Abstract

We demonstrate a Bayesian approach to tracking and characterizing colloidal particles from in-line digital holograms. We model the formation of the hologram using Lorenz-Mie theory. We then use a tempered Markov-chain Monte Carlo method to sample the posterior probability distributions of the model parameters: particle position, size, and refractive index. Compared to least-squares fitting, our approach allows us to more easily incorporate prior information about the parameters and to obtain more accurate uncertainties, which are critical for both particle tracking and characterization experiments. Our approach also eliminates the need to supply accurate initial guesses for the parameters, so it requires little tuning.

© 2016 Optical Society of America

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  1. F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta. 48, 109–115 (2009).
    [Crossref]
  2. R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
    [Crossref]
  3. J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
    [Crossref] [PubMed]
  4. D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
    [Crossref]
  5. B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29, S175–S184 (2000).
    [Crossref]
  6. F. C. Cheong, B. S. Rémi Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
    [Crossref] [PubMed]
  7. S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
    [Crossref] [PubMed]
  8. C. Wang, H. Shpaisman, A. D. Hollingsworth, and D. G. Grier, “Celebrating Soft Matter’s 10th Anniversary: Monitoring colloidal growth with holographic microscopy,” Soft Matter 11, 1062–1066 (2015).
    [Crossref]
  9. C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  12. A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
    [Crossref]
  13. J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
    [Crossref]
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  16. A. Yevick, M. Hannel, and D. G. Grier, “Machine-learning approach to holographic particle characterization,” Opt. Express 22, 26884–26890 (2014).
    [Crossref] [PubMed]
  17. T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express 18, 22527–22544 (2010).
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    [Crossref]
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    [Crossref] [PubMed]
  20. D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
    [Crossref]
  21. P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica® Support (Cambridge University, 2005).
    [Crossref]
  22. J. Goodman and J. Weare, “Ensemble samplers with affine invariance,” Comm. Appl. Math. Comp. Sci. 5, 65–80 (2010).
    [Crossref]
  23. D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013).
    [Crossref]
  24. D. J. Earl and M. W. Deem, “Parallel tempering: Theory, applications, and new perspectives,” PCCP 7, 3910–3916 (2005).
    [Crossref] [PubMed]
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2016 (1)

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

2015 (1)

C. Wang, H. Shpaisman, A. D. Hollingsworth, and D. G. Grier, “Celebrating Soft Matter’s 10th Anniversary: Monitoring colloidal growth with holographic microscopy,” Soft Matter 11, 1062–1066 (2015).
[Crossref]

2014 (3)

A. Yevick, M. Hannel, and D. G. Grier, “Machine-learning approach to holographic particle characterization,” Opt. Express 22, 26884–26890 (2014).
[Crossref] [PubMed]

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

T. G. Dimiduk, R. W. Perry, J. Fung, and V. N. Manoharan, “Random-subset fitting of digital holograms for fast three-dimensional particle tracking,” Appl. Opt. 53, G177–G183 (2014).
[Crossref] [PubMed]

2013 (1)

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013).
[Crossref]

2012 (4)

M. Seifi, C. Fournier, L. Denis, D. Chareyron, and J. L. Marié, “Three-dimensional reconstruction of particle holograms: a fast and accurate multiscale approach,” J. Opt. Soc. Am. A 29, 1808–1817 (2012).
[Crossref]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
[Crossref]

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (3)

2007 (2)

2005 (1)

D. J. Earl and M. W. Deem, “Parallel tempering: Theory, applications, and new perspectives,” PCCP 7, 3910–3916 (2005).
[Crossref] [PubMed]

2003 (1)

2000 (2)

B. Ovryn and S. H. Izen, “Imaging of transparent spheres through a planar interface using a high-numerical-aperture optical microscope,” J. Opt. Soc. Am. A 17, 1202–1213 (2000).
[Crossref]

B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29, S175–S184 (2000).
[Crossref]

1963 (1)

D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Amato-Grill, J.

Brenner, M. P.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[Crossref]

Chareyron, D.

Chaudhary, K.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Cheong, F. C.

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta. 48, 109–115 (2009).
[Crossref]

F. C. Cheong, B. S. Rémi Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[Crossref] [PubMed]

Deem, M. W.

D. J. Earl and M. W. Deem, “Parallel tempering: Theory, applications, and new perspectives,” PCCP 7, 3910–3916 (2005).
[Crossref] [PubMed]

Denis, L.

Dimiduk, T. G.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

T. G. Dimiduk, R. W. Perry, J. Fung, and V. N. Manoharan, “Random-subset fitting of digital holograms for fast three-dimensional particle tracking,” Appl. Opt. 53, G177–G183 (2014).
[Crossref] [PubMed]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
[Crossref]

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

Dixon, L.

Duarte, S.

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta. 48, 109–115 (2009).
[Crossref]

Earl, D. J.

D. J. Earl and M. W. Deem, “Parallel tempering: Theory, applications, and new perspectives,” PCCP 7, 3910–3916 (2005).
[Crossref] [PubMed]

Fink, H.-W.

Foreman-Mackey, D.

