Visual Plane
every picture states the point it is correct from
Concept

structure from motion — where it appears

Recovering a scene and the camera track that photographed it together, from a sequence and nothing else. It determines both up to a similarity and no further, so seven numbers remain free however long the sequence is.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

view 1view 6the scene, in plan — recovered points and cameras over the true onesreprojection 0.332 px · track 2.0e-3264 observations, 168 parameters

The track and the scene together

Six photographs go in and one hundred and sixty-eight numbers come out — every camera's position and orientation and every point's place in space, solved for at once. Nothing in the solve was ever told where a camera or a point was.

manyviews · Bundle 0.30.50.7125×10⁻⁴0.0010.0020.0050.010.02how finely each point is read (px, log scale)worst camera-centre error, as a fraction of the track's mean radius (log)chained pairsadjusted togetheradjustment is 4.0–9.2× better6 views · identical observations

A chain and an adjustment

Composing pairwise poses along a sequence is supposed to drift. Measured over five links it wanders instead — one chain ends closer to the truth than its own worst link — and the real cost of chaining turns out to be something else entirely.

manyviews · Drift 2468-1-0.50000.5001position along the family the two pictures leave freehow far the solid is from the scene's own shape (stretch ratio)the sceneevery member redraws both pictures to 7e-16 mthe ambiguity is a family of solids, not a tolerance

What two parallel views leave free

Two perspective pictures give shape and no size. Two parallel pictures do not even give shape: the metric upgrade has six unknowns, two views supply six equations, and the system comes out rank five — a one-parameter family of solids that redraw both pictures to 7e-16 m. A third view closes it, and the mirror image survives every count.

parallel · Parallelrecovery 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 adjustmentReprojection errorcamera trackConvergenceDepth reversalfield of viewLevenberg–MarquardtOrthographicAffine reconstructionBaselinedegrees of freedomDrift

All concepts