Quantitative Comparison of Algorithms for Tracking Single
Michael K. Cheezum, William F. Walker, and William H. Guilford
Department of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22908 USA
cell membranes to the movement of molecular motors. A plethora of computer algorithms have been developed to monitor
the sub-pixel displacement of fluorescent objects between successive video frames, and some have been claimed to have
“nanometer” resolution. To date, there has been no rigorous comparison of these algorithms under realistic conditions. In this
paper, we quantitatively compare specific implementations of four commonly used tracking algorithms: cross-correlation,
sum-absolute difference, centroid, and direct Gaussian fit. Images of fluorescent objects ranging in size from point sources
to 5 ?m were computer generated with known sub-pixel displacements. Realistic noise was added and the above four
algorithms were compared for accuracy and precision. We found that cross-correlation is the most accurate algorithm for
large particles. However, for point sources, direct Gaussian fit to the intensity distribution is the superior algorithm in terms
of both accuracy and precision, and is the most robust at low signal-to-noise. Most significantly, all four algorithms fail as the
signal-to-noise ratio approaches 4. We judge direct Gaussian fit to be the best algorithm when tracking single fluorophores,
where the signal-to-noise is frequently near 4.
Single particle tracking has seen numerous applications in biophysics, ranging from the diffusion of proteins in
Single particle tracking is the use of computer analysis of
video images to follow the sub-micron motion of individual
organelles, microspheres, and molecules under microscopic
observation. This technique has seen numerous applications
in biophysics, including the diffusion of proteins in cell
membranes (Ghosh and Webb, 1994), kinesin-driven move-
ment of beads on microtubules (Gelles et al., 1988), and the
myosin-driven movement of actin filaments in vitro (Work
and Warshaw, 1992). Using this technique, investigators
have been able to estimate the diffusion coefficients of
proteins in cell membranes and the step displacements gen-
erated by “molecular motors.” For an excellent overview of
the technique, see Saxton and Jacobson (1997).
A plethora of methods have been used for tracking single
particles. All include two basic steps. The first is segmen-
tation, in which multiple particles in a field of view are
identified and discriminated. Subsequently, an algorithm
tracks the particles individually to monitor their displace-
ment between successive video frames. The performance of
the tracking algorithm (rather than the segmentation algo-
rithm) defines the fundamental performance limit of the
Tracking algorithms used to date have included cross-
correlation of subsequent images (Gelles et al., 1988; Ku-
sumi et al., 1993; Guilford and Gore, 1995), calculating the
center-of-mass (centroid) of the object of interest (Ghosh
and Webb, 1994; Lee et al., 1991), and directly fitting
Gaussian curves to the intensity profile (Anderson et al.,
1992; Schu ¨tz et al., 1997). Some groups have claimed
“nanometer” resolution. Unfortunately, there have been no
studies quantitatively comparing the efficacy of these algo-
rithms under a variety of conditions. Complicating matters
is the fact that many laboratories develop custom-written
computer programs for analyzing the data, and incorporate
additional thresholds and filters to improve the consistency
of their results. This shortcoming is of particular concern
since the advent of single fluorophore imaging, in which
single fluorescent molecules are observed using intensified
video cameras (Sonnleitner et al., 1999; Schu ¨tz et al., 1997;
Goulian and Simon, 2000). The signal-to-noise ratio (S/N)
in these studies can be as low as 3 or 4 (Kubitscheck et al.,
2000). Thus, finding the best algorithm for use under these
conditions, and knowing its limitations, is vital.
Tracking algorithms may suffer two sorts of errors—
determinate and indeterminate. Determinate errors are the
result of inaccuracies inherent to the algorithm, systemati-
cally biasing the results toward incorrect values. Indetermi-
nate errors, as the name implies, cause the individual mea-
surements to fluctuate randomly, and generally result from
sensitivity to underlying noise in the data. Measures of these
two error types are colloquially referred to as the “accuracy”
and “precision” of the algorithm, respectively.
In general, investigators assess the efficacy of the track-
ing algorithm by tracking the position of a stationary object,
and taking fluctuations in the measured position as a mea-
sure of accuracy or precision. Although this technique may
be used to set a lower bound on the detectable motion of the
particle, to know the accuracy of a tracking algorithm, one
must compare the estimated to the actual position of the
object being tracked. Unfortunately, there is no way to
know the exact position of a real object relative to the
imaging array. One group simulated and tracked images of
an immobile point source at a fixed S/N ratio, under the
Received for publication 8 January 2001 and in final form 25 June 2001.
Address reprint requests to William H. O. Guilford, Biomedical Engineer-
ing, 1105 West Main St., Charlottesville, VA 22903. Tel.: 434-243-2740;
Fax: 804-982-3870; E-mail: email@example.com.
© 2001 by the Biophysical Society
2378Biophysical JournalVolume 81October 2001 2378–2388
assumption that the S/N fully determines the accuracy of the
apparatus and algorithm (Schu ¨tz et al., 1997). However, the
accuracy and precision of a tracking algorithm are depen-
dent upon the noise, the position of the source relative to the
imaging array, the shape and intensity of the object, and the
spatial resolution of the imaging system.
