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Denoising, Segmentation and Visualization of Diffusion Weighted MRI

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DENOISING,SEGMENTATIONANDVISUALIZATIONOFDIFFUSION WEIGHTEDMRI By TIME.MCGRAW ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2005

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Copyright2005 by TimE.McGraw

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Formywife,Jo,andmymother,Patti.

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ACKNOWLEDGMENTS Iamverygratefulfortheadvisementandencouragementofmy committee duringthepreparationofthiswork.Iwishtorstthankthec hairofmycommitteeDr.BabaC.Vemuriforhisacademicandprofessionalg uidancethrough theyears.Hisconsultationandadvicemadethisworkpossib le.ThanksgotoDr. ThomasMareciforproductiveweeklymeetingsandimportant clinicalinput.The guidanceandinputfromDr.AnandRagarajan,Dr.BenLokandD r.JeHoare muchappreciated.Thanksgotoeveryoneonmycommitteefora llofthetimeand energytheyhaveinvestedintomyeducationandresearch. IwishtoexpressmygratitudetoeverybodyintheImagingand Visualization departmentatSiemensCorporateResearch,especiallyDr.J imWilliamsandDr. MariappanNadarforgivingmetheopportunitytoworkwithth emasanintern fortwosummers.ThanksalsogotoTonyChenatSCRformakingt heinternship processrunsmoothly. ResearchersattheMcKnightBrainInstituteattheUniversi tyofFlorida haveprovidedvaluableclinicalinputandfruitfuldiscuss ions.Thanksgoto Evren Ozarslan,Dr.SteveBlackband,Dr.RobertYezierskiandDr. PaulReier. Specically,thanksgotoEvren OzarslanforthesyntheticdataandfastODF computationtechnique.WithouttheMRIdatathisresearchw ouldhavebeen iv

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impossible,soIalsowishtothankSaraBerensandRobertYez ierskiforproviding thespinalcorddatasampleandRonHayesforthebraindata. PresentandformerfacultymembersoftheCISEdepartmentha vebeen responsibleformanythought-provokingcoursesandstimul atingdiscussions, includingDr.JorgPeters,Dr.MeeraSitharam,Dr.Gerhard X.Ritter,Dr. JosephWilson,Dr.MarkSchmalz,Dr.AlperUngor,Dr.TimDav is,Dr.Richard Newman,Dr.DouglasDankel,Dr.DavidGu,andDr.Yan-HangLe e. FacultyintheMathdepartmentatUFhaveproveninvaluablet omystudies throughcoursestheyhaveoeredandthroughtheirfeedback andinput.Thanksgo toYunmeiChen,MuraliRao,JayGopalakrishnan,DavidGrois ser,andTimOlsen. FormerandpresentstudentsintheCVGMILabhavemadetheres earch experiencemostpleasantandalsoprovidedvaluableinputt omystudiesandresearch,includingJundongLiu,ZhizhouWang,FeiWang,Eric Spellman,Santhosh Kodipaka,JieZhangandHongyuGuo. Classmates,friends,andotherstudentsinthedepartmenth aveprovided camaraderieandwelcomedistraction.ManythanksgotoAndr esMendez-Vasquez, AndyShui,XiaobinWu,BrunoMaciel,LeweyGeselowitz,Ashi shMylesandmany others. TheocestaoftheCISEdepartmenthavealwaysbeenveryhel pfulwiththe administrativedetails.SpecialthanksgotoJohnBowersfo rmakingtheprocess mucheasierandmorepleasant.Thanksgotothesystemsstaf orkeepingallof thehardwareandsoftwarerunning. v

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Iwouldalsoliketothankmyfamily,especiallymywifeJo,fo rtheirsupport andpatience. ThisresearchwassupportedinpartbythegrantNIH-NS42075 andbyagrant fromSiemensCorporateResearch. vi

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .............................iv LISTOFFIGURES ................................ix ABSTRACT ....................................xii 1INTRODUCTION ...........................1 1.1DiusionMRI ..........................3 1.1.1OverviewofDiusion ..................5 1.1.2OverviewofMRImaging ................7 1.1.3DiusionWeightedImages ...............8 1.1.4DiusionTensorMRIAcquisition ...........9 1.1.5HighAngularResolutionDiusionImaging ......11 1.2Restoration ............................12 1.3Tractography ...........................14 1.4Anisotropy ............................15 1.5OverviewofOurModelingScheme ...............16 2DENOISING ..............................18 2.1TVNormMinimizationofS(x,y,z) ...............18 2.1.1ScalarTVNormMinimization .............19 2.1.2VectorTVNormMinimization .............20 2.2VariationalPrinciple ......................21 2.3SpatialLatticeSmoothingof S ( x ) ...............22 2.3.1Fixed-PointLagged-Diusivity .............23 2.3.2DiscretizedEquations ..................24 2.4FiniteElementSmoothingofS( ; ) ..............25 2.4.1ElementMatrices ....................27 2.4.2LocalElementCoordinates ...............30 2.4.3GlobalMatrices .....................34 3VISUALIZINGPROBABILITYFIELDS ...............36 3.1ComputingProbabilities ....................36 3.2GlyphVisualization .......................37 3.3ScalarMeasuresofAnisotropy .................38 3.4VisualizingtheDirectionalNatureofDiusion ........42 vii

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3.4.1Streamtubes .......................44 3.4.2LineIntegralConvolution ................45 4RESULTS ................................50 5SEGMENTATIONOFHARDI ....................58 5.1ModellingDiusion .......................58 5.1.1OtherModels .......................58 5.1.1.1GaussianMixtureModel ...........59 5.1.1.2SphericalHarmonics ..............60 5.1.1.3GeneralizedTensors ..............60 5.1.2vonMises-FisherMixtureModel ............61 5.1.2.1FittingthevMFMixture ...........64 5.1.2.2EntropyandDistance .............65 5.2TheSpaceofvMFDistributions ................67 5.2.1RiemannianGeometry ..................67 5.2.2RiemannianExpandLogMaps ............70 5.2.3SymmetricSpaces ....................72 5.2.4LieGroupsandHomogeneousSpaces .........73 5.2.5Results ..........................76 5.3TheSpaceofvMFMixtures ..................76 5.3.1RiemannianExpandLogMaps ............77 5.3.2IntrinsicMean ......................80 5.4SegmentationModels ......................80 5.4.1Mumford-ShahModel ..................82 5.4.2SpectralClustering ...................83 5.4.3MarkovianModelsForSegmentation ..........84 5.4.3.1MRFsforSegmentation ............85 5.4.3.2HMMFsforSegmentation ...........87 5.4.3.3Results .....................89 6VISUALIZATIONTECHNIQUESFORSEGMENTEDSURFACES 92 6.1SignedDistanceFunctions ...................93 6.2MultiresolutionMeshing ....................93 6.2.1MeshGeneration .....................93 6.2.2SurfaceSimplication ..................94 6.2.3AdaptiveSurfaceRenementWithErrorBounds ...96 6.2.4ComputingtheCutGraph ...............97 6.2.5MeshParameterization .................100 7CONCLUSION .............................103 REFERENCES ...................................105 BIOGRAPHICALSKETCH ............................114 viii

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LISTOFFIGURES Figure page 1.1OrientationalheterogeneityinDTI(left),andHARDI(r ight). ....2 1.2Diusionellipsoid .............................6 2.1TV( f 1) > TV( f 2)=TV( f 3) ......................19 2.2HARDgradientdirections( ; )correspondtotheverticesofasubdividedicosahedron. ..........................26 2.3Twodimensional(u,v)domainforFEM-basedsmoothingof HARDI data. ...................................26 2.4Mappingtobarycentriccoordinates ...................30 3.1OriginalODF(left),minimumprobabilitysphere(cente r),andsharpenedODF(right). ...........................37 3.2ODFglyphs ................................38 3.3FAimage(left),generalizedanisotropy(center),Shan nonanisotropy (HA)(right),fromcoronalsliceofratbrain .............38 3.4RaycastvolumevisualizationofHA. ...................40 3.5MIPvolumevisualizationofHA. ....................40 3.6ShannonanisotropydierencebetweenGaussianODFandg eneral ODF. ..................................41 3.7Anisotropycomputedfrom H 2 (top-left), H 5 (top-right), H 10 (bottomleft), H 20 (bottom-right). ........................42 3.8Renyientropydierences H 1 H 2 (top-left), H 1 H 5 (top-right), H 1 H 10 (bottom-left), H 1 H 20 (bottom-right). ..........42 3.9ColorFA. .................................43 3.10EDcolorsforsyntheticdistributioneld. ................44 3.11EDcolorsforsyntheticdistributioneld. ................44 3.12Streamtubesinaxialsliceofbrain. ...................45 ix

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3.13LICvisualizationofsyntheticeldwithkernelwidths 5,15,and55. .46 3.14LICbervisualizationinaxialsliceofbrain. ..............48 3.15LICbervisualizationincoronalsliceofbrain. .............48 3.16LICbervisualizationinsagittalslicesofbrain. ............49 4.1ODFprolesoverlaidonHAimage(left),andcoloranisot ropy(right) ofthesyntheticdataset. ........................50 4.2Multi-LICvisualizationofsyntheticdataset. ..............51 4.3SyntheticS(left),resultingODF(right). ................51 4.4SyntheticSwithnoiseadded(left),resultingODF(righ t). ......52 4.5Manifold(FEM)smoothingresultsforS(left),resultin gODF(right). 52 4.6Lattice(TV)smoothingresultsforS(left),resultingO DF(right). ..53 4.7ManifoldandlatticesmoothedS(left),resultingODF(r ight). ....53 4.8Originaldiusion-weightedimage(left),anddenoised (right)from spinalcorddata. ............................54 4.9Originalcoloranisotropyimage(left),anddenoised(r ight)fromspinal corddata. ................................55 4.10Original(left),anddenoised(right)HAimagesforcor onalslicesof ratbrain. ................................56 4.11Original(left),anddenoised(right)EDimagesforcor onalslicesof ratbrain. ................................57 5.1ExamplevMFdistributions( =1,5,10,15,25).Alldistributions havesamemeandirection, .....................62 5.2SamplevMFmixturesforvoxelswithoneandtwobers. .......63 5.3Intrinsicandextrinsicdistance ......................68 5.4Tangentspace ...............................69 5.5Riemannianexponentialmap .......................71 5.6Riemannianlogmap ...........................72 5.7PointsalongthegeodesicbetweentwovMFdistributions .......77 5.8Pointsalongthegeodesicbetweentwosetsofweights. .........79 x

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5.9 S 0 andeigenvectorimages. ........................84 5.10Firstorderneighborhood, N 1 ,andnearest-neighborcliques C 1 C 2 C 3 C 4 ..................................85 5.11HMMFsegmentationofsyntheticdata. .................90 5.12HMMFsegmentationofsyntheticdata. .................90 5.13HMMFsegmentationofsyntheticdata. .................91 5.14HMMFsegmentationofsyntheticdata. .................91 6.1Simplicationoperations. .........................95 6.2Examplecut-graphsfortorus,2-torusandtanglecube .........100 6.3Stretchminimizing(left)andconformalparameterizat ion(right). ..101 6.4Texture-mappedsegmentationofspinalcord(left)andb rain(right). .102 xi

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DENOISING,SEGMENTATIONANDVISUALIZATIONOFDIFFUSION WEIGHTEDMRI By TimE.McGraw August2005 Chair:BabaC.VemuriMajorDepartment:ComputerandInformationSciencesandEn gineering Despiteitsapparentsuccess,DiusionTensorMRI(DT-MRI) hassignicant shortcomingswhenthetissueofinteresthasacomplicateds tructure.Thisis duetotherelativelysimpletensormodelthatassumesaunid irectional{ifnot isotropic{localstructure.AsamoreviablealternativeTu chetal.haveproposed todothedataacquisitionsuchthatthediusionsensitizin ggradientdirections samplethesurfaceofasphere.Inthishighangularresoluti ondiusionimaging (HARDI)method,onedoesnothavetoberestrictedtothetens ormodel.Instead, itispossibletocalculatediusioncoecientsindependen tlyalongmanydirections. Thisimagingtechniquecanrevealwhitematterbercrossin gswhichwouldnotbe apparentinDTimages. xii

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Inthispaper,wepresentanovelvariationalformulationfo rrestoringthe HARDIdataandvisualizingthebersfromthisrestoreddata .Thisformulation involvessmoothingsignalmeasurementsoverthespherical domainandacrossthe 3Dimagelattice.Thesmoothingonthespheresateachlattic epointisachieved usingrstandsecondordersmoothnessconstraints,andacr ossthelatticeviaa totalvariationnormbasedscheme.Forthesmoothingproble monthesphere, weusetheniteelementmethod(FEM).Unlikethereportedwo rkonspherical harmonicbasisexpansionofthediusivityfunctiononthes phere,theFEMbasis functionshavelocalsupportandarebettersuitedforprese rvingdetailsinthe data.Tovisualizetheberpaths,theprobabilityvaluesfo rwatermoleculesto moveaparticulardistancealongdierentorientationswer ecalculatedusinga Laplaceseriesexpansionoftheseprobabilities.Wecomput etheShannonentropy ofthisdistributiontocharacterizetheanisotropyofdiu sion,withhigherentropy correspondingtoloweranisotropy.Theselocaldistributi onsarealsousedto computeavectorquantitycalledexpecteddirection.Surfa cesrenderedusingcolors correspondingtoexpecteddirectionrevealanisotropyand berdirectioninthe imagedtissue.Further,examplesarepresentedtodepictth eperformanceofthe HARDIdatarestorationandvisualizationschemesonratbra in,spinalcord,and syntheticdatasets. xiii

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CHAPTER1 INTRODUCTION Fundamentaladvancesinunderstandinglivingbiologicals ystemsrequire detailedknowledgeofstructuralandfunctionalorganizat ion.Thisisparticularlyimportantinthenervoussystem.Understandingfunda mentalstructural relationshipsisessentialtothedevelopmentandapplicat ionoftherapiestotreat pathologicalconditionssuchasdiseaseorinjury. Observingthedirectionaldependenceofwaterdiusioncan allowustoinfer structuralinformationaboutthesurroundingtissue.Whit ematterberbundles presentabarriertodiusion,causingrelativelyhighdiu sivityalongtheber direction,andlowerdiusivityacrosstheber. MRmeasurementscanbemadesensitivetothetranslationald iusionof watermoleculesbytheutilizationofmagneticeldgradien ts[ 78 ].Ingeneral, thesignalacquireddependsonthestrengthandthedirectio nofthesediusion sensitizinggradients.Repeatedmeasurementsofwaterdi usionintissuewith varyinggradientdirectionsprovideameanstoquantifythe levelofanisotropy aswellastodeterminethelocalberorientationwithinthe tissue.Inaseries ofpublications,Basserandcolleagues[ 8 7 9 ]haveformulatedanewimaging modalitycalled\diusiontensorMRI(DT-MRI)"thatemploy sasecondorder, positivedenite,symmetricdiusiontensortorepresentt helocaltissuestructure. 1

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2 Theyhaveproposedseveralrotationallyinvariantscalari ndicesthatquantify dierentaspectsofwaterdiusionobservedintissue,simi lartodierent\stains" usedinhistologicalstudies[ 5 ].Underthehypothesisthatthedirectionalongwhich thediusioncoecientislargestwillyieldthelocalbero rientation,onecan determinethedirectionalityofneuronalberbundles.Thi sfacthasbeenexploited togenerateber-tractmapsthatyieldinformationonstruc turalconnectionsin human[ 9 47 53 22 ]aswellasratbrains[ 59 94 88 57 56 ]andspinalcords[ 87 ]. Figure1.1:OrientationalheterogeneityinDTI(left),and HARDI(right). DT-MRIhassignicantshortcomingswhenthetissueofinter esthasacomplicatedgeometry.Thisisduetotherelativelysimpletens ormodelthatassumes aunidirectional{ifnotisotropic{localstructure.Inthe caseoforientationalheterogeneity,DT-MRItechniqueislikelytoyieldincorrect berdirections,and articiallylowanisotropyvalues.Thisisduetoviolation oftheassumptionof Gaussianprobabilitymodelcharacterizingthediusionim plicitinDTI.Inorderto overcomethesedicultiesseveralapproacheshavebeentak en.Q-spaceimaging,a techniquecommonlyusedtoexamineporousstructures[ 16 ],hasbeensuggestedas

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3 apossiblesolution[ 91 ].Howeverthisschemerequiresstronggradientstrengthsa nd longacquisitiontimes[ 6 ],orsignicantreductionintheresolutionoftheimages. AsamoreviablealternativeTuchetal.haveproposedtodoth eacquisitionsuch thatthediusionsensitizinggradientssamplethesurface ofasphere[ 85 84 ].In thishighangularresolutiondiusionimaging(HARDI)meth od,onedoesnothave toberestrictedtothetensormodel.Instead,itispossible tocalculatediusion coecientsindependentlyalongmanydirections.Thismeth od doesnot require morepowerfulhardwaresystemsthanthatrequiredbyDT-MRI .Severalgroups havealreadyperformedHARDIacquisitionsinclinicalsett ingsandhavereported 43to126dierentdiusionweightedimagesacquiredin20to 40minutesoftotal scanningtime[ 34 84 46 ]indicatingthefeasibilityofthehighangularresolution schemeasaclinicaldiagnostictool.SincetheHARDIdataac quisitionisvery nascent,notmanytechniquesofprocessingtheHARDIdataha vebeenreported inliterature.Inthefollowing,wewillbrieryreviewthefe wveryrecentlyreported techniquesofHARDIdatadenoising,whichmustbedoneprior tofurtheranalysis orvisualization. 1.1 DiusionMRI RecentlyMRmeasurementshavebeendevelopedtomeasurethe tensorof diusion.Thisprovidesacompletecharacterizationofthe restrictedmotionof waterthroughthetissuethatcanbeusedtoinfertissuestru ctureandhence bertracts.Inaseriesofpapers,Basserandcolleagues[ 10 8 7 4 9 70 ]have discussedindetailgeneralmethodsofacquiringandproces singthecomplete

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4 apparent-diusion-tensorofMRmeasuredtranslationalse lf-diusion.They showedthatdirectlymeasureddiusiontensorscouldberec astinarotationally invariantformandreducedtoparametricimagesthatrepres enttheaverage rateofdiusion(tensortrace),diusionanisotropy(rela tionshipofeigenvalues), andhowthediusionellipsoid(eigenvaluesandeigenvecto rs)canberelatedto thelaboratoryreferenceframe.Theparametricimages[ 4 69 ]ofvolumeratio, fractionalanisotropy,andlatticeanisotropyindexrepre sentscalarmeasuresof diusionthatareindependentofthelabreferenceframeand subjectorientation. Therefore,thesemeasurescanbeusedtocharacterizetheti ssuepathology,e.g., ischemia,independentofthespecicframeofreferenceuse dtoacquiretheimages. Thedevelopmentofdiusiontensoracquisition,processin g,andanalysismethods providestheframeworkforcreatingbertractmapsbasedon thiscomplete diusiontensoranalysis[ 22 47 53 59 ].Thishasbeenusedtoproducebertract mapsinratbrains[ 59 94 ]andtomapbertractsinthehumanbrain[ 47 ];then, therststepsweretakentorelatethisstructuralconnecti vitytofunction[ 22 ]. Thedirectionalpropertiesofdiusioncanbecharacterize dbyadiusion tensor,a3 3symmetricmatrixofrealvalues.Inordertocalculatethe6 independentcomponentsofthetensor,thesubjectisimagedin7d ierentdirections withseveralmagneticeldstrengths.Therelationship S = S 0 exp( P ij b ij D ij ) allowsthediusiontensor, D ,andtheT-2weightedimage S 0 tobecalculated giventhesamples S anddiusionweightingfactor, b .Previousworkhasconcentratedonsmoothingtheeldofeigenvectorsof D .Morerecentwork[ 20 ]has formulatedregularizationtechniquesforthetensoreldi tself,evenconstrainingthe

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5 resultingtensorstobepositive-denite.Wehaveprevious lytakentheapproachof smoothingtheobservedvector-valuedimage S priortocalculating D Insummary,theanisotropyofwatertranslationaldiusion canbeused tovisualizestructureinthebrainandprovidesthebasisfo ranewmethodof visualizingnervebertracts.Initialresultshavebeenve ryencouragingandsuggest thatthisapproachtobermappingmaybeappliedtoawideran geofstudiesin livingsubjects.However,itisessentialtooptimizetheac quisitionandprocessing algorithmsforbertractmappingandvalidatetheresultsr elativetoknown measuresofbertracts. 1.1.1 OverviewofDiusion Randommolecularmotion(Brownianmotion)cancausetransp ortofmatter withinasystem.Withinavolumeofwater,moleculesfreelyd iuseinalldirections.Thewaterabundantinbiologicalsystemsisalsosubj ecttosuchstochastic motion.Thepropertiesofthesurroundingtissuecanaectt hemagnitudeof diusion,andthedirectionalpropertiesaswell. Tissuecanformabarriertodiusion,restrictingmolecula rmotion.Within anorientedstructure,suchasabundleofaxonalbers,diu sioncanbehighly anisotropic.Thewhitematterofthebrainandspinalcordis characterizedby manysuchbundles. Diusionhasthepropertyofantipodalsymmetry:diusivit yisequalin oppositedirections.Thedirectionalpropertiesofdiusi oncan,insomecases,

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6 1 e 1 2 e 2 3 e 3 Figure1.2:Diusionellipsoid beapproximatedbyatensor.Thediusiontensor, D ,isasymmetric,positivedenite3 3matrix.Wewillmakeuseoftheeigenvaluesandeigenvector softhis tensor,sortingtheeigenvalues( 1 ; 2 ; 3 )fromlargesttosmallest,andlabelling thecorrespondinguniteigenvectors( e 1 ; e 2 ; e 3 ).Theeigenvaluesrepresentthe magnitudeofdiusioninthedirectionoftheircorrespondi ngeigenvector.For isotropicdiusion 1 = 2 = 3 .Theeigenvectorcorrespondingtothedominant eigenvalueiscommonlyreferredtoastheprincipaldiusio ndirection(PDD).A popularrepresentationfordepictinganisotropicdiusio nisthediusionellipsoid. Thisellipsoidistheimageoftheunitsphereunderthetrans formationdened bythetensor, D .Theeigenvectorsof D formanorthogonalbasis,representing theorientationoftheellipsoid.Thelengthofeachaxisoft heellipsoidisthe correspondingeigenvalue.Forisotropicdiusion,thedi usionellipsoidisasphere.

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7 Surroundingaxonalnervebersisaninsulatingmyelinshea th,whichis restrictivetothediusionofwatermolecules.Alossofblo odsupply(ishemia), suchaswouldoccurduringstroke,causesphysiologicalcha ngesinthenerve berswhichmaybedetectedaschangeinanisotropy.Disease s,suchasMultiple Sclerosis,whichresultindemyelinationofnerveberscan bedetectedaswell. 1.1.2 OverviewofMRImaging InthissectionwewillpresentabriefoverviewoftheMRIacq uisitionprocess. AdetailedtreatmentofthesubjectwasdonebyHaackeetal.[ 42 ]. Theprotonsinthenucleiofatomsaligntheiraxisofspinwit hthedirection ofanappliedmagneticeld.Themagneticeldalsoinducesa wobble,known asprecession,inthespinoftheprotons.Thisfrequency,th eresonantfrequency, isproportionaltothestrengthoftheappliedeld.Forprot onstheresonance frequencyliesintheRFrange. IntheMRIinstrument,astaticeld B 0 isappliedthroughouttheimaging process.Thedirectionofthiselddenestheaxialdirecti onoftheimage.Protons willabsorbenergyfromanRFpulseoftheresonancefrequenc yandtipaway fromthedirectioninducedby B 0 .Theamountoftipisproportionaltothepulse duration.TheRFpulsealsocausestheprotonstoprecessinp hasewitheachother. Thispulseiscalledthe B 1 eld. Whenthe B 1 transmitteristurnedo,theabsorbedenergyattheresonan t frequencyisre-emittedbytheprotons.Thisoccursasthesp ins,tippedby B 1 returntotheirprevious B 0 alignment.Thetimeconstantassociatedwiththis

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8 exponentialprocessisknownasthe T 1 relaxationtime.Theprotonsprecessions alsodephaseexponentiallywithtimeconstant T 2 .Thenalimagecontrastis inruencedbystrength,widthandrepetitiontimeoftheRFpu lsesinthe B 1 signal. Byspatiallyvaryingtheintensitiesof B 0 and B 1 ,positioninformationis encoded.Forinstance,speciallydesignedmagnetsaddagra dienteldto B 0 .This causestheprotonresonancefrequencytobeafunctionofaxi alposition.The frequencyof B 1 canthenbechosentotipprotonswithinachosenslice. Toencode x;y positionwithinaslice,twoadditionalgradienteldsare employed.Therstgradient, G y ,ispulsed,causingaphasevariation,justasin T 2 relaxation.Thephasevariationisafunctionofpositionin theydirection.A perpendiculargradient, G x ,isthenapplied,changingresonancefrequenciesinthe xdirection.A2DFouriertransformreconstructstheimageo feachslicefromthe datainthespatialfrequencydomain. 1.1.3 DiusionWeightedImages Bycarefullydesigninggradientpulsesequences,themeasu redsignalfrom protonsinwatermoleculesundergoingdiusioncanbeatten uated.Therst gradientpulseinducesaknownphaseshiftinprotonprecess ion.Aftersomedelay, asecondgradientpulseisapplied,inducingtheoppositeph aseshift.Protonswhich havenotmovedbetweenthetwogradientpulsesarereturnedt otheirprevious phase.Protonsbelongingtomoleculeswhichhavechangedlo cationhavesome netchangeinphase,changingtheir T 2 relaxationtime.Inthisworkwewillnot

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9 describethegradientpulsesequence G ( t )indetail,butwewillbemakinguseof severalquantitiesrelatedtothediusionsensitizinggra dient. Thevector g isthedirectionofthegradient.Thescalarquantity b ,called thediusionweightingfactororb-value,isafunctionofth egradientmagnitude, andthetiminganddurationofthepulsesequence.Thevector q isrelatedtothe integralof G ( t )overtime. Theattenuationofthediusionweightedimage(DWI), S ,relativetoanideal image S 0 acquiredwithnodiusionweightingisgivenby S = S 0 exp( bd )(1.1) where d istheapparentdiusioncoecient(ADC).Thisrelationisc alledthe Stejskal-Tannerequation[ 78 ].Wesay S 0 isanidealimagesincethegradients, G x and G y ,havesomenonzerodiusionsensitivity.Wecannotmeasure S 0 ,howeverit canbeestimatedfromtheStejskal-Tannerequationifweacq uireatleast2images withdierentb-values.TheimageSwillchangeas g changes.Themotivated readermayrefertotheworkofHaackeetal.[ 42 ]formoredetailsonthephysicsof acquisition. 1.1.4 DiusionTensorMRIAcquisition Wecanconstructamorecompletemodelofdiusionbyacquiri ngDWIs withmultiplegradientdirections.ByassumingaGaussianm odeldiusionwecan describethisdiusionwitharank-2tensor.Anotherformof theStejskal-Tanner

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10 equationrelatestheimages S and S 0 to g andtheapparentdiusiontensor, D S = S 0 exp( bg T Dg )(1.2) WhencomputingD,itiscommontorewriteEquation( 1.2 )inlog-linearized formas ln( S S 0 )= 3 X i =1 3 X j =1 b i;j D i;j (1.3) where b i;j arecomputedfrom b and g Thisimagingprocessmustbeperformedwithatleast7noncop lanargradient directionsinordertofullygenerateadiusiontensorimag eandrecover S 0 Multiplesamples,usually3or4,withvaryinggradientstre ngthsaretakenforeach gradientdirection.FormDWIswehavethesystem 26666664 ln S 1 ... ln S m 37777775 = 26666664 1 b 1xx b 1yy b 1zz 2 b 1xy 2 b 1yz 2 b 1xz ... ... ... ... ... ... ... 1 b mxx b myy b mzz 2 b mxy 2 b myz 2 b mxz 37777775 266666666666666666666664 ln S 0 D xx D yy D zz D xy D yz D xz 377777777777777777777775 (1.4) Theoverconstrainedlinearsystem,Eq.( 1.4 ),issolvedfor S 0 andtheelementsof thesymmetrictensor D byaleastsquareslinearregression.Alternatively,wemay estimate D bynonlinearregression.

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11 FortheGaussianmodelofdiusion,thediusiontensorisre latedtothe displacementprobabilityofawatermolecule.Foraxed(sm all)timeconstant wecandescribetheprobabilitythatamoleculeundergoesdi splacement, r ,bya Gaussianwithmeanzeroandcovariance2 t D 1.1.5 HighAngularResolutionDiusionImaging TheHARDIprocessproceedsbyacquiringdiusionweightedi mageswith manydiusionencodinggradientdirections,eectivelysa mplingasphericalshell oftheq-space.Itisdesiredthatthissamplingbeuniform,o rnearlyso.The gradientdirectionforeachimageisusuallychosentocorre spondtotheverticesof anicosahedronwhichhasbeenrepeatedlysubdivided.Since theprocessofdiusion isknowntobesymmetric,weneedonlysampleonehemisphereo fq-space.Inthe caseofourdata,weconsider81or46gradientdirections.Th egradientdirections correspondtotheverticesofthesurfaceshowninFigure 2.2 .Inadditionalow b-value(smalldiusionencodinggradientmagnitude)imag eisacquired. InHARDIwecanconsideranotherformofEquation( 1.1 ) S ( ; )= S 0 exp( bd ( ; ))(1.5) where and arethepolarandazimuthalangledescribing g Therandomprocessofdiusionofwatermoleculesisdescrib edbythe diusiondisplacementPDF p t ( r ).Thisistheprobabilitythatagivenmolecule hasadiusiondisplacementof r aftertime t .Therelationbetweenthemeasured

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12 image,andthediusiondisplacementPDFisgivenbyCallagh an[ 16 ]as p t ( r )= Z S ( q ) S 0 exp( 2 i q r ) d q (1.6) ThisissimplytheforwardFouriertransformof S ( q ) S 0 .Itisthemodesof p t ( r )that aretakentobetheunderlyingberdirections. Analternativeimagingscheme,calleddiusionspectrumim aging(DSI), acquiresimagebysamplingaCartesiangridinq-space.TheF ouriertransform of S ( q ) S 0 canthenbeevaluateddirectlytoyield p t ( r ).Theimageacquisitiontime for(DSI)isquitehigh,sincewemustsampleouttolargemagn itudesof q to accuratelyreconstruct p t ( r ). 1.2 Restoration ProcessingofHARDIdatasetshasreceivedincreasedattent ionlatelyand afewresearchershavereportedtheirresultsinliterature .Theuseofspherical harmonicexpansionshavebeenquitepopularinthiscontext sincetheHARDI dataprimarilyconsistsofscalarsignalmeasurementsonas pherelocatedat eachlatticepointona3Dimagegrid.Tuchetal.[ 85 84 ]developedtheHARDI acquisitionandprocessingandlaterFrank[ 34 ]showedthatitispossibletouse thesphericalharmonicsexpansionoftheHARDIdatatochara cterizethelocal geometryofthediusivityproles.Inhisworkhowever,the reisnodiscussion ofdenoising/restoringtheHARDIsignalmeasurements,whi chisessentialfor subsequentprocessingandinterpretation.Chenetal.[ 21 ]ndaregularized sphericalharmonicexpansionbysolvingaconstrainedmini mizationproblem.

