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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24461–24476
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Accuracy of the Gaussian Point Spread Function model in 2D localization microscopy

Sjoerd Stallinga and Bernd Rieger  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 24461-24476 (2010)
http://dx.doi.org/10.1364/OE.18.024461


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Abstract

The Gaussian function is simple and easy to implement as Point Spread Function (PSF) model for fitting the position of fluorescent emitters in localization microscopy. Despite its attractiveness the appropriateness of the Gaussian is questionable as it is not based on the laws of optics. Here we study the effect of emission dipole orientation in conjunction with optical aberrations on the localization accuracy of position estimators based on a Gaussian model PSF. Simulated image spots, calculated with all effects of high numerical aperture, interfaces between media, polarization, dipole orientation and aberrations taken into account, were fitted with a Gaussian PSF based Maximum Likelihood Estimator. For freely rotating dipole emitters it is found that the Gaussian works fine. The same, theoretically optimum, localization accuracy is found as if the true PSF were a Gaussian, even for aberrations within the usual tolerance limit of high-end optical imaging systems such as microscopes (Marechal’s diffraction limit). For emitters with a fixed dipole orientation this is not the case. Localization errors are found that reach up to 40 nm for typical system parameters and aberration levels at the diffraction limit. These are systematic errors that are independent of the total photon count in the image. The Gaussian function is therefore inappropriate, and more sophisticated PSF models are a practical necessity.

© 2010 Optical Society of America

1. Introduction

The exploitation of the physics of time dependent fluorescence has enabled the emergence of the field of far-field optical nanoscopy [1

1. D. Evanko, “Primer: fluorescence imaging under the diffraction limit,” Nat. Methods 6, 19–20 (2009). [CrossRef]

, 2

2. S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6, 24–32 (2009). [CrossRef] [PubMed]

]. This time dependence can be achieved by on/off switching of fluorophores, which is either intrinsically stochastic (blinking) or deliberately engineereed (photoswitching, saturation or depletion). Within the spectrum of optical nanoscopy techniques [3

3. S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316, 1153–1158 (2007). [CrossRef] [PubMed]

9

9. M. Heilemann, S. van de Linde, M. Schttpelz, R. Kasper, B. Seefeldt, A. Mukherjee, P. Tinnefeld, and M. Sauer, “Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes,” Angew. Chem. Int. Ed. Engl. 476172–6176 (2008). [CrossRef] [PubMed]

] localization microscopy holds the promise of ease-of-use because it uses a conventional widefield fluorescence microscopy set-up. One of the key factors in the success of localization microscopy is the Point Spread Function (PSF) model used for fitting the emitter positions in the sparse images, as the achievable localization accuracy is affected by the correctness of the PSF model. So far, mainly Gaussian model PSFs have been used, because of their conceptual simplicity and computational efficiency, despite the lack of physical foundation. High objective numerical aperture values require full vectorial modelling of the PSF. In addition, the orientation of the radiating dipoles plays a significant role in the observable spot shape. In many cases the emitter can either rotate or change its conformation freely during the lifetime of the excited state of the emitter, so for many excitation-emission cycles an average over randomly distributed emission dipole orientations will be observed. In other cases we must consider ensembles of fixed but randomly oriented dipoles, like fluorescent beads, and we need to average over the different dipole orientations as well, but now weighted with the excitation efficiency as a function of dipole orientation. In yet a third class of cases, the single emitter cannot rotate freely, such as in solid-state or cryofreezed samples or for emitters immobilized at a surface or in a matrix, and the fixed dipole orientation needs to be taken into account in the analysis, as it affects the observed spot shape [10

10. A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999). [CrossRef]

, 11

11. P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, “Orientational effects in the excitation and de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express 15, 3372–3383 (2007). [CrossRef] [PubMed]

].

The question how to accurately model the PSF in accordance with the laws of optics has been addressed by a number of authors. The starting point is the theory laid down by Wilson, Higdon, Török, and co-workers [12

12. T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997). [CrossRef]

15

15. M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008). [CrossRef] [PubMed]

] and adapted to single molecule localization by Enderlein and co-workers [16

16. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

] and recently by Mortensen and co-workers [17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

]. The latter two references restrict their analysis to the case of 2D Total Internal Reflection Fluorescence (TIRF) imaging without any aberrations.

