Abstract

The fast imaging speed and low-intensity requirement of structured illumination microscopy (SIM) have made it one of the most widely used imaging tools in live cell imaging. In order to obtain a high fidelity reconstructed image, a precise estimation of the phase of the illumination pattern is required, especially in those structured illumination based techniques that rely on high-order harmonics to improve the resolution. This can be achieved in one of two fundamental ways. The first is to build a high-end control system capable of shifting a sinusoidal pattern with high precision, while the second is to apply estimation algorithms to determine how patterns shift during post-processing. The latter method is preferred in low-cost super-resolution imaging systems; however, existing algorithms are either time-consuming or fail due to noise and a low modulation depth. In this paper, we introduce additional matrixes into the phase estimation algorithm and propose an inverse matrix based phase estimation method with which analytical solutions of the phases can be determined without iteration. The proposed algorithm was validated via simulation and experiments using a home-made total internal reflection fluorescent SIM system (TIRF-SIM). When tested, the method obtained the true phase even when the modulation depth was low. The source code is now available for download by researchers and others.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]

2018 (1)

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

2016 (3)

V. Perez, B.-J. Chang, and E. H. K. Stelzer, “Optimal 2D-SIM reconstruction by two filtering steps with Richardson-Lucy deconvolution,” Sci. Rep. 6(1), 37149 (2016).
[Crossref] [PubMed]

A. Lal, C. Shan, and P. Xi, “Structured Illumination Microscopy Image Reconstruction Algorithm,” IEEE J. Sel. Top. Quantum Electron. 22(4), 50–63 (2016).
[Crossref]

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
[Crossref] [PubMed]

2015 (1)

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

2014 (1)

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (1)

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

2011 (1)

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

2010 (1)

F. Wei and Z. Liu, “Plasmonic Structured Illumination Microscopy,” Nano Lett. 10(7), 2531–2536 (2010).
[Crossref] [PubMed]

2009 (2)

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” J. Opt. Soc. Am. A 26(2), 413–424 (2009).
[Crossref] [PubMed]

2008 (3)

R. Fiolka, M. Beck, and A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett. 33(14), 1629–1631 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

2007 (1)

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

2006 (2)

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] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

2005 (1)

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

2000 (3)

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

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. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. 25(1), 46–48 (2000).
[Crossref] [PubMed]

1999 (1)

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

1994 (1)

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Baird, M. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

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] [PubMed]

Beach, J. R.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Beck, M.

Best, G.

Betzig, E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Burke, B.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Cao, R.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

Cardoso, M. C.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Carlton, P. M.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Chang, B.-J.

V. Perez, B.-J. Chang, and E. H. K. Stelzer, “Optimal 2D-SIM reconstruction by two filtering steps with Richardson-Lucy deconvolution,” Sci. Rep. 6(1), 37149 (2016).
[Crossref] [PubMed]

Chen, B.-C.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Chen, Y.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

Chhun, B. B.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

Chung, E.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Cragg, G. E.

Cremer, C. G.

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Cui, Y.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Davidson, M. W.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Dyba, M.

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. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Egner, 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. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Fienup, J. R.

Fiolka, R.

Golubovskaya, I. N.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Griffis, E. R.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

Gustafsson, M. G.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

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

Gustafsson, M. G. L.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

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

Haase, S.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
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Hammer, J. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
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Heintzmann, R.

K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21(2), 2032–2049 (2013).
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R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
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Hell, S. W.

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. U.S.A. 97(15), 8206–8210 (2000).
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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).
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Hennig, S.

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
[Crossref] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
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Huang, E.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
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Hübner, W.

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
[Crossref] [PubMed]

Huser, T.

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
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Jakobs, S.

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. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Johansson, G. A.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

Kamps-Hughes, N.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

Kim, D.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Kim, Y.-H.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Kirchhausen, T.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

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. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Kner, P.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Kuang, C.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

Lal, A.

