## Accuracy of the Gaussian Point Spread Function model in 2D localization microscopy |

Optics Express, Vol. 18, Issue 24, pp. 24461-24476 (2010)

http://dx.doi.org/10.1364/OE.18.024461

Acrobat PDF (840 KB)

### 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

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]

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 de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express **15**, 3372–3383 (2007). [CrossRef] [PubMed]

## 2. Vectorial theory of imaging of a dipole emitter

### 2.1. Preliminaries and definitions

*n*

_{med}adjacent to a cover slip with refractive index

*n*

_{cov}. The emitter is imaged with an immersion objective lens with numerical aperture NA

_{ob}designed for an immersion fluid with refractive index

*n*

_{imm}. 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

*F*

_{ob}and focused by the tube lens with focal length

*F*

_{im}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*=

*F*

_{im}/

*F*

_{ob}, thus making the numerical aperture in image space NA

_{im}= NA

_{ob}/

*M*. In practice

*M*≫ 1, so that NA

_{im}≪ 1. The radius of the stop is given by

*R*=

*F*

_{ob}NA

_{ob}=

*F*

_{im}NA

_{im}. 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*/NA

_{ob}(object space) and

*λ*/NA

_{im}(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

*θ*

_{d}

*cos ϕ*, sin

_{d}*θ*

_{d}sin

*ϕ*

_{d}, cos

*θ*

_{d}), where

*θ*

_{d}is the polar angle, and

*ϕ*

_{d}is the azimuthal angle, and has magnitude

*d*

_{0}. The (2D) position of the emitter with respect to the focal point is

*r⃗*

_{d}= (

*x*

_{d},

*y*

_{d}). The scaled position is

*u⃗*

_{d}= NA

_{ob}

*r⃗*

_{d}/

*λ*.

### 2.2. Model for the PSF for arbitrary aberrations

*v⃗*= (

*v*,

_{x}*v*, 0) corresponds to the plane wave in object space with wavevector along (sin

_{y}*θ*

_{med}cos

*ϕ*, sin

*θ*

_{med}sin

*ϕ*, cos

*θ*

_{med}), so:

*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. 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]

*p*and

*s*basis polarization vectors

*p⃗*= (cos

*θ*

_{med}cos

*ϕ*, cos

*θ*

_{med}sin

*ϕ*, –sin

*θ*

_{med}) and

*s⃗*= (–sin

*ϕ*, cos

*ϕ*, 0) and on the Fresnel coefficients: 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]

*a*=

*p,s*and with

*c*=

_{p,l}*n*/ cos

_{l}*θ*and

_{l}*c*=

_{s,l}*n*cos

_{l}*θ*for

_{l}*l*= med, cov, imm.

*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: with: and:

*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: with:

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]

*u⃗*

_{p}is found by integrating over the pixel area

*A*

_{p}: with:

*d*

_{0}is given by: with

*α*the polarizability,

*E⃗*

_{ill}the electric field of the illumination and

*d⃗*

_{ex}a unit-vector along the excitation dipole. For a fixed dipole the excitation and emission dipole orientation are equal:

*d⃗*

_{ex}=

*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.

### 2.3. Orientational averaging

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]

### 2.4. The case with azimuthal symmetry

*w*(

_{jk}*u⃗*) can be expressed as: with the three integrals over functions containing the Bessel functions

*J*(

_{k}*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]

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]

*ψ*anymore.

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

*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. 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]

*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]

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

**7**, 377–381 (2010). [CrossRef] [PubMed]

*θ*

_{d}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 (

*θ*

_{d}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.

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]

*F*

_{1}(

*u*) and/or the function

*F*

_{0}(

*u*) +

*F*

_{2}(

*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

**14**, 8111 (2006). [CrossRef] [PubMed]

**7**, 377–381 (2010). [CrossRef] [PubMed]

_{ob}because the solid angle subtended by the aperture of the microscope objective increases, and for NA

_{ob}>

*n*

_{med}also by evanescent wave coupling into the marginal region of the objective aperture. This so-called supercritical fluorescence hugely increases the photon output [16

**14**, 8111 (2006). [CrossRef] [PubMed]

_{ob}= 1.45 and NA

_{ob}= 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

## 4. Numerical results

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]

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

*λ*/4NA

_{im}(Nyquist sampling). For a water immersion objective with NA

_{ob}= 1.25 and a wavelength

*λ*= 500 nm this implies a pixel size in object space of 100 nm. It was checked that the simulated error did not depend significantly on pixel size in the range [0.15–0.30] ×

*λ*/NA

_{im}. In addition, it appeared that convolution over the finite pixel size did not have a significant impact on the error either, so subsequently it was ignored in the simulation. Random configurations were generated with emitter positions distributed with a normal distribution centered on the central pixel of the block and with standard deviation of one pixel. The dipole orientation for each configuration was randomly selected from a uniform distribution over the full 4

*π*solid angle. After calculation of the ‘ground truth’ PSF an image was generated by distributing

