Abstract

The ultimate objective of a microscope of the highest resolution is to map the molecules of interest in the sample. Traditionally, linear imaging systems are characterized by their spatial frequency transfer function, which is given, in real space, by the point spread function (PSF). By extending the concept of the PSF towards the molecular contribution function (MCF), that quantifies the average contribution of a single fluorophore to the image, a straightforward concept for counting fluorophores is obtained. Using reversible saturable optical fluorescence transitions (RESOLFT), fluorophores are effectively activated only in a small, subdiffraction-sized volume before they are read out. During readout the signal exhibits an increased variance due to the stochastic nature of prior activation, which scales quadratically with the brightness of the active fluorophores while the mean of the signal scales only linearly with it. Using a two-state Markov model for the activation, showing comparable behavior to the switching kinetics of the switchable fluorescent protein rsEGFP2, we can approximate quantitatively the MCF of RESOLFT nanoscopy allowing to count the number of fluorophores within a subdiffraction-sized region of the sample. The method is validated on measurements of tubulin structures in Drosophila melagonaster larvae. Modeling and estimation of the MCF is a promising approach to quantitative microscopy.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
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)

Parallelized STED fluorescence nanoscopy

Pit Bingen, Matthias Reuss, Johann Engelhardt, and Stefan W. Hell
Opt. Express 19(24) 23716-23726 (2011)

2000-fold parallelized dual-color STED fluorescence nanoscopy

Fabian Bergermann, Lucas Alber, Steffen J. Sahl, Johann Engelhardt, and Stefan W. Hell
Opt. Express 23(1) 211-223 (2015)

References

  • View by:
  • |
  • |
  • |

  1. V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
    [Crossref]
  2. B. D. Engel, W. B. Ludington, and W. F. Marshall, “Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model,” J. Cell Biol. 187(1), 81–89 (2009).
    [Crossref]
  3. C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
    [Crossref]
  4. 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]
  5. T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
    [Crossref]
  6. M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. 102(37), 13081–13086 (2005).
    [Crossref]
  7. S. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21(11), 1347–1355 (2003).
    [Crossref]
  8. S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14(5), 599–609 (2004).
    [Crossref]
  9. 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]
  10. 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]
  11. S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
    [Crossref]
  12. G. Hummer, F. Fricke, and M. Heilemann, “Model-independent counting of molecules in single-molecule localization microscopy,” Mol. Biol. Cell 27(22), 3637–3644 (2016).
    [Crossref]
  13. R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
    [Crossref]
  14. M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nat. Methods 4(4), 319–321 (2007).
    [Crossref]
  15. M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
    [Crossref]
  16. L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
    [Crossref]
  17. H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
    [Crossref]
  18. 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]
  19. 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]
  20. 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]
  21. B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
    [Crossref]
  22. B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
    [Crossref]
  23. S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
    [Crossref]
  24. L. A. Amos and A. Klug, “Arrangement of subunits in flagellar microtubules,” Journal of cell science 14, 523–549 (1974).
  25. N. Joshi, R. Szeliski, and D. J. Kriegman, “PSF estimation using sharp edge prediction,” in Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, (IEEE, 2008), pp. 1–8.
  26. C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Machine Intell. 20(2), 113–125 (1998).
    [Crossref]
  27. A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
    [Crossref]
  28. E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses (Springer Science & Business Media, 2006).
  29. D. Bourgeois and V. Adam, “Reversible photoswitching in fluorescent proteins: a mechanistic view,” IUBMB Life 64(6), 482–491 (2012).
    [Crossref]
  30. D. E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A 10(6), 1938–1945 (1974).
    [Crossref]
  31. T. T. Cheng, “The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan,” Bull. Amer. Math. Soc. 55(4), 396–402 (1949).
    [Crossref]
  32. J. P. Romano and M. Wolf, “A more general central limit theorem for m-dependent random variables with unbounded m,” Stat. & probability letters 47(2), 115–124 (2000).
    [Crossref]
  33. P. Billingsley, Probability and Measure (Wiley, New York, 1979).
  34. A. W. van der Vaart, Asymptotic Statistics (Cambridge University, 1998).

2016 (4)

G. Hummer, F. Fricke, and M. Heilemann, “Model-independent counting of molecules in single-molecule localization microscopy,” Mol. Biol. Cell 27(22), 3637–3644 (2016).
[Crossref]

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[Crossref]

2015 (1)

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

2012 (4)

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]

S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
[Crossref]

D. Bourgeois and V. Adam, “Reversible photoswitching in fluorescent proteins: a mechanistic view,” IUBMB Life 64(6), 482–491 (2012).
[Crossref]

2011 (3)

V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
[Crossref]

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (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]

2009 (1)

B. D. Engel, W. B. Ludington, and W. F. Marshall, “Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model,” J. Cell Biol. 187(1), 81–89 (2009).
[Crossref]

2008 (2)

B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
[Crossref]

M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
[Crossref]

2007 (1)

M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nat. Methods 4(4), 319–321 (2007).
[Crossref]

2006 (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]

