Abstract

For more than a century, the wavelength of light was considered to be a fundamental limit on the spatial resolution of optical imaging. Particularly in light microscopy, this limit, known as Abbe's diffraction limit, places a fundamental constraint on the ability to image sub-cellular organelles with high resolution. However, modern microscopy techniques such as STED, PALM, and STORM, manage to recover sub-wavelength information, by relying on fluorescence imaging. Specifically, PALM/STORM acquire large sequences of fluorescence images from molecules attached to the organelles within the imaged specimen, such that in each frame only a small set of fluorophores are active. The position of each fluorophore can be found accurately in each frame, and the image is recovered by superimposing the points from all frames. The resulting grainy image is subsequently smoothed to produce the final super-resolved image with a resolution of tens of nano-meters. However, because PALM/STORM rely on many (>10,000) exposures, they suffer from poor temporal resolution. To address that, super-resolution optical fluctuation imaging (SOFI) was shown to produce sub-diffraction images with increased temporal resolution, by allowing for higher fluorophore density and exploiting the temporal statistics of the emissions. However, the improved temporal resolution of SOFI comes at the expense of its spatial resolution, which is not as high as that of PALM/STORM. Here, we present a new method called SPARCOM: sparsity-based super-resolution correlation microscopy, which combines a shorter integration time than previously reported with spatial resolution comparable to PALM and STORM. SPARCOM relies on sparsity in the correlation domain, exploiting the sparse distribution of fluorescent molecules and the lack of correlation between different emitters. We demonstrate our technique in simulations and in experiments and provide comparisons to state-of-the-art high density methods.

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

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Corrections

Oren Solomon, Maor Mutzafi, Mordechai Segev, and Yonina C. Eldar, "Sparsity-based super-resolution microscopy from correlation information: erratum," Opt. Express 26, 20849-20849 (2018)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-26-16-20849

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References

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2016 (6)

H. Deschout, T. Lukes, A. Sharipov, D. Szlag, L. Feletti, W. Vandenberg, P. Dedecker, J. Hofkens, M. Leutenegger, T. Lasser, and A. Radenovic, “Complementarity of PALM and SOFI for super-resolution live-cell imaging of focal adhesions,” Nat. Commun. 7, 13693 (2016).
[Crossref] [PubMed]

N. Gustafsson, S. Culley, G. Ashdown, D. M. Owen, P. M. Pereira, and R. Henriques, “Fast live-cell conventional fluorophore nanoscopy with ImageJ through super-resolution radial fluctuations,” Nat. Commun. 7, 12471 (2016).
[Crossref] [PubMed]

K. Agarwal and R. Macháň, “Multiple signal classification algorithm for super-resolution fluorescence microscopy,” Nat. Commun. 7, 13752 (2016).
[Crossref] [PubMed]

L. Weizman, Y. C. Eldar, and D. Ben Bashat, “Reference-based MRI,” Med. Phys. 43(10), 5357–5369 (2016).
[Crossref] [PubMed]

K. Jaganathan, Y. C. Eldar, and B. Hassibi, “STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms,” IEEE J. Sel. Top. Signal Process. 10(4), 770–781 (2016).
[Crossref]

D. Oren, Y. Shechtman, M. Mutzafi, Y. C. Eldar, and M. Segev, “Sparsity-based recovery of three-photon quantum states from two-fold correlations,” Optica 3(3), 226–232 (2016).
[Crossref]

2015 (8)

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39(2), 277–299 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

M. Mutzafi, Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based Ankylography for Recovering 3D molecular structures from single-shot 2D scattered light intensity,” Nat. Commun. 6(1), 7950 (2015).
[Crossref] [PubMed]

P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6(1), 8209 (2015).
[Crossref] [PubMed]

P. Schniter and S. Rangan, “Compressive phase retrieval via generalized approximate message passing,” IEEE Trans. Signal Process. 63(4), 1043–1055 (2015).
[Crossref]

V. I. Morgenshtern and E. J. Candès, “Stable Super-Resolution of Positive Sources : the Discrete Setup,” SIAM J. Imaging Sci. Imaging Sci. 9(2), 1–35 (2015).

