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Efficient, Tight Bounding Volumes for Subdivision Surfaces


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EFFICIENT,TIGHTBOUNDINGVOLUMESFORSUBDIVISIONSURFACE S By XIAOBINWU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2005

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Copyright2005 by XiaobinWu

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IdedicatethisworktomywifeFangwenChenandmyparents.

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ACKNOWLEDGMENTS IwishtothankmyadvisorJorgPetersforhissupportandmen toringthroughoutmyPh.D.study.Ialsothankhimforintroducingmeintoth eeldofgraphics andgeometricdesign.Hisremarkablededicationandexcell enceforresearchhas beenastronginspiration. IwishtothankmyvicecommitteechairMeeraSitharamforher guidance. ManyformaldiscussionsinseminarsandcasualtalkswithDr .Sitharamhaveled toexpandinganddeepeningofmyresearchgoals. IalsowishtothankmyPh.D.committeemembers:Dr.BabaC.Ve muri, DavidGroisserandDr.JayGopalakrishnanfortheircontinu oushelpandencouragement. Ithankmyfamilyforalwaysbeingthereforme.Withoutthem, thisworkwill notbepossible. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .............................iv LISTOFTABLES .................................vii LISTOFFIGURES ................................viii KEYTOABBREVIATIONS ...........................xi KEYTOSYMBOLS ................................xii ABSTRACT ....................................xiii CHAPTER1INTRODUCTION ..............................1 1.1SubdivisionSurfaces .........................2 1.2BoundingVolumes ..........................2 1.3Relatedwork .............................5 1.4Overview ................................6 2SUBDIVISIONSCHEMES ..........................9 2.1B-SplineSubdivision .........................9 2.2Catmull-ClarkSubdivision ......................11 2.3Doo-SabinSubdivision ........................12 2.4MidedgeSubdivision .........................12 2.5LoopSubdivision ...........................13 3NODALFUNCTIONSFORSUBDIVISIONSURFACES .........15 3.1NodalFunctions ............................15 3.2LinearIndependenceofNodalFunctions ..............15 3.3LinearIndependenceofLoopNodalFunctions ...........19 3.4LocalLinearIndependenceofLoopNodalFunction ........24 3.5LinearIndependenceofCatmullClarkNodalFunctions ......30 3.6LocalLinearIndependenceofCatmull-ClarkNodalFunct ions ..34 4TIGHTBOUNDINGVOLUMES ......................39 4.1IntervalTriangles ...........................39 4.2SubdivisionPatches ..........................40 v

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4.3BoundingBasisFunctions ......................40 4.3.1SubdivisionDomain ......................41 4.3.2BasisFunctions ........................42 4.3.3ComputingtheBoundsforBasisFunctions .........42 4.4ComponentBounds ..........................44 4.5ConstructingIntervalTriangles ...................45 4.6Semi-SharpCreasesandBoundaries .................45 4.7Convergence ..............................47 5APPLICATION ................................48 5.1CollisionDetectionUsingIntervalTriangles ............48 5.1.1PairwiseInterferenceDetection ................48 5.1.2IntersectionHierarchy .....................49 5.1.3PerformanceEvaluation ....................49 5.1.4ExperimentSetUp ......................50 5.1.5Results .............................51 5.2InnerandOuterHull .........................52 5.3AdaptiveSubdivision .........................53 6CONCLUSION ................................55 REFERENCES ...................................56 BIOGRAPHICALSKETCH ............................59 vi

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LISTOFTABLES Table page 5{1Numberoftrianglesneededforthegivenerrorbounds. ...51 5{2Intersectionhierarchy(OBBtree)creationtimeinmill iseconds. ....51 5{3Intersectioncostinmsfordierenttolerancesandroom sizes. ....52 vii

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LISTOFFIGURES Figure page 1{1ANURBSsurfaceanditscontrolpolygon.Notethe3-sided corneris dividedinto3quadrilateralsbecauseoftheNURBStopology constrain. ..................................1 1{2Theinputmesh,subdividedonce,twice,andthelimitsur face. ....2 1{3Interferencedetectionbasedonthecontrolmeshrather thanthelimit surfaceisneithersafenoraccurate.Thegreenandtheblues urfacesaredisjointbuttheircontrolmeshescollide.Thegre enand theorangesurfacescollidebuttheircontrolmeshesaredis joint. ..4 1{4ThecounterexampleoftheKobbeltetal.method ...........5 2{1AcubicB-spline. .............................10 2{2ThecubicB-splinecontrolpolygon(red)issubdividedt wotimes. ..11 2{3ThetensorproductbicubicB-splinesubdivision ............11 2{4StencilofCatmull-Clarkatirregularnodes,whereAhas valence n .11 2{5TheregularandirregularstencilsforDoo-Sabinsubdiv isionsurface. .12 2{6Thestencilformidedgesubdivision ...................13 2{73-Directionalboxspline.Theregulardomaingridandth equartic boxsplinefunction. ...........................14 2{8Loopsubdivisionstencils. .........................14 3{1OnenodalfunctionforLoopsubdivision ................15 3{2SummaryofndingsforLoopsubdivision.Domains G (shaded)and valence n forwhichthenodalfunctionswithsupporton G arelinearlyindependent. ...........................18 viii

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3{3Labelingofthesubmeshthatdenesatriangularsurface piece(schematicallyrepresentedbytheshadedarea)nearanextraordinar ynode (label1)( top,left ).Renedsubmesh, A p 0 ( top,right ).Renedsubmesh, A p 0 ,usedtoevaluatethenextsplinering( bottom,left ).The domain n ofthecompositetriangularsurfacepiececonsistsofan innitesequenceofquadrilateral(choppedtriangle)subd omains. Therstthreesuchsubdomains, n 1 n 2 n 3 ,areshaded( bottom,right ). ......................................20 3{4Thebox-splinecontrolpoints p 1 ;i (soliddots)usedtocertifythatpairs andtriplesofeigenfunctionsareindependent. ............27 3{5IndicesofCatmull-Clarknodesnearafacetwithoneextr aordinary node( n =5)( left ).Theindicesofthenewcontrolpointsafterone subdivision.Threequartersofthedomainnowhavewell-de ned tensorproductB{splinestructure( middle ).ThecompleterectangulardomainiscomposedofaninnitenumberofLshapedre-gions n ` ( right ) ............................31 3{6GloballineardependenceofCatmull-Clarksubdivision .Analternativerepresentationofthezerofunctionwith+indicationa nynonzero numberand itsnegativevalue( left ) Twocontrolnetswiththe connectivityofacubebutdierentnodepositions.Theygen erate thesameCatmull-Clarksurface( right )! ...............33 3{7Nonzeroinputcoecientsgeneratingthezerofunctiono n [ 1i =2 n i (shaded area). ..................................34 3{8TheB-splinecontrolpoints p 1 ; 2 ;p 1 ; 3 (redpoints)usedtocertifythat theeigenfunctionsassociatedwith u k and w k areindependent. ..35 3{9SummaryofndingsforCatmull-Clarksubdivision.Doma ins G (shaded) andvalence n forwhichthenodalfunctionswithsupporton G are linearlyindependent. .........................38 4{1Anintervalpoint .............................39 4{2Theintervalpolygondenedbythreeintervalpoints. .........40 4{3TheLoopsubdivisionpatch .......................40 4{4TheregularLooppatches(green)ofthevenusmodel .........41 4{5DomainlayoutofLoopsubdivisionpatches. ..............41 4{6Basisfunction b 3 for n =7anditscontrolpolygon( left ),Theupper andlowerbound(redpolygon)of b 3 (enlarged)( right ). .......43 ix

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4{7(a)Looppatch;(b)cornerintervalpoints i ;(c)convexhullofthe i forminganintervaltriangle(todisplaytheinside,thetop facet isremoved);(d)theenlargedpatchinsideitswire-framein terval triangle. .................................45 4{8Thesubdivisionsurface;thesurfaceanditsintervaltr iangles(with thetopfacetremoved);thesurfacewithsemi-transparenti nterval triangles(fromlefttoright). ......................46 4{9Subdivisionsurfacewithsemi-sharpcreases(red=crea sevalue1.0). .46 5{1Modelsusedforperformanceevaluation. ................50 5{2Thelimitsurface,theouterhullandasuperpositionoft helimitsurfaceandtheouterhullwithacutouttoshowthepositionofth e limitsurface(fromleft). ........................53 5{3Resultsofadaptivesubdivisiononthedeermodel.Input ;e=0.5%; e=0.1%;andthesurfacewithe=0.1%(fromlefttoright). .....54 5{4Ray-tracedimagesat800x600resolution.Inputmesh(67 1triangles), ray-tracingtime:8s( left).Adaptivesubdivisionwithe=0.2%.Resultingnumberoftriangles=9662andmaximumsubdivisionl evel: 4,ray-tracingtime:9s(middle).Uniformlysubdivided4ti mes.Resultingnumberoftriangles=171776,ray-tracingtime:12s (right). 54 x

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KEYTOABBREVIATIONS AABB:AxisAlignedBoundingBoxB-Rep:BoundaryRepresentationdop:DiscreteOrientationPolytopesNURBS:Non-UniformRationalB-SplineOBB:OrientedBoundingBox xi

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KEYTOSYMBOLS c ,controlpoints...................................43 n,subdivisionpatchparameterdomain.................. ....41 ,nodalfunctions.................................15 b ,basisfunctions..................................42 ,B-splinebasisfunctions............................ .9 xii

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy EFFICIENT,TIGHTBOUNDINGVOLUMESFORSUBDIVISIONSURFACE S By XiaobinWu August2005 Chair:JorgPetersMajorDepartment:ComputerandInformationScienceandEng ineering Subdivisionsurfacesprovideasimpleandyetpowerfultool forgraphics applicationsandcomputeraidedgeometricdesign.Theyll thegapbetween polyhedralandsplinemodelingandhavematuredtoanimport anthigh-end modelingmethod.Beingabletoboundthesubdivisionsurfac esiscrucialfor commontaskssuchasaccuraterenderingorintersectingsur faces. Thisthesisgivesaframeworkforconstructingtightboundi ngvolumesfor subdivisionsurfaces,bydecomposingthesurfaceintopatc hes,thepatchesinto combinationsofbasisfunctions,andpre-computingandtab ulatingofthebounds forthesebasisfunctions.Asubtlepointhereistoestablis hthebasispropertyof thefunctionsthatdenesubdivisionsurfacelocally. Weverifytheeectivenessoftheseboundingvolumeswithco ncretealgorithms forinterferencedetection,adaptivetessellationandren dering.Priortothisthesis, nocorrectandecientalgorithmexistedfortheseapplicat ions. xiii

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CHAPTER1 INTRODUCTION Theresearchareaofcomputergraphicsandcomputeraidedge ometricdesign (CAGD)revolvesaroundtherepresentationofobjectsinthe 3Dspace.Although mostlyphysicallysolid,objectsareoftenrepresentedbyt heirboundarysurfaces calledB-Rep.AB-Repmayconsistsofasetofpolygonsorofsm oothlycurved surfacesasshowninFigure 1{1 Complexcurvedobjectsarecalled"free-form"surfacesinc ontrasttofunctionalsurfacesandalsotopointouttheincreasedfreecont roloftheshape.A designercaneasilyeditthesurfacewhilemaintainingitss moothness. Themostcommonlyusedfree-formsurfacesinCADarecalledN URBS (Non-UniformRationalB-Spline)surfaces.NURBSsurfaces aredenedasthe imageofasmoothmappingfromaplanardomain,oftenrectang ular,intothe3D space.Consequently,anobjecthastoberstdividedintosu b-surfacesandcontrol structuresthatarehomeomorphictorectangles(seeFigure 1{1 ),aprocessthatis oftennotautomaticandhencecumbersomefordesigners. Figure1{1:ANURBSsurfaceanditscontrolpolygon.Notethe 3-sidedcorneris dividedinto3quadrilateralsbecauseoftheNURBStopology constrain. 1

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2 Figure1{2:Theinputmesh,subdividedonce,twice,andthel imitsurface. 1.1 SubdivisionSurfaces Subdivisioncontrolmeshesgivemorefreedomtothedesigne r,howeverata cost.Givenaninputmeshwith arbitrary connectivity,asurfaceisdenedasthe limitofarenementsequence.Thisrenementisoftenperfo rmedasacombination ofinsertingnewpointsandsplittingoldfaces.Thepositio nsofnewpoints(called controlnodes)arefunctionsoftheasetof"close-by"oldpo ints;thepositions ofoldpointscanalsobemodied.SeeFigure 1{2 foranexampleofsubdivision surface. SubdivisionsurfaceswererstintroducedbyDooandSabin[ 8 ]andCatmull andClark[ 4 ]in1978.Inthesetwopioneeringpapers,methodsaresugges tedto generalizetheuniformsubdivisionoftensor-productB-sp linessurfacestomeshes ofarbitrarytopology.Withsomedelay,duetoinsucientco mputingonthe 1980s,a"zoo"ofsubdivisionmethodshasbeencreatedbyres earchersforvarious applications.Insection 2 ,wegiveabriefintroductionandprovidetherulesand propertiesofthesesubdivisionschemes. 1.2 BoundingVolumes Free-formsurfacesproviderexiblemeanstomodelobjectsw ithsmooth boundarysothatdesignerscanconcentrateontheshapeofth eobjectwithout worryingabouttheunderlinesmoothnessofthesurface. However,theunderlyinghigherordermathematicalreprese ntationoffree-form surfaceimposesdicultiesonsomecommonoperations,fore xampledetecting

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3 theintersectionofsurfaces.Theproblemofndingtheinte rsectioncurvesoftwo bi-cubicBezierpatches,forexample,isequivalenttosol vingthreepolynomial equationswithfourvariablesanddegree6.Noanalyticsolu tioncanbesought ingeneral,andevennumericalmethodsneedtobecarefullyd esignedtoachieve acceptableresults. Forsubdivisionsurfaces,thesituationisyetmorecomplex :thesubdivision surfaceisdenedbasedonrecursionandlackofaclosed-for mformulationoverthe wholedomain.Althoughthereexistmethodstoevaluateapoi ntgivenanyparametervalue[ 30 29 ],itisnotpossibletowriteasubdivisionsurfaceasananal ytic expression:ndingtheintersectionleadstoaninnitenum berofequations. Boundingvolumesforthesubdivisionsurfacesoerasimple androbust solutiontotheproblem.A boundingvolume isasubsetin3Dspacethatencloses thesurfacepieceofinterest.Aboundingvolumeisgoodifit : hassimpleshape tightlyenclosesthesurface isecienttocompute isabletofurtherrene Thersttwopropertiesaddressthegeometricpropertiesan dtheshapequality ofaboundingvolume.Aboundvolumeiscreatedtosimplifyth ecomputation,so itsgeometryshouldbesimplerthantheoriginalsurface.An dwedonotwantto introducetoomucherrorwhileboundingthesurface,soitsh ould tightly enclose thesurface. TobeusefulingraphicsandCADapplications,itisessentia lthattheboundingvolumeisecientlycomputable.Finally,toprovideerr orcontrol,itisimportanttoallowfurtherrenementwhennecessary. Uptodate,therearefewseriousattemptstobuildboundingv olumesfor subdivisionsurfaces.Oneofthereasonsisthatformostapp licationsincomputer

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4 Control meshLimit surface Figure1{3:Interferencedetectionbasedonthecontrolmes hratherthanthelimit surfaceisneithersafenoraccurate.Thegreenandtheblues urfacesaredisjoint buttheircontrolmeshescollide.Thegreenandtheorangesu rfacescollidebut theircontrolmeshesaredisjoint.graphics,theaccuracyoftheoperationisnotaconcern.One wouldsubdividethe initialmeshseveralstepsandstopeitherwhenthevisualer rorisnolongervisible orthememoryandcomputingpowercannolongerprocessfurth errenements. However,suchaprocesstreatsthesubdividedmeshassimple collectionof polygonsandneglectsthefactthattherenedcontrolmeshi salinearcombination oftheinitialmeshandhasaninheritedstructure.Thisna veviewpointleadsto anunnecessarywasteoftimeandspace.Withaproperboundin gvolume,the computationcanbevastlyoptimized. Furthermore,tousethesubdivisionsurfacesinareassucha sCAD,users requireaccuratemeasureonthesmoothlimitsurface.Forex ample,ausermayask ifthetwosurfacesareawaybycertaindistance.Methodsbas edoncontrolmeshes willnolongerbecorrectbecausethecontrolmeshdoesnotgi veasafevolumethat containsthesurface!Figure 1{3 showsthatinterferenceofthecontrolmeshesdoes notimplythatthelimitsurfacesintersect,andseparation ofthecontrolmeshes doesnotimplythatthelimitsurfacesaredisjoint!

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5 Figure1{4:ThecounterexampleoftheKobbeltetal.method Weintroduceaboundingvolumeconstructionmethodforsubd ivisionsurfaces thathasproperties(a)(b)(c)(d)andhasbeenproventobee ectiveinnumberof applications. Thefollowingsectionsgiveashortpreviewontherelatedwo rkandsummary ofthenewmethod. 1.3 Relatedwork In1998,Kobbeltetal.[ 16 17 ]introducedaboundingvolumeforsubdivision surfacecalled boundingprism .Theylookedatthemaximumpositiveandnegative distancethatthesurfacecanvaryinacertaindirection.Af terwards,byshifting theinterpolatingtriangleintheaccordingdistance,they createdavolumein3D spacetoboundthesurface. Althoughtheircalculationofthemaximumdistancesiscorr ect,theyleftout theothertwodirectionsthatthesurfacecanchangetothesi desoftheprism. Astheresult,theirboundingprism fails toboundthesurfacewhenthesurface containshighcurvature.SeeFigure 1{4 foracounterexample. Fornon-interpolatingsubdivisionschemes,thesubdivisi onsurfaceisbounded bytheconvexhullofinputmesh.However,itisexpensivetoc omputetheconvex hullfora3Dmehs. GrinspunandSchroder[ 12 ]usedAxisAlignedBoundingBoxes(AABB) ontheinputmeshtoboundthesurface.AABBsgenerallyintro ducelargeerror

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6 marginandconvergeslowlyundersubdivision.Moreimporta ntly,theAABBsneed tobere-computedunderrotation. LutterkortandPeters[ 20 21 ]introducedtightenclosuresforunivariateand multivariateB-splinefunctions.MostrecentlyPetersand Wu[ 27 ]builtasafe boundingpolygonforunivariatecurve.Yetthegeneralboun dingvolumefor surfacesisnotclear.Thetightnessoftheirboundsforcubi cfunctionsisdiscussed inPetersandWu[ 26 ]. Thereareseveralmatureintersectionhierarchiesforpoly gonalmeshes. In1996,Gottschalketal.[ 9 ]introducedOrientedBoundingBoxes(OBB).It isproventoworkmoreecientlythanAABBsbecauseittightl yenclosesthe meshandconvergesfaster.In1998,Klosowskietal.[ 15 ]generalizedAABBand describedanotherboundinghierarchycalled K-dops .Thismethodworkswellfor objectswithcontinuousmotion. 1.4 Overview Theapproachintroducedinthisthesiscombinesideasfrome nvelopesfor ray-tracing[ 16 ],tightboundsonsplines[ 20 21 ]. Ateachmoment,weconsiderapartofthesurfacethatcorresp ondstoone particularcontrolface.Thispartofthesurface(called patch )onlydependsona neighborhoodofvertices.Wecreateaboundingvolume(call ed intervaltriangle seedenitioninsection 4 )toboundthispatchbasedonthepositionofthissetof relativevertices. WefollowKobbelt[ 16 ]inthatweuseamin-maxexpressionofinterval arithmetictocreatebounds.Ouroverallapproachdiers,h owever.Wedonot boundjustthenormaldirection;insteadweconsidereachco mponentin f x;y;z g as afunctionandweboundeachfunctionseparately. Wealsodonotperformtheeigendecompositionofthesubdivi sionmaskin Stam[ 30 ]andGrispunandSchroder[ 12 ].Inaddition,weareabletocreatea

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7 boundingvolumemuchtighterthanaxis-alignedboundingbo xes(AABBs)onthe controlmesh.Toimproveaccuracyandeciency,weremoveli nearcomponents fromthesubdivisionfunctionasinLutterkortandPeters[ 21 ]beforecomputingthe boundingvolume. Ourmethodhastwomajorsteps,explainedinSection 4.3 andSection 4.5 respectively.First,webuildthelocalparametrizationon thesubdivisionpatch, anduponthatweconstructasetofbasisfunctions.Foranygi veninputvalueson thecontrolmesh,thelimitfunctionwillbealinearcombina tionofthissetofbasis functions.Inthisstep,wealsondtheapairofupperandlow erboundsforeach basisfunctionandstorethemintoatable. IncontrasttoStam[ 30 ],whichparametrizesovertheunitsquare,weusethe parametrizationthathas n -gonsymmetrytosimplifyourcomputation. Inthesecondstep, intervaltriangles areconstructed.Theseintervaltriangles arecreatedbylinearlycombiningtheupperandlowerbounds ofthebasisthatwe storedinthetable.Eachintervaltriangleboundsonepiece ofthelimitsurface, andtheunionofintervaltrianglesenclosesthewholesurfa ce.Ifapre-dened accuracybound isgiven,theintervaltrianglesarelocally,adaptivelyre neduntil thethicknessislessthan We[ 32 ]usedourboundingvolumeintheapplicationofinterferenc edetection byintegratingitwiththewellusedintersectionhierarchy suchasOBBandKdops.WeonlymadeasmallchangetotheOBBtreeandk-dopstre ecode.In returnweachievedconsiderableimprovementontheinterse ctiontestbetweenthe subdivisionsurfaces. Therestofthethesisisorganizedasfollows:Chapter 2 reviewsthecommonlyusedsubdivisionschemes.Inchapter 3 ,weintroducethebasisfunctionsofsubdivisionsurfaces.Inchapter 4 ,weexplainhowtheintervaltriangleisconstructed.

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8 Inchapter 5 weexamineafewapplicationsthattakeadvantageofthecons tructed tightboundingvolume.Chapter 6 isasummary.

