Abstract

Single molecule super-resolution microscopy enables imaging at sub-diffraction-limit resolution by producing images of subsets of stochastically photoactivated fluorophores over a sequence of frames. In each frame of the sequence, the fluorophores are accurately localized, and the estimated locations are used to construct a high-resolution image of the cellular structures labeled by the fluorophores. Many methods have been developed for localizing fluorophores from the images. The majority of these methods comprise two separate steps: detection and estimation. In the detection step, fluorophores are identified. In the estimation step, the locations of the identified fluorophores are estimated through an iterative approach. Here, we propose a non-iterative state space-based localization method which combines the detection and estimation steps. We demonstrate that the estimated locations obtained from the proposed method can be used as initial conditions in an estimation routine to potentially obtain improved location estimates. The proposed method models the given image as the frequency response of a multi-order system obtained with a balanced state space realization algorithm based on the singular value decomposition of a Hankel matrix. The locations of the poles of the resulting system determine the peak locations in the frequency domain, and the locations of the most significant peaks correspond to the single molecule locations in the original image. The performance of the method is validated using both simulated and experimental data.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2015 (3)

2014 (2)

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, 4577 (2014).
[Crossref] [PubMed]

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

2012 (1)

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

2011 (4)

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. Express 19(18), 16963–16974 (2011).
[Crossref] [PubMed]

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59(5), 2182–2195 (2011).
[Crossref]

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(5), 1377–1393 (2011).
[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]

2010 (2)

T. A. Laurence and B. A. Chromy, “Efficient maximum likelihood estimator fitting of histograms,” Nat. Methods 7(5), 338–339 (2010).
[Crossref] [PubMed]

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

2006 (2)

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

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

2005 (2)

X. Lai, E. S. Ward, Z. Lin, and R. J. Ober, “Three-dimensional state space realization algorithm: noise suppression of fluorescence microscopy images and point spread functions,” Proc. SPIE 5701, 53–60 (2005).
[Crossref]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “State space realization of a three-dimensional image set with application to noise reduction of fluorescent microscopy images of cells,” Multidim. Syst. Sign. P. 16(1), 7–47 (2005).
[Crossref]

2004 (1)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

1996 (1)

T. McKelvey, H. Akçay, and L. Ljung, “Subspace-based multivariable system identification from frequency response data,” IEEE Trans. Automatic Control 41(7), 960–979 (1996).
[Crossref]

1995 (1)

J. M. Maciejowski, “Guaranteed stability with subspace methods,” Systems & Control Letters 26(2), 153–156 (1995).
[Crossref]

1992 (1)

Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,” IEEE Trans. Signal Process. 40(9), 2267–2280 (1992).
[Crossref]

Abraham, A. V.

Akçay, H.

T. McKelvey, H. Akçay, and L. Ljung, “Subspace-based multivariable system identification from frequency response data,” IEEE Trans. Automatic Control 41(7), 960–979 (1996).
[Crossref]

Berglund, A. J.

Byars, J. M.

Calderbank, A. R.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59(5), 2182–2195 (2011).
[Crossref]

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, 4577 (2014).
[Crossref] [PubMed]

Chao, J.

Chi, Y.

J. Huang, K. Gumpper, Y. Chi, M. Sun, and J. Ma, “Fast two-dimensional super-resolution image reconstruction algorithm for ultra-high emitter density,” Opt. Lett. 40(13), 2989–2992 (2015).
[Crossref] [PubMed]

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59(5), 2182–2195 (2011).
[Crossref]

Chromy, B. A.

T. A. Laurence and B. A. Chromy, “Efficient maximum likelihood estimator fitting of histograms,” Nat. Methods 7(5), 338–339 (2010).
[Crossref] [PubMed]

Ding, J.

Donoho, D. L.

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

Elnatan, D.

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

Fornasiero, E. F.

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

Gumpper, K.

Henriques, R.

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

Holden, S.

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, 4577 (2014).
[Crossref] [PubMed]

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]

Hua, Y.

Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,” IEEE Trans. Signal Process. 40(9), 2267–2280 (1992).
[Crossref]

Huang, B.

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

Huang, F.

Huang, J.

Huang, Z. L.

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]

Kirshner, H.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

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, 4577 (2014).
[Crossref] [PubMed]

Lai, X.

X. Lai, E. S. Ward, Z. Lin, and R. J. Ober, “Three-dimensional state space realization algorithm: noise suppression of fluorescence microscopy images and point spread functions,” Proc. SPIE 5701, 53–60 (2005).
[Crossref]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “State space realization of a three-dimensional image set with application to noise reduction of fluorescent microscopy images of cells,” Multidim. Syst. Sign. P. 16(1), 7–47 (2005).
[Crossref]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “A state space approach to noise reduction of 3D fluorescent microscopy images,” in Proc. IEEE International Conference on Image Processing (IEEE2004), pp. 1153–1156.

Laurence, T. A.

T. A. Laurence and B. A. Chromy, “Efficient maximum likelihood estimator fitting of histograms,” Nat. Methods 7(5), 338–339 (2010).
[Crossref] [PubMed]

Lelek, M.

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

Liddle, J. A.

Lidke, K. A.

Lin, Z.

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “State space realization of a three-dimensional image set with application to noise reduction of fluorescent microscopy images of cells,” Multidim. Syst. Sign. P. 16(1), 7–47 (2005).
[Crossref]

X. Lai, E. S. Ward, Z. Lin, and R. J. Ober, “Three-dimensional state space realization algorithm: noise suppression of fluorescence microscopy images and point spread functions,” Proc. SPIE 5701, 53–60 (2005).
[Crossref]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “A state space approach to noise reduction of 3D fluorescent microscopy images,” in Proc. IEEE International Conference on Image Processing (IEEE2004), pp. 1153–1156.

Liu, X.

Liu, Y.

Ljung, L.

T. McKelvey, H. Akçay, and L. Ljung, “Subspace-based multivariable system identification from frequency response data,” IEEE Trans. Automatic Control 41(7), 960–979 (1996).
[Crossref]

Ma, J.

Maciejowski, J. M.

J. M. Maciejowski, “Guaranteed stability with subspace methods,” Systems & Control Letters 26(2), 153–156 (1995).
[Crossref]

Manley, S.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

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, 4577 (2014).
[Crossref] [PubMed]

McClelland, J. J.

McKelvey, T.

T. McKelvey, H. Akçay, and L. Ljung, “Subspace-based multivariable system identification from frequency response data,” IEEE Trans. Automatic Control 41(7), 960–979 (1996).
[Crossref]

McMahon, M. D.

Mhlanga, M. M.

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

Min, J.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

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, 4577 (2014).
[Crossref] [PubMed]

Ober, R. J.

A. Tahmasbi, E. S. Ward, and R. J. Ober, “Determination of localization accuracy based on experimentally acquired image sets: applications to single molecule microscopy,” Opt. Express 23(6), 7630–7652 (2015).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

X. Lai, E. S. Ward, Z. Lin, and R. J. Ober, “Three-dimensional state space realization algorithm: noise suppression of fluorescence microscopy images and point spread functions,” Proc. SPIE 5701, 53–60 (2005).
[Crossref]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “State space realization of a three-dimensional image set with application to noise reduction of fluorescent microscopy images of cells,” Multidim. Syst. Sign. P. 16(1), 7–47 (2005).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “A state space approach to noise reduction of 3D fluorescent microscopy images,” in Proc. IEEE International Conference on Image Processing (IEEE2004), pp. 1153–1156.

Olivier, N.

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, 4577 (2014).
[Crossref] [PubMed]

Pengo, T.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

Pezeshki, A.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59(5), 2182–2195 (2011).
[Crossref]

Quan, T.

Ram, S.

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

Sage, D.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

Scharf, L. L.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59(5), 2182–2195 (2011).
[Crossref]

Schwartz, S. L.

Small, A.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Stahlheber, S.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Stuurman, N.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

Sun, M.

Tahmasbi, A.

