Abstract

We present an algebraic solution to the problem of localizing a single fluorescent particle with sub-diffraction-limit accuracy. The algorithm is derived and its performance studied experimentally. Isolated 20 nm fluorescent beads were imaged using a wide-field microscope at two different positions separated by 100 nm and at a range of signal-to-noise ratios (SNR). The data were analyzed using both the new algorithm and the standard approach of fitting the data to a Gaussian profile. Results indicate that the proposed approach is nearly as accurate as Gaussian fitting across a wide range of SNR while executing over 200 times faster. In addition, the new algorithm is able to localize at lower SNR than the fitting method.

©2008 Optical Society of America

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References

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  1. M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
    [Crossref]
  2. C. Kural, H. Balci, and P. R. Selvin, “Molecular motors one at a time: FIONA to the rescue,” J. Phys. Condens. Matt. 27, S3979–S3995 (2005).
    [Crossref]
  3. C. K. Payne, “Imaging gene delivery with fluorescence microscopy,” Nanomedicine 2, 847–860 (2007).
    [Crossref] [PubMed]
  4. W. E. Moerner, “New directions in single-molecule imaging and analysis,” P. Natl. Acad. Sci USA 104, 12,596–12,602 (2007).
    [Crossref]
  5. C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
    [Crossref] [PubMed]
  6. M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
    [Crossref] [PubMed]
  7. A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
    [Crossref]
  8. 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]
  9. R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2007).
    [Crossref]
  10. S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multdim. Syst. Sign. P. 17, 27–57 (2006).
    [Crossref]
  11. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
    [Crossref] [PubMed]
  12. S. B. Andersson, “Tracking a single fluorescent molecule with a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
    [Crossref]
  13. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
    [Crossref] [PubMed]
  14. S. Bancroft, “An algebraic solution of the GPS pseudorange equations,” IEEE T. Aero. Elec. Sys. AES-21, 56–59 (1985).
    [Crossref]
  15. T. Sun and S. B. Andersson, “Precise 3-D localization of fluorescent probes without numerical fitting,” in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181–4184.
    [PubMed]
  16. S. B. Andersson, “Precise localization of fluorescent probes without numerical fitting,” in Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI) (IEEE, 2007), pp. 252–255.
  17. S. B. Andersson, “Position estimation of fluorescent probes in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control (IEEE, 2007), pp. 4950–4955.
  18. R. Juškaitis, “Measuring the real point spread function of high numerical aperture microscope objective lenses,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 239–250.
    [Crossref]
  19. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Pure and Applied Mathematics (John Wiley and Sons, 1974).
  20. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

2008 (2)

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
[Crossref]

2007 (6)

C. K. Payne, “Imaging gene delivery with fluorescence microscopy,” Nanomedicine 2, 847–860 (2007).
[Crossref] [PubMed]

W. E. Moerner, “New directions in single-molecule imaging and analysis,” P. Natl. Acad. Sci USA 104, 12,596–12,602 (2007).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2007).
[Crossref]

T. Sun and S. B. Andersson, “Precise 3-D localization of fluorescent probes without numerical fitting,” in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181–4184.
[PubMed]

S. B. Andersson, “Precise localization of fluorescent probes without numerical fitting,” in Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI) (IEEE, 2007), pp. 252–255.

S. B. Andersson, “Position estimation of fluorescent probes in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control (IEEE, 2007), pp. 4950–4955.

2006 (2)

R. Juškaitis, “Measuring the real point spread function of high numerical aperture microscope objective lenses,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 239–250.
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multdim. Syst. Sign. P. 17, 27–57 (2006).
[Crossref]

2005 (4)

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
[Crossref] [PubMed]

S. B. Andersson, “Tracking a single fluorescent molecule with a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
[Crossref]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

C. Kural, H. Balci, and P. R. Selvin, “Molecular motors one at a time: FIONA to the rescue,” J. Phys. Condens. Matt. 27, S3979–S3995 (2005).
[Crossref]

2003 (1)

M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
[Crossref]

2002 (1)

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]

2001 (1)

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[Crossref] [PubMed]

1985 (1)

S. Bancroft, “An algebraic solution of the GPS pseudorange equations,” IEEE T. Aero. Elec. Sys. AES-21, 56–59 (1985).
[Crossref]

Andersson, S. B.

S. B. Andersson, “Precise localization of fluorescent probes without numerical fitting,” in Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI) (IEEE, 2007), pp. 252–255.

T. Sun and S. B. Andersson, “Precise 3-D localization of fluorescent probes without numerical fitting,” in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181–4184.
[PubMed]

S. B. Andersson, “Position estimation of fluorescent probes in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control (IEEE, 2007), pp. 4950–4955.

S. B. Andersson, “Tracking a single fluorescent molecule with a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
[Crossref]

Babcock, H. P.

M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
[Crossref]

Balci, H.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

C. Kural, H. Balci, and P. R. Selvin, “Molecular motors one at a time: FIONA to the rescue,” J. Phys. Condens. Matt. 27, S3979–S3995 (2005).
[Crossref]

Bancroft, S.

