Abstract

We introduce a method for determining the position and orientation of fixed dipole emitters based on a combination of polarimetry and spot shape detection. A key element is an effective Point Spread Function model based on Hermite functions. The model offers a good description of the shape variations with dipole orientation and polarization detection channel, and provides computational advantages over the exact vectorial description of dipole image formation. The realized localization uncertainty is comparable to the free dipole case in which spots are rotationally symmetric and can be well modeled with a Gaussian. This result holds for all dipole orientations, for all practical signal levels, and for defocus values within the depth of focus, implying that the massive localization bias for defocused emitters with tilted dipole axis found with Gaussian spot fitting is eliminated.

© 2012 OSA

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  1. 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, 1643–1645 (2006).
    [CrossRef]
  2. 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]
  3. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
    [CrossRef] [PubMed]
  4. J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
    [CrossRef] [PubMed]
  5. 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. Engl. 476172–6176 (2008).
    [CrossRef]
  6. S. Stallinga and B. Rieger, “Accuracy of the Gaussian Point Spread Function model in 2D localization microscopy,” Opt. Express 18, 24461–24476 (2010).
    [CrossRef] [PubMed]
  7. J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
    [CrossRef] [PubMed]
  8. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006).
    [CrossRef] [PubMed]
  9. T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
    [CrossRef] [PubMed]
  10. S. R. P. Pavani, J. G. DeLuca, and R. Piestun, “Polarization sensitive, three-dimensional, single-molecule imaging of cells with a double-helix system,” Opt. Express 17, 19644–19655 (2009).
    [CrossRef] [PubMed]
  11. M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008).
    [CrossRef] [PubMed]
  12. M. R. Foreman and P. Török, “Fundamental limits in single-molecule orientation measurements,” New J. Phys. 13, 093013 (2011).
    [CrossRef]
  13. R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of All Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982).
    [CrossRef]
  14. A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999).
    [CrossRef]
  15. P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, “Orientational effects in the excitation and de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express 15, 3372–3383 (2007).
    [CrossRef] [PubMed]
  16. F. Aguet, A. 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).
    [CrossRef] [PubMed]
  17. 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] [PubMed]
  18. T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
    [CrossRef]
  19. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small di-electric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
    [CrossRef]
  20. 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] [PubMed]
  21. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” 2nd ed. (Cambridge Univeristy Press, 1992).
  23. E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
    [CrossRef] [PubMed]
  24. 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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
    [CrossRef] [PubMed]
  25. L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, (053902), 1–3 (2007).
    [CrossRef]
  26. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
    [CrossRef] [PubMed]
  27. 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] [PubMed]

2011

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

2010

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] [PubMed]

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

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] [PubMed]

S. Stallinga and B. Rieger, “Accuracy of the Gaussian Point Spread Function model in 2D localization microscopy,” Opt. Express 18, 24461–24476 (2010).
[CrossRef] [PubMed]

2009

2008

M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008).
[CrossRef] [PubMed]

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] [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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

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. Engl. 476172–6176 (2008).
[CrossRef]

2007

L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, (053902), 1–3 (2007).
[CrossRef]

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[CrossRef] [PubMed]

P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, “Orientational effects in the excitation and de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express 15, 3372–3383 (2007).
[CrossRef] [PubMed]

2006

J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (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, 1643–1645 (2006).
[CrossRef]

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

2005

1999

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999).
[CrossRef]

1998

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small di-electric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

1997

T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

1986

1982

R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of All Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

Aguet, F.

Azzam, R. M. A.

R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of All Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

Balci, H.

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[CrossRef] [PubMed]

Bartko, A. P.

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999).
[CrossRef]

Bates, M.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

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

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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (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, 1643–1645 (2006).
[CrossRef]

Bewersdorf, 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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

Blehm, B. H.

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[CrossRef] [PubMed]

Bock, H.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (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, 1643–1645 (2006).
[CrossRef]

Bossi, M.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Churchman, L. S.

