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Simpli®cationEnvelopesJonathanCohenAmitabhVarshneyDineshManochaGregTurkHansWeberPankajAgarwalFrederickBrooksWilliamWrighthttp://www.cs.unc.edu/Ägeom/envelope.htmlAbstractWeproposetheideaofsimpli®cationenvelopesforgen-eratingahierarchyoflevel-of-detailapproximationsfora DepartmentofComputerScience,UniversityofNorthCarolina,ChapelHill,NC27599-3175.cohenj,weberh,manocha,turk,brooks,wright@cs.unc.eduDepartmentofComputerScience,StateUniversityofNewYork,StonyBrook,NY11794-4400.varshney@cs.sunysb.eduDepartmentofComputerScience,DukeUniversity,Durham,NC27708-0129.pankaj@cs.duke.edu1IntroductionWepresenttheframeworkofsimpli®cationenvelopes Figure1:Alevel-of-detailhierarchyfortherotorfromabrakeassembly.50 Suchanapproachhasseveralbene®tsincomputergraph-ics.First,onecanverypreciselyquantifytheamountofapproximationthatistolerableundergivencircumstances.Givenauser-speci®ederrorinnumberofpixelsofdevia-tionofanobject'ssilhouette,itispossibletochoosewhichlevelofdetailtoviewfromaparticulardistancetomaintainthatpixelerrorbound.Second,thisapproachallowsonea®necontroloverwhichregionsofanobjectshouldbeap-proximatedmoreandwhichonesless.Thiscouldbeusedforselectivelypreservingthosefeaturesofanobjectthatareperceptuallyimportant.Third,theuser-speci®abletol-eranceforapproximationistheonlyparameterrequiredtoobtaintheapproximations;®netweakingofseveralparam-etersdependingupontheobjecttobeapproximatedisnotrequired.Thus,thisapproachisquiteusefulforautomat-ingtheprocessoftopology-preservingsimpli®cationsofalargenumberofobjects.Thisproblemofscalabilityisim-portantforanysimpli®cationalgorithm.Oneofourmaingoalsistocreateamethodforsimpli®cationwhichisnotonlyautomaticforlargedatasets,buttendstopreservetheshapesoftheoriginalmodels.Therestofthepaperisorganizedinthefollowingman-ner:wesurveytherelatedworkinSection2,explainourassumptionsandterminologyinSection3,describetheen-velopeandapproximationcomputationsinSections4and5,presentsomeusefulextentionstoandpropertiesoftheapproximationalgorithmsinSection6,andexplainourim-plementationandresultsinSection7.2BackgroundApproximationalgorithmsforpolygonalmodelscanbeclassi®edintotwobroadcategories:Min-#Approximations:Forthisversionoftheap-proximationproblem,givensomeerrorbound,theobjectiveistominimizethenumberofverticessuchthatnopointoftheapproximationisfartherthandistanceawayfromtheinputmodelMin-Approximations:Herewearegiventhenum-berofverticesoftheapproximationandtheobjec-tiveistominimizetheerror,orthedifference,betweenandPreviousworkintheareaofmin-#approximationshasbeendoneby[6,20]wheretheyadaptivelysubdivideaseriesofbicubicpatchesandpolygonsoverasurfaceuntilthey®tthedatawithinthetolerancelevels.Inthesecondcategory,workhasbeendonebyseveralgroups.Turk[23]®rstdistributesagivennumberofverticesoverthesurfacedependingonthecurvatureandthenre-triangulatesthemtoobtainthe®nalmesh.Schroederetal.[21]andHinkerandHansen[13]operateonasetoflocalrulesÐsuchasdeletingedgesorverticesfromalmostcoplanaradjacentfaces,followedbylocalre-triangulation.Theserulesareappliediterativelytilltheyarenolongerapplicable.Asomewhatdifferentlocalapproachistakenin[18]whereverticesthatareclosetoeachotherareclusteredandanewvertexisgeneratedtorepresentthem.Themeshissuitablyupdatedtore¯ectthis.Hoppeetal.[14]proceedbyiterativelyoptimizinganenergyfunctionoverameshtominimizeboththedistanceoftheapproximatingmeshfromtheoriginal,aswellasthenumberofapproximatingvertices.Aninterestingandele-gantsolutiontotheproblemofpolygonalsimpli®cationbyusingwaveletshasbeenpresentedin[7,8]wherearbitrarypolygonalmeshesare®rstsubdividedintopatcheswithsubdivisionconnectivityandthenmultiresolutionwaveletanalysisisusedovereachpatch.Thiswaveletapproachpreservesglobaltopology.Incomputationalgeometry,ithasbeenshownthatcom-putingtheminimal-facet-approximationisNP-hardforbothconvexpolytopes[5]andpolyhedralterrains[1].Thus,algorithmstosolvetheseproblemshaveevolvedaround®ndingpolynomial-timeapproximationsthatareclosetheoptimal.bethesizeofamin-#approximation.Analgorithmhasbeengivenin[16]forcomputinganapproximationofsizelogforconvexpolytopes.ThishasrecentlybeenimprovedbyClarksonin[3];heproposesarandomizedalgorithmforcomputinganapprox-imationofsizeloginexpectedtimeforany0(theconstantofproportionalitydependson,andtendstotendsto0).In[2]BrÈonnimannandGoodrichobservedthatavariantofClarkson'salgorithmyieldsapolynomial-timedeterministicalgorithmthatcom-putesanapproximationofsize.Workingwithpoly-hedralterrains,[1]presentapolynomial-timealgorithmthatcomputesan-approximationofsizelogtoapolyhedralterrain.Ourworkisdifferentfromtheaboveinthatitallowsadaptive,genus-preserving,-approximationofarbitrarypolygonalobjects.Additionally,wecansimplifyborderedmeshesandmesheswithholes.Intermsofdirectcompari-sonwiththeglobaltopologypreservingapproachpresentedin[7,8],foragivenouralgorithmhasbeenempiricallyabletoobtainªreduced"simpli®cations,whicharemuchsmallerinoutputsize(asdemonstratedinSection7).Thealgorithmin[18]alsoguaranteesaboundintermsoftheHausdorffmetric.However,itisnotguaranteedtopreservethegenusoftheoriginalmodel.3TerminologyandAssumptionsLetusassumethatisathree-dimensionalcompactandori-entableobjectwhosepolygonalrepresentationhasbeengiventous.Ourobjectiveistocomputeapiecewise-linearapproximation.Giventwopiecewiselinearobjectsand,wesaythatandare-approximationsofeachotheriffeverypointoniswithinadistanceofsomepointofandeverypointoniswithinadistancesomepointof.Ourgoalistooutlineamethodtogeneratetwoenvelopesurfacessurroundinganddemonstratehowtheenvelopescanbeusedtosolvethefollowingpolygonalapproximationproblem.Givenapolygonalrepresentationofanobjectandanapproximationparameter,generateagenus-preserving-approximationwithminimalnum-berofpolygonssuchthattheverticesofareasubsetofverticesofWeassumethatallpolygonsinaretrianglesandthatisawell-behavedpolygonalmodel,i.e.,everyedgehaseitheroneortwoadjacenttriangles,notwotrianglesinter-penetrate,therearenounintentionalªcracks"inthemodel,noT-junctions,etc.Wealsoassumethateachvertexofhasasinglenormalvector,whichmustliewithin90ofthenormalofeachofitssurroundingtriangles.Forthepurposeofrendering,eachvertexmayhaveeitherasinglenormalormultiplenormals.Forthepurposeofgeneratingenvelopesurfaces,weshallcomputeasinglevertexnormalasacombinationofthenormalsofthesurroundingtriangles.Thethree-dimensional-offsetsurfaceforaparametricsurfaces;t)=(s;ts;ts;t whoseunitnormaltos;t)=(s;ts;ts;tisde®nedass;ts;ts;ts;t,wheres;ts;ts;tNotethatoffsetsurfacesforapolygonalobjectcanself-intersectandmaycontainnon-linearelements.Wede®neasimpli®cationenvelope(respectively)foranobjecttobeapolygonalsurfacethatlieswithinadis-tanceoffromeverypointinthesame(respectivelyopposite)directionasthenormalto.Thus,thesimpli-®cationenvelopescanbethoughtofasanapproximationtooffsetsurfaces.Henceforthweshallrefertosimpli®cationenvelopebysimplyenvelope.Letusrefertothetrianglesofthegivenpolygonalrepre-sentationasthefundamentaltriangles.Letbeanedgeof.Ifthenormalsatbothand,respectively,areidentical,thenwecanconstructaplanethatpassesthroughtheedgeandhasanormalthatisperpendiculartothatof.Thusandtheirnormalsallliealong.Suchaplanede®nestwohalf-spacesforedge,sayand(seeFig2(a)).However,ingeneralthenormalsandattheverticesandneednotbeidentical,inwhichcaseitisnotclearhowtode®nethetwohalf-spacesforanedge.Onechoiceistouseabilinearpatchthatpassesthroughandandhasatangentand.Letuscallsuchabilinearpatchforastheedgehalf-space.Letthetwohalf-spacesfortheedgeinthiscasebeand.ThisisshowninFig2(b). ev1v2 -e e+ e1n2nev1v21n2n -e e e Figure2:EdgeHalf-spacesLettheverticesofafundamentaltrianglebe,and.Letthecoordinatesandthenormalofeachvertexrepresentedasand,respectively.Thecoordinatesandthenormalofa-offsetvertexforavertexare:,and.The-offsetvertexcanbesimilarlyde®nedintheoppositedirection.TheseoffsetverticesforafundamentaltriangleareshowninFigure3.Nowconsidertheclosedobjectde®nedbyand3.Itisde®nedbytwotriangles,atthetopandbottom,andthreeedgehalf-spaces.Thisobjectcontainsthefundamentaltriangle(shownshadedinFigure3)andwewillhenceforthrefertoitasthefundamentalprism4EnvelopeComputationInordertopreservetheinputtopologyof,wedesirethatthesimpli®cationenvelopesdonotself-intersect.Tomeetthiscriterionwereduceourlevelofapproximationatcertainplaces.Inotherwords,toguaranteethatnointersectionsamongsttheenvelopesoccur,wehavetobe 1v2v3v3+v3-v-v-v+12v+21 1nn2n3 