Similarity-Guided Streamline Placement with Error Evaluation

Abstract—Most streamline generation algorithms either provide a particular density of streamlines across the domain or explicitly
detect features, such as critical points, and follow customized rules to emphasize those features. However, the former generally
includes many redundant streamlines, and the latter requires Boolean decisions on which points are features (and may thus suffer
from robustness problems for real-world data).
We take a new approach to adaptive streamline placement for steady vector fields in 2D and 3D. We define a metric for local
similarity among streamlines and use this metric to grow streamlines from a dense set of candidate seed points. The metric considers
not only Euclidean distance, but also a simple statistical measure of shape and directional similarity. Without explicit feature detection,
our method produces streamlines that naturally accentuate regions of geometric interest.
In conjunction with this method, we also propose a quantitative error metric for evaluating a streamline representation based on
how well it preserves the information from the original vector field. This error metric reconstructs a vector field from points on the
streamline representation and computes a difference of the reconstruction from the original vector field.
Index Terms—Adaptive streamlines, vector field reconstruction, shape matching.
1 I NTRODUCTION
Vector fields are commonly used to represent physical properties such
as particle velocity or magnetic field across some domain. Visualiza-
tion of vector fields is important for performing qualitative analysis
in areas such as astronomy, aeronautics, meteorology, and medicine.
The most common approaches are based on either line-integral con-
volution, which uses dense, highly local integration, or streamlines,
which are longer integral curves with a more discrete, global flavor.
One particular area of scientific interest to us is the study of the
turbulent behavior of ionized gases around a black hole. Velocity and
magneticfielddataareacquiredusingMHD(magnetohydrodynamics)
simulation on a spherical domain. Because the domain of interest is a
general 3D volume and is not restricted to a surface, sparse streamline
approaches are preferable to textured approaches such as line-integral
convolution.
As a sparse representation of a vector field, the quality of a set of
streamlines is highly dependent on their placement, which includes a
seed location and a length for each streamline.
If a specific type of feature of interest is known for the given appli-
cation area and is mathematically well-defined, a feature-guided algo-
rithm can tailor the placement to accentuate these features of interest.
However, the discrete classification of features may suffer from ro-
bustness problems in practice.
When a feature-based approach is not applicable, a density-guided
(or distance-guided) approach is typically employed. Density-guided
approaches place streamlines to enforce a user-specified density func-
tion across the domain. The function may be constant, producing
roughly evenly spaced streamlines, or it may be spatially varying (e.g.,
it may be mapped to vector magnitude or some other interesting scalar
function). Notice that in these techniques, the density of streamlines
is independent of the underlying flow function.
In this paper, we propose a similarity-guided placement strategy.
We define a similarity distance as a metric for computation of local
distance between streamlines. The similarity distance accounts not
only for the closest distance from a point on one streamline to another
streamline, but also accounts for their similarity of shape and direc-tion. This is accomplished by examining the statistics of the Euclidean
distance function as measured over a spatial window.
We employ this new similarity distance metric in the context of a
simple, greedy algorithm for streamline generation. A large number
of seed points are randomly ordered in a processing queue. Each seed
point is grown using Runge-Kutta integration until its similarity dis-
tance from any streamline would fall below a prespecified similarity
tolerance. This process may be performed in multiple passes to pro-
mote streamline fairness and to generate a multi-resolution streamline
representation. As in some other density-guided approaches, the simi-
larity tolerances may either be constant across the domain or spatially
varying. We show that by using the similarity distance as a metric,
even this simple placement strategy produces streamlines that natu-
rally accentuate regions of geometric interest, with sparser streamlines
in regions of more parallel flow.
In addition to visually demonstrating how the similarity-guided
placement algorithm behaves on simple and complex vector fields in
two and three dimensions, we introduce an error metric for streamline
representations. This error metric is based on the idea that the stream-
lines are an alternative representation for the vector field, and should
thus strive to preserve the information contained in the vector field.
Our approach is to reconstruct a vector field from points on the stream-
lines and compute a difference of the reconstructed vector field from
the original. We show that the similarity-guided placement strategy
produces a significantly lower error for a given number of streamlines
than a pure distance-guided strategy.
2 R ELATED W ORK
Flow visualization has a rich literature, with a number of surveys avail-
able [4, 7]. We consider here the most relevant work in streamline
visualization.