2. Vectorial theory of imaging of a dipole emitter

2.1. Preliminaries and definitions

The emitter dipole is located in a medium with refractive index nmed adjacent to a cover slip with refractive index ncov. The emitter is imaged with an immersion objective lens with numerical aperture NAob designed for an immersion fluid with refractive index nimm. The objective lens is assumed to be corrected for focusing onto the interface between the cover slip and the medium. The intersection of the optical axis with this interface is taken to be the origin of the coordinate system in object space. The emitted radiation is collected by the objective lens with focal length Fob and focused by the tube lens with focal length Fim onto the detector. Both lenses are assumed to be aplanatic and the imaging system is assumed to be telecentric, i.e. the aperture stop is located at the back focal plane of the objective lens, which coincides with the front focal plane of the tube lens. The magnification of the imaging system is M = Fim/Fob, thus making the numerical aperture in image space NAim = NAob/M. In practice M ≫ 1, so that NAim ≪ 1. The radius of the stop is given by R = FobNAob = FimNAim. Throughout this paper we use scaled coordinates. The pupil coordinates are scaled with the stop radius R scaling the pupil to the unit circle, the object and image coordiates are scaled with the diffraction lengths λ/NAob (object space) and λ/NAim (image space). We use ‘u’ for object and image coordinates and ‘v’ for Fourier/pupil coordinates throughout this text. The emitter dipole vector is oriented along the unit-vector d⃗ = (sin θd cos ϕd, sin θd sin ϕd, cos θd), where θd is the polar angle, and ϕd is the azimuthal angle, and has magnitude d0. The (2D) position of the emitter with respect to the focal point is r⃗d = (xd, yd). The scaled position is u⃗d = NAobr⃗d/λ.

2.2. Model for the PSF for arbitrary aberrations

The electric field in the pupil plane at point v⃗ = (vx, vy, 0) corresponds to the plane wave in object space with wavevector along (sin θmed cos ϕ, sin θmed sin ϕ, cos θmed), so:
vx=nmedsinθmedNAobcosϕ,
(1)
vy=nmedsinθmedNAobsinϕ.
(2)

The electric field in the pupil plane has only non-zero values for the Cartesian components j = x,y given by [12

12. T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997). [CrossRef]

15

15. M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008). [CrossRef] [PubMed]

]:
Epub,j(v,ud)=πd0ɛ0λ2Fob(1v2NAob2/nmed2)1/4k=x,y,zqjk(v)dke2πiudv,
(3)
and with the polarization vectors:
qxk=cosϕTppksinϕTssk,
(4)
qyk=sinϕTppk+cosϕTssk,
(5)
which depend on the p and s basis polarization vectors p⃗ = (cos θmed cos ϕ, cos θmed sin ϕ, –sin θmed) and s⃗ = (–sin ϕ, cos ϕ, 0) and on the Fresnel coefficients:
Ta=Ta,medcovTa,covimm,
(6)
for a = p, s and where the Fresnel coefficients for the two contributing interfaces (see in particular [14

14. O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11, 2964–2969 (2003). [CrossRef] [PubMed]

] for details of the effects of stratified media) are defined by:
Ta,12=2ca,1ca+1+ca+2,
(7)
for a = p,s and with cp,l = nl/ cosθl and cs,l = nl cos θl for l = med, cov, imm.

The field in the pupil plane is modified by the pupil function
C(v)={A(v)exp(iW(v)),|v|1,0,|v|>1,}
(8)
with A(v⃗) the amplitude and W(v⃗) the phase (aberration) of the pupil function, and the field in the image plane (on the detector) at scaled position u⃗ is found by subsequently applying a Fourier transform. As the numerical aperture of the tube lens is typically rather small compared to unity we may safely apply scalar diffraction theory to the x and y components of the field seperately. This gives:
Eim,j(u,ud)=E0k=x,y,zwjk(uud)dk,
(9)
with:
E0=π2NAobNAimd0iɛ0λ3,
(10)
and:
wjk(u)=1πd2vC(v)qjk(v)(1v2NAob2/nmed2)1/4exp(2πiuv).
(11)

The measured signal (PSF) is proportional to the z-component of the Poynting-vector. In view of the low numerical aperture on the image side this is simply proportional to the square of the modulus of the electric field. The PSF then follows as:
I(u,ud)=12ɛ0cj=x,y|Eim,j(u,ud)|2=I0k,l=x,y,z[j=x,ywjk(uud)wjl(uud)*]dkdl.
(12)
with:
I0=π4cNAob2NAim2|d0|22ɛ0λ6.
(13)

If there are no aberrations present the equations of refs. [16

16. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

, 17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

] are reproduced as a special case. The current theory generalizes on these prior results by including arbitrary aberrations, in particular asymmetric ones like astigmatism and coma.

