EUROGRAPHICS 2009/ M. Pauly and G. Greiner STAR – State of The Art Report
Over Two Decades of Integration-Based, Geometric Flow
Visualization
Tony McLoughlin1, Robert S. Laramee1, Ronald Peikert2, Frits H. Post3, and Min Chen1
1Visual and Interactive Computing Group
Department of Computer Science, Swansea University, United Kingdom
{cstony, r.s.laramee, m.chen}@swansea.ac.uk
2Institute of Computational Science
Swiss Federal Institute of Technology Zurich, Switzerland
peikert@inf.eth.ch
3Computer Graphics and CAD/CAM Group
Faculty of Electrical Engineering, Mathematics and Computer Science
F.H.Post@tudelft.nl
Abstract
Flow visualization is a fascinating sub-branch of scientific visualization. With ever increasing computing power, it
is possible to process ever more complex fluid simulations. However, a gap between data set sizes and our ability
to visualize them remains. This is especially true for the field of flow visualization which deals with large, time-
dependent, multivariate simulation datasets. In this paper, geometry based flow visualization techniques form
the focus of discussion. Geometric flow visualization methods place discrete objects in the vector field whose
characteristics reflect the underlying properties of the flow. A great amount of progress has been made in this field
over the last two decades. However, a number of challenges remain, including placement, speed of computation,
and perception. In this survey, we review and classify geometric flow visualization literature according to the most
important challenges when considering such a visualization, a central theme being the seeding object upon which
they are based. This paper details our investigation into these techniques with discussions on their applicability
and their relative merits and drawbacks. The result is an up-to-date overview of the current state-of-the-art that
highlights both solved and unsolved problems in this rapidly evolving branch of research. It also serves as a
concise introduction to the field of flow visualization research.
Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry
and Object Modeling
1. Introduction
Vector field visualization is a classic branch of scientific vi-
sualization. Its applications cover a broad spectrum ranging
from turbomachinery design to the modeling and simula-
tion of weather systems. The goal of flow visualization is
to present the behavior of vector fields in a meaningful man-
ner from which important flow features and characteristics
can be easily identified and analyzed.
Given the large variety of techniques currently utilized in
visualization applications, selecting the most appropriate vi-
sualization technique for a given data set is a non-trivial task.
Considerations have to be made taking into account the type
of information the user wishes to extract from the visualiza-
tion along with the spatial and temporal characteristics of
the data set being analyzed. Different approaches have to be
designed for different types of data. For example, visualiza-
tion of 2D data is much different from visualizing 3D data,
and many approaches are more suited to one spatial dimen-
sionality over others. On top of this is the further compli-
cation of temporal dimensionality, with varying techniques
more suited to steady flow compared to time-dependent flow
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fields and vice versa. To this end different tools have been
developed according to the different needs of the users and
the differing dimensionality of vector field data.
1.1. Challenges in Vector Field Visualization
Vector field visualization poses many challenges, the most
important of which we outline here.
Large Data Sets. A major technical issue arises from the
sheer volume of data that may be generated from complex
simulations. Vector data comprises scalar values for each
x;y;z vector component at each sample point within the data
domain. When coupled with several scalar data attributes
and consisting of many time-steps, a large amount of data
results. Advances in hardware are leading to more computa-
tional power and the ability to process larger, more complex
simulations with faster computation times. Therefore, flow
visualization algorithms must be able to handle this large
amount of data and present the results (ideally) at interactive
frame rates in order to be most useful in the investigation and
analysis of simulation data.
Interaction, Seeding and Placement. One of the main
challenges specific to geometric flow visualization is the
seeding strategy used to place the objects within the data
domain. Geometric vector visualization techniques produce
discrete objects whose shape, size, orientation, and position
reflect the characteristics of the underlying vector field. The
position of the objects greatly affects the final visualization.
Different features of the vector field may be depicted de-
pending on the final position and the spatial frequency of
the objects in the data domain. It is critical that the resulting
visualization captures the features of the vector field, e.g.,
vortices, turbulence, sources, sinks and laminar flow, which
the user is interested in. This aspect becomes an even greater
challenge in the case of 3D where a balance of field cover-
age, occlusion, and visual complexity must be maintained.
Time-dependent data also raises a challenge because the vi-
sualization then depends on when objects are seeded.
Computation Time and Irregular Grids. Another chal-
lenge stems from the computation time. Most of the visu-
alizations compute a geometry that is tangential to the vec-
tor field, e.g., the path a massless particle would take when
placed into the flow. Computing such curves through 3D, un-
structured grids is non-trivial. Thus much research has been
devoted to this and similar topics, see Section 2.4. A recent
trend to increase performance has been to move computa-
tions from the CPU and perform them on the GPU. This may
offer a significant improvement in performance as the vector
calculations are suited to the multiple execution units on the
GPU – resulting in high performance parallel processing.
Perception. A central challenge in vector field visualiza-
tion (and visualization in general) relates to perceptual chal-
lenges in visualizing 3D and 4D vector fields as well as
multi-variate data sets. If streamlines are used to visualize
flow in 3D, too many lines causes clutter, visual complexity,
and occlusion. If too few are rendered, important characteris-
tics may not be visualized. Thus an optimal balance between
coverage and perception must be achieved.
1.2. Contributions
In light of these challenges and the more than two decades
of research the main benefits and contributions of this paper
are:
 Here we review the latest developments in geometry-
based vector field visualization research.
 This is the first survey of its kind with a focus on geomet-
ric vector-field visualization techniques.
 We introduce a novel classification scheme based on chal-
lenges including seeding. This scheme lends itself to an
intuitive grouping of papers that are naturally related to
each other. This allows the reader to easily extract the rel-
evant literature without having to read the entire survey.
 Our classification highlights both unsolved problems in
the area of geometric flow visualization and mature areas
where many solutions have been provided.
 This survey is the most up-to-date presentation on this
popular topic. The last time this topic was touched upon
in the literature was over six years ago [PVH03].
 We provide a very concise introduction and overview in
the area of vector-field visualization for those that are new
to the topic and wishing to carry out research in this area.
We have made a great effort not to provide simply an enu-
meration of related papers in geometry-based flow visualiza-
tion, but to compare different methods, related to one another
and weigh their relative merits and weaknesses.
1.3. Classification
One of the main challenges of this survey is classifying
these approaches and presenting them in a meaningful or-
der. There are four general categories that into which vector-
field visualization approaches can be divided: direct, dense
texture-based, geometric, and feature-based. This paper fo-
cuses on the geometric approaches to vector field visualiza-
tion, which received little coverage in [HLD03, LHD04,
PVH03, LHZP07]. A large volume of research work has
been undertaken in geometry-based vector visualization. We
use four tiers of categorization. Our top level of classifica-
tion groups the literature by the dimensionality of the seed-
ing object used to place the resulting geometry, i.e. 0D for
points, 1D for line segments or curves, and 2D for planar
objects. We then subdivide the literature further according
to the spatial dimensionality of the data domain, i.e, 2D vec-
tor fields, 2.5D vector fields on surfaces and 3D volumes.
Temporal dimensionality is also used to group papers to-
gether i.e. steady vs. unsteady flow. And lastly, those papers
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belonging to the same sub-classification appear chronolog-
ically (See Table 1). Classifying the literature in this way
facilitates comparison of similar papers with one another.
It also highlights unaddressed challenges and problems for
which a range of solutions exist. We give a brief overview
and comparison of the four main categories before analyz-
ing the geometric approaches in more detail.
1.4. Direct, Texture-based, and Feature-based Vector
Field Visualization
Direct techniques are the most primitive methods of flow
visualization. Typical examples involve placing an arrow
glyph at each sample point in the domain to represent the
vector data or mapping color according to local velocity
magnitude. Direct techniques are simple to implement and
computationally inexpensive. They allow for immediate in-
vestigation of the flow field. However, direct techniques may
suffer from visual complexity and imagery that lacks in
visual coherency. They also suffer from serious occlusion
problems when applied to 3D data sets.
Dense, texture-based techniques, as the name implies, ex-
ploit textures to form a representation of the flow. The gen-
eral approach uses texture (generally a filtered noise pattern)
which is smeared and stretched according to the local prop-
erties of the vector field. Texture-based approaches provide a
dense visualization result, provide lots of detail, and capture
many flow characteristics even in areas of intricate flow such
as vortices, sources, and sinks. Texture-based methods gen-
erally cover the entire domain. They also share some of the
same weakness of 3D domain representation as direct meth-
ods and are generally more suited to 2D or 2.5D (surfaces
in 3D) domains. A thorough investigation of texture-based
flow visualization is presented by Laramee et al. [LHD04].
Feature-based algorithms focus the visualization on se-
lected features of the data such as vortices or topological
information rather than the entire data set. This may result
in a large reduction of the required data and thus they are
suited to large data sets that may consist of many time-steps.
