediction.ThepredictedmeaniscomputedastheweightedKarchermean, ^D(i)k+1=dXjwjD(i)k+1,j(5)ThepredictedcovarianceofthestatesiscomputedintheproductspaceT^D(1)k+1P3T^D(2)k+1P3, Pk+1,DD=XjwjUjUtj(5)whereUtj=[Log^D(1)k+1(D(1)k+1,j),Log^D(2)k+1(D(2)k+1,j)]isaconcatenationofthetwovectorsobtainedfromtheLog-mapofeachpredictedsample.ApplyingtheobservationmodeldenedinEquation 5 tothepredictedstatesampleswegetthepredictedvectorofMRsignalsfordifferentmagneticgradientsdenotedbySk+1,j.Becausethisisinavectorspace,wecanusestandardvectoroperationstocomputethepredictedmean^Sk+1astheaverageofSk+1,j.UsingtheobservationnoisecovarianceR,thepredictedobservationcovariancecanbecomputed 60

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as Pk+1,SS=R+Xjwj(Sk+1,j)]TJ /F8 11.955 Tf 14.55 2.52 Td[(^Sk+1)(Sk+1,j)]TJ /F8 11.955 Tf 14.54 2.52 Td[(^Sk+1)t(5)Alsothecross-correlationmatrixbetweentheobservationandthestatesisgivenby, Pk+1,DS=Xjwj(Uj(Sk+1,j)]TJ /F8 11.955 Tf 14.55 2.52 Td[(^Sk+1)t)(5)Intheupdatestep,theKalmangainiscomputedasKk+1=Pk+1,DSP)]TJ /F5 7.97 Tf 6.59 0 Td[(1k+1,SS.KnowingtheKalmangainwecanupdateofthestatesandcovariancewhicharegivenby: ^D(i)k+1=Exp^D(i)k+1z(i)k+1Pk+1,DD=Pk+1,DD)-222(Kk+1Pk,SSKtk+1(5)where[z(1),tk+1,z(2),tk+1]t=Kk+1(Sk+1)]TJ /F8 11.955 Tf 14.65 2.52 Td[(^Sk+1),andSk+1istheobservation(MRsignalvector)atstepk+1. 5.3ExperimentsTovalidateourtractography,weappliedIUKFtoHARDIscansofratcervicalspinalcordatC3C5.Inthisexperiment,8differentratswereincluded6ofthemhealthyand2injuredwiththeinjuryinthethoracicspinalcord.TheHARDIscanforeachratwasacquiredwith1s0image(takenwithbclosedtozero),and21differentdiffusiongradientswithb=1000s=mm2,=13.4msand=1.8ms.Thevoxelsizeofthescanis35m35m300m,andtheimageresolutionis128x128inthex)]TJ /F3 11.955 Tf 12.19 0 Td[(yplaneandinthez-directiontheresolutionis24to34.AllHARDIdatasetswherealignedintothesamecoordinatesystembyasimilaritytransformbeforetracking.Toinitializethealgorithm,foreachscanwerstplacedaseedpointateachvoxelofthegreymatter,andthena2ndordertensorestimationisemployedasaninitializationforthealgorithm.Intheexperiment,variousparametersweresetto:thestatetransitionnoisevarianceinEquation 5 Q1=Q2=0.1I,theobservationnoisevarianceR=0.03Iandthesizeofeachtackingstept=0.01mm.Thealgorithmstopsiftheanglebetweentwo 61

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Figure5-1. Fibertrackingresultsonrealdatasets.Figure(a)istheregionofinterestoverlayedwiththeS0image.Figure(b)&(c)arethebertrackingresultofahealthy(injured)ratoverlayedonS0wherethebersarecoloredbyitslocaldirectionwithxyzbeingencodedbyRGB.c[2011]IEEE consecutivetangentvectorsbecomeslargerthan60degreeorthebertractarrivesattheboundaryofthespinalcord.Theberbundleofinterestisthemotoneuronwhichstartsfromthegraymatterandendsattheboundaryofthespinalcord.Tovisualizethemotoneuronberbundle,wetookcorrespondingROIssuchthatonlytheberpassingthroughtheROIsaredisplayed.TheresultsareshowninFigure 5-2 ,wherewecanndberbundlesstartingfromthegraymatterandendattheboundaryofthespinalcord.Thedifferencesbetweentheinjuredandcontrolratsarenoteasilyseendirectly.Tovisualizethedifferencebetweenthehealthyandinjuredrats,werstcomputedtheaxonalberdensitymapforeachratbycountingthenumberofberspassingthroughthe3-by-3neighborhoodofeachvoxel.Wethennon-linearlydeformthedensitymaptoaspinalcordatlasderivedfromHARDIdata[ 20 ]anddovoxel-wiset-testanalysis.TheresultareshownintheFigure 5-2 ,wherewecanndsignicantdifferencesbetweenthehealthyandtheinjuredratsinthemotoneuronregion,whichdemonstratestheeffectivenessofourtrackingmethod. 62

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Figure5-2. Biomarkerscapturedbycomputingdensitymapforeachberbundle.Figure(a)&(b)showasamplesliceofberdensitymapsobtainedforeachcontrolandinjuredrats,respectively.Figure(c)istheregioninwhichthep-valueislessthan0.005,overlaidontheS0image.c[2012]IEEE 63

