Frames read up and down part a post's lean in one drive

A post read both ways sets its own lean aside found a way past the weak point of a rolling camera that reads a bend from its roadside posts. The camera rides at ten metres a second round a 50-metre bend, aimed at the bend’s centre, and reads its rows one after another over a thirtieth of a second, so every upright post in its picture leans by an amount set by the speed, the post’s range and the bend’s radius. A post that leans by itself adds its tilt to that, and one way round the bend the tilt and the curvature are indistinguishable: a timber pole standing a quarter of a degree off plumb reads the bend to ±11.7 metres. A second car driving the same road the other way sees the same tilt and the opposite shutter lean, and half the sum and half the difference of the two passes part them. A timber pole read both ways gave the bend to ±1.7 metres, better than a plumb steel column read one way.

That essay closed on a second way to reverse the shutter’s lean, one that needs no second car. The lean is the image’s motion across the picture times the time between reading the post’s head and reading its foot, and that time has a sign as well as the motion does. 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 question was whether a camera reading alternate frames top-down and bottom-up, as it rounds the bend once, separates the tilt from the bend as two passes did, and what that one drive costs and gains against the two.

It separates them, as well as two passes do for the same number of frames, and it is more robust in one respect that two passes cannot match.

Reading the rows the other way turns the shutter’s lean round #

A rolling shutter draws each row at its own moment. Every row is a different camera made that the starting point: a picture read row by row is a stack of thin pictures taken by a camera that kept moving while it took them. For an upright post the consequence is a lean. If the post’s image is sliding sideways across the picture, the row holding its head and the row holding its foot catch the slide at different points, and the post is drawn slanted by the slide’s speed times the interval between those two rows.

Driving the other way reverses the slide. Reading the rows the other way reverses the interval: bottom-up, the foot’s row comes first and the head’s after it, so a slide that drew the head behind the foot now draws it ahead. The size is the same, because the interval between the two rows is the same thirtieth of a second’s share and the slide is the same speed, and the sign is opposite. The post’s own tilt is untouched by either reversal. It is a property of where the post’s head stands in the world, drawn before the question of when each row is read arises at all.

The figure follows one post twelve metres out through sixteen frames, a thirtieth of a second apart, with the even frames read from the top down and the odd ones from the bottom up. The top-down frames lean the post by 43.41 thousandths at the first frame and 42.11 by the twelfth; the bottom-up frames by −34.67 and −33.22. Each kind on its own is the curve one pass of the earlier essay drew going round or coming back. Together they zigzag about a line that is neither: the post’s own tilt as drawn, 4.36 thousandths at the first frame and 4.21 by the last.

That last line deserves a word, because it is not quite the post’s tilt. A post leaning a quarter of a degree square to the first frame’s line of sight is drawn leaning by the tangent of a quarter of a degree, 4.36 thousandths, only while it stands on the picture’s centre column. As the car rounds the bend the post drifts seventeen degrees across the picture, and seen obliquely its lean is foreshortened, so the drawn tilt shrinks. The tilt a single fit reads, 0.2469 degrees, is the drawn tilt’s average over the frames. It is the earlier essay’s half-sum in another form, and that essay’s figure showed the same shrinking at its last place, 4.30.

The reading itself is a small linear fit rather than a subtraction. Each frame’s lean is the rolling factor for that frame’s own readout direction times one over the post’s range less one over the radius, plus a single tilt shared by all the frames. The rolling factor alternates in sign from frame to frame and the tilt does not, so the two unknowns are different directions of the problem from the first pair of frames onward. With exact leans the fit gives the bend as 50.05 metres. The same sixteen frames all read top-down, the post taken as plumb, give 76.4 metres, which is the one-way failure the earlier essays measured, and with the tilt left free instead they cannot be read at all: the curvature’s standard error grows ninety-six times, to seven times the curvature itself. The slider turns the lean through a quarter of a degree either way: the zigzag rides up and down with the tilt, and the bend read stays within a twentieth of a metre of 50.

Thirty-two frames of one drive match two passes of sixteen #

With real readings the question is a spread. Each 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, partly sideways and partly along the line of sight. A hundred and twenty posts are followed, each on its own, and the bend each reads is set beside the 50 metres it should be.

Sixteen alternate frames read a plumb post to ±2.6 metres, half the middle spread of the readings, and a timber pole at a quarter of a degree to ±2.8. That is a little worse than a plumb post read one way, ±2.3, because the tilt is now an unknown fitted beside the bend rather than assumed to be zero. Fitting it costs some precision even when the post happens to be plumb: half the frames now carry each sign, and the fit spends part of what they hold on the tilt. But a little worse than a perfectly plumb post is four times better than the same timber pole read one way, at ±11.7, and the timber pole is the post the roadside actually has.

