Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA...
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![Page 1: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/1.jpg)
CompressingHexahedral Volume Meshes
Martin IsenburgUNC
Chapel Hill
Pierre AlliezINRIA
Sophia-Antipolis
![Page 2: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/2.jpg)
Overview• Volume Meshes• Related Work• Compressing Connectivity
– Coding with Edge Degrees– Boundary Propagation– Adaptive Traversal
• Compressing Geometry– Parallelogram Prediction
• Demo
![Page 3: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/3.jpg)
Take this home:
“The connectivity of a hexahedral mesh can be coded through asequence of its edge degrees.”
“This encoding naturallyexploits the regularity commonly
found in such data sets.”
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Volume Meshes
![Page 5: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/5.jpg)
Volume Meshes
• scientific & industrial applications– thermodynamics
– structural mechanics
– …
• visualization & simulation• unstructured / irregular
( not on a grid )• tetrahedral, hexahedral, polyhedral
![Page 6: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/6.jpg)
Hexahedral Volume Meshes
• have “numerical advantages in finite element computations”
• challenging to generate
• their internal structure looks “nice” compared to tetrahedral meshes
![Page 7: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/7.jpg)
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
vtx1 ( x, y, z )vtx2 ( x, y, z )vtx3 ( x, y, z )
vtxv
hex1 1 3 6 4 7 8 9 2hex2 4 5 8 2 9 1 6 7hex3 7 5 …
hexh
8h * 32 bits
3v * 32 bits
71572 hexahedra78618 vertices
log2(v)
16
size: 3.23 MB
1.69
less than
84 KB
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Related Work
![Page 10: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/10.jpg)
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]
!! only for tetrahedral meshes !!
– 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]
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Surface / Volume Connectivity
a mesh with v vertices has maximalsurfaces:
2v-2 triangles ~ 6v indices
v-1 quadrilaterals ~ 4v indices
connectivity dominates geometry even more for volume meshes
volumes:
O(v2) tetrahedra ~ 12v indices
O(v2) hexahedra ~ 8v indices
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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]
4 5. . . . . .3 64 4 4 4 4 4 4 4
. . . 7 3 . . .644 56 7 6 6 6
compressed with arithmetic coder converges to entropy
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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 Distribution
2
3
4
56 7 8 9+
vertex degrees
3
4
5 6 7 8 9+
face degrees
[over a set of 11 polygonal meshes]
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“Worst-case” Distribution
…
3
4
5
6
78
9
vertex degrees
……
[Alliez & Desbrun, 01]
3.241… bpv
[Tutte, 62]
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Adaptation to Regularity
6
vertexdegrees
facedegrees
3
... ... ... ...
3
vertexdegrees
facedegrees
6
... ... ... ...
4
vertexdegrees
facedegrees
4
... ... ... ...
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Degree Coding for Volumes ?
tri tet quad hex
vertex degrees edge degrees
![Page 19: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/19.jpg)
Regular Volume Meshes?
• elements for regular 2D tiling – regular triangle
– regular quadrilateral
– regular hexagon
• elements for regular 3D tiling– regular tetrahedron
– regular hexahedron
![Page 20: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/20.jpg)
Compressing Connectivity
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Space Growing
similar in spirit to “region growing”:
algorithm maintains hull enclosing processed hexahedra
• pick incomplete face on hull• process adjacent hexahedra• record degrees of its free edges
1. initialize hull with a border face
2. iterate until done
![Page 22: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/22.jpg)
Coding with Edge Degrees
focus face
![Page 23: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/23.jpg)
Coding with Edge Degrees
focus face
![Page 24: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/24.jpg)
Coding with Edge Degrees
focus face
slots
![Page 25: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/25.jpg)
Coding with Edge Degrees
![Page 26: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/26.jpg)
Coding with Edge Degrees
![Page 27: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/27.jpg)
Coding with Edge Degrees
border faces
incompletefaces
![Page 28: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/28.jpg)
Coding with Edge Degrees
![Page 29: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/29.jpg)
Coding with Edge Degrees
![Page 30: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/30.jpg)
Coding with Edge Degrees
![Page 31: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/31.jpg)
Coding with Edge Degrees
![Page 32: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/32.jpg)
Coding with Edge Degrees
![Page 33: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/33.jpg)
Coding with Edge Degrees
edges onhull maintain
slot count
![Page 34: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/34.jpg)
Coding with Edge Degrees
edges witha slot countof zero are“zero slots”
zero slots
![Page 35: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/35.jpg)
Coding with Edge Degrees
![Page 36: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/36.jpg)
Coding with Edge Degrees
![Page 37: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/37.jpg)
Coding with Edge Degrees
![Page 38: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/38.jpg)
Coding with Edge Degrees
![Page 39: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/39.jpg)
Coding with Edge Degrees
![Page 40: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/40.jpg)
Coding with Edge Degrees
![Page 41: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/41.jpg)
Coding with Edge Degrees
![Page 42: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/42.jpg)
Coding with Edge Degrees
![Page 43: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/43.jpg)
Coding with Edge Degrees
![Page 44: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/44.jpg)
Coding with Edge Degrees
![Page 45: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/45.jpg)
Coding with Edge Degrees
![Page 46: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/46.jpg)
Coding with Edge Degrees
![Page 47: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/47.jpg)
Resulting Symbols
– border edge degrees
– interior edge degrees
. . . . . .
