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Multiscale ppr oach to D Scatter ed Data Inter polation with Compactly Supported Basis Functions utaka Ohtak Ale xander Belyae HansPeter Seidel Computer Graphics Group MaxPlanckInstitut ur Informatik

mpgde Fig 1 oin set surface the leftmost image and its coarseto57356ne hierarc of sets in terp olated with compactly supp orted basis functions Abstract In this paper we pr opose hier ar hical appr oac to 3D scatter ed data interpolation with compact

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Multiscale ppr oach to D Scatter ed Data Inter polation with Compactly Supported Basis Functions utaka Ohtak Ale xander Belyae HansPeter Seidel Computer Graphics Group MaxPlanckInstitut ur Informatik






Presentation on theme: "Multiscale ppr oach to D Scatter ed Data Inter polation with Compactly Supported Basis Functions utaka Ohtak Ale xander Belyae HansPeter Seidel Computer Graphics Group MaxPlanckInstitut ur Informatik"— Presentation transcript:

AMulti-scaleApproachto3DScatteredDataInterpolationwithCompactlySupportedBasisFunctionsYutakaOhtakeAlexanderBelyaevHans-PeterSeidelComputerGraphicsGroup,Max-Planck-InstitutfurInformatikStuhlsatzenhausweg85,66123Saarbrucken,GermanyE-mails:fohtake,belyaev,hpseidelg@mpi-sb.mpg.deFig.1.Apointsetsurface(theleftmostimage)anditscoarse-to- nehierarchyofsetsinterpolatedwithcompactlysupportedbasisfunctions.AbstractInthispaper,weproposeahierarchicalapproachto3Dscattereddatainterpolationwithcompactlysupportedba-sisfunctions.Ournumericalexperimentssuggestthattheapproachintegratesthebestaspectsofscattereddatat-tingwithlocallyandgloballysupportedbasisfunctions.Employinglocallysupportedfunctionsleadstoanefcientcomputationalprocedure,whileacoarse-to-nehierarchymakesourmethodinsensitivetothedensityofscattereddataandallowsustorestorelargepartsofmisseddata.Givenapointclouddistributedalongasurface,werstusespatialdownsamplingtoconstructacoarse-to-nehi-erarchyofpointsets.Thenweinterpolatethesetsstartingfromthecoarsestlevel.Weinterpolateapointsetofthehi-erarchy,asanoffsettingoftheinterpolatingfunctioncom-putedatthepreviouslevel.Fig.1showsanoriginalpointset(theleftmostimage)anditscoarse-to-nehierarchyofinterpolatedsets.Accordingtoournumericalexperiments,themethodisessentiallyfasterthanthestate-of-artscattereddataap-proximationwithgloballysupportedRBFs[9]andmuchsimplertoimplement.1IntroductionSincethepioneeringworksofRicci[32]andBlinn[4]geometricmodelingwithimplicitsurfacesremainstobeanactiveresearcharea[7].Recentdevelopmentsinthiseldincludelevelsetmethods[34],variationalimplicitsurfaces[33,35,36],andadaptivelysampleddistanceelds[17].Noveltrendsinimplicitsurfacemodelingarecloselyre-latedtointerpolatingandapproximatingpointsetsurfacesusinglevelsetmethods[40],viaRadialBasisFunctions(RBFs)[9,12,11,26],andbyMovingLeastSquares(MLS)[2,15,30],seealsoreferencestherein.Asdemonstratedin[9,10],implicitsurfacesareespeciallyusefulforrepair-ingincompletedatasincenotopologicalconstraintsarere-quired.InterpolationandapproximationofscattereddatawithRBFshasavariationalnature[18]whichsuppliesauserwitharichpaletteoftypesofradialbasisfunctions.ThebasicquestioniswhethertochooselocalorglobalRBFs.Fittingscattereddatabylocal,compactlysupported,RBFsleadstoasimplerandfastercomputationprocedure,whileapracticalusageofglobalRBFsisbasedonsophisti-catedmathematicaltechniquessuchasthefastmultipoleOnaleavefromUniversityofAizu,Aizu-Wakamatsu965-8580Japan1 