A knee in the ink limit quiets the shadow

A printer’s profile stores a separation — how much cyan, magenta, yellow and black to print for each colour — at a grid of nodes, and fills in between them by straight lines. Along the grey axis that means a table of seventeen greys from L* 92 to 14, about five units of lightness apart, and every grey between two nodes printed with coverages interpolated from the two. Where the separation curves, the straight line between nodes misses it, and the missed grey picks up a small cast. A run of essays has measured that cast. A little black ends the cast found black generation ending the warm cast a three-ink table leaves along the grey axis; the cast returns where black is steep found a small cast returning below black’s toe, set by how fast black has to take over the darkening and push the chromatic inks back down.

An ink limit takes the shadow’s cast away then capped the total. A press cannot lay down four solid inks — wet ink on wet ink stops drying — so a separation carries a cap, often between 240 and 300 per cent, and where the strategy’s own separation would exceed it black is raised until the four inks total the cap exactly, and the chromatic inks re-solved to print the grey. The cap took the shadow’s cast away on nine light-black strategies under four caps, rather than returning one. And it found a placement effect: the segment where the cap is first reached is quiet when the binding point sits inside it, and loud when the binding point sits within about a quarter of a segment of a node, because then the segment’s chord is the strategy’s alone or the cap’s alone, and the corner where the cap begins falls exactly on a node.

A real cap is rarely met at a corner. Profile-building software approaches its limit through a knee, easing black up over a few units of lightness so that the total arrives at the cap smoothly. The prediction was that a knee about as wide as a segment — five units on this table — would make the binding segment quiet wherever it fell, because every chord would then cross part of the knee; and that a knee much wider than a segment would turn the cap back into a steep, straight rise of black, whose shadow would cast like the heavy-black strategy’s at the same rate of rise.

A knee quiets the shadow, and more as it widens #

Through a five-unit knee the binding segment’s mean cast falls from 8.1 thousandths of a unit of chroma to 3.5, and six of the nine binding segments a hard cap leaves louder than a neighbour stay louder — every one with its binding point within about a quarter of a segment of a node. Wider knees go on quieting it, to 1.7 at ten units and 1.4 at fifteen. The whole capped shadow follows: its mean cast over every segment the cap reaches falls at every widening, from 11.5 thousandths met hard to 7.3, 4.5 and 3.0, against 49.9 for the same strategies uncapped. What a knee changes is how fast the chromatic inks are retired below the cap — the fastest rate falls from 16.8 thousandths of coverage per unit of lightness to 9.8 — and across every strategy, cap and knee that rate orders the shadow’s cast at a rank correlation of 0.73. Every separation stays within its cap and prints its grey.

• A segment-wide knee halves the binding segment’s cast but does not abolish the placement effect. A binding point near a node still makes its segment the loudest of its neighbours.
• A wide knee does not cast like steep black. It casts less than a narrow one, everywhere below the cap.
• The mechanism is the descent, as the earlier essays found it. A knee spreads the chromatic inks’ retirement over its width, and a table’s straight lines between nodes follow a slow retirement better than a fast one.

Each solid line is one cap, averaged over the strategies it binds on; the dashed line is the binding segment, averaged over all thirty-three strategy-and-cap pairs. Every line falls at every step. The capped shadows fall by different amounts — under the 240 per cent cap from 10.1 thousandths met hard to 1.5 at fifteen units, under the 260 from 12.7 to 2.0 — and the binding segment falls fastest of all, by a factor of nearly six. The 200 per cent cap falls least, from 10.7 to 5.0: it binds highest on the ramp, between L* 22 and 29, and a fifteen-unit knee centred there still leaves most of the segments below it at the cap’s own steep descent.

What a knee does to the separation #

A knee is a softened minimum of the strategy’s own total coverage and the cap: far above the cap the separation is the strategy’s own, far below it the total is the cap’s, and between, the total leaves the strategy’s own curve before reaching the cap and arrives at it smoothly, never exceeding it. Met hard, the 220 per cent cap is reached at L* 21.9, and there black turns up at a corner and cyan, which the strategy still had rising, turns down at a corner. Through a ten-unit knee black starts rising a few units earlier and bends up gently, and cyan’s fall begins earlier and is spread over the same width. The knee’s softness is set from the strategy’s own rate of rise at the binding point, so that the total crosses from the strategy’s to the cap’s across the knee’s width; at the bottom of the ramp the knee and the hard cap print the same separation.

