The peaks move the flanks

The last of the six departures does not filter anything. It moves a curve, and where a moved curve differs from an unmoved one is not where anybody looks for it.

The claim #

A peak shift is a derivative, so its effect is largest where the curve is steepest — on the flanks, which is exactly where display primaries sit.

• A three-nanometre shift changes a cone’s sensitivity at its own peak by about a tenth of a per cent and on its steepest flank by several per cent, because the derivative of a smooth maximum is zero.
• It costs 2.37 ΔE₀₀ on a red pigment under daylight, the second-largest of the six at that setting, and 1.39 at the median over forty-two surfaces.
• Under a three-laser projector it is 3.60, the largest single observer number this round produces, because three narrow lines sample the curves at three points and every point is on a flank.
• And the shift is not hypothetical. The long-wavelength pigment carries a common polymorphism worth several nanometres, and it is present in a substantial fraction of the population.

What a peak shift is #

A visual pigment’s absorbance is a function of λmax/λ rather than of λ — the template shape is fixed and slides in log-wavelength as the peak moves. So a shift is a horizontal displacement, and its effect on the value at any fixed wavelength is, to first order, the shift times the curve’s slope there.

At a maximum the slope is zero. Three nanometres of shift produce a second-order change, which is proportional to the curvature and to the square of the shift — a tenth of a per cent for the numbers here.

On a flank the slope is large. The middle-wavelength cone’s sensitivity falls by about a factor of ten between 541 and 620 nanometres, which is a mean logarithmic slope of about three per cent per nanometre, so a three-nanometre shift is roughly a nine per cent change in sensitivity there.

A hundredfold difference between the effect at the peak and the effect on the flank, from the same shift, and nothing about the parameter’s name suggests it. Calling it a peak wavelength describes where the curve’s maximum is and not where the parameter does its work.

Where the shift comes from #

The long-wavelength pigment carries a well-known polymorphism at position 180 of its opsin, where a serine or an alanine changes the peak by about four nanometres — 556.7 against 552.4 in the usual figures. The two alleles are both common, so a substantial minority of the population has a long-wavelength peak several nanometres from the majority’s.

This collection has an essay about what the pigments that remain can be measured to, and the polymorphism is the reason its precision matters: a Rayleigh match is sensitive to the separation between the long- and middle-wavelength peaks, and that separation is what the polymorphism moves.

Beyond the polymorphism there is ordinary variation. Reported standard deviations are about a nanometre and a half for the two long-wavelength pigments and a little less for the short one, so this round’s departure — three nanometres on the long, one and a half on the middle, 1.3 on the short — is roughly two standard deviations with the polymorphism on top.

That is a departure whose distribution in the population is bimodal on one parameter and Gaussian on the others, which is unlike any of the other five and is a complication no summary statistic carries.

Why narrow primaries are the worst case #

A broad source integrates a cone’s sensitivity over a wide band. Integration averages the flanks against the peak, and a shift that raises the sensitivity on one flank lowers it on the other, so a good deal of the effect cancels inside the integral.

A narrow source does not integrate. It samples, and it samples at one wavelength — which is somewhere on a flank for at least two of the three cones, because a primary at a cone’s own peak would be a badly-chosen primary.

The measurement bears that out. The peaks departure is 2.37 ΔE₀₀ under daylight, 2.84 under a three-emitter LED, and 3.60 under a three-laser projector: a monotone increase as the primaries narrow. And the identity underneath says why the projector is not worse still — a single line would give exactly zero, and it is the ratios between three lines that carry the disagreement.

So the ordering of a display technology’s observer risk follows its primary bandwidth, and the peaks departure is the term that follows it most cleanly. The macular departure follows the primaries’ position instead, and the lens follows the lamp’s overall blue content, so the three respond to three different design variables.

That figure is why the peaks departure had to be measured twice. On the site’s own five-nanometre grid the projector’s row is exactly zero, so the largest number this departure produces is invisible to every other figure in the round — and the departure’s own ladder, distribution and light table were all computed on that grid.

