Image-Guided Streamline Placement

Abstract
Accurate control of streamline density is key to producing several
effective forms of visualization of two-dimensional vector fields.
We introduce a technique that uses an energy function to guide the
placement of streamlines at a specified density. This energy func-
tion uses a low-pass filtered version of the image to measure the
difference between the current image and the desired visual den-
sity. We reduce the energy (and thereby improve the placement of
streamlines) by (1) changing the positions and lengths of stream-
lines, (2) joining streamlines that nearly abut, and (3) creating new
streamlines to fill sufficiently large gaps. The entire process is iter-
ated to produce streamlines that are neither too crowded nor too
sparse. The resulting streamlines manifest a more hand-placed ap-
pearance than do regularly- or randomly-placed streamlines. Ar-
rows can be added to the streamlines to disambiguate flow direc-
tion, and flow magnitude can be represented by the thickness, den-
sity, or intensity of the lines.1 Introduction
The need to visualize vector fields is common in many scientific
and engineering disciplines. Examples of vector fields include ve-
locities of wind and ocean currents (e.g., for weather forecasting),
results of fluid dynamics simulation (e.g., for calculating drag over
a body), magnetic fields, blood flow, components of stress and strain
in materials, and cell migration during embryo development. Ex-
isting techniques for vector field visualization differ in how well
they represent such attributes of the vector field as magnitude, di-
rection, and critical points.
This work was motivated by two recent innovations for displaying
vector fields: spot noise [van Wijk 91] and line-integral convolu-
tion (LIC) [Cabral & Leedom 93]. We wondered how to compare
the results of the techniques. What is the gauge that measures how
well a certain method depicts a vector field? Evidently the place-
ment of the graphical elements is tremendously important. The
graphical elements (e.g. coherent streaks) should follow the flow
direction, but they should not be spaced too close together or too far
apart. Both spot noise and LIC can produce images where stream-
aligned streaks are evenly distributed, but that is more an indirect
result than a guiding principle in the algorithms. How can the stream-
lines be positioned to explicitly satisfy a desired distribution?
The elegant hand-designed streamline drawings in physics texts (for
example in Figure 1a) provide ample inspiration for vector field
illustrations. The streamlines in such illustrations are placed so that
no region is devoid of streamlines and no region is overpopulated
with them. The eye is drawn to regions where the density of ink in
one place differs greatly from that of the surrounding region. When
the density of the streamlines is allowed to vary in such illustra-
tions, it is usually to represent field magnitude, where denser line
spacing shows greater field strength.
Bertin shows another effective hand-designed representation of flow
where the direction of ocean current is represented by chains of
arrows that are laid out end-to-end so that the eye connects arrows
into streamlines and thus gets a stronger sense of flow orientation
[Bertin 83]. The success of this representation depends on having
chosen proper endpoints for these chains so that nowhere does the
image become cluttered. The techniques presented in our paper
will permit designers of vector-field visualizations to control stream-
line-spacing automatically in order to achieve results that mimic
hand-drawn figures.
2 Previous Work
A streamline is an integral curve that is everywhere tangent to a
given vector field (see, for example, [Kundu 90]). Many research-
ers have examined how to effectively and accurately integrate
streamline paths through both regular and irregular meshes. To our
surprise, however, discussions of how best to place streamlines are
almost nonexistent in the visualization literature. We are aware of
three techniques that are used to “seed” streamlines within a vector
field: regular grids, random sampling, and user-specified seed points
for initiation of streamlines. Our knowledge of random and regular
grid seeding of streamlines is almost entirely limited to private com-
munications with visualization researchers. The one published tech-
nique that we have found uses particle traces on a 3D surface that
are terminated when they come too close to the paths of other par-
ticles [Max et al. 94]. The virtual wind tunnel (a 3D immersive
display system for flow visualization) allows users to initiate stream-
lines singly or in bundles [Bryson & Levit 91].
Recently there have been several exciting developments in display-
ing vector fields using texture synthesis. Line integral convolution
is a procedure that stretches a given image along paths that are dic-
tated by a vector field [Cabral & Leedom 93] [Forsell 94] [Stalling
& Hege 95]. Spot noise is a method of creating noise-like texture
by compositing many replicas of a shape [van Wijk 91] [de Leeuw
& van Wijk 95]. When the shapes that create spot noise textures are
stretched according to a given vector field, the resulting images
illustrate the vector field’s direction. Both line integral convolution
and spot noise are well-suited to depicting the fine detail of flow
orientation. They are somewhat less successful (in a single, static
image) at showing the flow magnitude; moreover, the local flow
direction is ambiguous in the sense that it can be interpreted to be
either of two directions that are 180 degrees apart.
