Correspondence — where it appears
Named by 19 essays across 8 fields — each of them below, with the objects they name alongside it.
The image of the other eye
Two photographs of one courtyard, and in each of them a point that is the other camera. It is computed from forty-four matched marks and nothing else, and it lands on the projection of the other eye to about a billionth of a pixel.
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.
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.
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.
Two rays that do not meet
Triangulation is described everywhere as the intersection of two rays, and two rays in space do not intersect. Read the same two marks to a whole pixel and they miss by 2.77 mm at seven metres, which is a real length and is the part a residual will not report.
Four cameras fit, and one of them can see
The essential matrix does not determine a camera pair. It determines four, all of which reproject every correspondence exactly, and the thing that picks one is not more algebra — it is the assumption that the photographer could see what was photographed.
Where the adjustment stops
Given exact marks the reprojection error falls to a hundredth of a billionth of a pixel, which is arithmetic. Given the same marks read to a whole pixel it falls to a third of a pixel and stays there, and a solver that reached zero on those would be fitting the rounding.
A wrong match is not a small error
Move one correspondence of forty-four by thirty pixels and the recovered geometry is wrong for every other point — the typical one by half a pixel, from a fit that was exact to a part in ten trillion. Least squares has nowhere to put a bad row except across all of them.
Two views give shape and no size
Every pairwise distance ratio in a courtyard recovered from two photographs matches the real one to fourteen digits. The courtyard's actual size is not merely uncertain — it is absent, and a reconstruction three and a half times larger fits the same two pictures exactly as well.
Counting the eyes needs the room
How many eyes made a picture is not a question the picture can be asked. Told what the room really measures, the rays refuse to meet and a second eye has been caught; told instead that the room is the one the picture is consistent with, the same rays meet exactly, at the first eye. The refusal is real and it belongs to the room.
Seven marks, three answers
Seven correspondences leave a two-dimensional nullspace, and the requirement that a fundamental matrix be singular is a cubic in the mix — one or three real roots. Here it has three, and all three satisfy every one of the seven marks to 8.9 × 10⁻⁹ pixels. The eighth mark, withheld, separates them by more than an order of magnitude.
One shutter, two views
A photograph with a mirror in it is a stereo pair, and a peculiarly well-behaved one. Its fundamental matrix is skew-symmetric, so both epipoles are the same point; that point is where the camera would see its own lens; and every line joining a mark to its reflection passes through it, to 1.4 × 10⁻¹² px. The baseline is twice the distance to the glass, which is the one number a single view cannot supply and a tape measure can.
Two matches are enough
A general fundamental matrix has seven free numbers and needs eight correspondences. A mirror pair's has two, and two correspondences fix it — with a straightedge, on a print, by drawing the line from each mark to its reflection and marking where the two cross. Given the same sixteen marks read to four tenths of a pixel, the constrained fit lands 4.8 times closer to the truth than the eight-point algorithm.
A mismatch on its own line needs a third eye
Slide one mark of a correspondence 30 px along the epipolar line the other mark fixes, and every test two photographs can run stays at the arithmetic floor — epipolar distance 2.2e-14 px, the two rays meeting to 1.5e-15 m, reprojection 1.1e-13 px — while the point is reported half a metre too near. A third picture exposes it by 21.4 px from a third eye two metres off the first line of sight, and by exactly nothing from an eye on that line.
The lamp is the second eye
One photograph, one lamp whose position is known, and a point's place in space comes back to 9e-16 m — the camera's ray through the point, the lamp's ray through the image of its shadow, and the intersection of two lines. It is triangulation with one of the two eyes replaced by a light, and it degrades exactly like a stereo pair: 5.9 mm of depth per pixel at 39° between the rays, 1 mm at 15.4°.
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 symmetric object is its own stereo pair
A building with a plane of symmetry photographed once gives fourteen correspondences whose joining lines meet at one point to 1.9 × 10⁻¹² pixels, a skew-symmetric matrix, and the object's whole shape to fifteen digits — with no mirror anywhere and no second exposure. What it does not give is the size, and the instrument that decides whether any of it applies is the same meeting point, which opens to 12.7 pixels when the symmetry is half a per cent out.
The drawing that gives the solid back
One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.
The drawing does not run out of lines
Every post supplies a line to every lamp, so a drawing of five posts offers ten lines to two lamps and twenty to four — the unknowns and the constraints grow together and two posts fix any number of lights. What runs out is the partition, whose margin falls from 251 pixels to six as the share of lines assigned correctly falls from all to just over half.
Named alongside it
The objects these essays reach for when they reach for this one.
BaselineFundamental matrixResidualEpipoledegrees of freedomEpipolar geometryTriangulationreconstruction ambiguityscale ambiguityConditioningDemonstrationleast squares