Abstract

Estimation of a parameter of interest from image data represents a task that is commonly carried out in single molecule microscopy data analysis. The determination of the positional coordinates of a molecule from its image, for example, forms the basis of standard applications such as single molecule tracking and localization-based super-resolution image reconstruction. Assuming that the estimator used recovers, on average, the true value of the parameter, its accuracy, or standard deviation, is then at best equal to the square root of the Cramér–Rao lower bound. The Cramér–Rao lower bound can therefore be used as a benchmark in the evaluation of the accuracy of an estimator. Additionally, as its value can be computed and assessed for different experimental settings, it is useful as an experimental design tool. This tutorial demonstrates a mathematical framework that has been specifically developed to calculate the Cramér–Rao lower bound for estimation problems in single molecule microscopy and, more broadly, fluorescence microscopy. The material includes a presentation of the photon detection process that underlies all image data, various image data models that describe images acquired with different detector types, and Fisher information expressions that are necessary for the calculation of the lower bound. Throughout the tutorial, examples involving concrete estimation problems are used to illustrate the effects of various factors on the accuracy of parameter estimation and, more generally, to demonstrate the flexibility of the mathematical framework.

© 2016 Optical Society of America

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References

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2015 (1)

J. Chao, T. Lee, E. S. Ward, and R. J. Ober, “Fluorescent microspheres as point sources: a localization study,” PLoS ONE 10, e0134112 (2015).
[Crossref]

2014 (6)

H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11, 253–266 (2014).
[Crossref]

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11, 267–279 (2014).
[Crossref]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113, 133902 (2014).
[Crossref]

A. Krull, A. Steinborn, V. Ananthanarayanan, D. Ramunno-Johnson, U. Petersohn, and I. M. Tolić-Nørrelykke, “A divide and conquer strategy for the maximum likelihood localization of low intensity objects,” Opt. Express 22, 210–228 (2014).
[Crossref]

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22, 16706–16721 (2014).
[Crossref]

Y. Wong, J. Chao, Z. Lin, and R. J. Ober, “Effect of time discretization of the imaging process on the accuracy of trajectory estimation in fluorescence microscopy,” Opt. Express 22, 20396–20420 (2014).
[Crossref]

2013 (4)

F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
[Crossref]

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

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[Crossref]

S. Liu, E. B. Kromann, W. D. Krueger, J. Bewersdorf, and K. A. Lidke, “Three dimensional single molecule localization using a phase retrieved pupil function,” Opt. Express 21, 29462–29487 (2013).
[Crossref]

2012 (5)

S. Stallinga and B. Rieger, “Position and orientation estimation of fixed dipole emitters using an effective Hermite point spread function model,” Opt. Express 20, 5896–5921 (2012).
[Crossref]

A. Agrawal, S. Quirin, G. Grover, and R. Piestun, “Limits of 3D dipole localization and orientation estimation for single-molecule imaging: towards Green’s tensor engineering,” Opt. Express 20, 26667–26680 (2012).
[Crossref]

X. Michalet and A. J. Berglund, “Optimal diffusion coefficient estimation in single-particle tracking,” Phys. Rev. E 85, 061916 (2012).
[Crossref]

H. Deschout, K. Neyts, and K. Braeckmans, “The influence of movement on the localization precision of sub-resolution particles in fluorescence microscopy,” J. Biophoton. 5, 97–109 (2012).
[Crossref]

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

2011 (2)

Z. Shen and S. B. Andersson, “Bias and precision of the fluoroBancroft algorithm for single particle localization in fluorescence microscopy,” IEEE Trans. Signal Process. 59, 4041–4046 (2011).
[Crossref]

M. R. Foreman and P. Török, “Fundamental limits in single-molecule orientation measurements,” New J. Phys. 13, 093013 (2011).
[Crossref]

2010 (8)

A. J. Berglund, “Statistics of camera-based single-particle tracking,” Phys. Rev. E 82, 011917 (2010).
[Crossref]

M. A. Thompson, M. D. Lew, M. Badieirostami, and W. E. Moerner, “Localizing and tracking single nanoscale emitters in three dimensions with high spatiotemporal resolution using a double-helix point spread function,” Nano Lett. 10, 211–218 (2010).
[Crossref]

