Off the grey axis a knee trades hue for chroma

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. Where the separation curves, the straight line misses it, and the colour printed between nodes is not quite the colour asked for — a miss a profile interpolates light found to be a bias rather than a scatter in the forward table, and the inverse table errs dark found running the other way in the table that separates. A run of essays has measured that miss along the grey axis, where it shows as a small cast. A little black ends the cast found black generation ending most of it; the cast returns where black is steep found it coming back wherever black must take over the darkening fast and retire the chromatic inks quickly.

An ink limit takes the shadow’s cast away capped the four inks’ total, as every press must, and found the cap quieting the shadow rather than adding a cast. A knee in the ink limit quiets the shadow approached the cap through a knee — black eased up over five to fifteen units of lightness instead of at a corner — and found the capped shadow’s cast falling with every wider knee, from 11.5 thousandths of a unit of chroma met hard to 3.0 at fifteen units. What a knee changes is how fast the chromatic inks are retired below the cap, and across every strategy, cap and knee that rate ordered the cast.

Its closing section pointed out that a grey is the one place where the chromatic inks carry nothing but the grey. 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 prediction was 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. The alternative named was that a knee moves the hue error into the segments above the cap, where it starts bending black early.

The knee quiets the hue and costs the chroma #

On a deep-blue and a brown ray of dark colours, every light-black strategy and cap that binds, the table’s mean hue error between nodes below the cap falls at every wider knee on both rays: the blue’s from 0.021 ΔEab units met hard to 0.006 through fifteen units, the brown’s from 0.026 to 0.015. A fifteen-unit knee takes the worst segment’s hue error to 0.28 of the hard cap’s on the blue and only to 0.68 on the brown. The hue error above the cap falls too. But below the cap the table’s main error is chroma: it prints the colours duller than asked, by 0.18 ΔEab on the blue and 0.14 on the brown, five to eight times the hue error; and every wider knee deepens that loss, to 0.19 and 0.15, so the table’s total error rises from 0.184 to 0.196 on the blue and from 0.151 to 0.160 on the brown. Met hard, the cap itself takes the blue’s error from 0.344 without a cap to 0.184, and leaves the brown’s chroma loss within a tenth of the uncapped.

• The prediction holds on the blue and fails on the brown. The knee quiets both rays’ hue; it halves the worst segment’s only where it slows the inks enough.
• The alternative does not happen. The hue error does not move above the cap; it falls there as well.
• The question was the wrong error. Off the grey axis the table’s miss is mostly chroma, and the knee buys hue with it.

The solid lines are the chroma lost, the dashed lines the hue error, and they go opposite ways. On the blue, the hue error falls from 21 thousandths to 14, 8 and 6 as the knee widens, while the chroma loss rises from 176 thousandths to 180, 186 and 188. The brown follows the same pattern more gently, its hue error falling from 26 to 15 thousandths and its chroma loss rising from 142 to 152. Neither change is visible to a reader — a ΔE*ab of a few hundredths is far below anything an eye would see in a print — and the census is not claiming that a knee spoils a shadow. It is reading what a knee does to the shape of a table’s error, and on the grey axis that shape was all cast.

Two rays of dark colours #

The rays are chosen to sit inside the press’s gamut at every lightness the census reads, and to carry their colour on different inks. The deep blue lies at the hue of solid cyan and magenta printed together; the brown at the hue of a dark warm mix of magenta and yellow with some cyan. On each, the colour asked for at lightness L* has a chroma of (L* − 14)(80 − L*)/60 — nought at the floor, rising to eighteen at L* 47 and back to nought at L* 80, above which the ray is the grey axis. The table is the grey census’s: seventeen nodes from L* 92 to 14, about five units of lightness apart.

At every node the strategy’s black is laid, and cyan, magenta and yellow are solved to print the colour asked for; under a cap, black is raised until the four inks total the cap, through the knee as the grey census raised it. Every node prints its colour within the solver’s tolerance and its total within the cap. Thirty-three strategy-and-cap combinations bind on each ray, of the nine strategies and four caps the grey census used.

