Martin Isenburg UC Berkeley Jack Snoeyink UNC Chapel Hill Early Split Coding of Triangle Mesh Connectivity.

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Presentation transcript:

Martin Isenburg UC Berkeley Jack Snoeyink UNC Chapel Hill Early Split Coding of Triangle 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 Connectivity v v v v v v v -3.3 – v v f f f f f f f f f f f Geometry each index uses log 2 ( v ) bits  6 log 2 ( v ) bpv  3.24 bpv enumeration of triangulated planar graphs [ 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 ] Computer Graphics “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 ] Planar Graph Coding

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 ] CRCRRS... RCRERC degree-based “Triangle Mesh Compression” [ Touma & Gotsman ‘98 ] S split offset

CBM and Edgebreaker

5 processed region unprocessed region compression boundary

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C C

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C C R

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C C C R C

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R C C R C R

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S C C R C R S offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C C R C R S L offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C C C R C R S L C offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C C R C R S L C R offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C C C R C R S L C R C offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C C R C R S L C R C R offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C C C R C R S L C R C R C offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R C C R C R S L C R C R C R offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R 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 processed region unprocessed region compression boundary C C R C R S L C R 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 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C C R C R S L C R C R C R R R L offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C C C R C R S L C R C R C R R R L C offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R C C R C R S L C R C R C R R R L C R offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R 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 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R 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 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R 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 processed region unprocessed region compression boundary C C R C 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 C R C R C R R R L C R R R S R offset = 3 offset = 11

CBM and Edgebreaker 5 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S E 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 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S E 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 processed region unprocessed region compression boundary C C R C R S L C R C R C R R R L C R R R S E 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 = = 4 = 12 C...

Subsequences of Labels are self-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 R 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 R 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 R 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 R 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 R 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 R E 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 R E 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 R E 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 R E 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 R E 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 R E 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 R E S R R R C L 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 R E S R R R C L 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 R E S R R R C L R 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 R E S R R R C L R 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 R E S R R R C L R RR C 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 R E S R R R C L R RR C 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 R E S R R R C L R RR C R C 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 R E S R R R C L R RR C R C 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 R E S R R R C L R RR C R C R …… C L we did not need split offsets !!!

TG coder

5 processed region unprocessed region compression boundary slot counts

TG coder 5 processed region unprocessed region compression boundary 7 7

TG coder 5 processed region unprocessed region compression boundary zero slot

TG coder 5 processed region unprocessed region compression boundary

TG coder 5 processed region unprocessed region compression boundary zero slot

TG coder 5 processed region unprocessed region compression boundary

TG coder 5 processed region unprocessed region compression boundary S offset = S

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S 7 offset = 18

TG coder 5 processed region unprocessed region compression boundary S S 7 offset = 18 zero slot

TG coder 5 processed region unprocessed region compression boundary S S 7 offset = 18 zero slot

TG coder 5 processed region unprocessed region compression boundary S S 7 offset = 18

TG coder 5 processed region unprocessed region compression boundary S S 7 6 offset = 18

TG coder 5 processed region unprocessed region compression boundary S S 7 6 offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18

TG coder 5 processed region unprocessed region compression boundary S S offset = 18 7 … E

Subsequences of Degrees are not self-contained Encodings

Subsequences are not self-contained … S E … 5 S

5 S Subsequences are not self-contained

… S E … 5 S Subsequences are not self-contained

… S E … 5 S Subsequences are not self-contained

… S E … 5 S Subsequences are not self-contained

… S E … 5 S Subsequences are not self-contained

Not unique without split offsets … S 4 3 E 4 3 E … …

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

R C R C C R The new label W

R C C R C C R

R C C R C C R C

R C C R C R C C R wart R C R C C R

The new label W R C R C C R C R C C R C R C C R wart zero slot

The new label W R C R C C R C W R C C R C R C C R wart

The new label W R C R C C R C W C R C C R C R C C R wart

The new label W R C R C C R C W C R R C C R C R C C R wart

The “early split” operation R R C R C

S R R C R C

S R R C R C R

S R R C R C R L

S R R C R C R L R

S R R C R C R L R L zero slot

The “early split” operation S R R C R C R L R L

S R R C R C R L R L

S R R C R C R L R L

S R R C R C R L R L

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

S R R C R C R L R L R R C R C C S zero slot

The “early split” operation S R R C R C R L R L R R C R C C S W zero slot

The “early split” operation S R R C R C R L R L R R C R C C W S W

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

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

S R R C R C R L R L R R C R C C W S W C R R zero slot

The “early split” operation S R R C R C R L R L R R C R C C W S W C R R W

Complete Example

One Scheme & Four Encodings label-based degree-based

Forward Decoding with offsets

V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E zero slot

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Forward Decoding with offsets V 5 V 4 V 5 V 5 V 6 V 6 V 4 V 8 V 5 V 5 S 1,0,0 C C C C R C R C R W C R C S 1 C R R E C R E

Reverse Decoding without offsets

Reverse Decoding w/o offsets E V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 V 3 E V 4 S V 5 V 5 W V 8 V 4 V 6 V 6 V 5 V 5 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 vertex must be at least 5 ( at least 6 for early-split ) degree of vertex must be at least 4 ( at least 5 for early-split )

Thank You