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On Sampling and Reconstruction of Distance Fields

Permanent Link: http://ufdc.ufl.edu/UFE0043003/00001

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Title: On Sampling and Reconstruction of Distance Fields
Physical Description: 1 online resource (69 p.)
Language: english
Creator: THAZHEVEETTIL,NITHIN P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: BCC -- CUBIC -- CUDA -- DISTANCE -- FIELDS -- GPU -- INTERPOLATION -- SAMPLING
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this thesis, we examine sampling and reconstruction of distance fields from surfaces represented by triangular meshes. Motivated by sampling theory, we explore the application of optimal sampling lattices in this context. To sample exact distance values from a triangular mesh, we propose a Graphics Processing Unit(GPU)implementation of the brute force approach and show how to adapt it to the Body Centered Cubic(BCC) lattice. The exact gradients of distance fields can be computed with relative ease and we believe that incorporating these values could improve the quality of reconstruction of discrete distance fields. Hence, we discuss ways of modifying our implementation to sample exact gradients with relatively few additional computations. The suitability of BCC as a sampling lattice for distance fields and the merits of using exact gradient data are evaluated by reconstructing and visualizing distance fields sampled from various triangular meshes. To reconstruct the data on BCC lattices, we introduce a cubic spline construction that is exactly interpolating at the lattice points and can utilize true gradient values where available. We also compare and contrast the images rendered from these datasets to those rendered using Catmull-Rom interpolation on distance fields sampled on Cartesian Cubic(CC) lattices.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by NITHIN P THAZHEVEETTIL.
Thesis: Thesis (M.S.)--University of Florida, 2011.
Local: Adviser: Entezari, Alireza.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-04-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043003:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043003/00001

Material Information

Title: On Sampling and Reconstruction of Distance Fields
Physical Description: 1 online resource (69 p.)
Language: english
Creator: THAZHEVEETTIL,NITHIN P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: BCC -- CUBIC -- CUDA -- DISTANCE -- FIELDS -- GPU -- INTERPOLATION -- SAMPLING
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this thesis, we examine sampling and reconstruction of distance fields from surfaces represented by triangular meshes. Motivated by sampling theory, we explore the application of optimal sampling lattices in this context. To sample exact distance values from a triangular mesh, we propose a Graphics Processing Unit(GPU)implementation of the brute force approach and show how to adapt it to the Body Centered Cubic(BCC) lattice. The exact gradients of distance fields can be computed with relative ease and we believe that incorporating these values could improve the quality of reconstruction of discrete distance fields. Hence, we discuss ways of modifying our implementation to sample exact gradients with relatively few additional computations. The suitability of BCC as a sampling lattice for distance fields and the merits of using exact gradient data are evaluated by reconstructing and visualizing distance fields sampled from various triangular meshes. To reconstruct the data on BCC lattices, we introduce a cubic spline construction that is exactly interpolating at the lattice points and can utilize true gradient values where available. We also compare and contrast the images rendered from these datasets to those rendered using Catmull-Rom interpolation on distance fields sampled on Cartesian Cubic(CC) lattices.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by NITHIN P THAZHEVEETTIL.
Thesis: Thesis (M.S.)--University of Florida, 2011.
Local: Adviser: Entezari, Alireza.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-04-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043003:00001


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ONSAMPLINGANDRECONSTRUCTIONOFDISTANCEFIELDSByNITHINPRADEEPTHAZHEVEETTILATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2011

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c2011NithinPradeepThazheveettil 2

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Idedicatethistomyparents. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,Dr.AlirezaEntezariforallhissupportandguidancewithoutwhichthisthesiswouldnothavebeenpossible.Iwouldalsoliketothankmythesiscommitteemembers,Dr.JorgPetersandDr.AnandRangarajanfortheirhelpandsuggestions.Finally,Ithankmyfamilyandfriendswhoseconstantsupportkeptmegoingwhenallelsefailed. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 1.1Motivation .................................... 9 1.2BodyCenteredCubicSamplingLattice .................... 11 1.3ComputeUniedDeviceArchitecture ..................... 13 1.4Contributions .................................. 16 2RELATEDWORK .................................. 18 2.1SamplingDistanceFields ........................... 18 2.2ReconstructionofVolumetricData ...................... 20 2.3SamplingandReconstructiononBodyCenteredCubicLattice ...... 22 3SAMPLINGDISTANCEFIELDS .......................... 24 3.1BruteForceMethod .............................. 24 3.2Point-TriangleDistance ............................ 25 3.3GraphicsProcessingUnitImplementation .................. 26 3.4GeneratingTrueGradients .......................... 32 3.5Performance .................................. 32 4CUBICINTERPOLATIONINBODYCENTEREDCUBICLATTICEUSINGHERMITEDATA ................................... 34 4.1CubicInterpolationinBodyCenteredCubicLattice ............. 34 4.1.1InterpolatingSplines .......................... 34 4.1.2SmoothnessAndApproximationOrder ................ 39 4.1.3IsotropicFinite-DifferencesontheLattice .............. 41 4.2TricubicInterpolationinCartesianCubicLattice ............... 46 5INTERPOLATIONOFDISTANCEFIELDSANDEXPERIMENTS ........ 49 6CONCLUSIONANDFUTUREWORK ....................... 62 REFERENCES ....................................... 64 BIOGRAPHICALSKETCH ................................ 69 5

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LISTOFTABLES Table page 3-1Triangularmeshesusedforperformancetesting ................. 33 3-2Comparisonofexecutiontimes.ThetimetakenbytheGPUimplementationisshowninthecolumn'GPU'andthatbytheCPUimplementationisshowninthecolumn'CPU'. ................................. 33 4-1TheL2normerrorinreconstructionofdatasetssampledataresolutionof111111222ontheBCClatticeusingtheproposedcubicinterpolation.Theweightedleast-squaresapproachisdesignedtouseazeromeanGaussianwith2=.5toestimatepartialderivativevalueswhichshowmarginalimprovementsintermsofreconstructionerror. ........................... 46 6

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LISTOFFIGURES Figure page 1-1TheBCCLattice. ................................... 12 1-2Hierarchyofthreads,warps,blocksandgrids. ................... 15 1-3CUDAmemorymodel. ................................ 15 3-1Thest-planepartitionedinto7regions. ....................... 26 3-2BlockdiagramoftheGPUimplementation. .................... 27 3-3TwowaysofadaptingtheGPUimplementationforBCClattices. ........ 31 4-1TheCarpshdataset. ................................ 40 4-2Weightsofthenite-differencingkernelonthe27-pointneighborhoodoftheBCClatticeforfxandfyz. .............................. 45 5-1Thetriangularmeshesusedinourexperiments. ................. 50 5-2Thesoccerballdatasetwithapproximately512,000samples. .......... 52 5-3TheStanforddragondataset(sideview)withapproximately512,000samples. 53 5-4TheStanforddragondataset(frontview)withapproximately512,000samples. 54 5-5Thebuddhadatasetwithapproximately512,000samples. ............ 55 5-6Thebunnydatasetwithapproximately615,000samples. ............ 56 5-7ThePawndatasetwithapproximately33,000samples. .............. 57 5-8ThePawndatasetwithapproximately200,000samples. ............. 58 5-9ThePawndatasetwithapproximately260,000samples. ............. 59 5-10ThePawndatasetwithapproximately512,000samples. ............. 60 5-11ThePawndatasetwithapproximately2,095,000samples. ............ 61 7

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceONSAMPLINGANDRECONSTRUCTIONOFDISTANCEFIELDSByNithinPradeepThazheveettilMay2011Chair:AlirezaEntezariMajor:ComputerEngineeringInthisthesis,weexaminesamplingandreconstructionofdistanceeldsfromsurfacesrepresentedbytriangularmeshes.Motivatedbysamplingtheory,weexploretheapplicationofoptimalsamplinglatticesinthiscontext.Tosampleexactdistancevaluesfromatriangularmesh,weproposeaGraphicsProcessingUnit(GPU)implementationofthebruteforceapproachandshowhowtoadaptittotheBodyCenteredCubic(BCC)lattice.Theexactgradientsofdistanceeldscanbecomputedwithrelativeeaseandwebelievethatincorporatingthesevaluescouldimprovethequalityofreconstructionofdiscretedistanceelds.Hence,wediscusswaysofmodifyingourimplementationtosampleexactgradientswithrelativelyfewadditionalcomputations.ThesuitabilityofBCCasasamplinglatticefordistanceeldsandthemeritsofusingexactgradientdataareevaluatedbyreconstructingandvisualizingdistanceeldssampledfromvarioustriangularmeshes.ToreconstructthedataonBCClattices,weintroduceacubicsplineconstructionthatisexactlyinterpolatingatthelatticepointsandcanutilizetruegradientvalueswhereavailable.WealsocompareandcontrasttheimagesrenderedfromthesedatasetstothoserenderedusingCatmull-RominterpolationondistanceeldssampledonCartesianCubic(CC)lattices. 8

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CHAPTER1INTRODUCTION 1.1MotivationAdistanceeldofanobjectisascalareldaroundtheobjectinwhicheverypointholdstheshortestdistancefromthatpointtothesurfaceoftheobject.1Itcanbeconsideredasanimplicitrepresentationoftheobject.Onthesurfaceoftheobject,theeldwillhaveavalueof0,providinganimplicitrepresentationoftheobjectasthezerolevelsetoftheeld.Inadditiontothedistancesthemselves,distanceeldscanalsoindicateotherpropertiesofagivenpoint,suchasthedirectionfromthatpointtothesurfaceandwhetherthepointisinsideoroutsidetheobject.Thegradientoftheeldatanypointgivesthedirectionfromthatpointtotheclosestpointonthesurface.Distanceeldscanbesignedorunsigned.Inasigneddistanceeld,everypointisassignedasigninadditiontothescalarvalue.Thissignindicateswhetherthepointisinsideoroutsidetheobject.Notethatforthistobemeaningful,thesurfacemustbeclosedandorientable,whiledistanceeldsingeneraldonotrequirethisproperty.Distanceeldshaveapplicationsinavarietyofeldslikecomputervision,physics,medicalimagingandcomputergraphics.Theyareusedforproximitycomputations[ 26 37 ],collisiondetection[ 48 49 ],morphing[ 5 8 ],skeletalanimation[ 21 ],pathplanning[ 29 32 ]andconstructivesolidgeometry(CSG)operations[ 6 39 ].Intheeldofvolumegraphics,distanceeldsareusedformodeling,manipulationandvisualizationofgeometricobjects.Techniquesofacceleratingvisualization,likeskippingemptyvoxelsduringraytracingbasedondistancevalues,alsomakedistanceeldrepresentationattractive.Inthisthesis,weareprimarilyconcernedwithusingdistanceeldstorepresentandvisualizesurfacesrepresentedbytriangularmeshes. 1PartsofChapters2,4,5and6areadaptedwithpermissionfromM.Mirzargar,N.Thazheveettil,W.YeandA.Entezari.CubicInterpolationOnTheBodyCenteredCubicLattice.SubmittedtoTransactionsonVisualizationandComputerGraphics. 9

