A tilted target pays for its tilt in perspective

A known target sharpens the fit and does not separate it replaced the straight edges this sequence had calibrated a lens with by printed circles of stated size, centred on the picture. A circle of known size reads the whole of the lens’s radial displacement at its radius, not only the bow a straight edge shows, and four circles reaching into the corners of the frame fixed the first radial coefficient far better than edges at the same number of marks: 4.77 thousandths with the target’s size in the picture fitted, 2.09 with it known. What they did not do was separate the two radial coefficients, whose correlation stayed between −0.979 and −0.9997 for every design of circles.

That target was held square to the lens. The essay ended by asking about the way targets are actually held, which is at an angle and usually in a hand. A circle photographed at an angle is an ellipse by perspective alone, and the lens then bends the ellipse, so the fit must recover the tilt and the distortion together. Two extra unknowns, the earlier essay reasoned, ought to cost precision. But a tilted target also does something a square one cannot: its near side is magnified more than its far side, so a single circle spans a range of radii rather than sitting at one — which was the straight edge’s advantage.

The question it posed had a sharp form: does the precision at the best tilt beat the square target’s 4.77 thousandths? It does, by nearly a factor of three, and the reason is exactly the one proposed.

What the lens draws #

The target is the same four concentric circles, at 60, 160, 260 and 360 pixels, photographed through the same barrel lens with its first coefficient at −0.28. Now it is tilted about a horizontal axis through its centre, its lower half toward the camera.

Tilted fifty degrees, every circle has become an ellipse, squashed top to bottom by the cosine of the tilt, and every ellipse is lopsided: its lower half, nearer the camera, is drawn larger than its upper half. The outer circles, which square on the frame cut to arcs in the corners, now fit more nearly whole because the squash brings their tops and bottoms inside.

The consequence the figure’s caption lists is the one that matters. Square on, the marks of the inner circle sat at one distance from the lens’s distortion centre, 60 pixels, and the other circles likewise each at one distance. Tilted, the innermost circle’s marks stand anywhere from 35 to 60 pixels from the centre, and the outermost’s from 136 to 310. A radial distortion is a function of distance from the centre, and a set of marks that samples many distances is informative about the shape of that function in a way that marks at a handful of distances cannot be.

The precision, tilt by tilt #

The measurement is the one the earlier essays used throughout: the inverse of the Fisher information for the lens’s parameters, from marks each read across the drawn edge to half a pixel, at a fixed budget of ninety-six marks. The unknowns are the two radial coefficients, the distortion centre, the target’s size in the picture, and now its tilt.

The fitted-tilt curve falls steadily from 4.77 thousandths square on to 1.70 at sixty degrees, and rises slightly beyond, to 1.74 at seventy. At its best the tilted target is 2.8 times more precise than the square one, from exactly the same number of marks. The two extra unknowns cost something — with the tilt given rather than fitted the precision at sixty degrees would be 1.43 — but what the tilt buys is far larger than what fitting it costs.

The third curve is the control that says where the gain comes from. It keeps the tilt but removes the perspective, so each circle is squashed by the cosine of the tilt into a symmetric ellipse with no near side and no far side. That target is no better than the square one at any angle: 5.1 to 6.5 thousandths, slightly worse than 4.77. So the gain is not the circles becoming ellipses, and not the outer circles fitting better in the frame, since both of those happen in the squashed control too. It is the perspective — the one thing the control lacks.

A band of radii, not a radius #

The figure below lays out, circle by circle, the range of distances from the distortion centre that each circle’s marks occupy, square on and at three tilts.

Square on, each bar is a point: every circle, whole or clipped, puts its marks at one distance from the centre. At twenty degrees the bars have become short bands; at sixty, the innermost circle spans 27 to 60 pixels and the outermost 102 to 310. The four bands overlap and together cover nearly the whole range from the centre to the corners.