Since they generally perform a search of the domain, these
techniques require considerably more processing before vi-
sualization in which the desired properties are extracted from
the data. A survey of feature-based approaches is presented
by Post et al. [PVH03].
1.5. Geometric Flow Visualization
Geometric methods define sets of seeding points from which
trajectories (streamlines or path lines) are computed. Trajec-
tories are then used for building geometric objects, in con-
trast to other methods where they are used for filtering or
advecting textures or for topological analysis.
Geometric approaches compute discrete objects within
the data domain. Velocity, v = dxdt , is a derivative quantity.
If we imagine tracking a massless particle through a vector
field, then the displacement of such a point can be described
by:
dx = v
dt (1)
where x is the position of the point, t is the time and v is the
vector field. In order to evaluate equation (1), we can express
it in integral form:
x(t;x0) =
Z t
0
v(t)dt (2)
The analytical solution is approximated using a numeri-
cal integration method. Thus geometry-based techniques are
also known as integration-based and characterize the flow
field with their geometry. The most common geometric tech-
nique is the streamline. A streamline is a curve that is every-
where tangent to the flow field. It is a non-trivial task to auto-
matically distribute the objects such that all of the important
features of the vector field are captured in the resulting visu-
alization.
The two main aspects of geometric flow visualization
each dominated research for a decade. In the first decade,
the focus was on particle tracing, i.e. the numerical com-
putation of trajectories for various types of data discretiza-
tion. In the second decade, interest shifted to particle seeding
strategies. Geometric visualization techniques are suited to
all spatial and temporal dimensions. However, without care-
ful use they are susceptible to visual clutter and occlusion
problems. These problems mainly arise from poor seeding
strategies and thus considerable effort has been put into re-
searching seeding strategies that provide clear, detailed visu-
alizations. We start out our survey of the literature with the
point-based seeding algorithms in 2D vector field domains.
Table 1 provides a concise overview of the literature
grouped according to our classification scheme. Literature is
divided up based upon both the dimensionality of the seed-
ing strategy used to place the geometry-based objects in the
domain and the dimensionality of the data domain itself. Or-
ganizing the literature in this way points out the mature areas
where many solutions are offered and those areas still rich
with unsolved problems.
2. Point-based Seeding Objects
All of the algorithms in this section use point-based (0D) ob-
jects to place streamlines in the vector field domain. Data is
usually obtained from flow simulations. Flow simulations, or
CFD (computational fluid dynamics) simulations, are com-
puted using methods such as the Navier-Stokes equations
[CSvS86] and are used to simulate experiments such as
wind-tunnel tests on cars and airplanes.
2.1. Point-based Seeding in 2D Steady-State Flow Fields
Here we group those methods restricted to two-dimensional,
steady state domains.
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2D Data Domain 2.5D Data Domain 3D Data Domain
Seeding Object Steady Unsteady Steady Unsteady Steady Unsteady
Dimensionality [TB96] [JL00] [vW92] [BS87]pt [Lan93]pt
[JL97a] [vW93a] [RBM87]pt [Lan94]pt
0D [JL97b] [MHHI98] [Bun89]pt [KL96]pt
[JL01] [BMP90]pt [TGE97]pt
[VKP00] [KM92]pt [TGE98]pt
[LJL04] [USM96]pt [TE99]pt
[MAD05] [LPSW96]pt [SGvR03]
[LM06] [SvWHP97]pt [KKKW05]pt
[LHS08] [SdBPM98]pt [BSK07]pt
[SRBE99]pt
[NJ99]pt
[VP04]pt
[HP93] [BL92]
[ZSH96] [WS05]
[FG98] [HE06]
[MT03] [GKT08]
[LWSH04]
[MPSS05]
[LGD05]
[LH05]
[YKP05]
[CCK07]
[LS07]
[Hul92] [STWE07]
1D [vW93b]
[BHR94]
[LMG97]
[SBH01]
[GTS04]
[LGSH06]
[SVL91] [BLM95]
2D [MBC93]
[XZC04]
Table 1: An overview and classification of geometric-based methods in vector field visualization. Research is grouped based
on the dimensionality of the seeding object and the dimensionality of the data domain. Each group is then split into techniques
that are applicable to steady or unsteady flow. Finally the entries are grouped into chronological order. References subscripted
with a pt denote research related to particle tracing. Each entry is also colored according to the main challenge, as outlined in
Section 1.1, that they address. The color coding scheme used is red for seeding strategies, green for techniques addressing
perceptual challenges and yellow for methods aimed at improving application performance. This table provides an at-a-glance
overview of research and highlights unsolved problems as well as challenges for which a range of solutions have been provided.
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Streamline placement for 2D flow fields greatly affects
the final image(s) produced by visualization applications.
Streamlines that are seeded in arbitrary locations may pro-
vide an unsatisfactory result. Critical features in the flow
field may be missed if there are regions containing only a
sparse amount of streamlines. Conversely, where there is a
large number of streamlines in a localized region, a cluttered
image may result making it difficult to distinguish flow be-
havior or, in the worst case, mis-interpreting the flow charac-
teristics and deriving incorrect information from the under-
lying phenomenon.
An image-guided streamline placement algorithm was in-
troduced by Turk and Banks in 1996 [TB96]. One of the
goals of this algorithm is to produce visualizations similar
to hand-drawn illustrations found in textbooks. Prior seed-
ing algorithms were simply based on regular grids, random
sampling or interactive seeding [BL92]. The seeding of the
streamlines is influenced by the resultant image. The goal is
to obtain a uniformly dense streamline coverage. It is for-
mulated as an optimization problem where the objective is
to minimize the variation of a low-pass filtered (blurred) im-
age. Starting from a random initial streamline seed, the prob-
lem is solved by iteratively performing one of the operations
move (displace a seed), insert, lengthen, shorten and com-
bine (connect two streamlines with suffieciently close end
points) on the set of streamlines. The operations are chosen
randomly and undone if they do not improve the objective
function. Figure 1 shows one result.
A follow-up technique was presented by Jobard and Lefer
[JL97a]. The motivation was to introduce a new stream-
line seeding strategy that was computationally efficient and
less costly than their previous streamline seeding strat-
egy [TB96] and allows the user to control the density of
the displayed streamlines. The authors introduce two user-
controlled parameters dsep and dtest . These parameters are
used to control the distance between adjacent streamlines.
Existing streamlines are used to seed new streamlines and
candidate seed points are chosen that are at a distance d =
dsep, from a streamline. All candidate points of one stream-
line are used before moving on to the next streamline. This
process stops when there are no candidate points generated.
The dtest parameter is a proportion of dsep. dtest is used to
control the closest distance that streamlines are allowed to
one another. Sufficient coverage (i.e., a minimum density) is
ensured by seeding streamlines at a distance dsep from one
another. The method was also combined with texture advec-
tion techniques [JL97b] for animating steady flow fields.
Jobard and Lefer [JL01] introduce a novel algorithm that
produces images of a vector field with multiple simulta-
neous densities of streamlines. The paper builds upon the
previous technique [JL97a]. The multi-resolution property
is ideal for vector field exploration as it allows for sparse
streamline placement for a quick overview of the vector
field, the streamline density can then be increased to allow
for a more detailed investigation into areas of interest. They
also demonstrate the use of the technique for zooming and
enrichment.
Lefer et al. extend upon the motion map to be perfectly
cyclical and produce variable-speed animations [LJL04].
This method encodes the motion information in the motion
map in the same way as the previous approach and a color ta-
ble is again utilized to animate the streamlines. Transparency
is also used, this allows the context of the flow field to be
visualized simultaneously with the streamlines. Seeding of
new streamlines is achieved by selecting a random cell, if
the cell is already covered by another streamline then an
adjacent cell is tested. The direction initially taken is re-
peated until an empty cell is found or the domain boundary
is reached. A new random cell is then selected and the pro-
cess repeated. Every time a new seed is placed a counter is
decremented, when this drops below a set value then seeding
is terminated.
Verma et al. present a novel method of streamline place-
ment that focuses on capturing flow patterns in the vicin-
ity of critical points [VKP00]. Templates are defined for
all types of critical points that may be present in 2D flow
fields. The algorithm begins by determining the location of
critical points within the field and its type. A critical point
is a location in the vector field where the velocity magni-
tude is zero. The behavior of the flow in the region around
the critical point is used to classify its type. Some exam-
ples of critical points are sources, sinks and saddle points.
Verma et al. use Voronoi partitioning around the critical
points that contain regions that exhibit similar flow behav-
ior. A random Poisson disk seeding strategy is finally used
to populate streamlines in sparse regions. Seeding in regions
of critical points first ensures that they are covered by a
sufficient amount of streamlines and that these streamlines
have a longer length. In this implementation FAST is used
to compute critical point locations and determine their na-
ture [BMP90], Voronoi diagrams are computed using tri-
angle [She96].
