Disparity — where it appears
Named by 28 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
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.
Two pictures on one screen
A stereoscopic display puts a point where two sightlines cross, so the depicted depth is b·D/(b−d) and the disparity that reaches infinity is exactly the separation of the reader's eyes — 63 mm, at any screen distance whatever. The depth budget is set by the width of a head and by nothing about the scene.
Turning the cameras inwards
A stereo pair made by rotating two cameras toward a common point puts the same world point at different heights in the two pictures — up to thirty pixels here, on a frame of four hundred. Two eyes level with each other see every point at the same height, so a pair with vertical difference is a pair of pictures of no scene at all.
The depth a pair calls zero
Two eyes verged on a point agree — the same coordinate in both pictures — not on a plane at the fixation distance but on a circle through both eyes and that point. Found by bisection along 121 azimuths and fitted rather than assumed, it is a circle to 0.0000 cm; at 26° off centre it lies 23 cm nearer than a flat wall does.
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.
Both coordinates agree on a circle and a line
Two eyes fixating a point straight ahead see their horizontal image coordinates agree on a whole vertical cylinder over the Vieth–Müller circle, the same radius at every height to the last bit. Their vertical coordinates agree on almost none of it — 7.35 px apart at 26° aside and 30 cm up, 29.26 px when the fixation is brought to 60 cm. The points where both agree are the circle and one vertical line, and the line is the axis of the motion that carries one eye onto the other.
A scroll through two slits ranges in a straight line
Draw a scroll twice, through a slit leaning 10° forward along the track and one leaning 10° back, and every point appears in both drawings on the same row, separated by 9.169 px for every metre of its depth — at four metres and at fifty-two. Depth is proportional to that separation rather than reciprocal to it, so a pixel of error costs 10.9 cm at every distance, averaging leaves no bias, and there is no range past which the depth runs off to infinity. The price is paid in roll: a 100 m scroll ranges nothing past 283.6 m.
A scroll round a bend loses its straight-line depth
Draw a scroll through two slits leaning ±10° from a track that bends, and the separation that was 9.169 px for every metre of depth stops being proportional. Outside a 100 m bend it is 653.8 px at 256 m where a straight track gives 2347, and it never passes 907.6 px however deep the point; inside a 200 m bend it runs nearly three times ahead of depth and no slit reaches past 165.3 m. The two drawings still share their rows, and the scale along the roll becomes a function of depth.
What the two eyes are sent
A reader's eyes are two seats sixty-three millimetres apart, so a curved screen delivers each of them a different map — and the part of the difference no homography absorbs is binocular evidence of the glass. Turned into a depth it comes back as the screen's own sag, 49 millimetres against 47 on a television, by a route that never saw the radius.
A scroll can be asked its own radius
The two marks a bend leaves separate exactly. The along-roll scale alone fixes the angle in the disparity, so one point and a neighbour at its depth give back the radius and the depth in closed form — 200 m and 40 m returned to a part in 10⁹, with no search. The two answers are not equally held: a scale read one per cent too large under-reads the depth by one per cent and over-reads the radius by tan(φ − α)/α, which is 50 for a point ten metres from a five-hundred-metre bend. And a painter who evens the scale out by eye reports a gentler bend, never a bend that was never there.
A set cut for one eye
Build a colonnade four metres deep and cut every column so that its picture is the picture of one eighteen metres deep. The taper is forced rather than chosen — height and width both scale as the real depth over the intended one — and the match from the design eye is exact to sixteen decimal places. What gives it away is the second eye, and not by the ratio anybody would predict.
An epipole in the picture leaves a blind disc
Step a camera half a metre straight forward and the image of the other eye sits in the middle of both pictures. Around it lies a disc where one pixel of reading costs a tenth of the depth or more — 20 px across a surface 2 m off, 247 px at 16 m — and at its centre no depth is recovered at any range.
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.
Rectification is a family, not an operation
Turn both pictures of a pair so their epipolar lines become shared rows. A turn about the line between the eyes and a focal length are left free, and every choice puts all 44 matches on common rows to a tenth of a trillionth of a pixel and every point back where it was. What the choices disagree about is the pixels — one stretches its pictures unevenly by 1.77, another by 4.86.
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.
A stereo picture is drawn for a level head
Every stereo pair is drawn for two eyes level with each other — a point's two images share a row and differ only across it. Tilt the head 10° in front of a television and that difference turns partly vertical, 14.46 arcminutes for anything drawn at infinity, and the two sightlines to a point stop meeting. At a desk monitor the same fifteen-arcminute limit arrives at 2.58°.
