“An efficient representation for irradiance environment maps”

We consider the rendering of diffuse objects under distant illumination, as specified by an environment map. Using an analytic expression for the irradiance in terms of spherical harmonic coefficients of the lighting, we show that one needs to compute and use only 9 coefficients, corresponding to the lowest-frequency modes of the illumination, in order to achieve average errors of only 1%. In other words, the irradiance is insensitive to high frequencies in the lighting, and is well approximated using only 9 parameters. In fact, we show that the irradiance can be procedurally represented simply as a quadratic polynomial in the cartesian components of the surface normal, and give explicit formulae. These observations lead to a simple and efficient procedural rendering algorithm amenable to hardware implementation, a prefiltering method up to three orders of magnitude faster than previous techniques, and new representations for lighting design and image-based rendering.

1. J. Arvo. Applications of irradiance tensors to the simulation of non-lambertian phenomena. In SIGGRAPH 95, pages 335-342, 1995.
2. R. Basri and D. Jacobs. Lambertian reflectance and linear subspaces. In International Conference on Computer Vision, 2001.
3. J. F. Blinn and M. E. Newell. Texture and reflection in computer generated images. Communications of the ACM, 19:542-546, 1976.
4. B. Cabral, N. Max, and R. Springmeyer. Bidirectional reflection functions from surface bump maps. In SIGGRAPH 87, pages 273-281, 1987.
5. B. Cabral, M. Olano, and P. Nemec. Reflection space image based rendering. In SIGGRAPH 99, pages 165-170, 1999.
6. P. Debevec, T. Hawkins, C. Tchou, H.P. Duiker, W. Sarokin, and M. Sagar. Acquiring the reflectance field of a human face. In SIGGRAPH 00, pages 145-156.
7. R. Epstein, P.W. Hallinan, and A. Yuille. 5 plus or minus 2 eigenimages suffice: An empirical investigation of low-dimensional lighting models. In IEEE Workshop on Physics-Based Modeling in Computer Vision, pages 108-116, 1995.
8. N. Greene. Environment mapping and other applications of world projections. IEEE Computer Graphics & Applications, 6(11):21-29, 1986.
9. G. Greger, P. Shirley, P. Hubbard, and D. Greenberg. The irradiance volume. IEEE Computer Graphics & Applications, 18(2):32-43, 1998.
10. P.W. Hallinan. A low-dimensional representation of human faces for arbitrary lighting conditions. In CVPR 94, pages 995-999, 1994.
11. J. Kautz, P. Vazquez, W. Heidrich, and H.P. Seidel. A unified approach to prefiltered environment maps. In EuroGraphics Rendering Workshop 00, pages 185-196, 2000.
12. P. Lalonde and A. Fournier. Filtered local shading in the wavelet domain. In EGRW 97, pages 163-174, 1997.
13. T.M. MacRobert. Spherical harmonics; an elementary treatise on harmonic functions, with applications. Dover Publications, 1948.
14. G. Miller and C. Hoffman. Illumination and reflection maps: Simulated objects in simulated and real environments. SIGGRAPH 84 Advanced Computer Graphics Animation seminar notes, 1984.
15. K. Proudfoot, W. Mark, S. Tzvetkov, and P. Hanrahan. A real-time procedural shading system for programmable graphics hardware. In SIGGRAPH 01, 2001.
16. R. Ramamoorthi and P. Hanrahan. On the relationship between radiance and irradiance: Determining the illumination from images of a convex lambertian object. To appear, Journal of the Optical Society of America A, 2001.
17. F. X. Sillion, J. Arvo, S. H. Westin, and D. Greenberg. A global illumination solution for general reflectance distributions. In SIGGRAPH 91, pages 187-196.
18. G. Ward and P. Heckbert. Irradiance gradients. In EGRW92, pages 85-98, 1992.
19. A. Wilkie, R. Tobler, and W. Purgathofer. Orientation lightmaps for photon radiosity in complex environments. In CGI 00, pages 279-286, 2000.