Abstract

The bidirectional reflectance distribution function (BRDF) represents the evolution of the reflectance with the directions of incidence and observation. Today BRDF measurements are increasingly applied and have become important to the study of the appearance of surfaces. The representation and the analysis of BRDF data are discussed, and the distortions caused by the traditional representation of the BRDF in a Fourier plane are pointed out and illustrated for two theoretical cases: an isotropic surface and a brushed surface. These considerations will help characterize either the specular peak width of an isotropic rough surface or the main directions of the light scattered by an anisotropic rough surface without misinterpretations. Finally, what is believed to be a new space is suggested for the representation of the BRDF, which avoids the geometrical deformations and in numerous cases is more convenient for BRDF analysis.

© 2007 Optical Society of America

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References

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  1. A. da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, "Multiple light scattering in multilayered media: theory, experiments," Physica B 338, 74-78 (2003).
    [CrossRef]
  2. M. Nadal and T. A. Germer, "Colorimetric characterization of pearlescent coatings," Proc. SPIE 4421, 757-760 (2001).
    [CrossRef]
  3. Q. Z. Zhu and Z. M. Zhang, "Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface," J. Heat Transfer 126, 985-993 (2004).
    [CrossRef]
  4. Vocabulaire International de la CIE, definition 845-04-70, 4th ed., publication CIE No. 50 (845), Bureau Central de la CIE (Paris, 1987).
  5. G. Obein, R. Bousquet, and M. E. Nadal, "New NIST reference goniospectrometer," Proc. SPIE 5880, 241-250 (2005).
  6. D. Hünerhoff, U. Grusemann, and A. Höpe, "New robot-based gonioreflectometer for measuring spectral diffuse reflection," Metrologia 43, S11-S16 (2006).
    [CrossRef]
  7. A. da Silva, M. Elias, C. Andraud, and J. Lafait, "Comparison between the auxiliary function method and the discrete-ordinate method for solving the radiative transfer equation for light scattering," J. Opt. Soc. Am. A 20, 2321-2329 (2003).
    [CrossRef]
  8. G. Meister, A. Rothkirch, H. Spitzer, and J. Bienlein, "Width of the specular peak perpendicular to the principal plane for rough surfaces", Appl. Opt. 40, 6072-6080 (2001).
    [CrossRef]
  9. R. Seve, "Problems connected with the concept of gloss," Color Res. Appl. 18, 241-252 (1993).
    [CrossRef]
  10. S. M. Rusinkiewicz, "New change of variables foe efficient BRDF representation," in Eurographics Rendering Workshop '98 (1998), pp. 11-22.
  11. K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 9, 1105-1114 (1967).
    [CrossRef]
  12. J. Blinn, "Models of light reflection for computer synthesized pictures," in Proceedings of SIGGRAPH'77 (1977), Vol. 11, 192-198.
  13. R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in Proceedings of SIGGRAPH'81 (1981), Vol. 15, pp. 301-316.
  14. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963), pp. 70-98.
  15. G. Ward, "Measuring and modelling anisotropic reflection," in Proceedings of SIGGRAPH'92 (1992), Vol. 26, pp. 265-272.

2006

D. Hünerhoff, U. Grusemann, and A. Höpe, "New robot-based gonioreflectometer for measuring spectral diffuse reflection," Metrologia 43, S11-S16 (2006).
[CrossRef]

2005

G. Obein, R. Bousquet, and M. E. Nadal, "New NIST reference goniospectrometer," Proc. SPIE 5880, 241-250 (2005).

2004

Q. Z. Zhu and Z. M. Zhang, "Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface," J. Heat Transfer 126, 985-993 (2004).
[CrossRef]

2003

2001

1993

R. Seve, "Problems connected with the concept of gloss," Color Res. Appl. 18, 241-252 (1993).
[CrossRef]

1987

Vocabulaire International de la CIE, definition 845-04-70, 4th ed., publication CIE No. 50 (845), Bureau Central de la CIE (Paris, 1987).

1967

K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 9, 1105-1114 (1967).
[CrossRef]

Andraud, C.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963), pp. 70-98.

Bienlein, J.

Blinn, J.

J. Blinn, "Models of light reflection for computer synthesized pictures," in Proceedings of SIGGRAPH'77 (1977), Vol. 11, 192-198.

Bousquet, R.

G. Obein, R. Bousquet, and M. E. Nadal, "New NIST reference goniospectrometer," Proc. SPIE 5880, 241-250 (2005).

Charron, E.

A. da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, "Multiple light scattering in multilayered media: theory, experiments," Physica B 338, 74-78 (2003).
[CrossRef]

Cook, R. L.

R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in Proceedings of SIGGRAPH'81 (1981), Vol. 15, pp. 301-316.

da Silva, A.

Elias, M.

Germer, T. A.

M. Nadal and T. A. Germer, "Colorimetric characterization of pearlescent coatings," Proc. SPIE 4421, 757-760 (2001).
[CrossRef]

Grusemann, U.

D. Hünerhoff, U. Grusemann, and A. Höpe, "New robot-based gonioreflectometer for measuring spectral diffuse reflection," Metrologia 43, S11-S16 (2006).
[CrossRef]

Höpe, A.

D. Hünerhoff, U. Grusemann, and A. Höpe, "New robot-based gonioreflectometer for measuring spectral diffuse reflection," Metrologia 43, S11-S16 (2006).
[CrossRef]

Hünerhoff, D.

D. Hünerhoff, U. Grusemann, and A. Höpe, "New robot-based gonioreflectometer for measuring spectral diffuse reflection," Metrologia 43, S11-S16 (2006).
[CrossRef]

Lafait, J.

Meister, G.

Nadal, M.

M. Nadal and T. A. Germer, "Colorimetric characterization of pearlescent coatings," Proc. SPIE 4421, 757-760 (2001).
[CrossRef]

Nadal, M. E.

G. Obein, R. Bousquet, and M. E. Nadal, "New NIST reference goniospectrometer," Proc. SPIE 5880, 241-250 (2005).

Obein, G.

G. Obein, R. Bousquet, and M. E. Nadal, "New NIST reference goniospectrometer," Proc. SPIE 5880, 241-250 (2005).

Rothkirch, A.

Rusinkiewicz, S. M.

S. M. Rusinkiewicz, "New change of variables foe efficient BRDF representation," in Eurographics Rendering Workshop '98 (1998), pp. 11-22.

Seve, R.

R. Seve, "Problems connected with the concept of gloss," Color Res. Appl. 18, 241-252 (1993).
[CrossRef]

Sparrow, E. M.

K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 9, 1105-1114 (1967).
[CrossRef]

Spitzer, H.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963), pp. 70-98.

Stout, B.

A. da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, "Multiple light scattering in multilayered media: theory, experiments," Physica B 338, 74-78 (2003).
[CrossRef]

Torrance, K. E.

K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 9, 1105-1114 (1967).
[CrossRef]

R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in Proceedings of SIGGRAPH'81 (1981), Vol. 15, pp. 301-316.

Ward, G.

G. Ward, "Measuring and modelling anisotropic reflection," in Proceedings of SIGGRAPH'92 (1992), Vol. 26, pp. 265-272.

Zhang, Z. M.

Q. Z. Zhu and Z. M. Zhang, "Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface," J. Heat Transfer 126, 985-993 (2004).
[CrossRef]

Zhu, Q. Z.

Q. Z. Zhu and Z. M. Zhang, "Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface," J. Heat Transfer 126, 985-993 (2004).
[CrossRef]

Appl. Opt.

Color Res. Appl.

