Abstract

In this paper, we present a model for BRDF that can be developed from a limited set of experimentally observed data, and which then can be used for predictive purposes in scene-generation or sensor-performance applications. The model is physics based and can be as detailed as desired, depending on the scope of experimental data available. Basic input parameters required are the complex refractive index of the material, or the directional hemispherical reflectivity (DHR) for s- and p-polarized radiation. At least one BRDF measurement is needed to determine the angular spread function. Incorporating BRDF measurements at several angles of incidence into the model yields better accuracy for describing behavior such as forward scatter, depolarization, the participation of volumetric and surface-scattering mechanisms in layered surfaces, diffuse-scatter coherence properties, and narrowing of the angular spread function which provides an indication of optical depth. Illustrations and demonstrations of the methodology are drawn from data sets measured on bead-blasted aluminum, automotive paint incorporating a clear-coat layer, and green low-gloss vehicle paint.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  3. E. L. Church and P. Z. Takacs, “Surface Scattering,” in Handbook of Optics, M. Bass, E. W. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 7.1–7.14.
  4. J. C. Stover, Optical Scattering Measurement and Analysis (McGraw-Hill, New York, 1990).
  5. J.-J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchoff’s law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
    [Crossref]
  6. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28(6), 1121–1138 (2011).
    [Crossref] [PubMed]
  7. M. T. Eismann, Hyperspectral Remote Sensing, SPIE Press, Bellingham, Washington (2012).
  8. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
    [Crossref]
  9. I. G. E. Renhorn, T. Hallberg, and G. D. Boreman, “Efficient polarimetric BRDF model,” Opt. Express 23(24), 31253–31273 (2015).
    [Crossref] [PubMed]
  10. M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info . Accessed on 2017–12–05.

2015 (1)

2011 (1)

2002 (1)

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

1998 (1)

Boreman, G. D.

Choi, N.

Greffet, J.-J.

Hallberg, T.

Harvey, J. E.

Krywonos, A.

Meier, S. R.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

Nieto-Vesperinas, M.

Priest, R. G.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

Renhorn, I. G. E.

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

Opt. Express (1)

Other (6)

M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info . Accessed on 2017–12–05.

M. T. Eismann, Hyperspectral Remote Sensing, SPIE Press, Bellingham, Washington (2012).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

E. L. Church and P. Z. Takacs, “Surface Scattering,” in Handbook of Optics, M. Bass, E. W. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 7.1–7.14.

J. C. Stover, Optical Scattering Measurement and Analysis (McGraw-Hill, New York, 1990).

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Figures (11)

Fig. 1
Fig. 1 The attenuation due to passing through the dielectric surface twice is illustrated with no depolarization in the diffuse scattering shown in A) and complete depolarization shown in B). Red is s-polarized, and blue is p-polarized radiation. The index of refraction was set to n = 1.5.
Fig. 2
Fig. 2 The generalized Gaussian distribution A) and the generalized Lorentzian distribution B) are shown. The scattering parameter is ρ ^ = 0.2 and the shape parameter is set to ν = (1,2,3).
Fig. 3
Fig. 3 Illustration of the similarity of Fresnel equation and DHR simulations for a diffuse paint.
Fig. 4
Fig. 4 BRDF measurements of bead-blasted Aluminum and initial model predictions based on limited amount of data. The difference between s-polarized (red) and p-polarized (blue) radiation is small. Data is missing ± 2 degrees around the angle of incidence due to obscuration effects.
Fig. 5
Fig. 5 BRDF measurements of bead-blasted Aluminum and model predictions based on initial data but a small correction for increased forward scattering is introduced.
Fig. 6
Fig. 6 BRDF measurements and model predictions at a logarithmic scale. The diffuse scattering is here assumed to be Lambertian.
Fig. 7
Fig. 7 Comparison of model with adjusted diffuse scattering parameters to observed polarimetric BDRF measurements of bead-blasted aluminum at the angle of incidence of 60 degrees. Different s-polarized and p-polarized diffuse scatter parameters has been added to the model. Red is s-polarized, and blue is p-polarized.
Fig. 8
Fig. 8 Angular spread function parameters are determined from scaled comparison to the BRDF measurement at an angle of incidence of 5 degrees.
Fig. 9
Fig. 9 Polarimetric BRDF measurements and initial model predictions. Red is s-polarized and blue is p-polarized radiation.
Fig. 10
Fig. 10 Comparison of measured and predicted BRDF curves at a log-scale at an angle of incidence of 60 degrees. Parameters have been fine tuned to fit observations.
Fig. 11
Fig. 11 BRDF measurements of red A) and green B) car paint at 633 nm. Observe the much higher diffuse scattering level of the red paint. The diffuse distribution was modelled using the GLorentzian distribution.