The purpose of this study is to quantitatively compare the
efficacy of four commonly used tracking algorithms: cross-
correlation, sum-absolute difference (SAD), centroid, and
direct Gaussian fit. Images of fluorescent particles with
diameters both greater than and less than the wavelength of
the emitted light (?) were computer generated, convolved
with an appropriate point-spread function, and resampled
with known sub-pixel displacement into a lower resolution
array representing a video camera. Shot noise was added to
give S/N ranging from ? to as low as 1.3. One thousand
iterations of image pairs were compared with specific im-
plementations of each of the four algorithms for every
combination of actual displacement and noise level. The
bias error (accuracy) and standard deviation (precision) of
the algorithm were determined by comparison to the known
displacement. We show that, for the case of a sub-wave-
length diameter particle, direct Gaussian fit to the intensity
distribution is the superior algorithm in terms of both accu-
racy and precision. However, all four algorithms fail as the
S/N approaches 4. Cross-correlation offers the best perfor-
mance of the four algorithms when the diameter of the
particle is ??. These data have important ramifications for
single fluorophore imaging, where the S/N is frequently
MATERIALS AND METHODS
To create an accurate model for a fluorescent object imaged with a
charge-coupled device (CCD), we first created a high resolution matrix
containing the initial, noise-free object function (i.e., image) of the particle
to be tracked (Fig. 1 A). Each cell in the matrix contains the corresponding
intensity of the target in space. Three target sizes were used: a point source
(d ? ? ?), a cell-sized object (d ? ?), and an object on the scale of the
wavelength of light (d ? ?). Assuming a fluorescence emission wavelength
of 570 nm tetramethylrhodomine isothiocyanate (TRITC), high-resolution
object functions were created of targets sized one pixel (9 nm), 0.5 ?m, and
5.0 ?m in diameter. The object functions were constructed by assigning a
“white level” of 10,000 to each element within the radius of the object, and
a “black level” of 0 to all other cells. These levels were later scaled.
Objects viewed through a microscope are distorted by the point-spread
function of the objective. To more faithfully model the distribution of
intensities in the high-resolution image, we convolved the high-resolution
object function with an appropriate point-spread function (PSF),
r is the distance from the origin, NA is the numerical aperture of the
objective (1.3), ? is the wavelength of light (570 nm), and J1is the Bessel
function (Young, 1996). The PSF has a radius (PSF ? 0) of approximately
0.27 ?m (?30 pixels in the high-resolution object function). We acquired
the convolved image by multiplication of the original image and the PSF
in Fourier space, and inverse transformation. No significant magnitudes
were found at the edges of the matrix prior to the inverse transform. The
point source, represented as one pixel in the high-resolution image, takes
on the scaled intensity distribution of the PSF. Larger objects retain their
basic original shape but appear diffracted.
CCD image construction
The CCD image was constructed by integrating over rectangular regions
corresponding to CCD pixels to form a smaller matrix representing a CCD
faceplate (Fig. 1 A). The CCD matrix was assumed to cover the same
physical dimensions as the high-resolution matrix, but with1⁄11 the reso-
lution. The factor of eleven was chosen so that the peak intensity of a small
object would be centered on a CCD matrix cell rather than a cell boundary.
By shifting the high-resolution matrix by one high-resolution element
relative to the CCD, and then integrating the underlying cells, we were able
to simulate relative displacements of less than one pixel on the CCD. We
assumed an objective of 100? magnification, resulting in 0.1 ?m/pixel in
the CCD matrix, which is typical of values reported in the literature.
A collection of noise-free CCD images with different relative displace-
ments was created by convolving object functions with point spread
functions and then integrating. These images were subsequently scaled to
and testing. A high-resolution image matrix is created that is subsequently
shifted relative to and integrated to generate a matrix representing a CCD
camera. For each of 1000 iterations of the tracking algorithms, shot noise
is applied to the CCD image. (B) Generated image of a point source object
with shot noise. The mean white level is 40 photoelectrons and the mean
background level is 10 photoelectrons. Each pixel is 100 nm in width.
(A) Schematic illustration of the process of image simulation
Algorithms for Single Particle Tracking 2379
Biophysical Journal 81(4) 2378–2388
give a black level of 10 (assumed to be the photoelectrons/pixel/frame),
and a white level between 15 and 1000.
Intensifiers for CCD cameras generate shot noise in the image (Ryan et al.,
1990). Shot noise is a Poisson process (Papoulis, 1984) where the noise
increases as?N, N being the number of detected photons or photoelec-
trons (Ryan et al., 1990). Shot noise was simulated in our experiments by
drawing a random value for each pixel from a Poisson distribution of mean
N (Press et al., 1997), where N is the level for that particular pixel (see
previous paragraph). This value was used as the measured intensity for that
The result is a realistic image representing a fluorescent particle of
known location imaged with an intensified CCD camera (Fig. 1 B). Noise
was generated independently for every trial of the tracking algorithm.
