Compressing Polygon Mesh Connectivity

1
CompressingPolygon Mesh Connectivity
withDegree DualityPrediction

• Martin Isenburg
• University of North Carolinaat Chapel Hill

2
Overview

• Background
• Connectivity Compression
• Coding with Degrees
• Duality Prediction
• Adaptive Traversal
• Example Run
• Conclusion

3
Background
4
Polygon Meshes

• connectivity
• geometry

face1 1 2 3 4 face2 3 4 3 face3 5
2 1 3 facef
vertex1 ( x, y, z ) vertex2 ( x, y, z
) vertex3 ( x, y, z ) vertexv
5
Mesh Compression

• 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
bits per vertex (bpv)
model
gain
Face Fixer
Degree Duality
triceratops galleon cessna tommygun
cow teapot
1.189 2.093 2.543 2.258 1.781 1.
127
2.115 2.595 2.841 2.611 2.213 1.
669
44 19 11 ... 14 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 withv vertices

6 log2 (v) bpv
3.24 bpv
11
Spanning Tree

• Succinct Representationsof Graphs Turan, 84
• Short encodings of planargraphs and maps
  Keeler Westbrook, 95
• Geometric CompressionthroughTopological
  Surgery Taubin Rossignac, 98

? 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

13
Classification

• code symbols are associated with edges, faces, or
  vertices

14
Edge-BasedCompression Schemes
15
Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region











unprocessedregion






focus
. . .
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Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region












unprocessedregion






. . .
17
Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region












unprocessedregion






. . .
18
Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region












unprocessedregion





. . .
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Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region












unprocessedregion





. . .
20
Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region






unprocessedregion



. . .
21
Edge-Based

• Dual Graph Approach, Lee Kuo, 99
• Face Fixer, Isenburg Snoeyink, 00

processed region







unprocessedregion



. . .
. . .
22
Face-BasedCompression Schemes
23
Face-Based

• Cut-Border Machine, Gumhold Strasser, 98
• Edgebreaker, Rossignac, 99

processed region







unprocessedregion



focus
. . .
24
Face-Based

• Cut-Border Machine, Gumhold Strasser, 98
• Edgebreaker, Rossignac, 99

processed region







unprocessedregion



. . .
25
Face-Based

• Cut-Border Machine, Gumhold Strasser, 98
• Edgebreaker, Rossignac, 99

processed region







unprocessedregion



. . .
26
Face-Based

• Cut-Border Machine, Gumhold Strasser, 98
• Edgebreaker, Rossignac, 99

processed region







unprocessedregion



. . .
27
Face-Based

• Cut-Border Machine, Gumhold Strasser, 98
• Edgebreaker, Rossignac, 99

processed region







unprocessedregion



. . .
. . .
28
Vertex-BasedCompression Schemes
29
Vertex-based

• Triangle Mesh Compression, Touma
  Gotsman, 98

processed region








unprocessedregion




focus
. . .
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Vertex-based

• Triangle Mesh Compression, Touma
  Gotsman, 98

processed region







unprocessedregion




. . .
31
Vertex-based

• Triangle Mesh Compression, Touma
  Gotsman, 98

processed region








unprocessedregion



. . .
32
Vertex-based

• Triangle Mesh Compression, Touma
  Gotsman, 98

processed region







unprocessedregion



. . .
33
Vertex-based

• Triangle Mesh Compression, Touma
  Gotsman, 98

processed region







unprocessedregion



. . .
. . .
34
Coding with Vertex and Face Degrees
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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
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add free vertex











processed region



unprocessed region









































. . .
. . .
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free vertex splits boundary





unprocessed region










































processed region








. . .
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free vertex merges boundary
boundary in stack





unprocessed region




processed region

stack focus






























processed region

















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Resulting Code

• two symbol sequences
• vertex degrees ( split / merge)
• face degrees
• compress with arithmetic coder
• ? converges to entropy

. . .
. . .
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41
Entropy

• for a symbol sequence of t types

t
1
?
Entropy pi log2( ) bits
pi
i 1
of type t
pi
total
42
Average Distributions
43
Adaptation to Regularity
44
Worst-case Distribution
3
3
Alliez Desbrun, 01 ? 3.241 bpv Tutte, 62
4
5
6
7
8
9



vertex degrees
face degrees
? ?
45
Compressing with Duality Prediction
46
Degree Correlation

• high-degree faces are likely to be
  surroundedby low-degreevertices
• and vice-versa
• ? mutual degree prediction

47
Face Degree Prediction

3

focus (widened)

4

3


average degree offocus vertices
3 4 3
fdc
3.333


3
face degree context
48
Vertex Degree Prediction



6




degree offocus face
vdc

vertex degree context
49
Compression Gain
without
with
model
triceratops galleon cessna tommygun
cow teapot
bits per vertex
bits per vertex
1.189 2.093 2.543 2.258 1.781 1.
127
1.192 2.371 2.811 2.917 1.781 1.
632
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
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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
processed region
60
Adaptive Traversal

• Valence-driven connectivity encoding for 3D
  meshes Alliez Desbrun, 01
• ? avoid creation of cavities

exitfocus

















61
Compression Gain
without
with
model
25 1.189 18 2.093 28 2.543
32 2.258 13 1.781 3 1.127
triceratops galleon cessna tommygun
cow teapot
53 1.311 78 2.309 172 2.882
131 2.449 154 2.313 10 1.167
bpv
splits
bpv
splits
min / max / average gain 4 / 23 / 10
62
Example Decoding Run
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Example Decoding Run
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Example Decoding Run
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Example Decoding Run

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Example Decoding Run

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Example Decoding Run

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Example Decoding Run



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Example Decoding Run


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Example Decoding Run
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Example Decoding Run

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Example Decoding Run


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Example Decoding Run
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Example Decoding Run

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Example Decoding Run








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Example Decoding Run


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Example Decoding Run


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Example Decoding Run






focus (widened)




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Example Decoding Run






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Example Decoding Run








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Example Decoding Run








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Example Decoding Run







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Example Decoding Run






focus (widened)






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Example Decoding Run








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Example Decoding Run





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Example Decoding Run




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Example Decoding Run









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Example Decoding Run



focus (widened)












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Example Decoding Run



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Conclusion
91
Summary

• degree coding for polygonal connectivity
• duality prediction
• adaptive traversal
• proof-of-conceptimplementationusing Shout3D

http//www.cs.unc.edu/isenburg/degreedualitycoder
/
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 ? Tuttes 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!