Abstract

A reflectance model that accurately predicts diffuse reflection from smooth inhomogeneous dielectric surfaces as a function of both viewing angle and angle of incidence is proposed. Utilizing results of radiative-transfer theory for subsurface multiple scattering, this new model precisely accounts for how incident light and the distribution of subsurface scattered light are influenced by Fresnel attenuation and Snell refraction at a smooth air–dielectric surface boundary. Whereas similar assumptions about subsurface scattering and Fresnel attenuation have been made in previous research on diffuse-reflectance modeling, the proposed model combines these assumptions in a different way and yields a more accurate expression for diffuse reflection that is shown to account for a number of empirical observations not predicted by existing models. What is particularly new about this diffuse-reflectance model is the resulting significant dependence on the viewing angle with respect to the surface normal. This dependence on the viewing angle explains distinctive properties of the behavior of diffuse reflection from smooth dielectric objects, properties not accounted for by existing diffuse-reflection models. Among these properties are prominent diffuse-reflection maxima effects occurring on objects when incident point-source illumination is greater than 50° relative to viewing, including the range from 90° to 180°, where the light source is behind the object with respect to viewing. For this range of incident illumination there is significant deviation from Lambertian behavior over a large portion of most smooth dielectric object surfaces, which makes it important for the computer vision community to be aware of such effects during incorporation of reflectance models into implementation of algorithms such as shape-from-shading. A number of experimental results are presented that verify the proposed diffuse-reflectance model.

© 1994 Optical Society of America

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References

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  1. L. B. Wolff, “Diffuse reflection,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992, pp. 472–478.
  2. L. B. Wolff, “A diffuse reflectance model for dielectric surfaces,” in Optics, Illumination, and Image Sensing for Machine Vision VII, Donald J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1822, 60–73 (1992).
    [CrossRef]
  3. R. Siegal, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).
  4. P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstriche,”Z. Tech. Phys. 12, 593 (1931).
  5. S. Orchard, “Reflection and transmission of light by diffusing suspensions,”J. Opt. Soc. Am. 59, 1584–1597 (1969).
    [CrossRef]
  6. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1. Theory,” Appl. Opt. 12, 1811–1815 (1973).
    [CrossRef] [PubMed]
  7. J. R. Aronson, A. G. Emslie, “Spectral reflectance and emittance of particulate materials. 2. Application and results,” Appl. Opt. 12, 2573–2584 (1973).
    [CrossRef] [PubMed]
  8. A. G. Emslie, J. R. Aronson, “Spectral reflectance and emittance of particulate materials. 1. Theory,” Appl. Opt. 12, 2563–2572 (1973).
    [CrossRef] [PubMed]
  9. E. Bahar, “Review of the full wave solutions for rough surface scattering and depolarization,”J. Geophys. Res. 92, 5209–5224 (1987).
    [CrossRef]
  10. G. Healey, T. O. Binford, “The role and use of color in a general vision system,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1987), pp. 599–613.
  11. G. Healey, “Using color for geometry-insensitive segmentation,” J. Opt. Soc. Am. A 6, 920–937 (1989).
    [CrossRef]
  12. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  13. J. H. Lambert, Photometria Sive de Mensura de Gratibus Luminis, Colorum et Umbrae (Eberhard Klett, Augsberg, Germany, 1760).
  14. B. K. P. Horn, M. J. Brooks, Shape From Shading (MIT Press, Cambridge, Mass., 1989).
  15. W. E. L. Grimson, “Binocular shading and visual surface reconstruction,” Comput. Vision Graphics Image Process. 28, 19–43 (1984).
    [CrossRef]
  16. G. B. Smith, “Stereo integral equation,” Proc. Am. Assoc. Artif. Intell. 6, 689–694 (1989).
  17. K. Torrance, E. Sparrow, “Theory for off-specular reflection from roughened surfaces,”J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  18. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  19. R. Cook, K. Torrance, “A reflectance model for computer graphics,”J. Comput. Graphics 15, 307–316 (1981).
    [CrossRef]
  20. G. Healey, T. O. Binford, “Local shape from specularity,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 151–160.
  21. L. B. Wolff, “Spectral and polarization stereo methods using a single light source,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 708–715.
  22. H. D. Tagare, R. J. P. deFigueiredo, “A theory of photometric stereo for a general class of reflectance maps,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 38–45.
    [CrossRef]
  23. S. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1990), pp. 185–212.
  24. X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH Proceedings (American Association for Artificial Intelligence, Menlo Park, Calif., 1991), pp. 175–186.
    [CrossRef]
  25. K. Torrance, Department of Mechanical Engineering, Cornell University, Ithaca, New York 14853 (personal communication).
  26. M. Oren, S. Nayar, “Diffuse reflection model for rough surfaces,” Tech. Rep. CUCS-057–92 (Columbia University, New York, 1992).
  27. D. Forsyth, A. Zisserman, “Reflections on shading,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 671–679 (1991).
    [CrossRef]
  28. S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  29. L. B. Wolff, “Polarization methods in computer vision,” Ph.D. dissertation (Columbia University, New York, 1991).
  30. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  31. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  32. L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
    [CrossRef]
  33. D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).
  34. L. B. Wolff, “On diffuse reflection and photometric stereo,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1992), pp. 437–448.
  35. B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18, 1770–1779 (1979).
    [CrossRef] [PubMed]
  36. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

