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Medical Image Segmentation and Diffusion Weighted Magnetic Resonance Image Analysis

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IwouldrstandforemostliketoexpressmydeepgratitudetomyadvisorprofessorYunmeiChen,foreverythingshehasdoneformeduringmydoctoralstudy.Thisdissertationwouldnotbepossiblewithouther.Sheprovidedinvaluableadviceonresearchandlife.Dr.Chenintroducedmetotheeldofmedicalimagingandalwayshelpedandencouragedme.Ihavebeenveryluckytoworkwithher.Herenthusiasmaboutresearchstimulatedmyinterests,herinsightandexperiencehaveguidedmethroughmyresearch.Iwouldalsoliketothankmyotherexcellentcommitteememebers,Dr.DavidGroisser,Dr.YijunLiu,Dr.MuraliRao,Dr.DavidWilsonforprovidingtheiradviceanddiscussion.IalsoappreciateDr.YijunLiu,Dr.GuojunHefromBrainInstituteandDr.ZhizhouWangforhappycooperation.Theyprovidedalotofinvaluableresearchmotivationanddata.IthankFengHuang,oneofmypeers,forhelpingmewithnumericalimplementationinmyearlystageresearch.Withouthishelp,Icouldnothavemyscienticcomputationability.IalsothankHemantTagarefromYaleUniversity,RachidDerichefromINRIA,SheshadriThiruvenkadam(whoisnowatUCLA)forexcellentcommentsanddiscussiononmywork.Ineverwouldhavebeenabletohavenishedthedissertationwereitnotfortheunwaveringloveandsupportofmyfamily:mymumZhongying,myhusbandQingguo,mybrotherWeidong,mysisterShuangxia.Theirsupportandencouragementweremysourceofstrength.Iwanttoespeciallythankmyhusbandwhoteachesmethenatureoftruelove.Heisanextremelysupportiveman,Ialwaysappreciatehisenthusiasmforlifeandresearch.Iamsogratefulaboutmy10-month-olddaughterTienna,whoteachesmehowwonderfullifeisbyhersunnysmiles.IwanttothanktheeditorintheEditorialoce,Dr.DavidGroisserandDr.DavidWilsonfortheircarefulandthoroughcorrectionstomythesis.Iwouldalsoliketothankthestaandthefacultymemberswhohavebeenverypatientandhelpfulduringmystudyhere. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1Background ................................... 12 1.1.1MedicalImageSegmentation ...................... 12 1.1.2DiusionWeightedImageanalysis ................... 19 1.1.3NeuronFiberTractsReconstructionbasedonSmoothTensorField 23 1.2Contributions .................................. 24 2USINGPRIORSHAPEANDPOINTSINMEDICALIMAGESEGMENTATION 29 2.1Descriptionoftheproposedmodel ....................... 29 2.2Numericalscheme ................................ 33 2.3ValidationandApplicationtoEchoCardiovascularUltrasoundImages .. 34 2.4Conclusion .................................... 37 3USINGNONPARAMETRICDENSITYESTIMATIONTOSMOOTHANDSEGMENTIMAGESSIMULTANEOUSLY .................... 41 3.1ProposedModel ................................. 41 3.2NumericalImplementation ........................... 45 3.3ValidationandApplicationtoT1MagneticResonanceImage ........ 49 3.4AnExistenceTheoremfortheModel ..................... 54 3.5Conclusion .................................... 59 4ESTIMATION,SMOOTHINGANDCHARACTERIZATIONOFAPPARENTDIFFUSIONCOEFFICIENT ............................ 61 4.1Introduction ................................... 61 4.2ModelDescription ............................... 62 4.3Characterizationofanisotropy ........................ 68 4.4NumericalImplementationIssues ....................... 69 4.5ValidationandApplicationtoDiusionWeightedImages(DWI) ...... 73 4.5.1Analysisofsimulateddata ....................... 73 4.5.2AnalysisofhumanMRIdata ...................... 74 4.6AnExistenceTheoremfortheModel ..................... 77 4.7Conclusion .................................... 82 5

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............. 87 5.1Introduction ................................... 87 5.2NewApproximationModelforADCProles ................. 88 5.3UseofCREtoCharacterizeAnisotropy .................... 92 5.4Conclusion .................................... 97 6RECONSTRUCTIONOFINTRA-VOXELSTRUCTUREFROMDIFFUSIONWEIGHTEDIMAGES ................................ 98 6.1DeterminationofFiberDirections ....................... 98 6.2ValidationandApplicationtoHARDWeightedImages ........... 99 6.3Conclusion .................................... 102 7RECONSTRUCTOFWHITEMATTERFIBERTRACESUSINGMULTI-TENSORDEFLECTIONINDWI ............................... 104 7.1Introduction ................................... 104 7.2RecoveryofMulti-TensorFieldinHARDMRI ................ 104 7.3WhiteMatterFiberTractography ....................... 106 7.4ExperimentalResults .............................. 107 7.5Conclusion .................................... 111 8FASTSEGMENTATIONOFWHITEMATTERFIBERTRACTSBASEDONGEOMETRICFLOWS ............................. 113 8.1Introduction ................................... 113 8.2Model ...................................... 113 8.3ExperimentalResults .............................. 116 8.3.1Syntheticresults ............................. 116 8.3.2Humanbrainresults .......................... 117 8.4Conclusion .................................... 117 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 127 6

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Table page 3-1SegmentationAccuracy ................................ 47 4-1ListofS0andAl;m'sfortworegions ......................... 73 7

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Figure page 2-1Comparingsegmentationresultsofasyntheticimage ............... 36 2-2Comparesegmentationresultsofonecardiacultrasoundimage. ......... 38 2-3Comparesegmentationresultsofanothercardiacultrasoundimage. ....... 39 3-1Compareresultsofmodel( 3{5 )basedontwochoicesofr(x) ........... 46 3-2Segmentationresultsofacleanimageandthatofanoisyversion ........ 48 3-3SegmentationresultofanoisyT1humanbrainimage ............... 49 3-4Comparethreemodelsinsegmentingimageswithlowerandhigherlevelofnoises 51 3-5ComparisonofgraphsofSAobtainedfromthreemodels ............. 52 3-6Comparesegmentationandsmoothingresultsbetweentwomodelswithxedandadaptiveradii .................................. 53 4-1Comparingshapesofd 77 4-2ComparisonofA20 83 4-3ImagesofFAandR2 84 4-4ZoomedFAandA20 85 4-5Classicationofvoxelsbasedond 86 5-1ComparisonofADC'sobtainedfromtwomodels .................. 91 5-2ImagesofR2bydierentmodels .......................... 93 5-3AnexampleforshapesofADCforisotropic,one-berandtwo-bervoxels ... 94 5-4Dierentmeasures .................................. 95 5-5Characterization ................................... 96 6-1Fiberdirectioncolormap .............................. 101 6-2Shapeofdwithorientations ............................ 102 6-3Fiberdirectioneld .................................. 103 7-1FAimageoftherstchannel ............................ 108 7-2ComparisonbetweenTENDandMTEND ..................... 109 7-3ComparisonbetweenMTENDandTENDatinternalcapsule ........... 110 8

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.................. 114 8-2Segmentationofa2-Dtensoreldusingtwomethods ............... 116 8-3Comparisonofcorpuscallosumsegmentationresultsfromtwomodels ...... 117 9

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Medicalimagesegmentationplaysanimportantroleindiagnosis,surgicalplanning,navigation,andvariousmedicalevaluations.Medicalimagesarefrequentlycorruptedbyhighlevelsofnoise,signaldropoutandpoorcontrastalongboundaries.Sometimes,theirintensitymighthavemulti-modaldistribution.Inthisdissertation,Iwillpresentonemethodtosegmentimagesthatarecorruptedbynoiseandsomedropout.Inthemodelpresented,priorpointstogetherwithpriorshapeinformationareincorporatedintoajointsegmentationandregistrationmodelinbothavariationalframeworkandinlevelsetformulation.Thistechniqueisappliedtosegmentcardiacultrasoundimages.Asecondmodel,whichisbasedonapplyingnon-parametricdensityapproximationtosimultaneouslysegmentandsmoothnoisymedicalimageswithoutaddingextrasmoothingterms,ispresented.Mygoalistodevelopapowerfulandrobustalgorithmtolocateobjectswithinteriorshavingacomplexmulti-modalintensitydistributionand/orhighnoiselevel.ThemodelwasappliedtotheproblemofsegmentingT1weightedmagneticresonanceimages. Diusionweightedimagesrendernon-invasiveinvivoinformationabouthowwaterdiusesintoa3Dintricaterepresentationoftissues.Myworkprovideshistologicalandanatomicalinformationabouttissuestructure,composition,architecture,andorganization.Ihaveproposedseveralmodelstoreconstructhumanbrainwhitematterbertracts,torecoverintra-voxelstructure,toclassifyintra-voxeldiusion,toestimate, 10

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1.1.1MedicalImageSegmentation 1 { 6 ])relyontheinformationoftheedges,suchashighmagnitudeofimagegradient.Theregion-basedmethods(e.g.[ 7 { 10 ])makeuseofhomogeneityonthestatisticsoftheregionsbeingsegmented.Thealgorithmdevelopedin[ 11 ]integratesgradientandregioninformationwithinadeformableboundaryndingframework.TheGeodesicActiveRegionmodelsproposedin[ 12 { 14 ]integratetheedgeandregion-basedsegmentationmethodsintoavariationalapproach. Theanalysisofmedicalimagesisfrequentlycomplicatedbynoise,dropout,confusinganatomicalstructures,motion,poorcontrastalongboundaries,non-uniformityofregionalintensitiesandmulti-modalintensitydistribution.Itishardtondonemodeltosegmentimageswithalltheproblemsmentionedabove.Wewillrstprovidesomegeneralliteraturereviewonimagesegmentationmodels,thenfocusonmodelsdealingwithimagesthatarecorruptedbynoiseandhavingmulti-modaldistribution.Finally,wereviewsomeexistingmodelsthatsegmentimagescorruptedbynoise,dropout,andpoorcontrastalongboundaries. LetI0:!Rbeagivenboundedimage-functiondenedonanopenandboundedregionwhichisassumedtobeasubsetofR2forthepurposeofillustration.Butanydimensionalcasecouldbesimilarlyconsidered.Thesegmentationproblemistondadecompositioni'sofandanoptimalpiecewisesmoothapproximationIofI0suchthatIvariessmoothlywithineachi,andrapidlyordiscontinuouslyacrosstheboundariesofi.LetCbeaclosesmoothcurveformedbyboundariesbetweeni's,andletthelength 12

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8 ]proposedtominimizethefollowingfunctional: Therstandsecondpartsarecalleddatadelityandsmoothnessterms,respectively.Iftheparametergoesto1,thentheapproximationIwillbeforcedtobeconstantCiineachi.PiecewisesmoothMSisthenreducedtopiecewiseconstantMS: Whilethefunctionaliselegantitself,itisdicultinpracticetondasolutionasthefunctionalisnon-convex.L.Ambrosioetal.[ 15 ]approximatedpiecewisesmoothMSvia-convergence.A.Tsaietal.[ 10 ]solvedpiecewisesmoothMSusingparametriccurveevolutiondirectly. ChanandVese[ 16 17 ]implementedbothpiecewiseconstantandpiecewisesmoothMSusingageometricimplicitframework-levelsetwhichwasinventedandcontinuouslyadvancedbyS.OsherandJ.Sethian[ 18 ].ThelevelsetmethodisusedtorepresentCimplicitlyandtoexpresseachsubregion.Themainadvantageofthelevelsetrepresentationisthattopologicalchanges,suchasmergingandpinchingoofcontourscanbecapturednaturallythroughsmoothchangestothelevelsetfunction. Forillustrationconciseness,weonlyconsiderthe2-phasecase:Iiscomposedofanobjectandbackground(i.e.i=1;2).Foranimagewithmorethan2phases,themodelcouldeasilybeextendedbyusingmorelevelsetfunctions.ThecurveCisrepresentedbythezerolevelsetofaLipschitzfunctionbyC=fxj(x)=0g.Theobject(A)andthebackground(nA)arerepresentedbyfxj(x)>0gandfxj(x)<0g,respectively.AHeavisidefunctiondenedasH()=1when>0and0elsewhereisusedtodistinguishAandnA.Thus,AcorrespondstotheregionwhereH()>0,while2correspondstotheregionwhere1H()>0.LetbetheDeltafunction. 13

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1{2 )(2-phasecase)istominimizethefunctional: wherethethirdtermisthetotalvariationofH(). Duetoitssimplicityandrobustness,theChan-Vesemodel( 1{3 )hasbecomequitepopularandhasbeenadoptedinmanyapplications.Ithasbeenexpandedtosegmentingvector-valuedimages,textureimagesetc.In[ 19 ],A.Yezzietal.alsousethelevelsetframeworktosegmentimages. ThevariationalformulationintheChan-Vesemodelisnon-convexandatypicalgradient-descentimplementationisnotguaranteedtoconvergetotheglobalminimumandthuscangetstuckinlocalminima.Onetrick[ 16 17 ]istosetecientinitials.Oneexampleisalargenumberofsmallclosecontoursuniformlydistributedintheimage.Inthisway,contoursareinitiatedeverywhereintheimagesuchthatchanceofcapturingtheglobalminimumisenhanced.ChanandEsedogluetal.[ 20 21 ]provideanothernovelandfundamentallydierentapproach.Itbasicallyconvexiestheobjectivefunctionbytakingadvantageoftheimplicitgeometricpropertiesofthevariationalmodels. ThetraditionallevelsetfunctionsareLipschitzcontinuous,andareinitializationprocedureissometimesneededtopreventthelevelsetfunctiontofrombecomingtooat.Recently,Lieetal.[ 22 ]representinterfacesusingpiecewiseconstantlevelsetfunctions,inwhicheachlevelsetfunctioncanonlytaketwovaluesatconvergence,e.g.,thefunctioncanonlyequal1or-1.Someofthepropertiesofstandardlevelsetmethodsarepreservedintheproposedmodel,whileothersarenot.Itisclaimedthatthenewmethodprovidesasgoodresultsasmethodsusingcontinuouslevelsetfunctions.TheReinitializationprocedureisremovedinthenewmethod.Taietal.[ 23 ]alsointroduceamodelthatonlyrequiresminimizingtheChan-Vesefunctional( 1{3 )withrespecttothelevelsetfunctions,withoutestimatingtheconstantsCi. 14

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17 ],butthiscanleadtodicultyinsettinginitials.Gaoetal.[ 24 ]recursivelyuseonelevelsetinahierarchicalway:rstsegmentintotworegions,thensegmenteachregionintotwonewregions,andsoon.See[ 25 ]forotherrelatedrecentdevelopments. AgeneralizationtopiecewiseconstantMSisthatinsteadofapproximatingIbyaconstantCiineachregioni,onecangenerallyapproximatetheintensityineachregionibyaprobabilisticmodelP(I(X)ji)withaparametervectori.ByBayesiananalysis,wehavetheposteriordistributionP(ijI(X))/P(I(X)ji)P(i).WhenchoosethepriordistributionP(i)tobeuniformdistribution,P(ijI(X))/P(I(X)ji). Thersttermin( 1{2 )isthenreplacedbysocalleddescriptioncostand( 1{2 )becomes: Zhuetal.[ 7 ]andRoussonetal.[ 26 ]choseP(Ii(X)ji)tobeaGaussiandistributionandpermitforeachregioninotonlyadierentmeanCibutalsoadierentvariancei: Wewouldliketomentionthatwhenallthei'sarethesame,model( 1{5 )isequivalentwithmodel( 1{2 ).Wecallmodel( 1{5 )aglobalGaussianprobabilitydensityfunction(pdf)basedmethodsinceitassumesallthepixelsinoneregionisharethesamemeanandvariance. Model( 1{4 )hasitsrestrictioninrealapplicationsinceitisbasedonaspecicassumptionoftheintensitydistribution,whileusuallywedonotknowwhetherthisassumptionisreasonable.Especially,whenthereexistshighlevelofnoiseand/orcomplexmulti-modalintensityfunction,onesingleglobalGaussianpdforanyothersingleparametricdistributionisnotenough.J.Kimetal.[ 27 ]presentedaninformation-theoretic 15

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28 ]usednonparametrickernel-basedapproximationoftheintensityprobabilitydensityfunctiontocapturetexture,thistextureinformationinthenincorporatedintoavariationalframeworkforimagesegmentation.Abd-Almageeetal.[ 29 ]introducedageneralframeworkfordrivinganactivedeformablemodelusingnonparametricestimationofthepdf.AParzenwindowapproachwithaGaussiankernelfunctionwasusedasaparameter-freeestimatorforthepdf'sforboththetargettobesegmentedandthesurroundingbackground.InChapter 3 ,anewmodelbasednonparametricdensityestimationisprovidedtosegmentandsmoothimagessimultaneously. Forimagesthathavenoise,dropout,andpoorcontrastalongboundaries,itisnotenoughtouseimageinformationitselftoobtainthedesiredresults.Recently,variousapproaches,includingdeterministicsettingandprobabilisticcontext,havebeendevelopedtousepriorshapeinformationinimagesegmentation(see[ 30 { 39 ]).Asurveyofmethods,whichincorporatepriorknowledgeintodeformablemodelsinmedicalimageanalysisisprovidedin[ 40 ].Theapproachespresentedinthepapers[ 8 9 41 { 43 ]arecloselyrelatedtoourcurrentworksincetheyincorporatethestatisticalshapeknowledgeintoeitheredge-basedorregion-basedsegmentationtechniques.Leventon,GrimsonandFaugeras[ 41 ]extendedthetechniqueofgeometricactivecontoursbyincorporatingshapeinformationintotheevolutionprocess.Asegmentationresultwasrstobtainedbythecurveevolutiondrivenbyaforcedependingonimagegradientandcurvature.Later,theshapepriorwas 16

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42 ]modiedthetechniquesdiscussedbyLeventonetal.[ 41 ]sothattheirvariationalmethodincorporatesbothhighimagegradientsandshapeinformationintotheenergytermofageodesicactivecontourmodel.Theshapetermcanalsobeusedtorecoverasimilaritytransformationthatmapstheevolvinginterfacetothepriorshape.Cremers,SchnorrandWeickert[ 9 ]incorporatedstatisticalshapeknowledgeintotheMumford-Shahsegmentationscheme[ 8 ]byminimizingafunctionalthatincludestheMumford-ShahenergyandtheshapeenergycorrespondingtotheGaussianprobability.Recently,in[ 43 ]RoussonandParagiosintroducedanenergyfunctionalthatconstrainsthelevelsetrepresentationstofollowashapeglobalconsistencywhilepreservingtheabilitytocapturelocaldeformation. Theexperimentalresultshaveshownthatallthesemodelshavetheirownstrengths,andprovidepromisingresultsforparticularapplications.However,duetotheaccuracyandeciencyrequirementformedicalimageanalysis,andthecomplexityofmedicalimages,aswellasthevariabilityoftheshapesofanatomicstructures,thequestionofhowbesttouseshapepriortogetabettersegmentationwillremainchallenging.Oneofthemostdicultproblemsistodeterminelocalshapevariationsfromthepriorshape.Onesolutionmaybetheuseofnonrigidregistrationtoassistsegmentation[ 44 45 ],butthisapproachrequiresreliableregionoredgeinformationintheimagetoestimatethevelocityeld.Thistechniqueisalsocomputationallyexpensive. Thegoalofthemethodpresentedinchapter 2 istondanimprovedwaytolocateboundarieswhentheymayvarywidelyfromapriorshapeandwhentheyaretobedetectedinanimagewhichmayhavesignicantsignalloss.Inparticular,wearemotivatedbytheproblemofdetectingtheboundaryoftheleftventricleoftheheartinechocardiographicimagesequences.Cardiacultrasoundimagesareplaguedwithnoise,signaldropout,andconfusingintracavitarystructures.Moreover,theshapeoftheboundaryofthemyocardiumvariesextensivelyfromonepatientimagetoanother,andis 17

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Togetadesirablesegmentationinsomecardiacultrasoundimagesmorepriorknowledgethantheexpectedshapeisneeded.Forinstance,in[ 46 ]DiasandLeitaousedtemporalinformationfromtimesequenceofimagestoassistthedeterminationoftheinnerandoutercontoursintheareasoflowcontrastinechocardiographicimages.Inthisnoteweconsiderthecasesinwhichusershavetheknowledgeaboutthelocationsofafewpointsontheboundaryoftheobjectofinterest.Wewillusethisinformationasaconstraintinadditiontoshapeconstrainttocontroltheevolutionofactivecontours.Ourbasicideaistoextendthesegmentationalgorithmdevelopedin[ 42 ]byincorporatingtheinformationonthelocationofafew\key"pointsinadditiontoshapepriorintogeometricactivecontours. Theideaofmatchingnonequivalentshapesbythecombinationofarigidtransformationandapoint-wiselocaldeformationdevelopedinthepapers[ 43 47 48 ]willbeappliedtoourformulation.In[ 48 ]SoattoandYezziviewageneraldeformationasthecompositionofanitedimensionalgroupaction(e.g.rigidoranetransformation)andalocaldeformation,andintroducedanotionof\shapeaverage"astheentitythatseparatesagroupactionfromadeformation.In[ 47 ]Paragios,RoussonandRameshproposedavariationalframeworkforglobalaswellaslocalshaperegistration.Theiroptimizationcriterionincludesaglobal(rigid,ane)transformationtogetherwithlocalpixel-wisedeformation.AsimilarideawasalsousedbyParagiosetal.[ 43 ]todeneshapepriormodelsintermsoflevelsetrepresentations.Theseideascanbeusedtosimultaneouslyapproximate,register,andtracknonequivalentshapesastheymoveanddeformthroughtime.However,thequestionofhowtodeterminethelocaldeformationhasnotbeenconsideredinthesepreviousworks. 18

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49 ]introducedaframeworkforuser-interactionwithinthepropagationofcurvesusinglevelsetrepresentations.Theuser-interactiontermisintroducedintheformofanevolvingshapepriorthattransformstheuser-editstolevelsetbasedpropagationconstraints.Theworkin[ 49 ]andinmystudyarebasedontheideathattheuserinteractiveeditscanbeusedasaconstrainttocorrectlocaldiscrepancies.However,theformulationsoftheconstraintinhisworkandmyworkaredierent(SeeChapter 2 fordetails). Inchapter 2 werstbrieyreviewgeometricactivecontoursandthemodeldevelopedin[ 42 ].Wethenproposethenewmodelforincorporatingbothpriorshapeandpointsinactivecontours. 50 { 53 ]).OnespecicexampleisthatGuptaetal[ 54 ]useDWItosearchandquantifytheextentofabnormalitybeyondtheobviouslesionsseenontheT2anduid-attenuationinversionrecovery(FLAIR)magneticresonanceimagesinpatientswithchronictraumaticbraininjurywithand 19

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55 ]usesDWItodetectacutemultiplebraininfarcts.TheyusethefactthatDWIissuperiortoconventionalMRIinidenticationofsmallnewischemiclesionsanddiscriminationofrecentinfarctsfromoldones. Thediusionofwatermoleculesintissuesoveratimeintervaltcanbedescribedbyaprobabilitydensityfunctionpt(r),whichgivestheprobabilitythatawatermoleculewilldiusionbyr.Sincept(r)islargestinthedirectionsofleasthindrancetodiusionandsmallerinotherdirections,theinformationaboutpt(r)revealsberorientationsandleadstomeaningfulinferencesaboutthemicrostructureoftissues. Thedensityfunctionpt(r)isrelatedtoDWIechosignals(q)viaaFouriertransformation(FT)withrespecttoq,whichrepresentsdiusionsensitizinggradient,by wheres0isMRIsignalintheabsenceofanygradient.Therefore,pt(r)canbeestimatedfromtheinverseFTofs(q)=s0.Recently,Tuchetal.[ 56 ]introducedthemethodofhigherangularresolutiondiusion(HARD)MRI,andWedeenetal.[ 57 ]succeedinacquiring500measurementsofs(q)ineachscantoperformafastFTinversion.However,thismethodrequiresalargenumberofmeasurementsofs(q)overawiderangeofqinordertoperformastableinverseFT. Amorecommonapproachtoestimatept(r)frommuchsparsersetofmeasurementss(q)isassumingpt(r)tobeaGaussian.ForGaussiandiusion,p(r;t)=1 1{6 )ityields 20

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58 ].Inthiscased(u)=uTDu: 2s wherei(i=1;2;3)aretheeigenvaluesofD,hasbecomethemostwidelyusedmeasureofdiusionanisotropy[ 53 ].Thisisknownasdiusiontensorimaging(DTI),andinparticularusefulforcreatingwhitematterbertracts[ 59 { 63 ]. However,ithasbeenrecognizedthatthesingleGaussianmodelisinappropriateforassessingmultiplebertractorientations,whencomplextissuestructureisfoundwithinavoxel[ 57 60 64 { 68 ].Asimpleextensiontonon-GaussiandiusionistoassumethatthemultiplecompartmentswithinavoxelareinslowexchangeandthediusionwithineachcompartmentisaGaussian[ 65 66 69 { 71 ].UndertheseassumptionthediusioncanbemodelledbyamixtureofnGaussians: wherefiisthevolumefractionofthevoxelwiththediusiontensorDi,fi0,Pifi=1,andtisthediusiontime.Inserting( 1{9 )intoequation( 1{6 )yields ToestimateDiandfi,atleast7n1measurementss(q)pluss0arerequired.In[ 69 { 71 ]themodelofamixtureoftwoGaussianswereusedtoestimatethePDF.Thisestimationrequiresatleast13diusionweightedimagesfrom13dierentdirections. 21

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where(;)(0<,0<2)representsthedirectionofq.ForGaussiandiusiond(u)=buTDu,whereuisthenormalizedq.Thetrace,eigenvaluesandfunctionsofeigenvaluesofDcanbeusedtocharacterizetheanisotropyanddirectionalpropertiesofthediusion.Fornon-GaussiandiusionthesphericalharmonicapproximationoftheADCprolesestimatedfromHARDdatahasbeenusedforcharacterizationofdiusionanisotropy.ThistechniquewasrstintroducedbyFrank[ 67 ],alsostudiedbyAlexanderetal.[ 72 ].Intheworkof[ 67 72 ]d(x;;)wascomputedfromHARDrawdataviathelinearformof( 1{11 ): andrepresentedbyatruncatedsphericalharmonicseries(SHS): whereYl;m(;):S2!CarethesphericalharmonicsseriesandCdenotesthesetofcomplexnumbers.Theodd-ordertermcoecientsintheSHSaresettobezero,sincetheHARDmeasurementsaremadebyaseriesof3-drotation,andd(;)isrealandhasantipodalsymmetry.Then,thecoecientsAl;m(x)'swereusedtocharacterizethediusionanisotropy.Intheiralgorithm,basically,thevoxelswiththesignicant4thorder(l=4)componentsinSHSarecharacterizedasanisotropicwithtwo-berorientations(shortenastwo-bers),whilevoxelswiththesignicant2ndorder(l=2)butnotthe4thordercomponentsareclassiedasanisotropicwithsingleberorientation(shortenasone-ber),whichisequivalenttotheDTImodel.Voxelswiththesignicant0thorder(l=0)butnotthe2ndand4thordercomponentsareclassiedasisotropic.The 22

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SincetheADCprolescanbeusedtocharacterizethediusionanisotropy,andtoestimatept(r)throughthecombinationof( 1{6 )and( 1{11 ),itisofgreatsignicancetodevelopmodelsforbetterestimationoftheADCprolesfromDWMRmeasurements.IngeneraltherawHARDMRIdataarenoisy.Computingthecoecientsdirectlyfromtherawdataoftenprovidespoorestimates.Asaresult,itwillleadtoinaccurateorfalsecharacterizationofthediusionandconsequentlyleadtoincorrectbertracking.ThegoalofChapter 4 andChapter 5 aretopresenttwonovelvariationalframeworksforsimultaneoussmoothingandestimationofnon-GaussianADCprolesfromHARDMRI. 73 ]andprovidesaverysensitiveprobefordetectingbiologicaltissuesarchitecture.Thekeyconceptthatisofprimaryimportancefordiusionimagingisthatdiusioninbiologicaltissuesreectstheirstructureandtheirarchitectureatamicroscopicscale.Forinstance,Brownianmotionishighlyinuencedinthepresenceoftissues,suchascerebralwhitematterortheannulusbrosusofinter-vertebraldiscs.Measuring,ateachvoxel,thatverysamemotionalonganumberofsamplingdirections(atleastsix,uptoseveralhundreds)providesanexquisiteinsightintothelocalorientationofbers. Therearecurrentlyseveraldierentapproachesforreconstructionofwhitemattertraces,theycanberoughlydividedintofourcategories:(1)linepropagationalgorithms;(2)surfacepropagationalgorithms;(3)globalenergyminimizationtondtheenergetically 23

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ThemainlocalinformationusedinmostclassicalalgorithmsforrecoveringbrainconnectivitymappingfromDTIdataistheprincipaleigenvector(PE)ofthediusiontensor[ 63 74 75 ].PEsuccessfullydeterminestheberdirectionincaseswherethereisasingleberdirectionineachvoxel,andisthereforeadequateforreconstructinglargetracesystems.However,withvoxelsizestypicalofdiusionacquisitions(1030mm3),thereissignicantpartialvolumeaveragingofberdirectioninanatomicalregionsofbothresearchandclinicalinterest,suchastheassociationbersnearthecortex.Moreover,imagenoisewillinuencethedirectionofthemajoreigenvector.Andasthedegreeofanisotropydecreases,theuncertaintyinthemajoreigenvectorincreases,inwhichsituationthetrackingmaybeerroneous.Whenadiusiontensorisplanarshaped,PEevendoesnotmakesense.Westinetal[ 76 ]andLazaretal[ 77 ]usedtheentiretensortodeecttheestimatedbertrajectory.Thisalgorithmiscalledtensordeection(TEND).ThedeectiontermisbetterthanPEinthesensethatthepreviousoneislesssensitivetoimagenoiseandislesserroneousinsituationofdegeneratedanisotropy.Butitstillhastheproblemofpartialvolumeaveragingofberdirection. Inchapter 2 wewillprovideamodeltosegmentimagesthatarecorruptedbynoise,dropoutandpoorcontrastalongboundaries.Wewillemploypriorshapeandpriorpointsinformationintoouralgorithmbyviewingtheevolutionofanactivecontourasadeformationoftheinterface.Thisdeformationconsistsofarigidtransformationandalocaldeformation.Wewillusethe\averageshape"todeterminetherigid 24

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Wemodifytheenergyfunctionofgeodesicactivecontoursothatitdependsontheimagegradientandpriorshape,aswellasafewpriorpoints.Weonlyneedafewpointssincewehaveinformationonexpectedshape.Themodiedenergyfunctionprovidesasatisfactorysegmentationdespitethepresenceofbothlargeshapedistortionsandimagedropout.Tocombinebothpriorshape(aglobalconstraint)andpriorpoints(alocalconstraint)intoasinglevariationalframework,weusealevelsetformulation.Further,wereportexperimentalresultsonsyntheticandultrasoundimages. Inchapter 3 ,amodelisdevelopedtosegmentimagesthathavecomplexmulti-modalintensitydistributionand/orhavehighlevelnoise.Inthissituation,itisnotwisetosetupmodelsbasedonaspecicparametricassumptionaboutimageintensities.Instead,weusenonparametricdensityestimationtocreateamodelthatisabletosimultaneouslysegmentandsmoothimageswithoutaddingextrasmoothingterms.Ateachvoxel,intensitiesofvoxelsinitsneighborhoodareusedtoobtainanonparametricestimateofitsintensitydistribution.Neighborhoodsizesarechosenadaptivelybasedonimagegradients.Wethencastthesegmentationproblemastheminimizationofthenegativeloglikelihood,subjecttoaconstraintonthetotallengthoftheregionboundaries.Imagesmoothingisautomaticallyfullledduetothenonparametricdensityestimation.RatherthanexpensiveGaussiankernels,aquadratickernelisappliedintheestimatetosavecalculation.Theoptimizationproblemissolvedbyderivingtheassociatedgradientowsandapplyingcurve 25

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First,inChapter 4 wepresentanewvariationalframeworkforsimultaneoussmoothingandestimationofapparentdiusioncoecient(ADC)prolesfromHARDMRI.ThemodelapproximatestheADCprolesateachvoxelbya4thordersphericalharmonicseries(SHS).ThecoecientsinSHSareobtainedbysolvingaconstrainedminimizationproblem.Thesmoothingwithfeaturepreservedisachievedbyminimizingavariableexponent,lineargrowthfunctional,andthedataconstraintisdeterminedbytheoriginalStejskal-Tannerequation.TheantipodalsymmetryandpositivenessoftheADCareaccommodatedinthemodel.WeusethesecoecientsandvarianceoftheADCprolesfromitsmeantoclassifythediusionineachvoxelasisotropic,anisotropicwithsingleberorientation,ortwoberorientations.TheproposedmodelhasbeenappliedtobothsimulateddataandHARDMRIhumanbraindata.TheexperimentsdemonstratedtheeectivenessofourmethodinestimationandsmoothingofADCprolesandinenhancementofdiusionanisotropy.Furthercharacterizationofnon-Gaussiandiusionbasedontheproposedmodelshowedaconsistencybetweenourresultsandknownneuroanatomy. Secondly,inChapter 5 wepresentanotherapproximationfortheADCofnon-Gaussianwaterdiusionwithatmosttwoberorientationswithinavoxel.TheproposedmodelapproximatesADCprolesbyproductoftwosphericalharmonicseries(SHS)uptoorder2fromHighAngularResolutionDiusion-weighted(HARD)MRIdata.ThecoecientsofSHSareestimatedandregularizedsimultaneouslybysolvingaconstrainedminimizationproblem.Anequivalentbutnon-constrainedversionof 26

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Inchapter 6 ,wepresentanewvariationalmethodforrecoveringtheintra-voxelstructureundertheassumptionthatpt(r)isamixtureoftwoGaussians:pt(r)=P2i=1fi((4t)3det(Di))1=2erTD1ir 1{10 )withn=2ineachisolatedvoxel,whichleadstoanill-posedproblem.Second,werecovertheADCproled(x;;)inSHrepresentationusingmethodintroducedinchapter 4 fromthenoisyHARDdatabeforeestimatingDi(x)andfi(x).Therecovereddandthevoxelclassicationondiusionanisotropyfromdareincorporatedintoourenergyfunctiontoenhancetheaccuracyoftheestimates.Third,weapplythebiGaussianmodeltoallthevoxelsintheeld,ratherthanthevoxelswheretheGaussianmodelonlytspoorly.Sinceboththeconstraintoff11ontheregionofstrongone-berdiusion,andtheregularizationforfiandDiarebuiltinthemodel,thesingleberandmulti-berdiusionscanbeseparated 27

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Inchapter 7 ,wewillprovideanewlinepropagationalgorithmbasedonsmoothmulti-tensoreld. Weassumethereareuptotwodiusionchannelsateachvoxel.Avariationalframeworkfor3Dsimultaneoussmoothingandrecoveringofmulti-diusiontensoreldaswellasanovelmulti-tensordeection(MTEND)algorithmforextractingwhitematterbertracesbasedonmulti-tensoreldareprovided.MTENDkeepsalltheadvantagesofTENDandhastwoadditionalgoodproperties:rst,problemofpartialvolumeaveragingisautomaticallysolvedasitisbasedonamulti-tensoreld;second,itusesadynamicallyadjustedstepsizetokeeptotalcurvatureoftraceslow,toappropriatelyterminatetrackingandtoincreasealgorithmeciency. FibertracesarecoloredusingLaplacianeigenmaps.Byapplyingtheproposedmodeltosyntheticdataandhumanbrainhighangularresolutiondiusionmagneticresonanceimages(MRI)dataofseveralsubjects,weshowtheeectivenessofthemodelinrecoveringintra-voxelmulti-berdiusionandinter-voxelbertraces.Superiorityoftheproposedmodeloverexistingmodelsarealsodemonstrated. Inchapter 8 ,a3Dgeometricowisdesignedtosegmentthemaincoreofbertractsindiusiontensormagneticresonanceimages.Theproposedmodeldesignedanewexternalforcethatdependedontwomagnitudes:similarityofdiusiontensorsandthecoincidenceleveloftheevolvingsurfacenormalwiththetensoreld.Thenewmodel,basedonthisnewexternalforce,wasabletosegmentwhitematterberbundlesinDT-MRImoreaccuratelyandeciently. 28

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Copyright[2003]lecturenotesoncomputerscience[ 78 ].Reprintedwithpermission. Tobeginthedescriptionoftheproposedmodel,werstbrieyreviewthegeodesicactivecontourmodelin[ 4 5 ],andtheactivecontourwithashapepriorin[ 42 ]. LetC(p)=(x(p);y(p))(p2[0;1])beadierentiableparameterizedcurveinanimageI.Thegeometricactivecontourmodelminimizestheenergyfunction: where 1+jrGIj2;(2{2) withaparameter>0,andG(x)=1 42.Theminimumofthisenergyfunctionaloccurswhenthetraceofthecurveisoverpointsofhighgradientintheimage.Becauseobjectboundariesareoftendenedbysuchpoints,theactivecontourbecomesstationaryattheboundary.Initslevelsetformulationthismodelcanhandletopologicalchange.However,sincethisalgorithmrequiresahighimagegradienttobepresentalongthe 29

