Abstract

We propose a model for predicting the reflectance and transmittance of multiple stacked nonscattering coloring layers that have different refractive indices. The model relies on the modeling of the reflectance and transmittance of a bounded coloring layer, i.e., a coloring layer and its two interfaces with neighboring media of different refractive indices. This model is then applied to deduce the reflectance of stacked nonscattering layers of different refractive indices superposed with a reflecting diffusing background that has its own refractive index. The classical Williams–Clapper model becomes a special case of the proposed stacked layer model.

© 2006 Optical Society of America

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References

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  1. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. 12, 593-601 (1931).
  2. F. C. Williams and F. R. Clapper, "Multiple internal reflections in photographic color prints," J. Opt. Soc. Am. 43, 595-599 (1953).
    [CrossRef] [PubMed]
  3. P. Kubelka, "New contributions to the optics of intensely light-scattering material. Part I," J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  4. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part II: Nonhomogenous layers," J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  5. J. L. Saunderson, "Calculation of the color pigmented plastics," J. Opt. Soc. Am. 32, 727-736 (1942).
    [CrossRef]
  6. J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).
  7. M. Elias, L. Simonot, and M. Menu, "Bidirectional reflectance of a diffuse background covered by a partly absorbing layer," Opt. Commun. 191, 1-7 (2001).
    [CrossRef]
  8. M. Hebert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).
  9. M. Elias and L. Simonot, "Bi-directional reflectance of a varnished painting part 1: influence of the refractive indices without using the approximations of Saunderson correction—exact computation," Opt. Commun. 231, 17-24 (2004).
    [CrossRef]
  10. M. Born and E. Wolfe, Principles of Optics, 6th ed. (Pergamon, 1987), Sec. 1.5.
  11. W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), pp. 7-13.
  12. D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

2004 (2)

M. Hebert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

M. Elias and L. Simonot, "Bi-directional reflectance of a varnished painting part 1: influence of the refractive indices without using the approximations of Saunderson correction—exact computation," Opt. Commun. 231, 17-24 (2004).
[CrossRef]

2001 (2)

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

M. Elias, L. Simonot, and M. Menu, "Bidirectional reflectance of a diffuse background covered by a partly absorbing layer," Opt. Commun. 191, 1-7 (2001).
[CrossRef]

1954 (1)

1953 (1)

1948 (1)

1942 (2)

J. L. Saunderson, "Calculation of the color pigmented plastics," J. Opt. Soc. Am. 32, 727-736 (1942).
[CrossRef]

D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

1931 (1)

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

Born, M.

M. Born and E. Wolfe, Principles of Optics, 6th ed. (Pergamon, 1987), Sec. 1.5.

Clapper, F. R.

Elias, M.

M. Elias and L. Simonot, "Bi-directional reflectance of a varnished painting part 1: influence of the refractive indices without using the approximations of Saunderson correction—exact computation," Opt. Commun. 231, 17-24 (2004).
[CrossRef]

M. Elias, L. Simonot, and M. Menu, "Bidirectional reflectance of a diffuse background covered by a partly absorbing layer," Opt. Commun. 191, 1-7 (2001).
[CrossRef]

Hebert, M.

M. Hebert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

Hersch, R. D.

M. Hebert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

Judd, D. B.

D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Kubelka, P.

McCluney, W. R.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), pp. 7-13.

Menu, M.

M. Elias, L. Simonot, and M. Menu, "Bidirectional reflectance of a diffuse background covered by a partly absorbing layer," Opt. Commun. 191, 1-7 (2001).
[CrossRef]

Munk, F.

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

Saunderson, J. L.

Shore, J. D.

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

Simonot, L.

M. Elias and L. Simonot, "Bi-directional reflectance of a varnished painting part 1: influence of the refractive indices without using the approximations of Saunderson correction—exact computation," Opt. Commun. 231, 17-24 (2004).
[CrossRef]

M. Elias, L. Simonot, and M. Menu, "Bidirectional reflectance of a diffuse background covered by a partly absorbing layer," Opt. Commun. 191, 1-7 (2001).
[CrossRef]

Spoonhower, J. P.

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

Williams, F. C.

Wolfe, E.

