Abstract

In fluorescence microscopy, high-speed imaging is often necessary for the proper visualization and analysis of fast subcellular dynamics. Here, we examine how the speed of image acquisition affects the accuracy with which parameters such as the starting position and speed of a microscopic non-stationary fluorescent object can be estimated from the resulting image sequence. Specifically, we use a Fisher information-based performance bound to investigate the detector-dependent effect of frame rate on the accuracy of parameter estimation. We demonstrate that when a charge-coupled device detector is used, the estimation accuracy deteriorates as the frame rate increases beyond a point where the detector’s readout noise begins to overwhelm the low number of photons detected in each frame. In contrast, we show that when an electron-multiplying charge-coupled device (EMCCD) detector is used, the estimation accuracy improves with increasing frame rate. In fact, at high frame rates where the low number of photons detected in each frame renders the fluorescent object difficult to detect visually, imaging with an EMCCD detector represents a natural implementation of the Ultrahigh Accuracy Imaging Modality, and enables estimation with an accuracy approaching that which is attainable only when a hypothetical noiseless detector is used.

© 2014 Optical Society of America

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References

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  1. W. Yang and S. M. Musser, “Visualizing single molecules interacting with nuclear pore complexes by narrow-field epifluorescence microscopy,” Methods 39(4), 316–328 (2006).
    [CrossRef] [PubMed]
  2. T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
    [CrossRef] [PubMed]
  3. X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
    [PubMed]
  4. V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
    [CrossRef] [PubMed]
  5. J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
    [CrossRef] [PubMed]
  6. M. P. Taylor, R. Kratchmarov, and L. W. Enquist, “Live cell imaging of alphaherpes virus anterograde transport and spread,” J. Vis. Exp. ( 78), e50723 (2013).
  7. J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
    [CrossRef] [PubMed]
  8. C. R. Rao, Linear Statistical Inference and its Applications (Wiley, 1965).
  9. 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]
  10. Y. Wong, Z. Lin, and R. J. Ober, “Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy,” IEEE T. Signal Proces. 59(3), 895–911 (2011).
    [CrossRef]
  11. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall PTR, 1993), Vol. I.
  12. 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]
  13. B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Optics 46(10), 1819–1829 (2007).
    [CrossRef]
  14. A. Santos and I. T. Young, “Model-based resolution: applying the theory in quantitative microscopy,” Appl. Optics 39(17), 2948–2958 (2000).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
    [CrossRef]
  16. A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345(3), 985–991 (2003).
    [CrossRef]

2013 (2)

M. P. Taylor, R. Kratchmarov, and L. W. Enquist, “Live cell imaging of alphaherpes virus anterograde transport and spread,” J. Vis. Exp. ( 78), e50723 (2013).

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
[CrossRef] [PubMed]

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]

2011 (1)

Y. Wong, Z. Lin, and R. J. Ober, “Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy,” IEEE T. Signal Proces. 59(3), 895–911 (2011).
[CrossRef]

2008 (1)

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

2007 (1)

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Optics 46(10), 1819–1829 (2007).
[CrossRef]

2006 (2)

W. Yang and S. M. Musser, “Visualizing single molecules interacting with nuclear pore complexes by narrow-field epifluorescence microscopy,” Methods 39(4), 316–328 (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]

2005 (3)

X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
[PubMed]

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
[CrossRef] [PubMed]

2003 (1)

A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345(3), 985–991 (2003).
[CrossRef]

2000 (1)

A. Santos and I. T. Young, “Model-based resolution: applying the theory in quantitative microscopy,” Appl. Optics 39(17), 2948–2958 (2000).
[CrossRef]

Basden, A. G.

A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345(3), 985–991 (2003).
[CrossRef]

Belmont, A. S.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

Born, M.

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

Chao, J.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
[CrossRef] [PubMed]

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]

Chen, P.

X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
[PubMed]

Dange, T.

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

Enquist, L. W.

M. P. Taylor, R. Kratchmarov, and L. W. Enquist, “Live cell imaging of alphaherpes virus anterograde transport and spread,” J. Vis. Exp. ( 78), e50723 (2013).

Ghigo, E.

J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
[CrossRef] [PubMed]

Gratton, E.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

Grünwald, A.

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

Grünwald, D.

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

Haniff, C. A.

