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CHF : A Scalable Topological Data Structure for Tetrahedral Meshes

This paper presents a new scalable data structure for tetrahedral meshes, focusing on memory efficiency and performance. It introduces manifold tetrahedral meshes at four levels, with generic containers and bitwise rules. The structure allows for efficient mesh visualization and traversal, providing explicit representation of cells and boundary surfaces. Additionally, it discusses future works on non-manifold meshes and vertex and edge singularities. Overall, it aims to enhance the efficiency and flexibility of working with tetrahedral meshes in various applications.

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CHF : A Scalable Topological Data Structure for Tetrahedral Meshes

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  1. 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. Sibgrapi 2005 - Natal

  2. Introduction • Topological Data Structures: MEMORY x PERFORMANCE • Scalable ? New Scalable Data Structure for manifold Tetrahedral meshes Sibgrapi 2005 - Natal

  3. Previous Works • For surfaces: • For 3-Manifolds: Sibgrapi 2005 - Natal

  4. Contributions • Manifold tetrahedral meshes • Four levels MEMORY x PERFORMANCE • Generic containers • Bitwise rules Sibgrapi 2005 - Natal

  5. Level 0 – Overview Characteristic: Tetrahedral “soup” Application: Mesh Visualization Sibgrapi 2005 - Natal

  6. Level 0 – Basics Sibgrapi 2005 - Natal

  7. Level 0 – Rules • Tetra(hf) := [hf/4] = hf>>2 • Nexthf(hf) := 4*Tetra(hf) + (hf+1)%4 = hf&(~3) + (hf|1)&3 • Midhf(hf) := 4*Tetra(hf) + (hf+2)%4 = hf&(~3) + (hf|2)&3 • Prevhf(hf) := 4*Tetra(hf) + (hf+3)%4 = hf&(~3) + (hf|3)&3 Sibgrapi 2005 - Natal

  8. Level 0 – Half-Edges • Nexthe(hf, he) := (hf, N[he%4][hf%4]) • Prevhe(hf, he) := (hf, N[hf%4][he%4] ) • Matehe(hf, he) := (Prevhe(hf, he), Nexthe(hf, he)) Sibgrapi 2005 - Natal

  9. Level 1 – Overview Characteristics: Neighborhood Information Application: • Traversal • Subdivision Sibgrapi 2005 - Natal

  10. Level 1 – Opposite Half-Face • Opposites half-faces: • same vertices • opposite orientation Sibgrapi 2005 - Natal

  11. Level 1 – Opposite Container Sibgrapi 2005 - Natal

  12. Level 1 – Edge Star • Radialhe(hf, he) := ( O[hf] , nexthe(hf, he) ) + • Matehe(hf, he) := ( prevhe(hf, he), nexthe(hf, he) ) Sibgrapi 2005 - Natal

  13. Level 2 – Overview Characteristics: Explicit representation of the cells Applications: • Attributes • Simplification Sibgrapi 2005 - Natal

  14. Level 2 – Extra Containers Sibgrapi 2005 - Natal

  15. Level 3 – Overview Characteristics: Explicit representation of the boundary surface Applications: • Topology • Direct Draw Sibgrapi 2005 - Natal

  16. Level 3 – Border CHE CHE: Compact half-edge • Version of CHF of surfaces • Four levels of structure MEMORY x PERFORMANCE • Generic containers • Arithmetic rules Sibgrapi 2005 - Natal

  17. Operations – Vertex Star • Level 0 • Level 1 • Levels 2 & 3 O(4*ntetra) O(2*ntetra) Θ(d°(v)) Sibgrapi 2005 - Natal

  18. Operations – Edge Star • Level 0 • Level 1 • Levels 2 & 3 O(4*ntetra) O(2*ntetra) Θ(d°(e)) Sibgrapi 2005 - Natal

  19. Future Works • Non-manifold meshes • Vertex andedge singularities Sibgrapi 2005 - Natal

  20. Thanks !!! Sibgrapi 2005 - Natal

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