The scale names the mistake when the camera has lost its place along the wall

A turn at the landing makes the mistake a scale followed a third camera up an outside stair beside a facade of identical windows, which a level stereo pair had matched some columns out and so rebuilt nearer than the wall. On a flight along the wall every rebuilt window slides through the camera’s picture by one shared amount; on a flight towards the wall the rebuilt facade’s picture becomes the real one’s scaled about the focus of expansion — the point the camera walks towards — by a factor each mistake sets differently. The scale is weak evidence, a fifth of what a step along the wall buys. But it has a property the slide does not: it depends only on how far the camera has come towards the wall, not on where along or up the wall it stands.

Every reading in that essay knew where the camera was, up to an error along one flight. It closed by asking what happens to a camera that knows its distance towards the wall and has lost the rest — a photographer who climbed a first flight they did not count, reached a landing they cannot place, and then paced the second flight towards the building. The slide is useless to such a camera, since every slide is a guess at its own place. Is the scale enough?

An unknown place is one uniform offset #

The facade is the one of the earlier essays: five columns of windows 2.4 m apart, three storeys of 3 m, twenty metres off, faced squarely by a pair 1.1 m apart with a focal length of 900 px. A $k$-column mistake rebuilds each window on the left eye’s ray through it at the depth ratio $\rho$ — 0.314 for one column out, 0.186 for two — so its rebuilt facade stands 6.3 m or 3.7 m from the pair.

A third camera moved by $(T_x, T_y, T_z)$ from the left eye draws a real window at $x$ and its rebuilt copy at $x' = e + s\,(x - e)$, with the scale $s = \rho(Z - T_z)/(\rho Z - T_z)$ and $e$ the focus of expansion, $c + f\,(T_x, -T_y)/T_z$. A matcher sees the offset $x' - x = (s - 1)\,x - (s - 1)\,e$, wrapped to the nearest real window — and the second term is the same for every window. Whatever the camera does not know about where along and up the wall it stands enters the picture as one uniform offset, two numbers, and two numbers can be fitted.

On the left, every rebuilt window is offset from the real window it lands on by very nearly the same 54 px: a slide, set by where the camera happens to stand. Remove the best uniform offset and what is left, on the right, is a spray of short arrows pointing away from the facade’s middle — the rebuilt facade’s picture scaled by 1.047 against the real one’s. That pattern depends only on how far the camera has come towards the wall. Each mistake predicts its own: the one-column reading scales by 1.047 at this distance, the two-column by 1.098, the three-column by 1.154. With each candidate’s uniform offset fitted freely, the one-column reading leaves nothing, the two-column 8.5 px of misfit, the three-column 18.

The honest reading is different in kind. It has no scale — its rebuilt facade is the facade — and so it predicts no offset at all, from any camera anywhere. It has nothing to fit. Any slide that is not a whole number of columns and storeys refutes it outright, which here is the 55 px of misfit it cannot remove.

Two questions the slide answered at once #

That splits what the straight stair’s slide used to answer into two questions, and the camera that has lost its place can still answer both, by different means.

Is the reading honest? Any offset says no. The slide is 98 px a metre for the one-column reading along the wall, and wherever the camera stands it is very rarely a whole lattice step. On the earlier essays’ straight stairs, the places where it was — the blind places — were where every window agreed with the honest reading at once. A camera that does not know where it stands cannot know whether it is at one, but it does not need to: if its picture shows no offset anywhere, it is either honest or at a blind place, and the scale decides between them.

Which mistake? That is what the slide’s size used to say, through the camera’s known position, and it is what an unknown position takes away. The scale says it instead.

How far towards the wall the scale has to come #

Where the slide happens to vanish, only the scale separates a mistake from the honest reading, and that is the worst case.

The part of a reading’s offsets that an unknown focus cannot absorb is $(s - 1)$ times the windows’ spread about the facade’s middle, 188 px across the fifteen windows. Ten centimetres towards the wall, that is 2.2 px for the one-column reading, 4.4 for the two-column, 6.7 and 9.0 for the three- and four-column. It grows in proportion to the distance paced, because $s - 1$ does: $(1 - \rho)\,T_z/(\rho Z - T_z)$ is very nearly $T_z/\rho Z$ times $(1 - \rho)$ while the walk is short against the rebuilt facade’s distance. The one-column reading, the commonest mistake, has the farthest rebuilt facade and the slowest scale, and it crosses a half-pixel reading error at about two and a half centimetres and a two-pixel error at about nine.