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13 Howevertheexpansionisatruncatedsphericalharmonicexp ansionoforder4,and hencethesolutioncanrepresentonly2berdirectionswith inavoxel.Jansons andAlexander[ 46 ]describedanewstatisticwhichwascalledpersistentangu lar structurethatwasobtainedfromthesamplesofa3Dfunction ,inthiscasethe displacementofwatermoleculesineachdirection.Thegoal intheirworkwasto resolvevoxelscontainingoneormorebers.However,there wasnodiscussionon howtorestorethenoisyHARDIdatapriortoresolutionofthe berpaths. Imagedenoisingcanbeformulatedusingvariationalprinci pleswhichinturn requiresolutionstoPDEs.ThetechniquedescribedbyChana ndShen[ 18 ]has beenemployedinmedicalimagingliteratureforsmoothingD T-MRIdatasets[ 24 ]. RecentworkbyTschumperleandDeriche[ 82 ]dealswithsmoothingtheeigenvector eldofthediusiontensorscomputedfromtherawechointen sityimagedata. Severalothermethodshavebeendevelopedforrestoringthe DT-MRIdatasets butmostofthemuseexistingrestorationschemesforscalar orvector-valued functionsfromimageprocessingliterature.Morerecently ,matrixvaluedfunction restorationwasintroduced[ 20 ]andappliedtotherestorationofnoisydiusion tensorelds.Aninterestingalternativetothevariationa lprincipleapproachwas takenbyWeickertandBrox[ 92 ]wherein,theydevelopedananisotropicdiusion lterthatachievesan\edge"preservingsmoothingofthepo sitivesemi-denite tensor-valuedimage.Alloftheaforementionedmethodssha reonecommonality i.e.,theyusealinearizedStejskal-Tannerequationasthe dataacquisitionmodel whichdoesnotaccuratelyrerectthephysicsofthedataacqu isition.Inrecently reportedwork,byWangetal.[ 90 89 ]analternativeapproachwhichovercomes

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14 thisweaknessbydirectlyestimatingasmoothpositivesemi denitetensoreld fromtherawdatausingtheactualnonlinearmono-exponenti almodelcharacterized bytheStejskal-Tannerequation[ 78 ]wasreported. 1.3 Tractography Waterinthebrainpreferentiallydiusesalongwhitematte rbers.By trackingthedirectionoffastestdiusion,asmeasuredbyM RI,noninvasiveber trackingofthebraincanbeaccomplished.InthecontextofD TIdata,ber tracksestimatedinreportedliteraturewereobtainedbyre peatedlystepping inthedirectionoffastestdiusion.Thedirectionalongwh ichthediusion isdominantcorrespondstothedirectionofeigenvectorcor respondingtothe largesteigenvalueofthetensor D .Thisapproachwastakenbymanyresearchers [ 23 59 93 71 24 67 87 56 ].Mostofthesetechniquesdoincorporatesome regularizationintheirstreamlineestimationschemesino rdertogeneratethe berpathways.Techniquesthatarequitedistinctfromthei deaofstreamline generationhavealsobeenreportedinliterature.Batchelo retal.[ 11 ]reporteda bertractmappingschemewhereintheyproduceamapindicat ingtheprobability ofaberpassingthrougheachlocationintheeld.However, nodiscussionon howtoestimatetheactualberswasdescribed.Analternati veapproachbased onsequentialimportancesamplingandregularizationtech niqueswasproposedby Bjornemoetal.[ 13 ],whichallowedpathstooriginatefromasinglelocationan d branchoutandproducedaprobabilitydistributionofthepa ths.O'Donnelletal. [ 61 ]describewaystoestimatetheconnectivityfromthegivent ensoreld.One

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15 approachtheysuggestedwastoestimatethegeodesicsinthe locallywarpedspace wherethewarpingisderivedfromthelocaltensor. InthecontextofHARDIprocessing, suggestions forberpathcomputation havebeenreportedintheliterature[ 34 46 62 ].Frank[ 34 ]describedaspherical harmonictransformrepresentationoftheHARDIdataandpoi ntedoutthat,due totheantipodalsymmetryonthesphere,onlytheeventermsi nthespherical harmonictransformcontributedtowardtherepresentation .Anynonzerooddterms wouldbeduetoartifactsinimagingsuchasnoise.Theberpa thwayswerenot computedinhisreportedwork;however,itwassuggestedtha tonecouldestimate thedirectionofthecrossingbersbyusingamulti-tensorm odelandnding theprincipaldiusiondirectionsofthismultitensorexpa nsion.Theapproachof usingHARDIdataandthenrevertingtoamulti-tensorapproa chseemssomewhat defeating.Theapproachdescribedby OzarslanandMareci[ 62 ]expressesthe diusivityfunction,afunctiondenedonthesphere,asage neralized(higherrank) Cartesiantensorandthenestimatestheprobabilitydistri butionofwatermolecule displacementoveralldirectionsusingtheFFT(fastFourie rtransform)ofthe signalmeasurementseitheronthesphereoraninterpolated Cartesiangrid.These distributionprolesarethensharpenedtodisplaytheposs ibleorientationofthe berswithcomplexlocalgeometries. 1.4 Anisotropy InDTItherearemanyscalarmeasureswhichcharacterizethe anisotropyof thediusionphenomenon.Mostusefularetherotationallyi nvariantmeasures,

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16 thoseindependentofthelaboratoryreferenceframe,sucha svolumeratio,rational anisotropyandfractionalanisotropy[ 5 ].Thesecanbecomputedintermsofthe eigenvaluesofthediusiontensor.Thesescalarindicesal onehavebeenusefulin clinicalstudies[ 79 ],soitisimportanttondanalogousmeasuresofanisotropy forHARDI.ThemeasuresforHARDIshouldovercometheweakne ssesofthe tensormodelwithregardtoregionsofcrossingbers.Inthe caseoforientational heterogeneity,DT-MRItechniqueyieldsincorrectberdir ectionsandarticially lowanisotropyvalues[ 64 ]. Frank[ 34 ]usesthecoecientsofthesphericalharmonicexpansionof the apparentdiusioncoecienttoquantifyanisotropy.The0t h,2ndand4thorder coecientsdescribeisotropicdiusion,single-berdiu sionandtwo-berdiusion respectively.Voxelscanbeclassiedusingtherelativema gnitudesofthese coecients. OzarslanandMareci[ 62 ]useahighranktensortodescribethediusion processandgeneralizetherank-2tensortracetohigherran ktensors.Thevariance ofthediusivityisthenexpressedintermsofthegeneraliz edtrace,andatransfer functionmapsthisvaluetothe[0 ; 1]range.Thisquantityiscalled"generalized anisotropy"(GA).Itwasshowntobeusefulindetectinganis otropyatber crossingswhicharenotdetectablebyFA. 1.5 OverviewofOurModelingScheme Therearetwomajorsubtasksinachievingthegoalofbertra ctmappingin thebrain.Firstly,therawnoisyDWIdatahavetoberestored andsecondly,the

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17 bertrackshavetobemappedfromthisrestoreddata.Thevar iationalprinciple involvessmoothing S valuesoverthesphereandacrossthe3Dimagelattice. Thekeyfactorthatcomplicatesthisproblemisthatthedoma inofthedataat eachvoxelinthevoxellatticeisasphere.Onemayusethelev el-settechniques developedbyTangetal.[ 80 ]toachievethissmoothing;however,whendatasets arelarge,itbecomescomputationallyimpracticaltoapply thelevel-settechnique ateachvoxelindependentlytorestorethesescalar-valued measurementsonthe sphere.Wearriveatacomputationallyecientsolutiontot hisproblembyusing theniteelementmethod(FEM)onthesphereandchoosingloc albasisfunctions forthedatarestoration.Unlikethereportedworkonspheri calharmonicbasis expansionofthediusivityfunctiononthesphere[ 33 62 ],theFEMbasisfunctions havelocalsupportandarebettersuitedforpreservingdeta ilsinthedata. Fromthedenoiseddatawewillcomputeaprobability, p t ( ; ),ofmolecular diusionoversphericaldirections.TheShannonentropyof thisdistributionwillbe usedtoquantifyanisotropy.

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CHAPTER2 DENOISING Wepresentanovelandeectivevariationalformulationtha twilldirectly estimateasmoothsignal S ( ; )andtheprobabilitydistributionofthewater moleculedisplacementoveralldirections p ( ; ),giventhenoisymeasurement ^ S ( ; )= S ( ; )+ (2.1) where ^ S isthesignalmeasurementtakenonaunitsphereoverall( ; )and isGaussiannoise. Theactualnoiseiscomplex-valuedRiciannoiseontheraw S imagesprior toFouriertransform.Sinceourdatahaverelativelyhighsi gnal-to-noiseratio,the noiseiswellmodeledbyadditiveGaussiannoiseonthemagni tudedata[ 40 ]. 2.1 TVNormMinimizationofS(x,y,z) Thetotalvariation(TV)normintroducedinFatemietal.[ 50 ]isapopular normusedforimagerestoration.Inthecaseofscalarvalued functions,minimizing theTV-normamountstominimizationofthe L 1 normoftheimagegradient. MinimizingtheTVnormproducesverysmoothimageswhileper mittingsharp discontinuitiesbetweenregions[ 50 ].Sincemostimagesconsistofsmoothregions separatedbydiscontinuities(edges),thisisausefulmode lforimagedenoising. 18

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19 2.1.1 ScalarTVNormMinimization TheTVnormforscalarimages, I ( x )isgivenby TV n; 1 ( I ( x ))= Z n jr I ( x ) j d x ; n R n (2.2) Thisnormrepresentsoscillation.Inourcase,imagenoisei sconsideredtobe \wrinkling"ofthesurfacedescribedbytheimage. Atdiscontinuities,theweakderivative DI ( x )tocalculatetheTVnorm.For apiecewisecontinuousfunction,theTVnormisthenthesumo ftheTVnorm ofeachcontinuouspieceplusthesumoftheabsolutevalueso fthe"jumps." Forexample,functions f 1and f 2inFigure 2.1 havethesameTVnorm.The oscillatoryfunction, f 1,hasahigherTVthanthefunctionwithadiscontinuity, f 3. Sincemostimagesconsistofpiecewisesmoothregionssepar atedbydiscontinuities (edges),thisisausefulmodelforimagedenoising. f 2 f 1 f 3 b a Figure2.1:TV( f 1) > TV( f 2)=TV( f 3)

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20 TheEuler-LagrangeconditionfortheminimizationofEquat ion( 2.2 ),written ingradient-descentform,is @I ( x ;t ) @t =div( r I ( x ;t ) jr I ( x ;t ) j ) I ( x ; 0)= I 0 ( x )(2.3) ThegradientdescentsolutiontotheTV-normminimizationp roblemwill convergeveryslowly.Wewillpresentanecientnumericalt echniquebelow. 2.1.2 VectorTVNormMinimization BlomgrenandChan[ 14 ]introducedtheTV n;m normforvectorvaluedimages. TV n;m ( I ( x ))= vuut m X i =1 [TV n; 1 ( I i )] 2 (2.4) For m =1,Equation( 2.4 )reducestothescalarTVnorm( 2.2 ).TheEulerLagrangeconditionfortheminimizationofEquation( 2.4 ),writteningradient descentformis @I i ( x;t ) @t = TV n; 1 ( I i ) TV n;m ( I ) r ( r I i jr I i j ) I ( x; 0)= I 0 ( x ) (2.5) Thiswasshowntobequiteeectiveforcolorimages,preserv ingedgesinthecolor spacewhileattenuatingnoise.However,formuchlargerdim ensionaldatasetsas intheworkproposedhere,theColorTVmethodbecomescomput ationallyvery intensiveandthusmaynotbethepreferredmethodinsuchapp lications.

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21 Bytreatingeachcomponentofthevectoreldasanindepende ntscalar eld,wecanproceedbysmoothingchannel-by-channel.This can,however,result inalossofcorrelationbetweenchannelsasedgesineachcha nnelmaymove independentlyduetodiusion.Topreventthis,theremustb ecouplingbetween thechannels. 2.2 VariationalPrinciple Thevariationalprincipleweproposeforestimatingasmoot h S ( x ;; )from theinitialdata ^ S ( x ;; )isgivenby min S E ( S )= Z n Z S 2 j S ( x ;; ) ^ S ( x ;; ) j 2 dSd x + Z S 2 jr ( ; ) S j 2 dS + Z S 2 ( S +2 S + S ) dS + Z n g ( x ) jr x S j d x (2.6) wherenistheimagespatialdomain,and S 2 isthesphericalimagedomainateach voxel,and dS denotesintegrationoverthesphere.ThersttermofEquati on( 2.6 ) isadatadelitytermwhichmakesthesolutiontobeclosetot hegivendata.The degreeofdatadelitycanbecontrolledbytheinputparamet er .Thesecond andthirdtermsarerstandsecondorderregularizationcon straintsenforcing smoothnessofthedataoverthesphericaldomainateachvoxe l.Thefourthtermis anotherregularizationtermwhichcausesthesolutiontobe smoothoverthespatial domain(the3Dvoxellattice).Tomakethesolutionoftheden oisingproblem feasible,weproposetoalternatelysolvethespatialandsp hericalsmoothing problems.

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22 2.3 SpatialLatticeSmoothingof S ( x ) Smoothingtherawvector-valueddata, S ( x ),isposedasavariationalprinciple involvingarstordersmoothnessconstraintonthesolutio ntothesmoothing problem.Notethatthedataateachvoxelisalargesetof S measurementsover asphereofdirectionsandcanbeassembledintoavectorafte rthesmoothingon thesphericalcoordinatedomainhasbeenaccomplished.Wep roposeaweighted TV-normminimizationforsmoothingthisvector-valuedima ge S .Thevariational principleforestimatingasmooth S ( x )isgivenby min S E ( S )= Z n ( g ( x ) m X i =1 jr S i j + 2 m X i =1 ( S i ^ S i ) 2 ) d x (2.7) where,nistheimagedomainand isaregularizationfactor.Therstterm hereistheregularizationconstraintonthesolutiontohav eacertaindegreeof smoothness.Thesecondterminthevariationalprinciplema kesthesolution faithfultothedatatoacertaindegree.Wehavepreviously[ 56 ]usedthecoupling function g ( x )=1 = (1+ FA ( x ))forsmoothingdiusiontensorimages,whereFA isthefractionalanisotropydenedbyBasserandPierpaoli [ 10 ].Thisselection criterionpreservesthedominantanisotropicdirectionwh ilesmoothingtherest ofthedata.Notethatsinceweareinterestedinthebertrac tscorrespondingto thestreamlinesofthedominantanisotropicdirection,iti sapttochoosesucha selectiveterm.ForHARDIdata,wemayreplaceFAwithgenera lizedanisotropy [ 62 ]sinceitcanbecomputedpriortotheODF.

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23 HerewehaveusedadierentTVnormthantheoneusedbyBlomgr enand Chan[ 14 ].TheTV n;m normisan L 2 normofthevectorofTV n; 1 normsforeach channel.Weusethe L 1 norminstead. Thegradientdescentformoftheaboveminimizationisgiven by @ S i @t =div g r S i kr S i k (S i ^ S i ) i =1 ;:::;m @S i @n j @ n R + =0 and S ( x ;t =0)= ^ S ( x )(2.8) TheuseofamodiedTV-norminEquation( 2.7 )resultsinaloosercoupling betweenchannelsthantheuseofthetrueTV n;m normwouldhave.Thisreduces thenumericalcomplexityofEquation( 2.8 )andmakessolutionforlargedatasets feasible. NotethattheTV n;m normappearsinthegradientdescentsolution,Equation ( 2.5 )ofthevector-valuedminimizationproblem.Considerthat ourdatasets consistofupto82images,correspondingto(magneticeld) gradientdirections. CalculatingtheTV n;m normbynumericallyintegratingoverthe3-dimensionaldat a setateachstepofaniterativeprocesswouldhavebeenprohi bitivelyexpensive.In contrast,usingthemodiedTV-normsuggestedearlierlead stoamoreecient solution. 2.3.1 Fixed-PointLagged-Diusivity Sincethe m Equations( 2.8 )arecoupledonlythroughthefunction g ,wecan dropthesubscripton S withnoambiguity(laterthesubscriptwillrefertospatial coordinates.)Inthissectionwewilldiscussthenumerical solutionforeachchannel,

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24 S ,ofthevector-valuedimage S .Equation( 2.8 )isnonlinearduetothepresenceof jr S j inthedenominatoroftherstterm.WelinearizeEquation( 2.8 )byusingthe methodof\lagged-diusivity"presentedbyChanandMulet[ 17 ].Byconsidering jr S j tobeaconstantforeachiteration,andusingthevaluefromt heprevious iterationwecaninsteadsolve 1 jr S t j ( r g r S t + g r 2 S t +1 )+ ( S t +1 S 0 )=0(2.9) Herethesuperscriptdenotesiterationnumber.Tomakethe xedpointiteration clearer,werewriteEquation( 2.9 )withallofthe S t +1 termsontheleft-handside r 2 S t +1 + jr S t j g S t +1 = jr S t j S 0 + r g r S t g (2.10) 2.3.2 DiscretizedEquations TowriteEquation( 2.10 )asalinearsystem( A S t +1 = f t ),discretizethe Laplacianandgradientterms.Usingcentraldierencesfor theLaplacianwehave r 2 S t +1 = S t +1 x 1 ;y;z + S t +1 x;y 1 ;z + S t +1 x;y;z 1 6 S t +1 x;y;z + S t +1 x +1 ;y;z + S t +1 x;y +1 ;z + S t +1 x;y;z +1 (2.11) Denethestandardcentraldierencestobe x S = 1 2 ( S x +1 ;y;z S x 1 ;y;z ) y S = 1 2 ( S x;y +1 ;z S x;y 1 ;z ) z S = 1 2 ( S x;y;z +1 S x;y;z 1 )(2.12)

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25 WecanrewriteEquation( 2.10 )indiscreteformusingthedenitionsinEquation ( 2.12 ) S x 1 ;y;z S x;y 1 ;z S x;y;z 1 +(6+ p ( x S t ) 2 +( y S t ) 2 +( z S t ) 2 g ) S x;y;z S x +1 ;y;z S x;y +1 ;z S x;y;z +1 = 1 g ( S 0 p ( x S t ) 2 +( y S t ) 2 +( z S t ) 2 + x g x S t + y g y S t + z g z S t )(2.13) Thisresultsinasparsebandedlinearsystemwith7nonzeroc oecientsperrow.2 6 6 6 6 6 6 6 6 6 6 46+ jr S t j 0 g 0 1 ::: 1 ::: 1 ::: 16+ jr S t j 1 g 1 1 ::: 1 ::: 1 0 16+ jr S t j 2 g 2 1 ::: 1 ::: ::: . . . . . . ::: . .3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4S t +1 0 S t +1 1 ... S t +1 n 33 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 4f t 0 f t 1 ... f t n 33 7 7 7 7 7 7 7 7 7 7 5(2.14) wheretheright-handsideofEquation( 2.13 )isdenotedby f t n .Thematrixin Equation( 2.14 )issymmetricanddiagonallydominant.Wehavesuccessfull yused conjugategradientdescenttosolvethissystem. ThesolutionofEquation( 2.14 )representsonexed-pointiteration.This iterationiscontinueduntil j S t S t +1 j
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26 Thediusion-encodinggradientdirectionsaretakenasthe verticesofa subdividedicosahedron,toachieveanearlyuniformsampli ngofsphericaldirection. Wemapthispiecewiselinearapproximationofaspheretothe planebytakingthe sphericalcoordinates( ; )oftheimaginggradientdirectionastheplanarglobal coordinates( u;v ).Thegradientdirections,andtheirembeddingintheplane are showninFigures 2.2 and 2.3 Figure2.2:HARDgradientdirections( ; )correspondtotheverticesofasubdividedicosahedron. Figure2.3:Twodimensional(u,v)domainforFEM-basedsmoo thingofHARDI data.

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27 Notethat,thedatacanbeseenasheightvaluesdenedonthes phereatthe coordinatesspeciedparametricallyby( u;v ).Thesmoothingwillbeappliedto theseheightmeasurements z ( u;v )usingthesmoothingfunctionalinEquation ( 2.15 ). E p = ZZ n ( ( j z u j 2 + j z v j 2 )+ ( j z uu j 2 +2 j z uv j 2 + j z vv j 2 )) dudv (2.15) Theweightonthemembranetermis andtheweightonthethin-platetermis Oncewehavecomputedasmooth z ( u;v ),theresultwillthenbemappedbackto theimageonthesphere, S ( ; ). Thedataenergyduetovirtualworkofthedataforcesis E d = ZZ n z ( u;v ) f ( u;v ) dudv (2.16) Therestorationateachvoxelisformulatedbytheenergymin imization min S E ( S )= E p ( S )+ E d ( S )(2.17) Weusepolynomialshapefunctions, N i asabasisforthedataoverthedomain ofsphericaldirections. z ( u;v )= n X i =1 q i N i ( u;v )= Nq (2.18) wherethe N isa(1 n )rowvector,and q isacolumnvectorofnodalvariables. 2.4.1 ElementMatrices Wewillsubdividethedomainintoelements,eachwiththeiro wnlocalshape functions.Foreachelement j z ( u;v )= N j ( u;v ) q j (2.19)

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28 for( u;v ) 2 n j .Thelocalenergyfunctionforeachelementisgivenby E j p = ZZ n j ( j z j u j 2 + j z j v j 2 + j z j uu j 2 +2 j z j uv j 2 + j z j vv j 2 ) dudv (2.20) Theglobalenergyisthesumoftheenergiesofeachelement E p = X j E j p (2.21) Theelementstrainvector,givenbyDhattandTouzot[ 27 ],is j = 2666666666666664 z j u z j v z j uu z j uv z j vv 3777777777777775 (2.22) j = 2666666666666664 ( N 1 ) u ::: ( N n ) u ( N 1 ) v ::: ( N n ) v ( N 1 ) uu ::: ( N n ) uu ( N 1 ) uv ::: ( N n ) uv ( N 1 ) vv ::: ( N n ) vv 3777777777777775 q j = Bq j (2.23) wherewehavedened B asthe( n 5)matrixofderivativesofthenodalbasis functions.Wecanthenwritetheelementstrainenergyas E j p = ZZ n j jT D j dudv (2.24)

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29 wherewedene D = 2666666666666664 0000 0 000 00 00 0002 0 0000 3777777777777775 (2.25) Since q j isconstantovereachelementwecanderivetheelementstin ess matrixintermsof D and B asfollows E j p = ZZ n j q jT B jT DB j q j dudv = q jT K j q j (2.26) Wewillmodelthedataconstraintasspringspullingeach z ( u;v )towardsome givenvalue z 0 ( u;v ).Theforceateachpointwillobey f = k ( z z 0 ),where k isthe springconstant. E j d = ZZ n j N j q j k ( N j q j z 0 ) dudv (2.27) E j d = k q jT ZZ n j N jT N j q j dudv + k q jT ZZ n j N jT z 0 dudv (2.28) Wewilldene F jk and f j suchthatthersttermofEquation( 2.28 )is q jT F jk q j and thesecondtermis q jT f j .Wecanthebalancedeformationenergyanddataenergy bysolvingthelinearsystem: ( K j + F jk ) q j = f j (2.29)

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30 2.4.2 LocalElementCoordinates Wenowpresentthecoordinatesystemforthelocalelements. Forlocal elements,triangularelementsareusedwithabarycentricc oordinatesystem ( r;; )sothateachcoordinateisin[0 ; 1]and r =1 u v ( u 0 ;v 0 ) ( u 1 ;v 1 ) ( u 2 ;v 2 ) 24 0 =0 0 =0 r 0 =1 35 24 1 =1 1 =0 r 1 =0 35 24 2 =0 2 =1 r 2 =0 35 Figure2.4:Mappingtobarycentriccoordinates Barycentriclocalcoordinatescanbemappedtoglobalcoord inates. u ( ; )=(1 ) u 0 + u 1 + u 2 v ( ; )=(1 ) v 0 + v 1 + v 2 (2.30) ThemappinginEquation( 2.30 )canberewrittenasalinearsystem. 2664 u v 3775 = 2664 u 1 u 0 u 2 u 0 v 1 v 0 v 2 v 0 3775 2664 3775 + 2664 u 0 v 0 3775 (2.31) DierentiatingEquation( 2.31 )weobtain 2664 du dv 3775 = 2664 @u @ @u @ @v @ @v @ 3775 2664 d d 3775 = J 2664 d d 3775 (2.32)

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31 Table2.1:Gauss-Radauweights i r i wi i i a i 1 0 : 0469100770 0 : 1184634425 0 : 0398098571 0 : 1007941926 2 0 : 2307653449 0 : 2393143353 0 : 1980134179 0 : 2084506672 3 0 : 5 0 : 2844444444 0 : 4379748102 0 : 2604633916 4 0 : 7692346551 0 : 2393143353 0 : 6954642734 0 : 2426935942 5 0 : 9530899230 0 : 1184634425 0 : 9014649142 0 : 1598203766 where J istheJacobianofthetransformationfromglobaltolocalco ordinates.We canconvertintegralsoverthe( u;v )domaintointegralsoverthelocal( ; )domain inthefollowingway: ZZ n j f ( u;v ) dudv = ZZ n j f ( u ( ; ) ;v ( ; ))det( J ) dd (2.33) UsingGauss-Radauquadraturerules,wecanapproximatethe integralin Equation( 2.33 )by 5 X i =1 5 X j =1 wi i wj j f ( u ( j ; i;j ) ;v ( j ; i;j ))det( J )(2.34) where i;j = r i (1 s j ), wj j = a j (1 j ), j ,and wi i aregiveinTable 2.1 Derivativesover( u;v )become @N @u = @N @ @ @u + @N @ @ @u @N @v = @N @ @ @v + @N @ @ @v (2.35) Thepartialderivativesof and withrespectto u and v canbecomputedby invertingtheJacobian 2664 d d 3775 = 2664 @ @u @ @v @ @u @ @v 3775 2664 du dv 3775 = J 1 2664 du dv 3775 (2.36)

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32 J 1 = 1 det( J ) 2664 v 2 v 0 ( u 2 u 0 ) ( v 1 v 0 ) u 1 u 0 3775 (2.37) WeusethefthorderelementshapefunctionsgivenbyDhatta ndTouzot[ 27 ]. Thiselementguarantees C 1 (surfacenormal)continuityacrosstriangles.Thebasis functionsaregivenbyN1= 2(10 15 2+6 3+30 ( + )) N2= 2(3 2 3 2+6 ) N3= 2(3 2 3 2+6 ) N4= 1 2 22(1 +2 ) N5= 2N6= 1 2 22(1+2 ) N7= 2(10 15 2+6 3+15 2 ) N8= 1 2 2( 8 +14 2 6 3 15 2 ) N9= 1 2 2 (6 4 3 3 2+3 ) N10= 1 4 2(2 (1 )2+5 2 ) N11= 1 2 2 ( 2+2 + + 2 ) N12= 1 4 22 + 1 2 32N13= 2(10 15 2+6 3+15 2 ) N14= 1 2 2(6 3 4 3 2+3 ) N15= 1 2 2( 8 +14 2 6 3 15 2 ) N16= 1 4 22 + 1 2 22

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33N17= 1 2 2( 2+ +2 + 2 ) N18= 1 4 2(2 (1 )2+5 2 )(2.38)Thequinticshapefunctionshavenodalvariableswhichcanb ewrittenin termsoflocalorglobalcoordinates q ; = 2666666666666664 z z z z z 3777777777777775 ; q u;v = 2666666666666664 z u z v z uu z uv z vv 3777777777777775 (2.39) whichwecanrelatetoeachotherby q u;v = 26666666666666666664 1000000 u u 000 0 v v 000 000 2 u 2 u u 2 u 000 u v ( u v + u v ) u v 000 2 v 2 v v 2 v 37777777777777777775 q ; (2.40)

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34 2.4.3 GlobalMatrices Wenowwishtoconstructglobalmatricessothattheenergyba lanceoverthe entireFEMmeshisgivenbythelinearsystem Kq = f (2.41) where K isa(6 n 6 n )matrixsincewehave6variablespernode. Wewillconsiderthesimplecaseof2elements.Expandingthe element Equation( 2.29 )intermsofnodalvariablesforelement0,weget 26666664 K 00 ; 0 K 00 ; 1 K 00 ; 2 K 01 ; 0 K 01 ; 1 K 01 ; 2 K 02 ; 0 K 02 ; 1 K 02 ; 2 37777775 26666664 q 00 q 01 q 02 37777775 = 26666664 f 0 0 f 0 1 f 0 2 37777775 (2.42) andlikewiseforelement1 26666664 K 13 ; 3 K 13 ; 2 K 13 ; 1 K 12 ; 3 K 12 ; 2 K 12 ; 1 K 11 ; 3 K 11 ; 2 K 11 ; 1 37777775 26666664 q 13 q 12 q 11 37777775 = 26666664 f 1 3 f 1 2 f 1 1 37777775 (2.43) whereeach q jl isa(6 1)columnvectorofnodalvariables.Weexpandeach K j tobe(6 n 6 n )byinsertingrowsandcolumnsofzeroscorrespondingtoeac hnode ofthemesh.Alsoexpand f j to(6 n 1).Theglobal K and q areobtainedby summingtheexpandedmatricesfromeachelementinthemesh. Forour2element

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35 examplewehave266666666664 K00 ; 0K00 ; 1K00 ; 20 K01 ; 0K01 ; 1+ K11 ; 1K01 ; 2+ K11 ; 2K11 ; 3K02 ; 0K02 ; 1+ K12 ; 1K02 ; 2+ K12 ; 2K12 ; 30 K13 ; 1K13 ; 2K13 ; 3377777777775 266666666664 q00q01q02q03377777777775 = 266666666664 f0 0f0 1+ f1 1f0 2+ f1 2f0 3377777777775 (2.44)ThegloballinearsystemEquation( 2.41 )issymmetric,andhasasparse bandedstructurewith18nonzerodiagonalbands.Wesolvefo r q byCholesky factorization.

