Abstract

We present the first modeling of the light scattered by a paint layer in a bidirectional configuration. The studied medium is composed of small concentrated pigments embedded in an oil binder. The color is modulated by changing the number of paint layers, called glazes. The radiative transfer equation is established for incoherent light scattered by the pigments with use of a collimated illumination. The equation is solved by use of the auxiliary function method. This new method, applied here for the first time to a practical case, allows for exact computations of the scattered flux for any incident and collected directions. Spectroscopic and goniometric measurements are implemented in bidirectional and backscattered configurations. The excellent agreement between the measurement and the simulation validates the assumptions used for the glaze model and proves the effectiveness of the auxiliary function method.

© 2004 Optical Society of America

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References

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  2. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
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    [CrossRef]
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    [CrossRef]
  10. M. Elias, G. Elias, “Radiative transfer in inhomogeneous stratified media using the auxiliary function method,” J. Opt. Soc. Am. A 21, 580–589 (2004).
    [CrossRef]
  11. L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
    [CrossRef]
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    [CrossRef]
  14. E. R. De La Rie, “The influence of varnishes on the appearance of paintings,” Stud. Conserv. 32, 1–13 (1987).
    [CrossRef]
  15. R. Molenaar, J. J. ten Bosch, J. R. Zijp, “Determination of Kubelka-Munk scattering and absorption coefficients by diffuse illumination,” Appl. Opt. 38, 2068–2077 (1999).
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  16. M. Elias, 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]
  17. M. Elias, M. Menu, “Experimental characterization of a random metallic rough surface by spectrophometric measurements in the visible range,” Opt. Commun. 180, 191–198 (2000).
    [CrossRef]

2004 (3)

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

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

M. Elias, 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]

2003 (2)

2002 (1)

2001 (1)

M. Elias, M. Menu, “Characterisation of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229 (2001).
[CrossRef]

2000 (1)

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

1999 (1)

1998 (1)

1988 (1)

1987 (1)

E. R. De La Rie, “The influence of varnishes on the appearance of paintings,” Stud. Conserv. 32, 1–13 (1987).
[CrossRef]

1984 (1)

1971 (1)

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Andraud, C.

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

A. da Silva, M. Elias, C. Andraud, 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]

Chandrasekhar, S.

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

Charron, E.

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

da Silva, A.

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

A. da Silva, M. Elias, C. Andraud, 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]

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).
[CrossRef]

Dran, J. C.

L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
[CrossRef]

Elias, G.

Elias, M.

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

A. da Silva, M. Elias, C. Andraud, 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]

L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
[CrossRef]

M. Elias, G. Elias, “New and fast calculation for incoherent multiple scattering,” J. Opt. Soc. Am. A 19, 894–905 (2002).
[CrossRef]

M. Elias, M. Menu, “Characterisation of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229 (2001).
[CrossRef]

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

Gouesbet, G.

Jayaweera, K.

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Lafait, J.

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

A. da Silva, M. Elias, C. Andraud, 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]

Letouzan, N.

Maheu, B.

Menu, M.

M. Elias, M. Menu, “Characterisation of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229 (2001).
[CrossRef]

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

Molenaar, R.

Mudgett, P. S.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Panofski, E.

E. Panofski, Les Primitifs Flamands (Hazan, Paris, 1992).

Richards, L. W.

Salomon, J.

L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
[CrossRef]

Simonot, L.

M. Elias, 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]

L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
[CrossRef]

Stamnes, K.

Stout, B.

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

ten Bosch, J. J.

Tsay, S.-C.

Vargas, W. E.

Wiscombe, W.

Zijp, J. R.

Zobelli, A.

L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
[CrossRef]

Appl. Opt. (5)

J. Mod. Phys. (1)

A. da Silva, C. Andraud, E. Charron, B. Stout, J. Lafait, “Multiple light scattering in multilayered media: theory, experiments,” J. Mod. Phys. 51, 313–332 (2004).

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

J. Trace Microprobe Tech. (1)

L. Simonot, A. Zobelli, M. Elias, J. Salomon, J. C. Dran, “Pigment distribution in art glazes,” J. Trace Microprobe Tech. 21, 35–48 (2003).
[CrossRef]

Opt. Commun. (2)

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

Stud. Conserv. (1)

E. R. De La Rie, “The influence of varnishes on the appearance of paintings,” Stud. Conserv. 32, 1–13 (1987).
[CrossRef]

Surf. Eng. (1)

M. Elias, M. Menu, “Characterisation of surface states on patrimonial works of art,” Surf. Eng. 17, 225–229 (2001).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (2)

E. Panofski, Les Primitifs Flamands (Hazan, Paris, 1992).

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

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Figures (6)

Fig. 1
Fig. 1

Goniospectrophotometry in the bidirectional configuration. θ and ϕ are the zenithal and azimuthal angles, respectively; Ω is the solid angle.

Fig. 2
Fig. 2

Painting cross section under scanning electronic microscope for 13 glaze layers on a white background. Image segmentation to distinguish Ca (dark pixels) and Cr (clear pixels) repartition obtained by energy dispersive x-ray spectrometry.

Fig. 3
Fig. 3

Estimation of the coefficients k and s derived from comparison between AFM results and experiments in backscattering configuration (22°) for one glaze layer (thickness 12 μm) and 13 layers (“infinite” thickness) on a white background. Refractive index of the medium, n = 1.5.

Fig. 4
Fig. 4

Notations of the different fluxes for a glaze layer.

Fig. 5
Fig. 5

Reflectance spectra for (a) white, (b) yellow, and (c) blue base layers and for 1, 2, 7, and 13 glaze layers, in backscattering configuration with θ = 22°. Experimental results are represented by curves, and simulations are represented by squares.

Fig. 6
Fig. 6

Reflectance factor as a function of the collection angle θ f , in the incident plane, for incidence angles θ i equal to (a) 30° and (b) 60° for 1, 2, and 13 glaze layers. Experimental results are represented by curves; and simulations are represented by squares.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

dw±μ, τdτ=±w±μ, τ|μ|±q4πμ1=01w+μ1, τ+w-μ1, τ2πdμ1μ1±q4πW+τμ0.
w+μ, τ=w+μ, 0exp-τ/μ+q2s=0τ tsexps-τ/μds, w-μ, τ=w-μ, hexpτ-h/μ+q2s=τh tsexpτ-s/μds.
fτ=q2s=0h Hτ, stsds+μ=01w+μ, 0exp-τ/μ+w-μ, hexpτ-h/μdμμ,
w+μ, 0=Rμw-μ, 0,
w-μ, h=ρπ Bμ,
fτ=q2s=0hHτ, s+Uτ, s+2ρ1-ρK MτMstsds+ρ1-ρKTμiexp-h/μ0π Mτ
Hτ, s=μ=01exp-|τ-s|/μdμμ, Uτ, s=μ=01 Rμexp-s+τ/μdμμ, Mτ=μ=01expτ-h/μ+Rμexp-h+τ/μdμ, K=2 μ=01 Rμμ exp-2h/μdμ.
w-μ, 0=ρπ Bμ exp-h/μ+q2s=0h tsexp-s/μds, with B=Tμiexp-h/μ0+πq s=0h ts Msds1-ρK.
ρμi, μ=ρmeasμin2Tair/paintμiTpaint/airμ+ρmeasμin2rpaint/air,
rpaint/air=2 01 μRpaint/airμdμ.

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