Compressing Polygon Mesh Geometry

1
CompressingPolygon Mesh Geometry
withParallelogramPrediction
















Martin Isenburg UNCChapel Hill
Pierre Alliez INRIASophia-Antipolis
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Take this home

• Non-triangular faces inthe mesh can be
  exploited for more efficientpredictive
  compressionof vertex positions.
• Non-triangular faces tend tobe planar and
  convex.

3
Overview

• Background
• Previous Work
• Linear Prediction Schemes
• within versus across
• Example Run
• Can we do better ?
• Conclusion

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Background
5
Polygon Meshes

• connectivity
• geometry

face1 1 2 3 4 face2 3 4 3 face3 5
2 1 3 facef
2832 vertices
vertex1 ( x, y, z ) vertex2 ( x, y, z
) vertex3 ( x, y, z ) vertexv
5647
? size 79296 bytes
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Mesh Compression

• Geometry Compression Deering, 95
• Fast Rendering
• Progressive Transmission
• Maximum Compression

Geometry Compression Deering, 95
Maximum Compression
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Mesh Compression

• Geometry Compression Deering, 95
• Fast Rendering
• Progressive Transmission
• Maximum Compression
• Connectivity
• Geometry

Geometry Compression Deering, 95
Maximum Compression
Geometry
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Mesh Compression

• Geometry Compression Deering, 95
• Fast Rendering
• Progressive Transmission
• Maximum Compression
• Connectivity
• Geometry
• Triangle Meshes
• Polygon Meshes

Geometry Compression Deering, 95
Maximum Compression
Geometry
Polygon Meshes
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Triangle Mesh Compression
Triangle Mesh Compression Touma Gotsman,
Graphics Interface 98

• Connectivity Coder
• stores the connectivity as sequence of vertex
  degrees

• Geometry Coder
• stores the geometry as sequence of vectors
  each corrects the prediction of a vertex position

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Not Triangles Polygons!
Face Fixer Isenburg Snoeyink, 00
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Generalization of TG coder

• Connectivity Coder

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Results
bits per vertex
model
TG
IA
triceratops galleon cessna tommygun
cow teapot
14.8 18.4 12.5 12.5 20.6 16.1
20.0 24.1 19.1 19.6 20.4 21.0
min / max / average 9 / 41 / 23
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Previous Work
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Previous Work

• Classic approaches 95 98
• linear prediction

Geometry Compression Deering, 95
Geometric Compression through topological
surgery Taubin Rossignac, 98
Triangle Mesh Compression Touma Gotsman, 98
15
Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

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Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

Spectral Compressionof Mesh Geometry Karni
Gotsman, 00
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Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

Progressive GeometryCompression Khodakovsky et
al., 00
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Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

Geometric Compressionfor interactive
transmission Devillers Gandoin, 00
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Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

Vertex data compressionfor triangle meshes Lee
Ko, 00
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Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

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Previous Work

• Classic approaches 95 98
• linear prediction
• Recent approaches 00 02
• spectral
• re-meshing
• space-dividing
• vector-quantization
• feature discovery
• angle-based

Angle-Analyzer A triangle-quad mesh
codec Lee, Alliez Desbrun, 02
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Linear Prediction Schemes
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Linear Prediction Schemes

1. quantize positions with b bits
2. traverse positions
3. linear prediction from neighbors
4. store corrective vector

24
Linear Prediction Schemes

1. quantize positions with b bits
2. traverse positions
3. linear prediction from neighbors
4. store corrective vector

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Linear Prediction Schemes

1. quantize positions with b bits
2. traverse positions
3. linear prediction from neighbors
4. store corrective vector

26
Linear Prediction Schemes

1. quantize positions with b bits
2. traverse positions
3. linear prediction from neighbors
4. store corrective vector

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Deering, 95

• Prediction Delta-Coding

processed region

unprocessed region





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Taubin Rossignac, 98

• Prediction Spanning Tree

processed region
unprocessed region
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Touma Gotsman, 98

• Prediction Parallelogram Rule

processed region
unprocessed region
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Parallelogram Rule




good prediction
badprediction
badprediction
non-convex
non-planar
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More good Predictions
Multi-way geometry encoding. Cohen-Or, Cohen
Irony, 02

• average multiple predictions

Optimized compression of triangle mesh
geometryusing prediction trees. Kronrod
Gotsman, 02

• search for best prediction
• direct the traversal (prediction tree)

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Polygon Meshes ?

• TG coder
• triangulate
• compress resulting triangle mesh

• IA coder
• do NOT triangulate
• use polygons for better predictions
• within versus across

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within versus across
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Non-triangular Faces

• Question Why would a mesh have a
  non-triangular face?

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Non-triangular Faces

• Question Why would a mesh have a
  non-triangular face?

