Two circles leave the eye on a circle

A circle reads a tilted camera’s column ended on a hope. A circle on the ground fixes the column and the distance from the eye to the horizon, and leaves the eye free to swing round that horizon on an arc. A building’s verticals cut the arc, but many pictures of the ground have no building: a car park, a square with its covers and kerbs, a field marked with rings. Two circles at different distances are drawn with different shapes, and that difference, the essay proposed, might state the eye’s height above the ground against its distance from the horizon, and so cut the arc without an upright in sight.

It does not, and the failure is exact. No number of circles at any distances does it, nor any other marks on the ground. The second circle does something else, and the something else is more useful than the hope was.

What a picture of the ground holds #

Take a camera with focal length $f$, $h$ above the ground, pitched down by $\theta$, whose foot stands at $Z_0$ along the ground’s depth axis. A ground point at lateral position $X$ and depth $Z$ lands at a distance below the horizon, and in a column away from the principal point’s, that come out with less in them than the six numbers suggest:

$$u - c = \frac{e\,X}{Z - d_0}, \qquad v - v_h = \frac{e\,k}{Z - d_0},$$

where

$$e = \frac{f}{\cos\theta},\quad k = \frac{h}{\cos\theta},\quad d_0 = Z_0 - h\tan\theta.$$

Those three, with the column $c$ and the horizon’s row $v_h$, are everything. The camera’s six numbers $f$, $h$, $\theta$, $Z_0$, $c$, and the principal point’s row collapse to five in the marks, whatever the marks are. $e$ is the quantity the earlier essay named, the distance from the eye to the horizon line. $k$ is new, and so is $d_0$, and both have a simple meaning. $d_0$ is the depth on the ground that would be drawn at infinity on the page: the place where a plane through the eye parallel to the picture meets the ground. $k$ is the distance from the eye to that same line.

Given those five, the sixth number is free. For any pitch $\theta'$ there is a camera with $f' = e\cos\theta'$, $h' = k\cos\theta'$, $Z_0' = d_0 + k\sin\theta'$ and its principal point’s row at $v_h + e\sin\theta'$, and it draws the same five numbers.

The check takes a few lines. Seven cameras pitched from 1° to 75° are given the five numbers read from one camera at 20°, and they project a mix of rings at 5, 8 and 16 metres through their own full projection, with no reference to the formula. The most any mark differs is $2.5\times10^{-13}$ of a pixel. That is the rounding of the arithmetic and it is not a small error; it means the pictures are the same. So the failure of the hope is not a matter of precision, and no quantity computable from these marks can tell the cameras apart.

The eye swings round a hinge in the ground #

Write the family’s eye positions in the side view. Its height is $k\cos\theta'$ and its distance ahead of $d_0$ is $k\sin\theta'$. That is a circle of radius $k$ about the point $d_0$ on the ground.

The earlier essay’s arc was a circle of radius $f/\cos\theta$ about the horizon, drawn on the picture’s side of things. This is its twin on the ground’s side. The two move together: as the eye swings round the hinge $d_0$ on the ground, the principal point swings round the horizon on the page, and the plane of the picture turns with it. This is the old draughtsman’s construction seen from the other direction. The arc every eye stands on folds the station point down into the picture plane about the horizon. A perspective drawing of the ground has always been a map between two planes hinged at a line, and any eye that keeps its distance from that line gets the same map, which is why a rabatment works at all. Here that freedom is not a convenience for the drawing board. It is a real ambiguity in the photograph.

There is a further way to read it, and it shows why the second circle could not have helped. Marks on the ground fix a homography from the ground to the page, which has eight numbers. Three of them only place and turn the ground’s own coordinates, so with no roll in the camera the five above are what remains. A homography is all that any arrangement of ground marks can ever reach: more circles improve its precision and cannot make it say anything else. Three conics are one conic and a choice of horizon arrives at the same limit from the conics’ side. The earlier essay’s observation that a near circle and a far circle have different shapes is correct, and those shapes are exactly the homography’s own.

Counting equations was the trap #

The earlier essay’s count was the natural one. A camera without roll has its focal length, its column, its principal point’s row and its pitch, which is four inner numbers. Each circle of known size and place, drawn as an ellipse, supplies five numbers, so two circles supply ten, and ten is far more than four. There should have been plenty to spare.

