Robust Clustering of Eye Movement Recordings for Quanti

In Eye Tracking Research and Applications (ETRA) Symposium 2004

Robust Clustering of Eye Movement Recordings
for Quantification of Visual Interest
Anthony Santella

Doug DeCarlo

Department of Computer Science
Center for Cognitive Science
Rutgers University

Abstract

These clusters can inform investigations into higher-level questions
about how an image is examined. The analysis is based upon
an existing algorithm that can be used for robust clustering—the
mean shift procedure [Fukunaga and Hostetler 1975; Comaniciu
and Meer 2002]—adapted for eye movement data. This procedure
decides the number and arrangement of clusters deterministically,
and hence is entirely data-driven. It is robust because the results
are not adversely affected by noise or outliers. We compare this
approach to methods based on other clustering algorithms.
A wealth of different measurements of eye movements are possible [Inhoff and Radach 1998]. Measuring the location of particular
objects or features that attracted the viewer’s focus is among the
more common objectives. Hand-coding of eye movement data to
realize such measurements produces sound results (i.e. [Harris et al.
1988]); a trained observer easily parses eye movement recordings
into fixations, gazes, or regions-of-interest (in the context of a particular experiment). The time-consuming nature of manual coding,
however, has led to the development of automated approaches.
The most common automatic techniques simply extract fixations
by removing saccades from a set of point-of-regard measurements.
The constrained nature of eye movements invites simple algorithms
based on applying velocity thresholds and enforcing lower-bounds
on fixation durations. These algorithms are quite effective in many
situations—see [Salvucci and Goldberg 2000] for a thorough description and comparative evaluation. Careful modeling of the processes involved can lead to more robust estimation procedures. For
instance, modeling eye movements using a state variable that declares each moment as either a fixation or saccade leads to the use
of Hidden Markov models (HMMs) as an estimation tool [Salvucci
and Goldberg 2000]. The success of all of these algorithms comes
from their dependence on the structure of fixations and saccades;
but this dependence is also what prevents their more general application in detecting higher-level features like regions-of-interest.
Higher-level measurements of viewer interest have remained, on
the whole, relatively simple. This hasn’t been a serious limitation—
the design of psychological experiments is often guided by the need
to take simple measurements. For instance, one methodology divides the viewing area into a regular grid, and tallies the time spent
inside each square [Mackworth and Morandi 1967]. Another common technique defines rectangular target regions (also described in
[Salvucci and Goldberg 2000]), and records the fixations by the
viewer inside each region. Such a method is well suited to situations such as reading, where each word is bounded by a rectangle
[Just and Carpenter 1980]. (Several commercial eye tracking systems provide this functionality.)
Data-driven algorithms work in terms of the point-of-regard locations themselves to define the regions-of-interest; these are more
flexible than measurements performed using a pre-defined grid or
box, and hence admit fine-grained analysis. Data-driven algorithms
have seen little deployment in the analysis of eye movements. But
they are required when there is no a priori structure that describes
the stimulus. In these cases, they can be used to simultaneously
locate areas of importance and quantify interest in them.

Characterizing the location and extent of a viewer’s interest, in
terms of eye movement recordings, informs a range of investigations in image and scene viewing. We present an automatic
data-driven method for accomplishing this, which clusters visual
point-of-regard (POR) measurements into gazes and regions-ofinterest using the mean shift procedure. Clusters produced using
this method form a structured representation of viewer interest, and
at the same time are replicable and not heavily influenced by noise
or outliers. Thus, they are useful in answering fine-grained questions about where and how a viewer examined an image.
Keywords: eye movement analysis, measures of visual interest,
clustering, mean shift

1

Introduction

Human eye movements provide strong evidence about the location of meaningful content in an image or scene [Mackworth and
Morandi 1967; Just and Carpenter 1976; Henderson and Hollingworth 1998]. And while the location of the meaningful content
changes based on the viewer’s task, so do their eye movements
[Yarbus 1967; Just and Carpenter 1976]. This fundamental fact
about human vision explains and motivates the need for algorithms
that concisely quantify areas of a viewer’s focus, so called regionsof-interest, in terms of a recording of their eye movements. Such
methods are effective and time-saving tools for conducting psychological research on eye movements, and are a necessary ingredient
in human-machine interfaces that use eye tracking technology.
This paper describes a clustering algorithm that processes pointof-regard (POR) measurements from an eye-tracker into collections
that are either:
• grouped spatially as in Figure 1(d), indicating the viewer’s
regions-of-interest; or
• grouped spatially and temporally—ranging from individual
fixations as in Figure 1(b), to gazes—sets of sequential fixations in a confined part of the viewing area [Just and Carpenter
1980], as in Figure 1(c).

