The spread a point gets

Another picture of the same sweep finds that going from three views to seven across the same sixty degrees leaves a reconstruction where it started, and at one point makes it worse. Its conclusion is that what a reconstruction is short of is angular spread rather than photographs.

That conclusion names a quantity — the spread — and quietly attributes it to the track. The track is not what has it.

Sixty degrees, and nobody gets sixty degrees #

The sequence is six stations along an arc of sixty degrees about a courtyard. Sixty degrees is a property of the track: it is the angle between the first station and the last, measured at the arc’s own centre.

No point of the scene stands at the arc’s centre, and the angle that matters to a point is the angle between the rays to it from the two stations furthest apart as seen from that point. Computed for every point of the courtyard, those angles run from 41 degrees to 89 — a factor of 2.2 across one scene, photographed by one track, in one sweep.

The near points get more than the track’s own sixty because they stand inside the arc’s radius, where the same chord subtends a wider angle. The far points get less. Neither is sixty.

The error follows the angle at the point #

If the per-point angle is the operative quantity, the errors should follow it and not the track’s number.

Every point of the courtyard was triangulated from the same six stations, with every mark read to half a pixel and fifty noise draws at each point. The cameras are held exact, so what is measured is the point’s own conditioning rather than the whole reconstruction’s — which is the right isolation for a claim about what a point gets out of a track.

The errors run from 1.5 millimetres to 8.4, a factor of 5.6. Against the per-point subtended angle they fall as a power law of exponent −1.68, with 94 per cent of the variation explained.

Against the track’s own sixty degrees there is nothing to plot. One number cannot have a slope against forty-four.

Where the exponent comes from, and what it is mixed with #

An exponent of −1.68 is worth taking apart rather than quoting, because it is not the exponent a simple argument predicts.

The uncertainty of a triangulated point across the line of sight is about the reading error times the range over the focal length, and along the line of sight it is that divided by the sine of the crossing angle. So the dominant error should go as range over sine — one power of the angle, not one and two thirds.

The extra comes from the range. For a fixed track the subtended angle at a point is about the chord divided by the range, so the two are not independent: a point that receives a small angle is also a far point, and its error carries both the sine in the denominator and the range in the numerator. Fitted against the range instead, the same errors give an exponent of +1.76 with 96 per cent explained — the two columns are nearly interchangeable predictors on one track, because on one track they are two readings of one thing.

That is the honest statement of the result, and it is stronger rather than weaker than the simple one. The subtended angle is the way the range enters, and reporting spread as a property of the track hides both.

The same quantity under three other names #

The angle at the point is not a new object. It is the quantity three other measurements in this collection are already about, and setting them together says how general it is.

For two views it is the crossing angle of the two rays, which two rays that do not meet measures the miss of and a third ray is worth what its picture is worth finds decides everything about how a bundle should be weighed: two rays spread over fifty-five degrees beat eight crowded into four, by a factor of 4.4.

For a stereo pair held at a fixed baseline it is the disparity, which is the same angle read as a difference of image coordinates; depth is a reciprocal is the account of what a fixed error in reading it does to the answer, and the reciprocal in its title is the same one over the range that appears here.

And for a mirror pair it is the angle at which the two rays cross, which square to the camera is the worst mirror finds runs from 23 degrees to 65 as the mirror turns — a factor of three in the crossing angle for a factor of three in the depth error, which is the same relation this essay fits an exponent to.

Four fields, four vocabularies, one angle. What this adds is that with more than two stations the angle is a property of the pair of a track and a point, and there is no way to report it as a property of either alone.

Reading a planned sequence before taking it #

The practical value of the per-point angle is that it can be computed in advance, and the arithmetic is a sentence.

Take the two stations of a planned track that are furthest apart, measure the straight-line distance between them — the chord — and divide by the distance to the object. That is the subtended angle in radians, near enough for anything under about forty degrees, and it is the number the conditioning depends on.

A room photographed from a two-metre sweep at three metres gives two thirds of a radian, which is 38 degrees. The same sweep for a facade at thirty metres gives 0.067 radians, 3.8 degrees, and by the power law measured here the error is some thirty times worse. Nothing about the sweep has changed and nothing about the camera has; the facade is simply further away.

That also says what a second object in the same picture is worth. A near foreground object and a far background one in one sequence are conditioned by the same chord and different ranges, so their errors differ by the ratio of their ranges to a power near one and two thirds — which is why a reconstruction of a room with a window in it is good about the room and poor about what is outside the window, in a way no amount of care about the sequence repairs.

What this changes about planning a sequence #

Three practical statements, and the first is the one that changes where a photographer stands.

A track is not good or bad; it is good or bad for a point. Two objects in one scene at different distances get different conditioning from the same sweep, by a factor of five in the error here. Planning a sequence by its arc is planning by a number that does not describe what any part of the scene receives.

To improve a particular point, move the track relative to it rather than adding stations along it — which is another picture of the same sweep’s finding, restated in a way that says which way to move: closer, so the same chord subtends more.

And a far object needs a longer track in proportion to its distance. Double the range and the subtended angle halves, so the arc has to double to hold the conditioning. That is why a survey of a distant facade needs a sweep that looks absurd beside one of a room, and it is a statement about the facade rather than about the survey.

Why the near points beat the far ones twice over #

The factor of 5.6 between the best-conditioned point and the worst is worth decomposing, because it is two effects and they run the same way.

A near point is closer, so a pixel of reading error covers a smaller length at it — the range over the focal length, which for the near points of this courtyard is 6.6 microns a pixel and for the far ones 14.1. That is a factor of 2.1 before any geometry.

