Abstract

In super-resolution imaging techniques based on single-molecule switching and localization, the time to acquire a super-resolution image is limited by the maximum density of fluorescent emitters that can be accurately localized per imaging frame. In order to increase the imaging rate, several methods have been recently developed to analyze images with higher emitter densities. One powerful approach uses methods based on compressed sensing to increase the analyzable emitter density per imaging frame by several-fold compared to other reported approaches. However, the computational cost of this approach, which uses interior point methods, is high, and analysis of a typical 40 µm x 40 µm field-of-view super-resolution movie requires thousands of hours on a high-end desktop personal computer. Here, we demonstrate an alternative compressed-sensing algorithm, L1-Homotopy (L1H), which can generate super-resolution image reconstructions that are essentially identical to those derived using interior point methods in one to two orders of magnitude less time depending on the emitter density. Moreover, for an experimental data set with varying emitter density, L1H analysis is ~300-fold faster than interior point methods. This drastic reduction in computational time should allow the compressed sensing approach to be routinely applied to super-resolution image analysis.

© 2013 Optical Society of America

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2013 (1)

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

2012 (5)

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy1(1), 6–10 (2012).
[CrossRef]

E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J.102(10), 2391–2400 (2012).
[CrossRef] [PubMed]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

2011 (5)

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods8(4), 279–280 (2011).
[CrossRef] [PubMed]

F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express2(5), 1377–1393 (2011).
[CrossRef] [PubMed]

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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

T. Quan, H. Zhu, X. Liu, Y. Liu, J. Ding, S. Zeng, and Z.-L. Huang, “High-density localization of active molecules using Structured Sparse Model and Bayesian Information Criterion,” Opt. Express19(18), 16963–16974 (2011).
[CrossRef] [PubMed]

S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011).
[CrossRef] [PubMed]

2010 (2)

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010).
[CrossRef] [PubMed]

B. Huang, H. P. Babcock, and X. Zhuang, “Breaking the diffraction barrier: Super-resolution imaging of cells,” Cell143(7), 1047–1058 (2010).
[CrossRef] [PubMed]

2009 (2)

E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput.31(2), 890–912 (2009).
[CrossRef]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences2(1), 183–202 (2009).
[CrossRef]

2008 (3)

H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008).
[CrossRef] [PubMed]

D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory54(11), 4789–4812 (2008).
[CrossRef]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008).
[CrossRef] [PubMed]

2007 (1)

M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007).
[CrossRef] [PubMed]

2006 (3)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J.91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

2005 (1)

D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP5, 733–736 (2005).

2000 (1)

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal.20(3), 389–403 (2000).
[CrossRef]

Babcock, H. P.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy1(1), 6–10 (2012).
[CrossRef]

E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J.102(10), 2391–2400 (2012).
[CrossRef] [PubMed]

B. Huang, H. P. Babcock, and X. Zhuang, “Breaking the diffraction barrier: Super-resolution imaging of cells,” Cell143(7), 1047–1058 (2010).
[CrossRef] [PubMed]

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008).
[CrossRef] [PubMed]

M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences2(1), 183–202 (2009).
[CrossRef]

Betzig, E.

H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008).
[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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Bi, G.-Q.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

Byars, J. M.

Cetin, M.

D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP5, 733–736 (2005).

Churchman, L. S.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010).
[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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Dempsey, G. T.

M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007).
[CrossRef] [PubMed]

Ding, J.

Donoho, D. L.

D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory54(11), 4789–4812 (2008).
[CrossRef]

Elnatan, D.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Flyvbjerg, H.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010).
[CrossRef] [PubMed]

Friedlander, M. P.

E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput.31(2), 890–912 (2009).
[CrossRef]

Galbraith, C. G.

H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008).
[CrossRef] [PubMed]

Galbraith, J. A.

H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008).
[CrossRef] [PubMed]

Ganesh, A.

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

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(11), 4258–4272 (2006).
[CrossRef] [PubMed]

Grünwald, D.

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

He, J.

S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011).
[CrossRef] [PubMed]

Heintzmann, R.

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. Methods9(2), 195–200 (2011).
[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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Hess, S. T.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J.91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

Holden, S. J.

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods8(4), 279–280 (2011).
[CrossRef] [PubMed]

Huang, B.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

B. Huang, H. P. Babcock, and X. Zhuang, “Breaking the diffraction barrier: Super-resolution imaging of cells,” Cell143(7), 1047–1058 (2010).
[CrossRef] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008).
[CrossRef] [PubMed]

M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007).
[CrossRef] [PubMed]

Huang, F.

