A plumb wall tells a slope from a horizon, faintly

A wrong model shows at its own level sorted the mistakes a single-view measurement can make by the level of the projective stratification they damage, and turned the sorting into two tests a reader can run on any picture. The depth test — two equal things across the view at different depths — catches a wrong plane. The square test — two equal things at right angles — catches a wrong shape. Both were run on the ground, through the camera closure, with marks read to 0.4 pixels.

The depth test could not say which of two wrong planes it had caught. A ground that slopes and a horizon found a few pixels out both give the closure the wrong vanishing line for the ground, and both stretch a receding ratio in the same way. They are different mistakes, though. A slope is a fact about the ground; a misplaced horizon is a fact about the reading, and every plane in the picture inherits it. The essay proposed placing a second test pair on a surface that is not the ground, so that the two mistakes would leave it in different states.

They do, and the separation is exact in kind. It is weak in size, for a reason that says something about walls, and one orientation of wall cannot see it at all.

A plumb wall does not care what the street does #

The picture is the one the earlier essays used: a camera 1.62 metres up with a 700-pixel focal length, looking down a stretch of marked ground. To it is added a wall standing plumb 8.5 metres away, facing the camera, with four level edges on it — the sill and head of a window below and of one above, at 0.4, 1.2, 2.0 and 2.8 metres. The ground’s test pair is the one from the earlier essay. The wall’s test pair is its lowest and highest level edges, which are of equal width.

To read the wall, the reader needs its plane, and a single picture gives it from two assumptions the reader would make anyway: that the wall is plumb, so its vertical is the closure’s vertical, and that its level edges run in the direction the closure’s camera assigns to their vanishing point. With both, the edges can be carried onto the wall and their widths compared. The ratio is scale-free, so the wall’s distance never enters, and neither does the one quantity the one thing a single view cannot give says no picture supplies.

A sloping ground changes nothing on the wall. The figure’s solid line runs along the ground’s axis from 2.6 per cent at half a degree to 14.5 per cent at three degrees, while the wall’s test stays at $10^{-16}$: the wall is plumb whatever the street does, the camera closure’s orientation is right, and a plane read from a right orientation is read right. A misplaced horizon moves both. Five pixels out, it reads 2.11 per cent on the ground and 0.20 per cent on the wall; fifteen pixels, 6.4 and 0.60. The two mistakes lie on two different lines through the origin, which is the separation the earlier essay was looking for. Drag the ringed horizon along its line and the proportion never changes: fifteen pixels low reads 6.10 per cent on the ground and 0.59 on the wall, five pixels high 2.11 and 0.20, the wall always a tenth of the ground and always on the same side of the axis as the error.

The lines part slowly. The horizon’s line leans away from the ground’s axis by a tenth — the wall moves a tenth as much as the ground for the same horizon — and the ellipse of the tests’ own scatter at 0.4 pixels is more than twice as tall, relative to its width, as that lean. A slope of one degree fails the ground’s test by 5.1 per cent and leaves the wall alone; the horizon that fails the ground’s test by the same 5.1 per cent moves the wall by about 0.48 per cent, three of the wall’s scatters with this wide a pair. Everything about how useful the separation is follows from that one ratio.

Why a wall is a weak witness of the horizon #

The reason the wall responds so little is geometric, and it is the same geometry that makes the ground respond so much.

A horizon found five pixels out is a camera read as pitched about 0.4 degrees more steeply than it is. Every ray the closure draws is turned by that angle about the camera’s level axis. For the wall, which faces the camera and stands up, the consequence is a lean: the reader’s plumb wall is the true wall tilted by 0.4 degrees about a level line, and two level edges 2.4 metres apart in height, eight and a half metres away, differ in reconstructed width by about the tilt times their height difference over their distance — a fifth of a per cent. For the ground, seen from 1.62 metres, the same turn changes the angle at which every ray meets the ground, and a ray meeting the ground at seven degrees has its landing point moved by the turn divided by seven degrees’ worth of slope. The ramp has its own horizon is the reason a ground seen at a grazing angle is so sensitive to where its horizon is; a wall seen face on is the opposite case.