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] [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, 1643–1645 (2006).
[CrossRef]

Dedecker, P.

DeLuca, J. G.

Dickson, R. M.

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999).
[CrossRef]

Eggeling, C.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Enderlein, J.

Engelhardt, J.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” 2nd ed. (Cambridge Univeristy Press, 1992).

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).
[CrossRef] [PubMed]

Fölling, J.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Foreman, M. R.

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

M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008).
[CrossRef] [PubMed]

Geissbühler, A.

Gosse, J. A.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

Gould, T. J.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

Gudheti, M. V.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

Gunewardene, M. S.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

Heilemann, M.

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. Engl. 476172–6176 (2008).
[CrossRef]

Hein, B.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Heintzmann, R.

Hell, S. W

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Hell, S. W.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

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, 1643–1645 (2006).
[CrossRef]

Hess, S. T.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small di-electric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Hofkens, J.

Holtzer, L.

L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, (053902), 1–3 (2007).
[CrossRef]

Hotta, J.

Hoyer, P.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

Huang, B.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

Jakobs, S.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Joseph, N.

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] [PubMed]

Jovin, T. M.

Juette, M. F.

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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

Juskaitis, R.

T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Kasper, R.

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. Engl. 476172–6176 (2008).
[CrossRef]

Keller, J.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

Lasser, T.

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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

Lidke, K. A.

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] [PubMed]

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]

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, 1643–1645 (2006).
[CrossRef]

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, 1643–1645 (2006).
[CrossRef]

Märki, I.

Meckel, T.

L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, (053902), 1–3 (2007).
[CrossRef]

Medda, R.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

Mortensen, K. I.

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] [PubMed]

Mukherjee, A.

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. Engl. 476172–6176 (2008).
[CrossRef]

Muls, B.

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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

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, 1643–1645 (2006).
[CrossRef]

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, 1643–1645 (2006).
[CrossRef]

Pavani, S. R. P.

Piestun, R.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” 2nd ed. (Cambridge Univeristy Press, 1992).

Reuss, M.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

Rieger, B.

Romero, C. M.

Rust, M. J.

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

Sauer, M.

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. Engl. 476172–6176 (2008).
[CrossRef]

Schmidt, T.

L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90, (053902), 1–3 (2007).
[CrossRef]

Schüttpelz, M.

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. Engl. 476172–6176 (2008).
[CrossRef]

Seefeldt, B.

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. Engl. 476172–6176 (2008).
[CrossRef]

Selvin, P. R.

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[CrossRef] [PubMed]

J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006).
[CrossRef] [PubMed]

Smith, C. S.

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] [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, 1643–1645 (2006).
[CrossRef]

Spudich, J. A.

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] [PubMed]

Stallinga, S.

Staudt, T.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” 2nd ed. (Cambridge Univeristy Press, 1992).

Tinnefeld, P.

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. Engl. 476172–6176 (2008).
[CrossRef]

Toprak, E.

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[CrossRef] [PubMed]

J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express 14, 8111 (2006).
[CrossRef] [PubMed]

Török, P.

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

M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020–1022 (2008).
[CrossRef] [PubMed]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small di-electric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

Unser, M.

van de Linde, S.

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. Engl. 476172–6176 (2008).
[CrossRef]

Verkhusha, V. V.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” 2nd ed. (Cambridge Univeristy Press, 1992).

Wang, W.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

Wilson, T.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small di-electric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Wurm, C. A.

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

Yin, S.-R.

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

Zauderer, E.

Zhuang, X.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

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

Angew. Chem., Int. Ed. Engl.

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. Engl. 476172–6176 (2008).
[CrossRef]

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), 1–3 (2007).
[CrossRef]

J. Mod. Opt.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small di-electric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. Chem. B

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237–11241 (1999).
[CrossRef]

Nano Lett.

