Visual Plane
every picture states the point it is correct from
Concept

plumb-line calibration — where it appears

Recovering a lens's law from the images of straight world lines alone, with no calibration target and no known angles. It identifies which rule a wide lens obeys once the field is wide enough, and below about fifty degrees the candidate rules fit inside the marking noise.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

fitted k₁ = -0.280000true -0.280000, off by 5e-15

Fitting a lens from straightness alone

No calibration target, no known scene, no camera. Only the knowledge that some edges in the picture were straight — and the coefficient comes back to fifteen digits. Then it comes back with a companion, and the two are correlated at −0.997.

lens · Plumbline three edges crowded on one side5.69e-3two on one side, one on the other5.66e-4three edges near the centre6.73e-3three edges placed by search9.57e-4the centre free · log scalebest: 5.66e-4

The lines that calibrate a lens

One straight edge through the centre of a picture says nothing about a lens's distortion, and one 180 px from the centre determines k₁ to 9.0 × 10⁻⁴ — the precision rises in proportion to the offset. But distance from the centre is not enough. Crowd three edges on one side and, the moment the distortion centre is also unknown, the coefficient is ten times worse, because a bend on one side looks like a moved centre; put one edge across the centre and it barely changes.

lens · Plumbline 0102030400.2000.4000.6000.800where along the edge, as a fraction of the visible lengthshare of the edge's response to the coefficient, in per cent93% outside the middle halfone edge, 82 marks, cut into tenthsmiddle two tenths: 1.4%

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.

lens · Plumbline 00.2500.5000.7501020406080angle off the optical axis (degrees)image radius, against the radius the same lens gives at 90°equidistantequal-areastereographicorthographicflat planeeach curve divided by its own radius at 90°54.0% apart at 45°

Which rule a fisheye obeys, from straightness alone

Four candidate rules for a fisheye lens part by 54.0 per cent at 45° off axis, and a plumb-line fit shown no scene, no camera and no calibration target can still name which one took a photograph — reliably from about 45° of half-field. Below that the four are indistinguishable in the marks, and naming one collapses to guessing.

curved · Fitlaw

Named alongside it

The objects these essays reach for when they reach for this one.

Camera calibrationResidualConditioningCorrelationleast squaresPrincipal pointerror propagationDemonstrationEquidistant projectionEquisolidfield of viewFisheye

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