Abstract

In a previous paper, the special visual appearance of art glazes was explained using the auxiliary function method (AFM) for solving the radiative transfer equation. Glazes are made of low concentrated colored scattering centers embedded in a transparent medium and the artist modulates the color by varying the number of glaze layers. A simple model of glazes and the new solving method have both been validated by comparison between flux measurements and modeling. The color of art glazes is analyzed here, and the study shows a spectacular maximum of saturation (purity) of the color that is never reached, to the best of our knowledge, with other techniques, such as pigment mixtures. This phenomenon is explained once more using the AFM that allows separation of the different contributions to the scattered fluxes. It is then shown that, on the one hand, single scattering never induces a maximum of saturation. On the other hand, multiple scattering has a typical increasing and decreasing behavior with an increasing number of glaze layers and thus participates to the maximum of saturation, just as the scattering by the diffuse base layer. A comparison between glazes and pigment mixtures, where the proportion of colored pigments with white pigments varies instead of the number of layers, shows that this maximum of saturation is much smaller with the second technique. To the best of our knowledge, we present a new development of the AFM that allows separation of the different origins of light scattering. We also show that it is possible to determine the optical properties of the scattering centers and of the base layer to create the required visual effect of a scattering medium.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. L. Simonot, M. Elias, and E. Charron, "Special visual effect of art glazes explained by the radiative transfer equation," Appl. Opt. 43, 2580-2587 (2004).
    [CrossRef] [PubMed]
  4. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. 12, 593-601 (1931).
  5. Z. C. Orel, M. K. Gunde, and B. Orel, "Application of the Kubelka-Munk theory for the determination of the optical properties of solar absorbing," Prog. Organ. Coatings , 30, 59-66 (1997).
    [CrossRef]
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    [CrossRef]
  10. D. R. Duncan, "The colour of pigment mixtures," Proc. Phys. Soc. 52, 390-400 (1940).
    [CrossRef]
  11. M. Elias and M. Menu, "Characterization of surface states on patrimonial works of art," Surf. Eng. 17, 225-229 (2001).
    [CrossRef]
  12. 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]
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2004 (2)

2003 (2)

2002 (1)

2001 (1)

M. Elias and M. Menu, "Characterization of surface states on patrimonial works of art," Surf. Eng. 17, 225-229 (2001).
[CrossRef]

1997 (1)

Z. C. Orel, M. K. Gunde, and B. Orel, "Application of the Kubelka-Munk theory for the determination of the optical properties of solar absorbing," Prog. Organ. Coatings , 30, 59-66 (1997).
[CrossRef]

1988 (1)

1984 (1)

1971 (1)

1940 (1)

D. R. Duncan, "The colour of pigment mixtures," Proc. Phys. Soc. 52, 390-400 (1940).
[CrossRef]

1931 (1)

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

Andraud, C.

Charron, E.

da Silva, A.

Dran, J. C.

L. Simonot, A. Zobelli, M. Elias, J. Salomon, and J. C. Dran, "Pigment distribution in art glazes," J. Trace Microprobe Techn. 21, 35-48 (2003).
[CrossRef]

Duncan, D. R.

D. R. Duncan, "The colour of pigment mixtures," Proc. Phys. Soc. 52, 390-400 (1940).
[CrossRef]

Elias, G.

Elias, M.

Gouesbet, G.

Gunde, M. K.

Z. C. Orel, M. K. Gunde, and B. Orel, "Application of the Kubelka-Munk theory for the determination of the optical properties of solar absorbing," Prog. Organ. Coatings , 30, 59-66 (1997).
[CrossRef]

Jayaweera, K.

Kubelka, P.

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

Lafait, J.

Letouzan, N.

Maheu, B.

Menu, M.

M. Elias and M. Menu, "Characterization of surface states on patrimonial works of art," Surf. Eng. 17, 225-229 (2001).
[CrossRef]

Mudgett, P. S.

Munk, F.

