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Efficient Sampling Geometries and Reconstruction Algorithms for Estimation of Diffusion Propagators

Permanent Link: http://ufdc.ufl.edu/UFE0044054/00001

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Title: Efficient Sampling Geometries and Reconstruction Algorithms for Estimation of Diffusion Propagators
Physical Description: 1 online resource (108 p.)
Language: english
Creator: Ye, Wenxing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: diffusion -- geometry -- hardi -- mri -- reconstruction -- sampling
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Diffusion weighted magnetic resonance imaging (DW-MRI) is a non-invasive techniques that can reveal the anatomical structure of living organism. Through measuring directional diffusion of water molecules in tissue rich in fiber structure, DW-MRI can provide invaluable in vivo information about the neuronal connectivity patterns. This kind of information is extremely important for many clinical applications, especially when brain areas which consist of abundant white matter fibers is concerned. Diffusion propagator is the most widely used measure to describe the water diffusion quantitatively. It gives the probability of water molecule motion along a vector in fixed MR setting. The main aim of this dissertation is to investigate better ways of sampling and reconstructing the diffusion propagator from a signal processing point of view. This dissertation shows a model free framework for diffusion propagator reconstruction with special focus on increasing the total reconstruction accuracy through optimizing the sampling geometry. The reconstruction framework is proposed through gradually introducing the concepts of optimal sampling lattice, interlaced mult-shell sampling scheme, multi-dimensional sinc basis representation and box-spline basis representation. The main difficulty in diffusion propagator reconstruction lies in the fact that sampling of diffusion signal is very time consuming and the time is linearly proportional to the number of sample points. In order to maintain the angular resolution, in practice, the diffusion signal is sampled on spherical shells. There are many ways to estimate the diffusion propagator, which is the Fourier transform of the diffusion signal, from the spherical samples. One of them is to set a Cartesian lattice in the space of the diffusion signal and estimate the values of diffusion signal on the lattice points. Then, the diffusion propagator can be easily estimated through a fast Fourier transform. One intuitive idea is to replace this Cartesian lattice with a body centered cubic (BCC) lattice. The interlaced mult-shell sampling means sampling the diffusion signal in an interlaced style across every alternating shell. A weighted sum of the shifted sinc or box-spline basis are used to represent the diffusion signal or the diffusion propagator and the weighting coefficients are estimated by solving an inverse problem. With these coefficients, a continuous form of the estimated diffusion propagator is obtained. Each of the concepts has its own way of improving the reconstruction accuracy. We have shown this through experimental comparison on both synthetic and real MRI data. As a whole framework, it seeks the more efficient way of sampling and reconstruction to achieve higher accuracy without increasing the sampling burden. Generally, the framework can take common spherical samples and generate accurate, model independent reconstructions. In the end, the concept of dictionary learning and non-local regularization were introduced for reconstructing a regularized field of diffusion propagators. By investigating the spatial correlation in the field, the algorithm can produce results more resilient to the noise compared with the voxel-wise reconstruction methods. This is the first such algorithm in the field and our future will involve combination of this technique with the optimal sampling framework briefly described above.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Wenxing Ye.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Vemuri, Baba C.
Local: Co-adviser: Entezari, Alireza.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044054:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044054/00001

Material Information

Title: Efficient Sampling Geometries and Reconstruction Algorithms for Estimation of Diffusion Propagators
Physical Description: 1 online resource (108 p.)
Language: english
Creator: Ye, Wenxing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: diffusion -- geometry -- hardi -- mri -- reconstruction -- sampling
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Diffusion weighted magnetic resonance imaging (DW-MRI) is a non-invasive techniques that can reveal the anatomical structure of living organism. Through measuring directional diffusion of water molecules in tissue rich in fiber structure, DW-MRI can provide invaluable in vivo information about the neuronal connectivity patterns. This kind of information is extremely important for many clinical applications, especially when brain areas which consist of abundant white matter fibers is concerned. Diffusion propagator is the most widely used measure to describe the water diffusion quantitatively. It gives the probability of water molecule motion along a vector in fixed MR setting. The main aim of this dissertation is to investigate better ways of sampling and reconstructing the diffusion propagator from a signal processing point of view. This dissertation shows a model free framework for diffusion propagator reconstruction with special focus on increasing the total reconstruction accuracy through optimizing the sampling geometry. The reconstruction framework is proposed through gradually introducing the concepts of optimal sampling lattice, interlaced mult-shell sampling scheme, multi-dimensional sinc basis representation and box-spline basis representation. The main difficulty in diffusion propagator reconstruction lies in the fact that sampling of diffusion signal is very time consuming and the time is linearly proportional to the number of sample points. In order to maintain the angular resolution, in practice, the diffusion signal is sampled on spherical shells. There are many ways to estimate the diffusion propagator, which is the Fourier transform of the diffusion signal, from the spherical samples. One of them is to set a Cartesian lattice in the space of the diffusion signal and estimate the values of diffusion signal on the lattice points. Then, the diffusion propagator can be easily estimated through a fast Fourier transform. One intuitive idea is to replace this Cartesian lattice with a body centered cubic (BCC) lattice. The interlaced mult-shell sampling means sampling the diffusion signal in an interlaced style across every alternating shell. A weighted sum of the shifted sinc or box-spline basis are used to represent the diffusion signal or the diffusion propagator and the weighting coefficients are estimated by solving an inverse problem. With these coefficients, a continuous form of the estimated diffusion propagator is obtained. Each of the concepts has its own way of improving the reconstruction accuracy. We have shown this through experimental comparison on both synthetic and real MRI data. As a whole framework, it seeks the more efficient way of sampling and reconstruction to achieve higher accuracy without increasing the sampling burden. Generally, the framework can take common spherical samples and generate accurate, model independent reconstructions. In the end, the concept of dictionary learning and non-local regularization were introduced for reconstructing a regularized field of diffusion propagators. By investigating the spatial correlation in the field, the algorithm can produce results more resilient to the noise compared with the voxel-wise reconstruction methods. This is the first such algorithm in the field and our future will involve combination of this technique with the optimal sampling framework briefly described above.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Wenxing Ye.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Vemuri, Baba C.
Local: Co-adviser: Entezari, Alireza.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044054:00001


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EFFICIENTSAMPLINGGEOMETRIESANDRECONSTRUCTIONALGORITHMSFORESTIMATIONOFDIFFUSIONPROPAGATORSByWENXINGYEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012WenxingYe 2

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Tomyparents 3

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ACKNOWLEDGMENTS Ioweagreatthankyoutomanypeoplewhohelpedmenishmydissertation.Firstandforemost,IwouldliketothankDr.BabaVemuri,myadvisor,whointroducedmeintotheeldofmedicalimageprocessingandguidedmethroughmyPh.D.studyusinghistirelessinspirationandrichacademicexperience.IamalsothankfulthatDr.VemuriofferedmefullresearchassistantshipwithwhichIcouldputallmyfocusontheresearchandnotworryaboutnancialproblems.Dr.Vemuribuilttheexampleofapassionateandseriousscholarforme.IwouldalsoliketoappreciateDr.AlirezaEntezari,myco-advisor,whoisalwaysavailablefordiscussionwhenIneedhistechnicaladvise.HeselesslysharedhisexperienceandknowledgesothatIavoidedmakingmanymistakes.Fromhim,Ilearnedtheimportanceofmathematicalthinkingandcarefulnessinsolvingaresearchproblem.Thememoriesarestillclearthatheworkedtogetherwithmefrommorningtothenightbeforetheconferencedeadlines.Itismylucktohavehimasapatientfriendandknowledgeablementor.Furthermore,IwouldliketoexpressmysincereappreciationtoDr.AnandRangarajan,Dr.JayGopalakrishnanandDr.StephenBlackbandforservingasmyPh.D.committeemembers.Theyspenthoursindiscussingdetailsofmyworks.ThecoursestaughtbyDr.RangarajanandDr.GopalakrishnanprovidedcriticalknowledgeandtrainingformylaterresearchandDr.Blackbandgavemanyadvisesfromaneuroscientistpoint.AlltheworksinthisdissertationareevaluatedonrealdiffusionMRIdataprovidedbytheMcKnightBrainInstitutionattheUniversityofFlorida.IowemydeepestgratitudetoSharonPortnoyandDr.MinSigHwangwhoworkedveryhardcollectingtestingdataforme.Theirtechnicalsuggestionsandfeedbacksalsohelpedalotinadjustingmystudiestomeettheclinicalrequirements.Inaddition,IamgratefulthatSharonhastakenthetimetocorrectthewordingofmymanuscripts. 4

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Moreover,IwanttothankallmyfriendsinthelaboratoryforComputerVision,Graphics,andMedicalImaging.IbenetedalotfromthetoolsandarticlesdevelopedbyDr.AngelosBarmpoutis,Dr.FeiWangandDr.BingJian.IalsogainedmanyenlightenmentsfromthediscussionwithGuangCheng,YuchenXie,TingChenandMeizhuLiu.Ithasbeenreallygoodtimeworkingwiththem.IalsowouldliketothankShannonChillingworth,thegraduatecoordinatoroftheElectricalandComputerEngineeringdepartment.Herprofessionalknowledgeandalwaysin-timeassistancesavedmelotsoftimeinhandlingnon-academicaffairsintheuniversity.Lastbutnotleast,IthankgenerousfundingforthisresearchfromtheNIHgrants,EB007082andNS066340tomyadvisor,Dr.Vemuri. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1DIFFUSIONPROPAGATORRECONSTRUCTIONFRAMEWORK ....... 13 1.1IntroductiontoDiffusionPropagatorReconstruction ............ 13 1.2CommonSamplingGeometries ........................ 14 1.3ExistingReconstructionMethods ....................... 16 1.3.1DiffusionTensorImaging ........................ 17 1.3.2HighAngularResolutionDiffusionImaging .............. 17 1.3.3Q-ballImaging ............................. 18 1.3.4DiffusionSpectrumImaging ...................... 19 1.3.5ReconstructionfromMulti-shellSampling .............. 19 1.4MainContributions ............................... 20 2MULTIVARIATELATTICESANDTHEOPTIMALINTERPOLATIONFUNCTIONS ..................................... 23 2.1SamplingMultivariateSignals ......................... 23 2.2MultidimensionalLattices ........................... 26 2.2.1PointLattices .............................. 26 2.2.2DicingLattices ............................. 27 2.2.3SpherePackingandCoveringLatticesForSampling ........ 27 2.3SincFunctionsOnMultidimensionalLattices ................ 29 2.3.1Zonotopes ................................ 31 2.3.2SpaceTesselationsandLatticeVoronoiPolytopes ......... 33 2.3.2.1Two-DimensionalLattices .................. 34 2.3.2.2Three-DimensionalLattices ................. 35 2.3.3FCCandBCCLattices ......................... 36 2.3.4MultivariateShannonWavelets .................... 38 2.3.5MultivariateLagrangeInterpolant ................... 40 2.3.6MultidimensionalLanczosWindowing ................ 42 2.4ExperimentalComparison ........................... 45 2.4.1ExperimentsSetup ........................... 45 2.4.2VisualComparison ........................... 48 2.4.3NumericalComparison ......................... 48 6

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3TOMOGRAPHICRECONSTRUCTIONOFDIFFUSIONPROPAGATORSUSINGOPTIMALSAMPLINGLATTICES .......................... 53 3.1Motivations ................................... 53 3.2AlgorithmandImplementation ........................ 54 3.3AlgorithmEvaluation .............................. 55 4RECONSTRUCTIONFROMINTERLACEDSAMPLING ............. 59 4.1InterlacedSamplingScheme ......................... 59 4.2ReconstructionAlgorithm ........................... 63 4.3Experiments .................................. 67 4.3.1Experimentsonsyntheticdata .................... 67 4.3.2Experimentsonrealdata ....................... 72 5BOXSPLINEBASEDRECONSTRUCTIONMETHOD .............. 79 5.1BoxSplinesandRadonTransform ...................... 80 5.2DetailedAlgorithm ............................... 82 5.3ExperimentalResultsoftheAlgorithm .................... 84 6RECONSTRUCTIONOFADIFFUSIONPROPAGATORFIELD ......... 88 6.1Overview:FromFixedBasistoDataDrivenDictionary ........... 88 6.2AdaptiveKernelsforMulti-berReconstruction ............... 89 6.3DictionarybasedReconstructionFramework ................ 91 6.4Results ..................................... 93 6.4.1SyntheticDataset ............................ 93 6.4.2RealDataset .............................. 95 7CONCLUSIONS ................................... 97 REFERENCES ....................................... 99 BIOGRAPHICALSKETCH ................................ 108 7

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LISTOFTABLES Table page 2-1DecompositionoftheBrillouinzonefortheBCClattice,BusedtoconstructthesincB. ....................................... 37 2-2DecompositionoftheBrillouinzonefortheFCClattice,FusedtoconstructthesincF. ....................................... 38 3-1SSEcomparisonofreconstructionsfromdifferentlatticeswithNr,N=12,N=13,=90 ................................... 56 3-2SSEcomparisonofreconstructionsfromdifferentlatticeswithN,N,Nr=10,=90 ...................................... 56 3-3SSEcomparisonofreconstructionsfromdifferentlatticeswith,Nr=10,N=12,N=13 ................................... 56 3-4SSEcomparisonofreconstructionsfromdifferentlatticeswith,Nr=10,N=12,N=13,=90 .............................. 57 4-1SamplingdirectionsusedinourexperimentsforRhombicTriacontahedronandIcosidodecahedron. ............................... 70 5-1MSEinpercentageofthereconstructionsforthesyntheticdata. ........ 85 8

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LISTOFFIGURES Figure page 1-1SamplingschemesusedindiffusionMRI. ..................... 15 2-1AlatticeLanditsfundamentaldomain(Voronoicell). ............... 26 2-23-DCartesian,BCCandFCClattices. ....................... 28 2-3HexagonallatticeanditsBrillouinzone. ...................... 34 2-4Thefundamentaldomainsofall3-Dlatticestessellatethespace. ........ 36 2-5Decompositionofarhombicdodecahedronandatruncatedoctahedron .... 39 2-6Brillouinzonesubdividedtoderiveamulti-scalefunctionforShannonwavelets. 40 2-7LagrangebasisforpolynomialinterpolationwhenthenodesareZconvergestosinc. ......................................... 41 2-8Lanczoslterasamultiplicationwiththemainlobeofthescaledsincfunction. 43 2-9PlotsofsincHanditsLanczoswindowedversioninthespacedomainandthefrequencydomain. .................................. 46 2-10Benchmarkvolumetricdatasets. .......................... 47 2-11VisualcomparisonofMLisosurfaceimagesrenderedfromCartesian,BCCandFCClattices. ................................... 49 2-12VisualcomparisonofcarpshisosurfaceimagesrenderedfromCartesian,BCCandFCClattices. ................................ 50 2-13VisualcomparisonofbonsaitreeisosurfaceimagesrenderedfromCartesian,BCCandFCClattices. ................................ 50 2-14TheRMSerrorcomparisonofBCC,FCCversus414141Cartesianlattice. 51 3-1VisualcomparisonofthereconstructedP(r). ................... 57 3-2Probabilitymapsreconstructedfromrealdataset. ................ 58 4-1Diagramofthereconstructionschemeusinginterlacedsampling. ........ 60 4-2Comparisonof2DstandardandinterlacedsamplingschemesinthecaseofCTreconstruction. .................................. 60 4-3ThestructureofBCClatticeas2-DCartesianlayersofsamplesshiftedonalternateZslices. .................................. 61 4-4Shapeofpolyhedrausedinsamplingschemes. .................. 62 9

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4-5Illustrationof3-Dinterlacedandstandardsamplinggeometries. ........ 64 4-6ReconstructionofP(r)inthetwo-bercase. ................... 69 4-7PlotsshowingthenormalizedMSEofthereconstructionforsyntheticdataoftwo-bercrossings. .................................. 71 4-8PlotsshowingthenormalizedMSEofthereconstructionforsinglebersyntheticdataofdifferentFAvalues. ............................. 72 4-9ReconstructionofP(r)inthethree-bercase. ................... 73 4-10PlotsshowingthenormalizedMSEofthereconstructionforsyntheticdataofthree-bercrossings. ................................. 74 4-11Reconstructionresultsondataset#1 ....................... 75 4-12Reconstructionresultsondataset#2 ....................... 76 4-13Reconstructionresultsondataset#3 ....................... 78 5-1Diagramoftheboxsplinebasedreconstructionprocess. ............. 80 5-2A2-DillustrationofRadontransformgeometry. .................. 82 5-3ThedesiredisosurfaceofP(r). ........................... 84 5-4Reconstructionresultsforthesyntheticdata. ................... 85 5-5Reconstructionresultsforaregionofthemulti-shellmousebraindatawithdifferentboxsplinebasisorders. .......................... 86 6-1Reconstructeddiffusionpropagatoreldsusingdifferentmethods. ....... 94 6-2Reconstructionerrorcomparisonandtheconvergenceplot. ........... 95 6-3S0imageoftheopticchiasmandtheROI. ..................... 95 6-4Reconstructedregionofinterestfromrealdatausingdifferentmethods. .... 96 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEFFICIENTSAMPLINGGEOMETRIESANDRECONSTRUCTIONALGORITHMSFORESTIMATIONOFDIFFUSIONPROPAGATORSByWenxingYeMay2012Chair:BabaC.VemuriCochair:AlirezaEntezariMajor:ElectricalandComputerEngineeringDiffusionweightedmagneticresonanceimaging(DW-MRI)isanon-invasivetechniquesthatcanrevealtheanatomicalstructureoflivingorganism.Throughmeasuringdirectionaldiffusionofwatermoleculesintissuerichinberstructure,DW-MRIcanprovideinvaluableinvivoinformationabouttheneuronalconnectivitypatterns.Thiskindofinformationisextremelyimportantformanyclinicalapplications,especiallywhenbrainareaswhichconsistofabundantwhitematterbersisconcerned.Diffusionpropagatoristhemostwidelyusedmeasuretodescribethewaterdiffusionquantitatively.ItgivestheprobabilityofwatermoleculemotionalongavectorinxedMRsetting.Themainaimofthisdissertationistoinvestigatebetterwaysofsamplingandreconstructingthediffusionpropagatorfromasignalprocessingpointofview.Thisdissertationshowsamodelfreeframeworkfordiffusionpropagatorreconstructionwithspecialfocusonincreasingthetotalreconstructionaccuracythroughoptimizingthesamplinggeometry.Thereconstructionframeworkisproposedthroughgraduallyintroducingtheconceptsofoptimalsamplinglattice,interlacedmult-shellsamplingscheme,multi-dimensionalsincbasisrepresentationandbox-splinebasisrepresentation.Themaindifcultyindiffusionpropagatorreconstructionliesinthefactthatsamplingofdiffusionsignalisverytimeconsumingandthetimeislinearlyproportional 11

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tothenumberofsamplepoints.Inordertomaintaintheangularresolution,inpractice,thediffusionsignalissampledonsphericalshells.Therearemanywaystoestimatethediffusionpropagator,whichistheFouriertransformofthediffusionsignal,fromthesphericalsamples.OneofthemistosetaCartesianlatticeinthespaceofthediffusionsignalandestimatethevaluesofdiffusionsignalonthelatticepoints.Then,thediffusionpropagatorcanbeeasilyestimatedthroughafastFouriertransform.OneintuitiveideaistoreplacethisCartesianlatticewithabodycenteredcubic(BCC)lattice.Theinterlacedmult-shellsamplingmeanssamplingthediffusionsignalinaninterlacedstyleacrosseveryalternatingshell.Aweightedsumoftheshiftedsincorbox-splinebasisareusedtorepresentthediffusionsignalorthediffusionpropagatorandtheweightingcoefcientsareestimatedbysolvinganinverseproblem.Withthesecoefcients,acontinuousformoftheestimateddiffusionpropagatorisobtained.Eachoftheconceptshasitsownwayofimprovingthereconstructionaccuracy.WehaveshownthisthroughexperimentalcomparisononbothsyntheticandrealMRIdata.Asawholeframework,itseeksthemoreefcientwayofsamplingandreconstructiontoachievehigheraccuracywithoutincreasingthesamplingburden.Generally,theframeworkcantakecommonsphericalsamplesandgenerateaccurate,modelindependentreconstructions.Intheend,theconceptofdictionarylearningandnon-localregularizationwereintroducedforreconstructingaregularizedeldofdiffusionpropagators.Byinvestigatingthespatialcorrelationintheeld,thealgorithmcanproduceresultsmoreresilienttothenoisecomparedwiththevoxel-wisereconstructionmethods.Thisistherstsuchalgorithmintheeldandourfuturewillinvolvecombinationofthistechniquewiththeoptimalsamplingframeworkbrieydescribedabove. 12

