Abstract

The performance of fluorescence microscopy and nanoscopy is often discussed by the effective point spread function and the optical transfer function. However, due to the complexity of the fluorophore properties such as photobleaching or other forms of photoswitching, which introduce a variance in photon emission, it is not trivial to choose optimal imaging parameters and to predict the spatial resolution. In this paper, we analytically derive a theoretical framework for estimating the achievable resolution of a microscope depending on parameters such as photoswitching, labeling densities, exposure time and sampling. We developed a numerical simulation software to analyze the impact of reversibly switchable probes in RESOLFT imaging.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Molecular contribution function in RESOLFT nanoscopy

Lars Frahm, Jan Keller-Findeisen, Philipp Alt, Sebastian Schnorrenberg, Miguel del Álamo Ruiz, Timo Aspelmeier, Axel Munk, Stefan Jakobs, and Stefan W. Hell
Opt. Express 27(15) 21956-21987 (2019)

Far-field optical nanoscopy with reduced number of state transition cycles

Thorsten Staudt, Andreas Engler, Eva Rittweger, Benjamin Harke, Johann Engelhardt, and Stefan W. Hell
Opt. Express 19(6) 5644-5657 (2011)

Pixel hopping enables fast STED nanoscopy at low light dose

Britta Vinçon, Claudia Geisler, and Alexander Egner
Opt. Express 28(4) 4516-4528 (2020)

References

  • View by:
  • |
  • |
  • |

  1. S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
    [Crossref]
  2. T. A. Klar, E. Engel, and S. W. Hell, “Breaking abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64(6), 066613 (2001).
    [Crossref]
  3. 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(5793), 1642–1645 (2006).
    [Crossref]
  4. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
    [Crossref]
  5. S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
    [Crossref]
  6. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994).
    [Crossref]
  7. S. Hell, S. Jakobs, and L. Kastrup, “Imaging and writing at the nanoscale with focused visible light through saturable optical transitions,” Appl. Phys. A 77(7), 859–860 (2003).
    [Crossref]
  8. T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
    [Crossref]
  9. T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
    [Crossref]
  10. T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
    [Crossref]
  11. O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
    [Crossref]
  12. I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
    [Crossref]
  13. L. Frahm, J. Keller-Findeisen, P. Alt, S. Schnorrenberg, M. del Álamo Ruiz, T. Aspelmeier, A. Munk, S. Jakobs, and S. W. Hell, “Molecular contribution function in resolft nanoscopy,” Opt. Express 27(15), 21956–21987 (2019).
    [Crossref]
  14. B. K. Cooper and A. G. York, “Photoswitching noise distorts all fluorescent images,” (2019).
  15. J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
    [Crossref]
  16. M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
    [Crossref]
  17. N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
    [Crossref]
  18. S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
    [Crossref]
  19. J. W. Goodman, Statistical Optics (John Wiley and Sons, 2000).
  20. A. Bodén, F. Pennacchietti, and I. Testa, “Three dimensional parallelized RESOLFT nanoscopy for volumetric live cell imaging,” preprint, Biophysics (2020).
  21. C. Wunsch, Time Series Analysis. A Heuristic Primer (Harvard, 2010).

2019 (3)

L. Frahm, J. Keller-Findeisen, P. Alt, S. Schnorrenberg, M. del Álamo Ruiz, T. Aspelmeier, A. Munk, S. Jakobs, and S. W. Hell, “Molecular contribution function in resolft nanoscopy,” Opt. Express 27(15), 21956–21987 (2019).
[Crossref]

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

2018 (1)

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

2013 (1)

N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
[Crossref]

2012 (2)

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

2011 (2)

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

2007 (1)

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref]

2006 (3)

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(5793), 1642–1645 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref]

2005 (1)

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

2003 (1)

S. Hell, S. Jakobs, and L. Kastrup, “Imaging and writing at the nanoscale with focused visible light through saturable optical transitions,” Appl. Phys. A 77(7), 859–860 (2003).
[Crossref]

2001 (1)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64(6), 066613 (2001).
[Crossref]

1994 (1)

Alt, P.

