Abstract

Fluorophores that are fixed during image acquisition produce a diffraction pattern that is characteristic of the orientation of the fluorophore’s underlying dipole. Fluorescence localization microscopy techniques such as PALM and STORM achieve super-resolution by applying Gaussian-based fitting algorithms to in-focus images of individual fluorophores; when applied to fixed dipoles, this can lead to a bias in the range of 5–20 nm. We introduce a method for the joint estimation of position and orientation of dipoles, based on the representation of a physically realistic image formation model as a 3-D steerable filter. Our approach relies on a single, defocused acquisition. We establish theoretical, localization-based resolution limits on estimation accuracy using Cramér-Rao bounds, and experimentally show that estimation accuracies of at least 5 nm for position and of at least 2 degrees for orientation can be achieved. Patterns generated by applying the image formation model to estimated position/orientation pairs closely match experimental observations.

© 2009 Optical Society of America

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S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009).
[CrossRef] [PubMed]

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

2008

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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (2008).
[CrossRef] [PubMed]

Z. Sikorski and L. M. Davis, "Engineering the collected field for single-molecule orientation determination," Opt. Express 16, 3660-3673 (2008).
[CrossRef] [PubMed]

M. R. Foreman, C. M. Romero, and P. Torok, "Determination of the three-dimensional orientation of single molecules," Opt. Lett. 33, 1020-1022 (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]

2006

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacio, 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. 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. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006).
[CrossRef] [PubMed]

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

2005

F. Aguet, D. Van De Ville, and M. Unser, "A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles," Opt. Express 13, 10,503-10,522 (2005).
[CrossRef]

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]

R. Schuster, M. Barth, A. Gruber, and F. Cichos, "Defocused wide field fluorescence imaging of single CdSe/ZnS quantum dots," Chem. Phys. Lett. 413, 280-283 (2005).
[CrossRef]

2004

M. Jacob and M. Unser, "Design of steerable filters for feature detection using Canny-like criteria," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1007-1019 (2004).
[CrossRef]

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

D. P. Patra, I. Gregor, and J. Enderlein, "Image analysis of defocused single-molecule images for threedimensional molecule orientation studies," J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

M. A. Lieb, J. M. Zavislan, and L. Novotny, "Single-molecule orientations determined by emission pattern imaging," J. Opt. Soc. Am. B 21, 1210-1215 (2004).
[CrossRef]

2003

M. Bohmer and J. Enderlein, "Orientation imaging of single molecules by wide-field epifluorescence microscopy," J. Opt. Soc. Am. A 20, 554-559 (2003).
[CrossRef]

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

O. Haeberle, "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]

M. S. Robbins and B. J. Hadwen, "The noise performance of electron multiplying charge-coupled devices," IEEE Trans. Electron Devices 50, 1227-1232 (2003).
[CrossRef]

2002

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

G. H. Patterson and J. Lippincott-Schwartz, "A photoactivatable GFP for selective photolabeling of proteins and cells," Science 297, 1873-1877 (2002).
[CrossRef] [PubMed]

2001

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[CrossRef] [PubMed]

2000

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

1999

A. Egner and S. W. Hell, "Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch," J. Microsc. 193, 244-249 (1999).
[CrossRef]

A. P. Bartko and R. M. Dickson, "Imaging three-dimensional single molecule orientations," J. Phys. Chem. B 103, 11,237-11,241 (1999).

1997

1993

S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

1991

1987

1986

1984

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

1959

E. Wolf, "Electromagnetic diffraction in optical systems—I. An integral representation of the image field," Proc. R. Soc. London A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems—II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Adelson, E. H.

W. T. Freeman and E. H. Adelson, "The design and use of steerable filters," IEEE Trans. Pattern Anal. Mach. Intell. 13, 891-906 (1991).
[CrossRef]

Aguet, F.

F. Aguet, D. Van De Ville, and M. Unser, "A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles," Opt. Express 13, 10,503-10,522 (2005).
[CrossRef]

Alivisatos, A. P.

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[CrossRef] [PubMed]

Axelrod, D.

Barth, M.