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CHAPTER3 VISUALIZINGPROBABILITYFIELDS Inthissectionwewillexaminetechniquesofscalar,vector andtensoreld visualization,andexplorepossibletechniquesforprobab ilityelds.Ofparticular interestarethetexture-basedmethods.Wemayconsiderthe hue,intensity,and textureofanimageasindependentchannelsforconveyingin formationaboutthese highdimensionaldatasets. 3.1 ComputingProbabilities TheprobabilityinEquation 1.6 cannowbeevaluatedbycomputingthequantity S ( q ) =S 0 andperformingtheFFT.Sinceweknowthatthesignal, S ,decays exponentiallyfromtheoriginofq-space(where S (0)= S 0 )wecaninterpolateand extrapolatesignalvaluesforarbitrary q .Weresamplefromsphericalcoordinatesto cartesianandperformtheFFTontheresampleddata.Theresu ltisaprobability ofwatermoleculedisplacementoverasmalltimeconstant.W eareinterestedin onlythedirectionofwaterdisplacement,soweintegrateou ttheradialcomponent of p t ( r )toget p t ( ; ).Thisiscommonlycalledthediusionorientationdistrib utionfunction(ODF).ComputingtheODFwiththismethodisc omputationally expensivesinceitrequiresa3DFFTateachvoxel,andthenan umericalintegrationforeachdirection.Inpracticeweuseanalternativeme thodgivenby Ozarslan in[]whichmakescomputingtheODFforlargedatasetsmorefe asible. 36

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37 3.2 GlyphVisualization Inhighdimensionaldatasets,itiscommontorepresentthed ataateach voxelwithsomeicon.InDT-MRIthisiconisoftensomesimple shapewhichis transformedbythediusiontensor.Forexample,thesphere willbetransformedby thediusiontensorintothediusionellipsoid. ToenhancethevisualimpactoftheODFweapplyasharpeningt ransformto theODFbysubtractingauniformdistribution(sphere)from eachdistribution,as showninFigure 3.1 .Theradiusofthesphereistheminimumoftheprobability overalldirections.Byperformingthisoperationthedirec tionsofhighestprobabilitybecomesmoreapparent.Theglyphsinregionsofisot ropicdiusionwill disappear. = Figure3.1:OriginalODF(left),minimumprobabilityspher e(center),andsharpenedODF(right). Whenappliedtosmallorsparseplanarregionsthismaybeaus efulvisualization.AsshownintheleftsideofFigure 3.2 .However,forlarge,denseregions thedisplaymaybecomeverycluttered,asintherightsideof Figure 3.2 .Dueto perspectiveorviewingangle,glyphspointingtowardthevi ewermaynotconvey

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38 Figure3.2:ODFglyphs anyusefulinformation.In3-dimensionalregionsofintere st,suchdisplayscanbe quiteawkward. 3.3 ScalarMeasuresofAnisotropy Indiusiontensorimageprocessing,thescalarquantitykn ownasfractional anisotropy(FA)isoftenconsideredausefulquantitytovis ualize.Anexampleof anFAimageisshowninFigure 3.3 .TheFAvalueateachvoxelcanbecomputed fromtheeigenvaluesofthetensoratthatvoxel.Thevalueso fFArangefrom0,for completelyisotropicdiusion,to1,forcompletelyanisot ropicdiusion.Thebright regionsofFigure 3.3 correspondtohighFAvalues.Inrank-2tensorimagesFAcan indicatethepresenceofasingleberdirectionwithinavox el. Figure3.3:FAimage(left),generalizedanisotropy(cente r),Shannonanisotropy (HA)(right),fromcoronalsliceofratbrain

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39 Sincewehaveadistributionateachvoxel,wecancomputethe Shannon entropyvalueateachvoxelasgivenby H ( p )= n X i =1 p ( i ; i )log p ( i ; i )(3.1) Consideringtheentropyofseveraltrivialdistributions, wecangetafeelforthe interpretationofentropyinthecontextofHARDI.Entropya ttainsitsmaximumvalue,log n ,forauniformdistribution.Inourcase,thiscorrespondst o isotropicdiusion.TheentropyofaGaussiandistribution decreasesasthevariance decreases.Avoxelwiththisdistributionhasorienteddiu siongreatestinthe directionofthemodeofthedistribution,implyingthepres enceofbrousstructure. Tuch[ 83 ]denedascalarmeasureofanisotropycalled"normalizede ntropy",which wasalsobasedonShannonentropy. TheanisotropyimageswepresentaremappedusingEquation( 3.2 )suchthat blackcolorcorrespondstohighentropy(isotropicdiusio n)andhigherintensity greycolorsrepresentlowentropy(anisotropicdiusion). Wedenotetheanisotropy measurecomputedfromShannonentropyas HA HA ( p )=(1 : 0 H ( p ) log n ) (3.2) Thisallowstheimagetobeinterpretedinthesamewayasafra ctional anisotropyimagewherethewhitecolorcorrespondstowhite matter,greycorrespondstogreymatterandblackcorrespondstocerebrospi nalruid.The parametercontrolsthecontrastbetweenwhiteandgreymatt er.Ourimagesin Figure 4.10 werecomputedusing =0 : 65.

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40 Figure3.4:RaycastvolumevisualizationofHA. Figure3.5:MIPvolumevisualizationofHA. TheHAindexisnotageneralizationofFA,andtheirvaluesca nnotbecomparedinameaningfulway.Inordertohighlightthedierenc einanisotropymeasuresforHARDIandDTI,aGaussianODFwascomputedfromthet ensordata, andtheShannonentropyofthesedistributionswascomputed .Theanisotropy valuesfortheGaussianODFscanthenbecomparedtotheaniso tropyvaluesfor thetrueODFs.ThedierenceimageisshowninFigure 3.6 .Itcanbeseenthat thereisastructuretothedierenceimage. Thisdierenceismorepronouncedin

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41 regionswherethetensormodelpredictsdiusionismoreiso tropicthanitactually is Figure3.6:ShannonanisotropydierencebetweenGaussian ODFandgeneral ODF. Shannonentropyisnottheonlymeasureofuncertaintyinara ndomvariable.Anotherdenitionofentropy,calledtheRenyientro pycanbeseenasa generalizationofShannonentropy[ 25 ].Renyientropyoforder isgivenby H ( p )= 1 1 log( n X i =1 p ( i ; i ) )(3.3) Theparameter, ,ofthisentropyformulationhasseveralinterestingprope rties: lim 1 H ( p )= H ( p )(Shannonentropy) H 0 ( p )=numberofnonemptybinsinthehistogramof p H 1 ( p )= log(max i ( p i ))(dependsonlyonthemodeof p ) Onemayinterprettheorder, ,asaparameterwhichchangestheshapeofthe distribution, p .For > 1,smallvaluesof p ( x )willshrinkclosertozero.As increases,thesesmallprobabilitiesapproachzeromorequ ickly.Forhigh ,events withhighprobabilityinruencetheentropymore.Thiscanbe seenascontrolling thecontrastbetweenwhiteandgraymatter.Wecanformulate ananisotropyindex

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42 basedonRenyientropiesjustaswedidforShannonentropy. Anisotropyimages computedfromRenyientropiesofdierentordersareprese ntedinFigure 3.7 Figure3.7:Anisotropycomputedfrom H 2 (top-left), H 5 (top-right), H 10 (bottomleft), H 20 (bottom-right). Instatisticalphysics,aquantitycalledstructuralentro pyhasusefulphysical interpretation[ 95 ].Thisentropyiscomputedasthedierence H 1 H 2 .It isunclearwhetherthereisanysuchphysicalmeaningforthi squantityinthe contextofdiusion,butitdoesprovideameansofapplyinga transferfunctionto entropyimageswhichhasarminformationtheoreticfootin g.Wepresententropy dierenceimagesforseveralvaluesof inFigure 3.8 Figure3.8:Renyientropydierences H 1 H 2 (top-left), H 1 H 5 (top-right), H 1 H 10 (bottom-left), H 1 H 20 (bottom-right). 3.4 VisualizingtheDirectionalNatureofDiusion Thescalarentropyvaluehasnodirectionalinformationhow ever.InDTI,there isprecedenceforusingcolorvaluestorepresentdirection .ColorFAimages,Figure

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43 3.9 ,areamappingoftheprincipaldiusiondirection(thedomi nanteigenvectorof thetensor)toahue,andthefractionalanisotropyvaluetoa nintensity. Figure3.9:ColorFA. Theseimagesareusefulfordistinguishingbetweenadjacen tanisotropicregions whichdierindirection.Sincewemayhavehighdiusioncoe cientsinseveral directions,weintegrateoverthespheretodeterminearepr esentativecolorforeach voxel.Wedenotethisvalue ED ,forexpecteddirection.ED =nXi =126666664 j cos isin ij j sin isin ij j cos ij 37777775 ( p ( i;i) pmin)(3.4)Theresultingvectorisinterpretedasred,green,andbluec olorcomponents. Thedirectionswiththehighestprobabilityofdiusionwil lhavetheircorrespondingcolorcontributemosttotheresultingcolor.Figure 3.10 depictsthecolors computedforasyntheticODFelddirectedinacurve.Figure 3.11 showsthe

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44 colorscomputedforODFsorientedinvariousdirections,bo thwithandwithout crossings. Figure3.10:EDcolorsforsyntheticdistributioneld. Figure3.11:EDcolorsforsyntheticdistributioneld. 3.4.1 Streamtubes Streamtubesareathree-dimensionalalternativetostream lines.Thestreamtubegeometryisgeneratedbysweepingacirclealongastrea mline.Byrendering thestreamtubeasalitsurfaceusingshadinganddepthcuein g,betterdirectional informationmaybeconveyedtotheviewerthanwithstreamli nes.Thestreamtube isnotatruestreamsurfacesincethevectoreldisnottange nttothetubesurface. Ingeneral,onlytheunderlyingstreamlinewhichisthecent erlineforthestreamtubeistangenttothevectoreld.Thestreamtubediameteri saparameterwemay usetoencodesomeadditionalinformationaboutthetensor eldbeingvisualized, suchastheFAvalue.Bydiscardingshortbers,asdetermine dbysomethreshold,

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45 wemayobtainalessclutteredviewofconnectivity.Fibertr acesmaybeseededon adenseorsparsegridspanningthewholedataset,oronlywit hinaregionofinterest.Thestreamtubecolormayalsoencodesomeusefulinf ormation.Wechoose tocolorthestreamtubestoindicatedirection.Bymappingd irectiontocolor,as inthecolorFAimageFigure 3.9 ,abettersenseof3-dimensionalbertrajectories canbeconveyedina2-dimensionalprojection.Previously, Laidlawetal.[ 51 ]has appliedthestreamtubevisualizationapproachtoDT-MRI.I tisimportanttonote thateachstreamtubedoesnotrepresentasingleberortrac t,butindicatesonly berdirection. Figure3.12:Streamtubesinaxialsliceofbrain. 3.4.2 LineIntegralConvolution Itisalsopossibletovisualizethe3Dvectoreldcorrespon dingtothedominanteigenvaluesofthediusiontensorusingothervisuali zationmethodssuchas thelineintegralconvolutiontechniqueintroducedbyCabr alandLeedom[ 15 ]{

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46 aconceptexploredinthisworkaswell.Theadvantageofthis visualizationtechniqueisthatitiswellsuitedforvisualizinghighdensityv ectoreldsanddoesnot dependontheresolutionofthevectoreld,moreover,itals ohastheadvantageof beingabletodealwithbranchingstructuresthatcausesing ularitiesinthevector eld.AnexampleofLICappliedtothesyntheticeld v [ xy ]=[ 1 : 0 cos( x ) ]isshownin Figure 3.13 Figure3.13:LICvisualizationofsyntheticeldwithkerne lwidths5,15,and55. Sincetheberdirectionisparalleltothedominanteigenve ctorofthediusion tensor,wecancalculateberpathsasintegralcurvesofthe dominanteigenvector eld.ThestoppingcriterionisbasedonFAvalue.WhenFAfal lsbelow0.17 weconsiderthediusiontobenearlyisotropicandstoptrac kingtheberat thispoint.Oncethediusiontensorhasbeenrobustlyestim ated,theprincipal diusiondirectioncanbecalculatedbyndingtheeigenvec torcorrespondingtothe dominanteigenvalueofthistensor.Thebertractsmaybema ppedbyvisualizing thestreamlinesthroughtheeldofeigenvectors. LICisatexture-basedvectoreldvisualizationmethod.Th etechnique generatesintensityvaluesbyconvolvinganoisetexturewi thacurvilinearkernel

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47 alignedwiththestreamlinethrougheachpixel,suchasby I ( x 0 )= Z s 0 + L s 0 L T ( ( s )) k ( s 0 s ) ds (3.5) where I ( x 0 )istheintensityoftheLICtextureatpixel x 0 k isalterkernelof width2 L T istheinputnoisetexture,and isthestreamlinethroughpoint x 0 Thestreamline, canbefoundbynumericalintegration,giventhediscretee ld ofeigenvectors.Theeectofincreasing L isillustratedinFigure 3.13 wherethe kernelwidthincreasesfromlefttorightinthe3images. Wecanthinkofthisprocessastexturesynthesis,wherewear especifyingthe orientationofthetexturetomatchthePDDateachpixel.The resultisatexture withhighlycorrelatedvaluesbetweennearbypixelsonthes amestreamline,and contrastingvaluesforpixelsnotsharingastreamline.Ino urcase,anFAvalue belowacertainthresholdcanbeastoppingcriterionforthe integrationsincethe diusioneldceasestohaveaprincipaldirectionforlowFA values.Stallingand Hege[ 76 ]achievesignicantcomputationalsavingsbyleveragingt hecorrelation betweenadjacentpointsonthesamestreamline.Foraconsta ntvaluedkernel, k theintensityvalueat I ( ( s + ds ))canbequicklyestimatedby I ( ( s ))+ ,where isasmallerrortermwhichcanbequicklycomputed.Previous ly,Chaingetal. [ 66 ]haveusedLICtovisualizeberdirectionfromdiusionten sorimagesofthe myocardium. Weadaptthistechniquetotensoreldvisualizationbyinco rporatingthe FAvalueateacheldlocationintotheLICtexture.Bymodula tingtheimage intensitywithanincreasingfunctionofFA,wehighlightth eareasofwhitematter

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48 Figure3.14:LICbervisualizationinaxialsliceofbrain. Figure3.15:LICbervisualizationincoronalsliceofbrai n. andde-emphasizeregionswithuncertaindirectionalitysu chasgreymatter andCSF.InFigures 3.14 3.15 ,and 3.16 wehaveusedcolortoindicatethe3dimensionalorientationofthetracts,somethingnotconve yedbythetexture information.

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49 Figure3.16:LICbervisualizationinsagittalslicesofbr ain.

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CHAPTER4 RESULTS Thedenoisingandrenderingtechniquesdescribedinthepre vioussection wereappliedtoasyntheticHARDIdataset.Thisdatasetwasg eneratedusingthe techniquedescribedby Ozarslanetal.[ 64 ].Thedatasetwasdesignedtodepicta regionofcurvingbers,aregionofstraightbers,andacro ssingbetweenthetwo. RepresentativesharpenedODFproleareshownoverlaidont heHAimageonthe leftsideofFigure 4.1 .Thecoloranisotropyvisualizationshownontherightside of Figure 4.1 Figure4.1:ODFprolesoverlaidonHAimage(left),andcolo ranisotropy(right) ofthesyntheticdataset. Asmallsampleofthesyntheticdata,takenfromnearthecros singregion, isshowninFigure 4.3 .ThesyntheticdatawerecorruptedwithGaussiannoise ofmeanzero,andvariance 2 =0 : 005.ThenoisydataareshowninFigure 4.4 50

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51 Figure4.2:Multi-LICvisualizationofsyntheticdataset. Thesamevoxelsareshownaftersmoothingoverthespherical manifoldinFigure 4.5 ,aftersmoothingoverthecartesianimagedomaininFigure 4.6 andafterboth techniqueshavebeenusedinFigure 4.7 .Theright-handsideofeachgureshows thesharpenedODFcomputedfromthe S valuesintheleft-handside. Figure4.3:SyntheticS(left),resultingODF(right). FromFigure 4.4 ,itcanbeseenthatthenoisehasalargeimpactonthe smoothnessoftheODF.Asexpectedfromthevariationalform ulation,thespikes ofnoisepresentintherawdatahavebeensmoothedwhilepres ervingtheoverall

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52 Figure4.4:SyntheticSwithnoiseadded(left),resultingO DF(right). Figure4.5:Manifold(FEM)smoothingresultsforS(left),r esultingODF(right). shapeofthe S prole.ThissmoothnessispropagatedtothecomputedODF proles. Wecancomparetheresultingdistributionswiththegroundtruthbyusing thesquarerootofJ-divergence(symmetricKL-divergence) asameasure.This divergenceisdenedas d ( p;q )= p J ( p;q )(4.1)

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53 Figure4.6:Lattice(TV)smoothingresultsforS(left),res ultingODF(right). Figure4.7:ManifoldandlatticesmoothedS(left),resulti ngODF(right). where J ( p;q )= 1 2 n X i =1 p ( i ; i )log p ( i ; i ) q ( i ; i ) + q ( i ; i )log q ( i ; i ) p ( i ; i ) (4.2) InTable( 4.1 )wecomparethedistances, d (^ p;p ),betweentheoriginalsynthetic data,(^ p ),andtheunrestoreddata,thedatarestoredonlyusingtheF EMmethod, thedatarestoredusingonlytheTV-normminimization,andt hedatarestored usingbothtechniques.Foreachtechniquethemeandistance ( d (^ p;p )),between

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54 Table4.1:Errorbetweenground-truthandrestoredsynthet icdata Method ( d (^ p;p )) 2 ( d (^ p;p )) p =NoRestoration 1 : 0409 0 : 0173 p =FEMRestoration 0 : 9088 0 : 0125 p =TVRestoration 0 : 7420 0 : 0119 p =FEM+TVRestoration 0 : 6576 0 : 0139 distributionsincorrespondingvoxelsispresented,andth evarianceofthese distances, 2 ( d (^ p;p ))incorrespondingvoxelsispresented. AsshowninTable( 4.1 ),theTVrestorationhadsuperiorperformancetothe FEMtechnique,bothintermsofthemeanerrorandvarianceof theerror.The combinationoftechniqueshadalowermeanerrorthaneither theFEMorTV restoration,howeverthevarianceoftheerrorwashigherth anthatwitheither techniquealone. Thedenoisingalgorithmwasappliedtoadatasetconsisting of47diusion weightedimagesofaratspinalcord.Axialslicesofonesuch image,beforeand afterdenoisingareshowninFigure 4.8 Figure4.8:Originaldiusion-weightedimage(left),andd enoised(right)from spinalcorddata.

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55 Theringingartifactsvisiblenearthesampleboundaryinth erawDWIin Figure 4.8 havebeennoticablydecreased.Notethattheedgesintheima gehave beenwellpreserved.FromthecoloranisotropyimagesinFig ure 4.9 ,itisclearthat thewhitematterbertractsarepredominantlyintheaxiald irection,whichis representedbythebluecolor. Figure4.9:Originalcoloranisotropyimage(left),andden oised(right)fromspinal corddata. Figures 4.10 and 4.11 showtheHAandcoloranisotropyvisualizationresults respectivelyforvariouscoronalslicesofaratbraindatas et.Thechosenslices showthecorpuscallosum,ananatomicalstructureknowntoh avelong-rangewhite mattertracts. TheHAimagesshowavisibledistinctionbetweengreyandwhi tematter.In addition,thedistinctionbetweenberdirectionsisevide ntinthecolor ED images. Redcorrespondstotheleft-rightdirectionandgreencorre spondstoup-downand blueisin-outofthepage.

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56 Figure4.10:Original(left),anddenoised(right)HAimage sforcoronalslicesofrat brain.

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57 Figure4.11:Original(left),anddenoised(right)EDimage sforcoronalslicesofrat brain.

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CHAPTER5 SEGMENTATIONOFHARDI 5.1 ModellingDiusion InDTI,dataaremostoftenmodelledintermsofthediusiont ensor.The apparentdiusioncoecientisaquadraticforminvolvingt hetensor,andthe diusiondisplacementpdfisaGaussianwithcovariancemat rixequaltoaconstant multipleoftheinverseofthetensor.ForHARDI,wewilladvo catemodeling neitherthediusivitynorthedisplacementpdf,butinstea dmodelingthediusion ODF.ComputingtheODFwasdiscussedinSection(3). AsimplewaytostorethediusionODFisasacollectionofsam plesofthe functionatregularlyspacedpoints,oralternatively,asa histogram.Thiswill requirealargenumberofdatapointstobestoredinordertor esolvecomplexber geometry.Inordertodesignecientalgorithms,wewishto ndaparametric modelfortheODFwithasmallnumberofparameters,whichisc apableof describingdiusioninthepresenceofIVOH. 5.1.1 OtherModels Toputourproposedmodelinperspectivewewillrstreviews omemodelsfor diusionusedinpreviousliterature. 58

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59 5.1.1.1 GaussianMixtureModel Gaussianmixturemodels(GMM)areonethemostcommonlyused modelfor multimodaldistributions.ThetrivariateGaussiandensit yfunctionwithmean andcovarianceisgivenby N ( x j ; )= 1 p (2 ) 3 det() exp( 1 2 ( x ) T 1 ( x ))(5.1) TheGMMgiveninEquation( 5.2 )candescribethe3-dimensionaldiusion displacementpdf P ( x )= m X i =1 w i N ( x j i ; i )(5.2) where m isthenumberofcomponentsinthemixture.EachGaussiancom ponent hasitsown3 3covariancematrix, i ,whichwillhave6independentelements. Fordiusiondata,allcomponentswillhave i =0. However,weareprimarilyconcernedwiththedirectionalch aracteristicsof diusion.Thiscanbecharacterizedbythemarginaldistrib ution, P ( ; )obtained byintegratingovertheradialcomponentof P ( x ).Thiswillallowustousea simplermodelfortheODF.Additionally,withtheGMM,wemus tbecarefulto imposethepositive-deniteconstraintonthecovariancem atrixofeachcomponent ofthemixture.PreviouslyFletcherandJoshi[ 30 ]havedescribedgeodesicanalysis onthespaceofdiusiontensors.Theanalysisincludesanal gorithmforcomputing theintrinsicmeanofdiusiontensors(thearithmeticmean ofSPDmatricesisnot, ingeneral,anSPDmatrix).Laterinthischapterwewilldesc ribeasimilaranalysis onthespaceofODFswhichwillresultinmuchsimpleralgorit hms.

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60 5.1.1.2 SphericalHarmonics Thesphericalharmonic(SH)expansionisausefulrepresent ationforcomplexvaluedfunctionsonthesphere.Wecanrepresentthediusio nwiththeexpansion d ( ; )= L X l =0 l X m = l a l;m Y l;m ( ; )(5.3) Notethatthecoecients a l;m arecomplex-valued,sothatthestoragerequirementisdoublethatofanequivalentmodelwithrealvariable s,andthearithmetic operationsaremorecostlyaswell.Frank[ 33 ]suggestsanexpansiontruncatedat order L =4(orhigher)todescribemultipleberdiusion.Thisrequ iresatleast 15complex-valuedcoecientspervoxel.Ingeneral,theord er L expansioncan describediusionwith L= 2berdirections. Ozarslan[ 65 ]hasdevelopedanextremelyfastalgorithmforcomputingaS H expansionfortheODFgivenaSHexpansionofthediusivity. 5.1.1.3 GeneralizedTensors Thediusiontensorimagingmodeldescribedpreviouslyrep resentsdiusion usingarank-2tensor.Diusionhasbeendescribedmoregene rallyby Ozarslanet al.[ 62 63 ]byconsideringtensorsofhigherrank.Acartesiantensoro frankIwill, ingeneral,have3 I .Duetosymmetry,thenumberofdistinctcomponentsinahigh rankdiusiontensorwillbemuchsmaller.Bygeneralizingt heconceptoftrace,it ispossibletoquantifytheanisotropyofdiusiondescribe dbytensorsofarbitrary rank[ 64 ].

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61 Sincetensorsofoddrankimplynegativediusioncoecient s,onlyevenrank tensorsareappropriatefordescribingdiusion.Fordius iontensorsofrank4,6, and8,thenumberofdistinctcomponentsare15,28,and45res pectively.Itisnot clearhowtoextractberdirectionsfromhigherranktensor s. 5.1.2 vonMises-FisherMixtureModel Manystatisticalapproachesinvolvedataover < n .Sincewearedealingwith multivariatedataoverthesphere, S 2 ,wewishtoexpressthedatausingdistributionsovertheappropriatedomain.Othercommonapplicat ionsofdirectional distributionsareintheeldsofgeologyandmeteorology,s incemanymeasurements inthoseeldsaretakenoverthesurfaceofasphere.Datainv olvingclockorcompassmeasurementsalsofrequentlyusedirectionaldistrib utions.Otherapplications aretextclustering[ 28 ]andgeneexpressionmapping.Thesedistributionsare discussedindetailbyMardiaandJupp[ 54 ]. InthissectionwewillpresentadirectionalmodelfortheOD Finterms ofvonMises-Fisherdistributions.Thismodelhasfarfewer variablesthanthe previouslydiscussedmodels,allowstheberdirectionsto beextractedeasily, involvesconstraintswhicharesimplertosatisfy,andlead stoaclosed-formfor severalusefulmeasures. ThevonMisesdistributionisthemostcommonlyuseddistrib utiononthe circle.Inpolarcoordinatesithastheform V ( j ; )= 1 2 I 0 ( ) exp( cos( )) ; 0 2 (5.4)

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62 ThisdistributioniscloselyrelatedtothewrappedGaussia ndistributiononthe circle.LiketheGaussian,itisunimodalandsymmetricabou tthemean. ThevonMisesdistributioncanbegeneralizedtospheresofa rbitrarygeometry bykeepingthelogofthedistributionlinearintherandomva riable x M p ( x j ; )= 2 p= 2 1 1 2 ( p= 2) I p= 2 1 ( ) exp( T x )(5.5) where j x j =1and j j =1, istheconcentrationparameterand I k denotesthe modiedBesselfunctionoftherstkind,order k .Theconcentrationparameter, ,quantieshowtightlythefunctionisdistributedaroundt hemeandirection For =0wehaveauniformdistributionoverthesphere.Thedistri butionsare rotationallysymmetricaroundthedirection For p =3thedistributioniscalledthevonMises-Fisher(vMF)dis tribution. M 3 ( x j ; )= 4 sinh( ) exp( T x )(5.6) Figure5.1:ExamplevMFdistributions( =1,5,10,15,25).Alldistributions havesamemeandirection, AusefulcharacteristicofthevMFdistributionisthatthep roductoftwo vMFsmayalsobewrittenasanunnormalizedvMF.Since exp( i Ti x )exp( j Tj x )=exp(( i i + j j ) T x )(5.7)

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63 wehave M 3 ( x j i ; i ) M 3 ( x j j ; j ) / M 3 ( x j ( i i + j j ( i ; j ; i ; j ) ) ; ( i ; j ; i ; j ))(5.8) where ( i ; j ; i ; j )= q 2i + 2j +2 i j ( i j )(5.9) Wewillusethisfactinthenextsection. TheODFmayhavemultiplemodesincasesofintravoxelorient ational heterogeneity,sowewishtomodeltheODFwithaparametricd istributionwhich iscapableofdescribingthiscomplexgeometry.SincethevM Fdistributionis unimodal,wecanchooseacombinationofthesedistribution s.Infact,sincethe ODFisantipodallysymmetric,wewillneedamixturetodescr ibediusionineven asingleberregion.Sincetheantipodalpairhave 1 = 2 ,wecanspecifya mixturewithonly3variablespercomponent:thetwospheric alcoordinateangles describing ,and Figure5.2:SamplevMFmixturesforvoxelswithoneandtwob ers. ODF ( x )= m X i =1 w i M 3 ( x j i ; i )(5.10)

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64 ChoosingaconvexcombinationofvMFdistributions,thewei ghtshavethe property m X i =1 w i =1 ;w i 0(5.11) Thisensuresthatthemixturestillhasnonnegativeprobabi lities,andwillintegrate to1. SincevMFdistributionsobeytheproperty( 5.7 ),theproductoftwovon Mises-FishermixturemodelsisalsoproportionaltovMFmix turemodel. 5.1.2.1 FittingthevMFMixture Inthissectionwedescribethegradientdescentalgorithmf orcomputingthe vMFmixturemodel.Wewillassumethatwehavebeengivenadis cretesetof samplesoftheODF.WeseekamixtureofvMFswhichagreeswith thesesamples intheleast-squaressense.Itisimportanttonotethatwear enotworkingwith randomsamplesinthestatisticalsense.Thesearepointsam plesofafunction,so wetakeasurfacettingapproach. Usingthesphericalcoordinatesfor x and x = 26666664 cos sin sin sin cos 37777775 ; = 26666664 cos sin sin sin cos 37777775 ; (5.12) wemaywritethevMFinpolarform: M 3 ( ; j ;; )= 4 sinh( ) exp( [cos cos +sin sin cos( )])(5.13)