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Sincedistanceeldsofgeometricmodelsarealmostalwaystoocomplextoberepresentedinanalyticalform,theyaresampledandstoredindiscreteform.Thesamplelocationsareusuallychosentobepointsofaregulargridthoughadaptiveschemesthatarebettersuitedforspecicapplicationshavebeensuggestedaswell[ 2 20 ].Non-regularsamplingschemesareattractivefortheadaptivityfeatures;however,efcientreconstructionisdifcultandexpensivescattered-datainterpolationtechniqueshavetobeemployed.Moreover,uniformgridrepresentationlendstoaneasywaytoanalyzesigneddistancefunctionsinFourierspacethroughFastFourierTransform(FFT)ofsampleddata.In3D,themostcommonlyuseduniformsamplinglatticeistheCartesianCubic(CC)lattice,thoughotherlatticesliketheBodyCenteredCubic(BCC)latticeandtheFaceCenteredCubic(FCC)latticehavebeenshowntobemoreefcientfromthesamplingtheoreticpointofview.InSection 1.2 ,wediscusstheCCandBCClatticesandtheirsamplingefciencies.Eachsamplepointinadiscretedistanceeldofasurfaceholdstheminimumdistancefromthatpointtothesurface.Computingthisdistanceinvolvesidentifyingthepointonthesurfacethatisclosesttothesamplepoint.Sincethisthesisfocusesonsurfacesrepresentedbytriangularmeshes('surface'or'object'hereafterreferstoasurfacerepresentedbytriangularmesh),thisclosestpointcanbeidentiedbyidentifyingthetriangleitispartof.Thus,adiscretedistanceeldcanbeconstructeddirectlyfromthetriangularmeshdatausingabruteforceapproach,i.e.computingthedistancefromeverysamplepointtoeverygeometricprimitiveintheobject.Whilethismethodisaccurate,itisalsoextremelyslow,makingitunsuitableformostapplications.Numerousotherapproacheshavebeensuggested,mostlyfocusingmoreonspeedthanaccuracy.WediscussmanyoftheseapproachesinSection 2.1 .However,asthereareapplicationsthatrequireaccuratevaluesofdistanceelds,wetrytoaddresstheissueofgeneratingaccuratedistanceeldsasfastaspossiblebypursuingtechniquesthatcanacceleratethebruteforcemethod. 10

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Distanceeldssogeneratedandtheobjectstheyrepresentareoftenconsumedvisually.Visualizationalsohelpsinevaluatingthequalityofvariousdiscreterepresentationsofdistanceelds.Hencethetopicofvisualizingdistanceeldsreceivesourattentionaswell.DistanceeldscaneitherbevisualizeddirectlyusingavolumevisualizationmethodlikeraycastingorbeconvertedtoapolygonalmeshusingtechniquesliketheMarchingCubesalgorithm(ortheMarchingTetrahedraalgorithm)andrenderedusingatraditionalsurfacerenderingtechnique.Ineithercase,thecontinuousdistanceeldmustbereconstructedfromthediscretevaluesrst,usingsomeinterpolationscheme.Whileawidevarietyofinterpolationschemescouldbeusedwithdistanceelds,weareparticularlyinterestedinthosethatmakeuseoftheavailabilityofitstruegradientvalues,whichcanbecomputedalongwiththedistancevaluesthemselves.ComingupwithsuchaschemefortheBCClatticeisanothertopicweaddressinthisthesis.Theprocessofconvertingapolygonalmeshtoadiscretedistanceeldandbackalmostalwaysintroduceserrorsinthemeshgeometry.Astheaccuracyofthisprocessisoftenakeyconsiderationindeterminingtheapplicationsitissuitablefor,wedevotesomeattentiontotheaccuracyofthesamplingandvisualizationschemesweintroduceinthisthesis.Therestofthischapterisorganizedasfollows.Section 1.2 talksaboutsamplingontheBCClattice.Section 1.3 givesabriefintroductiontoComputeUniedDeviceArchitecture(CUDA)andinSection 1.4 ,welistthemajorcontributionsofthisthesis. 1.2BodyCenteredCubicSamplingLatticeThegoalofoptimalsamplingistocapturetheentirespectrumoftheunderlyingsignalusingtheleastnumberofsamples.Foraspecicgivensignal,thereisauniquechoiceofoptimalsamplinglattice,whichdenesthepointsinspacewherethesignalissampled.Thislatticecanbecomputedbasedonthegeometricknowledgeofthespectrumofthesignal.Butforgenericdata,wheresuchknowledgeisnotavailable, 11

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regularlatticesareusedforsampling.In3D,themostcommonlyusedregularsamplinglatticeistheCartesianCubic(CC)lattice.TheCClatticeisformedbythetensor-productofuniformsamplinginthelowerdimensions.However,whilesimpleandpopular,theCClatticehasbeenshowntobeinefcientinsamplinggenericmultivariatesignals.BoththeBodyCenteredCubiclattice(BCC)andFaceCenteredCubic(FCC)lattice,whichare3Dcounterpartsofthehexagonallattice,exhibithighersamplingefciency(i.e.capturemoreinformationpersampletaken)thantheCClatticewiththeBCClatticeperformingbetteronsmoothsignals. A BFigure1-1. TheBCCLattice.(A)TheBCClatticeisformedbyaddingalatticepoint(inblue)tothecenterofeachcubicelementoftheCClattice.(B)TheuniquetetrahedralizationoftheBCClatticeiscomposedofthesemi-regularcongruenttetrahedra. TheBCClatticecanbeconstructedfromtheCClatticebyaddingalatticepointtothecenterofeachcubicelementformedby8neighboringlatticepoints(part(A)ofFigure 1-1 ).Itsrelativesuperiorityinsamplinggenericsignalscanbeexplainedbya 12

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frequencydomainanalysisofgenericsignalswhichweassumetobeisotropic(i.e.notbiasedinanydirection)andbandlimited.Samplingasignalinaregularmannerinthespatialdomaincorrespondstoperiodicallyreplicatingitsspectruminthefrequencydomain.Forthesamplingtobeoptimal,itshouldbedonesuchthatthespectrumisreplicatedasdenselyaspossiblewithoutoverlappinginthefrequencydomain.Thespectrumofanisotropicbandlimitedsignalhasasphericalsupport.So,theoptimalsamplinglatticeisthatwhosedualinfrequencydomainallowsspherestobepackedasdenselyaspossiblewithoutoverlap.Itcanbeseenthatamongthelatticesdiscussed,theFCClatticeenablesthemostoptimalspherepacking.Hence,itsdualinthespatialdomain,theBCClattice,performsbestasasamplinglattice.Everylatticepointinthelatticehasaneighboringregionwhereitistheonlylatticepoint.ThisregioncanbedenedusingitsDelaunydiagramoralternately,itsdual,theVoronoicells.AVoronoicellofalatticepointisthesetofpointsinitsneighborhoodthatareclosesttoit.Foragivenlatticepoint,theotherlatticepointswhoseVoronoicellsshareafacewithitsownVoronoicellformsits`rstring'ofneighborhood.InthecaseofBCClattice,thisneighborhoodformsarhombicdodecahedron,whichcanbedecomposedintocongruentDelaunaytetrahedra.AsthespaceofBCClatticeiscomposedofsuchrhombicdodecahedra,theentirespacecanbepartitionedintocongruentDelaunaytetrahedra.Therhombicdodecahedronanditstetrahedralpartitionareshowninpart(B)ofFigure 1-1 .ItisworthnotingherethattheDelaunaytetrahedralizationoftheBCClatticeisunique.Ifthetetrahedralizationisn'tunique,asisthecasewiththeCClattice,thereconstructionofthesampledsignalcoulddependonthechoiceofthetetrahedraandhencecouldresultinarbitraryandinconsistentreconstruction. 1.3ComputeUniedDeviceArchitectureComputeUniedDeviceArchitecture(CUDA)isageneralpurposeparallelprogrammingarchitecturedevelopedbyNVIDIA.Itmakesthemassiveparallel 13

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processingcapabilitiesofthegraphicsprocessingunit(GPU)accessibletotheuserforgeneralpurposecomputations.TheCUDAarchitectureallowsCUDAenabledGPUstobeprogrammedusingahighlevelprogramminglanguagecalledCforCUDAwhichisbasedonANSICwithafewCUDAspecicextensions.CUDAprogramscancontaincodesegmentstargetedtorunontheCPU(referredtoashost)aswellascodesegmentstargetedtorunontheGPU(referredtoasdevice).ThecodesegmentsthatrunonthedevicearewrittenasCstylefunctionscalledkernels.CUDAkernels,alongwithaccompanyinghostcode,arewrittenin.culeswhicharecompiledusingthenvcctool.AdditionalhostcodecanalsobewritteninseparateC/C++leswhichmustbecompiledseparatelyandlinkedusingastandardC/C++compilerandlinker.Whenakernelislaunched,auserspeciednumberofCUDAthreadsarecreatedtoexecutethekernel.Thenumberofthreadstobecreatedisspeciedintermsof3dimensionalarraysofthreadscalledthreadblocks.Allthreadswithinablockcansynchronizeexecutionusingabarrierprimitiveandsharedatathroughsharedmemory.Thereisalimitonthenumberofthreadsthatcanbeassignedtoablock.Multipleequally-shapedblocksarelaunchedtogeneratethenecessarynumberofthreadsforthekernel.Theseblocksarearrangedintoasingle2dimensionalgrid.Thenumberofthreadsassignedtoablock(alongwiththememoryresourceseachthreadrequires)inuenceshowwelltheprocessingpoweroftheGPUisutilized,indicatedbyapercentagevaluecalledoccupancy.Thus,thenumberofthreadsshouldbecarefullychosensoastomaximizeoccupancy.Threadsareexecutedingroupsof32calledwarps,inaSingleInstructionMultipleThread(SIMT)manner.Thatis,atanygiventime,everythreadinawarpexecutesthesameinstruction.Parallelismisachievedbyassigningdifferentdataunitstoeachthread.Thus,thisissimilartotheSingleInstructionMultipleData(SIMD)executionmodel.Therelationshipbetweenthreads,warps,blocksandgridsareshowninFigure 1-2 14

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Figure1-2. Hierarchyofthreads,warps,blocksandgrids. Figure1-3. CUDAmemorymodel. 15

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CUDAseparatesthehostmemoryspacefromthedevicememoryspace.Thedatathatakernelworkswithduringexecutionmustbeinthedevicememory.ACUDAenableddevicehasmultipletypesofmemoryasillustratedinFigure 1-3 .Global,constantandtexturememoryareaccessiblefromthehostandfromeverythreadonthedevice.Hence,datacanbetransferreddirectlyfromthehostmemorytothesememoryspacesandaccessedbyanythreadinthegrid.Allthreearepersistent(i.e.retaindata)acrosskernellaunchesbythesameapplication.However,globalmemorycanbewrittentobythethreadswhileconstantandtexturememoryareread-onlyfromthedevice.Sharedmemoryandregistersarefaston-chipmemory.Sharedmemoryisallocatedperthreadblock.Everythreadinathreadblockhasaccesstoitssharedmemorybutcannotaccessthesharedmemoryofotherblocks.Registersareallocatedperthreadandareusedforlocalvariablesdeclaredwithinthekernelandtemporaryvariablesusedforcomputations.Thenumberofregistersallocatedtoathreaddependsonthenumberofthreadsassignedtoablock.Oncetheregistersareusedup,anyadditionalvariablesarestoredinlocalmemory,whichisoff-chipandconsiderablyslower.Verylargearraysanddatastructuresgointothelocalmemoryaswell.Typically,dataisloadedintoregistersandsharedmemorybeforebeingprocessedtoavoidlargememorylatencies.ForamoredetaileddocumentationofCUDA,pleaserefertheCUDAprogrammingguide[ 1 ]. 1.4ContributionsTheprimarycontributionsofthisthesisareasfollows.Weintroduceanaccelerated,GPUbasedimplementationofthebruteforcemethodtoconstructadiscretesigneddistanceeldonaCClatticefromtriangularmeshes.WepresentwaysofadaptingthistoworkwithBCClatticesandalsodiscusscomputingthetruegradientvaluesofthedistanceeldatthelatticepoints.WeproposealocalcubicinterpolationschemeintrivariatesettingontheBCClatticethatusesHermitedata(i.e.functionvaluesanditspartialderivatives).Wealso 16

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brieysummarizealocaltricubicinterpolationschemeproposedbyLeikenandMarsden[ 31 ]thatusesHermitedataontheCClattice.Finally,wegeneratedistanceeldsforvarioustriangularmeshesonbothCCandBCClatticesusingthesamplingtechniqueproposedinthisthesis.Then,wecompareandcontrasttheresultsoftheirvisualizationusingthetwointerpolationschemesmentionedpreviouslyalongwiththewellknownCatmull-Romscheme. 17

