Abstract

An analytic expression is derived for the angle-dependent polarimetric bidirectional reflectance distribution function (BRDF) of a glossy coating consisting of a Lambertian reflector covered by a specular dielectric layer. This model conforms to all the requirements of a real BRDF and explains the angular dependence of the polarized BRDF measurements of a glossy paint. Two common assumptions about the properties of paint coatings are shown to be incorrect: Multiple scattering by paint pigments will seldom completely depolarize the reflected energy, and, even when it does, the first surface and the volume scattering cannot be completely separated based on polarization alone. Guidelines for using the model to separate first-surface and volume scattering components are proposed.

© 1996 Optical Society of America

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References

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  1. D. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975), pp. 420–438.
  2. R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graphics (Proc. SIGGRAPH), 15, 307–316 (1981).
    [CrossRef]
  3. J. R. Maxwell, S. F. Weiner, “Polarized emittance (Vol. I): Polarized bidirectional reflectance with Lambertian or non-Lambertian diffuse components,” Rep. CR-154 (Ballistics Vulnerability/Lethality Division of the Survivability/Lethality Analysis Directorate, Aberdeen Proving Ground, Md., formerly U.S. Army Ballistic Research Laboratories, 1975) (National Technical Information Service, accession number AD-782 178).
  4. D. P. Greenberg, “Light reflection models for computer graphics,” Science 244, 166–173 (1989).
    [CrossRef] [PubMed]
  5. K. E. Torrance, E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,”J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  6. T. S. Trowbridge, K. P. Reitz, “Average irregularity representation of a rough surface for ray reflection,”J. Opt. Soc. Am. 65, 531–536 (1975).
    [CrossRef]
  7. M. Minnaert, “The reciprocity principle in lunar photometry,” Astrophys. J. 93, 403–410 (1941).
    [CrossRef]
  8. B. Hapke, “A theoretical photometric function for the lunar surface,”J. Geophys. Res. 68, 4571–4586 (1963).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 42.
  10. K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, New York, 1979), p. 35.

1989

D. P. Greenberg, “Light reflection models for computer graphics,” Science 244, 166–173 (1989).
[CrossRef] [PubMed]

1981

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graphics (Proc. SIGGRAPH), 15, 307–316 (1981).
[CrossRef]

1975

1967

1963

B. Hapke, “A theoretical photometric function for the lunar surface,”J. Geophys. Res. 68, 4571–4586 (1963).
[CrossRef]

1941

M. Minnaert, “The reciprocity principle in lunar photometry,” Astrophys. J. 93, 403–410 (1941).
[CrossRef]

Bibby, J. M.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, New York, 1979), p. 35.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 42.

Cook, R. L.

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graphics (Proc. SIGGRAPH), 15, 307–316 (1981).
[CrossRef]

Greenberg, D. P.

D. P. Greenberg, “Light reflection models for computer graphics,” Science 244, 166–173 (1989).
[CrossRef] [PubMed]

Hapke, B.

B. Hapke, “A theoretical photometric function for the lunar surface,”J. Geophys. Res. 68, 4571–4586 (1963).
[CrossRef]

Judd, D.

D. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975), pp. 420–438.

Kent, J. T.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, New York, 1979), p. 35.

Mardia, K. V.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, New York, 1979), p. 35.

Maxwell, J. R.

J. R. Maxwell, S. F. Weiner, “Polarized emittance (Vol. I): Polarized bidirectional reflectance with Lambertian or non-Lambertian diffuse components,” Rep. CR-154 (Ballistics Vulnerability/Lethality Division of the Survivability/Lethality Analysis Directorate, Aberdeen Proving Ground, Md., formerly U.S. Army Ballistic Research Laboratories, 1975) (National Technical Information Service, accession number AD-782 178).

Minnaert, M.

M. Minnaert, “The reciprocity principle in lunar photometry,” Astrophys. J. 93, 403–410 (1941).
[CrossRef]

Reitz, K. P.

Sparrow, E. M.

Torrance, K. E.

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graphics (Proc. SIGGRAPH), 15, 307–316 (1981).
[CrossRef]

K. E. Torrance, E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,”J. Opt. Soc. Am. 57, 1105–1114 (1967).
[CrossRef]

Trowbridge, T. S.

Weiner, S. F.

J. R. Maxwell, S. F. Weiner, “Polarized emittance (Vol. I): Polarized bidirectional reflectance with Lambertian or non-Lambertian diffuse components,” Rep. CR-154 (Ballistics Vulnerability/Lethality Division of the Survivability/Lethality Analysis Directorate, Aberdeen Proving Ground, Md., formerly U.S. Army Ballistic Research Laboratories, 1975) (National Technical Information Service, accession number AD-782 178).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 42.

Wyszecki, G.

D. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975), pp. 420–438.

Astrophys. J.

M. Minnaert, “The reciprocity principle in lunar photometry,” Astrophys. J. 93, 403–410 (1941).
[CrossRef]

Comput. Graphics (Proc. SIGGRAPH)

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graphics (Proc. SIGGRAPH), 15, 307–316 (1981).
[CrossRef]

J. Geophys. Res.

B. Hapke, “A theoretical photometric function for the lunar surface,”J. Geophys. Res. 68, 4571–4586 (1963).
[CrossRef]

J. Opt. Soc. Am.

Science

D. P. Greenberg, “Light reflection models for computer graphics,” Science 244, 166–173 (1989).
[CrossRef] [PubMed]

Other

D. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975), pp. 420–438.