1


(a)

(b)

(c)

(d)

Figure 1: Clustering example (a) original image with recorded point-of-regard (POR) locations; (b) with estimated fixation locations (connected by lines); (c) with gazes (with lines connecting POR locations within each gaze); (d) with regions-of-interest (with POR)

histogram using k-means. This approach is less sensitive to noise.
Latimer also presents an more intricate approach based on an information theoretic criterion. Latimer notes the difficulty in producing consistent results automatically with k-means and similar
algorithms. (We discuss this further in Section 3.1.)

One common data-driven method of clustering uses a distance
threshold, and considers two points to be in the same cluster when
they are closer than this distance. A specific number of clusters is
obtained by simply choosing the appropriate distance threshold that
realizes it. For purposes of comparing real and simulated fixation
sequences, Privitera and Stark [2000] use such a method, choosing their threshold via k-means clustering [Duda et al. 2001]. The
clusters provide a means to relate the real and simulated fixations.
[Turano et al. 2003] proceeds similarly, but clusters quantities that
measure performance of salience models for purposes of comparison (and not the fixation locations themselves). While this simple
clustering algorithm is sufficient in situations where the clusters are
clearly delineated, its performance in more general settings would
be quite poor. Distance thresholds fail to resolve two dense but separate clusters if only two of their points happen to be close. As a
result, they cannot finely characterize gazes or regions-of-interest.

Clustering methods have also been used to extract gazes: spatial clusters of successive fixations. This is an easier problem, akin
to identifying fixations, so simpler techniques are reasonably effective. Successive fixations can be added to a cluster until the next
is too distant [Scinto and Barnette 1986]. Applying a threshold
on the distance between the next fixation location and the mean
of fixation points already in the cluster is more effective [Nodine
et al. 1992]. This technique tolerates points leading away from
the cluster (which was one difficulty with using simple thresholds).
One might imagine adapting this method for regions-of-interest by
choosing the spatially, rather than temporally closest neighbor as
the next candidate for addition. However, this would introduce a
dependence on the order of the fixations—an undesirable property
for data analysis.

More intricate algorithms exist that measure the regions-ofinterest. Latimer [1988] partitions data by forming a histogram of
fixation durations over the viewing area, and finds clusters of that

2

Other techniques provide visualizations of regions-of-interest by
adapting the original imagery. Latimer [1988] plots the distribution of fixations over stimuli. Wooding [2002] presents a technique
which darkens the image in uninteresting areas with a function defined using a mixture of Gaussians, each centered at a fixation.
Other systems create stylized transformations of images by modulating the level-of-detail in the image based on where the viewer
looked [DeCarlo and Santella 2002]. Indeed, such notions form
the basis of gaze-contingent displays that omit unnoticeable detail
[Baudisch et al. 2003].
In the next section, we start by describing our data-driven clustering method. This is followed in Section 3 by a demonstration
of our method, and a comparison of our clustering strategy with
alternative methods.

2

In general, a clustering process starts from a set of N points:
{x j | j ∈ 1..N},
and assigns one of K labels to each point:
{c j ∈ 1..K | j ∈ 1..N}.
Some algorithms produce fuzzy labelings, and may also provide
coarse descriptions such as means and covariances.
Perhaps the simplest clustering algorithm employs a distance
threshold d, and considers any two points to be part of the same
cluster when they are within distance d of each other. While this
algorithm has the desirable property of not requiring the number
of clusters to be known in advance, it is very fragile; any noise in
the data causes apparently distinct groups of points to be clustered
together.
Even so, this algorithm is salvageable. Mean shift clustering is
a robust version of distance-based clustering that includes a preprocessing stage. The entire process involves two steps:

Clustering eye movements

When an expert encodes gazes or regions-of-interest by hand, they
group nearby fixations together—they cluster the data. Although
this is a deceptively easy task for a human observer, it is a difficult
undertaking for a computer. There are many available algorithms
for clustering [Duda et al. 2001] that can automate this process.
A desirable algorithm for clustering eye movement data has three
characteristics:

