Abstract

A statistical ray method is developed for deriving reflection models of rough surfaces. Using this method, we derive generic equations of reflection due to single scattering. We have studied an isotropic surface with Gaussian statistics, and the derived bidirectional reflectance distribution function is found with four distinct regimes, namely, mirror reflection, grazing reflection, retroreflection, and normal reflection. An explicit form of self-shadowing for a surface with Gaussian statistics is also derived, and the result agrees well with computer simulation. Our solution is useful to describe the entire reflection of a highly or moderately smooth surface and offers a basis for studying multiple scattering. While focusing on optical reflection, the approach also applies to other waves if the assumptions are satisfied.

© 2007 Optical Society of America

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2004 (1)

2002 (2)

R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41, 988-993 (2002).
[CrossRef]

J. Caron, J. Lafait, and C. Andraud, 'Scalar Kirchhoff's model for light scattering from dielectric random rough surfaces," Opt. Commun. 207, 17-28 (2002).
[CrossRef]

1999 (2)

1998 (2)

B. van Ginneken, M. Stavridi, and J. J. Koenderink, "Diffuse and specular reflectance from rough surfaces," Appl. Opt. 37130-139 (1998).
[CrossRef]

L. B. Wolff, S. K. Nayar, and M. Oren, "Improved diffuse reflection models for computer vision," Int. J. Comput. Vis. 30, 55-71 (1998).
[CrossRef]

1997 (1)

1996 (1)

R. Bajcsy, S. W. Lee, and A. Leonardis, "Detection of diffuse and specular interface reflections and interreflections by color image segmentation," Int. J. Comput. Vis. 17, 241-271 (1996).
[CrossRef]

1994 (1)

C. Schlick, "A survey of shading and reflectance models," Comput. Graph. Forum 13, 121-131 (1994).
[CrossRef]

1991 (1)

S. K. Nayar, K. Ikeuchi, and T. Kanade, "Surface reflection: physical and geometrical perspectives," IEEE Trans. Pattern Anal. Mach. Intell. 13, 611-634 (1991).
[CrossRef]

1990 (2)

P. Strauss, "A realistic lighting model for computer animators," IEEE Comput. Graphics Appl. 10, 56-64 (1990).
[CrossRef]

E. Marx and T. V. Vorburger, "Direct and inverse problems for light scattered by rough surfaces," Appl. Opt. 29, 3613-3626 (1990).
[CrossRef] [PubMed]

1987 (1)

1985 (1)

D. H. Berman and J. S. Perkins, "Exponential substitution for Kirchhoff scattering from Gaussian rough surfaces," J. Acoust. Soc. Am. 78, 1024-1028 (1985).
[CrossRef]

1982 (2)

M. Nieto-Vesperinas, "Radiometry of rough surfaces," Opt. Acta 29, 961-971 (1982).
[CrossRef]

R. L. Cook and K. E. Torrance, "A reflection model for computer graphics," ACM Trans. Graphics 1, 7-24 (1982).
[CrossRef]

1979 (2)

J. C. Leader, "Analysis and prediction of laser scattering from rough-surface materials," J. Opt. Soc. Am. 69, 610-628 (1979).
[CrossRef]

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Relationship between surface scattering and microtopographic features," Opt. Eng. 18, 125-136 (1979).

1977 (1)

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360-374 (1977).

1976 (1)

P. J. Chandley, "Surface roughness measurements from coherent light scattering," Opt. Quantum Electron. 8, 323-327 (1976).
[CrossRef]

1975 (2)

1971 (1)

1969 (1)

M. I. Sancer, "Shadow-corrected electromagnetic scattering from a randomly rough surfaces," IEEE Trans. Antennas Propag. AP-17, 577-585 (1969).
[CrossRef]

1968 (1)

D. E. Barrick, "Rough surfaces scattering based on the specular point theory," IEEE Trans. Antennas Propag. 16, 449-454 (1968).
[CrossRef]

1967 (5)

