Shared nodes belong in the deep 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 draws straight lines between them; a profile is a fit between its nodes set out what that costs. In the shadows a press’s total-ink cap decides the separation, and a dark saturated colour printed between nodes comes out duller than asked: off the grey axis a knee trades hue for chroma found that dulling the table’s main error on a deep-blue and a brown ray of dark colours, and more nodes quarter the dulling and leave the knee a larger share found it falling with the square of the nodes’ spacing.

Three nodes at the cap’s corner cut the dulling by a third then placed nodes rather than adding them evenly. Three nodes one, two and three units of L* below where each colour’s ink reaches the cap cut the blue ray’s hard-cap dulling below the cap from 0.0097 to 0.0064, where three more even nodes cut it only to 0.0089 — because the hard cap leaves a corner in the separation, and a straight line between nodes misses a corner by one segment’s worth of error that a node placed on it removes.

Each colour in that essay had its own three nodes, placed at its own binding point. A profile cannot do that. Its nodes along lightness are planes of the grid, shared by every colour at that lightness, and the binding points of the census’s 66 strategies and caps run from L* 15 to 31. The closing section predicted that one shared placement — three nodes below the median binding point — keeps two thirds of the own-placement gain on the blue and all of it on the brown, because the knee’s bend is a dozen units wide and a node a few units from a colour’s own binding point still splits its steepest segments. If the shared placement gained little, a profile could not exploit the bend without a grid tied to each colour’s cap.

Two thirds on average, all or nothing case by case #

Three nodes placed once, at L 19.4, 18.4 and 17.4 — one to three units below the median binding point of all 66 cases — lower the blue ray’s mean hard-cap dulling below the cap from 0.0097 to 0.0076, keeping 63 per cent of what each case’s own placement gained, and the knee’s from 0.0131 to 0.0104, keeping 52 per cent. On the brown ray they keep 92 per cent of the hard cap’s gain and 85 of the knee’s. Case by case the average splits: every one of the ten cases whose cap binds more than two and a half units below the shared nodes keeps under five per cent of its own gain, and every one of the 29 whose cap binds more than two units above them keeps more than all of it — from 121 to 395 per cent. For every case binding above L 26, the dulling between L* 14 and 20 is more than one and three quarter times the dulling just below its binding point, and for half of them more than two and a half times. Three nodes spread across the binding points’ quartiles keep both rays’ dulling within a twentieth of what three more even nodes leave.**

• The prediction holds for the hard cap on the blue — 63 per cent against its two thirds — and nearly for the brown.
• It is wrong about why. The shared nodes are not catching the colours’ corners from a few units away; for most colours they are catching something else.
• The knee keeps more of its share. On the blue it rises to 37 per cent of the dulling with the shared nodes, where each case’s own nodes had lowered it to 25.

The picture is two populations, not one. To the left, colours whose caps bind below the shared nodes keep almost nothing: the nodes sit above their binding points, in lightnesses where their separations are smooth and their dulling small, and the corner the colour’s own nodes caught is untouched. To the right, colours whose caps bind above the shared nodes keep more than their own placement bought, and the further above, the more — up to four times. Between them, a handful of colours binding within a unit or two of the nodes keep about what the prediction expected. The average of 63 per cent is a blend of zeros and of ratios above one.

A placement that beats a colour’s own placement for that colour says the own placement was aimed at the wrong place. The earlier essay aimed every colour’s nodes at its corner, because that is where the knee’s addition gathered. It never asked where the hard cap’s own dulling lay for a colour whose cap binds high.

Where a high colour’s dulling lies #

For a brown strategy whose black starts at L* 40 and whose cap of 200 per cent binds at L* 31.4, the scallops between nodes are smallest just below the binding point — under a hundredth of a unit at their peaks — and grow towards the bottom of the range, to two hundredths near L* 15. Its own placed nodes split the three small scallops at the top. The shared nodes split three of the largest, near L* 17 to 19. Averaged over everything below the cap, the shared placement’s gain is about four times the own placement’s.

