The thread: Made by something finite — page 2
A camera on a bend is sharp on a circle
A camera car rounding a 50 m bend at 10 m/s, aimed into the bend, blurs the still world everywhere except at the bend's centre — and under a pixel from 42 m to 63 m along its axis. Off the axis the sharp place comes nearer as the cosine of the bearing, on the circle through the camera and the centre. Above the ground only the vertical line through the centre stays sharp. Aimed along the road, the camera has no sharp distance at all.
What a machine computesOne warped shadow map, and what it cannot reach
The lamp facing the eye needed a shadow map 9,530 texels square — 90.8 million texels — for none of them to land on the near floor larger than a pixel. Fitted to the floor and warped by one projective parameter, the same map needs 805 thousand; nothing can do better than one texel per pixel of the eye's picture, 148 thousand. What stays out of reach is not the cosine of a surface, which a warp absorbs, but two surfaces that want different densities along one ray from the lamp.
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.
Mirrors that are not camerasTwo people between mirrors see each other equally often
Put two people between a pair of mirrors and each sees some number of images of the other. The two numbers are always equal, and so is the apparent distance of each image against its partner — a path of light walked backwards is the same path. What is not shared is where each image appears, and what the shared count depends on turns out to be neither person's position but two quantities made of both: the difference of their angles about the mirrors' meeting line, and the sum.
What a pair is forA 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.
What each system gave upA strip keeps its ratio, not the end it began
A painter dividing a strip into bays tires as they go, and each bay comes out a little larger than the last. Divide the strip from its far edge instead of its near one and the tiring runs the other way — but the rows record none of it: a strip divided from the far edge by a tiring hand is, to the last digits, a strip divided from the near edge by a steadying one. A whole picture recovers every strip's direction anyway, because one hand shared one rate of tiring across strips of different splay.
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.
The rectangle behind the lensA rolling frame on a bend is right at its centre
Read a camera car's frame row by row as it rounds a bend and every still point moves — except the bend's centre, at every height, and the horizon row, at every depth. Each mark's displacement is its streak scaled by its row time, so the rolling frame is undistorted wherever the global one is sharp. And the usual repair makes it worse: a gyroscope that undoes the turn row by row puts 0.73 px back into the centre and is a net loss everywhere nearer than twice the bend's radius.
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 rectangle behind the lensA rolling frame on a bend is a coarse curvature gauge
Read row by row on a bend, a frame leans every vertical by an amount proportional to one over its range less one over the bend's radius, so posts at several ranges lie on a straight line that crosses zero at the radius — exactly, to a thousandth of a metre. Off the axis the zero moves to R·cos of the bearing, the circle on which a camera on a bend is sharp. As a gauge it is coarse: one frame's eight posts read a 50 m bend as 45 m (35–74) with tops placed to half a pixel, and sixteen frames as 51 m (47–55). The slower the sensor reads, the better it measures the road.
What a pair is forA 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.
The rectangle behind the lensA tracked post reads the bend, if the speed is known
A rolling camera aimed at a bend's centre passes across its roadside posts rather than towards them: in half a second a post twelve metres out moves from 12.0 to 12.8 metres away and its lean changes by five per cent. Followed through those sixteen frames it is one lean read sixteen times, and one lean is a ratio of the car's speed to the bend's radius. With the speed known it reads the radius as 49.9 metres (48.1–52.9), closer than eight posts in a single frame; with it unknown, it reads nothing. Aimed forty-five degrees ahead, the camera closes on the post, its lean doubles, and one post gives both.
What a machine computesHalving a picture moves its centre by a quarter of a pixel
Multiply a principal point by the scale of a resize and it comes out wrong by (s − 1)/2 of a new pixel, at any point and for every reduction. At a pyramid's fourth level that is nearly half a pixel, turning a recovered pose by half a degree. The picture, asked, reports the true map.
Where to standA design that lands in two rooms
Cast a design down a corridor with a doorway in it and four per cent of the picture goes through the opening and lands on a wall three metres beyond, at 9.4 metres from the eye against the end wall's 6.4. The picture is continuous across the edge of the doorway and its scale jumps by half again, which is a corner anamorph's discontinuity with a gap in the middle of it instead of a fold.
Mirrors that are not camerasA mirror's top edge waits for an eye above it
Between two upright mirrors a path of light from one eye to another, unfolded, climbs in a straight line from one eye's height to the other's, so every bounce lies between the two. While both eyes are inside the glass's height its top edge takes nothing at all. Above it, the images go in a fixed order — not the one with the most bounces, not the longest, but the one whose last bounce falls furthest along its path, in forty arrangements out of forty. The barber's corridor that sinks out of the glass is a mirror leaning a fraction of a degree, and its length goes as one over the square root of the lean.
