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

Disparity — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · depth
  2. 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.

    part 2 · depth
  3. -0.10000.1002.5057.5010depth reported from the stepped disparity (m)height reported for each image row (m)the floor as it is35 platesspacing 3.9 cm → 1.95 mslope 1.997

    Whole pixels cut space into shells

    A disparity read to whole pixels can report only the depths fB/k, so a stereo pair does not measure distance on a scale — it chooses among 113 shells between half a metre and twelve, 6.7 cm apart at two metres and 1.39 m apart at ten. A level floor comes back as 35 standing plates. And a finer step and a better reading are different purchases: at a quarter pixel with a quarter pixel of matcher error the pair prints 449 depths and can tell 149 apart.

    part 3 · depth
  4. where the eyes are aimedin plan, framed to the shells drawnzero at 0.92 m, 40° aside

    Vergence moves the shells and does not respace them

    Turn two eyes inward and the depths a whole-pixel reading can report stop being planes and become a family of near-circles through both eyes — the twenty-pixel shell standing at 0.74 m forty degrees aside where a parallel pair puts it at 3.82. The spacing between consecutive shells is the same to 0.07 per cent across the whole field, so vergence relabels the rays and does not sharpen them, and the resolution argument for turning the eyes in does not exist.

    part 4 · depth
  5. 0.2000.4000.6001.8022.20range, mheight above the plane of the eyes, mhorizontalverticalat 2 m, 40 cm off the plane79° between them

    The second disparity cuts cells

    A point off the plane of the eyes has a vertical disparity as well as a horizontal one, and quantising both, on an 86,400-point lattice of a room, gives 7,663 labels where one coordinate gives 179 — a count that belongs to the lattice rather than the room, as the essay after this one found. The gain is entirely vergence's — two eyes looking straight ahead have no vertical disparity at all, exactly — and it is largest where the first reading is already finest: 60.8 in the near metre and 3.7 in the far band.

    part 5 · depth
  6. 010203051015how far each camera is turned in, degreesper centwhat the vertical disparity addslabels merged by rectifyingof those, from the warpfocal length to merge noneeach pair rectified at its own 900 px4.0% for eyes at 1.2 m

    Rectifying a pair spends what its epipolar lines lean

    Counted from the two disparities alone, rectifying a verged pair looks as if it throws away at least 99.7 per cent of what the pair can tell apart. That cannot be true of a warp that loses no ray, and it is not: once the place in the picture is counted, the vertical disparity adds 4.1 per cent for eyes verged at 1.2 m, and rectifying at the same focal length gives back all but 4.0 of it.

    part 6 · depth
  7. a wall at 10 m0 px−0.25 m0.25 px+0.17 m0.5 px+0.64 m0.75 px−0.64 m7 m8 m9 m10 m11 m12 m13 m14 mf = 900 px, B = 65 mm, whole pixelsshift 0.150 px puts a shell on the wall

    A rectification's free shift is free only near the pair

    Sliding a rectified pair's principal points apart adds a constant to every disparity, and a constant changes nothing about where points are — on paper. A whole-pixel reading is not paper. The same slide moves every depth the reading can report, and for a wall ten metres away the choice between the best shift and the worst is 1.71 metres. Near the pair it is millimetres. A matcher with half a pixel of its own error erases the choice, and pays more than the worst shift did.

    part 7 · depth

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