The population rests on a template

Every observer on this site — the two hundred members of its population, the median member its lamp rankings use, the reference member its confusion points are derived from — is built out of one formula, and the formula was written down once.

The claim #

The template is auditable, the choice between published nomograms is worth about five per cent of a residual, and the obvious alternative to a template is not an alternative at all.

• Every cone here starts as a nomogram. Govardovskii’s 2000 formula for an A1 visual pigment, evaluated at a stated peak wavelength, then put through Beer–Lambert and the ocular media.
• The tabulated fundamentals cannot replace it, because a tabulated fundamental has no peak wavelength to move. Removing the argument removes the population.
• So the choice was never “template or fundamentals”. It was “a population or one observer”, and the template is the price of the first.
• Between the two published nomograms the median observer moves 0.052 ΔE2000, on a residual of about 0.95 — five per cent.
• And the simpler nomogram is the closer one, which is a result about what the residual is rather than an argument for changing template.

What a template is #

A visual pigment absorbs light, and how much it absorbs at each wavelength is a curve. Measuring one curve means microspectrophotometry on individual receptors, which has been done for a modest number of species and a small number of human cones.

A nomogram is a formula that produces the whole curve from one number: the wavelength of peak absorbance. The claim behind it is empirical and strong — that visual pigments, across an enormous range of animals and peaks from the ultraviolet to the far red, have essentially the same shape when plotted against a suitable variable, and differ only in where that shape sits.

Govardovskii’s 2000 formula is the current standard version. Its α-band is a sum of three exponentials in λmax/λ with a shape parameter that depends slightly on λmax, and it carries a second, smaller β-band below 400 nm. This collection computes rod sensitivity from it rather than tabulating one, and computes all three cone classes from it at stated peaks.

What follows the template is held in every comparison here: the same optical density through Beer–Lambert, so that a cone’s absorptance is broader than its pigment’s absorbance; the same lens and macular pigment at the population’s median values; and the same least-squares fit from cone responses to tristimulus values, refitted per template so that no comparison is measuring a mismatch of bookkeeping.

Why the fundamentals are not the alternative #

The choice this collection wrote down as unreached was that the population is a pigment template rather than the physiological fundamentals. Attempting to reach it is where the interesting part is.

The physiological fundamentals — the cone sensitivities themselves, as tabulated by the CIE — are a set of three curves. They are a measurement, they are what everything downstream of a cone should really be computed from, and using them here instead of a nomogram is impossible.

A tabulated fundamental has no peak wavelength. It is one curve, measured once, on whoever was measured. There is no λmax argument to it, because it is not a function of anything; it is a list of numbers.

That matters because this collection’s population is two hundred people who differ, and one of the five ways they differ is the peak wavelength of each pigment — a real, measured, polymorphic variation of several nanometres in the long-wave pigment especially. The template is what turns that variation into an observer. Remove the peak argument and the population has two hundred identical members.

So the choice named as unreached was misnamed, and correcting it is the useful thing this essay does. It was not template or fundamentals; it was a population or one observer, and the template is what a population costs. What remains auditable is which template, and that is the rest of this.

The menu #

Two published nomograms and two caricatures, all evaluated at the same peaks.

Govardovskii (2000) is the site’s: α-band plus β-band, with a peak-dependent shape parameter.

Lamb (1995) is what Govardovskii’s is a refinement of — the same functional form with fixed coefficients and no β-band. Worth having precisely because it is close: if two nomograms differing by a small correction and a secondary band moved everything, the choice would be a liability.

A Gaussian in wavelength has the same width and no asymmetry at all.

A Gaussian in wavenumber is the shape a chemist would draw, since an absorption band is roughly constant-width in energy — and in wavelength it is asymmetric the wrong way, with a long tail towards the red where a real pigment has one towards the blue.

Both caricatures are matched to Govardovskii’s own α-band width in wavenumber, measured rather than guessed, so neither can be dismissed for being the wrong size.

What the choice is worth #

The quantity to score them on is one this collection already publishes and never attributes: how far the median observer sits from the 1931 standard.

The statement in the machinery is that this site’s median member is not the 1931 observer and cannot be, because the template is a nomogram rather than the tabulated fundamentals. The residual is real, it is bounded rather than removed, and nobody had asked how much of it is Govardovskii’s.

Two published nomograms, 0.052 apart on a residual of 0.95 — about five per cent. Two caricatures, 0.16 and 0.79 away, which is three and fifteen times as far.

So the choice of published template is a small term and the choice to use a real pigment shape is a large one. That is the reassuring answer and it is the one a reader should take away: nothing in this collection’s population work turns on Govardovskii rather than Lamb.

