Squares of four print a pale tint like a 150-line screen

Brightened office paper glows: optical brighteners in the sheet turn the ultraviolet in daylight into visible blue, which is why some paper is brighter than white. A halftone tint printed on it carries the glow from the bare paper between its dots and, more faintly, from the paper under them — the light that, entering between dots and emerging under one, printing does not change the sign found carrying a tint part of the way from a mixture of light towards a mixture of paint. Adapting to a brightened sheet leaves the glow found a reader fully adapted to the sheet still seeing a tint’s hue turned against its solid ink, and a thin dot edge lets the glow back in traced part of that turn to the rim of each dot, where the ink film thins and the ultraviolet gets through to the brightener underneath. The residue is the turn the sheet’s own colour does not account for.

How much rim a tint has depends on the screen. Stochastic dots hold the glow at every tint found a stochastic screen’s dots, one small size at every coverage, staying 84 per cent rim and keeping the residue high along the whole ramp. Merged spots give the mid-tones half the glow back let those spots touch: where two spots share an edge the film runs full across it, so the rim falls as the spots merge, from 84 per cent of the ink to 50 by sixty per cent coverage, and merging gave back about half the stochastic screen’s mid-tone cost over a 150-line screen. The pale tints, printed as separate spots, kept all of it.

Its closing section proposed the screen that real platesetters offer for exactly this complaint. A second-order stochastic screen switches its spots on in small clusters from the first tint, so even a five-per-cent tint is printed as clumps — less edge for the same ink, at the price of a coarser pattern. The prediction was that clusters of four take the orange’s pale-tint residue down by about a degree, to within half a degree of a 300-line screen, while the sixty-per-cent tints stay where the merged first-order screen has them, because by then everything has merged anyway.

The squares print the palest tint like a 150-line screen #

With the spots switched on in squares of two by two — the squares placed by a dispersed order over the tile’s blocks — the ink of a five-per-cent tint is 52 per cent rim, against 84 for single spots. The orange’s residue falls from 8.94 degrees to 8.14, which is 0.02 above a 150-line screen’s and 0.81 below a 300-line screen’s; every one of eight pigments lands within a tenth of a degree of the 150-line screen and below the 300-line one. Pairs and Ls of three fall partway, to 8.57 and 8.43. At sixty per cent pairs and Ls come back to within three tenths of a degree of the merged single spots, as predicted, but squares keep the orange 0.9 degrees below them, at 6.00 against 6.90. From ten per cent on, the 150-line screen is quieter than the squares, by between 0.29 and 0.73 degrees.

• The pale tints gain about what was predicted, and land somewhere else. A 300-line screen is not a quiet reference in the palest tint; its tiny dots are nearly as much rim as stochastic spots.
• The mid-tones do not converge for squares. A cluster of four that merges with its neighbours is still a compact clump with less boundary than the maze single spots make.
• No stochastic screen here matches the 150-line screen past the palest tint. Its dots grow, and growing dots are the cheapest way to lose rim.

The six curves start in two groups and end in three. At five per cent the single spots and the 300-line screen sit together near nine degrees, and the squares and the 150-line screen together near eight, with pairs and Ls between. By sixty per cent the conventional screens have pulled ahead, the 150-line screen lowest at 5.67 degrees, and the stochastic screens have spread by cluster size: single spots 6.90, Ls 6.79, pairs 6.65, squares 6.00. Moved to other pigments, every curve scales with the pigment’s own residue — the red’s twelve degrees, the blue’s one and a half — and keeps its place.

What the clusters do to the ink #

The three panels hold the same number of spots. Laid singly in a dispersed order, they are a hundred and two separate squares; in pairs, fifty-one dominoes; in squares of four, about twenty-six clumps two spots across. The rim — the ink within six micrometres of bare paper — is 84 per cent of the single spots’ ink, 72 per cent of the pairs’ and 51 per cent of the squares’. Each twenty-micrometre spot alone is mostly rim because six micrometres in from each edge leaves only an eight-micrometre core; a clump forty micrometres across leaves a twenty-eight-micrometre core.

The placement of the clusters is the screen’s own. The tile’s 32 by 32 sites are read as 16 by 16 blocks of two by two, the blocks put in a dispersed order by the same best-candidate sampling the single-spot screen uses, and a screen of cluster size s switches on the first s spots of each block in that order — a pair, an L or the whole square — before going round again for the spots left. Up to ten per cent coverage no two clusters’ blocks share an edge, so each clump is still a separate island of ink; by fifteen per cent the pairs begin to touch.

