The floor a better camera cannot reach

Every measurement in this field has two sources of error and they behave completely differently.

The marks are read imperfectly, and reading them better — a longer lens, a steadier tripod, a higher-resolution sensor, a more careful hand — improves the answer without limit. The closure is a fact brought from outside the photograph, and it is wrong by whatever it is wrong by; no amount of care with the marks touches it.

Sweeping the first to zero separates them, and the shape of what happens is the essay.

The sweep #

Measure a length running into the picture, twice over. Once with the reference’s shape exactly right, and once with its aspect ratio wrong by six per cent — a paving slab assumed square that is not, a door assumed to a standard that has settled.

With the closure exact the error falls from 31 per cent at four pixels of marking error to 0.06 per cent at a hundredth — a factor of five hundred and fifty, and still falling.

With the closure six per cent wrong it falls from 34 per cent to 5.98, and stops. That is the six per cent, arriving as a floor.

The floor is the assumption, exactly #

The floor is not approximately the assumption’s error. It is the assumption’s error, to two decimal places, and that identity is what makes the sweep a diagnostic rather than a curiosity.

A reader who runs a measurement at two marking precisions and sees the answer stop improving has measured the size of their own wrong assumption, without knowing what the assumption is or where it entered. That is an unusually good position: model errors are normally invisible precisely because they are consistent, and this makes one of them report its own magnitude.

The sweep also gives its sign in an obvious way — the answer is consistently 6.0 per cent off in one direction rather than scattered around the truth — which is the ordinary distinction between a bias and a spread and is the reason repeating a measurement does not help.

The crossing, and what it decides #

The two curves cross where the two contributions are equal, and in this arrangement that is at about half a pixel of marking error.

Above the crossing, the marks are the limiting term and better reading is worth what it costs. Below it, the closure is the limiting term and better reading is worth nothing at all — a reader with a six per cent aspect error who moves from half a pixel to a hundredth of a pixel buys an improvement from 7.1 per cent to 5.98, which is a fiftyfold improvement in the marks for a sixth of a per cent in the answer.

That is a decision about equipment, and the useful thing about it is that the crossing can be computed before the photograph is taken. It needs only the geometry, an estimate of the marking precision and an estimate of how well the reference is known — all of which a reader has in advance.

Where the crossing is, and what moves it #

The crossing is described above as computable in advance. It is worth writing out, because where it sits depends on the arrangement in a way that reverses the advice between one photograph and another.

The total is $\sqrt{m^{2}+c^{2}}$ with $c$ the closure’s own error and $m = k\varepsilon$ the marking term, linear in the marking precision $\varepsilon$. The exact-closure curve fixes $k$ for this arrangement — 31 per cent at four pixels, so $k \approx 7.8$ per cent per pixel — and the crossing is at

$$\varepsilon_{\times} \;=\; \frac{c}{k},$$

which for a six per cent closure is 0.77 px, the half-pixel the section above quotes.

The useful part is what $k$ is made of. For a length running into the picture the field has already measured it: a pixel is worth $d^{2}/fh$ metres along the ray, so relative to a length $L$, $k \propto d^{2}/fhL$ and

$$\varepsilon_{\times} \;\propto\; \frac{c\,f\,h\,L}{d^{2}}.$$

The crossing moves as the inverse square of the distance, which is a strong dependence and it inverts the recommendation.

At long range, buy the instrument. Four times the distance makes the marking term sixteen times larger, so the crossing falls to a sixteenth of a pixel and effectively every measurement is mark-limited. A six per cent closure error is irrelevant to a length measured at fifty metres, because the marks are contributing far more than six per cent whatever is done to them.

At close range, buy the tape. The same expression at a metre puts the crossing far above any achievable marking precision, so the answer is the closure and nothing else. A carefully read measurement of a nearby object is a measurement of how well the reference is known.

