Abstract

We present an asymptotic analysis of the minimum probability of error (MPE) in inferring the correct hypothesis in a Bayesian multi-hypothesis testing (MHT) formalism using many pixels of data that are corrupted by signal dependent shot noise, sensor read noise, and background illumination. We perform our analysis for a variety of combined noise and background statistics, including a pseudo-Gaussian distribution that can be employed to treat approximately the photon-counting statistics of signal and background as well as purely Gaussian sensor read-out noise and more general, exponentially peaked distributions. We subsequently evaluate both the exact and asymptotic MPE expressions for the problem of three-dimensional (3D) point source localization. We focus specifically on a recently proposed rotating-PSF imager and compare, using the MPE metric, its 3D localization performance with that of conventional and astigmatic imagers in the presence of background and sensor-noise fluctuations.

© 2014 Optical Society of America

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2014

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods11, 267–279 (2014).
[CrossRef] [PubMed]

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nat. Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

S. Prasad, “Asymptotics of Bayesian error prability and 2D pair superresolution,” Opt. Express22, 16029–16047 (2014).

2013

2012

S. Prasad, “New error bounds for M-testing and estimation of source location with subdiffractive error,” J. Opt. Soc. Am. A29, 354–366 (2012).
[CrossRef]

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

J. Chao, E. Sally Ward, and R. Ober, “Fisher information matrix for branching processes with application to EMCCDs,” Multidimens. Syst. Signal Process.23, 349–379 (2012).
[CrossRef] [PubMed]

2011

2010

A. Pertsinidis, Y. Zhang, and S. Chu, “Subnanometre single-molecule localization, registration and distance measurements,” Nature466, 647–651 (2010).
[CrossRef] [PubMed]

2009

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

2008

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319, 810–813 (2008).
[CrossRef] [PubMed]

J. Yoon, A. Bruckbauer, W. Fitzgerald, and D. Klenerman, “Bayesian inference for improved single molecule fluorescence tracking,” Biophys. J.94, 4932–4947 (2008).
[CrossRef] [PubMed]

2006

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U. S. A.103, 4457–4462 (2006).
[CrossRef] [PubMed]

2005

S. Ram, E. Sally Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[CrossRef] [PubMed]

2004

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

2003

A. Yildiz, J. Forkey, A. McKinney, T. Ha, Y. Goldman, and P. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

2002

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

1997

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inform. Theory43, 280–282 (1997).
[CrossRef]

1995

1992

C. Anderson, G. Georgiou, I. Morrison, G. Stevenson, and R. Cherry, “Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera,” J Cell Sci.101, 415–425 (1992).

1986

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum.57, 1152–1157 (1986).
[CrossRef]

1969

J. Ziv and M. Zakai, “Some lower bounds on signal parameter estimation,” IEEE Trans. Inform. Theory15, 386–391 (1969).
[CrossRef]

Anderson, C.

C. Anderson, G. Georgiou, I. Morrison, G. Stevenson, and R. Cherry, “Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera,” J Cell Sci.101, 415–425 (1992).

Badieirostami, M.

Bates, M.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319, 810–813 (2008).
[CrossRef] [PubMed]

Bathe, M.

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

Bell, K.

H. van Trees and K. Bell, Detection, Estimation, and Modulation Theory (Wiley, 2013), Part I.

Bewersdorf, J.

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

Biteen, J.

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

Bobroff, N.

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum.57, 1152–1157 (1986).
[CrossRef]

Braeckmans, K.

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

Bruckbauer, A.

J. Yoon, A. Bruckbauer, W. Fitzgerald, and D. Klenerman, “Bayesian inference for improved single molecule fluorescence tracking,” Biophys. J.94, 4932–4947 (2008).
[CrossRef] [PubMed]

Burnette, D.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Chao, J.

J. Chao, E. Sally Ward, and R. Ober, “Fisher information matrix for branching processes with application to EMCCDs,” Multidimens. Syst. Signal Process.23, 349–379 (2012).
[CrossRef] [PubMed]

Cherry, R.

