Abstract

The purpose of the present manuscript is to present a polarimetric bidirectional reflectance distribution function (BRDF) model suitable for hyperspectral and polarimetric signature modelling. The model is based on a further development of a previously published four-parameter model that has been generalized in order to account for different types of surface structures (generalized Gaussian distribution). A generalization of the Lambertian diffuse model is presented. The pBRDF-functions are normalized using numerical integration. Using directional-hemispherical reflectance (DHR) measurements, three of the four basic parameters can be determined for any wavelength. This simplifies considerably the development of multispectral polarimetric BRDF applications. The scattering parameter has to be determined from at least one BRDF measurement. The model deals with linear polarized radiation; and in similarity with e.g. the facet model depolarization is not included. The model is very general and can inherently model extreme surfaces such as mirrors and Lambertian surfaces. The complex mixture of sources is described by the sum of two basic models, a generalized Gaussian/Fresnel model and a generalized Lambertian model. Although the physics inspired model has some ad hoc features, the predictive power of the model is impressive over a wide range of angles and scattering magnitudes. The model has been applied successfully to painted surfaces, both dull and glossy and also on metallic bead blasted surfaces. The simple and efficient model should be attractive for polarimetric simulations and polarimetric remote sensing.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Developing a generalized BRDF model from experimental data

Ingmar G. E. Renhorn and Glenn D. Boreman
Opt. Express 26(13) 17099-17114 (2018)

Four-parameter model for polarization-resolved rough-surface BRDF

Ingmar G. E. Renhorn, Tomas Hallberg, David Bergström, and Glenn D. Boreman
Opt. Express 19(2) 1027-1036 (2011)

Polarimetric bidirectional reflectance distribution function of glossy coatings

Kenneth K. Ellis
J. Opt. Soc. Am. A 13(8) 1758-1762 (1996)

References

  • View by:
  • |
  • |
  • |

  1. R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: Theory and measurements,” in Military Sensing Symposia (MSS) Specialty Group Meeting on Passive Sensors, (Infrared Information Analysis Center, Ann Arbor, MI, 2000). Available at www.dtic.mil , approved for public release, distribution unlimited.
  2. T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
    [Crossref]
  3. M. A. Culpepper, “Empirical bidirectional reflectivity model,” Proc. SPIE 2469, 159–168 (2001).
  4. 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]
  5. I. G. E. Renhorn, T. Hallberg, D. Bergström, and G. D. Boreman, “Four-parameter model for polarization-resolved rough-surface BRDF,” Opt. Express 19(2), 1027–1036 (2011).
    [Crossref] [PubMed]
  6. I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008).
    [Crossref] [PubMed]
  7. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. A 69(1), 31–47 (1979).
    [Crossref]
  8. J.-J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
    [Crossref]
  9. J. Qiu, W. J. Zhang, L. H. Liu, P. F. Hsu, and L. J. Liu, “Reflective properties of randomly rough surfaces under large incidence angles,” J. Opt. Soc. Am. A 31(6), 1251–1258 (2014).
    [Crossref] [PubMed]
  10. S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10(4), 210–218 (1985).
    [Crossref]
  11. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. Dissertation, University of Arizona (1976).
  12. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  13. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4(7), 767–773 (1965).
    [Crossref]
  14. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9(6), 1474–1475 (1970).
    [Crossref] [PubMed]
  15. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  16. J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).
  17. J. M. Bennet and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, DC, 1989).
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  19. G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50(18), 2855–2865 (2011).
    [Crossref] [PubMed]

2014 (1)

2011 (2)

2008 (1)

2004 (1)

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
[Crossref]

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]

2001 (1)

M. A. Culpepper, “Empirical bidirectional reflectivity model,” Proc. SPIE 2469, 159–168 (2001).

1998 (1)

1985 (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10(4), 210–218 (1985).
[Crossref]

1979 (1)

J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. A 69(1), 31–47 (1979).
[Crossref]

1970 (1)

1965 (1)

Bennett, J. M.

J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. A 69(1), 31–47 (1979).
[Crossref]

Bergström, D.

Boreman, G. D.

Chipman, R. A.

Crabtree, K.

Culpepper, M. A.

M. A. Culpepper, “Empirical bidirectional reflectivity model,” Proc. SPIE 2469, 159–168 (2001).

Elfouhaily, T. M.

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
[Crossref]

Elson, J. M.

J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. A 69(1), 31–47 (1979).
[Crossref]

Greffet, J.-J.

Guerin, C. A.

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
[Crossref]

Hallberg, T.

Hsu, P. F.

Liu, L. H.

Liu, L. J.

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]

Nicodemus, F. E.

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]

Qiu, J.

Renhorn, I. G.

Renhorn, I. G. E.

Shafer, S.

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10(4), 210–218 (1985).
[Crossref]

Yun, G.

Zhang, W. J.