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65 TheenergyfunctionwewillseektominimizeistheL2distanc ebetweenthemodel andthedata. min ;;;w n X l =1 ( m X i =1 w i M 3 ( l ; l j i ; i ; i ) p ( l ; l )) 2 (5.14) Equation( 5.14 )isnumericallydierentiatedtocomputethegradient t +1 = t E ( + ;;;w ) E ( ;;;w ) 2 t t +1 = t E ( ; + ;;w ) E ( ; ;;w ) 2 t t +1 = t E ( ;; + ;w ) E ( ;; ;w ) 2 t w t +1 = w t E ( ;;;w + w ) E ( ;;;w w ) 2 w t (5.15) Aftereachiterationwemustprojecttheweights, f w g ,backontothesimplex tomaintainthepropertythattheweightssumtoone. Itislikelythatmostvoxelswilltamixtureof4vMFdistrib utionsquite well.Inthiscasethemodelrequiresonly15real-valuedpar ameterstocompletely describe.OncewehavetthevMFmixturetotheODF,wecandir ectlyextract theberdirections, f g 5.1.2.2 EntropyandDistance Weseekclosed-formequationsforseveralmeasures,sincet hiswilleliminate theneedfortimeconsuming,andpossiblyinaccurate,numer icalintegration. Theentropyofthemixturemodel,whichweshowedcanbeuseda smeasure ofanisotropy,cantakeseveralanalyticalforms.Although theShannonentropyof thesedistributionscannotbewritteninclosed-form,some oftheRenyientropies

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66 may. H ( X )= 1 1 log( X x P ( x ) )(5.16) Inparticularly,theRenyientropiesofintegerorders > 1canbewrittenin closed-form.Sincetheexpressionwillhave m terms,wewillconcentrateonthe case =2. H 2 ( X )= log Z 2 0 Z 0 P ( x ) 2 sin dd (5.17) = log m X i =1 m X j =1 w i w j i j 4 sinh i sinh j sinh( ( i ; j ; i ; j )) ( i ; j ; i ; j ) Thenotionofdissimilaritybetweendistributionsisfunda mentaltothe problemofsegmentation.Inthesegmentationproblemwesee ktopartitionsome domainintoregionswhoseconstituentelementsaresimilar insomesense.Wecan obtainaclosed-formforthe L 2 distancebetweentwovMFmixures, P ( x j ; )and Q ( x j ; ). Z S 2 ( P Q ) 2 dS = Z S 2 P 2 dS + Z S 2 Q 2 dS 2 Z S 2 PQ 2 dS (5.18) = m X i =1 m X j =1 w i w j i j 4 sinh i sinh j sinh ( i ; j ; i ; j ) ( i ; j ; i ; j ) + m X i =1 m X j =1 w i w j i j 4 sinh i sinh j sinh ( i ; j ; i ; j ) ( i ; j ; i ; j ) 2 m X i =1 m X j =1 w i w j i j 4 sinh i sinh j sinh ( i ; j ; i ; j ) ( i ; j ; i ; j ) Relativeentropyisalsoacommoninformation-theoreticme asureofdissimilarity(divergence)betweendistributions. H ( f;g )=log ( R g 1 f ) 1 = (1 ) ( R g ) 1 = ( f ) 1 = ( (1 )) (5.19)

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67 For =2wehave H 2 ( f;g )=log ( R f 2 ) 1 = 2 ( R g 2 ) 1 = 2 R fg (5.20) Thismeasureisrelatedtothecosinesimilarityby H 2 ( f;g )=log( 1 cos ( f;g ) ) 5.2 TheSpaceofvMFDistributions ThevonMises-Fisherdistributionisparameterizedbytwov ariables:the concentrationparameter 2< + and 2 S 2 .Foreachpointin < + S 2 there isacorrespondingvMFdistribution.Thecurvedgeometryof thisspaceofvMF distributionswillinruencehowweformulatedistances,ge odesics,interpolation functionsandmeans.Ageneraltreatmentofthegeometryoft hespacesformedby parametricdistributionsisgivenbyAmari[ 1 2 ]. 5.2.1 RiemannianGeometry ThespaceofvMFdistributionsformsadierentiablemanifo ld,aspacewhich locallybehaveslikeEuclideanspace.Thismanifoldcanals obeconsideredto beembeddedinahigherdimensionalspace.Forexamplethesp here, S 2 canbe consideredtobeembeddedin < 3 .Theembeddingspaceinducesametriconthe manifoldwhichallowsdistancestobecomputedbetweenpoin tsonthemanifold. ThisRiemannianmetricisconsideredanintrinsicwaytomea suredistanceona manifold.Themetricmeasuresthelengthofacurvelyingont hemanifold.The extrinsicwayofmeasuringdistanceistotaketheEuclidean distancebetween twopointsintheembeddingspace.Asanexampleonthesphere ,theextrinsic distancebetweenthenorthandsouthpoleissimplythediame terofthesphere,the

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68 intrinsicdistanceisone-halfofthecircumference.Ifyou areconstrainedtomove onthesurfaceofthesphere,youwouldndtheintrinsicdist ancemoreuseful.This dierenceisillustratedinFigure 5.3 .HereweseetheEuclideandistance, d ( p;q ), comparedwithacurvesegment r :[ p;q ]lyingonthemanifold.If r :[ p;q ]hasthe shortestlengthofallcurvesbetween p and q ,wecanconsiderthislengthasthe intrinsicdistance. r d ( p;q ) p q Figure5.3:Intrinsicandextrinsicdistance Whenconsideringcurvesonamanifold,itwillbeusefultowo rkinthespace oftangentvectorstothesecurves.Ateachpointonamanifol dthereareaninnite numberofcurvesonthemanifoldwhichmaypassthroughthatp oint.Thespace ofthevelocitiesofthosecurvesisthetangentspaceat p ,denoted T p M .Foratwo dimensionalsurfaceembeddedin < 3 ,thetangentspaceisaplaneperpendicularto thenormalof M at p ,asshowninFigure 5.4 ARiemannianmanifoldisasmoothmanifoldsuppliedwithaRi emannian metric.Thismetrictakestheformofaninnerproduct, h v;w i p denedonthe tangentspace, T p M ,foreachpoint, p ,onthemanifold, M .Fortwodimensional

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69 T p M M p n Figure5.4:Tangentspace surfaces,wemaysimplyinheritthevectordotproductfromt heEuclideantangent space,butanyfunction h ; i satisfyingthefollowingpropertieswillbeacceptable: 1.Scaling: h v;w i = h v;w i 2.Linearity: h u + v;w i = h u;w i + h v;w i 3.Symmetry: h v;w i = h w;v i 4.Positivedeniteness: h v;v i 0and h v;v i =0 $ v =0. TheRiemannianmetricallowsustomeasurethelengthofacur ve, r ( t ) betweentwopoints, p;q on M L ( r )= Z q p ( h r 0 ( t ) ;r 0 ( t ) i r ( t ) ) 1 2 dt (5.21) Thenotionsofmetric,distance,geodesics,interpolation andmeanareall related.Ageodesicbetween p;q isacurve r forwhich L ( r )isminimized,and theminimumlengthisthedistance, d ( p;q ).Notethattheremaynotbeaunique geodesicbetweentwopoints.Onthesphere,forexample,all meridiansare geodesicsbetweenthenorthandsouthpole.However,pertur bingoneofthese

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70 pointswillresultinauniquegeodesic.Ingeneral,iftwopo intsarenearenough, thegeodesicisunique. Themeancanbedenedintermsofthedistance, d ,asthepoint, ,which satises min 2 M N X i =1 d 2 ( ;x i )(5.22) Interpolationcanbedenedintermsofaweightedmean,sowe caninterpolatein2Dbetween4distributionsbyminimizing min 2 M uvd ( ;x 00 )+(1 u ) vd ( ;x 10 )+ u (1 v ) d ( ;x 01 )+(1 u )(1 v ) d ( ;x 11 )(5.23) 5.2.2 RiemannianExpandLogMaps Let M besomemanifold,and T p M bethetangentspaceat p 2 M .Consider allgeodesicsgoingthroughthepoint, p ,on M .Givenatangentvector, v 2 T p M itisknownthatthereisauniquegeodesic, r ,suchthat r (0)= p ,and r 0 (0)= v Ifthemanifoldisgeodesicallycomplete,asitisinourcase ,theRiemannian exponentialmap,Exp p : T p M M ,canbedenedas Exp p ( v )= r (1)(5.24) Letusexaminetheexponentialmapsforafewsimplespaces.F or < n ,the geodesicsarestraightlines,soExp p ( v )= p + v .For < + theexponentialmapisthe exponentialfunction,soExp p ( v )= p exp( v ).Onthesphere, S 2 ,geodesicsaregreat circles.Parameterizethesphereby x ( ; )=(sin cos ; sin sin ; cos ).Ifwe considerapointatthenorthpole p =(0 ; 0 ; 1),thetangentvectorcanbewrittenas

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71 v =( v x ;v y ; 0).Inthiscasethegeodesicsinthedirectionof v arecurvesofconstant ,andwehavecos = v x jj v jj andsin = v y jj v jj .Wecanparameterizegeodesicswith speed, s ,as r ( )=( v x jj v jj sin( s ) ; v y jj v jj cos( s ) ; cos( s ))(5.25) Theexponentialmapisthengivenby Exp p ( v )=( v x jj v jj sin jj v jj ; v y jj v jj cos jj v jj ; cos jj v jj )(5.26) TpM M p r v Expp( v )Figure5.5:Riemannianexponentialmap Sincethegeodesichasconstantspeed,weknow d ( p; Exp p ( v ))= jj v jj TheRiemannianlogmapistheinverseoftheexponentialmap, Log p : Exp p ( v ) T p M .Ifthelogmap,Log p existsat q ,wecanwritetheRiemannian distancebetween p and q as d ( p;q )=Log p ( q )(5.27) Forthecaseofthesphere,givenapoint q =( q x ;q y ;q z ) 2 S 2 weknowthe lengthofthetangentvector,sincewecancomputethedistan ce d ( p;q )= = cos 1 ( p q ).Wealsoknowthedirectionofthetangentvectorisparalle lto( q x ;q y ).

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72 TpM M p r Logp( q ) qFigure5.6:Riemannianlogmap TheLogmapforthesphereisgivenby Log p ( q )=( q x sin ;q y sin )(5.28) 5.2.3 SymmetricSpaces Thedevelopmentoftheprevioussectionissucienttocompu tegeodesics atcertainpointsforsimplemanifolds,butourspaceofdist ributions(andthe spaceofmixtureswewillconsiderlater)aremorecomplicat ed.Inthissection wewilldiscusstheformalismrequiredtoworkwiththedirec tproductmanifold. Thenextsectionwillpresenttherequiredframeworkforcom putinggeodesicson thismanifold.ThisisthedevelopmentpresentedbyFletche randJoshi[ 31 ]for exploringthespaceofshapedrepresentedbymedialatoms. Asymmetricspace[ 49 ](orsymmetricRiemannianmanifold)isaconnected Riemannianmanifoldsuchthatforall p 2 M thereexistsanisometry(alengthpreservingmapping)

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73 p : M M (5.29) with T p p : T p M T p M = id ,where id istheidentitymapping.Suchamapping willhavetheeectofreversinggeodesicsthrough p ,andapplyingthemapping twicehasnoeectonthemanifold( 2 p = id ). For < 3 ,themappingisthererection2 p x .Forthesphere S 2 therequired transformationistherotationby180 aboutthenormalat p .For < + itisthe rerection p 2 x Ausefulpropertyofsymmetricspacesisthatthedirectprod uctofsymmetric spacesisalsoasymmetricspace.Knowing < + and S 2 aresymmetricspaces,we havethat( < + S 2 )isasymmetricspace,andeventhat( < + S 2 ) m isasymmetric space. If M 1 and M 2 aretwometricspacesand x 1 ;y 1 2 M 1 and x 2 ;y 2 2 M 2 ,thenthe metricfor M 1 M 2 is d (( x 1 ;x 2 ) ; ( y 1 ;y 2 )) 2 = d ( x 1 ;y 1 ) 2 + d ( x 2 ;y 2 ) 2 5.2.4 LieGroupsandHomogeneousSpaces InanearliersectionwecomputedgeodesicsandExpandLogma psforthe sphereataspecicpoint, p ,inthiscasethenorthpole.Onthesphereitseems intuitivelyobviousthatwecanrotatethesegeodesicsabou tthecenterofthe spheretoobtaingeodesicsthroughotherpoints.Observeth atwecanreachany otherpointonthespherebyapplyingrotationmatricestoth epoint, p ,andthat therotationtransformationdoesnotchangethelengthofth egeodesics.Inthis

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74 sectionwewillspecifytheconditionsunderwhichwemaycom putegeodesicson othersymmetricspacesinananalogousway. NowwecanconsideravMFdistributiontobeapointinsomesym metric space.Howeveragroupstructurewillsimplifysomefurther operations.Agroupis denedasaset, G ,withanoperator, whichobeysthese4properties: 1.Closure:if A;B 2 G ,then A B 2 G 2.Associativity:forall A;B;C 2 G A ( B C )=( A B ) C 3.Identity:thereexistsauniqueidentityelement, I ,suchthat A I = I A = A forall A 2 G 4.Inverse:foreach A 2 G ,thereexistsaninverse, A 1 2 G suchthat A A 1 = A 1 A = I ALiegroupisdenedasagroupwhichhasdierentiablemulti plicationand inverseoperators. ThespaceofvMFdistributions, M ,doesnothaveagroupstructuresince thereisnotauniqueidentityelement.Thereareaninniten umberofdistributions with =0.However,wecandeneaLiegroupwhichoperateson M SO (3)isthegroupofspecial(determinant=1)orthogonal3 3matrices. Thesematricesrepresentrotationsin3 d .Thegroup G = < + SO (3)acts smoothlyonM.Let g =( s;R ) 2 G ,and m =( ; ) 2 M .Thenthegroupaction g m =( s;R ).Thesearesimplythescalarandmatrix-vectormultiplica tion operations.Theisometrysubgroup G x isdenedasthesetofall g 2 G suchthat g x = x (i.e. x isaxedpointundertheactionof g ).Inthiscase G x =(1 ;Q ) where Q isthesetofallrotationsaboutthe axis.Ifweagainconsiderthebase

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75 point p tobethenorthpole, G p =(1 ;SO (2)),where SO (2)isthegroupofspecial orthogonal2 2matrices(rotationsintheplane). Theactionof G on M iscalledtransitiveifforanytwo x;y 2 M there existsa g 2 G suchthat g x = y .Ifthegroupactionistransitivethen M is isomorphicto G=G x .Foranarbitrarybasepoint, p ,wecanwrite M asasthe homogeneousspace M = G=G p .Wecanalsoseehowtheproductofsymmetric spacesisalsoasymmetricspace.Let M 1 = G 1 =H 1 and M 2 = G 2 =H 2 ,then M 1 M 2 =( G 1 =H 1 ) ( G 2 =H 2 )=( G 1 G 2 ) = ( H 1 H 2 ). If G istransitiveon M ,and G p isaconnectedcompactLiesubgroupof G then M hasG-invariantmetric.Thismeansthattheactionof G doesnotchange distanceson M : d ( g p;g q )= d ( p;q ). Geodesicsonsymmetricspacerepresentedas M = G=G p canbecomputedby applyingthegroupactiontogeodesicsthrough p .Soforthesphere,if r (0)= p and R 2 SO (3) =SO (2)suchthat Rp = q ,then Rr isageodesicthrough q Toexplicitlyndsuchan R whichtransformspoint p to q ,wecanforma rotationmatrixinthefollowingway.Therotationaboutana rbitraryaxis,[ x;y;z ] T byangle canbewritten R = 26666664 x 2 +cos (1 x 2 ) xy (1 cos ) z sin xz (1 cos )+ y sin xy (1 cos )+ z sin y 2 +cos (1 y 2 ) yz (1 cos ) x sin xz (1 cos ) yyz (1 cos )+ x sin z 2 +cos (1 z 2 ) 37777775 (5.30) Toalignpoints p and q ,werotateabouttheaxis p q jj p q jj bytheangle =cos 1 ( p q ). Toaligntangentvectors v p and w p wecanrotateabout p by =cos 1 ( v w ).

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76 5.2.5 Results Wehaveusedthefactthatthedirectproductofsymmetricspa cesisalso asymmetrictodeducethatthespaceofvMFdistributionsiss ymmetric.Now wewillusethisfacttocomputetheExpmapforvMFs.Forspace swhichare expressedasdirectproducts,wecanwritetheexponentialm apasthedirect productoftheexponentialmapsfortheconstituentspaces. ForasinglevMF,let p =( ; )representthedistribution M 3 ( x j ; ),and v =( a;u ) 2 T p M bethe tangentvector.Then Exp p ( v )=( exp( a ) ;Q 26666664 u x sin jj u jj jj u jj u y sin jj u jj jj u jj cos jj u jj 37777775 )(5.31) 5.3 TheSpaceofvMFMixtures Now,letusinvestigatethespaceofmixturesofvMFdistribu tions.The mixturemodelofmcomponentsisgiveninEquation( 5.10 ).Atrst,itmayseem thatwecansimplyextendtheresultsoftheprevioussection ,andconsiderthese mixturestocomefromthespace( < + < + S 2 ) m .However,consideringthesetof weightsasanpointin( < + ) m ignorestheconvexityconstraintontheweights.The space( < + ) m includeslinearcombinationsofvMFswhoseweightsdonotsu mto1. Instead,weconsiderthesquarerootsoftheweights, f p w 1 ::: p w m g .The convexityconstraintnowbecomes m X i =1 p w i 2 =1 ;w i > =0(5.32)

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77 (a) r (0) (b) r (0 : 1) (c) r (0 : 2) (d) r (0 : 3) (e) r (0 : 4) (f) r (0 : 5) (g) r (0 : 6) (h) r (0 : 7) (i) r (0 : 8) (j) r (0 : 9) (k) r (1) Figure5.7:PointsalongthegeodesicbetweentwovMFdistri butions. So,wecanconsiderthespaceofthesquarerootsoftheweight stobea hypersphere, S m 1 .Then,thespaceofmixtureswith m componentsis S m 1 ( < + S 2 ) m 5.3.1 RiemannianExpandLogMaps ForthevMFmixture,theexponentialmapisthedirectproduc tofthe exponentialmapsforeachvMF,andtheexponentialmapfor S m 1 .Sinceweare quiteunlikelytohavemorethan4berorientationspresent withinasinglevoxel, wewillconsiderfurtherthecaseofmixtureshaving4compon ents( m =4).Inthis case,thespaceofthesquarerootsof f w g istheunithypersphere S 3

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78 Fortunately,thespace S 3 iswellstudied,sincethisisequivalenttothespace ofunitquaternions.Itisalsocloselyrelatedtothespaceo forthogonalmatrices SO (3)(technically, S 3 isadouble-coveringof SO (3)).Infact, S 3 formsaLiegroup withrespecttothequaternionmultiplicationoperator. Quaternionoperationscanbedenotedcompactlybyobservin gthat S 3 isalso isomorphicto SU (2),thegroupofspecial(determinant=1)unitarymatrices Quaternionoperationsareusuallypresentedintermsof4-d imensionalcomplex numbers,butwecanwritethesameoperatorsintermsofmatri xoperations.The correspondencebetweenquaternions, q ,and Q 2 SU (2)isgivenby q =( a + bi + cj + dk ) $ Q = 2664 a di b + ci b + cia + di 3775 (5.33) Inthisnotation,theidentityquaternion, q =1,isrepresentedbythe2 2 identitymatrix,quaternionmultiplicationbecomesmatri xmultiplication,andthe quaternioninversionbecomesconjugatetransposition( q 1 $ Q H )sincethematrix isunitary.Thequaternion, q ,isoftenconsideredtohaveascalarpart,thereal component a ,andavectorpart[ b;c;d ] T Theexponentialmapfor S 3 is Exp p ( v )= sin( 1 2 jj v jj ) jj v jj v; cos( 1 2 jj v jj ) T (5.34) andthelogmapisgivenby Log p ( q )= 2cos 1 ( q w ) j q vec j q vec (5.35) where q vec and q w arethevectorandscalarpartsrespectivelyofthequaterni on q

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79 Incomputergraphics,therelationbetweenquaternionsand SO (3)isoften usedtocomputesmoothinterpolationsbetweenrotationmat ricesinanimation [ 75 ].Thecommonlyusedsphericallinearinterpolation(slerp )canbewrittenin termsoftheLogandExpmapas slerp t ( p;q )= p Exp( t Log( p 1 q ))(5.36) (a) r (0) (b) r (0 : 1) (c) r (0 : 2) (d) r (0 : 3) (e) r (0 : 4) (f) r (0 : 5) (g) r (0 : 6) (h) r (0 : 7) (i) r (0 : 8) (j) r (0 : 9) (k) r (1) Figure5.8:Pointsalongthegeodesicbetweentwosetsofwei ghts.

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80 5.3.2 IntrinsicMean Computingthe"center"ofagroupofdistributionshassever alapplications forourdata.Thersthasalreadybeenmentioned:interpola tion.Theother applicationistheconstructionofatlases.Byregistering multipledatasetsacrossa populationontoacommoncoordinatesystemwecancompareOD Fdistributions atcorrespondingvoxels. Previously,theintrinsicmeanproblemhasbeensolvedwith agradientdescent algorithm[ 68 32 48 ].ThegradientoftheenergyfunctioninEquation( 5.22 )can bewrittenintermsoftheLogmap.Thealgorithm,asgivenbyF letcherandJoshi [ 32 ]is Input: x 1 ;:::;x N 2 M Output: 2 M ,theintrinsicmean 0 = x 1 Do = N P Ni =1 Log t ( x i ) t +1 =Exp j ( ) While jj jj > 5.4 SegmentationModels Themean,variance,anddistanceformulationsdiscussedin theprevioussectioncanbequiteusefulinthecontextofmodel-basedsegmen tation.Inthissection wewillpresentresultsobtainedusingseveralpopularsegm entationframeworks.

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81 TheMumford-Shahfunctionalleadstoaregion-basedapproa chtosegmentation.Earlycurveevolutionmodelsforsegmentationwere basedontheimage gradientsalonganevolvingcurve.Thiscouldleadtothecur vebecomingtrapped inlocalminimaoftheenergyfunction.EvaluatingtheMumfo rd-Shahfunctional involvesintegratingovertheentireregionenclosedbythe curve. Spectralclusteringisausefulgroupingtechniquewhichis relatedtograph partitioningalgorithms.Thegraphunderconsiderationha sanodeforeachvoxel intheimage,andtheedgeweightsareanitiesbetweennodes .Thisanitycanbe computedfromsomedistancefunctionbetweenvoxels.Ifwep artitionthegraphso thateachsubgraphshasmaximum"association"withinthegr oup,wewillachieve asegmentationoftheunderlyingimage.Thispartitioncanb eapproximatedby solvingfortheeigensystemoftheanitymatrixofthegraph Astatisticalapproachtoimagesegmentationwillalsobeex plored.Inthe MarkovRandomFieldmodel,themeasuredimageisthesumofan oiseprocess andaparametricmodeldependingonalabeleld.Thevalueof thelabeleldat eachvoxelisstatisticallydependentononlynearbyneighb ors.Wewillsegmentthe imagebyndingthemostprobablelabeleldgiventheobserv edimage. Inthefollowingsectionwewillcoverthesesegmentationmo delsinmore detail,andpresentresultsobtainedfromeach.

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82 5.4.1 Mumford-ShahModel Wecanusetherobustintrinsicmeananddistancesdenedpre viouslyinthe Mumford-Shahmodel.Theinterpolationmethodsdescribedp reviouslymaybe usedtomodelpiecewisesmoothregions. TheMumford-Shahfunctionalisgivenby E ( ;C )= Z n d 2 ( ;I ) dx + Z n n C jr j 2 dx + r I C ds (5.37) wherenistheimagedomain,and C isthecurveformingtheboundarybetween tworegions,andisapiecewisesmoothmodelfortheimage, I .Sinceminimizationofthisfunctionalinvolvessolvingforand C ,thisprocesscanbeseenas simultaneousrestorationandsegmentation.Theminimizer ofthisfunctionalwill haveaboundarywhosesmoothnessiscontrolledbytheparame ter r PartitioningnintotworegionswerewriteEquation( 5.37 )as E ( R ; R c ;C )= ( Z R d 2 ( R ;I ) dx + Z R c d 2 ( R c ;I ) dx )+ ( Z R jr R j 2 dx + Z R jr R c j 2 dx )+ r I C ds (5.38) where R istheregionenclosedby C R c ,istheregionoutside C Thepiecewiseconstantmodelsimpliesthegeneralmodel. E ( R ; R c ;C )= ( Z R d 2 ( R I ) dx + Z R c d 2 ( R c I ) dx )+ r I C ds (5.39) and 1 andwhere arethemeanoverregions R and R c respectively. TheMumford-Shahmodelofsegmentationmaybeimplementedi nacurve evolutionframework,whereaparametricrepresentationof C rowstowarda

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83 minimizerofthefunctional.ThisistheapproachtakenbyTs aietal.[ 81 ].The alternative,presentedbyChanandVese[ 19 ]isalevelsetapproach,wherethe curveisrepresentedasthezerolevelsetofanembeddingfun ction. Forintensityimages, d 2 ( ;I )maybe( I ) 2 ,and maybethearithmetic mean.Wemayusethegeodesicdistanceandgeodesicmeantose gmentHARDI images. 5.4.2 SpectralClustering Thename"spectralclustering"issuggestiveofthefacttha ttheeigenvalues oftheanitymatrixcanrevealthegroupingstructureinduc edbytheanity function.BachandJordan[ 3 ]showhowspectralclusteringcanbeseenasa relaxationofthenormalizedcutproblem.Normalizedcutsa ndtheresulting eigenvalueproblemhavebeendiscussedbyShiandMalik[ 74 ]. Fundamentaltotheproblemofsegmentationistheconceptof distance betweenvoxels.Weseekregionswhoseconstituentvoxelsar enearbyinsome sense.Thechoiceofdistancemeasure,andthereforeanity function,willhave agreataectontheresultsofspectralclustering.Intheca seofagray-scale image,theabsolutevalueoftheintensitydierencesisani ntuitivechoiceforthe distancefunction.ForHARDIdata,wecanusetheclosed-for mequationforcosine similaritytoformtheanitymatrix.Sampleeigenvectorim agesobtainedfrom HARDIdataareshowninFigure 5.9

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84 Figure5.9: S 0 andeigenvectorimages. 5.4.3 MarkovianModelsForSegmentation Markovchainsareusedtomodelsequencesofvariableswhose valuedepends onlyaxednumberofpreviousvalues.TheMarkovRandomFiel d(MRF)isan extensionofthisconceptto2D(andhigher).Hereweconside rrandomvariables onalattice, L .Thevalueoftherandomvariableateachlatticepoint(orsi te)is dependentonlyonthevaluesofitsneighbors.InFigure 5.10 weshownanexample ofarst-orderneighborhood, N 1 ,wherethevalueateachsitedependsonlyonthe 4adjacentsitesin L Withineachneighborhoodwedenecliques,setsofsitessuc hthateachpairof sitesareneighbors.Whentheserandomvaluesareinterpret edasimageintensities wecanusethislocalitytoimposesmoothnessconstraintson theimage.Ingeneral, forlarger(higherorder)neighborhoodswecanachievesmoo therimages.

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85 (a) N 1 (b) C 1 (c) C 2 (d) C 3 (e) C 4 Figure5.10:Firstorderneighborhood, N 1 ,andnearest-neighborcliques C 1 C 2 C 3 C 4 Afamoustheorem,theHammersley-Cliordtheorem,statest hatif F isa MRFwithrespecttosomeneighborhoodsystem,then F isalsoaGibbsRandom Field.AproofofthistheoremisgivenbyBesag[ 12 ]. P ( f )= 1 Z exp( U ( f ) T )(5.40) where T isaconstant, Z isanormalizingconstant,and U isanenergyfunctional withtheform U ( f )= X c 2 C V c ( f )(5.41) where V c ( f )iscalledthecliquepotential. 5.4.3.1 MRFsforSegmentation BeforeconsideringtheHiddenMarkovMeasureField(HMMF)s egmentation model,itmaybeusefultodescribetheclassicalMRFmodel.I nthismodel,the

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86 observationmodelisgivenby I ( x )= M X k =1 ( x; k ) b k ( x )+ ( x )(5.42) where x isapixellocation,and I istheobservedimage,isaparametric imagemodelwithparameters .Thesubscript k rangesoverthe M imageclasses. Anadditivenoiseeldisdenoted .Theimageisassumedtobepartitionedinto disjointregions, R k .Theindicatorfunction, b k returnsvalue1when x 2 R k ,and0 otherwise. Theunknownlabeleld, f ( x ),whichitisourgoaltoestimate,isrelatedto theindicatorfunctionby b k ( x )= ( f ( x ) k ).Itisthislabeleldwhichisassumed tobeasamplefromaMRF,withpriorprobabilitygivenbytheG ibbsdistribution, asinEquation( 5.40 ).Therearemanymodelsforthecliquepotentialfunction. TheIsingmodelisasimpleandquitecommonmodelgivenby V C ( f i ;f j )= 8>>><>>>: if f i = f j and i;j 2 C + if f i 6 = f j and i;j 2 C (5.43) Theparameter canbechosebytheusertocontrolthesmoothnessofthe eld. TheMRFanalysishasanintermediategoalofndingthecondi tionalprobabilityoftheunknownsgiventheobservations,inourcase P ( f; j I ).Itmay seemnaturaltoseekthemodeofthisdistribution,( f; )whichhavemaximum probability.Howeverthisestimator,theMAP(maximumapos teriori)isbut onepossiblechoice.Themeanormedianoftheposteriormayb emoreusefulfor

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87 certainapplications. x ( k )= X f : f ( x )= k P ( f; j I )(5.44) UsingBayesrule,wecanwritetheposteriordistributionas P ( f; j I )= P ( I j f; ) P ( f; ) P ( I ) (5.45) Since f and areindependent,wehave P ( f; )= P ( f ) P ( ).Thelabeleldis Markovian,so P ( f )istheGibbsdistributiongiveninEquation( 5.40 ).Theprior forthemodelparameters, P ( )dependsonthemodel.Ifwehavenoinformation abouttheparametersvalueswemayassignauniformdistribu tion. P ( I )isthe probabilityofobservingimage I overthespaceofallpossibleimages.Thisisa constantwewilldenoteby Z Thelikelihoodoftheobservations,inthepresenceofindep endentnoisewith distribution is P ( f; j I )= Y x 2 L P ( I ( x ) j f ( x )= k; )= Y x 2 L ( I ( x ) ( x; k ))(5.46) SincewecanassumeGaussiannoise,weusethenoisemodel P ( I ( x ) j f ( x )= k; )= r r exp( r j I ( x ) ( x; k ) j 2 )(5.47) where r isproportionalto 1 (onedividedbythenoisevariance). 5.4.3.2 HMMFsforSegmentation Marroquinetal.[ 55 ]presentavariationontheMRFsegmentationmodel whichhasfewervariablesandcansolvedwithoutslowstocha sticmethods.The HMMFmodelusesadeterministicmodelforthelabels.Thelab eleldisassumed

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88 tocomefromaeldofdistributions.IntheclassicalMRFmod el,theindicator vectorreturns1foronlyasingleclassateachpixel,and0fo rallothers,resulting inahardsegmentation.IntheHMMFmodel,indicatorfunctio nisreplacedbythe peldwhichcanassignnonzerovaluestoseveralclassesper pixel. Theposteriorinthiscaseisgivenby P ( p; j I )= P ( I j p; ) P ( p ) P ( ) P ( I ) (5.48) Wecanwritethejointconditionalprobability P ( I ( x ) ;f ( r ) j p; )intermsofthe conditionalprobability P ( f ( x ) j p; ) P ( I ( x ) ;f ( r ) j p; )= P ( I ( x ) j f ( x ) ;p; ) P ( f ( x ) j p; )(5.49) Weobtain P ( I ( x ) j p; )bymarginalizingoverallpossiblelabels P ( I ( x ) j p; )= M X k =1 P ( I ( x ) j f ( x )= k;p; ) P ( f ( x )= k j p; )(5.50) Inthismodeltheimageobservationsdependonthelabeleld (thepeldis thehiddeneldreferredtointhenameHMMF),so P ( I ( x ) j f ( x )= k;p; )= P ( I ( x ) j f ( x )= k; ).Thesecondprobability, P ( f ( x )= k j p; ),issimplythe denitionofthepeld. P ( I ( x ) j p; )= M X k =1 P ( I ( x ) j f ( x )= k; ) p k ( x )(5.51) where P ( I ( x ) j f ( x )= k; )issameasfortheMRF,Equation( 5.47 ).

