Four primaries have a choice

Three primaries matching three tristimulus values is a square system. Solve it and there is one answer, exact, with nothing left over to argue about — which is why every display ever built has three primaries — colour being exactly three-dimensional — and why nobody has had to decide anything about how they are driven.

Add a fourth and the system stops being square. There is now a one-parameter family of drives that all produce the same three numbers, they are all exact, no colorimeter can tell them apart, and they are not the same colour to anybody who is not the observer they were solved for. Somebody has to choose, and the choice is worth a fifth of the mismatch.

The claim #

A tristimulus match is not a match. It is an acceptance rate, and with four primaries the rate is something a designer can choose.

Three measurements:

• An exact match comes apart for almost everybody. Three primaries two nanometres wide, solved so their sum has exactly D65’s tristimulus values for the reference member of a population, are seen as a different colour by that population at a median of ΔE00 9.3 and a ninety-fifth percentile of 17.9. Two members picked at random disagree with each other by 4.6 units at the median.
• The narrower the primaries, the worse it gets, monotonically. From forty-nanometre primaries to two, the ninety-fifth percentile rises from 11.3 to 17.9 — and display technology has been moving in that direction for thirty years, deliberately, for the gamut.
• And a fourth primary buys a quarter of it back for nothing. Sweeping the free parameter takes the ninety-fifth percentile from 17.3 to 13.7 — a factor of 1.26 — at no cost in anything a colorimeter can report, no change in gamut, and no change in the white.

What a population of observers is #

Every other essay on this site computes for an observer: the CIE 1931 average, or this site’s own pigment template at its nominal peaks. This one replaces that single point with two hundred people, and the people are not invented — the five things that differ between two pairs of eyes have all been measured, and all five have been arguments to the site’s own retinal machinery with defaults nobody had ever varied.

The lens, entered as an age, because that is what it is a function of: it yellows monotonically and its density roughly triples between twenty and seventy.

The macular pigment, with a mean near 0.35 and a spread of about a third of that, measured from nearly zero to above one in individuals.

The cone outer segment’s optical density, near 0.4 with a spread of about nine per cent, which changes the shape of a sensitivity curve rather than its height — the self-screening argument, with a population attached.

And the three pigment peaks, with standard deviations of about one and a half nanometres for the long and medium and a little less for the short.

One thing has to be said before any number from this population is quoted. Its median member is not the 1931 standard observer, and cannot be made into it: this site’s cone template is a nomogram at stated peaks rather than the fundamentals the CIE’s physiological observer is built from, and the best 3 × 3 matrix taking one to the other leaves the median member differing from the standard by ΔE00 0.95 on average and 1.58 at worst over twenty-four natural reflectances under D65.

So nothing here is quoted as a difference from the standard observer. Every result is a spread within the population or a disagreement between two of its members, and both are invariant to which member is called the reference.

Why narrow primaries are worse #

The mechanism is geometric and worth seeing clearly, because it is the reason the finding is a technological trend rather than a curiosity.

A metameric match is an agreement about three integrals of a spectrum against three sensitivity functions. Change the functions and the integrals change; how much they change depends on how much the two spectra differed in between.

A broad primary integrates the observer’s sensitivity over a wide band, so it samples an average of it. Two observers whose curves differ in shape but agree roughly in area produce similar integrals — the differences average out across the band.

A narrow primary samples the sensitivity at essentially one wavelength. There is nothing to average, so any difference between two observers at that wavelength passes straight through into the tristimulus value.

And the effect is worst in the blue, because that is where the lens and the macular pigment absorb and where both vary most between people. A narrow blue primary samples exactly the wavelength at which two observers are least likely to agree.

Where the disagreement comes from #

Turning one source of variation on at a time and holding the others at their medians gives a ranking, and the ranking is the surprise.

The largest single source of observer disagreement is age, and age is the one thing about an observer that is not random. It is predictable, it is monotone, it is written on a passport, and no colour specification in existence has a field for it.

The shares do not sum to the whole and are not meant to: the variates enter a nonlinear function of the spectrum, so what the table supports is an ordering and an order of magnitude. The ordering is what matters, and it says that the literature’s favourite variate — the pigment peaks, which is where the word polymorphism comes from — is third.

The four variates combine in quadrature, at the median #

The table of one-at-a-time contributions is offered with a caution — the shares do not sum to the whole and are not meant to — and the caution can be replaced with a measurement, because there is an obvious way for four independent sources to combine and it can be tested.

Adding the four in quadrature gives a median of 9.61 against the measured 9.3: three per cent over. Adding them linearly gives 17.2, nearly double. So at the median the four variates behave as four independent contributions almost exactly, which is what a population drawn with independent variates should produce and is a check that the sampler is doing what it says.

