A railing matched one along stands underground

A third eye that lands on the next post matched one post of a railing to its neighbour and found the pair reporting it at 19.8 metres instead of 9.0, every test two photographs can run sitting at the arithmetic floor. A third picture exposed it by hundreds of pixels almost everywhere — and at five azimuths confirmed it instead, because the wrong point landed on another post of the same railing.

That was one post. A matcher confused by one post of a regular railing is confused by all of them, and the natural failure is not a single wrong match but a whole row of them, each post taken to its neighbour. The earlier essay closed on exactly that case, and on the claim that no test on the railing alone can see it, since a railing at nineteen metres is a perfectly good railing. What would see it is the rest of the scene. The question it left was how far from the railing a unique feature has to be before the inconsistency it creates shows.

The answer turns out not to be a distance from the railing at all. A feature near the railing, however near, says nothing. A feature the railing touches says everything — and what decides how well it says it is how far that feature is from the cameras.

A railing that is wrong only about where it is #

The plan below is the same pair as before: two cameras 1.1 metres apart, nine hundred pixels of focal length, a railing of thirteen posts sixty centimetres apart at nine metres, its posts’ tops at the cameras’ height. Every post in the left picture is matched to the next post along in the right one.

The pair builds twelve posts at 19.80 metres, 1.32 metres apart, each lying on both of its rays to $8\times10^{-14}$ metres. It is a railing: straight, evenly spaced, perpendicular to the line of sight. It is the true railing scaled 2.2 times about the left eye, which is what it must be. Each wrong post lies on its true post’s ray from the left camera, since the left mark never moved, so the wrong railing is the true one seen from that eye and pushed back until its disparity is fifty pixels instead of a hundred and ten. Every length along it grows by the ratio of the depths — the move the one thing a single view cannot give describes, a world made larger and further away by the same factor, applied here to one object instead of the whole scene.

That is the sense in which nothing about it is wrong. A mismatch on its own line needs a third eye established that a match slid along its own epipolar line leaves every two-view test at the floor, and a railing slides every post along its line at once. A wrong match is not a small error found that one bad correspondence among good ones spreads its error across the whole fit; here there are no good ones among the posts to spread it across. The railing’s matches agree with each other perfectly, and they agree about a railing that is not there.

The ground is not part of the mistake #

The posts do not float. They stand on the ground, and a matcher that takes each post’s top to its neighbour’s takes each post’s foot to its neighbour’s foot as well, since the feet repeat exactly as the tops do.

Seen from the side, the consequence is plain. The ray from the left eye through a post’s foot descends from 1.1 metres to the ground at nine metres, and the wrong reconstruction carries it on to 19.8 metres, where it has fallen 1.32 metres further. The wrong post is 2.42 metres tall and stands with its foot 1.32 metres below the ground the true post stands on. Its top is still at eye height, because the ray through the top is level and scaling along it changes nothing.

Everything in the railing scales by 2.2 about the eye, and the ground does not, because the ground is not part of what was mismatched. That is the whole of the inconsistency: not a distance between two objects, but a relation — stands on — that the true railing satisfies and the scaled one cannot. A lamp hanging a few centimetres from a post, or a bird perched in front of it, is reconstructed correctly at nine metres and says nothing against a railing at nineteen, because nothing ties the two together. A stone on the ground a hundred metres off says a great deal, because the ground it lies on is the ground the posts stand on.

How far away a ground mark can be #

To use the ground the pair has to know where it is, and it knows only what its own marks tell it. The reading below takes unique marks on the ground — stones, cracks, anything that does not repeat — reads them in both pictures with a pixel of error, and reconstructs a ground from them. Each foot’s left mark is then carried along its ray to that ground and into the right picture, and the foot is convicted if the mark the matcher paired it with lies more than half a period, thirty pixels, from where the ground puts it.

With exact marks the test is exact: every shifted foot lands exactly one period — sixty pixels, the posts’ spacing as the right camera draws it — from where the ground puts it, and every honest foot lands exactly on it. With marks read to a pixel, the question is how long the ground’s own error stays below half that.

If the ground may be taken as level — a courtyard, a pavement, a floor — one mark fixes it, since its height is the only unknown. One mark read to a pixel convicts every foot in every one of a hundred pictures out to forty metres, 99 in a hundred at sixty, 82 at a hundred and twenty. Three marks carry it past a hundred and twenty metres. The railing is at nine metres. The marks that expose it can be thirteen times further away than it is.

