Abstract

Based on the transport theory, we present a modeling approach to light scattering in turbid material. It uses an efficient and general statistical description of the material’s scattering and absorption behavior. The model estimates the spatial distribution of intensity and the flow direction of radiation, both of which are required, e.g., for adaptable predictions of the appearance of colors in halftone prints. This is achieved by employing a computational particle method, which solves a model equation for the probability density function of photon positions and propagation directions. In this framework, each computational particle represents a finite probability of finding a photon in a corresponding state, including properties like wavelength. Model evaluations and verifications conclude the discussion.

© 2007 Optical Society of America

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References

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  1. A. Ishimaru, "Diffusion of light in turbid material," Appl. Opt. 28, 2210-2215 (1989).
    [CrossRef] [PubMed]
  2. G. L. Rogers, "Effect of light scatter on halftone color," J. Opt. Soc. Am. A 15, 1813-1821 (1998).
    [CrossRef]
  3. H. Granberg and P. Edström, "Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption," J. Pulp Pap. Sci. 29, 386-390 (2003).
  4. L. Yang, B. Kruse, and S. J. Miklavcic, "Revised Kubelka-Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media," J. Opt. Soc. Am. A 21, 1942-1952 (2004).
    [CrossRef]
  5. L. Yang and S. J. Miklavcic, "Revised Kubelka-Munk theory. III. A general theory of light propagation in scattering and absorptive media," J. Opt. Soc. Am. A 22, 1866-1873 (2005).
    [CrossRef]
  6. K. Simon and B. Trachsler, "A random walk approach for light scattering in material," in Discrete Random Walks, DRW'03, Vol. AC of DMTCS Proceedings 2003, C.Banderier and C.Krattenthaler, eds. (Discrete Mathematics and Computer Science, 2003), pp. 289-300.
  7. S. Mourad, "Improved calibration of optical characteristics of paper by an adapted paper-MTF model," J. Imaging Sci. Technol. 51, July/August (2007).
    [CrossRef]
  8. T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. 1 and 2.
  10. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).
  11. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  12. A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905), reprinted in Selected Papers on the Transfer of Radiation, D.H.Menzel, ed. (Dover, 1966), pp. 3-24.
    [CrossRef]
  13. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  14. M. I. Mishchenko, "Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electro-magnetics," Appl. Opt. 41, 7114-7134 (2002).
    [CrossRef] [PubMed]
  15. M. I. Mishchenko, "Microphysical approach to polarized radiative transfer: extension to the case of an external observation point," Appl. Opt. 42, 4963-4967 (2003).
    [CrossRef] [PubMed]
  16. S. B. Pope, Turbulent Flows (Cambridge U. Press, 2000).
  17. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).
  18. M.Hazewinkel, ed., Encyclopaedia of Mathematics (Springer, 1998).
  19. L. Wang, S. L. Jacques, and L. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  20. L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
    [CrossRef]

2007 (1)

S. Mourad, "Improved calibration of optical characteristics of paper by an adapted paper-MTF model," J. Imaging Sci. Technol. 51, July/August (2007).
[CrossRef]

2006 (1)

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

2005 (2)

L. Yang and S. J. Miklavcic, "Revised Kubelka-Munk theory. III. A general theory of light propagation in scattering and absorptive media," J. Opt. Soc. Am. A 22, 1866-1873 (2005).
[CrossRef]

L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
[CrossRef]

2004 (1)

2003 (2)

H. Granberg and P. Edström, "Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption," J. Pulp Pap. Sci. 29, 386-390 (2003).

M. I. Mishchenko, "Microphysical approach to polarized radiative transfer: extension to the case of an external observation point," Appl. Opt. 42, 4963-4967 (2003).
[CrossRef] [PubMed]

2002 (1)

1998 (1)

1995 (1)

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

1989 (1)

1931 (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

1905 (1)

A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905), reprinted in Selected Papers on the Transfer of Radiation, D.H.Menzel, ed. (Dover, 1966), pp. 3-24.
[CrossRef]

Buchholz, J.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Chandrasekhar, S.

