Uniform random points are honest in the least helpful way.

They keep the contract and still make a mess.

Drop 180 independent points into a square and they will do exactly what the math promises: sometimes crowd together, sometimes leave holes, never apologize for either. If those points are stars, grass clumps, stipple dots, shadow rays, bullet decals, sampling locations, or item spawns, the viewer may read the clumps as intention and the holes as neglect.

A grid fixes the holes and commits a different crime. It is too legible. Rotate the camera, slide the texture, march an enemy path across it, and the pattern starts whispering its lattice.

Blue noise is the compromise I reach for most often:

random enough to avoid a grid
repelled enough to avoid clumps
spectrally shaped
so mistakes become fine-grained noise

That last line is the whole trick. Blue noise is not just “nice random dots.” It is a claim about where the error lives in frequency space.

Eye Notices Slow Mistakes

Noise has color by analogy with light. White noise spreads power roughly evenly across frequencies. Blue noise pushes power toward higher frequencies and leaves a hole near the origin of the spectrum. In images, this is useful because broad, low-frequency mistakes look like blotches, bands, and cloudy structure. Small high-frequency mistakes often read as texture.

This is an old graphics lesson. Cook’s 1986 paper on stochastic sampling argued that regularly spaced samples are what turn high frequencies into structured aliasing; nonuniform samples can turn the same failure into noise that is less objectionable to vision.1 Ulichney’s 1988 work introduced blue-noise dithering as a way to render halftones with desirable high-frequency properties.2 Mitsa and Parker later designed blue-noise masks whose thresholded binary patterns keep the pleasant part of ordered dithering while avoiding periodic artifacts.3

The shared move is not “add randomness.” The shared move is “shape the randomness so its failures are harder to see.”

If a sampling pattern has too much low-frequency power, it has visible geography: large clumps and large empty islands. If it has narrow spectral spikes, it has a hidden grid. A good blue-noise point set is trying to have neither.

The Dot Lab

The lab below compares four point generators:

  • uniform random: independent points
  • jittered grid: one random point per mostly regular cell
  • best candidate: pick the farthest among (k) random proposals
  • Poisson disk: keep every pair at least (r) apart

For the default seed, 180 points, radius 54 / 1000, and 30 candidates, the numbers are deliberately visible. Uniform random gets a minimum gap of only 6.0 / 1000, low-frequency power around 0.89x, and a 95th-percentile void radius around 73.4 / 1000. The Bridson Poisson-disk run emits 180 points from a filled 215-point pool, keeps a 54.1 / 1000 minimum gap, and drops low-frequency power to about 0.19x.

The best-candidate pattern is competitive in this small two-dimensional lab: its low-frequency power is about 0.18x. But it paid for that by checking roughly 483k candidate-to-sample distances, while the Poisson-disk run used about 21.8k local grid checks. That bill is the algorithmic part of the story.

uniform random jittered grid best candidate Poisson disk

The spectrum panel computes a direct periodogram over integer spatial frequencies and radially averages it. Values below 1 near the origin mean low-frequency energy is suppressed relative to the pattern's average spectral power.

The Four Panels Are a Lie Detector

The first panel is only the bait. The dots should look plausible, but a pleasant scatter plot is not a diagnostic.

The nearest-neighbor histogram is the first audit. Uniform random points have a long left tail because independence allows arbitrarily close pairs. Jittered grids reduce the left tail, but adjacent cells can still put points near a shared edge. Best-candidate and Poisson-disk sampling both push the histogram rightward.

The spectrum panel is the second audit. In the lab, I use the point-set periodogram:

\[P(k)=\frac{1}{n}\left|\sum_{j=1}^n \exp(-2\pi i k\cdot x_j)\right|^2.\]

Then I average (P(k)) over rings of similar (\lVert k\rVert). A blue-noise pattern should have a low-frequency dip and then a noisy ring of energy at higher frequencies. That is the “blue” part: the power is not evenly distributed.

The void map is the third audit. Each small pixel asks how far it is from the nearest sample. Warm islands are places the pattern is underserving. This is the map a level designer secretly cares about when distributing rocks, enemies, trees, or loot. It is also the map a renderer cares about when deciding whether some part of an integral has not been sampled well.

Poisson Disk Draws a Small Boundary

“Poisson disk” is a slightly confusing phrase. A homogeneous Poisson process has independent counts and no minimum spacing. A Poisson-disk pattern is a hard-core pattern: every pair of samples must be at least distance (r) apart. The disk language comes from imagining a disk of radius (r/2) around every point. Those disks cannot overlap.

That one rule kills the worst clumps before they can become visible.

It does not make the pattern a grid. There is still randomness in angles, local packing, and boundary behavior. But it deletes the event that independent random sampling loves to produce: two samples almost on top of each other while some other region gets nothing.

This is why Poisson-disk points keep showing up in practical graphics and game systems:

trees that should not intersect
spawn points that should not overlap
texture stipples that should not band
shadow samples that should not alias into a lattice
particles that should not start as clumps

The rule is local. The relief is global.

Bridson Makes Dart Throwing Local

Naive dart throwing is simple:

draw a random point
keep it if it is at least r away
from every old point
repeat until tired

It works until the domain gets crowded. Then almost every new dart lands too close to something, and the algorithm spends most of its life rejecting.