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89 Thesmoothnessconstraintonthepeldisenforcedwiththeq uadratic potential V x;y ( p ( x ) ;p ( y ))= j p ( x ) p ( y ) j 2 = M X k =1 ( p k ( x ) p k ( s )) 2 (5.52) where isaparametercontrollingthesmoothnessoftheeld,and( x;y )areinthe sameclique. Theposteriordistributioncanbewrittenas P ( p; j I )= 1 Z exp[ U ( p; )](5.53) with U ( p; )= X x 2 L M X k =1 log( p k ( x ) P ( I ( x ) j f ( x )= k; ))+ X C V C ( p ) log P ( )(5.54) TheMAPestimators p and for p; arethosewhichmaximizetheposterior. Thesemaybeestimatedbyminimizingtheposteriorenergyin Equation( 5.54 ). Marroquin[ 55 ]presentsagradientprojectionNewtoniandescentalgorit hmforthis optimizationproblem.Thentheoptimallabeleld, f ,isdeterminedbytakingthe k forwhich p k ( x )isamaximumateachvoxel.Alternatively,wemayconsidert he eldofprobabilitiestobeasoftlabeling. 5.4.3.3 Results TheresultsoftheHMMFsegmentationusingthegeodesicdist anceappliedto syntheticHARDIdataarepresentedbelow.Thersttwodatas etsarepiecewise constantvMFeldswithtworegions.Theresultsarepresent edinFigure 5.11 Theleftimageshowsthesegmentationobtainedfromaeldwh erethetworegion

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90 Figure5.11:HMMFsegmentationofsyntheticdata. dierindirection.Intherightimage,theregionsdieronl yintheconcentration parameter, .Therearenoclassicationerrors. Figure5.12:HMMFsegmentationofsyntheticdata. InFigure 5.12 theresultsforsegmentationofvMFmixturesisshown.The dataconsistsofseveralpiecewiseconstantareasandacros sing.Herethealgorithm hascorrectlysegmentedeachregionandthecrossing. Nextthealgorithmwastestedoncurvedregions.Asynthetic datasetconsistingofacircularregionwithvMFsorientedtangentiallywas created.Theresults areshowninFigure 5.13 .Atworegionsegmentationwascomputedintheleft image,andathreephasesegmentationwascomputedintherig htimage.Notethat

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91 thetwophasesegmentationhasidentiednearlytheentirec ircularregion,even thoughthesegmentationmodelispiecewiseconstant.Three regionswassucient tosegmenttheentirecircularregion. Figure5.13:HMMFsegmentationofsyntheticdata. Finally,thealgorithmwastestedonadatasetwithcurvedge ometryand crossings.TheresultsareshowninFigure 5.14 .Inthiscase,thealgorithmwas Figure5.14:HMMFsegmentationofsyntheticdata. abletodiscriminatebetweenadjacentregionswithmultipl edirections.

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CHAPTER6 VISUALIZATIONTECHNIQUESFORSEGMENTEDSURFACES Inmedicalimagingweoftendealwithvolumedata,andvolume visualization techniquesareanaturalwaytovisualizethedata.Byusingt echniquessuchas raycasting,wecanobtainprojectionsofthedata.Thesepro jectsresembleX-ray imageswhichradiologistsandothercliniciansmaybefamil iarwithanalyzing. Forscalarvalueddatathistypeofvisualizationmaybesuc ient.Forvectorand tensorvalueddata,projectionisnotausefultechnique,an dthisdataisoften visualizedslice-by-slice. Wecanconsidertheslicetobearestrictionoftheeldtoapl anarsurface. Thisideacanbegeneralizedtorestrictingtheimagedatato ageneralsurfaceof interest.Havingperformedasegmentationwemaywishtovis ualizethesurfaceof interestalongwithsomeotherdiusionproperties.Wealre adyhaveseenhowto usetexturesynthesistechniquestoconveydiusionproper ties.Inthissectionwe willdescribeaframeworkforperformingthesamevisualiza tiononsurface.The frameworkwepresentisageneraltechniqueformultiresolu tionadaptivemeshing ofisosurface,andsimultaneouslyparameterizingthesurf acesothatismaybe texturemapped. 92

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93 6.1 SignedDistanceFunctions Aconvenientformforimplicitfunctionsisthesigneddista ncefunction(SDF). TheSDF, ,hastheproperty jr ( x ) j =1forall x .Wecanconvertanyimplicit functiontoanSDFby d dt =sign( )(1 jjr jj )(6.1) aslongasthezerolevelsetof representsthesurfaceofinterest,andthesign of iscorrecteverywhere.InimplementingEquation( 6.1 )itisimportanttouse upwindnitedierences,ortheGodunovschemesincewehave adiscontinuityat =0.Recoveryofthesigneddistancefunctionisdiscussedby Sethian[ 73 ]. 6.2 MultiresolutionMeshing Wewouldpreferatriangulationwithwithtwoproperties:tr iangleshould have"good"aspectratio(ratioofmaximumedgelengthtomin imumedgelength), andtrianglesshouldbedenserinregionsofmoredetail(hig hcurvature).Inthe followingsectionwewillreviewourmultiresolutionappro achtogeneratingsucha triangulation. 6.2.1 MeshGeneration Themostcommontechniquefortriangulatingimplicitsurfa cesisthemarching primitivealgorithm,wheretheprimitivemaybecubes[ 52 ],tetrahedra[ 41 ],or someothergeometricshapedependingonthegeometryofthed ataacquisition. Unfortunately,thesealgorithmsgeneratepoortriangulat ions.Wewillusemarching tetrahedratogenerateaninitial,low-resolutionmesh,an drenethismeshto obtaintothenaltriangulationoftheimplicitsurface.

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94 6.2.2 SurfaceSimplication Meshsimplicationcanachievetwoobjectives:improvingt hequalityofthe triangulation,andreducingthecomplexityofthetriangul ation.Simplication isimplementedthroughafewelementarytransformations,a ndthethreemost commonareshowninFigure 6.1 Fortrianglemeshsimplication,thebasicoperationisthe edgecollapse(or edgecontraction).Iftwo"skinny"trianglesshareashorte dge,thetriangulation maybeimprovedbyshrinkingthisedgetoapoint,eliminatin gtwotriangles. Theoperationiscomplicatedbythedecisionofwhichpointt ocollapsetheedge to.Wemaychoosefrombothendpointsandthemidpoint,depen dingonwhich congurationleadstoalowerenergystate.Alternatively, theslowerbutmore accuratesolutionistoperformthisenergyminimizationco ntinuously,allowingany pointalongtheedgetobechosen. Anotheroperationistheedgerip(oredgeswap).Twotriangl eswhichshare anedgeformaquadrilateral,withthesharededgebeingoned iagonalofthe quadrilateral.Theedgeripoperationreplacesthesetwotr iangleswiththetwo formedbytakingtheoppositediagonalofthequadrilateral .Thisoperationdoes notreducemeshcomplexity,butitmayimprovethemeshquali ty. Theedgesplitoperationreplacesanadjacentpairoftriang leswith4triangles bysplittingeachacrossthesharededge. Figure 6.1 ashowsanexamplemesh,withtheselectededgeshowninbold. Thesameareaofthemeshaftercollapseoftheselectededgei sshowninFigure

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95 (a)OriginalMesh (b)EdgeCollapse (c)EdgeSwap (d)EdgeSplit Figure6.1:Simplicationoperations. 6.1 b,afterrippingtheselectededgeisshowninFigure 6.1 c,andaftersplittingthe selectededgeisshowninFigure 6.1 d. Therearemyriadmeshsimplicationalgorithmswhichdier onlyinhowthey decidewhichedgestooperateonandwhichoperationtoapply .Suchalgorithms arepresentedbyGarlandandHeckbert[ 35 ],Hoppe[ 44 ],andHoppeetal.[ 45 ]. Allattempttokeepthesimpliedmeshneartheoriginalmesh .Itisimportant torecognizethattheseoperationsmayhaveundesirablecon sequences,suchas changingthetopologyofthemesh,orgreatlychangingdihed ralangles.For instance,intheedgeswapoperation,ifthequadrilateralf ormedbytwotrianglesis notconvex,theedgeripisinvalid. Forimplicitsurfacesrepresentedasanarrayofvaluessamp ledfroman implicitfunction,wecanperformamuchmoreecientsimpli cation.Thearray canbereducedinresolutioninthesamemannerasanimage.By repeatedly smoothingandsubsamplingtheimagewecanbuildanimagepyr amid.The

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96 triangulationsofthezero-valuedcontoursofeachimagein thepyramidcanbe consideredmultiresolutionhierarchyofmeshes. Wewilluseedge-ripsandedge-collapseoperationstoimpro vethequalityof thetriangulation. 6.2.3 AdaptiveSurfaceRenementWithErrorBounds Givenasimpletriangulationofreasonablequality,wenoww ishtorene themeshtorecoverthedetailinthehighestleveloftheimag epyramid.We accomplishthisbysubdividingthemesh,anddisplacingthe verticessothatthey lieonthezerolevelsetoftheimplicitfunction.Ourgoalis tominimize 2 ( x i )for allvertices f x g .Byallowingtheverticestorowalongthedirectionofthene gative gradientoftheSDF,wetthesubdividedmeshtotheimplicit surface. d x dt = r ( x )(6.2) Thesubdivisionprocessresultsinaregulartriangulation ,buttherowmay causetrianglestobunchup,resultinginapoorqualitytria ngulation.Tocounter thiseectwechoosetoaddasmallregularizingforcewhichp ullseachvertex towardthecentroidofallneighboringvertices.Themagnit udeofthisregularizing forceiscontrolledbytheparameter, k .Bymakingthevalue k datadependent,we canachieveanadaptivetriangulation.Sincewewantmorede tailwherethesurface changesmostrapidly,wemakethevalueof k afunctionofthelocalcurvatureof theimplicitsurface.Weestimatethiscurvaturebycomputi ngthecurvatureofthe

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97 SDFatthelocationofeachvertex. = r r jr j (6.3) = ( xx + yy ) 2 z 2 x y xy +( yy + zz ) 2 x 2 y z yz +( zz + xx ) 2 y 2 z x zx 2( 2 x + 2 y + 2 z ) 3 2 Letting k =1+ j j where isascaleparameterwillresultinshorteredgeinareas ofhighcurvature. ThevalueoftheSDFateachvertextellsushowfarthatvertex isfromthe implicitsurface.Usingthisfact,itispossiblecomputese veraldierenterror metrics,suchasmaximumerrorormeanerror.Therowisconti nueduntilthe errorisreducedbelowsomethreshhold. 6.2.4 ComputingtheCutGraph Ameshcanbetopologicallyalteredbyremovingvertices.Ac utgraphisthe setofedgeswhich,whenremovedfromthemesh,resultsinasu rfacetopologically equivalenttoadisk.Thediskcanthenbeparameterizedbyma ppingtheboundary ofthedisktotheboundaryoftheparameterspace(usuallyas quare),then mappingtheinteriorverticesofthedisktointeriorvertic esoftheparameterspace. Onemethodofcuttingasurfaceintoadiskistondthecanoni calpolygonal schemaofthemanifold.Thissetconsistsof2gcycleswhicha llpassthrougha commonpointofMsuchthatremovingtheedgefromMwillresul tinatopological disk.Amoregeneralstructureisthecutgraph.Thisisdene dasanarbitrary setofedgeswhoseremovalresultsinatopologicaldisk.Byr equiringallpathsto passthroughasinglepoint,thecanonicalschematacanresu ltinlongboundary

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98 edges.Itcanbeseenfromtheobviousconstructionofthecut graphfora2-holed torusthatboundarylengthcanbereducedwhenthe"commonpo int"restriction isdropped.Longercutlengthscanmaketheparameterizatio nmoredicult,and adverselyaectitsquality. Manymedicalimagingmodalitiesproduceimagevolumes,fro mwhichanatomicalstructurescanbesegmented.Thesesegmentationrouti nesoftenproduce,as output,asigneddistancefunctionwhosezerolevelsetrepr esentsthesurfaceofinterest.Thiseldcanbetriangulatedbymarchingcubes(ors omeothertechnique) forvisualization.Ourgoalinthisresearchtovisualizeth issurfacewithatexture appliedtothemesh.Thistexturemayrepresentotherscalar orvectorquantities whichwouldaidtheuser. Thissigneddistancefunctionmaysimplifymanysweepplane algorithms.By nature,theSDFrequiresnopresortingbasedoncoordinates .Wecananalyzeslices ofthiseldtodeterminetopologicalchangewhenconstruct ingtopologicalgraphs, asthetechniqueofSteinerandFischer[ 77 ]require. EricksonandHar-Peled[ 29 ]describeagreedyalgorithmwhichcanapproximatetheminimumlengthcutgraphinO(g 2 logn)time.Althoughourgoalisnot strictlytondtheshortestpath,thistechnique,basedonD ijkstra'ssinglesource shortestpathalgorithmisausefulreference.DeyandSchip per[ 26 ]presentaO(n) algorithmforndingpolygonalschema.Thisworkalsoprese ntsausefulalgorithm fordeterminingwhetheracycleiscontractibleinlinearti me.LazarusandVegter [ 86 ]ndcanonicalpolygonalschemainO(gn)time.Althoughwed onotwant

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99 canonicalpolygonalschemabecauseofthedistortionthese cutscause,thetechniquesinthispapermayproveuseful.SteinerandFisher[ 77 ]constructadistance functiononthemeshandatopologicalgraph.Cutsarestarte datbranchesofthe topologicalgraphandareguidedbytheiso-curves.Triangl esmaybesplitasthe iso-curvesmaynotalwaysbealongedges.Weprefernottobre akourtriangles, severalaspectsofthisapproachseemtoapplytoourproblem .Wecanconstruct geodesiciso-curvesusingthelevelsetmethodusingonlyth eSDF.Guetal.[ 37 ] presentamethodwhichincrementallyremovesfacesandedge sfromthemeshina waywhichleavesanumberofloopsandsomeextraneousedges. Thisinitialcutis thenrenedinawaythatminimizesthedistortionmetric.Ni etal.[ 60 ]useMorse theorytodeducethetopologicalstructureofamesh.Theyco nstructafunctionon thesurfacewhosecriticalpoints(pointswherethegradien tbecomeszero)lieon thecutpath,andcomputethepathbyfollowingthegradiento fthe"fair"Morse function.Hart[ 43 ]discussesMorsetheorywithregardstoimplicitsurfaces. An interestingandusefulndingisthattheEulercharacteris ticoftheimplicitsurface canbecomputedasthesumsofdierentclassesofcriticalpo intsoftheimplicit function. InFigure 6.2 weshowthecutgraphs(edgesingreen)forsurfacesofvaryin g genus.Thetorus(genus=1)requires2cutstobereducedtoad isk,the2-torus (genus=2)requires4cuts,andthetanglecube(genus=5)req uires10cuts.

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100 Figure6.2:Examplecut-graphsfortorus,2-torusandtangl ecube 6.2.5 MeshParameterization Theproblemofmeshparameterizationismorecomplicatedfo rhighergenus surfacesthanforsurfacesofgenuszero.Theconformalmapp ingofagenuszero surfacetothespherecanbeachievedbyaharmonicmap,andth espherecan betriviallymappedtotheplanebyexpressingthevertexpos itionsinspherical coordinates.Anymeshhomomorphictothespherecanbeparam eterizedinthis way.Thisprocessisoftenreferredtoassphericalparamete rization[ 72 ].Highgenus surfacemustbecutusingoneofthetechniquesdiscussedint heprevioussection. Theparameterizationcanbeusedformanygraphicsapplicat ions.The parameterspacecanbeconsideredtobethedomainofsomeima ge.Byassociating each3Dvertexwithits2Dparameterwecanapplythisimageto themesh bytexturemapping.Wemayalsousetheinversemapping,from 2Dto3D,to synthesizenormalmapsandgeometryimages[ 38 ].The3coordinatesofthenormal vectororvertexpositioncanbestoredinthe(R,G,B)channe lsofacolortexture. Themeshparameterizationcanalsobeusedtoobtainaregula rorsemi-regular remeshing,tocompressthemesh,ortosimplifythemesh.

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101 Therearetwometricswhichareusefulwhenmeasuringthequa lityofthe parameterization.Thestretchmetricistheratiooffacear eainparameterspaceto faceareaonthemanifold.Aconformalitymetriccanmeasure howwellanglesare preservedbythemapping.Thetopicofconformalmappinghas beencoveredby Gu[ 39 ].Sincethechoiceofmetricisdeterminedbyapplication,w ecanconsidera weightedsumofthetwometrics. Figure6.3:Stretchminimizing(left)andconformalparame terization(right). Meshparameterizationtechniquesmakeitpossibletogener atetexture coordinatesforsurfacesofarbitrarytopology.Manysucht echniquesaresurveyed byGotsman[ 36 ].Additionally,theresultingmappingfromthemanifoldto the planecanbeinverted,togenerategeometryimagesandnorma lmaps.Theinverted mappingmayalsobeusedtomapscalardataforthediusionte nsororODF eldontothesurfaceofinterest.Forexample,wemayperfor mLIConthesurface byseedingstreamlinesatpointsmappedtofromeachtexture elementinthe parameterspace. Wewillusethisconcepttotexturemapisosurfacescomputed fromsegmentingdiusionimages.Wewillseethatthisparameterization canbeeciently incorporatedintothemultiresolutionmeshingschemedisc ussedearlier.

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102 Previousmultiresolutionsolutionsinvolvesimplifyingt hethemeshusing asequenceofedgeripsandcollapses,andsavingthesequenc eofoperationsso thatthesimplicationmaybeincrementallyreversed.Thes impliedmeshis thenparameterized,thentheparameterizationisupdateda sthemeshisrened. Instead,weusetheSDFimagepyramid. AtthelowestleveloftheSDFpyramid,parameterizethesimp liedisosurface. Whensubdividingthemesh,alsosubdividethemeshrepresen tingtheimage ofthemapping. Afterreningthemesh,renetheparameterization. Figure6.4:Texture-mappedsegmentationofspinalcord(le ft)andbrain(right).

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CHAPTER7 CONCLUSION Inthissectionwewillreviewthecontributionsofthiswork Wehavepresentedanewvariationalformulationforrestori ngHARDIdata,a FEMtechniqueforimplementingtherestorationandtwotech niquesforvisualizing neuronalbersfromthisrestoreddata.Therestorationimp osestwotypesof smoothnessconstraints.Therstissmoothnessoverthesph ericaldomainofacquisitiondirections,andthesecondissmoothnessbetweennei ghboringvoxelsinthe cartesiandomain.Thesmoothingtechniqueiscapableofpre servingdiscontinuous detailinthedata.Thiswasdemonstratedonsyntheticandre alanatomicaldata. ByusingJ-divergenceasameasureofdistancebetweendistr ibutions,wewereable toshowquantitativelythatthecombinationofrestoration techniquesperforms betterthaneithertechniquealone. AstatisticalpropertyoftheODF,namelyentropy,wasshown tobeauseful indicatorofanisotropy.Resultsshowinghowcoloranisotr opyimagescomputed fromtheODFcouldbeusedtovisualizediusiondirectionan danisotropyin HARDIdatawerepresented.Theeectoftherestorationonth esemeasureswas showntoimprovetheclarityoftheimages. Wehaveintroducedanovelmodelfororientationaldiusion withmixtures ofvonMises-Fisherdistributions.Thismodelwasshowntol eadtoclosed-form 103

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104 expressionsfordistancesandanisotropymeasures.Ageode sicframeworkfor workingwiththismodelwasalsopresented.Theresultswere appliedwithin3 segmentationframeworks,andtheresultswerepresented. Finally,asurfacerenderingapproachtovisualizationofm edicalimages wasproposed,andappliedtoHARDI.Theproposedtechniques utilizedtexture mappingtoconveypropertiesofdiusion,suchasorientati onandanisotropy,on segmentedsurfaces.Aframeworkwaspresentedforecientl yimplementingtexture mappedsurfacerenderingformedicalimaging.Theframewor kisgeneralenoughto beusedinconjunctionwithavarietyofexistingtriangulat ionmethods,cut-graph algorithmsandmeshparameterizationschemes.

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BIOGRAPHICALSKETCH TimMcGrawwasborninKeyWest,Florida.HereceivedhisBach elorof SciencedegreefromtheMechanicalEngineeringDepartment attheUniversity ofFlorida.HewillreceivehisPh.D.degreeincomputerengi neeringfromthe UniversityofFloridainAugust2005.Hisresearchinterest sincludemedical imaging,imageprocessing,computervisionandcomputergr aphics. 114


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DENOISING, SEGMENTATION AND VISUALIZATION OF DIFFUSION
WEIGHTED MRI

















By
TIM E. MCGRAW


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005


































Copyright 2005

by
Tim E. McGraw


















For my wife, Jo, and my mother, Patti.
















ACK(NOWLED GMENTS


I am very grateful for the advisement and encouragement of my coninittee

during the preparation of this work. I wish to first thank the chair of my com-

nmittee Dr. Baha C. Venturi for his academic and professional guidance through

the years. His consultation and advice made this work possible. Thanks go to Dr.

Thomas Mareci for productive weekly meetings and important clinical input. The

guidance and input from Dr. Anand Ragarel ] .n Dr. Ben Lok and Dr. Jeff Ho are

much appreciated. Thanks go to everyone on my coninittee for all of the time and

energy they have invested into my education and research.

I wish to express my gratitude to everybody in the Imaging and Visualization

department at Siemens Corporate Research, especially Dr. Jini Williams and Dr.

Alariappan Nadar for giving me the opportunity to work with them as an intern

for two suniners. Thanks also go to Tomy Clu! 0! at SCR for making the internship

process run smoothly.

Researchers at the McE~night Brain Institute at the University of Florida

have provided valuable clinical input and fruitful discussions. Thanks go to

Evren Ozarslan, Dr. Steve Blackhand, Dr. Robert Yezierski and Dr. Paul Reier.

Specifically, thanks go to Evren Ozarslan for the synthetic data and fast ODF

computation technique. Without the AIRI data this research would have been











impossible, so I also wish to thank Sara Berens and Robert Yezierski for providing

the spinal cord data sample and Ron H .1-; a for the brain data.

Present and former faculty nienters of the CISE department have been

responsible for many thought-provoking courses and stimulating discussions,

including Dr. .Jorg Peters, Dr. Meera Sitharant, Dr. Gerhard X. Ritter, Dr.

.Joseph Wilson, Dr. Mark Schnmalz, Dr. Alper IUngor, Dr. Tint Davis, Dr. Richard

T. ein-~! ll. Dr. Douglas Dankel, Dr. David Gu, and Dr. Yan-Hang Lee.

Faculty in the Math department at ITF have proven invaluable to my studies

through courses they have offered and through their feedback and input. Thanks go

to Yunnmei C'I. in~ Murali Rao, .Jay Gopalakrishnan, David Groisser, and Tint Olsen.

Former and present students in the CVGMI Lab have made the research

experience most pleasant and also provided valuable input to my studies and re-

search, including .Jundong Liu, Zhizhou Wang, Fei Wang, Eric Spellnian, Santhosh

K~odipaka, .Jie Zhang and Hongyu Guo.

Classmates, friends, and other students in the department have provided

camaraderie and welcome distraction. Alany thanks go to Andres Mendez-Vasquez,

Andy Shui, Xiaohin Wu, Bruno Alaciel, Lewey Geselowitz, Ashish Myles and many

others.

The office staff of the CISE department have ak-- 1-< heen very helpful with the

administrative details. Special thanks go to .John Bowers for making the process

much easier and more pleasant. Thanks go to the systems staff for keeping all of

the hardware and software running.










I would also like to thank my family, especially my wife Jo, for their support

and patience.

This research was supported in part by the grant NIH-NS42075 and by a grant

from Siemens Corporate Research.



















TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... .. iv

LIST OF FIGUR ES ......... . .. ix

ABSTRACT ...... ...... .......... xii

1 INTRODUCTION ......... ... 1

1.1 Diffusion AIRI ......... .. :3
1.1.1 Overview of Diffusion .... .. .. 5
1.1.2 Overview of AIR Imaging ... .. 7
1.1.3 Diffusion Weighted Images ... .. .. 8
1.1.4 Diffusion Tensor AIRI Acquisition .. .. .. 9
1.1.5 High Angular Resolution Diffusion Imaging .. .. 11
1.2 Restoration ......... ... 12
1.3 Tractography ........ .. 14
1.4 Anisotropy ..... .... 15
1.5 Overview of Our Modeling Scheme ... .. .. .. 16

2 DENOISING . ...... ... 18

2.1 TV Norm Minintization of S(x,y,z) .. . .. 18
2.1.1 Scalar TV Norm Minintization ... .. .. 19
2.1.2 Vector TV Norm Minintization .. .. .. 20
2.2 Variational Principle . ... .. 21
2.3 Spatial Lattice Smoothing of S(.r) ... ... .. 22
2.3.1 Fixed-Point L .----d-Diffusivity ... .. .. 2:3
2.3.2 Discretized Equations .... .. .. 24
2.4 Finite Element Smoothing of S(0, ) ... .. .. .. 25
2.4.1 Element Matrices .... ... 27
2.4.2 Local Element Coordinates .... .. :30
2.4.3 Global Matrices ..... .. .. :34

:3 VISITALIZING PROBABILITY FIELDS ... .. .. ._ 36

:3.1 Computing Probabilities .... ... :36
:3.2 Glyph Visualization . .... .. .. :37
:3.3 Scalar Measures of Anisotropy ... .. .. .. 38
:3.4 Visualizing the Directional Nature of Diffusion .. .. .. 42












3.4. 1 Streamtubes . .. .. .. 44
3.4.2 Line Integral Convolution .. .. .. .. 45

4 RESULTS ........ . .. 50

5 SEGMENTATION OF HARDI ..... ... 58

5.1 Modelling Diffusion . .. .. .. 58
5.1.1 Other Models . . .. 58
5.1.1.1 Gaussian Mixture Model .. .. .. 59
5.1.1.2 Spherical Harmonics ... .. .. .. 60
5.1.1.3 Generalized Tensors .. .. .. 60
5.1.2 von Mises-Fisher Mixture Model .. .. .. 61
5.1.2.1 Fitting the vMF Mixture .. .. .. 64
5.1.2.2 Entropy and Distance ... .. .. .. 65
5.2 The Space of vMF Distributions .... .. .. 67
5.2.1 Riemannian Geometry ... .... .. 67
5.2.2 Riemannian Exp and Log Maps .. .. .. 70
5.2.3 Symmetric Spaces ..... .. 72
5.2.4 Lie Groups and Homogeneous Spaces .. .. .. .. 73
5.2.5 Results ......... .. 76
5.3 The Space of vMF Mixtures .... .. .. 76
5.3.1 Riemannian Exp and Log Maps .. .. .. 77
5.3.2 Intrinsic Mean . .... .. 80
5.4 Segmentation Models . .... .. 80
5.4. 1 Mumford-Shah Model ... .... .. 82
5.4.2 Spectral Cla ilI ty!! .... .. .. 83
5.4.3 Markovian Models For Segmentation .. .. .. .. 84
5.4.3.1 MRFs for Segmentation .. .. .. 85
5.4.3.2 HMMFs for Segmentation .. .. .. .. 87
5.4.3.3 Results ...... .. 89

6 VISUALIZATION TECHNIQUES FOR SEGMENTED SURFACES 92

6.1 Signed Distance Functions ..... ... .. 93
6.2 Multiresolution Meshing .. .. 93
6.2.1 Mesh Generation ...... .... 93
6.2.2 Surface Simplification ... .. . .. 94
6.2.3 Adaptive Surface Refinement With Error Bounds .. 96
6.2.4 Computing the Cut Graph ... .. .. 97
6.2.5 Mesh Parameterization .. .. .. 100

7 CONCLUSION ......... .. .. 103

REFERENCES ......... . .. 105

BIOGRAPHICAL SK(ETCH .....__. ... .. 114


















LIST OF FIGURES


Figure page

1.1 Orientational heterogeneity in DTI (left), and HARDI (right). .. 2

1.2 Diffusion ellipsoid .. ... .. 6

2.1 TV(fl) >TV(f2) =TV(f:3) .. ....... .... 19

2.2 HARD gradient directions (0, ~) correspond to the vertices of a sub-
divided icosahedron. .. ... .. 26

2.3 Two dimensional (usv) domain for FEM-hased smoothing of HARDI
data...... ................. 26

2.4 Mapping to harycentric coordinates ..... .. :30

:3.1 Original ODF (left), nmininiun probability sphere (center), and sharp-
ened ODF (right). ......... .. :37

:3.2 ODF glyphs ......... .. :38

:3.3 FA image (left), generalized anisotropy (center), Shannon anisotropy
(HA) (right), front coronal slice of rat brain ... .. .. :38

:3.4 R ii- I-r volume visualization of HA. .... .. 40

:3.5 MIP volume visualization of HA. ..... .. 40

:3.6 Shannon anisotropy difference between Gaussian ODF and general
ODF. ........ .... ...... .... 41

:3.7 Anisotropy computed from H2 (top-left), Hg (top-right), Hlo (hottoni-
left), H20 (bOttOm-right). . ...... .. 42

:3.8 Ri~nyi entropy differences H1 H2 (top-left), H1 Hg (top-right),
H1 Hlo (hottom-left), H1 H20 (bOttOm-right). .. .. .. 42

:3.9 Color FA. ......... . 4:3

:3.10 ED colors for synthetic distribution field. ... .. .. 44

:3.11 ED colors for synthetic distribution field. ... .. .. 44

:3.12 Streanitubes in axial slice of brain. .... .. .. 45










3.13 LIC visualization of synthetic field with kernel widths 5, 15, and 55.

3.14 LIC fiber visualization in axial slice of brain.

3.15 LIC fiber visualization in coronal slice of brain..

3.16 LIC fiber visualization in sagittal slices of brain.

4.1 ODF profiles overlaid on HA image (left), and color anisotropy (right)
of the synthetic dataset.

4.2 Multi-LIC visualization of synthetic dataset.

4.3 Synthetic S (left), resulting ODF (right).

4.4 Synthetic S with noise added (left), resulting ODF (right).

4.5 Manifold (FEM) smoothing results for S (left), resulting ODF (right).

4.6 Lattice (TV) smoothing results for S (left), resulting ODF (right).

4.7 Manifold and lattice smoothed S (left), resulting ODF (right).

4.8 Original diffusion-weighted image (left), and denoised (right) from
spinal cord data.

4.9 Original color anisotropy image (left), and denoised (right) from spinal
cord data.

4.10 Original (left), and denoised (right) HA images for coronal slices of
rat brain.

4.11 Original (left), and denoised (right) ED images for coronal slices of
rat brain.

5.1 Example vMF distributions (it = 1, 5, 10, 15, 25). All distributions
have same mean direction, p.

5.2 Sample vMF mixtures for voxels with one and two fibers.

5.3 Intrinsic and extrinsic distance.

5.4 Tangent space.

5.5 Riemannian exponential map .

5.6 Riemannian log map


5.7 Points along the geodesic between

5.8 Points along the geodesic between


two vMF distributions..

two sets of weightss.











