Abstract

Fluorescence lifetime imaging microscopy (FLIM) is a powerful imaging tool used to study the molecular environment of flurophores. In time domain FLIM, extracting lifetime from fluorophores signals entails fitting data to a decaying exponential distribution function. However, most existing techniques for this purpose need large amounts of photons at each pixel and a long computation time, thus making it difficult to obtain reliable inference in applications requiring either short acquisition or minimal computation time. In this work, we introduce a new nonparametric empirical Bayesian framework for FLIM data analysis (NEB-FLIM), leading to both improved pixel-wise lifetime estimation and a more robust and computationally efficient integral property inference. This framework is developed based on a newly proposed hierarchical statistical model for FLIM data and adopts a novel nonparametric maximum likelihood estimator to estimate the prior distribution. To demonstrate the merit of the proposed framework, we applied it on both simulated and real biological datasets and compared it with previous classical methods on these datasets.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. W. Becker, Advanced Time-correlated Single Photon Counting Techniques, vol. 81 (Springer, 2005).
  2. W. Becker, Advanced Time-correlated Single Photon Counting Applications, vol. 111 (Springer, 2015).
  3. D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
    [Crossref]
  4. A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
    [Crossref]
  5. M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
    [Crossref]
  6. S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
    [Crossref]
  7. S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
    [Crossref]
  8. K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
    [Crossref]
  9. K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
    [Crossref]
  10. M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
    [Crossref]
  11. D. A. Turton, G. D. Reid, and G. S. Beddard, “Accurate analysis of fluorescence decays from single molecules in photon counting experiments,” Anal. Chem. 75(16), 4182–4187 (2003).
    [Crossref]
  12. P. J. Verveer and P. Bastiaens, “Evaluation of global analysis algorithms for single frequency fluorescence lifetime imaging microscopy data,” J. Microsc. 209(1), 1–7 (2003).
    [Crossref]
  13. P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
    [Crossref]
  14. P. Barber, S. Ameer-Beg, S. Pathmananthan, M. Rowley, and A. Coolen, “A bayesian method for single molecule, fluorescence burst analysis,” Biomed. Opt. Express 1(4), 1148–1158 (2010).
    [Crossref]
  15. M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
    [Crossref]
  16. J. Kim, J. Seok, H. Lee, and M. Lee, “Penalized maximum likelihood estimation of lifetime and amplitude images from multi-exponentially decaying fluorescence signals,” Opt. Express 21(17), 20240–20253 (2013).
    [Crossref]
  17. M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
    [Crossref]
  18. B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
    [Crossref]
  19. M. Köllner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett. 200(1-2), 199–204 (1992).
    [Crossref]
  20. M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
    [Crossref]
  21. N. Krstajić, S. Poland, J. Levitt, R. Walker, A. Erdogan, S. Ameer-Beg, and R. K. Henderson, “0.5 billion events per second time correlated single photon counting using cmos spad arrays,” Opt. Lett. 40(18), 4305–4308 (2015).
    [Crossref]
  22. C. Guzmán, C. Oetken-Lindholm, and D. Abankwa, “Automated high-throughput fluorescence lifetime imaging microscopy to detect protein–protein interactions,” J. Lab. Autom. 21(2), 238–245 (2016).
    [Crossref]
  23. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2006).
  24. J. Kiefer and J. Wolfowitz, “Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters,” Ann. Math. Stat. 27(4), 887–906 (1956).
    [Crossref]
  25. B. G. Lindsay, “The geometry of mixture likelihoods: a general theory,” Ann. Statist. 11(1), 86–94 (1983).
    [Crossref]
  26. W. Jiang and C. Zhang, “General maximum likelihood empirical bayes estimation of normal means,” Ann. Statist. 37(4), 1647–1684 (2009).
    [Crossref]
  27. R. Koenker and I. Mizera, “Convex optimization, shape constraints, compound decisions, and empirical bayes rules,” J. Am. Stat. Assoc. 109(506), 674–685 (2014).
    [Crossref]
  28. B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
    [Crossref]
  29. H. Robinns, “Asymptotically subminimax solutions of compound decision problems,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, vol. 1950, (1951), pp. 131–148.
  30. C. Zhang, “Compound decision theory and empirical bayes methods,” Ann. Statist. 31(2), 379–390 (2003).
    [Crossref]
  31. B. Efron, “Two modeling strategies for empirical bayes estimation,” Statist. Sci. 29(2), 285–301 (2014).
    [Crossref]
  32. B. Kleijn and A. Van der Vaart, “The bernstein-von-mises theorem under misspecification,” Electron. J. Stat. 6, 354–381 (2012).
    [Crossref]
  33. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. Ser. B (methodological) 39(1), 1–22 (1977).
    [Crossref]
  34. R. Varadhan and C. Roland, “Simple and globally convergent methods for accelerating the convergence of any EM algorithm,” Scand. J. Stat. 35(2), 335–353 (2008).
    [Crossref]
  35. R. Koenker and J. Gu, “Rebayes: an r package for empirical bayes mixture methods,” Tech. Rep., Journal of Statistical Software 82(8), 1–30 (2017).
    [Crossref]
  36. J. V. Chacko and K. W. Eliceiri, “Autofluorescence lifetime imaging of cellular metabolism: Sensitivity toward cell density, ph, intracellular, and intercellular heterogeneity,” Cytometry, Part A 95(1), 56–69 (2019).
    [Crossref]

