Abstract

The ability to track single fluorescent particles in three-dimensions with sub-diffraction limit precision as well as sub-millisecond temporal resolution has enabled the understanding of many biophysical phenomena at the nanometer scale. While there are several techniques for achieving this, most require complicated experimental setups that are expensive to implement. These methods can offer superb performance but their complexity may be overwhelming to the end-user whose aim is only to understand the feature being imaged. In this work, we describe a method for tracking a single fluorescent particle using a standard confocal or multi-photon microscope configuration. It relies only on the assumption that the relative position of the measurement point and the particle can be actuated and that the point spread function has a global maximum that coincides with the particle’s position. The method uses intensity feedback to calculate real-time position commands that “seek” the extremum of the point spread function as the particle moves through its environment. We demonstrate the method by tracking a diffusing quantum dot in a hydrogel on a standard epifluorescent confocal microscope.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (1)

T. T. Ashley and S. B. Andersson, “Method for simultaneous localization and parameter estimation in particle tracking experiments,” Phys. Rev. E 92, 052707 (2015).
[Crossref]

2014 (3)

L. Cognet, C. Leduc, and B. Lounis, “Advances in live-cell single-particle tracking and dynamic super-resolution imaging,” Curr. Opin. Chem. Biol. 20, 78–85 (2014).
[Crossref] [PubMed]

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

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]

2013 (1)

F. Lindsten and T. B. Schön, “Backward simulation methods for Monte Carlo statistical inference,” Found. Trends Mach. Learn. 6, 1–143 (2013).
[Crossref]

2012 (3)

J. J. Han, C. Kiss, A. R. M. Bradbury, and J. H. Werner, “Time-Resolved, Confocal Single-Molecule Tracking of Individual Organic Dyes and Fluorescent Proteins in Three Dimensions,” ACS Nano 6, 8922–8932 (2012).
[Crossref] [PubMed]

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D Single Molecule Tracking with Multifocal Plane Microscopy Reveals Rapid Intercellular Transferrin Transport at Epithelial Cell Barriers,” Biophys. J. 103, 1594–1603 (2012).
[Crossref] [PubMed]

Z. Shen and S. B. Andersson, “Optimal measurement constellation of the fluorobancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[Crossref]

2011 (3)

S. B. Andersson, “A nonlinear controller for three-dimensional tracking of a fluorescent particle in a confocal microscope,” Appl. Phys. B 104, 161–173 (2011).
[Crossref]

T. B. Schön, A. Wills, and B. Ninness, “System identification of nonlinear state-space models,” Automatica 47, 39–49 (2011).
[Crossref]

A. J. Nichols and C. L. Evans, “Video-rate scanning confocal microscopy and microendoscopy,” J. Vis. Exp. 56, e3252 (2011).

2010 (1)

M. F. Juette and J. Bewersdorf, “Three-Dimensional Tracking of Single Fluorescent Particles with Submillisecond Temporal Resolution,” Nano Lett. 10, 4657–4663 (2010).
[Crossref] [PubMed]

2009 (1)

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

2008 (3)

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] [PubMed]

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

S. B. Andersson, “Localization of a fluorescent source without numerical fitting,” Opt. Express 16, 18714–18724 (2008).
[PubMed]

2007 (4)

O. Cappe, S. J. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential Monte Carlo,” P. IEEE 95, 899–924 (2007).
[Crossref]

B. Brandenburg and X. Zhuang, “Virus trafficking – learning from single-virus tracking,” Nat. Rev. Microbiol. 5, 197–208 (2007).
[Crossref] [PubMed]

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[Crossref] [PubMed]

H. Cang, C. S. Xu, D. Montiel, and H. Yang, “Guiding a confocal microscope by single fluorescent nanoparticles,” Opt. Lett. 32, 2729–2732 (2007).
[Crossref] [PubMed]

2006 (1)

T. Ragan, H. Huang, P. So, and E. Gratton, “3D Particle Tracking on a Two-Photon Microscope,” J. Fluoresc. 16, 325–336 (2006).
[Crossref] [PubMed]

2003 (2)

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning fcs, a novel method for three-dimensional particle tracking,” Biochem. Soc. T. 31, 997–1000 (2003).
[Crossref]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

2002 (1)

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

1994 (1)

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. 64, 1291–1300 (1994).
[Crossref]

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Allan, D. B.

