A post read both ways sets its own lean aside

A followed post’s own lean is read as the bend found the weak point of a rolling camera that reads a bend from its roadside posts. A camera whose rows are read one after another, carried at ten metres a second round a 50-metre bend and aimed at the bend’s centre, leans every upright post in its picture by a·(1/Z − 1/R): a slope a set by the speed and the sensor’s readout, times the difference between one over the post’s range Z and one over the bend’s radius R. Followed through sixteen frames with the speed known, a plumb post twelve metres out gives the radius back to a couple of metres. A post that leans by itself adds its tilt to every frame unchanged, and a constant added to the lean is a change of 1/R. A tenth of a degree moved the bend from 50 metres to 58. Fitting each post’s own lean as a free constant beside the bend did not help, because the tilt and the curvature were nearly one direction of the problem; the reading widened fourfold. A followed post had to be plumb to six hundredths of a degree, which is finer than a well-set steel column, and timber poles at a quarter of a degree helped only by the dozen.

The essay ended on a way round it. To a car going one way round the bend, the post’s tilt and the bend’s curvature are one constant. A car coming the other way sees the same post leaning by the same amount in the same direction, while the rolling shutter’s lean changes sign. The question was whether the sum and difference of the two passes’ leans separate the tilt from the curvature, how well each is read, and whether a camera that sees its posts from both directions reads the bend from timber poles as well as a one-way camera reads it from a plumb steel column.

They do, and it reads it better.

The shutter’s lean turns round and the post’s does not #

A rolling shutter leans an upright post because the post’s head and foot are drawn at different moments, which is the point every row is a different camera made of the shutter in general. The rows are read from top to bottom over a thirtieth of a second; in that time the image of a still point moves across the picture as the car travels and turns, and the head, read earlier, is drawn where the post was at the start of the readout while the foot is drawn where it had moved to by the end. The lean is the image’s speed across the rows times the time between the head’s row and the foot’s, divided by the post’s height in the picture. Its sign is the direction the image moves.

Drive the same piece of road the other way, and every still point’s image moves the other way across the picture. The camera stands at the same place and points the same way — at the bend’s centre — so the post is drawn at the same range and bearing and the same size, but the shutter now catches its head ahead of its foot instead of behind. The rolling lean has the same size and the opposite sign.

The post’s own tilt does none of that. A post leaning a quarter of a degree towards the north leans towards the north whichever way the car goes past it, and a camera at the same place pointing the same way draws that tilt identically on both trips. It is in the picture before any row is read, so the order in which the rows are read cannot touch it. A frame is an interval began from the observation that an exposure is a stretch of time rather than an instant; the post’s tilt is the one thing in this picture that the stretch of time has no part in.

The figure follows one post twelve metres out through sixteen places a thirtieth of a second’s drive apart, going round the bend and then coming back through the same places. Going round, the post’s lean in the picture runs from 43.41 thousandths down to 41.37 as the car closes on it; coming back, from −34.68 to −32.77. Neither curve looks like the plumb post’s 39.05 to 37.07, and the forward one read as if it were plumb gives the bend as 76.4 metres. But half their sum is 4.36 thousandths at every place — the tangent of a quarter of a degree, the post’s own tilt — and half their difference is 39.05 to 37.07, the plumb post’s lean exactly, from which the bend reads 50.03 metres. The slider turns the lean from a quarter of a degree one way to a quarter the other: the two passes’ curves move up and down together, the half-sum follows the tilt, and the half-difference does not move at all.

That is the whole of the method, and it is a line of arithmetic rather than a fit. It is not only the earlier essay’s free-lean fit with more data. One way, a free lean per post was a constant beside a·(1/Z − 1/R), and the a hardly changed from frame to frame, so the constant and the curvature traded almost freely. Both ways, the post’s constant enters both passes with the same sign and the curvature enters them with opposite signs, which makes them two directions of the problem rather than one.

A timber pole read both ways beats a plumb column read one way #

With real readings the question is a spread. Every post’s head is read to half a pixel in every frame, the speed is known, and each post leans by a stated amount in a random direction, so its tilt has some part sideways and some along the line of sight. A hundred and twenty such posts are followed, each by itself, and each reading of the bend is set beside the 50 metres it should be.

