A violet primary is no spike to the quantiser

A gamut margin moves each corner of a display’s colour solid inward by a multiple of itself, and the multiple depends on the unit the margin is written in. Three essays have read a display’s corners in four units. A display’s blue is sharp where the rotation is not found sRGB’s blue the sharpest corner in CIELAB; an appearance model flattens the blue by its hue found CAM16-UCS making it nearly a face; and a quantiser keeps the blue a spike found ΔEITP — the colour difference built for high-dynamic-range television on the ICtCp encoding — keeping it a spike, ×3.12 on sRGB and ×3.24 on BT.2020.

That essay traced the spike to two facts. Dimming the blue primary scales its three cone-like signals together, and because ICtCp’s chroma rows sum to nought, a response that compresses like a logarithm reads that as a step in intensity alone. And the two primaries that can be added to the blue both feed its L and M signals and leave its S alone, so they move it in nearly the same direction — 45 degrees apart on sRGB, where CAM16-UCS sets them 90 apart. A corner whose three directions lie close together is sharp.

The prediction was that a shorter-wavelength blue would make both facts stronger — a violet primary’s L and M fall further below its S — so that ΔEITP’s blue would sharpen monotonically towards the violet and pass four times a margin by 450 nm, its sharpness following the angle between the two added-primary steps, while CAM16-UCS’s stayed near a face. If not, the essay said, the spike would depend on where in the quantiser’s range the blue’s signals sat, rather than on their ratios.

The spike peaks at 485 nm and fades into the violet #

With red and green primaries at BT.2020’s 630 and 530 nm and a monochromatic blue moved from 490 nm to 440, ΔEITP’s blue corner is sharpest with the blue at 485 nm, ×4.37 per unit of margin, and falls at every step to ×1.23 at 440 — nearly a face. CIELAB runs the other way, bluntest at 480 nm, ×1.92, and ×11.8 at 440; ΔE₀₀ follows CIELAB to ×7.15; CAM16-UCS stays under ×1.55 throughout. ΔEITP’s sharpness follows the angle between its two added-primary steps at a rank correlation of −0.98, but that angle widens towards the violet, from 37 degrees at 480 nm to 111 at 440. Adding red always raises the blue’s L signal by the larger fraction; adding green raises its L by the larger fraction from 460 nm up and its M at 440 and 450, because a violet primary carries too little M of its own — and from there the two steps point opposite ways along ICtCp’s L–M axis. Neither of the essay’s alternatives holds: a logarithmic response blunts the violet further, to ×1.17, and a brighter display dulls it, ×1.28 at 100 cd/m² to ×1.16 at 4,000, while both sharpen the 480 nm blue.

• The prediction is wrong in sign. A violet primary is ΔEITP’s bluntest blue, not its sharpest.
• The units trade sides. At a display’s ordinary blue ΔEITP sides with CIELAB; at a violet it sides with the appearance model, and ΔE₀₀ crosses the other way.
• The account named the right quantity and assumed it fixed. The spike does follow the added-primary angle; the angle is set by the blue’s own L:M ratio, which falls towards the violet.
• The quantiser’s range is not the cause. Every response, including a pure logarithm, blunts the violet.

The four curves cross. ΔEITP’s rises from the violet to a peak at 485 nm and falls a little beyond; CIELAB’s falls from the violet to a trough at 480 nm and rises beyond. ΔE₀₀ follows CIELAB at the violet end, ×7.15 at 440 nm, and flattens towards a face, ×1.05 to ×1.15, between 470 and 485. CAM16-UCS stays between ×1.08 and ×1.50 from one end to the other. On BT.2020’s own primaries — its blue near 467 nm — the construction gives ΔEITP’s ×3.24, the earlier essay’s number exactly, and that blue sits on the rising slope between 465 and 470 nm, well short of the peak.

The two families of unit agree about almost nothing here. At 480 nm ΔEITP calls the blue the sharpest corner it has seen on any display, more than four times a margin, and CIELAB calls it almost a face. At 440 nm CIELAB calls it a spike nearly twelve times a margin and ΔEITP almost a face. A specification that wrote its gamut margin in one unit and checked it in the other would find the blue corner’s cost wrong by a factor of up to ten, in a direction that depends on which blue the display uses.

The added primaries split #

The account said the spike follows the angle between the two added-primary steps, and it does: across the eleven blues, ΔEITP’s sharpness falls as that angle widens at a rank correlation of −0.98. What the account did not expect is how the angle moves. Adding red and adding green lie 37 degrees apart with the blue at 480 or 485 nm, 45 at 470, 67 at 460, 95 at 450 and 111 at 440. The angles between either added primary and dimming the blue move by under twenty degrees over the whole range; dimming stays nearly pure intensity, as the zero-sum chroma rows require. The corner fails because its two other directions come apart.

