Early Split Coding of Triangle Mesh Connectivity

1
Early Split Coding ofTriangle Mesh Connectivity
Martin Isenburg UC Berkeley
Jack Snoeyink UNC Chapel Hill
2
Mesh Compression
Geometry Compression Deering, 95

• Efficient Rendering
• Progressive Transmission
• Compact Storage

main memory

• Surface Meshes
• triangular
• polygonal
• Volume Meshes
• tetrahedral, hexahedral

storage / network
3
Triangle Mesh Connectivity
v -9.7 1.6 -0.2 v -6.6 1.4 3.3 v -1.6 1.4
2.3 v -3.3 -1.4 3.0 v 5.6 1.7 4.5 v -6.6 1.4
6.1 v -3.3 4.4 3.7v -1.6 1.4 2.1 v -5.4
-1.6 6.4 ...f 1 2 3 f 3 2 4 f
2 1 5 f 2 5 4 f 6 4 5 f 6
5 7 f 5 1 8 f 4 8 7 f 7 8
9 f 9 8 10f 10 11 7 ...
Geometry
? 6 log2 (v) bpv
Connectivity
4
Compression Schemes
Cut-Border Machine Gumhold Strasser 98
Triangle Mesh Compression Touma Gotsman 98
Edgebreaker Rossignac 99
Face Fixer Isenburg Snoeyink 00
Valence-Driven Encoding Alliez Desbrun 01
Computer Graphics
Angle Analyzer Lee, Alliez Desbrun 02
Degree Duality Coder Isenburg 02
Near-optimal Coding Khodakovsky et al. 02
Out-of-Core Compression Isenburg Gumhold 03
Delphi Coding Coors and Rossignac 04
FreeLence Kälberer et al. 05
5
Connectivity Graph Traversal
split offset
6
Connectivity Graph Traversal
7
CBM Edgebreaker TG coder

• label-based

Cut-Border Machine Gumhold Strasser 98
Edgebreaker Rossignac 99
Early Split Coding this paper
8
CBM and Edgebreaker
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CBM and Edgebreaker






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CBM and Edgebreaker
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CBM and Edgebreaker
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CBM and Edgebreaker
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offset 11






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CBM and Edgebreaker
C C R C R S L C
offset 11






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CBM and Edgebreaker
C C R C R S L C R
offset 11






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CBM and Edgebreaker
C C R C R S L C R C
offset 11






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CBM and Edgebreaker
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CBM and Edgebreaker
C C R C R S L C R C R C
offset 11






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CBM and Edgebreaker
C C R C R S L C R C R C R
offset 11






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CBM and Edgebreaker
C C R C R S L C R C R C R R
offset 11






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CBM and Edgebreaker
C C R C R S L C R C R C R R R
offset 11






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CBM and Edgebreaker
C C R C R S L C R C R C R R R L
offset 11






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26
CBM and Edgebreaker
C C R C R S L C R C R C R R R L C
offset 11






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CBM and Edgebreaker
C C R C R S L C R C R C R R R L C R
offset 11






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28
CBM and Edgebreaker
C C R C R S L C R C R C R R R L C R R
offset 11




R


5

compression boundary
R



C


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unprocessed region





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29
CBM and Edgebreaker
C C R C R S L C R C R C R R R L C R R R
offset 11



R

R


5

compression boundary
R



C


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unprocessed region





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30
CBM and Edgebreaker
C C R C R S L C R C R C R R R L C R R R S
offset 11



R

R


5
S

compression boundary
R



C


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31
CBM and Edgebreaker
offset 3
C C R C R S L C R C R C R R R L C R R R S R
offset 11



R

R
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5
S

compression boundary
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C


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32
CBM and Edgebreaker
offset 3
C C R C R S L C R C R C R R R L C R R R S R E
offset 11



R

R
R


5
S

compression boundary
R

E


C


C

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unprocessed region





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33
CBM and Edgebreaker
offset 3
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



R

R
R


5
S

compression boundary
R

E

E

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C

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34
CBM and Edgebreaker
offset 3
1
3
4
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
...
1
-1
1
1
1
1
1
1
1
1
1
-1
1
3
3
-1
-1
-1
12
offset 11



R

R
R


5
S

compression boundary
R

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unprocessed region





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35
Subsequences of Labels areself-contained
Encodings
36
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible
37
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible

E


38
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible

E



E


39
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
E



E


40
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
E



E


41
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
E



E


42
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
E


E



43
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R

E
E




44
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
S

E
E



45
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R

R
S

E
E



46
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S

E
E



47
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E




48
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




49
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




L

50
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




L
R


51
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




L
R

R


52
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




L
R

R
R



53
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




L
R
C

R
R



54
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C




L
R
R

C

R
R



55
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C



C

L
R
R

C

R
R



56
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C



C
R

L
R

R

C

R
R



57
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible


R
R


R
S
R

E
E
C


C

C
R

L
R

R

C

R
R



58
Subsequences are self-contained
S L C R C R C R R R L C R R R S R E E


reversible
we did not needsplit offsets !!!