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013).
[Crossref]

Fournier, C.

Fung, J.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

T. G. Dimiduk, R. W. Perry, J. Fung, and V. N. Manoharan, “Random-subset fitting of digital holograms for fast three-dimensional particle tracking,” Appl. Opt. 53, G177–G183 (2014).
[Crossref] [PubMed]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
[Crossref]

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Gehri, F.

Goepfert, C.

Goodman, J.

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013).
[Crossref]

J. Goodman and J. Weare, “Ensemble samplers with affine invariance,” Comm. Appl. Math. Comp. Sci. 5, 65–80 (2010).
[Crossref]

Gregory, P.

P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica® Support (Cambridge University, 2005).
[Crossref]

Grier, D. G.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

C. Wang, H. Shpaisman, A. D. Hollingsworth, and D. G. Grier, “Celebrating Soft Matter’s 10th Anniversary: Monitoring colloidal growth with holographic microscopy,” Soft Matter 11, 1062–1066 (2015).
[Crossref]

A. Yevick, M. Hannel, and D. G. Grier, “Machine-learning approach to holographic particle characterization,” Opt. Express 22, 26884–26890 (2014).
[Crossref] [PubMed]

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta. 48, 109–115 (2009).
[Crossref]

F. C. Cheong, B. S. Rémi Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[Crossref] [PubMed]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
[Crossref] [PubMed]

Hannel, M.

Hogg, D. W.

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013).
[Crossref]

Hollingsworth, A. D.

C. Wang, H. Shpaisman, A. D. Hollingsworth, and D. G. Grier, “Celebrating Soft Matter’s 10th Anniversary: Monitoring colloidal growth with holographic microscopy,” Soft Matter 11, 1062–1066 (2015).
[Crossref]

Izen, S. H.

Kaz, D. M.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Kim, S.-H.

Kretzschmar, I.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Lang, D.

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013).
[Crossref]

Latychevskaia, T.

Lee, S.-H.

Lorenz, D.

Mani, M.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[Crossref]

Manoharan, V. N.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

T. G. Dimiduk, R. W. Perry, J. Fung, and V. N. Manoharan, “Random-subset fitting of digital holograms for fast three-dimensional particle tracking,” Appl. Opt. 53, G177–G183 (2014).
[Crossref] [PubMed]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
[Crossref]

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Marié, J. L.

Marquardt, D. W.

D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Martin, K. E.

McGorty, R.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Meng, G.

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

Meng, H.

Ovryn, B.

B. Ovryn and S. H. Izen, “Imaging of transparent spheres through a planar interface using a high-numerical-aperture optical microscope,” J. Opt. Soc. Am. A 17, 1202–1213 (2000).
[Crossref]

B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29, S175–S184 (2000).
[Crossref]

Perry, R. W.

T. G. Dimiduk, R. W. Perry, J. Fung, and V. N. Manoharan, “Random-subset fitting of digital holograms for fast three-dimensional particle tracking,” Appl. Opt. 53, G177–G183 (2014).
[Crossref] [PubMed]

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Philips, L. A.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

Pu, Y.

Razavi, S.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Rémi Dreyfus, B. S.

Roichman, Y.

Ruffner, D. B.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

Seifi, M.

Shpaisman, H.

C. Wang, H. Shpaisman, A. D. Hollingsworth, and D. G. Grier, “Celebrating Soft Matter’s 10th Anniversary: Monitoring colloidal growth with holographic microscopy,” Soft Matter 11, 1062–1066 (2015).
[Crossref]

Soulez, F.

Stutt, A.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

Thiébaut, E.

Thiébaut, É.

Trede, D.

van Blaaderen, A.

van Oostrum, P.

Wang, A.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Wang, C.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

C. Wang, H. Shpaisman, A. D. Hollingsworth, and D. G. Grier, “Celebrating Soft Matter’s 10th Anniversary: Monitoring colloidal growth with holographic microscopy,” Soft Matter 11, 1062–1066 (2015).
[Crossref]

Ward, M. D.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

Weare, J.

J. Goodman and J. Weare, “Ensemble samplers with affine invariance,” Comm. Appl. Math. Comp. Sci. 5, 65–80 (2010).
[Crossref]

Xiao, K.

Yang, S.-M.

Yevick, A.

Yi, G.-R.

Zhong, X.

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

Appl. Opt. (1)

Comm. Appl. Math. Comp. Sci. (1)

J. Goodman and J. Weare, “Ensemble samplers with affine invariance,” Comm. Appl. Math. Comp. Sci. 5, 65–80 (2010).
[Crossref]

Exp. Fluids (1)

B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29, S175–S184 (2000).
[Crossref]

Faraday Discuss. (1)

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discuss. 159, 211–234 (2012).
[Crossref]

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J. Quant. Spectrosc. Radiat. Transfer (2)

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging Multiple Colloidal Particles by Fitting Electromagnetic Scattering Solutions to Digital Holograms,” J. Quant. Spectrosc. Radiat. Transfer 113, 2482–2489 (2012).
[Crossref]

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D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
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Nat. Mater. (1)

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
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Figures (8)

Fig. 1
Fig. 1

A typical optical setup used to record inline digital holograms. We image with a collimated laser beam and record the hologram on a CMOS camera. The coordinate z is defined as the distance from the focal plane of the objective to the particle (z > 0 if the particle lies further from the objective than the focal plane). The reference wave is shown in red and the scattered wave in blue. x and y, not shown, represent the position of the particle in the plane normal to the optical axis. On the right, a measured, background-divided hologram of a 1-μm-diameter silica particle in water, taken using a 660 nm laser (hologram courtesy of Anna Wang).