A. Stogryn, "Electromagnetic scattering from rough, finitely conducting surfaces," Radio Sci. 2, 415-428 (1967).

R. J. Wagner, "Shadowing of randomly rough surfaces," J. Acoust. Soc. Am. 41, 138-147 (1967).
[CrossRef]

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

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

B. G. Smith, "Lunar surface roughness: shadowing and thermal emission," J. Geophys. Res. 72, 4059-4067 (1967).
[CrossRef]

1965 (2)

P. Beckmann, "Shadowing of random rough surfaces," IEEE Trans. Antennas Propag. AP-13, 384-388 (1965).
[CrossRef]

R. A. Brockelman and T. Hagfors, "Note on the effect of shadowing on the backscattering of waves from a random rough surface," IEEE Trans. Antennas Propag. AP-14, 621-626 (1965).

1964 (1)

T. Hagfors, "Backscattering from an undulating surface with applications to radar returns from the moon," J. Geophys. Res. 69, 3779-3784 (1964).
[CrossRef]

Andraud, C.

J. Caron, J. Lafait, and C. Andraud, 'Scalar Kirchhoff's model for light scattering from dielectric random rough surfaces," Opt. Commun. 207, 17-28 (2002).
[CrossRef]

Ashikhmin, M.

M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (ACM, Addison-Wesley, 2000), pp. 65-74.

Bajcsy, R.

R. Bajcsy, S. W. Lee, and A. Leonardis, "Detection of diffuse and specular interface reflections and interreflections by color image segmentation," Int. J. Comput. Vis. 17, 241-271 (1996).
[CrossRef]

Barrick, D. E.

D. E. Barrick, "Rough surfaces scattering based on the specular point theory," IEEE Trans. Antennas Propag. 16, 449-454 (1968).
[CrossRef]

Bass, F. G.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).

Beckmann, P.

P. Beckmann, "Shadowing of random rough surfaces," IEEE Trans. Antennas Propag. AP-13, 384-388 (1965).
[CrossRef]

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

Bennett, H. E.

J. M. Elson, H. E. Bennett, and J. M. Bennett, "Scattering from optical surfaces," in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1979), pp. 191-244.

Bennett, J. M.

J. M. Elson, H. E. Bennett, and J. M. Bennett, "Scattering from optical surfaces," in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1979), pp. 191-244.

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

Berman, D. H.

D. H. Berman and J. S. Perkins, "Exponential substitution for Kirchhoff scattering from Gaussian rough surfaces," J. Acoust. Soc. Am. 78, 1024-1028 (1985).
[CrossRef]

Bhandarkar, S. M.

M. Suk and S. M. Bhandarkar, Computer Science Workbench: Three-Dimensional Object Recognition from Range Image, T.L.Kunii, ed. (Springer-Verlag, 1992).
[CrossRef]

Blinn, J. F.

J. F. Blinn, "Models of light reflection for computer synthesized pictures," in Proceedings of ACM, SIGGRAPH 1977 (ACM, 1977), pp. 192-198.

Blunt, L.

K. J. Stout and L. Blunt, Three-Dimensional Surface Topography (Penton, 2000).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, 1975).
[PubMed]

Brockelman, R. A.

R. A. Brockelman and T. Hagfors, "Note on the effect of shadowing on the backscattering of waves from a random rough surface," IEEE Trans. Antennas Propag. AP-14, 621-626 (1965).

Cabral, B.

B. Cabral, N. Max, and R. Springmeyer, "Bidirectional reflectance functions from surface bump maps," in Proceedings of ACM SIGGRAPH (ACM, 1987), pp. 273-281.
[CrossRef]

Calvert, T. W.

Y. Sun, F. D. Fracchia, M. S. Drew, and T. W. Calvert, "Rendering iridescent colors of optical disks," in Proceedings of the 11th EUROGRAPHICS Workshop on Rendering (Springer-Verlag, 2000), pp. 341-352.

Caron, J.

J. Caron, J. Lafait, and C. Andraud, 'Scalar Kirchhoff's model for light scattering from dielectric random rough surfaces," Opt. Commun. 207, 17-28 (2002).
[CrossRef]

Chandley, P. J.

