Abstract

Superresolution optical fluctuation imaging (SOFI) is a simple and affordable super-resolution imaging technique, and attracted a growing community over the past decade. However, the theoretical resolution enhancement of high order SOFI is still not fulfilled. In this study, we identify “cusp artifacts” in high order SOFI images, and show that the high-order cumulants, odd-order moments and balanced-cumulants (bSOFI) are highly vulnerable to cusp artifacts. Our study provides guidelines for developing and screening for fluorescence probes, and improving data acquisition for SOFI. The new insight is important to inspire positive utilization of the cusp artifacts.

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

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References

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2019 (1)

2016 (3)

S. Jiang, Y. Zhang, H. Yang, Y. Xiao, X. Miao, R. Li, Y. Xu, and X. Zhang, “Enhanced SOFI algorithm achieved with modified optical fluctuating signal extraction,” Opt. Express 24(3), 3037–3045 (2016).
[Crossref]

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (2016).
[Crossref]

2015 (6)

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

A. M. Sydor, K. J. Czymmek, E. M. Puchner, and V. Mennella, “Super-resolution microscopy: from single molecules to supramolecular assemblies,” Trends Cell Biol. 25(12), 730–748 (2015).
[Crossref]

E. Betzig, “Single Molecules, Cells, and Super-Resolution Optics (Nobel Lecture),” Angew. Chem., Int. Ed. 54(28), 8034–8053 (2015).
[Crossref]

W. E. Moerner, “Single-Molecule Spectroscopy, Imaging, and Photocontrol: Foundations for Super-Resolution Microscopy (Nobel Lecture),” Angew. Chem., Int. Ed. 54(28), 8067–8093 (2015).
[Crossref]

S. W. Hell, “Nanoscopy with focused light,” Ann. Phys. 527(7-8), 423–445 (2015).
[Crossref]

2013 (2)

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

2012 (2)

P. Dedecker, G. C. Mo, T. Dertinger, and J. Zhang, “Widely accessible method for superresolution fluorescence imaging of living systems,” Proc. Natl. Acad. Sci. 109(27), 10909–10914 (2012).
[Crossref]

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

2010 (2)

T. Dertinger, R. Colyer, R. Vogel, J. Enderlein, and S. Weiss, “Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI),” Opt. Express 18(18), 18875–18885 (2010).
[Crossref]

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

2009 (1)

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[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]

2001 (1)

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

1994 (1)

1992 (1)

Ando, R.

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]

Battle, C.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Bawendi, M. G.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

Berclaz, C.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

Betzig, E.

E. Betzig, “Single Molecules, Cells, and Super-Resolution Optics (Nobel Lecture),” Angew. Chem., Int. Ed. 54(28), 8034–8053 (2015).
[Crossref]

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]

Bocchio, N. L.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[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]

Braun, G.

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

Brunetti, R.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Charles, P.

P. Charles, Digital Video and HDTV Algorithms and Interfaces (Morgan Kaufmann Publishers, 2003), 260, 630.

Chen, X.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Chizhik, A. M.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Cho, S.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Choi, M. C.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Colyer, R.

T. Dertinger, R. Colyer, R. Vogel, J. Enderlein, and S. Weiss, “Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI),” Opt. Express 18(18), 18875–18885 (2010).
[Crossref]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[Crossref]

Czymmek, K. J.

A. M. Sydor, K. J. Czymmek, E. M. Puchner, and V. Mennella, “Super-resolution microscopy: from single molecules to supramolecular assemblies,” Trends Cell Biol. 25(12), 730–748 (2015).
[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]

Dedecker, P.

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (2016).
[Crossref]

P. Dedecker, G. C. Mo, T. Dertinger, and J. Zhang, “Widely accessible method for superresolution fluorescence imaging of living systems,” Proc. Natl. Acad. Sci. 109(27), 10909–10914 (2012).
[Crossref]

Dekaliuk, M. O.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Dellagiacoma, C.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

Demchenko, A. P.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Dertinger, T.

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

P. Dedecker, G. C. Mo, T. Dertinger, and J. Zhang, “Widely accessible method for superresolution fluorescence imaging of living systems,” Proc. Natl. Acad. Sci. 109(27), 10909–10914 (2012).
[Crossref]

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

T. Dertinger, R. Colyer, R. Vogel, J. Enderlein, and S. Weiss, “Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI),” Opt. Express 18(18), 18875–18885 (2010).
[Crossref]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[Crossref]

Do Heo, W.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Duwé, S.