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CHAPTER2RELATEDWORK 2.1SamplingDistanceFieldsJonesetal.[ 27 ]presentanexcellentsurveyofthevariousmethodsusedtogeneratedistanceeldsalongwithacomparisonoftheirspeedandaccuracy.Wediscussaselectionofthoseapproachesthatarerelevanttothecontextofthisthesis.Thenaiveapproachtoconstructingadiscrete3Ddistanceeldfromatriangularmeshistoiteratethrougheverygridpoint/voxel,computetheshortestdistancefromittoeverytriangleinthemeshandstoretheminimumvalue.PayneandTogain[ 41 ]discussthisapproachandsuggestafewoptimizationstoaccelerateit,likeusinghierarchicalboundingboxesinatreestructuretoreducethenumberofcomputations.Theyalsodiscussanalgorithmtondtheshortestdistancefromapointtoatriangle.TheMeshsweeperalgorithmproposedbyGueziec[ 22 ]presentsadynamicalgorithmtondtheshortestdistancefromapointtoapolygonalmesh.Itusesahierarchyofmultilevelboundingboxeswiththeboundingboxesateachlevelcompletelyenclosingthemesh.Theseboundingregionsareindexedintoapriorityqueuebasedontheminimumdistancefromthepointtotheregion.Maush[ 34 35 ]presentedtheCharacteristic/ScanConversionmethodthatcomputesadistanceeldaroundapolygonalmeshuptoacertaindistanceusingscanconversion.Thepointonatriangularmeshclosesttoavoxelmustlieoneitherthefaceofatriangle,anedgeoravertex.Foreachofthesefeatures,theapproachconstructsapolyhedronthatholdsallthepointsin3Dspacethatareclosesttothatfeature,uptoacertaindistancefromthefeature.ThesepolyhedronsaresimilartotruncatedVoronoiregionsandarecalledcharacteristicsofthefeature.Then,foreachfeature,onlythedistancetothepointswithinthecorrespondingpolyhedronneedbecomputed.Thepolyhedronsarescanconvertedtodeterminethepointsinsidethemandthedistanceforthecorrespondingpointscomputed.Siggetal.[ 46 ]improvedthisalgorithmby 18

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usingthegraphicshardwaretoscanconvertslicesofthegrid.Thedistancevaluesarecomputedusingafragmentprogram.AstheslicingisdoneontheCPUwhichcanformabottleneck,thecharacteristicsforthetriangleface,edgesandverticesarecombinedtoformasinglepolyhedron.Sudetal.[ 47 ]presentsanotherhardwarebasedmethodwhichexploitspropertieslikeconnectivityandspatialcoherenceinVoronoiregionstocullthenumberofprimitivesconsideredfordistanceeldcomputationsforeachsliceandrestricttheregionofcomputationaroundeachprimitive.Thisreducesthenumberofdistancefunctionscomputedperslice.Adifferentapproachtocomputingdistanceelds,whichyieldsapproximatevalues,isDistanceTransforms.Here,thedistancevaluesforanarrowbandaroundthemeshsurfacearerstcomputedandthenpropagatedthroughtherestofthevolume.Mullikin[ 38 ]discussesapplyingoneparticulardistancetransformcalledtheVectorDistanceTransformto3Dimages.InVectorDistanceTransforms,thevectorconnectingapointtotheclosestpointontheobjectsurfaceiscomputedalongwiththedistancevaluesandthesevectorsarepropagatedtotheneighboringvoxelsandusedtocomputethedistancevaluesforthem.SatherleyandJones[ 42 ]introduceafasterandmoreaccurateVectorDistanceTransformanddiscusshowtogeneratedistanceeldsusingit.Breenetal.[ 6 ]presentawavefrontpropagationtechniquetogeneratedistanceeldsforCSGmodelswithsub-voxelaccuracy.TheycomputetheshortestdistanceandclosestsurfacepointforasetofpointsinthenarrowbandandpropagatethemtotherestofthevolumeusingaFastMarchingMethod[ 25 44 45 52 ].AcriticalreviewofvariousDistanceTransformmethodscanbefoundinCuisenaire[ 12 ].Finally,aCUDAbasedapproachtocomputingadaptivedistanceeldshasbeensuggestedrecently[ 40 ].Likeourimplementation,thisapproachalsousesaGPUimplementationofthebruteforcemethod(thenaivemethoddescribedatthebeginningofthissection)tocomputedistanceelds.EachCUDAthreadisassignedameshelement(triangle)andthesamplepointsarefedtotheGPUoneaftertheother. 19

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Whileeachthreadcouldbeassignedasamplepointinsteadofameshelement,thenon-uniformnatureofsamplinggridspromptstheauthorstopickmeshelementsasthefociforparallelization.Duringeachiteration,eachthreadcomputesthedistancefromtheinputsamplepointtothetriangleassignedtoit.Theshortestvalueamongtheseisthencomputedusingaparallelreductiontechnique[ 24 ].Thesignofeachdistancevalueiscomputedusingtheangle-weightedpseudonormalmethod[ 3 ].ComparedtoasinglecoreCPUimplementationofakd-treebasednearestneighborsearchalgorithm,theauthorsreportspeedupsrangingfrom10to65forvariousmeshesandsamplingresolution. 2.2ReconstructionofVolumetricDataWhileavastamountofliteratureisavailableonreconstructionofvolumetricdata,weareprimarilyinterestedinapproachesthatmakeuseofHermitedata,i.e.datavaluesanditsexactderivativevalues,toachieveimprovedreconstruction.Afewsuchapproachesarereviewedhere.MarchingCubes[ 33 ]isawellknownalgorithmforconstructingatriangularmeshrepresentationofasurfacefromagridbasedvolumetricrepresentation.Verticesofthetrianglesareformedbythepointsofintersectionofthesurfaceofinterestandtheedgesofthegrid.Theseintersectionpointsarefoundusinglinearinterpolationonthevaluesatthegridpoints.Then,foreachcubeinthegrid,theseverticesareconnectedaccordingtoapredenedcasetabletoformthetriangles.Onemajordrawbackofthismethodisthepoorreconstructionofsharpfeaturesinsidethegridcells.Kobbeltetal.[ 30 ]proposeanExtendedMarchingCubesalgorithm,alongwithanenhanceddistanceeldrepresentationtoimprovethereconstructionofsharpfeatures.Theenhancedrepresentationinvolvessamplingdirecteddistancevaluesinx,yandzdirectionsinsteadofjustascalardistancevalue.Thatis,ateachsamplepoint,thedistancestotheclosestsurfacepointineachofpositivex,yandzdirectionsarestored.Fortriangularmeshes,thesepointscanbefoundfromtheintersectionof 20

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thesurfacewiththecorrespondingedgeofthegrid.Duringreconstruction,thesurfacepointsobtainedfromthesevalueswillbemoreaccuratethanthoseobtainedfromlinearinterpolationoftheoriginalscalarvalues.TheExtendedMarchingCubesalgorithmtriestoidentifycubesthatholdsharpfeatures(cornersoredges).Forthis,atthepointsofintersectionofthesurfaceandedge,thegradientofthedistanceeldissampledandstored.Asthispointisonthesurface,fortriangularmeshes,thisgradientwillbethenormaltothetriangleatthatpoint.Duringreconstruction,theopeningangleoftheconeformedbythesegradientsisusedtodetectifthecubehasasharpfeature.Ifitdoesn't,thestandardMarchingCubestableisusedforthecube.Ifitdoes,thegradientsareusedtoconstructtangentstothesurfaceatthecorrespondingpointsandanadditionalvertexisinsertedattheintersectionofthesetangents.Atrianglefanisthenformedconnectingthisvertextoallotherverticesonthecubetotryandapproximatethesharpfeature.Apostprocessingstepofippingedgesisthenappliedtocorrectmeshconnectivityinthecaseofsharpedges.Juetal.[ 28 ]describeamethodforcontouringasignedgridthatimprovesupontheExtendedMarchingCubes(EMC)algorithm.Thisapproachusesanoctreeinsteadofauniform3Dgridandtheedgesoftheoctree'sleavesthathavesignchangesaretaggedusingexactintersectionandnormaldata.AQuadraticErrorFunction(QEF)isformedforeachleafcubeoftheoctreefromthenormaldata.Then,foreachcubethatexhibitsasignchange,avertexisplacedattheminimizeroftheQEF.Thisavoidshavingtoexplicitlyidentifycubesthathavesharpfeatures.TheQEFischosensuchthatthevertexthatminimizesitbestapproximatestheoriginalgeometry.Then,foreachedgeexhibitingsignchanges,theminimizervertexofthecubessharingtheedgeareconnectedtogenerateaquad.SimplicationstotheOctreearealsopresentedwhichavoidswastingofspacebycollapsingleafcubesthatarehomogeneous(i.e.havethesamesignforallvertices)andformingQEFsforinternalnodes. 21

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TheprevioustwoapproachesuseHermitedatatoconstructpolygonalmeshrepresentationsofsurfacesfromtheirdiscreteimplicitrepresentation.Hermitedatacanalsobeusedininterpolatingthesediscretevaluesmoreaccuratelytoaidinthevisualizationofthesurfaces.LeikenandMarsden[ 31 ]presentsuchaninterpolationschemeontheCClattice.Theirschemelocallyapproximatesthesampleddatausingtricubicsplinesthatareinterpolating.TheHermitedataassociatedwiththeverticesofaCClatticeisusedtoconstructalinearsystemofequationsfromwhichtheco-efcientsoftheinterpolatingpolynomialareobtained.Inreconstructingdistanceelds,theavailabilityofexactvaluesofpartialderivativesmakesapproachesusingHermitedataparticularlyattractive.Theinterpolationschemeweproposeinthisthesisisalsocapableofmakinguseofsampledderivativevalueswhereavailable. 2.3SamplingandReconstructiononBodyCenteredCubicLatticeMotivatedbythesamplingtheoreticadvantagesoftheBCClattice(Section 1.2 ),thisthesisexploresitsapplicationinsampling,andconsequentlyreconstructing,distanceelds.Inthissection,wediscusssomeofthepreviousworkcarriedoutinsamplingandreconstructingvolumetricdataontheBCClattice.Theuletal.[ 50 ]makeacaseforusingBCClatticeinvolumegraphicsbyshowingthatBCClatticescanachievethesameaccuracyasCClatticeswith29.3%fewersamples(or,inturn,samplesonaBCClatticeretainabout30%moreinformationthanthesamenumberofsamplesonaCClattice).TheyalsodemostrateimprovedrenderingratesonBCCusingasplattingtechniqueadaptedtoBCClattices.Entezarietal.[ 15 16 18 ]introduceasetofboxsplinesreconstructionschemeswhicharemoresuitedtothegeometryoftheBCClatticeandshowhowtoobtainitsoptimalapproximationorder[ 17 ]usingtheprincipleofquasi-interpolation[ 13 ].Mengetal.[ 36 ],conrmthattheseadvantagessignicantlyimprovethevisualqualityofthevisualizationpipelinebasedontheBCCsamplinglattice.Finkbeineretal.[ 19 ]haverecentlydevelopedaGPUimplementation 22

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ofafastalgorithmforconvolutionofBCCsampleddatawiththeabove-mentionedboxsplines.BasedontheideathattheBCClatticecanbeconsideredtobecomposedoftwooverlappingCClattices,CsebfalviproposesaGaussianreconstructiononBCCusingglobalpreltering[ 9 ]andaprelteredB-splinereconstructionschemeforquasi-interpolation[ 10 ].DecomposingBCClatticeintotwoCClatticesallowsforefcienthardwareimplementations[ 11 ].Butthisdisregardsthetopologicalstructureofthelattice[ 23 ]andhencetheneighborhoodofeachlatticepointisdistorted.Moreover,neitherthequasi-interpolationmethodsnortheboxsplineschemes(beyondthelinearC0case)areexactlyinterpolating.Ourinterpolationschemeaddressesboththeseshortcomings. 23

