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Every essay — page 42

Page 42 of 42, continuing through the fields in the same order.

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What it takes to deliver it

A colour that stays where it was measured is nobody's problem. This is the apparatus that carries one to somebody else — ink on a sheet, a profile, a rendering intent, a proof, and a room — and every stage of it is a map that loses something, chosen deliberately.

The error between a profile's nodes is a bias, not a scatter. Two quantities against the lattice size: the mean colour difference between the nodes, and the mean signed lightness error. If the interpolation erred in both directions the second would be near zero while the first was not. They lie on each other — 0.136 against 0.133 at a nine-step lattice — so the whole of what a profile does between its patches is to lighten. It errs light because a press's response is convex in ink coverage: the first drop of ink removes more light than the last, and a straight line between two points on a convex curve lies above it. At three steps 400 of 400 samples err light.

A profile interpolates light

A profile is exact at its patches and wrong between them, and how wrong has been measured twice. Which way it is wrong has not. At a nine-step lattice the mean signed lightness error between the nodes is +0.136 against a mean colour difference of 0.133 — the error is not a scatter but a bias, and it lightens. The repair costs nothing measured and is forbidden by how a profile is checked: let the table be wrong at its own patches.

7 figures
One instrument, one uncertainty, six places on the scale. What an absolute uncertainty of 0.001 in measured reflectance is worth in lightness, at six levels from a four-colour solid to the paper. Nothing about the instrument changes between the rows: what changes is the slope of the lightness function, which is a straight line of 903 units per unit of luminance factor below a luminance of 0.0089 and a cube root above it. At the solid the uncertainty is 0.903 lightness units and at the paper 0.043 — 21 times as much, for the same measurement.

The scale hangs from one measurement

Black point compensation is a straight line between two blacks, and the destination's is a measurement of one patch at the darkest place a spectrophotometer is ever asked to read. The lightness scale's slope is 903 units per unit of luminance factor there and 43 at the paper, so a thousandth of a reflectance is worth nine tenths of a lightness unit at the black and four hundredths at the white. That one number moves a mid grey by nearly a quarter of a delivery tolerance, and no specification names it.

7 figures
Two tables, two directions of error: 5-node ramps of four inks. For each ink, the mean signed lightness error between the nodes: of the ordinary forward table, which errs light, and of four ways to fill the inverse table, all of which err dark except the one refitted against its own objective. On cyan the forward table errs by +0.206; an inverse filled from the press by -0.317; one inverted from the ordinary table by -0.554, because it inherits the forward table's error on top of its own; one inverted from the refitted table by -0.367; and one refitted in its own right by -0.002.

The inverse table errs dark

A profile's forward table predicts a print lighter than the press makes, and refitting its nodes removes the bias. The table a colour engine actually uses to separate an image is the other one — the inverse, from colour to ink — and it errs the opposite way: filled exactly from the press it asks for too much ink and prints dark, and built by inverting the ordinary forward table it prints darker still. Inverting the refitted forward table removes the inherited part and leaves the inverse's own. The round trip through both tables improves only when each is refitted against its own error, and then the two are no longer each other's inverse.

5 figures
Where an unsharp mask errs, per pixel and as seen: print, 40 cm. The upper-left corner of a patch of skin against its shadow after an unsharp mask, as two maps of the colour difference between the result taken on stored values and taken on light. Left, pixel by pixel: the corner peaks at 16.8 and the middle of the edge at 16.6. Right, after the eye's three spatial channels at a print at 40 cm: the corner is seen at 17.1 and the edge at 7.1, a ratio of 2.40. Darker is larger, on one scale for both maps.

The eye counts a corner's error, not its peak

A Lanczos-magnified patch errs a fifth more at its corner than along its edges, pixel by pixel, and an unsharp mask errs almost exactly as much at its corner as along its edges. Filtered by the eye over the plane rather than along a line, the two swap: the magnified corner is seen exactly as its edge is, and on a printed page the sharpened corner is seen at 2.4 times its edge. What decides it is whether the error changes sign. Ringing averages away and a one-sided halo does not, and a corner is where two edges' halos land on the same patch of retina.

5 figures
Where a sharpened E errs, per pixel and as seen: black type on paper, 10 point. The letter E at 10 point on a 300-dot-an-inch page, black type on paper, sharpened by an unsharp mask on stored values and on light; the two maps show the colour difference between the results, pixel by pixel on the left and after the eye's three spatial channels at 40 centimetres on the right, on one scale. The thin line is the letter's outline and the dashed circles mark two pixels round each corner — 8 outer corners and stroke ends, 4 inside angles. The circled zones are 31% of the outline and hold 28% of the per-pixel error and 25% of the seen error.

Sharpened type errs on its dark side

A sharpened square hides its halo along its edges and shows it at its corners, so a sharpened page of type was predicted to show its error at the corners of its letters, several times out of proportion to their length. Black type does the opposite: its outer corners and stroke ends hold between two thirds and nine tenths of their share. As the eye leaves it, the error lies on the dark side of every edge, and a corner holds as much of it as it has dark ground around it — a quarter of a disc where a black corner points out into paper, three quarters where black wraps round a pale one.