J. Engelhardt, J. Keller, P. Hoyer, M. Reuss, T. Staudt, and S. W. Hell, “Molecular orientation affects localization accuracy in superresolution far-field fluorescence microscopy,” Nano Lett. 11, 209–213 (2010).
[CrossRef] [PubMed]

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[CrossRef] [PubMed]

Nat. Methods

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-100nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–530 (2008).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

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

J. Fölling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W Hell, “Fluorescence nanoscopy by ground-state depletion and single-molecule return,” Nat. Methods 5, 943–945 (2008).
[CrossRef] [PubMed]

T. J. Gould, M. S. Gunewardene, M. V. Gudheti, V. V. Verkhusha, S.-R. Yin, J. A. Gosse, and S. T. Hess, “Nanoscale imaging of positions and anisotropies,” Nat. Methods 5, 1027–1031, 2008.
[CrossRef] [PubMed]

New J. Phys.

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

Opt. Acta

R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of All Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

Opt. Commun.

T. Wilson, R. Juskaitis, and P. D. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Science

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, 1643–1645 (2006).
[CrossRef]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77,” 2nd ed. (Cambridge Univeristy Press, 1992).

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Figures (10)

Fig. 1
Fig. 1

Cross-sections of the PSF for a tilted dipole with defocus equal to the diffraction limit (left, (a)), and scatter plot of simulated localizations for a fixed dipole emitter with tilted dipole axis with defocus equal to the diffraction limit (boiling down to an axial object displacement of about 0.15 μm for the parameters assumed, NAob = 1.25, and λ = 500 nm) using a Gaussian PSF (right, (b)). The found positions, the 1 × σ confidence level and the Gaussian Cramer-Rao Lower Bound (CRLB) are plotted. The localization was done using Maximum Likelihood Estimation (MLE) with a Gaussian PSF, taking 500 detected signal photons, and 25 background photons, the photons distributed over an 11×11 pixel large Region Of Interest (ROI) according to Poisson statistics.

Fig. 2
Fig. 2

Schematic view of a light path for simultaneous acquisition of four polarization channels on a single camera based on the Azzam polarimeter architecture. Light from emitter E is captured by lenses L1 and imaged by lens L2 onto the camera. In the imaging branch the beam is split into two equal parts with a normal beam splitter BS. The two branches are color coded for the sake of clarity, it does not indicate wavelength. The ‘bottom’ (red) branch passes through a polarization rotating (over π/4) component R. Both the ‘top’ (blue) and ‘bottom’ (red) branches pass a polarizing beam splitter (PBS) splitting each branch into two orthogonally polarized parts. The horizontal (x) and vertical (y) polarized beams of both branches are directed towards the camera (EMCCD) via three mirrors M1, M2, and M3. The four channels Ix, Iy, Ix, and Iy are projected onto the four quadrants of the camera, thus generating four images of the emitter E.

Fig. 3
Fig. 3

The polarization PSFs in the four detection channels of the Azzam polarimeter architecture for horizontal dipoles with azimuthal angle π/4 (top row), tilted dipoles with azimuthal angle π/4 (middle row), and vertical dipoles (bottom row) dipoles for a field of view of size 2.2×λ/NA, a water objective (NA = 1.25 in a medium with nmed = 1.33), and diffraction limited defocus (72 mλ rms).

Fig. 4
Fig. 4

Cross-sections of realistic vectorial PSFs for diffraction limited defocus (72 mλ rms) and Hermite PSF model fits in the four polarization detection channels, arranged by rows, for three different dipole orientations (horizontal, tilted, vertical), arranged by columns. The horizontal and tilted dipoles have azimuthal angle π/4. Note that the vertical axes in the figures are adapted to the peak height of each image so as to make the spot shapes better visible. The x and y cross-sections for the two rotated polarization channels overlap exactly, thus making only two of the four curves visible in figures (g) to (i) (the red curve overlaps the blue curve, the green curve overlaps the magenta curve).