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CHAPTER3SAMPLINGDISTANCEFIELDSThischapterdiscusseshowthebruteforcemethodisusedtosampleaccuratedistanceeldsfromtriangularmeshesonCCandBCClattices.ItthendescribeshowtheparallelprocessingcapabilitiesoftheGPUcanbeutilizedtoacceleratetheimplementationofthealgorithm.Amethodtoobtainthetruegradientsofthedistanceeldateachlatticepointisalsodiscussed. 3.1BruteForceMethodThesimplestandmoststraightforwardwaytosampleaccuratedistanceeldsfromtriangularmeshesistoemploythebruteforcemethod.Beforewedescribethebruteforcemethodthough,itisimportanttoexplainthenotionofalatticepoint.Tosamplethedistanceeldofamesh,weoverlayalatticeovertheambientspaceinwhichthemeshlives.Thesamplesarethentakenatthepointsonthislattice,i.e.thelatticepoints.Constructingadiscretedistanceeldfromasurfaceinvolvescomputingtheminimumdistancefromeachlatticepointtothesurface.Foragivenlatticepoint,thiswouldbethedistancefromthelatticepointtothepointonthesurfacethatisclosesttoit.Astriangularmeshesarecomposedofanitenumberoftriangles,everypointonthesurfacefallsoneitherthefaceofatriangle,anedgeoravertex.Hence,everypointonthesurfacecanbeconsideredtobepartofoneormoretriangles.Then,thetaskofcomputingthedistancetotheclosestpointonthesurfacecanbesimpliedbyrstndingthetrianglethatholdstheclosestpointandthencomputingtheminimumdistancetoit.Asthenameimplies,thebruteforcetechniquedoesthisbycomputingtheminimumdistancefromeachlatticepointtoeverytriangleinthemeshandstoringtheminimumvaluescorrespondingtoeachlatticepoint.Tondtheminimumdistancefromalatticepointtoatriangle,weuseapoint-triangledistancealgorithmproposedbyEberlyin[ 14 ].ThisalgorithmisdescribedinSection 3.2 24

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Thoughsimpleandexact,thebruteforcetechniqueiscomputationallyveryexpensive.Itevaluatesthedistancebetweeneverypossiblelatticepoint-trianglepairing,leadingtoatimecomplexityofO(mn),wheremisthenumberoflatticepointsandnthenumberoftriangles.Asastraightforwardimplementationofthismethodcantakeaprohibitivelylongtime,weproposeanimplementationthatutilizestheparallelprocessingcapabilitiesofNVIDIA'smulti-coreGPUsusingtheCUDAdevelopmentplatform. 3.2Point-TriangleDistanceThissectiongivesabriefdescriptionofthePoint-TriangledistancemethodproposedbyEberly.TondtheminimumdistancebetweenapointPandatriangleT,TisrepresentedintheformT(s,t)=B+sE0+tE1for(s,t)2D=(s,t):s2[0,1],t2[0,1],s+t1whereBisoneoftheverticesofT,E0andE1arethevectorsfromBtotheothertwoverticesofTandsandtarescalars.Eachpairofvaluesforsandtsuchthat(s,t)2Ddescribesapointthatisonthetriangle,i.e.ontheface,edgesorverticesofthetriangle.When(s,t)=2D,thepointdescribedisonthesameplaneasthetriangle,butoutsideit.ThenourtaskistondthepointonTthatisclosesttoP.ThesquareddistancebetweenPandanypointonTisgivenbyQ(s,t)=jT(s,t))]TJ /F3 11.955 Tf 11.95 0 Td[(Pj2=as2+2bst+ct2+2ds+2et+ffor(s,t)2Dwherea=E0E0,b=E0E1,c=E1E1,d=E0(B)]TJ /F3 11.955 Tf 11.63 0 Td[(P),e=E1(B)]TJ /F3 11.955 Tf 11.63 0 Td[(P),andf=(B)]TJ /F3 11.955 Tf 11.95 0 Td[(P)(B)]TJ /F3 11.955 Tf 11.95 0 Td[(P).Then,thepointonTthatisclosesttoPisobtainedbyminimizingQoverD.Theminimumcanoccurinanyofthe7regionsasshowninFigure 3-1 .Iftheminimumoccursinanyofregions1)]TJ /F5 11.955 Tf 12.01 0 Td[(6,thecorrespondingpointontheboundaryofthetriangleiscomputed.Refer[ 14 ]foramoredetaileddescriptionofthealgorithmanditsimplementation. 25

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Figure3-1. Thest-planepartitionedinto7regions. 3.3GraphicsProcessingUnitImplementationThebruteforcemethoddescribedaboveperformsthesamesetofcomputationsoneverypoint-trianglepair.WeexploitthisinherentpotentialforparallelizationbyimplementingthesecomputationstoruninparallelonaCUDAcapableGPU.Forasmallnumberofpoint-trianglepairs,itispossibletoutilizetheavailableparallelismcompletelybyusingaseparatethreadforeachpair.However,withlargerdatasets,limitationsinmemoryandparallelprocessingpoweroftheGPUmakeitimpossibletoavoidsomeamountofserialization.Moreover,withmultiplethreadscomputingthedistancesfromthesamelatticepoints,identifyingandretainingtheshortestdistancewouldrequiremultiplethreadstocompareandwritetoacommonlocation.SinceatomicoperationsonoatsarenotavailableinCUDA(atthetimeofdesigningthisimplementation),implementingthisapproachbecomesquitecomplicatedandinefcient.Assigningatriangletoeachthreadandcomputingitsdistancetoeverylatticepointseriallywithinthethreadagaininvolvesmultiplethreadshandlingthesamelatticepoint,andsuffersfromthesamedrawbackmentionedabove.Hence,wefollowanapproachwhereeverythreadisresponsibleforaspeciclatticepointandcomputesthedistance 26

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betweeneverypairinvolvingthatpoint.Thesecomputationsareperformedinserialwithinthethread,makingiteasytolocallykeeptrackoftheshortestdistanceforthatpoint.Theonlydatasharedbetweenthethreadsisthetrianglecoordinates,whichareread-onlyasfarasthethreadsareconcerned.Withineachthread,aloopisusedtoiterateoverthetrianglesandcomputethedistancestothem.ThealgorithmdescribedinSection 3.2 isusedtocomputethedistances.Thisloop,alongwithafewmemoryoperations,formsourkernel(i.e.thefunctionthatisexecutedontheGPUbyeachthread).Ittakesthetriangledataandpointcoordinatesasinputandreturnstheshortestdistancefromthatpointtothemeshsurfaceastheoutput.AsthekernelcannotaccesstheCPUmemoryspace,thesevaluesmustbetransferredandstoredonGPUmemory.Figure 3-2 showsablockdiagramoftheGPUimplementationdiscussedhere. Figure3-2. BlockdiagramoftheGPUimplementation.EverylatticepointisassignedtoaCUDAthread.Thetrianglesareprocessedbythesethreadsinaloop,oneaftertheother. 27

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Thetriangledataisonlyreadfromwithinthekernelandeveryconcurrentthreadreadsthesametriangledataatanygivenpoint.ThismakestheconstantmemoryontheGPUagoodoptiontostoretriangledataon.Whilecomputingthedistancetoaparticulartriangle,thecoordinatesofthattrianglearefetchedfromconstantmemoryandstoredinsharedmemorysothatitisfastertoaccessthosevaluesduringcomputation.Theshortestdistancevalueforeachlatticepointisstoredinaregisterwithinitsthread.Tocomputethesignofthedistance,itisnecessarytokeeptrackofthetriangleandthepointonthetrianglecorrespondingtotheshortestdistance.Thesevaluesarestoredinregistersaswell.Onceallthetriangleshavebeenprocessedandtheshortestdistancefound,thisvaluealongwiththecorrespondingtriangledataistransferredtotheglobalmemorysothatitcanthenbetransferredtotheCPUmemoryspace.ThecoordinatesofthelatticepointcorrespondingtoeachthreadcanbecomputedwithinthethreadusingitsthreadIDandblockID.However,computationswithinthekernelusesregisters,andsometimes,itmightbenecessarytoreduceregisterusagetogetbetterefciency.Insuchcases,apartofthecomputationsforthelatticepointcoordinatescanbedoneontheCPUandtransferredtotheGPUviaconstantmemory.Wedothisbycomputingthebasex,yandzcoordinatesofeachthreadblockontheCPUandpassingthesevaluesasarraystotheGPU.Thesebasevaluesarethecoordinatesoftherstthreadinthecorrespondingthreadblock.Fortheotherthreadsintheblock,thelatticepointcoordinatesarecalculatedbyaddingthecoordinatesofthethreadIDtothesebasecoordinates.ThelimitedmemoryontheGPUlimitsthenumberoflatticepointsandtrianglesthatcanbeprocessedinasinglekernellaunch.Inthecaseoftriangles,thecapacityoftheconstantmemorylimitsthenumberoftrianglesthatcanbesenttotheGPUperlaunch.Ifthemeshhasmoretriangles,welaunchthekernelmultipletimes,untilalltriangleshavebeenprocessed.Sinceeachkernellaunchwillnditsownshortestdistance(from 28

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amongthesetoftrianglesitprocessed),weneedawaytondtheshortestdistanceamongthem.ThistaskissimpliedbythefactthatglobalmemoryontheGPUretainsdataacrosskernellaunches.Hence,atthetimeofaparticularlaunch,theglobalmemorywillbeholdingtheshortestdistanceforeachlatticepointfromthepreviouslaunch.Allthatneedstobedoneistofetchthesevaluesintoregistersatthestartofthekernelexecution,asthecurrentshortestdistance,sothatthedistancescomputedduringthatlauncharecomparedagainstthesevalues.ThenumberoflatticepointsthatcanbeprocessedperlaunchislimitedbytheglobalmemoryavailableontheGPU.ThisisbecauseeachlatticepointbeingprocessedduringalaunchstoresitsshortestdistanceandassociatedtriangledataintheglobalmemorybeforetransferringtotheCPU.Therefore,largelattices(orgrids)mustbesplitintosmallersectionsandprocessedovermultiplekernellaunches.Forsimplicity,weassumethatourlatticesareCartesiancubicandhaveresolutionsthatareamultipleof32alongeachaxis.Then,wesplitthelatticeintocubicalregionsofresolution32x32x32witheachregionbeingprocessedbyaseparatekernellaunch.Withsuchasplit,thecomputationoflatticepointcoordinatescanbegreatlysimpliedbychoosingthedimensionsofthethreadandblockIDsappropriately.Iftheresolutionisnotanexactmultipleof32,itcanbepaddedtothenexthighestmultiple.Theonlydrawbackisthatoneormorekernellauncheswillhaveafewthreadsthatarenotperformingnecessarywork.However,thiswillnotcauseasignicantperformancehit.Ifneitherthetrianglesnorthelatticepointstwithinasinglekernellaunch,thenthemultiplelaunchesarestructuredsothatallthetrianglesforaparticularregionofthegridisprocessedbeforemovingontothenextregion.ThisisdonesothatthedistancevaluesretainedbytheglobalmemoryontheGPUcanbeusedbythefollowingkernellaunches.Sincethelatticeissplitsuchthateachregionhas32,768latticepoints,therewillbeasmanythreadsperkernellaunch.Thesethreadsmustbegroupedintoblocks.Thenumberofthreadsablockcanhaveisrestrictedbythenumberofregistersusedby 29

