One wavelength is everyone's colour

The previous essay’s zero was an artefact of a grid. The identity underneath it is not, and it has a consequence that runs the opposite way from the artefact.

The claim #

A stimulus containing one wavelength is the same colour for every observer, exactly — and a stimulus containing three is where observers differ most.

• The identity is trivial and exact. Both the sample’s and the white’s cone responses are proportional to the same three sensitivity values, so the ratios are identical for any observer.
• Measured at 2 × 10⁻¹³ ΔE₀₀ between a twenty-year-old lens and a seventy-year-old one, at 545 nanometres.
• Three wavelengths break it comprehensively, at 3.78 ΔE₀₀ for the macular departure and 3.60 for the pigment peaks — the largest of any light in this round.
• And that is the whole account of why narrow primaries cost more. A broad primary integrates over a sensitivity curve and averages the differences between observers away; a narrow one samples the curve at a point and reports the difference in full.

The identity, in two lines #

A stimulus concentrated at a single wavelength λ₀ gives cone responses φ₀·lᵢ(λ₀) for i = 1, 2, 3. The white the observer is adapted to, being the same source, gives w₀·lᵢ(λ₀).

The relative excitations are φ₀/w₀ in every channel — the same number three times, and independent of lᵢ entirely. So the stimulus is a neutral of the appropriate lightness for every observer that has ever existed, and it is that exactly rather than approximately.

The identity is a special case of the neutral one and it comes from the other side. There, the sample was a scalar multiple of the light because the reflectance was constant. Here it is a scalar multiple of the light because both are supported at one wavelength, and there is nothing else for either to be.

Both are the pairing with the stimulus’s factor emptied, and they exhaust the ways of emptying it: either the sample does not modulate the light, or the light has nowhere to be modulated.

Three wavelengths are the opposite. Add a second line and the identity is gone, and the way it goes is instructive.

With two lines at λ₁ and λ₂ the relative excitations are

eᵢ = [φ₁ lᵢ(λ₁) + φ₂ lᵢ(λ₂)] / [w₁ lᵢ(λ₁) + w₂ lᵢ(λ₂)]

and the sensitivities no longer cancel. What decides the answer is the ratio lᵢ(λ₁)/lᵢ(λ₂) in each cone, which is a ratio of two point values of a curve — and a ratio of two point values is the quantity most sensitive to any change in that curve’s shape.

That is the sharpest available statement of why narrowband sources are hard. A broad source integrates each sensitivity over a wide band, and an integral of a curve is much less sensitive to the curve’s details than two samples of it are. Shifting a cone’s peak by three nanometres changes its integral against a broad phosphor by a fraction of a per cent and changes the ratio of its values at 532 and 638 nanometres by several per cent.

Integration averages an observer difference away and sampling does not, and a display is a device for making colours out of three samples.

The measurement #

On a quarter-nanometre grid the three-laser projector produces the largest departures in the round for the macular pigment at 3.78 ΔE₀₀ and for the pigment peaks at 3.60, against 1.71 and 2.37 under daylight.

The three-emitter LED, whose emitters are eighteen to thirty-three nanometres wide, sits between: 3.01 for the macular and 2.84 for the peaks. And the white LED, with a broad phosphor and a single narrow pump, gives 3.37 and 1.77 — high for the macular because its pump lands on the macular absorption band and ordinary for the peaks because its phosphor is broad where the cone peaks are.

That ordering is a clean monotone in bandwidth for the peaks departure and is not for the macular, and the difference between those two is worth having. The peaks departure is about the shape of the cone curves near their maxima, and it responds to how narrowly the source samples them. The macular departure is about a filter in a fixed band, and it responds to whether the source has power there at all.

Bandwidth predicts one departure and position predicts the other. A lamp designer choosing where to put a narrow emitter and how narrow to make it is trading against two different things.

What this says about display design #

The result is not new to the display industry, which has measured observer metamerism on wide-gamut and laser displays directly and found it to be a real production problem. What the decomposition adds is which knob does what.

Narrowing a primary raises the peaks departure and does so roughly monotonically. A primary of eighteen nanometres is already most of the way to a laser for this purpose; the difference between eighteen and a fifth of a nanometre is 2.84 to 3.60, a quarter, where the difference between a broad phosphor and eighteen nanometres is 1.77 to 2.84, a factor of one and a half.

