Where each closure enters the stratification

Five facts that close the same gap takes the gloss that one length has to come from outside the photograph and finds five options rather than one: a length on the ground, the camera’s own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise, and their spreads run from 0.36 to 3.60 per cent.

Read that way they are five instruments of different quality. Read against what a picture actually determines, they are not five of anything: two of them do a different job.

Four readings, one picture #

The projective stratification is four statements about what survives a class of maps, and each can be turned into a reading taken from a picture.

A cross-ratio of four collinear ground marks survives any projective map of the plane, so it is what a picture of a plane determines on its own, with no closure at all. A ratio of two segments along one direction survives an affine map, so it needs the plane’s vanishing line. An angle between two ground directions survives a similarity, so it needs the metric structure. And a length in metres survives nothing weaker than a rigid motion, so it needs the last level.

Those four are computed here through the ground map each closure supplies, on one picture, with one set of marks. The picture is the same one the earlier measurement used, and the marks are chosen so that the truth is known: the cross-ratio is 1.2833, the ratio 0.6957, the angle a right angle, the length 2.20 metres.

Three return all four, and two return three #

Run the five closures against the four readings and the table splits.

A length on the ground, the camera’s height, and a standing object of known height each return all four quantities exactly — the cross-ratio and the ratio to fourteen digits, the angle to thirteen, and the length to the arithmetic floor.

A repeated object of unstated size and the focal length with the horizon return the first three exactly and leave the fourth free. That is not a claim about precision; it is a claim about what is there. A repeat of unknown length fixes every ratio on the ground and says nothing about the metre, and a focal length with a horizon fixes the plane’s orientation and says nothing about its distance.

A wrong scale goes into exactly one reading #

Leaving the fourth reading out would only show that a number nobody supplied is unknown, which is not a measurement. So the two scale-free closures are given the missing scale ten per cent wrong instead.

The length then comes back 10.00 per cent wrong. The cross-ratio comes back right to two parts in ten thousand million million, the ratio to two parts in a hundred million million, and the angle to five parts in a hundred million million of a degree.

So the error does not leak. A reader who assumes a paving slab is 600 millimetres when it is 660 gets every proportion in the picture exactly right and every metre ten per cent wrong, and there is nothing in the first three readings to warn them.

Why the split is two and three rather than one and four #

It is tempting to read the five as a sequence, each buying one more level than the last. They are not: three of them jump straight to the top.

The reason is that the stratification’s levels are not bought one at a time by these particular facts. A length is a Euclidean fact and supplies the Euclidean level directly, taking the intermediate levels with it — because a plane whose metre is known has its angles and its ratios known too. The camera’s height is a length; a standing object’s height is a length; a measured distance on the ground is a length.

The other two are shape facts. A repeat says two things are the same size without saying what size, which fixes the plane’s similarity structure and stops. The focal length with the horizon fixes the plane’s orientation relative to the camera, which is the same structure reached by another route.

So the five divide into lengths and shapes, and the stratification’s four levels are not four steps anybody climbs — they are a classification of what a fact is, and every fact lands on the level its own units put it on.

A level is a claim about a class of maps, not about a quantity #

One thing is worth making explicit, because the word “level” invites the wrong picture.

A level is not a quantity that becomes known. It is a class of maps that the recovered ground map is known up to: projective, then affine, then similarity, then rigid. Each reading above is chosen as a quantity that is invariant under one class and not the next, so the reading is a test of which class the map has been narrowed to rather than a measurement of anything in the scene.

That is why a wrong scale shows up in one reading and no others. A map wrong by a scale is still in the similarity class, so every similarity invariant survives it untouched; only the quantity that distinguishes a similarity from a rigid motion — a length — can see it.

It also says how to design a check. Given a suspect closure, the reading that will catch it is the one invariant under everything above the level the closure is supposed to supply and not under that level itself. A closure claiming to give the metric structure is checked with an angle, and checked with nothing else; one claiming to give metres is checked with a length.

The exact subtraction, in numbers of parameters #

The stratification has an arithmetic that makes the division above exact rather than descriptive, and it is worth setting out because it says what each closure is worth in a currency that adds up.

The projective transformations of the plane form an eight-parameter group. The affine ones have six, the similarities four, the rigid motions three. So knowing the vanishing line is worth two parameters, knowing the metric structure two more, and one length the last one — 2 + 2 + 1 = 5, taking eight down to three.

A closure that supplies a length supplies the last parameter and, in practice, everything above it too, because the construction that uses it has already used the plane’s orientation. A closure that supplies a shape supplies the four above and not the last one.

The assumptions priced in conditioning rather than in kind runs the same stratification in the other currency — how well each level is determined rather than whether it is — and the two readings are independent. A level that is supplied can still be supplied badly, which is what the spreads of 0.36 to 3.60 per cent are about.

What a picture gives with no closure at all #

Before any of the five, a picture of a plane already gives something, and it is worth stating because it is the level everything else is built on.

Four collinear ground marks have a cross-ratio, and it comes out of the picture with no assumption whatever — no calibration, no horizon, no known length, not even a plane whose orientation is known. Four lines have a cross-ratio is the standing account of the invariant and the bias out of reach of what reading it costs in practice.

That is a genuine measurement and it is a narrow one. A cross-ratio compares four things on one line; it cannot compare two lines, cannot say whether a quadrilateral is a rectangle, cannot give an area, and cannot give a metre. Nearly everything a person wants from a photograph of a plane is above that level.

