Colour Science & Perception
what the eye settles, and what it cannot
Concept

Convexity — where it appears

The property of a set that contains the whole straight segment between any two of its points. A display's gamut is convex and a press's is not, which decides whether a path pushed back inside the set costs little or has to go around something.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Six starts for every held gradient, and where each one lands. Each row is one gradient held inside a gamut, relaxed from six starts: the straight line, the free shortest path, and the straight line bent towards and away from the neutral axis and up and down in lightness. Each dot is how much longer than the free shortest path that start's result is, on a logarithmic scale from a hundredth of a per cent to a thousand; the ring marks the best. The first 10 rows are gradients between colours on a coated CMYK press's boundary whose straight line leaves the press: their best routes cost a median of 0.12% and at most 1.1%, and on 5 of them some start lands at more than twice the free length. The next 3 are the display gradients held inside sRGB, whose best cost at most 1.3% and whose starts spread by at most 4.2%. The last 6 are press gradients that never leave, where no start is trapped.

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.

matching · Gamut Every corner of the press against its own tangent cone, at two samplings of black. For the nine end colours of the margin census, how far each moves per unit of a four-unit margin: as a perfect corner of its own tangent shape would, on the press walked at thirty-three black levels, and on the census's press of nine. At thirty-three every corner sits within seven per cent of its cone; at nine every edge moved a fifth or more further, and the half-black yellow 63 per cent further.

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.

matching · Gamut How sharp each corner of two displays is, in CIELAB and in ΔE₀₀. For the three primaries and three secondaries of sRGB and BT.2020, the move per unit of margin that a perfect corner of each one's tangent shape makes in CIELAB and carried into ΔE₀₀ by its full local matrix. The blue is ×3.35 and ×3.17 in CIELAB, against the press's blue at ×1.25; in ΔE₀₀ it is ×1.42 and ×1.26. ΔE₀₀ brings every corner of both displays under ×1.9.

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.

matching · Gamut What ΔE₀₀'s chroma length alone does to each corner of two displays. For the six corners of sRGB and BT.2020, the move per unit of margin their tangent shapes predict in CIELAB, with ΔE₀₀'s chroma length alone, and with its lengths along lightness, chroma and hue together. The chroma length alone brings every corner but the yellow under ×1.35; it leaves the yellow at ×1.78 and ×1.80, and the full metric at ×1.75 and ×1.78.

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.

matching · Gamut

Named alongside it

The objects these essays reach for when they reach for this one.

CIELABGamutToleranceCIEDE2000ΔEDisplay primariesModelling assumptionWorst caseGamut mappingGradientInk limitInterpolation

All concepts