A very different method of illustrating vector field data is to show
the important topological features of the flow. In general, stream-
lines that lie in a small neighborhood follow nearly-parallel paths.
The exceptions (in a continuously differentiable vector field) occur
in neighborhoods of points with zero-valued vectors. Several re-
searchers have developed techniques to identify these critical points
(sources, sinks, spirals, centers, and saddles) and the streamlines
that issue from them in eigen-directions [Globus et al. 91] [Helman
& Hesselink 91]. These particular points and curves partition a vec-
tor field into simpler regions where a texture-based method suffices
to display details of the vector field [Delmarcelle & Hesselink 94].
The remainder of this paper is organized as follows. In Section 3
we present a key concept in our work– a visual quality measure for
flow illustrations– and show how this measure can create visually
pleasing illustrations containing short arrows. In Section 4 we dem-
onstrate the creation of illustrations that contain well-placed long
streamlines. Section 5 discusses how these streamlines can be en-
hanced to produce a final illustration. We conclude by discussing
other applications that might use optimization based on a visual
quality measure.
3 Placement of Streamlets
Hedgehog illustrations (sometimes called vector plots) are perhaps
the most commonly used method of illustrating a two-dimensional
vector field. These are short field-aligned segments or arrows whose
base points lie on a regular grid (see Figure 2). The lengths of the
segments are often varied according to the field magnitude. The
popularity of hedgehog illustrations is almost surely due to their
ease of implementation. The resulting images can be slightly en-
hanced by using short streamlines (streamlets) that curve with the
flow instead of using straight lines. We use such streamlets in our
figures 2, 3, and 4.
Two artifacts are often present in hedgehog plots. First, the human
eye often picks out runs of adjacent arrows and groups them to-
gether visually, despite the fact that these groups are an artifact of
the underlying grid pattern and not related to the vector field being
illustrated. This effect can be seen in Figure 2a where three vertical
columns of streamlets erroneously suggest the presence of three
parallel field lines. One way to lessen this problem is to oversample
the seed points that produce the short segments. The drawback with
oversampling is that the resulting image becomes so filled with
streamlets that the eye can no longer discern individual elements. A
better solution to the sampling problem is to introduce noise, slightly
jittering the positions of the arrows to make their regularity less
noticeable [Crawfis & Max 92] [Dovey 95]. This strategy is illus-
trated in Figure 3a.
The second problem with hedgehogs is that as streamlets are placed
close together, portions of neighboring arrows come very close to
one another and may even overlap. Jittering the streamlets may in
fact make the overlaps more frequent (compare Figures 2a and 3a).
The twin problems of overlapped streamlets and grid regularity both
distract the viewer from the data being visualized; we would like to
reduce such distractions. We achieve this goal by using an energy
measure to guide streamlet placement and thus improve the quality
of the final image.
3.1 Optimization of Streamlet Positions
In the discussion that follows, S represents a collection of streamlets
s n for a given vector field V. The elements of S are idealized zero-
width curves, distinct from the geometric primitives (e.g., line seg-
ments or anti-aliased curves) employed to render them. We denote
by I(x,y) the idealized two-dimensional image of the streamlets in
S, with I(x,y) = 0 except along streamlines in S where it behaves
like the Dirac delta function.
Our method creates hedgehog plots by incrementally improving an
initial collection of streamlets. The initial collection can be created
by placing the streamlets either on a regular grid or in some random
fashion, and the final results appear to be independent of which
initialization method is chosen. An image may be improved by
selecting one streamlet at random and moving it a small amount in
a random direction. If the resulting image has a lower energy mea-
sure (lower energy means better quality) then that change is ac-
cepted. This process is repeated many times, terminating when the
energy reaches a threshold or when acceptance of random changes
become rare. Such a process is sometimes referred to as random
optimization or random descent. Figure 4a shows the result of this
algorithm applied to the same vector field as in Figures 2 and 3.
Notice how the streamlets of Figure 4 are more evenly spaced than
in Figures 2 and 3.
