Concept

Nullspace — where it appears

The set of vectors a linear map sends to zero, which for a fitting problem is the set of answers consistent with every measurement. One dimension means an answer; more than one means a family, and every member of it explains all the evidence exactly.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

left pictureright pictureevery one of them fits every mark1.6e-6 px

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.

twoviews · Planar
-2e-7-1e-70-0.800-0.700-0.600-0.500α, the mix of the two nullspace directionsthe determinant that a fundamental matrix must make zeroa cubic with three real roots3 matrices, all exact

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.

twoviews · Eightpoint
the lenstwo marks, a straightedge, no arithmetic2.8e-13 px

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.

mirrors · Mirrorpair

Named alongside it

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

Fundamental matrixCorrespondencedegrees of freedomEpipolar geometryEpipoleMinimal solverRankreconstruction ambiguityAmbiguityConditioningCritical surfaceDegeneracy

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