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CHAPTER1DIFFUSIONPROPAGATORRECONSTRUCTIONFRAMEWORK 1.1IntroductiontoDiffusionPropagatorReconstructionDiffusionMRIisanon-invasiveimagingtechniquethatexhibitssensitivitytoBrownianmotionofwatermoleculesthroughtissueinvivo.Watermoleculesexhibitpreferreddirectionaldiffusionthroughtissuerichinwhitematterbers.Thisdirectionalpreferenceallowsonetoinferconnectivitypatternsaswellaschangesinthemovertimethatcanbeusedinvariousclinicalapplications.Forexample,itisshowntobesensitivetotheevaluationofearlyischemicstagesoftheanimalbrain[ 86 ].Manyexperimentshavebeenexploitedtoinvestigatetheanisotropicmicro-structureofthebroustissueslikemuscleandwhitematterinthebrain[ 22 26 66 85 ].OneofthemajortasksofdiffusionMRIisthereconstructionofthe3-DdiffusionpropagatorP(r)characterizingthediffusionprocessofwatermoleculesinbroustissueswithprobabilitydensityfunction(PDF).ByintegratingP(r)fromjjrjj=0to1,diffusionorientationdistributionfunctions(ODF)isdenedandexploitedbymanyrecentworksasareplacementtoP(r)[ 2 36 59 107 110 ].However,ODFonlyprovidestheaveragedangularinformationaboutthediffusionprocess.Incontrast,thediffusionpropagator,P(r),providesbothradialandangularinformationanddescribesthediffusionprocessmoreaccurately.Becauseoftheadditionalradialinformation,P(r)canbeusedtoestimateextrafeaturessuchasprobabilityofdiffusionpermeabilityofthewalls,averagecellsize,axonaldiameterandotherfeaturesthatareusefulinsensingwhitematteranomalies[ 28 ].Underthenarrowpulseassumption,i.e.thedurationoftheapplieddiffusionsensitizinggradientsismuchsmallerthanthetimebetweenthetwopulses,thediffusionsignalE(q)inq-spaceandthediffusionpropagatorP(r)indistancespacearerelatedthroughFouriertransform[ 20 ]as: 13

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P(r)=ZE(q)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(2iqr)dq(1)whereE(q)=S(q)=S0,S0isthediffusionsignalwithzerodiffusiongradient(q=0),risthedisplacementvector,q=G=2isthereciprocalspacevector,isthegyromagneticratioandGisthegradientvector.Tobenoted,b=22jGj2tisanimportantquantity,calledb-value,thatcharacterizesthesensitivityofMRsequencestodiffusion.Here,t=)]TJ /F9 11.955 Tf 12.04 0 Td[(=3whereisthetimebetweendiffusiongradientsandisthedurationofdiffusiongradient.DiffusionpropagatorP(r)isnotdirectlymeasurable.Instead,theimagingspectrometermeasuresthediffusionsignalE(q).( 1 )statesthatthediffusionpropagatoristhe3-DFouriertransformofthediffusionsignalE(q)andprovidesthefoundationforreconstructingP(r).Theproblemseemsstraightforward.WecangetaccurateestimationofP(r)throughtakingFouriertransformonsamplesofE(q),aslongasenoughsamplesarecollected.However,samplingE(q)isverytimeconsumingandthetimecostislinearlyincreasingwiththenumberofsamplesdemanded.SohowtosampleE(q)andwhatpriorinformationtouseemergesasthetwocoreproblems.Therstproblemisthesamplinggeometryproblemandthesecondisthemodelselectionproblem(choosingamodelmeansassumingpriorinformationonthesignal).ThenalgoalistoachieveanaccurateestimationofP(r)fromveryfewnumberofsamplesofE(q). 1.2CommonSamplingGeometriesIndiffusionweightedMRI,thediffusionpropertiesofatissuearedeterminedbymeasuringitsresponsetoanorientedmagneticeldgradient.Byapplyinggradientsofdifferentstrengthsanddirections,weobtainsamplesofthediffusionsignalin3-Dq-spaceforeachvoxellocationwithinthetissue.Foreachq-spacesample,weneedtoaltertheorientationand/orstrengthofthemagneticeldgradientandmeasuretheresponsefortheentirevolume,q-spacesamplingcannotbeperformedinparallel14

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(a)(b)(c)Figure1-1. SamplingschemesusedindiffusionMRI.(a)Cartesiansampling,(b)Sphericalsampling,(c)Multi-shellsphericalsampling. thesamplingtimeincreaseslinearlywiththenumberofq-spacesamples.Therefore,withinthesignalacquisitionsstage,weseekanoptimalbalancebetweenthenumberofsamplesandreconstructionaccuracy.Thesamplingschemedetermineshowthesamplepointsaredistributedthroughout3-Dq-space.Anapproach,employedbyDSI,istoputthesamplesonaregularCartesianlattice(Figure 1-1 (a)).ThisisthemostconventionalschemeusedinsignalprocessingwhereShannonsamplingtheoryprovidesthetheoreticalframeworkforsamplingandreconstruction.Inaddition,P(r)canbereconstructedthroughtheFastFouriertransform(FFT).However,thisschemeistimeconsumingsinceitneedsalargenumberofq-spacesamplestoachieveareasonablyaccuratereconstruction.Inpractice,themostwidelyusedsamplingschemeindiffusionMRistotakeasampleattheorigintogetherwithuniformlydistributedsamplesonasphericalshellintheq-space(Figure 1-1 (b)).Allofthesamplesonthesphericalshelljqj=p b=t=2aretakenunderanappliedmagneticeldgradientofthesamestrengthbutwithdifferentdirections.Thebenetisthatvariationsinsignaltonoiseratioandotherfactorsrelatedtothestrengthofthemagneticeldareminimized.Withsomeglobalordirectional 15

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decaymodelsofthesignal,P(r)canbecomputedanalyticallythroughthecontinuousFouriertransform[ 37 57 ].Anotablevariationofthisschemeacquiressamplesfrommultipleshells(Figure 1-1 (c)),i.e.multiplebvalues,eachwiththesamedirectionaldistributionofsamples.Inthismulti-shellsetting,moresophisticatedmodelsinvolvingmoreparameterscanbeused.Thistypeofmodel-basedschemeneedsfewersamplesthanDSIandcanproducesmoothreconstructions,butitislimitedbytheassumptionsofproposedmodels[ 37 57 ]. 1.3ExistingReconstructionMethodsOncethesamplesarecollected,thenextproblemistoestimatethediffusionpropagatorfromthosesamples.Manytechniqueshavebeenproposedwithdifferentpriorassumptionsaboutthesignal.Makinganassumptionnallyturnsouttobechoosingamathematicalmodel.Differentmodelshavedifferentdegreesoffreedom(i.e.,numberofparameters)whichrequiresdifferentnumbersofsamplestobefullydetermined.Amodelwithhigherdegreeoffreedomismoreexibleinrepresentingmorecomplicatedsignals.Ofcourse,thepricetopayistheneedofmoresamples.TheextremecaseistheShannonsamplingwhichcanrepresentanybandlimitedsignalbutrequiresthemostnumberofsamples.Themostcommonassumptioninthediffusionpropagatorreconstructionscenarioisthattheqspacediffusionsignaldecaysalongtheradialdirectionexponentially.Thereareotherassumptionswhichimposesaglobalmathematicalmodeloverthe3-Dqspaceorassumesmulti-exponentialradialdecayingformulti-shelldatasamples.Attimes,introducingastrongmodel(i.e.,offewparameters)offerstheadvantagesofanalyticalreconstructionthroughFouriertransformation,smoothresultswhichareresistanttonoise,andalimitednumberofrequiredsamples.However,anymodel-basedtechniqueisinherentlybiased,asitwillalwaysproduceresultsthatobeytheunderlyingmodel.ThismodelbiasonthediffusionsignalwilltransfertothereconstructionerrorofdiffusionpropagatorthroughtheFouriertransform. 16

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Inthissection,sometypicalreconstructionmethodsarebrieyreviewed.Manyrelatedpublicationsaboutvariousapproachesarealsocitedtoprovideextrainformationfortheinterestedreaders. 1.3.1DiffusionTensorImagingDiffusiontensorimaging(DTI)proposedby[ 13 14 ]isasimpleyetcommonlyusedtechnique.ItassumesthediffusionsignalcanberepresentedwithanorientedGaussianprobabilitydensityfunctionas: S(G)=S0exp()]TJ /F6 11.955 Tf 9.29 0 Td[(bgTDg)(1)whereb=(G)2tisthediffusionweighting,t=)]TJ /F9 11.955 Tf 12.76 0 Td[(=3istheeffectivediffusiontime,Distheapparentdiffusiontensor,G=jGjandg=G=G.Itisalsoreferredtoasthesecondordertensormodel.Thismodelhas7coefcientswhichcanbexedwithonly7diffusionweightedimagestakenwithdifferentgradientdirections.Theprincipaleigenvectorofthediffusiontensorspeciestheberorientationatthegivenposition.ForDTI,theestimationissimple,thesamplingburdenislightandsomesuccessfulapplicationsinbertrackingwereexploitedintheregionofbrainandspinalcordwithsignicantwhitemattercoherence[ 12 29 54 67 84 119 ].However,sincetheassumedorientedGaussianPDFhasonlyoneprincipalorientation,Itisnowwellknownthatthismodelfailstocapturecomplexgeometriescausedbycrossing,kissingorsplayingbersthatresultinorientationalheterogeneityinavoxel[ 109 ].Thishasspurredthedevelopmentofimprovedacquisitiontechniquesandreconstructionmethods[ 2 10 11 37 56 57 64 89 106 ]. 1.3.2HighAngularResolutionDiffusionImaging[ 108 ]introducedahighangularresolutiondiffusionimaging(HARDI)methodwhichestimatesamoresophisticateddiffusionproleaccordingtomanyorientations,usuallymanymorethan7directionswhichareneededbyDTI.Withoutttingaglobalmodelfunctiontothedata,theoriginalHARDImethodcalculatesthediffusionproleusingthe 17

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Stejskal-Tannerexpression[ 99 ]: E(u)=exp()]TJ /F6 11.955 Tf 9.3 0 Td[(bD(u))(1)whereuisaunitvectordeningthediffusiondirection.TheresultsaregivenasanangulardistributionofapparentdiffusivitiesD(u)whichhasacomplicatedstructureinvoxelswithorientationalheterogeneity[ 109 114 ].SeveralstudieshavebeenproposedsuchasrepresentingthediffusivityfunctionD(u)withasphericalharmonicexpansion[ 4 50 ],generalizationofDTIusinghigherordertensors[ 10 88 90 ],modelingthediffusionsignalusingamixtureofGaussiandensities[ 109 ],combininghinderedandrestrictedmodelsofwaterdiffusion(CHARMED)[ 7 ],directestimationoftheberorientationusingsphericaldeconvolution[ 105 ],atensordistributionmodelintroducedin[ 57 ]anddiffusionorientationtransform(DOT)[ 89 ]whichanalyticallyevaluatestheFourierrelationinsphericalcoordinatesassumingthediffusionsignalbeingamixtureofexponentialdecayfunctions. 1.3.3Q-ballImagingTheoriginalversionofq-ballimaging(QBI)isanothermodelindependentreconstructionmethod[ 109 ].ThismethodtakesdiffusionsignalsamplesonasingleqspacesphereandcloselyresemblestheODFthroughsphericalRadontransform.SphericalRadontransform,alsoknownasFunk-Radontransform,takesafunctiononasphereandoutputsanothersphericalfunctionsthroughthefollowingintegration: (R[f])(u)=Zjxj=1f(x)(uTx)dx(1)whereuisaunitvector.Thismethodisefcientandmodelfree.ButtheresembledODF,supposedtobetheradialintegrationofthediffusionpropagator,isactuallytheintegrationoftherealdiffusionpropagatorconvolvedwitha0-orderBesselfunction.Thisunwantedconvolutiondegradesthedelityofthereconstructions.Arecentstudy[ 2 ]showsan 18

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improvedversionofQBIwhichismoreaccurateinmathematics,givessharperresultsandworksinmultipleq-shellsituations.However,QBIonlyworksforreconstructingODFwhichisacompromisedversionofdiffusionpropagator,notforthediffusionpropagatoritself. 1.3.4DiffusionSpectrumImagingDiffusionspectrumimaging(DSI)[ 116 ],i.e.q-spaceimaging(QSI)wasproposedtosamplethediffusionsignalonadense3-DCartesianlatticeandreconstructthediffusionpropagatorthrough3-DfastFouriertransform(FFT).Ithasbeenextendedthroughchangingthesamplingschemeintobodycenteredcubiclattice[ 24 ]andsomehybridlattice[ 118 ],orusingthetomographicreconstructionprinciple[ 92 ].ThesemethodshaveweakassumptionaboutthediffusionsignalandcanbeevaluatedpreciselythroughShannonsamplingtheorem.Buttheheavysamplingburdenmakestheacquisitiontime-intensiveandlimitsthewidespreadapplication. 1.3.5ReconstructionfromMulti-shellSamplingRecently,reconstructionmethodsusingmultipleq-shellacquisitionswereinvestigated.Undersuchmulti-shellsituation,[ 37 ]introducedadiffusionpropagatorreconstructionmethodassumingthatthediffusionsignalattenuationcanbeestimatedwiththe3-DLaplaceequation,[ 2 ]discussedaboutODFreconstructioninmulti-shellq-Ballimaging(QBI)withinconstantsolidangleassumingradialmulti-exponentialmodel,[ 89 ]extendedtheDOTmethodtothemulti-shelldatawithmulti-exponentialradialdecayingassumption.Hybriddiffusionimaging(HYDI)[ 118 ]wasproposedasanalternativetoDSI.Itsamplesdiffusionsignalonseveralsphericalshellsandestimatesthesignalvaluesonadense3-DCartesianlatticethroughlinearinterpolation.TheideaofinterpolatingsphericalsamplesontoadenseregularlatticeisalsousedinourproposedmethoddiscussedinChapter 3 andChapter 4 .However,HYDIacquiresmoresamplesonshells 19

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withhigherb-values(similartoCHARMED[ 6 ]),whereSNRdecreaseswhichleadstoalessaccuratereconstructionasdiscussedin[ 8 ]. 1.4MainContributionsCurrentreconstructionmethods,withorwithoutmodelassumptions,havetheirownproblemsinthesenseofaccuracy.Themodelbasedmethods(i.e.,withstrongassumptionaboutthesignal)sharethesamesignaldecayingassumptionstatedbyStejskal-Tannerequation.TheStejskal-Tannerequationisderivedfromamicroscopic,freediffusionphysicalmodelwhichhasnoboundaryconditions.Actually,thediffusionsignalcollectedinMRIisfromthecumulativemicroscopicdiffusionsofwatermoleculeswithinamillimetricvoxel.Inaddition,thediffusionofthesewatermoleculesareobviouslyrestrictedbytheberstructuresinthetissue.Thus,theexponentialsignaldecayingassumptioniscompromisedandthesemodelbasedmethodswillintroducefalsebiasintothereconstruction.Ontheotherhand,thecurrentmodelindependentmethods(i.e.,withveryweakassumptionaboutthesignal)havetheirowndrawbacks.QBIcanonlyreconstructcontaminated,radialintegratedversionofdiffusionpropagator.DSIhasveryheavysamplingburdenforacceptablereconstructionaccuracy.Thepurposeofthisworkistoproposeaframeworkofgettingreliableandaccuratediffusionpropagatorreconstructionsfromsmallnumberofsamples.IttriestomaintaintheadvantagesofDSItechniquefrommuchlesssphericalsamples.Thedesiredreconstructionframeworkshouldbeaccurate,reliableanddoesnotbringtoomuchburdeninthesamplingstage.Sowedecidedtostartwithsphericalsamplesandpushthemontoamuchdenserregularlatticeinqspace.Theextensionprocedureisfeasibleaslongasthediffusionsignalissmooth.ThisframeworkhastheadvantagesofQSI,butdoesnotrequiresomanysamples.Themainassumptionhereisthesmoothnessofthediffusionsignalwhichisnaturalandpractical.Therearethreeproblemsinthisframework:thechoiceoftheregularlattice,thedistributionof 20

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thesphericalsamplesandtheapproachtopushsphericalsamplestothelatticepoints.Inthisdissertation,wevisiteachoftheproblemsseparatelytomaketheframeworkcomplete.Themaincontributionscanbesummarizedasfollowing:Ageometricapproachtoexplicitlyderivethemultivariatesincfunctionsforgeneralsamplinglatticesisgiven.Itprovidesausefultoolforinterpolatingandreconstructingsignalssampledondifferentlattices.Chapter 2 givesthederivationandcomparesthesamplingefciencyofdifferentlattices.TheoptimalsamplinglatticeandthecorrespondingsincfunctionwillbeusedinChapter 3 andChapter 4 .Theideaofoptimalsamplinglatticeisintroducedintothetomographicreconstructionframeworktoestimatethediffusionpropagators.Chapter 3 showsthatsimplereplacementofthetraditionalCartesianlatticewithoptimalbody-centeredcubic(BCC)latticeleadstoincreasedreconstructionaccuracyofthetomographicmethod.Anovelinterlacedsamplinggeometryisproposedasabetterwayofdistributingdiffusionsignalsamplesonmultipleqshells.Theinterlacedgeometry,comparedwiththenon-interlacedsampling,preservesmoreangularinformationofthediffusionsignalasshowninChapter 4 .Inaddition,Section 4.2 givesthealgorithmofestimatingthesignalvaluesonregularlattices,CartesianorBCC,fromthesphericalsamples.ThisalgorithmisbasedonthesincfunctionsderivedearlierandshowsthefurtherincreaseofreconstructionaccuracyduetotheusageoptimalBCClattices.Forthersttime,boxsplinesareintroducedintothetomographicframeworkofdiffusionpropagatorreconstruction.Chapter 5 introducesanalgorithmwhichreconstructsP(r)inboxsplinebasisfromitsprojections.Experimentsshowthatusinghigherorderboxsplinesincreasesthereconstructionaccuracywiththesamenumberofsamples.Differentboxsplinescanbedenedfordifferentlattices,socombiningthisalgorithmwithinterlacedsamplinggeometryandoptimallatticescouldbeanalternativeversionofthetomographicreconstructionframeworkdiscussedwithinthisdissertation. 21

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Adictionary-basedframeworkisproposedforthereconstructionofadiffusionpropagatoreldwithnon-localregularization.Thisframeworkisshowntobeveryresilienttonoisecomparedwithvoxel-wisemethods.Initspreliminaryform,sphericalsplinebasisandsingleshellsamplingareused.Butthereisnodoubtthattheinterlacedsamplingcanplayaroleinpotentialaccuracyimprovement.DetailsareshowninChapter 6 22

