The wall under the paint

The viewing field has spent six essays on anamorphosis and every one of them takes the surface as given. A floor anamorph is three numbers because the floor is a plane; undoing a picture made on a curve needs the curve; how well the floor has to be known prices being wrong about it.

Turn that round. Suppose the surface is not known — a rubble floor, a wall that is not flat, a vault whose section nobody has. What does the painted design say about it?

An anamorph is an unusual calibration target #

A calibration target is an object of known shape photographed from an unknown place. A painted anamorph is the reverse and better: an object of unknown shape whose rays are known exactly.

The design is chosen before the paint goes on. Each mark is defined as the intersection of a stated ray from the eye with whatever surface is there, so the ray is known — not measured, known — and the only unknown is how far along it the paint landed.

One further photograph supplies the second ray, from a different place, and two rays meet in a point. So the wall comes back mark by mark, with nothing fitted and no model of the surface anywhere in the calculation.

It works on a floor with a step in it #

The recovery is at the arithmetic floor on all four of the site’s named receivers — a plane, a dish, a ridge and a step, which is two planes with a discontinuity between them.

The step is the case worth naming. Every fitting approach to this problem — assume a smooth surface, fit a curvature, fit a polynomial — either fails on it or returns a number that means nothing. Triangulation does not care: each mark is a separate pair of rays, and a discontinuity between two marks is not something the arithmetic has to represent.

That is the difference between a measurement and a fit, and this row of the field is about it twice over: the shadow’s curvature is a fit and returns a confident number for a floor whose curvature is zero, while this is a measurement and returns the step.

What it says between the marks: nothing, by the sagitta #

The recovery is exact at every mark, so the residual is zero and quoting it would say nothing. The honest number is a different one: how far can the surface wander between two marks without any mark noticing?

For a floor of the form $k(x^2+z^2)$ over a mark spacing $s$, that is the sagitta — the deviation of the parabola from the chord across it:

$$\text{unresolved} \;=\; \frac{k s^2}{4}$$

which falls as the square of the mark count. A seven-by-seven design over a three-metre patch leaves a little over four millimetres unresolved; a thirteen-by-thirteen leaves a quarter of that.

That is a resolution rather than an error bar and the distinction matters. An error bar says the answer might be somewhere else; this says the answer is exact there and unconstrained here, which is a different shape of ignorance and is repaired by a different action — more marks, not a better camera.

It also inverts what a reader would expect of the two inputs. The photograph’s quality does not set the resolution; the design’s density does. A blurry photograph gives the same marks in the same places, more noisily; a sparse design gives crisp marks and a coarse surface.

Why the step is the case that separates the methods #

It is worth staying with the step for a paragraph, because it is the cheapest way to see what kind of instrument this is.

Imagine three ways of answering the question. Assume a plane and fit it: the step returns a plane through the middle of both halves, with a residual that is large and does not say where the discontinuity is. Assume a smooth surface and fit a curvature: the step returns a number, with a small residual, for a surface whose curvature is zero at every point of it — which is the fit two rungs along and is the most dangerous of the three because it looks like success. Triangulate: the step returns the step.

The difference is where the model lives. The first two put a model of the surface inside the estimator, so the estimator can only report surfaces the model can express. Triangulation has no model at all — each mark is an intersection of two known lines — so the answer is whatever shape the marks are on.

That is a general point about recoveries and it is worth stating so it can be carried elsewhere: an estimator with a surface model cannot report a surface outside the model, and it will not say so. It reports the nearest member and a residual, and a small residual on a nearby member is indistinguishable from a correct answer.

The conditioning is one angle #

Two rays meet well when they cross at a decent angle and badly when they are nearly parallel. That angle — between the design ray and the camera ray, at the mark — is the whole of the conditioning here, and it is reported rather than assumed.

Over the design used in these figures the worst crossing is around twenty degrees, which is comfortable. It falls as the camera moves toward the eye and rises as it moves away, and it is not uniform across the design: marks near the middle, where both rays run nearly along the room, are the worst-conditioned.

Where the worst conditioning is, and it is not where a reader expects #

The angle between the two rays is the conditioning, and it varies over the design in a way worth reading off the figure rather than guessing.

A reader’s guess is that the marks furthest from the camera are the worst, because they are furthest away. They are not. The angle between two rays to a point depends on the baseline between the two viewpoints and on the distance to the point, and over a design a few metres across with a camera a couple of metres to one side, the distance barely changes — what changes is the direction.

The badly-conditioned marks are the ones whose design ray happens to run nearly parallel to the camera’s own line of sight, and where those fall depends entirely on where the camera was put. Move it and they move.

