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Efficient Visualization of
Triangulations
Efficient Visualization of Triangulations
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Motivation
We want to visualize Huge data sets in 3D.
NASA SRTM Mission: Elevation model of 30m
resolution between 60 and -60 degrees latitude
12Tb of raw data!
ASCI Project: Simulation of nuclear explosions.
Generates terabytes of raw data which need to be
visualized.
These amounts of data are not manageable even with the
Need intelligent methods!
world’s fastest computers
Efficient Visualization of Triangulations
2/28
Goals
For real-time rendering of triangle surfaces we want:
Fast rendering
Few triangles
Efficient data structures
Efficient algorithms
Good-looking meshes
Valid triangulation (ch.2)
Delaunay (ch.4)
Efficient Visualization of Triangulations
3/28
Means
To achieve our goals we will look at the following
Efficient data structures
Indexed triangulations.
Triangle strips/fans.
Data reduction.
Refinement/Decimation
View-dependent Level of Detail
Binary Tree Triangulations
Hierarchical TIN’s
Progressive Meshes
Efficient Visualization of Triangulations
4/28
Notes on 3D Rendering
All vertex data in the triangulation must be transformed
before visualization.
(x1,y1,z1)
(x2,y2,y3)
(x3,y3,z3)
M 00 M01 M02 M03
P00 P01 P02 P03
M 10 M11 M12 M13
P10 P11 P12 P13
M 20 M21 M22 M23
P20 P21 P22 P23
M 30 M31 M32 M33
P30 P31 P32 P33
GL_MODELVIEW_MATRIX
GL_PROJECTION_MATRIX
We want to do as few vertex transforms as possible!
Efficient Visualization of Triangulations
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Efficient Visualization of Triangulations
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(4,3,1)
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(2,6,0)
(0,4,0)
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Triangle rendering in OpenGL
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Indexed triangle structure
0
1
Idx
2
10
9
3
12
11
4
13
8
7
5
6
Efficient Visualization of Triangulations
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Coords
(−2,6,0)
(2,6,0)
(4,3,1)
(6,0,3)
(5,−3,5)
(2,−7,3)
(0,−7,3)
.....
Triangles
[0,10,1]
[1,10,2]
[2,10,12]
[2,12,3]
[3,12,4]
[4,12,13]
[4,13,5]
[13,6,5]
....
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Efficient Visualization of Triangulations
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Indexed rendering in OpenGL
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Efficient Visualization of Triangulations
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Indexed Rendering (cont.)
An even simpler way...
9/28
Shared Edges
Our triangles share
common edges
Efficient Visualization of Triangulations
Ideally: triangles
need only specify
vertices
When moving from one
triangle to its neighbor
we already know two of
the three vertices
10/28
Advantages
We already use an indexed structure with minimum vertex
transforms, so why bother?
Vertex sharing will probably not handle the entire
vertex array.
Index data takes resources too!
Memory usage
Bus traffic
Function calls
Efficient Visualization of Triangulations
11/28
Triangle Strips
Traverse the triangles in a zig-zag pattern:
2
0
4
Triangles
[0,1,2]
[2,1,3]
[2,3,4]
[4,3,5]
...
6
1
3
5
8
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Efficient Visualization of Triangulations
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Triangle Fans
Rotate about a central vertex
3
2
4
1
Triangles
0
[0,1,2]
[0,2,3]
[0,3,4]
[0,4,5]
[0,4,6]
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Efficient Visualization of Triangulations
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Triangle Strip Algorithms
Simple overview of an algorithm
1. Choose a starting triangle.
2. Choose a direction (crossing edge)
3. Extend the strip or fan in the chosen direction
4. Repeat from (2) until you reach a triangle with no forward
connections.
5. Repeat from (1) until all triangles have been visited.
We need a data structure with adjacency information (see ch. 3).
For strips we choose alternating right and left crossing edges.
For fans we always choose the same crossing edge direction.
Efficient Visualization of Triangulations
14/28
Choosing the start triangle
Two heuristics:
1. Choose the least connected triangle (fewest
neighbors).
2. Choose any triangle.
The first strategy seems to produce longer strips.
It has been shown that producing optimal length
strips from a general triangle mesh is an NP-hard
problem.
Efficient Visualization of Triangulations
15/28
Cutting Corners
Some times we may cheat a bit instead of starting a new triangle strip:
Two Strips (10 indices)
[0,1,2,3,4]
[4,3,5,6,7]
0
4
2
Triangles
[0,1,2]
[2,1,3]
[2,3,4]
[4,3,3]
[4,3,5]
[5,3,6]
[5,6,7]
5
1
3
7
Single Strip (9 indices)
[0,1,2,3,4,3,5,6,7]
6
The degenerate triangle will usually be skipped by the OpenGL
driver.
We save one vertex compared to starting a new strip.
Efficient Visualization of Triangulations
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Data Reduction
We want to display as much detail as possible with
as few triangles as possible.
For a surface model certain parts of the surface
usually have more detail than others.
Need to throw away input data that doesn’t contribute
significantly to the shape of the surface.
Efficient Visualization of Triangulations
17/28
Data reduction example
Efficient Visualization of Triangulations
18/28
Error Measure
How do we determine what input data points are
significant?
We need an Error Measure.
One simple error measure is the vertical distance from a
reference surface, for example the surface consisting of
all the input points except the one in question.
be the elevation on the reference surface, and
Let
let be the elevation of the data point in question.
indicates the error we do if we skip a certain data point.
Efficient Visualization of Triangulations
19/28
Error Tolerance
When we have an error measure we can set the
maximum error we can tolerate in our model. We denote
this the error tolerance, , and we include all data points
.
where
m, we need to use all data
I.e if we set tolerance
points that lie further than 2.0m from the reference
surface.
Efficient Visualization of Triangulations
20/28
Reducing triangulations
Two general approaches:
Refinement: Start with a coarse mesh, add details
where appropriate.
Decimation: Start with a detailed mesh. Remove
superfluous data.
a set of input points in
.
- points in the set
Some notation:
- Piecewise linear function over the current
triangulation.
Efficient Visualization of Triangulations
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Refinement Triangulation
Efficient Visualization of Triangulations
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Refinement Algorithm
then
if
do
repeat
for all
end if
end for
then
if
Add vertex to triangulation
end if
until
Efficient Visualization of Triangulations
23/28
Implementation notes
Start with the convex hull or a bounding rectangle of
the input points.
Keep an array of error values. Update only within the
influence area of the inserted point.
May also use max number of vertices as a stop
criterion instead of tolerance.
Efficient Visualization of Triangulations
24/28
Decimation Triangulation
Conceptually “inverse” of refinement.
Start with a full triangulation and remove points with
Efficient Visualization of Triangulations
25/28
Decimation Algorithm
then
if
do
from triangulation
repeat
for all
Remove
end if
Put back in triangulation
end for
then
if
Remove vertex from triangulation
end if
until
Efficient Visualization of Triangulations
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Comparison
Refinement depends on number of vertices in the
reduced mesh.
Decimation depends on number of vertices in the full
mesh.
Decimation is a heavier process.
Decimation generally gives better results (fewer
points at the same tolerance).
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Comparison(2)
(a) Refinement
(b) Decimation
δ>ε
1
δ<ε
2
δ>ε
δ>ε
3
Efficient Visualization of Triangulations
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