A tracked post reads the bend, if the speed is known

A rolling frame on a bend is a coarse curvature gauge found that a camera whose rows are read one after another, on a car rounding a bend, leans every upright post in its picture by an amount that lies on a straight line in one over the post’s range — crossing zero exactly at the range of the bend’s centre. Eight posts in one frame, their tops read to half a pixel, gave a 50-metre bend back as 44.6 metres with quartiles from 35 to 74; averaged over sixteen frames, 50.9 (47–55). That averaging pretended the posts kept their ranges while the car drove past them.

The essay ended by proposing the honest version. A moving car sees each post at a different range in each frame, so a single post tracked through many frames samples the line over a range of distances — including, perhaps, the far part where the line’s intercept is decided. If one post seen at many ranges is worth as much as many posts seen at one, a camera needs only one lamp post it can follow round the bend.

Whether it does depends on where the camera is pointing, and on whether the car knows its own speed.

What a followed post does #

The car is the earlier essay’s: a 50-metre bend at ten metres a second, a camera 1.5 metres up whose rows are read over a thirtieth of a second — every row is a different camera, each a little further round the bend — posts three metres tall. Each frame is read about its own moment, a thirtieth of a second after the last, and a frame is an interval, not an instant, which is why a post leans at all. A post standing still beside the road is seen in frame k from wherever the car is then, at a range and bearing of its own, and by the bend’s symmetry its lean in that frame is exactly the single-frame law at that range and bearing, which a direct trace of the rolling image confirms to nine decimal places.

The earlier essay’s camera looked straight at the bend’s centre, sideways out of the car. The slider shows what that means for a followed post: the car moves across the post’s line of sight, not along it. In half a second the post twelve metres out drifts from 12.0 to 12.8 metres away and seventeen degrees across the picture, and its lean falls from 39.1 thousandths to 37.2, five per cent. The post at 25 metres changes by two per cent; the one at 60, near the bend’s centre, by nothing. Followed through sixteen frames, a post seen by that camera does not sweep the line in one over range. It sits on one point of it.

Turned forty-five degrees towards the direction of travel, the camera looks along the road as well as across it, and the car closes on its posts. The twelve-metre post comes in to 8.9 metres and its lean more than doubles, to 50.5 thousandths. That is the sweep the earlier essay expected, and it happens only for a camera that looks ahead.

Why one camera sweeps and the other does not #

The difference between the two cameras is the angle at which the car’s travel meets the line of sight, the same angle that turning and travelling blur different worlds found deciding which kind of motion a picture records. A post’s range changes at the car’s speed times the cosine of that angle. Aimed at the bend’s centre, the line of sight is square to the direction of travel, the cosine is zero, and the range changes only at second order — by the square of the distance travelled over twice the range, which for five metres of travel and a post twelve metres out is about a metre. Aimed forty-five degrees ahead, the cosine is 0.71 and the car closes on the post at seven metres a second; in half a second that is three and a half metres, and a post at twelve metres comes in to under nine.

The lean depends on the range through one over it, so the change that matters is in the reciprocal, and that is largest for the nearest posts: the twelve-metre post’s reciprocal range grows by more than a third as the forward camera closes on it, while the sixty-metre post’s grows by six per cent. Far posts barely sweep for either camera. What sweeps the line is a near post seen by a camera that is approaching it.

One lean is one ratio #

A post’s lean is a·(1/Z − 1/R): a slope a, set by the car’s speed and the rows’ readout, times the difference between the post’s reciprocal range and the bend’s. One lean is one number, and it fixes the ratio of the speed to the radius, not either alone. The earlier essay’s eight posts fitted both, from eight points at eight ranges; one post at one range cannot.

The two clouds make the point. With the camera aimed at the centre, the sixteen readings of one post are sixteen readings of one lean, and a faster car on a gentler bend gives the same lean as a slower one on a tighter bend; the fits spread along that trade, and the radius comes back as 64.8 metres with an upper quartile at infinity. With the camera aimed ahead, the post’s sixteen leans are sixteen points along a stretch of the line, and a stretch of a line has a slope and an intercept: the radius comes back as 47.5 metres, quartiles 40.4 to 62.2 — about what eight posts in one frame gave, from one post.