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]

2005 (3)

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. 102(37), 13081–13086 (2005).
[Crossref]

L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
[Crossref]

A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
[Crossref]

2004 (1)

S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14(5), 599–609 (2004).
[Crossref]

2003 (1)

S. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21(11), 1347–1355 (2003).
[Crossref]

2000 (2)

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[Crossref]

J. P. Romano and M. Wolf, “A more general central limit theorem for m-dependent random variables with unbounded m,” Stat. & probability letters 47(2), 115–124 (2000).
[Crossref]

1998 (1)

C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Machine Intell. 20(2), 113–125 (1998).
[Crossref]

1994 (1)

1974 (2)

D. E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A 10(6), 1938–1945 (1974).
[Crossref]

L. A. Amos and A. Klug, “Arrangement of subunits in flagellar microtubules,” Journal of cell science 14, 523–549 (1974).

1949 (1)

T. T. Cheng, “The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan,” Bull. Amer. Math. Soc. 55(4), 396–402 (1949).
[Crossref]

Adam, V.

D. Bourgeois and V. Adam, “Reversible photoswitching in fluorescent proteins: a mechanistic view,” IUBMB Life 64(6), 482–491 (2012).
[Crossref]

Agasti, S. S.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

Alt, P.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

Amos, L. A.

L. A. Amos and A. Klug, “Arrangement of subunits in flagellar microtubules,” Journal of cell science 14, 523–549 (1974).

Avendaño, M. S.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[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]

Belov, V. N.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[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]

Billingsley, P.

P. Billingsley, Probability and Measure (Wiley, New York, 1979).

Bissantz, N.

A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
[Crossref]

Blom, H.

L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
[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]

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]

Bossi, M. L.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

Bourgeois, D.

D. Bourgeois and V. Adam, “Reversible photoswitching in fluorescent proteins: a mechanistic view,” IUBMB Life 64(6), 482–491 (2012).
[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]

Bustamante, C.

S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
[Crossref]

Cheng, T. T.

T. T. Cheng, “The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan,” Bull. Amer. Math. Soc. 55(4), 396–402 (1949).
[Crossref]

Coffman, V. C.

V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
[Crossref]

Dai, M.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

Dalal, R.

M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
[Crossref]

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]

Digman, M. A.

M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
[Crossref]

Dyba, M.

S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14(5), 599–609 (2004).
[Crossref]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[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. 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]

L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
[Crossref]

Egner, A.

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[Crossref]

Engel, B. D.

B. D. Engel, W. B. Ludington, and W. F. Marshall, “Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model,” J. Cell Biol. 187(1), 81–89 (2009).
[Crossref]

Feiger, Z.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

Freitag, G.

A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
[Crossref]

Fricke, F.

G. Hummer, F. Fricke, and M. Heilemann, “Model-independent counting of molecules in single-molecule localization microscopy,” Mol. Biol. Cell 27(22), 3637–3644 (2016).
[Crossref]

Gratton, E.

M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
[Crossref]

Grotjohann, T.

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[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]

Gustafsson, M. G.

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. 102(37), 13081–13086 (2005).
[Crossref]

Haltmeier, M.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Harke, B.

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
[Crossref]

Heilemann, M.

G. Hummer, F. Fricke, and M. Heilemann, “Model-independent counting of molecules in single-molecule localization microscopy,” Mol. Biol. Cell 27(22), 3637–3644 (2016).
[Crossref]

Hell, S.

S. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21(11), 1347–1355 (2003).
[Crossref]

Hell, S. W.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[Crossref]

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[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]

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]

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
[Crossref]

L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
[Crossref]

S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14(5), 599–609 (2004).
[Crossref]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[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]

Herzig, A.

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[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]

Horwitz, A. F.

M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
[Crossref]

Hummer, G.

G. Hummer, F. Fricke, and M. Heilemann, “Model-independent counting of molecules in single-molecule localization microscopy,” Mol. Biol. Cell 27(22), 3637–3644 (2016).
[Crossref]

Irie, M.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

Isacoff, E. Y.

M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nat. Methods 4(4), 319–321 (2007).
[Crossref]

Jakobs, S.

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[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]

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14(5), 599–609 (2004).
[Crossref]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[Crossref]

Joshi, N.

N. Joshi, R. Szeliski, and D. J. Kriegman, “PSF estimation using sharp edge prediction,” in Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, (IEEE, 2008), pp. 1–8.

Jungmann, R.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

Kastrup, L.

L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
[Crossref]

Keller, J.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
[Crossref]

Klar, T. A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[Crossref]

Klug, A.

L. A. Amos and A. Klug, “Arrangement of subunits in flagellar microtubules,” Journal of cell science 14, 523–549 (1974).

Koppel, D. E.

D. E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A 10(6), 1938–1945 (1974).
[Crossref]

Kriegman, D. J.

N. Joshi, R. Szeliski, and D. J. Kriegman, “PSF estimation using sharp edge prediction,” in Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, (IEEE, 2008), pp. 1–8.

Latuerbach, M. A.

