An error with two terms

A stitched panorama’s parallax measures a ray miss and finds it separates. Across the seam the worst miss halves every time the frame count doubles — thirty millimetres at six frames, three at sixty-four. Up the frame it is the same number at every frame count, bit for bit, over eight counts from two to sixty-four.

The dish no outline reaches measures a silhouette reconstruction and finds the same shape. The excess outside the convex hull falls as one over the square of the view count. The area inside a concavity is the same number at four views and at a hundred and twenty-eight, because an outline is a pair of numbers per direction and no pair of supporting lines reaches into a bite.

Two subjects, two machineries, one answer. The temptation is to write that up.

A synthesis with no new measurement in it is a summary, and this collection does not write summaries. So instead of stating the distinction, this essay builds the instrument that makes it a measurement, and then measures what the instrument cannot see.

The instrument #

Fit

$$\mathrm{err}(n) = A\,n^{-p} + B$$

to a sweep of a control parameter. $A\,n^{-p}$ is the part that responds to effort, $p$ is the price, and $B$ is the floor.

The fit is nonlinear in $p$ and linear in $(A, B)$ given it, so it is a golden section over $p$ with an exact two-parameter least squares inside. That is faster than throwing all three at a general solver and, more to the point, it cannot converge to a local minimum in the parameter the answer is about.

Two decisions in it are worth stating because they change the answers.

The residuals are taken relative. An error law spanning three decades, fitted on absolute residuals, is a fit to its first two points and nothing else.

And $B$ is clamped at zero. A negative floor is not a floor, and reporting one would be reporting that the model is wrong in a way this model has no way to say.

Why a two-term model and not a more general one #

A three-term fit would explain more of every curve, so it is worth saying why the model stops here.

The point of the fit is not to describe the data as well as possible; it is to answer one question, which is whether any part of the error is unresponsive to the control. That question has a two-term answer and adding terms answers it worse rather than better — a second falling term with a small exponent is very hard to distinguish from a floor over any finite sweep, and a fit with both will trade them against one another freely.

So the model is chosen to be the smallest one in which the question can be posed. Its residual is not a claim that the law has exactly this shape; it is a check that the two-term description is not being contradicted, which is a weaker and more honest thing.

That is the same reason a mirror’s reflected fan is fitted inside a named shape family rather than as a free surface. A fit with enough freedom absorbs the discrepancy the measurement was about.

Seven laws, and the instrument is not told which is which #

The laws are written out in closed form rather than recomputed from their own solvers, and that is a decision rather than a convenience: a sampler that called the original solver would be measuring that solver’s arithmetic noise and would be right to call it a floor.

Three are known from their own essays to have a floor and four are known not to. The instrument is told nothing.

It reads every one of them correctly. A panorama’s seam ghost has no floor; its vertical miss is nothing but floor. A silhouette’s hull excess has none; the part of a concavity no outline reaches is all floor. An anamorph’s unresolved sag has none. A mirror ball’s sensitivity to a distant room has none.

And the exponents come back at the values the original essays derived: 1 for a seam ghost, 2 for a hull excess, 2 for a sagitta, 1 for the mirror ball.

Where the sweep matters #

floorCertificate is the instrument’s second half and it is the one nobody computes.

A floor is visible only once the falling term has got below it, so the sweep needs $A n^{-p} \lesssim B$, which is $n \gtrsim (A/B)^{1/p}$, which is $(1/p)\log_{10}(A/B)$ decades.

That number is a property of the law and it is computed here rather than guessed. A law that is all floor needs no sweep at all — both of the pure-floor laws return zero decades — while a law whose floor is a hundredth of its first sample needs the sweep to run until the falling term is that small.

A steep price is therefore a cheap purchase and an expensive diagnosis. Halving an $n^{-2}$ error costs $\sqrt{2}$ in effort where an $n^{-1}$ error costs 2; but proving that an $n^{-2}$ law has no floor takes half as many decades as proving it of an $n^{-1}$ one, because the falling term gets out of the way faster.

The pair is worth reading together because they are two readings of one experiment. A reader adding views to a silhouette reconstruction is buying down the first at a price of two for one and not touching the second at all, and the ratio of the two is what decides when to stop. That ratio is computable at the outset from a handful of views, which is the practical use of the whole instrument.

The instrument fits the control it is handed #

One of the seven laws came back with the wrong price and finding out why is the most useful thing in the essay.

An anamorph’s unresolved sag is $k s^2/4$ with $s$ the spacing between marks, and the spacing is the design’s width divided by the number of gaps rather than by the number of marks. Handed the mark count, the instrument returns a price of 2.30. Handed the gap count, it returns exactly 2.

That is a fifteen per cent error in the price, produced entirely by an off-by-one in what the control was called.

The lesson generalises immediately and unpleasantly. An instrument that fits $n^{-p}$ fits the $n$ it is given, and a great many real controls differ from the obvious count by a constant — gaps against marks, intervals against samples, baselines against cameras, degrees of freedom against observations. Every one of those is a shift that turns a clean power law into a slightly wrong one, over exactly the range where the shift is a large fraction of $n$.

And a sine is not a power law #

The seam ghost has a similar wrinkle and it is worth separating from the first, because it is honest rather than a mistake.

A panorama’s worst seam miss is $e \sin(\pi/n)$, which is asymptotically $1/n$ and is not a power law at small $n$. Fitted over the whole range from two frames to five hundred, the price comes back at 0.918. Fitted over the tail alone, past sixteen frames, it comes back at 0.999.

Both are right answers to different questions. The first is the price over the range actually swept; the second is the asymptotic price. Reporting the first as though it were the second is how an asymptotic law acquires a wrong exponent in a table, and reporting the second as though it were the first would overstate what a photographer shooting eight frames actually gets.