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

Edström, P.

H. Granberg and P. Edström, "Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption," J. Pulp Pap. Sci. 29, 386-390 (2003).

Fogden, A.

L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
[CrossRef]

Granberg, H.

H. Granberg and P. Edström, "Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption," J. Pulp Pap. Sci. 29, 386-390 (2003).

Ishimaru, A.

A. Ishimaru, "Diffusion of light in turbid material," Appl. Opt. 28, 2210-2215 (1989).
[CrossRef] [PubMed]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. 1 and 2.

Jacques, S. L.

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

Kaser-Hotz, B.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Khan, T.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).

Kruse, B.

L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
[CrossRef]

L. Yang, B. Kruse, and S. J. Miklavcic, "Revised Kubelka-Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media," J. Opt. Soc. Am. A 21, 1942-1952 (2004).
[CrossRef]

Kubelka, P.

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Miklavcic, S. J.

Mishchenko, M. I.

Mourad, S.

S. Mourad, "Improved calibration of optical characteristics of paper by an adapted paper-MTF model," J. Imaging Sci. Technol. 51, July/August (2007).
[CrossRef]

Munk, F.

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Pauler, N.

L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
[CrossRef]

Pope, S. B.

S. B. Pope, Turbulent Flows (Cambridge U. Press, 2000).

Rogers, G. L.

Rothmaier, M.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Sävborg, Ö.

L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
[CrossRef]

Schuster, A.

A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905), reprinted in Selected Papers on the Transfer of Radiation, D.H.Menzel, ed. (Dover, 1966), pp. 3-24.
[CrossRef]

Selm, B.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Simon, K.

K. Simon and B. Trachsler, "A random walk approach for light scattering in material," in Discrete Random Walks, DRW'03, Vol. AC of DMTCS Proceedings 2003, C.Banderier and C.Krattenthaler, eds. (Discrete Mathematics and Computer Science, 2003), pp. 289-300.

Trachsler, B.

K. Simon and B. Trachsler, "A random walk approach for light scattering in material," in Discrete Random Walks, DRW'03, Vol. AC of DMTCS Proceedings 2003, C.Banderier and C.Krattenthaler, eds. (Discrete Mathematics and Computer Science, 2003), pp. 289-300.

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Unternährer, M.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Walt, H.

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Wang, L.

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

Yang, L.

Zheng, L.

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

Appl. Opt. (3)

Astrophys. J. (1)

A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905), reprinted in Selected Papers on the Transfer of Radiation, D.H.Menzel, ed. (Dover, 1966), pp. 3-24.
[CrossRef]

Comput. Methods Programs Biomed. (1)

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

J. Imaging Sci. Technol. (1)

S. Mourad, "Improved calibration of optical characteristics of paper by an adapted paper-MTF model," J. Imaging Sci. Technol. 51, July/August (2007).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Pulp Pap. Sci. (1)

H. Granberg and P. Edström, "Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption," J. Pulp Pap. Sci. 29, 386-390 (2003).

Nord. Pulp Pap. Res. J. (1)

L. Yang, A. Fogden, N. Pauler, Ö. Sävborg, and B. Kruse, "A novel method for studying ink penetration of a print," Nord. Pulp Pap. Res. J. 20, 423-429 (2005).
[CrossRef]

Photodiagn. Photodyn. Ther. (1)

T. Khan, M. Unternährer, J. Buchholz, B. Kaser-Hotz, B. Selm, M. Rothmaier, and H. Walt, "Performance of a contact textile-based light diffuser for photodynamic therapy," Photodiagn. Photodyn. Ther. 3, 51-60 (2006).
[CrossRef]

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Other (7)

M.Hazewinkel, ed., Encyclopaedia of Mathematics (Springer, 1998).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. 1 and 2.

