Abstract

From an acquired image, single molecule microscopy makes possible the determination of the distance separating two closely spaced biomolecules in three-dimensional (3D) space. Such distance information can be an important indicator of the nature of the biomolecular interaction. Distance determination, however, is especially difficult when, for example, the imaged point sources are very close to each other or are located near the focal plane of the imaging setup. In the context of such challenges, we compare the limits of the distance estimation accuracy for several high resolution 3D imaging modalities. The comparisons are made using a Cramer-Rao lower bound-based 3D resolution measure which predicts the best possible accuracy with which a given distance can be estimated. Modalities which separate the detection of individual point sources (e.g., using photoactivatable fluorophores) are shown to provide the best accuracy limits when the two point sources are very close to each other and/or are oriented near parallel to the optical axis. Meanwhile, modalities which implement the simultaneous imaging of the point sources from multiple focal planes perform best when given a near-focus point source pair. We also demonstrate that the maximum likelihood estimator is capable of attaining the limit of the accuracy predicted for each modality.

© 2009 Optical Society of America

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2009

2008

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

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] [PubMed]

S. R. P. Paving and R. Piston, "Three dimensional tracking of fluorescent micro particles using a photon-limited double-helix response system," Opt. Express 16, 22048-22057 (2008).
[CrossRef] [PubMed]

N. G. Walter, C. 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] [PubMed]

2007

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

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (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).
[CrossRef] [PubMed]

2006

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidimens. Syst. Sig. Process. 17, 27-57 (2006).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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

M. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

A. Sharonov and R. M. Hochstrasser, "Wide-field subdiffraction imaging by accumulated binding of diffusing probes," Proc. Natl. Acad. Sci. USA 103, 18911-18916 (2006).
[CrossRef] [PubMed]

2005

K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, "Superresolution by localization of quantum dots using blinking statistics," Opt. Express 13, 7052-7062 (2005).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

2004

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).
[CrossRef]

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. USA 101, 11298-11303 (2004).
[CrossRef] [PubMed]

M. P. Gordon, T. Ha, and P. R. Selvin, "Single-molecule high-resolution imaging with photobleaching," Proc. Natl. Acad. Sci. USA 101, 6462-6465 (2004).
[CrossRef] [PubMed]

2003

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

2000

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

1999

S. Weiss, "Fluorescence spectroscopy of single bimolecular," Science 283, 1676-1683 (1999).
[CrossRef] [PubMed]

1997

1995

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

1994

H. P. Kao and A. S. Verkman, "Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position," Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

1992

Abraham, A. V.

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).
[CrossRef]

Agrawal, A.

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

Alivisatos, A. P.

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

Bates, M.

M. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

Beane, G. L.

Bennett, B. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Bewersdorf, J.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, "Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy," Opt. Express 17, 8264-8277 (2009).
[CrossRef] [PubMed]

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Chao, J.

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).
[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] [PubMed]

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).
[CrossRef] [PubMed]

Chemla, D. S.

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Deo, R.

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

Gan, Z.

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Gibson, S. F.

Girirajan, T. P. K.

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

Gordon, M. P.

M. P. Gordon, T. Ha, and P. R. Selvin, "Single-molecule high-resolution imaging with photobleaching," Proc. Natl. Acad. Sci. USA 101, 6462-6465 (2004).
[CrossRef] [PubMed]

Gould, T. J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Ha, T.

M. P. Gordon, T. Ha, and P. R. Selvin, "Single-molecule high-resolution imaging with photobleaching," Proc. Natl. Acad. Sci. USA 101, 6462-6465 (2004).
[CrossRef] [PubMed]

Haeberlé, O.

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Heintzmann, R.

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Hess, S. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

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).
[CrossRef] [PubMed]

Hochstrasser, R. M.

A. Sharonov and R. M. Hochstrasser, "Wide-field subdiffraction imaging by accumulated binding of diffusing probes," Proc. Natl. Acad. Sci. USA 103, 18911-18916 (2006).
[CrossRef] [PubMed]

Holtzer, L.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Huang, C.

