A turn moves the picture, not the blind point

A sway gives the blind centre a depth, not a good one took three pictures from a camera driving forward, half a metre apart, and displaced the middle one sideways. The mark straight ahead — the one two pictures in a line cannot place at all, because it does not move between them — got a depth at once, but a depth resting on the sway alone: a pixel of reading moves it by the depth over the focal length times the sway, 144 per cent at eight metres for a centimetre of wobble. The hole closed; the disc around it stayed until the sway was a third of the step.

A vehicle that changes lane does not only sway. It turns, briefly, and a camera fixed to it turns with it — which moves the picture’s epipole, the image of the other camera, by the focal length times the tangent of the turn, about ten pixels a degree on this camera, without adding any baseline at all. The earlier essay ended by asking two things: whether a turn alone changes the depth of the centre mark at all, since it moves where the epipole sits without making the rays cross; and whether a turn and a sway together could put the blind point somewhere harmless — off the frame, or onto the sky — which a steered camera could then arrange on purpose.

A turn changes nothing about what the pictures can say about depth. What it changes is which pixels say it. And no aim puts the blind point anywhere harmless, because the blind point is not a place in the picture.

A turned picture, the same disc #

The figure draws the earlier essay’s disc — the curve inside which a pixel of reading in every picture costs more than a tenth of the depth of a surface eight metres away — for three arrangements of three pictures.

Three pictures in a line leave a disc 92 pixels in radius, blind at its centre. Turn the middle picture three degrees about the vertical — a vehicle’s brief yaw — and the disc is the same disc: its edge moves by seventeen thousandths of a pixel at most, and its centre is still blind. Turn it three degrees and sway it five centimetres, and the disc is 83 pixels with a centre good to 29 per cent a pixel, which is what the five-centimetre sway gave with no turn at all.

The reason is short. A turn is a rotation about the camera’s own centre. It changes the direction every ray from that camera points, and so where every point lands in its picture; it does not change where the camera stands. The three cameras still stand on one line, the rays through the mark straight ahead from all three are still one line, and a single line of rays meets itself everywhere. The depth along it is undetermined, whatever the pictures’ orientations.

The earlier essay’s measurement put this in its title — the centre’s depth rests on the sway alone — and a turn is precisely a change to a picture that is not a sway.

At the direction of travel, blind at every turn #

The mark straight ahead is the one the vehicle is driving toward, and it is the one the earlier essay’s question was about.

With no sway it is blind at every turn from none to twelve degrees. With a sway, its depth cost is the sway’s number at every turn: 71.9, 28.8 and 14.4 per cent a pixel for sways of two, five and ten centimetres, changing by at most 3.3 per cent across turns up to twelve degrees. So the answer to the earlier essay’s first question is exact: a turn alone does not change the depth of the centre mark, which stays undetermined, and a turn with a sway gives the sway’s depth.

The 3.3 per cent that does change has a small, specific cause. A turned picture sees the mark nearer its own edge, and near the edge of a pinhole picture a pixel subtends a little less angle than at the centre — by the square of the cosine of the angle off the axis. A pixel of reading error there is a slightly smaller angular error, and the mark’s depth is slightly better fixed. At twelve degrees of turn that is three per cent. It is a statement about the pixels, not about the geometry, and it would vanish for a camera whose pixels were equal in angle.

Where, not which way #

The same holds across the whole picture, and the figure below measures it where the earlier essay’s disc lives — every mark, not just the centre.

Sixty-six marks spread across the first picture, eight metres away, with the middle picture swayed five centimetres. Turning that picture changes the largest of the sixty-six depth costs by 0.06 per cent at one degree, 0.45 at six and 2.69 at fifteen. The pattern of good and bad depth across the picture — the disc, its edge, the slide the forward step gives everywhere else — is set by where the three pictures were taken. Where they pointed decides only which of each picture’s pixels do the measuring, and that matters only through the small difference in angle between a pixel at a picture’s centre and one near its edge.

