Tag: isotropic

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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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