T. Quan, S. Zeng, and Z.-L. Huang, “Localization capability and limitation of electron-multiplying charge-coupled, scientific complementary metal-oxide semiconductor, and charge-coupled devices for superresolution imaging,” J. Biomed. Opt. 15, 066005 (2010).
[Crossref]

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010).
[Crossref]

X. Michalet, “Mean square displacement analysis of single-particle trajectories with localization error: Brownian motion in an isotropic medium,” Phys. Rev. E 82, 041914 (2010).
[Crossref]

M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 107, 17864–17871 (2010).
[Crossref]

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

S. Stallinga and B. Rieger, “Accuracy of the Gaussian point spread function model in 2D localization microscopy,” Opt. Express 18, 24461–24476 (2010).
[Crossref]

2009 (5)

2008 (6)

M. Heilemann, S. van de Linde, M. Schüttpelz, R. Kasper, B. Seefeldt, A. Mukherjee, P. Tinnefeld, and M. Sauer, “Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes,” Angew. Chem. Int. Ed. 47, 6172–6176 (2008).
[Crossref]

N. G. Walter, C.-Y. Huang, A. J. Manzo, and M. A. Sobhy, “Do-it-yourself guide: how to use the modern single-molecule toolkit,” Nat. Methods 5, 475–489 (2008).
[Crossref]

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, 77–86 (2008).
[Crossref]

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref]

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100 nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–529 (2008).
[Crossref]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref]

2007 (5)

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

M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nat. Methods 4, 319–321 (2007).
[Crossref]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
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L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, 053902 (2007).
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2006 (8)

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA 103, 4457–4462 (2006).
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S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–57 (2006).
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S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
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M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
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A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
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2005 (2)

F. Aguet, D. Van De Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express 13, 10503–10522 (2005).
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S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE 5699, 426–435 (2005).
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2004 (3)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
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R. J. Ober, C. Martinez, X. Lai, J. Zhou, and E. S. Ward, “Exocytosis of IgG as mediated by the receptor, FcRn: an analysis at the single-molecule level,” Proc. Natl. Acad. Sci. USA 101, 11076–11081 (2004).
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P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
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2003 (4)

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1995 (1)

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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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A. G. Basden, C. A. Haniff, and C. D. Mackay, “Photon counting strategies with low-light-level CCDs,” Mon. Not. R. Astron. Soc. 345, 985–991 (2003).
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Bonifacino, J. S.

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H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11, 253–266 (2014).
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H. Deschout, K. Neyts, and K. Braeckmans, “The influence of movement on the localization precision of sub-resolution particles in fluorescence microscopy,” J. Biophoton. 5, 97–109 (2012).
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M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 107, 17864–17871 (2010).
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M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
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M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 107, 17864–17871 (2010).
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H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11, 253–266 (2014).
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Chao, J.

J. Chao, T. Lee, E. S. Ward, and R. J. Ober, “Fluorescent microspheres as point sources: a localization study,” PLoS ONE 10, e0134112 (2015).
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A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22, 16706–16721 (2014).
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Y. Wong, J. Chao, Z. Lin, and R. J. Ober, “Effect of time discretization of the imaging process on the accuracy of trajectory estimation in fluorescence microscopy,” Opt. Express 22, 20396–20420 (2014).
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J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
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J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidimens. Syst. Signal Process. 23, 349–379 (2012).
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J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
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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, 23352–23373 (2009).
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S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
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S. Ram, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “A novel approach to determining the three-dimensional location of microscopic objects with applications to 3D particle tracking,” Proc. SPIE 6443, 64430D (2007).
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P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
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M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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Cherry, R. J.

P. R. Smith, I. E. G. Morrison, K. M. Wilson, N. Fernández, and R. J. Cherry, “Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking,” Biophys. J. 76, 3331–3344 (1999).
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K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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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, 77–86 (2008).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
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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, 1642–1645 (2006).
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H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11, 253–266 (2014).
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H. Deschout, K. Neyts, and K. Braeckmans, “The influence of movement on the localization precision of sub-resolution particles in fluorescence microscopy,” J. Biophoton. 5, 97–109 (2012).
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H. Deschout, F. Cella Zanacchi, M. Mlodzianoski, A. Diaspro, J. Bewersdorf, S. T. Hess, and K. Braeckmans, “Precisely and accurately localizing single emitters in fluorescence microscopy,” Nat. Methods 11, 253–266 (2014).
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F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
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Fernández, N.

