A camera count needs a tolerance

Four surfaces, and no one camera that draws them ended with a number in pixels. A constructed picture of four splayed surfaces — a floor, a footstool, a table top and a book, each drawn with its sides spreading at its own rate — cannot be one camera’s picture of four parallel surfaces, and the nearest drawing that could be moves its far corners 17.6 px. That number was built to owe nothing to an assumed lens, and it does not.

It is a distance, though, and the question a reader actually has is a count. If one camera will not do, will two? The redraw forced every one of the four surfaces to a single meeting point. Relax that to several meeting points, and the smallest number of them that accounts for the picture is the number of cameras the picture needs.

That count turns out to exist, to be computable exactly, and to be less a property of the picture than it first appears.

The count is a function of two things #

A group of surfaces counts as one camera’s when the redraw bringing their sides to a common meeting point moves their far corners by no more than some tolerance. Every surface keeps its near edge and the height of its far edge, exactly as before; only the far corners move, and only sideways.

The tolerance is not optional and there is no natural value for it. Allow nothing at all and every picture drawn by a hand needs as many cameras as it has surfaces, because no two hand-drawn splays agree to the last pixel. Allow enough and every picture needs one, because a large enough correction reconciles anything. So the count describes the picture and the tolerance together, and the honest object is not a number but the whole step function.

That is not a defect peculiar to this measurement. Every reading this field has taken of a divergent picture has carried a free parameter of some kind: the assumed focal length in the tilt, the choice of which surfaces count as parallel in the redraw. The useful ones are those whose parameter is about something a reader can estimate, and a tolerance is about the marks.

The four surfaces’ sides meet at four heights on the page — 912, 590, 456 and 324 px down — and the step positions are 4.63 px, 11.11 px and 17.63 px. The last of those is the redraw distance four surfaces, and no one camera that draws them reported, which is as it should be: the tolerance at which one camera suffices is exactly the distance to the nearest one-camera drawing. The count and the distance are the same measurement read two ways, and the count adds the middle of the story that the distance skipped.

Why the partitions are exhausted rather than sorted #

Each surface has a meeting height, and it is tempting to think the groups must be runs of surfaces that are adjacent in that order. Sort the heights, cut the sorted list wherever the gaps are largest, and the partition falls out.

It does not, and the reason is worth holding onto. What a group costs is not the spread of its meeting heights. It is the root-mean-square sideways movement of its far corners, and how much corner movement a given change of meeting height costs depends on the surface — on how wide its near edge is, and on how far apart its near and far edges sit on the page. A small surface drawn shallow has to move its corners very little to change its meeting height by a hundred pixels; a wide surface drawn deep has to move them a lot.

So two surfaces whose meeting heights are far apart can be cheap to reconcile and two whose heights nearly agree can be dear. The optimal grouping is not an interval in meeting height, and a picture in this essay shows it directly: a constructed picture whose four surfaces meet at 640, 640, 300 and 300 px, read at a tolerance of 20 px, groups the floor, the table and the book together and leaves the footstool alone — a group that straddles the other one in height.

Because sorting cannot be trusted, the search exhausts the partitions. Four surfaces have fifteen ways of being grouped and eight have four thousand one hundred and forty, which is a picture’s worth either way, so the number returned is the answer rather than an attempt at it. That matters more here than it usually does: the whole claim is that a smallest number of cameras exists, and a heuristic that returned three when two would have done would be making the picture look worse than it is.

Does the count find groups that are really there? #

A measurement that can only be run on pictures whose answer nobody knows is not yet a measurement. The test is a picture built with a stated number of groups hidden in it.

Building one is arithmetic rather than fitting. A surface with its near edge at one height and its far edge at another meets at height h exactly when its far half-width is its near half-width times (h − far)/(h − near). Choose two heights, put two surfaces on each, and the picture has two groups in it by construction — not by having been fitted to two.

The search returns two groups and they are the right two. More useful than the single reading is the range over which it is right.

Two cameras account for it at a tolerance of zero — the groups share their meeting heights exactly, being built that way — and one camera does not account for it until 47.0 px. So anywhere in a window forty-seven pixels wide the search returns the number the picture was built with, and the window’s width is itself the useful quantity: it is how far apart the groups were placed, measured in the currency the search spends.

Two more controls sit either side of that one. A picture one camera really took needs one camera at a tolerance of 3 × 10⁻¹⁴ px, which is the arithmetic floor and not a tolerance at all. And the constructed stack, which was built with no groups in it — four surfaces splayed at four rates chosen to be unlike each other — has no window: its answer changes at 4.6, at 11.1 and at 17.6 px with no plateau wide enough to call a finding. A picture with groups in it announces them by holding one answer over a range, and a picture without them does not.