2019 (1)

J. V. Chacko and K. W. Eliceiri, “Autofluorescence lifetime imaging of cellular metabolism: Sensitivity toward cell density, ph, intracellular, and intercellular heterogeneity,” Cytometry, Part A 95(1), 56–69 (2019).
[Crossref]

2017 (3)

R. Koenker and J. Gu, “Rebayes: an r package for empirical bayes mixture methods,” Tech. Rep., Journal of Statistical Software 82(8), 1–30 (2017).
[Crossref]

K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
[Crossref]

B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
[Crossref]

2016 (4)

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

C. Guzmán, C. Oetken-Lindholm, and D. Abankwa, “Automated high-throughput fluorescence lifetime imaging microscopy to detect protein–protein interactions,” J. Lab. Autom. 21(2), 238–245 (2016).
[Crossref]

M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
[Crossref]

2015 (2)

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

N. Krstajić, S. Poland, J. Levitt, R. Walker, A. Erdogan, S. Ameer-Beg, and R. K. Henderson, “0.5 billion events per second time correlated single photon counting using cmos spad arrays,” Opt. Lett. 40(18), 4305–4308 (2015).
[Crossref]

2014 (2)

R. Koenker and I. Mizera, “Convex optimization, shape constraints, compound decisions, and empirical bayes rules,” J. Am. Stat. Assoc. 109(506), 674–685 (2014).
[Crossref]

B. Efron, “Two modeling strategies for empirical bayes estimation,” Statist. Sci. 29(2), 285–301 (2014).
[Crossref]

2013 (3)

J. Kim, J. Seok, H. Lee, and M. Lee, “Penalized maximum likelihood estimation of lifetime and amplitude images from multi-exponentially decaying fluorescence signals,” Opt. Express 21(17), 20240–20253 (2013).
[Crossref]

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

2012 (1)

B. Kleijn and A. Van der Vaart, “The bernstein-von-mises theorem under misspecification,” Electron. J. Stat. 6, 354–381 (2012).
[Crossref]

2011 (1)

M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
[Crossref]

2010 (1)

2009 (1)

W. Jiang and C. Zhang, “General maximum likelihood empirical bayes estimation of normal means,” Ann. Statist. 37(4), 1647–1684 (2009).
[Crossref]

2008 (1)

R. Varadhan and C. Roland, “Simple and globally convergent methods for accelerating the convergence of any EM algorithm,” Scand. J. Stat. 35(2), 335–353 (2008).
[Crossref]

2007 (1)

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

2005 (2)

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

2004 (1)

S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
[Crossref]

2003 (3)

D. A. Turton, G. D. Reid, and G. S. Beddard, “Accurate analysis of fluorescence decays from single molecules in photon counting experiments,” Anal. Chem. 75(16), 4182–4187 (2003).
[Crossref]

P. J. Verveer and P. Bastiaens, “Evaluation of global analysis algorithms for single frequency fluorescence lifetime imaging microscopy data,” J. Microsc. 209(1), 1–7 (2003).
[Crossref]

C. Zhang, “Compound decision theory and empirical bayes methods,” Ann. Statist. 31(2), 379–390 (2003).
[Crossref]

2001 (1)

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

1992 (1)

M. Köllner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett. 200(1-2), 199–204 (1992).
[Crossref]

1983 (1)

B. G. Lindsay, “The geometry of mixture likelihoods: a general theory,” Ann. Statist. 11(1), 86–94 (1983).
[Crossref]

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. Ser. B (methodological) 39(1), 1–22 (1977).
[Crossref]

1956 (1)

J. Kiefer and J. Wolfowitz, “Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters,” Ann. Math. Stat. 27(4), 887–906 (1956).
[Crossref]

Abankwa, D.

C. Guzmán, C. Oetken-Lindholm, and D. Abankwa, “Automated high-throughput fluorescence lifetime imaging microscopy to detect protein–protein interactions,” J. Lab. Autom. 21(2), 238–245 (2016).
[Crossref]

Abraham, B. G.

