Abstract

It appears that the development of machine vision may benefit from a detailed understanding of the imaging process. The reflectance map, showing scene radiance as a function of surface gradient, has proved to be helpful in this endeavor. The reflectance map depends both on the nature of the surface layers of the objects being imaged and the distribution of light sources. Recently, a unified approach to the specification of surface reflectance in terms of both incident and reflected beam geometry has been proposed. The reflecting properties of a surface are specified in terms of the bidirectional reflectance-distribution function (BRDF). Here we derive the reflectance map in terms of the BRDF and the distribution of source radiance. A number of special cases of practical importance are developed in detail. The significance of this approach to the understanding of image formation is briefly indicated.

© 1979 Optical Society of America

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References

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  1. J. van Diggelen, Bull. Astron. Inst. Neth. 11, No. 423 (1951).
  2. T. Rindfleisch, Photogramm. Eng. 32, 262 (1966).
  3. B. K. P. Horn, “Determining Shape from Shading,” in The Psychology of Computer Vision, P. H. Winston, Ed. (McGraw-Hill, New York, 1975), Chap. 4.
  4. B. K. P. Horn, Artific. Intell. 8, No. 11, 201 (1977).
    [CrossRef]
  5. R. J. Woodham, “A Cooperative Algorithm for Determining Surface Orientation from a Single View,” in Proceedings Fifth International Joint Conference on Artificial Intelligence, MIT, Cambridge, Mass., August1977, pp. 635–641.
  6. R. J. Woodham, “Reflectance Map Techniques for Analyzing Surface Defects in Metal Castings,” TR-457, Artificial Intelligence Laboratory, MIT, Cambridge, Mass. (1978).
  7. R. J. Woodham, “Photometric Stereo: A Reflectance Map Technique for Determining Surface Orientation from Image Intensity,” in Proceedings of SPIE’s 22nd Annual Technical Symposium (SPIE, New York, 1978), Vol. 155, pp. 136–143.
  8. B. K. P. Horn, R. J. Woodham, W. N. Silver, “Determining Shape and Reflectance Using Multiple Images,” A, I. Laboratory Memo 490, MIT, Cambridge, Mass. (August1978).
  9. F. H. Gilpin, Trans. Illum. Eng. Soc. 5, 854 (1910).
  10. W. E. Knowles Middleton, A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).
    [CrossRef]
  11. V. A. Fedoretz, Publ. Kharkov Obs. 2, 49 (1952).
  12. J. van Diggelen, Rech. Obs. Utrecht 14, 1 (1959).
  13. M. Minnaert, “Photometry of the Moon,” in Planets and Satellites, Vol. 3, G. Kuiper, B. Middlehurst, Eds. (U. Chicago Press, Chicago, 1961), pp. 213–248.
  14. V. Fesenkov, “Photometry of the Moon,” in Physics and Astronomy of the Moon, Z. Kopal, Ed. (Academic, New York, 1962), pp. 99–130.
  15. B. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
    [CrossRef]
  16. G. de Vaucouleurs, Icarus 3, 187 (1964).
    [CrossRef]
  17. J. van Diggelen, Planet. Space Sci. 13, 271 (1965).
    [CrossRef]
  18. P. Oetking, J. Geophys. Res. 71, 2505 (1966).
    [CrossRef]
  19. J. J. Rennilson, H. E. Holt, E. C. Morris, J. Opt. Soc. Am. 58, 747 (1968).
    [CrossRef]
  20. E. M. Patterson, C. E. Sheldon, B. H. Stockton, Appl. Opt. 16, 729 (1977).
    [CrossRef] [PubMed]
  21. C. J. Tucker, Appl. Opt. 16, 1151 (1977).
    [CrossRef] [PubMed]
  22. M. Minnaert, Astrophys. J. 93, 403 (1941).
    [CrossRef]
  23. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  24. B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
    [CrossRef]
  25. N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
    [CrossRef]
  26. B. Hapke, Astron. J. 71, 333 (1966).
    [CrossRef]
  27. K. E. Torrance, E. M. Sparrow, R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).
    [CrossRef]
  28. K. E. Torrance, E. M. Sparrow, J. Opt. Soc. Am. 57, 1105 (1967).
    [CrossRef]
  29. T. S. Trowbridge, K. P. Reitz, J. Opt. Soc. Am. 65, 531 (1975).
    [CrossRef]
  30. E. L. Simmons, Appl. Opt. 14, 1380 (1975).
    [CrossRef] [PubMed]
  31. E. L. Simmons, Opt. Acta 22, 71 (1975).
    [CrossRef]
  32. E. L. Simmons, J. Appl. Phys. 46, 344 (1975).
    [CrossRef]
  33. E. L. Simmons, Appl. Opt. 15, 603 (1976).
    [CrossRef]
  34. C. J. Tucker, M. W. Garratt, Appl. Opt. 16, 635 (1977).
    [CrossRef] [PubMed]
  35. G. N. Plass, G. W. Kattawar, J. A. Guinn, Appl. Opt. 16, 643 (1977).
    [CrossRef] [PubMed]
  36. H. Gouraud, “Computer Display of Curved Surfaces,” Technical Report 113, UTEC–CSC-71, Computer Science, U. Utah, Salt Lake City (1971).
  37. Bui Tuong-Phong, “Illumination for Computer-Generated Images,” Technical Report 129, UTEC-CSC-73, Computer Science, U. Utah, Salt Lake City (1973).
  38. J. F. Blinn, “Models of Light Reflection for Computer Synthesized Pictures,” in SIGGRAPH 77, Proceedings ACM, Computer Graphics (July1977), Vol. 11, No. 2, pp. 192–198.
    [CrossRef]
  39. W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).
  40. W. W. Wendlandt, Ed., Modern Aspects of Reflectance Spectroscopy (Plenum, New York, 1968).
    [CrossRef]
  41. G. Kortuem, Reflectance Spectroscopy, J. E. Lohr, translator (Springer, Berlin, 1969).
    [CrossRef]
  42. D. E. Spencer, E. G. Gaston, J. Opt. Soc. Am. 65, 1129 (1975).
    [CrossRef]
  43. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).
  44. M. J. Brooks, “Investigating the Effects of Planar Light Sources,” CSM 22, Department of Computer Science, Essex University, Colchester, England (1978).