The direction is the interesting part #

The uncomfortable reading is in the ordering. Lamb’s older, simpler nomogram puts the median observer closer to the standard than Govardovskii’s does, and removing Govardovskii’s secondary band brings it closer again.

The natural reading of the residual exists because the template is a nomogram is that the residual is a deficiency, and that a better template would shrink it. Both simplifications shrink it. So the natural reading is wrong.

The residual is structural. It is a mismatch between an observer built from pigments — three absorbance curves, self-screened, filtered by the media, then fitted to tristimulus values — and an observer built from matches, which is what the 1931 functions are: the average of what seventeen people did with three knobs. The two are different objects and a better pigment model does not make them the same object.

None of which is an argument for using Lamb. A nomogram is fitted to microspectrophotometry of individual receptors, not to colour matches, so agreement with the standard observer is not what either was trying to achieve and is not evidence about which is right. A coefficient that improves an unrelated agreement when moved off its measured value is a warning rather than an opportunity, and this collection has made that mistake’s shape explicit before.

What else the template decides #

Four things downstream, in increasing order of how much they depend on it.

The population’s spread. Two hundred members differing in five measured ways, and the spread between them is the quantity most of the population work reports. It depends on the template only through the shape each member’s curves take, and the five variates dominate.

The lamp rankings. Which lamp renders a set of surfaces best is computed over the population, and the median-to-standard offset is not the same for every lamp — which is the finding that the observer model does more work than the lamps being ranked.

The confusion points. A dichromat’s copunctal point is the null space of the two cone rows that remain, so it is derived from the fitted matrix and moves when the template does. The claim this collection restated in nanometres rests on it, and how far it moves is the subject of a separate essay.

And the fitted cone-to-tristimulus matrix. Refitted per template here, which is the step that makes the comparison fair. A template producing worse cones scored against a matrix fitted to the site’s template would be measured partly on the mismatch, which is bookkeeping rather than a property of the template.

The refit is not a detail #

One step in the comparison deserves separating, because getting it wrong would have produced a table that looked exactly as convincing and meant nothing.

Every observer here needs a 3×3 carrying its three cone responses to tristimulus values, and this collection fits one by least squares against the 1931 colour-matching functions on the reference member. That matrix is refitted for every template.

The alternative — fit it once on Govardovskii and use it for all five — is the obvious implementation and is wrong. A template producing differently-shaped cones, scored through a matrix fitted to a different set of cones, is being charged for the mismatch between the two, and the mismatch is bookkeeping rather than a property of the template. Under that construction the Gaussian in wavenumber scores several times worse than it does here, for a reason that has nothing to do with pigments.

The refit is what makes the comparison a comparison: each template gets the best matrix it can have, and what is left is the part no matrix can fix. That is also why the numbers here are lower bounds on the difference between templates — a real system would not refit, and would pay the mismatch too.

The general form is worth carrying: when comparing two constructions, refit everything downstream of the thing being compared, or the comparison measures the seam. It is the same discipline as scoring a camera profile on surfaces it was not fitted to, applied one level further up.

Where the model stops #

The menu has four members and only two of them are candidates anybody would use. A third published nomogram — Dartnall’s, or the Baylor–Nunn–Schnapf polynomial — would be a better third point than a Gaussian, and neither is here: the polynomial’s published coefficients do not reproduce a curve that peaks at its own λmax on this collection’s grid, and rather than debug somebody else’s coefficients the caricatures were built instead, which at least have a stated construction.

The scoring quantity is one number over twenty-four reflectances. It is the number this collection’s own machinery publishes, which is why it was chosen, and it is a mean over a small constructed set with all the caveats that carries.

And nothing here varies the peaks. The template takes λmax as an argument and the argument’s values — 566, 541 and 441 nanometres — are quoted from the literature and have their own uncertainty, which is a separate audit and a larger one.

The template has one coefficient that can be dialled rather than swapped, and it is the one the population argument actually rests on.

Who found it, and when #

The nomogram idea is Dartnall’s, 1953, from the observation that visual pigment spectra superimpose when plotted against wavenumber scaled by their peak. Lamb’s 1995 and Govardovskii’s 2000 formulae are successive refinements fitted to progressively larger sets of microspectrophotometric measurements, and the second is the one in current use.

The observation that a tabulated fundamental cannot support a population is not a finding so much as a consequence nobody writes down, because the two are used for different things: a fundamental is used to compute what one standard observer sees and a nomogram is used to model receptors across species. A collection that wants two hundred observers has to use the second, and this is a note on what that costs.

