Abstract

A new semi-empirical seven-parameter BRDF model is developed in the UV band using experimentally measured data. The model is based on the five-parameter model of Wu and the fourteen-parameter model of Renhorn and Boreman. Surface scatter, bulk scatter and retro-reflection scatter are considered. An optimizing modeling method, the artificial immune network genetic algorithm, is used to fit the BRDF measurement data over a wide range of incident angles. The calculation time and accuracy of the five- and seven-parameter models are compared. After fixing the seven parameters, the model can well describe scattering data in the UV band.

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2010

H. L. Zhang, Z. S. Wu, Y. H. Cao, and G. Zhang, “Measurement and statistical modeling of BRDF of various samples,” Opt. Appl. 40, 197–208 (2010).

2008

2007

2006

2002

Z. S. Wu, D. H. Xie, P. H. Xie, and Q. N. Wei, “Modeling reflectance function from rough surface and algorithms,” Acta Opt. Sin. 22, 897–901 (2002).

1999

K. J. Dana, B. van Ginneken, S. K. Nayar, and J. J. Koenderink, “Reflectance and texture of real-world surfaces,” ACM Trans. Graph. 18(1), 1–34 (1999).
[CrossRef]

1997

1996

M. P. Newell and R. A. M. Keski-Kuha, “Extreme ultraviolet BRDF measurements: instrumentation and results,” Proc. SPIE 2864, 453–464 (1996).
[CrossRef]

1995

T. Zurbuchen, P. Bochsler, and F. Scholze, “Reflection of ultraviolet light at 121.6 nm from rough surfaces,” Opt. Eng. 34(5), 1303–1315 (1995).
[CrossRef]

M. P. Newell, L. A. Whitlock, and R. A. M. Keski-Kuha, “Extreme ultraviolet scatter from particulate contaminated mirrors,” Proc. SPIE 2541, 174–185 (1995).
[CrossRef]

1994

1993

1992

J. L. Roujean, M. Leroy, and P. Y. Deschamps, “A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97(20), 455–468 (1992).

X. Li and A. H. Strahler, “Geometric-optical bidirectional reflectance modeling of the discrete crown vegetation canopy: effect of crown shape and mutual shadowing,” IEEE Trans. Geosci. Rem. Sens. 30(2), 276–292 (1992).
[CrossRef]

1985

1980

1967

1941

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

Amra, C.

Barnes, Y.

Y. Barnes and J. J. Hsia, “UV bidirectional reflectance distribution function measurements for diffusers,” Proc. SPIE 1764, 285–288 (1993).
[CrossRef]

Bennett, J. M.

Blad, B. L.

Bochsler, P.

T. Zurbuchen, P. Bochsler, and F. Scholze, “Reflection of ultraviolet light at 121.6 nm from rough surfaces,” Opt. Eng. 34(5), 1303–1315 (1995).
[CrossRef]

Boreman, G. D.

Bruel, L.

Campbell, G.

Cao, Y. H.

H. L. Zhang, Z. S. Wu, Y. H. Cao, and G. Zhang, “Measurement and statistical modeling of BRDF of various samples,” Opt. Appl. 40, 197–208 (2010).

Dana, K. J.

K. J. Dana, B. van Ginneken, S. K. Nayar, and J. J. Koenderink, “Reflectance and texture of real-world surfaces,” ACM Trans. Graph. 18(1), 1–34 (1999).
[CrossRef]

Deschamps, P. Y.

J. L. Roujean, M. Leroy, and P. Y. Deschamps, “A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97(20), 455–468 (1992).

Durand, G.

Elson, J. M.

Grèzes-Besset, C.

Hsia, J. J.

Y. Barnes and J. J. Hsia, “UV bidirectional reflectance distribution function measurements for diffusers,” Proc. SPIE 1764, 285–288 (1993).
[CrossRef]

Keski-Kuha, R. A. M.

M. P. Newell and R. A. M. Keski-Kuha, “Bidirectional reflectance distribution function of diffuse extreme ultraviolet scatterers and extreme ultraviolet baffle materials,” Appl. Opt. 36(22), 5471–5475 (1997).
[CrossRef] [PubMed]

M. P. Newell and R. A. M. Keski-Kuha, “Extreme ultraviolet BRDF measurements: instrumentation and results,” Proc. SPIE 2864, 453–464 (1996).
[CrossRef]

M. P. Newell, L. A. Whitlock, and R. A. M. Keski-Kuha, “Extreme ultraviolet scatter from particulate contaminated mirrors,” Proc. SPIE 2541, 174–185 (1995).
[CrossRef]

Koenderink, J. J.

K. J. Dana, B. van Ginneken, S. K. Nayar, and J. J. Koenderink, “Reflectance and texture of real-world surfaces,” ACM Trans. Graph. 18(1), 1–34 (1999).
[CrossRef]

Lavigne, C.

Leroy, M.

J. L. Roujean, M. Leroy, and P. Y. Deschamps, “A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97(20), 455–468 (1992).

