The wall convicts a pair set too wide

Two wide pairs are most of a symmetry read a partly symmetric facade from one photograph — twelve pairs of marks across a symmetric frame, eight lone marks in an asymmetric middle — and found that very little of an object has to be symmetric for the reading to work. It ended on the danger that remained. A pair that is not a pair, a window set a little wider than its partner, keeps every joining line through the mirror’s point to $2\times10^{-13}$ pixels, because a move along the symmetry plane’s normal slides a mark along its own joining line. The concurrence, the test a symmetric object is its own stereo pair built, is blind in exactly the direction a facade’s windows most often err.

The essay named the test that might cover the gap: the false pair’s recovered points lying off the wall the other pairs fix. That test turns out to be better than a complement. It sees every direction a pair can be wrong in but one, and the one it misses is a direction no photograph can see.

What the wall predicts #

The facade and the photograph are the earlier essay’s: a gabled front seen from 34 degrees round, twelve pairs of marks, a symmetry plane found from the pairs’ joining lines. The test works pair by pair. Fit the front wall to the recovered points of every other pair that lies on it. Carry the suspect pair’s left mark along its ray to that wall, reflect the point in the fitted symmetry plane, and project the reflection into the photograph. That is where the wall says the partner should be seen, and the test is how far the partner’s own mark lies from it.

With exact marks and every pair true, the wall predicts every partner exactly. Set the widest pair’s right half wider than its reflection and the two tests part company. The concurrence stays at $7\times10^{-13}$ pixels however far the half is moved, as the earlier essay found. The wall’s test grows in proportion: 1.21 pixels at two centimetres, 6.02 at ten, 11.96 at twenty — about 0.60 pixels for every centimetre. Drag the offset and the two rings part: the one on the concurrence stays on the floor of the plot at every setting, while the one on the wall’s test climbs a straight line away from it. The false pair’s two triangulated points lie 6.0 centimetres off the wall at a ten-centimetre offset, which is the depth error the offset introduced.

Moved higher instead, the pair fails both tests: the concurrence opens by 3.5 to 33.5 pixels and the wall’s test by 0.8 to 9.1. So the wall sees what the concurrence sees, and it also sees what the concurrence cannot.

The reason is what each test compares. The concurrence asks whether the pair’s joining line runs through the mirror’s point, which is a statement about the pair alone and the symmetry plane’s direction. The wall’s test asks whether the pair’s partner is where the rest of the facade says it must be, which uses the plane’s direction, its position through the fitted wall, and every other pair’s points. A move along the normal slides the partner along its joining line — invisible to a test about lines — but moves it off the place the wall predicts.

How small an offset it convicts #

With exact marks any offset shows. With marks read to 0.4 pixels the question is how large an offset must be before the test reads more on the false pair than it reads, by chance, on a true one.

The threshold is set where a true pair’s own reading exceeds it one picture in twenty: 1.35 pixels for the widest pair, 1.40 for a pair ninety centimetres wide. A pair set a centimetre too wide is convicted about one time in seven — little better than the one-in-twenty the threshold allows on true pairs. Two centimetres, about half the time. Three centimetres, eight or nine times in ten. Four and five centimetres, every time or nearly.

The test’s reach hardly depends on the pair’s own width, which is the opposite of the earlier essay’s finding for the symmetry plane itself. There, wide pairs fixed the plane far better than narrow ones, because a pair’s joining line is fixed by the distance between its marks. Here the pair’s own line hardly matters: the prediction of where its partner should be is made by the wall and the plane, which the other pairs fix. A narrow window is judged by the facade’s wide frame, and judged as well as a wide one.

That answers the question the earlier essay closed on. A window set five centimetres too wide lies measurably off its own wall, and one set three centimetres too wide usually does. For a facade read from one photograph at this distance and resolution, the two tests together leave no false pair of that size unexposed — with the one exception drawn at the end.