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (2016).
[Crossref]

Empedocles, S. A.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

Enderlein, J.

T. Dertinger, R. Colyer, R. Vogel, J. Enderlein, and S. Weiss, “Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI),” Opt. Express 18(18), 18875–18885 (2010).
[Crossref]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[Crossref]

Ganesan, P.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Geissbuehler, S.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

Gibson, S. F.

Gregor, I.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Gustafsson, M. G.

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

Heilemann, M.

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

Hell, S. W.

Hertel, F.

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (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]

Higgins, D. A.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Huss, A.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Ihee, H.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Iyer, G.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[Crossref]

Jang, J.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Jiang, S.

Kendall, M. G.

M. G. Kendall, Alan Stuart, and J. K. Ord, The Advanced Theory of Statistics (Wiley, 1968), Vol. 3.

Kim, M. W.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Kirkeminde, A. W.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Kisley, L.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Landes, C. F.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Lanni, F.

Lasser, T.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

Laurence, T. A.

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

Leatherdale, C. A.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

Lee, H.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Leutenegger, M.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

Li, R.

Li, W.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[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]

Ly, S.

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

Mennella, V.

A. M. Sydor, K. J. Czymmek, E. M. Puchner, and V. Mennella, “Super-resolution microscopy: from single molecules to supramolecular assemblies,” Trends Cell Biol. 25(12), 730–748 (2015).
[Crossref]

Miao, X.

Miyawaki, A.

Mo, G. C.

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (2016).
[Crossref]

P. Dedecker, G. C. Mo, T. Dertinger, and J. Zhang, “Widely accessible method for superresolution fluorescence imaging of living systems,” Proc. Natl. Acad. Sci. 109(27), 10909–10914 (2012).
[Crossref]

Moerner, W. E.

W. E. Moerner, “Single-Molecule Spectroscopy, Imaging, and Photocontrol: Foundations for Super-Resolution Microscopy (Nobel Lecture),” Angew. Chem., Int. Ed. 54(28), 8067–8093 (2015).
[Crossref]

Neuhauser, R. G.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[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]

Ord, J. K.

M. G. Kendall, Alan Stuart, and J. K. Ord, The Advanced Theory of Statistics (Wiley, 1968), Vol. 3.

Pallaoro, A.

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[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]

Peng, J.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Platen, M.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Puchner, E. M.

A. M. Sydor, K. J. Czymmek, E. M. Puchner, and V. Mennella, “Super-resolution microscopy: from single molecules to supramolecular assemblies,” Trends Cell Biol. 25(12), 730–748 (2015).
[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]

Sauer, M.

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

Schaap, I. A.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Shimizu, K. T.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

Shuang, B.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Son, S.

Song, C.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[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]

Stein, S.

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[Crossref]

Stuart, Alan

M. G. Kendall, Alan Stuart, and J. K. Ord, The Advanced Theory of Statistics (Wiley, 1968), Vol. 3.

Sun, Y.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Sydor, A. M.

A. M. Sydor, K. J. Czymmek, E. M. Puchner, and V. Mennella, “Super-resolution microscopy: from single molecules to supramolecular assemblies,” Trends Cell Biol. 25(12), 730–748 (2015).
[Crossref]

Tauzin, L. J.

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Vogel, R.

T. Dertinger, R. Colyer, R. Vogel, J. Enderlein, and S. Weiss, “Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI),” Opt. Express 18(18), 18875–18885 (2010).
[Crossref]

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

Weiss, S.

X. Yi, S. Son, R. Ando, A. Miyawaki, and S. Weiss, “Moments reconstruction and local dynamic range compression of high order Superresolution Optical Fluctuation Imaging,” Biomed. Opt. Express 10(5), 2430–2445 (2019).
[Crossref]

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

T. Dertinger, R. Colyer, R. Vogel, J. Enderlein, and S. Weiss, “Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI),” Opt. Express 18(18), 18875–18885 (2010).
[Crossref]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[Crossref]

X. Yi and S. Weiss. “Cusp-artifacts in high order superresolution optical fluctuation imaging.” bioRxiv: 545574 (2019).

Wichmann, J.