R. Schuster, M. Barth, A. Gruber, and F. Cichos, "Defocused wide field fluorescence imaging of single CdSe/ZnS quantum dots," Chem. Phys. Lett. 413, 280-283 (2005).
[CrossRef]

Bartko, A. P.

A. P. Bartko and R. M. Dickson, "Imaging three-dimensional single molecule orientations," J. Phys. Chem. B 103, 11,237-11,241 (1999).

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]

Betzig, E.

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

Bohmer, M.

M. Bohmer and J. Enderlein, "Orientation imaging of single molecules by wide-field epifluorescence microscopy," J. Opt. Soc. Am. A 20, 554-559 (2003).
[CrossRef]

Bonifacio, J. S.

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

Cheezum, M. K.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Cichos, F.

R. Schuster, M. Barth, A. Gruber, and F. Cichos, "Defocused wide field fluorescence imaging of single CdSe/ZnS quantum dots," Chem. Phys. Lett. 413, 280-283 (2005).
[CrossRef]

Cremer, C.

S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

Davidson, M. W.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

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

Davis, L. M.

Dickson, R. M.

A. P. Bartko and R. M. Dickson, "Imaging three-dimensional single molecule orientations," J. Phys. Chem. B 103, 11,237-11,241 (1999).

Egner, A.

A. Egner and S. W. Hell, "Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch," J. Microsc. 193, 244-249 (1999).
[CrossRef]

Enderlein, J.

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006).
[CrossRef] [PubMed]

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

D. P. Patra, I. Gregor, and J. Enderlein, "Image analysis of defocused single-molecule images for threedimensional molecule orientation studies," J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

M. Bohmer and J. Enderlein, "Orientation imaging of single molecules by wide-field epifluorescence microscopy," J. Opt. Soc. Am. A 20, 554-559 (2003).
[CrossRef]

Fetter, R. D.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Ford, G. W.

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

Foreman, M. R.

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Freeman, W. T.

W. T. Freeman and E. H. Adelson, "The design and use of steerable filters," IEEE Trans. Pattern Anal. Mach. Intell. 13, 891-906 (1991).
[CrossRef]

Galbraith, C. G.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Galbraith, J. A.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Gibson, S. F.

Gillette, J. M.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

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]

Goldman, Y. E.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (2008).
[CrossRef] [PubMed]

Gregor, I.

D. P. Patra, I. Gregor, and J. Enderlein, "Image analysis of defocused single-molecule images for threedimensional molecule orientation studies," J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

Gruber, A.

R. Schuster, M. Barth, A. Gruber, and F. Cichos, "Defocused wide field fluorescence imaging of single CdSe/ZnS quantum dots," Chem. Phys. Lett. 413, 280-283 (2005).
[CrossRef]

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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (2008).
[CrossRef] [PubMed]

Guilford, W. H.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (2008).
[CrossRef] [PubMed]

Ha, T.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

Hadwen, B. J.

M. S. Robbins and B. J. Hadwen, "The noise performance of electron multiplying charge-coupled devices," IEEE Trans. Electron Devices 50, 1227-1232 (2003).
[CrossRef]

Haeberle, O.

O. Haeberle, "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]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Heintzmann, R.

Hell, S. W.

S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009).
[CrossRef] [PubMed]

A. Egner and S. W. Hell, "Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch," J. Microsc. 193, 244-249 (1999).
[CrossRef]

S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

Hellen, E. H.

Hess, H. F.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacio, 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.

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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (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]

Hu, J.

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[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]

Jacob, M.

M. Jacob and M. Unser, "Design of steerable filters for feature detection using Canny-like criteria," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1007-1019 (2004).
[CrossRef]

Jovin, T. M.

Kanchanawong, P.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Lanni, F.

Larson, D. R.

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

Li, L.-s.

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[CrossRef] [PubMed]

Lidke, K. A.

Lieb, M. A.

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacio, 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.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

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

G. H. Patterson and J. Lippincott-Schwartz, "A photoactivatable GFP for selective photolabeling of proteins and cells," Science 297, 1873-1877 (2002).
[CrossRef] [PubMed]

Manley, S.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Manna, L.

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[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]

McKinney, S. A.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Novotny, L.

