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Bias and precision of the fluoroBancroftalgorithm for single particle localization
in fluorescence microscopy
Zhaolong Shen and Sean B. Andersson*Mechanical Engineering, Boston University, Boston, MA 02215, U.S.A.
Abstract: The fluoroBancroft algorithm is an analytical solution to theposition estimation problem in single molecule fluorescence microscopy. Ithas previously been shown through experiment to have an accuracy similarto Gaussian fitting in the two dimensional setting while executing twoorders of magnitude faster. In this paper we derive a theoretical descriptionof the bias and precision of the estimator for three dimensional estimationand illustrate the results through realistic simulations. The results indicatethat the algorithm exhibits a small bias that is driven primarily by modelingerror and is dependent on the location of the source particle relative tothe set of pixels used for estimation. In the shot noise limited case, theprecision scales approximately as the inverse square root of the number ofphotons detected and as the inverse of the number of photons detected inthe background noise limited case.
© 2010 Optical Society of America
OCIS codes: (180.2520) Microscopy: fluorescence microscopy; (180.6900) Microscopy:Three-dimensional microscopy; (100.2960) Image Processing: Image analysis.
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1. Introduction
The ability to localize the position of a single fluorescent particle with sub-diffraction limit ac-curacy is a powerful tool in molecular biology, allowing researchers to move beyond ensembleaverages and explore the distributions of the dynamics of individual molecules. Applications in-clude the study of virus trafficking [1], the exploration of mechanisms leading to slow diffusionin cell membranes [2], and the measurement of micromechanical properties of live cells [3].It is also a fundamental component of super-resolution microscopy [4]. Recent review articlesinclude [5, 6, 7].
Position estimation in two dimensions can be achieved by a variety of schemes, includingcentroid methods [8], Gaussian fitting [9], and the Gaussian mask [10]. A comparison of com-mon techniques was performed in [11] while the fundamental limits of localization based onoptimal estimation theory have been developed in [12]. There are also several techniques forestimation in 3-D, including centroid methods applied to a stack of 2-D images acquired at
different depths [13], off-focus imaging [14], astigmatism [15, 16], biplane detection [17, 18],interferometry [19], and polarization-sensitive double-helix systems [20] based on deconvolu-tion [21].
Despite their ongoing successes, these methods typically require extensive numerical cal-culations and often rely on reasonably complicated modifications to the imaging setup. Issuesrelated to the computation complexity are immaterial when doing offline position estimationin two-dimensions based on wide-field imaging using a CCD camera. In applications wherereal-time position information is desired, however, it is important to produce estimates as fastas possible. Moreover, when images are acquired using confocal or multi-photon techniques,or when three-dimensional estimation is to be done based on a z−stack of images, it is bene-ficial to have a technique that can provide accurate estimation using as few measurements aspossible.
In recent years, a few fast and accurate algorithms have been proposed, including a center-of-mass method [22] and the fluoroBancroft (FB) algorithm developed by one of the authors.The FB algorithm is an analytic solution to the localization problem and it has been shown ex-perimentally to have an accuracy similar to Gaussian fitting while executing two orders ofmagnitude faster [23]. It has also been shown to work well when only a small number ofmeasurements are available, with as few as four being required for localization in three di-mensions [24, 25]. The speed of the algorithm has been used to speed up the performance ofphotoactivation localization microscopy (PALM) by a factor of 200 while maintaining imageaccuracy [26].
In this work we describe a theoretical analysis of the bias and variance of the FB algorithm.While the derivation is done for the 3-D algorithm, the results also apply to estimation in twodimensions. We consider the effect of both shot and background noise in the intensity measure-ments. To illustrate the performance of the algorithm, we generated z−stacks of wide-fieldfluorescent images using a standard model for the point spread function, adding both shot andbackground noise. The position of the source molecule was varied as well as the intensity ofthe source and of the background. Localization was performed using the FB algorithm and theresults were compared to the theoretical predictions developed in this paper.
