One pair of frames reads the speed and the bend from the roadside

Frames read up and down part a post’s lean in one drive followed one roadside post through a drive round a 50-metre bend at ten metres a second, with a camera that read its rows from the top down on one frame and from the bottom up on the next. Reversing the readout reverses the rolling shutter’s lean and leaves the post’s own tilt alone, so the frames zigzag about the post’s tilt, and a fit of one tilt and one curvature parts them. Thirty-two frames read a timber pole leaning a quarter of a degree to ±1.8 metres — as well as two passes of sixteen frames by two cars driving the road in opposite directions.

Everything in that essay followed a single post through many frames. That essay ended by asking about the other way to use the roadside: not one post through time, but every post at once. A rolling frame on a bend is a coarse curvature gauge had found that the posts of a single frame give the bend only coarsely, and their own leans were much of the reason. In one frame the posts’ rolling leans lie on a straight line in one over the range — slope set by the speed, crossing set by the radius — and every post’s own tilt scatters them off it. A pair of frames read in opposite directions, a fifteenth of a second apart, removes every post’s tilt at once. Does one such pair read the speed and the bend from the roadside?

Half the difference is a straight line #

The car and the bend are the earlier essays’: a camera rounding a 50-metre bend at ten metres a second, aimed at the bend’s centre, its rows read in a thirtieth of a second, eyes 1.5 metres up and a 50° lens. Beside the road stand timber poles, between eight and forty metres out and spread across the view, each leaning a quarter of a degree in a direction of its own. The camera takes one frame reading from the top down and the next reading from the bottom up.

In the top-down frame each pole leans by its own tilt plus the shutter’s lean; in the bottom-up frame, by its own tilt minus the shutter’s lean. Half the difference of the two is the shutter’s lean alone, and that is the image’s speed across the picture times the time between reading the pole’s head and its foot. Every row is a different camera is where that came from, and the earlier essays measured it: a constant, set by the camera, times the car’s speed, times one over the pole’s range less one over the bend’s radius. So the half-differences of a row of poles lie on one straight line in one over the range. Its slope is the car’s speed, and it crosses zero where one over the range equals one over the radius — at the range of a pole that stands on the bend’s own centre line, which the shutter does not lean at all.

Read from exact leans, twelve poles give 10.03 metres a second and 49.9 metres. The small miss is each pole’s tilt drawn a little differently in the two frames, since in a fifteenth of a second the pole moves across the picture and is seen from a slightly different angle. The reading itself is a small linear fit: every frame’s lean is the shutter’s lean for that frame’s readout direction plus the pole’s own tilt, and the unknowns — the speed, the speed over the radius, and every pole’s tilt — enter linearly.

Speed and bend together, from a fifteenth of a second #

With every pole’s head read to half a pixel, the question is a spread.

Two poles read the bend to ±19 metres, half the middle spread of 120 roadsides; four to ±14; eight to ±9.0; sixteen to ±5.6. Each pole gives two equations, one per frame, and spends one of them on its own tilt, so a roadside of n poles gives n equations for the two numbers the poles share. The spread falls roughly as one over the square root of the count, with a little extra from the near poles a larger roadside adds.

The speed comes with it, and comes well. Four poles read it to ±0.64 metres a second, eight to ±0.38, sixteen to ±0.27 — three per cent of the ten metres a second the car is doing — with medians within a tenth of ten. The speed is the slope of the line, which the near poles fix: the shutter leans a pole at eight metres twenty times as far as one at forty, which stands close to the line’s zero.

That is the first answer to the question. One pair of frames reads the speed and the bend together, from the roadside alone, in a fifteenth of a second — and the bend it reads, from sixteen poles, is about twice as wide as a followed post’s.

Knowing the speed is worth a factor of two #

The earlier essays read the bend from a followed post with the speed given, and that is the fair comparison the question asked for.

With the speed known, the pair reads the bend to ±4.4 metres from two poles, ±3.2 from eight and ±2.4 from sixteen. One plumb post followed one way through sixteen frames, the speed known, reads it to ±2.7 metres. Sixteen leaning timber poles in two frames do a little better than one perfectly plumb post in sixteen frames, and they do it whatever the poles lean — the earlier essays found that the followed post read one way lost the bend entirely to a quarter of a degree of lean.

The speed costs the bend a factor of about two at every count, because the speed and the curvature are read from the same line: the slope and the crossing of a line fitted to points spread over a limited stretch of one over the range are correlated, and an error in the slope moves the crossing with it. A car whose speed is known from its own wheels should be given it.

The nearest pole decides #

Eight poles from 5 to 40 metres read the bend to ±7.2 metres and the speed to ±0.17 metres a second; from 8 to 40, ±9.0 and ±0.38; from 15 to 40, ±11.7 and ±1.42. Moving the far end barely matters — eight poles from 8 to 20 metres read the bend to ±8.6, from 8 to 80 to ±12.1. The line is in one over the range, and one over the range is spread almost entirely by the near poles: one over 5 metres is 0.2, one over 15 is 0.067, and one over 40 and one over 80 are both close to nothing. A roadside’s nearest poles are the ones that do the work, and a camera reading a bend from a pair of frames wants poles close to the road.

How far the poles lean does not matter #

Plumb poles read the bend to ±8.9 metres; poles leaning a quarter of a degree, ±9.0; a degree, ±8.9; two degrees, ±9.3. Each pole’s tilt is the same in both frames and is an unknown of its own, so it is taken out whole whatever its size. The one-way reading the earlier essays measured lost the bend to a quarter of a degree of lean — a timber pole read from one direction read the bend to ±11.7 metres — and the pair does not lose it to two degrees. That is the point of reversing the readout. A roadside is full of poles nobody has plumbed, and the pair does not need anyone to.

