Gamut mapping — where it appears
Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.
What a gamut costs
Three primaries reach a triangle and the visible region is not a triangle, so something has to give. Widening the primaries helps, has a price in precision and compatibility, and runs into a limit that is geometric rather than technological.
The triangle is a shadow
A display's gamut is drawn as a triangle on the chromaticity diagram, and coverage is quoted as a percentage of that triangle's area. Chromaticity has luminance divided out, so the triangle is a projection along the axis the eye cares most about — and the ratios computed on the solid are not the ratios computed on its shadow.
Neither gamut contains the other
Converting a picture to CMYK is described everywhere as a reduction, as though the press were a smaller version of the screen. Measured as solids, the press reaches eleven per cent of its volume outside sRGB while sRGB reaches fifty-four per cent of its volume outside the press — and in the cyans the press wins by seventy per cent of chroma.
No mapping preserves everything
More than half the sRGB solid is outside a press's gamut, so something has to be done with it, and there are two things that can be done. One leaves every reproducible colour exactly where it was and delivers a gradient as a flat area. The other keeps the gradient and moves every colour that needed no help by about three.
A difference is not a distance
Two earlier essays here said that CIEDE2000 violates the triangle inequality and left it at that. Searching for the violation finds a detour half the length of the direct route, a smaller one inside a five-unit ball, and a second defect nobody mentions — ΔE94 is not even symmetric.
A mean is not a difference
A specification for a proof, a profile or a print run is written against a set of colours and has to reduce that set to one number. The mean and the ninety-fifth percentile disagree about which of two reproductions is better in thirty per cent of cases, and both numbers are honest.
A delivery tolerance is three tolerances
A colour reaches a reader through a profile, a gamut mapping and a room, and each has a number somebody chose. Each has an essay of its own here, and none of those essays had put the three costs in one column. Put there, the stage nobody controls carries ninety-seven per cent of the total.
A budget drawn through one hue
The three-stage error budget this collection publishes for a colour-management chain is computed over twenty-four colours of a single hue at a single lightness. The quantity that actually varies with hue — how many distinguishable colours a rendering intent destroys — runs from nothing at all to more than a third, and the hue the budget uses is near the bottom of that range.
The budget adds two units
This collection publishes a three-stage error budget for a colour-management chain and prints its sum. Two of the three stages are colour differences between stimuli and the third is a distance between appearances, and the conversion between those is not a constant. Re-measured in one unit over the same colours the chain has four stages rather than three, its end-to-end error is 5.05 against a sum of 7.66, and the profile's contribution where a job actually lands is 3.2 times its published average.
An intent is not a function of the colour
A rendering intent is chosen per job and applied per pixel, which suggests it is a function of the colour. The colorimetric ones are, exactly — the same colour from two documents comes out identical to the floating-point floor. The perceptual one is not — it compresses the whole source solid into the destination, so the same colour delivered from a wide document and from a muted one lands 2.6 colour differences apart, on colours the destination could hold either way.
A second conversion is not a repeat
A colour converted to a device and back is described as lossy where the device cannot hold it and exact where it can, which suggests that a colour surviving one round trip survives any number. One of the four intents is idempotent to the floating-point floor. The other two are not — a perceptual conversion moves a ramp 3.70 colour differences on its first pass and another 2.73 on its second, and after three passes it is still moving by 2.42.
A chain measured in a unit that cannot add
A delivery chain's four stages, measured in the power-corrected appearance difference, sum to 7.66 while the chain end to end measures 5.05, and the shortfall was read as the stages partly cancelling. A chain whose four stages lay in a straight line and added exactly would still read 0.74 of its sum in that unit, because a distance raised to the power 0.63 cannot add. Measured in the model's own Euclidean space the same chain reaches 0.84 of its sum. Three quarters of the published shortfall was the exponent.
A gamut charges a gradient nothing
On red to green the shortest path happened to stay inside sRGB where the straight line did not, and that was called a coincidence of mechanism. Made into a measurement it is not a coincidence: constraining a path to stay inside the gamut costs less of the metric's own length than the relaxation's own noise, on every gradient tested — including the ones whose free shortest path is outside at fifteen of its twenty-one steps. The gamut's boundary and the metric's cheap region point the same way, and the price of the constraint is nothing.
White turns a hue, and the models part at blue
Mix a saturated light with white and its hue changes as well as its saturation — the Abney effect, and the reason lines of constant perceived hue curve in a chromaticity diagram. Asked what adding white does to the twenty-four most saturated colours a display makes, CIECAM16, CIELAB and Oklab all turn the hue. About reds they agree: CIECAM16 turns within a degree of Oklab, whose hue was fitted to observers' constant-hue judgements. About the display's blue they do not: Oklab turns sixteen degrees, CIECAM16 four, and CIELAB nine the other way.