If the ground may not be assumed level, it has to be fitted as a free plane, and then it needs marks spread across the view to fix its tilt. Three marks spread six metres across a ring convict every foot out to twenty metres and fall to three pictures in four at forty, and eight marks on the same ring do little better, because what fails is the tilt, and marks close together far away say little about it. The level assumption is worth a factor of several in reach — an assumption imported from outside the pictures, which is the currency every recovery in this subject is bought with.

Nearness to the railing is the wrong distance #

The figure carries a vertical rule at the railing’s depth, and the curves pass through it without noticing. What limits the ground test is not how far the marks are from the railing but how far they are from the cameras.

The reason is the one whole pixels cut space into shells gave for every depth a pair reports: a pixel of disparity is worth a length that grows as the square of the distance. Depth is a reciprocal puts it as an interval that is not centred on the answer. A mark at sixty metres is placed about forty times less precisely than one at nine, and its height — the ray to it descends only 1.1 metres over sixty — inherits that error at the slope of the ray. The height error grows in proportion to the mark’s distance, and carried back along a foot’s ray to nine metres it moves the foot’s predicted position in the right picture by a proportional number of pixels. When that reaches half a period, the ground stops separating a shifted foot from an honest one.

So a stone lying right beside a post at nine metres is a better witness than one at sixty, but only because it is nearer the cameras; a stone at four metres, in front of the railing, is better still. The question the earlier essay asked — how far from the railing — has a real answer, but it is in the wrong coordinate. The railing is convicted by the ground, and the ground is read best where the pair reads everything best: close to the cameras.

The railing’s length enters only through its ends #

The earlier essay wondered whether a long railing, dragged further to meet the rest of the scene, would be easier to catch than a short one. For the ground it makes no difference.

A ground read from marks carries one error, and that error is shared by every foot. Twelve feet do not give twelve independent chances to see the railing sunk; they give one chance, twelve times. One mark at forty metres convicts every foot in every picture whether the railing has five posts or forty-one. The railing’s length adds nothing to the ground’s testimony, because the ground has only one thing to say and says it about every post alike.

Length does enter through the ends. A railing short enough for both its ends to lie inside both pictures is a railing whose shifted reading leaves a post without a partner: the end post of the left picture would have to be matched to a post that does not exist, or to whatever unique thing stands beyond the railing’s end, which is not a post and does not look like one. Counting the posts each reading pairs, the honest reading wins — five against four, nine against eight.

Once the railing runs out of both frames the count reverses, and it reverses in the wrong direction. The right camera’s view is the left camera’s moved 1.1 metres along the railing, which is 1.83 periods; a reading that matches every post one along covers more of the right picture’s posts than the honest reading does. From eleven posts on, the shifted reading pairs ten posts and the honest one nine. Matching two along would pair eleven, but puts every post behind the cameras — a disparity of minus ten pixels — and cheirality refuses it. Among the readings that put the railing in front of the cameras, a matcher that trusts the number of matched posts prefers the wrong railing whenever the railing is longer than the frame.

A third picture has to confirm all of it #

The earlier essay’s third camera, swept round the middle post, confirmed a single mismatched post at five azimuths. A whole shifted railing asks more of a coincidence.

At the five azimuths where one post was confirmed, the other wrong posts do not follow: two of eight, one of seven, none of six land on a post. The whole railing lands entirely at one place on the sweep, 10.5 degrees, and that place is not a coincidence of the same kind. The sweep’s circle there passes within fifteen centimetres of the line through the two cameras, two baselines from the left eye.

A shifted railing is one object, and a third picture that sees it has to agree with all of it at once. For a single post, a third view is blind wherever one number — the exposure over the period — happens to be whole. For a whole railing it is blind only where that holds for every post simultaneously, and for a picture drawn through a centre that happens nowhere with any extent, except along one line.

The line where every third eye is blind #

That line is worth drawing on its own, because it is the one place where adding a picture adds nothing.