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2015).
[Crossref] [PubMed]

P. Pal and P. P. Vaidyanathan, “Pushing the Limits of Sparse Support Recovery Using Correlation Information,” IEEE Trans. Signal Process. 63(3), 711–726 (2015).
[Crossref]

2014 (5)

O. Bar-Ilan and Y. C. Eldar, “Sub-Nyquist Radar via Doppler Focusing,” IEEE Trans. Signal Process. 62(7), 1796–1811 (2014).
[Crossref]

T. Chernyakova and Y. Eldar, “Fourier-domain beamforming: the path to compressed ultrasound imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 61(8), 1252–1267 (2014).
[Crossref] [PubMed]

H. Jiang, G. Huang, and P. Wilford, “Multi-view in lensless compressive imaging,” APSIPA Trans. Signal Inf. Process 3, e152014.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62(4), 928–938 (2014).
[Crossref]

D. Cohen and Y. C. Eldar, “Sub-Nyquist Sampling for Power Spectrum Sensing in Cognitive Radios: A Unified Approach,” IEEE Trans. Signal Process. 62(15), 3897–3910 (2014).
[Crossref]

2013 (5)

2012 (6)

N. Wagner, Y. C. Eldar, and Z. Friedman, “Compressed beamforming in ultrasound imaging,” IEEE Trans. Signal Process. 60(9), 4643–4657 (2012).
[Crossref]

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2012).
[Crossref] [PubMed]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
[Crossref] [PubMed]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

W. T. Liu, T. Zhang, J. Y. Liu, P. X. Chen, and J. M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108(17), 170403 (2012).
[Crossref] [PubMed]

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A. 109(26), E1679–E1687 (2012).
[Crossref] [PubMed]

2011 (4)

Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express 19(16), 14807–14822 (2011).
[Crossref] [PubMed]

Y. Rivenson, A. Stern, and J. Rosen, “Compressive multiple view projection incoherent holography,” Opt. Express 19(7), 6109–6118 (2011).
[Crossref] [PubMed]

S. W. Ellingson, “Sensitivity of antenna arrays for long-wavelength radio astronomy,” IEEE Trans. Antenn. Propag. 59(6), 1855–1863 (2011).
[Crossref]

B. R. Rankin, G. Moneron, C. A. Wurm, J. C. Nelson, A. Walter, D. Schwarzer, J. Schroeder, D. A. Colón-Ramos, and S. W. Hell, “Nanoscopy in a living multicellular organism expressing GFP,” Biophys. J. 100(12), L63–L65 (2011).
[Crossref] [PubMed]

2010 (5)

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190(2), 165–175 (2010).
[Crossref] [PubMed]

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] [PubMed]

M. Mishali and Y. C. Eldar, “From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
[Crossref]

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
[Crossref] [PubMed]

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[Crossref] [PubMed]

2009 (5)

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

M. Mishali and Y. C. Eldar, “Blind multiband signal reconstruction: Compressed sensing for analog signals,” IEEE Trans. Signal Process. 57(3), 993–1009 (2009).
[Crossref]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

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. U.S.A. 106(52), 22287–22292 (2009).
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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).
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2008 (3)

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-Rate Far-Field Optical Nanoscopy Dissects Synaptic Vesicle Movement,” Science 320(5873), 246–249 (2008).
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2007 (2)

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
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2006 (5)

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, 1642–1645 (2006).

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E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
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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).
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1994 (1)

1991 (1)

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1964 (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
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1959 (1)

T. Főrster, “10th Spiers Memorial Lecture. Transfer mechanisms of electronic excitation,” Discuss. Faraday Soc. 27(0), 7–17 (1959).
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1953 (1)

D. L. Dexter, “A Theory of Sensitized Luminescence in Solids,” J. Chem. Phys. 21(5), 836–850 (1953).
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K. Agarwal and R. Macháň, “Multiple signal classification algorithm for super-resolution fluorescence microscopy,” Nat. Commun. 7, 13752 (2016).
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N. Gustafsson, S. Culley, G. Ashdown, D. M. Owen, P. M. Pereira, and R. Henriques, “Fast live-cell conventional fluorophore nanoscopy with ImageJ through super-resolution radial fluctuations,” Nat. Commun. 7, 12471 (2016).
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Baraniuk, R.

R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE Natl. Radar Conf. Proc.128–133, 2007.

Baraniuk, R. G.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).

Bar-Ilan, O.