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CHAPTER2 SUBDIVISIONSCHEMES Inthischapterwewillreviewafewcommonlyusedsubdivisio nschemes.The purposeofthissectionisalsotodenetheterminologythat wewilluseintherest ofthesis. 2.1 B-SplineSubdivision Weconsidertheunivariate(curve)case.AuniformB-spline functionisdened asthelinearcombinationofasetofuniformlyspacedbasis. Specically, f ( t ):= X i c i i ( t )(2.1) where c i sarescalarvaluescalled coecients and i sarefunctionscalled B-spline basisfunctions i aremerelythetranslationofbasis 0 atintegergrid i .Figure 2{1 showsanexampleofaB-splinefunction.Apairoffunctions x ( t )and y ( t ) denesacurvein2D. Theideaofsubdivisionistore-writeB-splinefunction f i intoarenedbasis 1 .Infact, 1 sarespacedhalfastheoriginal ,i.e. 1 2 .Itturnsoutthatwecan directlywriteeach intermsofthenewbasis 1 Fordegree2: 4 i = 12 i 1 +3 12 i +3 12 i +1 + 12 i +2 Fordegree3: 8 i = 12 i 2 +4 12 i 1 +6 12 i +4 12 i +1 + 12 i +2 9

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10 Figure2{1:AcubicB-spline. Nowwecanndthenewsetofcoecients c 0 s satisfying f ( t ):= X i c 0i 1i ( t ) bysubstituting by 1 inequation( 2.1 ). Fordegree2, 4 c 02 i := c i 1 +3 c i ; 4 c 02 i +1 :=3 c i + c i +1 Fordegree3(seeFigure 2{2 left ), 8 c 02 i := c i 1 +6 c i + c i +1 2 c 02 i +1 := c i + c i +1 Figure 2{2 right showshowthecubicB-splinecontrolpolygonissubdivided twotimes.Aswecansee,thesubdividedcontrolpolygonrapi dlyconvergetothe theblackcurve.

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11 6 1 1 1 1 Figure2{2:ThecubicB-splinecontrolpolygon(red)issubd ividedtwotimes. 1 1 1 1 6 6 1 1 1 1 1 36 1 6 1 6 6 1 6 Figure2{3:ThetensorproductbicubicB-splinesubdivisio n 2.2 Catmull-ClarkSubdivision CatmullandClark[ 4 ]extendedthesubdivisionformulaforcubicB-spline intosurfacebytensor-productingtherule(seeFigure 2{3 ).Thenumbersarethe weightsofeachvertexcontributingtothenewposition(blu e).Thenalresultis normalizedbythesumoftheweights.Thiscollectionofweig htsindicatingthe contributingverticesiscalledthe stencil forthenewvertex. Toapplythesubdivisionschemeonanarbitrarymesh,theysu ggestedastencil tothevertexwithotherthan4neighbors(seeFigure 2{4 ). A 6 6 6 6 6 1 1 1 1 1 A =4 n 2 7 n: Figure2{4:StencilofCatmull-Clarkatirregularnodes,wh ereAhasvalence n

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12 3 3 1 9 ... 0 1 2 n 1 Figure2{5:TheregularandirregularstencilsforDoo-Sabi nsubdivisionsurface. Suchanodewheretheregularsplinesubdivisionruledoesno tapplyiscalled extraordinary nodeor irregular node.InthecaseofCatmull-Clark,itisequivalent tothenodethatdoesnnothave4neighbors.Numberofneighbo rsiscalled valence .ThereforeforCatmull-Clark,valence 6 =4 irregular. Thelimitofthesubdivisioncreatesa C 2 surfaceexceptatirregularnodes, wherethesurfaceis C 1 2.3 Doo-SabinSubdivision DooandSabinsubdivisionisasubdivisionschemebasedonte nsorproduct bi-quadraticspline.Bytensoringtherulesforquadratics plinefromSection 2.1 ,we getthestencilinFigure 2{5 left Fortheirregularcasewherethefacedoesnothave4vertices (Figure 2{5 right ),DooandSabinsuggested 0 =1 = 4+5 = 4 n and i =(3+2 cos (2 i=n )) = 4 n where n isthenumberofverticesinthefaces. 2.4 MidedgeSubdivision In1997,PetersandReif[ 24 ]suggestedanevensimpliersubdivisionwith minimumstencilsize.Thenewvertexisthetheaverageofthe twomidpoints oftheneighboringedges(seeFigure 2{6 ).Thenewnodeonlydependsonthree verticesandnoirregularruleisneeded.Thissubdivisions chemeissometimecalled the simplest subdivisionscheme.

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13 2.5 LoopSubdivision In1987,Loop[ 18 ]introducedaschemebasedontriangulation.Loop'sscheme isbuiltonthe3-directionalboxspline(formoreonboxspli netheory,seedeBoor's book[ 6 ]).The3-directionalboxsplineisdenedonaregulartrian gulardomain whereeveryvertexhas6neighbors(seeFigure 2{7 ). Theregularrulecanbederivedfromtherecursionofbox-spl inebasis(see Figure 2{8 left ).Theirregular(valence 6 =6)rulesissuggestedbyLoopasinFigure 2{8 right ,where = n ( 64 40 (3+2cos(2 =n )) 2 ) 1). ... 1 1 2 Figure2{6:Thestencilformidedgesubdivision

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14 Figure2{7:3-Directionalboxspline.Theregulardomaingr idandthequarticbox splinefunction. . regular stencilirregular stencil 1 1 1 1 1 1 1 1 3 3 1 1 1 10 1 1 1 Figure2{8:Loopsubdivisionstencils.

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CHAPTER3 NODALFUNCTIONSFORSUBDIVISIONSURFACES Inthischapter,wedenenodalfunctionsforsubdivisionsu rfaces.Weexamine theirlinearindependencepropertiesoverdierenttypeof domains. 3.1 NodalFunctions Inmostgraphicsandgeometricapplications,eachcontrolp ointisgivenasa3dimensionalpointwith( x;y;z )coordinates.Sincesubdivisionrulesareappliedon eachdimensionindividually,wecanfocusononedimensiona time.Eachcontrol pointcanbeviewedasascalarvalueinsteadofavector. The nodalfunction i aredenedbysettingthescalarcontrolpoint a i to1 andallothersto0andapplyingsubdivision(see 3{1 ).Thenodalfunctionscanbe called Basisfunctions onlyaftertheyareshowntobelinearlyindependent. 3.2 LinearIndependenceofNodalFunctions Anumberofpublicationshavetacitlyassumedthatthenodal functions i arelinearlyindependent.Withoutproof,nodalfunctionsa recalledsubdivision basisfunctions[ 11 10 5 ],usedasscalingfunctionstoforma`basis'ofthecoarsest levelofamultiresolutionhierarchy[ 19 ],andusedto[ 23 ]tsubdivisionsurfacesby allowingoneinterpolationconditionforeachmeshnode. Figure3{1:OnenodalfunctionforLoopsubdivision 15

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16 Infact,forCatmull-Clarkthisassumptionisfalse.Forthe simplestquadrilateralcontrolmesh,acube,theeightnodalfunctionsareglob allylinearlydependent (seeLemma 6 ):ingeneral,wecannotteightarbitrarydatapointsbyadj usting thecoecients a i ofthecorrespondingsurface P 8i =1 a i i Forthewell-knowntensor-productsplinefunctions,globa llinearindependence maybeinterpretedaslinearindependenceovertheunionofd omainrectangles delineatedbytheknotlinesandjoinedbyidentifyingedges oftherectanglesinthe naturalfashion.Thisdenitiongeneralizestosubdivisio nsurfacesasfollows.Let n beaunitsquare(triangle)ifthe k thfacetofthecontrolmeshhas4vertices(3 vertices).Letbetheunionofalldomains( n ;k ),indexedbytheircontrolmesh facetindex,withedgestopologicallyidentied(setequal )ifthefacetsshareedges. Thisgivesthestructureofa2-manifoldhomeomorphictoth econtrolmesh. Globallinearindependenceislinearindependencewithres pectto. Denition1(Globallinearindependence) Asetofnodalfunctionsare globallylinearlyindependent iftheyareindependentoverthedomainmanifold Thatis,if 8 u 2 : P i a i i ( u )= 0 then a i =0 Whilesomeofthenumericalmethodsrequireonlystandard(g lobal)linear independence,others,suchaslocalHermiteinterpolation andlocalizedmultiresolution,relyonstrongernotionsofindependence.Wene edtoanalyzeindependenceoncertainring-shapedannuli A andonsubsets n i oftheunitsquareorunit triangle n .Thestrongestandmostsubtlenotionofindependenceisloc allinear independence.Denition2(Locallinearindependence) Asetofnodalfunctionsare locally linearlyindependent ifforanyboundedopen G ,allthenodalfunctionshavingsome supportin G arelinearlyindependenton G Remarkably,forbox-splinesandB{splines,thestandardno tionof(global) linearindependenceisequivalenttolocallinearindepend ence[ 6 ].Thatis,ifall

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17 coecients a i havetovanishsothat P i a i i = 0 (globallinearindependence), thenthecoecientsofallnodalfunctionsthatarenonzeroo veranyopenset G havetovanishif P i a i i vanisheson G (locallinearindependence).Since G can bearbitrarilysmall,locallinearindependenceisastrict errequirementonthe nodalfunctionsthangloballinearindependence.Wewillse ethatlocalandglobal independencearenotequivalentforsubdivisionnodalfunc tionsnearextraordinary nodesofhighervalence(numberofneighbors).Thisobserva tionthatprovides rareinsightintothestructuraldierencebetweensubdivi sionandsplinesurfaces. Specically,weshowthatfortheCatmull-ClarkandLoopsub division mathlist]val@ n ,valenceofanextraordinarynode (i)thenodalfunctionsaregloballylinearlyindependent; 1 (ii)thenodalfunctionsarelinearlyindependentoveranan nulus 1 suchasinFigure 3{2 left ; (iii)forvalence n higherthanthe`regular'valence,thenodalfunctionsaren ot locallylinearlyindependent;(iv)thenodalfunctionsarelinearlyindependentoneachdo main n naturally associatedwithonefacetofthecontrolnet. 1 TheabovecharacterizationisananalogueoftheSchoenberg -Whittney theoremofsplineinterpolation.Toillustratewhysuchade tailedcharacterization isusefulinpractice,considerthefollowingscenariosfor interpolationwithLoop subdivisionsurfaces.Interpolating12datapointsonadom ain n correspondingto ameshtrianglewhoseverticesallhavevalencesix,isawell -posedproblemwitha uniquesolution.However,ifoneoftheverticeshasvalence n =3thentheproblem 1 withoneexception:Catmull-Clarkappliedtonodeswithval ence n =3;see Lemma 6

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18 8 N (Lemma 1 ) N 6 (Lemma 3 ) N 7 (Conjecture 1 ) 8 N (Theorem 1 ) Figure3{2:SummaryofndingsforLoopsubdivision.Domain s G (shaded)and valence n forwhichthenodalfunctionswithsupporton G arelinearlyindependent.isoverconstrainedwhilefor n> 6itistypicallyunderconstrained.Ifwematch thenumberofinterpolationconditionstothenumberofnoda lfunctionsthatare nonzeroon n ,i.e.ifwespecify n +6interpolationpoints,wendthat,ifthe pointsbelongtoasubregion n 1 (shadedareainFigure 3{2 ,labelledConjecture 1 ),theproblemisoverconstrainedfor n> 6.InterpolationwiththeCatmull-Clark subdivisionfollowsasimilarpatternwithanadditionalco mplicationfor n =3. Theanalysisismadeeasierbythefactthatthecomponentfun ctionsofmost popularsubdivisionschemes,andinparticularofbothCatm ull-ClarkandLoop subdivisions,arevariationsofthewell-understoodbox-s plinesubdivision[ 6 ];much ofthesubdivisionlimitsurfaces,correspondingtoquadsw ith4-valentvertices, respectivelytriangleswith6-valentverticesare`regula r',i.e.aresplinesurfaces generatedbybox-splines.Thisbox-splineconnectionshou ldmakeuscautioussince theshiftsofbox-splinesare,ingeneral,notlinearlyinde pendent.Forexample,the four-direction(quincunx)subdivision,whichgivesriset o4-8subdivision[ 31 ],has dependentnodalfunctions.Catmull-Clarksubdivisionrul esgeneralizethetwodirectionbox-splinerules,i.e.therulesofthebicubicte nsor-productspline;and Loopsubdivisiongeneralizesathree-directionbox-splin e,theconvolutionofthe linear`hat'function,withitself.Fortunately,forboths plinesweknow[ 6 ]thatthe nodalfunctionsformabasis.Therefore,itsucestoanalyz esubmeshesthatdene theneighborhoodofextraordinarynodes,wheretheconnect ivityofthecontrol

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19 meshdiersfromtheregularconnectivityoftheboxspline, namelymeshnodesof valence n 6 =4forCatmull-Clarkmeshesandofvalence n 6 =6forLoopmeshes. ThuscharacterizingindependenceforLoopsubdivisionand Catmull-Clark subdivision,closesagapinthetheoryofgeneralizedsubdi visionandprovidesa basisforcomputationaluse.WerstdiscussLoopsubdivisi on,andthenCatmullClarksubdivision,sinceLoopsubdivisionisthesimplerof thetwoandtherefore showsthestructureoftheproofmoreclearly.Also,itisthe subdivisionscheme suggestedforcomputationalpurposesin[ 11 10 5 ]. 3.3 LinearIndependenceofLoopNodalFunctions ForLoopsubdivision,thereareonlytworules:tocomputene wnodes, correspondingtoedgesoftheoldmesh,andtocomputenewnod es,corresponding tooldnodes.Theserulesareexpressedbythetwostencilssh owninFigure 2{8 .A nodeofaLoopmeshis extraordinary ifitdoesnothavesixneighbors. Duetothesmallreachoftherules,asubmeshconsistingofon etriangleand alltrianglesattachedtoitdenes,bygoingtothelimit,at riangularpieceofthe surfaceadjacenttothelimitoftheextraordinarynode.Ifa llnodesofthecentral triangleareofvalencesix,thesurfaceisapolynomialpiec eofathree-directionbox splineanditspropertiesarewellunderstood.Sincenewedg enodeshavevalence six,extraordinarynodesaremoreandmoreisolatedunderre nement,andwecan focusontriangleswithoneextraordinarynodeofvalence n 6 =6.Inthefollowing, thesubscript0referstoameshwhereanytwoextraordinaryn odesareseparated byatleastonenodeofvalencesix.Thismaybetheresultofon esubdivision appliedtotheoriginalmesh. Therelevantsubmeshdeningthetriangularsurfacepiecec onsistsof K := n +6nodesthatcanbelabeledasinFigure 3{3 ( top,left ).Westorethesubmesh asavector p 0 :=( p 0 ; 1 ;:::p 0 ;K ) 2 R K :

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20 6 3 N+1 N+2 N+3 N+4 N+9 N+7 1 4 2 N N+4 5 N+5 N+6 N+2 N+1 N+3 3 N+2 N+8 N+10 N+11 32 N+3 N+5 N+1 N+4 1 N 6 5 4 N+6 N+12 N+10 N+11 N+12 2 4 N+6 1N+5 N 56 N+9 N+8 N+7 p 0 p 1 := A p 0 n 1 n 2 n 3 n A A u =1 ;v =0 u =0 ;v =1 u =0 ;v =0 Figure3{3:Labelingofthesubmeshthatdenesatriangular surfacepiece (schematicallyrepresentedbytheshadedarea)nearanextr aordinarynode(label1)( top,left ).Renedsubmesh, A p 0 ( top,right ).Renedsubmesh, A p 0 ,used toevaluatethenextsplinering( bottom,left ).Thedomain n ofthecompositetriangularsurfacepiececonsistsofaninnitesequenceofqua drilateral(chopped triangle)subdomains.Therstthreesuchsubdomains, n 1 n 2 n 3 ,areshaded ( bottom,right ). Subdivisiongeneratesanewsetof M := K +6controlverticesasshownin Figure 3{3 (top,right).Westorethosecontrolverticesinanewvector p 1 :=( p 1 ; 1 ;:::;p 1 ;K ;p 1 ;K +1 ;:::p 1 ;M ) : Ifwerepresenttheaveragingrulesasrowsofa M K matrix A (withrow sumone),thenthesubdivisionrulestocomputethevector p 1 from p 0 are p 1 = A p 0 where A := 0BBBB@ A 11 0 A 21 A 22 A 31 A 32 1CCCCA :

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21 Here A 11 isan( N +1) ( N +1)matrixthatcomputesthenewextraordinary nodeandtheverticesadjacenttoit(notethatthisalsohold sforanoptionalinitial renementtogenerate p 0 fromameshthathasneighboringextraordinarynodes); A 21 and A 22 determinetheveverticeswithindices N +4 ;N +3 ;N +2 ;N +5 ;N +6 ofthenextlayer;and A 31 and A 32 denethesixoutermostnodes.Thesizesof A 22 and A 32 are5 5and6 5respectively.Leavingoutthedirectneighborsofthe extraordinarynode, p 1 ; 4 ;p 1 ; 5 ;:::;p 1 ;n 1 ,theremainingcontrolpoints p box1 :=( p 1 ; 1 ;p 1 ; 2 ;p 1 ; 3 ;p 1 ;n ;:::p 1 ;M ) denethreetriangularpolynomialpiecesshownasshadedin Figure 3{3 (top,right). Tocomputethenodesofthenextsubdivisionstep,weneedonl ytherst K controlpointsof p 1 (seeFigure 3{3 bottomleft), ( p 1 ; 1 ;p 1 ; 2 ;:::;p 1 ;K )= A p 0 := 0B@ A 11 0 A 21 A 22 1CA p 0 : Byrepeatingtheprocess,aninnitesequenceofpiecewisep olynomialrings isgenerated.Wecanchoosetheirdomains n ` sothattheirunionllsoutthe triangulardomain n : n := f ( u;v ) j u + v + w =1 ;u;v;w 0 g = [ 1` =1 n ` n 1 := n n 1 2 n ; n ` +1 := 1 2 n ` : Thecontrolvertices p n after n subdivisionstepsthatdeterminethefunctionon n n are: p n = A ( A ) n 1 p 0 ;n 1 : (3.1) FromtherecursioninEquation 3.1 ,itisevidentthattheeigenstructureof A playsacrucialrulewhendeterminingthepropertiesofthes ubdivisionsurfaces suchasthecomputationofthelimitposition,tangentplane ,andshapeanalysis [ 8 1 28 25 14 ].

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22 UsingFouriertransform,itiseasytoderivethevectorofei genvalues 11 of A 11 11 :=[1 ; 5 8 ( n ) ;f (1) ;:::;f ( n 1)] where f ( k ):= 3+2cos(2 k=n ) 8 ; ( n ):= 5 8 f ( k ) f ( k ) ; and 22 of A 22 : 22 :=[ 1 8 ; 1 8 ; 1 8 ; 1 16 ; 1 16 ] : Exceptforthecase n =3, A canbediagonalizedbythematrix V ofits eigenvectors(detailsoftheeigenanalysisof A canbefounde.g.inStam[ 29 ]): A = VV 1 ; =diag( 11 ; 22 ) ; V = 0B@ U 0 0 U 1 W 1 1CA (3.2) wherethesubmatrices U 0 and W 1 aretheeigenvectorsof A 11 and A 22 ,respectively.For n> 3,thecolumnsof V arelinearlyindependentvectorsin R K Nowlettheinitialsubmesh p 0 := v i beaeigenvectorassociatedwitheigenvalue i and i thecorrespondinglinearcombinationofnodalfunctions.T hen, after n stepsofsubdivision, p n j p 0 = v i = A ( A ) n 1 v i = A n 1 i v i = n 1 i Av i = n 1 i p 1 j p 0 = v i : Therefore i ( n n +1 )isascaledmultipleof i ( n 1 ).Precisely, 8 ( u;v ) 2 n and 8 n 1 ;' i ( u 2 n ; v 2 n )= ni i ( u;v ) : (3.3) InStam[ 29 ]these K functions i arecalledeigenbasis.However,adjacentto anextraordinarynode,each i consistsofaninniteunionofpolynomialpieces. Thesubtlebutimportantpointtobesettledhereisthat,alt houghthecolumnsof V areindependent,thecorrespondingfunctionscanbedepend ent.Wetherefore callthefunctions i eigenfunctions .Tocharacterizesubdivisionnearextraordinary

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23 nodesassimilarto,butdistinctfromsplinerepresentatio ns,wewillshowthatthe eigenfunctionsarelinearlyindependentover n ,butlinearlydependentoncertain subsetsof n ToshowthatthenodalfunctionsofLoopsubdivisionare(glo bally)linearly independent,wefocusonsubdomainsthatformanannulussur roundingthe preimageofasequenceofextraordinarynodes.Withtheobvi oustopological identicationofedgestoinducethestructureofa2-manifo ldwithboundaries,we deneanannulusas n copiesof n 1 A := f 1 ;:::;n g n 1 : Lemma1 ThenodalfunctionsofLoopsubdivisionwithsupporton A arelinearly independentover A Proof.Assumethat f := P i p 0 ;i i iszeroonallof A .Recallthatthesubset ofnodes p box1 canbeinterpretedasaregularthree-directionbox-spline controlnet deningthreepolynomialpiecesneartheextraordinarynod e.Sincethebox-splines arelocallylinearlyindependent[ 6 ],allbox-splinecontrolpointsdening f on A are zeroand,inparticular, A p 0 .Sincealleigenvaluesof A arepositive, A isoffull rankandthereforeall p 0 ;i mustbezero.If p 0 wastheresultofonerenement,also theoriginalmeshnodesmustbezerosincethematrix A 11 isoffullrank. jjj Nowconsiderthenodalfunctionscorrespondingtothemesha fteronesubdivision.Lemma 1 proveslinearindependenceofthesenodalfunctionsontheu nion ofallannuli A associatedwithoriginalmeshnodes.Sincethematrix A 11 isoffull rank,alsotheoriginalcontrolnodesmustbezeroifthefunc tionvanishesonall annuli.Corollary1 ThenodalfunctionsofLoopsubdivisionaregloballylinear lyindependent.

