Following the obser vations of Joshi et al JMD 07 we show the advantage of having positive coordinates The control points of the deformation are the vertices of a cage enclosing the deformed mesh To de64257ne positive mean value coordinates for a ID: 23319
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Lipman&Kopf&Levin&Cohen-Or/PositiveMeanValueCoordinates Undeformed MVC(13.5sec) PMVC(18.5sec) HC(333.61sec,643voxels) Figure1:ComparisonbetweenMVC,PMVC,andHC.Theweaknessofnegativecordinatesisclearlyevident,whileHCtakesalongtimetocompute.ThePMVC,incontrast,havenonegativecoordinatesandarecomputedalmostasquicklyastheMVC. shownbyJoshietal.[ JMD07 ],themainawofMVCasatoolforsurfacedeformationisthattheyarenotnecessar-ilypositiveonnon-convexdomains.Thisinturnleadstocounter-intuitivedeformationwhenthecontrolmeshisnon-convex.AsillustratedinFigure 1 ,thenegativecoordinatesofaremotebranchofthecagedistortthegeometryinanon-intuitiveunexpectedway.Joshietal.[ JMD07 ]suggestusingharmonicfunctionswithKroneckerDelta-typeboundaryconditionstofurnishthedesiredpositivecoordinates.hereafterwewillrefertothesecoordinateswithHC.Theirsolutionismathematicallyelegantandguaranteepositiveness,however,ratherexpen-siveinpractice;thecoordinatesrequiressolvingtheLaplaceequationonthewholeinteriorofathreedimensionaldomain(theinteriorofthecage).ThesolutionoftheLaplaceequa-tionisanon-localexpensiveprocess.Thesolutionispracti-callycalculatedonagridinsidethecontrolmesh,wherethegridresolutionisdeterminedbyanerrorcriterion.Inthatcaseonehastobalanceaccuracywithstorageandcomputa-tion;increasingthegridresolutionbyoneleveloctuplicates(8X)thestorageandleadstosignicantlyslowercomputa-tion. Inthisworkweintroducepositivemeanvaluecoordinates(PMVC).UnliketheMVC,themodiedcoordinatesareun-conditionallypositive,andrequireonlyalocalcomputation.Wedemonstratetheadvantageofpositivecoordinatesinvar-iousexamplesofsurfacedeformation(e.g.,Figure 1 ).ThePMVCaretooinvolvedtobecomputedanalyticallyfromtheirclosedformformula,instead,weintroduceaGPU-assistedtechniquetocalculatenumericallythecoordinatesofagiveninputmesh.Aswewillshow,thecomputationofnewcoordinates,eitherasaresultofanewcageoranewmesh,requiresfewsecondsonly.2.PositiveMeanValueCoordinatesGivenashapetobedeformed,denotebyCacoarsecontrolmeshenclosingit.WewillrefertoCasa"cage"similartoJoshietal.[ JMD07 ].WedenotebyV=fvigi2IVthever-ticesofthecageC,whereIVisthecorrespondingindexset.Similarlytothecitedpreviousworks,thegoalistodeneasetoffunctionsli(x),i2VsuchthattheoperatorPf(x):=åi2Vli(x)f(vi);(1) c TheEurographicsAssociation2007. Lipman&Kopf&Levin&Cohen-Or/PositiveMeanValueCoordinates andthefunctionsli(x)satisfythefollowingproperties: 1. Afneinvariance:åili(x)=1. 2. Linearreproduction:Pf(x)=f(x)foralllinearfunc-tionsf(x). 3. Interpolation:limx!cPf(x)=f(c)wherec2¶C(theboundaryofC). 4. Smoothness:li(x)aresmooth. 