Merged spots give the mid-tones half the glow back

Brightened office paper glows: its optical brighteners turn the ultraviolet in daylight into visible blue. A halftone tint printed on it carries that glow from two places, the bare paper between the dots and the paper under them, where the ink’s vehicle stops much of the ultraviolet and the pigment some of the glow. Adapting to a brightened sheet leaves the glow found that a reader fully adapted to such a sheet still sees a tint’s hue turned against its solid, because added light is not something adapting to a white divides out. A thin dot edge lets the glow back in found the turn growing where a dot’s ink film thins at its edge and lets the ultraviolet through — the residue, the turn the sheet’s own colour does not account for.

Stochastic dots hold the glow at every tint then printed the tints on screens of different kinds. A conventional screen grows its dots as a tint darkens, so they go from mostly edge to mostly full film along the ramp. A stochastic screen prints every coverage with dots of one size, twenty micrometres or so across, and changes only how many there are. Kept separate, every stochastic dot is 84 per cent rim at every coverage, so the residue stays high along the whole ramp, and switching from a 150-line screen costs a sixty-per-cent orange three degrees of turn where it costs the palest tint one.

That census kept every dot separate, and said so. A real stochastic screen’s spots are not separate in the mid-tones: at half coverage half the paper is ink, and dots that fill half the paper with a dispersed pattern must touch. Two spots that share an edge share a boundary that is not a boundary of the ink — the film runs full across it. The prediction was that a merged screen follows the separate-dot curve up to about thirty per cent and then falls towards the conventional screens, giving back half the mid-tone cost by sixty per cent, so that the twenty-micrometre screen’s sixty-per-cent orange would sit near 7.5 degrees rather than 8.8.

Merging gives back half, as predicted #

Square spots twenty micrometres on a side, switched on in a dispersed order, stay apart up to a fifth coverage: no spot shares an edge with another. Near thirty per cent they begin to touch, a tenth of them; by forty, nearly half; by sixty, nine in ten. As they merge, the ink’s rim — the part within six micrometres of bare paper, where the film thins — falls from 84 per cent of the ink to 74 at forty per cent and 50 at sixty. The merged screen’s residue follows the separate spots’ to thirty per cent and falls at every step after: the orange keeps 8.9 degrees at a fifth coverage, 8.4 at forty and 6.9 at sixty, against the separate spots’ 8.6 and the 150-line screen’s 5.7 at sixty. That gives back 58 per cent of what separate spots cost over the 150-line screen at sixty per cent, and 40 per cent at fifty. Every one of eight pigments gets back between 55 and 58 per cent at sixty and between 38 and 41 at fifty.

• The prediction holds on all three counts: where the curves part, which way the merged one goes, and about how far.
• It lands a little lower than predicted. The orange’s sixty-per-cent residue is 6.9 degrees, not 7.5; merging gives back slightly more than half.
• The share is geometry. Eight pigments of very different residues get back the same fraction, because what merging changes is how much of the ink is edge.

The four curves fall into two pairs and a crossing. The separate square spots and the round twenty-micrometre dots of the earlier census lie together near the top, flat along the ramp; a separate square is as much rim as a round dot of the same width and leaves the same residue to within three tenths of a degree. The 150-line screen falls along the ramp from 8.1 degrees to 5.7. The merged spots run with the separate ones to thirty per cent and then turn down, crossing towards the conventional screen: 8.8 at thirty, 8.4 at forty, 7.8 at fifty, 6.9 at sixty. In the pale half of the ramp the stochastic screen’s cost is untouched by merging, because nothing has merged yet.

What the screen looks like #

A first-order stochastic screen is made by an imaging device that switches spots on or off on a fine grid, choosing which ones to switch on so that the pattern is as even as possible at every coverage — no clumps at low coverage, no holes at high. The census builds one the same way: a grid of square spots twenty micrometres on a side, in a tile of 32 by 32, switched on in an order found by best-candidate sampling, each next spot the one farthest from all those already on. At twenty per cent the spots sit apart, none sharing an edge; at forty, 47 per cent share an edge with at least one neighbour; at sixty, 92 per cent do, and the ink has become a maze of short runs and turns with bare paper threaded through it.