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24 3.4 LocalLinearIndependenceofLoopNodalFunction Inthissection,wecharacterizethelocallinearindepende nceofLoopsubdivisionnodalfunctions.Lemma2 Forgeneral n ,thenodalfunctionsofLoopsubdivisionare notlocally linearlyindependent .Specically,forany k thereexistsavalence n sothatthe nodalfunctionsofLoopsubdivisionwithsupporton n k i ;i =1 :::n +6 ,are locallylinearlydependent on n k andevenon [ k` =1 n ` Proof.Allnodalfunctionscorrespondingto p 0 arenonzerooneach n k .Each vector p boxk correspondingto n k has16entries.Forsucientlylargevalence,the nodalfunctionsmustthereforebedependenton n k andon [ k` =1 n ` fornite k jjj However,thenodalfunctionsarelocallylinearlyindepend entforsuciently lowvalence n Lemma3 For n 6 thenodalfunctions i ;i =1 :::n +6 ,arelocallylinearly independent. Proof.Denoteby P i ;i =1 ; 2 ; 3thethree12 ( n +12)pickingmatrices thatselectthebox-splinecoecientsofeachofthethreetr iangulardomainparts. Bysymboliccalculation,weverifythat P i A ;i =1 ; 2 ; 3isoffullrank,Therefore, ifallofits12controlpointsarezero,thenalso p 0 mustbezero.Locallinear independenceofthethree-directionboxsplinethenproves theclaimfor n 1 .Since thecontrolpointson n ` arecomputedfrom p 0 byapplying P i A ( A ) ` 1 locallinearindependenceon n 1 implieslocallinearindependenceon n ` jjj For n =7,thenodalfunctionsareindependenton n 1 andhenceon [ k` =1 n ` buttheyarenotlinearlyindependentonsubsetsof n 1 thatdonotstraddleall

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25 threepiecewisepolynomialdomains.Asthevalence n increases,asubtlepattern emerges.Conjecture1 For k := b ( n 6) = 2 c +1 > 1 ,thenodalfunctionsofLoopsubdivision i ;i =1 :::n +6 arelinearlyindependenton [ ki =1 n i andlinearlydependenton [ k 1 i =1 n i Weveriedtheconjecturesymbolicallyupto n =30whichshouldcover allcasesofpracticalinterest.However,toinvestigateli nearindependenceonthe naturaldomainscorrespondingtocontrolfacets,namelyon n = [ 1` =1 n ` ,weneed abetterstrategy.Weusethe`eigen'propertyoftheeigenfu nctions,thatadditional layersarescaledcopiesoftheearlierlayers.Lemma4 Forvalence n> 3 ,the eigenfunctions i i =1 ;:::;n +6 ,ofLoop subdivisionarelinearlyindependenton n Proof.Theproofisbycontradiction.Alleigenvalues i of A arepositive. Wesorttheeigenfunctions i ( i =1 ;:::;n +6)sothattheirassociatedeigenvalues i descendfromthelargesttothesmallest.Supposethereexis tscalars a 1 ;a 2 ;:::;a n +6 ,notallzero,suchthat n +6 X i =1 a i i =0 : Let j bethelargesteigenvaluesuchthat a j 6 =0.Thenwith w i := a i =a j ,we write j = n +6 X i = j +1 w i i :

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26 For 8 ( u;v ) 2 n and 8 n 1,( u 2 n ; v 2 n ) 2 n ,withtheaboveequationandequation ( 3.3 ),wehave j ( u 2 n ; v 2 n )= n +6 X i = j +1 w i i ( u 2 n ; v 2 n ) ) nj j ( u;v )= n +6 X i = j +1 w i ni i ( u;v ) ) j ( u;v )= n +6 X i = j +1 w i ( i j ) n i ( u;v ) Since( i j ) n 0as n !1 unless i = j j ( u;v )= X i 2f i j i = j g w i i ( u;v ) : musthold.Thereforetheeigenfunctionsassociatedwith j mustbelinearly dependent.Intheremainderoftheproof,weshowthistobefa lse.Inotherwords, theproblemofprovingthelinearindependenceofalleigenf unctionshasbeen reducedtotheindependenceoftheeigenfunctionswiththes ameeigenvalue. Becauseoftheeigenstructureof A ,themultiplicitiesofitseigenvaluesare small(atmostfour)anddonotincreasewith n .Recallthattheeigenvaluesof A are [1 ; 5 8 ( n ) ;f (1) ;:::;f ( n 1) ; 1 8 ; 1 8 ; 1 8 ; 1 16 ; 1 16 ] : where ( n ):= 5 8 (3+2cos(2 =n )) 2 64 ;f ( k ):= 3+2cos(2 k=n ) 8 : Tondtherepeatedeigenvalues,weobservethatfor k 2f 1 :::n 1 g 1. f ( k )= f ( n k ), 2.if n isevenand k = n= 2, f ( k )= 1 8 ;otherwise f ( k ) 62f 1 8 ; 1 16 g 3. f ( k ) 6 =1and f ( k ) 6 = 5 8 ( n ),

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27 p 1 ; 3 p 1 ; 2 C k 1 0 0 0 00 0 0 0 0 0 1 1 1 p 0 = u k p 0 = y 1 p 0 = z 1 p 1 ;N +8 p 1 ;N +9 p 1 ;N +10 p 1 ;N +8 p 1 ;N +10 Figure3{4:Thebox-splinecontrolpoints p 1 ;i (soliddots)usedtocertifythatpairs andtriplesofeigenfunctionsareindependent.Thatis,if isaneigenvalueof A withmultiplicitygreaterthanonethen = f ( k ) 6 = 1 8 ,or = 1 8 ,or = 1 16 .Inparticular,allrelevanteigenvaluesarenonzero. Welookateachcaseindividually. Case1: = f ( k ) 6 = 1 8 Inthiscase hasmultiplicity2andtheassociatedeigenvectors u k and w k aregivenin[ 29 ]: u Tk =(0 ; 1 ;C k ;C 2 k ;:::;C ( n 1) k ;::: )and w T k =(0 ; 0 ;S k ;S 2 k ;:::;S ( n 1) k ;::: ) where C k :=cos(2 k=n )and S k :=sin(2 k=n ).Toshowthetwoeigenfunctionsdenedby u k and w k arelinearlyindependent,weconsiderthetwo box-splineentriesof p 1 ; 2 and p 1 ; 3 (soliddotsinFigure 3{4 left )afterone stepofsubdivisionappliedtothemesh p 0 := u k andonestepappliedwith p 0 := w k .Thetwocorrespondingeigenfunctionsareindependentbec ause det 0B@ p 1 ; 2 j p 0 = u k p 1 ; 3 j p 0 = u k p 1 ; 2 j p 0 = w k p 1 ; 3 j p 0 = w k 1CA = 2 det 0B@ 1 C k 0 S k 1CA 6 =0 ; since S ( k ) 6 =0because f ( k ) 6 = 1 8 andhence k 6 = n 2 Case2: = 1 8

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28 Inthiscase canhavemultiplicityof3or4.Werstshowthattheeigenfunctionscorrespondingtorstthreecolumns y 1 ; y 2 ; y 3 of 0B@ 0 W 1 1CA are independent.Theeigendecomposition( 3.2 )of A 22 is[ 29 ]: W 1 = 0BBBBBBBBBB@ 0010010101100000111001000 1CCCCCCCCCCA : Theindependenceoftheeigenfunctionsfollowsfromtheind ependenceofthe threeboxsplinecontrolpoints p 1 ;n +8 ;p 1 ;n +9 ;p 1 ;n +10 (soliddotsinFigure 3{4 middle )afteronesubdivision: det 0BBBB@ p 1 ;n +8 j p 0 = y 1 p 1 ;n +9 j p 0 = y 1 p 1 ;n +10 j p 0 = y 1 p 1 ;n +8 j p 0 = y 2 p 1 ;n +9 j p 0 = y 2 p 1 ;n +10 j p 0 = y 2 p 1 ;n +8 j p 0 = y 3 p 1 ;n +9 j p 0 = y 3 p 1 ;n +10 j p 0 = y 3 1CCCCA = 1 8 3 det 0BBBB@ 440001144 1CCCCA 6 =0 : Ifthemultiplicityof 1 8 isthree,thenwearedone.Otherwise, = f ( n= 2)for n evenandwehaveoneadditionaleigenvector u k from 0B@ U 0 U 1 1CA .Afterone subdivision,thebox-splinecontrolpoint p 1 ; 3 iszerofor y 1 ; y 2 ; y 3 andnonzero for u k .Thisprovesindependenceofallfoureigenfunctions. Case3: = 1 16 Theeigenvectorsofthetwoeigenfunctionsassociatedwith 1 16 correspondto thelasttwocolumns z 1 and z 2 of 0B@ 0 W 1 1CA .Pairwiseindependencefollows fromtheindependenceofthetwoboxsplinecontrolpoints p 1 ;n +8 ;p 1 ;n +10

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29 (soliddotsinFigure 3{4 right ) det 0B@ p 1 ;n +8 j p 0 = z 1 p 1 ;n +10 j p 0 = z 1 p 1 ;n +8 j p 0 = z 2 p 1 ;n +10 j p 0 = z 2 1CA = 1 8 2 det 0B@ 0330 1CA 6 =0 : ThiscompletestheproofofLemma 4 jjj Wecannowaddressouroriginalgoalofshowingthatthenodal functions i arelinearlyindependent.Corollary2 For n> 3 ,thenodalfunctionsofLoopsubdivision, i ;i =1 :::n +6 arelinearlyindependenton n Proof.Recallthateachnodalfunction i isgeneratedbysubdivisionwhen settingcontrolpoint i to1andallothersto0.Theirindependencefollowsfrom [ 1 ;:::;' K ]= V [ 1 ;:::; K ] ;K = N +6 ; andthefactthat,for n> 3,thematrix V ofeigenvectorsisaninvertiblematrix. jjj Forthespecialcase n =3,thematrix A hasanon-trivialJordanblockand cannotbediagonalized.However,sincethenumberofthenod alfunctionsissmall, namelynine,weneednotdecomposeintotheeigenspace.Lemma5 For n =3 ,thenodalfunctionsofLoopsubdivision, i ;i =1 :::n +6 arelinearlyindependenton n Proof.Weexplicitlydeterminethe( n +12) 9matrix M thatmaps p 0 tothe box-splinecontrolpoints p box1 1 16 2666666664 73330000066220000062620000062260000026062000011011111002660002001111010011206600002060262000060026200062002600020660020000620062002600026 3777777775 :

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30 Since M hasfullrankandsincethebox-splinesassociatedwitheach of p 1 ; 1 ;p 1 ; 2 :::p 1 ;n +12 arelinearlyindependent,the i ;i =1 ::: 9arealsolinearlyindependent. jjj Together,Lemma 5 andCorollary 2 provethemainTheorem 1 Theorem1 ThenodalfunctionsofLoopsubdivision, i ;i =1 :::n +6 ,arelinearly independentover n Thetheoremsharplycharacterizesthelocalityoflinearin dependence.Onany niteunionof n ` thenodalfunctionsarelinearlydependentforsuciently highvalence.Onlyoncewetaketheuniontothelimit n ,doweobtainlinear independenceofthenodalfunctionsforallpossiblevalenc es. Lemma 4 andLemma 5 implytheanalogousresultforeigenfunctions. Corollary3 Forall n ,the eigenfunctions i i =1 ;:::n +6 ,ofLoopsubdivision arelinearlyindependentandformabasisfortheLoopsubdiv isionfunctionsover n Inparticular,wecannowcalltheLoopeigenfunctionsan eigenbasis 3.5 LinearIndependenceofCatmullClarkNodalFunctions Inthissection,weinvestigateanotherwidelyusedsubdivi sionscheme, Catmull-Clarksubdivision. TheCatmull-Clarkalgorithm[ 4 ]acceptsinputmeshesthathave m -sided facetsandverticeswith n neighbors.However,all m -sidedfacetsaresplitinto m quadrilateralsintherststepasfollows.Anewfacenodeis computedasthe averageofthefacetvertices;anewedgenodeastheaverageo ftheedgeendpoints andthetwonewfacenodesofthefacesjoinedbytheedge;anda newvertexnode ofvalence n iscomputedas ( Q +2 R +( n 3) S ) =n

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31 v u 2N+5 9 5 4 23 12N+7 8 6 7 2N+8 2N+1 2N+6 2N+2 2N+3 2N+4 6 1 23 45 7 82N+7 2N+82N+2 2N+3 2N+4 2N+132N+122N+112N+102N+9 2N+14 2N+15 2N+16 2N+17 2N+6 2N+5 2N+1 9 n 1 n 2 n 3 A Figure3{5:IndicesofCatmull-Clarknodesnearafacetwith oneextraordinary node( n =5)( left ).Theindicesofthenewcontrolpointsafteronesubdivisio n. Threequartersofthedomainnowhavewell-denedtensorpro ductB{splinestructure( middle ).Thecompleterectangulardomainiscomposedofaninnite number ofLshapedregions n ` ( right ) where Q istheaverageofthenewfacenodesofallfacesadjacenttoth eold vertex, R istheaverageofthemidpointsofalloldedgesincidentonth eoldvertex point,and S istheoldvertexpoint.Anewquadrilateralfacetthenconsi stsof consecutiveedgenode,vertexnode,edgenodeandthefaceno de. TherulesareconsistentwiththeCatmull-Clarkstencilsli stedinFigure 2{3 and 2{4 Ifeachnodeofaquadrilateralmeshfacethasvalence n =4,CatmullClarksubdivisionamountstotensorproductbi-cubicsplin esubdivision.Inthis case,thenodalfunctionsarethestandardtensorproductun iformB{splinebasis functionswhoseindependenceiswell-documented.Sinceth eextraordinarynodes (withvalence n 6 =4)arealwaysisolatedaftertwosubdivisionsteps,i.e.an ytwo extraordinarynodesareseparatedbyatleastonenodeofval encefour,wecan focusourlocalanalysisonsurfacepartsadjacenttoasingl eextraordinarynode. Thatis,thesubscript0referstoameshwithisolatedextrao rdinarynodes. Theindicesofthe K :=2 n +8adjacentcontrolpointsarestoredin p 0 asin Figure 3{5 left : p 0 :=( p 0 ; 1 ;:::p 0 ;K ) :

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32 Aftersubdivision,thenewsetof M := K +9controlverticesisorderedas showninFigure 3{5 middle andstoredinthevector: p 1 :=( p 1 ; 1 ;:::;p 1 ;K ;p 1 ;K +1 ;:::p 1 ;M ) : Thesubdivisionrulesareagaindenotedby p 1 = A p 0 where A := 0BBBB@ A 11 0 A 21 A 22 A 31 A 32 1CCCCA : Here A 11 isan(2 N +1) (2 N +1)matrixthatcomputesthenewextraordinary nodeandtheverticesadjacenttoit; A 21 and A 22 determinethesevenverticeswith indices2 N +2 ;:::; 2 N +8,inthemiddlevertexring;and A 31 and A 32 computethe lastnineverticeswithindices2 N +9 ;:::; 2 N +17.Wehaveenoughcontrolpoints in p 1 toevaluatethreeregularpatches(seeshadedareainFigure 3{5 middle ).The rst K controlpointsof p 1 ( p 1 ; 1 ;p 1 ; 2 ;:::;p 1 ;K )= A p 0 := 0B@ A 11 0 A 21 A 22 1CA p 0 ; areusedasthecontrolpointsforthenextsubdivisionstep. UnliketheLoopcase, A canalwaysbediagonalizedbyitseigenvectors V : A = VV 1 : Alleigenvaluesarenonzero,exceptfor n =3whenoneeigenvalueiszero.(The secondeigenvalueofthezeroFourierblock.)For n> 3,thelinearindependenceof thenodalfunctionson A follows,justasinthecaseofLoopsubdivision,fromthe locallinearindependenceoftensor-productsplinesandth efullrankof A .Thefull rankof A 11 implies globallinearindependence for n> 3. Thecase n =3meritscloserscrutiny.

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33 + + + + Figure3{6:GloballineardependenceofCatmull-Clarksubd ivision.Analternativerepresentationofthezerofunctionwith+indicatio nanynonzeronumber and itsnegativevalue( left ) Twocontrolnetswiththeconnectivityofacubebut dierentnodepositions.TheygeneratethesameCatmull-Cl arksurface( right )! Lemma6 ThenodalfunctionsofCatmull-Clarksubdivisioncorrespo ndingtothe graphinFigure 3{6 are(globally)linearlydependent. Proof.Giventhedisplayedchoiceofnonzerovaluesattheve rtices,allnew facenodeshavevalue0andallaveragesoftwooldnodesconne ctedbyanedge havevalue0.Thereforeallnewedgenodeshavevaluezeroand sodothenew vertexnodes:( Q +2 R +( n 3) S ) =n =(0+0+0 S ) = 3=0. jjj Figure 3{6 right ,illustratesdependenceasthenonuniquenessofthecontro lnet foragivensurface.Interestingly,anearlyversionoftheC atmull-Clarksubdivision algorithm,quotedbyDooandSabin[ 8 ],canbeshowntobelocallylinearly independentfor n =3.Hereanewvertexnodeofvalence n iscomputedas ( Q + R +2 S ) = 4. Thenodalfunctionsassociatedwiththemeshafteroneiniti alsubdivision steparelocallylinearlydependenton [ 1i =2 n i when N =3(seeFigure 3{7 ). However,thenodalfunctionsaregloballylinearlyindepen dentsincetheyare linearlyindependentover n 1 (SeeproofinLemma 8 ). Thereforewehave Lemma7 ThenodalfunctionsofCatmull-Clarksubdivisionwithsupp orton A arelinearlyindependentover A

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34 1 -11 1-1 -5 5 -5 1 -1-5 5 -5 25 0 00 0 00 0 0 0000 0 0 0 1 004 0 0 1 0 A Figure3{7:Nonzeroinputcoecientsgeneratingthezerofu nctionon [ 1i =2 n i (shadedarea). 3.6 LocalLinearIndependenceofCatmull-ClarkNodalFunction s Sincethevalence n canbearbitrarybuteachlayerofthesubdivisionfunction correspondingtoaregion n ` isdenedbyanitenumberofB-splinecontrol points,thenodalfunctionsofCatmull-Clarksubdivisionc annotingeneralbe locallylinearlyindependentoveranysubsetof n .Bysymboliccomputation,we canhowevershowindependenceforlowvalences.Thetwondi ngsarerecordedin thefollowingLemma.Lemma8 ThenodalfunctionsofCatmull-Clarksubdivisionarelocal lylinearly independentifandonlyif n =4 Proof.For n =3,wesymbolicallycheckedthat A isoffullrank,andhence thenodalfunctionsareindependentover n 1 .However,thenodalfunctionsarenot linearlyindependentover [ 1i =2 n i duetotheexamplegiveninFigure 3{7 If n =4,thelocallinearlyindependencefollowsfromthelocall inearly independenceoftensorproductB{splines. For n =5,thenodalfunctionsareindependenton n 1 ,(whichimplieslinear independenceon [ k` =1 n ` sincealleigenvaluesarepositive)andonanysubsetof n 1 thatstraddlesatleasttwoofthethreetriangularsubdomai nsof n 1 onwhich thesubdivisionsurfaceisasinglepolynomial.However,on anysingleoneofthe subdomains,thenodalfunctionsarelinearlydependent. For n> 5,thenodalfunctionsarelinearlydependenton n 1 jjj

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35 0 S k 1+ C k 4 k 1 p 1 ; 2 p 1 ; 3 p 1 ; 2 p 1 ; 3 p 0 = u k p 0 = w k Figure3{8:TheB-splinecontrolpoints p 1 ; 2 ;p 1 ; 3 (redpoints)usedtocertifythat theeigenfunctionsassociatedwith u k and w k areindependent. JustasforLoopsubdivision,forany k thereexistsavalence n sothatthe nodalfunctions i ofLoopsubdivisionwithsupporton n k arelocallylinearly dependenton n k andevenon [ k` =1 n ` .Thepatternisasfollows. Conjecture2 For k := n 4 > 0 ,thenodalfunctionsofCatmull-Clarksubdivision i ;i =1 ::: 2 n +8 arelinearlyindependenton [ ki =1 n i butlinearlydependenton [ k 1 i =1 n i Weveriedtheconjecturesymbolicallyupto n =20.Weshowthatthischaracterizationofthelocalnessoflinearindependenceissha rp:oncewetaketheunion ofregionstothelimit n ,thenodalfunctionsarelinearlyindependentregardlesso f valence.Asbefore,werstproveindependenceover n oftheeigenfunctionsdened bythecolumnvectorsin V .Thenweconcludeindependenceofthenodalfunctions forCCsubdivisionover n Lemma9 TheeigenfunctionsofCatmull-Clarksubdivisionarelinea rlyindependentover n Proof.For n> 3,analogoustotheproofofLemma 4 ,wecanreducethe problemtotheindependenceoftheeigenfunctionsassociat edwiththesame eigenvalue.