The consequence is a caveat on the numbers above rather than an error in them. Every peaks figure quoted from the five-nanometre grid is correct for the five lights the grid resolves, and the sixth had to be computed separately. Nothing else in the round needed that treatment, because no other departure has its maximum under a source the grid cannot hold.

The trade nobody can engineer away #

There is a structural reason the display industry cannot solve this, and it is worth stating because it looks like a solvable problem.

A wide gamut needs primaries far from the white point, which means far out along the spectral locus, which means at wavelengths where at least one cone’s sensitivity is small and falling. Small and falling is the definition of a flank.

A primary placed at a cone’s peak would be maximally observer-robust and would be a poor primary: the middle-wavelength cone’s peak is at 541 nanometres, which is a yellowish green with a chromaticity well inside the locus and a small gamut contribution.

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. A primary is chosen for four things and this is a fifth that pulls against the first, so the trade is real rather than an oversight.

The only lever that escapes it is a fourth primary. With four, a colour has more than one drive combination, and the combination that minimises the spread across a population can be chosen — which is one of the things a fourth primary is a design for and is not usually one of the things it is sold as.

What the shift does to a match rather than a colour #

There is a version of this that is much larger than any ΔE₀₀ in the table and it belongs to metamerism rather than to colour.

Two lights that match for one observer differ in spectral shape by a metameric black — a difference annihilated by all three of that observer’s sensitivities. Move a peak by three nanometres and the three sensitivities are different functions, so the same difference is no longer annihilated, and the two lights no longer match.

How badly depends on how much spectral structure the metameric black has near the flanks, and a metameric black constructed between two narrowband sources has all of its structure there. That is why observer metamerism is a display problem rather than a paint problem: paints differ from one another smoothly and displays differ from paints in lines.

The numbers in this round measure a colour difference between two observers looking at the same stimulus. A match failure is the same phenomenon evaluated on a pair, and it is larger, because the pair was constructed to be exactly balanced for one of the two observers and nothing balances it for the other.

A nanometre can be priced. The departure can be divided by its own cause to give a quantity a reader can carry, and it is a useful one.

Six nanometres of separation between the two observers’ long-wavelength peaks, three between their middle-wavelength ones and 2.6 between their short-wavelength ones produce 2.37 ΔE₀₀ under daylight. Divided through, that is roughly 0.4 ΔE₀₀ per nanometre of long-wavelength peak difference on an ordinary saturated sample under a smooth light, and about 0.6 under a three-emitter LED.

The polymorphism alone is about four nanometres, so it is worth one and a half to two and a half units between two people who differ in it — which is above every delivery tolerance in this collection and is decided by a single amino acid.

That number also puts the measurement literature in perspective. A Rayleigh match can locate a peak to a fraction of a nanometre, and what such a measurement is worth is now expressible: a tenth of a nanometre of precision is worth four hundredths of a colour difference, so the measurement is far more precise than any colour application needs and exactly as precise as the genetics requires.

The narrow span, explained #

The peaks departure spans a factor of eight across the surface family, which is the second-narrowest of the six after the rods, and that is the opposite of what the flank argument might suggest.

The resolution is that the family’s samples are all smooth. A smooth reflectance has structure spread over a hundred nanometres or more, so it overlaps every cone’s flanks to some degree and none of them selectively. There is no sample in the family that misses the flanks the way a red pigment misses the macular band.

So the peaks departure behaves like a broad departure when measured on broad samples and like a narrow one when measured on narrow lights. The localisation that matters is the localisation of the pairing, which combines the departure’s own shape with the stimulus’s, and either can supply the sharpness.

That is why the light table and the sample table give different answers about this departure, and why quoting one without the other misrepresents it: it is unremarkable across surfaces and it is the largest term in the round under a narrowband source.

The three-emitter LED is the one commercially widespread light under which the peaks departure leads, and that is worth more attention than the laser projector. Three-emitter and quantum-dot backlights are ordinary consumer hardware; three-laser projection is not.

