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CHFCHF: A Scalable Topological Data : A Scalable Topological Data Structure for Tetrahedral MeshesStructure for Tetrahedral Meshes

Marcos Lage¹, Thomas Lewiner¹Marcos Lage¹, Thomas Lewiner¹,,², Hélio Lopes¹, Luiz Velho³.², Hélio Lopes¹, Luiz Velho³.

¹ PUC-Rio, Dept. de Matemática, Matmídia Project, Rio de Janeiro – Brazil.¹ PUC-Rio, Dept. de Matemática, Matmídia Project, Rio de Janeiro – Brazil.² INRIA – Géométrica Project– Sophia Antipolis – France.² INRIA – Géométrica Project– Sophia Antipolis – France.

³ IMPA – Visgraf Project – Rio de Janeiro – Brazil.³ IMPA – Visgraf Project – Rio de Janeiro – Brazil.


IntroductionIntroduction

• Topological Data Structures:Topological Data Structures:

MEMORY x PERFORMANCEMEMORY x PERFORMANCE

• Scalable ?Scalable ?

New Scalable Data Structure for New Scalable Data Structure for manifold Tetrahedral meshesmanifold Tetrahedral meshes


Previous WorksPrevious Works

• For 3-Manifolds:For 3-Manifolds:

• For surfaces:For surfaces:


ContributionsContributions

• Manifold Manifold tetrahedral meshestetrahedral meshes

• Four levelsFour levelsMEMORY x PERFORMANCEMEMORY x PERFORMANCE

• Generic containersGeneric containers

• Bitwise rulesBitwise rules


Level 0 – OverviewLevel 0 – Overview

Characteristic:Characteristic:Tetrahedral “soup”Tetrahedral “soup”

Application:Application:Mesh VisualizationMesh Visualization


Level 0 – BasicsLevel 0 – Basics


Level 0 – RulesLevel 0 – Rules

• Tetra(hf) := Tetra(hf) := [hf/4][hf/4] = = hf>>2hf>>2

• NextNexthfhf(hf) := (hf) := 4*Tetra(hf) + (hf+1)%44*Tetra(hf) + (hf+1)%4 = hf&(~3) + (hf|1)&3= hf&(~3) + (hf|1)&3

• MidMidhfhf(hf) := (hf) := 4*Tetra(hf) + (hf+2)%44*Tetra(hf) + (hf+2)%4

= hf&(~3) + (hf|2)&3 = hf&(~3) + (hf|2)&3

• PrevPrevhfhf(hf) := (hf) := 4*Tetra(hf) + (hf+3)%44*Tetra(hf) + (hf+3)%4

= hf&(~3) + (hf|3)&3= hf&(~3) + (hf|3)&3


Level 0 – Half-EdgesLevel 0 – Half-Edges• NextNexthehe(hf, he) := (hf, N[he%4][hf%4])(hf, he) := (hf, N[he%4][hf%4])

• PrevPrevhehe(hf, he) := (hf, N[hf%4][he%4] )(hf, he) := (hf, N[hf%4][he%4] )

• MateMatehehe(hf, he) := (Prev(hf, he) := (Prevhehe(hf, he), Next(hf, he), Nexthehe(hf, he))(hf, he))

021

103

032

213

N



Characteristics:Characteristics:Neighborhood InformationNeighborhood Information

Application:Application:• TraversalTraversal• SubdivisionSubdivision


Level 1 – Opposite Half-FaceLevel 1 – Opposite Half-Face

Opposites half-faces:Opposites half-faces:– same verticessame vertices– opposite orientationopposite orientation


Level 1 – Opposite ContainerLevel 1 – Opposite Container


Level 1 – Edge StarLevel 1 – Edge Star

– RadialRadialhehe(hf, he)(hf, he) := ( O[hf] , next := ( O[hf] , nexthehe(hf, he) )(hf, he) )

++

– MateMatehehe(hf, he) := ( prev(hf, he) := ( prevhehe(hf, he), next(hf, he), nexthehe(hf, he) )(hf, he) )



Characteristics:Characteristics:Explicit representation Explicit representation of the cellsof the cells

Applications:Applications:• AttributesAttributes• SimplificationSimplification


Level 2 – Extra ContainersLevel 2 – Extra Containers



Characteristics:Characteristics:Explicit representation of Explicit representation of the boundary surfacethe boundary surface

Applications:Applications:• TopologyTopology• Direct DrawDirect Draw


Level 3 – Border CHELevel 3 – Border CHE

CHECHE: Compact half-edge: Compact half-edge• Version of Version of CHFCHF of surfaces of surfaces

• Four levels of structureFour levels of structure

MEMORY x PERFORMANCEMEMORY x PERFORMANCE

• Generic containersGeneric containers

• Arithmetic rulesArithmetic rules


Operations – Vertex StarOperations – Vertex Star• Level 0Level 0 • Level 1Level 1 • Levels 2 & 3Levels 2 & 3

O(4*nO(4*ntetratetra)) O(2*nO(2*ntetratetra)) ΘΘ(d°(v))(d°(v))


Operations – Edge StarOperations – Edge Star• Level 0Level 0 • Level 1Level 1 • Levels 2 & 3Levels 2 & 3

O(4*nO(4*ntetratetra)) O(2*nO(2*ntetratetra)) ΘΘ(d°(e))(d°(e))


Future WorksFuture Works

Non-manifold meshesNon-manifold meshes

Vertex andVertex andedge singularitiesedge singularities


Thanks !!!Thanks !!!

Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12,...

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