One picture with the focus free #

Over many pictures — a first flight of unknown pitch, 10° to 50°, and unknown length, 0.6 to 1.2 m, then a level pace of a stated distance towards the wall, forty pictures for the honest reading and for each of the four nearer mistakes, the interior windows only — the question is how often one picture names the mistake made.

With every mark read to half a pixel, fitting the focus freely names the mistake 69 times in 100 two centimetres towards the wall, 95 at five, and every time from ten centimetres on. Read to two pixels, it names it 54 times at five centimetres, 80 at ten and 97 at twenty. A camera that knows its whole position names it every time at every distance, because it also has the slide, and the slide is the strong evidence.

The interesting line is the third. A camera that knows how far it has paced and believes it knows the rest, but is out along the wall by a normal 20 cm — the size of error the earlier essay tried for a photographer who has miscounted a tread or two — names the mistake 76 to 83 times in 100, and pacing further does not help it. Its belief puts every candidate’s slide in the wrong place by up to 20 px, and the scale it could have read is buried under that. Fitting the focus costs less than believing a wrong position from five centimetres on, and costs nothing at all from ten. The earlier essay’s warning was that a step along the wall is strong evidence and an error in it expensive; the remedy is not to know the position better but to stop using it.

Every mistake the focus-free reading makes is between two mistakes, never between a mistake and the honest reading. At two centimetres towards the wall, over every picture drawn, the honest reading was named every time it was made and never named when it was not; the misreadings were two-column readings taken for one- or three-column ones, and three-column readings for one, two or four. The slide refutes honesty and the scale has to sort the rest, and sorting needs the scales of neighbouring mistakes to differ by more than the reading error.

Away from the wall reads as well #

The earlier essay found a flight running away from the wall the most forgiving: it shrinks the rebuilt facade’s picture towards the focus, so the rebuilt windows crowd in over the facade instead of off its edges.

With the focus free the two directions read alike. At two pixels, twenty centimetres towards the wall names the mistake 97 times in 100 and twenty centimetres away 92; at forty centimetres, 100 and 98; at a metre and a half, both every time. The scale grows with the distance paced whichever way the camera goes, since $1 - s$ changes sign but not size to first order. What away keeps is the windows: of two hundred pictures drawn at a metre and a half, 130 towards the wall still had three or more rebuilt windows on the facade to read, and 169 away from it. The direction matters for how long the reading lasts, not for how soon it starts.

The distance has to be paced #

The reading needs the distance towards the wall, because each mistake’s predicted scale is set by it. The earlier essay suggested the camera might have that distance from a rangefinder, or from the facade’s own known spacing. Neither will do, and the reason is instructive.

A rangefinder aimed at the wall measures the distance to the wall, and to turn that into how far the third camera has come towards it the reader needs the pair’s distance to the wall as well — which is the very thing in dispute. The honest reading puts the wall twenty metres off; the one-column reading puts it 6.3. A rangefinder that measured either distance would settle the mistake on its own, with no third picture at all. The windows’ known spacing is the same thing in another form: a rebuilt facade stands at $\rho$ of the wall’s distance with its windows $\rho$ of the true spacing apart, so a reader who knows the windows are 2.4 m apart already knows the one-column reading’s 0.75 m is wrong. A facade matched a column out comes forward found that, and a length fixes the scale where it lies is the general statement: one known length in the scene is a closure the reading can be checked against directly.

So the distance must come from the camera’s own motion — treads counted, a tape on the ground from the landing, a pace — and that is a distance with an error of its own.