Fig. 5
Fig. 5

Scatter plots of localizations for in focus images (left column) and for images with diffraction limited defocus (right column), for horizontal dipoles (top row), tilted dipoles (middle row), and vertical dipoles (bottom row).

Fig. 6
Fig. 6

Scatter plots of localizations for images of fixed dipole emitters with random dipole orientation in focus (left) and with diffraction limited defocus (right).

Fig. 7
Fig. 7

(a) Plot of the found ratio Dhor/Dver as a function of the ground truth ratio dhor/dver for a set of random fixed dipoles, and for a water immersion objective with NA = 1.25, and a linear fit through the data. (b) Plot of the found coefficient of linearity as a function of the average value of the squared asymmetry parameter for a water immersion objective (NA = 1.25) and an oil immersion objective (NA = 1.45).

Fig. 8
Fig. 8

Scatter plots of the variations of the found orientation around the ground truth orientation in the polar direction (Qp) and in the azimuthal direction (Qs) for in focus images (left column) and for images with diffraction limited defocus (right column), for horizontal dipoles (top row), tilted dipoles (middle row), and vertical dipoles (bottom row).

Fig. 9
Fig. 9

Scatter plots of the variations of the found orientation around the ground truth orientation for images of fixed dipole emitters with random dipole orientation in focus (left) and with diffraction limited defocus (right).

Fig. 10
Fig. 10

Plots of the rms localization error (RMSLE, top row), the rms orientational error in the s-direction (RMSOEs, middle row), and the rms orientational error (RMSOEp, bottom row) as a function of the signal photon count.

Equations (93)