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36 Accordingto[ 1 2 14 ],theeigenvaluesof A 11 k := 1 16 ( C k +5 p ( C k +9)( C k +1)) ;k =1 ;:::n 1 ; eachhavemultiplicitytwo.Wehave k 6 =0since( C k +5) 2 6 =( C k +9)( C k +1). When k 6 = n= 2,theassociatedeigenvectors u k and w k are[ 30 ] u k = 0BBBBBBBBBBBBBBBBBBBB@ 0 4 k 1 1+ C k (4 k 1) C k C k + C 2 k ... (4 k 1) C ( n 1) k C ( n 1) k +1 1CCCCCCCCCCCCCCCCCCCCA and w k = 0BBBBBBBBBBBBBBBBBBBB@ 00 S k (4 k 1) S k S k + S 2 k ... (4 k 1) S ( n 1) k S ( n 1) k 1CCCCCCCCCCCCCCCCCCCCA ; where C k :=cos(2 k=n )and S k :=sin(2 k=n ).Toshowthatthetwoeigenfunctionsdenedby u k and w k arelinearlyindependent,weconsiderthetensorproductB{splineentries p 1 ; 2 and p 1 ; 3 of p 1 j p 0 = u k and p 1 j p 0 = w k (soliddotsinFigure 3{8 ).Thetwoeigenfunctionsarelinearlyindependentoverthe shadedregionif theygenerateindependentB{splinecontrolpoints p 1 ; 2 and p 1 ; 3 ,i.e.if det 0B@ p 1 ; 2 j p 0 = u k p 1 ; 3 j p 0 = u k p 1 ; 2 j p 0 = w k p 1 ; 3 j p 0 = w k 1CA = 2k det 0B@ 4 k 11+ C k 0 S k 1CA 6 =0 :

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37 Infact, 4 k 1 6 =0 () 1 4 ( C k +5 p ( C k +9)( C k +1)) 6 =1 () ( C k +5) p ( C k +9)( C k +1)) 6 =4 () ( p ( C k +9)( C k +1)) 6 = 1 C k () ( C k +9)( C k +1) 6 =( 1 C k ) 2 () 8 C k +8 6 =0 () C k 6 = 1 : C k 6 = 1and S k 6 =0followsfrom k 6 = n= 2. When k = n= 2,theeigenvectorsof k = 1 4 are u Tk =(0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ;:::; 1 ; 0 ;::: )and w T k =(0 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ;:::; 0 ; 1 ;::: ) then det 0B@ p 1 ; 2 j p 0 = u k p 1 ; 3 j p 0 = u k p 1 ; 2 j p 0 = w k p 1 ; 3 j p 0 = w k 1CA = 2k det 0B@ 1001 1CA 6 =0 : Fortheeigenvaluesof A 22 f 1 8 ; 1 8 ; 1 16 ; 1 16 ; 1 32 ; 1 32 ; 1 64 g ,theeigenfunctionsarethe tensor-productpowerbasisfunctions([ 30 ]) f u 3 ;v 3 ;u 3 v;uv 3 ;u 3 v 2 ;u 2 v 3 ;u 3 v 3 g whosepairwiseindependenceiswellknown. Forthespecialcase n =3,thereisazeroeigenvalue.AsshowninFigure 3{7 ,theassociatedeigenfunctionhaszerovalueson [ 1i =2 n i butzon-zerovalues on n 1 .Wecansingleitoutbyrstlookingatthedomain [ 1i =2 n i .Therestof

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38 8 N N 4 N 5 8 N (Lemma 7 )(Lemma 8 )(Conjecture 2 )(Theorem 2 ) Figure3{9:SummaryofndingsforCatmull-Clarksubdivisi on.Domains G (shaded)andvalence n forwhichthenodalfunctionswithsupporton G arelinearlyindependent.eigenfunctionsarelinearlyindependenttheover [ 1i =2 n i ,withsimilarproofas above.Thatimpliestheoverallindependenceoftheeigenfu nctionsover n jjj Sincethetransformationbetweentheeigenfunctionsandno dalfunctionsare invertible,allthenodalfunctionsarealsolinearlyindep endentandformabasis. Theorem2 ThenodalfunctionsofCatmull-Clarksubdivisionthathave supporton n arelinearlyindependentover n

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CHAPTER4 TIGHTBOUNDINGVOLUMES Inthischapter,wedescribetheecientconstructionofint ervaltrianglesthat enclosethesubdivisionsurface:foreachpieceweboundthe x y and z component ofthelimitsurfaceseparately.Thiscomponentboundiscom putedbylinear combiningofpre-computedboundsofbasisfunctions.Avolu mecreatedfromthe componentboundstoboundthesubdivisionsurface. Atthismoment,wefocusonLoop'ssubdivisionalthoughtheu nderlying approachisindependentofthesubdivisionscheme. 4.1 IntervalTriangles An intervalpoint isanaxisalignedcubewithpossiblydierentsizesonthree dimensions.Itisa"large"pointdenedbythreeintervals: f [ x ;x + ] ; [ y ;y + ] ; [ z ;z + ] g justasaregularpointthatisdenedbythreevalues f x;y;z g Figure4{1:Anintervalpoint Atrianglecanbeviewedasaconvexcombinationofitsthreec ornerpoints. Similarly,an intervaltriangle istheconvexcombinationofthreeintervalpoints(see Figure 4{2 ).Geometrically,theintervaltriangleisapolyhedrallyb oundedvolume in3Dspace(seeFigure 4{2 right ). Weaimtoboundthesubdivisionsurfacewithasetofinterval triangles. 39

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40 Figure4{2:Theintervalpolygondenedbythreeintervalpo ints. 4.2 SubdivisionPatches A subdivisionpatch isthelimitsurfaceofatriangleanditsone-ringof neighborsundersubdivision(seeFigures 4{3 ).Givenaninputmesh,wecan associateeachtrianglewithonepatch.Theunionofallpatc hesisthecomplete subdivisionsurface. Figure4{3:TheLoopsubdivisionpatch Forsimplicityofcomputation,weassumethatatmostoneoft hethree verticeshas n 6 =6neighbors.Otherwise,astepoflocalsubdivisionisperf ormto enforethiscondition. Figure 4{4 showshowtheregularLooppatches(green)separatestheirr egular patches(holes)afteronesubdivision. 4.3 BoundingBasisFunctions Thissectionexplaintherststepofourmethod.Firstweexp lainthe2D domainsoftheLoopsubdivisionpatches.Thenwewilldenea setofbasis functionsforeachdomainlayout.Finallyweboundapairofu pperandlower boundsforthebasisfunctions.

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41 Figure4{4:TheregularLooppatches(green)ofthevenusmod el 3 4 7 8 5 6 11 9 10 0 12 7 2 1 8 0 9 5 7 6 4 2 1 0 8 3 119 10 12 3 46 5 10 k 7 k 5 n 7 n 5 Figure4{5:DomainlayoutofLoopsubdivisionpatches. Notethatwepre-computeandtabulatetheboundsanddomain, inprinciple, nouseroftheapproachneedstoeverbeconcernedwiththeder ivation.Onlythe secondstepneedstobeexecutedforaninputmeshandisvisib letousers. 4.3.1 SubdivisionDomain ThedomainofaregularLooppatch(allthreevertexhasvalen ce=6)is straightforward(Figure 4{5 left ).Thepointsarearrangedinregularpatternwhere everyedgehasthesamelengthandeveryanglehasthesomedeg ree( = 3). Whenoneofvertices(say u 0 )isanirregularnode,thedomainisarrangedin thefollowway: alltheedgesconnectedto u 0 havelength=1 alltheanglesadjacentto u 0 =2 =n u 4 extendsthelinesegment u 0 u 1 by k n

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42 u 6 extendsthelinesegment u 0 u 2 by k n u 5 isthemidpointof u 4 and u 6 u 3 and u 7 arethererectionof u 5 across u 0 u 1 and u 0 u 2 ,respectively. Thevalue k n ischosensothatthesubdivisionlimitofthis2Dmesh(shade d arean n inFigure 4{5 )tstightlyinsidethetriangle n withvertices u 0 u 1 and u 2 Thereasonfordoingthiswillbeevidentinthenextsection. Wewillproceed togivetheformulafor k n Theshapeofn n ,asshowninFigures 4{5 ,changeswith n .Becauseofthe built-insymmetry,thedomainalwaysstayintheareabetwee nray u 0 u 1 and u 0 u 2 Itagreeswith n exactlyif n =6.For n > 6,thedomainbulgesinward,towards theextraordinarynode.For n < 6,itbulgesoutward.Byaligningtheouter mostpointontheboundarycurveofthedomainwiththeline u 1 u 2 ,wehavethe followingformulafor k n : k n := 8>><>>: 4( c 2 2) = (1+2 c 2 )if n 6 ; 6(2 c 2 7) = (15+2 c 2 )if n < 6 ; c :=cos n : 4.3.2 BasisFunctions AsdenedinChapter 3 ,thenodalfunction b i foragivenirregularvalence n (n=6ifthereisnoirregularnode)isdenedasthelimitofth e3Dmeshthat consistsofvertices V j where V j has f x;y g = u j f x;y g and z = ( i;j ).Theindices i and j runfrom0to n +5.Thosenodalfunctionsarelinearlyindependentovern n thereforewecallthembasisfunctions. Figure 4{6 ( left )showsthe b 3 for n =7. 4.3.3 ComputingtheBoundsforBasisFunctions Nowwearereadytocreatetheupperandlowerboundsforbasis functions.

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43 Forthe(one-time)bounding,foreach n ,eachbasisfunctionissubdivided manytimes(wesubdivided7times).Duetotheconvexhullpro pertyofLoop's subdivision,weobtaincorrectlinearupper(lower)bounds ifwecanputaplane sothatitsitsabove(below)thesubdividedcontrolnet.The planesarecarefully chosensotheytightlyenclosethesubdividedcontrolnet(s eeFigure 4{6 right ). Sincethedomainofbasisfunctionn n sitsinsidethetriangle n withvertices u 0 u 1 u 2 ,weonlyneedtostorethe z valuesoftheplaneson u 0 u 1 u 2 Saythevaluesare f b +0 ;b +1 ;b +2 g and f b 0 ;b 1 ;b 2 g ,forupperandlowerbound respectively,thenwehavethefollowingLemma:Lemma10 forany ( u;v ) in n n ,thereexists ( s;t ) suchthat sb 0 + tb 1 +(1 s t ) b 2 b i ( u;v ) sb +0 + tb +1 +(1 s t ) b +2 and 0 s 1 0 t 1 0 (1 s t ) 1 Proof.( s;t )isthebarycentralcoordinateof( u;v )intriangle u 0 u 1 u 2 .Since n n n ,0 s 1,0 t 1,0 (1 s t ) 1. jjj Wethenstoreallthevalues b +0 ;b +1 ;b +2 and b 0 ;b 1 forevery i andpossible valence n intoatable.Wealsostorethe u i positionsforlateruse.Thisgeneration oftablesisonlydoneforonce. Figure4{6:Basisfunction b 3 for n =7anditscontrolpolygon( left ),Theupper andlowerbound(redpolygon)of b 3 (enlarged)( right ).

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44 4.4 ComponentBounds The x -componentoftheLooppatchcanbeexpressedas x ( u;v )= n +5 X i =0 c i b i ( u;v ) where b i ( u;v )istheLooppatchcorrespondingtothecontrolpolygonthat is1at i-thvertexand0elsewhere,and c i isthe x componentofthe i thcontrolpoint. Then,onn n x + ( u;v ):= n +5 X i =0 max f c i ; 0 g b +i ( u;v )+min f c i ; 0 g b i ( u;v ) isanupperboundfor x Sincelinearfunctionsaretheirownbestupperandlowerbou nd,wecan extractalinearfunction ` interpolating c 0 ;c 1 ;c 2 With d i thedierencebetween ` ( u i ;v i )and c i x ( u;v )= ` ( u;v )+ n +5 X i =3 d i b i ( u;v ) : Weremoved i =0 :: 2fromthesummationbecausebyconstruction, d i =0for i =0 ; 1 ; 2. Then,onn n x x x + where(4.1) x + := ` + P n +5 i =3 max f d i ; 0 g b +i +min f d i ; 0 g b i ; x := ` + P n +5 i =3 max f d i ; 0 g b i +min f d i ; 0 g b +i : The d i arelinearcombinationsofthecontrolpoints c i andcanbequickly computedusingthedomainposition u i fromsubsection 4.3.1 .The b +i and b i are directlyreadfromthepre-computedtables.Thelinearboun ds y y + z and z + aredeterminedanalogously.

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45 (c)(d) (a)(b) Figure4{7:(a)Looppatch;(b)cornerintervalpoints i ;(c)convexhullofthe i forminganintervaltriangle(todisplaytheinside,thetop facetisremoved);(d)the enlargedpatchinsideitswire-frameintervaltriangle. 4.5 ConstructingIntervalTriangles Hereisthemainresultsofthissection: Lemma11 Theintervalspoints f [ x i ;x +i ] ; [ y i ;y + i ] ; [ y i ;y + i ] g ,i=0..2,formsa intervaltriangleenclosesthesubdivisionpatch. Proof.Foreach( u;v ) 2 n n ,thereexists( s;t )suchthat sx 0 + tx 1 +(1 s t ) x 2 x ( u;v ) sx +0 + tx +1 +(1 s t ) x +2 where0 s 1,0 t 1,0 (1 s t ) 1.Thesameholdsfor f y ( u;v ) ;z ( u;v ) g .Sopoint f x,y,z g isenclosedbytheaconvexaverageofinterval points f [ x i ;x +i ] ; [ y i ;y + i ] ; [ y i ;y + i ] g ,thereforeenclosedbytheintervaltriangle. jjj Weshowhowtheintervaltrianglesenclosethesubdivisions urfaceandhow closetheyarebyafewexamplesinFigure 4{7 andFigure 4{8 4.6 Semi-SharpCreasesandBoundaries Forenhancedrealism,standardsubdivisionisoftenenhanc edwithdirectional, anisotropicsemi-sharpcreaserules.Forexample,DeRosee tal.[ 7 ]propose applying,alongmarkedmeshlines,afewstepsofthesharpcr easerulesfrom Hoppeetal.[ 13 ],followedbyoneoptionalstepofblendingandinnitelyma ny stepsofstandardsubdivision.

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46 Figure4{8:Thesubdivisionsurface;thesurfaceanditsint ervaltriangles(withthe topfacetremoved);thesurfacewithsemi-transparentinte rvaltriangles(fromleft toright). Figure4{9:Subdivisionsurfacewithsemi-sharpcreases(r ed=creasevalue1.0).

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47 Therearetwostrategiesforcreatingboundingintervaltri anglesforsurfaces withsuchsemi-sharpcreases.Therstistoboundthebasisf unctionscorrespondingtoalldierentcongurationsofcreaseedges.Thi sissimilartoBolzand Schroder[ 3 ],but,sinceweonlyneedupperandlower bounds ,weneednotgenerate boundsforeverycombinationoftwosubdivisionrules.Ifon eruleresultsconsistentlyinhighervaluesthantheother,sayasharpcreaserul eoraboundaryrule basedonunivariatesplines,andagenericsubdivisionrule ,thenitsucestobound theupperfunctionfromaboveandthelowerfunctionfrombel owtoenclosethe wholerangeofcombinations.Nevertheless,thisstrategyl eadstoalargenumberof tables. Thealternativestrategy,usedinFigure 4{9 ,istoapplysubdivisionwiththe sharpcreaseruleonthepatchesinruencedbythecreaseedge untilonlysmooth subdivisionstepsareleftandthestandardboundsapply.If aboundaryisthe resultofgeneratingoneextralayerofnodesonthery,i.e.b ycopyingverticesor rerectingvertices,orifboundariescontractbyonelayer, intervaltrianglescanbe usedwithoutmodication. 4.7 Convergence The thickness ofaintervaltriangleisthemaximumsizeofthethreeinterv al points.Lemma12 Thethicknessoftheboundingintervaltriangleforagivens ubdivision patchgoesto0underuniformsubdivision. Proof.Thethickness t P i d i C where C = max i f max f b +0 b 0 ;b +1 b 1 ;b +2 b 2 g d i 0undersubdivision,therefore t 0. jjj

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CHAPTER5 APPLICATION Wenowtakealookatapplicationsofthenewboundingvolume: interval triangles. 5.1 CollisionDetectionUsingIntervalTriangles 5.1.1 PairwiseInterferenceDetection Sinceoneintervaltrianglecanhaveupto19facets,(seeFig ure 4{7 d), comparingallfacetstodetectinterferenceisnotanecien tapproach.Instead, wereducethetasktoatriangle-triangleintersectiontest byslightlyenlargingthe intervaltriangletoanosettriangle. The basetriangle oftheosettriangleisdenedas ( x + x + 2 ; y + y + 2 ; z + z + 2 )(5.1) restrictedto n .Thelengthofthehalf-diagonalofthe i thcornercubeis l i := r ( x +i x i 2 ) 2 +( y + i y i 2 ) 2 +( z + i z i 2 ) 2 : Therefore,ifwerun(thecenterof)aspherewithradius :=max i =0 ; 1 ; 2 f l i g (5.2) overthebasetriangle,wecreateanosettrianglethatisgu aranteedtoenclosethe Looppatch.Theosettriangleisslightlyenlargestheinte rvaltrianglebutreduces thetasktoatriangle-triangleintersectiontestofthebas etriangles T 1 T 2 withan errorboundsettothesumofthetworadii, 1 2 : dist( T 1 ; T 2 ) 1 + 2 : 48

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49 Forthetest,weadaptMoller'smethod[ 22 ]:ifonetriangleliestoonesideofthe planecontainingtheothertrianglenon-intersectionisre ported;otherwisethe commonlinepassingthroughthetwotrianglesisexaminedan dnon-intersection reportediftheintersectionlineintervalsdonotoverlap. Weneedonlychangeto 1 + 2 + thetolerance thatMollerusestostabilizethecomputation.Thisadds threeadditionstotheapproximately100operationsofMol ler'stest. 5.1.2 IntersectionHierarchy Mostintersectionapplicationshaveanassociatedtoleran ce sothatsurfaces areconsidereddisjointiftheirdistanceismorethan .Toenforcethetolerance,we spliteachpatchintofourbylocalLoopsubdivision,untilt heosettrianglehasa radius lessthan /2.Itisstraightforwardtomodifyanytrianglehierarchyt ouse osettriangles.WemodiedtheOBBtreecodeofGottschalke taletal.[ 9 ]by enlargingthedimensionoftheOBBuntilitcontainseachbas etrianglewithinthe radiustolerance. 5.1.3 PerformanceEvaluation Amajorchallengewhenintroducinganewcollisiondetectio ntechniqueisto conductafairperformanceexperimenttomeasurespaceandt imerequirementsin arealisticratherthanjustaworstcasesetting.Ourtaskis madeeasier,inthatwe neednotcomparedierenttypesofhierarchiesbutonlythet est. Inourcontext,theperformancecriteriaarethenumberofpr imitivesneededto achieveagivenaccuracy(space),andthehierarchyinitial izationandtheaverage intersectioncost(time).Wecomparetheperformanceofano settriangle-based hierarchytoasimilarhierarchybasedonacontrolmeshthat hasbeenuniformly subdividedtoliewithinthesameprescribederrortoleranc e.Aspointedoutearlier (Figure 1{3 )suchameshdoesnotguaranteecorrectintersectiontestin gbutwould neverthelessbeacceptedinmanypracticalsituations.

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50 Figure5{1:Modelsusedforperformanceevaluation. Wedidnotcomparetoanadaptivelyrenedmeshbecausenoec ient adaptiverenementbasedonthemaximumnormisavailable.A ABBsoverestimate somuchthattheadaptiveschemeisnotcompetitive.Theeci entalternative istouseintervaltrianglestodriveadaptation{butthenwe arealmostbackto oursolution.Wealsoconrmedinourtestscenario,whichri gidlytransformsthe objects,thatAABBtreesbuiltontheconvexhullofthecontr olpolygondonot performwellcomparedtotheOBBtree[ 32 ]. 5.1.4 ExperimentSetUp Weusedsixdierentmodelsofsmalltomediumsizeasissuita bleforfurther subdivision.Themodelsrangefrom24triangles(Star)to14 18(Venus)andare showninFigure 5{1 .InthespiritofKlosowskietal.[ 15 ],weplacealwaystwo ofthesemodelsintoacubeorroom,eachwithrandomposition andrandom orientation.Allmodelsarescaledtotightlytintoabound ingboxofsize1. Theinterferencetestreturnsseparation,orpossiblecoll isionandonlytherst collisionpair.Sincetheintersectioncostdependsonthep ositionandorientationof theinputmodels,weran150,000randomtestsforeachpairan dreporttheaverage time.Wealsovariedtheroomsizetomodifythepercentageof intersectingcases.

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51 Table5{1:Numberoftrianglesneededforthegivenerrorbou nds. 2% 1% 0.5% i-t Loop i-t Loop i-t Loop Venus 5705 22688 6098 22688 7214 90752 Head 1499 3200 2474 12800 5990 51200 Pawn 1216 1216 2368 4864 4732 4864 Star 384 384 888 1536 1536 1536 Demon 1258 4384 2734 4384 4897 17536 Quake 1728 6528 4218 6528 6636 26112 Table5{2:Intersectionhierarchy(OBBtree)creationtime inmilliseconds. 2% 1% 0.5% i-t Loop i-t Loop i-t Loop Venus 180 561 180 551 240 2414 Head 40 70 80 310 180 1342 Pawn 30 20 60 110 140 110 Star 10 10 20 30 40 40 Demon 40 100 90 170 150 431 Quake 50 150 100 100 200 641 5.1.5 Results Inthefollowingtables,welabelthenewintervaltrianglebasedmethod"i-t" andtheintersectionbasedonsubdividedmeshes"Loop".The timingismeasured onasingleP42.4GHZCPUmachinewith1GRAM. Table 5{1 liststhenumberoftrianglesneededtoguaranteeagivenacc uracy. Thatis,apossibleintersectionisannouncedifthelimitsu rfaceswithinthecubicle arecloserthan2%,1%,respectively0.5%oftheobjectsize. Theosettrianglebasedmethodrequiresfewertrianglesinallcases,indicat ingthatthetightbounds payo. Table 5{2 liststhetimeusedtobuildtheOBBhierarchy.Thisincludes the computationofallintervaltrianglesinthecaseoftheose ttriangle-basedmethod. Nevertheless,theosettriangle-basedmethodrequiresle sstimeinmostcases.The averagesavingsare30%.

