Fast product importance sampling of environment maps

Environment maps have been used for decades in production path-tracers to recreate ambient lighting conditions captured from real world scenes. Stochastic sampling of the radiance integral can be very challenging however, as both the BSDF and the environment can have strong peaks that are not aligned with each other. Multiple importance sampling (MIS) between the environment and the BSDF is a common way to reduce variance by re-weighting each estimator, but can still result in wasted samples. Product importance sampling is an effective way to reduce the variance by drawing samples using a probability distribution built from the product of the BSDF and the environment map. To our knowledge, the most practical product sampling technique [Clarberg and Akenine-Möller 2008] is still relatively costly for production rendering because it approximates the BSDF by a sparse quad-tree built on the fly from a few hundred BSDF samples. Due to the high complexity of the multi-lobed models used in film rendering, this cost can be prohibitive.