. . . . . .
3 33 3 32 42
4 5 4 4 4 4 3 4
. . . . . .YN N N N N N N N N N N N N
– border ?
. . . . . .YN N N N N N NN N N N N N
– join ?
![Page 48: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/48.jpg)
Average Distributions
2
3
4 5 6+
border degrees
3
4
5 6 7+
interior degrees border?
2
join?
no
yes
no
yes
![Page 49: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/49.jpg)
Possible Configurations
![Page 50: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/50.jpg)
Possible Configurations
hut step bridge
roof
pit den
tunnel
gap
corner
![Page 51: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/51.jpg)
Configuration: “hut”
“hut”
hut
zero-slots: 0
free faces 5free edges 8free vertices 4
not azero slot
freeedge
free vertex
freeface
![Page 52: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/52.jpg)
Configuration: “step”
“step”
0
this is azero slot
step
zero-slots: 1
free faces 4free edges 5free vertices 2
![Page 53: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/53.jpg)
Configuration: “bridge”
“bridge”
00 bridge
zero-slots: 2
free faces 3free edges 2free vertices --
![Page 54: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/54.jpg)
“hut” or “roof”
for the free face of “hut” that potentially forms a roof: “join?” /Y N
hut
roof
?
“roof” “join” operation
![Page 55: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/55.jpg)
other “join” operations
• free vertex already on hull
• free edge already on hull
hut step
for every free vertex / edge: “join?” /Y N
![Page 56: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/56.jpg)
Adaptive traversalto avoid “join”operations
![Page 57: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/57.jpg)
Reason for “join” operations
hull
![Page 58: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/58.jpg)
Reason for “join” operations
hull
![Page 59: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/59.jpg)
Reason for “join” operations
hull
![Page 60: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/60.jpg)
Reason for “join” operations
hull
![Page 61: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/61.jpg)
Reason for “join” operations
processed region
unprocessed region
hull
![Page 62: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/62.jpg)
Reason for “join” operations
processed region
unprocessed region
hull
![Page 63: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/63.jpg)
Reason for “join” operations
processed region
unprocessed region
hull
![Page 64: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/64.jpg)
Reason for “join” operations
processed region
unprocessed region
hull
![Page 65: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/65.jpg)
Reason for “join” operations
processed region
unprocessed region
join
hull
![Page 66: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/66.jpg)
Adaptive Traversal
• Valence-driven connectivity encoding for 3D meshes [Alliez & Desbrun, 01]
avoid creation of cavities
focusface
pick face with largest # of zero slots
![Page 67: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/67.jpg)
Propagating theborder information
![Page 68: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/68.jpg)
Explaining Exampleborder faceborder face &
slot count = 1
![Page 69: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/69.jpg)
Explaining Exampleborder face
borderedges
![Page 70: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/70.jpg)
Explaining Example
interior edgesincomplete
faces
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Explaining Example
for everything else: “border?” /Y N
remainingelements
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hangerra
bump…
warpedhutch
c1
5.32.92.1
…0.20.30.6
Results (Connectivity)
raw
bits per hexahedron (bph)model compressed
72.0 80.0 88.0
… 112.0 112.0 136.0
1 : 141 : 281 : 42 ...1 : 6211 : 3611 : 226
ratio
average compression ratio = 1 : 163
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hangerra
bump…
warpedhutch
c1
5.32.92.1
…0.20.30.6
Results (Connectivity)
raw
bits per hexahedron (bph)model compressed
72.0 80.0 88.0
… 112.0 112.0 136.0
1 : 141 : 281 : 42 ...1 : 6211 : 3611 : 226
ratio
average compression ratio = 1 : 163