Under it the peaks departure is 2.84 ΔE₀₀ against the lens’s 2.57 and the macular’s 3.01 — so two of the three leaders are narrowband effects and the third is the blue pump again. A display technology has moved two departures to the top of the ladder that sit fourth and fifth under daylight, and it has done so for reasons entirely unconnected to vision.

What was computed, and how #

The shift is applied to the pigment template’s λmax argument, which slides the whole curve in the template’s own variable — the physically correct operation, since a visual pigment’s absorbance is a function of λmax/λ and not a curve that gets translated.

The three shifts are 3.0, 1.5 and 1.3 nanometres for the long-, middle- and short-wavelength pigments, applied in opposite directions for the two observers, so the comparison spans six, three and 2.6 nanometres respectively.

The fine-grid numbers use a quarter-nanometre tabulation for the reason the concealment essay establishes: on the site’s own five-nanometre grid a three-laser projector is a one-line spectrum and every observer departure it produces is exactly zero.

There is one more asymmetry in the flank argument that a display engineer would want. The two long-wavelength cones overlap heavily — their peaks are twenty-five nanometres apart and their curves are similar in shape — so a stimulus that lands on the middle-wavelength cone’s long flank is also landing on the long-wavelength cone’s short flank, in the opposite direction. Shifts in the two peaks therefore partly cancel for such a stimulus and add for one further out.

That is why the red primary is the least troublesome of the three for this departure and the green is the worst: at 625 nanometres both long-wavelength cones are falling together, and at 528 they are rising and falling against each other. A green primary is where the two curves’ derivatives have opposite signs, which is the maximum of the disagreement rather than an incidental feature of it.

Where the model stops #

The template is Govardovskii’s, and a shift within it is a shift of a fitted shape rather than of a measured curve. Real pigment variants differ slightly in shape as well as in position, and nothing here carries that.

The three shifts are applied together and in the same direction, which is not how the polymorphism works — it affects the long-wavelength pigment specifically, and the middle-wavelength one has its own separate variation. A single-parameter version, moving only the long peak, would be the more faithful model of the polymorphism and is not what this round computes.

And the population distribution is bimodal rather than Gaussian for the long-wavelength peak, so a two-standard-deviation comparison is a poor description of the population’s actual spread on that parameter.

A last note about the word peak, which has caused trouble in this collection before. A pigment template is parameterised by its maximum because that is the one feature of it anybody can measure directly, and the collection has an essay about how much of the template is in fact its tail rather than near its maximum. This essay is the same observation about the parameter rather than about the curve: the peak names the curve and the flanks carry the argument.

The generalisation #

The habit is about where a parameter does its work.

A parameter’s name usually describes where it is defined — a peak wavelength, a centre frequency, a mean — and rarely describes where changing it has an effect. For a displacement parameter those two places are systematically different: the definition is at the extremum and the effect is at the steepest point, and the two can be a long way apart.

The way to find out is to differentiate rather than to reason. The derivative of the output with respect to the parameter is a function of wavelength, and looking at that function says immediately where the sensitivity lives. It takes one numerical difference and it is skipped constantly in favour of the intuition that a peak parameter is about the peak.

The failure mode is to design a measurement at the place the parameter is defined. A protocol that characterises a pigment by measuring near its peak is measuring where the parameter has the least effect, which is the same inversion this round found about cone optical density and is a general hazard of parameterising by extrema.

Who found it, and when #

The serine–alanine polymorphism at position 180 was identified in the late 1980s and early 1990s through the molecular genetics of the long-wavelength opsin, and the associated shift in Rayleigh matches had been observed earlier without an explanation.

The template form — that a pigment’s absorbance is a function of λmax/λ — goes back to Dartnall’s nomogram of 1953 and is refined in Govardovskii’s 2000 template. That the effect of a peak shift is therefore concentrated on the flanks is an immediate consequence and is nowhere stated as such, presumably because it is obvious once written down and is not obvious before.

Where the ladder goes next #

Six departures have now been measured, decomposed and paired. What remains is to say what the object they depart from actually is, because it is not a description of anybody’s eye: a standard observer is a contract, and the round’s results are about what the contract costs.