Abstract

We present a reflection model for isotropic rough surfaces that have both specular and diffuse components. The surface is assumed to have a normal distribution of heights. Parameters of the model are the surface roughness given by the rms slope, the albedo, and the balance between diffuse and specular reflection. The effect of roughness on diffuse reflection is taken into account, instead of our modeling this component as a constant Lambertian term. The model includes geometrical effects such as masking and shadowing. The model is compared with experimental data obtained from goniophotometric measurements on samples of tiles and bricks. The model fits well to samples with very different reflection properties. Measurements of the sample profiles performed with a laser profilometer to determine the rms slope show that the assumed surface model is realistic. The model could therefore be used in machine vision and computer graphics to approximate reflection characteristics of surfaces. It could also be used to predict the texture of surfaces as a function of illumination and viewing angles.

© 1998 Optical Society of America

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References

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  1. J. H. Lambert, Photometria sive de mensura et gradibus luminus, colorum et umbrae (Eberhard Klett, Augsburg, Germany, 1760).
  2. K. Torrance, E. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  3. S. K. Nayar, K Ikeuchi, T Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991), Appendix D.
  4. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  5. D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449–454 (1968).
    [CrossRef]
  6. J. C. Leader, “Analysis and prediction of laser scattering from rough-surface materials,” J. Opt. Soc. Am. 69, 610–628 (1979).
    [CrossRef]
  7. X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
    [CrossRef]
  8. M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
    [CrossRef]
  9. M. S. Longuet-Higgins, “Reflection and refraction at a random moving surface. II: Number of specular points in a Gaussian surface,” J. Opt. Soc. Am. 50, 845–850 (1960).
    [CrossRef]
  10. M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
    [CrossRef]
  11. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
    [CrossRef]
  12. D. Middleton, Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).
  13. T. R. Thomas, Rough Surfaces (Longmans, London, 1982).
  14. P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
    [CrossRef]
  15. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
    [CrossRef]
  16. B. G. Smith, “Lunar surface roughness: Shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
    [CrossRef]
  17. R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–629 (1966).
    [CrossRef]
  18. D. J. Schertler, N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A 11, 2286–2297 (1994).
    [CrossRef]
  19. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).
  20. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, Berlin, 1969).
    [CrossRef]
  21. P. E. Gill, W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numeri. Anal. 15, 977–992 (1978).
    [CrossRef]
  22. T. S. Trowbridge, K. P. Reitz, “Average irregularity representation of a rough surface for ray reflection,” J. Opt. Soc. Am. 65, 531–536 (1975).
    [CrossRef]
  23. F. Abdellani, G. Rasigni, M. Rasigni, A. Lleberia, “Distributions of zero crossings for the profile of random rough surfaces,” Appl. Opt. 31, 4534–4539 (1992).
    [CrossRef] [PubMed]
  24. L. B. Wolff, “Diffuse-reflectance model for smooth dielectric surfaces,” J. Opt. Soc. Am. A 11, 2956–2968 (1994).
    [CrossRef]
  25. K. Lumme, J. Peltoniemi, W. M. Irvine, “Some photometric techniques for atmosphereless solar system bodies,” Adv. Space Res. 10, 1187–1193 (1990).
    [CrossRef]

1995 (1)

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

1994 (2)

1992 (1)

1991 (2)

S. K. Nayar, K Ikeuchi, T Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991), Appendix D.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
[CrossRef]

1990 (1)

K. Lumme, J. Peltoniemi, W. M. Irvine, “Some photometric techniques for atmosphereless solar system bodies,” Adv. Space Res. 10, 1187–1193 (1990).
[CrossRef]

1979 (1)

1978 (1)

P. E. Gill, W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numeri. Anal. 15, 977–992 (1978).
[CrossRef]

1975 (1)

1969 (1)

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

1968 (1)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449–454 (1968).
[CrossRef]

1967 (3)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

B. G. Smith, “Lunar surface roughness: Shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

K. Torrance, E. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
[CrossRef]

1966 (2)

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–629 (1966).
[CrossRef]

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
[CrossRef]

1965 (1)

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
[CrossRef]

1960 (1)

Abdellani, F.

