A known target sharpens the fit and does not separate it

No design separates the two coefficients spent a budget of ninety-six marks on straight edges every way it could think of and found that the two radial coefficients of a lens came out −0.98 correlated whatever was done. More edges or fewer, offsets crowded or spread, the pair behaved as one number reported twice. It ended by blaming the instrument. A plumb-line calibration knows nothing about the scene except that some edges were straight, so every edge covers the same kind of radial band and asks the same kind of question. A target that is known — a printed pattern of stated size — would be a different instrument, and a set of concentric circles centred on the picture would put every mark of each circle at one radius, which is precisely the arrangement no straight edge can produce.

The suggestion is natural and it is how most calibration is actually done: photograph a known pattern and fit the lens to it. It deserves a measurement at the same budget, the same lens and the same half-pixel reading as the edges were given, so that whatever comes out is a statement about the instrument and not about how much it was fed.

The answer divides into what the circles buy and what they do not, and both halves are sharper than the question.

The target, and the part of it the frame keeps #

The lens is the wide barrel this field has measured throughout, which pulls every point toward the centre by an amount growing as the square of its distance, and the frame is the same 690 by 400 picture the edges were drawn in. The target is four circles printed at stated sizes, centred on the picture, so that an undistorted lens would draw them at 60, 160, 260 and 360 pixels.

Two of the circles fit inside the frame and two do not. The frame is 400 pixels tall, so no circle wider than about 180 pixels in radius survives whole, and the outer two leave at the top and bottom and remain as arcs to the left and right, reaching into the corners. That looks like a flaw in the target and is the most important thing about it, because the corners are where the lens moves marks most, and it is the first thing the measurement finds.

Each mark is read the way an edge was read: across the edge, radially, to half a pixel. A printed circle’s edge says where it is across the circle and nothing about where along it, just as a straight edge’s did, so the comparison is fair mark for mark.

Circles inside the frame lose to a doorway #

The first comparison is at the same budget, ninety-six marks, with the distortion centre free and both coefficients fitted.

Eight straight edges fix the first coefficient to 7.99 × 10⁻³, the figure no design separates the two coefficients found for its best design. Three circles of known size at 60, 120 and 180 pixels — a tidy target, all of it visible — fix it to 1.65 × 10⁻², which is 2.1 times worse. A known target that fits inside the frame loses to four photographed door frames.

The reason is the one that essay found from the other side. An edge is not at a radius: it passes closest to the centre at its offset and runs out to the corner of the frame, so every straight edge reaches radii of over 320 pixels, where the barrel moves a mark tens of pixels. A circle is at a radius, which was the whole of its appeal, and a circle that fits inside the short side of the frame is confined to radii where the barrel barely moves anything. The edges were already getting the corners for nothing.

Push the target’s circles out to 360 pixels, so that the outer two survive as arcs in the corners, and the order reverses. Four circles at 60, 160, 260 and 360 fix the first coefficient to 4.77 × 10⁻³ when their size in the picture is fitted, and to 2.09 × 10⁻³ when it is known — 1.7 and 3.8 times better than the edges. The known shape is worth something, but only once the target reaches the same corners the edges were already reaching.

The correlation belongs to the two powers of the radius #

The question the previous essay asked was not about precision, though. It was whether a known target breaks the correlation.

It does not. Six designs — two circles, three, four, six, crowded near the centre or spread out to the corners — leave the two coefficients between 0.979 and 0.9997 correlated with their size known, and worse with it fitted. The band straight edges gave, from three edges to sixteen, runs from 0.980 to 0.991. The circles land in it or above it.

And there is a reason to expect exactly this, which is also a check on the calculation. With the target’s size known and each mark read radially, the fit is a straight regression: each circle is displaced outward or inward by an amount the lens model writes as $k_1\rho^3 + k_2\rho^5$ in units of the focal length, where $\rho$ is the circle’s radius. Two regression coefficients fitted to the same observations are correlated by minus the cosine of the angle between their columns. So the correlation should be the cosine between $\rho^3$ and $\rho^5$ over the radii the design uses, weighted by its marks, and nothing else.

The rings in the figure are that cosine, and the full computation agrees with it to 2.2 × 10⁻⁵ at every design. The correlation is a property of two functions, not of anything photographed.

