Concept

Convergence — where it appears

An iterative solver arriving at a place where its steps stop changing the answer. It says the descent has stopped, not that it stopped at the truth: a start on the wrong side of a saddle converges just as cleanly into a second minimum.

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

Also named here as levenberg–marquardt — the same set of essays touches all of them, so they are one junction rather than several.

-10-50012345iterationreprojection error (px, log scale)exact marksread to 1 px4.67 px → 0.3324 px in 5 iterationsexact marks reach 1.3e-11 px

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.

manyviews · Bundle
34384246505458626660° arc, 3° field60° arc, 25° field20° arc, 3° field20° arc, 10° fieldhalfway: the flattened scenelight: back to the scene · dark: into the twin · column: per cent of the way to the twinexact marks · one adjustment per cell68 starts

A start needs the sign of its depths, not their size

Started part of the way from a courtyard toward its inside-out twin, a bundle adjustment returns to the truth from every start less than 42 per cent of the way — a start with a sixth of the true relief, the right way round — and falls into the twin from every start past 56. Between, neighbouring starts settle in different answers. The band sits in the same place at a 3° field, where the twin misfits by under a pixel, and at 25°, where it misfits by seven and a half.

manyviews · Bundle
020406080100points: per cent of the way to the twin'scameras 0%cameras 10%cameras 20%cameras 30%cameras 40%cameras 50%cameras 60%cameras 70%cameras 80%cameras 90%cameras 100%solid: back to the scene · empty: into the twin · diagonal: the blend that moves both together60° arc, 3° field, exact marks121 adjustments

The cameras decide where a narrow view settles

Scatter a narrow-field bundle adjustment's starting depths and the answer it reaches stops following them: of eighty scattered starts whose depths had the right sign, twenty-six fell into the inside-out twin, and of a hundred with the wrong sign, forty-three came home. Split the start in two and the reason is plain. With the cameras where they are, the courtyard comes home from its own inside-out points; with the cameras on the twin's side, it falls in from the true ones.

manyviews · Bundle

Named alongside it

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

bundle adjustmentLevenberg–MarquardtReprojection errorDepth reversalfield of viewOrthographicstructure from motionCorrespondenceOverfittingResidual

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