There are two competing goals in streamline visualization. One is
to represent interesting features as well as possible. In principle, if
we are allowed to place streamlines densely enough, we will not miss
any information of the field. But practically, dense representations can
cause severe clutter problems, especially in 3D. Thus the second goal
is to have as little clutter as possible. Both feature-guided and density-
guided placements try to find a good balance between these two goals.
Feature-guided strategies begin by identifying particular features
in the field, that are of interest for a certain research or engineering
problem, such as boundary layers, separation lines, separation bub-
bles, critical points, shock waves and so on. For many applications,
the vicinity of critical points contains an interesting flow pattern. A
critical point is defined as a point in the vector field where all compo-
nents of the vector are zeros and the streamline slope is indeterminate.
So no streamline will pass through it. For example, in a velocity field,critical points are those with zero velocity magnitude. Critical points
arefurthercategorizedintodifferenttypes. Thefieldisthensegmented
into regions, each containing a single critical point. Different seeding
patterns may be used around the vicinity of the different critical points.
Finally, some additional seed points are randomly distributed around
the field.
Verma et al. [14] first proposed a feature-guided placement strat-
egy for 2D vector field visualization. Ye et al. [15] extended this idea
to 3D vector fields. They also provided a continuous mapping be-
tween seeding patterns and critical points, which accommodates the
variability of critical points in 3D. In some applications, the number
of critical points is small, so the number of streamlines generated by
feature-guided strategies is likewise relatively small, and the clutter
problem is minimal. But with a large number of critical points, the
feature-guided approach won’t necessarily produce a sparse stream-
line representation. Another drawback is the requirement of feature
identification, which is not always available or easy.
Density-based strategies are usually associated with some density
or distance metric. Turk et.al [13] proposed an image-guided place-
ment strategy in 2D, which defined the density as the gray level of
the low-pass filtered version of the image. Their optimization process
adjusts the streamline placement to achieve a desired density by itera-
tively adding, growing, shrinking, merging, or removing streamlines.
Jobard and Lefer [5] proposed an evenly-spaced streamline place-
ment algorithm in 2D where the density is defined by the Euclidean
distance between adjacent streamlines. Mebarki et al. [10] proposed a
farthest point seeding strategy for 2D vector fields. This greedy algo-
rithm places one streamline at a time, always at the farthest point away
from all existing streamlines to optimize continuity (i.e., it promotes
long streamlines). They also used the Euclidean distance between ad-
jacent streamlines as their density metric. Mattausch et al. [9] ex-
tended the idea of evenly spaced placement into 3D and used the 3D
Euclidean distance as the metric. Liu et. al [16] introduced double
queues for prioritizing topological seeding and favoring long stream-
lines. They also use cubic Hermite polynomial interpolation with
RK4-ASSEC (fourth order Runge-Kutta integrator with adaptive step
size and error control) to accelerate placement generation.
Li et al. [8] proposed a 3D image-space streamline placement
method. They control the seeding and generation of streamlines in
image space to avoid visual cluttering. This approach guarantees the
minimum spacing between adjacent streamlines on the image plane
and solves the clutter problem directly. But the quality of placement
is highly dependent on a precomputed depth map, which provides the
depth of 2D seeds in 3D object space. Schlemmer et al. [12] defined
the streamline density of a region as the ratio between the number of
occupied pixels by streamlines and the total number of pixels in the
region.
3 S IMILARITY D ISTANCE
3.1 Definition
The similarity distance is a window-based, point-to-streamline dis-
tance, where d sim (p,s i ,s j ) is the similarity distance from point p on
streamline s i to streamline s j . For point p on streamline s i , the win-
dow around p is a portion of streamline s i centered at p with a length
of w. The value of the similarity distance is computed as follows:
1. Locate windows: The first window in the comparison is cen-
tered at p. The other window is centered at q, the point on s j
with the smallest Euclidean distance to p. p and q are actual
sample points generated by the adaptive Runge-Kutta integration
process.
If i == j, we are checking for self-similarity. In that case, we
disqualify points that are too topologically close to p from con-
sideration; all points under consideration for q must be at least
d min away from p as measured along the streamline. Thus the
streamline i is only becoming self-similar if the window around
p is similar to the window around some topologically distant part
of s i
2. Sample windows: Center a window of size w about p (measured
along the streamline), and uniformly resample using m ordered
points, yielding p 0 ,…,p m−1 . Similarly, uniformly resample over
the window of size w about q using m ordered points, yielding
q 0 ,…,q m−1 .
3. Compute similarity: The result of the preceding resampling
process is m+1 pairs of corresponding points. We use the dis-
tances between these corresponding points to compute the simi-
larity between the two windows. Notice, as shown in Figure 1,
that the orientation is significant. Thus two parallel streamlines
are less similar if they the flow is going in opposite directions.
