A Reconstruction Framework for Common Sampling Lattices

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A Reconstruction Framework for Common Sampling Lattices
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Mirzargar, Seyyedeh Mahsa
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Entezari, Alireza
Committee Members:
Banerjee, Arunava
Ritter, Gerhard
Vemuri, Baba C
Blackband, Stephen J

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approximation -- quasi -- reconstruction -- splines -- voronoi
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
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Computer Engineering thesis, Ph.D.
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Abstract:
Sampling and reconstruction appear ubiquitously in many scientific and engineering applications. Conventionally, the Cartesian lattice has been used, while sampling on certain non-Cartesian lattices is known to capture signal’s information content more efficiently. In the trivariate case, the Body Centered Cubic lattice (BCC) and the Face Centered Cubic lattice (FCC) are considered to be superior to the Cartesian. However, developing suitable reconstruction schemes for these lattices has been a challenge. This dissertation is devoted to the introduction of Voronoi splines as an unbiased reconstruction framework for common sampling lattices. The main challenge for reconstruction on non-Cartesian lattices is due to the non-separable nature of these lattices. This challenge has been addressed by the geometric analysis of the Voronoi cell of a lattice and its relation to box spline theory to derive an exact and closed form solution which we call the generalization of B-splines, Voronoi splines. This reconstruction framework provides unbiased approximation schemes among various lattices. Various properties of this family of reconstruction kernels will be discussed in detail including their approximation properties and their Fourier transforms. Voronoi splines can be constructed for different degrees of continuity. Going beyond interpolating kernels, and using higher order kernels, the over-smoothing artifact results in degradation of the approximation. Over-smoothing artifact has been studied in terms of polynomial reproduction power of a reconstruction kernel. This artifact has been addressed through developing a quasi interpolation scheme tailored for Voronoi splines on BCC and FCC lattices to attain their asymptotic approximation error behaviour. In order to overcome the burden of box spline evaluation, clustering algorithms have been used to derive the exact polynomial representation of Voronoi splines of second order on BCC and FCC lattices. The second order Voronoi splines on these lattices are equivalent to the widely used trilinear interpolation. The efficiency and accuracy of reconstruction using Voronoi splines have been studied on different types of data including synthetic data and real life CT datasets. In addition, scattered data reconstruction using Voronoi splines has been studied through the introduction of a variational framework to approximate the underlying signal on different choices of uniform lattices. As a whole, this dissertation investigates the potential benefits of the Voronoi spline framework along with sampling on non-Cartesian sampling lattices.
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by Seyyedeh Mahsa Mirzargar.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Entezari, Alireza.
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ARECONSTRUCTIONFRAMEWORKFORCOMMONSAMPLINGLATTICESByMAHSAMIRZARGARADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012MahsaMirzargar 2

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Tomyfamily 3

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ACKNOWLEDGMENTS Finishingmygraduateschoolandwritingmydissertationwouldnothavebeenpossiblewithoutthesupportofseveralindividualswhoseguidanceandassistanceinonewayoranothercontributedinthepreparationandcompletionofthiswork.Firstandforemost,myutmostgratitudegoestomyPh.D.advisor,Dr.Entezari.Hispatience,encouragementandinspirationhelpedmetoovercometheobstaclesincompletionofthisresearchwork.Aspecialthankstomycommitteemembers(inalphabeticalorder):Dr.Banerjee,Dr.Blackband,Dr.RitterandDr.Vemurithatprovidedmethegreatopportunityofinteractinganddiscussingwiththem.Theirvaluableinsightshelpedmetounderstandmyownresearchareabetterandtothinkcriticallyaboutit.IwishtothankmycolleaguesandstaffmembersinCISEdepartment,inparticularmycollaboratorXieXuandmyfriendswhoalwaysbelievedinmeandhelpedmetogothroughdifculttimes.Lastbutnottheleast,Iamgratefulofmyfamilyfortheirunfailingsupportandlove. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTIONANDMOTIVATIONS ....................... 12 1.1CommonMulti-dimensionalSamplingLattices ................ 13 1.2OptimalSamplingLattices ........................... 13 1.3Non-SeparableLattices,BenetsandChallenges .............. 17 2RECONSTRUCTIONON(NON)-SEPARABLELATTICES ............ 20 2.1B-splinesandTheirMultivariateGeneralizations .............. 20 2.1.1ReviewofUnivariateB-splines .................... 20 2.1.2GeneralizationtotheMultivariateCase ................ 22 2.2BoxSplines ................................... 24 2.2.1BoxSplineDenition .......................... 24 2.2.2BoxSplineProperties ......................... 26 2.2.3BoxSplineReconstructionDiscussion ................ 27 2.3IdealReconstructionFrameworkforCommonSamplingLattices ..... 29 3VORONOISPLINES ................................. 31 3.1MotivationandDenition ............................ 31 3.1.1VoronoiSplinesDenition ....................... 32 3.1.2CharacterizationandRelationtoBoxSplines ............ 34 3.1.2.1Zonotopes .......................... 36 3.1.2.2Spacetessellations,Voronoipolytopesandzonotopes .. 37 3.2VoronoiSplinesonOptimalSamplingLattices ................ 39 3.3VoronoiSplines'PropertiesonaGeneralLattice .............. 42 3.3.1Continuity ................................ 44 3.3.2Support ................................. 45 3.3.3FourierTransformofVoronoiSplines ................. 46 4RECONSTRUCTIONUSINGVORONOISPLINES ................ 48 4.1VoronoiSplinesEvaluation,Experiments,ResultsandDiscussion .... 48 4.2QuasiInterpolationwithVoronoiSplines ................... 54 4.2.1OrthogonalProjectionvs.QuasiInterpolation ............ 55 4.2.2QuasiInterpolationUsingFourierTransform ............. 58 5

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4.2.2.1QuasiinterpolationforunivariatequadraticB-spline ... 59 4.2.2.2QuasiinterpolationforVoronoisplines ........... 60 4.2.3ResultsandComparisons ....................... 67 4.3EfcientEvaluationofVoronoiSplines .................... 71 4.3.1IdentifyingTheDomainPartitionForPolynomialPieces ...... 72 4.3.2TimeComparisons ........................... 75 4.4ScatteredDataReconstruction ........................ 78 4.4.1ProblemDenitionandRelatedWorks ................ 79 4.4.2ScatteredDataReconstructionUsingVoronoiSplines ....... 82 5CONCLUSIONANDFUTUREWORKS ...................... 93 APPENDIX:COMPLEMENTARYNOTEONQUASIINTERPOLATION ........ 95 REFERENCES ....................................... 99 BIOGRAPHICALSKETCH ................................ 106 6

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LISTOFTABLES Table page 2-1Propertiesofboxsplinesproposedin[ 32 36 48 49 ]. .............. 28 3-1ParallelepipeddecompositionofVoronoicellforFCClattice,intobuilding-blockboxsplines. ...................................... 40 3-2ParallelepipeddecompositionofVoronoicellforBCClattice,intobuilding-blockboxsplines. ...................................... 42 3-3ApproximationpropertiesofsplinespacesgeneratedbyVoronoisplines .... 45 4-1TheratioofthereconstructionerrorspresentinBCCandFCCsignalstotheCartesianreconstructedsignal. ........................... 54 4-2TheconvolutionbasedreconstructionusingunivariatequadraticB-spline. ... 60 4-3Reconstructionmeansquarederrorof10Krandomsamplepoints. ....... 70 4-4Thecoefcientsforpolynomialpartitions(asdocumentedinalgorithm 1 )toconstitutethesupportofthesecondorderVoronoisplineontheFCClattice. 77 4-5Thecoefcientsforpolynomialpartitions(asdocumentedinalgorithm 2 )toconstitutethesupportofthesecondorderVoronoisplineontheBCClattice. 77 4-6TherenderingtimemeasuredinsecondsonaquadcoreAMDAthlon(tm)processor(2.2GHz) ................................. 78 4-7TheevaluationtimefortheproposedefcientevaluationofthethesecondorderVoronoisplines. ................................ 78 7

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LISTOFFIGURES Figure page 1-1Ageneric2-Dlatticealongwithitsbasisu1,u2. .................. 14 1-2Thehyper-sphere(disk)packing. .......................... 17 1-3Cartesian,BCCandFCClattices. ......................... 18 2-1LinearandquadraticB-splines:convolutionsof0. ................ 21 2-2TheBCClatticecanbedecomposedinto2sub-Cartesianlattices. ....... 22 2-3A2-Dlatticealongwithitsfundamentaldomainand2differentbases. ..... 29 3-1Voronoisplinesforthehexagonallattice. ...................... 32 3-2Voronoisplines:sumofboxsplines. ........................ 34 3-3Ageneral2-Dlattice. ................................. 36 3-4Categorizationof3-Dlatticesbasedontheirfundamentaldomains. ....... 38 3-5VoronoicellsoftheFCCandBCClattices. .................... 41 3-6V-splinesaredenedastheselfconvolutionoftheindicatorfunctionoftheVoronoicellofthelattice. .............................. 43 4-1Marschner-Lobbfunctionusedasasyntheticbenchmark. ............ 50 4-2ReconstructionresultsofMarschner-LobbdatasetusingsecondorderVoronoisplines. ........................................ 52 4-3ReconstructionresultsofMarschner-LobbdatasetusingthirdorderVoronoisplines. ........................................ 53 4-4ThegroundtruthCarpshdataset. ......................... 54 4-5ReconstructionresultsforCarpshdataset. .................... 55 4-6Theeffectofusingquasiinterpolationinreducingtheover-smoothingartifact. 62 4-7Groundtruthvolumetricdatasets. .......................... 68 4-8QuasiinterpolationresultsforML(cutview). ................... 69 4-9QuasiinterpolationresultsforMLalongwithreconstructionerrorvisualizations. 87 4-10Quasiinterpolationresultsforcarpshdataset. .................. 88 4-11Quasiinterpolationresultsforbonsaitreedataset. ................ 89 8

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4-12Polynomialpartitionsconstitutingthesupportofthesecondorderhexspline. 90 4-13Theminimalregiondenedforthesupportofsecondorderhexspline(VH1). .. 90 4-14TheminimalregiondenedfortheBCCandtheFCClattices. .......... 91 4-15PartitioningoftheminimalregionforsecondorderV-splineintopolynomialpartitions. ....................................... 91 4-16PolynomialpartitionsformingtheminimalregionofVB1shownfrom2differentviewangles. ...................................... 92 4-17ScattereddatareconstructionusingsecondorderVoronoisplines. ....... 92 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyARECONSTRUCTIONFRAMEWORKFORCOMMONSAMPLINGLATTICESByMahsaMirzargarDecember2012Chair:AlirezaEntezariMajor:ComputerEngineeringSamplingandreconstructionappearubiquitouslyinmanyscienticandengineeringapplications.Conventionally,theCartesianlatticehasbeenused,whilesamplingoncertainnon-Cartesianlatticesisknowntocapturesignal'sinformationcontentmoreefciently.Inthetrivariatecase,theBodyCenteredCubiclattice(BCC)andtheFaceCenteredCubiclattice(FCC)areconsideredtobesuperiortotheCartesian.However,developingsuitablereconstructionschemesfortheselatticeshasbeenachallenge.ThisdissertationisdevotedtotheintroductionofVoronoisplinesasanunbiasedreconstructionframeworkforcommonsamplinglattices.Themainchallengeforreconstructiononnon-Cartesianlatticesisduetothenon-separablenatureoftheselattices.ThischallengehasbeenaddressedbythegeometricanalysisoftheVoronoicellofalatticeanditsrelationtoboxsplinetheorytoderiveanexactandclosedformsolutionwhichwecallthegeneralizationofB-splines,Voronoisplines.Thisreconstructionframeworkprovidesunbiasedapproximationschemesamongvariouslattices.VariouspropertiesofthisfamilyofreconstructionkernelswillbediscussedindetailincludingtheirapproximationpropertiesandtheirFouriertransforms.Voronoisplinescanbeconstructedfordifferentdegreesofcontinuity.Goingbeyondinterpolatingkernels,andusinghigherorderkernels,theover-smoothingartifactresultsindegradationoftheapproximation.Over-smoothingartifacthasbeenstudiedintermsofpolynomialreproductionpowerofareconstructionkernel.Thisartifacthasbeen 10

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addressedthroughdevelopingaquasiinterpolationschemetailoredforVoronoisplinesonBCCandFCClatticestoattaintheirasymptoticapproximationerrorbehaviour.Inordertoovercometheburdenofboxsplineevaluation,clusteringalgorithmshavebeenusedtoderivetheexactpolynomialrepresentationofVoronoisplinesofsecondorderonBCCandFCClattices.ThesecondorderVoronoisplinesontheselatticesareequivalenttothewidelyusedtrilinearinterpolation.TheefciencyandaccuracyofreconstructionusingVoronoisplineshavebeenstudiedondifferenttypesofdataincludingsyntheticdataandreallifeCTdatasets.Inaddition,scattereddatareconstructionusingVoronoisplineshasbeenstudiedthroughtheintroductionofavariationalframeworktoapproximatetheunderlyingsignalondifferentchoicesofuniformlattices.Asawhole,thisdissertationinvestigatesthepotentialbenetsoftheVoronoisplineframeworkalongwithsamplingonnon-Cartesiansamplinglattices. 11

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CHAPTER1INTRODUCTIONANDMOTIVATIONSSamplingandreconstructionarepervasiveinawiderangeofengineeringdisciplinessuchasscienticcomputing,biomedicalimaging,simulationandvisualization.Theefciencyofsamplingandaccuracyofreconstructioninuencethequality,usabilityandreliabilityoftheapplicationsusingthem.UnivariatesamplingandreconstructionhavebeentheoreticallystudiedforbandlimitedsignalsbyShannon-Nyquistsamplingtheory.Inthecaseofmultivariatesampling,inadditiontothesamplingrate,thegeometryusedforsamplingplaysamajorroleincapturinginformationmoreefciently.However,theCartesianlatticeisalmostalwaysthechoiceformultidimensionalsamplinginpracticalapplications.Theoretically,therehavebeensomeresultsofuniformsamplinggeometriesthatareverypromisingintermsofcapturing(i.e.,sampling)informationmoreefciently[ 15 ].Thediscussionwillstartwithaveryshortandbynomeanscomprehensivereviewofuniformsamplinggeometries.ThisintroductionisaprerequisitetothekeytrustofthisdissertationwhichistheintroductionofanewsamplingandreconstructionframeworkcalledVoronoisplines.WewillcontinuethediscussionwithstudyingthisframeworkmorecloselyfromapproximationtheoreticalpointofviewanddevelopanefcientevaluationapproachinanattempttomakethemcomputationallycomparabletowidelyusedB-splines.Ishouldbringtothereader'sattentionthatanumberofsectionsofthisdissertationareadirectandpermittedreprintsoftheauthor'srecentlypublishedworks.1 1c2011IEEE.Reprinted,withpermission,fromM.Mirzargar,A,Entezari,QuasiInterpolationWithVoronoiSplines,IEEETransactionsonVisualizationandComputerGraphics,Dec.2011c2010IEEE.Reprinted,withpermission,fromM.Mirzargar,A,Entezari,VoronoiSplines,IEEETransactionsonSignalProcessing,Sept.2010 12

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1.1CommonMulti-dimensionalSamplingLatticesStudyinguniformsamplingofmultivariatefunctionsinvolvesunderstandingthegeometryusedduringthesamplingandtherepresentationofthesamplesasvolumetricdataonagrid.Inparticular,theregulargridrepresentationofvolumetricdataisreferredtoasthesamplinglattice.Wewillstartthediscussionbyreviewingtheformaldenitionofalatticeandsomeofitsbasicpropertiesthatwillbeusedthroughtherestofthediscussion.ApointlatticecanbeconsideredasasubsetofEuclideanspacethatisformedbyperiodicarrangementofdiscretepointsandisrequiredtoincludeorigin.Becauseofthediscretenatureofthelattice,thereisaregioninwhichoriginistheonlylatticepoint.ThisregioniscalledtheVoronoiregionfortheorigin.Sincealatticeisclosedunderadditionandnegation,Voronoiregioncanbeassociatedtoanylatticepointapplyingatranslation.Hence,theVoronoiregionorformallyknownasVoronoicellofalatticeisuniqueandisreferredtoasfundamentaldomainorWigner-Seitzcell[ 72 ].Asamplinglatticecanbeformedbyintegermultiplesofa(non-unique)samplingmatrixL.Samplingpointsonthelatticecanberepresentedbyintegermultiplesofthecolumnsofthismatrix.UnliketheVoronoicell,thesamplingmatrixofasamplinglatticeinnotunique.Thereareinnitelymanysamplingmatricesforagivenlatticeandtheyarerelatedviaaunimodularmatrix[ 84 ](i.e.,L0=PL,withjdetPj=1).Forexample,thewidelyusedCartesianlattice,indimensiond,canbesimplyrepresentedbyddidentitymatrix.Ageneric2-DlatticehasbeendepictedinFigure 1-1 1.2OptimalSamplingLatticesSamplingamultidimensionalsignalonasamplinglatticegeneratedbyLresultsinthereplicationsofitsspectrumonthereciprocallatticeintheFourierdomain.Thereciprocallattice(alsoknownasdualorpolarlattice)toalatticerepresentedbyLisgeneratedusingthecolumnsofthematrix^L=2L)]TJ /F7 7.97 Tf 6.59 0 Td[(T.TheVoronoicellofthereciprocal 13

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Figure1-1. Ageneric2-Dlatticealongwithitsbasisu1,u2.Thevoronoicellhasbeendemonstratedforeachlatticepointsingray. latticeiscalledBrillouinzone[ 72 ]anditsboundarieswillspecifythemultidimensionalversionoftheNyquistfrequencyforthesamplinglatticeL.Intheunivariatecase,theNyquistfrequencywhichisdenedasthesupremumofthefrequenciespresentinthespectrumofaunivariatesignalspeciesthebestsamplingratefora1-Dbandlimitedsignal.Inotherwords,thesamplingratedictatedbyNyquisttheoremspeciestheminimumsamplingraterequiredtoexactlyrecoverthebandlimitedsignal.SamplingabandlimitedsignalbyitsNyquistsamplingratetranslatestotightlypackingthereplicationofitsspectrum(whichislimited)inthefrequencydomainwithoutintroducinganyoverlap.Allowingoverlapofthereplicascanreducethenumberofsamplingpointsinthespacedomainbutitwillcausethealiasingartifactinthereconstructionwhichisundesirable.Similarly,foramultidimensionalsignal,thegeometryofthespectrumofthesignalspeciestheNyquistfrequenciesforthatsignal.Thebestsamplinggeometry(lattice)foramultidimensionalsignalisgivenbytheoptimalsamplinglatticethatreplicatesthespectrumofthesignalasdenselyaspossiblewithnooverlap[ 30 64 67 84 ].Foragenericmultidimensionalsignal,becauseofthelackofanypriorknowledgeaboutthegeometryofthespectrumofthesignal,arealisticassumptionwouldbetoconsiderisotropictreatmentofallthefrequencies.Thisassumptionassuresthathighfrequencies 14

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alongalldirectionsarepreservedashighaspossible.Inthecaseofbandlimitedsignalsunderthisassumption,thefrequencyspectrumofthesignalformsahyper-sphere.Theproblemofndingtheoptimalsamplinglatticeisconnectedtothespherepackingproblem(inthefrequencydomain)whichhasbeenwidelystudiedindiscretegeometry[ 15 ].Thespherepackingproblemdealswithndingthebestarrangementoftouchinghyper-spheresllingeuclideanspacewhichminimizesthefreespaceremaining.Consideringthehyper-spherescenteredatlatticepoints,ndingoptimalsamplinglatticereducestospherepackingprobleminthefrequencydomain.Inthetwodimensionalcase,hexagonallatticeisknowntoprovidethetightesthyper-sphere(i.e.,disks)packingwhosereciprocallatticeisknowntobeanother(scaled/rotated)hexagonallattice.Theadvantagesofhexagonallatticeforincreasingaccuracyholdeveninthenon-bandlimitedsetting[ 14 ].Inthecaseofsamplingsignalsthatarenotbandlimited,wewanttominimizetheamountofoverlapbetweenthereplicasofthespectrumofthesignalinthefrequencydomain,asoverlapisunavoidable.Inthiscase,wearelookingforthelattice(inthefrequencydomain)thatresultsinthemostefcienttilingofthespacewiththereplicasofthespectrumofthesignal.Inotherwords,weareinterestedinthelattice(inthefrequencydomain)thatresultsintheleastamountofoverlapbetweenthereplicasofthespectrumofthesignalwhencenteredatthelatticepoints(i.e.,minimizingthealiasingartifact).Usingthehyper-sphereassumption(i.e.,disksin2-D)forthespectrumofthesignal,thisproblemtranslatestothewellknownspherecoveringproblem[ 15 ].Hexagonallatticehasbeenshowntobethesolutiontospherecoveringproblemin2-D[ 15 ].Therefore,thehexagonallatticeprovidestheleastamountofaliasinginthecaseofsamplinganon-bandlimitedfunction.MinimizingtheamountofthealiasingartifactisimportantinpracticalapplicationssuchasMRimaging.Theoverlapofthereplicasofthespectrumofthesignal(inthefrequencydomain)inMRimagingcanleadtomis-positioningofaportionoftheobjectoutoftheeldofviewintheimage[ 47 ].Thisproblemcanbeaddressedthroughexpandingtheeldofview 15

