A strip's scatter points to the end drawn last

A strip keeps its ratio, not the end it began closed a door. A painter dividing a splayed strip into bays tires as they go, and each bay comes out a little larger than the last — but a strip divided from its far edge by a tiring hand is, to the last digits, a strip divided from its near edge by a steadying one. The size of the bays records a ratio, and a ratio read from the other end is a ratio. The picture could still recover every strip’s direction, because one hand shared one rate of tiring across strips of different splay; no strip could do it alone.

That essay left one thing unmeasured and said so. A tiring hand is not only less accurate on average by the end of a strip. It is less consistent. The bays it draws last should scatter more about the creep than the bays it drew first, and scatter that grows along a strip is not symmetric under reversal: read from the wrong end, it shrinks. Where the rows sit cannot say which end came first. How much they scatter might.

It can, and the measurement below says how much. The short answer has three parts: a single strip carries a little of it, a strip of four bays carries none, and a picture recovers it well only where its strips share something — the hand’s one slip, or one edge.

Two slips that could have left the same rows #

Every figure here is drawn on the strips of the divergent interior the earlier essays used — floor, footstool, table and book — painted by a hand at a habit of 1, the even spacing down the page that the rows count hands, not cameras named. The hand’s slip is stated as a standard deviation on each row, in pixels of a stroke a hundred pixels tall: so much on the row it draws first, three times as much on the row it draws last, growing geometrically between.

The reading has two steps, and the first is the one the earlier essays already made. The habit and the creep are fitted to the strip’s rows by least squares, unweighted, exactly as before — and the fit is the same whichever edge the reader assumes, because a far-started creep is a near-started steadying. The residuals left over are therefore a fact about the strip and not about the reader’s guess. The check on that is exact: fitting the same rows as a strip divided from the near edge and as one divided from the far edge leaves residuals that agree to a millionth of a pixel.

The second step asks which of two slips explains those residuals better. One grows from the near edge — the hand started there, and its rows scatter more and more towards the far edge. The other is the same profile reversed. The overall size of the slip is fitted separately for each, so the comparison is only about where the large residuals sit, and it is scored as the difference in Gaussian log-likelihood between the two. A positive score names the near edge; a negative one, the far.

On the painting in the figure the large residuals sit towards the far edge’s end of the fitted rows — 1.40 px on the third interior row — but not cleanly; the first row, next to the near edge, carries 1.23. The score is −0.92, and it names the far edge, which is right. The slider draws other paintings of the same strip by the same hand, and some of them are read wrong: five residuals of a random process do not always put their largest members where the process is widest.

Two properties of the score are exact rather than measured. Reversing the residuals reverses its sign, to the last digit, because the two profiles are one another reversed. And a hand whose slip does not grow leaves a score of exactly zero on every strip, whatever its residuals: the two profiles are then the same profile, and there is nothing to choose between. That is the refusal, and it is the reason every reading below reports the growth of the slip beside its result.

A strip’s evidence is its rows less the fit #

How often is one strip read right? The answer depends on two things: how much the slip grows along the strip, and how many residuals the strip has left to show it.

At a growth of one, every strip is read at exactly one half — not approximately, because every score is zero and a tie is scored as half right. As the slip grows, the six-bay floor climbs: 75 per cent of paintings name the right edge when the last row slips three times as much as the first, 88 per cent at ten times. A strip twice as long does much better, 90 per cent at a tripling and every painting at ten times.

The four-bay footstool does not climb at all. It sits at chance for every growth, and the reason is the arithmetic of the fit rather than anything about its splay. A strip of four bays has three interior rows. The habit and the creep are two numbers, and fitting them takes two rows’ worth of freedom, leaving one residual. One residual has a size and no position along the strip — there is nothing to compare it with — and the score, which asks where the large residuals sit, has nothing to read.

The same count explains the others. The floor’s five interior rows leave three residuals; the twelve-bay strip’s eleven leave nine. A strip’s evidence about its starting edge is its interior rows less the two numbers that one strip cannot tell apart in the first place, and the reading improves roughly as that count grows. It is the same accounting the rows under a splay measure the bays did for the habit itself — each parameter the painter might have chosen costs one row — applied now to a second-order quantity, the pattern of the scatter rather than its centre.

So the answer to the question the earlier essay posed about a single strip is: sometimes. A six-bay strip divided by a hand that triples its slip names its edge three times in four. That is better than a coin and much worse than the certainty a reader of a real panel would want, and on the short strips a painter most often draws — a footstool, a step, a book — it is nothing. The strips that carry the reading are the long ones, which an inverse perspective is a leaning plane showed are usually the floor and the walls: the surfaces that recede furthest are divided into the most bays.