A near point also receives a wider angle from the same chord, so its two rays cross more squarely and the long axis of its uncertainty is shorter in proportion to the sine. That is the second factor of about 2.2.

Multiplied, those give something near five, which is the measured 5.6. And both factors are the ratio of the chord to the range read twice — which is why the fitted exponent of −1.68 sits between one and two rather than at either, and why a reader trying to attribute it to one mechanism will not be able to.

The practical form of that is unwelcome and worth stating. Distance costs a reconstruction roughly its square, not its first power, so a scene twice as deep is four or five times worse at its far end for reasons that have nothing to do with how carefully it is photographed.

Why the whole reconstruction does not simply follow #

The measurement holds the cameras exact, which is a deliberate narrowing, and it is worth saying what the full problem adds.

In a real reconstruction the cameras are recovered from the same marks, so a point’s error has the cameras’ errors in it, and a badly-conditioned point drags the cameras which then drag every other point. A chain and an adjustment measures what that coupling does along a sequence, and the coupling is the reason a bundle adjustment is not a set of independent triangulations.

The per-point reading here is therefore a floor rather than the whole of it. What it is good for is the comparison between points of one scene, which the coupling affects much less than it affects any of them absolutely: the near points are better conditioned than the far ones for the same reason whether or not the cameras are free.

And it is what a reader can compute in advance. The angle a planned track will subtend at a planned object needs nothing but two positions and a protractor, where the covariance of a reconstruction needs the reconstruction.

The one number that is a property of the track #

There is a quantity of the track alone that does predict something, and naming it separates the two ideas cleanly.

The track’s chord — the straight-line distance between its furthest stations — is the baseline, and it is what a point’s subtended angle is the chord divided by the range of. So the track supplies a length and the scene supplies a distance, and the conditioning is the ratio.

That is exactly the statement far enough away, a pair is one eye makes for two views: the parallax a single homography cannot explain falls as the distance to the power −0.968, which says the ratio of baseline to depth is the whole of it. A track is a pair with more stations in it, and the ratio is the same ratio.

So the vocabulary to use is not “how wide a sweep” but “how long a chord, against how far away” — and the second half is a fact about the scene that no amount of walking changes.

What the gauge has to do with it #

One quantity of a reconstruction is unaffected by any of this, and keeping it separate is what stops the measurement being confused with a different one.

A reconstruction from a track is determined only up to a similarity — seven numbers nothing in the pictures can fix — and seven numbers no picture can name is the standing account. That freedom is the same seven directions whatever the track’s arc and whatever angle any point receives: adding stations does not reduce it and widening the sweep does not reduce it, because it is not an uncertainty at all.

So there are two quite different things a reader might mean by “the reconstruction is poorly determined”. One is the gauge, which is exact, structural, and removed by choosing a frame. The other is the conditioning of the points inside that frame, which is what this measures and what the subtended angle governs.

Every error quoted above is measured after the similarity is divided out, which is why it can be quoted in millimetres at all. A comparison that skipped that step would report metres of error on a perfect reconstruction, which is the trap this whole field is about.

What this does not settle #

The scene is one cloud at similar depths. A courtyard spans four metres to ten, which is a factor of two in range and gives the factor of 2.2 in angle. A scene spanning a factor of ten would give a factor of ten, and whether the power law holds over that range is not measured.

Every point is seen by every station. Real sequences lose points to occlusion and to the frame’s edge, and a point seen by three of six stations has a smaller subtended angle than the geometry allows — so the operative quantity is the angle over the stations that see it, which is what is computed here but is not varied.

The track is an arc rather than a line. A straight dolly puts every station on one line, which makes the recovered track indistinguishable from a longer or shorter one and is a separate difficulty; an arc avoids it. Whether the arc’s curvature matters to a point’s conditioning, beyond fixing the chord, is not measured — and the chord argument says it should not, which is a prediction rather than a result.

And the reading error is the same everywhere. A mark near the edge of a frame, or on an oblique surface, is read worse, and that varies across a scene in its own pattern. The conditioning measured here is the geometry’s contribution alone.

Still open: what the direction of the error owes to the track #

The measurement reports how large each point’s error is. The covariance it comes from says more: the error is a needle rather than a ball, with its long axis roughly along the mean line of sight, and the needle’s direction is a second thing a track decides.

Two points with the same subtended angle can have their needles pointing quite differently — one along the depth of the scene, one across it — depending on where in the arc the stations that see them sit. That matters because a reconstruction is usually wanted for a purpose with a direction in it: a facade’s flatness is a measurement across one axis, a floor’s level across another.

The measurement that settles it computes each point’s covariance, resolves the long axis against the scene’s own axes, and asks whether a track that gives every point the same size of error can be arranged to give them all a usefully aligned direction — or whether the needle simply points along the mean line of sight everywhere, in which case the only control a photographer has is where they stand relative to the thing being measured, and the arc’s shape does not matter at all.

The short version #

Spread is not a property of a track. A sequence of six stations across sixty degrees gives the points of one courtyard between 41 degrees and 89 of subtended angle — a factor of 2.2 — and their triangulated errors run from 1.5 to 8.4 millimetres, a factor of 5.6.

The errors follow the angle at the point as its −1.68 power, with 94 per cent of the variation explained; against the track’s own sixty degrees there is nothing to fit, because one number cannot have a slope against forty-four. The exponent is not the −1 a crossing-angle argument predicts because on one track the angle and the range are two readings of one thing — the chord over the distance — and the point’s error carries both.