The second eye
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.
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.
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.
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.
An ambiguity is not an uncertainty
Eight marks to forty cuts a solid scene's pose error from 19.8° to 0.7° and leaves a flat one at 48°. The two failures look identical from inside — a confident answer, a residual at the floor — and they respond to opposite remedies, so telling them apart is worth more than either measurement.
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.
An epipole in the picture leaves a blind disc
Step a camera half a metre straight forward and the image of the other eye sits in the middle of both pictures. Around it lies a disc where one pixel of reading costs a tenth of the depth or more — 20 px across a surface 2 m off, 247 px at 16 m — and at its centre no depth is recovered at any range.
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.
Seeing the scene fences in the plane at infinity
A reconstruction made without the calibration does not know which of its planes is infinitely far away. Requiring every point to lie in front of both cameras rules out every candidate that would tear the courtyard, and what is left is a convex region — 32 per cent of a generous slice — that always holds the true plane and never shrinks to it.
A parallax length is a height over a depth
After a known plane's map, every raised mark's displacement points at the other camera's image, and its length carries the mark's height above the plane over its depth — but not as the ratio of lengths it looks like. That ratio departs from the point's own number by up to 45 per cent. Read as a coefficient on the epipole, the same length gives height over depth from the first camera to four parts in a hundred trillion, the same from every second picture.
A sway gives the blind centre a depth, not a good one
A camera driving straight forward cannot see how far away the thing it is driving toward is: the mark at the epipole does not move between pictures. Let one of three pictures sway sideways and the centre gets a depth at once — but a depth resting on the sway alone, which a pixel of reading moves by the focal length's reciprocal times the depth over the sway. For a centimetre of steering wobble at eight metres that is 144 per cent; for a tenth of the forward step, 29. The hole closes; the disc around it stays until the sway is a third of the step.
A plane's coefficient reaches as far as its parallax
After a known plane's map, every raised point's displacement is its height over its depth, read as a coefficient on the epipole — exactly, for any point either picture sees. The worry was that the number would be local, good only near the floor whose marks fixed the map. Read to a pixel, it is not a distance on the floor that runs out. It is a length in the picture: the point's error is about 260 per cent over its parallax in pixels, wherever the point stands.