A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345(3), 985–991 (2003).
[CrossRef]

Kalaidzidis, Y.

J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
[CrossRef] [PubMed]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall PTR, 1993), Vol. I.

Kratchmarov, R.

M. P. Taylor, R. Kratchmarov, and L. W. Enquist, “Live cell imaging of alphaherpes virus anterograde transport and spread,” J. Vis. Exp. ( 78), e50723 (2013).

Kubitscheck, U.

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

Levi, V.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

Lin, Z.

Y. Wong, Z. Lin, and R. J. Ober, “Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy,” IEEE T. Signal Proces. 59(3), 895–911 (2011).
[CrossRef]

Mackay, C. D.

A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345(3), 985–991 (2003).
[CrossRef]

Musser, S. M.

W. Yang and S. M. Musser, “Visualizing single molecules interacting with nuclear pore complexes by narrow-field epifluorescence microscopy,” Methods 39(4), 316–328 (2006).
[CrossRef] [PubMed]

Nan, X.

X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
[PubMed]

Ober, R. J.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
[CrossRef] [PubMed]

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]

Y. Wong, Z. Lin, and R. J. Ober, “Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy,” IEEE T. Signal Proces. 59(3), 895–911 (2011).
[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]

Olivo-Marin, J.-C.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Optics 46(10), 1819–1829 (2007).
[CrossRef]

Peters, R.

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

Plutz, M.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

Ram, S.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
[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]

Rao, C. R.

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

Rink, J.

J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
[CrossRef] [PubMed]

Ruan, Q.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

Santos, A.

A. Santos and I. T. Young, “Model-based resolution: applying the theory in quantitative microscopy,” Appl. Optics 39(17), 2948–2958 (2000).
[CrossRef]

Sims, P. A.

X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
[PubMed]

Taylor, M. P.

M. P. Taylor, R. Kratchmarov, and L. W. Enquist, “Live cell imaging of alphaherpes virus anterograde transport and spread,” J. Vis. Exp. ( 78), e50723 (2013).

Ward, E. S.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
[CrossRef] [PubMed]

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]

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]

Wolf, E.

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

Wong, Y.

Y. Wong, Z. Lin, and R. J. Ober, “Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy,” IEEE T. Signal Proces. 59(3), 895–911 (2011).
[CrossRef]

Xie, X. S.

X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
[PubMed]

Yang, W.

W. Yang and S. M. Musser, “Visualizing single molecules interacting with nuclear pore complexes by narrow-field epifluorescence microscopy,” Methods 39(4), 316–328 (2006).
[CrossRef] [PubMed]

Young, I. T.

A. Santos and I. T. Young, “Model-based resolution: applying the theory in quantitative microscopy,” Appl. Optics 39(17), 2948–2958 (2000).
[CrossRef]

Zerial, M.

J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
[CrossRef] [PubMed]

Zerubia, J.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Optics 46(10), 1819–1829 (2007).
[CrossRef]

Zhang, B.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Optics 46(10), 1819–1829 (2007).
[CrossRef]

Appl. Optics (2)

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Optics 46(10), 1819–1829 (2007).
[CrossRef]

A. Santos and I. T. Young, “Model-based resolution: applying the theory in quantitative microscopy,” Appl. Optics 39(17), 2948–2958 (2000).
[CrossRef]

Biophys. J. (1)

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89(6), 4275–4285 (2005).
[CrossRef] [PubMed]

Cell (1)

J. Rink, E. Ghigo, Y. Kalaidzidis, and M. Zerial, “Rab conversion as a mechanism of progression from early to late endosomes,” Cell 122(5), 735–749 (2005).
[CrossRef] [PubMed]

IEEE T. Signal Proces. (1)

Y. Wong, Z. Lin, and R. J. Ober, “Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy,” IEEE T. Signal Proces. 59(3), 895–911 (2011).
[CrossRef]

J. Cell Biol. (1)

T. Dange, D. Grünwald, A. Grünwald, R. Peters, and U. Kubitscheck, “Autonomy and robustness of translocation through the nuclear pore complex: a single-molecule study,” J. Cell Biol. 183(1), 77–86 (2008).
[CrossRef] [PubMed]

J. Phys. Chem. B (1)

X. Nan, P. A. Sims, P. Chen, and X. S. Xie, “Observation of individual microtubule motor steps in living cells with endocytosed quantum dots,” J. Phys. Chem. B 109(51), 24220–24224 (2005).
[PubMed]

J. Vis. Exp. (1)

M. P. Taylor, R. Kratchmarov, and L. W. Enquist, “Live cell imaging of alphaherpes virus anterograde transport and spread,” J. Vis. Exp. ( 78), e50723 (2013).