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52 Table5{3:Intersectioncostinmsfordierenttolerancesa ndroomsizes. 2% 1% 0.5% i-tLoop i-tLoopi-t Loop Roomsize6.0:3percentofthemodelscollide Venus/Head 2224 222622 27 Star/Pawn 1112 111212 12 Demon/Quake 1313 131414 14 Roomsize3.0:30percentofmodelscollide Venus/Head 147178 144178148 187 Star/Pawn 8191 849184 90 Demon/Quake 105116 107117106 118 Table 5{3 liststheaverageintersectiontimeforbothapproaches.Fo raroom ofsize6 3 ,osettrianglesimproveonlymarginallyovertheLoopmeth odbecause mostrejectionsoccurinhighlevelbox-boxtestsoftheOBBt ree.However,fora roomofsize3,morebox-boxtestsareneededforLoopthanfor osettriangles. Thatis,thesafeosettriangle-basedtestischeaperthant heunsafecontrol mesh-basedtest! 5.2 InnerandOuterHull Forintersectiontesting,itisnotnecessarytobuildanexp licitconforming innerandouterhull,sincetheunionoftheintervaltriangl esenclosesthesurface. However,forotherapplicationssuchasmanufacturingwith tolerances,intersection andoverlapoftheintervaltrianglesisnotacceptable.Ins tead,weneedapairof triangulationsthatsandwichthelimitsurface.Tothisend ,werstselectfromthe eightchoices p ;;r :=( x ;y ;z r ) ;;;r 2f ; + g ; computedinSection 4 ,foreachmeshtriangletwoplanesthattightlyenclosethe limitsurface.Then,weshrink-wraptheindividualplanesa roundeachnode. Thetwoplanesareselectedasfollows.Computethenormalof thelinear functions p ;;r .Ifthesignsofthe xyz -componentsofthenormalagreewith ( ;;r )thentheplaneistheouterplane.Ifthesignsagreewith( ; ; r )

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53 Figure5{2:Thelimitsurface,theouterhullandasuperposi tionofthelimitsurfaceandtheouterhullwithacutouttoshowthepositionofth elimitsurface(from left).thentheplaneistheinnerplane.Intheory,therecouldbeca seswhereweneedto subdividetoassureuniqueness,butinpracticewehaveneve rencounteredsucha case.Allthepointsinthetrianglecellliebetweenthesetw oplanesbecausetheir innerproductwiththeouterplaneisnegativeandtheirinne rproductwiththe innerplaneispositive. Inthesecondstep,wecompute,foreachnode,anoutertriang ulationvertex. Thevertexisthepointofleastdistancetothenode'slimitp ositionandsuch thatitliesoutsidetheouterplanesofthefacetssurroundi ngthenode.Thisisa simplelocalquadraticoptimizationproblemthatcanbesol vedbyenumeration. Connectingtheoutertriangulationverticesaccordingtot heinheritedconnectivity oftheinputmeshcompletestheconstruction(seeFigure 5{2 )[ 32 ][ 27 ]. 5.3 AdaptiveSubdivision Foragivenobject,Thedensityoftheverticesvariesfromre giontoregion.For exampleinFigure 1{2 secondpicture,theheadlightofthevehicleisdenseenough whilethebodyisyetcoarseandneedstobefurthersubdivide d. Anadaptivesubdivisiononlyperformtherenementatthepl aceswhere needed.Thethicknessofourintervaltriangles,giveanacc urateestimatehow

PAGE 67

54 Figure5{3:Resultsofadaptivesubdivisiononthedeermode l.Input;e=0.5%; e=0.1%;andthesurfacewithe=0.1%(fromlefttoright). Figure5{4:Ray-tracedimagesat800x600resolution.Input mesh(671triangles), ray-tracingtime:8s( left).Adaptivesubdivisionwithe=0.2%.Resultingnumber of triangles=9662andmaximumsubdivisionlevel:4,ray-trac ingtime:9s(middle). Uniformlysubdivided4times.Resultingnumberoftriangle s=171776,ray-tracing time:12s(right).theshapeofthepatchisawayfromarattriangle.Thereforea newadaptive subdivisioncanbebuilt,withourboundingvolumeasthe"te ststone". Figure 5{3 andFigure 5{4 showstheresultsoftheadaptivesubdivisionand theray-tracedimageofthesubdivisonsurfaces[ 33 ].

PAGE 68

CHAPTER6 CONCLUSION Asafe,accurateandecientboundingvolume:intervaltria ngleshasbeenintroduced.Itisappliedtointersectiontestingandadaptiv erenderingofsubdivision limitsurfacesandisproveneective.Alternativeapproac hes,suchascomparing renedcontrolmeshes,prisms[ 16 ]orhierarchiesofAABBsfaileithertobeecient,ortobecorrectorboth.Thealgorithmissimple:read pre-tabulateddata andformlinearcombinationsaccordingtoequation( 4.1 ).Thisapproachrequires onlyasimplemodicationofavailablesoftware. Thisthesisgivesaframeworkforconstructingsuchtightbo undingvolumes forsubdivisionsurfaces,bydecomposingthesurfaceintop atches,thepatchesinto combinationsofbasisfunctions,andpre-computingandtab ulatingofthebounds forthesebasisfunctions.Asubtlepointhereistoestablis hthebasispropertyof thefunctionsthatdenesubdivisionsurfacelocally. AcloserlookatSection 4 showsthattheapproachworksforanysurface parametrizationthatcanbeboundedandisadaptivelyrena ble.Thereforeitcan beappliedtoNURBSsurfaces(ifthedenominatorisboundeda wayfromzero)and tootherrenementschemes,sayinterpolatorysubdivision .Thekeyingredientin eachcase,istheone-timegenerationofaccurateboundsana logoustoSection 4.3 Thepropertiesthattheboundingvolumepossessesmakeitpr omisingfor manyotherapplications. 55

PAGE 69

REFERENCES [1]A.A.BallandD.J.T.Storry.Conditionsfortangentplan econtinuityover recursivelygeneratedB-splinesurfaces. ACMTrans.onGraphics ,7:83{102, 1988. [2]A.A.BallandD.J.T.Storry.Aninvestigationofcurvatu revariationsover recursivelygeneratedB-splinesurfaces. ACMTransactionsonGraphics 9(4):424{437,October1990. [3]JeBolzandPeterSchroder.Rapidevaluationofcatmul l-clarksubdivision surfaces.In ProceedingsoftheWeb3D2002Symposium ,pages11{18,New York,2002.ACMPress. [4]E.CatmullandJ.Clark.RecursivelygeneratedB-spline surfacesonarbitrary topologicalmeshes. ComputerAidedDesign ,10:350{355,1978. [5]FehmiCirak,MichaelOrtiz,andPeterSchroder.Subdiv isionsurfaces:Anew paradigmforthin-shellnite-elementanalysis. Internat.J.Numer.Methods Engineering ,47:2039{2072,2000. [6]C.deBoor.,K.Hollig,andS.Riemenschneider. Boxsplines ,volume98of AppliedMathematicalSciences .Springer-Verlag,NewYork,1993. [7]TonyDeRose,MichaelKass,andTienTruong.Subdivision surfacesin characteranimation.InMichaelCohen,editor, Siggraph1998conference proceedings ,pages85{94,NewYork,1998.ACMPress. [8]D.DooandM.Sabin.Behaviourofrecursivedivisionsurf acesnearextraordinarypoints. Computer-AidedDesign ,10:356{360,September1978. [9]S.Gottschalk,M.C.Lin,andD.Manocha.OBBTree:Ahiera rchicalstructure forrapidinterferencedetection. ComputerGraphics ,30:171{180,1996. [10]EitanGrinspun. TheBasisRenementMethod .PhDthesis,California InstituteofTechnology,May162003. [11]EitanGrinspun,PetrKrysl,andPeterSchroder.CHARM S:Asimple frameworkforadaptivesimulation.InJohnHughes,editor, Siggraph2002 ConferenceProceedings ,AnnualConferenceSeries,pages281{290,NewYork, 2002.ACMPress. 56

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57 [12]EitanGrinspunandPeterSchroder.Normalboundsfors ubdivision-surface interferencedetection.InThomasErtl,KenJoy,andAmitab hVarshney, editors, ProcVisualization ,pages333{340,NewYork,2001.IEEE. [13]HuguesHoppe,TonyDeRose,TomDuchamp,MarkHalstead, HubertJin, JohnMcDonald,JeanSchweitzer,andWernerStuetzle.Piece wisesmooth surfacereconstruction. ComputerGraphics ,28(AnnualConferenceSeries):295{ 302,July1994. [14]K.Karciauskas,J.Peters,andU.Reif.Shapecharacter izationofsubdivision surfaces{casestudies. Comput.AidedGeom.Design ,21(6):601{614,July 2004. [15]JamesT.Klosowski,JosephS.B.Mitchell,HenrySowizr al,andKarelZikan. Ecientcollisiondetectionusingboundingvolumehierarc hiesofk-DOPs. IEEETransactionsonVisualizationandComputerGraphics ,4(1):21{36, January1998. [16]L.Kobbelt.Tightboundingvolumesforsubdivisionsur faces.InBobWerner, editor, Pacic-Graphics'98 ,pages17{26,NewYork,1998.IEEE. [17]LeifKobbelt,KatjaDaubert,andHans-PeterSeidel.Ra ytracingofsubdivisionsurfaces.In RenderingTechniques'98(ProceedingsoftheEurographics Workshop) ,pages69{80,NewYork,1998.Springer-Verlag. [18]CharlesT.Loop.Smoothsubdivisionsurfacesbasedont riangles,1987. Master'sThesis,DepartmentofMathematics,Universityof Utah. [19]MichaelLounsbery,TonyD.DeRose,andJoeWarren.Mult iresolutionanalysis forsurfacesofarbitrarytopologicaltype. ACMTransactionsonGraphics 16(1):34{73,January1997. [20]D.LutterkortandJ.Peters.Optimizedrenableenclos uresofmultivariate polynomialpieces. Comput.AidedGeom.Design ,18(9):851{863,2001. [21]D.LutterkortandJ.Peters.Tightlinearboundsonthed istancebetweena splineanditsB-splinecontrolpolygon. NumerischeMathematik ,89:735{748, May2001. [22]TomasMoller.Afasttriangle-triangleintersection test. JournalofGraphics Tools:JGT ,2(2):25{30,1997. [23]A.H.Nasri.Polyhedralsubdivisionmethodsonfree-fo rmsurfaces. ACM TransactionsonGraphics ,6(1):29{73,1987. [24]JorgPetersandUlrichReif.Thesimplestsubdivision schemeforsmoothing polyhedra. ACMTransactionsonGraphics ,16(4):420{431,October1997.

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58 [25]JorgPetersandUlrichReif.Shapecharacterizationo fsubdivisionsurfaces{ basicprinciples. Comput.AidedGeom.Design ,21(6):585{599,July2004. [26]JorgPetersandXiaobinWu.Ontheoptimalityofpiecew iselinearmax-norm enclosuresbasedonslefes.In Proceedingsofthe2002StMaloConferenceon CurvesandSurfaces ,pages335{344,2003. [27]JorgPetersandXiaobinWu.Slevesforplanarspline curves. Comput.AidedGeom.Design ,21(6):615{635,2004. http://authors.elsevier.com/sd/article/S01678396040 00615. [28]UlrichReif.Auniedapproachtosubdivisionalgorith msnearextraordinary vertices. Comput.AidedGeom.Design ,12(2):153{174,1995. [29]JosStam.EvaluationofLoopsubdivisionsurfaces,199 8.SIGGRAPH98 Proceedingsnotes. [30]JosStam.Exactevaluationofcatmull-clarksubdivisi onsurfacesatarbitrary parametervalues.InMichaelCohen,editor, Siggraph98Proceedings ,pages 395{404,NewYork,1998.AddisonWesley. [31]LuizVelhoandDenisZorin.4-8subdivision. Comput.AidedGeom.Design 18(5):397{427,June2001. [32]XiaobinWuandJorgPeters.Interferencedetectionfo rsubdivisionsurfaces. ComputerGraphicsForum,Eurographics2004 ,23(3):577{585,2004. [33]XiaobinWuandJorgPeters.Anaccurateerrormeasuref oradaptivesubdivisionsurfaces.In ProceedingsofTheInternationalConferenceonShape ModelingandApplications ,pages51{57.IEEEComputerSociety,2005.

PAGE 72

BIOGRAPHICALSKETCH XiaobinWuwasborninZhejiang,China,onMay13th,1976.Her eceived hisB.E.fromUniversityofScienceandTechnologyofChinai n1996.In1999,he receivedhisM.S.fromChineseAcademyofSciences.

PAGE 73

IcertifythatIhavereadthisstudyandthatinmyopinionitc onformsto acceptablestandardsofscholarlypresentationandisfull yadequate,inscopeand quality,asadissertationforthedegreeofDoctorofPhilos ophy. JorgPeters,ChairProfessor,DepartmentofComputerandInformationScienceandEngineering IcertifythatIhavereadthisstudyandthatinmyopinionitc onformsto acceptablestandardsofscholarlypresentationandisfull yadequate,inscopeand quality,asadissertationforthedegreeofDoctorofPhilos ophy. MeeraSitharam,ViceChairAssociateProfessor,DepartmentofComputerandInformationScienceandEngineering IcertifythatIhavereadthisstudyandthatinmyopinionitc onformsto acceptablestandardsofscholarlypresentationandisfull yadequate,inscopeand quality,asadissertationforthedegreeofDoctorofPhilos ophy. DavidGroisserAssociateProfessor,DepartmentofMathematics IcertifythatIhavereadthisstudyandthatinmyopinionitc onformsto acceptablestandardsofscholarlypresentationandisfull yadequate,inscopeand quality,asadissertationforthedegreeofDoctorofPhilos ophy. BabaVemuri

PAGE 74

Professor,DepartmentofComputerandInformationScienceandEngineering IcertifythatIhavereadthisstudyandthatinmyopinionitc onformsto acceptablestandardsofscholarlypresentationandisfull yadequate,inscopeand quality,asadissertationforthedegreeofDoctorofPhilos ophy. JayGopalakrishnanAssistantProfessor,DepartmentofMathematics

PAGE 75

ThisdissertationwassubmittedtotheGraduateFacultyoft heCollegeof EngineeringandtotheGraduateSchoolandwasacceptedaspa rtialfulllmentof therequirementsforthedegreeofDoctorofPhilosophy.August2005 PramodP.KhargonekarDean,CollegeofEngineering WinfredM.PhillipsDean,GraduateSchool

PAGE 76

Ecient,TightBoundingVolumesforSubdivisionSurfacesXiaobinWu(352)392{1255DepartmentofComputerandInformationScienceandEnginee ring Chair:JorgPetersDegree:DoctorofPhilosophyGraduationDate:August2005 Smoothmathematicalsurfacesenablethedesingersandengi neerstomanufacturesomeofthemostimportantitemsinourdailylivessucha ssmoothcarbodies, fuelecientairplanesandevencomputermice. Subdivisionsurfacesisasuchkindofsimpleandyetpowerfu lmathematical toolforgraphicsapplicationsandcomputeraidedgeometri cdesign.Mostnoticeableusageofsubdivisionsurfacesarecharacterscreatedi nPixaranimationfeature lmssuchas"Abug'slife"or"Toystory". Thisthesisintroducesaframeworkforconstructingtightb oundingvolumes forsubdivisionsurfaces.Thecompletesurfacecanbeviewe dasacollectionof smoothlyjoinedpatcheswheretightboundingvolumesareco mputedeciently. Thetightboundingvolumescomputedarecrucialforgraphic sandgeometric algorithmssuchasinterferencedetection,adaptivetesse llationandrendering.


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EFFICIi T, TIGHT BOUNDING VOLULi L FOR SUBDIVISION SURFACE '


By

XIAOBIN WU















A Di i k, Z VTION PERF -TED TO THE GRADUATE .CiOL
OF THE UNIVi i i .' OF FLORIDA IN PARTIAL FLFI i i
OF THE REQUIREMENTS FOR TIE DEGRii OF
DO(C OR OF Pil i.OSOPi'".


UNIVERSITY OF FLORIDA


































Copyright 2005

by

Xiaobin \Vu
















1 dedicate this work to :, wife Fangwen Ci and ,: !' :'ents.















AC OWLEDGMENTS

I wish to thank i : advisor Jorg Peters for his support and mentoring 'I: 1:-

out : Ph.D. study. I also thank him for introducing me into the field <. !.cs

and geometric design. His remarkable dedication and excellence ... research has

been a strong i: : :

I wish to thank V vice committee chair Meera Sitharam for her guidance.

MIany formal .-: :: : in seminars and casual talks with Dr. Sitharam have led

to expanding and deepening of >-.- research goals.

I also wish to thank ;: Ph.D. committee members: Dr. Baba C. Vemuri,

David Groisser and Dr. -- Gopalakrishnan for their continuous help and encour-

agement.

I thank my family for always 1:. there for me. Without them, this work will

not be possible.















TABLE OF CONTENTS


ACKNOWLEDGMENTS .......

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

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

KEY TO ABBREVIATIONS .....

KEY TO SYMBOLS ..........

ABSTRACT .............

CHAPTER

1 INTRODUCTION ........

1.1 Subdivision Surfaces ..
1.2 Bounding Volumes ....
1.3 Related work .......
1.4 Overview ..........

2 SUBDIVISION SCHEMES ....

2.1 B-Spline Subdivision ..
2.2 Catmull-Clark Subdivision
2.3 Doo-Sabin Subdivision .
2.4 Midedge Subdivision ..


2.5 Loop Subdivision


3 NODAL FUNCTIONS FOR SUB

3.1 Nodal Functions .....


. . .


DIVISIONSURFACES.


3.2 Linear Independence of Nodal Functions ............
3.3 Linear Independence of Loop Nodal Functions .........
3.4 Local Linear Independence of Loop Nodal Function ......
3.5 Linear Independence of Catmull Clark Nodal Functions .
3.6 Local Linear Independence of Catmull-Clark Nodal Functions

4 TIGHT BOUNDING VOLUMES .. ................

4.1 Interval Triangles . . . . . . .
4.2 Subdivision Patches .. ....................


page


- -









4.3 Bounding Basis Functions .................. ..... 40
4.3.1 Subdivision Domain .................. ..... 41
4.3.2 Basis Functions .................. . .. 42
4.3.3 Computing the Bounds for Basis Functions . ... ..42
4.4 Component Bounds .................. ....... .. 44
4.5 Constructing Interval Triangles .................. .. 45
4.6 Semi-Sharp Creases and Boundaries ................ .. 45
4.7 Convergence .................. ........... .. 47

5 APPLICATION. .................. ............ .. 48

5.1 Collision Detection Using Interval Triangles . . 48
5.1.1 Pairwise Interference Detection ....... . . 48
5.1.2 Intersection Hierarchy ................ .. .. 49
5.1.3 Performance Evaluation ............. .. .. 49
5.1.4 Experiment Set Up .................. ..... 50
5.1.5 Results . . . . . . ... .. 51
5.2 Inner and Outer Hull .................. ..... .. 52
5.3 Adaptive Subdivision .................. ..... .. 53

6 CONCLUSION .................. ............. .. 55

REFERENCES ................... ... ... ........ .. 56

BIOGRAPHICAL SKETCH .................. ......... .. 59















LIST OF TABLES
Table page

5-1 Number of triangles needed for the given error bounds. .. 51

5-2 Intersection hierarchy (OBBtree) creation time in milliseconds ... 51

5-3 Intersection cost in ms for different tolerances and room sizes ... 52















LIST OF FIGURES
Figure page

1-1 A NURBS surface and its control polygon. Note the 3-sided corner is
divided into 3 quadrilaterals because of the NURBS topology con-
strain . .. . . .. .. ... ...... 1

1-2 The input mesh, subdivided once, twice, and the limit surface ... 2

1-3 Interference detection based on the control mesh rather than the limit
surface is neither safe nor accurate. The green and the blue sur-
faces are di -i ii but their control meshes collide. The green and
the orange surfaces collide but their control meshes are disjoint. .. 4

1-4 The counter example of the Kobbelt et al. method . . .. 5

2-1 A cubic B-spline. ............... ......... 10

2-2 The cubic B-spline control polygon (red) is subdivided two times. 11

2-3 The tensor product bicubic B-spline subdivision ......... 11

2-4 Stencil of Catmull-Clark at irregular nodes, where A has valence n. 11

2-5 The regular and irregular stencils for Doo-Sabin subdivision surface. 12

2-6 The stencil for midedge subdivision .................. 13

2-7 3-Directional box spline. The regular domain grid and the quartic
box spline function. .................. .... 14

2-8 Loop subdivision stencils. .................. ..... 14

3-1 One nodal function for Loop subdivision ............... .15

3-2 Summary of findings for Loop subdivision. Domains G (shaded) and
valence n for which the nodal functions with support on G are lin-
early independent. .................. .... 18









3-3 Labeling of the submesh that defines a triangular surface piece (schemat-
ically represented by the shaded area) near an extraordinary node
(label 1) (top,left). Refined submesh, Apo (top,right). Refined sub-
mesh, Apo, used to evaluate the next spline ring (1,.. /..I/ I. ft). The
domain Q of the composite triangular surface piece consists of an
infinite sequence of quadrilateral (chopped triangle) subdomains.
The first three such subdomains, fQ, f22, 3, are shaded (bottom,right).
. . . . . . . . . 2 0

3-4 The box-spline control points pl,i (solid dots) used to certify that pairs
and triples of eigenfunctions are independent. . . 27

3-5 Indices of Catmull-Clark nodes near a facet with one extraordinary
node(n = 5) (left). The indices of the new control points after one
subdivision. Three quarters of the domain now have well-defined
tensor product B-spline structure (middle). The complete rectan-
gular domain is composed of an infinite number of L shaped re-
gions fe (right) ............... ......... .. 31

3-6 Global linear dependence of Catmull-Clark subdivision. An alterna-
tive representation of the zero function with + indication any nonzero
number and its negative value (left)Two control nets with the
connectivity of a cube but different node positions. They generate
the same Catmull-Clark surface (right)! ............. .. 33

3-7 Nonzero input coefficients generating the zero function on Ui2 Qi (shaded
area). .................. ............... ..34

3-8 The B-spline control points p1,2,P1,3 (red points) used to certify that
the eigenfunctions associated with uk and Wk are independent. 35

3-9 Summary of findings for Catmull-Clark subdivision. Domains G (shaded)
and valence n for which the nodal functions with support on G are
linearly independent. .................. ..... 38

4-1 An interval point ............... .......... .. 39

4-2 The interval polygon defined by three interval points. . ... 40

4-3 The Loop subdivision patch ................ .... 40

4-4 The regular Loop patches (green) of the venus model . ... 41

4-5 Domain layout of Loop subdivision patches. .. . ..... 41

4-6 Basis function b3 for n = 7 and its control pI..l -on (left), The upper
and lower bound (red pI..l.-on) of b3 (enlarged) (right). ...... ..43









4-7 (a) Loop patch; (b) corner interval points Di; (c) convex hull of the
Oi forming an interval triangle (to di-pl iv the inside, the top facet
is removed); (d) the enlarged patch inside its wire-frame interval
triangle. .................. .............. ..45

4-8 The subdivision surface; the surface and its interval triangles (with
the top facet removed); the surface with semi-transparent interval
triangles (from left to right). ................ ..... 46

4-9 Subdivision surface with semi-sharp creases (red = crease value 1.0).. 46

5-1 Models used for performance evaluation. .............. 50

5-2 The limit surface, the outer hull and a superposition of the limit sur-
face and the outer hull with a cutout to show the position of the
limit surface (from left). .................. ..... 53

5-3 Results of adaptive subdivision on the deer model. Input ; e 0.5''
e=0.1 and the surface with e 0.1 (from left to right). . 54

5-4 R-Bi-traced images at 800x600 resolution. Input mesh (671 triangles),
r ,--tracing time: 8s (left). Adaptive subdivision with e=0.,.'. Re-
sulting number of ti,.:,l.' = .i'' .' and maximum subdivision level:
4, ,r ;-Iracing time: 9s (middle). Unifoi i,,nl subdivided 4 times. Re-
sulting number of t,',igl = 171776, ',r;,- racing time: 12s (right). 54















KEY TO ABBREVIATIONS


AABB: ; Aligned Bounding Box

B-P Boundary Representation

dop: Discrete Orientation P.' -topes

NURBS: Non-U :: .: : Rational B-Spline

OBB: Oriented Bounding Box















KEY TO SYMBOLS

c, control points . . . . . . . . 43

2, subdivision .I .h i. :'ameter domain ................... ... 41

, nodal functions ....................... . ..... 15

b, basis functions ..... ... 42

j), B .';: basis f actions .................... . ...... 9















Abstract of Dissertation Presented to the Graduate School
of the University of i iorida in Partial J : i : : ,, of the
ReI.'. .. f ;. for the Degree of Doctor of P1.'1 .1

S BO G VOL Li ; FOR SUBDIVI ON i ii \

By

Xiaobin Wu

August

Ci .': J6rg Peters
Major Department: Computer and Information Science and Engineering

Subdivision surfaces -e a simple and --- pf .p .tool for graphics

Sand : :' aided geometric design. Ti j: the gap between

polyhedral and spline modeling and have matured to an i... ....' ...1 high-end

modeling method. Being able to bound the subdivision surfaces is crucial for

common tasks such as accurate rendering or intersecting surfaces.

