The curvature a shadow reports

The floor that is not a plane measures a defect. A shadow on a flat floor is a homology, so four marks of an occluder’s outline determine the whole map and the rest of the outline comes back exactly; dish the floor and the same four marks mispredict the rest by millimetres, ridge it and by more, put a step in it and by a great deal more.

That essay reports the mispredict as a cost: what the receiver’s shape does to the projective description everything else in the light field depends on.

Turn it round. The mispredict is a function of the floor’s shape, so it is a measurement of it.

The fit, and what it is fitting #

The instrument is the same four-point predictor. Fit the map from four marks of the shadow to four marks of the occluder; ask the remaining twenty where they landed; take the worst.

On a plane that number is zero. On a dish of curvature $k$ it is some function of $k$, monotone and computable — so given an observed mispredict, search for the $k$ that produces it.

That is a fit of one parameter to one number, which is the simplest kind there is and is why it works so cleanly. The dished floor’s own curvature comes back to well inside a per cent.

The control, which is a flat floor #

A flat floor produces a mispredict of exactly nothing — the homology is exact, which is what the light field established — and the fit duly returns a curvature of zero.

That is the control that makes the previous paragraph mean something. A fit that returned a small positive curvature for a flat floor would be reporting its own noise, and every other number in this essay would be that noise plus a signal.

And the finding, which is a floor with a step in it #

Now the case that is the reason for the essay.

Take a floor made of two planes with a step of six centimetres between them. Its Gaussian curvature is zero at every point of it. Each half is exactly a homology; the surface is developable in the strongest sense, being flat; there is nothing anywhere on it that a curvature could be a curvature of.

Run the same fit and it returns 0.482 — eight times the dished floor’s own curvature — and matches the observed mispredict to ten microns.

Nothing in the output says anything is wrong. The residual is small — smaller, in fact, than a reader would demand of a successful fit. The number is precise to three figures. It is inside the range a real floor could plausibly have.

And it refers to nothing.

Where the phantom curvature comes from #

It is worth working out what the fit is actually seeing, because the answer says why the number came out large rather than small.

A step in the floor displaces the shadow abruptly at one place: everything on one side of the discontinuity lands where the near plane puts it, everything on the other side lands where the far plane does, and the two disagree by the step’s own projected size. So the mispredict is a jump.

A dish displaces the shadow smoothly, most at the edges of the region and least at the middle. Its mispredict is a bulge.

The fit is comparing one number — the worst mispredict — so it asks: how curved would a dish have to be to produce a bulge as large as this jump? On a six-centimetre step the answer is a dish far more sharply curved than anything in the arrangement, because a smooth bulge has to be very pronounced to reach the size a discontinuity attains immediately.

That is why the phantom is large rather than small, and it is the general shape of the failure: an estimator handed a defect of the wrong kind reports the amount of its own kind of defect that would produce the same headline number, and headline numbers are what estimators are usually written to match.

Why this is the dangerous kind of wrong #

A fit that fails loudly is a nuisance. A fit that returns a confident number for a quantity the object does not have is a different thing, and it is worth naming what makes it dangerous.

The estimator has a model — the floor is $k(x^2+z^2)$ — and it can only report members of that model. Handed data from outside it, the estimator does not detect that; it returns the nearest member, with a residual that measures the distance to that member and not the distance to the truth.

So the residual is answering the question how well does the best dish explain this? and it is being read as how well is this floor understood? Those are different questions and one of them has a good answer here.

That is the pair this row of the field is built on. A painted design plus one photograph is a measurement: each mark is an intersection of two known lines, nothing is assumed about the shape, and a step comes back as a step. A curvature fitted to a shadow is an estimate: it is cheaper, it needs no second view, and it can only report what its model can express.

Both are useful and they are not interchangeable, and an output of “0.482” gives a reader no way to tell which one produced it.

The detection floor, which is the other half of a measurement #

The fit above is exact on exact marks, so quoting its accuracy would be quoting the arithmetic. The number a reader actually needs is where the mispredict falls below what can be seen on a photograph.

At two millimetres of mark accuracy — a reasonable figure for a mark on a floor in a photograph of a room — the smallest reportable curvature is around 0.078.

Below that the floor is curved and the shadow reports nothing, which is a different statement from the floor being flat and is the one an error bar exists to make. Above it the fit is a measurement with a resolution.

That threshold moves with everything in the arrangement — the lamp’s height, the occluder’s size, how much of the floor the shadow covers — so it is not a constant of the method. It is a number to be recomputed for the arrangement in hand, which is the honest form and is why it is a function rather than a figure.

What the shadow does have that the two-view method does not #

The fit’s weakness is stated. Its advantage is real and worth putting beside it.

