Visual Plane
every picture states the point it is correct from
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The thread: Determined up to something

Every recovery on this site returns an answer with a qualifier attached. One picture gives ratios and no size. Two give shape and no size. Many give a scene and a camera track up to seven numbers no quantity of pictures supplies. The qualifier is not a shortcoming to apologise for; it is the exact statement of what the pictures contain, and here it is counted, walked along and shown to be free rather than merely small.
025507510010203040true depth (m)depth reported from the disparity, with a 1 px reading error5.86–7.33 m13.96–26.69 m23.76–126.49 mat 40 m: +86.5 m against −16.2 munbounded past 58.5 m What a pair is for

Depth is a reciprocal

Two eyes measure a shift in the picture, and depth is that shift divided into a constant. So a fixed error in what is read maps to an interval in what is reported that is not centred on the answer, and at forty metres runs sixteen metres nearer and eighty-six further.

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

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.

 123-2-10baseline (m, log scale)range past which 1 px of disparity error is unbounded (m, log scale)65 mm → 58.5 mrange = fB/δ — 58.5 m at a 65 mm baseline900 px focal length, 1 px reading What a pair is for

The range a pair cannot see past

A stereo rig has a distance beyond which it cannot say "no further than", and the distance is fixed before anything is built. It is the focal length times the baseline divided by the reading precision, and for a human pair of eyes it is fifty-eight and a half metres.

 × 4.6e+6seven flatand the rest stiffsingular value ÷ the largest, log scale, smallest firstσ₈/σ₇ = 4.6e+6168 parameters · 528 residuals Many pictures at once

Seven numbers no picture can name

Shift a whole reconstruction by a metre and a half, turn it half a radian, scale it by 2.7, and every photograph of it stays where it was to a hundredth of a billionth of a pixel. Move one point by fifty millimetres and they move by two thirds of a pixel.

 midpoint — 2.77 mm gapfrom the left eyefrom the right eyegap 2.77 mm at 7.22 mexact marks: 2.2e-16 m What a pair is for

Two rays that do not meet

Triangulation is described everywhere as the intersection of two rays, and two rays in space do not intersect. Read the same two marks to a whole pixel and they miss by 2.77 mm at seven metres, which is a real length and is the part a residual will not report.

 -3-2.50-2-0.600-0.400-0.20000.200how 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 Many pictures at once

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.

 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 The second eye

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.

 -2.80-2.6034567number of viewsworst camera-centre error (fraction of the track's mean radius, log scale)7 flat7 flat7 flat7 flat7 flat60° sweep, filled in12° per view, wideningfilled in: 1.1e-3 → 2.0e-3widened: 3.2e-3 → 1.8e-3 Many pictures at once

Another picture of the same sweep

Going from three views to seven across the same sixty degrees leaves the reconstruction exactly where it started, and at one point makes it worse. What a reconstruction is short of is angular spread, not photographs.

 as recovered0.308 across× 3.4 — same two pictures1.046 acrossworst ratio deviation 6.3e-14size fixed only by a supplied 0.80 m The second eye

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.

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