Algorithms for tracking the motion of single particles may be divided into
two basic categories. The first category is algorithms that estimate the
absolute positions of the particle in each image independently. This cate-
gory includes the center-of-mass, or centroid, algorithm, and direct fits of
Gaussian curves to the intensity profile. The second category includes
algorithms that estimate the change in position of a particle by comparing
an image to one subsequent. This category includes cross-correlation and
Comparing the center of mass or centroid of two successive images of a
particle is a computationally simple and efficient method for estimating the
distance an object has moved. For our purposes, an image is a matrix I of
intensities that contains both an object and a background. Eq. 3 gives the
centroid calculation for a single axis.
where xiis the coordinate of a pixel on the x axis, and Iijis the intensity of
that pixel. To calculate the distance an object has moved, Cxis calculated
for one image and subtracted from Cxfor a subsequent image. This
equation assumes that the intensities of the object have higher numerical
value than the background (not the case in all computer programs). Al-
though this approach is valid for asymmetric particles, the method is
especially susceptible to changes in particle shape and orientation between
It is vital to exclude as much of the image background as possible,
lest it strongly bias the centroid calculation to the center of the image.
This is accomplished by setting a threshold (expressed here as a fraction
of the peak image intensity) that a pixel must exceed to be included in
the calculation. There are two methods of handling thresholds. In
simple thresholding, values below the threshold level are assumed to be
zero, whereas those above threshold are unaltered (in the centroid
calculation, this is numerically equivalent to subtracting the threshold
value from all pixels, and setting negative values to zero). More
commonly, binary thresholding is used, where values below threshold
are taken to be zero, whereas those above are taken to be one. We tested
The equation of a two-dimensional (2D) Gaussian curve is of the general
G?x, y? ? A ? exp???x ? x0?2? ?y ? y0?2
where x0is the x coordinate of the center of the curve, and A and B are
constants. The peak of the point-spread function, and therefore the
intensity distribution of a point source, is well approximated by a
Gaussian. Thus, directly fitting the above equation to images of sub-
wavelength particles has become a common method of particle tracking
(Anderson et al., 1992; Schu ¨tz et al., 1997). We fit a 2D Gaussian using
a simplex algorithm with a least-squares estimator (Press et al., 1997),
allowing the constants A and B to float. As in centroid, independent
fitted values of x0and y0are subtracted to find the displacement
between any two images.
Correlation (COR) is more computationally intensive than the above tech-
niques (Gelles et al., 1988). This method compares an image (I) to a kernel
(K) of a successive image. K, which contains the object being tracked, is
shifted relative to I in one-pixel increments. For each increment, a corre-
lation value is calculated that describes how well the values in K match
those of the underlying image, I. At the relative shift where K and I are
most similar, one finds a maximum in the correlation matrix, X. The
cross-correlation between K and I is given by
where x and y describe the distance the kernel K has moved over the
original image I. If K and I are similar except that the object in the image
has translated along the x axis by p pixels, then the resulting correlation
matrix will have a maximum in cell Xp,0. The kernel dimensions were fixed
at 80 ? 80 in this study, irrespective of object diameter, to generate
unbiased comparisons among different object sizes.
Correlation tends to match the brightest regions of two images rather
than the best topographical fit, resulting in errors in some cases. To
alleviate this problem, one may use normalized correlation. Each value in
the correlation matrix is divided by the root mean square (RMS) of the
original image intensities, as shown in Eqs. 6 and 7.
n and m are the dimensions of K, and MKand MIx,yare the RMS values of
the kernel and the overlapping portion of the image, respectively.
Normalized covariance is an extension of this concept intended to deal
with situations where the image and the kernel have a relative offset in
intensity. In this method, one subtracts the mean of kernel K from each cell
2380Cheezum et al.
Biophysical Journal 81(4) 2378–2388
in K, and subtracts the mean in I in the overlapping region from each cell
in I. Eq. 10 describes the normalized covariance calculation.
n ? m
n ? m
m?1?Ix?i,y?j? I?x,y??Ki,j? K??
The SAD method determines the translation of I relative to K that mini-
mizes the sum of absolute differences between the overlapping pixels,
?Ix?i,y?j? Ki,j?. (11)
In contrast to the algorithms above, if the object in the image has translated
along the x axis by N pixels, then the resulting SAD matrix will have a
minimum in cell S?N,0. Although this algorithm has never been used for
tracking fluorescent particles, it is a standard algorithm for tracking the
motion of features in medical imaging (Bohs et al., 1993). We included
tests of this algorithm for completeness.
The centroid and Gaussian fit methods inherently return sub-pixel estimates of
distance moved because the position is calculated as an average over a set of
coordinates. However, methods that compare subsequent images return dis-
crete matrices, and thus offer only whole-pixel estimates. To achieve sub-pixel
resolution, the correlation, covariance, or sum-absolute difference matrices
must be interpolated to find the maximum or minimum.
The data in the SAD or correlation matrices form quasi-paraboloid
meshes, where the x, y, and z-coordinates are x and y distances moved, and
the corresponding sum-absolute difference or correlation values, respec-
tively. Three functions were used to interpolate our data: paraboloid,
consinusoid, and Gaussian. We used closed form solutions to fit parabo-
loids and cosinusoids to the maximum and four nearest neighbors in the
correlation matrix (or minimum in the SAD matrix). Briefly, a parabaloid
defining a 2D surface is described by
z ? a ? bx ? cy ? dx2? ey2. (12)
x and y designate the coordinates within the correlation matrix. The
coefficients in Eq. 12 are obtained from the cell containing the peak of the
correlation, z(x0, y0), and the four immediately surrounding points on the x
and y axes:
a ? z?x0,y0?