The work of Mebarki et al. [MAD05] builds upon pre-
vious research by Turk and Banks [TB96] and Jobard and
Lefer [JL97a]. The results produced are of comparable qual-
ity to the work of Turk and Banks [TB96] while being pro-
duced faster [MAD05]. The algorithm uses a farthest seed-
ing point strategy. When a new streamline is to be created,
the point that is furthest away from all of the current stream-
lines is used as the seed point for the subsequent stream-
line. Using a farthest point seeding strategy ensures that long
streamlines are produced. To determine the farthest point,
the points of the streamlines are inserted in a 2D Delauney
triangulation. The incident triangles of a newly integrated
point are used to generate a minimal circumdiameter. Any
diameter that is above the desired spacing distance and be-
low a saturation level is pushed onto a priority queue that
is sorted by length. The top circle is then popped out of the
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Figure 1: Arrows showing the wind direction and magnitude over Australia. The arrows are placed along streamlines generated
using the image-guided placement technique of Turk and Banks [TB96]. Image courtesy of Greg Turk.
queue, the center may then be used as the seeding point for
the next streamline.
The work of Liu et al. [LM06] introduces another evenly-
spaced streamline algorithm. It builds upon the work of Jo-
bard and Lefer [JL97a] and Mebarki et al. [MAD05]. Cu-
bic Hermite polynomial interpolation is used to create fewer
evenly-spaced samples in the neighborhood of each previ-
ous streamline in order to reduce the amount of distance
checking. Placement quality is enhanced by double queues
to favor long streamlines (this minimizes discontinuities).
The presented method is faster than that of Jobard and Lefer
[JL97a]. In addition, it incorporates the detection of stream-
line loops.
Li et al. [LHS08] introduced another novel method for
streamline placement. The algorithm is different from its
predecessors in that its goal is to generate the fewest num-
ber of streamlines possible while still capturing the most im-
portant flow features of the vector fields. The images pro-
duced use a small amount of streamlines and are intended
to be similar to hand-drawn diagrams. This is achieved by
taking advantage of spatial coherence and by using distance
fields to determine the similarity between streamlines. New
streamlines are only created when they represent flow char-
acteristics that are not already shown by neighboring stream-
lines. This way repetitive flow patters are omitted. Similarity
between streamlines is measured locally by the directional
difference between the original vector at each grid point and
an approximate vector is derived from nearby streamlines,
and globally by the accumulation of local dissimilarity at
every integrated point along the streamline path.
2.2. Point-based Seeds in a 2D, Time-Varying Domain
Vector fields are either steady (static) or unsteady (changing
over time). It is important to recognize the difference be-
tween steady and unsteady flows and that users are aware of
which one is being investigated by their application. There
are a variety of techniques that are usually more suited to one
temporal dimensionality over the another. Unsteady flow is
generally more challenging than steady flow. A natural way
of representing time-dependent flow is through animation,
which explicitly shows the changes over time. Animation
can also be used to visualize steady flows where the anima-
tion is used to indicate the downstream motion of the flow or
to depict local velocity magnitude.
Unsteady flow is not restricted to being visualized by an-
imation. Streaklines and pathlines, for instance, are gener-
ated by multiple tangent curves using, successive time steps
together so that multiple time-steps are displayed in a sin-
gle static image. A pathline or particle trace is the trajectory
that a massless particle takes in an unsteady fluid flow. A
streakline is the line joining a set of particles that have all
been seeded at the same spatial location (but at successive
times). If seeded at the same location in a steady flow field
streamlines, pathlines and streaklines will be identical.
Jobard and Lefer extend their evenly-spaced streamline
technique [JL97a] to unsteady flow [JL00]. Streamlines
across several time-steps represent the global nature of the
flow at each step and give insight into the evolution of the
vector field over time. However, simply generating a set of
streamlines at each step and cycling between them leads to
an incoherent animation. The authors present a set of pa-
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rameters that are used to choose a suitable set of streamlines
for the next time-step using the current set of streamlines
as a basis. A so-called feed forward method is used which
selects an appropriate subset of streamlines from the subse-
quent time-step that correlate with the current set. A tech-
nique is employed that quantitatively evaluates the corre-
sponding criterion between streamlines. The best candidates
are then used for the next time-step. Several methods are
used to improve upon the animation quality, such as giving
priority to circular streamlines and adding tapering and train
effects to the streamlines. A cyclical texture is also applied
and this is animated to indicate the downstream direction of
the flow on the streamlines.
2.3. Point-based Seeding on Surfaces
In practice, most vector field domains consist of either 2, 2.5,
or 3 spatial dimensions. We define 2.5D as data that is re-
stricted to a surface in 3D space. Some approaches are more
suitable for one spatial dimension over the others. Typically,
as we move from 2D to 3D, the complexity of algorithms
increases. This is due, in part, to the effort required to min-
imize visual clutter and occlusion and, the extra complexity
of adding the third spatial dimension into computations.
A novel visualization scheme based upon a particle sys-
tem is introduced by Van Wijk [vW92]. The particles can be
seeded from a variety of geometric objects, it suggested that
the most obvious objects are: points, lines, circles, rectan-
gles and spheres. The sources also have a temporal attribute
too, the particles can be injected at discrete time pulses or as
a continuous stream. A continuous point source will result
in streamlines being created by the particles and a stream
surface will be created if a curve-based continuous source is
used. The particles have a normal, which allows the lighting
equations to be used in order to apply shading and provide
greater depth cues. A Gaussian filter is used to smooth the
visualization, softening aliasing and strobing artifacts that
may otherwise.
Van Wijk [vW93a] builds upon his previous work in
[vW92]. Here an improved shading model is used to reduce
the aliasing and strobing artifacts that were found in [vW92].
A more detailed discussion of the seeding objects is also
presented detailing the flexibility of using the surface par-
ticles to emulate an array of visualization techniques such as
streamlines, stream surfaces and stream tubes. We classified
the work of Van Wijk [vW92, vW93a] into 0D objects in a
2.5D data domain because the focus of the research is on
how to effectively render particles on stream surfaces.
Mao et al. [MHHI98] present an evenly-spaced stream-
lines technique for curvilinear grids. They expand upon the
work of Turk and Banks [TB96] by applying the seeding
strategy to parameterized surfaces. This algorithm takes the
vectors from the 3D curvilinear surface and maps them to
computational space. An extended 2D image-guided algo-
rithm is then applied and streamlines of a desired density are
Figure 2: Evenly-spaced streamlines on the surface of a gas
engine simulation. Perspective foreshortening is utilized and
the density of streamlines further away from the viewpoint is
increased [SLCZ09].
generated. The streamlines are then mapped back onto the
3D curvilinear surface. However, curvilinear grids cells can
vary significantly in size. This means that streamlines dis-
tributed evenly in computational space won’t necessarily be
evenly spaced when they are mapped back to physical space.
This challenge is overcome by altering the computational-
space streamline density. The streamline density is locally
adapted to the inverse of the grid density in physical space
[MHHI98]. This is achieved by using Poisson ellipse sam-
pling which distributes a set of rectangular windows in com-
putational space. Figure 2 shows evenly-spaced streamlines
on the boundary surface of a gas engine simulation.
2.4. Efficient Particle Tracing in 3D
This subsection summarizes particle tracing strategies in 3D
space. The volume cell types used in simulations vary. The
cell types depend upon the model used to generate them.
The simplest grid type is a Cartesian grid. Curvilinear grids,
commonly used in flow simulations, contain the same cell
type but the grid is usually distorted (usually curving) so
that is fits around a geometry. Unstructured data may con-
tain several different cell types, tetrahedra and hexahedra
are commonly used. Unstructured grids are generally more
challenging than structured grids and some algorithms are
specifically aimed at particle tracing solutions on them.
This collection of papers focuses on computational meth-
ods, addressing challenge of providing fast, accurate results
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
that can be utilized by other visualization methods to im-
prove their performance. This is in contrast to the other
methods that directly provide novel visualizations. The fore-
runners to these techniques along with some of their appli-
cations can be found in [BS87, RBM87, Bun89].
2.4.1. Particle Tracing in Steady Vector Fields.
PLOT3D [BS87,WBPE90] was a command line driven pro-
gram for displaying results of CFD simulations on struc-
tured and unstructured grids. Besides a wide range of graph-
ics functionality, e.g., hidden line and hidden surface tech-
niques, PLOT3D offered 2D and 3D streamlines of the ve-
locity field, the vorticity field (vortex lines), and the wall
shear stress field (skin friction lines). The software was
designed to run on supercomputers, e.g., for computing
movies, but also on the first graphics terminals and work-
stations with hardware supported viewing transformations.