B. S. Schuster, L. M. Ensign, D. B. Allan, J. S. Suk, and J. Hanes, “Particle tracking in drug and gene delivery research: State-of-the-art applications and methods,” Adv. Drug Deliver. Rev. (2015).
[Crossref]

Andersson, S. B.

T. T. Ashley and S. B. Andersson, “Method for simultaneous localization and parameter estimation in particle tracking experiments,” Phys. Rev. E 92, 052707 (2015).
[Crossref]

Z. Shen and S. B. Andersson, “Optimal measurement constellation of the fluorobancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[Crossref]

S. B. Andersson, “A nonlinear controller for three-dimensional tracking of a fluorescent particle in a confocal microscope,” Appl. Phys. B 104, 161–173 (2011).
[Crossref]

S. B. Andersson, “Localization of a fluorescent source without numerical fitting,” Opt. Express 16, 18714–18724 (2008).
[PubMed]

Z. Shen and S. B. Andersson, “3-d tracking of fluorescent nanoparticles in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control and European Control Conference, pp. 5856–5861 (2011).

T. T. Ashley and S. B. Andersson, “A control law for seeking an extremum of a three-dimensional scalar potential field,” Proceedings of American Control Conference p. to appear (2016).

Ashley, T. T.

T. T. Ashley and S. B. Andersson, “Method for simultaneous localization and parameter estimation in particle tracking experiments,” Phys. Rev. E 92, 052707 (2015).
[Crossref]

T. T. Ashley and S. B. Andersson, “A control law for seeking an extremum of a three-dimensional scalar potential field,” Proceedings of American Control Conference p. to appear (2016).

Backer, A. S.

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]

Bennett, B. T.

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] [PubMed]

Berglund, A. J.

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[Crossref] [PubMed]

Bewersdorf, J.

M. F. Juette and J. Bewersdorf, “Three-Dimensional Tracking of Single Fluorescent Particles with Submillisecond Temporal Resolution,” Nano Lett. 10, 4657–4663 (2010).
[Crossref] [PubMed]

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] [PubMed]

Biteen, J. S.

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

Bradbury, A. R. M.

J. J. Han, C. Kiss, A. R. M. Bradbury, and J. H. Werner, “Time-Resolved, Confocal Single-Molecule Tracking of Individual Organic Dyes and Fluorescent Proteins in Three Dimensions,” ACS Nano 6, 8922–8932 (2012).
[Crossref] [PubMed]

Brandenburg, B.

B. Brandenburg and X. Zhuang, “Virus trafficking – learning from single-virus tracking,” Nat. Rev. Microbiol. 5, 197–208 (2007).
[Crossref] [PubMed]

Cang, H.

Cappe, O.

O. Cappe, S. J. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential Monte Carlo,” P. IEEE 95, 899–924 (2007).
[Crossref]

Cognet, L.

L. Cognet, C. Leduc, and B. Lounis, “Advances in live-cell single-particle tracking and dynamic super-resolution imaging,” Curr. Opin. Chem. Biol. 20, 78–85 (2014).
[Crossref] [PubMed]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Ensign, L. M.

B. S. Schuster, L. M. Ensign, D. B. Allan, J. S. Suk, and J. Hanes, “Particle tracking in drug and gene delivery research: State-of-the-art applications and methods,” Adv. Drug Deliver. Rev. (2015).
[Crossref]

Evans, C. L.

A. J. Nichols and C. L. Evans, “Video-rate scanning confocal microscopy and microendoscopy,” J. Vis. Exp. 56, e3252 (2011).

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Fujiwara, T.

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

Fujiwara, T. K.

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

Godsill, S. J.

O. Cappe, S. J. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential Monte Carlo,” P. IEEE 95, 899–924 (2007).
[Crossref]

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Gould, T. J.

