A photograph grades its posts across the sight, and not along it

A leaning post is worth its lean, not its length treated a street’s lamp posts and poles as witnesses to a camera’s horizon, and their leans as the nuisance that made them worse witnesses than a building’s plumb corners. Each lean was random from post to post and averaged away with the count, but each set a price: a timber pole leaning a quarter of a degree was worth about a sixty-fourth of a corner, and posts counted as the corners’ equals made the reading worse.

The essay closed by turning the nuisance round. The same picture that reads the horizon from a street’s verticals also reads each post’s own lean, once the vanishing point is known: the departure of the post’s line from the plumb line through its foot. For a camera that knows its orientation, the picture is a survey of how straight the street’s posts are. The question was what such a survey is worth — whether a single photograph with two plumb corners can grade a row of posts, and whether a lean along the line of sight is visible to it.

The reading: a post against the plumb line through its foot #

The picture is the standing camera of the earlier essays: 1.62 metres up, 700 pixels of focal length, pitched 6.78 degrees down, looking at a six-metre facade 8.5 metres away whose two corners are plumb. Eight posts 2.5 metres tall stand across the frame 7.5 metres away. Every line is sampled every eight pixels along its image and read across itself to 0.4 pixels, as before.

The survey takes three steps. The facade’s two corners, and nothing else, fix the vertical vanishing point, far below the frame. Through each post’s foot the line to that point is the plumb line a perfectly straight post standing there would draw. And the post’s own line is fitted; the angle between the two, divided by how much a sideways lean of one degree turns a post’s image at that place in the frame — 1.02 here, almost exactly one — is the post’s lean across the line of sight.

The street drawn has four posts leaning a quarter of a degree, each in a random direction, and four within a twentieth. Read against the two corners, the leans come back within about a tenth of a degree of the truth, and a reader who calls every post past 0.15 degrees leaning grades six of the eight rightly. The two it gets wrong are both leaning posts read as straight, and both lean mostly towards or away from the camera — the second half of the question, and the one the rest of the measurement keeps returning to.

A lean is read from the line’s own direction #

The reading’s precision is set by two things: how well each post’s own line is read, and how well the corners fix the vanishing point that every post is compared against.

For posts 2.5 metres tall, the post’s own line reads its lean to 0.060 degrees and the corners’ vanishing point brings the whole to 0.086. For posts a metre tall the own-line figure is 0.236 degrees and the whole 0.246. A lean is a direction, and the direction of a line read to 0.4 pixels across itself is known to about the reading error divided by its length, improved by the square root of how many samples lie along it; a short post is both short and sparsely sampled, and its lean is read four times worse than a tall one’s.

That reverses one of the earlier essay’s findings, which is worth seeing plainly. There, a lean was an angle that a tall post carried into the horizon as fully as a short one, so height bought a leaning post almost nothing as a witness. Here the lean is the thing being measured, and height is nearly the whole of what decides how well. A bollard’s lean is read to a quarter of a degree, which is the size of lean worth catching; a lamp column’s to a twelfth.

The corners add a roughly fixed amount on top of each post’s own reading — the difference between 0.086 and 0.060 is about 0.06 degrees in quadrature — and it matters only once the posts are tall enough to be read well. Giving the reading four plumb lines instead of two, the facade’s corners and two more corners of a building a little further back, brings the whole from 0.084 to 0.077: the extra plumb lines are further away and shorter in the picture, and fix the vanishing point less than the two near corners already did.

The far side of the street is guessed at #

Height in the picture is what a lean is read from, and distance takes it away.

Posts 2.5 metres tall have their leans read to 0.068 degrees at 5 metres, 0.086 at 7.5, 0.113 at 10 and 0.180 at 15. A post twice as far is drawn half as tall and sampled half as often, and its direction is read about three times worse; the corners’ share stays about the same at every distance. At 15 metres a post leaning a quarter of a degree sits less than one and a half standard deviations from straight, so a photograph taken from one pavement grades the posts on that side and guesses at those across the road. A survey of a whole street wants a picture from each side.

