Analysis and Synthesis of Digital Dyadic Sequences

TL;DR

This paper thoroughly characterizes digital dyadic nets and sequences, introduces a new family of efficient $\xi$-sequences, and demonstrates their advantages over Sobol sequences in computer graphics rendering.

Contribution

It provides a complete characterization of digital dyadic nets and sequences, and introduces a novel, efficient family of $\xi$-sequences with practical advantages.

Findings

Every digital dyadic net can be reordered into a sequence.

The $\xi$-sequences are highly efficient to sample and compute.

$\xi$-sequences outperform Sobol sequences in rendering tasks.

Abstract

We explore the space of matrix-generated (0, m, 2)-nets and (0, 2)-sequences in base 2, also known as digital dyadic nets and sequences. In computer graphics, they are arguably leading the competition for use in rendering. We provide a complete characterization of the design space and count the possible number of constructions with and without considering possible reorderings of the point set. Based on this analysis, we then show that every digital dyadic net can be reordered into a sequence, together with a corresponding algorithm. Finally, we present a novel family of self-similar digital dyadic sequences, to be named $\xi$-sequences, that spans a subspace with fewer degrees of freedom. Those $\xi$-sequences are extremely efficient to sample and compute, and we demonstrate their advantages over the classic Sobol (0, 2)-sequence.