The total power captured by a pixel centered at u⃗p is found by integrating over the pixel area Ap:
P(up,ud)=λ2NAim2Apd2uI(u,ud)=P0k,l=x,y,z[j=x,yApd2uwjk(uud)wjl(uud)*]dkdl,
(14)
with:
P0=π4cNAob2|d0|22ɛ0λ4.
(15)

The dipole magnitude d0 is given by:
d0=αEilldex,
(16)
with α the polarizability, E⃗ill the electric field of the illumination and d⃗ex a unit-vector along the excitation dipole. For a fixed dipole the excitation and emission dipole orientation are equal: d⃗ex = d⃗. So, apart from the bilinear dependence on dipole orientation explicit in Eqs. (12) and (14) the PSF also depends on dipole orientation implicitly via the excitation efficiency.

There is an interesting relation between the theory presented here and the theory of Richards and Wolf and many subsequent authors for the vectorial field in the focal region when a polarized plane wave in the pupil plane is focused by the objective lens. The exact same functions wjk(u⃗) turn out to determine the field close to focus. The electric field in the focal plane can be written as (see [20

20. S. Stallinga, “Axial birefringence and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001). [CrossRef]

, 21

21. S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004). [CrossRef]

] and references therein):
Efoc,k(u)=πFobiλj=x,yAjwjk(u),
(17)
where (Ax, Ay) is the unit-vector representing the polarization of the plane wave in the pupil plane.

2.3. Orientational averaging

There are two cases in which we must average over dipole orientation. The first case deals with fluorophores that have a rotational diffusion time that is much smaller than the lifetime of the excited state. In that case the emission dipole is fully uncorrelated to the excitation dipole and an independent average over both the excitation and emission dipole orientations must be taken. The second case in which averaging over dipole orientation is needed is the case of an ensemble of fixed but randomly oriented fluorescent molecules, for example as in a fluorescent bead. Now the excitation and emission dipoles are the same and the excitation probability poportional to |E⃗ill · d⃗|2 must be taken into account as a weight factor in the averaging prccedure over the different dipole orientations contributing to the PSF. In general this will lead to a different expression for the PSF. This has been explicitly shown for the TIRF-case by Mortensen et al. [17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

]. In the appendix we derive explicit expressions for the different orientationally averaged PSFs, for arbitrary states of polarization and coherence of the illumination electric field, and for arbitrary aberrations of the optical system. In the following we only report on results for fixed dipoles and for temporal orientationally averaged dipoles, the case of ensemble orientationally averaged dipoles is not considered any further from hereon.

The fixed and averaged dipole cases described here can be seen as limiting cases of a general distribution over dipole orientations characterized by a Probability Distribution Function (PDF), as described in ref. [22

22. M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express 15, 13597–13606 (2007). [CrossRef] [PubMed]

]. The fixed dipole case corresponds to an infinitely sharply peaked PDF, the averaged dipole case to a uniform PDF. Generalizing to an arbitary PDF would allow for a description of the effects of e.g. an arbitrary rotational diffusion time or of thermally driven fluctuations around a preferential orientation.

2.4. The case with azimuthal symmetry

In case of azimuthally symmetric aberrations (defocus, spherical aberration) the functions wjk(u⃗) can be expressed as:
wxx(u,ψ)=F0(u)+F2(u)cos(2ψ),
(18)
wxy(u,ψ)=F2(u)cos(2ψ),
(19)
wxz(u,ψ)=iF1(u)cosψ,
(20)
wyx(u,ψ)=F2(u)cos(2ψ),
(21)
wyy(u,ψ)=F0(u)F2(u)cos(2ψ),
(22)
wyz(u,ψ)=iF1(u)sinψ,
(23)
with the three integrals over functions containing the Bessel functions Jk(x), k = 0, 1, 2 well-known from the theory of Richards and Wolf and generalizations thereof [20

20. S. Stallinga, “Axial birefringence and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001). [CrossRef]

]:
F0(u)=201dvv(Ts+Tp1v2NAob2/nmed2)2(1v2NAob2/nmed2)1/4J0(2πuv)exp(iW(v)),
(24)
F1(u)=201dvv2TpNAob/nmed(1v2NAob2/nmed2)1/4J1(2πuv)exp(iW(v)),
(25)
F2(u)=201dvv(TsTp1v2NAob2/nmed2)2(1v2NAob2/nmed2)1/4J2(2πuv)exp(iW(v)).
(26)

