More nodes quarter the dulling and leave the knee a larger share

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. A profile is a fit between its nodes set out what that costs and what more nodes buy: exact at every node, a little wrong between them, and less wrong the closer the nodes. A profile interpolates light found the miss to be a bias rather than a scatter, and the inverse table errs dark found it running the other way in the table that separates.

A run of essays has followed the miss into the shadows, where a press’s total-ink cap decides the separation. A knee in the ink limit quiets the shadow found the cap approached through a knee — black eased up over fifteen units of lightness instead of at a corner — quieting the small cast a table prints between nodes along the grey axis. Off the grey axis a knee trades hue for chroma carried the knee to a deep-blue and a brown ray of dark colours and found the table’s main error there to be dulling: colours printed between nodes with less chroma than asked, five to eight times the hue error. The knee quieted the hue and deepened the dulling by about a hundredth.

That essay’s tables had seventeen nodes along the lightness axis, about five units of L* apart. The dulling is the scallop a straight line makes under a curved path, and for a smooth curve the scallop’s depth falls as the square of the spacing. The prediction was that doubling the nodes quarters the dulling — from 0.18 ΔE*ab at seventeen nodes to under 0.05 at 33 — and quarters the knee’s added cost with it, so that on a dense table the knee’s hue gain is the only effect left. If the dulling fell more slowly than that, some part of the separation bends faster than the nodes can follow.

The dulling quarters; the knee’s part of it does not #

On tables of 9, 17, 33 and 65 nodes, over every light-black strategy and cap that binds on each ray, the chroma lost below where the cap binds falls about fourfold with each doubling from seventeen nodes — capped hard, on the blue from 0.174 ΔE*ab to 0.043 and 0.0097, on the brown from 0.156 to 0.038 and 0.0097 — and as fast with no cap at all. The extra dulling a fifteen-unit knee adds falls more slowly: on the blue from 0.023 to 0.0098 and 0.0034, a share of the hard cap’s dulling that grows from 13 per cent to 23 and 35; on the brown from 8 per cent to 11, where it holds. The knee keeps the hue error under seven tenths of the hard cap’s at every density, but its whole colour difference, an eighth more than the hard cap’s at seventeen nodes on the blue, is a fifth more at 33 and a quarter more at 65.

• The prediction’s first half holds exactly. The dulling falls as the square of the spacing, under 0.05 at 33 nodes.
• Its second half fails. The knee’s added dulling falls more slowly than the dulling it adds to, so its share grows.
• A denser table makes the knee’s trade worse, not irrelevant. Every error shrinks, and the knee’s cost shrinks least.

On logarithmic axes a quantity that falls as the square of the spacing is a straight line of slope minus two, and from seventeen nodes on every curve runs parallel to the dotted reference. The hard cap’s dulling on the blue falls by 4.0 times from seventeen to 33 nodes and 4.4 from 33 to 65; the brown’s by 4.1 and 3.9; uncapped, by 3.6 and 4.0 on the blue, 4.0 and 3.8 on the brown. Between nine nodes and seventeen the curves are shallower, falling by three to four times on the capped tables and less on the uncapped blue: at nine nodes, ten units of lightness apart, the table is too coarse for the separation’s curvature to be the only thing the straight lines miss.

The three curves for each ray sit close together, and that is the picture of the earlier essay’s finding in another form. The cap halved the blue’s dulling at seventeen nodes, as an ink limit takes the shadow’s cast away had found a cap quieting the grey, and from seventeen nodes on it more than halves it at every density; the knee’s line runs just above the hard cap’s. What the logarithmic axis hides is the gap between those two lines growing in proportion, which is the knee’s share and the subject of the rest of this essay.

What the scallops look like as the nodes close up #

At nine nodes the printed chroma sags well below the asked curve between every pair of nodes, by just over a unit at the worst, and touches it only at the nodes themselves. Moved to seventeen nodes the sags are a quarter as deep and twice as many — 0.27 at the worst, then 0.07 at 33 nodes and 0.02 at 65; at 33 and 65 the printed curve is hard to tell from the asked one at this scale. The asked chroma is an arch, nought at the floor and eighteen at L 47, and every separation that prints it bends to follow; a straight line between two separations on a bending path falls inside the bend*, which here means duller, because the arch’s inside is towards grey.

The knee’s printed curve sits a little below the hard cap’s under the binding point at every density. That gap is what the hard cap’s own scallops do not explain. A knee raises black over a wider range below the cap and retires the chromatic inks more slowly there; every node of either table prints its colour exactly, so the gap is entirely in how the two separations bend between nodes.

