With Degree Duality Prediction Compressing Polygon Mesh Connectivity Martin Isenburg University of...
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![Page 1: With Degree Duality Prediction Compressing Polygon Mesh Connectivity Martin Isenburg University of North Carolina at Chapel Hill.](https://reader035.fdocuments.in/reader035/viewer/2022081516/56649d545503460f94a30296/html5/thumbnails/1.jpg)
withDegree Duality
Prediction
CompressingPolygon Mesh Connectivity
Martin IsenburgUniversity of North Carolina
at Chapel Hill
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Overview
• Background
• Connectivity Compression
• Coding with Degrees
• Duality Prediction
• Adaptive Traversal
• Example Run
• Conclusion
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Background
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Polygon Meshes
4
5
• 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
k v log2 (v)
:k ~ 4 :k ~ 6
24 ~ 96 v
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Mesh Compression
• Geometry Compression [Deering, 95]– Fast Rendering
– Progressive Transmission
– Maximum Compression
• Geometry
• Connectivity
– Triangle Meshes
– Polygon MeshesPolygon Meshes
Maximum Compression
Connectivity
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Not Triangles … Polygons!
Face Fixer [Isenburg & Snoeyink, 00]
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triceratopsgalleoncessna
…tommygun
cowteapot
1.1892.0932.543
…2.2581.7811.127
Results
Face Fixer
bits per vertex (bpv)model Degree Duality
2.1152.5952.841
…2.6112.2131.669
44 %19 %11 % ...14 %20 %33 %
gain
min / max / average gain [%] = 11 / 55 / 26
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Connectivity Compression
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Connectivity Compression
assumption
• order of vertices does not matter
advantage
• no need to “preserve” indices
approach
• code only the “connectivity graph”
• re-order vertices appropriately
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Connectivity Graphs
• connectivity of simple meshes is homeomorphic to planar graph
enumeration
asymptotic bounds
[William Tutte 62 /
63]number of planar triangulations withv vertices
3.24 bpv 6 log2 (v) bpv<<
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Spanning Tree• Succinct Representations
of Graphs
[Turan, 84]• Short encodings of planar
graphs and maps
[Keeler & Westbrook, 95]• Geometric Compression
throughTopological Surgery
[Taubin & Rossignac, 98]
extends to meshes of non-zero genus
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Region Growing• Triangle Mesh Compression
[Touma & Gotsman, 98]• Cut-Border Machine
[Gumhold & Strasser, 98]• Edgebreaker
[Rossignac, 99] • Simple Sequential Encoding
[de Floriani et al., 99] • Dual Graph Approach
[Lee & Kuo, 99]• Face Fixer
[Isenburg & Snoeyink, 00]
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boundary
focus
vertex-based
processed region
unprocessed region
boundary
focus
face-based
boundary
focus
edge-based
Classification
• code symbols are associated with edges, faces, or vertices:
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Edge-BasedCompression Schemes
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
processed region
unprocessedregion
F F R
focus
. . .
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
F
F
processed region
unprocessedregion
F F R. . .
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
F
F
F
processed region
unprocessedregion
FF F R. . .
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
F
F
R
R
processed region
unprocessedregion
FFF F R. . .
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
R
F
F
F
F
processed region
unprocessedregion
RFFF F R. . .
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
R
F
F
F
R
R
processed region
unprocessedregion
FRFFF F R. . .
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Edge-Based
• Dual Graph Approach, [Lee & Kuo, 99]
• Face Fixer, [Isenburg & Snoeyink, 00]
F
R
F
F
FR
F
. . .
processed region
unprocessedregion
RFRFF. . . F F R
5. . . 4 . . .3 6 4
? ? ? ? ?
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Face-BasedCompression Schemes
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Face-Based
• Cut-Border Machine, [Gumhold & Strasser, 98]
• Edgebreaker, [Rossignac, 99]
focus
processed region
unprocessedregion
C R. . .
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Face-Based
• Cut-Border Machine, [Gumhold & Strasser, 98]
• Edgebreaker, [Rossignac, 99]
processed region
unprocessedregion
C R C
C
. . .
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Face-Based
• Cut-Border Machine, [Gumhold & Strasser, 98]
• Edgebreaker, [Rossignac, 99]
processed region
unprocessedregion
C R C
R
C
R. . .
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Face-Based
• Cut-Border Machine, [Gumhold & Strasser, 98]
• Edgebreaker, [Rossignac, 99]
processed region
unprocessedregion
C R C R
R
C
R
R. . .
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Face-Based
• Cut-Border Machine, [Gumhold & Strasser, 98]
• Edgebreaker, [Rossignac, 99]
C
processed region
unprocessedregion
C R C R
R
C
R
R C . . .
5. . . 4 . . .3 6 4
. . .
? ? ? ? ?
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Vertex-BasedCompression Schemes
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focus
Vertex-based
• Triangle Mesh Compression, [Touma & Gotsman,
98]processed region
unprocessedregion
. . . 6
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Vertex-based
• Triangle Mesh Compression, [Touma & Gotsman,
98]processed region
unprocessedregion
56
5
. . .
