A crooked stick hides in a quarter pixel, and its residual finds it only below a tenth

Crooked sticks cancel each other’s bows slowly stood eight sticks in still water before three eyes, each bowed a millimetre in its own random direction, and read the water’s index with every stick held to a straight line. The bows pulled the index both ways and partly cancelled, and with a quarter pixel of reading error eight crooked sticks read the index to ±0.037, as well as one straight stick with its length taped.

That essay closed on the obvious next move. Sticks that disagree should show it: each stick’s own share of the combined fit’s residual ought to be larger when its bow pulls the index away from where the others put it. A reader could fit the set, find the stick whose rays miss worst, drop it and fit again, as a surveyor drops a reading that fails a test — and the case that matters is one genuinely bad pole in a jetty of fair ones.

At a quarter pixel the residual cannot find that pole. The reason is a ratio, and it is the subject of the first figure.

A bow pulls the index far more than it bends the rays #

The setting is the earlier essays’: three eyes 1.6 metres above still water, metre-long sticks leaning thirty degrees across the line of sight, the water’s index and its surface’s height fitted together with every stick held to a straight line. A bowed stick is read as a wrong index found how that goes wrong: a stick bowed by a millimetre is not straight, the fit insists that it is, and the cheapest way to make a bowed stick look straight is to bend the rays by a different index. The fit absorbs the bow into the index and leaves very little behind.

The figure measures how little. A stick 1.8 metres out, bowed a millimetre, moves the fitted index by up to 0.088, depending on which way round its axis the bow points; its rays, read exactly, then miss the fitted straight stick by at most 0.022 pixels. A stick 2.8 metres out moves the index by up to 0.048 and misses by at most 0.015 pixels. For each stick the miss is a fixed fraction of the pull, an index of about four for every pixel of miss at 1.8 metres and a little over three at 2.8. A stick that pulls hard does miss more.

But the miss is small against the reading. A careful photograph places every ray to about a quarter pixel, the dashed line, and the largest miss a millimetre of bow leaves is twelve times smaller. The same millimetre moves the index by more than the reading error of the whole set does: eight straight sticks at a quarter pixel read the index to ±0.026. A bow is a large effect on the answer and a small effect on the evidence.

That is the residual does not warn, measured stick by stick. A model error is absorbed by the fit that commits it, and what the fit cannot absorb is a small remainder; it is the remainder, not the error, that the residual sees.

Read exactly, the residual follows the pull #

The question about a set is different from the question about one stick, because in a set the fitted index is a compromise. Each stick’s bow pulls it its own way; the index settles where the pulls balance; and each stick’s residual is then its own small remainder plus whatever its pull disagrees with the compromise. The essay before this one expected the disagreement to make the bad sticks visible.

With exact readings it does. Twelve sets of eight, seven sticks bowed a millimetre and one bowed three at a random place, each in a random direction: the stick with the largest share of the residual is the three-millimetre one in nine sets of the twelve, and its mean rank, counting one for the worst, is 1.9. The dark dots climb with the pull. A stick whose bow alone would move the index far carries a large share of the set’s residual, since the compromise does not follow it.

With every ray read to a quarter pixel, the light dots, the shares crowd round an even eighth whatever the pull. A stick carries forty-eight rays, each missing by a quarter pixel of reading error in a random direction, and the sum of their squares is several times what any bow adds. The three-millimetre stick is ranked worst in two sets of the twelve, against the one and a half that chance would give, and its mean rank is 4.1, against chance’s 4.5.

So the information the reader wanted is in the residual, and it is drowned. It is not a matter of the residual pointing at the wrong stick; it points at no stick in particular.

The residual finds the bad stick down to a tenth of a pixel #

How fine must the reading be? The figure repeats the sets, sixteen at each reading error, and the answer has an unexpected shape. Read exactly, the bad stick is ranked worst in 69 per cent of sets. Read to a hundredth of a pixel, 69 per cent; to three hundredths, 69; to a tenth, still 69. Then, at a quarter pixel, 19 per cent.