1977 (5)

1976 (1)

1975 (5)

1968 (1)

1967 (1)

1966 (4)

B. Hapke, Astron. J. 71, 333 (1966).
[CrossRef]

K. E. Torrance, E. M. Sparrow, R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).
[CrossRef]

P. Oetking, J. Geophys. Res. 71, 2505 (1966).
[CrossRef]

T. Rindfleisch, Photogramm. Eng. 32, 262 (1966).

1965 (1)

J. van Diggelen, Planet. Space Sci. 13, 271 (1965).
[CrossRef]

1964 (1)

G. de Vaucouleurs, Icarus 3, 187 (1964).
[CrossRef]

1963 (3)

B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
[CrossRef]

B. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

1959 (1)

J. van Diggelen, Rech. Obs. Utrecht 14, 1 (1959).

1952 (2)

1951 (1)

J. van Diggelen, Bull. Astron. Inst. Neth. 11, No. 423 (1951).

1941 (1)

M. Minnaert, Astrophys. J. 93, 403 (1941).
[CrossRef]

1910 (1)

F. H. Gilpin, Trans. Illum. Eng. Soc. 5, 854 (1910).

Birkebak, R. C.

Blinn, J. F.

J. F. Blinn, “Models of Light Reflection for Computer Synthesized Pictures,” in SIGGRAPH 77, Proceedings ACM, Computer Graphics (July1977), Vol. 11, No. 2, pp. 192–198.
[CrossRef]

Brooks, M. J.

M. J. Brooks, “Investigating the Effects of Planar Light Sources,” CSM 22, Department of Computer Science, Essex University, Colchester, England (1978).

de Vaucouleurs, G.

G. de Vaucouleurs, Icarus 3, 187 (1964).
[CrossRef]

Fedoretz, V. A.

V. A. Fedoretz, Publ. Kharkov Obs. 2, 49 (1952).

Fesenkov, V.

V. Fesenkov, “Photometry of the Moon,” in Physics and Astronomy of the Moon, Z. Kopal, Ed. (Academic, New York, 1962), pp. 99–130.

Garratt, M. W.

Gaston, E. G.

Gilpin, F. H.