The energy measure that guides the optimization is based on a low-
pass-filtered (blurred) version of the image of S which is compared
against a uniform gray-level. Let L ∗ I represent a low-pass-filtered
version of the image I, where L is a given filter function. If t is the
target gray-scale value, then we define the energy measure E as the
squared error integrated over the domain:
illustrated. This effect can be seen in Figure 2a where three vertical
columns of streamlets erroneously suggest the presence of three
parallel field lines. One way to lessen this problem is to oversample
the seed points that produce the short segments. The drawback with
oversampling is that the resulting image becomes so filled with
streamlets that the eye can no longer discern individual elements. A
better solution to the sampling problem is to introduce noise, slightly
jittering the positions of the arrows to make their regularity less
noticeable [Crawfis & Max 92] [Dovey 95]. This strategy is illus-
trated in Figure 3a.
The second problem with hedgehogs is that as streamlets are placed
close together, portions of neighboring arrows come very close to
one another and may even overlap. Jittering the streamlets may in
fact make the overlaps more frequent (compare Figures 2a and 3a).
The twin problems of overlapped streamlets and grid regularity both
distract the viewer from the data being visualized; we would like to
reduce such distractions. We achieve this goal by using an energy
measure to guide streamlet placement and thus improve the quality
of the final image.
3.1 Optimization of Streamlet Positions
In the discussion that follows, S represents a collection of streamlets
s n for a given vector field V. The elements of S are idealized zero-
width curves, distinct from the geometric primitives (e.g., line seg-
ments or anti-aliased curves) employed to render them. We denote
by I(x,y) the idealized two-dimensional image of the streamlets in
S, with I(x,y) = 0 except along streamlines in S where it behaves
like the Dirac delta function.
Our method creates hedgehog plots by incrementally improving an
initial collection of streamlets. The initial collection can be created
by placing the streamlets either on a regular grid or in some random
fashion, and the final results appear to be independent of which
initialization method is chosen. An image may be improved by
selecting one streamlet at random and moving it a small amount in
a random direction. If the resulting image has a lower energy mea-
sure (lower energy means better quality) then that change is ac-
cepted. This process is repeated many times, terminating when the
energy reaches a threshold or when acceptance of random changes
become rare. Such a process is sometimes referred to as random
optimization or random descent. Figure 4a shows the result of this
algorithm applied to the same vector field as in Figures 2 and 3.
Notice how the streamlets of Figure 4 are more evenly spaced than
in Figures 2 and 3.
The energy measure that guides the optimization is based on a low-
pass-filtered (blurred) version of the image of S which is compared
against a uniform gray-level. Let L ∗ I represent a low-pass-filtered
version of the image I, where L is a given filter function. If t is the
target gray-scale value, then we define the energy measure E as the
squared error integrated over the domain:
filtered versions of Figures 2a and 3a. Locations where two
streamlets crowd together in Figures 2a and 3a appear as a high
intensity (black) spot in Figures 2b and 3b. Figure 4a and 4b show
the corresponding images after the optimization routine has been
run. The intensity level in Figure 4b is more uniform than in Fig-
ures 2b and 3b.
When the optimization process is animated it looks as though each
streamlet is pushing away other nearby streamlets, reminiscent of
methods that use repulsion between points to evenly distribute
samples on a surface [Turk 91] [Witkin & Heckbert 94]. This simi-
larity should come as no surprise, since both methods are designed
to minimize an energy term by making small changes in the posi-
tion of graphical elements. In fact, we too have implemented
streamlet-repulsion as a method for creating hedgehog plots. The
visual results of the repulsion method are very similar to the results
of random optimization, and the running times are also similar. We
pursued the random-descent technique rather than the repulsion
method because we expected random descent to be easily exten-
sible to the more complicated task of placing longer streamlines
within V (Section 4).
3.2 Implementation of Low-Pass Filter
This section describes the implementation details for efficiently
computing the energy term for a given set S of streamlets. There
are three components to this computation: the representation of the
blurred image, the low-pass filter used to perform the blur, and the
manner in which we apply the filter to calculate this blurred image.
It would be computationally prohibitive to calculate the energy term
E by actually filtering an entire image each time we consider a ran-
dom change to some streamlet s n . Instead, we associate with s n
certain information about how it affects the low-pass-filtered im-
age. The blurred image B contains pixel values for an image of S.