We now compute the overall similarity distance from p to s j :
3.2 Influence Factors
The two major terms of the similarity distance formulation Eq.(1), cor-
respond to the two major influence factors respectively: (a) transla-
tional distance and (b) shape and orientation.
3.2.1 Translational Distance
The first term contributes the effects of Euclidean distance to the sim-
ilarity distance. Given two streamlines s i and s j , translating them fur-
ther from each other without otherwise changing their shape or ori-
entation will monotonically increase the first term, while holding the
second term constant. Thus all local similarity distances between the
two streamlines will also increase monotonically as the translation in-
creases.
3.2.2 Shape and Orientation
The second term contributes the effects of shape and orientation to
the similarity distance. If the two streamline windows have identical
shape and orientation, the second term is zero. As we deform parallel
streamlines apart from each other without translating the window cen-
ters, the second term increases. Similarly, reversing the orientation of
one of the streamlines increases the second term.
The second term measures the average deviation of point pair dis-
tances from the center point pair (translational) distance. In fact, if we
replace the center point distance in the formula with the mean of point
pair distances, and replace the L 1 norm accumulation with an L 2 norm,
then the second term becomes the standard deviation of the point pair
distances. In fact, we have experimented with using the mean and the
standard deviation in the first and second terms, and achieved fairly
similar results. However, we prefer the formulation presented above,
because it gives special importance to the window center, which is the
point at which we are growing a new streamline.
The shape coefficient, α , determines the relative importance of the
two influence factors. In our examples, we have found values between
1 and 3 to be the most useful, with little noticeable change as we in-
crease beyond that (this may have some interesting significance if you
consider the second term to be a standard deviation). Notice that set-
ting either α or the window size to zero results in a pure, Euclidean
distance between p and s j .
Figure 2 illustrates how the similarity distance is affected by shape
and orientation. 2(a) through 2(c) with the same Euclidean distance
between center point pair (p,q) are in order of increasing similarity
distance. In 2(a), streamline i and j have similar shape and orientation.
In 2(b), the shapes are quite different, but they both point from right to
left. In 2(c), the shapes are the same as 2(b), but the direction of j is
reversed, resulting in a larger similarity distance.
4 S TREAMLINE P LACEMENT
We can use the similarity distance metric from Section 3 to place
streamlines over a domain. As in other distance-based streamline
placement algorithms, the user chooses a distance of separation, d sep ,
to enforce between streamlines. In the case of our similarity met-
ric, d sep will be the minimum distance between parallel streamlines.
Where thestreamlines are not parallel, the minimum separation will be
reduced by some amount as controlled by setting the shape coefficient,
α .
Figure 3 shows a comparison of streamlines generated using a Eu-
clidean distance metric and our similarity distance metric. Both im-
ages contain 50 streamlines, but the left image uses a window size of
zero, and the right image uses a window size that is six percent of the
domain width (d sep for the left image is set lower to achieve the same
streamline count). The right image uses the 50 streamlines to more
effectively portray the geometrically interesting regions, such as those
near the critical points. Although there may be some natural incli-
nation to judge the left image as prettier because it avoids the darker
regions of increased density, the right image clearly conveys more in-
formation about the flow about the critical points. Figure 4 shows a
4.1 Algorithm
Totesttheuseofsimilarityasametricforplacingstreamlines, wehave
developed a simple, greedy algorithm for streamline growth from seed
points.
The pseudocode for our basic placement algorithm is shown in Fig-
ure 5. Placement begins by inserting a densely-sampled set of seed
points into the Seeds queue in a random order. For each point in the
queue, we iteratively grow the streamline by applying adaptive fourth
order Runge-Kutta integration. As we grow the streamline, we check
its similarity distance to all the previously placed streamlines. When
the similarity distance falls below the user specified threshold, d Sep ,
or if we reach a boundary of the volume or if integration becomes
undefined due to a nearby critical point, we terminate the growth of
that streamline. In the case of self separation (comparing a streamline
to itself), we generally use a significantly smaller separation distance,
d Sel fSep , to allow for the case of streamlines that form a closed loop.
If the resulting streamline is longer than the desired minimum length,
it is added to Lines, the set of placed streamlines. Otherwise, it is
discarded.
