1. Farthest Point Seeding for Efficient Placement of Streamlines
Abdelkrim Mebarki
Pierre Alliez
Olivier Devillers
I.N.R.I.A. Sophia-Antipolis
GEOMETRICA

2. Outline
Introduction
State of the art
Algorithm
Results
Conclusion

3. Introduction

4. Modeling physical phenomena Visualization of vector fields.
Context
Modeling physical phenomena
Vector / direction fields.
Visualization of vector fields.

5. Scientific Visualization
Visualization of vector fields
Direct Visualization

6. Scientific Visualization
Visualization of vector fields
Texture-Based Vis.

7. Scientific Visualization
Visualization of vector fields
Feature-Based Vis.

8. Scientific Visualization
Visualization of vector fields
Geometric Visualization
Streamlines

9. Tangent to the field everywhere. integrated form a seed point.
Streamlines
Tangent to the field everywhere.
integrated form a seed point.
Successive integration => polyline.
Stop integrating when the new point:
reaches the boundaries.
encounters a critical point.
becomes too close to another streamline.

10. Placement of streamlines Critical stage:
Motivation
Placement of streamlines
Everywhere tangential to the field.
Critical stage:
Where to start integrating?
Properties sought after:
Quality.
Efficiency.
Robustness.
Few parameters.
Deterministic.

11. Qualifying a Placement
Validity:
Conformity
tangential to the field.
Domain saturation
according to a desired line density.
Quality:
Long streamlines
Less discontinuities in laminar areas.
Controllable density
Depicts an additional flow property (e.g. velocity).

12. State of the art

13. Image Guided Streamlines Placement Turk & Banks ’96
Create a set of streamlets
Iterate :
merge, delete, create, lengthen and shorten the streamlets.
until the desired placement is obtained.
Criterion :
Compare a filtered version of the image with an averaged gray level image (uniform density = constant filtered image).
+ Generate long streamlines.
- Need user intervention to stop the process.
- Slow convergence.

14. Creating Evenly Spaced Streamlines Jobard & Lefer ‘97
Fix the ideal separating distance between two adjacent streamlines.
Choose seed points at a fixed distance from the existing streamlines.
Seeding distance > Separating distance.
+ Single pass algorithm.
- Short streamlines.
- Some spaces left empty.

15. Creating Evenly Spaced Streamlines Jobard & Lefer ‘97

16. Creating Evenly Spaced Streamlines Jobard & Lefer ‘97

17. Creating Evenly Spaced Streamlines Jobard & Lefer ‘97

18. Flow Guided Streamline seeding strategy Verma, Kao & Pang ‘00
Emphasis on flow topology.
Seeding points near singularities.
+ Good interpretation of the topology.
- Seeding in turbulent areas?

19. Data Structures A typical placement: Used for: Commonly used:
28.6K streamlines
Total: 8.2M points
Used for:
Distance requests
Localization
Commonly used:
Regular grid
Boolean mask

20. Algorithm

21. Fixed seeding distance from existing streamlines [Jobard-Lefer 98]
Core Idea
Fixed seeding distance from existing streamlines [Jobard-Lefer 98]
Local criterion
Stopped by nearby lines.

22. Looking for long streamlines?
Core Idea
Looking for long streamlines?
Start integrating the farthest away from all already placed streamlines.
Choose seed points in the biggest cavities.

23. Start integrating the farthest away from already placed streamlines.
Core Idea
Start integrating the farthest away from already placed streamlines.
Criterion already known as
“Delaunay Refinement”
for meshing [Ruppert ‘95]
“Farthest Point”
for sampling [Lindenbaum et al. ‘96]
“Void & Cluster”
for halftonning [Ulichney ‘93]

24. Choice of seed points at centers of biggest cavities.
Data structure
Choice of seed points at centers of biggest cavities.
How to model cavities ?
Where is the biggest one ?
Empty circle property of the Delaunay Triangulation
All sample points are inserted to the triangulation.
The cavities are circumscribed circles of triangles.
Priority Queue :
Contains all triangles which circumcircle diameter is bigger than the seeding distance.