Andresen, M.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Aspelmeier, T.

Banterle, N.

N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

Beck, M.

N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
[Crossref]

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(5793), 1642–1645 (2006).
[Crossref]

Bock, H.

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

Bodén, A.

A. Bodén, F. Pennacchietti, and I. Testa, “Three dimensional parallelized RESOLFT nanoscopy for volumetric live cell imaging,” preprint, Biophysics (2020).

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(5793), 1642–1645 (2006).
[Crossref]

Brakemann, T.

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Bui, K. H.

N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
[Crossref]

Cáceres, R.

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

Castello, M.

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

Coceano, G.

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

Cooper, B. K.

B. K. Cooper and A. G. York, “Photoswitching noise distorts all fluorescent images,” (2019).

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(5793), 1642–1645 (2006).
[Crossref]

Deguchi, T.

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

del Álamo Ruiz, M.

Diaspro, A.

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

Dreier, J.

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Eggeling, C.

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

Enderlein, J.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Engel, E.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64(6), 066613 (2001).
[Crossref]

Frahm, L.

Girirajan, T. P.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley and Sons, 2000).

Grotjohann, T.

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Hell, S.

S. Hell, S. Jakobs, and L. Kastrup, “Imaging and writing at the nanoscale with focused visible light through saturable optical transitions,” Appl. Phys. A 77(7), 859–860 (2003).
[Crossref]

Hell, S. W.

L. Frahm, J. Keller-Findeisen, P. Alt, S. Schnorrenberg, M. del Álamo Ruiz, T. Aspelmeier, A. Munk, S. Jakobs, and S. W. Hell, “Molecular contribution function in resolft nanoscopy,” Opt. Express 27(15), 21956–21987 (2019).
[Crossref]

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref]

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

T. A. Klar, E. Engel, and S. W. Hell, “Breaking abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64(6), 066613 (2001).
[Crossref]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994).
[Crossref]

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(5793), 1642–1645 (2006).
[Crossref]

Hess, S. T.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref]

Hofmann, M.

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

Huhn, T.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Jakobs, S.

L. Frahm, J. Keller-Findeisen, P. Alt, S. Schnorrenberg, M. del Álamo Ruiz, T. Aspelmeier, A. Munk, S. Jakobs, and S. W. Hell, “Molecular contribution function in resolft nanoscopy,” Opt. Express 27(15), 21956–21987 (2019).
[Crossref]

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

S. Hell, S. Jakobs, and L. Kastrup, “Imaging and writing at the nanoscale with focused visible light through saturable optical transitions,” Appl. Phys. A 77(7), 859–860 (2003).
[Crossref]

Janke, T.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Kastrup, L.

S. Hell, S. Jakobs, and L. Kastrup, “Imaging and writing at the nanoscale with focused visible light through saturable optical transitions,” Appl. Phys. A 77(7), 859–860 (2003).
[Crossref]

Keller-Findeisen, J.

Klar, T. A.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64(6), 066613 (2001).
[Crossref]

Koho, S.

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

Lavoie-Cardinal, F.

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

Lemke, E. A.

N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
[Crossref]

Lemken, F.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Leutenegger, M.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

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(5793), 1642–1645 (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(5793), 1642–1645 (2006).
[Crossref]

Mason, M. D.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref]

Munk, A.

Nevskyi, O.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref]

Oppermann, A.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[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(5793), 1642–1645 (2006).
[Crossref]

Pennacchietti, F.

A. Bodén, F. Pennacchietti, and I. Testa, “Three dimensional parallelized RESOLFT nanoscopy for volumetric live cell imaging,” preprint, Biophysics (2020).

Plastino, J.

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

Plessmann, U.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Reuss, M.

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

Schnorrenberg, S.

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(5793), 1642–1645 (2006).
[Crossref]

Stein, S. C.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Stiel, A. C.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Sysoiev, D.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Testa, I.

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

A. Bodén, F. Pennacchietti, and I. Testa, “Three dimensional parallelized RESOLFT nanoscopy for volumetric live cell imaging,” preprint, Biophysics (2020).