Moving a primary changes which departure it excites. A blue primary at 452 nanometres sits in the macular band; one at 470 does not, and the macular departure falls accordingly. That is a design choice with a real cost — a longer-wavelength blue primary shrinks the gamut’s blue corner — and it is a trade nobody states in those terms.

And a fourth primary helps for a reason unrelated to gamut. A fourth primary is a design rather than a gamut increase, and one of the things it can be designed for is observer agreement: four primaries can render a colour with more than one drive combination, and choosing the combination that minimises the spread across a population is a free optimisation once the hardware exists.

There is a rule of thumb in that comparison worth extracting. Across the six departures the projector’s numbers are between 1.4 and 2.3 times the daylight ones on the same sample, and the three-emitter LED’s are between 1.1 and 1.4. Going from a broad source to a three-line one roughly doubles the observer spread, and most of the doubling has already happened by the time the emitters are twenty nanometres wide.

That is useful for anybody deciding how much to care. A display built from twenty-nanometre emitters — which is most of what is sold — is already close to the narrowband limit for this purpose, and narrowing further to lasers buys gamut without much additional observer cost. The large step is the one from a phosphor to a set of emitters, and it has already been taken across the whole industry.

Where the spectral locus comes into it #

There is a geometric reading of the identity that connects it to something every chromaticity diagram already shows.

The spectral locus is the set of chromaticities of monochromatic stimuli, and it is drawn on every diagram in this collection. The identity says that a monochromatic stimulus is observer-invariant in relative excitations, which is not the same as saying its chromaticity is observer-invariant — chromaticity is computed against a fixed white rather than against the stimulus itself, so the locus does move between observers.

The two statements are compatible and the difference between them is the whole of what an adapting white does. Judged against itself, a monochromatic light is a neutral for everybody. Judged against a different white, it is a saturated colour whose position depends on the observer, and the two standard observers’ loci differ visibly.

So the identity is not a claim that observers agree about spectral colours. It is a claim about a particular arrangement — a stimulus judged against itself — and the arrangement is the one that arises when a whole display is driven by lines.

Why the flanks matter more than the peaks #

That figure carries the mechanism and it is worth reading carefully, because the naive expectation is wrong.

Shifting a pigment’s peak by three nanometres moves the whole curve. At the maximum the curve is flat, so a three-nanometre shift changes the value there by almost nothing. On the flanks the curve is steep, so the same shift changes the value by several per cent.

A narrow primary placed at a cone’s peak therefore excites the peak departure hardly at all. One placed on a flank excites it strongly. And the flanks are where display primaries live, because a primary at the middle-wavelength cone’s peak would be a desaturated yellow-green rather than a green.

The primaries that make a wide gamut are the primaries on the steep parts of the sensitivity curves, and steepness is exactly what makes a peak shift matter. The gamut and the observer spread are being bought and sold in the same currency, which is why this trade cannot be engineered away.

What a population would say about it #

Everything above is two observers differing in one parameter, which is the round’s method and is not what a manufacturer needs.

This collection has the machinery for the other question and has had it for several rounds. A gamut has a population samples two hundred eyes from the same template with all four parameters varying together, and it reports what fraction of them accept a match made for the standard observer. That is the quantity a specification wants, and the single-parameter departures here are its decomposition.

The two agree about the direction and cannot be compared numerically, because a population statistic and a two-observer difference are different objects — one is a spread and the other is a distance. What the decomposition adds to the population result is which parameter to blame, and therefore which design change would help. A population number says a display is difficult; the departures say whether narrowing the primaries or moving them is the thing to change.

Putting the two identities on one figure closes the account of how a departure vanishes. There are exactly two ways to empty the stimulus’s factor of the pairing: make the sample fail to modulate the light, or make the light have only one place to be modulated. The first is a grey card and the second is a laser, and they are as far apart in appearance as two stimuli get.

The walk also gives the scale on which the second identity is fragile. A source that is nearly monochromatic — two lines at 545 and 546 nanometres, say — is nearly observer-invariant, because the sensitivity ratios between two adjacent wavelengths are nearly one for everybody. What matters is the spread of the lines rather than their number, which is why a three-line source at 465, 532 and 638 is the worst case and a three-line source clustered in the green would not be.