That is the whole of the relation between orientation and depth, and it is the same fact the image of the other eye rests on from the other side: the epipole is the image of the other camera’s centre, and the epipolar geometry that decides every depth is a matter of the two centres and the two pictures’ orientations together — but the precision along each ray is a matter of the angle between the rays, which only the centres set.

Aiming the camera moves the blind point across the picture #

The earlier essay’s second question was whether the blind point could be put somewhere harmless. The obvious lever is the camera’s aim: a camera mounted a little askew, or aimed into a bend, sees the direction of travel somewhere other than its centre.

Aim every picture twenty degrees off the road and the direction of travel falls 128 pixels from the picture’s left edge instead of at its centre: the focal length, 598 pixels, times the tangent of twenty degrees from the middle. Past thirty degrees — half the field of view — it leaves the picture altogether. The blind point has moved.

What it blinds has not. With the pictures in a line the direction of travel is blind at every aim: a pixel costs it more than a thousand times its depth, the arithmetic’s way of saying none. With a five-centimetre sway its depth costs between 29 and 25 per cent a pixel, whatever the aim. The blind point is not a place in the picture. It is a direction in the world — the direction the camera is travelling — and aiming the camera only chooses where in its picture that direction appears, or whether it appears at all.

So the arrangement the earlier essay hoped for — the blind point steered onto the sky, or off the frame — does not make anything visible that was invisible. On a level road the direction of travel is on the horizon, where the road ahead and whatever is on it are; aiming the camera off it moves that part of the scene out of the frame, blind spot and all. An epipole in the picture leaves a blind disc found the disc around the direction of travel; the disc goes where the direction goes.

What the two motions are #

The distinction between the two motions a vehicle makes between frames is worth stating as a rule, because the earlier essays have now measured both sides of it.

A sway moves where a picture is taken. It adds a baseline across the line of travel, and a baseline is what turns two rays into a triangle; the depth of every mark, including the one straight ahead, rests on the angle between rays, and a sway is the only one of the two motions that makes that angle. Depth is a reciprocal is the formula: a pixel costs the depth squared over the focal length times the baseline, and at the direction of travel the only baseline a forward-driving camera has is its sway.

A turn moves where a picture points. It adds no baseline, makes no angle between rays, and changes no depth — except through the second-order fact that pixels near a picture’s edge are finer in angle. It moves the epipole, the image of the other camera, across the turned picture by about ten pixels a degree on this camera, which is conspicuous and measures nothing.

A lane change is both, in a fixed proportion set by the steering: the vehicle turns to begin the change, sways as it travels at an angle, and turns back. What its camera can say about the road ahead during the change is set entirely by the sway. The track and the scene together solves for every camera’s position and orientation at once; its answer’s precision, at the direction of travel, comes from the positions alone.

A third picture is worth its position #

The result generalises an earlier one in a way worth making explicit. A third ray is worth what its picture is worth found that adding a third picture of a point helps in proportion to how differently it sees the point — arrangement beats count, two rays spread over fifty-five degrees beating eight inside four. The measure of “differently” there was the angle between rays, and the angle between rays is set by the cameras’ centres and the point, never by which way a camera faces. A third picture taken from where the second was, turned any amount, is worth nothing more than the second picture was; a third picture taken from somewhere else is worth what its position adds.

Whole pixels cut space into shells made the same point for a rectified pair, where a pixel of disparity is a shell of depth whose thickness is set by the baseline between the two centres; the turns that rectify the pair decide how the shells are counted in pixels, and the baseline decides how thick any shell can be. A turn of the vehicle is a turn of that second kind, and no turn is a baseline.

The sway’s own law, for comparison #

The turn has now been measured against the sway, and it is worth setting the sway’s own numbers beside the turn’s, since they are the numbers a turn cannot move.

