Brightness and the green sharpen the blue almost independently

A gamut margin moves each corner of a display’s colour solid inward, and how far it moves a corner per unit of margin is how sharp the corner is in the unit the margin is written in — the measure a margin costs a press its corners made on a press and a tolerance unit rounds the corners carried into ΔE₀₀. In ΔEITP the blue corner is the sharpest a display has, and the essays since a quantiser keeps the blue a spike have taken that spike apart one input at a time.

Three inputs have been moved so far. A violet primary is no spike to the quantiser moved the blue and found the spike peaking at 485 nm; along the way it raised the display’s white from 100 to 4,000 cd/m² and found the long blues sharpening, ×4.12 to ×4.79 at 480 nm. The green primary decides how sharp the blue is moved the green and found the 485 nm corner rising from ×2.39 with a 505 nm green to ×8.19 with a 555 nm one, at 320 cd/m². A green’s long tail sharpens the blue; its short tail does not found the corner reading the green through one number, its ratio of L to M signal.

Each of those was computed at one setting of the others. The prediction was that the green and the brightness compound: that the ratio between a 555 nm and a 505 nm green’s corner, 3.42 at 320 cd/m², grows past four by 4,000, because a brighter display narrows the angle between the two added primaries’ steps and the long green’s angle was narrowest to begin with; and that the turn — the blue at which adding green stops raising the blue’s M signal by the larger fraction — does not move with the peak at all, because it is a ratio of fractional rises the quantiser’s slope multiplies equally. The alternative named was that the two factors are independent, in which case a margin’s cost at the blue could be tabulated by green and by peak separately.

The factors multiply, and the turn moves #

On the green census’s displays balanced to D65 at 100, 320, 1,000, 4,000 and 10,000 cd/m², the 485 nm blue’s corner rises with the peak for every green, by 6% from 100 to 10,000 cd/m² with a 505 nm green, 14% with a 530 nm one and 17% with a 555 nm one. The 555 nm green’s lead over the 505 nm one grows from 3.30 to 3.42, 3.52, 3.60 and 3.63 — a tenth across a hundredfold range of brightness, and under four at 4,000. The corner on all thirty-five displays is the 530 nm green’s at that peak times the green’s own at 320 cd/m², over the 530 nm green’s at 320, to within 4.5%. The turn moves at every step, towards the violet: with a 505 nm green from 468.2 nm at 100 cd/m² to 462.5 at 10,000, with a 530 nm green from 454.8 to 451.6. Near 1,000 cd/m² it passes the blue at which adding green raises L and M by equal fractions, which no peak can move.

• The first half of the prediction fails by most of its size. The long green’s lead grows, but by a twentieth from 320 to 4,000 cd/m², not the sixth the prediction needed.
• The alternative is nearly right. Green and brightness are two factors, and their product misses by under five per cent anywhere in the census.
• The second half fails outright. The quantiser moves the turn, and the reason is a weighting the prediction overlooked.

On a logarithmic scale the five curves are nearly parallel. Each rises from the 505 nm green to the 555 nm one by a factor of about three and a half, and each lies a little above the dimmer one beneath it. What brightness adds to the corner is close to a fixed share, whatever the green, and what the green adds is close to a fixed share, whatever the brightness. The curves fan out a little towards the long greens — the 555 nm green’s corner gains more from the brighter displays than the 505 nm one’s — and that fanning is all the compounding the census finds.

The long green’s lead grows by a twentieth #

The prediction drew its number from a picture of the corner as two steps — adding red and adding green — pointing nearly the same way, and from the violet census’s finding that a brighter display narrows the angle between them. The angle does narrow, for every green: at the 485 nm blue, from 70 degrees at 100 cd/m² to 55 at 10,000 with a 505 nm green, and from 22 to 18 with a 555 nm one. But the angles narrow by much the same share whatever the green — a fifth for the 505 nm green, a sixth for the 555 nm one — and so every corner gains, and the ratio between two greens grows only by the difference between their gains: 6% for the 505 nm green across the whole range, 17% for the 555 nm one. The prediction treated the long green’s narrow angle as a lever that brightness would pull harder. The census finds the lever there, and short: a corner’s move is set by all three of its inward steps, not by the one angle, and the third — dimming the blue — swings away from the other two as the display brightens, from 82 degrees off adding red at 100 cd/m² to 102 at 10,000, for the short green and the long alike. That opening works against the closing angle between red and green, and it works equally on both greens, which is why their ratio barely moves.

The 545-over-515 ratio tells the same story with less leverage: 2.06 at 100 cd/m² and 2.18 at 10,000. Whichever pair of greens is compared, the ratio drifts upward by a few hundredths per tenfold of brightness. A display maker comparing two greens can quote their difference at the blue corner at any brightness and be within a tenth at any other.

