Each dark ray wants its own ease

A press’s total-ink cap protects the paper and costs the deepest colours their chroma: where cyan, magenta, yellow and black would add up past the limit, black is raised and the others cut, and the colour comes out duller than asked. Easing the cap reaches the deep shadow and no higher relaxed the cap by a tenth or a fifth on a straight ramp from L* 20 to the floor at L* 14 and found the deep blue’s dulling halved and the brown’s nearly removed for colours that bind low. Moving the ramp up, bending it and steepening it followed, each read by the mean chroma lost.

A steeper ramp cures the shadow by overshooting it spent the ease from L* 17 and found every group dulling less on that mean — and found why the mean could not be trusted any longer. Eased by a fifth from L 17, the brown’s low binders printed more chromatic than asked, by up to 0.07, and read by the size of the error — shortfall and excess both counted — the brown ray’s error went from 5.16 thousandths on the ramp from L 20 to 13.54. Eased by a tenth, the steep ramp was no worse anywhere, and better on three cases of four.**

Somewhere between a tenth and a fifth, the steep ramp’s faster return of ink should match the colours’ own fall in chroma. The prediction was that the ramp from L 17 eased by three twentieths leaves every ray’s error, hard and through the knee, below the ramp from L 20 eased by a fifth — the best of the earlier settings — and below the ramp from 20 eased by three twentieths.** If the brown’s low binders already overshot by more than 0.03 at three twentieths from L* 17, the overshoot would not be a matter of how much ease but of how steeply any of it is spent, and the ramp from L* 20 would be the steepest a fifth can safely be.

The brown still overshoots #

Eased by three twentieths from L 17, the brown’s deepest case prints 0.039 more chromatic than asked, and the brown ray’s error, hard, is 8.81 thousandths — against 5.16 on the ramp from L 20 at a fifth and 4.74 on the ramp from 20 at three twentieths. The blue’s, on the same setting, is 5.08, against 5.21 and 5.43.**

• The prediction holds for the blue and fails for the brown. The blue’s best setting is the one predicted; the brown’s is the gentle ramp at the same ease.
• The alternative arrives as stated. The brown overshoots by more than 0.03, and the ramp from L* 20 is the steeper of the two that serves it.
• The ramp from L 20 at three twentieths is the brown’s best setting yet*, better than a fifth on the same ramp, which every earlier essay had treated as the stronger cure.

The figure’s four lines cross. On the ramp from L* 20 both rays improve from a tenth to three twentieths; the blue keeps improving to a fifth, the brown turns back. On the ramp from L* 17 the blue is flat — 5.17, 5.08 and 5.19 thousandths, best in the middle and within a tenth of a thousandth either way — while the brown climbs from 6.23 to 8.81 to 13.54. The blue is indifferent to how steeply its ease is spent and wants it steep; the brown cannot take a steep ramp at any ease past a tenth.

The overshoot follows how fast ink comes back #

The two ramps and three eases are six settings, and they reduce to one number. A ramp from L* 17 spends its ease over three units of lightness and a ramp from 20 over six, so the rate at which the cap gives back ink as the colour darkens is the ease over the ramp’s length. For a 220 per cent cap, a tenth from L 17 returns 7.3 points of cap per unit of L, and so does a fifth from L* 20; they overshoot the brown by 0.013 and 0.019. Three twentieths from 17, at 11 points a unit, overshoots by 0.039; a fifth from 17, at 14.7, by 0.070. The three settings on the ramp from 20, at 3.7, 5.5 and 7.3 points a unit, overshoot by 0.001, 0.008 and 0.019.** Ordered by rate, the six overshoots deepen in order, and the two settings that share a rate share an overshoot within a factor of one and a half.

So the deep shadow’s cure and its overshoot are one mechanism, read at two settings of one number. A colour that binds low follows the cap closely, and as the cap falls the raised black falls with it and the chromatic inks come back; when they come back faster than the colour’s own chroma falls towards the floor, the print gains chroma it was never asked for. A bent ramp takes more ink and dulls more read the deep shadow’s cure as following how steeply the cap falls there. That was right, and it is also the overshoot’s cause. Steepness is a dose.

