Abstract

The applicability of widefield stochastic microscopy, such as PALM or STORM, is limited by their long acquisition times. Images are produced from the accumulation of a large number of frames that each contain a scarce number of super-resolved localizations. We show that the random and uneven distribution of localizations leads to a specific type of trade-off between the spatial and temporal resolutions. We derive analytical predictions for the minimal time required to obtain a reliable image at a given spatial resolution. We find that the image completion time scales logarithmically with the ratio of the image size to the spatial resolution volume, with second order corrections due to spurious localization within the background noise. We validate our predictions against experimental localization sequences of labeled microtubule filaments obtained by STORM. Our theoretical framework makes it possible to compare the efficiency of emitters, define optimal labeling strategies, and allow implementation of a stopping criterion for data acquisitions that can be performed using real-time monitoring algorithms.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (1)

Z. Zhang, Y. Nishimura, and P. Kanchanawong, “Extracting microtubule networks from superresolution single-molecule localization microscopy data,” Molec. Biol. Cell 28, 333–345 (2016).
[Crossref] [PubMed]

2015 (5)

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3D high- and super-resolution imaging using single-objective SPIM,” Nat. Methods 12, 641 (2015).
[Crossref] [PubMed]

H. Wolfenson, G. Meacci, S. Liu, M. R. Stachowiak, T. Iskratsch, S. Ghassemi, P. Roca-Cusachs, B. O’Shaughnessy, J. Hone, and M. P. Sheetz, “Tropomyosin controls sarcomere-like contractions for rigidity sensing and suppressing growth on soft matrices,” Nat. Cell Biol. 18, 33 (2015).
[Crossref]

R. Changede, X. Xu, F. Margadant, and M. P. Sheetz, “Nascent integrin adhesions form on all matrix rigidities after integrin activation,” Devel. Cell 35, 1–8 (2015).
[Crossref]

K. H. Biswas, K. L. Hartman, C.-H. Yu, O. J. Harrison, H. Song, A. W. Smith, W. Y. C. Huang, W.-C. Lin, Z. Guo, A. Padmanabhan, S. M. Troyanovsky, M. L. Dustin, L. Shapiro, B. Honig, R. Zaidel-Bar, and J. T. Groves, “E-cadherin junction formation involves an active kinetic nucleation process,” Proc. Natl. Acad. Sci. USA 112, 10932–10937 (2015).
[Crossref] [PubMed]

Z. Liu, L. D. Lavis, and E. Betzig, “Imaging live-cell dynamics and structure at the single-molecule level,” Mol. Cell 58, 644–659 (2015).
[Crossref] [PubMed]

2014 (3)

B. Hajj, J. Wisniewski, M. El Beheiry, J. J. Chen, A. Revyakin, C. Wu, and M. Dahan, “Whole-cell, multicolor superresolution imaging using volumetric multifocus microscopy,” Proc. Natl. Acad. Sci. USA 111, 17480–17485 (2014).
[Crossref] [PubMed]

G. Shtengel, Y. Wang, Z. Zhang, W. I. Goh, H. F. Hess, and P. Kanchanawong, “Imaging cellular ultrastructure by PALM, iPALM, and correlative iPALM-EM,” Methods Cell Biol. 123, 273–294 (2014).
[Crossref] [PubMed]

A. Martinez-Marrades, J.-F. Rupprecht, M. Gross, and G. Tessier, “Stochastic 3D optical mapping by holographic localization of Brownian scatterers,” Opt. Express 22, 29191–29203 (2014).
[Crossref] [PubMed]

2013 (3)

C. Bertocchi, W. I. Goh, Z. Zhang, and P. Kanchanawong, “Nanoscale imaging by superresolution fluorescence microscopy and its emerging applications in biomedical research,” Crit. Rev. Biomed. Eng. 41, 281–308 (2013).
[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. Methods 10, 557–562 (2013).
[Crossref] [PubMed]

M. Alexeyev, I. Shokolenko, G. Wilson, and S. Ledoux, “The maintenance of mitochondrial DNA integrity,” Cold Spring Harb. Perspect. Biol. 5, 1–17 (2013).
[Crossref]

2012 (2)

J. E. Fitzgerald, J. Lu, and M. J. Schnitzer, “Estimation theoretic measure of resolution for stochastic localization microscopy,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