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oroversampling.However,samplingonspherecoveringlattices(i.e.,hexagonalin2-D)inthefrequencydomainresultsintheminimumamountofaliasingartifactwithoutanyextracostintermsofthesamplingrateorexpansionoftheeldofview.Furthermore,ifonehaspriorknowledgeaboutthegeometryofthespectrumofasignal,theoptimalsamplinglatticeforthatspecicsignalcanbecomputed[ 52 ].Thesignalspecicoptimalsamplinglatticecorrespondstothedualofthelatticethatbestreplicatesthespectrumofthesignalinthefrequencydomain(i.e.,ifpackingisnotpossible,coveringthespacewithleastamountofoverlap).Amongmultidimensionallattices,theCartesianlatticecanbeconsideredasasimpleextensionoftheunivariatesamplingusingatensor-product.ThesimpleconstructionofhigherdimensionalCartesianlatticeanditsseparablenaturemakeitadesirablechoicetoextendmanyunivariatereconstructionschemestohigherdimensionsusingthetensor-product.However,theCartesianlatticehasbeenknowntobeaninefcientlatticefromthesampling-theoreticpointofview[ 64 67 ].TheCartesianlattice(i.e.,tensor-productsampling)leadstoaloosereplicationofthespectrumshowninFigure 1-2 forthe2-Dcaseandhenceitisinefcientforsamplingmultidimensionalsignals[ 10 15 31 ].Inthethreedimensionalcase,theFaceCenteredCubiclattice(FCC)isknowntoprovidethedensestspherepackingin3-D[ 15 ]andhence,itsreciprocallattice,whichiscalledBodyCenteredCubiclattice(BCC),isconsideredastheoptimallatticeforsamplingbandlimitedsignals[ 51 ].Moreover,BCClatticeisknowntoprovidethebestspherecoveringlatticewhosereciprocallatticeisFCC.Hence,FCClatticecanbeconsideredastheoptimallatticeforsamplingnonbandlimited,non-smoothsignals[ 51 ]usingthesamereasoningdescribedinthe2-Dcase.These2latticesalongwiththeCartesianlatticearethemainfocusofthisdissertation.Wewilldiscusslateronthatother3-DlatticesarenotanymorecomplicatedthanBCClatticeandonewouldbeabletogeneralizethediscussionhereforotherchoiceof3-Dlatticesandhigherdimensional 16

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ABFigure1-2. Thehyper-sphere(disk)packing.A)Thehyper-sphere(disk)packingoftheCartesianlatticealongwithitsVoronoicell(i.e.,acube).B)Thehyper-sphere(disk)packingofthehexagonallatticewithhexagonasitsVoronoicell.NotethathexagonallatticehasprovidedamuchdenserpackingofthediskscomparedtotheCartesianlatticewherethedisksinbothcasesareofthesamearea.Thearrowsdenoteonesetofbasistoformeachoftheselattices. latticesthatsatisfydicingpropertywhichwillbeintroducedlateron.Throughouttherestofthismanuscript,werefertobothoftheselatticesasoptimalsamplinglattices. 1.3Non-SeparableLattices,BenetsandChallengesHexagonallatticeanditshigher-dimensionalcounterpartsresultinadensereplicationofthespectrumofa(bandlimited)signalwhichtranslateintotheirabilitytoprovidealias-freesamplingforgenericsignalswithlessnumberofsamplescomparedtotheCartesianlattice[ 30 64 67 84 ].BodyCenteredCubic(BCC)andFaceCenteredCubic(FCC)lattices,showninFigure 1-3 ,havesamplingefciencyupto30%and27%respectivelycomparedtotheCartesianlattice.ThesesavingsgrowovertheconventionalCartesianlatticeasthedimensionincreases.TheBCCandFCClattices 17

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ABC DEFFigure1-3. Cartesian,BCCandFCClattices.TheVoronoicellshavebeenshowninred.A)Cube(VoronoicelloftheCartesianlattice).B)Rhombicdodecahedron(VoronoicellofFCClattice).C)Truncatedoctahedron(VoronoicellofBCClattice).D)Cubictiling(Cartesianlattice).E)Rhombicdodecahedraltiling(FCClattice).F)Truncatedoctahedraltiling(BCClattice). canberepresentedbythefollowingsamplingmatrices: B=266664)]TJ /F4 11.955 Tf 9.3 0 Td[(1111)]TJ /F4 11.955 Tf 9.3 0 Td[(1111)]TJ /F4 11.955 Tf 9.3 0 Td[(1377775,F=266664011101110377775.(1)Similarly,theCartesianlatticecanberepresentedbyC=I33,whereIdenotestheidentitymatrix.Resiliencetojitternoiseduringthesamplingoperation[ 53 ]makesoptimalsamplinglattices(i.e.,BCCandFCC)suitableforapplicationssuchasMRI[ 71 ].Thetheoreticaladvantagesofoptimalsamplinglatticesformultidimensionaldata,offerpromisingperspectivesforawidevarietyofapplications.Optimalsamplinghas 18

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beenproposedforuidowsimulations[ 3 68 ],medicalimageacquisition(e.g.,MRandCT)andprocessing[ 71 88 ],imageprocessing[ 1 44 85 ].Inparticular,forseismicimaging,theBCCandFCClatticeshavebeenshowntosignicantlyreduceacquisitiontimeandthecostsassociatedwithit[ 41 ].Fromthetheoreticalpointofview,theselatticeshavereceivedconsiderableattention[ 15 61 67 ].However,theyhavehardlybeendeployedinpracticalapplicationsmainlyduetothelackofpracticalsignalprocessingtoolsthatcanbetailoredtothegeometryofthesenon-Cartesianlattices.InChapter 2 ,abriefreviewofthereconstructionschemesproposedfortheCartesianlatticewillbediscussed.Moreover,reconstructionschemesthatwereproposedtobeusedonnon-separablelatticesandtheirshortcomingswillalsobereviewed.ThedrawbacksofthereviewedmethodsmotivatethenecessityofhavingageometricallytailoredframeworkforgeneralsamplinglatticesproposedinChapter 3 19

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CHAPTER2RECONSTRUCTIONON(NON)-SEPARABLELATTICESInChapter 1 ,theproblemofsamplingamultidimensionalsignalwasbrieyintroduced.Fromthesamplingpointofview,therearebetterchoicesthanwidelyusedCartesianlatticeforsamplingmultivariatefunctions.However,Cartesianlatticeisalwaystherstchoicewhenitcomestopracticalapplications.Thereasonforthissub-optimalchoicecomesfromthesimpleandinherent1-DnatureoftheCartesianlattice.Eveninhigherdimensions,theseparablestructureoftheCartesianlatticesimpliesthedesignofreconstructionschemesforthislatticesignicantly.ThediscussioninherestartsbyintroducingunivariateB-splinesasoneofthemostcommonlyusedreconstructionschemesontheCartesianlatticealongwithitsgeneralizationtohigherdimensions. 2.1B-splinesandTheirMultivariateGeneralizationsB-splines(Basicsplines)deneabasisforanapproximationspacecalledcardinalsplineswhereeachelementhasauniquerepresentation(i.e.,splinecoefcients)intermsoftheshiftsofthebasisfunctionsthatspanthesplinespace.Inotherwords,anarbitraryfunctioncanbeapproximatedwithanelementfromthesplinespacebyndingtheappropriatesetofsplinecoefcientsthatbestrepresentthefunction.Usingdifferentordersofthebasis(i.e.,B-splines),thesmoothnessandtheorderoftheapproximationspacecanbecontrolled.InordertohaveabetterunderstandingofhowB-splinescanbegeneralizedtotheframeworkintroducedinChapter 3 ,wepresenttheconstructionandpropertiesofB-splines(basicsplines)whicharepertinenttoVoronoisplinesinthissection.Interestedreadermayconsult[ 23 ]forfulldiscussionofB-splinesandtheirapproximationproperties. 2.1.1ReviewofUnivariateB-splinesThenearestneighborinterpolationistheconvolutionofdatawiththerst-orderB-splinewhichistheindicatorfunctionofthesamplinginterval,T=[)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2,1=2]:0(x):=T(x)(Figure 2-1 ).Notethat[)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2,1=2]containsallofthepointsthat 20

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Figure2-1. LinearandquadraticB-splines:convolutionsof0. areclosertothesampleattheoriginthanalltheotherintegerpointsontherealline;therefore,onecanconsiderTastheVoronoicellofthesamplepointattheorigin.Movingbeyondnearestneighborinterpolationwhichisadiscontinuous,piecewiseconstantinterpolationoftheoriginalfunction,higherorderB-splinesaredenedrecursively,asself-convolutionsofthesimplestB-spline,0,asin( 2 ).Figure 2-1 demonstratesrst,second,andthirdorderB-splines. 0(x)=T(x)=8>><>>:1x2)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 2,1 20otherwise.n=0n)]TJ /F7 7.97 Tf 6.59 0 Td[(1(2)TheunivariateB-splines,togetherwiththeirtensor-productextensiontothemultivariatesetting,havemanyapplicationsinapproximationtheory,imageandsignalprocessing[ 23 80 86 ].Afundamentaltheoremstatesthateverypiecewisepolynomialfunctionofagivendegree,smoothness,anddomainpartition,canberepresentedasalinearcombinationof(i.e.,convolutionwith)B-splinesofthatsamedegreeandsmoothness,andoverthatsamepartition[ 23 ].B-splineshavebeenproventobeaccurateandexceptionallyefcienttoolsfor1-Dsignalprocessing.TheadvantagesofB-splinesforsignalprocessing,particularlyforbiomedicalimaging,havebeenstudiedbymanyresearchers(e.g.,[ 79 81 83 ]). 21

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2.1.2GeneralizationtotheMultivariateCaseThegeneralizationofB-splinesisstraightforwardconsideringaseparablesamplinglattice(i.e.,Cartesian).InthecaseofhigherdimensionalCartesianlattice,eachdimensioncanbetreatedasa1-D(univariate)samplingspaceandunivariateB-splinecanbeusedineachdimensionwhichresultsinthetensor-productdenitionofB-splinesontheCartesianlattice.Whiletensor-productapproachtsthegeometryofCartesianlatticeexactly,othergenerallatticesdonotnecessarilyhaveaseparablenature.InparticularBCC(correspondingtothespherepackinglatticeinthefrequencydomain)andFCC(correspondingtospherecoveringlatticeinthefrequencydomain)arenotseparable. ABFigure2-2. TheBCClatticecanbedecomposedinto2sub-Cartesianlattices.A)The2sub-Cartesianlatticesshowninredandgreen.B)FindingthelargestpossiblecubeonaBCClatticeusinginterpolationtondthebluepointswhicharenotBCClatticepoints. Inspiteofthenon-separablenatureoftheoptimalsamplinglattices,trilinearinterpolationhasbeengeneralizedfortheBCClattice[ 56 76 ].TrilinearinterpolationforanarbitrarypointontheCartesianlatticeinvolvesthecornersofthecubethatthe 22

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interpolationpointfallsinside.Therefore,inordertodesignareconstructionschemebasedontrilinearinterpolation,theauthorsin[ 56 76 ]decomposedtheBCClatticeinto2sub-Cartesianlattices.ThisdecompositionhelpstoformthelargestpossiblecubeontheBCClatticeasshowninFigure 2-2 (b).Notethatthecubeusedforinterpolationhasonly2ofitscornersontheBCClatticeandtheremaining6corners(bluepointsinthepicture)needtoberesampledfromtheBCClatticepoints.Eventhough,thisapproachresultsinauniquecubeforanyinterpolationpoint,theinterpolationismorecomputationallyexpensivecomparedtotrilinearinterpolationontheCartesianlatticebecauseoftheresamplingstepusedtoformthecube.Moreover,theresamplingofthecornersofthecuberesultsinanisotropictreatmentofthedirectneighborsoftheinterpolationpoint.PrelteredGaussianreconstruction(PGR)[ 17 ]isanotherreconstructionschemebasedonthedecompositionofBCClatticeinto2Cartesianlattices.Thisapproachusestheconceptofgeneralizedinterpolationintroducedin[ 6 ]anddesignsatwostageinterpolationschemewherethereconstructionkernelisconsideredtobeatruncatedGaussianappliedonprelteredversionofthesamplepoints.TheinterpolationonBCClatticeismoreexpensivecomparedtotheCartesian.ForanyBCClatticepoint,2Gaussiankernelsshouldbeapplied,onefortheneighboringpointsonone(sub)CartesianlatticeandtheotherGaussianisusedtotakeintoaccountthepointsonthetheothersublattice.PGRcanprovideveryhighqualityreconstructionwhilestandarddeviationofthetruncatedGaussiankernelisafreeparameterforthedesignchoice.Thesmallerthestandarddeviation,themorethesmoothingartifactwillaffectthereconstruction[ 17 ].UnivariateB-splineshavealsobeengeneralizedbyusingsphericalreconstructionkernels[ 77 ].Thisapproachresultsinasphericalfootprintinthefrequencydomainandthereforetreatallfrequenciesisotropicallyandresultsinablurredreconstructionwhichisagainbecauseofnotconsideringtheunderlyinggeometryofthesamplinglattice 23

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completely.AmongthegeneralizationofB-splines,polyhedralsplines[ 42 ]whicharedenedasprojectionsofhigherdimensionalconvexpolyhedraintolowerdimensionshaveshownmorepromisingresultsformultidimensionalsignalprocessing.Dependingonthetypeofthepolyhedraprojected,differenttypesofsplineshavebeendened,suchasboxsplinesandsimplexsplines[ 27 60 ].Boxsplinesareaspecictypeofpolyhedralsplineswhichhavebeenshowntopossesattractivepropertiestodesignreconstructionschemesspecicallyforoptimalsamplinglatticesunderstudy[ 32 36 48 49 ].Boxsplinetheoryalsoshedlightondeningclosedformformulationfortheproposedreconstructionframeworkinthisdissertation.Therefore,wewouldtakesometimeandreviewboxsplinetheoryinmoredetailsinsection 2.2 2.2BoxSplinesAboxsplineisdenedasasmoothpiecewisepolynomialfunctioninRdwhichischaracterizedbyasetofcolumnvectors1,2,...,NgatheredinadbyNmatrix,,assumingNd.Thesetofdirections:=[12...N]fullydenestheboxspline'spropertiesinRd.Notethattheboxsplinediscussioninherebynomeansiscomprehensiveandinterestedreadercanreferto[ 27 ]forcompletediscussionofboxsplinetheory. 2.2.1BoxSplineDenitionGeometrically,aboxsplineisdenedastheprojectionorX-rayimageofahypercube(inRN)toalower-dimensionalspace,Rd(Nd).Theboxsplinerepresentsthedensityfunctionofthisprojection.Eachofthevectors,i,inthedirectionset,,istheshadowofanedgeoftheN-hypercubeadjacenttoitsorigin.Notethatthevectorsinthis(multi-)setneednotbedistinctastheycanappearwithsomemultiplicity.WhenN=d,thesimpleboxsplineisdenedtobethe(normalized)indicatorfunctionoftheparallelepipedformedbydvectorsinRd: 24

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M(x)=8>><>>:1 jdetj,x=Pdn=1tnnfor0tn10,otherwise, (2)Aboxsplineisnormalizedbydet()toensurethatitintegratestooneonitssupportanditwillaccountforavalidreconstructionkernel(i.e.,itsatisesthepartitionofunityrequirement).Inthegeneralcase,N>d,boxsplinescanbedenedrecursivelyusingdirectionalconvolutions: M[(x)=Z10M(x)]TJ /F5 11.955 Tf 11.96 0 Td[(t)dt.(2)Convolutioniscommutativeandtherefore,aboxsplineisindependentoftheorderofthecolumnvectorsinthedirectionmatrix.Notethatwhenthelowerdimensionalspace(i.e.,projectionspace)isR(i.e.,d=1),( 2 )correspondstoonedimensionalconvolutionandtherefore,itiseasytoconcludethatonedimensionalboxsplinescoincidewithunivariateB-splines.Moreover,ifthedistinctcolumnvectorsofareorthogonaltoeachother,boxsplinesamounttothetensor-productB-splines.Bythedenition,thedirections(i.e.,columnvectors)inthedirectionsetcanberepetitiveandhavingrepetitivedirectionsincreasesthesmoothnessoftheboxsplinealongtherepeateddirection.Convolutionofaboxsplinewithanotherboxsplineresultsinasmootherboxsplinewhosedirectionsetwouldbethesetunionofthedirectionsetoftheoriginalboxsplines(repetitionofdirectionsisallowed):M1M2=M1[2 (2)Forthesakeofreadability,anewnotationwillbeintroducedforboxsplineswithrepetitivedirections.Insteadofgatheringallthe(repetitive)directionsdeningaboxsplineinthedirectionmatrix,wedenearepetition-vector=[1,,N]alongwith 25

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thedirectionmatrixincludingonlythedistinctdirectionswhoserepetitionisencodedintherepetition-vector.Usingthisnotation,aboxsplineisdenedusingadirectionset,(whosecolumnsaredistinct)andarepetitionvector,.Theboxsplinewillbedenotedas,Mwhichcorrespondsto:M(x):=M[11| {z }1timesNN| {z }Ntimes](x) (2)Asanillustrativeexample,considerdenotesthe33identitymatrix.=[123]=266664100010001377775 (2)Asdiscussedabove,Mwoulddenotetheindicatorfunctionof3-Dcube(i.e.,therstordertensor-productB-spline).UsingconvolutionconstructionofhigherorderB-splines,thesecondorderB-splinecanbeconsideredastheconvolutionoftherstorderwithitself.Havingdenedtherepetitionvectorandusingtheconvolutionpropertyofboxsplines,thesecondorder(trivariate)B-splinecanbedenotedasaboxsplineMwithrepetitionvector=[2,2,2]orM2where2=[11,2,2,3,3].ThisconventionsimpliesintroductionofVoronoisplinesinChapter 3 2.2.2BoxSplinePropertiesAsmentionedbefore,thedirectionset,fullycharacterizestheboxsplineManditsapproximationproperties.Fromtherecursivedenitionofboxsplines,onecandirectlyinferthatboxsplinesarepositive,compactly-supportedfunctions.Support:Thesupportofaboxsplineisazonotope,whichistheMinkowskisumofNcolumnvectorsin.Zonotopesaregoingtobestudiedinmoredetailsinsection 3.1.2.1 ,wheretheirgeometricpropertieswillhelptowriteVoronoisplinesinclosedformintermsofboxsplines.ThecenterofthesupportofM(x)isgivenbyc:=1 2PNn=1n. 26

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Fouriertransform:TheFouriertransformofaboxsplineisalsogivenby: ^M(!)=NYn=11)]TJ /F4 11.955 Tf 11.96 0 Td[(exp()]TJ /F5 11.955 Tf 9.3 0 Td[(jh!,ni) jh!,ni=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(jh!,ci)NYn=1sinch!,ni,(2)where!:=(!1,...,!d)isthemultivariatefrequencyvectorandsinc(!):=sin(!=2)=(!=2).Fromapproximationtheoreticalpointofview,thelinearcombinationoftheshiftsofboxsplinesformthesplinespace: SM:=span(M()]TJ /F6 11.955 Tf 17.93 0 Td[(k))k2Zd.(2)whosecontinuityandapproximationpowerisfullydeterminedbasedonthepropertiesofthematrix.Continuity:IfistheminimalnumberofdirectionswhoseremovalfrommaketheremainingdirectionsdonotspanRd,thenallpolynomialsuptodegree)]TJ /F4 11.955 Tf 12.61 0 Td[(1arecontainedinSM[ 27 ,(II.59)]andthecontinuityoftheboxsplineis:M2C)]TJ /F7 7.97 Tf 6.58 0 Td[(2. (2)Approximationorder:Theapproximationpowerofareconstructionkernelwhosesupportislimitedwillspecifythepolynomialspacethatthekernelisideallycapabletoreproduceandtheapproximationordercorrespondstothedimensionalityofthispolynomialspace.TheapproximationorderofSMisshowntobe[ 27 ]. 2.2.3BoxSplineReconstructionDiscussionTheexibilityavailabletochoosedirectionsetofboxsplinesmakethempotentialcandidatesforreconstructiononnon-separablelattices,assumingthatonecanndappropriatedirectionset,.Non-separable,yetsymmetricgeometryofBCCandFCClatticesestablishedaconvenientrelationbetweenboxsplinesappropriatefortheselatticesandresultedintailor-madereconstructionkernelsontheselatticesasproposedin[ 32 36 48 49 ].Themainideaistouseone(ormultiple)layer(s)ofneighboringpointsofagenericlatticepointtoformapolyhedronthathappenstobethesupport 27