The printed chroma scallops below the asked curve between every pair of nodes: at a node the colour is printed exactly, and between nodes the interpolated coverages print it slightly duller. The scallops are the table’s main error on a saturated ray, and they are there with no cap at all: a chromatic colour’s separation curves in all four inks, and a straight line between two curved separations prints a colour nearer the paper’s or the black’s than the curve does. The cap binds at L* 19.9 for this strategy; below it, the knee deepens the scallops by about a hundredth. The knee’s curve sits only a little below the hard cap’s, which is why the census reads it as a mean over 33 combinations rather than off one picture. Moved to the brown, the handle shows the cap binding later, at L* 23.6, and the two printed curves lying closer still below it: the brown’s inks are retired more gently, and the knee has less to change.

The cap halves the blue’s error #

The grey census’s surprise was that a cap quietens the shadow, and on the blue ray it does so again. Without a cap the strategies print the dark blues between nodes at 0.344 ΔE*ab from the colours asked, and capped hard at 0.184 — the chroma loss falling from 0.325 to 0.176 and the hue error from 0.053 to 0.021. The cap decides the separation outright below where it binds, as the grey census found, and the separation it decides on curves less than the strategy’s own: with black taking a fixed share of a fixed total, the chromatic inks fall more evenly.

On the brown the cap does almost nothing to the error: 0.167 uncapped, 0.151 capped, its chroma loss within a tenth of the uncapped. The brown’s separation is already nearly as even as the cap would make it — presumably because its colour rides on magenta and yellow together, and the strategies’ black takes over from both at about the same rate. A knee, on either ray, then gives back a little of what the cap took away.

The blue’s worst hue halves; the brown’s does not #

The prediction was about the worst segment, where the grey census had found the cast concentrated at the binding point. On the blue a fifteen-unit knee takes the worst segment’s hue error from 0.030 to 0.008, 0.28 of the hard cap’s; on the brown from 0.038 to 0.026, 0.68. The blue meets the prediction for twenty-two of its thirty-three strategy-and-cap combinations, the brown for nine. The worst segment falls at every widening on both rays, so the direction is as predicted throughout; it is the size that divides them.

How fast the inks are retired #

The prediction’s mechanism was the grey census’s: a knee spreads the chromatic inks’ retirement over its width, and a table’s straight lines follow a slow retirement better than a fast one. On each ray, across every strategy, cap and knee, the fastest retirement of a chromatic ink orders the hue error below the cap at a rank correlation above 0.55. And the size of the effect follows the size of the slowing: a fifteen-unit knee keeps 0.59 of the blue’s fastest descent and 0.82 of the brown’s. The blue’s cyan and magenta are retired steeply when the cap binds, and a knee has a steep descent to slow; the brown’s inks are retired gently already, and a knee slows them less.

That is the prediction’s own clause — in proportion to how much it slows the ink that carries the hue — and it explains the split. What it does not explain is the chroma. Retiring the chromatic inks more slowly under a knee means more of the darkening falls to black over a wider range, and black darkens without colour: the table’s straight lines between nodes, now mixing a little more black into each interpolated colour, print it slightly duller. On the grey axis, where the asked colour has no chroma, there was nothing to dull.

At the gamut’s edge a knee asks for ink the colour cannot spare #

The rays sit well inside the press’s gamut, and the obvious next reading is the same census at its edge, where a saturated shadow is as saturated as the press can make it. Run with each ray’s chroma at nine tenths of the most the press prints at that hue and lightness, the census does not get as far as an error. Through a knee, the deep blue at L 17 cannot be printed at all: no amount of black, up to solid, brings the total the colour needs within the 176 per cent the knee allows there.*

The reason is the knee’s own shape. A knee meets the cap by a smooth minimum of the strategy’s total and the cap, and a smooth minimum lies below both across the knee’s width; near the binding point the total allowed is well under the cap itself. Inside the gamut that only means a little more black. At the edge a dark saturated blue needs nearly solid cyan and magenta to hold its chroma, and black cannot stand in for them: adding black darkens the colour without adding blue, so the solver must keep the chromatic inks and the total exceeds the knee’s allowance. Met hard, the cap is a corner the edge colours can reach; through a knee it is a curve that cuts into them.

That is a stronger form of the trade this essay measured inside the gamut. There the knee bought hue at the price of a hundredth of chroma; at the edge, on the colours a profile most needs to print, the price is the colours themselves, unless the knee is narrowed or its curve kept above the cap rather than below it.

