Fundamental matrix — where it appears
Named by 17 essays across 4 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.
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.
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.
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.
A flat scene fixes no second eye
Eight marks on one plane leave the eight-point design matrix two ranks short, so a two-parameter family of fundamental matrices satisfies every mark exactly — three of its members are 1.41 apart after normalisation and all three fit to 1.6 × 10⁻⁶ pixels. The same photographs determine the plane's own map from four marks, to 5.9 × 10⁻¹³ pixels.
Far enough away, a pair is one eye
Hold the baseline and walk the scene away, and the parallax a single homography cannot explain falls as the distance to the power −0.968 — one over the distance, which says the ratio of baseline to depth is the whole of it. The recovered translation direction follows it down, from 3.3° at four metres to 74.5° at two hundred and fifty-six.
How flat is flat enough
With exact marks the transition has no width at all — 84° of pose error at exactly coplanar and 0.000° at eight parts in ten thousand of relief. Put three tenths of a pixel of reading error in and the same sweep becomes a slope three decades wide, crossing into usefulness when the out-of-plane parallax reaches about ten times the marking error.
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.
The surface two pictures cannot separate
There is a quadric through both camera centres on which two genuinely different motions draw identical pictures. Built explicitly, forty-two marks satisfy both epipolar geometries to 2 × 10⁻¹³ pixels, and the two scenes they reconstruct place the same mark at 15.6 metres and 35.1. The design matrix's nullspace has two dimensions rather than one, which is the seven-point situation arrived at from the other side.
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.
Two marks off a known plane find the other eye
Map a courtyard's ground from one picture into the other, and every raised mark lands somewhere the map did not send it — displaced along a line through the image of the other camera, to a fifth of a billionth of a pixel. Two such marks put that image where it is, and with it the whole epipolar geometry.
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.
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
CorrespondenceEpipoleBaselineEpipolar geometryreconstruction ambiguityConditioningDegeneracydegrees of freedomHomographyStereo pairCritical surfaceDegenerate configuration