N. G. Walter, C. 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] [PubMed]

Jovin, T. M.

Juette, M. F.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, "Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy," Opt. Express 17, 8264-8277 (2009).
[CrossRef] [PubMed]

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Kao, H. P.

H. P. Kao and A. S. Verkman, "Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position," Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

Lacoste, T. D.

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

Lanni, F.

Lessard, M. D.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Lidke, K. A.

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Manzo, A. J.

N. G. Walter, C. 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] [PubMed]

Mason, M. D.

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

Meckel, T.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Mets, L.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. USA 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Michalet, X.

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

Mlodzianoski, M. J.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, "Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy," Opt. Express 17, 8264-8277 (2009).
[CrossRef] [PubMed]

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Moerner, W. E.

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

Nagpure, B.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

Nicholson, C.

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

Nie, S.

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

Ober, R. J.

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).
[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] [PubMed]

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).
[CrossRef] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidimens. Syst. Sig. Process. 17, 27-57 (2006).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Paving, S. R. P.

Pinaud, F.

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

Piston, R.

Prabhat, P.

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] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Qu, X.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. USA 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Ram, S.

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).
[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] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidimens. Syst. Sig. Process. 17, 27-57 (2006).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Rieger, B.

Rust, M.

M. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

Scherer, N. F.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. USA 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Schmidt, T.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Selvin, P. R.

M. P. Gordon, T. Ha, and P. R. Selvin, "Single-molecule high-resolution imaging with photobleaching," Proc. Natl. Acad. Sci. USA 101, 6462-6465 (2004).
[CrossRef] [PubMed]

Sharonov, A.

A. Sharonov and R. M. Hochstrasser, "Wide-field subdiffraction imaging by accumulated binding of diffusing probes," Proc. Natl. Acad. Sci. USA 103, 18911-18916 (2006).
[CrossRef] [PubMed]

Sobhy, M. A.

N. G. Walter, C. 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] [PubMed]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Tao, L.

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

Török, P.

Vaccaro, C.

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).
[CrossRef] [PubMed]

Varga, P.

Verkman, A. S.

H. P. Kao and A. S. Verkman, "Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position," Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

Walter, N. G.

N. G. Walter, C. 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] [PubMed]

Wang, G. D.

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

Wang, M. D.

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

Ward, E. S.

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).
[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] [PubMed]

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).
[CrossRef] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidimens. Syst. Sig. Process. 17, 27-57 (2006).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Weiss, S.

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

S. Weiss, "Fluorescence spectroscopy of single bimolecular," Science 283, 1676-1683 (1999).
[CrossRef] [PubMed]

Wu, D.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. USA 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Zhuang, X.

M. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Biophys. J.

H. P. Kao and A. S. Verkman, "Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position," Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

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

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] [PubMed]

IEEE Trans. Nanobiosci.

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).
[CrossRef]

J. Microsc.

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Multidimens. Syst. Sig. Process.

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidimens. Syst. Sig. Process. 17, 27-57 (2006).
[CrossRef]

Nat. Methods

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. 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] [PubMed]

M. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

N. G. Walter, C. 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] [PubMed]

Opt. Commun.

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).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Opt. Express

Proc. Natl. Acad. Sci. USA

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).
[CrossRef] [PubMed]

A. Sharonov and R. M. Hochstrasser, "Wide-field subdiffraction imaging by accumulated binding of diffusing probes," Proc. Natl. Acad. Sci. USA 103, 18911-18916 (2006).
[CrossRef] [PubMed]

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

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. USA 101, 11298-11303 (2004).
[CrossRef] [PubMed]

M. P. Gordon, T. Ha, and P. R. Selvin, "Single-molecule high-resolution imaging with photobleaching," Proc. Natl. Acad. Sci. USA 101, 6462-6465 (2004).
[CrossRef] [PubMed]

T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, "Ultrahigh-resolution multicolor colocalization of single fluorescent probes," Proc. Natl. Acad. Sci. USA 97, 9461-9466 (2000).
[CrossRef] [PubMed]

A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, "Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes," Proc. Natl. Acad. Sci. USA 105, 3298-3303 (2008).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Proc. SPIE

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).
[CrossRef] [PubMed]

Science

S. Weiss, "Fluorescence spectroscopy of single bimolecular," Science 283, 1676-1683 (1999).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Other

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M. Pluto, Advanced light microscopy, vol. 1: principles and basic properties (Elsevier, Amsterdam, 1988).

