A closer pose sharpens one coefficient and not the pair

A tilted target pays for its tilt in perspective found the first radial coefficient read about twice as well from a target at an angle as from one square to the lens, and found the two coefficients exactly as inseparable as they had always been: a correlation between them of −0.993 square on, and never better than −0.981 at any tilt. It ended with a question calibration practice has an answer to. Nobody calibrates a lens from one photograph. The target is held at many angles at many distances and the marks are pooled. Pooling adds information, and the essay asked whether what it adds has the right shape to separate the pair.

The hope was specific. A single pose always sees the same pair of nearly proportional functions, the cube and the fifth power of the radius, over the same band of radii. But a target photographed from twice as far lands at half the radius, so a set of poses might sample the lens at different radii, with the target’s printed size as the link between them. Distance was the promising thing to vary.

It turns out not to be, in either direction, and the way that became clear also corrected the essay that asked. A pool of poses reaching close enough to fill the frame’s outer band runs straight into a limit the single-tilt numbers had been ignoring.

What a pool is #

The lens and its centre are shared by all the poses: two radial coefficients and a distortion centre, four numbers. Each pose then brings its own: its scale in the frame and its tilt about each of two axes, three more numbers per photograph. A pool of eight poses has twenty-eight unknowns against several hundred marks. Each mark is read across the drawn curve, along its normal, to half a pixel, as before. The Fisher information is the sum of what each mark contributes, and its inverse gives every scatter and every correlation directly, with no noise simulated. These are the method’s limits rather than one experiment’s results.

The target is the same four concentric circles, at 60, 160, 260 and 360 px, through the same barrel lens with a first coefficient of −0.28.

A mark past the fold is not a mark #

Comparing a pool of one with the single-tilt figures should reproduce them, and at first it did: one pose at 50° came back to $1.95\times10^{-3}$, as that essay had it. Then a target at twice the scale reported marks 332 px from the centre, and that cannot happen. A barrel model folds at a radius it sets itself showed that a radial polynomial with a negative first coefficient stops being one-to-one where $\rho^2 = -1/(3k_1)$. For $k_1 = -0.28$ on this camera, with a focal length of 426 px, the undistorted radius there is 465 px and the drawn radius at that point is the largest the model can ever produce: 310 px. A printed point further out does not draw further out. The model turns back and draws it inside the frame, at a place no ray from the lens could reach.

Those marks were being counted. A test for whether a mark falls inside the frame cannot catch them, because they do. From 40° of tilt the near edge of the 360-px ring has points past the fold, 24 of them at 40°, 48 at 50°, 86 at 60°, and those points carried a large and quite fictitious lever on the coefficients. The earlier essay’s precision kept falling to $1.70\times10^{-3}$ at 60° on the strength of them, for an apparent gain of 2.8 times. With those marks excluded, the best tilt for this target is 30°, at $2.46\times10^{-3}$, a gain of 1.94 times. That essay has been corrected: its mechanism and control are untouched, and only its best tilt and the size of the gain have moved. So one pose at 50° reads $3.27\times10^{-3}$, not $1.95\times10^{-3}$, and every pool below has the same exclusion.

The tilted-target essay’s own control still stands, since a target squashed without perspective never comes near the fold. A fold is the sort of thing a calibration has no reason to hit in practice, since a lens has no rays beyond its field and a real target’s points out there simply do not appear. It took an idealised calculation to walk into it. It is still a warning: a mark inside the frame is not a mark from the lens unless the model’s map from world to page is still one-to-one at that mark.

Pooling tilts sharpens the fit and leaves the wall where it was #

Start with four tilted poses: 50° each way about each of the two axes, all at the usual scale.

The correlation goes from −0.982 to −0.975. Four views in four directions, with a fair amount of independent perspective in each, shift it by seven thousandths. At the same time the first coefficient’s precision goes from $3.27\times10^{-3}$ to $1.84\times10^{-3}$, a factor of 1.78 against the factor of two the marks alone would give. The centre, which a single tilted pose reads to $\pm 0.11$ px, comes to $\pm 0.05$ px.