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

S. B. Pope, Turbulent Flows (Cambridge U. Press, 2000).

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

K. Simon and B. Trachsler, "A random walk approach for light scattering in material," in Discrete Random Walks, DRW'03, Vol. AC of DMTCS Proceedings 2003, C.Banderier and C.Krattenthaler, eds. (Discrete Mathematics and Computer Science, 2003), pp. 289-300.

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Figures (24)

Fig. 1
Fig. 1

2D illustration of x s space with control volume Ω ¯ = Δ x × Δ s and expected photon fluxes across Ω ¯ .

Fig. 2
Fig. 2

Domain Ω with border regions and particles.

Fig. 3
Fig. 3

Isotropic scattering: (a) cumulative distribution function F ( θ ) ; (b) points x ̂ * of a corresponding evolution table.

Fig. 4
Fig. 4

Rayleigh scattering: (a) cumulative distribution function F ( θ ) ; (b) points x ̂ * of a corresponding evolution table.

Fig. 5
Fig. 5

Arbitrary scattering 1 with preferential forward scattering: (a) cumulative distribution function F ( θ ) ; (b) points x ̂ * of a corresponding evolution table.

Fig. 6
Fig. 6

Arbitrary scattering 2 with preferential sideward scattering: (a) cumulative distribution function F ( θ ) ; (b) points x ̂ * of a corresponding evolution table.

Fig. 7
Fig. 7

Sketch of the 3D test case with a diffuse light source.

Fig. 8
Fig. 8

F + and F based on diffuse light and isotropic scattering.

Fig. 9
Fig. 9

Comparison between the fluxes F + based on forward and isotropic scattering with diffuse light source.

Fig. 10
Fig. 10

Sketch of the 3D test case with a light beam.

Fig. 11
Fig. 11

PSFs of an isotropic scattering medium with γ a > 0 illuminated by a light beam with α = 0 .

Fig. 12
Fig. 12

PSFs of an isotropic scattering medium with γ a > 0 illuminated by a light beam with α = π 4 .

Fig. 13
Fig. 13

PSFs of an isotropic scattering medium with γ a > 0 illuminated by a light beam with α = π 3 .

Fig. 14
Fig. 14

Remission PSFs of two selected types of arbitrary scattering with γ a > 0 and α = 0 .

Fig. 15
Fig. 15

Remission PSFs of two selected types of arbitrary scattering with γ a > 0 and α = π 3 .

Fig. 16
Fig. 16

Transmission PSFs based on arbitrary sideward scattering of a light beam with α = π 3 .

Fig. 17
Fig. 17

Sketch of the 1D test case.

Fig. 18
Fig. 18

Kubelka–Munk: F + and F normalized by the incident flux. Analytical solution, solid curves; PDF simulations, shaded curves. They actually overlap.

Fig. 19
Fig. 19

Kubelka–Munk with six different γ s : F + normalized by the incident flux. Analytical solution, solid curves; PDF simulations, shaded curves.

Fig. 20
Fig. 20

Radial PSF computed with the new stencil method (solid curve) and the Monte Carlo implementation by Wang et al. [20] (gray circles); in both cases 10 7 particles were employed.

Fig. 21
Fig. 21

Absorption within the substrate computed with the new stencil method (solid curve) and the Monte Carlo implementation by Wang et al. [19] (gray circles); in both cases 10 6 particles were employed.

Fig. 22
Fig. 22

Dots, F simulated in a 3D regime for a diffusely illuminated, isotropic, and strongly scattering substrate (as in Fig. 8). Solid curve, same flux as obtained by the Kubelka–Munk J KM equation.

Fig. 23
Fig. 23

Predicted intensity deviations of the Kubelka–Munk solution for J KM from results of the 3D simulation. The gray and solid curves depict the deviations within scattering substrates with reflectances of 18% and 61%, respectively.

Fig. 24
Fig. 24

Comparison of the computational performance without (MC) and with (SA) the stencil technique as a function of the average number of single-scattering events per table lookup.