T :: thesis gives a : :::: )k for constructing tight bounding volumes for

sub(. on "... .-- decomposing the surface into 1 the .. into

combinations of basis : :tions, and pre-computing and tabulating of the bounds

for these basis functions. A subtle point here is to establish the basis i :w. crty of

the -.tions that define subdivision surface locally.

We :1 the : vcncss (." these bounding volumes with concrete algorithms

for interference detection, adaptive tessellation and rendering. Prior to this thesis,

no correct and t : ::. algorithm existed for these ::















CHAPTER 1
INTRODUCTION

The research area of computer graphics and computer aided geometric design

(CAGD) revolves around the representation of objects in the 3D space. Although

mostly physically solid, objects are often represented by their boundary surfaces

called B-Rep. A B-Rep may consists of a set of polygons or of smoothly curved

surfaces as shown in Figure 1-1.

Complex curved objects are called "free-form" surfaces in contrast to func-

tional surfaces and also to point out the increased free control of the shape. A

designer can easily edit the surface while maintaining its smoothness.

The most commonly used free-form surfaces in CAD are called NURBS

(Non-Uniform Rational B-Spline) surfaces. NURBS surfaces are defined as the

image of a smooth mapping from a planar domain, often rectangular, into the 3D

space. Consequently, an object has to be first divided into sub-surfaces and control

structures that are homeomorphic to rectangles (see Figure 1-1), a process that is

often not automatic and hence cumbersome for designers.













Figure 1-1: A NURBS surface and its control p" .Iv.-on. Note the 3-sided corner is
divided into 3 quadrilaterals because of the NURBS topology constrain.


















Figure 1-2: The input mesh, subdivided once, twice, and the limit surface.


1.1 Subdivision Surfaces

Subdivision control meshes give more freedom to the designer, however at a

cost. Given an input mesh with arbitrary connectivity, a surface is defined as the

limit of a refinement sequence. This refinement is often performed as a combination

of inserting new points and splitting old faces. The positions of new points (called

control nodes) are functions of the a set of "close-by" old points; the positions

of old points can also be modified. See Figure 1-2 for an example of subdivision

surface.

Subdivision surfaces were first introduced by Doo and Sabin [8] and Catmull

and Clark[4] in 1978. In these two pioneering papers, methods are -ii--. -1. 'I to

generalize the uniform subdivision of tensor-product B-splines surfaces to meshes

of arbitrary topology. With some delay, due to insufficient computing on the

1980s, a "zoo" of subdivision methods has been created by researchers for various

applications. In section 2, we give a brief introduction and provide the rules and

properties of these subdivision schemes.

1.2 Bounding Volumes

Free-form surfaces provide flexible means to model objects with smooth

boundary so that designers can concentrate on the shape of the object without

worrying about the underline smoothness of the surface.

However, the underlying higher order mathematical representation of free-form

surface imposes difficulties on some common operations, for example detecting









the intersection of surfaces. The problem of finding the intersection curves of two

bi-cubic B6zier patches, for example, is equivalent to solving three polynomial

equations with four variables and degree 6. No analytic solution can be sought

in general, and even numerical methods need to be carefully designed to achieve

acceptable results.

For subdivision surfaces, the situation is yet more complex: the subdivision

surface is defined based on recursion and lack of a closed-form formulation over the

whole domain. Although there exist methods to evaluate a point given any param-

eter value [30, 29], it is not possible to write a subdivision surface as an analytic

expression: finding the intersection leads to an infinite number of equations.

Bounding volumes for the subdivision surfaces offer a simple and robust

solution to the problem. A bounding volume is a subset in 3D space that encloses

the surface piece of interest. A bounding volume is good if it:

has simple shape

tightly encloses the surface

is efficient to compute

is able to further refine

The first two properties address the geometric properties and the shape quality

of a bounding volume. A bound volume is created to simplify the computation, so

its geometry should be simpler than the original surface. And we do not want to

introduce too much error while bounding the surface, so it should .,illl enclose

the surface.

To be useful in graphics and CAD applications, it is essential that the bound-

ing volume is efficiently computable. Finally, to provide error control, it is impor-

tant to allow further refinement when necessary.

Up to date, there are few serious attempts to build bounding volumes for

subdivision surfaces. One of the reasons is that for most applications in computer
























Control mesh Limit surface

Figure 1-3: Interference detection based on the control mesh rather than the limit
surface is neither safe nor accurate. The green and the blue surfaces are dli-i. .in
but their control meshes collide. The green and the orange surfaces collide but
their control meshes are disjoint.

graphics, the accuracy of the operation is not a concern. One would subdivide the

initial mesh several steps and stop either when the visual error is no longer visible

or the memory and computing power can no longer process further refinements.

However, such a process treats the subdivided mesh as simple collection of

]" Ivs.-ons and neglects the fact that the refined control mesh is a linear combination

of the initial mesh and has an inherited structure. This naive viewpoint leads to

an unnecessary waste of time and space. With a proper bounding volume, the

computation can be vastly optimized.

Furthermore, to use the subdivision surfaces in areas such as CAD, users

require accurate measure on the smooth limit surface. For example, a user may ask

if the two surfaces are away by certain distance. Methods based on control meshes

will no longer be correct because the control mesh does not give a safe volume that

contains the surface! Figure 1-3 shows that interference of the control meshes does

not imply that the limit surfaces intersect, and separation of the control meshes

does not imply that the limit surfaces are disjoint!











---- --^ ---



Figure 1-4: The counter example of the Kobbelt et al. method

We introduce a bounding volume construction method for subdivision surfaces
that has properties (a) (b) (c) (d) and has been proven to be effective in number of
applications.
The following sections give a short preview on the related work and summary
of the new method.
1.3 Related work
In 1998, Kobbelt et al. [16, 17] introduced a bounding volume for subdivision
surface called bounding prism. They looked at the maximum positive and negative
distance that the surface can vary in a certain direction. Afterwards, by shifting
the interpolating triangle in the according distance, they created a volume in 3D
space to bound the surface.
Although their calculation of the maximum distances is correct, they left out
the other two directions that the surface can change to the sides of the prism.
As the result, their bounding prism fails to bound the surface when the surface
contains high curvature. See Figure 1-4 for a counter example.
For non-interpolating subdivision schemes, the subdivision surface is bounded
by the convex hull of input mesh. However, it is expensive to compute the convex
hull for a 3D mehs.
Grinspun and Schroder [12] used Axis Aligned Bounding Boxes (AABB)
on the input mesh to bound the surface. AABBs generally introduce large error









margin and converge slowly under subdivision. More importantly, the AABBs need

to be re-computed under rotation.

Lutterkort and Peters [20, 21] introduced tight enclosures for univariate and

multivariate B-spline functions. Most recently Peters and Wu [27] built a safe

bounding polygon for univariate curve. Yet the general bounding volume for

surfaces is not clear. The tightness of their bounds for cubic functions is discussed

in Peters and Wu [26].

There are several mature intersection hierarchies for p" ..I vonal meshes.

In 1996, Gottschalk et al. [9] introduced Oriented Bounding Boxes (OBB). It

is proven to work more efficiently than AABBs because it tightly encloses the

mesh and converges faster. In 1998, Klosowski et al. [15] generalized AABB and

described another bounding hierarchy called K-dops. This method works well for

objects with continuous motion.

1.4 Overview

The approach introduced in this thesis combines ideas from envelopes for

r --tracing [16], tight bounds on splines [20, 21].

At each moment, we consider a part of the surface that corresponds to one

particular control face. This part of the surface(called patch) only depends on a

neighborhood of vertices. We create a bounding volume (called interval t, .: l.,/

see definition in section 4) to bound this patch based on the position of this set of

relative vertices.

We follow Kobbelt [16] in that we use a min-max expression of interval

arithmetic to create bounds. Our overall approach differs, however. We do not

bound just the normal direction; instead we consider each component in {x, y, z} as

a function and we bound each function separately.

We also do not perform the eigendecomposition of the subdivision mask in

Stam [30] and Grispun and Schroder [12]. In addition, we are able to create a









bounding volume much tighter than axis-aligned bounding boxes (AABBs) on the

control mesh. To improve accuracy and efficiency, we remove linear components

from the subdivision function as in Lutterkort and Peters [21] before computing the

bounding volume.

Our method has two 1i ii ri steps, explained in Section 4.3 and Section 4.5,

respectively. First, we build the local parametrization on the subdivision patch,

and upon that we construct a set of basis functions. For any given input values on

the control mesh, the limit function will be a linear combination of this set of basis

functions. In this step, we also find the a pair of upper and lower bounds for each

basis function and store them into a table.

In contrast to Stam [30], which parametrizes over the unit square, we use the

parametrization that has n-gon symmetry to simplify our computation.

In the second step, interval t, .:,,l,. are constructed. These interval triangles

are created by linearly combining the upper and lower bounds of the basis that we

stored in the table. Each interval triangle bounds one piece of the limit surface,

and the union of interval triangles encloses the whole surface. If a pre-defined

accuracy bound e is given, the interval triangles are locally, adaptively refined until

the thickness is less than e.

We [32] used our bounding volume in the application of interference detection

by integrating it with the well used intersection hierarchy such as OBB and K-

dops. We only made a small change to the OBB tree and k-dops tree code. In

return we achieved considerable improvement on the intersection test between the

subdivision surfaces.

The rest of the thesis is organized as follows: C'! lpter 2 reviews the c~. iri .11, -

used subdivision schemes. In chapter 3, we introduce the basis functions of subdi-

vision surfaces. In chapter 4, we explain how the interval triangle is constructed.







8

In chapter 5 we examine a few applications that take advantage of the constructed

tight bounding volume. ('! Ilpter 6 is a summary.














CHAPTER 2
SUBDIVISION SCHEMES

In this chapter we will review a few commonly used subdivision schemes. The

purpose of this section is also to define the terminology that we will use in the rest

of thesis.

2.1 B-Spline Subdivision

We consider the univariate (curve) case. A uniform B-spline function is defined

as the linear combination of a set of uniformly spaced basis. Specifically,



f(t) : C '(t) (2.1)

where cis are scalar values called coefficients and Qis are functions called B-spline

basis functions. Qi are merely the translation of basis Qo at integer grid i. Figure

2-1 shows an example of a B-spline function. A pair of functions x(t) and y(t)

defines a curve in 2D.

The idea of subdivision is to re-write B-spline function fi into a refined basis

'1. In fact, O1s are spaced half as the original Q, i.e. It turns out that we can

directly write each Q in terms of the new basis '1.

For degree 2:

40i = + 3 3 1 '1
21_1 + 3i + 3, [+ 1

For degree 3:


Si -= 1i_2 -+ 1 ',1 ,1
_Y2+' +16 +t it+ K I_.











2

1.5

1

0.5


1


Figure 2-1: A cubic B-spline.


Now we can find the new set of coefficients c's satisfying


f(t) : c'. (t)


by substituting Q by 1 in equation (2.1).

For degree 2,


4ci := ci- + 3ci;

4c2i+1 :=3ci + Ci+1


For degree 3 (see Figure 2-2 left),


8c2i := Ci-1 + 6c, + ci+l

2c'2i+ := Ci + Ci+


Figure 2-2 right shows how the cubic B-spline control polygon is subdivided

two times. As we can see, the subdivided control p" I. -on rapidly converge to the

the black curve.















0.5


1 6t 1 \

1 1



Figure 2-2: The cubic B-spline control p" ..I-on (red) is subdivided two times.

1 6 1
1-6-1
1-6-1 1--1
6 36- 6
1- 6- 1 1- 1
1 6- 1

Figure 2-3: The tensor product bicubic B-spline subdivision


2.2 Catmull-Clark Subdivision

Catmull and Clark[4] extended the subdivision formula for cubic B-spline

into surface by tensor-producting the rule (see Figure 2-3). The numbers are the

weights of each vertex contributing to the new position (blue). The final result is

normalized by the sum of the weights. This collection of weights indicating the

contributing vertices is called the stencil for the new vertex.

To apply the subdivision scheme on an arbitrary mesh, they --'i- -- I1 a stencil

to the vertex with other than 4 neighbors (see Figure 2-4).

1


\A
\ A- 6 A 4n2 7n.
l 6 1

Figure 2 4: Stencil of Catmull-Clark at irregular nodes, where A has valence n.









a0-- On- 1
9 3




3 -1 a2

Figure 2-5: The regular and irregular stencils for Doo-Sabin subdivision surface.


Such a node where the regular spline subdivision rule does not apply is called

extraol.:,.'.,i; node or .:,. jl;,1, node. In the case of Catmull-Clark, it is equivalent

to the node that doesn not have 4 neighbors. Number of neighbors is called

valence. Therefore for Catmull-Clark, valence / 4 = irregular.

The limit of the subdivision creates a C2 surface except at irregular nodes,

where the surface is C1.

2.3 Doo-Sabin Subdivision

Doo and Sabin subdivision is a subdivision scheme based on tensor product

bi-quadratic spline. By tensoring the rules for quadratic spline from Section 2.1, we

get the stencil in Figure 2-5 left.

For the irregular case where the face does not have 4 vertices (Figure 2-5

right), Doo and Sabin -it-i-. -. 1 c0o = 1/4 + 5/4n and ai = (3 + 2cos(2ir/n))/4n

where n is the number of vertices in the faces.

2.4 Midedge Subdivision

In 1997, Peters and Reif [24] -tr-.--- -i. an even simpler subdivision with

minimum stencil size. The new vertex is the the average of the two midpoints

of the neighboring edges (see Figure 2-6). The new node only depends on three

vertices and no irregular rule is needed. This subdivision scheme is sometime called

the simplest subdivision scheme.









2.5 Loop Subdivision

In 1987, Loop [18] introduced a scheme based on triangulation. Loop's scheme

is built on the 3-directional box spline (for more on box spline theory, see de Boor's

book [6]). The 3-directional box spline is defined on a regular triangular domain

where every vertex has 6 neighbors (see Figure 2-7).

The regular rule can be derived from the recursion of box-spline basis (see

Figure 2-8 left). The irregular (valence / 6) rules is -ii-,- -I .1 by Loop as in Figure

2-8 right, where a n= (3O2co(2/n)) 1)


2 1




1


Figure 2-6: The stencil for midedge subdivision































12345
1 2 3 4 5



Figure 2-7: 3-Directional box spline. The regular domain grid and the quartic box
spline function.


1-- 1


13



1


regular stencil


1 1


1 s 1




irregular stencil


Figure 2-8: Loop subdivision stencils.















CHAPTER 3
NODAL FUNCTIONS FOR SUBDIVISION SURFACES

In this chapter, we define nodal functions for subdivision surfaces. We examine

their linear independence properties over different type of domains.

3.1 Nodal Functions

In most graphics and geometric applications, each control point is given as a 3-

dimensional point with (x, y, z) coordinates. Since subdivision rules are applied on

each dimension individually, we can focus on one dimension a time. Each control

point can be viewed as a scalar value instead of a vector.

The nodal function Xi are defined by setting the scalar control point ai to 1

and all others to 0 and applying subdivision (see 3-1). The nodal functions can be

called Basis functions only after they are shown to be linearly independent.

3.2 Linear Independence of Nodal Functions

A number of publications have tacitly assumed that the nodal functions Xi

are linearly independent. Without proof, nodal functions are called subdivision

basis functions [11, 10, 5], used as scaling functions to form a 'basis' of the coarsest

level of a multiresolution hierarchy [19], and used to [23] fit subdivision surfaces by

allowing one interpolation condition for each mesh node.







Figure 3: One nodal function for Loop subdivision

Figure 3-1: One nodal function for Loop subdivision









In fact, for Catmull-Clark this assumption is false. For the simplest quadrilat-

eral control mesh, a cube, the eight nodal functions are globally linearly dependent

(see Lemma 6): in general, we cannot fit eight arbitrary data points by adjusting
the coefficients ai of the corresponding surface i 1 aiXi.

For the well-known tensor-product spline functions, global linear independence

may be interpreted as linear independence over the union of domain rectangles

delineated by the knot lines and joined by identifying edges of the rectangles in the

natural fashion. This definition generalizes to subdivision surfaces as follows. Let

f2 be a unit square (triangle) if the kth facet of the control mesh has 4 vertices (3

vertices). Let F be the union of all domains (2, k), indexed by their control mesh

facet index, with edges topologically identified (set equal) if the facets share edges.

This gives F the structure of a 2-manifold homeomorphic to the control mesh.

Global linear independence is linear independence with respect to F.

Definition 1 (Global linear independence) A set of nodal functions are

globally linearly independent if they are independent over the domain i,,.;.: ../.4 F.

That is, if Vu G F: a i(u) = 0 then ai = 0.

While some of the numerical methods require only standard (global) linear

independence, others, such as local Hermite interpolation and localized multi-

resolution, rely on stronger notions of independence. We need to analyze indepen-

dence on certain ring-shaped annuli A and on subsets Qi of the unit square or unit

triangle f2. The strongest and most subtle notion of independence is local linear

independence.

Definition 2 (Local linear independence) A set of nodal functions are locally

linearly independent if for :u; bounded open G, all the nodal functions having some

support in G are linearly independent on G.

Remarkably, for box-splines and B-splines, the standard notion of (global)

linear independence is equivalent to local linear independence [6]. That is, if all









coefficients ai have to vanish so that Zi ai = 0 (global linear independence),

then the coefficients of all nodal functions that are nonzero over any open set G

have to vanish if >i aixi vanishes on G (local linear independence). Since G can

be arbitrarily small, local linear independence is a stricter requirement on the

nodal functions than global linear independence. We will see that local and global

independence are not equivalent for subdivision nodal functions near extraordinary

nodes of higher valence (number of neighbors). This observation that provides

rare insight into the structural difference between subdivision and spline surfaces.

Specifically, we show that for the Catmull-Clark and Loop subdivision

mathlist]val@n, valence of an extraordinary node

(i) the nodal functions are globally linearly independent;1

(ii) the nodal functions are linearly independent over an annulus1 such as in Figure

3-2, left;

(iii) for valence n higher than the 'regular' valence, the nodal functions are not

locally linearly independent;

(iv) the nodal functions are linearly independent on each domain Q naturally

associated with one facet of the control net.1

The above characterization is an analogue of the Schoenberg-Whittney

theorem of spline interpolation. To illustrate why such a detailed characterization

is useful in practice, consider the following scenarios for interpolation with Loop

subdivision surfaces. Interpolating 12 data points on a domain Q corresponding to

a mesh triangle whose vertices all have valence six, is a well-posed problem with a

unique solution. However, if one of the vertices has valence n = 3 then the problem



1 with one exception: Catmull-Clark applied to nodes with valence n = 3; see
Lemma 6















VN N < 6 N < 7 VN
(Lemma 1) (Lemma 3) (Conjecture 1) (Theorem 1)

Figure 3-2: Summary of findings for Loop subdivision. Domains G (shaded) and
valence n for which the nodal functions with support on G are linearly indepen-
dent.

is overconstrained while for n > 6 it is typically underconstrained. If we match

the number of interpolation conditions to the number of nodal functions that are

nonzero on f2, i.e. if we specify n + 6 interpolation points, we find that, if the

points belong to a subregion f2 (shaded area in Figure 3-2, labelled Conjecture

1), the problem is overconstrained for n > 6. Interpolation with the Catmull-Clark

subdivision follows a similar pattern with an additional complication for n = 3.

The analysis is made easier by the fact that the component functions of most

popular subdivision schemes, and in particular of both Catmull-Clark and Loop

subdivisions, are variations of the well-understood box-spline subdivision [6]; much

of the subdivision limit surfaces, corresponding to quads with 4-valent vertices,

respectively triangles with 6-valent vertices are 'regular', i.e. are spline surfaces

generated by box-splines. This box-spline connection should make us cautious since

the shifts of box-splines are, in general, not linearly independent. For example, the

four-direction (quincunx) subdivision, which gives rise to 4-8 subdivision [31], has

dependent nodal functions. Catmull-Clark subdivision rules generalize the two-

direction box-spline rules, i.e. the rules of the bicubic tensor-product spline; and

Loop subdivision generalizes a three-direction box-spline, the convolution of the

linear 'hat' function, with itself. Fortunately, for both splines we know [6] that the

nodal functions form a basis. Therefore, it suffices to analyze submeshes that define

the neighborhood of extraordinary nodes, where the connectivity of the control









mesh differs from the regular connectivity of the box spline, namely mesh nodes of

valence n / 4 for Catmull-Clark meshes and of valence n / 6 for Loop meshes.

