Abstract

The computation of point spread functions, which are typically used to model the image profile of a single molecule, represents a central task in the analysis of single molecule microscopy data. To determine how the accuracy of the computation affects how well a single molecule can be localized, we investigate how the fineness with which the point spread function is integrated over an image pixel impacts the performance of the maximum likelihood location estimator. We consider both the Airy and the two-dimensional Gaussian point spread functions. Our results show that the point spread function needs to be adequately integrated over a pixel to ensure that the estimator closely recovers the true location of the single molecule with an accuracy that is comparable to the best possible accuracy as determined using the Fisher information formalism. Importantly, if integration with an insufficiently fine step size is carried out, the resulting estimates can be significantly different from the true location, particularly when the image data is acquired at relatively low magnifications. We also present a methodology for determining an adequate step size for integrating the point spread function.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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  1. R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
    [Crossref] [PubMed]
  2. W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104(31), 12596–12602 (2007).
    [Crossref] [PubMed]
  3. 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(5793), 1642–1645 (2006).
    [Crossref] [PubMed]
  4. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
    [Crossref] [PubMed]
  5. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
    [Crossref] [PubMed]
  6. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
    [Crossref]
  7. U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
    [Crossref] [PubMed]
  8. M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81(4), 2378–2388 (2001).
    [Crossref] [PubMed]
  9. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002).
    [Crossref] [PubMed]
  10. I. J. Cooper and C. J. R. Sheppard, “A matrix method for calculating the three-dimensional irradiance distribution in the focal region of a convergent beam,” Optik 114(7), 298–304 (2003).
    [Crossref]
  11. J. Markham and J.-A. Conchello, “Numerical evaluation of Hankel transforms for oscillating functions,” J. Opt. Soc. Am. A 20, (4)621–630 (2003).
    [Crossref]
  12. C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
    [Crossref] [PubMed]
  13. A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
    [Crossref] [PubMed]
  14. A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
    [Crossref]
  15. S. Ram, E. S. Ward, and R. J. Ober, “A novel resolution measure for optical microscopes: stochastic analysis of the performance limits,” in Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE, 2006), pp. 770–773.
  16. S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
    [Crossref]
  17. P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (Wiley, 2009).
    [Crossref]
  18. C. R. Rao, Linear Statistical Inference and its Applications (Wiley, 1965).
  19. J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Syst. Sign. P. 23, (3), 349–379 (2012).
    [Crossref]
  20. T. Sauer, Numerical Analysis (Pearson, 2005).

2012 (1)

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Syst. Sign. P. 23, (3), 349–379 (2012).
[Crossref]

2010 (1)

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
[Crossref] [PubMed]

2007 (1)

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104(31), 12596–12602 (2007).
[Crossref] [PubMed]

2006 (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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

2004 (1)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

2003 (2)

I. J. Cooper and C. J. R. Sheppard, “A matrix method for calculating the three-dimensional irradiance distribution in the focal region of a convergent beam,” Optik 114(7), 298–304 (2003).
[Crossref]

J. Markham and J.-A. Conchello, “Numerical evaluation of Hankel transforms for oscillating functions,” J. Opt. Soc. Am. A 20, (4)621–630 (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(5), 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(4), 2378–2388 (2001).
[Crossref] [PubMed]

2000 (1)

U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
[Crossref] [PubMed]

Abraham, A. V.

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

Bertaux, N.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
[Crossref] [PubMed]

Betzig, E.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Bonifacino, J. S.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

Chao, J.

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Syst. Sign. P. 23, (3), 349–379 (2012).
[Crossref]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

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(4), 2378–2388 (2001).
[Crossref] [PubMed]

Conchello, J.-A.

Cooper, I. J.

I. J. Cooper and C. J. R. Sheppard, “A matrix method for calculating the three-dimensional irradiance distribution in the focal region of a convergent beam,” Optik 114(7), 298–304 (2003).
[Crossref]

Davidson, M. W.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Girirajan, T. P. K.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

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(4), 2378–2388 (2001).
[Crossref] [PubMed]

Hess, H. F.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Hess, S. T.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Huber, P. J.

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (Wiley, 2009).
[Crossref]

Joseph, N.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref] [PubMed]

Kubitscheck, U.

U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
[Crossref] [PubMed]

Kückmann, O.

U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
[Crossref] [PubMed]

Kues, T.

U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
[Crossref] [PubMed]

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002).
[Crossref] [PubMed]

Lidke, K. A.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref] [PubMed]

Lindwasser, O. W.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Lippincott-Schwartz, J.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Marguet, D.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
[Crossref] [PubMed]

Markham, J.

Mason, M. D.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Moerner, W. E.

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104(31), 12596–12602 (2007).
[Crossref] [PubMed]

Ober, R. J.

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Syst. Sign. P. 23, (3), 349–379 (2012).
[Crossref]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A novel resolution measure for optical microscopes: stochastic analysis of the performance limits,” in Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE, 2006), pp. 770–773.

Olenych, S.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Patterson, G. H.

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Peters, R.

