The thread: What a second picture adds — page 3
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
What a pair is forThe 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.
The second projectionA 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°.
Mirrors that are not camerasA symmetric object is its own stereo pair
A building with a plane of symmetry photographed once gives fourteen correspondences whose joining lines meet at one point to 1.9 × 10⁻¹² pixels, a skew-symmetric matrix, and the object's whole shape to fifteen digits — with no mirror anywhere and no second exposure. What it does not give is the size, and the instrument that decides whether any of it applies is the same meeting point, which opens to 12.7 pixels when the symmetry is half a per cent out.
The second eyeA parallax length is a height over a depth
After a known plane's map, every raised mark's displacement points at the other camera's image, and its length carries the mark's height above the plane over its depth — but not as the ratio of lengths it looks like. That ratio departs from the point's own number by up to 45 per cent. Read as a coefficient on the epipole, the same length gives height over depth from the first camera to four parts in a hundred trillion, the same from every second picture.
What a pair is forRectifying 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.
The second projectionA 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.
The eye that movesA 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.
What a pair is forA sliding pair keeps its line only near the middle
Two eyes that roll by Listing's law lose the horopter's straight line as soon as they look up and aside, and the question worth a number is whether they lose it by much. Measured along the line they nearly keep, the disparity is exactly the slide seen by one eye — 3.20 px at 20° aside and 20° up, a metre and a bit away — and the slide does not shrink with distance. At arm's length the line survives to within a pixel only in a narrow cross through the middle of the field.
The second eyeA 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.
The eye that movesAn 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.
The second eyeA plane's coefficient reaches as far as its parallax
After a known plane's map, every raised point's displacement is its height over its depth, read as a coefficient on the epipole — exactly, for any point either picture sees. The worry was that the number would be local, good only near the floor whose marks fixed the map. Read to a pixel, it is not a distance on the floor that runs out. It is a length in the picture: the point's error is about 260 per cent over its parallax in pixels, wherever the point stands.
Through water and glassA third eye sees the water the pair cannot
A level stereo pair looking at a stick in water gets two rays that meet exactly and a stick 285 mm too short, with nothing to warn it. A third eye raised ten centimetres above the pair sees a disagreement of 2.6 pixels at the stick's tip. Asked to agree with the other two as if there were no water, it hides nearly all of that and makes the stick shorter still. Told only that there is a flat surface, the same three eyes find its refractive index and put the metre back.
The second projectionDrawing for the curve removes the curve, not the nearness
A stereo pair drawn for a flat screen and shown on a curved one sends a level head 31.8 arcminutes of vertical disparity before anyone tilts anything. Draw the pair for the curve and that vanishes exactly. What remains when the head rolls is the roll's own disparity, and on the curve it is 8.6 per cent worse than on a flat screen — not because of the curve's shape but because the arc brings its edges nearer the eyes, and a fixed separation seen from nearer is a larger angle.
Many pictures at onceSplit the track and the needles turn
A point triangulated from a track of cameras is not known equally well in every direction: its error is a needle, long along the direction its lines of sight barely constrain. That direction is not the mean line of sight but the lines of sight weighted by how near each camera is — and it can be turned. Two groups of cameras more than a right angle apart swing every needle across, and the same six pictures of a facade then measure its depth four times better.
What a pair is forA quarter of a degree of roll keeps the line
A Listing pair's horopter loses its straight line because the motion between the eyes slides along its own axis. Helmholtz's rule cures that by turning each eye several degrees. A roll a tenth that size, in opposite senses, cures it just as exactly, and its size is a short law: a quarter of the elevation times the vergence, with hardly any dependence on the azimuth.
What a pair is forA railing matched one along stands underground
Match every post of a railing to the next one along and two photographs build a perfect railing at 19.8 metres instead of 9.0 — straight, evenly spaced, every post on both its rays. Nothing near it can say so. What convicts it is what it stands on: its feet are 1.32 metres under the ground, and one ground mark read to a pixel shows every foot a whole period out from sixty metres away. Its length helps only through its ends, and once those leave the frame, counting matches prefers the wrong railing.
Mirrors that are not camerasTwo wide pairs are most of a symmetry
A facade with a symmetric frame and an asymmetric middle is still its own stereo pair. Two pairs whose halves stand wide apart find the symmetry plane to 0.40° with marks read to 0.4 px; twelve pairs find it to 0.26°; two narrow pairs only to 1.35°. The middle that pairs with nothing comes back too, to 0.8 per cent, carried along its rays to the wall the pairs have fixed. The risk is not too little symmetry but a false pair — and a window set wider than its partner is one the standard test cannot see.
The second eyeA turn moves the picture, not the blind point
A vehicle that yaws for a moment between frames points its camera somewhere new without taking it anywhere new, and that is exactly what a turn cannot help with. The mark it is driving toward stays blind at every turn; the disc around it moves by two hundredths of a pixel for a 3° yaw; with a sway, a turn changes a mark's depth cost by 0.2 per cent. Aiming the camera off the road moves the blind point across the picture — and off it past 30° — but the thing it blinds is still the thing the vehicle is heading for.
Through water and glassA water surface's height trades against its index
Three eyes that must find the water's surface as well as its index find the two along a valley: a surface assumed higher bends every ray sooner, and an index assumed weaker makes up for it. From one stick at a quarter pixel the index is known to ±0.14 and the surface to ±48 mm, and the fits do worse than that promises — they slide down the valley, and the metre-long stick comes back 881 mm. A second stick eight metres out narrows the valley to ±7.6 mm. Holding water's own index at 1.333 closes it: the surface to ±23 mm, the stick to 994.
The second eyeSecond pictures reach a floor unless they go round
A point with little parallax can be read against several second pictures, and each should count in proportion to the square of its parallax. Doing that does less than expected, because every second picture shares the same first reading. In one direction, no number of them goes below one pixel over the longest parallax. Spread them round the point and a fit can find out where the first reading really was, and the floor goes away.
What a pair is forA 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.
The second eyeSteering swings a camera too little to see ahead
A vehicle yaws about its rear axle, so a camera mounted ahead of the axle is swung sideways whenever the vehicle steers — a real baseline, where a turn alone gave the blind point ahead nothing. Half a degree of yaw swings a windscreen camera 1.3 centimetres and a bumper camera 3.1. Straight ahead at eight metres a pixel then costs 96 and 40 per cent of the depth. The cost falls as one over the arm times the yaw, and a usable 10 per cent needs 7.2 metre-degrees: two degrees of steering every frame on a bumper. Ordinary lane-keeping gives a fraction of that.
Mirrors that are not camerasThe wall convicts a pair set too wide
A window set wider than its partner's reflection keeps every joining line on the mirror's point, so the one test a symmetric photograph was known to carry cannot see it. The wall the other pairs fix can: it says where the partner should be seen, and a window set ten centimetres too wide misses by 6.0 pixels. With marks read to 0.4 px it convicts a 3-centimetre offset eight or nine times in ten and a 5-centimetre one every time. What no single photograph can see is a move along the camera's own ray to the window.