Abstract

A theoretical modeling of the light leaving a scattering medium with a rough interface is presented. It combines the calculation of the reflection and the transmission from a random Gaussian rough surface and the radiative transfer equation solved by the auxiliary function method. Diffuse and collimated illumination is considered. Numerical simulations with various roughness parameters lead to quantitative comparisons between the surface and the bulk scattered fluxes. The validity domain of the modeling is deduced from flux conservation. An application for a real pigment shows that the surface roughness does not significantly modify its reflectance spectrum because of bulk scattering and induces a simple translation of the global spectrum because of the surface scattering.

© 2010 Optical Society of America

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  1. L. Simonot and M. Elias, “Color change due to surface state modification,” Color Res. Appl. 28, 45-49 (2003).
    [CrossRef]
  2. G. Dupuis, M. Elias, and L. Simonot, “Pigment identification by fiber-optics diffuse reflectance spectroscopy,” Appl. Spectrosc. 56, 1329-1336 (2002).
    [CrossRef]
  3. M. Elias, C. Magnain, and J. M. Frigerio, “Contribution of surface state characterization to studies of works of art,” Appl. Opt. 49, 2151-2160 (2010).
    [CrossRef] [PubMed]
  4. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  5. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun's glitter,” J. Opt. Soc. Am. 44, 838-850 (1954).
    [CrossRef]
  6. M. Elias and M. Menu, “Experimental characterisation of a random metallic rough surface by spectrophotometric measurements in the visible range,” Opt. Commun. 180, 191-198 (2000).
    [CrossRef]
  7. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  8. Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431-442 (1994).
    [CrossRef] [PubMed]
  9. D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
    [CrossRef]
  10. S. V. Salinas and S. C. Liew, “Light reflection from a rough liquid surface including wind-wave effects in a scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 105, 414-424 (2007).
    [CrossRef]
  11. Y. Liang and L. X. Guo, “Study of the electromagnetic scattering from the rough sea surface with bubbles/foams by the modified two-scale method,” Acta Phys. Sin. 58, 6158-6166 (2009).
  12. M. M. R. Williams, “The scattering of radiation from a surface comprised of randomly distributed plates of differing reflective properties,” J. Quant. Spectrosc. Radiat. Transf. 109, 2182-2194 (2008).
    [CrossRef]
  13. A. B. Murphy, “Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D: Appl. Phys. 39, 3571-3581 (2006).
    [CrossRef]
  14. S. Mudaliar, “Some issues with using radiative transfer approach to scattering from layered random media with rough interfaces,” IEEE Trans. Antennas Propag. 57, 3646-3654 (2009).
    [CrossRef]
  15. C. Amra, C. Grezes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492-5503 (1993).
    [CrossRef] [PubMed]
  16. M. Elias and G. Elias, “Radiative transfer in inhomogeneous stratified media using the auxiliary function method,” J. Opt. Soc. Am. A 21, 580-589 (2004).
    [CrossRef]
  17. A. Da Silva, M. Elias, C. Andraud, and J. Lafait, “Comparison between the auxiliary function method and the discrete-ordinate-method for solving the radiative transfer equation for light scattering,” J. Opt. Soc. Am. A 20, 2321-2329 (2003).
    [CrossRef]
  18. G. Latour, M. Elias, and J. M. Frigerio, “Color modeling of stratified pictorial layers using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A 24, 3045-3053 (2007).
    [CrossRef]
  19. C. Magnain, M. Elias, and J. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method: The inverse problem,” J. Opt. Soc. Am. A 25, 1737-1743 (2008).
    [CrossRef]
  20. C. Lam and A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particules and random rough surfaces with moderate roughness,” J. Phys. A 260, 111-125 (1993).
  21. G. Latour, M. Elias, and J. M. Frigerio, “Determination of the absorption and scattering coefficients of pigments,” Appl. Spectrosc. 63, 604-610 (2009).
    [CrossRef] [PubMed]
  22. M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
    [CrossRef]

2010 (1)

2009 (4)

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Y. Liang and L. X. Guo, “Study of the electromagnetic scattering from the rough sea surface with bubbles/foams by the modified two-scale method,” Acta Phys. Sin. 58, 6158-6166 (2009).

