Measuring from one picture
A height, out of one photograph
Four points on a vertical, one cross-ratio, and the height of something nobody measured. The only metric input is the photographer's own eye height, because the horizon is at eye level and that is the one piece of perspective folklore that is exactly true.
Flattening a façade out of the photograph
Four corners of a rectangle whose proportions are known are enough to undo the projection of one plane. After that the plane can be measured with a ruler — lengths, angles, areas, all of it — in units of the rectangle's own width, and lengths the map was never given come back to fifteen digits.
The plan hidden in the photograph
Rectifying the ground is the same operation as rectifying a wall, and it turns a photograph into a site plan. The horizon is not an input to it and comes out as a consequence — the plan's points at infinity land on it, at first order, which is the check that the plan is a plan and not a plausible warp.
The one thing a single view cannot give
Make the world a hundred and thirty-seven times larger and move the eye a hundred and thirty-seven times further away, and the picture does not change by a measurable amount. Every ratio in a scene is recoverable from one photograph and no size is, and that is not a caveat about the method — it is the shape of the method.
How wrong a measurement from one picture can be
The formula divides by a difference of two nearly equal numbers when the object is tall, which looked like the instability and is not. Measured, a taller object is recovered better and a more distant one worse — half a per cent per pixel at six metres and eleven per cent at a hundred and twenty. The argument that was wrong is as much the finding as the curve that replaced it.
The wall under the paint
An anamorph's design is a set of rays from a known eye, fixed before any paint is applied — so the marks are a calibration target whose rays are known exactly and whose shape is not. One ordinary photograph supplies the second ray for each mark and the wall comes back at the arithmetic floor, on a dished floor, a ridged one, and a floor with a step in it. What it says between the marks is nothing, by the sagitta.
The curvature a shadow reports
A flat floor makes a shadow a homology, so four marks predict the rest exactly and a curved floor mispredicts them by millimetres. Turn that round and the mispredict is a measurement of the floor. It recovers a dished floor's own curvature to a per cent — and returns 0.482 for a floor made of two planes, whose curvature is zero everywhere, with a residual of ten microns.
One picture of a ball
The outline of a ball in a photograph fixes the cone of rays that graze it, so the cone's axis is the direction of the ball's centre and its half-angle is the arcsine of radius over distance — both exactly, and neither of them separately. A ball a hundred and thirty-seven times larger, a hundred and thirty-seven times further away, draws the identical conic. And the drawn ellipse's own centre is not the image of the ball's.
Two pictures of a ball
Two outlines give two tangent cones, and where their axes cross is the ball's centre — with the radius following from either half-angle. There is no correspondence problem, because a ball's outline in one picture and its outline in another are guaranteed to be the same ball; and there is a degeneracy, which is the two eyes and the ball in a line, and the solver refuses it rather than returning something plausible.
The dish no outline reaches
An outline is a pair of numbers per direction and nothing more, so a shape built from outlines has two errors that behave differently. The part outside the convex hull falls as one over the square of the view count. The part inside a concavity is the same area at four views and at a hundred and twenty-eight, because no pair of supporting lines ever reaches into a bite.
An area, out of one photograph
A patch of ground comes back at 5.205 m² from one photograph, to 1.8 × 10⁻¹⁴, through a homography built from four marks and their four known positions. What is worth knowing is how it degrades — the patch's extent across the picture is read with an error growing as the depth, its extent into the picture with an error growing as the depth squared, and at twenty-eight metres the two are sixteen times apart — which is the depth divided by the camera's height.
Counting is a measurement
A tiled floor gives its area with no reference length at all — count the tiles and multiply. The count is an integer, so it is exact wherever it can be made, which is a completely different error law from the rectifier's smooth decay. And the distance at which it fails is set by the tile's depth edge, which foreshortens as one over the depth squared, so 18.7 m for a 62 cm tile, where the across edge alone would have allowed 217.
An angle on the ground
An angle needs no length at all — it is scale-free, so the one thing a single view cannot give is not the thing an angle wants. What it wants instead is the plane's shape, and the difference is measurable — assume the reference rectangle's aspect a quarter wrong and the angle moves 11.5°, a length running into the picture is out by exactly that quarter, and a length along the reference's own direction does not move at all.
The answer is an ellipse
A mark read with a round error does not come back as a round region on the ground. The ray is grazing, so a pixel across the picture is worth a little and a pixel up it is worth a great deal, and the recovered point's uncertainty is an ellipse pointing at the camera — 5.07 to 1 at eight metres from a camera 1.62 m up, which is the depth over the height. Propagated and sampled agree to 0.5 per cent, and a ray that is not grazing gives a disc.
Which reference to measure from
Given four candidate scale bars in one photograph, the best is not the longest and not the nearest — it is the longest in the picture. A five-point-two metre bar near the horizon is the longest thing in the scene and the worst reference in it; a four metre bar close to the camera is the best. Walk one bar outward and the term it controls falls as one over its length in pixels, with a fitted exponent of −1.08.
The floor a better camera cannot reach
Sweep the marking error from four pixels down to a hundredth and the measurement's error falls thirty-fold and then stops — at 6.0 per cent, which is exactly the six per cent the reference's assumed shape was wrong by. With the closure exact the same sweep keeps falling to 0.06 per cent. The crossing is at half a pixel, and it can be computed before the photograph is taken, which makes it a decision about equipment rather than a discovery about it.
Five facts that close the same gap
The gloss that one length has to come from outside the photograph names a single option, and there are at least five — a length on the ground, the camera's own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise; their spreads run from 0.36 to 3.60 per cent, a factor of ten, and which one is available depends on the photograph rather than on the arithmetic.
Where each closure enters the stratification
The five facts that turn a photograph's ratios into metres do not all do the same job. Read on four quantities through the map each supplies — a cross-ratio, a ratio along a line, an angle, a length — three of them return all four exactly and two return only the first three. Set the free scale ten per cent wrong and the whole ten per cent appears in the length and nothing appears in the other three, to thirteen digits.