d1www.yyy13.com

Volumexxx,1998numberyyypp.
000000Metro:measuringerroronsimpliedsurfacesP.
Cignoniy,C.
RocchinizandR.
ScopignoxIstitutoperl'Elaborazionedell'Informazione-ConsiglioNazionaledelleRicerche,Pisa,ItalyTechnicalNoteShortcontributionAbstractThispaperpresentsanewtool,Metro,designedtocompensateforadeciencyinmanysimplicationmethodsproposedinliterature.
Metroallowsonetocomparethedierencebetweenapairofsurfacese.
g.
atriangulatedmeshanditssimpliedrepresentationbyadoptingasurfacesamplingapproach.
Ithasbeendesignedasahighlygeneraltool,anditdoesnoassuptionontheparticularapproachusedtobuildthesimpliedrepresentation.
Itreturnsbothnumericalresultsmeshesareasandvolumes,maximumandmeanerror,etc.
andvisualresults,bycoloringtheinputsurfaceaccordingtotheapproximationerror.
Keywords:surfacesimplication,surfacecomparison,approximationerror,scanconversion.
1.
IntroductionManyapplicationsproduceormanageextremelycom-plexsurfacemeshese.
g.
volumerendering,solidmod-eling,3Drangescanning.
Excessivesurfacecomplex-itycausesnoninteractiverendering,secondarytomainmemorybottleneckswhilemanaginginteractivevisualsimulations,ornetworksaturationin3Ddis-tributedmulti-mediasystems.
Inspiteofthecon-stantincreaseinprocessingspeed,theperformancesrequiredbyinteractivegraphicsapplicationsareinmanycasesmuchhigherthanthosegrantedbycur-renttechnology.
Substantialresultshavebeenreportedinthelastfewyears,aimedatreducingsurfacecomplexitywhileas-suringagoodshapeapproximation13;6.
Thetech-niquesproposedsimplifytriangularmesheseitherbymergingcollapsingelementsorbyre-samplingver-tices,usingdierenterrorcriteriatomeasurethet-nessoftheapproximatedsurfaces.
Anylevelofre-ductioncanbeobtainedwiththeseapproaches,ontheconditionthatasucientlycoarseapproximationthresholdissetanexampleisdrawninFigure1.
yEmail:cignoni@iei.
pi.
cnr.
itzEmail:rocchini@calpar.
cnuce.
cnr.
itxEmail:r.
scopigno@cnuce.
cnr.
itAgeneralcomparisonofthesimplicationap-proachesisnoteasy,becausethecriteriatodrivethesimplicationprocessarehighlydierentiatedandthereisnocommonwayofmeasuringerror;anat-tempthasbeenrecentlypresented3.
Infact,manysimplicationapproachesdonotreturnmeasuresoftheapproximationerrorintroducedwhilesimplifyingthemesh.
Forexample,giventhecomplexityreductionfactorsetbytheuser,somemethodstrytooptimize"theshapeofthesimpliedmesh,buttheygivenomea-sureontheerrorintroduced18;9;8.
Otherapproacheslettheuserdenethemaximalerrorthatcanbein-troducedinasinglesimplicationstep,butreturnnoglobalerrorestimateorbound17;7.
Someotherre-centmethodsadoptaglobalerrorestimate10;15;2;5orsimplyensuretheintroducederrortobeunderagivenbound4.
Buttheeldofsurfacesimplicationstilllacksaformalanduniversallyacknowledgedde-nitionoferror,whichshouldinvolveshapeapproxi-mationandhopefullypreservationoffeatureelementsandmeshattributese.
g.
color.
Forthesereasons,ageneraltoolthatwouldmea-suretheactualgeometricdierence"betweentheoriginalandthesimpliedmesheswouldbestrategicbothforresearchers,inthedesignofnewsimplica-tionalgorithms,andforusers,toallowthemtocom-paretheresultsofdierentsimplicationapproachescTheEurographicsAssociation1998.
PublishedbyBlackwellPublishers,108CowleyRoad,OxfordOX41JF,UKand238MainStreet,Cambridge,MA02142,USA.
2P.
Cignoni,C.
RocchiniandR.
ScopignoMetro:measuringerroronsimpliedsurfacesFigure1:Ameshsimplicationexample:theoriginalmesh7,960trianglesisontheleft,asimpliedone179trianglesisontheright.
onthesamemeshandtochoosethesimplicationmethodthatbestts"thetargetmesh.
