Counting the eyes needs the room

Two rungs have measured a two-centre picture and then taken the second centre away. Both are true and they read as contradictory, so it is worth settling what has actually been shown.

The marks are identical in both. Nothing in the drawing changed between the two answers.

What the fit is actually given #

Every measurement in a picture with two eyes in it takes a pair: a drawn mark, and the world point it is a mark of. From the pair it builds a line in space, and it asks for the point all the lines pass through.

The world points are the part the picture does not contain.

So the fit is not reading the picture. It is reading the picture and a claim about what is in front of it, and the residual it reports is the disagreement between the two.

The three questions, which are different #

Untangling this needs three questions kept apart.

Was this picture drawn from one centre? A question about the making of it. Unanswerable from the picture, and answerable if the scene is known.

Is this picture a projection of a stated scene from one centre? Answerable, and it is what the fit answers. The refusal is a refusal about the pair.

Is this picture a projection of anything from one centre? Answerable, and the answer is nearly always yes — because the absorbed scene exists and is an ordinary room.

Once separated, the apparent contradiction goes away. The previous two rungs answer the second question and the third, and neither of them answers the first.

What “nearly always” is hiding #

The third answer is not unconditionally yes, and the exceptions are worth having.

A picture with no depth in it cannot be caught at all. A plane seen from two centres differs by a homography of the picture, and a homography of a picture of a plane is a picture of the same plane from somewhere else. So a two-centre picture of a flat thing is a one-centre picture of that flat thing, with no shearing of anything, and the second eye leaves no trace whatever.

A picture whose parts do not overlap in depth is harder. The shear grows with depth, so parts that sit at nearly the same distance are displaced by nearly the same amount, and the absorbed scene is nearly a rigid copy of the real one.

And a picture of something whose shape is known is caught immediately. That is the case the fit is testing, and it is why the answer to the second question is a refusal: the room’s measurements are the extra information, and the moment they are supplied the two-centre picture has nothing left to hide behind.

How much room-knowledge it takes #

“Told the room” is doing a lot of work in the sentences above, and it can be made quantitative.

The fit does not need the whole room. It needs enough of it that the shear cannot be absorbed into a rigid motion, and that is a small requirement: the shear is three parameters — a direction and a rate — so any three independent metric facts about the scene that the shear would violate are enough to catch it.

One right angle spanning the seam is one such fact. Two lengths in different directions are two. A single known solid with three edge directions is three and to spare.

What is not enough is a great deal of information about a single plane. A floor covered in tiles of known size, however many, says nothing about the number of eyes, because the whole plane is absorbed by a homography and homographies of a plane are pictures of it.

So the useful summary is a shape rather than a count: catching a second eye needs metric knowledge that spans depth, and no quantity of knowledge confined to one surface will do it.

One fact catches it and three recover the room #

“Three independent metric facts” is the right order of magnitude and it conflates two different requirements, which are worth separating because they differ by a factor of three and by a great deal of practical difficulty.

The shear has three parameters: two for its direction and one for its rate. So:

To catch the second eye, one fact is enough. A single known right angle spanning the seam is one equation the absorbed room generically fails, and failing is all that catching requires. A reader who knows one corner of the room is square, and can see that corner drawn out of square, has the answer.

To recover the true room, three are needed. Undoing the shear means determining all three of its parameters, so three independent metric facts are the minimum — a box supplies them at once, which is why a box is the object this collection reaches for whenever a scene has to be pinned down.

Between the two lies a useful middle. With one fact the shear is confined to a two-parameter family and the room is known to be wrong without being known; with two, to a one-parameter family. So a reader with partial knowledge is in the position this collection is in everywhere else — holding a family rather than an answer — and the family’s dimension is three minus the count of facts.

The section above’s warning about planes can be sharpened by the same parameter count. A shear displaces each point along a fixed direction by an amount proportional to its depth, so every plane parallel to the picture plane is moved rigidly — translated, not distorted — and no metric fact confined to such a plane is violated by any shear whatever. That is a stronger statement than “a plane cannot tell them apart”: it identifies exactly which planes are useless, and they are the ones facing the reader.

A tilted plane is a different matter. A floor running away from the viewer is sheared within itself, so its square tiles come out as rhombi in the absorbed reading, and a reader who knows the tiles are square has one fact after all. The distinction is between a scene that is a plane — where the absorbed reading can be the same plane and nothing is violated — and a plane inside a scene with depth, where the shear is fixed by the rest of the scene and the plane then reports on it.

Which gives the practical instruction a shape. Look for a right angle between a depth direction and a cross direction, and do not look at anything flat and facing the camera. That is one glance rather than a survey, and it is available in almost every picture that has a floor and a wall in it.

It also says where the field’s own conventions sit. A carpet with its people upright puts its two grounds in planes at right angles, so the seam spans depth by construction and one right angle catches it immediately — which is why the convention has never fooled anybody about being a photograph. A parallel floor under a perspective room is worse off again: its absorbing map is projective rather than affine, so it violates ratios as well as angles, and a reader who knows nothing metric at all but can see that evenly spaced tiles ought to be evenly spaced has already caught it. And a two-ground panel sits between them, at an angle the shear’s own arctangent names.

So the conventions are ordered by how little a reader needs to know to catch them, and the ordering is the reverse of how strange they look.