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CHAPTER6ATLASCONSTRUCTIONFORHARDIDATASETREPRESENTEDBYGAUSSIANMIXTUREFIELDS 6.1BackgroundandPreviousWorkGroupwiseimageregistrationandimageatlasconstructionisaveryimportantandchallengingtaskinmedicalimageanalysis.Ithasmanyapplicationsinimagesegmentationandstatisticalanalysisofagroupofsubjectimages.Severalresearchergroupshavetackledvariationsofthisproblemandreportedtheirresultsinliterature[ 36 47 57 82 ].Mostoftheseareongroupwiseregistrationandatlasconstructionfromscalarimagesorsegmentedshapes.Diffusion-WeightedMRImaging(DW-MRI)isapowerfulnoninvasivetechniquethatcancapturetheinformationofwaterdiffusionintissuesandthusinferitsstructureinvivo.ServeralmethodshavebeenreportedinlitraturetomodelandestimatethediffusivityfunctionsfromtheMRIsignals.OnepopularmethodisthesocalledDiffusionTensorImaging(DTI)[ 12 ]whichapproximatesthediffusivityfunctionatavoxelbyapositivedenitematrix.ADTIbasedatlaswillobviouslyprovidemoreinformationthanconventionalscalarimagebasedatlas[ 55 ]sinceDTIcontainsbothscalaranddirectionalinformation.AtlasconstructionrequirestheDTIdatatobegroupwiseregisteredandinthisregard,untilrecently,mostoftheDTIregistrationtechniquesreportedinliteraturewerepairwiseregistrationmethods[ 10 15 86 88 ].SomeoftheexistingDTIbasedatlasesarebuiltbycoregistrationtechniquesasin[ 45 ]butaDTIbasedgroupwiseregistrationandatlasconstructionmethodswasreportedin[ 90 ].ItishoweverwellknownthattheDTImodelcannotresolvecomplextissuestructuresuchasbercrossings.Tohandlethisproblem,severalhigherordermodels[ 7 26 35 56 ]basedonHighAngularResolutionDiffusionImaging(HARDI)datasetwerereportedinliterature.ServeralrecentworkswerereportedfortheHARDIpairwiseregistrations[ 8 10 19 30 ],andshowntooutperformDTIbasedregistrationespeciallyinaligningbercrossingregions[ 10 ].Butveryfewworkshavebeenreportedin 64

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thegroupwiseregistrationforHARDIdataset,excepta4-thordertensoreldbasedgroupwiseregistrationreportedin[ 8 ]whichextendedtheunbiasedatlasconstructiontechniquein[ 36 ]tohandle4-thordertensoreldsbyusingnoveldistances.Inthisdissertation,wepresentanovelatlasconstructionmethodforHARDIdatasetsrepresentedbyaGaussainMixtureField(GMF)generatedbythealgorithmdescribedin[ 35 ].GMFisaeldofzeromean3DGaussianmixturemodelsoneeachateachlatticepointoftheeld.WeusetheL2distancebetweenGMFstomeasurethedissimilaritybetweentwoGaussianmixtureeldsaswasdenedin[ 19 ].Andwesignicantlyextendedtheframeworkin[ 36 ]toconstructtheatlasfromasetofGMFs.AnovelmeanGMFcomputationmethodisalsopresentedalongwiththegroupwiseregistrationprocess.Thekeycontributionsare:1.AGMFbasedgroupwiseregistrationisproposedwhichistherstofitskind;2.AnobjectivefunctioninvolvingtheL2distancebetweenGaussianmixturesisusedthatleadstoaclosedformexpressionforthedistanceandthegradientcomputation.3.Aminimaldistanceprojectionisdenedandusedtoobtainasharp(non-blurry)atlaswhichisusefulinatlasbasedsegmentation.Experimentsalongwithcomparisonsarepresentedtodemonstratetheperformanceofouralgorithm.Therestofthechapterisorganizedasfollows:ThemethodforatlasconstructiononGMFispresentedinSection 6.2 ,alongwiththeatlasconstructionframeworkinSection 6.2.1 ,followedbythedistancemetricinSection 6.2.2 ,andtheimplementationandmeanGMFcomputationinSection 6.2.3 .InSection 6.3 ,wepresentsyntheticandrealdataexperimentsalongwithcomparisonstootherexistingmethods. 6.2Methods 6.2.1ImageAtlasConstructionFrameworkAnatlasofasetofimages/shapesetc.iscommonlydenedasanaverageovertheset,whichistakentobearepresentativeoftheset.Theproblemwithsimplytakinganaverageastheatlasisthattheaveragetendstoberatherblurredandisnoteffective 65

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foruseintaskssuchas,atlas-basedsegmentationoratlas-basedregistrationetc.Mathematicallyspeaking,thiscanbecausedduetothefactthattheaveragemaynotnecessarilybelongtothesameabstractspace(e.g.,spaceofbrainimages)denedbytheoriginaldataset.Forinstance,thetechniquedescribedin[ 36 ]searchesfortheaverageintheimagespacewithoutaconstraintonthespaceofimages,whichcanleadtoablurredatlasimage.Tosolvethisproblem,in[ 47 ],theatlasisconstrainedtobedeformeddiffeomorphicallyfromasupertemplatewhichneedstobepre-selected;Andin[ 31 57 85 ]thestructureofthesubjectimagespaceislearnedfromthedataset,usingwhichtheatlasiscomputed.Thesemethodsneedregistrationsbetweenalltheimagepairsinthedataset(O(N2)registrations),whichmakestheapproachcomputationallyexpensiveforlargedatasets.Herewedenethespaceofimagesofinteresttous(spinalcordimages)tobespannedbyasetofGMFsfIngNn=1anddenotedbyS=SnO(In),whereO(In)=fJ:J=InT,T2DigistheorbitspannedbytheimageInandallthediffeomorphicdeformationsTn:!In,wheredenotesthedomainoftheimage.Thus,ndingtheatlasIcanbeviewedassolvingthefollowingproblemm,T1,...=argminm,T1,...XnE(InTn,ImTm)+(Tn)Andthenalatlascouldbedenedas^I=)]TJ /F5 7.97 Tf 6.58 0 Td[(1m[ImTm],wheremistheJacobianofthedeformationTm,and)]TJ /F5 7.97 Tf 6.59 0 Td[(1m[]denotesthere-orientationoperationdiscussedinSection 6.2.2 .SolvingthisproblemdirectlywouldmakethecomputationalcomplexitysimilartoO(N2)pairwiseregistrations.Whatwewouldliketodo,istoachieveanapproximatesolutionusingatwostepprocedure.Intherststep,wetrytondaintermidiateatlasinthespaceofallimages,whichcanbeviewedassolvingtheoptimizationproblem(similaras[ 36 ]butgeneralizedtoGMFs) I,T1,...=argminI,T1,...XnE(InTn,I)+(Tn)(6) 66

