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with Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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Page 1: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

withParallelogram

Prediction

CompressingPolygon Mesh Geometry

Martin IsenburgUNC

Chapel Hill

Pierre AlliezINRIA

Sophia-Antipolis

Page 2: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Take this home:

“Non-triangular faces inthe mesh can be exploited for

more efficientpredictive compression

of vertex positions.”

“Non-triangular faces tend tobe planar and convex.”

Page 3: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Overview

• Background

• Previous Work

• Linear Prediction Schemes

• “within” versus “across”

• Example Run

• Can we do better ?

• Conclusion

Page 4: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Background

Page 5: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Polygon Meshes

• connectivity

• geometry

vertex1 ( x, y, z )vertex2 ( x, y, z )vertex3 ( x, y, z )

vertexv

face1 1 2 3 4face2 3 4 3face3 5 2 1 3

facef

~ 4v 32 bits

3v * 32 bits

size: 79296 bytes

297365647

12

log2(v)

2832 vertices

Page 6: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

• Geometry Compression [Deering, 95]– Fast Rendering

– Progressive Transmission

– Maximum Compression

Geometry Compression [Deering, 95]

Mesh Compression

Maximum Compression

Page 7: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

• Geometry Compression [Deering, 95]– Fast Rendering

– Progressive Transmission

– Maximum Compression

• Connectivity

• Geometry

Geometry Compression [Deering, 95]

Mesh Compression

Maximum Compression

Geometry

Page 8: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

• Geometry Compression [Deering, 95]– Fast Rendering

– Progressive Transmission

– Maximum Compression

• Connectivity

• Geometry

– Triangle Meshes

– Polygon Meshes

Geometry Compression [Deering, 95]

Mesh Compression

Polygon Meshes

Maximum Compression

Geometry

Page 9: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Triangle Mesh Compression

• Connectivity Coder– stores the connectivity as sequence

of vertex degrees

Triangle Mesh Compression[Touma & Gotsman, Graphics Interface 98]

• Geometry Coder– stores the geometry as sequence of

vectors; each corrects the prediction of a vertex position

Page 10: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Not Triangles … Polygons!

Face Fixer [Isenburg & Snoeyink, 00]

Page 11: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

• Connectivity Coder

Generalization of TG coder

Compressing Polygon Connectivity withDegree Duality Prediction, [Isenburg, 02]

Near-optimal connectivity coding ofPolygon meshes, [Khodakovsky, Alliez,

Desbrun & Schroeder, 02]

this paper [Isenburg & Alliez, 02]

• Geometry Coder

Page 12: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

triceratopsgalleoncessna

…tommygun

cowteapot

14.818.412.5

…12.520.616.1

Results

TGbits per vertex

model IA20.024.119.1

…19.620.421.0

min / max / average [%] = 9 / 41 / 23

26 %26 %35 % ...36 %-1 %24 %

gain

Page 13: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Previous Work

Page 14: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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]

Java3D

MPEG - 4

Virtue3D

Page 15: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Previous Work

• Classic approaches [95 – 98]:– linear prediction

• Recent approaches [00 – 02]:– spectral

– re-meshing

– space-dividing

– vector-quantization

– feature discovery

– angle-based

Page 16: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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]

expensive numericalcomputations

Page 17: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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]

modifies mesh priorto compression

Page 18: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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]

poly-soups; complexgeometric algorithms

Page 19: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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]

local coord-system +vector-quantization

Page 20: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Previous Work

• Classic approaches [95 – 98]:– linear prediction

• Recent approaches [00 – 02]:– spectral

– re-meshing

– space-dividing

– vector-quantization

– feature discovery

– angle-based

certain 3D models +expensive matching

Compression of engineeringmodels by repeated feature

[Shikhare et al., 01]discovery

Page 21: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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]

dihedral + internal =heavy trigonometry

Page 22: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Linear Prediction Schemes

Page 23: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Linear Prediction Schemes

1. quantize positions with b bits

2. traverse positions

3. linear prediction from neighbors

4. store corrective vector

(1.2045, -0.2045, 0.7045) (1008, 68, 718)floating point integer

Page 24: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Linear Prediction Schemes

1. quantize positions with b bits

2. traverse positions

3. linear prediction from neighbors

4. store corrective vector

use traversal order implied bythe connectivity coder

Page 25: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Linear Prediction Schemes

1. quantize positions with b bits

2. traverse positions

3. linear prediction from neighbors

4. store corrective vector

(1004, 71, 723)apply prediction rule prediction

Page 26: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Linear Prediction Schemes

1. quantize positions with b bits

2. traverse positions

3. linear prediction from neighbors

4. store corrective vector0

10

20

30

40

50

60

70

position distribution

0

500

1000

1500

2000

2500

3000

3500

corrector distribution

(1004, 71, 723)(1008, 68, 718)position

(4, -3, -5)correctorprediction

Page 27: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Deering, 95

Prediction: Delta-Coding

A

processed regionunprocessed region

P

P = A

Page 28: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Taubin & Rossignac, 98

Prediction: Spanning Tree

A

BC D

E

processed regionunprocessed region

P

P = αA + βB + γC + δD + εE + …

Page 29: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Touma & Gotsman, 98

Prediction: Parallelogram Rule

processed regionunprocessed region

P

P = A – B + C

A

BC

Page 30: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Parallelogram Rule

“good”prediction

“bad”prediction

“non-convex”

“bad”prediction

“non-planar”

Page 31: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

More “good” Predictions

• average multiple predictions

Multi-way geometry encoding. [Cohen-Or, Cohen & Irony, 02]

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

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

average gain of 11 %

average gain of 8 % (smooth) & 42 % (CAD)

Page 32: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Polygon Meshes … ?

