Sequences with Low-Discrepancy Blue-Noise 2-D Projections

Abstract

Distributions of samples play a very important role in rendering, affecting variance, bias and aliasing in Monte-Carlo and Quasi-Monte Carlo evaluation of the rendering equation. In this paper, we propose an original sampler which inherits many important features of classical low-discrepancy sequences (LDS): a high degree of uniformity of the achieved distribution of samples, computational efficiency and progressive sampling capability. At the same time, we purposely tailor our sampler in order to improve its spectral characteristics, which in turn play a crucial role in variance reduction, anti-aliasing and improving visual appearance of rendering. Our sampler can efficiently generate sequences of multidimensional points, whose power spectra approach so-called Blue-Noise (BN) spectral property while preserving low discrepancy (LD) in certain 2-D projections. In our tile-based approach, we perform permutations on subsets of the original Sobol LDS. In a large space of all possible permutations, we select those which better approach the target BN property, using pair-correlation statistics. We pre-calculate such “good” permutations for each possible Sobol pattern, and store them in a lookup table efficiently accessible in runtime. We provide a complete and rigorous proof that such permutations preserve dyadic partitioning and thus the LDS properties of the point set in 2-D projections. Our construction is computationally efficient, has a relatively low memory footprint and supports adaptive sampling. We validate our method by performing spectral/discrepancy/aliasing analysis of the achieved distributions, and provide variance analysis for several target integrands of theoretical and practical interest.