Methods (1)

W. Yang and S. M. Musser, “Visualizing single molecules interacting with nuclear pore complexes by narrow-field epifluorescence microscopy,” Methods 39(4), 316–328 (2006).
[CrossRef] [PubMed]

Mon. Not. R. Astron. Soc. (1)

A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345(3), 985–991 (2003).
[CrossRef]

Multidim. Syst. Sign. P. (2)

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]

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]

Nat. Methods (1)

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10(4), 335–338 (2013).
[CrossRef] [PubMed]

Other (3)

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

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall PTR, 1993), Vol. I.

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

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Figures (6)

Fig. 1
Fig. 1

(a) Time discretization notation and terminology. An acquisition of Nf frames, spanning a total acquisition time Ttat over the time interval [t0, tNf], consists of the frame intervals [ti−1, ti], i = 1, 2,..., Nf. During each frame interval, the camera exposure begins at the start of the frame interval and stops at or before the end of the frame interval. More precisely, the exposure intervals are given by [ti−1, ei], eiti, i = 1, 2,..., Nf. (b) Schematic sketch of a linear trajectory. The trajectory is depicted as a line segment, with an arrowhead indicating the direction of movement. It is described by four parameters: the coordinates (x0, y0) of the starting position, the angle ϕ specifying the direction of movement with respect to the x-axis, and the speed v at which the object travels.

Fig. 2
Fig. 2

Limits of accuracy, shown as functions of the acquisition frame rate, for the estimation of (a) the coordinate x0 and (b) the coordinate y0 of the starting position, (c) the angle ϕ specifying the direction of movement with respect to the x-axis, and (d) the speed v, of a point source moving in a linear trajectory (see Fig. 1(b)). In each plot, the limits of accuracy correspond to imaging with an ideal detector (*), a hypothetical noiseless detector (□), a CCD detector (⋄), and an EMCCD detector (○). For each pixelated detector type, the pixel size is 16μm × 16μm, and an image consists of an 8×8 pixel array. The CCD detector adds readout noise with mean ηk = 0 e and standard deviation σk = 2 e to each pixel k. The EMCCD detector amplifies photon signals at an electron multiplication gain of g = 950, and adds readout noise with mean ηk = 0 e and standard deviation σk = 24 e to each pixel k. The absence of a background component is assumed. The 2D Gaussian profile that models the image of the point source has a standard deviation of σgauss = 84 nm, and the rate at which photons are detected from the point source is Λ0 = 2000 photons/s. The magnification of the microscope is M = 100. The values of the estimated parameters are x0 = y0 = −250 nm with respect to the optical (z-)axis which passes through the center of an image, ϕ = 30°, and v = 1500 nm/s. At any given frame rate, the total acquisition time is Ttat = 0.4 s, and is divided equally among all frames. The acquisition has no time gaps between successive exposures. The CCD limit of accuracy attains its best (i.e., lowest) value, in (a) and (d), at 15 fps, where the average photon signal level per frame and per pixel are 133 and 2.08 photons, and, in (b) and (c), at 10 fps, where the average photon signal level per frame and per pixel are 200 and 3.125 photons. In (a), (b), (c), and (d), the EMCCD limit of accuracy first attains a lower value than the CCD limit of accuracy at around 25 fps, where the average photon signal level per frame and per pixel are 80 and 1.25 photons.

