Jittering Samples using a kd-Tree Stratification

Abstract

Monte Carlo sampling techniques are used to estimate high-dimensional integrals that model the physics of light transport in virtual scenes for computer graphics applications. These methods rely on the law of large numbers to estimate expectations via simulation, typically resulting in slow convergence. Their errors usually manifest as undesirable grain in the pictures generated by image synthesis algorithms. It is well known that these errors diminish when the samples are chosen appropriately. A well known technique for reducing error operates by subdividing the integration domain, estimating integrals in each \emph{stratum} and aggregating these values into a stratified sampling estimate. Na\"{i}ve methods for stratification, based on a lattice (grid) are known to improve the convergence rate of Monte Carlo, but require samples that grow exponentially with the dimensionality of the domain. We propose a simple stratification scheme for $d$ dimensional hypercubes using the kd-tree data structure. Our scheme enables the generation of an arbitrary number of equal volume partitions of the rectangular domain, and $n$ samples can be generated in $O(n)$ time. Since we do not always need to explicitly build a kd-tree, we provide a simple procedure that allows the sample set to be drawn fully in parallel without any precomputation or storage, speeding up sampling to $O(\log n)$ time per sample when executed on $n$ cores. If the tree is implicitly precomputed ($O(n)$ storage) the parallelised run time reduces to $O(1)$ on $n$ cores. In addition to these benefits, we provide an upper bound on the worst case star-discrepancy for $n$ samples matching that of lattice-based sampling strategies, which occur as a special case of our proposed method. We use a number of quantitative and qualitative tests to compare our method against state of the art samplers for image synthesis.