A tone-mapped blue turns where it lands

A gamut margin moves each corner of a display’s colour solid inward, and how far the corner moves per unit of margin is how sharp it is in the unit the margin is written in. In ΔEITP the blue corner is the sharpest a display has, and a quantiser keeps the blue a spike traced that to the perceptual quantiser at the front of ICtCp, which reads each cone signal’s absolute level and compresses it. The green primary decides how sharp the blue is found the green setting how sharp, and a green’s long tail sharpens the blue; its short tail does not found it doing so through its ratio of L to M signal.

Brightness and the green sharpen the blue almost independently then raised the display’s peak, and found two things. The corner sharpened with the peak by the same share for every green, and the turn — the blue at which adding green stops moving the colour M-ward on ICtCp’s red–green axis and starts moving it L-ward — moved two to six nanometres towards the violet between 100 and 10,000 cd/m², because the axis weighs L’s rise against M’s by their encoded levels, and those levels rise with the peak.

A display rarely shows its blues at its own peak. A picture graded on a 1,000 cd/m² master and shown on a 100 cd/m² panel is tone-mapped: each channel’s level is compressed into the dimmer display’s range, and a blue graded at full drive reaches the panel well below the level it had on the master. 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, to within a few tenths of a nanometre, because the quantiser sees only the signal’s absolute level; and that the corner’s move differs between the two, because a dimmed blue’s red and green steps are taken against a corner that is no longer a vertex of the solid. The alternative was that the turn differs as well, in which case a margin computed on the solid’s vertices could not be carried to the colours a tone map actually delivers.

Where the blue lands decides both #

On displays balanced to D65 with a 630 nm red, every green from 505 to 545 nm and the census’s blues, a blue driven to a half, a quarter or a tenth of full on a 1,000 cd/m² panel turns within a hundredth of a nanometre of the full blue on a panel of 500, 250 or 100 cd/m², and its corner moves per unit of margin within half a per cent of that blue’s. The turn depends on the level the blue is delivered at alone, and moves towards the long wavelengths at every halving of it — with the 530 nm green from 451.6 nm at 10,000 cd/m² to 457.3 at 10. Through BT.2390’s tone curve to a 100 cd/m² display, a master’s blues graded at full, half and quarter drive all land between 92 and 100 cd/m² and turn within a tenth of a nanometre of one another, near 454.8 nm; blues graded below the curve’s knee, 28 cd/m², keep their turns exactly.

• The prediction’s first half holds, and more tightly than asked.
• Its second half fails. The dimmed blue’s corner is the dim panel’s corner.
• The alternative fails too. A margin computed on a display’s vertices carries to any colour the tone map delivers at the same level.

Drawn against the level the blue arrives at, the turn is one smooth curve per green, falling towards the violet as the level rises. With the 530 nm green it is 457.3 nm at 10 cd/m², 454.8 at 100, 452.9 at 1,000 and 451.6 at 10,000; with the 505 nm green it runs from 474.7 to 462.6 over the same range, and with the 545 nm green from 448.2 to 444.1. The earlier essay drew five points of each curve, one at each display’s peak; the curve itself says those five were samples of a single function of level, not five properties of five displays.

That reframes what a display’s turn is. A panel does not have a turn; a blue delivered at a level has one, and a panel’s full blue is only the brightest blue it can deliver. Every blue in every picture it shows turns somewhere on this curve.

The dimmed blue and the dim panel’s blue #

The prediction asked for agreement within a few tenths of a nanometre. Eighteen pairs — six greens, three drives — agree within 0.0094 nm, and the corner’s move per unit of margin within 0.4 per cent. What residue there is comes from the finite step the corner is read with: the step is a fixed fraction of full drive, so on a dimmed blue it is a larger share of the blue’s own level than on the dim panel’s full blue, and the two steps sample the quantiser’s curve over slightly different stretches.

The reason the agreement is exact in the limit is in what the turn and the corner are. Both are read from three small steps at the blue — red added, green added, blue taken away — and each step is reduced to a direction in ICtCp before anything is compared. A direction does not depend on the step’s length. A blue at a tenth of full drive on a bright panel and a full blue on a panel a tenth as bright put the same absolute signal into the quantiser, and the steps at that signal point the same ways whatever their lengths. That is the quantiser seeing only the absolute level, which was the prediction’s own reason, followed through to the corner as well as the turn.