What was computed, and how #

The identity is asserted directly: a Gaussian of a fifth of a nanometre at 545 nanometres, a red pigment, and two observers differing by fifty years of lens ageing. The assertion requires the difference below 10⁻¹⁰ ΔE₀₀ and it measures 2 × 10⁻¹³.

The multi-line results are computed on a quarter-nanometre grid, which is fifty times finer than the linewidth and is checked against the tabulation audit’s own tenth-nanometre reference for consistency on the shared lights.

The lights are the tabulation audit’s six, unchanged, so that the numbers in this essay are comparable with every other number in the round. That sharing is what made the concealment finding available at all, and it is the main methodological result of running two audits together.

The identity is also a design tool. Read forwards rather than backwards, the identity says something a display engineer can use, and it is the only genuinely constructive result in this part of the round.

A stimulus judged against itself is observer-invariant. A display’s white point is judged against itself, because the white is what the eye adapts to — so every observer agrees about a display’s white, whatever its primaries, exactly. The disagreement appears only for colours that are not the white, and it grows with distance from it.

That is why a laser projector’s white looks right to everybody and its saturated colours do not, which is the reported field experience and is usually attributed to the white being carefully calibrated. It is not the calibration; it is an identity, and no calibration was needed for it.

It also says where a calibration effort is worth spending. Adjusting a display’s white to please a population is adjusting something every member already agrees about. Adjusting its primaries’ drive at a saturated colour is where the population actually differs, and it is the adjustment no consumer display offers. A primary is chosen for four things and agreement across a population is not among them.

Where the model stops #

The three-laser projector is a construction and its lines are Gaussians of a stated width. Real laser projectors are speckle-reduced by deliberate broadening, by angular diversity or by multiple longitudinal modes, and a broadened source sits somewhere between this construction and a three-emitter LED.

The observer differences are one parameter at a time at literature strengths, and the largest numbers here are for single parameters. A real pair of observers differs in all of them, and the combination on a narrowband source has not been computed.

And nothing here is a measurement of people. Every number is what this collection’s pigment template predicts, and the template’s own residual against the tabulated observer is 1.42 ΔE₀₀ at the median — which is a third of the largest effect reported here.

There is a last practical corollary about measurement rather than display. An instrument that characterises a narrowband source has to resolve its lines, and an instrument that characterises a broad one does not — so the same colorimeter that is adequate for a phosphor lamp is inadequate for the display beside it, by the same argument that governs the observers. A tristimulus colorimeter with broad filters averages a display’s primaries the way a broad primary averages an observer’s cones, and the averaging hides the same information.

That is why display measurement moved to spectroradiometers as primaries narrowed, and it is the instrument-side twin of everything above. The two are the same fact about sampling a curve, appearing once in the eye and once in the meter.

The generalisation #

The habit is about the difference between sampling a function and integrating it.

Two systems that differ in a function’s shape will disagree about a sample of it and may agree about an integral of it, because integration is a smoothing operation and smoothing removes exactly the differences that are local. That is why broad instruments are robust and narrow ones are not, in every field where a response curve varies between instances.

The move that exploits it is to broaden deliberately when agreement matters more than efficiency. A wide filter, a broad primary, a long exposure: each of them trades a resource for insensitivity to a parameter nobody controls.

The failure mode is to treat narrowness as a free improvement. It buys resolution, gamut or efficiency depending on the setting, and it always buys sensitivity to whatever varies between the systems doing the measuring. Narrowing a source moves a system from averaging over a population to sampling one member of it, and the population is the thing that was being averaged.

Who found it, and when #

That monochromatic stimuli are the extreme case for observer metamerism is implicit in every treatment of the subject and is stated explicitly in the display literature of the last fifteen years, as laser and quantum-dot primaries reached production.

The identity itself is too simple to have an attribution. It is one line of algebra and it appears wherever anybody works out what a monochromatic source does to a colorimetric calculation — usually as a step towards something else, and usually without the observation that it makes the source observer-invariant.

Where the ladder goes next #

The departures have been measured against both factors of the pairing and against the arithmetic underneath. What remains is to take each one apart in its own terms, and the first is the one the CIE publishes two observers for: a field size is two changes rather than one, and they do not act in the same place.