A painter's closing can be read from one band

A band lifted by columns bends its lines described two ways a painter can close a curved seam between the bands of a several-station landscape. Shift each column of the far band up or down, as a rigid piece, by whatever that column needs. Or draw each column from its own eye, with a station that rises and falls across the picture, tracking the seam’s depth, by about thirty centimetres. Either closes the seam exactly. They differ in where the mismatch goes, and that essay followed it through a band’s lines and into the next join.

It ended with a claim it did not test. A painter’s choice ought to leave a signature in the marks, it said. A landscape that changes its rule halfway up recovered each band’s own station from its marks without being told any of them, to $3.6\times10^{-15}$ m, reading exact marks. A band closed by columns would report a station that varies across the picture, and in principle a reader could find the curve. But that reading was exact, and the thing to be found here is a thirty-centimetre ripple in a four-metre height. Reading to a pixel, is it there to be found? And given a band, can a reader tell which of the three things a painter did: nothing, a shift, or a station per column?

One column is one straight line #

Take a level eye at a height $H$ above flat ground. A ground point at depth $Z$ is drawn $fH/Z$ below the horizon. A band that has been lifted by $c$ draws it at

$$\text{row} = \frac{fH}{Z} - c.$$

So in one column, each road crossing the band at a known depth gives a point on a straight line, with the road’s row plotted against one over its depth. The slope of that line is $f$ times that column’s station height. Its intercept is minus that column’s lift. Fit the line and both numbers are in hand, column by column. There are three possible results.

• One lift for the whole band: one slope and one intercept. Every column has the same line.
• A shift per column: one slope and many intercepts. The columns’ lines are parallel.
• A station per column: many slopes and one intercept. The columns’ lines fan out from one point on the row axis.

The bent seam from the earlier essays, in a band with its station at 4 m, wandering between 19.6 and 22.3 m across the picture, has the following effect on the roads. Under a shift each road bends by 10.36 px, at every depth. Under a station per column the bend depends on the depth. It is 8.35 px at 26 m, 6.20 at 35, 4.34 at 50 and 3.10 at 70, a straight line through the origin when plotted against one over the depth. Under one lift, no road bends at all.

The test is therefore very direct. A reader needs at least two roads at different depths, and it is best to think of them as a lever, since a slope’s precision depends on how far apart they are.

What a pixel does to one column #

Take the four roads the earlier figures drew, at 26, 35, 50 and 70 m, and read each to one pixel. The errors in the slope and the intercept of a straight line through four points are standard results, and they say that a single column’s station comes back to $\pm 7.8$ cm, and its lift to $\pm 1.5$ px.

So one column gives a poor account of a thirty-centimetre curve, with a signal only four times its scatter at the very best. But there are many columns. The curve is smooth across the picture, while the reading errors are independent from column to column. A low-order curve fitted across 48 columns takes out most of the noise, and the ripple stands clear.

There is one subtlety, which makes the naive reading risky and shapes how the test is built. A column’s slope and intercept are strongly correlated. All four roads are to one side of the row axis, so raising the slope and lowering the intercept is very nearly the same thing, near the data, as doing nothing. The correlation is $-0.94$. So if each column is fitted for both a station and an offset, and each of those is tested on its own, a column-to-column wander in one looks like a wander in the other, and a shift can seem to be a station. A test built that way, which is the first test a person would write, would be badly confused.

The test used here avoids that. It asks about the variation across columns of the road rows themselves, and it compares the two possible laws. Under a station, the variation in a column is proportional to one over the road’s depth. Under a shift, it is the same at every road. Each law is fitted on its own, with the other absent, and the one that takes out more variation names the closing. Neither has to share the data with the other. If neither law finds a low-order curve above what the noise alone would produce, at about the one per cent level, the band is called one lift.

Three closings named #

With that test, here is how often each closing gets its correct name, reading the four roads to one pixel over 48 columns.

One lift is correctly called one lift 99.7 per cent of the time, as the threshold says it should be. A shift is correctly named in all 300 trials, and a station per column in all 300. So at a pixel, the answer to the earlier essay’s question is a plain yes, and with room to spare. Its second question was how small a station curve could be found.

Something is detected 39 per cent of the time at a range of 3.1 cm, 85 per cent at 4.6 cm and 99 per cent at 6.2 cm. Naming the thing correctly as a station takes a little more: 83 per cent at 6.2 cm and 98 per cent at 12.4 cm. The curve that the bent seam actually requires, at 31 cm, is about five times that floor. Reading to one pixel, therefore, a station curve would have to be very slight indeed to be hidden, about a sixth of what a seam of this size needs.

This has a useful consequence for the seam itself. A painter’s seam that wandered far less than the earlier essays’ seam would need a proportionally smaller station curve, so that a floor of 6 cm corresponds to a seam wandering by about half a metre in depth. A closing is detectable, in other words, for almost any seam worth closing at all, since where a seam is allowed to go already showed that seams much straighter than that do not need one.

A common error in the depths does nothing #

The reading assumed the roads’ depths are known, and for a painting they are not. A real road is on a painted landscape and is not labelled. Most of what is known about its depth comes from the painting’s own conventions, so that a reader would have to assume a scale or borrow one from a figure’s height.

That seems at first to be a serious difficulty. It turns out to be nearly irrelevant, as long as the misreading is common to all the columns. Multiply every depth by $1 + \epsilon$. A column’s slope changes by the same factor, so its reported station height is off by that factor in every column. The mean height of the band is wrong. But the curve across the columns is simply rescaled by a few per cent and keeps its shape. Whether a closing was by station or by shift is not affected at all, since each is a statement about how columns differ from each other.

