16
Aug
04

Perlin Noise

Perlin Noise is a popular technique for generating all kinds of random periodic textures such as artificial terrain, clouds, water, wood, fire, marble, etc.. It was invented by Ken Perlin. Most random number generators base subsequent random numbers on their previous numbers. In this way, highly complicated “random” scenes can be easily reconstructed from a single initial seed number.

(* runtime: 2 minutes *)
Interpolate[{y0_, y1_, y2_, y3_}, mu_] := (y3 - y2 - y0 + y1)mu^3 + (2 y0 - 2 y1 + y2 - y3) mu^2 + (y2 - y0) mu + y1;
m = 14; SeedRandom[0]; noise = Table[Random[], {m}, {m}];
Noise[x0_, y0_] := Module[{x = Mod[x0, 1], y = Mod[y0, 1], i0, j0}, i0 = Floor[m y]; j0 = Floor[m x]; Interpolate[Table[Interpolate[Table[noise[[Mod[i, m] + 1, Mod[j, m] + 1]], {j, j0 - 1, j0 + 2}], m x - j0], {i, i0 - 1, i0 + 2}], m y - i0]];
a = 2; b = 2; n = 275; image = Table[Sum[Noise[b^i x, b^i y]/a^i, {i, 0, 4}], {x, 0.0, 1.0, 1.0/n}, {y, 0.0, 1.0, 1.0/n}];
ListDensityPlot[image, Mesh -> False, Frame -> False];

Here is some Mathematica code that uses a triangle wave to make Perlin noise that looks like marble:
(* runtime: 4 seconds *)
ListDensityPlot[Table[t = 5j/n +image[[i, j]]; Mod[t, 1] - 2Mod[t + 0.5, 1]Floor[Mod[t, 1] + 0.5], {i, 1, n}, {j, 1, n}], Mesh -> False, Frame -> False, ColorFunction -> (RGBColor[1 - 0.5#, 0.5(1 - #), 0] &)];

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