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Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2010-05-31.

Permanent Link: http://ufdc.ufl.edu/UFE0021948/00001

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Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2010-05-31.
Physical Description: Book
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

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Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

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Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Chen, Yunmei.
Electronic Access: INACCESSIBLE UNTIL 2010-05-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021948:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021948/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2010-05-31.
Physical Description: Book
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Chen, Yunmei.
Electronic Access: INACCESSIBLE UNTIL 2010-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021948:00001


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IwouldrstandforemostliketoexpressmydeepestgratitudetomyadvisorprofessorYunmeiChen,foreverythingshehaddoneformeduringmydoctoralstudy.Thisdissertationwouldnotbepossiblewithouther.Sheprovidedwonderfulresearchopportunities,invaluableadvicesonresearchandlife.Dr.Chenintroducedmetotheeldofmedicalimagingandalwayshelpsandencouragesme.Ihavebeenveryluckytoworkwithher.Herenthusiasmaboutresearchstimulatesmyinterests,herinsightandexperiencehaveguidedmethroughmyresearch.Iwouldalsoliketothankmyotherexcellentcommitteemembers,Dr.JamesF.Dempsey,Dr.YijunLiu,Dr.JayGopalakrishnan,Dr.MuraliRaoforprovidingnumerousadviceanddiscussion.IespeciallyappreciateDr.JamesF.Dempsey,Dr.AnneyukoSaito,Dr.XiaodongLuandDr.JonathanG.Li,Dr.YijunLiuandDr.GuojunHeforhappycooperations.Theyprovidedmelotsofinvaluableresearchmotivationsandclinicaldata.IwouldliketothankWeihongGuoandFengHuang,forhelpingmewithnumericalimplementationinmyearlystageresearch,withouttheirhelp,Icouldnothavetoday'sscienticcomputationability.IwouldalsoliketothankDr.XuechengTaifromUniversityofBergen,Norway,Dr.RachidDerichefromINRIAforexcellentcommentsanddiscussiononmywork.Ineverwouldhavebeenabletohavenishedthedissertationwereitnotfortheunwaveringloveandsupportofmyfamily:mywifeWeihong,mymumMinLei,motherin-lawZhongying,brothersQingjunandWeidong,andsisterShuangxia.Theirsupportandencouragementaremysourceofstrength.Iwanttoespeciallythankmywifewhoteachesmethenatureoftruelove.Sheisanextremelysupportivelady,Ialwaysappreciateherenthusiasmforlifeandresearch.Ifeelsogratefulaboutmy22-month-olddaughterTienna,whoteachesmehowwonderfullifeisbyhersunnysmiles.IalsothankMr.ShengjingZhangandDr.QingHefortheirunforgettableguidancebeforemyPhD.ThankEditorialoceforthehelpIhavereceivedineditingthisdissertation. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 1.1ADCApproximation,DiusionAnisotropyAndDiusionDirectionCharacterizationInDiusionWeightedImage(DWI) ...................... 10 1.2NeuronFiberTractsReconstructionBasedOnSmoothTensorField .... 17 1.3MedicalImageRegistration .......................... 19 1.4Contributions .................................. 23 2ESTIMATION,SMOOTHINGANDCHARACTERIZATIONOFADCINHARDWEIGHTEDIMAGEUSINGSPHERICALHARMONICSERIESUPTOORDER4 ............................................ 28 2.1ModelDescription ............................... 28 2.2CharacterizationOfAnisotropy ....................... 34 2.3NumericalImplementationIssues ....................... 35 2.4ExperimentalResults .............................. 38 2.4.1Analysisofsimulateddata ....................... 38 2.4.2AnalysisofhumanMRIdata ...................... 40 2.5Existence .................................... 44 2.6Conclusions ................................... 48 3ESTIMATION,SMOOTHINGANDCHARACTERIZATIONOFADCINHARDWEIGHTEDIMAGEUSINGPRODUCTOFTWOSPHERICALHARMONICSERIESUPTOORDER2 ............................. 52 3.1NewApproximationModelForADCProles ................. 52 3.2UseOfCREToCharacterizeAnisotropy ................... 56 3.3Summary .................................... 61 4RECONSTRUCTIONOFINTRA-VOXELSTRUCTUREFROMHARDWEIGHTEDIMAGE ........................................ 62 4.1DeterminationOfFiberDirections ...................... 62 4.2ExperimentalResults .............................. 63 4.3Conclusion .................................... 66 5

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............................... 68 5.1RecoveryOfMulti-TensorFieldInHARDMRI ............... 68 5.2WhiteMatterFiberTractography ....................... 69 5.3ExperimentalResults .............................. 70 5.4Conclusion .................................... 74 6DEFORMABLEMEDICALIMAGEREGISTRATION .............. 76 6.1Same-ModalityDeformableImageRegistration ................ 76 6.2Multi-ModalityDeformableImageRegistration ............... 78 6.3NumericalIssues ................................ 80 6.4ExperimentResults ............................... 80 6.4.1Same-modalityexperiments ...................... 81 6.4.2Multi-modalityexperiments ...................... 85 6.5ConclusionAndFutureWork ......................... 89 7ACCURATEINVERSECONSISTENTNON-RIGIDIMAGEREGISTRATIONANDITSAPPLICATIONONAUTOMATICRE-CONTOURING ....... 90 7.1ProposedMethod ................................ 90 7.2ExperimentResults ............................... 95 7.3Conclusion .................................... 100 8FUTUREWORK ................................... 103 8.1DiusionWeightedMagneticResonanceImageAnalysis .......... 103 8.2DeformableImageRegistration ........................ 103 APPENDIX ASSDMODEL:EXISTENCEOFMINIMIZER ................... 104 BMODEL:EXISTENCEOFMINIMIZER .................... 105 CINVERSECONSISTENTMODEL:EXISTENCEOFMINIMIZER ....... 109 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 121 6

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Table page 2-1ListofS0andAl;m'sfortworegions ......................... 39 7-12Dsyntheticdatainverseinconsistencyerrors ................... 96 7-2ProstateMRIdatainverseinconsistencyerrors ................... 97 7-33DCTdatainverseinconsistencyerrors ...................... 101 7

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Figure page 2-1Comparingshapesofd 43 2-2ComparisonofA20 49 2-3ImagesofFAandR2 50 2-4ZoomedFAandA20 51 2-5Classicationofvoxelsbasedond 51 3-1SyntheticapproximationsforADC ......................... 55 3-2ImagesofR2bydierentmodels .......................... 57 3-3SpecialshapesofADCforisotropic,one-berandtwo-bervoxels ........ 57 3-4Dierentmeasures .................................. 59 3-5Characterization ................................... 60 4-1Fiberdirectioncolormap .............................. 65 4-2Shapeofdwithorientations ............................ 66 4-3Fiberdirectioneld .................................. 67 5-1FAimageoftherstchannel ............................ 71 5-2ComparisonbetweenTENDandMTEND ..................... 72 5-3ComparisonbetweenTENDandMTENDatinternalcapsule ........... 73 6-1Synthetic2Dregistrationresults ........................... 82 6-23DCTimageregistration .............................. 84 6-33DMRIimageregistration .............................. 86 6-4Noised3DMRIimageregistration ......................... 87 6-53DCTtoMRIimageregistration .......................... 88 7-1Synthetic2Dimageswithcontours ......................... 94 7-23DprostateMRIdatawith1002Dphases ..................... 99 7-3Inverseconsistenton3DCT ............................. 101 8

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MyPhDstudycoverstwomajortopics,oneconcentratesonDiusionWeightedMagneticResonanceImage(DW-MRI)analysis,andtheotherfocusesonmedicalimageregistration. MRIrendersnon-invasiveinvivoinformationabouthowwaterdiusesintoa3Dintricaterepresentationoftissues.Myworkprovidedhistologicalandanatomicalinformationabouttissuestructure,composition,architecture,andorganization.Ihaveproposedseveralmodelstoreconstructhumanbrainwhitematterbertracts,torecoverintra-voxelstructure,toclassifyintra-voxeldiusion,toestimate,smoothandcharacterizeapparentdiusioncoecientproles,andtoreconstructwhitematterbertraces. Medicalimageregistrationplaysanimportantroleindiagnosis,surgicalplanning,navigation,andvariousmedicalevaluations.Butmedicalimagesaregenerallymassiveandexpensivetoprocess,frequentlycorruptedbyhighlevelsofnoise,andmodelstoprocessthesedataareusuallysensitivetochoiceofparameters.Toconquerthesechallenges,Ipresentedonemethodtoalignlargevolumeimagesecientlybasedonanegativeloglikelihooddissimilaritymeasure,thismethodismorerobusttothepresenceofnoiseandislesssensitivetothechoiceofweightingparameters.Moreover,thismeasureisfurtherusedinanovelinverseconsistentdeformableimageregistrationmodel.Bothmodelsareappliedonautomaticre-contouringonComputedTomography(CT)imagesandMRI,andnumericalresultsshowedtheeectivenessofproposedapproaches. 9

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Newcomputertechnologiesarechangingthewaymedicalimagesishandledandprocessed,theyalsomakemedicalimageprocessingmoreandmoreinvolvedinresearchandclinicalapplications.Mystudyfocusontwotopicsofmedicalimageprocessing:DiusionWeightedMagneticResonanceImage(DW-MRI,alsoshortenasDWI)analysisandMedicalImageRegistration. Chapters 2 3 4 and 5 willcovermystudyonDW-MRI.Inchapters 2 and 3 ,IwillintroduceapproachesonapproximatingApparentDiusionCoecients(ADC)byusingSphericalHarmonicSeries.AmethodtoestimatetheprobabilityofwatermoleculediusionbasedonBi-Gaussianassumptionwillbeintroducedinchapter 4 ,basedonwhich,abertracereconstructionapproachisprovidedinchapter 5 .Chapters 6 and 7 covermyworkonmedicalimageregistration.Futureworkwillbediscussedinchapter 8 Intheremainingoftherstchapter,overallintroductionsforallthetopicscoveredinthisdissertationwillbegiven,followedbythemajorcontributionsforeachofchapter 2 throughchapter 7 10

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1 { 4 ]). Thediusionofwatermoleculesintissuesoveratimeintervaltcanbedescribedbyaprobabilitydensityfunctionpt(r)ondisplacementr.Sincept(r)islargestinthedirectionsofleasthindrancetodiusionandsmallerinotherdirections,theinformationaboutpt(r)revealsberorientationsandleadstomeaningfulinferencesaboutthemicrostructureoftissues. Thedensityfunctionpt(risrelatedtoDWIechosignals(q)viaaFouriertransformation(FT)withrespecttoq,whichrepresentsdiusionsensitizinggradient,by wheres0isMRIsignalintheabsenceofanygradient.Therefore,pt(r)canbeestimatedfromtheinverseFTofs(q)=s0.Recently,Tuchetal.[ 5 ]introducedthemethodofhigherangularresolutiondiusion(HARD)MRI,andWedeenetal.[ 6 ]succeedinacquiring500measurementsofs(q)ineachscantoperformafastFTinversion.However,thismethodrequiresalargenumberofmeasurementsofs(q)overawiderangeofqinordertoperformastableinverseFT. Amorecommonapproachtoestimatept(r)frommuchsparsersetofmeasurementss(q)isassumingpt(r)tobeaGaussian.ForGaussiandiusion,p(r;t)=1 1{1 )ityields whereu=q=jqj.Inthiscased(u)=uTDu:

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2s wherei(i=1;2;3)aretheeigenvaluesofD,hasbecomethemostwidelyusedmeasureofdiusionanisotropy[ 4 ].Thisisknownasdiusiontensorimaging(DTI),andinparticularusefulforcreatingwhitematterbertracts[ 7 { 11 ]. However,ithasbeenrecognizedthatthesingleGaussianmodelisinappropriateforassessingmultiplebertractorientations,whencomplextissuestructureisfoundwithinavoxel[ 6 8 12 { 16 ].Asimpleextensiontonon-GaussiandiusionistoassumethatthemultiplecompartmentswithinavoxelareinslowexchangeandthediusionwithineachcompartmentisaGaussian[ 13 14 17 { 19 ].UndertheseassumptionthediusioncanbemodelledbyamixtureofnGaussians: wherefiisthevolumefractionofthevoxelwiththediusiontensorDi,fi0,Pifi=1,andtisthediusiontime.Inserting( 1{4 )intoequation( 1{1 )yields whereu=q=jqj,andb=22jqj2(=3).Hereisthegyromagneticratio,andisthedurationoftwomagneticeldgradientpulseswithaseparationtimeintheuseofStejskal-Tannerpulsedgradientspinechomethod[ 20 ].ToestimateDiandfi,atleast7n1measurementss(q)pluss0arerequired.In[ 17 { 19 ]themodelofamixtureoftwoGaussianswereusedtoestimatethePDF.Thisestimationrequiresatleast13diusionweightedimagesfrom13dierentdirections. Oneofthealternativestoestimatept(r)andcharacterizediusionanisotropyisusingapparentdiusioncoecient(ADC)prolesd(x;;),whicharerelatedtoobservedDWI 12

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where(;)(0<,0<2)representsthedirectionofq,bisthediusion-weightingfactor.ForGaussiandiusiond(u)=buTDu,whereuisthenormalizedq.Thetrace,eigenvaluesandeigenfunctionsofDcanbeusedtocharacterizetheanisotropyanddirectionalpropertiesofthediusion.Fornon-GaussiandiusionthesphericalharmonicapproximationoftheADCprolesestimatedfromHARDdatahasbeenusedforcharacterizationofdiusionanisotropy.ThistechniquewasrstintroducedbyFrank[ 15 ],alsostudiedbyAlexanderetal.[ 21 ].Intheworkof[ 15 21 ]d(x;;)wascomputedfromHARDrawdataviathelinearizedversionof( 1{6 ): andrepresentedbyatruncatedSHS: whereYl;m(;):S2!CarethesphericalharmonicsandCdenotesthesetofcomplexnumbers.Theodd-ordertermsintheSHSaresettobezero,sincetheHARDmeasurementsaremadebyaseriesof3-drotation,d(;)isrealandantipodalsymmetry.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.Thetruncatedorderisgettinghigherasthestructurecomplexityincreases.Theirexperimentalresultsshowedthat 13

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SincetheADCprolescanbeusedtocharacterizethediusionanisotropy,andtoestimatept(r)throughthecombinationof( 1{1 )and( 1{6 ),itisofgreatsignicancetodevelopmodelsforbetterestimationoftheADCprolesfromDWMRmeasurements.IngeneraltherawHARDMRIdataarenoisy.Computingthecoecientsdirectlyfromtherawdataoftenprovidespoorestimates.Asaresult,itwillleadtoinaccurateorfalsecharacterizationofthediusionandconsequentlyleadtoincorrectbertracking.Theaimofthischapteristopresentanovelvariationalframeworkforsimultaneoussmoothingandestimationofnon-GaussianADCprolesfromHARDMRI. ThereisgrowinginterestindiusiontensordenoisingandreconstructionfromDTIdata.Therewereseveralpopularapproaches:(1).Smoothingtherawdatas(q)thenestimatingthediusiontensorfromthesmoothedrawdata([ 22 { 24 ]);(2).Smoothingtheprincipaldiusiondirectionafterthediusiontensorhasbeenestimatedfromtherawnoisymeasurements([ 25 { 29 ]);(3).Smoothingtensor-valueddatawithpreservingthepositivedenitepropertyofD[ 23 27 30 { 32 ]. However,veryfewresearchreportedinliteraturetodateonHARDdataanalysisconsidereddenoisingprobleminthereconstructionoftheADCproleswhiletheHARDrawdataisnoisy.Toimprovetheaccuracyoftheestimation,inchapter 2 ,wepresentanovelmodelthathastheabilityofsimultaneouslysmoothingandestimatingtheADCproled(x;;)fromthenoisyHARDmeasurementss(x;;)withpreservingtherelevantfeatures,andthepositivenessandantipodalsymmetryconstraintsofd(x;;).ThebasicideaofourapproachistoapproximatetheADCprolesateachvoxelbya4thorderSHS(lmax=4in( 1{8 ): 14

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1{9 )to15realvaluedfunctions:Al;0(x);(l=0;2;4);ReAl;m(x);ImAl;m(x);(l=2;4;m=1;:::;l): 2 elaboratesourworkpublishedat[ 33 34 ].Thismethoddiersfromtheexistingapproachesdevelopedin[ 15 ]and[ 21 ]mainlyintheaspectofthedeterminationoftheAl;m(x)'sin( 1{8 ).In([ 15 ])theAl;m(x)'s(liseven)aredeterminedby andin[ 21 ]theyareestimatedastheleast-squaressolutionsof Inchapter 2 ,theestimationofAl;m(x)'sisnotperformedindividuallyateachisolatedvoxel,butaprocessofjointestimationandregularizationacrosstheentirevolume.Thejointestimationandregularizationnotonlyguaranteesthewellposednessoftheproposedmodel,butalsoenhancestheaccuracyoftheestimationsincetheHARDdataarenoisy.Moreover,inchapter 2 ,weprovidemoredetailedmethodtocharacterizethediusionanisotropy,whichusesnotonlytheinformationofAl;m(x)'sasin([ 15 21 ]),butalsothevariationofd(;)aboutitsmean.OurexperimentalresultsshowedtheeectivenessofthemodelintheestimationandenhancementofanisotropyoftheADCprole.ThecharacterizationofthediusionanisotropybasedonthereconstructedADCprolesusingtheproposedmodelisconsistentwiththeknownberanatomy.Detailwillbeprovidedinchapter 2 15

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1{9 )toapproximatedin( 1{6 ),andhencetodetecttwo-berdiusion,atleast15diusionweightedmeasurementss(q)over15carefullyselecteddirectionsarerequired.However,tousethemixturemodel( 1{4 )withn=2todetecttwo-berdiusiononly13unknownfunctions:f,6entriesofeachofD1;D2needtobesolved.Thismotivatedustostudywhatistheminimumnumberofthediusionweightedmeasurementsrequiredfordetectingdiusionwithnomorethantwoberorientationswithinavoxel,andwhatisthecorrespondingmodeltoapproximatetheADCprolesinthiscase.Inchapter 3 (aworkpublishedin[ 35 ])weproposetoapproximatetheADCprolesfromHARDMRIbytheproductoftwouptothesecondordersphericalSHSinsteadofaSHSuptoorderfour.WealsoshowthattheproductoftwouptothesecondordersphericalSHSdescribesonlythediusionwithatmosttwoberorientations,whiletheSHSuptoorderfourmayalsorevealsthediusionwiththreeberorientations. Moreover,inchapter 3 (relatedworkwaspublishedin[ 35 ]),wewillintroduceaninformationmeasurementdevelopedin[ 36 ],andtermedasCumulativeResidualEntropy(CRE)(seedenition( 3{7 ))tocharacterizethediusionanisotropy.CREdiersfromShannonentropyintheaspectthatShannonentropydependsonlyontheprobabilityoftheevent,whileCREdependsalsoonthemagnitudeofthechangeoftherandomvariable.Weobservedthatisotropicdiusionhaseithernolocalminimumormanylocalminimawithverysmallvariationinthedenoiseds(q)=s0,i.e.,ebdprolesincomparingwithoneberortwo-berdiusions,whichimpliesthecorrespondingCREtobesmall.Wealsofoundthatoneberdiusionhasonlyonelocalminimumwithlargervariationinthes(q)=s0proles,whichleadstolargerCRE.Therefore,weproposetoproperlythresholdtheCREfortheregularizeds(q)=s0prolestocharacterizethediusionanisotropy.Detailwillbeprovidedinchapter 3 Inchapter 4 (relatedworkpublishedin[ 19 ])wepresentanewvariationalmethodforrecoveringtheintra-voxelstructureundertheassumptionthatpt(r)isamixtureoftwoGaussians.Ourapproachdiersfromtheexistingmethodsinthefollowingaspects.First, 16