P. J. Chandley, "Surface roughness measurements from coherent light scattering," Opt. Quantum Electron. 8, 323-327 (1976).
[CrossRef]

Chipman, J. W.

T. M. Lillesand, R. W. Kiefer, and J. W. Chipman, Remote Sensing and Image Interpretation (Wiley, 2004).

Church, E. L.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Relationship between surface scattering and microtopographic features," Opt. Eng. 18, 125-136 (1979).

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360-374 (1977).

E. L. Church and J. M. Zavada, "Residual surface roughness of diamond-turned optics," Appl. Opt. 14, 1788-1795 (1975).
[CrossRef] [PubMed]

E. L. Church and P. Z. Takacs, "Surface scattering," in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), pp. 7.1-7.14.

Cook, R. L.

R. L. Cook and K. E. Torrance, "A reflection model for computer graphics," ACM Trans. Graphics 1, 7-24 (1982).
[CrossRef]

Depew, C. A.

Drew, M. S.

Y. Sun, F. D. Fracchia, M. S. Drew, and T. W. Calvert, "Rendering iridescent colors of optical disks," in Proceedings of the 11th EUROGRAPHICS Workshop on Rendering (Springer-Verlag, 2000), pp. 341-352.

Elson, J. M.

J. M. Elson, H. E. Bennett, and J. M. Bennett, "Scattering from optical surfaces," in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1979), pp. 191-244.

Feiner, S. K.

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics: Principles and Practice, 2nd ed. (Addison-Wesley, 1996).

Foley, J. D.

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics: Principles and Practice, 2nd ed. (Addison-Wesley, 1996).

Foo, S.

E. P. F. Lafortune, S. Foo, K. E. Torrance, and D. P. Greenberg, "Non-linear approximation of reflectance functions," in Proceedings of SIGGRAPH (ACM, Addison-Wesley, 1997), pp. 117-126.
[CrossRef]

Fracchia, F. D.

Y. Sun, F. D. Fracchia, M. S. Drew, and T. W. Calvert, "Rendering iridescent colors of optical disks," in Proceedings of the 11th EUROGRAPHICS Workshop on Rendering (Springer-Verlag, 2000), pp. 341-352.

Freund, J. E.

J. E. Freund, Mathematical Statistics (Prentice Hall, 1962).

Fuks, I. M.

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Figures (15)

Fig. 1
Fig. 1

Geometry and notations for defining the BRDF.

Fig. 2
Fig. 2

Microscopic view of a rough surface and ray scattering. The path from 3 to 3 occurs if the medium is transparent. The path from 4 to 4 involves subsurface scattering, which occurs if the material is translucent.

Fig. 3
Fig. 3

(a) Object with a marked local illumination area Δ A . (b) The surface profile for Δ A with a marked microarea δ A . (c) The enlarged view of δ A . The size of the object is much larger than Δ A , which is much larger than δ A .

Fig. 4
Fig. 4

Minimal resolvable angle α m , length l m , and area a m D . Angle β is small if the object is located in front of the eye.

Fig. 5
Fig. 5

Surface normal h of a microarea δ A equally subdivides the lighting direction l and the viewing direction v.

Fig. 6
Fig. 6

Surface normal h and gradient direction s of microarea δ A and the associated angles. Directions h, s, and z are coplanar.

Fig. 7
Fig. 7

Ray starts from x = 0 at height ζ 1 at polar angle θ. The height at distance x is ζ 2 .

Fig. 8
Fig. 8

Relations between w and x. The curve of w ( x ) is approximated with the line x = σ w tan θ for w w 0 .

Fig. 9
Fig. 9

Determining k 0 in visibility function V ( θ ) based on cases (a) s = 1 , (b) s = 2 , (c) s = 4 , and (d) s = 10 . In each case the solid curves are V ( θ ) for k 0 equal to 0.5, 0.6, 0.7, 0.8 and 1, and the points in the diamond shape are from Brockelman and Hagfors’s computer simulation. In all cases, V ( θ ) with k 0 = 0.7 matches the simulation data very well.