P. R. Smith, I. E. G. Morrison, K. M. Wilson, N. Fernández, and R. J. Cherry, “Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking,” Biophys. J. 76, 3331–3344 (1999).
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Flyvbjerg, H.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7, 377–381 (2010).
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M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
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E. R. Fossum, “CMOS image sensors: electronic camera-on-a-chip,” IEEE Trans. Electron Devices 44, 1689–1698 (1997).
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W. E. Moerner and D. P. Fromm, “Methods of single-molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. 74, 3597–3619 (2003).
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P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
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Gibbons, S.

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
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S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
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Greengard, A.

Grover, G.

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, 77–86 (2008).
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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, 77–86 (2008).
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M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
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J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
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J. Chao, S. Ram, A. V. Abraham, E. S. Ward, and R. J. Ober, “A resolution measure for three-dimensional microscopy,” Opt. Commun. 282, 1751–1761 (2009).
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F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express 17, 6829–6848 (2009).
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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, 23352–23373 (2009).
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J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
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S. Stallinga and B. Rieger, “Accuracy of the Gaussian point spread function model in 2D localization microscopy,” Opt. Express 18, 24461–24476 (2010).
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A. Agrawal, S. Quirin, G. Grover, and R. Piestun, “Limits of 3D dipole localization and orientation estimation for single-molecule imaging: towards Green’s tensor engineering,” Opt. Express 20, 26667–26680 (2012).
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Figures (9)

Fig. 1.
Fig. 1.

Fundamental data model example. (a) Photon distribution profile f θ for an in-focus and stationary molecule whose image is modeled with the Airy pattern. This profile assumes that the molecule is located at ( x 0 , y 0 ) = ( 0    nm , 0    nm ) in the object space and that photons of wavelength λ = 520    nm are collected using an objective lens with a numerical aperture of n a = 1.4 and a magnification of M = 100 . While the profile is defined over the detector plane R 2 , the plot shows its values over the 150    μm × 150    μm region over which it is centered. (b) A fundamental image simulated according to the profile in (a), with each dot representing the location at which a photon is detected. The simulation assumes a constant photon detection rate of Λ 0 = 10 , 000    photons / s and an acquisition time of t t 0 = 0.05    s , such that the mean photon count in an image is Λ 0 · ( t t 0 ) = 500 . In this particular realization, N 0 = 516 photons are detected, of which 492 fall within the 150    μm × 150    μm region shown, over which the image is centered.

Fig. 2.
Fig. 2.

Poisson data model example. (a) The mean pixelated image of the in-focus and stationary molecule of Fig. 1, computed over an 11 × 11 -pixel region with a pixel size of 13    μm × 13    μm and a mean background photoelectron count of β θ , 0 = 10 per pixel (obtained as the product of the constant background photon detection rate Λ 0 b = 24 , 200    photons / s and the uniform spatial distribution f θ b ( x , y ) = 1 / 20 , 449    μm 2 , integrated over each pixel and the acquisition time of t t 0 = 0.05    s ). The molecule is located at ( x 0 , y 0 ) = ( 656.5    nm , 682.5    nm ) in the object space, which corresponds to (5.05 pixels, 5.25 pixels) in the image space, assuming (0, 0) in both spaces coincides with the upper-left corner of the 11 × 11 -pixel region. In terms of the photoelectron count due to the molecule, the mean image contains 476.15 out of the total of Λ 0 · ( t t 0 ) = 500 mean number of photoelectrons distributed over the detector plane. All other relevant parameters are as specified in Fig. 1. (b) A Poisson realization of the mean image in (a). The photoelectron count in each pixel is drawn from the Poisson distribution with mean given by the photoelectron count in the corresponding pixel in (a).

Fig. 3.
Fig. 3.