A streamlet maintains a list of pixels that it affects in B, together
with the values that it contributes to each of those pixels. To test
whether moving s n would improve the value of E, we first remove
the contribution of s n from its list of pixels in B and correct the
value of E based on the changes. Next, we add in the pixel contri-
butions for the new position of s n and recalculate E. We retain the
change to s n if the new value of E is better; otherwise we revert to
the old position for s n . The (un-blurred) image I is purely a concep-
tual aid, and at no time during optimization do we generate an ac-
tual representation of I.
Two criteria influence the choice of a filter to create the blurred
image B. First, the filter kernel should have compact support so
that filtering operations are fast to compute. Second, the point-
spread function should fall off smoothly so that the quality measure
changes smoothly with small changes in streamline position. This
allows the optimization to detect changes in E even for small changes
in the image.
We use the following circularly symmetric filter kernel (from a ba-
sis function of cubic Hermite interpolation) to blur the image:
This function has a similar shape to a two-dimensional Gaussian
filter, but it falls off to zero at a distance R away from its center.
The ideal density for a set of streamlines may be varied across the
image by stretching or shrinking the radius R of the filter.
We sample a streamlet s n at a finite number of points, resulting in a
piecewise-linear curve composed of zero-width line segments. We
calculate the filtered image of each segment by considering those
pixels in the filtered image that are within a distance R of the seg-
ment. The contribution of the line segment to a particular pixel in
the filtered image can quickly be computed by a variant of the tech-
nique used by Feibush for polygon anti-aliasing [Feibush et al. 80].
The line segment is rotated about the pixel center so that it lies
horizontally, and then two table-lookups based on the segment’s
endpoints are used to determine the kernel-weighted contribution
to the pixel. We have found that a very coarse low-pass filtered
image suffices to guide the placement of streamlines. Typically we
use a filter kernel that extends just two or three pixels in radius.
The filtered images in Figures 2, 3 and 4 were computed at a much
higher resolution than this for expository purposes.
4 Long Streamlines
This section describes how the optimization technique from Sec-
tion 3 can be extended to create images containing long, evenly-
distributed streamlines. One goal of this procedure is to enable fine
control over the distance between adjacent streamlines, whether that
target spacing be constant-valued or position-dependent. A second
goal is to avoid interrupting the streamlines. Since each endpoint
of a streamline distracts from the visual flow of the image, our im-
ages should favor fewer, longer streamlines over numerous, shorter
streamlets. It is not always possible to satisfy the two goals of uni-
form streamline separation and infrequent streamline breaks. In
places where the vector field converges (e.g. near a sink) these two
goals are at odds with one another. Our solution to the dilemma is
to let the energy function be the arbiter between uniform spacing
and long streamlines.
The optimization procedure for creating a hedgehog plot consists
of repeatedly considering small changes to the positions of the
streamlets, accepting only the changes that improve the measure E.
The procedure for creating a set of longer streamlines s n is similar.
We improve a set S of streamlines by considering several kinds of
changes to the streamlines. In addition to changes in a streamline’s
position, the algorithm also allows the operations of streamline in-
sertion/deletion, lengthening/shortening of streamlines, and com-
bination of two streamlines, end-to-end, into a single streamline.
We use the same quality measure E to determine which changes
will be accepted. In pseudo-code, the process for creating the col-
lection S of long streamlines is as follows.
4.1 The Allowable Operations
The primitive streamline operations that we employ to improve the
quality of an image are described in more detail below.
Move: Change the position of the seed point of the stream-
line. Each streamline is defined in terms of this seed
point and a length to travel forward and backward
through the flow.Insert: Create a new streamlet.
Delete: Remove a streamline entirely from S.
Lengthen: Add a positive value to the length of the streamline
(relative to the seed point) in the forward or back-
ward direction.
Shorten: Subtract from the length of the streamline (relative
to the seed point) in the forward or backward direc-
tion.
Combine: Connect two streamlines whose endpoints are suffi-
ciently close to one another. The location of the join
is a weighted average of the two endpoints based on
the relative lengths of the streamlines. The length
of the new streamline is the sum of the lengths of the
two parent streamlines.
Why do we allow so many kinds of changes during the optimiza-
tion process? Presumably we could create any possible collection
of streamlines using only insert and delete operations if we allow
newly-inserted streamlines to assume any length and position.