5.9 So and eigenvector images. . ...... .. 84

5.10 First order neighborhood, NI~, and nearest-neighbor cliques C1, C2,
C3,c 4.......... .. ........ 85

5.11 HMMF segmentation of synthetic data. .... .. 90

5.12 HMMF segmentation of synthetic data. .... .. 90

5.13 HMMF segmentation of synthetic data. .... .. 91

5.14 HMMF segmentation of synthetic data. .... .. 91

6.1 Simplification operations. ........ .. .. 95

6.2 Example cut-graphs for torus, 2-torus and tangle cube .. .. .. .. 100

6.3 Stretch minimizing (left) and conformal parameterization (right). 101

6.4 Texture-mapped segmentation of spinal cord (left) and brain (right). 102
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



DENOISING, SEGMENTATION AND VISUALIZATION OF DIFFUSION
WEIGHTED MRI




By

Tim E. McGraw

August 2005

C'I ny~: Baba C. Vemuri
M r~~ Department: Computer and Information Sciences and Engineering

Despite its apparent success, Diffusion Tensor MRI (DT-MRI) has significant

shortcomings when the tissue of interest has a complicated structure. This is

due to the relatively simple tensor model that assumes a unidirectional-if not

isotropic-local structure. As a more viable alternative Tuch et al. have proposed

to do the data acquisition such that the diffusion sensitizing gradient directions

sample the surface of a sphere. In this high angular resolution diffusion imaging

(HARDI) method, one does not have to be restricted to the tensor model. Instead,

it is possible to calculate diffusion coefficients independently along many directions.

This imaging technique can reveal white matter fiber crossings which would not be

apparent in DT images.











In this paper, we present a novel variational formulation for restoring the

HAR DI data and visualizing the fibers front this restored data. This formulation

involves smoothing signal measurements over the spherical domain and across the

3D image lattice. The smoothing on the spheres at each lattice point is achieved

using first and second order smoothness constraints, and across the lattice via a

total variation norm based scheme. For the smoothing problem on the sphere,

we use the finite element method (FEM). Unlike the reported work on spherical

harmonic hasis expansion of the diffusivity function on the sphere, the FEM hasis

functions have local support and are better suited for preserving details in the

data. To visualize the fiber paths, the probability values for water molecules to

move a particular distance along different orientations were calculated using a

Laplace series expansion of these probabilities. We compute the Shannon entropy

of this distribution to characterize the anisotropy of diffusion, with higher entropy

corresponding to lower anisotropy. These local distributions are also used to

compute a vector quantity called expected direction. Surfaces rendered using colors

corresponding to expected direction reveal anisotropy and fiber direction in the

imaged tissue. Further, examples are presented to depict the performance of the

HARDI data restoration and visualization schemes on rat brain, spinal cord, and

synthetic data sets.
















CHAPTER 1
INTRODUCTION


Fundamental advances in understanding living biological systems require

detailed knowledge of structural and functional organization. This is particu-

larly important in the nervous system. Understanding fundamental structural

relationships is essential to the development and application of therapies to treat

pathological conditions such as disease or injury.

Observing the directional dependence of water diffusion can allow us to infer

structural information about the surrounding tissue. White matter fiber bundles

present a barrier to diffusion, causing relatively high diffusivity along the fiber

direction, and lower diffusivity across the fiber.

MR measurements can be made sensitive to the translational diffusion of

water molecules by the utilization of magnetic field gradients [78]. In general,

the signal acquired depends on the strength and the direction of these diffusion

sensitizing gradients. Repeated measurements of water diffusion in tissue with

varying gradient directions provide a means to quantify the level of anisotropy

as well as to determine the local fiber orientation within the tissue. In a series

of publications, Basser and colleagues [8, 7, 9] have formulated a new imaging

modality called "diffusion tensor MRI (DT-MRI)" that employs a second order,

positive definite, symmetric diffusion tensor to represent the local tissue structure.










They have proposed several rotationally invariant scalar indices that quantify

different aspects of water diffusion observed in tissue, similar to different "stains"

used in histological studies [5]. Under the hypothesis that the direction along which

the diffusion coefficient is largest will yield the local fiber orientation, one can

determine the directionality of neuronal fiber bundles. This fact has been exploited

to generate fiber-tract maps that yield information on structural connections in

human [9, 47, 53, 22] as well as rat brains [59, 94, 88, 57, 56] and spinal cords [87].

















Figure 1.1: Orientational heterogeneity in DTI (left), and HARDI (right).


DT-AIRI has significant shortcomings when the tissue of interest has a com-

plicated geometry. This is due to the relatively simple tensor model that assumes

a unidirectional-if not isotropie-local structure. In the case of orientational het-

erogeneity, DT-AIRI technique is likely to yield incorrect fiber directions, and

artificially low anisotropy values. This is due to violation of the assumption of

Gaussian probability model characterizing the diffusion implicit in DTI. In order to

overcome these difficulties several approaches have been taken. Q-space imaging, a

technique commonly used to examine porous structures [16], has been -II__- -rb I1 as










a possible solution [91]. However this scheme requires strong gradient strengths and

long acquisition times [6], or significant reduction in the resolution of the images.

As a more viable alternative Tuch et al. have proposed to do the acquisition such

that the diffusion sensitizing gradients sample the surface of a sphere [85, 84]. In

this high angular resolution diffusion imaging (HARDI) method, one does not have

to be restricted to the tensor model. Instead, it is possible to calculate diffusion

coefficients independently along many directions. This method does not require

more powerful hardware systems than that required by DT-MRI. Several groups

have already performed HARDI acquisitions in clinical settings and have reported

43 to 126 different diffusion weighted images acquired in 20 to 40 minutes of total

scanning time [34, 84, 46] indicating the feasibility of the high angular resolution

scheme as a clinical diagnostic tool. Since the HARDI data acquisition is very

nascent, not many techniques of processing the HARDI data have been reported

in literature. In the following, we will briefly review the few very recently reported

techniques of HARDI data denoising, which must be done prior to further analysis

or visualization.

1.1 Diffusion MRI


Recently MR measurements have been developed to measure the tensor of

diffusion. This provides a complete characterization of the restricted motion of

water through the tissue that can be used to infer tissue structure and hence

fiber tracts. In a series of papers, Basser and colleagues [10, 8, 7, 4, 9, 70] have

discussed in detail general methods of acquiring and processing the complete










apparent-diffusion-tensor of MR measured translational self-diffusion. They

showed that directly measured diffusion tensors could be recast in a rotationally

invariant form and reduced to parametric images that represent the average

rate of diffusion (tensor trace), diffusion anisotropy (relationship of eigenvalues),

and how the diffusion ellipsoid (eigenvalues and eigenvectors) can be related to

the laboratory reference frame. The parametric images [4, 69] of volume ratio,

fractional anisotropy, and lattice anisotropy index represent scalar measures of

diffusion that are independent of the lab reference frame and subject orientation.

Therefore, these measures can be used to characterize the tissue pathology, e.g.,

ischemia, independent of the specific frame of reference used to acquire the images.

The development of diffusion tensor acquisition, processing, and analysis methods

provides the framework for creating fiber tract maps based on this complete

diffusion tensor analysis [22, 47, 53, 59]. This has been used to produce fiber tract

maps in rat brains [59, 94] and to map fiber tracts in the human brain [47], then,

the first steps were taken to relate this structural connectivity to function [22].

The directional properties of diffusion can be characterized by a diffusion

tensor, a 3 x 3 symmetric matrix of real values. In order to calculate the 6 inde-

pendent components of the tensor, the subject is imaged in 7 different directions

with several magnetic field strengths. The relationship S = So exp(- Ci bijDij)

allows the diffusion tensor, D, and the T-2 weighted image So to be calculated

given the samples S and diffusion weighting factor, b. Previous work has con-

centrated on smoothing the field of eigenvectors of D. More recent work [20] has

formulated regularization techniques for the tensor field itself, even constraining the











resulting tensors to be positive-definite. We have previously taken the approach of

smoothing the observed vector-valued image S prior to calculating D.

In suninary, the anisotropy of water translational diffusion can he used

to visualize structure in the brain and provides the basis for a new method of

visualizing nerve fiber tracts. Initial results have been very encouraging and -II__- -r

that this approach to fiber mapping may be applied to a wide range of studies in

living subjects. However, it is essential to optimize the acquisition and processing

algorithms for fiber tract mapping and validate the results relative to known

measures of fiber tracts.

1.1.1 Overview of Diffusion


Random molecular motion (Brownian motion) can cause transport of matter

within a system. Within a volume of water, molecules freely diffuse in all direc-

tions. The water abundant in biological systems is also subject to such stochastic

motion. The properties of the surrounding tissue can affect the magnitude of

diffusion, and the directional properties as well.

Tissue can form a barrier to diffusion, restricting molecular motion. Within

an oriented structure, such as a bundle of axonal fibers, diffusion can he highly

anisotropic. The white matter of the brain and spinal cord is characterized by

many such bundles.

Diffusion has the property of antipodal syninetry: diffusivity is equal in

opposite directions. The directional properties of diffusion can, in some cases,






























Figure 1.2: Diffusion ellipsoid


be approxiniated by a tensor. The diffusion tensor, D, is a syninetric, positive-

definite 3 x 3 matrix. We will make use of the eigfenvalues and eigfenvectors of this

tensor, sorting the eigenvalues (X1, 12, XS) frOml largest to smallest, and labelling

the corresponding unit eigenvectors (el, e2, e3). The eigenvalues represent the

magnitude of diffusion in the direction of their corresponding eigenvector. For

isotropic diffusion At = X2 : 4. The eigenvector corresponding to the dominant

eigenvalue is coninonly referred to as the principal diffusion direction (PDD). A

popular representation for depicting anisotropic diffusion is the diffusion ellipsoid.

This ellipsoid is the image of the unit sphere under the transformation defined

by the tensor, D. The eigenvectors of D form an orthogonal basis, representing

the orientation of the ellipsoid. The length of each axis of the ellipsoid is the

corresponding eigfenvalue. For isotropic diffusion, the diffusion ellipsoid is a sphere.










Surrounding axonal nerve fibers is an insulating myelin sheath, which is

restrictive to the diffusion of water molecules. A loss of blood supply (ishemia),

such as would occur during stroke, causes physiological changes in the nerve

fibers which may be detected as change in anisotropy. Diseases, such as Multiple

Sclerosis, which result in demyelination of nerve fibers can he detected as well.

1.1.2 Overview of AIR Imaging


In this section we will present a brief overview of the AIRI acquisition process.

A detailed treatment of the subject was done by Haacke et al. [42].

The protons in the nuclei of atoms align their axis of spin with the direction

of an applied magnetic field. The magnetic field also induces a wobble, known

as precession, in the spin of the protons. This frequency, the resonant frequency,

is proportional to the strength of the applied field. For protons the resonance

frequency lies in the RF range.

In the AIRI instrument, a static field Bo is applied throughout the imaging

process. The direction of this field defines the axial direction of the image. Protons

will absorb energy from an RF pulse of the resonance frequency and tip away

from the direction induced by Bo. The amount of tip is proportional to the pulse

duration. The RF pulse also causes the protons to process in phase with each other.

This pulse is called the B1 field.

When the B1 transmitter is turned off, the absorbed energy at the resonant

frequency is re-emitted by the protons. This occurs as the spins, tipped by B1,

return to their previous Bo alignment. The time constant associated with this










exponential process is known as the Ti relaxation time. The protons precessions

also dephase exponentially with time constant T2. The final image contrast is

influenced by strength, width and repetition time of the RF pulses in the B1 signal.

By spatially varying the intensities of Bo and B1, position information is

encoded. For instance, specially designed magnets add a gradient field to Bo. This

causes the proton resonance frequency to be a function of axial position. The

frequency of B1 can then be chosen to tip protons within a chosen slice.

To encode x, y7 position within a slice, two additional gradient fields are

emploi-r I The first gradient, G,, is pulsed, causing a phase variation, just as in

T2 T68XatiiOn. The phase variation is a function of position in the y direction. A

perpendicular gradient, G,, is then applied, changing resonance frequencies in the

x direction. A 2D Fourier transform reconstructs the image of each slice from the

data in the spatial frequency domain.

1.1.3 Diffusion Weighted Images


By carefully designing gradient pulse sequences, the measured signal from

protons in water molecules undergoing diffusion can be attenuated. The first

gradient pulse induces a known phase shift in proton precession. After some delay,

a second gradient pulse is applied, inducing the opposite phase shift. Protons which

have not moved between the two gradient pulses are returned to their previous

phase. Protons belonging to molecules which have changed location have some

net change in phase, changing their T2 T68XatiiOn time. In this work we will not










describe the gradient pulse sequence G(t) in detail, but we will be making use of

several quantities related to the diffusion sensitizing gradient.

The vector g is the direction of the gradient. The scalar quantity b, called

the diffusion weighting factor or b-value, is a function of the gradient magnitude,

and the timing and duration of the pulse sequence. The vector q is related to the

integral of G(t) over time.

The attenuation of the diffusion weighted image (DWI), S, relative to an ideal

image So acquired with no diffusion weighting is given by


S = So exp(-bd) (1.1)


where d is the apparent diffusion coefficient (ADC). This relation is called the

S' i-1: I1-Tanner equation [78]. We ;?-, So is an ideal image since the gradients, G,

and G,, have some nonzero diffusion sensitivity. We can not measure So, however it

can be estimated from the Steil1: I1-Tanner equation if we acquire at least 2 images

with different b-values. The image S will change as g changes. The motivated

reader may refer to the work of Haacke et al. [42] for more details on the physics of

acquisition.

1.1.4 Diffusion Tensor MRI Acquisition


We can construct a more complete model of diffusion by acquiring DWIs

with multiple gradient directions. By assuming a Gaussian model diffusion we can

describe this diffusion with a rank-2 tensor. Another form of the Steil1: I1-Tanner











equation relates the images S and So to g and the apparent diffusion tensor, D.


S = So exp(-bgTDg) (1.2)


When computing D, it is common to rewrite Equation (1.2) in log-linearized

form as

In ( ) = b,4 e (1.3)
i= 1 j= 1

where bioy are computed from b and g.

This imaging process must be performed with at least 7 noncoplanar gradient

directions in order to fully generate a diffusion tensor image and recover So.

Multiple samples, usually 3 or 4, with varying gradient strengths are taken for each

gradient direction. For m DWIs we have the system


In So



InS SI 1 -biz -b -biz -2b -2b z -2biz D,,


Dzz (1.4)

In Sm 1 -bmm -b" -bz 2b, 2bz 2bmz Da

Dz

Dz

The overconstrained linear system, Eq. (1.4), is solved for So and the elements of

the symmetric tensor D by a least squares linear regression. Alternatively, we may

estimate D by nonlinear regression.










For the Gaussian model of diffusion, the diffusion tensor is related to the

displacement probability of a water molecule. For a fixed (small) time constant

we can describe the probability that a molecule undergoes displacement, r, by a

Gaussian with mean zero and covariance 2tD.

1.1.5 High Angular Resolution Diffusion Imaging


The HARDI process proceeds by acquiring diffusion weighted images with

many diffusion encoding gradient directions, effectively sampling a spherical shell

of the q-space. It is desired that this sampling be uniform, or nearly so. The

gradient direction for each image is usually chosen to correspond to the vertices of

an icosahedron which has been repeatedly subdivided. Since the process of diffusion

is known to be symmetric, we need only sample one hemisphere of q-space. In the

case of our data, we consider 81 or 46 gradient directions. The gradient directions

correspond to the vertices of the surface shown in Figure 2.2. In addition a low

b-value (small diffusion encoding gradient magnitude) image is acquired.

In HARDI we can consider another form of Equation (1.1)


S(0, 4) = So exp(-bd(0, 4)) (1.5)


where 8 and 4 are the polar and azimuthal angle describing g.

The random process of diffusion of water molecules is described by the

diffusion displacement PDF p,(r). This is the probability that a given molecule

has a diffusion displacement of r after time t. The relation between the measured










image, and the diffusion displacement PDF is given by Callaghan [16] as


pt(r) = ~qS exp(-2r i q r)dq (1.6)


This is simply th~e forward Fourier transforms? of ~. It is th~e modes of p,(r) that

are taken to be the underlying fiber directions.

An alternative imaging scheme, called diffusion spectrum imaging (DSI),

acquires image by sampling a Cartesian grid in q-space. The Fourier transform

of can Ithen be evaluated dlirectlyl to yield pt(r.). The imnage acquisition time

for (DSI) is quite high, since we must sample out to large magnitudes of q to

accurately reconstruct pt().

1.2 Restoration

Processing of HARDI data sets has received increased attention lately and

a few researchers have reported their results in literature. The use of spherical

harmonic expansions have been quite popular in this context since the HARDI

data primarily consists of scalar signal measurements on a sphere located at

each lattice point on a 3D image grid. Tuch et al. [85, 84] developed the HARDI

acquisition and processing and later Frank [34] showed that it is possible to use

the spherical harmonics expansion of the HARDI data to characterize the local

geometry of the diffusivity profiles. In his work however, there is no discussion

of denoising/restoring the HARDI signal measurements, which is essential for

subsequent processing and interpretation. C'I. is et al. [21] find a regularized

spherical harmonic expansion by solving a constrained minimization problem.










However the expansion is a truncated spherical harmonic expansion of order 4, and

hence the solution can represent only 2 fiber directions within a voxel. Jansons

and Alexander [46] described a new statistic which was called persistent angular

structure that was obtained from the samples of a 3D function, in this case the

displacement of water molecules in each direction. The goal in their work was to

resolve voxels containing one or more fibers. However, there was no discussion on

how to restore the noisy HARDI data prior to resolution of the fiber paths.

Image denoisingf can be formulated using variational principles which in turn

require solutions to PDEs. The technique described by C'I I1. and Shen [18] has

been emploi-. I in medical imaging literature for smoothing DT-MRI data sets [24].

Recent work by Tschumperlii and Deriche [82] deals with smoothing the eigenvector

field of the diffusion tensors computed from the raw echo intensity image data.

Several other methods have been developed for restoring the DT-MRI data sets

but most of them use existing restoration schemes for scalar or vector-valued

functions from image processing literature. More recently, matrix valued function

restoration was introduced [20] and applied to the restoration of noisy diffusion

tensor fields. An interesting alternative to the variational principle approach was

taken by Weickert and Brox [92] wherein, they developed an anisotropic diffusion

filter that achieves an I Ir-; preserving smoothing of the positive semi-definite

tensor-valued image. All of the aforementioned methods share one commonality

i.e., they use a linearized S' i-i I1 .-Tanner equation as the data acquisition model

which does not accurately reflect the physics of the data acquisition. In recently

reported work, by Wang et al. [90, 89] an alternative approach which overcomes










this weakness by directly estimating a smooth positive semidefinite tensor field

from the raw data using the actual nonlinear mono-exponential model characterized

by the Stei-l: I1-Tanner equation [78] was reported.

1.3 Tractography


Water in the brain preferentially diffuses along white matter fibers. By

tracking the direction of fastest diffusion, as measured by AIRI, noninvasive fiber

tracking of the brain can he accomplished. In the context of DTI data, fiber

tracks estimated in reported literature were obtained by repeatedly stepping

in the direction of fastest diffusion. The direction along which the diffusion

is dominant corresponds to the direction of eigfenvector corresponding to the

largest eigenvalue of the tensor D. This approach was taken by many researchers

[2:3, 59, 9:3, 71, 24, 67, 87, 56]. Most of these techniques do incorporate some

regularization in their stream line estimation schemes in order to generate the

fiber patle- .1-<. Techniques that are quite distinct from the idea of stream line

generation have also been reported in literature. Batchelor et al. [11] reported a

fiber tract mapping scheme wherein they produce a map indicating the probability

of a fiber passing through each location in the field. However, no discussion on

how to estimate the actual fibers was described. An alternative approach hased

on sequential importance sampling and regularization techniques was proposed by

Bi~llre ine et al. [1:3], which allowed paths to originate from a single location and

branch out and produced a probability distribution of the paths. O'Donnell et al.

[61] describe v- a-s~ to estimate the connectivity from the given tensor field. One










approach they -II__- -1. .1 was to estimate the geodesics in the locally warped space

where the warping is derived from the local tensor.

In the context of HARDI processing, suggestions for fiber path computation

have been reported in the literature [34, 46, 62]. Frank [34] described a spherical

harmonic transform representation of the HARDI data and pointed out that, due

to the antipodal symmetry on the sphere, only the even terms in the spherical

harmonic transform contributed toward the representation. Any nonzero odd terms

would be due to artifacts in imaging such as noise. The fiber patle wRi~ were not

computed in his reported work, however, it was -II_t-r-- -1.. that one could estimate

the direction of the crossing fibers by using a multi-tensor model and finding

the principal diffusion directions of this multitensor expansion. The approach of

using HARDI data and then reverting to a multi-tensor approach seems somewhat

defeating. The approach described by Ojzarslan and Mareci [62] expresses the

diffusivity function, a function defined on the sphere, as a generalized (higher rank)

Cartesian tensor and then estimates the probability distribution of water molecule

displacement over all directions using the FFT (fast Fourier transform) of the

signal measurements either on the sphere or an interpolated Cartesian grid. These

distribution profiles are then sharpened to di-play~ the possible orientation of the

fibers with complex local geometries.

1.4 Anisotropy


In DTI there are many scalar measures which characterize the anisotropy of

the diffusion phenomenon. Most useful are the rotationally invariant measures,










those independent of the laboratory reference frame, such as volume ratio, rational

anisotropy and fractional anisotropy [5]. These can he computed in terms of the

eigfenvalues of the diffusion tensor. These scalar indices alone have been useful in

clinical studies [79], so it is important to find analogous measures of anisotropy

for HARDI. The measures for HARDI should overcome the weaknesses of the

tensor model with regard to regions of crossing fibers. In the case of orientational

heterogeneity, DT-AIRI technique yields incorrect fiber directions and artificially

low anisotropy values [64].

Frank [34] uses the coefficients of the spherical harmonic expansion of the

apparent diffusion coefficient to quantify anisotropy. The Oth, 2nd and 4th order

coefficients describe isotropic diffusion, single-fiber diffusion and two-fiber diffusion

respectively. Voxels can he classified using the relative magnitudes of these

coefficients.

Ojzarslan and Mareci [62] use a high rank tensor to describe the diffusion

process and generalize the rank-2 tensor trace to higher rank tensors. The variance

of the diffusivity is then expressed in terms of the generalized trace, and a transfer

function maps this value to the [0, 1] range. This quantity is called "generalized

sin-l i'vy(GA). It was shown to be useful in detecting anisotropy at fiber

crossings which are not detectable by FA.

1.5 Overview of Our Modeling Scheme


There are two 1!! Gi- ~ subtasks in achieving the goal of fiber tract mapping in

the brain. Firstly, the raw noisy DWI data have to be restored and secondly, the










fiber tracks have to be mapped from this restored data. The variational principle

involves smoothing S values over the sphere and across the :3D image lattice.

The key factor that complicates this problem is that the domain of the data at

each voxel in the voxel lattice is a sphere. One may use the level-set techniques

developed by Tang et al. [80] to achieve this smoothing; however, when data sets

are large, it becomes computationally impractical to apply the level-set technique

at each voxel independently to restore these scalar-valued measurements on the

sphere. We arrive at a computationally efficient solution to this problem by using

the finite element method (FEM) on the sphere and choosing local basis functions

for the data restoration. Unlike the reported work on spherical harmonic hasis

expansion of the diffusivity function on the sphere [:33, 62], the FEM hasis functions

have local support and are better suited for preserving details in the data.

From the denoised data we will compute a probability, pt(8, ~), of molecular

diffusion over spherical directions. The Shannon entropy of this distribution will be

used to quantify anisotropy.
















CHAPTER 2
DENOISING


We present a novel and effective variational formulation that will directly

estimate a smooth signal S(0, 4) and the probability distribution of the water

molecule displacement over all directions p(0, 4), given the noisy measurement


S(0, 4) = S(0, 4) + rl (2.1)


where S is the signal measurement taken on a unit sphere over all (0, 4) and rl

is Gaussian noise.

The actual noise is complex-valued Rician noise on the raw S images prior

to Fourier transform. Since our data have relatively high signal-to-noise ratio, the

noise is well modeled by additive Gaussian noise on the magnitude data [40].

2.1 TV Norm Minimization of S(x,y,z)

The total variation (TV) norm introduced in Fatemi et al.[50] is a popular

norm used for image restoration. In the case of scalar valued functions, minimizing

the TV-norm amounts to minimization of the L1 norm of the image gradient.

Minimizing the TV norm produces very smooth images while permitting sharp

discontinuities between regions [50]. Since most images consist of smooth regions

separated by discontinuities (edges), this is a useful model for image denoising.










2.1.1 Scalar TV Norm Minimization


The TV norm for scalar images, I(x) is given by


TV,,1(I(x)) C |VI(x)|lx, I1C (2.2)


This norm represents oscillation. In our case, image noise is considered to be

.--anklingt of the surface described by the image.

At discontinuities, the weak derivative DI(x) to calculate the TV norm. For

a piecewise continuous function, the TV norm is then the sum of the TV norm

of each continuous piece plus the sum of the absolute values of the "jumps."

For example, functions fl and f2 in Figure 2.1 have the same TV norm. The

oscillatory function, fl, has a higher TV than the function with a discontinuity, f3.

Since most images consist of piecewise smooth regions separated by discontinuities

(edges), this is a useful model for image denoising.


Figure 2.1: TV(fl) > TV(f2) = TV(f3)










The Euler-Lagrange condition for the minimization of Equation (2.2), written

in gradient-descent form, is

8I(x, t) VI(x, t)
= div()
iit |VI(x, t)|I

I(x, 0) = Io(x) (2.3)


The gradient descent solution to the TV-norm minimization problem will

converge very slowly. We will present an efficient numerical technique below.

2.1.2 Vector TV Norm Minimization


Blomgren and C'I .1. [14] introduced the TVn,m norm for vector valued images.


T~n~(I~)) = [TV,1(I)]2(2.4)
i= 1

For m = 1, Equation (2.4) reduces to the scalar TV norm (2.2). The Euler-

Lagrange condition for the minimization of Equation (2.4), written in gradient

descent form is


8lI (x, t) TV,,1(lI) VAi
dt TI~(I VI4|

I(x, 0) =Io(x) (2.5)


This was shown to be quite effective for color images, preserving edges in the color

space while attenuating noise. However, for much larger dimensional data sets as

in the work proposed here, the Color TV method becomes computationally very

intensive and thus may not be the preferred method in such applications.










By treating each component of the vector field as an independent scalar

field, we can proceed by smoothing channel-by-channel. This can, however, result

in a loss of correlation between channels as edges in each channel may move

independently due to diffusion. To prevent this, there must be coupling between

the channels.

2.2 Variational Principle

The variational principle we propose for estimating a smooth S(x, 8, ~) from

the initial data S(x, 8, 4) is given by


msn E(S) = ~ ~ p S(x,0, U) S(x,0, UI)|2d'ix

+ i a~ |(o,o)S|2dS + p/I (Soo +2So4 + S44)dS

+ g(x)|VzS dx (2.6)

where R is the image spatial domain, and S2 is the spherical image domain at each

voxel, and dS denotes integration over the sphere. The first term of Equation (2.6)

is a data fidelity term which makes the solution to be close to the given data. The

degree of data fidelity can be controlled by the input parameter p. The second

and third terms are first and second order regularization constraints enforcing

smoothness of the data over the spherical domain at each voxel. The fourth term is

another regfularization term which causes the solution to be smooth over the spatial

domain (the 3D voxel lattice). To make the solution of the denoising problem

feasible, we propose to alternately solve the spatial and spherical smoothing

problems .










