Triple Product Wavelet Integrals for 
All-Frequency Relighting


Ren Ng

Stanford University

Ravi Ramamoorthi

Columbia University

Pat Hanrahan

Stanford University


To appear at SIGGRAPH 2004






This paper focuses on efficient rendering based on pre-computed light transport, with realistic materials and shadows under all-frequency direct lighting such as 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.


Paper and Video


Paper: 300 dpi (1 MB) | Max quality (5 MB)


The video will be posted here as soon as available.





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High-res (zip) 


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High-res (zip)




Video (AVI, 3:28)


640x 480 (17.3 MB) | 320 x 240 (5.2 MB)





Powerpoint 2003 (11 MB)