Fig. 3
Fig. 3

Comparing the limits of accuracy, corresponding to imaging with CCD and EMCCD detectors at different levels of various noise sources and shown as functions of the acquisition frame rate, for the estimation of the coordinate x0 of the starting position of a point source moving in a linear trajectory. In (a), the limits of accuracy correspond to CCD imaging with a readout noise standard deviation (SD) of σk = 1 e (red ⋄), 2 e (black ⋄), and 6 e (blue ⋄) in each pixel k, and to EMCCD imaging (○) with an electron multiplication (EM) gain of g = 950 and a readout noise SD of σk = 24 e in each pixel k. In (b), the limits of accuracy correspond to EMCCD imaging with an EM gain of g = 950 and a readout noise SD of σk = 12 e (green ○), 24 e (black ○), 36 e (red ○), and 64 e (blue ○) in each pixel k, and to CCD imaging (⋄) with a readout noise SD of σk = 2 e in each pixel k. In (c), the limits of accuracy correspond to EMCCD imaging with an EM gain of g = 2000 (green ○), 950 (black ○), 300 (red ○), and 50 (blue ○), and a readout noise SD of σk = 24 e in each pixel k, and to CCD imaging (⋄) with a readout noise SD of σk = 2 e in each pixel k. In (a), (b), and (c), the absence of a background component is assumed. In (d), the limits of accuracy correspond to CCD imaging (⋄) and EMCCD imaging (○) with background noise levels of βk,i = 0 (black), 5 (red), and 10 (blue) photons in each pixel k of each frame i at 5 fps. (At each noise level, the background photons are assumed to be detected at a constant rate, and to be distributed uniformly over the detector.) For CCD imaging, readout noise with an SD of σk = 2 e in each pixel k is assumed. For EMCCD imaging, an EM gain of g = 950 and readout noise with an SD of σk = 24 e in each pixel k are assumed. In (a), (b), (c), and (d), the readout noise in all cases has a mean of ηk = 0 e in each pixel k. Other details of the acquisition setting and problem description, including the values of parameters not mentioned here, are as specified in Fig. 2.

Fig. 4
Fig. 4

Limits of accuracy, shown as functions of the acquisition frame rate, for the estimation of (a) the coordinate xc and (b) the coordinate yc of the center of the circular arc traversed by a point source, (c) the radius R of the circular arc, (d) the angular speed ω at which the point source travels, and (e) the angular offset ψ0 specifying the starting position of the point source with respect to the x-axis (see Fig. 5). In each plot, the limits of accuracy correspond to imaging with an ideal detector (*), a hypothetical noiseless detector (□), a CCD detector (⋄), and an EMCCD detector (○). For each pixelated detector type, the pixel size is 16μm × 16μm, and an image consists of an 8×8 pixel array. The CCD detector adds readout noise with mean ηk = 0 e and standard deviation σk = 2 e to each pixel k. The EMCCD detector amplifies photon signals at an electron multiplication gain of g = 950, and adds readout noise with mean ηk = 0 e and standard deviation σk = 24 e to each pixel k. The absence of a background component is assumed. The 2D Gaussian profile that models the image of the point source has a standard deviation of σgauss = 84 nm, and the rate at which photons are detected from the point source is Λ0 = 2000 photons/s. The magnification of the microscope is M = 100. The values of the estimated parameters are xc = yc = 0 nm with respect to the optical (z-)axis which passes through the center of an image, R = 250 nm, ω = 6 rad/s, and ψ0 = 20°. At any given frame rate, the total acquisition time is Ttat = 0.4 s, and is divided equally among all frames. The acquisition has no time gaps between successive exposures. The CCD limit of accuracy attains its best (i.e., lowest) value, in (a), at 5 fps, where the average photon signal level per frame and per pixel are 400 and 6.25 photons, and, in (b), (c), (d), and (e), at 15 fps, where the average photon signal level per frame and per pixel are 133 and 2.08 photons. In (a), (b), (c), (d), and (e), the EMCCD limit of accuracy first attains a lower value than the CCD limit of accuracy at around 25 fps, where the average photon signal level per frame and per pixel are 80 and 1.25 photons.

Fig. 5
Fig. 5

Schematic sketch of a circular arc trajectory. The trajectory is depicted as a circular arc, with an arrowhead indicating the direction of movement. It is described by five parameters: the coordinates (xc, yc) of the center of the circular arc, the radius R of the circular arc, the angular speed ω at which the object travels along the arc, and the angular offset ψ0 specifying the object’s starting position with respect to the x-axis.