The reason for the growing scallops is in the separation itself. Below a high binding point the cap holds the total at 200 per cent for sixteen units of lightness, and to keep darkening within that total the separation must keep trading chromatic ink for black — a trade that bends the inks continuously, all the way down, rather than once at a corner. Every case that binds high has that long bend under it, and the deep shadow is where it bends hardest.

What black does under a high cap #

The four inks make the bend visible. Above the binding point black is held near the strategy’s own level; below it, with the total fixed, black climbs from about 16 per cent at L* 30 to 76 at L* 14, and the three chromatic inks fall to make room. Black ends at more than twice the strategy’s own ceiling of 30 per cent, because a cap that forbids more total ink leaves black as the only way to darken. Each straight line between nodes cuts across that steady curve, and the curve is steepest in its last few units, where the lightness range ends.

That is the shadow’s own corner, at the floor of the table rather than at the cap, and every colour whose cap binds above it shares it. The earlier essay measured where the knee’s addition sat — just below the binding point — and placed nodes there; the hard cap’s dulling, which those nodes also cut, was mostly elsewhere for every colour whose cap binds high, and nobody had looked.

Where the caps bind #

The blue’s binding points cluster between L* 15 and 25 and the brown’s spread from 16 to 31. The shared placement sits one to three units below the median of all 66, at L* 17.4 to 19.4 — below most of the brown’s binding points and in the middle of the blue’s, which is why the brown keeps nine tenths and the blue two thirds. The ten colours that lose everything — six blues and four browns — are those whose caps bind lowest, between L* 15 and 17, where the shared nodes lie above them.

The quartile placement tries to serve every binding point at once, a node at L* 23.8, one at 20.4 and one at 17.5. It serves none of them: each node is a few units from most colours’ corners and above most colours’ deep shadow, and the result is what three more even nodes give.

Three placements against each other #

On the brown the shared placement is nearly as good as each colour’s own, 0.0076 against 0.0074; on the blue it is between the own placement’s 0.0064 and the even table’s 0.0097. The quartile placement and the 68 even nodes are indistinguishable on both rays, 0.0090 and 0.0089. Concentrating three nodes low in the shadow buys three or four times what spreading them buys, the same lesson the earlier essay drew for a single colour, now for all of them at once.

The knee does less well from sharing. Its addition gathers just below each colour’s binding point, which the shared nodes reach only for the colours that bind near them; on the blue the shared placement keeps half the knee’s own-placement gain and two thirds of the hard cap’s, and the knee’s share of the dulling rises to 37 per cent, above the even table’s 35. Shared nodes remove error the hard cap and the knee have in common — the deep shadow’s — and leave the knee’s own.

The deep shadow was in the earlier numbers #

The census had the numbers to see this before and did not read them that way. More nodes quarter the dulling and leave the knee a larger share reported the hard cap’s dulling as one mean below the cap, falling by four with each doubling of the nodes. A mean over sixteen units of lightness says nothing about which of those units carries the error, and the census never split it by lightness until this essay did; the inverse table errs dark had likewise read its table’s miss as a single signed bias. A mean below the cap was the right measure for comparing tables and the wrong one for placing nodes.

The essays on black itself had also pointed at the region. A black that is not black found the deepest colour a four-ink press can make at L* 2.4 and every printed shadow squeezed into the range above it, and an ink limit takes the shadows’ cast away found the cap forcing black up and the chromatic inks down wherever it is reached. Both describe the same trade the brown strategy’s inks show — black climbing under a held total — and that trade does not stop a few units below the binding point. It continues to the floor of the table, and its curvature is greatest where the lightness range runs out.

What the earlier corner essay measured, then, was the sum of two errors with different homes: a corner error at each colour’s binding point, which its own nodes caught, and a deep-shadow bend shared by every colour that binds high, which they did not. For the blue ray, whose caps bind low, the two homes are close together and one placement catches both. For the brown, whose caps mostly bind high, they are ten or more units apart, and a node at the corner leaves the larger error alone.

