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 vElectromagnetic field, E, BGradient 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 ChallengesGeneral 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 lineLines 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

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•

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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

An animation over time can give an illusion of motion

are computed by integrating with initial condition x(0) = x0

v(x,t) is a vector field where x is the position in space and t is time

Accuracy of numerical integration is crucial

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

dx

dt

v (

x , t)

•• • • • •

•

••

Initialposition

Finalposition

x i1

x i

v it simplest form

Runge-Kutta of higher order can be used

Streaklines A streakline is the locus at time t0 of all the

fluid particles that have previously passed through a specified point x0

Line connecting particles released from the same location

is computed by linking the endpoints of all trajectories between times ti and t0 for every ti such that 0 ≤ ti ≤ t0

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

•• • •

••

•

ti = 0

ti = t0

0 ≤ ti ≤ t0x0

All particles passed through point x0 in the

field

Streamlines or Fieldlines are integral curves satisfying

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

dx

ds

v (

x , t0)

curve s(x,t)

s v ds

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 parametersVorticity (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 cross-flow

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 cross-section 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.

A polygon is placed into a vector field at a specified point and then deformed according to local strain Align the polygon normal with the local vector Rigid body rotation about the local vector is thus the

streamwise vorticity

The radius of the polygon provides an additional degree of freedom and can be associated with some scalar value,

Normal strain

Shear strain

Rotation

Total deformation

Line Integral Convolution (LIC)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 algorithmDDA 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 DDA convolution example

l

Images generated by DDAImages generated by DDA

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Simple circular vector field Computational fluid dynamicsInput texture: white noise

DDA DrawbackDDA 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.

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.

Negative DirectionNegative Direction

Illustration of local stream line calculation 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 The entire LIC for output pixel F’(x, y)F’(x, y)

.

Local Stream line L 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 LThe effect of varying L

Application :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: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.

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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.