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wherethedatatermenergyfunctionE(,)isdenedasasumofsquaredvoxelwisedistance(detailsinSection 6.2.2 ) E(InTn,I)=ZIdist2n(InTn(x),I(x))dx(6)isthepenaltytermusedtoenforcethesmoothnessconstraintonthedeformation.Thedeformationcanbemodeledasadiffeomorphism,andparametrizedbyavelocityeld@Tn @t=vn(Tn(x,t),t).ThusthedeformationcanbecomputedasTn(x)=x+dn(x)=x+R10vn(x(t),t)dt,wherednrepresentsthedisplacementeld.Thesmoothnessconstraintweusehereisgivenby, (Tn)=log(det(n))(1)]TJ /F16 10.909 Tf 10.91 0 Td[(det(n))+ZIZ10jjLvn(x,t)jj2dt.(6)Where,Lisalinearoperator,andtherstterminEquation 6 imposesadditionalsmoothnessasin[ 87 ].Inthesecondstep,weprojecttheintermidiateatlastothespaceSbysolvinganotherdistanceminimization ^I=)]TJ /F5 7.97 Tf 6.58 0 Td[(1m[ImT]m,T=argminn,TE(InT,I)(6)andtheprojectionresult^Iisournalatlas. 6.2.2L2DistanceandRe-orientationforGMsWeusetheL2distanceasadissimilaritymeasurebetweentwoGaussianmixturedensities(GMs),whichcanbecomputedinaclosedform[ 19 ].Letf(r)=Mi=1iG(r;0,i)andg(r)=Nj=1jG(r;0,)]TJ /F4 7.97 Tf 6.94 -1.8 Td[(j)betwoGaussianmixturedensityfunctions,wherer2R3isthedisplacementvectorandi,jdenotethemixtureweightsofthecorrespondingGaussiancomponentsG(r;0,i)andG(r;0,)]TJ /F4 7.97 Tf 6.94 -1.79 Td[(j)withcovariancematricesiand)]TJ /F4 7.97 Tf 6.94 -1.79 Td[(jrespectively.TheL2distancebetweenfandgcanbewrittenasaquadraticfunctionofthemixtureweightsdist2(f,g)=tA+tB)]TJ /F8 11.955 Tf 12.5 0 Td[(2tC,where 67

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=(1,...,M)tand=(1,...,N)t,andAMM,BNNandCMNarethematricesgeneratedbytheGaussiancomponents,see[ 19 ]fordetails.Are-orientationoperationisneededforimagetransformation,whentheimagevalueateachpixel/voxelisrelatedtotheimagecoordinatesandisnotrotationallyinvariant[ 2 ].Inthiscase,theimagevalueshouldchangeaccordingtotheimagecoordinates.Otherwise,anartifactmightbeintroducedduetothistransformation.Asin[ 19 ],thePreservaionofPrincipalDirection(PPD)re-orientationisextendedtoGMs,anditistheonlyre-orientationstrategythatcancapturethechangeofanglebetweenbercrossingsduringanon-rigidtransformation.Inthisdissertation,weadoptthisre-orientationstrategy,andtheenergyfunctionwithreorientationwouldbedist2(f,g)=tA)]TJ /F15 5.978 Tf 5.75 0 Td[(1+tB)]TJ /F8 11.955 Tf 12.09 0 Td[(2tC)]TJ /F15 5.978 Tf 5.76 0 Td[(1,wherecomputationofA)]TJ /F15 5.978 Tf 5.75 0 Td[(1andC)]TJ /F15 5.978 Tf 5.75 0 Td[(1canbefoundin[ 19 ]. 6.2.3MeanGMFComputationWeemployan(iterative)greedyalgorithmtosolvetheprobleminEquation 6 .GiventheinitializationofatlasIandfTng,ineachiterationstep,werstxthedeformationsandupdatetheatlasbyoptimizingw.r.t.I.Inew=argminIPnRIdist2n(InTn(x),I(x))dxSinceweuseanL2distance,theglobalminimumcanbefoundasInew=P)]TJ /F15 5.978 Tf 5.75 0 Td[(1n[InT] N.However,fromthisformula,Inew(x)wouldhavemanymorecomponentsthanIn(x)wouldhavehad,whichwouldmakethealgorithmcomputationalexpensive.Tosolvethisproblem,wexedthegroupofmixturecomponentsofInew.Sinceeachmixturecomponentiszeromean,andcylindricallysymmetricwithxedeigenvalues.Allweneedtodoistodecideontheeigenvectors.Inthisdissertation,wediscretizethehemi-sphereusing46differentdirections,andusedthemastheeigenvectors(thesameapproachhasbeenusedinthereconstructionmethod[ 35 ]).Then,theonlythingleftistocomputethemixingweights,whichisequivalenttosolvingthelinearsystemateachvoxelA=PnC)]TJ /F15 5.978 Tf 5.76 0 Td[(1nn.ThiswouldbeeasytosolvesinceAislowdimensionalandfullrank. 68