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Toovercomethisproblemavariationalmodelwasproposedin[ 42 ]thatincorporatespriorshapeinformationingeodesicactivecontours.Thefeatureofthismodelisthecreationofashapetermintheenergyfunctional( 2{1 ).IfC(p)(p2[0;1])isacurverepresentingtheexpectedshapeoftheboundaryofinterest,thentheenergyfunctionaltobeminimizedin[ 42 ]is where(;R;T)aresimilaritytransformationparameters,anddC(x;y)isthedistanceofthepoint(x;y)toC: 2{3 )isthesameastheenergyfunctionalforgeodesicactivecontours,whichmeasurestheamountofhighgradientunderthetraceofthecurve.Thesecondtermistheshaperelatedenergy,thatmeasuresthedisparityinshapebetweentheinterfaceandtheprior.Theconstant>0isaparameter,whichbalancestheinuencefromtheimagegradientandshape.ThecurveCandthetransformationparameters,RandTevolvetominimizeE(C;;R;T).Atthestationarypoint,thecontourCliesoverpointsofhighgradientintheimageandformsashapeclosetoC,and,RandTdeterminethe\best"alignmentofCtoRC(p)+T. Theexperimentalresultsin[ 42 ]showedtheirmodelisabletogetasatisfactorysegmentationinthepresenceofgaps,evenwhenthegapsareasubstantialfractionoftheoverallboundary,iftheshapeofinterestissimilartotheexpectedshape.However,ifsomepartsoftheboundaryarenotvisible,andtheshapeofboundaryoftheobjecthasrelativelylargergeometricdistortionfromtheprior,asshownlater,model( 2{3 )cannotprovideadesiredsegmentation,sincetheknowledgeoftheexpectedshapedoesnotprovidecorrectinformationabouthowthegapsshouldbebridged. 30

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42 ].Similartotheapproachdevelopedin[ 42 ],togetasmoothcurveCthatcaptureshighergradientsweminimizethearc-lengthofCintheconformalmetricds=g(jrIj)(C(p))jC0(p)jdp,whereg(jrIj)isdenedin( 2{2 ).TocapturetheshapepriorC,wendacurveCandthetransformation(;R;T),suchthatthecurveRC+TandCare\best"aligned.Tocapturethepriorpointsweminimizethedistancesofeachpriorpointxi(i=1;:::;m)fromthecurveC. Wepresentourmodelinavariationallevelsetformulation.First,aswellknown,thelevelsetmethodinitiatedin[ 18 ]allowsforcusps,corners,andautomatictopologicalchanges,Secondly,itismoreconvenienttocomputethedistancesofthepriorpointstotheinterfacebyusingthelevelsetformoftheinterface. LetthecontourCbethezerolevelsetofaLipschitzfunctionusuchthatC=fx2RN:u(x)=0g,with(insideofC)=fx2RN:u(x)>0g,and(outsideC)=fx2RN:u(x)<0g.LetHbetheHeavisidefunctionH(z)=1,ifz0,otherwiseH(z)=0,andbetheDiracmeasureconcentratedat0(i.e.(z)=H0(z)inthesenseofdistribution).Then,thelengthofthezerolevelsetofuintheconformalmetricds=g(jrIj)jC0(p)jdpcanbecomputedbyRg(jrIj)jrH(u)j=R(u)g(jrIj)jDuj.ThedisparityinshapebetweenthezerolevelsetofuandCcanbeevaluatedbyR(u)d2C(Rx+T)dx,wherethedistancefunctiondCisthesameasthatin( 2{3 ).Moreover,theconstraintforcontourCpassingthroughthepriorpointsx1;:::;xmcanbesimplyrepresentedbyu(xi)=0,(i=1;:::;m). Nowletf(x)beasmoothfunctiondenedontheimagedomain,suchthat0f(x)1,f(x)=1forx=xi(i=1;:::;m).Thefunctionf(x)canbeobtained 31

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minu;;R;TE(u;;R;T)=minu;;R;TfZ(u)g(jrIj)jruj 2Zf(x)u2(x)gdx;(2{4) where>0and>0areparameters.Thersttwotermsinthisenergyfunctionalarethesameasthosein( 2{3 ),whichtendtoleadtheinterfacearrivingatalocationwherethemagnitudeofimagegradientishigh,andtoformashapesimilartotheprior.Thelasttermtendstoleadtheinterfacetopassthroughthegivenpoints,sinceminimizingthethirdtermin( 2{4 )withsucientlysmallleadsutobeclosetozeroatthegivenpoints.Notethatf(x)isnon-zeroonlyontheneighborhoodofthegivenpoints,sothethirdtermdoesn'taectmuchtheshapeofthecontouroutsidetheneighborhoodofthegivenpoints. Thismodelperformsajointsegmentationandregistration.Thesegmentationisassistedbytheregistrationbetweentheinterfaceandshapeprior.Thisregistrationisnon-rigidthatconsistsofaglobaltransformation(rigid)andalocaldeformation.Theglobaltransformationisdeterminedbyminimizingthesecondtermin( 2{4 ),whilethelocaldeformation,iscontrolledbyminimizingtherstandlasttermsin( 2{4 ). TheevolutionequationsassociatedwiththeEuler-Lagrangeequationsfor( 2{4 )are @t=(u)div((g+ @n=0;x2@;t>0;u(x;0)=u0(x);x2;(2{6) 32

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@t=Z(u)drd(Rx)jrujdx;t>0;(0)=0;(2{7) @t=aZ(u)drd(dR dx)jruj;t>0;(0)=0;(2{8) @t=Z(u)drdjrujdx;t>0;T(0)=T0;(2{9) whered=dC,Ristherotationmatrixintermsoftheangle,andthefunctiondisevaluatedatRx+T. 2{4 )byndingthesteadystatesolution(s)totheevolutionproblem( 2{5 )-( 2{9 ).Tosolvetheequations( 2{5 )-( 2{9 )numerically,asin[ 16 ],wereplacein( 2{4 )-( 2{9 )byaslightlyregularizedversionsofthem,denotedby:H"(z)=8>>>><>>>>:1ifz>0ifz<1 2[1+z +1 )]ifjzj (z)=H0(z)=8><>:0ifjzj>1 2[1+cos(z )]ifjzj 42 ].Lethbethestepsize,and(xi;yi)=(ih;jh)bethegridpoints,for1i;jM.Letuni;j=u(tn;xi;yj)beanapproximationofu(t;x;y).Thetimederivativeutat(i;j;tn)isapproximatedbytheforwarddierencescheme:ut(i;j;tn)=un+1i;juni;j Weadoptthealgorithmforthediscretizationofthedivergenceoperatorfrom[ 79 ],andtheimplicititerationfrom[ 80 ].Knowingun,wecomputeun+1byusingthefollowingdiscretizationandlinearizationschemeof( 2{5 ): 33

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Tokeepthesigneddistancefunctionnearthefront(zerolevelsetoftheevolvingu)thetechniqueofreinitializationdevelopedandappliedin[ 37 80 81 ]isalsousedinourcomputation.Thisprocedureismadebyusinganewfunctionv(x),whichisthesteadystatesolutiontotheequation@v @s=sign(u(;t))(1jrvj);v(;0)=u(;t); Theequation( 2{4 )-( 2{9 )arediscretizedasin[ 42 ]byusingnitedierence. Theaimofourrstexperimentistoverifythattheactivecontourwiththepriorshapeandpointscanllinthe\gaps"inaboundaryinameaningfulway. 34

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2-1 ashowsatypicalbinaryimageIwiththreepointsandanellipsesuperimposed.Theellipseandpointsareusedasthepriorshapeandpointsinthisexperiment,respectively.Theobjecttobesegmentedispartiallyoccluded,andtheshapeofitsboundaryisnotequivalenttothepriorshape.Wewanttodeterminewhetherornottheactivecontourwiththepriorshapeandpointscanusethepartialboundarytoaidtheprocessofllingintherest. Theactivecontourwasinitializedbythesolidcurvedisplayedingure 2-1 c.Evolvingtheactivecontouraccordingto( 2{5 )-( 2{9 )withtheparameters=250,=15,=0:5(ing(x)),0=1,0=0,T0=(0;0),wegetthestationarycontourC(thedottedone)ingure1c,andthetransformationparameters=0:91,=0:14,andT=(0:5;0:3)(pixels).WecanseethateventhoughcompletegradientinformationisnotavailablethecontourCcapturesthehighgradientintheimageI,passesthroughthreepriorpoints,andformsashapesimilarbutnotthesameasthepriorshape.Toshowtheadvantageofusingpriorpointswecomparedthesegmentationresultsobtainedbyusingmodel( 2{4 )and( 2{3 ).Figure 2-1 B)showsthesegmentationresultbyusingmodel( 2{3 ).Ingure 2-1 B)thesolidcontouristheinitialcontour,andthedottedoneisthesegmentationresult.Sincethepriorpointsarenotincorporatedinthemodel( 2{3 ),thesegmentedcontouronlycapturesthepriorshapeandhighgradients.Itcan'taccuratelycapturelocalshapevariations. Theaimofthesecondexperimentistosegmenttheendocardium(theinnerboundaryofthemyocardiumsurroundingtheleftventricle)inanapicaltwo-chamberimageoftheheart(seeFigure 2-2 A)foratypicalimage).Theendocardiumisnotcompletelyvisibleintheimage,anditsshapeisnotthesameasthe\averageshape"(theshapeprior).Ourtaskistodeterminetheendocardiumusing\averageshape"andvepointsgivenbyanexpert. Thepriorshapeiscreatedbythesamewayasthatin[ 42 ].Itisobtainedbyaveragingthealignedcontoursinatrainingset.ThealignmentoftwocontoursC1andC2

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(b) (c) Comparingsegmentationresultsofasyntheticimage.A)Animagewiththepriorshapeandthreepointssuperimposed.B)Thesegmentationresultusingmodel( 2{3 )(dotted)andtheinitialcontour(solid).C)Thesegmentationresultusingmodel( 2{4 )(dotted)andtheinitialcontour(solid). ismadebyndingthe\best"scalingconstant,rotationmatrixRandtranslationvectorTsuchthattheoverlappingareaoftheinteriorsofC1andRC2+Tismaximized.Iftheshapesofthecurvesinthetrainingsethavealargevariation,aclusteringtechniqueisrequiredtogroupthesecurvesintoseveralgroups.Theshapepriorsforeachgroupareobtainedusingthistechnique. Forthisparticularproblemtocreatethepriorshape,anexpertechocardiographertracedendocardialboundarieson112imagesequencesfor66patients.Aftertheboundarieswereclustered,theaveragewascomputed.Figure 2-2 bshowsthe\averagecontour"foroneoftheclusters(thedottedcontour),theendocardiumoutlinedbyanexpert(thesolidcontour),andvepointsontheexpert'scontour,Thepriorpointsareusuallygivenatthelocationwheretheimagegradientsarelow,andthelocalshapedistortionsarelarger.Figure 2-2 C)presentstheimagejrGIj.Fromthisimagewecanseethedropoutofimageinformationatseveralpartsoftheendocardium. TosegmenttheendocardiumintheimageshowninFigure 2-2 D)(itisthesameasinFigure 2-2 A),thetheactivecontourwasinitializedasthecontourshowninFigure 2-2 A).Thiscontourwasevolvedaccordingtotheequations( 2{5 )-( 2{9 )anditnallystoppedatthelocationofthedottedcontourinFigure 2-2 D).Wealsoobtainedthetransformationparameters=1:0024,=0:1710,andT=(24:6163;32:3165) 36

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2-2 D)istheexpert'sendocardium.Observethatthesegmentationisclosetotheexpert'scontour.Toseetheshapevariationbetweenthesolutionof( 2{4 )and\averageshape"wealignedthesolutionof( 2{4 )tothe\averageshape"usingthesolutions(;R;T)of( 2{4 ).Figure 2-2 E)showsthedisparityinshapebetweenthesetwocontours.Thedottedcontouristhetransformedsolutionof( 2{4 ),andthesolidoneisthe\averageshape".FromFigures 2-2 dand 2-2 E)wecanseethatouractivecontourformedashapedierentfromtheprioroneinordertocapturethehighgradientsandgivenpoints.Figure 2-2 F)providesthesegmentationresultobtainedbyusingmodel( 2{3 ).Inthisgurethedottedcontouristhesolutionof( 2{3 ),andthesolidcontouristheexpert'sendocardium.ComparingFigure 2-2 D)withFigure 2-2 F),notethatthesolutionof( 2{4 )isclosertotheexpert'scontourthanthesolutionof( 2{3 ).Figure 2-2 G)presentstheshapecomparisonbetweenthesolutionof( 2{3 )andpriorshape.InFigure 2-2 G)thesolidcontouristhe\averageshape",andthedottedoneisthetransformedsolutionof( 2{3 ),(thetransformationparametersarethesolutionof( 2{3 )).FromFigures 2-2 F)and 2-2 gweseethatthesolutionof( 2{3 )canonlycapturethehighimagegradientsandthe\averageshape",butitcan'tprovideasdesirableasegmentationresultastheexpert'sendocardium. Thelastexperimentisarepetitionofthesecondexperimentonasecondapical2-chambercardiacultrasoundimage.Welisttheguresbelowfortheresultsofthisexperimentinthesameorderasabove.ThesegmentationCisgiveningure 2-3 drepresentedbythedottedcontour,thetransformationparametersare=0:9883,=0:1981,andT=(28:7246;49:1484): 37

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(b) (c) (d) (e) (f) (g) Comparesegmentationresultsofonecardiacultrasoundimage.A)Atypical2-chamberultrasoundimagewithaninitialcontour;B).Expert'sendocardium(solidcontour),\averageshape"(dottedcontour),andvepointsontheexpert'scontour.C).TheimagejrGIj.D).Theendocardiumsegmentedbyusingmodel( 2{4 )(dotted)andtheexpert'scontour(solid).E)Thetransformedsolutionof( 2{4 )(dotted),andthe\averageshape".F).Theendocardiumsegmentedbyusingmodel( 2{3 )(dotted)andtheexpert'scontour(solid).G)Thetransformedsolutionof( 2{3 )(dotted),andthe\averageshape". 38

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(b) (c) (d) (e) (f) (g) Comparesegmentationresultsofanothercardiacultrasoundimage.A)atypical2-chamberultrasoundimagewithaninitialcontour.B)Expert'sendocardium(solidcontour),\averageshape"(dottedcontour),andvepointsontheexpert'scontour.C)TheimagejrGIj.D)Theendocardiumsegmentedbyusingmodel( 2{4 )(dotted)andtheexpert'scontour(solid).E)Thetransformedsolutionof( 2{4 )(dotted),andthe\averageshape".F)Theendocardiumsegmentedbyusingmodel( 2{3 )(dotted)andtheexpert'scontour(solid).G)Thetransformedsolutionof( 2{3 )(dotted),andthe\averageshape". 39

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40

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1{1 )isprovidedhere. Forarecapofnonparametricdensityestimation,letXf(x)bearandomvariable,fX1;X2;:::;XngbeasetofrealizationofX.Akernelmethodestimationoff(x)wouldbebf(x)=1 82 83 ]formoredetailedknowledgeaboutnonparametricmethods. Fornotationsimplicity,wedenotebyIitherestrictionoffunctionItoi.Toensureapplicabilityofthenonparametricdensityestimationidea,wemaketwoassumptions.Firstly,ateachvoxelx,wetreatIi(x)asarandomvariable.Thisisreasonableinthesensethatevenforthesamesubject,dierentscanswouldprovidedierentdataI0duetomotionofthesubject,noiseinvolvedinthedatacollectionprocessetc.RandomnessofI0leadstothatofIi(x)'s.Secondly,weassumethatIi(x)hasthesamedistributionasthatoftheI0(y)'s,forallythatareinasmallneighborhoodofxandareinsideregioni.Notethisismoregeneralthaninmodel( 1{4 ),whereIi(x)isassumedtohavethesamedistributionasthatoftheIi(y)'s,forallthey'sinsideregioni.Undertheabovetwoassumptions,weapproximatethedistributionofintensityIi(x)usingintensitiesofI0(y),forystaysintheintersectionofiandaneighborhoodofx,denotedbyB(x;r(x)),whichisaballcenteredatxwithradiusr(x).Theestimateddistributioniscalledbfi(x): 41

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Undertheindependenceassumption,probabilitydensityfunctionofIiinthedomainiwouldbeQx2bfi(x).JointprobabilitydensityfunctionofIinthewholedomainwouldbeQiQx2bfi(x).Westillcalllogarithmofthisfunctionasloglikelihoodeventhoughnoparametersareinvolved.TheproposedmodelistondC,andIi(x)'sforeachpointx2i,tominimizethefollowingfunctional: 22dydx+jCj Thersttermisnegativeloglikelihood,inwhichKhischosenasaGaussiankernelfunctionandbandwidthhissettobe1.Foreachpointxineachsubregioni,therstpartmaximizestheloglikelihoodforally'sthatlieintheintersectionofiandaneighborhoodB(x;r(x))ofx.ThisisequivalenttosomeextenttotryingtoforceIi(x)I0(y)tobe0,i.e.,intensityIi(x)tobeclosetoI0(y)'sforally'slyingintheintersectionofiandneighborhoodB(x;r(x))ofx.Roughlyspeaking,Ii(x)willbeclosetomeanvalueofI0inasmallneighborhoodthatisinsidei,whichwillsmoothimagesIi(x)withoutcrossingboundaries.Detailedproofofasimpliedversionwillbeprovidedinsection 3.4 .ThustherstpartbasicallysmoothseachIiinsideiwithoutcrossingtheboundary.Therefore,thersttermnotonlypreventsIi'sfrombeingtoofarawayfromI0,butalsosmoothsIi's.ThisimpliesitworkssimilarlytothersttwopartsinpiecewisesmoothMS( 1{1 ).Wewouldliketomentionthatconvolutionalsosmoothsimagethroughtakingaverage,butconvolutionwillblurimagesatboundariesandalsocauseshiftingofboundaries.ThereasonisthatIi(x)issettobeaverageofI0(y)'sforally'slyinginaneighborhoodofx,thisneighborhoodisnotrestrictedinsidei.Resultsofmodel( 3{3 )willnotcausetheseproblems. 42

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1{1 )toobtainasmoothboundary.Theproposedmodelisabletosimultaneouslysegmentandsmoothimageswithoutusingextrasmoothnessterms,andthisalsosavesannoyingworkonchoosingappropriateparametertobalancedatadelitytermandsmoothnessterm. Gaussiankernelcouldbereplacedbyanybell-shapedkernelsthataremaximizedat0.Bybell-shaped,wemeanthegraphofthekernelfunctionisincreasingonthelefthandsideof0andthendecreasingontherighthandsideof0.ComputationcomplexitywouldbedecreasedalotiftheGaussiankernelinvolvingexpensiveexponentialcomputationisreplacedbykernelsinvolvingsimplecalculation.Quadratickernelisagoodexample.OnexampleforquadratickernelisP(u)=3 4(1u2)[1;1](u),where[1;1](u)isthecharacteristicfunctionof[1;1].Analternativemodelwouldbetominimizethefollowingfunctionalwithrespectto0I1;I21andC: 4(1(Ii(x)I0(y))2)dydx+jCj ToforceIi(x)I0(y)tobeininterval[1;1],whichisthesupportofthequadratickernelmentionedabove,alineartransformation,whichdoesnotchangesegmentationresults,isrequiredtore-scalerangeoftheinitialimageI0to[a;b]forany1>b>a>0(a=:1;b=:9wouldbeachoice).Thisconditionisalsorequiredtoguaranteeexistenceofsolutions.Wecanshowthatitisnotenoughtore-scalerangeofI0to[0;1],ithastobeapropersubsetof[0;1].RangesofI1andI2arerestrictedto[0;1].Insection 3.4 ,wewillshowthat( 3{4 )isactuallyanon-constrainedproblem,i.e.,theconstraintswillbesatisedautomaticallyforsolutions. Obviously,thersttermisminimizedwhenforeachpointxineachsubregioni,Ii(x)isequaltoI0(y)'sforally'slyinginneighborhoodB(x;r(x))ofxandi.Henceitwouldworksimilarlyasmodel( 3{3 )butrequiremuchlesscomputation.Actually,( 3{4 )isequivalentto( 3{3 )whenallthe(x)'saresettobeaconstant.Notethisdoesnot 43

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3{4 )isreturnedtothepiecewiseconstantMS( 1{2 )eventhoughbothofthemhavenovarianceinvolved.( 3{4 )forcesIi(x)tobeclosetomeanofI0(y)'sforally'sintheintersectionofB(x;r(x))andi.ThisismuchmoregeneralandapplicablethanpiecewiseconstantMSwhichenforcesIi(x)tobeclosetoaconstantwhichturnsouttobemeanvalueofI0(y)'sforally'sinthewholei.Thisgeneralityenables( 3{4 )tohandleimageswhichinvolvemulti-modaldistributions. Thelevelsetformulationofmodel( 3{4 )isasfollows.ByusingHeavisidefunction,alltheintegralsinthemodelareover,soimplementationismucheasier: 4(1(I1(x)I0(y))2)dydxZ(1H((x)))logZB(x;r(x))(y)(1H((y)))3 4(1(I2(x)I0(y))2)dydx+Z((x))jr(x)jdx whereB(x;r(x))(y)isthecharacteristicfunctionofB(x;r(x)).Ballsizer(x)isadaptivelydependentonimagegradientatx:whenimagegradientishigh,i.e.,atlocationsnearboundaries,radiiaresmaller,whileradiiarebiggeratmorehomogeneousregions.Asaresult,smoothingspeedofI1;I2arehigherathomogenousregionsandlowernearboundariestokeepnestructure.Therefore,thesmoothedimageswillbesharp.Incomparison,in[ 84 ],ballsizesarexedforallx,soitishardertochooseoneuniformballsizethatworksforallthelocations;smoothingspeedwillbethesameatalllocations.Ifatoosmallballsizeischosen,noiseinthehomogenousregioncannotberemovedsuciently,whileifatoobigballsizeisselected,nestructuresnearboundarieswilldisappear.Similarly,model( 1{1 )alsosimultaneouslysmoothsandsegmentsimages,butitassignsaxedcoecientforallx,sosmoothingspeedwillbesimilartowhatwasobservedin[ 84 ].Acomparisonbetweenmodelsusingvariableradiiandxedradiiwillbeshowninsection 3.2 44

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3{5 ).First,radiusr(x)ischoseninthefollowingway: whereeI0isasmoothedversionofthegiveninitialimageI0obtainedthroughconvolvingitwithasmoothkernel.AnotherstraightforwardchoiceistoreplaceI0byI,smootherversionofI0thatisupdatingduringiterations: whereeIisasmoothedversionoftheupdatingimageIobtainedthroughconvolvingitwithasmoothkernel,westillconvolveIeventhoughitissmootherthanI0.Thereasonisduringtheiteration,especiallyintheearlystage,itisnotsmoothenough.InFigure 3-1 ,wecomparesegmentationandsmoothingresultsofmodel( 3{5 )withradiidenedasin( 3{6 )( 3{7 ).Figure 3-1 A)isacleanplaneimage,aspecklenoisewithparameter:05isaddedtoit,theresultingnoisyimageisshowninB).Thesecondandthethirdrowdemonstratessegmentationandsmoothingresultsbasedon( 3{6 )( 3{7 )respectively.Nosignicantdierenceinbothsegmentationandsmoothingresultsisvisualized.Therefore,wewilluse( 3{6 )forallthefollowingexperiments.Also,itiseasiertoproveexistenceofmodel( 3{5 )forr(x)denedasin( 3{6 ),section 3.4 providesdetails. 45

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Figure3-1. Compareresultsofmodel( 3{5 )basedontwochoicesofr(x)A)Acleanplaneimage(6090).B)anoisyplaneimageobtainedbyaddingspecklenoisewithparameter:05toA).C)-D):segmentationandsmoothingresultsofB)usingmodel( 3{5 )withradiusasdenedin( 3{6 )respectively.E)-F)segmentationandsmoothingresultsofB)usingmodel( 3{5 )withradiusasdenedin( 3{7 )respectively.Foralltheresults,timestepsize=.2,=:01,M=1,N=0,convergein20iterations. 46

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ThesecondissueistouseregularizedversionofHeavisidefunctionandDeltafunctiontomakethefunctionaldierentiablewithrespectto.Weusethosedenedin[ 16 ]. Thirdly,duetothedicultyofndingderivativeofthefunctionalwithrespecttodirectly,thefunctionalistotallydiscretized.Thediscretizedversionof( 3{5 )isthenimplementedusinganadditiveoperatorsplitting(AOS)algorithmwithboundaryconditions@ @n=0and@Ii 85 { 87 ]isasemi-implicitschemewhichisstableforlargestepsizes.Itisatleasttentimesmoreecientthanthewidelyusedexplicitschemes.Itguaranteesequaltreatmentofallcoordinateaxes,canbeimplementedeasilyinarbitrarydimensions,anditscomputationalcomplexityandmemoryrequirementarelinearinthenumberofpixels. Finally,initialI1;I2andcouldbechosenextremelyexible;evensimpleconstantinitialswouldworkformostoftheimageswehavetried.Thisremovestheworkofcreatingthedistancefunctionofaninitialcurveforinitialinregularlevelsetbasedmethod.Thealgorithmusuallyconvergesinfewerthan25iterations,makingreinitializationoflevelsetfunctionunnecessary.Parameteraectsexperimentalresultsandneedstobetunedforeachimage.Butitdoesnotvarymuchamongdierentimages. Table3-1: SegmentationAccuracy GaussianNoise 0.01 0.05 0.1 0.25 SA 0.996 0.979 0.957 0.868 SaltPepperNoise 0.05 0.15 0.25 0.50 SA 0.996 0.988 0.970 0.885 47

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Figure3-2. Segmentationresultsofacleanimageandthatofanoisyversion.A)-C)OriginalcartoonC-shapedimage,segmentationresultandthesmoothoutputresp.D)-F)AnimageobtainedbyaddingGaussiannoisewithvariance0.05toA),segmentationresultandsmoothedversion.G)-I)Animageobtainedbyadding0.03specklenoisetoA),segmentation,smoothedversion.Foralltheresults,timestepsize=.1,=:01,M=2,N=1,convergein20iterations. 48

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3{5 )tosimultaneouslysegmentandsmoothnoisyimages,andthesecondsetistocomparesegmentationabilityof( 3{5 )withthatofpiecewiseconstantMSandglobalGaussianpdfbasedmethod.Finally,segmentationandsmoothingcapabilityofmodel( 3{5 )thatusesadaptiveradiiiscomparedwithmodelin[ 84 ]thatusesxedradius.Nocomparisonbetween( 3{5 )andpiecewisesmoothMSisprovidedaswethinkmodelin[ 84 ]ismucheasiertoimplement,andinvolveslessparametersbutworkssomehowequivalentlyaspiecewisesmoothMSdoes. TherstsetofexperimentsisbasedonacleansyntheticC-shapedcartoonimagewithonlytwophases(Figure 3-2 A)).WeadddierentlevelsofrandomGaussiannoiseandsalt&peppernoiserespectivelytoFigure 3-2 A)tocreatedierentnoisyimages.Model( 3{5 )isthenappliedtoeachofthenoisyimageandthecleancartoonimagewithparameter=:2.Asegmentationaccuracy(SA)measureisdenedtoquantitativelyobservehowaccuratethemodelisinsegmentinganoisyimagecomparedtotheoriginalcartoonimage.Wetreatsegmentationresultofthecleancartoonimageasgroundtruth.ThendeneSAtobetheratioofnumberofpixelssharingthesamesegmentationwiththegroundtruthovertotalpixelnumber.SAisthenbetween0and1,thecloseritisto1,thebetterthemodelisinsegmentingthisnoisyimage. Figure3-3. SegmentationresultofanoisyT1humanbrainimage.A)AnoisyT1image.B)Segmentationresultbasedon( 3{5 ):black:backgroundandCSF;gray:graymatter;white:whitematter.C)RecoveredsmoothT1image. 49

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3-1 listsSA'scorrespondingtoimagesobtainedbyaddingrandomGaussiannoise,withmean0butdierentvariances,andsalt&peppernoisewithdierentparametersrespectively.EachoftheSAshownistheaveragevalueoftwoSA'scorrespondingtotwodierentimagesobtainedthroughaddingsametypeandstrengthofrandomnoiseattwodierenttimes.ItisobservedfromTable1thatwhenvarianceofGaussiannoiseislowerthan:1andparameterofsalt&peppernoiseislessthan0:25respectively,thesegmentationaccuracycouldbehigherthan95%.Andanobvioustrendisthatasnoiselevelincreases,SAdecreasesasexpected. Figure 3-2 showssegmentationandsmoothingresultsbasedonthecleanC-shapedcartoonimage(Figure 3-2 A))andtwonoisyversionsofit:(Figure 3-2 D))whichisobtainedbyaddingarandomGaussiannoisewithzeromeanandvariance0:05,(Figure 3-2 G))whichisobtainedbyaddingspecklenoisewithparameter0.03.Theirboundariesdepictedinredcurveswhicharefoundbyapplyingmodel( 3{5 )aresuperimposedonthemandshowninFigure 3-2 B)E)H)respectively.Itisobservedthattheyarealmostthesame,actually,theycoincideupto99:6%.Figure 3-2 C)F)I)demonstratethesmoothedversionofFigure 3-2 A)D)G)respectively.Therecoveredimagesaremuchsmootherthantheoriginalnoisyimages.Figure 3-3 demonstratessegmentationandsmoothingresultsonarealmedicalimage.Figure 3-3 A)isT1weightedMRIbrainimagethatinvolvesfourphases:Background,cerebralspinaluid(CSF),GrayMatterandWhiteMatter.Noisewithunknowntypeandlengthisinvolved,distributionofintensityisofunknowntypealso,model( 3{5 )isappliedtothisimage,segmentationandsmoothingresultsareshowninFigure 3-3 B)C)respectively.Thesegmentationresultisquitereasonable,anditisobviouslythatFigure 3-3 C)issignicantlysharperthanFigure 3-3 A).WeusehierarchicallevelsetmethodrecursivelyasGaoetal.mentionedin[ 24 ]. ThesecondsetofexperimentsisbasedonaplaneimageshowninFigure 3-4 A).WeaddrandomGaussiannoisewithmean0andvariance0:01,0:05toittocreatetwonoisyimageswhichareshowninFigure 3-4 B)andFigure 3-4 C)respectively.Weapplypiecewise 50

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Figure3-4. Comparethreemodelsinsegmentingimageswithlowerandhigherlevelofnoises.A)Aplaneimage.B)AnoisyplaneimageobtainedbyaddingarandomGaussiannoisewithmean0,variance0:01toA).C)AnoisyplaneimageobtainedbyaddingarandomGaussiannoisewithmean0,variance0:05toA).D)-F)Segmentationresults(redcurve)ofB)superimposedonB)basedonpiecewiseconstantMS,globalGaussianpdfand( 3{5 )respectively.G)-I)Segmentationresults(redcurve)ofC)superimposedonB)basedonpiecewiseconstantMS,globalGaussianpdfand( 3{5 )respectively.Foralltheresults,timestepsize=.1,=:01,M=1,N=0,convergein20iterations. 51

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3{5 )toFigure 3-4 B)respectivelyandobtainedresults(redcurves)superimposedonFigure 3-4 B)asshowninFigure 3-4 D)-F)respectively.Visually,notmuchdierenceamongthemisdetected.Thusallthethreemodelsareabletosegmentimageswithlowlevelnoise.ButafterapplyingthesethreemodelstothenoisierimageFigure 3-4 C),fromtheirresultssuperimposedonFigure 3-4 C)asshowninFigure 3-4 G)-I)respectively,hugedierencesareobserved.Moreover,itisobviousthatresultbasedon( 3{5 )isthebestasitsimultaneouslysegmentsandsmoothstheimagesothatboundariesofthenoisyspecklesareremovedandtheboundaryoftheplaneiskeptwell.ResultofpiecewiseconstantMSistheworst:inordertoremoveboundariesofspeckles,itpaysthepriceofobtainingbadplaneboundary.ResultofglobalGaussianpdfbasedmodelkeepsplaneboundarywellbutgainsmoreboundariesofspecklesasitallowsvariationofintensityini's. Figure3-5. GraphsofSAobtainedfromthreemodels:red:proposedmodel,green:globalGaussianpdfbasedmodel,blue:piecewiseconstantMSmodel. AquantitativecomparisonofthethreemodelsisprovidedthroughSA.WeaddrandomGaussiannoisewithmean0andvariance0:05100timestothecleanplaneimageFigure 3-4 B)toget100noisyimages.Thethreemodelsarethenappliedtothese100noisyimagestoobtain100SA'sforeachmodel,Figure 3-5 showsthegraphsofSAfor 52

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Figure3-6. ComparesegmentationandsmoothingresultsbetweentwomodelswithxedandadaptiveradiiA)-B)SegmentationandSmoothingresultsofFigure 3-4 B)usingxedradius0.C)-D)SegmentationandSmoothingresultsofgure 3-4 B)usingadaptiveradiidenedin( 3{6 ).E)-F)SegmentationandSmoothingresultsofgure 3-4 B)usingxedradius1.Foralltheresults,timestepsize=.1,=:01,convergein20iterations. 53

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Thirdly,wecomparemodel( 3{5 )withthemodelin[ 84 ]thatusesxedradius.InFigure 3-6 segmentationandsmoothingresultsofmodel( 3{5 )andthemodelin[ 84 ]withtwodierentradiiareshown.Therstrowcorrespondstothatofmodelin[ 84 ]withxedradius0,thesecondrowdepictsthatofmodel( 3{5 )withradiusequalsto1atrelativelyhomogenousregions,0nearboundaries.Thethirdrowshowsthatofmodelin[ 84 ]withxedradius1.Itisobviouslythatthesecondrowgivesgoodresultsinbothsegmentationandsmoothing.InFigure 3-6 B),regionsawayfromboundariesaresmoothedwell,nearboundaries,imagehasdiscontinuity.Incomparison,whenusethemodelin[ 84 ]withaxedsmallradius(seetherstrow),boundariesofnestructuresarekeptwellbutboundariesofthenoisybackgroundarealsocaught,recoveredimageisnotsmoothenougheither.Oppositely,whenabiggerxedradiusischosen(thethirdrow),imagesaresmoothedenough,however,butwepaythecostoflosingnestructures.Thisillustratestheadvantageofmodel( 3{5 ). 88 ],[ 89 ],[ 90 ]): Letbeanopen,boundedandconnectedsetinRN.Wesayu2L1()isafunctionofboundedvariation,denotedbyu2BV(),if 54

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ThespaceBV()isaBanachspaceendowedwiththenorm: 3{5 )concisely,weassumeimagedomainisdecomposedintotworegionsAandnA.Proofwouldbesimilariftherearemoresubregions,inwhichcase,therewillbejustmoresimilaradditivetermsintheenergyfunctionalE.LetA=fx2jH(x)>0g,thenA=H(),usingideasasin[ 16 ],model( 3{5 )couldbereformulatedas: 55

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4(1(I1(x)I0(y))2)dydxZ(1A(x))logZB(x;r(x))(1A(y))3 4(1(I2(x)I0(y))2)dydx wherethersttermisthelengthtermrepresentedastotalvariationofcharacteristicfunctionA.InBVspacenotation,itisjDAj().Euler-LagrangeequationsofE(A;I1;I2)withrespecttoI1andI2areI1(x)ZB(x;r(x))A(y)dyZB(x;r(x))A(y)I0(y)dy=0;x2A WemayassumeRB(x;r(x))A(y)dy>0forallx2A,thenI1(x)=RB(x;r(x))A(y)I0(y)dy IfRB(x;r(x))A(y)dy=0,whichiscalledthedegeneratecase,RB(x;r(x))A(y)I0(y)dywouldalsoequalto0,( 3{9 )issatisedautomaticallyforanychoiceofI1(x)between0and1. SimilaranalysisforI2(x)givesI2(x)=RB(x;r(x))(1A(y))I0(y)dy ThesolutionofI1;I2mustberelatedwithAby( 3{11 )( 3{12 ).Moreover,itisobservedfromequations( 3{11 )( 3{12 )thatifI0(x)2(:1;:9),forallx2,then,:1RB(x;r(x))A(y)dyRB(x;r(x))A(y)I0(y)dy:9RB(x;r(x))A(y)dy,fromwhichweget:1RB(x;r(x))A(y)I0(y)dy 56

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amongcharacteristicfunctionsofsetsAwithniteperimeterin,i.e.,characteristicfunctionsofboundedvariation([ 89 ]). Wenowstateourexistenceresult: 4(1(I1(x)I0(y))2)dydxjZjA(x)jjlogZB(x;r(x))jA(y)(1(I1(x)I0(y))2)jdyjdxZjlogjB(x;r(x))jjdxZjlogjjjdx=jjjlogjjj 4(1(I1(x)I0(y))2)dydxjZj1A(x)jjlogZB(x;r(x))j(1A(y))(1(I1(x)I0(y))2)jdyjdxZjlogjB(x;r(x))jjdxZjlogjjjdx=jjjlogjjj

