Rank-1 Lattices For Efficient Path Integral Estimation

We introduce rank-1 lattices as a quasi-random sequence to the numerical estimation of the high-dimensional path integral. Previous attempts at utilizing rank-1 lattices in computer graphics were very limited to low-dimensional applications, intentionally avoiding high dimensionality due to that the lattice search is NP-hard. We propose a novel framework that tackles this challenge, which was inspired by the rippling effect of the sample paths. Contrary to the conventional search approaches, our framework is based on recursively permuting the preliminarily selected components of the generator vector to achieve better pairwise projections and minimize the discrepancy of the path vertex coordinates in scene manifold spaces, resulting in improved rendering quality. It allows for the offline search of arbitrarily high-dimensional lattices to finish in a reasonable amount of time while removing the need to use all lattice points in the traditional definition, which opens the gate for their use in progressive rendering. Our rank-1 lattices successfully maintain the pixel variance at a comparable or even lower level compared to Sobol ' sampler, which offers a brand new solution to design efficient samplers for path tracing.