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[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]

Lee, A.

S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
[Crossref]

Lee, S.-H.

S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
[Crossref]

Lehmann, E. L.

E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses (Springer Science & Business Media, 2006).

Leutenegger, M.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[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]

Ludington, W. B.

B. D. Engel, W. B. Ludington, and W. F. Marshall, “Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model,” J. Cell Biol. 187(1), 81–89 (2009).
[Crossref]

Marshall, W. F.

B. D. Engel, W. B. Ludington, and W. F. Marshall, “Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model,” J. Cell Biol. 187(1), 81–89 (2009).
[Crossref]

Munk, A.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
[Crossref]

Neumann, D.

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

Nizamov, S.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[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]

Opazo, F.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Parthun, M. P.

V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
[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]

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]

Rodal, A.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

Romano, J. P.

J. P. Romano and M. Wolf, “A more general central limit theorem for m-dependent random variables with unbounded m,” Stat. & probability letters 47(2), 115–124 (2000).
[Crossref]

E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses (Springer Science & Business Media, 2006).

Roubinet, B.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[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]

Saka, S. K.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Schmied, J.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Schnorrenberg, S.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[Crossref]

Schönle, A.

Shin, J. Y.

S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
[Crossref]

Shojaei, H.

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

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]

Steger, C.

C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Machine Intell. 20(2), 113–125 (1998).
[Crossref]

Szeliski, R.

N. Joshi, R. Szeliski, and D. J. Kriegman, “PSF estimation using sharp edge prediction,” in Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, (IEEE, 2008), pp. 1–8.

Ta, H.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Testa, I.

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. 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]

Tinnefeld, P.

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Ulbrich, M. H.

M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nat. Methods 4(4), 319–321 (2007).
[Crossref]

Ullal, C. K.

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]

van der Vaart, A. W.

A. W. van der Vaart, Asymptotic Statistics (Cambridge University, 1998).

Vorbrüggen, G.

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[Crossref]

Wagner, T.

A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
[Crossref]

Westphal, V.

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]

Woehrstein, J. B.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

Wolf, M.

J. P. Romano and M. Wolf, “A more general central limit theorem for m-dependent random variables with unbounded m,” Stat. & probability letters 47(2), 115–124 (2000).
[Crossref]

Wu, J.-Q.

V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
[Crossref]

Wu, P.

V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
[Crossref]

Wurm, C. A.

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

Yin, P.

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

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]

Angew. Chem., Int. Ed. (1)

B. Roubinet, M. L. Bossi, P. Alt, M. Leutenegger, H. Shojaei, S. Schnorrenberg, S. Nizamov, M. Irie, V. N. Belov, and S. W. Hell, “Carboxylated Photoswitchable Diarylethenes for Biolabeling and Super-Resolution RESOLFT Microscopy,” Angew. Chem., Int. Ed. 55(49), 15429–15433 (2016).
[Crossref]

Biophys. J. (1)

M. A. Digman, R. Dalal, A. F. Horwitz, and E. Gratton, “Mapping the number of molecules and brightness in the laser scanning microscope,” Biophys. J. 94(6), 2320–2332 (2008).
[Crossref]

Bull. Amer. Math. Soc. (1)

T. T. Cheng, “The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan,” Bull. Amer. Math. Soc. 55(4), 396–402 (1949).
[Crossref]

Curr. Opin. Neurobiol. (1)

S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14(5), 599–609 (2004).
[Crossref]

eLife (2)

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]

S. Schnorrenberg, T. Grotjohann, G. Vorbrüggen, A. Herzig, S. W. Hell, and S. Jakobs, “In vivo super-resolution RESOLFT microscopy of Drosophila melanogaster,” eLife 5, e15567 (2016).
[Crossref]

IEEE Trans. Pattern Anal. Machine Intell. (1)

C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Machine Intell. 20(2), 113–125 (1998).
[Crossref]

IUBMB Life (1)

D. Bourgeois and V. Adam, “Reversible photoswitching in fluorescent proteins: a mechanistic view,” IUBMB Life 64(6), 482–491 (2012).
[Crossref]

J. Cell Biol. (2)

V. C. Coffman, P. Wu, M. P. Parthun, and J.-Q. Wu, “Cenp-a exceeds microtubule attachment sites in centromere clusters of both budding and fission yeast,” J. Cell Biol. 195(4), 563–572 (2011).
[Crossref]

B. D. Engel, W. B. Ludington, and W. F. Marshall, “Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model,” J. Cell Biol. 187(1), 81–89 (2009).
[Crossref]

J. Royal Stat. Soc. Ser. B (Statistical Methodol.) (1)

A. Munk, N. Bissantz, T. Wagner, and G. Freitag, “On difference-based variance estimation in nonparametric regression when the covariate is high dimensional,” J. Royal Stat. Soc. Ser. B (Statistical Methodol.) 67(1), 19–41 (2005).
[Crossref]

Journal of cell science (1)

L. A. Amos and A. Klug, “Arrangement of subunits in flagellar microtubules,” Journal of cell science 14, 523–549 (1974).

Mol. Biol. Cell (1)