Infact,evenboundedprecisionmethods10;15;2;5;4behavedier-entlyondierentmeshes.
Theygenerallyensuretheuserthattheapproximationwillnotbelargerthanagiventhreshold,butdinotgivedataontheactualerrordistributiononthemesh.
Anexampleisthefol-lowingquery:aretheresectionsofthemeshwhichholdanapproximationmuchbetterthanthegivenboundAnd,ifyes,whatistheirsizeanddistributionMetrohasbeendenedasatoolwhichisgeneralandsimpletoimplement.
Itcomparesnumericallytwotrianglemeshes,whichdescribethesamesurfaceatdierentlevelsofdetailLOD.
Metrorequiresnoknowledgeonthesimplicationapproachadoptedtobuildthereducedmesh.
Metroevaluatesthedierencebetweentwomeshes,onthebasisoftheapproximatedistancedenedinthefollowingsection.
2.
TerminologyWedeneheresometermsthatwillbeusedinthefollowingsectionactually,allthemeasuresevaluatedbyMetrofollowthedenitionsbelow.
Theapproximationerrorbetweentwomeshesmaybedened,asfollows,asthedistancebetweencor-respondingsectionsofthemeshes.
GivenapointpandasurfaceS;wedenethedistanceep;Sas:ep;S=minp02Sdp;p0wheredistheEuclideandistancebetweentwopointsinE3.
Theone-sideddistancebetweentwosurfacesS1;S2isthendenedas:ES1;S2=maxp2S1ep;S2:Notethatthisdenitionofdistanceisnotsymmetric.
ThereexistsurfacessuchthatES1;S26=ES2;S1.
Atwo-sideddistanceHausdordistancemaybeobtainedbytakingthemaximumofES1;S2andES2;S1.
Givenasetofuniformlysampleddistances,wede-notethemeandistanceEmbetweentwosurfacesasthesurfaceintegralofthedistancedividedbytheareaofS1:EmS1;S2=1jS1jZS1ep;S2dsIfthesurfaceS1isorientablewecanextendthedenitionofdistancebetweenapointpofS1andS2sothat,informallyspeaking,thisdistancee0ispositiveifthenearestpointp02S2isintheouterspacewithrespecttoS1,andnegativeotherwiseseeFigure2.
Or,inotherwords,ifNpisthevectornormaltoS1inthesampledpointpandp02S2isthenearestpoint,thenthesignofourdistancemeasureisthesignofNpp0,p.
ThisdenitionofsigneddistanceisintroducedtoletMetrodistinguishbetweenpositiveandnegativedis-tancesbetweentwosurfacesasfollows:E+S1;S2=maxp2S1e0p;S2cTheEurographicsAssociation1998P.
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RocchiniandR.
ScopignoMetro:measuringerroronsimpliedsurfaces3||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||p2p1d1d2S1S2Figure2:Signeddistanceevaluation;distanceisposi-tiveinp1andnegativeinp2S1isthesampledcurve.
E,S1;S2=jminp2S1e0p;S2jSigneddistancesareusedbyMetrotogiveaninde-pendentevaluationtothesectionsoftherstmeshwhichareintheinteriororintheexteriorspacewithrespecttothesecondmesh.
3.
TheMetroToolMetronumericallycomparestwotrianglemeshesS1andS2,whichdescribethesamesurfaceatdierentlevelsofdetail.
Itrequiresnoknowledgeofthesimpli-cationapproachadoptedtobuildthereducedmesh.
Metroevaluatesthedierencebetweenthetwomeshesonthebasisoftheapproximationerrormeasurede-nedintheprevioussection.
Itadoptsanapproximateapproachbasedonsurfacesamplingandthecomputa-tionofpointtosurfacedistances.
Thesurfaceoftherstmeshhereafterpivotmeshissampled,andforeachelementarysurfaceparcelwecomputethedis-tancetothenotpivotmesh.
Theideaisthereforetoadoptanintegrationprocessoverthesurface.
Surfacesamplingisachievedbyscanconvertingtriangularfacesunderauser-selectedsam-plingresolution.
Thesamplingresolutioncharacterizestheprecisionoftheintegration,andweobservedthatinmostcasesasucientlythinsamplingstepsizeis0.
1oftheboundingboxdiagonal.