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CHAPTER2MULTIVARIATELATTICESANDTHEOPTIMALINTERPOLATIONFUNCTIONSOneofthekeyideasinthisdissertationistheintroductionofdifferentlattices,BCClattice,intheqspaceontowhichwewanttopushthesphericalsamples.Later,wewillalsondthatthelatticedependentsincinterpolatingfunctionplaysaveryimportantroleinthereconstructionstage.DifferentfromtheCartesiancase,wherethesincfunctionissimplyatensorproductofunivariatesincfunctions,thesincfunctionsforBCCandFCC(facecenteredcubic)latticesarenotseparableandmorecomplicatedtoderive.Inthischapter,weareintroducingthereaderstothebasicsofmultivariatelatticesandgivethederivationsofthecorrespondingsincfunctions. 2.1SamplingMultivariateSignalsSamplingtheoremplaysapivotalroleinsignalprocessingandinformationtheoryasitiskeytothediscrete-continuousmodelforsamplingandreconstructionofunivariatesignals.ThecelebratedWhittaker-Shannon-Kotel'nikovtheorem[ 111 ]elegantlyintroducessinc(x):=sin(x)=(x)whoseshiftsprovideacountablebasisforthespaceofbandlimitedfunctions:aspace,PWT,offunctionswhoseFouriertransformsvanishoutsidetheopeninterval])]TJ /F9 11.955 Tf 9.29 0 Td[(T,T[.Thereconstructionformulapresentsacardinalseriesexpansionofanysuchfunctionf:f(x)=Xn2Zf(nT)sinc(x=T)]TJ /F6 11.955 Tf 11.96 0 Td[(n). (2)Inotherwords,thesamplingtheoremimpliesthatafunctionf2PWTiscompletelydeterminedbyitssamplesf(nT),n2Z.WhileShannon'ssamplingtheoremhashadanenormousimpactoncommunicationandvariousengineeringapplications,itservesasatheoreticalframework.Thebandlimitedassumptionis,sometimes,aprohibitivelystrongassumptionspeciallyforfunctionswithnite(compact)support.Non-bandlimitedsignalsareoftenpassedthroughapre-lteringstepthatmakesthembandlimited.Thispre-ltering 23

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operationcanbeelegantlyviewedasaprojectionontothespaceofbandlimitedfunctions[ 111 ].Moreover,fornon-bandlimitedfunctions,Slepianfunctions[ 95 97 ]allowforconcentratingtheenergyofasignaloveranitebandwidth.Thereareseveralgeneralizationsofsamplingtheorem[ 15 53 ]suchasforfunctionsdenedoverlocallycompactabeliangroups[ 62 ].Thesignicanceofsincfunction,fromthetheoreticalpointofview,isthatitprovidesanorthonormalbasisandareproducingkernelforthespaceofbandlimitedfunctions.Thesincfunctionservesasthetheoreticalbasefordevelopmentof1-Dsignalprocessingtools[ 17 111 ].Moreover,sincfunctionhasprovidedarichclassofapproximationsandnumericalanalysistoolsthatprovideefcientsolutionstoavarietyofcomputationalproblems[ 63 100 101 103 ].ForsamplingmultivariatefunctionstheCartesianlatticeisthecommonchoice.TheseparablestructureofCartesianlatticeallowsustoemploytensor-productlteringoperationsinmultidimensions.Forinstance,thesincfunction,canbeeasilyextendedtoamultidimensionalCartesiansamplinglatticeviaatensorproductofunivariatesincfunctions.TheseparablestructureofCartesianlatticemakesitthepreferredlatticeinapplications.TheCartesianlattice,fromthesamplingtheoryviewpoint,isinefcient.TheworksofMiyakawa[ 83 ]andPetersenandMiddleton[ 91 ]demonstratetheadvantagesofspherepackingandspherecoveringlatticesforsampling(seealso[ 48 ]).IncontrasttotheCartesiancase,theselatticesarenotseparable.Thetensorproductofsincfunctionorother1-Dltersfailtoadheretothenon-separablenatureoftheselattices.Althoughtheselatticesareinterestingfromthetheoreticalaspects[ 30 81 91 ],fortheiradoptioninapplicationsoneneedspecialsignalprocessingtoolstailoredtotheirnon-separablestructure.Thischapterpresentsanexplicitderivationofsincfunctionsonmultidimensionallatticestogetherwithanon-separablewindowingscheme,whicharefundamentaltoolsforgeneralmulti-dimensionallattices.Ourspecic 24

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contributionsinclude:Thischapterintroducesageometricapproachtoexplicitlyderivemultivariatesincfunctionsonsamplinglattices.Theframeworkalsoallowsonetoderivecompleteinterpolatorysequences(Rieszbasis)forthespaceoffunctionswhoseFouriertransformsaresupportedonzonotopes.Thisgeometricframeworkallowsustoderiveanon-separablegeneralizationofthe1-DShannonwaveletsinthemultivariatesetting.Moreover,intheunivariatesetting,thesincfunctionisknowntobeequivalenttotheLagrangebasisfunctionwhenthenodesareintegers.Usingthisgeometricframework,wedeviseamultidimensionalcounterpartwherethesincfunctionturnsouttobeasumofLagrangebasisfunctions.Theconstructionframeworkappliestoanymulti-dimensionallatticessatisfyingthedicing[ 46 ]property.Togiveconcreteexamples,wefocusoncharacterizingallbivariateandtrivariatesincfunctionsthatcoverall2-Dand3-Dlattices.Weintroduceanon-separable,windowingtechniquethatgeneralizestheconceptofLanczosltertothemultivariatesetting.ThiswindowingtechniqueallowsustoemploythesincfunctionsforreconstructionofvolumetricdatasetssampledonBCCandFCClatticesandperformcomparisonswithCartesian-sampleddatasets.Theapproximationspacesspannedbycountablymanytranslatesofasinglefunction(kernelorgenerator)arefoundationaltosplinetheoryandsamplingtheory(aswellaswaveletandradialbasisfunctiontheories).Whilethesplinetheoryoftenconsidersthecasewhenthekerneliscompactlysupported,theparadigminsamplingtheoryisthecompactnessofthesupportoftheFouriertransformofthekernel[ 35 ].Itisintriguingtonotethatthetwohaveasomewhatreciprocalrelationship,speciallywhenconsideringthestructureofLanczoswindowingfunctions.Asplineframeworkwasproposedformultidimensionallattices[ 82 ]andthecurrentchapteroffersthedualparadigmofthesincfunctionsforsamplingtheoryontheselattices.Theconstructionofthesincfunction,limitedtotheBCClattice,wasillustratedin[ 44 ](lackingapracticalimplementation).Inthecurrentchapterwegeneralize 25

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Figure2-1. AlatticeL=[u1,u2]anditsfundamentaldomain(Voronoicell)areillustratedontheleft.Thereciprocallattice^L=L>=[^u1,^u2]isshownontherightwhoseVoronoicellistheBrillouinzoneofL. theconstructiontozonotopesthatallowsustoexplicitlyderivethesincfunctionforgenerallattices.TheabstractiontozonotopesisalsoofinterestforderivingcompleteinterpolatingsequencesonPaley-Wienerspacessupportedonboundedconvexsets[ 9 71 ]. 2.2MultidimensionalLatticesInthissectionwereviewthegeometricaspectsofmultidimensionallatticesthatarerelevanttotheirapplicationinthecontextofsamplingtheorydiscussedlaterinSection 2.3 2.2.1PointLatticesApointlatticeisadiscretesubgroupoftheEuclideanspacethatincludestheorigin[ 96 ].Thesetofpointsbelongingtoalatticeareclosedunderadditionandnegation.Thisimpliesthateverylatticepointhasaneighborhoodwhichcontainsallpointstowhomthatlatticepointistheclosestlatticepoint.SuchaneighborhoodisdenedastheVoronoicellofalatticepoint.EverylatticepointissurroundedbycongruentVoronoicellsofitsneighbors.TheVoronoicellofalatticeisunique[ 19 ];andisusuallycalledthefundamentaldomainorWigner-Seitzcell[ 96 ]. 26

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Alatticecanbecharacterizedbyitssamplingmatrix,L,whosecolumnsareabasisforthelattice.Everylatticepointcanbegeneratedbyintegerlinearcombinationofthosecolumnvectors.Lisnotuniqueforagivenlattice.IfbothL'andLgeneratethesamelattice,theyarerelatedviaaunimodularmatrix[ 112 ],L'=PL,withjdetPj=1. 2.2.2DicingLatticesAfamilyofequallyspaced(parallel)hyperplanescutsthed-dimensionalspace,Rd,intoequi-thickbands(slabs).dsuchhyperplanefamiliesdiceRdintocongruentparallelepipedswhoseverticesformad-dimensionaldicinglattice.Generally,adicingisanarrangementofN(Nd)suchhyperplanefamiliesthatarenon-degenerateandvertexregular.Non-degeneracyofhyperplanesimplythatdoutoftheNfamiliesmusthavelinearlyindependentnormalvectorsandvertexregularityimpliesthatthereisonehyperplanefromeachfamilypassingthrougheachvertexofthedicing.See[ 46 ]foramoredetailedpresentationofdicinglattices. 2.2.3SpherePackingandCoveringLatticesForSamplingSimilartothedistributionaldenitionofcomb(generalized)function,qqT,onecanintroduceqqLtomodelsamplingofamultivariatesignalwithalattice.Thecombfunctionassociatedwithalatticecanbedenedas:qqL:=Xk2Zd()]TJ /F3 11.955 Tf 17.94 0 Td[(Lk),whichisformallydenedasadistributionwiththeaidoftestfunctions.Thenotionofthereciprocal(dualorpolar)latticeappearswhenoneconsiderstheFouriertransformofqqL.ThereciprocallatticetoalatticeLisgeneratedbythecolumnsofthematrix^L:=L>.The(distributional)FouriertransformofqqL,isgivenby1=jdetLjqq^L[ 44 ].ThenusingthemultivariatePoissonsumformulawecanobservethatsamplingamultidimensionalsignalonalattice,generatedbyL,thespectrumofthesignalisreplicatedonthereciprocallattice^L.Brillouinzone[ 96 ]istheunitcelloftheduallattice.Theboundary 27

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Figure2-2. 3-DCartesian,BCCandFCClattices. ofBrilouinzoneisthemultidimensionalcounterparttotheNyquistfrequencyforthatspecicmultidimensionallattice.Foralow-passbandlimited1-Dsignal,thebestsamplingrateisdeterminedbythesupremumfrequencyofthesignal'sspectrum,whichiscalledNyquistfrequency.Inmultidimensionalcases,thetermNyquistfrequenciesisrelatedtothegeometryofthesignal'sspectrum.Theoptimalsamplinglatticeisoptimalinthesensethatonceasignalissampledonthislattice,itsspectrumisperiodicallyreplicatedandpackedasdenselyaspossible[ 41 83 91 112 ].So,theoptimalsamplinglatticeforaspecicsignalisdependentonthegeometryofitsspectrumandcanbecomputedifthegeometricinformationisgiven[ 69 ].Ontheotherhand,withoutanypriorgeometricknowledgeaboutthespectrumofthesignal,spherepackingandcoveringlattices[ 30 ]outperformotherlatticesforsamplinggenericmultidimensionalsignals.Thereasoningisthatwithoutanypriorinformationaboutthedistributionofhigh-frequenciespresentinasignal,thebestchoiceistoassumethattheydistributeequallyalongeverydirectionandtoisotropicallypreservehighfrequenciesasmuchaspossible.Accordingtosuchisotropictreatment,thebestsamplinglatticeisdeterminedbythethebestspherepackinglatticeinthefrequencydomain.Becauseofthedenserspectrapackinginthefrequencydomain,the2-Dhexagonallatticeanditshigher-dimensionalcounterpartsrequirelessnumberofsamplesthantheCartesianlatticetoproducethealias-freesamplingofgenericsignals[ 41 83 91 112 ]. 28

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Comparedto3-DCartesianlattice,thesamplingefcienciesoftheFCCandBCClatticesare27%and30%higher.Moreover,recentresearchshowsthattheselatticesaremoreresilienttojitternoiseinthesamplingprocedure[ 70 ].ThispropertymakesthemsuitablealternativesforsamplingoperationinapplicationssuchasMRI[ 94 ].Inhigherdimensionalspace,theselatticesleadtomoresavingsoverthewidelyusedCartesianlattice.Thespherecovering(e.g.,BCC)latticeistheoptimalchoicetosamplesmoothandbandlimitedsignalwhilethespherepacking(e.g.,FCC)latticeisthebestchoiceforsamplingnon-smoothsignals[ 48 ]. 2.3SincFunctionsOnMultidimensionalLatticesAsdiscussedinSection 2.2.3 ,samplingamultivariatesignalfonthelatticeL,leadstoperiodicreplicationofthespectrum^fonthereciprocallattice^L:qqLf()1 jdetLjqq^L^fXk2Zd()]TJ /F3 11.955 Tf 17.93 0 Td[(Lk)f()1 jdetLjXk'2Zd^f()]TJ /F5 11.955 Tf 18.27 2.66 Td[(^Lk'),where,indicatesaFouriertransformpair.Thereconstructionformulaneedstorecoverthespectrumbyisolatingthemainperiodfromthereplica.Sincethereplicationinthefrequencyspaceoccursonthereciprocallattice,thefundamentaldomainofthereciprocallattice(i.e.,theBrillouinzone)identiesthemainperiodthatneedstobeisolatedforidealbandlimitedreconstruction.Therefore,thesincLfunctiononalatticeisdenedasafunctionwhoseFouriertransformistheindicatorfunctionoftheBrillouinzoneofthatlattice:sincL(x):=jdetLjZ^L(w)exp(2ihx,!i)dw, (2)wherex:=(x1,...,xd)Tand!:=(!1,...,!d)Tdenotethespaceandfrequencydomainvariables,^LdenotestheindicatorfunctionoftheVoronoicellofthelattice^Landh,i 29

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denotesthecommoninnerproduct.Thenthereconstructedfunction,fr,isobtainedby:fr=sincLqqLf()^fr=^Lqq^L^f. (2)TheconvolutionofthesampledfunctionqqLfwiththesincLleadstoalinearcombinationofshiftsofthesincLfunctiononthelatticeL:fr=Xk2Zdf(Lk)sincL()]TJ /F3 11.955 Tf 17.93 0 Td[(Lk). (2)IffisbandlimitedtotheBrillouinzone,then^f(!)=0forall!outsidetheBrillouinzone(i.e.,when^L(!)=0).Thereforetherighthandsideof( 2 )impliestheperfectrecoveryofthebandlimitedfunctions:^fr(!)=^L(!)Xk'2Zd^f(!)]TJ /F5 11.955 Tf 12.29 2.66 Td[(^Lk')=^f(!). (2)TheexplicitderivationofsincLfortheCartesianlatticeispossibleviathetensorproductoftheunivariatesincfunctionsasthemultivariateintegralin( 2 )separatesintoaproductofmultipleunivariateintegrals;thisisduetotheseparablegeometryoftheBrillouinzoneoftheCartesianlatticewhichisa(hyper)cube.However,suchderivationfornon-CartesianlatticesisnotpossiblesincetheirBrillouinzonesarenotseparableandintegrationin( 2 )doesnotlenditselftoaneasyderivationinthegeneralsetting.VanDeVilleetal.[ 113 ]derivedthesincfunctionforthecaseofhexagonallatticeexploitingthe12-foldsymmetryoftheregularhexagon(i.e.,theBrillouinzone).ThederivationmakesuseofafunctionwhosesupportisaconeinR2.Aspecicsuperpositionoftheconefunctionandanumberofitsrotationsleadstotheconstructionoftheindicatorfunctionofahexagon(i.e.,^L(!))whoseinverseFouriertransformisderivedusingthe(distributional)inverseFouriertransformoftheconefunction.Theapplicationofthisapproachforgenerallatticesbecomesdifcultasthechoiceofconefunctionanditssuperpositionsandrotationsbecomesincreasinglydifcultfor2-Dlatticeswithoutthe12-foldhexagonalsymmetryorhigherdimensionallattices. 30

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Onotherhand,theFouriertransformsoftheindicatorfunctionsofgeneralpolytopescanbederivedusingthedivergencetheorem[ 58 69 ]inarecursivefashion.Incontrast,ourapproach,whichisspecictozonotopes,leadstoadirectandexplicitexpansionintermsofsincfunctionswithoutanyrecursiveterms.WeexploitthegeometricpropertiesofzonotopestoderivetheinverseFouriertransformofindicatorfunctionsofgeneralzonotopesinanydimension.LyubarskiiandRashkovskii[ 71 ]derivedthecompleteinterpolatingsequencesforfunctionswhoseFouriertransformissupportedonsymmetricpolygonsinR2.Aswewillseeinwhatfollows,symmetricpolygonshappentobe2-Dzonotopesandourgeometricconstructionoffersanexplicitclosed-formrepresentation.WhileourproposedgeometricconstructionisgeneralforN-Dzonotopes(notnecessarilylimitedtolattices),wespecicallyfocusonthecaseofBrillouinzonesforall2-D/3-Dlatticesandprovidetherecipesforhigherdimensionallattices. 2.3.1ZonotopesAzonotopeisatypeofpolytopethatisobtainedbyprojectionofanN-dimensionalhypercubetothed-dimensions(Nd).Let1,...,N2Rdbeasetofvectorsthatareobtainedbyageometricprojectionofe1,...,eN2RN(i.e.,edgesoftheunithypercubeinRN)downtoRd.ThenthezonotopeistheconvexpolytopethatisformedbytheMinkowskisumofvectorsinRd.Hence,azonotope,,isuniquelyidentiedbythe(multi)set:=[1...N].Thecenterofazonotopeisgivenbyc:=1 2PNn=1n.Azonotopetogetherwithallofitsfaces(anyco-dimension)aresymmetrywithrespecttotheircenterpoints.Onezoneofazonotopeisdenedasthesetofallparalleledgesofthezonotope[ 52 ].Athree-dimensionalzonotopeisoftenreferredtoaszonohedron.Forinstance,theelongatedrhombicdodecahedronisave-zonezonohedron.OneofthezonesisillustratedwithredcolorinFigure 2-4 .Itturnsoutthatapolyhedronisazonohedronifandonlyifitsfacesare(center)symmetric[ 33 ].TheoriginalmotivationforstudyingzonohedraingeometryhasbeenthefactthattheVoronoicellsofanylatticearezonohedra[ 33 ]. 31

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TheelementsinthesetarevectorsinRdandasubsetBiscalledabasisifthevectorsinBareminimallyspanningRd.ThecollectionofallsuchbasesofthezonotopeisdenotedbyB().AzonotopecanbedecomposedintoanitenumberofparallelepipedsformedbythevectorsineachoneofthebasesinB().ThisstructureofzonotopesallowsustoexplicitlyderiveafunctionwhoseFouriertransformistheindicatorfunctionofazonotope. Theorem2.1. Azonotope,,istheessentiallydisjointunionof(shifted)parallelepipedseachofwhichisformedbythevectorsinabaseof.[ 34 ,I.53]LetdenotetheindicatorfunctionoftheunithypercubeinRd.ThentheindicatorfunctionofaparallelepipedformedbytheMinkowskisumofthedvectorsinB,abaseofthezonotope,isgivenby:B=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(B)]TJ /F13 7.97 Tf 6.58 0 Td[(1,anditsinverseFouriertransformis:jdetBjY2B1)]TJ /F5 11.955 Tf 11.96 0 Td[(exp(2ih,i) 2ih,i=jdetBjexp(2ihcB,i)Y2Bsinch,i()BHere,cB:=1 2P2Bdenotesthephase-shiftnecessarytotranslatetheparallelepipedsothatitscornercoincideswiththeorigin.Then,accordingtoTheorem 2.1 ,,theindicatorfunctionofthezonotope,maybewrittenas:=XB2B()B()]TJ /F21 11.955 Tf 17.94 0 Td[(B), (2)whereBisatranslationassociatedtothebaseBintheparallelepipeddecompositionof.Thistranslationissimplyasumofcertaindirectionsinthezonotope:B:=B. (2) 32