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

Alexandrov, Y.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Alibhai, D.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Ameer-Beg, S.

Ameer-Beg, S. M.

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Arteaga, C. L.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Barber, P.

Barber, P. R.

M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
[Crossref]

M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
[Crossref]

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Bastiaens, P.

P. J. Verveer and P. Bastiaens, “Evaluation of global analysis algorithms for single frequency fluorescence lifetime imaging microscopy data,” J. Microsc. 209(1), 1–7 (2003).
[Crossref]

Becker, W.

W. Becker, Advanced Time-correlated Single Photon Counting Techniques, vol. 81 (Springer, 2005).

W. Becker, Advanced Time-correlated Single Photon Counting Applications, vol. 111 (Springer, 2015).

Beddard, G. S.

D. A. Turton, G. D. Reid, and G. S. Beddard, “Accurate analysis of fluorescence decays from single molecules in photon counting experiments,” Anal. Chem. 75(16), 4182–4187 (2003).
[Crossref]

Bird, D. K.

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Chacko, J. V.

J. V. Chacko and K. W. Eliceiri, “Autofluorescence lifetime imaging of cellular metabolism: Sensitivity toward cell density, ph, intracellular, and intercellular heterogeneity,” Cytometry, Part A 95(1), 56–69 (2019).
[Crossref]

Cook, R. S.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Coolen, A.

M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
[Crossref]

P. Barber, S. Ameer-Beg, S. Pathmananthan, M. Rowley, and A. Coolen, “A bayesian method for single molecule, fluorescence burst analysis,” Biomed. Opt. Express 1(4), 1148–1158 (2010).
[Crossref]

Coolen, A. C.

M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
[Crossref]

Cotlet, M.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

de Jong, S.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

De Schryver, F. C.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. Ser. B (methodological) 39(1), 1–22 (1977).
[Crossref]

Dunsby, C.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Edens, R. J.

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Efron, B.

B. Efron, “Two modeling strategies for empirical bayes estimation,” Statist. Sci. 29(2), 285–301 (2014).
[Crossref]

Eickhoff, J.

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

Eliceiri, K. W.

J. V. Chacko and K. W. Eliceiri, “Autofluorescence lifetime imaging of cellular metabolism: Sensitivity toward cell density, ph, intracellular, and intercellular heterogeneity,” Cytometry, Part A 95(1), 56–69 (2019).
[Crossref]

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Erdogan, A.

Ezike, I.

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Foster, P. J.

B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
[Crossref]

French, P.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Gadella, T. W.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Gendron-Fitzpatrick, A.

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

Gensch, T.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

Gilbey, J. D.

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Goedhart, J.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Gu, J.

R. Koenker and J. Gu, “Rebayes: an r package for empirical bayes mixture methods,” Tech. Rep., Journal of Statistical Software 82(8), 1–30 (2017).
[Crossref]

Guzmán, C.

C. Guzmán, C. Oetken-Lindholm, and D. Abankwa, “Automated high-throughput fluorescence lifetime imaging microscopy to detect protein–protein interactions,” J. Lab. Autom. 21(2), 238–245 (2016).
[Crossref]

Henderson, R. K.

Herz, J.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Hicks, D. J.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Hofkens, J.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

Jalink, K.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Jiang, W.

W. Jiang and C. Zhang, “General maximum likelihood empirical bayes estimation of normal means,” Ann. Statist. 37(4), 1647–1684 (2009).
[Crossref]

Karp, M.

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

Katan, M.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Kaye, B.

B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
[Crossref]

Kedziora, K. M.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Keely, P. J.

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Kelly, D. J.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Kiefer, J.

J. Kiefer and J. Wolfowitz, “Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters,” Ann. Math. Stat. 27(4), 887–906 (1956).
[Crossref]

Kim, J.

Kleijn, B.

B. Kleijn and A. Van der Vaart, “The bernstein-von-mises theorem under misspecification,” Electron. J. Stat. 6, 354–381 (2012).
[Crossref]

Koenker, R.

R. Koenker and J. Gu, “Rebayes: an r package for empirical bayes mixture methods,” Tech. Rep., Journal of Statistical Software 82(8), 1–30 (2017).
[Crossref]

R. Koenker and I. Mizera, “Convex optimization, shape constraints, compound decisions, and empirical bayes rules,” J. Am. Stat. Assoc. 109(506), 674–685 (2014).
[Crossref]

Köllner, M.

M. Köllner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett. 200(1-2), 199–204 (1992).
[Crossref]

Krstajic, N.

Lafontant, A.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Laiho, L.

S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
[Crossref]

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. Ser. B (methodological) 39(1), 1–22 (1977).
[Crossref]

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2006).