S. Mudaliar, “Some issues with using radiative transfer approach to scattering from layered random media with rough interfaces,” IEEE Trans. Antennas Propag. 57, 3646-3654 (2009).
[CrossRef]

G. Latour, M. Elias, and J. M. Frigerio, “Determination of the absorption and scattering coefficients of pigments,” Appl. Spectrosc. 63, 604-610 (2009).
[CrossRef] [PubMed]

2008 (2)

C. Magnain, M. Elias, and J. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method: The inverse problem,” J. Opt. Soc. Am. A 25, 1737-1743 (2008).
[CrossRef]

M. M. R. Williams, “The scattering of radiation from a surface comprised of randomly distributed plates of differing reflective properties,” J. Quant. Spectrosc. Radiat. Transf. 109, 2182-2194 (2008).
[CrossRef]

2007 (2)

G. Latour, M. Elias, and J. M. Frigerio, “Color modeling of stratified pictorial layers using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A 24, 3045-3053 (2007).
[CrossRef]

S. V. Salinas and S. C. Liew, “Light reflection from a rough liquid surface including wind-wave effects in a scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 105, 414-424 (2007).
[CrossRef]

2006 (2)

A. B. Murphy, “Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D: Appl. Phys. 39, 3571-3581 (2006).
[CrossRef]

M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
[CrossRef]

2004 (1)

2003 (2)

2002 (1)

2000 (1)

M. Elias and M. Menu, “Experimental characterisation of a random metallic rough surface by spectrophotometric measurements in the visible range,” Opt. Commun. 180, 191-198 (2000).
[CrossRef]

1994 (1)

1993 (2)

C. Amra, C. Grezes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492-5503 (1993).
[CrossRef] [PubMed]

C. Lam and A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particules and random rough surfaces with moderate roughness,” J. Phys. A 260, 111-125 (1993).

1963 (1)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

1960 (1)

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

1954 (1)

Amra, C.

Andraud, C.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Bruel, L.

Chandrasekhar, S.

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

Charron, E.

M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
[CrossRef]

Chen, K. S.

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Cox, C.

Da Silva, A.

De La Rie, R.

M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
[CrossRef]

Delanay, J.

M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
[CrossRef]

Dupuis, G.

Elias, G.

Elias, M.

M. Elias, C. Magnain, and J. M. Frigerio, “Contribution of surface state characterization to studies of works of art,” Appl. Opt. 49, 2151-2160 (2010).
[CrossRef] [PubMed]

G. Latour, M. Elias, and J. M. Frigerio, “Determination of the absorption and scattering coefficients of pigments,” Appl. Spectrosc. 63, 604-610 (2009).
[CrossRef] [PubMed]

C. Magnain, M. Elias, and J. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method: The inverse problem,” J. Opt. Soc. Am. A 25, 1737-1743 (2008).
[CrossRef]

G. Latour, M. Elias, and J. M. Frigerio, “Color modeling of stratified pictorial layers using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A 24, 3045-3053 (2007).
[CrossRef]

M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
[CrossRef]

M. Elias and G. Elias, “Radiative transfer in inhomogeneous stratified media using the auxiliary function method,” J. Opt. Soc. Am. A 21, 580-589 (2004).
[CrossRef]

L. Simonot and M. Elias, “Color change due to surface state modification,” Color Res. Appl. 28, 45-49 (2003).
[CrossRef]

A. Da Silva, M. Elias, C. Andraud, and J. Lafait, “Comparison between the auxiliary function method and the discrete-ordinate-method for solving the radiative transfer equation for light scattering,” J. Opt. Soc. Am. A 20, 2321-2329 (2003).
[CrossRef]

G. Dupuis, M. Elias, and L. Simonot, “Pigment identification by fiber-optics diffuse reflectance spectroscopy,” Appl. Spectrosc. 56, 1329-1336 (2002).
[CrossRef]

M. Elias and M. Menu, “Experimental characterisation of a random metallic rough surface by spectrophotometric measurements in the visible range,” Opt. Commun. 180, 191-198 (2000).
[CrossRef]

Frigerio, J.

Frigerio, J. M.

Grezes-Besset, C.

Gui, Z.

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Guo, L. X.

Y. Liang and L. X. Guo, “Study of the electromagnetic scattering from the rough sea surface with bubbles/foams by the modified two-scale method,” Acta Phys. Sin. 58, 6158-6166 (2009).

Ishimaru, A.

C. Lam and A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particules and random rough surfaces with moderate roughness,” J. Phys. A 260, 111-125 (1993).

Jin, Z.

Lafait, J.

Lam, C.

C. Lam and A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particules and random rough surfaces with moderate roughness,” J. Phys. A 260, 111-125 (1993).

Latour, G.

Liang, D.

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Liang, Y.

Y. Liang and L. X. Guo, “Study of the electromagnetic scattering from the rough sea surface with bubbles/foams by the modified two-scale method,” Acta Phys. Sin. 58, 6158-6166 (2009).

Liew, S. C.

S. V. Salinas and S. C. Liew, “Light reflection from a rough liquid surface including wind-wave effects in a scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 105, 414-424 (2007).
[CrossRef]

Magnain, C.

Menu, M.

M. Elias and M. Menu, “Experimental characterisation of a random metallic rough surface by spectrophotometric measurements in the visible range,” Opt. Commun. 180, 191-198 (2000).
[CrossRef]

Mudaliar, S.