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HereBisaN1vector,associatedwithB,whoseelementsare0or1:B2f0,1gN[ 34 ].ThesumoftwovectorsBandcBtranslatetheparallelepipedthatcorrespondstothebaseBtotheappropriatelocationinthezonotope:B=B+cB=(B+1=2), (2)where1=2isaddedtoallelementsofBaccountingforthecenterofthebase,cB.Withthisparallelepipeddecomposition,wecanderivetheinverseFouriertransformoftheindicatorfunctionofazonotopeexplicitly: XB2B()jdetBjexp(2ihB,i)Y2Bsinch,i,(2) 2.3.2SpaceTesselationsandLatticeVoronoiPolytopesThereareonlyanitenumberofregulartesselationspossibleinanydimension.Therefore,variouslatticescanbecategorizedbasedontheirVoronoipolytopes(cells)sincetheVoronoicellstranslationallytile(tesselate)thespace. Theorem2.2. TheVoronoicellofanylatticeissymmetricwithrespecttoitscenter;moreover,allofitsfacets(i.e.,facesofco-dimension1)arecentrallysymmetric[ 40 ].TheVoronoicellsofall2-Dor3-Dlatticesarezonotopes[ 33 ].Moregeneral,theVoronoicellsofallthehigherdimensionallatticesthatsatisfytheDicingpropertyarealsozonotopes[ 46 ].ThistopiciswidelydiscussedingeometryandiscloselyrelatedtotheVoronoiconjecture[ 40 74 75 ].LetLdenotethezonotopethatiscongruentwiththeBrillouinzoneofthelatticeL.Sinceazonotopeisnotnecessarilycenteredattheorigin,andtheBrillouinzoneofalatticeissymmetricwithrespecttotheorigin,thezonotopeneedstobetranslatedtotheoriginbyatranslationcL:cL:=L()]TJ /F3 11.955 Tf 17.93 0 Td[(cL). 33

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Then^L=cLwhichissymmetricwithrespecttotheorigin:^L(!)=1 2(^L(!)+^L()]TJ /F21 11.955 Tf 9.3 0 Td[(!)).Using( 2 )and( 2 ),wehave: sincL(x)=jdetLjXB2B(L)jdetBjcos(2hB)]TJ /F3 11.955 Tf 11.96 0 Td[(cL,xi)Y2Bsinch,xi.(2)Therefore,todeterminethesincLfunctionforalatticeL,oneneedstoconsidertheBrillouinzoneasazonotope.ThebasesofthezonotopeB)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Lcanbeobtainedbyconsideringthefull-ranksubsetsofthezonesinthezonotope.TheshiftscBforeachbaseisdeterminedbythecenterofthatbase,andtheBtranslationsaredetermined,geometricallyorcombinatorially,basedonthe0=1vectorsB. 2.3.2.1Two-DimensionalLatticesRegulartesselationoftheplaneisonlypossiblewithequilateraltriangles,squaresandhexagons.Thelattertwotesselationsaretranslationaltilingsoftheplaneandhencetheycharacterizeall2-Dlattices.Inotherwords,theVoronoicellofany2-Dlatticeiscombinatoriallyequivalenttoasquareorahexagon(seeFigure 2-1 ). Figure2-3. Ontheleft:hexagonallatticeL=H.TheBrillouinzoneofthelatticeLisillustratedontheright(^L=L)thatisa3-zonezonogonwiththreebases. 34

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Thehexagonisa3-zonezonogonandcanbedecomposedintothreeparallelepipeds.Figure 2-3 showsuchadecompositionfortheBrillouinzoneofaregularhexagonallatticewhosesamplingmatrixis:H=[u1,u2]=1 226411p 3)]TJ 9.3 9.95 Td[(p 3375.TheBrillouinzoneofthislatticeisgivenbythefollowingzonotope(illustratedinFigure 2-3 right):H=[1,2,3]=2 3[u1,u2,)]TJ /F3 11.955 Tf 9.3 0 Td[(u1)]TJ /F3 11.955 Tf 11.95 0 Td[(u2]. (2)Thebasesofthiszonotopeare:B)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(H=f[1,2],[1,3],[2,3]g.AsillustratedinFigure 2-3 ,theparallelepipedsformingtheBrillouinzonearecorneredattheoriginandthetranslationsB=0forallbases.Hence,thesincHforthehexagonallatticecanbewrittenas: sincH(x)=1=3cos(h3,xi)sinch1,xisinch2,xi+1=3cos(h2,xi)sinch1,xisinch3,xi+1=3cos(h1,xi)sinch2,xisinch3,xi.(2)Thecaseofageneric2-Dlattice(Figure 2-1 )followssimilarlywiththeintroductionofnon-zerotranslationsB. 2.3.2.2Three-DimensionalLattices3-DspacecanbetranslationallytiledbyvecombinatoriallydifferentpolyhedracommonlyreferredtoastheFedorov'sparallelohedra[ 96 ].Cubeisoneofthem,whosecorrespondingsincfunctionispreciselytensor-productofthree1-Dsincfunctions.TheVoronoicellsofFCCandBCClatticesarerhombicdodecahedronandtruncated 35

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octahedronasshowninFigure 2-4 .Theresttwotilingpolyhedra,elongatedrhombicdodecahedronandhexagonalprism,areVoronoicellsofelongatedBCCandhexagonallattices.ThesefourtypesofVoronoicellsarezonohedra,sotheycanalsobecutintoparallelepipeds.TheVoronoicellsofalltheothernon-Cartesianlatticesin3-Dspacearecombinatoriallyequivalenttooneofthesefourtypes.Hence,weprovideapracticalapproachtoconstructsincfunctionsforall3-Dlattices.Thecubeisazonohedronwith3zones,rhombicdodecahedronandhexagonalprismhave4zones,elongatedrhombicdodecahedronhas5zonesandtruncatedoctahedronhas6zones.Sinceonezonotopecanbeconvertedtotheotherthroughsettingthelengthofalltheedgesinonezonetozero,thetruncatedoctahedroncanbetreatedasthemostgenericzonohedron.Actually,theVoronoicellofageneric3-Dlatticeiscombinatoriallyequivalenttothetruncatedoctahedron[ 70 ].TheVoronoicellsofCartesian,FCC,elongatedBCCandhexagonalprismlatticesaredegeneratecasesoftheVoronoicelloftheBCClattice(i.e.,truncatedoctahedron). Figure2-4. Thefundamentaldomainsofall3-Dlatticestessellatethespace.(Lefttoright):Rhombicdodecahedron(FCC'sVoronoicell),truncatedoctahedron(BCC'sVoronoicell),elongatedrhombicdodecahedron,andhexagonalprism.Theyallcanbedecomposedintoparalleograms. 2.3.3FCCandBCCLatticesSinceBCCandFCClatticesareofpracticalimportance[ 42 48 51 79 102 ],weconsidertheexplicitcharacterizationoftheirsincfunctions.ThesamplinglatticesofBCC 36

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Table2-1. DecompositionoftheBrillouinzonefortheBCClattice,BusedtoconstructthesincB. B)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(BZonesB B11230000B21240000B31340000B42340000 andFCCaregeneratedthecolumnsoftheirrespectivematrices: B=266664)]TJ /F5 11.955 Tf 9.3 0 Td[(1111)]TJ /F5 11.955 Tf 9.3 -.01 Td[(1111)]TJ /F5 11.955 Tf 9.3 0 Td[(1377775,F=266664011101110377775.(2)TheBCCandFCClatticesarereciprocaltoeachother;hence,theVoronoicellofoneserveastheBrillouinzoneoftheother.TheBrillouinzoneoftheBCClatticeisarhombicdodecahedronwhichisa4-zonezonohedron:B=[1...4]=1 42666641)]TJ /F5 11.955 Tf 9.29 0 Td[(1)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 9.29 0 Td[(111377775. (2)All3-subsetsofzonesinBarefull-rankandhencetherearefourbasesinthiszonotope.TheBshiftsareallzerosincealloftheparallelepipedsjoinattheoriginasillustratedinFigure 2-5 .ThebasesofthiszonotopeBandtheBdeterminingthetranslationvectorareshowninTable 2-1 .ThesincBfortheBCClatticedirectlyfrom( 2 )andvaluesinTable 2-1 .TheBrillouinzoneoftheFCClattice(i.e.,truncatedoctahedron)isa6-zonezonotopeF: F=[1...6]=1 42666641)]TJ /F5 11.955 Tf 9.3 0 Td[(1110011001)]TJ /F5 11.955 Tf 9.3 0 Td[(1001)]TJ /F5 11.955 Tf 9.3 0 Td[(111377775. (2) 37

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Table2-2. DecompositionoftheBrillouinzonefortheFCClattice,FusedtoconstructthesincF. B)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(FZonesB B1124001011B2156001100B3345000001B4234000011B5126001110B6356000100B7135000100B8146011010B9245000001B10123000110B11125000100B12456010000B13346010010B14256000100B15134010010B16236000110 UnlikethezonesinHorB,notevery3-subsetsofzonesinFisafull-rankmatrix.Fromthetotalof)]TJ /F13 7.97 Tf 5.48 -4.38 Td[(63=20choicesofthree-zonecombinations,fourofthemareco-planarandhencedonotforma3-Dparallelepiped.ThetruncatedoctahedronisdecomposedintotheremainingsixteenparallelepipedswhichareshowninthesecondrowofFigure 2-5 .Aswecansee,sincethereisoneparallelepipedcenteredattheorigin,allthesixteenparallelepipedshavetheircornersofftheorigin.Therefore,thetranslationsB6=0forallbases.ThechoicesofthebasesandtheBareshowninTable 2-2 .PluggingthevaluesinTable 2-2 into( 2 ),wecangetthesincFfortheFCClatticedirectly. 2.3.4MultivariateShannonWaveletsNon-separablewaveletsinthemultidimensionsaredifculttocomeby[ 55 ].Thespecial2-Dexamples[ 25 27 117 ],cannotbegenerallyextendedto3-Dorhigherdimensionsforvariouslattices.However,theparallelepipeddecompositionofgeneralBrillouinzonesoflattices,whichwasusedforexplicitderivationofsincfunctions,leads 38

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Figure2-5. Toprow:Arhombicdodecahedron(4-zones)canbedividedintofour3-Dparallelepipedsalljoiningtheorigin.Bottomrow:Decompositionofatruncatedoctahedron(6-zones)intosixteenparallelepipeds. toafrequencypartitioningthatleadstoawavelet-typedecompositionbasedonscaledsincLfunctions.Intheunivariatesetting,theShannonwavelets[ 72 ],areobtainedbyusingthesincasthescalingfunction.Themainobservationhereisthatthespectrum(i.e.,Brillouinzone)oflatticesaredecomposedintoparallelepipedswhichlendthemselvestoasubdivisionthatpartitionsthespectrum,inanon-separableway,tolow-bandsandhigh-bands.Thelow-bandfrequenciesfromeachparallelepipedformaself-similarBrillouinzonethatcanberecursivelypartitionedtolower-bandfrequencies,verysimilartothe1-Dlter-bankalgorithms.OnestepofthefrequencypartitioningisillustratedinFigure 2-6 thatleadstoaBrillouinzoneofhalfthesize.ThesincLasascalingfunctionhasamulti-scalerelationshipwhichisillustratedbythesubdivisionofparallelepipedsinFigure 2-6 .The 39

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Figure2-6. BrillouinzonefromFigure 2-3 ,subdividedtoderiveamulti-scalefunctionforShannonwavelets.Low-frequencybandsareindicatedwithdarkercolors. hexagonaftersubdivisionofparallelepipedswillbuildasmaller(centered)hexagon,ontherightimage,andthisoperationcanbecarriedrecursively.ThespectrumoftheShannon`mother'waveletiscomposedofninetermscorrespondingtothethree(bright)high-passsub-parallelepipedsforeachparallelepipedinFigure 2-6 (right).Onecanalsovieweachofthesub-parallelepipedsasaLH,HL,andHHpartitionsofthespectrumforeachparallelepiped.ThescalingfunctionforShannonwaveletsonthehexagonallatticewillbethesum: 'ShL(x)='Shred(x)+'Shgreen(x)+'Shblue(x),(2)where'Shred,similartotheothertwo,isconstructedbyasheartransformation,using1,and2froma2-Dtensor-productsincfunction. 2.3.5MultivariateLagrangeInterpolantTheequivalenceofsincinterpolationtoLagrangeinterpolationhasbeenknownasearlyasBorel[ 76 ].Givenaninnitenumberofequallyspacedsamplesontherealline,theLagrangebasispolynomialsconvergetoshiftsofthesincfunction.sinc(x)=Yn2Znf0g1)]TJ /F6 11.955 Tf 13.15 8.09 Td[(x n (2) 40

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Figure2-7. LagrangebasisforpolynomialinterpolationwhenthenodesareZconvergestosinc. Anargumentfordemonstratingtheequivalenceisthatanyanalyticfunctionisuniquelydeterminedbyitszerosanditsvalueatonepoint.Thesincfunctionandtherighthandsideof( 2 )crosszeroonallintegersexceptx=0wheretheyarebothequalto1(seealso[ 93 ]).ThemultivariatesincfunctionsdiscussedinSection 2.3.2 cannotbedirectlyrelatedtoaLagrangetypebasisforinterpolation.Theproductofsincfunctionsin( 2 )allowforaLagrange-basisinterpretation;thefollowingremarkeablecosineexpansion[ 1 ,p.85]allowsustowritethesincLasasumofLagrange-basisfunctionsonthelatticeL. cos(x)=1Yn=11)]TJ /F6 11.955 Tf 40.46 8.09 Td[(x2 2(n)]TJ /F5 11.955 Tf 11.95 0 Td[(1=2)2)cos(2x)=1Yn=11)]TJ /F5 11.955 Tf 25.25 8.09 Td[(16x2 (2n)]TJ /F5 11.955 Tf 11.96 0 Td[(1)2=1Yn=11)]TJ /F5 11.955 Tf 23.51 8.09 Td[(4x 2n)]TJ /F5 11.955 Tf 11.96 0 Td[(11+4x 2n)]TJ /F5 11.955 Tf 11.95 0 Td[(1=Yn2Z1)]TJ /F5 11.955 Tf 23.51 8.08 Td[(4x 2n)]TJ /F5 11.955 Tf 11.95 0 Td[(1.(2) 41

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Using( 2 )and( 2 )wecanwritethesincLforalatticeLasasumofLagrangeinterpolants: sincL(x)=jdetLjXB2B(L)jdetBj(1+4hB)]TJ /F3 11.955 Tf 11.95 0 Td[(cL,xi)Yn2Znf0g1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(4hB)]TJ /F3 11.955 Tf 11.95 0 Td[(cL,xi 2n)]TJ /F5 11.955 Tf 11.95 0 Td[(1Y2B1)]TJ 13.15 8.09 Td[(h,xi n.(2) 2.3.6MultidimensionalLanczosWindowingTheidealinterpolation,inthespaceofbandlimitedfunctions,demandsinnitecomputationsincethesupportofthesincfunctionisunbounded.Truncatingthesincfunction,producesartifactsonthereconstructedfunctionthatisoftenreferredasringing.Tominimizetheringingartifacts,thereareavarietyofwindowingtechniquesintheunivariatesetting(suchasHamming,Parzen,Blackman[ 87 ]).Inthemultivariatesetting,theproblemisconsiderablymoredifcultandthenon-separablemultidimensionalwindowfunctionsarenotwellstudied.Radialextensionofunivariatewindowingtechniques(i.e.,McClellantransform)providesanisotropictruncationofthesincfunctionwhichdisregardsthegeometryofthelattice.Moreover,theisotropicwindowingandtruncationinuencethefrequency-spacebehaviorofsincLdifferentlyforeachlatticeandmayleadtobiasesinonelatticeoveranother.Tensorproductof1-DwindowsisonlyappropriatefortheCartesianlattice.Anotherapproachistodesignadiscretizedwindowforaseparable(Cartesian)lattice,basedon1-Dwindows,anddownsampletothedesiredlatticeL[ 31 ];however,thismethodmayonlybeappliedinthediscretesettingutilizingadownsamplingoperationandcanonlyapproximatethetruesincfunctionuptothelevelaccommodatedbythediscretizationresolution.Amongdifferentunivariatewindowdesignmethods,theLanczoswindowisuniquelysuitableforthemultivariategeneralizationasitpicksthemainlobeofthesincfunctionasawindow.Alsoaswewillseeitsfrequencybehaviorisparticularlysuitableforthis 42

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(a)(b)Figure2-8. Lanczoslterasamultiplicationwiththemainlobeofthescaledsincfunction.(a)Mainlobeofthescaledsincfunctionandoriginalsincfunctions.(b)Lanczoswindowedsincfunction. generalization.TheLanczoswindow,alsocalledsincwindow,isessentiallythecentrallobeofascaledsincfunction,sinc(x=a),restrictedtothemainperiod)]TJ /F6 11.955 Tf 9.29 0 Td[(axa.SincethedenitionoftheLanczoswindowispurelydeterminedbythesincfunction,themultivariateLanczoswindowisdenednaturallyoneachlatticebythemainlobeofitssincL.TheboundaryofthesupportofthewindowistherstzerolevelsetofthecorrespondingsincLfunctionfromtheorigin.ThisapproachallowsthewindowfunctiontoinheritthesameanisotropicpropertiesfromthesincLfunction.Moreover,itdoesnotneedspecialtailoringfordifferentlatticeswhichmakesitanunbiasedapproachformultivariatewindowdesign.LetSfsincLgdenotethesupportofthemainlobeofsincL;theLanczoswindowedversionofsincLcanbewrittenas: La(x)=8><>:sincL(x)sincL(x a),x=a2SfsincLg0,otherwise.(2)Thescalingparameteraispickedasanintegervalue(usually2or3in1-Dapplications[ 16 ]),anddeterminesthesizeofthewindow. 43

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TheeffectofLanczoswindowedsincinthespacedomainistheincreasedsmoothnessofthetruncatedsincwhilemaintainingtheinterpolationproperty.Forexample,in1-Dcase,theoriginalsincfunctionandthemainlobeofthescaledsincfunctionareillustratedasFigure 2-8 (a),whiletheLanczoswindowedsincfunction,La,isshownasFigure 2-8 (b).TheeffectofLanczoswindowinthefrequencydomaincanbestudiedbyconsideringtheFouriertrasformsofsinc(x)andsinc(x=a)whicharebothboxfunctionsbutwithdifferentwidths.Sincesinc(x=a)correspondstoathinnerboxfunction(a>1),theFouriertransformoftheLanczoswindowedsincisbox(!)convolvedbyathinnerboxfunctionjajbox(a!).Theconvolutionofthesetworectangularfunctionsresultsinalineardropoffofthetransferfunctionwiththeincreasedsupportbeyondthecut-offfrequency.Theeffectinthefrequencyspaceistheincreasedcontinuityofbox(!)fromC)]TJ /F13 7.97 Tf 6.59 0 Td[(1toC0:whiletheoriginalsinclter'stransferfunctionwasadiscontinuousboxfunction,thetransferfunctionofsinc(!)sinc(!=a)hasalineardropoffandisacontinuousfunction.Figure 2-9 illustratesthesincHanditstransferfunctionforhexagonallattice,inthespaceandfrequencydomains.ThetransferfunctionofsincH(=a),isrelatedbyascaling(i.e.,shrinkedfora>1)tothatofsincH;hencewehave:sincHsincH(=a)()aHH(a). (2)ThesupportoftheconvolutionoftheindicatorfunctionsoftwopolytopesisdeterminedbytheMinkowskisumofthetwo[ 21 ].Therefore,thefrequencysupportoftherighthandsideisdeterminedbytheMinkowskisumofthesupportofHandH(a).ThesupportoftheHandgenerally^ListheVoronoicelloftheduallatticeandhenceisapolytopethatisconvexandsymmetric.Minkowskisumofaconvex,symmetric,setcontainingtheoriginwithascaledversionofitselfenlargesthatsetbyafactorof1+a[ 21 ].Hence,thesupportispreciselydeterminedbyenlargingtheBrillouinzone 44