So the wall’s test moves by 0.040 per cent a pixel, and it moves by that whatever its width. Width decides only the test’s scatter. Level edges a metre wide at 8.5 metres are about eighty pixels long, and marks read to 0.4 pixels place their widths to half a per cent each; the test scatters by 0.93 per cent and sees a horizon only when it is seventy pixels out. Edges six metres wide — a whole facade rather than one window — scatter by 0.15 per cent and see 11.5 pixels. The ground’s test, for comparison, sees 4.7 pixels. There is no width of wall in this picture that sees a horizon as well as the ground does, because the frame runs out first.

Which walls can see it #

The wall above faces the camera. Most walls in a street photograph do not: they run away from the camera along the street’s sides, and the question is whether they are witnesses at all.

Turned from facing the camera, the wall’s test gets steadily worse — 23 pixels at no turn, 34 at forty-five degrees, 74 at seventy, 147 at eighty. Its response to the horizon holds up, because its two level edges still lie at different heights and the pitch still leans it; what fails is the edges’ own length in the picture, which foreshortens as the wall turns, so that their scatter grows faster than the response.

At ninety degrees the test stops responding altogether, to $2\times10^{-15}$ a pixel, and the reason is not foreshortening. A horizon out is a turn about the camera’s level axis. For a wall that runs straight away from the camera, the camera’s level axis is the wall’s own normal, and a turn about a plane’s normal turns the plane within itself: the reader’s wall is the true wall, rotated in its own plane, and every ratio on it is exact. The side walls of a street, which are the walls most photographs have most of, cannot say whether the horizon is right. Only a wall that faces the camera across the view — a building at the end of a street, a facade across a square — can.

That also decides which test pair to use on the wall. The same argument says that two equal widths on one level line pass whatever is wrong, since a line parallel to the hinge of the error is scaled evenly along its length, and that two equal heights at different distances along the wall pass too. The pair has to be two equal level widths at different heights. A window’s head and sill will do, if the window is tall enough; a facade’s plinth and cornice do better.

When the ground cannot tell them apart #

The practical question is the one the earlier essay posed: given a picture whose ground fails the depth test, how often does the wall say correctly whether the ground slopes or the horizon is out?

For each slope the figure finds the horizon error that fails the ground’s test by exactly as much, so that the ground alone has nothing to go on: a quarter of a degree is matched by 3.1 pixels, one degree by 11.9, three degrees by 32.7. Then it paints a hundred pictures of each with marks read to 0.4 pixels and asks the wall. The rule is the plainest one: call it a horizon when the wall’s test has moved more than half the way to where that horizon would put it, and a slope otherwise.

With the whole facade, six metres wide, the verdict is right 65 times in a hundred at a quarter of a degree, 95 at one degree and every time from two. With a window pair a metre wide it is right 59 times in a hundred at one degree and 82 at four. So the answer to the question “at what slope does the answer become unambiguous” is: about one degree, with a wall as wide as a building, and never within the slopes a street has, with a wall as wide as a window.

One degree is not a small slope for a street — it is a gradient of 1.7 per cent, which a pavement often has and a square usually does not — but it is well above the 0.38 degrees the ground’s depth test detected in the earlier essay. There is a band of slopes, from a third of a degree to about one, that a single picture can see and cannot attribute: the ground’s test says a plane is wrong, and no wall in the picture can say whether it is the ground’s plane or the reader’s horizon.

Three tests and three mistakes #

The wall’s test was introduced to separate a slope from a horizon, but it is a test of the closure’s orientation, and a focal length is part of that orientation too.

A focal length ten per cent long — the kind of error focusing is a zoom warns of, when the number is read off the barrel of a lens focused close — with the horizon right leaves the ground’s vanishing line where it is, so the ground’s depth test passes it exactly, as the earlier essay found. It does not leave the wall alone: re-reading the camera’s pitch from the right horizon with the wrong focal length puts the vertical vanishing point in the wrong place, the reader’s plumb is no longer the wall’s plumb, and the wall’s test reads 0.58 per cent. So the three tests leave three distinct patterns. A slope fails the ground’s two tests and passes the wall’s. A focal length passes the ground’s depth test and fails the wall’s and the square test. A horizon fails all three.

That is a more complete diagnostic than either of the earlier essay’s tests alone, and it has the same weakness this essay found for the horizon: the wall is the faint witness in every pattern. A focal length has to be about eight per cent too long before a six-metre wall sees it, where the square test on the ground sees 6.8 per cent. What the wall adds is not sensitivity but attribution — it is the only test in the set that ignores the ground entirely, and so the only one that can say a failure was not the ground’s fault.