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Table2-1. Propertiesofboxsplinesproposedin[ 32 36 48 49 ].ThepropertiesoftheirB-splinecounterpartsontheCartesianlatticehavealsobeendocumented. ContinuityApproximationPolynomialDatapointsindegreeorderpiecesdegreethesupport TrilinearB-spline(Cartesian)C02384-directionalboxspline(BCC)C0214 TricubicB-spline(Cartesian)C24964Quinticboxspline(BCC)C24532 TriquadraticB-spline(Cartesian)C136276-directionalboxspline(FCC)C13316 Nocounterpart(Cartesian)----9-directionalboxspline(FCC)C33640 ofaboxspline.Usingthisapproach,a4-directionalboxsplinehasbeendesignedfortheBCClatticewhoseapproximationorderandsmoothnessmatchthatoftrilinearB-splineontheCartesianlattice[ 32 ].Theselfconvolutionofthisboxsplineresultsina8-directionalquinticboxsplinewhichisC2continuousandhasbeencomparedwithtricubicB-splineontheCartesianin[ 32 36 ].SimilartoBCC,a6and9directionalboxsplineshavebeenproposedforFCClattice[ 32 49 ].Thequalityofboxsplinereconstructionintroducedforoptimalsamplinglattices(i.e.,BCCandFCC)hasbeenstudiedin[ 32 36 48 49 ].TheexperimentsinthereferencesshowthehigherqualityofreconstructionusingboxsplinesonBCC/FCCversuscomparabletensor-productB-splinesontheCartesian.Moreover,fourdirectionalboxsplineonBCCin[ 32 36 ]and6-directionalboxsplineonFCCin[ 48 49 ]takeadvantageofefcientevaluationalongwithsmallerneighborhood.AsummaryofpropertiesoftheseboxsplineshasbeendocumentedinTable 2-1 .NotethattheseboxsplinesarecomparabletotheirB-splinecounterparts,intermsofcontinuityandapproximationpower.However,thepolynomialdegreeandtheirsupportsaredifferentonBCC/FCCcomparedtotheCartesian.Ononehand,smallersupportoftheseboxsplinesonBCC/FCCcanbeinterpretedasanadvantagetoachievemoreaccuracywithlessnumberofneighboringpoints.Ontheotherhand,itmakesthecomparisonofthereconstructionusingboxsplinesandB-splineson 28

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variouslatticesnotcompletelyfair.Yet,thereisanothershortcomingwithboxsplinereconstructionwhichmightnotbethatobviousatthebeginning.Ingeneral,ndingaboxsplineongenerallatticeswitharbitrarycontinuityandapproximationpowerisnotalwayspossible.Incontrast,Voronoisplinesintroducedinthesequelcanbedesignedforarbitrarylattices. 2.3IdealReconstructionFrameworkforCommonSamplingLattices Figure2-3. A2-Dlatticealongwithitsfundamentaldomainand2differentbases.Thefundamentaldomain(Voronoicell)ofthelatticeisillustratedinblack.Whilethefundamentaldomainofalatticeisunique,therearedifferentbasis(i.e.,L0=[u01,u02])foralattice.Differentbasesofalatticedenedifferentneighborhoodsforalatticepoint. Theconceptoftensor-productprovidesasuitablereconstructionapproachfortheCartesianlattice,whiletensoringwillfailwhenitcomestogenerallatticesthatdonothaveaseparablestructure.Theconceptoftensor-productisbasedononeparticularbasisthatwillgeneratealatticeandaswesawinChapter 1 ,thischoiceisnotuniqueandinfactcanbearbitrary.ThisarbitraryselectionofthebasisresultsinexclusionofneighboringpointsshowninFigure 2-3 .Therefore,thereconstructioncouldsufferfromdirectionalartifactsthataredirectresultofdiscardingneighboringpointsasaparticularbasissuggests.Incontrast,afairreconstructionkernelforagivenlatticeshouldconsideralltheneighborsofagenericlatticepointequally(i.e.,respectthegeometryofthelattice). 29

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Asdiscussedindetailsin[ 32 ],theconnectionofboxsplines'supportandgeometryoftheoptimalsamplinglatticesmotivatedthedesignofreconstructionkernelswiththisproperty.However,boxsplinescannotbedesignedforarbitraryanddesiredapproximationpropertiesfreely.Thesechallengesleadtotheneedfornon-separablereconstructionfunctionswhichcanbedesignedforarbitrarysamplinggeometriesfordesiredapproximationproperties.VoronoisplinesdiscussedindetailsinChapter 3 wereproposedtoaddressthesechallengesandwouldbethekeythrustofthisresearch.Oneofthemainadvantagesoftheproposedframeworkcanbeconsideredasalattice-invariantreconstructionscheme.Voronoisplinesreconstructioncanbeusedtoexamineandcomparetheinherentadvantagesthatoptimalsamplinglatticesprovideintermsoftheirsamplingefciencyinpracticewhichtothebestoftheauthor'sknowledgewasnotavailablebeforetheintroductionofVoronoisplines. 30

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CHAPTER3VORONOISPLINESB-splines(i.e.,Basicsplines)provideanadvantageousframeworkforsignalprocessingbecauseoftheirimplementationefciencyandsimplicity[ 82 ].Moreover,differentgeneralizationsofmultivariateB-splinesonnon-separablelatticeshavebeenproposedwhichhavebeenbrieysummarizedinChapterever,Voronoisplines,introducedhere,formtheveryrstreconstructionframeworkproposedthataregeometricallytailoredforvarioussamplinglattices(geometries)indifferentdimensions.VerysimilartoB-splines,Voronoisplinescanbeconstructedforarbitrarydegreesofsmoothnessandapproximationorder,whichwillmakethemexactcounterpartstoB-splinesfordifferentchoiceoflattices.Voronoisplinesformalattice-shift-invariantsplinespaceforapproximationandunlikeothermethodsdiscussedinChapteronoisplinesareconsideredasexiblereconstructionkernels. 3.1MotivationandDenitionTheideaofconstructinghigher-degreesmoothcompactlysupportedpiecewisepolynomialfunctionsbyconvolutionofcharacteristicfunctionsinthemultivariatecontextgoesbackatleasttoFredericksonandtoSabin[ 26 ].Forimagereconstructiononthe2-Dhexagonallattice,theideaofVoronoisplineswasrstproposedbyVanDeVille,etal[ 86 ],inwhichitwascalledhex-splines.Hex-splinesarepreciselytheVoronoisplinesforthehexagonallattice.Inthispioneeringwork,theyderivedFouriertransformsoftheirproposedhex-splinesusingasetofbuilding-blockfunctions.Thederivationofthebuilding-blockfunctionsbecomesextremelycomplexforgenerallatticesspeciallyin3-Dorhigherdimensionallatticesandhencenotpractical.Thisisperhapsthereasonthattheideabehindhex-splineshasnotyetbeengeneralizedtootherlattices.TheideaofVoronoisplinesasanextensionofhex-splinestoBCCandFCClatticeswererstconsideredin[ 35 ];however,aclosed-formsolutionfortheconvolutionoftheVoronoicellsremainedadifcultproblemtilltheintroductionofproposedmethodin[ 62 ].An 31

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ad-hocdiscretizationofBCC'sVoronoicellwasconsideredin[ 18 ]whereadiscreteconvolutionwasusedtoroughlyapproximatethecontinuous-domainconvolutionoftheBCC'sVoronoicell.Incontrast,theapproachproposedbytheauthorin[ 62 ]offersanexact,closed-form,solutionfortheBCCandFCClatticestogetherwithall2-Dand3-Dlattices1.Asitwillbediscussedinthesequel,Voronoisplinesleveragetheboxsplinetheoryforexactclosed-formcharacterization. 3.1.1VoronoiSplinesDenitionThenearestneighborinterpolationinthemultivariatesettinginvolvestheVoronoicellofthesamplinglattice(i.e.,allofpointsinsidetheVoronoicellofalatticepointareclosertothatlatticepointthananyotherlatticepoints).Inthemultivariatesetting,thesampledvalueatalatticesiteisassignedtoallofthepointsthatareinsidetheVoronoicellofthatlatticesite. Figure3-1. Voronoisplinesforthehexagonallattice.LdenotestheindicatorfunctionoftheVoronoicellofthelatticeL. 1Theself-convolutionoftheVoronoicelloftheFCClattice(i.e.,secondorderVoronoisplineonFCC)intermsofboxsplineswascarriedoutindependentlybyDr.M.Kimpreviously. 32

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HencewecanconstructVoronoisplinesasfollows:ChoosetheindicatorfunctionoftheVoronoicellofthelatticeasthesimplestVoronoisplineandthehigherorderVoronoisplinesareconstructedbyitssuccessiveself-convolutions: V0(x)=L(x)=8>><>>:1=jdetLjx2Voronoi(L),0otherwise.Vn=V0Vn)]TJ /F7 7.97 Tf 6.58 0 Td[(1(3)SincethevolumeoftheVoronoicellofalatticeisequivalenttojdetLj,thenormalizationensuresthatRV0=RVn=1.TheconvolutionconstructionimpliesthatVn=V0V0arecomposedofpiecewisepolynomialswhichareproducedbythemultidimensionalintegrationofconstantfunction(i.e.,V0)intheconvolution.Figure 3-1 demonstratesV0,V1andV2fortheexampleofahexagonallattice.Despitethesimpleconvolution-baseddenition,characterizingandevaluatingthepiecewisepolynomialsofVoronoisplinesareverycomplexsincetheboundariesofintegration(inconvolution)arecomplexandinvolvethegeometryofV0(i.e.,theVoronoicell).Unlikethe1-Dcase,wheretheconvolutionandintegrationcanbeeasilycomputedtoproducepiecewisepolynomialrepresentationofB-splines,piecewisepolynomialcharacterizationofVoronoisplinesarecomplex.TheexactpolynomialrepresentationofthesecondorderVoronoisplinesonBCCandFCChasbeencarriedoutanddiscussedinChapter 4 .ThehigherorderVoronoisplineshavehigherapproximationpoweranddegreesofcontinuityatthecostofcomputationalcomplexitysincethesupportofthekernelgrowswithn.SimilartoB-splines,theorderisadesignparameterthatisdecidedforagivenproblem,basedonthedesiredsmoothness,approximationpower,andcomputationalefciency.ForaCartesianlattice,thisconstructioncoincideswithtensor-productB-splines.Fornon-Cartesianlattices,thisconstructionprovidesnon-separablebasisfunctionsthatarepiecewisepolynomial. 33

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ABCFigure3-2. Voronoisplines:sumofboxsplines.A)VH0.B)M[1,1,2,2]orM[2,2,0].C)M[1,1,2,3]orM[2,1,1]. 3.1.2CharacterizationandRelationtoBoxSplinesThemainideahereistoexploitthegeometricpropertiesofVoronoicellsoflattices(i.e.,Theorems 3.3 ,and 3.2 )tomakethecharacterization,algebraandderivationofpolynomialpiecesfeasible.ThegeometricpropertiesofVoronoicellsmakethecomputationofVoronoisplinestractableforarbitrarylattices.Forillustrationpurposes,wedemonstratethisideaforthe2-Dhexagonallatticeandthenshowthatthistechniqueappliestoall2-D,3-Dandhigherdimensionallattices. Theorem3.1. ThenthorderVoronoisplineforthehexagonallatticeisasumofanitenumberof(upto)three-directionboxsplines. Proof. ThehexagonwhichistheVoronoicellofthehexagonallattice,hasthreezones(i.e.,setsofparalleledgesinsection 3.1.2.1 ),namely1,2and3: H=[1,2,3]=1 22640p 3=2)]TJ 9.3 9.95 Td[(p 3=21)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2375.(3)AsillustratedinFigure 3-2 thesezonescleavethehexagonintothreeparallelepipedseachofwhichisasimpleboxspline: VH0=H=1 3)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(M[1,2]+M[1,3]+M[2,3].(3) 34

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GiventwoboxsplinesM1andM2,wehave:M1M2=M[1[2].ForexamplethesecondorderVoronoispline,VH1=VH0VH0,canbewrittenas(Figure 3-2 ): VH1=1 32)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(M[1,2]+M[1,3]+M[2,3]2=1 9)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(M[1,1,2,2]+M[1,1,3,3]+M[2,2,3,3]+2 9)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(M[1,1,2,3]+M[1,2,2,3]+M[1,2,3,3].(3)Lettherepetition(row)vector=[1,2,3],andMdenotetheboxsplineinwhich1isrepeated1times,2isrepeated2times,and3isrepeated3times,thenthenthorderVoronoisplineforthehexagonallatticeisgivenby: VHn)]TJ /F7 7.97 Tf 6.58 0 Td[(1=)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(VH0n=1 3n)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(M[1,1,0]+M[1,0,1]+M[0,1,1]n.(3)Usingthemultinomialtheorem,higherorderVoronoisplinesforthehexagonallatticecanbeexpandedintermsofsumofthree-directionboxsplineswithmorerepetitionsofdirections: VHn)]TJ /F7 7.97 Tf 6.58 0 Td[(1=1 3nXjj=n0B@n1CAM[H],(3)wherejj=1+2+3,0B@n1CA=n! 1!2!3!andH=266664110101011377775.TherowsofHaretherepetitionsetinboxsplinedecompositionofVH0asin( 3 ).NotethateachrowofHcorrespondtooneparallelepipedinthedecompositionshowninFigure 3-2 .The1'sineachrowindicatethezonesofthehexagonthatformtheparallelepipedcorrespondingtothisrow. Inthebivariatesetting,theVoronoicellsarecombinatoriallyequivalenttoahexagon(Figure 3-3 )orasquarewhicharetheonlytwoobjectsthatcantessellatethe2-D 35

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Figure3-3. Ageneral2-Dlattice.Thefundamentaldomain(Voronoicell)ofthelatticeisillustratedanddividedintothreeparallelepipeds. space.Inotherwords,all2-DlatticeshaveVoronoisplinesthatcanbeconstructedbytheproposedapproach.Forhigherdimensionallattices,thefollowingremarkablegeometricresultsconnectingVoronoicells(Voronoipolytopes)andzonotopesentitleustogeneralizeTheorem 3.1 and( 3 ). 3.1.2.1ZonotopesPolytopesthatarethesupportoftheshadowofhigherdimensionalhypercubesarecalledzonotopes.Azonotopeisapolytopewherebothitselfaswellasallofitsfaces(anyco-dimension)exhibitpointsymmetrywithrespecttotheircenters(e.g.,arectangleissymmetricwithrespecttoitscenter,butatriangleisnot).Athree-dimensionalzonotopeisoftenreferredtoasazonohedron.Itturnsoutthatapolyhedronisazonohedronifandonlyifitsfacesare(center)symmetric[ 16 ].Thesetofedgesofazonohedroncanbepartitionedintozonesinsuchawaythatalledgesinonezoneareparallel[ 45 ].Theedgesinonezoneofazonohedronconstructabelt-likestructurearoundthepolyhedron.Forinstance,theelongatedrhombicdodecahedronisavezonezonohedron,oneofwhichhasbeenillustratedwithredcolorinFigure 3-4 (C).Azonohedroncanbespeciedbyarepresentativevectorfromeachzone.Theserepresentativevectorscan,inturn,beusedtoconstructaprojectionmatrixthatyields 36

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thepolyhedronfromthecorrespondinghypercube.Thedimensionofthehypercubeissimplythenumberofzonesinazonohedron. Theorem3.2. Anyzonotopecanbedecomposedintoanitenumberofparallelepipeds,thebasesofthezonotope.Thereisanexistenceprooftothistheoreminboxsplinestheory[ 27 ];hence,wecanwritetheindicatorfunctionoftheVoronoicell,whichisazonotope,asasumofsimpleboxsplines(squaredirectionmatrix). 3.1.2.2Spacetessellations,VoronoipolytopesandzonotopesCoxeter[ 16 ]wasthersttonotethattheVoronoicellsofall2-Dor3-Dlatticesarezonotopes.Erdahl[ 37 ]hasrecentlyextendedonthisobservationtohigherdimensionallattices.Allofthehigher(than3-D)dimensionallatticesthatsatisfytheso-calledDicingproperty2,theirVoronoicellsarealsozonotopes.ThistopiciswidelydiscussedingeometryandiscloselyrelatedtotheVoronoiconjecture[ 29 57 58 ]. Theorem3.3. TheVoronoicellofanylatticeissymmetricwithrespecttoitscenter;moreover,allofitsfacets(i.e.,facesofco-dimension1)arecentrallysymmetric[ 29 ].Forhigherdimensionallattices,giventheysatisfythedicingproperty[ 37 ](suchasAn,An,Dnandotherrootlattices),theirrespectiveVoronoicellsarezonotopesandaparallelepipeddecompositionexists;hence,wewillbeabletoconstructVoronoisplinesusingtheproposedboxsplineexpansion.Inthe3-Dsetting,therearevecombinatoriallydifferentVoronoicellsthattranslationallytile(tessellate)thespace.ThesepolyhedraarecommonlyreferredastheFedorov'sparallelohedra[ 72 ].TherstoneisacubewhosecorrespondingVoronoisplinesarepreciselytensor-productB-splines.Rhombicdodecahedronandtruncated 2Afamilyofequi-spacedhyperplanescutsEdintoslabsofequalthickness.Bytakingdsuchhyperplanefamiliesthatareindependent,Edisdicedintoequalparallelepipedswiththeverticesformingad-dimensionallattice.[ 37 ]whereEddenotesad-dimensionalspace. 37

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ABCD EFigure3-4. Categorizationof3-Dlatticesbasedontheirfundamentaldomainsthattessellate.A)Rhombicdodecahedron(VoronoicellofFCC).B)Truncatedoctahedron(VoronoicellofBCC).C)Elongatedrhombicdodecahedron.D)Hexagonalprism.Thesepolyhedraareallzonotohedraandcanbedecomposedintoparallelepipeds.E)Showsthedivisionoftherhombicdodecahedronandtruncatedoctahedronintofourandsixteen3-Dparallelepipedsrespectively. octahedronareVoronoicellsofFCCandBCClatticesandareshowninFigure 3-4 .ThetwoothertilingpolyhedraareelongatedrhombicdodecahedronandhexagonalprismwhichareVoronoicellsofelongatedBCCandahexagonallattice.ThesefourtypesofVoronoicellsarezonohedra,hencetheycanalsobecleavedintoparallelepipeds.Allothernon-Cartesian3-DlatticeshaveVoronoicellsthatarecombinatoriallyequivalenttooneofthesefourtypes;hence,wehaveapracticalmethodtoconstructVoronoisplinesforall3-Dlattices. 38

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Withthisknowledgeonecaneasilyverifythatthecubeisazonohedronwith3zones,rhombicdodecahedronandhexagonalprismhave4zones,elongatedrhombicdodecahedronhas5zonesandtruncatedoctahedronhas6zones.Sincezonotopescanbeconvertedtoeachotherbycollapsingalloftheedgesinonezone,thetruncatedoctahedroncanbeconsideredasthemostgenericzonohedron.Infact,theVoronoicellofagenericlatticein3-Discombinatoriallyequivalenttothetruncatedoctahedron[ 53 ],andtheVoronoicellsofFCC,Cartesian,hexagonalprismandelongatedBCClatticesaredegeneratecasesoftheVoronoicelloftheBCClattice(i.e.,truncatedoctahedron). 3.2VoronoiSplinesonOptimalSamplingLatticesWhileVoronoisplinesaredenedforgeneralmultidimensionallattices,wespecicallycharacterizethemhereforoptimalsamplinglatticesin3-dimensions.SinceBCCandFCClatticesareofpracticalimportance[ 32 44 51 59 74 ],weconsidertheexplicitcharacterizationoftheirVoronoisplines.TheBCCandFCClatticepointscanbegeneratedbyintegerlinearcombinationsofthecolumnsoftheirgeneratingmatrices: B=266664)]TJ /F4 11.955 Tf 9.3 0 Td[(1111)]TJ /F4 11.955 Tf 9.3 0 Td[(1111)]TJ /F4 11.955 Tf 9.3 0 Td[(1377775,F=266664011101110377775.(3) Theorem3.4. VoronoisplinesontheFCClatticeareformedbyasumof(upto)four-directionboxsplines. Proof. TheVoronoicelloftheFCClatticeisarhombicdodecahedronwhichisazonohedronof4zones(edgesinFigure 3-5 (A)).Thezonesofthiszonotopearethecolumnsofthefollowingmatrix: F=1 22666641)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 9.3 0 Td[(11)]TJ /F4 11.955 Tf 9.3 0 Td[(11)]TJ /F4 11.955 Tf 9.3 0 Td[(11)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 9.3 0 Td[(111377775.(3) 39