1. move the points into denser configurations until they can be
easily separated into clusters
2. apply a clustering algorithm that employs a distance threshold
The first stage—known as the mean shift procedure—is crucial, as
it makes the entire process robust. It proceeds by repeatedly moving
a point xi to a new location s(xi )—the weighted mean of nearby
points based on the kernel function k:

• it should produce consistent results (and not depend on a random initial guess);
• it should not need to know the number of clusters in advance;

s(x) =

• it should be robust (so the presence of outliers does not significantly affect distant clusters).

xj

∑j k x−xj

(1)

The kernel k is typically a multivariate Gaussian with zero mean
and covariance σ 2 I [Comaniciu and Meer 2002]. (Note that this
description of the kernel has been streamlined from [Comaniciu and
Meer 2002].) Robustness to extreme outliers is achieved by simply
limiting the support of the kernel, for instance by setting it to zero
for distances greater than 2σ .
The parameter σ describes the spatial extent of the weighted
mean computed by s: it provides a scale control for the algorithm.
For example, increasing the value of σ results in fewer, larger clusters which describe coarser structures in the data. Like the method
using a simple distance threshold, the mean shift method is nonparametric: it make no assumptions about the global shape of the
distribution of the data. This is typically an advantage, as such assumptions lead to garbled results when they do not apply.
The mean shift procedure proceeds as follows:

Commonly used techniques such as Expectation-Maximization
(EM) and k-means clustering [Duda et al. 2001] are initialized randomly, and so different runs on the same data can produce different local minima. Both EM and k-means also require knowing the
number of clusters in advance. While adaptations to these algorithms exist that aim to address these issues, they can only reduce
the problem. Such arbitrary behavior would detrimentally affect
any experimental analysis, and must be avoided.
Additionally, the technique should be robust. All of the places
a viewer fixates are not really regions-of-interest. Sustained viewing will produce clusters of fixations around particularly interesting features, jittered by noise and microsaccades, this is the kind
of noise all clustering methods seek to address. However, brief isolated fixations will also be present. These are outliers of the regionsof-interest—they are not just noise. They are isolated points that
commanded some focus, perhaps just a distraction of some sort.
Characterizing them might be valuable in particular situations. We
want results for the major clusters to be unaffected by their presence, but also we want to describe these outliers, in order to analyze or discard them. Here again, EM and k-means are not robust.
While their global solution characterizes outliers correctly, the local solutions provided by these algorithms in practice can include
outliers inside larger clusters, which distorts their statistics.

2.1

∑j k x−xj

I NITIALIZATION

for j ∈ 1..N
y0 = x j
j

I TERATION n

repeat until convergence
for j ∈ 1..N
yn = s(yn−1 )
j
j

It works by interpreting the points x j as samples from a distribution.
The mean shift property, established in [Fukunaga and Hostetler
1975], estimates the gradient of the density of these samples at x
as s(x) − x. Hence, the mean shift procedure iteratively moves all
points simultaneously towards locations of higher density (in the
gradient direction): towards the eventual points of convergence,
which are the modes of the distribution. (Proofs concerning convergence behavior are in [Comaniciu and Meer 2002].)
With all of the points collected at modes, a clustering method
that uses a distance threshold is both safe and successful. In this

Mean shift

Clustering based on the mean shift procedure meets the requirements above—it proceeds deterministically, does not require knowing the number of clusters in advance, and is robust to outliers. It
has its origins in the pattern recognition community [Fukunaga and
Hostetler 1975], but has only seen recent use—particularly in robust computer vision systems [Comaniciu and Meer 2002].

3

Viewer 1:

Viewer 2:

Gaze clusters with σs = 100, σt =

POR data

Regions-of-interest with σs = 100

1
3

Figure 2: Gaze and region-of-interest clusters for two different viewers
case, a distance threshold of σ is used—the scale parameter used to
specify the kernel. Finally, clusters containing a very small fraction
of the data are optionally discarded, as they are typically outliers.