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Afterupdatingtheatlas,theforceeldcanberstcomputedastherstordervariationofdatatermoftheobjectivefunctionplusthersttermontheright(enforcingadditionalsmoothness)inEquation 6 ;AndthenthevelocityeldisupdatedastheGaussainkernelbasedsmoothedversionoftheforceeld[ 8 ],andthedeformationeldT(x)isupdatedusingthefollowingupdateequationTnew(x)=Told(x+v).Thederivativeoftheobjectivefunctioniscomputedbyapplyingthederivativechainruleasin[ 19 ].AndthederivativeoftheregularizationterminEquation 6 canbecomputeddirectlyusingthederivativeforthedeterminent.Weemployacoarse-to-nestrategyinourregistrationalgorithmforanefcientimplementation.Withinitializationofthedeformationsettoidentity,thealgorithmyieldssatisfactoryresultsin200steps.AfterwegettheatlasI,wecanprojectittoStoget^IbyusingEquation 6 6.3Experiments 6.3.1SyntheticDataExperimentsTovalidatetheregistrationframeworkusedinouratlasconstructionmethod,werstapplyourmethodtothepairwiseregistrationproblem.Forthetwoimagescase,Equation 6 wouldreduceto T1,T2=argminT1,T2E(I1T1,I2T2)+(T1)+(T2).(6)Byusingthedisplacementeldinversemethod[ 25 ],wecangetT=T1T)]TJ /F5 7.97 Tf 6.59 0 Td[(12:I2!I1.Thuswehaveapairwiseregistrationalgorithm.WeappliedthisalgorithmtothesyntheticdatasetandthencomparedittoaGAbasedregistrationalgorithm(usinganSSDcost)andaDTIbasedregistrationalgorithmin[ 88 ]withthesamedataset.Togeneratethesyntheticdataset,a3Dsyntheticimage(64644)withtwocrossingberbundleswasgeneratedandthen,20randomlydeformedimagesweresynthesizedfromthisbyusingabspline-basednon-rigiddeformation.Themethoddescribedin[ 66 ]wasusedtogeneratethesimulatedMRsignalsfromtheberbundles. 69

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Figure6-1. Experimentalresultsonsyntheticdataset.Figure(a)and(b)arethemeanandstandarddeviationoferrorforthe20registrationsfromallthethreemethodsatdifferentnoiselevelsinthetwodifferentROIs.c[2011]IEEE Riciannoisewasaddedtosimulatedataat4differentnoiselevelswithSNR=50,20,10and5.Themethodin[ 35 ]wasusedtogeneratetheGMFfromtheMRsignalswith46Gaussiancomponentsateachvoxel.Afterthedatageneration,weregisteredeachoftherandomlydeformedimages(sourceimage)totheoriginalimage(targetimage)separately.Toevaluatetheregistration,theresultingdeformationobtainedfromtheregistrationwasappliedtothenoisefreesourceimage,andthenthedissimilaritybetweenthedeformedsourceandtargetimageswerecomputedastheerrorinregistration.ThedissimilaritymeasureweusedherewasHellingerdistancebetweenthedisplacementprobabilityproles(representedinsphericalharmoniccoefcients)atcorrespondinglatticepointsr=RS2(p f(x))]TJ /F10 11.955 Tf 12.76 10.39 Td[(p g(x))2dx.Also,wecomputedtheregistrationerrorintwodifferentregions:(1)thewholeimage,(2)theregionthatcontainsonlythecrossingbers.ThedatasetsandtheresultsaredisplayedinFigure 6-1 .Figure(a)and(b)showthatourmethodyieldsaslightlylowermeanandstandarddeviationoftheregistrationerrorsforthewholeimage,andmuchlowererrorinthebercrossingregionforallfournoiselevels.ThisdemonstratestheaccuracyofourHARDIregistrationmethod. 70

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6.3.2RealDataExperimentsFortherealdataexperiments,weapplyourmethodtotheHARDIscansofaratspinalcordatC3)]TJ /F8 11.955 Tf 13.05 0 Td[(5for7differentrats.Ineachscan21diffusiongradientswereused(withandthewassettobe13.4msand1.8ms)withb=1000s=mm2.Also,oneS0imageistakenwithbclosetozero.Theimageresolutioninthex)]TJ /F3 11.955 Tf 12.76 0 Td[(yplanewas128128,withthenumberofslicesvaryingfrom25to34.Thevoxelsizeis35m35m300m.[ 35 ]wasusedtogeneratetheGMFfromtheMRsignalswith46Gaussiancomponentsateachvoxel.Werstapplyasimilarityregistrationtoquotientoutthetranslation,rotationandscalingfactors,andapplyourgroupwiseregistrationalgorithmtothedatasettogetanatlasI.Inthefollowing,weprojectedItothespacespanedbythegivendatasamplesusingEquation 6 togetthesharp(non-blurry)atlas^I.TheresultsofouratlasconstructionmethodaredepictedinFigure 6-2 (viaS0imageseventhoughthealgorithmwasappliedtothetheGMFrepresentationoftheHARDIdata),wherethe(a)isthevoxelwisemeanofS0imagesbeforeregistration,and(b)afterregistration.Wecanseethat(a)isfuzzybecausethestructureisnotwellaligned,and(b)isnot.Thisindicatestheeffectivenessofourgroupwiseregistrationmethod.(c)istheIminEquation 6 ,and(d)isthenalatlas^I.Wecanseethat(b)ismuchmoreblurrythan(d),andtheshapeoftheboundarybetweenwhiteandgreymatterisnearlythesamefor(b)and(d).Thisindicatesthat^Icouldbeagoodrepresentativeforthewholedataset,andthusjustiestheeffectivenessofourmethod. 71

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Figure6-2. ExperimentalresultsonrealdatasetsdepictedusingS0images.Figure(a)-(d)areallS0images,with(a)thevoxelwisemeanbeforegroupwiseregistration,and(b)afterregistration.(c)istheS0imageforImand(d)istheS0imageforthenalatlas^I.Figure(f)isthediffusionproleof^I,andiscoloredbythemaxdirectionofthediffusionprole,withxyzdirecitionsmappedtoRGB.c[2011]IEEE 72

