Transcript Compressing Polygon Mesh Geometry with Parallelogram Prediction Martin Isenburg Pierre Alliez UNC Chapel Hill INRIA Sophia-Antipolis Take this home: “Non-triangular faces in the mesh can be exploited for more efficient predictive compression of vertex positions.” “Non-triangular.

Compressing
Polygon Mesh Geometry
with
Parallelogram
Prediction
Martin Isenburg
Pierre Alliez
UNC
Chapel Hill
INRIA
Sophia-Antipolis
Take this home:
“Non-triangular faces in
the mesh can be exploited for
more efficient
predictive compression
of vertex positions.”
“Non-triangular faces tend to
be planar and convex.”
Overview
•
•
•
•
•
•
•
Background
Previous Work
Linear Prediction Schemes
“within” versus “across”
Example Run
Can we do better ?
Conclusion
Background
Polygon Meshes
• connectivity
log2(v)
face1 1 2 3 4
face2 3 4 3
~ 4v 32 bits face3 5 2 1 3
facef
• geometry
2832 vertices
vertex1 ( x, y, z )
vertex2 ( x, y, z )
3v * 32 bits vertex3 ( x, y, z )
12
vertexv
5647
29736
 size: 79296 bytes
Mesh Compression
• Geometry Compression [[Deering,
Deering, 95]
– Fast Rendering
– Progressive Transmission
– Maximum
MaximumCompression
Compression
Mesh Compression
• Geometry Compression [[Deering,
Deering, 95]
– Fast Rendering
– Progressive Transmission
– Maximum
MaximumCompression
Compression
• Connectivity
• Geometry
Geometry
Mesh Compression
• Geometry Compression [[Deering,
Deering, 95]
– Fast Rendering
– Progressive Transmission
– Maximum
MaximumCompression
Compression
• Connectivity
• Geometry
Geometry
– Triangle Meshes
– Polygon
PolygonMeshes
Meshes
Triangle Mesh Compression
Triangle Mesh Compression
[Touma & Gotsman, Graphics Interface 98]
• Connectivity Coder
– stores the connectivity as sequence
of vertex degrees
• Geometry Coder
– stores the geometry as sequence of
vectors; each corrects the prediction
of a vertex position
Not Triangles … Polygons!
Face Fixer
[Isenburg & Snoeyink, 00]
Generalization of TG coder
• Connectivity Coder
Compressing Polygon Connectivity with
Degree Duality Prediction, [Isenburg, 02]
Near-optimal connectivity coding of
Polygon meshes,
[Khodakovsky, Alliez,
Desbrun & Schroeder, 02]
• Geometry Coder
 this paper
[Isenburg & Alliez, 02]
Results
bits per vertex
model
TG
IA
gain
triceratops
galleon
cessna
…
tommygun
cow
teapot
20.0
24.1
19.1
…
19.6
20.4
21.0
14.8
18.4
12.5
…
12.5
20.6
16.1
26 %
26 %
35 %
...
36 %
-1 %
24 %
min / max / average [%] = 9 / 41 / 23
Previous Work
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
Geometry Compression
Java3D
[Deering, 95]
Geometric Compression through topological
surgery
[Taubin & Rossignac, 98]
MPEG - 4
Triangle Mesh Compression
Virtue3D
[Touma & Gotsman, 98]
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
re-meshing
space-dividing
vector-quantization
feature discovery
angle-based
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
Spectral Compression
of Mesh Geometry
re-meshing
[Karni & Gotsman, 00]
space-dividing
vector-quantization
feature discovery
expensive numerical
computations
angle-based
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
Progressive Geometry
Compression
re-meshing
[Khodakovsky et al., 00]
space-dividing
vector-quantization
feature discovery
modifies mesh prior
to compression
angle-based
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
Geometric Compression
for interactive transmission
re-meshing
[Devillers & Gandoin, 00]
space-dividing
vector-quantization
feature discovery
poly-soups; complex
geometric algorithms
angle-based
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
Vertex data compression
for triangle meshes
re-meshing
[Lee & Ko, 00]
space-dividing
vector-quantization
feature discovery
local coord-system +
vector-quantization
angle-based
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
Compression of engineering
models by repeated feature
re-meshing
discovery [Shikhare et al., 01]
space-dividing
vector-quantization
feature discovery
certain 3D models +
expensive matching
angle-based
Previous Work
• Classic approaches [95 – 98]:
– linear prediction
• Recent approaches [00 – 02]:
–
–
–
–
–
–
spectral
Angle-Analyzer: A trianglequad mesh codec
re-meshing
[Lee, Alliez & Desbrun, 02]
space-dividing
vector-quantization
feature discovery
dihedral + internal =
heavy trigonometry
angle-based
Linear Prediction Schemes
Linear Prediction Schemes
1.
