Paper - Geometric Algorithms for Modeling, Motion, and Animation ...

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ut u0 u1 kt k1 k0 E 2 t E v0 l1 lt vt l v1 (a) Two deforming edges 1 2 E & E 1 t 0 ut kt E 2 t E vt ( l lt t kt) nt (b) Projected distance 1 between E and E Figure4: EEtestfilter:ToperformaEEtestbetweenthetwo edges E 1 and E 2 (definedby u0, v0and k0, l0at t = 0, u1, v1 and k1, l1at t = 1),weneedtocheckthecoplanarityconditions oftheseverticesbyfindingall t(t ∈ [0, 1])wheretheprojected distancebetween ltandthetriangledefinedby kt, ut,and vtis equaltozero,i.e., (lt − kt) · nt = 0. Soweobtainthesufficientconditionfornon-coplanar: Ifallthe fourscalarvaluesarenegative/positive,thevertices Ptwillnever beontheplanedefinedbythethreevertices(at, bt,and ct)ofthe deformingtriangle Tt. Thegeometricinterpretationofsymbol Aand Bisthattheyrepresenttheprojecteddistancesat t = 0and t = 1,respectively.From Equation(6),weconcludethatitisnotsufficienttoonlycheckthe signsof Aand B,thevaluesofothersymbols C-Falsoaffectthe coplanaritycondition. 3.3.1 ExtensiontoEEtests Inordertoperformanelementarytestbetweentwoedges E 1 and E 2 (definedby u0, v0and k0, l0at t = 0, u1, v1and k1, l1at t = 1),weneedtocheckthecoplanarityofthesefourpoints(Fig.4). Byreplacing Pt, at, bt, ctwith lt, kt, ut, vt,wecansymmetrically deduceanon-coplanartheoremforEEtests: Non-coplanarTheoremforEEtests: Fortwodeformingedges E 1 and E 2 definedbythestartandendpositionsduringtheinterval [0, 1],thesepositionsarelinearlyinterpolatedintheinterval withrespecttothetimevariable, t.Ifthefollowingfourscalarvalues: A ′ , B ′ , 2∗C′ +F ′ ,and 3 2∗D′ +E ′ havethesamesign, E 3 1 and E 2 willnotbecoplanarduringtheinterval: A ′ = (l0 − k0) · n ′ 0, B ′ = (l1 − k1) · n ′ 1 C ′ = (l0 − k0) · ˆn ′ , D ′ = (l1 − k1) · ˆn ′ E ′ = (l0 − k0) · n ′ 1, F ′ = (l1 − k1) · n ′ 0 And: n ′ 0 = (u0 − k0) × (v0 − k0), n ′ 1 = (u1 − k1) × (v1 − k1) ˆn ′ = (n′ 0 +n′ 1 −( vu− v k)×( vv− v k)) 2 vk = k1 − k0, vu = u1 − u0, vv = v1 − v0. 3.4 CCDAlgorithm We use the non-planarity filter defined above to perform coplanarity-basedculling.WeuseEquations(1)-(3)tocalculatethe valuesofthefourcontrolverticesdefinedinthoseequationsfora primitivepair.Ifthesevariableshavethesamesign,thispairwill beidentifiedasafalsepositive. 1 t t nt 2 t BVH Traversal Inter-object testing Triangle based bounding volume culling Intra-object testing Continuous normal cone PR-Trianges Orphan set Primitive based bounding volume culling Deforming Non- Penetration Filter Elementary tests Figure5: CCDalgorithmwithdeformingnon-penetrationfilter:Thedeformingnon-penetrationfilteriscomplementarytotheoverallCCDalgorithmandcanbeperformedrightbeforetheelementarytests. (a) Use of deforming filter (b) Only use of bounding volume tests Figure6: EdgesinvolvedbyEEtestswithdifferentculling methods:Thisfigureshowsedges(highlightedingreenandblue respectively)usedbyEEtestsduringaspecificframeoftheCloth benchmark(Fig.7(d))withcoplanarity-basedcullingandboundingvolumebasedcullingrespectively.Comparingtoonlyboundingvolumebasedculling,thenumberofEEelementarytestsare reducedby 80%withthedeformingnon-penetrationfilter. Thedeformingnon-penetrationfiltercanbecombinedwithhierarchicalrepresentations(Fig.5). DuringthetraversalofBVH,we performdifferenthigh-levelcullingmethodstoremoveredundant pairwisetestsorprimitivepairsthatarenotincloseproximityto oneanother.Next,thedeformingnon-penetrationfilterisusedas partoflow-levelcullingtofurtherremovethefalsepositives. Fig.6comparestheeffectsofboundingvolumebasedcullingand coplanarity-basedculling.ThegreenlinesegmentsinFig.6(a)and bluelinesegmentsinFig.6(b)describethedistributionofedgesinvolvedinEEtestswithdeformingnon-penetrationfilterandboundingvolumebasedcullingrespectively.Asshowninthefigure,comparedtothenumberoftestsintheboundingvolumebasedculling, weobservemorethan 80%reductioninthenumberofelementary testsonthisscene. 4 ImplementationandPerformance Inthissection,wedescribeourimplementationandhighlightthe performanceofouralgorithmonseveralbenchmarks.