The count treats those ten as independent, and they are not, because both ellipses are drawn by the same map. The first ellipse’s five numbers constrain the map, and the second ellipse’s five do too, but they constrain the same eight numbers and none of the camera’s numbers outside those eight. Equations are only useful to the extent they reach a new unknown. Here, past the fifth, each extra equation refines what the earlier ones already said, which is exactly what least squares reports as a falling scatter and which looks encouragingly like progress. A fit given ten equations and six free numbers will converge, report a tiny residual, and return a pitch. It returns whichever member of the family its start was nearest to, and it has no way to report that it had a free direction, unless the fit is asked for the rank of its own Jacobian.

That is why the trial above is careful to fit only five numbers. A six-number fit to the same marks does not produce a slightly worse answer. Its normal equations are singular, to the rounding of the arithmetic, and if a solver has been regularised to protect against that, its result is the regulariser’s opinion of the pitch.

What the second circle does #

The second circle does matter. It gives the hinge.

A circle of known size and place, on its own, reads the five numbers only weakly. Take numbered marks on one ring, sized to its distance, read to half a pixel. $k$ comes back to about $\pm 0.15$ m and $d_0$ to more than half a metre, because all the marks lie at nearly one depth and $d_0$ is what depth does to size. Add a second ring, twelve degrees away at the eye, and $k$ drops to $\pm 0.01$ m and $d_0$ to $\pm 0.03$ m. That is a factor of ten or twenty. So the near-and-far difference the earlier essay was reaching for is real and valuable; it is just aimed at the hinge rather than at the arc.

The figure below shows how much depth difference is needed, reading the pitch by the method of the next section with a camera at 30°.

A degree between the rings gives $\pm 6.7$°, four degrees $\pm 2.5$°, twelve $\pm 0.92$° and twenty-four $\pm 0.55$°. Roughly halving with each doubling of the gap, it behaves as one expects of a lever. The earlier essay also asked how far apart in depth the two must lie, and the answer has this form: not a threshold at which the pitch appears, but a rate at which the hinge sharpens, with the pitch available only through a length supplied from outside. A circle on the ground names the column has marks in the same style, and its half-pixel results for a level camera are the same order as those here.

One taped length closes it, and which one depends on the pitch #

Any one of the camera’s own numbers cuts the circle: the principal point’s row, the focal length, the height or the position of the foot. The first two are what the earlier essay assumed and what a calibrated camera knows. The last two are what somebody in the field can get with a tape measure, and they behave very differently.

A known height $h$ gives $\cos\theta = h/k$. A level line at that height across the eye’s circle cuts it. When the camera is nearly level the eye is near the top of the circle, where a level line crosses at a grazing angle, so the crossing is hard to place: the cosine is flat near zero, and an error $\delta h$ in the height moves the pitch by $\delta h/(h\tan\theta)$. A known foot $Z_0$ gives $\sin\theta = (Z_0 - d_0)/k$. An upright line at that position cuts the circle, nearly square-on near the top. Its error in pitch is $\delta Z_0/h$, at every pitch.

So at a 5° pitch one centimetre on the height means 4.09° and one centimetre on the foot means 0.36°, an eleven-to-one difference in favour of the length nobody thinks to measure. A photographer knows how high the tripod is. Almost nobody knows, and it is hardly ever recorded, how far along the ground from a reference mark the tripod stands. Yet that is the length that reads a near-level camera’s pitch. At 45° they tie, and past that, the height wins. The crossing has nothing to do with this particular camera. It is where the two ways of cutting a circle at a point are equally steep.

The trial brings the picture back into it. Now the marks have half a pixel of error, the fitted $k$ and $d_0$ have their own scatter, and at 5° the height reads the pitch to $\pm 3.79$° and the foot to $\pm 1.78$°. At 10° it is 2.50° and 1.05°, at 20° 1.33° and 0.97°, at 30° 1.05° and 0.92°, and at 45° 0.95° and 0.94°. The large ratio in the closed form has shrunk because the rings’ own reading now sets a floor of about 0.9°, and the foot’s advantage over the height shows only where the height’s lever is very poor. For a reading better than a degree, better marks are needed, not a better tape. That ordering of the error budget is itself a useful thing to know.