2.
3.
4.
quantize positions with b bits
traverse positions
linear prediction from neighbors
store corrective vector
floating point
(1.2045, -0.2045, 0.7045)
integer
(1008, 68, 718)
Linear Prediction Schemes
1.
2.
3.
4.
quantize positions with b bits
traverse positions
linear prediction from neighbors
store corrective vector
use traversal order implied by
the connectivity coder
Linear Prediction Schemes
1.
2.
3.
4.
quantize positions with b bits
traverse positions
linear prediction from neighbors
store corrective vector
apply prediction rule
prediction
(1004, 71, 723)
Linear
Predictioncorrector
Schemes
position distribution
distribution
3500
70
60
50
40
30
20
10
0
1.
2.
3.
4.
quantize positions with b bits
traverse positions
linear prediction from neighbors
store corrective vector
position
(1008, 68, 718)
3000
2500
2000
1500
1000
500
0
prediction
(1004, 71, 723)
corrector
(4, -3, -5)
Deering, 95
Prediction: Delta-Coding
P=A
P
A
processed region
unprocessed region
Taubin & Rossignac, 98
Prediction: Spanning Tree
P = αA + βB + γC + δD + εE + …
B
A
P
processed region
unprocessed region
C
D
E
Touma & Gotsman, 98
Prediction: Parallelogram Rule
P=A–B+C
A
P
B
C
processed region
unprocessed region
Parallelogram Rule
“good”
prediction
“bad”
prediction
“bad”
prediction
“non-convex”
“non-planar”
More “good” Predictions
Multi-way geometry encoding.
[Cohen-Or, Cohen & Irony, 02]
• average multiple predictions
average gain of 11 %
Optimized compression of triangle mesh geometry
using prediction trees.
[Kronrod & Gotsman, 02]
• search for best prediction
• direct the traversal (prediction tree)
average gain of 8 % (smooth) & 42 % (CAD)
Polygon Meshes … ?
TG coder
• triangulate
• compress resulting triangle mesh
IA coder
• do NOT triangulate
• use polygons for better predictions
• within versus across
“within” versus “across”
Non-triangular Faces
Question:
Why would a mesh have
a non-triangular face?
Non-triangular Faces
Question:
Answer:
Why would a mesh have
a non-triangular face?
Because there was no
reason to triangulate it!
This face was “convex”
and “planar”.
 use this info for “good” predictions
“within” versus “across”
across-prediction
within-prediction
 within-predictions avoid creases
 within-predictions often find existing
parallelograms ( quadrilaterals)
Bitrates: “within” vs. “across”
model
triceratops
galleon
cessna
…
tommygun
cow
teapot
bits per vertex
within
across
20.5
14.1
26.8
16.9
19.8
11.1
…
…
19.5
10.9
20.6
22.7
14.9
diff
31 %
37 %
45 %
...