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thethread.Atthesametime,ablockshouldideallyhaveatleast192threadstohidememorylatency.Consideringthesefactors,wegroupourthreadsinto32blocks,eachholding256threads.OnGPUsthatcanrunmorethan32,768threadsconcurrently,wecouldprocessmorethanoneregionofthelatticeperkernellaunchbyincreasingthenumberofthreadblocksperlaunch.Thismakessurethattheavailableparallelprocessingpoweriscompletelyutilized.WhiletheapproachdescribedsofarassumesaCClattice,itcanbeadaptedtosamplingonaBCClatticewithafewminormodications.TheeasiestwaytodothisisbysamplingonaCClatticethatistwicetheresolutionoftherequiredBCClatticeinxandydirections.Inotherwords,tosampleonaBCClatticeofresolutionRxxRyx2Rz(theresolutionalongzaxisisshownastwiceRztoaccountfortheadditionalpointsatthecenterofeachcubeonaBCClattice),wesampleonaCClatticeofresolution2Rxx2Ryx2Rz.ThiscanbeconsideredtobeaBCClatticeofresolutionRxxRyx2Rzwithadditionaldatapointsatthecenterofeveryfaceandedge.TheseadditionalvaluescanthenbethrownawaytoobtainthenecessarydistanceeldonaBCClattice.ThisistheapproachwehavefollowedandisillustratedinFigure 3-3 (A)withthepointstobediscardedshowninwhite.However,forlargeresolutions,thisapproachtakesalargenumberofunnecessarysamplesonlytobediscardedlater.Thiscanslowdowntheapplicationconsiderably.Toimproveperformance,wecansamplethedistanceeldontwoCClattices,ofresolutionsRxxRyxRzand(Rx)]TJ /F5 11.955 Tf 12.46 0 Td[(1)x(Ry)]TJ /F5 11.955 Tf 12.47 0 Td[(1)x(Rz)]TJ /F5 11.955 Tf 12.47 0 Td[(1),withthesecondgridbeingshiftedalongeachaxisbyhalfaunit.Inotherwords,foreachpoint(x,y,z)ontherstgrid,thecorrespondingpointonthesecondgridwillbelocatedat(x+h=2,y+h=2,z+h=2)wherehisthedistancebetweentwolatticepointsalonganyaxis.Itcanbeseeneasilythateachpointonthesecondgridfallsatthecenterofacubicalregionformedbyeightadjacentlatticepointsoftherstgrid.ThisisessentiallyaBCCgridofresolutionRxxRyx2Rz.Figure 3-3 (B)illustratesthismethod,withtheshiftedCClatticeshowninblue.Combiningthetwosetsofsamplesappropriatelyismore 30

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complicatedthansimplydiscardingasetofsamples.However,sinceweonlytakeasmanysamplesasnecessaryfortheBCClatticehere,theperformancewillbebetterforgridsofhigherresolution. A BFigure3-3. TwowaysofadaptingtheGPUimplementationforBCClattices.(A)SampleonCClatticeoftwicetheresolutionanddiscardtheadditional(white)points.(B)SampleontwoCClattices,thesecond(blue)shiftedalongeachaxisbyhalfacelllength. Thedistancevaluesobtainedfromthekernelareinfactsquareddistancevalues.Theseareconvertedtotheactualdistancevaluesbytakingthesquareroot.Then,thevaluescorrespondingtopointsinsidethetrianglemesharegivenanegativesign.Toidentifypointsthatareinsidethemesh,weuseamethodproposedbyBrentzenandAansin[ 3 ]thatusesangleweightedpseudo-normals(originallyproposedbyThurmerandWuthrich[ 51 ]andindependentlybySequin[ 43 ]).Finally,thesevaluesarescaledtotherange[0)]TJ /F5 11.955 Tf 12.16 0 Td[(255]suchthatavalueof127representspointsthatareonthesurfaceofthemesh.Valuesintherange[0)]TJ /F5 11.955 Tf 12.45 0 Td[(127)representpointsoutsidethemeshwithlower 31

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valuesindicatinglongerdistancesandvaluesintherange(127)]TJ /F5 11.955 Tf 12.29 0 Td[(255]indicatespointsinsidethemeshwiththevalueincreasingwithdistance. 3.4GeneratingTrueGradientsOnecharacteristicofdistanceeldsisthattheirtruegradientsatanypointcanbecomputedpreciselywithrelativelyfewextracomputations.Asthisgradientdataprovesusefulinreconstructingdistanceelds,wedescribehowthesevaluescanbecomputedaspartofourimplementationofthebruteforcemethod.Aspreviouslymentioned,thevalueofadistanceeldatanypointistheshortestdistancefromthatpointtothesurfaceofthetriangularmeshunderconsideration.Then,thetruegradientofthedistanceeldatthatpointisthevectorfromthepointtothepointonthemeshclosesttoit.Theaforementionedshortestdistanceisessentiallytheabsolutelengthofthisvector.Therstorderpartialderivativesofthedistanceeldalongeachaxisisthecomponentofthegradientvectoralongthataxis.Tocomputethegradientvalueforaparticularlatticepoint,weneedthepointonthemeshclosesttoit.Ourkernel,whichcomputestheshortestdistancecorrespondingtoalatticepoint,identiesthisclosestmeshpointduringthecourseofitscomputations.Recallthatalongwiththeshortestdistance,ourkernelalsotracksandreturnsassociatedtriangledataforeachlatticepoint.Thistriangledataincludesthetrianglethatholdstheclosestmeshpointandthesandtvalues(asdescribedinSection 3.2 )forthatpoint.Usingthisdata,theclosestmeshpointcanbeidentiedandthegradientvectorcomputed.Thisisthensplitintoitsx,yandzcomponentsandappropriatelyscaledtogivethetruevaluesoftherstorderpartialderivatives. 3.5PerformanceToanalyzethespeed-upachievedbyourimplementation,wecodedupapureCPUimplementationofthebruteforcealgorithm,andgenerateddistanceeldsatdifferentresolutionsfromthesoccerball,StanfordbunnyandStanforddragonmeshesusingbothimplementations.TheGPUimplementationwasexecutedonanNVIDIAGeForce 32

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GTX465whichhas352CUDAcores.Table 3-1 showsthedetailsofthesemeshes.TheresultingexecutiontimesaregiveninTable 3-2 .Forthedatasetstested,theGPUimplementationachievesspeedupsrangingfrom145to375.ItcanalsobeseenthattheperformancegainoftheGPUimplementationimprovesasthemeshsizeandresolutionincreases. Table3-1. Triangularmeshesusedforperformancetesting MeshNo.oftrianglesNo.ofvertices Soccerball3,5161,760Stanfordbunny69,66634,835Stanforddragon100,00050,000 Table3-2. Comparisonofexecutiontimes.ThetimetakenbytheGPUimplementationisshowninthecolumn'GPU'andthatbytheCPUimplementationisshowninthecolumn'CPU'. DatasetSamplePointsGPU(sec)CPU(sec)Speed-up Soccerball262k0.4412563.9975145Soccerball2,097k2.6365511.02194Stanfordbunny32k0.90175181.3325201Stanfordbunny262k5.762751451.54252Stanforddragon32k1.2325383.6311Stanforddragon262k8.164253063.5725375 33

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CHAPTER4CUBICINTERPOLATIONINBODYCENTEREDCUBICLATTICEUSINGHERMITEDATAInthischapter,weintroducealocalcubicinterpolationschemeontheBCClatticethatusesHermitedata,i.e.functionandderivativevalues.Thisschemeconstructscubicpolynomialinterpolantslocallyintetrahedralregionsofthelatticeusingdataandderivativevaluesintheneighborhoodoftheregion.WethendiscussatricubicinterpolationschemeusingHermitedataontheCClatticeproposedbyLeikenandMarsdenin[ 31 ]. 4.1CubicInterpolationinBodyCenteredCubicLattice 4.1.1InterpolatingSplinesConsiderafunctionfsampledonthelatticepointsofaBCClattice.WedescribeamethodtointerpolatethisfunctionusingpiecewisepolynomialsovertheBCClattice.Atthispoint,weassumethattherstandsecondorderderivativesoffatthelatticepointsarealsoavailabletous.Inthenextsection,wediscussanite-differencingschemetoestimatethederivativeswhentheyarenotavailable.AsdescribedpreviouslySection 1.2 ,theBCClatticecanbeuniformlypartitionedintocongruenttetrahedronswiththelatticepointsactingastheircorners.Withineachtetrahedron,theinterpolatingsplineisdenedbyapolynomialofdegreen,p2n.Inthetrivariatesetting,whichiswhatweareinterestedin,thiscanberepresentedasn(R3):=fp(x)=Pi+j+kni,j,k0aijkxiyjzkg.SuchapolynomialhasCn+3ncoefcientsandhencecanbeuniquelydeterminedby(n+3)(n+2)(n+1)=6constraints.Apolynomialofdegree1canbedetermineduniquelyby4constraintsandhencecanbeconstructedbyrestrictingthevalueittakesatthe4cornersofthetetrahedrontothecorrespondingvaluesoff:f(vi)=p(vi),whenvi2,fori=1...4,p21, (4)whereviindicatesverticesofatetrahedron. 34

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Thisgivesusalinearsystemofequationsin4variables,solvingforwhichyieldsthe4coefcientsofthepolynomial.ThisisessentiallylinearinterpolationwithineachtetrahedronandthesplineformedbythesepolynomialsisapiecewiselinearinterpolanttothedatagivenatlatticepointsApolynomialofdegree2requires10constraints.Asthedatavaluesatthecornerscanprovideonly4constraints,weusetherstorderpartialderivativesalongeachaxisatthecornerstoformtheremainingconstraints.However,itisnotpossibletochoose6constraintsfrom4verticesinanunbiasedmanner(bias,here,referstoanasymmetricchoiceofconstraintspervertex).Sincewerequireanisotropicchoiceofconstraints,weusedegree3polynomials,whichcanbeuniquelydenedusing20constraints.Thisgivesus5constraintspervertex.Thedatavalue( 4 )andthethreerstderivatives( 4 )ateachcornerform16constraints.fx(vi)=px(vi),fy(vi)=py(vi),fz(vi)=pz(vi), (4)wherefx,fyandfzdenotethepartialderivativeswithrespecttox,yandzrespectively.Fortheremaining4constraints,weusesecondorderpartialderivatives.Whileasinglesecondorderpartialderivative,like@2f @x2,canbechosenfromeachcornerofthetetrahedron,suchachoicewouldbebiasedalongspecicaxes.Thiscanbeavoidedbychoosingaconstraintbasedonasymmetricsumofindividualsecondderivativesateachcorner.Onesuchsumis(@2f @x2+@2f @y2+@2f @z2).However,aninterpolatingconstraintbasedonthissumisfoundtobelinearlydependentontheother16constraints.Theotherchoiceisaconstraintbasedon(@2f @x@y+@2f @y@z+@2f @x@z)whichislinearlyindependentandcanbeusedtodeterminethepolynomial.However,thiscombinationdoesnotrestricttheindividualsecondorderpartialderivativevaluestakenbythepolynomial.Astheinterpolationconstraintisenforcedonlyonthesum,thevaluestheindividualsecondderivativestakeateachcornercoulddisagreewiththecorrespondingvaluesoff.Thepolynomialsogeneratedcouldturn 35

