A growing car times the gap without its width

A car ahead costs a pair its speed put a stereo pair, two cameras thirty centimetres apart, on a car driving at fifteen metres a second, and asked how well its last second of pictures ranges the car in front, twenty metres ahead. Each old picture saw the car where it was at that moment, so the car’s speed had to be read with its distance, and reading it spent almost exactly what the old pictures were worth: a car keeping pace was ranged to 4.5 per cent of its distance for every pixel of reading error, the same as the still road. The speed itself came out to a metre and a half a second. And a single camera, which had a little depth from its drift across the lane when it looked at the road, lost nearly all of it when the thing ahead might be drifting too.

Every one of those readings treated the car ahead as a point. It is not one. It has a width, and the essay closed on what that width offers a single camera that the road did not. A car keeping pace keeps its size in the picture; a car being caught up grows. Its size gives its distance at once if its width is known, and its growth gives the time until the gap closes even if the width is not, because the distance and the rate at which it shrinks both scale with the same unknown width, and the time is their ratio. The question was how well that time is read from a second of growth, whether better than a pair reads it from disparity and speed, and what a pair gains by reading the growth as well.

Better, by about four and a half times. And a pair that reads both measures the car’s width, which turns the growth into a distance too.

The car’s width falls as one over the time to go #

The camera has a focal length of 598 pixels, and a car 1.8 metres wide twenty metres ahead is drawn 53.8 pixels wide: the focal length times the width over the distance. A second ago, if the gap is closing at five metres a second, the car was twenty-five metres off and drawn 43.0 pixels wide; two thirds of a second ago, 46.1; a third, 49.6. A car keeping pace is 53.8 pixels in every picture, and its picture says nothing about how far away it is unless somebody says how wide it is. The lamp’s size over its distance, and nothing else found the same of a lamp’s shadow: a size in a picture is a width over a distance, and scaling both together changes nothing.

The growth escapes that. One over the car’s drawn width is the distance over the focal length times the width, so it falls in a straight line as the gap closes, and the line reaches zero at the moment the gap would close. The time until it does is the line’s height now over its slope. The unknown width multiplies the height and the slope alike, so their ratio does not contain it. A narrow car and a wide one, at distances in proportion to their widths and closing at speeds in the same proportion, draw identical pictures and give the same answer, which is the right answer for both: the time to contact does not depend on which of them it is.

How well it is read is a matter of how well the edges are read. Each edge of the car is placed to a pixel in every picture, so its width is read to the square root of two of a pixel, and a straight line through thirty-one such readings over a second gives the time until the gap closes. The figure gives the error a pixel costs, as a share of the four seconds: 8.84 per cent from one camera that knows nothing of the car’s width. A pair thirty centimetres apart, reading the same time the way the earlier essay read the car — its distance from the disparity between its two pictures, its speed from how that distance changes, and the time as one over the other — reads it to 39.2 per cent. The slider changes the closing speed from two metres a second to fifteen; the growth steepens as the closing quickens, and the time is read better.

Over any length of look, the growth reads the time four and a half times better #

The time until the gap closes is a slope through readings, and a slope is read badly from a short look. From a tenth of a second of pictures the growth reads it to 146 per cent a pixel, which is no reading. From a quarter of a second, 45; from half, 20; from a second, 8.8; from two, 4.2. The error falls nearly as the look’s length to the power of one and a half, the rate the earlier essay found for the pair’s reading of speed: the look’s length counts once, and the number of pictures under it as its square root.

The pair’s reading of the same time falls at the same rate and stays above it by a nearly constant factor, from 4.3 at a tenth of a second to 4.5 at two. That is the ratio of two baselines. A pair reads distance from the disparity between two pictures thirty centimetres apart; the growth reads it from the two edges of a car 1.8 metres wide, a baseline six times longer, which happens to stand at the far end of the sight line rather than at the near one. Geometrically they are the same measurement, an angle subtended by a known or unknown length, and the longer length wins. It does not win the full factor of six, and the reason is a small advantage of the pair’s own. The car is straight ahead of one of its cameras, so the other sees it thirty centimetres to the side of its own line of travel, and with the car holding its lane that sideways place is fixed once for the whole look. Each picture from the second camera then reads the car’s distance from the size of a known offset to a full pixel, rather than from the difference of two readings to the square root of two of one. With the camera holding its lane the factor comes out at exactly six over the square root of two, 4.24; the drift across the lane adds the rest.