And improving the camera moves a reader across the crossing rather than along it. $\varepsilon_\times$ is proportional to $f$, so a longer lens raises the precision at which the closure starts to dominate — which is the correct and slightly deflating account of what better equipment buys: it converts a mark-limited measurement into a closure-limited one, and then stops.

One consolation falls out of the same algebra. At the crossing the total is $\sqrt{2}\,c$, only 41 per cent above the floor, so there is very little left to win by pushing past it. Reading the marks to the crossing captures most of what is available, and the last factor of fifty in marking precision that the sweep spends buys a sixth of a per cent. Effort is well spent up to the crossing and wasted after it, and the crossing is one division.

Which error is which, in this field’s own terms #

The two terms have names elsewhere in the collection and it is worth connecting them, because the distinction is old and this is a new instance rather than a new idea.

An error with two terms is the foundations field’s statement of the general shape: a total made of a part that falls with effort and a part that does not, with a crossover between them and a decision on either side. What a null result is worth in decades prices the same structure the other way round.

In the vocabulary of a fit, the marking error is variance and the closure error is bias. Variance falls as more or better measurements arrive; bias does not, and averaging makes it look more precise while leaving it exactly as wrong. The most misleading number a measurement can report is a small spread around a biased answer, and this sweep is a way of asking whether that is what is happening.

The closure that has no floor #

Not every measurement has one, and the exception is instructive about where the floor comes from.

A ratio of two lengths on the same plane, measured through the same rectifier, has no closure error at all — the scale cancels, so a reference six per cent wrong gives a ratio that is exactly right. Its sweep falls to the arithmetic limit and never flattens.

The same is true of a ratio of two areas, and it is why counting cloud by counting pixels can make a sky-fraction measurement with no reference at all. And it is true of an angle when the shape closure is right, which is the case an angle on the ground separates: an angle has no floor from a wrong reference length, only from a wrong reference shape.

So the floor is not a property of measuring from photographs. It is a property of quantities that require a closure, and the way to avoid it is to ask for a quantity that does not — which is often possible and rarely considered.

Where the floors come from in practice #

Four sources, in rough order of how often they dominate.

A reference whose stated size is wrong. A slab that is not the nominal size, a person who is not the assumed height, a car that is not the model assumed. This is the most common and it is the one the sweep above models.

A reference whose shape is wrong. The window that has settled, the tile that is not square. An angle on the ground prices this one separately, and its signature is different: it moves angles and lengths-into-the-picture and leaves lengths-across alone.

A plane that is not a plane. A floor that dishes, a lawn that slopes, a wall that bows. This one is worse than the others because it is not a single number: the error varies across the picture rather than scaling everything, so it produces a floor that differs from one part of the scene to another and cannot be corrected by a single factor.

A camera model that is wrong. Lens distortion left uncorrected is a bias in every mark, and a lens destroys the invariant measures what it does to the cross-ratio. That one is correctable and has been, which is why it comes last.

How to find the floor without knowing the assumption #

The sweep above knows which assumption is wrong because it was made wrong on purpose. A reader does not, and the useful question is what to do from the outside.

Read the marks twice, at different precisions. Once carefully and once carelessly, or once on a print and once on a magnified crop. If the answer improves in proportion, the marks are limiting. If it barely moves, the closure is, and the value it has settled on is the floor.

Close with two different references. Two closures that disagree by more than their spreads allow means at least one of them is wrong, and the disagreement is a lower bound on the floor. That is the check the previous rung recommends and it is the only one available from a single photograph.

Measure something known. A second object of stated size, measured rather than used, returns its own length with the same floor on it — which is a direct reading of the bias rather than an inference about it.

The third is the strongest and the least often possible. The first is always possible and costs nothing.

Why the floor is exactly the assumption, and not a multiple of it #

The identity is worth an argument, because “the floor equals the assumption’s error” is a stronger claim than it needs to be and it happens to be true here for a reason that does not always hold.

The assumed aspect enters the rectifier as a scaling of one world axis. A length running purely along that axis is therefore scaled by exactly the same factor, and its relative error is the factor’s relative error — six per cent in, six per cent out, with no geometry in between to amplify or dilute it.