A refit as the alternative #

Placing nodes is one way to aim a table’s accuracy; refitting is another. A strategy keeps the refit only where black prints found that adjusting a table’s node values against their own interpolation error — keeping the nodes where they are but moving what they store — halves an inverse table’s miss wherever the strategy prints black. The deep shadow is where black prints most, so a refit should do well there too, and it needs no change to the grid: every colour’s nodes move their stored inks to cut the scallops on either side.

The two remedies are not exclusive. A refit cannot remove a corner that falls between nodes — a straight line through two moved points still cuts across a kink — and placed nodes cannot remove the scallops of every segment a long bend crosses. A profile with three shared nodes in the deep shadow and a refit of all its nodes would attack the corner where the colours share it and the bend everywhere, and the census has not yet combined them.

What this means for a profile builder #

Put shared nodes low in the shadow, not at the caps’ corners. For a profile serving many colours, the dulling that matters is the deep shadow’s, where every high-binding colour’s separation bends hardest; three nodes near L* 17 to 19 cut it for most colours by more than their own corner nodes would. The colours whose caps bind below those nodes — the darkest blues — gain nothing and need their own treatment.

Do not spread the nodes to cover every corner. A placement that tries to serve every binding point serves none of them, and does no more than the same nodes spread evenly.

A knee is a separate question. Its addition lives at each colour’s own corner and a shared grid cannot follow it; a profile that keeps a knee keeps most of its cost.

How the tables were built and read #

The rays, strategies, caps and knee are the earlier essays’: dark colours along a deep-blue and a brown hue from L* 92 to 14, separated by light-black strategies under total-ink caps from 200 to 260 per cent, met hard or through a knee easing the total towards the cap over fifteen units. The shared placements add three nodes to 65 even nodes at fixed lightnesses for every case: one, two and three units below the median binding point of all 66 cases, L* 20.35, or at the binding points’ lower quartile, median and upper quartile. Each table interpolates every ink linearly between nodes and is read at the earlier census’s 481 test lightnesses below each case’s binding point; the chroma lost is the asked colour’s chroma less the printed one’s through the four-ink press model at D50. A case’s kept gain is its dulling on 65 even nodes less its dulling on the shared table, over the same less its dulling with its own placement.

What this does not reach #

Two rays. A real profile serves every hue and chroma; the census’s two dark rays stand in for them, and a palette weighted differently would put its median binding point elsewhere.

Mean dulling below the cap. The measure averages over every lightness below a colour’s binding point, which weights a high-binding colour’s long deep shadow heavily. A measure weighted by how often a picture uses each lightness would rank the placements by different arithmetic.

Lightness only. The tables are one-dimensional; a real grid places whole planes.

Still open: whether the deep shadow’s bend can be removed rather than sampled #

The deep shadow’s dulling comes from black climbing to more than twice the strategy’s ceiling under a held total, and every colour that binds high shares it. A strategy that let the total fall below the cap in the deepest shadows — so that black need not rise so steeply — would straighten that bend at its source.

The calculation is the census’s strategies with the total-ink cap lowered by ten and twenty per cent between L 20 and the floor, the separations re-solved, and the dulling below the cap read on 65 even nodes and with the shared placement.* The prediction is that easing the cap in the deep shadow halves the high-binding colours’ dulling on even nodes, more than the shared placement does, at a cost in the darkest colour the press can reach of a unit or two of L*. If the dulling barely moves, the deep shadow’s bend comes from the strategy’s black curve rather than from the cap, and nodes are the only remedy. Easing the cap reaches the deep shadow and no higher makes that calculation.

An average of two populations #

The habit here is about a mean that hides its parts.

The prediction asked for a single number — how much of the own-placement gain a shared placement keeps — and the census returned one close to what was predicted. Read case by case, the number is an average of colours that keep nothing and colours that keep up to four times their own gain, and neither population behaves as the prediction’s reasoning assumed.

The failure mode is to accept an average that matches a prediction without checking that the cases it averages behave alike. The match was a coincidence of proportions: as many cases far above the nodes as far below, weighted to land near two thirds. The test is to plot the quantity case by case against the variable the reasoning says controls it — here, where each colour’s cap binds against where the shared nodes sit — and to read the shape, not only the mean.