Figure1Leftoriginalmesh3 ID: 121440
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NormalMeshesIgorGuskovKirilVidimÿMississippiStateUniversityWimSweldensBellLaboratoriesPeterSchr¬ Figure1:Left:originalmesh(3ßoats/vertex).Middle:twostagesofouralgorithm.Right:normalmesh(1ßoat/vertex).(SkulldatasetcourtesyHeadus,Inc.) scribeanalgorithmwhichtakesanarbitrarytopologyinputmeshandproducesasemi-regularnormalmeshdescribingthesamege-ometry.Asidefromasmallamountofbasedomaininformation,ournormalmeshtransformconvertsanarbitrarymeshfroma3parameterrepresentationintoapurelyscalarrepresentation.Wedemonstrateouralgorithmbyapplyingittoanumberofmodelsandexperimentallycharacterizesomeofthepropertieswhichmakenormalmeshessoattractiveforcomputations.Thestudyofnormalmeshesisofinterestforanumberofrea-sons:theybringourcomputationalrepresentationsbacktowardstheÒÞrstprinciplesÓofdifferentialgeometry;areverystorageandbandwidthefÞcient,describingasurfaceasasuccinctlyspeciÞedbaseshapeplusahierarchicalnormalareanexcellentrepresentationforcompressionsinceallvari-anceisÒsqueezedÓintoasingledimension.1.2RelatedWorkEfÞcientrepresentationsforirregularconnectivitymesheshavebeenpursuedbyanumberofresearchers.Thisresearchismo-tivatedbyourabilitytoacquiredenselysampled,highlydetailedscansofrealworldobjects[19]andtheneedtomanipulatetheseef-Þciently.Semi-regularÑorsubdivisionconnectivityÑmeshesoffermanyadvantagesovertheirregularsettingdueoftheirwelldevel-opedmathematicalfoundationsanddatastructuresimplicity[23];manypowerfulalgorithmsrequiretheirinputtobeinsemi-regularform[21,22,25,1].Thishasledtothedevelopmentofanumberofalgorithmstoconvertexistingirregularmeshestosemi-regularformthroughremeshing.Ecketal.[9]useVoronoitilingandhar-monicmapstobuildaparameterizationandremeshontoasemi-regularmesh.KrischnamurthyandLevoy[15]demonstrateduserdrivenremeshingforthecaseofbi-cubicpatches,whileLeeetal.[18]proposedanalgorithmbasedonfeaturedrivenmeshreduc-tiontodevelopsmoothparameterizationsofmeshesinanautomaticfashion.Thesemethodsusetheparameterizationsubsequentlyforsemi-regularremeshing.Ourworkisrelatedtotheseapproachesinthatwealsoconstructasemi-regularmeshfromanarbitraryconnectivityinputmesh.However,inpreviousworkpredictionresiduals,ordetailvectors,werenotoptimizedtohavepropertiessuchasnormality.Themainfocuswasontheestablishmentofasmoothparameterizationwhichwasthensemi-regularlysampled.Thediscussionofparameterversusgeometryinformationorig-inatesintheworkdoneonirregularcurveandsurfacesubdivi-sion[4][13]andintrinsiccurvaturenormalßow[5].Thereitisshownthatunlessonehasthecorrectparametersideinformation,itisnotpossibletobuildanirregularsmoothsubdivisionscheme.Whilesuchschemesareusefulforeditingandtexturingapplica-tions,theycannotbeusedforsuccinctrepresentationsbecausetheparameterside-informationneededisexcessive.Inthecaseofnor-malmeshestheseissuesareentirelycircumventedinthatallpa-rameterinformationvanishesandthemeshisreducedtopurelyge-ometric,i.e.,scalarinthenormaldirection,information.Finally,wementiontheconnectiontodisplacementmaps[3],andinparticularnormaldisplacementmaps.Thesearepopularformodelingpurposesandusedextensivelyinhighendrender-ingsystemssuchasRenderMan.Inasensewearesolvingheretheassociatedinverseproblem.Givensomegeometry,Þndasim-plergeometryandasetofnormaldisplacementswhichtogetherareequivalenttotheoriginalgeometry.Typically,normaldisplacementmapsaresinglelevel,whereasweaimtobuildtheminafullyhi-erarchicalway.Forexample,singleleveldisplacementsmapswereusedin[15]tocapturetheÞnedetailofa3Dphotographymodel.Cohenetal.