Thus characterizing independence for Loop subdivision and Catmull-Clark

subdivision, closes a gap in the theory of generalized subdivision and provides a

basis for computational use. We first discuss Loop subdivision, and then Catmull-

Clark subdivision, since Loop subdivision is the simpler of the two and therefore

shows the structure of the proof more clearly. Also, it is the subdivision scheme

-.-., -i, ,1 for computational purposes in [11, 10, 5].

3.3 Linear Independence of Loop Nodal Functions


For Loop subdivision, there are only two rules: to compute new nodes,

corresponding to edges of the old mesh, and to compute new nodes, corresponding

to old nodes. These rules are expressed by the two stencils shown in Figure 2-8. A

node of a Loop mesh is extrao,.l.:,,; if it does not have six neighbors.

Due to the small reach of the rules, a submesh consisting of one triangle and

all triangles attached to it defines, by going to the limit, a triangular piece of the

surface .,.i ,i:ent to the limit of the extraordinary node. If all nodes of the central

triangle are of valence six, the surface is a polynomial piece of a three-direction box

spline and its properties are well understood. Since new edge nodes have valence

six, extraordinary nodes are more and more isolated under refinement, and we can

focus on triangles with one extraordinary node of valence n / 6. In the following,

the subscript 0 refers to a mesh where any two extraordinary nodes are separated

by at least one node of valence six. This may be the result of one subdivision

applied to the original mesh.

The relevant submesh defining the triangular surface piece consists of K

n + 6 nodes that can be labeled as in Figure 3-3 (top, left). We store the submesh

as a vector


po : (po,, ...PO,K) CRK.













A
-- ---I


u= O,v 0






u=0,v 1


u= ,v= 0


Figure 3-3: Labeling of the submesh that defines a triangular surface piece
(schematically represented by the shaded area) near an extraordinary node (la-
bel 1) (top,left). Refined submesh, Apo (top,right). Refined submesh, Apo, used
to evaluate the next spline ring (1, .11..'ii ft). The domain Q of the composite tri-
angular surface piece consists of an infinite sequence of quadrilateral (chopped
triangle) subdomains. The first three such subdomains, f21, f22, 23, are shaded
(bottom, right).


Subdivision generates a new set of M := K + 6 control vertices as shown in

Figure 3-3 (top, right). We store those control vertices in a new vector


pl : (Pl,i,... ,P1,K,P1,K+1, .. P1,M).


If we represent the averaging rules as rows of a M x K matrix A (with row

sum one), then the subdivision rules to compute the vector pl from po are


p, = Apo where A :


All 0

A21 A22

A31 A32)









Here All is an (N + 1) x (N + 1) matrix that computes the new extraordinary

node and the vertices .,1i ,i:ent to it (note that this also holds for an optional initial

refinement to generate po from a mesh that has neighboring extraordinary nodes);

A21 and A22 determine the five vertices with indices N+4, N+3, N+2, N+5, N+6

of the next l I.-r; and A31 and A32 define the six outermost nodes. The sizes of A22

and A32 are 5 x 5 and 6 x 5 respectively. Leaving out the direct neighbors of the

extraordinary node, p1,4,P1,5,... ,P1,n 1, the remaining control points


p x:= (Pl,1,pl,2,P1,3,Pl,n, .. .P1,M)

define three triangular polynomial pieces shown as shaded in Figure 3-3 (top,right).

To compute the nodes of the next subdivision step, we need only the first K

control points of p, (see Figure 3-3 bottom left),

(A1 0 )
(p1,i,p1,2, ,PI,K) = Aopo = Al Po.
A21 A22

By repeating the process, an infinite sequence of piecewise polynomial rings

is generated. We can choose their domains f2 so that their union fills out the

triangular domain 2:

1 1
Q : {(u,v)Iu+v+w= 1,u,v,w> 0}> Uo,, 1:= 2\-Q g+i :- -Q.
2 2

The control vertices p, after n subdivision steps that determine the function on Q2

are:

p = A(Ao)~-lpo, n > 1. (3.1)

From the recursion in Equation 3.1, it is evident that the eigenstructure of

A p1 i' a crucial rule when determining the properties of the subdivision surfaces

such as the computation of the limit position, tangent plane, and shape analysis

[8, 1, 28, 25, 14].









Using Fourier transform, it is easy to derive the vector of eigenvalues Anl of

An,
5
All := [1, a(n), f(1),..., f(n 1)]
8
where
3 + 2cos(27rk/n) 5
f(k) ), n) :a() f(k)f(k),
8 8
and A22 of A22:
11 1 1 1
A22 ti [ ]i
2 8' 8'8' 16' 16
Except for the case n = 3, A can be diagonalized by the matrix V of its

eigenvectors (details of the eigenanalysis of A can be found e.g. in Stam [29]):


A VAV-1, A diag(All, A22), V U (3.2)
Ui W1

where the submatrices Uo and Wi are the eigenvectors of All and A22, respec-

tively. For n > 3, the columns of V are linearly independent vectors in RK.

Now let the initial submesh po := vi be a eigenvector associated with eigen-

value Ai and pi the corresponding linear combination of nodal functions. Then,

after n steps of subdivision,


p.lov = A(AO)n-lv AA'-lv A Av, A 1-
Si P 10 pov.

Therefore pi(f[i+l) is a scaled multiple of (o(f2i). Precisely,


V(u, v) E and Vn > 1, p( ) = -v AX (u, v). (3.3)

In Stam [29] these K functions pi are called eigenbasis. However, .,i i:ent to

an extraordinary node, each pi consists of an infinite union of polynomial pieces.

The subtle but important point to be settled here is that, although the columns of

V are independent, the corresponding functions can be dependent. We therefore

call the functions pi .:'. ,. functions. To characterize subdivision near extraordinary









nodes as similar to, but distinct from spline representations, we will show that the

eigenfunctions are linearly independent over f2, but linearly dependent on certain

subsets of f2.

To show that the nodal functions of Loop subdivision are (globally) linearly

independent, we focus on subdomains that form an annulus surrounding the

preimage of a sequence of extraordinary nodes. With the obvious topological

identification of edges to induce the structure of a 2-manifold with boundaries, we

define an annulus as n copies of f2i,


A:= {1,...,n} x Q1.


Lemma 1 The nodal functions of Loop subdivision with support on A are linearly

independent over A.


Proof. Assume that f := p,,. i is zero on all of A. Recall that the subset

of nodes pi can be interpreted as a regular three-direction box-spline control net

defining three polynomial pieces near the extraordinary node. Since the box-splines

are locally linearly independent [6], all box-spline control points defining f on A are

zero and, in particular, Apo. Since all eigenvalues of A are positive, A is of full

rank and therefore all po,i must be zero. If po was the result of one refinement, also

the original mesh nodes must be zero since the matrix Anl is of full rank. III


Now consider the nodal functions corresponding to the mesh after one subdi-

vision. Lemma 1 proves linear independence of these nodal functions on the union

of all annuli A associated with original mesh nodes. Since the matrix All is of full

rank, also the original control nodes must be zero if the function vanishes on all

annuli.

Corollary 1 The nodal functions of Loop subdivision are ill,,1,ill.i linearly indepen-

dent.









For V(u, v) E Q and Vn > 1, ( #)

(3.3), we have

U V
2"' 2v)


^AT o(u, v)


==. Oj(u, )


E Q2, with the above equation and equation


n+6
S2 2n y
i j+1
n+6
A A'> (m, v)
i= +1
n+6

i j+1


Since ()' 0 as n -- o unless Ai Aj,


j(u, V) 1 / (", v)


must hold. Therefore the eigenfunctions associated with Aj must be linearly

dependent. In the remainder of the proof, we show this to be false. In other words,

the problem of proving the linear independence of all eigenfunctions has been

reduced to the independence of the eigenfunctions with the same eigenvalue.

Because of the eigenstructure of A, the multiplicities of its eigenvalues are

small (at most four) and do not increase with n. Recall that the eigenvalues of A

are
5 11 1 1 1
[1, a(),f (1),..., f(n 1), 1 1 1

where
5 (3 + 2cos(27/n))2 3 + 2cos(27k/n)
a() : ,f(k) :
8 64 8

To find the repeated eigenvalues, we observe that for k E {1... n- 1},

1. f(k)= f(n- k),

2. if n is even and k = n/2, f(k) = ; otherwise f(k) ( {1, 1}.

3. f(k) / 1 and f(k) / -(n),
















po = U P0 0 Yl Po Z1

Figure 3-4: The box-spline control points pl,i (solid dots) used to certify that pairs
and triples of eigenfunctions are independent.

That is, if A is an eigenvalue of A with multiplicity greater than one then A

f(k) / 1, or A = or A = In particular, all relevant eigenvalues are nonzero.
We look at each case individually.
Case 1: A f(k) / '
In this case A has multiplicity 2 and the associated eigenvectors uk and Wk

are given in [29]:

Uk( (0,1, Ck, C2k, C(n-)k,...) and

Wk = (0 0 k, S2k,..., S(n-l)k ***)

where Ck := cos(2rk/n) and Sk : sin(2rk/n). To show the two eigenfunc-

tions defined by Uk and Wk are linearly independent, we consider the two
box-spline entries of p1,2 and p1,3 (solid dots in Figure 3-4, left) after one
step of subdivision applied to the mesh po := uk and one step applied with

po := wk. The two corresponding eigenfunctions are independent because


det P1,21po=uk P1,3 pU det 1 Ck / 0,
P1,21po=wk P,31po=wk 0 Sk

since S(k) / 0 because f(k) / and hence k / "
Case 2: A -









In this case A can have multiplicity of 3 or 4. We first show that the eigen-
0
functions corresponding to first three columns yi, Y2, y3 of are
W1
independent. The eigendecomposition (3.2) of A22 is [29]:

00100

10101

Wi 1 0 0 0 0

01110
0 1 1 1 0

01000

The independence of the eigenfunctions follows from the independence of the

three box spline control points pi,n+8, p1,n+9, i,n+1o (solid dots in Figure 3-4

middle) after one subdivision:

Pl,n+8 po-=y P1,n+9 po=yi Pl,n+10 po=yi 4 4 0
1
det Pl,n+8 po=y2 P1,n+9po=y2 Pl,n+10po=y2 det 0 0 0.

Pl,n+8 po=Y3 P1,n+9 Po=y3 P1,n+10 po=y3 1 4 4i

If the multiplicity of is three, then we are done. Otherwise, A = f(n/2) for

n even and we have one additional eigenvector uk from After one
U1
subdivision, the box-spline control point p1,3 is zero for yi, y2, y3 and nonzero

for uk. This proves independence of all four eigenfunctions.

* Case 3: A = 1

The eigenvectors of the two eigenfunctions associated with correspond to

0
the last two columns zi and z2 of Pairwise independence follows
S i
from the independence of the two box spline control points pl,,+8,Pln+o10










(solid dots in Figure 3-4 right)


1 ,n+8 po=zi Pl,n+10p o=z\ 1 0 3

Pl,n+8 po=z2 P1,n+1po=z2 z3

This completes the proof of Lemma 4. 111

We can now address our original goal of showing that the nodal functions Xi

are linearly independent.

Corollary 2 For n > 3, the nodal functions of Loop subdivision, Xi, i = 1... n + 6,

are linearly independent on 2.

Proof. Recall that each nodal function Xi is generated by subdivision when

setting control point i to 1 and all others to 0. Their independence follows from


[1,-., PK]= V[Xi,...,XK], K N+6,


and the fact that, for n > 3, the matrix V of eigenvectors is an invertible matrix.

ill

For the special case n = 3, the matrix A has a non-trivial Jordan block and

can not be diagonalized. However, since the number of the nodal functions is small,

namely nine, we need not decompose into the eigenspace.


Lemma 5 For n = 3, the nodal functions of Loop subdivision, Xi, i = ... n + 6,

are linearly independent on 2.

Proof. We explicitly determine the (n + 12) x 9 matrix M that maps po to the

box-spline control points pox

7 3 3 3 00000
6 6 2 2 00000
6 2 6 2 00000
6 2 2 6 00000
2 6 0 6 20000
1101 1 1100
1 2 6 6 0 00200
1 1 11010011
16 2 0 6 6 00002
0 6 0 2 62000
0 6 0 0 26200
0 6 2 0 02600
0 2 0 6 60020
0 0 0 6 20062
-0 0 2 6 00026









Since M has full rank and since the box-splines associated with each of pli, p1,2 ... P1,n+12

are linearly independent, the X, i = 1... 9 are also linearly independent. |II

Together, Lemma 5 and Corollary 2 prove the main Theorem 1.

Theorem 1 The nodal functions of Loop subdivision, Xi, i 1... n + 6, are linearly

independent over f.

The theorem sharply characterizes the locality of linear independence. On any

finite union of Qf the nodal functions are linearly dependent for sufficiently

high valence. Only once we take the union to the limit f2, do we obtain linear

independence of the nodal functions for all possible valences.

Lemma 4 and Lemma 5 imply the analogous result for eigenfunctions.

Corollary 3 For all n, the eigenfunctions pi, i = 1,... n + 6, of Loop subdivision

are linearly independent and form a basis for the Loop subdivision functions over

Q.

In particular, we can now call the Loop eigenfunctions an eigenbasis.

3.5 Linear Independence of Catmull Clark Nodal Functions


In this section, we investigate another widely used subdivision scheme,

Catmull-Clark subdivision.

The Catmull-Clark algorithm [4] accepts input meshes that have m-sided

facets and vertices with n neighbors. However, all m-sided facets are split into

m quadrilaterals in the first step as follows. A new face node is computed as the

average of the facet vertices; a new edge node as the average of the edge endpoints

and the two new face nodes of the faces joined by the edge; and a new vertex node

of valence n is computed as


(Q + 2R + (n 3)S)/n









2N+1
9 2 3 2N+8

82N+7 A2 2-7 i

7 6 2N+6+ 4
2N 13 7N 12 2N 11 0 21 I9

2N+5 2N+4 2N+3 2N+2

Figure 3-5: Indices of Catmull-Clark nodes near a facet with one extraordinary
node(n = 5) (left). The indices of the new control points after one subdivision.
Three quarters of the domain now have well-defined tensor product B-spline struc-
ture (middle). The complete rectangular domain is composed of an infinite number
of L shaped regions f2 (right)


where Q is the average of the new face nodes of all faces .,.i i:ent to the old

vertex, R is the average of the midpoints of all old edges incident on the old vertex

point, and S is the old vertex point. A new quadrilateral facet then consists of

consecutive edge node, vertex node, edge node and the face node.

The rules are consistent with the Catmull-Clark stencils listed in Figure 2-3

and 2-4.

If each node of a quadrilateral mesh facet has valence n = 4, Catmull-

Clark subdivision amounts to tensor product bi-cubic spline subdivision. In this

case, the nodal functions are the standard tensor product uniform B-spline basis

functions whose independence is well-documented. Since the extraordinary nodes

(with valence n / 4) are albv-- isolated after two subdivision steps, i.e. any two

extraordinary nodes are separated by at least one node of valence four, we can

focus our local analysis on surface parts .,.i ,i:ent to a single extraordinary node.

That is, the subscript 0 refers to a mesh with isolated extraordinary nodes.

The indices of the K := 2n + 8 .1i i.:ent control points are stored in po as in

Figure 3-5, left:


po : (Po,0, 1...PO,K-)









After subdivision, the new set of M := K + 9 control vertices is ordered as

shown in Figure 3-5, middle and stored in the vector:


l := (P1,1,... ,PI,K,PI,K+I, ..P1,M).


The subdivision rules are again denoted by

All 0

pi = Apo where A := A21 A22

KA31 A32J

Here All is an (2N + 1) x (2N + 1) matrix that computes the new extraordinary

node and the vertices .,,li i,:ent to it; A21 and A22 determine the seven vertices with

indices 2N+ 2,..., 2N+8, in the middle vertex ring; and A31 and A32 compute the

last nine vertices with indices 2N + 9,..., 2N + 17. We have enough control points

in pi to evaluate three regular patches (see shaded area in Figure 3-5, middle). The

first K control points of pi,

(A1 0 )
(pI,1,p1,2, .. ,P1,K) = Aopo : A 0 po,
A21 A22

are used as the control points for the next subdivision step. Unlike the Loop case,

A can ahv--l- be diagonalized by its eigenvectors V:


Ao VAV-1.


All eigenvalues are nonzero, except for n = 3 when one eigenvalue is zero. (The

second eigenvalue of the zero Fourier block.) For n > 3, the linear independence of

the nodal functions on A follows, just as in the case of Loop subdivision, from the

local linear independence of tensor-product splines and the full rank of A0. The full

rank of All implies /gI /1.rl linear independence for n > 3.

The case n = 3 merits closer scrutiny.









+





Figure 3-6: Global linear dependence of Catmull-Clark subdivision. An alter-
native representation of the zero function with + indication any nonzero number
and its negative value (left)Two control nets with the connectivity of a cube but
different node positions. They generate the same Catmull-Clark surface (right)!

Lemma 6 The nodal functions of Catmull-Clark subdivision corresponding to the

g,,il'h in Figure 3 6 are I(,il.1,,lli) linearly dependent.

Proof. Given the di-1pl -i ,1 choice of nonzero values at the vertices, all new

face nodes have value 0 and all averages of two old nodes connected by an edge

have value 0. Therefore all new edge nodes have value zero and so do the new

vertex nodes: (Q + 2R + (n 3)S)/n = (0 + 0 + OS)/3 = 0. 11


Figure 3-6,right, illustrates dependence as the nonuniqueness of the control net

for a given surface. Interestingly, an early version of the Catmull-Clark subdivision

algorithm, quoted by Doo and Sabin [8], can be shown to be locally linearly

independent for n = 3. Here a new vertex node of valence n is computed as

(Q + R + 2S)/4.

The nodal functions associated with the mesh after one initial subdivision

step are locally linearly dependent on UiL2 Qi when N 3 (see Figure 3-7).

However, the nodal functions are globally linearly independent since they are

linearly independent over 21 (See proof in Lemma 8).

Therefore we have

Lemma 7 The nodal functions of Catmull-Clark subdivision with support on A

are linearly independent over A.









1 1 5

1 1 1 5 oA


5 5 25

Figure 3-7: Nonzero input coefficients generating the zero function on Uj2
(shaded area).


3.6 Local Linear Independence of Catmull-Clark Nodal Functions

Since the valence n can be arbitrary but each li. r of the subdivision function

corresponding to a region Qf is defined by a finite number of B-spline control

points, the nodal functions of Catmull-Clark subdivision can not in general be

locally linearly independent over any subset of 2. By symbolic computation, we

can however show independence for low valences. The two findings are recorded in

the following Lemma.

Lemma 8 The nodal functions of Catmull-Clark subdivision are 7.'.. ill linearly

independent if and only if n = 4.

Proof. For n = 3, we symbolically checked that A is of full rank, and hence

the nodal functions are independent over Q1. However, the nodal functions are not

linearly independent over Ui2 Qi due to the example given in Figure 3-7.

If n = 4, the local linearly independence follows from the local linearly

independence of tensor product B-splines.

For n = 5, the nodal functions are independent on i1, (which implies linear

independence on U.L-A since all eigenvalues are positive) and on any subset of

21 that straddles at least two of the three triangular subdomains of Q1 on which

the subdivision surface is a single polynomial. However, on any single one of the

subdomains, the nodal functions are linearly dependent.

For n > 5, the nodal functions are linearly dependent on Qi. |||


















po = uk P0 Wk
Figure 3-8: The B-spline control points p1,2,P1,3 (red points) used to certify that
the eigenfunctions associated with uk and Wk are independent.

Just as for Loop subdivision, for any k there exists a valence n so that the

nodal functions Xi of Loop subdivision with support on 2k are locally linearly

dependent on 2k and even on Uel-A1. The pattern is as follows.

Conjecture 2 For k := n- 4 > 0, the nodal functions of Catmull-Clark subdivision

Xi,i = 1... 2n + 8 are linearly independent on U lRi but linearly dependent on



We verified the conjecture symbolically up to n = 20. We show that this char-

acterization of the localness of linear independence is sharp: once we take the union

of regions to the limit Q2, the nodal functions are linearly independent regardless of

valence. As before, we first prove independence over Q of the eigenfunctions defined

by the column vectors in V. Then we conclude independence of the nodal functions

for CC subdivision over 2.

Lemma 9 The 1. :,,functions of Catmull-Clark subdivision are linearly indepen-

dent over Q2.

Proof. For n > 3, analogous to the proof of Lemma 4, we can reduce the

problem to the independence of the eigenfunctions associated with the same

eigenvalue.








According to [1, 2, 14], the eigenvalues of An,

Ak: (Ck+ 5 (Ck+9)(Ck+1)), k =,...n -,

each have multiplicity two. We have Ak / 0 since (Ck + 5)2 / (Ck + 9)(Ck + 1).
When k / n/2, the associated eigenvectors uk and Wk are [30]

0 0
4Ak- 1 0
1 + k Sk
(4Ak- 1)Ck (4Af- 1)Sk
Uk and Wk k =
Ck + C2 Sk + S2k


(4Ak 1)C(, )k (4Ak 1)S(_n-)k
C(n-l)k + 1 / S(n-l)k

where Ck := cos(27k/n) and Sk := sin(27k/n). To show that the two eigen-
functions defined by uk and Wk are linearly independent, we consider the tensor-
product B-spline entries p1,2 and p1,3 of pi p0o=u and p llo=wk (solid dots in Figure
3-8). The two eigenfunctions are linearly independent over the shaded region if
they generate independent B-spline control points p1,2 and p1,3, i.e. if


det P1,2 ouk P1,3 po=u Adet / 0.
l\,21po= w Pl,3po=wkj ) 0 Sk









In fact,

4Ak 1 / 0
1
S-(Ck + 5 (Ck + 9)(Ck +t))

S (Ck + 5) + (Ck + 9)(Ck+)) 4

S ( (Ck + 9)(Ck+)) Ck

= (Ck + 9)(Ck + ) (- Ck)2

S8Ck + 8 0

SCk -1.

Ck / -1 and Sk / 0 follows from k / n/2.

When k = n/2, the eigenvectors of Ak = are


u (0,1,0, 1,0, 1,0, ...-1,0,...) and

w T = (0,0,l,0,-1,0, ,..., 0,- ,. ..)

then

det P1,2 PoUk P1,3 po=Uk A det 0 ) 0.
Pl,2 po- w Pl,3 po=w 0 w~
For the eigenvalues of A22, { L, I }, the eigenfunctions are the
8 e 8e 16, 16, 32, 32, 64
tensor-product power basis functions ([30])


{u3, 3,3 u3, uv3 3v2 3, 32, 33v3}

whose pairwise independence is well known.

