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CDS 301Fall, 2009
Vector VisualizationChap. 6
October 7, 2009
Jie ZhangCopyright ©
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Outline
6.1. Divergence and Vorticity6.2. Vector Glyphs 6.3. Vector Color Coding6.4. Displacement Plots6.5. Stream Objects6.6. Texture-Based Vector Visualization6.7. Simplified Representation of Vector Fields
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Vector Function
) D-2:case(simpler
D)-3in (usually
22
33
RRf:
RRf:
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Vector versus Scalar
),,(
),,(
),,(
V
V
V
),,(
ˆˆˆ
:Vector
z
y
x
zyxf
zyxf
zyxf
V
or
VVVV
or
kVjViVV
V
z
y
x
zyx
zyx
),,(
s :Scalar
zyxfs
s
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Example in 2-D
x
y
),(
),(
V
V
:Vector
y
x
yxf
yxfV
V
V
y
x
yxs
es
yxfsyx
:exp
),(
s :Scalar
)( 22
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Gradient of a Scalar
)(
)(
)(
22
22
22
2
2
2D:Exp
),,(
vectora isscalar a ofGradient
yxy
yxx
yx
yey
sV
xex
sV
es
z
s
y
s
x
ssV
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Gradient of a Scalar
(1,1) (0,1), (0,0),at ector gradient v theDraw
2s and 1,s 0,s linecontour Draw
1
1
2D:Exp
y
sV
x
sV
yxs
y
x
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Divergence of a Vector
3D)in volumeand 2Din (area
Γby enclosed area theis|Γ|
3D)in surface closed and
2Din curve (closed
cehypersurfa closed is Γ
dsnVV )(||
1lim
0
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Divergence of a Vector•Divergence computes the flux that the vector field transports through the imaginary boundary Γ, as Γ0•Divergence of a vector is a scalar•A positive divergence point is called source, because it indicates that mass would spread from the point (in fluid flow)•A negative divergence point is called sink, because it indicates that mass would get sucked into the point (in fluid flow)•A zero divergence denotes that mass is transported without compression or expansion.
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Divergence of a Vector
x
V
x
V
x
V zyx
V
source :divergence Positive
211V
y)(x,V
:Exp
Free Divergence
000V
x)(y,V
:Exp
sink :divergence Negative
211V
(-x,-y)V
:Exp
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Divergence of a Vector
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Vorticity of a Vector
3D)in volumeand 2Din (area
by enclosed area theis|Γ|
3D)in surface and 2Din (curve
cehypersurfa closed is
)(||
1lim
0sdVV
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Vorticity of a Vector
•Vorticity computes the rotation flux around a point•Vorticity of a vector is a vector•The magnitude of vorticity expresses the speed of angular rotation•The direction of vorticity indicates direction perpendicular to the plane of rotation•Vorticity signals the presence of vortices in vector field
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Vorticity of a Vector
y
V
x
Vx
V
z
V
z
V
y
V
xy
zx
yz
V
000)V(
0)V(
0)V(
y)(x,VFor
0
0V
D-2in :Exp
y
x
z
z
z
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Vorticity of a Vector
y
V
x
Vx
V
z
V
z
V
y
V
xy
zx
yz
V
211)V(
0)V(
0)V(
x)(-y,VFor
0
0V
D-2in :Exp
y
x
z
z
z
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Vorticity of a Vector
Color:
Glyph:
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(Continued)
Vector VisualizationChap. 6
October 15, 2009
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Vector Glyph
))(,( xVkxxl
x
V
•Vector glyph mapping technique associates a vector glyph (or icon) with the sampling points of the vector dataset•The magnitude and direction of the vector attribute is indicated by the various properties of the glyph: location, direction, orientation, size and color
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Vector Glyph
Line glyph, or hedgehog glyph
Sub-sampled by a factor of 8(32 X 32)
Original (256 X 256)
Velocity Field of a 2D Magnetohydrodynamic Simulation
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Vector Glyph
Velocity Field of a 2D Magnetohydrodynamic Simulation
Line glyph, or hedgehog glyph
Sub-sampled by a factor of 4(64 X 64)
Original (256 X 256)
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Vector GlyphSub-sampled by a factor of 2(128 X 128)
Original (256 X 256)
Problem with a denseRepresentation using glyph: (1) clutter(2) miss-representation
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Vector Glyph
RandomSub-samplingIs better
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Vector Glyph: 3DSimulation box: 128 X 85 X 42; or 456,960 data point100,000 glyphsProblem: visual occlusion
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Vector Glyph: 3DSimulation box: 128 X 85 X 42; or 456,960 data point10,000 glyphs: less occlusion
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Vector Glyph: 3DSimulation box: 128 X 85 X 42; or 456,960 data point100,000 glyphs, 0.15 transparency: less occlusion
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Vector Glyph: 3DSimulation box: 128 X 85 X 42; or 456,960 data point3D velocity isosurface
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Vector Glyph•Glyph method is simple to implement, and intuitive to interpretation
•High-resolution vector datasets must be sub-sampled in order to avoid overlapping of neighboring glyphs.