[2]samplednormalÞeldsofgeometryandmaintainedtheseintexturemapsduringsimpliÞcation.WhiletheseapproachesalldiffersigniÞcantlyfromourinterestshere,itisclearthatmapsofthisandrelatednatureareofgreatinterestinmanycontexts.Inindependentwork,Leeetal.pursueagoalsimilartoours[17].Theyintroducedisplacedsubdivisionsurfaceswhichcanbeseenasatwolevelnormalmesh.Becauseonlytwolevelsareused,thebasedomaintypicallycontainsmoretrianglesthaninourcase.Alsothenormaloffsetsareoversampledwhileinourcase,thenormaloffsetsarecriticallysampled.2NormalPolylinesBeforewelookatsurfacesandnormalmeshes,weintroducesomeoftheconceptsusingcurvesandnormalpolylines.Acurveintheplaneisdescribedbyapairofparametricfunctions.Wewouldliketodescribethepointsonthecurvewithasinglescalarfunction.Inpracticeoneusespoly-linestoapproximatethefunction.Letbethelinearsegmentbetweenthepoints.AstandardwaytobuildapolylinemultiresolutionapproximationistosamplethecurveatpointsanddeÞnethethlevelapproximationasTomovefromweneedtoinsertthepoints(Figure2,left).Clearlythisrequirestwoscalars:thetwocoordi-natesof.Alternativelyonecouldcomputethedifferencebetweenthenewpointandsomepredictedpoint,saythemidpointoftheneighboringpointsThisdetailhasatangentialcomponentandanormalcom-.Thenormalcomponentisthegeometricinformationwhilethetangentialcomponentistheparameterinfor-mation.Thewaytobuildpolylinesthatcanbedescribedwithone Figure2:Removingonepointinapolylinemultiresolu-tionandrecordingthedifferencewiththemidpoint.Ontheleftageneralpolylinewherethedetailhasbothanormalandatangen-tialcomponent.Ontherightanormalpolylinewherethedetailispurelynormal.scalarperpoint,istomakesurethattheparameterinformationisalwayszero,i.e.,,seeFigure2,right.IfthetriangleisIsosceles,thereisnoparameterinformation.ConsequentlywesaythatapolylineisnormalifamultiresolutionstructureexistswhereeveryremovedpointformsanIsoscelestrian-glewithitsneighbors.Thenthereiszeroparameterinformationandthepolylinecanberepresentedwithonescalarperpoint,namelythenormalcomponentoftheassociateddetail.Forageneralpolylinetheremovedtrianglesarehardlyeverex-actlyIsoscelesandhencethepolylineisnotnormal.Belowwedescribeaproceduretobuildanormalpolylineapproximationforanycontinuouscurve.TheeasiestistostartbuildingIsoscelestri-anglesfromthecoarsestlevel.StartwiththeÞrstbaseseeFigure3.Nexttakeitsmidpointandcheckwherethenormaldirectioncrossesthecurve.Becausethecurveiscontinuous,therehastobeatleastonesuchpoint.Iftherearemultiplepickanyone. Figure3:Constructionofanormalpolyline.Westartwiththecoarsestlevelandeachtimecheckwherethenormaltothemidpointcrossesthecurve.Forsimplicityonlytheindicesoftheareshownandonlycertainsegmentsaresubdivided.Thepolylineisdeterminedbyitsendpointsandthreescalars,theheightsoftheIsoscelestriangles.CallthispointanddeÞnetheÞrsttriangle.Nowsplitthecurveintotwopartsandrepeattheprocedureoneachsubcurve.EachisfoundwherethenormaltothemidpointofcrossestheportionofthecurvebetweenThusanycontinuouscurvecanbeapproximatedarbitrarilycloselywithanormalpolyline.Theresultisaseriesofpolylinesallofwhicharenormalwithrespecttomidpointprediction.Effectivelyeachlevelisparameterizedwithrespecttotheonecoarserlevel.Becausethepolylinesarenormal,onlyasinglescalarvalue,thenormalcomponent,needstoberecordedforeachpoint.Wehaveapolylinewithnoparameterinformation.Onecanalsoconsidernormalpolylineswithrespecttofancierpredictors.Forexampleonecouldcomputeabasepointandnor-malestimateusingthewellknown4pointrule.Essentiallyanypredictorwhichonlydependsonthecoarserlevelisallowed.Forexampleonecanalsouseirregularschemes[4].Alsoonedoesnotneedtofollowthestandardwayofbuildinglevelsbydownsam-plingeveryotherpoint,butinsteadcouldtakeanyordering.ThisleadstothefollowingdeÞnitionofanormalpolyline:DeÞnition1Apolylineisnormalifaremovalorderofthepointsexistssuchthateachremovedpointliesinthenormaldirectionfromabasepoint,wherethenormaldirectionandbasepointonlyde-pendontheremainingpoints.Henceanormalpolylineiscompletelydeterminedbyascalarcom-ponentpervertex.Normalpolylinesarecloselyrelatedtocertainwellknownfrac-talcurvessuchastheKochSnowßake,seeFigure4.Hereeachtimealinesegmentisdividedintothreesubsegments.TheleftandrightgetanormalcoefÞcientofzero,whilethemiddlereceivesanormalcoefÞcientsuchthattheresultingtriangleisequilateral.Hencethepolylinesleadingtothesnowßakearenormalwithre-specttomidpointsubdivision. Figure4:FournormalpolylinesconvergingtotheKochsnowßake. NielsFabianHelgevonKoch(Sweden,1870-1924)Thereisalsoacloseconnectionwithwavelets.Thenormalco-efÞcientscanbeseenasapiecewiselinearwavelettransformoftheoriginalcurve.BecausethetangentialcomponentsarealwayszerotherearehalfasmanywaveletcoefÞcientsasthereareorigi-nalscalarcoefÞcients.Thusonesaves50%memoryrightaway.Inadditionofcoursethewaveletshavetheirusualdecorrelationprop-erties.Inthefunctionalcasetheabovetransformcorrespondstoanunliftedinterpolatingpiecewiselinearwavelettransformasintro-ducedbyDonoho[6].Thereitisshownthatinterpolatingwaveletswithnoprimal,butmanydualmomentsarewellsuitedforsmoothfunctions.Unlikeinthefunctionsetting,notallwaveletsfromthesamelevelhavethesamephysicalscale.HerethescaleofeachcoefÞcientisessentiallythelengthofthebaseofitsIsoseclestrian-3NormalMeshesWebeginbyestablishingterminology.Atrianglemeshisa,whereisasetofpointpositions,andisanabstractsimplicialcomplexwhichcontainsallthetopological,i.e.,adjacencyinfor-mation.Thecomplexisasetofsubsetsof .Thesesubsetscomeinthreetypes:vertices,edges ,andfaces .Twoverticesneighbors .The1-ringneighborsofavertexformaset WecanderiveadeÞnitionofnormaltrianglemeshesinspiredbythecurvecase.ConsiderahierarchyoftrianglemeshesbuiltusingmeshsimpliÞcationwithvertexremovals.Thesemeshesarenestedinthesensethat.Takearemovedvertex.ForthemeshtobenormalweneedtobeabletoÞndabasepointandnormaldirectionthatonlydependonsothatliesinthedirection.ThisleadstothefollowingDeÞnition2AmeshisnormalincaseasequenceofvertexremovalsexistssothateachremovedvertexliesonalinedeÞnedbyabasepointandnormaldirectionwhichonlydependsontheremainingvertices.ThusanormalmeshcanbedescribedbyasmallbasedomainandonescalarcoefÞcientpervertex.Asinthecurvecase,ameshisingeneralnotnormal.Thechancethatthedifferencebetweenaremovedpointandapredictedbasepointliesexactlyinadirectionthatonlydependsontheremainingverticesisessentiallyzero.Hencetheonlywaytoobtainanormalmeshistochangethetriangulation.Wedecidetousesemi-regularmeshes,i.e.,mesheswhoseconnectivityisformedbysuccessivequadrisectionofcoarsebasedomainfaces.Asinthecurvesetting,thewaytobuildanormalmeshistostartfromthecoarselevelorbasedomain.ForeachnewvertexwecomputeabasepointaswellasanormaldirectionandcheckwherethelinedeÞnedbythebasepointandnormalintersectsthesurface.Thesituation,however,ismuchmorecomplexthaninthecurvecasefortworeasons:(1)Therecouldbenointersectionpoint.(2)Therecouldbemanyintersectionpoints,butonlyonecorrectone.Incasetherearenointersectionpoints,strictlyspeakingnofullynormalmeshcanbebuiltfromthisbasedomain.Ifthathappens,werelaxthedeÞnitionofnormalmeshessomeandallowasmallnumberofcaseswherethenewpointsdonotlieinthenormaldi-rection.ThusthealgorithmneedstoÞndasuitablenon-normallo-cationforthenewpoint.IncasetherearemanyintersectionpointsthealgorithmneedstoÞgureoutwhichoneistherightone.Ifthewrongoneischosenthenormalmeshwillstartfoldingoveritselforleavecreases.Anyalgorithmwhichblindlypicksanintersectionpointisdoomed. ParameterizationInordertoÞndtherightpiercingpointorsuggestagoodalternate,oneneedstobeabletoeasilynavigatearoundthesurface.Thewaytodothisistobuildasmoothpa-rameterizationofthesurfaceregionofinterest.Thisisabasicbuildingblockofouralgorithm.Severalparameterizationmeth-odshavebeenproposedandourmethodtakescomponentsfromeachofthem:meshsimpliÞcationandpolarmapsfromMAPS[18],patchwiserelaxationfrom[9],andaspeciÞcsmoothnessfunctionalsimilartotheoneusedin[10]and[20].Thealgorithmwilluselo-calparameterizationswhichneedtobecomputedfastandrobustly.Mostofthemaretemporaryandarequicklydiscardedunlesstheycanbeusedasastartingguessforanotherparameterization.Consideraregionofthemeshhomeomorphictoadiscthatwewanttoparameterizeontoaconvexplanarregion,i.e.,Þndabijectivemap.ThemapisÞxedbyaboundarycon-andminimizesacertainenergyfunctional.Sev-eralfunctionalscanbeusedleadingto,e.g.,conformalorharmonicmappings.WetakeanapproachbasedontheworkofFloater[10].Inshort,thefunctionneedstosatisfythefollowingequationintheinterior:isthe1-ringneighborhoodofthevertexandthecomefromtheshape-preservingparameterizationscheme[10].ThemainadvantageoftheFloaterweightsisthattheyarealwayspositive,which,combinedwiththeconvexityoftheparametricregion,guaranteesthatnotriangleßippingcanoc-curwithintheparametricdomain.Thisiscrucialforouralgorithm.Notethatthisisnottrueingeneralforharmonicmapswhichcanhavenegativeweights.Weusetheiterativebiconjugategradientmethod[12]toobtainthesolutiontothesystem(1).Giventhatweoftenhaveagoodstartingguessthisconvergesquickly.Ouralgorithmconsistsof7stageswhicharede-scribedbelow,someofwhichareshownforthemoleculemodelinFigure5.Themoleculeisahighlydetailedandcurvedmodel.AnynaiveprocedureforÞndingnormalmeshesisveryunlikelytoTheÞrstfourstagesofthealgorithmpreparethegroundforthepiercingprocedureandbuildthenetofcurvessplittingtheoriginalmeshintotriangularpatchesthatareinone-to-onecorrespondencewiththefacesofthebasemesh,i.e.,thecoarsestlevelofthesemi-regularmeshwebuild.1.MeshsimpliÞcation:WeusetheGarland-Heckbert[11]simpliÞcationbasedonhalf-edgecollapsestocreateameshhierar-.Weusethecoarsestlevelasaninitialguessforourbasedomain.TheÞrstimageofFigure5showsthebasedomainforthemolecule.2.Buildinganinitialnetofcurves:Thepurposeofthisstepistoconnecttheverticesofthebasedomainwithanetofnonin-tersectingcurvesonthedifferentlevelsofthemeshsimpliÞcationhierarchy.ThiscaneasilybedoneusingtheMAPSparameteri-zation[18].MAPSusespolarmapstobuildabijectionbetweena1-ringanditsretriangulationafterthecentervertexisremoved.Theconcatenationofthesemapsisabijectivemappingbetweendiffer-entlevelsinthehierarchy.Thedesiredcurvesaresimplytheimageofthebasedomainedgesunderthismapping.Becauseofthebijectionnointersectioncanoccur.NotethatthecurvesstartandÞnishatavertexofthebasedomain,butneednotfollowtheedgesoftheÞnertriangulation,i.e.,theycancutacrosstriangles.ThesecurvesdeÞneanetworkoftriangularshapedpatchescorre-spondingtothebasedomaintriangles.LaterwewilladjustthesecurvesonsomeintermediatelevelandagainuseMAPStopropa-gatethesechangestootherlevels.ThetopmiddleimageofFigure5showsthesecurvesforsomeintermediatelevelofthehierarchy.3.Fixingtheglobalvertices:Anormalmeshisalmostcom-pletelydeterminedbythebasedomain.Onehastochoosethebasedomainverticesverycarefullytoreducethenumberofnon-normalverticestoaminimum.Thecoarsestlevelofthemeshsim-isonlyaÞrstguess.Inthissectionwedescribeaprocedureforrepositioningtheglobalvertices