C. Anderson, G. Georgiou, I. Morrison, G. Stevenson, and R. Cherry, “Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera,” J Cell Sci.101, 415–425 (1992).

Chu, S.

A. Pertsinidis, Y. Zhang, and S. Chu, “Subnanometre single-molecule localization, registration and distance measurements,” Nature466, 647–651 (2010).
[CrossRef] [PubMed]

Cover, T.

T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Cox, S.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Deschout, H.

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

Diaspro, A.

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

Faisal, M.

Fitzgerald, W.

J. Yoon, A. Bruckbauer, W. Fitzgerald, and D. Klenerman, “Bayesian inference for improved single molecule fluorescence tracking,” Biophys. J.94, 4932–4947 (2008).
[CrossRef] [PubMed]

Forkey, J.

A. Yildiz, J. Forkey, A. McKinney, T. Ha, Y. Goldman, and P. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Gahlmann, A.

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nat. Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

Georgiou, G.

C. Anderson, G. Georgiou, I. Morrison, G. Stevenson, and R. Cherry, “Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera,” J Cell Sci.101, 415–425 (1992).

Goldman, Y.

A. Yildiz, J. Forkey, A. McKinney, T. Ha, Y. Goldman, and P. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Guo, S.-M.

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

Ha, T.

A. Yildiz, J. Forkey, A. McKinney, T. Ha, Y. Goldman, and P. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

He, J.

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

Heintzmann, R.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Helstrom, C.

Hess, S.

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

Huang, B.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319, 810–813 (2008).
[CrossRef] [PubMed]

Johnson, D.

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inform. Theory43, 280–282 (1997).
[CrossRef]

Jones, G.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Jovanovic-Talisman, T.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Kay, S.

S. Kay, Fundamentals of Statistical Signal Processing: I. Estimation Theory (Prentice Hall, 1993), Chap. 3.

Klenerman, D.

J. Yoon, A. Bruckbauer, W. Fitzgerald, and D. Klenerman, “Bayesian inference for improved single molecule fluorescence tracking,” Biophys. J.94, 4932–4947 (2008).
[CrossRef] [PubMed]

Lanterman, A.

Larson, D.

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

Leang, C.

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inform. Theory43, 280–282 (1997).
[CrossRef]

Lee, M.

Lenart, P.

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

Lew, S.

Lippincott-Schwartz, J.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Liu, N.

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

Lord, S.

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

McKinney, A.

A. Yildiz, J. Forkey, A. McKinney, T. Ha, Y. Goldman, and P. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Mlodzianoski, M.

H. Deschout, F. Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods11, 253–266 (2014).
[CrossRef] [PubMed]

Moerner, W.

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nat. Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

M. Lee, S. Lew, M. Badieirostami, and W. Moerner, “Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects,” Opt. Lett.36, 202–204 (2011).
[CrossRef] [PubMed]

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

Monnier, N.

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

Monypenny, J.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Mori, M.

N. Monnier, S.-M. Guo, M. Mori, J. He, P. Lenart, and M. Bathe, “Bayesian approach to MSD-based analysis of particle motion in live cells,” Biophys. J.103, 616–626 (2012).
[CrossRef] [PubMed]

Morrison, I.

C. Anderson, G. Georgiou, I. Morrison, G. Stevenson, and R. Cherry, “Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera,” J Cell Sci.101, 415–425 (1992).

Ober, R.

J. Chao, E. Sally Ward, and R. Ober, “Fisher information matrix for branching processes with application to EMCCDs,” Multidimens. Syst. Signal Process.23, 349–379 (2012).
[CrossRef] [PubMed]

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U. S. A.103, 4457–4462 (2006).
[CrossRef] [PubMed]

S. Ram, E. Sally Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[CrossRef] [PubMed]

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

Pavani, S.

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

Pertsinidis, A.

A. Pertsinidis, Y. Zhang, and S. Chu, “Subnanometre single-molecule localization, registration and distance measurements,” Nature466, 647–651 (2010).
[CrossRef] [PubMed]

Piestun, R.

S. Pavani, M. Thompson, J. Biteen, S. Lord, N. Liu, R. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix PSF,” Proc. Natl. Acad. Sci. U. S. A.106, 2995–2999 (2009).
[CrossRef] [PubMed]

Prasad, S.