In our implementation, we accelerate the above algorithm by plac-
ing all the sample points of the streamlines into a uniform grid, where
the cell size is proportional to the separation distance. Because the
similarity distance between point p on streamline s i and point q on
streamline s j is always greater than or equal to the Euclidean distance
between p and q, there is no need to find the closest point on stream-
lines that are farther from p than the separation distance, nor do we
need to compute the actual similarity distance in that case. Of course,
more sophisticated space partitioning structures may be used in place
of the uniform grid, which has been sufficient for our test examples.
Although this simple placement algorithm seems to work reason-
ably well, it does have some shortcomings. It is unfair to the stream-
lines, favoring those with seeds early in the random queue over those
later in the queue. The earlier seeds have the opportunity to grow
longer, although there may be nothing particularly special about them.
One could order the seeds according to some importance function, but
often one cannot predict the importance of a streamline simply from
its seed position.
One way to mitigate this situation is to grow the streamlines in sev-
eral passes, with a schedule of decreasing separation distances. This is
more fair to all streamlines, but does tend to produce shorter stream-
lines overall.
We believe it may be possible to achieve even better results
by incorporating the similarity distance into a richer optimization
framework with more legal moves for exploring the space of place-
ments [13]. This is beyond the scope of this paper, but seems quite
promising for the future.
4.2 Parameters
At first it seems that there are a large number of parameters to be set in
our method. However, there is some logical interdependence among
these that allows most of them to be set automatically in most cases.
• d sep : This is the primary parameter for adjustment, which we set
as a fraction of the domain width. It varies from 2.0% to 10.0%
of the domain width in our examples shown in the paper.
• α : The most useful range for the shape coefficient is between 1
and 3. We typically get reasonable results by setting this between
2 and 3.
• w: The window width is also set as a fraction of the domain
width. We usually set w in the range of 1 to 5 times of d sep .
• d sel fsep : The self-separation distance is used for terminating
or closing a streamline due to self-proximity. This should be
smaller than d sep , and we generally set it to d sep /10.
• d min : The minimum streamline length away from the central
point for testing self-separation should be greater than d sel fsep
to prevent the two windows from having common samples (be-
cause of topological overlap). If it is too large, it will force small
loops to go around multiple times before closing. We generally
set this to be 3 to 10 times of d sel fsep .
• minLen: The minimum streamline length is largely a matter of
taste. Keeping more of the short streamlines will prevent some
potentially long streamlines from growing. It can be set as a
multiple of the w, d sep , or the domain width. We typically set it
to w∗2.
• numSeeds: In all of our tests, we use every sample in the vector
field as a seed. It is possible to use more or fewer seeds if you
know that the vector field is relatively over- or under-sampled
with respect to the features it contains.
We usually define the domain width as the smallest one among all
dimensions.
4.3 Favoring Interesting Features
In addition to Euclidean distance, the similarity distance measures the
difference in shape and orientation between streamlines. Thus it nat-
urally favors interesting regions, such as the vicinity of critical points
and separating planes.
4.3.1 Critical Points
Many applications consider the critical points of vector fields to
be interesting. As such, many feature-guided streamline placement
strategies[14, 15] explicitly choose streamlines which emphasize the
vicinityofthesecriticalpoints. InFigure6, weshowin2Dthatthedis-
tance between streamlines is naturally reduced around critical points
as compared to parallel flows (they may even visually touch when ren-
dered onto a discrete pixel grid). This is because the second term
of Eq. 1 grows when the streamline integration approaches a sink or
source.
We can also observe this behavior on fields with multiple critical
points in 2D and 3D (see Figures 3 and 4). In fact, we can see in Fig-
ure 7 that similarity-guided placement can produce results somewhat
comparable to those of [15], but without requiring explicit detection of
critical points or special case analysis based on type of critical point.
Note, however, that our algorithm does not guarantee emphasis of all
critical points, because it uses no such feature detection. Whether the
streamlines will miss critical points is dependent on the parameter val-
ues used as well as the spacing between critical points in the data. We
can capture all the critical points in Figure 7 with comparable clarity,
but it does require more streamlines than the feature-specific approach.
4.3.2 Separation
Separation is frequently studied in many fields, such as fluid dynamic
or aeronautics. However, it is harder to define in closed form than
a critical point, and it has higher dimension. Separating curves and
surfaces are global features which no streamline will cross. The sepa-
ration is either a closed curve/surface or intersects the boundary of the
domain.
Our similarity distance-based placement will emphasize these sep-
arations similarly to emphasizing critical points. Figure 8 shows two
sets of streamlines generated by a typical Euclidean distance method
and by our method. Our method allows one to see the presence of a
separating curve much more clearly.
