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

Control point — where it appears

A point of a scene whose position was measured by another instrument, such as a survey, and supplied to a reconstruction from pictures. Seven numbers of control fix where the reconstruction is, how it is turned and how large; more than seven make a claim about its shape.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

the picturethe same plane, rectifiedacross — 0.6000 widths (true 0.6000)along — 0.7667 widths (true 0.7667)diagonal — 1.2023 widths (true 1.2023)worst error 1e-15 relativethe probes were not used to build the map

The plan hidden in the photograph

Rectifying the ground is the same operation as rectifying a wall, and it turns a photograph into a site plan. The horizon is not an input to it and comes out as a consequence — the plan's points at infinity land on it, at first order, which is the check that the plan is a plan and not a plausible warp.

metrology · Rectify view 1, heldview 6one standard deviation, drawn 5× actual size — held: the first camerapoints 52.1–158.0 mm · cameras 0.0–104.8 mmmarks read to 1 px

An uncertainty is quoted from something

One adjustment, one set of marks read to a pixel, and its uncertainty written four ways. Held at the first camera, the last camera is 105 mm from certain; held at nothing, every camera is within 4 to 6 mm. A ratio of two distances carries 0.1465 per cent in all four, to two parts in a billion.

manyviews · Gauge controlhow the shape moved, drawn 200× actual size — held: twelve surveyed coordinatessurvey 20 mm out · shape moved up to 2.26 mmreprojection 3.83e-1 px

The eighth held number bends the scene

Four surveyed points, each 10 mm out in a different direction. Hold seven of their coordinates during an adjustment and the courtyard's shape moves by a trillionth of a millimetre; hold eight and it moves by 0.59 mm, because seven numbers choose a frame and the eighth makes a claim the pictures disagree with.

manyviews · Gauge start, heldworst: 86 along, 64 acrossellipses ×12 · worst camera 112 mmloop closed

A loop's far side is a length

The worst camera on a closed loop of twenty-four is 411 mm from certain along the line from the start and 64 mm across it. A picture taken across the ring from the start tripod measures directions and moves the worst camera from 418 mm to 405. One distance measured across the ring to 10 mm moves it to 74, and a four-metre length measured on the start wall to 112. What the far side of a loop lacks is not a connection but a length.

manyviews · Drift 0.111010010000.5125the accuracy the survey is stated to have (mm)shape error of the courtyard (mm)survey really 1 mmsurvey really 5 mmsurvey really 20 mmmarks read to 0.1 px · 200 trials eachlowest at the true accuracy

A survey is trusted at its own accuracy, unless its error has a shape

Entered into an adjustment with a stated accuracy, a survey whose errors are random gives the smallest shape error when the stated accuracy is the true one — at 1, 5 and 20 mm alike. Stated twenty times too tight it can cost thirteen times the error; twenty times too loose, almost nothing. But twelve surveys all 10 mm out in fixed directions are best trusted anywhere from 0.3 mm to 10 mm, and the size of their error cannot say which.

manyviews · Gauge 0.050.10.250.51240200400length of the scale bar at the start (m, log scale)worst camera, along the line from the start (mm)open walkloop closedopen, no barclosed, no bara scale bar at the start, measured to 0.1 mmopen: flat from 0.5 m

A scale bar is worth its ends, not its tape

A length measured at the start of a walk round a ring pins the walk's scale — but only to the precision with which the pictures place the length's two ends, 3.4 mm here, whatever the tape says. Measured to a thirtieth of a millimetre or to three millimetres, a one-metre bar leaves the worst camera at the same 89 mm. So a short length is not a long one measured well: a bar under about 13 cm tells the walk less about its scale than the walk already knows. And on an open walk no length at the start does better than 214 mm, because what is left there is drift, not scale.

manyviews · Drift 01002003004000123gap between two 25 cm bars stood on one wall (m)worst camera, along the line from the start (mm)two short barsone bar over the spanno barone short bartwo 25 cm bars on end, measured to 0.1 mm, gap unmeasuredloop closed

Two short bars are not one long one

A long scale bar is awkward to carry, so the natural substitute is two short ones set far apart on the same wall. It does not work. Two 25 cm bars 3.5 metres apart leave the worst camera of a closed walk at 219 mm; one bar over the same span leaves it at 72. The gap between the bars is read by the pictures like any other distance, so it carries no scale — and the same holds for a long length measured as separate pieces. What does work is a tape read at every mark: pieces that share their ends add up to the whole, and pieces that do not, do not.

manyviews · Drift 02004002610141822the wall the 1 m bar is laid on, by the camera that faces itworst camera, along the line from the start (mm)open walk, one barloop closed, one baropen, no barclosed, no bara 1 m bar measured to 0.1 mm, moved round the ringflat: the bar fixes one scale

A length fixes the scale where it lies, and the worst camera nowhere

Move one measured metre round a ring of twenty-four cameras and the worst camera hardly notices: 199 to 225 millimetres on the open walk, 80 to 103 on the closed loop, wherever the bar lies. What moves is the scale. On the open walk a length is known to 0.41 per cent beside the bar and 1.58 per cent twenty walls away, and no longer bar at the start brings the far end below 1.53. A survey that must measure lengths everywhere on an open walk needs its references spread along it; one that closes its loop can put a single reference anywhere.

manyviews · Drift

Named alongside it

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

bundle adjustmentgauge freedomCovarianceDriftLoop closurescale ambiguityReference lengthSimilarityerror propagationReprojection errorResidualcamera track

All concepts