Woo, W.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

Xi, P.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Xiao, Y.

Xu, P.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Xu, Y.

Yang, H.

Yi, X.

X. Yi, S. Son, R. Ando, A. Miyawaki, and S. Weiss, “Moments reconstruction and local dynamic range compression of high order Superresolution Optical Fluctuation Imaging,” Biomed. Opt. Express 10(5), 2430–2445 (2019).
[Crossref]

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

X. Yi, “SR_Simu3D” (2018), retrieved https://xiyuyi.github.io/SR_simu3D/ .

X. Yi and S. Weiss. “Cusp-artifacts in high order superresolution optical fluctuation imaging.” bioRxiv: 545574 (2019).

Yoon, T.-Y.

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[Crossref]

Zeng, Z.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Zhang, J.

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (2016).
[Crossref]

P. Dedecker, G. C. Mo, T. Dertinger, and J. Zhang, “Widely accessible method for superresolution fluorescence imaging of living systems,” Proc. Natl. Acad. Sci. 109(27), 10909–10914 (2012).
[Crossref]

Zhang, M.

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Zhang, X.

S. Jiang, Y. Zhang, H. Yang, Y. Xiao, X. Miao, R. Li, Y. Xu, and X. Zhang, “Enhanced SOFI algorithm achieved with modified optical fluctuating signal extraction,” Opt. Express 24(3), 3037–3045 (2016).
[Crossref]

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

Zhang, Y.

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]

ACS Nano (2)

X. Zhang, X. Chen, Z. Zeng, M. Zhang, Y. Sun, P. Xi, J. Peng, and P. Xu, “Development of a reversibly switchable fluorescent protein for super-resolution optical fluctuation imaging (SOFI),” ACS Nano 9(3), 2659–2667 (2015).
[Crossref]

L. Kisley, R. Brunetti, L. J. Tauzin, B. Shuang, X. Yi, A. W. Kirkeminde, D. A. Higgins, S. Weiss, and C. F. Landes, “Characterization of porous materials by fluorescence correlation spectroscopy super-resolution optical fluctuation imaging,” ACS Nano 9(9), 9158–9166 (2015).
[Crossref]

Angew. Chem. (1)

T. Dertinger, M. Heilemann, R. Vogel, M. Sauer, and S. Weiss, “Superresolution optical fluctuation imaging with organic dyes,” Angew. Chem. 122(49), 9631–9633 (2010).
[Crossref]

Angew. Chem., Int. Ed. (2)

E. Betzig, “Single Molecules, Cells, and Super-Resolution Optics (Nobel Lecture),” Angew. Chem., Int. Ed. 54(28), 8034–8053 (2015).
[Crossref]

W. E. Moerner, “Single-Molecule Spectroscopy, Imaging, and Photocontrol: Foundations for Super-Resolution Microscopy (Nobel Lecture),” Angew. Chem., Int. Ed. 54(28), 8067–8093 (2015).
[Crossref]

Ann. Phys. (1)

S. W. Hell, “Nanoscopy with focused light,” Ann. Phys. 527(7-8), 423–445 (2015).
[Crossref]

Biomed. Opt. Express (1)

Cell Rep. (1)

F. Hertel, G. C. Mo, S. Duwé, P. Dedecker, and J. Zhang, “RefSOFI for mapping nanoscale organization of protein-protein interactions in living cells,” Cell Rep. 14(2), 390–400 (2016).
[Crossref]

J. Microsc. (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

J. Opt. Soc. Am. A (1)

Nano Lett. (1)

A. M. Chizhik, S. Stein, M. O. Dekaliuk, C. Battle, W. Li, A. Huss, M. Platen, I. A. Schaap, I. Gregor, and A. P. Demchenko, “Super-resolution optical fluctuation bio-imaging with dual-color carbon nanodots,” Nano Lett. 16(1), 237–242 (2016).
[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]

Opt. Express (2)

Opt. Lett. (1)

Opt. Nanosc. (1)

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI),” Opt. Nanosc. 1(1), 4 (2012).
[Crossref]

Phys. Rev. B (1)

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63(20), 205316 (2001).
[Crossref]

Proc. Natl. Acad. Sci. (2)

