Three nodes at the cap's corner cut the dulling by a third

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 the cost: exact at every node, a little wrong between them, and less wrong the closer the nodes. In the shadows a press’s total-ink cap decides the separation, and a knee in the ink limit quiets the shadow found the cap approached through a knee — the ink total eased towards the cap over fifteen units of lightness instead of meeting it at a corner — quieting the cast a table prints between nodes along the grey axis.

Off the grey axis the knee has a price. Off the grey axis a knee trades hue for chroma found a table’s main error on dark saturated colours to be dulling — colours printed with less chroma than asked — and the knee deepening it. More nodes quarter the dulling and leave the knee a larger share then densified the table from 9 nodes to 65 and found the dulling falling about fourfold with each doubling, as a smooth curve’s scallops between straight lines should, but the knee’s addition falling more slowly, so that on 65 nodes it was 35 per cent of the hard cap’s dulling on a deep-blue ray.

The closing section predicted where the knee’s addition lives: in the two or three segments just below the lightness where the cap binds, where the separation turns from following the strategy’s own total to following the cap’s; and that a table with three extra nodes placed there takes the knee’s share back under a tenth. If instead the addition were spread over the whole shadow, the knee would change the separation’s curvature everywhere below the cap, and no placement of nodes would recover it more cheaply than density.

Placed nodes cut both, and the knee keeps a quarter #

On the blue ray, over every strategy and cap that binds, the knee’s added dulling on 65 nodes is largest one unit of L below where the cap binds, 0.0046 against the hard cap’s own 0.0086 there, and four fifths of it lies within seven units. Three nodes placed one, two and three units below the binding point lower the hard cap’s mean dulling below the bind from 0.0097 to 0.0064 and the knee’s from 0.0131 to 0.0080; three more evenly spaced nodes, 68 in all, lower them only to 0.0089 and 0.0120. The knee’s share falls from 34.7 per cent to 24.8 — not under the predicted tenth. A single node placed at the binding point lowers the hard cap’s dulling to 0.0085 and leaves the knee’s at 0.0120, so the knee’s share rises to 40.9 per cent. On the brown ray the knee’s share is 11.4 per cent on even nodes and 8.8 with three placed.*

• The prediction is right about where. The addition gathers just below the bind and fades within eight units.
• It is wrong about what placed nodes do. They help the hard cap almost as much as the knee, and the knee’s share falls by a third, not three quarters.
• The alternative is wrong too. The addition is not spread over the whole shadow; placed nodes recover far more than density does.

The two curves are the whole mechanism. Met hard, the ink total rises to the cap and stops in a corner: above the binding point it follows the strategy’s own total, below it the cap’s flat 220 per cent. Through the knee it bends instead, leaving the strategy’s total several units above the binding point and reaching the cap only gradually, still a few per cent short six units below it. A table that draws straight lines between nodes misses a corner by the scallop of one segment — the one the corner falls in — and misses a bend by a smaller scallop in each of the several segments the bend spans.

Where the addition gathers #

The hard cap’s dulling runs at about a hundredth of a unit at every depth below the bind — the even scalloping of a smooth separation sampled every 1.2 units of lightness. The knee’s addition is different: 0.0016 at the binding point itself, 0.0046 one unit below, then 0.0040, 0.0036, 0.0022, and under a thousandth from seven units down. It is where the prediction put it, and wider: not two or three segments but five or six, the width of the bend in the ink total. The binding point’s own unit carries little, because the knee has already begun easing the ink above it and the corner the hard cap has there is the hard cap’s error, not the knee’s.

On the brown ray the knee’s addition is under half the blue’s and spread as wide — under 0.002 at its peak — while the hard cap’s own dulling is about the same, which is why the knee’s share there was already near a tenth. Why the brown’s bend costs less between nodes the census does not separate; the ink totals along the two rays approach the cap differently, and a gentler approach is one a straight line between nodes follows more closely.