M. Born and E. Wolfe, Principles of Optics, 6th ed. (Pergamon, 1987), Sec. 1.5.

J. Imaging Sci. Technol. (2)

M. Hebert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

J. Opt. Soc. Am. (4)

J. Res. Natl. Bur. Stand. (1)

D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Opt. Commun. (2)

M. Elias, L. Simonot, and M. Menu, "Bidirectional reflectance of a diffuse background covered by a partly absorbing layer," Opt. Commun. 191, 1-7 (2001).
[CrossRef]

M. Elias and L. Simonot, "Bi-directional reflectance of a varnished painting part 1: influence of the refractive indices without using the approximations of Saunderson correction—exact computation," Opt. Commun. 231, 17-24 (2004).
[CrossRef]

Z. Tech. Phys. (1)

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

Other (2)

M. Born and E. Wolfe, Principles of Optics, 6th ed. (Pergamon, 1987), Sec. 1.5.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), pp. 7-13.

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Figures (2)

Fig. 1
Fig. 1

Reflection and transmission of light at an interface between two media of refractive indices n j and n k .

Fig. 2
Fig. 2

Reflection and refraction within a nonscattering coloring layer m 1 .

Equations (87)

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n j sin θ j = n k sin θ k .
R j k ( p ) ( θ j ) = tan 2 ( θ j θ k ) tan 2 ( θ j + θ k ) ,
R j k ( s ) ( θ j ) = sin 2 ( θ j θ k ) sin 2 ( θ j + θ k )
W i ( p ) = W i ( s ) = W i 2 .
W i = W i ( p ) + W i ( s ) .
R j k ( θ j ) = 1 2 [ R j k ( p ) ( θ j ) + R j k ( s ) ( θ j ) ] .
T j k * ( θ j ) = 1 R j k * ( θ j ) .
R j k * ( θ j ) = R k j * ( θ k ) ,
T j k * ( θ j ) = T k j * ( θ k ) .
d Ω k = ( n j n k ) 2 cos θ j cos θ k d Ω j .
L i ( p ) = L i ( s ) = L i 2 = W i 2 π .
d W i ( p ) ( θ j , ϕ j ) = L i ( p ) cos θ j d Ω j = L i ( p ) cos θ j sin θ j d θ j d ϕ j .
d W r ( p ) ( θ j , ϕ j ) = R j k ( p ) ( θ j ) L i ( p ) cos θ j sin θ j d θ j d ϕ j .
d W r ( s ) ( θ j , ϕ j ) = R j k ( s ) ( θ j ) L i ( s ) cos θ j sin θ j d θ j d ϕ j .
d W r ( θ j , ϕ j ) = R j k ( θ j ) W i π cos θ j sin θ j d θ j d ϕ j .
W r = ϕ j = 0 2 π θ j = 0 π 2 R j k ( θ j ) W i π cos θ j sin θ j d θ j d ϕ j .
W r = W i θ j = 0 π 2 R j k ( θ j ) sin 2 θ j d θ j .
r j k = θ j = 0 π 2 R j k ( θ j ) sin 2 θ j d θ j .
t j k = 1 r j k .
t = e α x .
W i ( p ) = W i ( s ) = W i 2 .
W r 0 * = R 01 * ( θ 0 ) W i * .
t 1 ( θ 1 ) = e α 1 h 1 cos θ 1 .
n 0 sin θ 0 = n 1 sin θ 1 = n 2 sin θ 2 .