The resulting electric field in the image plane turns out to be:
Eim,x(u,ψ)=E0[F0(u)sinθdcosϕd+F2(u)sinθdcos(2ψϕd)+iF1(u)cosθdcosψ,]
(27)
Eim,y(u,ψ)=E0[F0(u)sinθdsinϕd+F2(u)sinθdsin(2ψϕd)+iF1(u)cosθdsinψ],
(28)
and the resulting intensity is:
I(u,ψ)=I0[(|F0(u)|2+|F2(u)|2)sin2θd+|F1(u)|2cos2θd+2Im{(F0(u)+F2(u))F1(u)*}sin(2θd)cos(ψϕd)2Re{F0(u)F2(u)*}sin2θdcos(2ψ2ϕd)],
(29)

In the absence of aberrations, this expression for the PSF is equivalent to the results of [16

16. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

, 17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

]. For a dynamic dipole orientation the different terms average out to give:
I(u)=13I0(2|F0(u)|2+2|F2(u)|2+|F1(u)|2),
(30)
which does not depend on the azimuthal angle ψ anymore.

3. The effect of aberrations and dipole orientation on spot shape and localization accuracy

It is worthwile to study the expressions for the PSF for the fixed and free dipole cases in some detail before discussing the results of the numerical simulations.

The PSF for a freely rotating dipole is centrosymmetric but deviates significantly from the classical Airy-shape of scalar diffraction theory for the high NA values used in practice. Actually, the PSF effectively approaches a Gaussian shape with small shoulders at the location of the first Airy-rings (see Fig. 1). If the PSF is considered in the window of width 2 λ / NA it may be approximated with even better accuracy by a Gaussian plus a background [17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

,19

19. C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010). [CrossRef] [PubMed]

]. This change in PSF-shape is due to the various vectorial diffraction effects that smear out the fringe structure of the Airy-distribution. For a water immersion objective the best-fit Gaussian has a standard deviation approximately equal to 0.25 λ /NA, which is equal to one pixel for Nyquist sampling. The approximation of the PSF with a Gaussian has been analyzed in greater detail in ref. [23

23. B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. 46, 1819–1829 (2007). [CrossRef] [PubMed]

].

Fig. 1 Dipole orientation averaged PSF for a 100 ×/1.25 water immersion objective, an Airy-distribution for the same NA, and a Gaussian with standard deviation σ = 0.25 λ /NA.

For the case at hand, determining the location of a single emitter it may be anticipated that the use of a Gaussian model PSF does not introduce a very harmful model error. Even for symmetric aberrations like defocus and spherical aberration no large loss of localization accuracy is expected. However, for asymmetric aberrations like coma we may reasonably expect a significant impact on localization accuracy.

The PSF for a fixed emission dipole is generally non-centrosymmetric [16

16. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

, 17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

]. The spot shape can change dramatically of character for different polar dipole angles. For a planar dipole (θd close to π/2) the first and fourth term in Eq. (29) dominate. The first term is centrosymmetric, the fourth term makes the spot slighlty elliptical, similar to image spots suffering from astigmatism. For an axial dipole (θd close to zero) the second term dominates, giving rise to a doughnut spot (dark spot in the middle with bright ring), similar to a spot with significant defocus and spherical aberration. For intermediate polar angles the third term gains importance. This term makes the spot asymmetric and this resembles the effect of coma, when the spot becomes asymmetric with an increased side lobe in one direction.

Figure 2 shows simulated image spots for different fixed dipole orientations and for different aberrations. The qualitative similarity discussed before between the spot distortions that occur for different polar dipole angles and the spot distortions that occur for different types of aberrations is quite apparent. It is also clear that the similarity is only qualitative, as the spot shapes cannot be matched very well. The relation between dipole orientation induced spot distortions and conventional aberrations such as defocus, astigmatism, coma, and spherical aberration suggests that aberrations may be used to cancel the dipole orientation effects. Alas, it turns out that this does not work very well; aberrations tend to make things worse in all circumstances.

Fig. 2 Simulated spot shapes for different dipole orientations and aberrations. Left: Image spots for NAob = 1.45 (which is larger than nmed = 1.33) for a fixed dipole with polar angle θd = 0 (a), θd = π/4 (b), θd = π /2 (c). Right: Image spots for a free dipole aberrated by 300 m λ RMS defocus (d), 160 m λ RMS coma (e), and 100 m λ RMS astigmatism (f).

It may be anticipated that similar conclusions may be drawn regarding the symmetry or asymmetry of the spot, and type of distortions of spot shape, for more general optical imperfections, such as the polarization aberrations described by McGuire and Chipman [24

24. J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, 5080–5100 (1994). [CrossRef] [PubMed]

, 25

25. J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 2. Tilted and decentered optical systems,” Appl. Opt. 33, 5101–5107 (1994). [CrossRef] [PubMed]

].

