T样条教程.pptx

T样条教程.pptx

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T-splinesSpeaker:

周联2007.12.12MianworksSederberg,T.W.,Zheng,J.M.,Bakenov,A.,Nasri,A.,T-splinesandT-NURCCS.SIGGRAPH2003.Sederberg,T.W.,DavidL.C.,Zheng,J.M.,Lyche,T.,T-splineSimplicationandLocalRefinement.SIGGRAPH2004AuthorsThomasW.Sederberg,BrighamYoungUniversityJianminZheng,NanyangTechnologicalUniversityAlmazBakenov,EmbassyofKyrgyzRepublicWashington,D.C.AhmadNasri,AmericanUniversityofBeirutTomLyche,UniversityofOsloDavidL.Cardon,BrighamYoungUniversityOtherworksSongW.H.,YangX.N.,Free-formdeformationwithweightedT-spline,TheVisualComputer2005XinLi,JiansongDeng,FalaiChen,DimensionsofSplineSpacesOver3DHierarchicalT-Meshes,JournalofInformationandComputationalScience,Vol.3,No.3,487-501,2006.（EI）ZhangjingHuang,JiansongDeng,YuyuFeng,andFalaiChen,NewProofofDimensionFormulaofSplineSpacesoverT-meshesviaSmoothingCofactors,JournalofComputationalMathematics,Vol.24,No.4,501-514,2006JiansongDeng,FalaiChen,YuyuFeng,DimensionsofsplinespacesoverT-meshes,JournalofComputationalandAppliedMathematics,Vol.194,No.2,267-283,2006.XinLi,JiansongDeng,FalaiChen,SurfaceModelingwithPolynomialSplinesoverHierarchicalT-meshes,acceptedbyCAD/CG2007,andpublishedonTheVisualComputer,2007.JiansongDeng,FalaiChen,etal.,PolynomialsplinesoverhierarchicalT-meshes,submittedtoGraphicalModels,2006.10WhatareT-Splines?

T-Splines:

ageneralizationofnon-uniformB-splinesurfacesT-Splinesarethenextthing.Theyhaveopeneduppossibilitiestoworkwithsurfacesthatweresimplyimpossiblebefore.-EricAllen,ProductionManager,DAZhttp:

/useT-Splines?

AdddetailonlywhereyouneeditCreateeventhemostcomplexshapesasasingle,editablesurfaceCreatenaturaledgeflowandnon-rectangularWhyuseT-Splines?

FitsintoyourWorkflowT-Splinesvs.NURBSReducethenumberofsuperfluouscontrolpoints.T-Splinesvs.NURBSRemoveunwantedripples.T-Splinesvs.NURBSRemovegapOthermethodshierarchicalBsplinesD.Forsey,R.H.Bartels,HierarchicalB-splinerefinement,Comput.Graphics22（4）（1988）205212asplinespaceoveramoregeneralT-mesh,wherecrossing,T-junctional,andL-junctionalverticesareallowed.F.Weller,H.Hagen,Tensor-productsplinespaceswithknotsegments,in:

M.Dalen,T.Lyche,L.L.Schumaker（Eds.）,MathematicalMethodsforCurvesandSurfaces,VanderbiltUniversityPress,Nashville,TN,1995,pp.563572.PolarFormDefinition:

Examples:

PolarFormDefinition:

Ramshaw,L.1989.Blossomsarepolarforms.ComputerAidedGeometricDesign6,323-358.PolarFormKnotIntervalsKnotIntervalsPB-splineswhosecontrolpointshavenotopologicalrelationshipwitheachotheritispointbasedinsteadofgridbasedPB-splinesT-meshAT-splineisaPB-splinebymeansofacontrolgridcalledaT-mesh.T-meshInferknotvectorsfromT-gridTworulesforT-meshControlPointInsertion（2003）LocalknotinsertionCreatefeaturesExtractBezierPatchesMergeB-splinesintoaT-splineTraditionalwayUsecubicNURSSes（SIGGRAPH98）MergeB-splinesintoaT-splineMergeB-splinesintoaT-splineExamplesExamplesAproblemAlocalknotinsertionsometimesrequiresthatotherlocalknotinsertionsmustbeperformed.Aretherecasesinwhichtheseprerequisitescannotallbesatisfied?

T-splineLocalRefinement（2004）BlendingFunctionRefinementBlendingFunctionRefinementT-splineSpacesT-splineSpaces:

thesetofallT-splinesthathavethesameT-meshtopology,knotintervals,andknotcoordinatesystem.T-splineSpacesT-splineSpacesLocalRefinementAlgorithmLocalRefinementAlgorithmLocalRefinementAlgorithm=+LocalRefinementAlgorithmComparedwitholdonealwaysworkrequiresfarfewerunrequestedcontrolpointinsertionsT-splineSimplication12345ComparedwithB-splinewaveletdecompositionCreateaT-splinemodelConvertaT-splineintoaB-splinesurfaceStandardT-splinesSemi-StandardT-splinesSomeopenquestionsWhatT-meshconfigurationsyieldastandardT-spline?

AreT-splineblendingfunctionsalwayslinearlyindependent?

WhatarethefairnesspropertiesofPB-splines?

Free-formdeformationwithweightedT-splineWenhaoSong,XunnianYangTheVisualComputer（2005）Weighted3DPB-splinesT-latticeAT-latticeisarectangularparallelepiped,thatallowsT-junctions.Rule1.Ineachminimalcell,thesumofknotintervalsinthesamedirectionmustbeequal.Rule2.Anyedgemustbeacelledge.Rule3.TherearenozeroedgesinTlattices.AutomaticgenerationofT-lattice1.Definetheinitiallatticetobedeformed.2.Ifthecellcontainsanyvertexofthemodel,subdivideitbyapplyingtheoctreesubdivision.3.Repeatsteps2and3untilauser-specifiedthresholdisreached.AutomaticgenerationofT-latticeAutomaticgenerationofT-latticeAutomaticgenerationofT-latticeParametrizationNonlinearconjugategradientmethodDeformationalgorithmStep1.Definetheinitialregionofthemodeltobedeformed.Step2.GeneratethemultiresolutionlatticesandsetinitialweightsforT-splinevolumes.Step3.Calculatetheparametriccoordinates（u,v,