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

2011 (2)

G. T. Dempsey, J. C. Vaughan, K. H. Chen, M. Bates, and X. Zhuang, “Evaluation of fluorophores for optimal performance in localization-based super-resolution imaging,” Nat. Methods 8, 1027–1036 (2011).
[Crossref] [PubMed]

P. Annibale, S. Vanni, M. Scarselli, U. Rothlisberger, and A. Radenovic, “Identification of clustering artifacts in photoactivated localization microscopy,” Nat. Methods 8, 527–528 (2011).
[Crossref] [PubMed]

2010 (2)

S. van de Linde, S. Wolter, M. Heilemann, and M. Sauer, “The effect of photoswitching kinetics and labeling densities on super-resolution fluorescence imaging,” J. Biotechnol. 149, 260–266 (2010).
[Crossref] [PubMed]

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

2009 (1)

A. R. Small, “Theoretical limits on errors and acquisition rates in localizing switchable fluorophores,” Biophys. J. 96, L16–L18 (2009).
[Crossref] [PubMed]

2006 (3)

K. I. Willig, R. R. Kellner, R. Medda, B. Hein, S. Jakobs, and S. W. Hell, “Nanoscale resolution in GFP-based microscopy,” Nat. Methods 3, 721–723 (2006).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (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,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

2005 (1)

A. Triller and D. Choquet, “Surface trafficking of receptors between synaptic and extrasynaptic membranes: and yet they do move!” Trends in Neurosciences 28, 133–139 (2005).
[Crossref]

1998 (1)

C. E. Shannon, “Communication in the Presence Of Noise,” Proc. IEEE 86, 447 (1998).
[Crossref]

1995 (1)

1990 (1)

R. P. Stanley and H. S. Wilf, “Generatingfunctionology,” Am. Math. Monthly 97, 864 (1990).
[Crossref]

1979 (1)

B. Efron, “Bootstrap methods: another look at the jackknife,” Ann. Stat. 7, 1–26 (1979).
[Crossref]

1973 (1)

L. Flatto, “A limit theorem for random coverings of a circle,” Israel J. Math. 15, 167–184 (1973).
[Crossref]

1967 (1)

1961 (1)

P. Erdos and A. Renyi, “On a classical problem of probability theory,” Magyar Tudomanyos Akademia Matematikai Kutato Intezetenek Kozlemenyei 6, 215–220 (1961).

1960 (1)

D. J. Newman, “The double dixie cup problem”, The American Mathematical Monthly 67, 58–61 (1960).
[Crossref]

Alexeyev, M.

M. Alexeyev, I. Shokolenko, G. Wilson, and S. Ledoux, “The maintenance of mitochondrial DNA integrity,” Cold Spring Harb. Perspect. Biol. 5, 1–17 (2013).
[Crossref]

Annibale, P.

P. Annibale, S. Vanni, M. Scarselli, U. Rothlisberger, and A. Radenovic, “Identification of clustering artifacts in photoactivated localization microscopy,” Nat. Methods 8, 527–528 (2011).
[Crossref] [PubMed]

Aravind, A.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3D high- and super-resolution imaging using single-objective SPIM,” Nat. Methods 12, 641 (2015).
[Crossref] [PubMed]

Aufmkolk, S.

S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M.-C. Dabauvalle, S. V. D. Linde, M. Sauer, and S. van de Linde, “RapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[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. Methods 10, 557–562 (2013).
[Crossref] [PubMed]

G. T. Dempsey, J. C. Vaughan, K. H. Chen, M. Bates, and X. Zhuang, “Evaluation of fluorophores for optimal performance in localization-based super-resolution imaging,” Nat. Methods 8, 1027–1036 (2011).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
[Crossref] [PubMed]

Bertocchi, C.

C. Bertocchi, W. I. Goh, Z. Zhang, and P. Kanchanawong, “Nanoscale imaging by superresolution fluorescence microscopy and its emerging applications in biomedical research,” Crit. Rev. Biomed. Eng. 41, 281–308 (2013).
[Crossref] [PubMed]

Betzig, E.

Z. Liu, L. D. Lavis, and E. Betzig, “Imaging live-cell dynamics and structure at the single-molecule level,” Mol. Cell 58, 644–659 (2015).
[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,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

E. Betzig, “Proposed method for molecular optical imaging,” Opt. Lett. 20, 237 (1995).
[Crossref] [PubMed]

Biswas, K. H.