A length running at an angle to that axis is scaled by a factor between one and 1.06, depending on the angle, so its floor is somewhere between zero and six per cent. And a length running along the other axis has a floor of zero, which is the measurement an angle on the ground finds to be completely insensitive to the aspect.

So the general statement is that the floor is the projection of the assumption’s error onto whatever the measurement asks for, and this arrangement chooses a measurement that projects onto it entirely. That is the right choice for showing the effect, and a reader taking a real measurement should expect a floor somewhere below the assumption’s own error rather than equal to it.

The floor is where most published measurements sit #

A last observation, offered as an observation rather than as a measurement.

Photogrammetric results are conventionally reported with a spread — a residual, a reprojection error, a standard deviation over marks — and those quantities describe the term that falls. They say nothing whatever about the term that does not, and a report of “0.3 pixels of reprojection error” on a measurement closed with an assumed door height is a precise statement about the wrong quantity.

This collection’s own essays are not exempt, and the sweep here is the reason several of them quote their closure explicitly. A floor with a referent states its referent; the plan hidden in the photograph states which of its numbers are metric and which are ratios. The habit is worth the words: a result whose closure is unstated cannot have its floor estimated by anybody, including its author.

The boundary, stated #

One closure, one plane, one wrong assumption at a time.

Real measurements have several biases at once and they do not simply add: a wrong aspect and a sloping floor interact, because the rectifier is fitted to marks that are wrong in two ways and lands somewhere neither of them alone would put it. The sweep here isolates one, which is what makes the floor equal the assumption exactly, and a picture with three biases has a floor that is not any one of them.

The sweep also holds the geometry fixed while varying the marking precision, which is the right experiment for a decision about equipment and the wrong one for a decision about where to stand. Moving the camera changes both terms at once.

And the marking precisions at the fine end of the sweep — a hundredth of a pixel — are below what any real reading achieves. They are there to show that the floor is a floor rather than a slow decline, and a reader should not take the left-hand end as an achievable operating point.

A worked decision #

To make the crossing concrete, here is the decision it settles for one arrangement.

A reader is photographing a courtyard to recover its plan. The reference available is a paving slab of nominal size, which is a shape closure and a size closure at once, and slabs of that type are made to about two per cent. Marks on a print at the intended enlargement are readable to about half a pixel.

The sweep at two per cent of assumed error puts the crossing well to the left of half a pixel, which says the marks are the limiting term at the intended precision. So better reading is worth buying: a larger print, a magnified crop, a more careful hand, all of them earn until the answer approaches two per cent.

Change one thing — the only reference available is a doorway of assumed standard width, known to perhaps eight per cent — and the crossing moves right past half a pixel. Now the marks are already better than the closure, everything spent on reading them is wasted, and the entire effort should go into measuring the doorway with a tape.

Two arrangements, the same geometry, opposite advice, and the thing that decides is a number both readers could have computed on the way to the site.

What is measured here #

Four numbers.

With the closure exact, the relative error of a length running into the picture falls from 31.0 per cent at four pixels of marking error to 0.056 per cent at a hundredth — a factor of five hundred and fifty, with no sign of stopping. With the reference’s aspect six per cent wrong it falls from 34.5 to 5.98 and stops, and 5.98 against 6.00 is the assumption’s own error to two decimal places. The crossing between the two curves is at about half a pixel. And the whole shape holds at two per cent and at twenty per cent of assumed error, with the floor tracking the assumption in each case.

The short version #

A measurement from a photograph has a term that better reading improves without limit and a term it does not touch at all. Sweeping the marking error to zero separates them, and the level the total settles at is the closure’s own error — exactly, which makes the sweep a way of measuring a wrong assumption without knowing what it is.

The crossing between them is where buying a better instrument stops paying, and it is computable before the photograph is taken. A reader below the crossing should stop spending on optics and start spending on the reference, which is generally the cheaper of the two anyway.