W t 0 * = T 01 * ( θ 0 ) T 12 * ( θ 1 ) t 1 ( θ 1 ) W i * .
W r 1 * = T 01 * ( θ 0 ) R 12 * ( θ 1 ) T 10 * ( θ 1 ) t 1 2 ( θ 1 ) W i * .
W r k * = T 01 * ( θ 0 ) [ R 10 * ( θ 1 ) ] k 1 [ R 12 * ( θ 1 ) ] k t 1 2 k ( θ 1 ) T 10 * ( θ 1 ) W i * .
W r * = R 01 * ( θ 0 ) W i * + T 01 * ( θ 0 ) T 10 * ( θ 1 ) R 10 * ( θ 1 ) W i * k = 1 [ R 10 * ( θ 1 ) R 12 * ( θ 1 ) t 1 2 ( θ 1 ) ] k .
W r * = R 01 * ( θ 0 ) W i * + [ T 10 * ( θ 1 ) ] 2 R 12 * ( θ 1 ) t 1 2 ( θ 1 ) 1 R 10 * ( θ 1 ) R 12 * ( θ 1 ) t 1 2 ( θ 1 ) W i * .
R 012 * ( θ 0 ) = R 01 * ( θ 0 ) + [ T 10 * ( θ 1 ) ] 2 R 12 * ( θ 1 ) t 1 2 ( θ 1 ) 1 R 10 * ( θ 1 ) R 12 * ( θ 1 ) t 1 2 ( θ 1 ) .
W r = W r ( p ) + W r ( s ) = R 012 ( p ) ( θ 0 ) W i ( p ) + R 012 ( s ) ( θ 0 ) W i ( s ) .
W r = 1 2 [ R 012 ( p ) ( θ 0 ) + R 012 ( s ) ( θ 0 ) ] W i .
R 012 ( θ 0 ) = 1 2 [ R 012 ( p ) ( θ 0 ) + R 012 ( s ) ( θ 0 ) ] .
W t k * = T 01 * ( θ 0 ) [ R 10 * ( θ 1 ) ] k [ R 12 * ( θ 1 ) ] k [ t 1 ( θ 1 ) ] 2 k + 1 T 12 * ( θ 1 ) W i * .
W t * = T 01 * ( θ 0 ) T 12 * ( θ 1 ) t 1 ( θ 1 ) W i * k = 0 [ R 10 * ( θ 1 ) R 12 * ( θ 1 ) t 1 2 ( θ 1 ) ] k ,
W t * = T 01 * ( θ 0 ) T 12 * ( θ 1 ) t 1 ( θ 1 ) 1 R 10 * ( θ 1 ) R 12 * ( θ 1 ) t 1 2 ( θ 1 ) W i * .
T 012 * ( θ 0 ) = T 01 * ( θ 0 ) T 12 * ( θ 1 ) t 1 ( θ 1 ) 1 R 10 * ( θ 1 ) R 12 * ( θ 1 ) t 1 2 ( θ 1 ) .
W t = 1 2 [ T 012 ( p ) ( θ 0 ) + T 012 ( s ) ( θ 0 ) ] W i .
T 012 ( θ 0 ) = 1 2 [ T 012 ( p ) ( θ 0 ) + T 012 ( s ) ( θ 0 ) ] .
R 210 * ( θ 2 ) = R 21 * ( θ 2 ) + [ T 12 * ( θ 1 ) ] 2 R 10 * ( θ 1 ) t 1 2 ( θ 1 ) 1 R 12 * ( θ 1 ) R 10 * ( θ 1 ) t 1 2 ( θ 1 ) .
R 210 ( θ 2 ) = 1 2 [ R 210 ( p ) ( θ 2 ) + R 210 ( s ) ( θ 2 ) ] ,
T 210 * ( θ 2 ) = T 21 * ( θ 2 ) T 10 * ( θ 1 ) t 1 ( θ 1 ) 1 R 12 * ( θ 1 ) R 10 * ( θ 1 ) t 1 2 ( θ 1 ) ,
T 210 ( θ 2 ) = 1 2 [ T 210 ( p ) ( θ 2 ) + T 210 ( s ) ( θ 2 ) ] .
T 210 ( θ 2 ) = T 012 ( θ 0 ) .
d W i ( θ 0 , ϕ 0 ) = W i π cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
W r = ϕ 0 = 0 2 π θ 0 = 0 π 2 R 012 ( θ 0 ) W i π cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
r 012 = θ 0 = 0 π 2 R 012 ( θ 0 ) sin 2 θ 0 d θ 0 .
W t = ϕ 0 = 0 2 π θ 0 = 0 π 2 T 012 ( θ 0 ) W i π cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
t 012 = W t W i = θ 0 = 0 π 2 T 012 ( θ 0 ) sin 2 θ 0 d θ 0 .
R 0123 ( θ 0 ) = 1 2 [ R 0123 ( p ) ( θ 0 ) + R 0123 ( s ) ( θ 0 ) ] .
R 0123 * ( θ 0 ) = R 01 * ( θ 0 ) + [ T 01 * ( θ 0 ) ] 2 R 123 * ( θ 1 ) t 1 2 ( θ ) 1 R 10 * ( θ 1 ) R 123 * ( θ 1 ) t 1 2 ( θ ) .