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

E im , j ( u , u d ) = E 0 Σ k = x , y , z w j k ( u u d ) d k ,
E im , x ( u , ψ ) = E 0 [ F 0 ( u ) d x + i F 1 ( u ) cos ψ d z ] ,
E im , y ( u , ψ ) = E 0 [ F 0 ( u ) d y + i F 1 ( u ) sin ψ d z ] ,
E im , x ( u , ψ ) = E 0 [ F 0 ( u ) d x + i F 1 ( u ) cos ( ψ χ ) d z ] ,
E im , y ( u , ψ ) = E 0 [ F 0 ( u ) d y + i F 1 ( u ) sin ( ψ χ ) d z ] ,
d x = cos χ d x + sin χ d y ,
d y = sin χ d x + cos χ d y .
P S F x ( u , ψ ) = E 0 2 [ | F 0 ( u ) | 2 d x 2 + 2 Im { F 0 ( u ) F 1 ( u ) * } cos ( ψ χ ) d x d z + | F 1 ( u ) | 2 cos 2 ( ψ χ ) d z 2 ] ,
P S F y ( u , ψ ) = E 0 2 [ | F 0 ( u ) | 2 d y 2 + 2 Im { F 0 ( u ) F 1 ( u ) * } sin ( ψ χ ) d y d z + | F 1 ( u ) | 2 sin 2 ( ψ χ ) d z 2 ] ,
ψ n ( u ) = H n ( u ) exp ( u 2 ) ,
P S F x ( x , y ) = A 2 π σ 2 ( d x 2 + 1 2 β d z 2 ) Ψ 00 ( x , y ) + b / a 2 + A α β 2 π σ 2 d x d z ( cos χ Ψ 10 ( x , y ) + sin χ Ψ 01 ( x , y ) ) + A β 8 π σ 2 d z 2 ( cos 2 χ Ψ 20 ( x , y ) + 2 sin χ cos χ Ψ 11 ( x , y ) + sin 2 χ Ψ 02 ( x , y ) ) ,
P S F y ( x , y ) = A 2 π σ 2 ( d y 2 + 1 2 β d z 2 ) Ψ 00 ( x , y ) + b / a 2 + A α β 2 π σ 2 d y d z ( sin χ Ψ 10 ( x , y ) + cos χ Ψ 01 ( x , y ) ) + A β 8 π σ 2 d z 2 ( sin 2 χ Ψ 20 ( x , y ) 2 sin χ cos χ Ψ 11 ( x , y ) + cos 2 χ Ψ 02 ( x , y ) ) ,
Ψ lm ( x , y ) = ψ l ( x x 0 2 σ ) ψ m ( y y 0 2 σ ) ,
N x = d x d y ( P S F x ( x , y ) b / a 2 ) = A ( d x 2 + 1 2 β d z 2 ) ,
N y = d x d y ( P S F y ( x , y ) b / a 2 ) = A ( d y 2 + 1 2 β d z 2 ) ,
( D x , D y , D z ) = A ( d x , d y , β d z ) .
N = D x 2 + D y 2 + D z 2 ,
( d x , d y , d z ) = 1 D x 2 + D y 2 + D z 2 / β ( D x , D y , D z / β ) .
Q p = p 0 d ,
Q s = s 0 d ,
μ k l = x k a / 2 x k + a / 2 y k a / 2 y k + a / 2 d x d y P S F l ( x , y ) .
d ψ m ( u ) d u = ( d H m ( u ) d u 2 u H m ( u ) ) e u 2 = ( 2 m H m 1 ( u ) 2 u H m ( u ) ) e u 2 = H m + 1 ( u ) e u 2 = ψ m + 1 ( u ) .
μ k x = 1 π [ ( D x 2 + 1 2 D z 2 ) Σ 00 ( x k , y k ) + α D x D z ( cos χ Σ 10 ( x k , y k ) + sin χ Σ 01 ( x k , y k ) ) + 1 4 D z 2 ( cos 2 χ Σ 20 ( x k , y k ) + 2 sin χ cos χ Σ 11 ( x k , y k ) + sin 2 χ Σ 02 ( x k , y k ) ) ] + b ,
μ k y = 1 π [ ( D y 2 + 1 2 D z 2 ) Σ 00 ( x k , y k ) + α D y D z ( sin χ Σ 10 ( x k , y k ) + cos χ Σ 01 ( x k , y k ) ) + 1 4 D z 2 ( sin 2 χ Σ 20 ( x k , y k ) 2 sin χ cos χ Σ 11 ( x k , y k ) + cos 2 χ Σ 02 ( x k , y k ) ) ] + b ,
