Procedural texture generation creates images algorithmically using mathematical functions rather than captured photography or manual painting. The field is built on noise functions, mathematical algorithms that produce pseudo-random but visually coherent patterns suitable for simulating natural materials and organic phenomena.

Perlin noise, invented by Ken Perlin in 1983 for the film Tron, is the foundational algorithm for procedural generation. It works by placing pseudo-random gradient vectors at regular grid points and interpolating smoothly between them. For each sample point, the algorithm identifies the surrounding grid cell, computes dot products between the grid-point gradients and vectors from each grid point to the sample point, and interpolates these values using a smooth fade function (typically 6t^5 - 15t^4 + 10t^3). The result is a smooth, continuous function that varies between -1 and 1 with no sharp discontinuities, producing organic-looking patterns reminiscent of clouds, terrain, and natural textures. Ken Perlin won an Academy Award for Technical Achievement in 1997 for this contribution to computer graphics.

Simplex noise, also developed by Perlin in 2001, improves on the original algorithm in several ways. Instead of interpolating within a square grid (which requires evaluating 4 grid points in 2D or 8 in 3D), simplex noise uses a simplex grid (triangles in 2D, tetrahedra in 3D), requiring only 3 evaluations in 2D or 4 in 3D. This produces fewer directional artifacts (the subtle grid-aligned patterns visible in Perlin noise) and scales more efficiently to higher dimensions.

Fractional Brownian Motion (FBM) creates complex, natural-looking textures by layering multiple octaves of noise at different frequencies and amplitudes. The first octave provides the broad, low-frequency structure. Each subsequent octave adds finer detail at double the frequency (lacunarity) and reduced amplitude (persistence, typically 0.5). Summing 6-8 octaves produces rich, detailed patterns that exhibit self-similarity at different scales, a characteristic of fractals found throughout nature in coastlines, mountain profiles, cloud formations, and tree bark.

Voronoi diagrams, also called Worley noise or cellular noise, partition space based on proximity to a set of seed points. For each pixel, the algorithm finds the nearest seed point, and the resulting distance value creates patterns of cells and boundaries that resemble cracked mud, stone walls, cellular structures, scales, and crystalline formations. Variations using the distance to the second-nearest point, or the difference between first and second distances, produce different visual effects from rounded cells to sharp-edged cracks.