The main source of error in emitter localization can be expected to come from the asymmetry term. It appears from Eq. (29) that this term only contributes if the function F1(u) and/or the function F0(u) + F2(u) is complex. This can occur when the Fresnel coefficients are complex. This happens for a numerical aperture larger than the medium refractive index, e.g. when an oil immersion objective is used to collect light from emitters in a watery medium. Then evanescent waves can couple from the dipole emitter into the objective, and for these evanescent waves the Fresnel-coefficients are complex. This is exactly the case studied in [16

16. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

, 17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

], and indeed they have noticed the strong asymmetry due to the third term in the PSF. This dipole orientation induced asymmetry in the PSF can be eliminated by stopping down the objective to a numerical aperture smaller than the medium refractive index at the expense of a reduction in photon collection efficiency. The photon collection efficiency increases with NAob because the solid angle subtended by the aperture of the microscope objective increases, and for NAob > nmed also by evanescent wave coupling into the marginal region of the objective aperture. This so-called supercritical fluorescence hugely increases the photon output [16

16. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

] and the ratio of photon collection efficiency between NAob = 1.45 and NAob = 1.25 turns out to be a factor of about 3. It follows that either the acquisition time is increased with this factor of 3 or the statistical localization error is increased with a factor 3. This seems a small price to pay for avoiding large systematic localization errors due to spot shape distortions. Naturally, the amplitude of the evanescent wave contributions decay exponentially with the distance from the the cover slip. As a consequence, the spot asymmetry effect and the supercritical fluorescence based photon output gain only apply to the TIRF context, and vanish for emitters more than a few wavelengths away from the cover slip.

Fig. 3 Simulated spot shape for NAob = 1.25 (which is less than nmed = 1. 33) for a fixed dipole with polar angle θd = π /4, without aberrations (a) and with 72 m λ RMS defocus (b).

4. Numerical results

4.1. Effect of dipole orientation

Fig. 4 Estimated error as a function of detected number of photons for fixed/free dipole orientation and NAob values below and above the refractive index of the medium, and the theoretical Gaussian based CRLB.

4.2. Effect of aberrations

Figure 5 shows the error as a function of detected number of photons for free and fixed dipoles and for different aberration settings. We have investigated the effect of defocus, astigmatism, coma, and spherical aberration, taking values of 50% and 100% of Marechal’s diffraction limit (72 mλ RMS). An optical system is considered well-corrected if the total RMS aberation is less than the diffraction limit. In the simulations NAob = 1.25 and we take Nyquist sampling, so for the wavelength λ = 500 nm the pixel size is equal to the Nyquist unit λ /4NAob = 100 nm. For freely rotating dipoles aberrations hardly have an effect on localization accuracy, with the exception of coma, which gives rise to a systematic localization error of about several nm for the diffraction limit. For dipoles with a fixed orientation the situation is entirely different. Now the same level of aberrations introduce systematic localization errors (the plateau for large photon counts) of several tens of nm. For example, for a diffraction limited amount of defocus the x and y systematic error for diffraction limited defocus (the plateau for large photon count) is about 23 nm, so the total mean square error is 33 nm, in reasonable agreement with the estimate based on the average peak offset of the simulated spot shapes equal to 43 nm. Astigmatism and spherical aberration have a similar effect albeit with a smaller systematic localization error for the same level of aberrations. Naturally, coma gives foremost an effect in the asymmetry direction with systematic errors of 22 nm for the diffraction limited level.

Fig. 5 Estimated error as a function of detected number of photons for different aberration settings. Left (a to d): averaged over dipole orientation, right (e to f): randomly selected fixed dipole orientation. First row: defocus (a and e), second row: astigmatism (b and f), third row: coma (c and g), fourth row: spherical aberration (d and h).

4.3. Effect of background

We have noticed several subtleties regarding the fitted background. For a small photon count the fitted background is zero, for high photon count the fitted background is essentially non-zero. Somewhat suprisingly, the transition between the two regimes does not appear to be continuous. For intermediate photon counts, around 100, the MLE fit routine can converge to two qualitatively different local minima, one with zero background and one with non-zero background. The relative occurence of the two types of solutions gradually changes from the zero background type to the non-zero background type as the photon count increases. It appears that there is no correlation between the type of solution that is found and the dipole orientation or any other parameter. The small shoulder in the intermediate regime of the curves of Fig. 5 that can develop when aberrations are present is attributed to this effect. For the case of coma, the asymmetric aberration, another complication arises. Now the zero background solution appears to have a systematic offset whereas the non-zero background solution has not. For that reason the offset averaged over all configurations varies depending on the relative occurence of the two types of solutions, i.e. depending on photon count. This will introduced another systematic error if the variations in photon count per emitter are large. These are largely determined by shot noise variations, although the effect of dipole orientation on detection efficiency may be quite significant for the fixed dipole case. For a TIRF setup it is therefore advisable to use a circularly polarized illumination, as this minimizes the dependence of excitation efficiency on dipole orientation.