5. Positivity:li(x)0foralli.whereallthesepropertiesshouldholdforallx2C(in-sidethecage).Therstfourpropertieswereformulatedin[ JSW05 ],andthelastpropertyisintroducedin[ JMD07 ].ThePMVCisdenedasfollows:Pf(x)=Rs2Sxf(p(s))w(x;s)ds Rs2Sxw(x;s)ds;(2)whereSxisaunitspherecenteredatx,andp(s)istherstintersectionoftheline`(t):=x+(s�x)tandC,fort0,andw(x;s)=1=kx�p(s)k.NotetheinterestingrelationbetweenMVCandPMVC;inPMVCthesphereisprojectedonthecage,whileintheformerthecageisprojectedontothesphere.Givenf(vi);i2IV,letyi(x)beapiecewise-linearfunc-tionsuchthatyi(vi0)=di;i0,wheredi;i0=1iffi=i0,and0else.Denethepiecewiselinearfunctionåi2Vfiyi(x)ontheboundaryofC,andthentheapproximantis:Pf(x)=åi2VfiRs2Sxyi(p(s))w(x;s)ds Rs2Sxw(x;s)ds:Then,wedeneli(x):=Rs2Sxyi(p(s))w(x;s)ds Rs2Sxw(x;s)ds;(3)andtheinterpolantcanthenbewrittenintheform( 1 ).Theli(x)arecalledcoordinatefunctionsforthePMVCinterpolant.Thecoordinatesareusedfordeningatransfor-mationTfromtheinteriorofthecageCintoRd(d=2;3):T:interior(C)!Rd:ThetransformationTisdenedasfollows:Apointp2interior(C)canbewrittenasthefollowingafnecombina-tionduetothelinearreproductionpropertyp=åi2IVli(p)vi:Then,givenadeformedcage˜Cwithvertices˜V=f˜vigi2iVthetransformedpositionofpisdenedasT(p):=åi2IVli(p)˜vi:Therefore,thepropertiesofthetransformationTarede-rivedfromthepropertiesofthecoordinatefunctionsli(x).Letusprovesomeofthepropertieslistedabove. MVCPMVCFigure2:Acoordinatefunctionli(x)isdrawnforanon-convexpolygon.Thei-thvertexismarkedwithgreenpoint.Theredcolorstandsforpositivevaluesandthebluearenegativevalues. Theinterpolationandlinearreproductionpropertiescanbeunderstoodbytheargumentationof[ JSW05 ],butforthesakeofcompletenesswewilllayitherealso.First,theinter-polationisduetothefactthatw(x;y) Rs2Sxw(x;s)dsisconvergingtotheDeltafunctiononthesphereasx!cforc2¶C.Asforthelinearreproductionpropertyitresultsfromthesymmetryargument:Zs2Sxx�p(s) kx�p(s)kds=0:Andtherefore,x=Rs2Sxp(s)w(x;s)ds Rs2Sxw(x;s)ds:Thatis,thecoordinatefunctionsarereproduced,andfromthelinearityoftheoperator( 1 )andtheafneinvariancethepropertyresults.Astothenon-negativitypropertyoftheco-ordinatefunctions,thisreadilyresultsfromthefactthatthecoordinatesli(x)aredenedviaanintegrationofanon-negativefunctionoverasphereinEq.( 3 ).Forexample,seeFigure 2 forcomparisonwithMVC.SmoothnessOneofthestrongpropertiesoftheMVCisthattheyaresmooth.ThePMVCdenitioninvolvesvisibilityconsiderationwhichincurssingularitiesacrosssupportingplanesofreexvertices.ThesupportingplanespartitionthecageintoregionswithinwhichthePMVCaresmooth,whileacrossthesupportingplanessmoothnessisnotguaranteed.AnexampleofsuchscenarioisshowninFigure 4 ,wherethecoordinatefunctionassociatedwiththe`spike'vertexisnotsmooth.However,asdepictedinthatgure,minorre-nementofthecagealleviatesthisproblem.Inpractice,wefoundthatinmostcasestheresultisplausibleandsmooth.ThiscanbeobservedinFigure 3 ,whereashapeofachecker-boardpatternthatcrossesasupportinglineissmoothlybent.Theexampleshowstheeffectofeditingasinglevertexandconsequentlyofasinglecoordinateli(x). c TheEurographicsAssociation2007. Lipman&Kopf&Levin&Cohen-Or/PositiveMeanValueCoordinates UndeformedMVC(3.70sec)PMVC(3.03sec)HC(66.48sec) UndeformedMVC(5.92sec)PMVC(8.75sec)HC(28.61sec)Figure7:ComparisonbetweenMVC,PMVC,andHCdeformationsoftheArmadilloandHorsemodels.Notethatonmodelslikethese,wherethecagehaswellseperatedlimbs,thedeformationqualityissimilar. TherunningtimesofPMVCaregenerallyinthesameor-derofmagnitudeasMVC,andabout10–100timesfasterthanHC,dependingonmeshandcagecomplexity.Theper-formanceisroughlylinearinthenumberofcubemappix-els.Wefoundthatformostmeshes322cubemapshaveneenoughresolutionsothatnoquantizationartifactsareno-ticeable.Thehandmodel(Figure 1 )wascreatedwith642cubemaps.Table 1 providesdetailedtimingsforvariousex-amplesshowninthispaper.TheperformanceofPMVCisonlylooselyconnectedtocagecomplexity.Table 2 showstimingsforasingleobjectwithincreasinglymorecomplexcages.Astomemoryusage,itshouldbenotedthatPMVC,sim-ilartoMVC,computesthecoordinatesofeachembeddedshapepointdirectlyon-the-yduringsetup.5.ConclusionsWepresentedmeanvaluecoordinatesthatconsideronlythevisibleportionofthecagetoguarantee,likeharmoniccoor-dinates,thatthecoordinatevaluesarealwayspositive.ThekeypointisthatthepositivemeanvaluecoordinatescanbecomputedfastbyexploitingthereadilyavailablevisibilitycomputationoftheGPU.Furthermore,asweshowed,theGPUspeedturnsthecomputationpracticallyinsensitiveto thecageresolution.Thisallowsustorenethecageandimprovethequalityoftheinterpolationwithoutsignicantcost.Ourmethodsuccessfullybringstheideaofpositiveco-ordinatestothepointwhereitistrulyapracticalandusefultoolformeshdeformation.WebelievethatmoreresearchcanleadtoevenfastermethodstocomputelocalcoordinateswithassistanceoftheGPU.AcknowledgementsWethankScottSchaefferforprovidinguswiththecagesandmodelsoftheArmadilloandhorse,andHongboFuandKunZhouforthehandmodel.WealsothankTaoJuforhelpfulandinsightfulcomments.ThisworkwassupportedinpartbytheIsraelScienceFoundation.References [Coq90] COQUILLARTS.:Extendedfree-formdeformation:asculpturingtoolfor3dgeometricmodeling.ProceedingsofSIG-GRAPH'90(1990),187–196. [FKR05] FLOATERM.S.,KOSG.,REIMERSM.:Meanvaluecoordinatesin3d.ComputerAidedGeometricDesign22,7(2005),623–631. [Flo03] FLOATERM.S.:Meanvaluecoordinates.ComputerAidedGeometricDesign20,1(2003),19–27. c TheEurographicsAssociation2007. Lipman&Kopf&Levin&Cohen-Or/PositiveMeanValueCoordinates // Initializationsetalllv;i=0setallwsumv=0 // Renderingandintegrationforeachvertexvdo foreachcubemapfacefdo renderftorenderbuffersifrenderbufferfullorlastvertex-facepairthen // Integrationforeachrenderbufferpixel(x;y)do tri=getValue(x;y;3)w=getWeight(x;y)forc=0to2do i=vertexList[tri3+c]lv;i+=getValue(x;y;c)wendwsumv+=wendclearrenderbuffersendendend // Normalizationforeachvertexvdo foreachcubemapfacefdo lv;i==wsumvendend Algorithm1:OuralgorithmtocomputethePMVCco-ordinates. [HF06] HORMANNK.,FLOATERM.S.:Meanvaluecoordinatesforarbitraryplanarpolygons.ACMTransactionsonGraphics25,4(2006),1424–1441. [JMD07] JOSHIP.,MEYERM.,DEROSET.,GREENB.,SANOCKIT.:Harmoniccoordinatesforcharacterarticulation.TransactionsonGraphics26,3(Proc.SIGGRAPH)(2007). [JSW05] JUT.,SCHAEFERS.,WARRENJ.:Meanvalueco-ordinatesforclosedtriangularmeshes.vol.24,3(Proc.SIG-GRAPH),pp.561–566. [SP86] SEDERBERGT.W.,PARRYS.R.:Free-formdeforma-tionofsolidgeometricmodels.ProceedingsofSIGGRAPH'86(1986),151–160. Armadillo Horse Hand MeshVertices 15,002 48,485 24,795 CageVertices 110 51 252 MVC 3.70s 5.92s 13.50s PMVC322 3.03s 8.75s 6.64s PMVC642 9.82s 30.64s 18.50s HC643 66.48s 28.61s 333.61s HC1283 770.00s 305.93s 4413.66s Table1:Performancecomparison. CageVertices CageTriangles PMVCtiming 51 98 16.07s 102 200 16.32s 198 392 16.73s 402 800 17.44s 843 1682 18.70s 1523 3042 20.41s Table2:TheperformanceofPMVCisonlylooselycon-nectedtothecagecomplexity.Thetableshowstimingsforthehandmodel(24,795vertices)andsphericalcage.Thecagewassubdividedintoincreasingnumbersoftriangles. c TheEurographicsAssociation2007.