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

Orel, B.

Z. C. Orel, M. K. Gunde, and B. Orel, "Application of the Kubelka-Munk theory for the determination of the optical properties of solar absorbing," Prog. Organ. Coatings , 30, 59-66 (1997).
[CrossRef]

Orel, Z. C.

Z. C. Orel, M. K. Gunde, and B. Orel, "Application of the Kubelka-Munk theory for the determination of the optical properties of solar absorbing," Prog. Organ. Coatings , 30, 59-66 (1997).
[CrossRef]

Richards, L. W.

Salomon, J.

L. Simonot, A. Zobelli, M. Elias, J. Salomon, and J. C. Dran, "Pigment distribution in art glazes," J. Trace Microprobe Techn. 21, 35-48 (2003).
[CrossRef]

Simonot, L.

L. Simonot, M. Elias, and E. Charron, "Special visual effect of art glazes explained by the radiative transfer equation," Appl. Opt. 43, 2580-2587 (2004).
[CrossRef] [PubMed]

L. Simonot, A. Zobelli, M. Elias, J. Salomon, and J. C. Dran, "Pigment distribution in art glazes," J. Trace Microprobe Techn. 21, 35-48 (2003).
[CrossRef]

Stamnes, K.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Tsay, S. Chee

Wiscombe, W.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Zobelli, A.

L. Simonot, A. Zobelli, M. Elias, J. Salomon, and J. C. Dran, "Pigment distribution in art glazes," J. Trace Microprobe Techn. 21, 35-48 (2003).
[CrossRef]

Appl. Opt. (4)

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

J. Trace Microprobe Techn. (1)

L. Simonot, A. Zobelli, M. Elias, J. Salomon, and J. C. Dran, "Pigment distribution in art glazes," J. Trace Microprobe Techn. 21, 35-48 (2003).
[CrossRef]

Proc. Phys. Soc. (1)

D. R. Duncan, "The colour of pigment mixtures," Proc. Phys. Soc. 52, 390-400 (1940).
[CrossRef]

Prog. Organ. Coatings (1)

Z. C. Orel, M. K. Gunde, and B. Orel, "Application of the Kubelka-Munk theory for the determination of the optical properties of solar absorbing," Prog. Organ. Coatings , 30, 59-66 (1997).
[CrossRef]

Surf. Eng. (1)

M. Elias and M. Menu, "Characterization of surface states on patrimonial works of art," Surf. Eng. 17, 225-229 (2001).
[CrossRef]

Z. Tech. Phys. (1)

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

Other (1)

G. Wyszecki and W. S. Stiles, Color science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

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Figures (12)

Fig. 1
Fig. 1

Notations of the fluxes inside and outside the glaze layer.

Fig. 2
Fig. 2

Flux scattered by the base layer taking part in w B - ( μ , 0 ) .

Fig. 3
Fig. 3

Flux due to multiple scattering taking part in w M S - ( μ , 0 ) .

Fig. 4
Fig. 4

Flux due to single scattering taking part in w S S - ( μ , 0 ) .

Fig. 5
Fig. 5

Glaze layers on a white base layer. (a) Spectral variations of the total scattered flux for different numbers of glaze layers (1, 2, 3, 4, 5, 7, 9, 13, and ∞). (b) Colorimetric variations in the chroma C*/ lightness L* plane of the CIE-Lab colorimetric space as a function of the glaze layer number for the total scattered flux.

Fig. 6
Fig. 6

Spectral variations of each contribution for different numbers of glaze layers (1, 2, 3, 4, 5, 7, 9, 13, and ∞). (a) Flux scattered by the background. (b) Single scattering. (c) Multiple scattering.

Fig. 7
Fig. 7

Different contributions of the total diffuse flux at 509 nm as a function of the layer number.

Fig. 8
Fig. 8

Colorimetric variations in the chroma C*∕lightness L* plane of the CIE-Lab space as a function of the glaze layer number for the different contributions. (a) Flux scattered by the background. (b) Single scattering. (c) Multiple scattering.