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1{5 )withn=2ineachisolatedvoxel,whichleadstoanill-posedproblem.Second,werecovertheADCproled(x;;)inSHrepresentationusingmethodintroducedinchapter 2 fromthenoisyHARDdatabeforeestimatingDi(x)andfi(x).Therecovereddandthevoxelclassicationondiusionanisotropyfromdareincorporatedintoourenergyfunctiontoenhancetheaccuracyoftheestimations.Third,weappliedthebiGaussianmodeltoallthevoxelsintheeld,ratherthanthevoxelswheretheGaussianmodelonlytspoorly.Sinceboththeconstraintoff11ontheregionofstrongone-berdiusion,andtheregularizationforfiandDiarebuiltinthemodel,thesingleberandmulti-berdiusionscanbeseparatedautomaticallybythemodelsolution.Thisapproachshouldbelesssensitivetotheerrorinthevoxelclassication.Seedetailsinchapter 4 ,relatedworkwaspublishedin[ 19 37 ]. 38 ]andprovidesaverysensitiveprobefordetectingbiologicaltissuesarchitecture.Thekeyconceptthatisofprimaryimportancefordiusionimagingisthatdiusioninbiologicaltissuesreectstheirstructureandtheirarchitectureatamicroscopicscale.Forinstance,brownianmotionishighlyinuencedintissues,suchascerebralwhitematterortheannulusbrosusofinter-vertebraldiscs.Measuring,ateachvoxel,thatverysamemotionalonganumberofsamplingdirections(atleastsix,uptoseveralhundreds)providesanexquisiteinsightintothelocalorientationofbers. Wewillassumeptisamixtureof2Gaussians(n=2in( 1{4 )): 17

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wheref0;1f0areconsideredastheapparentvolumefractionsofdiusiontensorD1;D2respectively.Recently,Parkeretal[ 17 ]andTuchetal.[ 18 ]usedamixtureoftwoGaussiandensitiestomodelthediusionforthevoxelswheretheGaussianmodeltsthedatapoorly. AprimeprobleminrecoveringmultitensoreldDi(x);i=1;2andf(x)istheacquisitionnoisewhichcorruptsthedatameasurement.NeitherParkernorTuchconsideredremovingnoise.Inthisnotewepresentanewvariationalmethodwhichdiersfromtheexistingmethodsinthefollowingaspects.First,werecovertensoreldDi(x);i=1;2andf(x)globallybysimultaneoussmoothinganddatatting,ratherthanestimatingthemfrom( 1{12 )ateachisolatedvoxel,whichleadstoanill-posedproblemandisimpossibletogetasmoothmulti-tensoreld.Second,weappliedthebiGaussianmodeltoallthevoxelsintheeldwhileParkeretal[ 17 ]andTuchetal.[ 18 ]onlyappliedbiGaussianmodeltothevoxelswheretheGaussianmodeltsthedatabadly.SotheyneedtodopreprocessingtodistinguishvoxelsatwhichGaussianmodeltsthedatapoorlyfromthoseatwhichGaussianmodeltsdatawell.Inourapproach,thiskindofvoxelclassicationisnotrequiredandthusavoidstheerrorscomingfromit.Section 5.1 willexplainindetailshowtorecoversmoothmulti-diusiontensoreld. Regardingreconstructionofwhitemattertraces,currently,thereareseveraldierentapproacheswhichcanberoughlydividedintofourcategories:(1)linepropagationalgorithms;(2)surfacepropagationalgorithms;(3)globalenergyminimizationtondtheenergeticallymostfavorablepathbetweentwopredeterminedpixels;(4)solvingadiusionequation.Wewilldiscusslinepropagationalgorithmsindetails.Linepropagationalgorithmsuselocalinformationforeachstepofpropagation.Themaindierencesamongtechniquesinthisclassstemfromthekindoflocalinformationbeingconsideredandthewayinformationfromneighboringpixelsisincorporated. 18

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11 39 40 ].PEsuccessfullydeterminestheberdirectionincaseswherethereisasingleberdirectionineachvoxel,andisthereforeadequateforreconstructinglargetracesystems.However,withvoxelsizestypicalofdiusionacquisitions(1030mm3),thereissignicantpartialvolumeaveragingofberdirectioninanatomicalregionsofbothresearchandclinicalinterest,suchastheassociationbersnearthecortex.Moreover,imagenoisewillinuencethedirectionofthemajoreigenvector.Andasdegreeofanisotropydecreases,theuncertaintyinthemajoreigenvectorincreases,atwhichtrackingmaybeerroneous.Whendiusiontensorisplanarshaped,PEevendoesnotmakesense.Westinet.al[ 41 ]andLazaret.al[ 42 ]usedtheentiretensortodeecttheestimatedbertrajectory.Thisalgorithmiscalledtensordeection(TEND).ThedeectiontermisbetterthanPEinthesensethatthepreviousoneislesssensitivetoimagenoiseandislesserroneousinsituationofdegeneratedanisotropy.Butitstillhastheproblemofpartialvolumeaveragingofberdirection. Inchapter 5 section 5.2 ,wewillprovideanewlinepropagationalgorithmbasedonsmoothmulti-tensoreld.ItkeepsalltheadvantagesofTENDandhastwoadditionalgoodproperties:rst,problemofpartialvolumeaveragingisautomaticallysolvedasitisbasedonmulti-tensoreld;second,itusesdynamicallyadjustedstepsizetokeeptotalcurvatureoftraceslow,toappropriatelyterminatetrackingandtoincreasealgorithmeciency.Relatedworkwaspublishedin[ 43 { 45 ]. 46 ],Zitovaet.al.[ 47 ]andModersitzki[ 48 ].One 19

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49 ].Directmethodsinvolveestimatingthetransformationbetweentwogivenimagesfromtherawdata-withnofeatureextractionetc.OneexampleofdirectmethodistondadisplacementU,rigidornon-rigidbetweensourceimagesSandtargetimageTsuchthatthefollowingenergyisminimized: whereUisageometrictransformation,D()isadierencemeasure,F()isanintensitytransformation,R(U(X))isaregularitytermonTandisabalancingfactorbetweensimilarityandsmoothnessterms.Inrealapplications,howtobalancethesetwotermsisquitechallenging[ 48 ]. Forintra-modalitycases,DcouldsimplybechosenasL2normofintensitydierence,whichisalsousuallycalledSumofSquaredDierences(SSD).Forinter-modalityimages,ageneraldirectmeasureisnegativeMutualInformation(MI)(e.g.[ 50 { 53 ]),onecanalsotransformI1andI2intosamedomain,forinstance,applyFouriertransformtomovebothimagesintothefrequencydomain,thendeneDtobeL2normoffrequencydierence.Othertransforms,suchasHilbert[ 54 ],Gaborcouldbeappliedbasedonspecicapplications. Deformableimageregistrationallowsmorefreedomateachpoint,ithasbeenthesubjectofextensivestudyintheliterature(e.g.[ 47 55 { 63 ]).Lu.et.al.[ 58 ]presentatechniquedealingwithfastfree-formdeformationbetweentwoimagesofsamemodality.Inthiswork,L2normsareusedbothindierenceandregularityterms.Byusingcalculusofvariations,theminimizationproblemisreformedintoasetofnonlinearellipticpartialdierentialequations(PDEs).AGauss-Seidelnitedierenceschemeisusedtosolvethemiteratively.Vemuriet.al.[ 64 ]presentalevel-setcurveevolutiontechniquethatletsits 20

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AnothermethodwhichisreportedbyAyacheet.al.[ 65 ]usestheDemonsalgorithmofThirion[ 56 ]forachievingnon-rigidregistrationinmulti-modaldatasets,andtechniquesforestimatingintensityandgeometrictransformationsareprovided.OtherdirectmethodscouldbefoundinBajscyet.al.[ 66 ],Cachieret.al.[ 67 ]andHermosilloet.al.[ 52 ]. Inchapter 6 ,deformableimageregistrationforbothsame-modalityandmulti-modalitycaseswillbepresentindetails. Incertainapplicationssuchasimagingguidedradiationtherapy,itwouldbebettertohaveaone-to-oneandinverseconsistentdeformationeld,whilethemajorityofnon-rigidregistrationmethodsdonotguaranteesuchproperty.Theinverseconsistentmeansthatwhenthesourceandtargetimagesareswitchedinthemodel,thepointcorrespondencebetweenSandTdoesnotchange.Aninverseinconsistentdeformationeldcangeneratelargeerrorsintheprocesseslikeautore-contouring([ 68 ]),dosecalculate([ 69 { 73 ])inradiationtherapy.Anumberofworkhaveattemptedtomaketheregistrationinverseconsistent(e.g.[ 74 { 81 ]).Inthischapter,Iwillonlydiscusstwoofthemwhicharecloselyrelatedtomywork.In[ 75 ],ChristensenandJohnsonproposedthefollowingcoupledminimizationproblems: {z }E1+Zjhg1j2dxE(g)=M(S;T(g))+R(g)| {z }E2+Zjgh1j2dx(1{14) whereM(;)isadissimilaritymeasurebetweentwoimages,gisthebackwardmappingwhichdeformsTsuchthatT(g)isclosetoSundermeasureM,andgisexpectedtobetheinverseoftheforwardmappingh(i.e.hg=gh=id,whereidistheidentitymapping).R()isaregularitymeasureondeformationeldshandg,>0and>0areparametersbalancesthegoodnessofalignment,thesmoothnessofthedeformation, 21

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75 ],handgweresolvedbyusinggradientdescentalgorithm.In[ 79 ],theenergiesin( 1{14 )areaddedtogethertomaketheenergyfunctionsymmetric. Sincetheinverseconsistentconstraintsareaccommodatedbypenaltytermsintheenergyfunctionals,solutionshandgby( 1{14 )arenotexactlyinversetoeachother.Howhandgarecloselyinversetoeachotherdependsonhowlargetheparameteris,whichinpracticeishardtochooseandneedstobeadjustedcasebycase.Theoretically,handgareexactlyinversetoeachotheronlywhen!+1. In[ 76 ],Leowetal.proposedadierentapproach.Theyndhandgbythetimemarchingscheme: wheredtisthetimestep,1and2arevectoreldsrepresentinggradientdescentdirectionsofE1andE2in(1)respectively,i.e. Tomakethemodelinverseconsistent,1and2arechosenbythefollowingapproach. Supposeh(n)g(n)=idinthenthiteration,then2;1weredeterminedbytakingcareoftheinverseconsistentconstraintsh(n+1)g(n+1)=id,i.e. (h(n)+dt(1+2))(g(n)+dt(1+2))=id:(1{17) TakingtheTaylor'sexpansionof( 1{17 )withrespecttodtat0,andcollectinguptotherstordertermsofdt,onegets 22

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1{18 )maketheiterationsin( 1{15 )uni-directional,i.e.,updatingformulafortheforwardmappinghdoesnotdependonthebackwardmappingg,andviseversa. Inthisscheme,thedrivingforceforupdatingh(org)involvesbothforwardforcefromE1andbackwardforcefromE2,sotheschemealignstwoimagesfasterthanthemodelsinwhichtheforcedrivingthedeformationelddependsonE1orE2only([ 61 ]). However,h(n+1)andg(n+1)by( 1{15 to 1{18 )arenotexactlyinversetoeachotherevenh(n)andg(n)are.Sinceinthederivationof( 1{18 ),thehigherordertermsintheTaylorexpansionofthelefthandsideof( 1{17 )havebeendiscarded.Thisgeneratestruncationerrors,whichareaccumulatedandexaggeratedduringiterations.Startedwiththeidentitymappingforbothhandg,thesolutionshandgfrom( 1{15 to 1{18 )arenotinversetoeachother,aswewillshowinexperimentalresults. RegardingthedissimilaritymeasureM(;),aconventionallyusedoneforsamemodalityimageregistrationisSSD,whichissensitivetothepresenceofnoiseandoutliers(e.g.[ 46 54 ]).Moreover,thexedparameterin( 1{14 )balancingthesmoothnessofthedeformationeldandgoodnessofthealignmentisalwaysdiculttoselect,andaectstherobustnessofthemodeltothechoicetothisweightingparameter.Smallresultsanunstableanddiscontinuousdeformationeld,whilelargeleadstoinaccurateresult,andmayyieldanonphysicaldeformationeldduetounreasonablerestrictions.Inchapter 7 ,foraproposedmodel,IwillreplacetheSSDdissimilaritymeasurebyalikelihoodestimationthatisbasedontheassumptionofaGaussiandistributionoftheresidueimagebetweenthedeformedimageandxedimage. 2 throughchapter 7 willbeaddressedbriey. 23

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2 wepresentanewvariationalframeworkforsimultaneoussmoothingandestimationofapparentdiusioncoecient(ADC)prolesfromHARDMRI.ThemodelapproximatestheADCprolesateachvoxelbya4thordersphericalharmonicseries(SHS).ThecoecientsinSHSareobtainedbysolvingaconstrainedminimizationproblem.Thesmoothingwithfeaturepreservedisachievedbyminimizingavariableexponent,lineargrowthfunctional,andthedataconstraintisdeterminedbytheoriginalStejskal-Tannerequation.TheantipodalsymmetryandpositivenessoftheADCareaccommodatedinthemodel.WeusethesecoecientsandvarianceoftheADCprolesfromitsmeantoclassifythediusionineachvoxelasisotropic,anisotropicwithsingleberorientation,ortwoberorientations.TheproposedmodelhasbeenappliedtobothsimulateddataandHighAngularResolutionDiusion-weighted(HARD)MRIhumanbraindata.TheexperimentsdemonstratedtheeectivenessofourmethodinestimationandsmoothingofADCprolesandinenhancementofdiusionanisotropy.Furthercharacterizationofnon-Gaussiandiusionbasedontheproposedmodelshowedaconsistencybetweenourresultsandknownneuroanatomy. Secondly,inChapter 3 wepresentanotherapproximationfortheADCofnon-Gaussianwaterdiusionwithatmosttwoberorientationswithinavoxel.TheproposedmodelapproximatesADCprolesbyproductoftwosphericalharmonicseries(SHS)uptoorder2fromHARDMRIdata.ThecoecientsofSHSareestimatedandregularizedsimultaneouslybysolvingaconstrainedminimizationproblem.Anequivalentbutnon-constrainedversionoftheapproachisalsoprovidedtoreducethecomplexityandincreasetheeciencyincomputation.Comparetotherstmethod,thisonerequireslessmeasurementsbutprovidescomparableresults.MoreoverweusetheCREasameasurementtocharacterizediusionanisotropy.ByusingCREwecangetreasonableresultsusingtwothresholds,whiletheexistingmethodseitheronlycanbeusedtocharacterizeGaussiandiusionorneedmoremeasurements 24

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Inchapter 4 ,wepresentanewvariationalmethodforrecoveringtheintra-voxelstructureundertheassumptionthatpt(r)isamixtureoftwoGaussians:pt(r)=P2i=1fi((4t)3det(Di))1=2erTD1ir 1{5 )withn=2ineachisolatedvoxel,whichleadstoanill-posedproblem.Second,werecovertheADCproled(x;;)inSHrepresentationusingmethodintroducedinchapter 2 fromthenoisyHARDdatabeforeestimatingDi(x)andfi(x).Therecovereddandthevoxelclassicationondiusionanisotropyfromdareincorporatedintoourenergyfunctiontoenhancetheaccuracyoftheestimates.Third,weapplythebiGaussianmodeltoallthevoxelsintheeld,ratherthanthevoxelswheretheGaussianmodelonlytspoorly.Sinceboththeconstraintoff11ontheregionofstrongone-berdiusion,andtheregularizationforfiandDiarebuiltinthemodel,thesingleberandmulti-berdiusionscanbeseparatedautomaticallybythemodelsolution.Thisapproachshouldbelesssensitivetotheerrorinvoxelclassication. Inchapter 5 ,wewillprovideanewlinepropagationalgorithmbasedonsmoothmulti-tensoreld. Weassumethereareuptotwodiusionchannelsateachvoxel.Avariationalframeworkfor3Dsimultaneoussmoothingandrecoveringofmulti-diusiontensor 25

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FibertracesarecoloredusingLaplacianeigenmaps.Byapplyingtheproposedmodeltosyntheticdataandhumanbrainhighangularresolutiondiusionmagneticresonanceimages(MRI)dataofseveralsubjects,weshowtheeectivenessofthemodelinrecoveringintra-voxelmulti-berdiusionandinter-voxelbertraces.Superiorityoftheproposedmodeloverexistingmodelsarealsodemonstrated. Inchapter 6 ,afast,robustandaccuratedeformableimageregistrationtechniqueisprovided.Theproposedmethodconsidersthedierencebetweenthedeformedtemplateandthereferenceateachvoxelasindependentnormalrandomvariableswithzeromeanandavariancetobeoptimized.Thentheoptimaldeformationeldandvarianceareobtainedbyminimizinganenergyfunctionthatconsistsofthenegativelog-likelihoodfunctionandaregularizationtermforthedeformationeld.Sincetheproposedmodelaccommodatesthevarianceinthealignment,itismoregeneralandrobusttonoisethanthewidelyusedSumofSquareDistance(SSD)model,whichhaslimitationinregisteringnon-linearlyrelatedimages.Moreover,theweightbetweenthesmoothnessofthedeformationeldandgoodnessofthealignmentintheproposedmodelisadjustedautomaticallybytheupdatedvarianceduringiterations.Thisfeatureimprovestheaccuracyofthealignmentandspeedoftheconvergence.ToenhancealgorithmeciencyweusedtheAdditiveOperatorSplitting(AOS),afastsemi-implicitnitedierencescheme,andmulti-resolutionmethodinsolvingtheminimizationproblem.Theexperimentalresultsonsynthetic 26

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Chapter 7 providesanovelalgorithmforinvertiblenon-rigidimageregistration.Theproposedmodelminimizestwoenergyfunctionalscoupledbyanaturalinverseconsistentconstraint.Bothoftheenergyfunctionalsforforwardandbackwarddeformationeldsconsistasmoothnessmeasureofthedeformationeld,andasimilaritymeasurebetweenthedeformedimageandtheonetobematched.Inthisproposedmodelthesimilaritymeasureisbasedonmaximumlikelihoodestimationoftheresidueimage.Toenhancealgorithmeciency,theAdditiveOperatorSplitting(AOS)schemeisusedinsolvingtheminimizationproblem.Theinverseconsistentdeformationeldcanbeappliedtoautomaticre-contouringtogetanaccuratedelineationofRegionsOfInterest(ROIs).Theexperimentalresultsonsyntheticimagesand3Dprostatedataindicatetheeectivenessoftheproposedmethodininverseconsistencyandautomaticre-contouring.Relatedworkwillbepublishedin[ 82 ]. 27

