Stochastic dots hold the glow at every tint

Most office paper glows. It carries optical brighteners, which absorb the ultraviolet in daylight and give it back as visible blue, and some paper is brighter than white measured how far that puts a sheet above anything a non-fluorescent white can reach. Print a halftone tint on it and the glow comes from two places: the bare paper between the dots, and the paper under the dots, where the ink’s vehicle absorbs much of the ultraviolet on the way in and the pigment absorbs some of the glow on the way out. A solid puts most of it out; a pale tint keeps most of it.

Adapting to a brightened sheet leaves the glow found what that does to a reader. Adapted fully to the sheet’s white, a reader still sees a fifth-coverage orange turned in hue against its solid by ten degrees, where a sheet of the same colour without fluorescence leaves three, because added light is not something adapting to a white divides out. A thin dot edge lets the glow back in then gave each dot of a 150-line screen a rim over which its ink film thins to nothing. The thin edge lets ultraviolet in and glow out, and the turn the adapted reader keeps — the residue, beyond what the sheet’s colour accounts for — grew with the rim, most at the palest tints, where most of a dot is edge.

How much of a dot is edge depends on how large the dot is, and at a given coverage the dot’s size is set by the screen. A 200-line screen makes every dot three quarters the width of a 150-line dot. A stochastic screen does something different in kind: it prints every coverage with dots of one size, scattered in more or fewer numbers, typically twenty micrometres or so across. The prediction was that the residue rises with the ruling at every coverage, and that a stochastic screen holds its residue nearly flat across the pale half of the ramp at the level a 300-line screen reaches only at five per cent — so that switching a job to stochastic screening on brightened stock turns its pale tints a degree or two further, and no adaptation takes it back.

The stochastic screen is flat at the finest screen’s palest level #

Every dot given a six-micrometre rim, the residue rises with the ruling from 100 to 300 lines at every coverage and for every one of eight pigments: the fifth-coverage orange keeps 7.2 degrees on 100 lines and 7.9 on 300. A stochastic screen of twenty-micrometre dots holds the orange between 8.8 and 9.1 degrees from five per cent coverage to sixty and the red between 11.55 and 11.59, starting within fifteen hundredths of a degree of the 300-line screen’s five-per-cent tint — the prediction, on both counts. Its cost is not where the prediction put it. Against a 150-line screen the stochastic screen adds 0.9 degrees to the orange’s five-per-cent tint and 3.1 to its sixty-per-cent tint, the red 1.1 and 3.9: the switch costs the mid-tones three times what it costs the palest tint. On ten-micrometre dots, which are all edge, the orange’s residue rises along the ramp, from 10.4 degrees to 11.4. And at any one coverage the residue is a function of the dot’s diameter alone: across all seven screens, a smaller dot always leaves more.

• The ruling matters, and in the predicted direction. A finer screen makes every dot more edge and lets more glow back.
• A stochastic screen is flat because its dots are. Every coverage is printed with the same dot, so every coverage has the same share of edge.
• The cost of the switch is in the mid-tones. That is where conventional dots are largest and mostly full film, and where stochastic dots change most against them.
• The ramp’s shape belongs to the screen. The previous essay found the palest tint keeping the most residue; that is true of conventional screens and false on the finest stochastic ones.

The four solid curves are the conventional screens, stacked in order of ruling, each falling along the ramp from its palest tint to its sixty-per-cent tint by about two and a half degrees. The three dashed curves are the stochastic screens. The twenty-micrometre curve runs flat at the height the 300-line curve reaches only at its first point. The thirty-five-micrometre curve falls a little, from 8.3 degrees to 7.4, and sits between the 150- and 300-line screens at the pale end and above all of them at the dark end. The ten-micrometre curve rises. The family of conventional screens and the family of stochastic screens answer different questions about the ramp: one gives a pale tint more residue than a dark one, the other gives them about the same, or the reverse.

How much of a dot is edge #

The mechanism is the previous essay’s and it can be read off one number per screen and coverage: the share of a dot’s area lying within six micrometres of its edge, where its film thins.

On a conventional screen a dot grows as coverage rises, and the rim’s share of it falls. A five-per-cent dot on a 150-line screen is 43 micrometres across and 48 per cent rim; on a 300-line screen it is 21 micrometres across and 81 per cent rim. By sixty per cent coverage the 150-line dot is 148 micrometres across and 16 per cent rim. A stochastic screen’s dot is the same at every coverage, so its rim share is one number: a twenty-micrometre dot is 84 per cent rim — a little more than a 300-line five-per-cent dot, which is why the two start in the same place — a thirty-five-micrometre dot 57 per cent, and a ten-micrometre dot, whose radius is shorter than the rim, has no full film at all.