*N*photons over the entire image following Poisson statistics with the ‘ground truth’ PSF as rate parameter. The MLE with Gaussian model PSF was subsequently used to find the emitter locations as well as the best fit to the background and to the height and width of the Gaussian. About 20 iterations were sufficient for convergence of all fitted parameters. In addition to simulations with fixed, randomly selected, dipole orientation we have also done simulations with freely rotating dipoles giving the rotationally averaged PSF. The statistical error was evaluated from 500 random configurations. For a small photon count (less than 20 in the 11 × 11 block of pixels) outliers with a fitted photon count more than twice the ‘real’ photon count were removed before calculating the localization error. We wish to emphasize that similar results to ours are to be expected from other estimators than the MLE given a sufficient photon count, i.e. a good signal-to-noise-ratio [28

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]

### 4.2. Effect of aberrations

*λ*RMS). An optical system is considered well-corrected if the total RMS aberation is less than the diffraction limit. In the simulations NA

_{ob}= 1.25 and we take Nyquist sampling, so for the wavelength

*λ*= 500 nm the pixel size is equal to the Nyquist unit

*λ*/4NA

_{ob}= 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.

### 4.3. Effect of background

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]

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]

## 5. Conclusion

**7**, 377–381 (2010). [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]

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]

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]

## A. Temporal and ensemble orientational averaging

*A*defined by: and with the mutual coherence: where the angular brackets 〈...〉

_{kl}*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: with*

_{E}*A*

_{0}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:

*δ*Kronecker’s delta function (and a similar expression for the excitation dipole). For the ensemble orientational averaging we have

_{jk}*d⃗*

_{ex}=

*d⃗*and we can use the orientational average over the product of four dipole factors

*d*: (this expression can be easily derived from symmetry arguments), in order to obtain the averaged intensity:

_{k}## References and links

1. | D. Evanko, “Primer: fluorescence imaging under the diffraction limit,” Nat. Methods |

2. | S. W. Hell, “Microscopy and its focal switch,” Nat. Methods |

3. | S. W. Hell, “Far-Field Optical Nanoscopy,” Science |

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 |

5. | K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, “Superresolution by localization of quantum dots using blinking statistics,” Opt. Express |

6. | M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods |

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 |

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. |

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. |

10. | A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B |

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 |

12. | T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. |

13. | P. Torök, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. |

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 |

15. | M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. |

16. | J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express |

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 |

18. | R. J. Ober, S. Ram, and E. S. Ward, “Localization Accuracy in Single-Molecule Microscopy,” Biophys. J. |

19. | C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods |

20. | S. Stallinga, “Axial birefringence and the light distribution close to focus,” J. Opt. Soc. Am. A |

21. | S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A |

22. | M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express |

23. | B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. |

24. | J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. |

25. | J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 2. Tilted and decentered optical systems,” Appl. Opt. |

26. | S. Stallinga, “Compact description of substrate-related aberrations in high numerical aperture optical disk readout,” Appl. Opt. |

27. | J. L. Bakx, “Efficient computation of optical disk readout by use of the chirp z transform,” Appl. Opt. |

28. | A. van den Bos, |

29. | R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. |

30. | M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE |

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. |

32. | L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. |

33. | B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science |

34. | E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. |

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 |

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 |

**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

- D. Evanko, "Primer: fluorescence imaging under the diffraction limit," Nat. Methods 6, 19-20 (2009). [CrossRef]
- S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009). [CrossRef] [PubMed]
- S. W. Hell, "Far-Field Optical Nanoscopy," Science 316, 1153-1158 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. P. Bartko, and R. M. Dickson, "Imaging three-dimensional single molecule orientations," J. Phys. Chem. B 103, 11237-11241 (1999). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111 (2006). [CrossRef] [PubMed]
- 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]
- R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004). [CrossRef] [PubMed]
- 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]
- S. Stallinga, "Axial birefringence and the light distribution close to focus," J. Opt. Soc. Am. A 18, 2846-2859 (2001). [CrossRef]
- S. Stallinga, "Light distribution close to focus in biaxially birefringent media," J. Opt. Soc. Am. A 21, 1785-1798 (2004). [CrossRef]
- M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007). [CrossRef] [PubMed]
- 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]
- J. P. McGuire, Jr., and R. A. Chipman, "Polarization aberrations. 1. Rotationally symmetric optical systems," Appl. Opt. 33, 5080-5100 (1994). [CrossRef] [PubMed]
- J. P. McGuire, Jr., and R. A. Chipman, "Polarization aberrations. 2. Tilted and decentered optical systems," Appl. Opt. 33, 5101-5107 (1994). [CrossRef] [PubMed]
- S. Stallinga, "Compact description of substrate-related aberrations in high numerical aperture optical disk readout," Appl. Opt. 44, 949-958 (2005). [CrossRef]
- J. L. Bakx, "Efficient computation of optical disk readout by use of the chirp z transform," Appl. Opt. 41, 4879-4903 (2002). [CrossRef]
- A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley & Sons, New Jersey, 2007). [CrossRef]
- 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]
- 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]
- 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]
- L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007). [CrossRef]
- 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]
- 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]
- 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]
- 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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