Every essay — page 35
What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it
SeriesObserversNamed objectsRefutationsSearch
Matching and measuring
CIE XYZ, chromaticity, and gamut — a system for predicting when two lights match, frequently mistaken for a system that predicts how they look.
A press makes the cheap route a search
Holding a gradient inside a display's gamut cost nothing measurable, and the essay that found it credited the gamut's convexity. A press's gamut is not convex: 216 of 630 straight lines between colours on its own hue ring leave it. Holding gradients inside the press still costs little at best, a median of a tenth of a per cent. But the best route depends on where the relaxation starts, the free path is the best start on one gradient of ten, and on half of them some start is trapped at more than twice the free length. On the display no start is trapped.
A projection has no reason to detour
Holding a gradient inside a press by penalising the excursion turned the gamut's price from a number into a search: on five of ten crossing gradients some starting point leaves the relaxation trapped at more than twice the free length, and the spread across six starts runs to 271 per cent of the free path. Replacing the penalty with a projection — take the free step, then move each point to the nearest printable colour — leaves no trap on any gradient and a spread of 0.06 to 2.4 per cent.
A margin costs a press its corners
A gradient held inside a coated press by projection was predicted to tear if the press were first shrunk by a safety margin, at any pinch narrow enough for the shrinking to cut. The press has no such pinch: shrunk by any margin up to thirty CIELAB units it stays in one piece, and projected routes on it are neither trapped nor torn. What a margin costs is concentrated at the corners — the solid yellow moves nearly four times the margin to get inside, like the tip of a 31-degree spike.
A tolerance's own unit rounds the corners
Shrinking a coated press by a margin in CIELAB units made its corners pay several times the margin — the solid yellow nearly four. A print buyer states a margin in ΔE₀₀, and in that unit the corners pay about half as much: six of nine saturated end colours move within a third of the margin, and only the solid yellow is still sharper than a cube's corner. The prediction that the dark corners would stay sharp was wrong; they round too. What the tolerance's unit costs instead is volume — two ΔE₀₀ keep two thirds of the press where two CIELAB units kept four fifths.
The yellow stays sharp because it leans
A safety margin stated in ΔE₀₀ rounds almost every corner of a coated press, and leaves the solid yellow nearly as sharp as a cube's corner. The explanation offered was that the yellow's tongue points up towards lightness as well as out in chroma, so ΔE₀₀ — which shortens chroma at yellow five times and lightness one and a half — shortens it less than a pure chroma spike. Measured from the press's own cells, the tongue leans 23 degrees, less than predicted, and a cone about that axis rescaled by ΔE₀₀ accounts for the yellow's opening to within ten degrees; about a chroma axis it would open twenty-two degrees wider. ΔE₀₀'s own low hue weight at yellow, offered as the alternative, works the other way: it rounds the yellow. And the same account fails at every other corner.
Nine black levels sharpened the press's corners
Three essays measured how far a safety margin pushes the corners of a coated press inward, and found them sharper than anything a smooth solid would have: the solid yellow moving nearly four times the margin, every corner past a cube's. Read by their ink recipes, three of the nine corners are vertices and six are edges, and each has a shape the press model's own derivatives state exactly. Against that shape the edges moved a quarter to two thirds too far. The reason is not the press. It is how the press was built: as three-ink solids at nine levels of black with nothing between them, so that every face black helps to make was scalloped. Walked at thirty-three levels, every corner moves within seven per cent of its own tangent shape, three of the nine are rounder than a cube's corner, and the solid yellow moves 3.2 times the margin, not 3.8.
The tolerance unit opens every edge
On a finely sampled press every corner moves under a CIELAB margin as a perfect corner of its own shape would, the shape read off the press model's derivatives. A print buyer states the margin in ΔE₀₀, and the census in that unit had been explained one corner at a time. Carried into ΔE₀₀'s local metric, the same shapes explain all nine at once: each of the six edges' predicted move lies within a tenth of the margin of the census's, and each vertex's within a seventh. ΔE₀₀ opens the edges from 67–141 degrees to 123–157, nearly faces, because a unit along chroma is three to five CIELAB units at these colours. The solid yellow's cone opens only to 59 degrees and stays the one corner sharper than a cube's. Without its chroma term, ΔE₀₀ would leave every corner as sharp as CIELAB does, or sharper.