That is the whole difference between the two families. A conventional screen changes its dot as the tint darkens, from mostly edge to mostly film; a stochastic screen keeps its dot and changes how many there are. A halftone is not a mixture and a dot is larger than it was asked to be took the optical side of that difference apart, the light that scatters sideways in the paper under a dot’s edge; this is the chemical side, the ultraviolet that the edge’s thin ink does not stop.

The glow under each dot #

The glow let back under a dot follows its rim share. Under a five-per-cent orange dot on 150 lines, 15 per cent of the bare paper’s glow reaches the eye; on 300 lines, 26 per cent; under every twenty-micrometre stochastic dot, 27 per cent, and under every ten-micrometre one, 44, whatever the coverage. A full film passes about three per cent. The conventional curves fall along the ramp towards that floor as their dots fill in; the stochastic lines stay where they start. A sixty-per-cent tint on a twenty-micrometre stochastic screen carries four times as much glow under its dots as the same tint on 150 lines, which carries 6.6 per cent.

The residue a reader keeps is the turn that extra glow adds and the sheet’s colour does not account for. The glow is blue, the brightener’s emission, so it pulls a tint’s hue towards blue, and the solid it is compared with, under its full film, has almost none.

What the switch costs, and where #

The prediction located the switch’s cost at the pale end. The census puts it at the other end. For the orange the stochastic screen adds 0.94 degrees at five per cent coverage and 3.13 at sixty; for the red 1.10 and 3.86; for the magenta, cyan and green between twice and three times as much at sixty as at five. The palest tints cost little because a conventional screen’s pale dots are already mostly edge — a 150-line five-per-cent dot is half rim — and a stochastic dot is only somewhat more so. The mid-tones cost a great deal because a conventional mid-tone dot is mostly full film, a sixth rim, and a stochastic dot at the same coverage is still five sixths rim.

For a printer moving a job from conventional to stochastic screening on brightened stock, that is the opposite of where to look. The highlights, where a stochastic screen’s fine detail is usually the reason for the switch, move by a degree. The thirty to sixty per cent tints of the orange and red move by two to four degrees, beyond anything the reader’s adaptation to the sheet can take back, and they are the tints that make up most of a photograph’s skin and warm surfaces. A proof cannot glow found a soft proof unable to show a brightened sheet’s glow at all; it can show still less of a glow that depends on the screen.

An instrument reads the screen too #

The residue is a reader’s quantity, but the glow behind it is something an instrument measures, and measures differently depending on how it lights the sample. An instrument brings its own light found the four measurement conditions of ISO 13655 reading a brightened sheet up to seven and a half units apart, every instrument correct: one condition includes the ultraviolet that excites the brightener, one excludes it, and the others sit between. On an unbrightened sheet they agree. On a brightened one, how far they disagree for a tint is a measure of how much glow that tint carries.

That makes the screen a measurement variable as well as an appearance one. A sixty-per-cent orange printed on twenty-micrometre stochastic dots carries four times the glow under its dots that the same tint carries on 150 lines, so the gap between an ultraviolet-including and an ultraviolet-excluding measurement of it is wider on the stochastic screen; a press characterised on one screen and run on the other is characterised under a different glow at every coverage above the palest. The usual guidance — measure brightened stock with the ultraviolet the viewing light contains — holds whichever screen is used, and the census adds that the data must be measured on the screen it will be printed with, because the glow it contains is the screen’s.

The paper’s own colour is not the only thing that changes how a tint turns. Newsprint turns a tint with its colour, not its gain found an uncoated sheet’s yellow reversing an orange’s turn outright, and a reader sees the reversal until almost adapted found a real reader, part-adapted, seeing most of it. The glow is a third route by which the substrate enters a tint’s hue, and the only one of the three that depends on how the dots were drawn.

Every pigment, one order #

The screens fall in the same order for every pigment: at thirty per cent, 150 lines below 300 lines below twenty-micrometre stochastic dots below ten-micrometre ones. The size of the residue belongs to the pigment and runs over an order of magnitude, from 1.1 degrees for the blue on 150 lines to 9.2 for the red. A pigment whose own hue lies near the glow’s — the blue, the violet — keeps a small residue on every screen, because a blue added to a blue barely turns it; the orange and the red, opposite the glow in hue, keep the largest, and are the ones on which the choice of screen moves the residue most, by 2.0 and 2.3 degrees between 150 lines and twenty-micrometre dots at thirty per cent.

The ramp turns over on the finest dots #

The previous essay recorded that the residue is largest at the palest tint and falls towards the middle of the ramp, rim or no rim, against a prediction that it would grow towards the mid-tones. On every conventional screen here that still holds. On the stochastic screens it weakens and then reverses: the thirty-five-micrometre screen still falls, by a degree for the orange; the twenty-micrometre one is flat; and on ten-micrometre dots the orange’s residue rises from 10.4 degrees at five per cent to 11.4 at sixty, and the red’s from 13.1 to 14.6. The yellow and the violet rise too, more gently; the magenta, cyan, blue and green still fall, by under half a degree.