Unser, M.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

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, 4577 (2014).
[Crossref] [PubMed]

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]

Valtorta, F.

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

Vonesch, C.

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, 4577 (2014).
[Crossref] [PubMed]

Ward, E. S.

A. Tahmasbi, E. S. Ward, and R. J. Ober, “Determination of localization accuracy based on experimentally acquired image sets: applications to single molecule microscopy,” Opt. Express 23(6), 7630–7652 (2015).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17(26), 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
[Crossref]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “State space realization of a three-dimensional image set with application to noise reduction of fluorescent microscopy images of cells,” Multidim. Syst. Sign. P. 16(1), 7–47 (2005).
[Crossref]

X. Lai, E. S. Ward, Z. Lin, and R. J. Ober, “Three-dimensional state space realization algorithm: noise suppression of fluorescence microscopy images and point spread functions,” Proc. SPIE 5701, 53–60 (2005).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
[Crossref] [PubMed]

R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “A state space approach to noise reduction of 3D fluorescent microscopy images,” in Proc. IEEE International Conference on Image Processing (IEEE2004), pp. 1153–1156.

Ye, J. C.

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, 4577 (2014).
[Crossref] [PubMed]

Zeng, S.

Zhang, W.

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

Zhu, H.

Zhu, L.

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

Zimmer, C.

R. Henriques, M. Lelek, E. F. Fornasiero, F. Valtorta, C. Zimmer, and M. M. Mhlanga, “QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ,” Nat. Methods 7(5), 339–340 (2010).
[Crossref] [PubMed]

Biomed. Opt. Express (1)

Biophys. J. (1)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86(2), 1185–1200 (2004).
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R. J. Ober, X. Lai, Z. Lin, and E. S. Ward, “State space realization of a three-dimensional image set with application to noise reduction of fluorescent microscopy images of cells,” Multidim. Syst. Sign. P. 16(1), 7–47 (2005).
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S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Syst. Sign. P. 17(1), 27–57 (2006).
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Nat. Methods (6)

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
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X. Lai, E. S. Ward, Z. Lin, and R. J. Ober, “Three-dimensional state space realization algorithm: noise suppression of fluorescence microscopy images and point spread functions,” Proc. SPIE 5701, 53–60 (2005).
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Figures (13)

Fig. 1
Fig. 1

Analysis of the detection rate of the algorithm, applied to data sets in which each image contains one molecule, whose location in the image is chosen randomly according to a uniform probability distribution. For a given data set, the same mean photon count is used to simulate the molecule in each image. Different data sets differ by this mean photon count. For each mean photon count, 200 images of size 30 × 30 pixels were simulated using the parameters given in Section 5.1. The Hungarian algorithm with a search area of radius 100 nm is used to pair the localized molecules with the ground truth molecules.

Fig. 2
Fig. 2

Analysis of the error of location estimates obtained from a data set in which each frame contains one molecule whose location in the image is chosen randomly. Shown in the left and right plots are the differences between the x-estimates and the true x-values, and the differences between the y-estimates and the true y-values, respectively, for the true positives obtained with the algorithm. The data set consists of 1000 15 × 15-pixel images, each of a molecule with a mean photon count of 1500 photons whose location is randomly chosen from a uniform probability distribution that places the molecule between the 2nd and 14th pixel in both the x and y dimensions. The images were simulated using the parameters given in Section 5.1.

Fig. 3
Fig. 3

Analysis of the average of location estimates obtained from repeat images of one molecule. Shown in the left and right plots are the difference between the average of the x-estimates and the true x-value, and the difference between the average of the y-estimates and the true y-value, respectively, for data sets that differ by the mean photon count assumed for the molecule per image. For each mean photon count, the data set consists of 1000 images of size 15 × 15 pixels, simulated using the parameters given in Section 5.1.

Fig. 4
Fig. 4

Analysis of the standard deviation of location estimates obtained from repeat images of one molecule. (a) The standard deviations of the x- and y-estimates for nine of the data sets from Fig. 3. (b) The percentage difference between the standard deviation of the x-estimates and the limit of the x-localization accuracy, and the percentage difference between the standard deviation of the y-estimates and the limit of the y-localization accuracy. The percentage difference is the absolute difference between the standard deviation of the estimates and the corresponding limit of accuracy, expressed as a percentage of the limit of accuracy.