O. Bar-Ilan and Y. C. Eldar, “Sub-Nyquist Radar via Doppler Focusing,” IEEE Trans. Signal Process. 62(7), 1796–1811 (2014).
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Bates, M.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
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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).
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Beck, A.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62(4), 928–938 (2014).
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A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
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Becker, S.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
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Ben Bashat, D.

L. Weizman, Y. C. Eldar, and D. Ben Bashat, “Reference-based MRI,” Med. Phys. 43(10), 5357–5369 (2016).
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Betzig, E.

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

Bobin, J.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A. 109(26), E1679–E1687 (2012).
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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, 1642–1645 (2006).

Boyd, S. P.

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted L1 minimization,” J. Fourier Anal. Appl. 14(5), 877–905 (2008).
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Bullkich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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Burnette, D. T.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2012).
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Candes, E.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A. 109(26), E1679–E1687 (2012).
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Candès, E.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
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Candès, E. J.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39(2), 277–299 (2015).
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V. I. Morgenshtern and E. J. Candès, “Stable Super-Resolution of Positive Sources : the Discrete Setup,” SIAM J. Imaging Sci. Imaging Sci. 9(2), 1–35 (2015).

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted L1 minimization,” J. Fourier Anal. Appl. 14(5), 877–905 (2008).
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E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
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Carlini, L.

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2015).
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Chahid, M.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A. 109(26), E1679–E1687 (2012).
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Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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W. T. Liu, T. Zhang, J. Y. Liu, P. X. Chen, and J. M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108(17), 170403 (2012).
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Chernyakova, T.

T. Chernyakova and Y. Eldar, “Fourier-domain beamforming: the path to compressed ultrasound imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 61(8), 1252–1267 (2014).
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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).
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Cohen, D.

D. Cohen and Y. C. Eldar, “Sub-Nyquist Sampling for Power Spectrum Sensing in Cognitive Radios: A Unified Approach,” IEEE Trans. Signal Process. 62(15), 3897–3910 (2014).
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Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6(1), 8209 (2015).
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M. Mutzafi, Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based Ankylography for Recovering 3D molecular structures from single-shot 2D scattered light intensity,” Nat. Commun. 6(1), 7950 (2015).
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Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Efficient coherent diffractive imaging for sparsely varying objects,” Opt. Express 21(5), 6327–6338 (2013).
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A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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Cohen-Hyams, T.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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Colón-Ramos, D. A.

B. R. Rankin, G. Moneron, C. A. Wurm, J. C. Nelson, A. Walter, D. Schwarzer, J. Schroeder, D. A. Colón-Ramos, and S. W. Hell, “Nanoscopy in a living multicellular organism expressing GFP,” Biophys. J. 100(12), L63–L65 (2011).
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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).
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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. U.S.A. 106(52), 22287–22292 (2009).
[Crossref] [PubMed]

Cox, S.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2012).
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Culley, S.

N. Gustafsson, S. Culley, G. Ashdown, D. M. Owen, P. M. Pereira, and R. Henriques, “Fast live-cell conventional fluorophore nanoscopy with ImageJ through super-resolution radial fluctuations,” Nat. Commun. 7, 12471 (2016).
[Crossref] [PubMed]

Dahan, M.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A. 109(26), E1679–E1687 (2012).
[Crossref] [PubMed]

Dana, H.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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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, 1642–1645 (2006).

Dedecker, P.

H. Deschout, T. Lukes, A. Sharipov, D. Szlag, L. Feletti, W. Vandenberg, P. Dedecker, J. Hofkens, M. Leutenegger, T. Lasser, and A. Radenovic, “Complementarity of PALM and SOFI for super-resolution live-cell imaging of focal adhesions,” Nat. Commun. 7, 13693 (2016).
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Dertinger, T.

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).
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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. U.S.A. 106(52), 22287–22292 (2009).
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Deschout, H.

H. Deschout, T. Lukes, A. Sharipov, D. Szlag, L. Feletti, W. Vandenberg, P. Dedecker, J. Hofkens, M. Leutenegger, T. Lasser, and A. Radenovic, “Complementarity of PALM and SOFI for super-resolution live-cell imaging of focal adhesions,” Nat. Commun. 7, 13693 (2016).
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Dexter, D. L.

D. L. Dexter, “A Theory of Sensitized Luminescence in Solids,” J. Chem. Phys. 21(5), 836–850 (1953).
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Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
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Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
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Eisert, J.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
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Eldar, Y.