2?z?x1, y0? ? z?x?1, y0??
2?z?x0, y1? ? z?x0, y?1??
d ? ?z?x0, y0? ?1
2z?x1, y0? ?1
e ? ?z?x0, y0? ?1
2z?x0, y1? ?1
x and y are solved directly using the equations
xmax? b/2dymax? c/2e.(14)
Cosinusoidal interpolation was accomplished as in deJong et al. (1990).
The peak of the correlation function relative to the reference image is given
where ? is the angular frequency and ? the phase of a cosinusoid, given by
? ? cos?1?
? ? tan?1?
z?x0, y0? ? z?x?1, y0?
2z?x1, y0? ? sin ??.
z?x0, y0? ? z?x?1, y0?
The notation designating the cells in the COR/SAD matrix is the same as
in the description of parabolic interpolation, with z(x0, y0) designated as the
peak intensity in the matrix. The above implementation is for one-dimen-
sional fitting only.
Finally, one can use a Gaussian to fit the peak in the COR/SAD matrix.
We used a simplex algorithm (Press et al., 1997) with a least-squares
estimator to fit a Gaussian to the maximum and four nearest neighbors.
As previously indicated, the background level was held constant at 10
photoelectrons/pixel. Altering the maximum photoelectron count for the
object (white level) between 15 and 1000 photoelectrons/pixel changed the
signal-to-noise ratio of the image between 1.3 and 31.2. We also consid-
ered the deterministic case of S/N ? ? (no noise). The original high-
resolution matrix was displaced by one or more high-resolution cells over
the CCD matrix, resulting in known displacements of the object in multi-
ples of1⁄11 CCD pixels, or the equivalent of ?9 nm.
One thousand trials of each algorithm were performed for each condi-
tion of distance moved, S/N, and threshold level (for centroid) to obtain a
bias (B) and standard deviation ? for the condition,
B ? ?a ? a ˆ?
? ? ??a ? ?a??2?1/2.(18)
Error for the bias is equal to ? for the same condition. Error for ? was
determined by calculating ? for 10 sets of 100 successive iterations, and
taking the standard deviation of the resulting independent estimates. The
original high-resolution images were created using MatLab on an IBM
RS6000 43P workstation. All subsequent numerical experiments were
conducted on a Intel Pentium class microcomputer (Dell Optiplex) using
code written in Borland C??. Graphs were generated using SigmaPlot
Thresholding in centroid
For point source objects, the precision and accuracy of the
centroid algorithm were critically dependent upon the
threshold level applied to the image. As expected, lower
thresholds give poorer results at low S/N. With no threshold
applied to the image, bias remains high at all S/N levels
(Fig. 2 A). This is because all the background pixels in the
image are included in the center-of-mass calculation, bias-
Algorithms for Single Particle Tracking2381
Biophysical Journal 81(4) 2378–2388
ing the result toward the geometric center of the image.
Thus, the bias is always roughly equal in magnitude to the
actual displacement. Increasing the threshold (limiting the
pixels included in the centroid calculation to only the bright-
est) results in less bias at lower S/N. However, no signifi-
cant improvement is seen at any threshold level below 3.6
The relationship between ? and S/N is more complex
(Fig. 2 B). At 1.3 S/N, ? returned by all threshold levels are
low and roughly equal (?0.1 pixels). At such a low S/N,
even a moderately high threshold (expressed as a fraction of
the peak intensity in the image) includes many background
pixels in the centroid calculation. As a result, the back-
ground pixels in the image bias the result toward the geo-
metric center of the image, giving a consistent result in each
trial. At slightly higher S/N, many fewer background pixels
are included in the centroid calculation, which introduces
tremendous variability into the position estimates, even
though they become more accurate. This is reflected in a
peak in the ? versus S/N plot approaching 3 pixels for an
80 ? 80 region of interest, typically at ?4 S/N. Increasing
the threshold results in a shift of this peak to lower S/N,
suggesting that higher thresholds are preferable. However,
at a threshold of 0.8, the ? drops to zero above 13.4 S/N.
Indeed, at a S/N of 31 and thresholds 0.8 and 0.2, we find
? ? 0 and B ? ?(actual distance moved) for certain
sub-pixel displacements (not shown). These “discontinui-
ties” in the trend of ? versus S/N result from a strong bias
to the nearest pixel, giving the same result in each trial.
Obviously, sub-pixel performance is severely compromised
at these threshold levels. A threshold of 0.4 returned the
lowest bias and ? at moderate S/N while maintaining sub-
pixel performance (? ? 0). Thus, a threshold of 0.4 was
used for the remaining comparisons.
We compared two methods for thresholding, described in
Methods. We found that binary thresholding—converting
pixels above threshold to 1, and those below threshold to
0—resulted in higher biases and discontinuous relationships
among ?, S/N, and actual displacement (data not shown).
