A facade's corners see the horizon its widths could not

A plumb wall tells a slope from a horizon, faintly looked for a second witness to the camera’s orientation. A picture whose ground fails its depth test might have a sloping street or a misplaced horizon, and the ground alone cannot say which. A wall standing plumb can, because a slope of the street changes nothing about the wall while a misplaced horizon leans it. The trouble was size. A horizon five pixels out moved the ground’s test by 2.1 per cent and the wall’s by 0.2, and even a facade six metres wide saw the horizon only when it was 11.5 pixels out, where the ground saw 4.7.

The essay ended on a witness it had not tried. The wall’s test read the facade’s level edges and assumed its verticals. The verticals can be read too. Every plumb edge in the picture — a wall’s corners, a door’s jambs, a lamp post — runs towards one vanishing point below the picture, and where that point lies is fixed by the horizon and the focal length. Two readings of one camera orientation, which a picture with plumb lines in it can compare. The essay’s own arithmetic made the prospect look doubtful: for a camera pitched seven degrees down the vanishing point is about six thousand pixels below, and a vertical 240 pixels long must be extended twenty-five times its length to reach it.

The arithmetic was measuring the wrong quantity.

The horizon the verticals imply #

For a camera with no roll, the horizon and the vertical vanishing point stand on the principal column either side of the centre of the picture, and their distances from it multiply to the focal length squared: $(v_h - c_y)(v_v - c_y) = -f^2$. A reader who has the focal length and the plumb lines has a horizon, read off the lines’ convergence, without looking at the ground at all.

The picture is the standing camera of the earlier essays — 1.62 metres up, 700 pixels of focal length, pitched 6.78 degrees down at marked ground — with the six-metre facade 8.5 metres away and, for some readings, posts two and a half metres tall standing across the frame 7.5 metres away. Each vertical is read as a line: sampled every eight pixels along its image and each sample read across the line to 0.4 pixels, since a vertical edge has no feature along its own length. The lines are fitted, their common point found by least squares, and the horizon it implies compared with the horizon the reader assumed. The test is a number of pixels of horizon, directly.

Plumb lines lean away from the picture’s centre column, more the further out they stand, and every one of them points at the same place far below. The dots are the leans one picture delivers: from −3.1 to +3.1 degrees across the frame. The solid curve is what the true horizon predicts for each line, and the dots sit on it. A horizon read five pixels out predicts a vanishing point 376 pixels further down, and so leans that are all a little smaller — the dashed curve, which the dots do not follow. The picture’s lines put their own horizon 1.06 pixels from the true one; a reader who had read the horizon five pixels out would find the lines disagreeing with it by 3.9.

Two corners do what the whole facade’s widths could not #

The question the earlier essay put was how far the horizon must be out before the verticals see it at three times their own scatter.

The facade’s two corners alone see a horizon 4.2 pixels out. The same facade read by its level widths saw 11.5. Nothing about the wall has changed but which of its edges is read, and reading its sides rather than its sills makes it 2.7 times the better witness — as good as the ground’s own depth test, which sees 4.6.

Adding verticals helps in the obvious way. The corners and four posts see 2.4 pixels; eight posts across the frame without the corners, 2.4; sixteen, 1.9. The gain falls off roughly as the square root of the count, since each line is an independent reading of the same convergence.

What a vertical is worth depends on two things, and the last two bars isolate them. Eight posts a metre tall — bollards rather than lamp posts — see only 10.8 pixels, because a line’s direction is known to the reading error over its length, and a line two-fifths as long, carrying two-fifths as many samples, is known more than four times worse. Eight full-height posts crowded within a metre and a half of one another see 11.4, because their leans differ only by how far apart they stand, and lines that nearly coincide say little about where they meet. The best witnesses are long and far apart: a facade’s two corners, six metres apart, are exactly that, and a row of lamp posts along a street across the frame is better still.

Why the far vanishing point did not matter #

The worry in the earlier essay was that the vanishing point is very far away and a short line places it badly. Both halves are true. What they miss is that the horizon does not depend on the vanishing point’s distance but on its reciprocal.