K. H. Biswas, K. L. Hartman, C.-H. Yu, O. J. Harrison, H. Song, A. W. Smith, W. Y. C. Huang, W.-C. Lin, Z. Guo, A. Padmanabhan, S. M. Troyanovsky, M. L. Dustin, L. Shapiro, B. Honig, R. Zaidel-Bar, and J. T. Groves, “E-cadherin junction formation involves an active kinetic nucleation process,” Proc. Natl. Acad. Sci. USA 112, 10932–10937 (2015).
[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,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Changede, R.

R. Changede, X. Xu, F. Margadant, and M. P. Sheetz, “Nascent integrin adhesions form on all matrix rigidities after integrin activation,” Devel. Cell 35, 1–8 (2015).
[Crossref]

Chen, J. J.

B. Hajj, J. Wisniewski, M. El Beheiry, J. J. Chen, A. Revyakin, C. Wu, and M. Dahan, “Whole-cell, multicolor superresolution imaging using volumetric multifocus microscopy,” Proc. Natl. Acad. Sci. USA 111, 17480–17485 (2014).
[Crossref] [PubMed]

Chen, K. H.

G. T. Dempsey, J. C. Vaughan, K. H. Chen, M. Bates, and X. Zhuang, “Evaluation of fluorophores for optimal performance in localization-based super-resolution imaging,” Nat. Methods 8, 1027–1036 (2011).
[Crossref] [PubMed]

Choquet, D.

A. Triller and D. Choquet, “Surface trafficking of receptors between synaptic and extrasynaptic membranes: and yet they do move!” Trends in Neurosciences 28, 133–139 (2005).
[Crossref]

Dabauvalle, M.-C.

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

Dahan, M.

B. Hajj, J. Wisniewski, M. El Beheiry, J. J. Chen, A. Revyakin, C. Wu, and M. Dahan, “Whole-cell, multicolor superresolution imaging using volumetric multifocus microscopy,” Proc. Natl. Acad. Sci. USA 111, 17480–17485 (2014).
[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 313, 1642–1645 (2006).
[Crossref] [PubMed]

Dempsey, G. T.

G. T. Dempsey, J. C. Vaughan, K. H. Chen, M. Bates, and X. Zhuang, “Evaluation of fluorophores for optimal performance in localization-based super-resolution imaging,” Nat. Methods 8, 1027–1036 (2011).
[Crossref] [PubMed]

Dustin, M. L.

K. H. Biswas, K. L. Hartman, C.-H. Yu, O. J. Harrison, H. Song, A. W. Smith, W. Y. C. Huang, W.-C. Lin, Z. Guo, A. Padmanabhan, S. M. Troyanovsky, M. L. Dustin, L. Shapiro, B. Honig, R. Zaidel-Bar, and J. T. Groves, “E-cadherin junction formation involves an active kinetic nucleation process,” Proc. Natl. Acad. Sci. USA 112, 10932–10937 (2015).
[Crossref] [PubMed]

Efron, B.

B. Efron, “Bootstrap methods: another look at the jackknife,” Ann. Stat. 7, 1–26 (1979).
[Crossref]

El Beheiry, M.

B. Hajj, J. Wisniewski, M. El Beheiry, J. J. Chen, A. Revyakin, C. Wu, and M. Dahan, “Whole-cell, multicolor superresolution imaging using volumetric multifocus microscopy,” Proc. Natl. Acad. Sci. USA 111, 17480–17485 (2014).
[Crossref] [PubMed]

Erdos, P.

P. Erdos and A. Renyi, “On a classical problem of probability theory,” Magyar Tudomanyos Akademia Matematikai Kutato Intezetenek Kozlemenyei 6, 215–220 (1961).

Ester, M.

M. Ester, H. P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for discovering clusters in large spatial databases with noise,” in Proceedings of the 2nd International Conference on Knowledge Discovery and Data Mining (AAAI Press, 1996), pp. 226–231.

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2 (John Wiley & Sons Inc, 1968).

Fitzgerald, J. E.

J. E. Fitzgerald, J. Lu, and M. J. Schnitzer, “Estimation theoretic measure of resolution for stochastic localization microscopy,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

Flatto, L.