The blue corner has nothing to twist
A safety margin's cost at each corner of a coated press is predicted by the corner's own shape, read off its ink recipe and carried into whatever unit the margin is stated in — for nine corners, none of them blue. At a hue near 275 degrees ΔE₀₀ has a rotation term that couples chroma and hue, and the prediction was that leaving it out would miss the press's blue corner by more than a tenth of the margin. At the blue corner the term is strong: ΔE₀₀'s local matrix correlates a chroma step with a hue step by three quarters. And it changes the prediction by three hundredths. The blue corner and its edges are already blunt in CIELAB — none moves more than 1.3 times a margin — and a corner that is nearly a face has nothing for a shear to twist.
A display's blue is sharp where the rotation is not
ΔE₀₀'s rotation term shears chroma into hue near a CIELAB hue of 275 degrees, and at a press's blue corner it had nothing to act on because the corner was blunt. The prediction was that a display's blue — an additive primary — would be a sharp corner, and that there the rotation would change how far a safety margin moves it by more than a tenth. The first half holds: sRGB's blue moves 3.35 times a CIELAB margin and BT.2020's 3.17, against the press's 1.25, and insetting the curved gamut directly confirms it to the second decimal. The second half fails, twice. A display's blue sits at 306 degrees, thirty past the rotation's centre, where the term grips a quarter as hard; and ΔE₀₀ blunts the corner to 1.42 by compressing its chroma sevenfold. Along the edge from blue to cyan the hue does pass through 275 and the grip reaches −0.87 — where the edge is nearly a face. Nowhere near either display's blue does the rotation move a prediction by four hundredths.
The yellow is sharp where the tolerance unit does not compress
A safety margin moves each corner of a display's gamut inward by a multiple of itself that depends on the corner's shape. In CIELAB, sRGB's blue is its sharpest corner and its yellow among its bluntest; carried into ΔE₀₀, the order turns over, and the yellow becomes the sharpest corner of both sRGB and BT.2020. The prediction was that the yellow keeps its sharpness because its faces run along lightness, where ΔE₀₀'s weighting is largest at the yellow's L* of 97. Taking ΔE₀₀ apart length by length says otherwise. Its lightness weighting changes the yellow's move by nothing. Its chroma compression — the term that opens every other corner — does, on its own, bring every other corner of both displays under 1.35 times a margin, and leaves the yellow at 1.78. Of the yellow's three directions only one runs along chroma; at the other corners two or three do, and a metric that compresses chroma pulls those together and flattens the corner between them.
An appearance model flattens the blue by its hue
A safety margin moves a corner of a display's gamut inward by a multiple of itself set by the corner's shape, and the unit the margin is written in decides the shape. In ΔE₀₀ the yellow is the sharpest corner of an sRGB and a BT.2020 display, because its sharpness lies across hue where ΔE₀₀'s chroma compression cannot reach it. CAM16-UCS compresses colourfulness less steeply, and the prediction was that it would leave the blue sharper than the yellow. It leaves the yellow sharpest, at 1.66 times a margin on both displays, and flattens the blue to 1.17 and 1.10 — the bluntest corner of all twelve. Its colourfulness does what the prediction said: on its own it would sharpen the blue past the yellow. Its hue scale undoes that. At a display's blue one degree of CIELAB hue is three and a half degrees of the appearance model's, and a unit that counts hue that finely there spreads the blue's directions apart until the corner is nearly a face.
A quantiser keeps the blue a spike
A gamut margin moves a display's corners inward by multiples set by the unit it is written in. ΔE₀₀ and CAM16-UCS both blunt an sRGB or BT.2020 display's blue, CAM16-UCS to nearly a face, and the prediction was that ΔEITP — the difference built on ICtCp for high-dynamic-range television — would blunt it too. It does the opposite. With the white at 320 cd/m² the blue is ΔEITP's sharpest corner on both displays, moving 3.12 and 3.24 times a margin, about as far as in CIELAB; at the green, cyan, magenta and yellow the three tolerance units agree within a quarter, and at the blue they span a factor of three. ΔEITP's lengths alone would blunt the blue as ΔE₀₀ does. What keeps it sharp is the perceptual quantiser: ICtCp's chroma rows sum to nought, so under a response that compresses like a logarithm, dimming the blue primary is a step in intensity and nothing else. Replace the quantiser by a linear response and the blue falls to 2.15; by a logarithm, it rises to 3.39. A brighter display sharpens it further.