4.4 Variable Density
In our method, the similarity distance is defined locally. Thus it is pos-
sible to specify the separation distance, d sep , as a function of position
in the domain rather than as a constant. This allows us to achieve vari-
able density of streamlines. The separation distance is in this case a
scalar field over the domain. It can be specified either a function to be
evaluated or in sampled form (i.e., a texture image over the domain).
It can be chosen manually to highlight some particular spatial region
of interest, or it can be derived from the vector field or one of its as-
sociated scalar fields. Figure 9 shows a simple example of this effect,
where the separation distance decreases linearly from the left to the
right side of the domain.
4.5 Multiresolution
Exploring a complex vector field may require multiresolution stream-
lineplacement. Generally, aroughrepresentationonthewholedomain
is provided at the beginning. Then the user may want to define the re-
gion of interest and have more detailed visualization there. Jobard
and Lefer [6] proposed a nested streamline placements with increas-
ing density. Their method doesn’t give the streamline placed at a given
level the chance to grow further in the finer levels. So it does have the
limitation on producing long streamlines.In the method proposed by
Mebarki et.al[10], they favor long streamlines by elongating all previ-
ously placed streamlines before each new placement of streamlines at
all levels.
Themulti-passversionofourplacementalgorithm, describedabove
in Section 4.1, can be used as the basis for a simple multi-resolution
streamline representation. Each pass successively reduces the sepa-
ration distance, producing increasing levels of detail. By storing with
each streamline the range of samples to be used for each level of detail,
we can render at any of these levels of detail on demand. The result of
this can be seen in Figure 10. We see three increasing levels of detail.
With each new level, previously existing streamlines may grow, and
new streamlines may be added. For comparison, we also see the result
of generating the smallest separation distance streamline set using a
single pass.
In our method, the previously placed streamlines and new seed
points have a fairly similar chance to grow at each level.
5 E RROR F UNCTION
The goal of scientific visualization is to use graphical tools to reveal
underlying properties of the data field in an easy-to-understand way.
Given a set of relatively sparse streamlines, our brain may perform
some form of interpolation to understand the blank areas between
streamlines. So ideally, the result of such interpolation should agree
with the underlying field data. In other words, if the visualization suc-
cessfully represents most of information contained in the field, then
the field data could be fully reconstructed from the visualization. In-
spired by this, we define an error function for streamline-based vector
field visualization.
5.1 Definition
Given a vector field,V, containing n grid points,V 0 ..V n−1 , let S denote
a set of sampled streamlines. If V
? (S) is the vector field reconstructed
using S and resampled at the original n grid points, then we define the
error field as the residual magnitude:
We intentionally use the residual in our formulation rather than the
angle between the vectors because it is more general – it applies even
if the vectors are not unit length. This brings up the interesting point
that a basic streamline representation can only represent the directions
ofV, but not the magnitudes. In this case, we take the residual between
normalized versions of V and V
? (S). However, in some cases, stream-
line representations may logically include the magnitudes at each sam-
ple. This makes sense, for example, if the magnitudes are used to color
the streamlines. In that case, we can reconstruct a non-unit-length
V
? (S) and take the residual between the two non-unit-length vector
fields
5.2 Vector Field Reconstruction
Reconstruction of a vector field from streamline sample points is a
scattered data reconstruction problem. Although more sophisticated
solutions to this problem are known (e.g., [2, 3, 11]), we use a linear
interpolation approach as a proof of concept.
We first compute the Delaunay triangulation of the streamline sam-
ples in 2D or 3D (we use Qhull [1] in our current implementation). We
currently use only unconstrained triangulation because the constrained
Delaunay triangulation in 3D may be ill-posed, requiring the addition
of extra samples (i.e., Steiner points). Next, for every sample point,
we compute a tangent vector using local differences and normalize it.
Toperformthereconstructionandresamplingofthevectorfield, we
perform a point location query in the triangulation for each grid point,
then perform linear interpolation of the cell corners using barycentric
coordinates and normalize the result. In the case of streamlines with
magnitude values, it may be desirable to interpolate the vectors and
the magnitudes separately. For points outside the convex hull of the
sample points, we clamp to the nearest value on the convex hull as
a simple extrapolation technique. We currently accelerate the point
location queries using a uniform grid.
Figure 11 shows the residual magnitude fields for Euclidean
distance-based placement versus similarity-based placement with the
same number of streamlines (using the streamlines from Figure 3).