F. H. Gilpin, Trans. Illum. Eng. Soc. 5, 854 (1910).

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).

Gouraud, H.

H. Gouraud, “Computer Display of Curved Surfaces,” Technical Report 113, UTEC–CSC-71, Computer Science, U. Utah, Salt Lake City (1971).

Guinn, J. A.

Hapke, B.

B. Hapke, Astron. J. 71, 333 (1966).
[CrossRef]

B. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

Hapke, B. W.

B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

Hecht, H. G.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).

Holt, H. E.

Horn, B. K. P.

B. K. P. Horn, Artific. Intell. 8, No. 11, 201 (1977).
[CrossRef]

B. K. P. Horn, R. J. Woodham, W. N. Silver, “Determining Shape and Reflectance Using Multiple Images,” A, I. Laboratory Memo 490, MIT, Cambridge, Mass. (August1978).

B. K. P. Horn, “Determining Shape from Shading,” in The Psychology of Computer Vision, P. H. Winston, Ed. (McGraw-Hill, New York, 1975), Chap. 4.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).

Kattawar, G. W.

Knowles Middleton, W. E.

Kortuem, G.

G. Kortuem, Reflectance Spectroscopy, J. E. Lohr, translator (Springer, Berlin, 1969).
[CrossRef]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).

Melamed, N. T.

N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
[CrossRef]

Minnaert, M.

M. Minnaert, Astrophys. J. 93, 403 (1941).
[CrossRef]

M. Minnaert, “Photometry of the Moon,” in Planets and Satellites, Vol. 3, G. Kuiper, B. Middlehurst, Eds. (U. Chicago Press, Chicago, 1961), pp. 213–248.

Morris, E. C.

Mungall, A. G.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).

Oetking, P.

P. Oetking, J. Geophys. Res. 71, 2505 (1966).
[CrossRef]

Patterson, E. M.

Plass, G. N.

Reitz, K. P.

Rennilson, J. J.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).

Rindfleisch, T.

T. Rindfleisch, Photogramm. Eng. 32, 262 (1966).

Sheldon, C. E.

Silver, W. N.

B. K. P. Horn, R. J. Woodham, W. N. Silver, “Determining Shape and Reflectance Using Multiple Images,” A, I. Laboratory Memo 490, MIT, Cambridge, Mass. (August1978).

Simmons, E. L.

Sparrow, E. M.

Spencer, D. E.

Stockton, B. H.

Torrance, K. E.

Trowbridge, T. S.

Tucker, C. J.

Tuong-Phong, Bui

Bui Tuong-Phong, “Illumination for Computer-Generated Images,” Technical Report 129, UTEC-CSC-73, Computer Science, U. Utah, Salt Lake City (1973).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van Diggelen, J.

J. van Diggelen, Planet. Space Sci. 13, 271 (1965).
[CrossRef]

J. van Diggelen, Rech. Obs. Utrecht 14, 1 (1959).

J. van Diggelen, Bull. Astron. Inst. Neth. 11, No. 423 (1951).

Van Horn, H.

B. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

Wendlandt, W. W.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).

Woodham, R. J.

B. K. P. Horn, R. J. Woodham, W. N. Silver, “Determining Shape and Reflectance Using Multiple Images,” A, I. Laboratory Memo 490, MIT, Cambridge, Mass. (August1978).

R. J. Woodham, “A Cooperative Algorithm for Determining Surface Orientation from a Single View,” in Proceedings Fifth International Joint Conference on Artificial Intelligence, MIT, Cambridge, Mass., August1977, pp. 635–641.

R. J. Woodham, “Reflectance Map Techniques for Analyzing Surface Defects in Metal Castings,” TR-457, Artificial Intelligence Laboratory, MIT, Cambridge, Mass. (1978).

R. J. Woodham, “Photometric Stereo: A Reflectance Map Technique for Determining Surface Orientation from Image Intensity,” in Proceedings of SPIE’s 22nd Annual Technical Symposium (SPIE, New York, 1978), Vol. 155, pp. 136–143.