P. Dedecker, G. C. Mo, T. Dertinger, and J. Zhang, “Widely accessible method for superresolution fluorescence imaging of living systems,” Proc. Natl. Acad. Sci. 109(27), 10909–10914 (2012).
[Crossref]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. 106(52), 22287–22292 (2009).
[Crossref]

Q. Rev. Biophys. (1)

T. Dertinger, A. Pallaoro, G. Braun, S. Ly, T. A. Laurence, and S. Weiss, “Advances in superresolution optical fluctuation imaging (SOFI),” Q. Rev. Biophys. 46(2), 210–221 (2013).
[Crossref]

Sci. Rep. (1)

S. Cho, J. Jang, C. Song, H. Lee, P. Ganesan, T.-Y. Yoon, M. W. Kim, M. C. Choi, H. Ihee, and W. Do Heo, “Simple super-resolution live-cell imaging based on diffusion-assisted Förster resonance energy transfer,” Sci. Rep. 3(1), 1208 (2013).
[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]

Trends Cell Biol. (1)

A. M. Sydor, K. J. Czymmek, E. M. Puchner, and V. Mennella, “Super-resolution microscopy: from single molecules to supramolecular assemblies,” Trends Cell Biol. 25(12), 730–748 (2015).
[Crossref]

Other (5)

M. G. Kendall, Alan Stuart, and J. K. Ord, The Advanced Theory of Statistics (Wiley, 1968), Vol. 3.

P. Charles, Digital Video and HDTV Algorithms and Interfaces (Morgan Kaufmann Publishers, 2003), 260, 630.

X. Yi, “SR_Simu3D” (2018), retrieved https://xiyuyi.github.io/SR_simu3D/ .

e. a. Marcel Leutenegger, Balanced super-resolution optical fluctuation imaging, 2012.

X. Yi and S. Weiss. “Cusp-artifacts in high order superresolution optical fluctuation imaging.” bioRxiv: 545574 (2019).

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Figures (8)