Σ lm ( x k , y k ) = Δ ψ l 1 ( x k x 0 2 σ ) Δ ψ m 1 ( y k y 0 2 σ ) ,
Δ f ( u ) = f ( u + a 2 2 σ ) f ( u a 2 2 σ ) ,
ψ 1 ( u ) = π 2 erf ( u ) .
log L = l k [ n k l log μ k l μ k l log ( n k l ! ) ] .
G i = log L θ i = l k ( n k l μ k l 1 ) μ k l θ i ,
H i j = 2 log L θ i θ j = l k n k l μ k l 2 μ k l θ i μ k l θ j + l k ( n k l μ k l 1 ) 2 μ k l θ i θ j .
H i j s = l k n k l μ k l 2 μ k l θ i μ k l θ j .
θ θ [ H s + λ diag ( H s ) ] 1 G .
F i j = l k 1 μ k l μ k l θ i μ k l θ j .
w j k ( u ) = 1 π d 2 v C ( v ) q j k ( v ) ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 exp ( 2 π i u v ) ,
q x k = cos ϕ T p p k sin ϕ T s s k ,
q y k = sin ϕ T p p k + cos ϕ T s s k ,
w x x ( u , ψ ) = F 0 ( u ) + F 2 ( u ) cos ( 2 ψ ) ,
w x y ( u , ψ ) = F 2 ( u ) cos ( 2 ψ ) ,
w x z ( u , ψ ) = i F 1 ( u ) cos ψ ,
w y x ( u , ψ ) = F 2 ( u ) cos ( 2 ψ ) ,
w y y ( u , ψ ) = F 0 ( u ) F 2 ( u ) cos ( 2 ψ ) ,
w y z ( u , ψ ) = i F 1 ( u ) sin ψ ,
F 0 ( u ) = 2 0 1 d v v ( T s + T p 1 v 2 NA ob 2 / n med 2 ) 2 ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 J 0 ( 2 π u v ) exp ( i W ( v ) ) ,
F 1 ( u ) = 2 0 1 d v v 2 T p NA ob / n med ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 J 1 ( 2 π u v ) exp ( i W ( v ) ) ,
F 2 ( u ) = 2 0 1 d v v ( T s T p 1 v 2 NA ob 2 / n med 2 ) 2 ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 J 2 ( 2 π u v ) exp ( i W ( v ) ) .
μ k x x 0 = 1 2 π σ [ ( D x 2 + 1 2 D z 2 ) Σ 10 + α D x D z ( cos χ Σ 20 + sin χ Σ 11 ) + 1 4 D z 2 ( cos 2 χ Σ 30 + 2 sin χ cos χ Σ 21 + sin 2 χ Σ 12 ) ] ,
μ k y x 0 = 1 2 π σ [ ( D y 2 + 1 2 D z 2 ) Σ 10 + α D y D z ( sin χ Σ 20 + cos χ Σ 11 ) + 1 4 D z 2 ( sin 2 χ Σ 30 2 sin χ cos χ Σ 21 + cos 2 χ Σ 12 ) ] ,
μ k x y 0 = 1 2 π σ [ ( D x 2 + 1 2 D z 2 ) Σ 01 + α D x D z ( cos χ Σ 11 + sin χ Σ 02 ) + 1 4 D z 2 ( cos 2 χ Σ 21 + 2 sin χ cos χ Σ 12 + sin 2 χ Σ 03 ) ] ,
μ k y y 0 = 1 2 π σ [ ( D y 2 + 1 2 D z 2 ) Σ 01 + α D y D z ( sin χ Σ 11 + cos χ Σ 02 ) + 1 4 D z 2 ( sin 2 χ Σ 21 2 sin χ cos χ Σ 12 + cos 2 χ Σ 03 ) ] ,
μ k x σ = 1 π σ [ ( D x 2 + 1 2 D z 2 ) Λ 00 + α D x D z ( cos χ Λ 10 + sin χ Λ 01 ) + 1 4 D z 2 ( cos 2 χ Λ 20 + 2 sin χ cos χ Λ 11 + sin 2 χ Λ 02 ) ] ,
μ k y σ = 1 π σ [ ( D y 2 + 1 2 D z 2 ) Λ 00 + α D y D z ( sin χ Λ 10 + cos χ Λ 01 ) + 1 4 D z 2 ( sin 2 χ Λ 20 2 sin χ cos χ Λ 11 + cos 2 χ Λ 02 ) ] ,
μ k x D x = cos χ π [ 2 D x Σ 00 + α D z ( cos χ Σ 10 + sin χ Σ 01 ) ] ,
μ k y D x = sin χ π [ 2 D y Σ 00 + α D z ( sin χ Σ 10 + cos χ Σ 01 ) ] ,