So the earlier essay’s question has two answers, by camera. A forward-looking camera can read a bend from one followed post, because the car closes on it. A sideways camera cannot, because it passes it.

What knowing the speed buys #

A car knows its speed: its wheels count it. Given the speed, the slope of the lean’s line is known, and one lean gives the radius directly.

With the speed known, one post twelve metres out, read in a single frame, gives the radius as 48.9 metres with quartiles from 42.5 to 59.2 — already about as well as eight posts in one frame with the speed known, which give 49.5 (44.6–57.3). Followed through sixteen frames it gives 49.9 (48.1–52.9), and through thirty-two, 49.9 (48.0–51.3). Each frame is a fresh reading of nearly the same lean, and averaging fresh readings narrows the answer as the square root of their number.

Where the camera points hardly matters here. Aimed ahead, the post’s lean sweeps, and with the speed known the sweep adds little the averaging did not: 50.0 (48.4–52.4) from sixteen frames. The sweep is what separates the speed from the radius; once the speed is known, there is nothing left for it to separate.

A roadside against one post #

The sideways camera’s six cases line up into one rule. Without the speed, a camera needs posts at several ranges, and following them through frames is close to what averaging them did in the earlier essay: eight posts through sixteen frames give 50.0 metres (46.5–54.2), against 50.9 (47–55) from the earlier essay’s averaging of the same posts as if they stood still. The posts nearly keep their ranges, so following them changes nothing the averaging missed. With the speed, a camera needs one near post; the rest of the roadside improves the answer only from 49.9 (48.1–52.9) to 49.6 (48.0–51.7).

That answers the earlier essay’s last question in the camera’s favour, for a car that knows its speed. One lamp post, followed for half a second, reads a 50-metre bend to about three metres — better than a roadside of posts in a single frame.

What the earlier averaging got away with #

The earlier essay’s averaging read each post’s lean in every frame and averaged as if the post had stayed at the range it had in the first. For a sideways camera that was nearly true, and the measurement here says how nearly: the nearest of its eight posts, at twelve metres, drifts out to 12.8 and leans five per cent less by the sixteenth frame, so its averaged lean is about two and a half per cent below the first frame’s, attributed to the first frame’s range. The other seven drift less. A lean read a little low for its range is a slightly larger radius, and the averaging’s 50.9 metres sits on that side of the truth, where following the posts at their own ranges gives 50.0 — consistent with the drift, though a difference of under a metre is inside both spreads and is not proof of it.

For a forward camera the same averaging would not have got away with it. Its twelve-metre post’s lean doubles in half a second, and averaging sixteen frames of it at the first frame’s range would put a lean half again too large on a range a sixth too long. Following the posts at their own ranges is not a refinement for a camera that looks ahead; it is the only reading that is right.

Which post to follow #

The nearest post is the one to follow. At eight metres it reads the radius to ±1.5 metres, at twelve to ±2.4, at twenty-five to ±5.2 and at seventy to ±17. A near post leans most — its reciprocal range is furthest from the bend’s — and it stands tallest in the picture, so its top is read over the most rows, and the reading error is the smallest share of its lean. A post near the bend’s centre, at fifty metres here, hardly leans at all and says almost nothing about the radius, which is the same reason a rolling frame on a bend is right at its centre: there the rolling frame and a global one agree.

Aimed ahead, every post does a little better than aimed at the centre — ±1.2 metres at eight, ±2.0 at twelve — because the car closes on it and its lean grows during the half-second. The gain is a fifth at most. For a car that knows its speed, the camera’s aim is a small matter and the post’s distance a large one.

As long as a post stays in view #

Sixteen frames is a convenient half-second, but a post can be followed only while it stays in the picture, and a near post leaves sooner. Aimed at the centre, with a picture fifty degrees wide, the post eight metres out stays in view for fourteen frames, the twelve-metre post for twenty-three and the twenty-five-metre post for eighty-six, nearly three seconds. Aimed forty-five degrees ahead, the times are a little shorter: thirteen, twenty-one and fifty-six frames.

