Abstract

A common task in microscopy is to fit an image of a fluorescent probe to a point spread function (PSF) in order to estimate the position of the probe. The PSF is often approximated as a Gaussian for mathematical simplicity. We show that the separable property of the Gaussian PSF enables a reduction of computational time from O(L2) to O(L), where L is the width (in pixels) of the image. When tested on realistic simulated data, our algorithm is able to localize the probes with precision close to the Cramér–Rao lower bound.

© 2012 Optical Society of America

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Corrections

Rebecca Starr, Shane Stahlheber, and Alex Small, "Fast maximum likelihood algorithm for localization of fluorescent molecules: erratum," Opt. Lett. 37, 1967-1967 (2012)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-37-11-1967

References

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2010 (1)

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, Nature Methods 7, 373 (2010).
[CrossRef]

2009 (2)

2008 (2)

2007 (2)

2004 (1)

R. J. Ober, S. Ram, and E. S. Ward, Biophys. J. 86, 1185 (2004).
[CrossRef]

2002 (1)

R. E. Thompson, D. R. Larson, and W. W. Webb, Biophys. J. 82, 2775 (2002).
[CrossRef]

1992 (1)

L. Lucy, Astron. J. 104, 1260 (1992).
[CrossRef]

Abraham, A. V.

Andersson, S.

Berglund, A. J.

Chao, J.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Joseph, N.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, Nature Methods 7, 373 (2010).
[CrossRef]

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, Biophys. J. 82, 2775 (2002).
[CrossRef]

Liddle, J. A.

Lidke, K. A.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, Nature Methods 7, 373 (2010).
[CrossRef]

Lucy, L.

L. Lucy, Astron. J. 104, 1260 (1992).
[CrossRef]

McClelland, J. J.

McMahon, M. D.

Moerner, W.

W. Moerner, Proc. Natl. Acad. Sci. USA 104, 12596 (2007).
[CrossRef]

Ober, R.

S. Ram, E. Ward, and R. Ober, in Biomedical Engineering: Nano to Macro (IEEE, 2006), pp. 770.

Ober, R. J.

Olivo-Marin, J. C.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Ram, S.

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, Opt. Express 17, 23352 (2009).
[CrossRef]

R. J. Ober, S. Ram, and E. S. Ward, Biophys. J. 86, 1185 (2004).
[CrossRef]

S. Ram, E. Ward, and R. Ober, in Biomedical Engineering: Nano to Macro (IEEE, 2006), pp. 770.

Rieger, B.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, Nature Methods 7, 373 (2010).
[CrossRef]

Smith, C. S.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, Nature Methods 7, 373 (2010).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Thompson, R. E.

R. E. Thompson, D. R. Larson, and W. W. Webb, Biophys. J. 82, 2775 (2002).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Ward, E.

S. Ram, E. Ward, and R. Ober, in Biomedical Engineering: Nano to Macro (IEEE, 2006), pp. 770.

Ward, E. S.

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, Biophys. J. 82, 2775 (2002).
[CrossRef]

Zerubia, J.

Zhang, B.

Zhuang, X.

X. Zhuang, Nat. Photon. 3, 365 (2009).
[CrossRef]

Appl. Opt. (1)

Astron. J. (1)

L. Lucy, Astron. J. 104, 1260 (1992).
[CrossRef]

Biophys. J. (2)

R. E. Thompson, D. R. Larson, and W. W. Webb, Biophys. J. 82, 2775 (2002).
[CrossRef]

R. J. Ober, S. Ram, and E. S. Ward, Biophys. J. 86, 1185 (2004).
[CrossRef]

Nat. Photon. (1)

X. Zhuang, Nat. Photon. 3, 365 (2009).
[CrossRef]

Nature Methods (1)

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, Nature Methods 7, 373 (2010).
[CrossRef]

Opt. Express (3)

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

W. Moerner, Proc. Natl. Acad. Sci. USA 104, 12596 (2007).
[CrossRef]

Other (2)

S. Ram, E. Ward, and R. Ober, in Biomedical Engineering: Nano to Macro (IEEE, 2006), pp. 770.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

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Figures (2)

Fig. 1.
Fig. 1.

Localization precision (standard deviation of position estimate) vsersus fluorophore coordinates (x0,y0) (relative to ROI center) for ideal and nonideal cases. a=pixel edges width, d=center-to-center pixel spacing.

Fig. 2.
Fig. 2.

Positions of molecules in simulated images (red dots), mean estimated positions (centers of circles), and standard deviations of estimates (radius=2×standard deviation). 104 images were used for each circle.

Equations (4)

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

μ(x,y)=b+I0pixel areaeαk2((xx0)2+(yy0)2)dxdy,
μs(x)=bL+πI0L2αk2(Erf(αk(x+a/2x0))Erf(αk(xa/2x0))),
logL(Ss|θ)=xSs(x)logμs(x;θ)μs(x;θ)logSs(x)!,
xcm=xx(Ss(x)bL)/x(Ss(x)bL).

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