M. A. Lieb, J. M. Zavislan, and L. Novotny, "Single-molecule orientations determined by emission pattern imaging," J. Opt. Soc. Am. B 21, 1210-1215 (2004).
[CrossRef]

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Ober, R. J.

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

Olenych, S.

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

Patra, D. P.

D. P. Patra, I. Gregor, and J. Enderlein, "Image analysis of defocused single-molecule images for threedimensional molecule orientation studies," J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

Patterson, G. H.

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

G. H. Patterson and J. Lippincott-Schwartz, "A photoactivatable GFP for selective photolabeling of proteins and cells," Science 297, 1873-1877 (2002).
[CrossRef] [PubMed]

Petschek, R. G.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

Ram, S.

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

Reiner, G.

S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems—II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Rieger, B.

Robbins, M. S.

M. S. Robbins and B. J. Hadwen, "The noise performance of electron multiplying charge-coupled devices," IEEE Trans. Electron Devices 50, 1227-1232 (2003).
[CrossRef]

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]

Schuster, R.

R. Schuster, M. Barth, A. Gruber, and F. Cichos, "Defocused wide field fluorescence imaging of single CdSe/ZnS quantum dots," Chem. Phys. Lett. 413, 280-283 (2005).
[CrossRef]

Selvin, P. R.

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006).
[CrossRef] [PubMed]

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Shtengel, G.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Sikorski, Z.

Sougrat, R.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

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

Stelzer, E. H. K.

S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

Syed, S.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

Taekjip, H.

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Thompson, R. E.

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

Toprak, E.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, "Defocused orientation and position imaging (DOPI) of myosin V," Proc. Natl. Acad. Sci. USA 103, 6495-6499 (2006).
[CrossRef] [PubMed]

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006).
[CrossRef] [PubMed]

Torok, P.

Unser, M.

F. Aguet, D. Van De Ville, and M. Unser, "A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles," Opt. Express 13, 10,503-10,522 (2005).
[CrossRef]

M. Jacob and M. Unser, "Design of steerable filters for feature detection using Canny-like criteria," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1007-1019 (2004).
[CrossRef]

Van De Ville, D.

F. Aguet, D. Van De Ville, and M. Unser, "A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles," Opt. Express 13, 10,503-10,522 (2005).
[CrossRef]

Varga, R.

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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (2008).
[CrossRef] [PubMed]

Walker, W. F.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Wang, L.-w.

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[CrossRef] [PubMed]

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]

Ward, S.

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

Waterman, C. M.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, "Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure," Proc. Natl. Acad. Sci. USA 106, 3125-3130 (2009).
[CrossRef] [PubMed]

Webb, W. W.

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

Weber, W. H.

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

Winick, K. A.

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems—II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems—I. An integral representation of the image field," Proc. R. Soc. London A 253, 349-357 (1959).
[CrossRef]

Yang, W.

J. Hu, L.-s. Li,W. Yang, L. Manna, L.-w. Wang, and A. P. Alivisatos, "Linearly polarized emission from colloidal semiconductor quantum rods," Science 292, 2060-2063 (2001).
[CrossRef] [PubMed]

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, H. Taekjip, Y. E. Goldman, and P. R. Selvin, "Myosin V walks handover-hand: single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[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 molecular positions and anisotropies," Nat. Methods 5, 1027-1030 (2008).
[CrossRef] [PubMed]

Zavislan, J. M.

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]

Appl. Opt.

Biophys. J.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[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. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

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

Chem. Phys. Lett.

R. Schuster, M. Barth, A. Gruber, and F. Cichos, "Defocused wide field fluorescence imaging of single CdSe/ZnS quantum dots," Chem. Phys. Lett. 413, 280-283 (2005).
[CrossRef]

IEEE Trans. Electron Devices

M. S. Robbins and B. J. Hadwen, "The noise performance of electron multiplying charge-coupled devices," IEEE Trans. Electron Devices 50, 1227-1232 (2003).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

M. Jacob and M. Unser, "Design of steerable filters for feature detection using Canny-like criteria," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1007-1019 (2004).
[CrossRef]

W. T. Freeman and E. H. Adelson, "The design and use of steerable filters," IEEE Trans. Pattern Anal. Mach. Intell. 13, 891-906 (1991).
[CrossRef]

J. Microsc.

A. Egner and S. W. Hell, "Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch," J. Microsc. 193, 244-249 (1999).
[CrossRef]

S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem. A

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Proc. Natl. Acad. Sci. USA

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Figures (11)

Fig. 1.
Fig. 1.