Barrick, D. E.

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449–454 (1968).
[CrossRef]

Beckmann, P.

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Brockelman, R. A.

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–629 (1966).
[CrossRef]

George, N.

Gill, P. E.

P. E. Gill, W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numeri. Anal. 15, 977–992 (1978).
[CrossRef]

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
[CrossRef]

Hagfors, T.

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–629 (1966).
[CrossRef]

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
[CrossRef]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Ikeuchi, K

S. K. Nayar, K Ikeuchi, T Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991), Appendix D.

Irvine, W. M.

K. Lumme, J. Peltoniemi, W. M. Irvine, “Some photometric techniques for atmosphereless solar system bodies,” Adv. Space Res. 10, 1187–1193 (1990).
[CrossRef]

Kanade, T

S. K. Nayar, K Ikeuchi, T Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991), Appendix D.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, Berlin, 1969).
[CrossRef]

Lambert, J. H.

J. H. Lambert, Photometria sive de mensura et gradibus luminus, colorum et umbrae (Eberhard Klett, Augsburg, Germany, 1760).

Leader, J. C.

Lleberia, A.

Longuet-Higgins, M. S.

Lumme, K.

K. Lumme, J. Peltoniemi, W. M. Irvine, “Some photometric techniques for atmosphereless solar system bodies,” Adv. Space Res. 10, 1187–1193 (1990).
[CrossRef]

Middleton, D.

D. Middleton, Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).

Murray, W.

P. E. Gill, W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numeri. Anal. 15, 977–992 (1978).
[CrossRef]

Nayar, S. K.

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

S. K. Nayar, K Ikeuchi, T Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991), Appendix D.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Oren, M.

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

Peltoniemi, J.

K. Lumme, J. Peltoniemi, W. M. Irvine, “Some photometric techniques for atmosphereless solar system bodies,” Adv. Space Res. 10, 1187–1193 (1990).
[CrossRef]

Rasigni, G.

Rasigni, M.

Reitz, K. P.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Sancer, M. I.

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

Schertler, D. J.

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
[CrossRef]

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

B. G. Smith, “Lunar surface roughness: Shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

Sparrow, E.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Thomas, T. R.

T. R. Thomas, Rough Surfaces (Longmans, London, 1982).

Torrance, K.

Torrance, K. E.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
[CrossRef]

Trowbridge, T. S.

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
[CrossRef]

Wolff, L. B.

ACM Comput. Graphics (1)

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” ACM Comput. Graphics 25, 175–186 (1991).
[CrossRef]

Adv. Space Res. (1)

K. Lumme, J. Peltoniemi, W. M. Irvine, “Some photometric techniques for atmosphereless solar system bodies,” Adv. Space Res. 10, 1187–1193 (1990).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (5)

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
[CrossRef]

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449–454 (1968).
[CrossRef]

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–629 (1966).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. K. Nayar, K Ikeuchi, T Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991), Appendix D.

Int. J. Comput. Vision (1)

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

J. Acoust. Soc. Am. (1)

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
[CrossRef]

J. Geophys. Res. (1)

B. G. Smith, “Lunar surface roughness: Shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

J. Opt. Soc. Am. (4)

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

SIAM J. Numeri. Anal. (1)

P. E. Gill, W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numeri. Anal. 15, 977–992 (1978).
[CrossRef]

Other (6)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, Berlin, 1969).
[CrossRef]

D. Middleton, Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).

T. R. Thomas, Rough Surfaces (Longmans, London, 1982).

J. H. Lambert, Photometria sive de mensura et gradibus luminus, colorum et umbrae (Eberhard Klett, Augsburg, Germany, 1760).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

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Figures (7)

Fig. 1
Fig. 1

Coordinate system.

Fig. 2
Fig. 2

Measured radiance from sample 1 (a fairly smooth tile) with r = 0.12, determined by profilometer measurements. Radiance is given as a function of θ r , for θ i = 45° (◇), θ i = 60° (+), and θ i = 75° (×). The thick curve is the best fit for the present model when Eq. (35) is used with C = 15.1, ρ = 0.484, and n = 2.17. The thin curve (not visible because it coincides with the thick curve) is the best fit for the TSON model with C = 15.2, ρ = 0.485, and n = 2.18.