The stack whose surfaces disagree in sign #

The same essay also built a second stack, in which one surface is splayed hard enough that its sides meet above the picture’s centre rather than below it, so that under every lens three surfaces read as leaning one way and the fourth the other. Its redraw distance was 23.8 px against the first stack’s 17.6.

Every step sits further out: 10.48 px, 15.37 px and 23.76 px against 4.63, 11.11 and 17.63. The picture is harder to reconcile at every count, not only at one camera, which is what one would want a count to say about a picture whose surfaces disagree more sharply.

The groupings differ too. Both stacks put the floor on its own at two cameras, because the floor is the widest surface and the most expensive to move. It is the same asymmetry the redraw reported when it charged the smallest surface the largest correction: a common meeting height is placed where the large surfaces want it, and the small ones follow. At three the first stack pairs the footstool with the book and leaves the table alone; the mixed stack pairs the footstool with the table and leaves the book alone, the book being the surface splayed the other way. Neither pairing is what a reader would guess from the heights.

The floor under all of it belongs to the hand #

Everything above treats the picture as exact. A painted picture is not: a hand placing a far corner puts it somewhere near where it intended, and the scatter is a property of the brush, the support and the working distance rather than of the construction.

That scatter sets a floor under the whole measurement, and the floor is measurable. Take a picture one camera really drew, displace each far corner by a normal deviate of standard deviation σ, and ask what the count does.

That hand of six pixels’ scatter produces a picture that reads as one camera’s above 3.63 px of tolerance, as two cameras’ below that, and as three below 1.85 px. There is nothing in the picture to find. One camera drew it and the surfaces are parallel; what the search reports below 3.63 px is the brush.

Notice that the false structure is not noise-shaped. The search does not return a jumble; it returns a clean partition, with the same air of having found something that the built two-group picture had. A reader shown only the answer two cameras, worst group 2.8 px has no way to tell the two cases apart from the answer alone. What distinguishes them is the window — the built picture held its answer of two across forty-seven pixels, and this one holds each of its answers across barely a factor of two in tolerance.

Two things in that figure are worth separating, because one is arithmetic and the other is the finding.

A given hand’s threshold is exactly proportional to its own scatter. Displacing every corner by a deviate of standard deviation σ scales every quantity in the redraw by σ, so the threshold must scale too, and the single hand drawn across the sweep departs from its own straight line by 4 × 10⁻³ per cent — which is round-off and not fit. Nothing is learned by measuring that line and it would look equally perfect with one hand in it.

The constant of that line is a property of the hand and not of the arrangement. Across a hundred and twenty hands it averages 0.574, with deciles at 0.30 and 0.93 and a full range from 0.17 to 1.34. Two hands of identical scatter, drawing the same four surfaces, can have thresholds a factor of three apart, because the threshold depends on where each hand’s particular displacements happened to fall — whether they conspired to leave the four meeting heights nearly in line or scattered them.

That spread is the reason to report a band rather than a constant, and it is easy to miss. A first reading of this averaged five hands and got 0.683, which sits well inside the distribution and is not wrong about any one hand; it is simply five samples of a quantity whose deciles differ by a factor of three. The per-hand line’s exactness is what hides it — the fit is perfect however few hands are in it, so nothing in the goodness of the fit says the constant is under-sampled.

The constant is under one, on average, for a reason worth stating. A redraw does not have to undo the hand’s displacements; it has to find the single meeting height that leaves the smallest root-mean-square correction, and one free number absorbs part of the scatter. About four tenths of a four-surface picture’s jitter is absorbed that way and a little under six tenths survives it. The share depends on how many surfaces there are — with only two, one fitted height can do much more, and the constant falls to about 0.40; with six it settles near 0.59 — so a picture with more furniture in it has a higher floor, not a lower one, which is the opposite of what more evidence usually does.

So a reader with a picture, a count and a tolerance owes one more number, and owes it as a range: an estimate of the hand. Somewhere around half to one times the hand’s scatter, the count stops describing the construction and starts describing the brush. Nothing in the count itself says which side of that line a particular reading is on, and the width of the band means a reading close to the line cannot be rescued by knowing the hand’s scatter more precisely.

What a count is for, and what it cannot do #

Set against the redraw distance it grew out of, the count answers a different question and it is worth being clear which.

The distance asks how far a picture is from being one camera’s, in pixels of the page. It is a single number, it is comparable across pictures drawn at the same scale, and it says nothing about structure. The count asks how many rules the picture was made with, and it can find structure the distance is blind to — a picture whose surfaces fall into two consistent families reads to the distance as merely rather far from one camera, and to the count as exactly two.