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byafactorof1+a.Moreover,themultivariateconvolutionasin( 2 )increasesthedegreeofcontinuityby1[ 82 ].Theincreaseinthecontinuityofthetransferfunctionmakesthetruncation,inthespacedomain,lesssusceptibletotheringingphenomenon[ 65 ].Onecanapplythewindowfunctionseveraltimestofurtherreducetheringingartifactsthatarecausedbytruncation.Inthespacedomain,thewindowed-sincfunctionisgivenby: Lna(x)=8><>:sincL(x)sincnL(x a),x=a2SfsincLg0,otherwise.(2)Theparametersnandadeterminethesmoothnessandsizeofthekernel'ssupportrespectively.Itisinterestingtonotethat,unlikethemultidimensionalCartesiansetting,theLanczoslterkernelinthenon-separablesettinghasasupportwhichisnon-planarobject(i.e.,notapolytopeingeneral).Forexample,theLanczoswindowforhexagonallatticehastheshapewithhexagonalsymmetrybutwithcurvedfaces(seethesupportinFigure 2-9 (c)). 2.4ExperimentalComparisonOptimallatticessuchashexagonal,BCCandFCChavebeentheoreticallyconsideredtobesuperiortotheCartesianlatticeinthecontextofsamplingtheory[ 91 ].However,forpracticalapplications,thesamplingoperationtogetherwiththereconstructionstepinuencethesignalquality.Whilethereareseveralspline-typesolutionsforspeciclattices[ 47 60 ],thesincfunctionsofferafairandunbiasedframeworkforreconstructionacrossvarioussamplinglattices.Thisframeworkallowsustoexaminethepracticalaspectsofoptimalsamplingbyemployingtheidealinterpolationschemeoneachlatticeforthepursposeofsignalreconstruction. 2.4.1ExperimentsSetupInordertoshowtheresultsoftheoptimalsamplingschemescomparedtothecommonCartesianscheme,weimplementedaray-caster[ 115 ]torenderimagesfrom 45

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(a)(b) (c)(d)Figure2-9. PlotsofsincHanditsLanczoswindowedversioninspace(leftcolumn)andfrequency(rightcolumn)domains.sincHbyitselfin(a)spacedomainand(b)frequencydomain.LanczoswindowedsincHwitha=4andn=2in(c)spacedomainand(d)frequencydomain. thereconstructedsignalfromBCC,FCCandCartesiansampledvolumetricdatasets.The3-Ddatasetsconsistofsamplesfromafunctionf(Lk)wherek2Z3andListhesamplingmatrix.LisanidentitymatrixforCartesianlattice,BandFasin( 2 )forBCCandFCClattices.Thereconstructedsignal~f: ~f=fsincL=Xk2Z3f(Lk)sincL()]TJ /F3 11.955 Tf 17.93 0 Td[(Lk)(2)wherethesincfunctioniswindowedusing( 2 )forpracticalimplementation.Asabenchmarkforcomparison,wechoseafrequency-modulationsyntheticdatasetcalledML,shownasFigure 2-10 (a)whichwasrstproposedbyMarschnerandLobb[ 73 ].Thefunctionwassampledatthecriticalresolutionof414141onthe 46

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(a)(b)(c)Figure2-10. Benchmarkvolumetricdatasetsof(a)Marschner-Lobb,(b)carpshand(c)bonsaitree. Cartesianlattice,andequivalentsamplingresolutionsof323264ontheBCClatticeand2525100ontheFCClattice.Theresolutionof41waschosenasthecriticalresolution[ 73 ]becauseitcaptures98%oftheenergyofthespectrum.So,itistreatedasapracticalNyquistfrequencyforthisnon-bandlimitedsignal.InadditiontothesyntheticMLdataset,wehavefurtherexaminedourschemeonrealvolumetricdatasetsofcarpshandbonsaitreeshownasFigure 2-10 (b)andFigure 2-10 (c).SincetherearenoacquisitionsystemsdevelopedforBCC/FCCsampling,allthereal-lifedatasetsarescannedonCartesianlattice.Inordertotestthereconstructionperformanceonthethreelattices,wesimulatedthreedatasetsthroughsubsamplingtheoriginalCartesiandatasets,thatweredenselysampled,ontolow-resolutionCartesian,BCCandFCClatticeswithalmostidenticaldensities.Theoriginaldatasetswereattheresolutionof256256256andweresubsampled(to16%)onCartesian,BCCandFCClatticeswithresolutions140140140,111111222and8888352respectively.Forthereconstructionkernel,wechosethesecondorder,n=2,Lanczoswindowedsincinterpolantwithscalingfactora=3.Wehaveveriedinpracticethatthiskernel 47

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alwayscoversapproximatelythesamenumberofvoxelsinallthethreelattices,whichensuresthereconstructionfromdifferentlatticesusesthesameamountofinformation. 2.4.2VisualComparisonFigure 2-11 showstherenderedimagesfromtheMLdatasetondifferentlattices.ThisexperimentillustratesthatthereconstructionfromCartesianlatticeexhibitsthestrongestaliasingartifacts.ReconstructionsfromBCCandFCClatticesclearlyoutperformthatfromCartesianlattice.ThethirdrowinFigure 2-11 showstheangularerrorwhichisthedifferencebetweenthetruenormalandthereconstructednormalontheisosurface.Thedarkerpixelsdenotessmallererrorandthewhitepixelsindicatethemaximumangularerrorof30degrees.Again,weobservethatBCClatticeproducessmallesterrorwhileFCClatticeminimizesaliasingwhichmatchesthetheoreticalexpectation[ 48 ].Figure 2-12 andFigure 2-13 showtherenderedimagesofcarpshandbonsaitreedatasetsfromthethreedifferentlattices.WecanobservethatimagesrenderedfromBCC/FCCdatasetsalwayscontainsmoredetailsthanthoseoftheCartesiandatasets.ThesuperiorityofBCC/FCCdatasetsaremostlyvisibleintailnandbonesinthecarpshdatasetandtheconnectivityofthebranchesinbonsaidataset. 2.4.3NumericalComparisonToperformanumericalerroranalysisofourreconstructionschemes,weestimatedtherootmeansquare(RMS)error(i.e.,L2)presentinthereconstructedsignal~f,whencomparedwithf.Wecouldsimplycomparethenumericalerrorscalculatedfromthesamedatavolumesusedpreviouslyforrendering.Butwefounditismoreinterestingtoshowhowtheresolutionisinuencingtheerror.WepickedaCartesianvolumewithaxedresolution,e.g.414141fortheMLdataset.Wecallthisresolutionreference-resolutionandtreatitsnumberofsamplepointsasourreference,named100%.Then,wecancharacterizetheresolutionofaBCC/FCCdatavolumebythepercentageratioofitsnumberofsamplesoverthereference-resolution.Forinstance, 48

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(a)(b)(c)Figure2-11. VisualcomparisonofimagesrenderedfromCartesian,BCCandFCClattices.MLisosurfaceimagerenderedfrom(a)Cartesianlattice,(b)BCClatticeand(c)FCClattice.Thethirdrowshowstheangularerrorsoccurredingradientestimationontheisosurface.Blackindicateszeroerrorandwhitedenotesanangularerrorof30degrees. 49

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(a)(b)(c)Figure2-12. VisualcomparisonofimagesrenderedfromCartesian,BCCandFCClatticesusingLanczoswindowedsincinterpolants.Carpshisosurfaceimagesrenderedfrom(a)Cartesianlattice,(b)BCClattice,(c)FCClattice. (a)(b)(c)Figure2-13. VisualcomparisonofimagesrenderedfromCartesian,BCCandFCClatticesusingLanczoswindowedsincinterpolants.Bonsaitreeisosurfaceimagesrenderedfrom(a)Cartesianlattice,(b)BCClattice,(c)FCClattice.Thecloseupviewsoftheupperrightcornersareshowninthesecondrow. 50

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(a)(b)(c)Figure2-14. TheRMSerrorcomparisonofBCC,FCCversus414141Cartesianlatticeover(a)MLdataset(b)carpshdatasetand(c)bonsaitreedataset. ifwepickthe414141Cartesianvolumeasthe100%reference-resolution,the26.87%BCCvolumehasaresolutionof212142andthe114.24%FCCvolumehasaresolutionof2727108.Fornumericalcomparison,wechoseBCC/FCCvolumeswithresolutionvaryingfromaround30%to115%andplottedtheirRMSerrorsastwocurves.Duringthecomparison,onlyoneCartesianvolumewithresolution100%wasusedanditsRMSerrorwasplottedasahorizontallineconsideredthebenchmarkline.ThisbenchmarklineintersectstheBCCerrorcurveandFCCerrorcurverespectively.ThesetwointersectionpointstellusatwhatresolutiontheBCC/FCCvolumesproducethesameRMSerrorasthe100%Cartesianvolumedoes.Figure 2-14 (a)showshowtheRMSerrorofBCC/FCCreconstructionsonanMLdatasetvaryingwiththeirresolutions.Thereconstructionerrorfroma414141Cartesianvolumeisplottedasthebenchmarkline.Fromthegure,wecanseethatthecurveofBCC/FCCreconstructionerrorintersecttheCartesianbenchmarklineataround70%.Itmeansthatweonlyneed70%samplepointsforBCC/FCClattices,comparedwithCartesianlattice,toachievethesamereconstructionaccuracy.Thismatchesthetheoremprecisely.Finally,weshowthenumericalcomparisonsonRMSerrorsfromcarpshandbonsaitreedatasetsasFigure 2-14 (b)andFigure 2-14 (c).140140140Cartesianvolumeswerechosenasthe100%reference-resolutionforbothdatasets.Sincewedonothavethetrueanalyticalfunctionvaluestocalculatetheerror,weusetheinterpolated 51

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valuesfromtheoriginalhigh-resolutiondatavolumeasthetruth.Theseplotsalsoshowthatweneedaround70%numberofsamplestoachievethesamereconstructionaccuracyforBCC/FCClattices. 52

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CHAPTER3TOMOGRAPHICRECONSTRUCTIONOFDIFFUSIONPROPAGATORSUSINGOPTIMALSAMPLINGLATTICESThischapterexploitsthepowerofoptimalsamplinglatticesintomographybasedreconstructionofthediffusionpropagator.OptimalsamplingleadstoincreasedaccuracyofthetomographicreconstructionapproachintroducedbyPickalovandBasser[ 92 ].Alternatively,theoptimalsamplinggeometryallowsforfurtherreducingthenumberofsampleswhilemaintainingtheaccuracyofreconstructionofthediffusionpropagator.Theoptimalityoftheproposedsamplinggeometrycomesfromtheinformationtheoreticadvantagesofspherepackinglatticesinsamplingmultidimensionalsignals.Theseadvantagesareinadditiontothoseaccruedfromtheuseofthetomographicprincipleusedhereforreconstruction.WepresentcomparativeresultsofreconstructionsofthediffusionpropagatorusingtheCartesianandtheoptimalsamplinggeometryforsyntheticandrealdatasets. 3.1MotivationsInthischapter,weproposeamodel-freeapproachtoreconstructionofthediffusionpropagatorateachvoxel.TheapproachbuildsontheworkinPickalovandBasser[ 92 ],whereintheyexploittheFouriertransformrelationshipbetweenP(r)andS(q)todevelopatomographicreconstructionofthepropagatorateachvoxel.ByinterpolatingarelativelysmallnumberofsamplesfromtheDWsignalintheq-spaceontoaregulargrid,theirapproachallowsforatomographicreconstructionofP(r)usingtheFouriertransform.SinceP(r),ateachvoxel(tile),maycontainanisotropicfeaturesinanyarbitrarydirection,itbehoovesustochooseatilingofthespacewhereeachtilecapturesthemaximumradialcontentofP(r).OptimaltilingofthespaceresultsinvoxelsthatadmitalargerinscribingspherecomparedtothetraditionalCartesiantilingwithcubicvoxels.Themainideainthischapterexploitsoptimaltilingwhereeachvoxelisarhombicdodecahedronandadmitsalargerinscribingsphere;hence,abetterresolutionofP(r) 53

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(i.e.,theonewithalargerradiusr)isavailablewiththesamesamplingdensityaswithcubicCartesiantiling.TherhombicdodecahedraltilingcentersformtheFaceCenteredCubic(FCC)latticewhichisthedensestspherepackinglatticein3-D.Aswehaveseenin 2.2.1 ,thisamountstooptimalsamplingoftheq-spaceonaBodyCenteredCubic(BCC)latticewhichisthekeycontributionofourwork.Theoptimalsamplinginq-spaceallowsustoachieveabetterreconstructionduetosignicantlysmallerghostingeffectsinthereconstructedP(r).Whensamplingq-spaceE(q)onaBCClattice,theP(r)iscontainedinrhombicdodecahedralvoxelswhichadmitalargerinscribingspherethanthecommonlyusedCartesianvoxels.Theinscribingspheretotherhombicdodecahedralvoxelisabout30%largerthanthatofthecubicvoxelwhilethetwovoxelsareofthesameunitvolume.Therefore,whenreconstructingP(r)oneachindividualvoxel,areconstructionbasedonBCCsamplingofq-spaceyieldlargerresolutionofrwhilepreventingtheghostingartifacts.Theghostingartifactsisthespace-domainequivalentofaliasing.Whensamplingtheq-spacesignalinfrequencyspacewithacoarsesamplingrate,thereplicasinthespacedomainbleedintothemainvoxelarea.Foragivenxedsamplingresolutioninq-space,theghostingartifactissmallerfortherhombicdodecahedralvoxelcomparedtothecommonly-usedcubicvoxels.Additionally,byusingasignicantlyreducedsampleset,wecanmaintainthesameaccuracyasoneobtainedusingaCartesiansamplinglattice. 3.2AlgorithmandImplementationThealgorithmproposedbyPickalovandBasser[ 92 ]isamodiedversionoftheiterativeprocedurepresentedbyGerchbergandPapoulis(G-P).Theyassumetheoriginaldatasampleslieonradiallinesinq-space.Oneachradialline,severalsamplesareavailablecorrespondingtodifferentradii.Usinganinterpolator/extrapolator,theyobtainthedatavaluesonaCartesianlatticeintheq-space.Byimposingsomeconstrainsbothintheq-spaceanddisplacementprobabilityspace,theiralgorithmruns 54

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iterativelyviatheuseofdirectandinverseFouriertransforms.Ineachiteration,theoriginalsampleddataisimposedintotheq-spacevaluestoreinforcetheconsistencybetweenthereconstructedP(r)andthetruediffusionpropagatorsimpliedbythedatasamples.Aswesawin 2.2.1 ,wecanincreasetheaccuracyofreconstructionbychangingtheq-spacesamplingfromtheCartesianlatticetotheBCClattice.Therefore,wepushtheradiallysampleddata,ontotheBCClatticetogetherwithanapproachsimilartothealgorithmofPickalovandBasser.Sinceourpurposeistoinvestigatethetheoreticaladvantagesoftheoptimallattice,wetookthesameradialsamplesfromE(q)andpushedthemintobothCartesianandBCClatticeswiththesamenumberoflatticepoints.Whilethereareseveralesotericinterpolationmethods[ 47 ]forinterpolationintotheBCCandCartesianlattices,weusedanidenticalinterpolantinbothcasestoensurethattheonlydifferenceinthetworeconstructionsisthesamplinggeometry.Therefore,weusedacubicsplineinterpolantinbothcases,eventhoughinourexperiments,otherinterpolantsresultedinsimilarresults.AftertheCartesianandBCCre-samplingoftheq-spacedata,theP(r)reconstructionisobtainedthroughadirectFouriertransform.WhiletheusualFFTalgorithmissuitableforCartesiansampling,theweemployedthemodiedFFTalgorithm[ 5 ]fortheBCCsampleddata.Inordertoevaluateourmethodtheaccuracyofreconstructionineachcasewasmeasuredbycomparingthereconstructedsignalfromthetruesyntheticsignalbythemeansofsumofsquarederrors(SSE).Forthesyntheticdatasomevisualcomparisonisinsightfulanddiscussedin 3.3 3.3AlgorithmEvaluationWenowpresentpropagatorreconstructionexperimentsusingtheBCCandCartesianlatticesinasyntheticdataset(Figure 3-1 )andarealdataset(Figure 3-2 )fromaratopticchiasm. 55

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Table3-1. SSEcomparisonofreconstructionsfromdifferentlatticeswithNr,N=12,N=13,=90 Nr109876 Cartesian(10)]TJ /F13 7.97 Tf 6.58 0 Td[(5)2.192.312.633.344.04BCC(10)]TJ /F13 7.97 Tf 6.59 0 Td[(5)0.600.680.871.382.56 Table3-2. SSEcomparisonofreconstructionsfromdifferentlatticeswithN,N,Nr=10,=90 N,N12,1311,1210,119,108,9 Cartesian(10)]TJ /F13 7.97 Tf 6.58 0 Td[(5)2.193.916.675.403.36BCC(10)]TJ /F13 7.97 Tf 6.59 0 Td[(5)0.601.373.832.812.97 Foroursyntheticdataexperiments,wegeneratedsamplesfromamixtureoftwoGaussianfunctionsinthe3-Dq-spaceandexaminedtheSSEbetweenreconstructionsandthetruesignal.Intheexperiments,datasamplesaredistributedonradiallinesalong(r,,)inthesphericalcoordinatesystem.ThenumberofsamplesalongeachradiallineisdenotedbyNr,ofvaluesbyN,andthevaluesbyNrespectively.denotestheanglebetweenthetwoGaussiancomponents.Table 3-1 reportstheSSEdifferencesbetweentheBCCandCartesianreconstructionswithxed=90andvaryingNr.ItisevidentfromtheerrorsthattheBCC-basedreconstructionyieldssmallererrorsdespitethesamesamplingrateNrasintheCartesian-basedreconstruction;thisremainstobethecaseevenforvaryingsamplingresolutions.SimilarlybychangingthesamplingresolutionsinNandN,theadvantagesofBCCreconstructionismaintained(seeTable 3-2 ).Table 3-3 depictsthecomparisonbetweenBCCandCartesianreconstructionsbyvaryingtheanglebetweenthetwoGaussiancomponentssimulatingvariousanglesofber-crossings.Table 3-4 compares Table3-3. SSEcomparisonofreconstructionsfromdifferentlatticeswith,Nr=10,N=12,N=13 9080706050 Cartesian(10)]TJ /F13 7.97 Tf 6.58 0 Td[(5)2.193.754.164.042.92BCC(10)]TJ /F13 7.97 Tf 6.59 0 Td[(5)0.601.051.602.091.69 56

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Table3-4. SSEcomparisonofreconstructionsfromdifferentlatticeswith,Nr=10,N=12,N=13,=90 00.020.040.060.08 Cartesian(10)]TJ /F13 7.97 Tf 6.58 0 Td[(5)2.192.573.514.796.78BCC(10)]TJ /F13 7.97 Tf 6.59 0 Td[(5)0.601.052.313.846.28 CartesianBCCFigure3-1. VisualcomparisonofthereconstructedP(r).Row1:(1,2)=(20,100),Row2:(1,2)=(5,85).(1,2)arethedirectionsofthetwoGaussiancomponents. thereconstructionsunderRiciannoiseofdifferentnoiselevels.Wecanclearlyseethatunderallthetestconditions,reconstructionsfromBCClatticeachievesmallererrorscomparedtoreconstructionsfromtheCartesianlattice.Also,Table 3-1 andTable 3-2 showthatareconstructionbasedonasmallernumberofBCCsamples(e.g.,Nr=6)intheq-spaceiscomparabletoareconstructionwithlargernumberofCartesiansamples(e.g.,Nr=8)intheq-space.ThissuggestsastrategytofurtherreducethesamplesizeofE(q)andreducetheacquisitiontime.ForNr=10,N=12,N=13,=80and=0,Figure 3-1 showstheisosurfacesofthereconstructedP(r)fromCartesianandBCClatticesrespectively.WecanseethattheisosurfaceofP(r)fromCartesianlatticeexhibitssomeghostingartifactsatthetipsduetoleakagefromghostsintheneighboringperiod.ThereconstructionfromBCClatticeisnotinuencedbytheghostingartifactssincewhenwetakesampleson 57