A product of two numbers #

The test of independence is to predict each display’s corner from two tables — one indexed by green, read at 320 cd/m², one indexed by peak, read on the 530 nm green — and multiply. All thirty-five displays fall within 4.5% of the diagonal. The two that miss by more than three per cent are both the 505 nm green at the bright end, 3.8% at 4,000 and 4.5% at 10,000, where the product overstates a short green’s gain from brightness: the short green’s corner is the bluntest in the census and its angle the widest, and a wide angle narrows less per unit of compression.

For a colour engine this is a convenience with a stated price. A gamut margin’s cost at a display’s blue corner can be held as two short tables rather than a grid, and the error from doing so is under a twentieth of the cost — less than the difference between the 480 nm and 485 nm blues at any one setting. The green is the large factor, a threefold range; brightness is the small one, a sixth at most from 100 to 10,000 cd/m², and they interact at the second decimal.

The turn moves at every step #

The second half of the prediction rested on an argument that sounds exact. Adding a little green to a blue raises each cone signal by a fraction, the green’s signal over the blue’s; those fractions do not depend on how bright the display is, because doubling the display doubles both. The turn is where the two fractions are equal, so the turn cannot depend on brightness either.

The census finds the turn moving at every step of the peak, for every green that turns. With a 505 nm green it falls from 468.2 nm at 100 cd/m² to 466.3, 464.7, 463.3 and 462.5 at 10,000 — almost six nanometres. With a 530 nm green it falls 3.2 nm, from 454.8 to 451.6; with a 545 nm green 2.3, from 446.4 to 444.1. And each green’s turn crosses its own dotted line, the equal-fraction crossing, somewhere between 320 and 4,000 cd/m²: the 530 nm green’s at about 1,000, where the turn sits a third of a nanometre short of it.

So the argument is right about the equal-fraction crossing and wrong about the turn. The turn in this collection’s essays has always been read along ICtCp’s red–green axis, the direction the encoding uses — the same reading the green census found within two and a half nanometres of the equal-fraction crossing at 320 cd/m². Within two and a half nanometres is not the same, and the difference is what brightness moves.

What the quantiser weighs #

ICtCp does not compare cone signals; it compares their encoded values. A small fractional rise f in a signal x moves its encoded value PQ(x) by f times x times PQ’s slope there — by f times the encoded level times the curve’s logarithmic slope. Two equal fractional rises, one in L and one in M, therefore move the encoded values by different amounts whenever L and M sit at different levels on the curve, and a blue carries much less L than M: at 320 cd/m² a 485 nm blue’s L signal is about half its M.

For the 455 nm blue — near the 530 nm green’s turn — the quantiser alone weighs L’s fractional rise at 0.942 of M’s at 100 cd/m², climbing to 0.993 at 10,000: the perceptual quantiser grows more nearly logarithmic as the signal rises, and on a logarithmic scale a fractional rise is worth the same at any level. ICtCp’s red–green axis then adds its own small asymmetry, weighing encoded L by 17,933 and encoded M by 17,390, three per cent more. Along that axis the weight is 0.972 at 100 cd/m², 0.987 at 320, 1.001 at 1,000 and 1.024 at 10,000 — and the turn sits on the long side of the equal-fraction crossing at every peak where the weight is under one and on the short side wherever it is over, passing it where the weight passes one.

That is the whole of the turn’s movement. Below 1,000 cd/m² ICtCp undercounts L’s rise, so adding green must raise L by a somewhat larger fraction than M before the encoded step points L-ward, and that happens only at a blue with a little more M — a longer one. Above 1,000 it overcounts, and the turn moves past the crossing towards the violet. The prediction assumed the quantiser multiplies both fractions by the same slope; it multiplies each by its own signal’s encoded level, and those level-dependent weights converge only as the curve straightens towards a logarithm.

Brightness slides the spike #

The census turned up one more movement the prediction did not ask about. At 100 and 320 cd/m² the sharpest blue is the 485 nm one for every green; at 1,000 it is the 480 nm one for five greens of seven, and at 4,000 and 10,000 for all seven. With a 530 nm green the 480 nm corner gains on the 485 nm one at every step, from 0.98 of it at 100 cd/m² to 1.03 at 10,000, and the 490 nm corner falls away, from 0.96 of the sharpest to 0.89.

It is the turn’s movement seen from the other end of the range. The turn and the spike are both set by how the added-green step’s direction changes across the blues, and a weighting that moves the turn violet-ward by three nanometres moves the angle’s minimum violet-ward with it — from between 480 and 485 nm to just under 480. The green census’s summary that “the blue decides where the spike stands” holds at any one brightness; across brightness, the encoding moves it five nanometres.