methodemployedin[9].Ontheotherhand,globalRBFsareextremelyusefulinrepairingincompletedata[9]whileapproachesbasedoncompactlysupportedRBFsaresensi-tivetothedensityofinterpolated/approximatedscattereddataand,therefore,acarefulselectionofRBFinuencedo-mainscontrolledbycertainparametersisrequired.Apromisingwaytocombineadvantagesprovidedbylo-callyandgloballysupportedbasisfunctionsconsistsofus-inglocallysupportedbasisfunctionsinahierarchicalfash-ion.Tothebestofourknowledge,amulti-scaleapproachtottingrangedatawithbump-likebasisfunctionswasrstusedin[27].Atpresenthierarchicalmethodsforscattereddatattingquicklygainpopularityincomputationalmath-ematicsandcomputergraphicsresearchsocieties.Forex-ample,recentlyanRBF-basedmultilevelapproachtoscat-teredheightdatainterpolationwasemployedin[16](seealso[1,20]forveryrecentdevelopmentsinthisarea),hi-erarchicalGaussianswereusedin[24]forreconstructionandmodicationofmotionandimagedata.In[23]itwasdemonstratedthat,chosenanappropriatecarrierimplicitsurface,scattereddatattingwithlocallysupportedRBFscanbedoneveryfast.Thusahierarchicalapproachwithlocallysupportedbasisfunctionswheredatareconstructedatcoarserlevelsserveascarriersfornerlevelsmaysub-stantiallyacceleratescattereddatatting.Theapproachdevelopedinthispaperisanattempttoin-tegratethebestaspectsof3Dscattereddatattingwithlo-callyandgloballysupportedbasisfunctions.Weusecom-pactlysupportedfunctionstointerpolateagiven3Dpointsetsurfaceinahierarchicalway.Employinglocallysup-portedfunctionsleadstoanefcientcomputationalproce-dure,whileacoarse-to-nehierarchymakesourmethodinsensitivetothedensityofscattereddataandallowsustorestorelargepartsofmisseddata.Weproposetouseanewtypeofcompactlysupportedbasisfunctions:quadricsmultipliedbycompactlysupportedradialweights,wherethequadriccoefcientsaredeterminedvialocalweightedleastsquaresttingandviaaglobalinterpolationprocedure.Givenapointclouddistributedalongasurface,werstusespatialdownsamplingtoconstructacoarse-to-nehierar-chyofpointsets.Thenweinterpolatethesetsstartingfromthecoarsestlevel.Weinterpolateapointsetofthehierar-chybyanoffsetoftheinterpolatingfunctioncomputedatthepreviouslevel.Numericalexperimentssuggestthatourmethodisessentiallyfasterthanthestate-of-artscattereddataapproximationwithgloballysupportedRBFs[9].Inaddition,ourapproachismuchsimplertoimplementthanthatdevelopedin[9].Fig.2demonstratesareconstructionofanincompletedatabyourmulti-scalescattereddatainterpolationproce-dure.WesmoothedslightlytheangelmeshdatafromCal-tech3DGallery[8]andthenremovedallconnectivityin-formation.Fig.2.Top:anangelpoint-clouddata(40Kpoints).Bottom:ameshgeneratedbypolygonizingtheim-plicitsurface(zerolevel-setofthe3Dscalar eld)generatedbytheproposedmethod.Noticehowwellthemissedpartsare