Which is why the five closures matter at all, and why the division measured here is the useful thing to know about them. The bottom level is free and nearly useless; the second and third are bought by a shape fact; the fourth needs a length and nothing else will do.

Where the same split turns up with two photographs #

The division between shape facts and length facts is not a property of single-view metrology. It is the same division two views give shape and no size finds with two photographs, arriving from a different direction.

There, every pairwise distance ratio in a courtyard comes back to fourteen digits and the courtyard’s size is absent rather than uncertain; a reconstruction three and a half times larger fits the same two pictures exactly as well. The closure needed is one length, and the same three kinds supply it.

What two views add is the metric level for free: a calibrated pair fixes angles and ratios in three dimensions without any assumption about a plane, where a single view needs a shape fact to reach the same place. So the stratification is the same stratification and a second photograph is worth exactly the two middle levels — which is a clean way to say what a second exposure buys, and it is not the way it is usually said.

What a reader should take from the split #

Three practical statements, and the first is the one that changes what anybody does.

If proportions are what is wanted, two more closures are available than the catalogue suggests. A picture with a repeated feature in it, or a known focal length and a findable horizon, gives every ratio and every angle on the ground exactly — no tape measure, no known object. That is enough for a plan, a rectification, a comparison of two areas, or a shape fitted to a facade.

If metres are what is wanted, only three of the five will do, and the other two need a length supplied from somewhere anyway. A height, out of one photograph is the worked case of the cheapest of the three: the photographer’s own eye height, which costs nothing to know and is exactly what the horizon’s position encodes. Combining a repeat with a single measured length is the usual practical arrangement, and the repeat is then doing the shape work and the length the scale work — two facts of different kinds rather than two of the same kind.

And a wrong assumption about a free scale is invisible in every reading but one. That is worth carrying because the readings a person takes first — is the wall square, are these two bays the same, does this quadrilateral rectify to a rectangle — are all shape readings, and all of them look perfect on a picture whose metre is ten per cent out.

The two shape closures are not the same shape closure #

Both scale-free closures stop at the same level, and they are not interchangeable, which is worth separating out because the table above cannot show it.

A repeated object supplies the similarity structure through the plane’s own content: two instances of one thing, wherever they lie, fix the map that carries the plane to a Euclidean copy of itself up to scale. It needs nothing about the camera at all, and it survives a lens whose focal length nobody knows.

The focal length with the horizon supplies the same structure through the camera: the horizon gives the plane’s vanishing line and the focal length turns two image directions into two world directions, which fixes angles. It needs nothing in the scene to repeat.

So the two are available in different photographs. A brick wall supplies the first and no camera data; a photograph with a findable horizon and an EXIF focal length supplies the second and needs nothing to repeat. A reader with one of them has the metric level; a reader with both has it twice, and the disagreement between the two is a check — which is the only place in this catalogue where two closures can be compared without a third fact, and it is not measured here.

What this does not settle #

The ground is one plane and the marks are on it. Everything here is the metrology of a single plane, which is what the closure catalogue is about. A scene with several planes has a vanishing line for each and the levels are reached separately on each, and how a closure on one transfers to another is the question the assumptions priced in conditioning is closest to and does not answer either.

The readings are noiseless. The four readings above are computed on exact marks, so the errors reported are structural rather than statistical. Adding the marking error puts a spread on each, and the spreads differ by level — the cross-ratio is the best-conditioned of the four and the length the worst — which is a second table and not this one.

The closures are taken one at a time. A real reading usually has two or three available, and combining them is not the same as choosing the best: two closures that reach the same level give a check, and two that reach different levels give a reading neither gives alone. A floor with a referent is the case where a second fact is genuinely a different kind of fact, and nothing here measures what the combinations are worth.

And the focal length is taken as known. The one thing a single view cannot give states the scale ambiguity in its general form, and a focal length recovered from the picture rather than supplied carries its own error into the metric level, which would make that closure’s third reading inexact as well as its fourth.

Still open: which level a wrong assumption actually damages #

The measurement above sets the free scale wrong and finds the error confined to one reading. The more useful question is the same one for an assumption that is wrong in kind rather than in size.

A repeated object assumed to be a repeat when the two instances differ slightly; a horizon found a few pixels out; a focal length from a lens’s markings rather than a calibration; a ground assumed flat that slopes. Each of those is an assumption that is not merely mis-valued but mis-specified, and there is no reason such an error should stay on one level — a sloping ground, for instance, changes the vanishing line, which is the affine level, and the damage should therefore appear in the ratio reading as well as in the length.

The measurement that settles it perturbs each assumption in its own natural way, reads all four quantities, and reports which levels move. What would make it worth having is the possibility of a diagnostic: if a mis-specified assumption damages the ratio reading while a mis-valued one does not, then comparing a ratio read two ways in a picture would say which kind of mistake has been made — and the reader of a single photograph currently has no such test at all.

The short version #

The five closures of single-view metrology are not five instruments of one kind. Read through the ground map each supplies, three of them — a length on the ground, the camera’s height, a standing object of known height — return a cross-ratio, a ratio along a line, an angle and a length, all four exactly. The other two, a repeated object of unstated size and the focal length with the horizon, return the first three exactly and leave the fourth free.

Given the free scale ten per cent wrong, those two return the length 10.00 per cent wrong and the other three readings right to thirteen digits. So the division is between facts stated in lengths and facts stated in shapes, the first supplying the last level of the stratification and the second stopping one short — and a wrong assumption about the missing scale is invisible in every reading a person is likely to take first.