Invariant subspace
There are some standard constructions of low-discrepancy sequences.
The van der Corput sequence
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Let
be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ dk(n) < b. Set
Then there is a constant C depending only on b such that (gb(n))n ≥ 1 satisfies
where D*N is the star discrepancy.
The Halton sequence
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The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let s be an arbitrary dimension and b1, ..., bs be arbitrary coprime integers greater than 1. Define
Then there is a constant C depending only on b1, ..., bs, such that sequence {x(n)}n≥1 is a s-dimensional sequence with
The Hammersley set
Let b1,...,bs-1 be coprime positive integers greater than 1. For given s and N, the s-dimensional Hammersley set of size N is defined by
for n = 1, ..., N. Then
where C is a constant depending only on b1, ..., bs−1.
Poisson disk sampling
Poisson disk sampling is popular in video games to rapidly placing objects in a way that appears random-looking but guarantees that every two points are separated by at least the specified minimum distance.[1]
References
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- Quasi-Monte Carlo Simulations, http://www.puc-rio.br/marco.ind/quasi_mc.html
- ↑ Herman Tulleken. "Poisson Disk Sampling". Dev.Mag Issue 21, March 2008.