A. Lal, C. Shan, and P. Xi, “Structured Illumination Microscopy Image Reconstruction Algorithm,” IEEE J. Sel. Top. Quantum Electron. 22(4), 50–63 (2016).
[Crossref]

Leonhardt, H.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Li, D.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

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] [PubMed]

Liu, W.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

Liu, X.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

Liu, Z.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

F. Wei and Z. Liu, “Plasmonic Structured Illumination Microscopy,” Nano Lett. 10(7), 2531–2536 (2010).
[Crossref] [PubMed]

Lu, D.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Macklin, J. J.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

Mandula, O.

Milkie, D. E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Mönkemöller, V.

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
[Crossref] [PubMed]

Moses, B.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Müller, M.

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
[Crossref] [PubMed]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Pasham, M.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

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] [PubMed]

Perez, V.

V. Perez, B.-J. Chang, and E. H. K. Stelzer, “Optimal 2D-SIM reconstruction by two filtering steps with Richardson-Lucy deconvolution,” Sci. Rep. 6(1), 37149 (2016).
[Crossref] [PubMed]

Ponsetto, J. L.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Rego, E. H.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

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] [PubMed]

Schermelleh, L.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Sedat, J. W.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Shan, C.

A. Lal, C. Shan, and P. Xi, “Structured Illumination Microscopy Image Reconstruction Algorithm,” IEEE J. Sel. Top. Quantum Electron. 22(4), 50–63 (2016).
[Crossref]

Shao, L.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Shen, H.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Shroff, S. A.

So, P. T. C.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
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G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. 25(1), 46–48 (2000).
[Crossref] [PubMed]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Stelzer, E. H. K.

V. Perez, B.-J. Chang, and E. H. K. Stelzer, “Optimal 2D-SIM reconstruction by two filtering steps with Richardson-Lucy deconvolution,” Sci. Rep. 6(1), 37149 (2016).
[Crossref] [PubMed]

Stemmer, A.

Wan, W.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Wang, C. J. R.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Wei, F.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

F. Wei and Z. Liu, “Plasmonic Structured Illumination Microscopy,” Nano Lett. 10(7), 2531–2536 (2010).
[Crossref] [PubMed]

Wichmann, J.

Wicker, K.

Williams, D. R.

Winoto, L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Xi, P.

A. Lal, C. Shan, and P. Xi, “Structured Illumination Microscopy Image Reconstruction Algorithm,” IEEE J. Sel. Top. Quantum Electron. 22(4), 50–63 (2016).
[Crossref]

Xu, P.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Zhang, M.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Zhang, X.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “ADVANCED IMAGING. Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Zhang, Z.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

Zhu, D.

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

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] [PubMed]

Biophys. J. (2)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-Dimensional Standing Wave Total Internal Reflection Fluorescence Microscopy: Superresolution Imaging of Single Molecular and Biological Specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Lal, C. Shan, and P. Xi, “Structured Illumination Microscopy Image Reconstruction Algorithm,” IEEE J. Sel. Top. Quantum Electron. 22(4), 50–63 (2016).
[Crossref]

J. Biomed. Opt. (1)

Y. Chen, R. Cao, W. Liu, D. Zhu, Z. Zhang, C. Kuang, and X. Liu, “Widefield and total internal reflection fluorescent structured illumination microscopy with scanning galvo mirrors,” J. Biomed. Opt. 23(4), 1–9 (2018).
[Crossref] [PubMed]

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
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J. Opt. Soc. Am. A (1)

Nano Lett. (2)

F. Wei and Z. Liu, “Plasmonic Structured Illumination Microscopy,” Nano Lett. 10(7), 2531–2536 (2010).
[Crossref] [PubMed]

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide Field Super-Resolution Surface Imaging through Plasmonic Structured Illumination Microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Nat. Commun. (1)

M. Müller, V. Mönkemöller, S. Hennig, W. Hübner, and T. Huser, “Open-source image reconstruction of super-resolution structured illumination microscopy data in ImageJ,” Nat. Commun. 7, 10980 (2016).
[Crossref] [PubMed]

Nat. Methods (3)

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Other (1)

R. Cao, “Inverse matrix phase algorithm: SIM reconstruction software using inverse matrix based phase estimation method”, Zenodo (2018) [retrieved 17 September 2018], https://doi.org/10.5281/zenodo.1314351.