2.2

setting σt to infinity):
kspatial [xi , yi ] = exp −

The spatial scale parameter σs can be varied to produce predictable clusters on coarser or finer scales. For instance, one can determine if a viewer looked at the eye of a person, or merely whether
they looked at the entire face. A particular spatial scale choice does
not control the size of the cluster itself, it ensures that no clusters
exist which are closer than σs . Normally a σs below the distance
between two image features will resolve them into separate clusters,
assuming there is a drop-off in density in the space between them.
If both features are part of a unimodal blob of interest it will be impossible to resolve them. Given a particular experimental setup and
scale of question being posed, a single scale value will most likely
be sufficient for all data across a set of stimuli.
The value of the temporal scale σt has more to do with the average time separating fixations (we use a value of 1/3 second).
When clustering spatially and temporally, it is possible for nonconsecutive POR data to be clustered together. Although we have
seen no instances where this has occurred, these anomalies can easily be corrected during the second stage of mean shift clustering.
Given the choice of scale, the results of this algorithm are very
intuitive and predictable. Distant outliers (where distant is defined
by the scale choice) will become small clusters. Large regions of
uniform density will collapse to one cluster. Where density is locally bimodal, one cluster will result if the modes are within the
chosen scale of each other, and otherwise two clusters will result. (It
is worth contrasting this behavior which depends on the distance of
modes, with distance thresholding as in [Privitera and Stark 2000;
Turano et al. 2003] which will collapse the clusters if any of their
points are within the threshold distance of each other.) Mean shift

Formulations for POR data

Eye-trackers provide the location of a point on a particular planar surface (calibrated in advance), such as the pixel location on
a screen, and the time the measurement was taken. In this case,
xi = xi , yi ,ti . Clustering data spatially and temporally produces
clusters that describe fixations, or gaze points (groups of nearby,
sequential fixations), while spatial clustering produces regions-ofinterest. Both easily fit within the framework described in Section 2.1—the kernel function k from (1) enables this control.
A zero-mean Gaussian kernel is specified using a covariance
matrix—this matrix encodes the relative weighting of the dimensions when measuring distances. Diagonal matrices provide sufficient flexibility here, but independent control over the relative scale
between spatial and temporal dimensions is important. A small
temporal scale is called for when clustering points temporally (so
as not to cluster across non-consecutive fixations), but the spatial
scale is really application dependent. Using a spatial scale of σs
and temporal scale of σt , the kernel k is:
kspatiotemporal [xi , yi ,ti ] = exp −

x2 + y2
i
i
σs2

x2 + y2
t2
i
i
− i2
σs2
σt

(Note that the kernel does not need to be normalized given the denominator in (1).) For an analysis that ignores temporal information, we can simply exclude the temporal dimension (effectively

4

provides clusters of interest that are robust to both measurement
noise and small outlying points-of-regard, and provides a consistent and interpretable estimate of the regions-of-interest. The results are replicable, and similar shaped distributions of POR data
will produce similar clusters, making direct comparison possible.
This method should be particularly useful in characterizing the
regions-of-interest in aggregate data from a number of viewers
(over the same image). This is an interesting approach for trying to
capture general, viewer independent patterns of interest over an image. In this case, data from multiple subjects is collapsed together.
Robust clustering will extract patterns in the presence of noise and
variations in calibration. In Section 3 we show some results of our
method both on individual eye tracks and aggregate data, and argue
their value as a quantitative measure.

2.3

subjects who viewed the images as part of a larger experiment. The
eye movement recordings were performed using an ISCAN ETL500 table-top eye-tracker (with a RK-464 pan/tilt camera). The resulting point-of-regard data is expressed in pixel coordinates (the
images measure 1024 units horizontally; the forthcoming values
of σs are therefore expressed in pixel units). First, the POR data
was pre-processed to remove saccades using the dispersion-based
method of Widdel [1984] (also see [Salvucci and Goldberg 2000]).
We have also successfully used a threshold on velocity between
point samples to perform this filtering.
The examples here are representative of typical results of the proposed algorithm. The images in Figure 2 demonstrate gaze and
region-of-interest clustering for two different viewers of the same
image. On the left is the POR data superimposed on the image
(recorded for 8 seconds). At center are gaze clusters, and at the
right are regions-of-interest for this image. These locations, on the
whole, correspond to objects and groups of objects in the image.
The value of σt was 1/3 second for the each of these examples.