The cast lives in the corners. A straight line between two nodes misses a curve by an amount set by how sharply the curve bends between them, and a hard cap puts the sharpest bend in the whole separation at the binding point. A knee takes that bend and spreads it, and every segment it spreads into sees a gentler curve.

Segment by segment #

Read along the ramp for one strategy, the four tables agree far above the cap — the widest knee reaches a little higher and raises two segments near L* 65 by a few per cent, the price of beginning to bend black early — and part from L* 48 down. In the segments the cap reaches, each wider knee casts no more and mostly less, and in the last segment the fifteen-unit knee casts 5.1 thousandths against the hard cap’s 15.9. The segment holding the binding point is the quietest on the ramp in every version: 1.48 thousandths met hard, 0.20 at ten units, 0.21 at fifteen. The loud segments are the ones on either side of it, and those are the ones a knee quiets most — from 15.0 to 2.8 above the binding segment at fifteen units.

Where the binding point falls #

The placement effect is the earlier essay’s sharpest finding, and the knee does not remove it. Met hard, nine binding segments are louder than a neighbour, all with the binding point within a quarter of a segment of a node; through a five-unit knee six are, at positions 0.09 to 0.25 and 0.95 to 0.99 down their segments, and at fifteen units nine again, three of them now mid-segment. What changes is the scale. The loudest binding segment casts 40.9 thousandths met hard, 17.9 through five units and 5.6 through fifteen; a binding segment that is “loud” through a wide knee is louder than its neighbours only because they have become very quiet.

The prediction expected a segment-wide knee to make placement irrelevant, on the reasoning that every chord would cross part of the knee. Part of a knee is not enough: a knee centred at a node still puts the beginning of black’s rise in one segment and its end in the next, and a chord that spans only the beginning or only the end sees a bend the other does not. The knee halves the cost of a badly placed binding point; it does not make the placement free. A profile tool that can move its knee would still do best to centre it inside a segment.

The descent decides #

The earlier essays’ rule was that a returned cast follows how fast the chromatic inks have to fall: below a toe, set by black’s steepness; under a cap, by the cap’s. A knee is a way of choosing that rate. Across all hundred and thirty-two tables the fastest retirement of a chromatic ink orders the capped shadow’s cast at a rank correlation of 0.73, and a wider knee moves the points down and to the left together — mean rates of 16.8, 15.2, 12.3 and 9.8 thousandths of coverage per unit of lightness as the knee widens from nothing to fifteen units.

That is why the prediction’s second half fails. It expected a wide knee to make the cap a steep straight black and steep black to cast. But the cast was never about black’s steepness as such; it was about how fast the chromatic inks are pushed down, and a knee pushes them down more slowly the wider it is. The heavy-black strategy, whose shadow casts 0.9 thousandths below L 28, is quiet for the same reason*: it brings black in early and the chromatic inks fall gently all the way down. A fifteen-unit knee moves a light-black strategy’s capped shadow a long way towards that, to 3.0.

What the cap, the knee and the table each owe the press #

Three choices meet in a profile’s shadow, and the census separates what each does. The cap is the press’s: the fourth ink is not for colour found black buying depth at the bottom and a way to print a full-strength dark on a sheet that could not carry three inks’ worth of wet film, which is the cap’s whole reason. The strategy is the separator’s — how early black arrives and how far it climbs, a choice that exists at all because, as the separation is not unique showed, four inks can print each colour in a whole family of ways — and it decides how high on the ramp a given cap binds. A cap removes part of that freedom below the point it binds, and a knee decides how abruptly. The knee and the table are the profile builder’s, and they are the two this census shows interacting.

