1. Compressing Polygon Mesh Connectivity
with Degree Duality Prediction
Martin Isenburg
University of North Carolina at Chapel Hill

2. Overview Background Connectivity Compression Coding with Degrees
Duality Prediction
Adaptive Traversal
Example Run
Conclusion

3. Background

4. Polygon Meshes connectivity geometry :k ~ 4 :k ~ 6 face1 1 2 3 4
k v log2 (v)
:k ~ :k ~ 6
face
face
face
facef
The connectivity, super-linear … can be compressed constant.
The geometry, linear, quantization, predictive techiques
Here the vertices can be ordered arbitrarily, as long as they are referenced correctly. 4
24 ~ 96 v
vertex1 ( x, y, z )
vertex2 ( x, y, z )
vertex3 ( x, y, z )
vertexv 5

5. Mesh Compression Maximum Compression Connectivity Polygon Meshes
Geometry Compression [Deering, 95]
Fast Rendering
Progressive Transmission
Maximum Compression
Geometry
Connectivity
Triangle Meshes
Polygon Meshes
Maximum Compression
Connectivity
Polygon Meshes

6. Not Triangles … Polygons!
Face Fixer [Isenburg & Snoeyink, 00]

7. Results 2.115 1.189 44 % model gain triceratops galleon cessna …
bits per vertex (bpv)
model
gain
triceratops
galleon
cessna …
tommygun
cow
teapot
Face Fixer
Degree Duality
2.115
2.595
2.841 …
2.213
1.669
1.189
2.093
2.543 …
1.781
1.127
44 %
19 %
11 % %
20 %
33 %
min / max / average gain [%] = 11 / 55 / 26

8. Connectivity Compression

9. Connectivity Compression
assumption
order of vertices does not matter
advantage
no need to “preserve” indices
approach
code only the “connectivity graph”
re-order vertices appropriately

10. Connectivity Graphs
connectivity of simple meshes is homeomorphic to planar graph
 enumeration
 asymptotic bounds
[William Tutte 62 / 63]
number of planar triangulations with v vertices
<<
3.24 bpv
6 log2 (v) bpv

11. Spanning Tree  extends to meshes of non-zero genus
Succinct Representations of Graphs [Turan, 84]
Short encodings of planar graphs and maps [Keeler & Westbrook, 95]
Geometric Compression through Topological Surgery [Taubin & Rossignac, 98]
explicitly encode both spanning trees
 extends to meshes of non-zero genus

12. Region Growing Triangle Mesh Compression [Touma & Gotsman, 98]
Cut-Border Machine [Gumhold & Strasser, 98]
Edgebreaker [Rossignac, 99]
Simple Sequential Encoding [de Floriani et al., 99]
Dual Graph Approach [Lee & Kuo, 99]
Face Fixer [Isenburg & Snoeyink, 00]
Classified into three main approaches
similarities
Grow boundaries on the mesh !
Include face after face !
Describe change of boundary !
differences
What is a boundary ?
How can including a face change the boundaries ?

13. Classification
code symbols are associated with edges, faces, or vertices:
boundary
focus
edge-based
processed region
unprocessed region
boundary
focus
face-based
boundary
focus
vertex-based
We use the mesh element with which the description of a boundary change can be associated with to classify them as “face-based”, “edge-based, and “vertex-based”.

14. Edge-Based Compression Schemes

15. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region
unprocessed region
focus
. . . F F R

16. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region
unprocessed region F
. . . F F R F

17. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region F
unprocessed region F
. . . F F R F F

18. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region R F
unprocessed region F
. . . F F R F F R

19. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region F R F
unprocessed region F
. . . F F R F F R F

20. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region R F R F
unprocessed region F
. . . F F R F F R F R

21. Edge-Based Dual Graph Approach, [Lee & Kuo, 99]? ? ? ? ?
Dual Graph Approach, [Lee & Kuo, 99]
Face Fixer, [Isenburg & Snoeyink, 00]
processed region R F F R F
unprocessed region F
. . .
. . . F F R F F R F R F 5
. . . 4 3 6

22. Face-Based Compression Schemes

23. Face-Based Cut-Border Machine, [Gumhold & Strasser, 98]
Edgebreaker, [Rossignac, 99]
processed region
unprocessed region
focus
. . . C R

24. Face-Based Cut-Border Machine, [Gumhold & Strasser, 98]
Edgebreaker, [Rossignac, 99]
processed region
unprocessed region C
. . . C R C