The general shape this belongs to #

This is not a special situation. It is the standing shape of every single-view question on this site.

A single view fixes the geometry of a scene up to a similarity, so the one thing a single view cannot give is a length — and the way that is shown is to produce two scenes at different scales that draw the identical picture.

Two views fix shape and not size, and two views give shape and no size exhibits the family rather than asserting it. Many views leave seven directions untouched, and seven numbers no picture can name walks along them.

The number of centres is one more entry on that list. It is free, the family is a one-parameter group of shears, and what closes it is knowledge from outside the picture.

What it takes to close it, in practice #

Three things, in the order a person would reach for them.

A rigid object of known shape in the picture. A rectangle, a box, anything with a right angle. If it spans the seam it is caught at once; if it lies entirely inside one part it says nothing, because that part is an ordinary projection.

A second picture of the same scene. Two views of a scene drawn from two centres each are four rays per point and no consistent reconstruction, and the inconsistency shows without any prior knowledge of what is in the scene.

Or a straight line spanning the seam. A line drawn from one centre is a line; a line whose near half was drawn from one centre and far half from another is two lines with a kink in them, and the kink is visible.

That last one is the practically important one and it is why the seam matters. A two-centre picture is undetectable as long as nothing crosses the seam, and it is caught the moment something does — which is exactly the compositional rule the traditions that use the convention observe.

The convention’s side of it #

A tradition that draws a floor from above and its figures from in front is not attempting a photograph and failing. It is making a choice, and this rung says what the choice costs in the only currency this site deals in.

It costs the relation between the parts. Each part, on its own, is an ordinary projection of an ordinary thing: the carpet is a correct picture of a carpet, the figures are correct pictures of figures. What is not correct is where they stand relative to each other, and that is exactly the quantity a shear moves and a rigid motion does not.

Which is a good trade if what the picture is for is identifying what is present and how the floor is laid out, and a bad one if what it is for is measuring where anything stands. That is a statement about what a picture is for, and the geometry can price it and cannot decide it.

What this does not say #

It says nothing about whether a viewer notices. A composite picture is read by people as a picture and the geometry is not what they are reading. That is a fact about seeing and this site has no standing on it.

It says nothing about the order the parts were drawn in, or which eye came first. The construction picks one centre and absorbs the others onto it, and picking a different one gives a different absorbed scene drawing the identical picture. There is no first eye in the geometry; there is a first eye in the choice of which one to keep.

It says nothing about detection algorithms. What is described here is what information is present, not how to extract it, and a method for extracting it from an image would be a claim about image data — which this site’s own rulings put outside its scope.

And it does not say the absorbed room is a good description. It is a consistent one, which is a much weaker property, and the whole point of the previous rung’s measurements of what the shear costs is that consistency is cheap.

The refusal is real, and it is worth saying which one #

Two different things in this essay are called refusals and they should not be confused.

The fit’s refusal — the rays missing by two thirds of a metre — is a statement that a stated pair of scene and picture is inconsistent with any single centre. It is exact, it is measured in metres of the world, and it is what the previous rung’s slider drives.

The collection’s refusal — the strip that says a figure has no viewpoint — is a statement about a figure, and it is under control rather than available on request: exactly the figures on a named list may print it, every one of them must, and each one’s configuration has to be shown to have no centre by this same solver before it is allowed to.

The list now has four reasons on it — bent light, a moving eye, a curved reflector, and two centres — and the fourth is the only one that can be taken away without changing a drawn mark. That is what makes it the interesting entry and it is why the absorbed figure prints the opposite strip.

An exemption a figure could take silently would be a hole rather than an exemption. One that has to be named beside the number justifying it is a decision on the record, and the number is the same kind of number in all four cases because the same solver produces it.

The same question about the other three reasons #

It is worth running the essay’s own distinction over the other entries on the no-viewpoint list, because two of them answer differently and the difference is informative.

A refracted picture. Told the scene, the rays refuse. Told nothing, is there a scene whose ordinary projection this is? Not in general — the displacement a slab adds depends on how far away the point is, in a way no single projective map reproduces, so the refracted picture is not a projection of anything from anywhere. That is a stronger statement than the two-centre one and this collection has made it.

A handscroll. Same answer, same reason: the eye’s position varies continuously along the roll, so absorbing it would need a different map per column and there is no single map of space that does it.

A curved mirror. Same again — the reflected bundle is spread over a region rather than issuing from a point, and no rearrangement of the scene collects it.

So the two-centre picture is the only one of the four that can be absorbed, and the reason is that it is made of finitely many honest projections rather than of one dishonest one. That is worth having as the sharp version of this essay’s finding: it is not that a picture without a centre can generally be re-read as one with a centre, it is that a picture assembled from projections can.

The transferable form #

“Can this be detected?” is not a question about the thing. It is a question about what else is known, and the honest answer names the extra information rather than reporting a yes or a no.

The two-centre picture is undetectable given the picture, detectable given the room, and undetectable again given only that the room is some room. Three answers to what looks like one question, and the difference between them is which side information was on the table.

These essays have now met that shape three times over: a drawn rectangle’s proportion is undetermined until the centre of the picture is supplied; a ball’s outline gives a direction and no size until a length is supplied; and a picture’s number of eyes is free until a shape is supplied. In each case the free thing was a named object and the closing information was one number or one known solid, and in each case reporting a single figure without naming the assumption would have been the error.