Its mean rank tells the same story, from 2.3 exactly to 2.4 at a tenth of a pixel and then 3.9 at a quarter. What changes steadily with the reading error is the bad stick’s share of the residual, which falls from 45 per cent exactly to 38 at three hundredths of a pixel and 20 at a tenth: the reading error is filling every stick’s share, and the bad stick’s lead shrinks. Down to a tenth of a pixel its lead survives; between a tenth and a quarter it is lost.

The ceiling of 69 per cent with exact readings is the other half of the answer. Three sets in ten, even with perfect readings, the three-millimetre stick is not the worst. A bow’s pull depends on which way round the stick it points, and a three-millimetre bow pointing along the stick’s lean pulls less than a one-millimetre bow on a near stick pointing across it. The residual finds the stick that disagrees most, and that is not always the most crooked. The earlier essay found a near stick pulling two and a half times harder than a far one, and here that lever decides as much as the bow.

How bad a pole must be before a quarter pixel sees it #

A reader who cannot read to a tenth of a pixel can still ask how bad a pole must be before a quarter pixel finds it. The figure grows the bad stick’s bow. At one millimetre, the control, all eight sticks are alike and the “bad” one is ranked worst in two sets of sixteen, exactly one in eight, with a mean rank of 4.7. At two millimetres, 4.6: nothing. At three, 3.9. At five, 2.6, and ranked worst in half the sets. At ten millimetres, 1.9, and worst in three sets of four.

A bow’s miss grows with the bow and the reading error does not, so a bad enough pole stands out. But consider what the scale means. A two-millimetre bow on a near stick already moves the index by something like 0.15, six times what the reading error of the whole set costs, and the residual cannot tell it from its neighbours. It takes a bow of five millimetres — five times the bow that already spoils the reading — before the residual names the culprit half the time. On a jetty of real poles, the one a surveyor would want to drop is never the one the residual finds; the residual finds only poles that would be visible to the eye.

Dropping the stick the residual ranks worst #

The surveyor’s rule is to drop and refit, and the figure measures what that buys. With a quarter pixel of reading error and one stick bowed three millimetres, the eight sticks read the index to ±0.040; without the stick ranked worst, ±0.038; without one dropped at random, ±0.042; without the stick that really pulls hardest — a rule only a reader who could measure every bow could follow — ±0.049. With every stick bowed a millimetre the numbers are ±0.043, ±0.039, ±0.050 and ±0.038.

Twelve sets fix a root mean square to about a fifth of itself, so none of those differences is a difference. Dropping a stick removes its bow and also its forty-eight readings, and at a quarter pixel the two are worth about the same. Even dropping the truly hardest-pulling stick, with perfect knowledge, does not help reliably, because the hardest-pulling stick is often the nearest and most informative one.

Read exactly, where the residual does find the bad stick seven times in ten, dropping it still does not help: ±0.015 with all eight, ±0.017 without the worst-ranked, ±0.017 without a random one. With no reading error to average, the remaining seven sticks’ bows are the whole of the error, and seven random pulls cancel a little less well than eight, whichever one was removed. The bad stick’s bow was one pull among eight, larger than the rest, and with its direction random it was as likely to be cancelling the others as adding to them.

Why the residual cannot do the reader’s job here #

Put together: a crooked stick in a set of eight is not found by its share of the residual at the reading precision a photograph gives, and dropping the stick the residual names, or even the stick that truly pulls hardest, does not improve the index measurably.

The earlier essay put it as the step from a bias to noise: several sticks bowed independently turn one model error into something the residual can partly see. That step is real, and the dark dots of the scatter show it. What this measurement adds is the size of the “partly”. The residual sees the bows’ disagreement as a few hundredths of a pixel per ray, against a reading error of a quarter. It would take a photograph read ten times better than a careful one, or a pole crooked enough to see by eye, for the residual to act as a test.

