![Page 1: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/1.jpg)
Approximate Level-Crossing Probabilitiesfor Interactive Visualization of Uncertain Isocontours
Kai Pöthkow, Christoph Petz & Hans-Christian Hege
Zuse Institute Berlin
Working with Uncertainty Workshop, VisWeek 2011
![Page 2: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/2.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
2
Previous Work
Pfaffelmoser, Reitinger & Westermann: Visualizing the Positional and Geometrical Variability of Isosurfaces in Uncertain Scalar Fields,EuroVis 2011
Pöthkow, Weber & Hege: Probabilistic Marching Cubes, EuroVis 2011
![Page 3: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/3.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
3
Previous Work
Pfaffelmoser, Reitinger & Westermann (2011)
● First-crossing probabilities along rays
✔ Fast computation using lookup tables
✗ Fields with exponential correlation functions
✗ Depends on viewing-direction
Pöthkow, Weber & Hege (2011)
● Cell-wise level-crossing probabilities
✔ Arbitrary correlation structures
✗ Computationally expensive Monte-Carlo integration
![Page 4: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/4.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
4
Aims
● Fast and accurate approximation of cell-wise level-crossing probabilities
● Quantification of approximation errors
● Application to climate simulation data
![Page 5: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/5.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
5
Input Data
![Page 6: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/6.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
6
![Page 7: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/7.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
7
Level Crossing Probabilities
![Page 8: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/8.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
8
Level Crossing Probabilities
● Assume 'local' interpolation & extremal values at grid points (e.g., linear, bi- or trilinear)
● Define
● Level crossing probabilities (1D)
![Page 9: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/9.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
9
Level Crossing Probabilities
● Alternatively, use the complement
● Non-crossing case: faster for >2 vertices
![Page 10: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/10.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
10
Standardization for Lookup-Table
● Parameter Transformation
● Reformulated level-crossing probability
➔ 3D lookup-table for
![Page 11: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/11.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
11
Level Crossing Probabilities
● n-dimensional case
● High-dimensional integration necessary
![Page 12: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/12.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
12
Example: Triangular Cell
?
![Page 13: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/13.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
13
Approximation
![Page 14: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/14.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
14
Statistically Independent Vertices
● Approximate probability (cell c with n vertices):neglect correlation
● For positive correlations:overestimates crossingprobability
● Fast evaluation using1D lookup-table
![Page 15: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/15.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
15
Maximum Edge Crossing Probability
● Cell-crossing at least 2 edge-crossings
● Edge-crossing probabilities are lower bound forcell level-crossing probability
● Approximate probability (for m edges of a cell c)
● Fast evaluation using3D lookup-table
![Page 16: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/16.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
16
Can we do better?
● Max. edge probability only considers 2D marginal distributions
● Can we utilize the other parameters of the n-dimensional distribution?
● Idea: traverse the cell andpropagate the correlations
![Page 17: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/17.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
17
Linked-Pairs Approximation
● Approximate using
● No higher order conditionalsare considered
● Graphical model
spanning tree
● Evaluation using tables (1D and 2D CDFs)
![Page 18: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/18.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
18
Linked-Pairs Approximation
![Page 19: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/19.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
19
Spanning Trees
...
n trees (Cayley's formula)n-2
![Page 20: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/20.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
20
Linked-Pairs Approximation
● Induced approximate distribution
● Original expected values● Induced covariance matrix
with
![Page 21: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/21.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
21
Optimal Approximate Distribution
● Distribution depends on thespanning tree k
● Bhattacharyya distance to original distribution
● Choose
![Page 22: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/22.jpg)
Input: Find optimal spanning tree
isovalue LinkedPairs Approximation
CDF tables
prep roce ssingreal tim
e
![Page 23: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/23.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
23
Results
![Page 24: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/24.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
24
Climate Simulation Results
Monte Carlo integration (1000 samples)
independent vertices (no correlationconsideres)
absolute differences
![Page 25: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/25.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
25
Maximum Edge Approximation
Monte Carlo integration (1000 samples)
maximum edge approximation absolute differences
![Page 26: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/26.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
26
Linked-Pairs Approximation
Monte Carlo integration (1000 samples)
linked-pairs approximation absolute differences
![Page 27: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/27.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
27
Comparison
Monte-Carlo (ground truth)No correlation
Linked-pairsMax. edge
![Page 28: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/28.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
28
Optimal Approximate Distribution
random spanning tree optimal spanning tree
What is the impact of the optimization step?
![Page 29: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/29.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
29
3D Dataset
Monte Carlo integration
Maximum edge approximation Linked-pairs approximation
No correlation considered
![Page 30: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/30.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
30
Comparison (3D)
Monte-Carlo (ground truth)No correlation
Linked-pairsMax. edge
![Page 31: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/31.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
31
Performance
Time in seconds*
Monte-Carlo integration1000 samples/voxel
23
Max.-Edge method 0.17
Linked-Pairs method 0.11
* single-thread, i7, 2.6 GHz
![Page 32: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/32.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
32
Conclusions● Approximations allow fast evaluation of level-crossing probabilities
using lookup tables
● No Monte-Carlo noise
● Maximum edge method
➔ Simple concept
➔ Lower bound for true probability
● Linked-pairs method
➔ Considers complete random vector
➔ Induces approximate distribution
➔ Optimization w.r.t. the Bhattacharyya distance significantly increases accuracy
● Resulting visual impressions are close to the MC result
Thank you!
![Page 33: Approximate Level-Crossing Probabilities for Interactive ......Pfaffelmoser, Reitinger & Westermann (2011) First-crossing probabilities along rays Fast computation using lookup tables](https://reader035.vdocument.in/reader035/viewer/2022070922/5fba62ac8f697130b23c1300/html5/thumbnails/33.jpg)
Pöthkow, Petz & Hege: Approximate Level-Crossing Probabilities for Interactive Visualization of Uncertain Isocontours
33