Lec 5 Vector Field Visualization

Vector Field VisualizationLeif Kobbelt

2Visual Computing Institute | Prof. Dr. Leif Kobbelt

Computer Graphics and Multimedia

Geometry Processing

Types of Data

1D 2D 3D nD

C D E F

dimension of

domain

dimension of

data type

Geometry Processing

Characteristic Lines

• Types of characteristic lines in a vector field:

stream lines: tangential to the vector field

path lines: trajectories of massless particles

in the flow (non-static flow fields)

streak lines: trace of dye that is deposited

into the flow at a fixed position

time lines (time surfaces): propagation of

a line (surface) of massless elements in time

Geometry Processing

• stream lines

tangential to the vector field

stationary vector field (arbitrary, yet fixed time t)

stream line is a solution to the initial value

problem of an ordinary differential equation:

u0 0 Lv

initial value

(seed point x0)ordinary differential equation

Geometry Processing

• path lines

trajectories of massless particles in the flow

vector field can be time-dependent (unsteady)

path line is a solution to the initial value problem

of an ordinary differential equation:

t0 0 ,, Lv

Geometry Processing

• streak lines trace of dye that is released into the flow at a fixed

position

connect all particles that passed through a certain

position (non-stationary flow)

• time lines (time surfaces)propagation of a line (surface) of massless elements

over time

many point-like particles that are traced synchronously

connect particles that were deposited simultaneously

Geometry Processing

• comparison of path lines, streak lines, and stream lines

• path lines, streak lines, and stream lines are identicalfor stationary flows

t0 t1 t2 t3

path line streak line stream line for t3

Geometry Processing

Arrows and Glyphs

• visualize local features of the vector field: vector itself

vorticity

external data: temperature, pressure, etc.

• important elements of a vector:direction

magnitude

not: components of a vector

• approaches:arrow plots

glyphs

Geometry Processing

Arrows and Glyphs

• arrows visualization direction of vector field

orientation

magnitude: length of arrows

color coding

Geometry Processing

Arrows and Glyphs

• arrows

Geometry Processing

Arrows and Glyphs

• glyphs

can visualize more features of the vector field

(flow field)

Geometry Processing

Arrows and Glyphs

• pros and cons of glyphs and arrows:

+ simple

+ 3D effects

- heavy load in the graphics subsystem

- inherent occlusion effects

- poor results if magnitude of velocity changes

rapidly (use arrows of constant length and

color code magnitude)

Geometry Processing

Mapping Methods Based on Particle Tracing

• basic idea: trace particles

• characteristic lines

• local or global methods

• mapping approaches:

surfaces

individual particles

sometimes animated

Geometry Processing

• path lines

Geometry Processing

• stream balls

encode additional scalar value by radius

Geometry Processing

• streak lines

Geometry Processing

• stream ribbons trace two close-by particles

keep distance constant

Geometry Processing

• stream tubes specify contour, e.g. triangle

or circle, and trace it through

the flow

Geometry Processing

• LIC (Line Integral Convolution)

Geometry Processing

Numerical Integration of ODEs

• typical example of particle tracing problem (path

line):

• initial value problem for ordinary differential

equations (ODE)

• what kind of numerical solver?

t0 0 ,, Lv

Geometry Processing

• rewrite ODE in generic form

• initial value problem for:

• most simple (naive) approach: explicit Euler method

• based on Taylor expansion

• first order method

• increase accuracy by smaller step size

),()( tt xfx

),()()( ttttt xfxx

)()()()( 2tOttttt xxx

Geometry Processing

• Problem of explicit Euler method inaccurate

unstable

• Example:

divergence for t > 2/k

Geometry Processing

• Implicit Euler method

Approximation of derivative

Implicit timestep

Solution of linear system due to the derivative at the unknown

new position

• Also method of first order

txttxtt

)()()(x

)()()( ttttxttx x

Geometry Processing

• Midpoint method

1. explicit Euler step

2. Evaluation of f at midpoint

3. Complete step with value at midpoint

• Method of second order

),( tt xfx

mid)()( fx ttxtt

Geometry Processing

• Classical Runge-Kutta of fourth order

6336)()(

kkkkxx

Geometry Processing

• adaptive stepsize control change step size according to the error

decrease/increase step size depending on whether

the local error is high/low

higher integration speed in “simple” regions

good error control

• approaches:

stepsize doubling

embedded Runge-Kutta schemes

• further reading: WH Press, SA Teukolsky, WT Vetterling, BP Flannery: Numerical

Recipes

Geometry Processing

Particle Tracing on Grids

• vector field given on a grid

• solve

for the path line

• incremental integration

• discretized path of the particle

t0 0 ,, Lv

Geometry Processing

Particle Tracing on Grids

• most simple case: Cartesian grid

• basic algorithm:

Select start point (seed point)