For the special case n = 3, there is a zero eigenvalue. As shown in Figure

3-7, the associated eigenfunction has zero values on Uj2 but zon-zero values

on Q21. We can single it out by first looking at the domain Ui-2i. The rest of














VN N < 4 N <5 VN
(Lemma 7) (Lemma 8) (Conjecture 2) (Theorem 2)
Figure 3-9: Summary of findings for Catmull-Clark subdivision. Domains G
(shaded) and valence n for which the nodal functions with support on G are lin-
early independent.

eigenfunctions are linearly independent the over U 2 i, with similar proof as

above. That implies the overall independence of the eigenfunctions over f.





Since the transformation between the eigenfunctions and nodal functions are

invertible, all the nodal functions are also linearly independent and form a basis.

Theorem 2 The nodal functions of Catmull-Clark subdivision that have support on

2 are linearly independent over f.















CHAPTER 4
TIGHT BOUNDING VOLUMES

In this chapter, we describe the efficient construction of interval triangles that

enclose the subdivision surface: for each piece we bound the x, y and z component

of the limit surface separately. This component bound is computed by linear

combining of pre-computed bounds of basis functions. A volume created from the

component bounds to bound the subdivision surface.

At this moment, we focus on Loop's subdivision although the underlying

approach is independent of the subdivision scheme.

4.1 Interval Triangles

An interval point is an axis aligned cube with possibly different sizes on three

dimensions. It is a "1 ;, point defined by three intervals: {[x-, x+], [y-, y+], [z-, z+]}

just as a regular point that is defined by three values {x, y, z}.









Figure 4-1: An interval point


A triangle can be viewed as a convex combination of its three corner points.

Similarly, an interval .:, il 11.:is the convex combination of three interval points (see

Figure 4-2). Geometrically, the interval triangle is a polyhedrally bounded volume

in 3D space (see Figure 4-2 right).

We aim to bound the subdivision surface with a set of interval triangles.










1




Figure 4-2: The interval polygon defined by three interval points.

4.2 Subdivision Patches
A subdivision patch is the limit surface of a triangle and its one-ring of
neighbors under subdivision (see Figures 4-3). Given an input mesh, we can
associate each triangle with one patch. The union of all patches is the complete
subdivision surface.






Figure 4-3: The Loop subdivision patch

For simplicity of computation, we assume that at most one of the three
vertices has n / 6 neighbors. Otherwise, a step of local subdivision is perform to
enfore this condition.
Figure 4-4 shows how the regular Loop patches (green) separates the irregular
patches (holes) after one subdivision.
4.3 Bounding Basis Functions
This section explain the first step of our method. First we explain the 2D
domains of the Loop subdivision patches. Then we will define a set of basis
functions for each domain layout. Finally we bound a pair of upper and lower
bounds for the basis functions.


























Figure 4-4: The regular Loop patches (green) of the venus model

10
9
10 9
11 9
11 \0/ 8 10 0 0
1 122


S\ / I 2 /

4 5 6 4 5 6 4 5 6

Figure 4-5: Domain layout of Loop subdivision patches.


Note that we pre-compute and tabulate the bounds and domain, in principle,

no user of the approach needs to ever be concerned with the derivation. Only the

second step needs to be executed for an input mesh and is visible to users.

4.3.1 Subdivision Domain

The domain of a regular Loop patch (all three vertex has valence = 6) is

straightforward (Figure 4-5 left). The points are arranged in regular pattern where

every edge has the same length and every angle has the some degree (r/3).

When one of vertices (-o uo) is an irregular node, the domain is arranged in

the follow way:

> all the edges connected to uo have length 1

> all the angles .,.i ,i:ent to uo = 27/n

> u4 extends the line segment uoul by k,









> u6 extends the line segment uou2 by kn

> u5 is the midpoint of u4 and u6

> u3 and u7 are the reflection of u5 across uoul and uou2, respectively.

The value kn is chosen so that the subdivision limit of this 2D mesh (shaded

area Qn in Figure 4-5) fits tightly inside the triangle An with vertices uo, ul and

U2.

The reason for doing this will be evident in the next section. We will proceed

to give the formula for kn.

The shape of Qn, as shown in Figures 4-5, changes with n. Because of the

built-in symmetry, the domain aliv-, stay in the area between ray uoul and uou2.

It agrees with An exactly if n = 6. For n > 6, the domain bulges inward, towards

the extraordinary node. For n < 6, it bulges outward. By aligning the outer

most point on the boundary curve of the domain with the line ulu2, we have the

following formula for kn:


-4(c2 2)/(1+ 2c2) if n >6, r
kn := C :_ COS -.
-6(2c2 7)/(15 + 2c2) if n < 6,


4.3.2 Basis Functions

As defined in C'! lpter 3, the nodal function bi for a given irregular valence

n (n 6 if there is no irregular node) is defined as the limit of the 3D mesh that

consists of vertices Vj where Vj has {x, y } uj{x, y} and z = 6(i,j). The indices i

and j run from 0 to n + 5. Those nodal functions are linearly independent over Qn,

therefore we call them basis functions.

Figure 4-6 (left) shows the b3 for n = 7.

4.3.3 Computing the Bounds for Basis Functions

Now we are ready to create the upper and lower bounds for basis functions.









For the (one-time) bounding, for each n, each basis function is subdivided

many times (we subdivided 7 times). Due to the convex hull property of Loop's

subdivision, we obtain correct linear upper (lower) bounds if we can put a plane

so that it sits above (below) the subdivided control net. The planes are carefully

chosen so they tightly enclose the subdivided control net (see Figure 4-6 right).

Since the domain of basis function Qn sits inside the triangle An with vertices

uo, u1, u2, we only need to store the z values of the planes on u0, U1, u2.
i, the values are {b-+, b~ b`} and {b, ,bT, b }, for upper and lower bound

respectively, then we have the following Lemma:

Lemma 10 for r:,' (u,v) in Qn, there exists (s,t) such that

sb + tb + (1 s t)b2 < bi(u, v) < sb + tb+ (1 s t)b+

and 0 < 1, 0< t < 1 0 < (1 s t)< 1.

Proof. (s, t) is the barycentral coordinate of (u, v) in triangle u0, ui, u2. Since

Qn C An, 0
We then store all the values b+, bt, b+ and bo b for every i and possible

valence n into a table. We also store the ui positions for later use. This generation

of tables is only done for once.









Figure 4-6: Basis function b3 for n = 7 and its control p1 Iv.-on (left), The upper
and lower bound (red polygon) of b3 (enlarged) (right).









4.4 Component Bounds

The x-component of the Loop patch can be expressed as
n+5
x(u,v) -= Cibi(u,v)
i=0

where bi(u, v) is the Loop patch corresponding to the control p1 iv.-on that is 1 at

i-th vertex and 0 elsewhere, and ci is the x component of the ith control point.

Then, on Qn,
n+5
x(u, v) := max{ci, 0} bf (u, v) + min{c, } b0 (u,v)
i=0

is an upper bound for x.

Since linear functions are their own best upper and lower bound, we can

extract a linear function interpolating Co, ci, C2.

With di the difference between (ui, I ) and cs,

n+5
x(u, v) (u, v) + > dibiu, v).
i=3

We removed i = 0..2 from the summation because by construction, di = 0 for

i 0, 1, 2.

Then, on Qn,

x- < x < x+ where (4.1)


x := + Z I+5 max{di, 0} b + mind, 0} b ,

x- := Z+ +max{d, 0} b. + mind, 0} b+.


The di are linear combinations of the control points ci and can be quickly

computed using the domain position ui from subsection 4.3.1. The bV and b7 are

directly read from the pre-computed tables. The linear bounds y-, y+, z- and z+

are determined analogously.















(a) (b) (c) (d)

Figure 4-7: (a) Loop patch; (b) corner interval points Di; (c) convex hull of the Oi
forming an interval triangle (to display the inside, the top facet is removed); (d) the
enlarged patch inside its wire-frame interval triangle.

4.5 Constructing Interval Triangles

Here is the main results of this section:

Lemma 11 The intervals points {[x ,x,x, [yu ,y], [y~ ,yf]}, i 0..2, forms a

interval t, .:,l.' encloses the subdivision patch.

Proof. For each (u, v) E ,n there exists (s, t) such that


sx0 + tx. + (1 s t)x- < x(u, v) < sx+ + tx+ + (1 s t)x+


where 0 < s < 1,0 t < 1,0< (1 s -t) < 1. The same holds for

{y(u, v), z(u, v)}. So point {x,y,z} is enclosed by the a convex average of interval

points {[x x+], [y 7, y+], [y y+]}, therefore enclosed by the interval triangle. |II

We show how the interval triangles enclose the subdivision surface and how

close they are by a few examples in Figure 4-7 and Figure 4-8.

4.6 Semi-Sharp Creases and Boundaries

For enhanced realism, standard subdivision is often enhanced with directional,

anisotropic semi-sharp crease rules. For example, DeRose et al. [7] propose

applying, along marked mesh lines, a few steps of the sharp crease rules from

Hoppe et al.[13], followed by one optional step of blending and infinitely r' ',i-

steps of standard subdivision.
















Figure 4-8: The subdivision surface; the surface and its interval triangles (with the
top facet removed); the surface with semi-transparent interval triangles (from left
to right).


Figure 4-9: Subdivision surface with semi-sharp creases (red = crease value 1.0).


4









There are two strategies for creating bounding interval triangles for surfaces

with such semi-sharp creases. The first is to bound the basis functions corre-

sponding to all different configurations of crease edges. This is similar to Bolz and

Schroder [3], but, since we only need upper and lower bounds, we need not generate

bounds for every combination of two subdivision rules. If one rule results consis-

tently in higher values than the other, -- a sharp crease rule or a boundary rule

based on univariate splines, and a generic subdivision rule, then it suffices to bound

the upper function from above and the lower function from below to enclose the

whole range of combinations. Nevertheless, this strategy leads to a large number of

tables.

The alternative strategy, used in Figure 4-9, is to apply subdivision with the

sharp crease rule on the patches influenced by the crease edge until only smooth

subdivision steps are left and the standard bounds apply. If a boundary is the

result of generating one extra l1v. r of nodes on the fly, i.e. by (. iing vertices or

reflecting vertices, or if boundaries contract by one lIv. r, interval triangles can be

used without modification.

4.7 Convergence

The thickness of a interval triangle is the maximum size of the three interval

points.

Lemma 12 The thickness of the bo;,,:.,.: interval ',.:,.,,wjl: for a given subdivision

patch goes to 0 under a,,.:.rm subdivision.

Proof. The thickness t < Z diC where C = ,,.,, {max{b+ by, bt b, b -

b2}.


di 0 under subdivision, therefore t 0.














CHAPTER 5
APPLICATION

We now take a look at applications of the new bounding volume: interval

triangles.

5.1 Collision Detection Using Interval Triangles

5.1.1 Pairwise Interference Detection

Since one interval triangle can have up to 19 facets, (see Figure 4-7 d),

comparing all facets to detect interference is not an efficient approach. Instead,

we reduce the task to a triangle-triangle intersection test by slightly enlarging the

interval triangle to an offset triangle.

The base t,i .:;n,l. of the offset triangle is defined as

x- + X+ Y- + Y + z- + z+
(X +7 y Z Z) (5.1)
2 2 2

restricted to An. The length of the half-diagonal of the ith corner cube is

l ( := -( Z + (2 y 2 -+ )i )2.
2 2 2

Therefore, if we run (the center of) a sphere with radius


p : max {li} (5.2)
i=0,1,2

over the base triangle, we create an offset triangle that is guaranteed to enclose the

Loop patch. The offset triangle is slightly enlarges the interval triangle but reduces

the task to a triangle-triangle intersection test of the base triangles 'T, 'T with an

error bound set to the sum of the two radii, p1, p2:


dist (T, T2) < pi + p2









For the test, we adapt Moller's method [22]: if one triangle lies to one side of the

plane containing the other triangle non-intersection is reported; otherwise the

common line passing through the two triangles is examined and non-intersection

reported if the intersection line intervals do not overlap. We need only change to

Pi + P2 + c the tolerance c that Moller uses to stabilize the computation. This adds

three additions to the approximately 100 operations of Moller's test.

5.1.2 Intersection Hierarchy

Most intersection applications have an associated tolerance C so that surfaces

are considered dil-i, iii if their distance is more than c. To enforce the tolerance, we

split each patch into four by local Loop subdivision, until the offset triangle has a

radius p less than e/2. It is straightforward to modify any triangle hierarchy to use

offset triangles. We modified the OBB tree code of Gottschalk et al et al. [9] by

enlarging the dimension of the OBB until it contains each base triangle within the

radius tolerance.

5.1.3 Performance Evaluation

A major challenge when introducing a new collision detection technique is to

conduct a fair performance experiment to measure space and time requirements in

a realistic rather than just a worst case setting. Our task is made easier, in that we

need not compare different types of hierarchies but only the test.

In our context, the performance criteria are the number of primitives needed to

achieve a given accuracy (space), and the hierarchy initialization and the average

intersection cost (time). We compare the performance of an offset triangle-based

hierarchy to a similar hierarchy based on a control mesh that has been uniformly

subdivided to lie within the same prescribed error tolerance. As pointed out earlier

(Figure 1-3) such a mesh does not guarantee correct intersection testing but would

nevertheless be accepted in many practical situations.














S4-1





r i

Figure 5-1: Models used for performance evaluation.

We did not compare to an adaptively refined mesh because no efficient

adaptive refinement based on the maximum norm is available. AABBs overestimate

so much that the adaptive scheme is not competitive. The efficient alternative

is to use interval triangles to drive adaptation but then we are almost back to

our solution. We also confirmed in our test scenario, which rigidly transforms the

objects, that AABB trees built on the convex hull of the control polygon do not

perform well compared to the OBB tree [32].

5.1.4 Experiment Set Up

We used six different models of small to medium size as is suitable for further

subdivision. The models range from 24 triangles (Star) to 1418 (Venus) and are

shown in Figure 5-1. In the spirit of Klosowski et al. [15], we place alv-- two

of these models into a cube or room, each with random position and random

orientation. All models are scaled to tightly fit into a bounding box of size 1.

The interference test returns separation, or possible collision and only the first

collision pair. Since the intersection cost depends on the position and orientation of

the input models, we ran 150,000 random tests for each pair and report the average

time. We also varied the room size to modify the percentage of intersecting cases.









Table 5-1: Number of triangles needed for the given error bounds.

e I 1 0.5'
i-t Loop i-t Loop i-t Loop
Venus 5705 22688 6098 22688 7214 90752
Head 1499 3200 2474 12800 5990 51200
Pawn 1216 1216 2368 !1,. 1 4732 11. 1
Star 384 384 S, 1536 1536 1536
Demon 1258 :; 1 2734 4384 4897 17536
Quake 1728 6528 4218 1'..2' 6636 26112

Table 5-2: Intersection hierarchy (OBBtree) creation time in milliseconds.

e "[ I 1 0.5'.
i-t Loop i-t Loop i-t Loop
Venus 180 561 180 551 240 2414
Head 40 70 80 310 180 1342
Pawn 30 20 60 110 140 110
Star 10 10 20 30 40 40
Demon 40 100 90 170 150 431
Quake 50 150 100 100 200 641


5.1.5 Results

In the following tables, we label the new interval triangle-based method "i-t"

and the intersection based on subdivided meshes "Loop". The timing is measured

on a single P4 2.4G HZ CPU machine with 1G RAM.

Table 5-1 lists the number of triangles needed to guarantee a given accuracy.

That is, a possible intersection is announced if the limit surfaces within the cubicle

are closer than ''. 1 respectively 0.5'. of the object size. The offset triangle-

based method requires fewer triangles in all cases, indicating that the tight bounds

pl off.

Table 5-2 lists the time used to build the OBB hierarchy. This includes the

computation of all interval triangles in the case of the offset triangle-based method.

Nevertheless, the offset triangle-based method requires less time in most cases. The

average savings are tI' .









Table 5-3: Intersection cost in ms for different tolerances and room sizes.
e ". 1 .I 0.5'.
i-t Loop i-t Loop i-t Loop
Room size 6.0: 3 percent of the models collide
Venus/Head 22 24 22 26 22 27
Star/Pawn 11 12 11 12 12 12
Demon/Quake 13 13 13 14 14 14
Room size 3.0: 30 percent of models collide
Venus/Head 147 178 144 178 148 187
Star/Pawn 81 91 84 91 84 90
Demon/Quake 105 116 107 117 106 118


Table 5-3 lists the average intersection time for both approaches. For a room

of size 63, offset triangles improve only marginally over the Loop method because

most rejections occur in high level box-box tests of the OBB tree. However, for a

room of size 3, more box-box tests are needed for Loop than for offset triangles.

That is, the safe offset triangle-based test is cheaper than the unsafe control

mesh-based test!

5.2 Inner and Outer Hull

For intersection testing, it is not necessary to build an explicit conforming

inner and outer hull, since the union of the interval triangles encloses the surface.

However, for other applications such as manufacturing with tolerances, intersection

and overlap of the interval triangles is not acceptable. Instead, we need a pair of

triangulations that sandwich the limit surface. To this end, we first select from the

eight choices

Pa,3,7 W= :(x, y Z), a,j3, y7 {,+},

computed in Section 4, for each mesh triangle two planes that tightly enclose the

limit surface. Then, we shrink-wrap the individual planes around each node.

The two planes are selected as follows. Compute the normal of the linear

functions p,a,,. If the signs of the xyz-components of the normal agree with

(c, 3, 7) then the plane is the outer plane. If the signs agree with (-c, -p, -7)
























Figure 5-2: The limit surface, the outer hull and a superposition of the limit sur-
face and the outer hull with a cutout to show the position of the limit surface (from
left).


then the plane is the inner plane. In theory, there could be cases where we need to

subdivide to assure uniqueness, but in practice we have never encountered such a

case. All the points in the triangle cell lie between these two planes because their

inner product with the outer plane is negative and their inner product with the

inner plane is positive.

In the second step, we compute, for each node, an outer triangulation vertex.

The vertex is the point of least distance to the node's limit position and such

that it lies outside the outer planes of the facets surrounding the node. This is a

simple local quadratic optimization problem that can be solved by enumeration.

Connecting the outer triangulation vertices according to the inherited connectivity

of the input mesh completes the construction (see Figure 5-2) [32] [27]

5.3 Adaptive Subdivision

For a given object, The density of the vertices varies from region to region. For

example in Figure 1-2 second picture, the head light of the vehicle is dense enough

while the body is yet coarse and needs to be further subdivided.

An adaptive subdivision only perform the refinement at the places where

needed. The thickness of our interval triangles, give an accurate estimate how













I -

[7 _


'! J.
,
.
S '^' ""


A. =, _=,,, ,',,
"- *' .j- '


S a,:I
*'?3


Il


Figure 5-3: Results of adaptive subdivision on the deer model. Input ; e=0.5' .
e=0.1 .. and the surface with e 0.1 (from left to right).


Figure 5-4: RBi -traced images at 800x600 resolution. Input mesh (671 triangles),
r-,--tracing time: 8s (left). Adaptive subdivision with e=0.,.". Ri -A,]l',:u number of
I,.:,ill i = *111', and maximum subdivision level: 4, 'r-/racing time: 9s (middle).
Unifc, inlii subdivided 4 times. Resulting number of ,.:.ii.jl, = 171776, a r-/racing
time: 12s (right).


the shape of the patch is away from a flat triangle. Therefore a new adaptive

subdivision can be built, with our bounding volume as the "test stone".

Figure 5-3 and Figure 5-4 shows the results of the adaptive subdivision and


the r -,-traced image of the subdivision surfaces [33] .















CHAPTER 6
CONCLUSION

A safe, accurate and efficient bounding volume: interval triangles has been in-

troduced. It is applied to intersection testing and adaptive rendering of subdivision

limit surfaces and is proven effective. Alternative approaches, such as comparing

refined control meshes, prisms [16] or hierarchies of AABBs fail either to be effi-

cient, or to be correct or both. The algorithm is simple: read pre-tabulated data

and form linear combinations according to equation (4.1). This approach requires

only a simple modification of available software.

This thesis gives a framework for constructing such tight bounding volumes

for subdivision surfaces, by decomposing the surface into patches, the patches into

combinations of basis functions, and pre-computing and tabulating of the bounds

for these basis functions. A subtle point here is to establish the basis property of

the functions that define subdivision surface locally.

A closer look at Section 4 shows that the approach works for any surface

parametrization that can be bounded and is adaptively refinable. Therefore it can

be applied to NURBS surfaces (if the denominator is bounded away from zero) and

to other refinement schemes, ,v interpolatory subdivision. The key ingredient in

each case, is the one-time generation of accurate bounds analogous to Section 4.3.

The properties that the bounding volume possesses make it promising for

many other applications.















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BIOGRAPHICAL SKETCH

Xiaobin '..":: was born in 1: ::: ( ::: : : on I 1 ,1 1 lie received

his B.E. from Uni of Science and Technology of C I::. in n In i"-,- he

received his M.S. from C' Acadery of Sciences.









Efficient, Tight Bounding Volumes Subdivision Surfaces

Xiaobin ', .
( ) '-2 1 ,
Department < Computer and Information Science and Engineering
( :i : Jorg Peters
Degree: Doctor (- F':1 phy
( .. Date: August

Smooth mathematical surfaces enable the desingers and engineers to :::::

ture some of the most ':.... : I : items in our daily lives such as smooth car bodies,

.,:airplanes and even :. :. mice.

Subdivision surfaces is a such kind of '. i1.1 and yet powerful mathematical

tool for graphics applications and computer aided geometric design. Most notice-

able usage <. sub : )n : '.. c aare characters created in Pixar animation feature

S such as "A bug's life" or "Tby s! .

T:: thesis introduces a framework for constructing tight bounding volumes

for subdivision surfaces. Ti .. complete surface can be viewed as a collection of

smooth'l joined patches where tight bounding volumes are computed efficiently.

Ti tight bounding volumes computed are crucial .. graphics and geometric

algorithms such as :::: : :: detection, adaptive i : :on and rendering.