Appl. Opt. (3)

Color Res. Appl. (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10(4), 210–218 (1985).
[Crossref]

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

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 (2)

Proc. SPIE (1)

M. A. Culpepper, “Empirical bidirectional reflectivity model,” Proc. SPIE 2469, 159–168 (2001).

Waves Random Media (1)

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
[Crossref]

Other (7)

R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: Theory and measurements,” in Military Sensing Symposia (MSS) Specialty Group Meeting on Passive Sensors, (Infrared Information Analysis Center, Ann Arbor, MI, 2000). Available at www.dtic.mil , approved for public release, distribution unlimited.

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. Dissertation, University of Arizona (1976).

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

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).

J. M. Bennet and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, DC, 1989).

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1 Surface scattering materials.
Fig. 2
Fig. 2 Surface and volume scattering materials.
Fig. 3
Fig. 3 Clear-coat covered scattering materials.
Fig. 4
Fig. 4 Scattering factor using the generalized Gaussian distribution with shape factors 1, 2 and 4 where the factor equal to 2 is the normal distribution.
Fig. 5
Fig. 5 Pss (left) and Psp (right) for input direction α0 = 0.5.
Fig. 6
Fig. 6 Hemispherical (DHR) data for s- and p-polarizations at 60 degrees angle of incidence as a function of wavelength for green paint.
Fig. 7
Fig. 7 Hemispherical (DHR) data for s- and p-polarizations at the wavelength 3.39 µm as a function of angle of incidence for green paint. A small bias has been included when fitting to the Fresnel’s equations.
Fig. 8
Fig. 8 The fitting parameter of the DHR model are shown as a function of wavelength. A) Bias correlates strongly with the total reflectance. B) The scaling parameter σ shows anti-correlation to the total reflectance values. C) The index of refraction n is mostly centered on a value of 1.6 except in the 4-5.5 µm spectral region where the reflectance is high. D) The absorption coefficient k is zero except at a few narrow spectral regions.
Fig. 9
Fig. 9 pBRDF measurements and model fit to a bead-blasted aluminum surface. Diffuse scattering levels are different for s- and p-polarized radiation. Red is s-polarized and blue is p-polarized.
Fig. 10
Fig. 10 Figures shows the variation in magnitude (upper) and spread (lower) of the two peaks, one narrow (left) and one broad (right), as a function of angle of incidence.
Fig. 11
Fig. 11 pBRDF measurements and model fit to a bead-blasted aluminum surface. Scattering levels of p-polarized radiation are stronger than s-polarized radiation. Red is s-polarized and blue is p-polarized.
Fig. 12
Fig. 12 pBRDF measurements and model fit to diffuse green paint.
Fig. 13
Fig. 13 pBRDF measurements and model fit to diffuse green paint.
Fig. 14
Fig. 14 BRDF results for the red automotive paint. The specular peak at normal incidence is blocked. At other angles the peak is partly limited by the angular resolution of the instrument.
Fig. 15
Fig. 15 The diffuse scattering level is much lower for the green paint compared to red paint at 633 nm. The low scattering level reveals the deviation at the main peak tail. The specular peak is off scale by several orders of magnitude.

Equations (44)