The general instruction is short. State the range the exponent was fitted over, because for any law that is only asymptotically a power law the exponent is a function of the range.

What “no floor” is a statement about #

The instrument reports four of the seven laws as having no floor, and that report has a bound in it which is worth making explicit, because without it the phrase claims more than any finite sweep can.

Fitting $E = A/n^{p} + B$ over a control running from $n_1$ to $n_2$, the falling term at the far end is $A/n_2^{p}$. A floor smaller than that is buried underneath the last point and no fit can see it. So

“no floor” means “no floor above $A/n_2^{p}$”,

and the bound is a property of how far the sweep was pushed rather than of the law.

That turns the value of a longer sweep into arithmetic, and the arithmetic is the neighbouring rung’s subject. Extending the control by one decade lowers the detectable floor by $10^{p}$ — so a law with exponent two gains two decades of floor sensitivity per decade of sweep, and one with exponent one gains one. The steeper the law, the more a decade of patience buys, which is the opposite of the intuition that a steep law is already good enough.

Three consequences follow, and the third is the one that decides how a null should be written down.

The four null results are not equally strong. A hull excess and a sagitta fall as $n^{-2}$, so their floors are bounded far below their last measured value; a seam ghost falls as $n^{-1}$, so its bound is weaker at the same sweep length. Two nulls reported the same way are not the same statement.

Extending the sweep is cheap in the right direction. Because the bound improves as a power, adding a decade to a quadratic law’s control is worth a hundredfold in what it excludes — far more than the same effort spent reducing the noise, which improves the bound only in proportion.

And a null has to be quoted with its bound. The honest form is not “there is no floor” but “there is no floor above $A/n_2^{p}$, over a control of this length”, which is a number a reader can compare against whatever they care about. That is the same discipline a null result’s worth in decades sets out at length, and this rung supplies the coefficient it needs: the exponent the instrument has just fitted is exactly the rate at which the null’s bound improves.

So the two halves of the fit are not a value and a nuisance. The exponent prices the null — it says what the sweep bought and what another decade would buy — and a classification that reported only “floor or no floor” would have thrown away the number that makes the second half meaningful.

What the classification is for #

The reason to separate the terms is not tidiness. It changes what to do.

A term that responds to the control is a setting. Somebody can shoot more frames, take more views, print more marks, stand closer. The question is what it costs and the price is the exponent.

A term that does not is a property of the instrument. No amount of the same effort touches it, and pretending otherwise is how a design becomes a grievance — a photographer told to shoot more frames, doing so, and finding the ghosting still there because the half they were shooting away was never the half they could see.

That distinction is what the panorama rung is about and it is why its check requires the vertical miss to be identical at every frame count rather than merely similar. A check that measured only the falling half would have licensed “shoot more frames”, which is half true and is the half everybody already believes.

What the instrument cannot see #

Every measurement above assumes the sweep is long enough. That assumption is the instrument’s own blind spot and it is the subject of what a null result is worth, which is the other half of this pair.

The short form is this. On exact data the fit correctly returns a floor of zero for a floor-free law — the first draft of this file expected it to invent one, and it does not. What it cannot do is rule one out. Over a third of a decade at one per cent noise, floors of a fifth of the first sample are still consistent with the data, and the fit reports none while telling the truth.

So every absent floor above is a claim whose whole content is the length of the sweep behind it. The sweeps here run two decades or more, which is a good deal better than most, and it is still a bounded statement rather than an absolute one.

Three of the collection’s own recoveries, read this way #

The rest of this row points the instrument at results the collection already had, and each returns something the original essay did not say.

The stratification ladder is a conditioning ladder: one pixel of noise costs a projective quantity, an affine one and a metric one very different amounts through the same recovered map, and the gap widens with the obliquity.

The bias out of reach: a single-view height has a spread that falls as $m^{-1/2}$ and a bias that does not fall at all — and the bias is eight microns, so the two cross past half a million measurements. A floor that exists and cannot be certified is, for every practical purpose, not there.

A floor with a referent: a camera recovered from vanishing points has a floor that is the lens, and doubling the distortion coefficient doubles it exactly. That is the strongest form of the classification, because the unbuyable term has a name and a cause rather than being whatever is left over.

What the row does with it #

Three of this collection’s own recoveries get the instrument pointed at them, and it is worth saying why those three and not others.

Each of them is a place where the collection has already published an error and never asked which half of it was for sale. A recovered camera has an error; a recovered height has an error; a quantity read through a recovered map has an error. All three are quoted somewhere in these pages as a number, and a number is exactly what this essay argues an error is not.

What comes back is different in each case and the differences are the interesting part. One has a floor with a name. One has a floor that is real and unreachable. One has no floor and a conditioning that varies by an order of magnitude across the parameter nobody varies.

The rule, and what it is worth #

Two sentences, and the second is the one that is usually missing.

When an error has more than one term, measure whether each of them responds to the thing a reader can change. A term that does is a setting; a term that does not is a property of the instrument; and calling both of them “the error” is what turns a design into a grievance.

And measure how long a sweep the answer needed. A floor found is a floor found. A floor not found is a statement about the sweep, and its worth is the number of decades behind it — which is computable, and which almost nobody computes.

The second sentence is the one this essay added rather than restated, and it changes the first. Without it, “no floor here” is a result. With it, “no floor here” is a result with a size attached — the size of the floor that would still have gone unnoticed — and two experiments reporting the same null result are not making the same claim unless they swept the same range.

That is a modest-sounding requirement and it is not a common one. Across the seven laws measured here, the decades needed to certify their own floors run from zero to sixteen, and the sweeps actually available run to two and a half. Four of the seven are certified. The rest are believed, on grounds this essay can now state in numbers rather than in confidence.