Through the knee, each ray takes more #

Through the knee, the blue errs least at 6.88 thousandths on a fifth from L 17, and the brown least at 5.00 on a fifth from L 20; three twentieths from L* 17 leaves the brown at 6.97 and three twentieths from 20 at 6.93.** With the knee, both rays’ best setting moves to the fullest ease on its own ramp: the knee raises black gradually within fifteen units of the limit rather than all at once, so the cap’s release of ink is spread over the knee’s width and arrives more slowly than the ramp alone would deliver it.

That confirms the rate as the controlling quantity from another side, and it matches what a steeper ramp cures the shadow by overshooting it found when the knee kept the blue’s deepest overshoot to two thousandths on the steepest ramp. A knee slows the effective rate of return, and with it slowed, each ray can take a larger dose before it overshoots. More nodes quarter the dulling and leave the knee a larger share found the knee itself costing dulling that finer sampling could not remove; the same knee is what lets the eased cap be eased further.

The brown’s two groups pull apart #

The brown’s low binders err by 4.23 thousandths eased by three twentieths from L 20 and by 5.04 at a fifth; its high binders by 6.33 and 5.54.* Within the brown ray the two groups want different eases too: the colours that bind low overshoot at a fifth and do best at three twentieths, while those that bind above L* 26, whose deepest stretch begins well above the ramp, keep gaining to a fifth. On the ramp from L* 17 the low binders’ error at three twentieths is 9.63, more than twice their best.

A higher ramp trades one shadow for another met the same split from the other side, moving the ramp’s start upward and finding the high binders’ gain bought with the low binders’ loss. A cap that serves a ray as a whole is a compromise between groups that bind at different depths, and the compromise depends on how the ray’s colours are spread between them.

No one setting serves both rays #

Hard, the blue’s best setting, three twentieths from L 17, leaves the brown at 8.81 thousandths, 86 per cent above the brown’s own best of 4.74; the brown’s best, three twentieths from L 20, leaves the blue at 5.43, 7 per cent above its best of 5.08.** A table with one cap for every colour has to choose, and of the six settings three twentieths from L* 20 is the best compromise hard, its two rays summing to 10.17 thousandths against 10.37 for a fifth on the same ramp and 13.89 for the blue’s best.

The asymmetry is the useful part. The brown’s best costs the blue little because the blue is nearly indifferent to the ramp’s steepness — five of the six settings leave it between 5.08 and 5.43, and only a tenth on the gentle ramp, too little ease, leaves it at 6.44. The blue’s best costs the brown a great deal because the brown is sensitive to the rate and the blue’s best has a high one. A cap chosen for the colours that overshoot easily loses almost nothing on the colours that do not.

The blue’s deepest overshoot never passes 0.017 on any setting; the brown’s passes 0.03 on every setting from L 17 past a tenth.* The blue’s darkest colours, eased however far, overshoot by at most a quarter of the brown’s worst; it is the brown’s low binders that set the limit on how fast a cap can return ink, and a table that respects the brown’s limit serves the blue nearly as well as the blue’s own setting would.

How much of the overshoot is the table’s sampling #

Every figure here is read through the table, not through the separation: the table holds the separation at 65 nodes from L* 92 to 14, 1.2 units of lightness apart, and every lightness between nodes is printed by straight lines between them. A cap that falls 44 points in three units is sampled by two or three nodes, and the straight lines between them cannot follow the separation’s own bend where the cap stops binding and the inks come back. Some part of every overshoot here may be that interpolation rather than the separation.

The census has measured the table’s sampling of the deep shadow before, and both measurements bear on this. Three nodes at the cap’s corner cut the dulling by a third found that nodes placed just below where a colour’s ink reaches the cap take back a third of its dulling, because the dulling lives in the few segments where the separation turns; shared nodes belong in the deep shadow found that a profile’s nodes are shared by every colour and can be placed only once, so a placement good for one ray’s binding point is a compromise for another’s. A steep ramp moves the separation’s turn to the last few units above the floor, where 65 even nodes are as sparse as anywhere, and gives the interpolation a sharper corner to cut.

