A climbing eye names the column mistake, if it knows how high it is

A raised eye reads the storeys as the pair read the columns put a third camera on a fixed rod above the left one of a level stereo pair and asked what it made of the pair’s commonest mistake on a facade of identical windows. The pair, 1.1 metres apart and twenty metres from the wall, can match every window to the one a column along and rebuild a perfect facade at another depth — nearer, because the windows are 2.4 metres apart and wider than the baseline. The raised camera sees that rebuilt facade slide straight down its picture, and refuses it, except at a countable set of raises where the slide happens to be a whole number of storeys and every rebuilt window lands on a real one: 1.375 and 2.75 metres for the one-column mistake. Even there, the facade’s lowest storey slides off the lattice and lands on the paving.

The essay ended on a camera that does not stay put. A photographer on a staircase, a drone climbing, a camera on a mast being raised all pass through the blind raises on the way, and the question was whether that matters, and whether the frames of a climb can be pooled to do more than refuse: whether the rate at which the slide grows with the raise names which column mistake the pair made.

It names it, from a handful of frames taken low on the climb. The price is paid elsewhere — in where the climb begins, and in how well the camera knows its own height.

A slide that wraps at every storey #

The facade is the one of the raised-eye essay: five columns of windows 2.4 metres apart, three storeys of 3 metres, twenty metres off, the pair’s eyes 1.6 metres up and 1.1 metres apart, a focal length of 900 pixels. A matcher that jumps one column and then follows its neighbours rebuilds the windows at 6.3 metres; two columns, 3.7; three, 2.7; four, 2.1. Every rebuilt window lies on the left eye’s ray through its real window, so a camera straight above the left eye sees each rebuilt window in the same column as the real one, displaced only vertically.

The displacement is simple. A point at depth Z′ on the left eye’s ray, seen from a camera raised h above that eye, appears f·h·(1/Z′ − 1/Z) pixels below where the real window appears — a difference of reciprocal depths, as every disparity is, which is the observation depth is a reciprocal starts from. For a k-column mistake that is f·h·k·p/(B·Z): with these numbers, 98.2 pixels for every metre of raise and every column of the mistake. It is the same for every window of the facade, because none of the depths it depends on varies across a flat wall faced squarely.

A matcher does not see the displacement itself. It sees the rebuilt window against whichever real window is nearest, so the offset it reads wraps at every storey of 135 pixels, and the four mistakes draw four sawtooths at four slopes: 98, 196, 295 and 393 pixels a metre. Each line ends where every storey of the rebuilt facade has slid past the lattice’s lowest storey — at 3.44, 1.72, 1.15 and 0.86 metres — after which no rebuilt window has a real window within half a storey, and the matcher finds wall and paving where it expected glass.

The first half of the question the raised-eye essay asked is answered by the drawing. A frame accepts the mistake only where its line crosses zero, at the blind raises, and only within the band where a pixel of reading error hides the miss, about two centimetres wide for the one-column mistake and narrower for the others. A climb with frames at a few raises taken anywhere has to land every one of them in such a band to be fooled, which is a matter of choosing them that way. A climbing camera is safe by construction, as the essay expected.

The second half is in the slopes. Every frame reads one offset, and the four mistakes predict four different offsets at almost every raise. A single frame low on the climb already separates them; two frames almost anywhere do. The lines cross one another at the multiples of the blind raise divided by the difference in column count — at 1.375 metres all four mistakes and the honest reading coincide at zero — so a frame exactly there cannot tell them apart, but a second frame at any other height can. It is the climbing version of a third eye that lands on the next post: a single extra camera placed where the lattice repeats its view is blind, and a camera at any other place is not.

The slider moves the climb up the facade, four frames spread over one metre. Begun at level, the frames sit where every line is still on the lattice and well separated, and all four mistakes are named. Begun at two metres, three of the four lines have already ended: the frames refuse those three mistakes but cannot say which of them was made.

The bottom of the climb does the naming #

A frame above a mistake’s last raise reads no offset for it, only that every rebuilt window landed off the lattice. Every larger mistake has slid off too, so the frame says the same of each, and as evidence of which mistake was made it is worth nothing. The reading scores each candidate mistake by how far its predicted offsets sit from the ones the frames read, wrapped to a storey, and by whether it predicts a landing where the frames saw none; two candidates that predict the same thing at every frame tie, and a tie names neither.

The result is close to a step function, and the steps sit where the drawing said they would. With half a pixel of reading error and offsets that differ by tens of pixels, a climb either has a frame below a mistake’s last raise or it does not; when it does, the mistake is named in all two hundred climbs, and when it does not, in none. The one-column mistake survives the longest because it slides slowest; the four-column mistake, sliding at almost 400 pixels a metre, leaves the lattice before the camera has risen a metre.