Fig. 6
Fig. 6

Comparing the limits of accuracy, corresponding to imaging with detectors of different spatial resolutions, for the estimation of (a) the coordinate x0 and (b) the coordinate y0 of the starting position, (c) the angle ϕ specifying the direction of movement with respect to the x-axis, and (d) the speed v, of a point source moving in a linear trajectory (see Fig. 1(b)). In each plot, limits of accuracy as functions of the acquisition frame rate are shown which correspond to imaging with an ideal detector (*), a hypothetical noiseless detector (□), a CCD detector (⋄), and an EMCCD detector (○). For each pixelated detector type, the limits of accuracy correspond to imaging with a low resolution detector (—) having a 16μm × 16μm pixel size, and imaging with a high resolution detector (–.–) having an 8μm × 8μm pixel size. In either case, the size of an image is 128μm × 128μm, such that an image for the low resolution detector consists of an 8×8 pixel array, and an image for the high resolution detector consists of a 16×16 pixel array. Other details of the acquisition setting and problem description, including the values of parameters not mentioned here, are as specified in Fig. 2. Note that due to identical assumptions, the curves corresponding to the low resolution detector are the same as those shown in Fig. 2. For the high resolution detector, the CCD limit of accuracy attains its best (i.e., lowest) value, in (a), (b), and (d), at 10 fps, where the average photon signal level per frame and per pixel are 200 and 3.125 photons, and, in (c), at 5 fps, where the average photon signal level per frame and per pixel are 400 and 6.25 photons. Also for the high resolution detector, and in (a), (b), (c), and (d), the EMCCD limit of accuracy has a lower value than the CCD limit of accuracy at each frame rate shown. For analogous information on the CCD and EMCCD limits of accuracy for the low resolution detector, see Fig. 2.

Equations (23)

