Abstract

This work introduces a Markov Chain solution to model photon multiple scattering through turbid slabs via anisotropic scattering process, i.e., Mie scattering. Results show that the proposed Markov Chain model agree with commonly used Monte Carlo simulation for various mediums such as medium with non-uniform phase functions and absorbing medium. The proposed Markov Chain solution method successfully converts the complex multiple scattering problem with practical phase functions into a matrix form and solves transmitted/reflected photon angular distributions by matrix multiplications. Such characteristics would potentially allow practical inversions by matrix manipulation or stochastic algorithms where widely applied stochastic methods such as Monte Carlo simulations usually fail, and thus enable practical diagnostics reconstructions such as medical diagnosis, spray analysis, and atmosphere sciences.

© 2016 Optical Society of America

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References

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  1. E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski, and M. A. Linne, “Laser light scattering in turbid media Part I: Experimental and simulated results for the spatial intensity distribution,” Opt. Express 15(17), 10649–10665 (2007).
    [Crossref] [PubMed]
  2. P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
    [Crossref]
  3. A. A. Kokhanovsky, Light Scattering Media Optics (Springer Science and Business Media, 2004).
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    [Crossref] [PubMed]
  5. E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski, and M. A. Linne, “Laser light scattering in turbid media Part II: Spatial and temporal analysis of individual scattering orders via Monte Carlo simulation,” Opt. Express 17(16), 13792–13809 (2009).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. X. Li and L. Ma, “Scaling law for photon transmission through optically turbid slabs based on random walk theory,” Appl. Sci. 2(4), 160–165 (2012).
    [Crossref]
  8. X. Sun, X. Li, and L. Ma, “A closed-form method for calculating the angular distribution of multiply scattered photons through isotropic turbid slabs,” Opt. Express 19(24), 23932–23937 (2011).
    [Crossref] [PubMed]
  9. A. J. Welch and M. J. Van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, vol. 2 (Springer, 2011).
  10. X. Li and W. F. Northrop, “A Markov Chain-based quantitative study of angular distribution of photons through turbid slabs via isotropic light scattering,” Comput. Phys. Commun. 201, 77–84 (2016).
    [Crossref]
  11. J. R. Norris, Markov Chains (Cambridge University Press, 1998).
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    [Crossref] [PubMed]
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  14. L. Ma, X. Li, S. T. Sanders, A. W. Caswell, S. Roy, D. H. Plemmons, and J. R. Gord, “50-kHz-rate 2D imaging of temperature and H2O concentration at the exhaust plane of a J85 engine using hyperspectral tomography,” Opt. Express 21(1), 1152–1162 (2013).
    [Crossref] [PubMed]
  15. A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, and M. S. Feld, “Determination of particle size by using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19(4), 737–744 (2002).
    [Crossref] [PubMed]

2016 (1)

X. Li and W. F. Northrop, “A Markov Chain-based quantitative study of angular distribution of photons through turbid slabs via isotropic light scattering,” Comput. Phys. Commun. 201, 77–84 (2016).
[Crossref]

2014 (1)

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

2013 (1)

2012 (2)

A. Doronin and I. Meglinski, “Peer-to-peer Monte Carlo simulation of photon migration in topical applications of biomedical optics,” J. Biomed. Opt. 17(9), 0905041 (2012).
[Crossref] [PubMed]

X. Li and L. Ma, “Scaling law for photon transmission through optically turbid slabs based on random walk theory,” Appl. Sci. 2(4), 160–165 (2012).
[Crossref]

2011 (2)

2009 (1)

2007 (1)

2002 (1)

1993 (1)

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Backman, V.

Berrocal, E.

Bonner, R. F.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Caswell, A. W.

Dasari, R. R.

Doronin, A.

A. Doronin and I. Meglinski, “Peer-to-peer Monte Carlo simulation of photon migration in topical applications of biomedical optics,” J. Biomed. Opt. 17(9), 0905041 (2012).
[Crossref] [PubMed]

A. Doronin and I. Meglinski, “Online object oriented Monte Carlo computational tool for the needs of biomedical optics,” Biomed. Opt. Express 2(9), 2461–2469 (2011).
[Crossref] [PubMed]

Feld, M. S.

Gandjbakhche, A. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Gord, J. R.

Kalashnikov, M.

Keller, H. B.

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Knitter, R.

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Kolb, M.

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Kumar, R.

R. Kumar, A. Tomkins, S. Vassilvitskii, and E. Vee, “Inverting a steady-state,” in Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, (ACM, 2015), pp. 359–368.

Leys, O.

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Li, X.