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NoteI1n(x)=RB(x;r(x))Anj(y)I0(y)dy Regardingthesecondtermin( 3{8 ),Anj(y)3 4(1(I1n(x)I0(y))2)!D(y)3 4(1(I1(x)I0(y))2) stronglyasj!1andallthetermsarelessthanorequalto1,sobytheDominatedConvergenceTheorem, 4(1(I1n(x)I0(y))2)dy!ZB(x;r(x))D(y)3 4(1(I1(x)I0(y))2)dy

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4(1(I1n(x)I0(y))2)dyconvergestoD(x)logRB(x;r(x))D(y)3 4(1(I1(x)I0(y))2)dypointwise.Becauseallthetermsarelessthanorequaltojlog(jj)j,thenapplyDominantconvergencetheoremagain,weobtain limj!1RAnj(x)logRB(x;r(x))Anj(y)3 4(1(I1(x)I0(y))2)dydx=RD(x)logRB(x;r(x))D(y)3 4(1(I1(x)I0(y))2)dydx 3{14 )impliesE(D)liminfj!1E(Anj) HenceDisaminimizer. 3{8 )ismoregeneralandisabletosmoothandsegmentimagesthathavehighlevelnoiseinvolved. 59

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Experimentalresultsonseveralsetsofsyntheticandrealmedicalimageswithdierenttypesandlevelsofnoisedemonstratedthepotentialoftheproposedmodelinsimultaneouslysegmentingandsmoothingimages.Comparisonwithanotherthreemodelsshowedtheadvantageoftheproposedmodel. 60

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91 { 93 ]);(2).Smoothingtheprincipaldiusiondirectionafterthediusiontensorhasbeenestimatedfromtherawnoisymeasurements([ 94 { 98 ]);(3).Smoothingtensor-valueddatawhilepreservingthepositivedenitepropertyofD[ 92 96 99 { 101 ]. However,verylitteresearchreportedinliteraturetodateonHARDdataanalysisconsideresdenoisingprobleminthereconstructionoftheADCproleswhentheHARDrawdataisnoisy.Toimprovetheaccuracyoftheestimation,inthispaperwepresentanovelmodelthathastheabilityofsimultaneouslysmoothingandestimatingtheADCproled(x;;)fromthenoisyHARDmeasurementss(x;;)whilepreservingtherelevantfeatures,andthepositivenessandantipodalsymmetryconstraintsofd(x;;).ThebasicideaofourapproachistoapproximatetheADCprolesateachvoxelbya4thorderSHS(lmax=4in( 1{13 )): whosecoecientsaredeterminedbysolvingaconstrainedminimizationproblem.Thisminimizationproblemminimizesanon-standardgrowthfunctionaltoperformafeaturepreservedregularization,whileitminimizesthedatadelityterm.Notice,thereare15unknowncomplexvaluedfunctionsAl;minvolved.Sinced(;)isrealandYl;msatisesYl;m=(1)m 4{1 )to15realvaluedfunctions:Al;0(x);(l=0;2;4);ReAl;m(x);ImAl;m(x);(l=2;4;m=1;:::;l):

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67 ]and[ 72 ]mainlyintheaspectofthedeterminationoftheAl;m(x)'sin( 1{13 ).In([ 67 ])theAl;m(x)'s(liseven)aredeterminedby andin[ 72 ]theyareestimatedastheleast-squaressolutionsof Inthischapter,theestimationofAl;m(x)'sisnotperformedindividuallyateachisolatedvoxel,butaprocessofjointestimationandregularizationacrosstheentirevolume.Thejointestimationandregularizationnotonlyguaranteesthewellposednessoftheproposedmodel,butalsoenhancestheaccuracyoftheestimationsincetheHARDdataarenoisy.Moreover,inthispaperweprovidemoredetailedmethodtocharacterizethediusionanisotropy,whichusesnotonlytheinformationofAl;m(x)'sasin([ 67 72 ]),butalsothevariationofd(;)aboutitsmean.OurexperimentalresultsshowedtheeectivenessofthemodelintheestimationandenhancementofanisotropyoftheADCprole.ThecharacterizationofthediusionanisotropybasedonthereconstructedADCprolesusingtheproposedmodelisconsistentwiththeknownberanatomy. 1{11 ).Toexplainthebasicideaofourmethod,wefocusourattentiononthecaseswherethereareatmosttwoberspassingthrougha 62

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Thechallengeinregularizingdcomesfromtwoaspects.First,disdenedonS2ratherthanR2,hence,thederivativesfor(;)shouldbealongthesphere.Secondly,theregularizeddhastopreservetheantipodalsymmetrypropertywithrespectto(;).Consideringthesefactsweadopttheideadevelopedin[ 67 72 ]thatapproximatesdbyitsSHSconsistingofonlyevenordercomponentsuptoorder4,i.e. Theexpressionin( 4{4 )ensuresthesmoothnessandantipodalsymmetrypropertyofd(x;;)intermsof(;),thisiseasytoseefromthedenitionofYl;m(;).Forthecaseswherepossiblykberscrossinasinglevoxel,thesumin( 4{4 )shouldbereplacedbyPl=0;2;:::;2k. Nowtheproblemofregularizationandestimationofd(x;;)reducestothatforthe15complexvaluedfunctionsAl;m(x)(l=0;2;4andm=l;:::;l)in( 4{4 ).Sinced(;)ateachvoxelisarealvaluedfunction,andYl;msatisesYl;m=(1)mYl;m,Al;mshouldbeconstrainedbyAl;m=(1)mAl;m: 4{4 )to15realvaluedfunctions.Theyare Byusing( 4{5 ),wecanrewrite( 4{4 )as 63

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4{5 )simultaneously. Therearemanychoicesofregularizingoperatorstosmooththe15functionsin( 4{5 ).TotalVariation(TV)basedregularization,rstproposedbyRudin,OsherandFatemi[ 79 ],provedtobeaninvaluabletoolforfeaturepreservingsmoothing.However,itsometimescausesastaircaseeectmakingrestoredimageblocky,andevencontaining'falseedges'[ 102 103 ].Animprovement,thatcombinestheTVbasedsmoothingwithisotropicsmoothing,wasgivenbyChambolleandLions[ 103 ].TheirmodelminimizestheTVnormwhenthemagnitudeoftheimagegradientislarger,andtheL2normoftheimagegradientifitissmaller.However,thismodelissensitivetothechoiceofthethresholdwhichseparatestheTVbasedandisotropicsmoothing.TofurtherimproveChambolleandLions'modelandmakethemodelhavinganabilitytoselfadjustdiusionproperty,recently,certainnonstandarddiusionmodelsbasedonminimizingLp(x)normofimagegradienthavebeendeveloped[ 102 104 ].TorecoveranimageufromanobservedimageIin[ 102 ]thediusionwasgovernedbyminimizingminuZjrujp(jruj) 104 ]thediusionwasperformedthroughminimizingminuZ(x;Du) where(x;r):=1 1+kjrGIj2;

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102 ]and[ 104 ]areabletoselfadjustdiusionrangefromisotropictoTV-baseddependingonimagegradient.Atthelocationswithhigherimagegradients(p=1),thediusionisTVbasedandstrictlytangentialtotheedges([ 79 103 105 ]).Inhomogeneousregionstheimagegradientsareverysmall(p=2),thediusionisessentiallyisotropic.Atallotherlocations,theimagegradientforces1
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4{9 )wemaygetbetternumericalresults,sincepl;mwoulddependonupdatedAl;mratherthanthexedal;mintheiterationstominimize( 4{7 ).However,itgivesdicultyinthestudyoftheexistenceofsolutions. Sinced(x;;)isrelatedtotheHARDmeasurementss(x;;)ands0(x)throughtheStejskal-Tannerequation( 1{11 ),theestimationoftheAl;m'sisbasedontheoriginalStejskal-Tannerequation( 1{11 )ratherthanits(log)linearizedform( 1{12 ),thatis, 2ZZ20Z0js(x;;)s0(x)ebd(x;;)j2sindddx;(4{11) wheredisdeterminedin( 4{4 ).Asobservedin[ 106 ]whenthesignaltonoiseratioislowthelinearizedmodelgivesdierentresults. Finally,tosimultaneouslyregularizeandestimatetheADCd(x;;),ourmodelminimizestheenergyfunction withrespecttoAl;m(l=0;2;4andm=l;:::;l)inthespaceofBV(),(infact,only15functionsin( 4{5 )areneeded),andsubjecttotheconstraint: In( 4{12 ),( 4{13 ),s(x;;)ands0(x)arethenoisyHARDmeasurements(realvalued),d(x;;)istheSHSgivenin( 4{4 ),R3istheimagedomain,>0isaparameterwhichcouldbedierentfordierentAlm.E1andE2aregivenin( 4{7 )and( 4{11 ),respectively. BeforewederivetheEuler-Lagrangeequationsforourmodel( 4{12 ),( 4{13 ),wewouldliketopointoutthatifthemeasurementssatisfytheconditions(x;;)s0(x),thesolutionof( 4{12 )meetstheconstraint( 4{13 )automatically.Thereforewecantreatourmodelasanunconstrainedminimization.Thisisgiveninthefollowinglemma. 66

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theminimizerof( 4{12 )alwayssatisestheconstraint( 4{13 ). 4{12 )inBV(),andd(x;;)bethefunctiondenedin( 4{4 )associatedwiththeseoptimalAl;m(x)'s.Givenx2,ifd(x;;)<0forsome0<;0<2,thendene^d(x;;)=0,otherwise,dene^d(x;;):=d(x;;). Correspondingly,^Al;m(x):=Z20Z0^d(x;;)Yl;m(;)sindd: ^Al;m(x)=8><>:Al;m(x);ifd(x;;)0;80<;0<20;ifd(x;;)<0(4{15) Thisimpliesthatl;m(x;D^Al;m)l;m(x;DAl;m); 4{14 )holds.Fromthelasttwoinequalityabove,weobtainthatE(^d)E(d).Thiscontradictstothefactthatdminimizesenergyfunctional( 4{12 ). NowwegivetheevolutionequationsassociatedwiththeEuler-Lagrange(EL)equationsfor( 4{12 ):forl=0;2;4andm=l;:::;l, 67

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whereq(x)=p(x)ifjrjMl;m,andq(x)=1ifjrj>Ml;m. 67 ]thejAl;m(x)j(l=0;2;4andm=l;:::;l)inthetruncatedSHS( 4{4 )areusedtocharacterizethediusionanisotropyateachvoxelx.Ourexperimentalresults,however,indicatethisinformationaloneisinsucienttoseparateisotropicdiusion,one-berdiusion,andmulti-berdiusionwithinavoxel.WeproposetocombinetheinformationfromjAl;mjwiththevariancesofd(;)aboutitsmeanvaluetocharacterizethediusionanisotropy.Weoutlineouralgorithmasfollows: (1).If islarge,orthevarianceofd(;)aboutitsmeanissmall,thediusionatsuchvoxelsisclassiedasisotropic. (2).Fortheremainingvoxels,if 68

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4-3 D)presentsanintensity-codedimageofR2inabrainslicethroughtheexternalcapsule,animportantstructureofthehumanwhitematter.InFigure 4-3 D)thosevoxelsofahighintensity(brightregionsontheimage)arecharacterizedasone-berdiusion. (3).Foreachuncharacterizedvoxelaftertheabovetwosteps,searchthedirections(;),whered(;)attainsitslocalmaxima.Note,d(;)isantipodalsymmetric,i.e.,d(;)=d(;+),wemodoutthissymmetrywhencountthenumberoflocalmaxima.Thenwecomputetheweightsforthelocalmaxima(saywehave3localmaxima):Wi:=d(i;i)dmin 5-5 A)showsourclassicationofisotropicdiusion(darkregion),one-berdiusion(grayregion),andtwo-berdiusion(brightregion)inthesamesliceasinFigure 4-3 4{16 ),weuseAdditiveOperatorSplitting(AOS)algorithmforthediusionoperator(see[ 86 107 ]).Byusingthisalgorithm,thecomputationalandstoragecostislinearinthenumberofvoxels,andthecomputationaleciencycanbeincreasedbyafactorof10underrealisticaccuracyrequirements([ 86 ]).Thealgorithmisreadytobemodiedtoaparallelversion. 69

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4{16 )inthesystem,sinceeachequationhasthesamestructureasothers. Weusesemi-implicitnitedierencescheme: MqijrqijrX(n+1)i;j MqijdivrX(n+1)i;j HereXcanbereplacedbyoneofAl;m'swithl=0;2;4;m=ll,andfisafunctionofresultsfromlastiteration,namely,fisafunctionofallA(n)l;m's.q(x)=p(x)ifjrXjM,andq(x)=1ifjrXj>MforsomexedconstantM,whichwaschosenbasedoninitialvalueofX,soMmightbedierentfordierentAl;m's. Forsimplicityofformulas,wedene:4xXi;j=Xi;jXi1;j;4x+Xi;j=Xi+1;jXi;j;4xXi;j=Xi+1;jXi1;j4y+Xi;j=Xi;j+1Xi;j;4yXi;j=Xi;jXi;j1;4yXi;j=Xi;j+1Xi;j1Adoptingadiscretizationofthedivergenceoperatorfrom[ 16 ],onecanwrite( 4{21 )as: Mqij[4xqij;4yqij] 2hh4xX(n+1)i;j;4yX(n+1)i;ji=(2h) (2h)2+(4yX(n)i;j)2 (2h)2!2qij Mqijh2266644x0BBB@4x+X(n+1)i;j (2h)2!2qij (2h)2!2qij +(Ei;jHi;j)X(n+1)i;j1(Ei;j+Fi;j)X(n+1)i;j+(Fi;j+Hi;j)X(n+1)i;j+1(4{22) 70

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Mqijh2(X(n)i;jX(n)i1;j)2 2Di;j= Mqijh2(X(n)i+1;jX(n)i;j)2 2Ei;j= Mqijh2(X(n)i+1;j1X(n)i1;j1)2 2Fi;j= Mqijh2(X(n)i+1;jX(n)i1;j)2 2Gi;j=lnM(qi+1;jqi1;j) 2Hi;j=lnM(qi;j+1qi;j1) (2h)2(X(n)i+1;jX(n)i1;j)2 2 4{22 )wouldinvolvematrixinverseoperation,whichwouldbecomemoreandmorecomplicatedanddramaticallyexpensiveasdimensionincreasesifwesolveitdirectly.Instead,hereweuseAdditiveOperatorSplitting(AOS)algorithm,whichallowsustoreformatsystem( 4{22 )intothe2followingsystems: X(n+1)i;jX(n)i;j 2h(Ci;jGi;j)X(n+1)i1;j(Ci;j+Di;j)X(n+1)i;j+(Di;j+Gi;j)X(n+1)i+1;ji(4{23) X(n+1)i;jX(n)i;j 2h(Ei;jHi;j)X(n+1)i;j1(Ei;j+Fi;j)X(n+1)i;j+(Fi;j+Hi;j)X(n+1)i;j+1i(4{24) andX(n+1)i;j=X(n+1)i;j+X(n+1)i;j 71

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@n=0fortheMNmatrixX,oneneedstohave:X(n+1)1;j=X(n+1)2;j;X(n+1)M1;j=X(n+1)M;jX(n+1)i;1=X(n+1)i;2;X(n+1)i;N1=X(n+1)i;N 4{23 )and( 4{24 )correspondtolinearsystemsinmatrix-vectornotation:A1X 108 ]). 72

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4{12 )-( 4{13 )tosimulateddataandasetofHARDMRIdatafromthehumanbrain. 4-1 Table4-1: ListofS0andAl;m'sfortworegions Region 1 2 547 InFigure 4-1 wedisplayedthetrue,noisy,andrecoveredADCprolesd(x;;)forthesyntheticdatawithsize84.TheADCproled(x;;)wascomputedby( 4{6 )basedonthesesimulateddata,andthecorrespondingstrue(x;;)wasconstructedvia( 1{11 )withb=1000s=mm2.ThenthenoisyHARDMRIsignals(x;;)wasgeneratedbyaddingazeromeanGaussiannoisewithstandarddeviation=0:15.Figure 4-1 B)showsthe 73

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4{6 ),wherethecoecientsoftheSHSaretheleast-squaressolutionsof( 4{3 )withnoisys. Wethenappliedourmodel( 4{12 )-( 4{13 )tothenoisys(x;;)totesttheeectivenessofthemodel,with0;0=4;2;m=40(m=2:::2);4;m=60(m=4:::4).Bysolvingthesystemofequations( 4{16 )in2.5secondsoncomputerwithPIV2.8GHZCPUand2GRAMusingMatlabscriptcode,weobtained15reconstructedfunctionsasin( 4{5 ).UsingtheseAl;m(thesolutionsof( 4{16 ))wecomputedd(x;;)via( 4{6 ).Thereconstructedd(x;;)isshowninFigure 4-1 C).Comparingthesethreegures,itisclearthatthenoisymeasurementsshavechangedFigure 4-1 A),theoriginalshapesofd,intoFigure 4-1 B).Afterapplyingourmodel( 4{12 )-( 4{13 )toreconstructtheADCproles,theshapesofdinFigure 4-1 A)wererecovered,asshowninFigure 4-1 C).ThesesimulatedresultsdemonstratethatourmodeliseectiveinsimultaneouslyregularizingandrecoveringADCproles. TherawDWIdata,usuallycontainsacertainlevelofnoise,wereobtainedonaGE3.0Teslascannerusingasingleshotspin-echoEPIsequence.ThescanningparametersfortheDWIacquisitionare:repetitiontime(TR)=1000ms,echotime(TE)=85ms,theeldofview(FOV)=220mmx220mm.24axialsectionscoveringtheentirebrainwiththeslicethickness=3.8mmandtheintersectiongap=1.2mm.Thediusion-sensitizinggradientencodingisappliedinfty-vedirections(selectedfortheHARDMRIacquisition)withb=1000s=mm2.Thus,atotaloffty-sixdiusion-weightedimages,withamatrixsizeof256x256,wereobtainedforeachslicesection.Weappliedmodel( 4{12 )tothesedatatocomputetheADCprolesintheentirebrainvolume.Bysolvingasystemofequations( 4{16 )weobtainedallthecoecientsAl;m'sin( 4{5 ),anddeterminedd(x;;)using( 4{6 ). 74

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4{19 )and( 4{20 )respectively,aswellasthevariance(x)ofd(x;;)aboutitsmean:(x)=R0R20(d(x;;)P55i=1d(x;i;i)=55)2dd.BasedonresultsfromtheHARDMRIdataofthisparticularpatient,wecharacterizedthediusionanisotropyaccordingtothefollowingprocedure.IfR0(x)>0:856,or(x)<19:65thediusionatxisclassiedasisotropic.FortheremainingvoxelsifR2(x)>0:75,thediusionatsuchvoxelsisconsideredasone-berdiusion.Foruncharaterizedvoxelsfromthesetwostepswefurtherclassiedthembytheprinciplesstatedinthesection 4.3 .TheselectionofthethresholdsmentionedaboveforR0,R2andinvolvesexperts'inputandlargesampleexperiments.Experimentalresultsdenitelydependonthesethresholds,butnotsensitively. Figure 4-2 presentsA2;0(x),oneofthecoecientsin( 4{6 ),fortheparticularsliceinthevolume.TheimagesA2;0(x)inFigure 4-2 A)and 4-2 B)areestimatedbyusing( 4{2 )andsolving( 4{12 ),respectively. Figure 4-3 ComparesFAandthreeR2(x)'swithAl;m(x)'sobtainedfromthreedierentmodelsforthesamesliceasshowninFigure 4-2 .Figure 4-3 A)displaystheFAimageobtainedbyusingadvancedsystemsoftwarefromGE.TheAl;m(x)'susedtoobtainR2(x)inFigure 4-3 B)aredirectlycomputedfrom( 4{2 ).ThoseusedtoobtainR2(x)inFigures. 4-3 C)and 4-3 D)aretheleast-squaressolutionsof( 4{3 )andthesolutionsof( 4{12 ),respectively.InFigures. 4-3 C)and 4-3 D)thevoxelswithhighlevelsofintensities(red,yellow,yellow-lightblue)arecharacterizedasone-berdiusion. AlthoughtheFAimageinFigure 4-3 A)isobtainedbasedonaconventionalDTImodel( 1{7 ),itisstillcomparablewiththeR2map,sincesingletensordiusioncharacterizedbySHSrepresentationfromtheHARDimagesagreeswiththatcharacterizedbytheDTImodel.However,inDTIavoxelwithalowintensityofFAindicatesisotropicdiusion,whileusingouralgorithm,multi-bersdiusionmayoccuratthelocationwiththelowvalueofR2. 75

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4-3 A)4-3 D).Figure 4-3 B)indicatesagainthattheestimatesofAl;mdirectlyfromthelogsignalsusuallyarenotgood.Eventheleast-squaressolutionof( 4{3 )isnotalwayseective.ThiscanbeseenbycomparingtheanatomicregioninsidetheredsquareofFigures. 4-3 C)and 4-3 D),whicharezoomedinFigures. 4-4 A)and 4-4 B),respectively.Thereisadarkbrokenlineshowingonthemapoftheexternalcapsule(arrowtotherightonFigure 4-4 A),thissameregionwasrecoveredbytheproposedmodelandcharacterizedbythethirdstepinouralgorithmastwo-beranisotropicdiusion(arrowtotherightinFigure 4-4 B).(ThemodelsolutionsreducedthevalueofR0,increasedthevaluesofR1slightly,andmadethe3rdstepinourcharacterizationtobeapplied).Ourresultsalsoshowtheconnectioninacorticalassociativetract(arrowtotheleftinFigures. 4-4 B),however,thisconnectionwasnotmappedoutonFigure 4-3 C)orthezoomedimageinFigure 4-4 A).InfactthisconnectionwasnotmappedoutonFigures 4-3 A)-B)either.Allthesemappedconnectionsareconsistentwithknownneuroanatomy.Combinedtogether,ourresultsindicatethatourproposedmodelforjointrecoveryandsmoothingoftheADCproleshasanadvantageoverexistingmodelsforenhancingtheabilitytocharacterizediusionanisotropy. Figure 5-5 A)showsapartitionofisotropic,one-ber,andtwo-berdiusionforthesamesliceusedinFigure 4-4 .Thetwo-ber,one-ber,andisotropicdiusionregionswerefurthercharacterizedbythewhite,gray,andblackregions,respectively.TheregioninsidethewhitesquareinFigure 5-5 A),whichisthesameonesquaredinFigures. 4-3 C)and 4-3 D),iszoomedinFigure 4-4 C).ItisseenthatthetwoarrayedvoxelsinFigure 4-4 B)areclassiedastwo-berdiusion.Thecharacterizationoftheanisotropyonthevoxelsandtheirneighborhoodsisconsistentwiththeknownberanatomy. Figure 5-5 B)representstheshapesofd(x;;)atthreeparticularvoxels(upper,middleandlowerrows).Thedinallthreevoxelsiscomputedusing( 4{6 ).However,theAl;m(x)usedincomputingdontheleftcolumnaretheleast-squaressolutionsof( 4{3 ), 76

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Figure4-1. Comparingshapesofd.A)Trued.B)Thedgeneratedby( 4{6 ),withAl;m'stheleastsquaresolutionof( 4{10 )withthenoisymeasurements.C)Recovereddbyapplyingmodel( 4{12 ). whileintherightcolumntheyarethesolutionsoftheproposedmodel( 4{12 ).Therstandsecondrowsshowtwovoxelsthatcanbecharacterizedasisotropicdiusionbeforedenoising,butastwo-berdiusionafterapplyingmodel( 4{12 ).ThesetwovoxelsarethesamevoxelsasinFigure 4-4 directedbyarrows.ThelowerrowofFigure 5-5 B)showstheone-berdiusionwasenhancedafterapplyingourmodel. SolvingAl;m'sofsize15109868from4-Ddataofsize55109868takes46.2secondsforeachiterationoncomputerwithPIV2.8GHZCPUand2GRAMinMatlabscriptcode. 4{12 )usingtheideadevelopedin[ 104 ]. 77

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4{12 )overthefunctionsinBV(),werstneedtogiveaprecisedenitionforE1. 4{8 ),andRjDsAl;mjisthetotalvariationnormofAl;m. Then,ourenergyfunctional( 4{12 )isdenedasE(Al;m)=ZXl=0;2;4lXm=ll;m(x;rAl;m)+ZjDsAl;mj 2ZZ20Z0js(x;;)s0(x)ebd(x;;)j2sindddx:(4{25) Inthediscussionofexistence,withoutlossofgenerality,wesettheparameter=1in( 4{12 )andthresholdMl;m=1in( 4{8 )toreducethecomplexityintheformulation. Nextwewillshowlowersemi-continuityoftheenergyfunctional( 4{12 )inL1,i.e.ifforeachl;m(l=0;2;4andm=l;:::;l),ask!1,Akl;m!A0l;minL1(); Toprovethisweneedthefollowinglemma: 78

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Foru2BV()denote(u):=Z(x;Du); (u)=e(u)(4{28) Furthermore,islowersemi-continuousonL1(),i.e.ifuj;u2BV()satisfyuj!uweaklyinL1()asj!1then(u)liminfj!1(uj): Nextweshow( 4{28 ).Foru2BV(),wehavethatforeach2C10(;Rn),Zudivdx=Zrudx+ZDsu

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4{28 )itonlyremainstoshowthat Sinceany2L1(;Rn)canbeapproximatedinmeasureby2C10(;Rn),wehavethat sup2C10(;Rn)jj1Zrup(x)1 Choosing(x)=1fjruj1gjrujp(x)1ru 4{30 )isZ1 Toshowtheoppositeinequality,weargueasfollows.Forany2L1(;Rn),sincep(x)>1wehavethatforalmostallx,ru(x)(x)1 Ifjruj>1,noticingp(x)>1andjj1foralmostallxwehavethatru=jrujru

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Combining,( 4{30 ),( 4{31 ),( 4{32 ),and( 4{33 ),wehave( 4{29 ),andhenceforallu2BV(),e(u)=(u). Notethat(x;r)=l;m(x;r)ifp(x)=pl;m(x).AdirectconsequenceofthislemmaisthatwehavethatE1in( 4{7 )isweaklylowersemi-continuousinL1topologyonBV()norm. Furthermore,wecanshowthatE2in( 4{7 )islowersemi-continuousonL1().Indeed,whenAkl;m!A0l;m;inL1();ask!1; 4{26 )holds. Nowwecanproveourexistenceresults. 4{12 )overthespaceofBV(). 4{12 )inBV().Thenforeach(l;m)thesequenceAkl;misboundedinBV().FromthecompactnessofBV()thereexistsubsequencesofAkl;m(stilldenotedbyAkl;m)andfunctionsA0l;m2BV()satisfyingAkl;m!A0l;mstronglyinL1():

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4{26 )),wehaveE(A0l;m)liminfk!1E(Akl;m)infAl;m2BV()E(Al;m): 4{12 ). OurexperimentsonbothsyntheticdataandhumanHARDMRIdatashowedtheeectivenessoftheproposedmodelintheestimationofADCprolesandtheenhancementofthecharacterizationofdiusionanisotropy.Thecharacterizationofnon-Gaussiandiusionfromtheproposedmethodwasconsistentwithknownneuroanatomy. Thechoiceofthecurrentparameters,however,mayaecttheresults.Ourchoicewasmadebasedontheprinciplethatclassicationforone-berdiusionfromthemodelsolutionshouldagreewithaprioriknowledgeoftheberconnections. 82

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(b) ComparisonofA20.A)A20computedfrom( 4{2 ).B)A20obtainedfrommodel( 4{12 ). 83

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(b) (c) (d) ImagesofFAandR2.A)FAfromGEsoftware.B)-D)R2withtheAl;m'sasthesolutionsof( 4{2 ),least-squaressolutionsof( 4{3 ),andmodelsolutions,respectively. 84

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(b) (c) ZoomedFAandA20.A)-B)EnlargedportionsinsidetheredsquaresinFigures. 4-3 C)and 4-3 D),respectively.C)EnlargedportionsinsidethewhitesquaresinFigure 5-5 A). 85

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(b) Classicationofvoxelsbasedond.A)Classication:white,gray,andblackvoxelsareidentiedastwo-ber,one-ber,andisotropicdiusionrespectively.B)Shapesofd(x;;)atthreeparticularpoints(upper,middleandlowerrows).Thediscomputedvia( 4{6 ).Al;m(x)usedin( 4{6 )intheleftcolumnsaretheleast-squaressolutionsof( 4{3 ),whileintherightcolumnarethesolutionsfromourmodel. 86

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Copyright[2005]LectureNotesonComputerScience[ 109 ].Portionsreprinted,withpermission. 4{1 )toapproximatedin( 1{11 ),andhencetodetecttwo-berdiusion,atleast15diusionweightedmeasurementss(q)over15carefullyselecteddirectionsarerequired.However,tousethemixturemodel( 1{9 )withn=2todetecttwo-berdiusiononly13unknownfunctions:f,6entriesofeachofD1;D2needtobesolved.Thismotivatedustostudywhatistheminimumnumberofthediusionweightedmeasurementsrequiredfordetectingdiusionwithnomorethantwoberorientationswithinavoxel,andwhatisthecorrespondingmodeltoapproximatetheADCprolesinthiscase.InthischapterweproposetoapproximatetheADCprolesfromHARDMRIbytheproductoftwosecond-orderSHS'sinsteadofafourth-orderSHS.Wealsoshowthattheproductoftwosecond-ordersphericalSHS'sdescribesonlythediusionwithatmosttwoberorientations,whilethefourth-orderSHSmaydescribethediusionwiththreeberorientations. Moreover,inthischapterwewillintroduceaninformationmeasurementdevelopedin[ 110 ],andtermedasCRE(seedenition( 5{7 ))tocharacterizethediusionanisotropy.CREdiersfromShannonentropyintheaspectthatShannonentropydependsonlyontheprobabilityoftheevent,whileCREdependsalsoonthemagnitudeofthechangeoftherandomvariable.Weobservedthatisotropicdiusionhaseithernolocalminimumormanylocalminimawithverysmallvariationinthedenoiseds(q)=s0,i.e.,ebdproles,incomparingwithoneberortwo-berdiusions,whichimpliesthecorrespondingCREtobesmall.Wealsofoundthatoneberdiusionhasonlyonelocalminimumwithlargervariationinthes(q)=s0proles,whichleadstolargerCRE.Therefore,wepropose 87

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66 72 111 ]todetectthediusionwithatmosttwoberorientationtheADCproleswererepresentedbyatruncatedSHSuptoorder4intheformof( 4{1 ).In[ 66 ]thecoecientsAl;m's(liseven)weredeterminedbyinversesphericalharmonictransformof1 72 ]theywereestimatedastheleast-squaressolutionsof RegularizationontherawdataorAl;mwasn'tconsideredinthesetwowork.In[ 111 ]Al;m'swereconsideredasafunctionofx,andestimatedandsmoothedsimultaneouslybysolvingthefollowingconstrainedminimizationproblem:minAl;m(x);~s0(x)ZfXl=0;2;4lXm=ljrAl;m(x)jpl;m(x)+jr~s0(x)jp(x)gdx withtheconstraintd>0.Inthismodelpl;m(x)=1+1 1+kjrGAl;mj2,q(x)=1+1 1+kjrGs0j2,anddtakestheform( 4{1 ).Bythechoiceofpl;mandq,theregularizationistotalvariationbasednearedges,isotropicinhomogeneousregions,andbetweenisotropicandtotalvariationbaseddependingonthelocalpropertiesoftheimageatotherlocations.InthisworksincetheADCprolewasapproximatedby( 4{1 ),atleast15measurementsofs(q)wererequiredtoestimatethe15coecientsAl;m. However,themixturemodel( 1{9 )withn=2,whichisalsoabletodetecttwo-berdiusion,involvesonly13unknownfunctions.Thismotivatesustondamodelthatisabletodetectnon-Gaussiandiusionwiththeminimumnumberofunknowns.Inthispaperweonlydiscussthediusionwithnomorethantwoberorientationswithinavoxel.Thesignicanceofthisstudyisclear:asmallernumberofunknownsleadtoa 88

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OurbasicideaistoapproximatetheADCprolesbytheproductoftwosecondorderSHS'sinsteadofaSHSuptoorderfour.Thiscanbeformulatedas Inthismodelthereareonly12unknowns:bl;m,cl;m(l=0;2andlml). ToestimatetheADCprolefromtherawHARDMRIdata,whichusuallycontainsacertainlevelofnoise,weproposeasimultaneoussmoothingandestimationmodelsimilarto( 5{2 )forsolvingbl;m;cl;m,thatisthefollowingconstrainedminimizationproblem:minbl;m(x);cl;m(x);~s0(x)ZfXl=0;2lXm=l(jrbl;m(x)j+jrcl;m(x)j+jr~s0(x)jdx 2ZfZ20Z0js(x;q)~s0(x)ebd(x;;)j2sindd+j~s0s0j2gdx;(5{4) withconstraintd0,wheredisintheformof( 5{3 ).,areconstants.Therst3termsaretheregularizationtermsforbl;m,cl;mands0respectively.ThelasttwotermsarethedatadelitytermsbasedontheoriginalStejskal-Tannerequation( 1{11 ). Next,feasibilityofthismodelwillbeexplained.LetSAdenotethespaceofevenSHSoforder4,i.e.,SA=fd:d(;)=A=Pl=0;2;4Plm=lAl;mYl;mg,letSBCbethespaceofproductsoftwoevenSHSoforder2,i.e.,SBC=fd:d(;)=BC=Pl=0;2Plm=lbl;mYl;m(;)Pl=0;2Plm=lcl;mYl;m(;)g.SincetheproductoftwoSHSoforder2and2canbeexpressedasalinearcombinationofsphericalharmonicsoforderlessorequalto2+2=4,SBCSA.Butasimpledimensioncount,dim(SA)=15,whiledimensionofSBCislessthanorequalto12.ThenSBCisapropersubsetofSA.SofunctionsinSBCarelessgeneralthanfunctionsinSA.However,numerousexperimentsshowthatwhenavoxelisnotmorecomplicatedthantwo-berdiusion,itsADCiswell-approximatedbyafunctioninSBC.Thisisnottrueif3-berormorecomplicated 89

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5-1 ).Therefore,ifwefocusonlyoncharacterizingatmosttwo-berdiusion,whichisthemostcommoncase,model( 5{3 )isreasonableandsucienttorepresentADC. Model( 5{4 )isaminimizationproblemwithconstraintd(;)0forall0<;0<2,whichisusuallydiculttoimplement.ToimprovetheeciencyofcomputationweusedtheideathatanysecondorderSHSPl=0;2Plm=lbl;mYl;m(;)isequivalenttoatensormodeluTDuforsomesemi-positivedenite33matrixD,whereu(;)=(sincos;sinsin;cos).Thismeansthatthecoecientsbl;m,(l=0;2;m=l;:::;l)inSHSandtheentriesD(i;j);(i;j=1;:::;3)inDcanbecomputedfromeachotherexplicitly.Herearetwoexamples:b00=2 3p 4p Model( 5{4 )isthenreplacedbyminLjk1(x);Ljk2(x);~s0(x)Z(2Xi=13Xj=1jXk=1jrLj;kij+jr~s0j)dx 2ZfZ20Z0js~s0ebdj2sindd+j~s0s0j2gdx;(5{6) whered=(uL1LT1uT)(uL2LT2uT).Allthebl;m;cl;m;l=0;2;m=l:::laresmoothfunctionsofLjki,i=1;2;j=1;2;3;kj.SmoothnessofLjkiguaranteesthatofbl;m's,cl;m's.Therstterminmodel( 5{6 )thusworksequivalentlytothewaythersttwotermsinmodel( 5{4 )do,whilealltheothertermsarethesameasthoseremainingin( 5{4 ).Hence,( 5{6 )isequivalentto( 5{4 ),butitisanon-constrainedminimizationproblemand 90

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5{4 )canbeobtainedbytheonetoonerelationbetweenthem. Figure5-1. ComparisonoftheADC'sapproximatedby( 4{1 )and( 5{3 )infourcases:A)isotropicdiusion;B)one-berdiusion;C)two-berdiusion;D)three-berdiusion.InA)-D)fromlefttoright,toptobottom,weshowshapesofB,C,BC,andA,respectively. Weappliedmodel( 5{6 )toasetofhumanbrainHARDMRIdatatoreconstructandcharacterizeADCproles.Thedatasetconsistedof55diusionweightedimagesSk:!R;k=1;:::;55,andoneimageS0intheabsenceofadiusion-sensitizingeldgradient(b=0in( 1{11 )).24evenlyspacedaxialplaneswith256256voxelsineachsliceareobtainedusinga3TMRIscannerwithsingleshotspin-echoEPIsequence.Slicethicknessis3:8mm,gapbetweentwoconsecutiveslicesis1:2mm,repetitiontime(TR)=1000ms,echotime(TE)=85msandb=1000s=mm2.Theeldofview(FOV)=220mm220mm.Werstappliedmodel( 5{6 )tothedatatogetLi,andthenusedLitocomputebl;mandcl;m,l=0;2;m=l:::l,andtheADCd=BC.Forpurposeofcomparison,wealsousedthemodel( 5{2 )toestimateAl;mandgetA.ThecomparisonfortheshapesofADCintheformofBCandAisdemonstratedinFigure 5-1 A)-D)atfourspecicvoxels.Thediusionatthese4voxelsareisotropicA),one-berB),two-berC),andthree-berD),respectively.Ineachsubgure,theupleft,upright,downleft,downrightonesaretheshapesofB,C,BCandA,respectively.Itisevidentthatifthediusionisisotropic,one-berortwo-ber,BCandAarethesame.However,ifthediusionisthree-ber,Acan'tbewellapproximatedbyBC. 91