PLOT3D was the precursor of FAST (Flow Analysis Soft-
ware Toolkit) [BMP90], a modular redesign which added
a GUI and distributed processing. Visual2 [GH90] and Vi-
sual3 [HG91] were packages written by R. Haimes and
M. Giles for the visualization of 2D and 3D flow fields.
Linked to user-written main program, they provided inter-
active X-windows based visualization of steady or unsteady
flow fields given on unstructured grids. Visual3 was later
adapted to network computing and renamed to pV3 [Hai94].
The techniques for vector fields available in Visual3 include
streamlines and variants such as ribbons and tufts (or stream-
lets). Kenwright and Mallinson [KM92] give a new method
of particle tracing that does not rely on an approximation
produced using numerical integration methods. The stream-
lines here are created by intersections between stream sur-
faces. Stream surfaces by definition are tangent to the vector
field and therefore a line that is created from the intersection
of two stream surfaces also remains tangent to the field. This
is realized by using f-g diagrams.
An f-g diagram is a two dimensional plot of one stream
surface against another and provides a graphical means of vi-
sualizing the stream functions [KM92]. The f-g diagrams are
calculated for each cell rather than globally in order to avoid
problems with recirculating flows. The method requires ve-
locity and density data for computing mass fluxes, and it has
the benefit of being mass-conserving, as opposed to time-
stepping methods such as the popular fourth-order Runge-
Kutta integration with trilinear interpolation.
Ueng et al. [USM96] present an efficient method of
streamline construction in unstructured grids. This method
uses calculations performed in computational-space to re-
duce computational cost of the streamline generation. To
perform the calculation in computational-space the physical-
space coordinates of a cell and its corresponding vector
data must be transformed into canonical coordinates. A cell
searching strategy similar to that used in [KL96] which
takes advantage of the canonical coordinates to simplify and
speed up the operation. A specialized Runge-Kutta integra-
tor is also presented for use in the canonical coordinate sys-
tem which, based on the results presented, offered improved
computation times compared to the second- and fourth-order
Runge-Kutta integrators that perform in physical space.
The techniques used for streamtube and streamribbon
construction are also described in [USM96]. Streamtubes are
created by placing circular curves, oriented normal to the
flow, at the streamline integration points. The circular cross
sections are then connected to form an enclosed tube ob-
ject. The radius of the streamtube illustrates the local cross
flow divergence and is calculated at each streamline point,
i.e., when the circular glyphs are created. Streamribbons are
created using the streamline for one edge and then using a
constant length normal vector generate the position of the
opposite edge. The constant length normal rotates around the
initial streamline in order to depict local flow vorticity.
UFLOW is a system introduced by Lodha et al.
[LPSW96] to analyze the changes resulting from different
integrators and step-sizes used for computing streamlines.
A pair of streamlines are interactively seeded by the user
and each streamline is generated using a different integrator
and/or integration step-size. It is also possible to create a sin-
gle streamline and then trace it backwards from its end point
and compare it to the initial streamline.
A variety of techniques are used to visualize the differ-
ence between a pair of streamlines. Uncertainty glyphs de-
pict information through their size, shape and color. It is pos-
sible to animate an uncertainty glyph along the particle trace.
Strips and tubes are used to show a range of possible values
in the data and are ideal for illustrating min and max values.
A twirling baton method is also shown, the longer the ba-
ton the more uncertainty there is, and the speed at which it
is twirling represents the uncertainty in the orientation (See
Figure 3).
Sadarjoen et al. present a comparison of several algo-
rithms used for particle tracing on 3D curvilinear grids
[SvWHP97]. The particle tracing process is broken down,
with a brief description, into basic components: point-
location, locating which cell a point is in, interpolation,
and integration. A more thorough discussion and compari-
son of physical-space and computational-space algorithms
then ensues. Results for the implemented algorithms are
also given showing that physical-space computation algo-
rithms generally perform better then their computational-
space counterpart.
Sadarjoen et al. [SdBPM98] present a 6-tetrahedra de-
composition method for s-transformed grids. s-transformed
grids are structured hexahedral curvilinear grids in which the
x and y dimensions differ by 2-3 orders of magnitude from
the z dimension, thus resulting in very thin cells. The method
presented here is more accurate and allows for faster opera-
tions to be performed than the more common 5-tetrahedra
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decomposition that is usually employed [SvWHP97]. De-
composing the hexahedral cells into 6 tetrahedra prevents
a center tetrahedron covering the center of the cell and thus
makes point location much easier. This method also reduces
the chances of infinite loops between two cells when using
the 5-decomposition approach [SdBPM98].
Schulz et al. [SRBE99] present a set of flow visualization
techniques that are tailored for PowerFLOW, a lattice-based
CFD simulator. The grids in these simulations are multi-
resolution Cartesian grids, where finer voxels are used in
areas of interesting flow or boundary surface geometry. Par-
ticle tracing and collision detection are discussed, demon-
strating the need for collision detection between the parti-
cle and the object surface. When a collision occurs the par-
ticle tracing for that line may either terminate or follow a
path along the object boundary. The system has several seed-
ing types, that can be interactive manipulated, ranging from
rakes, planes and cubes. The visualization techniques pre-
sented are streaklines, arrow glyphs and color-coded planes
and they are demonstrated on a car simulation, modeling the
flow around the car body.
Nielson et al. introduce efficient methods for comput-
ing tangent curves for three-dimensional flow fields [NJ99].
This technique is an extension to 3D of their previous re-
search [NJS97]. The techniques are designed to be used on
tetrahedral grids and so a pre-processing step to decompose
a curvilinear grid into tetrahedra may be necessary. Incre-
mental methods are used for stepping along the analytic so-
lution of the streamline ODE and as a result produce exact
results. Techniques for both Cartesian and barycentric co-
ordinates are presented, allowing the user to use the tools
for the coordinate system that is most suited to the current
application. Several cases are defined which are dependent
upon the types of the eigenvalues found at a particular point,
these in turn present the values needed for the calculation of
a tangent curve through a tetrahedron. Results are presented
that compares the accuracy of the presented algorithms com-
pared to Euler and fourth-order Runga-Kutta integrators.
Verma and Pang present methods for comparing stream-
lines and streamribbons [VP04], and some of their methods
are loosely based on those that appear in the UFLOW sys-
tem [LPSW96]. Large CFD simulations are generally run
on supercomputers, however the applications used to visu-
alize these simulations are generally run on workstations.
Some of these simulations have to be approximated with
smaller data sets to make their use on workstations more fea-
sible. Different datasets are compared simultaneously, with
the second dataset being a subsampled version of the first
dataset. A metric for measuring difference is needed and
here the Euclidean distance between associated streamline
points is used. Associated points are connected by lines,
giving a ladder effect, which aids the visual representation
of the differences between the streamlines. Strip envelopes,
which fill in the ladder sections and spheres are also used
to depict the difference when comparing a pair of stream-
lines. A color mapping scheme is also applied, mapping the
difference value to a color scale. This approach is ideal for
comparison between few streamlines. When a dense set of
streamlines are used, a very cluttered visualization may re-
sult.
A method for the comparison of a dense set of stream-
lines is also introduced in [VP04]. This method allows for
flexibility when choosing a difference metric. A formulation
is given in order to quantitatively measure the difference and
create a scalar field of difference values. The scalar field gen-
erates a map of regions with high difference and can be visu-
alized using a variety of scalar field visualization techniques
such as direct volume rendering, isosurface extraction and
cutting planes.
Finally a method of comparing streamribbons is intro-
duced in [VP04]. The streamribbons are used over different
datasets in the same way that the streamline comparisons
were demonstrated. Simply using an overlay, approach and
rendering two different ribbons may not produce an ideal re-
sult as ribbons that are close together may make it difficult
to see the differences and relies more on the user to identify
the differences. Verma and Pang show a method of compar-
ison that involves the straightening out of one of the stream
ribbons. The second ribbon is then transformed such that
it maintains the relative distance and orientation to the first
streamribbon. When these transformed ribbons are overlaid
a more insightful representation of the differences between
them results.
Figure 3: Comparing streamlines of two datasets simulated
using different turbulence models. The streamlines are com-
pared using line glyphs, strip envelopes, and sphere glyphs
to highlight the differences between them. Image Courtesy of
Alex Pang [VP04].
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2.4.2. Particle Tracing in Unsteady Vector Fields.
Lane introduces a system for using streaklines (refer to Sec-
tion 2.2 for the visualization of unsteady flows) [Lan93].
Lane presents the numerical background for particle integra-
tion over many time-steps as well as integration over simu-
lations that involve a moving grid, a feature demonstrated
in very few systems. The two datasets visualized are in ex-
cess of 15GB and 64GB including both their grids and so-
lutions. Seeding points are positioned manually. Lane shows
that only two time-steps are needed to be loaded at once to
perform integration for one step, thus enabling this technique
to be applied to large datasets. The tools in this application
build upon similar systems such as the Virtual Wind Tun-
nel [BL92] (see also Section 2.6).
