A followed post's own lean is read as the bend

A tracked post reads the bend, if the speed is known followed one roadside post through a rolling camera’s frames as a car rounded a 50-metre bend. A camera whose rows are read one after another leans every upright post in its picture by a·(1/Z − 1/R): a slope set by the car’s speed and the sensor’s readout, times the difference between one over the post’s range and one over the bend’s radius. A single lean fixes only the ratio of speed to radius, but a car knows its speed, and with it one post twelve metres out, followed for half a second, gave the bend back as 49.9 metres with quartiles from 48.1 to 52.9 — better than a roadside of posts in one frame.

Every post in that reading was exactly plumb. The essay ended on the trouble with that. A steel column is plumb to a fraction of a degree, a timber pole or a sign post often to a degree or more, and a followed post carries its own lean into every frame unchanged. The questions were how large a lean a followed post can carry before its bias exceeds the reading’s own scatter, and whether a camera that sweeps its posts — one aimed ahead, closing on them — could fit each post’s lean along with the radius and so read the bend as well as plumb posts did.

The first answer is a very small number. The second is no, for a reason that is a line of algebra rather than a measurement.

A post’s tilt is a constant in every frame #

The car and the camera are the earlier essay’s: a 50-metre bend at ten metres a second, a camera 1.5 metres up aimed at the bend’s centre, rows read over a thirtieth of a second, posts three metres tall. The post now leans: its head is displaced from the vertical above its foot by the tangent of its lean, sideways — square to the first frame’s line of sight — or along that line, towards the camera, fixed in the world as the car drives past. Each frame is read exactly, by turning the world back about the bend’s centre by the car’s own turn, which puts every frame’s view into the first frame’s camera; with no lean this reproduces the plumb post’s leans frame for frame to nine decimal places.

The dashed line is the plumb post’s lean, falling a little as the car passes it. The solid line is the same post leaning a tenth of a degree sideways: the same curve lifted by 1.75 thousandths in every frame, which is the tangent of a tenth of a degree. A post’s own tilt is in its picture whatever the rolling shutter does to it, and it adds to the shutter’s lean the same amount each time.

The radius read from the solid line, with the speed known, is 58.0 metres. Leaning three tenths of a degree the other way, the same post reads the bend as 35.3. The arithmetic is direct. The lean’s law is a·(1/Z − 1/R); a constant ε added to every frame is the same as moving 1/R by −ε/a, and the slope a here is about 0.62 metres, so a tenth of a degree, 0.00175, moves the curvature by 0.0028 per metre against the bend’s own 0.02 — a seventh of it.

Every frame sees the same tilt, so no frame removes it #

The earlier essay found that following a post through more frames narrowed its reading as the square root of their number, because each frame is a fresh reading of the post’s top with a fresh error. A post’s own lean is not fresh. It is the same in every frame, so averaging frames averages it to itself, and the bias it puts into the radius is the same after one frame or a hundred. What frames remove is the reading error; what is left is the lean.

So the useful question is where the two cross: how much lean a followed post can carry before its bias is as large as the scatter its reading leaves.

A post twelve metres out, followed through sixteen frames, reads the curvature with a scatter of 0.0018 per metre — the quartiles of 47.8 to 52.7 metres. A sideways lean of 0.064 degrees biases it by as much. A post eight metres out reads better, 48.4 to 51.5 metres, and so tolerates less: 0.037 degrees. A post forty metres out reads worse and tolerates 0.169. The lean adds the same amount to every post’s curvature, ε over the same slope a, whatever its range; what changes with range is how much the reading scatters, and a nearer post, which is the one the earlier essay found best to follow, is the one whose lean shows first.

The tolerance grows almost in proportion to the range, about half a hundredth of a degree for every metre, and the reason is the size of the post in the picture. A post three metres tall stands over a number of rows that falls as one over its range, so its top’s displacement, read to half a pixel, is a lean read to an error that grows with the range; the curvature it implies scatters in proportion, and the lean that matches the scatter grows with it. A near post is read finely, so a small lean is already a large share of what it says.

At the far end the argument turns over. A post near the bend’s centre, fifty metres out here, has a rolling lean of nearly nothing, because one over its range is nearly one over the radius — the property a rolling frame on a bend is right at its centre found. Its picture says almost nothing about the bend, and whatever lean it shows is very nearly all its own. A reader who followed such a post would be measuring the post.

A well-set steel column is plumb to about a twentieth of a degree, and the dashed line puts it there: already past the tolerance of a post eight metres out and close to that of a post twelve metres out. A timber pole at a quarter of a degree is four times past the twelve-metre post’s tolerance, and its bias is four times the scatter. A leaning post is worth its lean, not its length found leans of this size setting the price of every post in a horizon read from verticals; here they set it far higher, because a horizon read from verticals is a direction, read from the lines’ convergence, while the bend is read from one post’s lean against an expectation of nothing.

Which way a post leans #

A sideways lean tilts the post’s image by the lean itself, so a tenth of a degree sideways moves the bend by eight metres for a camera aimed at the bend’s centre and by eleven for one aimed ahead, where the slope a is smaller. A lean along the line of sight barely tilts the image of a post standing on the picture’s centre column, as the leaning-post essay found for the verticals of a still picture; it tilts it in proportion to how far the post stands off the centre column, and a followed post drifts across the frame, so it costs a little: 1.3 metres for the sideways camera, 1.8 for the forward one.

That gives a reader a choice to make before following anything. A post standing on the camera’s axis, seen nearly square, carries its sideways lean in full and its lean towards the camera hardly at all; but which way a given post leans is not known in advance, and roadside posts lean whichever way the ground and the traffic pushed them. The sideways part is the one that matters, and for a lean pointing anywhere at random it is on average about two-thirds of the whole.