What this means for a profile builder #

A knee in the ink limit is a good choice for neutral shadows and a mixed one for saturated ones. Along the grey axis it removes almost all of the cast the cap leaves; in a deep blue it removes most of the hue error too, and in a brown about a third; and in both it adds a little chroma loss, enough that the table’s total error rises. None of these sizes is visible in a print, and a profile builder choosing a knee for the greys gives up nothing a reader would see in the saturated shadows. What the census does say is that a knee is not a free improvement off the axis, and that the right measure of a separation’s quality off the axis is the colour difference, not the hue alone.

The census also adds a reason to cap at all. The ink-limit essay found a cap quieting the grey; here it halves a saturated blue’s error between nodes, because it makes the separation curve less.

How the rays were separated and read #

The press is the census’s four-ink press at D50. A ray’s colour at lightness L* is (L*, C cos h, C sin h) with C = (L* − 14)(80 − L*)/60 below L* 80 and nought above, h the hue of solid cyan with magenta for the blue and of a dark warm mix for the brown. At each of seventeen nodes from L 92 to 14, black is laid by the strategy — none above its start, rising linearly to its ceiling at the floor — and cyan, magenta and yellow are solved by Newton’s method from several starts to print the colour within 0.05 ΔE*ab.* Where the four inks would exceed the cap, black is raised in hundredths and then by bisection until the total fits, through a knee by the grey census’s smooth minimum of the strategy’s own total and the cap, its softness set so the total crosses from one to the other over the knee’s width.

Between nodes the table interpolates the four coverages linearly, and at 241 lightnesses the printed colour is set against the colour asked for: ΔEab, its hue part ΔHab and its chroma part ΔCab. A segment’s error is the mean over the lightnesses it holds; the shadow’s is the mean over the segments below the binding point. The fastest retirement is the largest drop in any chromatic ink’s coverage per unit of L below the cap. The nine strategies are the grey census’s light-black ones, the caps 200 to 260 per cent.

What the census does not reach #

Two rays. A deep blue and a brown are two hues of many, and a dark green or a deep red carries its colour on other pairs of inks; how far the knee slows those inks decides how much it helps.

A one-dimensional table. A real profile interpolates in three dimensions, and its nodes off the grey axis are a lattice rather than a line; the miss between nodes on a ray is the line’s version of that.

One chroma. The rays sit well inside the gamut, and at the edge a knee cannot separate the darkest blues at all; whether a narrower knee, or one whose curve stays above the cap, can, is not computed.

Still open: whether more nodes take the chroma loss away #

Inside the gamut, the table’s main error off the grey axis is the scallop between nodes, and a scallop between straight lines on a curved path shrinks as the nodes close up. If the separation is smooth between nodes, the miss in each segment falls with the square of the spacing, so doubling the nodes should quarter it — and a knee, which deepens the scallops by changing the curvature below the cap, should then cost a quarter as much chroma too.

The calculation is this census on both rays with the table’s nodes at 9, 17, 33 and 65 along the same lightness range, every strategy, cap and knee, the chroma loss and hue error below the cap read at each density. The prediction is that the chroma loss falls by about four times with each doubling, from 0.18 ΔE*ab at seventeen nodes to under 0.05 at 33, and that the knee’s added chroma cost falls with it, leaving the knee’s hue gain as the only effect a dense table sees. If the loss falls more slowly than the square of the spacing, the separation has a corner — at the binding point, where the cap takes over — and only nodes placed on that corner will remove it — or a refit of the nodes against their own error, which a strategy keeps the refit only where black prints found halving an inverse table’s miss wherever the strategy prints black. More nodes quarter the dulling and leave the knee a larger share makes that calculation.

The error a fix is judged by #

The habit here is about carrying a measure from one case to the next.

The grey census measured cast, because a grey’s only possible error between nodes, besides lightness, is colour where there should be none. Carried to a saturated shadow, cast became hue error, and the prediction about it was largely right. But a saturated colour can also be printed too dull, and that error was five to eight times larger and moved the other way.

The failure mode is to judge a fix off its home ground by the error it was designed to remove there. The test is to measure the whole error in the new place, broken into its parts, before asking how the fix moves any one of them. Here that showed the knee doing what it was meant to do with the hue and, quietly, the opposite with the chroma.