Two wide pairs are most of a symmetry

A symmetric object is its own stereo pair found that one photograph of a building with a plane of symmetry gives fourteen correspondences whose joining lines meet at one point to $1.9\times10^{-12}$ pixels, and the object’s whole shape to fifteen digits, with no mirror and no second exposure. It then broke the symmetry — half a per cent out — and watched the meeting point open to 12.7 pixels. That is the test: the concurrence says whether the object is symmetric at all.

The question it left is the one a reader with a real photograph has. Real objects are approximately symmetric, and mostly only in part: a facade with a symmetric frame and an asymmetric middle, a car with a symmetric body and one wing mirror. How much of an object has to be symmetric before the reading is worth making — and is there a fraction below which the plane is so poorly fixed that the paired points do worse than no assumption at all?

The answer is that very little has to be symmetric, provided the symmetric part is wide. Two pairs do most of the work. The unpaired part is not lost. And the real danger is a pair that is not a pair, which is a different question from how many pairs there are.

A frame that pairs and a middle that does not #

The facade in the figure has a symmetric frame — the wall’s four corners, two windows either side of the centre, the eaves and ridge of a gable — and a middle that pairs with nothing: a door set twelve centimetres off the centre line, and a sign beside it on the other side. Twelve pairs, and eight lone marks, all on the same photograph three-quarters on.

The pairs do what the earlier essay found pairs do. Each pair’s two marks are the images of a point and its reflection, so the line joining them passes through the image of the reflection’s direction — the vanishing point of the symmetry plane’s normal — and all twelve lines meet there. Two matches are enough established why two pairs fix that point for a mirror: a skew fundamental matrix has two free numbers, and each pair gives one equation. The same holds for a symmetric object, because the object’s symmetry plane plays the mirror’s part exactly.

The lone marks do not pair, so they give the epipole nothing. But once the pairs have fixed the symmetry plane’s direction, each pair can be triangulated across it — the second mark read as seen from the camera reflected in that plane — and the frame comes back in three dimensions, up to one scale the picture does not hold. Several of the frame’s points lie on the front wall, so the wall is fixed too. A lone mark on that wall lies on its own ray from the camera, and where the ray meets the wall is where the mark is. With exact marks the whole facade — frame and middle together — comes back to $10^{-14}$ in shape.

So the earlier essay’s condition was stronger than it needed to be. An object does not have to be symmetric. It has to have a symmetric part that fixes the plane, and a surface that carries the rest.

Which pairs fix the mirror #

With exact marks, two pairs are as good as twelve. With marks read to 0.4 pixels, the question is how quickly more pairs help, and it turns out to depend more on which pairs than on how many.

Taken widest first — the wall’s corners, whose halves stand 2.6 metres apart — two pairs find the plane’s direction to 0.40 degrees, four to 0.33 and all twelve to 0.26. Taken narrowest first — the door’s neighbour, the gable’s inner marks — two pairs find it only to 1.35 degrees, and the narrow order needs ten pairs to catch up with the wide order’s two.

The reason is the joining line’s leverage. Each pair fixes one line through the epipole, and the line’s direction is fixed to the reading error over the distance between its two marks. A pair whose halves are two metres apart on the object, and so a few hundred pixels apart in the picture, pins its line to a fraction of a milliradian; a pair thirty centimetres apart pins it ten times less well. The epipole sits where the lines cross, so it is only as well placed as the lines. One shutter, two views drew the same leverage for a real mirror, where the pairs are a point and its reflection in the glass; on a symmetric object the widest pairs are the object’s outermost features, and they are the ones worth marking first.

Past the first few wide pairs, each added pair adds slowly — twelve pairs are 1.5 times better than two, not six times — because the error is shared. Every pair’s line runs toward the same distant point, most of them at similar angles, and adding lines at similar angles narrows the crossing only slowly. What would help more is a pair at a very different angle through the epipole, which on a facade means a pair at a very different height: the ridge against the plinth.

The middle rides on the wall #

The paired frame and the lone middle come back with different errors, and the difference says what the lone marks cost.

The frame’s shape, read as the root-mean-square departure of every distance’s ratio to the truth from their common ratio, is half a per cent or so whether four pairs are used or twelve. It does not improve as pairs are added, because each pair added after the widest is a narrower one, whose own short distances are read less well; the frame is a larger set of points but not a better-measured one.

The lone middle comes back to about 0.8 per cent against everything else: 0.96 per cent from two pairs, 0.79 from twelve. Its error is the wall’s error plus its own reading. The wall is fixed by the paired points that lie on it, so more pairs on the wall fix it better, and the lone marks improve slowly as they are added. What the lone marks never need is a partner. A door that sits off the centre line, a sign on one side, a drainpipe, a crack — any mark on a surface the symmetric frame fixes comes back as well as the frame does, a little worse, from the one photograph.

That answers the earlier essay’s worry about a fraction below which the reading is worse than nothing. On this facade there is no such fraction short of one pair. Two pairs — four marks of the facade’s thirty-two — give the whole facade to about one per cent. An object that is ninety per cent asymmetric is readable if its remaining tenth is wide and its asymmetric part lies on surfaces the symmetric part fixes.

Where to stand #

Everything above is measured from one place, 34 degrees round from square on, and the place matters in two directions at once.

Standing further round moves the epipole in from the edge of the picture — it is the vanishing point of the symmetry plane’s normal, and a normal that points more nearly at the camera vanishes nearer the middle of the frame — and a nearer epipole is fixed better by the same joining lines, because they cross at a wider angle. With all twelve pairs, the plane’s direction improves from 0.36 degrees at twelve degrees round to 0.26 at 34, 0.19 at 58 and 0.15 at 70. Square to the camera is the worst mirror found the mirror-pair version of this: a mirror facing the camera puts its second eye directly behind the first, the joining lines all but parallel, and the symmetric object facing the camera is the same bad case.