Which errors fall as the square #

A fourfold fall per doubling is what a smooth separation gives, and the figure lays the four errors against it. The hard cap’s dulling falls by 3.9 to 4.4 times at both doublings on both rays. The knee’s added dulling falls by 2.4 and 2.9 on the blue and by 2.8 and 3.9 on the brown. The hard cap’s hue error on the blue falls by 2.8 and 3.1; on the brown by 3.3 and 4.0. The knee’s own hue error falls by 2.8 to 4.4.

It is worth being precise about what a slower fall means, because the earlier essay’s closing section named it a corner and that is not quite right. A separation with a corner in it — a sudden change of slope at one lightness, where the cap takes over — is missed by a straight line by an amount proportional to the spacing, but only in the one segment holding the corner; spread over the whole shadow that one segment’s miss still falls as the square of the spacing. What falls more slowly is a bend that is sharp but not sudden: one that turns over a span of lightness narrower than the spacing between nodes. While the nodes are coarser than the bend, halving the spacing resolves more of it and the miss falls by less than four; once the nodes are finer than the bend, it falls by four like everything else.

So the hard cap’s hue on the blue, falling by 2.8 and then 3.1, is approaching the square from below — a bend, most likely where the cap takes over, a few units of lightness wide, which 65 nodes, 1.2 units apart, are only beginning to resolve. The knee’s added dulling on the blue is further from it, and on the brown reaches it only at the last doubling.

The knee’s hue gain survives every density #

The knee was introduced for its hue, and that part of the earlier finding holds at every density. On the blue the knee’s hue error is 0.41 of the hard cap’s at nine nodes, 0.22 at seventeen, 0.17 at 33 and 0.19 at 65; on the brown, 0.64, 0.56, 0.47 and 0.43. A denser table lowers both, and the knee’s stays the smaller by a wide margin. The prediction’s hope was that this ratio would be the knee’s only lasting effect, and it is a lasting effect; it is just not the only one.

The hue errors are also the smallest numbers in the census. At 65 nodes the blue’s hard-cap hue error is three thousandths of a unit and the knee’s six ten-thousandths. Neither is anything an eye would see in a print, and the same is true of every dulling here from 33 nodes on. The question is not whether a reader would notice the knee’s cost on a dense table but which way the table’s error budget tips, since a profile builder chooses a knee for its effect on the error a table actually makes.

A denser table leaves the knee a larger share #

Read as a share of the hard cap’s dulling, the knee’s added dulling climbs. On the blue it is 3 per cent at nine nodes, 13 at seventeen, 23 at 33 and 35 at 65; on the brown, 4, 8, 11 and 11. At nine nodes the hard cap’s own scallops are so deep that the knee’s addition is lost in them; as the table densens, the hard cap’s scallops shrink as the square of the spacing and the knee’s addition shrinks more slowly, so the knee becomes a larger part of what is left.

On the blue the knee’s added dulling actually grows from nine nodes to seventeen, from 0.019 to 0.023, before falling. A table with nodes ten units apart does not see a fifteen-unit knee as a bend at all; it samples the knee at one or two nodes and joins them with straight lines, and much of the knee’s effect on the separation falls between nodes where the table cannot represent it. The knee is a feature of the separation narrower than a coarse table can draw, which is the same reason its cost then falls slowly: a table must be fine enough to draw the knee before the knee’s scallops can shrink as the square.

On a dense table the knee costs more in all #

Put the hue and the chroma together and the knee’s trade worsens with density on the blue. Its whole error is 1.04 times the hard cap’s at nine nodes, 1.12 at seventeen, 1.19 at 33 and 1.27 at 65; on the brown, 1.04, 1.07, 1.10 and 1.10. The hue the knee saves is a shrinking part of a shrinking error, and the chroma it costs a growing part, so on the blue the balance moves steadily against it. The brown’s dulling and its knee cost reach the square of the spacing together by 65 nodes, and its ratio settles.

This is the opposite of what the prediction expected a dense table to do. It expected density to dissolve the knee’s chroma cost, leaving a pure hue gain; instead density dissolves the hard cap’s errors faster than the knee’s. A profile builder who has chosen a knee for its quiet greys, and builds a dense table to keep every error small, should expect the saturated shadows to show the knee’s trade more plainly, not less — in proportion, though never at a size a reader would see.