G. Hummer, F. Fricke, and M. Heilemann, “Model-independent counting of molecules in single-molecule localization microscopy,” Mol. Biol. Cell 27(22), 3637–3644 (2016).
[Crossref]

Nat. Biotechnol. (1)

S. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21(11), 1347–1355 (2003).
[Crossref]

Nat. Commun. (1)

H. Ta, J. Keller, M. Haltmeier, S. K. Saka, J. Schmied, F. Opazo, P. Tinnefeld, A. Munk, and S. W. Hell, “Mapping molecules in scanning far-field fluorescence nanoscopy,” Nat. Commun. 6(1), 7977 (2015).
[Crossref]

Nat. Methods (3)

R. Jungmann, M. S. Avendaño, M. Dai, J. B. Woehrstein, S. S. Agasti, Z. Feiger, A. Rodal, and P. Yin, “Quantitative super-resolution imaging with qPAINT,” Nat. Methods 13(5), 439–442 (2016).
[Crossref]

M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nat. Methods 4(4), 319–321 (2007).
[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]

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. A (1)

D. E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A 10(6), 1938–1945 (1974).
[Crossref]

Phys. Rev. Lett. (1)

L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005).
[Crossref]

Proc. Natl. Acad. Sci. (4)

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. 97(15), 8206–8210 (2000).
[Crossref]

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. 102(37), 13081–13086 (2005).
[Crossref]

C. A. Wurm, D. Neumann, M. A. Latuerbach, B. Harke, A. Egner, S. W. Hell, and S. Jakobs, “Nanoscale distribution of mitochondrial import receptor tom20 is adjusted to cellular conditions and exhibits an inner-cellular gradient,” Proc. Natl. Acad. Sci. 108(33), 13546–13551 (2011).
[Crossref]

S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, “Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM),” Proc. Natl. Acad. Sci. 109(43), 17436–17441 (2012).
[Crossref]

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

Stat. & probability letters (1)

J. P. Romano and M. Wolf, “A more general central limit theorem for m-dependent random variables with unbounded m,” Stat. & probability letters 47(2), 115–124 (2000).
[Crossref]

Other (4)

P. Billingsley, Probability and Measure (Wiley, New York, 1979).

A. W. van der Vaart, Asymptotic Statistics (Cambridge University, 1998).

E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses (Springer Science & Business Media, 2006).

N. Joshi, R. Szeliski, and D. J. Kriegman, “PSF estimation using sharp edge prediction,” in Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, (IEEE, 2008), pp. 1–8.

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 (10)

