Concept

Fundamental matrix — where it appears

5 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.
epipoleepipoleleft pictureright pictureepipole from 44 correspondences vs the projected eye: 1.1e-9 px2.60 m between the eyes

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.

twoviews · epipole
1234512345a point is chosen hereand must be on this linematch to its own epipolar line: 1.4e-13 px5 of 44 correspondences drawn

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.

twoviews · epipolar
-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
points in front of both camerasR₁, t0 of 44180° from the truthR₁, t reversed0 of 44180° from the truthR₂, t44 of 44chosenR₂, t reversed0 of 44180° from the truththe chosen pose is the true one to 0.0e+0°cheirality, not algebra, picks it

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.

twoviews · cheirality
00.50011.50010203040correspondenceepipolar error at every OTHER point, after one match is movedmedian 0.54 pxthe moved oneevery point wrong: median 0.54 px, worst 1.7 pxclean fit 2.3e-13 px

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.

depth · outlier

Named alongside it

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

CorrespondenceBaselineCamera matrixEpipolar geometryEpipoleResidualCheiralityConditioningdegrees of freedomdesign matrixeight-point algorithmEpipolar line

All concepts