Followed for as long as each stays in view, with the speed known, the three posts end up nearly level. The eight-metre post, over its fourteen frames, reads the radius as 50.0 metres with quartiles from 48.4 to 51.6; the twelve-metre post over twenty-three, 50.3 (48.0–52.2); the twenty-five-metre post over eighty-six, 50.0 (47.9–52.2). A near post is read best per frame and a far one is read for longer, and over a post’s whole passage across the picture the two roughly cancel. The practical rule is then not to wait for the ideal post but to follow whatever upright thing the picture holds for as long as it can be held.

A slower readout reads better #

The lean is made by the readout: the longer the sensor takes to read its rows, the further the car moves while it does, and the more each post leans. The reading error does not grow with it.

With the speed known, one post twelve metres out followed through sixteen frames reads the radius as 50.1 metres with quartiles from 43.3 to 64.0 when the frame is read in 8.3 milliseconds, (46.4–56.2) in 16.7, (48.1–52.9) in 33.3 and (49.0–51.4) in 66.7. The spread falls roughly in proportion to the readout time. The earlier essay found the same of eight posts in one frame: a slow sensor measures the road. A camera chosen to read bends from its own rolling shutter wants the slowest readout its pictures can tolerate, which is the opposite of what a camera is usually chosen for — and the opposite of what a turning frame can be straightened wanted, where the lean was an error to remove rather than a reading to keep.

What a car should do #

Put together, the measurements say how a car’s camera should read a bend. If the car knows its speed, which it does, the camera should pick the nearest upright thing it can see clearly — a post, a sign, a lamp standard — and follow it for as many frames as it stays in view; the radius comes out to a few per cent in half a second. If for some reason the speed is not trusted, a forward-looking camera can still read the bend from one post it is closing on, and a sideways camera needs posts at several ranges, as the earlier essay found.

A camera on a bend is sharp on a circle began this sequence by finding that a car’s motion blur vanishes on one circle; the earlier essay found that a rolling frame’s leans measure that circle coarsely. What following the posts adds is not more geometry but the car’s own knowledge of itself. A rolling frame measures the ratio of the car’s motion to the bend; the car supplies the motion, and the ratio becomes the bend.

What was assumed #

The posts are plumb and straight. As in the earlier essay, a lean is read off a post’s top against its foot. A post that leans by itself adds a constant to every frame’s lean, which following the post cannot average away; a near post that leans by a degree carries a bias of the same order as the lean being measured, and the reading would need posts known to be plumb or several posts whose own leans average out.

The car’s speed is known exactly. An odometer’s error of a per cent is a slope one per cent wrong, and since one over the radius is one over the post’s range less the lean over the slope, it moves one over the radius by a per cent of the difference between the two reciprocals — for the twelve-metre post, about three per cent of the radius, comparable to the spreads above. A followed near post needs the speed to a fraction of a per cent before its reading is limited by the posts rather than by the odometer.

The bend is a circle and the car follows it. A car that changes its line, or a bend that tightens, changes the slope and the intercept from frame to frame; half a second is short enough that neither matters much here, and long enough that a sharp change of steering would.

Each frame’s reading error is independent. The top of a post read in consecutive frames by the same feature tracker carries correlated errors; with correlation the averaging over frames gains less than the square root of their number.

Still open: what a post’s own lean costs a followed reading #

The reading here assumed every post exactly plumb. Real roadside posts are not: 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 measurement that settles what that costs gives each post a random lean of a stated size and asks two things. First, how large a lean a followed near post can carry before its bias exceeds the spread above — a single post with a lean is a constant error that no number of frames removes. Second, whether following several posts and fitting each one’s own lean along with the radius — possible only for a camera that sweeps them, since a sideways camera sees each post’s lean as one number — recovers the radius as well as plumb posts did, and so whether a forward-looking camera is the more robust instrument even for a car that knows its speed.