A Survey on Visualization of
Time-Dependent Vector Fields
by Texture-based Methods
Henry “Dan” Derbes
MSIM 842 ODU Main Campus
Outline
• Introduction – motivation, background terms
• Fundamental texture based methods
o Spot Noise
o Line Integral Convolution (LIC)
• Unsteady flow methods
o Unsteady Flow LIC (UFLIC)
o Dynamic LIC (DLIC)
o Lagrangian-Euler Advection (LEA)
o Image Base Flow Visualization (IBFV)
o Unsteady Flow Advection-Convolution (UFAC)
Motivation
• Survey goal: Explore ideas that led to techniques for
using texture and dye to visualize unsteady vector
fields in 3D
• Visualization of scalar and vector fields associated
with flow over surfaces has many applications
o Common scalar functions of two variables
o 3D distribution of pressure and velocity over a ship hull or
airplane wings
• Common goal: produce high-resolution images that
reveal flow field characteristics
o
o
o
o
Orientation
Direction
Magnitude
Rate of change
Background
• Spatial resolution of the vector field:
o Sampling with stream lines or particle traces
o Icons at every vector field coordinate
• Problem: these techniques depend critically on
placement
o Eddies or currents can be missed
o Icons do not miss data but use up a lot of spatial resolution
• Time-dependent methods progressively track
visualization results over time
• To achieve coherent animations, continuously track
visualization objects such as particles over time
Texture-based methods
• Texture-based methods offer higher
resolution outputs than previous
approaches
o Vector plots
o Particle tracing
o Stream surfaces
o Volume rendering
Terms
• Advection is the transport of a fluid
• Convolution is a mathematical operator which
takes two functions and produces a third function
• Volume rendering creates a 2D image from scalar
or vector datasets of multiple dimensions
• Particle tracing techniques place a set of insertion
points into a flow field. Particles are released from
the insertion points to trace the flow pattern
• Streamlines are tangent to a vector field at every
point
Terms
• Pathlines trace the trajectory of individual
particles
• Streaklines are the traces of a set of particles
emitted from the same insertion points
Pathlines, Streaklines, Streamlines are
identical for steady flows.
• Timelines link the particles emitted at the
same time from different insertion points.
Spot Noise
• In 1991,van Wijk proposed “spot noise”
o Texture is synthesized by addition of randomly
weighted and positioned spots.
o Local control is achieved by variation of the
spot.
o The spot is a useful primitive for texture
design, because the relationship between
spot and texture features is generally
straightforward.
Spot Noise
• van Wijk developed a convolution with a
white noise texture. Texture synthesis occurs
in two steps:
o Data corresponding to texture coordinates are
retrieved.
o Data are converted to parameter values using a
mapping scheme which expresses the variation.
• Spot noise is synthesized through the
convolution of a white noise grid and the
spot.
Spot Noise
• Variation of the texture for data visualization is
realized by variation of the spot.
• Spot size affects texture
Spot Noise
Disks work well for isotropic textures, but the
interesting parts of vector fields are anisotropic.
Elliptical spots demonstrate anisotropic texture
with the axis scaled to reflect the data vector .
The pattern also influences the texture
as does the shape.
Spot Noise
Visualization of velocity and pressure on a ship hull.
Line Integral Convolution (LIC)
• LIC advantages include: accuracy, locality of
calculation, simplicity, controllability and
generality.
• Streamline starts at the center of pixel (x, y)
and moves in the positive and negative
directions
o Only directional component of the vector field is
used in this advection.
o The magnitude can be added in post processing
LIC
This 2D vector field shows the integration path
for a local streamline starting in (x,y)
Convolution kernel for each
segment i of streamline (x,y)
LIC
• Algorithm maps an input vector field and texture to a filtered
version of the input texture.
• The dimension of the output texture is that of the vector
field.
• Several weaknesses with LIC.
o Flow orientation is not displayed.
o Velocity magnitude cannot be inferred from the final
output.
o Only Cartesian grids can be handled.
o The computational process is slow and real-time data
exploration is not possible.
o Unsteady vector fields can be visualized only as a
sequence of frames not time correlated
Unsteady Flow LIC (UFLIC)
• Simulates the advection of flow traces globally in
unsteady flow fields.
• White noise input texture advected over time to create
directional patterns of the flow at every time step.
• Convolution method is called time-accurate value
scattering scheme.
o Image value at every pixel is scattered forward following the
flow’s pathline trace
o Image value has a time stamp and particle have short lifespan
o Time-accurate value scattering process is repeated at every time
step.
• The resulting texture from the previous convolution step
is used to compute the new convolution after performing
high-pass filtering.
• Method acts as a low-pass filter diminishing contract
Unsteady Flow LIC (UFLIC)
• Output is highly coherent, both spatially
and temporally
• Weaknesses
o Paths are blurred in regions of rapid change
in direction
o Paths are thickest where flow is nearly
uniform
o High computational cost (3 to 5 particles per
pixel)
Dynamic LIC (DLIC)
• Has the outstanding resolution of LIC but is able to
generate animation sequences of time-varying fields
with temporal coherence.
