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

Aspect ratio — where it appears

The proportion of a rectangle's width to its height, whether of a sensor, a frame, a panel or a single pixel. It enters a camera's arithmetic twice, once as the shape of the frame and once as the shape of a pixel, and an unmodelled pixel aspect is absorbed into a confident wrong focal length.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

full frame · 39.6°APS-C · 26.6°Micro Four Thirds · 19.6°1 inch · 15.0°phone (1/1.7″) · 8.7°one 50 mm lens · the angle is a property of the rectangle behind itMicro Four Thirds: 2.00× diagonal, 2.08× wide, 1.85× tall50 mm across five formats39.6° down to 8.7°

A focal length is not an angle

Fifty millimetres means nothing until a rectangle is named behind it. The same lens is 39.6° across full frame, 26.6° across APS-C and 8.7° across a phone sensor — and the distance the resulting print is correct from depends on the ratio of the two, so two cameras matched on angle agree exactly whatever their formats.

sensor · Format the picturethe same plane, rectifieddoor — 0.5068 widths (true 0.5068)sill — 0.3026 widths (true 0.3026)diagonal — 1.1766 widths (true 1.1766)worst error 4e-16 relativethe probes were not used to build the map

Flattening a façade out of the photograph

Four corners of a rectangle whose proportions are known are enough to undo the projection of one plane. After that the plane can be measured with a ruler — lengths, angles, areas, all of it — in units of the rectangle's own width, and lengths the map was never given come back to fifteen digits.

metrology · Rectify pixels 2.00 : 1recovered focal length219.9 pxthe camera's actual one436.4 pxwhat the recovery returnsspread across three estimates 0.0e+0worst bundle residual 9.1e-13 pxboth are what a wrong picture would trippixel aspect 2.00 unmodelledfocal length 49.6% short, every diagnostic clean

The pixel that is not square

A camera with two focal lengths is a real thing — anamorphic cinema optics, non-square photosites, a stretched video format. Hand the round trip of camera recovery a picture from one and it returns a focal length 49.6% out, a principal point far from the truth, three independent estimates agreeing to 1e-16, and bundle residuals at the noise floor. Every alarm the site has stays silent.

sensor · Pixelaspect 010200.8000.90011.101.20the assumed aspect, as a multiple of the true onethe error — degrees for the angle, per cent for the lengththe angle, in degreesa length into the pictureone wrong assumption, three quantitiesshape and size are different facts

An angle on the ground

An angle needs no length at all — it is scale-free, so the one thing a single view cannot give is not the thing an angle wants. What it wants instead is the plane's shape, and the difference is measurable — assume the reference rectangle's aspect a quarter wrong and the angle moves 11.5°, a length running into the picture is out by exactly that quarter, and a length along the reference's own direction does not move at all.

metrology · Rectify 0102030401234the marking error, in pixelsthe error of the answer, in per centupper curve: the closure wrong · lower: the closure exactfloor 6.0%

The floor a better camera cannot reach

Sweep the marking error from four pixels down to a hundredth and the measurement's error falls thirty-fold and then stops — at 6.0 per cent, which is exactly the six per cent the reference's assumed shape was wrong by. With the closure exact the same sweep keeps falling to 0.06 per cent. The crossing is at half a pixel, and it can be computed before the photograph is taken, which makes it a decision about equipment rather than a discovery about it.

metrology · Uncertainty

Named alongside it

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

DemonstrationFocal lengthHomographyMetric rectificationResidualSensitivitysingle-view metrologyAffine structureAnisotropyAsymptoticsBiasCamera calibration

All concepts