CCD data model and EMCCD data model examples. (a) A CCD realization of the mean image in Fig. 2(a). The electron count in each pixel is the sum of a random number drawn from the Poisson distribution with mean given by the photoelectron count in the corresponding pixel in Fig. 2(a), and a random number drawn from the Gaussian distribution with mean η 0 = 0 electrons and standard deviation σ 0 = 8 electrons. In this particular realization, the Poisson random numbers are taken directly from the Poisson realization of Fig. 2(b). (b) An EMCCD realization of the mean image in Fig. 2(a). The electron count in each pixel is the sum of a random number drawn from the probability distribution of Eq. (14) with electron multiplication gain g = 950 and ν θ , k given by the photoelectron count in the corresponding pixel in Fig. 2(a), and a random number drawn from the Gaussian distribution with mean η 0 = 0 electrons and standard deviation σ 0 = 24 electrons.

Fig. 4.
Fig. 4.

Limits of accuracy under the fundamental data model. (a) Limit of accuracy δ x 0 for the estimation of the x 0 coordinate of an in-focus and stationary molecule from a fundamental image is plotted as a function of the expected number of photons detected in the image. Curves are shown that correspond to the detection of photons of wavelengths λ = 620    nm (dotted), 520 nm (solid), and 440 nm (dashed), which are collected by an objective lens with numerical aperture n a = 1.4 . The expected photon counts range from 100 to 10,000 and are obtained as the product of the constant photon detection rate Λ 0 = 10 , 000    photons / s and acquisition times t t 0 ranging from 0.01 to 1 s. The image of the molecule is described by the Airy image function. (b) Limit of accuracy δ Λ 0 for the estimation of the photon detection rate Λ 0 = 10 , 000    photons / s from a fundamental image is shown as a function of the expected number of photons detected in the image.

Fig. 5.
Fig. 5.

Limit of the accuracy for separation distance estimation under the fundamental data model. The limit of accuracy δ d for the estimation, from a fundamental image, of the distance d separating two in-focus and stationary molecules is plotted as a function of the distance d . Curves are shown that correspond to the detection of photons of wavelengths λ = 620    nm (⋄), 520 nm (*), and 440 nm (○), which are collected by an objective lens with numerical aperture n a = 1.4 . Photons are detected from each molecule at a rate of Λ 0 = 10 , 000    photons / s over an acquisition time of 0.3 s, such that Λ 0 · ( t t 0 ) = 3000 photons are, on average, detected from each molecule in a given image. The image of each molecule is described by the Airy image function. Inset shows the portion of the curves from d = 1    nm to d = 50    nm .

Fig. 6.
Fig. 6.

Comparing limits of accuracy corresponding to different data models and variations thereof. (a) Limit of accuracy δ x 0 for the estimation of the x 0 coordinate of an in-focus and stationary molecule from an image is plotted as a function of the effective pixel size for the Poisson data model (*) and the Poisson data model assuming the absence of background noise (+). Also shown are the limits of accuracy corresponding to the finite detection area variation of the fundamental data model (dashed line), the finite detection area variation of the fundamental data model assuming the absence of background noise (dotted line), and the fundamental data model (solid line), which are plotted as horizontal lines, as they do not depend on the effective pixel size. For the two cases involving the Poisson data model, the physical pixel size is 13    μm × 13    μm , and the different effective pixel sizes are obtained by varying the lateral magnification from M = 20 to M = 966.67 . The image consists of a 15 × 15 -pixel region at M = 100 and a proportionately scaled pixel region at each of the other magnifications. At each magnification, the molecule is positioned such that the center of its image, given by the Airy image function, is located at 0.05 pixels in the x direction and 0.25 pixels in the y direction with respect to the upper-left corner of the center pixel of the image. For the two cases involving the finite detection area variation of the fundamental data model, δ x 0 is computed using the unpixelated equivalent of the M = 100 scenario, though any other scenario can be used to obtain the same result. For all cases, photons are detected from the molecule at a rate of Λ 0 = 6000    photons / s over an acquisition time of t t 0 = 0.05    s , such that Λ 0 · ( t t 0 ) = 300 photons are, on average, detected from the molecule over the detector plane (and 289.5 photons are, on average, detected from the molecule over a given finite-sized image). For the two cases involving a background component, a background detection rate of Λ 0 b = 22 , 500    photons / s is assumed, along with a uniform background spatial distribution of f b ( x , y ) = 1 / A    μm 2 , ( x , y ) C , where C is the region in the detector plane corresponding to the image and A is the area of C . For example, for the M = 100 scenario, f b ( x , y ) = 1 / 38 , 025    μm 2 , such that there is a background level of β 0 = 5 photoelectrons per pixel over the 15 × 15 -pixel region. For all cases, it is assumed that photons of wavelength λ = 520    nm are collected by an objective lens with numerical aperture n a = 1.4 .