Which makes the obstacle plain. Over any set of positive radii, $\rho^3$ and $\rho^5$ both rise from nothing and $\rho^5$ is $\rho^3$ multiplied by something that only rises, so the two columns point in nearly the same direction. A target that sampled every radius from the centre to the corner equally would reach a cosine of 0.975, and no design inside a frame does meaningfully better, because the obstacle is the shape of the functions and a frame only ever offers them one range of radii. No design separates the two coefficients argued the same thing about straight edges from their shape; the circles show it holds whatever the marks are on.

A smaller sliver, pointing the same way #

What the known target changes is visible once the uncertainty is drawn rather than summarised.

All three instruments leave a sliver, and all three slivers lie along the same line in the plane of the two coefficients. The circles shrink it, and do not turn it. With their size known they fix the first coefficient 3.8 times better than the edges and the second 3.6 times better, and the combination of the two that the data determine best goes from 1.33 × 10⁻³ to 3.70 × 10⁻⁴.

That is the honest description of what a known target is for. It does not turn one number into two. It measures the one number the lens model can support much more precisely, and it drags the other along in proportion, so that the second coefficient is known better in absolute terms and exactly as badly relative to the first. A reader who wanted separate coefficients from a circle target in order to compare them with a manufacturer’s figure is in the position that essay described for straight edges, only with smaller error bars around the same ambiguity.

Known means known to a thousandth #

The 3.8 comes with a condition that is easy to state and easy to miss. The target’s printed size is known exactly, but its size in the picture is not: it depends on how far away the target stands and on the focal length, and the focal length is one of the things a calibration is usually trying to find. “Size known” in the figures above means the magnification is known. How well does it have to be known?

Very well. Enter the magnification as known to within a stated fraction and sweep the fraction. Known to a hundredth of a per cent, the fit is indistinguishable from knowing it exactly. By a tenth of a per cent half the advantage of knowing it has gone, and at one per cent the fit is 4.75 × 10⁻³ against 4.77 with nothing known at all.

A tenth of a per cent of a metre is a millimetre. A target a metre from the lens has to be placed to a millimetre, and the focal length known to the same fraction, before its known size counts for anything. Neither is usual. In ordinary practice the magnification is fitted along with everything else, and the target’s advantage over the edges is the 1.7 of the fitted case rather than the 3.8 of the known one.

This is the same trade the principal point is not the centre records for a different parameter: a quantity assumed known when it is not does not merely fail to help, it quietly moves every other number. Here the damage is milder, because leaving the scale free costs precision and biases nothing — but a calibration that assumes a scale it has only approximately will report the 2.09 and deliver something worse.

Counting radii before counting marks #

Fitting the target’s size has a second consequence, which is structural rather than a matter of precision.

Each circle, read all the way round, reports one number: how far its drawn radius is from its printed one. One circle is one number and cannot fix two coefficients, whatever else is known. Two circles of known size fix both — at 1.42 × 10⁻² for circles at 100 and 360 pixels, poorly, because two radii are the extreme case of the correlation above, 0.9997. Two circles of unknown size fix nothing at all, because the magnification is a third unknown against two radii. From three circles on, both kinds give an answer: 2.22 × 10⁻³ known and 5.51 fitted at three, 2.09 and 4.77 at four, 1.98 and 4.30 at six.

So a target of unknown size needs three radii to say two things about a lens, and after that more circles at the same budget buy little — the step from four to six is under ten per cent. That is the circles’ version of the finding that closed the straight-edge measurement, that the arrangement moves the answer by fractions and the number of marks moves it by factors, and it points the same way: past the minimum that makes the problem determined, spend on marks, not on pattern.

It is also why the refusal in the measurement is the case it is. One circle of unknown size reports a radius, and a change of magnification and a radial stretch at one radius are indistinguishable. A calibration routine that returned a coefficient from such a picture would be returning its own starting value, and such a picture is one a calibration should refuse rather than fit.

What the straight edges were doing all along #

Put side by side, the two instruments turn out to be doing the same thing with different nuisances.

A straight edge does not know where it is or which way it runs, so its offset and its angle are fitted along with the lens, and they absorb whatever part of the distortion looks like a shifted or turned line. What is left for the lens is the bow. A circle of unknown size does not know its magnification, and the magnification absorbs whatever part of the distortion looks like a uniform stretch. What is left is the departure from uniform stretching — which is again a curvature, this time in radius rather than along a line.