L. Flatto, “A limit theorem for random coverings of a circle,” Israel J. Math. 15, 167–184 (1973).
[Crossref]

Galland, R.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3D high- and super-resolution imaging using single-objective SPIM,” Nat. Methods 12, 641 (2015).
[Crossref] [PubMed]

Ghassemi, S.

H. Wolfenson, G. Meacci, S. Liu, M. R. Stachowiak, T. Iskratsch, S. Ghassemi, P. Roca-Cusachs, B. O’Shaughnessy, J. Hone, and M. P. Sheetz, “Tropomyosin controls sarcomere-like contractions for rigidity sensing and suppressing growth on soft matrices,” Nat. Cell Biol. 18, 33 (2015).
[Crossref]

Goh, W. I.

G. Shtengel, Y. Wang, Z. Zhang, W. I. Goh, H. F. Hess, and P. Kanchanawong, “Imaging cellular ultrastructure by PALM, iPALM, and correlative iPALM-EM,” Methods Cell Biol. 123, 273–294 (2014).
[Crossref] [PubMed]

C. Bertocchi, W. I. Goh, Z. Zhang, and P. Kanchanawong, “Nanoscale imaging by superresolution fluorescence microscopy and its emerging applications in biomedical research,” Crit. Rev. Biomed. Eng. 41, 281–308 (2013).
[Crossref] [PubMed]

Gradstein, I. S.

I. M. Ryzhik and I. S. Gradstein, Tables of Series, Products and Integrals (Academic Press, 2015).

Grenci, G.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3D high- and super-resolution imaging using single-objective SPIM,” Nat. Methods 12, 641 (2015).
[Crossref] [PubMed]

Gross, M.

Groves, J. T.

K. H. Biswas, K. L. Hartman, C.-H. Yu, O. J. Harrison, H. Song, A. W. Smith, W. Y. C. Huang, W.-C. Lin, Z. Guo, A. Padmanabhan, S. M. Troyanovsky, M. L. Dustin, L. Shapiro, B. Honig, R. Zaidel-Bar, and J. T. Groves, “E-cadherin junction formation involves an active kinetic nucleation process,” Proc. Natl. Acad. Sci. USA 112, 10932–10937 (2015).
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Supplementary Material (2)

NameDescription
» Code 1       Code for submission: Trade-offs between spatial and temporal resolutions in stochastic super-resolution microscopy techniquesJean-Francois Rupprecht, Ariadna Martinez-Marrades, Zhen Zhang, Rishita Changede, Pakorn Kanchanawong, and Gilles Tessier (D
» Dataset 1       Experimental datasets - STORM, TIRM, PALM

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Figures (9)

Fig. 1
Fig. 1

Structural reconstruction by stochastic super-localization microscopy. Probes (colored dots) are bound to a structure of interest (green line). (a) Circular patches representation: the diffraction pattern (left), from which super-resolution algorithms yield the coordinates of the center of the pattern. In the patch method representation (middle), each point coordinate is represented by a disk with a radius σ (blue circle: uncertainty of localization algorithm). Patches progressively accumulate as the acquisition time is increased, eventually covering the whole structure of interest (rightmost circle in (a): disks of different colors for separate time frames). (b–c) Box-filling representation: (b) The field of view is divided into N = 9 pixels among which F = 5 pixels contain probes. (c) Evolution of the map of the cumulative number of observations M j ( t ) for each pixel j and for each frame t. At t = 50, all pixels have been observed at least r = 10 times.

Fig. 2
Fig. 2

STORM imaging of the microtubule meshwork within a fibroblast cell, labelled by (left – a,c,e) Alexa-tubulin and (right – b,d,f) tdEos-tubulin fluorophores. (a–b) Accumulated number of observations per pixel of size σ = (160 nm)2. (c–d) Zoomed in region of crossed microtubules, in scatter (called patch-method) and density (called box-filling) representations (σ = (100 nm)2, error bar: 500 nm) (e) Analysis of the dispersion in the event probability density within the structure of interest. We fix a lower bound on the probability densities ρ = 2.4 ·10−6 nm−2.frame−1 for Alexa-tubulin and ρ = 1.2·10−6 nm−2.frame−1 for tdEos-tubulin. (f) Our theoretical expression predicts the centile t0.05 of the image acquisition time, expressed in number of frames. The solid lines represent the analytical result assuming an exponential dispersion (parameters are fixed by the fit of Fig. 2. (e) – see Sec. 4.2). The dashed line represent the coupon-collector scaling from Eq. (5), and the data point with error bars represent experimental results with bootstrap estimation of errors. See Dataset 1 (Files 1 and 2) for underlying values [38] and Code 1 [39].

