1. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal1 CHF: A Scalable Topological Data Structure for Tetrahedral Meshes Marcos Lage¹, Thomas Lewiner¹, ², Hélio Lopes¹, Luiz Velho³. ¹ PUC-Rio, Dept. de Matemática, Matmídia Project, Rio de Janeiro – Brazil. ² INRIA – Géométrica Project– Sophia Antipolis – France. ³ IMPA – Visgraf Project – Rio de Janeiro – Brazil.

2. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal2 Introduction Topological Data Structures:Topological Data Structures: MEMORY x PERFORMANCE Scalable ?Scalable ? New Scalable Data Structure for manifold Tetrahedral meshes

3. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal3 Previous Works For 3-Manifolds:For 3-Manifolds: For surfaces: For surfaces:

4. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal4 Contributions Manifold tetrahedral meshesManifold tetrahedral meshes Four levelsFour levels MEMORY x PERFORMANCE Generic containersGeneric containers Bitwise rulesBitwise rules

5. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal5 Level 0 – Overview Characteristic: Tetrahedral “soup” Application: Mesh Visualization

6. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal6 Level 0 – Basics

7. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal7 Level 0 – Rules Tetra(hf) := [hf/4]Tetra(hf) := [hf/4] = hf>>2 = hf>>2 Next hf (hf) := 4*Tetra(hf) + (hf+1)%4Next hf (hf) := 4*Tetra(hf) + (hf+1)%4 = hf&(~3) + (hf|1)&3 = hf&(~3) + (hf|1)&3 Mid hf (hf) := 4*Tetra(hf) + (hf+2)%4Mid hf (hf) := 4*Tetra(hf) + (hf+2)%4 = hf&(~3) + (hf|2)&3 = hf&(~3) + (hf|2)&3 Prev hf (hf) := 4*Tetra(hf) + (hf+3)%4Prev hf (hf) := 4*Tetra(hf) + (hf+3)%4 = hf&(~3) + (hf|3)&3 = hf&(~3) + (hf|3)&3

8. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal8 Level 0 – Half-Edges Next he (hf, he) := (hf, N[he%4][hf%4])Next he (hf, he) := (hf, N[he%4][hf%4]) Prev he (hf, he) := (hf, N[hf%4][he%4] )Prev he (hf, he) := (hf, N[hf%4][he%4] ) Mate he (hf, he) := (Prev he (hf, he), Next he (hf, he))Mate he (hf, he) := (Prev he (hf, he), Next he (hf, he))

9. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal9 Level 1 – Overview Characteristics: Neighborhood Information Application: TraversalTraversal SubdivisionSubdivision

10. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal10 Level 1 – Opposite Half-Face Opposites half-faces: – same vertices – opposite orientation

11. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal11 Level 1 – Opposite Container

12. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal12 Level 1 – Edge Star –Radial he (hf, he) := ( O[hf], next he (hf, he) ) + –Mate he (hf, he) := ( prev he (hf, he), next he (hf, he) )

13. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal13 Level 2 – Overview Characteristics: Explicit representation of the cells Applications: AttributesAttributes SimplificationSimplification

14. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal14 Level 2 – Extra Containers

15. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal15 Level 3 – Overview Characteristics: Explicit representation of the boundary surface Applications: TopologyTopology Direct DrawDirect Draw

16. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal16 Level 3 – Border CHE CHE: Compact half-edge Version of CHF of surfacesVersion of CHF of surfaces Four levels of structureFour levels of structure MEMORY x PERFORMANCE Generic containersGeneric containers Arithmetic rulesArithmetic rules

17. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal17 Operations – Vertex Star Level 0 Level 0 Level 1 Level 1 Levels 2 & 3 Levels 2 & 3 O(4*n tetra ) O(2*n tetra ) Θ(d°(v))

18. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal18 Operations – Edge Star Level 0 Level 0 Level 1 Level 1 Levels 2 & 3 Levels 2 & 3 O(4*n tetra ) O(2*n tetra ) Θ(d°(e))

19. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal19 Future Works Non-manifold meshes Vertex and edge singularities

20. Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Monday, October 12, 2015Sibgrapi 2005 - Natal20 Thanks !!!