A table is exact at its nodes and wrong between them, and a profile interpolates light found the error between nodes to be a bias rather than a scatter, lightening the forward table’s greys; the inverse table errs dark found the inverse, the table that actually separates an image, biased the other way. The cast measured here is the same kind of error, a chord missing a curve, in the colour of the grey rather than its lightness. Whatever a profile builder does to reduce the curvature between nodes reduces all three, and a knee is one of the cheapest such things: it changes no node’s colour, only the coverages that print it, and it changes them only near the cap.

The repair the earlier essays found for the inverse’s bias — refitting the nodes against their own error — keeps its gain only where black prints, as a strategy keeps the refit only where black prints showed. Under a cap, black prints everywhere below the binding point, so the shadow a knee has quieted is also the shadow a refit can act on. The two repairs are complementary: the knee reduces the curvature the table has to follow, and a refit removes what bias the remaining curvature leaves.

How the separations were made #

The press, the grey axis and the table are the earlier censuses’: a coated press model, a seventeen-node table from L* 92 to 14 along the neutral axis, its four coverages interpolated in straight lines between nodes, and each segment’s cast the mean chroma of the greys it prints at fifteen points. The strategies are the ink-limit census’s nine light-black strategies — black from L* 40, 50 or 60, rising to 0.3, 0.4 or 0.5 at the floor — under caps of 200, 220, 240 and 260 per cent. Met hard, black is raised wherever the strategy’s own total exceeds the cap until the total equals it. Through a knee, the allowed total is the smooth minimum −τ ln(e^(−own/τ) + e^(−cap/τ)) of the strategy’s own total and the cap, which never exceeds either; τ is the knee’s width in units of L* times the strategy’s own rate of rise in total coverage at the hard binding point, divided by four, so that the total crosses from the strategy’s to the cap’s across about the knee’s width. Where the allowed total is below the strategy’s own, black is raised by stepping and bisection until cyan, magenta and yellow printing the grey bring the total to it. The fastest retirement is the largest fall in any chromatic ink over half-unit steps from the knee’s top to the floor.

What the census does not reach #

One press, one table. A table with more nodes puts them closer than five units and moves the placement effect with them; a knee’s width would then be measured against a smaller segment.

A symmetric knee. Profile tools ease into a cap in their own ways, some starting black early and some bending the chromatic inks instead; the smooth minimum is one reasonable knee, not every one.

The grey axis only. The cap binds off the axis too, in saturated shadows, where the chromatic inks are not printing a grey and a knee’s effect on hue is a separate question.

Still open: whether a knee off the grey axis keeps a saturated shadow’s hue #

Along the grey axis a knee quiets the cast because it spreads the chromatic inks’ descent. In a saturated dark colour — a deep blue, a dark brown — the cap binds with the chromatic inks carrying the colour, and black replacing them changes the hue the table prints between nodes as well as its chroma.

The calculation is this census along a ray of dark saturated colours — the deep blues and browns of the press’s shadow — with the cap met hard and through the same knees, the error read as the hue and chroma the table prints between nodes against the colours asked for. The prediction is that a knee quiets the hue error as it quiets the grey’s cast, in proportion to how much it slows the retirement of the ink that carries the hue, and that a fifteen-unit knee cuts the worst segment’s hue error by more than half. If a knee instead moves the hue error into the segments above the cap, where it starts bending black early, a profile tool would need a narrower knee for saturated shadows than for greys.

A corner is a rate in disguise #

The habit here is about what a corner in a curve costs.

The earlier essay found the binding segment quiet or loud depending on where a corner fell among the nodes, and named the corner as the thing that mattered. The prediction followed it: spread the corner, and the placement stops mattering; spread it a lot, and the cap becomes a straight line whose steepness casts. Both readings treated the corner and the straight line as different kinds of thing. The census says they are one thing at two widths — a change of rate — and what a table between nodes cannot follow is a rate that changes faster than the nodes are spaced.

The failure mode is to treat a sharp feature and a steep one as different problems. Both are a curve the interpolation cannot keep up with, and the measure that orders them is how fast the thing the table has to follow changes, which here is how fast the chromatic inks are retired.