Computational concerns

For off-line analysis of data, computation time is not critical. However this technique could be useful for identifying ROI in interactive
systems—for example, in computer assisted examination of radiology slides [Mello-Thoms et al. 2002]. Here, computational concerns are more important. Though mean shift clustering is somewhat expensive, it is possible to adapt it to perform in near real-time
on eye movement data, under a range of circumstances.
One iteration of the mean shift procedure takes O(N 2 ). Although
the precise behavior depends on the data, we have empirically determined that 5 to 10 iterations of the mean shift procedure are sufficient for convergence to within a 0.1% change across iterations.
In this case, one minute of POR data (N = 3600) is processed in
about 3 seconds on a 2.4GHz Pentium 4 PC. Full convergence is
not necessary, however. What matters is that slightly overlapping
clusters are sufficiently separated. This should be tuned for a particular application.
The limits on the spatial and temporal extents of the kernel (discussed in Section 2.1) suggest that only the closest points are required for the computation of s; the use of a spatial hierarchy, or the
use of spatial coherence across iterations would reduce the running
time by a sizable constant.
But the structure of POR data lends itself to a particular optimization: the fixations are grouped quite tightly together. When
determining gaze points or regions-of-interest, one approximation
replaces each cluster of POR data representing a fixation, with a single point that is located at the mode of the cluster (the convergence
point), weighted by the number of points contained in that fixation.
This produces a set of fixations f j and weights w j , resulting in a
weighted version of (1):
sfix (f) =

∑ j w jk f − f j
∑ j w jk f − f j

fj

Figure 3: The image viewed by the tracking subjects
The next set of examples involve viewings of the image in Figure 3, and demonstrate region-of-interest clustering. The clustering results are presented by drawing each cluster using a different
color. (In examples with many clusters, there will be color re-use;
and there is no color correspondence across related examples.) As
some of the colors may appear similar on different displays, we additionally overlay covariance ellipses on large clusters (one for each
cluster that contains more than 2.5% of the total data); they are centered at the mean of the cluster, and mark 99% of the variance. Data
points in small clusters are colored grey. To show alignment with
the viewed image, the clusters are drawn over a set of linear features that were extracted (semi-automatically) from the photograph
in Figure 3.
The analysis demonstrates the algorithm on data from two individual viewers, as well as on data combined from six separate
viewers (all were 8 second recordings). Figure 4 shows the regionsof-interest gathered from two viewers, processed at three different
spatial resolutions (σs ). Small values of σs correspond largely to
isolated fixations, grouping together only fixations that overlapped
due to re-fixations of a particular feature. Clusters produced using
larger values of σs capture coherent regions that the viewer examined repeatedly (and not necessarily sequentially, as with gaze clusters). Comparisons between viewers seem quite reasonable with the
coarser scale clusters.
Similar behavior of the algorithm is seen for data collected together from six different viewers, which is displayed in Figure 5.

(2)

Iterating over fixations makes N effectively much smaller, and the
quadratic complexity becomes more manageable. In this case, one
minute of fixation data (with Nfix = 190) is processed in a under one
tenth of a second. For region-of-interest detection, this approximation is exact when the original POR points within fixations are perfectly aligned. Noise and drift in fixation location measurements,
as well as the temporal averaging that takes place for gaze detection
are the sources of error.

3

Results

This section demonstrates the clustering algorithm from Section 2
on a set of images viewed by either a single viewer or a group of
viewers. The data was picked at random from recordings of naive

5


4

As expected, the patterns of information in the image become
apparent—especially as σs is increased. Major clusters correspond
to both salient features in the POR data, to features in the single
user data in Figure 4, and for the most part, to features in the image. (This can be confirmed by comparing the cluster locations with
the original image in Figure 3.)
It should be noted that not all clusters correspond to regions of
significant interest. Particularly when data from multiple viewers
is combined, it is the clusters that hold the largest percentage of
the data that define the most important allocation of interest. This
is because important features are likely to be viewed longer, and
by more of the viewers. Very small clusters (colored in grey in
the examples) are outliers: they represent brief fixations of some
isolated feature. In some contexts, these may be of interest (in a
study of distracting features, for instance). In others, they can be
discarded. In either case, they are easily identified by the fact that
they contain a tiny proportion of the data.
Figure 6 repeats the computations from Figure 5, but using the
approximate solution from Section 2.3 that uses weighted fixations
in place of the raw POR data. The similarity in the output between
the approximation and full solution suggests this approximation
holds up well in these conditions.