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Copyright[2004]IEEE.Portionsreprinted,withpermission,from[ 93 ]. 1{6 ).Toexplainthebasicideaofourmethod,wefocusourattentiononthecaseswherethereareatmosttwoberspassingthroughasinglevoxel.Thesameideacanbeappliedtothecaseswheretherearemoreberswithinavoxel. Thechallengeinregularizingdcomesfromtwoaspects.First,disdenedonS2ratherthanR2,hence,thederivativesfor(;)shouldbealongthesphere.Secondly,theregularizeddhastopreservetheantipodalsymmetrypropertywithrespectto(;),sincetheHARDmeasurementsaremadebyaseriesof3-drotation.Consideringthesefactsweadopttheideadevelopedin[ 15 21 ]thatapproximatesdbyitsSHSconsistingofonlyevenordercomponentsuptoorder4,i.e. Theexpressionin( 2{1 )ensuresthesmoothnessandantipodalsymmetrypropertyofd(x;;)intermsof(;),thisiseasytoseefromthedenitionofYl;m(;).Forthecaseswherepossiblykberscrossinasinglevoxel,thesumin( 2{1 )shouldbereplacedbyPl=0;2;:::;2k. Nowtheproblemofregularizationandestimationofd(x;;)reducestothatforthe15complexvaluedfunctionsAl;m(x)(l=0;2;4andm=l;:::;l)in( 2{1 ).Sinced(;)ateachvoxelisarealvaluedfunction,andYl;msatisesYl;m=(1)mYl;m,Al;mshouldbe 28

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2{1 )to15realvaluedfunctions.Theyare Byusing( 2{2 ),wecanrewrite( 2{1 )as whereReFandImFrepresenttherealandimaginarypartofafunctionFrespectively.Nowtheproblemofregularizingandestimatingdreducestosmoothingandestimationof15functionsin( 2{2 )simultaneously. Therearemanychoicesofregularizingoperatorstosmooththe15functionsin( 2{2 ).TotalVariation(TV)basedregularization,rstproposedbyRudin,OsherandFatemi[ 83 ],provedtobeaninvaluabletoolforfeaturepreservingsmoothing.However,itsometimescausesastaircaseeectmakingrestoredimageblocky,andevencontaining'falseedges'[ 84 85 ].Animprovement,thatcombinestheTVbasedsmoothingwithisotropicsmoothing,wasgivenbyChambolleandLions[ 85 ].TheirmodelminimizestheTVnormwhenthemagnitudeoftheimagegradientislarger,andtheL2normoftheimagegradientifitissmaller.However,thismodelissensitivetothechoiceofthethresholdwhichseparatestheTVbasedandisotropicsmoothing.TofurtherimproveChambolleandLions'modelandmakethemodelhavinganabilitytoselfadjustdiusionproperty,recently,certainnonstandarddiusionmodelsbasedonminimizingLp(x)normofimagegradienthavebeendeveloped[ 84 86 ].Torecoveranimageufromanobserved 29

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84 ]thediusionwasgovernedbyminimizingminuZjrujp(jruj) 86 ]thediusionwasperformedthroughminimizingminuZ(x;Du) where(x;r):=1 1+kjrGIj2; 84 ]and[ 86 ]areabletoselfadjustdiusionrangefromisotropictoTV-baseddependingonimagegradient.Atthelocationswithhigherimagegradients(p=1),thediusionisTVbasedandstrictlytangentialtotheedges([ 83 85 87 ]).Inhomogeneousregionstheimagegradientsareverysmall(p=2),thediusionisessentiallyisotropic.Atallotherlocations,theimagegradientforces1
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2{5 ) 1+kjrGal;m(x)j2;(2{6) andal;mistheleast-squaressolutionof Inrealapplication,Ml;mispickedas90%percentileofr.Experimentalresultsarenotsensitivetothisparameter90%. AgainwewouldliketopointoutthatE1onlyneedstoinclude15termscorrespondingtothe15realvaluedfunctionsin( 2{2 ).HerewewriteitintermsofAl;minordertoshortentheexpressionoftheformula.IfusingAl;minsteadofal;min( 2{6 )wemaygetbetternumericalresults,sincepl;mwoulddependonupdatedAl;mratherthanthexedal;mintheiterationstominimize( 2{4 ).However,itgivesdicultyinthestudyoftheexistenceofsolutions. Sinced(x;;)isrelatedtotheHARDmeasurementss(x;;)ands0(x)throughtheStejskal-tannerequation( 1{6 ),theestimationoftheAl;m'sisbasedontheoriginalStejskal-tannerequation( 1{6 )ratherthanits(log)linearizedform( 1{7 ),thatis, 2ZZ20Z0js(x;;)s0(x)ebd(x;;)j2sindddx;(2{8) wheredisdeterminedin( 2{1 ).Asobservedin[ 88 ]whenthesignaltonoiseratioislowthelinearizedmodelgivesdierentresults. Finally,tosimultaneouslyregularizeandestimatetheADCd(x;;),ourmodelminimizestheenergyfunction 31

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2{2 )areneeded),andsubjecttotheconstraint: In( 2{9 ),( 2{10 ),s(x;;)ands0(x)arethenoisyHARDmeasurements(realvalued),d(x;;)istheSHSgivenin( 2{1 ),R3istheimagedomain,>0isaparameterwhichcouldbedierentfordierentAlm.E1andE2aregivenin( 2{4 )and( 2{8 ),respectively.WewouldliketopointoutthatforafunctionAl;m2BV,DAl;misameasure,thedenitionof( 2{4 )isnotobvious.Thiswillbediscussedinexistencesection 2.5 below. BeforewederivetheEuler-Lagrangeequationsforourmodel( 2{9 ),( 2{10 ),wewouldliketopointoutthatifthemeasurementssatisfytheconditions(x;;)s0(x),thesolutionof( 2{9 )meetstheconstraint( 2{10 )automatically.Thereforewecantreatourmodelasanunconstrainedminimization.Thisisgiveninthefollowinglemma. theminimizerof( 2{9 )alwayssatisestheconstraint( 2{10 ). 2{9 )inBV(),andd(x;;)bethefunctiondenedin( 2{1 )associatedwiththeseoptimalAl;m(x)'s.Ifforsomex2;0<;0<2,d(x;;)<0,wedene^d(x;;): ^d(x;;):=8><>:d(x;;);ifd(x;;)00;ifd(x;;)<0(2{12) and^Al;m(x)=:Z20Z0^d(x;;)Yl;m(;)sindd:

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^Al;m(x)=8><>:Al;m(x);ifd(x;;)00;ifd(x;;)<0(2{13) Thisimpliesthatl;m(x;D^Al;m)l;m(x;DAl;m); 2{11 )holds.Fromthelasttwoinequalityabove,weobtainthatE(^d)E(d).Thiscontradictstothefactthatdminimizesenergyfunctional( 2{9 ). NowwegivetheevolutionequationsassociatedwiththeEuler-Lagrange(EL)equationsfor( 2{9 ):forl=0;2;4andm=l;:::;l, withtheinitialandboundaryconditions:Al;m=al;m;onft=0g;(@rl;m)(x;DAl;m)n=0on@<+: 33

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whereq(x)=p(x)ifjrjMl;m,andq(x)=1ifjrj>Ml;m. 15 ]thejAl;m(x)j(l=0;2;4andm=l;:::;l)inthetruncatedSHS( 2{1 )areusedtocharacterizethediusionanisotropyateachvoxelx.Ourexperimentalresults,however,indicatethisinformationaloneisinsucienttoseparateisotropicdiusion,one-berdiusion,andmulti-bersdiusionwithinavoxel.WeproposetocombinetheinformationfromjAl;mjwiththevariancesofd(;)aboutitsmeanvaluetocharacterizethediusionanisotropy.Weoutlinedouralgorithmasfollows: (1).If islarge,orthevarianceofd(;)aboutitsmeanissmall,thediusionatsuchvoxelsisclassiedasisotropic. (2).Fortheremainingvoxels,if islarge,thediusionatsuchvoxelsischaracterizedasone-berdiusion.Figure 2-3 dpresentsanintensity-codedimageofR2inabrainslicethroughtheexternalcapsule,animportantstructureofthehumanwhitematter.InFigure 2-3 dthosevoxelsofahighintensity(brightregionsontheimage)arecharacterizedasone-berdiusion. (3).Foreachuncharacterizedvoxelaftertheabovetwosteps,searchthedirections(;),whered(;)attainsitslocalmaxima.Thenwecomputetheweightsforthelocalmaxima(saywehave3localmaxima):Wi=:d(i;i)dmin

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2-5 ashowsourclassicationofisotropicdiusion(darkregion),one-berdiusion(grayregion),andtwo-bersdiusion(brightregion)inthesamesliceasinFigure 2-3 2{14 ),weuseAdditiveOperatorSplitting(AOS)algorithmforthediusionoperator(see[ 89 90 ]).Byusingthisalgorithm,thecomputationalandstoragecostislinearinthenumberofvoxels,andthecomputationaleciencycanbeincreasedbyafactorof10underrealisticaccuracyrequirements([ 89 ]).InthisalgorithmtheprocessesofsolvingAl;mindierentdimensionsareindependentfromeachotherineveryiteration,thealgorithmisreadytobemodiedtoaparalleledversion. ToavoidthecomplicationofnotationsweuseXtorepresentanyAl;mintheEuler-Lagrangeequations,andonlywritethealgorithmforoneoftheequations( 2{14 )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)if 35

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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[ 91 ],onecanwrite( 2{19 )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(2{20) WhereC;D;EandFarefromdivergenceoperation,whileGandHaregeneratedbydotproduct,indetail: 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 2

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2Hi;j=lnM(qi;j+1qi;j1) (2h)2(X(n)i+1;jX(n)i1;j)2 2 2{20 )wouldinvolvematrixinverseoperation,whichwillbecomemoreandmorecomplicatedanddramaticallyexpensiveasdimensionincreasesifwesolveitdirectly.Instead,hereweuseAdditiveOperatorSplitting(AOS)algorithm,thensystem( 2{20 )canbereformattedintothefollowingtwosystemsfortherstandseconddimensionsofXrespectively: 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(2{21) 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(2{22) andX(n+1)i;j=X(n+1)i;j+X(n+1)i;j Toaccommodatetheboundarycondition@X @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 2{21 )and( 2{22 )correspondtolinearsystemsinmatrix-vectornotation:A1X

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92 ]). 2{9 2{10 )tosimulateddataandasetofHARDMRIdatafromthehumanbrain. 2-1 38

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ListofS0andAl;m'sfortworegions Region 1 2 547 InFigure 2.4.2 wedisplayedthetrue,noisy,andrecoveredADCprolesd(x;;)forthesyntheticdatawithsize84.TheADCproled(x;;)wascomputedby( 2{3 )basedonthesesimulateddata,andthecorrespondingstrue(x;;)wasconstructedvia( 1{6 )withb=1000s=mm2.ThenthenoisyHARDMRIsignals(x;;)wasgeneratedbyaddingazeromeanGaussiannoisewithstandarddeviation=0:15.Figure 2.4.2 bshowstheADCproledcomputedby( 2{3 ),wherethecoecientsoftheSHSaretheleast-squaressolutionsof( 1{11 )withthenoisys. Wethenappliedourmodel( 2{9 2{10 )tothenoisys(x;;)totesttheeectivenessofthemodel,with0;0=4;2;m=40(m=2:::2);4;m=60(m=4:::4).Bysolvingthesystemofequations( 2{14 )in2.5secondsoncomputerwithPIV2.8GHZCPUand2GRAMusingMatlabscriptcode,weobtained15reconstructedfunctionsasin( 2{2 ).UsingtheseAl;m(thesolutionsof( 2{14 ))wecomputedd(x;;)via( 2{3 ).Thereconstructedd(x;;)isshowninFigure 2.4.2 c.Comparingthesethreegures,itisclearthatthenoisymeasurementsshavechangedFigure. 2.4.2 a,theoriginalshapesofd,intoFigure 2.4.2 b.Afterapplyingourmodel( 2{9 2{10 )toreconstructtheADCproles,theshapes 39

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2.4.2 awererecovered,asshowninFigure 2.4.2 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( 2{9 )tothesedatatocomputetheADCprolesintheentirebrainvolume.Bysolvingasystemofequations( 2{14 )weobtainedallthecoecientsAl;m'sin( 2{2 ),anddeterminedd(x;;)using( 2{3 ). Then,weusedtheseAl;m(x)tocalculateR0andR2denedin( 2{17 )and( 2{18 )respectively,aswellasthevariance(x)ofd(x;;)aboutitsmean:(x)=Z0Z20(d(x;;)55Xi=1d(x;i;i)=55)2dd: 2.2 .TheselectionofthethresholdsmentionedaboveforR0,R2and 40

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Figure. 2.6 presentsA2;0(x),oneofthecoecientsin( 2{3 ),fortheparticularsliceinthevolume.TheimagesA2;0(x)inFigure. 2.6 aand 2.6 bareestimatedbyusing( 1{10 )andsolving( 2{9 ),respectively. Figure. 2-3 ComparesFAandthreeR2(x)'swithAl;m(x)'sobtainedfromthreedierentmodelsforthesamesliceasshowninFigure. 2.6 .Figure. 2-3 adisplaystheFAimageobtainedbyusingadvancedsystemsoftwarefromGE.TheAl;m(x)'susedtoobtainR2(x)inFigure. 2-3 baredirectlycomputedfrom( 1{10 ).ThoseusedtoobtainR2(x)inFigures. 2-3 cand 2-3 daretheleast-squaressolutionsof( 1{11 )andthesolutionsof( 2{9 ),respectively.InFigures. 2-3 cand 2-3 dthevoxelswithhighlevelsofintensities(red,yellow,yellow-lightblue)arecharacterizedasone-berdiusion. AlthoughtheFAimageinFigure. 2-3 aisobtainedbasedonaconventionalDTImodel( 1{2 ),itisstillcomparablewiththeR2map,sincesingletensordiusioncharacterizedbySHSrepresentationfromtheHARDimagesagreeswiththatcharacterizedbytheDTImodel.However,inDTIavoxelwithalowintensityofFAindicatesisotropicdiusion,whileusingouralgorithm,multi-bersdiusionmayoccuratthelocationwiththelowvalueofR2. Itisclearlyevidentthattheabilitytocharacterizeanisotropicdiusionisenhanced,asshowninFigures. 2-3 a2-3 d.Figure.3bindicatesagainthattheestimationsofAl;mdirectlyfromthelogsignalsusuallyarenotgood.Eventheleast-squaressolutionof( 1{11 )arenotalwayseective.ThiscanbeseenbycomparingtheanatomicregioninsidetheredsquareofFigures. 2-3 cand 2-3 d,whicharezoomedinFigures. 2-4 aand 2-4 b,respectively.Thereisadarkbrokenlineshowingonthemapoftheexternalcapsule(arrowtotherightonFigure. 2-4 a),thissameregionwasrecoveredbytheproposedmodelandcharacterizedbythethirdstepinouralgorithmastwo-bersanisotropicdiusion(arrowtotherightinFigure. 2-4 b).(Themodelsolutionsreducedthevalueof 41

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2-4 b),however,thisconnectionwasnotmappedoutonFigure. 2-3 corthezoomedimageinFigure. 2-4 a.InfactthisconnectionwasnotmappedoutonFigures 2-3 a-3beither.Allthesemappedconnectionsareconsistentwiththeknownneuroanatomy.Combinedtogether,ourresultsindicatethatourproposedmodelforjointrecoveryandsmoothingoftheADCproleshastheadvantageovertheexistingmodelsfortheenhancementoftheabilitytocharacterizethediusionanisotropy. Figure. 2-5 ashowsapartitionofisotropic,one-ber,andtwo-berdiusionforthesamesliceusedinFigure 2-4 .Thetwo-ber,one-ber,andisotropicdiusionregionswerefurthercharacterizedbythewhite,gray,andblackregions,respectively.TheregioninsidethewhitesquareinFigure 2-5 a,whichisthesameonesquaredinFigures. 2-3 cand 2-3 d,iszoomedinFigure. 2-4 c.ItisclearlytoseethetwoarrayedvoxelsinFigure. 2-4 bareclassiedastwo-berdiusion.Thecharacterizationoftheanisotropyonthevoxelsandtheirneighborhoodsisconsistentwiththeknownberanatomy. Figure. 2-5 brepresentstheshapesofd(x;;)atthreeparticularvoxels(upper,middleandlowerrows).Thedinallthreevoxelsiscomputedusing( 2{3 ).However,theAl;m(x)usedincomputingdontheleftcolumnaretheleast-squaressolutionsof( 1{11 ),whileontherightcolumntheyarethesolutionsoftheproposedmodel( 2{9 ).Therstandsecondrowsshowtwovoxelsthatcanbecharacterizedasisotropicdiusionbeforedenoising,butastwo-bersdiusionafterapplyingmodel( 2{9 ).ThesetwovoxelsarethesamevoxelsasinFigure 2-4 directedbyarrows.ThelowerrowofFigure 2-5 bshowstheone-berdiusionwasenhancedafterapplyingourmodel. SolvingAl;m'sofsize15109868from4-Ddataofsize55109868takes46.2secondsforeachiterationoncomputerwithPIV2.8GHZCPUand2GRAMinMatlabscriptcode. 42

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Figure2-1. (a).Trued,(b).Thedgeneratedby( 2{3 ),wheretheAl;m'saretheleastsquaresolutionof( 1{10 )withthenoisymeasurements,(c).Recovereddbyapplyingmodel( 2{9 ). 43

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2{9 )usingtheideadevelopedin[ 86 ]. Recallthatforafunctionu2BV(),Du=ruLn+Dsu 2{9 )overthefunctionsinBV(),werstneedtogiveaprecisedenitionfor( 2{9 ). 2{5 ),andRjDsAl;mjisthetotalvariationnormofAl;m. Then,ourenergyfunctional( 2{9 )isdenedasE(Al;m)=ZXl=0;2;4lXm=ll;m(x;rAl;m)+ZjDsAl;mj 2ZZ20Z0js(x;;)s0(x)ebd(x;;)j2sindddx:(2{23) Inthediscussionofexistence,withoutlossofgenerality,wesettheparameter=1in( 2{9 )andthresholdMl;m=1in( 2{5 )toreducethecomplexityintheformulation. Nextwewillshowthelowersemi-continuityoftheenergyfunctional( 2{9 )inL1,i.e.ifforeachl;m(l=0;2;4andm=l;:::;l),ask!1,Akl;m!A0l;minL1(); 44