Answer Because there was no reason to
triangulate it! This face was
convex and planar.
? use this info for good predictions
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within versus across
across-prediction
within-prediction

• ? within-predictions avoid creases

? within-predictions often find existing
parallelograms (? quadrilaterals)
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Bitrates within vs. across
bits per vertex
model
triceratops galleon cessna tommygun
cow teapot
within
across
20.5 26.8 19.8 19.5 20.6 22.7
14.1 16.9 11.1 10.9 - 14.9
min / max / average 13 / 47 / 32
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Maximizing the number of within-predictions
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Greedy Strategy

• always try to
• (A) pick a vertex whose position can be
  within-predicted
• otherwise
• (B) do an across-prediction, but pick a vertex
  that creates (A)for the next iteration

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Traversal Order

• were lucky
• ? process vertices in order dictated by our
  connectivity coder

Compressing Polygon Connectivity withDegree
Duality Prediction Isenburg, 02

• ? avoid splits by adaptive traversal

Valence-driven Connectivity Encodingfor 3D
meshes Alliez Desbrun, 01
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splits
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splits
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splits
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splits
45
splits
processed region
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splits
processed region
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splits
processed region
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splits
processed region
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splits
processed region
50
splits
processed region
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Adaptive Traversal






































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Adaptive Traversal






































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Adaptive Traversal






































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of within-predictions
prediction type
model
triceratops galleon cessna tommygun
cow teapot
within
across
last
center
2557 257 2 1 2007 324 24 12 3091 621 22 11
3376 678 78 39 0 2701 2 1 1016 170 2 1

min / max / average 74 / 91 / 84
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Example Decoding Run
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Example Decoding Run

• center-prediction
• no parallelogram rule possible
• predict this position as center ofthe bounding
  box

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Example Decoding Run

• last-prediction
• no parallelogram rule possible
• predict this position as the lastposition

0


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Example Decoding Run

• last-prediction
• no parallelogram rule possible
• predict this position as the lastposition

0


1

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Example Decoding Run

• across-prediction
• parallelogram rule possible
• predict across two polygons

0

2

1

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Example Decoding Run

• within-prediction
• use parallelogram rule
• predict within a polygon

3

0

2

1

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Example Decoding Run
4


3

0

2

1

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Example Decoding Run


4

5

3

0

2

1

63
Example Decoding Run
6


4

5

3

0

2

1

64
Example Decoding Run
6

7

4

5


3

0

2

1

65
Example Decoding Run
6

7

4

5

7

3

0

2


1

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Can we do better ?(and keep it simple)
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Keep it simple

• Constraints
• single linear prediction
• use connectivity traversal order
• Possibilities
• floating point coefficients
• use more than three vertices

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better within-predictions
assuming ideal (? regular) polygons
deg 4
A


B
C


perfect
P A B C
a 1, ß -1, ? 1
P aA ßB ?C
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Switch within Coefficients
if (deg 4) a 1.000 ß -1.000 ?
1.000 else if (deg 5) a 1.024 ß
-0.527 ? 0.503 else if (deg 6) a
1.066 ß -0.315 ? 0.249 else
How did we pick these numbers?
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Finding the Coefficients

• no obvious scientific way
• use Matlab ?
• for each degree separately
• sum all possible prediction errors as function
  of a and ß
• optimize a and ß for minimal error

N

A


C
B


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Switch within Coefficients
if (deg 4) a 1.000 ß -1.000 ?
1.000 else if (deg 5) a 1.024 ß
-0.527 ? 0.503 else if (deg 6) a
1.066 ß -0.315 ? 0.249 else
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better across-predictions

• assuming equal edge length

adjacent polygonsare coplanar















3?3
3?4
3?5
3?6
3?8




















4?3
4?4
4?5
4?6
4?8





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better across-predictions

• assuming equal edge length

adjacent polygonsare not coplanar















3?3
3?4
3?5
3?6
3?8





creaseangle 60















4?3
4?4
4?5
4?6
4?8





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Switch across Coefficients
if (deg 3?3) a 0.917 ß - 0.833 ?
0.916 else if (deg 3?4) a 0.621 ß -
0.504 ? 0.883 else if (deg 3?5) a
0.557 ß - 0.334 ? 0.777 else
else if (deg 4?3) a 1.153 ß - 0.354 ?
0.201 else if (deg 4?4) a 1.001 ß
- 0.648 ? 0.647 else
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Conclusion
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Summary (1)

• polygon information can improve predictive
  geometry compression
• using parallelogram rule within rather than
  across polygons
• average improvement of 23 over TG coder
• has simple and straight-forward implementation

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Summary (2)

• proof-of-conceptimplementationin form of a
  Webjava-applet
• compression software will soon be made available

http//www.cs.unc.edu/isenburg/pmc/
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Future Work

• a scientific way to find numbers for this
  coefficient switching

• polygonification
• turn triangle meshes into polygon meshes for
  better compression

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Acknowledgments

• funding
• ARC Télégéo grant of INRIA at Sophia-Antipolis
• logistics
• Jack Snoeyink
• Olivier Devillers
• Agnès Clément Bessière
• Jean-Daniel Boissonnat

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• Thank You!

http//www.cs.unc.edu/isenburg/pmc/