Fig. 1.
Fig. 1. The molecular contribution function (MCF) (center, $2\times$ magnified, with a total contribution of 20 photons of a single molecule the image) quantitatively relates the number and positions of molecules $n_0=50$ shown in the molecular map (left) with the total number and distribution of observed photons in the image (right). The number of molecules in a region could be estimated by adding the recorded photons and dividing by the integrated MCF.
Fig. 2.
Fig. 2. Image formation in RESOLFT nanoscopy. a: Two-state switching model with a fluorescent on-state and a non-fluorescent off-state. The on- and off-switching rates depend on the laser intensity. b: Time evolution of the on-state population for the model in (a) and for a given total switching rate $k_{{\mbox {on}}}+k_{{\mbox {off}}}$. c: Left: Profile of the doughnut-shaped deactivation light (calculation using vectorial diffraction theory, $\lambda$ = 491 nm, NA=1.4(oil), blue) and corresponding parabolic approximation around the focal center (red, dashed). Right: Population of the on-state after application of the on- and off-switching light (blue) and with spatially constant on-switching and parabolic off-switching approximations (red, dashed). The saturation level ($\sigma I t$) of the off-switching is quite high (equals 20 on the rim of the doughnut distribution). d: Left: Confocal readout PSF (calculation using vectorial diffraction theory, $\lambda _{\mbox {excitation}}$ = 491 nm, $\lambda _{\mbox {detection}}$ = 525 nm, NA=1.4(oil), blue) and Gaussian peak approximation of equal (red, dashed) Right: Resulting shape of the molecular contribution function ($\mbox {MCF} / b=p_{{\mbox {act}}}h_{{\mbox {read}}}$) with calculated illumination and detection distributions using diffraction theory (blue) and with approximations (red, dashed).
Fig. 3.
Fig. 3. Concept for counting in RESOLFT nanoscopy. Left: Depiction of a cluster of $n_0=20$ molecules activated with different activation probabilities. Right: Photon counting histogram of the number of photons measured for different activation probabilities, while keeping the average number of photons per fluorophore constant. The variance of the distribution depends on the molecular parameters non-linearly. Mean and variance allow to calibrate the brightness per fluorophore.
Fig. 4.
Fig. 4. Switching kinetics measurements for rsEGFP2. a: Single switching curve with an off-switching light intensity of 500 W/cm$^2$. Fits for exponential decay (red) and gamma distributed exponential decay (yellow) with residuals shown below. b: Measured switching rate $k$ (inverse of the time where the signal drops to $1/e$ of the initial value) in dependence of the off-switching light intensity. c: Equilibrium off-switching level (average level of the fluorescence in a at long times) in dependence of the off-switching light intensity.
Fig. 5.
Fig. 5. Measurement of rsEGFP2-$\alpha$-tubulin and estimation of the shape of the MCF. a: RESOLFT 2D image of dissected body wall muscles of wandering third instar Drosophila melanogaster larvae expressing rsEGFP2-$\alpha$-tubulin. b: Line detection of the structures in the image in a. c: Result of the PSF estimation (superposition of two Gaussian peaks). d: Calibration of the activation probability (setting the equilibrium level to the one observed for rsEGFP in kinetics measurements). e: Simulated image resembling the image of a by taking the estimated structure from b and uniformly drawing a certain number of molecules from it. With the estimated MCF in c, noisy images can be simulated. f: Histograms of estimated readout brightness and estimated total fluorophore number for 1000 simulations. The true readout brightness and number of molecules in the image were chosen to equal the estimated readout brightness and number determined from the measurement. Scale bars 1$\mu$m (a,b,e), 200nm (c).
Fig. 6.
Fig. 6. Western blot analysis of body wall muscle protein lysate of rsEGFP2-$\alpha$-tubulin expressing Drosophila melanogaster third instar larvae. a: Comparison of the signal from rsEGFP2-$\alpha$-tubulin (A) to the signal of unlabeled $\alpha$-tubulin (B) via labeling using antiserum against $\alpha$-tubulin. Ratio was determined on the total signal intensity of the respective peak. b: Control staining using antiserum against GFP.
Fig. 7.
Fig. 7. Simulation of isolated clusters of molecules. Relative standard deviation of the estimated number (a) and estimated brightness (b) for $n_0=20$ molecules in one cluster, with an activation probability of 20%. Above a minimum threshold brightness, the counting error depends mainly on the number of measurements (N). Full simulation results are plotted as dots, the calculated expressions of Eq. (30) are shown as lines.
Fig. 8.
Fig. 8. Influence of the pixel shift between the two images $Y_1$ and $Y_2$ after splitting the recorded image. Simulations of images with line-like structures parallel ($\parallel$) and perpendicular ($\perp$) to the scanning direction. The biased estimator simply ignores the pixel shift, while the unbiased estimator implements Eq. (33). Scale bars, 200nm.
Fig. 9.
Fig. 9. Estimation of the brightness in regions of the image of Fig. 5(a). The regions were chosen manually to adapt to the structure and include similar amounts of structure. Their boundaries are shown as green lines in a. In b the estimated brightness for each region using only data from this region is shown as color in the respective regions.
Fig. 10.
Fig. 10. Empirical power of the Welch test applied to two samples of the simulated images in Fig. 5(e) for different brightnesses. The first sample was taken from a set of four possible area from the upper half and the second sample from four possible areas from the lower half of the simulated images. The brightness of the first sample is $b_1$ and that of the second sample $b_2$. The nominal test level was $\alpha =0.01$.

Tables (1)

Tables Icon

Table 1. Number estimation of molecules in the regions A-E of Fig. 5(a).

Equations (94)

