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Quantitative Comparison of Algorithms for Tracking Single

Fluorescent Particles

Michael K. Cheezum, William F. Walker, and William H. Guilford

Department of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22908 USA

ABSTRACT

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

INTRODUCTION

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

method.

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: guilford@virginia.edu.

© 2001 by the Biophysical Society

0006-3495/01/10/2378/11$2.00

2378 Biophysical JournalVolume 81October 20012378–2388

Page 2

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

near 4.

MATERIALS AND METHODS

Image models

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),

PSF?r? ??

2J1?ra?

r?

2

, (1)

where

a ?2?NA

?

, (2)

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

FIGURE 1

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

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give a black level of 10 (assumed to be the photoelectrons/pixel/frame),

and a white level between 15 and 1000.

Noise model

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

pixel.

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.

Tracking Algorithms

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

SAD algorithms.

Centroid

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.

?xi? Iij???

Cx??

i?1

n?

j?1

m

i?1

n?

j?1

m

Iij, (3)

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

successive images.

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

both methods.

Gaussian fit

The equation of a two-dimensional (2D) Gaussian curve is of the general

form

G?x, y? ? A ? exp???x ? x0?2? ?y ? y0?2

B

?,(4)

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

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

Xx,y??

i?0

n?1?

j?0

m?1

Ix?i,y?j?Ki,j?,(5)

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.

XNx,y??

i?0

n?1?

j?0

m?1Ix?i,y?j?Ki,j?

MIx,y? MK

,(6)

MIx,y???

i?0

n?1?

j?0

m?1

?Ix?i,y?j?2

MK???

i?0

n?1?

j?0

m?1

?Ki,j?2.(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

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Biophysical Journal 81(4) 2378–2388

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in K, and subtracts the mean in I in the overlapping region from each cell

in I. Eq. 10 describes the normalized covariance calculation.

I?x,y??i?0

K???i?0

n ? m

n?1?j?0

m?1Ix?i,y?j

n ? m

n?1?j?0

, (8)

m?1Ki,j

, (9)

VNx,y??

i?0

n?1?

j?0

m?1?Ix?i,y?j? I?x,y??Ki,j? K??

MIx,y? MK

.(10)

Sum-absolute difference

The SAD method determines the translation of I relative to K that mini-

mizes the sum of absolute differences between the overlapping pixels,

SADx,y??

i?0

k?1?

j?0

k?1

?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.

Interpolation methods

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?

b ?1

2?z?x1, y0? ? z?x?1, y0??

c ?1

2?z?x0, y1? ? z?x0, y?1??

d ? ?z?x0, y0? ?1

2z?x1, y0? ?1

2z?x?1, y0?

e ? ?z?x0, y0? ?1

2z?x0, y1? ?1

2z?x0, y?1?

(13)

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

by

?max? ??/?,(15)

where ? is the angular frequency and ? the phase of a cosinusoid, given by

the equations,

? ? cos?1?

? ? tan?1?

z?x0, y0? ? z?x?1, y0?

2z?x1, y0?

2z?x1, y0? ? sin ??.

?,(16)

z?x0, y0? ? z?x?1, y0?

(17)

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.

Implementation

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

(SPSS Inc.).

RESULTS

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-

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Biophysical Journal 81(4) 2378–2388

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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

S/N.

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

studies.

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

manuscript.

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

FIGURE 2

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

2382 Cheezum et al.

Biophysical Journal 81(4) 2378–2388

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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-

placement.

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

FIGURE 3

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

FIGURE 4

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;

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Biophysical Journal 81(4) 2378–2388

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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.

Large objects

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

were trivial.

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.

DISCUSSION

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

TABLE 1

0.1 pixels and 1.0 pixels, respectively, for various algorithms

Approximate S/N at which bias and ? drop below

AlgorithmS/N0.1bias

S/N1.0?

Gaussian fit

Centroid

Sum-absolute difference

Cross-correlation

4.2

7.8

6.9

4.2

4.0

6.6

8.1

6.3

FIGURE 5

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; ?,

2384 Cheezum et al.

Biophysical Journal 81(4) 2378–2388

Page 8

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

imaging modality.

Limiting S/N

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.

2? ?o

2(Kubitscheck et

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

single fluorophores.

Algorithms for Single Particle Tracking2385

Biophysical Journal 81(4) 2378–2388

Page 9

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

Gaussianfit.Evenhere,oneshouldconsideradditionalsources

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

al., 1998).

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.

Limitations

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-

formly fluorescent.

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

Page 10

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

algorithmsfordeterminingthedisplacementofsingleparticles.

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

theirexistingsoftware.Wehighlyrecommendthealgorithmof

directGaussianfitwheneverafluorescentpointsourceisbeing

tracked, particularly when S/N is limiting.

This work was supported by National Institutes of Health grants AR45604

and HL64381.

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