S. Bancroft, “An algebraic solution of the GPS pseudorange equations,” IEEE T. Aero. Elec. Sys. AES-21, 56–59 (1985).
[Crossref]

Ben-Israel, A.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Pure and Applied Mathematics (John Wiley and Sons, 1974).

Berglund, A. J.

A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
[Crossref]

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
[Crossref] [PubMed]

Buranachai, C.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

Cheezum, M. K.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[Crossref] [PubMed]

Gratton, E.

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

Greville, T. N. E.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Pure and Applied Mathematics (John Wiley and Sons, 1974).

Guilford, W. H.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[Crossref] [PubMed]

Ha, T.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

Ishitsuka, Y.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

Joo, C.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

Juškaitis, R.

R. Juškaitis, “Measuring the real point spread function of high numerical aperture microscope objective lenses,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 239–250.
[Crossref]

Kural, C.

C. Kural, H. Balci, and P. R. Selvin, “Molecular motors one at a time: FIONA to the rescue,” J. Phys. Condens. Matt. 27, S3979–S3995 (2005).
[Crossref]

Lakadamyali, M.

M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
[Crossref]

Larson, D. R.

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]

Levi, V.

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

Liddle, J. A.

A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
[Crossref]

Mabuchi, H.

McClelland, J. J.

A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
[Crossref]

McMahon, M. D.

A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
[Crossref]

Moerner, W. E.

W. E. Moerner, “New directions in single-molecule imaging and analysis,” P. Natl. Acad. Sci USA 104, 12,596–12,602 (2007).
[Crossref]

Ober, R. J.

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2007).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multdim. Syst. Sign. P. 17, 27–57 (2006).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Payne, C. K.

C. K. Payne, “Imaging gene delivery with fluorescence microscopy,” Nanomedicine 2, 847–860 (2007).
[Crossref] [PubMed]

Ram, S.

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2007).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multdim. Syst. Sign. P. 17, 27–57 (2006).
[Crossref]

Ruan, Q.

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

Rust, M. J.

M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
[Crossref]

Selvin, P. R.

C. Kural, H. Balci, and P. R. Selvin, “Molecular motors one at a time: FIONA to the rescue,” J. Phys. Condens. Matt. 27, S3979–S3995 (2005).
[Crossref]

Sun, T.

T. Sun and S. B. Andersson, “Precise 3-D localization of fluorescent probes without numerical fitting,” in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181–4184.
[PubMed]

Thompson, R. E.

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]

Walker, W. F.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[Crossref] [PubMed]

Ward, E. S.

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2007).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multdim. Syst. Sign. P. 17, 27–57 (2006).
[Crossref]

Webb, W.W.

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]

Zhuang, X.

M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
[Crossref]

Annu. Rev. Biochem. (1)

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. 77, 51–76 (2008).
[Crossref] [PubMed]

Appl. Phys. B (1)

S. B. Andersson, “Tracking a single fluorescent molecule with a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
[Crossref]

Biophys. J. (4)

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[Crossref] [PubMed]

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]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2007).
[Crossref]

IEEE T. Aero. Elec. Sys. (1)

S. Bancroft, “An algebraic solution of the GPS pseudorange equations,” IEEE T. Aero. Elec. Sys. AES-21, 56–59 (1985).
[Crossref]

in Handbook of Biological Confocal Microscopy (1)

R. Juškaitis, “Measuring the real point spread function of high numerical aperture microscope objective lenses,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 239–250.
[Crossref]

in Proceedings of IEEE Conference on Decision and Control (1)

S. B. Andersson, “Position estimation of fluorescent probes in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control (IEEE, 2007), pp. 4950–4955.

in Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI) (1)

S. B. Andersson, “Precise localization of fluorescent probes without numerical fitting,” in Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI) (IEEE, 2007), pp. 252–255.

in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (1)

T. Sun and S. B. Andersson, “Precise 3-D localization of fluorescent probes without numerical fitting,” in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181–4184.
[PubMed]

J. Phys. Condens. Matt. (1)

C. Kural, H. Balci, and P. R. Selvin, “Molecular motors one at a time: FIONA to the rescue,” J. Phys. Condens. Matt. 27, S3979–S3995 (2005).
[Crossref]

Multdim. Syst. Sign. P. (1)

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multdim. Syst. Sign. P. 17, 27–57 (2006).
[Crossref]

Nanomedicine (1)

C. K. Payne, “Imaging gene delivery with fluorescence microscopy,” Nanomedicine 2, 847–860 (2007).
[Crossref] [PubMed]

Opt. Express (2)

A. J. Berglund, M. D. McMahon, J. J. McClelland, and J. A. Liddle, “Fast, bias-free algorithm for tracking single particles with variable size and shape,” Opt. Express 16, 14,064–14,075 (2008).
[Crossref]

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
[Crossref] [PubMed]

