Perlin noise (Ken Perlin, 1983)

Perlin noise is a function that gives a smoothly varying, random-looking value for every point in space. Ken Perlin made it in 1983 to give computer images a less mechanical surface. He described it in the SIGGRAPH paper "An Image Synthesizer" in 1985 and received an Academy Award for it in 1997.^1, 2, 3

Why it was made

Perlin began thinking about procedural textures, patterns computed rather than painted, while working on Tron at MAGI in Elmsford, New York, in 1981. "On some level I was frustrated by the fact that everything looked machine-like," he later said. "The first thing I did in 1983 was to create a primitive space-filling signal that would give an impression of randomness."¹

How it works

In two dimensions:

1. A grid with a direction at every point. The integer grid points each get a pseudo-random gradient, an arrow of length one. Perlin picks the arrow by passing the point's coordinates through a shuffled table of the numbers 0 to 255, so the same point always gets the same arrow.^4, 5
2. Four corner values. For a point inside a grid square, each of the square's four corners gives a number: the dot product of the corner's arrow with the line from the corner to the point. The number is large when the point lies in the direction the arrow points, negative when it lies behind, and zero at the corner itself.^2, 4
3. A smooth blend. The four numbers are mixed according to where the point lies in the square, through an S-shaped curve so that there are no kinks at the edges of the squares.⁴

Because every corner contributes zero at its own position, the noise is zero at every grid point.² In the sketch below this is visible: the thick line, where the value is exactly zero, passes through every grid point, at right angles to that point's arrow.

New drawing

Improved noise, 2002

In "Improving Noise" (SIGGRAPH 2002) Perlin corrected "two deficiencies in the original Noise algorithm".⁴ The original blending curve was 3t² − 2t³. Its second derivative, 6 − 12t, is not zero at the edges of a square, which shows up as visible creases when a surface shaped by noise is shaded. He replaced it with 6t⁵ − 15t⁴ + 10t³, and simplified the choice of gradients to a small fixed set of directions.^4, 5 His reference code in Java is on his NYU page.⁵

In generative art

The programming tool Processing has it built in: noise() "Returns the Perlin noise value at specified coordinates", adding several octaves together.⁶ In flow fields, "About 90% of the time, people use Perlin noise to initialize the vectors," according to the artist Tyler Hobbs.⁷ The noise is not random in the sense of a dice throw: the same input always gives the same output, and the randomness lies in the shuffled table. For a source of chance that comes from outside the program, see the note on SHA-256 as a random source; for the wider field, the editorial Generative Art.