The lens a pavement can hide

A photograph of a tiled floor is a drawing somebody might try to attribute, and it belongs in this row for a reason that is not obvious: it is the one procedure with no hand step at all. Whatever departure it shows comes from the instrument.

The instrument’s departure is a lens, and a lens is radial — it moves a mark along the line from the principal point, by an amount that depends on how far out the mark is and on nothing else. Straight lines that are not is where this collection measures it, and a lens destroys the invariant is where it establishes that the cross-ratio does not survive it.

So the reader’s projective test should convict a photograph, and the question is how strong the lens has to be before it does.

The answer, and it is a large number #

At a reader’s ruler of a fifth of a pixel, the threshold is a radial coefficient of about four tenths. Below it the photograph reads as a correct drawing.

Four tenths is not a subtle lens. It bows a straight oblique edge across this page by nearly six pixels, and a lens chart would print it as about twenty per cent distortion at the corner of the frame — which is a strong wide-angle, the kind whose barrel is visible in any photograph containing a horizon.

So an ordinary lens is invisible to this reading, and a conspicuous one is barely visible.

That is worth putting beside the count. Four marks before anything is said establishes that a short pavement makes almost no statements at all, and this essay establishes that even a long one makes its statements in the quietest part of the frame. The two limits compound: a reading of a small pavement in the middle of a photograph is a reading of very little.

Why a pavement is the worst place to look #

The reason is one line of geometry and it is worth having, because it says where to look instead.

A radial map displaces a mark by an amount proportional to the cube of its distance from the principal point — the mark moves by its radius times the coefficient times the square of the radius over the focal length. So the displacement grows fast with radius and is nearly nothing near the centre.

A pavement sits near the centre. Its transversals run from the ground line up to the horizon, and the horizon is at the principal point’s own height; on this panel the farthest transversal is forty-five per cent of the focal length from the centre, so the factor the coefficient is multiplied by is about a fifth. The corner of a frame is seven-tenths of the focal length out, where the factor is half.

A pavement therefore sees about two-fifths of the distortion a frame’s corner sees, and a reading confined to it is a reading conducted in the calmest part of the picture.

And the transversals are the one family it leaves straight #

There is a second reason, and it is sharper.

The transversals are read along the vertical through the centric point, which is a line through the principal point — and a radial map leaves such a line exactly straight, because it moves every point along the line rather than off it. The principal point is not the centre is where this collection establishes which point that is and why it is not the middle of the frame.

So the distortion does not bend the family the reading uses. It only slides the marks along it, and sliding marks along a line is the one thing that changes a cross-ratio without changing anything visible. The reading is therefore looking at exactly the residue of the distortion and none of its conspicuous part.

That is a general and slightly uncomfortable lesson: the most obvious symptom of a lens and the symptom a projective test is sensitive to are different symptoms, and a reading built on one says nothing about the other.

Two-fifths of what, exactly #

The comparison between the pavement and the frame’s corner is made twice above in two different currencies, and they give different numbers, so it is worth separating them.

As a fraction of its own radius, a mark moves by $k_{1}r^{2}/f^{2}$. At the farthest transversal, 0.45 of the focal length out, that factor is 0.20; at the corner, 0.70 out, it is 0.49. The ratio is 0.41 — the two-fifths quoted.

In pixels, a mark moves by $k_{1}r^{3}/f^{2}$, which is the quantity a reader’s ruler is laid against. At 0.45 that is $0.091\,k_{1}f$ and at 0.70 it is $0.343\,k_{1}f$, so the pavement’s farthest mark is displaced 27% as far as a corner mark — closer to a quarter than to two-fifths. Both numbers are right about different things, and the second is the one that decides what a reading in pixels can see.

Neither, though, is what the pavement’s test actually measures, and that is the sharper point. The reading is a cross-ratio along a line through the principal point, and the radial map slides marks along such a line without bending it. What survives is the departure the invariant essay derives: to first order, $k_{1}(r_{1}-r_{2})(r_{3}-r_{4})$, a product of two gaps rather than of two radii. Four transversal marks spanning radii from 0.15 to 0.45 have gaps of about a tenth of the focal length apiece, so the product is about $0.01\,k_{1}$ — four parts in a thousand at the threshold coefficient of 0.4.