UFAT [Lan94] is a system that is used to generate streak-
lines on datasets with a large number of time-steps. One
of the major challenges of unsteady flow is the size of the
datasets that may be produced. The size makes them diffi-
cult to store in memory. The system presented here solves
this by only storing two time-steps at any given point during
processing: the current time-step and the subsequent time-
step. This is also true when the grid of a time-dependent grid
is used. Examples of moving grids are shown such as en-
gine cylinder simulations with a moving piston, and turbine
simulations with rotating blades. A second-order Runge-
Kutta with adaptive step size is used to advect the particles
through the flow field. The system stores the streaklines at
each time on disk so that they can be recovered without re-
computation and used to create an animation. This system
builds upon work done on systems such as the Virtual Wind
Tunnel [BL92], pV3 [Hai94] and FAST [BMP90].
Kenwright and Lane [KL96] present a method that in-
creases the efficiency of particle tracing for simulations on
curvilinear grids. Many simulations output the vector data
on curvilinear grids. The grid generally curves according to
a geometry and the flow is simulated around this geome-
try. In this case, particle tracing can be calculated in phys-
ical space, i.e., on the curvilinear grid in its original state,
or in computational space, which transforms the curvilinear
grids coordinates into Cartesian space. Both methods have
their strengths and weaknesses. Calculations in computa-
tional space are easier to perform but tend to be less accurate
due to the transformation using approximated Jacobian ma-
trices. Physical space computation is more accurate but point
location, can be an expensive operation if done naively (e.g.
a brute force linear search in every cell). The authors over-
come this barrier by implementing a more efficient point-
location strategy for tetrahedral grids. However, this means
that the hexahedral cells of the curvilinear grid must be de-
composed in tetrahedral cells. This is performed on the fly
as the usual large size of unsteady simulations prohibit this
being performed as a pre-processing step.
Teitzel et al. [TGE97] describe an analysis of integra-
tion methods used in scientific visualization. The integration
methods investigated are both adaptive and non-adaptive
Runge Kutta integrators of orders 2, 3 and 4. A robust in-
tegration scheme is found by establishing the link in numer-
ical errors between the integration method and the linear in-
terpolation of the vector field values between the discretely
sampled grid points. Their approach is shown to be more ef-
ficient than that of [BLM95] and [KL95]. The authors also
describe implicit integration methods for use in stiff prob-
lems (areas of strong shear or vorticity).
Teitzel et al. [TGE98] introduce a particle tracing method
for sparse grids built upon their previous work [TGE97].
The main difficulty in this task is the interpolation opera-
tion to find the vector values along an integral path. On a
full grid the tri-linear interpolation is done as a local oper-
ation. To help with the efficiency the authors have used an
array to store the contributing coefficients of the sparse grid
as opposed to the binary tree which is normally used so that
the values can be accessed directly without traversing the
tree. Functions are added to calculate the contributing sam-
ples and to accumulate them over the different levels of the
sparse grid. The flow is visualized with color-coded streak
balls, streak tubes and streak bands (or ribbons). Streak balls
follow the same path as streaklines, however, the spacing be-
tween objects depicts acceleration and the size of the ball
depicts local flow convergence and divergence. Streak tubes
use a closed-curve seeding object resulting in a tube that fol-
lows a streakline path. The diameter of the tube depicts flow
divergence and convergence. A streak band uses a short line
segment as a seeding object. This results in a ribbon when
traced through unsteady flow whose twisting depicts the vor-
ticity (or swirling motion) of the flow.
Teitzel et al. [TE99] also introduce an improved method
to accelerate particle tracing on sparse grids and introduce
particle tracing on curvilinear sparse grids. An adaptive eval-
uation of the sparse grids is implemented. This is achieved
by omitting contribution coefficients with a norm below a
given error criterion during the interpolation process. The
combination technique is also used to improve the efficiency.
Streaklines, streak balls and streak tetrahedra are used to vi-
sualize flow on curvilinear sparse grids. Streak tetrahedra
attempt to combine the advantages of streaklines, ribbons,
tubes, and balls. The displacement of the tetrahedra along a
streakline path depicts acceleration, rotation depicts vortic-
ity, and volume reflects convergence and divergence.
2.5. Point-based Seeding in a 3D Steady-State Domain
This section surveys point-based seeding objects used to vi-
sualize 3D vector fields. Here, the challenges are perceptual.
Rendering too many field lines results in clutter, complex-
ity, occlusion, and other perceptual problems. Rendering too
few field lines may lead to missing important characteristics
of the data. Conveyance of depth and spatial orientation are
also challenges.
In 1993, Hin and Post introduced a method for depicting
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turbulent flow using a particle system [HP93]. Turbulence is
a common feature of flow fields, however, there a relatively
few techniques that are specifically focused on this flow fea-
ture. The model for turbulence was modeled on Reynolds’
decomposition [Rey95], which expresses turbulence by us-
ing a decomposition of flow into mean flow and fluctuation,
where the fluctuation represents local turbulent motion. This
was implemented using a stochastic process whereby a com-
pound velocity was composed of the mean velocity and a
random perturbation generated using random-walk models.
Tracing the random-walk particles over many steps leads to
an effect representing turbulent behavior. The seeding of par-
ticles is based on a uniform Cartesian grid aligned with the
domain boundary.
Zöckler et al. introduce a method of illuminating stream-
lines [ZSH96]. Graphics APIs such as OpenGL support
hardware acceleration for lighting when applied to surface
primitives. OpenGL uses the Phong reflection model which
typically uses the orientation of the surface (i.e., its normal)
with respect to the light direction and the viewing angle.
However, there is no native support for the lighting of line
primitives in these libraries, this is due to the fact that line
primitives have no unique normal vector.
From the set of possible normal vectors, the method
chooses the ones that maximize diffuse and specular re-
flection, respectively. For this, two products t1 = L T and
t2 = VT are computed from the light, tangent and view unit
vectors L;T and V on the vertices. By using specially con-
structed textures and t1 and t2 as texture coordinates, diffuse
and specular terms are obtained per pixel.
A streamline placement algorithm is also introduced. For
the placement technique a Monte-Carlo selection algorithm
is applied. This defines a degree of interest in each cell, the
interest in each cell is defined on some scalar value (i.e ve-
locity magnitude). An equalization strategy is then employed
to distribute the seed points more homogeneously.
Mattausch et al. [MT03] combine the illuminated
streamlines technique of [ZSH96] to an extension of evenly-
spaced streamlines seeding strategy of Jobard and Lefer
[JL97a] to 3D. With the 2D version of evenly-spaced stream-
lines presented by Jobard and Lefer [JL97a] the dsep param-
eter is used in connection to the current streamline point for
the candidate streamline seed points. In 2D there are only 2
possible positions for this new candidate seed position (one
on either side of the streamline). When this is extended to 3D
there are an infinite number of positions around a line at an
orthogonal distance of dsep. The authors simplify the exten-
sion to 3D by defining 6 points around a streamline that may
be used for the candidate seed point generation. A focus and
context tool, the magic volume, was also added that is based
on the multi-resolution model of Jobard and Lefer [JL01].
Mallo et al. present an improved illuminated lines tech-
nique [MPSS05]. This method builds upon the previous illu-
minated lines by Zöckler et al [ZSH96] and the cylinder av-
Figure 4: A Lorenz attractor visualized using streamlines.
The streamlines are illuminated using a cylinder averaging
presented by Mallo et al. [MPSS05].
eraging technique presented by Schussman and Ma [SM04].
This method calculates the diffuse and specular components
of lighting from the infinitesimal facets of a cylinder. The
authors take advantage of programmable GPU’s and imple-
ment shader programs. This technique improves upon Zöck-
ler et al’s technique which used maximal reflection due to
the fact that the maximal reflection technique produces bi-
directional lighting. The cylinder averaging technique does
not produce bi-directional lighting and thus provides clearer
orientation and depth information without having to use a
strong specular component. Figure 4 shows an example of
illuminated streamlines.
Fuhrmann and Gröller [FG98] present a technique aimed
at virtual environments that aims to reduce problems in visu-
alizing 3D data such as occlusion and visual clutter. The con-
cept of a dashtube is introduced. A dashtube is an animated,
opacity mapped streamline. The dashtubes are seeded us-
ing a straightforward extension of the evenly-spaced stream-
lines algorithm [JL97a] to 3D. For simplicity the tube por-
tions are set to either being fully opaque or fully transparent.