Why the stratification says the wall must see it #

The earlier essays placed each assumption on a level of the stratification and found that a wrong assumption damages its own level and every level above. The wall result is that rule with two planes in the picture instead of one. The ground and the wall have separate vanishing lines, and the reader builds each from a different set of assumptions. The ground’s comes from the horizon, the focal length and the ground being level — the facts five facts that close the same gap listed as ways of buying a plane’s levels. The wall’s comes from the horizon, the focal length and the wall being plumb. A mistake in an assumption the two share — the horizon, the focal length — damages both planes’ affine levels. A mistake in one plane’s own assumption — the ground being level — damages only that plane.

Where each closure enters the stratification described the camera closure as buying the ground’s affine and metric levels at once from a known camera. What this measurement adds is that the camera closure also buys every other plane’s levels from the same numbers, and so every other plane in the picture is a check on those numbers — a check that the ground’s own shape cannot contaminate. A height from one photograph uses a wall this way already, reading heights against the ground; the wall’s own test asks whether the reading that produced those heights was made with the right camera.

The weakness is the strength’s price. A wall is a good check because it does not depend on the ground, and it is a weak check because it stands up: an error that leans a vertical plane by a fraction of a degree changes its widths by a fraction of a per cent, while the same error lays a nearly horizontal plane at a noticeably different angle. The floor a better camera cannot reach is the other face of this. Sharper marks shrink every test’s scatter in proportion, and the wall’s test gains most from them, because its signal is small and fixed while its noise is what the marks set.

What a reader should do with it #

The procedure from the earlier essay gains one step. If the ground’s depth test fails, find a wall that faces the camera — not one along the street — with two equal level widths at different heights, as wide as the picture allows, and read it through the same closure. If it passes and it is wide enough to have seen the horizon that would explain the ground’s failure, the ground slopes. If it fails in proportion, the horizon is out. If it is only a window wide, it cannot say, and the reader is left with a wrong plane of unknown cause. Which reference to measure from made the same point about precision: the size of the thing in the picture decides whether it can do the job, and a wall’s job here needs the whole wall.

What was assumed #

The wall is plumb. A wall that leans — an old facade, a retaining wall battered back — carries its lean into the test exactly as a misplaced horizon would, and a reader who assumes it plumb will blame the horizon for the wall’s own fault. It is the wall’s version of the ground’s slope, and the same test cannot see it; a second wall facing another way would.

The wall’s level edges are level and equal. Two windows of one width on one facade are equal by design; a plinth and a cornice are equal only if the facade has no batter or setback between them.

Each mistake comes alone. A slope and a horizon together fail the ground’s test by their sum and the wall’s by the horizon’s share; the wall then estimates the horizon, and what the ground’s test has left is the slope. That is a fit rather than a test, and its precision is the wall’s, not the ground’s.

The camera closure is the reader’s. Every test here reads through a known camera height, a horizon found in the picture and a stated focal length. A reader using a reference object on the ground instead has no camera orientation to check, and the wall’s test has nothing to test.

Still open: whether the verticals witness the horizon better than the wall’s widths #

The wall’s test reads its level edges and assumes its verticals. The verticals can be read too. Every plumb line in the picture — a wall’s corners, a door’s jambs, a lamp post — runs towards one vanishing point far below the picture, and where that point lies is fixed by the horizon and the focal length: for a camera pitched down by a known angle, it sits a known distance below the principal point, and a horizon out puts it somewhere else. The horizon and the vertical vanishing point are two readings of one camera orientation, and a picture with plumb lines in it can compare them.

The measurement that settles whether that is the better witness takes the same picture, marks the plumb edges of the wall and of anything else standing in it, finds their vanishing point by least squares with the same 0.4-pixel marks, and asks how far that point has to move — how many pixels out the horizon has to be — before the disagreement is three times its own scatter. A camera pitched only seven degrees down puts the vertical vanishing point about six thousand pixels below the picture, where a horizon five pixels out moves it by three hundred; whether plumb lines two hundred pixels long can see that is the question, and whether many short verticals spread across the frame do better than one wide wall.