Ram, S.

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U. S. A.103, 4457–4462 (2006).
[CrossRef] [PubMed]

S. Ram, E. Sally Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[CrossRef] [PubMed]

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

Rosten, E.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. Burnette, J. Lippincott-Schwartz, G. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods9, 195–200 (2012).
[CrossRef]

Roussas, G.

G. Roussas, A Course in Mathematical Statistics (Academic, 1997).

Sally Ward, E.

J. Chao, E. Sally Ward, and R. Ober, “Fisher information matrix for branching processes with application to EMCCDs,” Multidimens. Syst. Signal Process.23, 349–379 (2012).
[CrossRef] [PubMed]

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U. S. A.103, 4457–4462 (2006).
[CrossRef] [PubMed]

S. Ram, E. Sally Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[CrossRef] [PubMed]

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

Selvin, P.

A. Yildiz, J. Forkey, A. McKinney, T. Ha, Y. Goldman, and P. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Small, A.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods11, 267–279 (2014).
[CrossRef] [PubMed]

Snyder, D.

Stahlheber, S.

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Figures (8)

Fig. 1
Fig. 1

(a) Doubly logarithmic plots of MPE vs. source brightness K for the conventional imager for two different values of the defocus phase, ζ, namely 0 and 16 radians. The plots for the various transverse super-localization factors are indicated explicitly both by means of their different colors, marker symbols, and labels, with the curves in the bottom half referring to ζ = 0 (in-focus) and those in the top half to ζ = 16. The corresponding asymptotic values of the MPE are connected by dashed line segments. The values of the MPE for 2x super-localization in the in-focus case are both too small and at too small K to appear in the figure, while the asymptotic results for the 8x and 16x super-localizations for ζ = 16 are suppressed for clarity. (b) The same plots are now shown on a linear vertical scale for ease of comparison with Fig. 2.

Fig. 2
Fig. 2

Plots of MPE vs. source brightness K, in dB, for the rotating-PSF imager for the same two values of the defocus phase, ζ, (a) 0 and (b) 16 radians, as in Fig. 1. The plots of the MPE for the various transverse super-localization factors are indicated explicitly both by means of their different colors, marker symbols, and labels. The corresponding asymptotic values are connected by dashed line segments.

Fig. 3
Fig. 3

Plots of MPE vs. the defocus phase, ζ, for the rotating-PSF imager (blue curves) and the conventional imager (green curves) for two different values of K. The three subfigures refer to the values (a) 2x; (b) 4x; and (c) 16x of the 2D super-localization factor. The dashed lines display the corresponding asymptotic results except for the rotating PSF imager at K = 103.

Fig. 4
Fig. 4

Plots of MPE vs. source brightness K, in dB, for the rotating-PSF imager for (a) ζ = 0 (in-focus); (b) ζ = 16 (defocused); and for the conventional imager (c) for ζ = 0 and 16. Three different values of M, namely 2, 4, and 8, were considered here. The solid line segments refer to the approximate pseudo-Gaussian distribution, while the dashed line segments refer to the Poisson distribution, which is exact for the present case of zero sensor noise.

Fig. 5
Fig. 5

Plots of Kmin vs. M 2 for the two imagers for four different values of the defocus phase, ζ, namely 0, 4, 8, and 16 radians. For the rotating-PSF imager, results shown for ζ = 0 and 16 radians bracket those for the intermediate values of ζ, not shown here. The dashed lines away from marker symbols for the conventional imager are extrapolations of our numerical results.

Fig. 6
Fig. 6

Plots of MPE vs. K, in dB, for two different values of defocus, (a) ζ = 0 and (b) ζ = 16, for two different axial and four different transverse localization enhancement factors for the rotating-PSF imager. The latter factors are indicated by different marker symbols, namely circle for 2x, + for 4x, square for 8x, and diamond for 16x, while the former are indicated by the line type, solid for 2x and dashed for 4x.

Fig. 7
Fig. 7

Same as in Figs. 6(a) and 6(b), except here for the conventional imager.