Electric field propagation in a microscope. The illustrated path is in direction of the azimuth angle ϕ.

Fig. 2.
Fig. 2.

High-resolution versions of the templates involved in the steerable decomposition of a dipole diffraction pattern. The example corresponds to a Cy5 dipole (λ = 660 nm) at an air/glass interface, imaged with a 100×, 1.45 NA oil immersion objective at 500 nm defocus. The template labels match the definitions given in Eq. (13).

Fig. 3.
Fig. 3.

Filterbank implementation of the steerable dipole filters.

Fig. 4.
Fig. 4.

Schematic outline of the proposed detection algorithm. The steerable filter-based component yields results that are accurate at the pixel level (*finer scale results can be obtained by applying shifted versions of the feature templates). Every update of the position estimates generates a new set of appropriately shifted templates, from which the orientation is estimated at little cost by making use of the steerable decomposition.

Fig. 5.
Fig. 5.

Cramer-Rao bounds for (a) xp and yp , (b) z, (c) θp , (d) ϕp , and (e) A. The two surfaces in (a) are the maximum and minimum values of the bound for xp , and vice-versa for yp ; the localization accuracy for these parameters varies as a function of ϕp . System parameters: ni = 1.515, ns = 1.00, λ = 565 nm. Average PSNR = 35 dB, background intensity level 20%.

Fig. 6.
Fig. 6.

Performance of the 3-D steerable filter-based estimation of (a) xp and yp , (b) z, (c) θp and ϕp , and (d) A. The solid lines show the CRB for each parameter, and the markers correspond to the standard deviation of the estimation over 250 realizations of noise for each point. Parameters: NA = 1.45, ni = 1.515, ns = 1.00, z = 400 nm, average PSNR = 34 dB, background intensity level 10%.

Fig. 7.
Fig. 7.

Frames from the experiment described in Table 1. Scale bar: 1 μm.

Fig. 8.
Fig. 8.

Frames from the experiment described in Table 2. Scale bar: 1 μm.

Fig. 9.
Fig. 9.

Detection of dipole orientations on simulated data, using the proposed steerable filters. (a) Dipole patterns at random, pixel-level positions and orientations. (b) High-resolution image generated using detected positions and orientations. Parameters: NA = 1.45, ni = 1.515, ns = 1.00, z = 400 nm, average PSNR = 25 dB, background intensity level 20%. Scale bar: 1 μm.

Fig. 10.
Fig. 10.

(a) Dipole diffraction patterns for ssDNA-bound Cy5 molecules at an air/glass interface. (b) Diffraction patterns rendered using the orientations and positions estimated from (a), using the proposed algorithm and image formation model. Scale bar: 1 μm.

Fig. 11.
Fig. 11.

Intensity distribution generated by a Cy3 dipole (λ = 565 nm) at an air/glass interface for different values of θp and z, imaged with a 100×, 1.45 NA objective. The azimuth angle is fixed at ϕp = 0. The intensities are normalized across every focus value z. Every row, as well as planar rotations of each pattern, can be generated using six unique templates. Scale bar: 500 nm.

Tables (2)

Tables Icon

Table 1. Mean μ and standard deviation σ for position, orientation, and defocus, measured over 22 images of a single Cy5 molecule. Three of these frames are shown in Fig. 7, along with the fitted model.

Tables Icon

Table 2. Mean μ and standard deviation σ for position and orientation measured over 4 images of two Cy5 molecules. These frames are shown in Fig. 8, along with the fitted model.