Fig. 5
Fig. 5

Analysis of the standard deviation of location estimates produced by the maximum likelihood estimator when the location estimates obtained with the algorithm are used as the initial conditions. (a) The standard deviations of the maximum likelihood x- and y-estimates for the same data sets as in Fig. 4, which comprise repeat images of one molecule. (b) The percentage difference between the standard deviation of the x-estimates and the limit of the x-localization accuracy, and the percentage difference between the standard deviation of the y-estimates and the limit of the y-localization accuracy.

Fig. 6
Fig. 6

Analysis of the detection rate of the algorithm, applied to data sets in which each image contains multiple molecules whose locations in the image are chosen randomly. For a given data set, the mean photon count is the same for each molecule in every frame. The location of each molecule is drawn from a uniform distribution that places it inside the image, with the constraint that the distance between each pair of molecules is not less than the minimum distance dmin. For each data set, we simulated 200 images of size 30 × 30 pixels using the parameters given in Section 5.1. For data sets in which there are two molecules per image, the precision and recall measures are shown as a function of dmin in (a), where the mean photon count is 2500 photons/molecule, and as a function of the mean photon count in (b), where dmin = 100 nm. For data sets in which there are three molecules per image, the precision and recall measures are shown as a function of dmin in (c), where the mean photon count is 2500 photons/molecule, and as a function of the mean photon count in (d), where dmin = 100 nm. The Hungarian algorithm with a search area of radius 100 nm is used to pair the localized molecules with the ground truth molecules.

Fig. 7
Fig. 7

Reconstruction of images containing two or three closely spaced molecules. (a) Images of size 60 × 60 pixels of 2, 3, and 2 closely spaced point sources separated from one another by a distance d of 300 nm, 250 nm, and 50 nm, respectively. The images are simulated using the parameters given in Section 5.1. (b) Mesh plots of the images shown in (a). (c) Mesh plots of the magnitude of the reconstructed image (algorithm result), showing the detection of 2, 3, and 2 single molecules in the image.

Fig. 8
Fig. 8

Analysis of the average of the location estimates obtained from sets of repeat images of two molecules. Shown in the left and right plots are the differences between the average of the x-estimates and the true x-value for the first and second molecules, respectively, for data sets comprising 15 × 15-pixel, 20 × 20-pixel, and 40 × 40-pixel images. For each image size, distances d between the two molecules are chosen around half of the side length of the square region occupied by the image in the object space. For a given data set, we simulated 500 images with a mean photon count of 2500 photons/molecule and the parameters given in Section 5.1. The results for d = s/2, where s = 65N nm is the side length of the square region occupied by an N × N-pixel image in the object space, are shown with filled symbols.

Fig. 9
Fig. 9

Analysis of the average and standard deviation of location estimates obtained from sets of repeat images of two molecules as a function of the mean photon count per molecule. Two scenarios are considered - one in which the distance d between the two molecules is 650 nm, and one in which d is 487.5 nm. For each scenario, the data sets differ by the mean photon count per molecule. For each mean photon count, the data set consists of 500 repeat images of size 20 × 20 pixels, simulated using the parameters given in Section 5.1. (a) Differences between the average of the estimated x-locations and the corresponding true x-coordinates for the two molecules. (b) The standard deviations of the estimated x-locations for the two molecules.

Fig. 10
Fig. 10

Result of the algorithm applied to an experimental super-resolution image. (a) Image of individual Alexa Fluor 647 molecules acquired using the microscopy setup described in Section 5.2. The pixel size and image size are 16 µm × 16 µm and 192 × 192 pixels, respectively. (b) The magnitude of the reconstructed image obtained with the algorithm.

