Lossless Basis Expansion for Gradient-Domain Rendering

Gradient-domain rendering utilizes difference estimates with shift mapping to reduce variance in Monte Carlo rendering. Such difference estimates are effective under the assumption that pixels for difference estimates have similar integrands. This assumption is often violated because it is common to have spatially varying BSDFs with material maps, which potentially result in a very different integrand per pixel. We introduce an extension of gradient-domain rendering that effectively supports such per-pixel variation in BSDFs based on basis expansion. Basis expansion for BSDFs has been used extensively in other problems in rendering, where the goal is to approximate a given BSDF by a weighted sum of predefined basis functions. We instead utilize lossless basis expansion, representing a BSDF without any approximation by adding the remaining difference in the original basis expansion. This lossless basis expansion allows us to cancel more terms via shift mapping, resulting in low variance difference estimates even with per-pixel BSDF variation. We also extend the Poisson reconstruction process to support this basis expansion. Regular gradient-domain rendering can be expressed as a special case of our extension, where the basis is simply the BSDF per pixel (i.e., no basis expansion). We provide proof-of-concept experiments and showcase the effectiveness of our method for scenes with highly varying material maps. Our results show noticeable improvement over regular gradient-domain rendering under both L¹ and L² reconstructions. The resulting formulation via basis expansion essentially serves as a new way of path reuse among pixels in the presence of per-pixel variation.