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CDS 301

Spring, 2013

Vector Visualization Chap. 6

March 7, 2013 – March 26, 2013

Jie Zhang Copyright ©

Outline

6.1. Divergence and Vorticity

6.2. Vector Glyphs

6.3. Vector Color Coding

6.4. Displacement Plots (skip)

6.5. Stream Objects

6.6. Texture-Based Vector Visualization

6.7. Simplified Representation of Vector Fields

(skip)

CH 6.1 Divergence and

Vorticity

Vector Function

) D-2:case(simpler

D)-3in (usually

22

33

RRf:

RRf:

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

Example in 2-D

x

y

),(

),(

V

V

:Vector

y

x

yxf

yxfV

V

V

y

x

yxs

es

yxfs

yx

:exp

),(

s :Scalar

)( 22

Gradient of a Scalar

)(

)(

)(

22

22

22

2

2

2D:Exp

),,(

vectora isscalar a ofGradient

yx

y

yx

x

yx

yey

sV

xex

sV

es

z

s

y

s

x

ssV

Gradient of a Scalar

1

1

2D:Exp

y

sV

x

sV

yxs

y

x

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


•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.


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

(March 07, 2013 Stops Here)

March 19, 2013

Review: 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

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


•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


y

V

x

V

x

V

z

V

z

V

y

V

xy

zx

yz

V

000)V(

0)V(

0)V(

0) y,(x,VFor

0

0V

ocityplanar vel 2Dfor :Exp

y

x

z

z

z


y

V

x

V

x

V

z

V

z

V

y

V

xy

zx

yz

V

211)V(

0)V(

0)V(

x,0)(-y,VFor

0

0V

ocityplanar vel 2Dfor :Exp

y

x

z

z

z


Stream line:

vector direction

Color: vorticity

CH 6.2 Vector Glyph

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: direction,

orientation, size and color

Vector Glyph

Line glyph,

or hedgehog glyph:

Orientation,

Length

and Color

Sub-sampled by

a factor of 8

(32 X 32)

Original (256 X 256)

Velocity Field of a 2D Magnetohydrodynamic Simulation

Vector Glyph

Velocity Field of a 2D Magnetohydrodynamic Simulation

Line glyph,

or hedgehog glyph

Sub-sampled by

a factor of 4

(64 X 64)


Vector Glyph Sub-sampled by

a factor of 2

(128 X 128)


Problem with a dense

Representation using

glyph:

(1) clutter

(2) miss-representation

Vector Glyph

Random

Sub-sampling

Is better

Vector Glyph: 3D Simulation box: 128 X 85 X 42; or 456,960 data point

100,000 glyphs

Problem: visual occlusion


10,000 glyphs: less occlusion


100,000 glyphs, 0.15 transparency: less occlusion


3D velocity isosurface

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

Matlab: Vector We will use Matlab to implement the vector visualization.

For one well-illustrated example, refer to “wind.m”

Matlab: Vector

Question

Use Matlab to visualize the 2-D vector function F, which is

defined as

Fx(x,y) = x

Fy(x,y) = y

“quiver” plot

%example

[x,y]=meshgrid(-2:0.2:2)

u = x % fx, or x-component of the vector

v = y % fy, or y-component of the vector

quiver(x,y,u,v)

“divergence” plot

%example

[x,y]=meshgrid(-2:0.2:2)

u = x % fx, or x-component of the vector

v = y % fy, or y-component of the vector

quiver(x,y,u,v)

%calculate and display the divergence

div=divergence(x,y,u,v)

hold on

imagesc([-2.0,2.0],[-2.0,2.0],div)

alpha(0.5)

“gradient” and “curl”

[FX, FY, FZ] = gradient(F)

[curlx, curly, curlz, cav] = curl (X, Y, Z, U, V, W)

March 21, 2013

(1) Gradient, Divergence and Vorticity

(2) Vector Glyph

Review

Review

%Author: Jie Zhang

%Date: March 20, 2013

%Purpose: CDS301 class CH6: Visualize the gradient, divergence and

%vorticity

%Scalar function: a double peak 2-D Gaussian function

%Vector function: the gradient of the Gaussian function

%Divergence function: divergence of the gradient

%Vorticity: the curl of the vector function

%clear the variables and figure space

clear

clf reset

%define the domain

%x: [-2.0, 2.0], and y: [-2.0,2.0]

[x,y]=meshgrid([-2.0:0.2:2.0],[-2.0:0.2:2.0]);

%define the Gaussian function

z=x.*exp(-x.^2-y.^2);

%surf(x,y,z) %get the idea of the data

“Gaussian_gradient_divergence.m”

Review

%display the image

hf_1=figure(1)

%hi_1=imagesc([-2.0,2.0],[-2.0,2.0],z)

hi_1=surface(x,y,z)

%get(hi_1) %find the handle values

set(hi_1,'EdgeAlpha',0) %remove the wireframe lines, or edge lines

set(hi_1,'FaceColor','interp') %smooth color from the surface

colorbar %colorbar indicates the values

%calculate the gradient

[gx,gy]=gradient(z,0.2,0.2) %note that spaceing between points is 0.1

hf_2=figure(2)

%surf(x,y,gy)

quiver(x,y,gx,gy)

“Gaussian_gradient_divergence.m” (continued)

Review

%calcuate the divergence

div=divergence(x,y,gx,gy)

hf_3=figure(3)

surf(x,y,div)

%calculate the vorticity

[curlz,cav]=curl(x,y,gx,gy)

hf_4=figure(4)

surf(x,y,curlz)

“Gaussian_gradient_divergence.m” (continued)

CH 6.3 Vector Color Coding

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 (e.g., no white color)

2-D Velocity

Field of the

MHD

simulation:

Orientation:

color

Magnitude:

brightness

Vector Color Coding

2-D Velocity

Field of the

MHD

simulation:

Orientation

only; no

magnitude

Vector Color Coding

Vector Color Coding

•Dense visualization

•Lacks of intuitive interpretation

• take time to be trained to interpret the image

CH 6.4 Stream Objects

Stream Objects

•Vector glyph plots show the straight trajectories over a

short time (fixed velocity) of trace particles released in the

vector fields

•Shows the direction of the flow at a given point

•Stream objects show the curved trajectories for longer

time intervals (varying velocity with time) of trace particles

released in the vector field.

•Follow the flow and trace out the flow

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

Streamlines

•All lines are traced up to the same time T

•Seed points (gray ball) are uniformly sampled

•Color is used to reinforce the vector magnitude

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

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

Stream tubes

Tracing

upstream: the

arrow heads

are on a regular

grid

Stream Objects in 3-D

Input:

128 X 85 X 42

Undersampling:

10 X 10 X 10

Opacity 1

Maximum

Length


Input:

128 X 85 X 42

Undersampling:

3 X 3 X 3

Opacity 1

Maximum

Length


Input:

128 X 85 X 42

Undersampling:

3 X 3 X 3

Opacity 0.3

Maximum Time


Stream tubes

Seed area at

the flow inlet

Stream Ribbons

Two

thick

Ribbons

Vorticity

is color

coded

Vector

Glyth

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

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

March 26, 2013

Vector

Color

Coding

Review

Stream

Tubes

Review

CH 6.6 Texture-Based Vector

Visualization

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

Vector

magnitude:

Color

direction:

Graininess


LIC principle: Line

Integrated

Convolution

Principle


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


•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.

End

of Chap. 6

Note: skip 6.7

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