lledanddataisreconstructed.Therestofthepaperisorganizedasfollows.InSec-tion2weexplainourscattereddatainterpolationprocedureatasinglelevel.InSection3wepresentamulti-levelinter-polationscheme.WedemonstrateanddiscussadvantagesandlimitationsofourapproachinSection4andconcludeinSection5.2Single-levelInterpolationInthissectionwedemonstratehowourscattereddatainterpolationprocedureworksatasinglelevel.LetusconsiderasetofNpointsP=fpigscatteredalongasurface.Weassumethatthepointsareequippedwithinnerunitnormalsnideninganorientation.Thenormalsareusuallycomputedduringtheshapeacquisitionstagefromrangeimages.Theycanalsobeestimateddi-rectlyfrompointsetdata[19].Wewanttogeneratea3D2 scalareldf(x)suchthatitszerolevel-setf=0interpo-latesP.Implicitsurfacef(x)=0separatesthespaceintotwoparts:f(x)�0andf(x)0.Letusassumethattheorientationnormalsarepointingintothepartofspacewheref(x)�0.Thusf(x)hasnegativevaluesoutsidethesurfaceandpositivevaluesinsidethesurface.WeinterpolatePbyfunction-valuedRBFsf(x)=pi2Pyi(x)=pi2P[gi(x)+li]fs(kxpik);(1)wherefs(r)=f(r=s),f(r)=(1r)4(4r+1)isWenda-land'scompactlysupportedRBF[38],sisitssupportsize,andgi(x)andliareunknownfunctionsandcoefcientstobedetermined.AnappropriatevalueofsisestimatedfromthedensityofP.Thefunctionsgi(x)andcoefcientsliarechosenviathefollowingtwo-stepprocedure.1.Ateachpointpiwedeneafunctiongi(x)suchthatitszerolevel-setgi(x)=0approximatestheshapeofPinasmallvicinityofpi.2.Wedeterminethecoefcientslifromtheinterpolationconditionsf(pj)=0=pi2Phgi(pj)+liifs(kpjpik):(2)Noticethat(2)canberewrittenaspi2PliFij=pi2Pgi(pj)Fij;Fij=fs(kpjpik)andtherefore(2)leadstoasparsesystemoflinearequationswithrespecttoli.SinceWendaland'scompactlysupportedRBFsarestrictlypositivedenite[38],theNNinterpo-lationmatrix=fFijgispositivedeniteifPconsistsofpairwisedistinctpoints.Foreachpointpi2Pwedeterminealocalorthogonalcoordinatesystem(u;v;w)withtheoriginofcoordinatesatpisuchthattheplane(u;v)isorthogonaltoniandthepositivedirectionofwcoincideswiththedirectionofni.WeapproximatePinavicinityofpibyaquadricw=h(u;v)Au2+2Buv+Cv2;wherethecoefcientsA,B,andCaredeterminedviathefollowingleast-squaresminimization(uj;vj;wj)=pj2Pfs(kpjpik)wjh(uj;vj)2!min:Nowwesetgi(x)=wh(u;v):(3)Thusthezerolevel-setofgi(x)coincideswiththegraphofw=h(u;v).pg + =0liiinisf=0liu,vw=npw=h(u,v)lshift:Fig.3.Geometricideabehindourapproachforscat-teredpointdatainterpolationatasinglelevel.AgeometricideabehindourinterpolatingprocedureisillustratedinFig.3.Fig.4showsthegraphofthe2Dversionofabasicfunctionyi(x)=[gi(x)+li]fs(kxpik);(4)asummandin(1).Fig.4.Graphof2Dversionofbasicfunctionyi(x)usedin(1).Zerolevelyi(x)=0(parabola)isdrawnbyboldline.Parameters,thesupportsizeoffs(),isestimatedfromthedensityofP.Westartanoctree-basedsubdivisionofaboundingboxofPandstopthesubdivisionifeachleafcellcontainsnomorethan8pointsofP.Thenwecomputetheaveragediagonaloftheleafcells.Finallywesetsequaltothreefourthofthataveragediagonal.Tosolvethelinearsystemcorrespondingto(2)weusethepreconditionedbiconjugategradientmethod[31]withtheinitialguessli=0.ThesizeofthelinearsystemisNN,whereN=jPjisthenumberofinterpolatingpoints.Notethatmethodsdevelopedin[33,35,26,9,36]requirealsointerior/exteriorconstraintswhichtogetherwithinter-polationconditionsf(pi)=0leadtoabiggersystemoflinearequations.Basisfunctions(4)usedin(1)aresimilartothesuretsintroducedbyPerlin[13].3 