valence-based connectivity coding

Valence-based Connectivity CodingValence-based Connectivity Coding

Hu Jianwei2007-11-07

Why Mesh CompressionWhy Mesh Compression

• Slow networks require data compression to reduce the latency

• Small storage devices need data compression to save space

Surface Based MeshesSurface Based Meshes

• Connectivity– How the vertices connect with each other

• Geometry– Coordinates of the vertices

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

facef

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

vertexv

ClassificationClassification

• Connectivity encoding

• Geometry encoding

• Single-rate compression

• Progressive compression

Valence-based CompressionValence-based Compression

• Triangle Mesh Compression [98 Touma & Gotsman]

• Valence-Driven Connectivity Encoding for 3D Meshes [01 Alliez & Desbrun]

• Near-Optimal Connectivity Encoding of 2-Manifold Polygon Meshes [02 Khodakovsky et al.]

• Compressing Polygon Mesh Connectivity with Degree Duality Prediction [02 Isenburg]

Key ObservationKey Observation

• A genus-0 manifold mesh is topologically equivalent to a planar graph

• The vertices incident on any mesh vertex may be ordered

98 Touma & Gotsman

Code WordsCode Words

• The topology may be encoded with:– add<degree>– split<offset>– merge<index><offset>

• Then entropy encoded– Huffman– arithmetic– run-length

98 Touma & Gotsman

Example TraversalExample Traversal98 Touma & Gotsman

Add dummy vertices98 Touma & Gotsman

Example TraversalTraversal98 Touma & Gotsman

Encoding98 Touma & Gotsman


Add 6; Add 7 ; Add 4 ;


Add 6; Add 7 ; Add 4 ; Add 4 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 (focus full) ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 (focus full) ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4 (focus full);


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4 ; (focus full)


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4;



Huffman codingrun-length

coding

Decoding98 Touma & Gotsman



Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 (focus full); Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 (focus full); Add 4 ; Add 4 ; Add Dummy 6 ; Add 4;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 (focus full); Add Dummy 6 ; Add 4;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 (list full, pop list); Add 4;


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4 (focus full);


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4; (focus full)


Add 6; Add 7 ; Add 4 ; Add 4 ; Add 8 ; Add 5 ; Add 5 ; Add 4 ; Add 5 ; Split 5 ; Add 4 ; Add 4 ; Add Dummy 6 ; Add 4; (focus full, list full)

ExampleExample98 Touma & Gotsman

ResultsResults

Model #tri bits/tri

Eight 1536 0.2

Triceratops 5666 1.4

Cow 5804 1.1

Beethoven 5028 1.4

Dodge 16646 0.9

Starship 8152 0.5

Average 0.9

98 Touma & Gotsman

Less than 1.5 bits per vertex on the average

ObstacleObstacle

• The number of split codes and the range of their offsets turn out to seriously impede the compression ratios

• This is a serious obstacle to guarantee a linear worst case bit-rate complexity

01 Alliez & Desbrun

Occurrence of “splits”Occurrence of “splits”01 Alliez & Desbrun


processed region

unprocessed region


unprocessed region

processed region


unprocessed region

split

processed region

The Best Pivot CandidateThe Best Pivot Candidate01 Alliez & Desbrun

How to ChooseHow to Choose

• The minimum free edges as the next pivot

• An adaptive mean number of free edges to select the next pivot vertex

01 Alliez & Desbrun

Getting Rid of the SplitsGetting Rid of the Splits01 Alliez & Desbrun

The Resulting ConquestThe Resulting Conquest01 Alliez & Desbrun

98 Touma & Gotsman

01 Alliez & Desbrun

Compression resultsCompression results01 Alliez & Desbrun

Not Triangles … Polygons!Not Triangles … Polygons!02 Isenburg

Add Free Vertices and FacesAdd Free Vertices and Faces

focusprocessed region

unprocessed region

boundary

boundary slots

45

3

4

34 45. . .. . .

focus(widened)

start slot

end slotfreevertices

5

5

4

4

3

3 exit focus

02 Isenburg

Free Vertex Splits BoundaryFree Vertex Splits Boundary

focus

processed region

unprocessed region

45

3

4

34 45. . .. . .

Ssplitoffset

02 Isenburg

Free Vertex Merges BoundaryFree Vertex Merges Boundary

stackfocus

processed region

processed region

unprocessed region

boundary in stack

mergeoffset

45

3

4

34 45. . .. . .