μ k x D y = sin χ π [ 2 D x Σ 00 + α D z ( cos χ Σ 10 + sin χ Σ 01 ) ] ,
μ k y D y = cos χ π [ 2 D y Σ 00 + α D z ( sin χ Σ 10 + cos χ Σ 01 ) ] ,
μ k x D z = 1 π [ D z Σ 00 + α D x ( cos χ Σ 10 + sin χ Σ 01 ) + 1 2 D z ( cos 2 χ Σ 20 + 2 sin χ cos χ Σ 11 + sin 2 χ Σ 02 ) ] ,
μ k y D z = 1 π [ D z Σ 00 + α D y ( sin χ Σ 10 + cos χ Σ 01 ) ] + 1 2 D z ( sin 2 χ Σ 20 2 sin χ cos χ Σ 11 + cos 2 χ Σ 02 ) ] ,
μ k x α = 1 π D x D z ( cos χ Σ 10 + sin χ Σ 01 ) ,
μ k y α = 1 π D y D z ( sin χ Σ 10 + cos χ Σ 01 ) ,
μ k x b = 1 ,
μ k y b = 1 ,
Λ lm = Δ ( x k x 0 2 σ ψ l + 1 ( x k x 0 2 σ ) ) Δ ( ψ m ( y k y 0 2 σ ) ) + Δ ( ψ l ( x k x 0 2 σ ) ) Δ ( y k y 0 2 σ ψ m + 1 ( y k y 0 2 σ ) ) .
x 0 Δ ψ l ( x k x 0 2 σ ) = 1 2 σ Δ ψ l + 1 ( x k x 0 2 σ ) ,
σ Δ ψ l ( x k x 0 2 σ ) = 1 σ Δ ( x k x 0 2 σ ψ l + 1 ( x k x 0 2 σ ) ) ,
d x d y P S F x ( x , y ) x = ( D x 2 + 1 2 D z 2 ) x 0 + α 2 D x D z σ ,
d x d y P S F x ( x , y ) y = ( D x 2 + 1 2 D z 2 ) y 0 ,
d x d y P S F y ( x , y ) x = ( D y 2 + 1 2 D z 2 ) x 0 ,
d x d y P S F y ( x , y ) y = ( D y 2 + 1 2 D z 2 ) y 0 + α 2 D y D z σ .
d x d y P S F x ( x , y ) x 2 = ( D x 2 + 1 2 D z 2 ) ( σ 2 + x 0 2 ) + D z 2 σ 2 + 2 α D x D z σ x 0 ,
d x d y P S F x ( x , y ) y 2 = ( D x 2 + 1 2 D z 2 ) ( σ 2 + y 0 2 ) ,
d x d y P S F y ( x , y ) x 2 = ( D y 2 + 1 2 D z 2 ) ( σ 2 + x 0 2 ) ,
d x d y P S F y ( x , y ) y 2 = ( D y 2 + 1 2 D z 2 ) ( σ 2 + y 0 2 ) + D z 2 σ 2 + 2 α D y D z σ y 0 .
M 0 j = k n k j ,
( M 1 x j , M 1 y j ) = k n k j ( x k , y k ) ,
( M 2 x j , M 2 y j ) = k n k j ( x k 2 , y k 2 ) .
x 0 v = M 1 x x + M 1 x y M 0 x + M 0 y ,
y 0 v = M 1 y x + M 1 y y M 0 x + M 0 y .
A 2 x j = M 2 x j 2 M 1 x j x 0 v + M 0 j x 0 v 2 ,
A 2 y j = M 2 x j 2 M 1 x j y 0 v + M 0 j y 0 v 2 ,
σ 2 = A 2 x y + A 2 y x M 0 x + M 0 y ,
D z 2 = A 2 x x + A 2 y y A 2 x y A 2 y x 2 σ 2 ,
D x 2 = M 0 x 1 2 D z 2 ,
D y 2 = M 0 y 1 2 D z 2 .
x 0 b = M 1 x y M 0 y ,
y 0 b = M 1 y x M 0 x .
x 0 = ( 1 ε ) x 0 b + ε x 0 v ,
MSLE = ( x 0 x true ) 2 = ( x 0 x 0 ) 2 + ( x 0 x true ) 2 = V + Δ 2 .
MSLE = ( 1 ε ) 2 V y + ε 2 V + ε 2 Δ 2 .
ε = V y V + V y + Δ 2 ,
MSLE = V y ( V + Δ 2 ) V + V y + Δ 2 .
κ = 4 M 0 x M 0 y ( M 0 x + M 0 y ) 2 .
α = 2 D x ( M 1 x x M 0 x x 0 ) + D y ( M 1 y y M 0 y y 0 ) ( D x 2 + D y 2 ) D z σ .
d u H n ( u ) H m ( u ) exp ( u 2 ) = π 2 n n ! δ n m .

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