One step is subtler than the drawing suggests. A climb begun at one metre has its lowest frame at 1.05 metres, above the four-column mistake’s last raise of 0.86, so no frame reads that mistake’s offset. It is named anyway, every time, by elimination. At 1.05 metres the three-column mistake still lands on the lattice and the four-column mistake does not; the frame reads no landing, and the only candidate predicting none is the four-column one. That works because this facade admits exactly four nearer mistakes. A reading that allowed any shift at all, including impossible ones, would lose this naming, and a facade with more columns would postpone it.

The practical rule is short. A climbing camera names a column mistake from frames taken before the rebuilt facade leaves the lattice, so the climb should begin low — level with the pair, ideally — and the frames that matter are its first.

What the camera must know of its own height #

Every offset the reading predicts is the mistake’s slide rate times the raise of the frame. A camera that believes it is at 70 centimetres when it is truly at 75 predicts an offset 5 centimetres of slide away from the one it reads — 5 pixels for the one-column mistake, 20 for the four-column one.

The climbs here are kept below 0.85 metres, where no mistake has left the lattice, so the only thing degrading them is the height error. Each mistake holds out until its slide over the error is a sizeable share of a storey, and the larger the mistake, the sooner that happens. The one-column mistake is named 94 times in a hundred with a 20-centimetre error and 26 times with 50; the four-column mistake 93 times with 5 centimetres and 41 with 10. A quarter of a storey is 34 centimetres of height error for the one-column mistake, 17 for two, 11 for three and 9 for four, and that is where each line in the figure turns down.

So the larger the mistake, the better the altimeter it needs, which is an uncomfortable order: the mistakes that rebuild the facade nearest and do most damage are the ones a sloppy climb is least able to name. Every climb in the figure still refuses every mistake. A refusal needs only that the offset is not zero, which a height error does not change; naming needs the offset’s size, which it does.

The camera can get its height from the facade it is looking at, up to a point. The real windows themselves slide down its picture as it climbs, by f·h/Z, 45 pixels a metre at twenty metres, so a window read to half a pixel gives the camera’s height to about a centimetre. But they slide past one another every storey, and the camera reading its height off the windows has the same ambiguity the pair had reading depth off them: it knows its height to a centimetre only modulo three metres. Something coarse has to settle which storey it is at — a barometer, a count of steps, a climb watched continuously from level — and the windows then refine it. A camera that knows its storey and reads its height from the facade knows it far better than any of these readings need.

A tall lattice keeps the slide in view #

A low facade lets the slide off the lattice quickly, and every frame above that is wasted on naming. A taller facade holds it longer.

A climb that begins two metres up is a climb that has already missed the bottom of a three-storey facade: on such a facade it names the one-column mistake and nothing else. On five storeys the same climb names all four, because the slide stays on the windows until it reaches four and a half storeys. On a facade of R storeys a mistake’s last raise is (R − ½) storeys over its slide rate, so each added storey adds 1.4 metres of useful climb for the one-column mistake and 34 centimetres for the four-column one.

This is the lattice’s edge working in reverse. In the raised-eye essay, the storey that slid off at a blind raise was what convicted the mistake there; the facade essay found the same of the column that fell off a facade shifted sideways. The railing essay found the same of a railing’s feet. An edge convicts. Here the same edge limits the naming: a frame past it can only say that every rebuilt window landed off the lattice, and every larger mistake says the same. A tower block of identical floors — the worst case for a pair, because its repeats are everywhere — is the best case for a climbing camera, because the slide never runs out of windows to be measured against.

Fine courses ask for a surveyor’s level #

The storeys are an easy repeat to read against: 135 pixels apart, so a reading error of half a pixel and a height error of a few centimetres are both small beside the period. A facade whose vertical repeat is fine — panels, boards, courses of brick, balcony bars — wraps the offset at its own period, and the slide rate does not change with it.

The slide rate is set by the columns, the baseline and the depth; the wrap is set by the vertical repeat. On a facade of 25-centimetre boards the period is 11 pixels and the four-column mistake slides through it every 3 centimetres of climb, so a camera whose height is known to a centimetre names the four mistakes 65 times in a hundred, averaged. On 5-centimetre courses — a period of 2¼ pixels, a slide through it every 6 millimetres of climb for the four-column mistake — even 3 millimetres of height error names the mistakes only 28 times in a hundred, and a centimetre puts the reading at 18, below the quarter a guess among four would get, because the ties count as failures.