Fig. 1.
Fig. 1. Demonstration of cusp artifact. Panels (i) and (ii) show two representations of same theoretical 3rd-order SOFI cumulant image of two emitters. (i) shows grayscale display, while (ii) shows g/r color code, with red (left lobe) representing positive cumulant values and green (right lobe) representing negative cumulant values; both have dynamic range of −0.048 to 0.048. On-time ratio of left emitter was set to 0.4 and that of right emitter to 0.6. Panels (iii) and (iv) show cross-sectional plots at dashed lines in (i) and (ii), respectively. Panel (v) shows plots of cumulants ωn as function of ρ, with n = 2–7. All cumulants (n > 2) oscillate between negative and positive values, and the number of zero crossings increases with the increase of cumulant order. The total number of zero crossings is always (n-2). As can be seen, ω3(0.4) > 0 and ω3(0.6) < 0, corresponding to virtual brightnesses shown in (i) and (ii). Cusp artifact for this example is highlighted by arrow in (iv).
Fig. 2.
Fig. 2. Demonstration (through simulations) of cusp artifacts for three adjacent blinking emitters spaced equally along line (spacing of 193 nm between nearest neighbors). Simulation parameters were: emission wavelength = 800 nm; numerical aperture (NA) = 1.4; and pixel size = 93.33 nm. For blinking statistics (simulated by Monte Carlo method), we set ρ to 0.831, 0.416, and 0.103 for emitters 1, 2, and 3 respectively. Orders of cumulants are shown in top row. Signs of virtual brightness values, as predicted in Fig. 1(v), are denoted by (+/-) signs (second row). Third row shows simulated SOFI images for cumulants with orders 2–7 (left to right). Fourth row (i) shows cross-sections for dotted white line in second row (in blue; red line denotes emitter positions). Fifth row shows absolute value cross-sections of |i| for dotted white line in second row (in green; red line denotes emitter positions). Sign oscillations and cusp artifacts are evident, except for 2nd order. Scale bars: 280 nm.
Fig. 3.
Fig. 3. Gamma-corrected high-order SOFI-processed experimental images displaying cusp artifacts. Fixed HeLa cells were labeled with QDs (emission wavelength = 800 nm) by immunostaining using primary antibody (eBioscience, Cat#: 14-4502-80) and secondary antibody conjugated to QD800 (ThermoFisher Scientific. Ref#: Q11071MP). Total of 2000 frames (exposure time of 30 ms) were processed to obtain SOFI cumulants with up to 7th order using both auto- and cross-correlations. In order to better illustrate source of cusp artifacts, final SOFI processing steps of deconvolution and Fourier reweighting were skipped. Each SOFI image of particular order is presented in three panels: large field-of-view (left), magnified absolute value SOFI image of box area (middle), and magnified positive/negative values SOFI image of box area (right). Positive/negative domains are color coded separately as shown by color bars for each panel, with color scheme shown at bottom. Cusp artifacts can be seen clearly for cumulants of orders greater than two: spatial distributions of cusps for cumulants of different orders differ and are located at boundaries between positive and negative domains. Scale bars: 3.2 µm (left) and 1.6 µm (middle/right). Image intensities are displayed with gamma correction to highlight the cusps, therefore resolution enhancement is not evident. Gamma values are the multiplicative inverse of the cumulant order. More comprehensive displays are available in Appendix 5 [28].
Fig. 4.
Fig. 4. Simulations (‘Simulation-1’) showing dependence of cusp artifacts on blinking statistics. (i) Four different populations of simulated emitters with different distributions of τon and τoff values, yielding four different distributions of ρ values (P1, P2, P3, and P4; dashed red, blue, black, and green curves respectively) are plotted on top in Fig. 1(v). (ii). Predicted signs of calculated virtual brightnesses for P1, P2, P3, and P4 for cumulants with orders of 2 to 7. (iii) SOFI processing of simulated data. Simulated filamentous morphology was populated with emitters from either P1, P2, P3, or P4. Signs of virtual brightnesses (red for positive, green for negative) of virtual emitters are mostly in keeping with predicted ones in (ii) for different orders, with exception of out-of-focus (P2) regions (details are given in Appendix 2 [28]) and case with imbalanced population of virtual brightnesses, where smaller portion is more attenuated (4th-order cumulant of P3 contains 14.5% positive virtual emitters, and positive virtual brightnesses are attenuated by small amplitude of cumulant). Scale bars: 933 nm. Image intensities are displayed with gamma correction, with gamma value equals to multiplicative inverse of the cumulant order.
Fig. 5.
Fig. 5. Simulations (‘Simulation-2’) to evaluate effects of bleaching and noise on cusp artifacts. (i) Total signal, summed over all pixels, as function of movie frame (time), after bleaching operator was used (before addition of background noise). (ii) Magnified version of curve in (i), showing intensity fluctuations. (iii) Example of single-emitter blinking trajectory that was “bleached” at approximately ∼17 s. (iv) Array of SOFI cumulant images for orders 2nd to 7th as functions of noise. Top row shows simulated images with no bleaching. Signs of virtual brightnesses are in keeping with predictions in Fig. 4(i) (bleaching correction factor, fbc, of 100% means there was no bleaching correction). Second row shows simulated images for bleaching correction factor, fbc, of 1% (see text). Bleaching correction algorithm was effective in restoring absolute value of virtual brightness distribution but not brightness signs. Bleaching correction protocol changed signs of cumulants, resulting in rapid sign changes in cases of 3rd, 5th, and 7th (odd) orders. Real background noise (recorded as empty frames with EMCCD camera) when added to simulated bleaching data severely degrades quality of images (background noise is always positive). If emitters’ blinking statistics yield pure negative virtual brightnesses, as shown for 4th-order cumulant in third row, significant enhancement in contrast results. Scale bars: 2.8 µm. Image intensities are displayed with gamma correction, with gamma value equals to multiplicative inverse of the cumulant order.
Fig. 6.
Fig. 6. Simulations (‘Simulation-3’) to elucidate dependence of cusp artifacts on slowly varying ρ. A semicircle is populated with emitters having ρ values ranging from 0.01 to 0.99, as indicated in left panel, with interval being 0.01. Cumulants of different orders are displayed using gamma scale (gamma = 1/n) in right panels as labeled. Color coding is represented by the bar at bottom right (green for negative values, red for positive values, black for 0). In each image panel, inset shows ground-truth virtual emitters of given order, with red indicating positive virtual emitters and green indicating negative virtual emitters. Number of zero crossings (green/red transitions) increased with order of cumulants along gradient of ρ values, which is in agreement with ground truths shown in insets. Scale bars: 1.4 µm. Image intensities are displayed with gamma correction, with gamma value equals to the multiplicative inverse of the cumulant order.
Fig. 7.
Fig. 7. Post-processing of SOFI reconstructions containing cusp artifacts (simulation). Amplitudes of reconstructions are shown in grayscale (each panel has different dynamic range). Background of each panel is always zero (and therefore should be used as reference). Negative pixel values have darker colors than background; positive pixel values have lighter colors than background. (i) Ground-truth virtual emitters with both positive and negative values. (ii) Corresponding 3rd-order cumulant image (convolved with PSF). (iii) Amplitude (absolute value) of (ii) cusps are clearly visible. (iv) Ideal deconvolution result obtained by dividing Fourier-transformed image by optical transfer function (OTF) and subsequently performing inverse Fourier transformation. (v) Ideal Fourier reweighting, where, in contrast to the case for ideal deconvolution, Fourier spectrum is multiplied by extending the OTF before followed with inverse Fourier transform. (vi) Deconvolution result obtained using “deconvlucy” function, which imposes positivity constraint, that could affect the deconvolution when the corresponding ground-truth contains a mixture of positive and negative virtual brightnesses. PSF is simulated as perfect Gaussian with standard deviation of 4 pixels, as shown by the isolated emitter at the bottom right corner on each panel.
Fig. 8.
Fig. 8. QD800-labeled microtubules (experimental data). For top two rows - first column: (i) average image of movie (2000 frames, 30 ms per frame) of QD800-labeled α-tubulin in fixed HeLa cell. (ii) magnified boxed region in sum image (i); second column: (iii) and (iv) show 2nd-order SOFI cumulants with extra pixels generated by cross-correlations (XC2), corresponding to panels (i) and (ii) respectively; third column: 6th-order moment-reconstruction with local dynamic range correction, corresponding to panels (i) and (ii) respectively; fourth column: 6th-order bSOFI corresponding to panels (i) and (ii). (ix) in last row: normalized intensity profiles of three green dashed lines (1—1’, 2—2’, and 3—3’) as labeled on top of each panel. Intensity profiles of Average, XC2, and M6 + ldrc are compared; legend is provided in left-most panel. Detailed analysis of moments reconstruction are available in the accompanying manuscript [21].