Why the threshold is about a pixel and a third #

The threshold is not a choice; it is what the photograph’s own reading error leaves, and it can be accounted for. The partner’s mark is read to 0.4 pixels in each direction. The prediction it is compared with is built from the left mark, read to the same 0.4 pixels, carried to a wall that the other pairs’ marks fix a little uncertainly and reflected in a plane that their joining lines fix a little uncertainly. Each of those adds its share, and the discrepancy on a true pair ends up with a typical size of about 0.6 pixels in each direction. A distance in the picture built from two such independent errors exceeds 1.35 pixels about one time in twenty, which is where the figure puts it.

Converted by the test’s own response, 0.60 pixels a centimetre, that threshold is a little over two centimetres. That is the test’s resolution at this distance and this focal length: a partner more than about two centimetres from where its reflection should be starts to show, and one more than four shows every time. For a facade seven metres away photographed on a sensor that resolves a centimetre of wall into about 0.9 pixels, it is roughly the size of a window reveal misjudged by a mason, or a frame fixed a finger’s width off its partner’s line — small errors, but the ones a symmetric facade actually has.

The same account says how the test scales. Everything in it is a length in pixels, so a photograph taken from half the distance, or with twice the focal length, doubles the response to a centimetre of offset and leaves the reading error in pixels where it was; the test’s resolution in centimetres halves. A photograph taken twice as far away needs twice the offset before the wall can see it. The one thing that does not scale is the direction the test is blind to, which is set by where the camera stands and not by how far.

The wall has to be there #

The test borrows everything from the other pairs, and so it is only as good as the wall they give it.

With five pairs, three of them on the front wall, the wall is fixed by four points besides the suspect pair’s, and a true pair’s reading scatters to 2.41 pixels one picture in twenty; a three-centimetre offset is convicted 28 times in a hundred. With eight pairs the threshold falls to 1.80 pixels and the conviction rate rises to 47. With all twelve, nine of them on the wall, it is 1.28 pixels and 82 per cent.

Two things improve together as pairs are added. The wall is fixed by more points, so its place along the suspect pair’s ray is better known; and the symmetry plane, which the prediction reflects through, is fixed by more joining lines. Two matches are enough found that two pairs fix the mirror’s point exactly with exact marks; with marks read to a fraction of a pixel, the wall’s test wants many more, and wants them on the wall.

Pairs that do not lie on the wall — the back corners of the building, the gable’s ridge — cannot be tested this way at all, since there is no fitted surface to carry their marks to. For those the concurrence is the only one-picture test, and the direction along the normal remains its blind spot.

Where to stand #

The earlier essay found that standing further round the facade fixes the symmetry plane better. For the wall’s test, the opposite holds, and for a reason that is easy to draw.

A move along the symmetry plane’s normal is a move across the facade. Seen from near square on, it is seen side-on, and the partner’s image moves by the most it can: 0.82 pixels a centimetre at twelve degrees round. Seen from far round, it is seen more nearly end-on — the move points increasingly along the line of sight — and the image moves less: 0.28 pixels a centimetre at seventy degrees. The threshold, meanwhile, is about 1.3 pixels from twelve to forty-six degrees and climbs to 2.2 at seventy, where the wall is foreshortened and fixed less well. So a three-centimetre false pair is convicted 96 times in a hundred from twelve degrees round, 82 from thirty-four and 13 from seventy.

Square to the camera is the worst mirror found that a symmetric object seen square on is read badly, because its joining lines are nearly parallel and cross far away. The best place to stand is therefore a compromise between two tests that want opposite things: the symmetry plane wants the camera well round, and the check on false pairs wants it near square. Somewhere between a quarter and a third of the way round does both reasonably, which is where the earlier essay’s photograph was taken.

Every direction a pair can be wrong in #

A real false pair is not wrong along a chosen axis; its half is wherever the builder put it. The last two figures ask which directions of wrongness the two tests catch between them.