Tortarolo, G.

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

Urban, N. T.

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

Urlaub, H.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Vicidomini, G.

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

Wahl, M. C.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Weber, G.

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Wichmann, J.

Willig, K. I.

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

Wöll, D.

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Wunsch, C.

C. Wunsch, Time Series Analysis. A Heuristic Primer (Harvard, 2010).

York, A. G.

B. K. Cooper and A. G. York, “Photoswitching noise distorts all fluorescent images,” (2019).

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

Appl. Phys. A (1)

S. Hell, S. Jakobs, and L. Kastrup, “Imaging and writing at the nanoscale with focused visible light through saturable optical transitions,” Appl. Phys. A 77(7), 859–860 (2003).
[Crossref]

Biophys. J. (1)

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref]

eLife (1)

T. Grotjohann, I. Testa, M. Reuss, T. Brakemann, C. Eggeling, S. W. Hell, and S. Jakobs, “rsegfp2 enables fast resolft nanoscopy of living cells,” eLife 1, e00248 (2012).
[Crossref]

J. Struct. Biol. (1)

N. Banterle, K. H. Bui, E. A. Lemke, and M. Beck, “Fourier ring correlation as a resolution criterion for super-resolution microscopy,” J. Struct. Biol. 183(3), 363–367 (2013).
[Crossref]

Nat. Biotechnol. (1)

T. Brakemann, A. C. Stiel, G. Weber, M. Andresen, I. Testa, T. Grotjohann, M. Leutenegger, U. Plessmann, H. Urlaub, C. Eggeling, M. C. Wahl, S. W. Hell, and S. Jakobs, “A reversibly photoswitchable GFP-like protein with fluorescence excitation decoupled from switching,” Nat. Biotechnol. 29(10), 942–947 (2011).
[Crossref]

Nat. Commun. (2)

S. Koho, G. Tortarolo, M. Castello, T. Deguchi, A. Diaspro, and G. Vicidomini, “Fourier ring correlation simplifies image restoration in fluorescence microscopy,” Nat. Commun. 10(1), 3103 (2019).
[Crossref]

J. Dreier, M. Castello, G. Coceano, R. Cáceres, J. Plastino, G. Vicidomini, and I. Testa, “Smart scanning for low-illumination and fast RESOLFT nanoscopy in vivo,” Nat. Commun. 10(1), 556 (2019).
[Crossref]

Nat. Methods (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

Nature (1)

T. Grotjohann, I. Testa, M. Leutenegger, H. Bock, N. T. Urban, F. Lavoie-Cardinal, K. I. Willig, C. Eggeling, S. Jakobs, and S. W. Hell, “Diffraction-unlimited all-optical imaging and writing with a photochromic GFP,” Nature 478(7368), 204–208 (2011).
[Crossref]

Neuron (1)

I. Testa, N. T. Urban, S. Jakobs, C. Eggeling, K. I. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64(6), 066613 (2001).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (1)

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

Science (2)

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(5793), 1642–1645 (2006).
[Crossref]

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref]

Small (1)

O. Nevskyi, D. Sysoiev, J. Dreier, S. C. Stein, A. Oppermann, F. Lemken, T. Janke, J. Enderlein, I. Testa, T. Huhn, and D. Wöll, “Fluorescent diarylethene photoswitches-a universal tool for super-resolution microscopy in nanostructured materials,” Small 14(10), 1703333 (2018).
[Crossref]

Other (4)

B. K. Cooper and A. G. York, “Photoswitching noise distorts all fluorescent images,” (2019).

J. W. Goodman, Statistical Optics (John Wiley and Sons, 2000).

A. Bodén, F. Pennacchietti, and I. Testa, “Three dimensional parallelized RESOLFT nanoscopy for volumetric live cell imaging,” preprint, Biophysics (2020).