2. Position estimation using the FB algorithm
We give here a brief overview of the FB algorithm. Details can be found in [23]. The spatialdistribution of the intensity of a point source with wavelength λ and located at the origin isgiven by the intensity point spread function (PSF) defined by (see, e.g. [27])
IPSF(u,v) = h(u,v)h∗(u,v) (1)
where the amplitude point spread function h is
h(u,v) =−i2πnAsin2
α
λe
iusin2(α)
∫ 1
0J0 (vρ)e−
iuρ22 ρ dρ. (2)
Here n is the index of refraction of the medium surrounding the source, nsinα is the numericalaperture of the objective lens, A is the area of the detection aperture, and J0 is the zeroth-orderBessel function. The normalized optical coordinates normal (u) and parallel (v) to the opticalaxis are given by
v =2πnr sinα
λ, u =
2πnzsin2 (α)λ
, (3)
where z is defined by the optical axis and r is the lateral distance from the optical axis. Thisfunction can be approximated by a 3-D Gaussian [13]
Ns(r) = me−12 (r−ro)T
Σ−1(r−ro) (4)
where ro = (xo,yo,zo)T is the position of the point source, r = (x,y,z)T is a point in space, andΣ = diag(σ2
x ,σ2y ,σ2
z ) is the spread of the PSF in the three directions. (Note that the positionsare expressed as column vectors and that T represents transpose.) Due to shot and backgroundnoise, the measured intensity at r is modeled by
I(r) = ηs +ηB (5)
where ηs and ηB are Poisson random variables with rates Ns(r) and NB, respectively.Given a collection of N intensity measurements {I1, . . . , IN} taken at positions {r1, . . . ,rN},
the fluoroBancroft algorithm calculates the position of the source particle using
ro = QB†α (6)
where
Q =
1 0 0 00 σy
σx0 0
0 0 σzσx
0
, α =
12
(‖r′1‖2 +P2
1)
...12
(‖r′N‖2 +P2
N) . (7)
Here r′i = (xi,σxσy
yi,σxσz
zi) is a scaled version of the position such that the axial spreads of the
PSF are the same, ‖r‖=√
rT r is the Euclidean norm,
P2i = 2σ
2x ln(Ii−NB) , (8)
and B† =(BT B
)−1 BT is the Moore-Penrose generalized inverse of
B =
r′T1 1...
...r′TN 1
. (9)
Note that the column of ones in B are understood to have units of length so that the elements ofB† are in units of length−1. The units of α are length2 and thus the estimated position, ro, hasunits of length.
To avoid problems with calculating the natural logarithm in (8), measurements below thebackground level are discarded.
As a model-based estimator, the performance of the fluoroBancroft algorithm depends onproper choice of the Gaussian parameters (σx,σy,σz) according to the optical parameters ofa given system. Note that the estimator is independent of the absolute intensity level definedby m. A detailed study of optimal choices of parameters for approximation of the PSF by aGaussian has been carried out in [28]. From those results, we set
σx = σy =√
2λ
2πN.A., σz =
2√
6nλ
2πN.A.2. (10)
3. Analysis
3.1. Effect of modeling error and pixelization
The actual measured intensity at a pixel is a Poisson random variable with a rate given by thebackground rate and the integral of the PSF defined in (1) over the pixel. The true noise-freerate in a square pixel with a center at ri and sides of length 2a is given by
NT (ri) =∫ xi+a
xi−a
∫ yi+a
yi−aIPSF ((u(zi),v(x,y))dxdy. (11)
We therefore define the error between the modeled and true noise-free rate as
Ne(ri) = NT (ri)−Ns(ri). (12)
This error depends explicitly on the pixel size through the integral in (11). Adding in the back-ground noise, the intensity measured in a pixel with center ri is a Poisson random process withmean given by
E[Ii] = Ns(ri)+Ne(ri)+NB4= NI(ri). (13)
The effect of this modeling error is considered explicitly in the calculations below.
3.2. Estimation bias
The bias of the estimator, denoted as δ ro, is defined as
δ ro = E[ro]− ro. (14)
From (6) we haveE[ro] = QB†E[α] (15)
and, from (7) and (8),
E[αi] =12(‖r′i‖2 +2σ
2x E [ln(Ii−NB)]
). (16)
Defineqi = E [ln(Ii−NB)]− ln(Ns(ri)) (17)
so thatE [ln(Ii−NB)] = ln(Ns(ri))+qi. (18)
Using these results in (16) together with the definition of the signal rate in (4) yields
E[αi] =12
(‖r′i‖2 +2σ
2x lnm−σ
2x (ri− ro)
TΣ−1 (ri− ro)+2σ
2x qi
)=−1
2‖r′o‖2 + r
′To r′i +σ
2x lnm+σ
2x qi. (19)
Using (19) in (15) yields
E[ro] = QB†
−12‖r′o‖2 + r
′To r′1
...− 1
2‖r′o‖2 + r
′To r′N
+σ2x lnmQB†
1...1
+σ2x QB†
q1...