How finely a picture grades #

The question was posed as a grading: can a single photograph tell the posts leaning a quarter of a degree from those within a twentieth?

For leans across the line of sight it can. Streets of eight posts, half leaning a quarter of a degree and half within a twentieth, each post called leaning when its read lean passes halfway between the two, are graded rightly 91 times in 100 with lines read to 0.4 pixels. Half a degree against a twentieth is graded every time. A tenth of a degree against a twentieth is graded 60 times in 100, which is little better than a coin: the two grades differ by less than the reading’s own scatter.

The reading error decides the rest. Lines read to a fifth of a pixel — careful work on a sharp photograph — grade a quarter of a degree 99 times in 100 and a tenth of a degree 75. Lines read to 0.8 pixels, a soft or compressed picture, grade a quarter of a degree only 70 times. So a single good photograph separates steel columns from timber poles that have been knocked; it does not separate a well-set pole from a slightly tired one.

Blind to a lean towards the camera #

The two posts the first street got wrong both leaned mostly along the line of sight, and that was not chance.

A quarter-degree lean across the line of sight is caught 86 to 93 times in 100 wherever the post stands. The same lean along the line of sight — the post’s top tipped towards the camera or away from it — is caught 4 and 5 times in 100 by the two posts beside the centre of the frame, 12 and 9 by the next pair out, and 32 and 37 at the edges. A post that tips towards the camera moves its top along the line of sight, and its image turns only by the part of that motion the picture sees across it, which is the post’s distance off the centre column over its distance from the camera. That share is 0.47 of the lean at the frame’s left edge and 0.07 beside the centre. Near the middle of the picture the post’s image shortens and does not turn at all.

The earlier essay found the same fact from the witness’s side: posts leaning along the line of sight hardly disturbed the horizon read from them, because their images hardly turned. It was a mercy there. Here it is a blindness, and it falls where a reader looks first — the posts in the middle of the frame, which are the posts the photograph was usually taken to show.

There is nothing a single picture can do about it. The lean along the line of sight is the one component of the post’s direction that projects onto nothing near the centre of the frame, and no reading of the lines recovers what the lines do not record. A second photograph from another direction sees that component across its own line of sight. It is the oldest loss in the subject, which what a projection destroys begins from: a picture keeps every direction across its rays and nothing along them, and a lean towards the camera near the middle of the frame is a direction along a ray.

What the posts say about each other #

The question’s title was what a street’s posts say about each other, and the answer turns on how the corners’ error reaches them.

A vanishing point that is a little out moves the plumb line at every post’s foot, and it moves all of them together: by a turn that is the same for every post — a sideways shift of the vanishing point is a roll of the plumb directions — plus a turn that grows steadily across the frame, from a shift along the column. So the corners’ error lays a straight slope across the row’s read leans, the same slope for every post. Read against the corners, eight plumb posts scatter by 0.071 to 0.102 degrees, worst at the edges; read against the true vanishing point, by their own lines’ 0.056 to 0.063.

The posts can remove that slope themselves. Fit a straight line across the row’s read leans, by column, and read each post against the line instead of against the corners: the shared slope goes with it, and the posts then scatter by 0.045 to 0.055 degrees — a little better than their own lines alone, because the fitted line takes a share of each post’s own error with it. Eight posts read against one another say which of them leans relative to the rest to better than the corners can say which of them leans at all.

What that reading cannot say is how the row as a whole leans. A street whose every post has settled the same way, which the earlier essay found to be the one error no count of posts averages away, is read against itself as perfectly straight. That is the division of labour the picture offers: the row, read against itself, grades each post against its neighbours; the corners, read against the row, say whether the neighbours themselves are plumb. Either alone answers half the question.

The blindness falls on the lean the horizon did not need #

There is a coincidence in the two findings that is worth stating, because it makes the survey and the earlier essay’s horizon reading one procedure rather than two.