What it does not take out is the part of a lean that changes between the two frames. A pole leaning towards or away from the camera is drawn with a lean that grows as the camera rounds the bend and sees it more obliquely, and a fifteenth of a second is long enough for a near pole’s view to change a little, and the medians, which wander a metre or so either side of fifty, carry that change.

The marks set the scale #

The reading is a linear fit to the leans, and its spread is in proportion to their reading error: eight poles read the bend to ±4.2 metres with every head read to a quarter of a pixel, ±9.0 to a half, ±19.6 to a pixel. Half a pixel is the reading the earlier essays used for a post’s head against the sky. A camera with twice the resolution, reading the same poles to the same fraction of a pixel, halves every figure here.

The poles’ tilts come with it, coarsely #

Half the sum of each pole’s two leans is that pole’s own tilt, so the pair reads the roadside’s tilts as well as the bend. It reads them coarsely. A three-metre pole at twenty metres is about a hundred and ten pixels tall, and its lean read from the head’s place to half a pixel is good to about a third of a degree; the half-sum of two such readings is little better, and over 120 roadsides of eight poles each pole’s sideways tilt comes back to ±0.34°. That is too coarse to grade the quarter-degree leans the roadside’s timber poles have — a photograph grades its posts across the sight needed a building’s plumb corners and a sharper reading to do that.

The difference is what each pole is asked for. For the bend, every pole contributes one equation to two shared numbers, and eight or sixteen poles average their reading errors down. For its own tilt, each pole contributes only its own two readings. The pair reads the bend from the roadside, not the roadside’s poles; a survey of the poles themselves would follow each of them through many frames, as the earlier essays did.

Why the speed costs a factor of two #

The line’s slope is the speed and its crossing is the bend, and the two are read from points spread over a limited stretch of one over the range. Tilt the line about the middle of that stretch and both the slope and the crossing move: a fitted line whose points lie between one over 40 and one over 8 has its slope and its crossing correlated, because the crossing — at one over 50, beyond the far end of the points — is an extrapolation. A known speed fixes the slope, and the crossing is then read from the line’s height alone, with no extrapolation along a guessed slope. That is the factor of two in the poles figure, and it is why a roadside with poles beyond the bend’s radius would do better: points on the far side of the crossing would turn the extrapolation into an interpolation. On a 50-metre bend those are poles more than fifty metres out, which a 50° lens sees small and reads coarsely.

The crossing has a name in the earlier essays. A pole at the bend’s radius is not leaned by the shutter at all, because its image does not move across the picture: a camera on a bend is sharp on a circle, and the circle passes through the bend’s centre. The pair reads where that circle is from poles that are not on it.

What the pair asks of the sensor and the road #

The sensor reverses its readout between frames. Some sensors can be told which way to read their rows; many cannot. A sensor that reverses must also wait one exposure between frames — the row it read last is the row it reads first, and that row needs light — and the earlier essay measured what a gap costs a followed post. For a pair it costs only the drift of each pole across the picture during the gap.

The two readouts last equally long. The half-difference cancels each tilt because the top-down and bottom-up readouts lean a plumb pole by equal and opposite amounts. A sensor whose bottom-up readout is slower than its top-down one leaves part of the shutter’s lean in every tilt and reads the shutter’s lean itself short, and the earlier essay measured that mismatch for a followed post: the two readout times have to agree to about a per cent.

The poles are seen in both frames. Every pole here stays in view. A pole entering or leaving the frame between the pair, or hidden by a passing car in one of them, has one equation and contributes nothing.

The camera’s aim is known. Every pole’s shutter lean per unit of speed and curvature depends on where the pole stands in the picture and on which way the camera points relative to the bend, and the reading computes it for a camera aimed at the bend’s centre. A camera aimed partly along the road leans the poles by different amounts — a turning frame can be straightened; a travelling one cannot is the general reason the turn and the travel lean things differently — and a reading that assumed the wrong aim would put the difference into the speed and the radius. A car’s camera is bolted on, and its aim is a number measured once.

The bend is one circle. A road that tightens or eases under the car changes the radius between the frames; a fifteenth of a second at ten metres a second is under a metre of road, and any real road holds its curvature over that.

A frame is an interval, and two are a measurement #

A frame is an interval began this line of reading with the observation that a rolling-shutter picture records a stretch of time, and a rolling frame on a bend is right at its centre and a tracked post reads the bend, if the speed is known turned that stretch into a measurement of the bend. Every reading since has needed something else to be known: the speed, or the post’s plumbness, or a second pass, or many frames of one post. The pair needs nothing but the roadside and a sensor that can read its rows both ways, and in exchange for a fifteenth of a second it returns both numbers the bend’s geometry has — and every pole’s tilt besides, as half the sum of its two leans. A post read both ways sets its lean aside found the same separation with two cars; this is the same separation with one camera, made once rather than followed.

Still open: whether a few pairs read a bend that changes #

The pair reads one bend, one speed, at one moment. A real road’s curvature changes as the car drives it — a transition curve eases from straight into the bend, its radius falling steadily — and a car’s speed changes too. A drive of alternating frames is a sequence of pairs, each reading its own speed and radius from the poles then in view, a fifteenth of a second apart.

The measurement that settles what that is worth drives the camera along a transition curve whose radius falls at a stated rate, reads every consecutive pair of frames for its own speed and bend from the roadside then in view, and asks how well the sequence of pairs follows the curvature as it changes; whether pairs that share a frame — top-down then bottom-up, then bottom-up then top-down — read twice as many moments as pairs that do not, or merely the same moments twice; and whether fitting the curvature’s rate of change across several pairs reads it better than differencing the pairs’ own radii, since a rate read from noisy radii is the noisier quantity.