A steeper ramp cures the shadow by overshooting it

A printing press can lay down only so much ink before the paper floods, and a separation table enforces that with a total-ink cap: wherever cyan, magenta, yellow and black would add up past the limit, black is raised and the other three cut until the sum fits. Raising black darkens and greys the colour, so the deepest colours a cap binds come out duller than asked. Easing the cap reaches the deep shadow and no higher let the cap fall by a tenth or a fifth on a straight ramp from L* 20 to the floor at L* 14 — an ink limit relaxed where the shadows are darkest, which presses tolerate — and found a fifth’s ease halving the deep blue’s dulling, and nearly removing the brown ray’s for colours that bind low, while doing nothing above L 20.*

Two later essays moved the ramp. A higher ramp trades one shadow for another started it at L* 26 and 32 and found the colours that bind high trading their upper dulling for deep-shadow dulling, total unchanged, and every group best from L* 20. A bent ramp takes more ink and dulls more spent a quarter of the ease above L* 20 and three quarters below, taking at least as much ink at every lightness as the straight ramp, and found every group dulling more — so the deep shadow’s cure appeared to follow how steeply the cap falls below L* 20, not how much ink is taken.

If steepness is what cures, a ramp steeper still should cure more. The prediction was that straight ramps from L 17 and 18.5 to the floor — the whole fifth spent over the last three or four and a half units — leave the low binders’ and the blue’s dulling lower than the ramp from L 20, and the high binders’ no higher.** If the ramp from 17 dulled more, the cure would need the cap to fall across the whole deep shadow rather than steeply, and L* 20 would be where the deep shadow’s bend begins rather than one choice among many.

Read as before, the prediction holds #

Eased by a fifth, the blue ray’s mean chroma lost falls from 4.89 thousandths on the ramp from L 20 to 3.74 from 18.5 and 2.37 from 17; the brown’s high binders’ from 5.19 to 4.93 and 4.33; the brown’s low binders’ from 0.06 to −3.17 and −8.95. Through the knee the pattern repeats, and at a tenth’s ease every group falls at each step too.*

• On the measure every earlier ramp was ranked by, the prediction holds. The steeper ramp dulls every group less, the high binders included.
• But one number has crossed nought. The brown’s low binders are dulled by less than nothing: on average their prints are more chromatic than the colours asked for.
• A negative dulling is not a cure. A print two hundredths too vivid is as wrong as one two hundredths too dull, and the signed mean counts the first as an improvement.

The figure is the one the earlier essays drew for their ramps, and on it the steeper ramps look like an unqualified success: three groups, three ramps, every bar shorter than the one above it. The brown’s low binders, whose dulling the ramp from L* 20 had already brought to nought, keep going and grow into a bar on the other side of the axis. Nothing in the figure says that bar is a failure.

Read by size, the brown’s error triples #

Read by the mean size of the chroma error — the shortfall where the print is duller and the excess where it is more vivid, both counted as error — the brown’s low binders err by 5.04 thousandths on the ramp from L 20, 8.05 from 18.5 and 15.72 from 17; the high binders by 5.54, 5.58 and 6.73; the blue by 5.21, 4.81 and 5.19. Over the brown ray as a whole the error’s size goes from 5.16 to 7.45 and 13.54 thousandths.* Through the knee the brown’s grows from 5.00 to 5.37 and 10.32.

On the measure that counts every error as an error, the ramp from L* 20 is the best of the three for the brown and as good as any for the blue. The ramp from 18.5 wins on the blue by four tenths of a thousandth and loses on the brown by half again. The ramp from 17, which the signed mean ranked best everywhere, is worst for the brown by a factor of two and a half and no better for the blue.

Even the brown’s high binders, which the prediction expected the steeper ramp to leave no worse, err more from L* 17 on this reading: 6.73 against 5.54. The signed mean had them at 4.33 against 5.19, because some of their darkest colours too had swung into overshoot, and the signed mean counted the swing as a cure.

