A car ahead costs a pair its speed, and a car that may drift costs a single camera everything

A stereo pair and a lane wander add as two readings of one depth put two cameras thirty centimetres apart across a car driving at fifteen metres a second and asked how well they range the road twenty metres straight ahead. From one moment’s two pictures, to 15.8 per cent of the depth for every pixel of reading error. From the last second’s sixty pictures, to 4.6, because every moment is another reading; and with the car’s ordinary drift of twenty centimetres either side of its lane, to 4.5, exactly what the pair’s look and the drift would give as two independent readings. A pair, it turned out, is already a moving pair, and its own history was worth more than the wander.

Every one of those readings was of a point on the road, which holds still. The essay ended on the thing a car most needs to range, which does not: the car in front. An old picture of a moving car is a picture of where it was, and the question was how many of the pair’s past pictures still help when each has to be read against the car’s place at its own moment — whether a pair that also reads the car’s speed from its pictures keeps most of its gain, how that falls as the gap closes faster, and whether the drift across the lane helps more or less with a moving target than with the road, since a moving target changes its own bearing and might lend the pictures a baseline of its own.

The pair keeps its gain, and pays for it with the car’s speed, almost exactly. The drift helps less, not more. And a moving target lends nothing, because its motion is one more thing to be read.

The car stood twenty metres from every picture #

The car and its pair are the earlier essay’s: fifteen metres a second, thirty frames a second, the lane wander of twenty centimetres either side every four seconds. Twenty metres straight down the road, where the earlier essay ranged the road surface, there is now another vehicle. Every picture in the last second saw it where it was at that picture’s moment, and the reading places it now by fitting all of them together, with as much of its motion as is not known fitted beside its position. The figure gives the depth error a pixel of reading in every picture costs, as a share of the twenty metres, for looks from a tenth of a second to two.

The dashed line is the still road, the earlier essay’s reading: 4.53 per cent from a second, 4.15 from two. The dotted line is a vehicle keeping pace, whose speed is known — told by the vehicle itself over a radio link, say. It is ranged far better than the road: 2.61 per cent from a second, 1.73 from two. The reason is the one the earlier essay found limiting its own gain. A picture taken a second ago stood fifteen metres further back, and the road twenty metres ahead was thirty-five metres from it; a pair’s reading of depth costs the square of the distance, as depth is a reciprocal found for every pair, so that old picture, whose reading’s variance grows as the fourth power of its distance, was worth about a tenth of a new one. A car keeping pace was twenty metres from every picture. Its whole second of pictures counts at full weight.

The solid line is the case a car actually has: the vehicle ahead keeping pace, but nobody has said so, and its speed along the road is read from the same pictures. It costs 4.49 per cent from a second — the still road’s figure to within a twentieth of itself. Reading the speed spends almost exactly what the constant distance gained. Over shorter looks the read speed costs more than the road: 6.06 per cent from half a second against 5.24. Over longer looks it costs less: 3.29 from two seconds against 4.15, because the road’s oldest pictures stand fifty metres from it while the car’s stand twenty.

The fourth line lets the vehicle drift across the lane at some steady speed too, which a car about to change lanes does. It costs 5.35 per cent from a second, a fifth more than the speed alone.

The slider changes how fast the gap closes. A car pulling away was nearer to the old pictures than to the new, and is ranged better; a car being caught up was further, and is ranged worse.

The faster the gap closes, the further back the old pictures saw #

The closing speed sets where the old pictures saw the car. A car pulling away at ten metres a second was ten metres nearer a second ago; a car standing still on the road was fifteen metres further, as the road itself was. With the speed known, a car pulling away at ten is ranged to 1.19 per cent from a second, one keeping pace to 2.61, a stopped one to 4.53 — the still road, as it must be.

With the speed read, the line is flatter: 3.22 per cent pulling away, 4.49 keeping pace, 5.77 standing still. Reading the speed costs most where the known-speed reading was best, nearly three times the error for a car pulling away and only a quarter again for one standing still. That is because the old pictures do both jobs. They place the car, and they are the only record of how its distance has changed, which is what a speed is. A car pulling away is well ranged by its old pictures because they saw it near; reading its speed from them asks the same pictures for something else — how the distance changed between them — and a depth that has to serve as one end of a slope cannot also be averaged with its neighbours as if they were readings of one number. The two uses compete, and the competition is fiercest where the pictures are best.