What this means for a profile builder #

More nodes are always better: every error here falls with every doubling, and the dulling the earlier essay found as a saturated shadow’s main error falls fourfold. A table with 33 nodes along its lightness axis prints the dark blues and browns to about five hundredths of a unit on average; 65 nodes, to about a hundredth. Those are numbers from a one-dimensional table along a ray; a profile’s three-dimensional lattice pays for each doubling eightfold in patches, which is why a profile is a fit between its nodes found accuracy so dearly bought.

A knee in the ink limit is a decision about greys, and its cost in saturated shadows does not go away with density. Along the grey axis the knee removes a cast; in the saturated shadows it trades hue for chroma, and the trade tips further towards chroma as the table densens. A profile builder can approach the cap through a knee for the neutral axis and through a narrower one, or hard, for the most saturated shadows, as the earlier essay suggested for other reasons, and the denser the table the stronger that case becomes. The cast returns where black is steep found the grey’s cast following how fast black takes over; here, too, the bend in the separation decides the error.

How the tables were built and read #

The press, rays, strategies, caps and knees are the earlier census’s: a four-ink press at D50, a deep-blue and a brown ray whose chroma at lightness L* is (L* − 14)(80 − L*)/60, nine light-black strategies, caps of 200 to 260 per cent, and knees by a smooth minimum of the strategy’s total and the cap. Each table holds 9, 17, 33 or 65 nodes evenly spaced from L 92 to 14, at each of which the separation is solved exactly as the census solves it; between nodes the four coverages are interpolated linearly, and at 481 lightnesses below where the cap binds the printed colour is set against the colour asked for.* The dulling is the mean chroma lost, the hue error the mean ΔHab and the whole error the mean ΔEab, each averaged over the 33 strategy-and-cap combinations that bind on each ray.

The average is over every lightness below the binding point rather than over whole segments, as the earlier census took it, so the seventeen-node numbers differ from its by a little: the blue’s hard-cap dulling is 0.174 here against 0.176, and its hue error 0.027 against 0.021, because the partial segment at the binding point, where the hue error is largest, counts by its length rather than as a whole. The uncapped table is read over the capped table’s lightnesses, so the two are compared over the same shadow.

What this does not reach #

One knee. Only the fifteen-unit knee is solved at every density; a narrower knee is a sharper bend and would, on this reading, lag the square for longer.

Rays, not a lattice. A profile interpolates in three dimensions, and its nodes off the grey axis form a lattice whose cells a ray crosses at an angle; the scallops along a ray are the line’s version of that.

Where the bend is. The slower fall locates a bend narrower than the spacing but not its place; whether it is the binding point, an ink reaching nought, or black reaching its ceiling is not read here.

Still open: where the knee’s bend sits #

The knee’s added dulling and the hard cap’s hue error on the blue both fall more slowly than the square, and both should reach it once the nodes are finer than whatever bend they are missing. Reading the error segment by segment would show where.

The calculation is the 65-node table on the blue, every strategy and cap, with each segment’s dulling and hue error read separately for the hard cap and the knee, and each segment’s error set against the separation’s second difference there — how sharply each ink bends — and against the lightness where the cap binds, where a chromatic ink reaches nought and where black reaches its ceiling. The prediction is that the knee’s added dulling sits in the two or three segments just below the binding point, where the smooth minimum turns from the strategy’s total to the cap’s, and that a table with three extra nodes placed there takes the knee’s share back under ten per cent. If the added dulling is spread over the whole shadow, the knee changes the separation’s curvature everywhere below the cap, and no placement of nodes recovers it more cheaply than density — though 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 where black prints, might. Three nodes at the cap’s corner cut the dulling by a third makes that calculation.

Density is a test, not only a cure #

The habit here is about what refining a discretisation can tell.

The earlier essay found a knee costing a hundredth of chroma on a seventeen-node table and suggested it might be an artefact of coarse sampling — a scallop that more nodes would remove. Refining the table was the right test, and it gave two answers at once: the hard cap’s dulling fell exactly as the theory of straight lines on curves says it should, and the knee’s did not. The theory’s prediction became a ruler against which the knee’s error could be measured, and the knee’s error fell short of it.

The failure mode is to refine a grid until the numbers are small and declare the question answered. The test is to compare how each number falls with how it should fall if the thing being sampled were smooth. A quantity falling as the theory says is a sampling error; one falling more slowly is a feature of the answer that the sampling has not yet reached, and it is worth finding.