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

Y ( x ) = ( n MCF ) ( x ) + ε ( x ) ,
n ( x ) = j = 1 n 0 δ ( x x j ) ,
S = R 2 MCF ( x ) d x .
n ^ i = 1 S X i Y ( x ) d x .
MCF ( x ) = p act ( x ) s read ( x ) .
p act ( x ) = p + ( p on p ) exp ( 4 ln ( 2 ) | x | 2 w RESOLFT 2 ) ,
w RESOLFT = 4 ln ( 2 ) σ λ / ( h c ) α I max t ,
s read ( x ) = b h read ( x ) .
h read ( x ) = exp ( 4 ln ( 2 ) | x | 2 w read 2 ) ,
A j ( s ) Bernoulli ( p act ( x j s ) ) .
B j ( s ) { Poisson ( b h read ( x j s ) ) , A j ( s ) ) = 1 0 , A j ( s ) ) = 0
E [ B j ( s ) ] = b h read ( x j s ) p act ( x j s ) Var [ B j ( s ) ] = E [ B j ( s ) 2 ] E [ B j ( s ) ] 2 = E [ B j ( s ) ] + b 2 h read 2 ( x j s ) p act ( x j s ) ( 1 p act ( x j s ) )
Y ( s ) = j B j ( s ) + D ( s ) ,
m ( s ) = b ( n h read p act ) ( s ) + d ( s ) , v ( s ) = m ( s ) + b 2 ( n h read 2 p act ( 1 p act ) ) ( s ) .
m ^ ( s ) = 1 N j N Y j ( s ) , v ^ ( s ) = 1 N 1 j N ( Y j ( s ) m ^ ( s ) ) 2 .
M = b n X H 1 + d total , V = M + b 2 n X H 2 ,
M ^ = s X m ^ ( s ) Δ p , V ^ = s X v ^ ( s ) Δ p .
b ^ = H 1 H 2 V ^ M ^ M ^ d total .
s ( t ) = A β α k α 1 Γ ( α ) exp ( β k ) exp ( k t ) d k + b = A β α ( β + t ) α + b ,
MCF ( r ) = b ( p exp ( 4 ln ( 2 ) r 2 w read 2 ) + ( p on p ) exp ( 4 ln ( 2 ) r 2 w eff 2 ) ) ,
w eff = ( w read 2 + w RESOLFT 2 ) 1 / 2 .
v ^ shift  ( s ) = 2 3 ( Y 1 ( s ) Y 2 ( s ) ) 2 ,
p on ( t ) = p + ( p 0 p ) exp ( k t ) ,
p = k on k on + k off .
k on = σ on λ / ( h c ) I , k off = σ off λ / ( h c ) I .
p on ( I t ) = p + ( p 0 p ) exp ( σ λ / ( h c ) I t ) .
m = b n 0 p on + d , v = m + b 2 n 0 p on ( 1 p on ) .
n ^ 0 = 1 p on p on ( m ^ d ) 2 v ^ m ^ , b ^ = 1 1 p on v ^ m ^ m ^ d .
( Δ m ) 2 = v / N , ( Δ v ) 2 = 2 v 2 N 1 .
Δ b b ( Δ v ) 2 ( v m ) 2 + ( Δ m ) 2 ( v d ) 2 ) ( v m ) 2 ( m d ) 2 , Δ n 0 n 0 ( Δ v ) 2 ( v m ) 2 + ( Δ m ) 2 ( 2 m d + 1 v m ) 2 .
Δ b b , Δ n 0 n 0 2 N 1 ( 1 + 1 b ( 1 p on ) ( 1 + 1 SBR ) ) .
SBR = m d d .
v ^ 1 ( s ) = ( Y 1 ( s ) 1 2 ( Y 2 ( s ) + Y 2 ( s + ) ) ) 2 .
v ^ shift  ( s ) = 2 3 ( Y 1 ( s ) Y 2 ( s ) ) 2 .
b ^ = H 1 H 2 1 | X | x X [ 2 3 ( Y ( x ) 1 2 Y ( x ) 1 2 Y ( x + ) ) 2 Y ( x ) ] 1 | X | x X ( Y ( x ) d ^ ( x ) ) ,
d ^ k ( x ) = 1 | X ~ k | s X ~ k Y k ( s ) .
b ^ k = H 1 ( k ) H 2 ( k ) 1 | X k | x X k ( 2 3 ( Y k ( x ) 1 2 Y k ( x ) 1 2 Y k ( x + ) ) 2 Y k ( x ) ) 1 | X k | x X k ( Y k ( x ) d ^ k ( x ) ) ,
lim k n 0 ( k ) | X k | = ρ .
lim k max x X k { Number of molecules at  x } = ρ ~ .
H 1 ( k ) = x X k h read ( x ) p act ( x ) , H 2 ( k ) = x X k h read 2 ( x ) p act ( x ) ( 1 p act ( x ) ) ,
lim k 1 | X k | x X k d ( x ) = D ¯ ( 0 , ) .
lim k E [ d ^ k ( x ) ] = D ¯ , Var [ x X k d ^ k ( x ) ] = C 1 | X k | + o ( | X k | ) , E [ d ^ k ( x ) 4 ] C 2 ,
w k ( x ) = E [ 2 3 ( Y k ( x ) 1 2 Y k ( x ) 1 2 Y k ( x + ) ) 2 Y k ( x ) ] , W k ( x ) = 2 3 ( Y k ( x ) 1 2 Y k ( x ) 1 2 Y k ( x + ) ) 2 Y k ( x ) w k ( x ) , z k ( x ) = E [ Y k ( x ) d ^ k ( x ) ] , Z k ( x ) = Y k ( x ) d ^ k ( x ) z k ( x ) ,
b ^ k := H 1 H 2 1 | X k | x X k W k ( x ) + w k ( x ) 1 | X k | x X k Z k ( x ) + z k ( x ) .