Fig. 8
Fig. 8

Plots of asymptotic values of the MPE vs. source strength for the astigmatic imager for (a) in-focus and (b) 8 rad out of focus 3D localization. The plots are labeled by the transverse super-localization factors in the same way as in Figs. 6 and 7. The corresponding rotating-PSF-imager results are shown in (c) and (d).

Equations (49)

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P e ( min ) = 1 𝔼 [ p ( m ^ MAP | X ) ] ,
m ^ MAP = argmax m = 1 , , M p ( m | X ) .
P e ( min ) = 1 m = 1 M p m m d x P ( x | m ) ,
m = { x | P ( x | m ) p m > P ( x | m ) p m , m m } ,
P e ( min ) = m = 1 M p m m m m d x P ( x | m ) .
m ˜ = argmax m m max x m { P ( x | m ) } ,
P e ( min ) = m = 1 M p m m ˜ d x P ( x | m ) .
P ( x | m ) = 1 ( 2 π σ 2 ) N / 2 exp [ ( 1 / 2 ) x x m 2 2 / σ 2 ] ,
m ˜ = argmin m m min { E m ( x ) | x m } ,
E m ( x ) x x m 2 2 2 σ 2 ln p m .
x x m 2 2 = x x m δ x m m 2 2 + 2 σ 2 ln ( p m / p m ) ,
δ x m m x m x m .
( x x m γ δ x m m 2 ) T δ x m m = 0 ,
γ = 1 + 2 σ 2 ln ( p m / p m ) δ x m m 2 2 .
F m m 2 γ 2 4 x m x m 2 2 .
m ˜ = argmin m m ( F m m 2 2 σ 2 ln p m ) .
P ( x | m ) = 1 ( 2 π σ 2 ) N / 2 exp [ ( 1 / 2 ) ( x t 2 + x 2 2 ) / σ 2 ] .
P e ( min ) = m = 1 M p m 1 ( 2 π σ 2 ) 1 / 2 F m m ˜ exp [ ( 1 / 2 ) x t 2 / σ 2 ] ,
d x exp [ ( 1 / 2 ) x 2 / σ 2 ] = ( 2 π σ 2 ) 1 / 2 ,
P e ( min ) = 1 2 m = 1 M p m erfc ( F m m ˜ / 2 σ 2 ) ,
erfc ( u ) 2 π u exp ( x 2 ) d x .
P e ( min ) 1 2 π m = 1 M p m σ F m m ˜ exp [ ( 1 / 2 ) F m m ˜ 2 / σ 2 ] = 1 2 π m = 1 M p m 2 σ γ x m ˜ x m 2 exp [ γ 2 x m ˜ x m 2 2 8 σ 2 ] .
P ( x | m ) = 1 ( 2 π ) N / 2 det 1 / 2 ( Σ m ) exp [ ( 1 / 2 ) ( x T x m T ) Σ m 1 ( x x m ) ] .
Σ m = diag ( σ 2 + x m ) ,
ln P ( x | m ) = ( 1 / 2 ) { ( x T x m T ) Σ m 1 ( x x m ) + ln [ ( 2 π ) N det Σ m ] } ,
P e ( min ) = 1 2 m p m erfc ( U m 2 / 2 ) ,
U m 2 = 1 2 i = 1 N ( σ 2 + x m i ) 1 / 2 ( σ 2 + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 [ i = 1 N 1 ( σ 2 + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 ] 1 / 2 .
ν lim N ln P e ( min ) N = lim 1 2 N min m U m 2 2 = 1 8 min m lim N [ 1 N i = 1 N ( σ 2 + x m i ) 1 / 2 ( σ 2 + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 ] 2 [ 1 N i = 1 N 1 ( σ 2 + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 ] ,