Simple thresholding, where all values below the threshold
were set to 0, was used through the remainder of these
Normalized correlation and covariance
Normalized correlation and normalized covariance per-
formed similarly under all conditions to non-normalized
cross correlation, except at S/N ? 3, where the differences
in performance were trivial. Thus, only non-normalized
correlation will be considered for the remainder of the
Comparison of algorithms
Bias as a function of distance moved
Bias was periodic for centroid, COR, and SAD as a function
of actual distance moved, with a period of 1 pixel (Fig. 3 A).
Qualitatively, bias above one pixel of actual movement did
not differ from bias below one pixel. For example, bias for
a given algorithm will be the same at 0.3, 1.3, 2.3, . . . pixels
actual displacement, except at very low S/N where the
algorithms cease to function. At these “limiting” S/N, bias
will often scale proportionately to the actual displacement.
However, for simplicity, we will only consider actual dis-
placements over the range of 0–1 pixel.
Only the highest S/N is shown in Fig. 3. Although the
magnitude of the bias increases with decreased S/N, the
results are readily generalized to lower S/N. For all algo-
rithms, bias is approximately zero at multiples of 0.5 pixels.
For COR and SAD, bias is toward the nearest whole pixel
with a maximum bias at ?0.3 pixels actual distance moved.
At high S/N (31.2 in Fig. 3 A), bias reaches a maximum of
?0.1 pixel in the case of SAD. COR was nearly an order of
source target. Actual displacement was 1 pixel. Negative bias is toward the
reference (unmoved) image. Errors are represented in the lower panel. (B)
Standard deviation versus S/N and threshold in centroid. Error bars indicate
one standard deviation. Threshold values are indicated in the graphs.
Standard deviation values of 0 are omitted.
(A) Bias versus S/N and threshold in centroid for a point
2382Cheezum et al.
Biophysical Journal 81(4) 2378–2388
magnitude less biased under the same conditions. These
values are not significantly different from the performance
at infinite S/N (data not shown). Peak bias was reduced by
25% by using cosinusoidal interpolation rather than para-
bolic, so all the remaining data use cosinusoidal interpola-
tion. In contrast, centroid was biased toward the midpoint
between adjacent pixels, reaching a maximum bias of
?0.08 pixels at high S/N. Direct Gaussian fit to the image
was effectively unbiased and independent of actual dis-
Precision as a function of actual distance moved
Standard deviation at the highest S/N was independent of
actual distance moved for direct Gaussian fit at ?0.02
pixels (Fig. 3 B). In contrast, ? returned by the three other
algorithms was highly dependent on actual distance moved.
For SAD and COR, ? was lowest at whole pixel displace-
ments, and maximal at the midpoint between pixels. Cen-
troid returned the lowest ? at whole and half pixel move-
ments, and highest at intermediate motions. COR returned
the lowest ? overall, whereas SAD and centroid returned the
highest. ? did not change significantly when cosinusoidal
interpolation was used instead of parabolic.
Bias as a function of S/N
Figure 4 shows bias and ? as a function of S/N for 0.27
pixels moved. An actual displacement of 0.27 pixels was
chosen because it results in the greatest overall bias (see Fig.
3). However, the results generalize readily to all displace-
ments. At low S/N, the relationship between bias and S/N is
relatively complex (Fig. 4 A). At S/N ? 1.3, the two com-
parative algorithms, COR and SAD, choose nearly random
locations within the matrix. Biases in this case may be
randomly high or low in any given round of simulations, but
fall toward zero at S/N ? 3.6. Similarly, Gaussian fit is free
to return values outside the actual image boundaries, which
can result in very high biases at low S/N.
In contrast, centroid begins with a high bias at limiting
S/N, and falls monotonically toward zero. Centroid tends to
target: F, Gaussian fit; E, centroid; ?, sum-absolute difference; and ƒ,
cross-correlation. S/N was 31.2. Note that bias in direct Gaussian fit is
largely independent of distance moved. Errors are represented in the lower
panel. (B) Standard deviation versus actual distance moved. Note that
variance in direct Gaussian fit is largely independent of distance moved.
Error bars fall within the symbols.
(A) Bias versus actual distance moved for a point source
E, centroid; ?, sum-absolute difference; and ƒ, cross-correlation. Actual
distance moved was 0.27 pixels. Errors are given in the lower panel. (B)
Standard deviation versus S/N. Error bars indicate one standard deviation.
(A) Bias versus S/N for a point source target: F, Gaussian fit;
Algorithms for Single Particle Tracking2383
Biophysical Journal 81(4) 2378–2388
return the center of the image matrix at limiting S/N, and so
the bias is approximately equal to the actual displacement
under these conditions.
As an arbitrary gauge of performance, we determined the
S/N at which the bias drops below 0.1 pixels on the de-
scending limb of the bias versus S/N curve (Table 1).
Gaussian fit and COR performed best, returning ?0.1 pixels
bias at S/N ? 4.2. In contrast, centroid did not return ?0.1
pixels bias until S/N ? 7.8. The method of interpolation had
a negligible effect on the S/N at which bias fell below 0.1
pixels in COR or SAD.