Fig. 11
Fig. 11

Results of the algorithm applied to an ROI from an experimental super-resolution image. (a) A 41 × 41-pixel ROI of the super-resolution image shown in Fig. 10. (b) The magnitude of the reconstructed image (algorithm result). (c) The image reconstructed using Eq. (36), in which the single molecule locations estimated using our algorithm are used in the computation of the Airy profile q in Eq. (37), and in which the mean photon counts Np,n and the parameter α : = 2 π n a λ are separately estimated with a maximum likelihood estimator. (d), (e), and (f) show the mesh plots of the images in (a), (b), and (c), respectively.

Fig. 12
Fig. 12

Analysis of the bias of location estimates obtained from repeat images containing exactly one molecule, simulated using the frequency response of a first-order system. (a) Image of a point source simulated using the frequency response of a first-order system, i.e., using Eq. (42) with h = 1, and a mean photon count of Np,1 = 1000. (b) Mesh view of the image shown in (a). (c) Difference between the average of the x-estimates and the true x-value, and difference between the average of the y-estimates and the true y-value for data sets that differ by the mean photon count per image assumed for the molecule. For each mean photon count, the data set consists of 1000 repeat images of size 20 × 20 pixels, simulated using the frequency response of a first-order system.

Fig. 13
Fig. 13

Analysis of the standard deviation of location estimates obtained from repeat images containing exactly one molecule, simulated using the frequency response of a first-order system. Shown in the left and right plots are the standard deviations of the x- and y-estimates and the limits of the x- and y-localization accuracy, respectively, for nine of the data sets from Fig. 12.

Tables (1)

Tables Icon

Table 1 Detection rate of the algorithm as a function of the threshold values used for the retention of singular values in the first, second and third SVDs.a

Equations (42)