J. R. Maxwell, S. F. Weiner, “Polarized emittance (Vol. I): Polarized bidirectional reflectance with Lambertian or non-Lambertian diffuse components,” Rep. CR-154 (Ballistics Vulnerability/Lethality Division of the Survivability/Lethality Analysis Directorate, Aberdeen Proving Ground, Md., formerly U.S. Army Ballistic Research Laboratories, 1975) (National Technical Information Service, accession number AD-782 178).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 42.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, New York, 1979), p. 35.

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Figures (4)

Fig. 1
Fig. 1

Reflection geometry for the BRDF derivation showing the multiple reflections between the top surface and the Lambertian substrate.

Fig. 2
Fig. 2

Zero-bistatic BRDF computed using the model with n = 1.55 and ρ0 = 0.75. The three curves correspond to illumination and receiver polarization combinations parallel (p) and perpendicular (s) to the plane of incidence. The p, s and s, p calculations are identical.

Fig. 3
Fig. 3

Near-zero-bistatic measurements of a glossy, white paint (solid curve) and model calculations for p, p (+), s, s (⃝), and p, s (×) polarization combinations. The measurements have an accuracy of ±5% of the value and a precision of 10−7 sr−1.

Fig. 4
Fig. 4

Comparison of sample reflectance data (⃝) and the best-fit Fresnel function for the polarization parallel to the plane of incidence (solid curve). The measurements have an accuracy of ±5% of the value. The refractive index value of n = 1.50 obtained from this fit was used in calculating the theoretical values plotted in Fig. 3.

Equations (28)

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R s ( θ ) = sin 2 ( θ θ * ) sin 2 ( θ + θ * ) ,
R p ( θ ) = tan 2 ( θ θ * ) tan 2 ( θ + θ * ) ,
T ( θ ) = 1 R ( θ ) ,
R ( θ r ) = R * ( θ r * ) ,
T ( θ r ) = T * ( θ r * )
sin θ r = n sin θ r * .
L spec ( θ i ) = L i p ( Ω i ) R p ( θ i ) ,
L 1 ( Ω d ) ( cos θ d ) d Ω d = T p ( θ i ) L i p ( Ω i ) ( cos θ i ) d Ω i ,
L 2 = ρ 0 π L 1 ( Ω d ) ( cos θ d ) d Ω d .
L 2 = ρ 0 π T p ( θ i ) L i p ( Ω i ) ( cos θ i ) d Ω i ,
L 3 ( Ω r ) = T ( θ r * ) L 2 ( cos θ r * ) d Ω r * ( cos θ r ) d Ω r = 1 n 2 T ( θ r * ) L 2 ,
L 3 ( θ r ) = L 2 2 n 2 [ T p * ( θ r * ) + T s * ( θ r * ) ] .
L 2 R ( θ d ) = L 2 R s + p * ( θ d ) ,
E 2 = L 2 R ( θ d ) ( cos θ d ) d Ω d = L 2 R s + p * ( θ d ) ( cos θ d ) d Ω d .
L 4 = ρ 0 π E 2 .
L 5 ( θ r ) = L 4 2 n 2 [ T p * ( θ r * ) + T s * ( θ r * ) ] ,
L 5 ( θ r ) = ρ 0 2 π n 2 [ T p * ( θ r * ) + T s * ( θ r * ) ] × L 2 R s + p * ( θ d ) ( cos θ d ) d Ω d .
L diff ( θ r ) = L 3 ( θ r ) + L 5 ( θ r ) + L 7 ( θ r ) + .
L diff ( θ r ) = 1 2 n 2 L 2 [ T p * ( θ r * ) + T s * ( θ r * ) ] × ( 1 + x + x 2 + ) ,
x = ρ 0 π R s + p * ( θ d ) ( cos θ d ) d Ω d .
n = 0 x n = 1 1 x .
L diff ( θ r ) = [ T p * ( θ r * ) + T s * ( θ r * ) ] T p ( θ i ) L i p ( Ω i ) ( cos θ i ) d Ω i 2 n 2 [ π ρ 0 R s + p * ( θ d ) ( cos θ d ) d Ω d ] .
f diff ( Ω i , Ω r ) L diff ( θ r ) L i p ( Ω i ) ( cos θ i ) d Ω i .
L i p ( Ω r ) = L 0 δ ( θ r θ i ) δ ( ϕ r ϕ i + π ) cos θ r sin θ r .
f p p ( Ω i , Ω r ) = R p ( θ i ) δ ( θ r θ i ) δ ( ϕ r ϕ i + π ) cos θ r sin θ r + T p ( θ i ) T p ( θ r ) 2 n 2 [ π ρ 0 R s + p * ( θ d ) ( cos θ d ) d Ω d ] ,
f p s ( Ω i , Ω r ) = T p ( θ i ) T s ( θ r ) 2 n 2 [ π ρ 0 R s + p * ( θ d ) ( cos θ d ) d Ω d ] ,
ρ 0 1 ρ 0 π R s + p * ( θ d ) ( cos θ d ) d Ω d ,
f p p ( Ω i , Ω r ) + f s s ( Ω i , Ω r ) f p s ( Ω i , Ω r ) f s p ( Ω i , Ω r ) = [ R p ( θ i ) + R s ( θ i ) ] δ ( θ r θ i ) δ ( ϕ r ϕ i + π ) cos θ r sin θ r + [ T p ( θ r ) T s ( θ r ) ] 2 2 n 2 [ π ρ 0 R s + p * ( θ d ) ( cos θ d ) d Ω d ] .

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