A loop's far side is a length

Closing a loop mends its ends walked twenty-four cameras once round a ring of walls and found that recognising the twenty-two points seen at both ends makes the last camera six and a half times better placed relative to the first, and the worst camera — across the ring — only a fifth better. The closure is a short route added where the walk begins and ends, and it helps each camera in proportion to how much it shortens that camera’s way back to the start.

The obvious next move follows from that reading: put a short route somewhere else. A connection across the middle of the ring should give the far side the short route the closure gave the end.

It does not, or not in the form that suggests itself first, and the reason is visible as soon as the uncertainty is drawn as what it is. The earlier measurement reported each camera’s uncertainty as one number, the size of a sphere. A covariance is not a sphere. Drawn as the ellipse it actually is, the far side of the loop turns out to be uncertain almost entirely in one direction, and that direction decides which connections can reach it.

The same covariance, drawn as needles #

The walk is the earlier essay’s, unchanged: facades 13 m from the ring’s centre, give or take two, twenty-four cameras on a circle 8 m out looking straight out at the wall through a 100° field, points kept when three pictures see them, marks read to a whole pixel. Nothing is solved. Every figure is the linearised covariance at the true configuration, in the gauge that holds the first camera and one coordinate of the second camera’s position for the scale — the gauge an uncertainty is quoted from something found right for the one question asked here, which is how far each camera is from where the walk began.

The only change is in the drawing. Each camera’s plan covariance is drawn as its one-standard-deviation ellipse, twelve times actual size.

Every ellipse is a needle, and every needle points at the start. Camera 13, straight across the ring, is 496 mm from certain along the line from the start and 126 mm across it — four to one. The cameras either side of it are the same shape, turned to keep pointing at the first camera.

That shape contradicts the explanation the earlier essay gave for why the far side is worst. It proposed a lever: a small error in orientation early in the walk turns everything after it about the first camera, so the camera furthest away moves furthest. A rotation about the first camera moves a far camera across the line joining them, and the across component is the small one here. What dominates is the other direction — how far the far camera is from the start, not which way it lies. The walk knows the bearing of its far side reasonably well. It does not know the distance.

The needles do not come from a lever, then, but from a length carried round the walk. The walk’s size is set where the gauge sets it, by one short step at the start — the mechanism a chain and an adjustment singled out as where drift genuinely bites — and every picture after that inherits its scale from the pictures before it through the points they share. A scale chain leans rather than wanders measured how a chain carries scale along a street; the same carrying, done round a ring by an adjustment instead of a chain, leaves the far side’s distance from the start as the weakest thing the pictures determine.

Closing the loop shrinks the needles at the ends to nearly nothing and leaves the far ones long: camera 13 is now 411 mm along and 64 mm across. The closure halved the across component and took a sixth off the along one. It mended the direction a closure can mend and barely touched the one that matters most.

Three connections, three different quantities #

With the shape in hand, a connection across the ring is not one experiment but several, because a connection can measure different things. Three are measured, each chosen to be something a surveyor could actually do.

A picture across the ring. From the start camera’s own tripod, one more photograph turned round to look across the courtyard at the far wall through a 40° lens. It sees 32 marks on points the far cameras also see. The tripod has not moved, so its centre is the start camera’s centre to a millimetre, and that is entered as a measurement. What it measures is directions from the start to the far wall.

A length at the start. Two points about four metres apart on the wall the start camera faces, their separation measured to 5 mm. What it measures is a distance, but near the start.

A tape across the ring. The distance from the start camera’s tripod to the tripod straight across the ring, measured to 10 mm. What it measures is a distance, and across the ring.

The picture across the ring does almost nothing. Added to the open walk it moves the worst camera from 520 mm to 492; added to the closed loop, from 418 to 405. A photograph that sees the far wall from the start, with its centre pinned, is worth three per cent to the camera it was taken to help.

A single measured length does more than the whole closure did. The four-metre length at the start, on the open walk, takes the worst camera from 520 mm to 325 — further than recognising all twenty-two shared points took it. With the closure as well, to 112. The tape across the ring does slightly better on the open walk, 317, and with the closure brings the worst camera to 74 mm, a sixth of what the closure alone left.

Which component each one reaches #

The ranking is predicted by the needles, and the prediction can be checked camera by camera. Resolving camera 13’s uncertainty along and across the line from the start, with the loop closed in every case:

The picture across the ring acts where a picture can: the across component falls from 64 mm to 53. The along component, 411 mm, falls to 400. A photograph measures directions, and the far camera’s direction from the start was the thing already known.

The length at the start leaves the across component exactly where the closure put it, 64 mm, and takes the along component from 411 to 86. It says nothing about direction. It says how big the walk is, near where the walk’s size was set, and the pictures carry that size round the ring more firmly than they carried the single step the gauge holds.

The tape across the ring takes the along component to 10 mm — the tape’s own accuracy — and again leaves the across component at 64. Measured directly between the two cameras, the far side’s distance from the start is no longer something the walk has to carry at all.

So the three connections sort cleanly by what they measure, and the needles say in advance which will work: a direction reaches the short axis, a length reaches the long one, and the long one is where the uncertainty is.

The ring with its distance measured #

Drawing the closed loop again with the tape added shows what is left when the length is no longer the walk’s to carry.

The needles are gone, and more than gone: across the ring the ellipses have turned through a right angle. With the distance fixed, what remains of the far cameras’ uncertainty is the across component the closure left — 64 mm — so the long axis now lies across the line from the start, which is where the earlier essay’s lever would have put it. The lever was there all along. It was simply six times smaller than the length and invisible behind it.