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] [PubMed]

Gratton, E.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D Particle Tracking on a Two-Photon Microscope,” J. Fluoresc. 16, 325–336 (2006).
[Crossref] [PubMed]

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning fcs, a novel method for three-dimensional particle tracking,” Biochem. Soc. T. 31, 997–1000 (2003).
[Crossref]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Han, J. J.

J. J. Han, C. Kiss, A. R. M. Bradbury, and J. H. Werner, “Time-Resolved, Confocal Single-Molecule Tracking of Individual Organic Dyes and Fluorescent Proteins in Three Dimensions,” ACS Nano 6, 8922–8932 (2012).
[Crossref] [PubMed]

Hanes, J.

B. S. Schuster, L. M. Ensign, D. B. Allan, J. S. Suk, and J. Hanes, “Particle tracking in drug and gene delivery research: State-of-the-art applications and methods,” Adv. Drug Deliver. Rev. (2015).
[Crossref]

Hell, S. W.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

Hess, S. T.

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] [PubMed]

Hirosawa, K. M.

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

Huang, H.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D Particle Tracking on a Two-Photon Microscope,” J. Fluoresc. 16, 325–336 (2006).
[Crossref] [PubMed]

Jacobson, K.

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

Jahn, R.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

Juette, M. F.

M. F. Juette and J. Bewersdorf, “Three-Dimensional Tracking of Single Fluorescent Particles with Submillisecond Temporal Resolution,” Nano Lett. 10, 4657–4663 (2010).
[Crossref] [PubMed]

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] [PubMed]

Kamin, D.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (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. 64, 1291–1300 (1994).
[Crossref]

Kasai, R. S.

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

Kim, D.

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D Single Molecule Tracking with Multifocal Plane Microscopy Reveals Rapid Intercellular Transferrin Transport at Epithelial Cell Barriers,” Biophys. J. 103, 1594–1603 (2012).
[Crossref] [PubMed]

Kis-Petikova, K.

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning fcs, a novel method for three-dimensional particle tracking,” Biochem. Soc. T. 31, 997–1000 (2003).
[Crossref]

Kiss, C.

J. J. Han, C. Kiss, A. R. M. Bradbury, and J. H. Werner, “Time-Resolved, Confocal Single-Molecule Tracking of Individual Organic Dyes and Fluorescent Proteins in Three Dimensions,” ACS Nano 6, 8922–8932 (2012).
[Crossref] [PubMed]

Kusumi, A.

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Lauterbach, M. A.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

Leduc, C.

L. Cognet, C. Leduc, and B. Lounis, “Advances in live-cell single-particle tracking and dynamic super-resolution imaging,” Curr. Opin. Chem. Biol. 20, 78–85 (2014).
[Crossref] [PubMed]

Lessard, M. D.

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] [PubMed]

Levi, V.

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning fcs, a novel method for three-dimensional particle tracking,” Biochem. Soc. T. 31, 997–1000 (2003).
[Crossref]

Lindsten, F.

F. Lindsten and T. B. Schön, “Backward simulation methods for Monte Carlo statistical inference,” Found. Trends Mach. Learn. 6, 1–143 (2013).
[Crossref]

Liu, N.

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

Lord, S. J.

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

Lounis, B.

L. Cognet, C. Leduc, and B. Lounis, “Advances in live-cell single-particle tracking and dynamic super-resolution imaging,” Curr. Opin. Chem. Biol. 20, 78–85 (2014).
[Crossref] [PubMed]

Mabuchi, H.

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[Crossref] [PubMed]

McHale, K.

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[Crossref] [PubMed]

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Mlodzianoski, M. J.

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] [PubMed]

Moerner, W. E.

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]

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

Montiel, D.

Moulines, E.

O. Cappe, S. J. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential Monte Carlo,” P. IEEE 95, 899–924 (2007).
[Crossref]

Murakoshi, H.

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

Nagpure, B. S.

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] [PubMed]

Nichols, A. J.