C. Wunsch, Time Series Analysis. A Heuristic Primer (Harvard, 2010).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. Schematic illustration of fluorophores exhibiting unidirectional switching. This model is commonly used to model the process of photobleaching. The lifetime of the fluorescent state is a Poisson distributed stochastic variable.
Fig. 2.
Fig. 2. Schematic illustration of fluorophores exhibiting bidirectional switching. This model is commonly used to model the process of photoswitching of the type characterising e.g. reversibly switchable fluorescent proteins.
Fig. 3.
Fig. 3. Figure showing a) an example sequence of different illumination patterns during a typical RESOLFT imaging scheme and b) the spatial distribution of fluorophores in the ON state, $p_{ON}(x,y)$, resulting from such a sequence at two different time points. Panel c) and d) shows the spatially dependent expectation and variance of the fluorescent photon emission after different read-out times.
Fig. 4.
Fig. 4. a) For the FRC resolution predictions a spectrally flat sample is assumed. b) After imaging, the power spectrum of the image reflects the shape of the transfer function. c) Graph showing the predicted resolution in blue line depending on the read-out time used. Yellow dots show the measured FRC resolution from simulated images. d) Plot shows the predicted resolution normalized for each curve between its lowest and highest value within the plotted range in order emphasize the shift in optimal read-out time. Values are plotted against the read-out time for four different fluorophore emission rates. e) Plot shows the predicted resolution depending on the power spectrum of the sample. Average sample density is kept constant. Yellow dots show the measured FRC resolution from simulated images. f) Plot shows the predicted resolution depending on read-out time for four different sets of parameters. For all four fluorophores, the product of emission rate and labeling density is kept constant. g) Lines in graph plot the predicted resolution depending on the saturation factor used for the OFF-switching light. Blue line predicts the resolution in the ideal case of zero intensity in the center of the OFF-illumination. Red line shows the imperfect case where the center of the OFF-illumination sees an illumination power 3% of the maximum illumination power. Yellow dots show the measured FRC resolution from simulated images with zero intensity in the center of the OFF-illumination. h) Plot shows the predicted resolution depending on the scanning step size used for two different sample powers.

Equations (61)

