Diffusion in Image Processing

Diffusion is simply stuff spreading out from where there's a lot of it to where there's less, like heat through metal or dye through water, and here the "stuff" is brightness in a photo, with bright pixels "hot" and dark ones "cold." The fancier version later on is called anisotropic diffusion, which just means it doesn't spread equally in every direction. It's good for cleaning up noisy photos and scans, and for spotting features at different sizes. The formula $I_t = k\,\nabla^2 I \iff I(\cdot, t) = G_\sigma * I_0$ looks dense, but it's two ways of saying one thing: on the left, $I_t$ (how fast a pixel's brightness changes) equals $k\,\nabla^2 I$ (how different it is from its neighbors); on the right, the result is $I_0$ (the original photo) smeared by $G_\sigma$ (a bell-shaped blur) of radius $\sigma = \sqrt{2kt}$ (how far each pixel's brightness has spread). It's the heat kernel from two lessons back, wearing a camera; bear with it.

Images as functions. A grayscale image is a grid of brightness values $I(x, y)$. That's just a temperature on a grid, so everything from finite differences carries over. The discrete Laplacian at a pixel is the sum of its four neighbors minus four times its own value, so one explicit heat step moves each pixel toward the average of its neighbors.

Why it removes noise. Noise is pixel-to-pixel flicker, which is the highest frequency an image can have. From the Fourier transform lesson, a wave of frequency $\omega$ fades like $e^{-\omega^2kt}$, so the flicker is the first thing to go while broad shapes barely change. Diffusion is a low-pass filter.

Scale space. Diffusing for longer blurs more: running the heat equation produces the whole family of blurred images at once, a standard tool for finding features at different sizes.

Blurs compose. Two Gaussian blurs with radii $\sigma_1$ and $\sigma_2$ equal one blur of radius $\sqrt{\sigma_1^2 + \sigma_2^2}$. Think about why: blurring for time $t_1$ and then $t_2$ is the same as blurring for $t_1 + t_2$, and since $\sigma^2 = 2kt$, adding times means adding the $\sigma^2$'s.

A common mistake. It's tempting to think two blurs of radius 3 and 4 make a blur of radius 7, because blurs pile up. They don't add like that. The squares add, so the total is $\sqrt{9 + 16} = 5$, the same rule that makes variances add in statistics.

Anisotropic diffusion. The plain heat equation can't tell noise from an edge: both are sharp changes in brightness, so both get smoothed. Perona and Malik fixed this by making the conductivity depend on the gradient: flat regions with small noisy wiggles diffuse freely, while strong edges, where the gradient is big, hardly diffuse at all. The result denoises without blurring outlines.

Photo editors, medical image denoising, computer vision feature detectors (SIFT) and the diffusion models that generate images all rely on diffusion equations.