On a screen whose dots are all edge, nothing about a dot changes along the ramp: each lets back 44 per cent of the paper’s glow, and darkening the tint only replaces bare paper, which glows fully, with dots that glow at 44 per cent. The census does not take apart why that raises the orange’s turn rather than lowering it; it shows that the shape of the ramp, which looked like a fact about pale tints, is a fact about how a screen grows its dots.

The dot’s diameter decides #

Rulings and stochastic dot sizes look like different kinds of choice, and at one coverage they are not. At every coverage, for every pigment, ordering the seven screens by the diameter of the dot each prints there orders their residues exactly the opposite way, without an exception: at twenty per cent the orange keeps 10.6 degrees on a ten-micrometre dot, 9.0 on twenty, 8.2 on thirty-five, 7.9 on a 300-line dot 43 micrometres across, and 7.2 on a 100-line dot 128 across. A 300-line screen and a stochastic screen are two ways of choosing a dot size, one that changes with coverage and one that does not, and the residue reads only the size.

That gives a specifier one number to ask for. A brightened-stock job whose mid-tones must not turn can be printed on any screen whose mid-tone dot is large; a stochastic screen with thirty-five-micrometre dots keeps the orange’s sixty-per-cent residue at 7.4 degrees, within two of a 150-line screen’s, where twenty-micrometre dots leave 8.8.

How the screens were printed #

The sheet, pigments, reader and film model are those of the dot-rim census: brightened office paper under daylight with its ultraviolet; eight process and spot pigments; a reader fully adapted to the sheet’s white by a Bradford transform, hue read in Oklab. Each dot is round, its film full in its centre and falling linearly to nothing over its last six micrometres. Through a film of relative thickness t the vehicle passes 10^(−A·t) of the ultraviolet the fluorophore absorbs, weighted by wavelength, and the pigment passes (R/0.9)^(t/2) of the glow on the way out; the glow under a dot is these averaged over its area in 240 rings. A conventional dot’s radius is that of a circle covering the coverage’s share of its cell, the cell 25.4 mm divided by the ruling; a stochastic dot’s radius is fixed at five, ten or seventeen and a half micrometres, the coverage set by how many there are. The tint’s reflected part is the census’s, a Yule–Nielsen mixture with one fixed gain, and the residue is the turn on the real sheet less the turn on a sheet dyed to its colour. At 150 lines the census reproduces the dot-rim census’s six-micrometre column exactly.

What the census does not reach #

One optical gain for every screen. A finer screen and a stochastic one both scatter more light sideways under their dots’ edges, which raises the tint’s optical gain, darkens it, and changes how much bare paper a reader effectively sees. That is the other half of switching screens and it is held fixed here.

One rim. The rim was held at six micrometres. A finer dot printed from a finer plate may have a narrower edge as well as a smaller body, which would offset some of what is computed here.

Round, isolated dots. Above about forty per cent coverage stochastic dots touch and merge into clusters, which a merged rim would make less edge than the census counts; the mid-tone figures for twenty-micrometre dots are the upper end of what such a screen could cost.

Still open: whether a merged stochastic dot gives the mid-tones back #

The stochastic screen’s mid-tone cost rests on every dot staying separate and keeping its rim. A real stochastic screen’s dots begin touching well before sixty per cent coverage, and two touching dots share a boundary that is no longer an edge: the film between them is full. The merged cluster is mostly film, like a conventional mid-tone dot.

The calculation is this census with stochastic dots placed at random at each coverage and merged where they overlap, the rim drawn only along the cluster’s outer boundary, at twenty micrometres, with the glow under the clusters and the residue along the ramp. The prediction is that the 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 sits near 7.5 degrees rather than 8.8. If it does not fall, the stochastic cost is real at every coverage a job uses; if it falls all the way, the cost is confined to the pale half of the ramp after all, and the prediction was right for the wrong reason.

A shape on a curve can belong to the instrument #

The habit here is about what a trend along an axis is a fact about.

The previous essay found the residue largest at the palest tint and recorded it as a fact about tints. It was a fact about tints printed on a conventional screen, where the palest tint has the smallest dots. Change how the dots are grown and the trend flattens, then reverses. Nothing about the paper, the ink or the reader changed; the axis along which the trend was read — coverage — was standing in for another one, dot size, which the conventional screen happens to tie to it.

The failure mode is to read a trend along one axis as a property of that axis, when the apparatus ties it to another. The test is to find an apparatus that unties them, and here it was the obvious alternative to the one in use.