2.3 Spatial Lattice Smoothing of S(x)

Smoothing the raw vector-valued data, S(x), is posed as a variational principle

involving a first order smoothness constraint on the solution to the smoothing

problem. Note that the data at each voxel is a large set of S measurements over

a sphere of directions and can be assembled into a vector after the smoothing on

the spherical coordinate domain has been accomplished. We propose a weighted

TV-norm minimization for smoothing this vector-valued image S. The variational

principle for estimating a smooth S(x) is given by


msinS(S) =(gx |~VS4| +(Si si)2)dx (2.7)


where, R is the image domain and p is a regularization factor. The first term

here is the regfularization constraint on the solution to have a certain degree of

smoothness. The second term in the variational principle makes the solution

faithful to the data to a certain degree. We have previously [56] used the coupling

function g(x) = 1/(1 + FA(x)) for smoothing diffusion tensor images, where FA

is the fractional anisotropy defined by Basser and Pierpaoli [10]. This selection

criterion preserves the dominant anisotropic direction while smoothing the rest

of the data. Note that since we are interested in the fiber tracts corresponding to

the streamlines of the dominant anisotropic direction, it is apt to choose such a

selective term. For HARDI data, we may replace FA with generalized anisotropy

[62] since it can be computed prior to the ODF.










Here we have used a different TV norm than the one used by Blomgren and

C'I 1.. [14]. The TVn,m norm is an L2 IlOrm of the vector of TV,,i norms for each

channel. We use the L1 norm instead.

The gradient descent form of the above minimization is given by


=~i div -llS~ p l(Si S) i= 1,...,m

-|anxa+ = 0 and S(x, t = 0) = S(x) (2.8)


The use of a modified TV-norm in Equation (2.7) results in a looser coupling

between channels than the use of the true TVn~ norm would have. This reduces

the numerical complexity of Equation (2.8) and makes solution for large data sets

feasible.

Note that the TVn,m norm appears in the gradient descent solution, Equation

(2.5) of the vector-valued minimization problem. Consider that our data sets

consist of up to 82 images, corresponding to (magnetic field) gradient directions.

Calculating the TVn,m norm by numerically integrating over the 3-dimensional data

set at each step of an iterative process would have been prohibitively expensive. In

contrast, using the modified TV-norm -II_t-r-- -1.. earlier leads to a more efficient

solution.

2.3.1 Fixed-Point L I__- d-Diffusivity


Since the m Equations(2.8) are coupled only through the function g, we can

drop the subscript on S with no ambiguity (later the subscript will refer to spatial

coordinates.) In this section we will discuss the numerical solution for each channel,








24

S, of the vector-valued image S. Equation (2.8) is nonlinear due to the presence of

|VS| in the denominator of the first term. We linearize Equation (2.8) by using the

method of "II__- d-diffusivity" presented by C'I I1. and Mulet [17]. By considering

|VS| to be a constant for each iteration, and using the value from the previous

iteration we can instead solve


(Vg VSt g2St+l) + p(St+l So) = 0 (2.9)
|VStl

Here the superscript denotes iteration number. To make the fixed point iteration

clearer, we rewrite Equation (2.9) with all of the St+l terms on the left-hand side


2St 1 pVStl St+1 p|1VSt|So+ Vg VSt (.0
9 9


2.3.2 Discretized Equations


To write Equation (2.10) as a linear system (ASt+l = t), discretize the

Laplacian and gradient terms. Using central differences for the Laplacian we have

V2St+] St+] St+] St+]
z-1,y,z z,y-1l,z z,y,z-1

-6;St + S ,] t1+, t/+ (2.11)


Define the standard central differences to be


az S = 2 (S3m+l,y,z S3m-l,y,z)

a,S = 2 (Sz,w+,z, Sz,w-,z)~

AzS=2(Sz,w,z+1 Sz,w,z-1) (2.12)










We can rewrite Equation (2.10) in discrete form using the definitions in Equation

(2.12)




p (Aczct)2+(Ayt)2+(Aztt)
+(6 +)S32,w,Z

-S3m+l,y,z S3z,y+l,z S3z,y,z+l




+AzgaSt + A,gA,St + az9azSt) (2.13)

This results in a sparse banded linear system with 7 nonzero coefficients per row.

6+ 1+ .., 1 ... 1 ... I S
0~i -1 6 (2.14)



where the right-hand side of Equation (2.13) is denoted by fr. The matrix in

Equation (2.14) is symmetric and diagonally dominant. We have successfully used

conjugate gradient descent to solve this system.

The solution of Equation (2.14) represents one fixed-point iteration. This

iteration is continued until | St St+l| < c, where c is a small constant.

2.4 Finite Element Smoothing of S(0, 4)


We will consider a deformation energy functional which is a weighted combina-

tion of thin-plate spline energy and membrane spline energy, that is commonly used

in computer vision literature for smoothing scalar-valued data in R3 (SeO Cillernley

and Terzopoulos [58]).










The diffusion-encoding gradient directions are taken as the vertices of a

subdivided icosahedron, to achieve a nearly uniform sampling of spherical direction.

We map this piecewise linear approximation of a sphere to the plane by taking the

spherical coordinates (0, 4) of the imaging gradient direction as the planar global

coordinates (u, v). The gradient directions, and their embedding in the plane are

shown in Figures 2.2 and 2.3.


Figure 2.2: HARD gradient directions (0, ~) correspond to the vertices of a subdi-
vided icosahedron.


Figure 2.3: Two dimensional (u,v) domain for FE1\-based smoothing of HARDI
data.










Note that, the data can be seen as height values defined on the sphere at the

coordinates specified parametrically by (u, v). The smoothing will be applied to

these height measurements z(u, v) using the smoothing functional in Equation

(2.15).

S,= ((z| 2 u |,,2 2)ud (2.15)

The weight on the membrane term is a~ and the weight on the thin-plate term is P.

Once we have computed a smooth z(u, v), the result will then be mapped back to

the image on the sphere, S(0, 4).

The data energy due to virtual work of the data forces is


Sa = (u, v) f u, vl)du dv (2.16)


The restoration at each voxel is formulated by the energy minimization


mm S(S) = S,(S) + Ed(S) (2.17)


We use polynomial shape functions, Nsi as a basis for the data over the domain

of spherical directions.

z u v 4 s(, =N q (2.18)
i= 1

where the N is a (1 x n) row vector, and q is a column vector of nodal variables.

2.4. 1 Element Matrices


We will subdivide the domain into elements, each with their own local shape

functions. For each element j


z (u, v) = N (u, v) q


(2.19)











for (u, v) E Rj. The local energy fumetion for each element is given by


(2.20)


83 = (al-z'|2 az 2 f Pz, 2 + 2/3|z',|2 $ cl 2)du dv


The global energy is the sum of the energies of each element


E, = 8'


12.21)


Dhatt and Touzot [27], is




v

uu


The element strain vector, given by






e"=


(2.22)


S=I (Ni;),, .. (i"')uu q = Bqj (2.23)






where we have defined B as the (n x 5) matrix of derivatives of the nodal basis

ftmetions. We can then write the element strain energy as


(2.24)


83 =b e*TDe da dv



















































(2.29)


where we define
c0 O

On c

D =I 0

0 0

0 0

Since q" is constant over each element

matrix in terms of D and B as follows


0 0

0 0

0 0

2/9 0

0 /9

can derive the element stiffness


(2.25)











(2.26)


iJ = qjTBjDB q du dt' = qjTK q


We will model the data constraint as springs pulling each x(u, t') toward some

given value xo(u, t'). The force at each point will obey f = k(x ro), where k is the

spring constant.


EI = -N q k(N q' coidu dt'


(2.27)


Sj = -kq N 'N q'i du d + kg ji~ N)'cdu dt' (2.28)

We will define F" and fj such that the first term of Equation (2.28) is qTF;7qj and

the second term is qj~f We can the balance deformation energy and data energy

by solving the linear system:


(K" + F;7)q~ = f










2.4.2 Local Element Coordinates


We now present the coordinate system for the local elements. For local

elements, triangular elements are used with a barycentric coordinate system

(y, (, rl) so that each coordinate is in [0, 1] and y = 1 ( rl.
rlo = 0










rl1 = 0 r/2
(Uo, to) L r = 0 1 2 = 0


Figure 2.4: Mapping to barycentric coordinates


Barycentric local coordinates can be mapped to global coordinates.


u(l, rl) = (1 ( rl)uo + (ui + rlu2

v(l, rl) = (1 ( rl)vo + (vi + rl02 (2.30)


The mapping in Equation (2.30) can be rewritten as a linear system.



V1-'II ~ -III 11 ::1(2.31)
U: U1 Un U2 9

Differentiating Equation (2.31) we obtain



d u viuiv3 iurdv~ l I rd J ~ rd ( 2 3 2 )











i ri wis ( ai
1 0.0469100770 0.1184634425 0.0398098571 0.1007941926
2 0.2307653449 0.2393143353 0.1980134179 0.2084506672
3 0.5 0.2844444444 0.4379748102 0.260 11 : :' 46
4 0.7692346551 0.2393143353 0.6954642734 0.2426935942
5 0.9530899230 0.1184634425 0.9014649142 0.1598203766


Table 2.1: Gauss-Radau weights


where J is the Jarob-ian of the transformation from global to local coordinates. We

can convert integrals over the (u, v) domain to integrals over the local ((, rl) domain

in the following way:


Sbi


f (u((, 9), v((, 9)) det(J)d( dy


f (u, v)du dv


(2.33)


Using Gauss-Radau quadrature rules, we can approximate the integral in

Equation (2.33) by


i= 1 j= 1


(2.34)


where rli,j = ri(1 sj), wjj = aj(1 (4), (;, and II'. are give in Table 2.1.

Derivatives over (u, v) become




+ (2.35)
8iv 8(~ 8iv 89~r 8iv

The partial derivatives of ( and rl with respect to u and v can be computed by

invertingf the Jacobian


J- ;
dv


(2.36)


dr Bu By dv













J -~- o (2.37)
det(J)
-(v1 vUo) U1 no

We use the fifth order element shape functions given by Dhatt and Touzot [27].

This element guarantees C1 (surface normal) continuity across triangles. The basis

functions are given by


N1 = X2(10A 15A2 + 6A3 + 30(q((~ + rl))

N2i X2(3 2A 3(2 + 6(ql)

N3 = rl2(3 2A 3r12 + 6(ql)

N4 2 2 LX(1 ( + 2rl)

Ns = (GAX2

N6 92 ~r2 (1 + 2( '7)

N7 =(2(0( 152 +6(3 + 15r12X

Ns= 2(-8( + 14(2 6(3 15r12X

N9 ~2r1(6 4( 3'1 3r12 3 9l)



1
N1o = 2(2X(~1 -(2+52
2




1 1
N12 ~ 2r2X _32
4 2











N17 = (qr2(-2 + (+2r+(

Nils = 92(28(1 q)2+ 5(2A) (2.38)




The quintic shape functions have nodal variables which can be written in

terms of local or global coordinates


zg z,

z, z,

980 zg R,v, = z,u (2.39)

zel zuv

zrlr zvv

which we can relate to each other by


1 0 0 0 0 0

0 (a rl O 0 0

0 (a rl O 0 0
qu,,,= q, (2.40)
0 0 0 2 (a9,

0 0 0 s( ((syl, + rlulv) rlurl

O 0 0 (~ 2 (a 9 9










2.4.3 Global Matrices


We now wish to construct global matrices so that the energy balance over the

entire FEM mesh is given by the linear system


Kq = f (2.41)


where K is a (6n x 6n) matrix since we have 6 variables per node.

We will consider the simple case of 2 elements. Expanding the element

Equation(2.29) in terms of nodal variables for element 0, we get




Ko,,q K KO, 90 1 (2.42)



and likewise for element 1


K), 3K, 2 K qf 3

K], K K] q = f2 (2.43)

K),3 K ,2 K q f

where each ql is a (6 x 1) column vector of nodal variables. We expand each Kj

to be (6n x 6n) by inserting rows and columns of zeros corresponding to each node

of the mesh. Also expand f" to (6n x 1). The global K and q are obtained by

summing the expanded matrices from each element in the mesh. For our 2 element











example we have


KO ,o K K], 0] qi foo

Ko,0 Ko K, KO2+K,2 K ,3 10 1
(2.44)
KoKP K1 K K K1 ,3 92 2

0 K K3, 2 K ,3 93



The global linear system Equation (2.41) is symmetric, and has a sparse

banded structure with 18 nonzero diagonal bands. We solve for q by Cle1.1 -l:y

factorization.
















CHAPTER :3
VISUALIZING PROBABILITY FIELDS


In this section we will examine techniques of scalar, vector and tensor field

visualization, and explore possible techniques for probability fields. Of particular

interest are the texture-based methods. We may consider the hue, intensity, and

texture of an image as independent channels for conveying information about these

high dimensional datasets.

:3.1 Computing Probabilities


The probability in Equation 1.6 can now he evaluated by computing the quan-

tity S(q)/So and performing the FFT. Since we know that the signal, S, decays

exponentially from the origin of q-space (where S(0) = So) we can interpolate and

extrapolate signal values for arbitrary q. We resample from spherical coordinates to

artesian and perform the FFT on the resampled data. The result is a probability

of water molecule displacement over a small time constant. We are interested in

only the direction of water displacement, so we integrate out the radial component

of p,(r) to get pt(0, ~). This is commonly called the diffusion orientation distrib-

ution function (ODF). Computing the ODF with this method is computationally

expensive since it requires a :3D FFT at each voxel, and then a numerical integfra-

tion for each direction. In practice we use an alternative method given by Ozarslan

in [] which makes computing the ODF for large datasets more feasible.










3.2 Glyph Visualization


In high dimensional data sets, it is common to represent the data at each

voxel with some icon. In DT-MRI this icon is often some simple shape which is

transformed by the diffusion tensor. For example, the sphere will be transformed by

the diffusion tensor into the diffusion ellipsoid.

To enhance the visual impact of the ODF we apply a sharpening transform to

the ODF by subtracting a uniform distribution (sphere) from each distribution, as

shown in Figure 3.1. The radius of the sphere is the minimum of the probability

over all directions. By performing this operation the directions of highest prob-

ability becomes more apparent. The glyphs in regions of isotropic diffusion will

disappear.











Figure 3.1: Original ODF (left), minimum probability sphere (center), and sharp-
ened ODF (right).


When applied to small or sparse planar regions this may be a useful visual-

ization. As shown in the left side of Figure 3.2. However, for large, dense regions

the display may become very cluttered, as in the right side of Figure 3.2. Due to

perspective or viewing angle, glyphs pointing toward the viewer may not convey



















Figure11 3.:O Fgyh
any useful infrmation. In 3-imensional re ion fintrssc pIcnb

quite awkward


3.3 SalarMeasues o Anistrop
In~~~~~~~~~~~~~~~~~~I difso esriaepoesntesaa unitykonafatoa
anisotroy (FA) s oftenconsiderd a useul quanity to isulieAnxapeo
















Figue 33: A iage(let),gen raied aniotop (cntr) Shnnn niotop




(HA (igt) frm orna sclc o create brainotop










Since we have a distribution at each voxel, we can compute the Shannon

entropy value at each voxel as given by


H(pi) = -j p(0s, r%) log p(0s, 9) (3.1)
i= 1

Considering the entropy of several trivial distributions, we can get a feel for the

interpretation of entropy in the context of HARDI. Entropy attains its maxi-

mum value, log n, for a uniform distribution. In our case, this corresponds to

isotropic diffusion. The entropy of a Gaussian distribution decreases as the variance

decreases. A voxel with this distribution has oriented diffusion greatest in the

direction of the mode of the distribution, implying the presence of fibrous structure.

Tuch [83] defined a scalar measure of anisotropy called "normalized el s..py'11 which

was also based on Shannon entropy.

The anisotropy images we present are mapped using Equation (3.2) such that

black color corresponds to high entropy isotropicc diffusion) and higher intensity

grey colors represent low entropy (anisotropic diffusion). We denote the anisotropy

measure computed from Shannon entropy as HA.


H(p)
HA(p) = (1.0 )" (3.2)
log n

This allows the image to be interpreted in the same way as a fractional

anisotropy image where the white color corresponds to white matter, grey cor-

responds to grey matter and black corresponds to cerebrospinal fluid. The a~

parameter controls the contrast between white and grey matter. Our images in

Figure 4.10 were computed using a~ = 0.65.



























Figure :3.4: R ii- I-r volume visualization of HA.


Figure :3.5: 1\lP volume visualization of HA.


The HA index is not a generalization of FA, and their values cannot he com-

pared in a meaningful way. In order to highlight the difference in anisotropy mea-

sures for HARDI and DTI, a Gaussian ODF was computed from the tensor data,

and the Shannon entropy of these distributions was computed. The anisotropy

values for the Gaussian ODFs can then he compared to the anisotropy values for

the true ODFs. The difference image is shown in Figure :3.6. It can he seen that

there is a structure to the difference image. This difference is more pronounced in











regions where the tensor model predicts diffusion is more isotropic than it .L;l
















Figure 3.6: Shannon anisotropy difference between Gaussian ODF and general
ODF.


Shannon entropy is not the only measure of uncertainty in a random vari-

able. Another definition of entropy, called the Riinyi entropy can be seen as a

generalization of Shannon entropy [25]. Ri~nyi entropy of order a~ is given by


Ho(P = 1log( p(0s jo)(3.3)
i= 1

The parameter, a~, of this entropy formulation has several interesting properties:

lim,,1 H,(p) = H(p) (Shannon entropy)

Ho(p) = number of nonempty bins in the histogram of p.

H,(p) = log(maxi(pi)) (depends only on the mode of p)

One may interpret the order, a~, as a parameter which changes the shape of the

distribution, p. For a~ > 1, small values of p(x) will shrink closer to zero. As a~

increases, these small probabilities approach zero more quickly. For high a~, events

with high probability influence the entropy more. This can be seen as controlling

the contrast between white and gray matter. We can formulate an anisotropy index










hased on Riinyi entropies just as we did for Shannon entropy. Anisotropy images

computed from Ri~nyi entropies of different orders are presented in Figure :3.7.










Figure :3.7: Anisotropy computed from H2 (top-left), Hg (top-right), Hlo (hottom-
left), H~o (hottom-right).


In statistical physics, a quantity called structural entropy has useful physical

interpretation [95]. This entropy is computed as the difference H1 H2* I

is unclear whether there is any such physical meaning for this quantity in the

context of diffusion, but it does provide a means of applying a transfer function to

entropy images which has a firm information theoretic footing. We present entropy

difference images for several values of a~ in Figure :3.8.










Figure :3.8: Riinyi entropy differences Hi H2 (top-left), Hi Hg (top-right), Hi -
Hlo (hottom-left), Hi H~o (hottom-right).


:3.4 Visualizing the Directional Nature of Diffusion


The scalar entropy value has no directional information however. In DTI, there

is precedence for using color values to represent direction. Color FA images, Figfure








43

3.9, are a mapping of the principal diffusion direction (the dominant eigenvector of

the tensor) to a hue, and the fractional anisotropy value to an intensity.

















Figure 3.9: Color FA.


These images are useful for distinguishing between .Illi Il:ent anisotropic regions

which differ in direction. Since we may have high diffusion coefficients in several

directions, we integrate over the sphere to determine a representative color for each

voxel. We denote this value ED, for expected direction.



| cos 04 sin OilI

ED = sin 94 sin 44| (PO,#) m (3.4)
i= 1
| cos 04|I


The resulting vector is interpreted as red, green, and blue color components.

The directions with the highest probability of diffusion will have their correspond-

ing color contribute most to the resulting color. Figure 3.10 depicts the colors

computed for a synthetic ODF field directed in a curve. Figure 3.11 shows the











colors computed for ODFs oriented in various directions, both with and without

crossmngs .















Figure :3.10: ED colors for synthetic distribution field.









Figure :3.11: ED colors for synthetic distribution field.


:3.4. 1 Streamtubes


Streamtubes are a three-dimensional alternative to streamlines. The stream-

tube geometry is generated by sweeping a circle along a streamline. By rendering

the streamtube as a lit surface using shading and depth cueingf, better directional

information may be conveyed to the viewer than with streamlines. The streamtube

is not a true streamsurface since the vector field is not tangent to the tube surface.

In general, only the underlying streamline which is the centerline for the stream-

tube is tangent to the vector field. The streamtube diameter is a parameter we may

use to encode some additional information about the tensor field being visualized,

such as the FA value. By discarding short fibers, as determined by some threshold,











we may obtain a less cluttered view of connectivity. Fiber traces may be seeded on

a dense or sparse grid spanning the whole data set, or only within a region of in-

terest. The streanitube color may also encode some useful information. We choose

to color the streanitubes to indicate direction. By mapping direction to color, as

in the color FA image Figure :3.9, a better sense of :$-dimensional fiber trajectories

can he conveyed in a 2-dintensional projection. Previously, Laidlaw et al. [51] has

applied the streanitube visualization approach to DT-1\RI. It is important to note

that each streanitube does not represent a single fiber or tract, but indicates only

fiber direction.


















Figure :3.12: Streanitubes in axial slice of brain.


:3.4.2 Line Integral Convolution


It is also possible to visualize the :3D vector field corresponding to the donli-

nant eigfenvalues of the diffusion tensor using other visualization methods such as

the line integral convolution technique introduced by Cabral and Leedon1 [15]










a concept explored in this work as well. The advantage of this visualization tech-

nique is that it is well suited for visualizing high density vector fields and does not

depend on the resolution of the vector field, moreover, it also has the advantage of

being able to deal with branching structures that cause singularities in the vector

field. An example of LIC applied to the synthetic field v [al = '1 ) s hwni


Figure 3.13.













Figure 3.13: LIC visualization of synthetic field with kernel widths 5, 15, and 55.


Since the fiber direction is parallel to the dominant eigfenvector of the diffusion

tensor, we can calculate fiber paths as integral curves of the dominant eigenvector

field. The stopping criterion is based on FA value. When FA falls below 0.17

we consider the diffusion to be nearly isotropic and stop tracking the fiber at

this point. Once the diffusion tensor has been robustly estimated, the principal

diffusion direction can be calculated by finding the eigenvector corresponding to the

dominant eigenvalue of this tensor. The fiber tracts may be mapped by visualizing

the streamlines through the field of eigenvectors.

LIC is a texture-based vector field visualization method. The technique

generates intensity values by convolving a noise texture with a curvilinear kernel










aligned with the streamline through each pixel, such as by


I(:ro) =T(a(s))k(so s)ds (:3.5)


where I(:ro) is the intensity of the LIC texture at pixel :ro, k is a filter kernel of

width 2L, T is the input noise texture, and a is the streamline through point :ro.

The streamline, a can he found by numerical integration, given the discrete field

of eigenvectors. The effect of increasing L is illustrated in Figure 3.1:3 where the

kernel width increases from left to right in the :3 images.

We can think of this process as texture synthesis, where we are specifying the

orientation of the texture to match the PDD at each pixel. The result is a texture

with highly correlated values between nearby pixels on the same streamline, and

contrasting values for pixels not sharing a streamline. In our case, an FA value

below a certain threshold can he a stopping criterion for the integration since the

diffusion field ceases to have a principal direction for low FA values. Stalling and

Hege [76] achieve significant computational savings by leveraging the correlation

between .Il11 Il-ent points on the same streamline. For a constant valued kernel, k,

the intensity value at I(a(s + ds)) can he quickly estimated by I(a(s)) + e, where

e is a small error term which can he quickly computed. Previously, C'I I!..g et al.

[66] have used LIC to visualize fiber direction from diffusion tensor images of the

myocardium.

We adapt this technique to tensor field visualization by incorporating the

FA value at each field location in to the LIC texture. By modulating the image

intensity with an increasing function of FA, we highlight the areas of white matter



























Figure :3.14: LIC fiber visualization in axial slice of brain.


Figure :3.15: LIC fiber visualization in coronal slice of brain,


and de-emphasize regions with uncertain directionality such as grey matter

and CSF. In Figures :3.14, :3.15, and :3.16 we have used color to indicate the :3-

dimensional orientation of the tracts, something not <..nri.; i-, I1 by the texture

information.












































Figure 3.16: LIC fiber visualization in sagfittal slices of brain.
















CHAPTER 4
RESULTS


The denoising and rendering techniques described in the previous section

were applied to a synthetic HARDI dataset. This dataset was generated using the

technique described by Ojzarslan et al. [64]. The dataset was designed to depict a

region of curving fibers, a region of straight fibers, and a crossing between the two.

Representative sharpened ODF profile are shown overlaid on the HA image on the

left side of Figure 4.1. The color anisotropy visualization shown on the right side of

Figure 4.1.

















Figure 4.1: ODF profiles overlaid on HA image (left), and color anisotropy (right)
of the synthetic dataset.


A small sample of the synthetic data, taken from near the crossing region,

is shown in Figure 4.3. The synthetic data were corrupted with Gaussian noise

of mean zero, and variance O.2 = 0.005. The noisy data are shown in Figure 4.4.

50
























Figure 4.2: Multi-LIC visualization of synthetic dataset.

The same voxels are shown after smoothing over the spherical manifold in Figure

4.5, after smoothing over the cartesian image domain in Figure 4.6 and after both

techniques have been used in Figure 4.7. The right-hand side of each figure shows

the sharpened ODF computed from the S values in the left-hand side.










Figur 4.3 Syt eti S (lf) reutn OD (rgh)







of nisepreentin the raw. daahaebensothed while preservsuing the overall.













2~
~CC~ 3.
;-Cc~ ~V.
1P ~bba~ -PPC~

rbilr ~i3[j~

tT U


ODF (right).


I


\t-


Cc
~c i


I
5


'E L:I
-rh I r

j?

(left), resulting


Figure 4.4: Synthetic S with noise added


~k b


6


r~

rS

BL

QtC~

(Fi~


~J

i/' i

II,
i L1

r, e
L II

~ ,C
L I


t

C,


1
;%
L


Figure 4.5: Manifold (FEM) smoothing results for S (left), resulting ODF (right).

shape of the S profile. This smoothness is propagated to the computed ODF

profiles .

We can compare the resulting distributions with the ground-truth by using

the square root of J-divergence (symmetric K(L-divergence) as a measure. This

divergence is defined as


d(p, q) =


(4.1)











~b ~C

~Yici a; ~L ~-


tr.;
r;


V

g
L/
.6
/

.I
ii'


C-'

g
rf.

~F
ii

S (left)










r:

O


,resulting ODF (right).


OC~



(Iti~-

~jlii~


Figure 4.6: Lattice (TV)


smoothing results for


Ib Y

~

~L; ..~ .

,C'


:e


Figure 4.7: Manifold and lattice smoothed S (left), resulting ODF (right).

where
1 (s, (0s, 4i)
A(P, ) = P(i i) log + q(0s, 4i) log (4.2)
ii= q(0s, 4i) p0,
In Table (4.1) we compare the distances, d(p, p) between the original synthetic

data, (p), and the unrestored data, the data restored only using the FEM method,

the data restored using only the TV-norm minimization, and the data restored

using both techniques. For each technique the mean distance, p(d(p, p)), between










Table 4.1: Error between ground-truth and restored synthetic data

Method p(d(p, p)) 0.2 ~,p>
p = No Restoration 1.0409 0.0173
p = FEM Restoration 0.9088 0.0125
p = TV Restoration 0.7420 0.0119
p = FEM + TV Restoration 0.6576 0.0139


distributions in corresponding voxels is presented, and the variance of these

distances, O.2 &p, p)) in COTTOSponding voxels is presented.

As shown in Table (4.1), the TV restoration had superior performance to the

FEM technique, both in terms of the mean error and variance of the error. The

combination of techniques had a lower mean error than either the FEM or TV

restoration, however the variance of the error was higher than that with either

technique alone.

The denoisingf algorithm was applied to a dataset consisting of 47 diffusion

weighted images of a rat spinal cord. Axial slices of one such image, before and

after denoisingf are shown in Figure 4.8.

















Figure 4.8: Original diffusion-weighted image (left), and denoised (right) from
spinal cord data.











The ringing artifacts visible near the sample boundary in the raw DWI in

Figure 4.8 have been noticably decreased. Note that the edges in the image have

been well preserved. From the color anisotropy images in Figure 4.9, it is clear that

the white matter fiber tracts are predominantly in the axial direction, which is

represented by the blue color.


















Figure 4.9: Original color anisotropy image (left), and denoised (right) from spinal
cord data.


Figures 4.10 and 4.11 show the HA and color anisotropy visualization results

respectively for various coronal slices of a rat brain dataset. The chosen slices

show the corpus callosum, an anatomical structure known to have long-range white

matter tracts.

The HA images show a visible distinction between grey and white matter. In

addition, the distinction between fiber directions is evident in the color ED images.

Red corresponds to the left-right direction and green corresponds to up-down and

blue is in-out of the page.





































































Figure 4.10: Original (left), and denoised (right) HA images for coronal slices of rat
brain.





































































Figure 4.11: Original (left), and denoised (right) ED images for coronal slices of rat
brain.
















CHAPTER 5
SEGMENTATION OF HARDI


5.1 Modelling Diffusion


In DTI, data are most often modelled in terms of the diffusion tensor. The

apparent diffusion coefficient is a quadratic form involving the tensor, and the

diffusion displacement pdf is a Gaussian with covariance matrix equal to a constant

multiple of the inverse of the tensor. For HARDI, we will advocate modeling

neither the diffusivity nor the displacement pdf, but instead modeling the diffusion

ODF. Computing the ODF was discussed in Section (3).

A simple way to store the diffusion ODF is as a collection of samples of the

function at regularly spaced points, or alternatively, as a histogram. This will

require a large number of data points to be stored in order to resolve complex fiber

geometry. In order to design efficient algorithms, we wish to find a parametric

model for the ODF with a small number of parameters, which is capable of

describing diffusion in the presence of IVOH.

5.1.1 Other Models


To put our proposed model in perspective we will first review some models for

diffusion used in previous literature.










5.1.1.1 Gaussian Mixture Model


Gaussian mixture models (GMM) are one the most commonly used model for

multimodal distributions. The trivariate Gaussian density function with mean p

and covariance E is given by

1 1
NV(xlp, E) = exp(- (x p) E 1(X p)) (5.1)
(2)3de(E 2

The GMM given in Equation (5.2) can describe the 3-dimensional diffusion

displacement pdf

P~x) e Nx~ps Es)(5.2)
i= 1

where m is the number of components in the mixture. Each Gaussian component

has its own 3 x 3 covariance matrix, Es, which will have 6 independent elements.

For diffusion data, all components will have ps = 0.

However, we are primarily concerned with the directional characteristics of

diffusion. This can be characterized by the marginal distribution, P(0, 4) obtained

by integrating over the radial component of P(x). This will allow us to use a

simpler model for the ODF. Additionally, with the GMM, we must be careful to

impose the positive-definite constraint on the covariance matrix of each component

of the mixture. Previously Fletcher and Joshi [30] have described geodesic analysis

on the space of diffusion tensors. The analysis includes an algorithm for computing

the intrinsic mean of diffusion tensors (the arithmetic mean of SPD matrices is not,

in general, an SPD matrix). Later in this chapter we will describe a similar analysis

on the space of ODFs which will result in much simpler algorithms.