X. Li and W. F. Northrop, “A Markov Chain-based quantitative study of angular distribution of photons through turbid slabs via isotropic light scattering,” Comput. Phys. Commun. 201, 77–84 (2016).
[Crossref]

L. Ma, X. Li, S. T. Sanders, A. W. Caswell, S. Roy, D. H. Plemmons, and J. R. Gord, “50-kHz-rate 2D imaging of temperature and H2O concentration at the exhaust plane of a J85 engine using hyperspectral tomography,” Opt. Express 21(1), 1152–1162 (2013).
[Crossref] [PubMed]

X. Li and L. Ma, “Scaling law for photon transmission through optically turbid slabs based on random walk theory,” Appl. Sci. 2(4), 160–165 (2012).
[Crossref]

X. Sun, X. Li, and L. Ma, “A closed-form method for calculating the angular distribution of multiply scattered photons through isotropic turbid slabs,” Opt. Express 19(24), 23932–23937 (2011).
[Crossref] [PubMed]

Linne, M. A.

Ma, L.

Matthes, J.

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Meglinski, I.

A. Doronin and I. Meglinski, “Peer-to-peer Monte Carlo simulation of photon migration in topical applications of biomedical optics,” J. Biomed. Opt. 17(9), 0905041 (2012).
[Crossref] [PubMed]

A. Doronin and I. Meglinski, “Online object oriented Monte Carlo computational tool for the needs of biomedical optics,” Biomed. Opt. Express 2(9), 2461–2469 (2011).
[Crossref] [PubMed]

Meglinski, I. V.

Northrop, W. F.

X. Li and W. F. Northrop, “A Markov Chain-based quantitative study of angular distribution of photons through turbid slabs via isotropic light scattering,” Comput. Phys. Commun. 201, 77–84 (2016).
[Crossref]

Nossal, R.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Paciaroni, M. E.

Plemmons, D. H.

Roy, S.

Sanders, S. T.

Sedarsky, D. L.

Sun, X.

Tomkins, A.

R. Kumar, A. Tomkins, S. Vassilvitskii, and E. Vee, “Inverting a steady-state,” in Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, (ACM, 2015), pp. 359–368.

Vassilvitskii, S.

R. Kumar, A. Tomkins, S. Vassilvitskii, and E. Vee, “Inverting a steady-state,” in Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, (ACM, 2015), pp. 359–368.

Vee, E.

R. Kumar, A. Tomkins, S. Vassilvitskii, and E. Vee, “Inverting a steady-state,” in Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, (ACM, 2015), pp. 359–368.

Waibel, P.

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Wax, A.

Weiss, G. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Yang, C.

Appl. Sci. (1)

X. Li and L. Ma, “Scaling law for photon transmission through optically turbid slabs based on random walk theory,” Appl. Sci. 2(4), 160–165 (2012).
[Crossref]

Biomed. Opt. Express (1)

Chem. Eng. Technol. (1)

P. Waibel, J. Matthes, O. Leys, M. Kolb, H. B. Keller, and R. Knitter, “High‐speed camera‐based analysis of the lithium ceramic pebble fabrication process,” Chem. Eng. Technol. 37(10), 1654–1662 (2014).
[Crossref]

Comput. Phys. Commun. (1)

X. Li and W. F. Northrop, “A Markov Chain-based quantitative study of angular distribution of photons through turbid slabs via isotropic light scattering,” Comput. Phys. Commun. 201, 77–84 (2016).
[Crossref]

J. Biomed. Opt. (1)

A. Doronin and I. Meglinski, “Peer-to-peer Monte Carlo simulation of photon migration in topical applications of biomedical optics,” J. Biomed. Opt. 17(9), 0905041 (2012).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Opt. Express (4)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Other (4)

A. A. Kokhanovsky, Light Scattering Media Optics (Springer Science and Business Media, 2004).

A. J. Welch and M. J. Van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, vol. 2 (Springer, 2011).

J. R. Norris, Markov Chains (Cambridge University Press, 1998).

R. Kumar, A. Tomkins, S. Vassilvitskii, and E. Vee, “Inverting a steady-state,” in Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, (ACM, 2015), pp. 359–368.

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Figures (5)

Fig. 1
Fig. 1 Schematic of multiple scattering modeled by Markov Chain model.
Fig. 2
Fig. 2 Phase functions (PDF) selected for this study.
Fig. 3
Fig. 3 Transmitted and reflected photons with different phase function distributions.
Fig. 4
Fig. 4 Transmitted/reflected photon angular distribution for photons exiting with different times of scatterings. Markov Chain theorem angular distribution predictions for # of scattering = 1, 2, and 3 are P’ × R, P’ × P × R, and P’ × P2 × R, respectively.
Fig. 5
Fig. 5 Markov Chain approximation for an absorbing medium. The scattering coefficient and absorption coefficient were set to be 9.5 cm−1 and 0.5 cm−1, respectively.

Tables (1)

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Table 1 Comparison of Total Transmittance with Different Methods

Equations (5)

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P(( z m , θ i ),( z n , θ j ))=P( z m , z n , θ i )P( θ i , θ j , z n )
P( z m , z n , θ i )=exp( k=m n Σ e,k Δz cos θ i )
α=arccos(cos θ i cos θ j cosφ+sin θ i sin θ j )
P( θ i , θ j , z n )= 0 2π Γ z n [arccos(cos θ i cos θ j cosφ+sin θ i sin θ j )]dφ
Q t =P'× P t1 ×R and Q total =P'×N×R=P'× (1P) 1 ×R

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