Appl. Opt. (6)

Artific. Intell. (1)

B. K. P. Horn, Artific. Intell. 8, No. 11, 201 (1977).
[CrossRef]

Astron. J. (1)

B. Hapke, Astron. J. 71, 333 (1966).
[CrossRef]

Astrophys. J. (1)

M. Minnaert, Astrophys. J. 93, 403 (1941).
[CrossRef]

Bull. Astron. Inst. Neth. (1)

J. van Diggelen, Bull. Astron. Inst. Neth. 11, No. 423 (1951).

Icarus (1)

G. de Vaucouleurs, Icarus 3, 187 (1964).
[CrossRef]

J. Appl. Phys. (2)

N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
[CrossRef]

E. L. Simmons, J. Appl. Phys. 46, 344 (1975).
[CrossRef]

J. Geophys. Res. (3)

B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

P. Oetking, J. Geophys. Res. 71, 2505 (1966).
[CrossRef]

B. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Acta (1)

E. L. Simmons, Opt. Acta 22, 71 (1975).
[CrossRef]

Photogramm. Eng. (1)

T. Rindfleisch, Photogramm. Eng. 32, 262 (1966).

Planet. Space Sci. (1)

J. van Diggelen, Planet. Space Sci. 13, 271 (1965).
[CrossRef]

Publ. Kharkov Obs. (1)

V. A. Fedoretz, Publ. Kharkov Obs. 2, 49 (1952).

Rech. Obs. Utrecht (1)

J. van Diggelen, Rech. Obs. Utrecht 14, 1 (1959).

Trans. Illum. Eng. Soc. (1)

F. H. Gilpin, Trans. Illum. Eng. Soc. 5, 854 (1910).

Other (16)

B. K. P. Horn, “Determining Shape from Shading,” in The Psychology of Computer Vision, P. H. Winston, Ed. (McGraw-Hill, New York, 1975), Chap. 4.

M. Minnaert, “Photometry of the Moon,” in Planets and Satellites, Vol. 3, G. Kuiper, B. Middlehurst, Eds. (U. Chicago Press, Chicago, 1961), pp. 213–248.

V. Fesenkov, “Photometry of the Moon,” in Physics and Astronomy of the Moon, Z. Kopal, Ed. (Academic, New York, 1962), pp. 99–130.

R. J. Woodham, “A Cooperative Algorithm for Determining Surface Orientation from a Single View,” in Proceedings Fifth International Joint Conference on Artificial Intelligence, MIT, Cambridge, Mass., August1977, pp. 635–641.

R. J. Woodham, “Reflectance Map Techniques for Analyzing Surface Defects in Metal Castings,” TR-457, Artificial Intelligence Laboratory, MIT, Cambridge, Mass. (1978).

R. J. Woodham, “Photometric Stereo: A Reflectance Map Technique for Determining Surface Orientation from Image Intensity,” in Proceedings of SPIE’s 22nd Annual Technical Symposium (SPIE, New York, 1978), Vol. 155, pp. 136–143.

B. K. P. Horn, R. J. Woodham, W. N. Silver, “Determining Shape and Reflectance Using Multiple Images,” A, I. Laboratory Memo 490, MIT, Cambridge, Mass. (August1978).

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

H. Gouraud, “Computer Display of Curved Surfaces,” Technical Report 113, UTEC–CSC-71, Computer Science, U. Utah, Salt Lake City (1971).

Bui Tuong-Phong, “Illumination for Computer-Generated Images,” Technical Report 129, UTEC-CSC-73, Computer Science, U. Utah, Salt Lake City (1973).

J. F. Blinn, “Models of Light Reflection for Computer Synthesized Pictures,” in SIGGRAPH 77, Proceedings ACM, Computer Graphics (July1977), Vol. 11, No. 2, pp. 192–198.
[CrossRef]

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).

W. W. Wendlandt, Ed., Modern Aspects of Reflectance Spectroscopy (Plenum, New York, 1968).
[CrossRef]

G. Kortuem, Reflectance Spectroscopy, J. E. Lohr, translator (Springer, Berlin, 1969).
[CrossRef]

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., October1977).

M. J. Brooks, “Investigating the Effects of Planar Light Sources,” CSM 22, Department of Computer Science, Essex University, Colchester, England (1978).

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Figures (12)

Fig. 1
Fig. 1

A typical reflectance map for a surface, with both a glossy and a matte component of reflection, illuminated by a point source. The coordinates are surface slope in the x and y directions, and the curves shown are contours of constant scene radiance.