Lee, H.

Lee, M.

Levitt, J.

Lindsay, B. G.

B. G. Lindsay, “The geometry of mixture likelihoods: a general theory,” Ann. Statist. 11(1), 86–94 (1983).
[Crossref]

Manning, H. C.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Margineanu, A.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Mastop, M.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Maus, M.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

Mishin, A. S.

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

Mizera, I.

R. Koenker and I. Mizera, “Convex optimization, shape constraints, compound decisions, and empirical bayes rules,” J. Am. Stat. Assoc. 109(506), 674–685 (2014).
[Crossref]

Munro, I.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Needleman, D. J.

B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
[Crossref]

Oetken-Lindholm, C.

C. Guzmán, C. Oetken-Lindholm, and D. Abankwa, “Automated high-throughput fluorescence lifetime imaging microscopy to detect protein–protein interactions,” J. Lab. Autom. 21(2), 238–245 (2016).
[Crossref]

Pathmananthan, S.

Pelet, S.

S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
[Crossref]

Petrich, J. W.

K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
[Crossref]

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

Poland, S.

Previte, M.

S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
[Crossref]

Ramanujam, N.

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Raspe, M.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Reid, G. D.

D. A. Turton, G. D. Reid, and G. S. Beddard, “Accurate analysis of fluorescence decays from single molecules in photon counting experiments,” Anal. Chem. 75(16), 4182–4187 (2003).
[Crossref]

Riching, K. M.

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

Robinns, H.

H. Robinns, “Asymptotically subminimax solutions of compound decision problems,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, vol. 1950, (1951), pp. 131–148.

Roland, C.

R. Varadhan and C. Roland, “Simple and globally convergent methods for accelerating the convergence of any EM algorithm,” Scand. J. Stat. 35(2), 335–353 (2008).
[Crossref]

Rowley, M.

Rowley, M. I.

M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
[Crossref]

M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
[Crossref]

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. Ser. B (methodological) 39(1), 1–22 (1977).
[Crossref]

Santala, V.

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

Santra, K.

K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
[Crossref]

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

Sarkisyan, K. S.

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

Schaffer, J.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

Seidel, C.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

Seok, J.

Skala, M. C.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

Smith, E. A.

K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
[Crossref]

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

So, P.

S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
[Crossref]

Song, X.

K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
[Crossref]

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

Talbot, C.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Tkachenko, N. V.

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

Turton, D. A.

D. A. Turton, G. D. Reid, and G. S. Beddard, “Accurate analysis of fluorescence decays from single molecules in photon counting experiments,” Anal. Chem. 75(16), 4182–4187 (2003).
[Crossref]

van den Broek, B.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Van der Vaart, A.

B. Kleijn and A. Van der Vaart, “The bernstein-von-mises theorem under misspecification,” Electron. J. Stat. 6, 354–381 (2012).
[Crossref]

Varadhan, R.

R. Varadhan and C. Roland, “Simple and globally convergent methods for accelerating the convergence of any EM algorithm,” Scand. J. Stat. 35(2), 335–353 (2008).
[Crossref]

Vaswani, N.

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

Vaughan, E. M.

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Verveer, P. J.

P. J. Verveer and P. Bastiaens, “Evaluation of global analysis algorithms for single frequency fluorescence lifetime imaging microscopy data,” J. Microsc. 209(1), 1–7 (2003).
[Crossref]

Vojnovic, B.

M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
[Crossref]

M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
[Crossref]

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Vrotsos, K. M.

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Walker, R.

Walsh, A. J.

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Warren, S. C.

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

White, J. G.

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Wolfowitz, J.

J. Kiefer and J. Wolfowitz, “Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters,” Ann. Math. Stat. 27(4), 887–906 (1956).
[Crossref]

Wolfrum, J.

M. Köllner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett. 200(1-2), 199–204 (1992).
[Crossref]

Yan, L.

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

Yoo, T.

B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
[Crossref]

Young, I. T.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Zhan, J.

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

Zhang, C.

W. Jiang and C. Zhang, “General maximum likelihood empirical bayes estimation of normal means,” Ann. Statist. 37(4), 1647–1684 (2009).
[Crossref]

C. Zhang, “Compound decision theory and empirical bayes methods,” Ann. Statist. 31(2), 379–390 (2003).
[Crossref]

Zhao, Q.