One way, a plumb post twelve metres out reads the bend to ±2.3 metres, half the middle spread of its readings. A steel column standing within a twentieth of a degree reads it to ±2.9, the column’s lean already showing. A timber pole at a quarter of a degree reads it to ±11.7: its lean is a bias several times the reading’s own scatter, different on every pole, and these are the numbers the earlier essay found.

Both ways, the plumb post reads ±1.6. That is the one-way plumb reading narrowed by the square root of two, because the half-difference averages thirty-two leans rather than sixteen. The steel column reads ±1.6 as well, and the timber pole ±1.7, and a pole leaning a full degree ±1.6. The tilt has been set aside before the bend is read, and what is left is the reading error alone, at twice the frames.

So the answer to the earlier essay’s last question is yes, with room to spare. A camera that sees its timber poles from both directions reads the bend better than a one-way camera reads it from the plumbest steel a road authority sets. The earlier essay’s requirement — plumb to six hundredths of a degree — was a requirement on one-way cameras, and it disappears. The medians say the same thing more quietly: 50.13 metres for the timber pole both ways, 50.20 for the pole at a degree, both within a tenth of the reading’s spread of the truth.

Every range gains the same #

The earlier essay found that a far post tolerates more lean than a near one, because a far post reads the bend worse in the first place and its lean is a smaller share of its error. The same comparison across range shows what both ways does there. One way and plumb, a post eight metres out reads ±1.5 metres and one forty metres out ±7.9: a far post is drawn smaller and leans less in the picture, and the same half-pixel of reading error is a larger share of a smaller lean. One way and leaning a quarter of a degree, every range reads about ±11 metres, the lean swamping the reading error near and far. Both ways and leaning a quarter of a degree, the readings run from ±1.1 to ±5.4, below the plumb one-way curve at every range by the same factor.

The gain is uniform because both of its causes are. Doubling the frames narrows every reading by the same square root of two, and removing the tilt removes a bias that does not depend on where the post stands. Nothing about the separation prefers near posts or far ones; the reading error still does, exactly as it did for a plumb post one way, and as it did for the eight posts of one frame in a rolling frame on a bend is a coarse curvature gauge, where the far posts decided the intercept and were read worst. A camera reading both ways should still follow its nearest posts, for the reason a tracked post reads the bend, if the speed is known gave: a near post’s lean is large and its reading error a small share of it.

The tilt comes out as well #

The half-sum is not thrown away. It is the post’s own tilt in the picture, frame by frame, and averaged over the sixteen places it reads how far the post leans.

Twelve to sixteen metres out, two passes read a post’s sideways lean to about three or four hundredths of a degree. That is a survey instrument’s figure: a road authority checking whether its poles still stand would want to know of any that has gone a quarter of a degree over, and these two passes see one that has gone a tenth. The reading worsens on both sides of that range, for two different reasons. A far post is drawn short, and its head’s half-pixel of error is a larger share of its height, so at forty metres the lean reads to ±0.078°. A near post sweeps across the picture — at eight metres it crosses 27 degrees of the view in half a second — and as soon as it stands off the picture’s centre column, a lean along the line of sight tilts its drawing sideways too: a quarter of a degree towards the camera reads as 0.065 degrees sideways at eight metres and 0.016 at forty. So at eight metres the lean reads to ±0.054°, worse than at twelve.

A photograph grades its posts across the sight read a street’s slope from the leans of its posts in a still picture, and found it could read only the part of a lean across the sight; a rolling camera driven both ways reads the same part, of every post it follows, as a by-product of reading the bend. A leaning post is worth its lean, not its length treated a street’s leaning posts as noise to be averaged away; here each one’s lean is measured and handed back.

The part of a tilt that does not cancel #

The separation has one leak, and it is worth knowing its size. A post leaning towards the camera has its head nearer than its foot. The rolling lean of a point depends on its range — a nearer point’s image moves faster across the picture — so a post whose head is nearer than its foot is drawn with a slightly larger rolling lean than a plumb post at the foot’s range. That extra part is still a rolling lean, so it changes sign with the direction of travel and survives the half-difference, where it is read as bend.