Fig. 2
Fig. 2

Undulations in a specularly reflecting surface causing scattering of incident rays into a variety of directions. The surface will not appear specular if it is imaged on a scale where the surface undulations are not resolved. It may instead have a glossy appearance.

Fig. 3
Fig. 3

Inhomogeneities in refractive index of surface layer components cause incident rays to be scattered into a variety of directions upon reflection. This kind of surface microstructure gives rise to matte reflection.

Fig. 4
Fig. 4

Compound surface illustrating more complex model of interaction of light rays with surface microstructure.

Fig. 5
Fig. 5

Point source illuminating a surface, illustrating basic radio-metric concepts.

Fig. 6
Fig. 6

Local geometry of incident and reflected rays needed for the definition of the bidirectional reflectance-distribution function (BRDF) (redrawn from Ref. 43).

Fig. 7
Fig. 7

Polar and azimuth angles used in double integrals over specified solid angles.

Fig. 8
Fig. 8

A simple image-forming system. Light collected by the lens from the surface patch of area dA0 is projected into the image patch of area dAp.

Fig. 9
Fig. 9

Viewer-oriented global coordinate system useful for specification of the distribution of source radiance Li.

Fig. 10
Fig. 10

Surface normal and direction to portion of the source shown in viewer-oriented coordinate system.

Fig. 11
Fig. 11

Spherical triangle extracted from previous figure and used in derivation of transformation equations between the surface normal, local coordinate system, and the viewer-oriented, global coordinate system.

Fig. 12
Fig. 12

Cross section through uniform hemispherical source and surface element, illustrating the horizon cutoff and the portion of extended source not visible from the surface.

Tables (1)

Tables Icon

Table I Radiometric Concepts

Equations (82)