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Foru2BV()denote(u):=Z(x;Du); (u)=e(u)(2{26) Furthermore,(u)islowersemi-continuousonL1(),i.e.ifuj;u2BV()satisfyuj!uweaklyinL1()asj!1then(u)liminfj!1(uj): Nextweshow( 2{26 ).Foru2BV(),wehavethatforeach2C10(;Rn),Zudivdx=Zrudx+ZDsu

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2{26 )itonlyremainstoshowthat Sinceany2L1(;Rn)canbeapproximatedinmeasureby2C10(;Rn),wehavethat sup2C10(;Rn)jj1Zrup(x)1 Choosing(x)=1fjruj1gjrujp(x)1ru 2{28 )isZ1 Toshowtheoppositeinequality,weargueasfollows.Forany2L1(;Rn),sincep(x)>1wehavethatforalmostallx,ru(x)(x)1 Ifjruj>1,noticingp(x)>1andjj1foralmostallxwehavethatru=jrujru

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Combining,( 2{28 ),( 2{29 ),( 2{30 ),and( 2{31 ),wehave( 2{27 ),andhenceforallu2BV(),e(u)=(u). Notethat(x;r)=l;m(x;r),ifp(x)=pl;m(x).AsadirectconsequenceofthislemmawehaveE1in( 2{4 )isweaklylowersemi-continuousonL1(). Furthermore,wecanshowthatE2in( 2{4 )islowersemi-continuousonL1().Indeed,whenAkl;m!A0l;m;inL1();ask!1; 2{24 )holds. Nowwecanprovetheexistenceresults. 2{9 )overthespaceofBV(). 2{9 )inBV().Thenforeach(l;m)thesequenceAkl;misboundedinBV().FromthecompactnessofBV()thereexistsubsequencesofAkl;m(stilldenotedbyAkl;m)andfunctionsA0l;m2BV()satisfyingAkl;m!A0l;mstronglyinL1():

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2{24 )),wehaveE(A0l;m)liminfk!1E(Akl;m)infAl;m2BV()E(Al;m): 2{9 ). OurexperimentsonbothsyntheticdataandhumanHARDMRIdatashowedtheeectivenessoftheproposedmodelintheestimationofADCprolesandtheenhancementofthecharacterizationofdiusionanisotropy.Thecharacterizationofnon-Gaussiandiusionfromtheproposedmethodisconsistentwiththeknownneuroanatomy. Thechoiceofthecurrentparameters,however,mayaecttheresults.Ourchoiceismadebasedontheprinciplethatclassicationforone-berdiusionfromthemodelsolutionshouldagreewithaprioriknowledgeoftheberconnections.Inthisarticle,wehavenotincludedtheworkfordeterminationofberdirectionsandthemethodforautomatedbertracking.Thestudyaddressingtheseproblemswillbereportedinseparatepapers. 48

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(b) (a).A20computedfrom( 1{10 ),(b).A20obtainedfrommodel( 2{9 ) 49

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(b) (c) (d) (a).FAfromGEsoftware,(b)-(d).R2withtheAl;m'sasthesolutionsof( 1{10 ),least-squaressolutionsof( 1{11 ),andmodelsolutions,respectively. 50

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(b) (c) (a)-(b).EnlargedportionsinsidetheredsquaresinFigures 2-3 cand 2-3 d,respectively.(c).EnlargedportionsinsidethewhitesquaresinFigure 2-5 a. (b) (a).Classication:white,gray,andblackvoxelsareidentiedastwo-ber,one-ber,andisotropicdiusionrespectively,(b).Shapesofd(x;;)atthreeparticularpoints(upper,middleandlowerrows).Thediscomputedvia( 2{3 ).Al;m(x)usedin( 2{3 )ontheleftcolumnsaretheleast-squaressolutionsof( 1{11 ),whileontherightcolumnarethemodelsolutions. 51

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Copyright[2005]SpringerLectureNotesonComputerSciencefrom[ 35 ].Portionsreprinted,withpermission. 14 21 93 ],todetectthediusionwithatmosttwoberorientation,theADCproleswererepresentedbyatruncatedSHSuptoorder4intheformof( 1{9 ).In[ 14 ]thecoecientsAl;m's(liseven)weredeterminedbyinversesphericalharmonictransformof1 21 ]theywereestimatedastheleast-squaressolutionsof RegularizationontherawdataorAl;mwasn'tconsideredinthesetwowork.In[ 93 ]Al;m'swereconsideredasafunctionofx,andestimatedandsmoothedsimultaneouslybysolvingaconstrainedminimizationproblem: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( 1{9 ).Bythechoiceofpl;mandq,theregularizationistotalvariationbasednearedges,isotropicinhomogeneousregions,andbetweenisotropicandtotalvariationbasedthatvariesdependingonthelocalpropertiesoftheimageatotherlocations,IntheseworksincetheADCprolewasapproximatedby( 1{9 ),atleast15measurementsofs(q)wererequiredtotoestimatethe15coecientsAl;m. However,themixturemodel( 1{4 )withn=2,whichisalsoabletodetecttwo-berdiusioninvolvesonly13unknownfunctions.Thismotivatesustondamodelthatis 52

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OurbasicideaistoapproximatetheADCprolesMRIbytheproductoftwosecondorderSHS'sinsteadofaSHSuptoorderfour.Thiscanbeformulatedas Inthismodelthereareonly12unknowns:bl;m,cl;m(l=0;2andlml). ToestimatetheADCprolefromtherawHARDMRIdata,whichusuallycontainsacertainlevelofnoise,weproposeasimultaneoussmoothingandestimationmodelsimilarto( 3{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;(3{4) withconstraintd0,wheredisintheformof( 3{3 ).,areconstants.Therst3termsaretheregularizationtermsforbl;m,cl;mands0respectively.ThelasttwotermsarethedatadelitytermsbasedontheoriginalStejskal-Tannerequation( 1{6 ). Next,feasibilityofthismodelwillbeexplained.DenoteB:=Pl=0;2Plm=lbl;mYl;m(;),C:=Pl=0;2Plm=lcl;mYl;m(;)andA:=Pl=0;2;4Plm=lAl;mYl;m.DenesetsSBC=fd:d(;)=BCg;SA=fd:d(;)=Ag.Sinceeachd(;)inSBCisafunctiondenedonS2,itcanbeapproximatedbySHS,simplecalculationshowsthatcoecientsoftheapproximatedSHSofevenorderlargerthan4areallzeros,soSBCSA.Ontheotherhand,numerousexperimentsshowthatwhenavoxelisnotmorecomplicatedthan2-berdiusion,itsADCisalwaysafunctioninsetSBC.Butifavoxel 53

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3-1 depictshowfunctionsinsetSBCandSAdierinrepresentingADC.ItisobservedthattheADCintheformAcannotbewellapproximatedbyBConlyin3-berdiusioncase(seeFig. 3-1 ).Therefore,ifwefocusonlyoncharacterizingatmosttwo-berdiusion,whichisthemostinterestingcase,model( 3{3 )isreasonableandsucienttorepresentADC. Model( 3{4 )isaminimizationproblemwithconstraintd(;)0forall0<;0<2whichisusuallydiculttoimplement.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 Furthermorewesubstitutedmodel( 3{4 )byminLjk1(x);Ljk2(x);~s0(x)Z(2Xi=13Xj=1jXk=1jrLj;kij+jr~s0j)dx 2ZfZ20Z0js~s0ebdj2sindd+j~s0s0j2gdx;(3{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( 3{6 )thusworksequivalentlyasthersttwotermsin 54

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3{4 )do,whilealltheothertermsarethesameasthoseleftin( 3{4 ).Hence,( 3{6 )isequivalentto( 3{4 ),butitisanon-constrainminimizationproblemandisthuseasytoimplement.AfterwegetL1andL2,bl;mandcl;min( 3{4 )canbeobtainedbytheonetoonerelationbetweenthem. (a)(b)(c)(d) Figure3-1. ComparisonoftheADC'sapproximatedby( 1{9 )and( 3{3 )infourcases:(a)isotropicdiusion,(b)one-berdiusion,(c)two-berdiusion,(d)three-berdiusion.In(a)-(d)fromlefttoright,toptobottom,weshowshapesofB,C,BC,andA,respectively. Weapplymodel( 3{6 )toasetofhumanbrainHARDMRIdatatoreconstructandcharacterizeADCproles.Thedatasetconsistsof55diusionweightedimagesSk:!R;k=1;:::;55,andoneimageS0intheabsenceofadiusion-sensitizingeldgradient(b=0in( 1{6 )).24evenlyspacedaxialplaneswith256256voxelsineachsliceareobtainedusinga3TMRIscannerwithsingleshotspin-echoEPIsequence.Slicethicknessis3:8mm,gapbetweentwoconsecutiveslicesis1:2mm,repetitiontime(TR)=1000ms,echotime(TE)=85msandb=1000s=mm2.Theeldofview(FOV)=220mm220mm.Werstappliedthemodel( 3{6 )tothedatatogetLi,andthenusedLitocomputebl;mandcl;m,l=0;2;m=l:::l,andtheADCd=BC.Ontheotherhand,weusedthemodel( 3{2 )toestimateAl;mandgetA.ThecomparisonfortheshapesofADCintheformofBCandAisdemonstratedinFig. 3-3 (a)-(d)atfourspecicvoxels.Thediusionatthese4voxelsareisotropic(a),one-ber(b),two-ber(c),andthree-ber(d),respectively.Ineachsubgure,theupleft,upright,downleft,downrightonesaretheshapesofB,C,BCandA,respectively.Itisevidentthatifthediusionisisotropic, 55

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ToshowtheeectivenessoftheproposedmodelinrecoveringADC,inFig. 3-2 (a)-(d)wecomparedimagesofR2(denedinsection 3.2 )withcoecientAl;mestimatedby4dierentmethods.ThevoxelswithhighervalueofR2wereconsideredasone-berdiusion.TheAl;m'sin(a),(b)and(c)wereestimatedusingleast-squaresmethodin[ 21 ],model( 3{2 ),andmodel( 3{6 )withthediusion-sensitizinggradientappliedto55directions,respectively.TheAl;m'sin(d)areestimatedbythesamewayasthatin(c),butfromtheHARDdatawith12carefullychosendirections.Themodel( 3{6 )appliedon55measurementsworkedasgoodasthemodel( 3{2 )ingettinghighervalueofR2.Bothofthemworkedbetterthantheleast-squaresmethodthatdoesnotconsiderregularization.Althoughtheresultfrom12measurementswasnotasgoodasthatfrom55measurements,theyarearestillcomparable.WewillshowinFig. 3-5 (a)and(b)thattheanisotropycharacterizationresultsbasedontheADCpresentedin(c)and(d)arealsoclose.Theseexperimentalresultsindicatedthatbyusingtheproposedmodelthevoxelswithtwo-berdiusioncanbedetectedreasonablywellfrom12HARDmeasurementsincarefullyselecteddirections. 93 ]Chenetal.realizedthatsuchavoxelcouldhaveisotropicorone-berdiusion.TheydenedR0:=jA0;0j 56

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CREisameasureofuncertainty/informationinarandomvariable.LetXbearandomvariableinR,CREofXisdenedby whereR+=fX2RjX0g. (a)(b)(c)(d) Figure3-2. (a)-(d)areimagesofR2withAl;m'scalculatedusingleast-squaresmethod,model( 3{2 ),model( 3{6 )appliedon55measurements,andmodel( 3{6 )appliedon12measurements,respectively. WeuseCREofebdratherthendtocharacterizediusionanisotropy,wheredisrecoveredfromHARDmeasurementsthrough( 3{6 ).ThemagnitudeofADCisusuallyin (a)(b)(c)(d) Figure3-3. (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. 57

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TheweakconvergencepropertyofCREprovedin[ 93 ]makesempiricalCREcomputationbasedonthesamplesconvergesinthelimittothetrueCRE.ThisisnotthecasefortheShannonentropy.WedeneempiricalCREofebdas wheref1<2<:::).Fig. 3-3 (b)showsthegraphsofF()atthreepre-classiedvoxels:isotropic(red),one-ber(green),two-ber(blue).ItisobservedthatthesupportandmagnitudeofF()arelargestatthevoxelwithone-berdiusion,andsmallestatthatwithisotropicdiusion.Fig. 3-3 (c)demonstratethegraphsofF()logF()atthesamethreevoxels.Itisevidentthattheareaunderthegreencurve(one-ber)ismuchlargerthanthatunderthebluecurve(two-ber),whiletheareaundertheredcurveisthesmallest.SinceCREisexactlytheareaundercurveF()logF(),wecanconcludethatmeasureCRE(ebd)isthelargestatthevoxelswithone-berdiusion,mediumwithtwo-berdiusion,andsmallestwithisotropicdiusion.ThusmeasureCRE(ebd)couldbeusedtodiscernisotopic,one-berandtwo-berdiusionwithtwothresholdsT1andT2,withT1
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Figure3-4. Imagesoffourmeasures:(a)R2,(b)FA,(c)CREofebd,(d)Varianceofebd. Fig. 3-3 (d)onsyntheticdataandFig. 3-4 onhumanbrainHARDMRIdatafurthershowthestrengthsofCREoverthethreepopularlyusedmeasuresR2,FAandvarianceincharacterizingdiusionanisotropy.ThehumandataisthesameasthatusedinFig. 3-2 .Thesyntheticdataisconstructedasfollows:SetD1andD2tobetwodiagonalmatrixwithdiagonalelement4102;102;2102and8102;102;3102,respectively.ThenxD1butrotateprincipleeigenvectorofD2byangletogetD2()LetB(;)=uTD1u,C(;):=uTD2()u.WecomputedR2,FAandCRE,varianceofebBCforvariousvaluesofandshowedtheminFig. 3-3 (d)inblue,yellowandblackrespectively.Whenvariesfrom0to=2,BCchangesfromatypicaloneberdiusiontoatwoberdiusion,andfrom=2toBCchangesbacktothesameshapeas=0.ThegraphofCREshowsthevalueofCREdecreaseswhenBCvariesfromone-berdiusiontotwo-berdiusion,andincreaseswhenBCgraduallychangesfromtwo-berdiusionbackstoone-berdiusion. 3-4 (a)and(b).ButCREdiersmuchfromR2andFA.InFig. 3-3 (d)thegraphCREismuchsteeperthantheothers.InFig. 3-4 ,visually,contrastofCREismuchbetterthanthatofFAandR2.Furthermore,thesmallnessofmagnitudeofR2orFAisunabletodistinguishbetweenisotropicandtwo-berdiusion,whilethatofCREdoesbetterjob.Note,CREiscomparabletoFAorRA2indetectingGaussiandiusion. 59

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94 ],E(jXE(X)j)2CRE(X).Inourcase,Xisebdwhosemagnitudeismultipleof102<1,sowehaveVar(X)=E(jXE(X)j2)E(jXE(X)j)2CRE(X).OurexperimentresultsshowthatmagnitudeofCREisalmost10timesofthatofVar(X).HighermagnitudeofCREmakesitlesssensitivetoroundingerrors.Onemoreresultfrom[ 94 ]isCRE(X)p 3-3 (d)wherethegraphofvarianceiswaybelowthatofCRE.Moreover,inFig. 3-4 (d),whichrepresentingthethevarianceofebd,theGenu/Spleniumofcorpuscallosumissobrightthatregionsbesidesitarenotclearlyvisualized,soCREismuchbetterthanvariancevisually. (a)(b)(c)(d) Figure3-5. (a)-(b).Characterization:black,gray,andwhiteregionsrepresentthevoxelswithisotropic,one-ber,andtwo-berdiusion,respectively.(a)using55measurements,(b)using12carefullyselectedmeasurements.(c)ImageofCREcalculatedfrom12measurements.(d)Characterizationresultsoftheregioninsidetheredboxin(a)usingCRE(top)andvariance(bottom)basedon55measurements.Redarrowspointtoavoxelthatiswronglycharacterizedasone-berdiusionbyusingvariancebutcorrectlyclassiedastwo-berdiusionusingCRE. Fig. 3-5 (a)showsapartitionofisotropic,one-berandtwo-berdiusionbasedonADCcalculatedfrom55measurements.Theblack,gray,whitevoxelsareidentiedasisotropic,one-berandtwo-berdiusion,respectively.Thecharacterizationisconsistent 60

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3-5 (b)representsthecharacterizationresultbasedontheADCestimatedfrom12measurements.Itisveryclosetothatfrom55measurements.CREbasedonADCestimatedfrom12measurements(Fig. 3-5 (c))isalsocomparabletothatfrom55measurements(Fig. 3-4 (c)).Thusourcharacterizationisnotsensitivetonumberofmeasurements.Fig. 3-5 (d)illustratesatwo-berdiusionvoxel(pointedbyredarrow)thatisincorrectlycharacterizedasone-berdiusionusingvariance(bottomimage)butcharacterizedastwo-bercorrectlyusingCRE(topimage).ThisfurtherverifysuperiorityofCREovervarianceincharacterizingdiusionanisotropy. OurexperimentsontwosetsofhumanbrainHARDMRIdatashowedtheeectivenessandrobustnessoftheproposedmodelintheestimationofADCprolesandtheenhancementofthecharacterizationofdiusionanisotropy.Thecharacterizationofdiusionfromtheproposedmethodisconsistentwiththeknownneuroanatomy. Inthisarticle,wehavenotincludedtheworkfordeterminationofberdirectionsandthemethodforautomatedbertracking.Thestudyaddressingtheseproblemswillbereportedinseparatepapers. 61

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Copyright[2004]IEEE.Portionsreprinted,withpermission,from[ 19 ]. 2{9 )and( 2{10 ).Then,fromtheSHrepresentationoftherecovereddwedeneR0(x)=jA0;0(x)j 1{4 )shouldbecloseto1. Undertheassumptionofpt(r)beingamixtureoftwoGaussians,thediusionismodelledby( 1{4 )withn=2.Thecombinationof( 1{4 )withn=2and( 1{6 )yields whereuT=(sincos;sinsin;cos).ToestimateDiandfiin( 1{5 )weminimizethefollowingfunction:minL1;L2;fZ(2Xi=1jrLijPi(x)+jrfjPf(x))dx+1Z1(f11)2dx withtheconstraintLm;mi>0.In( 4{2 )fori=1;2i>0isaparameter,pi(x)=1+1 1+kjrGrLij2,pf(x)=1+1 1+kjrGrfj2,LiisalowertriangularmatrixsuchthatDi=LiLTi,thatistheCholeskyfactorizationforDitoachievethepositivedeniteconstraintonDi(see[ 32 ]).jrLijp=P1m;n3jrLm;nijp, 62

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4{2 )aretheregularizationtermsBythechoiceofpi(x)(similarlyforpf),inthehomogeneousregionimagegradientsareclosetozeroandpi(x)2,thesmoothingisisotropic.Alongtheedges,imagegradientmakespi(x)1,thesmoothingisthetotalvariationbasedandonlyalongtheedges.Atallotherlocations,theimagegradientforces10:8416orR2>0:1823.ThesethresholdswereselectedusingthehistogramsofR0andR2.Then,we 63