This bypasses the challenge raised when using transparency
and blending, i.e., the problems encountered with rendering
primitives in the correct order with respect to depth. The
opacity mapping is achieved using textures with animation
taking place in texture space to improve efficiency and ease
of implementation. This method, like most texture based al-
gorithms, can suffer from aliasing problems. The authors
present two methods for resolving this. The first method
is a variation of well known mip-mapping, which instead
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
of filtering the mip-maps, produces sub-maps. The second
method uses a texture with bands of varying sizes for dif-
ferent sized regions on the streamline (regions further away
from the user appear smaller). The authors also present focus
and context techniques, these are magic lenses and magic
boxes. The magic lens is analogous to a magnifying glass.
The region within the lens contains a higher density of dash-
tubes and allows the user to investigate selected areas in
more detail. It also has the benefit of acting as a slicing plane,
removing all clutter between it and the user’s viewpoint. The
magic box shows a discrete volume which forms the focus
and works on the same principle as the lens while allowing
the user to change viewing position and orientation.
Laramee and Hauser present a set of geometric visualiza-
tion techniques including the introduction of two novel ap-
proaches: the streamcomet and a fast animating technique
[LH05]. These techniques are demonstrated in the context
of CFD simulation data. Oriented streamlines improve upon
standard streamlines by depicting the downstream direction
of the flow in a static image. This is achieved by rendering
the lines with a width, using polygons as opposed to line
primitives. These polygons are then given a transparency
value dependent upon the distance from the front of the
streamline. Streamlets are small streamline segments. An-
imation of streamlines is achieved by a stipple pattern. To
animate streamlines the stipple patterns are simply cycled,
which produces moving opaque segments of streamlines that
travel downstream. The streamcomet is a metaphor that of-
fers a large amount of flexibility and interaction from the
user. A streamcomet comprises of a head section and a tail
section. The streamcomet allows the user to choose a seed-
ing location, as well as the position of the head along the
integral line from the seed point, and the length of the tail of
the comet. The user may also alter the diameter of the head
and set the animation speed of the comet along the integral
path.
Ye et al. [YKP05] present a method for streamline place-
ment in 3D flow domains. This paper addresses the com-
mon goals of streamline placement, namely, the generation
of uncluttered visualizations and, sufficient coverage of the
domain to ensure that all important features are captured by
the visualization. Conceptually, this algorithm can be viewed
as an extension of Verma et al’s method [VKP00] to 3D.
This approach scans the vector field for critical points and
extracts them, identifying important areas of interest. Dif-
ferent seeding templates are defined a priori and positioned
around the vicinity of critical points. This approach also con-
tains an operation which detects the proximity of one critical
point is to another. A proximity map is then used to merge
the two most appropriate templates. Poisson seeding is used
to add streamlines to regions of low streamline density. Fil-
tering of the streamlines is then used to remove redundant
streamlines and to avoid visual clutter. The filtering pro-
cess is multi-staged and considers both geometrical and spa-
tial properties. First the streamlines with short lengths and
small winding angles are removed. The next step considers
the similarity of the remaining streamlines. The streamlines
are ranked in order of winding angle. The distance between
endpoints and centroids of streamlines with similar winding
angles are then considered. If the distance is below a pre-
defined threshold then one of them is filtered out. The final
filtering step checks cells with large numbers of streamlines
that also have high winding angles. The candidate stream-
lines are then grouped together according to their winding
angle. The distance between the endpoints of these lines are
tested and they are filtered out if they are too small.
Chen et al. [CCK07] present a novel method for the place-
ment of streamlines. Unlike many other streamlines place-
ment methods this technique does not rely solely on den-
sity placement or feature extraction. Streamline generation
methods relying on a density measure may contain redun-
dant streamlines. Strategies based on the extraction of criti-
cal points in the field require binary filtering of data based
on whether or not they describe a feature. This approach
is based on a similarity method which compares candidate
streamlines based on their shape and direction as well as
their Euclidean distance from one another.
A windowing method is exploited, whereby the window
is placed centered upon a point of the streamline. Points
within the streamline that are contained within the window
are then sampled [CCK07]. These subsets of points are used
to compare neighboring streamlines. The windows cannot
be placed arbitrarily. When a window has been placed on
one streamline, choosing which point on the next stream-
line (the one that is compared to the first) is done by find-
ing the point that has the shortest Euclidean distance to the
point used to place the window on the first streamline. Using
the distance between corresponding streamline points, sim-
ilarities in characteristics such as shape and direction can
be determined. These similarities are then tested to deter-
mine whether or not the point should remain in the final
visualization. Streamline placement is controlled by using
a dsep parameter, which is the closest distance for paral-
lel streamlines. The authors then introduce error evaluation
which quantitatively measures the reconstruction of the vec-
tor field using the streamlines.
Li et al. [LS07] present a streamline placement strat-
egy for 3D vector fields. The motivation is drawn from the
fact that streamlines that are generally well-organized in 3D
space may still produce a cluttered visualization when pro-
jected to the screen. This is the only approach of its kind
– where an image-based seeding strategy is used for 3D
flow visualization. The approach presented here places the
streamline seeds in image-space and then unprojects them
back onto object-space. This visualization result is driven by
how streamlines are positioned on the image-plane. The first
stage of this algorithm is to randomly select a seed point on
the image plane for the initial streamline. This position is
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
then unprojected back into object space. Switching between
image-space and object-space is made possible by storing a
depth map so that the depth position of every pixel on the
screen is known. Once the initial seed has been placed, the
streamline is integrated and placed into a queue. The old-
est streamline is then removed from the queue and used to
generate a new seed position for another streamline. There
are two candidate positions for the seeds, one on either side
of the streamline. The new streamline is integrated until it
is within a threshold, dsep, from other streamlines, it is then
placed in the queue. This process is then repeated. Compli-
cations arise when 3D streamlines in object-space then over-
lap in image-space. Halos are one of the tools used to address
this problem.
We point out that interactive, point-based seeding strate-
gies have been used in various modern, real-world applica-
tions including the investigation and visualization of engine
simulation data [LWSH04, LGD05] (See Figure 6).
2.6. Point-based Seeding in a 3D, Time-varying Domain
The following research focuses on point-based seeding in
unsteady, 3D vector fields. Bryson and Levit introduce the
Virtual Wind Tunnel [BL92]. The tunnel is a virtual en-
vironment for the exploration of vector fields. It utilizes a
mounted head-tracked stereoscopic display. This serves two
main purposes: The stereoscopic display provides depth in-
formation to the user and the head-tracking allows the user
to change their view point within the application by phys-
ically changing the position and orientation of their head.
This system also allows the user to interact and manipulate
objects (such as seed positions) in the system through the use
of a glove that is worn by the user and connected to the sys-
tem with input based on gestures from the user. It should be
noted that the virtual immersion of the user within the sim-
ulation has no effect on the local flow properties and so they
are free to explore the domain without changing the results
of the simulation. Visualization techniques that are used in-
clude tufts, streaklines, particle paths and streamlines. Per-
formance issues arise due to the nature of time-dependent
visualization, large data sets results and result in high band-
width and memory requirements when using techniques that
simultaneously depict many time-steps. This problem is ex-
aggerated more by the head-tracking feature, the application
needs to maintain a minimum execution performance rate (A
minimum of 10fps is recommended) to prevent the user from
losing co-ordination within the virtual environment.
Wiebel and Scheuermann present two methods for static
visualization of unsteady flow [WS05]. This is opposed to
using animation which is much more commonly used to vi-
sualize unsteady flow. The first method involves bundles of
streaklines and pathlines that pass through one point in space
(the eyelet) at different times. A surface construction method
is also introduced as an enhancement to the visualization. A
group of pathlines or streaklines passing through the eye-
let point (at different times) form the basis of a tangent sur-
face. This method is similar to the technique proposed by
Hultquist [Hul92], however in the cases of convergence, a
line trace is not terminated it is just simply ignored for the
purpose of surface construction. In the case of divergence a
test is made to see if there are any pathlines that are currently
being ignored and if so and they are in the correct place they
are then used again. If no appropriate pathline exists then a
new line must be traced from the eyelet. It is not adequate to
simply interpolate a new position between two pathline for
a seed point, this is because this new seed point won’t nec-
essarily pass through the eyelet. Regions of high activity are
of more interest for investigation in general and iso-surfaces
are used to separate regions of high activity from regions of
(nearly) steady flow. This effectiveness of this visualization
technique depends greatly upon the placement of the eye-
lets within the flow field. Positioning the eyelet is based on
sharp edges or corners of objects in the simulation, vortices,
critical points, and regions of high activity i.e., rapidly fluc-
tuating flow direction.
Helgeland and Elbroth present a hybrid geometric and
texture based method for visualizing unsteady vector fields
[HE06]. The seed positions for the field lines are computed
as a pre-processing step. A random initial seed position is
used to prevent visual artifacts that may arise when using a
uniform distribution of seed points. The seeding algorithm is
based upon the evenly-spaces streamline strategy introduced
by Jobard and Lefer [JL97a]. As a seed point is placed it is
advected both upstream and downstream a certain distance.