The two readings point the same way here, which is why the essay can lean on the rate. The two settings that share a rate — a tenth from L* 17 and a fifth from L* 20 — are sampled very differently, the first by two or three nodes and the second by five, and they overshoot within a factor of one and a half of each other. If the interpolation dominated, the steep setting would overshoot far more than the gentle one at the same rate. It overshoots less, 0.013 against 0.019. So the overshoot is mostly the separation’s, and the rate of return is a property of the separation that a finer table would not remove.

That does not make the sampling irrelevant. A table with nodes concentrated in the last few units of lightness would read every one of these settings more faithfully, and could shift where each ray’s best lies by a few thousandths. The ranking of settings — gentle for the brown, steep for the blue, three twentieths as the compromise — rests on differences of half a thousandth and more, which the census’s own sampling errors are unlikely to reverse, but the exact optimum is a property of this table as well as of the press.

What a separation engineer should take from this #

Set the cap’s ease by its rate of return, not by its size alone. The overshoot that limits an eased cap follows the points of cap returned per unit of lightness, and a tenth over three units overshoots as a fifth over six does. For these rays the safe rate is about five or six points a unit of L* for a 220 per cent cap.

For a single cap on every colour, ease gently over a longer ramp. Three twentieths from L* 20 is the best compromise here hard; through a knee, a fifth from L* 20.

If the separation can vary its cap by hue, give the blues a steeper ramp. The blue gains a little from L* 17 and never overshoots; the browns need the gentler one.

How the tables were separated and read #

The census’s press is four process inks at D50 on a coated stock, its separations solved for black by the census’s strategies under caps of 200 to 260 per cent, on two dark rays from the press’s blue and brown towards the floor at L* 14. The cap is eased at the floor by a tenth, three twentieths or a fifth on a straight ramp from L* 17 or 20, held above the ramp’s start. Where a separation would pass the eased cap, black is raised by bisection until it fits, hard or through a fifteen-unit knee; where black at full still does not fit, the chromatic inks are scaled down together. Each case’s table is sampled on 65 even nodes and read by straight lines between nodes at 481 lightnesses below its binding point. The error’s size is the mean of the absolute difference between the chroma asked and the chroma printed; the overshoot is the most the printed chroma exceeds the asked anywhere on a ray. A case binds high when its binding point lies above L* 26.

What this does not reach #

Eases between the three. The brown’s best on the ramp from L* 20 lies between a tenth and a fifth; whether its true best is nearer three twentieths or a fifth is read at three points only.

Hue-dependent caps. The census’s table has one cap for every colour; a cap that varies with the colour’s hue is a different table and is not built here.

Other rays. The two dark rays are the census’s; a green or a red ray darkening towards the floor would meet the cap at its own rate.

Still open: whether a cap can follow the hue #

The blue and the brown want different rates of return, and a single cap must choose between them. A separation table can carry a cap that depends on the colour as well as its lightness — easing more steeply where the colour’s chroma path can take it and less where it cannot.

The calculation is the census with the cap’s ramp start set by each case’s hue — from L 17 for colours within thirty degrees of the blue ray’s hue, from L 20 for the rest — eased by three twentieths, every case read by the size of its error.** The prediction is that the blue keeps its best of 5.08 thousandths and the brown its best of 4.74, so that the two rays together err by under 9.9, below any single setting’s 10.17. If the blend between the two ramps at intermediate hues creates new overshoots — colours on neither ray whose chroma path the hue’s ramp does not fit — a hue-dependent cap trades the two rays’ compromise for a third ray’s error, and the census needs a ray between them before the trade can be read.

A dose read as a direction #

The habit here is about reading a factor as a quantity rather than a direction.

The steeper ramp’s success on the signed mean was read as a direction — steeper is better — and the prediction asked only how much ease that better direction wanted. Laid out as rates, the six settings showed that steepness was never a direction: it was a dose of how fast ink returns, beneficial up to a point and harmful past it, with the point set by the colours and different for each ray.

The failure mode is to read an improvement from a larger setting as evidence that the setting should be larger, when the setting is a dose with an optimum. The test is to read it past where it seemed to help and find where it stops helping, for each group it acts on.