What the count cannot do is supply its own tolerance. Every use of it carries an argument about how much correction is allowed, and that argument is about the picture’s making rather than its geometry: the width of a brush line, the thickness of an incised guideline, the resolution of the surviving photograph. Two readers with different estimates of that will read the same picture as two cameras and as three, and both will be reading it correctly.

It also cannot see a group of one. A surface that shares its meeting height with nothing is its own group, and a picture of four surfaces in four groups is indistinguishable from a picture in which the painter simply splayed each thing by eye. The count says four either way. It becomes informative exactly when it is smaller than the number of surfaces and holds that value over a range, which is why the window matters more than the reading.

What this establishes and what it does not #

It establishes that the count exists and is exact. For any picture of one-point surfaces and any tolerance, there is a smallest number of groups, it is found by exhausting the partitions rather than approximating them, and the tolerances at which it steps are computed rather than sampled.

It establishes that the count recovers structure that is really there. A picture built with two groups gives back two, over a window forty-seven pixels wide, and a picture one camera drew gives back one at the arithmetic floor.

It establishes a floor, and the floor is about the drawing rather than the drawn. A one-camera picture made by a hand of scatter σ reads as more than one camera below about 0.57 σ on average — but the constant is that hand’s, spanning 0.17 to 1.34 across a hundred and twenty of them, so the floor is a band and not a line.

It does not establish that any painting has two cameras in it, or one, or four. Every picture measured here is constructed. Nothing about any tradition’s pictures is claimed, and the earlier caution stands in full: four meeting heights are evidence about cameras only if the four surfaces are parallel in the world depicted, which is a fact about the depicted world that a drawing does not supply.

And it inherits the earlier conditional. A count is a count of the cameras needed to draw surfaces taken to be parallel. A lectern really does lean differently from the floor it stands on, and a picture containing one honestly needs a group for it. The count cannot tell a second camera from a second kind of furniture, and no reading of the marks can.

Where this sits beside the rest of the field #

The tolerance here plays the part that the assumed focal length played in the single-surface reading, and it is worth noticing that the two are different kinds of free parameter. A focal length is a fact about an instrument that the drawing does not record; a reader who assumes one is guessing about the world. A tolerance is a fact about the drawing — how precisely its marks were placed — and a reader who assumes one is guessing about the page, which is at least the thing in front of them. That is why the count, unlike the tilt, has a floor that can be estimated from the picture itself rather than from a theory about the painter’s equipment.

It also sits beside the corrections that the cube that is a box, dividing depth by eye and the rule that draws another room measured, where a taught construction leaves one judgement to the hand and the hand’s error is the whole of the result. The difference is the direction of the reading. Those essays start from a rule and ask what a hand does to it; this one starts from marks and asks how few rules they could have come from. The same scatter appears in both, once as the error and once as the noise floor, and a picture drawn carelessly enough defeats both readings in the same way.

The comparison table’s rows are not involved. A row of that table is one rule applied uniformly, and a picture drawn by one rule throughout needs one camera at any tolerance above its own hand’s floor. What is being counted here is how many rules one picture contains, which is a question about pictures and not about systems — the same distinction drawn when the divergent construction was declined a row of its own, and the reason a centre and a measure are exclusive is a statement about rows rather than about any particular drawing.

One picture elsewhere in these essays provably contains more than one camera, with the count known in advance. Three distances in one landscape assembles a landscape from a low station for the near ground, a level one for the middle and a high one for the far, and a reader handed only its marks recovers each band’s own camera height without being told any of them. Its groups are its bands. They are separated by a great deal more than a brush line, so its window would be correspondingly wide, and it is the shape of picture this count was built to find, arrived at from the other direction. The contrast is the instructive part: there the change of station announces itself to a reader as a seam in the rate depth runs, and here the groups leave no seam at all. A count is worth having precisely where nothing shows.

Still open: whether the test is asking for a point or for a line #

Every reading above requires the surfaces of a group to share a meeting point, and for these pictures that is the same as requiring them to share a meeting height, because every surface in every stack here is drawn symmetric about one column of the page.

Those are not the same requirement in general, and the difference is not a technicality. One camera photographing several parallel surfaces gives them all the same vanishing line — but it gives them the same vanishing point only when they also share a receding direction, which is to say only when the furniture is all square to the same wall. Turn a footstool a few degrees and the picture is still one camera’s, and its surfaces no longer meet at one point.

The measurement that settles it takes one camera and several parallel surfaces turned by different angles in their own planes, computes what each test charges such a picture, and asks which of the two is the test one camera actually imposes. If the stricter one charges a picture a real camera took, then every count above has been using a test that is right for the pictures it was run on and wrong in general — and the count for a picture of a turned room would be too large for a reason that has nothing to do with how many cameras drew it.