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Anal. Chem. (2)

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. 73(9), 2078–2086 (2001).
[Crossref]

D. A. Turton, G. D. Reid, and G. S. Beddard, “Accurate analysis of fluorescence decays from single molecules in photon counting experiments,” Anal. Chem. 75(16), 4182–4187 (2003).
[Crossref]

Ann. Math. Stat. (1)

J. Kiefer and J. Wolfowitz, “Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters,” Ann. Math. Stat. 27(4), 887–906 (1956).
[Crossref]

Ann. Statist. (3)

B. G. Lindsay, “The geometry of mixture likelihoods: a general theory,” Ann. Statist. 11(1), 86–94 (1983).
[Crossref]

W. Jiang and C. Zhang, “General maximum likelihood empirical bayes estimation of normal means,” Ann. Statist. 37(4), 1647–1684 (2009).
[Crossref]

C. Zhang, “Compound decision theory and empirical bayes methods,” Ann. Statist. 31(2), 379–390 (2003).
[Crossref]

Biomed. Opt. Express (1)

Biophys. J. (1)

S. Pelet, M. Previte, L. Laiho, and P. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation,” Biophys. J. 87(4), 2807–2817 (2004).
[Crossref]

Cancer Res. (2)

D. K. Bird, L. Yan, K. M. Vrotsos, K. W. Eliceiri, E. M. Vaughan, P. J. Keely, J. G. White, and N. Ramanujam, “Metabolic mapping of MCF10A human breast cells via multiphoton fluorescence lifetime imaging of the coenzyme NADH,” Cancer Res. 65(19), 8766–8773 (2005).
[Crossref]

A. J. Walsh, R. S. Cook, H. C. Manning, D. J. Hicks, A. Lafontant, C. L. Arteaga, and M. C. Skala, “Optical metabolic imaging identifies glycolytic levels, subtypes, and early-treatment response in breast cancer,” Cancer Res. 73(20), 6164–6174 (2013).
[Crossref]

Chem. Phys. Lett. (1)

M. Köllner and J. Wolfrum, “How many photons are necessary for fluorescence-lifetime measurements?” Chem. Phys. Lett. 200(1-2), 199–204 (1992).
[Crossref]

Cytometry, Part A (1)

J. V. Chacko and K. W. Eliceiri, “Autofluorescence lifetime imaging of cellular metabolism: Sensitivity toward cell density, ph, intracellular, and intercellular heterogeneity,” Cytometry, Part A 95(1), 56–69 (2019).
[Crossref]

Electron. J. Stat. (1)

B. Kleijn and A. Van der Vaart, “The bernstein-von-mises theorem under misspecification,” Electron. J. Stat. 6, 354–381 (2012).
[Crossref]

J. Am. Stat. Assoc. (1)

R. Koenker and I. Mizera, “Convex optimization, shape constraints, compound decisions, and empirical bayes rules,” J. Am. Stat. Assoc. 109(506), 674–685 (2014).
[Crossref]

J. Biomed. Opt. (1)

M. C. Skala, K. M. Riching, D. K. Bird, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, P. J. Keely, and N. Ramanujam, “In vivo multiphoton fluorescence lifetime imaging of protein-bound and free nicotinamide adenine dinucleotide in normal and precancerous epithelia,” J. Biomed. Opt. 12(2), 024014 (2007).
[Crossref]

J. Lab. Autom. (1)

C. Guzmán, C. Oetken-Lindholm, and D. Abankwa, “Automated high-throughput fluorescence lifetime imaging microscopy to detect protein–protein interactions,” J. Lab. Autom. 21(2), 238–245 (2016).
[Crossref]

J. Microsc. (1)

P. J. Verveer and P. Bastiaens, “Evaluation of global analysis algorithms for single frequency fluorescence lifetime imaging microscopy data,” J. Microsc. 209(1), 1–7 (2003).
[Crossref]

J. Phys. Chem. A (1)

K. Santra, E. A. Smith, J. W. Petrich, and X. Song, “Photon counting data analysis: Application of the maximum likelihood and related methods for the determination of lifetimes in mixtures of rose bengal and rhodamine b,” J. Phys. Chem. A 121(1), 122–132 (2017).
[Crossref]

J. Phys. Chem. B (1)

K. Santra, J. Zhan, X. Song, E. A. Smith, N. Vaswani, and J. W. Petrich, “What is the best method to fit time-resolved data? a comparison of the residual minimization and the maximum likelihood techniques as applied to experimental time-correlated, single-photon counting data,” J. Phys. Chem. B 120(9), 2484–2490 (2016).
[Crossref]

J. Royal Stat. Soc. Ser. B (methodological) (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc. Ser. B (methodological) 39(1), 1–22 (1977).
[Crossref]

Nat. Methods (1)

M. Raspe, K. M. Kedziora, B. van den Broek, Q. Zhao, S. de Jong, J. Herz, M. Mastop, J. Goedhart, T. W. Gadella, I. T. Young, and K. Jalink, “siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data,” Nat. Methods 13(6), 501–504 (2016).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

