Abstract

General surface scattering is characterized through the bidirectional reflection distribution function (BRDF). The BRDF is a function of the directions of the incident and remitted beams and thus depends on four parameters. Under very general assumptions one shows that the BRDF is invariant under interchange of the incident and remitted beams, the so-called Helmholtz reciprocity. For isotropic surfaces the BRDF depends only on the absolute value of the difference between the azimuths of the incident and remitted beams. Since these exhaust the symmetries, the BRDF is a very complicated function. For many applications it would be advantageous to be able to summarize empirical data or to smooth and/or interpolate (often even extrapolate) BRDF data. We present a principled way to do this, exactly respecting the symmetry properties.

© 1998 Optical Society of America

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References

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  1. K. J. Dana, B. van Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” (Columbia University, New York, 1996).
  2. L. da Vinci, Treatise on Painting, A. P. McMahon, transl. (Princeton U. Press, Princeton, N.J., 1959).
  3. G. Kortüm, Reflectance Spectroscopy, J. E. Lohr, transl. (Springer-Verlag, Berlin, 1969).
  4. S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern. Anal. Mach. Intell. 13, 611–634 (1991).
    [CrossRef]
  5. S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
    [CrossRef] [PubMed]
  6. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 1977).
  7. H. L. F. von Helmholtz, Treatise on Physiological Optics (Dover, New York, 1962), Vol. I, p. 231.
  8. M. Minnaert, “The reciprocity principle in lunar photometry,” Astrophys. J. 93, 403–410 (1941).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  10. P. Beckmann, A. Spizzochino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  11. B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).
  12. B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18, 1770–1779 (1979).
    [CrossRef] [PubMed]
  13. M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
    [CrossRef]
  14. M. Oren, S. K. Nayar, “Generalization of Lambert’s reflectance model,” , 239–246 (1994).
  15. H. D. Tagare, R. J. P. deFegueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
    [CrossRef]
  16. K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  17. J. H. Lambert, Photometria sive de mensura de gradibus luminis, colorum et umbræ (Eberhard Klett, Augsburg, Germany, 1760).
  18. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
    [CrossRef]
  19. E. Öpik, “Photometric measures of the moon and the earth-shine” [Publications de L’Observatorie Astronomical de L’Université de Tartu (Estonia), 1924], Vol. 26, pp. 1–68.
  20. J. F. Blinn, “Models of light reflection for computer synthesized pictures,” , 542–547 (1977).
  21. R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” ACM Comput. Graphics 15, 309–328 (1982).

1995 (2)

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

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

1991 (2)

H. D. Tagare, R. J. P. deFegueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

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

1982 (1)

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” ACM Comput. Graphics 15, 309–328 (1982).

1979 (1)

1967 (2)

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

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

1941 (1)

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

Beckmann, P.

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

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” , 542–547 (1977).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Brooks, M. J.

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).

Cook, R. L.

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” ACM Comput. Graphics 15, 309–328 (1982).

da Vinci, L.

L. da Vinci, Treatise on Painting, A. P. McMahon, transl. (Princeton U. Press, Princeton, N.J., 1959).

Dana, K. J.

K. J. Dana, B. van Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” (Columbia University, New York, 1996).

deFegueiredo, R. J. P.

H. D. Tagare, R. J. P. deFegueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 1977).

Horn, B. K. P.

B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18, 1770–1779 (1979).
[CrossRef] [PubMed]

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 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).
[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).
[CrossRef]

Koenderink, J. J.

K. J. Dana, B. van Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” (Columbia University, New York, 1996).

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy, J. E. Lohr, transl. (Springer-Verlag, Berlin, 1969).

Lambert, J. H.

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

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 1977).

Minnaert, M.

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

Nayar, S. K.

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 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).
[CrossRef]

K. J. Dana, B. van Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” (Columbia University, New York, 1996).

M. Oren, S. K. Nayar, “Generalization of Lambert’s reflectance model,” , 239–246 (1994).

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 1977).

Öpik, E.

E. Öpik, “Photometric measures of the moon and the earth-shine” [Publications de L’Observatorie Astronomical de L’Université de Tartu (Estonia), 1924], Vol. 26, pp. 1–68.

Oren, M.

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

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

M. Oren, S. K. Nayar, “Generalization of Lambert’s reflectance model,” , 239–246 (1994).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 1977).

Sjoberg, R. W.

Smith, B. G.