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4{2 )bytheenergydecentmethod.Theinformationoff1on1wasalsoincorporatedintotheselectionoftheinitialf. Bysolving( 4{2 )weobtainedthesolutionsLiandf,andconsequently,Di=LiLTi(i=1;2).Fig. 4-1 arepresentsthemodelsolutionf.Functionf1onthedarkredregions.Thevoxelsintheseregionsareidentiedasisotropicorone-berdiusion.Thisisconsistenttotheknownneuroanatomy.Figs.1cand1dshowthecolorrepresentationofthedirectionsoftheprincipleeigenvectorsforD1(x)andD2(x),respectively.Bycomparingthecolor-codinginFigs. 4-1 cand1dwiththecolorpieshowninFig. 4-1 b,theberdirectionsareuniquelydetermined.TherepresentationinFig. 4-1 bisimplementedbyrelatingtheazimuthalangle()ofthevectortocolorhue(H)andthepolarangle(=2)tothecolorsaturation(S).Slightlydierentfrom[ 95 ],wedeneH==2,S=2()=,andValue(V)=1inSHV.Ifthedirectionoftheprincipleeigenvectorisrepresentedby(,),theberorientationcanbedescribedbyeither(;)or(;+).Weexpressthevectorsinthelowerhemisphere,i.e.=2.Theupperhemisphereisjustanantipodallysymmetriccopyofthelowerone.Thexyplaneistheplaneofdiscontinuity. Fig. 4-2 showstheshapesofd(x;;)togetherwiththeberdirectionsat4particularvoxels.TheblueandredarrowsindicatetheorientationsofthebersdeterminedfromtheprincipleeigenvectorsofD1andD2respectively.Thelastshapecorrespondstoisotropicdiusion.Figs. 4-1 andFig. 4-2 indicatethatourmodel( 4{2 )iseectiveinrecoveringtheintra-voxelstructure. Toexaminetheaccuracyofthemodelinrecoveringberdirections,weselectedaregionwherethediusionisknownasone-berdiusioninsidethecorpuscallosum.Foreachvoxelinthisregionwecomputedthedirectioninwhichdismaximized.ThisdirectionvectoreldisshowninFig. 4-3 a.Ontheotherhandwesolved( 4{2 )andobtainedthemodelsolutionf1onthisregion.Thedirectioneldgeneratedfromthe 64

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(c)(d) Figure4-1. (a).Modelsolutionf,(b).colorpie,(c).color-codingofthe1stberdirectionmapping,(d).color-codingofthe2ndberdirectionmapping. 65

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Shapeofd,andtheorientationsoftheprincipleeigenvectorsofD1(blue)andD2(red)at4particularvoxels principleeigenvectorofD1wasthenshowninFig. 4-3 b,inwhichthevectoreldisnotonlywellpreservedbutalsomoreregularizedduetotheregularizationtermsinthemodel. Thechoiceoftheparametersin( 4{2 )andthedeterminationoftheregion1wouldinuencetheresults.Ourchoiceismadebasedontheprinciplethattheone-berdirectionfromthemodelagreeswiththedirectioninwhichdismaximized.Wewill 66

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Figure4-3. Fiberdirectioneldobtainedby(a)maximizingd,(b).theprincipleeigenvectorofD(solutionof( 4{2 )) studyinthefuturehowtoincreasetheaccuracyintheestimationoftheorientationsofcrossingberswithinavoxel. 67

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43 ]. 1{12 ).Thegoalofthissectionistorecoverasmoothmulti-tensoreldfromthenoisyHARDMRIdata.ByCholeskyfactorizationtheorem,asymmetricmatrixDispositivedeniteifandonlyifD=LLT,whereLisalowertriangularmatrix.UniquenessofthefactorizationisensuredbypositivenessofthediagonalentriesofL.ToguaranteethepositivedenitenessofdiusiontensorD1;D2,letDi=LiLTi,fori=1;2,Liisalowertriangularmatrixwithpositivediagonalentries.Constraint0f1isfullledthroughvariablerelaxationmethod.Forexample,weletf(x)=:5+arctan(!(x)) minL1(x);L2(x);f(x)Z(2Xi=13Xm=1mXn=1jrLmni(x)j+jrf(x)j)dx+ (5{1)ZZ20Z0js0(x)(febuTL1LT1u+(1f)ebuTL2LT2u)s(x;;)j2sindddx 5{1 )willbeanill-posedproblemandthelasttermisthenonlineardatadelitytermbasedon( 1{12 ). 68

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5{1 ).Initialsoff;L1;L2arecarefullychosentoavoidstickingonlocalminimum.Forconciseness,Euler-Lagrangianequationsoff;L111onlyareshownasfollowings: @t=2s0(x)Z20Z0[s0(x)(febuTL1LT1u+(1f)ebuTL2LT2u)s(x;;)](ebuTL1LT1uebuTL2LT2u)sin()dd'+div(rf (5{2) (5{3) InDTIdata,Westinet.al[ 41 ]usedtheentiretensorDatlocationx(t)todeterminev(t+1)asDv(t).TheyalsoprovidedalesssensibleschemewhichdynamicallymodulatesPEe1oftensorDandtensordeectionDv(t)contributionstotracesteering: Whereandareuser-denedweightingfactorsthatvarybetween0and1,e1andv(t)arenormalizedbeforeused.SincePEdoesn'tmakesenseatvoxelswithdegenerated 69

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HereDv(t)isalsonormalizedbeforeused.Normalizationofitwasignoredin[ 41 ],butthisisessentiallynecessaryasforhumanbrainHARDMRIdata,normofDv(t)isusuallyintheorderof103,tensordeectionwithoutnormalizationwouldnotcontributeasmuchasexpected.Tomakefollowingexplanationconcise,letf1=f;f2=1f.Ateachvoxel,fori=1;2,apply( 5{5 )toDitogetvi(t+1).Denecorrespondingstepsizeasi=cfi(x(t))FAi(x(t))v(t)vi(t+1)withcaxedconstant,FAi(x(t))thefractionalanisotropy(FA)correspondingtotensorDi.Asweknow,iffiisverycloseto0,channelDicouldbeignored;ifFAiisverylow,anisotropyofDiislow;ifv(t)vi(t+1)islow,thereistoomuchbendingbetweenv(t)andvi(t+1).SobertrackingshouldbeterminatedatchannelDiwhenanyoneoftheabovequantitiesislow.Thiscouldsimplybedonebysettingathresholdtostepsizesothatchannelswithstepsizelessthanthisthresholdareterminated.Thethresholdisastatisticalvalueobtainedthroughalargesizeofexperiments.Thisselfadaptingstepsizeconstrainspropagationspeedinregionswithhighcurvatureandlowdiusionanisotropywhileincreasesspeedinregionswithlowcurvatureandhighdiusionanisotropy,italsoautomaticallyterminatesbertrackingatchannel(s)withextremelylowstepsize(s). Ourschemegeneralizesthetensorlinepropagationalgorithm.Wecallitmulti-tensorlinepropagation(MTEND).Thechallengingaspectofthismethodistheestimationofv(t+1)'satnon-gridpoints.Welinearlyinterpolatefand6entriesofD1;D2respectively,v(t+1)'sarethencalculatedusing( 5{5 )basedontheinterpolatedf;Di's. 70

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Figure5-1. 17 ])andproposedmodel( 5{1 )respectively.(c)ImagesofADCproleuTD1uofvoxelsinsidetheselectedregionsin(a)and(b),thetoponeandthebottomoneareobtainedusingD1calculatedusingproposedmodel( 5{1 )andParker'smethodrespectively. 5{1 )overParkeret.al'smethod([ 17 ])inrecoveringsmoothmulti-tensoreldaswellasthevolumefractionfusinghumanbrainHARDMRIdata.Thedatasetconsistsof33diusionweightedimagesaswellasoneimageintheabsenceofadiusion-sensitizingeldgradient.27evenlyspacedaxialplaneswith128128voxelsineachsliceareobtainedusinga3TMRIscannerwithasingleshotspin-echoEPIsequence.Slicethicknessis3:8mm,gapis0betweentwoconsecutiveslices,repetitiontime(TR)=1000ms,echotime(TE)=85msandb=1000s=mm2,andtheeldofview(FOV)=200mm200mm.Fig. 5-1 (a)(c)comparemapsofFAofthersttensoreldD1andapparentdiusioncoecient(ADC)prolesofD1,i.e.uTD1uusingsolutionsobtainedfromParkeret.al's([ 17 ])andproposedmodel( 5{1 ).ItisobviousthatalltheresultsobtainedfromourmodelaremuchmoresmoothandmorereasonablethanthatgotfromParker'smethod.Specically,Fig. 5-1 (b)(resultofproposedmodel( 5{1 ))givesareasonableFAimagefromwhichweareabletodistinguishvoxelswithhighanisotropy(redregion)fromthatwithlowanisotropy(blueregion).WhileinFig. 5-1 (a)(resultofParker'smethod)FAcollapsesexceptinregionsaroundthecorpuscallosum.InFig. 5-1 (c),ateachvoxelweshowshapeofADCproleuTD1uin4121directions,i.e.4121u's.InthetopimageofFig. 5-1 (c)(resultofproposedmodel( 5{1 )),shapesofADCprolechangesmoothly 71

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Figure5-2. 5-1 (c)(resultofParker'smethod)shapesofADCprolejumpalotfromvoxeltovoxel,especiallyinregionsoutsidethecorpuscallosum. ThesecondexperimentistoshowMTENDalgorithmoutdoesTENDalgorithminreconstructingbertracesincludingbifurcation.Thisisdoneonsimulateddata.Werstsimulatea20203multi-vectoreldshownasbluearrowsinFig. 5-2 (a)(b),wherevoxelwithonearrowownsonlyonetensorthathasthedirectionshownbythisbluearrowastheprincipleeigenvector,whilevoxelwithtwoarrowsownstwotensorsthathavethedirectionsshownbythetwobluearrowsastheprincipleeigenvectors.Secondly,weconstructamulti-tensoreldsothatthemulti-vectoreldisthecorrespondingprincipleeigenvectoreld.RawDTIdataarenallysimulatedbasedonthesimulatedmulti-tensoreldusing( 1{12 )withs0=400;b=1000,f=1atvoxelswithonevector,f=:5atvoxelswithtwovectors,and6u'swhichareuniformlydistributedonasphere.ApplyMTENDalgorithmtothemulti-tensoreldwhileapplyTENDalgorithmtothesingle-tensoreldobtainedfromDTIdata,wegetresultsshowninFig. 5-2 (a),(b)respectively.Thefourblackpointsaretheinitiations,i.e.seedsofthebertracking.Nicebifurcationsare 72

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Figure5-3. 5-2 (a),andtheytakeplaceinvoxelswith2diusiontensorsasexpected.Incomparison,nobifurcationisvisualizedfromFig. 5-2 (b)andonlythemostleftbertracegoesalmostalongthevectoreld,whiletheother3bertracesdonotmakesenseatall.ThisveriedtheeciencyofMTENDalgorithmbasedonmulti-tensoreldoutdoesTENDbasedonsingle-tensoreldinrecoveringberswithbranching. Next,experimentsonhumanbrainHARDMRIdataareshown.ThemainaimistoshowthatMTENDandTENDworksimilarlyinthecorpuscallosumregionwhereGaussiandiusionisdominant.Buttheydierinregionswithnon-Gaussiandiusion.Beforecomparingthem,weshowinFig. 5-2 (c)anaxialviewofthewholebrain'sbertraces.Theinitiationsoftrackingaresetatalltheanisotropicvoxelsinthewholebrainvolume.Dierentstrongberbundlesandbranchesareclearlyvisualized.Theyareconsistentwithknownneuroanatomy. Fig. 5-2 (d)showsaxialviewoftrackingresultsaroundthecorpuscallosumregionusingMTEND(top),TEND(bottom)algorithm.Trackingresultsareembeddedona2Danatomicimage.Trackingstartfromasmallportioninsidethecorpuscallosum.Nosignicantdierenceisobservedasincorpuscallosumregion,Gaussiandistributionisdominant,biGuassianmodelwithf'1andGaussianmodelworkequivalentlyinrecoveringasingle-tensoreld. 73

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5-3 (b))foranothersetofcomparison.Wesetalltheanisotropicvoxelsinthewholebrainvolumeasseeds,thenapplyMTENDbasedonmulti-tensoreldrecoveredusingmodel( 5{1 )andTENDbasedonsingle-tensoreldrecoveredusingGaussiandiusionmodeltoreconstructbertracesseparately.InMTENDweset=0:9,thresholdvalueofstepsizetobe0:1whichisobtainedfromalargesizeofexperiments.OnlythoseberspassingthroughtheROIsareretainedandshowninFig. 5-3 (a)(c)forTENDandMTENDrespectively.Clearly,MTENDmethodrecoversmorebranchingbersthanTENDmethoddoes.Specically,ithappensin3dierentlocations:oneisatthelowerrightpositionanddirectedbyorangearrows.BunchesofbertraceswithseveralbranchesarenicelyshownupinFig. 5-3 (c),buttheydonotappearinFig. 5-3 (a).Thesecondoneislocatedatthemiddleanddirectedbybluearrows.AstrongbundleconnectingtheleftportionandtherightportionisclearlyvisualizedinFig. 5-3 (c)butonlyonebertraceisshowninFig. 5-3 (a).Thethirdoneliesinthemostupperleftposition:Fig. 5-3 (c)looksthickerandincludesmorebersineachbranchingthanFig. 5-3 (a)does.ThemainreasonforthedierenceisthatvoxelsinvolvingbranchinginMTENDmethodarecharacterizedasisotropic,soTENDalgorithmterminatesatthesevoxels. 5{1 )forrecoveringmulti-tensoreldtogetherwithMTENDforreconstructionofwhitematterbertracesworkmoreaccuratelythanGaussiandiusionmodeltogetherwithTEND. TheproposedmodelisundertheassumptionthattheprobabilitydensityfunctionofdiusionisoflinearcombinationoftwoGaussians.Thisresultsin13unknownsateach 74

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75

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Asmentionedinsection( 1.3 ),onepromisingapproachfordeformableregistrationisndinganoptimaldeformationeldU(X)byminimizinganenergyfunctionalthatconsistsofasimilaritymeasurebetweenthereference(target)imageandtransformedtemplate(sourceimage)andasmoothnessmeasureforthedeformationeldU(X).krUkL2canbechosenastheregularityterm,butforcaseswhetherimagepairsareinsamemodalityorindierentmodalities,thesimilaritymeasurescanvaryalot.Herewejusttalkaboutthemseparately. 58 ]minimizesthefollowingenergy 2Zfjruj2+jS(x+u(x))T(x)j2gdx;(6{1) whereisaparameterbalancingthesmoothnessofthetransformationversusthesimilarityoftheimages.Theexistenceofminimizerfor( 6{1 )isprovedinappendix( A ).Theminimizationproblemwassolvedbyusingenergydecentmethod.ThatisiterativelysolvingthePDE: Theirnumericalresultsshowedtheeectivenessofthismodelinradiotherapyplanningandevaluationthatincorporatesinternalorgandeformation. However,theparameterinthismodeleectsregistrationresultsgreatly,andtheselectionofisalwaysadicultproblem. Onecommonlyusedmethodistosetasaconstant,theresultingdeformationeldisthenusuallysensitiveto:ifistoosmall,thedeformationwillbenotsmooth;ifis 76

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Intuitively,wewishislargerinthebeginningoftheiterationandreducedasthealignmentisgettingbetter.Moreoverthedissimilaritymeasurein( 6{1 )issensitivetothepresenceofmorenoisyandoutliers[ 46 ].Todealwiththeselimitationsweproposeasimplemodicationofthismodelthataccommodateslocaldegreesofvariabilityforthematching.OurideaistoconsiderS(x+u(x))T(x)asindependentnormalrandomvariablesindexedbythepixelxwithprobabilitydensityfunctionsp(S(x+u(x))T(x))=1 22: minu(x);E(u(x);):=minu(x);1 2Zfjruj2+1 22jS(x+u(x))T(x)j2gdx+jjln(6{3) Comparablesettingsappearedin[ 96 { 98 ]regardingopticalowestimationandin[ 99 ]regardingdeformableneuroanatomytextbookregistration.Proofofexistenceofaminimizerfor( 6{3 )isinappendix B .Bycalculusofvariation,wehavetheEuler-Lagrange(EL)equationsfor( 6{3 ): 22(S(x+u(x))T(x))rS(x+u(x))=0(6{4) 77

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6{5 )to( 6{4 )wegettheequationofu(x):u(x;t)+jj withtheboundaryandinitialconditions:@u @n=0;on@R+;u(x;0)=0;on: 6{3 )diersfromLuetal.'smodel( 6{2 )intheaspectthatitallowsthedeformedtemplatehavingavariancefromthereference.Thisisespeciallygoodforaligningtwoimageswhoseintensityarenotexactlyequalorlinearlyrelated.Moreover,theweightbetweenthesmoothnessandsimilarityin( 6{2 )isxedinallintegrations,whilethisweightin( 6{6 )variesateachiterationdependingonthestandarddeviationofS(x+u(x))T(x).TheweightonthesimilaritymeasureincreaseswhenthestandarddeviationofS(x+u(x))T(x)decreases. 52 100 { 102 ]). OurapproachisreplacingtheSSDmeasurein( 6{1 )bythemutualinformation(MI)betweenS(x+u(x))andT(x). MutualinformationbetweentworandomvectorsXandYisdenedasMI(X;Y)=H(X)+H(Y)H(X;Y);

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Consideranimageasindependentrandomvariablesindexedbythevoxelx.ThemutualinformationoftwoimagesS(x+u(x))andT(x)canbecomputedasfollows:Letf(x)=S(x+u(x));g(x)=T(x):MI(S(x+u(x));T(x))=:ZR2pf;g(i1;i2)logpf;g(i1;i2) UsingMIasasimilaritymeasureourmodelreadsasfollows: minu(x)E(u):=minu(x)1 2Zjruj2dxZR2pf;g(i1;i2)logpf;g(i1;i2) OurpdfestimatorisbasedonnormalizedGaussiankernelofvariance,i.e. forsomesmall>0.Then,itsmarginalsarecomputedbypf(i1)=ZRpf;g(i1;i2)di2;pg(i2)=ZRpf;g(i1;i2)di1:

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Then,thesteadystatesolutionofthefollowingevolutionproblemgivestheminimizerof( 6{7 ). withtheboundaryandinitialconditions:@u @n=0;on@R+;u(x;0)=0;on: 6{6 6{9 ),AdditiveOperatorSplitting(AOS)scheme,anecientsemi-implicitnitedierenceschemeisused.UnderAOSalgorithm,onecanmakebothcomputationalandstoragecostbelinearinthenumberofpixels,andcangainanincreaseofcomputationaleciencybyafactorof10underrealisticaccuracyrequirements([ 89 ]). Moreover,amulti-resolutionscheme(alsocalledpyramid)isused,inwhichhigherresolutioniterationsjusttaketheinterpolatedresultfrompreviousresolutionasinitial.Pyramidschemecannotonlyhelpuspreventbeingtrappedinlocalminima,butalsohelpusgetthesolutionecientlysincegettingsolutionforlowerresolutionimagestakesmuchlesstimecomparingwiththatoffullresolutionimages,thisisessentiallyhelpfulforregisteringlargedatapairs. 6{3 )comparingwithmodel( 6{1 ).Thesecondgroupofoneexampleistoshowthemultimodalityregistrationresultbasedonvariationalmethod. 80