A. J. Nichols and C. L. Evans, “Video-rate scanning confocal microscopy and microendoscopy,” J. Vis. Exp. 56, e3252 (2011).

Ninness, B.

T. B. Schön, A. Wills, and B. Ninness, “System identification of nonlinear state-space models,” Automatica 47, 39–49 (2011).
[Crossref]

Ober, R. J.

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D Single Molecule Tracking with Multifocal Plane Microscopy Reveals Rapid Intercellular Transferrin Transport at Epithelial Cell Barriers,” Biophys. J. 103, 1594–1603 (2012).
[Crossref] [PubMed]

Ortega, J. M.

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, (SIAM, 2000).
[Crossref]

Pavani, S. R. P.

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

Piestun, R.

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

Ragan, T.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D Particle Tracking on a Two-Photon Microscope,” J. Fluoresc. 16, 325–336 (2006).
[Crossref] [PubMed]

Ram, S.

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D Single Molecule Tracking with Multifocal Plane Microscopy Reveals Rapid Intercellular Transferrin Transport at Epithelial Cell Barriers,” Biophys. J. 103, 1594–1603 (2012).
[Crossref] [PubMed]

Rheinboldt, W. C.

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, (SIAM, 2000).
[Crossref]

Ritchie, K.

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

Rizzoli, S. O.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

Ruan, Q.

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning fcs, a novel method for three-dimensional particle tracking,” Biochem. Soc. T. 31, 997–1000 (2003).
[Crossref]

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Sahl, S. J.

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]

Schön, T. B.

F. Lindsten and T. B. Schön, “Backward simulation methods for Monte Carlo statistical inference,” Found. Trends Mach. Learn. 6, 1–143 (2013).
[Crossref]

T. B. Schön, A. Wills, and B. Ninness, “System identification of nonlinear state-space models,” Automatica 47, 39–49 (2011).
[Crossref]

Schuster, B. S.

B. S. Schuster, L. M. Ensign, D. B. Allan, J. S. Suk, and J. Hanes, “Particle tracking in drug and gene delivery research: State-of-the-art applications and methods,” Adv. Drug Deliver. Rev. (2015).
[Crossref]

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Shechtman, Y.

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]

Shen, Z.

Z. Shen and S. B. Andersson, “Optimal measurement constellation of the fluorobancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[Crossref]

Z. Shen and S. B. Andersson, “3-d tracking of fluorescent nanoparticles in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control and European Control Conference, pp. 5856–5861 (2011).

So, P.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D Particle Tracking on a Two-Photon Microscope,” J. Fluoresc. 16, 325–336 (2006).
[Crossref] [PubMed]

Suk, J. S.

B. S. Schuster, L. M. Ensign, D. B. Allan, J. S. Suk, and J. Hanes, “Particle tracking in drug and gene delivery research: State-of-the-art applications and methods,” Adv. Drug Deliver. Rev. (2015).
[Crossref]

Thompson, M. A.

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

Tsunoyama, T. A.

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

Twieg, R. J.

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

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. 64, 1291–1300 (1994).
[Crossref]

Ward, E. S.

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D Single Molecule Tracking with Multifocal Plane Microscopy Reveals Rapid Intercellular Transferrin Transport at Epithelial Cell Barriers,” Biophys. J. 103, 1594–1603 (2012).
[Crossref] [PubMed]

Werner, J. H.

J. J. Han, C. Kiss, A. R. M. Bradbury, and J. H. Werner, “Time-Resolved, Confocal Single-Molecule Tracking of Individual Organic Dyes and Fluorescent Proteins in Three Dimensions,” ACS Nano 6, 8922–8932 (2012).
[Crossref] [PubMed]

Westphal, V.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

Wills, A.

T. B. Schön, A. Wills, and B. Ninness, “System identification of nonlinear state-space models,” Automatica 47, 39–49 (2011).
[Crossref]

Xu, C. S.

Yang, H.

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Zhuang, X.