Set that against what the same coefficient is doing at the rim. A corner mark at 0.7 of the focal length is displaced by $0.137f$ — a seventh of the focal length, which on this panel is over a hundred pixels — and a long oblique edge is bowed by the sagitta $|k_{1}|\,d\,a^{2}/f^{2}$, several pixels across the page.

So the gap between what the pavement reports and what the picture contains is not a factor of two or of four. It is the difference between a fourth-order quantity built out of small differences and a third-order quantity built out of large radii, and that is why the threshold coefficient comes out at four tenths rather than at four hundredths.

A photograph and a rule are the same shape #

The uncomfortable part continues. Compare a photographed pavement with a pavement spaced by the taught constant-ratio rule and their residuals have the same shape.

Both are smooth: one hump, always the same way up, two sign changes on a pavement of any length. Measured by roughness — the statistic that separates a hand’s independent errors from a hand’s cumulative ones — the photograph gives 0.885 and the rule gives 0.841 on this panel, which is the same number to any precision worth quoting.

So the reading that separates a hand from a rule cannot separate a lens from a rule. Both are read as systematic and both are systematic; the reading’s granularity is honest and its granularity is not fine enough. One hand step each sets out what the roughness can and cannot say, and this is the case it cannot.

What separates them is a prediction #

Two explanations that fit the same data equally well are separated by asking each to predict something else, and here the something else is anywhere else on the page.

A lens has one number and that number has to explain the whole picture. Fit the coefficient from the pavement’s residual and it predicts, with no further freedom, how much any straight edge in the frame is bowed. A spacing rule has one number too — the ratio — and it predicts nothing at all about an edge, because it is a rule about transversals and says nothing about lines that are not transversals.

The lens bows the edge by seven pixels at a coefficient of a half. The rule leaves it straight to the arithmetic floor.

That is the test, and it is the only kind of test that could have worked. A residual is what a model failed to explain; two models that fail identically on one family of marks are distinguished by what they say about a family they were not fitted to.

Fitting the coefficient from the page instead #

There is a better reading available and it belongs in this field rather than in this row, and it is worth pointing at because it makes the pavement’s weakness concrete.

Fitting a lens from straightness alone recovers the coefficient from nothing but the knowledge that some drawn edges were straight in the world — no target, no scene, no camera. It uses the edges near the frame’s rim, where the distortion is largest, and it returns the coefficient to fifteen digits on synthetic data.

That reading and this one are asking the same question of the same photograph, and one of them is a hundred times more sensitive because it looks where the effect is. The pavement’s contribution is not the measurement; it is the agreement between what the pavement implies and what the edges imply, which is a consistency check a spacing rule cannot pass — the same shape of argument the rule that draws another room makes with the horizon instead of with an edge.

What a strong lens does to the horizon #

There is a third instrument in this row — the horizon the pavement implies against the horizon the panel drew — and it behaves interestingly under a lens.

A radial map leaves the principal point fixed and the horizon passes through it, so the drawn horizon is exactly where it was. The pavement’s implied horizon moves, by about three pixels at a coefficient of a seventh and by fifty at nine-tenths.

That puts a photograph in an awkward position between the two other rows. A hand at a pixel’s precision moves the implied horizon by one to twelve pixels; the taught spacing rule moves it by a hundred and seventy; a strong lens moves it by tens. So the horizon test, which separates a rule from a hand cleanly, puts a strong lens on the rule’s side and a moderate one on the hand’s — and there is a band of coefficients where it reports the wrong thing.

The band is stated rather than papered over. It is the reason the confusion matrix in this row is run at a coefficient deliberately above the threshold, and the reason an ordinary lens appears in it as a correct drawing.

The reading’s own model error #

Everything above measures a lens with one radial term, and real lenses have more.

Straight lines that are not works with the Brown–Conrady form, whose second radial coefficient matters at the rim and hardly at all near the centre. A pavement’s marks are all near the centre, so the second term contributes a displacement smaller than the first by the square of a small number — which is to say the pavement cannot see it at all.