Fig. 3
Fig. 3

Centile of the image completion time as a function of the redundant number of observations per pixel r, in the cases of (a) a homogeneous sample, for 3 values of the number of pixels within the ROI F (orange: F = 103; blue: F = 103; green F = 103); the coupon collector scaling (dashed lines) fits to the exact solution of image completion time (solid line) for sufficiently small value of r. (b) an exponential dispersion in the fluorophore densities, with a minimal probability of event p0m = 0.99 and F = 105: (upper triangle, dark blue) peaked distribution λ = 0 (circle, light orange) narrow dispersion λ = 0.01; (down triangle, magenta) wide dispersion λ = 0.1. A coupon-collector expression Eq. (5) in which the density is identified to the spatially averaged mean density 〈ρi〉 (dashed lines, magenta) significantly underestimates the image completion time.

Fig. 4
Fig. 4

Spatial inhomogeneities: estimation of the parameters Ci from Eq. (16) within the phase space (λ, p0,m): (a) C1, (b) C2 and (c) C3.

Fig. 5
Fig. 5

(a) Sketch of the evolution of the cumulative number of observations M j ( t ) in a given voxel j. (a, Inset) Trajectory of the particle around the voxel j (frame) labelled by the observation time. After t ≥ 6, the probe is not detected again. (b) Model: all correlated observations are collapsed into one single instantaneous event. (c–d) Evolution of the cumulative number of observations obtained for 4 pixels (among F = 100), either (c) from the experimental data set of [24], or (d) from Monte-Carlo simulations with time correlations. (e–f) Probability distribution of temporal correlations pk ∝ exp(−k/kc) (in log-scale): (e) from experiments (red circles), where the maximum likelihood estimator (MLE) of the exponential model provides the value kc = 2.9 ± 0.1 (black line); (f) from simulations (blue circles), with kc = 2.9 and with the same number of events as in the experiments. See Dataset 1 (File 3 - TIRM) for underlying values [38].

Fig. 6
Fig. 6

Comparison to the TIRM experimental data from [24]; the centile t0.05 of the image completion time is presented as a function of the required number of redundant observations per pixel r. (a) With F = 15; showing agreement between (red error bars) the centile estimation from experimental data (solid blue lines) our theoretical prediction from Eq. (5) with modelled time correlations, and (green crosses) simulations with time correlations. (b) With F = 4; showing agreement between (red error bars) the centile estimation from experimental data and (black dots) our theoretical prediction from Eq. (21). Error bars were estimated by bootstrapping [35]. See Dataset 1 (File 3 - TIRM) for underlying values [38].

Fig. 7
Fig. 7

STORM imaging at the entire cell level – the microtubule network is labelled with Alexa-tubulin fluorophores, see Fig. 2(a). (a) Distribution of spatial heterogeneities in the event probability 1 − p0 (σ = 160 × 160 nm2; F = 3.104): the tail of the experimental distribution (blue circles) is fitted by an exponential distribution (dashed black line). Inset: the field of view in terms of the number of collected observations per pixel. Under the hypothesis of a uniform event density, the distribution should follow a Gaussian distribution around its mean – indicated by the solid red line – with 95% of its statistical weight within the bounds indicated by the dashed line, but this is clearly not the case. (b) The centile t0.05 (in number of frames), where solid lines represent analytical result with p0,m = 0.986 and λ = 0.013, and the dashed line represent the coupon-collector scaling from 5, and the error bars represent the experimental results with bootstrap estimation of errors (see Sec. 2.6 for the fitting procedure and construction of the ROI; see Dataset 1 (File 1 - STORM Alexa647) for the underlying values [38].