This is the straight edge’s virtue, and the lines that calibrate a lens measured it when it found that one straight edge runs from near the centre to the edge of the frame and samples every radius on the way. What a straight edge lacked was a known shape: it reports only its bow, the part of the displacement that bends it, and not the displacement itself — straight lines that are not is the picture of that bow, and the bow is all an edge can ever say. A circle of known size reports the displacement, but square on it reports it at one radius. The tilted circle does both — it reports the displacement, because its shape is known, and it reports it across a band of radii, because perspective has drawn it lopsided.

The squashed control fails for a reason visible in the same terms. A symmetric ellipse also spans a band of distances from its centre, from its semi-minor to its semi-major axis; but the band is the same on both sides, and the squash is a linear change of the target’s shape which the fit’s magnification and tilt absorb. Perspective is not linear: it enlarges one half and shrinks the other by amounts that depend on each point’s depth, and no combination of size and tilt can mimic a radial distortion’s dependence on distance from the centre with it.

Only a target that fills the frame gains #

If the gain comes from the near half being drawn larger, it should depend on how large the target is against its distance — how much depth the tilt gives it. For a camera, that is how much of the frame the target fills.

The two designs that reach the corners — circles out to 340 or 360 pixels — gain 2.8 times at sixty degrees. The two that stay inside the frame’s short side, out to 180 pixels, gain at most 1.4 to 1.6 times, at thirty degrees, and less as the tilt grows, falling below their square-on precision by seventy degrees — 0.84 and 0.94 of it. A small target has little depth to give: tilting it barely enlarges one half against the other, and the tilt’s foreshortening then crowds the marks together without spreading them across radii.

So the usual advice to fill the frame with the target turns out to have two reasons rather than one. The earlier essay found the first: circles reaching the corners sample the radii where the lens moves marks most. This is the second: only a target large in the frame has a near side worth tilting.

The two coefficients stay inseparable #

Every essay in this sequence since fitting a lens from straightness alone has run into the same wall: the two radial coefficients are nearly perfectly correlated, so the second is known only badly and the first is known well only if the second is held fixed. The tilt spreads the marks across radii; it might have been expected to break the correlation as well.

It does not. Square on, the correlation is −0.9929; across every tilt it lies between −0.993 and −0.976. The tilt moves it by less than two hundredths. No design separates the two coefficients identified why: the coefficients multiply the cube and the fifth power of the radius, and over any band of radii inside one frame those two functions are nearly proportional. Spreading the marks across the band sharpens the combination the marks can see; it cannot make the two functions look different where they do not.

That makes the result’s shape precise. A tilted target is a better instrument for the first coefficient, with the second fitted alongside. It is not a better instrument for telling the two apart, and nothing about where a single target’s marks fall in a single frame will be.

What fitting the tilt costs #

The earlier essay’s expectation was that two extra nuisances would cost precision, and the cost can be measured directly: the ratio of the first coefficient’s precision with the tilt fitted to its precision with the tilt given.

Below thirty degrees the cost is under three per cent. At sixty degrees it is nineteen per cent, and at seventy-five, thirty-five. The fitted tilt itself comes back to a tenth of a degree at small tilts and to five hundredths at sixty. A tilt and a radial stretch move a circle’s marks in different patterns: the tilt squashes one direction and enlarges one half, while the stretch acts the same way all the way round, so the fit can tell them apart and pays little for having to.

That answers the earlier essay’s worry and settles the practical question it raised. A calibration target held at an angle in the hand does not need its angle measured; the angle comes back from the same marks, nearly for free. And holding it at an angle is not only a matter of covering the frame from several directions. It is information: at the same number of marks, a target tilted to sixty degrees fixes the first coefficient nearly three times better than the same target held square.

Better than knowing the size #

The earlier essay’s most practical finding was about the target’s size in the picture. Circles whose drawn size is known — because the target’s distance and the lens’s focal length are known some other way — fixed the first coefficient to 2.09 thousandths, against 4.77 with the size fitted; and half of that advantage was gone once the size was known only to a tenth of a per cent, a precision no tape measure offers. The size, it concluded, is worth knowing and nearly impossible to know well enough.