The tolerance is a proportion, because every mistake’s scale is very nearly proportional to the distance. Paced to within 5 cm, forty centimetres towards the wall names the mistake 92 times in 100 and ten centimetres 59; paced to within 20 cm, 61 and 53. At eighty centimetres a 5 cm pace is good for 99 and a 10 cm pace for 93. A pace wrong by a large share of itself scales every candidate together, and the one- and two-column readings, whose scales stand two to one, trade places once the share approaches a half. A photographer who paces a metre towards a building to within a stride’s tenth is well inside the tolerance; one who shuffles ten centimetres and guesses is not.

With no distance at all, each candidate may choose whatever distance fits it best, and the reading collapses — 17 times in 100 at ten centimetres, 43 at forty. The honest reading suffers most: a wrong reading allowed to choose a distance near zero predicts almost no scale, fits its uniform offset to the noise, and beats the honest reading on its own pictures. That is the control the measurement needs. The scale is evidence only about readings whose scales are fixed in advance, and they are fixed by the pace.

What the reading leans on #

The camera faces the wall squarely. A camera that also turns redraws everything through one warp, which the earlier essay found adds and loses nothing; but a turn the reader does not know is two more unknowns, and they are not a uniform offset. A camera whose turn is read from the facade’s own horizontal and vertical lines leaves the reading as it is here.

The facade is one plane. A projecting bay puts some windows nearer than others, and depth is a reciprocal is why the scale then differs from window to window even for the honest reading. The uniform offset that absorbs the camera’s place no longer absorbs everything that is not the mistake.

The mistake is a clean shift of whole columns. A wrong match is not a small error is the standing reason a coherent mistake has one $\rho$ and therefore one scale.

Each rebuilt window keeps its name. The offsets are compared window by window: the reading knows which rebuilt window it is predicting, because the pair rebuilt it from a named pair of marks, and it compares that window’s predicted offset with the offset of the same window in the third picture. What the matcher does not know is which real window a rebuilt one lands beside, and that is why every offset is wrapped. A reader that lost the names too would be comparing two clouds of offsets rather than two lists, and a cloud has no scale — only a spread, which a uniform offset cannot change and a scale enlarges by $s$. It would still read something, but the scale’s sign, which separates walking towards the wall from walking away, would be gone with the names.

The reading error is independent from window to window. A focal length wrong by a per cent scales every window’s picture about the principal point, which is a scale about a different centre — and a scale about any centre is a scale about the focus plus a uniform offset. Fitted freely, the focus absorbs the offset part and leaves a one per cent scale that every candidate must explain. A third ray is worth what its picture is worth: the focal length must be known to better than the scale being read, which at ten centimetres for the one-column reading is about one per cent.

What the camera needs to know about itself #

A mismatch on its own line needs a third eye began this line of reading with a third camera off the pair’s baseline. A staircase winds the mistake round the lattice and the essays on the climb found such a camera’s slide strong and its blind places few, provided it knew where it stood. The earlier essay found the scale weak and its blind places none. This one finds the scale needs to know only one thing about the camera — how far it has come towards the wall — and that a camera whose knowledge of the rest is merely good is better off forgetting it and fitting it from the picture.

That is a dolly zoom is a step and a zoom read the other way round. There, a step towards a subject moves every mark along the line from the point stepped towards, and a zoom moves them from the principal point; a subject at a known depth separates the two. Here the subject’s depth is the unknown, the step is known, and the point stepped towards is fitted. The geometry is one homothety, and it can be read for whichever of its three numbers the picture is not told.

Still open: whether a camera that turns as it paces can still fit its place away #

Every camera here faced the wall squarely, so that what it did not know about its position entered the picture as one uniform offset. A photographer pacing towards a building rarely holds the camera square: it pans a degree or two to keep the facade framed, and tips to keep the roofline in. A small turn about the camera’s own centre moves every window’s picture by nearly the same amount — a uniform offset again — plus a little perspective warp across the frame, and the warp is the same size as the scale being read.

The measurement that settles what that is worth gives the pacing camera an unknown pan and tilt of a stated size, fits them with the focus as four unknowns or reads them first from the facade’s own straight lines, and asks whether the scale then still names the mistake from one picture — how large a turn the four-unknown fit absorbs before the warp it leaves is mistaken for a scale, whether reading the turn from the facade’s lines first costs less than fitting it, and whether a camera that turns as it paces needs to pace further to make up for it.