The lone middle goes the other way past a point. Its marks are carried along their rays to the front wall, and the further round the camera stands, the more obliquely those rays meet the wall, so the same error in the wall’s placement moves a lone mark further along it. The middle’s shape error is 1.10 per cent at twelve degrees round, 0.81 at 34, 0.90 at 46, 1.19 at 58 and 2.14 at seventy. The best place for the whole facade is about a third of the way round — the frame wants the camera further round, the middle nearer square on — and the difference between the best place and a reasonable one is a factor of two in the part of the object that has no partner.

A false pair #

The risk that matters is not too few pairs. It is a pair that is not a pair: a window that is almost, but not quite, where its partner’s reflection would be.

Move one half of a pair twenty centimetres higher than its partner’s reflection and the pair’s joining line swings off the epipole: the lines miss their common point by 78 pixels, and the concurrence test the earlier essay built flags the object as asymmetric. Move it the same twenty centimetres outward — the window set wider than its partner, along the symmetry plane’s normal — and the lines still meet, to $2\times10^{-13}$ pixels, at every offset. The test says the object is symmetric. The shape comes back 2.2 per cent wrong.

The reason is geometric and exact. A pair’s joining line runs from a point’s image through the image of the normal’s direction, and moving the point’s partner along that same direction moves its image along that same line. The epipole cannot see a displacement along the normal, because the normal is what the epipole is. So the concurrence test catches every false pair except those whose falseness lies in the one direction the symmetry itself defines — and on a facade that is the commonest way for a window to be wrong: set a little further from the centre than its partner.

What the error does to the shape is to put that pair’s second point at the wrong depth, since depth across the mirror is what the normal direction carries. The rest of the facade is unaffected except through the wall, where the false pair’s points pull the fitted plane.

Why the symmetric test cannot be the only test #

The earlier essay’s concurrence was a real test and remains one. It is worth being exact about what it tests.

A symmetry broken in general — the earlier essay’s case, where every point of one half moves a little — has components across the normal, and the concurrence sees those. A symmetry broken along the normal alone is invisible to it, and the reason is not noise or conditioning but the structure of the problem: along the normal, a false pair and a true pair at a different depth are the same pictures. A photograph with a mirror in it is a stereo pair because it holds two views; the two views a false pair offers are two views of a point that is not where the symmetry says.

So a reader needs a second test for the one direction the first cannot see, and the facade supplies it. The lone marks and the wall give one: a false pair’s triangulated points lie off the wall that the other pairs fix, by the depth error the false pair introduced. A pair whose two recovered points do not lie on the wall they should lie on is suspect even when its joining line is perfect. Two views give shape and no size is the reminder that every such check is a check on shape — the size of the facade is absent throughout, and none of these tests can supply it.

What the unfolded shape shows #

The earlier essay drew the shape that a symmetric object gives back, unfolded into plan, against the object it came from.

That figure is the fully symmetric limit of this one. Everything here is the same reading with fewer pairs and a surface to carry the rest, and the difference in the answer is small: a per cent in shape from a sixth of the marks. The earlier essay treated symmetry as a property an object has or lacks, to be tested before the reading. On the evidence here it is better treated as a resource an object has some of — measured by how wide its widest pairs are, and checked in two directions rather than one.

What one photograph of a symmetric object is worth #

It is worth setting the reading against the obvious alternative, which is a second photograph. The image of the other eye recovers two cameras’ relation from forty-four marks and needs eight of them at the least; the symmetric reading needs two pairs, because the object’s own symmetry has already fixed everything about the second view except one direction. Two mirrors are three cameras found the same economy in glass — three views for five numbers — and a symmetric object is the one-mirror case of that economy, with the mirror built into the object.

What the second photograph buys that symmetry cannot is independence. A second view sees every point from another place, symmetric or not, and so it sees a false pair’s depth error directly, where symmetry sees it only through the wall. And what neither buys is size: the one thing a single view cannot give is still missing after the symmetric reading, and a wrong match is not a small error is the warning about what a single bad pair does to everything fitted from it — the false pair here is exactly such a match, made by the builder rather than the reader.

What was assumed #

The lone marks lie on a surface the pairs fix. A door and a sign on the front wall are carried to the wall; a chimney standing off the roof, or an extension behind the facade, is on no surface the pairs have fixed, and its depth is the one thing its ray does not say. Such a mark needs a surface of its own, or a second photograph.

The symmetry plane is the only mirror. A facade with two planes of symmetry — a square pavilion — gives two epipoles, and marks that pair across one plane and not the other must be assigned to the right one. Assigning a pair to the wrong plane is a false pair of a new kind.

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 wide pairs even more than an unweighted one does.

Still open: whether the wall exposes a pair set too wide #

The last test described above — a false pair’s recovered points lying off the wall — is the natural complement to the concurrence, and it has not been measured. It uses exactly the information the concurrence ignores: the depth a pair’s triangulation gives, compared with the depth the rest of the facade implies.

The measurement that settles what it is worth takes the facade with one pair set a stated distance too wide, fits the wall from every other pair on it, and asks how far the false pair’s recovered points lie off that wall, in centimetres and in pixels of reprojection, against the offset — and at what offset that departure exceeds what the reading noise alone produces on a true pair. If a window set five centimetres too wide lies measurably off its own wall, the two tests together cover every direction a false pair can be wrong in, and a reader of a single photograph of a partly symmetric object has a complete check. If it does not, the direction along the normal remains a blind spot of any one-picture symmetry reading, and the only remedy is a second photograph from another side.