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(a)(b)Figure3-2. Probabilitymapsreconstructedfromrealdataset:(a)fromCartesianlattice,(b)fromBCClattice. BCClatticeinq-space,thedistancebetweenthereconstructedP(r)anditsnearestghostreplicaislargerthanthatderivedfromCartesianlattice.Theoretically,thelargerdistanceimplieslessinuencefromtheghostreplica.Wenowpresentanexperimentwithrealdatafromaratopticchiasm,whichcontainssamplesmeasuredwith46differentdirectionswithjustonebvalue.Sinceouralgorithmneedssampleswithdifferentbvalues,forinterpolationpurposes,weusedthehighranktensormodelin[ 10 ]toprocessthedataandgetthevaluesfordifferentbsviare-sampling.Figure 3-2 depictstheprobabilitymapsreconstructedfromtheCartesianandBCClatticesrespectively.AcloseexaminationdepictsthatthereconstructionfromCartesianlatticeappearsdistortedcomparedtotheBCCreconstruction.ThisisduetotheresilienceoftheBCCsampling(ofq-space)totheghostingartifactsthatdistortthereconstructionofP(r). 58

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CHAPTER4RECONSTRUCTIONFROMINTERLACEDSAMPLINGInChapter 3 ,wehaveshownthatforthesamesamplingpattern,usingtheBCClatticeinsteadofCartesianlatticeinthetomographicreconstructionframeworkcanincreasethereconstructionaccuracythroughsuppressingtheghostingphenomenon.Inthischapter,weareintroducinganinterlacedsamplingschemeasansuperioralternatetothestandardmulti-shellscheme.Instandardmulti-shellsamplingscheme,samplepointsareuniformlydistributedonseveralconcentricsphericalshellsintheqspace.Thisdistributiondoesnotchangethroughdifferentshellsandisdeterminedbytheverticesofcertainpolyhedron.Actually,thisisnotthemostefcientwayofdistributingsamplepoints.Weproposeaninterlacedschemewheresamplepointsareplacedonverticesofapairofdualpolyhedraontheconsecutiveshells.Specically,weuserhombictriacontahedrononoddshellsandicosidodecahedrononevenshells.Thisschemeincreasestheangulardiscriminationbecauseitindeedsamplesmoredirectionsthanthestandardscheme.Togetherwiththesamplingscheme,reconstructionalgorithmsusingthelatticedependentsincinterpolantsareintroduced.Samplesmeasuredininterlacedmulti-shellpatternarere-sampled(interpolated)ontothedenseregularlattices,CaretesianorBCC.ThediagramisshowninFigure 4-1 .Thesamplingschemeandreconstructionalgorithmswereevaluatedonbothsyntheticdatasetandratbraindatacollectedfroma600MHz(14.1Tesla)Brukerimagingspectrometer. 4.1InterlacedSamplingSchemeInthiswork,weproposeaninterlacedsamplingschemeasanalternativetothestandardmulti-shellsamplingscheme.TheideaofinterlacedlatticecomesfromthestructureoftheBCClatticeasitcanbeviewedasastackof2-DCartesianlayersofpointswhereeveryalternatelayerisshiftedbyhalfofthesamplingdistance(inthe2-Dplane).Inthemulti-shellsampling,theproposedinterlacedlatticehastheinterleaving 59

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Figure4-1. Diagramofthereconstructionprocess.Themulti-shelldiffusionsignalacquisitionschemecanbestandard(top)orinterlaced(bottom).ThedenselatticecanbeCartesianorBCC. (a)(b)Figure4-2. Comparisonof2D(a)standardand(b)interlacedsamplingschemesinthecaseofCTreconstruction 60

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Figure4-3. ThestructureofBCClatticeas2-DCartesianlayersofsamplesshiftedonalternateZslices. structureinthesphericalcoordinateswhereeveryalternateshellisshiftedbyhalfoftheangularresolution.Asimilarideain2-Dinterlacedsamplinghasbeenexploredforcomputedtomography[ 49 ]whichcomparedtheperformanceofaninterlacedsamplingschemewiththestandardschemeinthecaseof2-Dcomputedtomography(CT)reconstruction.Fortheirinterlacedscheme,thedetectorarraywasshiftedbyonehalfofadetectorspacingwhengoingfromoneprojectiondirectiontothenext.Withsomegoodexperimentalresults,thisworkconcludedthattheinterlacedschemeallowsalmosttwicetheresolutionofthestandardschemewiththesamenumberofsamplesin2-D.Figure 4-2 showsthesampledistributionsofbothschemes.Intheinterlacedscheme,samplesontheoddandevencirclesformtwodifferentpolygons.Thesetwopolygonsareapairofdualpolygonswheretheverticesofonecorrespondtotheedgesoftheother.Byextendingtheideaofdualpolygonstodualpolyhedra(theverticesofonecorrespondtothefacesoftheother),weproducedathree-dimensionalinterlacedsamplingscheme.Toensurethatsamplesareuniformlydistributedoneachsphericalshell,onecouldusethesamplingdirectionsdenedbytheverticesofthecommonly-usedicosahedron(Figure 4-4 (a))anditsdual,thedodecahedron(Figure 4-4 (b)),alternatelyforoddandevenshells.IcosahedronanddodecahedronarebothPlatonicsolids,i.e.convexregularpolyhedrons,whicharehighlysymmetrical,beingedge-transitive,vertex-transitiveandface-transitive.Thispropertymakesthemsuitablechoicesasthe 61

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(a)(b) (c)(d)Figure4-4. Shapeofpolyhedrausedinthisscheme.(a)Icosahedron,12vertices,30edges,20faces,(b)Dodecahedron,20vertices,30edges,12faces,(c)Rhombictriacontahedron,32vertices,60edges,30faces,(d)Icosidodecahedron,30vertices,60edges,32faces. 62

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generatorsofsamplingdirections.However,thedodecahedronhas20verticeswhiletheicosahedronhas12.Therefore,aninterlacedsamplingschemebasedonthesetwopolyhedrawouldhaveeithergreaterorfewersamplesthanastandardmulti-shellschemeofcomparablesamplesize.Thisimbalancemakesthecomparisonofthetwoapproachesdifcult.Forafairercomparison,weusedaninterlacedschemebasedonanotherpairofdualpolyhedra:therhombictriacontahedron(Figure 4-4 (c))andicosidodecahedron(Figure 4-4 (d))whohaveaboutthesamenumberofvertices(32and30).Thevertexdirectionsoftherhombictriacontahedronandicosidodecahedrondeterminedthesamplingdirectionsforoddandevenshells,respectively.Inthestandardscheme,allshellsweresampledusingthevertexdirectionsoftherhombictriacontahedron.Sincearhombictriacontahedronhas32verticesandtheicosidodecahedronhasonly30,theinterlacedschemeusesslightlyfewersamplesthanthestandardscheme.Intheexperimentssectionwewillseethat,evenwithfewersamples,theinterlacedschemeachievesbetterreconstructionaccuracy.Ifnecessary,moresamplingdirectionscanbeaddedbysubdividingtheedgesofbothpolyhedra. 4.2ReconstructionAlgorithmInSection 4.1 ,weproposedtheuseofaninterlacedmulti-shellsamplingschemetosamplethediffusionsignalinq-space.Thisisnotauniformsamplingscheme,sowecannotestimatethediffusionpropagatordirectlythroughFFT.OnesolutionistoestimatethevaluesofP(r)bynumericallycomputingtheintegralin( 1 ).However,theirregulardistributionofsamplingpositionsmakesitdifculttodesignanaccuratenumericalintegrationalgorithm.Anothersolutionistodenearegularlatticeinq-spaceandestimatethevaluesonthisregularlatticethroughinterpolation/extrapolation.Inotherwords,wecanresamplethediffusionsignalonaregularlatticeusingthenonuniformlysampledvalues. 63

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(a)(b)Figure4-5. (a)Interlacedmulti-shellsamplingdirectionsdenedonrhombictriacontahedron(purple)andicosidodecahedron(copper)inanalternativemanner.(b)Standardmulti-shellsamplingdirectionsdenedonrhombictriacontahedronoveralltheshells.Rhombictriacontahedronhas32vertices,60edgesand30faces.Icosidodecahedronhas30vertices,60edgesand32faces. Shannon'ssamplingtheoryprovidesareconstructionformulaforabandlimitedfunction,f,fromitssamplesonauniformlattice,L,whenthesamplingrateishigherthantheNyquistfrequencyoff: f(x)=Xxk2Lf(xk)sincL(x)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)(4)sincL(x)istheidealinterpolationfunctionthatdependsonthesamplinglatticeL.Inthe3-DCartesianlatticecase,sincL(x)isthetensorproductofthree1-Dsincfunctionsdenedas: sincL(x)=sin(x) xsin(y) ysin(z) z.(4)( 4 )canbecomputedfromtheinverseFouriertransformoftheindicatorfunctionoftheBrillouinzoneacubeforCartesianlattice. 64

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FortheBCClattice,thecorrespondingsincLiscomputed,similarly,fromtheinverseFouriertransformoftheindicatorfunctionofitsBrillouinzonearhombicdodecahedronasshowninFigure 2-4 .Thisfunctioncanbeefcientlyevaluatedusingageometricapproach[ 120 ].TheexplicitformulaforBCClattice'ssincfunctionisgivenby: sincL(x)=1 44Xk=1[cos(Tkx)Ym6=ksinc(Tmx)](4)where[1...4]=1 42666641)]TJ /F5 11.955 Tf 9.29 0 Td[(1)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(11)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 9.29 0 Td[(111377775. (4)See( 2 )in 2.3 fordetailsofthederivation.Shannon'sreconstructionformula(e.g.,( 4 ))involvesinnitelymanyterms,whichmakeitsevaluationimpractical.Inpractice,onlynitetermsinvolvinglatticepoints,xk,withinaboundingboxareconsiderednon-zero.Whileonlynitetermsareconsideredinthesummation,thelinearcombinationofsincLshiftedtothesenitelatticepointsprovidesaninnitely-supportedapproximationtof: ^f(x)=Xxk2L,1kKf(xk)sincL(x)]TJ /F3 11.955 Tf 11.96 0 Td[(xk).(4)TheinverseFouriertransformofthisapproximationhasacompactsupportwhichwillapproximateP(r)inoursetting.Inourcase,thesignalunderinvestigationisthediffusionsignalE(q)inq-space.WearegivenNsamplemeasurementsE(qn)onmultiplesphericalshells,depictedbycopperandpurpleverticesinFigure 4-5 (a)andFigure 4-5 (b).ThedesiredestimatesareKvaluesE(xk)onaregularlatticexk2L,depictedbygraydotsinFigure 4-5 (a)andFigure 4-5 (b)fortheCartesiancase.Thediffusionsignalcanbeapproximatedaccordingtoequation( 4 )as^E(q).Bymatching 65

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^E(q)tothemeasurementsE(qn)atlocationsqn,weget: E(qn)=Xxk2L,1kKE(xk)sincL(qn)]TJ /F3 11.955 Tf 11.96 0 Td[(xk),n=1,2,...,N(4)whereNisthetotalnumberofmeasurementsofE(q).ConsideringE(xk)asunknowns,( 4 )areNequationsinKunknownswhichcanbewrittenasalinearsystemAe=b,whereAn,k=sincL(qn)]TJ /F3 11.955 Tf 11.96 0 Td[(xk),ek=E(xk)andbn=E(qn).Foraccuratelattice-basedreconstruction,theregularlatticeshouldbedense,soweusuallyhaveK>N.KNmeansthatthenumberofq-spacemeasurementsishigherthanthelatticeresolutionthatwearereconstructingonto.SincetheacquisitiontimeisdeterminedbyN,ahighsamplingrate(e.g.,DSI)isimpractical.Therefore,weconsiderK>Nthatourreconstructionlatticemoredensecomparedtotheacquiredsamplingrate.Thismeansthat( 4 )isanunderdeterminedlinearsystemwhichcanbesolvedintheleast-squaressenseusingnormalequationsandtheconjugategradientmethod.Toexpeditetheprocess,weprovideaninitialestimateofE(xk)usinglinearinterpolationonaDelaunaytriangulationofasetofsamplepointsqn,n=1,2,...N.OnecanalsoobtainsolutionswithdifferentpropertiesthroughreplacingtheL2normwithothernorms.Forexample,usingL1normonsometransformcoefcientsofleadstosparsereconstructionsasdiscussedin[ 23 77 78 ].OncethesignalE(qn)hasbeentheestimatedonaregularlattice,wegetacontinuousrepresentationofE(q)as: E(q)=Xxk2L,1kKE(xk)sincL(q)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)(4)TakingFouriertransformon( 4 ),weget: P(r)=box(r)Xxk2L,1kKE(xk)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(2ixkr)(4)wherebox(r)istheFouriertransformofsincL(q)whichisanindicatorfunctionwhosesupportisacubeforCartesianlatticeandarhombicdodecahedronforBCClattice. 66

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AsdiscussedinSection 2.2.3 ,tocapturethesameamountofinformationaboutthefunctionbeingsampled,theBCClatticeonlyneedsabout70%ofthesamplepointsneededfortheCartesianlattice.ThispropertymakesthenecessaryKvaluefortheBCClattice30%smallerthantheKvaluefortheCartesianlattice.Forsolvingtheunderdeterminedlinearsystem,smallerKispreferred,becauseitmeansthatthelinearsystemhasahigherrank.Thisusuallytranslatestosmalleruncertaintyinthenalsolution.Ontheotherhand,foraxedKvalue,resamplingonaBCClatticecanrevealmoredetailsabouttherealsignalbyallowingP(r)reconstructedbyalargerradiuswhileavoidingtheghostingeffects.Inotherwords,resamplingonaBCClatticeisaneffectivewayofimprovingtheaccuracywithoutincreasingtherankofthelinearsystemtobesolved. 4.3ExperimentsInthissection,wepresentseveralexperimentalresultsonsyntheticaswellasrealdatasets.First,thesyntheticdataexamplesarepresentedfollowedbytherealdataexperiments. 4.3.1ExperimentsonsyntheticdataInordertoshowtheadvantagesofoursamplingschemeandlatticeselection,werstdidquantitativecomparisonusingsyntheticdata.WeusedamixtureoftwoorientedGaussianfunctionsinthe3-Ddisplacementspacetosimulatethediffusionprobabilityofabercrossing.ThetwoGaussianfunctionsaretherotatedversionsofanorientedGaussiandistributionfunctionwithzeromeananddiagonalcovariancematrixC=diagf20,20,400gwhichhasafractionalanisotropyvalueof0.95.ThesetworotatedGaussianfunctionscanbespeciedbyitscovariancematricesC1andC2.Thediffusionpropagator,P(r),anddiffusionsignal,E(q),canthenbedenedanalytically: P(r)=1 21 (2)3=2"exp)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F13 7.97 Tf 10.5 4.71 Td[(1 2rTC)]TJ /F13 7.97 Tf 6.59 0 Td[(11r p detC1+exp)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F13 7.97 Tf 10.49 4.71 Td[(1 2rTC)]TJ /F13 7.97 Tf 6.58 0 Td[(12r p detC2#(4) 67

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E(q)=1 2exp)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(22qTC1q+exp)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(22qTC2q(4)WexedoneoftheGaussiancomponentsandrotatetheothercomponenttoformcrossinganglesfrom20to60witha5stepsize.Asdescribedearlier,q-spacedatawassampledonmultipleshellsusingboththestandardandinterlacedschemes.Forthestandardscheme,thevertexdirectionsoftherhombictriacontahedrondenedthesamplingdirectionsforallshells.Fortheinterlacedscheme,diffusiondirectionsonevenshellsweredeterminedfromtheverticesoftheicosidodecahedron.Duetothesymmetryofthediffusionsignal,i.e.S(q)=S()]TJ /F3 11.955 Tf 9.3 0 Td[(q),onlyhalfofthesamplingdirectionsarenecessary.Thus,onlythosedirectionswithqz0werechosen.Thesamplingdirectionsaresummarized(inpolarcoordinates)inTable 4-1 .TheCartesiancoordinatescanbecalculatedasx=sin()cos(),y=sin()sin()andz=cos()Forthesyntheticexperiments,wepicked7differentshellradiiuniformlydistributedbetween0andqmax=0.5p 1=20.Sowehadsampleson6shellsplustheorigin.Thetotalnumberofsampleswas193forthestandardschemeand187fortheinterlacedscheme,symmetricdirectionsincluded.Thedatawereinterpolatedontotwoembeddinglattices:the151515CartesianlatticeandtheequivalentBCClattice,whichconsistsoftwointerlacedCartesianlatticesofsize111111and121212.Thenumbersoflatticepointswere3375forCartesianand3059forBCC.Thereconstructeddiffusionpropagators,P(r)atdifferentjjrjjvalues,obtainedusingdifferentsamplingschemesandembeddinglatticesareshownasFigure 4-6 .Wecanseethattheinterlacedsamplingschemehashigherangulardiscriminationthanthestandardscheme.Intheinterlacedsamplingcase,thetwo-bercrossinggeometryiscorrectlyreconstructedwhenthecrossingangleisgreaterthanorequalto35.Inthestandardcase,thecrossinggeometryisnotreconstructeduntil45.Also,thereconstructionsfrominterlacedschemearesharperandofhigherdelitywithrespect 68

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30405060 30405060 (a) (b) Figure4-6. ReconstructionofP(r)inthetwo-bercase,evaluatedat(a)jjrjj=15and(b)jjrjj=25,usingdifferentsamplingschemeandembeddinglattices.Ineachgure,wehave:rstrow:truevaluesofP(r),secondrow:reconstructionsusingstandardschemeandCartesianlattice,thirdrow:usingstandardschemeandBCClattice,forthrow:usinginterlacedschemeandCartesianlattice,fthrow:usinginterlacedschemeandBCClattice. tothetruegeometryofP(r).WhenwecomparethereconstructionsfromCartesianandBCCembeddinglattices,thelatteralwayshaslessdistortionbecauseusingaBCCembeddinglatticeleadstosmalleraliasingeffectsonP(r).Inadditiontothevisualcomparison,somenumericalcomparisonofthereconstructionerrorsisalsonecessary.Inourmulti-shell,model-freescenario,meansquareerror(MSE)ofthereconstructed^P(r)comparedwiththetrueP(r)isagoodchoicebecauseitindicatesbothradialandangularreconstructionaccuracy.AccordingtoPlanchereltheorem,itisequivalenttotheMSEofthereconstructed^E(q)inq-space.Thus,weusetheMSEof^E(q)ontheembeddinglatticepointsasourerrormeasurement.This 69

