A strategy keeps the refit only where black prints

The inverse table errs dark measured the table a colour engine uses to separate an image — colour in, ink out — along a single ink, and found that it asks for too much ink between its nodes even when every node is exact, because it interpolates in lightness along a curve that bends. Refitting its nodes against its own error, rather than making them exact, halved the error and removed the bias.

That essay ended on the complication a single ink hides. A four-ink inverse does not have one answer to choose from. The separation is not unique showed that four inks and three coordinates leave a one-parameter family of ink combinations for almost every printable colour, and a separation strategy is the rule that picks one member: where black generation begins, how fast black rises, where grey balance is held. A refit that moves the nodes freely can move them off that rule, and a press room chose the rule for reasons — ink cost, drying, how the grey holds up on press — that a colour difference does not see.

The prediction was that a refit constrained to stay on the strategy would keep most of its gain where the strategy is smooth and lose it near the points where the strategy switches, such as the lightness at which black begins. The computation below makes that measurement along the grey axis, and the loss turns up somewhere else.

Where the constrained refit keeps its gain #

A refit held to the separation strategy keeps the whole of the free refit’s gain wherever the strategy prints black, and almost none of it where the strategy prints only cyan, magenta and yellow. Across five strategies, the share of the gain it keeps is the share of the grey ramp printed with black, to within three hundredths. The start of black is the table’s most accurate segment, not its least.

• Above the start, a free refit divides the table’s error by 2.0 to 2.3; a refit held to the strategy by 1.02 to 1.19.
• Below the start, the two refits gain alike — by 1.4 to 2.4 in each segment, within a few hundredths of each other.
• Above the start the error is half a warm cast, a chroma of about 0.05 in the printed grey, and the refit held to the strategy leaves the cast exactly as it was.
• The segment containing the start errs by 0.003 on average, a twentieth of the table’s mean, even though the table there asks for up to 0.039 of black at greys the strategy prints with none.
• The share kept is 0.33, 0.46, 0.62, 0.74 and 0.84 for strategies whose black covers 0.33, 0.46, 0.59, 0.72 and 0.85 of the ramp.

The table and its two refits #

The figure is the error along the grey ramp — a grey asked for at each lightness from 92, just under the paper, to 14, a deep shadow, looked up in a nine-node table, printed, and compared with what was asked for. The strategy begins black at L* 60, a mid-grey, and raises it linearly to nine tenths coverage at the floor.

The table filled exactly from the strategy errs by 0.073 on average and by 0.20 at worst, with the familiar shape of an interpolation error: zero at every node and a hump between each pair. The humps are larger in the deep shadows, where the curves the inks follow bend most, and they vanish in one place, the segment from 62.8 to 53 that contains the start.

Refitted freely, with each node’s four coverages moved to minimise the squared error over the whole ramp, the table errs by 0.036 — half. Refitted with each node sliding along the strategy, so that its four coverages are always the strategy’s own separation of some nearby grey, it errs by 0.050. The mean hides the shape. To the left of the dashed line, on the paper side of the start, the constrained refit lies almost on the filled table and gains nothing; to the right, it lies on the free refit and gains everything.

Two ways to refit a four-ink node #

The two refits differ only in what a node is allowed to be. A free node is any four coverages. A node on the strategy is one number — the lightness at which the strategy is asked for its separation — and its four coverages follow from that number. A node at L* 72.5 may become the strategy’s separation of L* 72.6, but never a separation the strategy would not produce for any grey.

The strategy’s own separation is the upper panel. Above L 60 it is three inks*, rising together from almost nothing at the paper to about a third each at the start, with cyan a little ahead of the other two. At the start black begins and rises linearly; cyan, magenta and yellow bend over and fall, each giving up what black now supplies. The fourth ink is not for colour found black’s whole colorimetric contribution below mid lightness, and this strategy uses it there and nowhere else.

The nodes, as dots, sit on the curves. A table interpolates straight between them. Where a curve is nearly straight between two nodes, the chord and the curve agree; where it bends, they part. At the start, all four curves bend at once, and the lower panel shows what that does to the black. The chord from the last node without black to the first node with it asks for black all the way across, up to 0.039 of coverage at L* 60, exactly where the strategy’s black is still zero. The ordinary table, filled exactly from the strategy, already leaves the strategy at its switch. No refit has moved anything yet.

Where the table errs, and in which direction #

The split between the two sides is not about how much the table errs but in which direction. A printed grey can miss its target in lightness, printing too dark or too light, or in chroma, printing a tinted grey — a cast.

Below the start, the table’s error is all lightness. Its mean cast is between 0.001 and 0.006, and its lightness error runs from 0.04 to 0.19, darkening in the deepest segment. Above the start, it is split. The lightness error there is 0.04 to 0.08, and there is also a cast of about 0.05, the same in all three segments, towards a slightly warm grey: a* and b* both positive by three or four hundredths.