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d ω = d A · cos θ / r 2
d Φ = I · d ω = I · d A · cos θ / r 2
E = d Φ / d A = I · cos θ / r 2
f r ( θ i , ϕ i ; θ r , ϕ r ) = d L r ( θ i , ϕ i ; θ r , ϕ r ; E i ) / d E i ( θ i , ϕ i )
d Φ i = L i · cos θ i · d ω i · d A = L i · d Ω i · d A = d E i · d A ,
d 2 Φ r = d L r · cos θ r · d ω r · d A = d L r · d Ω r · d A ,
f r ( θ i , ϕ i ; θ r ϕ r ) = ( d 2 Φ r / d Ω r ) / d Φ i = d L r / d E i
ω X d ω = - π π 0 π / 2 X sin θ d θ d ϕ Ω X d Ω = - π π 0 π / 2 X cos θ sin θ d θ d ϕ .
ω X d ω = - π π 0 π / 2 sin θ d θ ϕ = 2 π ,
Ω X d Ω = - π π 0 π / 2 ( ½ ) sin 2 θ d θ d ϕ = π .
M = Ω r L r d Ω r = L r π .
f r , i d = L r / E i = L r / M = 1 / π .
d E i ( θ i , ϕ i ) = L i ( θ i , ϕ i ) cos θ i d ω i .
L r = ( 1 / π ) ω L i ( θ i , ϕ i ) cos θ i d ω i .
E 0 = - π π 0 π / 2 L i sin θ i d θ i d ϕ i .
L i = E 0 δ ( θ i - θ 0 ) δ ( ϕ i - ϕ 0 ) / sin θ 0 .
δ [ f ( x ) - f ( x 0 ) = δ ( x - x 0 ) / f ( x 0 ) ,
L i = E 0 δ ( cos θ i - cos θ 0 ) δ ( ϕ i - ϕ 0 ) .
L r ( θ r , ϕ r ) = L i ( θ r , ϕ r + π ) .
L r = f r d E i = Ω i f r L i d Ω i .
L r = - π π 0 π / 2 f r L i cos θ i sin θ i d θ i d ϕ i .
f r , i s = δ ( θ i - θ r ) δ ( ϕ i - ϕ r + π ) / ( sin θ i cos θ i ) .
f r , i s = 2 δ ( sin 2 θ r - sin 2 θ i ) δ ( ϕ r - ϕ i + π ) .
d Φ L = d A 0 Ω r L r d Ω r ,
E p = d Φ L / d A p .
( d A 0 cos θ r ) / f 0 2 = ( d A p cos α ) / f p 2 .
E p = ( f 0 / f p ) 2 cos α ω r L r ( cos θ r / cos θ r ) d ω r .
E p = L r ( π / 4 ) ( d / f p ) 2 cos 4 α .
L r = Ω i f r L i d Ω i .
n = ( cos ϕ n sin θ n , sin ϕ n sin θ n , cos θ n ) .
( 1 , 0 , p ) × ( 0 , 1 , q ) = ( - p , - q , 1 ) .
n = ( - p , - q , 1 ) / ( 1 + p 2 + q 2 ) 1 / 2 .
sin θ n = ( p 2 + q 2 ) 1 / 2 / ( 1 + p 2 + q 2 ) 1 / 2 ; cos θ n = 1 / ( 1 + p 2 + q 2 ) 1 / 2 ; sin ϕ n = - q / ( p 2 + q 2 ) 1 / 2 ; cos ϕ n = - p / ( p 2 + q 2 ) 1 / 2 .
p = - cos ϕ n tan θ n ; q = - sin ϕ n tan θ n .
cos θ i = cos θ s cos θ n + sin θ s sin θ n cos ( ϕ s - ϕ n ) ;
sin θ i sin ( ϕ r - ϕ i ) = sin θ s sin ( ϕ s - ϕ n ) ;
sin θ i cos ( ϕ r - ϕ i ) = cos θ s sin θ n - sin θ s cos θ n cos ( ϕ s - ϕ n ) .
( θ i / θ s ) ( ϕ i / ϕ s ) - ( θ i / ϕ s ) ( ϕ i / θ s ) = ( sin θ s / sin θ i ) .
cos θ s = cos θ i cos θ r + sin θ i sin θ r cos ( ϕ r - ϕ i ) ;
sin θ s sin ( ϕ s - ϕ n ) = sin θ i sin ( ϕ r - ϕ i ) ;
sin θ s cos ( ϕ s - ϕ n ) = cos θ i sin θ r - sin θ i cos θ r cos ( ϕ r - ϕ i ) .
( θ s / θ i ) ( ϕ s / ϕ i ) - ( ϕ s / θ i ) ( θ s / ϕ i ) = ( sin θ i / sin θ s ) .
L r = Ω i f r L i d Ω i = ω i f r L i cos θ i d ω i
L r ( θ n , ϕ n ) = - π π 0 π / 2 f r ( θ i , ϕ i ; θ r , ϕ r ) L i ( θ s , ϕ s ) cos θ i sin θ i d θ i d ϕ i .
L r = - π π 0 π f r L i max [ 0 , cos θ i ] sin θ i d θ i d ϕ i .
L r = - π π 0 π f r L i max [ 0 , cos θ i ] sin θ s d θ s d ϕ s ,