If the field lines don’t maintain a minimum distance, di, from
all other field lines, the seed point is removed. The final set
of seedpoints are stored in a 3D texture. Particle advection
is implemented using a fourth-order Runge Kutta integra-
tor. Particles are added at inflow boundaries using the same
scheme as the initial seeding strategy. During the course of
the visualization, particles may cluster together producing
both regions of high particle density and regions of low par-
ticle density. To prevent this particles are removed in regions
of high density and new particles are injected into sparse
regions of particles. A texture-based approach is then used
to generate the field lines, followed by volume rendering to
draw and animate the scene.
3. Curve-based Seeding Objects in a 3D Domain
This section describes geometric methods involving 1D line
or curve-based seeding objects. Increasing the seeding ob-
ject dimensionality also increases the dimensionality of the
resulting integral object. Thus the geometric objects in this
section are surfaces. A stream surface for example, is a sur-
face that is everywhere tangent to the vector field (see Fig-
ure 5). Path surfaces and streak surfaces are extensions of
pathlines and streaklines that are obtained by seeding from a
curve instead of a single point. For a path surface, seeding is
done at a fixed time and particle positions are collected at all
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
times, while for a streak surface seeding is done continually
and particle positions correspond to a fixed time. Surfaces
have the added benefit of providing greater perceptual infor-
mation over line primitives, as shading provides better depth
cues. Surfaces also suffer to a lesser extent from visual clut-
ter when compared to line primitives as many lines can be
replaced by a single surface.
3.1. 1D, Line-based Objects in 3D Steady-State Domain
In 1992, Hulquist introduced a novel stream surface con-
struction algorithm [Hul92]. Streamlines are seeded from a
curve and are advanced through the vector field. The sam-
pling frequency is updated at the integration step if neces-
sary. This is achieved using distance tests for neighboring
streamline front points. For convergent flow the distance be-
tween neighboring points reduces and conversely for diver-
gent flow. In the case of divergent flow a new streamline is
seeded when the points exceed a pre-determined distance. In
the case of convergent flow, the advancement of a streamline
may be terminated if neighboring streamlines become too
close. These operations help control the density of the points
of the advancing front and maintain a sampling frequency
that accurately reconstructs the vector field. The streamline
points are used for the stream surface mesh. A locally-greedy
tiling strategy is used to tile the mesh with triangles to con-
struct the surface. The stream surfaces may also split apart
in order to visualize flow around highly divergent areas such
as the flow around an object boundary. The stream surfaces
are seeded using an interactive seeding rake.
In contrast to the local method of stream surface presented
by Hultquist [Hul92], Van Wijk presents a global approach
for stream surface generation [vW93b]. A continuous func-
tion f (x;y;z) is placed on the boundaries of the data set.
A scalar field is then computed throughout the domain by
streamlines placed at all grid points. An iso-surface extrac-
tion technique can then be used to construct the stream sur-
face. One drawback of this approach is that it only generates
stream surfaces that intersect the domain boundary.
Scheuermann et al. present a method of stream surface
construction on tetrahedral grids [SBH01] that builds upon
previous work introduced by Hultquist [Hul92]. This method
advances the surface through the grid one tetrahedra cell at
a time and calculates where the surface intersects with the
tetrahedra cell. When the surface leaves the tetrahedra the
end points are traced as streamlines. For each point on a
streamline, a line is added connecting it to its counter-point
on the other streamline. These are then clipped against the
faces of the tetrahedra cell and the result is the surface within
the cell. Due to the nature of this method, i.e., using the un-
derlying grid in the surface construction process, this method
is inherently compatible with multi-resolution grids and thus
benefits from the increased grid resolution in interesting flow
regions, such as near object boundaries within the flow field.
Brill et al. [BHR94] introduce the concept of a stream-
Figure 5: A stream surface visualizing a tornado simula-
tion [MLZ08].
ball and apply them to the visualization of steady and un-
steady flow fields. Streamballs are defined by a set of dis-
crete points in the vector field based on the metaballs of
Wyvill et al. [WMW86]. A streamball follows the same path
of a streamline, however acceleration and deceleration are
depicted by the amount of displacement between neighbor-
ing spheres. Other local properties of the flow can be mapped
to the radius of the sphere.
Using discrete streamball placement it is possible to con-
struct streamlines and pathlines by ensuring that the cen-
ter points for each streamball are close enough so that they
blend together and form an implicit surface [BHR94]. This
technique can also be applied to stream surface construc-
tion by advancing a set of streamballs that are seeded along
a curve. This produces a smooth surface where the stream-
balls merge automatically in areas of convergence and split
in areas of flow divergence.
Garth et al. [GTS04] introduce a stream surface tech-
nique that handles areas of intricate flow more accurately
than previous techniques such as the method presented by
Hultquist [Hul92]. The algorithm is demonstrated in the con-
text of vortex structures and is based upon an advancing
front with the insertion and deletion of points at each in-
tegration step in order to maintain sufficient resolution for
the construction of an accurate stream surface. In order to
handle more complex flow regions, a high order integrator is
used as well as streamline integration being based upon arc
length as opposed to parameter length, as used in Hultquist’s
method [Hul92]. This results in an improved triangulation
for the stream surface mesh, i.e. triangles are more regular
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
Figure 6: Stream surfaces showing the flow through an en-
gine cooling jacket. Either side of the surface is colored
differently to easily identify the orientation of the surface
[LGD05].
throughout the mesh. For the insertion of a new streamline
the authors also introduce a new rule based upon the angle
between neighboring triples of points on the stream surface
front. The angle must exceed a given threshold and not just
rely on the Euclidean distance between neighboring points.
Löffelmann et al. [LMG97] introduce an extension called
streamarrows for the enhancement of stream surfaces. Strea-
marrows are partitions that are removed from the surface
in order to convey inner flow structure within the surface.
The removal of the partition also help to reduce occlusion as
the area behind the removed portion is visible. In the orig-
inal streamarrows algorithm the arrows were placed on the
stream surface using regular tiling. However, this technique
may provide unsatisfactory results in regions of convergence
and divergence, as the arrows may become too small or too
large. The hierarchical extension overcome this shortfall by
generating a stack of streamarrow textures, where each tex-
ture contains a unique resolution of arrows. The most ap-
propriate texture is then chosen according to the size of the
stream surface, this ensures that the streamarrows all keep a
similar size. Two enhancements are also demonstrated. The
first enhancement removes the arrow segment from the sur-
face and displaces it so that it is slightly above the surface.
The second extension represents the borders of the streamar-
rows as tubes.
Laramee et al. [LGSH06] present a hybrid method by
applying texture advection to stream surfaces. This hybrid
technique enables the visualization to convey more informa-
tion about its inner flow structure. This technique has been
used on iso-surfaces [LSH04] but the textures can mislead
the user. If an iso-surface is generated using velocity magni-
tude the surface, there is no guarantee that the surface will
be everywhere tangent to the flow. This means there may be
a component of the flow vector that is, at least in part, or-
thogonal to the iso-surface and thus the advection may be
misleading. Stream surfaces don’t suffer from this weakness
as they are, by definition, always aligned with the flow field.
3.2. 1D, Line-based Objects in 3D Time-Dependent
Domain
More recently Schafhitzel et al. [STWE07] introduce
a point-based stream surface construction and rendering
method. This method is also applicable to unsteady flow
fields, for which it generates path surfaces. This method was
implemented to exploit graphics hardware acceleration and
therefore all data structures used lend themselves to being
stored in textures, the vector data is also stored in graph-
ics memory as a 3D texture. Like the method presented by
Hultquist [Hul92] this algorithm includes operations to ad-
just the density of the surface front, by adding or remov-
ing points this method updates the sampling frequency and
allows for accurate surface construction in flow exhibiting
local convergent or divergent behavior. A method and con-
ditions for splitting the surface are also implemented which
are similar to the technique Hultquist [Hul92] used in his
algorithm.
In order to render the surface, the particles are distributed
with a sufficient density (to cover the image space repre-
sented by the surface) so that point sprites can be used, this
results in a closed surface. Surface normals can be estimated
for each particle and this allows the surface to benefit from
shading and its associated advantages, i.e. greater depth cues
etc. A surface-based LIC algorithm was also applied in order
to provide internal visual structure for the surfaces.
Von Funck et al. present a novel technique for smoke
surface construction that provides nearly interactive frame-
rates by avoiding the expensive mesh re-triangulation at ev-
ery time step [vFWS08]. No re-triangulation of the mesh
leads to irregular triangles due to flow characteristics such
as divergence. This method exploits these irregular triangles
and maps the opacity of the triangle to its size and shape.