B. Brandenburg and X. Zhuang, “Virus trafficking – learning from single-virus tracking,” Nat. Rev. Microbiol. 5, 197–208 (2007).
[Crossref] [PubMed]

ACS Nano (1)

J. J. Han, C. Kiss, A. R. M. Bradbury, and J. H. Werner, “Time-Resolved, Confocal Single-Molecule Tracking of Individual Organic Dyes and Fluorescent Proteins in Three Dimensions,” ACS Nano 6, 8922–8932 (2012).
[Crossref] [PubMed]

Appl. Phys. B (1)

S. B. Andersson, “A nonlinear controller for three-dimensional tracking of a fluorescent particle in a confocal microscope,” Appl. Phys. B 104, 161–173 (2011).
[Crossref]

Automatica (1)

T. B. Schön, A. Wills, and B. Ninness, “System identification of nonlinear state-space models,” Automatica 47, 39–49 (2011).
[Crossref]

Biochem. Soc. T. (1)

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning fcs, a novel method for three-dimensional particle tracking,” Biochem. Soc. T. 31, 997–1000 (2003).
[Crossref]

Biophys. J. (2)

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. 64, 1291–1300 (1994).
[Crossref]

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D Single Molecule Tracking with Multifocal Plane Microscopy Reveals Rapid Intercellular Transferrin Transport at Epithelial Cell Barriers,” Biophys. J. 103, 1594–1603 (2012).
[Crossref] [PubMed]

Curr. Opin. Chem. Biol. (1)

L. Cognet, C. Leduc, and B. Lounis, “Advances in live-cell single-particle tracking and dynamic super-resolution imaging,” Curr. Opin. Chem. Biol. 20, 78–85 (2014).
[Crossref] [PubMed]

Found. Trends Mach. Learn. (1)

F. Lindsten and T. B. Schön, “Backward simulation methods for Monte Carlo statistical inference,” Found. Trends Mach. Learn. 6, 1–143 (2013).
[Crossref]

J. Cell Biol. (1)

T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, “Phospholipids undergo hop diffusion in compartmentalized cell membrane,” J. Cell Biol. 157, 1071–1082 (2002).
[Crossref] [PubMed]

J. Fluoresc. (1)

T. Ragan, H. Huang, P. So, and E. Gratton, “3D Particle Tracking on a Two-Photon Microscope,” J. Fluoresc. 16, 325–336 (2006).
[Crossref] [PubMed]

J. R. Stat. Soc. (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

J. Vis. Exp. (1)

A. J. Nichols and C. L. Evans, “Video-rate scanning confocal microscopy and microendoscopy,” J. Vis. Exp. 56, e3252 (2011).

Mechatronics (1)

Z. Shen and S. B. Andersson, “Optimal measurement constellation of the fluorobancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[Crossref]

Nano Lett. (2)

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[Crossref] [PubMed]

M. F. Juette and J. Bewersdorf, “Three-Dimensional Tracking of Single Fluorescent Particles with Submillisecond Temporal Resolution,” Nano Lett. 10, 4657–4663 (2010).
[Crossref] [PubMed]

Nat. Chem. Biol. (1)

A. Kusumi, T. A. Tsunoyama, K. M. Hirosawa, R. S. Kasai, and T. K. Fujiwara, “Tracking single molecules at work in living cells,” Nat. Chem. Biol. 10, 524–532 (2014).
[Crossref] [PubMed]

Nat. Methods (1)

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] [PubMed]

Nat. Rev. Microbiol. (1)

B. Brandenburg and X. Zhuang, “Virus trafficking – learning from single-virus tracking,” Nat. Rev. Microbiol. 5, 197–208 (2007).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

P. IEEE (1)

O. Cappe, S. J. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential Monte Carlo,” P. IEEE 95, 899–924 (2007).
[Crossref]

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

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

Phys. Rev. E (1)

T. T. Ashley and S. B. Andersson, “Method for simultaneous localization and parameter estimation in particle tracking experiments,” Phys. Rev. E 92, 052707 (2015).
[Crossref]

Phys. Rev. Lett. (1)

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]

Science (2)

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320, 246–249 (2008).
[Crossref] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[Crossref] [PubMed]

Other (5)

B. S. Schuster, L. M. Ensign, D. B. Allan, J. S. Suk, and J. Hanes, “Particle tracking in drug and gene delivery research: State-of-the-art applications and methods,” Adv. Drug Deliver. Rev. (2015).
[Crossref]

Z. Shen and S. B. Andersson, “3-d tracking of fluorescent nanoparticles in a confocal microscope,” in Proceedings of IEEE Conference on Decision and Control and European Control Conference, pp. 5856–5861 (2011).