qN
. (20)
It is shown in [24] that for any constant vector d, QB†d = 0. Thus, the middle term in (20) iszero. Explicitly evaluating B† and carrying out the matrix multiplications for the first term in(20) is a straightforward but tedious operation that yields
E [ro] = ro +σ2x QB†
q1...
qN
. (21)
Thus, from (14), the bias is given by
δ ro = σ2x QB†
q1...
qN
. (22)
The calculation of the term qi driving the bias can be done in several different ways. Under theassumption that the measured intensities are high enough such that the Poisson distribution iswell-approximated by a Gaussian, the statistics of qi can then be calculated using the propertiesof the lognormal distribution (see [29] for more on this approach). To allow for a wider rangeof intensity values, here we expand (17) in a Taylor series about the rate NT (ri)+NB, yielding
qi = ln(NT (ri))− ln(Ns(ri))+∞
∑k=1
(−1)k+1 E[(Ii−NB−NT (ri))
k]
k (NT (ri))k (23)
≈ ln(NT (ri))− ln(Ns(ri))−NI(ri)
2N2T (ri)
+NI(ri)
3N3T (ri)
(24)
where we have used the properties of the Poisson distribution (see, e.g. [30]) to explicitly eval-uate the expansion to third order. From the zeroth-order term of the expansion, we see that thebias is driven primarily by the modeling error.
3.3. Estimation precision
The variance of the fluoroBancroft estimator is given by
Var(ro) = E[(ro− (ro +δ ro))(ro− (ro +δ ro))
T]
= E[rorT
o]− (ro +δ ro)(ro +δ ro)
T . (25)
Evaluating the first term yields
E[rorT
o]= QB†E
[αα
T ]B†T QT . (26)
DefineΘ = E
[αα
T ]−E [α]E[α
T ] (27)
so that
Var(ro) = QB†E [α]E[α
T ]B†T QT +QB†ΘB†T QT − (ro +δ ro)(ro +δ ro)
T
= QB†ΘB†T QT . (28)
Consider now the definition of α in (7). By the independence of the intensity measurements,we have, for i 6= j,
E [αiα j] = E [αi]E [α j] . (29)
The off-diagonal terms of Θ are therefore zero. For the diagonal terms, consider the following.
(E [αi])2 =
(E[
12(‖r′i‖2 +P2
i)])2
=14
(‖r′i‖4 +2‖r′i‖2E
[P2
i]+(E[P2
i])2)
=14
(‖r′i‖4 +2‖r′i‖2E
[P2
i]+E
[P4
i]−E
[P4
i]+(E[P2
i])2)
=14
E[‖r′i‖4 +2‖r′i‖2P2
i +P4i]+
14
((E[P2
i])2−E
[P4
i])
= E[α
2i]+
14
((E[P2
i])2−E
[P4
i])
. (30)
Therefore,
Θ =14
diag{
E[P4
1]−(E[P2
1])2
, · · · ,E[P4
N]−(E[P2
N])2}
(31)
where diag(·) denotes a matrix whose main diagonal entries are as given and the off-diagonalelements are zero. Using the definition of Pi in (8) yields
Θii = σ4x
(E[ln2 (Ii−NB)
]− (E [ln(Ii−NB)])2
)(32)
where Θii indicates the (ii)th element of the matrix Θ. To evaluate the variance, we expand thefunctions in (32) in a Taylor series about the rate NT (ri)+NB. Using once again the propertiesof the Poisson distribution, to third order this yields
Θii ≈ σ4x
(NI(ri)N2
T (ri)− NI(ri)
N3T (ri)
)(33)
The precision is then the standard deviation of the estimator, given by the square root of (28).
