Houston TX 77005 sschaefericeedu and jwarrenriceedu Abstract We present a method for contouring an implicit function using a grid topologically dual to structured grids such as octrees By aligning the vertices of the dual grid with the features of t ID: 71147
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DualMarchingCubes:PrimalContouringofDualGridsScottSchaeferandJoeWarrenRiceUniversity6100MainSt.Houston,TX77005sschaefe@rice.eduandjwarren@rice.eduAbstractWepresentamethodforcontouringanimplicitfunctionusingagridtopologicallydualtostructuredgridssuchasoctrees.Byaligningtheverticesofthedualgridwiththefeaturesoftheimplicitfunction,weareabletoreproducethinfeaturesoftheextractedsurfacewithoutexcessivesub-divisionrequiredbymethodssuchasMarchingCubesorDualContouring.DualMarchingCubesproducesacrack-free,adaptivepolygonalizationofthesurfacethatrepro-ducessharpfeatures.Ourapproachmaintainstheadvan-tageofusingstructuredgridsforoperationssuchasCSGwhilebeingabletoconformtotherelevantfeaturesoftheimplicitfunctionyieldingmuchsparserpolygonalizationsthanhasbeenpossibleusingstructuredgrids.1.IntroductionImplicitmodelinghasbecomeapopularmethodforex-tractingsurfacesfromvolumetricinformationarisingfromsourcessuchasMRIdataandCATscans.Inthistechnique,avolumetricfunctionisgiven,whosevaluetypi-callyrepresentsdensityinformationordistancetoasurface.Toextractasurfacefromthisfunction,weusealevelsetof.Noticethatwecanalwaysreducethisprob-lemtoexaminingthezero-contourofthefunctionbysim-plysubtracting.Thisformofmodelinghasseveraladvantagesovertra-ditionalmodelingtechniques.Sinceencodesthesurface,modicationstothesurface(suchasCSGopera-tions)canbeperformedsimplybymodifyingtheunder-lyingfunctions.Thetopologyofthesurfaceisalsoimplicitlydenedbysotopologicalmodi-cationsareeasybecausethetopologydoesn'thavetobechangedexplicitly.Finally,thistechniqueisvolumetricandadditionalinformationcanbegleanedfromtheunderlyingfunctionsuchasdistancetothesurfacesimplybyevaluat-ing.Sincerepresentingfunctionsthatcontainarbitraryge-ometrycanbedifcult,manytechniquessamplethefunc-tiononastructuredgrid.Suchgridsaretypicallyaxis-alignedandhaveverticesfromauniformsamplingofspace(uniformgrids)orsomedyadicsampling(octrees).Themainadvantageofusingthesegridsisthatoperations(suchasCSG)areeasytoperformbecausethegridsbe-tweentwodifferentimplicitsurfacesalignandtheopera-tioncanbereducedtoanoperationontheverticesofthegrid.Methodsthatusestructuredgridsforimplicitmodelingarequiteabundant.PerhapsthemostpopularisMarchingCubes[9].MarchingCubestakesasinputauniformgridwhoseverticesaresamplesofthefunctionandextractsasurfaceasthezero-contour.Foreachcubeinthegrid,MarchingCubesexaminesthevaluesattheeightcor-nersofthecubeanddeterminestheintersectionofthesur-facewiththeedgesofthecube.ThenMarchingCubespro-videsalookuptableindexedbythesigncongurationattheeightcornersthatyieldsthetopologyofthesurfacein-sideofthatcube.Afterprocessingeachcubeinthegrid,thesurfaceiscomplete.TherehavebeenmanyattemptstoextendMarchingCubesfromuniformtoadaptivegridssuchasoctrees.However,thesemethodsallrequiresomesortofpatch-ingbetweencubesofdifferentresolution[12,11,5].Re-centlyseveraldualapproachestocontouringhavebeenintroducedthateffectivelyeliminatethispatchingprob-lem[2,10,4,13].Inthesemethods,eachcellinthegridisgivenarepresentativevertex.FormethodssuchasDualContouring,thisvertexisplacedatsharpfeaturesofthesur-face,whichallowsthemethodtoreproducesharpfeaturessuchasedgesandcorners.Polygonsarethengeneratedbyexaminingminimaledges(anedgethatcontainsnosmalleredge)intheoctree.Thenforeachminimaledgethesurfacepassesthrough,apolygonisgeneratedthatconnectsthever-ticesofthecellscontainingthatminimaledge.SincethesurfacesproducedbythesemethodsaretopologicallydualtothesurfacesproducedbyMarchingCubes,wecallthese