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X ( n ) = C A n 1 B , n = 1 , 2 , , N .
X ( n ) = C A n 1 B , n = 1 , 2 , , N .
X ˜ ( k ) = C ˜ ( e i 2 π k / N I A ˜ ) 1 B ˜ , k = 1 , 2 , , N ,
X ( n ) : = ( I D F T ( X ˜ ) ) ( n ) = 1 N k = 1 N X ˜ ( k ) e i 2 π k n / N , n = 1 , 2 , , N ,
X ( n ) = C A n 1 B , n = 1 , 2 , , N .
X ˜ ( k ) = ( D F T ( X ) ) ( k ) = n = 1 N X ( n ) e i 2 π k n / N = C B e i 2 π k / N + C A B e i 4 π k / N + + C A N 1 B e i 2 π k N / N = C e i 2 π k / N ( I + A e i 2 π k / N + + A N 1 e i 2 π k ( N 1 ) / N ) B = C e i 2 π k / N [ n = 0 N 1 ( A e i 2 π k / N ) n ] B .
X ˜ ( k ) = C e i 2 π k / N ( I A e i 2 π k / N ) 1 ( I A N ) B = C ( e i 2 π k / N I A ) 1 ( I A N ) B = C ˜ ( e i 2 π k / N I A ˜ ) 1 B ˜ ,
X ( n 1 , n 2 ) : = ( I D F T 2 D ( X ˜ ) ) ( n 1 , n 2 ) = 1 N 1 N 2 k 1 = 1 N 1 k 2 = 1 N 2 X ˜ ( k 1 , k 2 ) e i 2 π ( k 1 n 1 / N 1 + k 2 n 2 / N 2 ) ,
X ( n 1 , n 2 ) = X 1 ( n 1 ) X 2 ( n 2 ) , n i = 1 , 2 , , N i , i = 1 , 2 ,
X i ( n i ) : = C i A i n i 1 B i , n i = 1 , 2 , , N i .
X ˜ ( k 1 , k 2 ) = X ˜ 1 ( k 1 ) X ˜ 2 ( k 2 ) , k i = 1 , 2 , , N i , i = 1 , 2 ,
X ˜ j ( k j ) : = C ˜ j ( e i 2 π k j / N j I A ˜ j ) 1 B ˜ j ,
X ( N 1 , n 2 ) : = ( I D F T 2 D ( X ˜ ) ) ( n 1 , n 2 ) = 1 N 1 N 2 k 1 = 1 N 1 k 2 = 1 N 2 X ˜ ( k 1 , k 2 ) e i 2 π ( k 1 n 1 / N 1 + k 2 n 2 / N 2 ) ,
Q : = [ X ( 1 , 1 ) X ( 1 , 2 ) X ( 1 , N 2 ) X ( 2 , 1 ) X ( 2 , 2 ) X ( 2 , N 2 ) X ( N 1 , 1 ) X ( N 1 , 2 ) X ( N 1 , N 2 ) ] .
[ X 1 ( 1 ) X 1 ( 2 ) X 1 ( N 1 ) ] : = U Σ 1 / 2 , [ X 2 ( 1 ) X 2 ( 2 ) X 2 ( N 2 ) ] : = Σ 1 / 2 V .
X ( n 1 , n 2 ) = X 1 ( n 1 ) X 2 ( n 2 ) , n i = 1 , 2 , , N i , i = 1 , 2 .
X i ( n i ) = C i A i n i 1 B i , n i = 1 , 2 , , N i .
X ˜ ( k 1 , k 2 ) = ( D F T 2 D ( X ) ) ( k 1 , k 2 ) = n 1 = 1 N 1 n 2 = 1 N 2 X ( n 1 , n 2 ) e i 2 π ( k 1 n 1 / N 1 + k 2 n 2 / N 2 ) = ( n 1 = 1 N 1 X 1 ( n 1 ) e i 2 π k 1 n 1 / N 1 ) ( n 2 = 1 N 2 X 2 ( n 2 ) e i 2 π k 2 n 2 / N 2 ) = X ˜ 1 ( k 1 ) X ˜ 2 ( k 2 ) , k i = 1 , 2 , , N i , i = 1 , 2 ,
X ˜ j ( k j ) : = C ˜ j ( e i 2 π k j / N j I A ˜ j ) 1 B ˜ j ,
X ˜ ( k 1 , k 2 ) = X ˜ 1 ( k 1 ) X ˜ 2 ( k 2 ) , k i = 1 , 2 , , N i , i = 1 , 2 ,
X ˜ j ( k j ) : = C ˜ j ( e i 2 π k j / N j I A ˜ j ) 1 B ˜ j ,