4. Numerical simulations
4.1. CCD image construction
Using (1) to model the spatial intensity of a diffraction-limited spot, a collection of noise-freeCCD image stacks was created by integrating (1) over square pixels with sides of 50 nm inlength. Intensity was varied by scaling the resulting values. An example of such noise-freeimages is shown in Fig. 4.1 (at -500 nm relative to the focal plane) and Fig. 4.1 (in the focalplane). The data was scaled such that the peak intensity value in the center image was 362counts. To simulate actual CCD images containing both background and shot noise, measuredpixel values were generated by sampling from a Poisson distribution with a rate given by thenoise-free value in the pixel plus a background level. Example images generated from the noise-free images with a background noise level of 10 are shown in Figs. 4.1 (-500 nm relative to thefocal plane) and 4.1 (focal plane). Define the signal-to-noise ratio (SNR) in an image to be
SNR =Imax√
Imax + IB, (34)
where Imax is the maximum intensity level in the image above the background and IB is the meanbackground level. Because in this work we are generating the images from known mean rates,we use the exact noise-free value for Imax. Using this formula, the SNRs in the noisy images inFig. 1 were 13.4 in the image at -500 nm and 18.8 in the focal plane image. We define SNR ofan image stack to be the maximum SNR of the images in the stack.
For estimation, image stacks of three images at (-100,0,100) nm relative to the focal planewere generated. On each data set, estimation of the source position was performed using thecenter 5× 5 subset of pixels. To illustrate the effect of background and signal levels on theperformance of the FB algorithm, image sets were created for a range of background levelsranging from 0 to 20 and maximum intensities ranging from approximately 30 to 1500, corre-sponding to SNRs ranging from approximately 4 to 38. In order to illustrate the effect of therelative position of the source in the subset of pixels used in estimation, multiple image setswere created with shifts of the source position from -100 nm to 100 nm in the x direction andfrom -300 nm to 300 nm in the z direction for a fixed intensity and background level. To obtainstatistics on the estimator, 2500 image sets were created for each set of parameters.
In the discussion below, results in the y direction were similar to those in the x direction andare omitted.
4.2. Bias
From (24), the bias of the FB algorithm depends on the modeling error, the total number ofphotons used in the estimate, and the relative locations of the measurements (pixels) to the
(a) Noise-free image at z =−500 nm (b) Noise-free image at z = 0 nm
(c) Noisy image at z =−500 nm (d) Noisy image at z = 0 nm
Fig. 1. (a)-(b) noise-free and (c)-(d) corresponding noisy images at -500 nm and 0 nmrelative to the focal plane of the source particle. The noisy images were created from thenoise free by generating pixel values as samples from a Poisson distribution with rate givenby the noise-free value plus a background rate of 10 photons per pixel. Note that the scalesare different in the two planes. Using (34), the SNRs of the images were 6.02 in image (c)and 4.83 in image (d).
source particle. Fig. 2 presents the theoretical and simulation results for the bias in the x andz directions as a function of the total number of photons (summed over all pixels used in theestimation) and for a range of background intensities ranging from 0 to 20. The number ofphotons used at a fixed background rate was adjusted by scaling the noise-free image. Theresulting SNRs of the image stacks ranged from 4 to 38. The source particle was located at theorigin. The theoretical bias in both the lateral and axial directions in this configuration, shownas the solid lines in the figures, was zero. The simulation results show a small scatter aroundzero, with the range of scatter diminishing with increasing number of photons. The bias wasindependent of the background rate.
(a) Bias in x (b) Bias in z
Fig. 2. Bias of the FB estimate in the (a) x and (b) z directions as a function of the totalnumber of photons used for estimation, corresponding to SNRs from approximately 4 to38. Estimation was performed using a 5×5 set of pixels in each of the three images of thez−stack. Theoretical results were calculated using (22) together with the third-order expan-sion in (24). Simulation results with different background noise levels are plotted as dots ofdifferent colors. Simulation results show a small scatter in both the axial and lateral direc-tions around the theoretical value of zero bias with the scatter decreasing with increasingnumber of photons. The bias in both directions was independent of the background rate.
The bias of the FB algorithm in the x and z directions as a function of the particle positionalong the x−axis is shown in Fig. 3. In these results, the simulation parameters were set toa background level of 10 and signal intensities varied to produce SNRs ranging from 5.33 to14.4. The results illustrate that shifts of the source particle in the plane do not affect the biasof the estimate along the optical axis but do introduce a bias in the plane. This bias was nearlylinear over a shift of 50 nm (one pixel), increasing from 0 to approximately ±5 nm over thatrange for all SNR values. The theoretical curves and simulation results diverged somewhatat shifts larger than one pixel with the error larger at lower SNR. This is due primarily totwo effects. The first is error introduced by the finite expansions used in the calculation ofthe standard deviation. The second is due to discarding measurements below the backgroundlevel. Discarding measurements based on their values changes the probability density of themeasurements from a Poisson distribution to a conditional distribution (the distribution giventhat the value is above the background level). At large shifts, the measured intensities approachthe background level and thus more measurements are discarded. As a result, the differencesbetween the conditional and the Poisson distribution become more pronounced. In general, (24)shows that the bias can be reduced by decreasing the modeling error. This can be achieved bychoosing a smaller set of pixels for estimation (c.f. Fig. 8), though at the cost of reducing the
total number of photons used and thus the precision of the algorithm. (See also Sec. 4.4.)
