A green's long tail sharpens the blue; its short tail does not

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 reading a margin costs a press its corners made on a printing press and a tolerance unit rounds the corners carried into the units a print buyer writes. In ΔEITP, the difference built for high-dynamic-range television on the ICtCp encoding, a display’s blue corner is the sharpest it has; a quantiser keeps the blue a spike traced that to the two primaries that can be added to a blue — red and green — moving it in nearly the same direction.

Two essays since have asked what decides how nearly. The green primary decides how sharp the blue is moved a single-wavelength green from 505 to 555 nm and found the 485 nm blue’s corner rising from 2.39 times a margin to 8.19: a longer green adds more L signal, so its step points more nearly along the red’s. A broad green sharpens the blue; a broad blue dulls it then gave the primaries Gaussian spectra, and found a green forty nanometres wide at 530 nm acting on the corner as a single line at 535.8 nm — further than its dominant wavelength, 533, would say.

A quantum-dot green is not a Gaussian. Its emission rises steeply on the short-wavelength side and trails away on the long side, towards the yellow, because the dots in a film are not all the same size and the larger ones emit longer. The prediction was that a tail on the long side, holding a third of the green’s power, moves the equivalent line further than the symmetric forty-nanometre green does, past 538 nm, because all of its spread is on the L side. The alternative named was that the corner answers to the green’s lost purity rather than its L, in which case a tail would move it less than a symmetric width of the same spread.

That alternative has a test the prediction did not name: put the same tail on the other side. A tail on the short side spreads the green just as far from its peak, and costs it more purity, because the short side of 530 nm runs towards the cyan, where the spectral locus bends away from white. If the corner reads purity, the short tail should move it at least as far. If it reads L, the short tail should move it the other way.

The long tail passes the prediction; the short tail stands still #

With a green whose core is twenty nanometres wide at 530 nm and a third of whose power sits in an exponential tail twenty-five nanometres long on its long side, the 485 nm blue’s ΔEITP corner moves 5.70 times a margin — the corner of a single-wavelength green at 540.7 nm, past the predicted 538 and five nanometres beyond the symmetric forty-nanometre green’s 535.8. The same tail on the short side gives ×4.39, a line at 530.2, against the untailed core’s ×4.51 at 531.3. Across eighteen tails — fifteen, twenty-five and forty nanometres long, holding a fifth, a third or a half of the power, on either side — the long tails move the equivalent line by up to eighteen nanometres and the short ones by under three, while the short tails cost the green more purity at every setting. For every long tail, the single wavelength with the green’s own ratio of L to M signal lies within 0.31 nm of the line the corner treats it as.

• The prediction holds, with room. A third of the power in a long tail takes the green past 538 nm at every tail length the census draws — 537.2 at fifteen nanometres is the one exception, and it is the shortest tail.
• The alternative fails outright. The short tail is the purity test and the corner ignores it.
• What the corner reads is a ratio. Not the spectrum’s centroid, not its dominant wavelength: the L signal it adds against the M signal it adds.

The two lines leave the same point — the untailed twenty-nanometre core, at ×4.51 — and part at once. Each fifth of the power moved to the long side adds two thirds of a margin or more to the corner, almost linearly to half the power, where it reaches ×6.34. The short side takes under a tenth of a margin per fifth and is still within a twentieth of the core at half. The symmetric forty-nanometre green, ×5.05, is passed by the long tail at under a fifth of its power. A display maker reading a datasheet would describe both the forty-nanometre Gaussian and the long-tailed dot as a broad green near 530 nm; to the blue corner they are as different as a 536 nm laser and a 541 nm one.

Three greens with the same peak #

The tail is built as the physical one is: the core’s own Gaussian smeared along one direction by an exponential, so that each slice of the dot population emits the core’s shape at a longer — or, for the control, shorter — wavelength, with fewer dots the further the shift. A third of the power moved this way lowers the peak to about four fifths of the core’s and leaves it at 530 nm. All three spectra peak at the same wavelength; a spectrometer’s peak-finding would report them as one green. Their centroids are 530.0, 538.2 and 522.0 nm, their dominant wavelengths 531, 540 and 523.

That symmetry in construction is what makes the control fair. The long and short tails move the same power the same distance from the peak, and differ only in which way. Anything the corner does differently between them is a property of which way the spectrum spreads — of where on the cone fundamentals the moved light lands — and not of how much it spreads.

The corner ignores the green’s purity #

Read the three bars in each row against each other. On the long side they move together: half the power in the tail takes the equivalent line to 545.2 nm, the centroid to 542.2 and the dominant wavelength to 544. The corner runs a few nanometres ahead of the colour, as the symmetric broad green’s did, but in the same direction and by a similar amount. On the short side the three come apart. Half the power in the short tail takes the centroid to 517.9 nm and the dominant wavelength to 518 — a green thirteen nanometres shorter by either reading — and the corner treats it as a 529.5 nm line, two nanometres from where it started.