T. T. Ashley and S. B. Andersson, “A control law for seeking an extremum of a three-dimensional scalar potential field,” Proceedings of American Control Conference p. to appear (2016).

T. T. Ashley and S. B. Andersson, SMC-EM. https://github.com/andersson-sean/SMC-EM

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, (SIAM, 2000).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Schematic illustrating the confocal microscope used in this work. A 488 nm diode laser is expanded and collimated to overfill the back aperture of the objective lens. The beam reflects off the dichroic and passes through the objective lens which focuses it onto the fluorescent sample. A three-axis piezoelectric nanopositioner displaces the sample relative to the fixed beam. The resulting fluorescence is imaged by the objective, passed through the dichroic, and focused onto a 75 μm-diameter pinhole and avalanche photodiode. To assist with diagnostics, a pellicle beam splitter divided one-third of the fluorescence onto a CCD camera; the CCD was used for finding regions of interest and was not used during tracking. The control algorithm was implemented on the FPGA of a real-time embedded controller which sampled the photodiode pulses and calculated commands for the nanopositioner.
Fig. 2
Fig. 2 Three-dimensional PSF measurements (left) and the corresponding Gaussian model (right), calculated by a nonlinear least-squares fit. Three planes (magenta, orange, and green) are shown as cross sections in Fig. 3. These planes were determined by the ZYX rotation in Eq. (4.1) with the three rotations ( ψ ^ x, ψ ^ y, ψ ^ z) given by the least-squares fit. The intensity values in the measured PSF are normalized by the maximum measured intensity value, and the intensity values in the model PSF are normalized by the peak intensity value calculated by least-squares fit.
Fig. 3
Fig. 3 Two-dimensional point spread function measurements (top row) and their corresponding Gaussian models (bottom row) through the three planes depicted in Fig. 2.
Fig. 4
Fig. 4 The inferred three-dimensional position of a quantum dot (blue) diffusing in a hydrogel relative to the position of the focal volume (black) which followed the particle in real-time using the extremum seeking method described in this work. The particle position was inferred by employing the SMC-EM algorithm. The left three plots show the (top) x, (middle) y, and (bottom) z time-series while the right three plots show the (top) measured intensity, (middle) estimated intensity, and (bottom) residual intensity, given by the difference between the measured and estimated intensity values. The mean of the residual, shown as a solid blue line, was −1.24 counts / ms.
Fig. 5
Fig. 5 The inferred three-dimensional trajectory of a quantum dot diffusing in a hydrogel as parametrized by time and subsampled from the estimated trajectory by a factor of five. Three planar cross-sections are also shown. The quantum dot was tracked in a confocal microscope using the method presented in this work; the resulting particle position was inferred using the SMC-EM algorithm.
Fig. 6
Fig. 6 The inferred three-dimensional position of a quantum dot (blue) diffusing in a hydrogel relative to the position of the focal volume (black) which followed the particle in real-time using the extremum seeking method described in this work. The particle position was inferred by employing the SMC-EM algorithm. The left three plots show the (top) x, (middle) y, and (bottom) z time-series while the right three plots show the (top) measured intensity, (middle) estimated intensity, and (bottom) residual intensity, given by the difference between the measured and estimated intensity values. The mean of the residual, shown as a solid blue line, was 2.97 counts / ms.
Fig. 7
Fig. 7 The inferred three-dimensional trajectory of a quantum dot diffusing in a hydrogel as parametrized by time and subsampled from the estimated trajectory by a factor of two. Three planar cross-sections are also shown. The quantum dot was tracked in a confocal microscope using the method presented in this work; the resulting particle position was inferred using the SMC-EM algorithm.
Fig. 8
Fig. 8 The inferred three-dimensional position of a quantum dot (blue) diffusing in a hydrogel relative to the position of the focal volume (black) which followed the particle in real-time using the extremum seeking method described in this work. The long tracking run was broken into 20 s segments, the particle position was inferred on each segment using the SMC-EM algorithm, and then the segments stiched together. The left three plots show the (top) x, (middle) y, and (bottom) z time-series while the right three plots show the (top) measured intensity, (middle) estimated intensity, and (bottom) residual intensity, given by the difference between the measured and estimated intensity values. The mean of the residual, shown as a solid blue line, was 2.97 counts / ms.
Fig. 9
Fig. 9 Four sections of the inferred three-dimensional trajectory of a quantum dot diffusing in a hydrogel as parametrized by time and subsampled from the estimated trajectory by a factor of four. Three planar cross-sections are also shown. The quantum dot was tracked in a confocal microscope using the method presented in this work; the resulting particle position was inferred using the SMC-EM algorithm.
Fig. 10
Fig. 10 SMC-EM estimates of (left) diffusion coefficients and (right) velocities for each of the 20 s segments of the run shown in Figs. 8. The overall motion parameters were (0.0082 ± 0.0014; 0.0082 ± 0.0015; 0.0134 ± 0.0019) μm2/s for the diffusion and (0.006 ± 0.0342, −0.0027 ± 0.0308, 0.0005 ± 0.0401) μ/s for the velocities in x, y, and z, respectively.