R
R


R
S
R

E
E
C


C

L
C
R

L

R

R

C

R
R



59
TG coder
60
TG coder






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TG coder
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TG coder
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TG coder
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TG coder
7 6 5






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compression boundary











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65
TG coder
7 6 5






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compression boundary











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66
TG coder
7 6 5 S






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compression boundary







S




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67
TG coder
7 6 5 S
offset 18






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compression boundary







S




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68
TG coder
7 6 5 S 7
offset 18






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compression boundary




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S




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TG coder
7 6 5 S 7
offset 18






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compression boundary




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S




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TG coder
7 6 5 S 7
offset 18






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compression boundary




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S




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TG coder
7 6 5 S 7
offset 18






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compression boundary




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S




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TG coder
7 6 5 S 7 6
offset 18






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compression boundary




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unprocessed region

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TG coder
7 6 5 S 7 6
offset 18






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compression boundary




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TG coder
7 6 5 S 7 6 5
offset 18






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compression boundary




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TG coder
7 6 5 S 7 6 5
offset 18






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compression boundary




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TG coder
7 6 5 S 7 6 5
offset 18






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compression boundary




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TG coder
7 6 5 S 7 6 5
offset 18






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compression boundary




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7




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78
TG coder
7 6 5 S 7 6 5
offset 18






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compression boundary




7


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5










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unprocessed region

7




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79
TG coder
7 6 5 S 7 6 5 6
offset 18






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compression boundary




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80
TG coder
7 6 5 S 7 6 5 6
offset 18






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compression boundary




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81
TG coder
7 6 5 S 7 6 5 6
offset 18






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6
compression boundary




7


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unprocessed region

7




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82
TG coder
7 6 5 S 7 6 5 6
offset 18






5

6
compression boundary




7


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unprocessed region

7




processedregion



83
TG coder
7 6 5 S 7 6 5 6
offset 18






5

6
compression boundary




7


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S




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unprocessed region

7




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84
TG coder
7 6 5 S 7 6 5 6
offset 18






5

6
compression boundary




7


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S




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unprocessed region

7




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85
TG coder
7 6 5 S 7 6 5 6 7

offset 18






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compression boundary




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86
Subsequences of Degrees arenot self-contained
Encodings
87
Subsequences are not self-contained
S 7 6 5 6 E




5






S












88
Subsequences are not self-contained
S 7 6 5 6 E




5




S















89
Subsequences are not self-contained
S 7 6 5 6 E




5





S

















90
Subsequences are not self-contained
S 7 6 5 6 E




5





S



















91
Subsequences are not self-contained
S 7 6 5 6 E




5




S























92
Subsequences are not self-contained
S 7 6 5 6 E





5
S




























93
Not unique without split offsets
6 6 5 6 6 5 6
5 7 5 6 6 6 5














S 4 3 E 4 3 E
94
Early Split Coder
95
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

96
The new label W














C

C
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R
C
R

97
The new label W







C







C

C
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98
The new label W







C
C







C

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99
The new label W







C
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C

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R

100
The new label W


wart





C
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wart










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C

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R

101
The new label W


wart





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C

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C
R
wart








W


C





C

C
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R
C
R

102
The new label W







C
W


C





C

C
R
R
C
R

103
The new label W







C
W
R


C





C

C
R
R
C
R

104
The early split operation





















C
R
C
R
R
105
The early split operation














S







C
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C
R
R
106
The early split operation














S
R







C
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C
R
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107
The early split operation












L


S
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C
R
C
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R
108
The early split operation












L
R


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C
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C
R
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109
The early split operation








L




L
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C
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110
The early split operation








L




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C
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111
The early split operation








L




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C
R
C
R
R
112
The early split operation








L




L
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S
R







C
R
C
R
R
113
The early split operation








L




L
R


S
R







C
R
C
R
R
114
The early split operation








L




L
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C
R
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C







C
R
C
R
R
115
The early split operation








L




L
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R
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R
R








S










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C
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C
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116
The early split operation








L




L
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C
R
C
R
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W

S










C







C
R
C
R
R
117
The early split operation








L




L
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C
R
C
R
R







W

W
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C







C
R
C
R
R
118
The early split operation








L




L
R


S
R







C
R
C
R
R







W

W
S








C


C







C
R
C
R
R
119
The early split operation








L




L
R


S
R







C
R
C
R
R







W

W
S








C


R
C







C
R
C
R
R
120
The early split operation








L




L
R


S
R







C
R
C
R
R







W

W
S








C
R


R
C







C
R
C
R
R
121
The early split operation








L




L
R


S
R







C
R
C
R
R







W
W

S


W






C
R


R
C







C
R
C
R
R
122
Complete Example
123
One Scheme Four Encodings
124
Forward Decodingwith offsets
125
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
126
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
127
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
128
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
129
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
130
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
131
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
132
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
133
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
134
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
135
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
136
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
137
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
138
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
139
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
140
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
141
Forward Decoding with offsets
C C C C R C R C R W C R C S1 C R R E C R E
V5 V4 V5 V5 V6 V6 V4 V8 V5 V5 S1,0,0
142
Reverse Decodingwithout offsets
143
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
144
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
145
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
146
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
147
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
148
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
149
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
150
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
151
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
152
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
153
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
154
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
155
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
156
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
157
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
158
Reverse Decoding w/o offsets
E R C E R R C S C R C W R C R C R C C C C
E V3 E V4 S V5 V5 W V8 V4 V6 V6 V5 V5
159
Results
160
Compression Results
161
Improving the TG coder






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Valence-Driven Encoding Alliez Desbrun 01
162
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