Concept

Conditioning — where it appears

5 essays name this object, across 5 fields. What follows is each of them, and 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
-1012-0.500-0.25000.2500.500how finely each point is read off the picture (px, log scale)worst epipolar error against the exact geometry (px, log scale)raw pixelscentred and scaled1.1× apart at 0.25 px, 29.9× at 4 pxspread 102 against 2.5e+5

Eight points and the basis they are read in

The linear system that recovers a fundamental matrix is written in whatever coordinates the marks were read in, and pixel coordinates are a bad choice. Centring and scaling them first is worth nothing at a quarter-pixel reading and a factor of thirty at four.

twoviews · eightpoint
-10-50-2-10baseline (m, log scale)worst deviation in the recovered shape (log scale)exact pointsread to 1 pxat 2 mm: 4.7e-9 exact, 2.9e+1 read to 1 px6e+9× apart

A turn of the head is not a step sideways

The textbook says a short baseline makes reconstruction ill-conditioned. Measured, the algebra does not notice — a two-millimetre baseline recovers the courtyard to nine digits from exact marks. What fails is the ratio of baseline to reading error, and it fails without refusing.

depth · baseline
05101550100150200distance from the camera (metres)error in the recovered height, per pixel of click error (%)6 m — 0.55%25 m — 2.27%60 m — 5.43%120 m — 10.84%one pixel, on a 690 px picturelinear in distance

How wrong a measurement from one picture can be

The formula divides by a difference of two nearly equal numbers when the object is tall, which looked like the instability and is not. Measured, a taller object is recovered better and a more distant one worse — half a per cent per pixel at six metres and eleven per cent at a hundred and twenty. The argument that was wrong is as much the finding as the curve that replaced it.

metrology · height
-2.80-2.6034567number of viewsworst camera-centre error (fraction of the track's mean radius, log scale)7 flat7 flat7 flat7 flat7 flat60° sweep, filled in12° per view, wideningfilled in: 1.1e-3 → 2.0e-3widened: 3.2e-3 → 1.8e-3

Another picture of the same sweep

Going from three views to seven across the same sixty degrees leaves the reconstruction exactly where it started, and at one point makes it worse. What a reconstruction is short of is angular spread, not photographs.

manyviews · spread

Named alongside it

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

BaselineResidualbundle adjustmentCamera calibrationcamera trackCorrelationCorrespondenceCross ratioDegeneracydesign matrixeight-point algorithmEntrance pupil

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