This provides a fair approximation of the optical model for
smoke resulting in surfaces that give a smoke-like effect. A
number of enhancements are also given, showing how sim-
ple modifications to the core algorithm can be used to simu-
late smoke injection from nozzles and wool tufts attached to
the boundary surface of geometries within the flow domain.
4. Planar-based Seeding Objects
This section describes geometric methods involving 2D sur-
face or planar-based seeding objects. Once more, increasing
the dimensionality of the seeding object increases the dimen-
sionality of the integral object. The result is a geometric ob-
ject that sweeps a volume. The volume of these objects can
be used to depict flow characteristics such flow convergence
and divergence.
4.1. 2D Planar-based Seeding in a 3D Steady-State
Domain
Schroeder et al. introduce the Stream Polygon [SVL91]. The
stream polygon is a regular n-sided polygon that is oriented
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
normal to the vector field. The stream polygon can by used
by placing a new polygon for each point of a streamline or
it may be swept along the streamline to form a tube. The
polygon is deformed according to the local flow properties.
Rotation of the polygon reflects the local vorticity of the flow
field. There are no constraints on the polygon maintaining a
rigid body structure, therefore deformation of the polygon is
used to illustrate the local strain of the flow field.
Max et al. introduce flow volumes [MBC93]. A flow
volume is the volumetric equivalent of a streamline. This
method draws inspiration from experimental flow visualiza-
tion, using a tracer material released into a fluid flow. As the
trace propagates through the flow it forms into a flow vol-
ume. The flow volume is divided into a set of tetrahedra,
the projection of which are divided into triangles. Color and
opacity are computed for each tetrahedra using the density
emitter model of Sabella [Sab88]. The contributing pixel val-
ues can be composited in an arbitrary manner, thus negating
the need for a complex sorting algorithm for the volumetric
cells. This is suitable as it produces a reasonable approxima-
tion of the tracer material effect that the authors are aiming
for. An interactive seeding object is used which is always
oriented normal to the local flow field and allows the user to
change attributes of the seed object such as: position, color
and opacity of the smoke and the number of sides for the
seeding polygon.
Xue et al. introduce implicit flow volumes [XZC04]. This
idea builds upon flow volumes introduced by Mat et al
[MBC93] and the implicit stream surface technique pre-
sented by Van Wijk [vW93b]. Two techniques are presented
for the rendering of the implicit flow volume, a slice-based
3D texture mapping and interval volume rendering. The first
approach renders the flow field directly without the inflow
mapping to a scalar field, as used by Van Wijk [vW93b].
This method takes advantage of graphics hardware acceler-
ation. Volume shaders can be used to change the appearance
Figure 7: A flow volume created using the tornado dataset.
Image from https://graphics.llnl.gov/flow.html [MBC93].
and representation of the flow volume. This method allows
for high levels of interactivity and fine texture detail in all
regions of the flow volume. The second approach utilizes a
flow mapping that produces a scalar field, the flow volume
created from the interval volume enclosed between two iso-
surfaces. The rendering of the volume is then achieved by
using a tetrahedra-based technique.
4.2. 2D Planar-based Seeding in a 3D Time-Dependent
Domain
Becker et al. extend the flow volume technique [MBC93] for
the use with unsteady vector fields [BLM95]. Flow volumes
in steady flow fields are created using a set of streamlines
seeded from a polygon oriented normal to the local flow. To
extend this to unsteady flow Becker et al. construct the flow
volumes using streaklines. As in the steady case, the vol-
ume is divided up into tetrahedra and volume rendered us-
ing hardware. Using streaklines instead of streamlines intro-
duces several complications to the initial flow volume strat-
egy. In the steady case, only the end points of each stream-
line are advected and a new layer is added to the at the end
flow volume in the downstream direction during each in-
tegration step. However, when streaklines are used, every
point on the streakline must be advected (not just the end
points). This may result in the flow volume geometry chang-
ing over time. The subdivision strategy used in [MBC93]
was performed only at the end of the flow volume, but the
changing geometry here requires a subdivision strategy that
operates anywhere in the volume. A subdivision created in a
previous time step may be unnecessary in future time steps.
Subdivision in time is also required, if all particles within a
given layer exceed a given distance threshold from the previ-
ous layer, a new layer is inserted between them. The reverse
is also applicable with the possibility that a layer may be
removed.
5. Discussion and Conclusions
In this section we reflect on the previous research and pro-
vide an assessment of areas for potential future work. A
variety of techniques have been discussed, each with their
own relative merits and shortcomings. As clearly illustrated
in this literature, there is no single visualization tool which
provides optimal results for all given phenomena. The most
appropriate method is dependent upon several factors such
as the data dimensionality (both spatial and temporal). The
size of the simulation output and the goal of the user are also
factors, i.e., is the visualization to be used for detailed inves-
tigation in specific regions, is the visualization intended fast
exploration of the vector field or for high quality presenta-
tion purposes.
In the context of geometric approaches, a large volume
of effort has been placed on streamlines. Streamlines are a
highly effective tool and coupled with an effective seeding
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T. McLoughlin & R. S. Laramee & R. Peikert & F. H. Post & M. Chen / Over Two Decades of Integration-Based, Geometric Flow Visualization
strategy may produce some highly insightful visualizations.
The success of streamlines comes partly due their ease of
implementation and the quality of the results produced by
them. Streamline enhancements tend to fall in one of two
categories, particle tracing or seeding algorithms. The parti-
cle tracing algorithms also have their own subset of classi-
fications, with their contributions differentiating them from
other tracing algorithms. Most effort has been undertaken
on providing ever faster tracing, while other methods have
focused tracing on specific grid types, while others on ac-
curacy, producing exact results rather than approximations
when using a numerical integration method. Particle tracing
methods were a popular area of research in the 1990’s with
comparatively very little work been undertaken recently.
This may be due to the efficiency of current methods, making
it more difficult to obtain further significant gains in perfor-
mance.
Seeding strategies have been heavily researched and are
still an area of active research. Many algorithms are based on
providing aesthetic, insightful visualizations in image-space.
The focus of these papers tends to be on producing unclut-
tered visualizations that avoid bombarding the user with vi-
sual overload which may lead to longer times being under-
taken in order to interpret the results and/or key features of
the vector field being obscured or misinterpreted. The major-
ity of seeding algorithms have been targeted at 2D domains
with only a handful being extended or specifically aimed at
higher spatial dimensions. We believe that seeding in 3D do-
mains is still a fruitful area of research and seeding strate-
gies may be extended to providing efficient algorithms for
unsteady flow fields.
During the writing of this survey we identified an inter-
esting trend. As the dimensionality of the seeding object
increases the volume of research decreases, this is evident
from Table 1. We believe that this is due to the added com-
plexity of creating higher dimensional geometric objects and
ensuring that they maintain an accurate representation of the
underlying vector field. We have already stated the advan-
tages that surfaces and volumes present over line primitives,
but these advantages are countered by the difficulty of creat-
ing a suitable mesh for a surface or volume. Stream surfaces
are a very useful tool for vector field visualization however,
research and their implementation in industrial applications
is limited. While several algorithms exist for stream surface
construction, their complexity is often a barrier for a devel-
oper. We believe better construction methods that are both
efficient and simple are possible. The extension to unsteady
flow fields is also an attractive prospect with few papers ex-
plicitly showing a time-dependent implementation using sur-
faces. We also note that there are very few strategies that fo-
cus on automatic placement of these structures in the flow
field.
Many of the techniques here are used in commercial ap-
plications. The array of tools is too vast for all of them to
be combined into a single package and so the most appro-
priate subset must be chosen. This will be more successful if
the application designer, works closely with a CFD engineer
who has an intricate knowledge of the simulations they will
be creating and the phenomena they wish to investigate.
Some of the key areas we identified needing additional
work are:
 Higher dimensional (both spatial and temporal) data do-
main seeding strategies.
 Uncertainty visualization tools for geometric techniques.
 Comparative visualization tools for geometric techniques.
 Improved surface and volume construction methods.
 Surfaces for visualizing unsteady flow fields.
 Automatic seeding for surfaces and volumes.
A large amount of success has been gained for 2D vector
fields and more recent techniques are offering a less signifi-
cant improvement over current methods. Three-dimensional
vector fields have also attained a high levels of progress,
however due to the added challenges, there is still room
for significant improvement in many areas. Similarly, many
problems remain unsolved when visualizing unsteady flow
and the barriers are constantly pushed as simulations are in-
creasing the size of their output and ever more efficient meth-
ods are always required to address this expansion.
6. Acknowledgements
This work was funded in part by EPSRC Research Grant
EP/F002335/1 on the project Interactive, High-Dimensional
Flow Visualization Using Image Space-Based Approaches.
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