A water surface’s height trades against its index found the valley the fit slides along, and a straight stick tests the index only across the sight found the lean at which straightness is evidence. A bow moves the fit along the same valley, which is exactly why the fit can absorb it so cheaply: the valley is the set of answers the readings cannot tell apart, and a bow of a millimetre lands in it.

Where a residual has told the truth, and why not here #

Residuals have been asked for verdicts before, and the cases where they answered share one feature that the crooked stick lacks. A third eye sees the water the pair cannot: a level pair looking at a stick in water had two rays that met exactly and a stick 285 millimetres too short, and a third eye raised ten centimetres saw a disagreement of 2.6 pixels at the tip. A mismatch on its own line needs a third eye found the same of a correspondence slid along its epipolar line: invisible to two pictures, exposed by 21 pixels in a third placed off the line. In both, the error was large in the answer and, once the right view was added, large in the evidence as well: pixels, against a reading error of a fraction of one.

The crooked stick is the opposite case. Its error is large in the answer — a millimetre of bow on a near stick moves the index by more than the whole set’s reading error — and small in the evidence, a few hundredths of a pixel, because the three eyes already see every stick and the fit has already found the cheapest way to absorb it. No extra view changes that, since the bow is absorbed along the valley that all three eyes share. The ratio of the error’s effect on the answer to its effect on the residual is what decides whether a residual can be a test, and for a bowed stick held to a line it is about four units of index for every pixel of miss.

A fit weighted by the miss trusts only the surface found the reverse failure, a residual taken too seriously: weight every point by how well its rays meet, and the fit hands nearly all its trust to the one point where the rays meet by construction. A ranking of sticks by residual share is a coarse weighting of the same kind, and at a quarter pixel it rewards the sticks whose reading errors happened to fall kindly.

What a reader with crooked poles should do instead #

The practical rule changes from the earlier essay’s in one respect. That essay said count eight or tape one. This one says do not try to clean the eight by their residuals: keep them all. A test by residual removes good sticks as often as bad ones at a quarter pixel, and when it does remove the bad one, the index does not improve.

What helps is evidence the residual does not carry. A stick sighted along its length by eye shows a bow of a few millimetres. A stick laid against a straightedge shows a millimetre. And what a ray does at a surface is the reminder that the whole fit rests on one flat surface; a ripple is a bow in the water rather than in the stick, and the residual is as blind to it.

What the measurement assumes #

Every stick leans thirty degrees across the sight. That is where a bow is worth least of the index’s own curvature, and where the earlier essays placed them. Sticks leaning more nearly along the sight pull harder for their bow and would show more residual, and the ranking might do better there.

The bows are random in direction and independent. A set bowed all one way shares its pull, and then no stick disagrees with the compromise; the residual sees nothing at all, at any reading precision.

The reading error is the same on every ray and independent. A camera whose error is correlated along a stick’s drawing — a slightly blurred stick, a slightly misplaced edge — puts a bow-like miss into the residual on its own, and the ranking would find the blurred sticks.

Twelve or sixteen sets a point. The finding rates are fixed to about a tenth either way, which is enough to separate a tenth of a pixel from a quarter and five millimetres from two, and not enough to rank the rules of dropping.

Still open: whether a bow is read as a sixth number of the stick #

The sticks here were held to straight lines, and the bow was the model’s error. A reader who suspects bows can fit them instead: a stick allowed one bow of unknown size and direction is two more numbers per stick, and a set of eight then has sixteen numbers more than it had, read from the same rays. The bows would no longer be absorbed into the index; they would be read, each to some precision, and the index left alone.

The measurement that settles what that is worth fits each stick as a shallow arc rather than a line, asks how well a quarter pixel reads each bow and whether the index then comes back as well as from straight sticks, and asks how many bowed sticks it takes for the extra numbers to cost less than the bows did — since a stick that is allowed to bend is a stick that tests the index less.