Fig. 7.
Fig. 7.

Comparing limits of accuracy corresponding to the CCD and EMCCD data models. (a) For the CCD data model, the limit of accuracy δ x 0 for the estimation of the x 0 coordinate of an in-focus and stationary molecule from an image is plotted as a function of the effective pixel size. Limits of accuracy are shown for three different levels of readout noise in each pixel, characterized by standard deviations of σ 0 = 6 electrons (•), 1 electron (⋄), and 0.5 electrons (○). In each case, the mean of the readout noise in each pixel is η 0 = 0 electrons. The limit of accuracy for the Poisson data model (with background noise) from Fig. 6 (*) is shown as a reference for comparison because the CCD limits of accuracy are computed by adding readout noise to the same assumed conditions. All details not mentioned are therefore exactly as given in Fig. 6 for the Poisson data model. The inset provides a clearer view of some data points for five relatively large effective pixel sizes. (b) Zoomed-in version of (a) with the addition of δ x 0 corresponding to the EMCCD data model with readout noise of mean η 0 = 0 electrons and standard deviation σ 0 = 48 electrons in each pixel and an electron multiplication gain of g = 1000 .

Fig. 8.
Fig. 8.

Comparing noise coefficients corresponding to the CCD and EMCCD data models. The noise coefficient α k for the k th pixel of an image is plotted as a function of the mean photoelectron count ν θ , k in the pixel. Noise coefficients are shown for two EMCCD detectors operated at an electron multiplication gain of g = 1000 , with readout noise of standard deviations σ 0 = 24 electrons (□) and 48 electrons (+) in each pixel, and for two CCD detectors, with readout noise of standard deviations σ 0 = 0.5 electrons (○) and 1 electron (⋄) in each pixel. The readout noise mean is η 0 = 0 electrons in all cases.

Fig. 9.
Fig. 9.

Limits of the axial coordinate estimation accuracy for a conventional microscopy setup and a two-plane MUM setup. The limit of accuracy δ z 0 for the estimation of the z 0 coordinate of an out-of-focus and stationary molecule from a conventional image (*) or a pair of two-plane MUM images (○) is plotted as a function of the coordinate z 0 . Focal planes 1 and 2 of the MUM setup are denoted by vertical dashed lines and are located at z 0 = 0    nm and z 0 = 450    nm , respectively. The value of z 0 is specified with respect to focal plane 1, which coincides with the focal plane of the conventional setup. In both setups, an image consists of an 11 × 11 -pixel region, acquired using a CCD detector with a pixel size of 13    μm × 13    μm and readout noise with mean η 0 = 0 electrons and standard deviation σ 0 = 6 electrons at each pixel. The image of the molecule is described by the Born and Wolf image function. For the conventional setup, the molecule is laterally positioned such that the center of its image is located at 5.05 pixels in the x direction and 5.25 pixels in the y direction. Photons are detected from the molecule at a rate of Λ 0 = 10 , 000    photons / s over an acquisition time of t t 0 = 0.1    s , such that Λ 0 · ( t t 0 ) = 1000 photons are on average detected from the molecule over the detector plane (and 952.3 photons are on average detected from the molecule over a given image when the molecule is in focus). A background detection rate of Λ 0 b = 24 , 200    photons / s is assumed, along with a uniform background spatial distribution of f b ( x , y ) = 1 / 20 , 449    μm 2 , such that there is a background level of β 0 = 20 photoelectrons per pixel over the 11 × 11 -pixel image. Photons of wavelength λ = 520    nm are collected by an objective lens with magnification M = 100 and numerical aperture n a = 1.4 . The refractive index of the immersion medium is n = 1.518 . For the MUM setup, the collected molecule and background photons are split equally between the images corresponding to the two focal planes. Therefore, the photon detection rate is Λ 0 = 5000    photons / s per image, and the background detection rate is Λ 0 b = 12 , 100    photons / s per image. For the image corresponding to focal plane 1, all other details are as given for the conventional setup. For the image corresponding to focal plane 2, the magnification is M = 98.18 , calculated using a formula from [76] for a plane spacing of Δ z f = 450    nm and a tube length of 160 mm. The lateral position of the image of the molecule and the background spatial distribution are accordingly adjusted based on this slightly smaller magnification.