U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
[Crossref] [PubMed]

Ram, S.

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A novel resolution measure for optical microscopes: stochastic analysis of the performance limits,” in Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE, 2006), pp. 770–773.

Rao, C. R.

C. R. Rao, Linear Statistical Inference and its Applications (Wiley, 1965).

Rieger, B.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref] [PubMed]

Rigneault, H.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
[Crossref] [PubMed]

Ronchetti, E. M.

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (Wiley, 2009).
[Crossref]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

Sauer, T.

T. Sauer, Numerical Analysis (Pearson, 2005).

Sergé, A.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
[Crossref] [PubMed]

Sheppard, C. J. R.

I. J. Cooper and C. J. R. Sheppard, “A matrix method for calculating the three-dimensional irradiance distribution in the focal region of a convergent beam,” Optik 114(7), 298–304 (2003).
[Crossref]

Smith, C. S.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref] [PubMed]

Sougrat, R.

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(5793), 1642–1645 (2006).
[Crossref] [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(5), 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(4), 2378–2388 (2001).
[Crossref] [PubMed]

Ward, E. S.

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Syst. Sign. P. 23, (3), 349–379 (2012).
[Crossref]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A novel resolution measure for optical microscopes: stochastic analysis of the performance limits,” in Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE, 2006), pp. 770–773.

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

Biophys. J. (5)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

U. Kubitscheck, O. Kückmann, T. Kues, and R. Peters, “Imaging and tracking of single GFP molecules in solution,” Biophys. J. 78(4), 2170–2179 (2000).
[Crossref] [PubMed]

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81(4), 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(5), 2775–2783 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Multidim. Syst. Sign. P. (2)

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Syst. Sign. P. 23, (3), 349–379 (2012).
[Crossref]

Nat. Methods (3)

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref] [PubMed]

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing to probe spatiotemporal cartography of cell membranes,” Nat. Methods 5(8), 687–694 (2008).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

Opt. Express (1)

Optik (1)

I. J. Cooper and C. J. R. Sheppard, “A matrix method for calculating the three-dimensional irradiance distribution in the focal region of a convergent beam,” Optik 114(7), 298–304 (2003).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104(31), 12596–12602 (2007).
[Crossref] [PubMed]

Science (1)

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(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Other (5)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A novel resolution measure for optical microscopes: stochastic analysis of the performance limits,” in Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE, 2006), pp. 770–773.

T. Sauer, Numerical Analysis (Pearson, 2005).

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (Wiley, 2009).
[Crossref]

C. R. Rao, Linear Statistical Inference and its Applications (Wiley, 1965).

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Figures (7)

Fig. 1
Fig. 1

Fineness of integration of the image profile. (a) Sampling scheme for the 2D trapezoidal method of integration over a square pixel with side length l. The integration step size ss is defined as l divided by the pixel integration factor (PIF). The notation xmin : ss : xmax denotes the positions {xmin, xmin +ss, xmin +2 · ss, …, xmax} along the x-axis at which the image profile is evaluated, and the notation ymin : ss : ymax denotes analogous positions along the y-axis. (b) Examples showing the location of the sampling points within a pixel for different values of the PIF. A PIF value of n, where n = 1,2,…, means that the image profile is evaluated at (n + 1)2 evenly spaced points covering the pixel area.

Fig. 2
Fig. 2

Airy profile results: difference d(x0) between the median of the x0 estimates and the true value x0 (left-hand side plots), and percentage difference δ (x0) between the standard deviation (with respect to the median) of the x0 estimates and the PLAM for x0 (right-hand side plots), as functions of the PIF used to integrate the image profile during estimation. For each scenario of Table 1, results are shown for a 63× (cyan⋄) and a 100× (red○) data set, simulated at PIF = 13 using the Airy profile with parameters given in Section 4. Each data set is fitted with an Airy profile by an ML estimator, with PIF values ranging from 0 to 13. In all plots the horizontal line denotes 0, and the inset shows the results for PIF 3.

Fig. 3
Fig. 3

2D Gaussian profile results: difference d(x0) between the median of the x0 estimates and the true value x0 (left-hand side plots), and percentage difference δ (x0) between the standard deviation (with respect to the median) of the x0 estimates and the PLAM for x0 (right-hand side plots), as functions of the PIF used to integrate the image profile during estimation. For each scenario of Table 1, results are shown for a 63× (cyan⋄) and a 100× (red○) data set, simulated at PIF = 13 using the 2D Gaussian profile with parameters given in Section 4. Each data set is fitted with a 2D Gaussian profile by an ML estimator, with PIF values ranging from 0 to 13. In all plots the horizontal line denotes 0, and the inset shows the results for PIF 3.

Fig. 4
Fig. 4

Obtaining a guideline for determining an appropriate fineness of integration. The maximum percentage difference, as defined in Section 6.4, between the image profile computed using a given PIF value and the reference image profile computed using PIF = 13, is shown as a function of the PIF value for the former for (a) the Airy image profile and (b) the 2D Gaussian image profile. In both (a) and (b), the profiles are computed using the relevant parameters given in Section 4, and the maximum percentage difference is shown for both the 63× (cyan ⋄) and the 100× (red ○) magnifications. The inset shows the maximum percentage difference for PIF values ranging from 4 to 13.

