Abstract

Single-molecule, localization-based, wide-field nanoscopy often suffers from low time resolution because the localization of a single molecule with high precision requires a low emitter density of fluorophores. In addition, to reconstruct a super-resolution image, hundreds or thousands of image frames are required, even when advanced algorithms, such as compressive sensing and deep learning, are applied. These factors limit the application of these nanoscopy techniques for living cell imaging. In this study, we developed a single-frame, wide-field nanoscopy system based on ghost imaging via sparsity constraints (GISC), in which a spatial random phase modulator is applied in a wide-field microscope to achieve random measurement of fluorescence signals. This method can effectively use the sparsity of fluorescence emitters to enhance the imaging resolution to 80 nm by reconstructing one raw image using compressive sensing. We achieved an ultrahigh emitter density of ${143}\;\unicode{x00B5} {{\rm m}^{ - 2}}$ while maintaining the precision of single-molecule localization below 25 nm. We show that by employing a high-density of photo-switchable fluorophores, GISC nanoscopy can reduce the number of sampling frames by one order of magnitude compared to previous super-resolution imaging methods based on single-molecule localization. GISC nanoscopy may therefore improve the time resolution of super-resolution imaging for the study of living cells and microscopic dynamic processes.

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

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References

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    [Crossref]

2018 (4)

Y. M. Sigal, R. Zhou, and X. Zhuang, “Visualizing and discovering cellular structures with super-resolution microscopy,” Science 361, 880–887 (2018).
[Crossref]

E. Nehme, L. E. Weiss, T. Michaeli, and Y. Shechtman, “Deep-STORM: super-resolution single-molecule microscopy by deep learning,” Optica 5, 458–464 (2018).
[Crossref]

S. Oren, M. Maor, S. Mordechai, and Y. C. Eldar, Sparsity-based super-resolution microscopy from correlation information,” Opt. Express 26, 18238–18269 (2018).
[Crossref]

S. Han, H. Yu, X. Shen, H. Liu, W. Gong, and Z. Liu, “A review of ghost imaging via sparsity constraints,” Appl. Sci. 8, 1379 (2018).
[Crossref]

2016 (3)

K. Kyrus and C. K. W. Clifford, “High-order ghost imaging using non-Rayleigh speckle sources,” Opt. Express 24, 26766 (2016).
[Crossref]

Z. Liu, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

2015 (2)

J. Schneider, J. Zahn, M. Maglione, S. J. Sigrist, J. Marquard, J. Chojnacki, and S. W. Hell, “Ultrafast, temporally stochastic STED nanoscopy of millisecond dynamics,” Nat. Methods 12, 827–830 (2015).
[Crossref]

J. Wang, “Support recovery with orthogonal matching pursuit in the presence of noise: a new analysis,” IEEE Trans. Signal Process. 63, 5868–5877 (2015).
[Crossref]

2014 (3)

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30, 2389–2390 (2014).
[Crossref]

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

X. F. Liu, X. H. Chen, X. R. Yao, W. K. Yu, G. J. Zhai, and L. A. Wu, “Lensless ghost imaging with sunlight,” Opt. Lett. 39, 2314–2317 (2014).
[Crossref]

2013 (1)

F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
[Crossref]

2012 (8)

S. H. Shim, C. Xia, G. Zhong, H. P. Babcock, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. USA 109, 13978–13983 (2012).
[Crossref]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster storm using compressed sensing,” Nat. Methods 9, 721–723 (2012).
[Crossref]

E. A. Mukamel and M. J. Schnitzer, “Unified resolution bounds for conventional and stochastic localization fluorescence microscopy,” Phys. Rev. Lett. 109, 168102 (2012).
[Crossref]

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

Y. Shih, “The physics of ghost imaging: nonlocal interference or local intensity fluctuation correlation?” Quantum Inf. Process. 11, 995–1001 (2012).
[Crossref]

W. Gong and S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376, 1519–1522 (2012).
[Crossref]

H. Wang, S. Han, and M. I. Kolobov, “Quantum limits of super-resolution of optical sparse objects via sparsity constraint,” Opt. Express 20, 23235–23252 (2012).
[Crossref]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

2011 (1)

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods 8, 279–280 (2011).
[Crossref]

2009 (3)

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, 339–342 (2009).
[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. USA 106, 22287–22292 (2009).
[Crossref]

P. Zhang, W. Gong, X. Shen, D. Huang, and S. Han, “Improving resolution by the second-order correlation of light fields,” Opt. Lett. 34, 1222–1224 (2009).
[Crossref]

2008 (1)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

2007 (3)

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[Crossref]

M. A. Figueiredo, T. R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[Crossref]

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[Crossref]

2006 (3)

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

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

A. Sharonov and R. M. Hochstrasser, “Wide-field sub-diffraction imaging by accumulated binding of diffusing probes,” Proc. Natl. Acad. Sci. USA 103, 18911–18916 (2006).
[Crossref]

2005 (3)

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

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

D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30, 2354–2356 (2005).
[Crossref]

2004 (1)

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[Crossref]

2003 (1)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
[Crossref]

1994 (1)

Babcock, H. P.