A stopped car standing in the road is the still road read by someone who has not been told it is still. It costs 5.77 per cent against the road’s 4.53, a quarter more: the price of not knowing that the thing ahead is parked.

The speed is read as well, in a second #

The speed comes out of the reading as well as costing it. From a tenth of a second of pictures, a pair thirty centimetres apart reads the car’s speed to thirty metres a second a pixel, which is no reading at all. From a second, to 1.4 metres a second; from two, to half a metre a second. The error falls nearly as the look’s length to the power of one and a half, the rate at which a straight line fitted through more and more points over a longer and longer stretch narrows its slope: the stretch counts once, and the number of points under it counts as its square root. A wider pair reads it better in proportion to its width — 2.8 metres a second from a second’s look for a pair twelve centimetres apart, 0.69 for one sixty centimetres apart — because each moment’s depth is read in proportion to the width, and the speed is a slope through those depths.

A metre and a half a second is not fine. It is the difference between a car keeping pace and one losing a car-length every three seconds, and a pixel of matching error is a modest one; a matcher that reads to a fifth of a pixel reads the speed to three tenths of a metre a second from the same second. But it is enough to say whether the gap is closing quickly, which is the question a car following another asks most often, and it comes from the same pictures that range the car rather than from any extra instrument.

A target that may drift takes a single camera’s wander away #

The earlier essay’s last question supposed that a moving target, by changing its own bearing, might lend the pictures a baseline of its own. It does not, and the reason is short. A baseline is a known separation between the places pictures were taken from. A target’s motion is not known; it has to be read, and every part of it that has to be read is a direction in which the pictures can explain what they see without the depth.

The figure measures that for three rigs. One camera alone, wandering, ranges the still road to 22.8 per cent a pixel from a second — the drift essay’s figure. A car ahead with its speed read costs it 38.0, because one camera’s depth comes from how the thing ahead shifts across the picture as the wander carries the camera sideways, and the car’s own change of distance has to be read from the same small shifts. Let the car drift across the lane at some steady speed as well, and one camera’s reading falls apart: 184.5 per cent. Its whole depth came from a sideways movement of the camera, and a sideways movement of the target, which nobody has measured, explains almost all of it.

A pair loses far less, because each moment’s two pictures range the car whatever it is doing; no motion of the target touches the disparity between two pictures taken at the same instant. A pair twelve centimetres apart goes from 10.5 per cent to 12.0 when the car may drift, one thirty centimetres apart from 4.49 to 5.35. What the pair loses is the pairings between moments — the left camera now against the right camera a moment ago — which the earlier essay found adding to the pair when the wander was large. Those pairings are where a target’s drift and the camera’s drift can be confused, and with the drift free they stop counting.

That leaves the wander itself with less to do. For the still road, a pair thirty centimetres apart went from 4.61 per cent with no wander to 3.89 with a wander of half a metre either side. For a car ahead with its speed read, the same pair goes from 4.40 to 4.20: the wander’s sideways movement buys a twentieth, where it bought a sixth. A narrow pair, twelve centimetres apart, gains more for the road — 11.5 to 5.5 — and less for the car: 11.0 to 7.4. The drift was a bonus for the still road and is a smaller one for the moving car, the reverse of what the question expected.

A braking car caps the look #

The readings above assumed the car ahead keeps a steady speed over the look. A car ahead that brakes does not, and a reading that assumes it does is wrong by more the further back it looks. A car braking at three metres a second squared — firm braking for traffic, not an emergency — was three metres a second faster a second ago, and the pictures taken then saw it at places a steady car would not have been. The steady-speed reading fits them anyway and places the car wrongly.

