Abstract

The diffuse reflectance spectra and the trichromatic coordinates of diffusing stratified paints are modeled. Each layer contains its own pigments, and their optical properties are first determined from experiments. The radiative transfer equation is then solved by the auxiliary function method for modeling the total light scattered by the stratified systems. The results are in good agreement with experimental spectra and validate the modeling. The calculations are then applied on the same stratified systems to study the influence of the observation angle in a bidirectional configuration and to study the influence of the thickness of the layers in a given configuration. In both cases, the reflectance spectra and the trichromatic coordinates are calculated and compared.

© 2007 Optical Society of America

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References

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2007

2006

2004

2003

2002

2001

J. J. Joshi, D. B. Vaidya, and H. S. Shah, "Application of multiflux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films," Color Res. Appl. 26, 234-245 (2001).
[CrossRef]

1988

1984

1971

1954

1948

Andraud, C.

Chandrasekhar, S.

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

Charron, E.

da Silva, A.

De La Rie, E. R.

E. R. De La Rie, "The influence of varnishes on the appearance of paintings," Stud. Conserv., 32, 1-13 (1987).

Elias, G.

Elias, M.

Frigerio, J.-M.

Gouesbet, G.

Hebert, M.

Hersch, R. D.

Jayaweera, K.

Joshi, J. J.

J. J. Joshi, D. B. Vaidya, and H. S. Shah, "Application of multiflux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films," Color Res. Appl. 26, 234-245 (2001).
[CrossRef]

Kubelka, P.

Lafait, J.

Letouzan, J. N.

Magnain, C.

Maheu, B.

Mudgett, P. S.

Richards, L. W.

Shah, H. S.

J. J. Joshi, D. B. Vaidya, and H. S. Shah, "Application of multiflux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films," Color Res. Appl. 26, 234-245 (2001).
[CrossRef]

Simonot, L.

Stamnes, K.

Tsay, S. C.

Vaidya, D. B.

J. J. Joshi, D. B. Vaidya, and H. S. Shah, "Application of multiflux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films," Color Res. Appl. 26, 234-245 (2001).
[CrossRef]

Wiscombe, W.

Appl. Opt.

Color Res. Appl.

J. J. Joshi, D. B. Vaidya, and H. S. Shah, "Application of multiflux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films," Color Res. Appl. 26, 234-245 (2001).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

E. R. De La Rie, "The influence of varnishes on the appearance of paintings," Stud. Conserv., 32, 1-13 (1987).

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

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Figures (10)

Fig. 1
Fig. 1

Definition and notation of the flux for the RTE.

Fig. 2
Fig. 2

Absorption and scattering coefficients of the viridian green pigment.

Fig. 3
Fig. 3

Reflectance spectra of the viridian green on ultramarine blue dark sample and of each single layer.

Fig. 4
Fig. 4

Experimental (solid curve) and modeled (dotted curve) reflectance spectra for (a) vermilion on blue, (b) green on blue, and (c) blue on yellow paintings according to model 1 (perfect diffusing background) and to model 2 (use of the experimental reflectance spectrum of the background).

Fig. 5
Fig. 5

Modeling of the reflectance spectra for different observation angles in the case of (a) vermilion on blue, (b) green on blue, and (c) blue on yellow paintings.

Fig. 6
Fig. 6

Lightness as a function of the observation angle for vermilion on blue, green on blue, and blue on yellow paintings.

Fig. 7
Fig. 7

Chromatic coordinates ( a * , b * ) for different observation angles in the case of green on blue, vermilion on blue, and blue on yellow paintings.

Fig. 8
Fig. 8

Reflectance spectra for different thicknesses of the upper layer in the case of (a) green on blue and (b) blue on yellow paintings.

Fig. 9
Fig. 9

Lightness as a function of the upper layer thickness in the case of (a) green on blue and (b) blue on yellow paintings.

Fig. 10
Fig. 10

Chromatic coordinates ( a * , b * ) for different thicknesses of the upper layer in the case of (a) green on blue and (b) blue on yellow paintings ( ) and deduced from experiment ( ) .

Tables (1)

Tables Icon

Table 1 Standard Deviation between the Experimental and the Modeled Reflectance Spectra for Nine Samples, According to the Background Model (Model 1: Perfect Diffusing Background and Model 2: Use of the Experimental Reflectance Spectrum of the Background)

Equations (17)

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d f ± ( u , z ) d z = ( k + s ) f ± ( u , z ) cos θ ± s 4 π 4 π f ( u 1 , z ) cos θ 1 p ( u , u 1 ) 2 π sin θ 1 d θ 1 ± s 4 π F ± ( z ) p ( u , u 0 ) ,
f ( u 1 , z ) = f + ( u 1 , z ) + f ( u 1 , z ) .
d f ± ( μ , τ ) d τ = f ± ( μ , τ ) μ ± ϖ 4 π μ μ [ μ 1 = 0 1 f + ( μ 1 , τ ) + f ( μ 1 , τ ) μ 1 2 π d μ 1 + F ± ( τ ) ] .
A ( τ ) = 0 1 f + ( μ 1 , τ ) + f ( μ 1 , τ ) μ 1 d μ 1 ,
A ( τ ) = 0 1 f + ( u 1 , τ ) + f ( u 1 , τ ) μ 1 d μ 1 .
d f ± ( μ , τ ) d τ = f ± ( μ , τ ) μ ± ϖ 2 t ( τ ) .
f + ( μ , τ ) = f + ( μ , 0 ) exp ( τ μ ) + ϖ 2 s = 0 τ t ( s ) exp ( s τ μ ) d s ,
f ( μ , τ ) = f ( μ , τ h ) exp ( τ τ h μ ) + ϖ 2 s = τ τ h t ( s ) exp ( τ s μ ) d s ,
A ( τ ) = ϖ 2 s = 0 τ h H ( τ , s ) t ( s ) d s + μ = 0 1 [ f + ( μ , 0 ) exp ( τ μ ) + f ( μ , τ h ) exp ( τ τ h μ ) ] d μ μ ,
H ( τ , s ) = μ = 0 1 exp ( τ s μ ) d μ μ
f ( μ , τ h ) = R p a i n t b a c k g r o u n d ( μ ) f + ( μ , τ h ) + F + ( τ h ) .
f ( μ , 0 ) = R p a i n t b a c k g r o u n d ( μ ) [ f + ( μ , τ h ) + F + ( τ h ) ] exp ( τ h μ ) + ϖ 2 s = 0 τ h t ( s ) exp ( s μ ) d s
or f ( μ , 0 ) = ρ g π B μ exp ( τ h μ ) + ϖ 2 s = 0 τ h t ( s ) exp ( s μ ) d s .
ρ f ( μ i , μ f ) = π F 0 f f ( μ f ) μ f ,
ρ f ( μ i , μ f ) = π F 0 T ( μ ) n 2 μ f ( μ , 0 ) .
ρ = 1 ρ g ( a b coth ( b S h ) ) a + b coth ( b S h ) ρ g ; a = K + S S
b = a 2 1 .

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