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Table3-1. ParallelepipeddecompositionofVoronoicellforFCClattice,intobuilding-blockboxsplines.TheboxsplinesareconstructedbyzonevectorsineachparallelepipedPi.Eachrowoftherepetitionmatrix,F,denoteswhichthreezonesfromFformthatparallelepiped. P)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(FF P11110P21101P31011P40111 ThisVoronoicellcanbedecomposedinto3-DparallelepipedswhichareformedbyanythreeofthefourzonesinF.TheFCCVoronoicellparallelepipeddecomposition,asillustratedinFigure 3-4 (E),canbeencodedbytherowsoftherepetitionmatrix,F,documentedinTable 3-1 :Hence,therstorderVoronoisplineiswrittenas:VF0=1 4)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(M[1,1,1,0]+M[1,1,0,1]+M[1,0,1,1]+M[0,1,1,1], (3)wherethesubscriptrowvectorindicatestherepetitionofdirections2F.Similarto( 3 ),usingthemultinomialtheorem,thenth-orderVoronoisplinecanbeexpandedbythefour-directionboxsplines:VFn)]TJ /F7 7.97 Tf 6.59 0 Td[(1=1 4nXjj=n0B@n1CAM[F]. (3)Notethatjj=P4k=1k.ThesupportoftherstthreeordersofVoronoisplinesonFCClatticeshasbeenshowninFigure 3-6 (B). Similarlythetruncatedoctahedron,theVoronoicelloftheBCClattice,iscomposedofthefollowing6zones(edgesinFigure 3-5 (B)): B=1 22666641)]TJ /F4 11.955 Tf 9.29 0 Td[(1110011001)]TJ /F4 11.955 Tf 9.3 0 Td[(1001)]TJ /F4 11.955 Tf 9.3 0 Td[(111377775.(3) 40

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ABFigure3-5. VoronoicellsoftheFCCandBCClattices.A)Rhombicdodecahedron(VoronoicellofFCClattice).B)Truncatedoctahedron(VoronoicellofBCClattice). UnlikethezonesinF,noteverythree-zoneofBformaparallelepiped.Fromthetotalof0B@631CA=20three-zonesofB,fourofthemareco-planarandhencedonotforma3-Dparallelepiped(e.g.,1,4and5).Therefore,truncatedoctahedronisdecomposedinto16parallelepipedwhichareformedbytheremainingnon-co-planarthree-zones.Theparallelepipeddecompositionofthetruncatedoctahedronismoreinvolvedsincesomeoftheparallelepipedsneedtobeshiftedfromtheorigin.Therequiredtranslationcanbeencodedascertainf0,1gcombinationofthezonesontheBCClattice,B.Thesef0,1gcombinationsaregivenasarowvectors,,inTable 3-2 alongwiththerepetitionvectors. Theorem3.5(GeneralTrivariateVoronoiSplines). VoronoisplinesontheBCClatticeareformedbyasumof(upto)six-directionboxsplines.Sinceageneral3-DlatticehasaVoronoicellthatiscombinatoriallyequivalenttothatoftheBCClattice,alltrivariateVoronoisplinesareidentiedbyboxsplinesofuptosixdirections. 41

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Table3-2. ParallelepipeddecompositionofVoronoicellforBCClattice,intobuilding-blockboxsplines.TheboxsplinesareconstructedbyzonevectorsineachparallelepipedPi.Eachrowoftherepetitionmatrix,B,indicateswhichthreezonesfromBformthatparallelepiped. P)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(BB P1110100001011P2100011001100P3001110000001P4011100000011P5110001001110P6001011000100P7101010000100P8100101011010P9010110000001P10111000000110P11110010000100P12000111010000P13001101010010P14010011000100P15101100010010P16011001000110 Proof. Sincethereare16parallelepipedsthatformthetruncatedoctahedron,therepetitionrowvectorhas16elementseachofwhichdenotestherepetitionofathree-directionsimpletrivariateboxspline(e.g.,M[1,2,4]).Thereare6zones(directions)thatarecommontoall16ofthesethree-directionboxsplineswhicharegivenbycolumnsofB.TheVoronoisplinesontheBCClatticearegivenbysix-directionboxsplinesspeciedbyrepetitionsofdirectionsgivenbythismatrix.VBn)]TJ /F7 7.97 Tf 6.59 0 Td[(1=1 16nXjj=n0B@n1CAM[B]()]TJ /F6 11.955 Tf 17.93 0 Td[(B()T). (3)Notethatjj=P16k=1k.SimilartoFCCcase,thesupportoftherstthreeordersofVoronoisplineshasbeendepictedinFigure 3-6 (A). 3.3VoronoiSplines'PropertiesonaGeneralLatticeDeterminationoftheapproximationorderoftheshift-invariantspacesgeneratedbyasingletonoranitenumberofgeneratorsisanintricatesubjectandoneofthemaintopicsinshift-invariantapproximationtheory.TheclassicalresultofStrang-Fix[ 75 ]fordeterminingtheapproximationorderofaspacehasbeenrevisedandgeneralizedinseveraldirections([ 24 25 46 ]forsurveys).SinceVoronoisplinesarebuiltbasedon 42

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ABFigure3-6. V-splinesaredenedastheselfconvolutionoftheindicatorfunctionofthelattice.A)Truncatedoctahedron(i.e.,theVoronoicellofBCC).B)Rhombicdodecahedron(i.e.,theVoronoicellofFCC). successiveconvolutionsofV0,theylendthemselvestoaneasyanalysisofStrang-FixconditionsbasedonthoseofV0.SincetheVoronoicellofalatticetessellatesRd,the(local)partitionofunityconditionistriviallymetbythecompactlysupportedV0:jdetLjXk2LZdV0()]TJ /F6 11.955 Tf 17.93 0 Td[(k)=1. (3)Convolving( 3 )withV0showsthatthe(local)partitionofunityisalsosatisedforV1(alsoforVn).ApplyingthePoissonsumformula(whichcanbeappliedwithoutacontinuityrequirementusingtestfunctions[ 20 ])toV0,weobserve:jdetLjXk2LZdV0(k)=1=Xk02^LZd^V0(k0). (3) 43

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Since^V0(0)=RV0=1,weconcludethat:Xk02^LZdf0g^V0(k0)=0. (3)ThepartitionofunityandthePoissonsumformulacanalsobeappliedtoV1,toconclude:Xk02^LZdf0g^V1(k0)=Xk02^LZdf0g^V20(k0)=0, (3)whichtogetherwith( 3 ),allowsustoconcludethat^V0(k0)=0,forallk02^LZd)-221(f0g. (3)ThisestablishesthefactthatV0hasatleastonevanishingmomentonaliasingfrequencies,onthereciprocallattice,^LZd)-131(f0g.Todemonstratethatthereisexactlyonevanishingmomentonaliasingfrequencies,werstobservethatV0cannotreproducea(nontrivial)linearpolynomial.ThisisapparentasSV0isaspaceofpiecewise-constantfunctions.Then,sinceV0isacompactlysupportedfunction,itsvanishingmomentscannotexceedthedegreeoftheleastdegreepolynomialthatisnotcontainedinSV0[ 28 ,Theorem3.3].Bythemultiplication-convolutiontheorem,Vnhas(n+1)vanishingmomentsonthealiasingfrequencies,hencetheshift-invariantsplinespacegeneratedbyshiftsofVnonthelatticeLhas(n+1)thorderofapproximation[ 20 25 ]. 3.3.1ContinuityByconstructionitiseasytoverifythattherstorderVoronoisplineisdiscontinuousV02C)]TJ /F7 7.97 Tf 6.59 0 Td[(1.ThecontinuityofVncanbederivedbytheexpansionintermsofboxsplines.SinceVnisasumofanumberofboxsplines,itscontinuityisequivalenttotheleastcontinuousboxsplineintheexpansion(e.g.,( 3 )).Asdiscussedin( 2 ),thecontinuityofaboxsplineMdependsontheminimumnumberofdirectionsthatneedtoberemovedfromsothattheremainingdirectionsdonotspanRd.Amongthe 44

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differentrepetitionvectorsthatdeterminetheboxsplinesthatconstituteVnundertheconstraintofjj=n,therepetitionvector=[n,0,...],orgenerally=[0...,n,0,...]identifyboxsplineswithfewestdistinctdirections(e.g.,Figure 3-2 (B)).Therefore,theserepetitionsetsidentifyboxsplineswiththegreatestnumberofdirectionstoberemovedforthecontinuitytest(i.e.,( 2 ));hencetheseboxsplinesaretheleastcontinuousintheVoronoisplineexpansion.Theseleastcontinuousboxsplinesareshearedtensor-productB-splinessincetheirdirectionsetcontainonlythedirectionsthatarefromasingleparallelepipedfromtheVoronoicell'sparallelepipeddecomposition.Therefore,thecontinuityofVnisthesameasasimpleboxspline(N=ddirections)whosedirectionsarerepeatedbyn;hence,Vn2Cn)]TJ /F7 7.97 Tf 6.58 0 Td[(1. 3.3.2SupportThesupportofVnispreciselythesupportoftheVoronoicell(i.e.,V0)scaledupbyafactorofn+1.ForinstancethesupportofV1andV2aregivenbythesupportofV0scaledby2and3respectively.ThisisduetothefactthatthesupportoftheconvolutionoftheindicatorfunctionsoftwopolytopesistheMinkowskisumofthetwopolytopes[ 9 ].Minkowskisumofaconvex,symmetric,setcontainingtheoriginwithitselfenlargesthatsetbyafactorof2.SinceVoronoicellsareconvexandsymmetricwithrespecttotheircenters(origin),thesupportofV1istwiceaslargeasV0.HencethesupportofVnisenlargedbyafactorofn+1fromthesupportofV0.Thismeansthatthe(hyper)volumeofthesupportofVnisgivenby(n+1)djdetLj,sincetheVoronoicellofalatticehasvolumeofjdetLj.SincetheVoronoisplines,coincidewithtensor-productB-splineswhentheunderlyinglatticeLisaCartesianlattice,theapproximationpropertiesofgeneralVnagreewiththatofthetensor-productB-splines. Table3-3. ApproximationpropertiesofsplinespacesgeneratedbyVoronoisplines SplineSpaceDegreeApproxOrderContinuityFootprint SVnndn+1Cn)]TJ /F7 7.97 Tf 6.59 0 Td[(1(Rd)(n+1)djdetLj 45

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3.3.3FourierTransformofVoronoiSplinesThemainresultofsection 3.1.2 isestablishingthefactthatVoronoisplinescanbeviewedasasumofboxsplinesormulti-boxsplines.Therefore,FouriertransformsofVoronoisplines(^Vn)canbeexplicitlyderived,foreachlattice,bytheFouriertransformsoftheconstitutingboxsplinesusing( 2 ).SincetheVoronoicellofalatticeissymmetric(theorem 3.3 ),^Vnarereal-valuedfunctionsasVnhavecompactsupport.TherstorderV-spline,^VL0,canbeexplicitlyderivedasthesumoftheFouriertransformsofbuilding-blockboxsplines.SincehigherorderV-splinesareconstructedbytheconvolutionoftherstorderV-splinebyitself,wecanusetheconvolutiontheoremtoderivethefrequencydomainrepresentationofVLn.WerstrecallthattheFouriertransformoftheboxfunctionisgivenbythesincfunction:box(x),sinc(!).FortheCartesianlattice,theFouriertransformofVC0,theindicatorfunctionofthecentered-unitcube,isgivenbythetensor-product:^VC0(!)=sinche1,!isinche2,!isinche3,!i (3)Letdenotetheindicatorfunctionoftheunitcube,corneredattheorigin.Thentheindicatorfunctionofaparallelepipedthatisformedbythree(zone)vectors2Pis3obtainedbyalineartransformationof:P(x)=(P)]TJ /F7 7.97 Tf 6.59 0 Td[(1x).Usingthemultivariatescalingtheoremwehave:f(P)]TJ /F7 7.97 Tf 6.59 0 Td[(1x),jdetPj^f(PT!).Therefore,weestablishtheFouriertransformofthebuilding-blockboxsplinesofVoronoisplinesas: ^P(!)=jdetPjexp(2jhcP,!i)Y2Psinch,!i,(3) 3PcorrespondstozonesgivenbyarowofFinTable 3-1 orBinTable 3-2 46

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where,cP:=1 2P2PisthecenteroftheparallelepipedPthatcorrespondstothephase-shift.Hence,wehave: ^VF0(!)=1 4XP2P(F)jdetPjexp(2jhcP,!i)Y2Psinch,!i(3)SinceVF0issymmetricwithrespecttotheorigin(apropertyoftheVoronoicellsharedbyallzonotopes),itsFouriertransformmustberealvalued.Therefore,wehave:^V0(!)=1 2(^V0(!)+conj^V0(!)): ^VF0(!)=1 4XP2P(F)jdetPjcos(2hcP,!i)Y2Psinch,!i.(3)Similarly,theFouriertransformforthecaseintheBCClatticecanbedenedas: ^VB0(!)=1 16XP2P(B)jdetPjcos(2hP,!i)Y2Psinch,!i.(3)Here,PencodeboththeshiftsrequiredtoformthetruncatedoctahedronrequiredforeachparallelepipedPandadditionalshifttopositionitscentertotheorigin: P=cP+BPT.(3)ThisallowsustoderivetheFouriertransformofhigherorderV-splinesfortheselatticesby:^VLn(!)=^VL0(!)n+1. 47

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CHAPTER4RECONSTRUCTIONUSINGVORONOISPLINESWhilethesampling-theoreticalfeaturesofBCC,FCCandCartesianlatticeshavebeendiscussedonatheoreticallevel[ 32 51 64 67 ],theirperformanceforsamplingandreconstructionoftrivariatefunctionscanonlybetestedwithacomparablereconstructionframeworkacrosstheselattices.TherehasbeencomparisonsofBCCversusCartesianin[ 33 36 ]andFCCversusCartesianin[ 49 ].Eventhoughthereconstructionmethodsusedoneachlatticehavebeenproventobeappropriateforthatlattice,weneedreconstructionmethodsthatareidenticalamongtheselattices;V-splinesallowustocomparethesampling-theoreticperformanceoftheselatticesbycomparingthesignalsthatarereconstructedfromeachlattice.Vnoneachlatticehasthesamesmoothness,approximationorder,supportandpolynomialdegree;hence,weconsiderthecomparisonbasedonV-splinestheleastbiasedexaminationofsampling-theoreticfeaturesofBCC,FCCandCartesianlattices.ForinstanceV1ontheBCC(VB1)andFCC(VF1)latticesconstitutethebestmatchtothetrilinearB-splineontheCartesianlatticesincetheyareallconstructedbyconvolvingthelatticeVoronoicellwithitselfonce.Forasingleevaluationpoint,allthreetake8samplepoints(fromthelattice)forreconstructioninaC0,2nd-orderspace.SimilarlyVB2andVF2arethecounterpartstotriquadraticB-splineontheCartesianlatticeandtake27pointsfortheirrespectivereconstructionsinC1,3rd-orderspaces. 4.1VoronoiSplinesEvaluation,Experiments,ResultsandDiscussionSinceV-splinesarecomposedofanumberofboxsplines,wecanresorttoboxsplineevaluationmethodstoevaluateV-splines.Aspointedoutearlier,VFniscomposedofthree-directionboxsplines(i.e.,shearedtensor-productB-splines)andfour-directionboxsplineswithdifferentrepetitions.VBniscomposedofthree-directionboxsplines(i.e.,shearedtensor-productB-splines),four-direction,ve-directionandsix-directionboxsplineswithvariousrepetitionsets.Therefore,onecancharacterizethepolynomial 48

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piecesoftheseboxsplinesforefcientevaluation[ 12 36 ].ItisimportanttonotethateventhoughthenumberofboxsplinesconstitutingVnincreaseswithn,thepolynomialpartitionofthesupportofVndoesnotbecomemorecomplex.Thisisduetothefactthatthe(hyper)planesthatdelineatethesupportofboxsplines([ 27 ,(I.37)])inVnarecomposedofthezonesinV0,andincreasingndoesnotintroducenewzones.BuildingexplicitpiecewisepolynomialrepresentationofVoronoisplinesispossibleandwillbediscussedindetailsinthesequel.OnecanalsoresorttostandardevaluationtechniquesforboxsplinesandevaluatethemseparatelyforaV-splineevaluation.ThemostcommonlyusedmethodforevaluatinggenericboxsplinesatarbitrarypointsisthroughdeBoor-Holligrecurrencerelation[ 27 ].Therecursiveevaluationofboxsplinesispronetonumericalinstabilities[ 22 ].Kobbelt'sevaluationapproachminimizestheinstabilityissues[ 50 ].Moreover,varioussubdivisionschemes[ 27 ]canbeusedforevaluationofboxsplinesonaregularmeshthatcanbeusedforresamplingoperations.InordertoexaminethereconstructionusingV-spline,wehavemodiedvuVolumeray-caster[ 87 ]torenderimagesfromthereconstructedsignalfromFCC,BCCandCartesiansampledvolumetricdatasets.Thevolumetricdatasetswerereconstructedasasplines2SVn:s=cVn=Xk2LZ3ckVn()]TJ /F6 11.955 Tf 17.93 0 Td[(k) (4)whoseisosurfaceswererenderedintheray-caster.LdenotesCartesian(identitymatrix),BCC(B)orFCC(F)lattice.Thesplinecoefcientswerechosentobethesamplevaluesck=f(k),asuboptimalchoicewhichiscommoninrenderingandimageprocessing[ 54 ].Thesplinecoefcientscanbechosentooptimizetheapproximationerrorortoguaranteereproductionoflow-orderpolynomials[ 13 34 ]whichwillbediscussedindetailsinsection 4.2 .Wehavechosenafrequency-modulationsyntheticdataset,calledML,rstproposedbyMarschnerandLobbin[ 54 ]asabenchmarkforourcomparisons. 49

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Figure4-1. Marschner-Lobbfunctionusedasasyntheticbenchmark. TheFigure 4-1 wasrenderedbyevaluatingtheexplicitfunctiondiscussedin[ 54 ].Forvisualizationpurposestheisosurfaceof.5waschosenforextractingasurfacefromthisvolumetricdataset.Thefunctionwassampledatthecriticalresolutionof414141ontheCartesianlatticeandatanalmost1equivalentsamplingontheBCClatticeof323264andavolumeof2525100ofFCCsamples.Thisresolutionischosenasthecriticalsamplingratesince98%oftheenergyofitsspectrumiscapturedbythissamplingfrequency;henceitservesasapracticalNyquistfrequencyforthissignal[ 54 ].TheimagesinFigure 4-9 arereconstructedandrenderedusingtheV1oneachlattice.Whiletheeffectofcriticalsamplingresolutionisvisibleonallthreelattices,theCartesianlatticeexhibitsthestrongestaliasingartifacts.Figure 4-3 illustratesthesamedatasetsreconstructedwithV2oneachlattice.ThealiasingartifactsareminimizedbytheFCClattice.BothBCCandFCClatticesperformbetterthanCartesianintermsoftheerrorpresentinthereconstruction,withaslightadvantageforBCC.TheimagesinthethirdrowinFigure 4-3 documentthecorrespondingerrorimagesthatareobtained 1SinceanitesamplingofavolumecannotproducetheexactsamenumberofsamplesfortheBCC/FCCandCartesiansamplingpatterns,forourdiscreteresolutions,wechosetheresolutionsconservativelyinfavoroftheCartesiansampling.Therefore,theactualsamplingdensityintheCartesiansampleddatasetsisslightlyhigherthantheBCC/FCCsamplingdensity. 50