Fig. 10
Fig. 10

Curves of the visibility function V ( θ ) with k 0 = 0.7 against the polar angle for different values of surface smoothness s.

Fig. 11
Fig. 11

Curves of D ( θ h ) for different values of s. When s 1 (rough surfaces), D ( θ h ) shows a peak at a large θ h . When s 5 (smooth surfaces), D ( θ h ) shows a peak at θ h = 0 . The curves for 1 s 5 show how D ( θ h ) varies gradually from the case of rough to smooth surfaces.

Fig. 12
Fig. 12

Slope angle and the corresponding polar angle within the incident plane.

Fig. 13
Fig. 13

Derived BRDF excluding F ¯ ( α , λ ) against the slope angle γ v for various values of surface smoothness parameter s. The incident direction is γ l = 3 π 4 . The curves show how the BRDF changes as s increases.

Fig. 14
Fig. 14

Derived BRDF excluding F ¯ ( α , λ ) as a function of the outgoing direction ( θ v , φ v ) . The incident direction is given by θ l = π 4 and ϕ l = π . (a) s = 0.2 (normal reflection regime). (b) s = 1 (back reflection regime). (c) s = 4 (grazing reflection regime). (d) s = 10 (mirror-reflection regime).

Fig. 15
Fig. 15

Microscopic view of a rough surface and ray scattering. Parameters σ and τ are the surface height deviation and correlation length.

Tables (1)

Tables Icon

Table 1 Comparison of the Approaches for Deriving Reflection Models

Equations (124)