Huang, Z.-L.

Jones, G. E.

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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

Jones, S. A.

S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011).
[CrossRef] [PubMed]

Jovanovic-Talisman, 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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

Kapanidis, A. N.

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods8(4), 279–280 (2011).
[CrossRef] [PubMed]

Lidke, K. A.

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express2(5), 1377–1393 (2011).
[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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

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. Methods9(2), 195–200 (2011).
[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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Liu, X.

Liu, Y.

Ma, Y.

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

Malioutov, D. M.

D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP5, 733–736 (2005).

Mason, M. D.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J.91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

Monypenny, J.

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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

Mortensen, K. I.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010).
[CrossRef] [PubMed]

Mukamel, E. A.

E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J.102(10), 2391–2400 (2012).
[CrossRef] [PubMed]

Nieuwenhuizen, R. P. J.

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. Methods10(6), 557–562 (2013).
[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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Osborne, M. R.

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal.20(3), 389–403 (2000).
[CrossRef]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Presnell, B.

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal.20(3), 389–403 (2000).
[CrossRef]

Puig, D. L.

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

Quan, T.

Rieger, B.

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

Rosten, E.

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. Methods9(2), 195–200 (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. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Sastry, S. S.

A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010).
[CrossRef]

Schwartz, S. L.

Shankar, S. S.

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

Shim, S.-H.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011).
[CrossRef] [PubMed]

Shroff, H.

H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008).
[CrossRef] [PubMed]

Sigal, Y. M.

H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy1(1), 6–10 (2012).
[CrossRef]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Spudich, J. A.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010).
[CrossRef] [PubMed]

Stallinga, S.

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences2(1), 183–202 (2009).
[CrossRef]

Tsaig, Y.

D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory54(11), 4789–4812 (2008).
[CrossRef]

Turlach, B. A.

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal.20(3), 389–403 (2000).
[CrossRef]

Uphoff, S.

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods8(4), 279–280 (2011).
[CrossRef] [PubMed]

van den Berg, E.

E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput.31(2), 890–912 (2009).
[CrossRef]

Vaughan, J. C.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

Wang, W.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008).
[CrossRef] [PubMed]

Wang, X.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

Willsky, A. S.

D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP5, 733–736 (2005).

Xia, C.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

Xu, C.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

Yang, A. Y.

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010).
[CrossRef]

Zeng, S.

Zhang, W.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Zhong, G.

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

Zhou, Z.

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

Zhu, H.

Zhu, L.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Zhuang, X.

H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy1(1), 6–10 (2012).
[CrossRef]

E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J.102(10), 2391–2400 (2012).
[CrossRef] [PubMed]

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011).
[CrossRef] [PubMed]

B. Huang, H. P. Babcock, and X. Zhuang, “Breaking the diffraction barrier: Super-resolution imaging of cells,” Cell143(7), 1047–1058 (2010).
[CrossRef] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008).
[CrossRef] [PubMed]

M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Biomed. Opt. Express (1)

Biophys. J. (2)

E. A. Mukamel, H. P. Babcock, and X. Zhuang, “Statistical Deconvolution for Superresolution Fluorescence Microscopy,” Biophys. J.102(10), 2391–2400 (2012).
[CrossRef] [PubMed]

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J.91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

Cell (1)

B. Huang, H. P. Babcock, and X. Zhuang, “Breaking the diffraction barrier: Super-resolution imaging of cells,” Cell143(7), 1047–1058 (2010).
[CrossRef] [PubMed]

ICASSP (1)

D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” ICASSP5, 733–736 (2005).

IEEE Trans. Image Process. (1)

A. Y. Yang, Z. Zhou, A. Ganesh, S. S. Shankar, and Y. Ma, “Fast L1-Minimization Algorithms For Robust Face Recognition,” IEEE Trans. Image Process.22(8), 3234–3246 (2012).