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Vertex-based
• Triangle Mesh Compression, [Touma & Gotsman,
98]processed region
unprocessedregion
56
5
. . .
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Vertex-based
• Triangle Mesh Compression, [Touma & Gotsman,
98]processed region
unprocessedregion
56
5
. . .
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Vertex-based
• Triangle Mesh Compression, [Touma & Gotsman,
98]processed region
5. . . 4 . . .3 6 4
56
5
6
? ? ? ? ?
unprocessedregion
6
. . .. . .
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Coding with Vertex and Face Degrees
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Coding with Degrees
while ( unprocessed faces )move focus to a face face degreefor ( free vertices )
caseswitch ( case )“add”: vertex degree“split”: offset“merge”: index, offset
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Example Traversal
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focus
“add” free vertex
processed regionunprocessed region
boundary
boundary slots
45
3
4
34 45. . .
. . .
focus(widened)
start slot
end slotfree
vertices
5
5
4
4
3
3 exit focus
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focus
free vertex “splits” boundary
processed region
unprocessed region
45
3
4
34 45. . .
. . .
Ssplit
offset
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stackfocus
free vertex “merges” boundary
processed region
processed region
unprocessed region
boundary in stack
mergeoffset
45
3
4
34 45. . .
. . .
M
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Resulting Code
• two symbol sequences–vertex degrees (+ “split” / “merge”)
–face degrees
• compress with arithmetic coder
converges to entropy
4 5
3. . . . . .64 4 44 M 5 44
. . . . . .3 64 4 4 4 4 4 4 4
S
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Entropy
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
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Average Distributions
2
3
4
56 7 8 9+
vertex degrees
3
4
5 6 7 8 9+
face degrees
add
mergesplit
case
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Adaptation to Regularity
6
vertexdegrees
facedegrees
3
... ... ... ...
3
vertexdegrees
facedegrees
6
... ... ... ...
4
vertexdegrees
facedegrees
4
... ... ... ...
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“Worst-case” Distribution
…
3
4
5
6
78
9
vertex degrees
3
face degrees
……
i = 3
pi = 1
i = 3
i • pi = 6
[Alliez & Desbrun, 01]
3.241… bpv
[Tutte, 62]
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Compressing with Duality Prediction
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Degree Correlation
• high-degree faces are “likely” to be surroundedby low-degreevertices
• and vice-versa
mutual degree prediction
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Face Degree Prediction
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
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Vertex Degree Prediction
6
=
degree offocus face
“vertex degree context”
vdc 6
vdc 6
vdc = 3
vdc = 4
vdc = 5
4
5
3
6
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Compression Gain
triceratopsgalleoncessna
…tommygun
cowteapot
without
bits per vertexmodel
min / max / average gain [%] = 0 / 31 / 17
with
bits per vertex
1.1892.0932.543
…2.2581.7811.127
1.1922.3712.811
…2.9171.7811.632
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Reducing the Number of Splits
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Occurance of “splits”
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Occurance of “splits”
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Occurance of “splits”
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Occurance of “splits”
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Occurance of “splits”
processed region
unprocessed region
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Occurance of “splits”
processed region
unprocessed region
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Occurance of “splits”
processed region
unprocessed region
![Page 58: With Degree Duality Prediction Compressing Polygon Mesh Connectivity Martin Isenburg University of North Carolina at Chapel Hill.](https://reader035.fdocuments.in/reader035/viewer/2022081516/56649d545503460f94a30296/html5/thumbnails/58.jpg)
Occurance of “splits”
processed region
unprocessed region
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Occurance of “splits”
processed region
unprocessed region
split
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focus
Adaptive Traversal
• Valence-driven connectivity encoding for 3D meshes [Alliez & Desbrun, 01]
avoid creation of cavities
exitfocus
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25 1.18918 2.09328 2.543… …32 2.25813 1.7813 1.127
Compression Gain
triceratopsgalleoncessna
…tommygun
cowteapot
bpvmodel splits bpvsplits
53 1.31178 2.309
172 2.882… …
131 2.449154 2.31310 1.167
without with
min / max / average gain [%] = 4 / 23 / 10
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Example Decoding Run
![Page 63: With Degree Duality Prediction Compressing Polygon Mesh Connectivity Martin Isenburg University of North Carolina at Chapel Hill.](https://reader035.fdocuments.in/reader035/viewer/2022081516/56649d545503460f94a30296/html5/thumbnails/63.jpg)
Example Decoding Run
45
3 . . .64 4 4 45 2
. . .3 64 4
6 3 5 444
3 5
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Conclusion
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Summary
• degree coding for polygonal connectivity
• duality prediction
• adaptive traversal
• proof-of-conceptimplementationusing Shout3D
http://www.cs.unc.edu/~isenburg/degreedualitycoder/
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Similar Result
• Near-Optimal Connectivity Coding of 2-manifold polygon meshes [Khodakovsky, Alliez, Desbrun, Schröder]
Martin France
analysis of worst-case face degree and vertex degree distribution
entropy Tutte’s bounds
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Current Work (w. Pierre Alliez)
• use polygons for better predictive geometry coding
• extend degree coding to volume mesh connectivity
“edge degrees”
“fairly planar & convex”
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Thank You!