Li, X.

X. Li and A. H. Strahler, “Geometric-optical bidirectional reflectance modeling of the discrete crown vegetation canopy: effect of crown shape and mutual shadowing,” IEEE Trans. Geosci. Rem. Sens. 30(2), 276–292 (1992).
[CrossRef]

Minnaert, M.

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

Nayar, S. K.

K. J. Dana, B. van Ginneken, S. K. Nayar, and J. J. Koenderink, “Reflectance and texture of real-world surfaces,” ACM Trans. Graph. 18(1), 1–34 (1999).
[CrossRef]

Newell, M. P.

M. P. Newell and R. A. M. Keski-Kuha, “Bidirectional reflectance distribution function of diffuse extreme ultraviolet scatterers and extreme ultraviolet baffle materials,” Appl. Opt. 36(22), 5471–5475 (1997).
[CrossRef] [PubMed]

M. P. Newell and R. A. M. Keski-Kuha, “Extreme ultraviolet BRDF measurements: instrumentation and results,” Proc. SPIE 2864, 453–464 (1996).
[CrossRef]

M. P. Newell, L. A. Whitlock, and R. A. M. Keski-Kuha, “Extreme ultraviolet scatter from particulate contaminated mirrors,” Proc. SPIE 2541, 174–185 (1995).
[CrossRef]

Norman, J. M.

Rahn, J. P.

Renhorn, I. G. E.

Roblin, A.

Roujean, J. L.

J. L. Roujean, M. Leroy, and P. Y. Deschamps, “A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97(20), 455–468 (1992).

Scholze, F.

T. Zurbuchen, P. Bochsler, and F. Scholze, “Reflection of ultraviolet light at 121.6 nm from rough surfaces,” Opt. Eng. 34(5), 1303–1315 (1995).
[CrossRef]

Sparrow, E. M.

Strahler, A. H.

X. Li and A. H. Strahler, “Geometric-optical bidirectional reflectance modeling of the discrete crown vegetation canopy: effect of crown shape and mutual shadowing,” IEEE Trans. Geosci. Rem. Sens. 30(2), 276–292 (1992).
[CrossRef]

Thennadil, S. N.

Torrance, K. E.

van Ginneken, B.

K. J. Dana, B. van Ginneken, S. K. Nayar, and J. J. Koenderink, “Reflectance and texture of real-world surfaces,” ACM Trans. Graph. 18(1), 1–34 (1999).
[CrossRef]

Velazco-Roa, M. A.

Walthall, C. L.

Wei, Q. N.

Z. S. Wu, D. H. Xie, P. H. Xie, and Q. N. Wei, “Modeling reflectance function from rough surface and algorithms,” Acta Opt. Sin. 22, 897–901 (2002).

Welles, J. M.

Whitlock, L. A.

M. P. Newell, L. A. Whitlock, and R. A. M. Keski-Kuha, “Extreme ultraviolet scatter from particulate contaminated mirrors,” Proc. SPIE 2541, 174–185 (1995).
[CrossRef]

Wu, Z. S.

H. L. Zhang, Z. S. Wu, Y. H. Cao, and G. Zhang, “Measurement and statistical modeling of BRDF of various samples,” Opt. Appl. 40, 197–208 (2010).

Z. S. Wu, D. H. Xie, P. H. Xie, and Q. N. Wei, “Modeling reflectance function from rough surface and algorithms,” Acta Opt. Sin. 22, 897–901 (2002).

Xie, D. H.

Z. S. Wu, D. H. Xie, P. H. Xie, and Q. N. Wei, “Modeling reflectance function from rough surface and algorithms,” Acta Opt. Sin. 22, 897–901 (2002).

Xie, P. H.

Z. S. Wu, D. H. Xie, P. H. Xie, and Q. N. Wei, “Modeling reflectance function from rough surface and algorithms,” Acta Opt. Sin. 22, 897–901 (2002).

Zhang, G.

H. L. Zhang, Z. S. Wu, Y. H. Cao, and G. Zhang, “Measurement and statistical modeling of BRDF of various samples,” Opt. Appl. 40, 197–208 (2010).

Zhang, H. L.

H. L. Zhang, Z. S. Wu, Y. H. Cao, and G. Zhang, “Measurement and statistical modeling of BRDF of various samples,” Opt. Appl. 40, 197–208 (2010).

Zurbuchen, T.

T. Zurbuchen, P. Bochsler, and F. Scholze, “Reflection of ultraviolet light at 121.6 nm from rough surfaces,” Opt. Eng. 34(5), 1303–1315 (1995).
[CrossRef]

ACM Trans. Graph.

K. J. Dana, B. van Ginneken, S. K. Nayar, and J. J. Koenderink, “Reflectance and texture of real-world surfaces,” ACM Trans. Graph. 18(1), 1–34 (1999).
[CrossRef]

Acta Opt. Sin.