At the violet end the corner is a different shape altogether. With adding red and adding green 111 degrees apart and each about a right angle from dimming, the three directions spread over most of a hemisphere, and a corner like that is barely a corner: a margin moves it 1.23 times itself.

What splits them #

ICtCp’s second chroma axis, Cp, is mostly an L-minus-M axis, and it weights a difference between the two signals several times as heavily as intensity weights their sum. Adding red raises the blue’s L signal by three to five times the fraction it raises its M at every blue, so its step lies more than nine tenths along that axis, pointing L-ward. Adding green raises both. Which it raises by the larger fraction depends on how much of each the blue already carries, because the same added green is a larger fraction of a smaller signal. With the blue at 480 nm, adding green raises its L by 3.0 per unit of drive and its M by 2.2; the step points L-ward along the axis, the same way as adding red, and the two lie 37 degrees apart. With the blue at 440 nm, adding green raises its L by 12.0 and its M by 14.5; the step points M-ward, the opposite way, and the two lie 111 degrees apart. The turn happens between 450 and 460 nm.

The reason is in the blue primary’s own signals. At 480 nm the blue carries 1.9 times as much M as L; at 440 nm only 1.15 times, and both have fallen to about a sixteenth of its S. A green primary adds more M than L in absolute terms — ICtCp’s LMS matrix gives green 2,951 parts of M to 2,146 of L — and against a blue rich in M that is still the smaller fraction; against a violet with nearly equal, tiny L and M, it is the larger. The account was right that a violet’s L and M fall further below its S. It missed that they fall unequally: M falls faster, and the spike depends on M.

No response rescues it #

The essay’s alternative was that the spike depends on where in the quantiser’s range the blue’s signals sit: a violet primary’s L and M are small, and the perceptual quantiser behaves more like a power law at low signals and more like a logarithm at high ones. Two tests settle it.

A pure logarithm, which has no range at all, blunts the violet below the quantiser — ×1.17 at 440 nm against ×1.23 — while it sharpens the longer blues beyond it, ×4.71 at 480 nm against ×4.34. The cube root, the least compressive of the three, leaves the violet sharpest, ×1.62, and the longer blues bluntest. The response’s compression amplifies whatever geometry the matrices give: where the two added-primary steps agree, compressing sharpens the corner, and where they disagree, compressing blunts it. The quantiser’s range is not what decides which.

The brightness test agrees. At 480 nm a brighter display sharpens the corner, ×4.12 at a white of 100 cd/m² to ×4.79 at 4,000; at 440 nm it dulls it, ×1.28 to ×1.16, and the curves cross between 455 and 460 nm — where adding green’s step turns from L-ward to M-ward. A brighter display pushes every signal up the quantiser towards its logarithmic end, and so does what the logarithm does: more of the same geometry.

The violets were already the quantiser’s odd case #

This is not the first time ΔEITP has been found treating violets differently from every other colour. The hue disagreement is the quantiser’s found that as a display brightens ΔEITP shifts violets towards lightness and yellow-greens towards chroma, where CAM16-UCS shifts every colour alike, and traced it to the same curve: replace the quantiser with a pure power law and the disagreement vanishes exactly. How far a colour moved followed the spread of its three signals, and which way followed where its S signal sat against the other two. At the gamut’s edge the reds move as far then walked those shifts out to a BT.2020 display’s boundary and found the dark blues’ shifts capped by how little chroma BT.2020’s blue edge reaches.

The violet corner is the same story read through a different question. A violet primary is the most extreme set of signals a display can make — an S signal sixteen to nineteen times its L and M, and L and M nearly equal — and every construction that runs those signals through a compressive response and an opponent rotation gives a verdict on it that nothing else on the display tests. On colour differences between pairs, the verdict showed up as a shift towards lightness; on the shape of the solid, as a corner that is a spike in one unit and a face in the next. In both, CIELAB and CAM16-UCS, which have no quantiser, did not follow.

The blue’s hue turn was lost with a matrix found the appearance model’s blues decided by the cone space its compression acts in; here the deciding quantity is again a property of the cone signals before any compression — the blue’s own L:M ratio — and the compression only amplifies what that ratio has already set.

