Lossless Compression of Floating-Point Geometry

Lossless Compressionof

Floating-Point Geometry

Martin IsenburgUNC

Chapel Hill

Peter LindstromLLNL

Livermore

Jack SnoeyinkUNC

Chapel Hill

Overview

• Motivation

• Geometry Compression

• Predictive Schemes

• Corrections in Floating-Point

• Results

• Conclusion

127 MB

12,748,510 triangles11,070,509 vertices

Meshes and Compression

• connectivityface1 1 2 3 face2 2 1 4face3 3 2 5

facem

3m * 32 bits

3n * 32 bits

146 MB

UNCpower-plant

3 MB

25 MB 20uniform quantization

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

vertexn

• geometry

Floating-Point Numbers

• have varying precision

• largest exponent least precise

- 4.095 190.974

0 1286432

x - axis

1684

23 bits ofprecision

23 bits ofprecision

23 bits ofprecision

Samples per Millimeter16 bit 18 bit 20 bit 22 bit 24 bit

97 388 15532.7 m

327 1311 20 cm

5 22 86195 m

No quantizing possible …

• stubborn scientists / engineers – “Do not touch my data !!!”

• data has varying precision– specifically aligned with origin

• not enough a-priori information– no bounding box

– unknown precision

– streaming data source

• Geometry Compression [Deering, 95]– Fast Rendering

– Progressive Transmission

– Maximum Compression

Geometry Compression [Deering, 95]

Mesh Compression

Maximum Compression




• Connectivity

• Geometry


Mesh Compression

Maximum Compression

Geometry




• Connectivity

• Geometry

– lossy

– lossless


Mesh Compression

lossless

Maximum Compression

Geometry

Overview

• Motivation




• Results

• Conclusion

Geometry Compression

• 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



• Recent approaches [00 – 03]:– spectral

– re-meshing

– space-dividing

– vector-quantization

– angle-based

– delta coordinates




– re-meshing

– space-dividing


– angle-based


Spectral Compressionof Mesh Geometry

[Karni & Gotsman, 00]

expensive numericalcomputations




– re-meshing

– space-dividing


– angle-based


Progressive GeometryCompression

[Khodakovsky et al., 00]

modifies mesh priorto compression




– re-meshing

– space-dividing


– angle-based


Geometric Compressionfor interactive transmission

[Devillers & Gandoin, 00]

poly-soups; complexgeometric algorithms




– re-meshing

– space-dividing


– angle-based


Vertex data compressionfor triangle meshes

[Lee & Ko, 00]

local coord-system +vector-quantization




– re-meshing

– space-dividing


– angle-based


Angle-Analyzer: A triangle-quad mesh codec

[Lee, Alliez & Desbrun, 02]

dihedral + internal =heavy trigonometry




– re-meshing

– space-dividing


– angle-based


High-Pass Quantization forMesh Encoding

[Sorkine et al., 03]

basis transformationwith Laplacian matrix

Overview

• Motivation




• Results

• Conclusion

Predictive Coding

1. quantize positions with b bits

2. predict from neighbors

3. compute difference

4. compress with entropy coder

(1.314, -0.204, 0.704) (1008, 68, 718)floating point integer

Predictive Coding





prediction rule(1004, 71, 723)

prediction

(1.276, -0.182, 0.734)predictionprediction rule

Predictive Coding





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

(4, -3, -5)correctorprediction

(1.314, -0.204, 0.704)position

(1.276, -0.182, 0.734)prediction

?

Predictive Coding





0

10

20

30

40

50

60

70

position distribution

0

500

1000

1500

2000

2500

3000

3500

corrector distribution

Arithmetic Entropy Coder

for a symbol sequence of t types

# of type tpi =

i = 1

t

Entropy = pi • log2( ) bitspi

1

# total

2.0 bits1.3 bits0.2 bits

Deering, 95

Prediction: Delta-Coding

A

processed regionunprocessed region

P

P = A

Taubin & Rossignac, 98

Prediction: Spanning Tree

A

BC D

E


P

P = αA + βB + γC + δD + εE + …

Touma & Gotsman, 98

Prediction: Parallelogram Rule


P

P = A – B + C

A

BC

Talk Overview

• Motivation




• Results

• Conclusion

Corrective Vectors ?