Precision as a function of S/N
For all but one algorithm (centroid), ? decreased monoton-
ically as S/N increased (Fig. 4 B). Standard deviation at 1.3
S/N for COR and SAD are equal, and fully explained as the
variance one would predict by choosing random locations
within the COR or SAD matrix. Gaussian fit at low S/N
exhibits very high ?, as the curve-fitting algorithm may
return random values that fall outside the image. In contrast,
centroid returned low ? at 1.3 S/N, indicating that the
algorithm reproducibly finds the center of the image rather
than the center of the object. For centroid, ? increases as
S/N rises to 5.6 S/N, and then falls. Once again, as an
arbitrary gauge of performance, we determined the S/N at
which ? drops below 1 pixel on the descending limb of the
? versus S/N curve (Table 1). Gaussian fit performed the
best, reaching ? ? 1 pixel at S/N ? 4.
When the object is much larger than the wavelength of light
(d ? ?), the profile of pixel intensities is no longer de-
scribed by a Gaussian. Gaussian fit has therefore been
omitted from this portion of the analysis. Among the three
remaining algorithms, correlation still performed the best in
terms of bias error (Fig. 5 A). Bias versus actual distance
moved is reminiscent of that for d ? ? ? (Fig. 3 A). How-
ever, both COR and SAD remained at ?0.1 pixels bias at all
S/N tested. Centroid, in contrast, did not fall below 0.1
pixels bias until S/N ? 4.7. Overall, the bias for each of the
algorithms at low S/N was improved relative to tracking the
point-spread object, whereas, at high S/N, improvements
For SAD and COR, ? was highest at 0.5 pixels moved,
whereas centroid displayed uniform ? with respect to dis-
tance moved at high S/N (data not shown). ? was approx-
imately one order of magnitude lower at each S/N compared
to d ? ? ? (Fig. 5 B). This is expected because a larger target
has a higher total energy, making motion estimation less
susceptible to noise. The results at d ? ? were qualitatively
similar and quantitatively intermediate to those at d ? ? ?
and d ? ?, and will therefore not be discussed.
We have demonstrated significant differences between
common implementations of particle tracking algorithms
both in terms of their accuracy and precision. For particles
that are smaller than one wavelength in diameter, direct fit
of a Gaussian curve to the intensity profile may be judged
the best method by several criteria. This method achieves a
given precision at lower S/N compared to the other algo-
rithms tested. Further, both the bias and the variance are
0.1 pixels and 1.0 pixels, respectively, for various algorithms
Approximate S/N at which bias and ? drop below
sum-absolute difference; and ƒ, cross-correlation. Errors are given in the
lower panel. Actual distance moved was 0.27 pixels. The scaling is the
same as in Fig. 4 A. (B) Standard deviation versus S/N.
(A) Bias versus S/N for a 5-?m target: E, centroid; ?,
2384Cheezum et al.
Biophysical Journal 81(4) 2378–2388
independent of the actual distance moved—a characteristic
unique among the algorithms tested. When direct Gaussian
fit cannot be used, such as when the particle is much larger
than one wavelength in diameter, cross-correlation appears
to be the best choice.
Why does Gaussian fit operate with less bias than COR?
Gaussian fit is similar to COR, in that a kernel (function) is
matched to an underlying matrix containing pixel intensi-
ties. However, in Gaussian fit, the kernel is perfect—that is,
noise-free and known to fit the intensity distribution across
the object. Thus, Gaussian fit operates at a lower effective
S/N than COR (see Walker and Trahey, 1994). Alterna-
tively, when COR is used with a second or subsequent
image as a kernel, one would expect the error in the esti-
mates to increase by at least ?2 (Walker and Trahey,
1994). Although this is often a trivial increase, it becomes
important when S/N is limiting, as we have demonstrated.
One might also generalize that, when tracking point source
objects, COR cannot match the precision of Gaussian fits,
whatever the exact implementation of the algorithm.
Our data comparing COR, normalized COR, and SAD do
not agree with a related study by Friemel et al. (1995).
These same three algorithms were used to track synthetic
patterns representing speckle in ultrasound images. Friemel
found that normalized COR and SAD performed similarly
at all S/N, judged in terms of ?, whereas non-normalized
COR performed significantly worse and converged with the
others only as S/N became very high. However, there are
two significant differences between our two studies. First,
speckle patterns are continuous and relatively uniform. The
net effect may be to increase the effective spatial sampling
density per object relative to a discrete object. Indeed,
tracking 5-?m objects increases sampling density per object
relative to a point source, and we see that SAD and COR
converge in their performance (Fig. 5). Second, Friemel
dealt with additive noise rather than shot noise. Whatever
the exact source of the discrepancies, the differences be-
tween these two studies highlight the importance of under-
standing tracking algorithm performance in each unique
One of the most important indices of algorithm performance
is the limiting S/N. That is, what minimum S/N is necessary
for the algorithm to function? We have rather arbitrarily
defined “limiting” as that S/N at which the worst-case bias
falls to 0.1 pixels and the standard deviation to 1 pixel at
0.27 actual pixels moved. Although the choice of these
limits is completely arbitrary, they do have some practical
relevance. At a standard deviation of 1 pixel, or 1 pixel
RMS, 67% of the time the algorithm can be expected to
return the correct value to within 1 pixel. This then is a
reasonable gauge of the S/N needed for 1-pixel precision.