A camera on the line through the two cameras, at two baselines from the left eye, sees every disparity doubled. The true railing’s disparity becomes 220 pixels and the wrong railing’s 100, and the difference, 120 pixels, is exactly two periods: the wrong railing lands on the true one slid two posts along, to $5\times10^{-12}$ pixels. At three baselines it is three periods. Between the whole numbers the wrong posts fall between the true ones, up to 34 pixels away, and the mistake is plain.

The reason it holds only on this line is a fact about pencils of rays in the plane. The true railing and the wrong one both lie in the plane of the eyes; the left eye’s rays carry one onto the other post for post, and the right eye’s rays carry each wrong post onto the next true one. A projection from a centre that carries one row of posts onto another, post for post, is fixed by where it sends three of them, and the centres that do it for a shift of a whole number of posts are exactly the points of the cameras’ line at whole numbers of baselines. A third eye placed there is a repetition of the pair, not a new witness — the same shape of failure as an epipole in the picture leaves a blind disc, where a second picture taken along the first one’s line adds nothing near its own epipole. A third eye placed anywhere else — above the line, in front of it, round to the side — sees the whole railing disagree.

What convicts a repeated structure #

The pieces now fit into one statement. A structure matched one period out everywhere at once reconstructs as the true structure scaled about the first eye. Nothing it contains can contradict that, because scaling a structure about an eye keeps every relation inside the structure. What contradicts it is a relation between the structure and something that was not scaled: the ground it stands on, the wall it is fixed to, the end where it meets something that does not repeat.

Nearness is not such a relation. Two rays that do not meet is the reminder that the pair’s evidence about a point is its two rays and nothing else, and a unique object near a post has rays of its own, which it satisfies at its own correct depth whatever the post is doing. Only a physical tie — stands on, is fixed to, ends at — carries information from the correctly reconstructed scene into the railing, and the ground is the tie a railing almost always has. Where the ground is visible, the railing cannot be shifted whole without sinking into it.

That turns the practical advice round. A third eye that lands on the next post found that the structure hardest to match is the structure a third picture is least likely to rescue, and suggested a constraint along the structure instead. For a whole shifted railing the better constraint is across it: check the feet against the ground. One stone read to a pixel does it, from sixty metres, for every post at once.

What was assumed #

The railing stands on visible ground. A railing along the top of a wall, a balcony edge over a drop, a row of windows in a facade: each repeats with nothing visible beneath it, and each has only its ends and its fixings to meet the rest of the scene. For those, the count of matched posts is the only witness, and it favours the wrong reading once the structure is longer than the frame.

The railing runs parallel to the cameras’ baseline, or meets its line. Only then does a row of posts lie in one plane with the two cameras, so that every post’s image falls on one epipolar line and a match one post along is consistent in both pictures. The tops here are at the cameras’ height for convenience; any row parallel to the baseline shares a plane with it, and the argument holds at any height. A railing turned away from the baseline at another height puts its posts on different epipolar lines, and the pair catches the shift without help.

The matcher shifts everything by one post. A matcher that shifts part of the railing and not the rest leaves a break in the middle — a post matched twice or not at all — which is its own witness, and a different failure.

Marks are read with independent errors of a pixel. A ground read from marks that are themselves on a repeated pattern — paving slabs, a tiled floor — is as exposed to the same mistake as the railing, and may be shifted with it.

Still open: whether a facade shifted whole is held by its corners #

The first assumption is the one a real scene most often breaks, and the case it leaves is the commonest repeated structure in a photograph: a facade of identical windows, matched one column out. The facade’s windows reconstruct as a facade 2.2 times further away and 2.2 times wider, with nothing beneath it to sink into. What it does have is corners — the building’s edges, where the wall turns and the windows stop, which are unique and are matched correctly.

The measurement that settles what a corner is worth builds a facade of windows in a wall whose two corners are visible, shifts the windows one column, and asks two things. First, whether the wall between the windows and the corners — the plain stretch of masonry at each end, which carries no marks and so is not matched at all — is enough of a tie: the windows then sit in a plane nineteen metres away inside a wall whose edges are at nine, and the question is whether anything in the two pictures says the windows and the wall must be one surface. Second, whether a single mark on that plain stretch — a drainpipe, a sign, a crack — is enough to pin the plane of the wall, and how far from the corner it can be before the windows’ plane, fitted from the corner and the mark, stops separating the shifted reading from the honest one.