Fig. 2
Fig. 2

Comparison of results obtained from least-squares fitting using the Levenberg-Marquardt method, as described in section 2, to those obtained from the Bayesian inference method described in this paper. Units of x, y, z and r are micrometers. The results from fitting are shown by the dashed orange lines with orange regions showing one-sigma confidence intervals calculated from the parameter covariance matrix. For α (“alpha”) the width of the interval exceeds the width of the plot. Fully marginalized distributions obtained from Markov-chain Monte Carlo (MCMC) sampling are shown along the diagonal, while the off-diagonal contour plots show the joint distributions of each pair of parameters, represented as Gaussian kernel density estimates. We explain the Bayesian approach and the MCMC technique later in the text.

Fig. 3
Fig. 3

Plots of walker position as a function of MCMC time step. In both cases, the walker positions are chosen from a uniform distribution with lower bound at zero. In the plot on the left, the upper bound is 5 μm, whereas on the right it is 8 μm. As can be seen from the plot on the right, walkers that are further away from the maximum in the posterior distribution get stuck and do not reach steady state.

Fig. 4
Fig. 4

Comparison of posterior probability distributions of the parameter z, as obtained through MCMC calculations at three subset fractions. Top row shows a hologram of a 1-μm-diameter polystyrene sphere in water (same hologram in all three columns). The red pixels are the randomly chosen pixels used in the calculations. The bottom row shows the posterior probabilities at the different random-subset fractions f. The peak in the probability gets sharper as the fraction increases, but it is present even at the lowest fraction, which represents only 10 pixels in the hologram.

Fig. 5
Fig. 5

The random-subset tempering procedure involves refining position estimates at successively higher fractions, starting from coarse guesses. Top row shows the initial walker z positions for each fraction and bottom row the posterior, as obtained from MCMC ensemble sampling. The intial position distribution for the second and third stage are chosen from the peak in the posterior for the previous stage. The final posterior distribution is much narrower than the distribution used for the first stage; note the change in scale in the horizontal axis, which shows the z position in micrometers.

Fig. 6
Fig. 6

Results for particle position estimation from a single frame of a time-series. The marginalized posteriors for x, y, and z obtained from our MCMC scheme are shown along the diagonal, while the off-diagonal contour plots show the covariances (Gaussian kernel density estimates). Orange lines are results from least-squares fitting.

Fig. 7
Fig. 7

Particle trajectory inferred from analysis of a 300-frame time-series (3 seconds total). The blue curve shows the position estimate (determined from the median of the posterior) at each frame, and the shaded magenta region indicates the uncertainty (68% credible interval).

Fig. 8
Fig. 8

Posterior distributions of particle properties (refractive index n and radius r) inferred from analysis of one frame (top row) and ten successive frames (bottom row) of a time-series of holograms. Plots are Gaussian kernel density estimates of MCMC samples.

Equations (15)

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H ( x , y , z , n , r , α ) = | α E scat ( x , y , z , r , n ) + E ref | 2
h = H ( x , y , z , r , n , α ) | E ref | 2 = | α E scat ( x , y , z , r , n ) E ref + 1 | 2 .
θ = arg min θ χ 2 = arg min θ i , j [ h i j h i j M ( θ ) ] 2
p ( θ | h , M , I ) = p ( θ | M , I ) p ( h | θ , M , I ) p ( h | M , I ) .
p ( θ | h , M , I ) = p ( θ | h , M , I ) p ( h | M , I ) = p ( θ | M , I ) p ( h | θ , M , I ) .
h i j = h i j M + u i j
p ( h | θ , M , I ) = i , j p ( u i j | θ , M , I )
p ( u i j | θ , M , I ) = 1 2 π σ i j exp { [ h i j h i j M ( θ ) ] 2 2 σ i j 2 } .
p ( h | θ , M , I ) = 1 ( 2 π ) N / 2 i , j σ i j exp { i , j [ h i j h i j M ( θ ) ] 2 2 σ i j 2 }
χ 2 ( θ ) = i , j [ h i j h i j M ( θ ) ] 2 σ i j 2 .
i , j σ i j = σ N .
p ( θ | h , M , I ) = p ( θ | M , I ) ( 2 π ) N / 2 σ N exp [ χ 2 ( θ ) 2 ] .
p ( x , y , z | h , M , I ) = d r d n d α p ( x , y , z , r , n , α | h , M , I )
p ( n , r | h , M , I ) = d x d y d z d α p ( x , y , z , n , r , α | h , M , I )
σ = std ( b ) / mean ( b ) .

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