A pair that reads both its disparities and the car’s growth reads the time to 6.18 per cent from a second, about as much better than the growth alone as a second, independent reading of somewhat lower quality would make it. The disparity is not wasted; it is simply the weaker of two ways to the same number.

There is one place where this matters more than anywhere else. An epipole in the picture leaves a blind disc found that a single camera driving straight forwards learns nothing about the depth of whatever lies straight ahead, because every picture stands on the line to it. A car in front sits exactly there, and a camera holding its lane perfectly reads its distance to nothing at all. It reads the time until it reaches the car to 8.84 per cent a pixel all the same. The blind disc is blind to distance, and the growth never asks for one.

The faster the gap closes, the better its time is read #

The figure holds the car at twenty metres and the look at a second, and changes how fast the gap closes. At a metre a second, twenty seconds to go, the car grows by a twentieth over the second, two and a half pixels, and the growth reads the time to 34 per cent a pixel. At five metres a second, 8.8 per cent. At fifteen metres a second, which is a car standing still in the road ahead of one doing fifty-four kilometres an hour, the time is 1.3 seconds and the growth reads it to 4.85 per cent.

The share error falls because the growth grows. A time measured as a height over a slope is read best when the slope is steep, and a fast closing makes it steep. That is the convenient direction: the times that matter most, a second or two, are the ones read best. For a car two seconds away the growth reads the time to about six per cent a pixel and the pair’s disparity to twenty-six; for one twenty seconds away, which nobody need act on yet, 34 and 152.

The pair’s disparity follows the same curve at the same factor of four and a half. Both are reading a speed, the gap’s rate of closing, and a speed read from one second of pictures is a small change read against a large distance. The growth reads its small change against a larger baseline.

A car further off is timed much worse #

Distance costs the growth heavily. Ten metres off, closing at five metres a second, the car is two seconds away and the growth reads that time to 2.90 per cent a pixel. Twenty metres off, 8.84; forty, 30.2; sixty, 64.3. Every doubling of the distance multiplies the error by a little over three, and the reason is three halvings at once. A car twice as far is drawn half as wide; it grows, in pixels a second, a quarter as fast, because its growth is its width times the closing speed over its distance; and the time it is read against is twice as long. A share error of the time rises by the product of those, less the help of more of the car’s readings counting.

The pair’s disparity falls away at the same rate, from 12.8 per cent at ten metres to 286 at sixty, and the factor between them hardly moves. That is what the baseline argument predicts. A pair and a car’s width are both lengths seen from the camera, and their angular size falls with distance alike; it was far enough away, a pair is one eye that first found a pair’s depth giving out with distance faster than the distance grows, and the car’s width does the same, from a better start.

So the growth is a near instrument. At sixty metres a single camera cannot say whether a car ahead closing at five metres a second will be reached in twelve seconds or in twenty, and nothing short of a much longer look or a much finer edge reading will tell it. At ten to twenty metres, where a decision about braking is made, it is a good one.

The distance needs the width, and a pair reads the width #

The time until the gap closes is the growth’s whole gift. The distance is not in it: the growth fixes the distance over the width and the distance over the closing speed, and neither ratio is the distance. A single camera that is told the car’s width — its own, to the centimetre — reads the distance from the car’s size in every picture, to 1.02 per cent a pixel from a second, five times better than the pair’s disparity at 4.97. That is the same factor of baselines, applied this time to depth rather than to the time.

But no camera knows the width of the car in front of it. Cars run from about 1.6 to 2.0 metres, and a camera told only that a car is a car — 1.8 metres, give or take five per cent — carries the five per cent straight into the distance: 4.68 per cent, no better than the pair. A single camera that drifts across its lane, as the earlier essays let it, reads a little of the distance from the drift, and together with the growth reaches 11.0 per cent; the drift’s small baseline is all the absolute size it has.

A pair needs telling nothing. Its disparity gives the distance at every moment, coarsely; the growth ties every moment’s distance to the car’s width, finely; and between them the pair reads the width, to 2.95 per cent a pixel from a second. With that width the growth’s distance becomes available, and the pair ranges the car to 3.02 per cent instead of 4.97. Told the width outright, it would reach 0.72. Two views give shape and no size found that a reconstruction from pictures alone has no scale until one length in it is known; here the car’s width is that length, and the pair is the instrument that measures it, so that from the second second of looking onward the growth does the ranging and the disparity supervises it.