What an overshoot looks like #

For the brown case whose overshoot is largest, the ramp from L 20 leaves the chroma error within two hundredths of nought all the way to the floor, dipping to −0.014 at the last; the ramp from L 17 leaves the error as the held cap left it down to L* 17, where its ramp begins, and then drives it to −0.070 at the floor — a print seven hundredths of a chroma unit more vivid than the colour asked.** The overshoot is concentrated in the last three units of lightness, exactly where the steep ramp spends its ease.

The cap’s shape explains it. The ramp from L 17 holds a 220 per cent cap until L 17 and then gives up its 44 points over three units of lightness, 14.7 a unit; the ramp from 20 spends the same 44 points over six, 7.3 a unit.** At every lightness above the floor the steep ramp’s cap is at or above the gentle one’s, so the steep ramp takes less extra ink everywhere; but between L* 17 and the floor it releases ink twice as fast. For a colour that binds low, the separation follows the cap closely, and when the cap rises steeply the black that had been raised to fit it falls back steeply too, returning the chromatic inks faster than the colour’s own darkening asks for. The print overshoots in chroma because it has been given its ink back faster than its lightness falls.

So the deep shadow’s cure does not follow steepness without limit. A bent ramp takes more ink and dulls more found a gentler fall below L* 20 dulling more; a steeper one dulls less and then overshoots, and the best fall lies between, where the colour’s own path in chroma and the cap’s release of ink keep pace.

The blue never crosses nought #

The two rays part company on the steep ramp, and the difference says where the overshoot comes from. Eased by a fifth from L 17, the blue ray’s deepest case overshoots by only 0.017 hard and 0.002 through the knee, and its error’s size through the knee falls from 7.98 thousandths on the ramp from 20 to 7.45 from 18.5 and 6.88 from 17 — the steep ramp is the best of the three for the blue through the knee, and within a hundredth of a thousandth of the best hard.* The brown, eased the same way, overshoots by 0.070 hard and 0.046 through the knee.

Every blue case binds at or below L* 26, as do most of the brown’s, so the difference is not where the cap binds. A plausible reading, not traced here, is how far each colour’s own separation stands above the cap as it darkens: easing the cap reaches the deep shadow and no higher found the deep blue still well short of its asked chroma after a fifth’s ease, so returned ink has somewhere to go, while the brown’s low binders had been brought nearly to their target already, so the same returned ink carries them past it. A ramp that suits one ray’s darkest colours can overshoot another’s, and the table has one cap for both.

The knee softens the overshoot on both rays because it does not let a separation meet the cap all at once: within fifteen units of the limit black is raised gradually, and the cap’s fast release on the steep ramp is spread over the knee’s width. More nodes quarter the dulling and leave the knee a larger share found the knee’s own dulling the part that finer sampling could not remove; here it is the part that keeps the steep ramp from overshooting, and a knee and a steep ramp together give the blue its best reading of any ramp yet.

Eased by a tenth, the steep ramp is never worse #

Eased by a tenth, the ramp from L 17 leaves the blue’s error at 5.17 thousandths hard and 9.36 through the knee, against the ramp from 20’s 6.44 and 10.08, and the brown’s at 6.23 and 7.39 against 6.23 and 8.71.* At half the ease the steep ramp beats or ties the gentle one on every case, by up to a fifth, and nowhere overshoots by more than 0.014. The blue’s hard error with a tenth from L* 17 is within half a thousandth of a fifth from L* 20.

That is the useful reading of the prediction. A steeper ramp is a more efficient way to spend an ease — the same improvement for less ink, concentrated where it is needed — and it can afford to spend less. Combined with a fifth’s ease it spends too much in too few units. The two choices interact: how far to ease and over how many units are not separate settings, and the steep ramp’s advantage at a tenth is the evidence that the ease and the ramp should be chosen together.

