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

Metric upgrade — where it appears

The step that pins a reconstruction known only up to a projective or affine transformation down to a similarity. Angles and ratios of lengths mean something after it and not before. It needs facts brought from outside the pictures, and what stays free afterwards is one overall scale.

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

2468-1-0.50000.5001position along the family the two pictures leave freehow far the solid is from the scene's own shape (stretch ratio)the sceneevery member redraws both pictures to 7e-16 mthe ambiguity is a family of solids, not a tolerance

What two parallel views leave free

Two perspective pictures give shape and no size. Two parallel pictures do not even give shape: the metric upgrade has six unknowns, two views supply six equations, and the system comes out rank five — a one-parameter family of solids that redraw both pictures to 7e-16 m. A third view closes it, and the mirror image survives every count.

parallel · Parallelrecovery a length on the ground3.22%1.00 m across the referencethe camera's height0.36%1.62 m above the grounda repeated object, size unknown3.60%the answer in units of the repeata standing object of known height0.41%1.75 m, upright, anywhere on the …the focal length and the horizon1.94%700 px, and where the ground's li…the spread of the answer, per closureshorter is better

Five facts that close the same gap

The gloss that one length has to come from outside the photograph names a single option, and there are at least five — a length on the ground, the camera's own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise; their spreads run from 0.36 to 3.60 per cent, a factor of ten, and which one is available depends on the photograph rather than on the arithmetic.

metrology · Scale a cross-ratioa ratio along a linean anglea length in metressuppliesa length on the groundexactexactexactexactthe camera's heightexactexactexactexacta standing object of known heightexactexactexactexacta repeated object, size unknownexactexactexact10%the focal length with the horizonexactexactexact10%one picture, one set of marks, four readingsthe free scale set 10% wrong

Where each closure enters the stratification

The five facts that turn a photograph's ratios into metres do not all do the same job. Read on four quantities through the map each supplies — a cross-ratio, a ratio along a line, an angle, a length — three of them return all four exactly and two return only the first three. Set the free scale ten per cent wrong and the whole ten per cent appears in the length and nothing appears in the other three, to thirteen digits.

metrology · Scale cross-ratioratio acrossratio recedingright angle45° anglelengtha length 10% wrongexactexactexactexactexact10.0%a repeat's shape 10% wrongexactexactexactexact2.73°exacta repeat 3° out of squareexactexactexact3.00°1.46°exacta focal length 10% wrongexactexactexactexact2.69°0.1%a horizon 5 px outexactexact2.1%0.35°0.35°2.0%ground sloping 3°exactexact14.3%2.30°2.35°2.0%a road crowned 10 cm2.3%1.8%exact0.64°1.59°1.3%one picture, one set of marks, six readingsexact to the arithmetic floor, or not

A wrong model shows at its own level

Give a single-view measurement the wrong length and the error stays in the length. Give it the wrong model — a slope where it assumed level ground, a horizon five pixels out, a focal length off the barrel — and the error lands on the level that model belongs to and on every level above it, never below. That is a diagnostic. Two equal things at different depths catch a wrong plane; two equal things at right angles catch a wrong shape; marks read to 0.4 px see a slope of 0.38°. A wrong length, alone, nothing in one picture can see.

metrology · Scale

Named alongside it

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

Projective stratificationsingle-view metrologyVanishing lineAffine structureCross-ratioMetric structureReference lengthscale ambiguityAffine reconstructionBiasCamera calibrationConditioning

All concepts