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

Sparrow, E. M.

Spizzochino, A.

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

Tagare, H. D.

H. D. Tagare, R. J. P. deFegueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

Torrance, K. E.

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” ACM Comput. Graphics 15, 309–328 (1982).

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

van Ginneken, B.

K. J. Dana, B. van Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” (Columbia University, New York, 1996).

von Helmholtz, H. L. F.

H. L. F. von Helmholtz, Treatise on Physiological Optics (Dover, New York, 1962), Vol. I, p. 231.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

ACM Comput. Graphics (1)

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” ACM Comput. Graphics 15, 309–328 (1982).

Appl. Opt. (1)

Astrophys. J. (1)

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

IEEE Trans. Antennas Propag. (1)

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

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

H. D. Tagare, R. J. P. deFegueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[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).
[CrossRef]

Int. J. Comput. Vis. (1)

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

J. Opt. Soc. Am. (1)

Science (1)

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

Other (12)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (National Bureau of Standards, Washington, D.C., 1977).

H. L. F. von Helmholtz, Treatise on Physiological Optics (Dover, New York, 1962), Vol. I, p. 231.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

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

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).

M. Oren, S. K. Nayar, “Generalization of Lambert’s reflectance model,” , 239–246 (1994).

K. J. Dana, B. van Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” (Columbia University, New York, 1996).

L. da Vinci, Treatise on Painting, A. P. McMahon, transl. (Princeton U. Press, Princeton, N.J., 1959).

G. Kortüm, Reflectance Spectroscopy, J. E. Lohr, transl. (Springer-Verlag, Berlin, 1969).

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

E. Öpik, “Photometric measures of the moon and the earth-shine” [Publications de L’Observatorie Astronomical de L’Université de Tartu (Estonia), 1924], Vol. 26, pp. 1–68.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” , 542–547 (1977).

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Figures (6)

Fig. 1
Fig. 1

Contour density plots of the orthonormal basis functions Knl(θ, ϕ) for even orders n up to 4. From left to right and top to bottom, the plots show K00, K2-2, and K20 (top row); K22, K4-4, and K4-2 (center row); K40, K42, and K44 (bottom row). The odd orders are shown in Fig. 2.

Fig. 2
Fig. 2

Contour density plots of the orthonormal basis functions Knl(θ, ϕ) for odd orders n up to 4. From left to right and top to bottom, the plots show K1-1, K11, and K3-3 (top row); K3-1, K31, and K33 (bottom row). The even orders are shown in Fig. 1.

Fig. 3
Fig. 3

Example of an IBSSS, illustrating a layout that permits a quick overview of the spectral composition of an isotropic BRDF. The order n runs from left to right (the numbers on the top row), and the associated order m runs from top to bottom (the numbers in the column on the right). The azimuthal order l also runs from left to right but starts anew for every new order n. The magnitude of the spectral components is given by the gray level; medium gray (background) defines the zero level.

Fig. 4
Fig. 4

Truncated series of the ideal mirror with the addition of a small Lambertian component. Orders 0, 1, and 2 are depicted on the top row, orders 4, 6, and 8 on the bottom row. The angle of incidence is 45° for all plots. Notice the pronounced specular lobe for orders 2 and higher, growing narrower with increasing order. Also notice the ringing due to abrupt truncation of the series (the small spurious lobes in directions other than that of the reflected ray). Such ringing can be damped through a more gradual attenuation of the amplitudes, just as with the familiar Fourier series.

Fig. 5
Fig. 5

Truncated series of the ideal mirror with the addition of a small Lambertian component for only terms of order 8 and lower. The angle of incidence is 67.5° (left), 45° (middle), and 22.5° (right). Notice that the approximation neatly satisfies the reflection laws and that the precise structure of the ringing depends on the geometry (angle of incidence).

Fig. 6
Fig. 6

Truncated series of the ideal retroreflector with the addition of a small Lambertian component for only terms of order 8 and lower. The angle of incidence is 67.5° (left), 45° (middle), and 22.5° (right). Notice that the approximation neatly satisfies the ideal backscatter laws and that the precise structure of the ringing depends on the geometry (angle of incidence).