The steep ramps lighten the deep shadow #

Eased by a fifth, the blue ray’s mean lightness error rises from 7.7 thousandths of a unit of L on the ramp from 20 to 8.9 from 18.5 and 10.8 from 17; the brown’s from 5.7 to 6.5 and 7.9. The floor itself does not move: every ramp prints the deepest colour at L 14.** The steeper ramp gives its colours back their chroma partly by giving back less black, and the print lightens a little where the ramp acts.

The lightness errors are a hundredth of a unit, too small to see, and they matter only as a check on the mechanism: the overshoot in chroma comes with a small excess of lightness, as it would if black were being withdrawn faster than the colour darkens. Three nodes at the cap’s corner cut the dulling by a third and shared nodes belong in the deep shadow found the table’s sampling of the deep shadow mattering as much as its cap; a steep ramp asks more of that sampling, since the cap changes faster between nodes, and the 65 even nodes here, 1.2 units of L* apart, place only two or three of them inside the ramp from L* 17.

What a separation engineer should take from this #

Read a cap’s effect by the size of the error, not its sign. A mean of signed chroma lost ranks a ramp that overshoots as a cure. Count shortfall and excess alike, or report both.

Spend a steep ramp with a small ease. A tenth’s ease on a ramp from L* 17 matches or beats a tenth on the ramp from 20 everywhere; a fifth on the ramp from 17 overshoots the brown’s low binders by up to seven hundredths.

Choose the ease and the ramp together. How far the cap falls and over how many units it falls set one quantity between them — how fast ink is returned in the deep shadow — and either can be traded for the other.

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 — black starting at L* 40 to 60 and reaching 0.3 to 0.4 at the floor — under caps of 200, 220, 240 and 260 per cent, on two dark rays from the press’s blue and brown towards the floor at L* 14. The cap is eased by a tenth or a fifth at the floor on a straight ramp from L* 17, 18.5 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 from L* 92 to 14 and read by straight lines between nodes at 481 lightnesses below the binding point, the print’s chroma and lightness against the colour asked. A case binds high when its binding point lies above L* 26.

What this does not reach #

Eases between a tenth and a fifth. The steep ramp beats the gentle one at a tenth and loses badly at a fifth; where between them the brown’s overshoot begins is not read.

Finer nodes. Sixty-five even nodes leave two or three inside the steepest ramp; a table with nodes concentrated in the last few units of lightness would follow the cap’s fall more closely and might overshoot differently.

Other rays. The blue and brown are the census’s two dark rays; a ray whose colour darkens towards the floor along a different chroma path would meet the cap’s release of ink differently.

Still open: what ease a steep ramp wants #

A ramp from L* 17 eased by a tenth beats the ramp from 20 everywhere, and eased by a fifth overshoots the brown and triples its error. Somewhere between, the steep ramp’s faster return of ink should match the colours’ own fall in chroma.

The calculation is the census on straight ramps from L 17 and from L 20, each eased by three twentieths — 0.15 — at the floor, every case read by the size of its chroma error as here.** The prediction is that the ramp from 17 eased by 0.15 leaves every ray’s error, hard and through the knee, below the ramp from 20 eased by a fifth — the best of the earlier ramps — because at 0.15 the steep ramp still releases ink no faster than the low binders darken, and below the ramp from 20 eased by 0.15 too. If the brown’s low binders already overshoot by more than 0.03 at 0.15 from L* 17, the overshoot is not a matter of how much ease but of how steeply any of it is spent, and the ramp from L* 20 is the steepest a fifth can safely be.

A signed measure that ranked an overshoot first #

The habit here is about what a summary statistic can see.

Every ramp before this one left the brown’s low binders duller than asked, so a mean of the chroma lost was a measure of how far each fell short, and ranking ramps by it was sound. The steepest ramp pushed some of them past the target, and the same mean, unchanged, scored the overshoot as a further improvement. The measure had not been wrong for the earlier ramps; it had been right only because nothing in them changed sign.

The failure mode is to keep a signed summary as a figure of merit after the quantity it summarises can take either sign. The test is to report the summary’s absolute version beside it whenever any case crosses zero, and to rank by that.