The fair comparison with two passes is at equal frames. Two passes of sixteen frames each read the timber pole to ±1.7 metres; one drive of thirty-two alternate frames reads it to ±1.8. A pole leaning a full degree, one drive of thirty-two, reads ±2.2, the extra from the part of a lean that points along the line of sight, which neither method cancels and which the earlier essay measured at nine tenths of a metre for a degree. So a sensor that reverses its readout is worth, for this purpose, a second car driving the road the other way, provided it takes as many frames in one drive as the two cars took between them.

Those thirty-two frames last a little over a second, against half a second for each pass, and that is the only reason the one drive reads a few per cent worse. Over the longer drive the post closes on the camera and drifts further across the picture, so the later frames see it obliquely and read its lean less cleanly. The separation itself is no weaker. The medians of the one drive’s readings, 49.64 to 49.70 metres, sit a third of a metre short, under a fifth of the reading’s spread.

Every range keeps the same order #

The earlier essay found that two passes improve on one way by the same factor at every range, because both causes of the gain — twice the frames and the tilt removed — are indifferent to where the post stands. One drive of alternate frames inherits that. At eight metres it reads the timber pole to ±1.3 metres against two passes’ ±1.1; at forty, ±6.0 against ±5.4; at every range between, within a sixth of the two passes and well below the one-way reading of a plumb post, which runs from ±1.5 at eight metres to ±7.9 at forty.

The reading error still prefers near posts, for the reason a tracked post reads the bend, if the speed is known gave: a near post’s lean is large and the half-pixel of error in its head is a small share of it, while a far post is drawn short and leans little. Nothing about reversing the readout changes that. What changes is that a post’s plumbness no longer enters the choice. One way, a camera had to follow posts it could trust to stand straight, and a followed post’s lean is read as the bend found that meant plumb to six hundredths of a degree, which no timber pole is. With the readout reversing, the camera follows its nearest posts whatever they are made of.

A sway is shared by frames a thirtieth of a second apart, not a lap apart #

The earlier essay listed, among the things its two passes take from outside the pictures, that the post stands still. A sign post in the wind sways, and two passes see it at different moments of its sway. The tilt is then not shared between the passes, and the part that differs is read as bend exactly as a lean was read one way. Alternate frames of one drive compare tilts a thirtieth of a second apart, and a post sways far more slowly than that.

The figure sways the post by a tenth of a degree about its quarter-degree lean, at periods from a fifth of a second to four seconds, with eight starting points in the sway at each period and the second pass meeting the sway at a different point from the first. Two passes are moved by 2.2 metres at a one-second sway, 3.3 at two seconds and 3.6 at four. A slow sway is almost constant through each half-second pass, so within a pass it behaves like a lean, and the difference between the passes’ leans goes straight into the bend. A tenth of a degree of difference, one way, is worth about eight metres; here it is diluted by averaging over where in the sway each pass begins, but it stays several metres. A fast sway averages out within each pass and costs less, a few tenths of a metre.

One drive with a single tilt is moved by between 0.08 and 0.31 metres across the same range of periods. A post’s sway changes its tilt only a little between neighbouring frames, and since neighbouring frames carry opposite shutter leans, a slow change in the tilt is nearly orthogonal to the alternating pattern the fit reads the bend from. Nearly, not wholly: within each pair, the tilt has moved by a small amount, and half of that amount enters the half-difference with the same sign for as long as the sway is moving one way. Pairing only neighbouring frames does not help, because it is exactly the within-pair change that leaks. Allowing the tilt to drift in a straight line within each group of four frames does help: the leak falls below three hundredths of a metre for any sway slower than a fifth of a second, at a cost of between a sixth and a third in the reading’s spread, since the fit now spends more of what the frames hold on the tilt.

There is a general shape here, and it is worth naming. The alternating readout imposes a pattern on the shutter’s lean at the highest rate the frames allow, flipping every frame, and everything the fit must reject — the post’s tilt, its sway, the tilt’s apparent shrinking as the post drifts off the centre column — changes slowly. The bend is read from the fast pattern and the rest is left in the slow one. Two passes impose the same flip once, between two runs that may be minutes apart, and anything that changes in that time falls on the wrong side of it. The rule the earlier essay drew from one pass reads the road’s slope as well as its spring — a second look separates what has the same shape in the first and changes differently between them — holds here with a refinement: the second look separates best when it comes soonest, because then the only thing that has changed between the looks is the one the reversal changed.

