A tone curve per channel turns a sky towards cyan

A display’s blue corner in ΔEITP has a turn: the blue primary at which adding a little green stops moving the colour towards M on ICtCp’s red–green axis and starts moving it towards L. A tone-mapped blue turns where it lands found that turn, and the corner’s sharpness with it, set by the level the blue is delivered at and by nothing else. A blue at a tenth of full drive on a 1,000 cd/m² panel turned within a hundredth of a nanometre of a 100 cd/m² panel’s full blue, and a tone map that carried a picture from a bright master to a dim display changed a blue’s turn only by moving its level.

That was a pure primary. The blue of a sky, a stage light or a logo is a mixture: mostly the display’s blue, with some green and a little red. Tone curves differ in how they treat such a mixture. ITU-R BT.2390’s curve, applied to each channel’s level separately, compresses whatever lies above its knee and leaves whatever lies below it alone; a mixture’s bright blue channel is compressed and its dim green channel, a tenth as bright, may not be. The same curve applied to the largest channel, with all three scaled by the same factor, compresses the mixture as one colour.

The prediction was that the per-channel curve raises the mixture’s green share by about half, turning its hue several degrees towards cyan, and moves its added-green step’s turn a nanometre or more from where the scaled curve puts it; the scaled curve keeping the hue and turning the mixture as a pure blue at its level. The alternative was that the two curves turn the mixture alike, which would mean the turn belongs to the blue channel’s level alone even in a mixture, and the choice of curve does not reach it.

Twenty-three degrees, not several #

A blue of the 530 nm green’s display mixed with a tenth of green, graded at full on a 1,000 cd/m² master and shown on a 100 cd/m² display through BT.2390 applied to each channel, arrives with its green at 69 per cent of its blue, its ICtCp hue turned 23.4 degrees towards the green, and the turn of its added-green step at 453.5 nm. Through the same curve scaled across the channels it arrives with its green at the tenth it was graded at, its hue within two degrees of the master’s and its turn at 455.2 nm. Across green shares from a twentieth to a half, on displays of 100 and 300 cd/m², the per-channel curve turns the hue between 4 and 23 degrees towards the green, and at a tenth of green or more puts the turn 1.6 to 6.5 nanometres from the scaled curve’s; the scaled curve keeps every hue within three degrees.

• The prediction’s turn holds: more than a nanometre between the two curves at a tenth of green and beyond.
• Its hue is far too small. The share is multiplied nearly sevenfold, not raised by half, and the hue turns by twenty degrees, not several.
• The alternative fails. The curve reaches the turn through the mixture’s other channels.

Read across green shares, the per-channel curves form a dip and the scaled ones a flat line. On the 100 cd/m² display the per-channel hue change is −16.9 degrees at a twentieth of green, −23.4 at a tenth and at a quarter, and −14.9 at a half; on the 300 cd/m² display −4.2, −9.9, −18.8 and −14.3. The dip has a floor because a per-channel curve can at most bring the green up to the blue’s own level, and once the graded green is large enough to be compressed too, the two channels approach each other and the hue stops moving further. The scaled curves never move more than three degrees, and what they do move is the quantiser’s own small dependence of hue on level, the same at every share.

The 300 cd/m² display shows the knee’s part directly. Its knee sits at 161 cd/m², so a twentieth of green — 50 cd/m² on the master — passes it untouched and a tenth, at 100, passes it too, while the blue is cut only from 1,000 to 300. The green’s share there rises from a twentieth to 0.17 and from a tenth to 0.33, about threefold rather than sevenfold, and the hue turns 4 and 10 degrees rather than 17 and 23. The brighter the display, the smaller the ratio between the cut to the blue and the cut to the green, and the smaller the change; a display as bright as the master would change nothing.

Why the green is barely dimmed #

The mechanism is the knee. For a 100 cd/m² display, BT.2390’s curve leaves every channel level below 28 cd/m² unchanged and compresses the rest into the display’s top. A blue graded at full is 1,000 cd/m² in its own channel and is brought down to 100 — a tenfold cut. A tenth of green is 100 cd/m² in the green channel, just above the knee, and is brought down only to 69.5; a twentieth is 50 cd/m² and arrives at 46. Relative to the blue, the green has gone from a tenth to seven tenths.