Weimposetheconstraintthattheneedstocoincidewithsomevertexoftheoriginalmesh,butnotnecessarilyTherepositioningistypicallydoneonsomeintermediatelevelTakeabasedomainvertex.Webuildaparameterizationfromthepatchesincidenttovertextoadiskintheplane,seeFig-ure6.Boundaryconditionsareassignedusingarclengthparame-terization,andparametercoordinatesareiterativelycomputedforeachlevelvertexinsidetheshadedregion.Itisnoweasytore-placethepointwithanylevelpointfromintheshadedregion.Inparticularweletthenewbethepointofthatintheparam-eterdomainisclosesttothecenterofthedisk.Theexactcenterofthedisk,ingeneral,doesnotcorrespondtoavertexofthemesh.Onceanewpositionischosen,thecurvescanberedrawnbytakingtheinversemappingofstraightlinesfromthenewpointintheparameterplane.Onecankeepiteratingthisprocedure,butwefoundthatifsufÞcestocycleoncethroughallbasedomainvertices.Wealsoprovideforausercontrolledrepositioning.Thentheusercanreplacethecentervertexwithanypointintheshadedregion.Thealgorithmagainusestheparameterizationtorecomputethecurvesfromthatpoint.ThetoprightofFigure5showstherepositionedvertices.Noticehowsomeofthemliketherightmostonehavemovedconsiderably. cd c Figure6:Basedomainvertexrepositioning.Left:originalpatchesaround,middle:parameterdomain,right:repositionednewpatchboundaries.Thisisreplacedwiththevertexwhosepa-rametercoordinatearetheclosesttothecenter.Theinversemap-ping(right)isusedtoÞndthenewpositionandthenewcurves.4.Fixingtheglobaledges:TheimageoftheglobaledgesontheÞnestlevelwilllaterbethepatchboundariesofthenormalmesh.Forthisreasonweneedtoimprovethesmoothnessoftheas-sociatedcurvesattheÞnestlevel.Weuseaproceduresimilarto[9].Foreachbasedomainedge weconsidertheregionformedontheÞnestlevelmeshbyitstwoincidentpatches.Letbetheopposingglobalvertices.Wethencomputeaparameterfunctionwithinthediamond-shapedregionofthesurface.Theboundaryconditionissetaswithlinearvariationalongtheedges.Wethencomputetheparam-eterizationandletitszerolevelsetbeournewcurve.Againonecoulditeratethisproceduretillconvergencebutinpracticeonecy-clesufÞces.ThecurvesofthetoprightimageinFigure5aretheresultofthecurvesmoothingontheÞnestlevel.Notethatasimilarresultcanbeachievedbyallowingtheusertopositiontheglobalverticesanddrawtheboundariesofthepatchesmanually.Indeed,thefollowingstepsofthealgorithmdonotde-pendonhowtheinitialnetofsurfacecurvesisproduced. Figure5:Theentireprocedureshownforthemoleculemodel.1.Basedomain.2.Initialsetofcurves.3.Globalvertexrepositioning4.InitialParameterization5.Adjustingparameterization6.Finalnormalmesh.(HIVproteasesurfacemodelcourtesyofArthurOlson,TheScrippsResearchInstitute)5.Initialparameterization:OncetheglobalverticesandedgesareÞxed,onecanstartÞllingintheinterior.ThisisdonebycomputingtheparameterizationofeachpatchtoatrianglewhilekeepingtheboundaryÞxed.Theparametercoordinatesfromthelaststagecanserveasagoodinitialguess.Wenowhaveasmoothglobalparameterization.Thisparameterizationisshowninthebot-tomleftofFigure5.Eachtriangleisgivenatriangularchecker-boardtexturetoillustratetheparameterization.6.Piercing:Inthisstageofthealgorithmwestartbuildingtheactualnormalmesh.Thecanonicalstepisforanewvertexofthesemi-regularmeshtoÞnditspositionontheoriginalmesh.Inquadrisectioneveryedgeoflevelgeneratesanewvertexonlevel.WeÞrstcomputeabasepointusinginterpolatingButter-ßysubdivision[8][24]aswellasanapproximationofthenormal.ThisdeÞnesastraightline.Thislinemayhavemultipleintersec-tionpointsinwhichcaseweneedtoÞndtherightone,oritcouldhavenone,inwhichcaseweneedtocomeupwithagoodalternate.Supposethatweneedtoproducethenewvertexthatlieshalfwayalongtheedgewithincidenttriangles,seeFigure7.Letthetwoincidentpatchesformthere-BuildthestraightlinedeÞnedbythebasepointpredictedbytheButterßysubdivisionrule[24]andthedirectionofthenormalcomputedfromthecoarserlevelpoints.WeÞndalltheintersectionpointsofwiththeregionbycheckingalltrianglesinside.Ifthereisnointersectionwetakethepointthatliesmidwaybetweenthepointsintheparameterdomain:.Thisisthesamepointastandardparameteri-zationbasedremesherwoulduse.Notethatinthiscasethedetailvectorisnon-normalanditsthreecomponentsneedtobestored.Inthecasewhenthereexistseveralintersectionsofthemeshre-withthepiercinglinewechoosetheintersectionpointthatisclosesttothepointintheparameterdomain.Letusdenotebytheparametriccoordinatesofthatpiercingpoint. b