1. Vector Field Visualization
* T. Moeller, H. Shen 의 slide를 이용함.

2. Vector Field Visualization
A vector field: F(U) = V
U: field domain (x,y) in 2D (x,y,z) in 3D
V: vector (u,v) or (u,v,w)
- Like scalar fields, vectors are defined at discrete points

3. Flow Visualization Flow visualization – classification
– Dimension (2D or 3D)
– Time-dependency: steady vs. unsteady
– Grid type

4. Examples

7. Vector Field Visualization Challenges
General Goal
Display the field’s directional information
Domain Specific
Detect certain features
e.g. vortex cores

8. Vector Field Visualization Challenges
A good vector field visualization is difficult to get:
Displaying a vector requires more visual attributes
(u,v,w): direction and magnitude
Displaying a vector requires more screen space
more than one pixel is required to display an arrow
=> It becomes more challenging to display a dense vector field

9. Vector Field Visualization Techniques
Local technique
Particle Advection
Display the trajectory of a particle starting from a particular location
Global technique
Hedgehogs, Line Integral Convolution etc
Display the flow direction everywhere in the field

10. Characteristic of Lines
Streamline
tangential to the vector field
Pathline
trajectories of massless particles in the flow
Streakline
trace of dye that is released into the flow at a fixed position
Time line (Time surface)
propagation of a line (surface) of massless elements in time
Streamline, pathline and streakline are different
only when the flow is unsteady (time dependent)

11. Streaklines
Trace of dye that is released into the flow at a fixed position
Connect all particles that passed through a certain position

12. Time lines (Time surfaces)
Propagation of a line (surface) of massless elements in time
Idea: “consists” of many point-like particles that are traced
Connect particles that were released simultaneously t0 t1 t2
Timelines or time surfaces can show the evolution of a flow.

13. Example : streamline, pathline, streakline
The red particle moves in a flowing fluid;
Its pathline is traced in red; the tip of the trail of blue
ink released from the origin follows the particle, but
unlike the static pathline (which records the earlier
motion of the dot), ink released after the red dot
departs continues to move up with the flow.
(This is a streakline.) The dashed lines represent
contours of the velocity field (streamlines), showing
the motion of the whole field at the same time.

14. Comparison of Lines

15. Pathline

16. Stream ball
Use radii to visualize additional properties (e.g. divergence and acceleration in a flow)

17. streaklines

18. Stream ribbons
The area swept out by a deformable line segment along a streamline
Visualizes rotational behavior of a 3D flow

19. streamtubes
Thick tube shaped streamline whose radial extent shows the expansion of flow

20. Local technique - Particle Tracing
Visualizing the flow directions by releasing particles and
calculating a series of particle positions based on the vector
field
The motion of particle: dx/dt = v(x)
x: particle position (x,y,z)
v(x): the vector (velocity) field
Use numerical integration to compute a new particle position
x(t+dt) = x(t) + Integration( v(x(t)) dt )

21. Numerical Integration
First Order Euler method:
x(t+dt) = x(t) + v(x(t)) * dt
- Not very accurate, but fast
- Other higher order methods are available: Runge-Kutta
second and fourth order integration methods (more
popular due to their accuracy)
Result of first order
Euler method

22. Numerical Integration (2)
Second Runge-Kutta Method
x(t+dt) = x(t) + ½ * (K1 + K2)
k1 = dt * v(x(t))
k2 = dt * v(x(t)+k1)
½ * [v(x(t))+v(x(t)+dt*v(x(t))]
x(t+dt)
x(t)

23. Numerical Integration (3)
Standard Method: Runge-Kutta fourth order
x(t+dt) = x(t) + 1/6 (k1 + 2k2 + 2k3 + k4)
k1 = dt * v(t); k2 = dt * v(x(t) + k1/2)
k3 = dt * v(x(t) + k2/2); k4 = dt * v(x(t) + k3)

24. Streamline
Displaying streamlines is a local technique because you can only visualize the flow directions initiated from one or a few particles
When the number of streamlines is increased, the scene becomes cluttered
You need to know where to drop the particle seeds
Streamline computation is expensive

25. Arrows and Glyphs Visualize local features of the vector field:
Vector itself
Extern data: temperature, pressure, etc.
Important elements of a vector:
Direction
Magnitude
Not: components of a vector
Approaches:
Arrow plots
Glyphs

26. Arrows Arrows visualize Direction of vector filed Magnitude
Length of arrows
Color coding

27. Glyphs
Can visualize more features of the vector field

28. Flow Volume Construction of the flow volume
Seed polygon (square) is used as smoke generator
Constrained such that center is perpendicular to flow
Square can be subdivided into a finer mesh
Volume rendering
Opacity is inversely proportional to the tetrahedra’s volume

29. Global Techniques Display the entire flow field in a single picture
Minimum user intervention
Example: Hedgehogs (global arrow plots)

30. LIC : Line Integral Convolution
Given :
Vector field and texture image
Texture image is normally white noise
Output
Colored field correlated in the flow direction
Convolve a random texture
along the streamlines

31. LIC Visualize dense flow fields by imaging its integral curves
Cover domain with a random texture (so called ‚input texture‘, usually stationary white noise)
Blur (convolve) the input texture along the path lines using a specified filter kernel

32. LIC : Idea Global visualization technique Dense representation
Start with random texture
Smear out along stream lines

33. example

34. LIC Algorithm

35. LIC

36. OLIC : Oriented Line Integral Convolution
Visualizes orientation (in addition to direction)
Use
Sparse texture
Anisotropic convolution kernel

37. OLIC

38. 세부적인 이슈 소개
Illumination
Seed Placement

39. LIC on photograph

40. Illuminated Streamline

41. Comparisons
Fully opaque
Use transparency
With Illumination

42. Seed Placement for Streamline
The placement of seeds directly determines the visualization quality
Too many: scene cluttering
Too little: no pattern formed
It has to be the right number at the right places!!!

43. Seed Placement

44. Image-Guided Streamline Placement
Main idea: the distribution of ink on the screen should be even [turk 96]

45. Use Energy Function as a Metric
An energy function is defined to measure whether ‘ink’ is distributed evenly
I(x,y): image with streamlines
L: A low pass filter (Gaussian or Hermit interpolation)
t: A user specified threshold for average image intensity
S S ( L*I (x,y) - t ) x y 2

46. Iterative Algorithm
Place an initial set of seeds and compute streamlines
Iteratively minimize the energy function (random decent) with the following operations on the seeds:
Move
Insert
Delete
Lengthen
Shorten
Combine
Iterate until the energy function converges

47. Results
Seeds at regular grid
After energy minimization