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5-2 A)-D)wecomparedimagesofR2(denedinsection 5.3 )withcoecientAl;mestimatedby4dierentmethods.ThevoxelswithhighervalueofR2wereconsideredasone-berdiusion.TheAl;m'sinA),B)andC)wereestimatedusingleast-squaresmethodin[ 72 ],model( 5{2 ),andmodel( 5{6 )withthediusion-sensitizinggradientappliedto55directions,respectively.TheAl;m'sinD)areestimatedbythesamewayasthatinC),butfromtheHARDdatawith12carefullychosendirections.Themodel( 5{6 )appliedon55measurementsworkedasgoodasthemodel( 5{2 )ingettinghighervalueofR2.Bothofthemworkedbetterthantheleast-squaresmethodthatdoesnotconsiderregularization.Althoughtheresultfrom12measurementswasnotasgoodasthatfrom55measurements,theyarearestillcomparable.WewillshowinFigure 5-5 A)andB)thattheanisotropycharacterizationresultsbasedontheADCpresentedinC)andD)arealsoclose.Theseexperimentalresultsindicatedthatbyusingtheproposedmodelthevoxelswithtwo-berdiusioncanbedetectedreasonablywellfrom12HARDmeasurementsincarefullyselecteddirections. 111 ]Chenetal.realizedthatsuchavoxelcouldhaveisotropicorone-berdiusion.TheydenedR0:=jA0;0j 92

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whereR+=fX2RjX0g. Figure5-2. A)-D)areimagesofR2withAl;m'scalculatedusingleast-squaresmethod,model( 5{2 ),model( 5{6 )appliedon55measurements,andmodel( 5{6 )appliedon12measurements,respectively. WeuseCREofebdratherthandtocharacterizediusionanisotropywhendisrecoveredfromHARDmeasurementsthrough( 5{6 ).ThemagnitudeofADCisusuallyontheorderof103,whilethemagnitudeofebdisintheorderof101,whichislargerthanthatofADCitself.Moreover,ebdisasmoothapproximationofthedatas=s0. TheweakconvergencepropertyofCREprovedin[ 111 ]makesempiricalCREcomputationbasedonthesamplesconvergesinthelimittothetrueCRE.ThisisnotthecasefortheShannonentropy.WedenetheempiricalCREofebdas wheref1<2<:::
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Figure5-3. AnexampleforshapesofADCforisotropic,one-berandtwo-bervoxels.A)ShapesofADCatanisotropic(rstrow),one-ber(secondrow)andtwo-ber(lastrow).B)-C)GraphsofF(),F()logF()atthreeparticularvoxels:isotropic(red),one-ber(green),two-ber(blue).D)R2(blue),CRE(yellow),variance(black)asfunctionsofrotationangleusedinconstructingsyntheticdata. whyCREisthelargestforone-ber,mediumfortwo-berandsmallestforisotropicdiusionvoxels.Inourexperiment,wechooseM=1000uniformlydistributeddirections(;)in( 5{8 ). DenethedecreasingdistributionfunctionF():=P(ebd>).Figure 5-3 B)showsthegraphsofF()atthreepre-classiedvoxels:isotropic(red),one-ber(green),two-ber(blue).ItisobservedthatthesupportandmagnitudeofF()arelargestatthevoxelwithone-berdiusion,andsmallestatthatwithisotropicdiusion.Figure 5-3 C)demonstratesthegraphsofF()logF()atthesamethreevoxels.Itisevidentthattheareaunderthegreencurve(one-ber)ismuchlargerthanthatunderthebluecurve(two-ber),whiletheareaundertheredcurveisthesmallest.SinceCREisexactlytheareaundercurveF()logF(),wecanseethatthemeasureCRE(ebd)isthelargestatthevoxelswithone-berdiusion,mediumwithtwo-berdiusion,andsmallestwithisotropicdiusion.ThusmeasureCRE(ebd)couldbeusedtodistinguishisotropic,one-berandtwo-berdiusionwithtwothresholdsT1andT2,withT1
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Figure5-4. Imagesoffourmeasures:A)R2;B)FA;C)CREofebd;D)Varianceofebd. variance,incharacterizingdiusionanisotropy.ThehumandataarethesameasthatusedinFigure 5-2 .Thesyntheticdataareconstructedasfollows:SetD1andD2tobetwo33diagonalmatriceswithdiagonalelements4102;102;2102and8102;102;3102,respectively.ThenxD1butrotateprincipaleigenvectorofD2aboutxyaxisbyangletogetD2().LetB(;)=uTD1u,C(;):=uTD2()u.WecomputedR2,FAandCRE,varianceofebBCforvariousvaluesofandshowedtheminFigure 5-3 D)inblue,yellowandblackrespectively.Whenvariesfrom0to=2,BCchangesfromatypicaloneberdiusiontoatwoberdiusion,andfrom=2toBCchangesbacktothesameshapeas=0.ThegraphofCREshowsthevalueofCREdecreaseswhenBCvariesfromone-berdiusiontotwo-berdiusion,andincreaseswhenBCgraduallychangesfromtwo-berdiusionbackstoone-berdiusion. 5-4 A)andB).ButCREdiersmuchfromR2andFA.InFigure 5-3 D)thegraphCREismuchsteeperthantheothers.InFigure 5-4 ,visually,contrastofCREismuchbetterthanthatofFAandR2.Furthermore,thesmallnessofmagnitudeofR2orFAisunabletodistinguishbetweenisotropicandtwo-berdiusion,whilethatofCREdoesbetterjob.Note,CREiscomparabletoFAorR2indetectingisotropicandone-berdiusion. 95

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112 ],E(jXE(X)j)2CRE(X).InourcaseXisebd,whosemagnitudeisamultipleof102<1,sowehaveVar(X)=E(jXE(X)j2)E(jXE(X)j)2CRE(X).OurexperimentalresultsshowthatmagnitudeofCREisalmost10timesofthatofVar(X).HighermagnitudeofCREmakesitlesssensitivetoroundingerrors.Moreover,inFigure 5-4 D),whichrepresentingthethevarianceofebd,theGenu/Spleniumofcorpuscallosumissobrightthatregionsbesidesitarenotclearlyvisualized,soCREismuchbetterthanvariancevisually. Figure5-5. A)-B).Characterization:black,gray,andwhiteregionsrepresentthevoxelswithisotropic,one-ber,andtwo-berdiusion,respectively.InA)weused55measurements,inB)weused12carefullyselectedmeasurements.C)ImageofCREcalculatedfrom12measurements.D)CharacterizationresultsoftheregioninsidetheredboxinA)usingCRE(top)andvariance(bottom)basedon55measurements.Redarrowspointtoavoxelthatiswronglycharacterizedasone-berdiusionbyusingvariancebutcorrectlyclassiedastwo-berdiusionusingCRE. Figure 5-5 A)showsapartitionofabrainregionintoisotropic,one-berandtwo-berdiusionbasedonADCcalculatedfrom55measurements.Theblack,gray,whitevoxelsareidentiedasisotropic,one-berandtwo-berdiusion,respectively.Thecharacterizationisconsistentwithknownberanatomy.Figure 5-5 B)representsthecharacterizationresultbasedontheADCestimatedfrom12measurements.Itiscomparablewiththatfrom55measurements.CREbasedonADCestimatedfrom 96

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5-5 C))isalsocomparabletothatfrom55measurements(Figure 5-4 C)).Thusourcharacterizationappearsnottobesensitivetonumberofmeasurementsaslongasatleast12measurementsareused.Figure 5-5 D)illustratesatwo-berdiusionvoxel(indicatedbyredarrow)thatisincorrectlycharacterizedasone-berdiusionusingvariance(bottomimage)butcharacterizedastwo-bercorrectlyusingCRE(topimage).ThisfurtherveriesthesuperiorityofCREovervarianceincharacterizingdiusionanisotropy. OurexperimentsontwosetsofhumanbrainHARDMRIdatashowedtheeectivenessandrobustnessoftheproposedmodelintheestimationofADCprolesandtheenhancementofthecharacterizationofdiusionanisotropy.Thecharacterizationofdiusionfromtheproposedmethodwasconsistentwithknownneuroanatomy. 97

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Copyright[2006]IEEE[ 71 ].Portionsreprinted,withpermission. 4{12 )and( 4{13 ).Then,fromtheSHrepresentationoftherecovereddwedeneR0(x)=jA0;0(x)j 1{9 )shouldbecloseto1. Undertheassumptionofpt(r)beingamixtureoftwoGaussians,thediusionismodelledby( 1{9 )withn=2.Thecombinationof( 1{9 )withn=2and( 1{11 )yields whereuT=(sincos;sinsin;cos).Fornotationconciseness,denotef1byf.ToestimateDiandfin( 1{10 )weminimizethefollowingfunctionw.r.tL1;L2;f:Z(2Xi=1jrLijPi(x)+jrfjpf(x))dx+1Z1(f1)2dx withtheconstraintsLm;mi>0.In( 6{2 )fori=1;2,i>0isaparameter,pi(x)=1+1 1+kjrGrLij2,pf(x)=1+1 1+kjrGrfj2,andLiisalowertriangularmatrix.DiisrecoveredfromLibyDi=LiLTi(i.e.,LiLTiistheCholeskyfactorizationofDi).WritingthecostfunctionintermsofLiratherthandirectlyintermsofDi

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101 ]).AlsowehavejrLijp=P1m;n3jrLm;nijp. Thersttwotermsin( 6{2 )aretheregularizationterms.Bythechoiceofpi(x)(similarlyforpf),inthehomogeneousregionimagegradientsareclosetozeroandpi(x)2,sothesmoothingisapproximatelyisotropic.Alongtheedges,imagegradientmakespi(x)1,sothesmoothingisapproximatelytotal-variation-basedandisalmostonlyalongtheedges.Atallotherlocations,theimagegradientforces10:8416or 99

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6{2 )bytheenergydescentmethod.Theinformationoff1on1wasalsoincorporatedintotheselectionoftheinitialf. Bysolving( 6{2 )weobtainedthesolutionsLiandf,andconsequently,Di=LiLTi(i=1;2).Figure 6-1 A)representsthemodelsolutionf.Functionf1onthedarkredregions.Thevoxelsintheseregionsareidentiedasisotropicorone-berdiusion.Thisisconsistentwithknownneuroanatomy.Figures 6-1 C)and 6-1 D)showthecolorrepresentationofthedirectionsoftheprincipaleigenvectorsforD1(x)andD2(x),respectively.Bycomparingthecolor-codinginFigures 6-1 C)and 6-1 D)withthecolorpieshowninFigure 6-1 B),theberdirectionsareuniquelydetermined.TherepresentationinFigure 6-1 B)isimplementedbyrelatingtheazimuthalangle()ofthevectortocolorhue(H)andthepolarangle(=2)tothecolorsaturation(S).Slightlydierentfrom[ 113 ],wedeneH==2,S=2()=,andValueV=1.Ifthedirectionoftheprincipaleigenvectorisrepresentedby(;),theberorientationcanbedescribedbyeither(;)or(;+).Toresolvethisambiguity,wechoosetheeigenvectortolieinthelowerhemisphere,i.e.=2.Theupperhemisphereisjustanantipodallysymmetriccopyofthelowerone.Thexyplaneistheplaneofdiscontinuity. Figure 6-2 showstheshapesofd(x;;)togetherwiththeberdirectionsat4particularvoxels.TheblueandredarrowsindicatetheorientationsofthebersdeterminedfromtheprincipaleigenvectorsofD1andD2respectively.Thelastshapecorrespondstoisotropicdiusion.Figs. 6-1 andFigure 6-2 indicatethatourmodel( 6{2 )iseectiveinrecoveringtheintra-voxelstructure. Toexaminetheaccuracyofthemodelinrecoveringberdirections,weselectedaregioninsidethecorpuscallosumwherethediusionisknowntobeofone-ber.Foreachvoxelinthisregionwecomputedthedirectioninwhichdismaximized.ThisdirectionvectoreldisshowninFigure 6-3 A).Ontheotherhandwesolved( 6{2 )andobtainedthemodelsolutionf1onthisregion.Thedirectioneldgeneratedfromtheprincipal 100

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C)D) Figure6-1. Fiberdirectioncolormap.A)Modelsolutionf.B)colorpie.C)Color-codingofthe1stberdirectionmapping.D)Color-codingofthe2ndberdirectionmapping. 101

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Shapeofd,andtheorientationsoftheprincipaleigenvectorsofD1(blue)andD2(red)at4particularvoxels eigenvectorofD1isshowninFigure 6-3 B),inwhichthevectoreldisnotonlywellpreservedbutalsomoreregularizedduetotheregularizationtermsinthemodel. Thechoiceoftheparametersin( 6{2 )andthedeterminationoftheregion1inuencetheresults.Ourchoicewasmadebasedontheprinciplethattheone-berdirectionfromthemodelagreeswiththedirectioninwhichdwasmaximized. 102

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Figure6-3. Fiberdirectioneld.A)Thatobtainedbymaximizingd.B)TheprincipaleigenvectorofD(solutionof( 6{2 )) 103

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Copyright[2006]IEEE[ 114 ].Portionsreprinted,withpermission. 1{9 )): wheref0;1f0areconsideredastheapparentvolumefractionsofdiusiontensorD1;D2respectively.Recently,Parkeretal.[ 69 ]andTuchetal.[ 70 ]usedamixtureoftwoGaussiandensitiestomodelthediusionforthevoxelswheretheGaussianmodeltsthedatapoorly. AprimeprobleminrecoveringmultitensoreldDi(x);i=1;2;andf(x)istheacquisitionnoisewhichcorruptsthedatameasurement.NeitherParkernorTuchconsideredremovingnoise.Inthisnotewepresentanewvariationalmethodwhichdiersfromtheexistingmethodsinthefollowingaspects.First,werecovertensoreldDi(x);i=1;2;andf(x)globallybysimultaneoussmoothinganddatatting,ratherthanestimatingthemfrom( 7{1 )ateachisolatedvoxel,whichleadstoanill-posedproblemandisimpossibletogetasmoothmulti-tensoreld.Second,weappliedthebiGaussianmodeltoallthevoxelsintheeldwhileParkeretal[ 69 ]andTuchetal.[ 70 ]onlyappliedbiGaussianmodeltothevoxelswheretheGaussianmodeltsthedatabadly.SotheyneedtodopreprocessingtodistinguishvoxelsatwhichGaussianmodeltsthedatapoorlyfromthoseatwhichGaussianmodeltsdatawell.Inourapproach,thiskindofvoxelclassicationisnotrequiredandthusavoidstheerrorscomingfromit.Section 7.2 willexplainindetailhowtorecoversmoothmulti-diusiontensoreld. 7{1 ).Thegoalofthissectionistorecoverasmooth 104

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(7{2)ZZ20Z0js0(x)(febuTL1LT1u+(1f)ebuTL2LT2u)s(x;;)j2sindddx 7{2 )wouldbeanill-posedproblem,andthesecondintegralisthenonlineardatadelitytermbasedon( 7{1 ). Weemployagradientdescentschemetosolvetheminimizationproblem( 7{2 ).Ourinitialf;L1;L2arecarefullychosentoavoidgettingcaughtinlocalminima.Forconciseness,weshowthegradient-descentonlyforf;L111asfollowing: 105

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@t=2s0(x)Z20Z0[s0(x)(febuTL1LT1u+(1f)ebuTL2LT2u)s(x;;)](ebuTL1LT1uebuTL2LT2u)sin()dd'+div(rf (7{3) (7{4) InDTIdata,Westinetal[ 76 ]usedtheentiretensorDatlocationx(t)todeterminev(t+1)asDv(t).Theyalsoprovidedaschemewhichdynamicallymodulatestheprincipaleigenvector(PE)e1oftensorDandthetensordeectioncontributionDv(t)totracesteering: whereandareuser-denedweightingfactorsthatvarybetween0and1,ande1andv(t)arenormalizedbeforeused.However,( 7{5 )cannothandlevoxelswithdegenerateanisotropy,sincePEPEdoesnotmakesensethere.Thereforewedenev(t+1)as HereDv(t)isalsonormalizedbeforeused.Normalizationofitwasignoredin[ 76 ],butthisisessentiallynecessaryasforhumanbrainHARDMRIdata,normofDv(t)is 106

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7{6 )toDitogetvi(t+1).Denecorrespondingstepsizeasi=cfi(x(t))FAi(x(t))v(t)vi(t+1)withcaxedconstant,FAi(x(t))thefractionalanisotropy(denedin 1{8 )correspondingtotensorDi.Asweknow,iffiisverycloseto0,channelDicouldbeignored;ifFAiisverylow,anisotropyofDiislow;ifv(t)vi(t+1)islow,thereistoomuchbendingbetweenv(t)andvi(t+1).SobertrackingshouldbeterminatedatchannelDiwhenanyoneoftheabovequantitiesislow.Thiscouldsimplybedonebysettingathresholdtostepsizesothatchannelswithstepsizelessthanthisthresholdareterminated.Thethresholdisastatisticalvalueobtainedthroughalargesizeofexperiments.Thisselfadaptingstepsizedecreasespropagationspeedinregionswithhighcurvatureofbertracksandlowdiusionanisotropy,increasesspeedinregionswithlowcurvatureandhighdiusionanisotropy,anditalsoautomaticallyterminatesbertrackingatchannel(s)withextremelylowstepsize(s). Ourschemethusgeneralizestheusualtensorlinepropagationalgorithmbasedonasingletensor.Wecallitmulti-tensorlinepropagation(MTEND).Thechallengingaspectofthismethodistheestimationofv(t+1)'satnon-gridpoints.Welinearlyinterpolatefand6entriesofD1;D2respectively,v(t+1)'sarethencalculatedusing( 7{6 )basedontheinterpolatedf;Di's. Ourrstexperimentcomparetheproposedmodel( 7{2 )toParkeret.al'smethod([ 69 ])inrecoveringsmoothmulti-tensoreldandthevolumefractionfusinghumanbrainHARDMRIdata.Thedatasetconsistsof33diusionweightedimagesaswellasoneimageintheabsenceofadiusion-sensitizingeldgradient.27evenlyspacedaxialplaneswith128128voxelsineachsliceareobtainedusinga3TMRIscanner 107

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Figure7-1. FAandADCproleoftherstchannel.A)-B)FAmapsofthersttensoreldD1obtainedusingParkeret.al's([ 69 ])andproposedmodel( 7{2 )respectively.C)ImagesofADCproleuTD1uofvoxelsinsidetheselectedregionsinA)andB),thetoponeandthebottomoneareobtainedusingD1calculatedusingproposedmodel( 7{2 )andParker'smethodrespectively. withasingleshotspin-echoEPIsequence.Slicethicknessis3:8mm,gapis0betweentwoconsecutiveslices,repetitiontime(TR)=1000ms,echotime(TE)=85msandb=1000s=mm2,andtheeldofview(FOV)=200mm200mm.Figure 7-1 69 ])andproposedmodel( 7{2 ).ItisclearthatalltheresultsobtainedfromourmodelaremuchmoresmoothandseemmorereasonablethanthoseobtainedbyParker'smethod.Specically,Figure 7-1 7{2 ))givesareasonableFAimagefromwhichweareabletodistinguishvoxelswithhighanisotropy(redregion)fromthatwithlowanisotropy(blueregion),whileinFigure 7-1 7-1 7-1 7{2 )),shapesofADCprolechangesmoothlyfromvoxeltovoxel,andintheregionbelowcorpuscallosum,voxelswhicharemostlylikelytobeofisotropicdiusionhavesphere-shapedADC.ButinthebottomimageofFigure 7-1 108

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Figure7-2. ComparisonbetweenTENDandMTEND.A)TracesrecoveredbyusingMTENDbasedonsimulatedmulti-tensoreld,theblackpointsatthebottomareseeds.B)TracesrecoveredbyusingTENDbasedonsimulatedDTIdata.C)AxialviewofbertrackingresultofthewholebrainbyMTENDalgorithm.D)AxialviewofbertrackingresultsusingMTEND(top),TEND(bottom)algorithm. voxel,especiallyinregionsoutsidethecorpuscallosum.ThusourmodelappearssuperiortoParkeretal.'sintheseaspects. OursecondexperimentistocomparetheMTENDalgorithmandtheTENDalgorithminreconstructingbertracesthatincludebifurcation.Thisisdoneonsimulateddata.Werstsimulatea20203multi-vectoreldshownbybluearrowsinFigure 7-2 A)B),whereeachvoxelwithonearrowownsonlyonetensorthathasthedirectionshownbythisbluearrowastheprincipaleigenvector,whileeachvoxelwithtwoarrowsownstwotensorsthathavethedirectionsshownbythetwobluearrowsastheprincipaleigenvectors.Second,weconstructamulti-tensoreldsothatthemulti-vectoreldisthecorrespondingprincipaleigenvectoreld.RawDTIdataarenallysimulatedbasedonthesimulatedmulti-tensoreldusing( 7{1 )withs0=400;b=1000,f=1atvoxelswithonevector,f=:5atvoxelswithtwovectors,and6u'swhichareuniformlydistributedonasphere.ApplyingMTENDalgorithmtothemulti-tensoreldwhileapplyingTENDalgorithmtothesingle-tensoreldobtainedfromDTIdata,weobtaintheresultsshowninFigure 7-2 A),B)respectively.Thefourblackpointsaretheinitiations,i.e.seedsofthebertracking.NicebifurcationsareobservedinFigure 7-2 A),andthey 109

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Figure7-3. ComparisonbetweenTENDandMTENDatinternalcapsule.A)C)aretrackingresultsusingTEND,MTENDmethodrespectively.B)Anatomicimageofoneslicewithredregionsastheseedsoftracking. takeplaceinvoxelswithtwodiusiontensorsasexpected.Incomparison,nobifurcationisvisualizedfromFigure 7-2 B)andonlytheleft-mostbertracegoesalmostalongthevectoreld,whiletheother3bertracesdonotmakesenseatall.ThisindicatedthattheMTENDalgorithmoutdoestheTENDalgorithminrecoveringberswithbranching. ResultsofexperimentsonhumanbrainHARDMRIdataarealsoshowninFigure 7-2 andFigure 7-3 .TheseshowthatMTENDandTENDworksimilarlyinthecorpuscallosumregion,whereGaussiandiusionisdominant,buttheydierinregionswithnon-Gaussiandiusion.Figure 7-2 C)showsanaxialviewofthewholebrain'sbertraces.Theinitiationsoftrackingaresetatalltheanisotropicvoxelsinthewholebrainvolume.Dierentstrongberbundlesandbranchesareclearlyvisualized,andareconsistentwithknownneuroanatomy. Figure 7-2 D)showsaxialviewoftrackingresultsaroundthecorpuscallosumregionusingtheMTEND(top),andTEND(bottom)algorithms.Trackingresultsareembeddedona2Danatomicimage.Trackingstartsfromasmallportioninsidethecorpuscallosum.Nosignicantdierenceisobservedbecauseinthecorpuscallosumregion,Gaussiandistributionisdominant,andthereforebiGuassianmodelwithf'1andtheGaussianmodelworkjustaswellinrecoveringasingle-tensoreld. 110

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7-3 B))foranothersetofcomparisons.Wesetalltheanisotropicvoxelsinthewholebrainvolumeasseeds,thenapplyMTENDbasedonmulti-tensoreldrecoveredusingmodel( 7{2 )andTENDbasedonsingle-tensoreldrecoveredusingGaussiandiusionmodeltoreconstructbertracesseparately.InMTENDweset=0:9andthethresholdvalueofstepsizetobe0:1;theseareobtainedfromalargesizeofexperiments.OnlythoseberspassingthroughtheROIsareretainedandshowninFigure 7-3 A)C)forTENDandMTENDrespectively.Clearly,MTENDmethodrecoversmorebranchingbersthanTENDmethoddoes.Specically,thishappensinthreedierentlocations.Oneisatthelowerrightpositionandisindicatedbyorangearrows.BunchesofbertraceswithseveralbranchesarenicelyshowninFigure 7-3 C),butdonotappearinFigure 7-3 A).Thesecondlocationisatthemiddleandisindicatedbybluearrows.AstrongbundleconnectingtheleftportionandtherightportionisclearlyvisualizedinFigure 7-3 C)butonlyonebertraceisshowninFigure 7-3 A).Thethirdlocationliesinthemostupperleftposition:Figure 7-3 C)looksthickerandincludesmorebersineachbranchingthanFigure 7-3 A)does.ThemainreasonforthedierenceisthatvoxelsinvolvingbranchinginMTENDmethodarecharacterizedasisotropic,soTENDalgorithmterminatesatthesevoxels. 7{2 )forrecoveringmulti-tensoreldtogetherwithMTENDforreconstructionofwhitematterbertracesworkmoreaccuratelythantheGaussiandiusionmodeltogetherwithTEND. TheproposedmodelassumesthattheprobabilitydensityfunctionofdiusionisoflinearcombinationoftwoGaussians.Thisresultsin13unknownsateachvoxel,andhence 111

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112

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Ageneralowfora3Dclosedsurfacecanbedescribedas @t=(F+H)N;(8{1) whereFisanimagebasedspeedfunction,HisanintrinsicspeeddependingonthecurvatureofthesurfaceS,Nisthenormalofthesurfaceandtistime. Tosolvethistimedependentpartialdierentialequation,weusethelevelsetmethodintroducedbyOsherandSethian(1988),wheretheevolvingsurfaceisconsideredas0levelsetofafunctionofonedimensionhigher().Bydoingthis,anumericallystablealgorithmthateasilyhandlestopologychangesoftheevolvingsurfaceisobtained.Finally,( 8{1 )becomes @t=(F+H)jrj(8{2) ThekeypointofanewmodelistodesignanecientandsensefulexternalforceorspeedfunctionF. 115 ]introducedasimilaritybasedfrontpropagationalgorithm.Theydened 113

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Figure8-1. Illustrationofsurfacenormalandevolvingcurve.A)Choiceofadjacentvoxelswithrespecttothenormalofthesurface,courtesyof[ 115 ].B)Evolvingcurve(black)superimposedinsideasemi-circleshaped2Dtensoreld. wherethenotationisasfollows.Diisthediusiontensorinthecurrentvoxel.Forp=1;2,Dipisthediusiontensorinthegridelementfoundbyfollowingthenormaltothesurfacepvoxelsbackwardsfromtheoriginalvoxeli,asillustratedinFigure 8-1 A),andthenselectingthenearestneighbor.NTSP(Di;Di1)[ 116 ]istheNormalizedTensorScalarProductdenedas withDi:Di1=Trace(DiDi1),NTSP(Di;Di2)isdenedsimilarly.Thefundamentalassumptionofthesegmentationtechniquein[ 115 ]isthatadjacentvoxelsinatracthavesimilardiusionproperty.Theproposedmodelmaintainsthisassumption,butthegoalistoobtainafasterandmoreaccurategeometricow. Figure 8-1 D)showsanexampleof2Dtensoreldtodemonstratemotivationoftheproposedmodel.InFigure 8-1 D)theblackellipsedepictstheevolvingcurve,redarrowsandgreenarrowsshownormaldirectionsatcorrespondinglocations.Intuitively,evolutionspeedalonggreenarrowsshouldbelargerthanthatalongtheredarrows,thisisbecausetheevolvingcontourisclosetotheboundaryatlocationswithredarrows,whileprettyfarawayfromtheboundaryatlocationswithredarrows.Mathematically,evolutionshouldbefasteratlocationwherethenormaldirectionismorecoincidentwiththetensoreld. 114

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8{3 ) whereCOINisameasureofcoincidenceofnormaldirectiontotheevolvingcurve(surfacein3D)andthetensoreld.Wehavetriedtwocandidates,therstofwhichis: withPEtheprincipaleigenvectorofthecurrentdiusiontensorDi,andNisthenormaldirection.TheabsolutevalueistoguaranteethepositivityofCOINmeasure. Thesecondcandidateis withFAthefractionalanisotropyvalueoftensorDi.ThedenitionofCOIN1isquiteintuitive:ifnormaldirectioniscoincidentwiththeprincipledirectionoftensor,theevolutionspeedishigher.ThedenitionofCOIN2ismotivatedbyMarianaLazarel.al[ 117 ].COIN2islesssensitivetoimagenoiseandexperimentalresultsshowthatitismoreecientthanCOIN1. Notethatindenition( 8{5 ),similarityisdenedusingDi+1andDi+2,whichareneighborsofDialongthepositivenormaldirectioninsteadofnegativenormaldirectionusedin( 8{3 ).Thisistoavoidovershooting. Weterminateevolutionofthecurve/surfaceatlocationswherethesimilaritymeasureNTSPislessthanathresholdorthenormofthegradientofNTSPislessthanathreshold.Ourexperimentsindicatethatthesetwocriteriacombinedtogethercanaccuratelycatchtheboundary.In[ 115 ]onlytherstoneisused,butthisdoesnotworkforsegmentationofavector/tensoreldcomposedofhomogeneousregions.Inthiscase,thesimilaritymapofthevector/tensoreldisapiecewiseconstantfunction,andtheboundarybetweenonehomogeneousregionandanotherissignaledbyahighgradient 115

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Segmentationofa2-Dtensoreldusingtwomethods.Redcontour:Segmentationresultoftheproposedmodel;Bluecontour:segmentationresultofL.Jonasson'smodelin[ 115 ];Greencurve:initialcontour. ofthesimilaritymeasure;whileathresholdofsimilarityalonecouldnottellwheretheboundaryis. 8-2 comparessegmentationqualityandspeedofproposedmethodandmethodin[ 115 ].Theblackarrowsshowtheprincipaleigenvectorsofa2Ddiusiontensoreld.Thegreencurveistheinitialcurve,andtheblueandtheredoneshowtheresultofmodelin[ 115 ]andtheproposedmodel,respectively.Intheproposedmodel,weset=10,itcosts25iterationsand53.45secondsinacomputerwithPIV2.8GHZCPUand2GRAMusingMatlabscriptcode,whileformodelin[ 115 ],itcosts50iterationsand105.55seconds.Itshowsthattheproposedmodelismorethantwiceasfastandtheresultisevenmoreaccurate. 116

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Figure8-3. Comparisonofcorpuscallosumsegmentationresultsfromtwomodels.A)-B)Segmentationofcorpuscallosumobtainedbyusingtheproposedmodelandtheonein[ 115 ]respectively.) 8-3 showcomparisonofsegmentationofcorpuscallosumobtainedusingtheproposedmodelandthemodelin[ 115 ].Figure 8-3 B)showstwosetsofsegmentationresultsobtainedfromtwothresholdvaluesforNTSP.Thetoponecorrespondstoahigherthreshold,whilethebottomonecorrespondstoalowerthreshold.Itisclearthatwhenahigherthresholdischosen,onlypartofthecorpuscallosumissegmented,butwhenalowerthresholdischosen,regionsotherthancorpuscallosumarealsosegmented.Incomparison,theproposedmodel(Figure 8-3 A))capturesmoreaccuratecorpuscallosumdetails. 115 ]. 117

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[1] M.Kass,A.Witkin,andD.Terzopoulos,\Snakes:Activecontourmodels,"InternationalJournalofComputerVision,vol.1,no.4,pp.321{331,1988. [2] V.Caselles,F.Catte,T.Coll,andF.Dibos,\Ageometricmodelforactivecontoursinimageprocessing,"NumerischeMathematik,vol.66,pp.1{31,1993. [3] R.Malladi,J.Sethian,andB.Vemuri,\Shapemodelingwithfrontpropagation:Alevelsetapproach,"IEEETrans.PatternAnal.MachineIntell.,vol.17,pp.158{175,1995. [4] V.Caselles,R.Kimmel.,andG.Sapiro,\Ongeodesicactivecontours,"Intel.JournalofComputerVision,vol.22,no.1,pp.61{79,1997. [5] S.Kichenassamy,A.Kumar,P.Olver,A.Tannenbaum,andA.J.Yezzi,\Gradientowsandgeometricactivecontourmodels,"Proc.ICCV'95,pp.810{815,1995. [6] A.Yezzi,S.Kichenassamy,A.Kumar,P.J.Olver,andA.Tannenbaum,\Ageometricsnakemodelforsegmentationofmedicalimagery,"IEEETrans.MedicalImaging,vol.16,pp.199{209,1997. [7] S.C.ZhuandA.Yuille,\Regioncompetition:unifyingsnakes,regiongrowing,andBayes/MDLformultibandimagesegmentation,"IEEEPAMI,vol.18,pp.884{90,1996. [8] D.MumfordandJ.Shah,\Optimalapproximationbypiecewisesmoothfunctionsandassociatedvariationalproblems,"Comm.PureAppl.Math.,vol.42,pp.557{685,1989. [9] D.Cremers,F.Tischhuser,J.Weickert,andC.Schnrr,\Diusion-snakes:IntroducingstatisticalshapeknowledgeintotheMumford-Shahfunctional,"In-ternationalJournalofComputerVision,vol.50,no.3,pp.295{315,2002. [10] A.Tsai,A.YezziJr.,,andA.S.Willsky,\CurveevolutionimplementationoftheMumford-Shahfunctionalforimagesegmentation,denoising,interpolation,andmagnication,"IEEETransactionsonImageProcessing,vol.10,no.8,pp.1169{1186,2001. [11] A.Chakraborty,H.Staib,andJ.Duncan,\Deformableboundaryndinginmedicalimagesbyintegratinggradientandregioninformation,"IEEETransactionsonMedicalImaging,vol.15,no.6,pp.859{870,1996. [12] N.ParagiosandR.Deriche,\Geodesicactiveregionsforsupervisedtexturesegmentation,"ICCV,Cofu,Greece,pp.926{932,1999. [13] N.Paragios,\Geodesicactiveregionsandlevelsetmethods:contributionsandapplicationsinarticialvision,"Ph.D.thesis,SchoolofComputerEngineering,UniversityofNice/SophiaAntipolis,2000. 118

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[14] N.Paragios,\AvariationalapproachforthesegmentationoftheleftventricleinMRcardiacimages,"Proceedings1stIEEEWorkshoponVariationalandLevelSetmethodsinComputerVision,Vancouver,B.C.,Canada,pp.153{160,13July2001. [15] L.Ambrosio,N.Fusco,andD.Pallara,\Partialregularityoffreediscontinuitysets,"Ann.ScuolaNorm.Sup.PisaCl.Sci.,vol.24,pp.39{62,1997. [16] T.F.ChanandL.A.Vese,\Activecontourswithoutedges,"IEEETrans.ImageProcessing,vol.10,no.2,pp.266{277,2001. [17] L.A.VeseandT.F.Chan,\Amultiphaselevelsetframeworkforimagesegmentationusingthemumfordandshandmodel,"Int.JournalofComputerVision,vol.50(3),pp.271{293,2002. [18] S.OsherandJ.A.Sethian,\Frontspropagatingwithcurvature-dependentspeed:algorithmbasedonHamilton-Jacobiformulation,"JournalofComputationalPhysics,vol.70,pp.12{49,1988. [19] A.Yezzi,S.Soatto,A.Tsai,andA.Willsky,\TheMumford-Shahfunctional:Fromsegmentationtostereo,"MathematicsandMultimedia,2002. [20] T.F.ChanandS.Esedoglu,\AspectsoftotalvariationregularizedL1functionapproximation,"SIAMJ.Appl.Math.,vol.65(5),pp.1817{1837,2005. [21] T.F.Chan,S.Esedoglu,andM.Nikolova,\Algorithmsforndingglobalminimizersofimagesegmentationanddenoisingmodels,"SIAMJ.Appl.Math.,vol.66(5),pp.1632{1648,2006. [22] J.Lie,M.Lysaker,andX.C.Tai,\Avariantofthelevelsetmethodandapplicationstoimagesegmentation,"Math.Comp.,vol.75,pp.1155{1174,2006. [23] X.C.TaiandC.H.Yao,\ImagesegmentationbypiecewiseconstantMumford-Shahwithoutestimatingtheconstants,"J.Comput.Math.,vol.24(3),pp.435{443,2006. [24] S.GaoandT.D.Bui,\Anewimagesegmentationandsmoothingmodel,"Proc.ofInt.Sym.onBiomed.Img.,pp.137{140,2004. [25] TonyChan,BertaSandberg,andMarkMoelich,\Somerecentdevelopmentsinvariationalimagesegmentation,"UCLACAMReports,,no.06-52,Sep.2006. [26] M.RoussonandR.Deriche,\Avariationalframeworkforactiveandadaptivesegmentationofvectorvaluesimages,"Proceedingsofworkshoponmotionandvideocomputing,pp.56{61,Dec.2002. [27] J.Kim,J.W.FisherIII,A.Yezzi,M.Cetin,andA.S.Willsky,\Anonparamtericstatisticalmethodforimagesegmentationusinginformationtheoryandcurveevolution,"IEEETransactionsonImageProcessing,vol.14:10,pp.1482{1502,Oct.2005.