Figure1.Athin-walledroomdenedviaCSG(upperleft).PolygonalapproximationsweregeneratedbyMarchingCubes(lowerleft,67Kpolys),DualContouring(lowerright,17Kpolys)andDualMarchingCubes(upperright,440polys).UsingDualMarchingCubes,thesizeofthecontourmeshisinsensitivetothethicknessofthewalls.methodsdualcontouringmethodsandcube-basedmethodsprimalcontouringmethods.However,allofthesemethodsarelimitedintheirex-pressivitybecausetheyarebasedonstructuredgrids.Forinstance,torepresentverythinfeatures,thesemethodsre-quirethatavertexofthegridliesinsidethesurfaceandan-other,adjacentvertexoutsidethesurface.Sincethegridsarestructuredinnature,extractingverysmall,thinfeaturesmayrequireaverynegrid.Figure1(top,left)showsaroomgeneratedbyCSGoperationswithverythinwalls.Threedifferentcontouringmethodshavebeenusedtopro-ducethesurfacefromtheCSGtree.BothMarchingCubesandDualContouringrelyonstructuredgridsandrequirenegrids(and,consequently,largeamountsofpolygons)toreproducethecorrecttopologyoftheroom.WhileDualContouringisamulti-resolutioncontouringalgorithmthatcanadaptivelysimplifythesurface,theamountofsimpli-cationisstilllimitedbythethicknessofthewalls.Balmellietal[1]attempttosolvethisproblembycon-structingauniformgridwhoseverticesareallowedtocon-formtothefeaturesofthesurfaceusinganimportancemap.Whilethisapproachyieldsbettertessellations,itisstilllim-itedbyuniformsamplingandcannotreconstructsharpfea-turesaccurately.Furthermore,performingoperationssuchasCSGbetweentwoobjectsbecomesdifcultbecausetheverticesofthetwogridsnolongeralignandresamplingmustbeperformed.Varadhanetal[13]provideanalternatesolutionbasedoffofDualContouring[4]anddirecteddistanceelds[6].Inthatpapertheauthorsprovideatechniquethatcanre-constructatwo-sheetedsurfaceinacelloftheoctree.Thisallowstheirmethodtoreproduceverythinwalls.However,theiralgorithmisstillbasedoffofaprimalgridandrequirestheisosurfacetointersecttheedgesoftheprimalgridinor-dertobereconstructedfaithfully.ContributionsWeproposeafundamentallydifferentapproachtocon-touring,whichweentitleDualMarchingCubes.Thegridthatweperformcontouringonwillbetopologicallydualtothestructuredgridsusedbyothertechniques.Therefore,wedenotethistypeofgridasadualgridandstructuredgridsasprimalgrids.Ourunderlyingdatastructureisanoctreethatadaptivelysamples.Togenerateasurface,weex-tractadualgridtothisprimaloctree.Theverticesofthisdualgridarepositionedatthefeaturesoftheimplicitfunc-tioninsideeachcelloftheoctree.Usingthisdualgridthatconformstothefeaturesof,wegeneratethesurfaceusingageneralizedversionofMarchingCubes.ContouringonthisdualgridyieldsseveraladvantagesSinceweuseastructuredgridtosample,wemaintaintheadvantagesassociatedwithstructuredgridsforoperationssuchasCSG.Theverticesofourdualgridalignwiththefeaturesof,whichallowsustoreproducesharpfeaturessimilartothereproductionachievedby[6,4].Thesurfacesproducedbythismethodareadaptivepolygonalizationsthatarecrack-freeandtopologicallymanifold.Duetotheuseofadualgridforcontouring,wecanreproducesmall,thinfeaturesinthesurfacesuchaswallsortubeswithoutexcessivesubdivisionoftheoc-tree.First,wedescribethecreationofthedualgridthatweperformcontouringon,whichproceedsintwosteps:fea-tureisolationandtopologycreation.Featureisolationtakesanoctreeasinputandgeneratesasinglevertexforeachcellatafeatureofinsidethatcell.Afterbuildingtheverticesofthedualgrid,weconstructthetopologyofthegridusingasimpleextensionofthetraversalalgorithmofJuetal.