25. Face-Based Cut-Border Machine, [Gumhold & Strasser, 98]
Edgebreaker, [Rossignac, 99]
processed region R
unprocessed region C
. . . C R C R

26. Face-Based Cut-Border Machine, [Gumhold & Strasser, 98]
Edgebreaker, [Rossignac, 99]
processed region R R
unprocessed region C
. . . C R C R R

27. Face-Based Cut-Border Machine, [Gumhold & Strasser, 98]? ? ? ? ?
Cut-Border Machine, [Gumhold & Strasser, 98]
Edgebreaker, [Rossignac, 99]
processed region R C R
unprocessed region C
. . .
. . . C R C R R C 5
. . . 4 3 6

28. Vertex-Based Compression Schemes

29. Vertex-based Triangle Mesh Compression, [Touma & Gotsman, 98] . . . 6
processed region
unprocessed region
Processing a face means
focus
. . . 6

30. Vertex-based Triangle Mesh Compression, [Touma & Gotsman, 98] . . . 5
processed region 5
unprocessed region
. . . 6 5

31. Vertex-based Triangle Mesh Compression, [Touma & Gotsman, 98] . . . 5
processed region 5
unprocessed region
. . . 6 5

32. Vertex-based Triangle Mesh Compression, [Touma & Gotsman, 98] . . . 5
processed region 5
unprocessed region
. . . 6 5

33. Vertex-based Triangle Mesh Compression, [Touma & Gotsman, 98] . . .? ? ? ? ?
Triangle Mesh Compression, [Touma & Gotsman, 98]
processed region 6 5
unprocessed region
. . . 6 5 6
. . . 5
. . . 4 3 6

34. Coding with Vertex and Face Degrees

35. Coding with Degrees
while ( unprocessed faces ) move focus to a face  face degree for ( free vertices )  case switch ( case ) “add”:  vertex degree “split”:  offset “merge”:  index, offset

36. Example Traversal
Their scheme grows a region on the mesh by including triangle after triangle into the compression boundary.
Three cases can arise:
A new vertex is added, the boundary is split, or two boundaries are merged.
The decoder simply replays what happens during encoding.
The only data structure that need to be maintained by are these compression boundaries.

37. “add” free vertex . . . . . . focus (widened) 3 exit focus end slot
boundary
focus
(widened)
focus 3
processed region
exit focus
end slot
unprocessed region
free
vertices 5
boundary slots 4
start slot
. . . 4 5 3 4 3 4 3
. . . 5 4 4 5

38. free vertex “splits” boundary
unprocessed region S
split offset
focus
processed region
. . . 4 5 3 4 3
. . . 5 4 4

39. free vertex “merges” boundary
boundary in stack
unprocessed region
stack
focus
processed region M
merge offset
processed region
. . . 4 5 3 4 3
. . . 5 4 4

40. Resulting Code two symbol sequences compress with arithmetic coder
vertex degrees (+ “split” / “merge”)
face degrees
compress with arithmetic coder
 converges to entropy
. . . 4 3 4 4 4 6 5 S M 4 4
. . .
. . . 3 4 4 4 5 4 4 4 6 4 4 4
. . .

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

42. Average Distributions
add
merge
split
case 3 4 5 6 7 8 9+
face degrees 2 3 4 5 6 7 8 9+
vertex degrees

43. Adaptation to Regularity6
vertex degrees
face degrees 3
... 3
vertex degrees
face degrees 6
... 4
vertex degrees
face degrees
...

44. “Worst-case” Distribution3 3 
i = 3 
pi = 1
[Alliez & Desbrun, 01]
 3.241… bpv
[Tutte, 62] 4 
i = 3 
i • pi = 6 5 6 7 8 9 … … …
vertex degrees
 
face degrees

45. Compressing with Duality Prediction

46. Degree Correlation
high-degree faces are “likely” to be surrounded by low-degree vertices
and vice-versa
 mutual degree prediction

47. Face Degree Prediction
fdc  3.3
3.3  fdc  4.3
4.3  fdc  4.9
4.9  fdc 3
focus
(widened) 4 3
average degree of focus vertices
fdc = =
3.333 3
“face degree context”

48. Vertex Degree Prediction
vdc  6
vdc = 3
vdc = 4
vdc = 5 4 5 3 6 6
degree of focus face
vdc = 6
“vertex degree context”

49. Compression Gain 1.192 1.189 model triceratops galleon cessna …
without
with
model
triceratops
galleon
cessna …
tommygun
cow
teapot
bits per vertex
bits per vertex
1.192
2.371
2.811 …
1.781
1.632
1.189
2.093
2.543 …
1.781
1.127
min / max / average gain [%] = 0 / 31 / 17