Equations (9)

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F ( r , t ) = k = 1 N ϵ k b k ( t ) U ( r r k ) ,
δ F ( r , t ) = F ( r ) F ( r ) t = k = 1 N ϵ k δ b k ( t ) U ( r r k ) ,
C 2 ( τ ) = δ F ( r , t ) δ F ( r , t + τ ) = k = 1 N j = 1 N δ b k ( t ) δ b k ( t + τ ) t U ( r r k ) U ( r r j ) .
δ b k ( t ) δ b j ( t + τ ) = { = 0 , for k j = 1 , for k = j .
C 2 ( τ ) = k = 1 N ϵ k 2 ω 2 , k ( τ ) U 2 ( r r k ) ,
C n ( r , τ 1 , , τ ) = k = 1 N ϵ k n ω n , k ( τ 1 , τ 2 , , τ n 1 ) U 2 ( r r k ) .
U n ( r x , r y , r z ) = exp ( ( r x 2 + r y 2 2 ( σ x y n ) 2 + r z 2 2 ( σ z n ) 2 ) ) .
ρ = τ o n τ o n + τ o f f .
ω 2 ( ρ ) = ρ ρ 2 ω 3 ( ρ ) = ρ 3 ρ 2 + 2 ρ 3 ω 4 ( ρ ) = ρ 7 ρ 2 + 12 ρ 3 6 ρ 4 ω 5 ( ρ ) = ρ 15 ρ 2 + 50 ρ 3 60 ρ 4 + 24 ρ 5 ω 6 ( ρ ) = ρ 31 ρ 2 + 180 ρ 3 390 ρ 4 + 360 ρ 5 120 ρ 6 ω 7 ( ρ ) = ρ 63 ρ 2 + 602 ρ 3 2100 ρ 4 + 3360 ρ 5 2520 ρ 6 + 720 ρ 7 .

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