S. Inoué, "Foundations of confocal scanned imaging in light microscopy," in "Handbook of Biological Confocal Microscopy," J. B. Pawley, ed. (Springer Science+Business Media, LLC, New York, 2006), 3rd ed.

C. R. Rao, Linear statistical inference and its applications (Wiley, New York, USA, 1965).

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, 64430D1-D7 (2007).

T. Sun and S. B. Andersson, "Precise 3-D localization of fluorescent probes without numerical fitting," in Proceedings of the International Conference of IEEE Engineering in Medicine and Biology Society (IEEE, 2007) pp. 4181-4184.

D. L. Snyder and M. I. Miller, Random point processes in time and space (Springer Verlag, New York, USA, 1991), 2nd Ed.

"EstimationTool," http://www4.utsouthwestern.edu/wardlab/estimationtool.

"FandPLimitTool," http://www4.utsouthwestern.edu/wardlab/fandplimittool.

S. Zacks, The theory of statistical inference (John Wiley and Sons, New York, USA, 1971).

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Figures (9)

Fig. 1.
Fig. 1.

Simulated images of a pair of like point sources in 3D space as acquired by (a) the SIM-SNG, (b) the SEP-SNG, (c) the SIM-MUM, and (d) the SEP-MUM imaging modalities. The point sources are deliberately simulated to be separated by a relatively large distance of 500 nm to clearly show the presence of a pair. The exposure time of each image is the same regardless of the modality. (a) The SIM-SNG modality abstracts the conventional fluorescence imaging setup which produces a single image of both point sources. (b) The SEP-SNG modality detects the two point sources separately in time and hence produces two images each capturing only one of the point sources. (c) The SIM-MUM modality uses two cameras to simultaneously detect the point source pair from two distinct focal planes, and hence produces two images each capturing both point sources. (d) The SEP-MUM modality combines separate detection with two-plane imaging and therefore produces a total of four images - two of one point source from the two distinct focal planes, and two of the other point source at a different time, but from the same two focal planes. In (c) and (d), the images are dimmer because the collected photons are split between the two focal planes.

Fig. 2.
Fig. 2.

Two point sources P 1 and P 2, located respectively at (x 01, y 01, z 01) and (x 02, y 02, z 02), in 3D space. In formulating the distance estimation problem, the 3D geometry of the pair is equivalently described by the six parameters d, ϕ, ω, sx , sy , sz . The parameter d is the distance that separates P 1 and P 2. The coordinates sx, sy, and sz specify the location of the midpoint between P 1 and P 2. The parameter ω is the angle between the line segment P 1 P 2 and the positive z-axis. The parameter ϕ is the angle between the xy-plane projection of P 1 P 2 and the positive x-axis.

Fig. 3.
Fig. 3.