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[109] Y.Chen,W.Guo,Q.Zeng,X.Yan,andY.Liu,\ApparentdiusioncoecientapproximationanddiusionanisotropycharacterizationinDWI,"LectureNotesinComputerScience,Proceedingsofinternationalconferenceoninformationprocessinginmedicalimaging,pp.246{257,2005. [110] M.Rao,Y.Chen,B.C.Vemuri,andF.Wang,\Cumulativeresidualentropy:Anewmeasureofinformation,"IEEETrans.onInfo.Theory,vol.50,pp.1220{1228,2004. [111] Y.Chen,W.Guo,Q.Zeng,X.Yan,F.Huang,H.Zhang,G.He,B.CVemuri,andY.Liu,\Estimation,smoothing,andcharaterizationofapparentdiusioncoecientprolesfromhighangularresolutionDWI,"Proc.ofCVPR,pp.588{593,2004. [112] M.Rao,\Moreonanewconceptofentropyandinformation,"Jour.ofTheo.Prob.,vol.18(4),pp.967{981,2004. [113] D.StrongandT.Chan,\Spatialandscaleadaptivetotalvariationbasedregularizationandanisotropicdiusioninimageprocessing,"UCLA-CAMRe-port,vol.46,1996. [114] W.Guo,Q.Zeng,Y.Chen,andY.Liu,\Reconstructwhitematterbertracesusingmulti-tensordeectioninDWI,"ProceedingsofInternationalSymposiumonBiomedicalImage,pp.69{72,2006. [115] L.Jonasoon,X.Bresson,P.Hagmann,O.Cuisenaire,R.Meuli,andJ.Thiran,\WhitematterbertracesegmentationinDT-MRIusinggeometricows,"Med.Img.Ana.,vol.9,pp.223{236,2005. [116] D.Alexander,J.Gee,andR.Bajcsy,\Similaritymeasuresformatchingdiusiontensorimages,"Proc.BMCV,pp.93{102,1999. [117] M.Lazar,D.Weinstein,andJ.Tsurudaet.al,\Whitemattertractographyusingdiusiontensordeection,"HumanBrainMapping,vol.18,pp.306{321,2003.

PAGE 127

WeihongGuowasborninHeiFei,AnhuiProvince,P.R.China.ShegotherbachelorofsciencedegreefromCentralUniversityforNationalities,P.R.China,in1999.ShethenwasadmittedtoBeijingNormalUniversitygraduateprogramwithouttakinganadmissionexam.SheearnedherDoctorofPhilosophydegreeinAppliedMathematicsandMasterofStatisticsdegreefromUniversityofFloridainMay2007.Herresearchinterestsincludemathematicalmodelingandalgorithmdevelopinginmedicalimageprocessing,statisticalimageprocessing,medicalimageanalysisandnumericalpartialdierentialequation. 127


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Title: Medical Image Segmentation and Diffusion Weighted Magnetic Resonance Image Analysis
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MEDICAL IMAGE SEGMENTATION AND
DIFFUSION WEIGHTED MAGNETIC RESONANCE IMAGE ANALYSIS



















By

WEIHONG GUO


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

2007
































2007

by

Weihong Guo



































To my family









ACKNOWLEDGMENTS

I would first and foremost like to express my deep gratitude to my advisor professor

Yunmei C'! i, for everything she has done for me during my doctoral study. This

dissertation would not be possible without her. She provided invaluable advice on research

and life. Dr. C'!, i1 introduced me to the field of medical imaging and ahv--,b helped and

encouraged me. I have been very lucky to work with her. Her enthusiasm about research

stimulated my interests, her insight and experience have guided me through my research. I

would also like to thank my other excellent committee members, Dr. David Groisser, Dr.

Yijun Liu, Dr. Murali Rao, Dr. David Wilson for providing their advice and discussion. I

also appreciate Dr. Yijun Liu, Dr. Guojun He from Brain Institute and Dr. Zhizhou Wang

for happy cooperation. They provided a lot of invaluable research motivation and data.

I thank Feng Huang, one of my peers, for helping me with numerical implementation

in my early stage research. Without his help, I could not have my scientific computation

ability. I also thank Hemant Tagare from Yale University, Rachid Deriche from INRIA,

Sheshadri Thiruvenkadam(who is now at UCLA) for excellent comments and discussion on

my work.

I never would have been able to have finished the dissertation were it not for the

unwavering love and support of my family: my mum Zhongying, my husband Qingguo, my

brother Weidong, my sister Shuangxia. Their support and encouragement were my source

of strength. I want to especially thank my husband who teaches me the nature of true

love. He is an extremely supportive man, I ahv--, appreciate his enthusiasm for life and

research. I am so grateful about my 10-month-old daughter Tienna, who teaches me how

wonderful life is by her sunny smiles.

I want to thank the editor in the Editorial office, Dr. David Groisser and Dr. David

Wilson for their careful and thorough corrections to my thesis. I would also like to thank

the staff and the faculty members who have been very patient and helpful during my

study here.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 7

LIST OF FIGURES .................................... 8

ABSTRACT . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION ...................... .......... 12

1.1 Background . . . . . . . .. 12
1.1.1 Medical Image Segmentation .......... ............ 12
1.1.2 Diffusion Weighted Image analysis ....... ........... 19
1.1.3 Neuron Fiber Tracts Reconstruction based on Smooth Tensor Field 23
1.2 Contributions .................. ............... .. 24

2 USING PRIOR SHAPE AND POINTS IN MEDICAL IMAGE SEGMENTATION 29

2.1 Description of the proposed model .................. .. 29
2.2 Numerical scheme ........ . . .............. 33
2.3 Validation and Application to Echo Cardiovascular Ultrasound Images .34
2.4 Conclusion . .. . . . ... . . .. 37

3 USING NONPARAMETRIC DENSITY ESTIMATION TO SMOOTH AND
SEGMENT IMAGES SIMULTANEOUSLY ......... . . 41

3.1 Proposed M odel .................. ............ .. .. 41
3.2 Numerical Implementation .................. ........ .. 45
3.3 Validation and Application to T1 Magnetic Resonance Image . ... 49
3.4 An Existence Theorem for the Model ................... ... .. 54
3.5 Conclusion . .. . . . .... . . .. 59

4 ESTIMATION,SMOOTHING AND CHARACTERIZATION OF APPARENT
DIFFUSION COEFFICIENT .................. ......... .. 61

4.1 Introduction .................. ................ .. 61
4.2 Model Description .................. ............ .. 62
4.3 C!i i.terization of anisotropy ......... ........... .. 68
4.4 Numerical Implementation Issues ..... . . .... 69
4.5 Validation and Application to Diffusion Weighted Images(DWI) . 73
4.5.1 Analysis of simulated data .................. .. 73
4.5.2 Analysis of human MRI data .................. ..... 74
4.6 An Existence Theorem for the Model ................ .. .. 77
4.7 Conclusion . .. . . . .... . . .. 82









5 ESTIMATION,SMOOTHING AND CHARACTERIZATION OF APPARENT
DIFFUSION COEFFICIENT(A SECOND APPROACH) ....... ...... 87

5.1 Introduction .............. .. ........... ...... 87
5.2 New Approximation Model for ADC Profiles ............. 88
5.3 Use of CRE to C'!I ii i.terize Anisotropy ......... ........... 92
5.4 Conclusion . .. . . . . . . .. 97

6 RECONSTRUCTION OF INTRA-VOXEL STRUCTURE FROM DIFFUSION
WEIGHTED IMAGES ................... ......... 98

6.1 Determination of Fiber Directions ......... .......... .. 98
6.2 Validation and Application to HARD Weighted Images ........... 99
6.3 Conclusion ..................... ............ 102

7 RECONSTRUCT OF WHITE MATTER FIBER TRACES USING MULTI-TENSOR
DEFLECTION IN DWI ................... ........ 104

7.1 Introduction .................. .......... ...... 104
7.2 Recovery of Multi-Tensor Field in HARD MRI ....... ....... 104
7.3 White Matter Fiber Tractography ......... ........ ..... 106
7.4 Experimental Results .............................. 107
7.5 Conclusion ..................... ............ 111

8 FAST SEGMENTATION OF WHITE MATTER FIBER TRACTS BASED
ON GEOMETRIC FLOWS ................... ....... 113

8.1 Introduction ...................... ........... 113
8.2 Model ...................... .............. 113
8.3 Experimental Results ................... ........ 116
8.3.1 Synthetic results ................... ....... 116
8.3.2 Human brain results .......... ............... 117
8.4 Conclusion ..................... ............ 117

REFERENCES ....................................... 118

BIOGRAPHICAL SKETCH .................. ............. 127









LIST OF TABLES
Table page

3-1 Segmentation Accuracy .................. ............. .. 47

4-1 List of So and Al,i 's for two regions .................. ...... .. 73











Figu

2-1

2-2

2-3

3-1

3-2

3-3

3-4

3-5

3-6


4-1

4-2

4-3

4-4

4-5

5-1

5-2

5-3

5-4

5-5

6-1

6-2

6-3

7-1

7-2


Comparing shapes of d . ...........

Comparison of A20 .. ...............

Images of FA and R2 .. ...............

Zoomed FA and A20 . .............

Classification of voxels based on d . .....

Comparison of ADC's obtained from two models .

Images of R2 by different models . ......

An example for shapes of ADC for isotropic, one-fiber

Different measures . ..............

C( i .:terization . . . . .

Fiber direction color map . ..........

Shape of d with orientations . .

Fiber direction field . .............

FA image of the first channel . ........

Comparison between TEND and MTEND . .


and two-fiber


LIST OF FIGURES
re

Comparing segmentation results of a synthetic image . ......

Compare segmentation results of one cardiac ultrasound image . .

Compare segmentation results of another cardiac ultrasound image. .

Compare results of model(3-5) based on two choices of r(x) . .

Segmentation results of a clean image and that of a noisy version .

Segmentation result of a noisy T1 human brain image . .....

Compare three models in segmenting images with lower and higher level

Comparison of graphs of SA obtained from three models . .

Compare segmentation and smoothing results between two models with fi
and adaptive radii ...................... . .....


voxels .


7-3 Comparison between MTEND and TEND at internal capsule


of noises



xed


page

36

38

39

46

48

49

S51

52


53

77

83

84

85

86

91

93

94

95

96

101

102

103

108

109









8-1 Illustration of surface normal and evolving curve ................. .114

8-2 Segmentation of a 2-D tensor field using two methods .............. .116

8-3 Comparison of corpus callosum segmentation results from two models ..... ..117









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

MEDICAL IMAGE SEGMENTATION AND
DIFFUSION WEIGHTED MAGNETIC RESONANCE IMAGE ANALYSIS

By

Weihong Guo

May 2007

C'!I i': Yunmei Chen
Major: Mathematics

Medical image segmentation pl 'i, an important role in diagnosis, surgical pl'1 i 1i:_.

navigation, and various medical evaluations. Medical images are frequently corrupted

by high levels of noise, signal dropout and poor contrast along boundaries. Sometimes,

their intensity might have multi-modal distribution. In this dissertation, I will present

one method to segment images that are corrupted by noise and some dropout. In the

model presented, prior points together with prior shape information are incorporated

into a joint segmentation and registration model in both a variational framework and in

level set formulation. This technique is applied to segment cardiac ultrasound images.

A second model, which is based on applying non-parametric density approximation to

simultaneously segment and smooth noisy medical images without adding extra smoothing

terms, is presented. My goal is to develop a powerful and robust algorithm to locate

objects with interiors having a complex multi-modal intensity distribution and/or high

noise level. The model was applied to the problem of segmenting T1 weighted magnetic

resonance images.

Diffusion weighted images render non-invasive in vivo information about how water

diffuses into a 3D intricate representation of tissues. My work provides histological

and anatomical information about tissue structure, composition, architecture, and

organization. I have proposed several models to reconstruct human brain white matter

fiber tracts, to recover intra-voxel structure, to classify intra-voxel diffusion, to estimate,









smooth and characterize apparent diffusion coefficient profiles. A geometric flow is

designed to segment the main core of white matter fiber tracts in diffusion tensor images.









CHAPTER 1
INTRODUCTION

1.1 Background

1.1.1 Medical Image Segmentation

Segmentation of anatomical structures from medical images has very important

applications in diagnosis, surgical pl1 -,,,i,..r navigation, and medical image analysis.

A multitude of segmentation algorithms have been proposed to tackle these types of

problems. They might be classified as edge-based, region-based, or a combination of these

two. The edge-based methods (e.g. [1-6]) rely on the information of the edges, such as

high magnitude of image gradient. The region-based methods (e.g. [7-10]) make use of

homogeneity on the statistics of the regions being segmented. The algorithm developed

in [11] integrates gradient and region information within a deformable boundary finding

framework. The Geodesic Active Region models proposed in [12-14] integrate the edge

and region-based segmentation methods into a variational approach.

The analysis of medical images is frequently complicated by noise, dropout, confusing

anatomical structures, motion, poor contrast along boundaries, non-uniformity of regional

intensities and multi-modal intensity distribution. It is hard to find one model to segment

images with all the problems mentioned above. We will first provide some general

literature review on image segmentation models, then focus on models dealing with

images that are corrupted by noise and having multi-modal distribution. Finally, we

review some existing models that segment images corrupted by noise, dropout, and poor

contrast along boundaries.

Let Io : -- R be a given bounded image-function defined on an open and bounded

region Q which is assumed to be a subset of R2 for the purpose of illustration. But any

dimensional case could be similarly considered. The segmentation problem is to find a

decomposition fi's of Q and an optimal piecewise smooth approximation I of lo such that

I varies smoothly within each i, and rapidly or discontinuously across the boundaries of

fi. Let C be a close smooth curve formed by boundaries between fi's, and let the length









of C be ICI To solve this problem, Mumford-'! ,i!,I\!S) [8] proposed to minimize the

following functional:



( I- Io)2dX + a VI|2dX + 3C (1-1)

The first and second parts are called data fidelity and smoothness terms, respectively.

If the parameter a goes to oo, then the approximation I will be forced to be constant Ci

in each Qi. Piecewise smooth MS is then reduced to piecewise constant MS:



S (C o)2dX + PIC (1-2)

While the functional is elegant itself, it is difficult in practice to find a solution as

the functional is non-convex. L. Ambrosio et al.[15] approximated piecewise smooth MS

via F-convergence. A. Tsai et al.[10] solved piecewise smooth MS using parametric curve

evolution directly.

C'!i i, and Vese [16, 17] implemented both piecewise constant and piecewise smooth

MS using a geometric implicit framework-level set which was invented and continuously

advanced by S. Osher and J. Sethian[18]. The level set method is used to represent

C implicitly and to express each subregion. The main advantage of the level set

representation is that topological changes, such as merging and pinching off of contours

can be captured naturally through smooth changes to the level set function.

For illustration conciseness, we only consider the 2-phase case: I is composed of an

object and background (i.e. i = 1, 2). For an image with more than 2 phases, the model

could easily be extended by using more level set functions. The curve C is represented by

the zero level set of a Lipschitz function Q by C = {x|l(x) = 0}. The object(A) and the

background(Q\A) are represented by {xl((x) > 0} and {xl((x) < 0}, respectively. A

Heaviside function defined as H(O) = 1 when Q > 0 and 0 elsewhere is used to distinguish

A and Q\A. Thus, A corresponds to the region where H(O) > 0, while f2 corresponds to

the region where 1 H(O) > 0. Let 6 be the Delta function.









C'!h i,-Vese model, the level set formulation of model (1 2)(2-phase case) is to

minimize the functional:



j H(0)(Ci-Io)2dx + f(1- H())(C2 Io)2+dx /3 DH(Q)|dx, (1-3)

where the third term is the total variation of H(0).

Due to its simplicity and robustness, the ('!h ,i-Vese model(1-3) has become quite

popular and has been adopted in many applications. It has been expanded to segmenting

vector-valued images, texture images etc. In [19], A. Yezzi et al. also use the level set

framework to segment images.

The variational formulation in the ('!: ,I-Vese model is non-convex and a typical

gradient-descent implementation is not guaranteed to converge to the global minimum

and thus can get stuck in local minima. One trick [16, 17] is to set efficient initials. One

example is a large number of small close contours uniformly distributed in the image. In

this way, contours are initiated everywhere in the image such that chance of capturing the

global minimum is enhanced. ('C! io and Esedoglu et al. [20, 21] provide another novel and

fundamentally different approach. It basically convexifies the objective function by taking

advantage of the implicit geometric properties of the variational models.

The traditional level set functions are Lipschitz continuous, and a reinitialization

procedure is sometimes needed to prevent the level set function to from becoming too flat.

Recently, Lie et al. [22] represent interfaces using piecewise constant level set functions, in

which each level set function can only take two values at convergence, e.g., the function Q

can only equal 1 or -1. Some of the properties of standard level set methods are preserved

in the proposed model, while others are not. It is claimed that the new method provides

as good results as methods using continuous level set functions. The Reinitialization

procedure is removed in the new method. Tai et al. [23] also introduce a model that only

requires minimizing the ('!h ,i-Vese functional(1-3) with respect to the level set functions,

without estimating the constants Ci.









There are several v- -v- to handle images with multiple phases. One widely used idea

is to use log2 n level sets to distinguish n phases in an image [17], but this can lead to

difficulty in setting initials. Gao et al. [24] recursively use one level set in a hierarchical

way: first segment into two regions, then segment each region into two new regions, and so

on. See [25] for other related recent developments.

A generalization to piecewise constant MS is that instead of approximating I by a

constant Ci in each region Qi, one can generally approximate the intensity in each region

Qi by a probabilistic model P(I(X)|AX ) with a parameter vector A\. By B i, -i i, analysis,

we have the posterior distribution P(AiI (X)) oc P(I(X)|A) P(Ai). When choose the prior

distribution P(A,) to be uniform distribution, P(AII(X)) o P(I(X)|A,).

The first term in (1-2) is then replaced by so called description cost and (1-2)

becomes:



log P(AjLIi(X))dX + 3|C (1-4)

Zhu et al.[7] and Rousson et al. [26] chose P(IJ(X)IAi) to be a Gaussian distribution

and permit for each region Qi not only a different mean Ci but also a different variance ai:


1 -( (X) C- ')2
P( I(X) C, 1a) = exp 2 (1-5)

We would like to mention that when all the ji's are the same, model(1-5) is

equivalent with model (1-2). We call model(1-5) a global Gaussian probability density

function(pdf) based method since it assumes all the pixels in one region Qi share the same

mean and variance.

Model (1-4) has its restriction in real application since it is based on a specific

assumption of the intensity distribution, while usually we do not know whether this

assumption is reasonable. Especially, when there exists high level of noise and/or

complex multi-modal intensity function, one single global Gaussian pdf or any other single

parametric distribution is not enough. J. Kim et al.[27] presented an information-theoretic









approach to image segmentation. They cast the segmentation problem as the maximization

of the mutual information between the region labels and the image pixel intensities,

subject to a constraint on the total length of the region boundaries. They first assume

that the probability densities associated with the image pixel intensities with each region

are completely unknown priors, then formulate the problem based on nonparametric

density estimates. Due to the nonparametric structure, the model did not require the

image regions to have a particular type of probability distribution and did not require

the extraction and use of a particular statistic. Huang et al. [28] used nonparametric

kernel-based approximation of the intensity probability density function to capture

texture, this texture information in then incorporated into a variational framework for

image segmentation. Abd-Almagee et al. [29] introduced a general framework for driving

an active deformable model using nonparametric estimation of the pdf. A Parzen window

approach with a Gaussian kernel function was used as a parameter-free estimator for the

pdf's for both the target to be segmented and the surrounding background. In Chapter 3,

a new model based nonparametric density estimation is provided to segment and smooth

images simultaneously.

For images that have noise, dropout, and poor contrast along boundaries, it is not

enough to use image information itself to obtain the desired results. Recently, various

approaches, including deterministic setting and probabilistic context, have been developed

to use prior shape information in image segmentation (see [30-39]). A survey of methods,

which incorporate prior knowledge into deformable models in medical image analysis is

provided in [40]. The approaches presented in the papers [8, 9, 41-43] are closely related

to our current work since they incorporate the statistical shape knowledge into either

edge-based or region-based segmentation techniques. Leventon, Grimson and Faugeras [41]

extended the technique of geometric active contours by incorporating shape information

into the evolution process. A segmentation result was first obtained by the curve evolution

driven by a force depending on image gradient and curvature. Later, the shape prior was









used to make a correction by maximizing a posterior estimate of shape and pose. C'!. i1 et

al. [42] modified the techniques discussed by Leventon et al. [41] so that their variational

method incorporates both high image gradients and shape information into the energy

term of a geodesic active contour model. The shape term can also be used to recover a

similarity transformation that maps the evolving interface to the prior shape. Cremers,

Schnirr and Weickert [9] incorporated statistical shape knowledge into the Mumford-Shah

segmentation scheme [8] by minimizing a functional that includes the Mumford-Shah

energy and the shape energy corresponding to the Gaussian probability. Recently, in

[43] Rousson and Paragios introduced an energy functional that constrains the level set

representations to follow a shape global consistency while preserving the ability to capture

local deformation.

The experimental results have shown that all these models have their own strengths,

and provide promising results for particular applications. However, due to the accuracy

and efficiency requirement for medical image analysis, and the complexity of medical

images, as well as the variability of the shapes of anatomic structures, the question of how

best to use shape prior to get a better segmentation will remain challenging. One of the

most difficult problems is to determine local shape variations from the prior shape. One

solution may be the use of nonrigid registration to assist segmentation [44, 45], but this

approach requires reliable region or edge information in the image to estimate the velocity

field. This technique is also computationally expensive.

The goal of the method presented in chapter 2 is to find an improved way to locate

boundaries when they may vary widely from a prior shape and when they are to be

detected in an image which may have significant signal loss. In particular, we are

motivated by the problem of detecting the boundary of the left ventricle of the heart

in echocardiographic image sequences. Cardiac ultrasound images are plagued with

noise, signal dropout, and confusing intracavitary structures. Moreover, the shape of the

boundary of the myocardium varies extensively from one patient image to another, and is









not equivalent to the ,i., i ,; '" of a set of training shapes. More importantly, for patients

with cardiac disease the shape distortion of the myocardium from the i i. in I shape"

cannot be ignored.

To get a desirable segmentation in some cardiac ultrasound images more prior

knowledge than the expected shape is needed. For instance, in [46] Dias and Leitao used

temporal information from time sequence of images to assist the determination of the

inner and outer contours in the areas of low contrast in echocardiographic images. In this

note we consider the cases in which users have the knowledge about the locations of a

few points on the boundary of the object of interest. We will use this information as a

constraint in addition to shape constraint to control the evolution of active contours. Our

basic idea is to extend the segmentation algorithm developed in [42] by incorporating the

information on the location of a few "key" points in addition to shape prior into geometric

active contours.

The idea of matching nonequivalent shapes by the combination of a rigid transformation

and a point-wise local deformation developed in the papers [43, 47, 48] will be applied to

our formulation. In [48] Soatto and Yezzi view a general deformation as the composition

of a finite dimensional group action (e.g. rigid or affine transformation) and a local

deformation, and introduced a notion of -! pe ox, i ;,' as the entity that separates

a group action from a deformation. In [47] Paragios, Rousson and Ramesh proposed a

variational framework for global as well as local shape registration. Their optimization

criterion includes a global (rigid, affine) transformation together with local pixel-wise

deformation. A similar idea was also used by Paragios et al. [43] to define shape prior

models in terms of level set representations. These ideas can be used to simultaneously

approximate, register, and track nonequivalent shapes as they move and deform through

time. However, the question of how to determine the local deformation has not been

considered in these previous works.









Paragios [49] introduced a framework for user-interaction within the propagation

of curves using level set representations. The user-interaction term is introduced in

the form of an evolving shape prior that transforms the user-edits to level set based

propagation constraints. The work in [49] and in my study are based on the idea that the

user interactive edits can be used as a constraint to correct local discrepancies. However,

the formulations of the constraint in his work and my work are different( See C'! Ilpter 2

for details).

In chapter 2 we first briefly review geometric active contours and the model developed

in [42]. We then propose the new model for incorporating both prior shape and points in

active contours.

1.1.2 Diffusion Weighted Image analysis

Diffusion-weighted magnetic resonance imaging( I RI) (DW-MRI, shorten as DWI)

adds to conventional MRI the capability of measuring the random motion of water

molecules in tissue, referred as diffusion. The motion of water molecules can be free or

restricted depending on the tissue structures. In tissues containing a large number of

fibers (such as cardiac muscle and brain white matter) water diffusion is fastest along the

direction that a fiber is 1p."iil i_ but slowest in the direction perpendicular to it. This

characteristic of the diffusion is termed as anisotropy. In tissues that contain few fibers

water diffuses isotropically. DWI renders non-invasively such complex in vivo information

about how water diffuses into a 3D intricate representation of tissues, and provides

profound histological and anatomical information about tissue structure, composition,

architecture, and organization. ('!Ci i;, in these tissue properties can often be correlated

with processes that occur in development, degeneration, disease, and aging, so this

technique has become more and more widely applied ([50-53]). One specific example is

that Gupta et al [54] use DWI to search and quantify the extent of abnormality beyond

the obvious lesions seen on the T2 and fluid-attenuation inversion recovery (FLAIR)

magnetic resonance images in patients with chronic traumatic brain injury with and









without epilepsy. Another example is that Roh et al. [55] uses DWI to detect acute

multiple brain infarcts. They use the fact that DWI is superior to conventional MRI in

identification of small new ischemic lesions and discrimination of recent infarcts from old

ones.

The diffusion of water molecules in tissues over a time interval t can be described by

a probability density function pt(r), which gives the probability that a water molecule will

diffusion by r. Since pt(r) is largest in the directions of least hindrance to diffusion and

smaller in other directions, the information about pt(r) reveals fiber orientations and leads

to meaningful inferences about the microstructure of tissues.

The density function pt(r) is related to DWI echo signal s(q) via a Fourier transformation

(FT) with respect to q, which represents diffusion sensitizing gradient, by


s(q) so I (r)e-iqdr, (16)

where so is MRI signal in the absence of any gradient. Therefore, pt(r) can be estimated

from the inverse FT of s(q)/so. Recently, Tuch et al. [56] introduced the method of higher

angular resolution diffusion(HARD) MRI, and Wedeen et al. [57] succeed in acquiring

500 measurements of s(q) in each scan to perform a fast FT inversion. However, this

method requires a large number of measurements of s(q) over a wide range of q in order

to perform a stable inverse FT.

A more common approach to estimate pt(r) from much sparser set of measurements

s(q) is assuming pt(r) to be a Gaussian. For Gaussian diffusion,

1 -rTD-1r
p(r, t) = ex1p{ },
'(4t)3det(D) 4t

where D is called the diffusion tensor. Inserting this to equation (1-6) it yields


s(q) soe-buTDu (1 7)









where u = q/lql, diffusion weighting factor b = 72621q2(A 6/3). Here 7 is the

gyromagnetic ratio, and 6 is the duration of two magnetic field gradient pulses with a

separation time A in the use of S' i1; il-Tanner pulsed gradient spin echo method [58]. In

this case

d(u) = TDu.

The principal eigenvector of D indicates diffusion direction of the diffusion. The fractional

anisotropy (FA) defined as

3 (Ai A2)2 + (A2 A3)2 + (A3 1)2
F2 (A, + A+ A3)2' (

where i (i = 1, 2, 3) are the eigenvalues of D, has become the most widely used measure

of diffusion anisotropy [53]. This is known as diffusion tensor imaging (DTI), and in

particular useful for creating white matter fiber tracts [59-63].

However, it has been recognized that the single Gaussian model is inappropriate for

assessing multiple fiber tract orientations, when complex tissue structure is found within a

voxel [57, 60, 64-68]. A simple extension to non-Gaussian diffusion is to assume that the

multiple compartments within a voxel are in slow exchange and the diffusion within each

compartment is a Gaussian [65, 66, 69-71]. Under these assumption the diffusion can be

modelled by a mixture of n Gaussians:

-rTD 1r
pt(r) = f4((4rt)3det(Dj))-1/2e 4t (1-9)
i=

where fi is the volume fraction of the voxel with the diffusion tensor Di, fi > 0, i fi = 1,

and t is the diffusion time. Inserting (1-9) into equation (1-6) yields


s(q)= so fie-buTD, (1 10)
i= 1

To estimate Di and fi, at least 7n 1 measurements s(q) plus so are required. In [69-71]

the model of a mixture of two Gaussians were used to estimate the PDF. This estimation

requires at least 13 diffusion weighted images from 13 different directions.









One of the alternatives to estimate pt(r) and characterize diffusion anisotropy is using

apparent diffusion coefficient(ADC) profiles d(O, Q), which are related to observed DWI

signals through the Stei-l: il-Tanner equation:


s(q) = soe-b (1 11)

where (0, Q) (0 < < < 0 < < 27) represents the direction of q. For Gaussian diffusion

d(u) = buTDu, where u is the normalized q. The trace, eigenvalues and functions of

eigenvalues of D can be used to characterize the anisotropy and directional properties of

the diffusion. For non-Gaussian diffusion the spherical harmonic approximation of the

ADC profiles estimated from HARD data has been used for characterization of diffusion

anisotropy. This technique was first introduced by Frank [67], also studied by Alexander

et al. [72]. In the work of [67, 72] d(x, 0, 0) was computed from HARD raw data via the

linear form of (1-11):
1 s(q)
d(q) log (112)
b so
and represented by a truncated spherical harmonic series(SHS):


d(x,0,, = Al,m(x)Y,m(0, ), (1-13)
l=0,2,...,lmax m= -l

where Y,m(0, 4) : S2 -i C are the spherical harmonics series and C denotes the set of

complex numbers. The odd-order term coefficients in the SHS are set to be zero, since

the HARD measurements are made by a series of 3-d rotation, and d(0, 4) is real and

has antipodal symmetry. Then, the coefficients Al,m(x)'s were used to characterize the

diffusion anisotropy. In their algorithm, basically, the voxels with the significant 4th order

(1 = 4) components in SHS are characterized as anisotropic with two-fiber orientations

(shorten as two-fibers), while voxels with the significant 2nd order (1 = 2) but not the

4th order components are classified as anisotropic with single fiber orientation (shorten

as one-fiber), which is equivalent to the DTI model. Voxels with the significant Oth

order (1=0) but not the 2nd and 4th order components are classified as isotropic. The









truncated order is getting higher as the structure complexity increases. Their experimental

results showed that non-Gaussian profiles arise consistently in various brain regions where

complex tissue structure is known to exist.

Since the ADC profiles can be used to characterize the diffusion anisotropy, and to

estimate pt(r) through the combination of (1-6) and (1-11), it is of great significance to

develop models for better estimation of the ADC profiles from DW MR measurements.

In general the raw HARD MRI data are noisy. Computing the coefficients directly from

the raw data often provides poor estimates. As a result, it will lead to inaccurate or false

characterization of the diffusion and consequently lead to incorrect fiber tracking. The

goal of C(i lpter 4 and C!i lpter 5 are to present two novel variational frameworks for

simultaneous smoothing and estimation of non-Gaussian ADC profiles from HARD MRI.

1.1.3 Neuron Fiber Tracts Reconstruction based on Smooth Tensor Field

The assessment of connectivity and the reconstruction of 3D curves representing

fiber traces are useful for basic neuroanatomical research and for disease detection.

Most normal brain functions require that specific cortical regions communicate with

each other through fiber pathi--,v-. Diffusion imaging is based on magnetic resonance

imaging technique which was introduced in the mid 1980s [73]and provides a very sensitive

probe for detecting biological tissues architecture. The key concept that is of primary

importance for diffusion imaging is that diffusion in biological tissues reflects their

structure and their architecture at a microscopic scale. For instance, Brownian motion is

highly influenced in the presence of tissues, such as cerebral white matter or the annulus

fibrosus of inter-vertebral discs. A i-ii ii:- at each voxel, that very same motion along a

number of sampling directions (at least six, up to several hundreds) provides an exquisite

insight into the local orientation of fibers.

There are currently several different approaches for reconstruction of white matter

traces, they can be roughly divided into four categories: (1) line propagation algorithms;

(2) surface propagation algorithms; (3) global energy minimization to find the energetically









most favorable path between two predetermined pixels; (4) solving a diffusion equation.

We will discuss line propagation algorithms in details. Line propagation algorithms use

local information for each step of propagation. The main differences among techniques

in this class stem from the kind of local information being considered and the way

information from neighboring pixels is incorporated.

The main local information used in most classical algorithms for recovering brain

connectivity mapping from DTI data is the principal eigenvector(PE) of the diffusion

tensor[63, 74, 75]. PE successfully determines the fiber direction in cases where there is

a single fiber direction in each voxel, and is therefore adequate for reconstructing large

trace systems. However, with voxel sizes typical of diffusion acquisitions(10 30mm3),

there is significant partial volume averaging of fiber direction in anatomical regions of both

research and clinical interest, such as the association fibers near the cortex. Moreover,

image noise will influence the direction of the 1 ii' r eigenvector. And as the degree of

anisotropy decreases, the uncertainty in the 1 i, i"r eigenvector increases, in which situation

the tracking may be erroneous. When a diffusion tensor is planar shaped, PE even does

not make sense. Westin et al [76] and Lazar et al[77] used the entire tensor to deflect

the estimated fiber trajectory. This algorithm is called tensor deflection(TEND). The

deflection term is better than PE in the sense that the previous one is less sensitive to

image noise and is less erroneous in situation of degenerated anisotropy. But it still has

the problem of partial volume averaging of fiber direction.

1.2 Contributions

Using prior shape and points in medical image segmentation

In chapter 2 we will provide a model to segment images that are corrupted by noise,

dropout and poor contrast along boundaries. We will employ prior shape and prior

points information into our algorithm by viewing the evolution of an active contour

as a deformation of the interface. This deformation consists of a rigid transformation

and a local deformation. We will use the ,v, i ;,'. shape" to determine the rigid









transformation that better maps the interface to the prior shape, and use the image

gradient and a few "key" points to determine the local deformation that provides

more accurate segmentation. Based on these thoughts we propose a variational

framework that is able to incorporate prior knowledge of the expected shape and a

few points that boundary should pass through.

We modify the energy function of geodesic active contour so that it depends on the

image gradient and prior shape, as well as a few prior points. We only need a few

points since we have information on expected shape. The modified energy function

provides a satisfactory segmentation despite the presence of both large shape

distortions and image dropout. To combine both prior shape (a global constraint)

and prior points (a local constraint) into a single variational framework, we use

a level set formulation. Further, we report experimental results on synthetic and

ultrasound images.

SA nonparametric adaptive method for simultaneous image smoothing and segmentation

In chapter 3, a model is developed to segment images that have complex multi-modal

intensity distribution and/or have high level noise. In this situation, it is not wise

to set up models based on a specific parametric assumption about image intensities.

Instead, we use nonparametric density estimation to create a model that is able

to simultaneously segment and smooth images without adding extra smoothing

terms. At each voxel, intensities of voxels in its neighborhood are used to obtain a

nonparametric estimate of its intensity distribution. Neighborhood sizes are chosen

adaptively based on image gradients. We then cast the segmentation problem as

the minimization of the negative log likelihood, subject to a constraint on the total

length of the region boundaries. Image smoothing is automatically fulfilled due to

the nonparametric density estimation. Rather than expensive Gaussian kernels, a

quadratic kernel is applied in the estimate to save calculation. The optimization

problem is solved by deriving the associated gradient flows and applying curve









evolution techniques. We use level set methods to implement the resulting evolution.

We demonstrate the superiority of proposed model over other models by showing

segmentation results from various images with different levels and types of noise.

The proposed model is able to solve a variety of challenging image segmentation.

* Estimation, smoothing and characterization of apparent diffusion coefficient using

two different methods

First, in C'! lpter 4 we present a new variational framework for simultaneous

smoothing and estimation of apparent diffusion coefficient (ADC) profiles from

HARD MRI. The model approximates the ADC profiles at each voxel by a 4th order

spherical harmonic series (SHS). The coefficients in SHS are obtained by solving

a constrained minimization problem. The smoothing with feature preserved is

achieved by minimizing a variable exponent, linear growth functional, and the data

constraint is determined by the original S-' i-i.: -Tanner equation. The antipodal

symmetry and positiveness of the ADC are accommodated in the model. We use

these coefficients and variance of the ADC profiles from its mean to classify the

diffusion in each voxel as isotropic, anisotropic with single fiber orientation, or two

fiber orientations. The proposed model has been applied to both simulated data and

HARD MRI human brain data. The experiments demonstrated the effectiveness of

our method in estimation and smoothing of ADC profiles and in enhancement of

diffusion anisotropy. Further characterization of non-Gaussian diffusion based on the

proposed model showed a consistency between our results and known neul, .1 ii., ':11.

