One square-on picture ties the two groups a third row only steadies

A wall’s twist can be tested against the error the adjustment reports took a flatness survey of a facade — six cameras in two groups, sixty degrees either side of the wall’s normal, every pose and every mark found in one bundle adjustment — and asked whether the adjustment’s own covariance could be trusted to say when the reconstructed wall’s twist was real. It could. A flat wall’s fitted twist cleared twice its reported error one time in twenty at every layout tried. What the layout decided was how large a real twist had to be before it was seen: ten marks in two rows reported the twist to ±5.05 millimetres and saw a five-millimetre twist seventeen times in a hundred; fifteen marks in three rows reported ±2.31 and saw it fifty-six.

The third row helps, the essay before that one found, because the two groups are tied to each other only through the marks both see, and a third row tightens the tie. It ended by naming the other way to tighten it. A few pictures taken square to the wall see every mark the two oblique groups see, and tie each group to the middle rather than to each other. The question was whether three square-on pictures, which cost a photographer a few seconds, buy what the third row buys — so that a plain wall with only a string course and a cornice to mark could still be surveyed for a saddle.

They buy more than the row does, and one picture buys almost all of it.

One picture against a row #

The survey is the earlier essays’ throughout: a wall 6 metres wide, marked in two rows of five from half a metre to two and a half metres up; three cameras either side, 55, 60 and 65 degrees off the normal, 8 metres from the wall’s middle, with 50-degree lenses; every mark read in every picture to half a pixel; seven numbers held to fix the gauge, as seven numbers no picture can name requires. The twist is the saddle: the marked corners pushed out of the wall’s plane, opposite corners the same way, the middle left where it is. Its reported error is the standard deviation the adjustment’s covariance implies for it.

The two groups alone report the twist to ±5.05 millimetres from sixty views — six cameras, ten marks. A third row of five marks adds thirty views and brings it to ±2.31. One picture taken square to the wall, from the same 8 metres, adds ten views and brings it to ±1.25. Three square-on pictures spread over ten degrees bring it to ±1.13, and adding the third row as well changes the third decimal place: ±1.12.

In the currency the adjustment itself counts in — information, the reciprocal of the variance — the single picture adds 4.1 times what the row adds, from a third as many views. And the two fixes are not additive. Once a square-on picture is in, the third row has almost nothing left to do, which says the two were remedies for one and the same weakness and the picture is the stronger remedy.

The twist is the two groups pitching against each other #

The adjustment’s covariance says what the weakness is. Every quantity it fits is correlated with every other, and the twist’s correlations with the cameras’ poses are the place to look.

In the two-group adjustment the fitted twist moves with every camera’s pitch — its rotation about its own horizontal axis — with a correlation of 0.94 or 0.95, negative for the three cameras on the left and positive for the three on the right. The twist is the left group tilting down a little and the right group tilting up, or the reverse. Each group reads the wall’s depth through its own oblique view, and a camera turned 60 degrees from the normal that is pitched by a hair reads its marks’ depths sheared across the wall: higher marks pushed one way, lower marks the other. The two groups pitched in opposite senses shear in opposite senses, and two opposite shears seen from opposite sides meet in a saddle. The marks both groups see constrain this only weakly, because a mark’s depth is the quantity each group reads worst.

A picture taken square to the wall belongs to neither group. It is not pitched with the left group or with the right, and in it every mark is drawn where it is across and up the wall almost regardless of its depth — so it says, with its full precision, where each mark is in the wall’s plane. A counter-pitch of the two groups moves the marks’ reconstructed positions in that plane as well as in depth — more, in fact: read off the same covariance, each millimetre of twist at the corners carries the marks 2 to 3.5 millimetres across the wall, the lower row one way and the upper row the other, and up to 1.6 millimetres up or down. Those positions no longer agree with what the square-on picture drew, and it reads them to a fraction of a millimetre. With one such picture added the correlations fall to at most 0.12. The loose mode is gone, and the twist is left with whatever error the readings themselves set.

That also explains why the third row helps less. A third row gives both groups more marks through which to hold their relative pitch, but each of those marks is again read in depth by two oblique groups. It tightens the tie through the same weak channel. The square-on picture ties through a different one.