The figure sets the bias against the reading’s own error. At half a second’s look the bias is 0.31 per cent of the depth, six centimetres, and the error a pixel costs is 6.06 per cent: the braking is invisible. At a second the bias is 1.39 per cent, twenty-eight centimetres, against 4.49. At a second and a half they are equal, 3.60 against 3.76. At two seconds the bias is 8.63 per cent, nearly two metres, two and a half times the reading error. The bias grows as about the square of the look, because a braking car’s departure from a steady one grows as the square of the time; the reading error falls only as about its square root. So there is a longest useful look, and it is about a second and a half for firm braking and a second for hard braking at six metres a second squared, where the bias at two seconds is a third of the depth.

A car can do better by reading the braking too, as one more unknown beside the speed — which would cost the reading what the speed did, and more — or by holding a look of about a second, which is where the pair’s gain had mostly arrived anyway: two seconds took the still road only from 4.53 per cent to 4.15, the arithmetic by which far enough away, a pair is one eye. The earlier essay found the second second of pictures nearly worthless for the road. For a car that may brake, it is worse than worthless.

What the pair’s history is worth for a moving car #

Put together, a stereo pair’s past pictures are worth nearly as much for the car ahead as for the road, provided the car’s speed is read with them. The gain the earlier essay found — from 15.8 per cent a pixel at one moment to 4.5 from a second — survives for a car keeping pace, because a car keeping pace was at the same distance from every picture, and the speed it costs to read is paid from that. It shrinks for a car being caught up and grows for one pulling away. It shrinks by a fifth more if the car may be drifting across the lane. It must stop at about a second if the car may be braking. And the lane wander, which helped the still road, helps the moving car little.

The image of the other eye began this sequence with two pictures of a still courtyard and the point in each that is the other camera. The blind point of a car driving forwards, which an epipole in the picture leaves a blind disc found, is blind because every later picture stands on the same line as the first, and every essay since has been about breaking that line with a sideways separation of known size: a sway, a turn, a lever arm, a drift, a bracket. A moving target turns the question inside out. Its motion is a separation too, but not a known one, and an unknown separation is not a baseline; it is one more thing for the pictures to explain. What the pictures can still do is read it — the speed to a metre and a half a second from a second’s look — and that reading is paid for out of the depth.

A single camera, which steering swings a camera too little to see ahead and the drift essay tried to give a baseline, comes out of this worst. For the road its drift gave it 23 per cent a pixel; for a car whose motion is not known, 38; for one that may drift, nothing usable. A car that wants to range the car in front needs a bracket, because a bracket’s baseline is the one separation in the problem that nothing the other car does can mimic.

What the reading takes from outside the pictures #

The car’s own motion is known. As in the earlier essay, every picture is placed where the camera was, from the car’s odometry and steering. A moving target makes that knowledge more important, not less: an error in the car’s own speed reads as an error in the other car’s speed, and the two are confused exactly as the target’s drift and the camera’s drift were.

The car ahead is a point. A vehicle is a body three or four metres long with a back that faces the pair. Its depth is read from many marks on that back, every one moving with it, which helps as more pixels help; it also turns when it changes lanes, and a turning back is not a point moving across the road.

The speed and the drift are steady over the look. The braking figure measured what a steady-speed reading costs when that fails along the road; nothing here measured a drift that starts mid-look, which is what a lane change is.

Every reading has a pixel of independent error. A matcher that follows the car ahead from frame to frame carries correlated errors, which make sixty pictures worth fewer than sixty readings, and which the speed reading — a slope through correlated depths — would feel first.

Still open: whether the car ahead’s own size reads its distance #

A pair ranges the car in front from the disparity between its two pictures. A single camera has something the road did not offer: the car ahead has a size, and a car keeping pace keeps it, while a car being caught up grows in the picture. Its size gives its distance directly if the size is known, and its growth gives the time until the gap closes — the distance over the closing speed — even if the size is not, because both the distance and its rate of change scale with the same unknown size.

The measurement that settles what that is worth reads a single camera’s pictures of a car ahead of stated width, closing at stated speeds, and asks how well the time until the gap closes is read from the car’s growth over a second, whether it is read better than a pair reads distance over closing speed from its disparities, and how much a pair gains by using the car’s growth as well as its disparity — since the growth needs no baseline at all, and the blind point straight ahead, where the earlier essays found no depth, is exactly where a car ahead sits.