Tip the camera from twelve degrees down to two and the corners’ vanishing point moves from 3,300 pixels below the centre of the picture to 20,000. A point twenty thousand pixels away, fixed by two lines 240 pixels long, is placed along the lines to within thousands of pixels. And the horizon the corners imply is known to the same precision at every pitch: 4.18 pixels at three times its scatter at two degrees, 4.22 at 6.8, 4.23 at eight.

The reason is in the product. The horizon’s distance above the centre is the focal length squared divided by the vanishing point’s distance below it — the focal length squared times the reciprocal of that distance. And the reciprocal is what the lines measure directly: a plumb line one column-width away from the centre leans by one over the vanishing point’s distance, so the lines’ lean per column is that reciprocal, read off a slope that the lines’ own length and spread fix. A vanishing point twice as far away is placed four times worse along the lines, and its reciprocal, which is all the horizon uses, is placed exactly as well. The earlier essay’s estimate — a horizon five pixels out moving the vanishing point by three hundred — was right, and it was the reason a far point looked hopeless; it is equally the reason the reading of the horizon is not.

The only thing that changes with pitch is how much of each corner the frame shows. A camera pitched more steeply cuts the facade’s tops off, and at twelve degrees the corners are 199 pixels long instead of 241; their threshold rises with it, to 5.6 pixels. The ground’s test changes even less, from 4.5 to 4.8 across the same range. So the two witnesses are equals at every pitch a standing photographer uses.

A far point is a near reciprocal #

The same shape of mistake has been met before, in another guise. Depth is a reciprocal found a stereo pair’s disparity measuring one over the distance, so that a point twice as far is known four times worse in depth and exactly as well in disparity; the error that looks catastrophic in metres is a constant in the quantity the instrument actually reads. A vanishing point is a point at a distance too — a distance on the picture plane rather than in the scene — and plumb lines read it the same way a pair reads depth: through an angle, which is a reciprocal.

The practical rule that follows is to ask, of any quantity a construction needs, whether it needs the far point itself or only its reciprocal. A drawing that must place the far point on the page to run lines to it — a perspective construction with a vanishing point off the board — needs the point, and suffers from its distance. A reading that uses the point only through a product or a ratio, as the horizon does here, needs the reciprocal and does not suffer at all. The worry the earlier essay voiced about six thousand pixels was the first kind of worry applied to the second kind of use.

Telling a slope from a horizon #

The earlier essay’s practical question was whether a second witness can say, of a picture whose ground fails its depth test, whether the street slopes or the horizon is out. It found that the facade’s widths could at about one degree of slope and a window’s could not within the slopes a street has.

For each slope the figure takes 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 matched by 3.1 pixels, half a degree by 6.1, one degree by 11.9. A sloping street leaves plumb lines plumb, so a picture of a slope reads the verticals’ noise alone; a picture of a misplaced horizon reads that horizon’s error in full. The rule is the one the earlier essay used: call it a horizon when the verticals disagree with the horizon read by more than half of what the matched horizon would give.

The facade’s corners get the answer right 72 times in a hundred at a quarter of a degree, 99 at half a degree and every time from one degree up. With four posts beside them, 96 times at a quarter of a degree. The facade’s widths, on the same pictures, reached 65 per cent at a quarter of a degree and 95 at one. The band the earlier essay found — slopes from a third of a degree to one, which the ground’s test could see and nothing could attribute — is closed by the verticals: from half a degree a single picture with a facade’s corners in it says which of the two it is.

What the verticals cannot separate #

A plumb line reads the camera’s orientation, and the camera’s orientation is read through two assumptions, the horizon and the focal length. The ground’s test responds to the horizon and, as the triad in the earlier essay found, not to the focal length. The verticals respond to both.

A focal length ten per cent long moves the horizon the verticals imply by 17.5 pixels, because the implied horizon’s distance from the centre is the focal length squared over the vanishing point’s distance, and a focal length ten per cent long is a square twenty-one per cent larger. The ground’s depth test does not move at all. A horizon fifteen pixels out moves both: the verticals by fifteen pixels and the ground by six per cent.