So the first half of the question has its answer. Pooling views helps exactly as pooling usually does, at roughly the square root of the number of views. It sharpens what the marks already read well, and the combination they cannot read stays unread. No design separates the two coefficients explains why: over any band of radii inside one frame, the cube and the fifth power are nearly proportional, and four tilted poses still lie within that band.

Moving the target further away does nothing #

Now the proposed separation. Add four more tilted poses at half the image scale. The target’s circles then land at half the radius, in a band the first set of poses scarcely touched, carrying the same printed sizes with a fitted scale.

The correlation moves from −0.9745 to −0.9723. The first coefficient goes from $1.84\times10^{-3}$ to $1.76\times10^{-3}$. A further set at a quarter of the scale, for 716 additional marks in all, changes the correlation in the fifth decimal and the precision in the fourth. That is nothing a person could ever quote.

The cause is arithmetic. A radial displacement is $k_1 r^3 + k_2 r^5$, so a coefficient’s lever on a mark grows as the cube of its radius, and a mark at half the radius has an eighth of the lever on $k_1$ and a thirty-second of the lever on $k_2$. Half-scale poses reach only to 187 px on this lens. Alone, four of them read the first coefficient to $17.5\times10^{-3}$, nearly ten times worse than the same poses at the usual scale, and four at a quarter of the scale read it to $184\times10^{-3}$. They are not wrong. They are just very quiet.

The hope also rested on a misconception about what the correlation measures. Different radii would separate the two functions only if the functions had a different shape over those radii, and both are powers. Over a band from zero to any radius $R$, rescaling the radius gives each function the same form, so the shape relating them does not change with the band’s width. Half-scale poses, looked at in their own band, see the same pair of curves with the same near-proportionality. A set of bands all starting from zero contains no contrast between them, however different their sizes are.

Bringing it closer, and where that stops #

The same reasoning says which way to go for precision, and which way not to go for separation. A pose larger in the frame fills the outer radii, moving the outer printed circles out of the field entirely and bringing the inner ones to where the outer ones were.

A second set at the same scale gives what doubling marks should: $1.84\times10^{-3}$ to $1.30\times10^{-3}$. At one and a half times the scale, $1.15\times10^{-3}$. At twice, $1.01\times10^{-3}$. Then it stops: $1.01\times10^{-3}$ at two and a half times, $1.01\times10^{-3}$ at three. The floor has a clear cause.

The fold is a hard outer limit, and it is reached already at the usual scale. So moving in cannot widen the band of radii. It can only move marks within it, towards the outside, where each carries more lever. Once they are all out there, there is nothing further to be gained by moving in.

Meanwhile the correlation is moving the wrong way: −0.975 at the same scale, −0.983 at twice, −0.988 at three times. The marks have concentrated in a narrow band near the fold, and over a narrow band the two functions look more alike than they do over a wide one. So the pool’s two jobs are in direct opposition. Marks near the edge read the first coefficient well, and they read the two coefficients apart less well.

A single tilted pose shows the same exchange at its plainest. At the usual scale, one pose at 50° reads the first coefficient to $3.27\times10^{-3}$ with a correlation of −0.982. At twice the scale the same tilt reads it to $1.99\times10^{-3}$ with a correlation of −0.989. Where a mark helps most is in the same place the response is at the ends described for a straight edge: a coefficient’s lever grows quickly toward the frame’s outside, and a target that spends all its budget there reads the combination extremely well, while the difference between the two functions has almost no room to show.

Taking a pose-for-pose budget, four poses at the usual scale read the first coefficient to $1.84\times10^{-3}$, and four at twice the scale read it to $1.25\times10^{-3}$, at the same four photographs: a gain of about a half, with a correlation of −0.986 against −0.975. A session spending eight frames should spend most of them close if the first coefficient is what it wants. It should not expect that to help with the second, and it should not expect the gain to continue once its marks have reached the fold.

What knowing the pose would buy #

Everything so far has fitted each pose’s scale and tilts as extra unknowns, as a calibration does. If the distance were given, or the target’s pose measured some other way, how much would that help?