5.1.1.2 Spherical Harmonics


The spherical harmonic (SH) expansion is a useful representation for complex-

valued functions on the sphere. We can represent the diffusion with the expansion



l=0 m= -1

Note that the coefficients at,m are complex-valued, so that the storage require-

ment is double that of an equivalent model with real variables, and the arithmetic

operations are more costly as well. Frank [33] -11---- -r- an expansion truncated at

order L = 4 (or higher) to describe multiple fiber diffusion. This requires at least

15 complex-valued coefficients per voxel. In general, the order L expansion can

describe diffusion with L/2 fiber directions.

Ojzarslan [65] has developed an extremely fast algorithm for computing a SH

expansion for the ODF given a SH expansion of the diffusivity.

5.1.1.3 Generalized Tensors


The diffusion tensor imaging model described previously represents diffusion

using a rank-2 tensor. Diffusion has been described more generally by Ozarslan et

al. [62, 63] by considering tensors of higher rank. A cartesian tensor of rank I will,

in general, have 31. Due to symmetry, the number of distinct components in a high

rank diffusion tensor will be much smaller. By generalizing the concept of trace, it

is possible to quantify the anisotropy of diffusion described by tensors of arbitrary

rank [64].











Since tensors of odd rank imply negative diffusion coefficients, only even rank

tensors are appropriate for describing diffusion. For diffusion tensors of rank 4,6,

and 8, the number of distinct components are 15, 28, and 45 respectively. It is not

clear how to extract fiber directions from higher rank tensors.

5.1.2 von Mises-Fisher Mixture Model


Many statistical approaches involve data over sRn. Since we are dealing with

multivariate data over the sphere, S2, We Wish to express the data using distri-

butions over the appropriate domain. Other coninon applications of directional

distributions are in the fields of geology and meteorology, since many measurements

in those fields are taken over the surface of a sphere. Data involving clock or com-

pass measurements also frequently use directional distributions. Other applications

are text clustering [28] and gene expression mapping. These distributions are

discussed in detail by Mardia and Jupp [54].

In this section we will present a directional model for the ODF in terms

of von Mises-Fisher distributions. This model has far fewer variables than the

previously discussed models, allows the fiber directions to be extracted easily,

involves constraints which are simpler to satisfy, and leads to a closed-fornt for

several useful measures.

The von Mises distribution is the most coninonly used distribution on the

circle. In polar coordinates it has the form


V(|p m 2los exp(a cos(8 p)) O < 0 < 2xr (5.4)










This distribution is closely related to the wrapped Gaussian distribution on the

circle. Like the Gaussian, it is unimodal and symmetric about the mean.

The von Mises distribution can be generalized to spheres of arbitrary geometry

by keeping the log of the distribution linear in the random variable x.

(a v/2-1
1(~ (xi m exp (sp x) (5.5)


where |x| = 1 and |p| = 1, a is the concentration parameter and Ik denotes the

modified Bessel function of the first kind, order k. The concentration parameter,

N, quantifies how tightly the function is distributed around the mean direction p.

For a = 0 we have a uniform distribution over the sphere. The distributions are

rotationally symmetric around the direction p.

For p = 3 the distribution is called the von Mises-Fisher (vMF) distribution.


if x~, ) f exp ( p x) (5.6)
4xr smh(s)










Figure 5.1: Example vMF distributions (a = 1, 5, 10, 15, 25). All distributions
have same mean direction, p.

A useful characteristic of the vMF distribution is that the product of two

vMFs may also be written as an unnormalized vMF. Since


exp(sip x) exp(Kyp x) = exp((asps + sypy) x)


(5.7)










we have


if (~ps Ke.T (xpymy) c -T (|( ipi+ Kpy ) P(i, y, i, y)) (5.8)


where




We will use this fact in the next section.

The ODF may have multiple modes in cases of intravoxel orientational

heterogeneity, so we wish to model the ODF with a parametric distribution which

is capable of describing this complex geometry. Since the vMF distribution is

unimodal, we can choose a combination of these distributions. In fact, since the

ODF is antipodally symmetric, we will need a mixture to describe diffusion in even

a single fiber region. Since the antipodal pair have p1 = -p2, We CRI1 Specify a

mixture with only 3 variables per component: the two spherical coordinate angles

describing p, and m.














Figure 5.2: Sample vMF mixtures for voxels with one and two fibers.





ODF(x) ij~~if xllps, s) (5.10)
i= 1











C'!s .. .-!in a convex combination of vMF distributions, the weights have the

property

re- = r > 0 (5.11)
i= 1

This ensures that the mixture still has nonnegative probabilities, and will integrate

to 1.

Since vMF distributions obey the property (5.7), the product of two von

Mises-Fisher mixture models is also proportional to v31F mixture model.

5.1.2.1 Fitting the vMF Mixture


In this section we describe the gradient descent algorithm for computing the

v31F mixture model. We will assume that we have been given a discrete set of

samples of the ODF. We seek a mixture of v31Fs which agrees with these samples

in the least-squares sense. It is important to note that we are not working with

random samples in the statistical sense. These are point samples of a function, so

we take a surface fittingf approach.

Using the spherical coordinates for .r and p


cos 8 sin 4 cos a~ sin /9

.r = sin 8 sin 4 I- = sincn sinp / (5.12)




we may write the v31F in polar form:


Mi (0, |0~, 7, ) = exp([cos cos /9 + sin sin /9 cos(8 c0)]) (5.13)
47r sinh(s)








65

The energy function we will seek to minimize is the L2 distance between the model

and the data.


ap,~,w1)(01. C~))~
I=1 i-i

Equation (5.14) is numerically differentiated to compute the gradient


G(c~ ac~, p, ~, W) E(c~ ac~, p, ~, W)
Qt+l = Qt at
2ac~
E(c~, p + ap, ~, W) E(c~, p ap, ~, W)
Pt+l = Pt at
aap
E(Q, p, K + aK, W) E(Q, p, K a~, W)
Kt+l = Kt at

G(c~, p, ~, w aw)- E(c~, p, ~, w aw)
I,', I i = Wt at
2aw


(5.14)


(5.15)


After each iteration we must project the weights, {w}, back onto the simplex

to maintain the property that the weights sum to one.

It is likely that most voxels will fit a mixture of 4 vMF distributions quite

well. In this case the model requires only 15 real-valued parameters to completely

describe. Once we have fit the vMF mixture to the ODF, we can directly extract

the fiber directions, {p}.

5.1.2.2 Entropy and Distance


We seek closed-form equations for several measures, since this will eliminate

the need for time consuming, and possibly inaccurate, numerical integration.

The entropy of the mixture model, which we showed can be used as measure

of anisotropy, can take several analytical forms. Although the Shannon entropy of

these distributions cannot be written in closed-form, some of the Renyi entropies










may.

Ho(X) = 1 Jlog(C P(xj") (5.16)

In particularly, the Renyi entropies of integer orders a~ > 1 can be written in

closed-form. Since the expression will have me terms, we will concentrate on the

case a~ = 2.


H2 (X) =-lolg~le 6 Pl(x)2 Slin Odi (5.17)


log
4xr smh ai sinh my pasy s y

The notion of dissimilarity between distributions is fundamental to the

problem of segmentation. In the segmentation problem we seek to partition some

domain into regions whose constituent elements are similar in some sense. We can

obtain a closed-form for the L2 distance between two vMF mixures, P(x|K, p) and





/ (P- Q2 d P2 dS + Q2 dS -2 P2 (5.18)



4xr sinh Ai sinh AspAAy j y
i= 1 j= 1
I, ,. .,X s smh p(Xi, Ay, ps, vy)
4xr sinh Xisinh As p(Xi, Ay ps, vy)
i= 1 j= 1



Relative entropy is also a common information-theoretic measure of dissimilar-

ity (divergence) between distributions.


Hx~f, g) = S log ~ l(-X( )l/ (5.19)










For A = 2 we have


H2(f g) = log )'"S "'/S (5.20)


Th~is mca~sure is related to the cosine similarity byi H2 f, y) = log( )f~

5.2 The Space of v31F Distributions


The von Mises-Fisher distribution is parameterized by two variables: the

concentration parameter a E sR+ and pe E 2. For each point in sR+ x ,92 there

is a corresponding vMF distribution. The curved geometry of this space of v31F

distributions will influence how we formulate distances, geodesics, interpolation

functions and means. A general treatment of the geometry of the spaces formed by

parametric distributions is given by Amari [1, 2].

5.2.1 Riemannian Geometry


The space of vMF distributions forms a differentiable manifold, a space which

locally behaves like Euclidean space. This manifold can also be considered to

be embedded in a higher dimensional space. For example the sphere, ,92 0811 be

considered to be embedded in sRS. The embedding space induces a metric on the

manifold which allows distances to be computed between points on the manifold.

This Riemannian metric is considered an intrinsic way to measure distance on a

manifold. The metric measures the length of a curve lying on the manifold. The

extrinsic way of measuring distance is to take the Euclidean distance between

two points in the embedding space. As an example on the sphere, the extrinsic

distance between the north and south pole is simply the diameter of the sphere, the










intrinsic distance is one-half of the circumference. If you are constrained to move

on the surface of the sphere, you would find the intrinsic distance more useful. This

difference is illustrated in Figure 5.3. Here we see the Euclidean distance, d(p, q),

compared with a curve segment y : [p, q] lying on the manifold. If y : [p, q] has the

shortest length of all curves between p and q, we can consider this length as the

intrinsic distance.















Figure 5.3: Intrinsic and extrinsic distance


When considering curves on a manifold, it will be useful to work in the space

of tangent vectors to these curves. At each point on a manifold there are an infinite

number of curves on the manifold which may pass through that point. The space

of the velocities of those curves is the tangent space at p, denoted T! 11 For a two

dimensional surface embedded in sR3, the tangent space is a plane perpendicular to

the normal of M~ at p, as shown in Figure 5.4.

A Riemannian manifold is a smooth manifold supplied with a Riemannian

metric. This metric takes the form of an inner product,(v, w), defined on the

tangent space, T .T for each point, p, on the manifold, M~. For two dimensional
























Figure 5.4: Tangent space

surfaces, we may simply inherit the vector dot product front the Euclidean tangent

space, but any function (-, -) satisfying the following properties will be acceptable:

1. Scaling: (a to w) = c (<', w)

2. Linearity: (u + t', w) = (u, w) + (v, w)

3. Syninetry: (', w) = (w, t)

4. Positive definiteness: (<', t') > 0 and (<', t') = 0 ta t = 0.

The Rieniannian metric allows us to measure the length of a curve, q*(t)

between two points, p, q on M~.


L~q* = (q*'t),q*'()); dt(5.21)


The notions of nietric, distance, geodesics, interpolation and mean are all

related. A geodesic between p, q is a curve ]* for which L(q*) is nxinintized, and

the nxininiun length is the distance, d(p, q). Note that there may not he a unique

geodesic between two points. On the sphere, for example, all meridians are

geodesics between the north and south pole. However, perturbing one of these










points will result in a unique geodesic. In general, if two points are near enough,

the geodesic is unique.

The mean can be defined in terms of the distance, d, as the point, p, which

satisfies

mm 2 i (5.22)
i= 1

Interpolation can be defined in terms of a weighted mean, so we can interpo-

late in 2D between 4 distributions by minimizing


mma vd(p, xoo) +(1 -u)vd(p, xio) +u(1 -v)d(p, xol) +(1 -u)(1 -v)d(p, xll) (5.23)
pLEM


5.2.2 Riemannian Exp and Log Maps


Let M~ be some manifold, and ( TT be the tangent space at pe M Consider

all geodesics going through the point, p, on M~. Given a tangent vector, v E T .T T

it is known that there is a unique geodesic, y, such that y(0) = p, and y'(0) = v.

If the manifold is geodesically complete, as it is in our case, the Riemannian

exponential map, Exp, : T .11 M~, can be defined as


Exp,(v) = y(1) (5.24)


Let us examine the exponential maps for a few simple spaces. For s9", the

geodesics are straight lines, so Exp,(v) = p + v. For sR+ the exponential map is the

exponential function, so Exp,(v) = pexp(v). On the sphere, S2, geOdesics are great

circles. Parameterize the sphere by x(0, 4) = (sin 4 cos 8, sin 4 sin 8, cos 4). If we

consider a point at the north pole p = (0, 0, 1), the tangent vector can be written as










v = (v,, v,, 0). In this case the geodesics in the direction of v are curves of constant

8, and we have cos 8 = g and sin 8 = We can parameterize geodesics with

speed, s, as

r(0) = ( vesin(a ), "- co(s(s#), cos(s#)) (5.25)

The exponential map is then given by



TU, M


~~Exp,(v) =( snIvl1 ~ sIll~o ll>(.6











Figure 5.5: Riemannian exponential map

Since the geodesic has constant speed, we know d(p, Exp,(v)) = ||v||.

The Riemannian log map is the inverse of the exponential map, Log,:

Exp,(v) T1 M. If the log map, Log, exists at q, we can write the Riemannian

distance between p and q as

d(p, q) = Log,(q) (5.27)

For the case of the sphere, given a point q = (q,, q,, qz) E S2 We know the

length of the tangent vector, since we can compute the distance d(p, q) = 0 =

cos l(p q). We also know the direction of the tangent vector is parallel to (q,, qu).























Figure 5.6: Riemannian log map

The Log map for the sphere is given by


Log,() = q, ., ty. ) 5.28)


5.2.3 Symmetric Spaces

The development of the previous section is sufficient to compute geodesics

at certain points for simple manifolds, but our space of distributions (and the

space of mixtures we will consider later) are more complicated. In this section

we will discuss the formalism required to work with the direct product manifold.

The next section will present the required framework for computing geodesics on

this manifold. This is the development presented by Fletcher and Joshi [31] for

exploring the space of shaped represented by medial atoms.

A symmetric space [49] (or symmetric Riemannian manifold) is a connected

Riemannian manifold such that for all p EM1 there exists an isometry (a length-

preserving mapping)













e, : M M(5.29)


with T4, : T. T T, -TT= -id, where id is the identity mapping. Such a mapping

will have the effect of reversing geodesics through p, and applying the mapping

twice has no effect on the manifold (a,2 d).

For sR3, the mapping is the reflection 2p x. For the sphere S2 the required

transformation is the rotation by 1800 about the normal at p. For sR+ it is the

reflection 13

A useful property of symmetric spaces is that the direct product of symmetric

spaces is also a symmetric space. Knowing sR+ and S2 arT Symmetric spaces, we

have that (sR+ x S2) is a Symmetric space, and even that (sR+ x S2) s IS Symmetric

space.

If Mr and M. 1 are two metric spaces and x yl E Mr and x2, Y2 E Mi ., then the

metric for My x M. 1 is d((xlxa, (2 9, Ya2) 2 1~l Y1) 2 2(a Y2 2

5.2.4 Lie Groups and Homogeneous Spaces


In an earlier section we computed geodesics and Exp and Log maps for the

sphere at a specific point, p, in this case the north pole. On the sphere it seems

intuitively obvious that we can rotate these geodesics about the center of the

sphere to obtain geodesics through other points. Observe that we can reach any

other point on the sphere by applying rotation matrices to the point, p, and that

the rotation transformation does not change the length of the geodesics. In this











section we will specify the conditions under which we may compute geodesics on

other syninetric spaces in an analogous way.

Now we can consider a vMF distribution to be a point in some syninetric

space. However a group structure will simplify some further operations. A group is

defined as a set, G, with an operator, which obeys these 4 properties:

1. Closure: if ,4, B E G, then ,4 B E G.

2. Associativity: for all ,4, B, C E G, ,4 (B C) = (,4 B) C.

:3. Identity: there exists a unique identity element, I, such that ,4 I = I ,4 = ,4

for all ,4 E G.

4. Inverse: for each ,4 E G, there exists an inverse, ,4-l EG uCh that'^C'~

4 ,4-l = ,4-l = I.

A Lie group is defined as a group which has differentiable multiplication and

inverse operators.

The space of vMF distributions, Af, does not have a group structure since

there is not a unique identity element. There are an infinite number of distributions

with a = 0. However, we can define a Lie group which operates on Af.

SO(:$) is the group of special (determinant = 1) orthogonal :3 x :3 matrices.

These matrices represent rotations in :3d. The group G = sR+ x SO(:$) acts

smoothly on AI. Let g = (s, R) E G, and ni = (n, p) E Af. Then the group action

g ni = (as, Rp). These are simply the scalar and niatrix-vector multiplication

operations. The isonietry subgroup G., is defined as the set of all g EG such that

gx -;,, .r,, =r(ie riafidpont unde the~, action;,, of T) n this case G.( (1, Q,,)

where Q,, is the set of all rotations about the p axis. If we again consider the base










point p to be the north pole, G, = (1, SO(2)), where SO(2) is the group of special

orthogonal 2 x 2 matrices (rotations in the plane).

The action of G on M~ is called transitive if for any two x, y EM1 there

exists a g EG such that g x = y. If the group action is transitive then M~ is

isomorphic to G/G,. For an arbitrary base point, p, we can write M~ as as the

homogeneous space M~ = G/G,. We can also see how the product of symmetric

spaces is also a symmetric space. Let My = GI/H1 and .ll = G2/H2, then

My x l. = (GI/Hl) x (G2/H2) = (GI x G2)/(HI x H2 -

If G is transitive on M~, and G, is a connected compact Lie subgroup of G,

then M~ has G-invariant metric. This means that the action of G does not change

distances on M~ : d(g p, g q) = d(p, q).

Geodesics on symmetric space represented as M~ = G/G, can be computed by

applying the group action to geodesics through p. So for the sphere, if y(0) = p,

and R E SO(3)/SO(2) such that Rp = q, then Ry is a geodesic through q.

To explicitly find such an R which transforms point p to q, we can form a

rotation matrix in the following way. The rotation about an arbitrary axis, [x, y, z]T

by angle 8 can be written

X2 COS 8 1 X2) 11 1 COS 8) Z Sin 8 ZZ 1 COS 8) y Sin 8

R = ,; (1 cos 8) + z sin 0 Y2 COS 8 1 Y2) yZ 1 COS 8) x Sin 8

xz(1 cos 8) y yz(1 cos 8) + x sin 8 z2 COS 8 1 Z2)
(5.30)

To align points p and. q, we~ rotate about the axis Ig~ by1 the angle 0= cos-l(p q).

To align tangent vectors v, and w, we can rotate about p by 8 = cos-l(v w).










5.2.5 Results


We have used the fact that the direct product of symmetric spaces is also

a symmetric to deduce that the space of vMF distributions is symmetric. Now

we will use this fact to compute the Exp map for vMFs. For spaces which are

expressed as direct products, we can write the exponential map as the direct

product of the exponential maps for the constituent spaces. For a single vMF, let

p = (s, p-) represent the distribution if (xlx, p-), and v = (a, u) E T.TT be the

tangent vector. Then

sin ||U|I


Exp,(v) = (a exp(a), Q sinll"I (5.31)

cos ||U||I


5.3 The Space of vMF Mixtures


Now, let us investigate the space of mixtures of vMF distributions. The

mixture model of m components is given in Equation (5.10). At first, it may seem

that we can simply extend the results of the previous section, and consider these

mixtures to come from the space (sR+ x sR+ x S2)m. However, considering the set of

weights as an point in (sRt)m ignores the convexity constraint on the weights. The

space (sR+)m includes linear combinations of vMFs whose weights do not sum to 1.

Instead, we consider the square roots of the weights, {@/W...97}.) The

convexity constraint now becomes


iE- i2 = w >= 0 (5.32)
i= 1

















(a) y(0) (b) y(0.1) (c) y(0.2) (d) y(0.3)








(e) y(0.4) (f) y(0.5) (g) y(0.6) (h) y(0.7)








(i) y(0.8) (j) y(0.9) (k) y(1)

Figure 5.7: Points along the geodesic between two vMF distributions.

So, we can consider the space of the square roots of the weights to be a

hypersphere, Sm-l. Then, the space of mixtures with m components is Sm-]

(s9+ x S2 m

5.3.1 Riemannian Exp and Log Maps

For the vMF mixture, the exponential map is the direct product of the

exponential maps for each vMF, and the exponential map for Sm-l. Since we are

quite unlikely to have more than 4 fiber orientations present within a single voxel,

we will consider further the case of mixtures having 4 components (m = 4). In this

case, the space of the square roots of {w} is the unit hypersphere S3










Fortunately, the space S3 is well studied, since this is equivalent to the space

of unit quaternions. It is also closely related to the space of orthogonal matrices

SO)(:) (technically, S" is a double-covering of SO)(:)). In fact, S" forms a Lie group

with respect to the quaternion multiplication operator.

Quaternion operations can he denoted compactly by observing that S3 is also

isomorphic to SU(2), the group of special (determinant = 1) unitary matrices.

Quaternion operations are usually presented in terms of 4-dimensional complex

numbers, but we can write the same operators in terms of matrix operations. The

correspondence between quaternions, q, and Q E SU(2) is given by



q = (a + bi + cj + dk) ta Q = (5.:3:3)+ c
b +ci a +di (.3

In this notation, the identity quaternion, q = 1, is represented by the 2 x 2

identity matrix, quaternion multiplication becomes matrix multiplication, and the

quaternion inversion becomes conjugate transposition (q-1 t H) since the matrix

is unitary. The quaternion, q, is often considered to have a scalar part, the real

component a, and a vector part [b, c, d] .

The exponential map for S" is


Exp;>(c) = n, cos( || C||)1 (5.34)


and the log map is given by

2 cos-lq,
Log>(q) = Gree, (5.35)


where q,-ec and q,, are the vector and scalar parts respectively of the quaternion q.










In computer graphics, the relation between quaternions and SO(3) is often

used to compute smooth interpolations between rotation matrices in animation

[75]. The commonly used spherical linear interpolation (slerp) can be written in

terms of the Log and Exp map as


slerpt (p, q) = p Exp (t Log (p lq))


(5.36)


(a) y(0)








(e) y(0.4)


(b) y(0.1)








(f) y(0.5)


(c) y(0.2)


(d) y(0.3)








(h) y(0.7)


(g) y(0.6)


(i) y(0.8) (j) y(0.9) (k) y(1)

Figure 5.8: Points along the geodesic between two sets of weights.










5.3.2 Intrinsic Mean


Computing the "center" of a group of distributions has several applications

for our data. The first has already been mentioned: interpolation. The other

application is the construction of atlases. By registering multiple datasets across a

population onto a common coordinate system we can compare ODF distributions

at corresponding voxels.

Previously, the intrinsic mean problem has been solved with a gradient descent

algorithm [68, 32, 48]. The gradient of the energy function in Equation (5.22) can

be written in terms of the Log map. The algorithm, as given by Fletcher and Joshi

[32] is

Input: xl,..., xr EM1

Output: pe M the intrinsic mean

I-o = xl

Do


apl = C( E Logmt(x4)


Pt+1 = Exp,,(a p)l

While ||ap|| >

5.4 Segmentation Models


The mean, variance, and distance formulations discussed in the previous sec-

tion can be quite useful in the context of model-based segmentation. In this section

we will present results obtained using several popular segmentation frameworks.










The Mumford-Shah functional leads to a region-based approach to segmen-

tation. Early curve evolution models for segmentation were based on the image

gradients along an evolving curve. This could lead to the curve becoming trapped

in local minima of the energy function. Evaluating the Mumford-Shah functional

involves integrating over the entire region enclosed by the curve.

Spectral clustering is a useful grouping technique which is related to graph

partitioning algorithms. The graph under consideration has a node for each voxel

in the image, and the edge weights are affinities between nodes. This affinity can be

computed from some distance function between voxels. If we partition the graph so

that each subgraphs has maximum "association" within the group, we will achieve

a segmentation of the underlying image. This partition can be approximated by

solving for the 6,_ n-i--1b in of the affinity matrix of the graph.

A statistical approach to image segmentation will also be explored. In the

Markov Random Field model, the measured image is the sum of a noise process

and a parametric model depending on a label field. The value of the label field at

each voxel is statistically dependent on only nearby neighbors. We will segment the

image by finding the most probable label field given the observed image.

In the following section we will cover these segmentation models in more

detail, and present results obtained from each.










5.4. 1 Mumford-Shah Model


We can use the robust intrinsic mean and distances defined previously in the

Mumford-Shah model. The interpolation methods described previously may be

used to model piecewise smooth regions.

The Mumford-Shah functional is given by


E(@,C) p 2 2d + ds(5.37)


where R is the image domain, and C is the curve forming the boundary between

two regions, and # is a piecewise smooth model for the image, I. Since minimiza-

tion of this functional involves solving for # and C, this process can be seen as

simultaneous restoration and segmentation. The minimizer of this functional will

have a boundary whose smoothness is controlled by the parameter y.

Partitioning R into two regions we rewrite Equation (5.37) as



E( ~JR JRc JR JR, JC(Ti lr ~~V,',~ Pi i~l;

(5.38)

where R is the region enclosed by C, Rc, is the region outside C.

The piecewise constant model simplifies the general model.


JR J5c JC


and p-1 and where p are the mean over regions R and Rc respectively.

The Mumford-Shah model of segmentation may be implemented in a curve

evolution framework, where a parametric representation of C flows toward a










minimizer of the functional. This is the approach taken by Tsai et al. [81]. The

alternative, presented by ('I! I1. and Vese [19] is a level set approach, where the

curve is represented as the zero level set of an embedding function.

For intensity images, d2(@, I) may be (4 I)2, and p- may be the arithmetic

mean. We may use the geodesic distance and geodesic mean to segment HARDI

images .

5.4.2 Spectral ('!.1-1. i llig~


The name "spectral
of the affinity matrix can reveal the grouping structure induced by the affinity

function. Bach and Jordan [3] show how spectral clustering can be seen as a

relaxation of the normalized cut problem. Normalized cuts and the resulting

eigenvalue problem have been discussed by Shi and Malik [74].

Fundamental to the problem of segmentation is the concept of distance

between voxels. We seek regions whose constituent voxels are nearby in some

sense. The choice of distance measure, and therefore affinity function, will have

a great affect on the results of spectral clustering. In the case of a gi li---- 1.

image, the absolute value of the intensity differences is an intuitive choice for the

distance function. For HARDI data, we can use the closed-form equation for cosine

similarity to form the affinity matrix. Sample eigenvector images obtained from

HARDI data are shown in Figure 5.9.






























Figure 5.9: So and eigfenvector images.


5.4.3 Alarkovian Models For Segmentation


Markov chains are used to model sequences of variables whose value depends

only a fixed number of previous values. The Markov Random Field (jl!RF) is an

extension of this concept to 2D (and higher). Here we consider random variables

on a lattice, L. The value of the random variable at each lattice point (or site) is

dependent only on the values of its neighbors. In Figure 5.10 we shown an example

of a first-order neighborhood, Ni~, where the value at each site depends only on the

4 ..111 Il-ent sites in L.

Within each neighborhood we define cliques, sets of sites such that each pair of

sites are neighbors. When these random values are interpreted as image intensities

we can use this locality to impose smoothness constraints on the image. In general,

for larger (higher order) neighborhoods we can achieve smoother images.

















(a) N~








(b) C1 (c) C2 (d) Os (e) C4

Figure 5.10: First order neighborhood, NI~, and nearest-neighbor cliques C1, C2, C3
04 -

A famous theorem, the Hammersley-Clifford theorem, states that if F is a

MRF with respect to some neighborhood system, then F is also a Gibbs Random

Field. A proof of this theorem is given by Besag [12].

1 -U(f)
P(f)= ex() (5.40)
Z T

where T is a constant, Z is a normalizing constant, and U is an energy functional

with the form

U/( f) = Ve( f) (5.41)
cEC

where Ve( f) is called the clique potential.

5.4.3.1 MRFs for Segmentation


Before considering the Hidden Markov Measure Field (HMMF) segmentation

model, it may be useful to describe the classical MRF model. In this model, the










observation model is given by


I(x) = (,O~kx ~)(5.42)


where x is a pixel location, and I is the observed image, O is a parametric

image model with parameters 0. The subscript k ranges over the M~ image classes.

An additive noise field is denoted rl. The image is assumed to be partitioned into

dl;i 1~! regions, Rk. The indicator function, bk, TeturnS Value 1 When x E Rk, and 0

otherwise.

The unknown label field, f(x), which it is our goal to estimate, is related to

the indicator function by bk(x) = 6(f(x) k). It is this label field which is assumed

to be a sample from a MRF, with prior probability given by the Gibbs distribution,

as in Equation (5.40). There are many models for the clique potential function.

The Ising model is a simple and quite common model given by



vc~f~fj) -p if fi = fj and i,j jC (.

+p if fi / fj and i, j EC

The parameter can be chose by the user to control the smoothness of the

field.

The MRF analysis has an intermediate goal of finding the conditional prob-

ability of the unknowns given the observations, in our case P(f, 0|I). It may

seem natural to seek the mode of this distribution, ( f, 8) which have maximum

probability. However this estimator, the MAP (maximum a posteriori) is but

one possible choice. The mean or median of the posterior may be more useful for










certain applications.

?i,(k)j = ):P( f |I) (5.44)
f :f (.)=k
Using Bayes rule, we can write the posterior distribution as

P(| f 8)P( f, 8)
P( f, 0|I) =(5.45)
P (I)

Since f and I are independent, we have P(f, 0) = P(f)P(0). The label field is

Markovian, so P( f) is the Gihhs distribution given in Equation (5.40). The prior

for the model parameters, P(0) depends on the model. If we have no information

about the parameters values we may assign a uniform distribution. P(I) is the

probability of observing image I over the space of all possible images. This is a

constant we will denote hv Z.

The likelihood of the observations, in the presence of independent noise with

distribution if is


P(f 0|I P I.)|f(r) = k, 8 = y(I(.r) (., Ok)) (5.46)
.rEL .rEL

Since we can assume Gaussian noise, we use the noise model


P(I(r)|f (r) k,8) exp-q*l(.) -#(., O 2)(5.47)


where ]* is proportional to ( one divided by the noise variance).

5.4.3.2 HMAIFs for Segmentation


Alarroquin et al. [55] present a variation on the AIRF segmentation model

which has fewer variables and can solved without slow stochastic methods. The

HMMF model uses a deterministic model for the labels. The label field is assumed