It needs one photograph. There is no second camera, no pose to establish between two viewpoints, no correspondence problem. A picture of a shadow on a floor, with the occluder’s outline known, is enough.

And the occluder’s outline is often known for nothing — a stencil, a card, a window frame, a leaf. A shadow can be un-cast establishes that the map from occluder to shadow is invertible on a plane, and everything here is that map’s failure being read rather than repaired.

So the arrangement is genuinely cheap, and cheapness is why the trap matters. An expensive method gets checked; a method that needs one photograph gets used, and the number it returns gets written down.

What the shape of the mispredict would have said #

The section above says the fit collapses a shape to a number. It is worth being concrete about what the shape carries, because the repair is available and cheap.

The four-point predictor produces a mispredict per mark — twenty numbers, one for each point of the outline that was not used to fit the map. The fit above takes the worst of them and throws the rest away.

Those twenty numbers are a signature. On a dish they vary smoothly and are largest at the outline’s extremes. On a ridge they are largest along one direction and near zero along the other, because a ridge is developable and its curvature has a direction. On a step they are near zero for every mark on one side of the discontinuity and large for every mark on the other, with the change happening between two adjacent marks.

So a reader with the twenty numbers can distinguish the three floors without fitting anything at all, and a reader with one number cannot distinguish them at any residual. That is the whole of what “test the model rather than the parameter” means in this case, and it costs nothing but keeping the intermediate quantity the fit already computed.

It is also why the essay’s headline number is worth quoting with its residual attached. Ten microns is a very good fit, and reporting the two together is what makes the point: the residual is not a defence, and a reader who takes it as one has read the fit’s own question as the question they asked.

The right way to use it #

Three things, and they follow from the sections above rather than from taste.

Report the residual and the model together. “Curvature 0.482” is not a result; “curvature 0.482 under a paraboloidal model, residual ten microns” is, because the second half tells a reader what was assumed.

Test the model rather than the parameter. A dished floor and a stepped floor produce mispredicts with different shapes — where on the outline the error is largest, and whether it is smooth — and the shape is what separates them. The fit above collapses that shape to one number, which is what loses the information.

Or use two views. If the answer matters, the previous rung’s method has no model in it at all and returns the step.

Why the shadow carries any shape information at all #

There is a prior question worth answering, because a reader may not accept that a shadow says anything about a floor.

A shadow is a projection from the lamp. On a flat floor that projection is a homology — a projective map of the plane with a centre and an axis — and a homology has eight parameters, of which four correspondences use all eight. So on a plane the shadow’s outline carries exactly as much information as four points and no more: everything else is predicted.

That is the sense in which a flat floor is transparent to the measurement. The map is fully determined by a handful of marks, so nothing is left over to say anything about the floor, because the floor’s flatness is what made the map a homology in the first place.

Curve the floor and the map is no longer projective. The four correspondences still fit something — a homology exists through any four points — and the remaining marks are now surplus data, constrained by the surface rather than by the map. That surplus is where the shape information is.

So the measurement is available because of a mismatch: eight parameters of map against forty-eight numbers of outline, on a surface that does not oblige. On a plane the mismatch vanishes and there is nothing to measure, which is why the control returns zero and returns it exactly.

The species, met three times #

This is the fourth instance on this site of a quantity that is computed correctly and refers to nothing, and the four are worth listing because the family resemblance is the useful part.

A cross-ratio test that measured nothing, because it was evaluated at the one input where it cannot fail. A conformality test that gave the cylinder a perfect score, because the tangent basis chosen happened to be the cylinder’s own. A sensitivity computed in a variable nobody perturbs. And a curvature fitted to a surface that has none.

The first three are checks that pass when they should fail. This one is an estimate that succeeds when it should refuse, which is the same defect seen from the other side: in every case the machinery is answering a narrower question than the one being asked, and nothing in the output says which question it answered.

What this does not settle #

It does not treat the receiver’s own effect on the shadow’s other properties — a seam, a kink, a shadow that leaves the receiver altogether — each of which is a different signature and a different measurement.

It does not treat a floor with two parameters. A dish plus a tilt, or an ellipsoidal dish, would fit better and would have the same defect at one more remove.

It does not treat noise properly. The threshold above is a worst-case mispredict against a stated mark accuracy, not a statistical statement, and turning it into one is a question about estimation rather than about projection.

And it does not say the fit should not be used. It says what it can express, what it does when handed something it cannot, and what the alternative costs — which is what a reader needs in order to decide.

A fitted parameter is a statement about the model as much as about the object, and a small residual measures the distance to the nearest member of the model rather than to the truth. An estimator handed a surface its model cannot express returns a number and does not mention it.