The vertex that was not needed #

The prediction’s second half rested on the dimmed blue not being a vertex. At full drive the blue can only be taken away; at partial drive it could be added as well, and the colour sits on an edge of the solid rather than at its corner. That is true of the solid, and irrelevant to what the corner measures. The corner’s move per unit of margin is read from the directions the colour can move in from where it sits, and a tone map that moves the whole picture down leaves the dimmed blue’s three directions exactly what a dim panel’s full blue offers.

The census therefore reads the corner as a property of a delivered colour and its neighbourhood, not of a display. The hue disagreement is the quantiser’s found ΔEITP’s hue behaviour set by where the quantiser sits on its curve, and the same holds for the corner: the quantiser’s position is the colour’s absolute level, and a corner computed at the level a tone map delivers is the corner the viewer meets.

The tone curve’s knee #

BT.2390’s curve, applied to each channel in the quantiser’s own encoding, passes levels unchanged below a knee and compresses the rest into the display’s top. The knee is 28 cd/m² for a 100 cd/m² display, 161 for a 300 and 464 for a 600, and the master’s full level lands on the display’s peak in each case. A blue graded below the knee arrives at the display at the level it was graded at; a blue above it arrives lower, and the brighter it was graded, the more it is pulled down.

Because the turn depends on the delivered level alone, that is all a tone map does to it. Below the knee the turn is the master’s; above it the turn is the one at the compressed level, read off the curve in the first figure. A curve applied per channel, as here, or one that scales the three channels together, delivers a pure blue to the same level and gives it the same turn.

A ramp of graded blues #

Followed along a ramp of graded blues, the tone curve’s effect is a flattening. On the master the blues graded at a quarter, a half and full drive turn at 453.9, 453.3 and 452.9 nm; on a 100 cd/m² display, after the curve, all three land between 92 and 100 cd/m² and turn within a tenth of a nanometre of 454.8. On a 300 cd/m² display the flattening starts higher, at the 161 cd/m² knee, and the brightest blues turn near where a 300 cd/m² panel’s full blue does; below the knee every display’s curve joins the master’s.

For a display maker this means a margin designed at the master’s peak is wrong for the brightest blues on every dimmer display, and right for the darker ones. A margin is placed against the corner the viewer meets, and on a dim display the master’s bright blues all meet nearly the same corner — the dim display’s own full-blue corner — however differently they were graded.

A blue that changes side #

A turn moving with level means that for a blue primary near the turn, its green step moves the colour one way across the red–green axis when the blue is bright and the other way when it is dim. With the 530 nm green, the 455 nm blue changes side at 80 cd/m²; with the 505 nm green, the 465 nm blue at 814 and the 470 nm blue at 40; with the 525 nm green, the 455 nm blue at 1,645 and the 460 nm blue at 11. Every green with a turn has at least one census blue that changes side somewhere between 10 and 10,000 cd/m².

A display with a 505 nm green and a 465 nm blue, showing a 1,000 cd/m² master’s full blue on its own 1,000 cd/m² panel, adds green to that blue on one side of the axis; showing the same picture tone-mapped to a 100 cd/m² panel, on the other. A gamut-mapping rule that steers out-of-gamut blues by adding a little green, tuned on the master, changes the hue it steers towards on the dimmer panel. Which side the step falls on is a matter of the delivered level, and the level is the tone map’s decision.

The corner, level by level #

The corner’s sharpness follows the level the same way. With the 530 nm green the 485 nm corner moves ×4.19 per unit of margin at 100 cd/m² and ×4.53 at 1,000; with the 555 nm green ×7.78 and ×8.57; with the 505 nm green ×2.36 and ×2.43. A tone map that delivers a master’s full blue at 100 cd/m² gives it the dim panel’s corner, about seven per cent less sharp than the master’s with the 530 nm green, and a margin computed for the master’s corner overprotects the dim display’s.

Four inputs, one of them a level #

The run of essays on the blue corner has now moved four things. A violet primary is no spike to the quantiser moved the blue primary and found the spike peaking at 485 nanometres; a broad green sharpens the blue; a broad blue dulls it widened the primaries; the green’s position and tail came after; and the earlier essay raised the peak. Each was a property of the display, fixed when it was built.