Supplementary Material (1)

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» Code 1       SIM reconstruction software using inverse matrix based phase estimation algorithm.

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Figures (5)

Fig. 1
Fig. 1 Comparison of different modulation frequency estimation methods. The peak inside the green boxes in (a)–(d) is located at k 0 , while the peak inside the blue box is located at k 0 . The intensity of (a)–(d) was normalized by the corresponding maximum in each image. The correlation results were obtained when the phase shift was set as 2 π / 3 . Because the actual phase shift differed from the desired 2 π / 3 , the redundant component appeared in the cross-correlation and phase-only correlation results. A high-pass mask was applied in (a) to exclude components whose spatial frequency was smaller than 0.8 k c . (e) The peak intensities are located at k 0 (green square) and k 0 (blue circle). Abbreviations: AC, auto-correlation; CC, cross-correlation; CCwD, cross-correlation with deconvolution; PoC, phase-only correlation.
Fig. 2
Fig. 2 Simulations of the proposed inverse matrix based phase estimation algorithm versus the auto-correlation and cross-correlation algorithms. The period of the illumination pattern was 195 nm and the minimum detectable period in this simulation was 214 nm. As the radial pattern consisted of certain frequencies, the auto-correlation algorithm failed to retrieve the correct phase under low modulation depth conditions even though the number of photons was large. The value of m in the lower left of each subfigure demotes the modulation depth. In this simulation, Poisson noise was introduced, the minimum detectable number of photons was 10, and 24 phases were simulated when calculating the mean phase error.
Fig. 3
Fig. 3 (a) Simulation of the ground truth, (b) uncorrected reconstruction, and (c) phase corrected result using the inverse matrix based algorithm. The Fourier transform of the reconstructed images (b) and (c) are shown in (d). The green background pattern in (d)–(f) is the Fourier transform of the ground truth sample. (e) and (f) show the corresponding areas in (d), and the white dashed square in (e) highlights part of the redundant Fourier component that appears due to the error in the phase. The reconstruction results using (g) the auto-correlation, (h) cross-correlation, and (i) inverse matrix based method employed the same set of data that was also applied in Fig. 2 (m = 0.3, number of photons = 3162, three images were used).
Fig. 4
Fig. 4 Experimental results: (a) the filtered widefield results, (b) the reconstructed results without correction, and (c) the results with phase correction via the inverse matrix based method (note the brightness was increased by 40% to improve the visibility). (d) The phase-only correlation results of the reconstructed image. Lower left: without correction, and upper right: with correction. The intensity profile along the cutline indicated by the arrows in (a)–(c) is shown in (e) using the results without brightness enhancement.
Fig. 5
Fig. 5 Filtered widefield result (a), SIM results with the auto-correlation (b), cross-correlation (c), and inverse matrix (d) based phase estimation methods. The profile along the line segment in each subfigure is shown in (e). The profile was normalized by the corresponding maximum along each line segment. COS-7 cells were used in this experiment. Abbreviation: WF, widefield; AC, auto-correlation method; CC, cross-correlation method; IM, inverse matrix based method.

Equations (18)