Simulations with ‘ground truth’ images with an added background reveal another effect of interest. For an added background of one photon per pixel the bistable effect described above has disappeared, but now a new complication is observed. For small photon counts (total number of ‘signal’ photons in the block of 11 × 11 pixels is less than the total number of background photons) the MLE fit routine converges to Gaussians with a significantly larger width than the physical PSF width and a significant fraction of the background photons is counted as signal. We believe this is due to randomly distributed shot noise peaks in the background that are mistakenly identified as the image spot. As a consequence, the localization uncertainty is larger than the CRLB for a Gaussian PSF with the width of the physical PSF, even if the effect of the added background is incorporated in the CRLB-curve [19

19. C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010). [CrossRef] [PubMed]

]. The increase in localization error in this regime is evident from Fig. 6. The deviations of the MLE curve from the CRLB in the low photon count regime can possibly be described by efficiency metrics such as the Barankin bound [30

30. M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768, 153–160 (1992). [CrossRef]

] or even more elaborate measures [31

31. S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005). [CrossRef] [PubMed]

].

Fig. 6 Estimated error as a function of detected number of photons for fixed dipole orientation and NAob = 1.25, for different defocus settings in presence of a background of one photon per pixel, and the theoretical Gaussian based CRLB with this background.

5. Conclusion

It may be concluded from the foregoing analysis that the use of a Gaussian PSF does not compromise the localization accuracy provided the emitter dipole orientations are not fixed, even for aberrations that are within the usual tolerances of optical design, with the exception of coma. For fixed emitter dipole orientations the use of a Gaussian PSF is no longer justified. In case the aberrations are corrected well below the diffraction limit the use of a water immersion objective instead of an oil immersion objective is quite beneficial, but for aberration levels that can occur in practice the image spots are distorted to such an extent that the use of Gaussian model PSFs introduces systematic errors of up to about 40 nm, even for water immersion objectives.

There are several ways forward to alleviate the problem for fixed dipoles. First of all, we may follow the lines of [17

17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

] and use a fully vectorial PSF model, including aberrations, in the estimator for the emitter position. Second, we can also try a computationally efficient set of basic functions to expand the PSF, and use the expansion coefficients to fit a possibly asymmetric spot shape. Possibly, a limited set of extra shape parameters can be accommodated, at the expense of a moderate increase in the photon count required for a good quality fit of all parameters involved. Finally, the experimental setup can be modified with polarization optics [10

10. A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999). [CrossRef]

, 11

11. P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, “Orientational effects in the excitation and de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express 15, 3372–3383 (2007). [CrossRef] [PubMed]

, 15

15. M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008). [CrossRef] [PubMed]

] for measuring dipole orientation parameters instead of estimating them.

Aberration induced spot deformations are also anticipated to be problematic for 3D localization microscopy, outside of the context of TIRF imaging. The measurement of the axial coordinate requires the introduction of astigmatism with a cylinder lens [32

32. L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, 053902 (2007). [CrossRef]

, 33

33. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008). [CrossRef] [PubMed]

] or making two images with different defocus [34

34. E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007). [CrossRef] [PubMed]

, 35

35. M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008). [CrossRef] [PubMed]

] (see also [36

36. M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, “Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy,” Opt. Express 17, 8264–8277 (2009). [CrossRef] [PubMed]

] for an experimental comparison). In addition, the different axial positions introduce large amounts of defocus. This level of aberrations introduces spot distortions that are so large that they must be dealt with in the position estimator with a more sophisticated model PSF than the simple, straightforward, and convenient Gaussian.

A. Temporal and ensemble orientational averaging

Here we show how to do the averaging for both temporal and ensemble orientational averaging, for arbitrary states of polarization and coherence of the illumination electric field, and for arbitrary aberrations of the optical imaging system. For the sake of notational convenience we drop all functional dependencies from the equations and write Eq. (12) as:
I=ijklCijAkldex,idex,jdkdl,
(31)
with the matrix Akl defined by:
Akl=I0lwjlwkl*,
(32)
and with the mutual coherence:
Cij=Re(Eill,iEill,j*)E,
(33)
where the angular brackets 〈...〉E denotes the averaging over the stochastic occurences of the electric field (state of coherence). The electric field of the illumination can always be written as:
Eill=12A0(p+iq),
(34)
with A0 a (complex) amplitude, and p⃗ and q⃗ two real unit vectors that span the polarization ellipse. For example, for a linear polarization p⃗ and q⃗ are equal, while for a circular polarization they are mutually orthogonal. The mutual coherence matrix can be expressed in terms of these vectors as:
Cij=12|A0|2pipj+qiqjE.
(35)

For fully incoherent illumination we have Cij=(|A0|2/3)δij.