1994 (1)

L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
[CrossRef]

1991 (1)

D. Forsyth, A. Zisserman, “Reflections on shading,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 671–679 (1991).
[CrossRef]

1989 (2)

G. B. Smith, “Stereo integral equation,” Proc. Am. Assoc. Artif. Intell. 6, 689–694 (1989).

G. Healey, “Using color for geometry-insensitive segmentation,” J. Opt. Soc. Am. A 6, 920–937 (1989).
[CrossRef]

1987 (1)

E. Bahar, “Review of the full wave solutions for rough surface scattering and depolarization,”J. Geophys. Res. 92, 5209–5224 (1987).
[CrossRef]

1985 (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

1984 (1)

W. E. L. Grimson, “Binocular shading and visual surface reconstruction,” Comput. Vision Graphics Image Process. 28, 19–43 (1984).
[CrossRef]

1981 (1)

R. Cook, K. Torrance, “A reflectance model for computer graphics,”J. Comput. Graphics 15, 307–316 (1981).
[CrossRef]

1979 (1)

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

1973 (3)

1969 (1)

1967 (1)

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstriche,”Z. Tech. Phys. 12, 593 (1931).

Aronson, J. R.

Bahar, E.

E. Bahar, “Review of the full wave solutions for rough surface scattering and depolarization,”J. Geophys. Res. 92, 5209–5224 (1987).
[CrossRef]

Beckmann, P.

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

Binford, T. O.

G. Healey, T. O. Binford, “The role and use of color in a general vision system,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1987), pp. 599–613.

G. Healey, T. O. Binford, “Local shape from specularity,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 151–160.

Brooks, M. J.

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

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Clarke, D.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Cook, R.

R. Cook, K. Torrance, “A reflectance model for computer graphics,”J. Comput. Graphics 15, 307–316 (1981).
[CrossRef]

deFigueiredo, R. J. P.

H. D. Tagare, R. J. P. deFigueiredo, “A theory of photometric stereo for a general class of reflectance maps,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 38–45.
[CrossRef]

Emslie, A. G.

Forsyth, D.

D. Forsyth, A. Zisserman, “Reflections on shading,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 671–679 (1991).
[CrossRef]

Ginsburg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

Grainger, J. F.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH Proceedings (American Association for Artificial Intelligence, Menlo Park, Calif., 1991), pp. 175–186.
[CrossRef]

Grimson, W. E. L.