T. Chernyakova and Y. Eldar, “Fourier-domain beamforming: the path to compressed ultrasound imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 61(8), 1252–1267 (2014).
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Eldar, Y. C.

L. Weizman, Y. C. Eldar, and D. Ben Bashat, “Reference-based MRI,” Med. Phys. 43(10), 5357–5369 (2016).
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K. Jaganathan, Y. C. Eldar, and B. Hassibi, “STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms,” IEEE J. Sel. Top. Signal Process. 10(4), 770–781 (2016).
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D. Oren, Y. Shechtman, M. Mutzafi, Y. C. Eldar, and M. Segev, “Sparsity-based recovery of three-photon quantum states from two-fold correlations,” Optica 3(3), 226–232 (2016).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6(1), 8209 (2015).
[Crossref] [PubMed]

M. Mutzafi, Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based Ankylography for Recovering 3D molecular structures from single-shot 2D scattered light intensity,” Nat. Commun. 6(1), 7950 (2015).
[Crossref] [PubMed]

O. Bar-Ilan and Y. C. Eldar, “Sub-Nyquist Radar via Doppler Focusing,” IEEE Trans. Signal Process. 62(7), 1796–1811 (2014).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62(4), 928–938 (2014).
[Crossref]

D. Cohen and Y. C. Eldar, “Sub-Nyquist Sampling for Power Spectrum Sensing in Cognitive Radios: A Unified Approach,” IEEE Trans. Signal Process. 62(15), 3897–3910 (2014).
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Y. Shechtman, E. Small, Y. Lahini, M. Verbin, Y. C. Eldar, Y. Silberberg, and M. Segev, “Sparsity-based super-resolution and phase-retrieval in waveguide arrays,” Opt. Express 21(20), 24015–24024 (2013).
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Y. Shechtman, Y. C. Eldar, O. Cohen, and M. Segev, “Efficient coherent diffractive imaging for sparsely varying objects,” Opt. Express 21(5), 6327–6338 (2013).
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N. Wagner, Y. C. Eldar, and Z. Friedman, “Compressed beamforming in ultrasound imaging,” IEEE Trans. Signal Process. 60(9), 4643–4657 (2012).
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A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express 19(16), 14807–14822 (2011).
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Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
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M. Mishali and Y. C. Eldar, “From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
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M. Mishali and Y. C. Eldar, “Blind multiband signal reconstruction: Compressed sensing for analog signals,” IEEE Trans. Signal Process. 57(3), 993–1009 (2009).
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S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
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S. W. Ellingson, “Sensitivity of antenna arrays for long-wavelength radio astronomy,” IEEE Trans. Antenn. Propag. 59(6), 1855–1863 (2011).
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L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
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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] [PubMed]

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. U.S.A. 106(52), 22287–22292 (2009).
[Crossref] [PubMed]

Feletti, L.

H. Deschout, T. Lukes, A. Sharipov, D. Szlag, L. Feletti, W. Vandenberg, P. Dedecker, J. Hofkens, M. Leutenegger, T. Lasser, and A. Radenovic, “Complementarity of PALM and SOFI for super-resolution live-cell imaging of focal adhesions,” Nat. Commun. 7, 13693 (2016).
[Crossref] [PubMed]

Flammia, S. T.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[Crossref] [PubMed]

Fleischer, A.

P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6(1), 8209 (2015).
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Forster, T.

T. Főrster, “10th Spiers Memorial Lecture. Transfer mechanisms of electronic excitation,” Discuss. Faraday Soc. 27(0), 7–17 (1959).
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Friedman, Z.

N. Wagner, Y. C. Eldar, and Z. Friedman, “Compressed beamforming in ultrasound imaging,” IEEE Trans. Signal Process. 60(9), 4643–4657 (2012).
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Gazit, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
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Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
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S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
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Golay, M. J. E.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
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Proc. Natl. Acad. Sci. U.S.A. (2)

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. U.S.A. 106(52), 22287–22292 (2009).
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M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).

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J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2015).
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O. Solomon, Y. C. Eldar, M. Mutzafi, and M. Segev, “SPARCOM: Sparsity Based Super-Resolution Correlation Microscopy,” arXiv Prepr. arXiv: 1707.09255, 2017.

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O. Solomon, M. Mutzafi, X. Yi, S. Weiss, Y. C. Eldar, and M. Segev, “Sparsity-based super-resolution optical fluctuation imaging,” in CLEO: Applications and Technology, 2016.