Equations (28)

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α=sin( θ )cos( φ )
β=sin( θ )sin( φ )
γ=cos( θ ),
L r ( α o , β o ;λ )= f BRDF ( α i , α o , β o ;λ ) L i ( α i ;λ ) γ i dΩ,
f BRDF pol ( α i , α o , β o ;λ )= types a type pol H type pol ( α i , α o , β o ;λ ) F type pol ( α i , α o , β o ;λ ) G type ( α i , α o , β o ;λ ),
F Diffuse =1.
F Fresnel s ( α i , α o , β o ;λ )= R p ( β o 2 ;λ ) R p ( 0;λ ) R s ( α i + α o 2 ;λ )
F Fresnel p ( α i , α o , β o ;λ )= R s ( β o 2 ;λ ) R s ( 0;λ ) R p ( α i + α o 2 ;λ )
R s ( α,n,k )= | ( nik ) 1 α 2 ( nik ) 2 1 α 2 ( nik ) 1 α 2 ( nik ) 2 + 1 α 2 | 2
R p ( α,n,k )= | ( nik ) 1 α 2 1 α 2 ( nik ) 2 ( nik ) 1 α 2 + 1 α 2 ( nik ) 2 | 2 .
H Surface =1.
H Fresnel s ( α i , α o , β o ;λ )=( 1 R s ( α i ;λ ) )( ( 1 a 2 )( 1 R s ( α o ;λ ) )+ a 2 ( 1 R p ( α o ;λ ) ) )
H Fresnel p ( α i , α o , β o ;λ )=( 1 R p ( α i ;λ ) )( ( 1 a 2 )( 1 R p ( α o ;λ ) )+ a 2 ( 1 R s ( α o ;λ ) ) ).
G Diffuse ( α o , β o , ν D ;λ )= 1+ ν D /2 π ( 1 α o 2 β o 2 ) ν D /2 ,
G Gaussian ( α i , α o , β o , ρ ^ ,ν;λ )=nor m G ( α i , ρ ^ ,ν;λ )Exp( | α o α i | ν + | β o | ν ( ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ν ),
nor m G ( α i , ρ ^ ,ν;λ )=1/ α o =1 α o =1 β o = 1 α o 2 β o = 1 α o 2 Exp( | α o α i | ν + | β o | ν ( ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ν )d α o d β o
G Lorentzian ( α i , α o , β o , ρ ^ ,ν;λ )=nor m L ( α i , ρ ^ ,ν;λ ) ( 1+ ( α o α i ) 2 + β o 2 ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ( ν+1 )/2 ,
nor m L ( α i , ρ ^ ,ν;λ )=1/ α o =1 α o =1 β o = 1 α o 2 β o = 1 α o 2 ( 1+ ( α o α i ) 2 + β o 2 ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ( ν+1 )/2 d α o d β o .
f BRDF pol = a F,D,D s,p H Fresnel pol F Diffuse G Diffuse + a S,F,G s,p H Surface F Fresnel pol G Gaussian ,
f BRDF pol = a 1 H Fresnel pol G Diffuse + a 2 F Fresnel pol G Gaussian ,
f DHR pol ( α i ;λ )= α o =1 α o =1 β o = 1 α o 2 β o = 1 α o 2 f BRDF pol d α o d β o
f DHR pol ( α i ;λ )= 1+σ 1 α i 2 1+σ F Fresnel pol ( α i , α i ,0;λ )+ σ D ,
f BRDF pol = a S,D,D pol H Surface F Diffuse G Diffuse + a S,F,G1 s,p H Surface F Fresnel pol G 1,Gaussian + a S,F,G2 s,p H Surface F Fresnel pol G 2,Gaussian ,
f BRDF pol = a 1 pol G Diffuse + a 2 F Fresnel pol G 1,Gaussian + a 3 F Fresnel pol G 2,Gaussian ,
f BRDF pol = a F,D,D s,p H Fresnel pol F Diffuse G Diffuse + a S,F,G s,p H Surface F Fresnel pol G Gaussian + a S,F,G1 s,p H Surface F Fresnel pol G 1,Gaussian Volume + a S,F,G2 s,p H Surface F Fresnel pol G 2,Gaussian Volume ,
f BRDF pol = a 1 H Fresnel pol G Diffuse + a 2 F Fresnel pol G Gaussian + a 3 F Fresnel pol G 1,Gaussian Volume + a 4 F Fresnel pol G 2,Gaussian Volume
f BRDF pol = a F,D,D s,p H Fresnel pol F Diffuse G Diffuse + a S,F,G s,p H Surface F Fresnel pol G Gaussian + a S,F,L s,p H Surface F Fresnel pol G Lorentzian + a F,F,L s,p H Fresnel pol F Fresnel pol G Lorentzian + a F,D,L s,p H Fresnel pol F Diffuse G Lorentzian ,
f BRDF pol = a 1 H Fresnel pol G Diffuse + a 2 F Fresnel pol G Gaussian + a 3 F Fresnel pol G Lorentzian + a 4 H Fresnel pol F Fresnel pol G Lorentzian + a 5 F Fresnel pol G Lorentzian ,

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