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6{3 )and( 6{1 ),weusenitedierenceschemewithsteepestgradientdescentmethod.Bothiterationschemeshavetwoparts:TherstpartisLaplacianoperatorcomesfromtheregularityterm,andthesecondpartisdeformationforcetermwhichcomesfromthedelitytermofmodel.Iftheforcetermisdominant,thenlargedeformationwillbemoreeasytoachievebutthevectoreldmightbenotsmooth.Iftheregularitytermisdominant,theschemewillgeneratesmoothdeformationeldbutlargedeformationishardtoget.Togetgoodregistrationresults,wewouldliketomakethesetwopartsreasonablybalanced.IfwedenearatiooftheFidelitytermOvertheRegularityterm(FOR)as: Formodel( 6{1 ) 6{3 ) 22RjS(x+u(x))T(x)j2dX Therstexampleistoapplybothschemesonsyntheticimagesofsize6565,herebothsourceandtargetaresmoothimagesgeneratedbydistancefunction.InthesourceimageI1,thereisacirclewithcenter(33;33)andradius30,theintensityisthedistancetothecircleifapixellocatesinsideofthecircleandis0otherwise,whileinthetargetimageI2,thereisasquarewithcenterat(33;33)andwithwidth61,similarlytheintensityis0outsideofthesquareandis30d(x)insidethesquarewhered(x)isthedistancefromxtothenearestside.Sincetheimageisverysmall,weapplyouralgorithmsonthefullresolutionimagesdirectlywithoutusingpyramidscheme. Afterapplyingbothmodel( 6{3 )andmodel( 6{1 )onthesyntheticimages,onecanseethatthedeformedresults,eitherbymodel( 6{3 )inFig 6-1 (b)orbymodel( 6{1 )in 81

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(d)(e)(f) (g)(h)(i) (j) Figure6-1. 6{3 )with=:1;dt=1:0after500itera-tionsandmodel( 6{1 )with=:001;dt=50:0after1000iterationsrespectively;(e,f),correspondingdierenceimage;(g)FOR;(h,i)correspondingdeformationgrids;(j)correlation;

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6-1 (c),areclosetothetargetimage(d).Fig 6-1 (e)showsthatFORapproachesto0rapidlyintheiterationsofmodel( 6{1 )(bluesolidline),whichmakestheforcetodrivethedeformationapproachto0duringiterations,hencescheme( 6{2 )makethevectorelddeformslowlyafterfewiterations,whileFORinmodel( 6{3 )cankeepthetwotermsinareasonableratio(reddashedline),thismakesscheme( 6{6 )convergesfasterthan( 6{2 ). BothschemesgivegooddeformationgridsFig 6-1 (h;i).Inthisexperiment,thegroundtruthisknown:sincewearetrytoaligntwoimagesbasedonintensityonly,thecorrespondingintensitylevelsetsintheimagepairshouldbematched,namely,totalevelcurvewhichformasaclosedsquareinI2,thecorrespondingpartinI1shouldbethesameintensitylevelsetwhichformsacircle,thisisobservedinbothFig 6-1 (h)and 6-1 (i).Deformedgridfrom( 6{3 )in500iterationsisevenslightlybetterthangridgotin1000iterationsfrom( 6{1 ),thiscomparisonagainshowsfasterconvergenceofmodel( 6{3 ). Fig 6-1 (j)showsthecrosscorrelationbetweendeformedimageandthetargetimageduringrst500iterations.OnecanseethatCCofscheme( 6{6 )approachesto1inabout50iterationswhilescheme( 6{2 )takesabout500iterations.Fasterconvergenceofmodel( 6{3 )isagainconcludedhere. Inaword,Fig 6-1 showsthatbothmodelscangetgoodresultsifdataisgood,butmodel( 6{3 )convergesfasterthanmodel( 6{1 ). Fig 6-2 istocompareperformanceofschemeson3DCTlungdata.LargedeformationisobservedfromCTlungvolumestakenbetweenthestatusofexhale(Fig 6-2 (a))andthestatusofinhale(Fig 6-2 (b)),whichalsogeneratemotionoftumor(theisolatedwhitespotincoronalview).Withinlimitednumberofiterations,usingmodel( 6{1 )canonlycatchaclosesolutiontothislargedeformationbecauseoftheregularitytermhindersthelargedeformation(Fig 6-2 (d,f)),anotherwaytosayisthatthedeformationforcedecreasedramaticallybecauseFORapproachesto0rapidlyinscheme( 6{2 ),whilethemodel( 6{3 )canmakethelargedeformationhappenwhilemaintainthesmoothdeformation 83

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(c)(d) (e)(f) (g)(h) Figure6-2. 6{3 )and( 6{1 )respectively;(e,f)volumedier-encewithtargetbysubtractingresultsbymodel( 6{3 )and( 6{1 )respectively;(g)correlation;(h)Originalvolumedierence.

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6-2 (c,f)).Fig 6-2 (e)shows,insenseofCC,scheme( 6{6 )convergesfasterthanscheme( 6{2 ). Fig 6-3 istocompareperformanceofschemeson3DMRIvolumeswithsize12018751.Thesetwodatawerecollectedintwoadjacentdays,andinthedata,largemotionisobservedbetweenFig 6-3 (a)and(b).After20iterations,model( 6{1 )givesadeformationwhichisnotquitgood(d,f,g),whilethemodel( 6{3 )behaviorsbetterinFig 6-3 (c,e,g).Fig 6-3 (g)againshowsscheme 6{3 convergesfasterthanscheme 6{2 fromthecurveofCC.Thedierenceimagesshowtheadvantagesofmodel( 6{3 )thanmodel( 6{1 )bycomparingFig 6-2 (e,f,g). Fig 6-4 istocompareperformanceofschemesonnoised3DMRIvolumes,byaddingGaussiannoisewith0meanandvariance.005toimagesshowninFig 6-3 (a,b).After20iterations,model( 6{1 )adeformationwhichisfarfromgoodenough(d,f),whileresultsin 6-4 (c,e)showsthemodel( 6{3 )behaviorsaswellasinFig 6-3 (ce).Fig 6-4 (g)againshowsscheme( 6{3 )convergesfasterthanscheme( 6{2 )fromthecurveofCC.Inoneword,Fig( 6-4 )showsthatmodel( 6{3 )canhandlenoisemuchbetterthanmodel( 6{1 ). Toeliminatetheeectofpyramidscheme,hereallthesefourexamplesaredoingiterationsonfullresolutiondirectly.AftercombiningpyramidschemeandAOSalgorithm(whichisparallelizable),allthese3Dexperimentscanbedonewithin10to20seconds.Afterparallelization,thetimecanbereducedtoseconds,whichisveryclosetoclinicalapplications. 6{7 ).Sincethecomputationofjointpdfforimagepair( 6{8 )involvesexponentialfunction,whichisveryexpensivewhenthevolumeshavelargesizes.Toimproveeciencyofthescheme 6{9 ,bothpyramidschemeandAOSalgorithmareoccupiedhere. InFig 6-5 ,model( 6{7 )isappliedtoregister3DCTvolumeto3DMRIvolumeofsize12018751byapplyingpyramidschemeandAOSalgorithm,20equallyspaced 85

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(c)(d) (e)(f) (g)(h) Figure6-3. 6{3 )andmodel( 6{1 )respectively.(e,f)dierenceT(X)S(X+U(X))basedonregistrationresultsbymodel( 6{3 )andmodel( 6{1 )respectively.(g)correlation;(h)originaldierenceT(X)S(X)

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(c)(d) (e)(f) (g)(h) Figure6-4. 6{3 )andmodel( 6{1 )respectively.(e,f)dierenceT(X)S(X+U(X))basedonregistrationresultsbymodel( 6{3 )andmodel( 6{1 )respectively.(g)correlation;(h)originaldierenceT(X)S(X)

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(c)(d) Figure6-5. 6{7 ).(a,b)Source(CT)andtarget(MRI)imagesrespectively;(c)CurveofMIduringiterations;(d)Deformedresultinthreeviews.

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6{3 )isfeaturedatallowingthedeformedtemplatehavingavariancefromthereference.Thisisespeciallygoodforaligningtwoimageswhoseintensityarenotexactlyequalorlinearlyrelated,orcorruptedbynoise.Moreover,theweightbetweenthesmoothnessandsimilaritymeasuresisadjustedautomaticallyinourmodelbythevarianceateachiteration.Theweightonthegoodnessofmatchingincreaseswhenthestandardvariancedecreases.Thisfeaturemakesmodel( 6{3 )lesssensitivethanthemodel( 6{1 )tothechoicesoftheparameter,thiswillbeinvestedinthenearfuture.Allthesefeatureshavecontributedtotheimprovementofrobustnessandaccuracyinregistrationgreatly,asshowninexperiments. Duetoclinicalapplicationrequirement,ecientalgorithmsareneededtospeedupthecomputationofMIinthemultimodalitycase.Futureworkwillfocusonmodelingandalgorithmsforrobustandecientregistration. 89

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Themaincontributionofthischapterisontheimprovementofinverseconsistency,anditsapplicationtotheradiationtherapy,inparticular,togetmoreaccurateautore-contouring.ThebasicideaisminimizingE1andE2in( 1{14 )coupledbytheinverseconsistentconstraintsdenedinthenextsection.Applicationsoftheseinverseconsistentdeformationsonautore-contouringwillbediscussedinexperimentalresults. 82 ].Portionsreprinted,withpermission. Inthissection,wewillrstintroducea"natural"formulationofinverseconsistentconstraints,whichcanbeusedtocorrectthetruncationerrorsin( 1{17 ).Thenwewillcombinethiswiththedissimilaritymeasuresbasedonthelikelihoodoftheresidueimageintoourenergyfunctionalstoimprovetheaccuracy,robustnessandinverseconsistentofthedeformableimageregistration. Letuandvbetheforwardandbackwarddisplacementeldsrelatedtodeformationhandgby Thentheinverseconsistentconstrainth(g(x))=xcanbewrittenintermsofuandvas Therefore, Similarly,theconstraintg(h(x))=xcanberepresentedby 90

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7{3 )and( 7{4 )ashardconstraintsinourproposedenergyminimizationmethod. Asinthechapter 6 ,toaccommodatecertaindegreeofvariabilityintheimagematching,weconsidertheresiduebetweenthedeformedsourceimageandtargetimageS(h(x))T(x)ateachpointasanindependentrandomvariablewithGaussiandistributionofmeanzeroandavariancetobeoptimized.Bytheindependencyassumption,thejointpdfofalltheserandomvariables,whichisthelikelihoodoftheresidualimagegivenparameter,becomes 22:(7{5) Thenthenegativelog-likelihoodfunctionis ReplaceM(S(h);T)inE1of( 1{14 )by( 7{6 ),andreplaceM(S;T(g))inE2inasimilarmanner,E1andE2canberewrittenintermsofuandvas: AfterchoosingR()=j5j2L2()withboundaryandinitialconditions: @~n(x;t)=0;@v @~n(x;t)=0;on@R+u(x;0)=0;v(x;0)=0;on;(7{8)1and2in( 1{16 )become 91

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7{7 ),weget Byusingthisdissimilaritymeasure,theresidualimagenolongerneedstobepointwiselyclosetozerotomaketheL2normsmall.Instead,thenewmeasureonlyforcesthemeanoftheresiduetobezero,andallowstheresiduehavingavariancetoaccommodatecertainvariability.Thisisespeciallygoodforaligningtwoimageswhoseintensitiesarenotexactlyequalorlinearlyrelated,andmakesthemodelmorerobusttonoiseandartifacts. Moreover,thelikelihoodbasedapproachislesssensitivetothechoiceoftheparameter.InSSDmodels,isprexed,sothebalanceofthedissimilaritymeasureandregularitymeasuredoesnotchangeduringiterations.Thismakestheselectionofverydicultandaectsregistrationresult.Intheproposedmodelthebalancingfactorofthesetwomeasuresis,infact,2ratherthanalone.Therefore,evenisprexed,theweightbetweenthesetwomeasuresvariesateachiterationasthevarianceupdates.Astheiterationsgraduallyapproachtoconvergencestage,theresiduemagnitudebecomessmaller,hence,thevariancereduces,andconsequently,theweightonsmoothingdeformationeldversusmatchingimagesautomaticallydecreases. Combiningtheseideas,weproposeanewmodeltoimprovetheinverseconsistencyof( 1{14 )byusingthehardconstraints( 7{3 )and( 7{4 )toreplacethepenaltytermsin( 1{14 ),andtoimprovetheeciencyofalignmentbyusingtheproposedsimilaritymeasure.Moreprecisely,weproposetominimizeacoupledminimizationproblem 7{7 )subjecttou(x)+v(x+u(x))=08x2v(x)+u(x+v(x))=08x2(7{11) 92

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103 ]),thuswemaynotbeabletogetitsinversethrough( 7{3 ).Bychoosing1and2asin( 7{9 ),thesolutionsoftheconstrainedcoupledminimizationproblem( 7{11 )areobtainedviathefollowingalgorithm Toenhancetheeciencyoftheproposedalgorithm,theAdditiveOperatorSplittingscheme([ 89 104 ])isusedtospeedupournumericalcomputation.Wepresentourresultsbasedonsyntheticimages,and3DprostateMRIdata. 93

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(c)(d) (e)(f)(g) Figure7-1. 1{15 ),Algorithm1withandwithoutconstraintsrespectively.

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Basedon( 7{3 )and( 7{4 ),ifhandgareinversetoeachother,thenbothoftheinverseinconsistenterroreldsu(x)+v(x+u(x))andv(x)+u(x+v(x))shouldbe0.Thus,theircomponentsandnormsshouldalsobe0.Thispropertywillbeusedtocomparetheinverseinconsistencyerrorsbetweenthreemethods:Scheme( 1{15 ),Algorithm1andModel( 7{11 )withoutconstraints. Therstexperimentisaimedtoexamandcomparetheinverseconsistencyofthesethreemethodsonsyntheticdata.Fig 7-1 (a)and(b)presentthesourceimageSandtargetimageT,respectively,withtheboundariesoftheobjectssuperimposed.Objectshaveintensity1.0inbothimages,andtheirbackground/holeshaveintensity0.Thethreemethodsareappliedseparatelywiththesameparametersdt=:05;=5:0for800iterations,andthecorrespondingresultsareshowninFig 7-1 (c-g).FromFig 7-1 (c),onecanseethatcorrelationbyallmethodsconvergestoabout1.0similarly.HoweverinFig 7-1 (d),themeansofnormsforbothinverseinconsistenterroreldsu(x)+v(x+u(x))andv(x)+u(x+v(x))bybothScheme( 1{15 )andnon-constrainedModel( 7{11 )areincreasedtoabout0.4pixelsinaverage,i.e.,themeanvalueoftheirinverseinconsistencyerrorsareabout0.4pixels,whilethosefromAlgorithms1aremaintainedinanegligiblelowlevel(about0.01pixel).Wealsocomparetheerroreldu(x)+v(x+u(x))inFig 7-1 (e,f,g)byapplyingresultsfromScheme( 1{15 ),Algorithm1andModel( 7{11 )withoutconstraintsrespectivelyonaregulargridmesh.Fig 7-1 (f)showsthattheerrorbyAlgorithm1is0almosteverywhere(almostnodisplacementintheregulargridmesh),while(e)and(g) 95

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Method 0.7396 0.0051 0.7876 0.0059 0.7879 0.0085 Scheme( 1{15 ) 0.9915 0.2390 1.1055 0.2320 1.2272 0.3526 Noconstraints 0.9916 0.2300 1.0979 0.2255 1.2202 0.3414 inconsistencyerrorv(x)+u(x+v(x))on Algorithm1 0.4870 0.0047 0.5814 0.0053 0.7287 0.0076 Scheme( 1{15 ) 0.8705 0.2290 1.2224 0.2240 1.2304 0.3398 NoConstraints 0.8303 0.2199 1.2670 0.2174 1.2729 0.3284 inconsistencyerroru(x)+v(x+u(x))oncontoursinS 0.2810 0.0431 0.2769 0.0502 0.3742 0.0719 Scheme( 1{15 ) 0.7237 0.1445 0.8047 0.1878 0.8073 0.2577 NoConstraints 0.7403 0.1461 0.8374 0.1908 0.8410 0.2612 inconsistencyerrorv(x)+u(x+v(x))oncontoursinT 0.3833 0.0412 0.4058 0.0484 0.4059 0.0697 Scheme( 1{15 ) 0.5937 0.1390 0.9308 0.1661 0.9715 0.2361 NoConstraints 0.6180 0.1410 0.9735 0.1695 1.0171 0.2402 96

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Algorithm1 0.5155 0.0105 0.4010 0.0082 0.5549 0.0146 Scheme( 1{15 ) 0.4936 0.0457 0.6750 0.0793 0.7154 0.1020 inconsistencyerroreldv(x)+u(x+v(x))on Algorithm1 0.5937 0.0121 0.4766 0.0087 0.6230 0.0164 Scheme( 1{15 ) 1.6315 0.0583 1.5860 0.0951 1.9917 0.1231 inconsistencyerroru(x)+v(x+u(x))oncontoursinS 1.6165 0.0524 1.9630 0.0459 2.3577 0.0770 Scheme( 1{15 ) 4.7525 0.1036 6.2510 0.1292 6.3180 0.1843 showerrorsfromothertwomethodsarelarger,especiallyintheregioncorrespondingtotheboundariesoftwoholesinSandT. Toquantitativelyvalidatetheimprovedinverseconsistencybytheproposedalgorithm,wecomparethemaximumandmeanvaluesofcomponentsandnormsofu(x)+v(x+u(x))andv(x)+u(x+v(x))inTable 7-1 ,whereXmax,Xmeandenotethemaximumandmeanvaluesoftherstcomponentoftheerroreldsrespectively,andYmax,Ymeanarethoseofthesecondcomponents.Thequantitativecomparisonsareperformedintworegions:oneisintheimagedomain,andtheotherisonthecontourswhicharetheboundariesoftheobjectsintheimagesshowninFig 7-1 (a,b)respectively.Table1showstheproposedalgorithmyieldsmuchsmallererrorsinallaspects.Particularly,itsmeanerrorisaboutonefortiethofthosefrombothscheme( 1{15 )andnon-constrainedmodel( 7{11 )in,andaboutonethirtiethoftheirsatthecontourregions. Thesecondexperimentistovalidatetheimprovementinaccuracyoftheproposedalgorithmon3Dprostatedata,whichconsistsof100phasesof2Dimagesfocusingonprostatearea,whereROIshavelargeinternalmotions.ThesourcevolumeSistherstphase,andtheboundariesofROIsinSaredelineatedbycontoursandsuperimposedinFig 7-2 (a),andtheother99phasesaretargets,andmethodsScheme( 1{15 )andAlgorithm1areappliedtonddeformationsbetweenthe1stphaseandeachofthe 97

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7-2 (b),andtheautore-contouringresultbyAlgorithm1issuperimposedonit. ComparisonontheconvergenceandinverseinconsistencybythersttwomethodsareshowninFig 7-2 (c,d).Fig 7-2 (c)comparesCCbetweendeformedimagesandtargetimagesbythesetwomodels.BothScheme( 1{15 )andAlgorithm1improvetheinitialCCbetweenSandeachofother99phasestoasimilarlevel.However,fromFig 7-2 (d)wecanobservethatthenormofinverseinconsistencyerrorbyscheme( 1{15 )ismuchhigherinaveragethanthatofAlgorithm1. Thequantitativecomparisononinverseinconsistencyerrorsbetweenthe1stphaseandthe21stphaseforthisexperimentislistedinTable 7-2 fordemonstration.Besidecomparingtheerroreldson,wealsoevaluateerrorv(x)+u(x+v)atpointsofallgivencontoursonS.BycomparingthecorrespondingcomponentsandnormsoftheinverseinconsistencyerroreldsinTable2,wendthaterrorsgeneratedbyproposedalgorithmismuchlowerthanthatbyscheme( 1{15 ),thisindicatesthatthepointcorrespondenceandautomaticre-contouringresultsaremoreaccurate. Thethirdexperimentistovalidatetheimprovedaccuracyofproposedalgorithmson3DCTdata.ThesourcevolumeS,whichistreatedasplanningdatainaradiationtherapy,isshowninFig 7-3 (a)inthreeviews(theaxial,sagittalandcoronalviews).TheboundariesoforgansinSaredrawnslicebysliceinaxialviewasshowninFig 7-3 (a).ThetargetvolumeTisshowninFig 7-3 (b).BothSandTareofdimension25625664.Weapplythethreealgorithmsonthisdataasintherstexperiment.Amulti-resolutionapproachisusedinourcomputation.Westartfromresolution646416withparameters=10;dt=0:4for40iterations,thenincreasetheresolutionto12812832withdt=:2for20iterations,andnallytotheoriginalresolutionwithdt=:1for10steps. 98

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(c)(d) Figure7-2.