Fig. 3
Fig. 3

Measured radiance from sample 2 (a rough pink tile) with r = 0.51, determined by profilometer measurements. Radiance is given as a function of θ r , for θ i = 45° (◇), θ i = 60° (+), and θ i = 75° (×). The thick curve is the best fit for the present model when Eq. (34) is used with C = 39.8 and g = 0.00501. The thin curve (not visible because it coincides with the thick curve) is the best fit for the TSON model with C = 45.6 and g = 0.0903.

Fig. 4
Fig. 4

Measured radiance from sample 3 (a grey concrete brick) with r = 0.49, determined by profilometer measurements. Radiance is given as a function of θ r , for θ i = 45° (◇), θ i = 60° (+), and θ i = 75° (×). The thick curve is the best fit for the present model when Eq. (34) is used with C = 68.5 and g = 0.0200. The thin curve is the best fit for the TSON model with C = 75.07 and g = 0.0385.

Fig. 5
Fig. 5

Measured radiance from sample 4 (a red brick thinly painted with matte white latex paint) with r = 0.43, determined by profilometer measurements. Radiance is given as a function of θ r , for θ i = 45° (◇), θ i = 60° (+), and θ i = 75° (×). The thick curve is the best fit for the present model when Eq. (34) is used with C = 126 and g = 0.00527. The thin curve is the best fit for the TSON model with C = 134 and g = 0.0102.

Fig. 6
Fig. 6

Illuminated and visible portion of the surface as a function of θ r for various values of θ i . The values computed from the profiles of sample 2, measured by a profilometer, are denoted ◇. For these profiles a rms slope of r = 0.51 was computed. The bistatic shadowing function for this r is plotted as a solid curve. Computed values for computer-generated profiles with r = 0.51 are denoted +.

Fig. 7
Fig. 7

Left, texture pattern generated by use of the present theory (θ i = 75°, θ r = 0°, r = 0.50, g = 0). Right, image taken with a CCD camera from the white painted brick (sample 4) for θ i = 75°, θ r = 0°. For this sample r = 0.43, determined by profilometer measurements; and g = 0.0143, determined by model fitting.

Tables (1)

Tables Icon

Table 1 Statistical Measures of Sample Surfaces

Equations (38)