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

E [ P ] = r f l E [ Φ ]
V a r [ P ] = r f l E [ Φ ] + r f l 2 V a r [ ϕ ]
E [ Φ ] = 1 r O F F ( 1 e T r O F F )
V a r [ Φ ] = 1 r O F F 2 ( 2 ( ( 1 e T r O F F ) T r O F F e T r O F F ) ( 1 e T r O F F ) 2 ) .
p O N ( t 0 + t ) = r O N r O N + r O F F + ( p O N ( t 0 ) r O N r O N + r O F F ) e ( r O N + R O F F ) ( t t 0 ) .
E [ Φ ] = t 0 T p O N ( t 0 ) d t = T r O N r O N + r O F F + p O N ( t 0 ) r O N r O N + r O F F 1 e ( r O N + r O F F ) ( T t o ) r O N + r O F F .
E [ E m ( x , y ) ] i , j = E [ P x , y ] d ( x i , y j )
V a r [ E m ( x , y ) ] i , j = V a r [ P x , y ] d ( x i , y j )
E [ I P 1 ( k , l ) ] i , j = ( E [ E m ( x , y ) ] i , j P S F ( x , y ) ) ( k , l ) .
E [ S ( i , j ) ] = k , l p ( k , l ) ( E [ E m ( x , y ) ] i , j P S F ( x , y ) ) ( k , l ) d k d l
E [ S ( i , j ) ] = x , y ( p ( k , l ) P S F ( k , l ) ) ( x , y ) E [ E m ( x , y ) ] i , j d x d y .
g ( x , y ) = ( p ( k , l ) P S F ( k , l ) ) ( x , y ) .
E [ S ( i , j ) ] = x , y g ( x , y ) E [ E m ( x , y ) ] i , j d x d y .
V a r [ S ( i , j ) ] = x , y | g ( x , y ) | V a r [ E m ( x , y ) ] i , j d x d y
E [ S ( i , j ) ] = x , y g ( x , y ) E [ P x , y ] d ( x i , y j ) d x d y
V a r [ S ( i , j ) ] = x , y | g ( x , y ) | V a r [ P x , y ] d ( x i , y j ) d x d y .
h E ( x , y ) = g ( x , y ) E [ P x , y ]
h V a r ( x , y ) = | g ( x , y ) | V a r [ P x , y ]
E [ S ( i , j ) ] = x , y h E ( x , y ) d ( x i , y j ) d x d y = ( h E ( x , y ) d ( x , y ) ) ( i , j )
V a r [ S ( i , j ) ] = x , y h V a r ( x , y ) d ( x i , y j ) d x d y = ( h V a r ( x , y ) d ( x , y ) ) ( i , j ) .
F { E [ S ( i , j ) ] } = S ~ E ( f x , f y ) = H E ( f x , f y ) D ( f x , f y )
F { V a r [ S ( i , j ) ] } = S ~ V a r ( f x , f y ) = H V a r ( f x , f y ) D ( f x , f y )
F R C ( r k ) = r r k f ~ 1 ( r ) f ~ 2 ( r ) r r k f ~ 1 2 ( r ) r r k f ~ 2 2 ( r )
C F 1 , F 2 ( r ) = E [ F ~ 1 ( r ) F ~ 2 ( r ) ] E [ F ~ 1 ( r ) 2 ] E [ F ~ 2 ( r ) 2 ] = ϕ F 1 , F 2 ( r ) ϕ F 1 , F 1 ( r ) ϕ F 2 , F 2 ( r )
F R C ( r k ) = C ^ F 1 , F 2 ( r k ) = ϕ ^ F 1 , F 2 ( r k ) ϕ ^ F 1 , F 1 ( r k ) ϕ ^ F 2 , F 2 ( r k )
I [ m , n ] = S ( m Δ i , n Δ j ) = I S [ m , n ] + I N [ m , n ] .
I ~ S [ u , v ] = D F T { I S [ m , n ] }
I ~ N [ u , v ] = D F T { I N [ m , n ] }
C I ( r k ) = E [ | I ~ S ( r k ) | 2 ] E [ | I ~ S ( r k ) | 2 ] + E [ | I ~ N ( r k ) | 2 ] = ϕ S ( r k ) ϕ S ( r k ) + V a r [ I ~ N ( r k ) ]
| I ~ S [ u , v ] | 2 = 1 ( Δ i Δ j ) 2 | S ~ E ( u Δ i , v Δ j ) | 2 = 1 ( Δ i Δ j ) 2 | H E ( u Δ i , v Δ j ) | 2 | D ( u Δ i , v Δ j ) | 2 .
ϕ S ( r k ) = 1 N u 2 + v 2 r k | I ~ S [ u , v ] | 2 = 1 N ( Δ i Δ j ) 2 u 2 + v 2 r k | H E ( u Δ i , v Δ j ) | 2 | D ( u Δ i , v Δ j ) | 2