The earlier essay found that the direction of a post’s lean decided how much it disturbed the horizon read from the verticals. Sixteen posts leaning a quarter of a degree only along the line of sight put the horizon 3.4 pixels out, hardly worse than the 1.9 they gave plumb; the same posts leaning only across it put the horizon 8.1 pixels out. The lean along the sight barely turns a post’s image, so it barely moves the vanishing point the posts imply. The survey cannot see that component, for the same reason. What it can see — the lean across the line of sight — is exactly the component that made a post a bad witness.

So the order of work is short. Fix the vanishing point from the plumb corners alone; read every post’s lean against it; then add to the horizon fit only the posts graded straight, each with an allowance of a tenth of a degree rather than the half degree the earlier essay recommended for posts of unknown straightness. A post leaning towards the camera slips through the grading, and does almost no harm when it does. That is which reference to measure from carried one step further: not only which kind of line to trust, but which particular lines, chosen from the same picture.

The two errors no grading removes are still there. A lean shared by every post is, to the horizon, the camera pitched or rolled, and a wrong model shows at its own level is the earlier finding that such a misreading lands exactly where a wrong camera would put it; the corners, being plumb, are the only check. And a street that slopes leaves its posts plumb and its ground tilted, which figures on a street that slopes found moves the horizon read from people’s heads and leaves the one read from plumb lines alone. The survey grades verticals; it says nothing about the ground they stand on.

What a single photograph surveys #

Put together, one photograph with two plumb corners in it is a survey of its posts’ leans across the line of sight, read to about a twelfth of a degree for a post 2.5 metres tall at 7.5 metres, and to a quarter of a degree for a bollard. It grades quarter-degree leans from straight ones nine times in ten, tenth-degree leans hardly at all. It reads the near side of a street and guesses at the far side. And it is blind, near the centre of the frame, to the lean along the line of sight, which is half of every lean.

The floor a better camera cannot reach separated an error that shrinks with care from one shared by every reading. The survey has both, in a particular arrangement: each post’s own line is the first kind, the corners’ vanishing point is the second, and the second has a shape — a slope across the row — which reading the posts against each other removes. The answer is an ellipse is the general point that an error’s shape is information, and here the shape is the difference between grading a post against the world and grading it against its street.

What the survey takes on trust #

The camera’s roll and focal length are known. The question assumed a camera that knows its orientation, and the reading uses the vanishing point only for the plumb direction at each foot. A rolled camera whose roll is not known turns every plumb line by the roll and reads every post as leaning by it — a shared lean, which reading the posts against each other cannot see and the corners can.

The posts are straight. A bowed pole read as a straight line has a lean that depends on how much of it is visible, which the earlier essay suggested catching by reading each post’s upper and lower halves separately; a survey should do that before believing a lean.

The foot is where the post meets the ground. The plumb line is drawn through the post’s foot, and a foot hidden behind a parked car or a kerb is read from wherever the visible line ends. That moves the comparison point along the post’s own line, which costs nothing if the post is straight and the line is long; it costs a short post its reading.

The corners are plumb. A facade’s corners see the horizon took a building’s corners as exactly vertical, and they are to a small fraction of a degree. A corner that is not would turn the vanishing point and every post’s lean with it, and reading the posts against each other would not notice.

Still open: whether a second photograph from across the street reads the lean along the sight #

One picture reads each post’s lean across its own line of sight and none of the lean along it near the centre. A second picture taken from across the street, or from further along it, looks at each post from another direction, and the component one picture is blind to is, for the other, partly across its line of sight.

The measurement that settles what a second picture is worth places two cameras at stated positions — across the street from each other, or a few paces apart along the same pavement — reads each post’s lean in both against each picture’s own plumb corners, and combines the two readings into a lean in two components. The question with a number in it is how far apart the two viewpoints must be before the lean along the first camera’s line of sight is read as well as the lean across it, whether two pictures taken a few steps apart along one pavement, which see each post from directions only a few degrees apart, are any use at all, and whether each picture needs plumb corners of its own or one picture’s corners can fix both once the two have been tied together by the posts they share.