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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How to evenly distribute points on a sphere more effectively than the canonical Fibonacci Lattice

Mapping the Fibonacci lattice (aka Golden Spiral, aka Fibonacci Sphere) onto the surface of a sphere is an extremely fast and effective approximate method to evenly distribute points on a sphere. I show how small modifications to the canonical implementation can result in notable improvements for nearest-neighbor measures. 

Figure 1. A simple modification to the canonical Fibonacci lattice can result in an improvement of up to 8.3% in packing distance (maximum nearest neighboring distance).

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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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Maximal Poisson disk sampling:
an improved version of Bridson’s algorithm

Bridson’s Algorithm (2007) is a very popular method to produce maximal ‘blue noise’ sample point distributions such that no two points are closer than a specified distance apart. In this brief post we show how a minor modification to this algorithm can make it 20x faster and allows it to produce much higher density blue noise sample point distributions.

Figure 1. Poisson disc sampling based on a modified version of Bridson’s algorithm. This modified algorithm runs in linear time and is up to 20x faster than the original algorithm

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an improved version of Bridson’s algorithm”

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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sample point sets with uniform projections”

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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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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Lissajous Curves

This post illustrates and explains the beautiful Lissajous Curves – trajectories of points whose coordinates follow sinusoidal movements.  A simple but timeless classic curve that has numerous applications as well as artistic elegance.

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Multiple Pendulums

This fun post illustrates the phenomenon of multiple uncoupled pendulums whose periods are all rational multiples of each other.

Figure 1. Multiple uncoupled pendulums. All periods are rationally related. This entire motion has a period of precisely 60 seconds.


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