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ComparisonontheconvergenceandinverseinconsistencybythethreemethodsareshowninFig 7-3 (c,d).Fig 7-3 (c)comparescrosscorrelation(CC)ofthesethreemodels.Scheme( 1{15 )convergesslightlyfasterthantheothertwo,butnallytheCCtendstothesamelevel.However,fromFig 7-3 (d)wecanobservethatthenormofinverseinconsistencyerrorbyscheme( 1{15 )ismuchhigherinaveragethanthatofAlgorithm1.Thisagainshowstheimprovementofinverseconsistencyoftheproposedmethods. ThequantitativecomparisononinverseinconsistencyerrorsforthethirdexperimentislistedinTable 7-3 .BycomparingthecorrespondingcomponentsandnormsoftheinverseinconsistencyerroreldsinTable 7-3 ,wendthaterrorsgeneratedbyproposedalgorithmsisaboutonefthofthatbyscheme( 1{15 ). Thedissimilaritymeasureusedinthisworkisthenegativelog-likelihoodoftheresidualimagebetweenthedeformedsourceandtarget.Thisdissimilaritymeasureisabletoaccommodatecertainvariabilityinthematching.Hence,themodelismorerobusttonoisethanSSD,moreover,itislesssensitivetothechoiceoftheparameterthatbalancesthesmoothnessofthedeformationeldandgoodnessofmatching. 100

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Method 8.9336 0.0151 3.0239 0.0154 1.4882 0.0084 9.0485 0.0265 Scheme( 1{15 ) 6.6532 0.0740 3.6780 0.0765 2.2982 0.0734 7.1051 0.1523 inconsistencyerroreldv(x)+u(x+v(x))on Algorithm1 5.3978 0.0135 2.8936 0.0148 2.9918 0.0079 5.4209 0.0247 Scheme( 1{15 ) 6.3512 0.0724 2.7737 0.0760 4.7697 0.0722 7.9680 0.1499 inconsistencyerroreldv(x)+u(x+v(x))onallcontoursinS 1.9935 0.0210 1.0991 0.0169 0.8705 0.0138 2.5338 0.0320 Scheme( 1{15 ) 2.5347 0.0853 1.3701 0.0770 0.8306 0.0984 3.0055 0.1491 (a)(b) (c)(d) Figure7-3. Experimenton3DCTvolume.(a)Sourcevolumewithcontourssupperimposed,(b)targetvolume,(c)correlationscorr(S(h);T),andcorr(S;T(g))duringiterations,(d)meanofjju(x)+v(x+u(x))jjandjjv(x)+u(x+v(x))jjduringiterations. 101

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102

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Inthischapter,Iwillbrieydiscussmyfutureresearchdirectionsrelatedwithmyworkpresentedinpreviouschapters. 105 { 107 ]).In[ 105 ],TuchproposedQ-BallImaging,wheretheODFisestimateddirectlyfromtherawHARDMRImeasurementsonasinglespherebytheFunk-RadonTransform(FRT),whichwaslatersolvedanalyticallyandecientlybyDescoteauxet.al.in[ 106 ]byexpressingtheHARDMRIsignalasasphericalharmonicseriesuptosomeorderandbyusingtheFunk-Hecketheorem[ 108 ].McGrawet.al.[ 107 ]proposedavonMises-FishermixturemodeloftheODF.Jianet.al.in[ 109 ]reconstructedmulti-berbyusingMixtureofWishartsandsparsedeconvolution.RecentworkregardingbertracereconstructionbasedonODFcanbefoundin[ 110 111 ]. 74 112 { 114 ]),vianovelmodelsoreventhroughdierenthardwarearchitectures(e.g.[ 115 116 ]). StillanotherdirectionofmyfutureworkisondiusionweightedMRIregistration(e.g.[ 117 118 ]),whichcanbedonebychoosingappropriatedissimilaritymeasures(e.g.[ 119 120 ])andtranslations. 103

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minu2W1;2()E(u):=1 2Zfjruj2+jS(x+u(x))T(x)j2gdx;(A{1) A{1 )hasatleastonesolution. A{1 ){

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B{1 )hasatleastoneminimizer(u;)2W1;2()(0;1). Byassumption,there9">0suchthatRjS(x+u(x))T(x)j2dx";8u2W1;2(),thus1 2;where"="holdsas2=": 2(1+ln")(B{2) soE(u;)isboundedfrombelow. 2.Sothereexistsaminimizingsequencefuk;kgW1;2()(0;1)s.t. limk!1E(uk;k)=infu2W1;2();>0E(u;):=m:(B{3) 105

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By( B{3 ),thereexistsK02Nsuchthat8kK0,wehavemE(uk;k)m+1: lim!0+;1ZjS(x+u(x))T(x)j2 Henceforthegivenminimizingsequencef(uk;k)gW1;2()(0;1),k90andk9+1.i.e.,kisboundedbelowandabove,thereexist2(0;1)suchthatk2[;1 3.SetK2=max(K0;K1),then8kK2,kisbounded,soislnk,while wehavef5ukgL2()isbounded:

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limk!1ZjS(x+uk(x))T(x)j2 limk!1jjlnkm2(B{7) limk!1 Clearly,sinceln(x)iscontinues,lnk!ln,( B{7 )isproved. B{8 )isproved. Theonlythingleftistoprove( B{6 ). 4.Fromcompactnesstheorem,thereexistsasubsequencefukj;kjg(westilldenoteitasfuk;kg)suchthat ByDominatedConvergenceTheorem, 107

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2k2jZjS(x+uk(x))T(x)j2dxZjS(x+u(x))T(x)j2dxj+ZjS(x+u(x))T(x)j2dxj1 2k21 22j1 22jZjS(x+uk(x))T(x)j2dxZjS(x+u(x))T(x)j2dxj+ZjS(x+u(x))T(x)j2dxj1 2k21 22j!0:(B{11) Thus( B{6 )isproved.Thus(u;)isaminimizerof( B{1 ). 108

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andZjS(x)T(x+v(x))j2dx";8v2W1;2(); C{2 )hasatleastonesolutionsuchthatu;vinA:=fu;v2W1;2()ju(x)=v(x+u(x)a:e:;v(x)=u(x+v(x)a:e)g. B ,wehaveaminimizingsequencef(uk;1k;vk;2k)g1k=1withf(uk;vk)g1k=12Asuchthat 109

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Usingthecompactnesstheory,wededucefrom( C{4 )and( C{7 )that Sincefuk;vkg2A,fu;vg2A,thusE1(u;1)=m1andE2(v;2)=m2,i.e.,fu;vgisaminimizerfor( C{2 ). 110

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[1] D.LeBihanandP.J.Basser,\Moleculardiusionandnuclearmagneticresonance,"Diusionandperfusionmagneticresonanceimaging,pp.5{17,1995. [2] M.E.Moseley,Y.Cohen,J.Mintorovitch,J.surudaL.Chileuitt,D.Norman,andP.Weinstein,\Evidenceofanisotropicself-diusionincatbrain,"Proc.ofthe8thAnnualMeetingofInternationalSocietyforMagneticResonanceinMedicine,pp.136{136,1989. [3] M.E.Moseley,J.Kucharczyk,H.S.Asgari,andD.Norman,\AnisotropyindiusionweightedMRI,"Magn.Reson.Med.,vol.19,pp.321{326,1991. [4] P.J.BasserandC.Pierpaoli,\Microstructualandphysiologicalfeaturesoftissueselucidatedbyquantitativediusiontensormri,"Magn.Reson.Med.,vol.111(B),pp.209{219,1996. [5] DSTuch,RMWeissko,JWBelliveau,andVJWedeen,\Highangularresolutiondiusionimagingofthehumanbrain,"inProc.ofthe7thannualmeethingofIn-ternationalSocietyforMagneticResonanceinMedicine(ISMRM'99),Philadelphia,1999,p.321. [6] VJWedeen,TGReese,DSTuchandMRWeigel,J-GDou,andRMWeisskoandD.Chesler,\MappingberorientationspectraincerebralwhitematterwithfouriertransformdiusionMRI.,"Proc.ofthe8thAnnualMeetingofInternationalSocietyforMagneticResonanceinMedicineISMRM'00,pp.82{82,2000. [7] P.J.Basser,J.Mattiello,andD.Lebihan,\Estimationoftheeectiveself-diusiontensorfromthenmr,"SpinEcho.J.Magn.Reson.,vol.seriesB103,pp.247{254,1994. [8] PJBasser,JMattiello,andDLeBihan,\MRdiusiontensorspectroscopyandimaging,"Biophys,vol.66:259,pp.267,1994. [9] TLChenevert,JABrunberg,andJGPipe,\Anisotropicdiusioninhumanwhitematter:demonstrationwithMRtechniquesinvivo,"Radiology,vol.177,pp.401{405,1990. [10] EWHsuandS.Mori,\AnalyticalexpressionfortheNMRapparentdiusioncoecientsinananisotropysystemandasimpliedmethodfordetermingberorientation,"MagnResonMed,vol.34,pp.194{200,1995. [11] RachidDeriche,DavidTschumperle,andChristopheLenglet,\DT-MRIestimation,regularizationandbertractography,"inProc.of2ndIEEEInternationalSympo-siumonBiomedicalImaging:FromNanotoMacroISBI'04,WashingtonD.C,2004,pp.9{12. [12] P.J.Basser,J.Mattiello,andD.LeBihan,\MRdiusiontensorspectroscopyandimaging,"Biophys,vol.66,pp.259{267,1994. 111

PAGE 112

[13] A.L.Alexander,K.M.Hasan,M.Lazar,J.S.Tsuruda,andD.L.Parker,\Analysisofpartialvolumeeectsindiusion-tensorMRI,"Magn.Reson.Med.,vol.45,pp.770{780,2001. [14] L.Frank,\CharacterizationofanisotropyinhighangularresolutiondiusionweightedMRI,"inProceedingsofthe9thAnnualMeetingofInternationalSocietyforMagneticResonanceinMedicine(ISMRM'01),Glasgow,Scotland,2001,p.1531. [15] L.Frank,\Anisotropyinhighangularresolutiondiusion-weightedMRI,"MagnResonMed,vol.45,pp.935{939,2001. [16] D.S.Tuch,R.M.Weissko,J.W.Belliveau,andV.J.Wedeen,\Highangularresolutiondiusionimagingofthehumanbrain,"Proc.ofthe7thAnnualMeetingofInternationalSocietyforMagneticResonanceinMedicine(ISMRM'99),p.321,1999. [17] G.J.M.ParkerandD.C.Alexander,\Probabilisticmontecarlobasedmappingofcerebralconnectionsutilisingwhole-braincrossingberinformation,"inInformationProcessinginMedicalImaging,AmblesideUK,072003,pp.684{696. [18] D.S.Tuch,T.G.Reese,M.R.Wiegell,N.Makris,J.W.Belliveau,andV.J.Wedeen,\Highangularresolutiondiusionimagingrevealsintravoxelwhitematterberheterogeneity,"MagnResonMed,vol.48,pp.577{582,2002. [19] Y.Chen,W.Guo,Q.Zeng,Y.Liu,andB.CVemuri.et.al.,\Recoveryofintra-voxelstructurefromhighangularresolutiondiusion(HARD)MRI,"Proc.of2ndIEEEInternationalSymposiumonBiomedicalImaging:FromNanotoMacro(ISBI'04),pp.1028{1031,2004. [20] E.O.StejskalandJ.E.Tanner,\Spindiusionmeasurements:Spinechoesinthepresenceofatime-dependenteldgradient,"Chem.Phys.,vol.42,pp.288{292,1965. [21] D.C.Alexander,G.J.Barker,andS.R.Arridge,\Detectionandmodelingofnon-gaussianapparentdiusioncoecientprolesinhumanbraindata,"Magn.Reson.Med.,vol.48,pp.331{340,2002. [22] G.J.M.Parker,J.A.Schnabel,M.R.Symms,D.J.Werring,andG.J.Baker,\Nonlinearsmoothingforreductionofsystematicandrandomerrorsindiusiontensorimaging,"Magn.Reson.Med.,vol.11,pp.702{710,2000. [23] B.C.Vemuri,Y.Chen,M.Rao,T.McGraw,Z.Wang,andT.Mareci,\FibertractmappingfromdiusiontensorMRI,"inProc.ofIEEEWorkshoponVariationalandLevelSetMethodsVLSM'01,Philadelphia,2001,pp.81{88. [24] C.Feddern,J.Weickert,andB.Burgeth,\Level-setmethodsfortensor-valuedimages,"inProceedingsofthe9thAnnualMeetingofInternationalSocietyforMagneticResonanceinMedicine(ISMRM'03),Nice,2003,pp.65{72.

PAGE 113

[25] C.Poupon,J.F.Mangin,C.A.Clark,V.Frouin,J.Regis,D.LeBihan,andI.Block,\TowardsinferenceofhumanbrainconnectivityfromMRdiusiontensordata,"Med.ImageAnal.,vol.5,pp.1{15,2001. [26] D.TschumperleandR.Deriche,\Regularizationoforthonormalvectorsetsregularizationwithpde'sandapplications,"InternationalJournalofComputerVision,vol.50(3),pp.237{252,2002. [27] C.Chefd'hotel,D.Tschumperle',andOlivierD.Faugeras,\Constrainedowsofmatrix-valuedfunctions:Applicationtodiusiontensorregularization,"inEuropeanConferenceonComputerVisionECCV'02,2002,vol.1,pp.251{265. [28] SinisaPajevicandCarloPierpaoli,\Colorschemestorepresenttheorientationofanisotropictissuesfromdiusiontensordata:Applicationtowhitematterbertractmappyinginthehumanbrain,"MagnResonMed,vol.42,pp.526{540,1999. [29] D.TschumperleandR.Deriche,\Tensoreldvisualizationwithpde'sandapplicationtoDT-MRIbervisualization,"inProc.ofIEEEWorkshoponVaria-tionalandLevelSetMethodsVLSM'03,Nice,2003,pp.256{26. [30] J.WeickertandT.Brox,\Diusionandregularizationofvector-andmatrix-valuedimages,"ContemporaryMathematics,vol.313,pp.251{268,2002. [31] Z.Wang,B.C.Vemuri,Y.Chen,andT.Mareci,\SimultaneoussmoothingandestimationofthetensoreldfromdiusiontensorMRI,"inProc.ofIEEEConferenceonComputerVisionandPatternRecognitionCVPR'03,Wisconxin,2003,vol.2. [32] Z.Wang,B.C.Vemuri,Y.Chen,andT.Mareci,\Aconstrainedvariationalprinciplefordirectestimationandsmoothingofthediusiontensoreldfromdwi,"inProc.ofInformationProcessinginMedicalImagingIPMI'03,Ambleside,UK,2003,pp.660{671. [33] Y.Chen,W.Guo,Q.Zeng,andY.Liu,\Anonstandardsmoothinginreconstructionofapparentdiusioncoecientprolesfromdiusionweightedimages,"InverseProblemsandImaging2008,toappear. [34] Y.Chen,W.Guo,Q.Zeng,andY.Liu,\Classicationofintra-voxeldiusionfromHARDMRI,"inProc.ofthe12thannualmeethingofInternationalSocietyforMagneticResonanceinMedicine(ISMRM'04),2004,p.252. [35] Y.Chen,W.Guo,Q.Zeng,X.Yan,M.Rao,andY.Liu,\ApparentdiusioncoecientapproximationanddiusionanisotropycharacterizationinDWI,"19thInternationalConferenceonInformationProcessinginMedicalImagingIPMI'05,pp.246{257,10-15July2005. [36] M.Rao,Y.Chen,B.C.Vemuri,andF.Wang,\Cumulativeresidualentropy:Anewmeasureofinformation,"IEEETrans.onInfo.Theory,vol.50,pp.1220{1228,2004.