W. E. L. Grimson, “Binocular shading and visual surface reconstruction,” Comput. Vision Graphics Image Process. 28, 19–43 (1984).
[CrossRef]

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH Proceedings (American Association for Artificial Intelligence, Menlo Park, Calif., 1991), pp. 175–186.
[CrossRef]

Healey, G.

G. Healey, “Using color for geometry-insensitive segmentation,” J. Opt. Soc. Am. A 6, 920–937 (1989).
[CrossRef]

G. Healey, T. O. Binford, “Local shape from specularity,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 151–160.

G. Healey, T. O. Binford, “The role and use of color in a general vision system,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1987), pp. 599–613.

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).

Howell, J. R.

R. Siegal, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

Ikeuchi, K.

S. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1990), pp. 185–212.

Kanade, T.

S. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1990), pp. 185–212.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstriche,”Z. Tech. Phys. 12, 593 (1931).

Lambert, J. H.

J. H. Lambert, Photometria Sive de Mensura de Gratibus Luminis, Colorum et Umbrae (Eberhard Klett, Augsberg, Germany, 1760).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstriche,”Z. Tech. Phys. 12, 593 (1931).

Nayar, S.

M. Oren, S. Nayar, “Diffuse reflection model for rough surfaces,” Tech. Rep. CUCS-057–92 (Columbia University, New York, 1992).

S. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1990), pp. 185–212.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

Orchard, S.

Oren, M.

M. Oren, S. Nayar, “Diffuse reflection model for rough surfaces,” Tech. Rep. CUCS-057–92 (Columbia University, New York, 1992).

Reichman, J.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

Shafer, S.

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Siegal, R.

R. Siegal, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH Proceedings (American Association for Artificial Intelligence, Menlo Park, Calif., 1991), pp. 175–186.
[CrossRef]

Sjoberg, R. W.

Smith, G. B.

G. B. Smith, “Stereo integral equation,” Proc. Am. Assoc. Artif. Intell. 6, 689–694 (1989).

Sparrow, E.

Spizzichino, A.

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

Tagare, H. D.

H. D. Tagare, R. J. P. deFigueiredo, “A theory of photometric stereo for a general class of reflectance maps,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 38–45.
[CrossRef]

Torrance, K.

R. Cook, K. Torrance, “A reflectance model for computer graphics,”J. Comput. Graphics 15, 307–316 (1981).
[CrossRef]

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

K. Torrance, Department of Mechanical Engineering, Cornell University, Ithaca, New York 14853 (personal communication).

Torrance, K. E.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH Proceedings (American Association for Artificial Intelligence, Menlo Park, Calif., 1991), pp. 175–186.
[CrossRef]

van de Hulst, H. C.

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

Wolff, L. B.

L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
[CrossRef]

L. B. Wolff, “Spectral and polarization stereo methods using a single light source,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 708–715.

L. B. Wolff, “Diffuse reflection,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992, pp. 472–478.

L. B. Wolff, “Polarization methods in computer vision,” Ph.D. dissertation (Columbia University, New York, 1991).

L. B. Wolff, “A diffuse reflectance model for dielectric surfaces,” in Optics, Illumination, and Image Sensing for Machine Vision VII, Donald J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1822, 60–73 (1992).
[CrossRef]

L. B. Wolff, “On diffuse reflection and photometric stereo,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1992), pp. 437–448.

Zisserman, A.

D. Forsyth, A. Zisserman, “Reflections on shading,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 671–679 (1991).
[CrossRef]

Appl. Opt. (4)

Color Res. Appl. (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

W. E. L. Grimson, “Binocular shading and visual surface reconstruction,” Comput. Vision Graphics Image Process. 28, 19–43 (1984).
[CrossRef]

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

D. Forsyth, A. Zisserman, “Reflections on shading,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 671–679 (1991).
[CrossRef]

J. Comput. Graphics (1)

R. Cook, K. Torrance, “A reflectance model for computer graphics,”J. Comput. Graphics 15, 307–316 (1981).
[CrossRef]

J. Geophys. Res. (1)

E. Bahar, “Review of the full wave solutions for rough surface scattering and depolarization,”J. Geophys. Res. 92, 5209–5224 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Natl. Bur. Stand. Monogr. (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsburg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. Monogr. 160 (1977).