That is the shape the separate-dot census could not represent. A maze is ink with a great deal of boundary, but much less than the same number of separate spots would have: every shared edge between two spots is twenty micrometres of what would have been rim on both sides and is now full film.

How much of the ink is edge #

Separate, the spots are 84 per cent rim at every coverage. Merged, they are 84 per cent to a fifth coverage, 82 at thirty, 74 at forty, 62 at fifty and 50 at sixty. The 150-line screen’s dots, round and growing, go from 48 per cent rim at five per cent coverage to 16 at sixty. So even fully merged at sixty per cent, the stochastic ink is three times as much edge as the conventional screen’s, which is why merging gives back only part of the cost.

The rim falls faster than the share of touching spots rises would suggest at first, because a spot that touches on two or three sides loses rim on all of them; at sixty per cent most touching spots touch on more than one side. The census reads the film’s thickness at each micrometre of the ink as its distance to the nearest bare paper, so a spot surrounded on four sides by ink has no thin film at all.

The glow under the ink #

The glow follows the rim. Under separate spots, 26 per cent of the bare paper’s glow reaches the eye through an orange tint’s ink at every coverage; under merged spots, 25 per cent at thirty, 22 at forty, 18 at fifty and 15 at sixty; under the 150-line screen’s dots, 6.6 per cent at sixty. A full film passes about three per cent. The residue a reader keeps is the turn this extra glow adds to the tint and the sheet’s colour does not account for, and the brightener’s glow is blue, so the turn is largest for the pigments opposite it — the orange and the red.

Every pigment gets the same share back #

The earlier census found the residue’s size belonging to the pigment, over an order of magnitude — a degree for the blue, ten for the red — and the screens’ order the same for every pigment. Merging behaves the same way, and more tightly. At sixty per cent coverage every pigment gets back between 55 and 58 per cent of what separate spots cost it over a 150-line screen; at fifty, between 38 and 41; at forty, between 19 and 21. The red gets back 2.1 degrees of its 3.6; the blue 0.29 of its 0.53. The same fraction, of very different amounts.

That uniformity is the census saying what merging is. It changes nothing about the ink’s colour or the paper’s glow; it changes only how much of the ink is thin, and every pigment’s residue responds to that in proportion. A specifier choosing between a separate-dot and a merging stochastic screen is choosing a fraction of the cost, and the fraction can be read off the screen’s geometry alone.

Why the pale half cannot be helped #

A dispersed screen keeps its spots apart as long as geometry allows, and on a square grid geometry allows it up to a quarter coverage: a checkerboard of every other site on every other row is a quarter of the sites with no two sharing an edge. The census’s order reaches twenty per cent with no spot touching another, twenty-five with fewer than one in a hundred touching, and only past that does merging begin — a tenth of the spots at thirty per cent, a quarter at thirty-five. Below a quarter, then, a first-order stochastic screen prints separate spots whatever its ordering, and every one of them is 84 per cent rim. The pale tints’ cost is the separate spots’ whole cost, and no first-order screen of this spot size can reduce it.

That puts a floor under the choice of screen. On brightened stock, the residue a reader adapted to the sheet keeps on a pale stochastic tint is set by the spot’s size and the rim’s width alone: twenty-micrometre spots and a six-micrometre rim leave an orange 8.9 degrees of turn up to a quarter coverage, where a 150-line screen leaves 7.4 to 8.1. Some paper is brighter than white measured how much glow a brightened sheet carries; this says the screen decides how much of it survives under the ink, and in the pale tints a stochastic screen decides it at the separate spot’s rate.

What an instrument would read #

The glow under the ink is something an instrument measures as well as something a reader sees, and an instrument brings its own light found the standard measurement conditions disagreeing on a brightened sheet by up to seven and a half units, according to how much ultraviolet each lets reach the sample. A tint whose ink lets back more glow widens that disagreement. A sixty-per-cent orange on merged stochastic spots carries 15 per cent of the paper’s glow under its ink, against 26 per cent under separate spots and 6.6 under a 150-line screen — so a press characterised with one kind of screen and run with another is characterised under a different glow at every mid-tone, by an amount the merging halves but does not remove.