Dependence of the 3D resolution measure (i.e., the best possible standard deviation for distance estimation) for (a) the SIM-SNG, (b) the SEP-SNG, (c) the SIM-MUM, and (d) the SEP-MUM modality on the distance of separation d and the axial position sz of the imaged point source pair. In (a), (b), (c), and (d), the image of each point source in the pair is assumed to be described by the 3D point spread function of Born and Wolf. Each point source emits photons of wavelength λ=655 nm, which are detected at a rate of Λ12=5000 photons per second. The image acquisition time is set to 1 second per image. The refractive index of the object space medium is set to n=1.515, and the numerical aperture and the magnification of the objective lens are set to na =1.4 and M=100, respectively. A single image consists of a 21×21 array of 13 µm by 13 µm pixels, and the position of the point source pair in the xy-plane is set to sx =sy =1365 nm, such that its image is centered on the pixel array given the 100-fold magnification. The point source pair is oriented such that it forms a 45° angle with the positive z-axis (ω=45°) and projects at a 60° angle from the positive x-axis in the xy-plane (ϕ=60°). In (c) and (d), the spacing between the two focal planes is set to Δzf =500 nm. The first focal plane corresponds to the focal plane of the modalities in (a) and (b), and is located at sz =0 nm with an associated magnification of 100. The second focal plane is located above the first at sz =500 nm, and its associated magnification is set to M′=97.98 (computed using Eq. (12) by assuming a standard tube length of L=160 mm). The collected photons are assumed to be split 50:50 between the two focal planes. In (a) and (b), the mean of the background noise in each image is assumed to be a constant and is set to β (k)=80 photons per pixel. In (c) and (d), however, it is set to β (k)=40 photons per pixel due to the equal splitting of the collected fluorescence. In (a), (b), (c), and (d), the mean and the standard deviation of the readout noise in each image are respectively set to ηk =0 e- and σk =8 e- per pixel.

Fig. 4.
Fig. 4.

Dependence of the 3D resolution measures for the SIM-SNG (*), the SEP-SNG (◦), the SIM-MUM (×), and the SEP-MUM (◇) modalities on (a) the distance of separation d of, and (b) the expected photon count detected from, the imaged point source pair. In (a), the curves shown correspond to the 2D slice sz =367.2 nm from each of the four 3D plots of Fig. 3. In (b), the resolution measures shown pertain to the point source pair with a distance of separation of d=50 nm from (a). The expected photon count per point source is varied from 1000 to 100000 photons. In both (a) and (b), all experimental, noise, and other point source pair parameters are as given in Fig. 3.

Fig. 5.
Fig. 5.

Dependence of the 3D resolution measure for (a) the SIM-SNG, (b) the SEP-SNG, (c) the SIM-MUM, and (d) the SEP-MUM modality on the distance of separation d and the orientation angle ω of the imaged point source pair. In (a), (b), (c), and (d), the point source pair is axially centered at sz =400 nm. Except for d and ω which are varied in these plots, all experimental, noise, and other point source pair parameters are as given in Fig. 3.

Fig. 6.
Fig. 6.

(a) Correlation of the 3D resolution measure (i.e., limit of the distance estimation accuracy) (*) for a point source pair with the limit of the z-localization accuracy of each of the point sources P 1 (◦) and P 2 (◇) in the pair. The resolution measure curve corresponds to the 2D slice d=200 nm from Fig. 3(b) (i.e., the SEP-SNG modality). (b) Dependence of the 3D resolution measures for the SIM-SNG (*), the SEP-SNG (◦), the SIM-MUM (×), and the SEP-MUM (◇) modalities on the axial position sz of the imaged point source pair. The curves shown correspond to the 2D slice d=200 nm from each of the four 3D plots of Fig. 3. The dashed vertical line at sz =0 nm marks the focal plane of the SIM-SNG and the SEP-SNG modalities, and focal plane 1 of the SIM-MUM and the SEP-MUM modalities. The dashed vertical line at sz =500 nm marks focal plane 2 of the SIM-MUM and the SEP-MUM modalities. In both (a) and (b), all experimental, noise, and other point source pair parameters are as given in Fig. 3.

Fig. 7.
Fig. 7.

Dependence of the 3D resolution measure for (a) the SIM-SNG, (b) the SEP-SNG, (c) the SIM-MUM, and (d) the SEP-MUM modality on the axial position sz and the orientation angleω of the imaged point source pair. In (a), (b), (c), and (d), the two point sources are separated by a distance of d=200 nm. Except for sz and ω which are varied in these plots, all experimental, noise, and other point source pair parameters are as given in Fig. 3.

Fig. 8.
Fig. 8.

Dependence of the 3D resolution measures for the SIM-SNG (*), the SEP-SNG (◦), the SIM-MUM (×), and the SEP-MUM (◇) modalities on the orientation angle ω of the imaged point source pair. The curves shown correspond to the 2D slice d=200 nm from each of the four 3D plots of Fig. 5. All experimental, noise, and other point source pair parameters are as described in Fig. 5.