R. Seve, "Problems connected with the concept of gloss," Color Res. Appl. 18, 241-252 (1993).
[CrossRef]

J. Heat Transfer

Q. Z. Zhu and Z. M. Zhang, "Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface," J. Heat Transfer 126, 985-993 (2004).
[CrossRef]

J. Opt. Soc. Am.

K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 9, 1105-1114 (1967).
[CrossRef]

J. Opt. Soc. Am. A

Metrologia

D. Hünerhoff, U. Grusemann, and A. Höpe, "New robot-based gonioreflectometer for measuring spectral diffuse reflection," Metrologia 43, S11-S16 (2006).
[CrossRef]

Physica B

A. da Silva, C. Andraud, E. Charron, B. Stout, and J. Lafait, "Multiple light scattering in multilayered media: theory, experiments," Physica B 338, 74-78 (2003).
[CrossRef]

Proc. SPIE

M. Nadal and T. A. Germer, "Colorimetric characterization of pearlescent coatings," Proc. SPIE 4421, 757-760 (2001).
[CrossRef]

G. Obein, R. Bousquet, and M. E. Nadal, "New NIST reference goniospectrometer," Proc. SPIE 5880, 241-250 (2005).

Other

S. M. Rusinkiewicz, "New change of variables foe efficient BRDF representation," in Eurographics Rendering Workshop '98 (1998), pp. 11-22.

Vocabulaire International de la CIE, definition 845-04-70, 4th ed., publication CIE No. 50 (845), Bureau Central de la CIE (Paris, 1987).

J. Blinn, "Models of light reflection for computer synthesized pictures," in Proceedings of SIGGRAPH'77 (1977), Vol. 11, 192-198.

R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in Proceedings of SIGGRAPH'81 (1981), Vol. 15, pp. 301-316.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963), pp. 70-98.

G. Ward, "Measuring and modelling anisotropic reflection," in Proceedings of SIGGRAPH'92 (1992), Vol. 26, pp. 265-272.

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Figures (13)

Fig. 1
Fig. 1

Angular notations.

Fig. 2
Fig. 2

Representation of the BRDF of a surface in a Fourier plane, here a semiglossy white paper. Direction of illumination: θ I = 45 ° , φ I = 180 ° . A peak is visible around the specular reflection direction ( θ S = 45 ° , φ S = 0 ° ) .

Fig. 3
Fig. 3

Evolution of the BRDF of an isotropic surface, here a semiglossy black paper. Left: θ I = 20 ° , middle: θ I = 40 ° , right: θ I = 60 ° ( φ I = 180 ° ) . The white curve is the isoreflectance curve at half-maximum. It appears that the shape of the peak is not circular and undergoes a deformation when the zenith of illumination changes.

Fig. 4
Fig. 4

Evolution of an isoreflectance curve with the zenith of the direction of illumination, projected in the BRDF space ( φ I = 180 ° ) . It shows the distortions inducted by the geometrical configuration and the BRDF space itself. Here θ N = 5 ° and θ I varying from 0° to 75° with a step of 15°.

Fig. 5
Fig. 5

(Color online) Four possible planes for the calculations of a perpendicular width. Example with θ I = 45 ° and φ I = 180 ° . Left, the four different planes of cut in the BRDF space. Right, the same planes in three dimensions.

Fig. 6
Fig. 6

(Color online) Angular notation for the calculation of the perpendicular width 2 δ .

Fig. 7
Fig. 7

(Color online) A scratched sample can be assimilated to an anisotropic surface with parallel straight lines. The reference plane is the plane perpendicular to the straight lines on the sample. The reference plane is set at φ = 0 ° .

Fig. 8
Fig. 8

Evolution of the BRDF of an anisotropic surface (scratched silicon wafer) for different azimuths of the incident light according to the reference plane. Left: φ I = 180 ° , middle: φ I = 165 ° , right: φ I = 142 ° ( θ I = 45 ° ) .

Fig. 9
Fig. 9

Theoretical evolution of the scattering plane for a brushed surface. Left, the scattering planes for different values of φ I and an incident angle θ I = 45 ° . Right, the scattering planes when φ I = 45 ° for different angles of incidence θ I .

Fig. 10
Fig. 10

Evolution of the BRDF of an isotropic surface in the new BRDF space, here the same measurements as those proposed in Fig. 3. The deformation of the specular peak is not due to the representation. The white curve is the isocurve at half-maximum.