S. H. Shim, C. Xia, G. Zhong, H. P. Babcock, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. USA 109, 13978–13983 (2012).
[Crossref]

Baird, M. A.

F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
[Crossref]

Bates, M.

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

Bernex, R.

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

Bewersdorf, J.

F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
[Crossref]

Borkovec, J.

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30, 2389–2390 (2014).
[Crossref]

Boyd, R. W.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

Candès, E. J.

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[Crossref]

Chen, M.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Chen, X. H.

Cheng, J.

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[Crossref]

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, 339–342 (2009).
[Crossref]

Chojnacki, J.

J. Schneider, J. Zahn, M. Maglione, S. J. Sigrist, J. Marquard, J. Chojnacki, and S. W. Hell, “Ultrafast, temporally stochastic STED nanoscopy of millisecond dynamics,” Nat. Methods 12, 827–830 (2015).
[Crossref]

Clifford, C. K. W.

Colyer, R.

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. USA 106, 22287–22292 (2009).
[Crossref]

Davidson, M. W.

F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
[Crossref]

de Rooi, J. J.

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

Dedecker, P.

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

Dertinger, T.

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. USA 106, 22287–22292 (2009).
[Crossref]

Devos, O.

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

Duim, W. C.

F. Huang, T. M. P. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, and J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10, 653–658 (2013).
[Crossref]

Duwé, S.

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

Eggeling, C.

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

Eilers, P. H. C.

S. Hugelier, J. J. de Rooi, R. Bernex, S. Duwé, O. Devos, M. Sliwa, P. Dedecker, P. H. C. Eilers, and C. Ruckebusch, “Sparse deconvolution of high-density super-resolution images,” Sci. Rep. 6, 21413 (2016).
[Crossref]

Eldar, Y. C.

S. Oren, M. Maor, S. Mordechai, and Y. C. Eldar, Sparsity-based super-resolution microscopy from correlation information,” Opt. Express 26, 18238–18269 (2018).
[Crossref]

M. Mutzafi, Y. Shechtman, Y. C. Eldar, and M. Segev, “Single-shot sparsity-based sub-wavelength fluorescence imaging of biological structures using dictionary learning,” in Conference on Lasers and. Electro-Optics (CLEO) (2015), Vol. 3, paper STh4K.5.

Elnatan, D.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster storm using compressed sensing,” Nat. Methods 9, 721–723 (2012).
[Crossref]

Enderlein, J.

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. USA 106, 22287–22292 (2009).
[Crossref]

Fernandez-Granda, C.

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

Figueiredo, M. A.

M. A. Figueiredo, T. R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[Crossref]

Gilbert, A. C.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[Crossref]

Girirajan, T. P.

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

Gong, W.

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Appl. Phys. Lett. (1)

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
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Appl. Sci. (1)

S. Han, H. Yu, X. Shen, H. Liu, W. Gong, and Z. Liu, “A review of ghost imaging via sparsity constraints,” Appl. Sci. 8, 1379 (2018).
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Bioinformatics (1)

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Biophys. J. (1)

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Commun. Pure Appl. Math. (1)

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IEEE J. Sel. Top. Signal Process. (1)

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IEEE Signal Process. Mag. (2)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
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E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
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IEEE Trans. Inf. Theory (1)

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[Crossref]

IEEE Trans. Signal Process. (1)

J. Wang, “Support recovery with orthogonal matching pursuit in the presence of noise: a new analysis,” IEEE Trans. Signal Process. 63, 5868–5877 (2015).
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Inverse Probl. (1)

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
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Nat. Methods (6)