Five centimetres in four of the five directions, the wall’s test convicts every time. The concurrence, read at 0.4 pixels, is much the weaker test: with twelve pairs of different widths, the narrow pairs’ joining lines are fixed poorly, and a true set of pairs misses its common point by up to 23.9 pixels one picture in twenty. It catches a pair moved five centimetres straight up 87 times in a hundred and almost nothing else. On this facade, with a wall to test against, the wall’s test does nearly all the work.

The fifth direction, wider and deeper, is caught only half the time by either test, and that is not an accident of the facade.

Swept round the level plane, the wall’s test convicts a five-centimetre move every time from −90 degrees through straight wider to ten degrees past it, and again from seventy degrees to straight into the wall. Between, it dips to 52 per cent at forty degrees and 57 at fifty, and the concurrence adds almost nothing there. The camera’s ray to the moved half runs at forty-five degrees in this sweep.

A point moved along the ray through it lands on the same pixel. Its photograph has not changed at all, and nothing read from that photograph can say it moved — not the concurrence, not the wall, not any test that could be invented. A five-centimetre move exactly along the ray is invisible, and moves near it are seen only by their small component across it. The image of the other eye is where this collection’s reading of two photographs begins, and it is the reason a second photograph exists: a second eye’s rays cross the first’s, and a move along one eye’s ray is a move across the other’s.

What the two tests are #

It is worth being exact about what has been built. The concurrence is a test the symmetry supplies from the pairs alone, with no surface and no assumption beyond the symmetry; it sees any move that takes a pair off its joining line and none that slides a pair along it. The wall’s test borrows two assumptions — that the pairs on the front wall lie on one plane, and that the suspect pair is on it too — and in exchange it sees any move that changes where the partner is seen, which is every move but one.

The one it cannot see is set by the camera, not by the facade. One shutter, two views described a mirror’s photograph as two views taken at once; a symmetric object is the same pair of views with the mirror built in, and a pair of views of one point can disagree about the point only in the direction both views look along. For a mirror pair, the two views of the partner are the camera’s ray to it and the reflected camera’s ray to its reflection. A move of the partner along the camera’s own ray changes neither view that the photograph contains.

So the earlier essay’s worry has a precise answer. A single photograph of a partly symmetric object, read against its wall, has a complete check for false pairs up to one direction per pair — the camera’s ray to the pair’s second half. A window set wider than its partner is caught unless it happens to be set wider along that ray, and the facade seen from about thirty degrees round makes that a direction some forty-five degrees away from the one a builder’s error usually takes. Two mirrors are three cameras is where a second mirror gives a third view; a second photograph from another side gives the same, and closes the last direction.

What was assumed #

The front wall is flat and carries the suspect pair. The test carries the left mark to the wall the other pairs fix, and a window set in a projecting bay, or a pair of balconies, is off that wall by design; the test then convicts an honest pair by its relief, as the corners’ plane did for the shifted windows of a facade matched a column out.

One pair is false. A facade with several false pairs pulls the fitted wall and plane towards them, and a threshold set on true pairs is then too generous. Testing each pair against a wall fitted without it, as here, limits the damage but does not remove it.

Marks are read independently to 0.4 pixels. The corners of a real facade are read better than the middle of a window’s edge, and a reading weighted by each mark’s own precision would favour the wall’s test further, since the wall is fixed mostly by corners.

Still open: whether a false pair can be named rather than only detected #

The tests here say that a pair is false. They do not say which of its two halves is wrong, nor by how much. A false pair has two marks, either of which could be the one out of place, and the wall’s prediction runs from left to right; run it from right to left as well and there are two discrepancies, one for each half.

The measurement that settles whether the pair’s fault can be located takes the facade with one half of one pair moved by a stated vector, computes the wall’s prediction of each half from the other, and asks whether the two discrepancies — their sizes and directions in the picture — identify which half moved and recover the move itself, in centimetres, up to its component along the camera’s ray. If they do, a reader of one photograph can not only reject a false pair but correct it, and keep the pair’s information rather than throwing the pair away.