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Table4-1. SamplingdirectionsusedinourexperimentsforRhombicTriacontahedronandIcosidodecahedron.istheelevationandistheazimuth. RhombicTriacontahedronIcosidodecahedron 0058.282528837.377428858.28257237.37747231.717532437.3774031.71753663.434932458.2825063.43493631.717518063.434925258.282521663.434910858.282514479.187728831.717525279.18777231.717510863.4349180909037.3774216905437.37741449012679.187709016279.1877216901879.1877144 errorisfurtherdividedbythemeanvalueofthetrueE2(q)asanormalizationsothatitisnotdependentonthemagnitudeofthesignal.WerepeatedthereconstructionprocedurewithRiciannoiseofdifferentlevels,n.TheresultsareshowninFigure 4-7 .ItisobviousthattheinterlacedschemegiveslesserrorthanthestandardschemeandtheBCCembeddinglatticeevenfurtherreducestheerror.ThebenetofBCClatticeoverCartesianlatticeisrootedinitsabilitytocaptureextrahighfrequencyinformationwithoutintroducingaliasing.Butwhenthisinformationistoonoisy,thebenetsgraduallydisappear.Assuch,thereconstructionerrorsusingCartesianandBCClatticesapproacheachotherwhenthenoiselevelincreases.Wehaveshownthevisualandnumericalcomparisonofdifferentreconstructionschemeson2-bercrossingsyntheticdata.Toexploredifferentbercongurations,werepeatthecomparisonon3-bersyntheticdata.ThevisualcomparisonsareshowninFigure 4-9 .ThenumericalcomparisonsareshowninFigure 4-10 .Intheend, 70

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Figure4-7. PlotsshowingthenormalizedMSEofthereconstructionforsyntheticdataoftwo-bercrossings.Inthelegend,SCmeansstandardschemewithCartesianlattice.SBmeansstandardschemewithBCClattice.ICmeansinterlacedschemewithCartesianlattice.IBmeansinterlacedschemewithBCClattice. 71

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Figure4-8. PlotsshowingthenormalizedMSEofthereconstructionforsinglebersyntheticdataofFAvalues0and0.6.FAvalueequalsto0meansthesignalisisotropic. weprovidethenumericalerrorcomparisonsfor1-bersyntheticdatawithdifferentFAvaluesof0and0.6inFigure 4-8 .Itisstillobviousthatourproposedschemeoutperformsthestandardscheme. 4.3.2ExperimentsonrealdataTofurthertestourscheme,weacquiredrealMRIdatausingboththestandardandinterlacedmulti-shellsamplingschemesandcomparedthereconstructionresults.Allmagneticresonanceimagingwasperformedona600MHz(14.1Tesla)Brukerimagingspectrometer,usingaconventionaldiffusionweightedspinechopulsesequence.Thesamplingdirectionsarethesameasthoseusedinthesyntheticexperiments.Threemulti-shelldatasets,whichcaptureddifferentregionsofthemousebrainwereacquired:1)acoronalsetatthelevelofthecorpuscallosum,2)acoronalsetthroughthebrainstematthelevelofthesuperiorcolliculus,and3)asagittalsetthroughthemidline.Allthedatasetswerepre-processedwithnon-localmeanslteringtoreducethenoise[ 32 39 ]. 72

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30405060 30405060 (a) (b) Figure4-9. ReconstructionofP(r)inthethree-bercase,evaluatedat(a)jjrjj=10and(b)jjrjj=15.Twoxedbersareof120crossingandthethirdberisrotatingtoformdifferentangletotheverticalber.ThearrangementofthegureisthesameasFigure 4-6 Dataset#1wasacquiredwith:slicethickness=0.7mm,1.21.2cm2eld-of-view,128128datamatrix,and94min-planeresolution.Diffusionparametersincluded:diffusiontime,=15msec,diffusiongradientduration,=1msec,andb-valuesof500,1000,2000,3000,4000and5000s=mm2(kqk=29.1,41.1,58.1,71.2,82.2,and91.9mm)]TJ /F13 7.97 Tf 6.59 0 Td[(1). Dataset#2wasacquiredwith:slicethickness=0.3mm,1.21.2cm2eld-of-view,192192datamatrix,62.5min-planeresolution,=12msec,=1msec,andb-valuesof187,750,1687,and3000s=mm2.Thenonuniformspacingbetweenb-valueswaschosentoprovideroughlyequalspacingbetweenq-values(kqk=20.2,40.3,60.5,and80.7mm)]TJ /F13 7.97 Tf 6.59 0 Td[(1). Dataset#3wasacquiredwith:slicethickness=0.35mm,1.80.9cm2eld-of-view,256128datamatrix,and70.3min-planeresolution.Diffusionparameterswereidenticaltothoseusedfordataset#2. 73

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Figure4-10. PlotsshowingthenormalizedMSEofthereconstructionforsyntheticdataofthree-bercrossings. Thechoiceofmaximumb(orkqk)isacompromisebetweensignal-to-noiseratio(SNR)andresolutionofthediffusionpropagator,P(r).Intheexperimentaldata,theSNRofthehighestb-valueimagewas15,16and25fordatasets1,2,and3,respectively.SNRwascomputedfrommagnitudeimages,bydividingthemeanofthesignalovertheentireobjectbythestandarddeviationofthenoiseinanartifact-freebackgroundregion.Thevaluesprovidedrepresenttheminimumacrossallofthesampleddiffusiondirections. 74

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Figure4-11. Reconstructionresultsondataset#1evaluatedatkrk=8.0m.(a)S0imageofthesliceundertestwheretheregionofinterest(ROI)isshowninthebluebox.(b)ReconstructionusingstandardschemeandCartesianlattice.(c)ReconstructionusingtheproposedschemeandBCClattice.Zoom-inviewsofreconstructionsonseveralvoxelsusingstandardschemewithCartesianlattice(SC)(d),standardschemewithBCClattice(SB)(e),interlacedschemewithCartesianlattice(IC)(f),interlacedschemewithBCClattice(IB)(g). FurtherincreasesinkqkorbwereavoidedtolimittheRiciannoisebiaswithinthedata[ 38 ].ThiscomesattheexpenseofP(r)resolution,whichisequaltotheinverseofthemaximalq-value[ 28 ].Whileadditionalresolutionisdesirable,acquiringnoisydataathighq-(orb-)valuesmaynotrepresentanefcientuseoftheavailableimagingtime.Instead,moretimewasspentonhigherSNRacquisitionsatintermediateb-values.Thisallowsfornersamplingofthediffusionsignal,E(q),whichreducesaliasingdistortioninP(r).Thetotalimagingtimefortheexperimentaldataacquiredwiththestandardsamplingschemewasapproximately34,35and28hoursfordatasets1,2,and3respectively.Theinterlacedschemeresultedinroughlya1hourdecreaseintotalscantime. 75

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Figure4-12. Reconstructionresultsondataset#2evaluatedatkrk=9.8m.(a)S0imageofthesliceundertestwheretheROIisshowninthebluebox.(b)ReconstructeddiffusionpropagatorusingSCscheme.(c)ReconstructionusingtheIBscheme.Zoom-inviewsofreconstructionsonseveralvoxelsusingSCscheme(d),SBscheme(e),ICscheme(f),IBscheme(g). Figure 4-11 showsthereconstructionresultsofdataset#1.Theboxedregionofinterestcontainsintersectingberbundlesfromcingulumandcorpuscallosum.OurproposedschemeprovidessharperreconstructionsofthebercrossingsintheidentiedROI. 76

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Figure 4-12 showsthereconstructionresultsofdataset#2.Wepickedthisregionofinterestbecauseithasbeenvalidatedthatthereareplentyofin-planecrossingbersinthisregion[ 68 ].Theresultsshowthatourproposedschemecanrecoverthecrossingsmoreaccurately.Figure 4-13 showsthereconstructionresultsofdataset#3.Thebranchingstructureinthehighlightedregionofinterestisobvious.Ourproposedschemeprovidessharperreconstructionsofthosebercrossings.Thecomparisonsofthereconstructeddiffusionpropagatorsusing4differentsettings,standardorinterlacedschemewithCartesianorBCClattice,areshowninFigure 4-11 ,Figure 4-12 andFigure 4-13 as(d),(e),(f)and(g).TheinterlacedschemewithBCClatticeistheclearwinnerforitsabilitytoaccuratelyreconstructsharpcrossings. 77

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Figure4-13. Reconstructionresultsondataset#3evaluatedatkrk=10.0m.(a)S0imageofthesliceundertestwheretheROIisshowninthebluebox.(b)ReconstructionusingSCscheme.(c)ReconstructionusingIBscheme.Zoom-inviewsofreconstructionsonseveralvoxelsusingSCscheme(d),SBscheme(e),ICscheme(f),IBscheme(g). 78

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CHAPTER5BOXSPLINEBASEDRECONSTRUCTIONMETHODInChapter 3 andChapter 4 ,wediscussedthesamplinggeometriesofthediffusionsignalandtheapproachestore-samplingthediffusionsignalonregularlattices,CartesianandBCC,intheqspace.Inthischapter,wedevelopanindependentapproachtotomographicreconstructionofdiffusionpropagatorswhicharerepresentedinboxsplinebasis.Thediffusionpropagatorsaredirectlyreconstructedonapre-denedregularlatticeinthedisplacementspace.Inthisapproach,thediffusionpropagatorsareanalyticallyrelatedtothesamplesofdiffusionsignalsintheqspacethroughRadontransformandFourierslicetheorem.Thus,thereisnoneedtore-samplediffusionsignalsonaqspacelatticeortodoexplicitFouriertransform.However,itcanstillbenetfromtheinterlacedsamplinggeometrydiscussedinChapter 4 .ThekeypropertyofboxsplinesthatmakesthemparticularlysuitableforthetomographicreconstructionisthattheyareclosedunderX-RayandRadontransforms[ 45 ].Inotherwords,asignalwhichisrepresentedinboxsplinebasiscanberepresentedexactly(i.e.,withnoapproximationordiscretizationoftheforwardmodel)inthesinogramspace.Therefore,fromtheX-RayorRadondataonecanformulatethetomographicreconstructionprocessexactly.TheboxsplineapproachoffersanexactinversionofX-RayorRadontransform,similartoFilteredBackProjectionalgorithm;however,theboxsplineapproachachievestheexactinversionprocesswithniteamountofcomputationanddata,incontrasttotheFBPsolution.Fromtheapproximation-theoreticpointofview,theboxsplineapproachcanbeconsideredasageneralizationofthe(square)pixelbasisapproachthatallowsforbasisfunctionswithhigherapproximationorder(thanthepixelbasis).Theincreaseinapproximationorderofferedbythecompactlysupportedboxsplinescanleadtosignicantsavingsinthecomputationalcostofreconstruction. 79

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Figure5-1. Diagramofthereconstructionprocess.MeasurementsofE(q)areonq-spaceshells.P(r)isrepresentedwithshiftedsumofboxsplinessittingonaregularlattice,Cartesianinthiscase.Thereconstructionproblemturnsouttobetheleastsquarettingtotheprojecteddata(sinogramdata). Wepresentsyntheticandrealmulti-shelldiffusion-weightedMRdataexperimentsthatdemonstratetheincreasedaccuracyofP(r)reconstructionastheorderofbasisfunctionsisincreased. 5.1BoxSplinesandRadonTransformAboxsplineisasmoothpiecewisepolynomial,compactly-supported,function(denedonR2,R3orgenerallyinRd),thatisassociatedwithasetofvectorsthatareusuallygatheredinamatrix:=[1...N][ 34 ].Fromthesignalprocessingpointofview,boxsplinesareconstructedbyrepeatedconvolutionofelementaryline-segmentdistributionsalongeachvectorin.Specically,wehave:M(x)=)]TJ /F6 11.955 Tf 5.47 -9.68 Td[(M1MN(x), (5)wheretheelementaryboxsplines,Mn,areDirac-likelinedistributionssupportedoverx=tnwitht2[0,1].Theseelementaryboxsplinesareindirectgeometric 80

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correspondence(viaarotationandaproperscaling)withtheprimaryboxsplineM~e1(x)=box(x1)(x2,,xd) (5)where(x2,,xd)isthe(d)]TJ /F5 11.955 Tf 11.95 0 Td[(1)-dimensionalDiracdistributionandbox(x)=8>><>>:10x10otherwise.Moreover,theyintegrateto1whichisapropertythatissharedbyallboxsplines(andalsopreservedthroughconvolution).Basedon( 5 ),onedirectlyinfersthattheboxsplinesarepositive,compactly-supportedfunctions.Theirsupportisazonotope,whichistheMinkowskisumofNvectorsin.Forinstance,ina3-Dsetting,apixelbasis(i.e.,voxelbasis)canberepresentedbyaboxsplinewhosedirectionmatrix,=I3isthe33identitymatrix.Moregenerally,annth-ordertensor-productB-splinecanberepresentedasaboxsplinewhosedirectionmatrixcontainsthedirectionsinI3,eachofwhichisrepeatedbyn-times.ThekeypropertyofboxsplinesthatisusedinourtomographicapproachisthattheRadontransformofaboxsplinealongaparticulardirectionspeciedby(,)isaunivariate(1-D)boxsplinewhosedirectionvectorsarethegeometricprojectionoftheoriginalboxsplinedirections[ 45 ].LetP(,)denotetheprojectionmatrixthatgeometricallyprojectsapointR3tothelinespeciedbythedirection(,).Then,theRadontransformofatrivariateboxsplineassociatedwithamatrixisa1-Dboxsplinespeciedby'=P(,)(seeFigure 5-2 ).Thispropertysuggeststhatfortomographicreconstructionapplications,boxsplinesaresuitablebasisfunctionsforrepresentingthesourcesignal.Thischoiceofrepresentationofthesourcesignal,leadstoanexactforward-modelthatcanbeusedtomatchthesinogramdata.Thispropertyisexploitedinthefollowingsectionforourreconstructionalgorithm. 81

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Figure5-2. A2-DillustrationofRadontransformgeometry.Ontheright:Radontransformofapixel(tensor-productrst-orderB-spline)leadstoa1-Dboxsplinewhosedirectionsarethegeometricprojectionofthedirectionsofthesourceboxspline.Thispropertydirectlygeneralizesto3-Dforvoxelorhigher-orderbasis. 5.2DetailedAlgorithmInpractice,diffusionpropagatorP(r)isreconstructedfromsamplesofdiffusionsignalE(q).Theq-spacesamplesusuallylieonradiallinesthroughtheoriginofdifferentorientations.AccordingtotheFouriertransformrelationshipin( 1 )andtheFourierslicetheorem,the1-DinverseFouriertransformoftherestrictionofE(q)toaradiallineequalstotheRadontransformofP(r)alonglineinrspaceofthesameorientation.Hence,thereconstructionproblemisequivalenttoinvertingtheRadontransformthattranslatesintoreconstructingtheP(r)fromitsprojections,(i.e.,Radondata).Thisproblemproblemhaswidelybeenstudiedinthetomographicreconstructionliterature.Ourcontributionisthatwegeneralizethenaturalpixelbasis,whichistherstordertensorproductB-spline,tobasisfunctionsofhighapproximationpowerusingthe 82

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frameworkofnon-separableboxsplines.Themoregeneralclassofboxsplinesincludehigher-ordertensor-productB-splinesasaspecialcase,butalsoincludenon-separablebasisfunctions.ThisgeneralizationsignicantlysimpliestheinversionofRadontransformsincethemoregeneralclassofboxsplineshappentobeclosedunderX-RayandRadontransform.WeassumethatP(r)canberepresentedinboxsplinebasis: P(r)=cM=Xk2Z3ckM(r)]TJ /F3 11.955 Tf 11.96 0 Td[(k).(5)Whenisthe33identitymatrix,theboxsplinecorrespondstothecubevoxelbasisandwehaveapiecewise-constantapproximationofP(r).However,ahigherordertensor-productB-splineorgenerallyanon-separableboxsplinecanbeusedintheaboveapproximationtoachieveahigher-orderapproximationtothetrueP(r).Thereconstructionproblemisnowthetaskofndingsuitablecoefcientsckin( 5 )suchthatP(r)agreeswiththegivendata(i.e.,samplesofdiffusionsignalE(q)).Accordingtothediscussionin 5.1 ,theRadontransformofP(r)foragivendirection(,)is: P(,)(r)=Xk2Z3ckM')]TJ /F6 11.955 Tf 5.48 -9.69 Td[(r)]TJ /F3 11.955 Tf 11.96 0 Td[(P(,)k'=P(,)(5)whereP(,)isthe3-Dto1-Dprojectionmatrixontothedirection(,).NotethattheCartesiangridshiftsk2Z3,arealsotransformedbyP(,).Inotherwords,theRadon-transformdatacanberepresentedbyP(,)kshiftsofthe1-Dboxsplinewhichisassociatedwiththematrix'=P(,).Denotingthesamplesofrealprojecteddataasd(,)(r)andenforced(,)(r)=P(,)(r)atallthesamplepoints,wecanbuildalinearequationsetexpressedasAc=dwherecisthevectorcomposedofcoefcientsckin( 5 ),disthevectorofsampleddataandeachrowofAistheevaluationsofMP(,)(r)ateachsampleposition.Inourapplication,thenumberofsamplesisusuallysmallerthanthenumberofshiftedbox 83

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(a)(b)Figure5-3. ThedesiredisosurfaceofP(r):(a)of90crossing(b)of60crossing splines,sothelinearsystemisunder-determined.Inordertogetreasonablesolutionofcoefcients,weintroduceasmoothnessregularizationandsearchfortheleastsquaresolution.Hence,wesolvethefollowingoptimizationproblem: mincjjAc)]TJ /F3 11.955 Tf 11.96 0 Td[(djj2+jjLcjj2(5)whereListhematrixof3-DdiscreteLaplacianoperatoronc,istheweightsforthesmoothnessregularizationtermjjLcjj2. 5.3ExperimentalResultsoftheAlgorithmWerstevaluatedouralgorithmusingsyntheticdata.WegeneratedsamplesfromamixtureoftwoGaussianfunctionsin3-Dq-spacesimulatingthediffusionsignaloftwo-bercrossings.Crossinganglesof90and60asshowninFigure 5-3 weretested.Thedatasamplesareuniformlydistributedonradiallinesspeciedbysphericalcoordinates(r,,)with0r35.Wepicked19pointsoneachradiallineand81directionsthatcorrespondtoverticesofasubdividedicosahedronapproximatingaunithemisphere.Thebasisfunctionofchoiceweretensor-productB-splinesshiftedona252525Cartesianlattice. 84

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(a)(b)(c)(d)Figure5-4. Reconstructionresultsforthesyntheticdata.Firstrow,90crossing.Secondrow,60crossing.(a)rstorderboxspline,(b)secondorderboxspline,(c)thirdorderboxspline,(d)fourthorderboxspline. Table5-1. MSEinpercentageofthereconstructionsforthesyntheticdata.isthecrossingangle.nisthenoiselevel.nistheorderoftheboxspline. nn=1n=2n=3n=4 3*90016.037.023.342.80.0419.16.7910.17.255.25.154.53.14.0827.871.3119.88.6211.34.4610.38.313*60011.677.423.572.95.0415.05.5410.45.265.37.164.66.14.0822.97.8519.99.8011.38.3310.56.39 Figure 5-4 showtheisosurfacesofthereconstructedP(r)withdifferentB-splinebasisorders.Weobservethatthereconstructionbecomessmootherandmoreaccurateastheorderofthebasisfunctionincreases.Increasingtheapproximationorderabovethethirdorderleadstolesssignicantimprovementastheapproximationerrorbecomesverysmallforthisparticulardataset.NumericalcomparisonswithdifferentnoiselevelnareshowninTable 5-1 .Themeansquareerror(MSE)normalizedbytheenergyofthedesiredsignalwasusedasthemeasurement.Thenumericalresultsalsosupporttheexpectedadvantagesofhigherorderbasisfunctions.Wealsotestedouralgorithmonrealmulti-shelldataconsistingofmid-sagittalmousebrainscansacquiredatdifferentbvalues.16orientationsand5differentb 85

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Figure5-5. Reconstructionresultsforaregionofthemulti-shellmousebraindatawithdifferentboxsplinebasisorders. values,includingb=0,wereused.Allmagneticresonanceimagingwasperformedona600MHz(14.1Tesla)Brukerimagingspectrometer,usingaconventionaldiffusionweightedspinechopulsesequence.Thedatasetwasacquiredwith:slicethickness=0.35mm,1.80.9cm2eld-of-view,256128datamatrix,and70.3mmin-planeresolution,=12msec,=1msec,andb-valuesof187,750,1687,and3000s=mm2.Thenonuniformspacingbetweenb-valueswaschosentoprovideroughlyequalspacingbetweenq-values(kqk=20.2,40.3,60.5,and80.7mm)]TJ /F13 7.97 Tf 6.58 0 Td[(1). 86

