Category: Low Discrepancy

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The Unreasonable Effectiveness of Quasirandom Sequences

I present a new low discrepancy quasirandom sequence that offers many substantial improvements over other popular sequences such as the Sobol and Halton sequences.

Figure 1a. Comparison of the various low discrepancy quasirandom sequences. Note that the newly proposed $R$-sequence produces more evenly spaced points than any of the other methods. Furthermore, all other current methods require careful selection of basis parameters, and if not chosen carefully can lead to degeneracy (eg top right).

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An alternate canonical grid layout with uniform projections


When points are placed in a canonical grid layout, they are well-separated and their projections are uniform. I present a simple canonical grid layout which  offers better closest-neighbor characterisics than the two most common contemporary canonical layouts.

Figure 1. A canonical grid layout whose projections are uniform and its closest neighbor distance characteristics are optimal.

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A new method to construct isotropic blue-noise sample point sets with uniform projections


I describe a how a small but critical modification to correlated multi-jittered sampling can significantly improve its blue noise spectral characteristics whilst maintaining its uniform projections. This is an exact and direct grid-based construction method that guarantees a minimum neighbor point separation of at least $0.707/n$ and has an average point separation of $0.965/n$.

Figure 1. Examples of point sets with $n^2$ points for $n=8,12,16,24,32,40$.
The minimum nearest neighbor distance is guaranteed to exceed $0.707/n$ and the average distance is $0.965/n$. Their blue noise sample point distributions are isotropic and their 1-D projections are exactly uniformly distributed. Futhermore each set can be directly and exactly constructed through just two permutations of the $n$-dimensional vector $\{1,2,3,…,n\}$. This therefore, presents a new way of directly constructing tileable isotropic blue noise without the need for iterative methods such as best candidate or simulated annealing

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A simple method to construct isotropic quasirandom blue noise point sequences


I describe a simple method for constructing a sequence of points, that is based on a low discrepancy quasirandom sequence but exhibits enhanced isotropic blue noise properties. This results in fast convergence rates with minimal aliasing artifacts.

Figure 1. The first 100, 200, 500, 1000, 2000 and 5000 sample points of the proposed point sequence (eqn 11) that is progressive, non-stochastic, exhibits near isotropic blue noise characteristics with fast QMC convergence rates with reduced artifacts. It is based on a new simple low discrepancy quasirandom sequence, $R_2$.

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Evenly Distributing Points in a Triangle.

Most  two dimensional quasirandom methods focus on sampling over a unit square. However, sampling evenly over the triangle is also very important in computer graphics.  Therefore,  I describe a simple and direct construction method for a point sequence to evenly cover an arbitrary shaped triangle. 

Figure 1. A new direct method for constructing an open (infinite) low discrepancy quasirandom sequence over an triangle of arbitrary shape and size. Shown are the point distributions for twelve random triangles for the first 150 points.

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A probabilistic approach
to fractional factorial design


I describe a probabilistic alternative to fractional factorial design based on the Sobol’ low discrepancy quasirandom sequence. This method is robust to aliasing  (confounders), is often simpler to implement than traditional fractional factorial sample designs, and produces more accurate results than simple random sampling.

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to fractional factorial design”