c b a Figure7:Upperleft:piercing,theButterßypointis,thesurfaceispiercedatthepoint,theparametricallysuggestedpointliesonthecurveseparatingtworegionsofthemesh.Right:parameterdo-main,thepiercedpointfallsinsidetheapertureandgetsaccepted.Lowerleft:theparameterizationisadjustedtoletthecurvepassthroughWeacceptthispointasavalidpointofthesemi-regularmeshif,whereisanÒapertureÓpa-rameterthatspeciÞeshowmuchtheparametervalueofapierced pointisallowedtodeviatefromthecenterofthediamond.Oth-erwise,thepiercingpointisrejectedandthemeshtakesthepointwiththeparametervalue,resultinginanon-normaldetail.7.Adjustingtheparameterization:Oncewehaveanewpiercingpoint,weneedtoadjusttheparameterizationtoreßectthis.Essentially,theadjustedparameterizationshouldbesuchthatthepiercingpointhastheparameters .Whenimpos-ingsuchanisolatedpointconstraintontheparameterization,thereisnomathematicalguaranteeagainstßipping.Hencewedrawanewpiecewiselinearcurvethroughintheparameterdomain.Thisgivesanewcurveonthesurfacewhichpassesthrough,seeFigure7.Wethenrecomputetheparameterizationforeachofthepatchesontoatriangleseparately.Weuseapiecewiselinearbound-aryconditionwiththehalfpointatonthecommonedge.Whenallthenewmidpointsfortheedgesofafaceoflevelarecomputed,wecanbuildthefacesoflevel.Thisisdonebydrawingthreenewcurvesinsidethecorrespondingregionoftheoriginalmesh,seeFigure8.Beforethatoperationhappensweneedtoensurethatavalidparameterizationisavailablewithinthepatch.Thepatchisparameterizedontoatrianglewiththreepiecewiselin-earboundaryconditionseachtimeputtingthenewpointsatthemidpoint.Thenthenewpointsareconnectedintheparameterdo-mainwhichallowsustodrawnewÞnerlevelcurvesontheoriginalmesh.Thisproducesametameshsimilarto[16],sothatthenewnetofcurvesreplicatesthestructureofthesemi-regularhierarchyonthesurfaceoftheoriginal.Theconstructionofthesemi-regularmeshcanbedoneadaptivelywiththeerrordrivenprocedurefromMAPS[18].AnexampleofparameterizationadjustmentaftertwolevelsofadaptivesubdivisionisshowninthebottommiddleofFig-ure5.Notethatastheregionsforwhichwecomputeparameteriza-tionsbecomesmaller,thestartingguessesarebetterandthesolverconvergencebecomesfasterandfaster. Figure8:Facesplit:Quadrisectionintheparameterplane(left)leadstothreenewcurveswithinthetriangularpatch(right).TheapertureparameterofthepiercingprocedureprovidescontroloverhowmuchoftheoriginalparameterizationispreservedintheÞnalmeshandconsequently,howmanynon-normaldetailswillappear.Atwebuildameshentirelybasedontheoriginalglobalparameterization.Atweattempttobuildapurelymeshindependentoftheparameterization.Inourexperience,thebestresultswereachievedwhentheaper-turewassetlow(0.2)atthecoarsestlevels,andthenincreasedto0.6onÞnerlevels.OntheveryÞnelevelsofthehierarchy,wherethegeometryofthesemi-regularmeshescloselyfollowstheorigi-nalgeometry,onecanoftensimplyuseanaivepiercingprocedurewithoutparameteradjustment.Onemaywonderifthecontinuousreadjustmentofparameteri-zationsisreallynecessary.Wehavetriedthenaivepiercingpro-cedurewithoutparameterizationfromthebasedomainandfoundthatittypicallyfailsonallmodels.AnexampleisFigure9whichshows4levelsofnaivepiercingforthetorusstartingfroma102vertexbasemesh.Clearly,thereareseveralregionswithßippedandself-intersectingtriangles.Theerrorisabout20timeslargerthanthetruenormalmesh. Figure9:Naivepiercingprocedure.Clearly,severalregionshaveßippedtrianglesandareself-intersecting. DatasetSizeBaseNormalNotnormal%Timemeshsize(%)error(min) Feline4986415640346729(1.8%).0154Molecule10028379521270(2.8%).0751.5Rabbit16760338235196(2.4%).0372Torus35884985294421(8.0%).033Skull2000211225376817(3.2%).022.5Horse4848523459319644(1.1%).0046.8 Table1:Summaryofnormalmeshingresultsfordifferentmodels.Thenormalmeshiscomputedadaptivelyandcontainsroughlythesamenumberoftrianglesastheoriginalmesh.TherelativeerrorsarecomputedwiththeI.E.I.