The prediction’s “about half” came from imagining the green near the knee and the blue well above it, with the curve’s compression between them modest. The compression above the knee is not modest when the master is ten times the display: the top of the master’s range is squeezed into the top of the display’s, and a channel at full is cut by the whole ratio. A channel near the knee is cut by almost nothing. The ratio of the two cuts is the factor the green’s share is multiplied by, and on a 100 cd/m² display that ratio is nearly seven.

Drives on the dim display #

Drawn as drives, the difference is stark. Per channel, a mixture graded with a twentieth, a tenth, a quarter and a half of green arrives with its green at 0.46, 0.69, 0.92 and 0.99 of full; scaled, at 0.05, 0.1, 0.25 and 0.5. Every per-channel mixture with a tenth or more of green arrives as something closer to cyan than to the blue it was graded as — at a quarter of green, the display shows nearly equal blue and green.

That is a colour change a grader would see. A sky graded as a deep blue on the master appears on a dim display through a per-channel curve nearer cyan, and the change is largest for blues with a little green — enough for the green channel to sit just above the knee once the master is bright — which is where many pictures’ skies, water and screens lie.

What the turn does #

The turn follows the mixture rather than the blue. On the master itself, the turn moves towards the violet with every added share of green — 452.9 nm with none, 451.7 at a tenth, 449.5 at a quarter, 444.0 at a half. A blue with more green in it already has more M signal, and the added-green step’s balance between L and M shifts accordingly. Under the scaled curve on the 100 cd/m² display every mixture’s turn stays within half a nanometre of the pure blue’s 454.8 nm, because the scaled mixture keeps its share and moves to the dim display’s level as a whole.

Under the per-channel curve the turn lies between the two: 453.5 nm at a tenth of green, 452.2 at a quarter, against the scaled curve’s 455.2. The two curves put the mixture’s turn 1.6 nanometres apart at a tenth of green and 3.1 at a quarter; on the 300 cd/m² display, 1.9 and 6.5. A display whose blue primary sits between the two turns has its added-green step on one side of the red–green axis under one curve and on the other under the second — the change of side a tone-mapped blue turns where it lands found between levels, here found between tone curves at one level.

Brighter as well as greener #

The per-channel curve keeps the mixture brighter. At a tenth of green its intensity is 318, against 236 for the scaled curve and 391 on the master; at a quarter, 335 against 267. The green the per-channel curve leaves uncompressed is light the scaled curve takes away, and a picture tone-mapped per channel keeps its skies brighter as well as greener than the scaled curve’s.

That is part of why per-channel curves are used: they keep highlights brighter and saturated colours from collapsing towards the display’s peak. The cost is the hue, and for the blues it is large. A tone curve that scales channels together and a curve that acts per channel make different trades between brightness and hue, and ΔEITP’s blue corner reads both.

Hue as a direction #

Placed as directions in ICtCp’s chroma plane, the scaled mixtures lie beside the master’s, each a degree or three round, and the per-channel ones swing towards the green. The per-channel mixtures with a tenth and a quarter of green point past even the master’s half-green mixture: −86.5 and −93.3 degrees against its −80.2. The per-channel curve does not just add some green to each mixture; it remaps a family of blues graded with different amounts of green into a tighter family near cyan, compressing the hue differences the grader made along with the levels.

The hue disagreement is the quantiser’s found ΔEITP’s hue behaviour set by where the quantiser sits on its curve. Here the quantiser is not the cause: the per-channel curve changes the colour’s chromaticity before ICtCp sees it. Any colour space would report the hue turning; ICtCp reports it in degrees that can be set beside the earlier essays’ turns.

The corner a grader never sees #

The essays on the blue corner have treated it as a property of a display: a quantiser keeps the blue a spike found it sharp in ΔEITP because of where the quantiser sits, the green primary decides how sharp the blue is found the green setting how sharp, and a broad green sharpens the blue; a broad blue dulls it found the primaries’ widths mattering too. Each of those is fixed when the display is built, and a margin computed from them is computed once.

The tone curve is applied afterwards, to every picture, and the per-channel curve changes which colour reaches the corner’s neighbourhood. A grader working on the master sees a blue with a tenth of green near the master’s blue corner; a viewer of the 100 cd/m² display is shown a colour seven tenths of the way to cyan, near a different part of the solid’s edge, with a different turn. The corner the margin was designed for is the one the grader saw and the viewer does not.