WealsoimplementedaMontecarloapproachgener-aterandomkpointsintheinteriorofeachface,withthenumberkofsamplesproportionaltothefacetarea,whichgavesimilarresultsintermsofprecision.
Moreover,theadoptionofMontecarlosamplingmakesnotpossibletheerrorvisualizationviaerror-texturemapping,becausethelatterrequiresaregular,rastersampling.
InanearlyversionofourtoolMetrov.
1aray-castingapproachwasadoptedtocomputepointtosurfacedistances.
InordertoimproveperformancesandprecisionweadoptedadierentapproachinthecurrentreleaseofMetro,v.
2.
Distancesfromthesam-plingpointandthenon-pivotmesharenowcomputedecientlybyusingabucketeddatastructure.
UniformgridUGtechniquesareveryeectiveingeometriccomputationsbecauseinmanycaseselementswhicharefarapartgenerallyhavelittleornoeectoneachother1.
Localprocessingcan,therefore,highlyreduceempiricalcomplexityformanygeometricproblems.
A3DuniformgridisusedinMetrov.
2asanindexingschemeforthefastsearchofthenearestfacetothesamplingpoint.
TheboundingboxofmeshS2ispar-titionedintocubiccellsfollowingaregularpattern.
Then,westoreineachcellcijkthelistoffacesofS2whichintersectcijk.
Foreachsamplingpointp,rstlywecomputethedistancebetweenpandallthefacesofthenon-pivotmeshS2containedinthesamegridcellofp.
Then,adjacentgridcellsareprocessed,inorderofincreasingdistancefromp,untilwendthatallnottestedcellsarefartherthanthecurrentnearestface.
ThedistancebetweenpandasinglefaceofS2iscom-putedusinganoptimizedalgorithmcontainedinthesourcecodeofthePOVray-tracer12.
Thestrategyadoptedimpliesthatuniquenessofthenearestpointisnotensured.
Accordingtothedeni-tioninSection2,wemightndmultiplefacesatmin-imaldistancefromthecurrentsamplingpoint.
But,ifwearelookingforunsignedapproximationerror,thenuniquenessisnotaproblembecauseweareinterestedonlyinthevalueofthisdistance.
Conversely,inthecaseofsignedapproximationerrorevaluation,havingpointsatthesamedistancebutholdingdierentsignforcesMetrotooperatearandomchoiceandintro-ducesapotentialimprecision.
TheworstcasecomputationalcomplexityofMetrodependsonthesurfaceareaAS1ofthepivotmeshmeasuredinsquaredsamplingstepunitstimesthenumbernfoffacesofthenon-pivotmesh.
Theresult-ingcomplexityisOAS1nf.
But,ifweuseanUG,thenwecanexpectthatamuchlowernumberoffaceswillbetestedtocomputetheminimaldistanceforeachsamplingpoint.
Wemeasuredinanumberofrunsthatthemeannumberoffacesevaluatedforeachsam-plingpointisonlyfewtensaspresentedinTable1.
InTable1wereportalsotherunningtimesandthenumberofsamplesexecutedbyMetroonthreedier-entpairsofmeshes.
Timesareinseconds,measuredonaSGIO2workstationR5000180Mhz,96MBRAM.
AnoptionisprovidedbyMetrotocomputeasym-metricevaluationofthemaximalerror.
AttheendcTheEurographicsAssociation19984P.
Cignoni,C.
RocchiniandR.
ScopignoMetro:measuringerroronsimpliedsurfacesS1S2samplingstepsamplesno.
testedfacesno.
timefacesno.
facesno.
persamplesec.
4,00169,4510.
2365,30730.
3292,86728,3220.
1540,66729.
324.
76,36967,6070.
11,670,42024.
889.
8Table1:Numberofsamplingpoints,samplingstepsize,timeandnumberoffacestestedpersampleonthreedierentmeshes.
ofthesamplingprocess,ifthe,soptionisset,thenMetroswitchesthepivotandnotpivotmeshesandexecutessamplingagain.
Givenasamplingstep,themeshmaycontaintrian-gleswhichhaveanareasmallerthanthesquaredsam-plingstep.
Metromanagesthisspecialcasebyadopt-ingarandomchoice:arandomvariableisgenerated,withtheprobabilityofitsTRUEvalueequaltothera-tiobetweenthetriangleareaandthesquaredsamplearea.