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D i = { S [ 1 + m cos ( 2 π k 0 r + φ i ) ] } h d e ,
[ D ˜ 1 ( k ) D ˜ 2 ( k ) D ˜ 3 ( k ) ] = M ( 1 , 1 , 1 ; φ 1 i l l , φ 2 i l l , φ 3 i l l ) [ S ˜ ( k ) H ˜ d e ( k ) S ˜ ( k k 0 ) H ˜ d e ( k ) S ˜ ( k + k 0 ) H ˜ d e ( k ) ] , where M ( μ 1 , μ 2 , μ 3 ; α 1 , α 2 , α 3 ) = [ μ 1 m 2 e i α 1 m 2 e i α 1 μ 2 m 2 e i α 2 m 2 e i α 2 μ 3 m 2 e i α 3 m 2 e i α 3 ]
D ˜ i u p d a t e = c o n j ( H ˜ d e ) D ˜ i | H ˜ d e | + ε 1 ,
M 1 ( 1 , 1 , 1 ; γ 1 , γ 2 , γ 3 ) M ( 1 , 1 , 1 ; φ 1 i l l , φ 2 i l l , φ 3 i l l ) = [ 1 c 0 c o n j ( c 0 ) 0 r 1 e i ϕ 1 r 2 e i ϕ 2 0 r 2 e i ϕ 2 r 1 e i ϕ 1 ] ,
[ R ˜ 1 ( k ) R ˜ 2 ( k ) R ˜ 3 ( k ) ] = M 1 ( 1 , 1 , 1 ; γ 1 , γ 2 , γ 3 ) [ D ˜ 1 ( k ) D ˜ 2 ( k ) D ˜ 3 ( k ) ] = [ 1 c 0 c o n j ( c 0 ) 0 r 1 e i ϕ 1 r 2 e i ϕ 2 0 r 2 e i ϕ 2 r 1 e i ϕ 1 ] [ S ˜ ( k ) H ˜ d e ( k ) S ˜ ( k k 0 ) H ˜ d e ( k ) S ˜ ( k + k 0 ) H ˜ d e ( k ) ] ,
R ˜ 2 ( k ) = r 1 e i ϕ 1 S ˜ ( k k 0 ) H ˜ d e ( k ) + r 2 e i ϕ 2 S ˜ ( k + k 0 ) H ˜ d e ( k ) ,
C ( k ) = W ˜ ( k ) | W ˜ ( k ) | + ε 2 R ˜ 2 ( k ) | R ˜ 2 ( k ) | + ε 2 = c o n j ( W ˜ ( k ) | W ˜ ( k ) | + ε 2 ) R ˜ 2 ( k + k ) | R ˜ 2 ( k + k ) | + ε 2 d k ,
C ( k 0 ) = [ R ˜ 1 ( k ) | R ˜ 1 ( k ) | + ε 2 R ˜ 2 ( k ) | R ˜ 2 ( k ) | + ε 2 ] ( k 0 ) = c o n j ( R ˜ 1 ( k ) | R ˜ 1 ( k ) | + ε 2 ) R ˜ 2 ( k + k 0 ) | R ˜ 2 ( k + k 0 ) | + ε 2 d k = c o n j [ S ˜ ( k ) + c 0 S ˜ ( k k 0 ) + c o n j ( c 0 ) S ˜ ( k + k 0 ) ] | R ˜ 1 ( k ) | + ε 2 [ r 1 e i ϕ 1 S ˜ ( k ) + r 2 e i ϕ 2 S ˜ ( k + 2 k 0 ) ] | R ˜ 2 ( k + k 0 ) | + ε 2 c o n j [ H ˜ d e ( k ) ] H ˜ d e ( k + k 0 ) d k c o n j [ S ˜ ( k ) ] | R ˜ 1 ( k ) | + ε 2 r 1 e i ϕ 1 S ˜ ( k ) | R ˜ 2 ( k + k 0 ) | + ε 2 c o n j [ H ˜ d e ( k ) ] H ˜ d e ( k + k 0 ) d k
ψ p e a k = arg { [ W ˜ R ˜ 2 ] ( k 0 ) } = arg { c o n j [ W ˜ ( k ) ] R ˜ 2 ( k + k 0 ) d k } =arg { c o n j [ S ˜ ( k ) H ˜ ( k ) ] [ r 1 e i ϕ 1 S ˜ ( k ) + r 2 e i ϕ 2 S ˜ ( k + 2 k 0 ) ] H ˜ d e ( k + k 0 ) d k } , arg { r 1 e i ϕ 1 c o n j [ S ˜ ( k ) ] S ˜ ( k ) c o n j [ H ˜ ( k ) ] H ˜ d e ( k + k 0 ) d k } ϕ 1