Why the scatter does not fade #

There is a respect in which this reading is better than the creep’s, and it is worth drawing separately because it inverts the usual relation between a signal and its noise.

The earlier reading finds each strip’s edge from the creep: one hand’s shared rate of tiring has to be read with the right sign on every strip, or the strips disagree about it. That reading is a comparison of where rows sit, and the slip is its noise. As the slip grows, it fades: on this interior, with the slip tripling along each strip, it names 98 per cent of strips’ edges at an eighth of a pixel on the first row, 81 per cent at one pixel and 54 per cent at eight, where it is nearly guessing.

The slip reading does not fade at all. It names 68 per cent of the strips’ edges at an eighth of a pixel and 70 per cent at eight pixels — the same, within the scatter of the sample — because it never compares the residuals with anything outside them. It fits the slip’s size afresh and asks only how the residuals are distributed along the strip, and a pattern of residuals multiplied by any factor is the same pattern. The noise is the signal. That makes this reading most useful exactly where the creep’s is least: a loose hand, slipping a pixel or more, whose rows no longer sit precisely enough for the creep’s sign to be read.

The two readings cross between two and four pixels on the first row here. Below that the rows’ positions say more about each strip’s edge than their scatter does; above it, the scatter says more. Weighted together — every row weighted by the slip that each pattern of edges implies, and the slip’s likelihood added to the creep’s — they beat either alone at every slip: every strip at an eighth of a pixel, 91 per cent at one pixel, 77 per cent at eight, where the creep alone has fallen to 54.

What a workshop’s strips pool #

A single strip reads its edge poorly because it has few residuals, and the natural remedy is more strips. It works, but only for the part of the question the strips share.

Suppose a workshop always divided its strips from the far edge — ruled the back line of a floor first, say, and worked forward. Two grounds and what the second costs is a reminder that the surfaces of one divergent picture were often laid out by different rules; a workshop habit is the opposite case, one rule applied to all of them. Then every strip in a picture carries the same answer, and their residuals can be pooled. Tripling slip, one floor strip names the edge 79 per cent of the time, four strips 87 per cent, six strips 95 per cent. Two strips — the floor and the footstool — are no better than one, because the footstool adds a residual that carries nothing and a scale that has to be fitted.

What makes pooling work is fitting one slip for the whole hand. It is one hand, so the size of its slip on its first row is one number, whatever strip it is painting. Fitting that scale once across every row of every strip gives the figure’s upper curves. Fitting it strip by strip and adding the scores does worse — 85 per cent from six strips instead of 95 — because a strip whose residuals happen to be tiny at one end then produces a large score of its own and outvotes the rest. The rule is the one one camera means one horizon found for a picture’s geometry: what the parts share should be fitted once, and every number fitted per part is a place for chance to hide.

This picture has no creep in it at all. The earlier reading of a picture’s edges depended entirely on a hand that tires in the sizes of its bays; with no creep, it recovers nothing, and the essay said so as its refusal. The scatter needs no creep. A hand that places its bays at a perfectly steady spacing but gets less precise as it works still leaves its direction in the picture, and a workshop habit of dividing from one edge is recovered from six strips nineteen times in twenty.

What pooling cannot do is recover a direction that differs from strip to strip. If the floor was divided from its near edge and the book from its far one, each strip’s scatter speaks only for itself, and no amount of pooling across strips adds to any one strip’s evidence. That is the same division the earlier essays found for the creep, in a different quantity: a picture fixes what its parts share, and each part keeps its own secrets.

When the edges differ from strip to strip #

For a picture whose strips were divided from different edges, the creep remains the main reading, and the question is how much the scatter adds to it.

The joint reading tries every pattern of edges. For each, it knows where the hand started on every strip, so it knows which rows should scatter most; it weights each row by the inverse of the slip that pattern implies, fits the one habit and one creep, and scores the pattern by its whole likelihood — the creep’s fit and the slip’s profile together. The best pattern among those whose creep is a tiring one is the reading.

Across a whole picture of four strips it lifts the share of pictures in which every edge comes back from 88 to 96 per cent at a quarter of a pixel on the first row, from 70 to 83 per cent at half a pixel, from 44 to 58 per cent at a pixel, and from 18 to 44 per cent at two pixels. The gain grows as the hand loosens, which is the previous section’s crossing seen from the other side: the creep’s evidence falls away with the slip and the scatter’s does not.