A margin costs a press its corners began this run of essays by measuring how far a margin moves a corner; the measure assumed the colours near the corner arrive there. Under a per-channel curve the blues near the master’s corner are carried away from it on the dim display, and the margin’s cost there is paid on colours that are no longer present. Under a scaled curve they arrive where the corner analysis expects them, at the dim display’s level.

What a display maker should take from this #

Tone-map blues as colours, not as channels. A curve that scales the three channels together keeps every mixture’s hue within three degrees and its turn within half a nanometre of a pure blue’s at its level; a curve applied to each channel multiplies a sky’s green share up to sevenfold and turns its hue by twenty degrees.

A margin computed on the pure blue carries only under a scaled curve. Under a per-channel curve a mixture’s turn moves with its green share as well as its level, and a gamut-mapping rule tuned on pure primaries misplaces the mixtures that make up a picture’s blues.

Check a per-channel curve on mixtures near the knee. The worst hue changes are for blues with a twentieth to a quarter of green, whose green channel sits just above the knee once the master is bright.

How the mixtures were computed #

Each display has a red line at 630 nm, a green at 530 and a blue from the census’s 440 to 490 nm, balanced to D65, and ICtCp is computed from absolute XYZ through BT.2020 RGB and the perceptual quantiser. A mixture is the blue at full drive with a share of the green’s drive — nought, a twentieth, a tenth, a quarter or a half — on a 1,000 cd/m² master. BT.2390’s curve maps the master with black at nought to a display of 100 or 300 cd/m², either applied to each channel’s level, each channel’s level in cd/m² being its drive times the master’s peak, or applied to the largest channel’s level with all three scaled by the factor that curve gives it. Hue is the angle of the mixture’s Cp against its Ct, read with the 465 nm blue; the turn is where the added-green step’s red–green direction crosses nought across the census’s blues, with the mixture’s drives the same at every blue, read as the earlier essays read it.

What this does not reach #

No red. Every mixture is blue and green; a picture’s blues carry some red as well, whose channel would sit lower still and be compressed less again.

One master. The master is 1,000 cd/m². A 4,000 cd/m² master would put more of every mixture’s green above the knee and cut its blue by more, and the hue change would grow.

Hue in ICtCp. The hue angles are ICtCp’s; CIELAB or an appearance model would report the per-channel curve’s change in other degrees, though every model would report one.

Still open: whether the hue change is visible as one #

A twenty-three-degree turn in ICtCp’s hue angle at a blue is large on the scale, but ICtCp’s hue scale is not uniform around the circle, and an appearance model flattens the blue by its hue found CIECAM16 compressing hue differences near the blue.

The calculation is each mixture’s colour on the master and on the 100 cd/m² display through each curve, converted to CIECAM16 under the display’s white at the display’s adapting luminance, the hue quadrature and the ΔEITP between master and display read for each. The prediction is that the per-channel curve’s change at a tenth of green is still more than twenty CIECAM16 hue-quadrature units — a fifth of the way from blue to green — and more than ten ΔEITP units, while the scaled curve’s stays under three of each. If CIECAM16 shrinks the per-channel change below ten quadrature units, the blue’s hue scale absorbs the turn, and the per-channel curve’s cost lies in the green’s brightness more than in its hue.

A curve that acts on the parts #

The habit here is about applying a function to a sum’s parts.

A tone curve is a function of level. Applied to a colour as a whole — its largest channel, with the rest scaled alike — it changes the level and keeps the proportions. Applied to each channel, it changes each part by its own amount, and wherever the parts sit on different stretches of the curve, the proportions change. The prediction knew this and guessed the size from the curve’s slope near the knee; the size was set instead by the ratio of the curve’s slopes at the two parts’ levels, which for a bright master and a dim display is nearly the whole compression ratio.

The failure mode is to estimate a nonlinear function’s effect on a mixture from its effect near one part. A per-channel curve’s change to a colour is the ratio of what it does to each channel, and the channels may be far apart on the curve. The test is to put each part’s level on the curve before guessing — here, the blue at 1,000 cd/m² and the green at 100, either side of a 28 cd/m² knee.