That is convenient for the arithmetic and it is a warning about the conclusion. The threshold quoted here is a threshold for a one-term lens. A real lens whose first term is below it and whose second is large would still be invisible to the pavement, because both terms are invisible to the pavement; but the coefficient recovered by fitting the pavement alone is not the lens’s coefficient, it is whatever single number best explains the calmest fifth of the frame.

Quoting it as a measurement of the lens would be the model error this collection names elsewhere: a number that is wrong because the model does not contain the thing being measured, rather than because the measurement was noisy.

Where a lens moves a measurement, and where it does not #

The row this essay sits in is about attribution, and there is a neighbouring question worth separating from it: whether a hidden lens damages the measurements a reader takes off the pavement, as against whether it is detectable.

The two answers are different and neither follows from the other. A lens below the threshold above is undetectable in the pavement, and it is also nearly harmless there — the same smallness of the displacement near the centre does both jobs. A lens above it is detectable in the pavement and is doing real damage at the frame’s edge, where the reader was probably not measuring anyway.

What that means practically is that the pavement’s own readings are safer than they look. How wrong a measurement from one picture can be prices the chain from a mark to a height; a lens’s contribution to that chain, for marks near the principal point, is the smallest term in it, and the dominant terms are the reader’s own mark placement and the assumed proportion.

The place a hidden lens is genuinely dangerous is the opposite one: a height read from a figure standing near the frame’s edge, transferred to the floor by the horizon. There the mark is far out, the displacement is large, and height error uncorrected is the size of what it costs.

The instrument and the drawing hand, side by side #

It is worth putting the four procedures’ departures on one scale, because the row’s whole structure depends on them being comparable and they arrive from completely different places.

A hand at a pixel’s precision leaves a worst departure of a pixel or two on an eight-braccio pavement. The taught spacing rule leaves nine and a half. A lens at the threshold leaves a fifth of a pixel in the reading and nearly six pixels of bow on an edge elsewhere.

So the ordering by conspicuousness in the pavement is: the rule, then a hand, then a lens. The ordering by conspicuousness in the picture as a whole is nearly the reverse: a strong lens is the first thing anybody notices, a hand’s slips are invisible without measurement, and the rule is invisible until the two halves of the drawing are compared.

That reversal is the practical content of this row. The departures a reader notices and the departures a projective reading is sensitive to are almost disjoint sets, and a reading that does not say which of the two it is reporting is not saying much.

What a photograph would have to be to be caught here #

To finish the case, it is worth naming what kind of photograph the pavement does convict.

A coefficient of four tenths corresponds to a lens with strong, uncorrected barrel — a wide-angle from before software correction, a fisheye used at a modest field, or a cheap zoom at its short end. Modern photographs are usually corrected in the camera, and a corrected photograph’s residual coefficient is a fiftieth of the threshold.

So the honest conclusion for a modern photograph is that it is a correct perspective as far as any pavement can tell, which is the same conclusion the reading reaches for a well-executed classical construction. The two are indistinguishable, and they are indistinguishable because they are both projections — which is a satisfying place for the argument to end rather than a failure of it.

What this means for attributing a photograph #

Three statements, and the third is the one that matters.

A photograph of a floor cannot be told from a drawing of a floor by the floor. Not with an ordinary lens, and barely with a strong one. Any claim that a picture must be photographic because its perspective is exact has the argument backwards: exactness is what a photograph and a good construction share.

A photograph can be told from a drawing by anything near the frame’s edge. One long straight edge running obliquely across the picture is worth more than the whole pavement, and most photographs contain several.

And the pavement’s contribution is a cross-check rather than a measurement. Fit the coefficient at the rim, predict the pavement’s residual, and see whether it lands. If it does, one instrument explains the whole picture; if it does not, something in the picture is not a projection, and the spacing is the first place to look.

The short version #

A pavement can hide a radial coefficient of about four tenths at a reader’s own precision — a lens that bows a straight edge across the page by nearly six pixels and prints as twenty per cent distortion at the frame’s corner.

It hides it because a pavement lives near the principal point, where a radial map does almost nothing, and because the family the reading uses is the one family a radial map leaves exactly straight. What convicts a lens is a second family of marks somewhere else in the frame, and the test is a prediction made on one part of the picture and checked on another — which is also the only thing that separates a lens from a taught spacing rule.