IEEE Trans. Inf. Theory (1)

D. L. Donoho and Y. Tsaig, “Fast Solution of the L1-norm Minimization Problem When the Solution May be Sparse,” IEEE Trans. Inf. Theory54(11), 4789–4812 (2008).
[CrossRef]

IMA J. Numer. Anal. (1)

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal.20(3), 389–403 (2000).
[CrossRef]

Nat. Methods (8)

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods8(4), 279–280 (2011).
[CrossRef] [PubMed]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

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. Methods9(2), 195–200 (2011).
[CrossRef] [PubMed]

S. A. Jones, S.-H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods8(6), 499–505 (2011).
[CrossRef] [PubMed]

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. Methods10(6), 557–562 (2013).
[CrossRef] [PubMed]

H. Shroff, C. G. Galbraith, J. A. Galbraith, and E. Betzig, “Live-cell photoactivated localization microscopy of nanoscale adhesion dynamics,” Nat. Methods5(5), 417–423 (2008).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010).
[CrossRef] [PubMed]

Opt. Express (1)

Optical Nanoscopy (1)

H. P. Babcock, Y. M. Sigal, and X. Zhuang, “A high-density 3D localization algorithm for stochastic optical reconstruction microscopy,” Optical Nanoscopy1(1), 6–10 (2012).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

S.-H. Shim, C. Xia, G. Zhong, H. P. Babcock, J. C. Vaughan, B. Huang, X. Wang, C. Xu, G.-Q. Bi, and X. Zhuang, “Super-resolution fluorescence imaging of organelles in live cells with photoswitchable membrane probes,” Proc. Natl. Acad. Sci. U.S.A.109(35), 13978–13983 (2012).
[CrossRef] [PubMed]

Science (3)

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,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

M. Bates, B. Huang, G. T. Dempsey, and X. Zhuang, “Multicolor super-resolution imaging with photo-switchable fluorescent probes,” Science317(5845), 1749–1753 (2007).
[CrossRef] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science319(5864), 810–813 (2008).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (1)

E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput.31(2), 890–912 (2009).
[CrossRef]

SIAM Journal on Imaging Sciences (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences2(1), 183–202 (2009).
[CrossRef]

Other (3)

M. C. Grant and S. P. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.0 beta,” http://cvxr.com/cvx (2013).

M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab Software for Disciplined Convex Programming, Version 1.0 beta 3,” Recent Advances in Learning and Control}, 95-110 (2006).

A. Y. Yang and S. S. Sastry, “Fast l1-minimization algorithms and an application in robust face recognition: a review,” Proceedings of 2010 IEEE 17th International Conference on Image Processing, 1849–1852 (2010).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic depiction of STORM imaging using compressed sensing. A subset of fluorescent emitters (red dots; top left) from a labeled sample (bottom left) are activated stochastically. The activated emitters are close enough that the individual emitters are not distinguishable in a 4 pixel × 4 pixel conventional image (top left). Using compressed sensing, a high resolution grid of fluorophore locations is reconstructed (top right) from the low resolution image (top left) under the constraint that this reconstruction contains the smallest number of emitters that can reproduce the measured conventional image up to a given accuracy. This process is repeated and the individual reconstructed frames are summed to produce the final high resolution reconstruction (lower right) of the original sample.

Fig. 2
Fig. 2

One-dimensional (1D) illustration of the properties of the solution space exploited by L1-Homotopy. (a) A simulated 1D image with two emitters before (blue) and after (red) convolution with a Gaussian point-spread-function. (b) Results of the L1H analysis of the image shown in (a) displayed as a kymograph of the amplitude of each up-sampled pixel, i.e. potential fluorophore location (rows), as a function of the homotopy parameter, λ. Note that as λ decreases, increasingly favoring accuracy over sparsity, the single initial peak splits into two peaks, representing two fluorophore localizations. (c) The amplitude of two adjacent up-sampled pixels, e.g. emitter locations, as a function of λ for the left emitter (top) and the right emitter (bottom). The amplitude of these pixels, depicted in red, blue, green and black symbols, corresponds to the pixels marked by red, blue, green and black arrows in (b). The amplitude of these pixels are piece-wise linear functions of λ. The break-points (dashed grey lines), where the slopes change, correspond to the addition, removal, or movement of a possible emitter, i.e. where the amplitude of a pixel changes from zero to non-zero or vice versa (marked by gray circles). Note that “movement” of an emitter from one up-sampled pixel to another is actually accomplished by removing it from the solution and then adding another emitter at a new location.

Fig. 3
Fig. 3

Analysis of a sub-region of the image by L1-Homotopy. The wide-field fluorescence image detected by the camera is first divided into partially overlapping, sub-regions of 7 × 7 camera pixels in size (a camera pixel is represented by a black box). One such region is shown here. In the reconstruction of this region, we allow emitters to be placed within an 8 × 8 camera pixel space, corresponding to a 1/2 pixel extension in each direction from the 7 × 7 camera pixel region, and the emitter positions are allowed on a finer, 8-fold up-sampled grid (red dots). To further limit errors that might arise due to edge effects, only emitter localizations located within a central 5 pixel × 5 pixel block defined by the thicker black line are kept. Multiple of these non-overlapping, 5-pixel wide, up-sampled images are then stitched together to form the final STORM image.

