Abstract

A novel method to calculate the reflectance of light from a turbid medium is presented. The method takes an approach similar to that of the Beer–Lambert law, where the intensity of light is attenuated by an exponential factor involving the path length and the absorption coefficient. Due to scatter, however, there are many path lengths; in the present method all possible path lengths are weighted by their probabilities and summed over. A path length probability density is derived by considering a photon random walk through the medium. The result is a simple expression for the reflectance based on the physical properties of the medium.

© 2008 Optical Society of America

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References

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2007

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka--Munk theory,” J. Phys. D: Appl. Phys. 40, 2210-2216 (2007).
[CrossRef]

2005

2002

A. A. Kokhanovsky, “Statistical properties of a photon gas in random media,” Phys. Rev. E 66, 037601 (2002).
[CrossRef]

2001

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

1998

G. H. Weiss, J. M. Porr, and J. Masoliver, “The continuoustime random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268-276 (1998).
[CrossRef]

1997

G. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643-656 (1997).

1996

1995

R. F. Lutomirski, A. P. Ciervo, and G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125-7136 (1995).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

1994

R. A. Bolt and J. J. ten Bosch, “On the determination of optical parameters for turbid materials,” Waves Random Media 4, 233-242 (1994).
[CrossRef]

1993

1987

1983

1971

1943

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1-89 (1943).
[CrossRef]

1931

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

Berns, R. S.

R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology, 3rd ed. (Wiley, 2000).

Bolt, R. A.

R. A. Bolt and J. J. ten Bosch, “On the determination of optical parameters for turbid materials,” Waves Random Media 4, 233-242 (1994).
[CrossRef]

Bonner, R.

Bonner, R. F.

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Caze, C.

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1-89 (1943).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Ciervo, A. P.

de Mul, F.

Dupont, D.

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Ferwerda, H. A.

Gandjbakhche, A.

Greve, J.

Groenhuis, R. A. J.

Hall, G. J.

Havlin, S.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Kokhanovsky, A. A.

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka--Munk theory,” J. Phys. D: Appl. Phys. 40, 2210-2216 (2007).
[CrossRef]

A. A. Kokhanovsky, “Statistical properties of a photon gas in random media,” Phys. Rev. E 66, 037601 (2002).
[CrossRef]

Kolinko, V.

Kubelka, P.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

Lutomirski, R. F.

Masoliver, J.

G. H. Weiss, J. M. Porr, and J. Masoliver, “The continuoustime random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268-276 (1998).
[CrossRef]

Miklavcic, S.

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

Nossal, R.

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Porr, J. M.

G. H. Weiss, J. M. Porr, and J. Masoliver, “The continuoustime random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268-276 (1998).
[CrossRef]

Priezzhev, A.

Redner, S.

S. Redner, A Guide to First-Passage Processes (Cambridge U. Press, 2001).

Richards, L. W.

Rogers, G.

G. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643-656 (1997).

Ross, S. M.

S. M. Ross, Stochastic Processes (Wiley, 1983).

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 1982).

ten Bosch, J. J.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, 1980).

Van Kampen, N. G.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, 1981).

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Weiss, G. H.

G. H. Weiss, J. M. Porr, and J. Masoliver, “The continuoustime random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268-276 (1998).
[CrossRef]

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423-432 (1987).
[CrossRef] [PubMed]

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 1982).

Yang, L.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt.

Comput. Methods Programs Biomed.

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

J. Imaging Sci. Technol.

G. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643-656 (1997).

J. Opt. Soc. Am. A

J. Phys. D: Appl. Phys.

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka--Munk theory,” J. Phys. D: Appl. Phys. 40, 2210-2216 (2007).
[CrossRef]

Opt. Commun.

G. H. Weiss, J. M. Porr, and J. Masoliver, “The continuoustime random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268-276 (1998).
[CrossRef]

Opt. Eng. (Bellingham)

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Phys. Rev. E

A. A. Kokhanovsky, “Statistical properties of a photon gas in random media,” Phys. Rev. E 66, 037601 (2002).
[CrossRef]

Rev. Mod. Phys.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1-89 (1943).
[CrossRef]

Waves Random Media

R. A. Bolt and J. J. ten Bosch, “On the determination of optical parameters for turbid materials,” Waves Random Media 4, 233-242 (1994).
[CrossRef]

Z. Tech. Phys. (Leipzig)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.

R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology, 3rd ed. (Wiley, 2000).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, 1980).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 1982).

S. M. Ross, Stochastic Processes (Wiley, 1983).

S. Redner, A Guide to First-Passage Processes (Cambridge U. Press, 2001).

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, 1981).

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Figures (2)

Fig. 1
Fig. 1

Path length probability, μ ( ξ ) , as given by Eq. (12) (curve) and a Monte Carlo calculation for I ( n ) (circles), the probability of a random walk of n steps.

Fig. 2
Fig. 2

Reflectance as a function of the albedo given by Eq. (16) (curve) and the Monte Carlo calculation (circles).

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

L n = k = 1 n l k ,
w ( l ) = γ s exp ( γ s l ) ,
w n ( L ) = γ s n L n 1 ( n 1 ) ! exp ( γ s L ) ,
L n = 0 L w n ( L ) d L ,
L n = n γ s 1 .
P n ( x , y , z ) d x d y d z = ( 2 π n l 2 3 ) 3 2 exp { 3 [ ( ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 ) ] ( 2 n l 2 ) } d x d y d z .
P n ( z ) = γ s 4 π n 3 exp [ 3 γ s 2 ( z z 0 ) 2 4 n ] .
W n ( z z 0 ) = γ s 4 π n 3 { exp [ 3 γ s 2 ( z z 0 ) 2 4 n ] exp [ 3 γ s 2 ( z + z 0 ) 2 4 n ] } .
j n ( z z 0 ) = D d d z W n ( z z 0 ) ,
D = z n 2 2 n .
z n 2 = 2 n 3 γ s 2 ,
D = ( 3 γ s 2 ) 1 .
j n ( z z 0 ) = γ s 2 4 π n 3 3 { ( z z 0 ) exp [ 3 γ s 2 ( z z 0 ) 2 4 n ] ( z + z 0 ) exp [ 3 γ s 2 ( z + z 0 ) 2 4 n ] } .
j n ( z z 0 ) = z 0 γ s 4 π n 3 3 exp [ 3 ( γ s z 0 ) 2 4 n ] .
ξ L n + 1 .
μ ( ξ ) d ξ j n + 1 ( 0 z 0 ) ,
μ ( ξ ) d ξ = 1 2 π γ s ξ 3 3 exp [ 3 ( 2 γ s ξ ) ] d ξ ,
0 μ ( ξ ) d ξ = 1 .
d I ( ξ ) = μ ( ξ ) exp ( γ a ξ ) I 0 d ξ ,
R = 0 μ ( ξ ) exp ( γ a ξ ) d ξ .
R = 1 2 π γ s 3 0 ξ 3 2 exp [ 3 ( 2 γ s ξ ) γ a ξ ] d ξ ,
R = exp ( 6 γ a γ s ) .
R = exp ( α 3 γ a γ s )

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