25. Input & Parameters Input Parameters Vectors
Discrete values (e.g. defined on a regular grid).
Vector Field
Dense representation obtained by interpolation (e.g. bilinear interpolation).
Parameters
Density
The ideal separating distance between streamlines.
Saturation ratio
The saturation criterion.

26. Integrate first streamline
The Algorithm
Initialization
Insert points in the triangulation along a square bounding the domain (guaranteed domain saturation).
Integrate first streamline
from the center of the domain.
While (queue not empty)
Extract valid triangle;
Integrate streamline from circumscribed circle center;
insert streamline to the flow;
Insert newly created triangles to the queue;

27. The Algorithm

28. The Algorithm

29. The Algorithm

30. The Algorithm

31. The Algorithm

32. The Algorithm

33. The Algorithm

34. The Algorithm

35. The Algorithm

36. The Algorithm

37. Overall 2000 lines of code using standard components:
Implementation
Overall 2000 lines of code using standard components:
1600 for the core part of the algorithm
STL Priority queue:
2D Delaunay triangulation from CGAL

38. Distance between P and its neighbors in the triangulation
Distance Requests
Efficient measure of the distance between a point P and all other streamlines?
Distance between P and its neighbors in the triangulation
except for points in the same streamline.
consider all points in spiral streamlines.

39. Distance requests
Over all triangles incident to P.
Find the maximum circumscribed circle diameter.

40. Integrate a streamline. For all sample points
Priority queue
Integrate a streamline.
For all sample points
Find triangles which the circumscribed circle diameter is bigger than the seeding distance.
Insert it in the priority queue.
All possible seed points = Too much elements
Much elements will be invalid.
Optimization
Follow the evolution of the circumscribed circle radii on each side of the streamline.
keep only triangles corresponding to local maxima.
Less invalid points

41. Optimization: Priority Queue

42. Sub-Sampling of Streamlines
Insert points in the triangulation
Localization and proximity requests.
Modeling empty cavities.
Small integration step
Too much points are inserted in the triangulation.
More computations and thus less efficiency.
Optimization
Insert a subset of sample points in the triangulation.
insert one point each n integrated points

43. Controllable Density
Density field as input

44. Multi Resolution Placement
First level
Apply basic algorithm;
Next levels
Increase density;
Lengthen existing streamlines;
Add new streamlines;

45. Results

46. Results
density: 4.8%
2.4%
1.2%

47. Results
density: 4.8%
2.4%
1.2%

48. Results Wind in Japan 522 non-uniform sensor locations
data courtesy Weathernews Inc., Japan.

49. Farthest Point Seeding (Separating distance : 3.36% of image width)
Results
Turk & Banks
Jobard & Lefer
Farthest Point Seeding
(Separating distance : 3.36% of image width)

50. Farthest Point Seeding (Separating distance :1.68% of image width)
Results
Turk & Banks
Jobard & Lefer
Farthest Point Seeding
(Separating distance :1.68% of image width)

51. Farthest Point Seeding (Separating distance : 3.36% of image width)
Results
Turk & Banks
Jobard & Lefer
Farthest Point Seeding
(Separating distance : 3.36% of image width)

52. Timings measured on a 2GHz Pentium IV.
Separating distance
Our algorithm
Jobard & Lefer
(Our implementation)
Turnk & Banks
0.84%
240 ms
790 ms
47 s
1.68%
140 ms
290 ms
36 s
3.36%
80 ms
130 ms
20 s
Timings measured on a 2GHz Pentium IV.

53. Conclusion

54. Placement of streamlines
Conclusion
Placement of streamlines
Farthest point paradigm: favors long streamlines.
Comparable results with [Turk-Banks 96].
200 times faster.
Features
Quality.
Efficiency.
Extends to multiresolution.
Download the demo!
Submitted as CGAL Package

55. Thank you.