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

L λ ( r,s )= d 2 ϕ λ dA  s z  dΩ
I λ ( s )= L λ ( r,s )  s z  dA.
L λ r ( r,s )=  ρ( r,s,S )  L λ inc ( r,S ) S z  dΩ
W( r 1 , r 2 ;λ )= U * ( r 1 ;λ ) U( r 2 ;λ )
M( r;λ )=W( r,r;λ ).
W (0) ( r 1 , r 2 ;λ ) M ( 0 ) ( r 1 + r 2 2 ;λ )  μ (0) ( r 2 r 1 ;λ )
W ˜ (0) ( s , s ' ;λ ) M ˜ ( 0 ) ( k  s ' )  μ ˜ (0) ( k  s )
L λ ( r,s )= k 2  | s z |  M ( 0 ) ( r )  μ ˜ (0) ( k  s ).
f BRDF pol ( s,S;λ )=nor m pol g( s,S;λ )  f DHR pol ( S;λ )  Q pol ( s,S;λ )
f DHR pol ( S;λ )= σ rel   R α pol ( S, n eff , k eff )+ D α pol
Q s R s ( s )  R s ( S )
Q s = f DHR s ( α+ α 0 2 ;λ )  f DHR p ( β 2 ;λ ) f DHR s ( α 0 ;λ )  f DHR p ( 0;λ ) Q p = f DHR p ( α+ α 0 2 ;λ )  f DHR s ( β 2 ;λ ) f DHR p ( α 0 ;λ )  f DHR p ( 0;λ )
R s ( α,n,k )= ( ( ni k )  1 α 2 ( ni k ) 2 1 α 2 ) ( ( n+i k )  1 α 2 ( n+i k ) 2 1 α 2 ) ( ( ni k )  1 α 2 ( ni k ) 2 + 1 α 2 ) ( ( n+i k )  1 α 2 ( n+i k ) 2 + 1 α 2 ) R p ( α,n,k )= ( ( ni k )  1 α 2 1 α 2 ( ni k ) 2 ) ( ( n+i k )  1 α 2 1 α 2 ( n+i k ) 2 ) ( ( ni k )  1 α 2 + 1 α 2 ( ni k ) 2 ) ( ( n+i k )  1 α 2 + 1 α 2 ( n+i k ) 2 )
g G ( α,β, α 0 , ρ ^ ,ν ;λ )=Exp[   | α α 0 | ν + | β | ν ( ( 1 α 0 2 + 1 α 2 β 2 )  ρ ^ ) ν ]
α=1 α=1 β= 1 α 2 β= 1 α 2 nor m pol  g( α,β, α 0 , ρ ^ )  Q pol ( α,β, α 0 ,n,k ) dα dβ=1.
D( α,β )= σ D   ( 1 α 2 β 2 ) ν D /2
D ss ( α,β )= σ D   ( 1 α 2 β 2 ) ν D /2  ( 1 R s ( α 0 ,n,k ) )  1 R s ( α 2 + β 2 ,n,k ) 2 D sp ( α,β )= σ D   ( 1 α 2 β 2 ) ν D /2  ( 1 R s ( α 0 ,n,k ) )  1 R p ( α 2 + β 2 ,n,k ) 2 D pp ( α,β )= σ D   ( 1 α 2 β 2 ) ν D /2  ( 1 R p ( α 0 ,n,k ) )  1 R p ( α 2 + β 2 ,n,k ) 2 D ps ( α,β )= σ D   ( 1 α 2 β 2 ) ν D /2  ( 1 R p ( α 0 ,n,k ) )  1 R s ( α 2 + β 2 ,n,k ) 2
f BRDF pol ( α,β, α 0 , ρ ^ ;λ )= i=1 N a i   f BRDF pol ( α,β, α i , ρ ^ i ;λ )+ D pol ( α,β )
f DHR pol ( α 0 , ρ ^ ;λ )= α=1 α=1 β= 1 α 2 β= 1 α 2 f BRDF pol ( α,β, α 0 , ρ ^ ;λ ) dα dβ.
α C = Erf[ k α α 0 ] Erf[ k α ]
E out = O out   J surface   O in 1   E in
O in 1 =[ s x,in s y,in s z,in p x,in p y,in p z,in k x,in k x,in k x,in ]
O out =[ s x,out p x,out k x,out s y,out p y,out k y,out s z,out p z,out k z,out ]
s in = k in  × k out | k in  × k out |
p in = k in  × s in
s out = s in
p out = k out  × s out
s 0 ={ 0,1,0 }
p 0 ={ 1 α 0 2 ,0, α 0 }
k 0 ={ α 0 ,0, 1 α 0 2 }
s f = 1 α 2 + β 2  { β,α,0 }
p f = 1 α 2 + β 2  { α  1 α 2 β 2 ,β  1 α 2 β 2 , α 2 β 2 }
k f ={ α,β, 1 α 2 β 2 }
s in =  { 1 α 0 2  β,α  1 α 0 2 α 0   1 α 2 β 2 , α 0  β } α 0 2   β 2 +( 1 α 0 2 )  β 2 + ( α  1 α 0 2 α 0   1 α 2 β 2 ) 2
p in = { α+α  α 0 2 α 0   1 α 0 2   1 α 2 β 2 ,β,α  α 0 1 α 0 2 α 0 2   1 α 2 β 2 } α 2 + α 0 2 2  α 2   α 0 2 + β 2 α 0 2   β 2 +2 α  α 0 1 α 0 2   1 α 2 β 2  
k in = k 0
s out = s in
p out = { α 0 α 2   α 0 +α  1 α 0 2   1 α 2 β 2 ,α  α 0  β+ 1 α 0 2  β  1 α 2 β 2 , α 2   1 α 0 2 1 α 0 2   β 2 α  α 0   1 α 2 β 2 } α 2 + α 0 2 2  α 2   α 0 2 + β 2 α 0 2   β 2 +2 α  α 0 1 α 0 2   1 α 2 β 2  
k out = k f
P ss = ( s f . O out . O in 1 . s 0 ) 2
P sp =1 P ss
P ps =1 P ss
P pp = P ss
P ss = ( α 3 +α  α 0 2 2  α 3   α 0 2 +α  β 2 + α 0   β 2 2 α  α 0 2   β 2 +2  α 2   α 0   1 α 0 2   1 α 2 β 2 + α 0   1 α 0 2  β  1 α 2 β 2 ) 2 ( α 2 + β 2 )  ( α 2 + α 0 2 + β 2 +2 α  α 0   1 α 0 2   1 α 2 β 2 α 0 2  ( 2  α 2 + β 2 ) ) 2

Metrics