-CNRMetrotool.Thetimesarereportedona700MHzPentiumIIImachine.4ResultsWehaveimplementedthealgorithmsdescribedintheprecedingsection,andperformedaseriesofexperimentsinwhichnormalmeshesforvariousmodelswerebuilt.ThesummaryoftheresultsisgiveninTable1.Aswecanseefromthetable,thenormalsemi-regularmesheshaveveryhighaccuracyandhardlyanynonnormalOneinterestingfeatureofournormalmeshingprocedureisthefollowing:whilethestructureofpatchescomesfromperformingsimpliÞcationtherearefarfewerrestrictionsonhowcoarsethebasemeshcanbe.NoteforexamplethattheskullinFigure1wasmeshedwiththetetrahedronasbasemesh.Thisislargelyduetotherobustmeshparameterizationtechniquesusedinourapproach.Figure10showsnormalmeshesforrabbit,torus,feline,andskull,aswellasclose-upoffeline(bottomleft)normalmesh.Notehowsmooththemeshesareacrossglobaledgesandglobalvertices.Thissmoothnessmostlycomesfromthenormality,nottheparam-eterization.Itisthusanintrinsicquantity.Oneofthemostinterestingobservationscomingfromthisworkisthatlocallythenormalmeshesdonotdiffermuchfromthenon-normalones,whileofferinghugebeneÞtsintermsofefÞciencyofrepresentation.Forexample,Table2showshowtheÒaperturepa-thatgovernstheconstructionofnormalmeshesaffectsthenumberofdetailcoefÞcientswithnon-trivialtangentialcom-ponentsforthemodelofthethreeholetorus(thesenumbersaretypicalforothermodelsaswell).Inparticular,weseethatalreadyaverymodestacceptancestrategy()getsridofmorethan90%ofthetangentialcomponentsintheremeshedmodel,andthemoreaggressivestrategiesofferevenmorebeneÞtswithoutaffect-ingtheerroroftherepresentation.5SummaryandConclusionInthispaperweintroducethenotionofnormalmeshes.Normalmeshesaremultiresolutionmeshesinwhichverticescanbefoundinthenormaldirection,startingfromsomecoarselevel.Henceonlyonescalarpervertexneedstobestored.Wepresentedarobust normalerror( 00%1.020.291.9%1.050.492.4%1.04best98.3%1.02 Table2:Therelationbetweentheacceptancestrategyduringthepiercingprocedureandthepercentageofperfectlynormaldetailsinthehierarchy.Theoriginalmodelhas5884vertices,allthenor-malmesheshave26002vertices(4levelsuniformly),andthebasemeshcontained98vertices.ThebeststrategyinthelastlineusedontheÞrstthreelevelsandafterwardalwaysacceptedthepiercingcandidates.algorithmforcomputingnormalsemi-regularmeshesofanyinputmeshandshowedthatitproducesverysmoothtriangulationsonavarietyofinputmodels.Itisclearthatnormalmesheshavenumerousapplications.Webrießydiscussafew.Usuallyawavelettransformofastandardmeshhasthreecomponentswhichneedtobequantizedandencoded.In-formationtheorytellsusthatthemorenonuniformthedistributionofthecoefÞcientsthelowertheÞrstorderentropy.Having2/3ofthecoefÞcientsexactlyzerowillfurtherreducethebitbudget.Fromanimplementationviewpoint,wecanalmostdirectlyhookthenor-malmeshcoefÞcientsuptothebestknownscalarwaveletimagecompressioncode.Ithasbeenshownthatoperationssuchassmoothing,enhancement,anddenoisingcanbecomputedthroughasuitablescalingofwaveletcoefÞcients[7].Inanormalmeshanysuchal-gorithmwillrequireonly1/3asmanycomputations.AlsolargescalingcoefÞcientsinastandardmeshwillintroducelargetangen-tialcomponentsleadingtoßippedtriangles.Inanormalmeshthisismuchlesslikelytohappen.TexturingNormalsemi-regularmeshesareverysmoothinsidepatches,acrossglobaledges,andaroundglobalverticesevenwhenthebasedomainisexceedinglycoarse,cf.theskullmodel.Theim-pliedparameterizationsarehighlysuitableforalltypesofmappingNormalmapsareaverypowerfultoolfordecora-tionandenhancementofotherwisesmoothgeometry.Inparticularinthecontextofbandwidthbottlenecksitisattractivetobeabletodownloadanormalmapintohardwareandonlysendsmoothco-efÞcientupdatesfortheunderlyinggeometry.Thenormalmeshtransformeffectivelysolvestheassociatedinverseproblem:con-structanormalmapforagivengeometry.Theconceptofnormalmeshesopensupmanynewareasofre-OuralgorithmusesinterpolatingsubdivisiontoÞndthebasepoint.Buildingnormalmesheswithrespecttoapproximatingsubdivisionisnotstraightforward.Thetheoreticalunderpinningsofnormalmeshesneedtobestudied.Docontinuousvariablenormaldescriptionsofsurfacesexist?Whataboutstability?Whataboutconnectionswithcur-vaturenormalßowwhichactstoreducenormalinformation?Weonlyaddressedsemi-regularnormalmesheshere,whilethedeÞnitionallowsforthemoreßexiblesettingofprogressiveir-regularmeshhierarchies.Purelyscalarcompressionschemesforgeometryneedtobecomparedwithexistingcoders.Generalizenormalmeshestohigherdimensions.ItshouldbepossibletorepresentadimensionalmanifoldinsionswithvariablesasopposedtotheusualThecurrentimplementationonlyworksforsurfaceswithoutboundariesanddoesnotdealwithfeaturecurves.Wewillad-dresstheseissuesinourfutureresearch.AcknowledgmentsThisworkwassupportedinpartbyNSF(ACI-9624957,ACI-9721349,DMS-9874082,DMS9872890),AliasWavefront,aPackardFellow-ship,andaCaltechSummerUndergraduateResearchFellowship(SURF).SpecialthankstoNathanLitkeforhissubdivisionlibrary,toAndreiKhodakovsky,MathieuDesbrun,AdiLevin,ArthurOlson,andZo¬eWoodforhelpfuldiscussions,ChrisJohn-sonfortheuseoftheSGI-UtahVisualSupercomputingCenterresources,andtoCiciKoenigforproductionhelp.DatasetsarecourtesyCyberware,Headus,TheScrippsResearchInstitute,andUniversityofWashington.References[1]CERTAIN,A.,POPOVIC,J.,D,T.,D,T.,SALESIN,D.,TUETZLE,W.InteractiveMultiresolutionSurfaceViewing.ProceedingsofSIGGRAPH96(1996),91Ð98.[2]C,J.,O,M.,,D.Appearance-PreservingSimpli-ProceedingsofSIGGRAPH98(1998),115Ð122.[3]C,R.L.Shadetrees.ComputerGraphics(ProceedingsofSIGGRAPH84),3(1984),223Ð231.[4]DAUBECHIES,I.,GUSKOV,I.,WELDENS,W.RegularityofIrregularSubdivision.Constr.Approx.15(1999),381Ð426.[5]DESBRUN,M.,M,M.,S,P.,,A.H.ImplicitFair-ingofIrregularMeshesUsingDiffusionandCurvatureFlow.ProceedingsofSIGGRAPH99(1999),317Ð324.[6]D,D.L.Interpolatingwavelettransforms.Preprint,DepartmentofStatistics,StanfordUniversity,1992.[7]D,D.L.UnconditionalBasesareOptimalBasesforDataCompressionandforStatisticalEstimation.Appl.Comput.Harmon.Anal.1(1993),100Ð115.[8]D,N.,L,D.,REGORY,J.A.AButterßySubdivisionSchemeforSurfaceInterpolationwithTensionControl.ACMTransactionsonGraphics,2(1990),160Ð169.[9]E,M.,D,T.,D,T.,H,H.,LOUNSBERY,M.,TUETZLE,W.MultiresolutionAnalysisofArbitraryMeshes.ProceedingsofSIGGRAPH95(1995),173Ð182.[10]FLOATER,M.S.ParameterizationandSmoothApproximationofSurfaceTri-ComputerAidedGeometricDesign14(1997),231Ð250.[11]G,M.,ECKBERT,P.S.SurfaceSimpliÞcationUsingQuadricErrorMetrics.InProceedingsofSIGGRAPH96,209Ð216,1996.[12]G,G.H.,OAN,C.F.V.MatrixComputations,2nded.TheJohnHopkinsUniversityPress,Baltimore,1983.[13]GUSKOV,I.,SWELDENS,W.,,P.MultiresolutionSignalPro-cessingforMeshes.ProceedingsofSIGGRAPH99(1999),325Ð334.[14]KHODAKOVSKY,A.,S,P.,SWELDENS,W.ProgressiveGeometryProceedingsofSIGGRAPH2000[15]KRISHNAMURTHY,V.,EVOY,M.FittingSmoothSurfacestoDensePolygonMeshes.ProceedingsofSIGGRAPH96(1996),313Ð324.[16]L,A.W.F.,D,D.,SWELDENS,W.,,P.Multireso-lutionMeshMorphing.ProceedingsofSIGGRAPH99(1999),343Ð350.[17]L,A.W.F.,MORETON,H.,H,H.DisplacedSubd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