IftherandomvalueisTRUE,asinglepointtosurfacedistanceiscomputed;otherwise,Metrostartsthescanconversionofthenextface.
MetroInputMetrohasacommand-lineinputinterface.
Theop-tionsavailableareshown,asusual,bytyping:metro-h.
TheoptionsavailableareshowninFigure3.
ThedataformatsacceptedininputareeithertheOpenInventor19formatorarawindexedrepresen-tationalistofvertexcoordinates,andalistoftrian-gularfaces,denedbythethreeindicestothevertexlist.
Thetwomeshesshouldhavesimilarshapesasinmul-tiplelevelofdetailrepresentation.
Iftheshapesdiertoomuch,withthedisappearanceofsignicantfea-tures,thecomputationoftheerrormightbelocallyimprecise.
Metroconsidersexcessivethedierencebe-tweentwomeshesiftheirboundingboxdiagonalsdif-ferinlengthbymorethan10.
Ifthesurfacestobecomparedarenotorientableormultiple-connected,thenitwouldbeimpossibletodis-tinguishbetweenpositiveandnegativeerrorsi.
e.
ifthelowdetailmeshpassesbeloworabovethehighdetailmesh.
MetroOutputMetroreturnsbothnumericalandvisualevaluationsofsurfacemesheslikeness"Figure5showsasnap-shotofitsGUI.
TheformatofthenumericalresultsisreportedinFigure4.
Itcontainsdataoninputmeshescharacteris-ticstopology,size,surfacearea,meshvolume,featureedgestotallength,diagonaloftheminimalbound-ingbox,diameteroftheminimalboundingsphere;themeanandmaximumdistancesbetweenmeshesreturnedusingabsolutemeasuresandasapercent-agesofthediagonalofthemeshboundingbox;andaveryroughapproximationofthepositive,negativeandtotalvolumeofthedierencebetweenthetwomeshesi.
e.
thetotalvolumeVtisthevolumeofS1,S2S2,S1.
Allthepositivenegativemeasuresfollowstheden-itionsinSection2,andcanbecomputedonlyiftheinputsurfacesareorientableandsingle-connected.
Errorisalsovisualizedbycoloringthepivotmeshwithrespecttotheevaluatedapproximationerror.
Twodierentcolormappingmodalitiesareavailable:per-vertexmapping:foreachvertex,wecomputetheerroroneachmeshvertexasthemeanoftheer-rorsontheincidentfaces,andassignacolorpro-portionaltothaterror.
Thefacesarethencoloredbyinterpolatingvertexcolors;error-texturemapping:foreachface,argb-textureiscomputedwhichstoresthecolor-codederrorseval-uatedoneachsamplingpointmappedonacolorscale.
Theerror-texturemappingapproachgivesvisualre-sultswhichingeneralaremoreprecise,butwhosevi-sualizationdependsonthesamplingstepsizeusedbyMetro.
SeeforexampleinFigure6thedierentvisualrepresentationofthesamemeshzone.
Inbothcases,ahistogramreportingtheerrordistrib-utionisalsovisualizedontheleftoftheMetrooutputwindowFigure5.
Whentheerror-texturemappingisused,wecanalsovisualizetheerrorbyconsideringitssign:zeroerrormapstogreen,negativeandpositivetoredandblueseeFigure7.
LimitednumericalprecisionmanagementTheerrorevaluatedbyMetromaybeaectedbythelimitednumericalprecision,althoughdoubleprecisionisadoptedinnumericalcomputations.
Anadhoc"managementhasbeenprovidedforanumberofdan-gerouscases,suchasnearlycoincidentvertices,facetscTheEurographicsAssociation1998P.
Cignoni,C.
RocchiniandR.
ScopignoMetro:measuringerroronsimpliedsurfaces5Usage:Metrofile1file2-a-e-h-l-s-r-q|v-b|bs|tfile1,file2:inputmeshestobecompared;-acreaseanglesettingforfeatureedgesdetectionandclassification.
