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Vector Visualization
Mengxia Zhu
Vector Data

A vector is an object with direction and length
v = (vx,vy,vz)

A vector field is a field which associates a vector with
each point in space

The vector data is 3D representation of direction and
magnitude

Examples include
Fluid flow, velocity v
Electromagnetic field, E, B
Gradient of any scalar field A = T
Vector Field Visualization Techniques
Local technique: Advection based methods Display the trajectory starting from a
particular location
Global technique: Hedgehogs, Line Integral Convolution,
Texture Splats etc.
Display the flow direction everywhere
in the field
Vector Field Viz Applications
Computational Fluid Dynamics
Weather modeling
Vector Field Visualization
Challenges
General Goal:
Display the field’s directional information
Domain Specific: Detect certain features
Vortex cores, Swirl
Point Icons


Draw point icons at selected points of the vector field
•
A natural approach is to draw an oriented, scaled line for •
•
each vector v = (vx,vy,vz)
 The line begins at the point with which the vector is
associated and is oriented in the direction of the vector
components
Arrows are added to indicate the direction of the line
•
Lines may be colored according to the vector magnitude or
type of the quantity (e.g. temperature)

Oriented glyphs (such as triangle or cone) can be used

To avoid visual clutter, the density of displayed icons must
be kept very low
 Point icons are useful in visualizing 2D slices of 3D
vector fields
Line Icons

Line icons provide a continuous representation of the
vector field, thus avoiding interpolation of point icons

Line icons are particles traces, streaklines, and
streamlines

In general, these three families of trajectories are distinct
from each other
 turbulent flows where flow pattern is time dependent

But they are equivalent to each other in steady flows
Air Flow over Windshield
Air flow coming from
a dashboard vent
and striking the
windshield of an
automobile
http://www-fp.mcs.anl.gov/fl
Particle Traces or Pathlines

are trajectories traversed by fluid particles over time


Final
position
•
An animation over time can give an illusion of motion
are computed by integrating



dx
 v (x,t)
dt
with initial condition x(0) = x0
v(x,t) is a vector field where x is the position
in space and t is time
•
• •
•
•
•
•
Initial
position
of numericalintegration is crucial
xi1  xi  vit simplest form
Runge-Kutta of higher order can be used
 Accuracy


give a sense of complete time evolution of the flow
are experimentally visualized by injecting
instantaneously a dye or smoke in the flow and taking
a long exposure photograph
•
Streaklines

A streakline is the locus at time t0 of all the
fluid particles that have previously passed
through a specified point x0


ti = t0
•
Line connecting particles released from the
same location
is computed by linking the endpoints of all
x0
trajectories between times ti and t0 for
•0
every ti such that 0 ≤ ti ≤ t0
ti =

gives information on the past history of the
flow

is experimentally observed by injecting
continuously at the point x0 a tracer such as
hydrogen bubbles and by taking a short
exposure photograph at time t0
•
•
•
•
•
0 ≤ ti ≤ t0
All particles passed
through point x0 in the
field
Streamlines or Fieldlines

are integral curves satisfying
dx
 v (x,t 0 )
ds

where s is a parameter measuring
the distance along the path

are everywhere tangent to the steady flow

provide an instantaneous picture of the flow
at time t0

are experimentally visualized by injecting a
large number of tracer particles in the flow
and taking a short-time exposure
photograph at t0

curve s(x,t)
s
 vds
t
Streamlines through a Room
Streamlines are color coded to display pressure variation
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
Streamribbons

Are narrow surfaces defined by two adjacent
streamlines and then bridging the lines with a
polygonal mesh

Provide information about two flow parameters
Vorticity (w)- a measure of rotation of the vector
field

streamwise vorticity (Ω) is rotation of the
vector field around the streamline and is
depicted by the twist of the ribbon.
Divergence - a measure of the spread of the
flow


Changing width is proportional to the crossflow divergence
Adjacent streamlines in divergent flows tend to
spread. If too wide and must be discarded
because the twist rate in the widely separated
streamlines no longer reflect the vorticity along
the trajectory
Streamtube

The combined effects of divergence and vorticity
can be visualized in streamtubes which are
effectively formed by linking a number (N) of
streamribbons. Another way of generating the
tube is to sweep an N-sided polygon along the
streamline.

