experimental flow vispascucci/classes/2009_fall/files/lect07.pdfexperimental flow vis. experimental...
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Experimental Flow Vis
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Experimental Flow Vis
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Experimental Flow Vis
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Why would we notstick with these?
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-A vector field: F(U) = V
U: field domain (x,y) in 2D
V: vector (u,v)
-Like scalar fields, vectors are defined at discrete points:
-interpolation issues
Vector Field Visualization
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Visualization techniques
Geometry-based methods: rendering primitives built from particle trajectories
Glyphs
streamlines
pathlines
streaklines
topology
LIC
…….
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Streamlines
A curves that connect all the particle positions
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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 (in 2D (x1, x2) position vector)
v(x): the vector (velocity) field
Use numerical integration to compute a new particle position
x(t) = x(t-dt) + Integration( v(x(t-dt)) dt )
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Numerical Integration
First Order Euler method: x(t) = x(t-dt) + v(x(t-dt)) * 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
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Euler’s MethodAssume flow = f(t)
X
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Euler Integration Error
• Error = X - area under flow curve
XError =
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Numerical Integration (2)
Second Runge-Kutta Method
x(t) = x(t-dt) + ½ * (K1 + K2) k1 = dt * v(x(t-dt))k2 = dt * v(x(t-dt)+k1)
½ * [v(x(t))+v(x(t)+dt*v(x(t))]
x(t+dt)
x(t)
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Runge-Kutta 2Assume flow = f(t)
Like Trapezoid Method.
X
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RK2 Integration Error
• Error = X - area under flow curve
Error =X
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Numerical Integration (3)
Standard Method: Runge-Kutta fourth order
x(t) = x(t-dt) + 1/6 (k1 + 2k2 + 2k3 + k4)
k1 = dt * v(t-dt); k2 = dt * v(x(t-dt) + k1/2)
k3 = dt * v(x(t-dt) + k2/2); k4 = dt * v(x(t-dt) + k3)
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Runge-Kutta 4Assume flow = f(t)
X
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What Method to Use?
• RK2 and RK4 are more accurate for same dtthan Euler
• RK2 and RK4 work well for continuous systems
• Euler works poorly for oscillatory systems
• RK2 and RK4 work poorly with discretesystems
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Pathlines, Timelines, and Streaklines
-Extension of streamlines for time-varying data
Pathlines:
Timelines:
T=1
T=2
T=3 T=4
T=5
T = 1 T = 2 T = 3
timeline
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Streaklines
- Continuously injecting a new particle at each time step,
advecting all the existing particles and connect them
together into a streakline
b.t. =5
b.t. =4
b.t. =3
b.t. =2 b.t. =1
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When should we expect self-intersections?
- Streamlines- Pathlines
- Streaklines
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• Streamlines do not cross• Streaklines still never cross• Pathlines do cross
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• Streamlines not cross• Streaklines still cross • Pathlines do cross
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Seed Placement
• 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!!!
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Rendering - LIC
• embed a noise texture under the vector field
• integrates along a streamline
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Line Integral Convolution (LIC)
• LIC:• convolve a• random texture• along the • streamlines
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Line Integral Convolution (LIC)
• Assume input texture, vector and output images are all the same resolution.
• For each output pixel/voxel, generate a streamline both forwards and backwards of a fixed length.
• Integrate the intensity that the streamline passes through
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Line Integral Convolution (LIC)
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Line Integral Convolution (LIC)
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Comparison (LIC and Streamlines)