For the temporal orientational averaging we need independent averaging over the excitation and emission dipole, which gives a PSF:
I=19klCkkAll,
(36)
where it is used that:
djdk=13δjk,
(37)
with δjk Kronecker’s delta function (and a similar expression for the excitation dipole). For the ensemble orientational averaging we have d⃗ex = d⃗ and we can use the orientational average over the product of four dipole factors dk:
didjdkdl=115(δijδkl+δilδjk+δikδjl),
(38)
(this expression can be easily derived from symmetry arguments), in order to obtain the averaged intensity:
I=115kl(CkkAll+2CklAkl).
(39)

The second term between brackets makes this ensemble orientationally averaged PSF generally different in character from the temporal orientationally averaged PSF. For the special case of fully incoherent illumination, however, it is found that both cases of orientational averaging give the same PSF.

References and links

1.

D. Evanko, “Primer: fluorescence imaging under the diffraction limit,” Nat. Methods 6, 19–20 (2009). [CrossRef]

2.

S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6, 24–32 (2009). [CrossRef] [PubMed]

3.

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316, 1153–1158 (2007). [CrossRef] [PubMed]

4.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313, 1643–1645 (2006). [CrossRef]

5.

K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, “Superresolution by localization of quantum dots using blinking statistics,” Opt. Express 13, 7052–7062 (2005). [CrossRef] [PubMed]

6.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006). [CrossRef] [PubMed]

7.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008). [CrossRef]

8.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, “Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters,” Biophys. J. 93, 3285–3290 (2007). [CrossRef] [PubMed]

9.

M. Heilemann, S. van de Linde, M. Schttpelz, R. Kasper, B. Seefeldt, A. Mukherjee, P. Tinnefeld, and M. Sauer, “Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes,” Angew. Chem. Int. Ed. Engl. 476172–6176 (2008). [CrossRef] [PubMed]

10.

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999). [CrossRef]

11.

P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, “Orientational effects in the excitation and de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express 15, 3372–3383 (2007). [CrossRef] [PubMed]

12.

T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997). [CrossRef]

13.

P. Torök, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998). [CrossRef]

14.

O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11, 2964–2969 (2003). [CrossRef] [PubMed]

15.

M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008). [CrossRef] [PubMed]

16.

J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]

17.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010). [CrossRef] [PubMed]

18.

R. J. Ober, S. Ram, and E. S. Ward, “Localization Accuracy in Single-Molecule Microscopy,” Biophys. J. 86, 1185–1200 (2004). [CrossRef] [PubMed]

19.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010). [CrossRef] [PubMed]

20.

S. Stallinga, “Axial birefringence and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001). [CrossRef]

21.

S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004). [CrossRef]

22.

M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express 15, 13597–13606 (2007). [CrossRef] [PubMed]

23.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. 46, 1819–1829 (2007). [CrossRef] [PubMed]

24.

J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, 5080–5100 (1994). [CrossRef] [PubMed]

25.

J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 2. Tilted and decentered optical systems,” Appl. Opt. 33, 5101–5107 (1994). [CrossRef] [PubMed]

26.

S. Stallinga, “Compact description of substrate-related aberrations in high numerical aperture optical disk readout,” Appl. Opt. 44, 949–958 (2005). [CrossRef]

27.

J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. 41, 4879–4903 (2002). [CrossRef]

28.

A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley & Sons, New Jersey, 2007). [CrossRef]

29.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002). [CrossRef] [PubMed]

30.

M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768, 153–160 (1992). [CrossRef]

31.

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005). [CrossRef] [PubMed]

32.

L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, 053902 (2007). [CrossRef]

33.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008). [CrossRef] [PubMed]

34.