PAGE 114

[37] Q.Zeng,Y.Chen,W.Guo,andY.Liu,\Recovermulti-tensorstructurefromHARDMRIunderbi-gaussianassumption,"MultiscaleOptimizationMethodsandApplications(NonconvexOptimizationandItsApplications),pp.379{386,2004. [38] D.LeBihan,E.Brethon,andD.Lallemandet.al,\MRimagingofintravoxelincoherentmotions:Applicationtodiusionandperfusioninneurologicdisorders,"Radiology,vol.161,pp.401{407,1986. [39] B.C.Vemuri,Y.Chen,M.Rao,Z.Wang,TMcGraw,T.Mareci,S.J.Blackband,andP.Reier,\Automaticbertractographyfromdtianditsvalidation,"inProc.of1stIEEEInternationalSymposiumonBiomedicalImaging,2002,pp.505{508. [40] C.Pierpaoli,P.Jezzard,P.J.Basser,A.Barnett,andG.DiChiro,\DiusiontensorMRimagingofthehumanbrain,"Radiology,vol.201(3),pp.637{648,1996. [41] C.-F.Westin,S.E.Maier,B.Khidir,P.Everett,F.A.JoleszH.,andR.Kikinis,\Imageprocessingfordiusiontensormagneticresonanceimaging,"inLecturenotesincomputerscience:InternationalSocietyandConferenceSeriesonMedicalImageComputingandComputer-AssistedInterventionMICCAI'99,Cambridege:Springer,1999,pp.441{452. [42] M.Lazar,D.M.Weinstein,J.S.Tsuruda,KhaderM.Hasan,K.Arfanakis,M.Meyerand,B.Badie,H.Rowley,V.Haughton,A.Field,andA.Alexander,\Whitemattertractographyusingdiusiontensordeection,"HumanBrainMapping,vol.18,pp.306{321,2003. [43] W.Guo,Q.Zeng,Y.Chen,,andY.Liu,\Usingmultipletensordeectiontoreconstructwhitematterbertraceswithbranching,"3rdIEEEInternationalSymposiumonBiomedicalImaging:MacrotoNano(ISBI'06),pp.69{72,6-9April2006. [44] Q.Zeng,Y.Chen,W.Guo,andY.Liu,\Whitematterbertrackingbasedonmulti-directionalvectoreld,"inProc.ofthe13thannualmeethingofInternationalSocietyforMagneticResonanceinMedicine(ISMRM'05),2005,p.218. [45] Q.Zeng,Y.Chen,W.Guo,andY.Liu,\Whitematterbertrackingbasedonmulti-directionalvectoreld,"inProc.ofthe11thAnnualScienticMeetingoftheOrganizationofHumanBrainMapping,2005,p.1649. [46] D.Hill,P.Batchelor,M.Holden,andD.Hawkes,\Topicalreview:medicalimageregistration,"PhysicsinMedicineandBiology,vol.46,pp.1{45,2001. [47] B.ZitovaandJ.Flusser,\Imageregistrationmethods:asurvey,"ImageVis.comput.,vol.21,pp.977{1000,2003. [48] J.Modersitzki,Numericalmethodsforimageregistration,Oxford,2004. [49] J.MaintzandM.Viergever,\Asurveyofmedicalimageregistration,"Medicalimageanalysis,vol.2(1),pp.1{36,1998.

PAGE 115

[50] JPWPluim,JBAMaintz,andMAViergever,\Mutual-information-basedregistrationofmedicalimages:asurvey,"IEEETrans.Med.Imaging,vol.22,pp.986{1004,2003. [51] PaulA.ViolaandWilliamM.WellsIII,\Alignmentbymaximizationofmutualinformation.,"inIEEEInternationalConferenceonComputerVisionICCV'95,1995,pp.16{23. [52] G.Hermosillo,C.C.Hotel,andO.Faugeras,\Variationalmethodsformultimodalimagematching,"Int.J.ComputerVision,vol.50(3),pp.329{343,2002. [53] F.Maes,A.Collignon,D.Vandermeulen,G.Marchal,andP.Suetens,\Multi-modalityimageregistrationmaximizationofmutualinformation,"inMM-BIA'96:Proceedingsofthe1996WorkshoponMathematicalMethodsinBiomedicalImageAnalysis(MMBIA'96),Washington,DC,USA,1996,p.14,IEEEComputerSociety. [54] B.Jian,B.Vemuri,andJ.Marroqun,\Robustnonrigidmultimodalimageregistrationusinglocalfrequencymaps.,"in19thInternationalConferenceonInformationProcessinginMedicalImaging,2005,pp.504{515. [55] RuzenaBajcsyandStanislavKovacic,\Multiresolutionelasticmatching.,"Com-puterVision,Graphics,andImageProcessing,vol.46,no.1,pp.1{21,1989. [56] J.P.Thirion,\Imagematchingasadiusionprocess:ananalogywithmaxwell'sdemons,"Medicalimageanalysis,vol.2(3),pp.243{260,1998. [57] RogeljPandKovacicS,\Similaritymeasuresfornon-rigidregistration,"Proc.SPIE,vol.4322,pp.569{78,2001. [58] W.Lu,M.Chen,G.Olivera,K.Ruchala,andT.Mackie,\Fastfree-formdeformableregistrationviacalculusofvariations,"PhysicsinMedicineandBiology,vol.49,pp.3067{3087,2004. [59] CoselmonMM,BalterJM,McShanDL,andKesslerML,\Mutualinformationbasedctregistrationofthelungatexhaleandinhalebreathingstatesusingthin-platesplines,"Med.Phys.,vol.31,pp.2942{8,2004. [60] RohlngT,MaurerCRJ,O'DellWG,andZhongJ,\Modelinglivermotionanddeformationduringtherespiratorycycleusingintensity-basednonrigidregistrationofgatedmrimages,"Med.Phys.,vol.31,pp.427{32,2004. [61] H.Wang,L.Dong,J.O'Daniel,R.Mohan,A.Garden,K.Ang,D.Kuban,M.Bonnen,J.Chang,andR.Cheung,\Validationofanaccelerated'demons'algorithmfordeformableimageregistrationinradiationtherapy,"Phys.Med.Biol.,vol.50,no.12,pp.2887{2905,June2005.

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[62] BrockKK,SharpeMB,DawsonLA,KimSM,andJarayDA,\Accuracyofniteelementmodel-basedmulti-organdeformableimageregistration,"Med.Phys.,vol.32,pp.1647{59,2005. [63] SchreibmannEandXingL,\Narrowbanddeformableregistrationofprostatemagneticresonanceimaging,magneticresonancespectroscopicimaging,andcomputedtomographystudies,"Int.J.Radiat.Oncol.Biol.Phys.,vol.62,pp.595{605,2005. [64] B.C.Vemuri,J.Ye,Y.Chen,andC.M.Leonard,\Imageregistrationvialevel-setmotion:Applicationtoatlas-basedsegmentation,"Medicalimageanalysis,vol.7,pp.1{20,2003. [65] N.Ayache,A.Guimond,A.Roche,andJ.Meunier,\Threedimensionalmultimodalbrainwarpingusingthedemonsalgorithmandadaptvieintensitycorrection,"IEEETrans.Med.Imag.,vol.20(1),pp.58:69,2001. [66] R.BajscyandS.Kovacic,\Multiresolutionelasticmatching,"Comput.Vision.Graph.ImageProcess,vol.46,pp.1{12,1989. [67] P.CachierandX.Pennec,\3dnon-rigidreigistrationbygradientdescentonagaussian-windowsimilaritymeasureusingconvolutions,"IEEEworkshoponmathematicalmethodsinbiomedicalimageanalysis,pp.182{189,2000. [68] W.Lu,G.H.Olivera,Q.Chen,MChen,andK.Ruchala,\Automaticre-contouringin4dradiotherapy,"PhysicsinMedicineandBiology,vol.51,pp.1077{1099,2006. [69] YanD,JarayDA,andWongJW,\Amodeltoaccumulatefractionateddoseinadeformingorgan,"Int.J.Radiat.Oncol.Biol.Phys.,vol.44,pp.665{75,1999. [70] LuW,MackieTR,KellerH,RuchalaKJ,andOliveraGH,\Ageneralizationofadaptiveradiotherapyandtheregistrationofdeformabledosedistribution,"Proc.13thInt.Conf.ontheUseofComputersinRadiationTherapy(ICCR'00),pp.521{3,2000. [71] OliveraGH,RuchalaK,LuW,KapatoesJ,ReckwerdtP,JerajR,andMackieR,\Evaluationofpatientsetupandplanoptimizationstrategiesbasedondeformabledoseregistration,"Int.J.Radiat.Oncol.Biol.Phys.,vol.57,pp.S188{9,2003. [72] P.J.Keall,J.V.Siebers,S.Joshi,andR.Mohan,\Montecarloasafour-dimensionalradiotherapytreatment-planningtooltoaccountforrespiratorymotion,"Phys.Med.Biol.,vol.49,pp.3639{3648,2004. [73] W.Lu,G.H.Olivera,andT.R.Mackie,\Motion-encodeddosecalculationthroughuence/sinogrammodication,"Med.Phys.,vol.32,pp.118{27,2005. [74] PLorenzen,BDavis,andSJoshi,\Modelbasedsymmetricinformationtheoreticlargedeformationmulti-modalimageregistration,"inIEEEInternationalSympo-siumonBiomedicalImaging:MacrotoNano,ISBI'04,2004,pp.720{723.

PAGE 117

[75] G.ChristensenandH.Johnson,\Consistentimageregistration,"IEEETransactionsonMedicalImaging,vol.20,pp.568{582,2001. [76] A.D.Leow,S.C.Huang,A.Geng,J.Becker,S.Davis,A.Toga,andP.Thompson,\Inverseconsistentmappingin3ddeformableimageregistration:itsconstructionandstatisticalproperties,"InformationProcessinginMedicalImagingIPMI'05,pp.493{503,2005. [77] M.F.BegandA.Khan,\Symmetricdataattachmenttermsforlargedeformationimageregistration,"MedImg,vol.26,no.9,pp.1179{1189,September2007. [78] G.ChristensenandH.Johnson,\Invertibilityandtransitivityanalysisfornonrigidimageregistration,"JournalofElectronicImaging,vol.12,pp.106{117,jan2003. [79] GEChristensen,JHSong,WLu,NaqaIEl,andDALow,\Trackinglungtissuemotionandexpansion/compressionwithinverseconsistentimageregistrationandspirometry,"MedPhys,vol.34,no.6,pp.2155{63,2007. [80] BrianAvantsandJamesC.Gee,\Geodesicestimationforlargedeformationanatomicalshapeaveragingandinterpolation,"NeuroImage,vol.23Supplement1,pp.S139{S150,2004. [81] BrianB.Avants,MurrayGrossman,andJamesC.Gee,\Symmetricdieomorphicimageregistration:Evaluatingautomatedlabelingofelderlyandneurodegenerativecortexandfrontallobe,"BiomedicalImageRegistration,pp.50{57,2006. [82] QZengandYChen,\Accurateinverseconsistentnon-rigidimageregistrationanditsapplicationonautomaticre-contouring,"LectureNotesofComputerScience,proceedingsof4thInternationalSymposiumonBioinformaticsResearchandApplications(ISBRA'08),2008toappear. [83] L.Rudin,S.Osher,andE.Fatemi,\Nonlineartotalvariationbasednoiseremovalalgorithm,"PhysicaD,vol.60,pp.259{268,1992. [84] P.Blomgren,T.Chan,P.Mulet,andC.K.Wong,\Totalvariationimagerestoration:Numericalmethodsandextensions,"ProceedingofIEEEInt'lConferenceonImageProcessing,vol.3,pp.384{387,1997. [85] A.ChambolleandP-L.Lions,\Imagerecoveryviatotalvariationminimizationandrelatedproblems,"NumerischeMathematik,vol.1(76),pp.167{188,1997. [86] Y.Chen,S.Levine,andM.Rao,\Variableexponent,lineargrowthfunctionalsinimagerestoration,"SIAMJournalonAppliedMathematics,vol.66,no.4(1),pp.383{406,2006. [87] P.BlomgrenandT.Chan,\Colortv:totalvariationmethodsforrestorationofvector-valuedimages,"IEEETrans.onImageProcessing,vol.7(3),pp.304{309,1998.

PAGE 118

[88] Z.Wang,B.C.Vemuri,Y.Chen,andT.Mareci,\AconstrainedvariationalprinciplefordirectestimationandsmoothingofthetensoreldfromcomplexDWI,"IEEETMI,vol.23:8,pp.930{939,2004. [89] J.Weickert,B.Romeny,andM.Viergever,\Ecientandreliableschemesfornonlineardiusionltering,"IEEETrans.onImg.Proc.,vol.7,no.3,pp.398{410,March1998. [90] T.Lu,P.Neittaanm,andX.Tai,\Aparallelsplittingupmethodanditsapplicaitontonavier-stokesequations,"AppliedMathematicsLetters,vol.4(2),pp.25{29,1991. [91] T.F.ChanandL.A.Vese,\Activecontourswithoutedges,"IEEETrans.ImageProcessing,vol.10,no.2,pp.266{277,2001. [92] S.D.ConteandC.DeBoor,ElementaryNumericalAnalysis,McGraw-Hill,NewYork,1972. [93] Y.Chen,W.Guo,Q.Zeng,X.Yan,F.Huang,H.Zhang,G.He,B.CVemuri,andY.Liu,\Estimation,smoothing,andcharaterizationofappranetdiusioncoecientprolesfromhighangularresolutiondwi,"Proc.ofIEEEConferenceonComputerVisionandPatternRecognitionCVPR'04,pp.588{593,2004. [94] M.Rao,\Moreonanewconceptofentropyandinformation,"JournalofTheoreti-calProbability.,vol.18,no.4,pp.967{981,2005. [95] D.StrongandT.Chan,\Spatialandscaleadaptivetotalvariationbasedregularizationandanisotropicdiusioninimageprocessing,"UCLA-CAMRe-port,vol.46,1996. [96] DJFleetandYWeiss,\Opticalowestimation,"MathematicalmodelsforComputerVision:TheHandbook,(eds.)N.Paragios,Y.ChenandO.Faugeras,vol.chapter15,pp.239{258,2005. [97] EPSimoncelli,EHAdelson,andDJHeeger,\Probabilitydistributionsofopticalow,"IEEEComputerSocietyConferenceonComputerVisionandPatternRecognition,(CVPR'91),pp.310{315,Jun1991. [98] YWeissandDJFleet,\Velocitylikelihoodsinbiologicalandmachinevision,"ProbabilisticModelsoftheBrain:PerceptionandNeuralFunction,(eds.)R.P.N.Rao,B.A.OlshausenandM.S.Lewicki,pp.81{100,2001. [99] GEChristensen,RDRabbit,andMIMiller,\Adeformableneuroanatomytextbookbasedonviscousuidmechanics,"Proceedingsofthe1993ConferenceonInforma-tionSciencesandSystems,JohnsHopkinsUniversity,March24-26,pp.211{216,1993. [100] P.Viola,\Alignmentbymaximizationofmutualinformation,"PhDthesis,1995.

PAGE 119

[101] A.Collignon,F.Maes,D.Vandermeulen,P.Suetens,andG.Marchal,\Automatedmulti-modalityimageregistrationbasedoninformationtheory,"InformationProcessinginMedicalImaging,1995. [102] G.Hermosillo,\Variationalmethodsformultimodalimageregistration,"INRIA,France,2002. [103] MChen,WLu,QChen,KJRuchala,andGHOlivera,\Asimplexed-pointapproachtoinvertadeformationeld,"MedPhys.,vol.35,no.1,pp.81{88,Jan2008. [104] T.Lu,P.Neittaanmki,andX.-C.Tai,\Aparallelsplitting-upmethodforpartialdierentialequationsanditsapplicationtonavier-stokesequations,"RAIROMath.Model.andNumer.Anal,vol.26,no.6,pp.673{708,1992. [105] D.Tuch,\DiusionMRIofcomplextissuestructure,"HarvardUniversityandMassachusettsInstituteofTechnology,vol.PhDthesis,2002. [106] M.Descoteaux,E.Angelino,S.Fitzgibbons,andR.Deriche,\Afastandrobustodfestimationalgorithminq-ballimaging,"3rdIEEEInternationalSymposiumonBiomedicalImaging:MacrotoNano(ISBI'06),pp.81{84,6-9April2006. [107] TMcGraw,BCVemuri,BYezierski,andTMareci,\vonMises-FishermixturemodelofthediusionODF,"3rdIEEEInternationalSymposiumonBiomedicalImaging:MacrotoNano(ISBI'06),pp.65{68,6-9April2006. [108] G.E.Andrews,R.Askey,andR.Roy,SpecialFunctions,CambridgeUniversityPress,1999. [109] BingJianandBabaC.Vemuri,\Multi-berreconstructionfromdiusionmriusingmixtureofwishartsandsparsedeconvolution,"inProc.ofInformationProcessinginMedicalImagingIPMI'07,2007,pp.384{395. [110] J.S.W.Campbell,PSavadjiev,KSiddiqi,andGBPike,\ValidationandregularizationindiusionMRItractography,"3rdIEEEInternationalSympo-siumonBiomedicalImaging:MacrotoNano(ISBI'06),pp.351{354,6-9April2006. [111] RDericheandMDescoteaux,\Splittingtrackingthroughcrossingbers:multidirectionalq-balltracking,"4thIEEEInternationalSymposiumonBiomedicalImaging:MacrotoNano(ISBI'07),pp.756{759,12-15April2007. [112] JPByrne,PEUndrill,andRPPhillips,\Featurebasedimageregistrationusingparallelcomputingmethods,"ProceedingsoftheFirstConferenceonVisualizationinBiomedicalComputing,pp.304{310,May1990. [113] XGeng,DKumar,andGEChristensen,\Transitiveinverse-consistentmanifoldregistration,"InfProcessMedImagingIPMI'05,pp.468{479,2005.

PAGE 120

[114] AJarc,PRogelj,andSKovacic,\Texturefeaturebasedimageregistration,"4thIEEEInternationalSymposiumonBiomedicalImaging:FromNanotoMacroISBI'07,pp.17{20,April2007. [115] N.Gupta,\AVLSIarchitectureforimageregistrationinrealtime,"IEEETransactionsonVeryLargeScaleIntegration(VLSI)Systems,vol.9,no.15,pp.981{989,2007. [116] R.Strzodka,M.Droske,andM.Rumpf,\FastimageregistrationinDX9graphicshardware,"JournalofMedicalInformaticsandTechnologies,vol.6,pp.43{49,Nov2003. [117] HZhang,PAYushkevicha,andJCGeea,\Registrationofdiusiontensorimages,"Proceedingsofthe2004IEEEComputerSocietyConferenceonComputerVisionandPatternRecognition(CVPR'04),vol.1,pp.I{842{I{847,2004. [118] HZhang,PAYushkevicha,DCAlexanderb,andJCGeea,\DeformableregistrationofdiusiontensorMRimageswithexplicitorientationoptimization,"MedicalImageAnalysis,vol.10,no.5,pp.764{785,2006. [119] WGuo,YChen,andQZeng,\Ageometricowbasedapproachfordiusiontensorimagesegmentation,"PhilosophicalTransactionsA,toappear. [120] LZhukov,KMuseth,DBreenAHBarr,andRWhitaker,\LevelsetsegmentationandmodelingofDT-MRIhumanbraindata,"J.ElecImag.,vol.12,pp.125{133,2003.

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QingguoZengwasborninXiangyang,HubeiProvince,P.R.Chinain1978.HegothisBachelorofSciencedegreein1999andMasterofEngineeringin2002fromBeijingNormalUniversity,Beijing,China.HeearnedhisPh.D.degreeinAppliedMathematicsfromUniversityofFloridainMay2008.Hisresearchinterestsincludemathematicalmodelingandalgorithmdevelopinginmedicalimageprocessing,statisticalimageprocessing,parallelcomputing,optimization,numericalanalysisandnumericalPartialDierentialEquations. 121