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University, 2012).

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[Crossref]

R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE Natl. Radar Conf. Proc.128–133, 2007.

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Figures (19)

Fig. 1
Fig. 1 The difference between SMV, MMV and correlation based MMV. All models (a)-(c) have the same measurement matrix A. (a) Single measurement vector (SMV) model. For measurement y 1 of length m, a sparse signal x 1 with up to m/2 non-zero entries can be recovered. (b) Multiple measurement vector (MMV) model. Assume that x 1 ,, x T share a common support, denoted by their colored rectangles. From T measurements, denoted by y 1 ,, y T , each of length m, x 1 ,, x T with sparsity level of at most O( m 2 ) can be recovered. (c) MMV with additional assumption of uncorrelated entries of x 1 ,, x T . In this model, support recovery of the emitters up to O( m 2 ) can be achieved.
Fig. 2
Fig. 2 Simulation results. Upper row: (a) Ground truth: high resolution image of simulated microtubules. (b) Diffraction-limited image. (c)-(d) SRRF reconstructions from high and very-high density movies of 361 and 60 frames (6 times denser), respectively. Lower row: (e)-(f) MUSICAL reconstructions from high and very-high density movies of 361 and 60 frames, respectively. (g)-(h) SPARCOM reconstructions from high and very-high density movies of 361 and 60 frames, respectively. Comparing the reconstructions for the same number of frames (e.g., the magnified regions in the red boxes), clearly the ability of SPARCOM to separate between closely adjacent subwavelength features, outperforms both SRRF and MUSICAL, even with as few as 60 frames. These separations closely match those in the ground truth.
Fig. 3
Fig. 3 Intensity profiles comparing the performance of SRRF, MUSICAL and SPARCOM. Normalized cross-sections taken along the solid yellow line (upper) and the dashed yellow line (lower) of Fig. 2, comparing the ground truth (dash-dot green, Fig. 2(a)), diffraction-limited image (dot blue, Fig. 1(b)), SRRF image recovered from a movie of 60 frames (solid yellow, Fig. 2(c)), MUSICAL image recovered from a movie of 60 frames (dash red, Fig. 2(f)), and SPARCOM image recovered from the high density movie of 60 frames (purple circle-head, Fig. 2(h)). These panels substantiate the visual conclusions drawn from Fig. 2: SPARCOM recovers the profiles of sub-diffraction spaced microtubules from a high emitters' density movie clearly and with a good agreement to the ground truth, while SRRF and MUSICAL do not achieve such high resolution between close features.
Fig. 4
Fig. 4 Experimental results. Panels (a)-(b) illustrate a diffraction-limited image of microtubules and a single noisy frame (from a movie of 500 frames). (c)-(d) SRRF reconstructions from a movie of 500 frames and a 10-times denser movie of 500 frames, respectively. (e)-(f) MUSICAL reconstructions for the same movies. (g)-(h) SPARCOM reconstructions for the same movies. Red boxes in the lower left corners of each panel are enlarged regions of the small red boxes in the upper left corner of each panel, respectively. Judging visually, both SPARCOM and MUSICAL exhibit clear separation of the microtubules within the enlarged boxes, while the SRRF reconstructions do not, for both movies. (i, j) Intensity cross-sections (normalized) taken along the solid (i) and dashed (j) yellow lines. Panel (i) shows that in regions of high emitters' density, SPARCOM clearly achieves better spatial resolution than both SRRF and MUSICAL, depicting a clear bifurcation, which is absent in MUSICAL and SRRF. On the other hand, panel (j) shows that in areas of low emitters' density, the spatial resolution of SPARCOM is similar to SRRF and better than MUSICAL.
Fig. 5
Fig. 5 Fourier Ring Correlation (FRC) analysis. (a)-(b) FRC analysis from the simulated movies of 361 frames and 60 frames of increased emitter density from Fig. 2, respectively. (c) FRC analysis from the movie of 500 frames of the experimental data set in Fig. 4. Colored curves are the FRC curves and the horizontal black lines represent the fixed 1/7 resolution threshold. Resolution corresponds to the intersection points of the curves with the horizontal threshold lines. In both (a) and (b), the resolution presented by SPARCOM matches that of ThunderSTORM (perfect reconstruction), even when the number of frames decreases to 60, and is higher than the other methods. For example, the spatial resolution of FALCON in both cases is lower by a factor of ~2 compared with SPARCOM. In (c), both SPARCOM and SRRF demonstrate similar resolution, while MUSICAL shows decreased resolution.
Fig. 6
Fig. 6 Block diagram of the main elements of SPARCOM. Each frame from the diffraction limited movie is converted into the Fourier domain using the fast Fourier transform (FFT2 in Matlab). Afterwards, the auto-correlation matrix (for a pre-chosen time-leg) of the entire Fourier transformed data set is estimated empirically (displayed here in logarithmic scale). Using an estimate of the PSF, a sparse solver (based on a tailored FISTA implementation) recovers the super-resolved image on a denser grid. All SPARCOM reconstructions provided in this work are generated on a high-resolution grid 8 times denser than the low-resolution grid of the input movies.
Fig. 7