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I ( θ ) = i = 1 N f I i ( θ ) = i = 1 N f k = 1 N p 1 υ θ , k , i ( μ θ , k , i θ ) T ( μ θ , k , i θ ) , θ Θ ,
μ θ , k , i = 1 M 2 t i 1 e i C k Λ ( τ ) q z θ ( τ ) , o θ ( τ ) ( x M x θ ( τ ) , y M y θ ( τ ) ) d x d y d τ ,
β k , i = 1 M 2 t i 1 e i C k Λ b ( τ ) b τ ( x M , y M ) d x d y d τ ,
I ( θ ) = i = 1 N f I i ( θ ) = i = 1 N f k = 1 N p ( μ θ , k , i θ ) T ( μ θ , k , i θ ) × ( e 2 υ θ , k , i 2 π σ k 2 1 p θ , k , i ( z ) ( l = 1 υ θ , k , i l 1 ( l 1 ) ! e 1 2 ( z l η k σ k ) 2 ) 2 d z 1 ) ,
p θ , k , i ( z ) = e υ θ , k , i 2 π σ k l = 0 [ υ θ , k , i ] l l ! e 1 2 ( z l η k σ k ) 2 , z ,
I ( θ ) = i = 1 N f I i ( θ ) = i = 1 N f k = 1 N p ( μ θ , k , i θ ) T ( μ θ , k , i θ ) × ( e 2 υ θ , k , i 2 π σ k 2 g 2 1 p θ , k , i ( z ) ( l = 1 e 1 2 ( z l η k σ k ) 2 j = 0 l 1 ( l 1 j ) ( 1 1 g ) l j 1 j ! ( g υ θ , k , i ) j ) 2 d z 1 ) ,
p θ , k , i ( z ) = e υ θ , k , i 2 π σ k [ e 1 2 ( z η k σ k ) 2 + l = 1 e 1 2 ( z l η k σ k ) 2 j = 0 l 1 ( l 1 j ) ( 1 1 g ) l j 1 ( j + 1 ) ! ( g υ θ , k , i ) j + 1 ] , z ,
I ( θ ) = i = 1 N f I i ( θ ) = i = 1 N f t i 1 e i Λ 2 ( τ ) V θ T ( τ ) × [ ( q z θ ( τ ) , o θ ( τ ) ( x , y ) p ( τ ) ) T ( q z θ ( τ ) , o θ ( τ ) ( x , y ) p ( τ ) ) Λ ( τ ) q z θ ( τ ) , o θ ( τ ) ( x , y ) + Λ b ( τ ) b τ ( x , y ) d x d y ] V θ ( τ ) d τ ,
q ( x , y ) = 1 2 π σ gauss 2 e x 2 + y 2 2 σ gauss 2 , ( x , y ) 2 ,
δ x 0 = δ y 0 = 2 σ gauss T e 2 + 3 2 T e ( T tat 1 F ) + ( T tat 2 3 T tat 2 F + 1 2 F 2 ) Λ 0 F T tat T e ( T e 2 + T tat 2 1 F 2 ) , δ ϕ = 2 σ gauss v 3 Λ 0 F T tat T e ( T e 2 + T tat 2 1 F 2 ) , δ v = 2 σ gauss 3 Λ 0 F T tat T e ( T e 2 + T tat 2 1 F 2 ) ,
δ x 0 = δ y 0 = 2 σ gauss Λ 0 T tat , δ ϕ = 2 3 σ gauss v T tat Λ 0 T tat , δ v = 2 3 σ gauss T tat Λ 0 T tat ,
δ x 0 = δ y 0 = 1 γ b 3 b 1 b 3 b 2 2 , δ ϕ = 1 γ v b 1 b 1 b 3 b 2 2 , δ v = 1 γ b 1 b 1 b 3 b 2 2 ,
b 1 = i = 1 N f t i 1 e i Λ ( τ ) d τ , b 2 = i = 1 N f t i 1 e i Λ ( τ ) ( τ t 0 ) d τ , b 3 = i = 1 N f t i 1 e i Λ ( τ ) ( τ t 0 ) 2 d τ .
δ x 0 = δ y 0 = 2 γ T e 2 + 3 2 T e ( T tat 1 F ) + ( T tat 2 3 T tat 2 F + 1 2 F 2 ) Λ 0 F T tat T e ( T e 2 + T tat 2 1 F 2 ) , δ ϕ = 2 γ v 3 Λ 0 F T tat T e ( T e 2 + T tat 2 1 F 2 ) , δ v = 2 γ 3 Λ 0 F T tat T e ( T e 2 + T tat 2 1 F 2 ) ,
I ( θ ) = 4 π 0 r 3 q ˜ ( r 2 ) ( q ˜ ( r 2 ) r 2 ) 2 d r i = 1 N f t i 1 e i Λ ( τ ) [ x θ ( τ ) θ y θ ( τ ) θ ] T [ x θ ( τ ) θ y θ ( τ ) θ ] d τ ,
I ( θ ) = γ 2 i = 1 N f t i 1 e i Λ ( τ ) [ x θ ( τ ) θ y θ ( τ ) θ ] T [ x θ ( τ ) θ y θ ( τ ) θ ] d τ ,
x θ ( τ ) θ = [ 1 0 v ( τ t 0 ) sin ϕ ( τ t 0 ) cos ϕ ] , y θ ( τ ) θ = [ 0 1 v ( τ t 0 ) cos ϕ ( τ t 0 ) sin ϕ ] ,
I ( θ ) = γ 2 [ b 1 0 b 2 v sin ϕ b 2 cos ϕ 0 b 1 b 2 v cos ϕ b 2 sin ϕ b 2 v sin ϕ b 2 v cos ϕ b 3 v 2 0 b 2 cos ϕ b 2 sin ϕ 0 b 3 ] ,
b 1 = i = 1 N f 0 e i t i 1 Λ 0 d τ = i = 1 N f 0 T e Λ 0 d τ = N f Λ 0 T e ,
b 2 = i = 1 N f 0 e i t i 1 Λ 0 ( τ + t i 1 t 0 ) d τ = Λ 0 N f T e 2 2 + Λ 0 T e i = 1 N f ( t i 1 t 0 ) ,
b 3 = i = 1 N f 0 e i t i 1 Λ 0 ( τ + t i 1 t 0 ) 2 d τ = Λ 0 N f T e 3 3 + Λ 0 T e 2 i = 1 N f ( t i 1 t 0 ) + Λ 0 T e i = 1 N f ( t i 1 t 0 ) 2 .
i = 1 N f ( t i 1 t 0 ) = T tat ( N f 1 ) 2 , i = 1 N f ( t i 1 t 0 ) 2 = T tat 2 ( N f 1 ) ( 2 N f 1 ) 6 N f .
b 2 = Λ 0 N f T e 2 2 + Λ 0 T e T tat ( N f 1 ) 2 , b 3 = Λ 0 N f T e 3 3 + Λ 0 T e 2 T tat ( N f 1 ) 2 + Λ 0 T e T tat 2 ( N f 1 ) ( 2 N f 1 ) 6 N f .

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