The first picture does the work #

If the square-on picture is a tie rather than a reading, the first one should do nearly everything and the rest little.

So it is. The first picture takes the reported twist from ±5.05 to ±1.25 millimetres. The second takes it to ±1.17, the third to ±1.13, and six reach ±1.07. The twist’s share of the wall’s whole flatness variance — the measure the bundle keeps the needles turned used to show that one shape dominated the error — falls from 71 per cent to 17 with the first picture and stays there with the rest. After the first picture, no single shape dominates: the error is spread over the wall’s departures the way the readings spread it.

The same picture closes most of the distance between the bundle and a survey whose camera poses were known beforehand. With the two groups alone, the bundle’s root-mean-square flatness error was 4.23 millimetres against 2.38 with every pose known; with one square-on picture, 2.28 against 2.21. The adjustment then knows the wall almost as well as a survey with a calibrated rig — which is the ceiling, since where the adjustment stops is that recovering poses from the pictures can only add error.

The third group need only be well clear of the other two #

The argument was that a square-on picture breaks the counter-pitch because it belongs to neither group. If that is right, what matters is not squareness but separation, and a third group standing some way off the normal should work almost as well.

Three pictures standing 10 degrees off the normal report the twist to ±1.16 millimetres; 20 degrees off, ±1.28; 30, ±1.53; 40, ±2.02, still better than the third row. At 50 degrees, ten degrees from the group at 60, they report ±3.04 and have become a fourth, fifth and sixth camera of that group, pitched with it. A single picture follows the same curve a little higher and beats the row up to 30 degrees off.

So the practical instruction is loose. A photographer who has walked round two sides of a wall at sixty degrees should take a picture or two from anywhere within twenty or thirty degrees of square-on, and the instruction it replaces — mark a third row — is both more work and less effective.

From four times as far, still better than a row #

The square-on picture need not be taken from where the groups stand either.

With the same 50-degree lens, taken from 12 metres the picture reports the twist to ±1.42 millimetres, from 16 to ±1.59, from 24 to ±1.94, and from 32 metres — where the wall is a quarter the size in the frame and every mark’s position is read four times less finely in metres — to ±2.29, still a hair better than the third row. A distant picture is a weaker reading, but it is no less a member of neither group, and the tie it makes survives the loss of resolution until the marks are only a few pixels apart. A picture from across a street, taken because the photographer could not stand any nearer, is worth taking.

The other direction is closed by the lens rather than the geometry. Nearer than 8 metres, a 50-degree lens no longer holds the six-metre wall in its frame: at 6 metres six of the fifteen marks of a three-row wall fall outside it, and at 3 metres fourteen. A closer square-on picture needs a wider lens, and was not measured here.

What a surveyor now sees #

The point of reporting the twist’s error was to say how large a real twist has to be before it is seen. With the error reported honestly — which the earlier essay established at every layout — the answer is the two tails of a normal distribution at each reported error.

A 5-millimetre twist is called real 98 times in 100 with one square-on picture, 56 with a third row and 16 with neither. A 3-millimetre twist, 66, 24 and 8. A 2-millimetre twist, 35, 13 and 6. On a flat wall every layout calls a twist real 5 times in 100, as a test at twice its reported error should. The plain wall of the earlier question, with only a string course and a cornice to mark — two rows — was a wall on which a five-millimetre saddle went unseen five times in six; with one more picture, taken square from wherever the photographer can stand, it is seen every time but twice in a hundred.

Neither kind of picture surveys the wall alone #

It would be a misreading to conclude that the square-on pictures are the survey and the groups a formality. Three pictures taken square to the wall and nothing else measure its flatness to a root-mean-square 37 millimetres with their poses known, and 94 with their poses found from the pictures — useless for finding a saddle of five. A square-on camera reads each mark’s depth along its own line of sight, which is the direction a single camera reads worst, and three cameras ten degrees apart barely separate their sight lines. Six of them spread over twenty degrees still give 16 and 25 millimetres.