(1008, 68, 718)position

(1004, 71, 723)prediction

(4, -3, -5)correctorcompute difference vector

with integer arithmetic

(1.314, -0.204, 0.704)position

(1.276, -0.182, 0.734)prediction

(0.038, -0.022, -0.030)correctorcompute difference

vector withfloating-point arithmetic ?

32-bit IEEE Floating-Point

0

- 1-½

- 2.0- 4.0- 8.0-16.0

½1

2.04.0 8.0 16.0

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

1 bit sign 8 bit exponent 23 bit mantissa

1 82 …1 81 …

1 80 …

1 7E …

1 7D … 0 7D …

0 7E …

0 80 …

0 81 …0 82 …

1 7a 75c28f-0.06

position

1 82 4ccccd-12.8

prediction

corrector

0 82 4bd70812.74

corrector

01999a

32-bit IEEE Floating-Point

0

- 1-½

- 2.0- 4.0- 8.0-16.0

½1

2.04.0 8.0 16.0


1 bit sign 8 bit exponent 23 bit mantissa

1 82 …1 81 …

1 80 …

1 7E …

1 7D … 0 7D …

0 7E …

0 80 …

0 81 …0 82 …

821 75c2847a1 49999a820

1 82 01999a-8.1

1 81 7ccccd-7.9

position prediction

-0.2

1 7a 75c284-0.06

0 7c 0f5c290.14

position prediction

-0.2

0 82 49999a12.6

0 82 4ccccd12.8

position prediction

-0.2

Floating-Point Corrections (1)

0 82 49999a12.6

position

0 82 4ccccd12.8

prediction

0 82 49999a 0 82 4ccccd

0 0 0

49999a 4ccccd -033333use context 82

use context 8282

compress


1 82 01999a-8.1

position

1 81 7ccccd-7.9

prediction

1 82 01999a 1 81 7ccccd

1 1 0

01999a 0 01999ause context 82

use context 8182

compress


1 7a 75c284-0.06

position

0 7c 0f5c290.14

prediction

1 7a 75c284 0 7c 0f5c29

1 0 1

75c284 0 75c284use context 7a

use context 7c7a

compress

Talk Overview

• Motivation




• Results

• Conclusion

gzip -9buddha

david

power plantlucy

55.856.123.778.7

Results [bpv]

rawmodel

96.0 96.0 96.0 96.0

ours

49.934.729.144.4

0.90

0.62

1.21

0.56

ratio

Percentage of Unique Floats

buddha david lucypowerplant

xy

zx

yz

x

y

z

x y z

49.0

%

82.4

%

77.2

%

40.2

% 28.9

%

42.0

%

53.4

%

37.6

%

22.2

%

4.4

%

1.5

%

2.0

%

228,402different

z-coordinates

543,652 vertices

226,235different

z-coordinates

11,070,509 vertices

Completing, not competing !!!

buddha

david

power plantlucy

model 16 bit

bit-rate [bpv]

buddha

david

power plantlucy

54393144

45262430

61473454

52363046

compression ratio [%]

20 bit

32.223.218.526.5

21.8 12.5 11.6 14.6

24 bit

44.034.124.239.1

lossless

49.934.729.144.4

Simpler Scheme (Non-Predictive)

A X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

buddha

david

power plantlucy

A ours

76.553.961.768.1

78.958.467.369.5

76.4 52.356.468.6

49.934.729.144.4

B Cmodel

32 symbols, each ranging from 0 to 1

B X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X

8 symbols, each ranging from 0 to 15

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX XC4 symbols, each ranging from 0 to 63

Simpler Scheme (Predictive)

A

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX

X

XC



BX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X

buddha

david

power plantlucy

55.638.238.5

45.4

59.444.044.948.8

53.136.133.644.8

49.934.729.144.4

A oursB Cmodel

Talk Overview

• Motivation




• Results

• Conclusion

Summary

• efficient predictive compression of floating-point in a lossless manner

• completing, not competing

– predict in floating-point

– break predicted and actual float into several components

– compress differences separately

– use exponent to switch contexts

Current / Future Work

• investigate trade-offs– compression versus throughput

• improve scheme for “sparse” data• quantize without bounding box

– guarantee precision

– learn bounding box

• optimize implementation– run faster / use less memory

Acknowledgments

funding:• LLNL

• DOE contract No. W-7405-Eng-48

models:• Digital Michelangelo Project

• UNC Walkthru Group

• Newport News Shipbuilding

Thank You!

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Lossless Compression of Floating-Point Geometry

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