Find cell that contains start point point locationWhile (particle in domain) do

Interpolate vector field at interpolationcurrent position

Integrate to new position integrationFind new cell point locationDraw line segment between latest

particle positions

Endwhile

Geometry Processing

Line Integral Convolution

• a global method to visualize vector fields

Geometry Processing

• line Integral Convolution (LIC) visualize dense flow fields by imaging its integral curves

vover domain with a random texture (so called ‚input texture‘, usually

stationary white noise)

blur (convolve) the input texture along the path lines using a specified filter

kernel

• look of 2D LIC images intensity distribution along path lines shows high correlation

no correlation between neighboring path lines

Geometry Processing

• idea of Line Integral Convolution (LIC)

global visualization technique

start with random texture

smear out along stream lines

Geometry Processing

• algorithm for 2D LIC let t F0(t) be the path line containing the point (x0,y0)

T(x,y) is the randomly generated input texture

compute the pixel intensity as:

• kernel: finite support [-L,L]

normalized

often simple box filter

often symmetric (isotropic)

dttTtkyxIL

L))(()(),( 000

1kernel

1dttkL

convolution with

kernel

Geometry Processing

• algorithm for 2D LIC

convolve a random texture

along the stream lines

Geometry Processing

Convolution

Input noise Vector field

Final image

1kernel

1dttkL

Geometry Processing

• fast LIC

• problems with LICnew stream line is computed at each pixel

convolution (integral) is computed at each pixel

• improvement: compute very long stream lines

reuse these stream lines for many different pixels

incremental computation of the convolution integral

Geometry Processing

• fast LIC: incremental integration discretization of convolution integral

summation

assumption: box filter

Geometry Processing

• fast LIC: incremental integration discretization of convolution integral

summation

assumption: box filter

T-L )12/( LII L-1L01 TT

Geometry Processing

• fast LIC: incremental integration for constant kernel

stream line x-m,..., x0,..., xn with m,n L

given texture values T-m,...,T0,...,Tn

what are results of convolution: I-m+L,..., I0,..., In-L?

for box filter (constant kernel):

incremental integration:

i10 TI

jL-1jL12L1

ji1ji12L1

j1j TTTT

Geometry Processing

• fast LIC: Algorithm data structure for output: Luminance/Alpha image

luminance = gray-scale output alpha = number of streamline passing through that pixel

For each pixel p in output image

If (Alpha(p) < #min) ThenInitialize streamline computation with x0 = center of pCompute convolution I(x0)Add result to pixel pFor m = 1 to Limit M

Incremental convolution for I(xm) and I(x-m)Add results to pixels containing xm and x-m

End for

End if

End for

Normalize all pixels according to Alpha

Geometry Processing

• oriented LIC (OLIC): visualizes orientation (in addition to direction)

sparse texture

anisotropic convolution kernel

acceleration: integrate individual drops and compose them to final

1 anisotropic

convolution kernel

Geometry Processing

• oriented LIC (OLIC)

Geometry Processing

• LIC - Line Integral Convolution

Geometry Processing

Lic-Applications: length of convolution integral with respect to magnitude of vector field

Geometry Processing

Lic and color coding of velocity magnitude

Geometry Processing

• summary:dense representation of flow fields

convolution along stream lines correlation along

stream lines

for 2D and 3D flows

stationary flows

extensions:

Unsteady flows

Animation

Texture advection

Geometry Processing

3D Vector Fields

• most algorithms can be applied to 2D and 3D

vector fields

• main problem in 3D: effective mapping to

graphical primitives

• main aspects:

occlusion

amount of (visual) data

depth perception

Geometry Processing

3D Vector Fields

• approaches to occlusion issue:

sparse representations

animation

color differences to distinguish separate objects

continuity

• reduction of visual data:

sparse representations

clipping

importance of semi-transparency

Geometry Processing

3D Vector Fields

• missing continuity

Geometry Processing

3D Vector Fields

• color differences to identify connected structures

Geometry Processing

3D Vector Fields

• reduction of visual data

3D LIC

Geometry Processing

3D Vector Fields

• improving spatial perception:

depth cues

perspective

occlusion

motion parallax

stereo disparity

color (atmospheric, fogging)

orientation of structures by shading (highlights)

Geometry Processing

3D Vector Fields

• illumination

Geometry Processing

3D Vector Fields

• illuminated streamlines [Zöckler et al. 1996]

model: streamline is made of thin cylinders

problem

no distinct normal vector on surface

normal vector in plane perpendicular to tangent: normal space

cone of reflection vectors

Geometry Processing

3D Vector Fields

• illuminated streamlines (cont.)

light vector is split in tangential and normal parts

Idea: Represent f() by 2D texture

Access pre-computed f() during

rendering

)(1)(1))((

))(())((

TVTLTVTL

NVNLTVTL

NNLTTLVLLVRV

Lec 5 Vector Field Visualization

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