| X k | ( b ^ k E [ b ^ k ] ) D N ( 0 , σ 2 )
Y ~ k ( x ) N ( E [ Y k ( x ) ] , Var [ Y k ( x ) ] ) ,     x X k ,
σ ~ 2 = 1 ( ρ b H 1 H 2 ) 2 ( ρ b ( H 1 + b H 2 ) 3 + D ¯ ( H 1 + b H 2 ) 2 + H 1 2 | X k | x X k ( 35 9 v k ( x ) 2 5 9 v k ( x ) m k ( x ) 2 + v k ( x ) m k ( x ) ) ) ,
σ ~ | X k | = 0.041 , σ ~ b | X k | = 0.049 ,
Σ := lim k 1 | X k | ( Var [ x X k W k ( x ) ] Cov ( x X k W k ( x ) , x X k Z k ( x ) ) Cov ( x X k W k ( x ) , x X k Z k ( x ) ) Var [ x X k Z k ( x ) ] )
1 | X k | ( x X k W k ( x ) x X k Z k ( x ) ) D N ( ( 0 0 ) , Σ )
t T Σ t > 0
t T Σ k t C
Σ k := 1 | X k | ( Var [ x X k W k ( x ) ] Cov ( x X k W k ( x ) , x X k Z k ( x ) ) Cov ( x X k W k ( x ) , x X k Z k ( x ) ) Var [ x X k Z k ( x ) ] ) .
1 | X k | ( t 1 x X k W k ( x ) + t 2 x X k Z k ( x ) ) D N ( 0 , t T Σ t )
t T Σ t = t 1 2 Σ 1 , 1 + t 2 2 Σ 2 , 2 + 2 t 1 t 2 Σ 1 , 2 = lim k 1 | X k | Var [ x X k t 1 W k ( x ) + t 2 Z k ( x ) ] .
W k , i := W k ( x i ) ,     Z k , i := Z k ( x i )
E [ | t 1 W k , i + t 2 Z k , i | 2 + δ ] C 1 <      for all  i = 1 , , | X k | ,      and all  k N ,
Var [ i = a a + L 1 t 1 W k , i + t 2 Z k , i ] C 2 L      for all  a = 1 , , | X k | L + 1 ,   L = 2 , , | X k |      and all  k N ,
Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] C 3 | X k |      for all  k N ,
1 Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] i = 1 | X k | t 1 W k , i + t 2 Z k , i D N ( 0 , 1 )
1 Var [ x X k t 1 W k ( x ) + t 2 Z k ( x ) ] x X k t 1 W k ( x ) + t 2 Z k ( x ) D N ( 0 , 1 )
E [ | t 1 W k , i + t 2 Z k , i | 4 ] E [ W k , i 4 ] + 4 | E [ W k , i 3 z k , i ] | + 6 E [ W k , i 2 Z k , i 2 ] + 4 | E [ W k , i Z k , i 3 ] | + E [ Z k , i 4 ] C max j = i , i 1 , i 2 ( E [ Y k ( x j ) 8 ] + E [ d ^ k ( x i ) 4 ] ) ,
Var [ i = a a + L 1 t 1 W k , i + t 2 Z k , i ] = i = a a + L 1 Var [ t 1 W k , i + t 2 Z k , i ] + + 2 i = a a + L 1 Cov [ t 1 W k , i + t 2 Z k , i , t 1 W k , i 1 + t 2 Z k , i 1 ] + + 2 i = a a + L 1 Cov [ t 1 W k , i + t 2 Z k , i , t 1 W k , i 2 + t 2 Z k , i 2 ] i = a a + L 1 Var [ t 1 W k , i + t 2 Z k , i ] + + 2 i = a a + L 1 Var [ t 1 W k , i + t 2 Z k , i ] Var [ t 1 W k , i 1 + t 2 Z k , i 1 ] + + 2 i = a a + L 1 Var [ t 1 W k , i + t 2 Z k , i ] Var [ t 1 W k , i 2 + t 2 Z k , i 2 ] 5 i = a 2 a + L 1 Var [ t 1 W k , i + t 2 Z k , i ]
Var [ t 1 W k , i + t 2 Z k , i ] C 2
Var [ t 1 W k , i + t 2 Z k , i ] t 1 2 Var [ W k , i ] + t 2 2 Var [ Z k , i ] + 2 t 1 t 2 Cov [ W k , i , Z k , i ] 2 ( Var [ W k , i ] + Var [ Z k , i ] )
Z k , i = Y k ( x i ) d ^ k ( x i ) E [ Y k ( x i ) d ^ k ( x i ) ] ,
Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] = Var [ i = 1 | X k | t 1 W k , i + t 2 Y k ( x i ) ] + t 2 2 Var [ i = 1 | X k | d ^ k ( x i ) ] + 2 t 2 Cov ( i = 1 | X k | t 1 W k , i + t 2 Y k ( x i ) , i = 1 | X k | d ^ k ( x i ) ) = 0 ,
Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] = Var [ i = 1 | X k | t 1 W k , i + t 2 Y k ( x i ) ] 0 + t 2 2 Var [ i = 1 | X k | d ^ k ( x i ) ] ϵ 2 C | X k |
Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] = t 1 2 Var [ i = 1 | X k | W k , i ] + t 2 2 ( Var [ i = 1 | X k | Y k ( x i ) ] + Var [ i = 1 | X k | d ^ k ( x i ) ] ) + + 2 t 1 t 2 Cov ( i = 1 | X k | W k , i , i = 1 | X k | Y k ( x i ) ) .
Var [ i = 1 | X k | Y k ( x i ) ] | X k | ρ ( b H 1 + b 2 H 2 ) + | X k | D ¯
| Cov ( i = 1 | X k | W k , i , i = 1 | X k | Y k ( x i ) ) | Var [ i = 1 | X k | W k , i ] Var [ i = 1 | X k | Y k ( x i ) ] max { Var [ i = 1 | X k | W k , i ] , Var [ i = 1 | X k | Y k ( x i ) ] } = Var [ i = 1 | X k | W k , i ]
Var [ i = 1 | X k | W k , i ] Var [ i = 1 | X k | Y k ( x i ) ]
Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] t 1 2 Var [ i = 1 | X k | W k , i ] + t 2 2 C | X k | 2 | t 1 t 2 | | Cov ( i = 1 | X k | W k , i , i = 1 | X k | Y k ( x i ) ) | t 1 2 Var [ i = 1 | X k | W k , i ] + t 2 2 C | X k | 2 | t 2 t 2 | Var [ i = 1 | X k | W k , i ] = | t 1 | ( ( | t 1 | 2 | t 2 | ) Var [ i = 1 | X k | W k , i ] + t 2 2 C | X k |
t 1 = 1 t 2 2 > 1 ϵ 2 = 5 6 ,
t 1 ( t 1 2 t 2 ) > t 1 ( 5 6 2 6 ) t 1 0.2 6 > 0.05.
Var [ i = 1 | X k | t 1 W k , i + t 2 Z k , i ] 0.05 Var [ i = 1 | X k | W k , i ] + t 2 2 C | X k | .
| X k | ( b ^ k b ) D N ( 0 , σ 2 ) ,
σ 2 = 1 ( ρ b H 1 H 2 ) 2 ( H 1 2 Σ 1 , 1 + b 2 H 2 2 Σ 2 , 2 2 b H 1 H 2 Σ 1 , 2 ) ,
α k := x X k W k ( x ) + w k ( x ) | X k | , β k := x X k Z k ( x ) + z k ( x ) | X k | ,
α := lim k x X k w k ( x ) | X k | ,       β := lim k x X k z k ( x ) | X k | ,
| X k | ( ( α k β k ) ( α β ) ) D N ( ( 0 0 ) , Σ ) .
| X k | ( α k β k α β ) D N ( 0 , σ ¯ 2 )
σ ¯ 2 = ( 1 β , α β 2 ) Σ ( 1 β , α β 2 ) T ,
α = ρ b ( H 1 + b H 2 ) + D ¯ ,           β = ρ b H 1 .
Y ~ k ( x ) N ( E [ Y k ( x ) ] , Var [ Y k ( x ) ] ) ,     x X k ,
m k ( x ) = E [ Y k ( x ) ] ,       v k ( x ) = Var [ Y k ( x ) ] .
W ~ k ( x ) = 2 3 ( Y ~ k ( x ) 1 2 Y ~ k ( x ) 1 2 Y ~ k ( x + ) ) 2 Y ~ k ( x ) .
Σ 2 , 2 = lim k 1 | X k | Var [ x X k Y ~ k ( x ) d ^ k ( x ) ] = ρ b ( H 1 + b H 2 ) + D ¯ .
1 | X k | Cov ( x X k W ~ k ( x ) , x X k Y ~ k ( x ) d ^ k ( x ) ) = = 1 | X k | x X k v k ( x ) + 1 | X k | x i X k v k ( x i ) ( m k ( x i ) + 1 3 m k ( x i 2 ) 4 3 m k ( x i 1 ) ) + 1 | X k | x i X k m k ( x ) ( v k ( x i ) + 1 3 v k ( x i 2 ) 4 3 v k ( x i 1 ) ) 1 | X k | x X k v k ( x )
Σ 1 , 2 = lim k 1 | X k | Cov ( x X k W ~ k ( x ) , x X k Y ~ k ( x ) d ^ k ( x ) ) ρ b ( H 1 + b H 2 ) D ¯
T k ( x i ) := m k ( x i ) + 1 3 m k ( x i 2 ) 4 3 m k ( x i 1 ) ,
Var [ x X k W ~ k ( x ) ] = x X k v k ( x ) + 2 v k ( x ) 2 + x i X k v k ( x i ) ( T k ( x i ) 2 T k ( x i ) m k ( x i ) + 4 9 m k ( x i 1 ) m k ( x i 2 ) ) + x i X k v k ( x i 1 ) ( 16 9 m k ( x i ) 2 + 16 9 v k ( x i ) 8 3 m k ( x i ) ( T k ( x i 1 ) + m k ( x i 1 ) 1 ) ) + x i X k v k ( x i 2 ) ( 1 9 m k ( x i ) 2 + 1 9 v k ( x i ) 8 9 m k ( x i ) m k ( x i 1 ) + + 2 3 m k ( x i ) ( T k ( x i 2 ) + m k ( x i 2 ) 1 ) ) .
1 | X k | Var [ x X k W ~ k ( x ) ] 1 | X k | x X k v k ( x ) + 1 | X k | x X k ( 35 9 v k ( x ) 2 5 9 v k ( x ) m k ( x ) 2 + v k ( x ) m k ( x ) ) ,
Σ 1 , 1 ρ b ( H 1 + b H 2 ) + D ¯ + lim k 1 | X k | x X k ( 35 9 v k ( x ) 2 5 9 v k ( x ) m k ( x ) 2 + v k ( x ) m k ( x ) ) ,

Metrics