Analysis and Synthesis of Digital Dyadic Sequences

TL;DR

This work studies digital dyadic nets and sequences in base $2$, providing a complete design-space characterization, an algorithm to reorder any digital dyadic net into a digital dyadic sequence, and a new family of self-similar $\xi$-sequences that are fast to sample and invert. It presents a rigorous linear-algebraic framework using progressive and dyadic pairs, derives exact dimension counts for the net/sequence design spaces, and demonstrates practical constructions (Hammersley, LP, Gray) with validation and performance benefits. The introduction of $\xi$-sequences yields improved sampling properties and GPU-friendly implementations that outperform traditional Sobol-based methods in rendering contexts, while maintaining compact memory footprints. Collectively, the paper links theory and practice to enable efficient, high-quality sampling for Monte Carlo rendering and related applications, and sets the stage for extending these ideas to higher dimensions and more complex sampling 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 $ξ$-sequences, that spans a subspace with fewer degrees of freedom. Those $ξ$-sequences are extremely efficient to sample and compute, and we demonstrate their advantages over the classic Sobol (0, 2)-sequence.

Figures (10)

• Figure 1: An example 4-bit 16-point dyadic net (i.e., $m=4$, $N=16$) showing all $m + 1 = 5$ possible stratifications: $\frac{1}{2^4}\times 1$, $\frac{1}{2^3}\times \frac{1}{2}$, $\frac{1}{2^2}\times \frac{1}{2^2}$, $\frac{1}{2}\times \frac{1}{2^3}$, $1\times \frac{1}{2^4}$.
• Figure 2: The hierarchical structure of a dyadic sequence is visualized as a tree. Note that the individual points also represent (0, 0, 2)-nets (i.e., 1-point dyadic nets). All sub-sequences corresponding to an internal (or leaf) node in the tree are required to be dyadic nets. Here the gray midlines are shown for better appearance.
• Figure 3: A visualization to support the proof of Theorem \ref{['th-digital dyadic sequence']}. If an $m \times m$ generating matrix for a 1D digital dyadic sequence is multiplied with the matrix that encodes the sequence of integers $[0, 1, 2, 3, \ldots]$ in binary, the output matrix must ensure that specific blocks in the output form $2^1, 2^2, \ldots, 2^m$-point nets. However, to establish the net property for a $2^k$-point net, we only need to look at the first $k$ bits ($k$ rows) of the output. In the output we visualize the blocks of bits that need to be checked to verify the $2^1$-point ( blue color), $2^2$-point ( cyan color), and $2^3$-point ( green color) net property.
• Figure 4: The first 16-point nets of 256-point (a) Hammersley (b) LP (c) Gray sequences, with possible indexings (in hexadecimal) of the points.
• Figure 5: Structure, generating matrices, points, and periodograms of various self-similar $\xi$-sequences. The plots to the left depict the first few points; larger circles show the points closer to the beginning of the sequence. The top plots show the $\xi_0$-sequence generated by $X=Y=(1,0,\dots,0)$.

Theorems & Definitions (36)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• theorem 1
• Definition 3.6
• theorem 2
• theorem 3
• theorem 4