![Page 74: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/74.jpg)
Compressing Geometry
![Page 75: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/75.jpg)
Predictive Compression
1. quantize positions with b bits
(1.2045, -0.2045, 0.7045) (1008, 67, 718)floating point integer
![Page 76: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/76.jpg)
Predictive Compression
1. quantize positions with b bits
2. traverse positions
(1.2045, -0.2045, 0.7045) (1008, 67, 718)floating point integer
![Page 77: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/77.jpg)
Predictive Compression
1. quantize positions with b bits
2. traverse positions
3. predict position from neighbors
(1.2045, -0.2045, 0.7045) (1008, 67, 718)floating point integer
(1004, 71, 723)prediction
![Page 78: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/78.jpg)
Predictive Compression
1. quantize positions with b bits
2. traverse positions
3. predict position from neighbors
4. store corrective vector
(1.2045, -0.2045, 0.7045) (1008, 67, 718)floating point integer
(4, -3, -5)corrector
(1004, 71, 723)prediction
![Page 79: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/79.jpg)
Parallelogram Rule
across“non-convex”
triangles
across“non-planar”
triangles
within“planar” & “convex”
quadrilateral
Triangle Mesh Compression, [Touma & Gotsman, 98]
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8
Position Predictions
init
hut
0
4
12
3
56
7
0
12
3
v0 0v1 v0
v2 v1
v3 v0 - v1 + v2
v4 2v0 – v8 (or v0 )v5 v1 – v0 + v4
v6 v2 – v1 + v5
v7 v3 – v2 + v6
vertex prediction rule
![Page 81: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/81.jpg)
hangerra
bump…
warpedhutch
c1
23.230.824.4
…10.519.9
5.9
Results (Geometry)
raw
bits per vertex (bpv)model compressed
48.0 48.0 48.0
… 48.0 48.0 48.0
1 : 2.11 : 1.61 : 2.0
...1 : 4.61 : 2.41 : 8.1
ratio
average compression ratio = 1 : 3.7
![Page 82: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/82.jpg)
Demo
![Page 83: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/83.jpg)
Summary
• degree coding for volume mesh connectivity– edge degrees
– boundary propagation
– adaptive traversal
• parallelogram prediction for volume mesh geometry– “within” predictions
![Page 84: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/84.jpg)
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
![Page 85: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/85.jpg)
Acknowledgements
data sets• Alla Sheffer
• Steven Owen
• Scott Mitchell
• Claudio Silva
financial support
• ARC TéléGéo grant from INRIA
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Thank You!
![Page 87: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/87.jpg)
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 hotelbring :
• map• address card of your hotel• ~100 yuan (smaller bills for subway / bus)
![Page 88: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/88.jpg)
![Page 89: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/89.jpg)
![Page 90: Compressing Hexahedral Volume Meshes Martin Isenburg UNC Chapel Hill Pierre Alliez INRIA Sophia-Antipolis.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d3e5503460f94a169eb/html5/thumbnails/90.jpg)
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Configurations: “roof”
“roof”
zero-slots: 0adjacent faces: 1
free faces 4free edges 4free vertices --
roof
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Configurations: “tunnel”
tunnel
“tunnel”
zero-slots: 2adjacent faces: 4
free faces 2free edges --free vertices --
00 00
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corner
Configuration: “corner”
“corner”
zero-slots: 2adjacent faces: 3
free faces 3free edges 3free vertices 1
0
00
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gap
Configuration: “gap”
“gap”
zero-slots: 3adjacent faces: 4
free faces 2free edges 1free vertices --
0
0
0
00
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Configurations: “pit”
0
0
0 0 pit
zero-slots: 4adjacent faces: 5
free faces 1free edges --free vertices --
“pit”
00
0 0
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den
Configurations: “den”
zero-slots: 4adjacent faces: 6
free faces --free edges --free vertices --
“den”
0
0
0 00
0