After the refit on the strategy, the lightness error is down to a hundredth or two in every segment. The cast above the start is not: it is 0.054, 0.051 and 0.052 before, and 0.053, 0.051 and 0.052 after.

A node on the strategy cannot correct a cast. Sliding it along the strategy changes which grey it separates, and every separation the strategy produces is, by construction, the grey it was asked for — a colour with no cast at all. So the node can move lightness, by becoming a lighter or darker grey, and it cannot move chroma, because every place it can go prints neutral. Where the table’s error is lightness, the slide fixes it. Where the error is partly a cast, the slide fixes the lightness half and nothing else.

The free refit has no such restriction. It fixes the cast by changing the balance of cyan against magenta and yellow at a node — the node then prints a very slightly tinted grey at its own lightness, tinted against the cast that the interpolation between it and its neighbour will produce. That is a profile interpolates light’s repair arriving in a new dimension: let the table be wrong at its own nodes so that it is right between them. And it is exactly what a strategy holding grey balance forbids.

Why three inks cast and black does not #

The cast comes from where it was predicted to be least likely: the smooth part of the strategy, where nothing switches.

Above the start the grey is a balance of three chromatic inks, and each ink’s coverage follows its own curve against lightness, with its own bend. The table interpolates all three linearly, so between two nodes each ink is off its curve by a slightly different fraction, and a slightly unbalanced mixture of three chromatic inks is a tinted grey. In the upper panel the three curves are close but not parallel, and the chords are off them by up to 0.0015 of coverage in each segment above the start — enough to put a twentieth of a unit of chroma into the grey.

Below the start, the extra ink is black, and the three chromatic inks fall along curves that bend together. An error in black is an error in lightness, because black absorbs nearly evenly across the spectrum; the fourth ink is not for colour found a black-built grey moving only in lightness under press variation while a three-ink grey moved in hue, and the table’s interpolation error is the same phenomenon arriving from arithmetic instead of from the press.

What each refit gains, segment by segment #

In the three segments above the start, the free refit gains 2.3, 2.3 and 2.0 times; the constrained refit 1.02, 1.06 and 1.19. In the four segments below the segment containing the start, the two refits gain 1.8 and 1.8, 1.4 and 1.4, 2.3 and 2.4, 2.3 and 2.3. The constrained refit’s slightly larger gain in the third of those is a descent landing in a slightly different place, not a real advantage; the two differ by less than any tolerance would notice.

In the segment containing the start, both refits lose, from 0.003 to between 0.02 and 0.03. That is not a failure of either. The filled table is almost exact there, both of that segment’s nodes are also the nodes of their neighbours, and a refit that moves them to help the neighbours spends a segment with nothing to lose. The error both refits leave there is still well under the table’s average.

So the loss is not near the switch. It is on the whole of one side of the switch — the side where the strategy prints no black. The prediction had the right answer to “where does a constraint cost” in the wrong place: it expected the cost where the strategy changes, and the cost is where the strategy is simplest, because the constraint forbids the one correction that side needs.

The share kept is the share with black #

If the refit on the strategy gains fully on the black side and nothing on the three-ink side, then the share of the free refit’s gain it keeps should be the share of the ramp on the black side. That is a prediction about five different strategies from a mechanism measured on one.

Every strategy lies within three hundredths of the diagonal. A strategy that starts black late, at L* 40, prints black on a third of the ramp and keeps 0.33 of the gain. One that starts black at L* 80, just below the highlights, prints black on 0.85 of the ramp and keeps 0.84 — the heavy grey-component replacement a press room chooses when it wants its greys to hold on press. Between them the kept share runs 0.46, 0.62 and 0.74 against 0.46, 0.59 and 0.72.

That turns the question the lead asked — will a strategy-bound refit survive a separation strategy? — into a choice the press room already makes for other reasons. A heavy black strategy makes the refit free to take; a light one leaves most of the ramp in three inks, where a refit that respects grey balance has almost nothing to do.

At three table sizes #

A nine-node table along the grey axis is fine; profiles are built at nine to thirty-three steps per axis, and the grey axis passes diagonally through the cube, so its effective spacing varies. The split holds at the coarsest and finest lattices tried.

On the three-ink side, the refit on the strategy gains 1.04 to 1.09 times at every size, and the free refit 1.86 to 2.20. On the black side the two agree at every size — 1.49 and 1.49 at five nodes, 2.06 and 2.09 at nine, 1.99 and 1.97 at seventeen. The absolute errors fall steeply with the lattice, from a mean of 0.28 at five nodes to 0.02 at seventeen — the spacing law a profile is a fit between its nodes found for the forward direction — but the ratio between the two sides does not move. The cast is a fixed fraction of the three-ink error at every spacing, so a finer lattice shrinks it without changing who can fix it.