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CHAPTER7DISCUSSIONANDCONCLUSIONSTrackingonthemanifoldofPnisanimportantresearchproblemandhasmanyapplicationsincomputervisonandmedicallearning.Inthisdisseration,threenovelintrinsictrackingalgorithmsinPnarepresented.TherstoneisanovelrecursiveKarcherexpectationestimator(RKEE).TheunbiasednessandL2-convergenceunderthesymmetricdistributionisalsopresented.RealandsyntheticdataexperimentsshowedthesimilaraccurarcybetweentheRKEEandtheclassicalbatchKarcherexpectationestimatorKarchermean,buttheRKEEismuchmoreefcientthanKarchermeanespeciallyforrecursiveestimationproblem.Thesecondoneisanovelintrinsicrecursivelter(IRF)forcovariancetracking,whichprovedtobemorerobusttonoisethanexistingmethodsreportedinliterature.IRFisbasedontheintrinsicgeometryofthespaceofcovariancematricesandaGL-invariantmetricthatareusedindevelopingthedynamicmodelandtherecursivelter.ThethirdoneisanovelintrinsicunscentedKalmanlterwhichthenusedtorecursivelyestimatetheGaussianmixturemodelonDWMRdatasets.Incombinationofthestreamlinetrackingstrategy,thisformsanovelbertrackingalgorithm.Experimentsonthehumanbrainandratspinalcorddemonstratedtheaccuracyoftheproposedalgorithm.AgeneralizationofthenormaldistributiontoPnispresentedandusedtomodelthesysteminIRFandtheobservationnoiseintheIRF.SeveralpropertiesofthisdistributioninPnwerealsopresentedinthisdissertation,whichtothebestofourknowledgehaveneverbeenaddressedintheliterature.NotethatourgeneralizationofthenormaldistributiontoPnisrotationallyinvariant,andthevarianceofthedistributioniscontrolledbyascalar(!2inEquation 4 )ratherthanavariancecontroltensorwhichisamoregeneralform.OnemainreasonforusingthisspecicformisthatthescalarvariancecontrolparameterisGL-invariant,whilethevariancecontroltensorisnotasshownthroughthefollowingsimplecalculation.SupposeV2TXPnisatangentvector(whichis 73

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asymmetricmatrix)atpointX2Pn,andisavariancecontroltensor.ThevalueofthedensityfunctiononexpX(V)woulddependuponthequadraticformvec(V)t)]TJ /F5 7.97 Tf 6.59 0 Td[(1vec(V)wherevec()isthevectorizationoperationonthematrixargumentandisasecondordertensor.InpracticeXwouldbetheKarcherexpectationoftheaforementioneddistributionandVwouldbethetangentvectorcorrespondingtothegeodesicfromXtoasamplepointfromthedistribution.IfwechangethecoordinatesbyusingaGLoperationg,theKarcherexpectationbecomesgXgt,thevectorbecomesgVgt,andthequadraticformbecomesvec(gVgt)t)]TJ /F5 7.97 Tf 6.59 0 Td[(1vec(gVgt).Ifwewanttokeepthevalueofthedensityunchanged,weneedtochangeaccordingtogwhichmeansthatisnotGL-invariant.However,incontrast,itiseasytoshowthat!2inEquation 4 isGL-invariant.Further,theIRFisquitedifferentfromtheKalmanlterwhichisknowntobeanoptimallinearlterbasedonanadditiveGaussiannoiseassumption.OnereasonfortheKalmanltertobeoptimalisthatitactuallytracksthedistributionoftheobjectstate(posterior)basedonaBayesiantrackingframework.Ifalterdoesn'ttrackthewholedistribution,usuallyitwouldexplicitlyorimplicitlyapproximatetheposteriorbasedonthestatevariablesithastracked.However,theapproximationerrormightaccumulateinthesystem.Fromageometricpointofview,KalmanlterishighlydependentongeometricpropertiesoftheEuclideanspace.ThisisbecauseKalmanlterisbasedonthefactthattheconvolutionoftwoGaussiansisaGaussian.AndthispropertyoftheGaussianstemsfromthefactthattheGaussianisthelimitdistributioninthecentrallimittheorem.OnekeyprobleminextendingtheKalmanlterintrinsicallytoPnisndingtwodensitiespA(X;A),pB(XjY)withfollowingproperties pA(X;A)=ZPnpB(XjY)pA(Y;)[dY](7)whereAistheparameterofdensitypA.pAhereisusuallytheposteriorandpBisthestatetransitionnoisedistribution.Theequationabovemeansthatafterthestatetransitiontheformoftheposteriorremainsthesame.Withoutthisproperty,evenifthe 74

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wholedistributionistracked,thelterisimplicitlyapproximatingthetruedistributionafterthestatetransitionbyusingthesameformastheposteriorfromthelaststepwhichstillwouldleadtoerrorsbeingaccumulatedinthesystem.However,itisnon-trivialtondsuchdistributionsonPn.In[ 33 70 ],acentrallimittheoremwaspresentedinPnforrotationallyinvariantprobabilitymeasuresbasedontheHelgason-Fouriertransform[ 34 ].However,currentlytheprobabilitymeasureinthelimitdoesnothaveaclosedforminthespacedomain.Thus,intrinsicallyextendingtheKalmanltertoPnisstillanopenproblem.IRFinsteadonlytracksthemodeofthedistribution.Itisnotanoptimallter,butisintrinsicandmathematicallyconsistentwithrespecttothenoisemodelused.Wealsopresentedarealtimecovariancetrackingalgorithmbasedonthislterwhichiscombinedwithanexistingparticlepositiontrackerfromliterature[ 5 ].Finally,experimentsonsyntheticandrealdatafavorablydemonstratedtheaccuracyofourmethodoverrivalmethods. 75