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oftheangularerrorthatisthedifferencebetweenthetruenormalandthereconstructednormal(ontheisosurface).Thegrayvalueof255(white)denotesanangularerrorof.3radiansbetweenthecomputednormalandtheexactnormal.ItisclearfromthisgurethattheBCCresultsinthemostaccuratereconstructionamongthethree(darkererrorimage),andFCCminimizesthealiasingeffects.ThisperfectlymatchesthetheoreticalexpectationthattheduallatticesforBCCandFCCaretheoptimalspherepackingandspherecoveringlatticesrespectively.Table 4-1 documentstheerrorspresentinthereconstructedsignalcomputedovertheentirevolume(asopposedtothesurfacesillustratedinFigure 4-3 ).TheerrorsweremeasuredinL2normanddemonstratetheincreasedaccuracyinreconstructionsofferedbyBCCandFCClattices.TypicalvolumetricdataarescannedonandreconstructedfromtheCartesianlattice.Asuitable(anti-aliasing)pre-lteringstepisappliedtolimitthespectrumofthesampleddatawithintheNyquistregion,whichistheVoronoicellofthereciprocallattice.FortheBCC/FCClattices,thesecellsareclearlydifferentfromtheoneoftheCartesianlattice.Therefore,theultimatetestoftheBCC/FCCreconstructiononreal-lifedatasetscannotbeperformeduntilthereareacquisitionsystemsdevelopedforBCC/FCCsampling.Nevertheless,weconstructedcomparableBCC,FCCandCartesiandatasetsbymerelysubsamplingafairlydenselysampledCartesiandatasetontolow-resolutionsoneachlattice.Thedatageneration(subsampling)schemeensuresthatallthreelow-resolutiondatasetsareofthenearly-identicalsamplingdensities[ 36 ].Asapracticalcase,weexaminedthecarpshdataset.Theoriginaldatasethasaresolutionof256256256(Figure 4-4 )whichrepresentsthegroundtruthandthelow-resolutiondatasetshaveabout16%ofthehighresolutiondataontheirrespectivelattices.ThesubsampledCartesianvolumehasaresolutionof140140140andthesubsampledBCCvolumehasaresolutionof111111222,whiletheFCCvolume 51

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AVC1BVB1CVF1 DVC1EVB1FVF1Figure4-2. ReconstructionresultsofMarschner-LobbdatasetusingsecondorderVoronoisplines.TheMarschner-Lobbdatasetdirectlysampledatacriticalrate[ 54 ]withcomparableresolutionsontheCartesian,theBCCandtheFCClatticesreconstructedwithV1.ImagesA-C(rstrow)illustratehead-oneviewandimagesD-F(secondrow)showaclose-uprenderingofthesamedataset. hasaresolutionof8888352.ThesevolumeswerealsorenderedwiththeV1oneachlatticeandtheresultsareshownintherstrowofFigure 4-5 ,whilethesecondrowdisplaystheresultsofreconstructionusingV2oneachlattice.Again,theseresultsshowthesuperiorityoftheBCC/FCCsamplingschemesincetheCartesiansubsampleddatasetmissesthetailnsandmostoftheribs.Table 4-1 documentstheerrorspresentinthereconstructedsignalcomputedovertheentirevolumesinFigure 4-5 52

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AVC2BVB2CVF2 DVC2EVB2FVF2 GVC2HVB2IVF2Figure4-3. ReconstructionresultsofMarschner-LobbdatasetusingthirdorderVoronoisplines.TheerrorimagesG-I(thirdrow)documentstheangularerrorwhichisthedifferencebetweenthereconstructednormal(ontheisosurface)fromtheexactnormal.Thegrayvalueof255(white)denotesanangularerrorof.3radiansandblackpixelsdenotezeroerror.WhileBCChasthesmallestreconstructionerror(darkest),theFCCerrorimageshowsresiliencetoaliasingpatterns. 53

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ABFigure4-4. ThegroundtruthCarpshdataset.TheoriginalCartesiandataset,with16,777Kpoints,reconstructedusingdifferentreconstructionkernels.A)TrilinearB-spline(V1).B)TriquadraticB-spline(V2). Table4-1. TheratioofthereconstructionerrorspresentinBCCandFCCsignalstotheCartesianreconstructedsignal.ReconstructionerrorsmeasuredovertheentirevolumewiththeL2normforCartesian,BCC,andFCCdatasetssampledfromtheMLbenchmarkandtheCarpdatasetinFigure 4-4 .TheBCCsignalswerereconstructedwithVB1andVB2,theFCCsignalswerereconstructedwithVF1andVF2andCartesiandatasetswerereconstructedwithtrilinearB-spline(V1)andtriquadraticB-spline(V2).SmootherreconstructionofferedbyV2(non-interpolating)increasestheoverallerror,buttherelativeadvantagesofBCC/FCCoverCartesiandatasetsaremostlymaintained.MLCarpV1V2V1V2 BCC38.8063.410.561.28FCC39.2765.070.601.30Cartesian52.3378.950.941.55 BCC/Cartesian74%80%59%82%FCC/Cartesian75%82%63%83% 4.2QuasiInterpolationwithVoronoiSplinesTheapproximationorderofasplinespacecanbestudiedbyquasi-interpolation[ 20 78 ].Quasiinterpolationconvertsthesampleddatatosplinecoefcients.Giventhesampleddatacomesfromanypolynomialofuptocertaindegree,quasiinterpolationwouldguaranteeexactreconstruction.Polynomialreproduction,asmanyasthegeneratorfunctionallows,guaranteestheoptimalasymptoticbehavioroftheapproximation 54

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AVC1BVB1CVF1 DVC2EVB2FVF2Figure4-5. ReconstructionresultsforCarpshdataset.TheoriginalCartesiansampledCarpshdataset(Figure 4-4 )issubsampledto16%andreconstructedondifferentlattices.ThetailnsandribsaremoreaccuratelyreconstructedinBCCandFCCdatasets. error(i.e.,optimaln)whenthegeneratorfunctioniscompactlysupported[ 21 ].Incontrast,theoptimalapproximationerror,foraxedh,isonlyattainedbytheorthogonalprojection.Sincetheorthogonalprojectionisoftennotpractical,thequasiinterpolation,viapolynomialreproduction,offersanefcientalternativethatmaintainsthesameasymptoticdecayintheapproximationerror. 4.2.1OrthogonalProjectionvs.QuasiInterpolationInthissection,wediscussfurtherindetailswhyorthogonalprojectionisnotpracticalandwhyquasiinterpolationisanappropriatealternativewhenusingV-splinesasthereconstructionkernels.Before,wedescribetheapproximationorderasdenedin[ 21 ],oneshouldnotethatapproximationorderdenesboundsonthebehavioroftheapproximationerror(forthesplinespace)asthesamplingstephgetssmallerandsmaller. 55

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Fixingthesamplingstephandusinggeometricintuition,itiseasytoconcludethatthelowestapproximationerrorwouldbeachievedifonechoosestheelementofthesplinespace(i.e.,approximatedfunction)thatcorrespondstotheorthogonalprojectionoftheunknownfunctionontothesplinespace.Inotherwords,theorthogonalprojectionwillbetheclosestpointintheapproximationspacetotheunknownfunction(i.e.,thebestapproximationfortheunknownfunction).Notethatorthogonalprojectionofanarbitraryvector,vontoanothervector,uisgivenby:Pv=hv,ui hu,uiu.Consideringtheorthogonalprojectionofanunknownfunctionf(x)ontoasplinespacetobedenotedasPf(x),wecanwrite:Pf(x):=XkckVn(x)]TJ /F6 11.955 Tf 11.96 0 Td[(k)=cVn (4)WherecdenotesthesplinecoefcientscorrespondingtotheorthogonalprojectionandVncorrespondstotheV-splinesofordernwhoseshiftsarethebasisforthesplinespace.IfwecanspecifythecoefcientvectorcrepresentingPf,wecanconstructtheorthogonalprojectionofthefunctionfusing( 4 ).TherelationbetweenfandthecoefcientvectorofitsorthogonalprojectionintothesplinespacecanbebetterobservedintheFourierdomain.Theconvolutioninthespacedomaincorrespondstothemultiplicationinthefrequencydomain:cVn$^c^Vn.Therefore,theorthogonalprojectionofafunctionfontoasplinespacewhosebasisisVnisgivenby[ 21 ]: ^Pf=[^f,^Vn] [^Vn,^Vn]^Vn,(4)where,[^f,^Vn]denotesthebracketproductof^f,^Vnandisdenedas:[^f,^g]:x7!Pk22Zd^f(x+k)^g(x+k),consideringbothf,garerealfunctions.Bycomparing( 4 )and( 4 ),onecanconcludethat[^f,^Vn] [^Vn,^Vn]denotestheFouriertransformofthesplinecoefcientsoftheorthogonalprojection.Inapproximationtheory,thefunction^Vn [^Vn,^Vn]is 56

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denotedasdualbasistoVn[ 7 11 ]: ^Vdn(!)=^Vn(!) [^Vn(!),^Vn(!)]=^Vn(!) Pkj^Vn(!+2k)j2(4)Notethatthedenominatorin( 4 )isaperiodicfunction.NowwecancompactlywritetheFouriertransformofthesplinecoefcientsfortheorthogonalprojectionas:^c=[^f,^Vdn].Thereare2mainchallengesthatmakeorthogonalprojectionimpracticalwhichcanbeinspectedfrom( 4 ): Forgeneralchoiceofbasisfunctions(e.g.,B-splines,boxsplinesandV-splines),thedualbasisisnotnecessarilycompactlysupportedandneedstobetruncatedforpracticalpurposes. TheFouriertransformofthe(orthogonalprojection)splinecoefcientsincludestheFouriertransformofthefunctionbutinpracticetheonlyavailableinformationabouttheunderlyingfunctionfisitssamplepoints.Now,wecanintroducetheformaldenitionoftheapproximationorderofasplinespaceandtheimportanceofquasiinterpolationasapracticalequivalenttotheorthogonalprojection.TheapproximationorderofasplinespaceSh(withsamplingdistanceh)iskgiven[ 21 ]: 1. Forallsmoothf,dist(f,Sh)=O(hk)(lowerbound) 2. Forsomesmoothf,dist(f,Sh)6=o(hk)(upperbound),wherethedistanceismeasuredintheLp-norm(1p1).Fromtherstcriterion,onecanconcludethattheapproximationerrordecaysatleastasfastorfasterthanhkandthesecondcriterionmandatestheapproximateerrortodecaynofasterthanhk.Asin[ 7 11 21 ]theupperbounddiscussedabove,canbeonlysatisedbyorthogonalprojectionwhichisnotpractical.However,quasiinterpolationprovidesanalternativetotheorthogonalprojection(i.e.,bestapproximation)whichwillasymptoticallyattainthelowerboundoftheapproximationorderforcompactlysupportedkernels[ 21 ].ThemainideaofquasiinterpolationistondthesplinecoefcientsbyprelteringsamplepointstominimizetheL2normerrorbetweentheactualfunctionandthereconstructedfunction. 57

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Eventhough,quasiinterpolationdonotnecessarilymeettheupperbounddiscussedin[ 21 ]atagivenresolution,butasymptoticallyquasiinterpolationmeetstheoptimalapproximationorder. 4.2.2QuasiInterpolationUsingFourierTransformAquasi-interpolant,Q,involvesthetransformationoftheinputfunctionftofnviaaconvolution:fn(x):=(nf)(x),whereisacontinuousgeneralizedfunction(suchasDiracdeltaorsumofitsderivatives)withaboundedandlocalsupport.ThenQisdenedasalinearmapintotheLSIspaceoftheform:Qf(x):=Xkfn(k)Vn(x)]TJ /F6 11.955 Tf 11.96 0 Td[(k). (4)IfQreproducesallpolynomialsofdegreeslessthann+1(i.e.,Qp=pforanyp2n),thentheoptimalapproximationorderisattainedbyQ[ 27 ].Inthemultivariatesetting,thereareseveralapproachestoderivequasiinterpolationforageneralreconstructionkernel,suchasmethodsbasedonNeumannseries,AppellsequencesviaMarsden'sidentity,orFouriertransform[ 27 ].SincetheFouriertransformsofV-splinesareexplicitlyavailableinclosed-form(( 3 ),( 3 )),wechoosethismethodforthegeneralizationoftheFouriermethod,whichwasoriginallydevelopedforboxsplines,totheclassofV-splines[ 27 ].Inordertohaveabetterunderstanding,howpolynomialreproductionhelpsustodesignaquasiinterpolationschemewepresentasimple1-dimensionalexampleherewiththequadraticB-spline(i.e.,2).Thisexamplemotivateshowtoimprovethereconstructionaccuracyandattaincloserapproximationerrormandatedbyapproximationorderofthereconstructionkernel. 58

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4.2.2.1QuasiinterpolationforunivariatequadraticB-splineAnanalyticalsmoothfunctioncanberepresentedasitsTaylorseriesexpansion.Intheunivariatecase,theTaylorseriesexpansionofafunctioniswrittenas: f(x)=f(0)+f0(0)x+f00(0)x2+O(x3)(4)Inordertoapproximateafunctionfromitssamples,weneedtorecoverasmanytermsofthisexpansionaswecan.NotethattheapproximationorderoftheunivariatequadraticB-splineis3andhencequadraticB-splineiscapableofrecoveringonlytherstthreetermsoftheTaylorseriesexpansion.Inotherwords,ifquadraticB-splineisusedforreconstructionofmonomials,itshouldbeabletoexactlyrecoverconstant,linearandquadraticmonomials.Usingsamplesofmonomialsofdifferentdegreeasthesplinecoefcients,quadraticB-splineperfectlyrecoversconstantandlinearterm,whileitfailstorecoveraquadraticterm.Table 4-2 summarizesthemonomialreproductionresultsusingquadraticB-splineasthereconstructionkernel.ThequadraticB-splineisformedbyquadraticpolynomialpiecesandthereforeshouldbeabletorecoveraquadraticmonomial.AsshowninTable 4-2 ,usingfunctionsamplepointsastheinputtothereconstructionalgorithmresultsinanextra(smoothing)termwhenreproducingthenormalizedquadraticmonomial.Intermsofreconstructingagenericfunction,thisdeciencytranslatestotheintroductionofanextratermbasedonthesecondorderderivativeoftheunderlyingfunctionalongwithapproximationerrorof2whichisoforder3.Asitwillbediscussedtheoreticallyinthesequel,theextrasmoothingtermiscompletelyspeciedbytheFourieranalysisofthereconstructionkernel.Inordertoremovethissmoothingterminthereconstruction,onecansimplyfeedthereconstruction(i.e.,convolution)withtheappropriatecoefcientswhichisthelteredversionofthesamplepoints.Usingprelteringbeforemonomialreconstructionresultsintheapproximationerrorexpected,asshowninthelastrowofTable 4-2 forthecaseof 59

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Table4-2. TheconvolutionbasedreconstructionusingunivariatequadraticB-spline.Eachrow'sleftentrycorrespondstothedesiredfunctiontoberecoveredandtherightentrycorrespondstothereconstructedfunctionusingtheformulagivenintherstrow.hintheformulacorrespondstothesamplingdistance. f(x)~f(x)=Pk2hZf(k)2(x)]TJ /F5 11.955 Tf 11.96 0 Td[(k) ccxxx2 2x2 2+1 8h2f(x)f(x)+1 8f00(x)h2+O(h3)f(x))]TJ /F7 7.97 Tf 13.15 4.71 Td[(1 8f00(x)f(x)+O(h3) 2.InFigure 4-6 ,wehavealsoshowntheresultofapplyingquasiinterpolationforthereconstructionofaunivariaterandomfunctionf(x)usingtheconvolutionformula:f(x)~f(x)=Xk2Zck2(x)]TJ /F5 11.955 Tf 11.96 0 Td[(k) (4)whereckrepresentsthekthconvolutioncoefcient.Wecanclearlyobservetheover-smoothingartifactpresentinthereconstructionusingfunctionsamplesasconvolutioncoefcients,ck,hasbeenimprovedbyusingquasiinterpolation.Notethatsincewedonothavedirectaccesstothefunctionderivatives,weneedtousenitedifferencingoforder2tocalculatethecoefcientsck.Thiswillnotintroduceanyfurtherapproximationerror,becausenitedifferencingoforder2isexactforpolynomialsofuptodegree2(i.e.,<3).Insummary,loworderpolynomialreproductionprovidestheinsightonhowtoderivetherightsplinecoefcientsfromsamplepointstobeusedintheconvolution-basedreconstruction. 4.2.2.2QuasiinterpolationforVoronoisplinesInthissection,weusethesameideaandexplainitconcretelyforagenericreconstructionkernel.Forthesimplicityofthediscussioninthemultivariatecase,weintroduceafewnotationhere. 60

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Letthe:=(1,2,3),jj:=1+2+3and[[x]]denotethenormalizedmonomialinthetrivariatesetting,denedas:[[x]]:=x !=x1y2z3 1!2!3!.Moreover,wedeneadifferentialoperatorDthattakespartialderivesofordersgivenbythevector.ThisnotationallowsustorewritetheTaylorseriesexpansionofafunctionas:f(x)=X[[x]]Df(0).Theoptimalapproximationorderisachievedby( 4 )wheneveralllow-orderpolynomials(jjn)intheTaylorexpansionarereproducedexactlybythequasiinterpolationasdescribedinthesequel.ThisschemeisoptimalsincepolynomialsintheTaylorexpansionthathavehigherpower(jj>n)cannotbereproducedbythe( 4 ),sincenispickedtobethelargestthatcanbeaffordedbythereconstructionkernelVn.ConsideringtheapproximationorderofageneralreconstructionkernelVnisn+1,weknow[ 62 ]thatanypolynomialinncanbereproducedbythelattice-shift-invariant(LSI)spacewhichisformedbylinearcombinationofshiftsofthenthorderV-splines(i.e.,semi-discreteconvolution):LSIn:fsjs(x)=c0VLn(x)=Xk2LZ3ckVLn(x)]TJ /F6 11.955 Tf 11.96 0 Td[(k)g, (4)Theabilityto(exactly)reproducenmeansthatconvolutionwithVnisaninvertiblelineartransformationonthispolynomialspace.Therefore,ifwestudytheactionofVnonsomebasisofthisspace,oneisabletosystematicallyinvertit.Thisideagivesusarecipeforconstructionofacontinuous(generalized)functionnintroducedin( 4 )thatinvertsthetransformationthatVnperformsonn.Inother 61

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ck=f(k)ck=5 4f(k))]TJ /F7 7.97 Tf 13.15 4.71 Td[(1 8(f(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+f(k+1))ABFigure4-6. Theeffectofusingquasiinterpolationinreducingtheover-smoothingartifact.Reconstructionofa1-DrandomfunctionusingquadraticB-spline.Thefunctionsamplepointsareshownasredpoints.A)Thereconstructionoftherandomfunctionwithoutapplyinganyquasiinterpolation.Theconvolutioncoefcients(ck)arechosentobethesamplepoints.B)Thereconstructionwhiletheconvolutioncoefcients(ck)havebeencalculatedusingthequasiinterpolationdescribed.Thebluecurve(i.e.,reconstructedfunction)whileusingquasiinterpolationshowsnoticeableimprovementcomparedtothecasewherenoquasiinterpolationhasbeenapplied.ThegreencurvesshowshiftsofthequadraticB-splinecenteredatthesplinecoefcients,ck. words,weneedtondnsuchthat: (Vnnf)(x)=f(x),(4)wheneverf2n.Forgeneralfunctionsf,thisschemereproducestherstfewpolynomialterms(asmanyasVncanafford)oftheTaylorseries.Therefore,nisdesignedinsuchawaythatitinvertstheactionofVnonn.Fordesigningn,weconsiderthetransformationonnormalizedmonomialsthatareabasisforn.Letg(x)beapolynomialthatisthetransformationof[[x]]withtheactionofagenericn. g(x)=n(x)[[x]].(4) 62

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Whenn(x)=(x),thenwehavetheidentitytransformationonthepolynomialspacethatcorrespondstosamplingthefunctionitself(f=f)in( 4 ).Generally,ntransformsmonomialstoapolynomialgthatcontainsotherterms;wenowconsiderderivativesofDiracthathelpusbuildnthatgivesusatransformationof[[x]]toageneralpolynomial(i.e.,g(x)).Thisgeneralpolynomialhasinterestingpropertiesthathasbeendiscussedindetailsintheappendix.Let()(x)denotethederivativeofDiracdeltafunction,formallydenedbyitsactiononatestfunctionfthatisapoint-evaluationofthederivativeofthetestfunction:()f:=Z()(x)f(x)dx=(Df)(0). (4)Then,convolutionwiththederivativeofDiracsimplyreducesthepowerofthenormalizedmonomial:()(x)[[x]]=[[x]])]TJ /F26 7.97 Tf 6.59 0 Td[(.Nowwecandenenbyitscoefcientvectorc=[c](foralljjn): n(x)=Xjjnc()(x).(4)Forexample,intheunivariatesetting,convolutionwithn(x)=(x)+1=8(1)(x),transformsthenormalizedcubicmonomialton(x)[[x]]3=[[x]]3+1=8[[x]]2.Thisnhascoefcientsc0=1andc1=1=8.Foragiven,jjnthegeneralcoefcientvectorcofn(x)allowsustodeneg(x),initsmonomialexpansionbysubstituting( 4 )in( 4 ): g(x)=Xc[[x]])]TJ /F26 7.97 Tf 6.59 0 Td[(.(4) 63