The count is strict. A picture in which three of four edges come back is scored as a failure, and guessing would name every edge one time in eight among the patterns whose creep is a tiring one. The joint reading at two pixels, 44 per cent, is well above that, and above a pixel it is the difference between a reading that works about half the time and one that does not.

The reader does not need the growth #

Every reading so far assumed the reader knew how much the hand’s slip grows along a strip. A reader of a real panel does not, and the question is whether a wrong guess matters.

It barely does. A floor strip whose slip truly triples is read right 73 per cent of the time by a reader who assumes it grows by half, 75 per cent by one who assumes it triples, and 74 per cent by one who assumes six times. The same holds at the other true growths, to within two points. The rate is set by how much the hand really worsens, and the assumed growth decides only how sharply the score weights the ends — never, in practice, which end it favours.

The reason is in the shape of the score. Both profiles grow monotonically, one each way, so for any assumed growth greater than one, the score is positive when the residuals are larger towards the far end in the weighted sense and negative otherwise. Changing the assumed growth reweights the rows but rarely moves the balance point across a given set of residuals. So the reading needs only the qualitative assumption — that the hand gets worse as it goes, not better — and not the rate. That is the same assumption the earlier essay used to break its last tie: a hand that tires rather than steadies. Fatigue is the one thing a reader may take for granted.

What the two directions record #

It is worth being clear about what the two readings are reading, because they are not two estimates of one quantity. The creep records how the sizes of the bays change down a strip, and it is closed under reversal: it says which way the ratio runs, not which end the painter began at. The scatter records how the precision of the bays changes, and it is not closed under reversal: a hand that worsens is worse at the end it reached last. So the scatter answers directly the question the creep answered only through the other strips.

That makes the scatter a reading of a different kind of fact. A tiring panel keeps its order, not its direction found that the order of the strips came back only as a line, and every reading of that family left a global reversal free. The scatter has no global reversal. A panel whose every strip scatters most at its far edge was divided from its near edges, whatever the creep says, because a hand does not grow more precise as it tires. The ambiguity the whole sequence has carried — every direction reversed together with a steadying hand — is broken by a second kind of evidence rather than by an assumption.

The earlier essays drew the picture’s reach as a ledger: what one strip keeps, what a picture keeps, what nothing keeps. The ledger now has an entry the size of the bays could not make. A strip’s own scatter keeps its direction weakly, a workshop’s pooled scatter keeps its common direction well, and the combination keeps each strip’s direction better than either. The exclusion is two conditions, not ten rows closed a different question about these pictures by finding its structure; this one closes by finding a second signal where the first was provably blind.

What was assumed #

The slip grows along the order of drawing, geometrically, from the first row to the last. A hand might instead worsen in steps — after a rest, or on a new brush — or its slip might track the size of the bay rather than its place in the order. A slip proportional to the bay’s size would put the growth in the creep’s direction and so be closed under reversal, like the creep: that hand would leave no direction in its scatter at all.

Slips are independent from row to row. A hand that sets each row by eye from the last one makes correlated errors, which accumulate along the order of drawing and grow in exactly the way this reading looks for, even when the hand does not tire. Such a strip would be read correctly by accident; a strip whose accumulated error was reset by measuring from both edges would not show it.

The edges are ruled first and do not slip. Every residual here is on an interior row, and the two edges hold still, as in the earlier essays. A painter who ruled the far edge last, after dividing, would put the largest slip of all on an edge the reading treats as exact.

The rows are measured to much better than the slip. A real reading of a panel measures its rows with its own error, which is the same at every row and so behaves like a hand that does not tire. It adds a flat floor under the growing slip and pulls every rate here towards one half in proportion to its size.

Still open: whether a hand’s accumulating error looks like fatigue #

The second assumption above is the one that matters most for a real panel, because it names a different mechanism that produces the same signature. A painter who marks each division by measuring from the one before — a compass stepped along the strip, or dividers opened to the last bay — accumulates error: every row inherits the slip of the row before it plus its own. The spread of an accumulated error grows as the square root of the number of steps, from wherever the stepping began, and that is a scatter growing along the order of drawing with no fatigue in it at all.

The measurement that separates the two paints strips both ways — an independent slip that grows geometrically, and a constant slip that accumulates from the starting edge — and asks two things. First, whether the reading here names the starting edge of an accumulating strip as well as it names a tiring one, which the shape of the growth suggests it should. Second, and more usefully, whether the two mechanisms can be told apart: accumulated errors are correlated from row to row, so neighbouring residuals should lean the same way, while independent slips do not. If the correlation between neighbouring residuals is readable on a strip of six bays, a divided strip records not only the edge it was begun from but whether its painter stepped the divisions or set each one fresh.