How a model makes a picture by removing noise. · Source

To make a picture, it starts from pure static.

Every image generator faces the same wall: a million pixels that must all agree at once. Diffusion models dodge it with a strange move — they start from a screen of pure random static and take it away, a little at a time, until a picture nobody has ever seen is standing there. The first thing they learned wasn't how to create. It was how to destroy.

First, the easy direction: how to wreck a picture.

q(x_t \mid x_{t-1}) = \mathcal{N}\!\left(x_t;\ \sqrt{1-\beta_t}\,x_{t-1},\ \beta_t \mathbf{I}\right)

Reversing chaos is hard; making it is trivial — so start there. Take a real photo and stir in a little random noise, step after step, until nothing's left but grey snow. Like a drop of ink in still water: it spreads a touch more each moment, never un-spreads, until the glass is one even grey. In plain words: each step keeps a hair less of the picture and blends in a hair more pure static — tiny, exact, and easy to learn.

You can skip to any amount of ruin in one shot.

x_t = \sqrt{\bar{\alpha}_t}\,x_0 + \sqrt{1-\bar{\alpha}_t}\,\varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \mathbf{I})

Adding noise one slow step at a time would be tedious. Happily, all those little steps collapse into one formula: any noised version is just the original turned down and pure static turned up, mixed by a single knob. Like a translucent scrim drawn across a stage: the further you pull it, the more the scene dims and the white takes over — and exactly how far you've pulled is the only number you need. So you can jump to any moment of decay at once.

Now the real trick: teach a net to spot the static.

\mathcal{L}_{\text{simple}} = \mathbb{E}_{x_0,\,\varepsilon,\,t}\left[\,\left\lVert \varepsilon - \varepsilon_\theta(x_t,\,t) \right\rVert^2\,\right]

Here's the whole magic. You can't un-mix ink by hand — but a network can estimate what was stirred in. Show it a noisy image and ask one question: which part of this is the static I added? Grade it on how close its guess is, across millions of examples. Like an art restorer lifting grime off an old painting: it doesn't repaint the scene — it learns to see only the dirt, so that what remains is the picture underneath.

Why hundreds of tiny steps, not one big leap?

x_{t-1} = \frac{1}{\sqrt{\alpha_t}}\left(x_t - \frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t}}\,\varepsilon_\theta(x_t,\,t)\right) + \sigma_t z, \quad z \sim \mathcal{N}(0, \mathbf{I})

If it can spot the noise, why not strip it all at once? Because one giant guess asks it to imagine the whole picture from grey — too much. So it removes a thin slice, looks again, and repeats, hundreds of times. In plain words: subtract a little of the predicted static, stir in a whisper of fresh randomness, repeat. Like a stonemason splitting granite: not one wild blow but a row of small, sure taps — each easy, the stone cleaving true only because no single hit had to do it all.

But how does it draw a cat and not just… anything?

\tilde{\varepsilon}_\theta(x_t, c) = \varepsilon_\theta(x_t, \varnothing) + w\left(\varepsilon_\theta(x_t, c) - \varepsilon_\theta(x_t, \varnothing)\right)

Left alone, denoising drifts toward some plausible image — not yours. So at every step you also hand it your words, and lean its guess toward them. The cleanest way: compare its denoise with the prompt against its denoise blind, and exaggerate the gap. Like a magnet under a sheet of iron filings: the prompt is the field; crank its strength and the scattered grains snap harder into the shape you named.

Static in, a picture out — destruction, run backward.

Step back and see the whole engine. It taught itself to ruin pictures, learned to name the ruin, and now runs that knowledge in reverse — from a fresh field of static, steered by your words, one gentle step after another. Like a print rising in the developer tray: the blank sheet was never blank; under the right bath, in its own time, the image simply surfaces. Nothing was drawn. The noise was sculpted.

🌱 Did it create the picture — or just uncover it?

Change the opening static and you get a different picture; the same static always gives the same one. So every image it can make is, in a way, already implied — folded into some grain of noise, waiting. Like a face the sculptor swears is already inside the marble: remove what isn't it, and there it stands. So is the model making these pictures — or only carving away everything they are not?