Equations (46)

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

x ˙ s = R ( ω 1 sin θ cos ϕ + ω 2 cos θ sin ϕ ) ,
y ˙ s = R ( ω 1 sin θ sin ϕ ω 2 cos θ cos ϕ ) ,
z ˙ s = ω 1 R cos θ ,
θ ˙ = ω 1 ( 1 K p d I d t ) , ϕ ˙ = ω 2 ,
Δ x ( t ) = R cos ( ω 1 t + α ¯ ) cos ( ω 2 t + β ¯ ) ,
Δ y ( t ) = R cos ( ω 1 t + α ¯ ) sin ( ω 2 t + β ¯ ) ,
Δ z ( t ) = R sin ( ω 1 t + α ¯ ) ,
x k + 1 = x k + Δ t R ( ω 1 sin θ k cos ϕ k + ω 2 cos θ k sin ϕ k ) ,
y k + 1 = y k Δ t R ( ω 1 sin θ k sin ϕ k ω 2 cos θ k cos ϕ k ) ,
z k + 1 = z k + Δ t R ( ω 1 cos θ k ) ,
θ k + 1 = θ k + ω 1 Δ t K p ω 1 ( I k I k 1 ) , ϕ k + 1 = ϕ k + ω 2 Δ t .
θ ^ ML = arg min θ log p θ ( I 1 : N , x s , 1 : N ) ,
x p , k + 1 p θ Mot ( x p , k + 1 | x p , k ) ,
I k p θ Obs ( I k | x p , k ) ,
x p , 1 p θ Init ( x p , 1 ) ,
p V x , D x ( x p , k + 1 | x p , k ) = N [ x p , k + 1 ] ( x p , k + V x Δ t , 2 D x Δ t ) ,
p G ( I k | x p , k ) = P [ I k ] ( G F P S F ( x s , k , x p , k ) + N bgd ) ,
p μ , ( x p , 1 ) = N [ x p , 1 ] ( μ , ) .
θ ^ = arg max θ log p θ ( I 1 , , I N )
Q ( θ , θ ^ e ) log [ p θ ( x p , 1 : N , I 1 : N ) ] p θ ^ e ( x p , 1 : N | I 1 : N ) d x p , 1 : N .
Q ( θ , θ ^ e ) = Q 1 + Q 2 + Q 3 ,
Q 1 log [ p θ Init ( x p , 1 : ) ] p θ ^ e ( x p , 1 | I 1 : N ) d x p , 1 ,
Q 2 k = 1 N 1 log [ p θ Mot ( x p , k + 1 | x p , k ) ] p θ ^ e ( x p , k + 1 , x p , k | I 1 : N ) d x p , k + 1 ,
Q 3 k = 1 N log [ p θ Obs ( I k | x p , k ) ] p θ ^ e ( x p , k | I 1 : N ) d x p , k .
θ ^ e + 1 = arg max θ Q ( θ , θ ^ e ) .
p θ ^ e ( x p , k | I 1 : N ) i = 1 M w k | N , e i δ ( x p , k x k | N , e i ) ,
p θ ^ e ( x p , k , x p , k + 1 | I 1 : N ) i = 1 M j = 1 M w k | N , e i j δ ( x p , k x k | N , e i , x p , k + 1 x k + 1 | N , e j ) ,