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Figure 5-5 showsthereconstructedP(r)forasmallregionoftherealdatasetwithdifferentboxsplineorders.Wecanobservethesmoothnessimprovementgoingfromtherstorderonetothethirdorder.Fortherstorderboxsplinebasis(i.e.,voxelbasis),thereconstructedP(r)isdiscontinuous;hence,theP(r)values(evaluatedonthesphereforvisualization)appearspikyasshowninFigure 5-5 Order1. 87

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CHAPTER6RECONSTRUCTIONOFADIFFUSIONPROPAGATORFIELDAllthediscussionbeforewasfocusedonthereconstructionofasinglediffusionpropagatoratonevoxel.Inreality,thediffusionMRIsignalisalmostalwayscollectedasavolumetricdatacontaininglotsofvoxelsinthe3Dspace.Sothedesiredreconstructionresultisadiffusionpropagatoreld.Byassumingindependenceamongdifferentvoxels,wecanreconstructthewholeeldthroughapplyingtheaforementionedmethodsontoallthevoxelsonebyone.However,thereconstructioncanbemuchmoreresilienttonoisebyconsideringthecoherenceamongthediffusionprolesoftheadjacentvoxels.Inthischapter,Wepresentadictionarylearningframeworkforachievingasmoothdiffusionpropagatorreconstructionacrosstheeldwherein,thedictionaryatomsarelearnedfromthedataviaaninitialregressionusingadaptivesplinekernels.TheformulationinvolvesoptimizingforasparsedictionaryusingaK-SVDbasedupdatingandanon-localmeansbasedregularizationacrosstheeld.ThenoveltyliesinadictionarybasedreconstructionaswellasanNLM-basedregularizationthathelpspreservingfeaturesinthereconstructedeld.Wedocumentexperimentalresultsonsyntheticdatafromcrossingbersandrealopticchiasmdatasetthatdemonstratetheadvantagesoftheproposedapproach. 6.1Overview:FromFixedBasistoDataDrivenDictionaryTheDW-MRIdatasetsareusuallyprovidedasaeldofS(q,x)wherexspeciesspatiallocations.MostoftheexistingreconstructionmethodsreconstructP(r,x)ateachlocationxindependentlywhichdonotconsiderthespatialcoherencythatinherentlyexistsinthedata.Inarecentstudy[ 80 ],aspatiallyregularizedreconstructionapproachwasdevelopedthatexploitssparserepresentationofP(r)insphericalridgeletbasis.Inthisframework,sphericalridgelettransformwasappliedateveryvoxelonS(q,x)toobtainasparserepresentation.Thenthesparsityoftransform-domaincoefcientsaswellasthetotalvariationofthereconstructedsignaleldS(q,x),withrespecttoxwere 88

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usedtoformulatethereconstructionasthefollowingoptimizationproblem: minC1 2jjAC)]TJ /F3 11.955 Tf 11.95 0 Td[(Sjj2F+Xijjcijj1+TVfACg.(6)Inthisformulation,Aisthesphericalridgelettransformmatrix,Ccontainsthetransformcoefcientsatallvoxelsintheeldofinterestwhereeachcolumn,ci,isonesetoftransformcoefcientsatithvoxel.ACrepresentsthereconstructeddiffusionsignaleldandTVfACgisthetotalvariationofthereconstructedeld.Thethirdtermenforcescorrelationsamongcifromneighbouringvoxelswhichmakesthesolutionofthecoefcienteldspatiallyregularized.ThenalstepistocalculatethediffusionpropagatorfromthereconstructionsasP=fAgCwherefAgdenotestheFouriertransformofthebasisfunctions(i.e.,sphericalridgelets).Aintheaboveformulationisxedtobethesphericalridgeletbasestruncateduptoacertaindegree.Incontrastweintroduceadictionary-basedmethodwherewelearnthebasisfunctionsinAfromdataexamplesthatwillprovideasparserepresentation.ThroughupdatingbothAandCduringtheoptimizationprocess,weobtainadictionarylearnedfromthespecicdatasetaswellasthecorrespondingsparserepresentationofthereconstruction.Thegloballydeneddictionaryplaysanimplicitroleofregularizingthereconstructionsoverdifferentvoxels.Wealsointroducethesphericaldeconvolutionmodelwithadaptivekernelstocontrolthewaythedictionarygetsupdated.Inaddition,weuseanNLM-basedregularizationwhichfurthersuppressesthenoise.WewillbrieyintroducetheadaptivekernelsinSection 6.2 andgivethedictionarylearningframeworkinSection 6.3 .Section 6.4 willshowtheexperimentresults. 6.2AdaptiveKernelsforMulti-berReconstructionGivenadiffusion-weightedMRdataset,therearemanymethodsemployingdifferentsphericaldeconvolutionkernelstoreconstructthemulti-berdiffusionprole.Inthesphericaldeconvolutionframework,theDW-MRIsignalisconsideredastheconvolution 89

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ofakernelfunctionkwithaprobabilitydensityfunctionfoverthesphere[ 56 ]: E(b,g)=Zf(p)k(b,gjp)dp(6)wherebisthediffusionweighting,jjgjj=1andqp bg.Theintegrationisoverthedomainofparameterp.Manywellknownreconstructiontechniquesturnouttobethespecialcasesof( 6 )bypickingcertainkernelfunctionsk(b,gjp)andmixingdensitiesf(p)[ 56 ].Forexample,kcanbemultivariateGaussianfunctionk(b,gjD)=exp()]TJ /F6 11.955 Tf 9.3 0 Td[(bgTDg)[ 104 ].Thechoicesofsuchx-shapedkernelsareusedtorepresentthediffusionpropertyoftheunderlyingbers;howevertheyalsoimposesomeunnecessaryassumptionswhichmaynotholdfortherealDW-MRIdataset.Adaptivesplinekernels[ 11 ]wereproposedasaexiblekernelmodeltotdatasetswithvarietiesofdifferentdiffusionpatterns.Thedensityfunctionfisre-parametrizedonthesphereasf(p)=PNj=1wj(pjvj)wherev1,...,vNisasetofunitvectorsuniformlydistributedonthehemisphere.BylettingthereconstructionkernelK(b,gjvj)=R(pjvj)k(b,gjp)dpandconsideringthecaseofaconstantb-value(whichisquitecommoninHARDIacquisition),wehave: E(g)=NXj=1wjK(gjvj)=NXj=1wjPXk=1ck k(jgvjj),(6)whereKisrepresentedinsplinebases k.Theshapeofthekernelisexibleandisdeterminedbythecontrolpointsck.WhenwepluganumberofmeasurementsS(gi),i=1,...,Mandthecorrespondinggradientdirectionsgiinto( 6 ),weobtainMlinearequationswithrespecttounknownswjandck.wjandckarethenestimatedthroughnon-negativeleastsquarettinginanalternativepattern.Inotherwords,wjisestimatedwhileckarexedandthenck'sareestimatedwhilewjarexed.See[ 11 ]fordetailsofthesearchalgorithm. 90

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Finally,thediffusionpropagatoriscalculatedbytheestimatedwjandckparametersandbyapplyingtheFouriertransformonbothsidesof( 6 ).Since kareknownsplinebases,theirFouriertransformscanbecalculatedbeforehand. 6.3DictionarybasedReconstructionFrameworkIntheafore-mentionedsphericaldeconvolutionframework,diffusionsignalismodelledasweightedsumofkernelfunctionseachofwhichrepresentsthediffusionpropertiesofasingleber.Thenumberofparameters,ck,inkernelfunctionsandtheweights,wj,isusuallyverylargewhichmakesthereconstructionproblemill-posed.Fortunately,thefactthatthenumberofbersateachvoxelislimitedinrealdatasetsallowsustoexploitthesparsityofweightingcoefcientstosolvethereconstructionproblem.Theproblemofsearchingfortheproperkernelfunctionaswellasthesparseweightingcoefcientscanbesolvedbyadictionarylearningparadigm.Foragivenvoxelv,wedenetheweightsasavectorwvwhosejthelementisdenotedbywj,themeasurementsasvectorevwhoseithelementisdenotedbyE(gi),andKvwhichdenotesthekernelKv(i,j)=K(gijvj).Withthesenotations( 6 )becomesev=KvwvwhereKisexiblebychangingcoefcientsck.Modellingsignalwithinasinglevoxelcanbeinterpretedasadictionarylearningproblemwithoneobservationanddictionaryatomsformedbyoursplines.Insteadofttingtheadaptivekerneltothesignalateachvoxelindividually,hereweuseanovercompletedictionaryKMD,D>Nforallthevoxels.WedeneE=[e1,...,eV]asthemeasurementsfromallthevoxelsintheeldofinterest,W=[w1,...,wv]asthecorrespondingweightsrespecttothenewglobaldictionaryK.Now,wecanformulatethemodellingofthewholeeldastheoptimizationproblem: minK,WjjE)]TJ /F3 11.955 Tf 11.95 0 Td[(KWjj2Fs.t.8v,jjwvjj0T0(6)whereT0specieshowmanynon-zeroweightsareallowed.Therearemanyalgorithmswhichsolvethisdictionarylearningproblem,wepicktheK-SVDalgorithm[ 3 ]because 91

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ofitssimplicityandefciency.TheglobaldictionaryKimplicitlybringsinsomespatialregularizationsuchthatthereconstructedeldKWdoesnotchangeindependentlyateachvoxel.ThedictionarysizeDandsparsityconstraintT0indirectlycontrolthesmoothnessofthereconstructions.Inordertofurthersuppressthenoiseduringthereconstruction,weintroduceanexplicitregularizationterm: minK,WjjE)]TJ /F3 11.955 Tf 11.96 0 Td[(KWjj2F+MfKWgs.t.8v,jjwvjj0T0(6)whereMfKWgisaroughnessmeasureofthereconstructionKW.In( 6 ),Missettobethetotalvariation.Inthischapter,weproposetousethedeviationofKWfromitsnon-localmeansNLfKWg[ 18 ],i.e.MfKWg=jjKW)]TJ /F6 11.955 Tf 11.95 0 Td[(NLfKWgjj2F.Adirectsolutiontothisproblemisdifcultbecauseofthecomplicatedregularizationterm.SowesolveititerativelythroughassumingthatthecomponentNLfKWghasxedvalueestimatedfromlastiteration.Then,ateachiterationt,wearesolvingthefollowingoptimizationproblem: fKt,Wtg=argminK,WjjE)]TJ /F3 11.955 Tf 11.96 0 Td[(KWjj2F+jjKW)]TJ /F6 11.955 Tf 11.96 0 Td[(NLfKt)]TJ /F13 7.97 Tf 6.59 0 Td[(1Wt)]TJ /F13 7.97 Tf 6.59 0 Td[(1gjj2Fs.t.8v,jjwvjj0T0(6)whichisequivalentto: fKt,Wtg=argminK,WjjEt)]TJ /F13 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(KWjj2Fs.t.8v,jjwvjj0T0(6)whereEt)]TJ /F13 7.97 Tf 6.59 0 Td[(1=1 1+(E+NLfKt)]TJ /F13 7.97 Tf 6.58 0 Td[(1Wt)]TJ /F13 7.97 Tf 6.59 0 Td[(1g)andE0=E.Then,theproblemcanbesolvedthroughiterativelyapplyingK-SVDalgorithm.ToinitializethedictionaryK,weindividuallyttheadaptivesplinekerneltoeveryvoxelintheeldandpickDofthekernelswiththelargestweightsacrosstheeld.Andateachiterationt,werettheadaptivekerneltoKttoensurethattheatomsofourdictionarycanstillbeexpressedin( 6 ). 92

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6.4ResultsInthissection,weevaluateourproposedreconstructionframeworkbycomparingtothereconstructionofP(r)onindividualvoxels,andourregularizedframework.Theexperimentsdemonstratetheadvantagesoftheregularizationfromthelearnedglobaldictionaryaswellasthenon-localsmoothnessregularizer. 6.4.1SyntheticDatasetWerstevaluatethereconstructionperformancewithoursyntheticdataeldusingthesimulationmodelproposedin[ 98 ]withcylindricalberradiusof5m,length5mmanddiffusionweightingb=1500s=mm2.Allthedataweresimulatedusing81gradientdirectionswithdifferentnoiselevelchangingfrom0to0.3.Wecomparedourresultsagainstthevoxel-wiseindividualreconstruction,voxel-wisereconstructiononthesmootheddataeldusingthenon-localmeandenoisingalgorithmandthereconstructionusingdictionarylearningwithoutsmoothnessregularization.Fortheindividualadaptivekernel,wepickedN=321,P=5, ktobea3rd-orderB-splinebasisfunction.Forourdictionarylearningframework,wesetdictionarysizeD=100,T0=4,=2and=10wherecanbeestimatedinrealdataset.ForNLM,wesettheradiusoflocalpatchtobe3,theradiusofneighbourhoodsearchwindowtobe5.Inourexperiments,weobservedtheadvantagesofourmethodisnotsensitivetochoicesofand.Thereconstructeddiffusionpropagatorelds(=0.2)areshowninFigure 6-1 .WealsoshowtheerrorcomparisonamongthesemethodsunderdifferentnoiselevelinFigure 6-2 (a)andtheconvergenceplotinFigure 6-2 (b).Theresultsshowthattheproposedmethodismuchmoreaccuratewhenthenoiselevelishighsincethevoxel-wisereconstructionmethoddoesnottakeadvantageofthesmoothglobalberstructureandismorevulnerabletothenoise.Wealsoseethatthedictionarylearningmethoditselfwithoutsmoothnessregularizationcanprovidesomedegreesofresiliencetothenoise.Thisisbecausethedictionarylearnedissupposedtorepresenttheconsistentcomponentsoverthewholedatavolume. 93

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(a)(b) (c)(d)Figure6-1. Reconstructedeld(=0.2)using(a)voxel-wiseindividualreconstruction,(b)voxel-wisereconstructiononNLMdenoiseddataeld,(c)reconstructionusingdictionarylearningwithoutsmoothnessregularization,(d)theproposedmethod. 94

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(a)(b)Figure6-2. (a)Comparisonoftheangularreconstructionerrorswithdifferentnoiselevel.(b)Plotshowingtheconvergenceofthecostfunctionvaluesoveriterations. Figure6-3. S0imageoftheopticchiasmandtheROI. 6.4.2RealDatasetWealsoperformedanevaluationoftheproposedmethodwithrealdatafromaratopticchiasm,whichcontainssamplesmeasuredwith46differentdirectionswithb-valuearound1240s=mm2.TheS0imageaswellastheregionofinterest,markedwithabluebox,areshowninFigure 6-3 .ThereconstructedP(r)eldattheregionofinterestisshowninFigure 6-4 .Weobservethattheproposedmethodgeneratesasmoothreconstructionwhilekeepingtheunderlyingberstructure.Theparametersettingisthe 95

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(a)(b)Figure6-4. Reconstructedregionofinterestusing(a)voxel-wiseindividualreconstruction,(b)theproposedmethod.Thezoomedinviewsshowthattheresultfromtheproposedmethodismoreregularizedandthestructuresarepreserved. sameasthoseusedinthesyntheticexperimentsexceptthatwepickedadifferentvalueofT0=12andestimatedthevalueof=0.17fromthehomogeneous(noisy)areasoftheimage. 96

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CHAPTER7CONCLUSIONSInthisdissertation,anovelsamplingschemetogetherwithseveralreconstructionmethodsforreconstructingdiffusionpropagatorswerepresented.Topursuethegoalofmoreaccuratereconstructionfromlesssamplepoints,conceptsofoptimalsamplingtheory,idealinterpolatingsincfunction,boxsplinebasisanddictionarylearningwerebroughtintothereconstructionframework.AnewinterlacedsamplingschemetogetherwithseveralreconstructionmethodswereproposedandevaluatedwithbothsyntheticandrealdiffusionMRIdata.Chapter 2 addressedtheoptimalsamplingandinterpolatingprobleminmultivariatecasesandprovidedthetheoreticalfoundationforthereconstructionmethodsintroducedinChapter 3 andChapter 4 .Chapter 3 rstintroducedtheuseofoptimalsamplinglatticeinthetomographicreconstructionofthediffusionpropagatorsinDW-MRI.Thebenet,increasedaccuracy,ofreplacingthetraditionalCartesianlatticewiththeBCClatticeduringinterpolationprocedurewasvalidated.Then,theinterlacedsamplinggeometry,whichsharestheinterleavingstructureofBCClattice,wasbroughtintotheframeworkinadditiontotheoptimalinterpolationlattice.Sincethesamplinggeometriesandtheinterpolationlatticesareindependent,wecouldpickeitherinterlacedornon-interlacedgeometryandCartesianorBCClatticetoformthenalreconstruction(totalof4combinations).Ourexperimentsshowedtheimprovementsofthereconstructionaccuracy,withnomoresamples,frombothofthetechniques.Anotherdirectionthattheproposedframeworkcanbeappliedisforthereductionofacquisitiontime.Comparedtothestandard(non-interlaced)dataacquisitionscheme,wherewecanachievethesamereconstructionaccuracy,withfewersamples.Chapter 5 isnotadirectextensionofChapter 3 andChapter 4 .Itisalsorootedfromthetomographicreconstructionframework,butthefocusmovedfromthesamplinglatticestohigherorderboxsplinebasesfortherepresentationofP(r).Thesimple 97

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voxel(cube)basisachievesarstorderapproximationinthecontextoftomographicreconstruction.Theboxsplineframeworkallowsonetoemployhigher-orderbasisfunctionsthatsignicantlyincreasetheaccuracyofreconstruction.Chapter 6 extendsthepixel-wisereconstructionframeworkontothescenarioofestimatingthewholediffusionpropagatoreldatthesametime.Anon-localregularizeddictionarylearningframeworkwaspresentedtoimprovethereconstruction.Throughlearningadictionaryfromthegivendatavolumeandenforcingthenon-localregularization,thisapproachgeneratedreconstructionsatvoxelsthatarerobusttonoiseandpreserveberstructuresatthesametime.Differentconceptswerebroughttogetherintothediffusionpropagatorreconstructionproblem,addressingdifferentaspectsoftheprocess.Combiningtheseideaswillgeneratesurprising,newideaswhichhavenotbeenfullyexploitedyet.Forexample,adirectgeneralizationistoadapttheinterlacedgeometryandoptimalsamplinglatticesontotheboxsplineframework.DifferentboxsplineshavealreadybeendenedonBCCandFCClattices[ 43 61 ],soitwouldbeinterestingtoseetheeffectsofcombiningboxsplinebasis,BCClatticeandinterlacedsampling.Moreover,bringingthespatialinformationfromthediffusionpropagatoreldintothetomographicreconstructionframeworkandusingdictionarylearningtechniquestoguidethesamplingprocessareanothertwointerestingdirectionstofollow. 98

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BIOGRAPHICALSKETCH WenxingYereceivedhisB.E.degreefromtheDepartmentofAutomationatUniversityofScienceandTechnologyofChina,Hefei,Chinain2004.ThenhewasenrolledinthegraduateprogramatGraduateUniversityoftheChineseAcademyofSciences,associatedtotheInstituteofautomation.In2006,hechosetopursuehisPh.D.degreeintheDepartmentofElectricalandComputerEngineeringattheUniversityofFlorida.HehasbeenworkingintheMultimediaCommunicationsandNetworkingLabandjointthelaboratoryforComputerVision,Graphics,andMedicalImagingin2009.Hisresearchinterestsincludemedicalimageprocessing,machinelearningandimageanalysis.Hewouldliketomakecontributionstothesocietythroughinnovativeandcontinuousresearch. 108