“Generalization of Lambert’s reflectance model” by Oren and Nayar

Lambert’s model for body reflection is widely used in computer graphics. It is used extensively by rendering techniques such as radiosity and ray tracing. For several real-world objects, however, Lambert’s model can prove to be a very inaccurate approximation to the body reflectance. While the brightness of a Lambertian surface is independent of viewing direction, that of a rough surface increases as the viewing direction approaches the light source direction. In this paper, a comprehensive model is developed that predicts body reflectance from rough surfaces. The surface is modeled as a collection of Lambertian facets. It is shown that such a surface is inherently non-Lambertian due to the foreshortening of the surface facets. Further, the model accounts for complex geometric and radiometric phenomena such as masking, shadowing, and interreflections between facets. Several experiments have been conducted on samples of rough diffuse surfaces, such as, plaster, sand, clay, and cloth. All these surface demonstrate significant deviation from Lambertian behavior. The reflectance measurements obtained are in strong agreement with the reflectance predicted by the model.

1. P. Beckmann. Shadowing of random rough surfaces. IEEE Transactions on Antennas and Propagation, AP-13:384-388, 1965.
2. P. Beckmann and A. Spizzichino. The Scattering of Electro-magnetic Waves from Rough Surfaces. Pergamon, New York, 1963.
3. J. F. Blinn. Models of light reflection for computer synthe-sized pictures. ACM Computer Graphics (SIGGRAPH 77), 19(10):542-547, 1977.
4. D. Buhl, W. J. Welch, and D. G. Rea. Reradiation and thermal emission from illuminated craters on the lunar surface. Journal of Geophysical Research, 73(16):5281-5295, August 1968.
5. B. Cabral, N. Max, and R. Springmeyer. Bidirectional re-flection functions from surface bump maps. ACM Computer Graphics (SIGGRAPH 87), 21(4):273-281, 1987.
6. S. Chandrasekhar. Radiative Transfer. Dover Publications, 1960.
7. M. F. Cohen and D. P. Greenberg. The hemi-cube, a radiosity solution for complex environments. ACM Computer Graphics (SIGGRAPH 85), 19(3):31-40, 1985.
8. R. L. Cook and K. E. Torrance. A reflection model for com-puter graphics. ACM Transactions on Graphics, 1(1):7-24, 1982.
9. D. Forsyth and A. Zisserman. Mutual illumination. Proc. Conf. Computer Vision and Pattern Recognition, pages 466- 473, 1989.
10. R. Hall. Illumination and Color in Computer Generated Im-agery. Springer-Verlag, 1989.
11. P. Hanrahan and W. Krueger. Reflection from layered surfaces due to subsurface scattering. Computer Graphics Proceedings (SIGGRAPH 93), pages 165-174, 1993.
12. B. W. Hapke, R. M. Nelson, and W. D. Smythe. The opposition effect of the moon: The contribution of coherent backscatter. Science, 260(23):509-511, April 1993.
13. B. W. Hapke and Huge van Horn. Photometric studies of complex surfaces, with applications to the moon. Journal of Geophysical Research, 68(15):4545-4570, August 1963.
14. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg. A comprehensive physical model for light reflection. ACM Computer Graphics (SIGGRAPH 91), 25(4):175-186, 1991.
15. R. G. Hering and T. F. Smith. Apparent radiation proper-ties of a rough surface. AIAA Progress in Astronautics and Aeronautics, 23:337-361, 1970.
16. M. Jakob. Heat Transfer. Wiley, 1957.
17. J. T. Kajiya. Anisotropic reflection model. ACM Computer Graphics (SIGGRAPH 91), 25(4):175-186, 1991.
18. J. J. Koenderink and A. J. van Doorn. Geometrical modes as a general method to treat diffuse interreflections in radiometry. Journal of the Optical Society of America, 73(6):843-850, 1983.
19. Y. Kuga and A. Ishimaru. Retroreflectance from a dense dis-tribution of spherical particles. Journal of the Optical Society of America A, 1(8):831-835, August 1984.
20. J. H. Lambert. Photometria sive de mensure de gratibus lumi-nis, colorum umbrae. Eberhard Klett, 1760.
21. M. Minnaert. The reciprocity principle in lunar photometry. Astrophysical Journal, 93:403-410, 1941.
22. S. K. Nayar, K. Ikeuchi, and T. Kanade. Shape from in-terreflections. International Journal of Computer Vision, 6:3:173-195, 1991.
23. F. E. Nicodemus, J. C. Richmond, and J. J. Hsia. Geometrical Considerations and Nomenclature for Reflectance. National Bureau of Standards, October 1977. Monograph No. 160.
24. P. Oetking. Photometric studies of diffusely reflecting surfaces with application to the brightness of the moon. Journal of Geophysical Research, 71(10):2505-2513, May 1966.
25. E. Opik. Photometric measures of the moon and the moon the earth-shine. Publications de L’Observatorie Astronomical de L’Universite de Tartu, 26(1):1-68, 1924.
26. N. S. Orlova. Photometric relief of the lunar surface. Astron. Z, 33(1):93-100, 1956.
27. P. Poulin and A. Fournier. A model for anisotropic reflection. ACM Computer Graphics (SIGGRAPH 90), 24(4):273-282, 1990.
28. T. Shibata, W. Frei, and M. Sutton. Digital correction of solar illumination and viewing angle artifacts in remotely sensed images. Machine Processing of Remotely Sensed Data Sym-posium, pages 169-177, 1981.
29. R. Siegel and J. R. Howell. Thermal Radiation Heat Transfer. Hemisphere Publishing Corporation, third edition, 1972.
30. B. G. Smith. Lunar surface roughness: Shadowing and thermal emission. Journal of Geophysical Research, 72(16):4059-4067, August 1967.
31. K. Torrance and E. Sparrow. Theory for off-specular reflec-tion from rough surfaces. Journal of the Optical Society of America, 57:1105-1114, September 1967.
32. L. Tsang and A. Ishimaru. Backscattering enhancement of random discrete scatterers. Journal of the Optical Society of America A, 1(8):836-839, August 1984.
33. S. Upstill. The RenderMan Companion. Addison Wesley, 1989.
34. R. J. Wagner. Shadowing of randomly rough surfaces. Journal of the Acoustical Society of America, 41(1):138-147, June 1966.
35. H.W. Westin, J.R. Arvo, and K.E. Torrance. Predicting re-flectance functions from complex surfaces. ACM Computer Graphics (SIGGRAPH 92), 26(2):255-264, 1992.
36. T. Whitted. An improved illumination model for shaded dis-play. Communications of the ACM, 23(6):343-349, 1980.