Field

What a pair is for

Depth from two views is a reciprocal, so a fixed error in what is read maps to an interval that is not symmetric and eventually is not bounded. What a stereo pair can and cannot measure, computed rather than described, including the range past which one pixel reaches infinity.
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

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.

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

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.

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

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.

-10-50-2-10baseline (m, log scale)worst deviation in the recovered shape (log scale)exact pointsread to 1 pxat 2 mm: 4.7e-9 exact, 2.9e+1 read to 1 px6e+9× apart

A turn of the head is not a step sideways

The textbook says a short baseline makes reconstruction ill-conditioned. Measured, the algebra does not notice — a two-millimetre baseline recovers the courtyard to nine digits from exact marks. What fails is the ratio of baseline to reading error, and it fails without refusing.

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.

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