Along the ramp the rim tells the residue’s story in advance. At five per cent single spots are 84 per cent rim, squares 52, the 150-line screen 48 and the 300-line screen 81. The 300-line screen’s five-per-cent dot is 21 micrometres across, the size of a stochastic spot, and that is why it sits with the single spots in the palest tint. Its dots grow with the tint and its rim falls fast; by a fifth coverage it has passed the squares. The squares’ own rim stays near a half until their clusters start to merge, then falls to 26 per cent by sixty — half the single spots’ — which is why they keep their advantage there.

The palest tint, pigment by pigment #

For every pigment, the squares’ five-per-cent residue lies within a tenth of a degree of the 150-line screen’s — 0.004 for the yellow, 0.045 for the cyan, 0.02 for the orange — and below the 300-line screen’s. They close between 95 and 98 per cent of the single spots’ gap to the 150-line screen. The pigments differ by an order of magnitude in how much residue they carry at all, from the yellow’s one and a half degrees to the red’s eleven, and every one gets the same geometry: what clustering changes is how much of the ink is edge, and the glow follows the edge.

So the prediction’s size was about right — the orange falls by 0.8 degrees, “about a degree” — and its reference was the wrong neighbour. The stochastic-dot census had found the finer the conventional screen, the more it glows, at every coverage; the 300-line screen is the louder of the two conventional screens everywhere on the ramp, and in its palest tint as loud as single stochastic spots. A target of half a degree from it asked for almost nothing. The squares overshoot it and land on the coarser, quieter screen instead.

Sixty per cent: the pairs converge, the squares do not #

The prediction’s second half reasoned that by sixty per cent everything has merged, so how the spots were switched on stops mattering. For pairs and Ls it holds: they come back to within seven per cent of the merged single spots for every pigment, the orange to a quarter of a degree. For squares it does not: they stay twelve to twenty-five per cent below the single spots for every pigment, the orange 0.9 degrees below.

The reason is in the rim chart. At sixty per cent single spots merged into a maze are still half rim, because a dispersed maze of single spots has bare paper threaded through it everywhere; the squares, merged with their neighbours, form fatter clumps with fewer, wider channels of paper between, a quarter rim. A pair is a narrow domino whose merged maze looks much like the single spots’. Merging erases the history of how the spots were laid only when the laid shapes are thin; a square’s compactness survives it.

The L is the worst shape to merge #

Read through the mid-tones, the three cluster sizes do not stay in order. At thirty per cent the residue falls with the cluster’s size, as it did in the pale tints: 8.15 degrees for pairs, 8.11 for Ls of three, 7.82 for squares. By sixty per cent the Ls have overtaken the pairs and nearly caught the single spots: 6.79 degrees against the pairs’ 6.65 and the single spots’ 6.90, and the ink under them is 52 per cent rim — more than the single spots’ 50 and the pairs’ 47.

The reason is in the order the spots go down. An L of three is a square with a corner missing, and a screen of Ls lays its third spot in every block before it lays any block’s fourth; at sixty per cent coverage 614 spots have gone down, which is 205 blocks’ worth of Ls and no completed squares at all. Every block holds a notched shape, and every notch is a short stretch of boundary between ink and bare paper inside a region that is otherwise ink. Merging removes the edges between clusters and keeps the notches, so the L screen’s merged maze carries an inside corner in every block. The pairs at sixty per cent have finished their dominoes and begun adding third spots to some blocks, and a domino merged with its neighbours along both long sides leaves straighter channels of paper than a field of notched Ls does.

The squares avoid both. At sixty per cent 153 blocks are complete squares and the rest empty, and complete squares merged edge to edge make solid regions whose only boundary is their outline. That is why squares alone keep their advantage into the mid-tones, and why the advantage is about the clusters’ shapes once merged, not about their count: pairs, Ls and squares lay the same number of spots at every coverage. For a printer, the consequence is that a cluster shape which leaves gaps in its own block is worse than single spots in the mid-tones on brightened stock, whatever it saves in the highlights.