L r ( θ n , ϕ n ) = - π π 0 π f r ( θ i , ϕ i ; θ r , ϕ r ) L i ( θ s , ϕ s ) × max [ 0 , cos θ i ] sin θ s d θ s d ϕ s .
L i = E 0 δ ( θ s - θ 0 ) δ ( ϕ s - ϕ 0 ) / sin θ 0 ,
L r = - π π 0 π ( E 0 / π ) δ ( θ s - θ 0 ) δ ( ϕ s - ϕ 0 ) × max [ 0 , cos θ i ] ( sin θ s / sin θ 0 ) d θ s d ϕ s .
L r = ( E 0 / π ) max [ 0 , cos θ i ] ,
cos θ i = cos θ r cos θ 0 + sin θ r sin θ 0 cos ( ϕ 0 - ϕ n )
cos ( ϕ 0 - ϕ n ) = cos ϕ 0 cos ϕ n - sin ϕ 0 sin ϕ n .
R ( p , q ) = ( E 0 / π ) max [ 0 , ( 1 + p 0 p + q 0 q ) ( 1 + p 2 + q 2 ) 1 / 2 ( 1 + p 0 2 + q 0 2 ) 1 / 2 ] ,
p 0 = - cos ϕ 0 tan θ 0 , q 0 = - sin ϕ 0 tan θ 0 .
L r = - π π 0 π / 2 ( L 0 / π ) cos θ i sin θ i d θ i d ϕ i .
L r = L 0 0 π / 2 sin 2 θ i d θ i = L 0 .
L i ( θ s , ϕ s ) = L 0             for θ s < π / 2 , L i ( θ s , ϕ s ) = 0             for θ s > π / 2.
cot θ i = - tan θ r cos ( ϕ r - ϕ i ) .
L r = - π π 0 π / 2 f r L i cos θ i sin θ i d θ i d ϕ i ,
L r = ( L 0 / π ) - π π 0 min [ θ i , π / 2 ] cos θ i sin θ i d θ i d ϕ i .
L r = ( L 0 / π ) - π / 2 π / 2 0 π / 2 cos θ i sin θ i d θ i d ϕ + ( L 0 / π ) - π / 2 π / 2 0 θ i cos θ i sin θ i d θ i d ϕ .
0 π / 2 cos θ i sin θ i d θ i = ½ ,
0 θ i cos θ i sin θ i d θ i = ( 1 - cos 2 θ i ) / 4 = sin 2 θ i / 2 ,
sin 2 θ i = 1 / [ 1 + tan 2 θ r cos 2 ( ϕ r - ϕ i ) ] .
( L 0 / 2 π ) - π / 2 π / 2 1 / ( 1 + tan 2 θ r cos 2 ϕ ) d ϕ ,
( L 0 / 2 π ) [ cos θ r tan - 1 ( cos θ r tan ϕ ) ] - π / 2 π / 2 = ( L 0 / 2 ) cos θ r .
L r ( θ n , ϕ n ) = ( L 0 / 2 ) ( 1 + cos θ n ) = L 0 cos 2 ( θ n / 2 ) .
R ( p , q ) = ( E 0 / 2 ) [ 1 + 1 / ( 1 + p 2 + q 2 ) 1 / 2 ] .
f r = δ ( θ i - θ r ) δ ( ϕ i - ϕ r + π ) / ( sin θ i cos θ i ) .
L r = - π π 0 π / 2 ( L 0 / sin θ 0 ) δ ( θ i - θ r ) δ ( ϕ i - ϕ r + π ) × δ ( θ s - θ 0 ) δ ( ϕ s - ϕ 0 ) d θ i d ϕ i ,
L r = L 0 δ ( θ s - θ 0 ) δ ( ϕ s - ϕ 0 ) / sin θ 0 ,
L r = L 0 δ ( 2 θ r - θ 0 ) δ ( ϕ n - ϕ 0 ) / sin θ 0
L r ( θ n , ϕ n ) = ( L 0 / 2 ) δ ( θ n - θ 0 / 2 ) δ ( ϕ n - ϕ 0 ) / sin θ 0 .
δ [ f ( x , y ) - f ( x 0 , y 0 ) ] δ [ g ( x , y ) - g ( x 0 , y 0 ) ] = δ ( x - x 0 ) δ ( y - y 0 ) / J ( x 0 , y 0 ) ,
J ( x , y ) = ( f / x ) ( g / y ) - ( f / y ) ( g / x ) .
J ( p , q ) = 1 / [ ( p 2 + q 2 ) 1 / 2 ( 1 + p 2 + q 2 ) ] .
p 1 = - cos ϕ 0 tan θ 0 / 2 , q 1 = - sin ϕ 0 tan θ 0 / 2.
sin θ 0 = 2 ( p 1 2 + q 1 2 ) 1 / 2 / ( 1 + p 1 2 + q 1 2 ) ,
R ( p , q ) = ( E 0 / 4 ) δ ( p - p 1 ) δ ( q - q 1 ) ( 1 + p 1 2 + q 1 2 ) 2 .
p 1 = p 0 [ ( 1 + p 0 2 + q 0 2 ) 1 / 2 - 1 ] / ( p 0 2 + q 0 2 ) , q 1 = q 0 [ ( 1 + p 0 2 + q 0 2 ) 1 / 2 - 1 ] / ( p 0 2 + q 0 2 ) .
L r ( θ n , ϕ n ) = L 0 for θ n < π / 4 , L r ( θ n , ϕ n ) = 0 for θ n > π / 4.
R ( p , q ) = L 0 for p 2 + q 2 < 1 , R ( p , q ) = 0 for p 2 + q 2 > 1.

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