So the two tests together place each mistake on its own line through the origin. A focal length off moves the verticals and leaves the ground; a horizon out moves both, in a fixed proportion; and a slope, which the verticals ignore, moves the ground alone. Three mistakes, three directions, and the pair of tests reads them apart with a sensitivity set by the better of the two witnesses rather than, as with the wall’s widths, by the worse. What the verticals cannot do on their own is say which of the camera’s two assumptions is wrong, since both move the implied horizon, and focusing is a zoom is the reminder that the focal length is the one more likely to be.

Why the widths were the wrong edges #

The earlier essay explained why the wall’s widths were a weak witness: a horizon out leans a plumb wall by a fraction of a degree about a level line, and two level edges a couple of metres apart in height differ in reconstructed width by the lean times their height difference over their distance, a fraction of a per cent. That explanation was right, and it says what to read instead. The lean is the signal. The widths turn it into a second-order quantity, a difference of two lengths each measured to the reading error; the wall’s own sides are the lean, read directly as an angle in the picture.

Where each closure enters the stratification placed the camera closure as buying every plane’s affine and metric levels from a known camera, and the earlier essay added that every other plane in the picture is then a check on the camera’s numbers. The verticals are that check at its sharpest, because they are not a plane at all. They are a direction — the one direction every plumb object in the picture shares — and a direction’s vanishing point is the closure’s own quantity. The one thing a single view cannot give is a distance; the vertical vanishing point is not one, and the picture gives it freely.

What a reader should do with it #

The procedure the earlier essays built gains a better second step. When the ground’s depth test fails, read the plumb lines rather than a facade’s widths: the corners of any building that faces the camera, door jambs, lamp posts, sign poles. Read each along its whole visible length, prefer lines that stand far apart across the frame, and read the horizon their convergence implies with the same focal length the closure used. If it agrees with the horizon read to within a few pixels, the ground slopes. If it disagrees in proportion to the ground’s failure, the horizon is out. If it disagrees and the ground passes, the focal length is wrong.

The facade’s widths still have a use, which the earlier essay found: they ignore the ground and they ignore everything about the lens except through the orientation. But as a witness of the horizon they are a weak reading of the same lean the corners carry, and which reference to measure from is again the lesson — the precision is in the choice of what to read.

What was assumed #

The plumb lines are plumb. A building’s corners are vertical to a small fraction of a degree; a lamp post often leans by more, and a sign pole or a tree by much more. A vertical that leans by a quarter of a degree towards or away from the camera shifts its own contribution to the vanishing point by the same order as a horizon several pixels out, and many leaning posts add scatter rather than bias. How much a street’s worth of real posts costs was not measured here; the corners of buildings are the safer witnesses.

The camera has no roll. A rolled camera moves the vertical vanishing point off the principal column, and the product rule holds along the line through the centre perpendicular to the horizon rather than down the column. The reading generalises by fitting the roll from the verticals’ vanishing point’s sideways position, which the lines supply; it was not measured here.

The principal point is the centre of the picture. The product rule is written about the principal point. A lens shifted to keep verticals parallel — the principal point is not the centre — has moved it by design, and a cropped picture has moved it by accident. Both change the implied horizon by the shift, a first-order error in exactly the quantity being read.

The lines are straight. Radial distortion bends a vertical near the edge of the frame, and a bent line fitted as straight has a lean that depends on how much of it is visible. The corners here are well inside the frame; posts at its edge, where they help most, are also where a lens bends most.

Still open: what a street’s leaning posts cost #

The picture here has plumb lines that are exactly plumb. The verticals most photographs have in quantity — lamp posts, signs, bollards, tree trunks — are not, and the ones that are, the corners of buildings, are usually few.

The measurement that settles what that trade is worth gives each post a random lean of a stated size, from a twentieth of a degree for a well-set steel column to half a degree for a timber pole, and asks how many leaning posts are worth one exactly plumb corner — whether sixteen posts leaning a tenth of a degree still see a horizon better than the facade’s two corners, or whether leans of that size put a floor under the reading that no number of posts goes below. A lean is random from post to post, so it should average down like the reading error does; but it does not shrink with the post’s length as the reading error does, and a tall post that leans is worth no more than a short one.