Opt. Eng. (1)

L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
[CrossRef]

Proc. Am. Assoc. Artif. Intell. (1)

G. B. Smith, “Stereo integral equation,” Proc. Am. Assoc. Artif. Intell. 6, 689–694 (1989).

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstriche,”Z. Tech. Phys. 12, 593 (1931).

Other (20)

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

G. Healey, T. O. Binford, “The role and use of color in a general vision system,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1987), pp. 599–613.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

J. H. Lambert, Photometria Sive de Mensura de Gratibus Luminis, Colorum et Umbrae (Eberhard Klett, Augsberg, Germany, 1760).

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

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

L. B. Wolff, “On diffuse reflection and photometric stereo,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1992), pp. 437–448.

L. B. Wolff, “Diffuse reflection,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992, pp. 472–478.

L. B. Wolff, “A diffuse reflectance model for dielectric surfaces,” in Optics, Illumination, and Image Sensing for Machine Vision VII, Donald J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1822, 60–73 (1992).
[CrossRef]

R. Siegal, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

L. B. Wolff, “Polarization methods in computer vision,” Ph.D. dissertation (Columbia University, New York, 1991).

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

G. Healey, T. O. Binford, “Local shape from specularity,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 151–160.

L. B. Wolff, “Spectral and polarization stereo methods using a single light source,” in Proceedings of the IEEE First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 708–715.

H. D. Tagare, R. J. P. deFigueiredo, “A theory of photometric stereo for a general class of reflectance maps,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 38–45.
[CrossRef]

S. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” in Proceedings of the DARPA Image Understanding Workshop (Defense Advanced Research Projects Agency, Arlington, Va., 1990), pp. 185–212.

X. D. He, K. E. Torrance, F. X. Sillion, D. P. Greenberg, “A comprehensive physical model for light reflection,” in SIGGRAPH Proceedings (American Association for Artificial Intelligence, Menlo Park, Calif., 1991), pp. 175–186.
[CrossRef]

K. Torrance, Department of Mechanical Engineering, Cornell University, Ithaca, New York 14853 (personal communication).

M. Oren, S. Nayar, “Diffuse reflection model for rough surfaces,” Tech. Rep. CUCS-057–92 (Columbia University, New York, 1992).

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Figures (14)

Fig. 1
Fig. 1

Geometric refraction of incident and subsurface scattered light by a dielectric with a smooth surface boundary.

Fig. 2
Fig. 2

Scattering angle θ for single scattering of light off a particle.

Fig. 3
Fig. 3

Experimental apparatus used for verifying the newly proposed diffuse-reflectance model.

Fig. 4
Fig. 4

Comparison of theoretical curves derived from measurements with an optically smooth piece of white ceramic for the newly proposed diffuse-reflectance model (solid curve), Lambert’s law (dashed curve), and empirical data (open circles) for various angles of incidence (normal viewing).

Fig. 5
Fig. 5

Same as Fig. 4 but for various viewing angles (normal light incidence).

Fig. 6
Fig. 6

Ceramic cup illuminated by an approximate point light source incident from the left-hand side perpendicular to viewing.

Fig. 7
Fig. 7

Comparison of theoretical curves for the newly proposed diffuse-reflectance model, Lambert’s law, and empirical data for distribution of diffuse reflection across the ceramic cup shown in Fig. 6: (a) with illumination incident at 90° as a function of surface orientation relative to viewing, (b) with illumination incident at 90° as a function of image pixels, (c) with illumination incident at 60° as a function of surface orientation relative to viewing, and (d) with illumination incident at 120° as a function of surface orientation relative to viewing.

Fig. 8
Fig. 8

(a) Real image of an illuminated billiard ball, (b) computer-graphics rendering with Lambert’s law, and (c) computer-graphics rendering with the proposed model.