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outtobeapoorapproximationoff.Inourexperiments,theindividualsecondderivativevalueshadlargedeviationsfromcorrespondingvaluesoff,leadingtosevereartifactsinrenderedimages.Togetabetterapproximationoffandavoidsuchartifacts,eachindividualsecondorderpartialderivativemustbeconstrainedseparately.Enforcinginterpolatingconstraintsoneverysecondderivativevalueatthecornersgivesus3constraintspercorner,makingitanover-determinedsystem.Hence,werelaxtheinterpolationconstraintonthesevaluesandopttominimizetheL2normoftheirerroroverthesetofallverticesofthetetrahedroninstead.Tothisend,wedeneanerrorfunctionoverthespaceofallcubicpolynomialsp23asfollows. E(p):=4Xi=1k@2f @x@y(vi))]TJ /F6 11.955 Tf 17.57 8.08 Td[(@2p @x@y(vi)k2+k@2f @x@z(vi))]TJ /F6 11.955 Tf 17.42 8.08 Td[(@2p @x@z(vi)k2+k@2f @y@z(vi))]TJ /F6 11.955 Tf 20.71 8.08 Td[(@p @y@z(vi)k2.(4)Here,eachtermisthesquarederrorinaspecicsecondorderpartialderivativeataspeciccorner.Also,thissetofpartialderivativesisinvariantalongx,yandzdirections.Thus,thereisnobiasalongaparticulardirection.Minimizingthiserrorfunctionwithrespecttothecoefcientsofpaccountsfortheremaining4degreesoffreedomindeterminingthepolynomial.Thisisaconstrainedminimizationproblem,wheretheerrorfunctionactsastheobjectiveandthe16constraintsareformedbytheinterpolatingconstraintsdenedin( 4 )and( 4 ).Thisproblemissolvedforthe20coefcientsfromwhichthepolynomialisconstructed.Theerrorfunctiondenedaboveisquadraticintermsofthepolynomialcoefcientsinp,thatcanberepresentedbya201vectora=[a1,...,a20]T.Theconstrainedminimizationofthiserrorfunctioniscarriedoutwithrespecttothecoefcientsofthecubicinterpolantp,withconstraintsdenedin( 4 )and( 4 ).Sincetheseinterpolationconstraintsarelinearintermsofpolynomialcoefcients,wecanmodelouroptimization 36

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problemasaspeciccaseofquadraticprogrammingproblem,knownasEqualityQP[ 4 ]:minimizea2R20E(a)=aTGa+hTa+b (4)subjecttoMa=f. (4)Weintroduceavectornotationforrepresentingourpolynomialsthatallowsustotransformthefunction( 4 )tothequadraticformin( 4 )andtheinterpolationconstraints( 4 )and( 4 )intotheconstraintsin( 4 ).Thepolynomialpcanberepresentedasaninnerproduct:p(x)=hm,ai(x),inwhichthecolumnvector,a,encodesthecoefcientsofourpolynomial.misacolumnvectorinwhicheachelementisoneofthemonomials(invariablex)oftheformxyzwith,,>0and++<4thatspan3.Theinnerproductresultsinatypicalpower-formrepresentationofcubicpolynomialwhichcanbeevaluatedatpointx=vi(i.e.,oneofthecornersoftetrahedron).Asasimpleexample,onecanwriteagenericquadraticpolynomialevaluatedatsamplepoint2as:(a1+2a2+4a3)=h[1,x,x2]T,[a1,a2,a3]Ti(2).Interpolationconstraintsintroducedin( 4 )and( 4 )cannowbewrittenintermsofthecoefcientvectora: f(vi)=hm,ai(vi)fx(vi)=hmx,ai(vi)fy(vi)=hmy,ai(vi)fz(vi)=hmz,ai(vi),(4)thataredenedforeachvertexoftheBCCtetrahedronvi2,i=1...4.These16equationsformalinearsystemofconstraintsin( 4 ),wherethe1620matrixMisformedbythemonomialsinmandtheirpartialderivativesevaluatedattheverticesofthetetrahedron,vi2.Inotherwords,eachrowofMcorrespondstomoranyofitsthreepartialderivativesevaluatedatavertexvi.Asmentionedbefore,columnvectora 37

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representstheunknowncoefcientsofpandnally,fisholdingthesamplevaluesoftheunderlyingfunctionanditspartialderivatives.Moreover,bysimplelinearalgebraonecanreformulatetheerrorfunctiondenedin( 4 )intheformof( 4 )usinguvTtodenotetheouterproductoftwocolumnvectoruandv: G=4Xi=1)]TJ /F16 11.955 Tf 5.48 -9.68 Td[(mxymTxy+mxzmTxz+myzmTyz(vi)h=)]TJ /F5 11.955 Tf 9.3 0 Td[(24Xi=1(fxymxy+fxzmxz+fyzmyz)(vi)b=4Xi=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(f2xy+f2xz+f2yz(vi).(4)Solvingtheunder-determinedlinearsystemofequations,( 4 ),onecanndaparticularsolution(e.g.,normalequationsonM),thatwecalla0.Anysolutiontothesystem( 4 )canbewrittenasasumoftheparticularsolution,a0,andanarbitraryelementinthenullspaceofM:a=a0+Zt,wherecolumnsofZformabasisforthenullspaceofMandt2R4sinceMisfullrank(i.e.,16)fortheBCCtetrahedron.Thebasisforthenullspacecanbecomputedbythereduced-rowechelonformorviasingularvaluedecomposition.SubstitutingthisrelationinE(a)allowsustore-writetheminimizationprocessasafunctionoft,E(t).TheminimizercanthenbeexplicitlyderivedbysolvingthelinearsystemofequationsthatisobtainedfromdifferentiatingE(t):Et(t)=0. (4)Theuniqueminimizertotheerrorfunctionalis,then,obtainedby:)]TJ /F16 11.955 Tf 5.48 -9.68 Td[(ZTGZt=)]TJ /F16 11.955 Tf 9.3 0 Td[(ZT1 2h+Ga0. (4) 38

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ThislinearsystemhasauniquesolutionsinceZTGZispositivedenite(asGissymmetricpositivedenite).SincetheBCClatticecanbetetrahedralizedtocongruenttetrahedra,asingletparameterfortheoptimalcubicinterpolantcanbepre-computedforthegeometryofintheBCClattice.Hence,wecanpre-computethecoefcientsofthecubicinterpolantintermsofthesamplesoftheunderlyingfunctionanditsrst-orderpartialderivativeasspeciedin( 4 ).Inotherwords,thecomputationoftheoptimalcubicinterpolantcanbeimplementedasafastlterbyconsideringthetetrahedronintheBCClatticethatcontainstheinterpolationpoint.Insummary,20degreesoffreedomforacubicinterpolantforeachtetrahedronaresatisedby16(exact)interpolationconstraintsforthefunctionvaluesandtherst-orderpartialderivativevaluesatthelatticesites(i.e.,verticesofcontainingtetrahedron).Theadditional4degreesoffreedomarechosenoptimallytominimizeinterpolationerrorsonthesecond-orderpartialderivativesonthelatticepoints.ThequadraticprogrammingproblemhasauniqueminimizerthatweusetoconstructtheoptimalcubicinterpolantforthegeometryoftheBCClattice.Wetestedthisapproachonthecarpdataset.Theoriginaldatasethasaresolutionof256256256(Figure 4-1 (A))whichrepresentsthegroundtruthandthelow-resolution,sub-sampled,datasetshaveabout16%ofthehighresolutiondataontheBCCandtheCClattices.ThesubsampledCCvolumehasaresolutionof140140140(Figure 4-1 (B))andthesubsampledBCCvolumehasaresolutionof111111222(Figure 4-1 (C)).TheBCCdatasetwasrenderedwithourcubicsplinesandtheCCdatasetwasrenderedwiththetricubicCatmull-Romsplines.TheribareahasbeenmostlydistortedintheCCimagebutarebetterpreservedintheBCCimage. 4.1.2SmoothnessAndApproximationOrderThesplines,constructedasdescribedintheprevioussection,isC1smoothacrossthefacesofthetetrahedra.First,wewillnotethatthevaluesofsanditsrstderivatives 39

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A B CFigure4-1. TheCarpshdataset.Theground-truthCarpshdataset,(A),with16,777kpointsissub-sampledto16%ontheCartesian(B)andBCC(C)latticesforcomparison.TheCartesiandataisinterpolatedwithCatmull-RomandtheBCClatticeisinterpolatedwithourcubicspline.ThetailnsandribareashavepreservedtheirconnectivityintheBCCdatasetwhilehavebeendistortedintheCatmull-Romcase. onafaceofatetrahedrondependsonlyontheverticesonthatface.ThisbecomesevidentwhenthepolynomialwithinthetetrahedronisrepresentedinBernstein-Bezierbasisform,asthebarycentriccoordinatesofthecorneroppositethefaceis0ontheface.Now,considerafacesharedbytwotetrahedra,T1andT2.We'lldenotethepolynomialpiecesofswithinthemasP1andP2respectively.Atthethreeverticesofthesharedface,P1andP2havethesamevaluesandrstderivatives,equaltothecorrespondingvaluesoff,theunderlyingfunction.Thus,thevaluestakenbyP1andP2onthesharedfacearethesame,makingsC1smoothacrosstheface.Theorderofapproximationcanbethoughtofasreferringtotheaccuracyofanapproximation,oralternatively,theorderofmagnitudeoftheerrorinapproximation.Whenaninterpolatingpolynomialpisusedtorepresentafunctionf,anorderofapproximationofnmeansthattheapproximationerrorcanberepresentedintermsofthesamplingdistancehasO(hn).TheclassicalStrang-FixconditionrelatestheapproximationorderofaSplinespaceSnwithitsabilitytopreciselyreproducepolynomialsofdegreeupto,and 40

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including,()]TJ /F5 11.955 Tf 13.37 0 Td[(1).Inthemultivariatesetting,thenotionoflocalreproductionofpolynomialsisneededforprovingapproximationorder.Wenowshowthatourcubicinterpolantcanexactlyreproduceallpolynomialsuptodegree3withinthecorrespondingtetrahedron.Rememberthatthecoefcientsofthepolynomialarerecoveredbysolvingaconstrainedminimizationproblem,wherethe16constraintsareformedbyenforcinginterpolatingconstraintsonthevaluesthepolynomialanditsrstderivativestakeatthe4verticesofthetetrahedron.Theoriginalpolynomialf23(R3)canberecoveredbyrecoveringits20coefcientsfromthisproblem.Itiseasytoseethatthecoefcientsoffsatisfyall16aforementionedconstraints.Now,considertheerrorfunctionin( 4 ).Thisisasummationofsquaredtermsandhencecannottakeanegativevalue.Usingthecoefcientsoffinthisfunctionwillmakeeachindividualterm0,thusreturningtheminimumvalueofthefunction.Thus,p=fisavalidsolutionoftheconstrainedminimizationproblem.Sincethefunctiontobeminimizedisquadratic,itssolutionhastobeunique,provingthatthesolutionobtainedwillbep=f.Sinceourinterpolatingsplinecanlocallyreproduceallpolynomialsuptodegree3,ithasanorderofapproximationof=4.Thisorderofapproximationisonlypossiblewhenexactpartialderivativesoffareavailable.Oneareaofapplicationwheretheexactrstderivativesareavailableisthesamplingandreconstructionofsigneddistanceelds,whichisdiscussedinthenextchapter.Forscalarelddata,whereonlythefunctionvaluesareavailable,weneedanite-differencingschemethatmeetstheapproximationorderofourconstruction.Thisisdiscussedinthenextsection. 4.1.3IsotropicFinite-DifferencesontheLatticeWhenthepartialderivativesofthefunctionfisknown(e.g.,Hermitedata),thesplineconstructioninterpolatesfanditsrstorderpartialderivativesexactly.However,forthescalar-elddatawhereonlyfunctionvaluesareknown,weneedtoemploynite-differencestoapproximatepartialderivativesoff(usedin( 4 )).Thisapproach 41