The tone map is the first input set by the picture rather than the panel, and it turns out to act through the one input the earlier essays had already found: the level. That makes it the cheapest of the four to account for. A display’s primaries fix the curve of turn against level; the picture and its tone map only choose where on that curve each blue sits. A display’s blue is sharp where the rotation is not found the same blue corner sharp in CIELAB and blunted sevenfold by ΔE₀₀’s compression of chroma: how sharp a corner is depends on what the unit’s front end does at that colour. For ΔEITP the front end is the quantiser, a function of level, and the corner is wherever the level puts it.

What a display maker should take from this #

Compute margins at the level the blue is delivered at. The turn and the corner depend on the blue’s absolute level and on nothing else about how it got there; a dimmed blue on a bright panel and a dim panel’s full blue are the same to ΔEITP.

A tone map changes a blue’s turn only through the level. Below BT.2390’s knee it changes nothing; above it the brightest blues are gathered near the display’s peak and share that peak’s turn.

Check every blue primary’s flip level against the levels the display will deliver. A blue whose green step changes side within the display’s range will be steered in opposite directions by one gamut-mapping rule depending on how bright the picture is.

How the displays and tone curve were computed #

Each display has line primaries — a red at 630 nm, a green at 505 to 555 nm and a blue at 440 to 490 — combined into a matrix that balances full drive to D65, and ICtCp is computed from absolute XYZ through BT.2020 RGB and the perceptual quantiser. At a blue driven to a fraction d of full on a panel of peak P, the three steps are a ten-thousandth of full drive of red or green added and of blue taken away; each step’s ICtCp change is reduced to a unit vector, the turn is where the green step’s red–green component crosses nought between the census’s blues by straight lines, and the corner’s move per unit of margin is the vertex ratio of the three unit vectors, as the earlier essays computed it. BT.2390’s tone curve maps a 1,000 cd/m² master with black at nought to each display: the master’s level is encoded by the quantiser and normalised to the master’s peak, passed unchanged below the knee at one and a half times the display’s normalised peak less a half, and carried above it by the recommendation’s Hermite spline to the display’s peak. The flip level of a blue primary is found by bisection in the logarithm of the level.

What this does not reach #

Pure blues. Every blue here is the primary alone; a picture’s blues are mixtures with a little red and green, and a tone map applied per channel compresses those parts by different amounts, changing the mixture’s chromaticity as well as its level.

One tone curve. BT.2390’s is one of several in use; a curve with a different knee moves the boundary between the blues it leaves alone and those it gathers.

ΔEITP alone. The turn is a property of ICtCp’s red–green axis; another colour-difference space with a different front end would put the blue’s corner and turn elsewhere.

Still open: whether a mixed blue keeps its turn under a per-channel curve #

A pure primary is delivered to one level by any tone curve. A picture’s blue is not pure: a sky’s blue carries some green and a little red, each at a lower level than the blue, and a curve applied to each channel separately leaves the weaker channels below the knee while it compresses the blue above it.

The calculation is a blue of the 530 nm green’s display mixed with a tenth and a quarter of green, graded at full drive on the 1,000 cd/m² master and shown on a 100 cd/m² display through BT.2390’s curve applied per channel and through the same curve applied to the maximum channel with the three scaled together, the delivered chromaticity and the turn of the mixture’s added-green step read in each. The prediction is that the per-channel curve raises the mixture’s green share by about half, moving its hue several degrees towards cyan, and that its added-green step turns a nanometre or more from where the scaled curve’s does; the scaled curve keeps the mixture’s hue and turns it as a pure blue at its level. If the two curves turn the mixture alike, the turn belongs to the blue channel’s level alone even in a mixture, and the choice of tone curve does not reach it. A tone curve per channel turns a sky towards cyan makes that calculation.

Where the argument was exact #

The habit here is about reading a geometric quantity for what it depends on.

The prediction treated the dimmed blue and the dim panel’s blue as alike in one respect, their level, and unlike in another, their position in the solid. It then asked which quantity followed which. Both followed the level, because both were defined from directions at a point, and a direction carries no memory of how far the point is from the solid’s edge.

The failure mode is to attribute to a quantity a dependence on something its definition never reads. The corner was called a corner because, at full drive, it is one, and the name invited the thought that being a vertex mattered. The test is to write down what the quantity is computed from — here, three unit vectors at one absolute signal — and to ask of each proposed influence whether it enters that computation at all.