7 figures
A four-ink inverse table's error along the grey axis, 9 nodes, black from L* 60. The colour difference between a grey asked for and the grey printed, from L 92 to 14, for a 9-node inverse table whose separation strategy begins black generation at L 60 (dashed line). Three tables: filled exactly from the strategy, mean 0.073; refitted with each node sliding along the strategy, 0.050; refitted freely, 0.036. To the left of the dashed line the strategy-bound refit lies close to the filled table; to the right it lies close to the free refit. Every table is exact at its nodes, marked on the axis.

A strategy keeps the refit only where black prints

Refitting an inverse table's nodes against their own error halves it, and in a four-ink table the refit has to leave the separation strategy's choice of black alone. Held to the strategy, it keeps the whole of its gain wherever the strategy prints black and almost none of it where the strategy does not — and the share it keeps equals the share of the grey ramp printed with black, to within three hundredths. The start of black generation, predicted to be where the refit would fail, is where the table needs it least.

6 figures
Where a sharpened serif letter's error is seen, in black type and reversed, 10 point. The serifed letter E at 10 point, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres, in black type on paper and in paper-coloured type on black, on one scale. Dashed circles mark the serifs' corners: their tips and their brackets against the stems. In black type the error lies along the strokes' dark side and the serifs' corners hold little of it; reversed, it gathers round the serif tips, which hold 11 per cent of the letter's seen error against 3 in black type.

A serif moves the error and adds little

When a scanned page is sharpened on its stored values, the error the eye sees lies on the dark side of every edge, so black sans-serif type does not gather it at its corners. The worry was that a serif changes that: every stroke end gains a bar and every bar an inside angle, the configuration where large letters did gather error. It does not. Black serif type's brackets hold three quarters of their share of the error and its serif tips half. Reversed, the serif tips hold two and a half times theirs. Across both, serifs lengthen the outline by a tenth and the error by a few per cent — they move it, and in black type away from the corners.

6 figures
The table's cast along the grey axis, with and without a toe. A seventeen-node four-ink inverse table filled from a strategy that begins black at L* 60: the mean chroma of the grey it prints, segment by segment. Above the start every strategy is the same three inks and the cast is about 0.0126. Below it the cast falls under a tenth of that 2, 7, 12, 12 units below the start for the kinked start and toes of 10, 20 and 30 — later with a wider toe, but by less than the toe's width, and nowhere near where black takes over.

A little black ends the cast

A four-ink inverse table errs by a warm cast along the grey axis wherever the separation strategy prints only three inks, and almost only in lightness wherever it prints black. Real strategies start black gently, and the gentle start was expected to keep the cast until black carried much of the grey. It does not. The cast falls to a tenth of its three-ink size once black supplies a tenth to a sixth of the grey's darkening, at every toe from nothing to thirty units wide — because inside the toe the table's chord runs along the trade of black against the other three inks, which errs in lightness. What the toe does cost is the segment where black begins, which goes from the table's best to its worst.

6 figures
Where a sharpened E with hairline serifs is seen to err, 48 point. The capital E with hairline serifs, a tenth of a stem thick, set at 48 point on a 300-dot page, sharpened on stored values rather than on light, as the eye's three spatial channels leave the difference at forty centimetres: black type on the left, reversed on the right, on one scale. Circles mark the serif tips and the junctions where the middle arm meets the stem. In black type the error follows the ink and gathers in the junctions' inside angles, and the serif tips hold 2 per cent of it; reversed, it spreads into the dark ground and the tips hold 6 per cent.

A hairline spills its error onto the paper

Sharpening a scanned page on its stored values errs on the dark side of every edge, which is why black type's serifs hold less than their share of the error at text sizes. A hairline serif, a tenth of a stem, was expected to change that at display sizes: its dark side is so thin that the halo would wrap its corners. It does not. From 12 to 48 point, black type's hairline tips hold a third to three fifths of their share, less than slab serifs' tips, and the share falls as the letter grows. What gathers at display sizes is the junction of bar and stem, with or without serifs — and where the scan lands the letter against the pixels moves a tip's share more than its size does.

6 figures
The cast segment by segment under three toes of width 20. The seventeen-node table's cast from where black begins to the floor, for the three toes and the kinked start. Above the toe's end at L 40 the shapes differ; below it the parabola and the smoothstep draw one line. Between the toe's end and L 20 the returned cast averages 1.94, 0.94, 1.94 thousandths for the parabola, the eased cubic and the smoothstep.

The cast returns where black is steep

Below a black-generation toe a four-ink table's grey picks up a small cast again, up to a third of the three-ink cast at the widest toe, and the explanation offered was the jump in black's curvature where the toe ends. It is not. A toe whose curvature is nothing at its end returns exactly the same cast, segment for segment, and a kink put into straight black casts in its own segment once in twelve tries. What the returned cast follows is how steep black has to be once the toe is done — how far the chromatic inks climbed while black was held back, and how fast they then have to come down.

6 figures