Fig. 5
Fig. 5

Airy profile results for a different single molecule location: difference d(x0) between the median of the x0 estimates and the true value x0 (left-hand side plots), and percentage difference δ(x0) between the standard deviation (with respect to the median) of the x0 estimates and the PLAM for x0 (right-hand side plots), as functions of the PIF used to integrate the image profile during estimation. For each scenario of Table 1, results are shown for a 63× (cyan ⋄) and a 100× (red ○) data set, simulated at profile PIF = 13 using the Airy with parameters given in Section 4, except the molecule is placed at 5.5 pixels in both the x and y directions within the 11×11 image array. Each data set is fitted with an Airy profile by an ML estimator, with PIF values ranging from 0 to 13. In all plots the horizontal line denotes 0, and the inset shows the results for PIF 3.

Fig. 6
Fig. 6

2D Gaussian profile results for a different single molecule location: difference d(x0) between the median of the x0 estimates and the true value x0 (left-hand side plots), and percentage difference δ(x0) between the standard deviation (with respect to the median) of the x0 estimates and the PLAM for x0 (right-hand side plots), as functions of the PIF used to integrate the image profile during estimation. For each scenario of Table 1, results are shown for a 63× (cyan ⋄) and a 100× (red ○) data set, simulated at Gaussian PIF = 13 using the 2D profile with parameters given in Section 4, except the molecule is placed at 5.5 pixels in both the x and y directions within the 11×11 image array. Each data set is fitted with a 2D Gaussian profile by an ML estimator, with PIF values ranging from 0 to 13. In all plots the horizontal line denotes 0, and the inset shows the results for PIF 3.

Fig. 7
Fig. 7

Mean percentage difference between the image profile computed using a given PIF value and the reference image profile computed using PIF = 13, shown as a function of the PIF value for the former for (a) the Airy image profile and (b) the 2D Gaussian image profile. For each PIF value, the value shown is the mean of the same 121 percentage differences for which the maximum is shown in Fig. 4, and the error bar represents the standard deviation (with respect to the mean) of the 121 percentage differences. In both (a) and (b), the mean percentage difference and the corresponding standard deviation are shown for both the 63× (cyan ⋄) and the 100× (red ○) magnifications. The inset shows the mean percentage difference and the corresponding standard deviation for PIF values ranging from 4 to 13. In (a), the standard deviation as a percentage of the mean, for the 63× magnification, is 28% for PIF = 0 and 1, and 32% for PIF 2. For the 100× magnification, it is ∼5% and 6% for PIF = 0 and 1, and ∼7% for PIF 2. In (b), the standard deviation as a percentage of the mean, for the 63× magnification, is 38% and 24% for PIF = 0 and 1, and 28% for PIF 2. For the 100× magnification, it is 5% and 2% for PIF = 0 and 1, and 3% for PIF 2.

Tables (1)

Tables Icon

Table 1 Signal and additive noise levels for the four data scenarios considered

Equations (13)

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I θ , k = S θ , k + B k + W k , k = 1 , , K p i x , θ Θ ,
μ θ , k = N M 2 C k q z 0 ( x M x 0 , y M y 0 ) d x d y , θ Θ ,
q z 0 ( x , y ) = q ( x , y ) = J 1 2 ( α x 2 + y 2 ) π ( x 2 + y 2 ) , ( x , y ) 2 , ( Airy profile )
q z 0 ( x , y ) = q ( x , y ) = 1 2 π σ 2 e x 2 + y 2 2 σ 2 , ( x , y ) 2 . ( 2 D Gaussian profile )
L ( θ | z 1 , , z K p i x ) = k = 1 K p i x p θ , k ( z k ) , θ Θ ,
θ ^ M L = arg max θ ( k = 1 K p i x log ( p θ , k ( z k ) ) ) .
d ( x 0 ) = m e d i a n ( x ^ 0 ) x 0 ,
I ( θ ) = k = 1 K p i x κ θ , k v θ , k ( μ θ , k θ ) T μ θ , k θ , θ Θ ,
δ ( x 0 ) = sd ( x ^ 0 ) PLAM ( x 0 ) PLAM ( x 0 ) × 100 ,
p θ , k ( z k ) = 1 2 π σ w , k l = 0 [ v θ , k ] l e v θ , k l ! . e ( z k l η k ) 2 2 σ w , k 2 , z k , k = 1 , , K p i x ,
p θ , k ( z k ) = [ v θ , k ] z k e v θ , k z k ! , z k = 0 , 1 , , k = 1 , , K p i x .
κ θ , k = v θ , k . [ ( 1 2 π σ w , k l = 1 [ v θ , k ] l 1 e v θ , k ( l 1 ) ! . e ( z l η k ) 2 2 σ w , k 2 ) 2 p θ , k ( z ) d z 1 ] , θ Θ ,
κ θ , k = 1 , θ Θ .

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