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REFERENCES [1] Alexander,D.C.MaximumEntropySphericalDeconvolutionforDiffusionMRI.IPMI.2005,76. [2] Alexander,D.C.,Pierpaoli,C.,Basser,P.J.,andGee,J.C.SpatialTransformationsofDiffusionTensorMagneticResonanceImages.IEEETrans.Med.Imag.20(2001).11:1131. [3] Arsigny,V.,Fillard,P.,Pennec,X.,andAyache,N.FastandSimpleCalculusonTensorsintheLog-EuclideanFramework.MICCAI.vol.3749.2005.115. [4] .Log-Euclideanmetricsforfastandsimplecalculusondiffusiontensors.Magn.Reson.Med.56(2006):411. [5] Arulampalam,M.S.,Maskell,S.,Gordon,N.,andClapp,T.Atutorialonparticleltersforonlinenonlinear/non-GaussianBayesiantracking.SignalProcessing,IEEETransactionson50(2002):174. [6] Ballmann,W.Manifoldsofnonpositivecurvature.ArbeitstagungBonn1984.eds.FriedrichHirzebruch,JoachimSchwermer,andSilkeSuter,vol.1111ofLectureNotesinMathematics.SpringerBerlin/Heidelberg,1985.261.10.1007/BFb0084594. [7] Barmpoutis,A.,Hwang,M.,Howland,D.,Forder,J.R.,andVemuri,B.C.Regularizedpositive-denitefourthordertensoreldestimationfromDW-MRI.NeuroImage45(2009):153. [8] Barmpoutis,A.andVemuri,B.C.GroupwiseRegistrationandatlasconstructionof4th-ordertensoreldsusingtheR+Riemannianmetric.MICCAI.2009. [9] .Auniedframeworkforestimatingdiffusiontensorsofanyorderwithsymmetricpositive-deniteconstraints.InProceedingsofISBI10:IEEEInterna-tionalSymposiumonBiomedicalImaging(2010):1385. [10] Barmpoutis,A.,Vemuri,B.C.,andForder,J.R.RegistrationofHighAngularResolutionDiffusionMRIImagesUsing4thOrderTensors.MICCAI.2007. [11] Barmpoutis,A,Vemuri,BabaC,Shepherd,TM,andForder,JR.TensorSplinesforInterpolationandApproximationofDT-MRIWithApplicationstoSegmentationofIsolatedRatHippocampi.IEEETrans.Med.Imag.26(2007). [12] Basser,P.,Mattiello,J.,andLeBihan,D.Estimationoftheeffectiveself-diffusiontensorfromtheNMRspinecho.J.Magn.Reson.B103(1994):247. [13] Basser,P.,Pajevic,S.,Pierpaoli,C.,Duda,J.,andAldroubi,A.InVivoFiberTractographyUsingDT-MRIData.MRM44(2000):625. 76

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[14] Behrens,T.,Berg,H.,Jbabdi,S.,Rushworth,M.,andWoolricha,M.Probabilisticdiffusiontractographywithmultiplebreorientations:Whatcanwegain?Neu-roImage34(2007):144. [15] Cao,Y.,Miller,M.,Mori,S.,Winslow,R.,andYounes,L.DiffeomorphicMatchingofDiffusionTensorImages.CVPR.2006. [16] Caselles,V.,Kimmel,R.,andSapiro,G.GeodesicActiveContours.Intl.Journ.ofCompu.Vision22(1997).1:61. [17] Chan,T.andVese,L.ActiveContourswithoutEdges.IEEETrans.onImageProc.10(2001).2:266. [18] Cheng,G.,Salehian,H.,Vemuri,B.C.,Hwang,M.S.,Howland,D.,andForder,J.R.ANovelIntrinsicUnscentedKalmanFilterforTractographyfromHARDI.ISBI.2012. [19] Cheng,G.,Vemuri,B.C.,Carney,P.R.,andMareci,T.H.Non-rigidRegistrationofHighAngularResolutionDiffusionImagesRepresentedbyGaussianMixtureFields.MICCAI.2009. [20] Cheng,G.,Vemuri,B.C.,Hwang,M.S.,Howland,D.,andForder,J.R.AtlasConstructionforHighAngularResolutionDiffusionImagingDataRepresentedbyGaussianMixtureFields.ISBI.2011. [21] Cherian,A.,Sra,S.,Banerjee,A.,andPapanikolopoulos,N.EfcientSimilaritySearchforCovarianceMatricesviatheJensen-BregmanLogDetDivergence.ICCV.2011. [22] Chikuse,Y.StateSpaceModelsonSpeicalManifolds.JournalofMultivariateAnalysis97(2006):1284. [23] Comaniciu,D.andMeer,P.MeanShift:ARobustApproachTowardFeatureSpaceAnalysis.IEEETransactionsonPatternAnalysisandMachineIntelligence24(2002):603. [24] Comaniciu,D.,Ramesh,V.,andMeer,P.Real-timetrackingofnon-rigidobjectsusingmeanshift.CVPR.2000. [25] Crum,W.R.,Camara,O.,andHawkes,D.MethodsforInvertingDenseDisplacementFields:EvaluationinBrainImageRegsitration.MICCAI.2007. [26] Descoteaux,M.,Angelino,E.,Fitzgibbons,S.,andDeriche,R.Apparentdiffusioncoefcientsfromhighangularresolutiondiffusionimaging:Estimationandapplications.MRM56(2006):395. [27] Feddern,C.,Weickert,J.,andBurgeth,B.Level-setMethodsforTensor-valuedImages.Proc.2ndIEEEWorkshoponVariational,GeometricandLevelSetMethodsinComp.Vis..2003,65. 77