Equations (68)

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ε s = p , e p s e p s + p , e s e s ,
ε a = p , e p s t p ( 1 ) t p ( 2 ) e p a + p , e s t s ( 1 ) t s ( 2 ) e s ,
t p ( l ) = 2 n l cos θ l n l + 1 cos θ l + n l cos θ l + 1
t s ( l ) = 2 n l cos θ l n l cos θ l + n l + 1 cos θ l + 1 ,
e p s = ( cos θ s cos ϕ , cos θ s sin ϕ , sin θ s )
= ( 1 n s n s 2 n i 2 sin 2 θ cos ϕ , 1 n s n s 2 n i 2 sin 2 θ sin ϕ , n i n s sin θ )
e p a = ( cos ϕ , sin ϕ , 0 )
e s = ( sin ϕ , cos ϕ , 0 ) ,
k = k n i ( cos ϕ sin θ , sin ϕ sin θ , cos θ ) ,
ε = i A 0 λ 0 2 π 0 α ε a e i k , x e i k Λ ( θ ; τ ) cos θ sin θ d θ d ϕ
= i A 0 λ 0 2 π 0 α ε a e ikr n i sin θ cos ( ϕ ϕ d ) e i k n i z cos θ e ik Λ ( θ ; τ ) cos θ sin θ d ϕ ,
τ = ( n i , n i * , n g , n g * , n s , t i * , t g , t g * )
Λ ( θ , z ; z p , τ ) = ( z p z + n i ( z p n s t g n g + t g * n g * + t i * n i * ) ) n i cos θ
+ z p n s 2 n i 2 sin 2 θ + t g n g 2 n i 2 sin 2 θ
t g * n g * 2 n i 2 sin 2 θ t g * n i * 2 n i 2 sin 2 θ .
ε = i [ I 0 + I 2 cos ( 2 ϕ d ) I 2 sin ( 2 ϕ d ) 2 i I 1 cos ( ϕ d ) I 2 sin ( 2 ϕ d ) I 0 I 2 cos ( 2 ϕ d ) 2 i I 1 ( ϕ d ) ] p ,
I 0 ( x ; x p , τ ) = 0 α B 0 ( θ ) ( t s ( 1 ) t s ( 2 ) + t p ( 1 ) t p ( 2 ) 1 n s n s 2 n i 2 sin 2 θ ) d θ
I 1 ( x ; x p , τ ) = 0 α B 1 ( θ ) t p ( 1 ) t p ( 2 ) n i n s sin θ d θ
I 2 ( x ; x p , τ ) = 0 α B 2 ( θ ) ( t s ( 1 ) t s ( 2 ) t p ( 1 ) t p ( 2 ) 1 n s n s 2 n i 2 sin 2 θ ) d θ
B m ( θ ) = cos θ sin θ J m ( k r n i sin θ ) e i k Λ ( θ ; z ; z p , τ ) .
h θ p , ϕ p ( x ; x p , τ ) = ε 2
= sin 2 θ p ( I 0 2 + I 2 2 + 2 cos ( 2 ϕ p 2 ϕ d ) e { I 0 * I 2 } )
2 sin ( 2 θ p ) cos ( ϕ p ϕ d ) m { I 1 * ( I 0 + I 2 ) } + 4 I 1 2 cos 2 θ p
= p T Mp .
m 11 = I 0 2 + I 2 2 + 2 e { I 0 * I 2 } cos 2 ϕ d
m 12 = 2 e { I 0 * I 2 } sin 2 ϕ d
m 13 = 2 cos ϕ d m { I 1 * ( I 0 + I 2 ) }
m 22 = I 0 2 + I 2 2 + 2 e { I 0 * I 2 } cos 2 ϕ d
m 23 = 2 sin ϕ d m { I 1 * ( I 0 + I 2 ) }
m 33 = 4 I 1 2 .
h ( x ; x p , τ ) = 0 2 π 0 π h θ p , ϕ p ( x ; x p , τ ) θ p d θ p d ϕ p
= 8 π 3 ( I 0 2 + 2 I 1 2 + I 2 2 ) .
q ¯ ( x ; x p , τ ) = c · ( A h θ p , ϕ p ( x ; x p , τ ) + b ) ,