The rectangle behind the lensA followed post's own lean is read as the bend
A rolling camera on a bend reads the bend from how its roadside posts lean, and a post that leans by itself adds its own tilt to every frame, unchanged. That tilt is indistinguishable from the bend's curvature: a tenth of a degree on a post twelve metres out turns a 50-metre bend into 58. No number of frames removes it, and fitting each post's own lean along with the bend loses the bend — a sideways camera's reading widens from ±7 metres to ±29. A followed post has to be plumb to six hundredths of a degree, finer than a well-set steel column; timber posts at a quarter of a degree help only in their dozens.
What a pair is forA climbing eye names the column mistake, if it knows how high it is
A level pair that matches a facade a column out rebuilds it nearer, and a third camera climbing above the left one watches the rebuilt windows slide down its picture at 98 pixels a metre for every column of the mistake. Read against the real windows, that slide is a ruler: a metre of climb from level names a one-, two-, three- or four-column mistake every time. What it costs is not frames but knowledge — the climb must begin before the rebuilt facade slides off the lattice, and the camera must know its own height to about a quarter storey over the mistake's slide, 34 centimetres for one column and 9 for four.
What a machine computesA rounded viewport has two focal lengths
Compute the aspect ratio from a window's requested size and draw into a buffer of whole device pixels, and the picture has two focal lengths that differ by up to one part in three hundred. It looks like a defect that should grow worse on small screens and better on large ones. At the frame's edge it does neither, because the error is a bounded displacement of a fraction of a pixel on any window at all.
What a pair is forA staircase winds the mistake round the lattice
A third camera climbing a stair along a facade sees a stereo pair's column mistake slide through its picture along the stair's own direction, wrapping at the columns across and the storeys up — a line wound round a torus. Such a line comes back onto a real window only when the stair's rise over its run is a ratio of whole storeys to whole columns; a degree off the diagonal, no two readings come within 3 px. The stair names the mistake from one frame, as a vertical climb does, but its readings sit twice as far apart in a frame, so a camera that knows where it is only to 10 cm still names it 95 times in 100 against 82, and a 30° stair keeps naming it a metre and a half along, where a vertical climb has lost half.
Mirrors that are not camerasA corridor of mirrors counts their angle, not their leans
Between two facing mirrors that both lean, the barber's corridor of self-images ends where it leaves the glass, and the count depends on the two leans only through their sum — the angle between the mirrors. Forty-two pairs of leans fall on one curve, within an image. Counted from near either mirror the number is the same, so counting twice cannot say which mirror leans, and two mirrors tilted together run the corridor on for ever. A photograph can: each round trip lowers the image by the sum of the leans, and where the row of images starts shares the sum between the two — provided the camera knows which way is level.
The rectangle behind the lensA post read both ways sets its own lean aside
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend: a quarter of a degree leaves a 50-metre bend read to ±12 metres. Followed again by a car coming the other way, the same post shows the same tilt and the opposite shutter lean. Half the sum of the two passes is the tilt and half the difference is the bend, and a timber pole read both ways gives the bend to ±1.7 metres — better than a plumb steel column read one way, at ±2.3.
Light and mirrorsA floor is read along curves
Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.
What a pair is forA turn at the landing makes the mistake a scale, and a scale is weak evidence
On a straight stair along a facade, a stereo pair's column mistake slides through a third camera's picture as one offset shared by every window. Turn the stair towards the wall at a landing and the offset stops being shared: the rebuilt facade's picture becomes the real one's scaled about the point the camera is walking towards, a different scale for every mistake, so no picture on that flight is blind. But the scale grows as the walk over the rebuilt facade's distance, and 10 cm towards the wall separates the one-column reading from the honest one by 1.8 px where 10 cm along the stair gives 9.8. Turning the camera in place adds nothing at all. What the turn does buy is tolerance: an error in how far towards the wall the camera has come costs almost nothing, for the same reason.
Mirrors that are not camerasTwo mirrors nearly facing bend the corridor two ways, and each bend runs out at its own edge
Between two mirrors that lean and are also turned a little about the vertical — the barber's shop, seen past the shoulder — the corridor of self-images steps down by the sum of the leans and sideways by the sum of the turns on every round trip. Its count is not a reading of the angle between the mirrors: each bend runs out against its own edge of the glass, and the corridor ends at whichever does first. A quarter of a degree between the planes gives 22 images when it is all lean, 21 when it is all turn and 25 when it points between. A photograph reads both bends from how the images step, with no reference at all, and needs gravity only to share the lean between the mirrors and the room only to share the turn.