It is not surprising that most noticeable errors occur near the critical
points, and these tend to be reduced by our approach.
6 R ESULTS
So far, we have shown results on fairly simple, synthetic data as a
way of validating our approach. Now we demonstrate on two more
complicateddatasetsfromrealscientificsimulations, thenreportsome
data from all the models.
Figure 14 shows a 2D turbulent flow from a simulation of a swirling
jet. It simulates the situation with an inflow into a steady medium. The
domain is a structured grid with 251x159 cells. The topology of the
flow has a very complicated structure and contains about 300 critical
points. The extremely dense distribution of critical points makes it
hard to seed streamlines using feature-guided strategies. Figure 14(a)
shows the result using the typical Euclidean distance placement with-
out detecting self closure. It contains some obvious visual clutter ar-
tifacts. The computational time is roughly three times longer than the
others in Figure 14 because those closed streamlines will only stop
integration when the integration length is beyond some user-defined
maximum length, consuming unnecessary computational time. In Fig-
ure 14(b), we still use the Euclidean distance, but terminate the in-
tegration of closed curves. Figure 14(c) shows the result from our
similarity-guided strategy with self closure detection. The separation
distance is reduced from (a) and (b) so that we get the same num-
ber of streamlines in all three. Figure 14(c) favors interesting regions
with longer and more streamlines. We can see more structure in (c)
than in (b) almost anywhere we look closely. For comparison, Fig-
ure 14(d) uses the Euclidean distance based method with the same dis-
tance threshold of Figure 14(c), so it uses fewer streamlines overall,
and captures even less structure than (b) or (c).
Figure 13 shows a 3D magnetic field from an MHD simulation of
ionized gases around a black hole. The field is symmetric, so the
streamlines may be traced through a spherical volume, though the data
is in one sphere quadrant (minus some small cutouts at the center and
down the polar axis). The curvilinear grid has a huge range of cell
sizes, with the smallest ones at the smallest radii and closest to the
equatorial plane. The magnetic field has no critical points. In Fig-
ure 13(a), we use a variable density function to emphasize the region
close to the polar axis. Streamlines in this region tend to follow a
fairly tight coil up the axis. In Figure 13(b), we emphasize the stream-
lines near the equatorial plane, which tend to wind around in a much
broader pattern.
We have implemented the streamline generation algorithm on a
Linux PC equipped with 2.4 GHz dual AMD Opteron 250 CPU, 8GB
RAM. Table 1 presents some timing data for several test models. We
use one seed point per grid sample. The 1.vec data (Figure 10) is
a simple 2D synthetic data set provided with Abdelkrim Mebarki’s
streamline placement package. The om08 (Figure 14 is a complex
simulation of a swirling jet with an inflow into a steady medium. The
3d.vec (Figure 4) is a synthetic data set we generated to contain six
critical points. KdPhrg (Figure 13) is a magnetic field resulting from
a relativistic MHD simulation of ionized gas in proximity to a black
hole. Figure 12 demonstrates the relation between the processing time
and the number of streamlines generated. Since the similarity distance
check is the most crucial operation in our algorithm, the computing
time grows fairly linearly with the number of generated streamlines.
Thereisalsoasmallfactorcorrespondingtothenumberofseedpoints.
Table 2 shows the reconstruction errors of our method and Eu-
clidean distance based method on 2D and 3D data sets.
7 C ONCLUSION AND F UTURE W ORK
We have proposed the idea of a similarity metric for measuring stream-
line proximity, and demonstrated a streamline placement algorithm us-
ing similarity to drive the selection of streamlines and their lengths.
This new approach favors streamlines near interesting flow features,
such as critical points and separations, without the need to explicitly
enumerate these features.
We have also introduced an error metric for streamline representa-
tions to measure how well they preserve the information in the orig-
inal vector field. Using the metric, we can see that similarity-guided
streamline placement outperforms Euclidean distance-guided place-
ment not only visually, but quantitatively as well.
The placement algorithm still has some room for improvement. It
may benefit from more carefully chosen order of seed point growth,
such as the farthest point formulation of [10]. It may also be beneficial
to add more types of moves to our optimization process, as in [13],
making the initial placement order less crucial and less sensitive to a
particular randomization. A more efficient spatial search structure can
be used to improve the speed of generating the streamlines.
We choose to use a very intuitive and simple reconstruction method
for vector field reconstruction in our error measurement, which con-
sists of Delaunay triangulation and linear interpolation. It would be
interesting to see some other reconstruction methods, which may be
more suitable for this application.