4.1 Implementation Wehaveimplementedouralgorithmonastandard2.4GHzIntel Pentiummachinewith4GBRAMon32-bitWindows/XPplatform. Theperformanceismeasuredusingasinglethread.Weusek-DOPs (specifically16-DOPs)asboundingvolumesbecausetheyprovide agoodbalancebetweentightfittingandrapidupdating. Weuserestructuringandrefittingtoupdatethehierarchyfordeformablemodels.IntelSSE/SSE2instructionsareusedtoacceleratetheupdatingofandoverlaptestingbetweenboundingvolumes. WeuseanimplementationoftheICCDalgorithmbasedonnormal conesforhigh-levelculling[Tangetal.2009a]asabaselinefor comparison.WealsointegrateourfilterswiththeR-Trianglealgorithm[Curtisetal.2008],asitspendsalargefractionofthequery timeontheelementarytests. 4.2 Benchmarks&Performance Inordertotesttheperformanceofouralgorithm,weusedsixdifferentbenchmarks,arisingfromdifferentsimulationswithdifferent characteristics. • Lion:Forthisbreakingbenchmarkwith1.6Mtriangles(Figure1),ouralgorithmreducesthenumberofelementarytest by 10xascomparedtopriorCCDalgorithms. • Balls:Ascenewithhundredsofballs(34Ktriangles)thatare collidingwitheachother(Figure7(a)).Ourdeformingnonpenetrationfilterreducesthenumberofelementarytestsby 17x. • FallingAlphabets: Multipledeformingcharacters(5Ktriangles)fallintoabowlandbreakintopieces(Figure7(b)). OurCCDalgorithmreducesthenumberofelementarytests by 58x. • Princess: Thismodel(40Ktriangles)hasmanyinter-and intra-objectcollisions(Figure7(c)). Ouralgorithmreduces thenumberofelementarytestsby 20.5x. • Cloth: Acloth(92Ktriangles)hasahighnumberofselfcollisions(Figure7(d)).Ouralgorithmreducesthenumberof elementarytestsby 11.5x. • Flamenco:Thisbenchmark(49Ktriangles)hasmanyinterandintra-objectcollisions(Figure7(e)).TheCCDalgorithm withthedeformingnon-penetrationfilterreducesthenumber ofelementarytestsby 17x. Fig.8highlightsthecullingefficiencyofouralgorithmbycomparingthenumberofelementarytestsperformed.Asshowninthefigure,theelementarytestsaredramaticallyreducedby 10 − 58.7x. Also,bypreventingthecomputationofthesefalsepositives,we achieve 1.5−3.5ximprovementonoverallperformanceincomparisonwiththeICCDalgorithm[Tangetal.2009a],and 2.4 − 8.2x improvementoverR-Triangles[Curtisetal.2008](Table.1). For BenchmarkFlamenco,itsaveragerunningtimeperframeinR- Triangles,ICCD,andoursystemareshowninFigure9. 5 AnalysisandComparison Inthissection,weanalyzeourresultsandcompareagainstprior methods. 5.1 Analysis Althoughtheperformanceofourdeformingnon-penetrationfilter varieswithdifferentbenchmarks,weareabletoobtainhighculling (a) Balls (b) Falling Alphabets (d) Cloth (c) Princess (e) Flamenco Figure7:Benchmarks:Allthebenchmarkshavemultiplesimulationsteps.WeperformCCD,includingself-collisions,betweendiscretestepsofthesimulationandcomputethefirsttime-of-contact. efficiencyinmanyofthem.Thebasicfiltertestproposedinthenoncoplanartheoremsisconservative. Intuitively,ournon-coplanar theoremeliminatestheneedtoperformexacttestsonpairswhere no‘penetrations’occuralongthecontinuoustrajectory. Inpractice,duetothetemporalcoherencebetweensubsequentframes,the ’penetrations’onlyoccurinfrequently. Fig.10showsthechangingofratiosofprimitivepairsinwhichno‘penetration’occurs duringthetimeinterval. Inthefigure,theratiosstayabove 80%, andarechangingsmoothly.Duetocoherence,ourdeformingnonpenetrationfiltercanbequiteeffective. WeusetheInterval-Newtonmethodtosolvethecubicequations. Anelementarytesttakesaboutroughly155additions,217multiplications,and6divisionsonaverage(allforfloatpointvalues). Ontheotherhand,thedeformingnon-penetrationfilteronlyneedto perform29additionsand40multiplicationsonaverage.Inpractice, thespeedofadeformingnon-penetrationfilterisabout5.5-10.2x fasterthananexacttest. 5.2 Comparison Inthissection,wecompareournovelalgorithmwithpriorculling algorithms. Bounding volume based culling: Bounding volumes (e.g. spheres [Hubbard 1993; Palmer and Grimsdale 1995; BradshawandO’Sullivan2004],AABBs[vandenBergen1997],k-
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