A wrong model shows at its own level

Where each closure enters the stratification placed the five facts that turn a photograph’s ratios into metres on the projective stratification. Read on four quantities — a cross-ratio, a ratio along a line, an angle, a length — three of the five returned all four exactly and two returned only the first three. Then it set the free scale ten per cent wrong and found the whole ten per cent in the length and nothing anywhere else, to thirteen digits.

That was an assumption wrong in size: the right model with a wrong number in it. The essay closed on the more useful question of an assumption wrong in kind — a repeated object assumed a repeat when the two instances differ, a horizon found a few pixels out, a focal length taken from a lens’s markings, a ground assumed flat that slopes. There is no reason such an error should stay on one level. And if a wrong model damages a level that a wrong number does not, then comparing two readings of one picture might say which kind of mistake has been made — a test a reader of a single photograph has never had.

The answer is a rule, and the rule gives the test.

Seven wrong assumptions, six readings #

The picture is the one the earlier essays used: a camera 1.62 metres up, looking down at a stretch of flat ground with marks on it whose true positions are known. The readings are the four of the stratification plus two that split them further. A ratio along a line is read twice, once on a line across the view and once on a line running into it, because a wrong plane treats the two differently. An angle is read twice, once as a right angle along the ground’s own axes and once as a forty-five-degree angle between them, because a wrong shape treats those differently too.

The table sorts itself. A length ten per cent wrong — the earlier essay’s case — moves the length by exactly ten per cent and nothing else. A repeated object whose shape is assumed ten per cent wrong, one assumed square that is three degrees out, and a focal length ten per cent wrong leave every ratio along every line exact, the cross-ratio exact, and bend angles and lengths. A horizon five pixels out and ground sloping three degrees leave the cross-ratio and the ratio across the view exact, and stretch the ratio along a receding line as well as everything above it. A road crowned by ten centimetres moves the cross-ratio itself.

That is the rule. An assumption belongs to a level — it is the fact that buys that level — and when the fact is wrong, the error lands on its level and every level above, and on nothing below. A length buys only the Euclidean level, so a wrong length damages only lengths. A shape or a focal length buys the metric level, the upgrade from parallel to square, so a wrong one damages angles and lengths. A plane’s orientation buys the affine level, the vanishing line, so a wrong one damages ratios along receding lines too. And the projective level is bought by the ground being a plane at all, so only a ground that is not one damages the cross-ratio.

A wrong plane stretches only what recedes #

The slope is the case the earlier essay predicted would reach the ratio reading, and it does — but only in one direction, and the direction is the point.

As the ground tilts away from level, a ratio of two segments along a line running into the view goes wrong fast — 2.55 per cent at half a degree, 9.79 at two, 22.65 at five. A ratio along a line running across the view stays exact at every slope, and so does the cross-ratio. The length, which mixes directions, goes wrong more slowly, 3.34 per cent at five degrees.

The reason is the geometry of the mistake. The camera closure assumes the ground is level and 1.62 metres below the eye, and carries every mark down its ray to that imagined plane. The real ground is tilted about a level line across the view. The map from the real ground to the imagined one is then a perspective from the eye between two planes that meet in that level line, and such a map sends every line parallel to the meeting line onto a line parallel to it, scaled by one factor along its whole length. Ratios along such a line survive. A receding line is stretched unevenly — its near end hardly moved, its far end moved a great deal — because the vanishing line the closure assumed is not the ground’s vanishing line. The ramp has its own horizon: a tilted plane vanishes along its own line, and a closure that reads it with the level ground’s horizon has the wrong affine structure; a horizon five pixels out does the same thing as a slope, by the same mechanism.

So the earlier essay’s guess was right in kind and wrong in detail. A sloping ground does reach the ratio reading, but a ratio read across the view would never show it. The level that is damaged is damaged only in the directions its wrong vanishing line distorts.