Figure3:TheFundamentalPrismcontentatcertainplaceswiththedistancebetweenandtheenvelopebeingsmallerthan.Thisishowsimpli®cationenvelopesdifferfromoffsetsurfaces.Weconstructourenvelopesuchthateachofitstrian-glescorrespondstoafundamentaltriangle.Weoffsetvertexoftheoriginalsurfaceinthedirectionofitsnormalvectortotransformthefundamentaltrianglesintothoseoftheenvelope.Ifweoffseteachvertexbythesameamount,togettheoffsetverticesand,theresultingenvelopes,and,maycontainself-intersectionsbecauseoneormoreoffsetverticesareclosertosomenon-adjacentfundamentaltriangle.Inotherwords,ifwede®neaVoronoidiagramoverthefundamentaltrianglesofthemodel,theconditionfortheenvelopestointersectisthattherebeatleastoneoffsetvertexlyingintheVoronoiregionofsomenon-adjacentfundamentaltriangle.ThisisshowninFig-ure4bymeansofatwo-dimensionalexample.Inthe®gure,theoffsetverticesandareintheVoronoiregionsofedgesotherthantheirown,thuscausingself-intersectionoftheenvelope. Figure4:OffsetSurfacesOncewemakethisobservation,thesolutiontoavoidself-intersectionsbecomesquitesimpleÐjustdonotallowanoffsetvertextogobeyondtheVoronoiregionsofitsadjacentfundamentaltriangles.Inotherwords,determinethepositiveandnegativeforeachvertexsuchthattheverticesanddeterminedwiththisnewdonotlieintheVoronoiregionsofthenon-adjacentfundamentaltriangles.Whilethisworksintheory,ef®cientandrobustcom-putationofthethree-dimensionalVoronoidiagramofthefundamentaltrianglesisnon-trivial.Wenowpresenttwomethodsforcomputingthereducedforeachvertex,the®rstmethodanalytical,andthesecondnumerical. 4.1AnalyticalComputationWeadoptaconservativeapproachforrecomputingtheeachvertex.Thisapproachunderestimatesthevaluesforthepositiveandnegative.Inotherwords,itguaranteestheenvelopesurfacesnottointersect,butitdoesnotguar-anteethattheateachvertexisthelargestpermissibleWenextdiscussthisapproachforthecaseofcomputingthepositiveforeachvertex.Computationofnegativefollowssimilarly.Considerafundamentaltriangle.Wede®neaprism,whichisconceptuallythesameasitsfundamentalprism,butusesavalueof2insteadofforde®ningtheenvelopevertices.Next,consideralltrianglesthatdonotshareavertexwith.Ifintersectsabove(thedirectionsaboveandbelowaredeterminedbythedirectionofthenormalto,aboveisinthesamedirectionasthenormalto),we®ndthepointonthatlieswithinandisclosestto.Thispointwouldbeeitheravertexofortheintersectionpointofoneofitsedgeswiththethreesidesoftheprism.Oncewe®ndthepointofclosestapproach,wecomputethedistanceofthispointfromThisisshowninFigure5. 1v2v3 ittp Figure5:ComputationofOncewehavedonethisforall,wecomputethenewvalueofthepositiveforthetriangle minIftheverticesforthistrianglehavethisvalueofpositivetheirpositiveenvelopesurfacewillnotself-intersect.Oncethevaluesforallthetriangleshavebeencomputed,theforeachvertexissettobetheminimumofthevaluesforallitsadjacenttriangles.Weuseanoctreeinourimplementationtospeeduptheidenti®cationoftrianglesthatintersectagivenprism.4.2NumericalComputationTocomputeanenvelopesurfacenumerically,wetakeanit-erativeapproach.Ourenvelopesurfaceisinitiallyidenticaltotheinputmodelsurface.Ineachiteration,wesequen-tiallyattempttomoveeachenvelopevertexafractionofthedistanceinthedirectionofitsnormalvector(ortheoppositedirection,fortheinnerenvelope).Thiseffectivelystretchesorcontractsallthetrianglesadjacenttothevertex.Wetesteachoftheseadjacenttrianglesforintersectionwitheachotherandtherestofthemodel.Ifnosuchintersectionsarefound,weacceptthestep,leavingthevertexinthisnewposition.Otherwisewerejectit,movingthevertexbacktoitspreviousposition.Theiterationterminateswhenallverticeshaveeithermovedorcannolongermove.Inanattempttoguaranteethateachvertexgetstomoveareasonableamountofitspotentialdistance,weuseanadaptivestepsize.Weencourageavertextomoveatleast(anarbitraryconstantwhichisscaledwithrespecttoandthesizeoftheobject)stepsbyallowingittoreduceitsstepsize.Ifavertexhasmovedlessthanstepsanditsmoveisbeenrejected,itdividesitsstepsizeinhalfandtriesagain(withsomemaximumnumberofdividesallowedonanyparticularstep).Noticethatifavertexmovesstepsandisrejectedontheststep,weknowithasmovedatleast%ofitspotentialdistance,sowhichisalowerboundofsorts.Itispossible,thoughrare,foravertextomovelessthansteps,ifitscurrentpositionisalreadyquiteclosetoanothertriangle.Eachvertexalsohasitsowninitialstepsize.We®rstchooseaglobal,maximumstepsizebasedonaglobalprop-erty:eithersomesmallpercentageoftheobject'sboundingboxdiagonallengthor,whicheverissmaller.Nowforeachvertex,wecalculatealocalstepsize.Thislocalstepsizeissomepercentageofthevertex'sshortestincidentedge(onlythoseedgeswithin90oftheoffsetdirectionareconsidered).Wesetthevertex'sstepsizetotheminimumoftheglobalstepsizeanditslocalstepsize.Thismakesitlikelythateachvertex'sstepsizeisappropriateforastepgiventheinitialmeshcon®guration.Thisapproachtocomputinganenvelopesurfaceisro-bust,simpletoimplement(ifdif®culttoexplain),andfairtoallthevertices.Ittendstomaximizetheminimumoff-setdistanceamongsttheenvelopevertices.Itworksfairlywellinpractice,thoughtheremaystillbesomeroomforimprovementingeneratingmaximaloffsetsforthinobjects.Figure6showsinternalandexternalenvelopescomputedforthreevaluesofusingthisapproach.Asintheanalyticalapproach,asimpleoctreedatastruc-turemakestheseintersectiontestsreasonablyef®cient,es-peciallyformodelswithevenlysizedtriangles.5GenerationofApproximationGeneratingasurfaceapproximationtypicallyinvolvesstart-ingwiththeinputsurfaceanditerativelymakingmodi®ca-tionstoultimatelyreduceitscomplexity.Thisprocessmaybebrokenintotwomainstages:holecreation,andhole®lling.We®rstcreateaholebyremovingsomeconnectedsetoftrianglesfromthesurfacemesh.Thenwe®lltheholewithasmallersetoftriangles,resultinginsomereductionofthemeshcomplexity.Wedemonstratethegeneralityofthesimpli®cationen-velopeapproachbydesigningtwoalgorithms.Thehole®llingstagesofthesealgorithmsarequitesimilar,buttheirholecreationstagesarequitedifferent.The®rstalgorithmmakesonlylocalchoices,creatingrelativelysmallholes,whilethesecondalgorithmusesglobalinformationaboutthesurfacetocreatemaximally-sizedholes.Thesedesignchoicesproducealgorithmswithverydifferentproperties.Webeginbydescribingtheenvelopevaliditytestusedtodeterminewhetheracandidatetriangleisvalidforinclusionintheapproximationsurface.Wethenproceedtothetwoexamplesimpli®cationalgorithmsandadescriptionoftheirrelativemerits.5.1ValidityTestcandidatetriangleisonewhichweareconsideringforinclusioninanapproximationtotheinputsurface.Validcandidatetrianglesmustliebetweenthetwoenvelopes.Becauseweconstructcandidatetrianglesfromtheverticesoftheoriginalmodel,weknowitsverticesliebetweenthetwoenvelopes.Therefore,itissuf®cienttotestthecandidatetriangleforintersectionswiththetwoenvelope InnerEnvelopesOuterEnvelopesFigure6:Simpli®cationenvelopesforvarioussurfaces.Wecanperformsuchtestsef®cientlyusingaspace-partitioningdatastructuresuchasanoctree.Avalidcandidatetrianglemustalsonotcauseaself-intersectioninoursurface,Therefore,itmustnotintersectanytriangleofthecurrentapproximationsurface.5.2LocalAlgorithmTohandlelargemodelsef®cientlywithintheframeworkofsimpli®cationenvelopesweconstructavertex-removal-basedlocalalgorithm.Thisalgorithmdrawsheavilyontheworkof[21],[23],and[14].Itsmaincontributionsaretheuseofenvelopestoprovideglobalerrorboundsaswellastopologypreservationandnon-self-intersection.Wehavealsoexploredtheuseofamoreexhaustivehole-®llingapproachthananypreviousworkwehaveseen.Thelocalalgorithmbeginsbyplacingallverticesinaqueueforremovalprocessing.Foreachvertexinthequeue,weattempttoremoveitbycreatingahole(remov-ingthevertex'sadjacenttriangles)andattemptingto®llit.Ifwecansuccessfully®llthehole,themeshmodi®cationisaccepted,thevertexisremovedfromthequeue,anditsneighborsareplacedbackinthequeue.Ifnot,thevertexisremovedfromthequeueandthemeshremainsunchanged.Thisprocessterminateswhentheglobalerrorboundseven-tuallypreventtheremovalofanymorevertices.Oncethevertexqueueisemptywehaveoursimpli®edmesh.Foragivenvertex,we®rstcreateaholebyremovingalladjacenttriangles.Webeginthehole-®llingprocessbygeneratingallpossibletrianglesformedbycombinationsoftheverticesontheholeboundary.Thisisnotstrictlynecessary,butitallowsustouseagreedystrategytofavortriangleswithniceaspectratios.We®lltheholebychoos-ingatriangle,testingitsvalidity,andrecursively®llingthethree(orfewer)smallerholescreatedbyaddingthattrian-gleintothehole(see®gure7).Ifaholecannotbe®lledatanyleveloftherecursion,theentirehole®llingattemptisconsideredafailure.Notethatthisisasingle-passhole®llingstrategy;wedonotbacktrackorundoselectionofatrianglechosenfor®llingahole.Thus,thisapproachdoesnotguaranteethatifatriangulationofaholeexistswewill®ndit.However,itisquitefastandworksverywellinpractice. Figure7:Hole®lling:addingatriangleintoaholecreatesuptothreesmallerholesWehavecomparedtheaboveapproachwithanexhaus-tiveapproachinwhichwetriedallpossiblehole-®llingtri-angulations.Forsimpli®cationsresultingintheremovalofhundredsofvertices(likehighlyoversampledlaser-scannedmodels),theexhaustivepassyieldedonlyasmallimprove-mentoverthesingle-passheuristic.Thissortofcon®rma-tionreassuresusthatthesingle-passheuristicworkswellinpractice.5.3GlobalAlgorithmThisalgorithmextendsthealgorithmpresentedin[3]tonon-convexsurfaces.Ourmajorcontributionistheuseofsimpli®cationenvelopestoboundtheerroronanon-convexpolygonalsurfaceandtheuseoffundamentalprismstoprovideageneralizedprojectionmechanismfortestingforregionsofmultiplecovering(overlaps).Wepresentonlyasketchofthealgorithmhere;see[24]forthefulldetails.Webeginbygeneratingallpossiblecandidatetrianglesforourapproximationsurface.Thesetrianglesareall3-tuplesoftheinputverticeswhichdonotintersecteitheroftheoffsetsurfaces.Nextwedeterminehowmanyverticeseachtrianglecovers.Werankthecandidatetrianglesinorderofdecreasingcovering.Wethenchoosefromthislistofcandidatetrianglesinagreedyfashion.Foreachtrianglewechoose,wecreatealargeholeinthecurrentapproximationsurface,removingalltriangleswhichoverlapthiscandidatetriangle.Nowwebegintherecursivehole-®llingprocessbyplacingthiscandidatetriangleintotheholeand®llingallthesubholeswithothertriangles,ifpossible.Onefurtherrestrictioninthisprocessisthatthecandidatetrianglewearetestingshouldnotoverlapanyofthecandidatetriangleswehavepreviouslyaccepted.5.4AlgorithmComparisonThelocalsimpli®cationalgorithmisfastandrobustenoughtobeappliedtolargemodels.Theglobalstrategyisthe-oreticallyelegant.Whiletheglobalalgorithmworkswellforsmallmodels,itscomplexityrisesatleastquadratically, envelope curveenvelope curveoriginal curveapproximating curve Figure8:Curveatlocalminimumofapproximationmakingitprohibitiveforlargermodels.Wecanthinkofthesimpli®cationproblemasanoptimizationproblemaswell.Apurelylocalalgorithmmaygettrappedinalocalªmin-imumºofsimpli®cation,whileanidealglobalalgorithmwillavoidallsuchminima.Figure8showsatwo-dimensionalexampleofacurveforwhichalocalvertexremovalalgorithmmightfail,butanalgorithmthatgloballysearchesthesolutionspacewillsuc-ceedin®ndingavalidapproximation.Anyoftheinteriorverticesweremovewouldcauseanewedgetopenetrateanenvelopecurve.Butifweremovealloftheinteriorvertices,theresultingedgeisperfectlyacceptable.Thisobservationmotivatesawiderangeofalgorithmsofwhichourlocalandglobalexamplesarethetwoextremes.Wecaneasilyimagineanalgorithmthatchoosesnearbygroupsofverticestoremovesimultaneouslyratherthansequentially.Thiscouldpotentiallyleadtoincreasedspeedandsimpli®cationperformance.However,choosingsuchsetsofverticesremainsachallengingproblem.6AdditionalFeaturesEnvelopesurfacesusedinconjunctionwithsimpli®cationalgorithmsarepowerful,general-purposetools.Aswewillnowdescribe,theyimplicitlypreservesharpedgesandcanbeextendedtodealwithborderedsurfacesandperformadaptiveapproximations.6.1PreservingSharpEdgesOneoftheimportantpropertiesinanyapproximationschemeisthewayitpreservesanysharpedgesornormaldiscontinuitiespresentintheinputmodel.Simpli®cationenvelopesdealgracefullywithsharpedges(thosewithasmallanglebetweentheiradjacentfaces).Whenthetol-eranceissmall,thereisnotenoughroomtosimplifyacrossthesesharpedges,sotheyareautomaticallypreserved.Asthetoleranceisincreased,itwilleventuallybepossibletosimplifyacrosstheedges(whichshouldnolongerbevis-iblefromtheappropriatedistance).Noticethatitisnotnecessarytoexplicitlyrecognizethesesharpedges.6.2BorderedSurfacesAborderedsurfaceisonecontainingpointsthatarehome-omorphictoahalf-disc.Forpolygonalmodels,thiscorre-spondstoedgesthatareadjacenttoasinglefaceratherthantwofaces.Dependingonthecontext,wemaynaturallythinkofthisastheboundaryofsomeplane-likepieceofasurface,oraholeinanotherwiseclosedsurface.Thealgorithmsdescribedin5aresuf®cientforclosedtrianglemeshes,buttheywillnotguaranteeourglobaler-rorboundformesheswithborders.Whiletheenvelopesconstrainourerrorwithrespecttothenormaldirectionofthesurface,borderedsurfacesrequiresomeadditionalconstraintstoholdtheapproximationborderclosetotheoriginalborder.Withoutsuchconstraints,theborderoftheapproximationsurfacemayªcreepin,ºpossiblyshrinkingthesurfaceoutofexistence.Inmanycases,thecomplexityofasurface'sbordercurvesmaybecomealimitingfactorinhowmuchwecansimplifythesurface,soitisunacceptabletoforgosimpli-fyingtheseborders.Weconstructasetofbordertubestoconstraintheerrorindeviationofthebordercurves.Eachborderisactuallyacyclicpolyline.Intuitivelyspeaking,abordertubeisasmooth,non-self-intersectingsurfacearoundoneofthesepolylines.Removingabordervertexcausesapairofborderedgestobereplacedbyasingleborderedge.Ifthisnewborderedgedoesnotintersecttherelevantbordertube,wemaysafelyattempttoremovethebordervertex.Toconstructatubewede®neaplanepassingthrougheachvertexofthepolyline.Wechooseacoordinatesystemonthisplaneandusethattode®neacircularsetofvertices.Weconnecttheseverticesforconsecutiveplanestocon-structourtube.Ourinitialtubeshaveaverynarrowradiustominimizethelikelihoodofself-intersections.Wethenexpandthesenarrowtubesusingthesametechniqueweusedpreviouslytoconstructoursimpli®cationenvelopes.Thedif®culttaskistode®neacoordinatesystemateachpolylinevertexwhichencouragessmooth,non-self-intersectingtubes.ThemostobviousapproachmightbetousetheideaofFrenetframesfromdifferentialgeometrytode®neasetofcoordinatesystemsforthepolylinevertices.However,Frenetframesaremeantforsmoothcurves.Forajaggedpolyline,atubesoconstructedoftenhasmanyself-intersections.Instead,weuseaprojectionmethodtominimizethetwistbetweenconsecutiveframes.LiketheFrenetframemethod,weplacetheplaneateachvertexsothatthenormaltotheplaneapproximatesthetangenttothepolyline.ThisiscalledthenormalplaneAtthe®rstvertex,wechooseanarbitraryorthogonalpairofaxesforourcoordinatesysteminthenormalplane.Forsubsequentvertices,weprojectthecoordinatesystemfromthepreviousnormalplaneontothecurrentnormalframe.Wethenorthogonalizethisprojectedcoordinatesystemintheplane.Forthenormalplaneofthe®nalpolylinevertex,weaveragetheprojectedcoordinatesystemsofthepreviousnormalplaneandtheinitialnormalplanetominimizeanytwistinthe®naltubesegment.6.3AdaptiveApproximationForcertainclassesofobjectsitisdesirabletoperformanadaptiveapproximation.Forinstance,considerlargeter-raindatasets,modelsofspaceships,orsubmarines.Onewouldliketohavemoredetailneartheobserverandlessdetailfurtheraway.Apossiblesolutioncouldbetosub-dividethemodelintovariousspatialcellsanduseadif-ferent-approximationforeachcell.However,problemswouldariseattheboundariesofsuchcellswheretheapproximationforonecell,sayatavalueneednotnec-essarilybecontinuouswiththe-approximationfortheneighboringcell,sayatadifferentvalueSinceallcandidatetrianglesgeneratedareconstrainedtoliewithinthetwoenvelopes,manipulationoftheseen-velopesprovidesonewaytosmoothlycontrolthelevelofapproximation.Thus,onecouldspecifytheatagivenvertextobeafunctionofitsdistancefromtheobserverÐthelargerthedistance,thegreateristheAsanotherpossibility,considerthecasewherecertain featuresofamodelareveryimportantandarenottobeapproximatedbeyondacertainlevel.Suchfeaturesmighthavehumanperceptionasabasisfortheirde®nitionortheymighthavemathematicaldescriptionssuchasregionsofhighcurvature.Ineithercase,ausercanvarytheassociatedwitharegiontoincreaseordecreasethelevelofapproximation.ThebunnyinFigure9illustratessuchanapproximation. Figure9:Anadaptivesimpli®cationforthebunnymodel.variesfrom1/64%atthenoseto1%atthetail.7ImplementationandResultsWehaveimplementedbothalgorithmsandtriedoutthelocalalgorithmonseveralthousandobjects.Wewill®rstdiscusssomeoftheimplementationissuesandthenpresentsomeresults.7.1ImplementationIssuesThe®rstimportantimplementationissueiswhatsortofinputmodeltoaccept.Wechosetoacceptonlymanifoldtrianglemeshes(orborderedmanifolds).Thismeansthateachedgeisadjacenttotwo(oneinthecaseofaborder)trianglesandthateachvertexissurroundedbyasingleringoftriangles.Wealsodonotacceptotherformsofdegeneratemeshes.Manymeshdegeneraciesarenotapparentoncasualin-spection,sowehaveimplementedanautomaticdegener-acydetectionprogram.Thisprogramdetectsnon-manifoldvertices,non-manifoldedges,slivertriangles,coincidenttriangles,T-junctions,andintersectingtrianglesinapro-posedinputmesh.Notethatcorrectingthesedegeneraciesismoredif®cultthandetectingthem.Robustnessissuesareimportantforimplementationsofanygeometricalgorithms.Forinstance,theanalyticalmethodforenvelopecomputationinvolvestheuseofbi-linearpatchesandthecomputationofintersectionpoints.Thecomputationofintersectionpoints,evenforlinearel-ements,isdif®culttoperformrobustly.Thenumericalmethodforenvelopecomputationismuchmorerobustbe-causeitinvolvesonlylinearelements.Furthermore,itrequiresanintersectiontestbutnotthecalculationofinter-sectionpoints.Weperformallsuchintersectiontestsinaconservativemanner,usingfuzzyintersectionteststhatmayreportintersectionsevenforsomeclosebutnon-intersectingelements.Anotherimportantissueistheuseofaspace-partitioningschemetospeedupintersectiontests.Wechosetouseanoctreebecauseofitssimplicity.Ourcurrentoctreeim-plementationdealsonlywiththeboundingboxesoftheelementsstored.Thisworkswellformodelswithtrian-glesthatareevenlysizedandshaped.ForCADmodels,whichmaycontainlong,skinny,non-axis-alignedtriangles,asimpleoctreedoesnotalwaysprovideenoughofaspeed-up,anditmaybenecessarytochooseamoreappropriatespace-partitioningscheme.7.2ResultsWehavesimpli®edatotalof2636objectsfromtheauxiliarymachineroom(AMR)ofasubmarinedataset,picturedinFigure10totestandvalidateouralgorithm.Wereproducethetimingsandsimpli®cationsachievedbyourimplemen-tationfortheAMRandafewothermodelsinTable1.Allsimpli®cationswereperformedonaHewlett-Packard735/125with80MBofmainmemory.Imagesofthesesimpli®cationsappearinFigures11and12.Itisinterest-ingtocomparetheresultsonthebunnyandphonemodelswiththoseof[7,8].Forthesameerrorbound,weareabletoobtainmuchimprovedsolutions.Wehaveautomatedtheprocesswhichsetsthevalueforeachobjectbyassigningittobeapercentageofthediagonalofitsboundingbox.WeobtainedthereductionspresentedinTable1fortheAMRandFigures11and12byusingthisheuristic.FortherotorandAMRmodelsintheaboveresults,thelevelofdetailwasobtainedbysimplifyingthelevelofdetail.Thiscausestototaltobethesumofallprevious's,sochoosingof1,2,4,and8percentresultsintotalof1,3,7,and15percent.Therearetwoadvantagestothisscheme:(a)Itallowsonetoproceedincrementally,takingadvantageoftheworkdoneinprevioussimpli®cations.(b)Itbuildsahierarchyofdetailinwhichtheverticesatthelevelofdetailareasubsetoftheverticesatthelevelofdetail.Oneoftheadvantagesofthesettingtoapercentoftheobjectsizeisthatitprovidesanawaytoautomatetheselectionofswitchingpointsusedtotransitionbetweenthevariousrepresentations.Toeliminatevisualartifacts,weswitchtoamorefaithfulrepresentationofanobjectwhenprojectstomorethansomeuser-speci®ednumberofpixelsonthescreen.Thisisafunctionofthethatapproximation,theoutputdisplayresolution,andthecorrespondingmaximumtolerablevisibleerrorinpixels.8FutureWorkTherearestillseveralareastobeexploredinthisresearch.Webelievethemostimportantofthesetobethegenerationofcorrespondencesbetweenlevelsofdetailandthemovingofverticeswithintheenvelopevolume. Bunny Phone % #Polys Time % #Polys Time % #Polys Time % #Polys Time 0 69,451 N/A 0 165,936 N/A 0 4,735 N/A 0 436,402 N/A 1= 44,621 9 1=64 43,537 31 1=8 2,146 3 1 195,446 171 1=32 23,581 10 1=32 12,364 35 1=4 1,514 2 3 143,728 61 1=16 10,793 11 1=16 4,891 38 3=4 1,266 2 7 110,090 61 1=8 4,838 11 1=8 2,201 32 =4 1 15 87,476 68 1=4 2,204 11 1=4 1,032 35 =4 1 31 75,434 84 1=2 1,004 11 1=2 544 33 =4 1 1 575 11 1 412 30 153 674 1 Table1:Simpli®cation'sandruntimesinminutes8.1GeneratingCorrespondencesAtruegeometrichierarchyshouldcontainnotonlyrepre-sentationsofanobjectatvariouslevelsofdetail,butalsosomecorrespondenceinformationabouttherelationshipbetweenadjacentlevels.Theserelationshipsareneces-saryforpropagatinglocalinformationfromoneleveltothenext.Forinstance,thisinformationwouldbehelpfulforusingthehierarchicalgeometricrepresentationtoperformradiositycalculations.Itisalsonecessaryforperforminggeometricinterpolationbetweenthemodelswhenusingthelevelsofdetailforrendering.Notethattheenvelopetech-niquepreservessilhouetteswhenrendering,soitisalsoagoodcandidateforalphablendingratherthangeometricinterpolationtosmoothouttransitionsbetweenlevelsofdetail.Wecandeterminewhichelementsofahigherlevelofdetailsurfacearecoveredbyanelementofalowerlevelofdetailrepresentationbynotingwhichfundamentalprismsthiselementintersects.Thisisnon-trivialonlybecauseofthebilinearpatchesthatarethesidesofafundamentalprism.Wecanapproximatethesepatchesbytwoormoretrianglesandthentetrahedralizeeachprism.Giventhistetrahedralizationoftheenvelopevolume,itispossibletostabeachedgeofthelowerlevel-of-detailmodelthrthetetrahedronstodeterminewhichonestheyintersect,andthuswhichtrianglesarecoveredbyeachlowerlevel-of-detailtriangle.8.2MovingVerticesTheoutputmeshgeneratedbyeitherofthealgorithmswehavepresentedhasthepropertythatitssetofverticesisasubsetofthesetofverticesoftheoriginalmesh.Ifwecanaffordtorelaxthisconstraintsomewhat,wemaybeabletoreducetheoutputsizeevenfurther.Ifweallowtheverticestoslidealongtheirnormalvectors,weshouldbeabletosimplifypartsofthesurfacethatmightotherwisebeimpossibletosimplifyforsomechoicesofepsilon.Wearecurrentlyworkingonagoal-basedapproachtomov-ingverticeswithintheenvelopevolume.Foreachvertexwewanttoremove,weslideitsneighboringverticesalongtheirnormalstomakethemlieascloselyaspossibletoatangentplaneoftheoriginalvertex.Intuitively,thisshouldincreasethelikelihoodofsuccessfullyremovingthevertex.Duringthiswholeprocess,wemustensurethatnoneoftheneighboringtriangleseverviolatestheenvelopes.Thisapproachshouldmakeitpossibletosimplifysurfacesusingsmallerepsilonsthanpreviouslypossible.Infact,itmayevenenableustousetheoriginalsurfaceandasingleen-velopeasourconstraintsurfacesratherthantwoenvelopes.Thisisimportantforobjectswithareasofhighmaximalcurvature,likethincylinders.9ConclusionWehaveoutlinedthenotionofsimpli®cationenvelopesandhowtheycanbeusedforgenerationofmultiresolutionhi-erarchiesforpolygonalobjects.Ourapproachguaranteesnon-self-intersectingapproximationsandallowstheusertodoadaptiveapproximationsbysimplyeditingthesim-pli®cationenvelopes(eithermanuallyorautomatically)intheregionsofinterest.Itallowsforaglobalerrortoler-ance,preservationoftheinputgenusoftheobject,andpreservationofsharpedges.Ourapproachrequiresonlyoneuser-speci®ableparameter,allowingittoworkonlargecollectionsofobjectswithnomanualinterventionifsode-sired.Itisrotationallyandtranslationallyinvariant,andcanelegantlyhandleholesandborderedsurfacesthroughtheuseofcylindricaltubes.Simpli®cationenvelopesaregen-eralenoughtopermitbothsimpli®cationalgorithmswithgoodtheoreticalpropertiessuchasourglobalalgorithm,aswellasfast,practical,androbustimplementationslikeourlocalalgorithm.Additionally,envelopespermiteasygen-erationofcorrespondencesacrossseverallevelsofdetail.10AcknowledgementsThankstoGregAngelini,JimBoudreaux,andKenFastatElectricBoatforthesubmarinemodel,RichRiesen-feldandElaineCohenoftheAlpha 1groupattheUni-versityofUtahfortherotormodel,andtheStanfordComputerGraphicsLaboratoryforthebunnyandtele-phonemodels.ThankstoCarlMueller,MarcOlano,andBillYakowenkoformanyusefulsuggestions,andtotherestoftheUNCSimpli®cationGroup(RuiBastos,CarlErikson,MerlinHughes,andDavidLuebke)forprovid-ingagreatforumfordiscussingideas.ThefundingforthisworkwasprovidebyaLinkFoundationFellowship,AlfredP.SloanFoundationFellowship,AROContractP-34982-MA,AROMURIgrantDAAH04-96-1-0013,NSFGrantCCR-9319957,NSFGrantCCR-9301259,NSFCa-reerAwardCCR-9502239,ONRContractN00014-94-1-0738,ARPAContractDABT63-93-C-0048,NSF/ARPACenterforComputerGraphicsandScienti®cVisualization,NIHGrantRR02170,anNYIawardwithmatchingfundsfromXeroxCorp,andaU.S.-IsraeliBinationalScienceFoundationgrant.References[1]P.AgarwalandS.Suri.Surfaceapproximationandgeometricpar-titions.InProceedingsFifthSymposiumonDiscreteAlgorithmspages24±33,1994. 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Figure10:Lookingdownintotheauxiliarymachineroom(AMR)ofasubmarinemodel.Thismodelcontainsnearly3,000objects,foratotalofoverhalfamilliontriangles.Wehavesim-pli®edover2,600oftheseobjects,foratotalofover430,000triangles. Figure11:AnarrayofbatteriesfromtheAMR.Allpartsbuttheredaresimpli®edrepresentations.Atfullresolution,thisarrayrequires87,000triangles.Atthisdistance,allowing4pixelsoferrorinscreenspace,wehavereduceditto45,000triangles. (a)bunnymodel:69,451triangles(b)phonemodel:165,936triangles(c)rotormodel:4,736triangles(a)16%,10793triangles(b)32%,12364triangles(c)8%,2146triangles(a)4%,2204triangles(b)16%,4891triangles(c)4%,1266triangles(a)1%,575triangles(b)1%,412triangles(c)4%,716trianglesFigure12:Level-of-detailhierarchiesforthreemodels.Theapproximationdistance,,istakenasapercentageoftheboundingboxdiagonal.