Bridson’s 2007 sketch gives the version that became a workhorse in graphics code.4 It keeps an active list of samples that may still have room nearby. For an active point, it proposes candidates in the annulus between radius (r) and (2r). A background grid with cell width at most (r/\sqrt{2}) makes the distance test local: in two dimensions, each grid cell can hold at most one sample, so only nearby cells need to be checked.

In pseudocode:

start with one random point
while the active list is not empty:
    pick an active point x
    try k candidates near x
    accept the first candidate
        at least r from its grid neighbors
    if all k candidates fail, retire x

Bridson’s analysis is pleasingly small. Each loop either emits a sample or retires one, so the number of active-list iterations is linear in the number of samples when (k) is treated as a constant.

The implementation in this post fills the domain first. If the filled pool has more points than the slider asks for, it takes a seeded subset. That detail is important. If you stop the Bridson growth the instant you reach n, you do not get a uniform pattern over the whole square; you get a growing colony around the initial point with a large empty frontier. I made that mistake while building the lab, and the void map caught it immediately. The empty frontier looked like a weather system.

Best Candidate Has Taste and a Bill

Mitchell’s best-candidate idea is seductive.5 To add the next point, draw (k) random candidates and keep the one farthest from the existing set. The algorithm is almost a sentence:

many proposals enter
the least crowded proposal leaves

For small (n), it works beautifully. It is easy to animate. It is easy to explain. In the default lab run, it slightly beats the Poisson-disk subset on low-frequency power.

The cost is that “farthest from the existing set” is a global question unless you add more spatial data structures. With 180 points and 30 candidates, the lab performs 483,300 candidate-to-sample distance checks. That is still tiny in JavaScript. It is not tiny if you are generating many patterns per frame, many dimensions per sample, or many worlds in a build pipeline.

Poisson disk wins not because it is morally better. It wins because it turns a global aesthetic into a local rejection test.

Jitter Fixes Half the Problem

Jittered grids deserve more respect than they sometimes get.

One random point per cell removes the worst voids of independent sampling. It is also extremely cheap and easy to stratify across pixels, frames, or dimensions. Cook discussed jitter as a practical stochastic sampling scheme because it reduces aliasing while keeping a useful amount of regularity.1

But a jittered grid remembers the grid. Depending on the task, this memory can be good or bad. If you are integrating one sample per pixel, stratification is often exactly what you want. If you are placing mushrooms in a forest, the viewer might eventually sense the invisible checkerboard. If you are sampling motion blur or a glossy lobe, the grid may interact with other dimensions in ways that are not visible in a two-dimensional scatter plot.

This is why “blue noise” is not a universal replacement for stratification, low-discrepancy sequences, multiple importance sampling, or domain-specific design. Sampling patterns are instruments. Each one has a frequency response.

What the Blue Does Not Buy You

Blue noise is a useful aesthetic and numerical bias. It is not magic.

It does not promise independent samples. In fact, it achieves its look by destroying independence.

It does not promise the lowest variance for every integral. If the integrand has structure aligned with the pattern’s remaining spectral energy, another sampler may win.

It does not automatically handle nonuniform density. If the desired density changes across the map, the radius should change too, and then the simple uniform grid argument needs care.

It does not settle game design. A fair spawn system may need team constraints, visibility constraints, travel-time constraints, and anti-camping memory. A Poisson-disk sampler only says “not too close.” That is a geometric sentence, not a design document.

It also has boundary effects. Samples near the edge have less annulus available, and filled finite domains are not the same as infinite stationary processes. For tileable textures, you may want periodic distance tests. For streaming worlds, you may need chunks whose samples agree across chunk boundaries.

The best use of blue noise is modest: reach for it when low-frequency clumps are more annoying than high-frequency grain.

My Rule of Thumb

When I need plausible randomness in a visual system, I ask three questions:

Can two samples be too close?
Can a regular grid become visible?
Can broad holes be worse than fine grain?

If the answer to all three is yes, I reach for blue noise or Poisson-disk sampling first.

Then I check the spectrum.

Not because the spectrum is more real than the picture, but because the picture is too persuasive. A point set can look natural for one seed and fail for the next. A grid can hide in a scatter plot and announce itself in motion. A random pattern can look fine until the one clump appears exactly where the player is staring.

Randomness is not a vibe. It is a distribution, a geometry, and a spectrum.

Blue noise is the version that tells the dots: be random, but give each other a little room.

  1. Robert L. Cook, “Stochastic Sampling in Computer Graphics”, ACM Transactions on Graphics 5(1), 1986. ↩ ↩2

  2. Robert A. Ulichney, “Dithering with Blue Noise”, Proceedings of the IEEE 76(1), 1988. ↩

  3. Theophano Mitsa and Kevin J. Parker, “Digital halftoning technique using a blue-noise mask”, Journal of the Optical Society of America A 9(11), 1992. ↩

  4. Robert Bridson, “Fast Poisson Disk Sampling in Arbitrary Dimensions”, SIGGRAPH 2007 sketch. ↩

  5. Don P. Mitchell, “Spectrally Optimal Sampling for Distribution Ray Tracing”, SIGGRAPH 1991. ↩