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cos   θ i = i ˆ , a ˆ = cos   ϕ a sin   θ i sin   θ a + cos   θ i cos   θ a ,
cos   θ r = r ˆ , a ˆ = cos ϕ a - ϕ r sin   θ r sin   θ a + cos   θ r cos   θ a .
P z z , σ d z = 1 2 π σ exp - z 2 2 σ 2 d z ,
P ϕ a d ϕ a = 1 2 π d ϕ a .
P z x z x , r d z x = 1 2 π r exp - z x 2 2 r 2 d z x ,
P z m z m , r d z m = z m r 2 exp - z m 2 2 r 2 d z m .
P θ a θ a , r d θ a = sin   θ a r 2 cos 3   θ a exp - tan 2   θ a 2 r 2 d θ a .
P d ω a θ a , r d ω a = 1 π U ( - 1 / 2 , 0 , 1 / 2 r 2 ) r 2 cos 3   θ a × exp - tan 2   θ a 2 r 2 d ω a ,
if   θ a π 2 - θ i , then - π < ϕ a π , else   a < ϕ a < b , where   a = - π + arccos cot   θ i cot   θ a , b = π - arccos cot   θ i cot   θ a .
if   θ a π 2 - θ r , then - π < ϕ a π , else   a < ϕ a < b , where   a = ϕ r - π + arccos cot   θ r cot   θ a , b = ϕ r + π - arccos cot   θ r cot   θ a .
P ill + vis = P ill P vis | ill ,
P ill = exp - 0   g τ d τ ,
g τ = cot θ i d z x z x - cot θ i P τ z = z + τ cot θ i - d z x - z + τ cot θ i d z P τ ,
P ill θ i , r , z , σ d z = 1 - 1 2 erfc z 2 σ Λ r , θ i d z ,
Λ r , θ i = r 2 π cot | θ i | exp - cot 2   θ i 2 r 2 - 1 2 erfc cot | θ i | 2 r .
P ill θ i , r = -   P ill θ i , r , z , σ P z z , σ d z = 1 1 + Λ r , θ i .
P ill + vis θ i , θ r , ϕ r = 0 , r = 1 1 + Λ ( r , max θ i , θ r ) .
P ill + vis θ i , θ r , ϕ r π , r 1 1 + Λ r , θ i + Λ r , θ r .
P τ = C τ exp - z 2 2 1 + erf z + τ   cot   θ i - ρ z 2 1 - ρ 2 1 / 2 ,
ρ = R d = exp - d 2 T 2 ,
d = 2 τ   sin ϕ r 2 ,
P z | ill θ i , z , σ , r d z = 1 + Λ r , θ i × 1 - 1 2 erfc z 2 σ Λ r , θ i P z d z .
P ill + vis θ i , θ r , ϕ r , r 1 1 + Λ r , max θ i , θ r + α Λ r , min θ i , θ r .
α = 4.41 ϕ r 4.41 ϕ r + 1 ,
θ a   spec = arccos cos   θ i + cos   θ r cos   ϕ r sin   θ r + sin   θ i 2 + sin 2   ϕ r sin 2   θ r + cos   θ i + cos   θ r 2 - 1 / 2 .
d ω a = d ω r 4   cos   θ i .
L rs θ i , θ r , ϕ r = E 0 cos   θ i cos   θ a d A   cos   θ r d ω r .
L rs θ i , θ r , ϕ r , r = C s P ill + vis θ i , θ r , ϕ r , r cos   θ r cos 4   θ a   spec × exp - tan 2   θ a   spec 2 r 2 ,
C s = E 0 4 π U ( - 1 / 2 , 0 , 1 / 2 r 2 ) .
L rd θ i , θ r , ϕ r , θ a , ϕ a = ρ π   E 0 cos   θ i   cos   θ r cos   θ a d A   cos   θ r ,
L rd θ i , θ r , ϕ r , r = P vis + ill θ i , θ r , ϕ r , r × 0 π / 2 a b   L rd θ i , θ r , ϕ r , θ a , ϕ a P ϕ a d ϕ a P θ a θ a , r d θ a .
a b   L rd θ i , θ r , ϕ r , θ a , ϕ a P ϕ a d ϕ a = C d c 1 2 + c 5 b - a + c 1 4 sin   2 b - sin   2 a + c 2 4 cos   2 a - cos   2 b + c 3 sin   b - sin   a + c 4 cos   a - cos   b ,
C d = ρ E 0 2 π 2 cos   θ r cos   θ a , c 1 = sin   θ i sin 2   θ a cos   ϕ r sin   θ r , c 2 = sin   θ i sin 2   θ a sin   ϕ r sin   θ r , c 3 = sin   θ a cos   θ a sin   θ i cos   θ r + cos   θ i cos   ϕ r sin   θ r , c 4 = cos   θ i cos   θ a sin   ϕ r sin   θ r sin   θ a , c 5 = cos   θ i cos   θ r cos 2   θ a ,
L r θ i , θ r , ϕ r , r , g , C = C gL rs θ i , θ r , ϕ r , r + 1 - g L rd θ i , θ r , ϕ r , r
L r θ i , θ r , ϕ r , r , C , n , ρ = C F n , θ i × L rs θ i , θ r , ϕ r , r + 1 - F n , θ i × L rd θ i , θ r , ϕ r , r , ρ .
F n , θ i = 1 2 sin 2 θ i - θ t sin 2 θ i + θ t + tan 2 θ i - θ t tan 2 θ i + θ t ,
with   θ t = arcsin sin   θ i n .
R a = -   | z | P z z , σ d z = - | z | 2 π σ exp - z 2 2 σ 2 d z = 2 π   σ 0.80 σ .

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