The ninety-fifth percentile does not behave the same way. Quadrature predicts 20.53 and the measurement is 17.9 — fifteen per cent over, five times the median’s discrepancy. So the tail is shorter than independence would give, which is the signature of a metric that compresses at large differences: ΔE00 divides a chroma difference by the chroma it was measured at, so four large departures combining produce less than four independent large departures would.

That gives the table a use it did not have. The shares are not merely an ordering — they are a decomposition, valid at the median to three per cent, and a reader can therefore ask what fraction of the variance each variate carries:

Age carries about seventy per cent of the variance on both statistics. That is a much stronger statement than the largest single source, and it survives the tail’s compression: the lens’s share is identical at the median and at the ninety-fifth, while the other three grow as the tail is approached. The one variate a specification could actually record is the one that would remove seven tenths of the problem.

The fourth primary recovers exactly half of what narrowing cost #

The essay’s two headline improvements are measured against different baselines and are worth putting on one scale, because together they say what a designer is trading.

Broadening the primaries from two nanometres to forty takes the ninety-fifth percentile from 17.9 to 11.3, a factor of 1.58. The fourth primary, at two-nanometre primaries, takes it from 17.3 to 13.7, a factor of 1.26.

In logarithms the second is 0.51 of the first. A fourth primary gives back almost exactly half of what the move to narrow primaries cost — half, on a log scale, which is the scale these ratios compose on since they multiply rather than add.

That is the trade stated in one number, and it is the number a display maker would want. Narrow primaries buy gamut and cost 1.58 in observer agreement; a fourth primary costs an emitter and a driver and buys 1.26 back. Whether that is worth doing depends on the price of the fourth emitter against the price of the gamut the narrow primaries bought, and both are commercial rather than colorimetric — but the exchange rate is now a number rather than a direction.

Doubling the population is the check that the width of the band is the eyes rather than the sample, and the fourth primary’s freedom is where it matters most.

What the twenty-five survivors say about the sweep #

One small reconstruction, because the essay’s most interesting structural claim rests on a count.

Twenty-five of thirty-three candidates survive the positivity constraint, and the best sits at a share of 0.60, described as the largest the positivity constraint admits. A sweep from zero to 0.8 in steps of 0.025 puts step 24 at exactly 0.600 and gives twenty-five survivors and eight refusals, which reproduces both counts.

So the constraint bites at about 0.6 and the sweep looked as far as 0.8 — far enough to establish that the boundary is real rather than the sweep’s own edge. The claim that the answer is as much fourth primary as the other three can spare is therefore supported by the sweep having gone past the boundary and found nothing, which is the distinction between a limit and a cap that this collection draws elsewhere and does not draw here.

It also bounds how much more is available. The improvement runs from 17.3 to 13.7 over shares from zero to 0.6; if the curve continued at the same rate, a device with headroom to reach 0.8 would gain a further third of the same amount. A little more headroom in the other three channels is worth about one more unit, which is small enough to say that the positivity limit is not costing much and is not nothing.

What was computed, and how #

The match is a solve. Three primaries give three spectra; each spectrum’s tristimulus values under the reference member are a column; the three columns and the target make a 3 × 3 system, solved by Gaussian elimination. A solution with a negative weight is refused rather than clipped, because a negative amount of a primary is not a display setting.

The population is deterministic. Two hundred members from a stated seed, so a number quoted in this essay is reproducible rather than sampled. assertThePopulationIsReproducible requires the same seed to give the same people and a different seed to give different ones, which sounds trivial and is the assertion that catches a change to the sampler.

Each member is adapted to their own view of the ambient white by CAT16 before the difference is taken, which is the comparison the visual system actually makes. Doing it the cruder way — handing CIELAB the member’s own white point and letting its cube roots do the scaling — moves every number here by four per cent. That the two constructions agree is not obvious beforehand, since they differ by a full matrix, and it is asserted rather than assumed.

And the four-primary family is the same solve with one column held. The fourth primary’s power is fixed as a share of the target’s luminance; the remaining three are solved exactly against what is left; members needing a negative weight are dropped. Twenty-five survive out of thirty-three tried, and every one of them matches to better than 10⁻¹⁴.

The best member is at share 0.60 — the largest the positivity constraint admits. That is itself a result: the answer to “how much fourth primary” is as much as the other three can spare, so what stops the improvement is not colour but arithmetic, and a device with a little more headroom in its other channels would do better still.

The width of the primaries is the other lever, and pushing it down to two nanometres reaches the laser projector rather than stopping at a quantum dot.

Where the model stops #

The fourth primary’s wavelength is chosen, not solved. It sits at 590 nm because a sweep over candidate wavelengths put the best result there, and a sweep is not an optimisation: the joint problem — four wavelengths, four widths and a share, minimising a percentile over a population — has not been solved here and would need a different kind of search.

