Barrel distortion — where it appears
Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.
Straight lines that are not
Everybody says the edges of a wide-angle frame bow. Nothing is special about the edge. A radial map moves every point along its own radius, so the only line it leaves straight is one through the principal point, and the bend of every other is decided by how far it passes from that one place.
Wide angle is not distortion
A wide lens stretches shapes at the edge of the frame by exactly 1/cos θ — 3% at 28° across, 41% at 90°. Every bit of that is what a correct rectilinear projection must do, and every bit of it disappears if the picture is viewed from the point it was made for. Nobody views it from there.
The render is distorted on purpose
A headset renders a bent picture so its lens can straighten it, which is a lens's distortion polynomial run backwards, and the one case in which distortion is introduced deliberately. The round trip closes to a thousandth of a millionth of a pixel, and the price is that one rendered pixel becomes 0.493 delivered pixels at the edge of the field and one at the centre.
A barrel model folds at a radius it sets itself
The polynomial every calibration fits to a wide lens stops increasing at a radius fixed by its own first coefficient — 47.49° of field at k₁ = −0.28 — and past it two directions land on one picture radius. The routine that undistorts pictures with it does not refuse there. It hands back wrong directions from 46.75°, by as much as 106.5°, and refuses only at 65.5°: a fifth of the field returned silently wrong.
A model that inverts has a horizon instead of a fold
The polynomial every calibration fits turns around at a finite radius and stops being a map from direction to picture. The division model, chosen because it inverts in closed form, never turns around — it rises for ever toward a horizon at one over the root of its own coefficient, so the whole hemisphere of directions lands inside a finite disc. Fitted to the four fisheye laws it follows every one of them more closely than the polynomial at every field from forty degrees to eighty — a hundred times more closely for the equidistant law at forty, and the stereographic law exactly.
A stereographic fisheye is a division model
The division model divides the picture radius by one plus a coefficient times its square. The fits that compared it with the polynomial were not of that model: they multiplied instead, and the model they measured has neither a fold nor a horizon. Fitted as it is written, the division model follows every fisheye law more closely than the polynomial at every field from forty degrees, and the stereographic law it follows exactly — the law is the model, with a coefficient of minus a quarter. Its horizon then turns out to sit beyond the lens's own ninety degrees, and pinning it there is a trade rather than a free constraint.
Named alongside it
The objects these essays reach for when they reach for this one.
field of viewRadial distortionBrown–ConradyCamera calibrationInverse projectioninstrument limitFisheyeFocal lengthInvertibilityModel errorAnamorphosisAnisotropy