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[28] Fletcher,P.andJoshi,S.PrincipalGeodesicAnalysisonSymmetricSpaces:StatisticsofDiffusionTensors.ComputerVisionandMathematicalMethodsinMedicalandBiomedicalImageAnalysis.eds.MilanSonka,IoannisKakadiaris,andJanKybic,vol.3117ofLectureNotesinComputerScience.2004.87. [29] Fletcher,P.T.,Lu,C.,Pizer,S.M.,andJoshi,S.Principalgeodesicanalysisforthestudyofnonlinearstatisticsofshape.MedicalImaging,IEEETransactionson23(2004).8:995. [30] Geng,X.,Ross,T.J.,Zhan,W.,Gu,H.,Chao,Y.,Lin,C.,Christensen,G.E.,Schuff,N.,andYang,Y.DiffusionMRIRegistrationUsingOrientationDistributionFunctions.IPMI.2009. [31] Gerber,S.,Tasdizen,T.,Joshi,S.,andWhitaker,R.OntheManifoldStructureoftheSpaceofBrainImages.MICCAI.2009. [32] Goh,A.andVidal,R.Segmentingberbundlesindiffusiontensorimages.EuropeanConferenceonComputerVision(ECCV).2008,238. [33] Graczyk,P.Acentrallimittheoremonthespaceofpositivedenitesymmetricmatrices.Ann.Inst.Fourier42(1992):857. [34] Helgason,S.DifferentialGeometry,LieGroups,andSymmetricSpaces.NewYork:AcademicPress,2001. [35] Jian,BingandVemuri,BabaC.AUniedComputationalFrameworkforDeconvolutiontoReconstructMultipleFibersFromDWMRI.IEEETMI26(2007):1464. [36] Joshi,S.,Davis,B.,Jomier,B.M.,andGerig,G.Unbiaseddiffeomorphicatlasconstructionforcomputationalanatomy.NeuroImage23(2004):151. [37] Kailath,T.LinearSystems.Prentice-Hall,1980. [38] Karcher,H.Riemanniancenterofmassandmolliersmoothing.Comm.PureAppl.Math30(1977):509. [39] Kwon,J.andPark,F.C.VisualTrackingviaParticleFilteringontheAfneGroup.TheInternationalJournalofRoboticsResearch29(2010):198. [40] Lenglet,C.,Rousson,M.,andDeriche,R.DTIsegmentationbystatisticalsurfaceevolution.IEEETransactionsonMedicalImaging25(2006).6:685. [41] Lenglet,C.,Rousson,M.,Deriche,R.,andFaugeras,O.StatisticsontheManifoldofMultivariateNormalDistributions:TheoryandApplicationtoDiffusionTensorMRIProcessing.JournalofMathematicalImagingandVision25(2006):423.10.1007/s10851-006-6897-z. 78

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[42] Li,M.,Chen,W.,Huang,K.,andTan,T.VisualTrackingviaIncrementalSelftuningParticleFilteringontheAfneGroup.CVPR.2010. [43] Li,Xi,Hu,Weiming,Zhang,Zhongfei,Zhang,Xiaoqin,Zhu,Mingliang,andCheng,Jian.VisualTrackingViaIncrementalLog-EuclideanRiemannianSubspaceLearning.CVPR.2008. [44] Liu,Y.,Li,G.,andShi,Z.Covariancetrackingviageometricparticleltering.EURASIPJ.Adv.SignalProcess2010(2010):22:1:9. [45] Liu,Z.,Zhu,H.,Marks,B.,Katz,L.,Goodlett,C.,Gerig,G.,andStyner,M.VOXEL-WISEGROUPANALYSISOFDTI.ISBI.2009. [46] Lurie,Jacob.LecturenotesonthetheoryofHadamardspaces(metricspacesofnonpositivecurvature).http://www.math.harvard.edu/lurie/papers/hadamard.pdf,???? [47] Ma,J.,Miller,M.I.,Trouve,A.,andYounes,L.Bayesiantemplateestimationincomputationalanatomy.NeuroImage42(2008):252. [48] Malcolm,J.G.,Shenton,M.E.,andRathi,Y.FilteredMultitensorTractography.TMI29(2010):1664. [49] Malladi,R.,Sethian,J.A.,andVemuri,B.C.ShapeModelingwithFrontPropagation:ALevelSetApproach.IEEETrans.onPAMI17(1995).2:158. [50] McGraw,T.,Vemuri,B.,Chen,Y.,Rao,M.,andMareci,T.DT-MRIdenoisingandneuronalbertracking.MedIA8(2004):95. [51] McGraw,T.,Vemuri,B.C.,Yezierski,B.,andMareci,T.SegmentationofHighAngularResolutionDiffusionMRIModeledasaFieldofvonMises-FisherMixtures.ECCV.2006. [52] Metha,M.L.RandomMatricesandthestatisticstheoryofEnergylevels.AcademicPress,NewYorkandLondon,1967. [53] Moakher,M.ADifferentialGeometricApproachToTheGeometricMeanOfSymmetricPositive-deniteMatrices.SIAMJ.MATRIXANAL.APPL.26(2005):735. [54] Moakher,M.andBatchelor,P.G.SymmetricPositive-DeniteMatrices:FromGeometrytoApplicationsandVisualization.VisualizationandProcessingofTensorFields,Springer,2006. [55] Mori,S.,Oishi,K.,andFariaab,A.V.Whitematteratlasesbasedondiffusiontensorimaging.CurrOpinNeurol.22(2009):362. 79

PAGE 80

[56] Ozarslan,E.,Shepherd,T.M.,Vemuri,B.C.,Blackband,S.J.,andH.Mareci,T.Resolutionofcomplextissuemicro-architectureusingthediffusionorientationtransform.Neuroimage31(2006):1086. [57] Park,H.,Bland,P.,Hero,A.,andMeyer,C.Leastbiasedtargetselectioninprobabilisticatlasconstruction.MICCAI.2005. [58] Pennec,X.IntrinsicStatisticsonRiemannianManifoldsBasicToolsforGeometricMeasurements.JournalofMathematicalImagingandVision25(2006):127. [59] Pennec,X.,Fillard,P.,andAyache,N.ARiemannianframeworkfortensorcomputing.InternationalJournalofComputerVision66(2006).1:41. [60] Porikli,F.Learningonmanifolds.Proceedingsofthe2010jointIAPRinter-nationalconferenceonStructural,syntactic,andstatisticalpatternrecognition.SSPR&SPR'10.Berlin,Heidelberg:Springer-Verlag,2010,20. [61] Porikli,F.,Tuzel,O.,andMeer,P.CovarianceTrackingusingModelUpdateBasedonLieAlgebra.CVPR.vol.1.2006,728. [62] Prados,E.,Lenglet,C.,Wotawa,N.,Deriche,R.,Faugeras,O.,andSoatto,S.ControlTheoryandFastMarchingTechniquesforBrainConnectivityMapping.CVPR.2006. [63] Savadjiev,P.,Rathi,Y.,Malcolm,J.G.,Shenton,M.E.,andWestin,C-F.AGeometry-basedParticleFilteringApproachtoWhiteMatterTractography.MIC-CAI.2010. [64] Schwartzman,A.Randomellipsoidsandfalsediscoveryrates:Statisticsfordiffusiontensorimagingdata.Ph.D.thesis,StanfordUniversity,2006. [65] Snoussi,H.andRichard,C.MonteCarloTrackingontheRiemannianManifoldofMultivariateNormalDistributions.DSPSPE.2009. [66] Soderman,O.andJonsson,B.RestrictedDiffusioninCylindiricalGeometry.J.Magn.Reson.B117(1995):94. [67] Sorenson,H.W.,ed.KalmanFiltering:TheoryandApplication.IEEEPress,1985. [68] Srivastava,A.andKlassen,E.BayesianandGeometricSubspaceTracking.AdvancesinAppliedProbability36(2004):43. [69] Subbarao,R.andMeer,P.NonlinearMeanShiftoverRiemannianManifolds.InternationalJournalofComputerVision84(2009):1.10.1007/s11263-008-0195-8. [70] Terras,A.Harmonicanalysisonsymmetricspacesandapplications.Springer-Verlag,1988. 80