The figures fall below chance on fine courses for a reason worth stating. With the period that small, a reading error of half a pixel is already a fifth of a period, and the offsets the frames read are nearly uniform whatever the mistake. Then several candidates fit about equally, and a reading that refuses to name a mistake when two fit equally names nothing. The honest output of such a climb is not a wrong mistake but no mistake named — and it still refuses the facade, because a rebuilt window that sits nowhere near a course in one frame and somewhere else in the next is not a wall.

The raised-eye essay’s courses were the opposite case. There a fixed raise convicted a course shifted one along at every period down to five centimetres, using the roofline and the foot as its references. Conviction is cheap: it needs one edge the repeat does not cover. Naming is expensive: it needs the repeat’s own period to be large compared with everything uncertain about the frames. The climb that names a mistake against storeys from a barometer’s reading needs a surveyor’s level to name one against bricks.

What the climb adds to the fixed third eye #

Put together, a climbing third camera does what the fixed one did and one thing more. It refuses the pair’s column mistake in every frame not inside a two-centimetre band about a blind raise, which nearly every frame of nearly any climb is not. And from its lowest frames it names the mistake: the rebuilt windows slide down its picture at 98 pixels a metre for each column, a frame reads that slide against the nearest real window, and the four mistakes the facade admits predict four different sawtooths. A metre of climb from level names every one of them; a metre begun two metres up names only the slowest.

A wrong match is not a small error began this sequence by observing that a mismatch moves a point somewhere else rather than slightly off its own place, and every essay since has been about what refuses such a move. Naming the mistake is different in kind. Refusal needed only that a third picture disagreed; naming needs the disagreement measured, and a measurement inherits every uncertainty in the frame it was made from. That is why the cost is in the camera’s knowledge of its own height rather than in the number of frames. The title of a third ray is worth what its picture is worth states the rule for a third camera’s ray, and it holds for a third camera’s frames: each is worth what is known about where it was taken.

There is also a familiar shape in the result. The climbing camera reads the facade the way any moving camera reads a repeated scene: with frames close enough together, nothing jumps a whole period between them and the slide can be followed; with frames far apart it cannot. A mismatch on its own line needs a third eye found that the ambiguity of a repeat along one baseline is broken by any baseline not parallel to it. The climb breaks it by having many baselines at once, all vertical, of every length from zero up, and it is the short ones near the start that carry the mistake’s name.

The geometry the reading leans on #

The camera climbs straight up and faces the wall squarely. That makes the slide the same for every window and purely vertical. A camera that pitches down to keep the facade framed, as a hand-held one does, turns each frame about a horizontal axis; the rebuilt windows then slide by amounts that vary up the picture, and the reading would have to model that frame by frame. The arithmetic of the slide rate is unchanged, since it depends on where the camera is and not where it points.

The pair’s mistake is a clean shift of whole columns. A matcher that errs window by window rebuilds a scatter of depths, and no single slide rate describes it; every frame then refuses the scatter, but no candidate names it. This is the same assumption the fixed third eye made.

The facade admits a finite list of mistakes. With five columns there are four nearer readings, and the naming by elimination at the top of a short climb depends on that. A facade running out of frame on both sides admits more, and the elimination is lost; the naming from the low frames, which reads the slide itself, is not.

The reading error is half a pixel and independent from frame to frame. A correlated error — a camera whose calibration drifts as it warms, or a lens whose focal length is wrong — scales every offset together, which is indistinguishable from a different slide rate and so from a different mistake. How large a calibration error the naming survives was not measured here.

Still open: a staircase climbs sideways as it rises #

Everything here climbs straight up, and a vertical baseline from the left eye sees the facade’s windows slide only vertically, wrapping at the storeys. A photographer on an outside staircase rises along the wall as well as up it, and a drone rarely climbs vertically.

A third eye displaced up and along has a slanted baseline with the left eye, and the pair’s column mistake then slides through its picture both down and across: the vertical part wraps at the storeys and the horizontal part at the columns, so a frame reads the offset on a two-dimensional lattice rather than a line. The measurement that settles what that is worth takes a climb along a stair of stated pitch, carries the pair’s one- to four-column mistakes into each frame, and asks whether the two wraps together name a mistake from fewer frames than the vertical climb needed, or whether particular pitches line the slide up with a diagonal of the lattice and reintroduce blind directions — a stair whose rise over its run equals the storey over the column spacing would carry every rebuilt window along the lattice’s own diagonal, and the question with a number in it is how far off that pitch a stair has to be before its frames name the mistake as well as a vertical climb does.