What a display maker should take from this #

A long green costs a ΔEITP margin about as much more at the blue on a bright display as on a dim one. The factor between greens is set by their L to M ratios and barely moves with peak luminance, so a decision about the green can be made once. Brightness then adds its own share, a sixth at most across a hundredfold range — real, but smaller than a ten-nanometre step in the green.

A display’s blue is sharpest at a slightly shorter wavelength the brighter the display. The difference between a 480 and a 485 nm blue is a few per cent of a margin either way, and a maker choosing a blue for a 1,000 cd/m² panel loses nothing by choosing 480. The violet escape the earlier essays found works better at high brightness, not worse: a brighter display dulls the violet and moves the turn towards it.

A turn read from cone fractions alone is a turn at about 1,000 cd/m². Any argument that locates the turn by comparing L and M before encoding — as the green census’s account did — is exact at one peak and off by up to three nanometres at the others, in a known direction. The hue disagreement is the quantiser’s found the same encoding tilting ΔEITP’s balance between lightness and chroma as a display brightens; here it tilts the balance between L and M, by the same mechanism of unequal signals at unequal levels on one curve.

How the displays were built and read #

Each display has monochromatic primaries at 630 nm, at a green from 505 to 555 nm and at a blue from 440 to 490 in five-nanometre steps, their chromaticities read off the CIE 1931 functions and balanced to D65. A drive is carried to absolute XYZ with the white at 100, 320, 1,000, 4,000 or 10,000 cd/m², then through BT.2020’s primaries, ICtCp’s LMS matrix, the perceptual quantiser and the opponent matrix, ΔEITP’s space being 720 times I, half of Ct and Cp; at 320 cd/m² every corner is the green census’s to a part in a billion. A corner’s move per unit of margin is the shortest vector at least one unit inside every face of the cone its three inward steps span. The turn is interpolated between five-nanometre blues where adding green’s unit step changes sign along Cp; the equal-fraction crossing where adding green raises L and M before encoding by the same fraction. The quantiser’s weight on a signal is its change in encoded value per unit fractional rise, read by a millionth’s step.

What this does not reach #

Line primaries only. Every green here is a single wavelength. A broad green sharpens the blue; a broad blue dulls it found a broad blue costing the spike a tenth of its height, and the tail essay found the corner reading a broad green through its L to M ratio at 320 cd/m²; that a broad green’s equivalent line is the same at every brightness is likely, because the ratio is computed before encoding, and is not computed.

One unit. CAM16-UCS’s response to brightness is set by its own luminance adaptation, not by an encoding curve, and an appearance model flattens the blue by its hue found it following the green at half ΔEITP’s rate; whether it separates green and brightness as cleanly is open.

A corner, not a picture. The display’s peak is the luminance of its white. A picture rarely puts its blue at full drive on a 10,000 cd/m² panel, and a margin applied to a tone-mapped blue sits at the encoded level the tone map gives it.

Still open: whether the tone map moves the turn as the peak does #

The turn moves because ICtCp’s red–green axis weighs L’s rise against M’s by their encoded levels, and those levels rise with the display’s peak. A high-dynamic-range picture shown on a dimmer panel is tone-mapped: its blues reach the encoding at lower levels than the content was graded at.

The calculation is the 530 nm green’s display at 1,000 cd/m² with its blue corner read not at full drive but at drives of a half, a quarter and a tenth — the levels a tone map would leave a graded blue at — and the turn, the spike’s position and the corner’s move read at each, against the same display at peaks of 500, 250 and 100 cd/m². The prediction is that a blue at a tenth’s drive on a 1,000 cd/m² panel turns where the full blue does on a 100 cd/m² one, 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 the dimmed blue’s red and green steps are taken against a corner that is no longer a vertex of the solid. If the turn differs as well, the level is not the whole of what the encoding responds to at a corner, and a margin computed on the solid’s vertices cannot be carried to the colours a tone map actually delivers. A tone-mapped blue turns where it lands makes that calculation.

When the exact argument is about the wrong quantity #

The habit here is about the gap between a quantity and the reading of it.

The prediction’s second half was a sound argument: fractional rises in cone signals do not depend on brightness, so a crossing of fractional rises cannot either. The argument was exact. It was exact about the equal-fraction crossing, which the census confirms does not move; the turn the essays have always reported is read along the encoding’s axis, a nearby quantity the green census found within two and a half nanometres of the first and treated as the same.

The failure mode is to prove a claim about one quantity and apply it to another that has agreed with it so far. Two quantities that agree at one setting can be separated by the setting the next calculation changes. The test is to compute both every time the setting moves — here, the turn and the equal-fraction crossing side by side at every peak — and to report which one an argument is about.