P. Natl. Acad. Sci USA (2)

W. E. Moerner, “New directions in single-molecule imaging and analysis,” P. Natl. Acad. Sci USA 104, 12,596–12,602 (2007).
[Crossref]

M. Lakadamyali, M. J. Rust, H. P. Babcock, and X. Zhuang, “Visualizing infection of individual influenza viruses,” P. Natl. Acad. Sci USA 100, 9280–9285 (2003).
[Crossref]

Other (2)

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Pure and Applied Mathematics (John Wiley and Sons, 1974).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

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Figures (7)

Fig. 1.
Fig. 1. Range-based estimation in the plane. Each measurement yields an estimate of the range to the source (yellow star) and thus a circle of possible locations for that source. With two measurements from two different locations, the source must lie at the intersection of the two circles, leading to two possible solutions. In the absence of noise a third measurement produces a unique solution given by the single common point of intersection of the circles.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Fluorescence image of a 20-nm diameter microsphere embedded with “Nile Red” fluorophores. Axis labels are pixel indices and grayscale represents fluorescence intensity.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Results for images acquired with an SNR of 8.49. The mean of the error is indicated with a black dotted line and the standard deviation with a red dashed line. (a) Error in displacement when the fluoroBancroft algorithm was used for position estimation. The mean was 44.5 nm and the standard deviation was 24.5 nm. (b) Error in displacement when a Gaussian fit was used for position estimation. The mean of 37.0 nm was better than that obtained by fluoroBancroft while the standard deviation of 25.5 nm was slightly worse.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. Biasedness of the fluoroBancroft estimator. If the estimator were unbiased, then the mean and variance of the displacement estimates would be related by (21). That the difference, shown here, is not zero indicates the algorithm exhibits some bias in the position estimates.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. Comparison of the standard deviation of the position estimate using fluoroBancroft (blue solid line) and Gaussian fit (red dashed line). (a) Standard deviation of the x-position estimate. (b) Standard deviation of the y-position estimate. In the x-direction, the two estimators are very similar while in the y- direction the Gaussian fit exhibits consistently better performance at SNRs above 6.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6. Comparison of the standard deviation of the errors in the calculated displacement. (a) Standard deviation of the error for each estimator. (b) Difference between Gaussian fit and fluoroBancroft results.

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Fig. 7.
Fig. 7. Ratio of run times.

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Equations (24)

Equations on this page are rendered with MathJax. Learn more.

I ( x , y ) = m e − ( x − x o ) 2 2 σ x 2 − ( y − y o ) 2 2 σ y 2 + η B + η shot
I = m e − ( x − x o ) 2 2 σ x 2 − ( y − y o ) 2 2 σ y 2 + η shot
〈 I 〉 = m e − ( x − x o ) 2 2 σ x 2 − ( y − y o ) 2 2 σ y 2
σ x = σ y = 0.6 λ 2 N . A .
I ( x , y ) = m e − ( x − x o ) 2 2 σ x 2 − ( y ˜ − y ˜ ′ o ) 2 2 σ x 2 + η B + η shot
= m e − r 2 2 σ x 2 + η B + η shot
r = ( x − x o ) 2 + ( y ˜ − y ˜ ′ o ) 2 .
r 2 = 2 σ x 2 ln ( m ) − 2 σ x 2 ln ( 〈 I 〉 − N B ) .
b = 2 σ x 2 ln ( m ) , P i 2 = 2 σ x 2 ln ( I i − N B ) ,
α i = 1 2 ( x i 2 + y ˜ i 2 + P i 2 ) , Λ = 1 2 ( x o 2 + y ˜ o 2 ) .
0 = α i + Λ − ( x i         y ˜ i         1 )   ( x o y ˜ o b ) .
B ( x 0 y ˜ 0 b ) = α + Λ e
α = ( α 1 ⋮ α n ) , e = ( 1 ⋮ 1 ) , B = ( x 1 y ˜ 1 1 ⋮ x N y ˜ n 1 )
δ = B ( x 0 y ˜ 0 b ) − ( α + Λ e ) ,
( x o y ˜ o b ) = B † ( α + Λ e )
B † e = ( 0 ⋮ 0 1 ) .
𝕀 = ( B T B ) − 1 ( B T B ) = ( B T B ) − 1 B T ( A           e ) = ( ( B T B ) − 1 B T A         ( B T B ) − 1 B T e ) = ( B † A           B † e ) .
Q = ( 1 0 0 0 σ x σ y 0 ) .
( x o y o ) = Q B † α .
SNR = I 0 σ BG 2 + σ I 0 2
I ( x , y ) = m e − ( x − x o ) 2 2 σ x 2 − ( y − y o ) 2 2 σ y 2 .
f d ( d ) = d σ e 2 e − d 2 2 σ e 2 U ( d ) .
〈 d 〉 = σ e π 2 , Var [ d ] = ( 2 − π 2 ) σ e 2 .
Var [ d ] = ( 4 π − 1 ) 〈 d 〉 2 .

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