[4].WethendescribehowMarchingCubescanbeappliedtothisdualgridtogenerateasurface. 2.FeatureisolationGivenafunctionandanoctree,ourgoalinfeatureisolationistoconstructavertexforeachcellintheoctreethatwillbecomeavertexofthedualgrid.Incon-trasttomethodssuchasDualContouring,thisvertexisnotalignedwiththefeaturesofthesurfacebutwithfea-turesoftheimplicitfunction.Therefore,eachvertexwillnotonlycontainapositionbutascalarvalueindicatingtheestimatedvalueofatthatvertex.OurapproachtodetectingfeaturesfromimplicitfunctionsissimilartothetechniqueofGarland[3]fordetectingfea-turesofsurfaces.Todeterminethevertexthatapproximatesthefeatureofinsideofacell,weusequadraticerrorfunctions(QEFs)(asdevelopedin[3]).TogeneratetheseQEFs,wecomputetangentplanestothegraphofonagridofpointssampledover.Atasam-plepoint,thetangentplanetohastheequationwhereandisthegradientof.TobuildtheQEF,wesquarethisequationandsumoverallsamplepointsyielding(1)Thedenominatorofthisexpressionnormalizesthecontri-butionofeachtangentplanetohaveequalweight.Next,weminimizethisquadraticfunctionoverthecelltondthevertexofthedualgrid.Ifthemin-imizerisunderdetermined,wefollowthemethodofLind-strom[7]andusethepseudo-inversetopositionthemini-mizerascloseaspossibletothecenterof.Noticethatourdescriptionofthisfeatureisola-tionphasedoesnotspecifywhatformatmusttakeon.Specically,ourdescriptionallowsforawideva-rietyofpossibleinputs.Forinstance,uniformscalargridstypicallyassociatedwithMarchingCubescanbeusedwhereeachcubeinthegridisregardedasatrilinearfunc-tionsothatandarewelldened.Al-ternatively,wecanrestrictoursamplingabovetothegridpointsofthisuniformgridandusedivideddiffer-encestoestimateatthesevertices.Thelat-terstrategyhastheadvantageofdeningasinglenormalatthegridverticesinsteadofseveraldiscon-tinuousnormalsastheformermethoddoes.Directeddis-tanceeldsprovideanalternativetouniform,scalareldsandcanprovidemoreaccurategradientstohelprepro-ducesharpfeatures.CSGtreesareanotherpossibleinputtoouralgorithm.CSGperformssetoperationsonsolidswheretheleavesFigure2.Connectivityofthedualgrid(thickblack)foraprimalquadtree(thinblue).oftheCSGtreesaresolidsandinteriornodesareopera-tionssuchasunionandintersection.TousesuchCSGtreesinouralgorithm,wemustconvertthetreestoscalarval-uedfunctionsthatcontainawelldenedgradi-ent.AttheleavesoftheCSGtree,wereplacethesolidswiththeirsignedEuclideandistancefunctionsforprimitivessuchasplanes,cylinders,etc...Atinteriornodes,wereplaceunionandintersectionoperationswithMinandMaxoperationsrespectively.EvaluationoftheCSGtreeatapointsimplyinvolvesevaluationofthedistancefunctionsofeachleafatandperformingthecorre-spondingoperationsattheinteriornodesoftheCSGtree.Tocomputethegradient,isevaluatedatthecor-respondingpointateachoftheleavesoftheCSGtreeandpassedupwardsduringtheMin/Maxoperationsalongwiththevalueofthefunctionatthatpoint.Figures1and6wereeachgeneratedbyevaluatingaCSGtree.2.1.OctreeconstructionOurcurrentdescriptionoffeatureisolationhasassumedthatanoctreeispresenttopartitionspaceintocells.How-ever,sometimesitisdesirabletoapproximatetoagiventolerance.Toperformthisapproximation,wepresentatop-downoctreeconstructionalgorithm.Thisoctreeconstructionalgorithmproceedsbystartingwithasinglecellastheoctree.Weusethesamplingalgo-rithmabovetosamplenelyonauniformgridover(arandomsamplingstrategycouldalsobeused).Next,weconstructaQEFforusingthosesamplepointsandminimizetheQEF.Iftheerrorfromequa-tion1isgreaterthan,thenwesubdividethecellintoeightsub-cellsandproceedrecursively.Theprocedurestopswhenalloftheleavesoftheoctreehaveerrorlessthanandconstructsaminimaloctreetoapproximate. Figure3.RecursivefunctionsfaceProc(black),edgeProc(darkgray)andvertProc(lightgray)usedinenumeratingcellsofthedualgrid.3.TopologycreationGivenanoctree,featureisolationgeneratesverticesofthedualgridthatthesurfacewillbeextractedfrom.Topol-ogycreationactuallygeneratesthetopologyofthedualgrid.Thisdualgridistopologicallydualtotheoctree.Therefore,foreachvertexintheoctree,acellinthedualgridwillbecreatedwhoseverticesarethefeatureverticesinsideofeachcubeintheoctreecontainingthatvertex.Figure2showsthetopologyofadualgridcreatedfromanexamplequadtreewhereeachvertexofthedualgridisplacedatthecenterofitscube.Althoughouralgorithmop-eratesonoctrees,thediscussioninthissectionwillfocusonquadtreesforsimplicity;however,thealgorithmsdescribedherenaturallyextendtooctrees.Givenaquadtree,ourtaskistoenumerateeachcellofthedualgridinanefcientmannerwithoutanyexplicitneighborndinginthequadtree.Oursolutionisarecur-sivetraversalofthatenumeratestuplesofallleafsquaresthatshareacommonvertexandisasimpleextensionofthequadtreetraversalusedin[4].Thetraversalinvolvesthreerecursivefunc-tionsfaceProc[ ],edgeProc[ ,]andvertProc[ ,, , ].Givenaninteriornode inthequadtree,faceProc[ ]recursivelycallsitselfonthefourchildrenof aswellascallingedgeProconallfourpairsofedge-adjacentchildrenofandonecalltovert-Proconitsfourchildren.Givenapairofedge-adjacentin-teriornodesand,edgeProc[,]recursivelycallsitselfonthetwopairsofedge-adjacentchildrenspan-ningthecommonedgebetweenandaswellasmak-ingasinglecalltovertProconthefourchildrenof andthattouchthemidpointofthiscommonedge.Fi-Figure4.ThedualgridoftheMaxoftwolinearfunctionswiththeplanedrawntrans-parently(left).Thezero-contourofthisfunc-tion(bold)aswellastheprojectionsofthedualgridontothecontourplane(right).nally,givenfourinteriornodes ,, and thatshareacommonvertex,vertProc[ ,, , ]recur-sivelycallsitselfonthefourchildrenthatmeetatthecom-monvertex.Duringthesecalls,ifoneoftheisaleaf,thenitschildrencannotbeusedforsubsequentrecur-sivecalls.Inthiscase,acopyofispassedtotherecursivecallforeachchildofrequiredbytherecursion.Fig-ure3depictsthemutuallyrecursivestructureofthesethreefunctions.Therecursivecallstothesefunctionsterminatewhenallareleavesofthequadtree.Atthispointinvert-Proc,wehavefourleavessharingthesamecorner.Wethenconstructacellofthedualgridbyconnectingthever-ticesgeneratedbyfeatureisolationforeachcelltotopolog-icallyformasquare.Inadaptivecases,oneofthemayactuallyberepeated,whichcausesthesquaretogeometri-callyformatriangle.NotetherunningtimeofthismethodislinearinthesizeofthequadtreesincethereisonecalltofaceProcforeachsquareinthequadtree,onecalltoedgeProcforeachedgeinthequadtreeandonecalltovertProcforeachvertexinthequadtree.Asdescribedabove,callingfaceProc[ ]generatesonlycellsthatareinteriortotherootofthequadtree.Toex-tendthedualgridtotheboundaryofthequadtree,wetreatthequadtreeasbeingthecentersquareinadegener-aterectangulargrid.Foursquaresinthisgriddegeneratetoedgesofthequadtreewhiletheremainingfoursquaresde-generatetotheverticesofthequadtree.CallingedgeProcwiththerootofthequadtreeandeachofitsdegenerateedgeneighborsgeneratesthosetilestouchingthecorrespondingedgeofthequadtree.Likewise,callingvertProcateachvertexoftherootwithitsthreedegeneratevertexneigh-borsgeneratesthesingledualcelltouchingthevertex.Fig-ure2showsadualgridgeneratedusingthismethodthatcontainsverticesontheedgesandcornersofthegrid. Figure5.Contouredspherewithoutsliverelimination(left)andwithsliverelimination(right).4.ContouringdualgridsAftergeneratingthedualgrid,weextractthesurfaceus-ingasimpleextensionofMarchingCubestothesedualgrids.MarchingCubeswasoriginallydesignedtooperateonlyonaxis-alignedcubes.However,thecellsofourdualgridhavearbitraryverticesinspace.Oursolutionistono-ticethateachcellinthedualgridgeneratedbytherecursivealgorithminsection3istopologicallyequivalenttoacube(althoughsomeverticesmayberepeatedduetothemulti-resolutionstructureoftheoctree).Usingthescalarvaluesattheverticesofthedualgrid,wecomputeedgeintersec-tionpointsforeachedgethatcontainsasignchange.Thenweusethelook-uptableprovidedbyMarchingCubestogeneratethetopologyofthesurfaceinteriortothatcell.Sincetheverticesofthedualgridlieonthefeaturesof,thecontoursofthisdualgridalsoexhibitsharpfeaturessimilartothoseproducedbyExtendedMarchingCubesandDualContouring.TheprincipleadvantageofDualMarchingCubesoverthesemethodsisthattheunder-lyingoctreeusedtoproduceanequivalentcontourismuchsparser.Tounderstandthisphenomena,weconsidera2Dexam-ple.Figure4(left)containsthedistancefunctionfortwocloselyspace,parallellinesshownontherightoftheg-ure.ThesetwolinearfunctionsareformedbytakingtheMaxofthedistancefunctionforeachline.Theleftofthegureshowsthedualgridofwhichconsistsoffourquadswheretheheightofthevertexisformedbythescalarvalue.Contouringeachofthesequadsyieldsfourlinesegmentsindependentofthespacingofthetwoparallellines.InmethodssuchasEMCandDualContouring,thedensityoftheprimalgridusedtoresolvethetwoseparatecontourlinesofthisfunctiondependsontheseparationdis-tanceofthetwocontours.WithDualMarchingCubes,thegridusedincontouringthefunctionadaptstofeaturesofthedistancefunctionasopposedtofeaturesofthecontour.Thisdistinctionoftenavoidstheneedforhighlevelsofre-nementtoseparatecloselyspacedfeaturesofthecontour.Asamorecomplex3Dexample,considerthethin-walledroomofgure1.Thisroomisdesignedasase-quenceofCSGoperationsbasedonplanarprimitives.ThelowerleftmeshistheMarchingCubescontourofthisdis-tancefunctiononanegridtoreproducethethinwalls(notetheroundingofsharpedges).Thelowerrightmesh,producedbyDualContouring,exactlyreproducestheshapeoftheroombutcannotsimplifythenumerousatregionsinthemeshduetothethinnessoftheroom'swalls.OurmethodcomputesanoctreetoapproximatetheCSGtreeusingthemethodofsection2.1.TheresultingsurfaceontheupperrightwasgeneratedbyDualMarchingCubesbycontouringthedualgridofthatoctree.Thissurfaceexactlyreproducestheshapeoftheroombutusesfarfewerpoly-gons.Notethatasthethicknessofthewallsofthisroomde-creases,thenumberofpolygonsusedtocontourthisroomusingMarchingCubesandDualContouringincreasesdra-maticallywhilethenumberofpolygonsproducedbyDualMarchingCubesremainsroughlyconstant.OnedrawbackwithusingMarchingCubestocontourthesedualgridsisthatittendstoproducenumeroussliverpolygonswhentheverticesofthedualgridlieclosetotheplane.Thesolutiontothisproblemistopositiontheverticesofthedualgridtolieexactlyonwhenpossi-ble.Theeffectontheresultingcontouristoeliminatemostsliverpolygons.Toachievethispositioning,werstcom-putetheminimizeroftherestrictedQEF.Iftheresidualatislessthan,isusedasthevertexforthatcube.Otherwise,weperformfeatureisolationasbe-fore.Figure5showsthecontoursgeneratedbyDualMarch-ingCubesillustratingsliverelimination.Figure6depictsanexampleofarocketconstructedasasequenceofCSGoperationsandtheapproximatingmod-elsusingDualMarchingCubes,MarchingCubesandDualContouring.Therocketisaninterestingexamplebecausethethinnsonthebottomoftheshapeexhibitve-foldsymmetry.Whileprimalgridsperformwellwhenfeaturesareaxisaligned,thesensdonotconformtoparameterlinesontheprimalgrid.Consequently,methodssuchasMarchingCubesandDualContouringhaveadifculttimereproducingthesethinfeatures.Ontheotherhand,DualMarchingCubesconstructsadualgridthatconformstothensandcanreconstructthemaccuratelydespitethefactthatthensdonotalignwiththeoctree.Figure7illustratesadaptivesurfaceextractionusingahorseandadragon.Onthehorse,thepolygonsaremorecoarseonthebodythanalongtheneckandthelegsofthehorse.Despitethefactthatthelegsarethincomparedtotherestofthehorse,theyarereproducedaccurately.Thedragonalsoshowsanadaptivepolygonalizationwhereitsfaceandfeetcontainnerpolygonsthantherestofitsbody.ThoughwehavesimplyrunMarchingCubesonthesemod-elswithoutanymulti-resolutionextensions,thestructure Figure6.CSGmodelofarocket(upperleft)andmodelsapproximatingtheshapeusingthesamenumberofpolygons.DualMarchingCubes(upperright),MarchingCubes(lowerleft)andDualContouring(lowerright).ofthedualgridnaturallygeneratesthesemulti-resolutioncontours.Furthermore,sinceweuseMarchingCubestocontourthesemodels,theresultingsurfaceistopologicallymanifold.5.ImplementationWhenimplementingthismethod,weabstractoutthedif-ferentpossibleinputstoouralgorithmasafunctionthatwecanevaluatetodeterminethescalarvalueandgradientatoursamplepoints.SinceweareestimatingsharpfeaturesusingQEFs,thesameproblemsariseforvolumesasfoundinsurfacemethods.Forinstance,whenminimizingequa-tion1,thefeaturevertexmaylieoutsideofthecubethatgeneratedit.Inthiscase,weminimizetheQEFovertheboundaryofthecubeasinLindstrom[8].Also,noticethatthereisnothingthatrequiresweplaceverticesatthefeaturesofthedistancefunction.WhilethisFigure7.AdaptivesurfacesextractedusingDualMarchingCubes.placementreproducessharpfeatures,onecouldmodifythefeatureisolationphasetosimplysampleatthecenterofeachcube.Theresultingalgorithmisacontouringalgorithmsimilarto[10]exceptnospecialcasesaregener-atedinadaptivecongurationsandamanifoldsurfaceisal-waysproduced.Asfarasperformanceisconcerned,contouringdualgridstakesnolongerthancontouringprimalgrids.Sincedualgridsconformbettertothefeaturesof,sparsergridscanbeusedtoextractsurfaces,whichcanac-tuallydecreasethetimetakentocontour.However,dualgridgenerationisstillsomewhatslow.Featureisolationtooksecondsforgure1,secondsforgure6andseveralminutesforgure7usingtheoctreeconstructional-gorithmfromsection2.1onaGHzPentium.Onewayofsolvingthisproblemistorealizethatmostofthedualgriddoesnotdirectlycontributetotheex-tractedsurface.Onlycellsofthedualgridthatcontainasignchangewillcontainapieceofthesurface.Therefore,itmaybepossibletodevelopanalgorithmthatrestrictsthefeatureisolationandtopologycreationphasestoonlythosecellsthatcontainapieceoftheextractedsurface.Weantic-ipatethespeedofthisoptimizedmethodtobecomparablewithotherpopularmethods.6.FutureWorkWebelievethattheconceptofdualgridgenerationhasapplicationsoutsideofcontouring.Infact,thisgridgener-ationactuallyextractsapiecewiselinearapproximationtoagivenfunctionoveracubicaldomain.Withthis 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