Secondly, in C!i lpter 5 we present another approximation for the ADC of non-Gaussian

water diffusion with at most two fiber orientations within a voxel. The proposed

model approximates ADC profiles by product of two spherical harmonic series (SHS)

up to order 2 from High Angular Resolution Diffusion-weighted (HARD) MRI data.

The coefficients of SHS are estimated and regularized simultaneously by solving a

constrained minimization problem. An equivalent but non-constrained version of









the approach is also provided to reduce the complexity and increase the efficiency

in computation. Compare to the first method, this one requires less measurements

but provides comparable results. Moreover we use the Cumulative Residual Entropy

(CRE) as a measurement to characterize diffusion anisotropy. By using CRE we

can get reasonable results using two thresholds, while the existing methods either

only can be used to characterize Gaussian diffusion or need more measurements

and thresholds to classify anisotropic diffusion with two fiber orientations. The

experiments on HARD MRI human brain data indicate the effectiveness of the

method in the recovery of ADC profiles. The characterization of diffusion based

on the proposed method shows a consistency between our results and known

neuroanatomy.

* Reconstruction of Intra-voxel Structure from Diffusion Weighted Images

In chapter 6, we present a new variational method for recovering the intra-voxel

structure under the assumption that pt(r) is a mixture of two Gaussians: pt(r)

Sfi((4t)det(D))-1/2e 4t Our approach differs from the existing methods
in the following aspects. First, we recover each field Di(x) or fi(x) globally by

simultaneous smoothing and data fitting, rather than estimating them from

(1-10) with n = 2 in each isolated voxel, which leads to an ill-posed problem.

Second, we recover the ADC profile d(x, 0, 0) in SH representation using method

introduced in chapter 4 from the noisy HARD data before estimating Di(x) and

fi(x). The recovered d and the voxel classification on diffusion anisotropy from d

are incorporated into our energy function to enhance the accuracy of the estimates.

Third, we apply the biGaussian model to all the voxels in the field, rather than

the voxels where the Gaussian model only fits poorly. Since both the constraint of

fl t 1 on the region of strong one-fiber diffusion, and the regularization for fi and

Di are built in the model, the single fiber and multi-fiber diffusions can be separated









automatically by the model solution. This approach should be less sensitive to the

error in voxel classification.

* Reconstruct White Matter Fiber Traces Using Multi-Tensor Deflection In DWI

In chapter 7, we will provide a new line propagation algorithm based on smooth

multi-tensor field.

We assume there are up to two diffusion channels at each voxel. A variational

framework for 3D simultaneous smoothing and recovering of multi-diffusion tensor

field as well as a novel multi-tensor deflection(i\ TEND) algorithm for extracting

white matter fiber traces based on multi-tensor field are provided. MTEND keeps all

the advantages of TEND and has two additional good properties: first, problem of

partial volume averaging is automatically solved as it is based on a multi-tensor field;

second, it uses a dynamically adjusted step size to keep total curvature of traces low,

to appropriately terminate tracking and to increase algorithm efficiency.

Fiber traces are colored using Laplacian eigenmaps. By applying the proposed

model to synthetic data and human brain high angular resolution diffusion magnetic

resonance images(\ I 1I) data of several subjects, we show the effectiveness of the

model in recovering intra-voxel multi-fiber diffusion and inter-voxel fiber traces.

Superiority of the proposed model over existing models are also demonstrated.

* Diffusion Tensor Image Segmentation

In chapter 8, a 3D geometric flow is designed to segment the main core of fiber tracts

in diffusion tensor magnetic resonance images. The proposed model designed a new

external force that depended on two magnitudes: similarity of diffusion tensors and

the coincidence level of the evolving surface normal with the tensor field. The new

model, based on this new external force, was able to segment white matter fiber

bundles in DT-MRI more accurately and efficiently.









CHAPTER 2
USING PRIOR SHAPE AND POINTS IN MEDICAL IMAGE SEGMENTATION

Copy right [2003] lecture notes on computer science [78]. Reprinted with permission.

2.1 Description of the proposed model

In this section we present our variational approach for joint segmentation and

registration using a prior shape and the locations of a few points. The main idea of our

model is to propagate a curve/surface by a velocity that depends on the image gradients,

prior shape and locations, so that the propagation stops when the active contour/surface

forms a shape similar to the shape prior, arrives at high gradients, and passes through

the prior points. The notion of shape is independent of translation, rotation, and scaling.

In particular, two contours C, and C2 have the same shape, if there exists a scale p, a

rotation matrix R (with respect to an angle 0), and a translation T, such that C1 coincides

with pRC2 + T.

To begin the description of the proposed model, we first briefly review the geodesic

active contour model in [4, 5], and the active contour with a shape prior in [42].

Let C(p) = (x(p), y(p)) (p E [0, 1]) be a differentiable parameterized curve in an image

I. The geometric active contour model minimizes the energy function:


E(C)= g( VI7)(C(p)) C'(p) dp, (2-1)


where





1 .2
1 P|VG, 22'

with a parameter f > 0, and G,(x) = e 4,. The minimum of this energy functional

occurs when the trace of the curve is over points of high gradient in the image. Because

object boundaries are often defined by such points, the active contour becomes stationary

at the boundary. In its level set formulation this model can handle topological change.

However, since this algorithm requires a high image gradient to be present along the









boundary of the object to stop the curve evolution, it may "leak" through the 'p-' in

the boundary, where it is not salient.

To overcome this problem a variational model was proposed in [42] that incorporates

prior shape information in geodesic active contours. The feature of this model is the

creation of a shape term in the energy functional (2-1). If C*(p) (p E [0, 1]) is a curve

representing the expected shape of the boundary of interest, then the energy functional to

be minimized in [42] is


E(C,p,R,T) = {(VI)(C(p)) + d 2(pRC(p) +T)} C'(p) dp, (2-3)

where (p, R, T) are similarity transformation parameters, and dc (x, y) is the distance of

the point (x, y) to C*.

The first term in the energy functional (2-3) is the same as the energy functional

for geodesic active contours, which measures the amount of high gradient under the trace

of the curve. The second term is the shape related energy, that measures the disparity

in shape between the interface and the prior. The constant A > 0 is a parameter,

which balances the influence from the image gradient and shape. The curve C and the

transformation parameters pt, R and T evolve to minimize E(C, p, R, T). At the stationary

point, the contour C lies over points of high gradient in the image and forms a shape close

to C*, and /i, R and T determine the "b. -I alignment of C to pRC(p) + T.

The experimental results in [42] showed their model is able to get a satisfactory

segmentation in the presence of gaps, even when the gaps are a substantial fraction of

the overall boundary, if the shape of interest is similar to the expected shape. However,

if some parts of the boundary are not visible, and the shape of boundary of the object

has relatively larger geometric distortion from the prior, as shown later, model (2-3) can

not provide a desired segmentation, since the knowledge of the expected shape does not

provide correct information about how the gaps should be bridged.









In this note we intend to incorporate extra prior knowledge of the location of a few

points on the boundary into segmentation process. These points are given at the location

where the edge information is not salient and the local shape has relatively large variation

from the prior one. Our idea is the creation of a new energy term, that measures the

distances of the prior points from the interface, into the energy functional of the active

contours in [42]. Similar to the approach developed in [42], to get a smooth curve C

that captures higher gradients we minimize the arc-length of C in the conformal metric

ds = g( VI|)(C(p)) C'(p) dp, where g(IVI|) is defined in (2-2). To capture the shape prior

C*, we find a curve C and the transformation (p, R, T), such that the curve pRC + T and

C* are "b. -I aligned. To capture the prior points we minimize the distances of each prior

point xi (i 1,..., m) from the curve C.

We present our model in a variational level set formulation. First, as well known,

the level set method initiated in [18] allows for cusps, corners, and automatic topological

changes, Secondly, it is more convenient to compute the distances of the prior points to

the interface by using the level set form of the interface.

Let the contour C be the zero level set of a Lipschitz function u such that C = {x e

RN : u(x) = 0}, with (inside of C) = {x E RN : u(x) > 0}, and (outside C) = {x E RN

u(x) < 0}. Let H be the Heaviside function H(z) = 1, if z > 0, otherwise H(z) = 0, and

6 be the Dirac measure concentrated at 0 (i.e. 6(z) H'(z) in the sense of distribution).

Then, the length of the zero level set of u in the conformal metric ds = g(\VI\)CC'(p)ldp

can be computed by f, g(|VII) VH(u)l = S6(u)g(|VI|) Dul. The disparity in shape

between the zero level set of u and C* can be evaluated by f' 6(u)d,. (pRx + T)dx,

where the distance function dc. is the same as that in (2-3). Moreover, the constraint

for contour C passing through the prior points x,,..., Xm can be simply represented by

u(x)= O0, (i 1,... ,m).
Now let f,(x) be a smooth function defined on the image domain Q, such that

0 < f,(x) < 1, f,(x) = for x = xi (i = ,... ,m). The function f,(x) can be obtained









by the convolution of f and q],, where f is a function taking value one on the points

xl,..., x, and zero elsewhere, and r, is a mollifier with parameter 7 such that f,(x) is

only nonzero in neighborhood of xi's. Now we can formulate our new variational approach

as


mmin E(u, R,T) mm {a 6(u)g( VII)IVu
U,p,R,T U,p,R,T JQ

+ 6(u)d (pRx + T)Vu + f,(x)u2(x)}dx, (2-4)

where a > 0 and 3 > 0 are parameters. The first two terms in this energy functional are

the same as those in (2-3), which tend to lead the interface arriving at a location where

the magnitude of image gradient is high, and to form a shape similar to the prior. The

last term tends to lead the interface to pass through the given points, since minimizing the

third term in (2-4) with sufficiently small 7 leads u to be close to zero at the given points.

Note that f,(x) is non-zero only on the 7 neighborhood of the given points, so the third

term doesn't affect much the shape of the contour outside the 7 neighborhood of the given

points.

This model performs a joint segmentation and registration. The segmentation is

assisted by the registration between the interface and shape prior. This registration is

non-rigid that consists of a global transformation (rigid) and a local deformation. The

global transformation is determined by minimizing the second term in (2-4), while the

local deformation, is controlled by minimizing the first and last terms in (2-4).

The evolution equations associated with the Euler-Lagrange equations for (2-4) are

6(u)div((ag + d2) ) fu, (2-5)
at 2 |Vu|


au
= 0, x e OQ, t > 0; u(x, 0) = uo(x), x e Q, (2-6)
an










p= 6u)dVd (Rx)|Vuldx, t> 0, p(0) -o, (2-7)



o8 / dR
= -a 6(u)pdVd -(d x) Vu t > 0, 0(0) = 0o, (2-8)
at Q do


= 6(u)dVd|Vuldx, t > 0, T(0) To, (2-9)
at Q
where d = dc., R is the rotation matrix in terms of the angle 0, and the function d is
evaluated at pRx + T.
2.2 Numerical scheme

We solve the proposed model (2-4) by finding the steady state solutions) to the
evolution problem (2-5)-(2-9). To solve the equations (2-5)-(2-9) numerically, as in [16],
we replace 6 in (2-4)-(2-9) by a slightly regularized versions of them, denoted by:

1 ifz > e
He(z) = 0 ifz < -e

I [l+ Z+sm()] iflzl

S= H(z) = 0 if z >
I [1 + cos(w)] iflzl <
To discretize the equation of u, here we use a implicit finite difference scheme rather
than the one used in [42]. Let h be the step size, and (xi, yi) = (ih,jh) be the grid
points, for 1 < i, j < M. Let uj = u(t,, xi, yj) be an approximation of u(t, x, y).
The time derivative ut at (i, t,) is approximated by the forward difference scheme:
n+l_
Ut(i,j,tn) = At', where At is the time step.
We adopt the algorithm for the discretization of the divergence operator from [79],
and the implicit iteration from [80]. Knowing u", we compute u"+1 by using the following
discretization and linearization scheme of (2-5):










un+1 lU Vu n+1
+At( (u {(U ) an+ + 2id d )div( U' 1 )

(Vg. Vu)t1 (nV u)sd.
+a 11 +1 v-Y(Un '
17VU n+i 1vUn In+ I
where

Vun+l AXun+l1 Ay Hun+1
div( 1 Ax( + i,j)Ay + i,j
Ilu+ -( A ')2 + (A ,+l1)2 )(Ax'+1)2 + (A + -)2
V + iyj V i ) 71+1 'ty


When we compute u + 1 and t were replaced by U+lj and +l

respectively, since they are unknown. We use the forward or backward finite difference

schemes adaptively to approximate Vd Vu and Vg Vu. That is

(Vd Vu) (max(Axd j,0)A>2. + min(Axdj ,<0)Au


+max(A ij ( 0)AU .j + m(in(Ayd', O))A u`).

The term Vg Vu is approximated in the same way.

To keep the signed distance function near the front(zero level set of the evolving u)

the technique of reinitialization developed and applied in [37, 80, 81] is also used in our

computation. This procedure is made by using a new function v(x), which is the steady

state solution to the equation

agv
a sign(u((.,t))(- IVv ), v(.,0) u(.,t),

as u(., t) for the next iteration t + At.

The equation (2-4)-(2-9) are discretized as in [42] by using finite difference.

2.3 Validation and Application to Echo Cardiovascular Ultrasound Images

In this section we report our experimental results on both synthetic and ultrasound

images.

The aim of our first experiment is to verify that the active contour with the prior

shape and points can fill in the *, 1'-" in a boundary in a meaningful way.









Figure 2-la shows a typical binary image I with three points and an ellipse

superimposed. The ellipse and points are used as the prior shape and points in this

experiment, respectively. The object to be segmented is partially occluded, and the shape

of its boundary is not equivalent to the prior shape. We want to determine whether or not

the active contour with the prior shape and points can use the partial boundary to aid the

process of filling in the rest.

The active contour was initialized by the solid curve di-l 'i. t in figure 2-1c. Evolving

the active contour according to (2-5)-(2-9) with the parameters a = 250, f = 15,

a = 0.5 (in g(x)), po = 1, 0o = 0, To = (0, 0), we get the stationary contour C (the

dotted one) in figure Ic, and the transformation parameters p = 0.91, 0 = -0.14, and

T = (-0.5, 0.3)(pixels). We can see that even though complete gradient information is

not available the contour C captures the high gradient in the image I, passes through

three prior points, and forms a shape similar but not the same as the prior shape. To show

the advantage of using prior points we compared the segmentation results obtained by

using model (2-4) and (2-3). Figure 2-1 B) shows the segmentation result by using model

(2-3). In figure 2-1 B) the solid contour is the initial contour, and the dotted one is the

segmentation result. Since the prior points are not incorporated in the model (2-3), the

segmented contour only captures the prior shape and high gradients. It can't accurately

capture local shape variations.

The aim of the second experiment is to segment the endocardium (the inner boundary

of the myocardium surrounding the left ventricle) in an apical two-chamber image of the

heart (see Figure 2-2 A) for a typical image). The endocardium is not completely visible

in the image, and its shape is not the same as the i.- I .,.- shape" (the shape prior). Our

task is to determine the endocardium using ,i, I,. shape" and five points given by an

expert.

The prior shape is created by the same way as that in [42]. It is obtained by

averaging the aligned contours in a training set. The alignment of two contours C1 and C2


















(a) (b) (c)

Figure 2-1. Comparing segmentation results of a synthetic image. A) An image with the
prior shape and three points superimposed. B) The segmentation result using
model (2-3) (dotted) and the initial contour (solid). C) The segmentation
result using model (2-4) (dotted) and the initial contour (solid).


is made by finding the "b, -1 scaling constant p, rotation matrix R and translation vector

T such that the overlapping area of the interiors of C1 and pRC2 + T is maximized. If

the shapes of the curves in the training set have a large variation, a clustering technique

is required to group these curves into several groups. The shape priors for each group are

obtained using this technique.

For this particular problem to create the prior shape, an expert echocardiographer

traced endocardial boundaries on 112 image sequences for 66 patients. After the

boundaries were clustered, the average was computed. Figure 2-2b shows the i,, I -

contour" for one of the clusters (the dotted contour), the endocardium outlined by an

expert (the solid contour), and five points on the expert's contour, The prior points are

usually given at the location where the image gradients are low, and the local shape

distortions are larger. Figure 2-2 C) presents the image |VG, I|. From this image we can

see the dropout of image information at several parts of the endocardium.

To segment the endocardium in the image shown in Figure 2-2 D) (it is the same as

in Figure 2-2 A), the the active contour was initialized as the contour shown in Figure

2-2 A). This contour was evolved according to the equations (2-5)-(2-9) and it finally

stopped at the location of the dotted contour in Figure 2-2 D). We also obtained the

transformation parameters p = 1.0024, 0 -0.1710, and T = (-24.6163, 32.3165)









(pixels). The solid contour in Figure 2-2 D) is the expert's endocardium. Observe that

the segmentation is close to the expert's contour. To see the shape variation between the

solution of (2-4) and ,'. i ,- shape" we aligned the solution of (2-4) to the i, i,

shape" using the solutions (p, R, T) of (2-4). Figure 2-2 E) shows the disparity in shape

between these two contours. The dotted contour is the transformed solution of (2-4),

and the solid one is the i.... I.- shape". From Figures 2-2d and 2-2 E) we can see that

our active contour formed a shape different from the prior one in order to capture the

high gradients and given points. Figure 2-2 F) provides the segmentation result obtained

by using model (2-3). In this figure the dotted contour is the solution of (2-3), and the

solid contour is the expert's endocardium. Comparing Figure 2-2 D) with Figure 2-2 F),

note that the solution of (2-4) is closer to the expert's contour than the solution of (2-3).

Figure 2-2 G) presents the shape comparison between the solution of (2-3) and prior

shape. In Figure 2-2 G) the solid contour is the i, I ,'. shape", and the dotted one is the

transformed solution of (2-3), (the transformation parameters are the solution of (2-3)).

From Figures 2-2 F) and 2-2g we see that the solution of (2-3) can only capture the high

image gradients and the ,,- i ,',. shape", but it can't provide as desirable a segmentation

result as the expert's endocardium.

The last experiment is a repetition of the second experiment on a second apical

2-chamber cardiac ultrasound image. We list the figures below for the results of this

experiment in the same order as above. The segmentation C is given in figure 2-3d

represented by the dotted contour, the transformation parameters are p = 0.9883,

0 = -0.1981, and T = (-28.7246, 49.1484).

2.4 Conclusion

In this chapter we proposed the addition of prior points to an active contour with

shape in a variational framework and in level set formulation. The key idea was to

introduce an energy term which measured the image gradients, the closeness of the shape

between the active contour and a prior shape, as well as the distance of the prior point



















(a) (b) (c)


(d) (e)


(f)


Figure 2-2.


Compare segmentation results of one cardiac ultrasound image. A) A typical
2-chamber ultrasound image with an initial contour; B). Expert's endocardium
(solid contour), i,. I ,,. shape" (dotted contour), and five points on the
expert's contour. C). The image |VG, II. D). The endocardium segmented
by using model (2-4) (dotted) and the expert's contour (solid). E) The
transformed solution of (2-4) (dotted), and the ,v- i ,_;,'- shape". F). The
endocardium segmented by using model (2-3) (dotted) and the expert's
contour (solid). G) The transformed solution of (2-3) (dotted), and the
i,. i ;,'- shape".



















(a) (b) (c)


(d) (e)


(f) (g)


Figure 2-3.


Compare segmentation results of another cardiac ultrasound image. A) a
typical 2-chamber ultrasound image with an initial contour. B) Expert's
endocardium (solid contour), i I i ,,. shape" (dotted contour), and five
points on the expert's contour. C) The image IVG, II. D) The endocardium
segmented by using model (2-4) (dotted) and the expert's contour (solid).
E) The transformed solution of (2-4) (dotted), and the i., I ',; shape". F)
The endocardium segmented by using model (2-3) (dotted) and the expert's
contour (solid). G) The transformed solution of (2-3) (dotted), and the
i,. i ;,'- shape".









from the active contour. We used an implicit numerical scheme to solve the minimization

problem. In the experiments with application to cardiac ultrasound images, the active

contour could segment images in which the complete boundary was missing and the shape

of the boundary has relatively large distortion from the prior. Besides the segmentation,

the algorithm also provides estimates of translation, rotation, scale that map the active

contour to the prior shape. These estimates are useful in aligning images.









CHAPTER 3
USING NONPARAMETRIC DENSITY ESTIMATION TO SMOOTH AND SEGMENT
IMAGES SIMULTANEOUSLY

3.1 Proposed Model

A new model which could simultaneously segment and smooth images without using

smoothness term as the second part in (1-1) is provided here.

For a recap of nonparametric density estimation, let X ~ f(x) be a random variable,

{X1, X2, ..., X,} be a set of realization of X. A kernel method estimation of f(x) would

be f(x) Kh(x X(- Xi), where Kh = h) and K(-)is a kernel function symmetric

about y axis and maximized at x = 0, and h is called bandwidth. Please refer [82, 83] for

more detailed knowledge about nonparametric methods.

For notation simplicity, we denote by Ii the restriction of function I to i To ensure

applicability of the nonparametric density estimation idea, we make two assumptions.

Firstly, at each voxel x, we treat Ii(x) as a random variable. This is reasonable in the

sense that even for the same subject, different scans would provide different data 1o due

to motion of the subject, noise involved in the data collection process etc. Randomness

of 1o leads to that of i(x)'s. Secondly, we assume that i(x) has the same distribution as

that of the lo(y)'s, for all y that are in a small neighborhood of x and are inside region Qi.

Note this is more general than in model (1-4), where i(x) is assumed to have the same

distribution as that of the li(y)'s, for all the y's inside region Qi. Under the above two

assumptions, we approximate the distribution of intensity i(x) using intensities of lo(y),

for y I i,,-; in the intersection of Qi and a neighborhood of x, denoted by B(x, r(x)), which

is a ball centered at x with radius r(x). The estimated distribution is called fi(x):



fi K ) B(x, rx)) n (3 1)
j=1









In real implementation, sum and integral are treated equivalently, so for notation

simplicity, we denote

fi(x oc Kh(Ij(x) lo(y))dy (3-2)
JB(x,r(x))nrm
Under the independence assumption, probability density function of Ii in the domain

Qi would be Hxen fi(x). Joint probability density function of I in the whole domain Q

would be HI nHen f(x). We still call logarithm of this function as log likelihood even

though no parameters are involved. The proposed model is to find C, and Ii(x)'s for each

point x CE i, to minimize the following functional:



(1 (I1() Io(y+))2
Sj log ( 22 dy dx + |C| (3-3)

The first term is negative log likelihood, in which Kh is chosen as a Gaussian kernel

function and bandwidth h is set to be 1. For each point x in each subregion 2i, the

first part maximizes the log likelihood for all y's that lie in the intersection of 2i and

a neighborhood B(x, r(x)) of x. This is equivalent to some extent to trying to force

Ii(x) Io(y) to be 0, i.e., intensity Ii(x) to be close to lo(y)'s for all y's lying in the
intersection of a and neighborhood B(x, r(x)) of x. Roughly speaking, Ii(x) will be close

to mean value of o1 in a small neighborhood that is inside 2i, which will smooth images

li(x) without crossing boundaries. Detailed proof of a simplified version will be provided
in section 3.4. Thus the first part basically smooths each Ii inside 2i without crossing the

boundary. Therefore, the first term not only prevents 1i's from being too far away from

1o, but also smooths li's. This implies it works similarly to the first two parts in piecewise
smooth MS(1 1). We would like to mention that convolution also smooths image through

taking average, but convolution will blur images at boundaries and also cause shifting of

boundaries. The reason is that li(x) is set to be average of lo(y)'s for all y's lying in a

neighborhood of x, this neighborhood is not restricted inside Qi. Results of model (3-3)

will not cause these problems.









The second term works the same as the third term in (1-1) to obtain a smooth

boundary. The proposed model is able to simultaneously segment and smooth images

without using extra smoothness terms, and this also saves annoying work on choosing

appropriate parameter to balance data fidelity term and smoothness term.

Gaussian kernel could be replaced by any bell-shaped kernels that are maximized at

0. By bell-shaped, we mean the graph of the kernel function is increasing on the left hand

side of 0 and then decreasing on the right hand side of 0. Computation complexity would

be decreased a lot if the Gaussian kernel involving expensive exponential computation

is replaced by kernels involving simple calculation. Quadratic kernel is a good example.

On example for quadratic kernel is P(u) = (1 u2)X[-1,1](), where X[_,](u) is the

characteristic function of [-1, 1]. An alternative model would be to minimize the following

functional with respect to 0 < I1, 2 < 1 and C:




o(1 (x) JBo(Yr(x)) i 4 (34)

To force li(x) lo(y) to be in interval [-1, 1], which is the support of the quadratic

kernel mentioned above, a linear transformation, which does not change segmentation

results, is required to re-scale range of the initial image lo to [a, b] for any 1 > b > a >

0(a = .1, b = .9 would be a choice). This condition is also required to guarantee existence

of solutions. We can show that it is not enough to re-scale range of lo to [0, 1], it has to be

a proper subset of [0, 1]. Ranges of I1 and I2 are restricted to [0, 1]. In section 3.4, we will

show that (3-4) is actually a non-constrained problem, i.e., the constraints will be satisfied

automatically for solutions.

Obviously, the first term is minimized when for each point x in each subregion Qi,

li(x) is equal to lo(y)'s for all y's lying in neighborhood B(x, r(x)) of x and Qa. Hence it
would work similarly as model (3-3) but require much less computation. Actually, (3-4)

is equivalent to (3-3) when all the a(x)'s are set to be a constant. Note this does not









mean (3-4) is returned to the piecewise constant MS (1-2) even though both of them have

no variance involved. (3-4) forces li(x) to be close to mean of lo(y)'s for all y's in the

intersection of B(x, r(x)) and Qj. This is much more general and applicable than piecewise

constant MS which enforces li(x) to be close to a constant which turns out to be mean

value of lo(y)'s for all y's in the whole Qj. This generality enables (3-4) to handle images

which involve multi-modal distributions.

The level set formulation of model (3-4) is as follows. By using Heaviside function, all

the integrals in the model are over Q, so implementation is much easier:



H(O(x))log XB(x,r(x))(y)H( (y))3(1 (II(x) y))2 dx

(1 H(O(x)))log XB(x,r())(y)(1 H(O(y))) (1 (2(X)- 0)) dx

+V (0 (x)) V1 (x)|dx (3-5)

where XB(x,r(z))(y) is the characteristic function of B(x, r(x)). Ball size r(x) is

adaptively dependent on image gradient at x: when image gradient is high, i.e., at

locations near boundaries, radii are smaller, while radii are '-i.--. r at more homogeneous
regions. As a result, smoothing speed of I, I2 are higher at homogenous regions and lower

near boundaries to keep fine structure. Therefore, the smoothed images will be sharp. In

comparison, in [84], ball sizes are fixed for all x, so it is harder to choose one uniform ball

size that works for all the locations; smoothing speed will be the same at all locations.

If a too small ball size is chosen, noise in the homogenous region can not be removed

sufficiently, while if a too big ball size is selected, fine structures near boundaries will

disappear. Similarly, model (1-1) also simultaneously smooths and segments images, but

it assigns a fixed coefficient a for all x, so smoothing speed will be similar to what was

observed in [84]. A comparison between models using variable radii and fixed radii will be
shown in section 3.2.









3.2 Numerical Implementation

We will discuss some numerical implementation details of (3-5). First, radius r(x) is

chosen in the following way:


r(x) (3-6)
1+ NVIo(xr)
where lo is a smoothed version of the given initial image 1o obtained through

convolving it with a smooth kernel. Another straightforward choice is to replace 1o by

I, smoother version of lo that is updating during iterations:


r(x) =- (3-7)
1 + NVI(x)
where I is a smoothed version of the updating image I obtained through convolving it

with a smooth kernel, we still convolve I even though it is smoother than 1o. The reason

is during the iteration, especially in the early stage, it is not smooth enough. In Figure3-1,

we compare segmentation and smoothing results of model (3-5) with radii defined as

in (3-6)(3-7). Figure3-1A) is a clean plane image, a speckle noise with parameter .05

is added to it, the resulting noisy image is shown in B). The second and the third row

demonstrates segmentation and smoothing results based on (3-6)(3-7) respectively. No

significant difference in both segmentation and smoothing results is visualized. Therefore,

we will use (3-6) for all the following experiments. Also, it is easier to prove existence of

model (3-5) for r(x) defined as in (3-6), section 3.4 provides details.

M is an integer chosen based on noise level of the input image lo: M is chosen to

be higher if the noisy level is high, and chosen smaller otherwise. N is chosen to be

small all the time to keep and enhance boundaries while smoothing. s is set to be the

85th percentile of |VIo(x) This means we assume 15'. of the points belong to the

boundaries. 15'. is a very conservative number; usually, in one image, number of points

on the boundary will not be that many. Experimental results are not very sensitive to this










































E) F)


Figure 3-1.


Compare results of model(3-5) based on two choices of r(x) A) A clean plane
image(60 x 90). B) a noisy plane image obtained by adding speckle noise with
parameter .05 to A). C)-D): segmentation and smoothing results of B) using
model(3-5) with radius as defined in (3-6) respectively. E)-F) segmentation
and smoothing results of B) using model(3-5) with radius as defined in (3-7)
respectively. For all the results, time step size = .2,3 = .01, M = 1, N 0,
converge in 20 iterations.









parameter; the general rule to adjust it is: set it to be higher if you prefer to keep and

enhance fine structures more than to smooth images, while make it smaller if there are not

that many fine structures in a quite noisy image.

The second issue is to use regularized version of Heaviside function and Delta function

to make the functional differentiable with respect to 4. We use those defined in [16].

Thirdly, due to the difficulty of finding derivative of the functional with respect

to Q directly, the functional is totally discretized. The discretized version of (3-5) is

then implemented using an additive operator splitting(AOS) algorithm with boundary

conditions = 0 and = 0 for i = 1, 2. AOS [85-87] is a semi-implicit scheme which

is stable for large step sizes. It is at least ten times more efficient than the widely used

explicit schemes. It guarantees equal treatment of all coordinate axes, can be implemented

easily in arbitrary dimensions, and its computational complexity and memory requirement

are linear in the number of pixels.

Finally, initial 1, I2 and Q could be chosen extremely flexible; even simple constant

initials would work for most of the images we have tried. This removes the work

of creating the distance function of an initial curve for initial Q in regular level set

based method. The algorithm usually converges in fewer than 25 iterations, making

reinitialization of level set function 0 unnecessary. Parameter 3 affects experimental

results and needs to be tuned for each image. But it does not vary much among different

images.

Table 3-1: Segmentation Accuracy
Gaussian Noise 0.01 0.05 0.1 0.25
SA 0.996 0.979 0.957 0.868
Salt Pepper Noise 0.05 0.15 0.25 0.50
SA 0.996 0.988 0.970 0.885

























A) B)


G) H)


Figure 3-2.


Segmentation results of a clean image and that of a noisy version. A)-C)
Original cartoon C-shaped image, segmentation result and the smooth output
resp. D)-F) An image obtained by adding Gaussian noise with variance 0.05
to A),segmentation result and smoothed version. G)-I) An image obtained by
adding 0.03 speckle noise to A),segmentation ,smoothed version. For all the
results, time step size = .1, = .01, M = 2, N 1, converge in 20 iterations.









3.3 Validation and Application to T1 Magnetic Resonance Image

We provide three sets of experimental results. The goal of the first set is to test the

ability of model (3-5) to simultaneously segment and smooth noisy images, and the second

set is to compare segmentation ability of (3-5) with that of piecewise constant MS and

global Gaussian pdf based method. Finally, segmentation and smoothing capability of

model (3-5) that uses adaptive radii is compared with model in [84] that uses fixed radius.

No comparison between (3-5) and piecewise smooth MS is provided as we think model

in [84] is much easier to implement, and involves less parameters but works somehow

equivalently as piecewise smooth MS does.

The first set of experiments is based on a clean synthetic C-shaped cartoon image

with only two phases(Figure 3-2A)). We add different levels of random Gaussian noise

and salt & pepper noise respectively to Figure3-2A) to create different noisy images.

Model (3-5) is then applied to each of the noisy image and the clean cartoon image with

parameter f = .2. A segmentation accuracy(SA) measure is defined to quantitatively

observe how accurate the model is in segmenting a noisy image compared to the original

cartoon image. We treat segmentation result of the clean cartoon image as ground truth.

Then define SA to be the ratio of number of pixels sharing the same segmentation with

the ground truth over total pixel number. SA is then between 0 and 1, the closer it is to

1, the better the model is in segmenting this noisy image.











A) B C)
Figure 3-3. Segmentation result of a noisy T human brain image. A noisy T image.
B) Segmentation result based on (3-5): black: background and CSF; gray:gray
matter; white:white matter. C) Recovered smooth T1 image.









Table 3-1 lists SA's corresponding to images obtained by adding random Gaussian

noise, with mean 0 but different variances, and salt & pepper noise with different

parameters respectively. Each of the SA shown is the average value of two SA's

corresponding to two different images obtained through adding same type and strength

of random noise at two different times. It is observed from Table 1 that when variance

of Gaussian noise is lower than .1 and parameter of salt & pepper noise is less than 0.25

respectively, the segmentation accuracy could be higher than 95'. And an obvious trend

is that as noise level increases, SA decreases as expected.

Figure3-2 shows segmentation and smoothing results based on the clean C-shaped

cartoon image (Figure3-2A)) and two noisy versions of it: (Figure3-2D)) which is obtained

by adding a random Gaussian noise with zero mean and variance 0.05, (Figure3-2G))

which is obtained by adding speckle noise with parameter 0.03. Their boundaries depicted

in red curves which are found by applying model (3-5) are superimposed on them and

shown in Figure3-2B)E)H) respectively. It is observed that they are almost the same,

actually, they coincide up to 99.'.- Figure3-2C)F)I) demonstrate the smoothed version

of Figure3-2A)D)G) respectively. The recovered images are much smoother than the

original noisy images. Figure3-3 demonstrates segmentation and smoothing results on

a real medical image. Figure3-3A) is T1 weighted MRI brain image that involves four

phases: Background, cerebral spinal fluid(CSF), Gray Matter and White Matter. Noise

with unknown type and length is involved, distribution of intensity is of unknown type

also, model (3-5) is applied to this image, segmentation and smoothing results are shown

in Figure3-3B)C) respectively. The segmentation result is quite reasonable, and it is

obviously that Figure3-3C) is significantly sharper than Figure3-3A). We use hierarchical

level set method recursively as Gao et al. mentioned in [24].

The second set of experiments is based on a plane image shown in Figure3-4A). We

add random Gaussian noise with mean 0 and variance 0.01,0.05 to it to create two noisy

images which are shown in Figure3-4B) and Figure3-4C) respectively. We apply piecewise






































G) H)


Figure 3-4.


Compare three models in segmenting images with lower and higher level of
noises. A) A plane image. B) A noisy plane image obtained by adding a
random Gaussian noise with mean 0,variance 0.01 to A). C) A noisy plane
image obtained by adding a random Gaussian noise with mean 0,variance 0.05
to A). D)-F) Segmentation results(red curve) of B) superimposed on B) based
on piecewise constant MS, global Gaussian pdf and (3-5) respectively. G)-I)
Segmentation results(red curve) of C) superimposed on B) based on piecewise
constant MS, global Gaussian pdf and (3-5) respectively. For all the results,
time step size = .1, = .01, M = 1, N 0, converge in 20 iterations.











constant MS, global Gaussian pdf based method and proposed model (3-5) to Figure3-4B)

respectively and obtained results(red curves) superimposed on Figure3-4B) as shown in

Figure3-4 D)-F) respectively. Visually, not much difference among them is detected. Thus

all the three models are able to segment images with low level noise. But after applying

these three models to the noisier image Figure3-4C), from their results superimposed

on Figure3-4C) as shown in Figure3-4G)-I) respectively, huge differences are observed.

Moreover, it is obvious that result based on (3-5) is the best as it simultaneously segments

and smooths the image so that boundaries of the noisy speckles are removed and the

boundary of the plane is kept well. Result of piecewise constant MS is the worst: in order

to remove boundaries of speckles, it pI i, the price of obtaining bad plane boundary.

Result of global Gaussian pdf based model keeps plane boundary well but gains more

boundaries of speckles as it allows variation of intensity in fi's.




099

098
SProposed model, mean 9901, variance 1 88e-6
097- Global pdf model, mean 9879, variance 1 37e-6
Piecewise constant, mean 9446, variance 7 37e-6
096

095

094

093
0 10 20 30 40 50 60 70 80 90 100
Trials


Figure 3-5. Graphs of SA obtained from three models: red: proposed model, green: global
Gaussian pdf based model, blue: piecewise constant MS model.



A quantitative comparison of the three models is provided through SA. We add

random Gaussian noise with mean 0 and variance 0.05 100 times to the clean plane image

Figure3-4B) to get 100 noisy images. The three models are then applied to these 100

noisy images to obtain 100 SA's for each model, Figure3-5 shows the graphs of SA for











































E) F)


Figure 3-6.