S. Mudaliar, “Some issues with using radiative transfer approach to scattering from layered random media with rough interfaces,” IEEE Trans. Antennas Propag. 57, 3646-3654 (2009).
[CrossRef]

Munk, W.

Murphy, A. B.

A. B. Murphy, “Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D: Appl. Phys. 39, 3571-3581 (2006).
[CrossRef]

Salinas, S. V.

S. V. Salinas and S. C. Liew, “Light reflection from a rough liquid surface including wind-wave effects in a scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 105, 414-424 (2007).
[CrossRef]

Simonot, L.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Stamnes, K.

Tsang, L.

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Williams, M. M. R.

M. M. R. Williams, “The scattering of radiation from a surface comprised of randomly distributed plates of differing reflective properties,” J. Quant. Spectrosc. Radiat. Transf. 109, 2182-2194 (2008).
[CrossRef]

Xu, P.

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Acta Phys. Sin. (1)

Y. Liang and L. X. Guo, “Study of the electromagnetic scattering from the rough sea surface with bubbles/foams by the modified two-scale method,” Acta Phys. Sin. 58, 6158-6166 (2009).

Appl. Opt. (3)

Appl. Spectrosc. (2)

Color Res. Appl. (1)

L. Simonot and M. Elias, “Color change due to surface state modification,” Color Res. Appl. 28, 45-49 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Mudaliar, “Some issues with using radiative transfer approach to scattering from layered random media with rough interfaces,” IEEE Trans. Antennas Propag. 57, 3646-3654 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. A (1)

C. Lam and A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particules and random rough surfaces with moderate roughness,” J. Phys. A 260, 111-125 (1993).

J. Phys. D: Appl. Phys. (1)

A. B. Murphy, “Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically-rough surfaces,” J. Phys. D: Appl. Phys. 39, 3571-3581 (2006).
[CrossRef]

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

S. V. Salinas and S. C. Liew, “Light reflection from a rough liquid surface including wind-wave effects in a scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 105, 414-424 (2007).
[CrossRef]

M. M. R. Williams, “The scattering of radiation from a surface comprised of randomly distributed plates of differing reflective properties,” J. Quant. Spectrosc. Radiat. Transf. 109, 2182-2194 (2008).
[CrossRef]

Opt. Commun. (2)

M. Elias and M. Menu, “Experimental characterisation of a random metallic rough surface by spectrophotometric measurements in the visible range,” Opt. Commun. 180, 191-198 (2000).
[CrossRef]

M. Elias, R. De La Rie, J. Delanay, and E. Charron, “Leveling of varnishes over rough substrates,” Opt. Commun. 266, 586-591 (2006).
[CrossRef]

Prog. Electromagn. Res. (1)

D. Liang, P. Xu, L. Tsang, Z. Gui, and K. S. Chen, “Electromagnetic scattering by rough surfaces with large heights and slopes with applications to microwave remote sensing of rough surface over layered media,” Prog. Electromagn. Res. 95, 199-218 (2009).
[CrossRef]

Other (2)

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

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

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Figures (14)

Fig. 1
Fig. 1

Geometry and notation.

Fig. 2
Fig. 2

Reflected flux f s = f r 0 , 1 as a function of the observation angle for different ratio h l in the case of diffuse lighting.

Fig. 3
Fig. 3

Reflected flux f r 1 , 0 as a function of the observation angle for different ratio h l in the case of diffuse lighting.

Fig. 4
Fig. 4

Total flux F r 01 + F t 01 and F r 10 + F t 10 for diffuse lighting as a function of the ratio h l .

Fig. 5
Fig. 5

Reflected flux f s as a function of the observation angle for a collimated incident light at θ i = 20 ° and for different ratio h l .

Fig. 6
Fig. 6

Total reflected flux F r 01 , transmitted flux F t 01 , and the sum F r 01 + F t 01 for a plane interface and a rough one with h l = 0.1 illuminated by collimated light at θ i = 20 ° as a function of the observation angle.

Fig. 7
Fig. 7

Total reflected flux F r 10 , transmitted flux F t 10 , and the sum F r 10 + F t 10 for a plane interface and a rough one with h l = 0.1 illuminated by collimated light at θ i = 20 ° as a function of the observation angle.

Fig. 8
Fig. 8

Bulk scattered flux f b as a function of the observation angle for different ratio h l in the case of diffuse lighting.

Fig. 9
Fig. 9

Sum of the total calculated surface reflected F s and bulk scattered F b flux in the cases of diffuse and collimated light.

Fig. 10
Fig. 10

Variation of the corrected bulk scattered flux relative to the plane case Δ f b corr as a function of the observation angle for different ratio h l in the case of diffuse lighting.