What a nomogram is asserting #

The template is not only a convenience, and the assertion behind it is worth stating because it is unusually strong for something used as machinery.

A nomogram says that every visual pigment has the same shape, once the wavelength axis is scaled by the peak. Not approximately the same, and not the same within a family: the same across pigments peaking from about 350 to about 620 nanometres, across fish, birds, reptiles and mammals, with the same three exponentials and the same coefficients.

That is a claim about photochemistry rather than about vision — the chromophore is the same molecule, 11-cis-retinal, in all of them, and what differs is the protein it is bound to and the electrostatic environment that protein provides. The environment shifts the energy of the transition, which shifts the peak, and the shape of the absorption band is a property of the chromophore.

Which is why the caricatures fail in the way they do. A Gaussian in wavenumber is the shape a simple two-level absorber would have, and a visual pigment is not one: its long tail towards the short wavelengths is vibrational structure — transitions to excited vibrational states of the excited electronic state — and it is present in every pigment because it is present in the molecule.

So the template’s asymmetry is not a fitted detail. It is the one feature of the curve with a physical cause, and it is the feature the next essay measures.

The tail ratio does not explain the β-band #

The closing section promises that what separates the caricatures from the real nomograms is one property of the curve’s shape, put on an axis and measured. The table above already carries that axis, and read across all five rows it does most of the work and not all of it.

Ranking the five templates by tail ratio and by residual gives a rank correlation of −0.90: one inversion in five rows. The inversion is between the two Govardovskii variants, and it is the informative part, because those two differ by exactly one thing.

The β-band moves the residual by 1.8 per cent and the tail ratio by 0.9 per cent, in opposite directions. Removing it takes the mean from 0.952 to 0.935 — an improvement — while taking the tail ratio from 1.16 to 1.15, which on the axis’s own logic should have made things worse. That is not noise in a fifth-row measurement; it is a feature the axis cannot see, and the reason is geometric: the β-band sits below 400 nanometres, and the tail ratio is a measure of the α-band’s asymmetry about its own peak. A secondary band a hundred and fifty nanometres away is not part of the quantity being measured.

So the axis separates pigment-shaped from not pigment-shaped, which is the comparison it was built for and which it makes decisively — the gap from 1.15 to 0.97 is six times the gap between any two real templates. It does not order the two published nomograms, and it puts the site’s own choice on the wrong side of its own α-band. One number captures the large effect and is silent about the small one, which is worth saying before the next essay is read as having settled both.

The distribution widens faster than it shifts #

The table’s third column is not a repetition of its second, and dividing one by the other says something neither says alone.

The ratio rises monotonically down the table, with a rank correlation of exactly +1.00 against the mean. From Lamb to the Gaussian in wavenumber the mean rises by a factor of 1.93 and the worst surface by 2.61.

A worse template is not a template that is uniformly further away. If it were, every surface would move by roughly the same factor and the ratio would be flat. What happens instead is that the error concentrates: some of the twenty-four reflectances are hit two and a half times harder while others are barely touched, which is the signature of a spectral mismatch rather than a global one. A curve with the wrong tail is wrong at particular wavelengths, so the surfaces that fail are the ones with structure there.

That has a practical consequence for how the five per cent between Lamb and Govardovskii should be read. On the mean the two published nomograms are 5.7 per cent apart; on the worst surface they are 8.9 per cent apart; and their worst-to-mean ratios differ by three per cent. The reassuring headline is a mean, and the quantity a reader would care about if one particular surface mattered is half again as sensitive. It is still small, and it is not as small as the headline.

The cost is convex in the departure #

One last reading of the same five rows. Taking Lamb as the origin and plotting each template’s excess residual against its shortfall in tail ratio gives ratios of 0.85, 1.70, 0.97 and 2.10 — a curve that steepens rather than a line.

The Gaussian in wavenumber is 0.40 of tail ratio away from Lamb and costs 0.840 of residual; the Gaussian in wavelength is 0.22 away and costs 0.214. Halving the departure from a real pigment shape buys back rather more than half the error, which is the opposite of what a linear sensitivity would give and is the reason the two published nomograms sit so close together despite differing in their coefficients throughout.

It is also the reason the audit comes out reassuring. The published templates are both in the flat part of a convex curve, where the choice between them is worth five per cent, and a reader would have to leave the class of real pigment shapes entirely before the choice started to matter.

Where the ladder goes next #

The two caricatures cost two and four times what either real nomogram costs, and they are matched to the real ones in width. What separates them is one property of the curve’s shape, and it can be put on an axis and measured — which turns “a real pigment shape matters” into a number.