Martin IsenburgUC Berkeley

Jack SnoeyinkUNC Chapel Hill

Early Split Coding ofTriangle Mesh Connectivity

Geometry

Connectivity

– Efficient Rendering

– Progressive Transmission

– Compact Storage

Mesh Compression“Geometry Compression” [Deering, 95]

• Surface Meshes– triangular

– polygonal

• Volume Meshes– tetrahedral, hexahedral

storage / network

main memory

Connectivity

Triangle Mesh Connectivityv -9.7 1.6 -0.2v -6.6 1.4 3.3v -1.6 1.4 2.3v -3.3 -1.4 3.0v 5.6 1.7 4.5v -6.6 1.4 6.1v -3.3 –4.4 3.7v -1.6 1.4 2.1v -5.4 -1.6 6.4...

f 1 2 3f 3 2 4f 2 1 5f 2 5 4f 6 4 5f 6 5 7f 5 1 8f 4 8 7f 7 8 9f 9 8 10f 10 11 7...

Geometry

each index uses log2 (v) bits

6 log2 (v) bpv

3.24 bpv

enumeration of triangulatedplanargraphs

[Tutte 62]

“Out-of-Core Compression” [Isenburg & Gumhold ‘03]

“Delphi Coding” [Coors and Rossignac ‘04]

“FreeLence” [Kälberer et al. ‘05]

Compression Schemes“Cut-Border Machine” [Gumhold & Strasser ‘98]

“Triangle Mesh Compression” [Touma & Gotsman ‘98]

“Edgebreaker” [Rossignac ‘99]

“Face Fixer” [Isenburg & Snoeyink ‘00]

“Angle Analyzer” [Lee, Alliez & Desbrun ‘02]

“Degree Duality Coder” [Isenburg ‘02]

“Near-optimal Coding” [Khodakovsky et al. ‘02]

“Valence-Driven Encoding” [Alliez & Desbrun ‘01]ComputerGraphics

“Succinct encodings of planar graphs” [He, Yao, & Hu ‘99]

“Short encodings of planar graphs” [Keeler & Westbrook ‘95]

“Succinct Representations of Graphs” [Turan ‘84]

“Optimal Coding” [Poulahlon & Schaeffer ‘03]

PlanarGraphCoding

Connectivity Graph Traversal

split offset

Connectivity Graph Traversal

CBM – Edgebreaker – TG coder• label-based

split offset

“Cut-Border Machine” [Gumhold & Strasser ‘98]

“Edgebreaker” [Rossignac ‘99]

“Early Split Coding” [this paper]

C R C R R S. . . R C R E R C . . .

• degree-based

“Triangle Mesh Compression” [Touma & Gotsman ‘98]

3. . . . . .64 4 44 5 44S

split offset

CBM and Edgebreaker

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CBM and Edgebreaker

5

processedregion

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compressionboundary

C

C

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

C C

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

R

C C R

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

C C R C

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

R

C C R C R

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

RS

C C R C R S

12

3

4

5

6

7

89

10

11

offset = 11

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

RS

L

C C R C R S L

offset = 11

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

RS

L C

C C R C R S L C

offset = 11

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

RS

L CR

C C R C R S L C R

offset = 11

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

RS

L CR C

C C R C R S L C R C

offset = 11

CBM and Edgebreaker

5

processedregion

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CC

RC

RS

L CR C

R

C C R C R S L C R C R

offset = 11

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

RS

L CR C

R C

C C R C R S L C R C R C

offset = 11

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

RS

L CR C

R C

R

C C R C R S L C R C R C R

offset = 11

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

RS

L CR C

R C

R R

C C R C R S L C R C R C R R

offset = 11

CBM and Edgebreaker

5

processedregion

unprocessed region

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CC

RC

RS

L CR C

R C

R R

R

C C R C R S L C R C R C R R R

offset = 11

CBM and Edgebreaker

5

processedregion

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CC

RC

RS

L CR C

R C

R R

RL

C C R C R S L C R C R C R R R L

offset = 11

CBM and Edgebreaker

5

processedregion

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CC

RC

RS

L CR C

R C

R R

RL

C

C C R C R S L C R C R C R R R L C

offset = 11

CBM and Edgebreaker

5

processedregion

unprocessed region

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CC

RC

RS

L CR C

R C

R R

RL

CR

C C R C R S L C R C R C R R R L C R

offset = 11

CBM and Edgebreaker

5

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unprocessed region

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CC

RC

RS

L CR C

R C

R R

RL

CR

R

C C R C R S L C R C R C R R R L C R R

offset = 11

CBM and Edgebreaker

5

processedregion

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CC

RC

RS

L CR C

R C

R R

RL

CR

RR

C C R C R S L C R C R C R R R L C R R R

offset = 11

CBM and Edgebreaker

5

processedregion

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CC

RC

RS

L CR C

R C

R R

RL

CR

RR

S

C C R C R S L C R C R C R R R L C R R R S

1

2

offset = 3

offset = 11

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

R

C C R C R S L C R C R C R R R L C R R R S R

S

L CR C

R C

R R

RL

CR

RR

SR

offset = 3

offset = 11

CBM and Edgebreaker

5

processedregion

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compressionboundary

CC

RC

RS

L CR C

R C

R R

RL

CR

RR

SE

R

C C R C R S L C R C R C R R R L C R R R S R E

offset = 3

offset = 11

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

RS

L CR C

R C

R R

RL

CR

RR

SE

R

E

C C R C R S L C R C R C R R R L C R R R S R E E

offset = 11

offset = 3

CBM and Edgebreaker

5

processedregion

unprocessed region

compressionboundary

CC

RC

RS

L CR C

R C

R R

RL

CR

RR

SE

R

E

C C R C R S L C R C R C R R R L C R R R S R E E C

offset = 11

offset = 3

+1 +3

+1 -1 +1 +1 +1 +1 +1 +1 +1 +1 +1 -1 +1 +3 +3-1 -1 -1

= 4

= 12

C

...