Fig. 9
Fig. 9

Pigment mixture. (a) Spectral variations of the total diffuse flux for mixtures with different concentrations of green pigment (c = 0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1). (b) Colorimetric variations in the chroma C*∕lightness L* plane of the CIE-Lab colorimetric space as a function of the green pigment concentration for the total diffuse flux.

Fig. 10
Fig. 10

Spectral variations of each contribution for different concentrations of pigment mixture (c = 0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1). (a) Single scattering. (b) Multiple scattering.

Fig. 11
Fig. 11

Colorimetric variations in the chroma C*∕lightness L* plane of the CIE-Lab space as a function of the green pigment concentration for each contribution. (a) Single scattering. (b) Multiple scattering.

Fig. 12
Fig. 12

Comparison of colorimetric variations in the chroma C*∕lightness L* plane of the CIE-Lab space for the glaze and the pigment mixture and for the total flux.

Equations (23)

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d w ± ( μ , τ ) d τ = w ± ( μ , τ ) | μ | ± q 4 π μ 1 = 0 1 [ w + ( μ 1 , τ ) + w ( μ 1 , τ ) ] 2 π d μ 1 μ 1 ± q 4 π W + ( τ ) μ o ,
d w ± ( μ , τ ) d τ = w ± ( μ , τ ) | μ | ± q 2 [ f ( τ ) + g ( τ ) ]
w ( μ , 0 ) = ρ π A μ exp ( h / μ ) + q 2 s = 0 h t ( s ) × exp ( s / μ ) d s ,
A = T ( μ i ) exp ( h / μ o ) + π q s = 0 h t ( s ) B ( s ) d s 1 ρ K ,
B ( τ ) = μ = 0 1 [ exp ( ( τ h ) / μ ) + R ( μ ) exp ( ( h + τ ) / μ ) ]  d μ ,
K = 2 μ = 0 1 R ( μ ) exp ( 2 h / μ ) μ d μ .
w ( μ , 0 ) = w B     ( μ , 0 ) + w M S               ( μ , 0 ) + w S S             ( μ , 0 ) .
C ( x ) = μ = 0 1 R ( μ ) exp ( x / μ ) d μ ,
D ( x ) = μ = 0 1 R ( μ ) exp ( x / μ ) μ d μ ,
E ( x ) = μ = 0 1 exp ( x / μ ) d μ .
w B     ( μ , 0 ) = Λ T ( μ i ) exp ( h / μ o ) 2 π μ exp ( h / μ )
Λ = 2 ρ 1 2 ρ D ( 2 h ) .
w MS - ( μ , 0 ) = q 2 0 h [ exp ( s / μ ) + Λ μ [ C ( h + τ ) + E ( h τ ) ] exp ( h / μ ) ] f ( s ) d s .
w SS - ( μ , 0 ) = q 2 0 h [ exp ( s / μ ) + Λ μ [ C ( h + τ ) + E ( h τ ) ] exp ( h / μ ) ] g ( s ) d s .
k m = c k g + ( 1 c ) k w ,
s m = c s g + ( 1 c ) s w ,
X = K S ( λ ) x ¯ ( λ ) R ( λ ) d λ ,
Y = K S ( λ ) y ¯ ( λ ) R ( λ ) d λ ,
Z = K S ( λ ) z ¯ ( λ ) R ( λ ) d λ ,
K = 100 S ( λ ) y ¯ ( λ ) d λ .
L * = f ( Y Y B ) , a * = 500 116 [ f ( X X B ) f ( Y Y B ) ] , b * = 200 116 [ f ( Y Y B ) f ( Z Z B ) ] ,
f ( A ) = 116 A 1 / 3 16         if        f ( A ) 8 ,
f ( A ) = ( 29 3 ) 3 A         if        f ( A ) 8 ( Pauli correction ) .

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