The purity numbers say the same from the other side. A third of the power in the long tail costs the green a little purity, from 0.98 of the way to the spectral locus to 0.95; the same share on the short side costs it a quarter, to 0.77, and half the power to 0.70. Desaturating a primary that much is a large change in what a display can show — the green corner of its gamut retreats a long way towards white. The blue corner’s sharpness does not register it. Whatever the margin’s cost at the blue depends on, it is not how saturated the green is.

That is the alternative the earlier essay named, and it was the natural reading. A broad green is less pure, and a less pure primary moves the corners of a gamut inward and flattens its faces; it would have been unsurprising if a corner flattened with it. The short-tail control shows the corner was never reading that. A broad symmetric green sharpens the blue because of what its long side adds, not because it is broad.

What the corner reads instead #

The account carried from the green census was that adding green raises a blue’s L and M signals each by the green’s own signal in that cone, and that whichever the blue has least of, relative to the green, rises by the larger fraction. If that is the whole of it, the corner should depend on the green only through its ratio of L to M — the direction its step points in the L–M plane — and two greens with the same ratio should give the same corner whatever their spectra.

For the long tails it is the whole of it. The single wavelength whose ICtCp L to M ratio equals the tailed green’s lies within 0.31 nm of the line the corner treats the green as, across all nine long tails, from a fifth of the power in a fifteen-nanometre tail to half in a forty-nanometre one. The spectrum’s centroid misses by up to three nanometres, always short, because the L to M ratio rises faster than linearly across 530 to 550 nm, so light moved to the long side raises it by more than its mean wavelength suggests. That is why the symmetric forty-nanometre green, too, acted longer than its colour: its long half adds more L than its short half takes away.

The short tails sit above the diagonal, by up to four and a half nanometres. Their L to M ratio says they should act as 527 to 530 nm greens; they act as 528.6 to 531.4, and the further the tail reaches the further above it they sit. The difference is the third cone. A short tail moves light towards 500 nm and below, where the S cone still responds, and the green’s S signal rises from 0.12 of its M with no tail to 0.29 with half the power in a twenty-five-nanometre short tail and 0.40 with a forty-nanometre one. Adding that green to a blue now raises the blue’s S signal too, by a small fraction — the blue has a great deal of S — but in a direction along ICtCp’s blue–yellow axis that partly cancels the M-ward swing the lower ratio would bring.

So a short tail fails to move the corner for two reasons that add. It lowers the green’s L to M ratio by less than its colour suggests, because the ratio changes slowly below 530 nm; and the S signal it brings pushes back against what change there is. At forty nanometres long the two cancel almost exactly: half the power in the tail moves the equivalent line by eight hundredths of a nanometre.

A longer tail, on one side only #

Real quantum-dot films differ in how far their tails reach, and the census draws three lengths. On the long side the effect scales with the length as well as the share: at a third of the power, a fifteen-nanometre tail makes the green a 537.2 nm line, twenty-five nanometres a 540.7 nm one and forty a 543.8 nm one; at half the power the forty-nanometre tail reaches 549.1, close to the 555 nm green that made the census’s sharpest corner. The tail’s length and its share multiply, because what matters is how much L the moved light carries, and light moved further along the long side carries more.

On the short side the three lengths fan out the other way, and the longest is the flattest. A fifteen-nanometre short tail holding half the power makes the green a 528.6 nm line, a twenty-five-nanometre one 529.5 and a forty-nanometre one 531.4 — above the untailed core. The further the short tail reaches, the more of its light lands where the S cone answers, and the more of the M-ward swing it undoes. No short tail the census draws moves the green by three nanometres, against eighteen for the long ones.

The turn follows the corner’s green #

The green census found a second quantity set by the green: the blue at which adding green stops raising the blue’s M signal by the larger fraction — the turn, 453.7 nm with a 530 nm line, moving about half a nanometre for every nanometre the green moves. Every long tail turns where its equivalent line would, within two tenths of a nanometre for each one whose equivalent the green census spans: half the power in a twenty-five-nanometre long tail turns the blue at 445.8 nm, where a 545 nm line puts it.

The short tails come off the census’s line. Their corners barely move, but their turns move later, to 455.6 nm with half the power in a forty-nanometre tail — two and a half nanometres from the untailed core’s 453.1. That is the ratio account again: the turn is where adding green raises L and M by equal fractions along ICtCp’s L–M axis, so it follows the green’s L to M ratio, which the short tail does lower; the corner’s height also counts the S signal, which the short tail adds. The two numbers the green census tied together come apart for a green whose spread brings S, and they come apart in the direction the ratio account says they should.

What a display maker should take from this #

A quantum-dot green’s tail is a longer green to a ΔEITP margin at the blue. A film with a third of its power in a twenty-five-nanometre tail is, for this purpose, a 541 nm green, and the 485 nm blue corner on that display moves a quarter again as far per unit of margin as on a display with an untailed twenty-nanometre green. A colour engine that keeps colours a fixed number of ΔEITP units inside the blue will carve away more on the first display than its datasheet’s peak wavelength or its chromaticity’s dominant wavelength would predict — by about five nanometres of green either way.

