Convolution on Programmer.ie: Modern AI programming

Cellular Automata From First Principles 52: Use FFTs for Large Neighborhoods

Mon, 10 Aug 2026 20:55:00 +0100

Small local neighborhoods are cheap to evaluate directly.

Large smooth kernels are different.

Lenia taught us that a neighborhood may cover dozens of cells in every direction. At that scale, direct convolution can become expensive.

The Fourier transform gives us another route.

For periodic domains:

convolution in space
        ↕
multiplication in frequency

So instead of sliding a large kernel over every location, we can transform both arrays, multiply them, and transform back.

Mon, 10 Aug 2026 19:45:00 +0100

A neighborhood does not have to be a list of nearby coordinates.

It can be a spatial weighting function.

That lets us say:

cells near this radius matter a lot
cells closer in matter less
cells farther away do not matter at all

The standard programming tool for applying that same weighted neighborhood everywhere is convolution.

import numpy as np

kernel = np.array([
    [0.0, 0.1, 0.0],
    [0.1, 0.6, 0.1],
    [0.0, 0.1, 0.0],
])

Normalize it: