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.

Instant transport maps on 2D grids

“…The cost of generating one adapted sample is thus the sum of generating the random sample itself plus the cost of one BVH search that has a O (log(n)) complexity, with n the number of grid cells. For the actual generation, we implemented a simple tiling strategy of a pre-computed uniform pattern of 1024 samples, but for practical Monté-Carlo integration some recent work can stream high-quality uniform samples with very high efficiency [Perrier et al 2018]. This procedure is analogue to the classical inverse transform sampling method based on conditional CDFs [Pharr et al 2016].…”

Analysis of Sample Correlations for Monte Carlo Rendering

“…Thus, among the many blue noise generation and distribution algorithms, only a few, such as ART [ANHD17] and the approach by Perrier et al [PCX∗18], seem relevant for rendering. The first one provides adaptive sampling with a stratification property, while the later uses a complex hierarchical scrambling principle to obtain a higher dimensional sequential sampling pattern with blue noise and low discrepancy properties on 2D projections.…”

“…The first one provides adaptive sampling with a stratification property, while the later uses a complex hierarchical scrambling principle to obtain a higher dimensional sequential sampling pattern with blue noise and low discrepancy properties on 2D projections. As discussed by Perrier et al [PCX∗18], enforcing the sequential property in higher dimensions leads to lower quality in the desired 2D projections, as compared to the one obtained by 2D only samplers such as BNOT or LDBN. Christensen et al [CKK18, CFS∗18] considered both samplers in recent comparisons for Monte Carlo integration.…”

Progressive Multi‐Jittered Sample Sequences

“…Some of our sequences are similar to ART, but have better stratification. Perrier et al [PCX*18] modified the Sobol’ sequence to get a blue noise spectrum; their LDBN sequences are stratified in all elementary intervals when the sample count is a power of 16, but for in‐between sample counts their samples are not particularly evenly distributed.…”

“…It seems that the strict stratification of pmj02 samples does not leave much “wiggle room” to increase the nearest‐neighbor distances ‐ at least with our current algorithm. See Perrier et al [PCX*18] for a different trade‐off between stratification and blue noise.…”