Fig. 4
Fig. 4

L1-Homotopy produces nearly identical reconstructions to interior point methods. (a-c; left) Simulated, high-density, 7 × 7 camera pixel single-frame conventional image. The locations of individual emitters are marked with the red crosses. (a-c; middle) The CVX reconstruction of the emitter locations. (a-c; right) The L1H reconstruction of the same images. The arrows in (c) highlight a rare difference between the two solutions. (d) Histogram of the distance between an emitter in the CVX reconstruction and the nearest emitter in the L1H reconstruction for the same simulated data. The histogram is plotted for three different emitter densities. (e) The average distance between every emitter in the CVX solution to the nearest emitter in the L1H solution as a function of emitter density. (f) The average fractional difference in the number of emitters found in the CVX solution and the L1H solution as a function of emitter density. (g) The average percent difference between the L1 norm of the CVX solution and the L1H solution. (f) The average percent difference between the residual image error of the CVX and the L1H solution. (e) – (f) Both the mean (blue) and median (red) are provided. Error bars represent standard deviation measured directly (blue) or estimated from the inter-quartile range (red).

Fig. 5
Fig. 5

Comparison of the reconstructed emitter density and localization error derived from L1H, CVX, a single-emitter fitting algorithm, and a multi-emitter fitting algorithm. (a) The density of reconstructed emitters as a function of the density of simulated emitters for a single-emitter fitting algorithm (green squares), a multi-emitter fitting algorithm (blue pluses), L1H (red crosses), and CVX (black diamonds). The dashed black line has the slope of 1. (b) The XY localization error for each algorithm labeled as in panel (a). The two panels in (a) and (b) cover different density ranges.

Fig. 6
Fig. 6

L1-Homotopy reconstructs images with a substantially higher speed than interior point methods, but is slower than the emitter fitting algorithms. (a) The average analysis time for a 256 × 256 camera pixel image as a function of emitter density for the two compressed sensing algorithms CVX (black diamonds) and L1H (red crosses) as well as a multi-emitter fitting algorithm (blue pluses) and a single-emitter fitting algorithm (green squares). (b) The ratio of the analysis time per frame for CVX to L1H as a function of emitter density.

Fig. 7
Fig. 7

L1-Homotopy analysis of experimental STORM data. (a) A sub-area of a single frame of a high-emitter-density STORM data set acquired from Alexa-647-labeled microtubules in BS-C-1 cells. Individual molecules found with L1H (green circles), CVX (blue crosses), and a single-emitter localization algorithm (red crosses) are plotted. (b) Average analysis time per frame (256 × 256 camera pixel) for L1H and CVX estimated from the analysis time for the first 10 frames of a 5000 frame STORM movie of microtubules in a BSC-1 cell. CVX takes 340-fold longer than L1H to analyze these frames. (c) The full reconstructed STORM image of this data set using a single-emitter localization algorithm. (d) The same data set reconstructed with L1H. The red arrows and arrow heads indicated two regions with high and low microtubule density, respectively. (e) A zoom-in of the area outlined by the red box in (c). (f) A zoom-in of the same area outlined by the red box in (d). The image reconstructed by the L1H compressed sensing algorithm is much smoother than that by the single-emitter localization algorithm because roughly 4-fold more fluorophores are localized by the L1H algorithm. Scale bars in (c,d) are 10 µm and 1 µm in (a, e, f).

Fig. 8
Fig. 8

A comparison of different analysis methods on simulated STORM data. (a) A single frame of the STORM movie. (b) The true locations of the emitters used for the simulation. (c) The STORM image reconstructed using a single-emitter fitting algorithm that can only handle sparse emitter densites (d) The STORM image reconstructed using a multi-emitter fitting algorithm, DAOSTORM, that can handle moderate-to-high emitter densities. (e) The STORM image reconstructed using the deconSTORM algorithm. (f) The STORM image reconstructed using the L1H algorithm. All scale bars are 500 nm. The simulated STORM movie had an average density of 5 emitters/μm2 and was 200 frames long.

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Minimize: x 1 Subject to: Axb 2 ε.
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