Fig.5.TheStanfordbunnyandIgeamodelrecon-structedfromscatteredpointdataaspolygonizedzerolevel-setsof(1).Fittingtimeis6secondsfortheStanfordbunny(35Kpoints)and47secondsfortheIgeamodel(134Kpoints).Afterrescalingboththemodelsinorderto tthemintoaunitcubeweuses=0:02ands=0:0125fortheStanfordbunnyandIgeamodel,respectively.AsdemonstratedinFig.5,theaboveinterpolationpro-cedureisquitefast.Howeverusingcompactlysupportedbasisfunctionsimpliesseveralessentiallimitations.Ithasnoabilityofrepairingincompletedata,inpartic-ularinterpolatingirregularlysampleddata(seeFig.6)andllingholes(seetherightimageofofFig.7).En-largingthesupportsizeparametersinordertoxthesedrawbacksslowsdownthereconstructionpro-cessessentially.Theinterpolatingimplicitsurfacehasanarrowbandsupport(theleftimageofFig.7).Itrequires,forex-ample,thepolygonizationgridtobesmallerthanthewidthofthesupportband.Fig.6.Left:IrregularlysamplingpointsonthebellypartoftheStanfordBuddha.Right:Meshgeneratedfromthezerolevel-setofthesingle-levelcompactlysupportedimplicitfunction.Implementingamulti-scaleinterpolationprocedurede-scribedinthenextsectioneliminatestheseproblems.Fig.7.Left:ThebottompartoftheStanfordbunnyshowntheleftimageofFig.5.Theholesarenot lled.Right:thewhiteregionindicatesthesupportoftheimplicitfunctionusedtoreconstructtherightimageofFig.5.3Multi-levelInterpolationToovercomeproblemsmentionedattheendofthepre-vioussectionwebuildamulti-scalehierarchyofpointsetsfP1;P2;:::;PM=PgandinterpolateapointsetPm+1ofthehierarchybyoffsettingtheinterpolationfunctionusedinthepreviousleveltointerpolatePm.Fig.8demonstratesthemainstepsofourmulti-levelinterpolationapproach.3.1ConstructionofPointSetHierarchyToconstructthemulti-scalehierarchyofpointsetsfP1;P2;;PM=PgwersttPintoaparal-lelepipedandthensubdivideitanditspartsrecursivelyintoeightequaloctants.PointsetPisclusteredwithrespecttothecellsofthebuiltoctree-basedsubdivisionoftheparal-lelepiped.ForeachcellweconsiderthepointsofPcon-tainedinthecellandcomputetheircentroid.Aunitnormalassignedtothecentroidisobtainedbyaveragingthenor-malsassignedtothepointsofPinsidethecellandnor-malizingtheresult.SetP1correspondstothesubdivisionoftheboundingparallelepipedintoeightequaloctants.3.2MultilevelInterpolationviaOffsettingAfterconstructinghierarchyfP1;P2;;PM=Pg,ourmulti-levelinterpolationprocedureproceedsinthecoarse-to-neway.Firstwedeneabasefunctionf0(x)=1andthenrecursivelydenethesetofinterpolatingfunctionsfk(x)=fk1(x)+ok(x)(k=1;2;:::;M);wherefk(x)=0interpolatesPk.Anoffsettingfunctionokok(x)=pk2Pkhgk(x)+lkiifsk(kxpkk):hastheformusedintheprevioussectionforthesingle-levelinterpolation.Inparticular,localapproximationsgk(x)are4 level1level2level3level4level5level6level7Fig.8.Multi-scaleinterpolationoverview.Weuseamonkmodel(60Kpoints)obtainedasalaserscannerdata.Toprow:multi-scalehierarchyofpointswheretheradiiofthespheresateachlevelkareproportionaltosk,thesupportsizeofRBFsusedfortheinterpolation(theactualsizesthespheresare