50. Reducing the Number of Splits

51. Occurance of “splits”

52. Occurance of “splits”

53. Occurance of “splits”

54. Occurance of “splits”

55. Occurance of “splits”
unprocessed region
processed region

56. Occurance of “splits”
unprocessed region
processed region

57. Occurance of “splits”
unprocessed region
processed region

58. Occurance of “splits”
unprocessed region
processed region

59. Occurance of “splits”
unprocessed region
split
processed region

60. Adaptive Traversal
Valence-driven connectivity encoding for 3D meshes [Alliez & Desbrun, 01]
 avoid creation of cavities
exit focus
focus

61. Compression Gain model triceratops galleon cessna … tommygun cow
without
with
model
triceratops
galleon
cessna …
tommygun
cow
teapot
… …
splits
… …
bpv
splits
bpv
min / max / average gain [%] = 4 / 23 / 10

62. Example Decoding Run

63. Example Decoding Run 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

64. Example Decoding Run 4 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

65. Example Decoding Run . . . . . . focus 6 4 6 6 3 3 4 5 4 5 4 4 2 4 4 3

66. Example Decoding Run . . . . . . 3 free vertex 4 6 6 3 3 4 5 4 5 4 4 2

67. Example Decoding Run . . . . . . exit focus 6 4 6 6 3 3 4 5 4 5 4 4 2

68. Example Decoding Run . . . . . . free vertices 5 4 6 6 3 3 4 5 4 5 4 42 4 4
. . . 3 3 5 5
. . . 4 4 4 6

69. Example Decoding Run 3 5 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

70. Example Decoding Run 3 5 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

71. Example Decoding Run . . . . . . exit focus 4 4 6 6 3 3 4 5 4 5 4 4 2

72. Example Decoding Run 4 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5
. . . 4 4 4 6 5

73. Example Decoding Run 5 4 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

74. Example Decoding Run . . . . . . 4 exit focus 4 6 6 3 3 4 5 4 5 4 4 2

75. Example Decoding Run . . . . . . 5 3 end slot focus (widened)
start slot 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5
. . . 4 4 4 6 5

76. Example Decoding Run 4 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5
. . . 4 4 4 6 5

77. Example Decoding Run . . . . . . exit focus 4 5 4 6 6 3 3 4 5 4 5 4 42 4 4
. . . 3 3 5
. . . 4 4 4 6 5

78. Example Decoding Run . . . . . . focus (widened) 4 6 6 3 3 4 5 4 5 4 42 4 4
. . . 3 3 5
. . . 4 4 4 6 5

79. Example Decoding Run 4 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5
. . . 4 4 4 6 5

80. Example Decoding Run . . . . . . exit focus focus 4 4 6 6 3 3 4 5 4 52 4 4
. . . 3 3 5
. . . 4 4 4 6 5

81. Example Decoding Run 3 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5
. . . 4 4 4 6 5

82. Example Decoding Run . . . . . . 4 exit focus 4 6 6 3 3 4 5 4 5 4 4 2

83. Example Decoding Run . . . . . . focus (widened) 4 6 6 3 3 4 5 4 5 4 42 4 4
. . . 3 3 5
. . . 4 4 4 6 5

84. Example Decoding Run 6 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5
. . . 4 4 4 6 5

85. Example Decoding Run 2 6 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

86. Example Decoding Run 4 6 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

87. Example Decoding Run . . . . . . 4 exit focus 4 6 6 3 3 4 5 4 5 4 4 2

88. Example Decoding Run . . . . . . focus (widened) 4 6 6 3 3 4 5 4 5 4 42 4 4
. . . 3 3 5
. . . 4 4 4 6 5

89. Example Decoding Run
. . . 5 4 6 6 3 3 4 5 4 5 4 4 2 4 4
. . . 3 3 5 5
. . . 4 4 4 6

90. Conclusion

91. Summary degree coding for polygonal connectivity duality prediction
adaptive traversal
proof-of-concept implementation using Shout3D

92. Similar Result
Near-Optimal Connectivity Coding of 2-manifold polygon meshes [Khodakovsky, Alliez, Desbrun, Schröder]
 analysis of worst-case face degree and vertex degree distribution
entropy  Tutte’s bounds
Martin  France

93. Current Work (w. Pierre Alliez)
use polygons for better predictive geometry coding
extend degree coding to volume mesh connectivity
“fairly planar & convex”
“edge degrees”

94. Thank You!