1. Martin Isenburg Jack Snoeyink University of North Carolina Chapel Hill Mesh Collapse Compression

2. Introduction A novel scheme for encoding triangle mesh topology.

3. Introduction A novel scheme for encoding triangle mesh topology. –triangle mesh

4. Introduction A novel scheme for encoding triangle mesh topology. –triangle mesh –encoding

5. Introduction A novel scheme for encoding triangle mesh topology. –triangle mesh –encoding –topology

6. Triangle Mesh Computer Graphics Surface description Hardware support Used everyday Used everywhere Used by everybody Increasingly complex

7. Encoding Compressed representation Decrease storage and transmission time 011101101011...

8. Topology geometry versus topology x 0 y 0 z 0 0 1 2 x 1 y 1 z 1 2 3 0 x 2 y 2 z 2 2 3 1 x 3 y 3 z 3 2 4 3 x 4 y 4 z 4 5 4 3..... x n y n z n.....

9. Overview previous work simple meshes video example general meshes results summary

10. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ]

11. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Geometry Compression

12. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Short Encodings of Planar Graphs and Maps 4.6 bpv (4.6)

13. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Topological Surgery 2.2 ~ 4.8 bpv (--)

14. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Triangle Mesh Compression 0.2 ~ 2.9 bpv (--)

15. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Edgebreaker 3.4 ~ 4.0 bpv (4.0)

16. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] A Simple and Efficient Encoding for Triangle Meshes 4.2 ~ 5.4 bpv (6.0)

17. Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Face Fixer 3.9 ~ 4.1 bpv (6.0)

18. Simple Meshes (1) the mesh is a surface composed of triangles (2) the mesh has no boundary (3) the mesh has no holes (4) the mesh has no handles

19. Simple Mesh

20. Mesh Collapse Compression i Input:Output: a simple mesh- a sequence of code words - a permutation of vertices

21. Compression Scheme (1) initialize - declare arbitrary directed mesh edge as current edge

22. Compression Scheme (2) loop until done (and usually): - contract current edge - record the removed vertex - record the degree - select new current edge

23. Compression Scheme (2) loop until done (but occasionally): - divide mesh along current edge - record start symbol - continue on one mesh part - record end symbol - continue on other mesh part

24. Digons A triangulation with exception of the outer face that is bound by only two edges.

25. mc-edge From Mesh to Digon mc-vertex cut and openrearrange new mc-edge

26. Various Digons simple trivialcomplex dividing edge

27. Encode Algorithm encode( Mesh mesh, Codec codec ) { stack.push( digonify( mesh ) ); while ( not stack.empty( ) ) { digon = stack.pop( ); while ( not digon.trivial ( ) ) { if ( digon.complex() ) { subdigon = mc-divide( digon ); stack.push( subdigon ); codec.pushCode( “S” ); } else { degree = mc-contract( digon ); codec.pushCode( degree ); } codec.push( “E” ); }

28. mc-contract contract mc-edge removeselectcontract new mc-edgeloop

29. mc-divide divideselect dividing edgemc-edge

30. Video

31. Example

33. mc-vertex

34. Example mc-edge

35. Example digonify mc-edge

36. Example

37. mc-contract degree of removed vertex removed vertex

38. Example

39. mc-contract removed vertex degree of removed vertex

40. Example

41. mc-contract removed vertex degree of removed vertex

42. Example

43. mc-contract removed vertex degree of removed vertex

44. Example

45. mc-divide divided vertex

46. Example mc-divide divided vertex

47. Example push on stack

48. Example

49. mc-contract degree of removed vertex

50. Example

51. trivial digon trivial digon

52. Example trivial digon

53. Example pop from stack

54. Example

55. mc-contract degree of removed vertex

56. Example

57. mc-contract degree of removed vertex removed vertex

58. Example

59. trivial digon trivial digon

60. Example trivial digon

61. Example

62. mc-tree

63. Example mc-tree embedded mc-tree

64. Example mc-tree

65. Example Vertex permutation Code word sequence mc-encodingmc-tree

66. Decode algorithm decode( Mesh mesh, Codec codec ) { while ( not codec.empty( ) ) { code = codec.popCode( ); if ( code == “E” ) { stack.push( digon ); digon = new Digon( ); } else if ( code == “S” ) { subdigon = stack.pop( ); mc-join( digon, subdigon ); } else { mc-expand( digon, code ); } mesh = undigonify ( digon ); }

67. General Meshes (1) the mesh is a surface composed of triangles (2) the mesh can have a boundary (3) the mesh can have holes (4) the mesh can have handles

68. Simple Mesh

69. Boundary

70. Holes

71. Handle

72. Handle & Boundary

73. Handle & Holes

74. Encoding a Boundary mc-vertex mc-edgeadded edge

75. Encoding a Hole holeadded vertex

76. Encoding a Handle digon in stack After mc-contract: M + index + [0..5] After mc-divide: M + index + [0..1] Hiding in a trivial digon: M + index + [0..7]

77. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

78. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

79. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

80. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

81. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

82. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

83. Results vertices/triangles bpvmesh 250/496 1.4 ~ 1.7bishop 1524/3044 2.7 ~ 2.8bunny 1502/3000 3.1 ~ 3.4dragon 2832/5660 2.4 ~ 2.7triceratops 2655/5030 2.5 ~ 2.7beethoven 2562/5120 1.1 ~ 1.2shape

84. Summary We presented a novel scheme for encoding the topology of triangular meshes. The achieved compression rates of 1.1 ~ 3.4 bits/vertex compete with the best results known today. Currently we work on a progressive version of mc-compression.

85. Acknowledgements University of British Columbia at Vancouver, Canada Gene Lee and his RASP tools International Computer Science Institute (ICSI) at Berkeley, USA funding: NSERC, IRIS, UGF