A false pair is named by its neighbours, not by itself

The wall convicts a pair set too wide gave a single photograph of a partly symmetric facade a test for a false pair. The other pairs fix the symmetry plane and the front wall; carry one half of a suspect pair along its ray to that wall, reflect it in the symmetry plane and project it, and the partner’s mark should be there. A window corner set ten centimetres wider than its partner missed by six pixels while every joining line stayed concurrent to a trillionth, and at 0.4 pixels of reading error a three-centimetre fault was convicted most of the time.

A conviction throws the pair away. The essay ended by asking whether it could do better: run the prediction in the other direction as well, from right to left, and ask whether the two misses — their sizes and directions — say which half is wrong and by how much. If they do, the reader keeps the pair’s information instead of discarding it.

They do not, and the reason is short enough to state before anything is drawn. What they give instead is the move itself, and a way to name the half from the facade’s other marks.

Two predictions, one residual #

The wall and the symmetry plane together are a map between the two halves of the facade’s picture. A mark on the left half, carried to the wall and reflected, lands at one place on the right half; carried the other way, a mark on the right lands at one place on the left. That map is a homography — the wall’s picture sent to the wall, reflected, and sent back to the picture — and it is invertible. A pair is consistent exactly when its right mark is the image of its left mark under that map.

So the forward prediction measures how far the right mark is from the image of the left, and the reverse prediction measures how far the left mark is from the image of the right under the inverse map. Those are one residual, seen once in each half of the picture. A pair whose right half has moved and a pair whose left half has moved the mirror way produce the same residual, because each is a pair of marks one of which is not the other’s image; the only difference between the two stories is where on the wall the true pair stands, and a single pair’s marks do not say where on the wall a true pair ought to stand.

The figure makes the pair false by moving its right half six centimetres straight out from the centre line and reads it against the other eleven pairs — the tested pair is left out of the symmetry plane’s and the wall’s fits, so that the facade judges it rather than it judging itself. The right reading recovers the move exactly: the pair stands where the facade was designed, and its right half has moved six centimetres outwards. The left reading explains the same two marks just as exactly: the pair stands six centimetres further out, where the moved right mark says, and its left half has moved six centimetres back in, towards the centre.

The slider turns the move. Moved at forty-five degrees — outwards and up by 4.2 centimetres each — the right reading gives (−4.2, 4.2) across and up, and the left reading (−4.2, −4.2). The left reading is always the right one mirrored in the centre line and turned back: what one half’s move looks like when it is blamed on the other. Two predictions give two readings, but they are one fact about the pair — its marks disagree with the symmetry by this much — assigned to one half or the other.

This is not a weakness of the particular test. It is a count. A true pair on the wall has two numbers of freedom, its place on the wall. Its two marks carry four. The two left over are the residual, and they measure how far the pair is from being true; they cannot also say which half it is, because a move of either half by the right amount uses exactly those two numbers.

Where to read the one residual #

That the two predictions are one residual does not make them equally good places to read it. The residual is one fact on the wall, and it is seen in the picture at two places, one in each half’s image, through the camera’s own foreshortening at each. On this facade the camera stands on the left half’s side of the centre line, so the left half is nearer and seen larger.

For the window corner moved three centimetres straight outwards, the forward prediction misses the right mark by 1.87 pixels and the reverse prediction misses the left mark by 2.71: the same residual, seen 1.45 times larger in the nearer half. Moved at forty-five degrees the two are 1.83 and 3.07; straight up, 2.42 and 2.86. And the pattern is symmetric in the way the two stories are: a right half moved at forty-five degrees gives the same pair of misses as a left half moved at a hundred and thirty-five, which is that move mirrored.

So a reader convicting a pair should read its residual in the half seen larger, where the same fault is more pixels and the same reading error is a smaller share of it. That is a gain in sensitivity of up to a half on this facade, with no new information: it is the difference between measuring one length with a finer ruler and measuring it twice. The second prediction was never a second witness, but it is a better-placed copy of the first.

The move comes back, whichever half it was #

The half is not named, but the move is recovered. Its size and direction are the same in both readings, up to the mirror, and so they are known whichever half turns out to be wrong.

At a reading error of 0.4 pixels the move comes back to about a centimetre either way at every size — at five centimetres the middle eight pictures in ten recover between 3.7 and 6.1 — and below about two centimetres it is lost in the reading. The right and left readings have the same length in every picture, since one is the other mirrored, so this is the precision of the correction whichever half it is applied to. The wall convicts a pair set too wide found the test convicting three-centimetre faults most of the time; this says the fault, once convicted, is measured to about a third of itself.

So a reader who knows which half is wrong can correct it, to that precision, and keep the pair. What remains is to know which half.

A line names what crosses it #

The pair’s own marks cannot name the half, but the facade has other marks, and a real facade’s marks are not scattered. A window’s corner shares a vertical line with the window’s upper corner and a level line with the next window’s corner. A half that has moved breaks the lines it shares with its own neighbours, and leaves its partner’s lines intact.

That is the reading the next figure measures. Every mark on the front wall is carried to the wall along its ray. For the suspect pair, each half’s distance across from the mark above it on the same side is compared, and the half whose column is broken more is named.

The column names the moved half 93 times in a hundred at two centimetres, 99 at three, and every time from five. At half a centimetre and one, below the size at which the move itself is recovered, it names it about two times in three — better than a guess, since even a move lost in the pair’s own residual still shifts one mark relative to its column mate.

