Radial distortion — where it appears
Named by 13 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.
A lens destroys the invariant
The cross-ratio is the one thing a projection preserves, and nearly everything checkable about a photograph is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.
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.
The lens a pavement can hide
A photographed pavement reads as a correct drawing up to a radial coefficient of about four tenths — a lens strong enough to bow a straight edge across the page by nearly six pixels and to print as twenty per cent distortion at the frame's corner. The reason is that a pavement sits near the principal point, which is the one part of the frame a radial map barely touches.
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 tilted sensor is not a distortion
Tilt a sensor 3° out of square with its lens and every point of the picture moves — up to 7.5 px on the frame drawn here — yet every straight line stays straight to 10⁻¹³ px and the cross-ratio survives to 10⁻¹⁶. The picture is an ordinary pinhole picture whose principal point has moved 22.30 px. A calibration that frees its principal point absorbs it exactly; one that holds the principal point and reaches for tangential distortion terms leaves 1.87 px, and used as a correction it bends straight rows by 4 px.
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.
The response is at the ends and the information is not
A radial map bows a straight edge by an amount that grows as the square of the distance along it, so 93 per cent of an edge's response to the coefficient lies in its outer quarters. Spending the marks there is 16 per cent worse than spreading them evenly, because two clusters say nothing a shifted, tilted line could not say. What identifies the coefficient is a curvature, which needs three places — both ends and the middle, which beats an even spread by 11 per cent.
No design separates the two coefficients
Separating a squared term from a fourth-power one was supposed to need marks at radii far apart, which is a statement about where edges are placed. It is not: one straight edge already runs from 20 px to 326. Spreading ninety-six marks over three edges or sixteen changes the answer by 23 per cent, spreading the offsets makes it 19 per cent worse, and the correlation stays at −0.98 whatever is done. What a plumb-line calibration determines is one number, to 1.52 thousandths, and which number depends on the model.
A known target sharpens the fit and does not separate it
Printed circles of stated size were supposed to break the −0.98 correlation between a lens's two radial coefficients, because a circle puts every mark at one radius and no straight edge can. They do not: every design of circles leaves the pair 0.979 to 0.9997 correlated, and for circles of known size the figure is exactly the cosine between r³ and r⁵. What a known target buys is precision — 3.8 times the straight edges' at the same budget — and only if its size in the picture is known to about a thousandth.
A wedge moves the centre, not the lens
A wedge of glass in front of a lens deflects every ray by a little more the further off the axis it goes, which looks like the shape a radial distortion coefficient describes. Fitted together, the two are nearly independent — correlated at 0.16 at most — and a calibration that knows nothing of the glass does not invent a lens: it reports a coefficient of about 0.002, moves its principal point by 4.5 to 14 pixels, and leaves a swirling residual that no radial model takes. The blame goes to the camera's centre, and the residual says so from twenty-two degrees off the axis.
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.
A tilted target pays for its tilt in perspective
A calibration target of printed circles held square to the lens fixes a lens's first radial coefficient to 4.77 thousandths from ninety-six marks. Tilt it sixty degrees, fit the tilt along with everything else, and the same ninety-six marks fix it to 1.70 — nearly three times better. Squash the circles by the same angle without perspective and nothing is gained. What pays is the near half of each circle being drawn larger than the far half, which spreads a circle's marks across a band of distances from the lens's centre.
Named alongside it
The objects these essays reach for when they reach for this one.
Camera calibrationBrown–ConradyPrincipal pointResidualBarrel distortionConditioningerror propagationfield of viewFocal lengthInverse projectionModel errorCorrelation