TG coder

• triangulate

• compress resulting triangle mesh

IA coder

• do NOT triangulate

• use polygons for better predictions

• within versus across

Page 33: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“within” versus “across”

Page 34: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Non-triangular Faces

Question: Why would a mesh have

a non-triangular face?

Page 35: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Non-triangular Faces

Question: Why would a mesh have

a non-triangular face?Answer: Because there was noreason to triangulate it!

This face was “convex”and “planar”.

use this info for “good” predictions

Page 36: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

within-predictions often find existing parallelograms ( quadrilaterals)

“within” versus “across”

within-predictions avoid creases

within-predictionacross-prediction

Page 37: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

triceratopsgalleoncessna

…tommygun

cowteapot

20.526.819.8

…19.520.622.7

Bitrates: “within” vs. “across”

withinbits per vertex

model across

14.116.911.1

…10.9

-14.9

min / max / average [%] = 13 / 47 / 32

31 %37 %45 % ...44 %

-

34 %

diff

Page 38: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Maximizing the number of within-predictions

Page 39: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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

Page 40: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

we’re lucky:

process vertices in order dictated by our connectivity coder

Traversal Order

Compressing Polygon Connectivity withDegree Duality Prediction [Isenburg, 02]

Valence-driven Connectivity Encodingfor 3D meshes [Alliez & Desbrun, 01]

avoid “splits” by adaptive traversal

Page 41: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

Page 42: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

Page 43: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

Page 44: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

Page 45: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

processed region

Page 46: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

processed region

Page 47: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

processed region

Page 48: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

processed region

Page 49: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

processed region

Page 50: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

“splits”

processed region

split

Page 51: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Adaptive Traversal

Page 52: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Adaptive Traversal

Page 53: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Adaptive Traversal

Page 54: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

% of within-predictions

triceratopsgalleoncessna

…tommygun

cowteapot

within

prediction type:model

min / max / average [%] = 74 / 91 / 84

across

2557 257 2 1

2007 324 24 123091 621 22 11 … … … …3376 678 78 39

0 2701 2 11016 170 2 1

last center

90 %85 %83 % …81 %0 %

85 %

within% of

Page 55: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

Page 56: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

center-prediction:• no parallelogram rule possible• predict this position as center of

the bounding box

center

Page 57: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

0last

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

position

Page 58: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

1

0

last

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

position

Page 59: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

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

across

1

0

2

Page 60: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

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

within

1

0

2

3

Page 61: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

1

0within

4

3

2

Page 62: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

3

1

2

across5

4

0

Page 63: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

3

1

0

2

within6

45

Page 64: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

7

5

1

0

2

within6

4

3

Page 65: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Example Decoding Run

7

6

5

1

0

4

within

7

2

3

Page 66: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Can we do better ?(and keep it simple)

Page 67: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Keep it simple

Constraints:

• single linear prediction

• use connectivity traversal order

Possibilities:

• floating point coefficients

• use more than three vertices

Page 68: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

better within-predictions

perfect systematic error

deg = 4

assuming ideal ( regular) polygons

P = A – B + C

P = αA + βB + γCα = 1, β = -1, γ = 1

deg = 6

A

B C

deg = 8

B C

A

CB

A

deg = 5

A

CB

Page 69: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Switch “within” Coefficients

if (deg == 4) {α = 1.000; β = -1.000; γ = 1.000;

} else if (deg == 5) {α = 1.024; β = -0.527; γ = 0.503;

} else if (deg == 6) {α = 1.066; β = -0.315; γ = 0.249;

} else …

How did we pick these numbers?

Page 70: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Finding the Coefficients

• no obvious “scientific” way• use Matlab

– for each degree separately– sum all possible prediction errors as

function of α and β

– optimize α and β for minimal error

Err = len (N – αA + βB + γC)with 1 = α + β + γ B

A

C

N

Page 71: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Switch “within” Coefficients

if (deg == 4) {α = 1.000; β = -1.000; γ = 1.000;

} else if (deg == 5) {α = 1.024; β = -0.527; γ = 0.503;

} else if (deg == 6) {α = 1.066; β = -0.315; γ = 0.249;

} else …

on average 70 % of predictions

parallelogram prediction within a quadrilateral is optimal

Page 72: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

assuming equal edge length

better across-predictions

33 34 35 36 38

43 44 45 46 48

adjacentpolygons

arecoplanar

Page 73: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

assuming equal edge length

better across-predictions

33 34 35 36 38

43 44 45 46 48

adjacentpolygonsare not

coplanar

creaseangle:

60°

Page 74: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Switch “across” Coefficientsif (deg == 33) {

α = 0.917; β = - 0.833; γ = 0.916;} else if (deg == 34) {

α = 0.621; β = - 0.504; γ = 0.883;} else if (deg == 35) {

α = 0.557; β = - 0.334; γ = 0.777;} else … …

} else if (deg == 43) {α = 1.153; β = - 0.354; γ = 0.201;

} else if (deg == 44) {α = 1.001; β = - 0.648; γ = 0.647;

} else … …

Page 75: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Conclusion

Page 76: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

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

Page 77: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Summary (2)

• proof-of-conceptimplementationin form of a Webjava-applet

• compression software will soon be made available

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

Page 78: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Future Work

• a scientific way to find numbers for this “coefficient switching”

• polygonification:– turn triangle meshes into polygon

meshes for better compression

Page 79: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Acknowledgments

funding:• ARC Télégéo grant of INRIA at

Sophia-Antipolis

logistics:• Jack Snoeyink

• Olivier Devillers

• Agnès Clément Bessière

• Jean-Daniel Boissonnat

Page 80: With Parallelogram Prediction Compressing Polygon Mesh Geometry Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis

Thank You!

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