Where today’s displays sit #

The displays most screens are built to — sRGB, Display P3 and BT.2020 — put their blue primaries between about 460 and 470 nm in dominant wavelength, on the rising slope of ΔEITP’s curve and well short of its peak. They sit where the verdicts are most sensitive to the choice: five nanometres either way moves ΔEITP’s blue corner by half a margin, from ×2.10 at 460 nm to ×3.50 at 470, while CIELAB’s moves from ×4.64 to ×2.69 in the other direction. A display maker who nudged a blue LED’s peak from 465 to 460 nm to widen the gamut would, by the ΔEITP measure, have made its blue corner a quarter cheaper to margin, and by the CIELAB measure nearly a third dearer.

What a display maker would take from this #

A laser or narrow-band display picks its blue primary from a short list of practical sources, and the choice has always been argued in terms of gamut area and efficiency. A narrow primary buys a disagreement added the observer: narrower primaries raise how much observers disagree. This adds the tolerance unit. A blue near 465 to 485 nm makes the blue corner the costliest corner of the display for a ΔEITP margin and nearly free for a CAM16-UCS one; a violet near 440 to 450 nm reverses CIELAB’s verdict and ΔEITP’s at once. A high-dynamic-range specification stated in ΔEITP is, on these numbers, kindest to a display with a violet blue — which is exactly the display whose blue CIELAB and ΔE₀₀ call the sharpest corner they have ever measured.

None of the units is wrong. Each is a different claim about which differences are equal, and a display’s corners are where those claims are tested hardest, because a primary is the most extreme stimulus the display can make. Four primaries have a choice found a display with a fourth primary able to choose among exact matches; one that chose its blue for the corner a margin costs least would have to choose its tolerance unit first.

How the corners were read #

Each display’s primary matrix is built from the CIE 1931 chromaticities of monochromatic light at 630, 530 and the blue’s wavelength, on the observer’s 5 nm table, with D65 white; on BT.2020’s own chromaticities the construction reproduces the earlier census’s blue corner to six decimal places. A drive is carried to absolute XYZ with the white at the stated peak, then through BT.2020’s primaries, ICtCp’s fixed LMS matrix, the perceptual quantiser and the opponent matrix exactly as the broadcast specification defines them, ΔEITP’s space being 720 times I, half of Ct and Cp. CIELAB is against the display’s white; ΔE₀₀ acts through its full local matrix at the corner; CAM16-UCS is at the display’s white, 64 cd/m² and an average surround. A corner’s three directions are the finite-difference images of its drive moved by a ten-thousandth, and its move per unit of margin the shortest vector whose inward component on every face is at least one. The fractional rises are each added primary’s change to the blue’s L and M signals, before the quantiser, over the signals themselves, per unit of drive. The response test keeps ICtCp’s matrices and replaces the quantiser by ln(1 + Y) or the cube root of Y.

What the census does not reach #

Monochromatic primaries. A real laser is nearly monochromatic; an LED or quantum-dot primary has a width that mixes neighbouring wavelengths and would sit between the census’s neighbours.

Fixed red and green. A display with a different green — a 520 nm laser, say — would change how much M an added green brings, and with it where the turn falls.

Small margins. The tangent shapes hold to a few per cent at the margins the earlier essays tested; the violet’s CIELAB spike, at nearly twelve times a margin, is narrow enough that a margin of four units would already reach where its faces curve.

Still open: whether a green primary decides where the turn falls #

The turn between 450 and 460 nm is where adding green stops raising the blue’s L by the larger fraction, and that depends on the green as much as the blue. A shorter green adds more M against L; a longer one, more L.

The calculation is this census with the green primary moved too — 515, 530 and 545 nm, the blue from 440 to 490 at each — with ΔEITP’s blue corner, the added-primary angle, and the wavelength at which adding green’s step turns from L-ward to M-ward. The prediction is that the turn moves with the green: a 545 nm green, adding relatively more L, keeps adding green L-ward down to about 445 nm and keeps the violet sharp to there, while a 515 nm green turns it near 470 nm and dulls even BT.2020’s blue. If the turn does not move, the blue’s own L:M ratio decides it alone, and a display maker choosing a green has no say in what a ΔEITP margin costs at the blue.

A mechanism found at one point is a statement about a neighbourhood #

The habit here is about extrapolating a mechanism along the variable it depends on.

The account of the spike was right at the point where it was found: at a display’s blue the two added primaries move it in nearly one direction because both raise L and M. The extrapolation took one feature of that point — L and M small against S — and pushed it further, without asking whether the other feature the mechanism needed, that adding green raise L by the larger fraction, survived the push. It did not, and the turn came within twenty nanometres.

The failure mode is to extend a mechanism by strengthening the condition it was noticed under, without checking the conditions it silently depended on. The test is to write down every inequality the mechanism needs and follow each one along the variable, which here took one more column in the same table.