Abstract

This paper proposes a cascading algorithm (CSR) based on compressed sensing, which aims to reduce intensive computations in super-resolution imaging of fluorescence microscopy. Performance of existing algorithms such as CVX and L1H drop sharply when applied to obtain finer images with high density molecules. CSR fully exploits the extreme sparsity property of molecules in the compressed sensing model and progressively restricts solution space stage by stage. We perform a comprehensive study of existing algorithms and the proposed algorithm under different resolutions and molecules’ densities. Simulation and experimental results confirm the performance advantage of CSR when applied to recover dense molecules.

© 2015 Optical Society of America

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References

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2014 (1)

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Rep. 45388 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

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

2011 (4)

E. J. Candés, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. A. 31, 59–73 (2011).
[Crossref]

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] [PubMed]

S. Wolter, U. Endesfelder, S. Van de Linde, and M. Heilemann, “Measuring localization performance of super-resolution algorithms on very active samples,” Opt. Express 19, 7020–7033 (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. Express 2, 1377–1393 (2011)
[Crossref] [PubMed]

2008 (1)

D. L. Donoho and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE T. Inform. Theory 54, 4789–4812 (2008).
[Crossref]

2006 (4)

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52, 489–509 (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] [PubMed]

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

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

2003 (1)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

2002 (1)

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[Crossref] [PubMed]

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Aufmkolk, S.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

Babcock, H. P.

Barsic, A.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Rep. 45388 (2014).
[Crossref] [PubMed]

Bates, M.

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

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 5793, 1642–1645 (2006).
[Crossref]

Bonifacino, J. S.

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

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, (1999).
[Crossref]

Byars, J. M.

Candés, E. J.

E. J. Candés, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. A. 31, 59–73 (2011).
[Crossref]

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52, 489–509 (2006).
[Crossref]

Cao, Y.

Dabauvalle, M. C.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[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,” Science 5793, 1642–1645 (2006).
[Crossref]

Donoho, D. L.

D. L. Donoho and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE T. Inform. Theory 54, 4789–4812 (2008).
[Crossref]

Eldar, Y. C.

E. J. Candés, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. A. 31, 59–73 (2011).
[Crossref]

Elnatan, D.

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

Endesfelder, U.

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Girirajan, T. P. K.

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

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Grover, G.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Rep. 45388 (2014).
[Crossref] [PubMed]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Heilemann, M.

Hess, H. F.

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

Hess, S. T.

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

Holden, S. J.

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] [PubMed]

Holm, T.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

Huang, B.

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

Huang, F.

Kapanidis, A. N.

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] [PubMed]

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[Crossref] [PubMed]

Lidke, K. A.

Lindwasser, O. W.

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

Lippincott-Schwartz, J.

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

Löschberger, A.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Mason, M. D.

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

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Moffitt, J. R.

Needell, D.

E. J. Candés, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. A. 31, 59–73 (2011).
[Crossref]

Ober, R. J.

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” in Proceedings of the National Academy of Sciences of the United States of America (2006), pp. 4457–4462.
[Crossref]

Olenych, S.

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

Patterson, G. H.

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

Piestun, R.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Rep. 45388 (2014).
[Crossref] [PubMed]

Ram, S.

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” in Proceedings of the National Academy of Sciences of the United States of America (2006), pp. 4457–4462.
[Crossref]

Randall, P.

E. J. Candés, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. A. 31, 59–73 (2011).
[Crossref]

Romberg, J.

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52, 489–509 (2006).
[Crossref]

Rust, M. J.

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

Sauer, M.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

Schwartz, S. L.

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Sougrat, R.

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

Tao, T.

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52, 489–509 (2006).
[Crossref]

Thompson, R. E.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[Crossref] [PubMed]

Tsaig, Y.

D. L. Donoho and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE T. Inform. Theory 54, 4789–4812 (2008).
[Crossref]

Uphoff, S.

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] [PubMed]

van de Linde, S.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

S. Wolter, U. Endesfelder, S. Van de Linde, and M. Heilemann, “Measuring localization performance of super-resolution algorithms on very active samples,” Opt. Express 19, 7020–7033 (2011).
[Crossref] [PubMed]

Ward, E. S.

