Projective Blue‐Noise Sampling

Abstract

We propose projective blue-noise patterns that retain their blue-noise characteristics when undergoing one or multiple projections onto lower dimensional subspaces. These patterns are produced by extending existing methods, such as dart throwing and Lloyd relaxation, and have a range of applications. For numerical integration, our patterns often outperform state-of-the-art stochastic and low-discrepancy patterns, which have been specifically designed only for this purpose. For image reconstruction, our method outperforms traditional blue-noise sampling when the variation in the signal is concentrated along one dimension. Finally, we use our patterns to distribute primitives uniformly in 3D space such that their 2D projections retain a blue-noise distribution.

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

“…In higher dimensions (dimension 4 for numerical comparisons) the Sobol sequence and its scrambling are also considered. For anti-aliasing evaluation, we have compared to Projective Blue Noise (PBN) [RRSG16] that also aims to control projections. Reinert et al have defined two variants of PBN; the first one relies on a Dart Throwing (DT) strategy and the second one on a Lloyd's relaxation approach.…”

“…Projective Blue‐Noise Sampling The idea of having s ‐D sampling with good blue‐noise projections, has been presented in [RRSG16], where the metric of distance between points used in any optimization process has been modified in order to keep track of the projective distance. Although the goal of getting projective BN properties has been achieved that paper does not guarantee uniformity in higher‐dimensional space, which could be potentially harmful for integration tasks, especially when the number of samples grows.…”

“…They were the first to propose stochastic sampling as an alternative to regular grid-based sampling, arguing that unstructured noise is better perceived than visually perturbing aliasing. Since then, two optimization-based approaches have been developed and presented in numerous papers: (1) on-line optimization [MF92, DH06, LD08, BSD09, BWWM10, EMP * 12, CYC * 12, SGBW10, SHD11, Fat11, dGBOD12, ZHWW12, OG12, HSD13, RRSG16], and (2) off-line optimization [ODJ04, KCODL06, Ost07, WPC * 14, APC * 16,ANHD17], where the near-optimal solution is prepared in Projective Blue-Noise Sampling The idea of having s-D sampling with good blue-noise projections, has been presented in [RRSG16], where the metric of distance between points used in any optimization process has been modified in order to keep track of the projective distance. Although the goal of getting projective BN properties has been achieved that paper does not guarantee uniformity in higher-dimensional space, which could be potentially harmful for integration tasks, especially when the number of samples grows.…”

Low-discrepancy blue noise sampling

“…The idea is that the projections of the samples on each axis are uniformly distributed. Saka et al [2007] studied Latinization as a post-processing of an optimized set, whereas Reinert et al [2016] integrated the idea within the optimization process. Our technique also produces uniform distributions of projection, but with provable LD.…”

“…We prefer the latter option, since it imposes less bias (over-or under-crowding) near the edges of the strata, and is also easier to implement. We also occasionally Latinized the point-set during the optimization process in the spirit of Reinert et al [2016]. This process almost circumvents the bias to the edges.…”