Q ( θ , θ ^ e ) Q ^ ( θ , θ ^ e ) = Q ^ 1 + Q ^ 2 + Q ^ 3 ,
Q ^ 1 i = 1 M w 1 | N , e i log [ p θ Init ( x 1 | N , e i ) ] ,
Q ^ 2 k = 1 N 1 i = 1 M j = 1 M w k | N , e i j log [ p θ Mot ( x k + 1 | N , e i | x k | N , e i ) ] ,
Q ^ 3 k = 1 N i = 1 M w k | N , e i log [ p θ Obs ( I k | x k | N , e i ) ] ,
x ˜ 1 | 1 , e i p θ ^ e Init ( x p , 1 ) .
x ˜ k | k , e i p θ ^ e Mot ( x p , k | x k 1 | k 1 , e i ) ,
w ˜ k | k , e i p θ ^ e Obs ( I k | x ˜ k | k , e i ) ,
( x k | k , e j = x ˜ k | k , e i ) = w ˜ k | k , e i ,
w k | N , e i = w k | k , e i m = 1 M w k + 1 | N , e m p θ ^ e Mot ( x k + 1 | k + 1 , e m | x k | k , e i ) v k m ,
v k m j = 1 M w k | k , e j p θ ^ e Mot ( x k + 1 | k + 1 , e m | x k | k , e j ) ,
w k | N , e i j = w k | k , e i w k + 1 | N , e j p θ ^ e Mot ( x k + 1 | k + 1 , e j | x k | k , e i ) l = 1 M w k | k , e l p θ ^ e Mot ( x k + 1 | k + 1 , e j | x k | k , e l ) .
θ ^ e + 1 = arg max θ Q ^ ( θ , θ ^ e ) .
μ ^ x , e + 1 = i = 1 M w 1 | N , e i x 1 | N , e i , σ ^ x , e + 1 2 = i = 1 M w 1 | N , e i ( x 1 | N , e i μ ^ x , e + 1 ) 2 ,
V ^ x , e + 1 = 1 N Δ t k = 1 N 1 i = 1 M j = 1 M w k | N , e i j ( x k + 1 | N , e j x k | N , e i ) ,
D ^ x , e + 1 = 1 2 N Δ t k = 1 N 1 i = 1 M j = 1 M w k | N , e i j ( x k + 1 | N , e j x k | N , e i V ^ x , e + 1 Δ t ) 2 ,
k = 1 N i = 1 M w k | N , e i λ k , e i ( I k G ^ e + 1 λ k , e i + N bgd 1 ) = 0 ,
λ k , e i F PSF ( x s , k , x k | N , e i ) .
F PSF ( x , x c ) = exp ( 1 2 ( x , x c ) T R ( ψ ) T 1 R ( ψ ) ( x x c ) ) ,
R ( ψ ) = [ cos ψ z sin ψ z 0 sin ψ z cos ψ z 0 0 0 1 ] [ cos ψ y 0 sin ψ y 0 1 0 sin ψ y 0 cos ψ y ] [ 1 0 0 0 cos ψ x sin ψ x 0 sin ψ x cos ψ x ] ,

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