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.
A tone-mapped blue turns where it lands
In ΔEITP a display's blue corner, and the turn at which adding green to it changes which side of ICtCp's red–green axis it moves the colour to, both shift with the display's peak, because the quantiser reads absolute levels. A high-dynamic-range picture graded on a bright master and shown on a dimmer panel is tone-mapped, its blues delivered below the levels they were graded at. The prediction was that a blue at a tenth of full drive on a 1,000 cd/m² panel turns where a 100 cd/m² panel's full blue does, and that the corner's move differs, because the dimmed blue no longer sits at a vertex. The turn agrees within a hundredth of a nanometre. So does the corner, within half a per cent: the steps both are read from are directions at one absolute signal, whatever drive delivered it. The tone map's whole effect on the blue is where it puts the blue. Through ITU-R BT.2390's curve, a master's bright blues are gathered near a dim display's peak and share one turn there — two nanometres from where the master turned them — while blues under the curve's knee keep their turns exactly.
A tone curve per channel turns a sky towards cyan
A pure blue tone-mapped from a bright master to a dim display turns, in ΔEITP, wherever its delivered level puts it, and any tone curve that delivers it to the same level gives it the same turn. A picture's blue is not pure. The prediction for a blue mixed with a little green was that a tone curve applied to each channel separately raises the green's share by about half — the blue channel compressed above the curve's knee, the dimmer green left near it — turning the mixture's hue several degrees towards cyan and its turn a nanometre or more from where a curve that scales all three channels together puts it. The turn does move, by 1.6 to 3 nanometres on a 100 cd/m² display and up to 6.5 on a 300. But the share is not raised by half: through BT.2390 to a 100 cd/m² display, a blue graded with a tenth of green arrives with its green at 69 per cent of its blue, nearly seven times as much, and its ICtCp hue turns 23 degrees towards the green. The scaled curve keeps every mixture's hue within three degrees and its turn within half a nanometre of the pure blue's.
A curve on intensity cannot see a blue's brightness
ITU-R BT.2390's tone curve applied to each channel of a bright master turns a tone-mapped sky's appearance fifty-two hue-quadrature units towards green on a 100 cd/m² display; applied to the largest channel with all three scaled, it keeps hue and lightness to a quarter of a unit. BT.2390 also describes applying the curve to ICtCp's intensity and scaling its chroma, which keeps ICtCp's hue by construction, and the prediction was that it keeps the appearance's hue within three units and its lightness within two. It keeps the hue, within 2.3 units. It cannot keep anything on the display: a saturated blue's intensity is the PQ level of 79 cd/m², under the 300 cd/m² display's knee, so the curve barely touches it and every sky asks for two to seven times the blue the display has. Clipped channel by channel, the sky turns as far as the per-channel curve turned it; scaled back into the gamut, it lands where the scaled curve put it. The step back into the gamut decides the appearance, not the curve.
Lowering only the chroma turns the sky white and its hue with it
A tone curve applied to ICtCp's intensity keeps a high-dynamic-range sky's hue, but a saturated blue's intensity lies under the curve's knee, so the colour it asks for lies outside every display's gamut. The prediction was that bringing it back by lowering its chroma alone — intensity and ICtCp hue held — keeps the CIECAM16 hue within three units, lifts the lightness by ten and halves the chroma: a paler blue of the right hue. The ICtCp hue and intensity are kept to a hundredth. Everything else fails. The chroma must fall to a fifth or less on a 100 cd/m² display, the sky rises from a lightness of 22 to 76 and higher, and its hue quadrature moves eight to seventeen units toward green — a pure blue's by 17. Holding ICtCp's hue while lowering chroma does not hold the hue a viewer sees, and the intensity the curve chose was never a blue's.
The sky a dim display can show best is the master's own
A tone curve applied to ICtCp's intensity maps a saturated high-dynamic-range sky to a colour no display can show, and bringing it back by lowering only its ICtCp chroma turns it pale and toward green. The prediction was that a return along CIECAM16's own hue line, at the lightness the curve chose, holds the hue by construction, needs a deeper chroma cut, and still lies at least twice as far from the master as simply scaling the drives. It holds the hue to a hundredth, keeps more chroma than ICtCp's return rather than less, and lies two and a half to four times as far as scaling — because holding the true hue removes only one or two of seventeen units of difference. The rest is the lightness the curve held. And the nearest a dimmer display can come to the master's appearance turns out to be the master's drives, shown unmapped.
Named alongside it
The objects these essays reach for when they reach for this one.
CIELABGamutColour differenceSpecificationColour managementThe ICC profileToleranceCIEDE2000ConstraintDisplay gamutDynamic rangeProcess inks