vetimeslargerthanthatusedforvisualization).Middlerow:interpolatingimplicitsurfacespolygonizedateachlevelofthehierarchy.Bottomrow:cross-sectionsoftheinterpolating;theboldblacklinescorrespondtothezerolevelsetsofthefunctions.determinedsimilarto(3)vialeastsquarettingappliedtoPk.Theshiftingcoefcientslkiarefoundbysolvingthefollowingsystemoflinearequationsfk1(pk)+ok(pk)=0:(5)Similartothesingle-levelinterpolationcaseweusethepre-conditionedbiconjugategradientmethod[31]todeterminelki.Thesupportsizeskisdenedbysk+1=sk2;s1=cL;whereListhelengthofadiagonaloftheboundingparal-lelogramandtheparametercischosensuchthatanoctantoftheboundingboxisalwayscoveredbyaballofradiuss1centeredsomewhereintheoctant.Inpracticeweuse5 c=0:75.Finally,thenumberofsubdivisionlevelsMisdeter-minedbys1ands0wheres0isthesupportsizeforthesingle-levelinterpolation.Accordingtoourexperience,M=dlog2s0=(2s1)eproducesgoodresults.4ResultsandDiscussionThedevelopedmethodcanbeappliedtopointsetsur-facesconsistingofseveralhundredsthousandpointsonstandardPCs.AllexamplespresentedinthispaperwerecomputedonaPentium4M1.6GHzPC.Fig.9presentsresultsofinterpolatingandapproximatingalargeandcomplexpointsetsurface,theStanfordBuddhamodel.Theleftimageshowstheresultofthecompletein-terpolationoftheoriginalStanfordBuddhadatawhiletherightimagedemonstratesanincompletettingprocedure:only(M1)levelsofthemulti-scalehierarchyofpointsetswereused.Novisualdifferencebetweentheleftandrightimagesisobserved.Fig.10showsaresultofinterpolationofirregularlysam-pledpoints.FirsttherightpartoftheoriginalIgeameshwas90%decimated,thenallconnectivityinformationofthemodelwasremoved.Noticethatthesharpdropofthesam-plingdensityproducenovisualartifactsintheimplicitsur-facereconstructedbyourmethodfromthecloudofpoints.ComparisonwithFastRBFInFig.11wecompareourapproachwiththeFastRBFmethod[9].Forthecompari-sonweuseafreeversionofFastRBFtoolbox,release1.2,availablefrom[14].Weusethetoolboxwiththe-direct-accuracy=0.25optionswhichmeanthataRBFcenterreductionprocedureisnotappliedandapproximationaccu-racyisequalto0.25.AccordingtoFig.11,thereisnovi-sualdifferenceintheHandmodelreconstructionbytheFastRBFmethod[9]andourmulti-scalettingtechnique.How-everourapproachworksapproximatelyfourtimesfasterthentheFastRBFmethod.FastRBFworksapproximatelytwotimesfasterwiththecenterreductionprocedurethanwithoutitfortheHandmodel.Ourmethodisstillfaster,moreoveritcanbeacceleratedifasimilarcenterreductionprocedureisimplemented.UnfortunatelywewereunabletotestFastRBFonmorecomplexmodelsbecausethefreeversionofFastRBFhaslimitedcapabilities.VisualizationTovisualizeimplicitsurfaceswepolygo-nizethem.ThemodelsdisplayedinFig.1,Fig.8andFig.9werepolygonizedbyBloomenthal'smethod[5,6].Othermodelsconsideredinthepaperwerepolygonizedbythedual-contouringmethod[21].Ofcourse,otherpolygoniza-tionmethodssuchastheMarchingCubes[25]andextendedMarchingCubes[22]canbeused.AsapostprocessingstepFig.9.Polygonizedimplicitsurfacesinterpolating(left)andapproximating(right)theStanfordBuddhapointdata(544Kpoints).Ninelevelsofthepointsethierarchyweregenerated.Theleftmodelwasgener-atedbymulti-level