Knowing every scale takes the first coefficient from $1.76\times10^{-3}$ to $0.95\times10^{-3}$ and moves the correlation from −0.972 to −0.949. Knowing the tilts as well takes those to $0.86\times10^{-3}$ and −0.939. That is the furthest any arrangement here moves the pair, and it was not done with the poses at all. A radial scaling is the term most like a radial coefficient’s own effect, so a fit with a free scale spends much of what it knows on telling them apart. A known target sharpens the fit found a similar factor for a target square on, and it also found how hard the price is to pay, since half the advantage was gone once the size was known only to a tenth of a per cent. That finding stands, and it is the reason this figure is not an instruction to buy a rail. Still, among the things a calibration could plausibly change, this is where the correlation moves most, and it is not much. At −0.94 the second coefficient still costs the first about a factor of three, compared with what it would be alone.

Why this is not a failure of practice #

The usual advice in calibration guides is to cover the frame with the target, at as many positions and angles as possible, including the corners. On this measurement that advice is right, and closer poses are a cheap way to take it. The advice sometimes also suggests a few shots from further away. These have almost no support as sources of information on the two coefficients, though they cost only a little time, and they are often taken for reasons this model has nothing to say about: checking that distortion is stable with focus, for example.

Two limits should be stated. First, this treats the lens’s model as true. If the real lens has a term that a two-coefficient model cannot represent, a pooled fit will spread the disagreement across its coefficients, and a wider spread of poses can reveal a fault that a tighter one hides. Reading a model’s failure is a different job from reading its parameters. Second, these are the lower bounds of an idealised fit. A real pose carries a blur and a focus change with distance, and a close tilted pose has the most depth of all to hold: its near edge is much nearer than its far edge, and the near edge’s outer marks, where the lever is, are also the likeliest to be soft. That is the condition under which fitting a lens from straightness alone is at its most fragile as well. The half-pixel reading should be measured separately for the closest poses, from the fit’s own residuals split by distance, and not assumed.

What the wall is, after all this #

A whole sequence of essays has now come up against this wall from different directions. Straight lines could not get past it, nor could printed circles, tilts, pools or distances, and a known size moves it a little. A limit met that many ways begins to look like a theorem, and it should: in one frame the two functions cover the same band, and any weighting of marks on that band has a limit that no design separates already calculated.

So it is worth asking what practice does about it, and it does the one sensible thing. It stops wanting the two coefficients. It reports the combination, the radial displacement at each radius, which the fit reads with a precision all these figures suggest should be excellent. A pair of coefficients at a correlation of −0.98 can still produce a displacement curve that is well known everywhere in the frame, and that curve is what undoing the distortion actually needs. The wall affects it only in an extrapolation past the marks, a calibration from a mid-frame target used on a corner, or when one coefficient is clipped or rescaled by another piece of software. That is the correct reading of −0.98. The two numbers should travel together, and neither should be quoted alone. It also fixes what the closer poses bought. They narrowed the scatter of the first coefficient, which is a real change in what the fit can report, and they are no evidence that the second coefficient has become separately known.

Still open: how precisely the curve itself is known #

Reporting the displacement rather than the coefficients changes the question that matters. The scatter of $k_1$ alone is what every figure here reports, because that is what the earlier comparisons used. The scatter of the displacement at a stated radius is a different number, with its own dependence on the pool, and it might not care about many of the distinctions drawn here, since a correlation near minus one is what a well-determined curve looks like when written as two poorly determined coefficients.

The measurement that settles it propagates each pool’s full covariance into the displacement at every radius, in pixels, and plots the resulting scatter from the frame’s centre out to the fold. It asks where in the frame each pool’s curve is known worst; whether the close poses, which won clearly on the coefficient and lost on the correlation, win or lose on the curve; and whether, at the middle of the frame, where their marks actually sit, the far poses that looked useless on the coefficient turn out to contribute anything at all.

A lens’s two coefficients are read best from close and from a stated distance, and they are separated by neither. The number to want is the curve they draw together.