T 0123 ( θ 0 ) = 1 2 [ T 0123 ( p ) ( θ 0 ) + T 0123 ( s ) ( θ 0 ) ] ,
T 0123 * ( θ 0 ) = T 01 * ( θ 0 ) T 123 * ( θ 1 ) t 1 ( θ 1 ) 1 R 10 * ( θ 1 ) R 123 * ( θ 1 ) t 1 2 ( θ 1 ) .
R 0123 * ( θ 0 ) = R 012 * ( θ 0 ) + [ T 012 * ( θ 0 ) ] 2 R 23 * ( θ 2 ) t 2 2 ( θ 2 ) 1 R 210 * ( θ 1 ) R 23 * ( θ 2 ) t 2 2 ( θ 2 )
T 0123 * ( θ 0 ) = T 012 * ( θ 0 ) T 23 * ( θ 2 ) t 2 ( θ 2 ) 1 R 210 * ( θ 1 ) R 23 * ( θ 2 ) t 2 2 ( θ 2 ) ,
t 2 ( θ 2 ) = e α 2 h 2 cos θ 2 .
R 0 k + 1 ( θ 0 ) = 1 2 [ R 0 k + 1 ( p ) ( θ 0 ) + R 0 k + 1 ( s ) ( θ 0 ) ] ,
T 0 k + 1 ( θ 0 ) = 1 2 [ T 0 k + 1 ( p ) ( θ 0 ) + T 0 k + 1 ( s ) ( θ 0 ) ] .
R 0 k + 1 * ( θ 0 ) = R 0 k * ( θ 0 ) + [ T 0 k * ( θ 0 ) ] 2 R k , k + 1 * ( θ k ) t k 2 ( θ k ) 1 R k 0 * ( θ k ) R k , k + 1 * ( θ k ) t k 2 ( θ k )
T 0 k + 1 * ( θ 0 ) = T 0 k * ( θ 0 ) T k , k + 1 * ( θ k ) t k ( θ k ) 1 R k 0 * ( θ k ) R k , k + 1 * ( θ k ) t k 2 ( θ k ) ,
t k ( θ k ) = e α k h k cos θ k .
r 0 k + 1 = θ 0 = 0 π 2 R 0 k + 1 ( θ 0 ) sin 2 θ 0 d θ 0
t 0 k + 1 = θ 0 = 0 π 2 T 0 k + 1 ( θ 0 ) sin 2 θ 0 d θ 0 .
w 1 = ρ g T 012 ( θ 0 ) W i .
w 2 = ρ g r 210 w 1 = ( ρ g r 210 ) ρ g T 012 ( θ 0 ) W i .
W g = k = 0 ( ρ g r 210 ) k ρ g T 012 ( θ 0 ) W i .
W g = ρ g T 012 ( θ 0 ) 1 ρ g r 210 W i .
W r = t 210 ρ g T 012 ( θ 0 ) 1 ρ g r 210 W i .
R m 1 = t 210 ρ g T 012 ( θ 0 ) 1 ρ g r 210 .
L r ( θ 0 ) = d 2 Φ d s cos θ 0 d Ω 0 .
d 2 Φ = T 012 ( θ 0 ) d 2 Φ g .
d Ω 2 = ( n 0 n 2 ) 2 cos θ 0 cos θ 2 d Ω 0 .
W g π = d 2 Φ g d s cos θ 2 d Ω 2 .
L r ( θ 0 ) = W g π T 012 ( θ 0 ) cos θ 2 d Ω 2 cos θ 0 d Ω 0 .
L r ( θ 0 ) = 1 π ( n 0 n 2 ) 2 T 012 ( θ 0 ) T 012 ( θ 0 ) ρ g 1 ρ g r 210 W i .
BRDF ( θ 0 , θ 0 ) = L r ( θ 0 ) W i = 1 π ( n 0 n 2 ) 2 T 012 ( θ 0 ) ρ g 1 ρ g r 210 .
R ( θ 0 , θ 0 ) = ( n 0 n 2 ) 2 T 012 ( θ 0 ) ρ g 1 ρ g r 210 .
R m 1 k = t k + 1 0 ρ g T 0 k + 1 ( θ 0 ) 1 ρ g r k + 1 0 .
R ( θ 0 , θ 0 ) = ( n 0 n 2 ) 2 T 0 k + 1 ( θ 0 ) T 0 k + 1 ( θ 0 ) ρ g 1 ρ g r k + 1 0 .
r 110 = θ = 0 π 2 R 10 ( θ ) t 1 2 ( θ ) sin 2 θ d θ ,
t 110 = θ = 0 π 2 T 10 ( θ ) t 1 ( θ 1 ) sin 2 θ d θ .
R ( θ 0 , θ 0 ) = ( n 0 n 1 ) 2 ρ g T 01 ( θ 0 ) T 01 ( θ 0 ) t 1 ( θ 1 ) t 1 ( θ 1 ) 1 ρ g r 110 .
R m 1 = t 110 ρ g T 01 ( θ 0 ) t 1 ( θ 1 ) 1 ρ g r 110 .
ρ = t 20 ρ g T 02 ( θ 0 ) 1 ρ g r 20 ,
R ( θ 0 , θ 0 ) = ( n 0 n 2 ) 2 T 02 ( θ 0 ) T 02 ( θ 0 ) ρ g 1 ρ g r 20 .
ρ g = ρ t 20 T 02 ( θ 0 ) + r 20 ρ
ρ g = R ( θ 0 , θ 0 ) ( n 0 n 2 ) 2 T 02 ( θ 0 ) T 02 ( θ 0 ) + r 20 R ( θ 0 , θ 0 ) .

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