Fig. 9.
Fig. 9.

Dependence of the 3D resolution measures for the SIM-SNG (*), the SEP-SNG (◦), the SIM-MUM (×), and the SEP-MUM (◇) modalities on (a) the distance of separation d, (b) the axial position sz , and (c) the orientation angle ω of the imaged point source pair. The plots in (a), (b), and (c) are analogous to the plots of Figs. 4(a), 6(b), and 8, respectively, but with the object space medium refractive index changed to n=1.33, and the numerical aperture and the magnification of the objective lens changed to na =1.2 and M=63, respectively. (In order that the image of the point source pair is still centered on the 21×21 array of 13 µm by 13 µm pixels given the new magnification, the position of the point source pair in the xy-plane has also been changed to sx =sy ≈2167 nm. Also, the magnification associated with focal plane 2 of the SIM-MUM and the SEP-MUM modalities is accordingly changed to M′=62.42.) All other experimental, noise, and point source pair parameters are as described in the corresponding Figs. 4(a), 6(b), and 8.

Tables (1)

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Table 1. Results of maximum likelihood estimations

Equations (15)

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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 , ( x , y ) 2 , z 0 ,
1 n ( ( θ z 1 , , z N p ) ) = Σ k = 1 N p ln ( P θ , k ( z k ) ) ,
Λ θ ( τ ) = Λ 1 ( τ ) + Λ 2 ( τ ) , τ t 0 ,
f θ , τ ( x , y ) = 1 M 2 [ ε 1 ( τ ) q z 01 , 1 ( x M x 01 y M y 01 ) + ε 2 ( τ ) q z 02 , 2 ( x M x 02 , y M y 02 ) ] ,
μ θ ( k ) = t 0 t C k Λ θ ( τ ) f θ , τ ( x , y ) dxdyd τ ,
I ( θ ) = Σ k = 1 N p ( μ θ ( k ) θ ) T μ θ ( k ) θ ( ( Σ l = 1 [ v θ ( k ) ] l 1 e v θ ( k ) ( l 1 ) ! · 1 2 π σ k e 1 2 ( z l η k σ k ) 2 ) 2 P θ , k ( z ) dz 1 ) ,
P θ , k ( z ) = 1 2 π σ k Σ l = 0 [ v θ ( k ) ] l e v θ ( k ) l ! e 1 2 ( z 1 η k σ k ) 2 , z .
Cov ( θ ̂ ) I 1 ( θ ) ,
μ θ ( 1 ) ( k ) = 1 M 2 t 0 t 1 C k Λ 1 ( τ ) · q z 01 , 1 ( x M x 01 , y M y 01 ) dxdyd τ ,
μ θ ( 2 ) ( k ) = 1 M 2 t 2 t 3 C k Λ 2 ( τ ) · q z 02 , 2 ( x M x 02 , y M y 02 ) dxdyd τ ,
I ( θ ) = I ( 1 ) ( θ ) + I ( 2 ) ( θ ) ,
M = M L L · M 2 · Δ z f n · L + M 2 · Δ z f L ,
f θ , τ ( x , y ) = 1 ( M′ ) 2 [ ε 1 ( τ ) q z 01 Δ z f , 1 ( x M′ x 01 , y M′ y 01 ) + ε 2 ( τ ) q z 02 Δ z f , 2 ( x M′ x 02 , y M′ y 02 ) ] ,
μ θ ( 1 ) ( k ) = 1 ( M′ ) 2 t 0 t 1 C k Λ 1 ( τ ) · q z 01 Δ z f , 1 ( x M′ x 01 , y M′ y 01 ) dxdyd τ
μ θ ( 2 ) ( k ) = 1 ( M′ ) 2 t 2 t 3 C k Λ 2 ( τ ) · q z 02 Δ z f , 2 ( x M′ x 02 , y M′ y 02 ) dxdyd τ ,

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