Fitting each post’s lean loses the bend #

The earlier essay hoped that a camera which sweeps its posts could fit each post’s own lean along with the radius. The algebra says why not. With the speed known, each frame’s lean is a/Z − a·(1/R) + c: the term a/Z is known from the post’s range, so what the frames measure is −a·(1/R) + c, a constant times the curvature plus the post’s own constant. If a were the same in every frame, the two would be one number and no fit could separate them. a changes a little from frame to frame — it depends on the post’s bearing in the picture, which drifts as the car passes — and that small change is the only thing a fit has to work with.

With the leans ignored — every post read as if plumb — eight posts leaning a random quarter of a degree read the bend to ±7.0 metres, the posts’ own leans partly averaging out. With a lean fitted for each, the same posts read it to ±29 metres. The fit has spent the posts’ information on their own constants and kept almost none for the bend, because the bend’s curvature is, to within the small drift of a, one more constant shared by every post.

The forward camera does better at separating them: aimed forty-five degrees ahead, its posts’ bearings change more as the car closes on them, a varies more, and fitting the leans costs less — from ±9.5 metres to ±15 with eight posts, ±10 with two. But it never does better than ignoring the leans. A camera that closes on its posts changes a too little in half a second to tell a post’s tilt from the bend’s curve, and a fit that tries pays more in lost precision than it gains in removed bias.

So the earlier essay’s last hope does not survive. A forward-looking camera is the more robust instrument when the speed is unknown, because it can separate the speed from the radius; it is not more robust against leaning posts, because no aim separates a post’s tilt from the bend’s curvature.

Many leaning posts instead of one plumb one #

What does help is the property the leans have in common with the reading error: they are different from post to post. Each post’s own lean is a fresh error in the curvature, so posts read as if plumb average their leans down as frames average the reading down.

One timber post at a quarter of a degree reads the bend to ±12.4 metres, six times worse than a plumb post’s ±2.0. Thirty-two of them, at ranges from eight to forty metres, read it to ±3.9 — still twice the single plumb post. The medians stay near fifty metres, because the leans are as likely to go one way as the other and they do not shift the answer on average; they widen it. At a twentieth of a degree the price is small: one steel column reads ±2.7 metres, four read ±1.8.

The averaging is slower than the square root of the count would suggest, because the posts added later stand further out and each reads the curvature worse; a roadside of posts is not thirty-two copies of the best one. The practical rule is the one the leaning-post essay reached for a horizon, with a sharper edge. A followed reading of a bend wants one post known to be plumb — a building’s corner, a well-set lamp column checked against it — more than it wants many posts of unknown lean, and it should read a roadside of timber poles only in their dozens and only as an average.

The camera cannot tell the road from its furniture #

Put together, a post’s own lean is, to a rolling camera following it round a bend, part of the bend. It adds a constant to every frame’s lean, the constant moves the curvature by the lean over the slope of the lean’s law, and nothing in the frames separates the two: not the number of frames, not the camera’s aim, not a fit that tries. A followed post has to be plumb to a few hundredths of a degree for its lean to matter less than its reading — six hundredths at twelve metres, four at eight — which is plumber than most roadside posts are set.

A rolling frame on a bend is a coarse curvature gauge began this reading by noticing that a rolling shutter turns a car’s motion into a lean of every vertical, and every row is a different camera is why. What this adds is the limit of that gauge: it measures the lean of every vertical, and some of that lean was there before the car arrived. A camera on a bend is sharp on a circle found the circle on which a car’s motion vanishes from its picture, and the rolling leans read the same circle; a leaning post, followed alone, reports a circle of its own. A frame is an interval made the same point about blur — a picture records how the scene moved during the exposure, not how it was — and a post that was already tilted is a scene that looks, to a rolling frame, as if it had moved.

There is a way out, and it is not in the camera. A plumb reference in the picture — a building’s corner, whose verticality a facade’s corners see the horizon relied on — gives the camera a vertical it can trust, and its lean in the rolling frame is pure bend. One such corner followed through half a second is worth a roadside of poles.

What the reading assumes about posts and frames #

A post’s lean is fixed and its shaft straight. A post that sways in the wind adds a lean that changes from frame to frame, which averages like reading error if the sway is fast and like a fixed lean if it is slow. A bowed timber pole has a tilt that depends on how much of it is seen, and its top’s displacement reads a lean that is not the lean of its foot.

The car’s speed is known exactly. As the earlier essay found, an odometer’s error of a per cent moves the curvature by a per cent of the difference between the post’s reciprocal range and the bend’s — for the twelve-metre post about three per cent of the radius, comparable to a lean of a few hundredths of a degree.

Each post’s lean is independent of the others’. A row of posts set by one crew, or pushed by one wind, may lean together. A shared lean is a bias that averaging over posts cannot remove, and the leaning-post essay found the same for verticals read in a still picture.

The reading error is half a pixel and independent. A feature tracker that follows a post’s top carries correlated errors from frame to frame, which make the reading error behave more like a lean than the independent errors measured here.

Still open: whether a post’s lean can be read on the way back #

A post’s own lean and the bend’s curvature are one constant to a car going one way round the bend. A car coming the other way sees the same post leaning by the same amount, in the same direction in the world, while the rolling shutter’s lean, which depends on the car’s direction of travel and the side of the road, changes sign.

The measurement that settles what that is worth follows the same post from two passes — one each way round the bend, or a forward and a rear camera on the same car — and asks whether the sum and the difference of the two passes’ leans separate the post’s tilt from the bend’s curvature, how well each is then read for a post twelve metres out, and whether a camera that sees its posts from both directions can read the bend from timber poles as well as a one-way camera reads it from a plumb steel column.