The check runs that case with a 5 per cent error in all the depths and exact marks: each column’s station, relative to the mean, comes back as $1.05$ times what it should be, to nine digits. So the closing can be named with the scale of the world unknown, and only the amount of the rise and fall inherits the scale’s error. A reader who says “this band’s station rises and falls by about 8 per cent of its height, in the pattern of a seam” has made a claim with no size in it, which is exactly what can be supported. The one thing a single view cannot give was never a limit on that.

A different error is much more of a problem. If the roads are not actually straight and level, the test sees a bent road and attributes it to a closing. The test can only say that the marks behave as if drawn by a certain law across columns. Roads laid straight across a valley are a modelling convenience. In a real painted landscape, the things that are known to run straight across the picture at one depth are few, and this is the largest gap between this measurement and any use of it on a real work.

A hand is not a pixel #

Computed lines are read to a pixel. A painter’s are not. The marks in a real band would have an error of a few pixels in a picture this size, and a good deal more for a loose brush. So the reading’s practical limit is more interesting than its theoretical one.

At 3 px a station per column is still named 96 per cent of the time. At 5 px, 85 per cent. At 10 px, only 26 per cent. So the reading’s limit lies somewhere near 5 to 8 px, about a fiftieth of a figure’s height on a scene of this size, which is a reasonable estimate of a careful hand and is not out of reach. A shift has a larger bend, so it holds up better, at 73 per cent at 10 px. One lift stays at 100 per cent, and this matters as much as anything here. A test with a small false-alarm rate does not find closings where none were made, however bad the marks. What it does at a worse hand is what it should do: it says nothing, and calls the band a single lift.

These numbers depend on having four roads at the right depths. Which roads count is itself a useful question.

Four roads from 26 to 70 m give 96 per cent, and seven give 99. Just two at 26 and 70 m give 91 per cent, so most of the power is in that pair. Two at the far end, at 60 and 70 m, give 39 per cent, and two at the near end, at 26 and 30 m, give 60 per cent. The ratio of the depths is what counts, in both cases, and the near road’s greater bend gives it an advantage. That rule is the familiar one for fitting any slope, and it is also the rule for a cross-ratio read off a picture. A reader planning such a measurement on a real band would look first for a road or a riverbank at the band’s near edge and another at its far edge, and would then ignore most of what lay between.

What the reading can and cannot say about a painter #

Nothing here says a painter ever drew a band that way. The earlier essays described a closing that can be done, and the present one says it can be detected. No Chinese or Japanese handscroll or any other multi-station landscape has been measured here, and the stations are also a staircase showed how very differently a station’s picture can be read, so any such measurement would be careful and slow, and an honest result might well be that no closing of any kind can be told from a careful hand at the reading available.

What it does supply is a form of statement about such a picture that is strong. A painter of such a landscape had a practical problem at each seam. Either the paint was left discontinuous, or something had to give: a hill was moved up, the far road was bent a little, a cloud was placed there. A conventional account of that problem speaks in terms of composition and style. The reading here converts it into a three-way choice, with a test. It also says what the test needs: a pair of roads, known to be straight, with a large ratio of depths. A later scroll might use one closing throughout, or different ones in different places. If so, that is a fact that could be recovered without knowing anything about the painter, the period or the depth scale.

There is also something to be said for a null result. A band reporting one lift throughout, on the test above, has been found to have no curve above a few centimetres, on the condition that the seam it stands above did wander. So that band’s painter simply accepted the gap. A seam that bends trades a turn for a gap priced that gap. It would be a positive statement about the painter’s practice, not merely an absence of information.

Level eyes, straight roads and one closing at a time #

Level eyes. Each column’s station varies only in height. Any change of pitch or of horizon across the band adds a term that this model does not contain, and it would mostly be taken up as an offset. So a painter whose horizon drifts by a few pixels across a band would look like a mild shift.

Straight roads at one depth each. As above, the largest assumption here. A seam breaks direction and not size has a road crossing a join in the same idealised way.

A known seam shape was not needed, but a smooth one was. The test takes a low-order curve across the columns, on the assumption that a closing varies slowly. A seam that changes abruptly from column to column would escape it, though it would also be visible by eye.

One closing at a time. A painter might do both: a little shift and a little station. A mixture sits between the two laws and would be named for whichever part takes out more of the variation, and the test as built would not report the mixture. Fitting both together would run into the correlation described above, and it would need many more roads to come apart.

Still open: whether a mixture can be split #

The test here chooses one law. A real solution to a seam may well be mixed, partly a lift that varies with the column and partly a station that does, and nothing about a painter’s practice says the two should be kept apart. A mixed closing has a bend at each depth equal to a constant plus a term in one over the depth, and with four roads across a wide range, those two terms are separately visible in principle, though with the large correlation of $-0.94$ standing between them.

The measurement that settles it draws bands closed by a stated mixture of the two, at each share from all shift to all station, and fits the two laws together rather than apart, across all the columns at once. It asks how well the share comes back at one pixel and at five; how many roads the split needs before the correlation stops dominating it; and whether a low-order fit across the columns, which tamed that correlation for the choice between two laws, tames it for the estimate of a mixture as well.

A closing is a decision about where a mismatch goes, and where it goes can be read. A road’s bend against one over its depth says which of three things a painter did, with no size in the statement and no need to know the painter’s scale.