That ordering is worth keeping, because it says which of the two stories to reach for first on any walk of this kind. A loop of pictures read to a pixel, round a courtyard twenty-six metres across, is uncertain first in size and only second in turn. The turn is what a closure and a picture across the ring can reach; the size is what they cannot. A measurement programme built on the second story — more pictures, better recognition, more careful overlap — works hard on the smaller error.

The worst camera with both the closure and the tape is 74 mm from certain, and the last camera, beside the start, 14. The far side is still the worst place on the walk, but by the across component now — by the turn the closure could reach and did not finish, rather than by the length it could not reach at all.

Why a picture across cannot reach the length #

It is fair to ask why a photograph of the far wall from the start carries no distance, since photographs of a scene from two places are exactly how this whole field measures distances. The track and the scene together recovers every point’s position from pictures alone.

The answer is in what the picture across is paired with. Its rays run from the start to the far wall, twenty-one metres, and the only other pictures of that wall are the far cameras’ own, which look at it from five metres in the opposite direction. A far wall point is therefore seen along nearly the same line from both sides, and two rays along one line fix a direction and not a depth along it. The spread a point gets measured exactly this: a point’s error follows the angle its rays subtend, and rays from the start and from the far camera subtend almost nothing at the far wall. The picture across is a very long baseline pointed the wrong way.

A pair of pictures whose rays cross at the far wall would do better, and a pair from tripods a quarter of the way round was tried for that reason. It gained less than the four-metre length did, because those tripods stand a quarter of the way round the walk, and the walk has already carried its uncertain scale to where they stand. A picture can only measure a length in units the walk has already carried to where the picture was taken.

Why a length anywhere helps the far side #

The length at the start is the result worth dwelling on, because it is not a connection across the ring at all. It is a measurement taken at the one place the walk was already well determined, and it cuts the far side’s distance from the start by nearly five times.

That is what a scale error looks like. A position error that grows with the walk because of accumulated turning has to be corrected where it accumulated. A position error that grows because the size of the whole reconstruction is uncertain can be corrected by fixing the size anywhere, since every length in the reconstruction moves together. The start was well determined relative to itself — its cameras are close to each other in the walk — and badly determined in size, because one short step fixed its scale and every mark near it was read to a pixel.

Seven numbers no picture can name established that no photograph can say how big a scene is: scale is one of the seven. A walk of pictures does not have one unknown scale so much as a scale set once and then passed hand to hand round the loop, and the needles are the record of how badly it was passed. A measured length re-states it in metres, and it re-states it for every camera at once.

This is also where the connection differs from control. The eighth held number bends the scene found that holding more than seven surveyed numbers makes claims the pictures can disagree with. A single length is one number and makes no such claim about shape: it fixes the scale the gauge fixed more weakly, and leaves the pictures to say everything else.

What this changes about a closure #

The earlier essay read the closure as a short route added at one point of the loop. That reading survives, and the needles sharpen it: a closure is a short route in direction and in position near the join, and a long route in scale everywhere else. Recognising that the last picture sees the first picture’s wall pins the last camera’s place beside the first. It does not tell the adjustment how far away the far wall is, because both routes to the far wall carry the same scale from the same short step.

Three practical statements follow.

A walk that must be accurate across its far side needs a length, not another picture. Where the adjustment stops found the floor a pixel’s reading puts under an adjustment’s fit; the needles are that floor’s shape on a loop, and a length is the only thing measured here that lowers it where it is deepest. Adding photographs — across the ring, from a second pass, from a longer lens — mends directions the walk already knew. Another picture of the same sweep found the same thing along a single arc: what a reconstruction lacks is rarely photographs.

The length does not have to be where the error is. A four-metre length at the start did most of what a tape across the ring did for the far side. What matters is that it is long compared with the step that set the walk’s scale, and measured better than the walk carried that scale.

And the closure is still worth making. With the length at the start, adding the closure takes the worst camera from 325 mm to 112, because the closure fixes the ends in direction and the length fixes the size, and the two are different deficiencies.

What these needles do not settle #

One walk, one gauge. The loop, the scene, the rule for keeping points and the seed are the earlier essay’s. The gauge holds the first camera and one coordinate of the second’s position; a gauge holding nothing would draw the needles about the walk’s centroid rather than about its start, and their proportions would change.

Linearised, and random only. The covariance is first-order at the truth and describes how reading error spreads. A scale that is systematically wrong — the lean the scale-chain essay measured — is not in a covariance, and whether a measured length corrects a lean as well as it corrects a spread is a question about a solve on biased data, and no solve was run.

The connections are ideal. The tripod is taken to return to a millimetre, the length and the tape to be measured between exactly the points the adjustment carries. A tape that is 10 mm out systematically would put its own error into every camera’s distance from the start, and the along component would stop at that error rather than at the tape’s precision.

The picture across the ring is one design. A picture taken from somewhere other than the start tripod, or a pair of pictures across the ring whose rays cross each other at a wide angle, measures more than a bearing, and could reach the along component by triangulation. Two pictures from tripods a quarter of the way round, aimed at the far wall, were tried beside these and gained less than a length did; their full range of placements is not measured.

Still open: how short a length still re-states the scale #

The four-metre length and the tape across the ring gave almost the same result on the open walk, which says the far side’s gain is not about the length’s position but about its size against the step that set the scale. That invites an exact question.

The measurement sweeps the measured length from a few centimetres to the diameter of the ring, at a fixed measuring accuracy, and records the along component of the far camera’s uncertainty at each; then sweeps the accuracy at a fixed length. If the along component depends only on the ratio of accuracy to length — on how many parts per thousand the scale is stated to — then a short length measured very well is as good as a long one measured roughly, and a walk’s scale can be fixed with a ruler at the start. If it depends on the length itself, because a short length is too local for the pictures to carry, then there is a shortest useful length, and it is a property of how far apart the walk’s own points are.