Compare segmentation and smoothing results between two models with fixed
and adaptive radii A)-B) Segmentation and Smoothing results of Figure3-4B)
using fixed radius 0. C)-D) Segmentation and Smoothing results of figure3-4B)
using adaptive radii defined in (3-6). E)-F) Segmentation and Smoothing
results of figure3-4B) using fixed radius 1. For all the results, time step size
.1, = .01, converge in 20 iterations.









the three models. Mean value and variance of the 100 SA's obtained from the proposed

model (shown in red contour), global Gaussian model(shown in green contour) and

piecewise constant model(shown in blue contour) are (.9901, 1.88 10-6), (.9446, 7.37

10-6), (.9879, 1.37 10-6). The proposed model has the highest mean and smaller variance,

while the piecewise constant has the lowest mean and smallest variance. It is obvious that

the proposed model and the Global Gaussian pdf based model are significantly better

than piecewise constant MS, while the proposed model is slightly better than the Global

Gaussian pdf based model. But an advantage of the proposed model over the Global

Gaussian pdf based model is that the former one can recover smooth image but the later

cannot.

Thirdly, we compare model (3-5) with the model in [84] that uses fixed radius. In

Figure3-6 segmentation and smoothing results of model (3-5) and the model in [84] with

two different radii are shown. The first row corresponds to that of model in [84] with

fixed radius 0, the second row depicts that of model (3-5) with radius equals to 1 at

relatively homogenous regions, 0 near boundaries. The third row shows that of model in

[84] with fixed radius 1. It is obviously that the second row gives good results in both

segmentation and smoothing. In Figure3-6B), regions .1i--v from boundaries are smoothed

well, near boundaries, image has discontinuity. In comparison, when use the model in [84]

with a fixed small radius (see the first row), boundaries of fine structures are kept well

but boundaries of the noisy background are also caught, recovered image is not smooth

enough either. Oppositely, when a 1i.-.,--r fixed radius is chosen( the third row), images are

smoothed enough, however, but we p iv the cost of losing fine structures. This illustrates

the advantage of model (3-5).

3.4 An Existence Theorem for the Model

To prove existence, we will use some facts about BV space([88],[89],[90]):

Let Q be an open, bounded and connected set in RN. We uv- u e L1() is a function

of bounded variation, denoted by u E BV(Q), if










IDul(Q) sup{ udivfdxf GC C (; RN), f < 1} < .
J.
IDul(Q) is also denoted by f |VXn C '!i :teristic function of a set A C RN is
in BV(Q) if and only if its perimeter perQ(A) :=DXA I(A ) is finite, where XA is the
characteristic function of A.

The space BV(2) is a Banach space endowed with the norm:


IU lBv(Q) = IUILI(Q) + IDu|(2).

Compactness theorem in BV
Assume Q is an open, bounded and connected set with Lipschitz boundary. If {u,})>l

is a bounded sequence in BV(2), then there exists a subsequence {u1j} of {u,} in BV(2)

and a function u E BV(2), such that uj u strongly in L1() as j -00o.
Lower Semi-Continuity of BV-norm in L1

If u, {uj} C BV(2) and uj -- u in L1(2), then


IDu|() < liminfj,,~ DuT |().

To prove the existence of solutions) to model (3-5) concisely, we assume image
domain 2 is decomposed into two regions A and 2 \ A. Proof would be similar if there

are more subregions, in which case, there will be just more similar additive terms in the
energy functional E. Let A = {x E 2lH(x) > 0}, then XA = H(O), using ideas as in [16],

model (3-5) could be reformulated as:











min E(XA, II, 2) =3 DXA
XAEBV(Q),O
XA(X)f09 XA(3 _I 1h x ) ) 2o)dy dx
JQ (\ B(x,r()) X )

(1 XA(X))og ( (1- XA(Y)) 3( (I2 ( ())dyj dx (3-8)
SfB(x,r(x)) 4

where the first term is the length term represented as total variation of characteristic
function XA. In BV space notation, it is IDXAI(Q). Euler-Lagrange equations of
E(XA, 1, 12) with respect to I1 and 12 are

(X) f XA(y)dy f XA(y)Io(y)dy = 0, x A (3-9)
JB(x,r(x)) JB(x,r(2))
2 (- XA ())dy X (-A(y))Io(y)dy 0,x E \A (3-10)
JB(x,r(x)) JB(x,r(x))

We may assume fB(x,r(x)) XA(y)dy > 0 for all x E A, then

Il(X) A(Y) x EA (3-11)
W(x,r(x)) XA(y)dy

If fB(x,r(x)) XA(y)dy = 0, which is called the degenerate case, fB(x,r(x)) XA(y)Io(y)dy
would also equal to 0, (3-9) is satisfied automatically for any choice of Ii(x) between 0
and 1.
Similar analysis for 12(x) gives

I2(x) (= x CE ( \ A (3-12)
fB(X,r(x))-( XA(y))dyA
The solution of I,2 must be related with XA by (3-11)(3-12). Moreover, it is
observed from equations (3-11)(3-12) that if lo(x) E (.1,.9), for all x E Q, then,

.1 fB(,r(x)) XA(Y)dy < SB(x,r(x)) XA(Y)Io(y)dy < .9 f(x,r()) XA(y)dy, from which we get
J SB(s,rt()) XA(Y)IO(y)dy .Btx,rtx)) XA(y)dy
.1< B(,j(,))XA(y)dy -< J B(x, ( y)) XA(y)dy < .9, i.e., 0.1 < I (x) < .9, therefore 0 < I,(x) < 1.

We can also prove 0 < 12(x) < 1. Hence Ii(x),12(x) will be in (0,1) automatically for all
x E 2. Thus, we do not need to worry about the constraints of I,, 12.








Because the unknown functions I1 and I2 have explicit representations as functions
of the unknown XA, we can consider the energy E in XA only: E(XA, I1(XA), I2(XA)).
Therefore, we obtain the minimization problem:


minE(XA),XA E {0, l}dx a.e.
XA


(3-13)


among characteristic functions of sets A with finite perimeter in 2, i.e., characteristic
functions of bounded variation([89]).
We now state our existence result:
Theorem
If 2 is open, bounded and connected with Lipschitz boundary, 0.1 < Io(x) < .9 for all
x E 2, then infXA E(XA), XA E BV(2), XA E {0, 1}dx a.e has a minimizer.
Proof: Observe that


I- XA(x) log (0 XA(y)(1 (II(x)- 0 (y))2)dy) dx

< IXA(X) 0log (j / XA(y)( -(I (X)- Io(y)) 2) ydx
J o \JB(x,r(x)) /
< |10og\B(x,r(x))||dx < |10og | ||dx = | | |logI|Q||
Jo Jo


Similarly,


j(- (1- XA(x))10g ((1- XA (y)) 3 (II 0 (y))2)dy dx\

< I1 XA(x) -log f (1 XA(y))(- (I(x)- (y))2) ) dx
Jo \JB(x,r(x)) /
< 0IlogIB(x,r(x))|ldx < |Ilog\ l | dx= | | |Ilogl|l |

Thus the energy functional E(XA) is bouned below, so there exists a minimizing
sequence {XA)} such that


lim E (XA,)


liminf E(XA).
XAEBV(Q)









By boundness of {E(XAn)} we have {IDXA A|} bounded above because the second and the
third terms in {E(XA,)} are bounded below. Moreover, Jo XAndx < Jo tdx = 121 < oO,

so {XAn} is bounded in L1(Q). Therefore {XA)} is a bounded sequence in BV(Q), so
by compactness of BV space and lower semi-continuity of the BV-norm in L1 topology,
there exists a subsequence {XAn } and a u c BV(Q), such that XA, -- u strongly in
L1(Q) as j oo. From this we deduce that there exists a subsequence, still denoted by
A,,, such that XAn, -- u a.e. in 2, and since XAn, takes values of either 0 or 1, u = XD
for some D with finite perimeter in 2. By compactness theorem in BV space and lower
semi-continuity in L1, we have

IDuI() < liminf,, |DA,( I(). (3 14)

Note


(fB(,r(x)) (1 XA^W (y)dy
fB L(xr x) t /XA~ (y))loy) dy
12, x E s \ AX,
I 1 ()1B( z,) ) (t XA.j (]) d]
Since XAn, XA pointwisely a.e, and .1 < Io < .9, we have 0.1 < XA, (y)Io(y) < .9, so
JB rr, rx()) XD(y)Io(y)dy
by the Dominanted Convergence Theorem, i, (x) converges to Ii(x) X:= .( xa(y-)dy
The range of Ii(x) is then obviously in (0, 1) because .1 < Io < .9. It is straightforward to
show that I2(x), defined as --(c,,(xc))(1l(XD)(y)dy (limit of o2n (x)) has range in (0,1) as well.
Regarding the second term in (3-8),

3 3
XAn(y) ( (IX)- I0(y))2) XD(Y) 3 (II (X)- o(0 ))2)
4 4

strongly as j o0 and all the terms are less than or equal to 1, so by the Dominated
Convergence Theorem,



XA(Y)(- (I(X)- I)))dy XD(Y) (1 (II(X) Io(y))2)dy
JB(x,r(x)) 4 JB(x,r(x)) 4









pointwise. This implies XA (x) log (B(x,r(x)) XA, (/)3( -- (I (x) (o (/2)dy converges

to XD(x) log (B()()) XD(y)I ( (x) oQ ))2)dy) pointwise. Because all the terms
are less than or equal to I log(| l|), then apply Dominant convergence theorem again, we
obtain

lim,__ fo XA (x)log (fB(x,r(x)) XA, (?)3(I (Ii (x) lo(y))2))dy dx

f- J XD(X)log (fB(x,r(x)) XD()3 (1 (II(x)- Io(y))2)d dx
Similarly we have continuity of the third term in the energy functional. Continuity of

these two terms together with (3-14) implies

E(XD) < liminfjoE(XAnj)

Hence XD is a minimizer.

Remark: Ii(x), 12(x) turn out to be mean values of 1o in a neighborhood of x

that is restricted to lie inside subregions A and 2 \ A respectively. This is comparable

but different from piecewise constant Mumford-Shah model in which intensity in each

subregion is approximated by a constant that is in fact the mean value of the original

image intensity in the whole subregion. In comparison, model (3-8) is more general and is

able to smooth and segment images that have high level noise involved.

3.5 Conclusion

A new model that utilized both intensity and edge information was introduced

to simultaneously segment and smooth images without using extra smoothness

terms. Due to lack of knowledge regarding image intensity distribution, we applied

nonparametric method to estimate intensity distribution, using intensity information

from neighboring points. Sizes of neighborhood were chosen adaptively based on image

gradient. Segmentation and smoothing results are obtained through minimizing negative
log likelihood and also the length of the objects boundaries. The minimization problem

was implemented using semi-implicit iterative scheme. During the iteration, segmentation

results were updated based on better smoothing images, smoothing results were then









updated by using the better edge information derived from the better segmentation

results. Segmentation and smoothing procedures were improving each other in one single

model. The adaptive neighborhood sizes allowed for variate smoothing levels for different

locations, such that edges were enhanced.

Experimental results on several sets of synthetic and real medical images with

different types and levels of noise demonstrated the potential of the proposed model in

simultaneously segmenting and smoothing images. Comparison with another three models

showed the advantage of the proposed model.









CHAPTER 4
ESTIMATION,SMOOTHING AND CHARACTERIZATION OF APPARENT
DIFFUSION COEFFICIENT

4.1 Introduction

There is growing interest in diffusion tensor denoising and reconstruction from DTI

data. There are several popular approaches: (1). Smoothing the raw data s(q) then

estimating the diffusion tensor from the smoothed raw data ([91-93]); (2). Smoothing the

principal diffusion direction after the diffusion tensor has been estimated from the raw

noisy measurements ([94-98]); (3). Smoothing tensor-valued data while preserving the

positive definite property of D [92, 96, 99-101].

However, very little research reported in literature to date on HARD data analysis

considered denoising problem in the reconstruction of the ADC profiles when the HARD

raw data is noisy. To improve the accuracy of the estimation, in this paper we present a

novel model that has the ability of simultaneously smoothing and estimating the ADC

profile d(x, 0, 0) from the noisy HARD measurements s(x, 0, 0) while preserving the

relevant features, and the positiveness and antipodal symmetry constraints of d(x, 0, 4).

The basic idea of our approach is to approximate the ADC profiles at each voxel by a 4th

order SHS(l1,a 4 in (1-13)):


d(x,0,e) S S Al,,(x)Yl,.(0,) (4 1)
1=0,2,4 m=-1

whose coefficients are determined by solving a constrained minimization problem. This

minimization problem minimizes a non-standard growth functional to perform a feature

preserved regularization, while it minimizes the data fidelity term. Notice, there are 15

unknown complex valued functions Al,n involved. Since d(0, 4) is real and Y,in satisfies

Y,-,n = (-1)nYiz,n, each complex valued Al,, is constrained by A,1-_ (-1)mA,mr,

where Al,m denotes the complex conjugate of Al,.. This constraint transforms the 15

unknown complex valued functions in (4-1) to 15 real valued functions: Ao,0(x), (1

0, 2,4), ReA,mr(x), I mA,,(x), (1 2,4;m r 1,...,1).









This chapter elaborates a work on how to obtain all those 15 coefficients at each

voxel. This method differs from the existing approaches developed in [67] and [72] mainly

in the aspect of the determination of the Al,m(x)'s in (1-13). In ([67]) the Al,m(x)'s (1 is

even) are determined by
/027 f0 1 s(q) -2)
A,,m(x) j j log1)Y (O, ,) i'i,'i ', (4-2)

and in [72] they are estimated as the least-squares solutions of

log = (q) A, ,m(, ). (4-3)
l=0 m=-1

In this chapter, the estimation of Al,m(x)'s is not performed individually at each

isolated voxel, but a process of joint estimation and regularization across the entire

volume. The joint estimation and regularization not only guarantees the wellposedness of

the proposed model, but also enhances the accuracy of the estimation since the HARD

data are noisy. Moreover, in this paper we provide more detailed method to characterize

the diffusion anisotropy, which uses not only the information of Al,m(x)'s as in ([67, 72]),

but also the variation of d(0, 4) about its mean. Our experimental results showed the

effectiveness of the model in the estimation and enhancement of anisotropy of the ADC

profile. The characterization of the diffusion anisotropy based on the reconstructed ADC

profiles using the proposed model is consistent with the known fiber anatomy.

4.2 Model Description

In this section we will present a variational framework for simultaneous smoothing

and estimation of the ADC profile d(x, 0, 0) from the noisy HARD measurements s(q).

This model minimizes an energy functional consisting of two parts: one is the cost for

regularizing d, and the other is the cost for fitting d to the HARD measurements through

the original Stei-l: il-Tanner equation (1-11). To explain the basic idea of our method,

we focus our attention on the cases where there are at most two fibers passing through a









single voxel. The same idea can be applied to the cases where there are more fibers within

a voxel.

The challenge in regularizing d comes from two aspects. First, d is defined on Q x S2

rather than Q x R2, hence, the derivatives for (0, 0) should be along the sphere. Secondly,

the regularized d has to preserve the antipodal symmetry property with respect to (0, 0).

Considering these facts we adopt the idea developed in [67, 72] that approximates d by its

SHS consisting of only even order components up to order 4 i.e.
I
d(x,0,A) S S A,(x)Yl,(0,). (4 4)
1=0,2,4 m=-l

The expression in (4-4) ensures the smoothness and antipodal symmetry property of

d(x, 0, 0) in terms of (0, 0), this is easy to see from the definition of Yl,m(0, 0). For the

cases where possibly k fibers cross in a single voxel, the sum in (4-4) should be replaced

by EC 0,2,...,2k'

Now the problem of regularization and estimation of d(x, 0, 0) reduces to that for the

15 complex valued functions Al,m(x) (1 0, 2, 4 and m -1,..., 1) in (4-4). Since d(0, 0)

at each voxel is a real valued function, and Y,m satisfies Yl,- = (-1)mYm, Ai,m should be

constrained by

A,-, ( 1 tmAA*

This constraint reduces the number of the unknown coefficients Az,m in (4-4) to 15 real

valued functions. They are


Alo(x), (1 0,2,4), ReAi,m(x), ImAi,m(x), (1 = 2, 4 and m 1,..., 1). (4-5)

By using (4-5), we can rewrite (4-4) as

I
d(x, 0,) = AA,o(x)Y,o(O, 0)+2 (ReAl,m(x)ReY, (0, Q)-ImA,m(x)rnImY,m )),
1=0,2,4 1=2,4 m=1
(4-6)









where ReF and ImF represent the real and imaginary part of a function F respectively.

Now the problem of regularizing and estimating d reduces to smoothing and estimation of

15 functions in (4-5) simultaneously.

There are many choices of regularizing operators to smooth the 15 functions in (4-5).

Total Variation (TV) based regularization, first proposed by Rudin, Osher and Fatemi[79],

proved to be an invaluable tool for feature preserving smoothing. However, it sometimes

causes a staircase effect making restored image blocky, and even containing 'false edges'

[102, 103]. An improvement, that combines the TV based smoothing with isotropic

smoothing, was given by ('!i i:lhoolle and Lions [103]. Their model minimizes the TV

norm when the magnitude of the image gradient is larger, and the L2 norm of the image

gradient if it is smaller. However, this model is sensitive to the choice of the threshold

which separates the TV based and isotropic smoothing. To further improve ('!ii i)hlolle

and Lions' model and make the model having an ability to self adjust diffusion property,

recently, certain nonstandard diffusion models based on minimizing Lp(') norm of image

gradient have been developed [102, 104]. To recover an image u from an observed image I

in [102] the diffusion was governed by minimizing


min |V7|l(I- )
JI

where p(s) is monotonically decreasing function and limso p(s) = 2, limso, p(s) = 1. In

[104] the diffusion was performed through minimizing


min j (x, Du)
u JI

where Q(x, r) := |rp() if Ir < a, and Q(x,r) := r ap x)-'P( if Ir > a for threshold

a. In this model p(x) is chosen as


p(x) = p(|VI|) = +
1 + k|VG, |2'









where k, a > 0 are parameters, G, is the Gaussian kernel. Both models in [102] and

[104] are able to self adjust diffusion range from isotropic to TV-based depending on

image gradient. At the locations with higher image gradients (p = 1), the diffusion is TV

based and strictly tangential to the edges ([79, 103, 105]). In homogeneous regions the

image gradients are very small (p = 2), the diffusion is essentially isotropic. At all other

locations, the image gradient forces 1 < p < 2, and the diffusion is between isotropic and

total variation based and varies depending on the local properties of the image. This self

adjusting ability enables these models to effectively preserve features when images are

smoothed.

Applying the idea of minimizing functionals with variable exponent to the problem of

regularizing the coefficients Al,m's in (4-4) we propose to minimize


Ei(Ai,m) := Si,(x, DAi,), (4-7)
S=0,2,4 m=-l

where
S r 1 |,m(x), |r| < M] ,Tn

r I | (1 ) r| > MITn
We would like to point out that for a function Ai,m E BV, DAI,m is a measure, the

definition of (4-7) is not obvious. This will be discussed in existence section 4.6 below.

In (4-8)
1
pl,m = 1 + (4-9)
p 1 + 1+kVG, *a l,(x)2'
and a,im is the least-squares solution of

log s(x, 00)= a ,m((x)Y,,(O0, ). (4-10)
S=0,2,4 m= -

In real application, MiL, is picked as 90th percentile of r. Experimental results are

not sensitive to this parameter.

Again we would like to point out that El only needs to include 15 terms

corresponding to the 15 real valued functions in (4-5). Here we write it in terms of









Az,m in order to shorten the expression of the formula. If using Az,i instead of al,m in

(4-9) we may get better numerical results, since pli, would depend on updated Az,i rather

than the fixed al,m in the iterations to minimize (4-7). However, it gives difficulty in the

study of the existence of solutions.

Since d(x, 0, 0) is related to the HARD measurements s(x, 8, 0) and so(x) through

the Stei-1: il-Tanner equation (1 11), the estimation of the Al,,'s is based on the original

Stei-1: il-Tanner equation (1 11) rather than its (log) linearized form (1 12), that is,

1 I r27x rv
E2(Am) = i (, 0, ) so(xe-)bd(x,0,)2 .1. (4- 1)
2 Jn Jo Jo

where d is determined in (4-4). As observed in [106] when the signal to noise ratio is low

the linearized model gives different results.

Finally, to simultaneously regularize and estimate the ADC d(x, 0, 0), our model

minimizes the energy function


E(Ai,m) := AEi(Ai) + E2(Ai,m), (4 12)

with respect to Az,i (1 0,2,4 and m -1,..., 1) in the space of BV( ), (in fact, only 15

functions in (4-5) are needed), and subject to the constraint:


d(x,80, ) > 0. (4 13)

In (4 12),(4-13), s(x, 0, 0) and so(x) are the noisy HARD measurements (real valued),

d(x, 0, 0) is the SHS given in (4-4), 2 C R3 is the image domain, A > 0 is a parameter

which could be different for different Am. El and E2 are given in (4-7) and (4-11),

respectively.

Before we derive the Euler-Lagrange equations for our model (4 12),(4-13), we would

like to point out that if the measurements satisfy the condition s(x, 0, 0) < so(x), the

solution of (4 12) meets the constraint (4-13) automatically. Therefore we can treat our

model as an unconstrained minimization. This is given in the following lemma.









Lemma: Under the assumption that


s(x, 0, 4) < so(x), for all x e Q, < 0 < T, 0 < < 27 (4-14)

the minimizer of (4-12) al-,v- satisfies the constraint (4-13).

Proof: Let Al,m(x) (1 = 0, 2, 4 and m -1,... ,1) be the minimizer of (4-12) in

BV(Q), and d(x, 0, 4) be the function defined in (4-4) associated with these optimal

Al,m(x)'s. Given x E Q, if d(x, 0, ) < 0 for some 0 < 0 < 7, 0 < Q < 27, then define

d(x, 0, ) 0, otherwise, define d(x, 0,) := d(x, 0, ).

Correspondingly,

Ai,m(x) :- O d(x, 06,' O) .(6, 0) :,,,, ,,

Then, using the orthonormality of the spherical harmonics and the definition of d, we have

,( Ai,m(x), if d(x, 0, ,) > 0, V 0 < 0 < 7, 0 < < 27r
Al,(x) (4-15)
0, if d(x, 0, ) < 0

This implies that

Oi,m(x, DAl,m) < (l,m(x,DAl,m),

hence

Ei(Ai,m) < Ei(A,,m).

Moreover, it is easy to obtain

E2(d) < E(d),

if (4-14) holds. From the last two inequality above, we obtain that E(d) < E(d). This

contradicts to the fact that d minimizes energy functional (4-12).

Now we give the evolution equations associated with the Euler-Lagrange (EL)

equations for (4-12): for 1 0,2,4 and m= -1,...,1,

A, Adiv(,01,m)x, DAi,m) b se-bd(S soe-bd)Y, .'.:,,1 (4-16)
at Jo Jo









with the initial and boundary conditions:

Al,m = al,m, on Q x {t = 0},

(a,01,m))(x, DAi,m) n= 0 on 90 x +.

In the above EL equation n is the unit outward normal to the boundary of Q, and

0,r0,m(x, r) is a continuously differentiable function in r, and

M1 p(x)-'r, Ir< ML,m
0,,,(x, r) > M) (4 17)
M I Irl

9,rl,m can also be written as

1,m(x, r) : 1 (4-18)

where q(x) = p(x) if Irl < M,,, and q(x) = 1 if Ir| > M-,.

4.3 Characterization of anisotropy

In [67] the IAlz,(x)l (1 = 0,2,4 and m -,...,1) in the truncated SHS (4-4) are

used to characterize the diffusion anisotropy at each voxel x. Our experimental results,

however, indicate this information alone is insufficient to separate isotropic diffusion,

one-fiber diffusion, and multi-fiber diffusion within a voxel. We propose to combine the

information from |Az,,i with the variances of d(b, 0) about its mean value to characterize

the diffusion anisotropy. We outline our algorithm as follows:

(1). If

Ro : |Ao,ol/ |A z,n, (4-19)
1=0,2,4 m= -
is large, or the variance of d(0, p) about its mean is small, the diffusion at such voxels is

classified as isotropic.

(2). For the remaining voxels, if
m= 2 1
R2 := |A2,j|/ |AI,,| (4-20)
m=-2 1=2,4 m=--









is large, the diffusion at such voxels is characterized as one-fiber diffusion. Figure 4-3 D)

presents an intensity-coded image of R2 in a brain slice through the external capsule, an

important structure of the human white matter. In Figure 4-3D) those voxels of a high

intensity (bright regions on the image) are characterized as one-fiber diffusion.

(3). For each uncharacterized voxel after the above two steps, search the directions

(0, 0), where d(0, 0) attains its local maxima. Note, d(0, Q) is antipodal symmetric,
i.e., d(0, 0) = d(7 0, + 7), we mod out this symmetry when count the number of

local maxima. Then we compute the weights for the local maxima (-w we have 3 local

maxima):

wi. 3d(Oi, 0t) -dun
1 d(0i, Oi) dmi

where (0O, 0i) (i = 1, 2, 3) are the directions in which d attains its local maxima. If one of

the weights is significant, it is considered as one fiber diffusion. If two weights are similar

but much larger than the third one, it is viewed as two-fiber diffusion, if all three weights

are similar, d can be considered either three-fiber diffusion or isotropic diffusion. In our

experiment we restrict ourselves to the cases where we only distinguish isotropic, one-fiber

or two-fiber diffusions. Under this restriction if three weights are similar, we include this

voxel in the class of isotropic diffusion. Figure 5-5A) shows our classification of isotropic

diffusion (dark region), one-fiber diffusion (gray region), and two-fiber diffusion (bright

region) in the same slice as in Figure 4-3.

4.4 Numerical Implementation Issues

To efficiently solve the Euler-Lagrange equations (4-16), we use Additive Operator

Splitting(AOS) algorithm for the diffusion operator (see [86, 107] ). By using this

algorithm, the computational and storage cost is linear in the number of voxels, and

the computational efficiency can be increased by a factor of 10 under realistic accuracy

requirements([86]). The algorithm is ready to be modified to a parallel version.









To avoid the complicated notation, we use X to represent any Al,m in the

Euler-Lagrange equations, and write the algorithm for only one of the equations (4-16) in

the system, since each equation has the same structure as others.

We use semi-implicit finite difference scheme:

X(n+l)- X (n) VX (n+1)
-1 X--- f (XW ) + Adiv (- -2q
S1 .1|X(TOVX |2-q

-AlnM Vqij- VX7 n+) A d VXy(1)
f(X )) + i 2- +qiv I i(4-2
Mij M 1 |VX \) |2-qij 1 |VX\ |2-qi

Here X can be replaced by one of Al,,'s with = 0, 2, 4, m -1... 1, and f is a

function of results from last iteration,namely, f is a function of all At's. q(x) = p(x) if

|VX| < M, and q(x) = 1 if |VX| > M for some fixed constant M, which was chosen based

on initial value of X, so M might be different for different Al,m's.

For simplicity of formulas, we define:


AXj = Xij Xilj, A Xi,j = Xi+,j X AXj X, Ai+l,j Xil,j


A+Xi,j = Xi,j+l Xi,j, AYXi,j = Xi,j Xi,j_, AYXi,j = Xij+l Xi,j_1

Adopting a discretization of the divergence operator from [16], one can write (4-21)

as:
Xyn+l)_X n) ) [ X aX+I) ,A Xy+ )/(X2h) .
2"= fI Y() lM i(X ) M i 2'i Mi
(Azx ( )2 (aYX )2) 2
(2h)T

/x x(n+l) Ay + n+1)
Ax +2-7 + ... .. 2-
S 2 i *2 qij
21 f ..(- G ) ) D )X + ( ) ( ) +2

f(X(T)) + (Cj G ij)Xl)f (Ci ,+ D ,j)Xi (Di,) + +Gj)X +


+ (Eij -H i,j)Xl (Ei,j + F,)X + (F.i, + Hij, )X


(4-22)










Where C, D, E and F are from divergence operation, while G and H are generated by dot

product, in detail:

qi-1,j-2
S(X i ( )2 (X ) ~)2
C^ _ij 2 A | i~j h^i 1,q + 1,j+1 ) 1j l |
F2 M(/2i h2 h 2 (2h )2

qij-1- 2

(A (()_-X 2 ( X( -( X )2 2
SMqij)h2 (h) 2 (2hi)2 1
(X(") -X(") 2 (X) X( ) 2 2

A i i+l,j i- ,j i ,j+
S- Mqijh2 (2h~)2 (h)2
iij -2
( I (q -) )24) ( )2 (1() _'(X)'2 2
Fi A i+ji- J, ) 1 +1 i j j

qij 2
H (2h _x) (xx)2 (x x )2 2





Solving (4-22) would involve matrix inverse operation, which would become more and

more complicated and dramatically expensive as dimension increases if we solve it directly.



reformat system (4-22) into the 2 following systems:
(n+1) X )) ()
i _,j A I qizl.--qi; 1 ) fi+l,j~X )i,j +,'
-- 1 ,' ,- (2h)2 (2h+)2

















2(Ci,j Gij)X(_ (Ci j + Dj)X ) + (D+ + Gjj)X- (4) (4-23)
j I f \ (Xy() -X(n) X2 (X (n)1 )2/
', = f (X i l, j 1,j

Solving (4-22) would involve matrix inverse operation, which would become more and

more complicated and dramatically expensive as dimension increases if we solve it directly.

Instead, here we use Additive Operator Splitting (AOS) algorithm, which allows us to

reformat system (4-22) into the 2 following systems:







(n+l) (n+1 (n)





2 [(Ei, HJ)X (E + Fj)X (F, + G/^) (4-24)


(n++) (nj) + X+)
X (+1) __ ,3 2 ,3
r, 2











To accommodate the boundary condition 0 for the M x N matrix X, one needs

to have:

Xt"+1) x(X+1) X(n+l) X(n+")
l,j 2,j M -,j Mj

X(n+l) y (nr+l) y(n+l) X(n+l)
i, i,2 i,N-1 i,N

Then (4-23) and (4-24) correspond to linear systems in matrix-vector notation:


A1X_1 ='rf(X[A ) + X

A (n+l) ((n)) (n)
A2X (+) rf(X () + X()


where X and X are (M 2)(N 2) x 1 vectors formed by columns and transpose of

rows of the original matrix X respectively, both A1 and A2 are (M 2)(N 2) x (M -

2)(N-2) matrices, specifically, A1 is a tri-diagonal matrix that repeats a (M-2) x (M-2)

tri-diagonal matrix (N 2)2 times diagonally, and A2 is a tri-diagonal matrix that repeats

a (N 2) x (N 2) tri-diagonal matrix (M 2)2 times. They are defined as:
A1 = I 2.

-D2,2 -G2,2 ,2 + G2,2 0
3,2 -G3,2 -C3,2 D3,2 D3,2 + G3,2
0


0
DM-2,N-1 + GM-2,N-1
0 CM-1,N- GM1,N-1 -CM-1,N-1 +GM-1,N-1

A2 = I 27.

F2,2 H2,2 F2,2 + H2,2 0
E2,3 H2,3 E2,3 F2,3 2,3 + H2,3 0
0


0
FM-1,N-2 + HM-1,N-2
0 EMi1,-1- HM-1,N-1 -EM-1,Nl1+- HM1,N1
X [X2,2 X3,2 ... XM-1,2 2,3... XM-2,N-1 XM-1,N-]T
S= [X2,2 X2,3 ... X2,N-1 X3,2 ... XM-1,N-2 M-1,N-1]T

Since both A1 and A2 are tri-diagonal matrices, one can get their inverses efficiently

by using Thomas Algorithm([108]).









4.5 Validation and Application to Diffusion Weighted Images(DWI)

In this section we present our experimental results on the application of the proposed

model (4-12)-(4-13) to simulated data and a set of HARD MRI data from the human

brain.

4.5.1 Analysis of simulated data

The aim of our experiment on the simulated data is to test whether our model can

efficiently reconstruct a regularized ADC profile from the noisy HARD measurements.

We simulated an ADC profile on a 2D lattice of size 8 x 4. The volume consists of two

homogeneous regions, values of So and all the Az,m's were shown in table 4-1.

Table 4-1: List of So and Az,m's for two regions
Region 1 2
So 414 547
Ao,o 5.21 x 10-3 1.43 x 10-2
A2,0 -1.17 x 10-3 -2.68 x 10-3
ReA2,1 -4.37 x 10-5 0
ReA2,2 1.43 x 10-3 0
ImA2,1 3.64 x 10-5 0
ImA2,2 3.28 x 10-5 0
A4,0 -3.15 x 10-5 8.4 x 10-6
ReA4,1 -1.56 x 10-4 0
ReA4,2 1.02 x 10-4 0
ReA4,3 6.30 x 10-5 0
ReA4,4 -8.54 x 10-5 -1.73 x 10-3
ImA4,1 -8.01 x 10-5 0
ImA4,2 0.9961.55 x 10-4 0
ImA4,3 1.41 x 10-5 0
ImA4,4 3.63 x 10-5 0


In Figure 4-1 we di-1-1 h- ,1 the true, noisy, and recovered ADC profiles d(x, 0, ) for the

synthetic data with size 8 x 4. The ADC profile d(x, 0, ) was computed by (4-6) based on

these simulated data, and the corresponding strue(x, 0, 0) was constructed via (1-11) with

b = 1000s/mm2. Then the noisy HARD MRI signal s(x, 0, 4) was generated by adding

a zero mean Gaussian noise with standard deviation = 0.15. Figure 4-1B) shows the









ADC profile d computed by (4-6), where the coefficients of the SHS are the least-squares

solutions of (4-3) with noisy s.

We then applied our model (4-12)-(4-13) to the noisy s(x, 0, 4) to test the

effectiveness of the model, with Ao,o = 4, A2,m = 40(m = -2... 2), A4, = 60(m = -4... 4).

By solving the system of equations (4-16) in 2.5 seconds on computer with PIV 2.8GHZ

CPU and 2G RAM using Matlab script code, we obtained 15 reconstructed functions

as in (4-5). Using these A,,m (the solutions of (4-16)) we computed d(x, 0, 4) via (4-6).

The reconstructed d(x, 0, 4) is shown in Figure 4-1C). Comparing these three figures, it

is clear that the noisy measurements s have changed Figure 4-1A), the original shapes

of d, into Figure 4-1B). After applying our model (4-12)-(4-13) to reconstruct the ADC

profiles, the shapes of d in Figure 4-1A) were recovered, as shown in Figure 4-1C). These

simulated results demonstrate that our model is effective in simultaneously regularizing

and recovering ADC profiles.

4.5.2 Analysis of human MRI data

The second test is to reconstruct and characterize ADC profiles d(x, 0, 0) from human

HARD MRI data.

The raw DWI data, usually contains a certain level of noise, were obtained on

a GE 3.0 Tesla scanner using a single shot spin-echo EPI sequence. The scanning

parameters for the DWI acquisition are: repetition time (TR) 1000ms, echo time

(TE) -Soi,- the field of view (FOV)=220 mm x 220 mm. 24 axial sections covering

the entire brain with the slice thickness 3.8 mm and the intersection gap=1.2 mm. The

diffusion-sensitizing gradient encoding is applied in fifty-five directions (selected for the

HARD MRI acquisition) with b = 1000s/mm2. Thus, a total of fi'l ---i:; diffusion-weighted

images, with a matrix size of 256 x 256, were obtained for each slice section. We applied

model (4-12) to these data to compute the ADC profiles in the entire brain volume. By

solving a system of equations (4-16) we obtained all the coefficients Az,m's in (4-5), and

determined d(x, 0, 4) using (4-6).









Then, we used these Al,m(x) to calculate Ro and R2 defined in (4-19) and

(4-20) respectively, as well as the variance a(x) of d(x, 0, 0) about its mean:

7(x) =- (d(x, 0, ) d(x, 0i, (ii)/55)2d0d. Based on results from the

HARD MRI data of this particular patient, we characterized the diffusion anisotropy

according to the following procedure. If Ro(x) > 0.856, or a(x) < 19.65 the diffusion

at x is classified as isotropic. For the remaining voxels if R2(x) > 0.75, the diffusion at

such voxels is considered as one-fiber diffusion. For uncharaterized voxels from these two

steps we further classified them by the principles stated in the section 4.3. The selection

of the thresholds mentioned above for Ro, R2 and a involves experts' input and large

sample experiments. Experimental results definitely depend on these thresholds, but not

sensitively.

Figure 4-2 presents A2,o(x), one of the coefficients in (4-6), for the particular slice in

the volume. The images A2,0(x) in Figure 4-2A) and 4-2B) are estimated by using (4-2)

and solving (4 12), respectively.

Figure 4-3 Compares FA and three R2(x)'s with Al,,(x)'s obtained from three

different models for the same slice as shown in Figure4-2. Figure 4-3A) di- p'-1' the FA

image obtained by using advanced system software from GE. The Al,m(x)'s used to obtain

R2(x) in Figure 4-3B) are directly computed from (4-2). Those used to obtain R2(x)
in Figures. 4-3C) and 4-3D) are the least-squares solutions of (4-3) and the solutions of

(4 12), respectively. In Figures. 4-3C) and 4-3D) the voxels with high levels of intensities

(red, yellow, yellow-light blue) are characterized as one-fiber diffusion.
Although the FA image in Figure 4-3A) is obtained based on a conventional DTI

model (1-7), it is still comparable with the R2 map, since single tensor diffusion

characterized by SHS representation from the HARD images agrees with that

characterized by the DTI model. However, in DTI a voxel with a low intensity of FA

indicates isotropic diffusion, while using our algorithm, multi-fibers diffusion may occur at

the location with the low value of R2.