PLoS One (4)

B. G. Abraham, K. S. Sarkisyan, A. S. Mishin, V. Santala, N. V. Tkachenko, and M. Karp, “Fluorescent protein based fret pairs with improved dynamic range for fluorescence lifetime measurements,” PLoS One 10(8), e0134436 (2015).
[Crossref]

M. I. Rowley, A. Coolen, B. Vojnovic, and P. R. Barber, “Robust bayesian fluorescence lifetime estimation, decay model selection and instrument response determination for low-intensity FLIM imaging,” PLoS One 11(6), e0158404 (2016).
[Crossref]

B. Kaye, P. J. Foster, T. Yoo, and D. J. Needleman, “Developing and testing a bayesian analysis of fluorescence lifetime measurements,” PLoS One 12(1), e0169337 (2017).
[Crossref]

S. C. Warren, A. Margineanu, D. Alibhai, D. J. Kelly, C. Talbot, Y. Alexandrov, I. Munro, M. Katan, C. Dunsby, and P. French, “Rapid global fitting of large fluorescence lifetime imaging microscopy datasets,” PLoS One 8(8), e70687 (2013).
[Crossref]

Proc. SPIE (2)

M. I. Rowley, P. R. Barber, A. C. Coolen, and B. Vojnovic, “Bayesian analysis of fluorescence lifetime imaging data,” Proc. SPIE 7903, 790325 (2011).
[Crossref]

P. R. Barber, S. M. Ameer-Beg, J. D. Gilbey, R. J. Edens, I. Ezike, and B. Vojnovic, “Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM,” Proc. SPIE 5700, 171–181 (2005).
[Crossref]

Scand. J. Stat. (1)

R. Varadhan and C. Roland, “Simple and globally convergent methods for accelerating the convergence of any EM algorithm,” Scand. J. Stat. 35(2), 335–353 (2008).
[Crossref]

Statist. Sci. (1)

B. Efron, “Two modeling strategies for empirical bayes estimation,” Statist. Sci. 29(2), 285–301 (2014).
[Crossref]

Tech. Rep., Journal of Statistical Software (1)

R. Koenker and J. Gu, “Rebayes: an r package for empirical bayes mixture methods,” Tech. Rep., Journal of Statistical Software 82(8), 1–30 (2017).
[Crossref]

Other (4)

H. Robinns, “Asymptotically subminimax solutions of compound decision problems,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, vol. 1950, (1951), pp. 131–148.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2006).

W. Becker, Advanced Time-correlated Single Photon Counting Techniques, vol. 81 (Springer, 2005).

W. Becker, Advanced Time-correlated Single Photon Counting Applications, vol. 111 (Springer, 2015).

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Figures (11)

Fig. 1.
Fig. 1. The data structure of fluorescence-lifetime imaging microscopy photon counting data.
Fig. 2.
Fig. 2. A typical example of empirical prior distribution estimated from data.
Fig. 3.
Fig. 3. Flow chart of the non-parametric empirical bayesian framework for FLIM data (NEB-FLIM).
Fig. 4.
Fig. 4. Ground truth of $\tau _1$, $\tau _2$, and $a$ in simulation. All lifetimes in the figures are shown in picosecond(ps).
Fig. 5.
Fig. 5. Performance of lifetime prior distribution estimation at different $n$ and $L$: the average error $D(\pi ^{\ast }(t),\hat {\pi }^{\ast }(t))$ against logarithm of number of photons per pixel $\log n$. Different colors represent different numbers of intervals $L$.
Fig. 6.
Fig. 6. Pixel-wise recovery performance comparisons between pixel-wise analysis, global analysis, and NEB-FLIM: the plots are of mean square error across the image against the number of photons per pixel. All results of lifetimes in the figures are shown in ps$^2$. Left is plot of $\tau _1$; middle is plot of $\tau _2$ and right is plot of $a$.
Fig. 7.
Fig. 7. Average estimated mean of weighted averaged lifetime $\tau _m^\ast$ with error bar of double standard deviation for each imaging duration and cell type. The plot is summarized from results of intergal property inference of NEB-FLIM on 100 randomly chosen regions.
Fig. 8.
Fig. 8. Comparisons of pixel-wise recovery result by pixel analysis, global analysis, and empirical bayesian analysis on real datasets: pixel-wise physical component contribution $A_{2}$. Top right is MCF10A cells and bottom left is MDA-MB-231 cells. The imaging time from top to bottom is 20s, 60s, 120s and 240s.
Fig. 9.
Fig. 9. Comparisons of pixel-wise recovery result by pixel analysis, global analysis, and empirical bayesian analysis on real dataset: pixel-wise weighted average lifetime $\tau _m$ in ps. Top right is MCF10A cells and bottom left is MDA-MB-231 cells. The imaging time from top to bottom is 20s, 60s, 120s and 240s.
Fig. 10.
Fig. 10. Density plot of estimated weighted averaged lifetime $\tau _m^\ast$ of MCF10A cell for different pixel-wise lifetime recovery methods and imaging time: NEB=pixel-wise lifetime estimated by NEB-FLIM, PA=pixel-wise lifetime estimated by pixel-wise analysis, and GA=pixel-wise lifetime estimated by global analysis.
Fig. 11.
Fig. 11. Average estimated mean of weighted averaged lifetime $\tau _m^\ast$ with error bar of standard deviation for different methods, imaging time and cell types. The results are summarized from estimation results of NEB-FLIM on 100 randomly chosen regions for each combination of method, imaging time and cell type.

