Blur Is the Heat Equation

Blurring a photo spreads each pixel's brightness to its neighbors, exactly like heat spreading through metal. Running the heat equation longer gives a stronger blur.

How it works

Diffusion and the heat kernel

In one sentence

Blurring with a Gaussian of variance $2\alpha t$ is the same as running the heat equation for time $t$ : $u(x, t) = (4\pi\alpha t)^{-1/2}e^{-x^2/(4\alpha t)}$ is the heat kernel.

General formula

{\htmlData{sym=0}{u}}({\htmlData{sym=2}{x}}, {\htmlData{sym=4}{t}}) = {\htmlData{sym=1}{\frac{1}{\sqrt{4\pi\alpha t}}}}\,e^{-\frac{{\htmlData{sym=2}{x}}^2}{4{\htmlData{sym=3}{\alpha}} {\htmlData{sym=4}{t}}}}

The bell is how one bright pixel's light spreads: wider and lower as time goes on, with the same total.

What the symbols mean

$u$
Spread brightness
Brightness that has spread out from one bright pixel.
$\frac{1}{\sqrt{4\pi\alpha t}}$
Normalizer
Keeps the total brightness constant as it spreads out.
$x$
Distance
How far from the original bright pixel.
$\alpha$
Diffusivity
How fast brightness spreads to neighboring pixels.
$t$
Blur time
How long the heat equation runs. Longer means a stronger blur.

A single bright pixel spreads into a Gaussian bump called the heat kernel.
Its width grows like $\sqrt{t}$ , so blur strength is just diffusion time.
Gaussian blur in photo editors applies this kernel to every pixel.

Learn the math

Convolution IntegralsDiff Eq · Lesson 29, Laplace Transforms The Fourier Transform and the Infinite RodPDE · Lesson 17, Beyond the Basics Diffusion in Image ProcessingPDE · Lesson 19, Beyond the Basics