Compressing Hexahedral Volume Meshes

1
CompressingHexahedral Volume Meshes

• Martin Isenburg
• UNCChapel Hill

Pierre Alliez INRIASophia-Antipolis
2
Overview

• Volume Meshes
• Related Work
• Compressing Connectivity
• Coding with Edge Degrees
• Boundary Propagation
• Adaptive Traversal
• Compressing Geometry
• Parallelogram Prediction
• Demo

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Take this home

• The connectivity of a hexahedral mesh can be
  coded through asequence of its edge degrees.
• This encoding naturallyexploits the regularity
  commonlyfound in such data sets.

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Volume Meshes
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Volume Meshes

• scientific industrial applications
• thermodynamics
• structural mechanics
• visualization simulation
• unstructured / irregular ( ? not on a
  grid )
• tetrahedral, hexahedral, polyhedral

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Hexahedral Volume Meshes

• have numerical advantages in finite element
  computations
• challenging to generate
• their internal structure looks nice compared to
  tetrahedral meshes

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Ingredients

• geometry positions of vertices
• connectivity which vertices form a hexahedron
• properties attached to vertices density,
  pressure, heat, ...

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Standard Representation

• connectivity
• geometry

hex1 1 3 6 4 7 8 9 2 hex2 4 5 8 2 9 1 6 7 hex3
7 5 hexh
less than84 KB
vtx1 ( x, y, z ) vtx2 ( x, y, z ) vtx3 ( x, y,
z ) vtxv
71572 hexahedra 78618 vertices
? size 3.23 MB
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Related Work
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Surface Mesh Compression

• Geometry Compression, Deering, 95
• Topological Surgery, Taubin Rossignac, 98
• Cut-Border Machine, Gumhold Strasser, 98
• Triangle Mesh Compression, Touma Gotsman, 98
• Edgebreaker, Rossignac, 99
• Spectral Compression of Geometry, Karni
  Gotsman, 00
• Face Fixer, Isenburg Snoeyink, 00
• Valence-driven Connectivity Coding, Alliez
  Desbrun, 01
• Near-Optimal Coding, Khodakovsky, Alliez,
  Desbrun
• Degree Duality Coder, Isenburg, 02
• Polygonal Parallelogram Prediction, Isenburg
  Alliez, 02

Schröder, 02
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Volume Mesh Compression

• Grow Fold, Szymczak Rossignac, 99
• Cut-Border Machine, Gumhold, Guthe Strasser,
  99

• Rendering of compressed volume data, Yang et
  al., 01

• Simplification
• Simplification of tetrahedral meshes, Trotts et
  al., 98
• Progressive Tetrahedralizations, Staadt Gross,
  98

• Progressive Compression
• Implant Sprays, Pajarola, Rossignac Szymczak,
  99

!! only for tetrahedral meshes !!
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Surface / Volume Connectivity

• a mesh with v vertices has maximal
• surfaces
• 2v-2 triangles ? 6v indices
• v-1 quadrilaterals ? 4v indices

volumes O(v2) tetrahedra ? 12v
indices O(v2) hexahedra ? 8v indices
? connectivity dominates geometry even more
for volume meshes
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Degree Coding for Connectivity

• Triangle Mesh Compression, Touma Gotsman, 98
• Valence-driven Connectivity Coding, Alliez,
  Desbrun, 01
• Degree Duality Coder, Isenburg, 02
• Near-Optimal Connectivity Coding, Khodakovsky,
  Alliez, Desbrun, Schröder, 02

compressed with arithmetic coder ?
converges to entropy
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Entropy

• for a symbol sequence of t types

t
1
?
Entropy pi log2( ) bits
pi
i 1
of type t
pi
total
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Average Distribution
over a set of 11 polygonal meshes
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Worst-case Distribution
3
4
5
Alliez Desbrun, 01 ? 3.241 bpv Tutte, 62
6
7
8
9



vertex degrees
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Adaptation to Regularity
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Degree Coding for Volumes ?
tri ? tet
vertex degrees ? edge degrees
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Regular Volume Meshes?

• elements for regular 2D tiling
• regular triangle
• regular quadrilateral
• regular hexagon

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Compressing Connectivity
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Space Growing

• similar in spirit to region growing
• algorithm maintains hull enclosing processed
  hexahedra

?

1. initialize hull with a border face
2. iterate until done

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Coding with Edge Degrees
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Coding with Edge Degrees
focus face
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Coding with Edge Degrees
slots
focus face
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Coding with Edge Degrees
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Resulting Symbols

• border edge degrees
• interior edge degrees

. . .
. . .
. . .
. . .
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Average Distributions
no
no
4
3
2
3
4
5
5
6
6
7
2
yes
yes
border degrees
interior degrees
border?
join?
49
Possible Configurations
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Possible Configurations
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Configuration hut
hut
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Configuration step
step
0
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Configuration bridge
bridge
0
0
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hut or roof
roof ? join operation
?
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other join operations

• free vertex already on hull
• free edge already on hull

hut
step
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Adaptive traversalto avoid joinoperations
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Reason for join operations
hull
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Reason for join operations
hull
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Reason for join operations
hull
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Reason for join operations
hull
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Reason for join operations
unprocessed region
hull
processed region
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Reason for join operations
unprocessed region
hull
processed region
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Reason for join operations
unprocessed region
hull
processed region
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Reason for join operations
unprocessed region
hull
processed region
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Reason for join operations
unprocessed region
hull
processed region
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Adaptive Traversal

• Valence-driven connectivity encoding for 3D
  meshes Alliez Desbrun, 01

? avoid creation of cavities


















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Propagating theborder information
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Explaining Example
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Explaining Example
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Explaining Example
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Explaining Example
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Results (Connectivity)
bits per hexahedron (bph)
model
ratio
raw
compressed
hanger ra bump warped hutch c
1
5.3 2.9 2.1 0.2 0.3 0.6
72.0 80.0 88.0 112.0 112.0
136.0
1 14 1 28 1 42 ... 1 621 1
361 1 226
average compression ratio 1 163
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Results (Connectivity)
bits per hexahedron (bph)
model
ratio
raw
compressed
hanger ra bump warped hutch c
1
5.3 2.9 2.1 0.2 0.3 0.6
72.0 80.0 88.0 112.0 112.0
136.0
1 14 1 28 1 42 ... 1 621 1
361 1 226
average compression ratio 1 163
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Compressing Geometry
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Predictive Compression

1. quantize positions with b bits

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Predictive Compression

1. quantize positions with b bits
2. traverse positions

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Predictive Compression

1. quantize positions with b bits
2. traverse positions
3. predict position from neighbors

prediction
(1004, 71, 723)
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Predictive Compression

1. quantize positions with b bits
2. traverse positions
3. predict position from neighbors
4. store corrective vector

corrector
prediction
(4, -3, -5)
(1004, 71, 723)
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Parallelogram Rule
Triangle Mesh Compression, Touma Gotsman, 98




across non-convex triangles
across non-planar triangles
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Position Predictions
init
vertex prediction rule
v0 0 v1 v0 v2 v1 v3 v0 - v1 v2 v4
2v0 v8 (or v0 ) v5 v1 v0 v4 v6
v2 v1 v5 v7 v3 v2 v6
1
2
3
0
hut
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Results (Geometry)
bits per vertex (bpv)
model
ratio
raw
compressed
hanger ra bump warped hutch c
1
23.2 30.8 24.4 10.5 19.9 5.9
48.0 48.0 48.0 48.0 48.0 48.0

1 2.1 1 1.6 1 2.0 ... 1
4.6 1 2.4 1 8.1
average compression ratio 1 3.7
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Demo
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Summary

• degree coding for volume mesh connectivity
• edge degrees
• boundary propagation
• adaptive traversal
• parallelogram prediction for volume mesh geometry
• within predictions

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Current / Future Work

• Mixed Volume Meshes
• hex tet prism pyramid cells
• Universal Connectivity Coder
• face, vertex, and edge degrees
• tri / quad / poly surfaces
• tet / hex / poly volumes
• surface mesh cell of volume mesh
• bit-rate like specialized coder

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Acknowledgements

• data sets
• Alla Sheffer
• Steven Owen
• Scott Mitchell
• Claudio Silva
• financial support
• ARC TéléGéo grant from INRIA

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• Thank You!

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Old Beijing Duck

• Whoever had expressed interest in going to eat
  duck in the ancient-style hutong area
• meet me 10-15 minutes after
• end of PG in front of hotel
• bring
• map
• address card of your hotel
• 100 yuan (smaller bills for subway / bus)

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Configurations roof
roof
zero-slots 0 adjacent faces 1 free
faces 4free edges 4free vertices --
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Configurations tunnel
tunnel
0
0
0
0
zero-slots 2 adjacent faces 4 free
faces 2free edges --free vertices --
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Configuration corner
corner
0
0
0
zero-slots 2 adjacent faces 3 free
faces 3free edges 3free vertices 1
94
Configuration gap
gap
0
0
0
0
0
zero-slots 3 adjacent faces 4 free
faces 2free edges 1free vertices --
95
Configurations pit
pit
0
0
0
0
0
0
0
0
zero-slots 4 adjacent faces 5 free
faces 1free edges --free vertices --
96
Configurations den
den
0
0
0
0
0
0
zero-slots 4 adjacent faces 6 free
faces --free edges --free vertices --