Abstract

A physics-based model is developed for rough surface BRDF, taking into account angles of incidence and scattering, effective index, surface autocovariance, and correlation length. Shadowing is introduced on surface correlation length and reflectance. Separate terms are included for surface scatter, bulk scatter and retroreflection. Using the FindFit function in Mathematica, the functional form is fitted to BRDF measurements over a wide range of incident angles. The model has fourteen fitting parameters; once these are fixed, the model accurately describes scattering data over two orders of magnitude in BRDF without further adjustment. The resulting analytical model is convenient for numerical computations.

© 2008 Optical Society of America

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References

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2007 (1)

2006 (1)

2004 (1)

T. Elfouhaily and C. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

2001 (2)

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

R. Watson and P. Raven, "Comparison of measured BRDF data with parameterized reflectance models," Proc. SPIE 4370, 159-168 (2001).
[CrossRef]

1999 (2)

J. Harvey, C. Vernold, A. Krywonos and P. Thompson, "Diffracted radiance: a fundamental quantity in non-paraxial scalar diffraction theory, Appl. Opt. 38, 6469-6481 (1999).
[CrossRef]

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

1998 (2)

1996 (1)

1994 (1)

1991 (1)

1983 (1)

Blasband, C.

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

Elfouhaily, T.

T. Elfouhaily and C. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Gamiz, V.

Greffet, J.

Guérin, C.

T. Elfouhaily and C. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Harvey, J.

Hilgers, J.

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

Hoover, B.

Jafolla, J.

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

Jakeman, E.

Jordan, D.

Krywonos, A.

Lewis, G.

Llebaria, A.

Nee, S.

Nieto-Vesperinas, M.

Palmari, J.

Rasigni, G.

Rasigni, M.

Raven, P.

R. Watson and P. Raven, "Comparison of measured BRDF data with parameterized reflectance models," Proc. SPIE 4370, 159-168 (2001).
[CrossRef]

Reynolds, B.

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

Saillard, M.

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

Sánchez-Gil, J.

Sentenac, A.

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

Sun, Y.

Thomas, D.

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

Thompson, P.

Varnier, F.

Vernold, C.

Watson, R.

R. Watson and P. Raven, "Comparison of measured BRDF data with parameterized reflectance models," Proc. SPIE 4370, 159-168 (2001).
[CrossRef]

Wolff, L.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Proc. SPIE (2)

J. Jafolla, D. Thomas, J. Hilgers, B. Reynolds, and C. Blasband, "Theory and measurement of bidirectional reflectance for signature analysis," Proc. SPIE 3699, 2-15 (1999).
[CrossRef]

R. Watson and P. Raven, "Comparison of measured BRDF data with parameterized reflectance models," Proc. SPIE 4370, 159-168 (2001).
[CrossRef]

Waves Random Media (2)

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

T. Elfouhaily and C. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Other (6)

J. Goodman, Statistical Optics (Wiley, 1985).

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, 1963).

J. Stover, Optical Scattering, Measurement and Analysis (SPIE Press, 1995).
[CrossRef]

J. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).

J. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1999).

A. Ngan, F. Durand, and W. Matusik, "Experimental analysis of BRDF models," in Eurographics Symposium on Rendering, K. Bala and P. Dutré, eds., (Eurographics Association, 2005), pp. 117-126.

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Figures (3)

Fig. 1.
Fig. 1.

Definition of coordinates for scattering geometry (adapted from [15]).

Fig. 2.
Fig. 2.

Plot of masking/shadowing function Vm as a function of scattering angle, for two angles of incidence.

Fig. 3.
Fig. 3.

Rough-surface BRDF as a function of in-plane scattering angle, parameterized on angle of incidence in degrees.

Equations (20)

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g ( x , y ) = σ g 2 e ( x 2 + y 2 ) ρ 2
G ( α , β ) = σ g 2 2 ρ e 1 4 ( α 2 + β 2 ) ρ 2
g ( x , y ) = σ g 2 e ( x + y ) ρ
G ( α , β ) = 2 π ρ 2 σ g 2 α 2 + β 2 + ρ 2 .
BRDF = σ N 1 ( α + α 0 ) 2 + β 2 + ρ 2
N = 1 π log [ α 0 2 + ρ 2 + ρ 4 + 2 α 0 2 ρ 2 + 2 ρ 2 + ( 1 α 0 2 ) 2 + 1 2 ρ 2 ] .
BRDF = σ N 1 sin 2 ( θ i ) 2 cos ( ϕ ) sin ( θ ) sin ( θ i ) + ρ 2 + sin 2 ( θ )
N = 1 π log [ sin 2 ( θ i ) + ρ 2 + ρ 4 + 2 sin 2 ( θ i ) ρ 2 + 2 ρ 2 + ( 1 sin 2 ( θ i ) ) 2 + 1 2 ρ 2 ] .
BRDF = σ N 1 [ ( α α 0 ) 2 + β 2 + 1 4 ( 1 α 0 2 + 1 α 2 β 2 ) 2 ρ 2 ]
V ( χ , w , n ) = Γ ( 1 + n , ( 1 + n ) χ 2 w 2 ) n !
V ( χ , w , 0 ) = exp ( χ 2 w 2 ) .
BRDF = σ N ( α α 0 ) 2 + β 2 + 1 4 ( 1 α 0 2 + 1 α 2 β 2 ) 2 V ( α 0 , w , n ) V ( α 2 + β 2 , w , n ) ρ 2 .
V m ( ζ , ζ 0 , w a , w b , γ , δ ) = 1 2 { 1 exp ( γ ζ 0 e δ ζ 0 2 ) } [ Sgn ( ζ ) + 1 ]
+ exp ( w a ζ e w b α 2 ) { 1 2 [ 1 Sgn ( ζ ) ] + 1 2 exp ( γ ζ 0 e δ ζ 0 2 ) [ Sgn ( ζ ) + 1 ] }
BRDF = σ N V m ( ζ , ζ 0 , w a , w b , γ , δ ) ( α α 0 ) 2 + β 2 + 1 4 ( 1 α 0 2 + 1 α 2 β 2 ) 2 V ( α 0 , w , n ) V ( α 2 + β 2 , w , n ) ρ 2 .
R s ( α ) = 1 α 2 ( n eff + ik eff ) 1 α 2 ( n eff + i k eff ) 2 1 α 2 + ( n eff + ik eff ) 1 α 2 ( n eff + i k eff ) 2 2
R p ( α ) = ( n eff + i k eff ) 1 α 2 + 1 α 2 ( n eff + i k eff ) 2 ( n eff + ik eff ) 1 α 2 + 1 α 2 ( n eff + i k eff ) 2 2 .
BRDF total = BRDF surface + BRDF bulk + BRDF retro
σ bulk = [ 1 R ( α 0 ) R ( α , β ) ] bulk = ( 1 σ surface ) bulk
σ retro = [ 1 R ( α 0 ) R ( α , β ) ] retro = ( 1 σ surface ) retro

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