However, an actual implementation of the optimization process us-
ing such a restricted set of operations would be prohibitively slow
to converge. We use the larger complement of operations so that
the optimization procedure can move smoothly through the space
of all collections of streamlines. For example, suppose that joining
two particular streamlines would greatly improve the measure E.
The optimization routine could choose at random to remove each
of these streamlines and, also at random, create another streamline
that fills the void left by the two that were removed. It is very
unlikely that these three independent events would happen by
chance. Explicitly providing a combine operation makes this small
change in visual appearance much more likely to occur.
We find candidate pairs of streamlines for the combine operation
by querying a data structure that maintains the positions of stream-
line endpoints and can return pairs of endpoints whose distance is
less than a given tolerance. There are several ways in which we can
favor joining together streamlines. We could add a term to the en-
ergy function that gives a higher energy to those images that con-
tain more streamlines. Instead of this approach, however, we choose
to accept combine operations if they result in a new value of E that
is no greater than the old energy value plus a tolerance.
We can animate the optimization process by displaying the collec-
tion of streamlines every time a favorable change occurs. An ani-
mation of the optimization indicates the role of each operation. First,
streamlets are inserted throughout the image. After this initial phase
is finished the result looks much like a hedgehog plot using a jit-
tered grid, reminiscent of a Poisson-disk distribution of points. Next,
many of the streamlines gradually lengthen. As streamline end-
points approach one another, pairs of streamlines combine to form
longer streamlines. This dual process of lengthening and joining
creates many longer streamlines that typically follow nearly-paral-
lel trajectories. Gradually the changes in the image become minor,
and many of the changes at this stage are streamlines moving a
small distance, evening the spacing between neighbors. Changes
are accepted with decreasing frequency, and the process is termi-
nated when accepted changes become sufficiently rare.
4.2 Acceleration Using an Oracle
The stochastic optimization produces good results, but it spends
considerable time entertaining changes that are unlikely to improve
the image. The method can be accelerated by using an oracle that
suggests changes that are likely to decrease the energy function E.
An oracle is only effective if it can be consulted quickly and its
answers are generally reliable. The oracle described in this section
typically speeds up the convergence of the optimization by a factor
of three to five.
There are two systematic ways for an oracle to select changes to
propose: an image-based approach, and a streamline-based ap-
proach. Our oracle uses a combination of the two. The image-
based approach examines the blurred image B to identify places
where the streamlines are too sparse. The oracle makes insert sug-
gestions in these places. The streamline-based approach examines
the neighborhood of each individual streamline to decide if an op-
eration applied to the streamline is likely to improve the image.
The oracle uses information gathered from around a streamline to
decide whether to suggest a lengthen, shorten or move operation.
More precisely, the oracle keeps a running measure of how “ener-
getic” a given streamline is, and it maintains a priority queue that
orders the streamlines based on their individual level of energy.
When consulted, the oracle returns one of the most energetic stream-
lines, along with a suggestion of how to lessen its measure of en-
ergy.
The energy of a streamline is the sum on three factors: “desire” to
lengthen, “desire” to shorten, and “desire” to move. Each of these
factors is calculated by sampling the image B at a small number of
positions near the streamline. The desire to lengthen is computed
by comparing the target gray level t with the image values a short
distance beyond the endpoints of the streamline. The lower these
values are with respect to t, the greater the streamline desires to
grow into this empty region. The desire to shorten is found by
sampling B on either side of the streamline endpoints. If these val-
ues are too high, the streamline desires to shrink. The desire to
move is computed by comparing the image values on one side of
the streamline with the values on the other side. The greater the
difference between these two values, the more the streamline de-
sires to change its position. We typically consult 20 samples of the
image B to determine each of the three factors that determine a
streamline’s energy. This sampling is an inexpensive task in com-
parison to creating the entire path of a streamline and then low-
pass-filtering the resulting curve.
The oracle need not bother suggesting that a streamline be deleted.
Every time the optimization routine attempts to modify a stream-
line it can easily check whether entirely removing the streamline
improves the total energy measure E of the image. This is done by
evaluating E after the contribution of the streamline to the image B
is removed and before the altered streamline’s effect is added to B.
The oracle is important for improving efficiency, but it is the en-
ergy measure E that drives the optimization. The oracle is used
purely as a source of suggestions for how to reduce E, not as a
source of directives that are applied blindly. The oracle’s sugges-
tions are only accepted if the change improves the image quality.
