The rim was never most of the glow

Office paper is brightened: a fluorescent agent in the sheet absorbs ultraviolet and re-emits it as blue, and a reader adapted to the sheet takes that blue as white. Printed ink puts out the glow beneath it, so a tint — partly paper, partly ink — glows where its solid does not, and adapting to a brightened sheet leaves the glow found a pale tint’s hue turned by several degrees that no amount of adaptation removes. A thin dot edge lets the glow back in then gave every dot a rim where the ink thins and ultraviolet passes, and found the residue growing with it: an orange at a fifth coverage from 6.8 degrees with a uniform film to 7.9 with a twelve-micrometre rim.

Four essays since have worked on that rim. Stochastic dots hold the glow at every tint found a dispersed screen’s separate spots all rim; merged spots give the mid-tones half the glow back, squares of four print a pale tint like a 150-line screen and a growing clump prints like the screen with as many dots clustered and grew them until the residue followed the number of clumps on the tile — thirteen clumps the 150-line screen, twenty-six the 200-line — and concluded that only coarser patterns lower it.

The proposal was to be coarse only where it matters most: six clumps on the tile in the pale tints, where a halftone’s pattern is faintest and its glow worst, doubling to thirteen at twenty per cent and twenty-six at forty. The prediction was that six clumps beat the 150-line screen by more than half a degree below twenty per cent, sitting with the 100-line screen there, and match the finer screens where their counts match. The alternative named was that splitting the clumps costs more rim than the coarse highlights save, so that a screen should hold its count.

A fifth of a degree, and a floor beneath it #

Below twenty per cent, six growing clumps leave the orange’s residue 0.11 to 0.17 degrees below the 150-line screen’s — 7.97 against 8.12 at five per cent — and within a fifth of a degree of the 100-line screen’s. Against a fixed thirteen clumps the schedule saves 0.17, 0.33 and 0.22 degrees at five, ten and fifteen per cent, and its twenty-six clumps give back 0.14 to 0.19 from forty per cent; over the ramp it saves 0.22 degrees in all. A single clump on the tile — the coarsest pattern the tile holds — leaves 7.48 degrees at five per cent and 5.30 at sixty, nine tenths or more of the 150-line screen’s residue at every coverage. Going from thirteen clumps to one moves the five-per-cent residue by 0.66 degrees, under a tenth of it. For every pigment of eight, one clump keeps at least three quarters of the thirteen clumps’ residue.

• The prediction fails by two thirds. Coarse clumps help, and a sixth of a degree is what help amounts to.
• The alternative is half right. The splits do cost more than the coarse tints save from forty per cent, but neither is large.
• The finding is the floor. Everything the rim essays chased is the top tenth of the residue.

The pairs of bars are the whole argument. At every coverage the single clump’s bar is nearly as long as the thirteen clumps’: 92 per cent of it at five per cent, 94 at sixty. A single clump is about as coarse as a halftone can be — one solid patch of ink in a field of bare paper, its rim a thin ring round a disc — and it keeps almost all the turn. The rim essays were right that the rim adds to the residue and right about what lowers the rim’s part. The rim’s part was never most of it.

The schedule along the ramp #

The five curves fall together from about eight degrees at five per cent to five and a half at sixty, and the space between them is never more than three quarters of a degree. The scheduled screen starts between the 100- and 150-line screens, joins the thirteen clumps from twenty to thirty per cent, and from forty runs above them, its twenty-six clumps behaving like the 200-line screen they have as many dots as. The single clump runs below everything, a floor a third to two thirds of a degree under the others along the whole ramp, and it is the floor the others approach as their clumps coarsen.

How much a halving buys #

Plotted against the clump count, the residue rises gently and flattens towards one clump. Each halving of the clumps below thirteen lowers it by a tenth to a fifth of a degree at five per cent — the first, from twenty-six to thirteen, by 0.43 — and from one clump to twenty-six the whole span is 1.09 degrees at five per cent and 0.54 at sixty. The rule the growing-clump essay found — the clump count sets the residue — holds; its slope is shallow. A screen designer who halved the clumps in the pale tints, accepting a pattern a third coarser to the eye, would buy a fifth of a degree at most of hue that a reader cannot adapt away.