Equations (40)

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f θ , τ ( x , y ) = 1 M 2 q z 0 , τ ( x M x 0 , τ , y M y 0 , τ ) ,
q ( x , y ) = J 1 2 ( 2 π n a λ x 2 + y 2 ) π ( x 2 + y 2 ) , ( x , y ) R 2 ,
q ( x , y ) = 1 2 π σ g 2 · e x 2 + y 2 2 σ g 2 , ( x , y ) R 2 ,
q z 0 ( x , y ) = 4 π n a 2 λ 2 | 0 1 J 0 ( 2 π n a λ x 2 + y 2 ρ ) e j π n a 2 z 0 n λ ρ 2 ρ d ρ | 2 ,
Λ θ ( τ ) = Λ θ 1 ( τ ) + Λ θ 2 ( τ ) , θ Θ , τ t 0 ,
f θ , τ ( x , y ) = Λ θ 1 ( τ ) Λ θ ( τ ) f θ , τ 1 ( x , y ) + Λ θ 2 ( τ ) Λ θ ( τ ) f θ , τ 2 ( x , y ) ,
f θ , τ ( x , y ) = 1 M 2 ( Λ θ 1 ( τ ) Λ θ ( τ ) q 1 ( x M x 01 , y M y 01 ) + Λ θ 2 ( τ ) Λ θ ( τ ) q 2 ( x M x 02 , y M y 02 ) ) ,
H θ , k = S θ , k + B θ , k , θ Θ ,
μ θ , k = t 0 t Λ θ ( τ ) C k f θ , τ ( x , y ) d x d y d τ , θ Θ ,
β θ , k = t 0 t Λ θ b ( τ ) C k f θ , τ b ( x , y ) d x d y d τ , θ Θ ,
ν θ , k = μ θ , k + β θ , k , θ Θ ,
H θ , k = S θ , k + B θ , k + W k , θ Θ .
H θ , k = S θ , k a + B θ , k a + W k , θ Θ .
p θ , k a ( l ) = { e ν θ , k , l = 0 , e ν θ , k j = 0 l 1 ( l 1 j ) ( 1 1 g ) l 1 j ( ν θ , k g ) j + 1 ( j + 1 ) ! , l = 1 , 2 , ,
p θ , k a ( u ) = { e ν θ , k , u = 0 , e ( u g + ν θ , k ) ν θ , k u / g I 1 ( 2 ν θ , k u / g ) u , u > 0 ,
Cov ( θ ^ ) I 1 ( θ ) ,
Var ( θ ^ i ) [ I 1 ( θ ) ] i i ( [ I ( θ ) ] i i ) 1 , i = 1 , , N ,
I ( θ ) = E [ ( θ ln p θ ( w ) ) T ( θ ln p θ ( w ) ) ] , θ Θ ,
p θ ( w 1 , , w N 0 ) = e t 0 t Λ θ ( τ ) d τ ( k = 1 N 0 f θ , τ k ( r k ) ) ( k = 1 N 0 Λ θ ( τ k ) ) ,
I ( θ ) = t 0 t R 2 1 Λ θ ( τ ) f θ , τ ( x , y ) ( [ Λ θ ( τ ) f θ , τ ( x , y ) ] θ ) T × ( [ Λ θ ( τ ) f θ , τ ( x , y ) ] θ ) d x d y d τ , θ Θ .
δ x 0 = [ I 1 ( θ ) ] 11 = λ 2 π n a Λ 0 · ( t t 0 ) ,
δ y 0 = [ I 1 ( θ ) ] 22 = λ 2 π n a Λ 0 · ( t t 0 ) ,
δ Λ 0 = [ I 1 ( θ ) ] 33 = Λ 0 t t 0 .
δ d = I 1 ( θ ) = 1 4 π · Λ 0 · ( t t 0 ) · Γ 0 ( d ) · λ n a ,