Three in the right place against three anywhere #

Three more nodes spread evenly — 68 instead of 65 — shorten every segment by a twentieth and lower the dulling by about a tenth, as a square law says they should. Three nodes placed one, two and three units below the binding point lower the hard cap’s dulling by a third and the knee’s by nearly two fifths. They are worth three or four times as much as the same nodes spread evenly, because they split the segments where both curves bend hardest: the hard cap’s corner sits in the first of them and the knee’s bend is steepest across the first three.

That is the alternative refuted, and it agrees with what a profile interpolates light found of tables generally: their errors are biases with a shape, and a shape can be aimed at. The knee’s addition is not spread through the shadow; nodes placed where it gathers recover much of it, and more cheaply than density — three nodes placed do what about fifteen more even ones would, by the square law’s arithmetic. The prediction’s mistake was about the hard cap. It assumed the placed nodes would leave the hard cap’s dulling where it was and remove the knee’s, but the hard cap’s corner sits exactly where the nodes went.

The knee’s share falls by a third #

Read as a share, the placed nodes move the knee from 34.7 per cent of the hard cap’s dulling to 24.8 on the blue, and from 11.4 to 8.8 on the brown. The even 68-node table leaves both shares exactly where 65 nodes had them, which is the square law again: an even refinement scales every scallop by the same factor. A single node placed at the binding point does the reverse of what the prediction hoped for the three: it catches the hard cap’s corner, lowering its dulling by an eighth, misses the knee’s bend almost entirely, and raises the knee’s share to 40.9 per cent. Whatever helps the hard cap more than the knee raises the knee’s share, and the corner is the hard cap’s.

What one strategy’s scallops look like #

For one strategy the scallops are visible one by one. Every segment between nodes loses chroma in an arch, touching zero at each node, and the knee’s arches are taller than the hard cap’s in the first three segments below the bind and the same height below that. The placed nodes split the first three arches into six smaller ones, and the knee’s curve there drops beneath the hard cap’s on even nodes. Below the placed nodes the even table resumes, and so do its arches, with nothing to tell the knee’s from the hard cap’s.

A corner and a bend, under a straight line #

Why one node catches a corner and three cannot straighten a bend is a matter of where a straight line’s error comes from. Between two nodes the table draws a straight line in each ink, and the printed colour misses the asked one by however far the true separation curves away from that line. For a smooth curve the miss in a segment grows with the segment’s length squared times the curve’s bending, which is why doubling the nodes quarters the dulling everywhere a separation is smooth. A corner is not smooth: all its bending happens at one lightness, and the segment that contains it misses by an amount that grows only with its length, not its square — the one segment that straddles the corner carries an error out of proportion to the rest.

That is the hard cap’s error at the binding point. Averaged over the whole shadow its share still shrinks as the square of the spacing — only one segment ever holds the corner, and that segment shortens with the rest — so a density census cannot single it out; but it is the one scallop that does not shrink in shape as its neighbours do, and a node placed at the corner removes it at once: on either side of the node the separation is smooth again, and the neighbouring segments fall back to the square law. The single node at the binding point did exactly this, cutting the hard cap’s dulling by an eighth with one node.

A knee replaces the corner with a bend — the same total change of slope spread over a dozen units of lightness. Its error is the square-law kind everywhere, a little larger in each of the dozen units than a smooth stretch with no bend would give. One node removes one segment’s share of that; three remove three segments’ shares; the rest of the bend’s segments keep theirs. Placed nodes cut the knee’s error by roughly the fraction of its bend they cover, and three nodes against a bend twelve units wide cover the steepest third.

What this does to the knee’s case #

The knee entered the profile-table thread as a fix for the grey axis. An ink limit takes the shadows’ cast away and the knee essay after it found that a table printing greys near the cap’s corner picks up a small cast, because the corner sits between nodes and the straight lines on either side of it are drawn through inks that turn sharply there. A knee smooths the turn and the cast goes quiet. Seen through this essay, that cast was the corner’s one-segment error, and the knee’s remedy was to trade one large local error for several small ones spread over the bend.

