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Vector Field Visualization

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. Vector Field Viz Applications. Computational Fluid Dynamics. Weather modeling.

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Vector Field Visualization

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  1. 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

  2. Vector Field Viz Applications Computational Fluid Dynamics Weather modeling

  3. Vector Field Visualization Challenges General Goal: Display the field’s directional information Domain Specific: Detect certain features Vortex cores, Swirl

  4. 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

  5. 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, Texture Splats etc - Display the flow direction everywhere in the field

  6. 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 )

  7. 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 avilable: Runge-Kutta second and fourth order integration methods (more popular due to their accuracy) Result of first order Euler method

  8. x(t+dt) x(t) ½ * [v(x(t))+v(x(t)+dt*v(x(t))] 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)

  9. 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)

  10. Streamlines A curves that connect all the particle positions

  11. Streamlines (cont’d) - 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

  12. T=3 T=4 T=2 T=5 timeline T = 2 T = 1 T = 3 Pathlines, Timelines, and Streaklines • Extension of streamlines for time-varying data • Pathlines: • Timelines: T=1

  13. b.t. =1 b.t. =2 b.t. =3 b.t. =4 b.t. =5 Streaklines - Continuously injecting a new particle at each time step, advecting all the existing particles and connect them together into a streakline

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

  15. Hedgehogs - Dense 2D fields or 3D fields are difficult

  16. Line Integral Convolution (LIC)

  17. LIC Example

  18. 3D LIC

  19. Comparison (LIC and Streamlines)

  20. LIC on 3D surfaces and volumes

  21. More global techniques Texture Splats

  22. Spot Noise

  23. Line bundles

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