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Thecoefcientsofanypolynomial,includingg,canbeextractedbytakingderivativesandevaluatingtheresultingpolynomialatzero: c=()]TJ /F26 7.97 Tf 6.58 0 Td[()(x)g(x)x=0=D)]TJ /F26 7.97 Tf 6.58 0 Td[(g(0)=Z(j!))]TJ /F26 7.97 Tf 6.59 0 Td[(^g(!)d!.(4)Thelatterequationisduetothefactthattheevaluationofafunctionatzero,amountstotheintegrationofitsFouriertransforminthefrequencyspace(i.e.,DCvalueinthespacedomain).TheFouriertransformofD)]TJ /F26 7.97 Tf 6.58 0 Td[(g(x)isgivenby(j!))]TJ /F26 7.97 Tf 6.59 0 Td[(^g(!)duetothederivativetheorem.Now,wearereadytodesignnbychoosingitscoefcientscinsuchawaythattheactionofVntransformsgfunctionsbacktonormalizedmonomialsinvertingnonn: [[x]]=Vn(x)g(x).(4)Equivalently,inthefrequencydomain,wehave: ^g(!)=j()(!) ^Vn(!).(4)Substituting( 4 )into( 4 ),wehave:c=Zj(j!))]TJ /F26 7.97 Tf 6.58 0 Td[( ^Vn(!)()(!)d!=jD(j!))]TJ /F26 7.97 Tf 6.59 0 Td[( ^Vn(!)!=0.Thelatterequationisduetothedistributionaldenitionof()denedin( 4 ).Inparticular,wehave: c=Dj ^Vn(!)!=0.(4)ThisequationgivesusarecipeforconstructionofnthatinvertsVn,usingthecoefcientvectorcand( 4 ).Essentiallythecoefcientsofnarethederivativesof1=^Vn(!),theinverseoftheFouriertransformofthereconstructionkernel,evaluatedattheDCfrequency.SinceweknowtheFouriertransformsofV-splinesinclosedform(section 3.3.3 ),thismethodallowsustoderivequasiinterpolationformulationforanyorderV-splinefor 64

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anylattice.Wehaveperformedderivativecalculationof( 3 )and( 3 )inasymbolicenvironment(Mathematica).Theonlynon-zerocoefcientscforjj2(i.e.forthirdorderV-spline)onBCC,FCCandCartesianlatticesinvolvesecondorderpartialderivativesoftheDiracfunctionfrom( 4 ):(2,0,0),(0,2,0),(0,0,2).Thismeansthat2ontheselatticestakesthesecond-orderpartialderivativesoftheinputfunctionftobuildf2tobeusedinquasiinterpolationasin( 4 ).SpecicallyfortheCartesian,BCCandFCClatticesthe2thatinvertstheoperationsofthirdorderV-splinesontheirrespectivelatticesareobtainedfrom( 4 ),( 3 )and( 3 ): B2(x):=(x))]TJ /F7 7.97 Tf 15.37 4.71 Td[(3 16)]TJ /F8 11.955 Tf 5.48 -9.69 Td[((2,0,0)(x)+(0,2,0)(x)+(0,0,2)(x)F2(x):=(x))]TJ /F7 7.97 Tf 13.15 4.71 Td[(19 64)]TJ /F8 11.955 Tf 5.48 -9.69 Td[((2,0,0)(x)+(0,2,0)(x)+(0,0,2)(x)C2(x):=(x))]TJ /F7 7.97 Tf 13.15 4.7 Td[(1 8)]TJ /F8 11.955 Tf 5.48 -9.68 Td[((2,0,0)(x)+(0,2,0)(x)+(0,0,2)(x)(4)Essentially2computedaweightedLaplacianoneachcasetoderivef2fromf.Therefore,weneedtoevaluatesecondorderderivativesoffalongX,YandZaxis.However,ontheBCCandFCClattices,theLaplacianinvolvesdirectionalderivativesalongthelatticedirectionsdenedas:Df(x):=limh!0f(x+h))]TJ /F17 7.97 Tf 6.59 0 Td[(f(x) h.TheconversionofdirectionalderivativestoderivativesalongX,YandZispossibleviaalineartransformationthatinvolveslatticedirections.Latticedirections,oneachlattice,areobtainedfromthedirectionsthatareformedbyconnectingalatticepointtoitsDelaunayneighbors[ 34 49 ].ForexamplefortheFCClattice,thelatticedirectionsare:[16]:=2666641)]TJ /F4 11.955 Tf 9.29 0 Td[(1110011001)]TJ /F4 11.955 Tf 9.3 0 Td[(1001)]TJ /F4 11.955 Tf 9.3 0 Td[(111377775. (4) 65

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Usingthelatticedirections,onecanwritetherstorderdirectionalderivativesalongFCClatticedirectionsasfollows:D1=D(1,0,0)+D(0,1,0)D2=)]TJ /F5 11.955 Tf 9.3 0 Td[(D(1,0,0)+D(0,1,0)D3=D(1,0,0)+D(0,0,1)D4=D(1,0,0))]TJ /F5 11.955 Tf 11.96 0 Td[(D(0,0,1)D5=D(0,1,0)+D(0,1,0)D6=)]TJ /F5 11.955 Tf 9.3 0 Td[(D(0,1,0)+D(0,1,0) (4)Squaringtheaboveequationsandsummingupthesixofthem,wehave:X2FD2=4)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(D(2,0,0)+D(0,2,0)+D(0,0,2) (4)Similarly,secondorderdirectionalderivativesonBCCcanbecalculated,whichgivesthesameformula: P2BD2=4)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(D(2,0,0)+D(0,2,0)+D(0,0,2),(4)Notethatthesums,onthelefthandside,aretakenoverthenearestDelaunayneighborsthatdenotethelatticedirectionsoneachlattice.Inordertoemploythesedirectionalderivativesonsampledvolumetricdata,weneedtoreplacethedirectionalderivativeswithdirectionalnitedifferences.Thenumericalorderofnitedifferencingschemeischosenaccordingtotheoptimalorderthatisdesiredbythequasiinterpolationmethod[ 34 ].ThirdorderV-splinesreproducepolynomialsuptodegreetwo(2).Thenitedifferencingoforder2isexactoverthispolynomialspace:D2f=2f=f(+))]TJ /F4 11.955 Tf -435.09 -23.91 Td[(2f()+f()]TJ /F25 11.955 Tf 19.08 0 Td[().Therefore,wecanemploynitedifferencingtoimplement( 4 )for 66

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sampledvolumetricdatasetas: B2(x):=51 32(x))]TJ /F7 7.97 Tf 15.36 4.7 Td[(19 256P2B)]TJ /F8 11.955 Tf 5.48 -9.68 Td[((x)]TJ /F25 11.955 Tf 11.96 0 Td[()+(x+)F2(x):=25 16(x))]TJ /F7 7.97 Tf 15.37 4.71 Td[(3 64P2F)]TJ /F8 11.955 Tf 5.48 -9.69 Td[((x)]TJ /F25 11.955 Tf 11.96 0 Td[()+(x+)C2(x):=7 4(x))]TJ /F7 7.97 Tf 13.15 4.71 Td[(1 8P2C)]TJ /F8 11.955 Tf 5.48 -9.69 Td[((x)]TJ /F25 11.955 Tf 11.95 0 Td[()+(x+)(4)ThisimpliesthatthequasiinterpolationcanbeimplementedasaFIRlterwherethedataatagivensampledpointisreplacedbyaweightedaverageofdataattheneighboringlatticepoints.TheFIRlterisactuallyahigh-passlterandpreservehighfrequencieswithoutintroducinganyerrorinthereconstruction. fB2(x):=51 32f(x))]TJ /F7 7.97 Tf 15.37 4.71 Td[(19 256P2B)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(f(x)]TJ /F25 11.955 Tf 11.95 0 Td[()+f(x+)fF2(x):=25 16f(x))]TJ /F7 7.97 Tf 15.37 4.71 Td[(3 64P2F)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(f(x)]TJ /F25 11.955 Tf 11.95 0 Td[()+f(x+)fC2(x):=7 4f(x))]TJ /F7 7.97 Tf 13.15 4.71 Td[(1 8P2C)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(f(x)]TJ /F25 11.955 Tf 11.95 0 Td[()+f(x+).(4) 4.2.3ResultsandComparisonsTheFIRlterdiscussedinsection 4.2.2 hasbeenappliedonvolumetricdataasapreprocessingstepinMATLAB.Thispreprocessingsteptakesonlyafewsecondsforallofthedatasetswehadreportedhere.WehaveexaminedtheproposedquasiinterpolationapproachonvolumetricdatasampledonCartesian,BCCandFCClatticesusingtheircorrespondingthirdorderV-splines.InthecaseofCartesianlatticeasmentionedbefore,V-splinescoincidewiththetensor-productB-splines.ToevaluatethequalityofthirdorderV-splinereconstructionusingquasiinterpolationproposedinsection 4.2.2 ,bothvisualandnumericalcomparisonshavebeenconsideredandoriginallyreportedin[ 63 ]. VisualComparisons.WehaveexaminedourreconstructionapproachesonthesyntheticMarschner-Lobbdataset[ 54 ],thecarpshdatasetandthebonsaidatasetasrealtestcases.ThegroundtruthdatasetsinourexperimentsareshowninFigure 4-7 67

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ABCFigure4-7. Groundtruthvolumetricdatasets.A)Marschner-Lobb(ML).B)Carpsh.C)Bonsai. thatareusedforvisualandnumericalevaluationofthereconstructionaccuracy.Thesamplingresolutionsarechosentobethesameasresolutionsreportedinsection 4.1 forcomparisonpurpose.TheimagesintherstcolumnofFigure 4-8 arerenderedusingV2withoutanyprelteringondifferentlattices,whilethesecondcolumnshowstheresultofapplyingquasiinterpolationwithV-splineforreconstruction.Asdemonstratedbytheimages,theringsarebetterpreservedandarevisuallytallerthantherstcolumn.Reductionofheightinthereconstructedringsareduetoover-smoothingofthereconstructionkernel[ 54 65 ].Thequasiinterpolationschemehasreducedthisproblemandhasprovidedamoreaccuratereconstructionoftherings.InFigure 4-9 wehavealsodocumentedtheimprovementintheaccuracyofMLreconstructionusingquasiinterpolationbyrenderingthetopviewofMarschner-Lobbdatasetalongwitherrorimages.TherstrowshowsthereconstructionofMLdatasetandthefollowingrowsshowrenderingoftheangularerrorinreconstructingnormalsoftherenderedisosurface.Theerrorencodestheangulardifferencebetweentheestimatednormalandthetruenormaloftheisosurface.Grayscaleimagesvisualizetheerrorwherethemaximumerrorof.3radiansismappedtowhiteandzeroerror 68

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AVC2withoutquasiBVC2withquasi CVB2withoutquasiDVB2withquasi EVF2withoutquasiFVF2withquasiFigure4-8. QuasiinterpolationresultsforML(cutview).Thevolumetricdatahasapproximatelythesameresolution(64,000samples)ondifferentlattices.Asshownintheimages,theringsinthecaseofquasiinterpolationaretallerthatdemonstratelessover-smoothing.Moreover,theringsarelessjaggedinthecaseofBCCandFCCcomparedtotheCartesiancase. ismappedtoblack.Thedarkerimagesinthethirdrowconrmsthemoreaccuratereconstructionofferedbyquasiinterpolation.WhiletheBCCreconstructionisthemostaccurate(darkest)amongthethree,theFCCcloselyfollowsintermsofaccuracy.However,intermsofaliasingeffects,theFCCshowsnoticeablysmallerpresenceofaliasinginthesamplingprocess.Thisexperimentconrmsthetheoreticalexpectations[ 51 ]thattheFCClatticeisagoodcandidatewhen 69

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Table4-3. Reconstructionmeansquarederrorof10Krandomsamplepoints.Thesamplepointsarechosenovertheentirevolumeoftherendereddatasets.ApplyingquasiinterpolationbeforeV2reconstructionhasdecreasedtheerrorsignicantly. MLCarpBonsaiNon-QuasiQuasiNon-QuasiQuasiNon-QuasiQuasi BCC8.324.822.11.254.083.11FCC8.14.552.081.294.073.07Cartesian8.95.492.231.44.323.41 samplingprocesshasinherentaliasing.WealsonotethatthisvalidationisnowpossiblewiththeV-splinesframeworksincetheygiveanunbiasedreconstructionacrosstheselatticeswhichisconrmedbythestudyofover-smoothingandpost-aliasingpropertiesofV-splinesondifferentchoiceoflatticesin[ 62 ].InFigure 4-10 andFigure 4-11 thesubsampledversionsofcarpshandbonsaitreedatasetshavebeenvisualizedondifferentlatticesusingV2.TherstcolumninFigure 4-10 andthersttwocolumnsinFigure 4-11 correspondtothereconstructionwithoutquasiinterpolationandtheremainingsshowtheeffectofapplyingquasiinterpolation.Inthecaseofcarpshdataset,thesmoothingartifactisclearlyvisibleintheribsandthetailsectionintherstcolumn,whichisnotpresentinthesecondcolumn.Forbonsaidataset,manybrancheshavebeendisconnectedduringthereconstructionwhiletheyhavebeenreconnectedbythequasiinterpolation.InsecondandforthcolumnsofFigure 4-11 ,azoomedinviewofbonsaidatasetisshowninwhichabranchisalmostpreservedbyquasiinterpolationwhichisnotpresentinthereconstructionswithoutquasiinterpolation.WehavealsodocumentedthenumericalerroranalysisofquasiinterpolationwithV2reconstruction,bymeasuringmeansquarederrorinthereconstructedsignalovertheentirevolumesusedforrendering.WehavesummarizedtheresultsinTable 4-3 whichshowsthattheaccuracyofreconstructionhasbeenimprovedbyapplyingquasiinterpolation. 70

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4.3EfcientEvaluationofVoronoiSplinesAlthough,box-spline-basedevaluationschemeforV-splinesisexactandcanbeextendedforanyotherlatticewhoseVoronoicellisazonohedra(all2-Dand3-DlatticesandDicinglatticesinhigherdimensions[ 62 ]),itiscomputationallyveryexpensive.AsinglepointevaluationofagivenV-spline,writtenintermsofsummationofboxsplines,involvesevaluatingalloftheboxsplinesinthesummation.However,theboxsplineexpansionalsoshowsthatV-splinesaresmoothpiecewisepolynomialsdenedovertheirsupport.Therefore,onecanutilizetheboxsplineexpansiontopre-computethepiecewisepolynomialrepresentationofV-splines.Oncethepiecewisepolynomialrepresentationisestablished,itcanbeusedforfastevaluationduringtherenderingprocess.Inwhatfollowswedemonstratethatanexact(andefcient)evaluationschemecanbedevelopedbasedonndingthepolynomialpartitionsconstitutingthesupportofthesecondorderV-splines.Inthepre-computationstep,weutilizethe(inefcient)pointevaluationsofV-splinesintheboxsplineexpansiontondthepolynomialpartitionsformingthesupportofVL1forFCCandBCC.Forillustrationpurposes,wediscusstheproblemin2-DforVH1(i.e.,secondorderV-splineonhexagonallattice,H).Wegeneralizethesameideaforoptimalsamplinglatticesin3-D.TheVoronoicellofthehexagonallatticehasthreezonevectors(paralleltotheedgesoftheVoronoicell)gatheredinHandintroducedin( 3 ).Fromthediscussioninsection 3.1.2 ,VH1canbewrittenasthesummationof6distinctboxsplines[ 62 ]whosesupportshavebeenshowninFigure 3-2 : VH1=1 32)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(M[1,2]+M[1,3]+M[2,3]2=1 9)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(M[1,1,2,2]+M[1,1,3,3]+M[2,2,3,3]+2 9)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(M[1,1,2,3]+M[1,2,2,3]+M[1,2,3,3].(4)ThesupportofeachoftheseboxsplinescanbedelineatedintopolynomialpartitionsshowninFigure 4-12 .Thereaderconversantwiththeboxsplinetheorywill 71

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noticetherelationbetweenthesupportofaboxsplineanditspolynomialpartitions.Thepolynomialpartitionsformingthesupportofaboxsplinearespeciedbythedirectionsetformingtheboxspline[ 27 ],whicharethe2-subsetsofthethreezonesofahexagon.TheexpansionintermsofboxsplinesestablishesthattheboundariesofthepolynomialpartitionsdelineatingthesupportofVH1willnotbearbitrarylines.Morespecically,ascanbeobservedinFigure 4-12 ,allthepotentialboundarylinesforpolynomialpartitions(shownasdashedlineinFigure 4-12 )aredeterminedbythezonesofthehexagon.Similarly,thesupportofVB1andVF1willbecleavedintopartitionsusingplanesin3-D.Theseplanesaredeterminedbythepolynomialpartitioningoftheconstitutingboxsplines.Therefore,thesetofallpotentialplanesisspeciedfromthezonedecompositionofBCCandFCClatticereportedin( 3 )and( 3 ).ThesetofallplanesformedbycolumnvectorsofB(i.e.,zonesofVoronoicellofBCClattice)contains)]TJ /F7 7.97 Tf 5.48 -4.38 Td[(62=15planes.SimilarlyforFCClattice,thenumberofdistinctplanespotentiallydelineatingthesupportofVF1is:)]TJ /F7 7.97 Tf 5.48 -4.38 Td[(42=6.TodeterminethepartitionsformingthesupportofVF1andVB1,wehavedesignedaclusteringalgorithmwhichwillbediscussedinsection 4.3.1 4.3.1IdentifyingTheDomainPartitionForPolynomialPiecesToderivethepiecewisepolynomialschemeforevaluationofVF1andVB1,weneedtodeterminethepolynomialpartitionsformingtheirsupports.Asmotivatedbythehexsplineexample,theefcientevaluationofthesecondorderV-splinesinvolves: 1. Specifyingthepolynomialpartitionsthatwillconstitutethesupport. 2. Determiningtheexactpolynomialpieceinsideeachpartition.Thisprocesswillbecarriedoutasapre-computation(performedonce).Usingthebox-spline-basedevaluationschemediscussed,wecanrstevaluateVF1andVB1onadenseanduniformpopulationofpointsfallinginsidetheirsupports.Thesepointswillbeusedtospecifytheboundariesdelineatingpolynomialpartitions.Foreachpoint,we 72

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cantasinglepolynomialandassignthepointalabelbasedon(coefcientsof)thepolynomialtted.WeusetheselabelstoclusterthepointsintopartitionsdelineatingthesupportofthesecondorderV-splineintopolynomialpartitions.Moreover,consideringthesymmetryofthesupportofV-spline,wecansimplifytheproblemevenfurther.Wecanrestricttheproblemtoasub-regionofthepositiveoctantdenotedas:minimalregion.ItiseasiertoillustratetheideaforVH1in2-DwhilethesameideahasbeencarriedoutforVF1andVB1in3-Danddocumentedattheendofthissection.ThesupportofVH1hasbeendemonstratedinFigure 4-13 wheretheminimalregionisshowningray.Inwhatfollowswedescribeamethodtondtheboundaryofthepiecesinsidetheminimalregion.ThesupportofVH1restrictedtoeachpieceisexactlyasinglepolynomial.ThesupportofVH1iscomposedofbivariatepolynomialpiecesoftotaldegree2[ 62 ].Therefore,thepolynomialineachpiecehasanexpansionoftheformp(x)=P+2,0axy,whichisuniquelyspeciedbyits6coefcients.ThesecoefcientscanbedeterminedbysamplingVH1at6pointsinsidethepartitionandinvertingthecorrespondinglinearsystem(i.e.,polynomialregression).Thepointsinsidetheminimalregioncanthenbeclusteredbasedontheirlabels(determinedfromthepolynomialcoefcients)asdemonstratedinFigure 4-13 bydifferentcolors.NotethatotherthanthemainclusterswhichcorrespondtotheactualpolynomialpartitioningofthesupportofVH1,theclusteringapproachwillintroduceunpopular(i.e.,clusterswithsmallpopulations)alongtheboundariesofthepartitionsshowninFigure 4-13 .Thisisduetothefactthatthepointslayingontheboundarybetweentwopiecesresultinanapproximatetsincethettingpolynomialtothesepointsisbasedonpointsonthetwosidesoftheboundary(i.e.,twodistinctpolynomials).However,wecanprunetheseclusterseasilyconsideringtheirsmallpopulation.Notethatpruningunpopularclusterswillnotintroduceanyambiguitysimplybecausewealreadypredeterminedthesetoflinesfromwhichtheboundariesofthe 73