What it means for the press #

The earlier essay’s practical point was that switching a brightened-stock job from conventional to stochastic screening costs the mid-tones, not the pale tints: the thirty to sixty per cent oranges and reds, where skin and warm surfaces live. A real first-order stochastic screen softens that by a little over half at sixty per cent and by two fifths at fifty, because its spots merge, and leaves the pale half untouched. The cost that remains in the mid-tones is still two to three degrees of turn for the orange and red beyond what a conventional screen leaves — more than a reader adapted to the sheet can lose — and a soft proof, which a proof cannot glow found unable to show a brightened sheet’s glow at all, cannot show it either.

The optical side of the same switch, which this census holds fixed, runs the same way. A halftone is not a mixture and a dot is larger than it was asked to be found light scattering sideways under a dot’s edge and darkening a tint, which a screen with more edge does more of; merging reduces that too, for the same reason.

How the screen was built and read #

The tile is 32 by 32 sites of twenty micrometres, 640 micrometres square, wrapped at its edges so the pattern repeats without seams. The order is best-candidate sampling on that torus: each next site the farthest from every site already on among twenty random candidates, seeded so the order is the same every run. A coverage of k/1,024 switches on the first k sites; the census reads the dot-rim census’s coverages from five to sixty per cent. The ink is rasterised at one micrometre. At each inked point the film’s relative thickness is its distance to the nearest bare paper, less half a micrometre for the raster, over the six-micrometre rim, capped at one; merged, bare paper is anything not inked, and separate, it is anything not inked by the same spot. The glow under the ink is the dot-rim census’s film model — the vehicle passing 10^(−A·t) of the ultraviolet the brightener absorbs, the pigment passing (R/0.9)^(t/2) of its glow — averaged over the ink’s thickness histogram in sixtieths of the rim. The tint, the reader and the residue are that census’s: brightened office paper, a Yule–Nielsen mixture with a fixed gain, a reader fully adapted to the sheet’s white, hue in Oklab, and the residue the turn on the real sheet less the turn on a sheet dyed to its colour.

What the census does not reach #

One spot size and one ordering. A second-order stochastic screen clusters its spots on purpose, to print more stably, and its spots merge earlier; a larger spot is less rim from the start. Both would move the curve, in directions the census’s numbers predict.

Square spots with sharp corners. A real spot’s corners are rounded by the plate and the press; a rounded spot is a little less rim at its corners, which the census’s separate squares already show, matching round dots within three tenths of a degree.

One optical gain. Merging also changes how much light scatters sideways under the ink’s edge, which the tint’s fixed Yule–Nielsen factor does not follow.

Still open: whether a second-order screen keeps the pale tints quiet as well #

A first-order stochastic screen keeps its spots apart as long as it can, so its pale tints are all edge and its cost there is the separate spots’ whole cost. A second-order screen clusters its spots into small groups at every coverage, so that even a pale tint is printed as clumps of two to four spots — less edge from the start, at the price of a coarser pattern.

The calculation is this census with the spots switched on in clusters — the dispersed order applied to groups of two, three and four adjacent spots — at every coverage, with the residue along the ramp. The prediction is that clusters of four take the pale tints’ residue down by about a degree for the orange, to within half a degree of a 300-line screen, while leaving the sixty-per-cent tints where the merged first-order screen has them, because by then everything has merged anyway; so that a clustered screen moves the stochastic cost from the whole ramp to nowhere in particular. If clustering costs the pale tints instead — clusters with a ragged outline being more edge than their spot count suggests — then for brightened stock the first-order screen’s dispersion is the better choice after all.

A prediction that holds is a measurement too #

The habit here is about what to take from a prediction that comes out right.

The earlier essay predicted where the merged screen would part from the separate one, which way it would go and about how far, and all three came out within the census’s resolution. It would be easy to record that as a confirmation and move on. But the prediction was a half, and the census says 58 per cent at sixty and 40 at fifty — numbers a printer can use, which the prediction did not supply — and it says something the prediction did not consider at all: that the share is the same for every pigment, which is what identifies the mechanism as geometric.

The failure mode is to treat a confirmed prediction as having taught nothing new. A prediction states a direction and a rough size; the census that tests it returns the exact size, its dependence on everything the prediction ignored, and sometimes — as here — a regularity that names the cause.