•Glyph method is a sparse visualization: does not represent all points
•Occlusion
•Subsampling artifacts: difficult to interpolate
•Alternative: color mapping method is a dense visualization
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Vector Color Coding
•Similar to scalar color mapping, vector color coding is to associate a color with every point in the data domain
•Typically, use HSV system (color wheel)•Hue is used to encode the direction of the vector, e.g., angle arrangement in the color wheel
•Value of the color vector is used to encode the magnitude of the vector
•Saturation is set to one
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2-D Velocity Field of the MHD simulation:
Orientation,Magnitude
Vector Color Coding
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2-D Velocity Field of the MHD simulation:
Orientation only; no magnitude
Vector Color Coding
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Vector Color Coding
•Dense visualization
•Lacks of intuitive interpretation; take time to be trained to interpret the image
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Stream Objects
•Vector glyph plots show the trajectories over a short time of trace particles released in the vector fields
•Stream objects show the trajectories for longer time intervals for a given vector field
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Streamlines
•Streamline is a curved path over a given time interval of a trace particle passing through a given start location or seed point
point seed the,)0(
)()
T]} [0, ),({
0
0
pp
where
dtpV(τp
pS
t
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Streamlines
All lines are traced up to the same maximum time TSeed points (gray ball) are uniformly sampledColor is used to reinforce the vector magnitude
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Streamlines: Issues•Require numerical integration, which accumulates errors as the integration time increases
tVpp
where
tipVdtpV(τp
iii
t
it
11
/
00
)()()
nintegratioEuler
•Euler integration: fast but less accurate•Runge-Kutta integration: slower but more accurate•Need to find optimal value of time step Δt•Choose number and location of seed points•Trace to maximum time or maximum length•Trace upstream or downstream•Saved as a polyline on an unstructured grid
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Stream tubes
Tracing downstream: the seed points are on a regular grid
•Add a circular cross section along the streamline curves, making the lines thicker
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Stream tubes
Tracing upstream: the arrow heads are on a regular grid
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Stream Objects in 3-D
Input: 128 X 85 X 42
Undersampling:10 X 10 X 10
Opacity 1
Maximum Length
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Stream Objects in 3-D
Input: 128 X 85 X 42
Undersampling:3 X 3 X 3
Opacity 1
Maximum Length
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Stream Objects in 3-D
Input: 128 X 85 X 42
Undersampling:3 X 3 X 3
Opacity 0.3
Maximum Time
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Stream Objects in 3-D
Stream tubes
Seed area at the flow inlet
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Stream Ribbons
Two thick Ribbons
Vorticity is color coded
Vector Glyth
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Stream Ribbons•A stream ribbon is created by launching two stream lines from two seed points close to each other. The surface created by the lines of minimal length with endpoints on the two streamlines is called a stream ribbon
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Stream Surface•Given a seed curve Γ, a stream surface SΓ is a surface that contains Γ and its streamlines
•Everywhere tangent to the vector field•Flow can not cross the surface
•Stream tube is a particular case of a stream surface: the seed curve is a small closed curve
•Stream ribbon is also a particular case of a stream surface: the seed curve is a short line
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Texture-Based Vector Vis.
•Discrete or sparse visualizations can not convey information about every point of a given dataset domain
•Similar to color plots, texture-based vector visualization is a dense representation
•The vector field (direction and magnitude) is encoded by texture parameters, such as luminance, color, graininess, and pattern structure
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Vector magnitude: Color
Vector direction: Graininess
Texture-Based Vector Vis.
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LIC principle: Line Integrated Convolution Principle
Texture-Based Vector Vis.
function blurring of width :
function blurringor weighting:
Ppoint seed of streamline :
texturenoise :
)(
)(
)()),(()(
2
L
k(s)
S(p,s)
N
esk
dssk
dsskspSNpT
s
L
L
L
L
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Texture-Based Vector Vis.
•LIC is a process of blurring or filtering the texture (noise) image along the streamlines •Due to blurring, the pixels along a streamline are getting smoothed; the graininess of texture is gone •However, between neighboring streamlines, the graininess of texture is preserved, showing contrast.
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Endof Chap. 6
Note: skip 6.7