Fig. 8
Fig. 8

PALM imaging of a silicon wafer with quasi-uniform coating in fluorophores (the ROI corresponds to the whole field of view, with N = F = 2500). (a) Non-uniform fluorophore density leads to spatial heterogeneities in p0: the experimental distribution (blue boxes) is fitted by a Gaussian distribution (dashed red line). Inset: the field of view in terms of the number of collected observations per pixel. (b) The centile t0.05 for two distinct samples (blue and red) with identical concentration of fluorophores. The solid lines represent analytical results and the error bars represent experimental results with bootstrap estimation of errors. See Dataset 1 (Files 4 and 5) for the underlying values [38] and Code 1 [39].

Fig. 9
Fig. 9

Simulation and real-time imaging methods, with F = 100 and an event probability per frame and per pixel p1 = 10−2. (a) Evolution of the mean value of (t) (solid blue curve) and of P((t) = F) (solid magenta curve), where error bars indicate the expected standard deviation to those curves when considering a single random realization – this shows that probability estimated over a single experiments closely follow the true probability of image completion P((t) = F). (b) Evolution in terms of the number of frames t of the estimator t 0.05 ^ ( 5 ) (solid error bars), which converges to the exact centile t0.95 = 760 (vertical red line).

Tables (1)

Tables Icon

Table 1 List of notations used in the paper.

Equations (21)

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T ~ 1 σ ρ { ln ( S σ θ ) + ( r + γ D ) ln ln ( S σ θ ) } ,
𝕇 [ T t 0 ] = 𝕇 [ min j ( M j ( t θ ) ) r ] = 1 θ ,
p 1 , i [ p ^ 1 , i ± 1.96 ( p ^ 1 , i ( 1 p ^ 1 , i ) / T t ) 1 / 2 ] .
t θ ~ 1 1 p 0 ln ( F θ ) ~ 1 ρ σ ln ( S θ σ ) .
t θ ~ 1 1 p 0 { ln ( F / θ ) + ( r 1 ) ln ln ( F / θ ) } ,
𝕇 ( M m ( t ) = j ) = t ! ( t j ) ! j ! p 1 j ( 1 p 1 ) t j , j r 1 ,
𝕇 ( M m ( t ) = r ) ~ 1 t r 1 p 0 t ( r 1 ) ! ( p 1 p 0 ) r 1 for 1 t .
1 ( 1 θ ) 1 / F = p 0 t θ ( r 1 ) ! ( t θ p 1 p 0 ) r 1 .
ln ( p 0 ) t θ + ( r 1 ) ln ( p 1 p 0 t θ ) = ln ( θ F ) + ln [ ( r 1 ) ! ] ,
t θ = { ln ( F θ ) + ( r 1 ) ln [ p 1 p 0 ( ln ( p 0 ) ) ln ( F θ ) ] } ln ( p 0 ) ,
𝕇 ( M i ( t ) r ) = 𝕇 ( M i r p 0 , i = q ) = 0 1 d q 𝕇 ( M i r p 0 , i = q ) ψ ( q ) ,
ψ ( q ) = 𝕇 ( p 0 , i = q ) = Θ ( p 0 , m q ) exp ( ( q p 0 , m ) / λ ) λ ( 1 exp ( p 0 , m λ ) ) ,
t θ ~ 1 1 p 0 , m { ln ( F / θ ) + ( r 1 ) ln ln ( F / θ ) C 1 } ,
𝕇 ( M i ( t ) r ) = 0 1 d q 𝕇 ( M i r p 0 , i = q ) 𝕇 ( p 0 , i = q ) .
( 0 1 d q 𝕇 ( M 1 ( t θ ) r p 0 , i = q ) 𝕇 ( p 0 , 1 = q ) ) F = 1 θ .
t θ ( F , r ) = 1 p 1 { C 3 ln ( F / θ ) + C 2 ( r 1 ) ln ( ln F ) C 1 } .
q j ( t ) = t ! ( t j u ) ! j r ! p 0 t ( p 1 p 0 ) j 1 ( p r p 0 ) j r ,
𝕇 ( M m ( t ) r ) ~ 1 t r 1 p 0 t ( r 1 ) ! ( p 1 p 0 ) r 1 ,
r ( ρ ROI + ρ BG ρ ROI ρ BG ) 2 when ρ ROI ~ ρ BG .
𝕇 [ T t ] = 2 1 / F { 1 erf ( r μ t / F 2 Σ 2 t / F ) } F ,
t θ ~ F r μ + [ 2 r Σ 2 μ log ( F 2 2 π θ ) ] 1 / 2 ,

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