The flatness is the floor seen another way. A rim’s share of the ink shrinks as the square root of the clump count falls, and the residue would follow it towards the uniform-film value if the rim were most of it; it does follow, and the uniform-film value is close.

What the schedule saves and spends #

The schedule’s ledger is small on both sides. Six clumps save a sixth of a degree at five per cent and a third at ten; twenty-six clumps give back a seventh to a fifth from forty. Over the ramp it saves 0.22 degrees in total, about a fortieth of the pale tints’ residue. The prediction’s half-degree advantage over the 150-line screen would have been worth a pattern change; a sixth of a degree is not.

There is also a cost the residue does not count. A screen that changes its clump count along the ramp must split its clumps somewhere, and a split shows as a step in texture between two neighbouring tints — exactly the kind of artefact a screening schedule is designed to hide. The census reads each tint on its own and does not price the step.

What the floor is #

The four tiles hold the same ink, and the single clump makes the floor’s origin plain: a disc of solid ink, its rim a thin ring, and four fifths of the tile bare paper. The paper glows; the ink beneath its film does not; the solid the reader compares the tint with is ink everywhere. A tint is therefore a mixture of the sheet’s own glowing white and the solid’s dark, unglowing colour, in the proportion of its coverage, and its hue sits on a line between them that the solid’s own hue does not lie on. The uniform-film census measured that line with no rim at all, 6.8 degrees at a fifth coverage; the single clump, with a little rim left, gives 7.01. That is the floor, and its cause is the coverage, not the pattern.

No screen changes coverage — that is what the tint is. So the levers that move the floor are the ink and the paper: an ink film that lets some glow through everywhere, as a thinner or more transparent ink would; a paper whose brightener sits deeper, under the ink’s reach; or a tint printed as a thin layer over all the paper rather than full-strength ink over part of it, which is what the comparison with a dyed tint measures.

Why the floor stayed out of sight #

Each of the rim essays measured its screen against the screen before it, and against a 150-line conventional screen as the standard of comparison. Both comparisons are differences between screens, and every screen sits on the same floor. A difference between two numbers of eight degrees each, both mostly floor, reports the rim’s tenth faithfully and hides the nine tenths beneath it, because the nine tenths cancel. The 150-line screen was never a baseline of no glow; it was a baseline of about the same glow.

The first essay of the thread had the floor in hand. A thin dot edge lets the glow back in began from a census with a uniform film and no rim, found 6.8 degrees there, and added the rim to reach 7.9; the next essays inherited the rim’s 1.1 degrees as their subject and the 6.8 as a fixed background. Nothing was wrong with that inheritance except its scale: by the time the thread asked how coarse a screen must be to quiet the glow, the background had become five essays old and the question was being asked about the part on top.

A single clump is the simplest test that would have shown it, and it costs nothing: one centre on the tile instead of thirteen. It was not computed because no essay in the thread was looking for the bottom, only for the next improvement. Asking where a quantity goes at the limit of the lever being studied is the cheapest calculation there is, and it is often the one that says whether the lever is worth pulling.

What the residue measures #

The residue is defined by a comparison that makes the floor’s cause visible. It is the hue turn a fully adapted reader sees between a tint and its solid on brightened paper, less the turn the same tint makes on a sheet of the same colour whose brightener does not fluoresce. On the non-fluorescent sheet, adapting to the paper removes the paper’s colour, and what is left is the ordinary turn every halftone makes as its paper and its ink mix. On the brightened sheet, adapting removes the paper’s colour including its glow — but only for the paper. The ink’s film puts the glow out wherever ink lies, so the solid, all ink, has none of it, and a tint has it in proportion to its bare paper.

That makes the floor a statement about where the glow is, not how the ink is arranged. Some paper is brighter than white found that a fluorescent sheet has no reflectance curve at all, because its light leaves at wavelengths it did not arrive at; a halftone on it is a mixture of two surfaces of which only one behaves that way, and any mixture of a surface that fluoresces and one that does not carries the difference between them in proportion to their areas. The rim is a third surface, ink thin enough to let some glow through, and its area is what screening controls. The two big areas — bare paper and full ink — are fixed by the coverage, and they set the floor.