The rotation of the edges found in the tube
represents streamwise vorticity, and the crosssection encodes cross flow divergence.
Streamtube Through a Room
A single streamtube with diameter of the tube indicating
flow-velocity, and color representing pressure variation.
A thinner tube indicates faster flow.
Streampolygons

streampolygon is a regular polygon with n sides and
radius r.

The goal of the streampolygon technique is to
represent information about local strain (shear and
convergence or divergence) and rotation in addition to
the usual flow direction and velocity.
Normal strain

A polygon is placed into a vector field at a specified
point and then deformed according to local strain
Shear strain


Align the polygon normal with the local vector
Rigid body rotation about the local vector is thus the
streamwise vorticity
Rotation

The radius of the polygon provides an additional
degree of freedom and can be associated with some
scalar value,
Total deformation
Line Integral Convolution (LIC)
LIC represents a new and general method for imaging two and
three dimensional vector fields. The LIC Algorithm takes as :

Input:


An input texture image
A vector field

Output: generates an output image by filtering the input image
along local stream lines defined by the vector field .

LIC emulates the effect of a strong wind blow across a fine
sand.
DDA Convolution

This approach is a generalization of traditional line
drawing techniques and the spatial convolution
algorithms given by Van Wijk and Perlin.
DDA Convolution algorithm

Each vector in the field is used to define a long, narrow,
DDA generated filter kernel that is tangential to the vector
and going in the positive and negative vector direction
some fixed distance, L.

A texture is then mapped one-to-one onto the vector field.

The input texture pixels under the filter kernel are summed,
normalized by the length of the filter kernel, 2L, and placed
in an output pixel image for the vector position.
DDA convolution example

l
Images generated by DDA
.
Simple circular vector field
Computational fluid dynamics
Input texture: white noise
DDA Drawback

Symmetry
 This algorithm is very sensitive to symmetry of the DDA
algorithm and filter.
 If the algorithm weights the forward direction more than the
backward direction, the circular field appear to spiral inward
implying a vortical behavior that is not present in the field.

Accuracy
 This algorithm assumes that the local vector field can be
approximated by a straight line.
 For complex structures smaller than the length of the DDA line,
the local radius of curvature is small and is not well
approximated by a straight line.
 In a sense, DDA convolution renders the vector field unevenly,
treating linear portions of the vector field more accurately than
small scale vortices.
LIC


The LIC algorithm is a derivative of the DDA technique that,
instead of using a vector, uses a local streamline to generate the
filter.
The local behavior of the vector field can be approximated by
computing a local stream line that starts at the center of pixel (x,
y) and moves out in the positive and negative directions.
Negative Direction
As with the DDA algorithm, it is important to maintain
symmetry about a cell. Hence, the local stream line is also
advected backwards by the negative of the vector field as
shown in equation (3).
Primed variables represent the negative direction counterparts to
the positive direction variables and are not repeated in
subsequent definitions. As above Dsi, is always positive.
Illustration of local stream line calculation



Continuous sections of
the local stream line — i.e.
the straight line segments
can be thought of as
parameterized space
curves.
The input texture pixel
mapped to a cell can be
treated as a continuous
scalar function of x and y.
It is then possible to
integrate over this scalar
field along each
parameterized space curve.
The entire LIC for output pixel F’(x, y)
.
Local Stream line L
•The length of the local stream line, 2L, is
given in unit pixels. Depending on the input
pixel field, F, if L is too large, all the resulting
LICs will return values very close together for
all coordinates (x, y).
• On the other hand, if L is too small then an
insufficient amount of filtering occurs. Since
the value of L dramatically affects the
performance of the algorithm, the smallest
effective value is desired.
The effect of varying L
Application :



The LIC implementation is a module in a data
flow system like that found in a number of public
domain and commercial products.
This implementation allows for rapid exploration
of various combinations of operators. The
algorithm can be used as a data operator in
conjunction with other operators.
Specifically, both the texture and the vector field
can be preprocessed and combined with post
processing on the output image.
Post processing:
 The
output of the LIC algorithm can
be operated on in a variety of ways.
In this section several standard
techniques are used in combination
with LIC to produce novel results.
.
The fixed normalization fluid dynamics field is
multiplied by a color image of the magnitude of the
vector field.
A wind velocity visualization is created by compositing an image of
North America under an image of the velocity field rendered using
variable length LIC over 1/f noise.
Reference
This set of slides are developed from the
lecture slides used by Prof. Karki at
Louisiana State University.
 Also from Prof. Jian Huang at University of
Tennessee at Knoxville.