M 1 ( 1 , 1 , 1 ; γ 1 , γ 2 , γ 3 ) M ( 1 , 1 , 1 ; φ 1 i l l , φ 2 i l l , φ 3 i l l ) = [ 1 m 2 e i γ 1 m 2 e i γ 1 1 m 2 e i γ 2 m 2 e i γ 2 1 m 2 e i γ 3 m 2 e i γ 3 ] 1 [ 1 m 2 e i φ 1 i l l m 2 e i φ 1 i l l 1 m 2 e i φ 2 i l l m 2 e i φ 2 i l l 1 m 2 e i φ 3 i l l m 2 e i φ 3 i l l ] = [ x 1 x 2 1 x 1 x 2 b 1 e i β 1 b 2 e i β 2 b 3 e i β 3 b 1 e i β 1 b 2 e i β 2 b 3 e i β 3 ] [ 1 m 2 e i φ 1 i l l m 2 e i φ 1 i l l 1 m 2 e i φ 2 i l l m 2 e i φ 2 i l l 1 m 2 e i φ 3 i l l m 2 e i φ 3 i l l ] = [ 1 c 0 c o n j ( c 0 ) 0 r 1 e i ϕ 1 r 2 e i ϕ 2 0 r 2 e i ϕ 2 r 1 e i ϕ 1 ]
m 2 ( b 1 e i φ 1 i l l + i β 1 + b 2 e i φ 2 i l l + i β 2 + b 3 e i φ 3 i l l + i β 3 ) = r 1 e i ϕ 1 r 1 e i ψ p e a k .
[ b 1 sin ( φ 1 i l l + β 1 ) + b 2 sin ( φ 2 i l l + β 2 ) + b 3 sin ( φ 3 i l l + β 3 ) ] cos ( ψ p e a k ) = [ b 1 cos ( φ 1 i l l + β 1 ) + b 2 cos ( φ 2 i l l + β 2 ) + b 3 cos ( φ 3 i l l + β 3 ) ] sin ( ψ p e a k ) .
b 1 sin ( θ 1 φ 1 i l l ) + b 2 sin ( θ 2 φ 2 i l l ) + b 3 sin ( θ 3 φ 3 i l l ) = 0 , where θ i = ψ p e a k β i .
M 1 ( σ 1 , σ 2 , σ 3 ; γ 1 , γ 2 , γ 3 ) M ( 1 , 1 , 1 ; φ 1 i l l , φ 2 i l l , φ 3 i l l ) = [ σ 1 m 2 e i γ 1 m 2 e i γ 1 σ 2 m 2 e i γ 2 m 2 e i γ 2 σ 3 m 2 e i γ 3 m 2 e i γ 3 ] 1 [ 1 m 2 e i φ 1 i l l m 2 e i φ 1 i l l 1 m 2 e i φ 2 i l l m 2 e i φ 2 i l l 1 m 2 e i φ 3 i l l m 2 e i φ 3 i l l ] = [ x 1 x 2 x 3 b 1 e i β 1 b 2 e i β 2 b 3 e i β 3 b 1 e i β 1 b 2 e i β 2 b 3 e i β 3 ] [ 1 m 2 e i φ 1 i l l m 2 e i φ 1 i l l 1 m 2 e i φ 2 i l l m 2 e i φ 2 i l l 1 m 2 e i φ 3 i l l m 2 e i φ 3 i l l ] = [ w 0 c 0 c o n j ( c 0 ) w C r 1 e i ϕ 1 r 2 e i ϕ 2 c o n j ( w C ) r 2 e i ϕ 2 r 1 e i ϕ 1 ] ,
R ˜ 2 ( k ) = w C S ˜ ( k ) H ˜ d e ( k ) + r 1 e i ϕ 1 S ˜ ( k k 0 ) H ˜ d e ( k ) + r 2 e i ϕ 2 S ˜ ( k + k 0 ) H ˜ d e ( k ) .
R ˜ 2 u p d a t e d = R ˜ 2 w C 1 n i = 1 n D ˜ i .
g ( φ 1 i l l , φ 2 i l l , φ 3 i l l ) = t | b 1 e i φ 1 i l l + i β 1 + b 2 e i φ 2 i l l + i β 2 + b 3 e i φ 3 i l l + i β 3 | b 1 e i φ 1 i l l + i β 1 + b 2 e i φ 2 i l l + i β 2 + b 3 e i φ 3 i l l + i β 3 | e i ψ p e a k | ,
φ s i l l = sin 1 [ ( sin φ s i l l ) e s t ( sin φ s i l l ) e s t 2 + ( cos φ s i l l ) e s t 2 ]

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