ϕ S ( r k ) = ( | H E ( r k ) | Δ i Δ j ) 2 1 N u 2 + v 2 r k | D ( u Δ i , v Δ j ) | 2 = ( | H E ( r k ) | Δ i Δ j ) 2 ϕ D ( r k )
ϕ D ( r k ) = 1 N u 2 + v 2 r k | D ( u Δ i , v Δ j ) | 2 .
V a r [ I ~ N ( r k ) ] = E [ I ~ N ( r k ) 2 ] = m , n E [ I N [ m , n ] 2 ] = 1 Δ i Δ j S ~ V a r ( 0 , 0 ) = 1 Δ i Δ j H V a r ( 0 , 0 ) D ( 0 , 0 ) .
F R C ( r k ) = ( | H E ( r k ) | Δ i Δ j ) 2 ϕ D ( r k ) ( | H E ( r k ) | Δ i Δ j ) 2 ϕ D ( r k ) + H V a r ( 0 , 0 ) D ( 0 , 0 ) + σ d e t 2
d ( x , y ) = β κ ( x , y )
D ( 0 , 0 ) = β K ( 0 , 0 ) .
ϕ D ( r ) = β 2 ϕ K ( r )
F R C ( r k ) = ( | H E ( r k ) | Δ i Δ j ) 2 β 2 ϕ K ( r k ) ( | H E ( r k ) | Δ i Δ j ) 2 β 2 ϕ K ( r k ) + | H V a r ( 0 , 0 ) | β K ( 0 , 0 ) + σ d e t 2 .
d P ( s =" O F F " | s 1 =" O N " ) = r O F F d t
d P ( s =" O N " | s 1 =" O F F " ) = r O N d t
P ( s =" O N " , t + d t ) = P ( s =" O F F " , t ) r O N d t + P ( s =" O N " , t ) ( 1 r O F F ) d t =
= P ( s =" O N " , t ) + ( 1 P ( s =" O N " , t ) ) r O N P ( s =" O N " , t ) r O F F ) d t
P ( s =" O N " , t + d t ) P ( s =" O N " , t ) d t = d P ( s =" O N " , t ) d t
d P ( s =" O N " ) d t = ( 1 P ( s =" O N " , t ) ) r O N P ( s =" O N " , t ) r O F F =
= r O N P ( s =" O N " , t ) ( r O N + r O F F )
P ( s =" O N " , t ) = r O N r O N + r O F F A e ( r O N + r O F F ) t
P ( s =" O N " , t ) = r O N r O N + r O F F + ( p r O N r O N + r O F F ) e ( r O N + r O F F ) t
C I 1 , I 2 ( r ) = E [ I ~ 1 ( r ) I ~ 2 ( r ) ] E [ I ~ 1 ( r ) 2 ] E [ I ~ 2 ( r ) 2 ] = ϕ I 1 , I 2 ( r ) ϕ I 1 , I 1 ( r ) ϕ I 2 , I 2 ( r )
C I 1 , I 2 ( r ) = E [ ( I ~ S ( r k ) + I ~ N 1 ( r k ) ) ( I ~ S ( r k ) + I ~ N 2 ( r k ) ) ] E [ | I ~ S ( r k ) + I ~ N 1 ( r k ) | 2 ] E [ | I ~ S ( r k ) + I ~ N 2 ( r k ) | 2 ]
ϕ I 1 , I 2 ( r ) = E [ ( I ~ S ( r k ) + I ~ N 1 ( r k ) ) ( I ~ S ( r k ) + I ~ N 2 ( r k ) ) ] =
= E [ I ~ S ( r k ) I ~ S ( r k ) ] + E [ I ~ S ( r k ) I ~ N 2 ( r k ) ] + E [ I ~ N 1 ( r k ) I ~ S ( r k ) ] + E [ I ~ N 1 ( r k ) I ~ N 2 ( r k ) ]
ϕ I 1 , I 2 ( r ) = E [ I ~ S ( r k ) I ~ S ( r k ) ] = ϕ S ( r k )
ϕ I 1 , I 1 ( r ) = E [ ( I ~ S ( r k ) + I ~ N 1 ( r k ) ) ( I ~ S ( r k ) + I ~ N 1 ( r k ) ) ] =
= E [ I ~ S ( r k ) I ~ S ( r k ) ] + E [ I ~ S ( r k ) I ~ N 1 ( r k ) ] + E [ I ~ N 1 ( r k ) I ~ S ( r k ) ] + E [ I ~ N 1 ( r k ) I ~ N 1 ( r k ) ] =
= ϕ S ( r k ) + V a r [ I ~ N 1 ( r k ) ]
C I ( r k ) = ϕ S ( r k ) ϕ S ( r k ) + V a r [ I ~ N ( r k ) ]
P N ( u , v ) = E [ | I ~ N ( u , v ) | 2 ] = E [ | m , n I N [ m , n ] e j 2 π m u M e j 2 π n v N | 2 ] =
E [ ( m , n I N [ m , n ] e j 2 π m u M e j 2 π n v N ) ( m , n I N [ m , n ] e j 2 π m u M e j 2 π n v N ) ] =
= m , n m , n E [ I N [ m , n ] I N [ m , n ] ] e j 2 π ( m m ) u M e j 2 π ( n n ) v N =
P N ( u , v ) = m , n σ m , n 2

Metrics