Fig. 11
Fig. 11

Bulk scattered flux f b as a function of the observation angle for different ratio h l in the case of collimated lighting at θ i = 20 ° .

Fig. 12
Fig. 12

Variation of the corrected bulk scattered flux relative to the plane case Δ f b corr as a function of the observation angle for different ratio h l in the case of collimated lighting at θ i = 20 ° .

Fig. 13
Fig. 13

Corrected bulk reflectance spectrum R b ( λ ) in backscattering configuration at θ i = 20 ° for different ratio h l .

Fig. 14
Fig. 14

Total reflectance spectrum R ( λ ) in backscattering configuration at θ i = 20 ° for different ratio h l .

Equations (31)

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P ( N ) δ Ω N = P ( θ N , ϕ N ) δ Ω N = δ s S .
P ( N ) = P ( θ N ) = 1 2 π ( l h ) 2 1 cos 3 θ N exp ( 1 2 ( l h ) 2 tan 2 θ N ) .
N = u r u i u r u i .
f r a b ( u r ) δ Ω r = F i R a b ( n a , n b , γ i ) P ( N ) δ Ω N ,
δ Ω N δ Ω r = 1 4 cos γ i ,
f r a b ( u r ) = F i R a b ( n a , n b , γ i ) 4 cos γ i P ( N ) .
ρ r a b ( u r , u i ) = R a b ( n a , n b , γ i ) 4 cos γ i P ( N ) ,
f r a b ( u r ) = Ω i ρ r a b ( u r , u i ) f i ( u i ) d Ω i .
N = n a u i n b u t n a u i n b u t
f t a b ( u t ) δ Ω t = F i T a b ( n a , n b , γ i ) P ( N ) δ Ω N ,
f t a b ( u t ) = F i ρ t a b ( u t , u i )
ρ t a b ( u t , u i ) = n b 2 cos γ t ( n b cos γ t n a cos γ i ) 2 T a b ( n a , n b , γ i ) P ( N ) ,
f t a b ( u t ) = Ω i ρ t a b ( u t , u i ) f i ( u i ) d Ω i .
d d τ f 1 ± ( μ , ϕ , τ ) = f 1 ± ( μ , ϕ , τ ) μ ± ω ( τ ) 4 π 0 2 π d ϕ 0 1 d μ f 1 ( μ , ϕ , τ ) μ p ( μ , ϕ , μ , ϕ , τ ) ,
f 1 ± ( μ , ϕ , τ ) = m = 0 ν max f 1 ± m ( μ , τ ) cos m ϕ .
p ( u , u , τ ) = l = 0 ν max p l ( τ ) P l ( cos γ ) ,
A l m ( τ ) = 0 1 [ f 1 + m ( μ , τ ) + ( 1 ) l + m f 1 m ( μ , τ ) ] P l m ( μ ) d μ μ ,
f 1 i + ( u ) = ρ t 01 ( u , u i ) F i .
f 1 i + ( u ) = Ω i ρ t 01 ( u , u i ) f i ( u i ) d Ω i ,
f 1 + ( μ , ϕ , 0 ) = f 1 i + ( μ , ϕ ) + μ = 0 1 ϕ = 0 2 π ρ r 10 ( μ , μ , ϕ , ϕ ) f 1 ( μ , ϕ , 0 ) d μ d ϕ .
ρ r 10 ( μ , μ , ϕ , ϕ ) = m = 0 ρ r 10 , m ( μ , μ ) cos m ( ϕ ϕ ) .
f 1 + m ( μ , 0 ) = f 1 i + m ( μ ) + μ = 0 1 ρ r 10 , m ( μ , μ ) f 1 m ( μ , 0 ) d μ ,
f 1 + m ( μ , 0 ) = f 1 t + m ( μ ) + R 10 ( n 1 , n 0 , μ ) f 1 m ( μ , 0 ) .
f 1 ( μ , ϕ , τ e ) = μ ϱ b π W τ e ,
W τ e = 0 2 π 0 1 f 1 + ( μ , ϕ , τ h ) d ϕ d μ = 2 π 0 1 f 1 + 0 ( μ , τ h ) d μ .
f 1 m ( μ , τ h ) = μ ϱ b π W τ h δ m , 0 .
f s ( u ) = ρ r 01 ( u , u 0 i ) F 0 i
f s ( u ) = Ω i ρ r 01 ( u , u i ) f i ( u i ) d Ω i ,
f b ( u ) = Ω i ρ t 10 ( u , u 1 ) f 1 ( u 1 , τ = 0 ) d Ω 1 .
Δ f b ( θ , h l ) = f b ( θ , h l ) f b ( θ , h l = 0 ) .
R s , b ( λ ) = π cos θ i f s , b ( λ )

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