Z. S. Wu, D. H. Xie, P. H. Xie, and Q. N. Wei, “Modeling reflectance function from rough surface and algorithms,” Acta Opt. Sin. 22, 897–901 (2002).

Appl. Opt.

Astrophys. J.

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

IEEE Trans. Geosci. Rem. Sens.

X. Li and A. H. Strahler, “Geometric-optical bidirectional reflectance modeling of the discrete crown vegetation canopy: effect of crown shape and mutual shadowing,” IEEE Trans. Geosci. Rem. Sens. 30(2), 276–292 (1992).
[CrossRef]

J. Geophys. Res.

J. L. Roujean, M. Leroy, and P. Y. Deschamps, “A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97(20), 455–468 (1992).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Appl.

H. L. Zhang, Z. S. Wu, Y. H. Cao, and G. Zhang, “Measurement and statistical modeling of BRDF of various samples,” Opt. Appl. 40, 197–208 (2010).

Opt. Eng.

T. Zurbuchen, P. Bochsler, and F. Scholze, “Reflection of ultraviolet light at 121.6 nm from rough surfaces,” Opt. Eng. 34(5), 1303–1315 (1995).
[CrossRef]

Opt. Express

Proc. SPIE

M. P. Newell and R. A. M. Keski-Kuha, “Extreme ultraviolet BRDF measurements: instrumentation and results,” Proc. SPIE 2864, 453–464 (1996).
[CrossRef]

Y. Barnes and J. J. Hsia, “UV bidirectional reflectance distribution function measurements for diffusers,” Proc. SPIE 1764, 285–288 (1993).
[CrossRef]

M. P. Newell, L. A. Whitlock, and R. A. M. Keski-Kuha, “Extreme ultraviolet scatter from particulate contaminated mirrors,” Proc. SPIE 2541, 174–185 (1995).
[CrossRef]

Other

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).

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Figures (6)

Fig. 1
Fig. 1

Angle definition of BRDF.

Fig. 2
Fig. 2

A schematic diagram of the instrument.

Fig. 3
Fig. 3

Comparison of the seven- and five-parameter models of sample #1 for various angles of incidence when wavelength of incidence is λ=266nm .

Fig. 4
Fig. 4

Comparison of the seven- and five-parameter models of sample #1 for various angles of incidence when wavelength of incidence is λ=369nm .

Fig. 5
Fig. 5

Comparison of the seven- and five-parameter models of sample 2# for various angles of incidence when wavelength of incidence is λ=266nm .

Fig. 6
Fig. 6

Comparison of the seven- and five-parameter models of sample 2# for various angles of incidence when wavelength of incidence is λ=369nm .

Tables (2)

Tables Icon

Table 1 Comparison of the Five- and Seven-Parameter Models of Sample #1

Tables Icon

Table 2 Comparison of the Five- And Seven-Parameter Models of Sample #2

Equations (16)

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cosα= ( cos θ i +cos θ r ) / (2cosγ)
cos 2 γ= 1 2 ( cos θ i cos θ r +sin θ i sin θ r cos φ r +1 )
f r ( θ i , φ i , θ r , φ r )= d L r ( θ i , φ i , θ r , φ r ) d E i ( θ i , φ i )
f r = P s / ( P i cos θ r Ω s )
f rs ( θ i , φ i ; θ r , φ r ,λ) f rr ( θ i , φ i ; θ r , φ r ,λ) = V s V ref
BRD F surface = f r ( θ i , θ r ,φ,λ) = k b k r 2 cosα 1+( k r 2 1 )cosα exp[ b ( 1cosγ ) a ] V( θ i , θ r ,φ,λ ) cos θ i cos θ r + k d cos θ i
g(x,y)= σ g 2 exp[ ( x 2 + y 2 ) ρ 2 ]
G(ξ,η)= σ g 2 2 ρ exp[ 1 4 ρ 2 ( ξ 2 + η 2 ) ]
g(x,y)= σ g 2 exp[ (| x |+| y |)ρ ]
G(ξ,η)= 2 π σ g 2 ξ 2 + η 2 + ρ 2
BRDF= σN ( ξ ξ 0 ) 2 + η 2 + ρ 2
N= 1 πlog[ ξ 0 2 + ρ 2 + ρ 4 +2 ξ 0 2 ρ 2 +2 ρ 2 + (1 ξ 0 2 ) 2 +1 2 ρ 2 ]
BRDF= σN (ξ ξ 0 ) 2 + η 2 + 1 4 ( 1 ξ 0 2 + 1 ξ 2 η 2 ) 2 ρ 2
BRD F total =BRD F surface +BRD F bulk +BRD F retro
BRD F bulk = N (ξ ξ 0 ) 2 + η 2 + 1 4 ( 1 ξ 0 2 + 1 ξ 2 η 2 ) 2 ρ 1 2
BRD F retro = N (ξ ξ 0 ) 2 + η 2 + 1 4 ( 1 ξ 0 2 + 1 ξ 2 η 2 ) 2 ρ 2 2

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