Fig. 11
Fig. 11

(Color online) Two possible planes for the calculations of a perpendicular width represented in the new BRDF space. The perpendicularity of case c is not present in this space contrary to case d. Example with θ I = 45 ° and φ I = 180 ° . Left, the planes of cut (related to cases c and d) in the new BRDF space. Right, the representation of the planes in three dimensions.

Fig. 12
Fig. 12

Variation of the peak width 2 δ in terms of θ N for θ I = 45 ° for cases c and d.

Fig. 13
Fig. 13

Evolution of the BRDF of an anisotropic surface in the new BRDF space for different azimuths of the incident light according to the scattering plane, here the same measurements as those proposed in Figure 8. The straight lines give the orientations of the scratches.

Equations (110)

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

( sr 1 )
θ I
φ I
θ S = θ I
φ S = π φ I
θ R
φ R
2 ξ
ξ = 1 2   arccos [ cos θ R cos θ I + sin θ R sin θ I cos ( φ I φ R ) ] ,
θ N = arccos ( cos   θ R + cos θ I 2 cos ξ ) ,
φ N = φ I ± arccos ( cos ξ cos θ N cos θ I sin θ N sin θ I ) .
ξ = arccos [ cos θ N cos θ I + sin θ N sin θ I cos ( φ I φ N ) ] ,
θ R = arccos ( 2 cos θ N cos ξ cos θ I )
φ R = φ I ± arccos ( cos 2 ξ cos θ R cos θ I sin θ R sin θ I ) .
θ I
φ I
θ R
φ R
θ I
φ I
θ R
φ R
θ R
φ R
10 × 18 × 36 = 6480
θ R
φ R
θ R
φ R
X = θ R cos φ R
Y = θ R  sin  φ R .
I = H ( θ I = 0 ° )
θ N = constant
0 ° < φ N < 360 °
θ N = 5 °
θ I = constant
θ I
θ I = 0 °
θ R = θ I .
θ I < θ N
θ R cos φ R = θ I .
( θ R = 90 ° ; φ R = 90 ° )
( θ R = 90 ° ; φ R = 270 ° )
( θ S = θ I , φ S = 0 ° )
sin θ R cos φ R = tan θ I cos θ R .
φ N = ± 90 ° .
R HM
R HM I S
cos δ = 1 2 cos 2 θ I 2 cos 2 ξ + 4 cos θ I cos ξ cos θ N .
φ R HM
R HM
θ R HM
φ R HM
cos δ = cos θ I cos θ R HM + sin θ R HM sin θ I cos ( φ I φ R HM ) .
cos ξ
cos ξ
2 δ 4 θ N cos θ I .
4 θ N
φ = 0
θ I
φ I
φ I = 180 °
φ I 180 °
φ N
0 ° < θ N < 90 °
( θ S = θ I , φ S = 0 ° )
θ I = 0 °
φ R = φ N
φ I = φ N = 0 °
( θ I , φ I , θ R , φ R )
( θ N , φ N , ξ , ψ N )
( θ I , φ I , θ R , φ R )
( θ I , φ I , θ N , φ N )
( θ R cos φ R ; θ R sin φ R )
X = θ N cos φ N ,
Y = θ N  sin  φ N .
θ R = 90 °
( X = 0 )
φ I = 180 °
cos δ = 1 2 sin 2 θ N cos 2 θ I .
θ N
θ I
( θ R , φ R )
N ( θ N , φ N )
θ I = 45 °
φ I = 180 °
( θ S = 45 ° , φ S = 0 ° )
θ I = 20 °
θ I = 40 °
θ I = 60 °
( φ I = 180 ° )
( φ I = 180 ° )
θ N = 5 °
θ I
θ I = 45 °
φ I = 180 °
2 δ
φ = 0 °
φ I = 180 °
φ I = 165 °
φ I = 142 ° ( θ I = 45 ° )
φ I
θ I = 45 °
φ I = 45 °
θ I
θ I = 45 °
φ I = 180 °
2 δ
θ N
θ I = 45 °

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