So the practical rule is not “get closer” or “use a wider design”; it is put the camera where its rays cross the design’s rays squarely, which for a floor design means off to one side and at a different height. That is the ordinary advice for any triangulation and it arrives here with the design’s own geometry deciding what “squarely” means.

And the camera that recovers nothing #

There is a place the second camera must not stand, and it is exactly the place a photographer would choose.

Put the camera on one of the design’s own rays — behind the eye, looking along the same line — and that mark is seen along the same direction from both places. Two coincident rays do not meet in a point; they meet in a line, and where along it the paint sits is unknown.

The solver refuses rather than returning something plausible. closestPointToRays asserts that the bundle spans enough directions to pin a point, and a pair of parallel rays fails it.

That refusal is the same one epipolar geometry makes about a point on the baseline, and it has the same practical form: the design should be photographed from off to one side, and a camera standing behind the intended eye is the one arrangement that is useless.

What is known and what is assumed #

It is worth being exact about what this arrangement needs, because the answer is less than a reader might expect and more than nothing.

It needs the eye’s position — the point the anamorph is designed for. That is not a measurement; it is a decision made before the design existed, and it is written down.

It needs the camera’s pose relative to the eye. That is a genuine input, and getting it is an ordinary photogrammetric problem: photograph something of known geometry in the same frame, or use the site’s own machinery to recover the camera from the picture it drew.

It needs no model of the surface at all, which is the whole point.

And it inherits the ordinary two-view scale ambiguity: with the eye-to-camera distance known in metres the answer is in metres, and without it the answer is a shape and not a size. That is the site’s own recurring limit arriving in a new place, and it arrives without anything special being done to it.

Why the design’s rays are free #

The thing that makes this work is worth isolating, because it is not obvious that anything unusual is happening.

An ordinary two-view reconstruction has to match points: find the same feature in both pictures, which is the hardest part of the problem and the source of most of its failures. A wrong match is not a small error; it is a point somewhere else entirely.

Here there is nothing to match. The design says which ray each mark is on, by construction, so the correspondence is given rather than found. What remains is arithmetic that cannot go wrong.

That is a genuinely unusual position to be in and it comes from the anamorph’s own definition. A picture painted to be right from one place is a labelled set of rays; the labels are the design, and the design is a document.

A design is a document, and that is the unusual part #

The claim that makes this work bears restating in a different register, because it is a claim about what kind of thing an anamorph is rather than about geometry.

Most objects that get photographed carry no record of how they were made. A wall is a wall; a statue is a statue; recovering anything about them from pictures means recovering it from the pictures alone, with all the ambiguity that entails.

An anamorph is different because its making was a computation. Somebody chose an eye, chose a picture the eye should see, and let the rays fall where they fell. The design — the list of rays — is a written record of half the geometry, and it existed before the object did.

So a painted anamorph is a photograph’s worth of information plus a document’s worth, and the document half is exact. That is why one extra view is enough where an ordinary object would need two and a matching problem, and it is the reason this essay is short on caveats and long on arithmetic.

The same observation runs the other way and is the more useful direction. A photograph of an anamorph on a surface nobody measured contains, in principle, both the design and the surface — and the design is recoverable because the thing the anamorph is a picture of is usually obvious. A skull on a floor is a skull; knowing that fixes the rays, and fixing the rays fixes the wall.

The round trip, and what it is for #

This site’s habit is the round trip: recover the camera from the drawing it made and check it against the camera that made it. Every earlier instance recovers a camera — a focal length, a principal point, a pose — from a scene made of planes and points.

This recovers a shape, and the check is the same: build a floor, paint an anamorph on it, photograph it, recover the floor, and compare. Four floors, four recoveries, all at the arithmetic floor.

What it is for is a real question and the honest answer is narrow. Nobody needs to survey a floor by painting an anamorph on it — there are easier instruments. What the result establishes is that the design carries information about the surface, which matters for the reverse problem: a photograph of an existing anamorph, on a surface nobody measured, whose design can be inferred from what it is a picture of.

What this does not settle #

It does not recover the surface where there is no paint. The design’s marks are the measurement, and the space between them is unconstrained by anything but the sagitta above.

It does not recover the eye. That is a different problem the field has already answered, and the answer there is that the marks name the place and not the height — so eye and surface cannot both be unknown.

And it does not deal with a surface the design’s rays miss. A ray that leaves the room recovers nothing, and paintedMarks refuses a design whose rays do not all land.

A recovery is exact where it has data and silent where it does not, and the useful number is not the residual — which is zero — but how far the answer could move in the gaps. Reporting a residual for a measurement that interpolates is reporting the wrong quantity.