With exact leans, the figure measures it. A post leaning sideways moves the bend by 0.03 metres at a quarter of a degree and 0.22 at two degrees: about a ninth of a metre a degree, because a sideways lean moves the head across the sight and hardly changes its range. A post leaning towards the camera moves the bend by 0.22 metres at a quarter of a degree and 1.82 at two: about nine tenths of a metre a degree, eight times as much. The leak grows in proportion to the lean, and since a real post’s lean points any way, from one post to the next it is a spread rather than a bias. At a quarter of a degree it is a fifth of a metre against the reading’s own ±1.7, which is why it did not show in the trials. At two degrees towards the camera it is nearly two metres, which would — a post visibly leaning towards the road is the one to leave out.

Reading twice the road, one way and the other #

Twice now a second pass has been proposed as the way to separate two things a single pass confuses, and the two cases come out differently for a clear reason. One pass reads the road’s slope as well as its spring drove a sprung vehicle over the same rough road at two speeds, hoping the spring’s swelling, which moves with the speed, could be told from the road’s unevenness, which does not. It could not be told any better than one pass already told it, because the road’s spectrum and the spring’s swelling already had different shapes in one pass. Here the post’s tilt and the bend’s curvature have the same shape in one pass: each is a constant added to sixteen nearly equal leans, and nothing a single pass sees can part them. The second pass is worth its whole value only where the first could not tell the two apart at all. That is the general rule for a second look: it separates what has the same shape in the first and changes differently between them.

It also explains why the earlier essay’s free-lean fit could not help. A free constant beside a curvature that changes by a few per cent across sixteen frames is a constant beside a nearly constant; the fit had a shape to work with, but only the few per cent by which the bend’s term changed as the car closed on the post. Driving the other way changes the bend’s term by two hundred per cent.

What the two passes take from outside the pictures #

The return pass goes through the same places. The reading paired each forward frame with the return frame at the same place, so that half their difference is the rolling lean at one range and bearing. A car coming the other way is in the other lane, three and a half metres further from or nearer to the bend’s centre, and its frames see every post at slightly different ranges. Both passes then need their own a·(1/Z − 1/R), each at its own ranges, with one tilt per post shared between them; that is a small linear fit rather than a subtraction, and the separation should survive it, since the shared tilt and the opposite-signed curvature are still two directions. It was not measured.

The speed is known on both passes. The rolling lean is proportional to the speed, and the bend is read from the half-difference with the speed taken as known. A return pass at a different speed has a different a; the half-difference is then no longer the plumb lean, but a fit with each pass’s own known speed still separates the tilt, because the tilt does not scale with the speed.

The camera points at the bend’s centre both times. A camera fixed to the car looking out of one side faces the bend’s centre going round and away from it coming back, and sees the post from the other side of the road, mirror-wise; a car would need cameras on both sides, or one that turns. A rolling frame on a bend is right at its centre and the essays after it all aimed the camera at the centre, and a camera looking away from it reads posts on the outside of the bend, beyond R, where a·(1/Z − 1/R) changes sign; that reading is its own question.

The post stands still. A sign post that sways in the wind leans differently on the two passes, and a sway is read as bend in exactly the way a lean was, only it is not shared between the passes and so does not cancel.

Still open: whether a sensor that reverses its readout does it in one drive #

The two passes worked because the rolling lean has a sign set by the direction the image moves across the rows, and driving the other way reversed it. There is a second way to reverse it that needs no second drive. The lean is the image’s motion times the time between the head’s row and the foot’s, and that time has a sign too: rows read from the top down draw the head first, rows read from the bottom up draw the foot first. Some sensors can be told which way to read.

The measurement that settles what that is worth reads alternate frames top-down and bottom-up as the car rounds the bend once, follows each post through both kinds of frame, and asks whether the half-sum and half-difference of neighbouring frames separate the tilt from the bend as well as two passes did — at half the frames for each sign, but at places a thirtieth of a second apart rather than a lap apart — and whether the readings of the two kinds of frame, which are not at the same places, need the small fit the lane offset would need, or nothing at all.