A sensor that reverses must leave the row it read last some light #

Reversing the readout has a price that one-way reading does not pay. A row is exposed from the moment it is reset until the moment it is read, and in an ordinary rolling shutter the reset sweeps down the picture one exposure ahead of the read, so each row’s light is gathered while the previous frame is still being read further down. A frame read top-down ends on the bottom row, and the next frame, read bottom-up, begins on it. That row has just been read; it needs a full exposure of fresh light before it can be read again. So consecutive frames must be an exposure apart at the turn, and the sensor has two choices: take its frames that much less often, or shorten its readout by the exposure so as to keep the frame rate.

The figure measures both. Taking the frames less often costs nothing: the bend reads ±1.8 metres with no gap and ±1.85 with a sixtieth of a second between frames, because the shutter’s lean is set by the readout time, which has not changed, and thirty-two frames merely take half a second longer to collect. Shortening the readout costs a great deal. The rolling lean is proportional to the readout time, so a readout cut from a thirtieth to a sixtieth of a second leans every post half as much, and the same half-pixel of reading error weighs twice as heavily against it: the bend reads ±3.8 metres. At the short exposures a bright road allows, a thousandth or a two-hundred-and-fiftieth of a second, either choice is nearly free. In dim light it matters, and the sensor should give up frame rate rather than readout time.

That is a reversal of the usual engineering preference. A rolling shutter’s readout is normally made as short as the hardware allows, because the lean it draws is a defect, and a turning frame can be straightened is about undoing it after the fact. A frame is an interval argued that an exposure is a stretch of time to be read rather than an instant to be approximated, and here the readout is the signal itself: the longer it is, the more every post leans, and the better the bend is read.

Both readout directions must take the same time #

Two passes with one camera have one property the alternating sensor has to earn: both passes read their rows the same way, at the same rate, so the shutter’s lean in one is exactly the negative of the other’s at the same place. A sensor that reverses its readout relies on reading upward exactly as fast as downward. If the bottom-up readout runs slower by some fraction, its frames lean every post more than the fit expects, the two directions’ leans no longer cancel in the shared tilt, and the excess divides between the tilt and the bend.

The figure gives the size. With the readouts equal the bend reads 50.05 metres. A mismatch of one part in a thousand moves it to 50.13; three parts in a thousand, to 50.28; one per cent, to 50.83; three per cent, to 52.46, more than the reading’s whole spread. The tilt read shrinks with it, from 0.247 degrees to 0.214. A sensor’s row timing is set by a clock rather than by anything mechanical, and the two directions are normally driven by the same line period, so a mismatch of a per cent would be a design fault rather than a tolerance. But it is a fault the method cannot see from its own frames, since a mismatch and a curvature both scale the alternating pattern, and it has to be known from the sensor rather than read from the road.

What one drive takes from outside the pictures #

The speed is known. As in the earlier essays, the shutter’s lean is proportional to the speed and the bend is read with the speed taken from elsewhere. The alternating frames leave the speed and the curvature as entangled as they were for a plumb post one way, and a tracked post reads the bend, if the speed is known found that following one post cannot untangle them.

The post’s lean along the line of sight still leaks. A post leaning towards the camera has its head nearer than its foot, and the shutter leans a nearer point more; that extra part flips sign with the readout just as it did with the direction of travel, so it survives into the bend. It is the same nine tenths of a metre a degree, and it is why the pole leaning a full degree read ±2.2 rather than ±1.8.

The tilt read is the tilt as drawn. The fit returns the post’s tilt averaged over the frames as it appears in the picture, and that is the true sideways lean only for a post near the picture’s centre column. Over a long drive the post drifts and the drawn tilt shrinks, which is harmless to the bend and a small bias in the tilt; a survey of the roadside’s plumbness from one drive would correct each frame for the post’s bearing.

The camera points at the bend’s centre. Every essay on this bend aimed the camera there, from a rolling frame on a bend is right at its centre onward, and the alternating frames were measured only that way. Unlike the two passes, though, one drive needs no camera on the far side of the car, since nothing about the reversal turns the car round.

Still open: whether one pair of frames reads the speed and the bend from the whole roadside #

Everything above followed a single post through many frames, because one way round the bend that was how the speed and the curvature were read: a rolling frame on a bend is a coarse curvature gauge had found that the posts of a single frame, read at once, give the bend only coarsely, and their own leans were a large part of why. In one frame, the leans of posts at different ranges lie on a straight line in one over the range, whose slope is set by the speed and whose crossing by the radius, and every post’s tilt scatters it.

Two neighbouring frames read in opposite directions remove every post’s tilt at once, a fifteenth of a second apart. The measurement that settles what that is worth takes a row of timber poles at eight to forty metres, reads one top-down frame and the next bottom-up, fits the line through the half-differences, and asks whether that one pair reads the speed and the bend together from the roadside as well as a followed plumb post reads the bend alone with the speed given — and how many poles in the pair it takes.