“Triple product wavelet integrals for all-frequency relighting” by Ng, Ramamoorthi and Hanrahan

This paper focuses on efficient rendering based on pre-computed light transport, with realistic materials and shadows under all-frequency direct lighting such an environment maps. The basic difficulty is representation and computation in the 6D space of light direction, view direction, and surface position. While image-based and synthetic methods for real-time rendering have been proposed, they do not scale to high sampling rates with variation of both lighting and viewpoint. Current approaches are therefore limited to lower dimensionality (only lighting or viewpoint variation, not both) or lower sampling rates (low frequency lighting and materials). We propose a new mathematical and computational analysis of pre-computed light transport. We use factored forms, separately pre-computing and representing visibility and material properties. Rendering then requires computing triple product integrals at each vertex, involving the lighting, visibility and BRDF. Our main contribution is a general analysis of these triple product integrals, which are likely to have broad applicability in computer graphics and numerical analysis. We first determine the computational complexity in a number of bases like point samples, spherical harmonics and wavelets. We then give efficient linear and sublinear-time algorithms for Haar wavelets, incorporating non-linear wavelet approximation of lighting and BRDFs. Practically, we demonstrate rendering of images under new lighting and viewing conditions in a few seconds, significantly faster than previous techniques.

1. AGARWAL, S., RAMAMOORTHI, R., BELONGIE, S., AND JENSEN, H. 2003. Structured importance sampling of environment maps. ACM Transactions on Graphics 22, 3, 605–612. Google ScholarDigital Library
2. ARVO, J., TORRANCE, K., AND SMITS, B. 1994. A framework for the analysis of error in global illumination algorithms. In SIGGRAPH 94, 75–84. Google ScholarDigital Library
3. CABRAL, B., OLANO, M., AND NEMEC, P. 1999. Reflection space image based rendering. In SIGGRAPH 99, 165–170. Google ScholarDigital Library
4. DANA, K., GINNEKEN, B., NAYAR, S., AND KOENDERINK, J. 1999. Reflectance and texture of real-world surfaces. ACM Transactions on Graphics 18, 1 (January), 1–34. Google ScholarDigital Library
5. DEBEVEC, P. E., AND MALIK, J. 1997. Recovering high dynamic range radiance maps from photographs. In SIGGRAPH 97, 369–378. Google ScholarDigital Library
6. DEVORE, R. 1998. Nonlinear approximation. Acta Numerica 7, 51–150.Google ScholarCross Ref
7. DORSEY, J., ARVO, J., AND GREENBERG, D. 1995. Interactive design of complex time dependent lighting. IEEE Computer Graphics and Applications 15, 2 (March), 26–36. Google ScholarDigital Library
8. GERSHBEIN, R., SCHRÖDER, P., AND HANRAHAN, P. 1994. Textures and radiosity: Controlling emission and reflection with texture maps. In SIGGRAPH 94, 51–58. Google ScholarDigital Library
9. INUI, T., TANABE, Y., AND ONODERA, Y. 1990. Group theory and its applications in physics. Springer Verlag.Google Scholar
10. KAUTZ, J., AND MCCOOL, M. 1999. Interactive rendering with arbitrary BRDFs using separable approximations. In Eurographics Rendering Workshop 99, 247–260. Google ScholarDigital Library
11. KAUTZ, J., SNYDER, J., AND SLOAN, P. 2002. Fast arbitrary brdf shading for low-frequency lighting using spherical harmonics. In Eurographics Rendering Workshop 2002, 291–296. Google ScholarDigital Library
12. LEHTINEN, J., AND KAUTZ, J. 2003. Matrix radiance transfer. In Symposium on Interactive 3D graphics, 59–64. Google ScholarDigital Library
13. MACROBERT, T. 1948. Spherical harmonics; an elementary treatise on harmonic functions, with applications. Dover Publications.Google Scholar
14. MALLAT, S. 1999. A Wavelet Tour of Signal Processing. Academic Press. Google ScholarDigital Library
15. MEYER, Y., COIFMAN, R., AND SALINGER, D. 1997. Wavelets: Calderon-Zygmund and Multilinear Operators. Cambridge Univ. Press.Google Scholar
16. NG, R., RAMAMOORTHI, R., AND HANRAHAN, P. 2003. All-frequency shadows using non-linear wavelet lighting approximation. ACM Transactions on Graphics 22, 3, 376–381. Google ScholarDigital Library
17. NIMEROFF, J., SIMONCELLI, E., AND DORSEY, J. 1994. Efficient re-rendering of naturally illuminated environments. In Eurographics Workshop on Rendering 94, 359–373.Google Scholar
18. RAMAMOORTHI, R., AND HANRAHAN, P. 2002. Frequency space environment map rendering. ACM Transactions on Graphics (SIGGRAPH 02 proceedings) 21, 3, 517–526. Google ScholarDigital Library
19. RUSINKIEWICZ, S. 1998. A new change of variables for efficient BRDF representation. In Eurographics Rendering Workshop 98, 11–22.Google ScholarCross Ref
20. SLOAN, P., KAUTZ, J., AND SNYDER, J. 2002. Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. ACM Transactions on Graphics 21, 3, 527–536. Google ScholarDigital Library
21. SLOAN, P., HALL, J., HART, J., AND SNYDER, J. 2003. Clustered principal components for precomputed radiance transfer. ACM Transactions on Graphics 22, 3, 382–391. Google ScholarDigital Library
22. SLOAN, P., LIU, X., SHUM, H., AND SNYDER, J. 2003. Bi-scale radiance transfer. ACM Transactions on Graphics 22, 3, 370–375. Google ScholarDigital Library
23. STOLLNITZ, E., DEROSE, T., AND SALESIN, D. 1996. Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann. Google ScholarDigital Library
24. THORNBER, K., AND JACOBS, D. 2001. Broadened, specular reflection and linear subspaces. Tech. Rep. TR#2001-033, NEC.Google Scholar