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polynomialpartitionsarechosenfrom(i.e.,zonesofahexagon).TakingintoaccountthesymmetryofthesupportofVH1(andpropertransformations)weareabletorecoverthepartitionsfortheentiresupport.TheminimalregionfortheBCCandFCClatticeshasbeenshowninFigure 4-14 .Theremainingpartsofthesupportcanbetransformedtotheminimalregionbyapplyingsymmetrictransformations.Similarly,thepolynomialpartitionsconstitutingthesupportofVF1andVB1canbedeterminedandhavebeendemonstratedinFigure 4-15 .TheminimalregionforVF1canbecleavedinto4polynomialpartitionsallofwhichhappentobetetrahedra.However,polynomialpartitionsformingthesupportofVB1aremoreinvolved.TheminimalregionofVB1canbecleavedinto3mainregionsasshowninFigure 4-15 .Eachmainregionfurtherdividestoform11polynomialpartitionsconstitutingtheminimalregionofVB1showninFigure 4-16 .Havingdenedtheboundariesofthepolynomialpartitions,weneedtodeterminethepolynomialpiecesineachpartition.ThedegreeofthepolynomialpiecesresidingineachpartitiononlydependsontheorderofV-splineandthedimensionofthelatticethatithasbeendesignedfor(Table 3-3 reportedfrom[ 62 ]).InthecaseofsecondorderV-splineforBCCandFCClattices,similartothetrilinearB-spline,thepolynomialsareofdegree3(cubic):p(x)=++3X,,0axyz, (4)p(x)canbeuniquelydeterminedintermofits)]TJ /F7 7.97 Tf 5.48 -4.38 Td[(3+33=20coefcients:a.Aregressionusing20pointevaluationsinsideeachpartitionallowsustodeterminethecoefcientsofeachpiece.HavingthepartitionsandpolynomialpiecesforVF1andVB1,wecanwritethepseudocodeforefcientevaluationofthesecondorderV-splinesonBCCandFCC.Thepseudocodehasbeendocumentedinalgorithm 1 andalgorithm 2 where 74

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thepolynomialpartitionsandtheircorrespondingcoefcientsarereportedinTable 4-4 andTable 4-5 Algorithm1EfcientevaluationofthesecondorderV-splinefor(x,y,z)insideminimalregionofVF1. plane1 x+z plane2 z)]TJ /F5 11.955 Tf 11.95 0 Td[(x plane3 x+y if(1plane1)then partition1 else if(1plane2)then partition2 else if(1plane3)then partition3 else partition4 endif endif endif 4.3.2TimeComparisonsAlongwithvisualcomparisonsshowninsection 4.1 ,theefcientevaluationschemeintroducedwillenableustoperformtimecomparisonsbetweenreconstructionusingV-splinesontheBCCandFCClatticesandB-splinesontheCartesianwhichhasbeensummarizedinTable 4-6 .ConsideringtheV-splinesondifferentlattices,trilinearinterpolationisstilltwiceasfastasV-splineonoptimalsamplinglatticesbecauseofthemorecomplicatedboundariesforpolynomialpartitionsonoptimalsamplinglattices.Box-spline-basedevaluationreportedin[ 62 63 ]tookseveraldaysforanimagetorenderwhereasthenewevaluationmethodiscomparabletotrilinearinterpolationonCartesianlatticeintermsofcomputationalefciency.Inordertocomparethecomputationalefciencyoftheproposedmethodversusbox-spline-basedevaluationscheme,bothapproacheshavebeenimplementedinC++wheretheboxsplineshavebeencomputedbytheapproachproposedin[ 32 ].Table 4-7 summarizesthe 75

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Algorithm2EfcientevaluationofthesecondorderV-splinefor(x,y,z)insideminimalregionofVB1. plane1 x)]TJ /F5 11.955 Tf 11.96 0 Td[(y+zplane3 x+y)]TJ /F5 11.955 Tf 11.95 0 Td[(zplane5 x plane2 x)]TJ /F5 11.955 Tf 11.96 0 Td[(y)]TJ /F5 11.955 Tf 11.95 0 Td[(zplane4 x+y+zplane6 z if(plane20)&(1plane5)then //redmainregion if(plane11)then partition1 else partition2 endif elseif(plane20)&(2plane4)then //bluemainregion if(plane11)&(plane31)then if(1 2z)then partition3 else partition4 endif else if(plane11)then partition5 else partition6 endif endif else //greenmainregion if(plane51.5)j(plane31)then if(2plane1)then partition7 else partition8 endif else if(plane12)&(1plane6)then partition9 else if(plane20)then partition10 else partition11 endif endif endif endif 76

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Table4-4. Thecoefcientsforpolynomialpartitions(asdocumentedinalgorithm 1 )toconstitutethesupportofthesecondorderVoronoisplineontheFCClattice. partition1zz2z3yyzyz2y2y2zy3xxzxz2xyxyzxy2x2x2zx2yx3 11 2)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 201 12)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 2)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 81 24000000)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 8)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 8022 3)]TJ /F4 11.955 Tf 9.29 0 Td[(11 2)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 12000)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 81 24000000)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 81 8031)]TJ /F4 11.955 Tf 9.29 0 Td[(11 40)]TJ /F4 11.955 Tf 9.3 0 Td[(13 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 81 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 80)]TJ /F4 11.955 Tf 9.3 0 Td[(13 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 83 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 81 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 8)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 8047 12)]TJ /F7 7.97 Tf 10.49 4.71 Td[(3 41 40)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 81 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 8)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 81 24)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 81 4)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 8)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 41 40)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 41 801 24 Table4-5. Thecoefcientsforpolynomialpartitions(asdocumentedinalgorithm 2 )toconstitutethesupportofthesecondorderVoronoisplineontheBCClattice. partition1zz2z3yyzyz2y2y2zy3xxzxz2xyxyzxy2x2x2zx2yx3 11 4)]TJ /F7 7.97 Tf 10.5 4.7 Td[(1 8)]TJ /F7 7.97 Tf 12.71 4.7 Td[(3 641 64)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 83 32)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(3 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 641 64)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 83 32)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 643 321 32)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(3 64)]TJ /F7 7.97 Tf 12.72 4.7 Td[(1 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 641 64249 192)]TJ /F7 7.97 Tf 10.5 4.7 Td[(1 8)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 165 192)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 81 8)]TJ /F7 7.97 Tf 12.71 4.7 Td[(3 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 161 641 192)]TJ /F7 7.97 Tf 10.49 4.7 Td[(1 81 161 641 8)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 321 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(1 161 64)]TJ /F7 7.97 Tf 12.71 4.7 Td[(3 645 192325 96)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 32)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 81 96000)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 161 320)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 41 161 160001 16)]TJ /F7 7.97 Tf 12.72 4.71 Td[(1 3200449 1920)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 165 96000)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 161 320)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 41 161 160001 16)]TJ /F7 7.97 Tf 12.72 4.71 Td[(1 320051 4)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 96)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 3201 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 321 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 96)]TJ /F7 7.97 Tf 12.72 4.71 Td[(7 321 161 321 1601 321 32)]TJ /F7 7.97 Tf 12.72 4.71 Td[(1 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 321 96613 480)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 321 48000)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 3201 48)]TJ /F7 7.97 Tf 12.72 4.71 Td[(9 3201 32001 323 3200)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 9673 8)]TJ /F7 7.97 Tf 10.5 4.71 Td[(1 8)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 641 64)]TJ /F7 7.97 Tf 10.49 4.71 Td[(1 83 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 64)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 64)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 641 64)]TJ /F7 7.97 Tf 10.5 4.71 Td[(13 323 321 643 32)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 321 649 64)]TJ /F7 7.97 Tf 12.72 4.71 Td[(1 64)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 64)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 6489 16)]TJ /F7 7.97 Tf 10.5 4.71 Td[(3 81 160000)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 1600)]TJ /F7 7.97 Tf 10.5 4.71 Td[(21 325 16)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 32001 321 4)]TJ /F7 7.97 Tf 12.72 4.71 Td[(1 160)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 32965 192)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 16)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 165 192)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 161 8)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 64)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 161 641 192)]TJ /F7 7.97 Tf 12.72 4.71 Td[(5 163 161 640)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 321 641 16)]TJ /F7 7.97 Tf 12.72 4.71 Td[(3 641 641 1921027 64)]TJ /F7 7.97 Tf 10.5 4.71 Td[(3 81 161 192001 64)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 161 641 192)]TJ /F7 7.97 Tf 10.49 4.71 Td[(3 85 16)]TJ /F7 7.97 Tf 12.71 4.71 Td[(3 640)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 321 641 16)]TJ /F7 7.97 Tf 12.72 4.71 Td[(3 641 641 1921127 64)]TJ /F7 7.97 Tf 10.5 4.71 Td[(3 81 160000)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 1600)]TJ /F7 7.97 Tf 10.49 4.71 Td[(3 85 16)]TJ /F7 7.97 Tf 12.71 4.71 Td[(1 32001 321 16)]TJ /F7 7.97 Tf 12.72 4.71 Td[(1 1601 96 77

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Table4-6. TherenderingtimemeasuredinsecondsonaquadcoreAMDAthlon(tm)processor(2.2GHz).Fortheresolutionof51ksamplesforMLandaround6millionssamplepointsfortheotherdatadatasets. ReconstructionKernelMLCarpBonsaiAneurism TrilinearInterpolation(Cartesian)7.514.84.96.6SecondOrderV-spline(BCC)22.7101.125.933.3SecondOrderV-spline(FCC)23.5113.429.236.9 Table4-7. TheevaluationtimefortheproposedefcientevaluationofthethesecondorderVoronoisplines.Theproposedefcientevaluationisuptothreeordersofmagnitudefasterthantheoriginalbox-spline-basedevaluationalgorithmin[ 62 ]onFCCandBCC.Thepointevaluationtimefor1millionrandompointsinsidethesupportofsecondorderV-splinesonBCCandFCCmeasuredontheprocessormentionedinTable 4-6 VB1VF1 Box-spline-basedEvaluation(sec.)145.212.16OurEfcientEvaluation(sec.)0.1480.156 CPUtimerequiredtoevaluate1millionrandomsamplepointsinsidethesupportofsecondorderV-splinesonBCCandFCClattices.Notethatbox-spline-basedevaluationforVB1ismuchslowerthanVF1becauseofsignicantlymorenumberofboxsplinesinvolved.However,theefcientevaluationofbothkernelswouldonlyinvolvendingtheappropriatepolynomialpartitionandacubicpolynomialevaluationwhichshouldresultincomparablecomputationtime.However,whenusedforinterpolation,evaluationofV-splinesonnon-CartesianlatticeswillinvolvethemorecomplicatedkernelsupportandwouldresultinaslightlylessefcientevaluationasshowninTable 4-6 4.4ScatteredDataReconstructionAlongwithuniformsamplingthathasreceivedalotofattention[ 80 ],scattereddatainterpolationalsoappearsfrequently.Inmanyapplications,scattereddataisunavoidablesuchasgeophysics,astronomy,medicalimagingandtelecommunication[ 2 ].Asanexample,dataincomputerizedtomographyarecollectedonapolarsamplinggrid.Dataforgeophysicsandastronomycanbecapturedatnon-uniformtimeinstancesduetoweatherconditionsorradiations.Moreover,uniformdataisalsopronetodatalossandnoiseespeciallyincommunicationchannelswhichresultinarbitrary 78

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placedsamples.[ 69 ].Althoughsometimesnotdesirable,non-uniformsamplingcanhavemanyapplications.Imageandvideocompressiontechniquescanbedesignedbasedonscattereddatasamplingtoreducetherequirednumberofsampleswithoutlosingimportantdetailsforperception[ 70 ].Interestedreadercanreferto[ 2 69 ]forcomprehensivereviewofapproachestonon-uniformsamplingandreconstructionandtheirapplications. 4.4.1ProblemDenitionandRelatedWorksIrregular(orscattereddata)reconstructionproblemdealswithrecoveringtheband-limitedsignalf(x)fromwhichonlyascatteredsetofsamplepointsisavailable.Scattereddata,ff(xi)g,isusuallyreferredtothefunctionvaluesatasetofarbitrarilypositionedpoints,X=fx1,,xNg.Oneofthemostpopularstrategiesforscattereddatareconstructionisgridding[ 69 ].Inthisapproach,non-uniformlysampleddataisrstapproximated(resampled)onaregulargridandthentheapproximateduniformdataisinterpolatedtoapproximatetheunderlyingfunction.TheapproachesbasedonresamplingthescattereddataontoauniformgridtrytogeneralizetheShannon'ssamplingtheorytonon-uniformcases.ExactreconstructioninthesecasesisguaranteedwhenthemaximumgapbetweensamplepointsisboundedbytheNyquistsamplinginterval[ 38 ].Approximation-Projection(A-P)isanexampleofiterativemethodsproposedtondtheexactreconstructionforabandlimitedfunctionffromitsarbitrarilypositionedsamplepoints[ 2 ].Thebasicideaistostartwithanapproximation:f0andimprovetheapproximationineachstepbyanadditivefactor.TheadditivefactorisspeciedusinganoperatorAappliedontheapproximationerrorinthatstep:fn+1=fn+A(f)]TJ /F5 11.955 Tf 11.95 0 Td[(fn) (4) 79

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fninthisrelationrepresentstheapproximationinstepn.Givenanappropriateoperatorhasbeenusedineachstep(i.e.,aprojector),frametheorywillprovideaconvergenceguaranteetothetruefunctionwithaknownconvergencerate[ 69 ].Voronoimethod[ 55 ]isasimpleandyeteffectiveiterativemethodtorecoverbandlimitedsignalsfromnon-uniformsamplesassumingthattheextentofthespectrumofthesignalisknownbeforethereconstruction.IntheVoronoimethod,thenearestneighborinterpolationisusedtoprovidetherstapproximationfromsamplepoints.Inotherwords,discretesamplepointsarereplacedbyastepfunctionthatisformedbyconstantpiecesbetweensuccessivemidpointsofthesamplingsequencewhichisequaltothesamplepoint'svalue.Thisstepcorrespondstoresamplingthescattereddataintoauniformlattice.TheoperatorAisalowpasslterwhosepassbandisdeterminedbasedonthespectrumofthesignal.Thelowpasslterisusedtotruncatethespectrumoftherstapproximation(i.e.,nearestneighborinterpolationofnon-uniformsamples).Inotherwords,therstguessisprojectedintothespaceofbandlimitedsignals.Thisprojectionwillresultinasmoothfunctionthatnolongerinterpolatestheoriginalnon-uniformsamplingpositions.Theapproximationusingnearestneighborinterpolationfollowedbylowpasslteringwillbepreformedontheapproximationerroruntilwefullyrecovertheunderlyingsignal.ThisprocedureisguaranteedtoconvergetotheunderlyingbandlimitedsignalifthemaximumgapbetweenscatteredsamplesissmallerthantheNyquistlimit[ 55 ].Thislimitdeterminesthethresholdofsamplingdensityintheuniformcasewithregardstothemaximumfrequencypresentinthespectrumofthebandlimitedsignal.Theapproximationspaceforscattereddatareconstructioncanbeextendedtospacesotherthanthebandlimitedfunctions.Anothercommonapproachforscattereddataapproximationisbasedonniteelementmethods.Generallyspeaking,niteelementmethodsapproximateasolutionforasetofequations,denedonregionsofinterest(elementsordatasites)thatareconnectedthroughdiscretepoints.Having 80

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deneddatasitesandboundaryconditions,niteelementmethodscanbeusedforunstructuredgeometries.Delaunaytriangulationin2-Dandtetrahedralizationin3-Darewidelyusedforvolumerenderingandraytracing.Theycanbeusedtodenedatasitesforapproximationfromscattereddata[ 4 ].Havingtheconnectivityinformationfordatasites(i.e.,simplices),thepolynomialpiecesineachsitecanbeapproximated.Asanexample,linearinterpolationcanbedenedusingbarycentriccoordinatestolinearlyinterpolatethefunctionvaluesfromthesamplepoints(i.e.,verticesofthesimplex).Higherorderinterpolationcanbeachievedthroughimposingsmoothingconstraintsatthecornersandfacesofthedatasites(i.e.,simplices).Although,thismethodprovidesapolynomialfunctionforeachregionwithchosensmoothnessattheboundaries,yettheaccuracyofthereconstructionusingthisapproachhighlydependsonthequalityofthemeshingschemeusedtodenethedatasitesfromthesamplepoints[ 73 ].Inordertoavoiddeningdatasitesasniteelementmethodsrequire,onecanusebasisfunctionsthattakesintoaccountthedistanceofneighboringpoints.RadialBasisFunctions(RBF)areamongtheapproachesthatareinherentlysuitableforscattereddatainterpolationandcanbeusedtosolvetheestimationproblemdirectlyusingscattereddatasamples[ 66 ].Bydenition,RBFestimateafunction,f(x),bythelinearcombinationofshiftedbasisfunctions,(kk),wherekkdenotestheEuclideandistance:f(x)=NXi=1i(kx)]TJ /F6 11.955 Tf 11.95 0 Td[(xik) (4)Inthisrelation,i2Rdenotesthecoefcientsofthebasisfunctionsandxi2Rddenotestheithsamplepointfromthescattereddata.Sincethebasesarerealfunctionsofdistance,theyareradiallysymmetric.Differentchoicesofbasisfunctionsarepossible,asanexample:thin-platespline((x)=x2log(x))orGaussian((x)=exp()]TJ /F8 11.955 Tf 9.3 0 Td[(x2))aretwoofthepopularchoicesfor(x).Byxingthebasisfunctionand 81

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applyingtheinterpolationconstraintsatthesamplingpoints,onecanformalinearsystemofequations.Thecoefcientsin( 4 )canbecomputedsolvingthelinearsystemofequations:Ac=f (4)whereA=f(kxi)]TJ /F6 11.955 Tf 12.99 0 Td[(xjk)g(alsoknownasinterpolationmatrix),xidenotestheithsamplepoint,fandcdenotethecolumnvectorsforsamplepointsandcoefcientsrespectively.theinterpolationmatrixformedbyRBFisusuallyinvertible,given(x)ischosenappropriately[ 66 ].However,dependingonthedecayingpropertiesofthebasisfunctionschosen,ndingthecoefcientsusingthismethodcanposecomputationalchallenges.Consideringbasisfunctionsthatdecayfast,thesupportofthebasisforeachpointcanbeconsideredlocalandasaresultinterpolationmatrixwillbesparse.Havingasparseinterpolationmatrixisbenecialtosolvetheinterpolationproblemmoreefciently.Ontheotherhand,inordertopreventtheinterpolationmatrixtobeill-posed,thedistributionofthesamplepoints(themaximumgapallowed)shouldbefurtherrestricted. 4.4.2ScatteredDataReconstructionUsingVoronoiSplinesApproximationusingunivariatesplinesisalsooptimalfromavariationalpointofview,wherethesmoothingsplineestimatorwillresultincubicsplines[ 23 ].Inthissection,avariationalframeworkisintroducedanddiscussedforreconstructionusingV-splines.Leastsquareserrorminimizationhasbeenwidelyusedforapproximationandreconstruction[ 5 ].ItsapplicationforimagereconstructiononCartesianandhexagonallatticesusingsecondordersplineapproximation(2-DcounterpartsofVB1,VF1)hasbeenstudiedin[ 39 ].Theauthorsin[ 39 ]haveformulatedthescattereddatareconstructionasaminimizationproblemwherethecostfunctionincludedtwoterms:thesquarederror 82

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inapproximatingthevaluesatthesamplingpoints(i.e.,delityterm)andaquadratictermthatpenalizesthesolutionswhichlacksmoothness(i.e.,regularizationterm).Intheunivariatecase,onecanwritetheregularizedleastsquarederrorproblemas:mins(x)Xk2Zjf[k])]TJ /F5 11.955 Tf 11.96 0 Td[(s(k)j2+Z1js0(x)j2dx (4)Wheres(x)denotesthesplineapproximatingfands0(x)denotestherstorderderivativeofthespline.Inthetrivariatecase,regularizedleastsquareshasbeenstudiedforsincfunctionondifferentlatticesin[ 89 ]wheretruncatedsincfunctionwasusedforthereconstructionalongwitharegularizationfactorformedbysecondorderderivativesoftheapproximatedfunction.WehavealsoexploitedtheregularizedleastsquareserrorminimizationframeworktocompareCartesian,BCCandFCClattices.ThesplinespaceformedbyshiftsofV-splineshasexactlythesamesizeofsupportandapproximationproperties[ 62 ]ondifferentlatticesandhencetheapproximationspaceisconsistentaswecomparethesubstratelattices.Thespline,s(x),evaluatedatanarbitrarypointx,formedbyshiftsofV-splinesonthereconstructionlatticeLwasintroducedin( 4 ).Notethats(x)canbeevaluatedatanyarbitrarypointx.Consideringthescattereddateff(xi)gandthesetof(non-uniform)samplingpointsX=fx1,,xNg,wecanformasystemofequationsbyformingasplinecoefcientcolumnvectordenotedascandforminganinterpolationmatrixwhoseithrowwillcorrespondtotheevaluationoftheshiftedbasisatsamplepointxi.Notethatweonlyrequiretoconsidertheshiftsofourbasisfunctions(V-splinesinthiscase)onlyforthelatticepointsfallinginsidetheboundingboxofthegivensamplingpoints.Forthesimplicityofthediscussion,thesetoflatticepointsfallinginsidethisboundingboxisindexed(linearly)using:K:=fk1,,kMg,whereMdenotesthenumberoflatticepointsandtheapproximatedsplinecanbewrittenas:s(x)=MXm=1cmVLn(x)]TJ /F6 11.955 Tf 11.95 0 Td[(km)=[VLn(x)]TJ /F6 11.955 Tf 11.95 0 Td[(k1)VLn(x)]TJ /F6 11.955 Tf 11.95 0 Td[(km)][c1cM]T| {z }c (4) 83