Fig. 7 Experimental reconstruction of ER in a U2OS cell. Panels (a)-(c) illustrate a diffraction-limited image, a single noisy frame and the SPARCOM recovery, under sparsity assumption in the wavelet domain. SPARCOM achieves a clear and smooth depiction of the ER, while rejecting most of the background noise in the raw movie (b).
Fig. 8
Fig. 8 Additional simulation results. Upper row: (a) Ground truth: high resolution image of simulated microtubules. (b) Diffraction-limited image. (c) Smoothed ThunderSTORM recovery from a low-density movie of 12000 frames. (d) 4th SOFI recovery (absolute value, zero time-lag) from a high-density movie of 361 frames. Lower row: (e)-(f) FALCON reconstructions from movies of 361 frames and 60 frames of increased densities, respectively. (g)-(h) SPARCOM reconstructions for the 361 and 60 frames movies, respectively. Comparing the reconstructions for the same number of frames (e.g., the magnified regions in the red boxes), clearly the ability of SPARCOM to separate between closely adjacent subwavelength features is superior to FALCON and SOFI. In this example, SPARCOM has an acquisition rate 33 (361 frames) and 200 (60 frames) times faster than PALM/STORM.
Fig. 9
Fig. 9 Intensity profiles comparing the performance of ThunderSTORM, FALCON and SPARCOM. Normalized cross-sections taken along the solid yellow line (upper) and the dashed yellow line (lower) of Fig. 8, comparing the ground truth (dash-dot green, Fig. 8(a)), diffraction-limited image (dot blue, Fig. 8(b)), smoothed ThunderSTORM recovered from the low density movie of 12000 frames (solid yellow, Fig. 8(c)), FALCON image recovered from the high density movie of 60 frames (dash red, Fig. 8(f)), and SPARCOM image recovered from the high density movie of 60 frames (purple circle head, Fig. 8(h)). These panels substantiate the visual conclusions drawn from Fig. 8: SPARCOM recovers the profiles of sub-diffraction spaced microtubules from a high emitters' density movie clearly and with a good agreement to the ground truth and ThunderSTORM, while FALCON does not achieve such high resolution.
Fig. 10
Fig. 10 Simulation results comparing between SPARCOM, single molecule ThunderSTORM, SOFI and the ground truth in a movie with high emitters' density. Upper row: (a) Ground truth: high resolution image of simulated subwavelength structure. (b) Diffraction limited image created by summing the 1000 frames of high the emitters' density movie. (c) A single diffraction limited frame from the raw data of the high density movie. Clearly most emitters are active simultaneously and their far-fields are overlapping. (d) Smoothed ThunderSTORM reconstruction from a low emitters' density movie with 4000 frames. Comparing visually, the ThunderSTORM image is very similar to the ground truth. Middle row: (e) Smoothed ThunderSTORM, recovered from the movie of 1000 frames of high emitters' density. (f) Correlation SOFI (zero time-lag), recovered from the movie of 1000 frames of high emitters' density. (g) 4th order SOFI (in absolute value, zero time-lag), recovered from the movie of 1000 frames of high emitters' density. (h) SPARCOM, recovered from 200 frames of the high density movie. Lower row: SPARCOM reconstructions from varying number of frames. All reconstructions were performed from the high emitters' density movie. (i) SPARCOM, recovered from 100 frames. (j) SPARCOM, recovered from 500 frames. (k) SPARCOM, recovered from 600 frames. (l) SPARCOM, recovered from 1000 frames. All SPARCOM reconstructions were smoothed with the same kernel as in (d) and (e). Comparing panels (d) with (h)-(l) visually, it can be seen that the spatial resolution of SPARCOM is similar to that of single molecule ThunderSTORM recovered from a low density movie, in which clear separation of emitters in each frame is possible. The SPARCOM resolution is comparable to that of the ground truth image, performed only with 200 frames instead of 4000 frames as in the low density ThunderSTORM recovery, leading to a 20 times faster acquisition rate. Additionally, panel (b) shows the diffraction limited features with out-of-focus features (thicker objects). Correlation SOFI (panel (f)) did not eliminate this interference, while all SPARCOM and ThunderSTORM images successfully reject these features.
Fig. 11
Fig. 11 Comparative simulation intensity profiles taken from Fig. 10. Normalized cross-sections along the solid line (upper) and the dashed line (lower) of Fig. 10, comparing the ground truth (dash blue, Fig. 10(a)), diffraction-limited image (dash-dot green, Fig. 10(b)), smoothed ThunderSTORM reconstructed from the low density movie of 4000 frames (solid dot purple, Fig. 10(d)), smoothed ThunderSTORM reconstructed from the high density movie of 1000 frames (solid dot cyan, Fig. 10(e)), 4th order cumulant SOFI reconstructed from the high density movie of 1000 frames, absolute value, zero time-lag (dot black, Fig. 10(g)), and SPARCOM (solid red, Fig. 10(h)), reconstructed from 200 frames of the high density movie. We omit cross-sections from SPARCOM reconstructions performed with more frames, since they are similar to the SPARCOM cross-section from 200 frames. Considering both panels, it is clear that SPARCOM manages to recover the two adjacent sub-diffraction features with comparable width to the ground truth and the ThunderSTORM reconstruction from the low-density movie, while SOFI and ThunderSTORM recovered from the high density movie do not recover these features correctly.