Although the choice of 0.1 pixel’s worst-case bias would
seem to be an order of magnitude a more stringent require-
ment for accuracy, both requirements give conveniently
similar estimates of the minimum required S/N.
By these definitions, we can estimate that the limiting
S/N for direct Gaussian fit is approximately 4. For compar-
ison, limiting S/N for cross-correlation, SAD, and centroid
are very roughly 6.3, 8.1, and 6.6, respectively. Quantita-
tively, this agrees well with predicted values from Walker
and Trahey (1994). None of these algorithms should be
expected to return believable results if the S/N is less than
the corresponding limit.
Properly estimating the S/N is critical to application of
these guidelines, but is most often calculated incorrectly.
Shot noise is proportional to the square root of the number
of photoelectrons (? ??N). As a result, the noise level is
higher in pixels containing a fluorescent object than in the
background, even though the noise may be more visually
apparent in the background. Failure to account for this
relationship may lead to very large errors in estimated S/N.
For example, assume a fluorescent object on an imperfectly
dark background. Assume that the mean photoelectrons/
frame/pixel detected for the object is 144, while, for back-
ground, it is 16. The mean RMS noise in these two regions
of the image is expected to be?144 and?16, or 12 and 4,
respectively. S/N is calculated as the difference in mean
intensity between the object (Io) and background (Ib), di-
vided by a representative noise level (?),
S/N ? ?Io? Ib?/?.(19)
In our example, choosing for ? the background RMS (?b)
(as in Sonnleitner et al., 1999) would yield a S/N estimate of
(144 ? 16)/4, or 32. However, the RMS variation in the
background is an inappropriate choice for ?, because it is
the lowest noise region in the image. Thus, the noise over
the object (?o) yields a more accurate estimate of S/N
(?11). This was the method used to determine the S/N in
this study. A more conservative estimate can be obtained by
propagating the noise error, as??b
al., 2000), which, in this example, yields S/N ? 10.
Implications for single molecule imaging
These limits are of critical importance for the emerging field
of single fluorophore imaging. Whether detected using an
intensified CCD camera (Goulian and Simon, 2000; Kubits-
check et al., 2000) or a spot-confocal system (Warshaw et
al., 1998), the S/N when detecting single fluorescent mol-
ecules in real time is generally limited to approximately
3–4. Although a S/N of 4.0 is just barely acceptable by our
criteria, a drop in S/N from 4 to 3 increases bias by approx-
imately four-fold, and variance by two orders of magnitude
when Gaussian fit is used! Thus, taking steps to maximize
S/N is crucial when particle tracking is to be used with
Algorithms for Single Particle Tracking 2385
Biophysical Journal 81(4) 2378–2388
Claims of sub-pixel resolution
Claims of nanometer or even sub-pixel resolution should be
viewed with some skepticism. It is true that, given our
conditions and at the highest S/N examined (31.2), direct
Gaussian fit has a predicted mean accuracy (under our
assumed conditions) of 1 nm and a precision of ?2 nm.
However, all these algorithms degrade rapidly in perfor-
mance below 10 S/N. Gaussian fit, for example, degrades to
10-nm precision at ?9 S/N, and to only 100-nm precision (1
pixel) at 4.0 S/N. These values are in reasonable agreement
with Kubitscheck et al. (2000). Reasonable precision can be
expected from all four algorithms at high S/N, though
predicted biases of several nanometers persist for all but
of noise and inaccuracy that will undoubtedly decrease accu-
racy and precision in all algorithms (see Limitations).
Implicit in these statements is that the tracking algorithm
has been properly optimized. The centroid algorithm, for
example, has seen many variations. The size of the image
kernel, the threshold level, the option of converting the
image to binary before thresholding, and image averaging
(Goulian and Simon, 2000) can all affect the accuracy and
precision of the algorithm. Assume, for example, that one
used centroid, and chose the threshold level to give the
lowest variability in the position of a fixed object. Although
this is certainly an intuitive approach, the likely cause of the
low variability is that the investigator has chosen a threshold
that biases the algorithm consistently to the center of the
nearest pixel. Though the variability in estimated positions
might imply nanometer precision, the accuracy would ac-
tually be on the order of 1 pixel (typically 100 nm).
It is common to track the position of a fixed particle and
use the variance of the positions as an indicator of both
precision and accuracy. However, such empirical tests of
tracking algorithm efficacy are lacking in two respects.
First, one does not know if the particle is truly motionless
with respect to the imaging system. Second and more im-
portantly, one does not know the actual position of the
particle relative to the elements in the imaging system.
Without this knowledge, one cannot estimate the bias in the
tracking algorithm, which indicates the true accuracy of the
measurement. Claims of sub-pixel accuracy and precision
(or even whole-pixel precision in very low contrast images)
must be demonstrated in numerical simulations or more
rigorous experiments than those performed to date. These
might include tracking of microspheres simultaneously
through video and an alternative, higher resolution tech-
nique, such as back-focal plane interferometry (Allersma et
Origins of bias
At very low S/N, very large biases can result. At low S/N,
centroid tends to weight toward the geometric center of the
region of interest, regardless of where the tracked feature
resides. Gaussian fits to a noisy image can return values
completely outside the image unless artificially constrained.