A percentile is not a design objective. Minimising the ninety-fifth percentile is one choice among several; minimising the median gives a different member, and minimising the worst case gives a third. A manufacturer would have to say which, and none of them says.

The three do not even agree here. The member that minimises the ninety-fifth percentile sits at the top of the admissible range; the median falls fastest a little before it; and the worst case is noisier than either because it is one person out of two hundred. Choosing between them is choosing whose complaint matters, which is a commercial decision rather than a colorimetric one — and it is the same decision, in a different trade, as the one a tolerance’s parametric factors encode.

And the population is a model of variation, not a sample of people. Five variates with quoted spreads, drawn independently, is not the same object as a measured cohort — real observers’ parameters are correlated, and the correlations are not in the literature this site has read. Independent draws will overstate the spread in some directions and understate it in others.

The white is also the easy case, and it is the only case computed. Both spectra here are broad and bright and near neutral, which is where adaptation does most work and where the three channels are all well fed. A saturated blue, where one of the two spectra is a single narrow primary and the short-wavelength channel is carrying the comparison nearly alone, is a different measurement and there is no reason to expect it to be smaller.

The generalisation #

The sentence worth carrying is: an underdetermined system is an opportunity, and colour keeps producing them.

A four-ink separation is not unique and the freedom is spent on ink cost and shadow stability. A metameric pair can be constructed to order because the metameric black space is enormous. And now: four primaries matching three numbers leave one degree of freedom, and it can be spent on making the match work for more people.

In every case the extra dimension is invisible to the measurement that defined the problem. A colorimeter cannot see which member of a separation family was used, cannot see which metamer is on the page, and cannot see where in this family a display is driven. The freedom is real, it has consequences somebody notices, and the instrument that certifies the result is blind to all of it.

The surprising part is the direction. Multiprimary displays are built and sold for gamut — a fourth or fifth emitter reaches colours three cannot. The observer-robustness gain measured here comes with the same hardware, costs nothing, requires no change to the content pipeline, and as far as this site can tell is not something anybody quotes.

Who found it, and when #

Observer metamerism has been known since the standard observer was published, and quantified since the CIE’s 1989 special metamerism index for observer differences. It became an industrial problem rather than a curiosity when narrow-primary displays arrived: two people looking at the same laser projector or quantum-dot panel disagree visibly about the white, and the disagreement is reported on the shop floor rather than in a laboratory.

The CIE’s 2006 physiological observer and the individual observer models built on it are where the five variates and their spreads come from. Their purpose was exactly this: to replace one average with a family whose members can be sampled.

And the multiprimary display literature is largely about gamut, from the six-primary projectors of the early 2000s onward. The degree of freedom is discussed there — it has to be, since somebody must write the conversion from three numbers to four or more drives — and it is usually spent on power efficiency or on keeping the drives away from their limits.

What has not been done, as far as this site can tell, is to spend it on the observer. The arithmetic is one sweep, the objective is a percentile over a published population model, and the answer is a number a manufacturer could put in a calibration file. It is the kind of gap that stays open because the two literatures — display engineering and individual colorimetry — publish in different places and use the freedom for different things.

The same population can be asked about a tolerance rather than a match, which is the form the question takes for anybody writing a specification.

What the pictures cannot show #

Nothing in this family draws a colour patch, and that is deliberate. The subject is a disagreement between observers, and a patch printed on this page arrives at one reader’s retina through one reader’s lens. A swatch purporting to show “what the other observer sees” would be the one figure on this site that cannot be honest — it would be a stimulus, chosen by this site’s median observer, delivered to the reader’s own eyes, which are neither.

And the reader cannot be placed in the distribution. A reader who knew their own lens density, macular pigment and pigment peaks could be located on the histogram. Nobody knows those things about themselves, three of the four require an instrument that is not in ordinary use, and the fourth — the age — is the one everybody has and nobody quotes.

Where the ladder goes next #

The nearest unfinished piece is the joint search: four wavelengths, four widths and a share, against a stated objective over the population. That is an optimisation rather than a sweep, and it would produce a designed four-primary display rather than a chosen point in one family.

The second is the correlations. Independent draws from five quoted spreads is the simplest population that can be built from published numbers, and it is certainly wrong in detail — lens density and macular pigment are both age-dependent, so drawing them independently produces people who do not exist. The repair needs a covariance nobody has published.

And the third is the one with the most consequence: everything here concerns a white. A display’s white is the easiest case, because both spectra are broad and bright. The same measurement on a saturated colour, where one of the two spectra is a single primary, is a different and probably worse number, and it is one line of code away.