Fig. 12
Fig. 12 Experimental results. Panels (a)-(c) illustrate a diffraction-limited microtubule image, a single noisy frame and 4th order SOFI (absolute value, zero time-lag) recovery from a 500 frames movie, respectively. Judging visually, FALCON (500 frames, panel e and 50 frames, panel f) and SPARCOM (500 frames panel g, and 50 frames, panel h) depict a clearer image than ThunderSTORM (500 frames, panel d). Zoomed in red boxes show that FALCON requires hundreds of frames, while SPARCOM requires only tens of frames to resolve the filaments. (i, j) Intensity cross-sections (normalized) taken along the solid (i) and dashed (j) yellow lines. Panel (i) shows that in regions of high emitters' density, SPARCOM achieves better spatial resolution than the other methods, depicting the filaments' bifurcation clearly (FALCON missed this bifurcation). On the other hand, panel (j) shows that in areas of low emitters' density, the spatial resolution of SPARCOM is similar to that of single molecule localization and FALCON.
Fig. 13
Fig. 13 Multi-emitter fitting comparisons on both simulated and experimental data sets. (a), (d) Diffraction limited images of the experimental and simulated movies, respectively. (b), (e) Multi-emitter ThunderSTORM fitting using movies of 500 and 361 frames of experimental and simulated data sets, respectively. (c), (f) Multi-emitter ThunderSTORM fitting using movies of 50 and 60 frames of experimental and simulated data sets, respectively. Red boxes indicated enlarged areas in lower right corner of each panel. As the number of frames is reduced, while preserving the total number of emitters, reconstruction quality degrades.
Fig. 14
Fig. 14 Qualitative comparison between SPARCOM and 3B for high emitters’ density data sets. (a), (f) are diffraction limited images of a selected area from the simulated and experimental data sets presented in Figs. 2 and 4 in the main paper, respectively. Panels (b) and (c) show 3B and SPARCOM reconstructions using 361 frames, respectively, while panels. (d) and (e) show 3B and SPARCOM reconstructions for 60 frames, respectively (by summing every 6 consecutive frames in the 361 frames movie). Panels (g) and (h) present 3B and SPARCOM reconstructions for 500 frames, while panels (i) and (j) compare 3B and SPARCOM reconstructions of the same object from a movie of 50 frames (by summing every 10 consecutive frames in the 500 frames movie). The 3B reconstructions seem very aggregated and broken, compared with SPARCOM, in all cases.
Fig. 15
Fig. 15 Simulation of the minimal separation distance for SPARCOM. (a) and (c) show diffraction limited spots created by summing 50 frames of two and four simulated fluctuating emitters, respectively. Purple dots represent true positions of the emitters. (b) and (d) illustrate SPARCOM recoveries of the two and four emitters, respectively, alongside their true locations, marked by the purple dots. Clear separation is achieved even when the emitters are 75nm apart, as shown in panel (b).
Fig. 16
Fig. 16 Reconstruction comparison between SPARCOM and a sparsity-based frame-by-frame recovery. Panels (a) and (b) show an overlay of diffraction limited spots (grey), SPARCOM (purple) and frame-by-frame sparsity-based reconstruction (green) reconstructions for two different emitters' densities. Each movie contains 1000 frames. Clearly, SPARCOM exhibits more details and better localization of the emitters than by performing a sparsity-based frame-by-frame recovery. SPARCOM recoveries were performed using λ=0.0001, while frame-by-frame recoveries were performed using λ=0.1, such that each method produces the best possible result.
Fig. 17
Fig. 17 Qualitative comparison between the ability of SPARCOM and ThunderSTORM to correctly identify positions of emitters with increasing emitters' density. (a)-(c) Overlay of diffraction limited spots (grey), SPARCOM (green) and ThunderSTORM (purple) reconstructions for different emitters' densities. White dots indicate that both reconstructions overlap. All images are presented in logarithmic scale, for clarity. Comparisons are made on a linear scale. (d)-(e) Correct support identification (“hit rate”) and erroneous support identification (“false alarm rate”), respectively. (d)-(e) show that when high emitters' density is used, SPARCOM identifies the correct emitters' locations substantially better than single molecule fit, while preserving a low false-alarm rate.
Fig. 18
Fig. 18 Resolution enhancement versus increasing labeling density. Each curve represents the resolution enhancement achieved by performing FRC analysis over movies of increasing labeling densities for MUSICAL, SRRF and SPARCOM. Each point on each curve is the result of FRC analysis between the ground truth (simulated) image and the reconstruction for each method. This figure clearly shows that the resolution gain of SPARCOM is better than SRRF and MUSICAL, especially as the labeling density increases.
Fig. 19
Fig. 19 Resolution enhancement versus SNR. Each curve represents the resolution gain achieved by performing FRC analysis over movies of increasing SNR values (the lower the curve, the higher the spatial resolution). Each point on a curve is the result of FRC analysis between the ground truth (simulated) image and the reconstruction for each method. This figure clearly shows that the resolution gain of SPARCOM is better than the other methods, for all SNR values considered here.

