Epipolar line — where it appears
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
A point is a line over there
Knowing where a mark sits in one photograph does not say where it sits in the other. It says which line it is on, which removes one of the two unknowns and leaves one — and the five lines drawn here meet at a point to within a thirtieth of a billionth of a pixel.
The two pencils keep one number
Four lines through the image of the other eye in one picture, and the four epipolar lines they become in the other. The angles between them change by up to 11.4°; their cross-ratio is 3.012836 on both sides, to eight parts in a trillion. Three pairs of lines fix the map between the pencils, and the fourth is predicted to a third of a billionth of a pixel.
Rectification is a family, not an operation
Turn both pictures of a pair so their epipolar lines become shared rows. A turn about the line between the eyes and a focal length are left free, and every choice puts all 44 matches on common rows to a tenth of a trillionth of a pixel and every point back where it was. What the choices disagree about is the pixels — one stretches its pictures unevenly by 1.77, another by 4.86.
A third eye that lands on the next post
Match one post of a railing to its neighbour and the pair reports it at 19.8 metres instead of 9.0, with every test two photographs can run at the arithmetic floor. A third picture usually exposes that by hundreds of pixels — but at five azimuths in seventy-eight degrees the wrong point lands within three pixels of another post, and the third view confirms the mistake. Narrow the railing to twenty centimetres and those places cover 28 per cent of the arc.
A railing matched one along stands underground
Match every post of a railing to the next one along and two photographs build a perfect railing at 19.8 metres instead of 9.0 — straight, evenly spaced, every post on both its rays. Nothing near it can say so. What convicts it is what it stands on: its feet are 1.32 metres under the ground, and one ground mark read to a pixel shows every foot a whole period out from sixty metres away. Its length helps only through its ends, and once those leave the frame, counting matches prefers the wrong railing.
A facade matched a column out comes forward, not back
A railing matched one post along is rebuilt 2.2 times further away. A facade of windows 2.4 metres apart cannot be: a one-column shift that way needs a negative disparity, and the only mistake left in front of the cameras brings the facade to 6.3 metres from twenty, a third its size, hanging 1.1 metres off the ground with nothing under it to sink into. Its corners hold it only if the wall is assumed flat. Two corners then convict it at every period; one corner and a drainpipe need the pipe half a metre in for windows and four and a half for a fine repeat.
A raised eye reads the storeys as the pair read the columns
Put a third camera 3.5 metres above the left one and its epipolar lines run up the facade: the storeys repeat along them as the columns repeated along the level pair's rows, the roofline and the foot carry depth where the corners did, and a raise shorter than a storey can only pull the windows nearer. The three pictures together refuse the level pair's one-column mistake at every raise except those where the storey over the raise equals the period over the baseline — 1.375 and 2.75 metres here — and even there the facade's lowest storey lands on the ground.
A climbing eye names the column mistake, if it knows how high it is
A level pair that matches a facade a column out rebuilds it nearer, and a third camera climbing above the left one watches the rebuilt windows slide down its picture at 98 pixels a metre for every column of the mistake. Read against the real windows, that slide is a ruler: a metre of climb from level names a one-, two-, three- or four-column mistake every time. What it costs is not frames but knowledge — the climb must begin before the rebuilt facade slides off the lattice, and the camera must know its own height to about a quarter storey over the mistake's slide, 34 centimetres for one column and 9 for four.
A staircase winds the mistake round the lattice
A third camera climbing a stair along a facade sees a stereo pair's column mistake slide through its picture along the stair's own direction, wrapping at the columns across and the storeys up — a line wound round a torus. Such a line comes back onto a real window only when the stair's rise over its run is a ratio of whole storeys to whole columns; a degree off the diagonal, no two readings come within 3 px. The stair names the mistake from one frame, as a vertical climb does, but its readings sit twice as far apart in a frame, so a camera that knows where it is only to 10 cm still names it 95 times in 100 against 82, and a 30° stair keeps naming it a metre and a half along, where a vertical climb has lost half.
Named alongside it
The objects these essays reach for when they reach for this one.
CorrespondenceDegenerate configurationOutlierrobust estimationTriangulationDisparityBaselineEpipoleFundamental matrixResidualSamplingscale ambiguity