ttingusingallninelevels:com-putationaltimeis19.1min.,maximalRAMusedis332Mb,themodelisrepresentedbythesumof901Kbasisfunctions.Therightmodelwasgeneratedus-ingeightlevelsonly:computationaltimeis7.5min.,maximalRAMusedis198Mb,themodelisrepre-sentedbythesumof362Kbasisfunctions.Fig.10.Interpolationofirregularlysampleddata(73Kpoints,38sec.).6 Fig.11.TocompareourmethodwithFastRBF[9]weusetheHandmodel(13,348points).Left:thesurfaceisapproximatingbyFastRBF(computationaltimeis30sec.,26,696RBFsareusedtorepresentthemodel).Right:thesurfaceisinterpolatedbyourmethod(computationaltimeis7sec.,themodelisrepresentedbythesumof18,647basisfunctions).wecanalsoemployamethodproposedin[28]inordertoimprovethemeshquality.Extrazero-setsAllthemodelsconsideredinthispaperexceptonedisplayedintherightimageofFig.12werein-terpolatedandpolygonizedbyinspectingtheirboundingboxesandnoextrazerolevelsetswereobserved.How-ever,ifaninterpolatingpointsetsurfacehasthinparts,ex-trazerolevel-setsmaybegeneratednearthesurface,asseenintherightimageofFig.12.Theseextrazerolevel-setswillnotbepolygonizedifapolygonizationprocedurestartsfromseedpointschosenontheinterpolatedpointsetsurface,see[5]whereanappropriateimplicitsurfacepoly-gonizationprocedurewasdeveloped.Howeverextrazerolevel-setsmaybeharmfulforthebooleanoperationswithimplicits.Onepossiblewaytosolvethisproblemofextrazerolevel-setsistouseamoresophisticateddownsamplingprocedurerespectingtopologyand/orcomplexgeometryoftheinterpolatingpointsetsurface.Somehintstosolvethisproblemcanbealsofoundin[30].Robustnesswithrespecttoqualityofnormals.Thenor-malsofapointsetsurfaceareeithercomputedduringashapeacquisitionprocessorestimateddirectlyfromthepoints.ThenormalsaremorepronetonoisethanpointsFig.12.Interpolationofpointsetsurfacesrepresent-ingcomplextopologicalobjects.Left:noextrazerolevel-setsaregenerated.Right:extrazerolevel-setsaregenerated.themselves.Accordingtoourexperience,ourmultiscaleapproachisquiterobustwithrespecttoqualityofnormals.Inparticularitisimpliedbyasmoothingeffectofthedown-samplingprocedureweuse.Ifthenormalassociatedwithpointpiiszero(sometimesithappensduetoerrorsduringtheshapeacquisitionpro-cess)wecannotdecidethelocalshapeorientationatpi.AcommonwaytohandlesuchcasewithinthestandardRBFapproach[35,9,26,36]consistsofnotusingnormalcon-straintsatpi.Similarlywesetgi(x)=0ifthenormalatpiiszero.Fig.13demonstratesrobustnessofourmethodwithre-specttothequalityofnormals.Fig.13.Left:point-renderedrabbitmodel.Middle:point-renderedrabbitmodelwithnoisynormals,2%normalsaresettozero.Right:polygonizedimplicitmodelreconstructedfromthenoisyrabbitbyourmulti-scaleapproach.7 Approximationvs.InterpolationIfscattereddataisnoisy,approximationprocedureispreferableoverinterpo-lation.Thisiswhyinweinterpolateasmoothedversionoftheoriginalangeldata,seeFig.2.Boundariesofrangedataareusuallymorecorruptedbynoisethaninnerparts.Thusitisreasonabletointroduceadelitymeasureanduseanapproximationprocedurewhichtakesthatmeasureintoaccount.ConclusionWehavepresentedamuti-scaleapproachtointerpolatingpointsetsurfacesbyimplicitsurfaces.Ourmethodgener-ateimplicitsolidsthatcanbefurtherusedformorphing,surfacecarvingandotherimplicitsurfaceprocessingop-erations,asseeninFig.15.Ourcontributionistwofold.Besidesanewschemetobuildahierarchyof3Dscattereddatasetswehaveintroducedanewtypeofcompactlysup-portedbasisfunctions.Theinterpolationproceduredevel-opedinthepaperdemonstratesagoodperformancewhileworkingwithirregularlysampledand/orincompletedata.Usingcompactlysupportedbasisfunctionsmakesourap-proachfasterthanthosebasedongloballysupportedbasisfunctions.Fig.14.Left:asurfaceandfeaturepoints(ridgeandravinepoints)detectedonit.Middle:onlythefea-turepointsarekept.Right:surfacereconstructionfromthefeaturepointsonly.Inthefuture,wehopetoimproveourapproachinor-dertohandlelargepointsetsurfaces(severalmillionsofpoints).Wehopethatourmethodcanbeeasilyadoptedtoscattereddataapproximationwhichisimportantforpro-cessingnoisyscattereddata[12,9].Scattereddataapprox-imationwillalsoallowustousefewerpoints[39,9].Weareplanningtocombineourmethodwithfeatureextractionprocedures[3,37,29]inordertoadaptitforprocessingveryincompletedata,seeFig.14forourrststepsinthisdirection.Reconstructionofscattereddatawithsharpfea-tures[11]alsointriguesusinthisarea.Fig.15.CSGoperationswithimplicitsolidsrecon-structedusingourapproach.Left:torusissubtractedfrombunny.Right:blendeddragons.AcknowledgmentsWeareindebtedtoVladimirSavchenkowhohasreadanearlyversionofthispaperandgivenusseveralvalu-ablecommentsandtoAlexanderPaskoforafruitfuldis-cussion.Wearegratefultoanonymousreviewersfortheirusefulcommentsandsuggestions.TheangelmodelisfromCaltech3DGallery[8].Thebunny,Buddha,anddragonmodelsarecourtesyofStanford3Dscanningrepository.TheIgeaandrabbitmodelsareduetoCyberware.WearegratefultoFarFieldTechnologyLtd[14]foritsfreeversionoftheFastRBFtoolkitandtheHandmodel.References[1]M.Alexa.Hierarchicalpartitionofunityapproximation.Technicalreport,TUDarmstadt,August2002.[2]M.Alexa,J.Behr,D.Cohen-Or,S.Fleishman,D.Levin,andC.T.Silva.Pointsetsurfaces.InIEEEVisualization2001,pages2128,October2001.[3]A.G.BelyaevandY.Ohtake.Animageprocessingapproachtode-tectionofridgesandravinesonpolygonalsurfaces.InEurographics2000,ShortPresentations,pages1928,2000.[4]J.F.Blinn.Ageneralizationofalgebraicsurfacedrawing.ACMTransactionsonGraphics,1(3):235256,July1982.[5]J.Bloomenthal.Polygonizationofimplicitsurfaces.Computer-AidedGeometricDesign,5(4):341349,1982.[6]J.Bloomenthal.Animplicitsurfacepolygonizer.GraphicsGemsIV,pages324349,1994.[7]J.Bloomenthal,editor.IntroductiontoImplicitSurfaces.MorganKaufmann,1997.[8]J.-Y.BouguetandP.Perona.3Dphotographyonyourdesk.http://www.vision.caltech.edu/bouguetj/ICCV98/gallery.html.[9]J.C.Carr,R.K.Beatson,J.B.Cherrie,T.J.Mitchell,W.R.Fright,B.C.McCallum,andT.R.Evans.Reconstructionandrepresentationof3Dobjectswithradialbasisfunctions.InComputerGraphics(ProceedingsofACMSIGGRAPH2001),pages6776,August2001.[10]J.Davis,S.R.Marschner,M.Garr,andM.Levoy.Fillingholesincomplexsurfacesusingvolumetricdiffusion.InFirstInternationalSymposiumon3DDataProcessing,Visualization,andTransmission,pages428438,Padua,Italy,June2002.8 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