Tables (3)

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Table 1 Algorithm 1: SPARCOM via FISTA for minimizing (9)

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Table 1 FRC resolution gains for panels a and b of Fig. 5, showing the gain at the intersection points of the FRC lines with the fixed 1/7 threshold, depicted as a black vertical line.

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Table 2 FRC resolution gains for panel c of Fig. 5, showing the gain at the intersection points of the FRC lines with the fixed 1/7 threshold, depicted as a black vertical line.

Equations (13)

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f[m Δ L ,n Δ L ,t]= k=0 L1 u[m Δ L m k ,n Δ L n k ] s k (t) ,
f[m Δ L ,n Δ L ,t]= k=0 L1 u[m Δ L i k Δ H ,n Δ L j k Δ H ] s k (t) .
f[mP,nP,t]= i=0 N1 j=0 N1 u[mPi,nPj] s ij (t).
F[ k m , k n ,t]= m=0 M1 n=0 M1 f[mP,nP,t] W N k m m W N k n n ,
F[ k m , k n ,t]=U[ k m , k n ] i=0 N1 j=0 N1 W N k m i W N k n j s ij (t),
y(t)=As(t),
H =diag{U[0,0],...,U[M1,M1]}.
R y (τ)=A R s (τ) A H ,
R y (τ)= 1 Tτ t=1 Tτ (y(t) y ¯ ) (y(t+τ) y ¯ ) H ,
min x0 λ||Wx| | 1 +f(x),
f(x)= 1 2 R y (0) l=1 N 2 a l a l H x l F 2 .
f(x)=Mxv,
SN R dB =10 log 10 Powe r signal Powe r noise .

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