Γ 0 ( d ) = R 2 1 J 1 2 ( γ r 01 ) r 01 2 + J 1 2 ( γ r 02 ) r 02 2 ( ( x + d 2 ) J 1 ( γ r 01 ) J 2 ( γ r 01 ) r 01 3 ( x d 2 ) J 1 ( γ r 02 ) J 2 ( γ r 02 ) r 02 3 ) 2 d x d y ,
I ( θ ) = t 0 t Λ ( τ ) d τ × R 2 1 f θ ( x , y ) ( f θ ( x , y ) θ ) T ( f θ ( x , y ) θ ) d x d y ,
I ( θ ) = t 0 t C 1 Λ θ ( τ ) f θ , τ ( x , y ) + Λ θ b ( τ ) f θ , τ b ( x , y ) × ( [ Λ θ ( τ ) f θ , τ ( x , y ) + Λ θ b ( τ ) f θ , τ b ( x , y ) ] θ ) T × ( [ Λ θ ( τ ) f θ , τ ( x , y ) + Λ θ b ( τ ) f θ , τ b ( x , y ) ] θ ) d x d y d τ ,
p θ ( z 1 , , z K ) = k = 1 K p θ , k ( z k ) , θ Θ .
I ( θ ) = k = 1 K E [ ( θ ln p θ , k ( z ) ) T ( θ ln p θ , k ( z ) ) ] ,
I ( θ ) = k = 1 K ( ν θ , k θ ) T ( ν θ , k θ ) · E [ ( ln p θ , k ( z ) ν θ , k ) 2 ] ,
p θ , k ( z ) = ν θ , k z e ν θ , k z ! , z = 0 , 1 , , θ Θ .
I ( θ ) = k = 1 K ( ν θ , k θ ) T ( ν θ , k θ ) · 1 ν θ , k , θ Θ .
p θ , k ( z ) = e ν θ , k 2 π σ k l = 0 ν θ , k l l ! e ( z l η k ) 2 2 σ k 2 , z R , θ Θ .
I ( θ ) = k = 1 K ( ν θ , k θ ) T ( ν θ , k θ ) × ( ( e ν θ , k 2 π σ k l = 1 ν θ , k l 1 ( l 1 ) ! e ( z l η k ) 2 2 σ k 2 ) 2 p θ , k ( z ) d z 1 ) ,
p θ , k ( z ) = e ν θ , k 2 π σ k [ e ( z η k ) 2 2 σ k 2 + l = 1 e ( z l η k ) 2 2 σ k 2 × j = 0 l 1 ( l 1 j ) ( 1 1 g ) l 1 j ( ν θ , k g ) j + 1 ( j + 1 ) ! ] , z R ,
I ( θ ) = k = 1 K ( ν θ , k θ ) T ( ν θ , k θ ) × ( ( e ν θ , k 2 π σ k l = 1 e ( z l η k ) 2 2 σ k 2 j = 0 l 1 ( l 1 j ) ( 1 1 g ) l 1 j ν θ , k j j ! g j + 1 ) 2 p θ , k ( z ) d z 1 ) ,
p θ , k ( z ) = e ν θ , k 2 π σ k [ e ( z η k ) 2 2 σ k 2 + 0 e ( z u η k ) 2 2 σ k 2 u g × ν θ , k u g I 1 ( 2 ν θ , k u g ) u d u ] , z R ,
I ( θ ) = k = 1 K ( ν θ , k θ ) T ( ν θ , k θ ) × ( ( e ν θ , k 2 π σ k g 0 e ( z u η k ) 2 2 σ k 2 u g I 0 ( 2 ν θ , k u g ) d u ) 2 p θ , k ( z ) d z 1 ) ,
α k = E [ ( ln p θ , k ( z ) ν θ , k ) 2 ] 1 ν θ , k = ν θ , k · E [ ( ln p θ , k ( z ) ν θ , k ) 2 ] ,
I set ( θ ) = k = 1 N i m I k ( θ ) , θ Θ .

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