M

02 Isenburg

Resulting CodeResulting Code

• Two symbol sequences

– vertex degrees (+ “split” / “merge”)

– face degrees

• Compress with arithmetic coder

02 Isenburg

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

4 5. . . . . .3 64 4 4 4 4 4 4 4

Example Decoding RunExample Decoding Run02 Isenburg

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


focus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

6


3

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

freevertex


exitfocus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

6


5

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

free vertices


5

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

3


exitfocus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


4

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


4

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

5


exitfocus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


focus(widened)

startslot

endslot

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

5

3

3


4

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


exitfocus

4

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

5


focus(widened)

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


4

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


focus

exitfocus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


3

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


exitfocus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


focus(widened)

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


6

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


6

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

2


6

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


exitfocus

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

4


focus(widened)

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5


5

45

3 . . .64 4 4 45 2

. . .3 64 4

6 3 5 444

3 5

. . . . . .

Average Distributions Average Distributions

2

3

4

56 7 8 9+

vertex degrees

3

4

5 6 7 8 9+

face degrees

add

mergesplit

case

02 Isenburg

Degree CorrelationDegree Correlation02 Isenburg

Face Degree Prediction Face Degree Prediction 02 Isenburg

focus(widened)

4

3

3

3 + 4 + 3=

33.333

average degree offocus vertices

“face degree context”

fdc =

fdc 3.3

3.3 fdc 4.3

4.3 fdc 4.9

4.9 fdc

Vertex Degree Prediction Vertex Degree Prediction 02 Isenburg

6

=

degree offocus face

“vertex degree context”

vdc 6

vdc 6

vdc =

vdc = 4

vdc = 5

4

5

3

6

Compression Gain Compression Gain 02 Isenburg

triceratopsgalleoncessna

…tommygun

cowteapot

withoutbits per vertexmodel

min / max / average gain [%] = 0 / 31 / 17

withbits per vertex

1.1892.0932.543

…2.2581.7811.127

1.1922.3712.811

…2.9171.7811.632


vs. Face Fixer coder [00 Isenburg & Snoeyink]

Reducing the Number of SplitsReducing the Number of Splits02 Isenburg

• A simple version of [Alliez & Desbrun]’s stratege – Only move the focus if the smallest number of slots is 0

or 1

focus

exitfocus

Valence-based CompressionValence-based Compression

• Valence-based coding is widely considered to be the best connectivity coding method available

• If valence-based coding is not optimal, it is probably not far from this

Arithmetic CodingArithmetic Coding

• Arithmetic coding is a method for lossless data compression

• It is a form of variable-length entropy encoding

• Arithmetic coding encodes the entire message into a single number, a fraction n where (0.0 ≤ n < 1.0)

Arithmetic Coding ExampleArithmetic Coding Example

• Four-symbol model– 60% chance of symbol NEUTRAL – 20% chance of symbol POSITIVE– 10% chance of symbol NEGATIVE– 10% chance of symbol END-OF-DATA

• Decoding of 0.538


• The interval for NEUTRAL would be [0, 0.6)

• The interval for POSITIVE would be [0.6, 0.8)

• The interval for NEGATIVE would be [0.8, 0.9)

• The interval for END-OF-DATA would be [0.9, 1)


0.538 falls into the sub-interval for NEUTRAL, [0, 0.6)• The interval for NEUTRAL would be [0, 0.36) -- 60% of [0, 0.6)

• The interval for POSITIVE would be [0.36, 0.48) -- 20% of [0, 0.6)

• The interval for NEGATIVE would be [0.48, 0.54) -- 10% of [0, 0.6)

• The interval for END-OF-DATA would be [0.54, 0.6). -- 10% of [0, 0.6)

A diagram showing decoding of 0.538


0.538 falls into the sub-interval for NEGATIVE, [0.48, 0.54)• The interval for NEUTRAL would be [0.48, 0.516) -- 60% of [0.48, 0.54)

• The interval for POSITIVE would be [0.516, 0.528) -- 20% of [0.48, 0.54)

• The interval for NEGATIVE would be [0.528, 0.534) -- 10% of [0.48, 0.54)

• The interval for END-OF-DATA would be [0.534, 0.540) -- 10% of [0.48, 0.54)



0.538 falls into the sub-interval for END-OF-DATA, [0.534, 0.540)

Decoding is complete


The same message could have been encoded by the equally short fractions .534, .535, .536, .537 or .539.

valence-based connectivity coding

Documents

polygon mesh connectivity

manifold mesh

mesh vertex

data compression

z vertex2 x

z vertex3 x

mesh compressionslow

manifold polygon