P q ( x ; x p , τ ) ( q ) = e q ¯ ( x ; x p , τ ) q ¯ ( x ; x p , τ ) q q ! .
( θ p * ( x ) , ϕ p * ( x ) ) = arg max ( θ p , ϕ p ) f ( x ) * g θ p , ϕ p ( x ) ,
J LS ( x ; θ p , ϕ p ) = Ω ( A h θ p , ϕ p ( v ; x p , τ ) f ( x v ) ) 2 d v
= A h θ p , ϕ p ( x ; x p , τ ) 2 + Ω f ( x v ) 2 d v
2 A h θ p , ϕ p ( x ; x p , τ ) * f ( x ) ,
( h θ p , ϕ p * f ) ( x ) = i j a i j ( θ p , ϕ p ) ( m i j * f ) ( x )
= p T M f p ,
A h θ p , ϕ p ( x ; x p , τ ) 2 = A 2 u θ p T E u θ p ,
E = [ m 11 2 m 11 m 13 m 11 m 33 m 11 m 13 m 13 2 m 13 m 33 m 11 m 33 m 13 m 33 m 33 2 ] .
J ( x ; θ p , ϕ p ) = A 2 u θ p T E u θ p 2 A p T M f p .
θ p J ( x ; θ p , ϕ p ) = 2 A ( A u θ p T E θ p u θ p 2 p T M f θ p p )
ϕ p J ( x ; θ p , ϕ p ) = 4 A p T M f ϕ p p
A ̂ = p T M f p u θ p T E u θ p .
F i j = Ω 1 q ¯ q ¯ ϑ i q ¯ ϑ j d x
Var ( ϑ ̂ i ) [ F 1 ] i i ,
Var ( ϕ ̂ p ) 1 / Ω 1 q ¯ ( q ¯ ϕ p ) 2 d x .
tan 4 θ p ( f 13 cos ( ϕ p ) + f 23 sin ( ϕ p ) A e 12 )
+ tan 3 θ p ( f 33 ( f 11 cos ( ϕ p ) 2 + f 12 sin ( 2 ϕ p ) + f 22 sin ( ϕ p ) 2 ) + A ( e 11 e 13 2 e 22 ) )
+ 3 tan 2 θ p A ( e 12 e 23 )
+ tan θ p ( f 33 ( f 11 cos ( ϕ p ) 2 + f 12 sin ( 2 ϕ p ) + f 22 sin ( ϕ p ) 2 ) + A ( e 13 e 33 + 2 e 22 ) )
( f 13 cos ( ϕ p ) + f 23 sin ( ϕ p ) A e 23 )
= 0
tan 4 ϕ p ( f 12 2 sin 2 θ p f 13 2 cos 2 θ p )
+ tan 3 ϕ p ( f 13 f 23 cos 2 θ p + f 12 ( m 11 f 22 ) sin 2 θ p )
+ tan 2 ϕ p ( ( ( f 11 f 22 ) 2 2 f 12 2 ) sin 2 θ p ( f 13 2 + f 23 2 ) cos 2 θ p )
+ tan ϕ p ( f 13 f 23 cos 2 θ p f 12 ( f 11 f 22 ) sin 2 θ p )
+ f 12 2 sin 2 θ p f 23 2 cos 2 θ p
= 0
h θ p , ϕ p x p = sin 2 θ p ( 2 e { I 0 * I 0 x p + I 2 * I 2 x p } + 2 cos ( 2 ϕ p 2 ϕ d ) e { I 0 * I 2 x p + I 2 * I 0 x p } )
2 sin ( 2 θ p ) cos ( ϕ p ϕ d ) m { I 1 * ( I 0 x p + I 2 x p ) I 1 x p ( I 0 * + I 2 * ) }
+ 8 cos 2 θ p e { I 1 * I 1 x p }
h θ p , ϕ p θ p = sin 2 θ p ( I 0 2 + I 2 2 + 2 cos ( 2 ϕ p 2 ϕ d ) e { I 0 * I 2 } 4 I 1 2 )
4 cos ( 2 θ p ) cos ( ϕ p ϕ d ) m { I 1 * ( I 0 + I 2 ) }
h θ p , ϕ p ϕ p = 4 sin 2 θ p sin ( 2 ϕ p 2 ϕ d ) e { I 0 * I 2 }
+ 2 sin ( 2 θ p ) sin ( ϕ p ϕ d ) m { I 1 * ( I 0 + I 2 ) } .

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