The tilt is an easier purchase. A target tilted sixty degrees with its size fitted fixes the first coefficient to 1.70 thousandths, better than the square target with its size known exactly. No measurement of distance is needed, only an angle the fit recovers for itself. And the two advantages compound: tilted sixty degrees with the size also known, the same ninety-six marks give 0.68 thousandths, three times better than the square target’s best and seven times better than its usual.

That reorders the advice a calibration manual might give. Measuring the target’s distance precisely is hard and buys a factor of two. Tilting the target is free and buys nearly three. A calibration that must choose where to spend its effort should tilt first.

Why a tilted target is not a tilted sensor #

One confusion is worth heading off, because the words are alike. A tilted sensor is not a distortion found that a sensor tipped inside the camera leaves a picture that is still an exact projection — a pinhole with its principal point moved — and that a distortion fit which treats the tilt as a lens defect finds one that is not there. That is a tilt of the picture surface. The tilt here is of the object photographed, and it changes nothing about the camera at all: the lens, the sensor and the pinhole are exactly as they were. What changes is which distances from the distortion centre the target’s marks land at.

The difference matters for the fit. A sensor’s tilt is a parameter of the camera and belongs in the model of every picture the camera takes; a target’s tilt is a parameter of one photograph and is fitted afresh for each. Both come back from the marks, and they must not be confused with each other or with the distortion, whose defining property — a lens destroys the invariant that a pinhole keeps — is what every mark on the target is ultimately measuring.

Why perspective is the part that pays #

It helps to state the mechanism once in the plainest terms. A lens’s radial distortion moves each point along the line from the distortion centre by an amount that depends on its distance from the centre. To measure that dependence, marks are needed at many distances whose true positions are known. A straight edge gives many distances and no known positions, only straightness — and the response is at the ends, the information is not found how little of that a straight edge’s far ends add on their own. A circle square to the lens gives known positions and one distance. A circle tilted in perspective gives known positions at many distances, because its near half is drawn larger than its far half by an amount set by the depth each point lies at — a depth the fit recovers as the tilt.

The calibration by straightness this sequence began with worked by looking past perspective, since a pinhole keeps a straight line straight however it is tilted. Here perspective is the thing that makes the calibration work. The same property that makes a tilted circle a lopsided ellipse — that nearer things are drawn larger — is what spreads its marks across the lens’s radii. A target with no perspective, an orthographic projection of the same tilt, has no near side and teaches the fit nothing new.

What was assumed #

One target, one picture. Every precision here is from a single photograph of a single target. Real calibrations combine many photographs at many poses, and the information adds; whether it adds in the directions that matter is the question below.

The tilt is about a horizontal axis through the target’s centre, and the target is centred on the distortion centre. A target held off-centre, or tilted about another axis, puts its near side elsewhere in the frame and spans a different band of radii. The mechanism is the same; the numbers are for this arrangement.

Marks are read across each drawn edge to half a pixel. A tilted target’s far half is drawn smaller and its edges are foreshortened, and in a real photograph they are read less precisely than the near half’s. The figures assume every mark is equally good, which flatters the steepest tilts.

Still open: whether several poses break the correlation #

One tilted target fixes the first coefficient well and leaves the two coefficients as inseparable as ever, because in one frame the cube and the fifth power of the radius are nearly proportional over any band of radii the marks can reach. Calibration practice photographs the target many times, at different tilts about different axes and at different distances, and pools the marks.

Pooling adds information, but only in the directions each pose supplies, and every pose supplies the same nearly proportional pair of functions. What might differ between poses is the target’s size in the picture: the same printed circle photographed from twice as far lands at half the radius, so a set of poses at different distances samples the lens’s radii in a way no single target can, with the circles’ true sizes linking them. The measurement that settles it pools the Fisher information of the four-circle target over a set of poses — tilted either way about both axes, and at two distances — and asks whether the correlation between the two coefficients finally moves away from minus one, or whether every pose, at every distance, still sees the same two functions over the same frame.