Affine double- and triple-product wavelet integrals for rendering

Many problems in computer graphics involve integrations of products of functions. Double- and triple-product integrals are commonly used in applications such as all-frequency relighting or importance sampling, but are limited to distant illumination. In contrast, near-field lighting from planar area lights involves an affine transform of the source radiance at different points in space. Our main contribution is a novel affine double- and triple-product integral theory; this generalization enables one of the product functions to be scaled and translated. We study the computational complexity in a number of bases, with particular attention to the common Haar wavelets. We show that while simple analytic formulae are not easily available, there is considerable sparsity that can be exploited computationally. We demonstrate a practical application to compute near-field lighting from planar area sources, that can be easily combined with most relighting algorithms. We also demonstrate initial results for wavelet importance sampling with near-field area lights, and image processing directly in the wavelet domain.