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.
A curved screen tilts a stereo pair both ways at once
A stereo pair drawn for a flat screen and shown on a curved one gets a vertical disparity from the curve alone, with the head level — opposite in sign either side of the middle, and 31.8 arcminutes at the corners of a curved desk monitor for anything drawn at infinity. A rolled head adds its own, of one sign everywhere. The two add in one pair of corners and cancel in the other, so the curved screen is kinder than a flat one in two corners and harsher in the two that decide.
A scroll of a climbing road measures its grade
Every reading of the two-slit scroll has leaned on its two drawings of a point sharing a row, because the eye is at one height at both moments. On a road that climbs they do not — and what parts them is the height climbed between the two moments over the reach, which on a straight climb is 2·f·g·sin φ for every point at every depth and height. The scroll does not lose its rows to a hill. It gains a third mark, a gradient meter that a level bend cannot counterfeit.
A sway gives the blind centre a depth, not a good one
A camera driving straight forward cannot see how far away the thing it is driving toward is: the mark at the epipole does not move between pictures. Let one of three pictures sway sideways and the centre gets a depth at once — but a depth resting on the sway alone, which a pixel of reading moves by the focal length's reciprocal times the depth over the sway. For a centimetre of steering wobble at eight metres that is 144 per cent; for a tenth of the forward step, 29. The hole closes; the disc around it stays until the sway is a third of the step.
An eye that pitches with the road keeps its rows
An upright eye climbing a road parts each point's two drawings by the same few rows, and that offset reads the grade. Fix the eye to the vehicle instead, so it pitches with the road, and the offset vanishes exactly — for every point, at every depth and height. The grade has not gone. It has moved into the posts, which now lean by an amount that grows with their depth, and into one drawing, which can now read the grade on its own.
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.
A rig is right on one surface
Several cameras looking outward have several centres, and no warp registers all of a scene. The residual is a disparity, so the depth the stitch is computed for has an exact optimum — the harmonic midpoint of the depth range, not the middle of it — and the arithmetic middle costs a factor of 2 z_far over the sum, which tends to two.
A facade matched a column out comes forward, not back
A railing matched one post along is rebuilt 2.2 times further away. A facade of windows 2.4 metres apart cannot be: a one-column shift that way needs a negative disparity, and the only mistake left in front of the cameras brings the facade to 6.3 metres from twenty, a third its size, hanging 1.1 metres off the ground with nothing under it to sink into. Its corners hold it only if the wall is assumed flat. Two corners then convict it at every period; one corner and a drainpipe need the pipe half a metre in for windows and four and a half for a fine repeat.
A vehicle's pitch lags the road by its wheelbase
A camera fixed to a vehicle does not pitch with the road under it; it pitches with the chord between its wheels. Over a step from level to six per cent, a two-slit scroll's rows — silent on any steady grade — depart by up to 2.18 rows over 6.5 metres of road for a 2.7-metre wheelbase, against 4.26 over 4.0 for a camera that pitched at the point. The excursion's width is the eye's own chord plus about the wheelbase, and one line of posts reads the wheelbase back to ±8 centimetres. A vertical curve does not silence the rows either; it shrinks them as one over its length.
A raised eye reads the storeys as the pair read the columns
Put a third camera 3.5 metres above the left one and its epipolar lines run up the facade: the storeys repeat along them as the columns repeated along the level pair's rows, the roofline and the foot carry depth where the corners did, and a raise shorter than a storey can only pull the windows nearer. The three pictures together refuse the level pair's one-column mistake at every raise except those where the storey over the raise equals the period over the baseline — 1.375 and 2.75 metres here — and even there the facade's lowest storey lands on the ground.
A scroll camera rings with its vehicle's suspension
A vehicle does not pitch with the chord between its wheels; its body follows that chord through springs, lags it, overshoots and swings back. Over a step to six per cent at ten metres a second, a body sprung at 1.3 hertz parts a scroll's rows by up to 2.6 pixels where a rigid vehicle parted them by 2.2, and leaves them parted eight metres past the step. No rigid vehicle can take the difference, and one line of posts read to a tenth of a pixel gives the suspension back: 1.302 ± 0.011 hertz, damped 0.303 ± 0.008.
Named alongside it
The objects these essays reach for when they reach for this one.
Depth uncertaintyBaselineinstrument limitrectified pairTriangulationMoving viewpointPushbroomStereo pairBinocular disparityHandscrollVergenceParallax