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ρ ( θ l , φ l , θ v , φ v , λ ) d L v ( θ v , φ v , λ ) d E l ( θ l , φ l , λ ) = d L v ( θ v , φ v , λ ) L l ( θ l , φ l , λ ) cos θ l d ω l ,
Φ surf = Φ single + Φ multiple .
h = l + v 2 cos α ,
L v ( θ v , φ v ) = F ¯ ( α , λ ) L l ( θ l , φ l ) δ ( h , v , l ) .
δ ( h , v , l ) = { 1 when h = l + v 2 cos α 0 otherwise .
δ Φ ( θ v , φ v ) = L v ( θ v , φ v ) cos α δ A d ω l = F ¯ ( α , λ ) L l ( θ l , φ l ) δ ( h , v , l ) cos α δ A d ω l .
Φ Δ A ( θ v , φ v ) = L l ( θ l , φ l ) cos α d ω l Δ A F ¯ ( α , λ ) V ( δ A ) δ ( h , v , l ) δ A .
L Δ A ( θ v , φ v ) = Φ Δ A ( θ v , φ v ) Δ A cos θ v d ω v = L l ( θ l , φ l ) cos α F ¯ ( α , λ ) d ω l Δ A cos θ v d ω v Δ A V ( δ A ) δ ( h , v , l ) δ A .
ρ single ( θ l , φ l , θ v , φ v , λ ) = cos α F ¯ ( α , λ ) Δ A cos θ l cos θ v d ω v Δ A V ( δ A ) δ ( h , v , l ) δ A .
( δ A ) = δ A cos θ h ,
( δ A ) δ ( h , v , l ) Δ A = p ( ζ , ζ x , ζ y ) d ζ d ζ x d ζ y ,
p ( ζ ) = p ( ζ , ζ x , ζ y ) d ζ x d ζ y ,
p ( ζ x ) = p ( ζ , ζ x , ζ y ) d ζ d ζ y ,
p ( ζ y ) = p ( ζ , ζ x , ζ y ) d ζ d ζ x ,
ρ single ( θ l , φ l , θ v , φ v , λ ) = cos α F ¯ ( α , λ ) cos θ h cos θ l cos θ v d ω v d ζ x d ζ y d ζ p ( ζ , ζ x , ζ y ) V ( ζ , l , v ) .
ζ = h ( x , y ) ;
ζ x = h ( x , y ) x , ζ y = h ( x , y ) y .
r = ( x , y , h ( x , y ) ) .
f ( x , y , z ) z h ( x , y ) ,
h = f f ,
f ( x , y , z ) = ( f ( x , y , z ) x , f ( x , y , z ) y , f ( x , y , z ) z ) = ( ζ x , ζ y , 1 ) .
ζ = h ( x , y ) h ( x , y ) R s = tan θ h ,
ζ x = h ( x , y ) x = R s x h ( x , y ) R s = cos φ s ζ ,
ζ y = h ( x , y ) y = R s y h ( x , y ) R s = sin φ s ζ .
ζ x 2 + ζ y 2 = ζ 2 ,
ζ y ζ x = tan φ s .
d ζ x d ζ y = ζ d ζ d φ s = tan θ h d ( tan θ h ) d φ h = sin θ h d θ h d φ h cos 3 θ h = d ω h cos 3 θ h ,
d ω h = d ω v ( 4 cos α ) .
ρ single ( θ l , φ l , θ v , φ v , λ ) = F ¯ ( α , λ ) 4 cos θ l cos θ v cos 4 θ h d ζ p ( ζ , ζ x , ζ y ) V ( ζ , l , v ) .
p ( ζ , ζ x , ζ y ) = p ( ζ ) p ( ζ x ) p ( ζ y ) ,
ρ single ( θ l , φ l , θ v , φ v , λ ) = F ¯ ( α , λ ) p ( ζ x ) p ( ζ y ) 4 cos θ l cos θ v cos 4 θ h V ( l , v ) ,
V ( l , v ) V ( ζ , l , v ) = d ζ p ( ζ ) V ( ζ , l , v )
cos θ h = cos θ l + cos θ v 2 cos α = cos θ l + cos θ v 2 + 2 cos ( 2 α ) ,
cos 2 α = l v = sin θ l sin θ v cos ( φ l φ v ) + cos θ l cos θ v .
ζ = h ( x ) .
d T ( x ) = T ( x + d x ) T ( x ) = k T ( x ) Q ( x ) d x ,
