Manifold Dual Contouring Scott Schaefer Tao Ju Joe War - PDF document

ManifoldDualContouringScottSchaefer,TaoJu,JoeWarren—DualContouringisafeature-preservingiso-surfacingmethodthatextractscrack-freesurfacesfrombothuniformandadaptiveoctreegrids.WepresentanextensionofDualContour-ingthatfurtherguaranteesthatthemeshgeneratedisamanifoldevenunderadaptivesimplication.Ourmaincontributionisanoctree-based,topology-preservingvertexclusteringalgorithmforadaptivecontouring.Thecontouredsurfacegeneratedbyourmethodcontainsonlymanifoldverticesandedges,preserves ofthegridisprocessed.Theiso-surfaceiscontouredusingMCandsimpliÞcationisbasedonedgecontractionsonthepolygonalsurface.ByenforcingtheÒlinkconditionsÓpro-posedbyDeyetal.[25]duringsimpliÞcation,thesimpliÞedsurfaceisalwaysmanifoldandpreservesthetopologyoftheoriginaliso-surface.WhileAttaliÕsmethodavoidsstoringtheentire,Þne-leveliso-surface,thespeedofthemethodremainsslowsincethisÞnesurfacestillsneedstobegeneratedandthensimpliÞed.Aswewillsee,ourtopology-preservingmodiÞcationtoDCsimpliÞesaniso-surfaceinmuchlesstimesincenopolygonisgenerateduntilafterthegridissimpliÞed.III.CONTOURINGONAUNIFORMGRIDWestartbydescribingasimplemodiÞcation,ÞrstproposedbyNielson[13],totheoriginalDCalgorithm[3].OneofthelimitationsofDCisthatitallowsnomorethanonevertexwithineachgridcell.Onauniformgrid,DCleadstonon-manifoldverticesandedgesforalloftheambiguoussignconÞgurationsintheoriginalMarchingCubesalgorithm[1].Tocombatthiseffect,NielsonÕsmodiÞcationallowsmultipleverticestobeplacedinasinglecell.Inparticular,NielsonassociatesonevertexwitheachcycleofamodiÞedMarchingCubestable[26].Sinceeachcycleconsistsofalistofedgesonthecubiccell,eachvertexisassociatedwithasetofedgesandeachedgeisassociatedwithexactlyonevertex.Tocreatepolygons,thealgorithmconstructsonepolygonconnectingtheverticesassociatedwiththatedgeinthefouradjacentcells.ThisalgorithmcreatesaquadrilateralsurfacethatisthedualofthesurfacecreatedusingMarchingCubes(andwasthereforegiventhenameÒDualMarchingCubesÓ).Furthermore,thissurfaceisalwaysamanifoldbecausetheoriginalMarchingCubesalgorithmalwaysconstructsamanifoldandthedualpreservesthetopologyofthesurface.OneoftheadvantagesofDCoveratraditionalcontouringmethod,suchasMarchingCubes,isitscapabilityofrepro-ducingsharpfeaturesinthepresenceofHermitedata.ToincorporateHermitedataintoNielsonÕsDualMarchingCubesalgorithm,wesimplyconstructaQuadraticErrorFunction(QEF)[11]foreachvertexusingtheHermitedataontheedgesassociatedwiththatvertex.Weplacethisvertexatthelocationthatminimizesthaterrorfunction.Figure2showsacomparisonin2Dofthedifferentmethods.MarchingCubesalwaysproducesamanifoldbutdoesnotreproducesharpfeatures.DualContouringreproducessharpfeaturesbutthetopologymaybenon-manifoldinsomecon-Þgurations.TheHermiteextensiontoDualMarchingCubesalwaysproducesatopologicalmanifoldandcanreproducesharpfeaturesaswell.IV.ADAPTIVECONTOURINGIntheprevioussection,weconsideredconstructingmanifoldiso-surfacesfromuniformgridsthatpreservessharpfeatures.However,formodelswithrelativelyßatregions,theuniformcontouringalgorithmproducesalargenumberofpolygons (a)(b) (c)(d)Fig.2.Comparisonofcontouringwithhermitedata.Cellwithhermitedataonedges(a),MarchingCubes(b),DualContouring(c)andHermiteDualMarchingCubes(d).coveringtheseßatregions.Ideallythecontouringalgorithmwouldextractasurfacewherethenumberofpolygonsadaptstothelocalpropertiesofthesurface(i.e;fewerpolygonsinßatregions).DCprovidessuchanalgorithmtoconstructmulti-resolutioniso-surfaces.Themethodessentiallyperformsvertexclusteringwheretheverticesofthechildcellsintheoctreecollapsetoasinglevertexinatopologicallysafemanner.However,sinceonlyonevertexwasallowedper-cellinDC,thecollapsewasveryrestrictive.HerewedevelopanewcontoursimpliÞcationmethodviaoctree-basedvertexclustering,whichallowsforanarbitrarynumberofverticespercell.Furthermore,wedescribeapolygongenerationalgorithmforconstructingsurfacesfromtheseadaptivelyclusteredvertices.A.VertexclusteringGivenanerrorthreshold,thevertexclusteringphasecreatesavertextreestartingwiththeverticesattheÞnestleveloftheoctree.EachvertexcontainsaparentpointeraswellastheQEFassociatedwiththisvertexandthevalueoftheQEFevaluatedatthisvertex(i.e;theerrorassociatedwiththisvertex).Furthermore,avertexismarkedasbeingiftheerrorassociatedwiththevertexislessthanourgiventhreshold.Initially,weßagallverticesascollapsibleandsettheirparentindicestoNULL.Whensimplifyingtheoctree,weonlyclusterverticestogetherthataretopologicallyconnectedonthesurface.NotethatthisapproachissimilartoZhangetal.[7],butitisnotsufÞcienttoguaranteethatwemaintainthemanifoldpropertiesofthesurfaceundersimpliÞcation(whichwillbeaddressedinSectionV). 6 (a)(b)(c)Fig.7.(a):Aspringmodelcontouredonauniformgrid.(b):Amodelsim-pliÞedusingourmethodallowseachsurfacetobesimpliÞedindependently.(c):DualContouringrestrictssimpliÞcationevenforseparatesurfaces.foraclusteredvertex,weobservethatthesurfacetheunionofsurfacesaretheverticesclusteredtogethertoformfromthechildcellsof.Tocompute,wesimplysumthenumberofintersectionsofeachalongedgesof,shownasthinlinesinFigure6(bottom).WecancomputeinanequallyefÞcientmannerusinganinductiveformulathatrelates(seeproofinAppendixII): denotesthesumofthenumberofintersectionsalongtheinternaledgesof,shownasthickenedlinesinFigure6(bottom).Figure6showsanexamplewhere(bottom)isbuiltfromtensurfaces(top).ForeachchildcellwedisplaythequantitiesforeachinFigure6(top).Observethat10and36,andhence1byequation2,whichisthecorrectEulercharacteristicofthedisk-likesurfaceTointegratethetopologyconstraintintotheadaptivecontour-ingalgorithmintheprevioussection,werequirethatavertexiscollapsibleiftheassociatedQEFerrorisbelowthegivensatisfythetwoconditionsinProposition1.Figure4(c)showstheresultofadaptivecontouringwithtopologyconstraint,whichpreservesallthethreadsofthespider-webwithmanifoldverticesandedges.VI.RESULTSComparedwithothercontoursimpliÞcationalgorithmssuchastheoriginalDualContouringmethodortheextendedDualContouringmethodbyZhangetal.[7],ouralgorithmismuchlessrestrictiveinthetypesofsimpliÞcationsallowed.First,multiplecontourcomponentswithinasameoctreecellsimplifyinanindependentmanner,henceallowingßatregionstomaximallycollapseevenifinthevicinityofothergeometry(seeFigure7).Second,unlike[7],ourmethodputsnorestrictiononthenumberofcontourintersectionsoneachoctreecelledge,asourvertextreeisseparatefromtheoctree.Thisallowsustosimplifymultiplelayersofthingeometry.A2DexampleisshowninFigure8,whereourproposedmethodiscapableofsimplifyingnearbylayersofcontoursmuchbetterthanbothDCandExtendedDualContouring[7]. (a)(b) (c)(d)Fig.8.ComparisonbetweenDualContouring(b),ExtendedDualContouring[7](c),andtheproposedManifoldDualContouring(d)insimplifyinga2Dcontour(a).Verticesandedgesonthecontouraredrawnasrounddotsandlines,andoctreecellsinwhichtheclusteredverticesareformedareshown.Furthermore,mostcontoursimpliÞcationalgorithms[3],[7]stopsimplifyingsurfacecomponentsassoonasanunsafesimpliÞcationisencountered,whichlimitstheamountofsimpliÞcationpossible.Incontrast,ourmanifoldcriterionmaybeabletodeterminethatasafesimpliÞcationoccurslaterinvertexclusteringevenifunsafecollapsesoccurredpreviously.ThismethodallowsforextremesimpliÞcationswhereevenverydensemodelssuchasFigure9collapsetoextremelysimpleshapes.Figures10and11showtwoothercomplexscannedmodelsthathavebeensimpliÞedusingourmethodbyvaryingtheerrorthreshold(thehermitevolumerepresentationsofeachmodelwereobtainedusingthePolyMendertool[10]).Eachmodelistopologicallyequivalentto(i.e,havingthesamegenusas)theoriginalanddoesnotcontainanynon-manifoldedgesorvertices.Oneattractivefeatureofourvertexclusteringalgorithmisthat,oncethevertextreeisconstructed,simpliÞedpolygonscanbegeneratedefÞcientlyoffthevertextreegivendifferentuser-speciÞedQEFerrorthresholds.ThisisdonesimplybyrevisingtheÒcollapsibleÓtagofeachclusteredvertexaccordingtothenewerrorthreshold,andthereisnoneedtorebuildthevertextree.Incontrast,methodsthatÞrstbuildtheÞne-levelcontourfollowedbymeshsimpliÞcation(suchas[24])wouldneedtore-runtheentiresimpliÞcationprocesswhentheerrorthresholdischanged.Suchfeatureofouralgorithmcouldbeuseful,forexample,inrealtimenavigationofacomplexvolume.Intheseapplications,theQEFerrorthresholdsarehigherinoctreecellsthatarefurtherawayfromtheviewerÕslocation,resultinginmoredetailedgeometryintheviewerÕsvicinityandcoarserpolygonsatdistances.Figure12shows 8 Octree Base ClusteringTime(sec) ClusteringTime(sec) Polygon SimpliÞed Depth Polygons Without With Generation Polygons ManifoldCriterion ManifoldCriterion Time(sec) Fig7 6 28740 0.254 0.259 0.060 1042 Fig4 7 44784 0.459 0.465 0.097 3672 Fig9 9 476184 5.58 5.76 1.12 78 Fig10 9 611476 6.65 6.71 1.42 9944 Fig11 9 878368 10.89 10.99 2.01 30002 TABLEIIMPLIFICATIONTIMEINSECONDSFORTHEVARIOUSSTAGESLUSTERINGANDOLYGONENERATIONCOMPARINGCLUSTERINGWITHANDWITHOUTTHEMANIFOLDCRITERION (a)(b)(c)(d)Fig.12.SimplifyingthecontourbasedontwodifferentviewerÕslocations(markedasbluedots),nearthehead(a,b)andnearthebase(c,d),shownwithpolygonedges(a,c)andwithoutedges(b,d).NotethatthesurfacefurtherawayfromtheviewpointissimpliÞedmore.VII.CONCLUSIONSANDWehavepresentedanextensiontoDualContouringthatpreservessharpfeaturesandalwaysconstructsamanifoldsurface.Furthermore,wedevelopedasimplecriterionforvertexclusteringinanoctreethatisguaranteedtopreservethegenusoftheoriginalsurfaceandalwaysproducea2-manifoldwithoutanynon-manifoldverticesoredges.Thoughthesurfacesweproducearetopologicallymanifold,theymaystillcontainintersectingpolygons.Forexample,inourHermiteextensiontoNielsonÕsDualMarchingCubesalgorithm,wemayplacemultipleverticesinsideofacell.ItispossiblethattheHermitedataalongthecelledgescausestheverticestobepositionedsuchthatthesurfacesintersectwithinthecell.Notethatintersectingpolygonsmayariseevenwhenasinglevertexisplacedinsideacell,asobservedin[28].Asaresult,theoriginalDCalgorithmaswellasitsvariantsareallsubjecttosuchgeometricerrors.Anaiveapproachfordetectingintersectingpolygonsgener-atedbyDC-likemethodsinvolvestime-consumingneighbor-Þndingontheoctreeaseachpolygonspansmultipleoctreecells.Instead,[28]presentedanefÞcient,intersection-freemodiÞcationtotheoriginalDCmethodbydevisingasetofsimplegeometricteststoidentifypotentiallyintersectingpoly-gons,whicharethentessellatedintosmaller,non-intersectingtriangles.Whilethemethodof[28]isrestrictedtosinglevertexperoctreecell,inthefuturewewouldliketoextendsuchmethodandexplorecriteriaforplacingmultipleverticeswithinacellthatbothreproducessharp-featuresandavoidsintersectionsevenunderadaptivesimpliÞcation.GiventhatthesimpliÞediso-surfaceusingourapproachpreservesthetopologyoftheoriginalmodel,aninterestingdirectionthatworthinvestigatingishowourmethodcanbecombinedwithtopology-repairalgorithmsforlargemeshes,andinparticular,thegrid-basedmethodssuchas[29],[30].WeanticipatethatageometricallysimpliÞedyettopologicallyequivalentsurfacewouldgreatlyacceleratetheprocessoflocatingtopologicalerrorsinthesemethods.AcknowledgementsWewouldliketothanktheStanford3DScanningRepositoryfortheDragonandThaiStatuemodels,CindyGrimmfortheSpider-webmodel,andVanDuzanfortheQueenmodel.Wewouldalsoliketothanktheanonymousreviewersfortheircommentsandsuggestions.[1]W.E.LorensenandH.E.Cline,ÒMarchingcubes:Ahighresolution3dsurfaceconstructionalgorithm,ÓinComputerGraphics(ProceedingsofSIGGRAPH87),vol.21,no.4,Anaheim,California,July1987,pp.[2]L.P.Kobbelt,M.Botsch,U.Schwanecke,andH.-P.Seidel,ÒFeature-sensitivesurfaceextractionfromvolumedata,ÓinProceedingsofSIG-GRAPH2001,ser.ComputerGraphicsProceedings,AnnualConferenceSeries.ACMPress/ACMSIGGRAPH,August2001,pp.57Ð66.[3]T.Ju,F.Losasso,S.Schaefer,andJ.Warren,ÒDualcontouringofhermitedata,ÓACMTransactionsonGraphics,vol.21,no.3,pp.339Ð346,July2002,iSSN0730-0301(ProceedingsofACMSIGGRAPH[4]R.Shekhar,E.Fayyad,R.Yagel,andJ.F.Cornhill,ÒOctree-baseddecimationofmarchingcubessurfaces,ÓinVISÕ96:Proceedingsofthe7thconferenceonVisualizationÕ96.LosAlamitos,CA,USA:IEEEComputerSocietyPress,1996,pp.335Ðff. 10 (a)(b)(c)Fig.13.Centerplanesofacell(a),centerlinesofacell(b),centerlinesofeachcellface(c).Next,as istheunionofall ,weconsiderasthesetofverticesandedgesthatarecontainedinmorethanone Thekeyobservationisthatliesonthe12internalfacesoftheoctreecell(seeFigure13(a)).Furthermore,weusetodenoterespectivelythesetofverticesinlyingonthecenterlinesof(seeFigure13(b))andonthecenterlinesoffacesof(seeFigure13(c)).Observethateachvertexiniscontainedinexactly4 ,whereaseachotherelementofiscontainedinexactly2 .Accordingtoformula1, Svk( ))+Ontheotherhand,sinceeach isa2-manifold,avertexiscontainedinexactly2edgesofexceptforthoseverticesin,eachcontainedin4edges,andthoseineachcontainedin1edge.Hencewehave,))+Substitutingequation4intoequation3yields Svk( Svk)2V(MfvV(Mcv) Equation5yieldsequation2,because Sv),(Svk( ,andeachvertexincontributestooneedgeintersectioninfor2and4 ScottSchaeferisanAssistantProfessorintheCom-puterSciencedepartmentatTexasA&MUniversity.HegraduatedfromTrinityUniversityin2000withaB.S.degreeinComputerScienceandMathematics,receivedanM.S.degreefromRiceUniversityin2003andaPh.D.fromRiceUniversityin2006.HisresearchinterestsincludeComputerGraphics,GeometricModelingandScientiÞcVisualization. TaoJugraduatedfromTsinghuaUniversityin2000withaBAdegreeinEnglishandaBSdegreeinComputerScience.HereceivedhisPh.DdegreeinComputerSciencefromRiceUniversityin2005.TaoiscurrentlyanassistantprofessorintheDepartmentofComputerScienceandEngineeringatWashing-tonUniversityinSt.Louis.Hisresearchinterestsareintheareasofmeshprocessing,visualization,geometricmodeling,andbiomedicalapplications. JoeWarren,aProfessorofComputerScienceatRiceUniversity,isoneoftheworldÕsleadingex-pertsonsubdivision.Hehaspublishednumerouspapersofthistopicanditsapplicationstocomputergraphics.ThesepublicationshaveappearedinsuchforumsasSIGGRAPH,TransactionsonGraphics,Computer-AidedGeometricDesignandTheVisualComputer.Hehasalsoorganizedandparticipatedinanumberofinternationalworkshops,shortcoursesandminisymposiaonthetheoryandpracticeofsubdivision.ProfessorWarrenÕsrelatedareasofex-pertiseincludecomputergraphics,geometricmodelingandvisualization.