A braking car makes the time too long #

The time read from the growth assumes the gap has been closing steadily over the look. When the car in front brakes, it has not: the gap was closing more slowly a second ago than it is now. A reading that fits a steady closing to the whole look measures the closing speed averaged over it, which is slower than the present speed by about half the braking times the look, and divides the distance by that. It reads more time than there is.

That is the dangerous direction, and the figure gives its size. With the car ahead braking at three metres a second squared, firm braking for traffic, a quarter of a second’s look reads the time 7.94 per cent too long; half a second’s, 14.9; a second’s, 30.6; two seconds’, 67.4. At six metres a second squared, hard braking, the same looks read 16.0, 30.3, 64.4 and 166 per cent too long. Against that, the error a pixel costs falls from 44.9 per cent at a quarter of a second to 8.84 at one. The bias grows with the look and the error shrinks, and they cross at a little over half a second for firm braking and under half for hard.

The earlier essay found the same cap on the pair’s reading of distance, at about a second and a half, because the bias in a distance grows as the square of the look. The time is capped sooner, because its bias grows as the look itself: it is a speed’s bias, not a position’s. A camera that times the car ahead should look back about half a second, and accept the twenty per cent a pixel that buys, or read the braking as one more unknown. Either way the second second of growth, which narrowed the time so handsomely for a steady car, is worse than useless for one that may brake, since braking is exactly the moment the time is wanted.

What the growth adds to the pair’s history #

Put together, a single camera watching the car ahead grow reads the time until it reaches that car four and a half times better than a pair thirty centimetres wide reads it from disparity, at every length of look, every closing speed and every distance, and needs to know nothing about the car to do so. It reads it straight ahead, in the blind disc where it reads no distance at all. What it cannot read is the distance, which needs the car’s width; a pair reading both its disparities and the growth measures that width and ranges the car better than its disparities alone.

The image of the other eye began this sequence with the point in a picture where the other camera is, and every essay since has been about separations of known size: a pair’s baseline, a sway, a drift, the swing of a steered camera, a lever arm. The earlier essay found that the car ahead’s motion is a separation too, but not a known one, and so not a baseline. Its width is the reverse case: a separation of unknown size that never changes, carried along with the thing to be ranged. An unknown fixed length is a scale waiting to be set, and a time needs no scale at all, which is why the growth reads one perfectly well with the scale unset.

What the reading takes from outside the pictures #

The car’s back is a rigid width square to the road. The edges read are the car’s sides as seen from behind, and a car that turns, changes lanes or sits on a bend shows a width that changes for reasons other than distance. A car turning by a few degrees narrows its back by the cosine of the turn, which is small, but its side comes into view, and an edge finder that jumps from the back corner to the front one reads a sudden growth.

The edges are read to a pixel each, independently. A tracker that follows the car’s outline from picture to picture carries correlated errors, which make thirty-one readings worth fewer than thirty-one, and the slope through them — the time — feels that first.

The closing is steady, or the look is short. The braking figure measured the cost when it is not; a reading that fits the braking too would trade some of the error for none of the bias.

The camera’s own speed is not needed. It enters only through the pair’s reading of the gap’s closing; the growth reads the time between the two cars without it. That is a strength of the growth, and it is also why the growth alone cannot say how fast the car in front is going: only how soon it will be reached.

Still open: whether the road’s own marks scale the growth into a distance #

A single camera’s growth reads the time until the gap closes and nothing else, and a pair was needed to give that time a distance. There is a length on the road that a single camera can see and that is known to some precision without asking the car in front: the lane’s own width, or the spacing of the dashes painted between lanes, which a road authority fixes and repeats. A car ahead sits on that road, and its foot meets the road at a row in the picture that, with the camera’s height known, gives its distance on a flat road, resting on the camera’s height as a height from one photograph rests on the photographer’s.

The measurement that settles what that is worth reads, from one camera, the row where the car ahead meets a flat road and the car’s growth together, and asks how well the road’s geometry scales the growth into a distance and a width — whether that beats the pair, and how much a road that is not flat, rising or falling by a degree or two, takes back.