Q ( x ) = Prob ( h ( x ) > ζ 1 + x cot θ )
Q ( x ) = ζ 1 + x cot θ + p ( ζ 2 ζ 1 ) d ζ 2 ,
p ( ζ 2 ζ 1 ) = p ( ζ 1 , ζ 2 ) p ( ζ 1 ) ,
d ln T ( x ) = d T ( x ) T ( x ) = k Q ( x ) d x .
T ( x ) = exp [ k 0 x Q ( x ) d x ] .
V ( ζ , d , l ) exp [ k 0 l Q ( x ) d x ]
V ( ζ , d ) = lim l V ( ζ , d , l ) = exp [ k 0 Q ( x ) d x ] .
V ( l , v ) = V ( θ l ) V ( θ v ) ,
V ( θ ) = V ( d ) = V ( ζ , d ) = d ζ p ( ζ ) V ( ζ , d ) .
p ( ζ ) = 1 2 π σ exp ( ζ 2 2 σ 2 ) = 1 σ g ( ζ σ ) ,
σ = 2 σ τ ,
p ( ζ x ) = 1 2 π σ exp ( ζ x 2 2 σ 2 ) = 1 σ g ( ζ x σ ) ,
p ( ζ y ) = 1 2 π σ exp ( ζ y 2 2 σ 2 ) = 1 σ g ( ζ y σ ) .
p ( ζ x ) p ( ζ y ) = 1 2 π σ 2 exp ( ζ 2 2 σ 2 ) = τ 2 4 π σ 2 exp ( τ 2 tan 2 θ h 4 σ 2 ) ,
ρ single ( θ l , φ , θ v , φ v , λ ) = τ 2 F ¯ ( α , λ ) exp ( τ 2 tan 2 θ h 4 σ 2 ) 16 π σ 2 cos θ l cos θ v cos 4 θ h V ( l , v ) .
p ( ζ 1 , ζ 2 ) = 1 2 π σ 2 1 C 2 exp [ ζ 1 2 2 ζ 1 ζ 2 C + ζ 2 2 2 σ 2 ( 1 C 2 ) ] .
p ( ζ 1 ζ 2 ) = 1 2 π σ 1 C 2 exp [ ( ζ 2 ζ 1 C ) 2 2 σ 2 ( 1 C 2 ) ] .
Q ( x ) = 1 2 π σ 1 C 2 ζ 1 + x cot θ + exp [ ( ζ 2 ζ 1 C ) 2 2 σ 2 ( 1 C 2 ) ] d ζ 2 = 1 G ( w ( x ) ) ,
w ( x ) ζ 1 ( 1 C ) + x cot θ σ 1 C 2 .
0 x Q ( x ) d x = x Q ( x ) 0 x x d Q ( x ) ,
0 x Q ( x ) d x = x [ 1 G ( w ( x ) ) ] 0 x x d ( 1 G ( w ( x ) ) ) = x [ 1 G ( w ( x ) ) ] + w 0 w ( x ) x g ( w ) d w ,
w 0 w ( 0 ) = τ cot θ 2 σ ,
V ( ζ 1 , θ ) = T ( x ) x .
0 Q ( x ) d x = w 0 x ( w ) g ( w ) d w ,
V ( ζ 1 , θ ) = exp [ k w 0 x ( w ) g ( w ) d w ] .
V ( θ ) V ( ζ 1 , θ ) = V ( ζ 1 , θ ) p ( ζ 1 ) d ζ 1 .
V ( θ ) V ( ζ 1 = 0 , θ ) ,
w ( x ) = x cot θ σ 1 C 2 .
w x cot θ σ .
x = σ w tan θ .
0 Q ( x ) d x = σ tan θ w 0 w g ( w ) d w = σ tan θ 2 π exp ( w 0 2 2 ) .
V ( θ ) = exp [ k σ tan θ 2 π exp ( τ 2 4 σ tan 2 θ ) ] .
k = 2 π k 0 τ ,
V ( θ ) = exp [ k 0 σ tan θ τ exp ( τ 2 4 σ 2 tan 2 θ ) ]
V ( θ ) = exp [ k 0 tan θ s exp ( s 2 4 tan 2 θ ) ] ,
s = τ σ .
r 1 s = σ τ .
ρ single ( θ l , φ l , θ v , φ v , λ ) = s 2 F ¯ ( α , λ ) exp ( s 2 tan 2 θ h 4 ) 16 π cos θ l cos θ v cos 4 θ h V ( θ l ) V ( θ v ) .
cos θ h = cos θ l + cos θ v 2 cos α = cos θ l + cos θ v 2 + 2 cos ( 2 α ) ,
cos 2 α = 1 v = sin θ l sin θ v cos ( φ l φ v ) + cos θ l cos θ v .
V ( θ ) = exp [ k 0 tan θ s exp ( s 2 4 tan 2 θ ) ] ,
ρ single ( θ l , φ l , θ v , φ v , λ ) = F ¯ ( α , λ ) χ ( θ l , φ l , θ v , φ v ) D ( θ h ) V ( θ l ) V ( θ v ) .
D ( θ h ) = s 2 exp ( s 2 tan 2 θ h 4 ) 4 π cos 3 θ h .