The specification that predicts it is the green’s ratio of L to M signal. Two numbers a display maker already has — the green’s chromaticity, carried through ICtCp’s LMS matrix — give it within a third of a nanometre for any green whose spread lies on its long side. A green whose spread lies on its short side, as an organic emitter’s sometimes does, needs its S signal counted too, and the chromaticity carries that as well. The earlier census’s warning that a specification in chromaticities is not enough was about matching by dominant wavelength; matched by the right function of the chromaticity, it is enough here, for the corner. It is not enough for the disagreement between observers a narrow primary buys, which depends on the spectrum’s shape in ways no single chromaticity can carry.

Desaturating the green is not a way to blunt the blue. A display that wanted ΔEITP to treat its blue corner more gently would gain nothing by broadening its green on the short side and losing saturation for it. The escapes the earlier essays found are the ones that move the ratio: a shorter green, or the violet blue that a violet primary is no spike to the quantiser found splits the two added steps whatever the green.

How the greens were built and read #

Each display has a red line at 630 nm, a blue line at every five nanometres from 440 to 490, and a green whose spectrum is sampled at one nanometre: a Gaussian core twenty nanometres wide at half height, centred on 530 nm, plus a tail. The tail is the same Gaussian shifted by every whole nanometre from nought to four hundred along one direction and weighted by an exponential of the shift over the tail’s length — fifteen, twenty-five or forty nanometres — then normalised to carry its stated share of the power. With no tail the green is the broad-primary census’s twenty-nanometre green to a part in a hundred million at every blue corner. The green’s chromaticity is integrated over the CIE 1931 colour-matching functions at one nanometre, and the white is D65 at 320 cd/m².

At each blue corner the three inward directions — adding red, adding green, dimming the blue — are carried into ICtCp through BT.2020’s matrices, the perceptual quantiser and ICtCp’s rotation, and the corner’s move per unit of margin is the distance to the nearest point at least one unit inside every face of the cone they span, as in the earlier corner essays. The equivalent single-wavelength green is interpolated between the green census’s lines at 505 to 555 nm by the 485 nm corner’s move. The L to M ratio is ICtCp’s own LMS — BT.2020’s matrix carried to XYZ — at the green’s chromaticity; its equivalent line is read at one nanometre. Purity is the green’s distance from D65’s white over the distance to the spectral locus along the same ray, which also gives the dominant wavelength.

What this does not reach #

One shape of tail. Every tail is the core smeared by an exponential. Real films’ tails are set by their size distributions and can be shoulders rather than exponentials; the ratio account predicts any shape’s effect from its chromaticity, but only the exponential has been computed.

One core. The core is twenty nanometres wide at 530 nm. A narrower core with a longer tail, or a core at 525, would sit elsewhere on the same ratio curve; the census has not drawn them.

One unit. Every corner is read in ΔEITP. An appearance model flattens the blue by its hue found CAM16-UCS treating the same blue corner as nearly a face, following the green at under half ΔEITP’s rate, and a display’s blue is sharp where the rotation is not found ΔE₀₀ sharp at the blue for a different reason; whether either unit reads a tail by the green’s L to M ratio is not computed.

One white and one peak. The display is balanced to D65 at 320 cd/m². The perceptual quantiser’s slope changes with luminance, and the violet census found a brighter display sharpening the long blues; whether a long green’s sharpening scales with the display’s peak is not computed.

Still open: whether a brighter display multiplies the green’s sharpening #

The violet census moved the display’s peak from 100 to 4,000 cd/m² with the green held at 530 nm, and found the long blues sharpening as the peak rose. The quantiser compresses the blue’s large S signal more at high luminance, and the corner’s sharpness is a contest between the added steps’ L–M swing and the S they leave alone — so a green whose L–M swing is already small should lose more of its remaining separation as the peak rises.

The calculation is the green census’s single-wavelength greens from 505 to 555 nm, and the long-tailed green holding a third of its power, on displays balanced to D65 at 100, 320, 1,000 and 4,000 cd/m², the 485 nm corner and the turn read at each. The prediction is that the corner’s ratio between a 555 nm and a 505 nm green — 3.4 at 320 cd/m² — grows with the peak, past four at 4,000, because a brighter display narrows every angle and the L-heavy green’s was narrowest to begin with; and that the turn does not move with the peak at all, because it is a ratio of fractional rises the quantiser’s slope multiplies equally. If the ratio between greens is the same at every peak, the green’s effect and the display’s brightness are independent factors, and a margin’s cost at the blue can be tabulated by green and by peak separately. Brightness and the green sharpen the blue almost independently makes that calculation.

The control that names the variable #

The habit here is about choosing the comparison an alternative cannot survive.

The broad-green essay named two readings of its result — the green’s L, or the green’s lost purity — and the obvious next calculation, the tail it proposed, could not tell them apart: a long tail adds L and loses purity at once. Its mirror image adds no L and loses more purity. One case moves both suspected causes together; its reflection moves them apart.

The failure mode is to test a prediction only in the direction it was made. The long tail would have confirmed the prediction and left the alternative standing, and every later calculation built on it would have inherited both. The test is to find the case where the two readings disagree most sharply and compute that one too — here, a tail that is the same in every respect but its side.