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Compressing Hexahedral Volume Meshes

Learn about compressing hexahedral volume meshes using edge degrees, a method that takes advantage of the regularity commonly found in such data sets. Explore the process of coding with edge degrees, boundary propagation, adaptive traversal, and more.

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Compressing Hexahedral Volume Meshes

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  1. CompressingHexahedral Volume Meshes Martin Isenburg UNCChapel Hill Pierre Alliez INRIASophia-Antipolis

  2. Overview • Volume Meshes • Related Work • Compressing Connectivity • Coding with Edge Degrees • Boundary Propagation • Adaptive Traversal • Compressing Geometry • Parallelogram Prediction • Demo

  3. Take this home: “The connectivity of a hexahedral mesh can be coded through asequence of its edge degrees.” “This encoding naturallyexploits the regularity commonlyfound in such data sets.”

  4. Volume Meshes

  5. Volume Meshes • scientific & industrial applications • thermodynamics • structural mechanics • … • visualization & simulation • unstructured / irregular (not on a grid ) • tetrahedral, hexahedral, polyhedral

  6. Hexahedral Volume Meshes • have “numerical advantages in finite element computations” • challenging to generate • their internal structure looks “nice” compared to tetrahedral meshes

  7. Ingredients • geometry :positions of vertices • connectivity :which vertices form a hexahedron • properties:attached to vertices density, pressure, heat,...

  8. log2(v) 8h*32 bits 16 3v *32 bits 1.69 Standard Representation • connectivity • geometry hex1 1 3 6 4 7 8 9 2 hex2 4 5 8 2 9 1 6 7 hex3 7 5 … hexh less than84 KB vtx1 ( x, y, z ) vtx2 ( x, y, z ) vtx3 ( x, y, z ) vtxv 71572 hexahedra 78618 vertices size: 3.23 MB

  9. Related Work

  10. Surface Mesh Compression • Geometry Compression, [Deering, 95] • Topological Surgery, [Taubin & Rossignac, 98] • Cut-Border Machine, [Gumhold & Strasser, 98] • Triangle Mesh Compression, [Touma & Gotsman, 98] • Edgebreaker, [Rossignac, 99] • Spectral Compression of Geometry, [Karni & Gotsman, 00] • Face Fixer, [Isenburg & Snoeyink, 00] • Valence-driven Connectivity Coding, [Alliez & Desbrun, 01] • Near-Optimal Coding, [Khodakovsky, Alliez, Desbrun & • Degree Duality Coder, [Isenburg, 02] • Polygonal Parallelogram Prediction, [Isenburg & Alliez, 02] Schröder, 02]

  11. Volume Mesh Compression • Grow & Fold, [Szymczak & Rossignac, 99] • Cut-Border Machine,[Gumhold, Guthe & Strasser, 99] • Rendering of compressed volume data, [Yang et al., 01] Simplification: • Simplification of tetrahedral meshes,[Trotts et al., 98] • Progressive Tetrahedralizations,[Staadt & Gross, 98] Progressive Compression: • Implant Sprays,[Pajarola, Rossignac & Szymczak, 99] !! only for tetrahedral meshes !!

  12. Surface / Volume Connectivity a mesh with v vertices has maximal surfaces: 2v-2 triangles  ~ 6v indices v-1 quadrilaterals  ~ 4v indices volumes: O(v2) tetrahedra  ~ 12v indices O(v2) hexahedra  ~ 8v indices  connectivity dominates geometry even more for volume meshes

  13. 4 4 6 6 6 6 6 7 7 5 4 4 4 4 4 4 4 4 4 . . . . . . 3 6 3 . . . . . . 5 Degree Coding for Connectivity • Triangle Mesh Compression, [Touma & Gotsman, 98] • Valence-driven Connectivity Coding, [Alliez, Desbrun, 01] • Degree Duality Coder, [Isenburg, 02] • Near-Optimal Connectivity Coding, [Khodakovsky, Alliez, Desbrun, Schröder, 02] compressed with arithmetic coder converges to entropy

  14. 0.2 bits 1.3 bits 2.0 bits Entropy for a symbol sequence of t types t 1  Entropy = pi• log2( ) bits pi i =1 # of type t pi= # total

  15. 4 4 3 3 5 6 5 7 8 8 6 2 9+ 7 9+ vertex degrees face degrees Average Distribution [over a set of 11 polygonal meshes]

  16. “Worst-case” Distribution 3 4 5 [Alliez & Desbrun, 01]  3.241… bpv [Tutte, 62] 6 7 8 9 … … … vertex degrees

  17. 6 3 3 6 4 4 ... ... ... ... ... ... ... ... ... ... ... ... vertexdegrees vertexdegrees vertexdegrees facedegrees facedegrees facedegrees Adaptation to Regularity

  18. quad  hex Degree Coding for Volumes ? tri  tet vertex degrees  edge degrees

  19. elements for regular 3D tiling • regular tetrahedron • regular hexahedron Regular Volume Meshes? • elements for regular 2D tiling • regular triangle • regular quadrilateral • regular hexagon

  20. Compressing Connectivity

  21. pick incomplete face on hull • process adjacent hexahedra • record degrees of its free edges Space Growing similar in spirit to “region growing”: algorithm maintains hull enclosing processed hexahedra  • initialize hull with a border face • iterate until done

  22. focus face Coding with Edge Degrees

  23. Coding with Edge Degrees focus face

  24. Coding with Edge Degrees slots focus face

  25. Coding with Edge Degrees

  26. Coding with Edge Degrees

  27. incomplete faces border faces Coding with Edge Degrees

  28. Coding with Edge Degrees

  29. Coding with Edge Degrees

  30. Coding with Edge Degrees

  31. Coding with Edge Degrees

  32. Coding with Edge Degrees

  33. edges on hull maintain slot count Coding with Edge Degrees

  34. edges with a slot count of zero are “zero slots” zero slots Coding with Edge Degrees

  35. Coding with Edge Degrees

  36. Coding with Edge Degrees

  37. Coding with Edge Degrees

  38. Coding with Edge Degrees

  39. Coding with Edge Degrees

  40. Coding with Edge Degrees

  41. Coding with Edge Degrees

  42. Coding with Edge Degrees

  43. Coding with Edge Degrees

  44. Coding with Edge Degrees

  45. Coding with Edge Degrees

  46. Coding with Edge Degrees

  47. 2 2 4 3 3 3 3 3 5 4 4 4 4 4 4 • border ? 3 . . . . . . • join ? N N N N N N N N N N N N N N N N N N N N N N N N N N Y Y . . . . . . Resulting Symbols • border edge degrees • interior edge degrees . . . . . . . . . . . .

  48. Average Distributions no no 4 3 2 3 4 5 5 6+ 2 6 7+ yes yes join? borderdegrees interiordegrees border?

  49. Possible Configurations

  50. step gap hut bridge roof corner tunnel pit den Possible Configurations

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