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intheunivariatesettingleadsdirectlytotheCatmull-RomsplineswhichareessentiallyHermiteinterpolationwhenderivativesareapproximatedwithnite-differences.Furthermore,inordertomaintaintheapproximationorder,thenite-differencingschememustbeexactonthepolynomialspaceofinterest.Inotherwords,weneedtodesignfourth-ordernitedifferencesontheBCClatticethatprovideexactpartialderivativeswheneverf23(R3).Intheunivariatesetting,derivativeestimationiseasilyderivedusingTaylorseriesexpansion.Theideabehindcentraldifferencingistousetheexpansiontoevaluatefatvarioussmalldistances,h,fromxtogetagoodestimateoff0atx.f(x+h))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)]TJ /F3 11.955 Tf 11.95 0 Td[(h)=2hf0(x)+h3 3f000(x)+O(h4) (4)TheTaylorseriesanalysisshowsthatthecentraldifferencingestimateoftheunivariatederivativeisasecondorderapproximant.Higherordersofapproximationtothederivativef0areobtainedbyemployingatechnique,calledRichardson'sextrapolation[ 7 ],thatscalesh:f(x+2h))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)]TJ /F5 11.955 Tf 11.95 0 Td[(2h)=4hf0(x)+8h3 3f000(x)+O(h4). (4)Onecaneliminatef000(x)termamong( 4 )and( 4 )andobtainahigher-orderapproximant.Therefore,thewell-knownve-pointstencilforapproximatingthederivativeisoforderfour:f0(x)=8f(x+h))]TJ /F5 11.955 Tf 11.95 0 Td[(8f(x)]TJ /F3 11.955 Tf 11.96 0 Td[(h))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x+2h)+f(x)]TJ /F5 11.955 Tf 11.95 0 Td[(2h) 12h+O(h4).Thisapproachcanberepeatedtoobtainseven-pointandnine-pointstencilsthatconstitutelterswithincreasingapproximationordersforderivativeestimation.Themainobservationhereisthatdesigninghigh-ordernitedifferencinginvolvesaTaylorseriesexpansionofthefunctionlocally.Thepolynomialisformedbytherstfewtermsinthe 42

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Taylorexpansion.Thenthepolynomial'sderivativeattheexpansionpointapproximatesthederivativeoftheoriginalfunctionwithanorderofaccuracywhichisonegreaterthanthedegreeofthepolynomial.Thispolynomialcanbeconstructedbyapolynomialinterpolationusingtheneighboringsamplepoints(i.e.,f(xh),f(x2h),...).Thelargertheneighborhoodis,thelargerthedegreeofthepolynomialiswhichdeterminestheorderofaccuracyforderivativeestimation.Theactualderivativeofthispolynomialinterpolant,then,constitutesthenite-differenceapproximationtothetruederivative.ExtendingRichardson'sextrapolationtothemultivariatesettinginvolvesthemultivariateTaylorseriesexpansion.Ontheotherhand,wecanemployRichardson'sextrapolationmethodontheBCC,oranyotherlattice,leveragingtheequivalenceofRichardson'sextrapolationwithpolynomialinterpolationonalocalneighborhood.Theideabehindourapproachistoemployapolynomialinterpolationschemethatbuildsaninterpolantonanisotropicneighborhoodofalatticepoint.ThisinterpolantagreeswiththetermsinTaylorseriesexpansionuptoitsdegree.Iftheunderlyingfunctionfisapolynomialitself(ofthesamedegreeastheinterpolant),thentheuniquepolynomialinterpolantagreeswiththeunderlyingfunctionandthederivativeestimationwillbeexact.Hence,iff23(R3),thenalocalcubicpolynomialinterpolationatalatticepointwillprovidetheexactderivative.Inthisapproximationscheme,thepartialderivativesareestimatedinanon-separablefashionandonecanchooseanisotropiccombinationoftheneighborsofalatticepoint.ForaBCClatticepoint,thereare8neighborpointsatoffsetsof(1,1,1),and6neighborpointsatoffsetsof(2,0,0),(0,2,0)and(0,0,2).Thenextringofneighborsarelocatedatoffsetsof(2,2,0),(2,0,2)and(0,2,2)thattogetherwiththeoriginallatticepointforma27-pointneighborhoodforit(seeFigure 4-2 ).Consideringthataninterpolatingpolynomialin3(R3)needs20datapoints,the27-pointneighborhoodover-determinesthepolynomialinterpolationproblem.Theover-determinedsystemofequationscanbesolvedusingaleast-squaresmethod(i.e., 43

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Normalequations)whichisdetailedbelow.Whentheoriginalfunctionf23(R3),theleast-squaressolutioncoincideswithfandhence,thederivativeestimationbecomesexact.Letx1,x2,...,x27denotethe27-pointneighborsofaBCClatticepoint.Wetacubicpolynomialinterpolantp(x)=hm,ai(x)onthis27-pointneighborhood.Heremis,again,avectorthatcontainsmonomialsuptocubicsandadenotesthecoefcientsofp.Thenwecansetupaninterpolationproblemtodeterminethelocalpolynomialttothefunctionfbysolvingtheminimizationproblemas:mina27Xi=1wi(fi)]TJ /F3 11.955 Tf 11.96 0 Td[(p(xi))2. (4)Thescalarvalue,fi,here,isthesamplevalueofthefunctionfatthelatticepointxi,andwiisaweightthatallowsustocontroltheinterpolationerroratlatticepointxi.Letdenotetheinterpolationmatrixwhichisofdimension2720.Thentheweightedleast-squaressolutiontothelinearsystemisgivenby:(TW)a=TWy (4)whereWisadiagonalmatrixwithWi,i=wi,yi=fiandi,j=mj(xi).WecansetWi,i=1fornormalleast-squaressolutionorotherchoicesforweightedleast-squaressolution.Sincemonomialtermsmj(x)areknown,andthelocalcoordinatesofthe27-pointneighborhoodofaBCClatticepointarexed,wecansolveforthecoefcientsafrom( 4 ).Whenwewanttoestimatethederivativesofthefunctionatanygivenlatticepoint,weperformalocaltandusethederivativesofthatlocalpolynomialatthatlatticepoint(whichisx=0inthelocalcoordinatesystem)astheapproximatedderivatives.Forexample,theestimatedrstorderderivativealongxatagivenlatticepointxisthenapproximatedbypx(0). 44

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ABFigure4-2. Weightsofthenite-differencingkernelonthe27-pointneighborhoodoftheBCClatticefor(A)fxand(B)fyz.Theillustratedcoefcientsaredividedby24in(A)anddividedby72in(B).The27-pointneighborhoodincludesthered,blue,greenandgraylatticepointsandexcludestheyellowpointsatthecorners. Whenweconsiderthederivativewithrespecttox,px(x)=hmx,ai(x),wecanconstructthenite-differencingweightsbyevaluatingpx(0):px(0)=bT(TW))]TJ /F4 7.97 Tf 6.59 0 Td[(1TWy=Kxy. (4)Inthisnotation,yisavectorthatcontainsthefunctionvaluesfromtheneighboring27pointsandb=mx(0).ThenitedifferencingkernelKxisamatrixwhoseconvolutionwithygivesthepartialderivativeoftheunderlyingfwithrespecttox.Thenitedifferencingweightsfortheotherpartialderivativescanbeobtainedsimilarly.ThespatialdistributionofnitedifferencingweightsasthekernelvaluesforfxisshowninFigure 4-2 (A)andthatofthekernelvaluesforfyzisshowninFigure 4-2 (B).Similarly,bychangingtheorderoftheaxes,wecangetthekernelforotherrstandsecondorderderivativesasneeded. 45

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Table4-1. TheL2normerrorinreconstructionofdatasetssampledataresolutionof111111222ontheBCClatticeusingtheproposedcubicinterpolation.Theweightedleast-squaresapproachisdesignedtouseazeromeanGaussianwith2=.5toestimatepartialderivativevalueswhichshowmarginalimprovementsintermsofreconstructionerror. DatasetLSQEWeightedLSQE ML8.9398.725Carp0.740.731Bonsai5.7415.721Lobster8.5388.467 Intuitively,thelatticepointsthatareclosertothecenteraremoreimportant(withrespecttotheresidualsintheinterpolationconditionsin( 4 ))thanthelatticepointsfurtherfromthecenter.Therefore,onecanassignhigherweightstotheerrortermscorrespondingtothelatticepointsclosertothecenter.Thechoiceoftheweightingfunctionisveryexible.AnisotropicchoicefortheweightingfunctionistheGaussianfunction: w(xi)=expkxik2 2,(4)wherekxikisthedistanceoftheneighborpointxifromthecenterpoint,0,and2isthevariancewhichdeneshowfasttheweightingfunctiondecays.When2isverylarge,theweightingfunctiondegeneratesintoaconstantfunctionwhichgivestheunweightedsolution.Smaller2valuemeanshigherweightsareassignedtotheresidualsclosertothecenter.2=.5showedtheminimumerrorinestimatingrstandsecondorderpartialderivatives,butimprovementintheperformanceofinterpolationusingGaussianweightedleast-squareswasmarginalinourexperiments.Table 4-1 summarizestheimprovementsobtainedbytheweightedleast-squaresapproach. 4.2TricubicInterpolationinCartesianCubicLatticeThissectionprovidesasummaryofthelocaltricubicinterpolationschemeproposedbyLeikenandMarsdenintheirpaper[ 31 ].ThisschemeusesHermitedatatoachievefullC1interpolationofagivenfunctionsampledonaCClattice.Chapter 5 discussesthe 46

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applicationofthismethodininterpolatingdistanceeldssampledonCClattices.Pleasenotethatthissectiononlyservestobrieydescribewhathasalreadybeenproposedin[ 31 ]andthecontributionofthisthesisisinapplyingitintheareaofdistanceelds.Consideratrivariatefunctionfsampledattheverticesofaregulargrid.Theinterpolantisapiecewisepolynomialwhichcanberepresentedbythegeneralformp(x,y,z)=PNi,j,k=0aijkxiyjzkwithineachcubiccellofthegrid.Astheinterpolantistri-cubic,Ntakesthevalue3andthepolynomialhas64coefcientsgivenbyaijk.ThesecoefcientsmustbedeterminedinawaythatachievesC1continuityacrossallfacesofthecube.Tothatend,interpolationconstraintsareenforcedonthevaluestakenbyPanditsthreerstderivativesatthe8cornersofthecube,giving32constraints.Torecover64coefcients,anadditional32constraintsarerequired.Theseconstraintsarechosensuchthattheyareisotropic,i.e.invariantunderrotationoftheaxes,andinamannerthatfavorssmoothnessoveraccuracy.Smoothnessisimprovedbyusinginterpolatingconstraintsonhigherorderderivativesofp.Thus,weneed4higherorderderivativesfromeachcornerfortheadditionalconstraints.Thereareonlytwosuchsetsthatareisotropic.Ofthese,theset(@2f @x2,@2f @y2,@2f @z2,@3f @x@y@z)islinearlydependentontherst32constraintsandhencecannotbeused,leavinguswiththeset(@2f @x@y,@2f @y@z,@2f @x@z,@3f @x@y@z).Thus,64constraintsareformulatedbyrestrictingthevaluestakenbythefunctionsinthefollowingsetateachcornerofthecubetothecorrespondingvaluesoff.(p,@p @x,@p @y,@p @z,@2p @x@y,@2p @y@z,@2p @x@z,@3p @x@y@z)Thisgivesalinearsystemof64equationswith64unknowncoefcients.ThiscanberepresentedinmatrixformasBx=bwhere'x'isavectorofthe64coefcientsand'b'isavectorofthevaluestakenbyfanditsderivativesateachofthe8cornersofthecube.Asthe64x64matrix'B'hasadeterminantof1,itsinversecanbecomputedandthelinearsystemcanbesolvedasx=B)]TJ /F4 7.97 Tf 6.58 0 Td[(1b.Thisgivesthevaluesforthecoefcientsfromwhichtheinterpolantcanbeconstructed. 47

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Foradetaileddescriptionofthemethod,themotivationbehinditandvariousproofs,wereferyouto[ 31 ]. 48