A placed node is a different remedy for the same error: keep the corner and put a node on it. It costs nothing off the grey axis, where the knee costs dulling, and it does on the grey axis what the knee did. The earlier refit essays offer a third remedy — a strategy keeps the refit only where black prints found refitting a table’s nodes against their own error halving its miss — and a refit, unlike a placement, can move nodes the builder did not think to place. Which of the three a profile builder should prefer depends on the format’s freedom: a fixed grid allows only the knee or the refit, a format with uneven spacing allows the placement, and only the placement costs nothing anywhere.

What this means for a profile builder #

Place nodes where the separation bends, not evenly. Three nodes just below where a dark colour’s ink reaches the cap buy what fifteen spread evenly would; a profile format that allows uneven node spacing along lightness — as one-dimensional curves in a profile do — can use them.

Placed nodes help the hard cap more than they help the knee. The knee’s case was that it quiets the cast a table prints at the cap’s corner. Put nodes at the corner and the corner’s error falls on its own, and the knee’s remaining advantage on the grey axis shrinks while its cost off the axis stays a quarter of the dulling. A builder who can place nodes may not need the knee at all; one who cannot will find the knee’s cost growing as the table densifies.

On a brown, none of this matters much. The knee’s share was a tenth there and stays under one.

How the tables were built and read #

The rays, strategies and separations are the earlier census’s: dark colours along a deep-blue and a brown hue from L* 92 to 14, separated by light-black strategies — black starting at L* 50, 60 or 70 and rising to at most 40, 60 or 80 per cent — under total-ink caps from 200 to 260 per cent, met hard or through a knee that eases the ink total towards the cap over fifteen units of lightness. Each table holds the separation at its nodes and interpolates every ink linearly between them; 65 and 68 even nodes span L* 92 to 14 evenly, and the placed nodes add three at one, two and three units of L* below where each case’s cap binds, or one at the binding point. Each table is read at the 481 test lightnesses below the binding point, its interpolated inks printed through the census’s four-ink press model at D50, and the chroma lost is the asked colour’s chroma less the printed one’s. The 65-node readings reproduce the node census’s to a part in a billion.

What this does not reach #

One placement. The three nodes go one, two and three units below the bind. Nodes placed to the knee’s own bend — spread over its twelve units, or chosen to minimise the error — would do better, and how much better is not searched.

Lightness only. The tables here are one-dimensional, along a ray, and they are the separating direction’s tables; the inverse table errs dark found that direction’s errors running opposite to the forward table’s, so a placement tuned here is not a placement for the forward table. A real profile’s nodes are a three-dimensional grid, and a node placed along lightness is a whole plane of nodes placed.

The binding point known. Every case’s placed nodes are put where its own cap binds. A profile serves many colours whose caps bind at different lightnesses, and one set of placed nodes must serve them all.

Still open: whether placed nodes can serve every colour at once #

The cap binds at a different lightness for every hue and chroma — earlier for a deep blue, later for a brown — and a profile’s nodes along lightness are shared by every colour at that lightness. Three nodes placed for the blue’s binding point sit in the wrong place for the brown’s.

The calculation is a table whose three extra nodes are placed once, at the lightnesses where the cap binds for the median case across both rays, every strategy and cap read on it, and the dulling compared with each case’s own placement and with even nodes. The prediction is that a shared placement keeps two thirds of the own-placement gain on the blue and all of it on the brown, because the knee’s bend spans a dozen units of lightness and a node a few units from a case’s own binding point still splits its steepest segments. If the shared placement gains little, a profile cannot exploit the bend without a node grid tied to each colour’s cap, and density is the only general cure. Shared nodes belong in the deep shadow makes that calculation.

The corner belonged to the baseline #

The habit here is about what a comparison’s baseline is doing while the comparison is being made.

The knee’s share was a ratio: its addition over the hard cap’s dulling. The prediction treated the denominator as fixed and asked how to shrink the numerator, and placed nodes where the numerator was largest. The denominator had its own error there — the hard cap’s corner — and the placed nodes cut it too, by nearly as much.

The failure mode is to optimise a ratio by attacking its numerator without checking what the attack does to its denominator. The knee’s share fell by a third, and both of the quantities it divides fell by a third or more. The test is to report both quantities alongside any share — here, the hard cap’s and the knee’s dulling for every layout — so that a share’s movement can be read for what it is.