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Usingthisnotation,theinterpolationmatrixis:W:=266664VLn(x1)]TJ /F6 11.955 Tf 11.96 0 Td[(k1)VLn(x1)]TJ /F6 11.955 Tf 11.96 0 Td[(kM)......VLn(xN)]TJ /F6 11.955 Tf 11.95 0 Td[(k1)VLn(xN)]TJ /F6 11.955 Tf 11.95 0 Td[(kM)377775 (4)Moreover,wecanwritec=[c1,,cM]Tandthesamplingpointsvalueasf:=f(X)=[f(x1)f(xN)]T.EachcolumnvectorofmatrixWdenotestheshiftsofV-splinesonthe(linearizedindexingof)reconstructionlatticeL.Theinterpolationproblemfornon-uniformsamplingpointscannowbeformulatedasalinearsystemofequations:Wc=f (4)InthecaseofsecondorderV-splines,eachrowofmatrixWhasonly8non-zerovalues(i.e.,thenumberofsamplingpointsfallinginthesupportofsecondorderV-spline).Thenon-uniformsamplingpointscanhavearbitrarydistances;hence,someofthesplinecoefcientsmightnothaveanyconstraintsonthemwhichresultsinanunder-determinedsystemofequations.Moreover,exactinterpolationofsamplingpointsisnotalwaysdesirable.Therefore,insteadofsolvingthelinearsystemofequationsintroducedabove,aregularizedleastsquaresproblemwillbeusedtondthesplinecoefcients:minsNXn=1(s(xn))]TJ /F5 11.955 Tf 11.96 0 Td[(f(xn))2+M1(s) (4)ThesmoothingregularizationfactorM1(s)providesatrade-offbetweensmoothnessofthesolutionandthedelityoftheinterpolationandiscontrolledbyparameter.Theregularizationisbasedontherstordersemi-normoftheapproximatedsplines(x)[ 5 ]:M1(s):=Zkrs(x)k2dx=Z(s2x(x)+s2y(x)+s2z(x))dx (4) 84

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wheresx(x),sy(x)andsz(x)denotethepartialderivativesofthesplinealongX,YandZdirectionsrespectively.Therstordersemi-normcanbedenotedastheinnerproductoftwofunctionsas:hf,gi:=Rf(x)g(x)dx.Usinginnerproduct,wecanrewritetheregularizationfactorasaninnerproductintheBeppo-Levispaceas:M1(s)=hs,siBL=hsx,sxi+hsy,syi+hsz,szi (4)Usingsplineexpansionin( 4 ),wecanrewriteM1(s):M1(s)=hs,siBL=hMXm=1cmVLn(x)]TJ /F6 11.955 Tf 11.96 0 Td[(km),MXm=1cmVLn(x)]TJ /F6 11.955 Tf 11.96 0 Td[(km)iBL (4)Splittingthecoefcientsetfromtheshiftsofthebasisfunctionasbefore,wecanrewriteaclosedformforM1(s)as:M1(s)=cTGc,whereMMmatrixGisspeciedby:G=hVLn()]TJ /F6 11.955 Tf 17.93 0 Td[(km),VLn()]TJ /F6 11.955 Tf 17.93 0 Td[(km)iBL (4)ThematrixGencapsulatestheinnerproductofshiftsofthederivativeofthekernel.UsingthepolynomialrepresentationofthesecondorderV-splinesdiscussedinsection 4.3 ,thismatrixcanbeprecalculatedinclosedformandprecomputedonceforaxedlatticeresolution.Therefore,( 4 )canberewrittenas:minc2RMkWc)]TJ /F6 11.955 Tf 11.96 0 Td[(fk2+cTGc (4)Inpractice,largesystemofequationsliketheoneintroducedabovecanbesolvediterativelyusingnumericalmethods(i.g.,kaczmarz,conjugategradient).Byfurthersimplication,onecanconcludethatthisisaquadraticfunctionintermsoftheunknowncoefcientsetcandcanbesolvedbyquadraticprogramming[ 8 ].Fortheexperimentsinthissection,weusedifferentchoiceoflatticesforresamplingnon-uniformdatapointsandusetheregularizedleastsquareserrorminimizationtocomputethesplinecoefcients.Wehavecomparedtheefciencyofdifferentlatticesinreconstructingnon-uniformdatabyreportingtheSNR(signaltonoiseratio)which 85

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iscalculatedasthelogarithmoftheratioofthepowerinthesignal(Ps)overthepowerofthenoise(Pn):SNR:=10logPs Pn.UsingtheMarschner-Lobbsyntheticdataset,wehaveexaminedrandomsamplingasshowninFigure 4-17 .Inthisexperiment,1.2613randomdatapointshavebeenresampledtodifferentlatticeswithresolution613.Theresultshavebeenreportedfordifferentregularizationparameters().TheresilienceofdifferentsamplinglatticesinpresenceofjitternoiseisanothersetofexperimentsthatwehaveconductedonrealCTdatasets.Jitternoisecanbecausedbycalibrationproblemsordeciencyoftheacquisitionsystem.Tosimulatejitternoiseforrealdata,valuesofauniformlysampleddataisconsideredasthevaluesforajittered(perturbed)versionoftheunderlyinglattice.TheSNRcorrespondingtoresamplingthisjittereddatabackontothe(uniform)latticesusingdifferentregularizationfactorhasbeendepictedinFigure 4-17 foraneurismdata.Aswecanobservefromtheplots,inalltheexperiments,randomsamplingandjitternoiseexperiments,bothBCCandFCCcurvesareconsistentlyabovetheCartesiancase. 86

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AVC2BVB2CVF2 DVC2EVB2FVF2 GVC2withquasiHVB2withquasiIVF2withquasi Figure4-9. QuasiinterpolationresultsforMLalongwithreconstructionerrorvisualizations.TheMarschner-Lobbsampleddatasetrenderedfromthetop.ImagesA-C(rstrow)showV-splinereconstructiononCartesian,BCCandFCCwithoutapplyinganypreltering.ImagesD-I(thesecondandthirdrow)illustratetheangularerrorinthereconstructionofnormalsontheisosurfaceingrayscale.White(255)correspondstoangularerrorof.3radiansandblackshowsangularerrorofzeroradian.ImagesD-F(secondrow)illustratetheerrorinreconstructionofvolumetricdatapresentedinimagesA-C(rstrow)usingV2interpolationonCartesian,BCCandFCClatticewithoutapplyinganypreltering.ThedarkererrorimagesinimagesG-I(thirdrow)indicatemoreaccuratereconstructioncomparedtousualreconstructionforV-splinereconstruction. 87

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AVC2BVC2withquasi CVB2DVB2withquasi EVF2FVF2withquasiFigure4-10. Quasiinterpolationresultsforcarpshdataset.16%subsampledfromgroundtruthCarpshdatasetonCartesian,BCCandFCClattice.ThevolumetricdatasetshavebeenrenderedusingV2interpolationoncorrespondinglattices.Applyingquasi-interpolationhaspreservedmoredetailsinthetailandribssectionofcarpdatasetincaseofV-splinereconstructionwithapplyingquasiinterpolation,speciallyonBCCandFCC. 88

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AVL2withoutquasionCartesianDVL2withquasionCartesian BVL2withoutquasionBCCEVL2withquasionBCC CVL2withoutquasionFCCFVL2withquasionFCC Figure4-11. Quasiinterpolationresultsforbonsaitreedataset.ThebonsaitreedatasetreconstructedusingV2havingthesameresolutionondifferentlatticesasofcarpshdatasetsinFigure 4-5 .ThezoomedinviewinimagesA-C(secondcolumn)showsthedisconnectedbranchesbecauseofsmoothingartifact.TheconnectivityofbranchesismorevisibleinthezoomedinviewprovidedinimagesD-F(lastcolumn).Asshowninthegure,theconnectivityofthebrancheshasbeenpreservedbetteronBCCandFCClattices. 89

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ABCFigure4-12. Polynomialpartitionsoftheboxsplinesconstitutingthesupportofthesecondorderhexspline.A)ThepolynomialpartitionsofboxsplinesinFigure 3-2 (A).B)ThepolynomialpartitionsofboxsplinesinFigure 3-2 (B).C)Super-positioningofthesepolynomialpartitionsformthepotentialpartitioningofthesupportofVH1. ABCFigure4-13. Theminimalregionandtheclusteringresultsforthesupportofthesecondorderofhexspline(VH1).A)MinimalregionofthesupportofVH1.B)Clusteringofthepointsinsidetheminimalregionresultsin3mainclustersandalargenumberofmis-classiedpoints(incolors).C)PolynomialpartitionsformingthesupportofVH1. 90

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ABFigure4-14. TheminimalregiondenedfortheBCCandtheFCClattices.A)Fortruncatedoctahedron(i.e.,VoronoicellofBCClattice).B)Forrhombicdodecahedron(i.e.,VoronoicellofFCClattice). ABFigure4-15. PartitioningofminimalregionofthesecondorderV-splineintopolynomialpartitions.A)FortheminimalregionoftheBCC.B)FortheminimalregionoftheFCC. 91

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Figure4-16. PolynomialpartitionsformingtheminimalregionofVB1shownfrom2differentviewangles. ABFigure4-17. ScattereddatareconstructionusingsecondorderVoronoisplines.A)MLreconstructionwithuniformlatticedensityof613andthenon-uniformsampledensityof1.2613.B)Aneurismreconstructionfromjitteredsamplesonresolution1403withperturbationfactorof.25. 92

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CHAPTER5CONCLUSIONANDFUTUREWORKSThisdissertationintroducedVoronoisplinesasanewfamilyofmultivariatesplinesforall2-Dand3-D(andhigherdimensional)lattices.Voronoisplinesarenotlimitedtoanydimensionsandcanbeconstructedforhigherdimensionallatticeswiththedicingproperty.Asanapproximationscheme,Voronoisplinesprovideunbiasedreconstructionapproachtocomparedifferentchoiceoflatticesintermsoftheirsamplingefciency.ThisframeworkwasdenedinclosedforminChapter 3 whereithasbeenmorecloselystudiedforBCCandFCClattices,astheyareproventoprovidebettersamplingefciencycomparedtoCartesianlattice.ThegeometricpropertiesofthesupportofthesekernelswasusedtodevelopanexactevaluationschemeontheselatticesusingexpansionofVoronoisplinesintermsofconstitutingboxsplines.Inaddition,theapproximationpropertiesofVoronoisplinesalongwiththeirFouriertransformswerestudiedinmoredetails.FouriertransformofVoronoisplinesenabledustointroduceaquasiinterpolationapproachasapreprocessingstepforreconstructiontoovercometheover-smoothingartifactassociatedwithhigherorderreconstructionkernels.ThequasiinterpolationproposedwillasymptoticallyprovidethebestapproximationerroraffordablebyVoronoisplinesandshowsvisualandnumericalimprovementinthereconstruction.ThecomputationalcostofVoronoisplinescanbeimprovedsignicantly,giventhepolynomialpartitionsconstitutingthesupportareknowninthepolynomialform,ratherthanrecursiveboxsplineevaluation.ThisproblemhasbeensolvedforthecaseofsecondorderVoronoisplinesonBCCandFCC,whichprovidestheexactcounterpartinterpolationschemefortrilinearinterpolationonCartesian.Theadvantageofusingoptimalsamplinglattices(i.e.,BCCandFCC)isnotlimitedtothecaseofsamplingonauniformgridoralattice.Moreover,inpracticalapplications,almostnoacquisitionsystemisperfectandthesamplingprocessmightbepronetojitternoisewhichintroducesinaccuracyifthesamplesareconsidered 93

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onaregulargrid.Toinvestigatetheadvantageofusingoptimalsamplinglatticesforscattereddatareconstruction,avariationalapproachhasbeenproposed.TheproblemofapproximationfromscattereddatainthesplinespaceformedbysecondorderVoronoisplinesasitsbasishasbeenstudiedinChapter 4 .Avariationalframeworkwasusedtoresamplethescattereddataontodifferentchoiceoflatticesandtondthecorrespondingsplinerepresentation.Similartotheuniformcase,theoptimalsamplinglatticesshowedmoreaccuratereconstructionscomparedtotheCartesianlattice.TheintroductionofVoronoisplinesdiscussedindetailsinthisdissertation,showstheknowntheoreticaladvantagesofoptimalsamplinglatticesinvolumetricdatavisualizationapplication.ThediscussionhereshedlightsonthepotentialadvantagesofusingoptimalsamplinglatticesforsamplingalongwithreconstructionusingVoronoisplinesfordifferentapplicationsandtheneedforfurtherstudyofthisframework.Although,thepolynomialrepresentationofsecondorderVoronoisplineshasbeenstudiedinhere,drivinganefcientandpotentiallyautomaticevaluationschemeforhigherordersofVoronoisplinesisstillanopenproblem.Moreover,hardwareacceleratedreconstructionltershavebeenstudiedandprovidedimprovementinthereconstructiontimeusingGPU-basedtechniques[ 19 40 ].HavingGPUimplementationofVoronoisplines,thereconstructionefciencycanbeimproved.Inaddition,ThemathematicalandgeometricalfoundationsofVoronoisplinesprovidearelationtoreconstructionfromprojectionsandtomographicreconstructionwhichcanbestudiedfurther. 94

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APPENDIXCOMPLEMENTARYNOTEONQUASIINTERPOLATION Inthisappendix,weshowthattheconvolutionofVoronoisplinesofdifferentorderswithmonomialsresultsinspecicpolynomialfunctions,g(x),introducedinsection 4.2 .Inaddition,westudythepropertiesofg(x)(i.e.,theimageof[[x]]undertheconvolutionwithVn)inmoredetails.Forsimplicity,wewillusef(2)=D2f=f00throughoutthisappendixinterchangeably.Wewillworkwithdistributionsanditisworthtakingamomentandreviewthedenitionofmultiplicationofadistributionwithafunction:Iffisadistributionandhisasmoothfunction,theproductoffandhisdenedtobethedistributionhf:!hf,hi,forall,whichisa(smooth)testfunction[ 43 ].Nowwearereadytoshowthatanormalizedmonomialrepresentedas:[[x]],convolvedwithVnwillresultinapolynomial.FromthedenitionofdistributionalFouriertransform,wecanwrite: [[x]]Vn(x)()()(!)^Vn(!)(A)Tosimplifytheargumentandwithoutlossofgenerality,consider=2whichmeanswewillfocusonnormalizedquadraticmonomialconvolvedwithVoronoisplineofdegreen.Wewanttostudythisconvolutionmorecloselyinthefrequencydomain: [[x]]2Vn(x)()(2)(!)^Vn(!)(A)Usingthedistributiondenitionofthedeltafunction(h,'i='(0))andthedenitionintroducedformultiplicationofasmoothfunction(i.e.,Voronoispline)andadistribution(i.e.,Diracdeltafunction)withthehelpofasmoothtestfunction,,wecanwrite: h(2),^Vni=)]TJ /F4 11.955 Tf 7.34 -7.03 Td[(^Vn(2)(!)j!=0=)]TJ /F4 11.955 Tf 7.35 -7.03 Td[(^V(2)n+2(1)^V(1)n+(2)^Vn(!)j!=0(A) 95

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Nowbypluggingin!=0,wehave: h(2),^Vni=)]TJ /F4 11.955 Tf 7.34 -7.02 Td[(^V(2)n(0)(0)+2(1)(0)^V(1)n(0)+(2)(0)^Vn(0)=ah,i+bh(1),i+ch(2),i(A)DerivativesofVnevaluatedat0(i.e.,^V()n(0))areconstantnumbersandthereforewehavedenotedthemasa,bandc.Therefore,wecanrewritethisrelationintermsoftheactionofanotherdistribution.Notethat2distributionsarethesameiftheiractiononthetestfunctionisthesameandthatwouldhelpustoshowthattheconvolutionwillresultinapolynomial[ 43 ].Therefore,wecanrewritetheequationaboveas: h(2)^Vn,i=ha+b(1)+c(2),i(A)Thecoefcientsa,bandcaretheconstantnumbersfrom^V()n.Notethata+b(1)+c(2)correspondstoFouriertransformofaquadraticpolynomial(a+bx+cx2).Theaboveexampleshowsthatifweconvolveanormalizedmonomial[[x]]withVnwewillgetbackapolynomial: [[x]]Vn=Xcx(A)Itisimportanttonotethatthecoefcientscorrespondingtothispolynomialareonlydependentonthekernelusedfortheconvolution.Onecanusethesepolynomialsintroducedinsection 4.2.2 asg(x)tondthequasiinterpolationforspecicorderofV-splinesasdiscussedinsection 4.2.2 .Thepolynomialfunctionghavesomeinterestingpropertiesthathasbeendiscussedin[ 27 ].Wewanttoreviewsomeofthepropertiesofthisclassofpolynomialinhere.Wealreadyknowthateachnormalizedmonomialwillbeassignedtoapolynomialbyconvolution,nowletuslookatdifferentordersofnormalizedmonomialsandconsiderthecorrespondinggandstudytheirproperties.Notethatnormalizedmonomialsofdifferentdegreecanbeusedtoformasequencef[[x]]g.Thesequenceformedby 96

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thesenormalizedmonomialsarecalledAppellsequence[ 27 ]:OnepropertyforAppellsequenceisthat,takingthethderivativeofthethterminthesequence,givesthe()]TJ /F8 11.955 Tf 11.96 0 Td[()thelementinthatsequence.Thisiseasytoproveforthecaseofnormalizedmonomialsfromthedenition: D[[x]]=[[x]])]TJ /F30 7.97 Tf 6.59 0 Td[((A)Nowwehavethefollowingcorrespondencebetweengandnormalizedmonomial[[x]]: [[x]]=Vng(A)Takingthethderivativeoftheequationabovewehave:[[x]])]TJ /F30 7.97 Tf 6.58 0 Td[(=D[[x]]=Vn(Dg).Moreover,wealreadyknowthat[[x]])]TJ /F30 7.97 Tf 6.58 0 Td[(correspondstog)]TJ /F30 7.97 Tf 6.58 0 Td[(andtherefore,onecanconcludethatDg=g)]TJ /F30 7.97 Tf 6.59 0 Td[(andgalsoformanAppellsequence.NotethatVoronoisplinessatisfypartitionofunityconditionasdiscussedinChapter 3 .Usingthepartitionofunityconditionandsubstituting[[x]]0whichisequaltotheconstantfunction1in( A ),wecanconcludethatg0=1.Applyingthederivativerelationbetweendifferentordersofgandg0onecanconstructgiasfollows: g0(x)=1g1(x)=[[x]]+ag2(x)=[[x]]2+a[[x]]+bg3(x)=[[x]]3+a[[x]]2+b[[x]]+cg4(x)=[[x]]4+a[[x]]3+b[[x]]2+c[[x]]+d(A) 97

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Wenoticethatthecoefcientsarerepeatedwhichgivesthegeneralformulag=Pc()]TJ /F8 11.955 Tf 12.46 0 Td[()[[x]].Tondeachcoefcient,alloneneedstodoistondg(0)whichgivesc(becauseofthewaythecoefcientsrepeatedlyshowupinthenextpolynomial).Therefore,foreachorderofmonomials([[x]])wejustneedtondccorrespondingtotheconstantvalueofg,sincetheothercoefcientshavealreadybeenspeciedforg,.In( 4 )wehavefoundarelationbetweencandthederivativeDj ^Vn(!)evaluatedatzero.Therefore,wecancalculateallofthecoefcientsofgusingthecorrespondingderivativevalues.NotethatnoneofthesecoefcientsarenecessarilyzerobutsinceB-splinesandVoronoisplinesarerealvaluedfunctionsandtheirFouriertransformisanevenfunction,therstorderderivativeoftheirFouriertransformisalwayszero. 98

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BIOGRAPHICALSKETCH MahsaMirzargarreceivedherbachelor'sdegreeincomputerengineeringmajorininformationandcommunicationtechnologyfromSharifUniversityofTechnology,Tehran,Iran,in2008.ShewasadmittedtothePh.D.programincomputerengineeringatUniversityofFloridaandjoinedtheDepartmentofComputerandInformationScienceandEngineering(CISE),UniversityofFloridainAugust2008.Shehasreceivedhermaster'sdegreeinDecember2011andreceivedherPh.D.inDecember2012.Herresearchinterestsincludesignalandimageprocessing,visualization,machinelearning,medicalimaginganddatacommunication. 106