Tables (4)

Tables Icon

Table 1. Accuracy comparisons between different integral property inference methods: PI-NEB=direct integral property inference in NEB-FLIM, PBA-NEB=mean of pixel-wise lifetime estimated by NEB-FLIM, PA=mean of pixel-wise lifetime estimated by pixel-wise analysis, and GA=mean of pixel-wise lifetime estimated by global analysis. The error criteria is square root of mean square error $e(\tau _k)$ for $k=1,2$. All results in the table are shown in ps.

Tables Icon

Table 2. Computation speed comparisons between different integral property inference methods: NEB-400, NEB-800=direct integral property inference in NEB-FLIM with $L=400$ and 800, PA=plugin estimator of pixel-wise lifetime estimated by pixel-wise analysis, and GA=plugin estimator of pixel-wise lifetime estimated by global analysis. The computation time in the table is shown in seconds.

Tables Icon

Table 3. Summarized information of the biological data set estimated by direct integral property inference of NEB-FLIM: the average number of photons per pixel $\bar {n}$, the mean of statistical component contribution $a^\ast$, mean of lifetime of the first component $\tau _1^\ast$, mean of lifetime of the second component $\tau _2^\ast$, mean of physical component contribution (after normalization) $A_1^\ast$, $A_2^\ast$ and mean of weighted averaged lifetime $\tau _m^\ast$ All results of lifetime in the table are shown in picosecond.

Tables Icon

Table 4. Comparisons between different property inference methods on real data: PI-NEB=direct integral property inference in NEB-FLIM, PBA-NEB=mean of pixel-wise lifetime estimated by NEB-FLIM, PA=mean of pixel-wise lifetime estimated by pixel-wise analysis, and GA=mean of pixel-wise lifetime estimated by global analysis. All results of lifetime in the table are shown in picosecond.

Equations (41)