The row does not help with this move at all. The window corner and its neighbour share a height, and a move straight across slides the corner along that level line without breaking it; the row names the half 58 times in a hundred at every size, a small preference that has nothing to do with the move. A line can only name a move that crosses it.

Every direction, with two lines #

A move straight across is the kind the wall essay was built around — a window set wider than its partner — but a real fault can go any way. The next figure turns a three-centimetre move from straight across to straight up and back.

Each line reads the part of the move that crosses it. The column names a move across every time and a move straight up 57 times in a hundred, since a move straight up slides the corner along its column; at sixty degrees it still names it nine times in ten. The row is the column’s mirror: 54 per cent across, every time straight up. Summed — each half’s break over its column and its row together — the two lines name the moved half 99 or 100 times in a hundred in every direction measured.

A window corner is the best case for this, because it has a mark above it and a mark beside it by construction. The same reading works for any mark that shares a line with another mark on the same side: a wall’s end with its other end, a gable’s foot with the eaves, a door’s jamb with its head. A mark that shares no line with anything — a sign fixed at an arbitrary place, a stain — cannot be named this way, and a false pair made of such marks can only be convicted and discarded, as before.

What a single picture does not see #

The move the readings recover is the move as the camera sees it. The wall essay found the test blind along the camera’s own ray to the partner, and the recovered move has the same blind direction.

A move straight across the wall is recovered at its full five centimetres. A move at right angles to that, in the plane that also holds the camera’s ray — which takes the corner partly out of the wall — is recovered as 4.76 centimetres, the in-wall move that would draw the same mark. And a move along the camera’s own ray to the corner is recovered as nothing at all, to fourteen decimal places, because a mark moved along its own ray lands on the same pixel and the picture holds no trace of it.

That is the limit on the correction. What the reader keeps is the pair’s mark moved back to where the symmetry says it should be seen, which is exactly what a reconstruction from this one picture needs, since a reconstruction from one picture places every mark somewhere along its ray anyway. What the reader does not learn is where along its ray the moved half really stands. A second photograph from elsewhere would say; the facade’s own lines, which name the half, cannot, because they too are seen through the same camera.

The blind direction matters less in practice than it sounds. The faults a symmetric facade actually carries — a window set a little wider than its partner, a sill a course higher, a door hung off centre — lie in the wall’s plane, and a move in the wall’s plane has no part along the camera’s ray unless the camera looks along the wall, which is exactly the view in which the wall essay found the test weakest anyway. A fault out of the plane — a bay window projecting on one side only, a reveal deeper on one side — is recovered as the in-wall move that would draw the same marks, and a reader who applies that correction gets the picture right and the depth of the projection wrong, which is the most one photograph can do.

Correct, then keep #

Put together, the answer to the wall essay’s closing question is in three parts. The two predictions, left to right and right to left, are one residual and cannot name the half, because a true pair’s place on the wall is exactly what the two stories disagree about and a single pair does not fix it. The move itself is recovered to about a centimetre at 0.4 pixels, the same whichever half it belongs to, and blind only along the camera’s ray. And the half is named by the facade’s other marks — by a line the moved half shared with a neighbour — which a window’s corners supply in both directions.

The procedure a reader can follow is therefore: convict with the wall, recover the move, name the half by its lines, move it back, and keep the pair. Two wide pairs are most of a symmetry found that the widest pairs carry most of the reconstruction’s precision, and the widest pairs on a real facade — its corners, its outer windows — are also the ones with the most neighbours on their lines. They are the pairs most worth keeping and the easiest to correct.

A symmetric object is its own stereo pair began this line of argument by reading a symmetric object as a second camera, and every essay since has leaned on that: a pair is a correspondence, and a false pair is a false correspondence. What this adds is the same thing a facade matched a column out comes forward, not back found for a stereo pair’s repeated windows — the evidence against a mistaken match is never in the match itself, and always in something it should have lined up with.

What was assumed #

Only one half of one pair is false. The other eleven pairs are true, so the symmetry plane and the wall are fixed by them. A facade with several false pairs needs them found first — the wall test does that one pair at a time — and one whose false pairs share a line would see those lines broken on both sides.

The facade’s lines are true lines. The column test assumes the window’s lower corner and upper corner stand on one vertical in the design. A window built out of true, or a gable whose eaves are not level, breaks a line with no pair at fault, and the half it names is then the one whose neighbour is wrong.

The moved half stays on its ray’s side of the wall. A move large enough to take a mark behind the wall, or round a corner, is no longer read as an in-wall move; the readings here are small moves on a flat front wall.

Marks are read with independent errors of 0.4 pixels. A mark on a window’s corner is found where two edges meet, and its error is not isotropic; along a well-defined edge it is smaller than across it. The column test uses only the across part, which is the part a vertical edge fixes best, so it is likely to do slightly better on real corners than on the isotropic marks measured here.

Still open: whether a facade’s lines can replace some of its pairs #

This essay used the facade’s lines only to name a half. They are also constraints in their own right: marks on one vertical line share their distance from the symmetry plane, marks on one level line share a height, and neither fact needs a partner on the other side.

The measurement that settles what they are worth recovers the facade from fewer pairs — three, four, six — with and without the constraint that marks known to share a line do share it, and asks how much of the precision twelve pairs gave is bought back by lines alone. Two wide pairs are most of a symmetry found three wide pairs close to twelve for the symmetry plane; if lines are as good as pairs for the wall, a facade that is barely symmetric at all — one door off-centre, one window on each side — could still be recovered whole from its straight edges and a single pair.