Tables (3)

Tables Icon

Table 1 Outline of PDF Algorithm

Tables Icon

Table 2 Outline of Algorithm to Generate Evolution Tables

Tables Icon

Table 3 Correspondence to the Random Walk Approach

Equations (36)

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d I ( x , s ) d s = γ t I ( x , s ) + γ t 4 π 4 π p ( s , s ) I ( x , s ) d ω ,
1 4 π 4 π p ( 0 , s ) d ω = γ s γ t = ϖ .
F ( x ) = 4 π I ( x , s ) s d ω
F ( x ) = ρ ( x ) c e ̂ p 4 π f s ̂ ( s ; x ) s d ω = ρ ( x ) c e ̂ p s ̂ ( x ) ,
I ( x , s ) = ρ ( x ) c e ̂ p f s ̂ ( s ; x ) .
ψ ( x ) = 4 π I ( x , s ) d ω .
ψ ( x ) = ρ ( x ) c e ̂ p .
F ( s ; x ) = ρ ( x ) f s ̂ ( s ; x ) ,
n Ω ¯ = F ( s ; x ) Ω ¯ ,
1 Ω ¯ d n Ω ¯ d t = n Ω ¯ Ω ¯ c γ a i = 1 3 { 1 Δ x i [ d x ̂ i d t s ; x i + F ( s ; x i + ) d x ̂ i d t s ; x i F ( s ; x i ) ] } i = 1 3 { 1 Δ s i [ d s ̂ i d t s i + ; x F ( s i + ; x ) d s ̂ i d t s i ; x F ( s i ; x ) ] } ,
F t + x i [ d x ̂ i d t s ; x F ] + s i [ d s ̂ i d t s ; x F ] = c γ a F
ρ t + c ρ s ̂ i x i = c γ a ρ .
ρ t + ρ U i x i = c γ a ρ .
F j t + c 2 e ̂ p ρ s ̂ j s ̂ i x i = c 2 γ a e ̂ p ρ s ̂ j + c e ̂ p ρ d s ̂ j d t
ρ U j t + ρ U j U i x i = ρ u j u i x i c γ a ρ U j + ρ d U j d t = p x j π i j x i c γ a ρ U j + ρ d U j d t
p δ i j and π i j = ρ u i u j p δ i j ,
p = 1 3 ρ u k u k = 1 3 ρ ( c 2 U k U k )
E int = 1 2 u k u k = 1 2 ( c 2 U k U k ) ,
p = 2 3 ρ E int .
w * n + 1 = w * n e d t τ a .
x ̂ * n + 1 = x ̂ * n + T x ̂ * n + 1 ,
s ̂ * n + 1 = T s ̂ * n + 1
T = [ e 1 1 e 1 2 e 1 3 e 2 1 e 2 2 e 2 3 e 3 1 e 3 2 e 3 3 ] .
Φ ̃ n + 1 = μ Φ ̃ n + ( 1 μ ) Φ ,
ψ i c e ̂ p Ω i particles Ω i { w * } ,
F i c e ̂ p Ω i particles Ω i { w * s ̂ * } .
1 c ( F + + F ) t + ( F + F ) x 1 = γ a ( F + + F ) ,
1 c ( F + F ) t + ( F + + F ) x 1 = γ a ( F + F ) + 1 c 2 ( F + + F ) d U 1 d t .
F + x 1 = γ a F + + F + + F 2 c 2 d U 1 d t ,
F x 1 = γ a F + F + + F 2 c 2 d U 1 d t .
d U 1 d t = 2 m m Δ t U 1 = 2 m c m Δ t F + F F + + F
F + x 1 = γ a F + γ s ( F + F ) = γ s F ( γ a + γ s ) F + ,
F x 1 = γ a F γ s ( F + F ) = ( γ a + γ s ) F γ s F + ,
T MC = p t s E ( S tot )
T SC < k t s [ 1 + j E ( S tot ) ] .
T SA = T SC + d a T MC .

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