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CHAPTER5INTERPOLATIONOFDISTANCEFIELDSANDEXPERIMENTSInChapter 4 ,weproposedapiecewisecubicinterpolationmethodontheBCClatticeanddescribeditsconstruction,smoothnessandapproximationorder.WealsosummarizedalocaltricubicinterpolationproposedbyLeikenandMarsdenin[ 31 ].BoththesemethodsuseHermitedataassociatedwiththefunctionbeinginterpolated,i.e.thevaluesofthefunctionanditspartialderivativesattherespectivelatticepoints.InSection 4.1.3 ,wedescribedanite-differencingtechniquetoestimatethederivativesofthefunctionfromthefunctionvaluessampledatthelatticepoints.However,whenthetruevaluesofthefunctionderivativesareavailable,thesecanbeincorporateddirectlyintobothinterpolationschemestoprovideamoreaccuratereconstructionoftheunderlyingsurface.Ithasalreadybeenmentionedthatthetruegradientsofdistanceeldscanbecomputedwithrelativeease.Whilesamplingadistanceeld,itispossibletoalsocomputethetruerstderivativesoftheeldatthelatticepoints.ThiscanbedoneonbothCCandBCClatticesusingthetechniqueexplainedinSection 3.4 andrequiresalmostnoadditionalcomputation.Theaforementionedinterpolationschemescanthenbeemployed,alongwiththetruederivativevalues,onthesedistanceeldstoreconstructorvisualizetheoriginalfunction.Inthischapter,westudytheeffectsofapplyingtheproposedinterpolationschemeondistanceeldssampledonBCClattices.Thesedistanceeldsareconstructedatdifferentresolutionsfromtriangularmeshesofvarious3Dmodels(Figure 5-1 ),usingtheGPUacceleratedsamplingmethoddescribedinChapter 3 .Weusearaycastertovisualizetheresultsoftheinterpolation.WearealsointerestedinassessinghowwelltheBCCsamplinglatticedoescomparedtotheCClatticeinthecontextofdistanceelds.Hence,wesamplethesametriangularmeshesonCClatticesandusethetricubicschemetorenderthem.TheresolutionoftheCCandBCClatticesarechosen 49

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A B C D EFigure5-1. Thetriangularmeshesusedinourexperiments.Thesoccerball(A)has3,516triangles,thebunny(D)has69,666triangles,thebuddha(B)anddragon(C)have100,000triangleseachandthepawn(E)has304triangles. 50

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suchthatthetotalnumberofsamplepointsareapproximatelythesameinbothcases.Boththeinterpolationschemesareemployedtwiceoneachdataset,onceusingtruerstorderderivativesandonceusingestimatedvalues.Sincetruevaluesofthehigherorderderivativesarenotavailable,theyarealwaysestimated.OnBCC,thisisdoneusingthemethoddescribedinSection 4.1.3 andonCC,asimplenite-differencingschemeisused.Asabasecaseforcomparison,wealsorendertheCCversionofeachdatasetusingCatmull-Rominterpolation.Theresultsoftheseexperimentsaregivenbelow,alongwithrelevantobservations.Eachsetofimagesarearrangedasfollows.TherstrowhastheCatmull-Romimage,theimageusingcubicinterpolationonBCCwithtruederivativesandtheimageusingthesameschemewithestimatedderivatives,inthatorder.ThesecondrowhasimagesrenderedusingtricubicinterpolationonCCwiththerstimageusingtruederivativesandthesecondusingestimatedderivatives.TheimagesinFigure 5-2 wererenderedfromdistanceeldssampledfromthesoccerballdataset.Theresolutionsusedwere80x80x80forCCand64x64x128forBCC.ComparedtotheCatmull-Romimage,thestitchesontheballappearmuchsharperinthetwoimagesthatusetruerstderivativeswiththeinterpolationschemesweareinterestedin.Intheimagesthatusethesameinterpolationschemeswithestimatedderivatives,thestitchesareaboutasblurredasintheCatmull-Romcase,andtheseimagesarecomparableinqualitytoeachother.ImagesinFigure 5-3 andFigure 5-4 wererenderedfromtheStanforddragondatasetatresolutionsof80x80x80forCCand64x64x128forBCC.NoticethescalesonthesurfaceofthebodyandthenedetailsontheheadinFigure 5-3 andtheridgesonthebodyjustbelowtheheadinFigure 5-4 .TheCatmull-RomimageinFigure 5-4 hasadisconnectedsurfaceneartheearaswell.Theseimagesshowthatthesharperfeaturesarereproducedmuchbetterbythetwointerpolationmethodsusingtruederivativevalues.Amongtheotherthreeimages,i.e.theonesthatdonotusetruederivatives,the 51

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A B C D EFigure5-2. Thesoccerballdatasetwithapproximately512,000samples.Thestitchesontheballaresignicantlysharperintheimagesusingtruederivatives(B,D)whiletheimagesusingestimatedderivatives(C,E)arecomparableinqualitytotheCatmull-Romimage(A). narrowareasbehindthehead(Figure 5-3 )andtheridgesonthebody(Figure 5-4 )arereproducedbetterintheBCCimage.ThesephenomenacanbeobservedinFigure 5-5 aswell,whichshowstheStanfordbuddhadatasetsampledatthesameresolutionsastheprevioussets.Thisisamodelthathasalargeamountofnedetailsonitssurfaceandthevaryingaccuracytowhichthesedetailsarereproducedbythedifferentmethodsisobviousfromtheimages.Thebunnydataset,renderedatresolutionsof85x85x85forCCand68x68x136forBCC,isshowninFigure 5-3 .Here,allCCimagesshowstaircaseartifactsontheearofthebunny.ThesameareasintheBCCimagesaremostlyartifactfree. 52

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A B C D EFigure5-3. TheStanforddragondataset(sideview)withapproximately512,000samples.Thedetailsontheheadandscalesonthebodyaremorevisibleintheimagesusingthetruederivatives(B,D). Figure 5-7 showsimagesrenderedfromthepawndataset.Theseweresampledattheextremelylowresolutionsof32x32x32forCCand26x26x52forBCC.Attheseresolutions,itcanbeseenthatthecubicandtricubicschemesusingtruederivativesdoamuchbetterjobofretainingthebasicshapeofthepawn.Figures 5-8 5-9 5-10 and 5-11 showthepawndatasetsampledatincreasingresolutions.Astheresolutionincreases,thedifferenceinqualitybetweenthedifferentinterpolationschemesdiminishesuntilalltheimagesarecomparableinqualityasseeninthenalset.Thisshowsthattheadvantageofusingthetruederivativesismoreprominentatlowerresolutionswherethesampleddistanceeldmightnothaveenoughdatatoproduceareasonablyaccuratereconstruction. 53

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A B C D EFigure5-4. TheStanforddragondataset(frontview)withapproximately512,000samples.Theridgesontheundersideofthebellyarebetterreproducedbytheimagesusingthetruederivatives(B,D).Moreover,thesurfaceneartheearisdisconnectedintheCatmull-Romimage(A). 54

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A B C D EFigure5-5. Thebuddhadatasetwithapproximately512,000samples.Imageswithtruederivatives(B,D)showmuchmoredetailcomparedtotherest. 55

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A B C D EFigure5-6. Thebunnydatasetwithapproximately615,000samples.ThestaircaseartifactsseenontheearintheCCimages(A,B,C)aremostlyabsentintheBCCimages(D,E). 56

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A B C D EFigure5-7. ThePawndatasetwithapproximately33,000samples. 57

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A B C D EFigure5-8. ThePawndatasetwithapproximately200,000samples. 58

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A B C D EFigure5-9. ThePawndatasetwithapproximately260,000samples. 59

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A B C D EFigure5-10. ThePawndatasetwithapproximately512,000samples. 60

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A B C D EFigure5-11. ThePawndatasetwithapproximately2,095,000samples. 61

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CHAPTER6CONCLUSIONANDFUTUREWORKInthisthesis,westudiedsamplingandreconstructionofdistanceeldsforsurfacesrepresentedbytriangularmeshes.WeexaminedtheideaofoptimalsamplinglatticesinthiscontextanddiscussedthesamplingtheoreticmotivationforusingBCClattices.Forsamplingdistanceeldsfromtriangularmeshes,weproposedaGPUimplementationofthebruteforceapproach.Sincethebruteforceapproachiteratesoverallpossiblepoint-trianglepairs,thedistancevaluesobtainedareexact.WhileaCPUimplementationofthebruteforceapproachisprohibitivelytimeconsuming,ourGPUimplementation,basedontheCUDAarchitecture,usestheparallelprocessingcapabilitiesoftheGPUtoachievesignicantacceleration.WethendiscussedwaystoadaptourimplementationtoBCClattices.Inadditiontothedistancevalues,ourimplementationalsocalculatestheexactgradientsofthedistanceeldatthelatticepointswithrelativelyfewadditionalcomputations.WhileourGPUimplementationisfairlybasic,webelievethatitpavesthewayforfutureimplementationsthatcanachieveevenhigherspeedupsbyutilizingthelatestGPUhardwareandarchitectures,andbyusingacceleratingtechniqueslikehierarchicalspacepartitioning.WeintroducedalocalcubicinterpolationschemeontheBCClatticethatcanbeusedtoreconstructandvisualizediscretedistanceelds.Theconstructedsplinesareexactlyinterpolatingatthelatticepointsandthecubicsplinespaceleadstoafourth-ordermethodwithC1continuity.Ourinterpolationschemeutilizestheexactderivativevaluesavailablefordistanceeldstogiveamoreaccuratereconstruction.Whereexactvaluesofthederivativesarenotavailable,itusesanite-differencingschemetoestimatethederivatives,whichprovidesageneralizationtoCatmull-Romsplinesinthenon-separablesetting.Wealsodevisedanite-differencingschemeontheBCClatticethatguaranteestheorderofaccuracy.Unlikesuper-splines,oursplineconstructionispossiblewithoutintroducingintermediate(i.e.non-lattice)points.Its 62

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lowdegreehasadvantagesfromthepolynomialttingpointofview.Itisalsosimpletoimplementandefcienttocompute.Finally,weevaluatedthemeritsofusingtheBCClatticetosampledistanceeldsbyconductingaseriesofexperimentsondistanceeldssampledonCCandBCClatticesofcomparabledensity(i.e.numberofsamples).Themeasureofqualityusedwasthevisualqualityofimagesrenderedfromthesedistanceelds.OurcubicinterpolationwasusedtoreconstructthesamplesontheBCClatticewhilebothCatmull-Romandthetricubicinterpolationmethod(LeikenandMarsden[ 31 ])wereusedontheCClattice.Wealsoexaminedtheeffectsofusingexactderivativevaluesonthequalityofreconstruction.OurexperimentsshowedthattheBCCdatasetsusingtruederivativeswereabletoreproducesharpdetailsontheoriginalsurfacesmuchmorefaithfullythantheCatmull-RommethodontheCClattice.Withoutexactderivatives,theBCCdatasetsproducedimagesofqualitycomparableto,orbetterthan,bothCatmull-Romandtricubic(withoutexactderivatives)images. 63

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[49] M.Teschner,S.Kimmerle,B.Heidelberger,G.Zachmann,L.Raghupathi,A.Fuhrmann,M.Cani,F.Faure,N.Magnenat-Thalmann,W.Strasser,etal.Collisiondetectionfordeformableobjects.InComputerGraphicsForum,volume24,pages61.WileyOnlineLibrary,2005. [50] T.Theul,T.Moller,andM.Groller.Optimalregularvolumesampling.InVisualiza-tion,2001.VIS'01.Proceedings,pages91.IEEE,2009.ISBN0780372018. [51] G.ThurmerandC.Wuthrich.Computingvertexnormalsfrompolygonalfacets.JournalofGraphicsTools,3(1):43,1998.ISSN1086-7651. [52] J.Tsitsiklis.Efcientalgorithmsforgloballyoptimaltrajectories.IEEETransactionsonAutomaticControl,40(9):1528,1995.ISSN0018-9286. 68

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BIOGRAPHICALSKETCH NithinPradeepThazheveettilreceivedhisbachelor'sinElectricalandElectronicsEngineeringfromTKMCollegeofEngineering,Kollam,Indiain2005.HethenworkedasasoftwaredeveloperatTataConsultancyServicesLtd,Bangalore,Indiafor3yearsbeforejoiningUniversityofFlorida,GainesvilletodohisMSincomputerengineering.HehasbeendoingresearchintheeldofComputerGraphicsaspartofhismaster'sprogramandisexpectedtograduateinMay2011. 69