Fig. 9
Fig. 9

Gray-level representations of isophote curves corresponding to images shown in Fig. 8.

Fig. 10
Fig. 10

(a) Polar scattering plots of Chandrasekhar diffuse reflection from normal incident light of unit radiance for various single-scattering albedos ρ. (b) Chandrasekhar diffuse reflection for various angles of incidence of light of unit radiance at normal viewing for various single-scattering albedos ρ.

Fig. 11
Fig. 11

Snell refraction at a planar dielectric surface boundary.

Fig. 12
Fig. 12

Distortion of an incident external infinitesimal angle dω to an internal infinitesimal angle d ω ¯ caused by Snell refraction.

Fig. 13
Fig. 13

(a) Transmission curve of externally incident light into a dielectric as a function of the external angle of incidence. This is also the transmission curve of the internally incident light from within the dielectric out into the air as a function of the external angle of emittance. (b) Transmission curve of externally incident light into a dielectric as a function of the internal angle of emittance. This is also the transmission curve of the internally incident light from within the dielectric out into the air as a function of the internal angle of incidence.

Fig. 14
Fig. 14

Prediction of total diffuse albedo ρ as a function of single-scattering albedo ρ according to Eq.(B15).

Tables (1)

Tables Icon

Table 1 Assumptions Made in the Derivation of Expression (3)

Equations (38)