A wrong shape keeps every ratio #

The focal length is the opposite case: an assumption that sounds as though it should affect everything, and affects less than the slope.

A reader who takes the focal length from the barrel when the lens is focused close — focusing is a zoom, and the barrel’s number is the lens at infinity — but finds the horizon correctly in the picture has the ground’s vanishing line exactly right. Every ratio along every line then survives, receding or not, because ratios along lines are what a vanishing line fixes. What the wrong focal length changes is the angle the ground makes with the picture: re-reading the camera’s pitch from the right horizon with the wrong focal length gives the wrong pitch, and the ground comes out stretched in depth against its width. That is an affine distortion of the ground, which keeps parallel lines parallel and ratios along them exact, and bends every angle and every comparison between lengths in different directions.

It is the same level as a repeated object whose shape is assumed wrong. A repeated object of unstated size buys the metric level by being assumed square, or of a known aspect; assumed wrong, it gives the same kind of stretch. The table shows one wrinkle worth keeping: a wrong aspect stretches the ground along its own axes, so a right angle along those axes stays square and the forty-five-degree angle bends. A skewed repeat bends both. What they share is that no ratio along a line moves.

Two tests a reader can run #

The rule becomes a diagnostic because the levels can be checked separately, and each check needs only something the picture already contains.

The depth test reads two equal lengths running across the view at different depths — two paving slabs, two windows of one row, two parked cars — and asks whether they come out equal. It is a test of the affine level: a closure with the right vanishing line gives equal parallel lengths equal readings wherever they are, and one with the wrong vanishing line does not. Ground sloping three degrees fails it by 14.5 per cent; a horizon five pixels out by 2.1. A wrong shape, a wrong focal length and a wrong length pass it exactly.

The square test reads two equal lengths at right angles from one corner — the two sides of a square slab, the width and depth of a known object — and asks whether they come out equal and square. It is a test of the metric level. A wrong focal length fails it by 9.0 per cent, a wrong aspect by 9.1, a three-degree skew by 5.2, a crowned road by 13. A wrong plane fails it too, since the metric level sits above the affine one, and the depth test is what separates the two: a picture that passes the depth test and fails the square test has the right plane and the wrong shape.

A wrong length passes both, exactly, at any size — fifty per cent wrong, and both tests read one to the last digit. That is not a weakness of the tests. It is the one thing a single view cannot give: every ratio is in the picture and no size is, so a wrong size leaves every ratio untouched and nothing in the picture can object to it. The diagnostic separates the kinds of mistake that change ratios and is silent, by necessity, on the kind that does not.

How small a mistake each test sees #

A test is only worth running if it sees mistakes smaller than the ones that matter, and the marks’ own reading error sets how small.

With every mark read to 0.4 pixels, the depth test through the camera closure scatters by 0.65 per cent and the square test by 2.1 per cent, and a mistake is seen when it moves a test three times further than that. The depth test then sees ground sloping 0.38 degrees and a horizon 4.6 pixels out. The square test sees a focal length 6.8 per cent wrong and a road crowned by 4.9 centimetres. Those are useful thresholds: a slope of a third of a degree is a ramp nobody would notice underfoot, and a focal length seven per cent off is what close focusing does to a standard lens.

Through a small reference object — a slab a metre by sixty centimetres — every reading is noisier, because four marks close together fix a map poorly, and the square test then sees only a shape twenty-five per cent wrong or a corner eleven and a half degrees out of square. The lesson is the one which reference to measure from drew for precision: the size of a reference in the picture decides how well it does its job, and here the job includes testing itself.

What stays exact, and why that is the test #

The diagnostic rests on the readings that do not move, and it is worth being plain about why they do not.

Each level of the stratification is a class of maps, and a reading belongs to a level if every map in the class leaves it alone. A wrong assumption replaces the true map from picture to ground with a wrong one, and the wrong one differs from the truth by some map. If that difference belongs to a class — a similarity, an affinity, a projectivity — then every reading that class leaves alone is left alone. A wrong length differs by a similarity; a wrong shape or focal length by an affinity; a wrong plane or horizon by a projectivity that is not affine. The table is those three classes read off one picture.