44 %
34 %
min / max / average [%] = 13 / 47 / 32
Maximizing the number of
within-predictions
Greedy Strategy
always try to:
(A) pick a vertex whose position
can be within-predicted
otherwise:
(B) do an across-prediction, but
pick a vertex that creates (A)
for the next iteration
Traversal Order
we’re lucky:
 process vertices in order dictated
by our connectivity coder
Compressing Polygon Connectivity with
Degree Duality Prediction
[Isenburg, 02]
 avoid “splits” by adaptive traversal
Valence-driven Connectivity Encoding
for 3D meshes
[Alliez & Desbrun, 01]
“splits”
“splits”
“splits”
“splits”
“splits”
processed region
“splits”
processed region
“splits”
processed region
“splits”
processed region
“splits”
processed region
“splits”
split
processed region
Adaptive Traversal
Adaptive Traversal
Adaptive Traversal
% of within-predictions
model
triceratops
galleon
cessna
…
tommygun
cow
teapot
prediction type:
within across last center
2557
2007
3091
257
324
621
2
24
22
1
12
11
…
…
…
…
3376
0
1016
678
2701
170
78
2
2
39
1
1
% of
within
90 %
85 %
83 %
…
81 %
0%
85 %
min / max / average [%] = 74 / 91 / 84
Example Decoding Run
Example Decoding Run
center
center-prediction:
• no parallelogram rule possible
• predict this position as center of
the bounding box
Example Decoding Run
last-prediction:
• no parallelogram rule possible
• predict this position as the last
position
0
last
Example Decoding Run
last-prediction:
• no parallelogram rule possible
• predict this position as the last
position
0
1
last
Example Decoding Run
across-prediction:
• parallelogram rule possible
• predict across two polygons
across
0
2
1
Example Decoding Run
within-prediction:
• use parallelogram rule
• predict within a polygon
within
3
0
2
1
Example Decoding Run
4
3
within
0
2
1
Example Decoding Run
across
4
5
3
0
2
1
Example Decoding Run
6
within
4
5
3
0
2
1
Example Decoding Run
6
within
7
4
5
3
0
2
1
Example Decoding Run
6
7
4
7
5
3
within
0
2
1
Can we do better ?
(and keep it simple)
Keep it simple
Constraints:
• single linear prediction
• use connectivity traversal order
Possibilities:
• floating point coefficients
• use more than three vertices
better within-predictions
assuming ideal ( regular) polygons
deg = 5
deg = 4
deg = 8
deg = 6
A
A
A
A
B
C
B
C
perfect
P=A–B+C
P = αA + βB + γC
B
C
B
C
systematic error
α = 1, β = -1, γ = 1
Switch “within” Coefficients
if (deg == 4) {
α = 1.000; β = -1.000; γ = 1.000;
} else if (deg == 5) {
α = 1.024; β = -0.527; γ = 0.503;
} else if (deg == 6) {
α = 1.066; β = -0.315; γ = 0.249;
} else …
How did we pick these numbers?
Finding the Coefficients
• no obvious “scientific” way
• use Matlab 
– for each degree separately
– sum all possible prediction errors as
function of α and β
N
Err = len (N – αA + βB + γC)
with 1 = α + β + γ
A
B
C
– optimize α and β for minimal error
Switch “within” Coefficients
if (deg == 4) {
α = 1.000; β = -1.000; γ = 1.000;
} else if (deg == 5) {
α = 1.024; β = -0.527; γ = 0.503;
} else if (deg == 6) {
α = 1.066; β = -0.315; γ = 0.249;
} else …
parallelogram prediction within
a quadrilateral is optimal
on average 70 % of predictions
better across-predictions
assuming equal edge length
adjacent
polygons
are
coplanar
33 34
35
36
38
43
45
46
48
44
better across-predictions
assuming equal edge length
adjacent
polygons
are not
coplanar
33 34
35
36
38
43
45
46
48
crease
angle:
60°
44
Switch “across” Coefficients
if (deg == 33) {
α = 0.917;
β = - 0.833;
} else if (deg == 34) {
α = 0.621;
β = - 0.504;
} else if (deg == 35) {
α = 0.557;
β = - 0.334;
} else …
…
γ = 0.916;
γ = 0.883;
γ = 0.777;
} else if (deg == 43) {
α = 1.153;
β = - 0.354;
} else if (deg == 44) {
α = 1.001;
β = - 0.648;
} else …
…
γ = 0.201;
γ = 0.647;
Conclusion
Summary (1)
• polygon information can improve
predictive geometry compression
• using parallelogram rule “within”
rather than “across” polygons
• average improvement of 23 %
over TG coder
• has simple and straight-forward
implementation
Summary (2)
• proof-of-concept
implementation
in form of a Web
java-applet
• compression software will soon
be made available
http://www.cs.unc.edu/~isenburg/pmc/
Future Work
• a scientific way to find numbers
for this “coefficient switching”
• polygonification:
– turn triangle meshes into polygon
meshes for better compression
Acknowledgments
funding:
• ARC Télégéo grant of INRIA at
Sophia-Antipolis
logistics:
•
•
•
•
Jack Snoeyink
Olivier Devillers
Agnès Clément Bessière
Jean-Daniel Boissonnat
Thank You!
http://www.cs.unc.edu/~isenburg/pmc/