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g ( t ) := n = 0 1 τ e t + n T p τ = 1 τ ( 1 e T p / τ ) e t τ , when   0 < t < T p .
f ( t ) := α T p I ( 0 < t < T p ) + ( 1 α ) ( g h ) ( t ) , when   0 < t < T p .
P j ( τ ) = B j f ( t ) d t .
a 1 τ 1 e t τ 1 + ( 1 a ) 1 τ 2 e t τ 2 , when   t > 0.
π 1 ( t ) = 1 | I | i I δ τ i 1 and π 2 ( t ) = 1 | I | i I δ τ i 2 ,
τ i 1 π 1 ( t ) and τ i 2 π 2 ( t ) N i Multi ( n i , a i P ( τ i 1 ) + ( 1 a i ) P ( τ i 2 ) )
τ l π ( t ) j l Multi ( 1 , P ( τ l ) ) , l = 1 , , n := i I n i .
π ( t ) = 1 n i I [ n i a i δ τ i + n i ( 1 a i ) δ τ i 2 ] .
1 | I | i I f 1 ( n i ) f 2 ( a i ) f 3 ( τ i k ) = ( 1 | I | i I f 1 ( n i ) ) ( 1 | I | i I f 2 ( a i ) ) ( 1 | I | i I f 3 ( τ i k ) )
π ( t ) = a π 1 ( t ) + ( 1 a ) π 2 ( t ) ,
M := ( M 1 , , M m ) Multi ( n , P ( t ) d π ( t ) ) ,
π Δ ( t ) = l = 1 L p l δ h l
f ( p 1 , , p L ) := n ! M 1 ! M m ! j = 1 m ( l = 1 L p l P j ( h l ) ) M j ,
min ( p 1 , , p L ) j = 1 m M j log ( l = 1 L p l P j ( h l ) ) s.t.   l = 1 L p l = 1 and p l 0 ,   l = 1 , , L .
π ^ ( t ) = l = 1 L p ^ l δ h l .
sup { t : π 1 ( t ) > 0 } < inf { t : π 2 ( t ) > 0 } .
t T = argmax r { h 1 , , h L 1 } a ^ ( r ) ( 1 a ^ ( r ) ) [ τ ^ 1 ( r ) τ ^ 2 ( r ) ] 2
a ^ ( r ) = l = 1 L p ^ l I ( h l r ) , τ ^ 1 ( r ) = l = 1 L h l p ^ l I ( h l r ) a ^ ( r ) and τ ^ 2 ( r ) = l = 1 L h l p ^ l I ( h l > r ) 1 a ^ ( r ) .
π ^ 1 ( t ) = l = 1 L p ^ l δ h l I ( h l t T ) a ^ , π ^ 2 ( t ) = l = 1 L p ^ l δ h l I ( h l > t T ) 1 a ^ ,
a ^ = l = 1 L p ^ l I ( h l t T ) .
p ( τ i , a i | N i ) ( j = 1 m N i j ) ! N i 1 ! N i m ! j = 1 m ( a i P j ( τ i 1 ) + ( 1 a i ) P j ( τ i 2 ) ) N i j k = 1 2 π k ( τ i k ) .
min τ i 1 , τ i 2 , a i j = 1 m N i j log ( a i P j ( τ i 1 ) + ( 1 a i ) P j ( τ i 2 ) ) k = 1 2 log ( π ^ k ( τ i k ) ) .
j i s | z i s Multi ( 1 , P ( τ i z i s ) ) and P ( z i s = 1 ) = 1 P ( z i s = 2 ) = a i , l = 1 , , n i ,
γ i j ( t ) = P ( z i s = 1 | j i s = j ) = a i ( t ) P j ( τ i 1 ( t ) ) a i ( t ) P j ( τ i 1 ( t ) ) + ( 1 a i ( t ) ) P j ( τ i 2 ( t ) ) .
Q ( τ i , a i | τ i ( t ) , a i ( t ) ) = j = 1 m N i j [ γ i j ( t ) log ( a i P j ( τ i 1 ) ) + ( 1 γ i j ( t ) ) log ( ( 1 a i ) P j ( τ i 2 ) ) ] + k = 1 2 log ( π ^ k ( τ i k ) ) .
a i ( t + 1 ) = j = 1 m N i j γ i j ( t ) j = 1 m N i j , τ i 1 ( t + 1 ) = argmax τ i 1 j = 1 m N i j γ i j ( t ) log P j ( τ i 1 ) + log ( π ^ 1 ( τ i 1 ) ) ,
τ i 2 ( t + 1 ) = argmax τ i 2 j = 1 m N i j ( 1 γ i j ( t ) ) log P j ( τ i 2 ) + log ( π ^ 2 ( τ i 2 ) ) .
F k ( g ) = T L T U g ( t ) d π k ( t ) , k = 1 , 2.
τ k := t d π k ( t ) and v ( τ k ) := t 2 d π k ( t ) ( t d π k ( t ) ) 2 .
A 1 := 1 | I | i I A i 1 = a 1 t d π 1 ( t ) a 1 t d π 1 ( t ) + ( 1 a ) 1 t d π 2 ( t ) and A 2 = 1 A 1 .
F ^ k naive ( g ) = 1 | I | i I g ( τ ^ i k ) , k = 1 , 2.
F ^ k NEB ( g ) = g ( t ) d π ^ k ( t ) , k = 1 , 2.
τ m := 1 | I | i I ( A i 1 τ i 1 + A i 2 τ i 2 ) .
τ ^ 1 = t d π ^ 1 ( t ) and τ ^ 2 = t d π ^ 2 ( t ) ,
A ^ 1 = a ^ 1 t d π ^ 1 ( t ) a ^ 1 t d π ^ 1 ( t ) + ( 1 a ^ ) 1 t d π ^ 2 ( t ) and A ^ 2 = 1 A ^ 1 ,
τ ^ m = A ^ 1 τ ^ 1 + A ^ 2 τ ^ 2 .
a 1 τ 1 e t τ 1 + ( 1 a ) 1 τ 2 e t τ 2 .
D ( π ( t ) , π ^ ( t ) ) = T L T U ( F π ( t ) F ^ π ( t ) ) 2 d t ,
F π ( t ) = 1 | I | i I [ a i I ( τ i 1 t ) + ( 1 a i ) I ( τ i 2 t ) ] and F ^ π ( t ) = l = 1 L p l I ( h l t ) .
1 | I | i I ( r ( τ ^ i , a ^ i ) r ( τ i , a i ) ) 2 ,
e ( τ k ) := 1 H h = 1 H ( τ ^ k h τ k ) 2 ,