A crowned road differs by no map of the plane at all, because the crowned ground is not a plane, and so nothing is safe — even the cross-ratio of four marks in a line across the road moves, by 2.3 per cent for a ten-centimetre crown. That is the one mistake the four levels cannot localise, and the square test, which reads it at 4.9 centimetres, sees it only because it sees everything above the projective level.

This is why the round trip through the stratification was worth making. The earlier essay placed each closure on a level by what it buys. The same placement says what each closure’s failure costs, and a reading that should be invariant under a class of maps becomes a test of whether the true map and the assumed one differ by that class.

Where the stratification has met these mistakes before #

None of the seven mistakes is new to this collection; what is new is seeing them sorted by one rule. A height from one photograph rests on the camera closure and so on a level ground and a right horizon, and inherits exactly the affine sensitivity the slope sweep draws: a height read against a receding reference is as wrong as the ground’s vanishing line. Two views give shape and no size is the same stratification with two photographs, where the fact that buys each level is different — a known motion, a known calibration — and a wrong one damages the same readings for the same reason.

The shape of the rule also explains why the floor a better camera cannot reach is where it is. Sharper marks shrink every reading’s scatter, and they shrink the tests’ scatter with it, which lowers every threshold in the last figure in proportion. What they cannot shrink is a wrong model’s error, which is a fixed amount set by the mistake; so the floor that essay found — better marks stop buying a better answer at the assumption’s own error — is also the point at which the mistake stops being hidden by the noise. Once the marks are good enough to reach it, one of the two tests can say so — unless the mistake is a wrong length.

What a reader should do with it #

The practical procedure is short. Before trusting a single-view measurement, find two equal things at different depths and two equal things at right angles, and read both pairs through the same closure as the measurement. If the depth test fails, the plane is wrong — a slope, or a misplaced horizon — and every ratio along a receding line is suspect. If the depth test passes and the square test fails, the plane is right and the shape is wrong — a focal length, or an assumed aspect — and every angle is suspect while every ratio along a line is sound. If both pass, the picture has told the reader everything it can, and the remaining risk is the one no picture can report: that the length used to set the scale is itself wrong.

What was assumed #

The marks are on the plane the closure reads. Every mark here lies on the ground. A mark on a kerb or a step is off the plane by construction, and reads as a crowned road does: as damage the levels cannot localise.

Each mistake comes alone. A slope and a wrong focal length together fail both tests, and the depth test alone cannot then say how much of the square test’s failure is the focal length. Two mistakes of different levels can still be separated by fitting both — a slope from the depth test, then a focal length from what the square test has left — but that is a fit, not a test.

The test pairs are known to be equal. Two slabs of one pavement are equal by manufacture; two cars are equal only if they are the same model. A test pair that is not what it is assumed to be is itself a wrong assumption, and fails the test for the wrong reason.

Still open: whether one test pair can separate a slope from a horizon #

The depth test sees a slope and a misplaced horizon alike, because both give the closure the wrong vanishing line. They are not the same mistake. A slope is a fact about the ground, and a second picture of other ground, or the same ground further away, would show a different vanishing line. A misplaced horizon is a fact about the reading, and every plane in the picture inherits it.

The measurement that settles whether one picture can tell them apart places a second test pair on a surface that is not the ground — two equal windows on a facade, whose vanishing line is set by the wall’s orientation rather than the ground’s — and asks whether a misplaced horizon, which moves every vanishing line the reader constructs by the same amount, and a sloping ground, which moves only the ground’s, leave the facade’s test in different states. If the facade passes while the ground fails, the ground slopes; if both fail by related amounts, the horizon is wrong. The number to find is how precisely the two failures can be told apart with marks read to 0.4 pixels, and at what slope the answer becomes unambiguous.