What cannot be taped #

A known length supplied to this method has to be in the same units as the marks. Both $k$ and $d_0$ are in the ground’s units, since the rings have known sizes. If the rings are of unknown size, a whole scale is lost and $k$ with it, and the method fails for a reason that has nothing to do with the arc: the one thing a single view cannot give is a size. The height and the foot both also assume the ground is flat over the whole span from the camera to the far ring. Since it is the camera’s foot that is being taped, a slope between the camera and the rings is a bias of exactly the kind that this reading cannot see: a ground tilted by $\alpha$ puts the eye’s height above the plane in place of the tape’s, and $k$ follows it.

That is the assumption doing the most work here, and a field square is flat to better than the half-degree most of these numbers are quoted to. A car park on a slope is not.

Why this is an old fact in new clothes #

The result is close to something every student of perspective is told in another form: a perspective view of the ground plane has one free parameter once the horizon and the scale are given, and it is the viewer’s distance. Plates in the old treatises sit on that freedom. Straightening does not move the eye uses a related form of it. The novelty here is only in how exact the failure is in a modern setting, where the camera has six numbers, a fitting routine happily returns all six, and the reasonable-looking route of adding more circles adds no information about the sixth.

There is also a more cautious reading of what a calibration from ground marks can claim. A routine that fits a full camera to ground marks alone has to take its sixth number from somewhere. It takes it from its start, or from a prior, or from a principal point it assumed at the frame’s centre. In the last case the reported pitch is $\theta$ exact if the assumption is true and wrong by the whole arc if it is not, and there is no residual to warn of this. A camera with lens shift or a crop off-centre breaks that assumption without any visible effect. This is the same trap a circle reads a tilted camera’s column found from the arc’s side, and a principal point recovered from the ground is not available to get out of it.

The mechanism may also be turned to good use. If it is the map that matters, as it is for rectifying a floor plan or a road survey, the sixth number is simply not needed, and any member of the family gives the same plan. A surveyor who asks only what is on the ground need not know the pitch. Only when the picture is being used to say something off the ground, about a wall, or the height of a post, or a point in the air, does the eye’s actual position become needed. Then it has to come from a building, a tape or a data sheet, since the ground cannot supply it.

The assumption in the numbered marks #

The trials above give each mark on a ring its place: the fit knows which mark on the ring it is looking at, as if the ring were a dial. A painted circle has no such numbers, only an outline, and an outline carries five numbers per conic rather than two per mark. Two conics give ten numbers, so the five of the ground are still over-determined, and the family is exactly as free as before because it draws the outlines in the same place. But the precision quoted will be better than an outline reading of the same ring. Rings of numbered studs, a parking grid, the corners of paving: those supply what the trial supposes. For a bare painted ring, the figures should be taken as a lower limit.

And the marks here are read with no lens distortion. A real wide lens bows a ring’s picture and the error is far larger than half a pixel, so a bow must be removed first or fitted alongside; a circle reads a tilted camera’s column kept to the same restriction, and this section adds nothing to it.

Still open: what a slope in the ground costs #

The trials fix a flat plane, and a tape runs along it. Real ground has a slope, and the slope is what the picture has no way to see, since a tilted plane is itself a plane and draws a perfectly valid homography. A camera pitched 20° over a ground sloping by 2° is the same map as one pitched 18° or 22° over a level one, depending on the direction of the slope.

The measurement that settles what that costs supplies a slope of stated size and direction, generates the rings on it, and reads the pitch by each of the two lengths, measuring the height from the ground under the camera and the foot along the true surface. It asks three things: whether the height reading or the foot reading is biased more for a given slope; whether the two taped lengths together, read from one picture, reveal the slope by disagreeing, since each gives its own pitch; and whether that disagreement, set against the half-degree scatter of the reading, is large enough to detect a slope of the size a car park actually has.

More marks on the ground can make a map sharper but cannot move the eye off the circle the map allows. The sixth number has to arrive from somewhere the ground is not.