3.1

Discussion and Conclusion

The clustering method presented here draws upon robust methods
that have deterministic behavior. It is simple to understand and implement. As a result, we expect it will become a useful tool for
detecting and analyzing the regions-of-interest of a viewer; both
for use in experimental analysis, and to automate content decisions
in interactive systems using eye trackers. This approach applies
most in situations involving unconstrained image or scene viewing.
More traditional approaches are still in order for specific domains,
such as reading (which successfully employs boxes around words).
We have not performed a formal evaluation of this algorithm as
it applies to analyzing eye movement recordings. Comparing the
results to hand-labeled data (using a range of algorithms, including mean-shift) is the most obvious course of action. Alternatively,
validation can proceed by replicating known effects established by
other means. In this case, the benefits of robustness and orderindependence would be demonstrated should lower variances be
seen in measuring these effects (when compared to analyses using
algorithms such as EM).
Improvements to the proposed clustering algorithm can come on
a number of fronts. Automatic methods for scale-selection [Lindeberg 1994] would choose optimal values of σs —the extracted clusters would be very stable under a range of scale values around the
selected value. Varying σs over the extent of the viewing area could
accommodate stimuli containing varying scales. Formulating an
on-line approximate version of the mean shift procedure, perhaps
taking advantage of the structure in POR data, would further facilitate its introduction into interactive systems.
Perhaps most interesting are the new experimental protocols and
interactive systems that will become possible with such an automated technique. The method seems to have the sensitivity and robustness to be applied to a wide variety of problems. The locations
and properties of the estimated clusters can be used to compare
the similarity of interest across viewers, tasks, or different versions
of images. This could be done spatially by comparing the cluster
centroids produced by different viewers or groups of viewers. Or,
clusters could be used to label fixations for a string matching comparison [Privitera and Stark 2000]. The clusters themselves can
also be analyzed further. The relative number of data points inside
a cluster indicates its importance. We can examine the percentages
of different viewers POR samples which are included in a cluster,
and so determine if the location is of general interest or only interesting to one or a few idiosyncratic viewers. Alternatively the time
stamps of gazes can be used to determine the percentage of interest that represents initial or reoccurring fixations. This flexibility
should make the method a useful technique for quantifying visual
interest.

Comparison of clustering methods

Eye movement data is complex and noisy, both in terms of measurement and individual behavior. Other clustering algorithms such as
k-means clustering or EM [Duda et al. 2001] tend to produce erratic
results in this case. The sources of such irregular behavior include
the choice of the number of clusters (K), randomization in the initial
guess, and statistical properties of the data. Strategies for determining an optimal K exist, and may involve running EM many times
(across a range of K values), and selecting the best fit. However,
many difficulties remain. These are algorithms with a significant
random component which makes the clusters difficult to reproduce:
running the algorithm multiple times on the same data can produce
significantly different results. To combat this, these algorithms are
in practice always run multiple times with different random initializations and the best result selected. This encourages convergence
to a global minimum, though it provides no guarantees.
Figure 7 shows three different results from such an EM based
algorithm (which tested a range of values of K, using 5 different
initializations for each, and selected the best result): it produced
K = 6 once and K = 7 twice. The randomness inherent in the algorithm clearly shows through—different runs produced quite different answers. More initializations would make the results more
consistent at increasing expense, with no principled way of knowing how many are sufficient.
We can compare the output from Expectation-Maximization,
shown in Figure 7, with the mean shift clustering (of the same
data) in Figure 5 (specifically, when σs = 75). Because these EM
estimates represent local minima, the clusters do not always correspond to features or regions in the image as consistently as the
mean shifted clusters do. Similar effects are observed when using
k-means clustering. The most obvious failure in the EM results is
robustness. This can be seen in the leftmost cluster in Figure 7 in
both the K = 6 result and the second K = 7 result. Such clusters
contain distant fixations (outliers), and are substantially distorted.
Again, more random initializations may improve the results, but
there are no guarantees.
All of this seriously impairs the interpretability and the usability
of the EM clusters. More consistent and robust clusters are necessary to answer fine-grained questions, such as whether two viewers
had similar interest in some set of image features.

Acknowledgments
Thanks to Andrew Duchowski, Eileen Kowler and Peter Meer. This
research was supported by the NSF (SGER 0227337).

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