At the centre, the cost is one constant times the depth over the sway — close to one over the focal length in pixels — at four, eight and sixteen metres alike. A turn does not appear in that expression at all, which is the whole of the result above written as a formula. Its only trace in the measurement is the small change in the constant as the turned picture sees the mark nearer its edge, where the focal length’s reciprocal is not quite the angle a pixel subtends.

The sway’s law also says what a vehicle would have to do to close the disc by steering. A tenth of the depth at eight metres needs a sway of fourteen centimetres between frames half a metre apart, which is a lateral velocity of more than a quarter of the forward one — a lane change, not a wobble. No amount of turning in place substitutes for it; a turn that is not accompanied by the sideways motion it would eventually produce buys nothing at all.

The disc shrinks with the sway and not with the turn #

The earlier essay’s second measurement, the disc’s radius against the sway, is the one the turn map above repeated at three degrees.

The disc shrinks from 92 pixels with no sway to 83 at five centimetres, 59 at ten, and closes at fifteen. Its radius is a statement about how fast depth precision improves away from the direction of travel, and away from that direction the forward step itself provides a slide that grows with the angle — so the disc’s edge is set by the step and the sway together, and the turn enters neither. The turn map’s rings for a turned and an unturned middle picture lie on each other to hundredths of a pixel because both are drawn by the same three camera centres.

Why the epipole’s motion is a distraction #

The epipole moving across a turned picture looks like evidence. In the turned picture, the image of the first camera is ten pixels a degree from where it was, and a matcher or a person looking for the blind spot finds it has moved. But the blind spot that matters is in the first picture — the one whose marks are being placed in depth — and in the first picture the epipole is the image of the other cameras’ centres, which a turn does not move.

This is the same trap two rays that do not meet guards against in a different form: an effect on the pictures is not an effect on the rays. Every quantity the depth depends on is a statement about rays — their origins and the angles between them — and a picture’s orientation is a choice of coordinates for its rays that leaves both unchanged. The epipole’s travel across the turned picture is a change of coordinates, visible and irrelevant.

What was assumed #

The turn is about the camera’s own centre. A camera mounted ahead of or behind the vehicle’s axis of yaw is carried sideways when the vehicle turns, by its lever arm times the turn. That is a sway, and it gives the centre a depth by exactly the sway’s rule: a camera a metre ahead of the axis, turned three degrees, sways five centimetres. A turn of the vehicle is a turn of the camera only for a camera on the axis.

Pixels are equal on the picture, not in angle. The small effects of a turn — 3.3 per cent at the centre, 2.7 across the picture — come from a pinhole picture’s pixels subtending less angle toward its edge. A camera whose pixels were equal in angle, an equidistant fisheye, would show none.

The surface is eight metres away. Every cost here is quoted at one depth; the earlier essay found that the centre’s cost grows in proportion to the depth, and nothing about a turn changes that.

Still open: whether the lever arm is a sway worth having #

The first assumption is where a turn stops being useless. A camera mounted a distance ahead of the vehicle’s yaw axis is swept sideways whenever the vehicle turns, and every small steering correction then buys the centre mark a baseline it would otherwise lack. The earlier essay found that a centimetre of sway buys a depth good only to 144 per cent at eight metres; a camera on the front of a long vehicle, two or three metres ahead of the rear axle, turns every degree of yaw into several centimetres of sway.

The measurement that settles what that is worth mounts the camera a stated distance ahead of the yaw axis, drives the vehicle straight with small random yaw corrections of a stated size between frames, and asks how the depth at the direction of travel, and the radius of the blind disc, depend on the mounting distance and the yaw’s size — and whether a camera placed at the front of a vehicle, rather than behind its windscreen near the driver, sees the road ahead in depth materially better for the same steering. If a metre of lever arm and ordinary steering noise close the disc at the ranges a vehicle cares about, the answer to where a forward camera should go is a geometric one; if not, the blind point ahead remains something no mounting can remove.