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” in Proceedings of the National Academy of Sciences of the United States of America (2006), pp. 4457–4462.
[Crossref]

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, (1999).
[Crossref]

Wolter, S.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

S. Wolter, U. Endesfelder, S. Van de Linde, and M. Heilemann, “Measuring localization performance of super-resolution algorithms on very active samples,” Opt. Express 19, 7020–7033 (2011).
[Crossref] [PubMed]

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Zhang, W.

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

Zhu, L.

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

Zhuang, X.

H. P. Babcock, J. R. Moffitt, Y. Cao, and X. Zhuang, “Fast compressed sensing analysis for super-resolution imaging using L1-homotopy,” Opt. Express 21, 28583–28596 (2013).
[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] [PubMed]

Appl. Comput. Harmon. A. (1)

E. J. Candés, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. A. 31, 59–73 (2011).
[Crossref]

Biomed. Opt. Express (1)

Biophys. J. (2)

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

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[Crossref] [PubMed]

IEEE T. Inform. Theory (2)

D. L. Donoho and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE T. Inform. Theory 54, 4789–4812 (2008).
[Crossref]

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52, 489–509 (2006).
[Crossref]

J. Soc. Ind. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Nat. Methods (4)

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M. C. Dabauvalle, S. van de Linde, and M. Sauer, “Rapid-STORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[Crossref] [PubMed]

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] [PubMed]

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

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] [PubMed]

Opt. Express (2)

Sci. Rep. (1)

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Rep. 45388 (2014).
[Crossref] [PubMed]

Science (2)

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

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science 5628, 2061–2065 (2003).
[Crossref]

Other (5)

Nature Education, “The relative scale of biological molecules and structures,” http://www.nature.com/scitable/content/the-relative-scale-of-biological-molecules-and-14704956 .

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, (1999).
[Crossref]

M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined convex programming,” http://cvxr.com/cvx/ .

M. S. Asif and J. Romberg, “L1 Homotopy,” http://users.ece.gatech.edu/sasif/homotopy/

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” in Proceedings of the National Academy of Sciences of the United States of America (2006), pp. 4457–4462.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 An example for the cascading design which aims to obtain the fine-resolution image with RF equal to 8. Here, the cascading stages for one raw pixel with three molecules inside are shown.
Fig. 2
Fig. 2 Performance of algorithms CVX, L1H and CSR. (a, c, e) Computation time per 7×7 pixel window averaged over 16 windows per frame and 30 frames per simulation for refining factors of 8, 12 and 16 respectively. (b, d, f) Recall rate averaged over 30 frames per simulation for refining factors 8, 12 and 16 respectively. The horizontal axis of each sub-figure represents a series of discrete value of molecules’ density from 1/μm2 to 12 μm2 while the vertical axis shows either the recall rate or computation time. The speed of L1H drops a lot with the increase of density while CSR remains stable. Also, CSR shows higher recall rate than L1H for high density molecules.
Fig. 3
Fig. 3 Performance comparison with different refining factors and image results display. (a) One instance from the random simulations. (b, c) Computation time and recall rate for refining factors from 2 to 18 given the density of 5 molecules/μm2.
Fig. 4
Fig. 4 A hollow three-ring pattern is used to check the ability of algorithms on reconstructing images with different densities. Color arrows point to areas with different densities. The green arrow indicates the highest density with three rings crossing together while the white arrows show the crossing areas of two rings. The hollow areas are clearly recovered under RF equal to 8 but slightly vague under RF equal to 16 with the proposed algorithm CSR.
Fig. 5
Fig. 5 Reconstruction results for real STORM images by RapidSTORM, CVX and the proposed CSR. (a) A raw image frame. (b) Images recovered by RapidSTORM, CVX and CSR with RF equal to 8 respectively. (c) Images recovered by RapidSTORM, CVX and CSR with RF equal to 16 respectively. (d) The whole image reconstructed by RapidSTORM from 5000 raw images. Scale bars are 1μm.
Fig. 6
Fig. 6 Time and quality analysis for the reconstructed results of real image series. (a) Timing cost taken by three algorithms for different resolutions. (b) The profile of the image patch pointed by the white line in Fig. 5(c).

Tables (2)

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Algorithm 1: Cascading Super-Resolution (CSR) Algorithm

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Table 1 Performance comparison of four algorithms to recover the three-ring pattern with different resolutions.

Equations (2)

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min x 1 s . t . A x b 2 ε Σ b j & x 0
min A x b 2 2 + λ x 1 s . t . x 0

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