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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of experimental setup and imaging process. (a) Experimental setup: A random phase modulator with a low magnification objective (${10} \times $) is set before sCMOS in a conventional inverted microscope to form speckle patterns of fluorescence signals. Fluorescence images are directly recorded. RPM, Random phase modulator; L, Lens; I, Iris; M, Mirror; DM, Dichroic mirror; EX, Excitation filter; EM, Emission filter; PBS, Polarization beam splitter; OL, Objective lens; and NP, Nanopositioning stage. (b) Calibration and imaging processes: All speckle patterns generated from each position of the sample plane are recorded as the random measurement matrix in the calibration process. One speckle image from the actual imaged sample is obtained in the imaging process and then a super-resolution image can be reconstructed via compressive sensing.
Fig. 2.
Fig. 2. Imaging resolution and its influencing factors for GISC nanoscopy. (a) Normalized mutual correlation curve to compare the performance for the spatial resolution of the speckle pattern and PSF. The resolution of speckles can be increased by a factor of $\sqrt 2 $ compared to that of the PSF. (b) Diffraction limited rings with different spacings (80, 120, and 240 nm) are created and the single-frame imaging results reconstructed by the GPSR algorithm are shown in the middle column; the scale bar is 500 nm long, corresponding to the wide-field images shown in left column (160 nm/pixel); the spatial resolution of 80 nm is determined by the Rayleigh criterion shown in the right column; and the black and red lines represent the normalized intensity distribution along the blue line in the wide field and the reconstructed images, respectively. (c) Effect of different SNRs and Cs on the resolution. The inset images show the reconstruction images at ${\rm SNR} = {20}$ and $C = {0.75}$, and ${\rm SNR} = {10}$ and $C = {0.75}$; the right image represents the threshold for accurate recovery using compressive sensing; and the left image and the dashed lines indicate the results and the extent of inaccurate recovery using compressive sensing, respectively.
Fig. 3.
Fig. 3. Capability of GISC nanoscopy to identify molecules efficiently at a high density and the effect of the SNR. (a) Comparison of reconstruction results (white grids) to the true molecule positions (red crosses) at ${50.7}\,\,\unicode{x00B5} {{\rm m}^{ - 2}}$; scar bar of 100 nm. The smaller figure (upper left corner) corresponds to the wide-field image (160 nm/pixel). (b) Density of identified molecules and localization precision versus different densities and different SNRs obtained with the OMP localization algorithm ($C = {0.75}$).
Fig. 4.
Fig. 4. Experimental results of nanometer rulers. (a) Reconstruction results of the images of the 270 (upper row) and 160 (lower row) nm rulers at ${\rm SNR} = {20}$ and $C = {0.75}$ and the corresponding wide-field images; the yellow boxes represent the reconstruction results of diffraction-limited rulers, corresponding to “+” in the wide-field images. The scale of the wide-field images is 160 nm per pixel; the scale bars in the other images indicate 500 nm. (b) Histogram and normalized intensity distribution along the white line in the reconstructed images of the 270 (upper row) and 160 (lower row) nm rulers. (c) Statistical analysis of accuracy for 40 reconstructed images of the 270 (upper row) and 160 (lower row) nm rulers.
Fig. 5.
Fig. 5. Simulation and experimental results that demonstrate the capability of GISC-STORM to identify molecules efficiently at high densities. (a) Comparison of the performances of GISC-STORM, CS-STORM, and the single-molecule fitting method for molecule identification. The dashed line indicates the case where the number of identified molecules equals the number of molecules present. (b) Comparison of localization precisions. (c) Reconstruction results of the ring with a spacing of 60 nm from 4000, 500, and 10 frames by the single-molecule fitting, CS-STORM, and GISC-STORM, respectively. (d) Reconstructed result (upper right corner) and histogram for one representative DNA origami structures at a designed distance of 40 nm from 100 frames. The histogram shows the accumulated intensity of the red line in the inset image (20 nm/pixel). Scale bar: 500 nm.

Equations (4)

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Δ G ( 2 ) ( r i , j ) = I r ( r i , j ) I t ( r i , j ) 1 M i M j i = 1 M i j = 1 M j I ( i , j ) r ( i , j ) I ( i , j ) t { T i ( r i , j ) { { ( π a 2 λ f ) [ 2 J 1 ( 2 π a λ f r i , j ) 2 π a λ f r i , j ] } exp { 2 [ 2 π ω ( n 1 ) / 2 π ω ( n 1 ) λ λ ] 2 { 1 exp [ ( z 2 r i , j ( z 1 + z 2 ) ζ ) 2 ] } } } 2 } ,
m i n x 0 s . t . Y = A X ,
M C 0 | K | μ 2 ( A ) log ( N / δ ) ,
ρ e r r o r C κ 2 δ 2 K 1 / 2 i f S N R κ δ 2 K 3 / 4 ,

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