• Extends LIC to time-dependent fields making it
possible to visualize the evolution of streamlines.
• The vector field varies arbitrarily over time with the
motion of streamlines describes by a second “motion”
vector field.
• Each frame is rendered using LIC.
• The input texture is generated by advecting a dense
collection of particles over time and adjusting them to
maintain the appropriate level of detail.
Dynamic LIC (DLIC)
• Texture Coverage Map
• Texture generation from particles
Lagrangian-Eulerian Advection
(LEA)
• Motion of a dense collection of particles (one per pixel)
• High spatio-temporal correlation
• Interactive frame rates through spatial locality and
instruction pipelining
• Lagrangian approach: the trajectory of each particle is
computed separately.
• Eulerian approach: particles lose their identity however,
the particle property, viewed as a field, is known for all
time at any spatial coordinate.
• LEA is a hybrid method. For each time step:
– Particle coordinates are calculated through Lagrangian
integration
– Advection of particle property through an Eulerian method
Lagrangian-Eulerian Advection
(LEA)
• Issues common to texture advection:
– Flow divergence: LEA avoids this issue by
regenerating particles for each time step, through
image blending
– Edge effects: LEA eliminates the need to test for
boundaries by adding a buffer zone with a random
noise texture. Due to vector inflow, some of these
values are advected into the image
– Arbitrary domains: Areas where flow isn’t defined
(island in a river), LEA interprets these areas as zero
velocity. Resulting stationary noise is hidden by image
masking
Lagrangian-Eulerian Advection
(LEA)
• LEA achieves spatial correlation by blending the
previous image and the current advected image
Icurrent = α Icurrent +(1 - α) Iprevious
• Disadvantages of LEA:
– Suboptimal temporal correlation in the form of noisy
and rather short spatial patterns after convolution.
– LEA is limited to white noise input textures.
Image Based Flow Visualization
(IBFV)
• Single framework to generate: particle tracing and
streamlines, moving textures, topological images
• Features: handles unsteady flow, efficiency and
ease of implementation
• Method:
– Warp the image in response to a vector field
– Blend the image with background noise
• Blended images eliminate need for post
processing
• Method takes advantage of graphics hardware
Image Based Flow Visualization
(IBFV)
• Image blending: convex combination of
current image F and another image G
F ( pk : k )  (1   ) F ( pk 1 : k  1)  G( pk : k )
where pk is position at time k
• G is a random noise image
• α can vary by position and time, range [0,1]
• Eliminating the recurrency term gives:
k 1
F ( pk : k )  (1   ) F ( p0 : 0)    (1   ) G ( pk 1 : k  1)
k
i
i 0
Image Based Flow Visualization
(IBFV)
• First term is contribution of the first image
and can be ignored if the first image is black
or if k is large.
k 1
F ( pk : k )    (1   )i G ( pk 1 : k  1)
i 0
• Then pk is result of a LIC of a sequence of
images G with an exponential decay filter
 (1   )
i
Image Based Flow Visualization
(IBFV)
• Top image is noise
• Second image is unclamped
velocity
– Source, saddle point and direction
of flow are indicated
• Third image is clamped velocity
– Direction of flow not indicated
• Fourth image shows artifacts
resulting from unconstrained
maximum velocity and long
integration intervals
Unsteady Flow AdvectionConvolution (UFAC)
• Provides the user with separate control over temporal and
spatial coherence
– Allows use of more advanced visualization techniques
• Dense representation of time-dependent vector field
• Takes white noise, filtered noise, color as texture input
• Method:
– First, continuous trajectories are constructed in spacetime to
guarantee temporal coherence.
• Performs time evolution of unsteady fluid flows using pathlines
– Second, convolutions along another set of paths through the above
spacetime result in spatially correlated patterns.
• Builds spatial correlation according to instantaneous streamlines.
• Length of the streamlines is related to the degree of unsteadiness of the
vector field.
Unsteady Flow AdvectionConvolution (UFAC)
• A texture-based approach:
– Spatial slices of the property field are constructed from trajectories
– Trajectories for each texel backward in time are iteratively computed
using Lagrangian methods
– Combine spatial slices to build the spacetime domain
– Compute a convolution along each pathline in the 3D property field
• Particles have limited lifetime
• Common Issues
– Edge effects: input texture is larger than output creating a boundary
region
– Divergence: limited lifespan of particles, continuous injection of new
particles
– Arbitrary domains: not addressed
Unsteady Flow AdvectionConvolution (UFAC)
• UFAC cannot solve the fundamental dilemma
of inconsistency between spatial and
temporal patterns, but it explicitly addresses
the problem and directly controls the length
of the spatial structures. It maximizes the
length of spatial patterns and the density of
their representation while retaining temporal
coherence.
Conclusion
• Texture based methods support visualization
of complex dynamic fluid flows
• Interactive visualizations have been
demonstrated but need to continue to
improve
• Display of 3D dynamic vector fields has
much room for development
– Sparse representations
– Semi-transparency
– Feature extraction