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007). [CrossRef] [PubMed]

35.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008). [CrossRef] [PubMed]

36.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, “Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy,” Opt. Express 17, 8264–8277 (2009). [CrossRef] [PubMed]

OCIS Codes
(100.6640) Image processing : Superresolution
(110.2990) Imaging systems : Image formation theory
(180.2520) Microscopy : Fluorescence microscopy

ToC Category:
Imaging Systems

History
Original Manuscript: August 17, 2010
Revised Manuscript: October 18, 2010
Manuscript Accepted: October 24, 2010
Published: November 9, 2010

Citation
Sjoerd Stallinga and Bernd Rieger, "Accuracy of the Gaussian Point Spread Function model in 2D localization microscopy," Opt. Express 18, 24461-24476 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24461


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References

  1. D. Evanko, "Primer: fluorescence imaging under the diffraction limit," Nat. Methods 6, 19-20 (2009). [CrossRef]
  2. S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009). [CrossRef] [PubMed]
  3. S. W. Hell, "Far-Field Optical Nanoscopy," Science 316, 1153-1158 (2007). [CrossRef] [PubMed]
  4. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006). [CrossRef]
  5. K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, "Superresolution by localization of quantum dots using blinking statistics," Opt. Express 13, 7052-7062 (2005). [CrossRef] [PubMed]
  6. M. J. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006). [CrossRef] [PubMed]
  7. J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008). [CrossRef]
  8. A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007). [CrossRef] [PubMed]
  9. M. Heilemann, S. van de Linde, M. Schttpelz, R. Kasper, B. Seefeldt, A. Mukherjee, P. Tinnefeld, and M. Sauer, "Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes," Angew. Chem. Int. Ed. Engl. 47, 6172-6176 (2008). [CrossRef] [PubMed]
  10. A. P. Bartko, and R. M. Dickson, "Imaging three-dimensional single molecule orientations," J. Phys. Chem. B 103, 11237-11241 (1999). [CrossRef]
  11. P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, "Orientational effects in the excitation and deexcitation of single molecules interacting with donut-mode laser beams," Opt. Express 15, 3372-3383 (2007). [CrossRef] [PubMed]
  12. T. Wilson, R. Juskaitis, and P. D. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997). [CrossRef]
  13. P. Török, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998). [CrossRef]
  14. O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, "The point spread function of optical microscopes imaging through stratified media," Opt. Express 11, 2964-2969 (2003). [CrossRef] [PubMed]
  15. M. R. Foreman, C. M. Romero, and P. Török, "Determination of the three-dimensional orientation of single molecules," Opt. Lett. 33, 1020-1022 (2008). [CrossRef] [PubMed]
  16. J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]
  17. K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, "Optimized localization analysis for single molecule tracking and super-resolution microscopy," Nat. Methods 7, 377-381 (2010). [CrossRef] [PubMed]
  18. R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004). [CrossRef] [PubMed]
  19. C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, "Fast, single-molecule localization that achieves theoretically minimum uncertainty," Nat. Methods 7, 373-375 (2010). [CrossRef] [PubMed]
  20. S. Stallinga, "Axial birefringence and the light distribution close to focus," J. Opt. Soc. Am. A 18, 2846-2859 (2001). [CrossRef]
  21. S. Stallinga, "Light distribution close to focus in biaxially birefringent media," J. Opt. Soc. Am. A 21, 1785-1798 (2004). [CrossRef]
  22. M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007). [CrossRef] [PubMed]
  23. B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "Gaussian approximations of fluorescence microscope point-spread function models," Appl. Opt. 46, 1819-1829 (2007). [CrossRef] [PubMed]
  24. J. P. McGuire, Jr., and R. A. Chipman, "Polarization aberrations. 1. Rotationally symmetric optical systems," Appl. Opt. 33, 5080-5100 (1994). [CrossRef] [PubMed]
  25. J. P. McGuire, Jr., and R. A. Chipman, "Polarization aberrations. 2. Tilted and decentered optical systems," Appl. Opt. 33, 5101-5107 (1994). [CrossRef] [PubMed]
  26. S. Stallinga, "Compact description of substrate-related aberrations in high numerical aperture optical disk readout," Appl. Opt. 44, 949-958 (2005). [CrossRef]
  27. J. L. Bakx, "Efficient computation of optical disk readout by use of the chirp z transform," Appl. Opt. 41, 4879-4903 (2002). [CrossRef]
  28. A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley & Sons, New Jersey, 2007). [CrossRef]
  29. R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002). [CrossRef] [PubMed]
  30. M. F. Kijewski, S. P. Müller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153-160 (1992). [CrossRef]
  31. S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005). [CrossRef] [PubMed]
  32. L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007). [CrossRef]
  33. B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008). [CrossRef] [PubMed]
  34. E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, "Three-dimensional particle tracking via bifocal imaging," Nano Lett. 7, 2043-2045 (2007). [CrossRef] [PubMed]
  35. M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008). [CrossRef] [PubMed]
  36. M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, "Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy," Opt. Express 17, 8264-8277 (2009). [CrossRef] [PubMed]

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