COR and SAD may return random whole-pixel estimates of
location within the COR/SAD matrix. Further, the interpo-
lation function may give a value that lies outside the matrix
boundaries. These effects can result in randomly high or low
apparent biases on any given round of simulations.
At moderate-to-? S/N, centroid, SAD, and COR can still
be biased, though the underlying cause is not as apparent.
Centroid is biased because the centroid of a continuous
function need not equal the centroid of a sampled function
(like an image). It has been proposed that sub-pixel bias in
COR and SAD results from the failure of the interpolation
algorithm to accurately account for the shape of the peak in
the COR/SAD matrix (deJong, 1990). Indeed, we have no a
priori knowledge of the shape of the peak. The fact that
cosinusoidal interpolation is less biased than parabolic an-
ecdotally supports this conclusion, as does visual inspection
of parabolic and cosinusoidal fits to our COR and SAD
matrices (not shown). Direct Gaussian fit to the image is
presumably unbiased because it accurately reflects the
shape of the peak of the PSF.
To eliminate assumptions about the shape of the peak in
the correlation matrix, others have estimated the centroid of
the peak (Gelles et al., 1988) instead of resorting to inter-
polation. This again requires the use of thresholds to isolate
the peak in the centroid calculation—a parameter whose
effects must be carefully assessed. Further, the correlation
matrix is a sampled function, so it is probable that the
centroid of this matrix will be biased (see previous para-
graph). However, we did not extensively test this particular
implementation of correlation.
This study focused on the simplest case of particle track-
ing—a point-source. The situation becomes much more
complex when the “particle” becomes greater in dimension
than ?, and when the particle is no longer symmetric. One
example is the tracking of cell movements through laminar
flow chambers (Smith et al., 1999). Although our data on
fluorescent objects ?? in diameter suggests excellent be-
havior both in terms of bias and ?, we did not simulate the
entire range of variables expected of a real, moving cell.
Cells are not of predefined shape, can change apparent
shape over short time spans, and are not ordinarily uni-
For the centroid calculations, we expressed the threshold
as a fraction of the noise-free peak image intensity, and it
was therefore constant for any given S/N. However, should
thresholds be thus applied to real images, the threshold
value for any given image will vary due to variations in the
peak intensity, and this may randomly vary the bias. If the
background level of a series of images is relatively constant,
2386Cheezum et al.
Biophysical Journal 81(4) 2378–2388
it may be adequate to set an absolute intensity for the
threshold. Otherwise, setting a threshold relative to the
mean pixel intensity may be a better choice.
Another example of single particle tracking applied to
relatively large, asymmetric particles is the tracking of
myosin-driven movement of fluorescent actin filaments in
the in vitro motility assay (Work and Warshaw, 1992).
Measuring the frame-to-frame velocity of actin filaments
has proven a very fruitful technique. Although a number of
techniques have been developed for tracking actin fila-
ments, centroid is the most common underlying algorithm.
This is cause for concern, because the S/N in these exper-
iments, which effectively involve single molecule imaging,
is ?5, and because the actin filament can change shape and
orientation as it progresses randomly across the myosin-
coated surface. If an asymmetric particle changes shape or
orientation during tracking, centroid will not always repre-
sent the same point in successive frames, and is not there-
fore a wholly valid measure (Uttenweiler et al., 2000). Actin
filaments are only 7 nm wide, and can therefore be consid-
ered a long string of closely spaced point sources. Thus, if
our data on centroid performance with point-source objects
is applicable to actin filaments, then we can assume that the
algorithm is both significantly biased, and on the ascending
limb of its variance versus S/N relationship. The resulting
bias error may explain recent observations of periodic mo-
tion of actin filaments (deBeer et al., 1997), which can make
an actin filament appear to “jump” from pixel to pixel in the
image. The leading and trailing ends of the filament would
appear to move at the same overall rate, but with different
phases depending on their positions relative to the imaging
array. Only through simulations of these particular cases
will we identify the best tracking method. However, our
preliminary data on tracking actin filaments suggests that
SAD may be the algorithm of choice among those tested in
this study. A recent study has also shown excellent results
using optical flow techniques (Uttenweiler et al., 2000).
We have extensively compared the efficacy of specific
Our study is not a comprehensive comparison of the many
different specific implementations of these algorithms, nor
have we accounted for two other important aspects of particle
tracking. First, we did not examine additional sources of noise
and error. These might include noise resulting from analog or
digital video encoding (such as interlace), spatially correlated
noise common in ICCDs, and optical aberrations causing
asymmetrical point spread functions. All these may differen-
tially affect particle-tracking algorithms. Neither did we exam-
ine the plethora of techniques for image segmentation—iden-
tification and discrimination of many particles in a single set of
images. Nonetheless, the algorithm that determines particle
displacement defines the practical limit of particle tracking.
The limiting bias and S/N values reported here should serve as
a useful guide to the investigator both in choosing an algorithm
and determining the performance limits and consequences of
tracked, particularly when S/N is limiting.
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