(a) Bias in x (b) Bias in z
Fig. 3. Bias of the FB estimate in the (a) x and (b) z directions as a function of particleposition along the x−axis. Simulation parameters were set to a background rate of 10 withSNRs from 5.33 to 14.4. Estimation was performed using a 5×5 set of pixels in each of thethree images of the z−stack. Theoretical results were calculated using (22) together withthe third-order expansion in (24). Bias in the axial direction was nearly independent of theshift of the source in the plane while the planar estimate exhibited a shift-dependent bias.Within one pixel (50 nm) the bias was typically less than 5 nm. See text for a discussion ofthe error at large shifts and low SNR between the simulation and theoretical values.
The bias of the FB algorithm in the x and z directions as a function of the shift of the particlealong the z−axis is shown in Fig. 4. As before, the background level was set to 10 and simula-tions were performed at different SNR values. The shift of the source along the optical axis didnot affect the bias of the estimate in the plane but did create a bias along the optical axis. Forthe settings used in these simulations, the bias was on the order of a few nm throughout shiftsof up to 100 nm and increased only up to 10-20 nm at shifts of 300 nm. Note that unlike shiftsof the source molecule in the plane, shifts along the optical axis do not move the particle awayfrom the center of the 5 × 5 set of pixels in each plane for estimation. As a result, fewer pixelswere discarded, leading to a good match between theory and simulation results at all shifts.
4.3. Precision
Fig. 5 shows the theoretical and simulation results for the precision in the x and z directions asa function of the total number of photons used in the estimation and for a range of backgroundintensities ranging from 0 to 20. As before, the SNRs of the image stacks ranged from 4 to38 and the source particle was located at the origin. At high numbers of photons (equivalently,high SNRs), the curves approach a 1/
√N power law where N is the total number of photons,
indicating that shot noise dominates. At lower SNR levels, the curves approach a 1/N powerlaw, indicating that background noise dominates. Precision in the x−direction is significantlybetter (by approximately an order of magnitude) than in the z−direction, reflecting both thesmaller width of the PSF and the better approximation of the Gaussian model in the lateraldirection over the axial direction.
The precision of the FB algorithm in the x and z directions as a function of the shift ofthe source particle along the x−axis is shown in Fig. 6. The background level was 10 andsimulations performed at SNRs ranging from 5.33 to 14.4. At all SNRs, precision both in theplane and along the optical axis decreased as the source particle was shifted away from the
(a) Bias in x Bias in z
Fig. 4. Bias of the FB estimate in the (a) x and (b) z directions as a function of particleposition along the z−axis. Simulation parameters were set to a background rate of 10 withSNRs of 5.33 to 14.4. Estimation was performed using a 5×5 set of pixels in each of thethree images of the z−stack. Theoretical results were calculated using (22) together with thethird-order expansion in (24). Bias in the plane was independent of the shift of the sourceparticle along the optical axis while the z−estimate exhibited a small shift-dependent bias.
(a) Std. dev. in x (b) Std. dev. in z
Fig. 5. Standard deviation in the FB estimate in the (a) x and (b) z directions as a functionof the total number of photons used for estimation (across the image stack), correspondingto SNRs from 4 to 38. Estimation was performed using a 5× 5 set of pixels in each ofthe three images of the z−stack. Theoretical results were calculated using the third-orderexpansion in (33). At low SNR, background noise dominated, leading to a 1/N dependencewhere N is the total number of photons. At high SNR, shot noise dominated, leading to a1/√
N dependence. Precision in the plane was roughly an order of magnitude better thanalong the optical axis.
center of the set of pixels used in the estimation algorithm. As before, at larger shifts, thetheoretical values began to diverge from the simulation results.
(a) Std. dev. in x (b) Std. dev. in z
Fig. 6. Standard deviation as a function of particle position along the x−axis. Simulationparameters were set to a background rate of 10 with SNRs ranging from 5.33 to 14.4.Estimation was performed using a 5× 5 set of pixels in each of the three images of thez−stack. Theoretical results were calculated using the third-order expansion in (33). Preci-sion in the plane was significantly better than the axial direction for all shift values. Bothshow a dependence on the relative shift of the source position in the plane.