X ˜ ( k 1 , k 2 ) = j = 1 2 C ¯ j ( e i 2 π k j / N j I A ¯ j ) 1 B ¯ j = C ¯ 1 [ b 1 1 c 1 2 ( e i 2 π k 1 / N 1 a 1 1 ) ( e i 2 π k 2 / N 2 a 1 2 ) b 1 1 c 2 2 ( e i 2 π k 1 / N 1 a 1 1 ) ( e i 2 π k 2 / N 2 a 2 2 ) b 1 1 c s 2 2 ( e i 2 π k 1 / N 1 a 1 1 ) ( e i 2 π k 2 / N 2 a s 2 2 ) b 2 1 c 1 2 ( e i 2 π k 1 / N 1 a 2 1 ) ( e i 2 π k 2 / N 2 a 1 2 ) b 2 1 c 2 2 ( e i 2 π k 1 / N 1 a 2 1 ) ( e i 2 π k 2 / N 2 a 2 2 ) b 1 1 c s 2 2 ( e i 2 π k 1 / N 1 a 2 1 ) ( e i 2 π k 2 / N 2 a s 2 2 ) b s 1 1 c 1 2 ( e i 2 π k 1 / N 1 a s 1 1 ) ( e i 2 π k 2 / N 2 a 1 2 ) b s 1 1 c 2 2 ( e i 2 π k 1 / N 1 a s 1 1 ) ( e i 2 π k 2 / N 2 a 2 2 ) b s 1 1 c s 2 2 ( e i 2 π k 1 / N 1 a s 1 1 ) ( e i 2 π k 2 / N 2 a s 2 2 ) ] B ¯ 2 = l = 1 s 1 j = 1 s 2 c l 1 b l 1 c j 2 b j 2 ( e i 2 π k 1 / N 1 a l 1 ) ( e i 2 π k 2 / N 2 a l 2 ) .
x t 2 : = Δ x w t 2 2 N 1 2 M π + Δ x 2 M , y t 1 : = Δ y w t 1 1 N 2 2 M π + Δ y 2 M ,
X ˜ b s ( k 1 , k 2 ) : = X ˜ ( k 1 , k 2 ) β ^ , k i = 1 , 2 , , N i , i = 1 , 2 .
X ( n 1 , n 2 ) : = ( I D F T 2 D ( X ˜ b s ) ) ( n 1 , n 2 ) , n i = 1 , 2 , , N i , i = 1 , 2 .
Q : = [ X ( 1 , 1 ) X ( 1 , 2 ) X ( 1 , N 2 ) X ( 2 , 1 ) X ( 2 , 2 ) X ( 2 , N 2 ) X ( N 1 , 1 ) X ( N 1 , 2 ) X ( N 1 , N 2 ) ] .
[ X 1 r ( 1 ) X 1 r ( 2 ) X 1 r ( N 1 ) ] : = U ^ Σ ^ 1 / 2 , [ X 2 r ( 1 ) X 2 r ( 2 ) X 2 r ( N 2 ) ] : Σ ^ 1 / 2 V ^ .
H i : = [ X i r ( 1 ) X i r ( 2 ) X i r ( N i 1 ) X i r ( N i ) 0 X i r ( 2 ) X i r ( 3 ) X i r ( N i ) 0 0 X i r ( N i ) 0 0 0 0 0 0 0 0 0 ] , i = 1 , 2 ,
U ^ i = [ U ¯ 1 i U ¯ N i i U ¯ N i + 1 i ] , i = 1 , 2 ,
U ^ i : = [ U ¯ 2 i U ¯ N i + 1 i ] , U ^ i : = [ U ¯ 1 i U ¯ N i i ] , i = 1 , 2 .
x ^ t 2 : = Δ x w t 2 2 N 1 2 M π + Δ x 2 M , y ^ t 1 : = Δ y w t 1 1 N 2 2 M π + Δ y 2 M , t i = 1 , 2 , ; s i , i = 1 , 2 ,
r ^ = min r = 1 , , K { r : E r E K > τ } ,
l ^ i = min l i = 1 , , N i { l i : E l i E N i > τ i } ,
x ^ n : = Δ x 2 M w ¯ n 2 N 1 π + Δ x 2 M , y ^ n : = Δ y 2 M w ¯ n 1 N 2 π + Δ y 2 M ,
h ^ = arg min h = min ( s 1 , s 2 ) , s i = 1 , , l ^ i , i = 1 , 2 ( k = 1 N p i x ( z k μ θ ^ h ( k ) ) 2 ) ,
μ θ ^ h ( k ) : = n = 1 h N p , n M 2 C k q ( x M x ^ n , y M y ^ n ) d x d y , θ ^ h 2 h , h = min ( s 1 , s 2 ) ,
q ( x , y ) : = J 1 2 ( 2 π n a λ x 2 + y 2 ) π ( x 2 + y 2 ) , ( x , y ) 2 ,
θ ^ m l e = arg min θ Θ ( L ( θ | z 1 , , z N p i x ) ) ,
L ( θ | z 1 , , z N p i x ) = k = 1 N p i x log ( 1 2 π σ k l = 0 ( [ μ θ ( k ) + β k ] l e [ μ θ ( k ) + β k ] l ! e 1 2 ( z k l η k σ k ) 2 ) ) .
μ θ ( k ) : = N p M 2 C k q ( x M x 0 , y M y 0 ) d x d y , k = 1 , , N p i x ,
P R E : = T P F P + T P , R E C : = T P F N + T P .
μ θ ^ h ( k 1 , k 2 ) : = 1 C | n = 1 h N p , n ( e i 2 π k 1 / N 1 a ¯ n 1 ) ( e i 2 π k 2 / N 2 a ¯ n 2 ) | , k i = 1 , , N i , i = 1 , 2 ,

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