What a profile maker can do with it #

Three things follow, none of them expensive.

Refit on the strategy wherever the strategy prints black, and nowhere is it cheaper. On that side of the ramp a node sliding along the strategy recovers everything a free refit does, and it leaves every node a separation the press room would have chosen.

Treat the three-ink side separately. There the only correction is a deliberate small imbalance at the nodes, which is a decision about grey balance rather than about interpolation. It is worth about a factor of two on an error that is already small — a mean of a tenth of a unit at a nine-node lattice — and whether it is worth the imbalance is a judgement about how the press holds grey. One thing argues for it: the budget adds two units found a chain’s stages adding to less than their sum because their errors point in different directions, and this one always points the same way, towards a warm grey, so it is the kind of error that gets no such discount.

And stop worrying about the switch. Interpolation across the start of black leaves the strategy in ink and not in colour, because trading a few hundredths of black against the other three inks is close to a metameric substitution — one member of the separation family swapped for its neighbour. The table asks for black the strategy does not print, and the grey comes out right. What that costs is ink, not colour, and it is a cost on a single segment.

How the table was computed #

The press is the spectral four-ink model the separation essays use: the sixteen overprints on coated paper weighted by Demichel’s equations after a mechanical dot gain, combined with a Yule–Nielsen exponent of 1.8, with ink trapping, and read under D50 by the 1931 observer. The target grey at each lightness is L* with a* and b* zero. The strategy sets black at zero above the start and linearly to 0.9 at L* 14 below it, and cyan, magenta and yellow are solved by Newton’s method to hit the target to two thousandths of a unit given that black.

The table holds the strategy’s separation at evenly spaced lightnesses from 92 to 14 — the grey axis of the lattice a profile is a table describes — and interpolates the four coverages linearly between them. Its error is ΔE₀₀ between the target grey and the print of the interpolated coverages, at 241 lightnesses. Both refits are coordinate descents with shrinking steps on the mean squared error over those 241 greys, accepting only moves that lower it; the free refit moves each node’s four coverages, and the refit on the strategy moves each node’s lightness and takes the strategy’s separation there.

What this leaves out #

The grey axis is one line through a three-dimensional table. It is where a separation strategy is defined and where grey balance is held, so it is where the constraint bites hardest; off the axis the three-ink side’s error is a colour error in any case and the distinction between lightness and cast is less clean. A census over the whole table, with the strategy’s rule for chromatic colours stated, is the natural extension and is not made here.

The strategy is one shape: a linear rise of black from a start. Real strategies start black with a smooth toe, which removes the kink and would spread the switch over several segments. That should make the switch segment less exact than it is here and the black side’s refit no less effective, but neither is measured.

The errors are small. A nine-node grey axis errs by a tenth of a unit, and the refits’ gains are gains on that. The finding is about the direction of an error and who can correct it, and it holds at five nodes where the errors are near a unit; but at a fine lattice the whole question is a question about hundredths.

Still open: whether the cast survives a toe #

The one feature of the strategy that made the switch segment exact was its kink — black starting abruptly at one lightness, so that the table’s chord across the kink was a straight trade of black for the other three inks. Real black generation curves start with a toe, a gentle rise over ten or twenty units of lightness, and in that stretch the grey is made of three chromatic inks and a little black together.

The prediction is that the toe region behaves like the three-ink side, not the black side — that a small amount of black does not remove the chromatic inks’ unequal bends, so the cast persists until black carries a substantial share of the grey. If that is right, the share kept should track the share of the ramp where black carries more than some fraction of the absorption, not the share where it is merely present, and the diagonal in the census would shift by the width of the toe. The computation is this one with a smooth start of black at several toe widths, and the quantity worth reporting is the black share at which a segment’s cast falls below a tenth of its lightness error.

What the constraint forbids #

The habit is about reading a constraint by what it removes, not by where it binds.

A constraint on a refit looks as though it will cost most where it changes, because that is where it is visibly doing something — the corner of the strategy, the switch into black. It costs most where it forbids the one correction the error needs. The refit on the strategy can make any node a lighter or darker grey and cannot make it a tinted one, so it can fix an error in lightness anywhere and an error in chroma nowhere, and the question of where it fails became the question of where the table errs in chroma. That was the smooth, three-ink part of the ramp, where nothing about the strategy changes at all.

The failure mode is to locate a constraint’s cost at its most conspicuous feature. The feature that decides the cost is the dimension the constraint removes, and the place it costs is wherever the error happens to point along that dimension. Asking which direction the error has before asking where the constraint bends found the loss on the other side of the ramp from the prediction.