We have found it effective for the oracle to propose 50% of the
changes, and for the other changes to be chosen completely at ran-
dom. Thus any change to the collection of streamlines is possible,
which makes it unlikely that the optimization will overlook a worth-
while improvement arising from any systematic bias of the oracle.
4.3 Intensity Tapering at Streamline Ends
Some streamlines must terminate within a region of converging flow
or else the target density of the image cannot be preserved there.The resulting break of the streamline is visually jarring if it is ren-
dered as a rectangular end cap. We make the termination less abrupt
by gradually decreasing the width or intensity of the streamline near
its endpoint. We can gently fade a streamline by allowing yet an-
other operation, namely streamline tapering. Each streamline car-
ries with it (in addition to its center and length) two positions alongits length that indicate where to begin linearly fading to the back-
ground color at either endpoint. This intensity tapering is used to
weight the contribution of the streamline to the filtered image B.
Streamline tapering allows the optimization to find an even closer
match to the ideal gray-scale value in regions near the streamline
ends. In practice we have found it most effective to let intensity
tapering be a separate optimization phase, after the streamlines have
settled into their final position. In this final phase each streamline
is allowed to perform only two operations: 1) changes in length,
and 2) changes in the locations at which to begin intensity tapering.
Performing the intensity tapering after long streamlines have been
formed avoids the possibility that the optimization will produce many
short, intensity-tapered streamlines to satisfy the target density.
Figures 6 and 8 are rendered using the tapering information to modu-
late streamline width and intensity, respectively.
Saito and Takahashi have demonstrated a similar tapering effect for
drawing contour lines of a scalar field [Saito & Takahashi 90]. They
use information about the gradient of the scalar field to guide the
fading out of the contour lines. Their technique can also be used for
drawing streamlines of vector fields where the divergence is zero
everywhere, but it has no obvious generalization when the diver-
gence is non-zero (e.g. fields with sources and sinks).
4.4 Optimization Issues
Two recent techniques in computer graphics provided inspiration
for the optimization approach described here. The first of these is
the work by Andrew Witkin and Paul Heckbert for distributing par-
ticles over an implicit surface [Witkin & Heckbert 94]. In their
constrained optimization method, they let a small number of seed
particles repel one another in order to distribute themselves evenly
over a surface. They found that it is helpful to allow the initial
particles to grow, split, shrink or die to accommodate any change in
surface area when the surface geometry is being edited. Their op-
erations on particles are analogous to our operations on streamlines.A second source of inspiration was the mesh optimization work by
Hugues Hoppe and co-workers [Hoppe et al. 93]. Their technique
uses three fundamental operations to automatically simplify a po-
lygonal mesh: edge split, edge collapse, and edge swap. They used
an energy measure to guide the optimization by random descent.
The high quality of the results produced by this method encouraged
us to try random descent in streamline optimization.
One frequently-voiced concern about optimization techniques is that
the behavior of the system is highly sensitive to the values of many
parameters. An example of such a parameter for streamline optimi-
zation is the maximum distance a streamline can move. The fear is
that the system may require a large amount of “parameter tweak-
ing.” Happily, we have found it unnecessary to change our param-
eter settings between datasets. The single parameter that we specify
for an illustration is the desired distance between neighboring stream-
lines (which can even be position-dependent). Other parameters
are derived from this target-distance. We believe that researchers
who implement the techniques described here will not have diffi-
culty replicating our results. To relieve the burden of re-imple-
menting our technique, we are making our source code publicly
available at http://www.cs.msstate.edu/~banks/IGSP.
5 Binding Visual Attributes to Streamlines
There is an important distinction between a streamline (a zero-width
integral curve) and the geometric elements associated with its dis-
play. A simple approach for displaying a streamline is to draw an
anti-aliased curve that connects vertices sampled along the stream-
line, but such a constant-width, constant-intensity curve is not nec-
essarily the best way to visualize the flow. For example, the curves
are unchanged if all the vectors reverse direction in the underlying
vector field; that is, the sense of flow direction is ambiguous in a
simple streamline display. Arrows can be inserted into the image to
disambiguate the flow direction. We apply two different techniques
to bind arrow-shaped glyphs to streamlines. The first technique is
to traverse the streamlines and deposit an arrow whenever the inte-
grated arc-length along a streamline is sufficient to accommodate
the arrow’s length. Such an object-order traversal is appropriate
for binding a long chain of glyphs onto a streamline. The second
approach is to distribute arrow-glyphs uniformly throughout the
image and then snap them to the nearest point on a streamline. Such
an image-order traversal is appropriate for images with only a few
scattered arrows serving as reminders of the flow direction.