Knowing the target removes that nuisance and lets the fit see the whole displacement, which is why the known case is so much more precise; it is also why the known case is so fragile, since a nuisance assumed away has to be truly away. What knowing the target does not do is change the two functions the displacement is written in, and those decide the correlation. Fitting a lens from straightness alone found the −0.997 on a single edge; the edges’ best design reached −0.98; the circles reach −0.979. The number is the model’s, measured three ways.

What would separate them #

If the obstacle is two functions that look alike over a frame’s radii, three things could move it, and the measurements above say what each is worth.

A wider range of radii would not. The cosine between $\rho^3$ and $\rho^5$ is unchanged when every radius is scaled by the same factor, so a lens with a wider field or a bigger sensor offers the same pair of functions over the same proportional range. The obstacle is the shape of the range, from zero to the corner, not its size.

A different pair of functions would, by definition. Reparameterise the model so that its two coefficients are the stiff combination and the soft one, and the correlation is zero — but the soft one is as poorly known as before, since nothing about the data changed. That is bookkeeping, and a model that inverts has a horizon is about a genuinely different family, the division model, which bends the same way with one coefficient.

And more information in the soft direction would. The soft direction is the one the known target shrinks in proportion rather than preferentially, so the only lever is more marks, read more precisely, reaching further into the corners — the same lever no design separates the two coefficients found to be the only one with slope in it.

Where a circle target stops being the answer #

The circles are centred. Every target here sits with its centre on the distortion centre. An off-centre circle is drawn as an oval whose shape depends on the lens in a way a centred one’s does not, and a grid of dots — the usual printed target — is many small off-centre features at once. Whether their oval shapes carry information the radii do not is a separate calculation.

The target is square to the lens. A tilted target draws its circles as ellipses by perspective before the lens touches them, and separating the two effects adds the target’s orientation to the nuisances. The lines that calibrate a lens found that freeing a nuisance can cost a factor of ten in the wrong arrangement; tilt has not been priced here.

One lens, one frame, half a pixel. As in the essays before it, every number is for the wide barrel at −0.28, a 690 by 400 frame, and marks read to half a pixel. The shapes of the comparisons are the finding; the magnitudes scale with the lens.

And the information is a prediction. Every precision here is the inverse Fisher information, which a fit achieves when its residual is quadratic and its noise normal. The closed form agrees with it to five decimals where both apply, which checks the arithmetic, not the assumption; a seeded simulation of the circle fit has not been run.

The circles, weighed #

Concentric circles of known size were proposed as the instrument that would separate a lens’s two radial coefficients, because they put every mark of a circle at one radius. At the same ninety-six marks, the same lens and the same half-pixel reading as straight edges, they do not: six designs leave the pair 0.979 to 0.9997 correlated, against 0.980 to 0.991 for edges, and with the target’s size known the correlation is exactly minus the cosine between $r^3$ and $r^5$ over the radii used — a property of the model, agreeing with the full computation to 2.2 × 10⁻⁵.

What the target buys is precision, and only on conditions. Circles that fit inside the frame are 2.1 times worse than straight edges, because edges reach the corners and small circles do not. Circles reaching the corners are 1.7 times better with their size fitted and 3.8 times better with it known, and the region the two coefficients are known within shrinks by that factor without turning. Knowing the size means knowing it to about a tenth of a per cent — a millimetre at a metre — and a target of unknown size needs three radii before it says anything about the lens at all.

Still open: whether a tilted target pays for its own tilt #

Real targets are rarely square to the lens, and a calibration that can use a tilted target can use one held in the hand. A circle photographed at an angle is an ellipse by perspective alone, and the lens then bends the ellipse; the fit must recover the tilt and the distortion together.

That adds two nuisances, which on the evidence above ought to cost precision. But a tilted target also does something a square one cannot: it spreads each circle across a range of depths, so its near side is magnified more than its far side, and a single circle then spans a range of radii rather than sitting at one — which is the straight edge’s advantage, recovered by a known shape.

The measurement that settles it tilts the four-circle target through a range of angles about a horizontal axis, fits the tilt, the magnification and both coefficients, and asks whether the precision at the best tilt beats the square target’s 4.77 × 10⁻³ with the size fitted. If it does, the usual advice to hold a target at several angles is advice about information, not just about coverage; if it does not, the square target is the best a known circle can do and the tilt is pure cost.