D ( θ h ) d ω h = s 2 exp ( s 2 tan 2 θ h 4 ) 4 π cos 3 θ h sin θ h d θ h d φ h = s 2 4 exp ( s 2 tan 2 θ h 4 ) d tan 2 θ h = 1 .
D ( θ h ) 1 2 π δ ( 1 cos θ h ) .
χ ( θ l , φ l , θ v , φ v ) 1 4 cos θ l cos θ v cos θ h ,
ρ single ( θ l , φ l , θ v , φ v , λ ) = s 2 F ¯ ( α , λ ) 16 π cos θ l cos θ v cos 4 θ h exp ( s 2 tan 2 θ h 4 ) ,
V ( θ l ) V ( θ v ) exp [ k 0 tan θ l s ] exp [ k 0 tan θ v s ] .
Φ l = L l cos θ l Δ A Δ ω l .
L v ( θ v , φ v ) = d L v = ρ single ( θ l , φ l , θ v , φ v , λ ) L l cos θ l d ω l = ρ single ( θ l , φ l , θ v , φ v , λ ) L l cos θ l Δ ω l .
Φ v = Δ A L v ( θ l , θ v , λ ) cos θ v d ω v = Δ A L l cos θ l Δ ω l ρ single ( θ l , φ l , θ v , φ v , λ ) cos θ v d ω v .
Φ v Φ l 1 .
ρ single ( θ l , φ l , θ v , φ v , λ ) cos θ v d ω v 1 .
ρ single ( θ l , φ l , θ v , φ v , λ ) cos θ v d ω v 1 4 cos θ l δ ( 1 cos θ h ) 2 π cos θ h d ω v = 1 2 π cos θ l cos α δ ( 1 cos θ h ) cos θ h d ω h = 1 cos θ l cos α δ ( 1 cos θ h ) cos θ h sin θ h d θ h = 1 cos θ l cos α δ ( 1 cos θ h ) cos θ h d cos θ h = 1 cos θ l cos θ l = 1 .
ρ single ( θ l , φ l , θ v , φ v , λ ) cos θ v d ω v exp [ k 0 tan θ v s ] sin θ v d θ v 0
θ v = { π 2 γ v when φ v = 0 γ v π 2 when φ v = π .
ζ = h ( R ) = h ( x , y ) ,
p ( ζ ) d ζ = 1 ,
P ( z ) = z p ( ζ ) d ζ ,
p ( ζ ) = d P ( ζ ) d ζ .
f ( ζ ) = f ( ζ ) p ( ζ ) d ζ ,
ζ 0 ζ = ζ p ( ζ ) d ζ ,
ζ 0 = 0 .
p ( ζ ) = 1 2 π σ exp ( ζ 2 2 σ 2 ) ,
σ ζ 2 = [ ζ 2 p ( ζ ) d ζ ] 1 2 .
g ( t ) = 1 2 π exp ( t 2 2 )
G ( t ) t g ( t ) d t = 1 2 π t exp ( t 2 2 ) d t
p ( ζ ) = 1 σ g ( ζ σ ) ,
P ( z ) = z p ( ζ ) d ζ = G ( z σ ) .
G ( + ) = 1 , P ( + ) = 1 .
C ( R 1 , R 2 ) h ( R 1 ) h ( R 2 ) σ 2 ,
C ( R ) = h ( R 1 ) h ( R 1 + R ) σ 2 ,
C ( R ) = h ( R 1 ) h ( R 1 + R ) σ 2 ,
C ( 0 ) = 1 .
C ( ) = 0 .
C ( R ) = exp ( R 2 τ 2 ) ,
C ( R ) = exp ( R τ ) .
ζ ( R 1 ) ζ x ( R 2 ) h ( R 1 ) h ( R 2 ) x = x h ( R 1 ) h ( R 2 ) .
R = R 2 R 1 ,
R = R = [ x 2 + y 2 ] 1 2 .
ζ ( R 1 ) ζ x ( R 2 ) = x σ 2 C ( R ) = σ 2 x C ( R ) .
x C ( R ) = R x R C ( R ) = 2 x τ 2 exp ( R 2 τ 2 ) ,
x C ( R ) 0 when R 0 .
ζ ( R 1 ) ζ x ( R 2 ) 0 when R 2 R 1 .
ζ ( R 1 ) ζ y ( R 2 ) = σ 2 y C ( R ) = 2 y σ 2 τ 2 exp ( R 2 τ 2 ) ,
ζ x ( R 1 ) ζ y ( R 2 ) = 2 x y h ( R 1 ) h ( R 2 ) = σ 2 2 x y C ( R ) = 4 x y σ 2 τ 4 exp ( R 2 τ 2 ) ,
p ( ζ , ζ x , ζ y ) = p ( ζ ) p ( ζ x ) p ( ζ y ) .
ξ = σ tan θ τ ,

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