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

ρ _ L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] ( 1 - F { sin - 1 [ sin ( ϕ ) n ] , 1 / n } ) d ω
1 π L ρ cos ( ψ ) d ω ,
unit sphere P [ cos ( θ ) ] d ω 4 π = ρ 1 ,
ρ _ 1 L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω
ρ _ L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] K N - 1 [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω ,
K = 0 π / 2 F ( ϕ , 1 / n ) C ρ [ cos ( ϕ ) , cos ( ϕ ¯ ) ] 2 π sin ( ϕ ) d ϕ .
ρ _ L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω ,
ρ _ = i = 1 ρ _ 1 K i - 1 = ρ _ 1 1 - K ,
F ( , n ) ρ _ [ 1 - F ( , n ) ] [ 1 - F ( , 1 / n ) ] d ω ,
C ρ ( μ inc , μ ref ) = ρ 4 π L μ inc μ ref + μ inc H ρ ( μ inc ) H ρ ( μ ref ) ,
H ρ ( μ ) = 1 μ 1 μ n i = 1 N ( μ + μ i ) α ( 1 + κ α μ ) ,
1 = j = 1 N a j ρ 1 - κ 2 μ j 2 .
ψ ¯ = sin - 1 [ sin ( ψ ) n ] ,             ϕ ¯ = sin - 1 [ sin ( ϕ ) n ] ,
sin ( ψ ) = n sin ( ψ ¯ ) cos ( ψ ) d ψ = n cos ( ψ ¯ ) d ψ ¯ .
d ω = sin ( ψ ) d ψ d θ ,
d ω ¯ = sin ( ψ ¯ ) d ψ ¯ d θ = cos ( ψ ) n 2 cos ( ψ ¯ ) d ω ,
cos ( ψ ¯ ) d ω ¯ = ( 1 / n 2 ) [ cos ( ψ ) d ω ] .
F ( ψ , n ) = ½ ( F + F ) ,
F ( ψ , n ) = a 2 - 2 a cos ( ψ ) + cos 2 ( ψ ) a 2 + 2 a cos ( ψ ) + cos 2 ( ψ ) , F ( ψ , n ) = a 2 - 2 a sin ( ψ ) tan ( ψ ) + sin 2 ( ψ ) tan 2 ( ψ ) a 2 + 2 a sin ( ψ ) tan ( ψ ) + sin 2 ( ψ ) tan 2 ( ψ ) × F ( ψ , n ) , a = [ n 2 - sin 2 ( ψ ) ] 1 / 2 .
[ 1 - F ( ψ , n ) ] = ( 1 - F { sin - 1 [ sin ( ϕ ) n ] , 1 / n } )             when ψ = ϕ .
[ 1 - F ( ψ , n ) ] L cos ( ψ ) d ω cos ( ψ ¯ ) d ω ¯ = [ 1 - F ( ψ , n ) ] L n 2 ,
C ρ [ cos ( ψ ¯ ) , cos ( ϕ ¯ ) ] cos ( ψ ¯ ) .
[ 1 - F ( ψ , n ) ] n 2 C ρ [ cos ( ψ ¯ ) , cos ( ϕ ¯ ) ] cos ( ψ ¯ ) cos ( ψ ¯ ) d ω ¯ = [ 1 - F ( ψ , n ) ] n 2 C ρ [ cos ( ψ ¯ ) , cos ( ϕ ¯ ) ] cos ( ψ ) n 2 cos ( ψ ¯ ) d ω = [ 1 - F ( ψ , n ) ] C ρ [ cos ( ψ ¯ ) , cos ( ϕ ¯ ) ] cos ( ψ ) cos ( ψ ¯ ) d ω .
[ 1 - F ( ϕ ¯ , 1 / n ) ] L ¯ cos ( ϕ ¯ ) d ω ¯ cos ( ϕ ) d ω = [ 1 - F ( ϕ ¯ , 1 / n ) ] L ¯ n 2 .
1 n 2 [ 1 - F ( ψ , n ) ] cos ( ψ ) cos ( ψ ¯ ) C ρ [ cos ( ψ ¯ ) , cos ( ϕ ¯ ) ] × [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω .
ρ _ 1 L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω ,
0 π / 2 [ 1 - F ( ψ , n ) ] C ρ [ cos ( ψ ¯ ) , cos ( ϕ ) ] cos ( ψ ) cos ( ψ ¯ ) d ω × { F ( ϕ , 1 / n ) C ρ [ cos ( ϕ ) , cos ( ϕ ¯ ) ] 2 π sin ( ϕ ) d ϕ } ,
ρ _ 1 L [ 1 - F ( ψ , n ) ] cos ( ψ ) d ω 0 π / 2 F ( ϕ , 1 / n ) × C ρ [ cos ( ϕ ) , cos ( ϕ ¯ ) ] 2 π sin ( ϕ ) d ϕ .
K = 0 π / 2 F ( ϕ , 1 / n ) C ρ [ cos ( ϕ ) , cos ( ϕ ¯ ) ] 2 π sin ( ϕ ) d ϕ
ρ _ 1 L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] K { [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω } .
ρ _ 1 L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] K N - 1 { [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω } .
i = 1 ρ _ 1 L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] K i - 1 { [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω } = ρ _ L [ 1 - F ( ψ , n ) ] [ cos ( ψ ) ] { [ 1 - F ( ϕ ¯ , 1 / n ) ] d ω } ,
ρ _ = i = 1 ρ _ 1 K i - 1 = ρ _ 1 1 - K .
L ( λ ) ρ _ [ ρ ( λ ) ] d λ ,
1 π L ( λ ) ( 1 - R S ) C ( ψ , λ ) ( 1 - r i ) [ R ( λ ) - D ( ψ ) ] 2 [ 1 - r i R ( λ ) ] d ω .
R = 2 - ρ ( λ ) - 2 [ 1 - ρ ( λ ) ] 1 / 2 ρ ( λ ) , C ( ψ , λ ) = ρ ( λ ) [ cos ( ψ ) ] [ 2 cos ( ψ ) + 1 ] 1 - 4 [ 1 - ρ ( λ ) ] cos 2 ( ψ ) , D ( ψ ) = 2 cos ( ψ ) - 1 2 cos ( ψ ) + 1 .
r i = 1.4399 - 0.7099 n + 0.3319 n 2 - 0.0636 n 3 n 2 ,
1 π L ( 1 - R S ) cos ( ψ ) d ω .

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