Tables (2)

Tables Icon

Table 1 Isotropic Surface-Scattering Modes up to Order 4

Tables Icon

Table 2 Values of n, m, l for Isotropic Surface-Scattering Modes up to Order 4

Equations (46)

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

Unm(ρ, ϕ)=2(n+1)π Rnm(ρ)cos mϕ,m0,
Unm(ρ, ϕ)=2(n+1)π Rnm(ρ)sin mϕ,m<0,
Rn±m(ρ)
=s=0(n-m)/2(-1)s (n-s)!s!n+m2-s!n-m2-s!ρn-2s.
D2Unk(ρ, ϕ)Uml(ρ, ϕ)dA=δnmδkl,
azm(ϕ)=-sin mϕ,m<0,
azm(ϕ)=12,m=0,
azm(ϕ)=cos mϕ,m>0.
-π+πazn(ϕ)azm(ϕ)dϕ=πδnm.
azn(ϕ+π)=azn(ϕ),neven-azn(ϕ),nodd.
2 sin θ2d2 sin θ2=sin θdθ,
0π/2Rnk2 sin θ2Rmk2 sin θ2sin θdθ=δnmn+1.
ϴ(θ)=12R2 sin θ2,
Ψ(ϕ)=Φ(ϕ)
D2U(ρ, ϕ)dA=HS2W(θ, ϕ)dΩ.
Knl(θ, ϕ)=n+1π Rnl2 sin θ2azl(ϕ).
HS2Knk(θ, ϕ)Kml(θ, ϕ)dΩ=δnmδkl.
f(θi, ϕi, θr, ϕr)=dNr(θr, ϕr)dHi(θi, ϕi),
g(θi, ϕi, θr, ϕr)=f(θi, ϕi, θr, ϕr)cos θi,
f(θi, ϕi, θr, ϕr)=mknlanmkl Knk(θi, ϕi)Kml(θr, ϕr)+Knk(θr, ϕr)Kml(θi, ϕi)(2+2δnmδkl)1/2.
Hnmkl(θi, ϕi, θr, ϕr)
=Knk(θi, ϕi)Kml(θr, ϕr)+Knk(θr, ϕr)Kml(θi, ϕi)(2+2δnmδkl)1/2,
Hnmkl(θi, ϕi, θr, ϕr)=1π(n+1)(m+1)2+2δnmδkl1/2
×Rnk2 sin θi2Rml2 sin θr2azk(ϕi)azl(ϕr)+Rml2 sin θi2Rnk2 sin θr2azl(ϕi)azk(ϕr).
Hs2×Hs2Hnmkl(θi, ϕi, θr, ϕr)
×Hnmkl(θi, ϕi, θr, ϕr)dΩidΩr=δnnδmmδkkδll.
anmkl=Hs2×Hs2f(θi, ϕi, θr, ϕr)×Hnmkl(θi, ϕi, θr, ϕr)dΩidΩr.
Inml(θi, θr, Δϕir)=12π(n+1)(m+1)Anml1/2×Rnl2 sin θi2Rml2 sin θr2+Rml2 sin θi2Rnl2 sin θr2×cos lΔϕir.
Anml=4if(n=0)or((n=m)and(l=0))2if((n=m)or(l=0))1otherwise.
f(θi, θr, Δϕir)=nmlanmlInml(θi, θr, Δϕir),
anml=Hs2×Hs2f(θi, θr, Δϕir)×Inml(θi, θr, Δϕir)dΩidΩr.
n0,0mn,0lm
(n-l),(m-l)even;
g(θi, ϕi, θr, ϕr)=δ(θr-θi)δ(ϕi-ϕr+π)sin θr.
Hs2×Hs2g(ri, rr)Hnmkl(ri, rr)dΩidΩr.
1π(n+1)(m+1)(2+2δnmδkl)1/2
×0π/2 sin θRnk2 sin θ2Rml2 sin θ2dθ
×02π(azk(ϕ)azl(ϕ-π)+azl(ϕ)azk(ϕ-π))dϕ
=(-1)kπδnmδkl.
R2n0(ρ)=Pn(2ρ2-1).
g(θi, ϕi, θr, ϕr)=δ(θr-θi)δ(ϕi-ϕr)sin θr.
HS2×HS2g(ri, rr)Hnmkl(ri, rr)dΩidΩr
=1π(n+1)(m+1)(2+2δnmδkl)1/2 0π/2 sin θRnk2 sin θ2×Rml2 sin θ2dθ
×202πazk(ϕ)azl(ϕ)dϕ=1πδnmδkl.
f(θi, ϕi, θr, ϕr)=k+12π(cos θi cos θr)k-1
(0k1).

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