The precision of the FB algorithm in the x and z directions as a function of the shift of thesource particle along the z−axis is shown in Fig. 7. The background level was set to 10 and theSNR varied from 5.33 to 14.4. As with the shift in the plane, the precision both in the plane andalong the optical axis decreased as the source particle was shifted away from the center planeof the z−stack, though the dependence on the shift was weaker than for the shift in the plane.
4.4. Effect of the size of the estimation set
The Gaussian model is a good approximation to the planar PSF but is less accurate for the fullthree-dimensional PSF. In general, the model is more accurate close to the source. As a result,measurements that are much lower than the peak intensity tend to reduce the performance ofthe FB algorithm. Using a small pixel set can help to ensure that only pixels with high valuesare selected, thus reducing modeling error. The bias is driven primarily by modeling error, andtherefore reducing the size of the pixel set can reduce the bias. To illustrate this, the simulationswere repeated using a 3×3 pixel set in each image. Fig. 8(a) shows the bias in the x−directionas a function of the particle position along the x−axis and should be compared to Fig. 3(a).Reducing the pixel set from 5× 5 to 3× 3 reduced the bias by approximately half. Fig. 8(b)shows the bias in the z−direction as a function of the particle position along the z−axis andshould be compared to Fig. 4(b). Reducing the pixel set changed the overall shape of the bias asa function of shift and actually increased the bias at the larger shifts. This was caused primarilyby the reduction in the total number of photons at the larger shifts.
As seen from (33) and the results in Fig. 5, the precision improves with increasing totalnumber of photons and thus reducing the size of the pixel set can degrade the performance.This is illustrated in Fig. 9 which shows the precision in the x− and z− directions as a functionof particle position along those axes when a 3×3 pixel set is used. Performance is degraded inboth instances over Figs. 6(a) and 7(b), respectively, where a 5×5 pixel set was used
(a) Std. dev. in x (b) Std. dev. in z
Fig. 7. Standard deviation as a function of particle position along the z−axis. Simulationparameters were set to a background rate of 10 with SNRs from 5.33 to 14.4. Estimationwas performed using a 5×5 set of pixels in each of the three images of the z−stack. Theo-retical results were calculated using the third-order expansion in (33). Precision in the planewas significantly better than the axial direction for all shift values. Both show a dependenceon the relative shift of the source position along the axis, though the dependence is weakerthan with the planar shift shown in Fig. 6.
(a) Bias in x (b) Bias in z
Fig. 8. Bias of the FB estimate in the (a) x direction as a function of shift along the x−axis and in the (b) z direction as a function of particle position along the z−axis. Simula-tion parameters were set to a background rate of 10 with SNRs ranging from 5.33 to 14.4.Estimation was performed using a 3× 3 set of pixels in each image of the z−stack. The-oretical results were calculated using (22) together with the third-order expansion in (24).(a) should be compared with Fig. 3(a) and (b) with Fig. 4(b). While reducing the pixel setcan reduce the bias, as in (a), it also reduces the total photon count. In (b), the lower inten-sities created by the larger shifts along the optical axis, the smaller total number of photonsincreased the bias.
(a) Std. dev. in x (b) Std. dev. in z
Fig. 9. Standard deviation in the (a) x direction as a function of shift along the x− axisand in the (b) z direction as a function of particle position along the z−axis. Simulationparameters were set to a background rate of 10 and runs performed at SNRs ranging from5.33 to 14.4. Estimation was performed using a 3× 3 set of pixels in each image of thez−stack. Theoretical results were calculated using the third-order expansion in (33). (a)should be compared with Fig. 6(a) and (b) with Fig. 7(b). In both cases the precision of thealgorithm was degraded due to the fewer total number of photons used in the estimate.
5. Conclusions
This paper has presented a theoretical analysis of the bias and precision of the fluoroBan-croft position estimation algorithm for use in localizing single sub-diffraction limit fluores-cent sources. Simulation results illustrate the results and help to indicate where the theoreticalexpressions are valid. These results can help guide parameter choices when using the fluo-roBancroft algorithm as well as to understand the accuracy of estimation when applying thealgorithm.
Acknowledgements
This work was supported in part by the National Science Foundation through grants DBI-0649823 and CMMI-0845742.