What this means for a print job on brightened stock #

A second-order stochastic screen with square clusters of four is, in the palest tints on brightened office paper, as good as a 150-line screen at keeping the paper’s glow out of the hue a reader sees, and better than a 300-line screen. That is the tint range where stochastic screening’s usual advantages — no rosette, no moiré, fine detail — are most prized, and where the first-order screen pays most for them. It is also the range where the white a halftone is mixed with has moved its hue most before any glow is added, the turn a tint in paint turns the other way found in every tint made by adding white light. Pairs and Ls buy part of the gain, and squares buy nearly all of it.

From ten per cent on, a conventional 150-line screen is quieter still, and by thirty per cent the 300-line screen has passed the squares as well. No stochastic screen in the census reaches the conventional screens through the mid-tones because a screen whose dots grow loses rim faster than one whose dots only multiply and merge. A printer choosing a screen for a brightened-stock job is choosing where along the ramp to pay: clustered stochastic screening pays in the mid-tones and not the highlights; conventional screening the reverse in the faintest tints only.

How the screens were built #

The tile, spots, rim, film and glow are the merged census’s: a periodic tile of 32 by 32 sites twenty micrometres apart, square spots twenty micrometres on a side, the ink film thinning linearly over the last six micrometres to its boundary with bare paper, read on a one-micrometre raster; the glow under the ink is the dot-rim census’s film model averaged over the ink, and the residue is the turn in Oklab a reader fully adapted to the sheet still sees, less the turn on a sheet dyed to its colour. Merged spots share edges that are not boundaries.

For the clustered screens the tile is read as 16 by 16 blocks of two by two sites. The blocks are ordered by best-candidate sampling on the 16-block torus — each next block the farthest, of twenty random candidates, from every block already chosen — with a fixed seed. A screen of cluster size two, three or four switches on, block by block in that order, the first two, three or four of the block’s spots in a fixed order — a pair, then an L, then the square — and once every block holds its cluster, switches on the remaining spots block by block. A tint at coverage a is the first a × 1,024 spots. The 150- and 300-line screens are the dot-rim census’s round dots.

What the census does not reach #

The pattern’s own visibility. Clusters of four print a coarser pattern — forty-micrometre clumps — and at reading distance that is still finer than the eye resolves, but the census does not compute the pattern’s graininess, which is the cost a printer weighs against the glow.

Dot gain. Every spot is printed at its nominal size. On press a twenty-micrometre spot spreads, and clusters, with less edge, gain less ink than single spots do, which would move the comparison in the squares’ favour.

One rim and one stock. The film thins over six micrometres on one brightened office sheet; a thicker film or a brighter sheet scales every residue.

Still open: whether a cluster that grows keeps the pale tints and catches up in the mid-tones #

Squares win the palest tints and lose to the 150-line screen through the middle of the ramp, because a conventional dot grows and a stochastic cluster does not. A hybrid screen does both: it lays clusters of a few spots in the pale tints and grows each cluster as the tint darkens, until in the mid-tones it prints as a fine conventional screen.

The calculation is this census with a growing-cluster screen — clusters started as squares of four at five per cent, each grown by its neighbouring spots in order of distance from the cluster’s centre as coverage rises, so that by sixty per cent the tile carries the same number of clumps it started with — its rim and residue along the ramp beside the fixed squares and the 150-line screen. The prediction is that the growing clusters keep the squares’ five-per-cent residue and reach within a tenth of a degree of the 150-line screen by thirty per cent, because a growing clump’s rim falls as the square root of its area; and that it stays slightly above the 150-line screen through the mid-tones, because its clumps are placed by a dispersed order rather than on a lattice and so begin merging earlier. If the growing clusters fall below the 150-line screen, a stochastic screen can beat a conventional one on brightened stock at every coverage. A growing clump prints like the screen with as many dots makes that calculation.

A reference point has to be measured where it is used #

The habit here is about the benchmark a prediction is written against.

The prediction set a clustered stochastic screen against the 300-line screen, the finest conventional screen in the census, because a stochastic screen is fine-grained and the natural comparison is with the finest thing like it. But the quantity in question is rim, and in rim a 300-line screen’s five-per-cent dot is a stochastic spot — 21 micrometres across — while a square of four is a 150-line dot. Grain size and edge share are different properties, and the comparison was drawn on the first when the claim was about the second.

The failure mode is to choose a benchmark by its resemblance to the candidate rather than by the property being measured. The test is to place every candidate and every reference on the axis the claim is about — here the share of ink that is edge — and compare there. That put the squares beside the 150-line screen, not the 300-line one, and turned a prediction of “nearly as good as the finer screen” into a finding of “as good as the coarser one”.