P ( x | m ) = exp [ L m ( x ; N ) ] ,
P e ( min ) = 1 2 m p m max s P ( x * s | m ) ,
x m = s m + b ¯ , Σ m = diag ( σ s 2 + s m + b ¯ ) ,
min x ( x T x m T ) Σ m 1 ( x x m ) λ [ ( x T x m T ) Σ m 1 ( x x m ) ( x T x m ˜ T ) Σ m ˜ 1 ( x x m ˜ ) ] ,
( 1 λ ) Σ m 1 ( x * x m ) + λ Σ m ˜ 1 ( x * x m ˜ ) = 0 .
( x * x m ) = diag [ 1 + ( 1 λ ) λ ( σ 2 + x m ˜ ) ( σ 2 + x m ) ] 1 δ x m ˜ m .
( x * x m ˜ ) = ( x * x m ) δ x m ˜ m = diag [ 1 + λ ( 1 λ ) ( σ 2 + x m ) σ 2 + x m ˜ ] 1 δ x m ˜ m .
δ x m ˜ m T diag { λ 2 ( σ 2 + x m ) ( 1 λ ) 2 ( σ 2 + x m ˜ ) [ λ ( σ 2 + x m ) + ( 1 λ ) ( σ 2 + x m ˜ ) ] 2 } δ x m ˜ m = ln p m 2 det Σ m ˜ p m ˜ 2 det Σ m .
r m = ( 2 λ 1 ) ( σ 2 + x m ) ( 1 λ ) 2 δ x m ˜ m [ λ ( σ 2 + x m ) + ( 1 λ ) ( σ 2 + x m ˜ ) ] 2 = ( 2 λ 1 ) ( σ 2 + x m ) [ 1 ( 3 λ 1 ) ( 1 λ ) ( 2 λ 1 ) δ x m ˜ m ( σ 2 + x m ) + O ( δ 2 ) ] ,
λ = 1 2 + 1 2 i = 1 N δ x m ˜ m i ( σ 2 + x m i ) + 2 ln p m p m ˜ i = 1 N δ x m ˜ m i 2 ( σ 2 + x m i ) .
v Σ m 1 / 2 ( x x m ) ,
n * = * ln P ( x * | m ) * ln P ( x * | m ˜ ) * ln P ( x * | m ) * ln P ( x * | m ) 2 = Σ m 1 ( x * x m ) Σ m ˜ 1 ( x * x m ˜ ) Σ m 1 ( x * x m ) Σ m ˜ 1 ( x * x m ˜ ) 2 ,
n * = Σ m 1 ( x * x m ) Σ m 1 ( x * x m ) 2 = Σ m 1 ( x * x m ) [ ( x * T x m T ) Σ m 2 ( x * x m ) ] 1 / 2 ,
v = n * n * T v = n * n * T Σ m 1 / 2 ( x x * ) + n * n * T Σ m 1 / 2 ( x * x m ) = u m + U m and v = v v
u m = n * n * T Σ m 1 / 2 ( x x * ) ; U m = n * n * T Σ m 1 / 2 ( x * x m ) ,
P ( x | m ) = 1 [ ( 2 π ) N det Σ m ] 1 / 2 exp [ ( 1 / 2 ) ( v T v + v T v ) ] .
m ˜ P ( x | m ) d x = 1 ( 2 π ) 1 / 2 U m 2 exp [ ( 1 / 2 ) v 2 2 ] d v 2 = 1 2 erfc ( U m 2 / 2 ) ,
P e ( min ) = 1 2 m p m erfc ( U m 2 / 2 ) .
U m 2 = ( x * T x m T ) Σ m 3 / 2 ( x * x m ) [ ( x * T x m T ) Σ m 2 ( x * x m ) ] 1 / 2
U m 2 = 1 2 δ x m ˜ m T diag [ ( σ 2 + x m ) 2 ( σ 2 + x ¯ m ) 2 ] Σ m 3 / 2 δ x m ˜ m { δ x m ˜ m T diag [ ( σ 2 + x m ) 2 ( σ 2 + x ¯ m ) 2 ] Σ m 2 δ x m ˜ m } 1 / 2 ,
U m 2 = 1 2 i = 1 N ( σ 2 + x m i ) 1 / 2 ( σ 2 + x ¯ m i ) 2 ( δ x ˜ m ˜ m i ) 2 [ i = 1 N 1 ( σ 2 + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 ] 1 / 2 .

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