The ability to characterize anisotropic diffusion is enhanced by this algorithm, as

shown in Figures. 4-3A)-4-3D). Figure 4-3B) indicates again that the estimates of Al,,

directly from the log signals usually are not good. Even the least-squares solution of (4-3)

is not alv--,v- effective. This can be seen by comparing the anatomic region inside the

red square of Figures. 4-3C) and 4-3D), which are zoomed in Figures. 4-4A) and 4-4B),

respectively. There is a dark broken line showing on the map of the external capsule

(arrow to the right on Figure 4-4A), this same region was recovered by the proposed

model and characterized by the third step in our algorithm as two-fiber anisotropic

diffusion (arrow to the right in Figure 4-4B). (The model solutions reduced the value of

R0, increased the values of R1 slightly, and made the 3rd step in our characterization to be

applied). Our results also show the connection in a cortical associative tract (arrow to the

left in Figures. 4-4B), however, this connection was not mapped out on Figure 4-3C) or

the zoomed image in Figure 4-4A). In fact this connection was not mapped out on Figures

4-3A)-B) either. All these mapped connections are consistent with known neuroanatomy.

Combined together, our results indicate that our proposed model for joint recovery and

smoothing of the ADC profiles has an advantage over existing models for enhancing the

ability to characterize diffusion anisotropy.

Figure 5-5A) shows a partition of isotropic, one-fiber, and two-fiber diffusion for the

same slice used in Figure 4-4. The two-fiber, one-fiber, and isotropic diffusion regions were

further characterized by the white, gray, and black regions, respectively. The region inside

the white square in Figure 5-5A), which is the same one squared in Figures. 4-3C) and

4-3D), is zoomed in Figure 4-4C). It is seen that the two arre', '1 voxels in Figure 4-4B)

are classified as two-fiber diffusion. The characterization of the anisotropy on the voxels

and their neighborhoods is consistent with the known fiber ,_an l. -m:v.

Figure 5-5 B) represents the shapes of d(x, 0, 4) at three particular voxels (upper,

middle and lower rows). The d in all three voxels is computed using (4-6). However, the

A,m((x) used in computing d on the left column are the least-squares solutions of (4-3),









V lop. 1% Vp 1




















A)


W% *b4 E



Br )
i p P u












B)


tib A t *' B


Figure 4-1. Comparing shapes of d. A) True d. B) The d generated by (4-6), with
Al,m's the least square solution of (4-10) with the noisy measurement s. C)
Recovered d by applying model (4-12).

while in the right column they are the solutions of the proposed model (4-12). The first

and second rows show two voxels that can be characterized as isotropic diffusion before

d. I,,i-ii.-r but as two-fiber diffusion after applying model (4-12). These two voxels are the

same voxels as in Figure 4-4 directed by arrows. The lower row of Figure 5-5 B) shows the

one-fiber diffusion was enhanced after applying our model.

Solving Al,m's of size 15 x 109 x 86 x 8 from 4-D data of size 55 x 109 x 86 x 8

takes 46.2 seconds for each iteration on computer with PIV 2.8GHZ CPU and 2G RAM in

Matlab script code.

4.6 An Existence Theorem for the Model

In this section we will discuss the existence of a solution to our minimization problem

(4-12) using the idea developed in [104].









Recall that for a function u E BV(Q),


Du Vu.- + D'u

is a Radon measure, where Vu is the density of the absolutely continuous part of Du
with respect to the n-dimensional Lebesgue measure / and Dsu is the singular part. To

minimize (4-12) over the functions in BV(Q), we first need to give a precise definition for

El.
Definition: For Al,m e BV(Q), define


j/ i,m(x, DAi,) :- I im(x, VAi,m)dx + D'Ai,m

where 0i,m is defined as in (4-8), and fo ID'Ai,mI is the total variation norm of A,m,.
Then, our energy functional (4-12) is defined as
I
E(AI^m A / Y Y. 0l,,m(x, VAim) + A DSAim
S=0,2,4 m= -1

+ I s(xt, )- so(x)e-bd(x,0,6) p2 ", 1, (4-25)
2 Jn Jo Jo
In the discussion of existence, without loss of generality, we set the parameter A = 1
in (4-12) and threshold Mi,m = 1 in (4-8) to reduce the complexity in the formulation.
Next we will show lower semi-continuity of the energy functional (4-12) in L1, i.e. if

for each 1,m (/ = 0,2,4 and m = -,...,1), as k oc,

Akn- Aom in L1(Q),

then

E(Ao,) < liminfkoE(A kT), (4 26)

To prove this we need the following lemma:









Lemma: Let
( 1 ,p(x), | ^ < 1
(x, r)= P(X) r- (4-27)
r ( ), ) rl >l1

For u e BV(Q) denote

()(u) : / (xDu),

and
~/ p(x)- 1 pe(r
S(U) sup -udivM P- ()- dx.
EC'(O,R) Q p(x)
SI<1
Then,

4 (u) (u) (4-28)

Furthermore, 4 is lower semi-continuous on L1('), i.e. if uj, u e BV(Q) satisfy uj u

weakly in L1(Q) as j oo then


4)(u) < liminf 4)(uj).
J-00
j---o

Proof First note that for each E c C'(Q, R"), the map

f p(x) 1 p(),
u! I udtMb Ax) 0- PW- dx
Jn p(x)

is continuous and affine on L1(Q). Therefore, 4 is convex and lower semi-continuous on

L1(Q) and the domain of 4, {u I 4(u) < oo}, is precisely BV(Q).

Next we show (4-28). For u e BV(Q), we have that for each e C( o, Rn),


udivdx Vu dx + I Du b


and so
S/ p(x) 1 p()
S(u) = sup IVu 1.- ||P()-Idx + D -u.
Ec (nR) p(x)
i <1









Since the measures dx and D"u are mutually singular, by a standard argument we can


have


sup (Vu.
I1ECI(',R")
l l <1


p(x) -1 P) u.
p(x) )-)d + Du.


To prove (4-28) it only remains to show that


I O(x, Vu)dx
J


sup (Vu -
bEC1 (Q,Rn")
Sl<1


p(x) -
p(x)


1 p(x)
-| I -1)dx.


Since any p E L' (R, R") can be approximated in measure by zb E C' (, R"), we have that


p(x) 1 pe,)
p(x) 10 Per)-I dx


1 p(-)
p-pe)-l dx. (4 -30)


sup Vu-p-p(X)-
pEL- (Q,Rn) j x)
pl<1


l{Ivu.lilVu1'uP()-l1 + l{ivul>i}jj where XE is the indicator function


on E, we see that the right hand side of (4-30) is


I O(x, Vu)dx.
Jn


> f p(x) | Vu i 1{v } + |IU


(4-31)


To show the opposite inequality, we argue as follows. For any p E L"(Q, R"), since

p(x) > 1 we have that for almost all x,


17 + p(x) -
Vu(x) p(x) Vu p(x)
p(x) ~ x


1 PWr
-p(x) I I~)-


In particular, if |Vu| < 1,


Vu(x) p(x)


p(x) -
p(x)


1 p(z)
Ip(x) Ip|x)l <


p x) UIP X
p(x)


If |Vul > 1, noticing p(x) > 1 and |p| < 1 for almost all x we have that

V Vu 1 p(.r)- 1l
p(Vu ) p(x)


#(u)


(4-29)


jn


sup
bEc1 (n,
k <1


(4-32)


Ch... p(Xr)


p(--1 1{Ivul>i}dx









and so


p(x) 1 (Ws 1 p(x) 1 W p(x) 1
Vu-p- pp 1< IVu +(Vul-1) lpl 1 < lVulp (4-33)
p(x) p(x) p(x) p(x)

Combining, (4-30), (4-31), (4-32), and (4-33), we have (4-29), and hence for all

u E BV (Q), (u) 4(u).

Note that O(x, r) = 0,m((x, r) if p(x) = pl,m(x). A direct consequence of this lemma is

that we have that El in (4-7) is weakly lower semi-continuous in L1 topology on BV(Q)

norm.

Furthermore, we can show that E2 in (4-7) is lower semi-continuous on L1(Q).

Indeed, when

Ak, -- A,mr, in L'(Q), as k ,

for all 1 0,2,4 and m -1,...,1), At m A, a.. on Then, if s(x, O ) e L2(xS2

and so(x) e L2(Q), by the Dominated Convergence Theorem we have

E2(A ) E2 (A,).


Therefore, E = El + E2 is lower semi-continuous in L1 topology on BV(Q) norm, and

(4-26) holds.

Now we can prove our existence results.

Theorem: Let Q be a bounded open set of R". Assume that s(x, 0, 0) E L2(Q x S2)

and so(x) E L2(2). Then, there exists a solution consisting of functions Ao, (1 = 0, 2, 4

and m -1,...,1) to the minimization problem (4-12) over the space of BV().

Proof: Let Aik ( = 0, 2,4 and = -1,... 1) be the minimizing sequences of

(4-12) in BV(). Then for each (1, m) the sequence A," is bounded in BV(Q). From

the compactness of BV(Q) there exist subsequences of Ak,, (still denoted by Al, ) and

functions Ao,, E BV(Q) satisfying

Ak ---A Jl,.i,,, in L'l().
IrZ IrZ









By the lower semi-continuity of the energy functional on L1(Q) (see (4-26)), we have


E(Am) < liminfk-E(Am)1 < ifnfAmeBV(Q)E(A,m).


Hence, all these Am (1 = 0,2,4 and m = -1,...,1) together form a solution to the

minimization problem (4-12).

4.7 Conclusion

A novel variational framework was introduced for simultaneous smoothing and

estimation of ADC profiles in the form of truncated SHS based on HARD MRI. Features

of this model included minimizing a nonstandard growth function with nonlinear data

fitting. Moreover, the constraints on the positivity and antipodal symmetry properties of

d was also accommodated in the model. We also demonstrated our algorithm for using

the variance of d from its mean and the coefficients of its truncated SHS approximation to

characterize diffusion anisotropy.

Our experiments on both synthetic data and human HARD MRI data showed

the effectiveness of the proposed model in the estimation of ADC profiles and the

enhancement of the characterization of diffusion anisotropy. The characterization

of non-Gaussian diffusion from the proposed method was consistent with known

neuroanatomy.

The choice of the current parameters, however, may affect the results. Our choice

was made based on the principle that classification for one-fiber diffusion from the model

solution should agree with a priori knowledge of the fiber connections.


























Wir
./". A


Ii


. "1


I.

A,

- VA



(b)


Figure 4-2. Comparison of A20. A) A20 computed from (4-2). B) A20 obtained from
model(4-12).





































v,}


Figure 4-3. Images of FA and R2. A) FA from GE software. B)-D) R2 with the Al,m's as
the solutions of (4-2), least-squares solutions of (4-3), and model solutions,
respectively.


\Ui








































a) (b) c

Figure 4-4. Zoomed FA and A20. A)-B) Enlarged portions inside the red squares in
Figures. 4-3 C) and 4-3D), respectively. C) Enlarged portions inside the white
squares in Figure 5-5A).











C)
a

It,


IL"


r


7)


( db)


k I


Figure 4-5.


Classification of voxels based on d. A) Classification: white, gray, and black
voxels are identified as two-fiber, one-fiber, and isotropic diffusion respectively.
B) Shapes of d(x, 0, 4) at three particular points (upper, middle and lower
rows). The d is computed via (4-6). Al,,(x) used in (4-6) in the left columns
are the least-squares solutions of (4-3), while in the right column are the
solutions from our model.









CHAPTER 5
ESTIMATION,SMOOTHING AND CHARACTERIZATION OF APPARENT
DIFFUSION COEFFICIENT(A SECOND APPROACH)

Copyright [2005] Lecture Notes on Computer Science [109]. Portions reprinted, with

permission.

5.1 Introduction

To use SHS model (4-1) to approximate d in (1-11), and hence to detect two-fiber

diffusion, at least 15 diffusion weighted measurements s(q) over 15 carefully selected

directions are required. However, to use the mixture model (1-9) with n=2 to detect

two-fiber diffusion only 13 unknown functions: f, 6 entries of each of D1, D2 need to be

solved. This motivated us to study what is the minimum number of the diffusion weighted

measurements required for detecting diffusion with no more than two fiber orientations

within a voxel, and what is the corresponding model to approximate the ADC profiles in

this case. In this chapter we propose to approximate the ADC profiles from HARD MRI

by the product of two second-order SHS's instead of a fourth-order SHS. We also show

that the product of two second-order spherical SHS's describes only the diffusion with at

most two fiber orientations, while the fourth-order SHS may describe the diffusion with

three fiber orientations.

Moreover, in this chapter we will introduce an information measurement developed

in [110], and termed as CRE(see definition (5-7)) to characterize the diffusion anisotropy.

CRE differs from Shannon entropy in the aspect that Shannon entropy depends only on

the probability of the event, while CRE depends also on the magnitude of the change of

the random variable. We observed that isotropic diffusion has either no local minimum or

many local minima with very small variation in the denoised s(q)/so, i.e., e-bd profiles,

in comparing with one fiber or two-fiber diffusions, which implies the corresponding CRE

to be small. We also found that one fiber diffusion has only one local minimum with

larger variation in the s(q)/so profiles, which leads to larger CRE. Therefore, we propose









to properly threshold the CRE for the regularized s(q)/so profiles to characterize the

diffusion anisotropy.

5.2 New Approximation Model for ADC Profiles

In[66, 72, 111] to detect the diffusion with at most two fiber orientation the ADC

profiles were represented by a truncated SHS up to order 4 in the form of (4-1). In [66]

the coefficients Al,m's (1 is even) were determined by inverse spherical harmonic transform

of -Ilogsq) and in [72] they were estimated as the least-squares solutions of

S / max
log s = Ai,mYim(, 4). (5-1)
S1=0 m=-l

Regularization on the raw data or Ai,m wasn't considered in these two work. In [111]

Al,m's were considered as a function of x, and estimated and smoothed simultaneously by
solving the following constrained minimization problem:


minm { VA,(x) Wm(x) + IVo(x)p(X}dx
Aj,m(x),so(x) ,,
l=0,2,4 m= -l

+ { I2 j Is(x, q) so(x)e- bd(x,0,)12 2: *'.1'. + Io -So 2}dx, (5 2)

with the constraint d > 0. In this model p,,,(x) 1 + 1+kVGAimF, q(x) 1 +

1+kIVGa*,802, and d takes the form (4-1). By the choice of pt,, and q, the regularization
is total variation based near edges, isotropic in homogeneous regions, and between

isotropic and total variation based depending on the local properties of the image at other

locations. In this work since the ADC profile was approximated by (4-1), at least 15

measurements of s(q) were required to estimate the 15 coefficients Ai,,.

However, the mixture model (1-9) with n = 2, which is also able to detect two-fiber

diffusion, involves only 13 unknown functions. This motivates us to find a model that is

able to detect non-Gaussian diffusion with the minimum number of unknowns. In this

paper we only discuss the diffusion with no more than two fiber orientations within a

voxel. The significance of this study is clear: a smaller number of unknowns lead to a









smaller number of required HARD measurements. This will significantly reduce the scan
time and thus is important in clinical applications.
Our basic idea is to approximate the ADC profiles by the product of two second order
SHS's instead of a SHS up to order four. This can be formulated as
I I
d(x, ) ( bzm(x)Ym,)) (Z c ,(x)Ym(,)). (5-3)
1=0,2 m=-l 1=0,2 m=-l

In this model there are only 12 unknowns: bl,m, cl, (1 0, 2 and -1 < m < ).

To estimate the ADC profile from the raw HARD MRI data, which usually contains a
certain level of noise, we propose a simultaneous smoothing and estimation model similar

to (5-2) for solving bli,, cl,m, that is the following constrained minimization problem:

mmin { Y a( Vbi,m(x)| + |Vc,m(x)| + 0 |Vo(x) dx
b,m (x),Cl (x),-so(x) '1J 0,2=-n1



+ j f{ Is(x, q) o(x)e-bd(x,,6) 12,. + Io s2}dx, (5 4)

with constraint d > 0, where d is in the form of (5-3). a, 3 are constants. The first 3
terms are the regularization terms for bl6,, cl, and so respectively. The last two terms are
the data fidelity terms based on the original Stei-l:. i-Tanner equation(1-11).
Next, feasibility of this model will be explained. Let SA denote the space of even
SHS of order 4, i.e., SA {= d : d(O, ) = A = o=0,2,4 _=- A1,,Y},, let SBC be
the space of products of two even SHS of order 2, i.e., SBC = {d : d(O, 0) = B C

i=0,2 tm=- 1bm Yim(0, Q) C1=,2 ~~m=-I c,Y,T(0, Q)}. Since the product of two SHS
of order 2 and 2 can be expressed as a linear combination of spherical harmonics of order
less or equal to 2 + 2 = 4, SBC c SA. But a simple dimension count, dim(SA)=15, while
dimension of SBC is less than or equal to 12. Then SBC is a proper subset of SA. So
functions in SBC are less general than functions in SA. However, numerous experiments

show that when a voxel is not more complicated than two-fiber diffusion, its ADC is
well-approximated by a function in SBC. This is not true if 3-fiber or more complicated









diffusion is involved(see Figure5-1). Therefore, if we focus only on characterizing at

most two-fiber diffusion, which is the most common case, model (5-3) is reasonable and

sufficient to represent ADC.

Model (5-4) is a minimization problem with constraint d(0, ) > 0 for all 0 < 0 <

7r, O < 0 < 27r, which is usually difficult to implement. To improve the efficiency of

computation we used the idea that any second order SHS Y =0,2 Inm=- bi,mYi,m(0, 0)

is equivalent to a tensor model uTDu for some semi-positive definite 3 x 3 matrix D,

where u(0, 0) = (sin0cos,, sin0sin,, cos0). This means that the coefficients bl,m,(l

0, 2, m -1,..., 1) in SHS and the entries D(i,j), (i,j = 1,..., 3) in D can be computed

from each other explicitly. Here are two examples: boo = Ir(D(1, 1) + D(2, 2) + D(3, 3)).

D(1, 1) Vb20-2boo- 3Re(eb2) where Re(b22) is the real part of b22. Hence, we could

let B = uDluT,C = uD2uT, and d = (uD1T)(uD2UT). Then for i = 1,2we can write

Di LiL with Li a lower triangular matrix to guarantee semi-positiveness of Di. The

ADC is finally approximated by


d(x,0, -)= [u(0, )LI(x)L ) TU(0, ) T][(u(0, )L2(x)L2(x)Tu(0, )T]. (5-5)

Model (5-4) is then replaced by


mm (a kL + |V X
Ljre(x),Lj' (x),o(x-) 1 Lj + l ol) dx
L1 (x),L (x),so(x) J i1 j=1 k 1

1 2x
j{ |j -I soe-bd 2 -' 0 + S2}dx, (5-6)
2 oJo
where d = (uLLLTuT)(uL2LTuT). All the bl,m,cl,m,l = 0,2,m = -1...1 are smooth

functions of Li ,i 1, 2; j 1, 2, 3; k < j. Smoothness of Li guarantees that of bl,m's,

cl,m's. The first term in model (5-6) thus works equivalently to the way the first two terms

in model (5-4) do, while all the other terms are the same as those remaining in (5-4).

Hence, (5-6) is equivalent to (5-4), but it is a non-constrained minimization problem and









is thus easy to implement. After we get L1 and L2, bl,m and cl,m in (5-4) can be obtained

by the one to one relation between them.







A) B) C) D)

Figure 5-1. Comparison of the ADC's approximated by (4-1) and (5-3) in four cases: A)
isotropic diffusion; B) one-fiber diffusion; C) two-fiber diffusion; D) three-fiber
diffusion. In A)-D) from left to right, top to bottom, we show shapes of B, C,
B C, and A, respectively.


We applied model(5-6) to a set of human brain HARD MRI data to reconstruct

and characterize ADC profiles. The data set consisted of 55 diffusion weighted images

Sk : Q -- R, k = 1,..., 55, and one image So in the absence of a diffusion-sensitizing

field gradient(b 0 in (1-11)). 24 evenly spaced axial planes with 256 x 256 voxels in

each slice are obtained using a 3T MRI scanner with single shot spin-echo EPI sequence.

Slice thickness is 3.8mm, gap between two consecutive slices is 1.2mm, repetition time

(TR) = 1000ms, echo time (TE) = S-,, and b = 1000s/mm2. The field of view (FOV)

220mm x 220mm. We first applied model(5-6) to the data to get Li, and then used Li

to compute bl,m and cl,m, = 0, 2, m = -1...1, and the ADC d = B C. For purpose of

comparison, we also used the model (5-2) to estimate Al,m and get A. The comparison

for the shapes of ADC in the form of B C and A is demonstrated in Figure5-1A)-D) at

four specific voxels. The diffusion at these 4 voxels are isotropic A), one-fiber B), two-fiber

C), and three-fiber D), respectively. In each sub figure, the up left, up right, down left,

down right ones are the shapes of B, C, B C and A, respectively. It is evident that if

the diffusion is isotropic, one-fiber or two-fiber, B C and A are the same. However, if the

diffusion is three-fiber, A can't be well approximated by B C.









To show the effectiveness of the proposed model in recovering ADC, in

Figure5-2A)-D) we compared images of R2 (defined in section 5.3) with coefficient Am,,

estimated by 4 different methods. The voxels with higher value of R2 were considered

as one-fiber diffusion. The Al,'s in A), B) and C) were estimated using least-squares

method in [72], model (5-2), and model(5-6) with the diffusion-sensitizing gradient

applied to 55 directions, respectively. The A,,'s in D) are estimated by the same way as

that in C), but from the HARD data with 12 carefully chosen directions. The model (5-6)

applied on 55 measurements worked as good as the model (5-2) in getting higher value

of R2. Both of them worked better than the least-squares method that does not consider

regularization. Although the result from 12 measurements was not as good as that from

55 measurements, they are are still comparable. We will show in Figure5-5A) and B) that

the anisotropy characterization results based on the ADC presented in C) and D) are also

close. These experimental results indicated that by using the proposed model the voxels

with two-fiber diffusion can be detected reasonably well from 12 HARD measurements in

carefully selected directions.

5.3 Use of CRE to Characterize Anisotropy

As mentioned, FA is only able to detect Gaussian diffusion. For non-Gaussian

diffusion, Frank and Alexander et al. used the order of significant components in SHS to

characterize anisotropy. They considered voxels with significant 4th order components as

two-fiber diffusion. In [111] C'!i, 1 et al. realized that such a voxel could have isotropic or

one-fiber diffusion. They defined Ro := A,2, 2 2 A2, Higher
=0,2,4 m=-1l ,m E=0,2,4 im=- A,mI
values of Ro and R2 are correspond to isotropic and one-fiber diffusion, respectively.

For other points, the number of local maxima of ADC, together with the weights of

the variances at the local maxima, were used to classify voxels as isotropic, one-fiber or

two-fiber diffusion. This procedure is more precise, but there are many criteria involved

and thus more thresholds needed to be set subjectively. In this section, we will introduce a

simple scheme using only one measurement, CRE, and two thresholds.









CRE is a measure of uncertainty/information in a random variable. If X is a random

variable in R, the CRE of X is defined by


CRE(X) P(X > A)logP(X > A)dA, (5-7)
JR+

where R+ = {X RX > 0}.










A) B) C) D)

Figure 5-2. A)-D) are images of R2 with Alz,'s calculated using least-squares method,
model (5-2), model (5-6) applied on 55 measurements, and model (5-6)
applied on 12 measurements, respectively.


We use CRE of e-bd rather than d to characterize diffusion anisotropy when d is

recovered from HARD measurements through(5-6). The magnitude of ADC is usually on

the order of 10-3, while the magnitude of e-bd is in the order of 10-1, which is larger than

that of ADC itself. Moreover, e-bd is a smooth approximation of the data s/so.

The weak convergence property of CRE proved in [111] makes empirical CRE

computation based on the samples converges in the limit to the true CRE. This is not the

case for the Shannon entropy. We define the empirical CRE of e-bd as
M
CRE(e-bd(0)) P bd(0,6) > Xi)logPe-bd(0, > A,)JAA (5-8)
i= 2

where {AI < A2 < ... < AM is range of e-bd at voxel x. AAi = i Ai-1 is the absolute

difference between two .,1i ,i:ent values of e-bd; note this term is not shown in Shannon

entropy. In most of the cases, the variation of e-bd is the largest for one-fiber diffusion

voxels, smaller for two-fiber diffusion and smallest for isotropic voxels. This also explains















Figure 5-3.
Figure 5-3.


w* i


A) B) C) D)

An example for shapes of ADC for isotropic, one-fiber and two-fiber voxels.
A) Shapes of ADC at an isotropic (first row),one-fiber (second row) and
two-fiber (last row). B)-C) Graphs of F(A), -F(A)logF(A) at three particular
voxels:isotropic (red),one-fiber (green),two-fiber (blue). D) R2 (blue),CRE
(yellow),variance (black) as functions of rotation angle b used in constructing
synthetic data.


why CRE is the largest for one-fiber, medium for two-fiber and smallest for isotropic

diffusion voxels. In our experiment, we choose M 1000 uniformly distributed directions

(0, ) in (5-8).

Define the decreasing distribution function F(A) := P(e-bd > A). Figure5-3B)

shows the graphs of F(A) at three pre-classified voxels: isotropic (red), one-fiber(green),

two-fiber(blue). It is observed that the support and magnitude of F(A) are largest at the

voxel with one-fiber diffusion, and smallest at that with isotropic diffusion. Figure5-3

C) demonstrates the graphs of -F(A)logF(A) at the same three voxels. It is evident

that the area under the green curve(one-fiber) is much larger than that under the blue

curve(two-fiber), while the area under the red curve is the smallest. Since CRE is exactly

the area under curve -F(A)logF(A), we can see that the measure CRE(e-bd) is the

largest at the voxels with one-fiber diffusion, medium with two-fiber diffusion, and smallest

with isotropic diffusion. Thus measure CRE(e-bd) could be used to distinguish isotropic,

one-fiber and two-fiber diffusion with two thresholds T1 and T2, with T1 < T2. Set up 3

intervals: (0, TI), (T, T2), (T2, oo). Voxels with CRE that fall into the first, second, and

third intervals are classified as isotropic, two-fiber and one-fiber diffusion respectively.

Figure5-3D) on synthetic data and Figure5-4 on human brain HARD MRI data

further show the strengths of CRE over the three popularly used measures, R2, FA and


















A) B) C) D)

Figure 5-4. Images of four measures: A)R2; B) FA; C) CRE of e-bd; D) Variance of e-bd


variance, in characterizing diffusion anisotropy. The human data are the same as that used

in Figure5-2. The synthetic data are constructed as follows: Set D1 and D2 to be two 3 x 3

diagonal matrices with diagonal elements 4x 10-2, 10-2, 2x 10-2 and 8x 10-2, 10-2, 3x 10-2,

respectively. Then fix D1 but rotate principal eigenvector of D2 about xy axis by angle )

to get D2()). Let B(O, ) -= TDiu, C(0, 0) := UD2(Q)u. We computed R2, FA and

CRE, variance of e-bB'C for various values of b and showed them in Figure5-3D) in blue,

yellow and black respectively When i varies from 0 to r/2, B C changes from a typical

one fiber diffusion to a two fiber diffusion, and from r/2 to 7 B C changes back to the

same shape as = 0. The graph of CRE shows the value of CRE decreases when B C

varies from one-fiber diffusion to two-fiber diffusion, and increases when B C gradually

changes from two-fiber diffusion backs to one-fiber diffusion.

R2 cannot detect multi-fiber diffusion since it measures the significance of only

the second order components in SHS. N. i.-i3iii 1Ii, difference between R2 and FA is

observed from the images in Figure5-4A) and B). But CRE differs much from R2 and FA.

In Figure5-3D) the graph CRE is much steeper than the others. In Figure5-4, visually,

contrast of CRE is much better than that of FA and R2. Furthermore, the smallness of

magnitude of R2 or FA is unable to distinguish between isotropic and two-fiber diffusion,

while that of CRE does better job. Note, CRE is comparable to FA or R2 in detecting

isotropic and one-fiber diffusion.









Next we discuss from the theoretical point of view why CRE beats variance in

characterizing diffusion anisotropy. Let X be a random variable, Var(X) its variance.

According to the proof in [112], E(IX E(X)I) < 2CRE(X). In our case X is e-bd

whose magnitude is a multiple of 10-2 < 1, so we have Var(X) = E(IX E(X) 2) <

E(IX E(X)|) < 2CRE(X). Our experimental results show that magnitude of CRE is

almost 10 times of that of Var(X). Higher magnitude of CRE makes it less sensitive to

rounding errors. Moreover, in Figure5-4D), which representing the the variance of e-bd,

the Genu/Splenium of corpus callosum is so bright that regions besides it are not clearly

visualized so CRE is much better than variance visually.


Figure 5-5.


A) B) C) D)

A)-B). ('C! i:terization: black, gray, and white regions represent the voxels
with isotropic, one-fiber, and two-fiber diffusion, respectively. In A) we used
55 measurements, in B) we used 12 carefully selected measurements. C) Image
of CRE calculated from 12 measurements. D) Characterization results of
the region inside the red box in A) using CRE (top) and variance (bottom)
based on 55 measurements. Red arrows point to a voxel that is wrongly
characterized as one-fiber diffusion by using variance but correctly classified
as two-fiber diffusion using CRE.


Figure5-5A) shows a partition of a brain region into isotropic,one-fiber and two-fiber

diffusion based on ADC calculated from 55 measurements. The black, gray, white

voxels are identified as isotropic, one-fiber and two-fiber diffusion, respectively. The

characterization is consistent with known fiber anatomy. Figure5-5B) represents the

characterization result based on the ADC estimated from 12 measurements. It is

comparable with that from 55 measurements. CRE based on ADC estimated from









12 measurements (Figure5-5C)) is also comparable to that from 55 measurements

(Figure5-4C)). Thus our characterization appears not to be sensitive to number of

measurements as long as at least 12 measurements are used. Figure5-5D) illustrates a

two-fiber diffusion voxel (indicated by red arrow) that is incorrectly characterized as

one-fiber diffusion using variance (bottom image) but characterized as two-fiber correctly

using CRE (top image). This further verifies the superiority of CRE over variance in

characterizing diffusion anisotropy.

5.4 Conclusion

In this chapter, we presented a novel variational framework for simultaneous

smoothing and estimation of ADC profiles depicted by two diffusion tensors. To our

knowledge this was the first attempt to use the least amount of measurement to detect

two-fiber diffusion from human brain HARD MRI data. We also demonstrated our

algorithm for using CRE of e-bd to characterize the diffusion anisotropy.

Our experiments on two sets of human brain HARD MRI data showed the

effectiveness and robustness of the proposed model in the estimation of ADC profiles

and the enhancement of the characterization of diffusion anisotropy. The characterization

of diffusion from the proposed method was consistent with known neuim n ,Ii w,:.








CHAPTER 6
RECONSTRUCTION OF INTRA-VOXEL STRUCTURE FROM DIFFUSION
WEIGHTED IMAGES
Copyright [2006] IEEE [71]. Portions reprinted, with permission.
6.1 Determination of Fiber Directions

In our approach of determining fiber directions, the first step is to recover the ADC
profiles d from noisy HARD data by using model (4-12) and (4-13). Then, from the SH
representation of the recovered d we define

|Ao,o(x) EZ 22 A2,m X)
Ro0(x) = A0,( R2(X) = A
A(x) A(x)

where A(x) = Y=0,2,4 Ym=-1 A,m |(x). The voxels with significant Ro and R2 are
identified as strong isotropic diffusion and one-fiber diffusion, respectively. The union of all
these voxels is denoted by Q1. On Q1, flin (1-9) should be close to 1.
Under the assumption of pt(r) being a mixture of two Gaussians, the diffusion is
modelled by (1-9) with n = 2. The combination of (1-9) with n = 2 and (1-11) yields

2
e-bd(x,0,y) ie-buT Di(x), (6 1)
i=1
where uT = (sinOcosQ, sinOsinQ, cosO). For notation conciseness, denote fl by f. To
estimate Di and f in (1-10) we minimize the following function w.r.t L1, L2, f:
2
( IVL iP(x) + IV/f (X))dx +i f (f 1)2dx

27 7Tr 2
+ A2 1 1e-f-buTL0 LLu -bd 1 2.1.1.,.x (6-2)
A2 I J0 0 eie I.,l, Ix
with the constraints Lm > 0. In (6-2) for i = 1, 2, Ai > 0 is a parameter,

pj(x) 1 + 1+kVGVLil2, pf(X) 1 + 1+kVG aVfl2, and Li is a lower triangular
matrix. Di is recovered from Li by Di = LiLT( i.e., LiLT is the C'!i. -l:y factorization
of Di). Writing the cost function in terms of Li rather than directly in terms of Di









ensures that the positive-definite constraint on Di is met (see [101]). Also we have

IVLJ = Zl1m,n<3 VILmn l P

The first two terms in (6-2) are the regularization terms. By the choice of pi(x)

(similarly for py), in the homogeneous region image gradients are close to zero and

pi(x) 2 2, so the smoothing is approximately isotropic. Along the edges, image gradient
makes pi(x) t 1, so the smoothing is approximately total-variation-based and is almost

only along the edges. At all other locations, the image gradient forces 1 < p < 2, and the

diffusion is between isotropic and total variation based, and varies depending on the local

properties of the image. Therefore, the smoothing governed by this model well preserves

relevant features in these images. The third term in (6-2) is forcing f 1 on Qr. The last

term is the nonlinear data fidelity term based on (6-1).

The fiber orientations at each voxel are determined by the directions of the principal

eigenvectors of D1 and D2. For the voxels where f (or 1 f) is significantly large, we

consider 1 f (or f) as zero, and (1-10) with n = 2 reduces to the Gaussian diffusion

model.

6.2 Validation and Application to HARD Weighted Images

We applied model (6-2) to a set of HARD MRI human data. The raw HARD MR

images were obtained using a single shot spin-echo EPI sequence. The imaging parameters

for the DW-MRI acquisition are repetition time (TR) = 1000ms, echotime(TE)

S ,. Diffusion-sensitizing gradient encoding is applied in fifty-five directions with

b = 1000s/mm2. Thus, a total of fifl i--i:: DW images with the matrix size =256 x 256

were obtained for each slice, and images through the entire brain are obtained by 24

slices. The slice is transversally oriented and the thickness is 3.8mm, and intersection gap

between two contiguous slices is 1.2mm. The field of view (FOV) =220mm x 220mm.

We will show our experimental results from a particular subject in a brain slice

through the external capsule. In this experiment we first recovered the ADC profiles

d using (4-12)-(4-13), and defined QR as the set of the voxels where Ro > 0.8416 or









R2 > 0.1823. These thresholds were selected using the histograms of Ro and R2. Then, we

solved the minimization problem (6-2) by the energy descent method. The information of

f t 1 on R1 was also incorporated into the selection of the initial f.

By solving (6-2) we obtained the solutions Li and f, and consequently, Di

LiL (i = 1, 2). Figure6-1A) represents the model solution f. Function f m 1 on

the dark red regions. The voxels in these regions are identified as isotropic or one-fiber

diffusion. This is consistent with known neui.l." i- ,' wl Figures 6-1C) and 6-1D) show the

color representation of the directions of the principal eigenvectors for D (x) and D2(x),

respectively. By comparing the color-coding in Figures6-1C) and 6-1D)with the color pie

shown in Figure6-1B), the fiber directions are uniquely determined. The representation in

Figure6-1B) is implemented by relating the azimuthal angle (4) of the vector to color hue

(H) and the polar angle (0 > r/2) to the color saturation (S). Slightly different from [113],

we define H = /2r, S = 2(7r 0)/7, and Value V = 1. If the direction of the principal

eigenvector is represented by (0, 4), the fiber orientation can be described by either (0, 4)

or (r 0, 0 + 7r). To resolve this ambiguity, we choose the eigenvector to lie in the lower

hemisphere, i.e. 0 > r/2. The upper hemisphere is just an antipodally symmetric copy of

the lower one. The xy plane is the plane of discontinuity.

Figure6-2 shows the shapes of d(x, 8, 0) together with the fiber directions at 4

particular voxels. The blue and red arrows indicate the orientations of the fibers

determined from the principal eigenvectors of D1 and D2 respectively. The last shape

corresponds to isotropic diffusion. Figs.6-1 and Figure6-2 indicate that our model (6-2) is

effective in recovering the intra-voxel structure.

To examine the accuracy of the model in recovering fiber directions, we selected a

region inside the corpus callosum where the diffusion is known to be of one-fiber. For each

voxel in this region we computed the direction in which d is maximized. This direction

vector field is shown in Figure6-3A). On the other hand we solved (6-2) and obtained

the model solution f t 1 on this region. The direction field generated from the principal