PAGE 81

[71] Tosato,D.,Farenzena,M.,Cistani,M.,andMurino,V.ARe-evaluationofPedestrianDetectiononRiemannianManifolds.PatternRecognition,Interna-tionalConferenceon0(2010):3308. [72] Tsai,A.,Yezzi,A.Jr.,andWillsky,A.S.CurveEvolutionImplementationoftheMumford-ShahFunctionalforImageSegmentation,Denoising,Interpolation,andMagnication.IEEETrans.onImageProc.10(2001).8:1169. [73] Tuzel,O.,Porikli,F.,andMeer,P.RegionCovariance:AFastDescriptorforDetectionandClassication.ECCV.2006. [74] .PedestrianDetectionviaClassicationonRiemannianManifolds.IEEETransactionsonPatternAnalysisandMachineIntelligence30(2008):1713. [75] Tyagi,A.andDavis,J.W.ArecursivelterforlinearsystemsonRiemannianmanifolds.CVPR.2008,1. [76] Tyagi,A.,Davis,J.W.,andPotamianos,G.Steepestdescentforefcientcovariancetracking.IEEEWorkshoponMotionandvideoComputing.2008. [77] Vemuri,B.C.,Liu,M.,Amari,S.I.,andNielsen,F.TotalBregmandivergenceanditsapplicationstoDTIanalysis.IEEETransactionsonMedicalImaging30(2011).2:475. [78] Wang,Z.andVemuri,B.C.DTIsegmentationusinganinformationtheoretictensordissimilaritymeasure.IEEETransactionsonMedicalImaging24(2005).10:1267. [79] Wang,ZhizhouandVemuri,BabaC.TensorFieldSegmentationUsingRegionBasedActiveContourModel.ECCV(4).2004,304. [80] Weldeselassie,Y.T.andHamarneh,G.DT-MRIsegmentationusinggraphcuts.SPIEMedicalImaging.vol.6512.2007. [81] Wong,Y.SectionalCurvaturesofGrassmannManifolds.Proc.Nati.Acad.Sci.USA60(1968):75. [82] Wu,G.,Yap,P.,Wang,Q.,andShen,D.GROUPWISEREGISTRATIONFROMEXEMPLARTOGROUPMEAN:EXTENDINGHAMMERTOGROUPWISEREGISTRATION.ISBI.2010. [83] Wu,Y.,Cheng,J.,Wang,J.,andLu,H.Real-timeVisualTrackingviaIncrementalCovarianceTensorLearning.ICCV2009.2009. [84] Wu,Y.,Wu,B.,Liu,J.,andLu,H.ProbabilisticTrackingonRiemannianManifolds.ICPR.2008. [85] Xie,Y.,Ho,J.,andVemuri,B.ImageAtlasConstructionviaIntrinsicAveragingontheManifoldofImages.CVPR.2010. 81

PAGE 82

[86] Yang,J.,Shen,D.,Davatzikos,C.,andVerma,R.DiffusionTensorImageRegistrationUsingTensorGeometryandOrientationFeatures.MICCAI.2008. [87] Yanovsky,I.,Thompson,P.M.,Osher,S.,andLeow,A.D.TopologyPreservingLog-UnbiasedNonlinearImageRegistration:TheoryandImplementation.CVPR.2007. [88] Yeo,B.,Vercauteren,T.,Fillard,P.,Peyrat,J.,Pennec,X.,Golland,P.,Ayache,N.,andClatz,O.DT-REFinD:DiffusionTensorRegistrationWithExactFinite-StrainDifferential.IEEETrans.Med.Imaging,28(2009):1914. [89] Zhang,F.,Hancock,E.R.,Goodlett,C.,andGerig,G.ProbabilisticwhitematterbertrackingusingparticlelteringandvonMisesFishersampling.MedIA13(2009):5. [90] Zhang,H.,Yushkevich,P.,Rueckert,D.,andGee,J.Unbiasedwhitematteratlasconstructionusingdiffusiontensorimages.MICCAI.2007. [91] Ziyan,U.,Tuch,D.,andWestin,C.F.SegmentationofthalamicnucleifromDTIusingspectralclustering.MedicalImageComputingandComputer-AssistedIntervention(MICCAI)(2006):807. 82

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BIOGRAPHICALSKETCH GuangChengwasborninXingjiang,Chinain1981.HegraduatedfromTsinghuaUniversitywithaBachelorofEngineeringdegreeinautomationin2003andaMasterofEngineeringdegreeinconstrolscienceandengineeringin2006.HecametotheUniversityofFloridatopursueaDoctorofPhylosophydegreeincomputerengineeringinAugust2006.GuangmetYandiFanatUF,andtheyweremarriedinDecemberof2007.Theirrstdaughter,LerongCheng,wasborninAugust2010. 83