Every pigment has its floor #

The floor’s size varies by pigment as much as the residue does — from about a degree for the yellow and the blue to nearly ten for the red — and the share the rim carries varies less. For every pigment the single clump keeps at least three quarters of the residue, the cyan least, at 78 per cent, and the red, orange and yellow most, at nine tenths or more. A pigment whose own colour sits far from the brightener’s blue suffers most from the glowing paper beside it, and no screen alters which pigments those are.

What a printer should take from this #

Screen choice is not the lever for brightened paper. The brightener is being used up found a sheet’s glow fading with exposure, so the floor itself moves over a print’s life — faster than any screen could move it. Between the coarsest pattern a tile can hold and a 200-line screen, a pale orange’s residue moves by about a degree out of eight. A printer who wants a pale tint on brightened stock to look as its solid predicts must change the ink or the stock.

The rim essays’ rankings stand. Merged spots beat separate ones, clusters beat spots in the pale tints, and fewer clumps beat more; each ranking is a statement about the rim’s tenth, and the tenth is real. The schedule here confirms them at their true scale.

How the screens were read #

Every screen is the growing-clump construction: N clump centres at the first N blocks of the squares’ dispersed block order on a torus of 32 × 32 twenty-micrometre sites, every site assigned to its nearest centre, each clump switched on in order of distance from its centre, a tint at coverage a switching on a × 1,024 sites one clump at a time. At thirteen and twenty-six clumps it is the growing-clump census’s screen exactly. Each tint’s ink film is thinned over a six-micrometre rim where it meets bare paper, the brightener’s glow passes the film in proportion to its thickness, and the residue is the hue turn of the tint against its solid seen by a reader fully adapted to the sheet, less the turn the same tint makes on a sheet whose brightener does not fluoresce — the census’s measure throughout. The scheduled screen reads each coverage at its scheduled count; a real screen would have to split clumps between counts, which is not modelled.

What this does not reach #

One ink film. The film’s thickness, its transparency to ultraviolet and the rim’s width are the census’s. A more transparent ink lowers the floor and the rim’s share with it.

The split. A screen that changes its count must split each clump in two somewhere between counts, and the tints there carry clumps of two sizes; the census reads only the tints at which every clump has the scheduled size.

Texture. Fewer clumps make a coarser pattern a reader may see as grain. The census prices hue only.

Still open: whether a light ink lowers the floor #

The floor is a tint’s bare paper glowing beside full-strength ink. A light ink — the same pigment at a quarter of its strength, as inkjet printers carry for pale tints — prints the same pale colour over four times the area, so that a five-per-cent tint becomes a twenty-per-cent tint of a weaker ink and much less of the paper is left bare.

The calculation is the census’s tint at each pale coverage printed instead with the pigment at a half and a quarter of its strength over twice and four times the area, matched to the full ink’s colour on non-fluorescent paper, and the residue read as here. The prediction is that the quarter-strength ink lowers the five-per-cent orange’s residue by a third or more, to about five degrees — a larger change than any screen produced — because three quarters of the bare paper is covered and a weak ink’s thin film puts out less of the glow beneath it. If the residue barely moves, the floor is set by the colour the tint must reach rather than by how it is laid down, and brightened paper limits every pale tint the same way. A light ink can cancel the glow at one strength makes that calculation.

A tenth chased through five calculations #

The habit here is about measuring the size of a component before optimising it.

The rim entered this thread as the explanation for a surprise — the residue grew when the dots got an edge — and every essay since asked how to make less of it. Each answer was right, and each was measured against the essay before rather than against the whole residue. A single clump on the tile, the limit every rim essay was moving towards, was never computed.

The failure mode is to optimise the part of a quantity a study has learned to move, without asking what share of the quantity it is. The test is to compute the limit — here, a screen with almost no rim — and set the whole residue against it. The rim carried a tenth; four essays’ worth of real findings fit inside that tenth, and the floor beneath it calls for a different lever.