Theanglevalue""isgivenindegrees,from0alledgesareclassified'featureedge'to180degrees.
itisusedtomeasurethetotallengthofthefeatureedges;-bshowerrorusing"error-texture"modeDEFAULTis"per-vertex"mode-bsshowerrorusing"signederror-texture"modegreen==error=0;-esetthemaximalabsoluteerrorinthehistogramscaleandcolormapping;itisusefultocomparevisuallytheresultsoftwodifferentrunsofMetro;-hshowtheMetrocommandsyntaxandtheoptionsavailable;-lselectthescanconversionstepvalue"":percentageofthemeshboundingbox;-quse"quiet"i.
e.
verysyntheticoutput;-ruse"Montecarlo"samplingDEFAULT:usescanconversion;-scomputesymmetricmaximumdistancedoublerun;-tsettextmodeonly,donotvisualizeresultsunderOpenInventor;-vverboseoutput.
Example:metro-vmeshcomp.
ivmesh.
iv-l0.
5-a45Figure3:Metroinputoptions.
Figure5:TheMetrographicoutputwindow.
withsmallarea,andveryelongatedtriangles.
Anotherproblemmaybethecomputationofthesumofhundredsofthousandsofnearlyzerovalues.
Tomin-imizeroundingerrorsinthecomputationofthesum,weusedafaninalgorithmbinarytreestructuredsum11.
4.
ConcludingRemarksWehaveintroducedanewtool,Metro,toallowsim-plecomparisonsbetweensurfaces.
Itsmainuseisintheevaluationoftheerrorintroducedinthesimpli-cationofsurfaces.
Metroreturnsbothnumericalandvisualevaluationsofthemeshes'likeness.
Thesemea-suresarecomputedusinganerrordenedasanap-cTheEurographicsAssociation19986P.
Cignoni,C.
RocchiniandR.
ScopignoMetro:measuringerroronsimpliedsurfacesFigure6:Dierentcolormappingmodality:per-vertexmappingontheleft,anderror-texturemappingontheright.
proximationofthesurfacetosurfacedistance.
Theerrorisevaluatedby:1scanconvertingtherstmeshfaceswithauserspeciedsamplingstep,and2computingapointtosurfacedistanceforeachscanconvertedpoint.
Thetooladoptswellknowntechniquesandcanbesimplyimplemented.
WetestedwithMetrothesimpliedmeshesob-tainedwithsomepublicdomainsoftware.
Inthecaseofaboundedprecisionmethod,theSimplicationEn-velopes4,weobtainederrorvaluesverysimilartothethresholdset;ingeneral,aslightlylowererrorismea-sured:0.
759forameshsimpliedundertargeterror0.
77,or0.
0884fortherelativetargeterror0.
0895.
ButtheaddedvalueofMetrointhecaseofaboundederrormethodistogivethepossibilitytoviewthedistribu-tionoftheerroronthemeshFigure5.
Animportantpointtobeconsideredintheeval-uationofasurfacesimplieristowhatextentitpreservesfeatureedges.
ThecurrentimplementationofMetrodetectsfeatureedgesandreturns,foreachmesh,theirtotallength.
Butthismaynotbesu-cient:eventwomesheswithnearlyequaltotallengthofthefeatureedgesmightdieralot.
Metrocouldbeeasilyextendedtogetridofthislim-itation.
GiventwosetoffeatureedgesF1andF2,wemightapplyagainasamplingapproach.
Foreachfea-tureedgee2F1andeachsamplingpointspi2e,letusevaluatetheminimaldistancebetweenpiandtheedgesinF2.
Theseminimaldistancescanthenbeusedtocomputethemaximumandmeandisplacementsbe-tweenthesetoffeatureedgesoralsothemaximumandmeananglesbetweenpairsofcorrespondingfea-tureedges.
AlimitationofMetroregardsthetopologychangesthatsomesimplicationalgorithmscanintroduceinthesimpliedsurfaces5;14;16.
Metrocanonlypar-tiallycoverthisissue.
Itreturnsthenumberofcon-nectedcomponentsofeachmeshandalsoiftheyareorientableandclosed,andthereforeinmanycaseswemaydetectifatopologychangehasoccurred.
Butamoresophisticatedapproachisneededtodetecteachsinglechangeoftopologyandtomeasuretheassoci-atedimpactonmeshesdisparity.
5.
AcknowledgementsMetrov.
2isavailableaspublicdomainsoft-wareattheVisualComputingGroupwebsiteoftheCNUCEandIEI,C.
N.
R.
InstitutesatPisahttp:miles.
cnuce.
cnr.
itcgmetro.
html.
ThisworkwaspartiallynancedbytheProgettoFi-nalizzatoBeniCulturalioftheItalianNationalRe-searchCouncilCNR.
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