Subsequences of Labels areself-contained Encodings

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

reversible

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

reversible

E

E

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

L

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

R

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

C

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

CR

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

CR

C

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

CR

CR

… …

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

CR

CR

… …

C

S L C R C R C R R R L C R R R S R E E

Subsequences are self-contained

E

reversible

RE

S

R R

R

C

LR

RR

CR

CR

… …

CL

we did not needsplit offsets !!!

TG coder

TG coder

5

processedregion

unprocessed region

compressionboundary

slot counts

TG coder

5

processedregion

unprocessed region

compressionboundary

7

7

TG coder

5

processedregion

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compressionboundary

7

6

7 6

zero slot

TG coder

5

processedregion

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compressionboundary

7

6

7 6

TG coder

5

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unprocessed region

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7

6

5

7 6 5

zero slot

TG coder

5

processedregion

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7

6

5

7 6 5

TG coder

5

processedregion

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compressionboundary

7

6

5

S 24

6

7

8

11

12

1315

16

18

offset = 18

7 6 5 S

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7 6 5 S

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

7 6 5 S 7

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

7 6 5 S 7

offset = 18

zero slot

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

7 6 5 S 7

offset = 18

zero slot

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

7 6 5 S 7

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

7 6 5 S 7 6

offset = 18

TG coder

5

processedregion

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compressionboundary

7

6

5

S

7

6

7 6 5 S 7 6

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

7 6 5 S 7 6 5

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

7 6 5 S 7 6 5

offset = 18

TG coder

5

processedregion

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compressionboundary

7

6

5

S

7

6

5

7 6 5 S 7 6 5

offset = 18

TG coder

5

processedregion

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compressionboundary

7

6

5

S

7

6

5

7 6 5 S 7 6 5

offset = 18

TG coder

5

processedregion

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compressionboundary

7

6

5

S

7

6

5

7 6 5 S 7 6 5

offset = 18

TG coder

5

processedregion

unprocessed region

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7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6

offset = 18

TG coder

5

processedregion

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compressionboundary

7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6

offset = 18

TG coder

5

processedregion

unprocessed region

compressionboundary

7

6

5

S

7

6

5

6

7 6 5 S 7 6 5 6 7

offset = 18

7

…

E

Subsequences of Degrees arenot self-contained Encodings

Subsequences are not self-contained

… S 7 6 5 6 E …

5

S

… S 7 6 5 6 E …

5

S

Subsequences are not self-contained

… S 7 6 5 6 E …

5

S

Subsequences are not self-contained

… S 7 6 5 6 E …

5

S

Subsequences are not self-contained

… S 7 6 5 6 E …

5

S

Subsequences are not self-contained

… S 7 6 5 6 E …

5

S

Subsequences are not self-contained

Not unique without split offsets

… S 4 3 E 4 3 E

4

3

34 4

34

3

6 6 5 6 6 5 6 … 5 7 5 6 6 6 5 …

Early Split Coder

Early Split Coder• encode with labels or degrees

• traverse triangles in same order – new label W

– earlier split operations

• one-to-one mapping insights:

– how to turn degree subsequences into self-contained encodings

– decoding strategy decides success

RCR

CCR

The new label W

R

C

CRCC

R

The new label W

R

C

CRCC

R

The new label W

C

R

CC R

CRCC

R

The new label W

wart

wart

RCR

CCR

The new label W

RCR

CCR

C

R

CC R

CRCC

R

wart

wart

zero slot

The new label W

RCR

CCR

CW

R

CC R

CRCC

R

wart

wart

The new label W

RCR

CCR

CWC

R

CC R

CRCC

R

wart

wart

The new label W

RCR

CCR

CWC R

R

CC R

CRCC

R

wart

wart

The “early split” operation

RR

CRC

The “early split” operation

S

RR

CRC

The “early split” operation

S

RR

CRC

R

The “early split” operation

S

RR

CRC

RL

The “early split” operation

S

RR

CRC

RL R

The “early split” operation

S

RR

CRC

RL R

Lzero slot

The “early split” operation

S

RR

CRC

RL R

L

The “early split” operation

S

RR

CRC

RL R

L

The “early split” operation

S

RR

CRC

RL R

L

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

Szero slot

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

SW

zero slot

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

WS W

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

WS W

C

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

WS W

CR

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

WS W

CR

R

zero slot

The “early split” operation

S

RR

CRC

RL R

L

RR

CRC

C

WS W

CR

R

W

Complete Example

One Scheme & Four Encodings

label-based

degree-based

Forward Decodingwith offsets

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

zero slot

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

zero slot

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

zero slot

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

zero slot

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Forward Decoding with offsets

V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0

C C C C R C R C R W C R C S1 C R R E C R E

Reverse Decodingwithout offsets

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Reverse Decoding w/o offsets

E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5

E R C E R R C S C R C W R C R C R C C C C

Results

Compression Results

Improving the TG coder

5

“Valence-Driven Encoding” [Alliez & Desbrun ‘01]

degree of vertexmust beat least 5

(at least 6 for early-split)

degree of vertexmust beat least 4

(at least 5 for early-split)

Thank You