The two groups are what measure depth. Their sight lines cross the wall’s normal at sixty degrees from either side, so between them they read each mark’s departure from the wall across their lines of sight, which is the direction every camera reads well. What they cannot do is hold themselves to each other. The square-on picture does exactly that and nothing else of consequence: it reads where each mark is in the wall’s plane, which is the one thing the counter-pitch disturbs that the groups read badly. The survey’s flatness comes from the groups, and its freedom from a saddle comes from the picture. Each kind of picture supplies what the other lacks, and the combination reaches 2.28 millimetres, within a tenth of a millimetre of what the same cameras give with every pose known in advance.

Opening the groups does less than one picture #

The earlier essay found a second lever: opening the two groups further from the normal shrinks the twist, because steeper views read each mark’s depth with more of the reading’s precision. It is worth setting that against the picture. Groups at 45 degrees either side report the twist to ±5.83 millimetres; at 60, ±5.05; at 75, ±3.30. With one square-on picture added, the same three openings give ±1.71, ±1.25 and ±1.03. At every opening the picture does more than opening the groups by thirty degrees does, and the two levers combine: a steep pair of groups with a square-on picture between them is the best of the arrangements tried. A third row at the same three openings gives ±3.38, ±2.31 and ±1.47 — always between the groups alone and the groups with a picture.

That ordering is the same at every opening because the weakness is the same at every opening. However steep the two groups, they are two directions, and two directions leave a counter-rotation free; only the steepness of the views changes how much each rotation costs in the marks’ depths. A third direction removes the freedom rather than raising its price.

Why a picture is cheaper than a mark #

There is a general point here about what a survey’s design is spending. Split the track and the needles turn found that splitting cameras into two groups either side of a wall turns each mark’s error needle across the wall, which is what makes the two-group survey good at flatness in the first place. The price is that the groups see the wall from two directions only, and anything that rotates one direction against the other is weakly held. Marks add constraints within those two directions. A camera from a third direction adds a direction.

That is why a single picture outperforms a whole row of marks. The row’s thirty new views are thirty more readings of the same two kinds; the picture’s ten are readings of a third kind. A survey is trusted at its own accuracy, unless its error has a shape warned that a reported accuracy is a fair summary only when the error is spread evenly; the two-group survey’s error had a shape — 71 per cent of it in one saddle — and the square-on picture is what removes the shape rather than shrinking it.

The linear world the numbers live in #

Every number is from the adjustment’s covariance at the true solution. That is the first-order description of a least-squares fit, exact for small reading errors. Half a pixel is small for these cameras, but the earlier essay drew its reconstructions from the same linear model, so neither it nor this one checks what larger errors do to the adjustment’s own nonlinearity.

Every picture sees every mark. The square-on picture is placed where the wall fits its frame, and the groups at 8 metres see all marks. A picture that sees only part of the wall ties only the part it sees, and a third group that sees only the middle of a long wall would leave its ends to the two groups’ counter-pitch.

The marks are matched correctly in every picture. A square-on view of a regular facade is the view most prone to matching a mark to its neighbour, the failure a wrong match is not a small error is about; the earlier essay showed a mismatch big enough to fake a twist shows in its own residual, and the same test applies to the square-on picture’s marks.

The lens is the same in every picture and known. A square-on picture taken with another camera, of another focal length, adds its own unknown calibration, and whether the tie survives an unknown focal length in the picture that makes it was not measured.

Still open: whether the square-on picture can come from a different camera #

A photographer surveying a wall with a calibrated camera may not have it for the last picture, or may take the square-on view with a phone from across the street. That picture’s focal length and principal point are then unknowns of the adjustment too, and a focal length is exactly the kind of number that trades against a depth.

The measurement that settles what such a picture is worth adds one square-on picture whose focal length — and, separately, whose principal point — the adjustment must find for itself, and asks how much of the tie survives: whether a picture square to the wall, which reads the marks’ positions in the wall’s plane, fixes its own focal length from the wall’s known width between the two groups’ readings, or whether the unknown focal length lets the picture scale itself along with the counter-pitch, so that the twist returns. The question with a number in it is how far the reported twist climbs back from ±1.25 millimetres towards ±5.05, and whether a second uncalibrated picture from another place, or a known distance between two marks, recovers it.