Often some important scalar quantity is associated with a vector
field. The scalar value might be the temperature or density in a
fluid flow, or it might be the magnitude of the vector field at each
point. We would like to bind visual attributes to display such a sca-
lar quantity along with the streamlines. The thickness and the
grayscale-intensity of a streamline offer two convenient visual at-
tributes to convey a scalar quantity. Figure 6 shows a vector field
whose magnitude is bound to the width of the streamlines and where
the streamlines themselves have been placed so that the scalar field
determines the distance between neighboring streamlines.
6 Results
In this section we show some examples of images constructed us-
ing the optimization method for positioning long streamlines. The
first example is Figure 1b, which illustrates a numerical simulation
of flow around a cylinder. The arrowheads in this figure disam-
biguate flow orientation in the eddies. Figure 7 shows computed
wind velocity in the vicinity of Australia. First, the long streamline
optimization method placed streamlines through the image. Thenarrows were bound to these streamlines. The size of the arrow indi-
cates the wind magnitude. The arrows line up head-to-tail so that
the eye can easily follow from one to the next, as is favored by
illustrators [Bertin 83]. Human-subject studies have shown that if
a graphical stroke varies in width from large to small, people have a
strong sense that the direction is towards the larger end [Fowler &
Ware 89]. This guided our choice of tapered arrows in Figure 7.
Another application of the streamline-placement technique is to cre-
ate iso-intensity contours that are evenly spaced. Consider the ef-
fect of highlighting several discrete intensity levels in a gray-scale
image: even if the intensity-values are chosen in equal increments,
the resulting contours are likely to clump together in some regions
and spread apart in others. Our optimization technique provides a
convenient way to adaptively sample the intensity values so that
the curves are uniformly distributed in the image. Figure 8 shows
how the technique can be applied to a color photograph. We con-
verted the image to monochrome, blurred it, and then calculated its
gradient vector field. We ran the optimization on the gradient vec-
tor field and on a vector field orthogonal to it (and thus aligned with
the iso-value lines). The two sets of streamlines that resulted were
combined and used as a mask to apply the original color values to
the grayscale image. The effect is akin to weaving, with constant-
intensity thread being used along the contours.
Our streamline optimization program was written in C++, and the
calculations for the figures herein were performed on a Silicon
Graphics Indigo2 with an R4400 processor operating at 250 MHz.
Figures 4 (a) and 5 (b) were created in under one minute, and the
streamlines for Figures 6, 7 and 8 required roughly 15 minutes each.
We expect that fine-tuning the code would improve the speed by a
factor of two to four.
7 Conclusion and Future Work
There are several logical extensions to the streamline optimization
method presented in this paper. This same process can be used to
create streamlines on curved surfaces by running the optimization
in the parametric space of the surface and correcting for mappingdistortions. The technique could also be used to create streamlines
in three dimensions, although computational efficiency will prob-
ably become an issue. The density of 3D streamlines could be made
dependent on additional properties of the vector field, such as prox-
imity to vortex cores. Another research area is in creating illustra-
tions that reveal different levels of detail when the viewer is at vari-
ous distances.
We expect that the notion of guiding the placement of graphical
elements by a visual measure of quality will have applications be-
yond vector field visualization. For instance, a similar optimiza-
tion method might prove useful in placing graphical elements in a
texture. Another potential use for such techniques is for computer
generation of illustrations that have a hand-drawn appearance [Saito
& Takahashi 90] [Winkenbach & Salesin 94].
8 Acknowledgments
We thank Glenn Wightwick of IBM Australia and Lloyd Treinish
of the IBM T. J. Watson Research Center for the Australia wind
data. Earth image is courtesy of Geosphere, Inc. The fluid flow
data of Figure 1 was provided courtesy of David Rudy, NASA Lan-
gley Research Center. We thank Peggy Wetzel and Mary Whitton
for help in making video of this work. Funding for this work was
provided in part by the NSF Science and Technology Center for
Computer Graphics and Scientific Visualization. Travel support
was provided by ICASE and the NSF Engineering Research Center