The price is also the person

A number computed for one observer and described as a property of a construction is a number about a person, and the only way to find out which is to change the person.

The claim #

The seventy per cent the confusion-point construction costs is a statement about one pair of eyes. Across a population it runs from twenty-six per cent to a hundred and thirty, and the spread is larger than the entire range of the published transforms it is being compared against.

• The population runs 1.223 to 2.245 ΔE00, median 1.780, standard deviation 0.200, against an unconstrained floor of 0.974.
• The published transforms occupy 1.140 to 1.584 — a range of 0.444, against the population’s span of 1.021.
• Seventeen and a half per cent of members are better served by their own receptors than by Hunt–Pointer–Estévez; two per cent than by CAT16; none than by Bradford.
• The observer this site quotes sits at 1.651, which is below the population median and inside the lower half of the distribution.
• And the ordering of the published transforms does not change. What changes is whether the receptor construction has a place in that ordering at all, and it does not: it has a distribution.

What was claimed, and what it rests on #

The argument this corrects is a good one and is not being withdrawn. A von Kries gain is exactly invariant to the scale of each row of its basis, so three of the nine numbers a colour match leaves free do nothing to an adaptation model; the six that remain are exactly what three dichromat confusion points supply; so the receptor basis is determined by dichromat data rather than fitted, with nothing left over. That algebra is exact and this essay does not touch it.

What was then measured is what the determination costs: 1.651 ΔE00 averaged over the census of illumination changes, against 0.974 for a basis free to be any nine numbers — seventy per cent — with Bradford at 1.140 and CAT16 at 1.312 in between.

Every one of those numbers is computed with COPUNCTAL, three pairs of numbers quoted in this collection since its first commit:

protan (0.7465, 0.2535) · deutan (1.4, −0.4) · tritan (0.1748, 0.0)

They are among the small number of things here that are not computed from something else. They are the copunctal points of a population, and nothing in this collection propagated the variation between one pair of eyes and another through an argument that turns entirely on them.

How the population is built #

Not by deriving each member’s points from a pigment model, which does not work and is worth knowing why: a copunctal point is a quotient of small differences between nearly parallel rows, and a template accurate to a fraction of a per cent puts the protanope’s point a quarter of the way across the diagram from where it is measured.

Instead the template supplies the difference and the measurement supplies the position. For each member, fit the 3×3 that reads their cone catches off standard tristimulus values; take G as that matrix times the reference member’s inverse; apply G to the basis the published points give. On the reference member G is the identity to 2 × 10⁻¹⁷ and the construction returns the published points exactly, which is the check that this is a perturbation of somebody’s measurement rather than a second model.

The members are drawn from five reported spreads — the lens’s density entered as an age between twenty and seventy, the macular pigment’s peak density, the cone outer segment’s axial density, and the peak wavelength of each of the three pigments.

The distribution #

Two hundred members, each with their own receptor basis, each scored on the same census:

• minimum 1.223, fifth percentile 1.475, median 1.780, ninety-fifth percentile 2.078, maximum 2.245;
• standard deviation 0.200.

Against a floor of 0.974 that is an excess running from 26 per cent to 130, with a median of 83.

The reference observer’s 1.651 is at about the twenty-fifth percentile. That is not a coincidence and not a bias in the draw: the reference member has every variate at its median, and the cost is a convex function of the parameters, so the cost of the median member is below the median cost. The observer whose points are quoted is a typical observer and their cost is not a typical cost, which is a distinction worth having in general and is invisible from a single evaluation.

What that does to the comparison #

The published transforms occupy 1.140 to 1.584 on this axis. The population occupies 1.223 to 2.245. The two overlap substantially, and the overlap is the result.

Seventeen and a half per cent of members would be better served, on this census, by the basis their own confusion points determine than by Hunt–Pointer–Estévez — a matrix built specifically to make a von Kries gain behave. Two per cent would be better served by their own receptors than by CAT16. None beats Bradford, which is the best of the five on this census by a clear margin.

So the sentence the receptor basis costs seventy per cent becomes three sentences, and all three are needed:

1. For the observer whose points are published, it costs seventy per cent.
2. Across a population, the median cost is eighty-three per cent and the spread is wider than the differences between the published transforms.
3. And for a substantial minority of observers, the construction beats a transform that was fitted.

The third is the one that changes the argument’s shape. The original conclusion — that the diagonal model wants axes the eye does not have — is about the optimum being far from the receptors, and that stands: even the best member of the population, at 1.223, is well above the free floor of 0.974, and the assertion that no member reaches it is in the build. What does not stand is treating 1.65 as a fixed price.

The other objective moves further #

The same members scored on how nearly a lightness–chroma space built on their own basis makes the discrimination contours circles give a much wider spread: 2.09 to 10.52, median 2.815, against the reference observer’s 2.596.

The distribution is strongly skewed — the fifth percentile is 2.387 and the ninety-fifth is 7.006 — because the anisotropy is a ratio of extremes and a basis that puts a row near a degenerate direction produces very large ratios, which is what an axis ratio does near a circle’s opposite. A handful of members have receptor bases that are catastrophic for colour difference and unremarkable for adaptation.

That asymmetry is worth naming. The two objectives are not equally robust to who is looking, and a comparison made on one of them across a population is not a comparison made on the other.

Which of the five variates does it #

The width has to come from somewhere, and taking the population apart one variate at a time says which of the five measurements a reader should care about.

Run with only the pigment peaks varying and the rest at their medians, the three confusion points move by 54, 66 and 21 per cent of their full spread. Run with only the macular pigment varying: 69, 60 and 69 per cent. Only the lens age: 58, 22 and 77. Only the axial density: 13, 12 and 16.

Two things follow. The axial density does almost nothing — a sixth of the spread on every point — so the one variate that is a property of the receptor rather than of what sits in front of it is the least consequential. And the other three are each carrying more than half of at least one point’s spread, which means the width of every distribution in this essay is set by three numbers rather than by one.

The four shares combine in quadrature to within seven per cent of the whole on every point, so they can be read as independent contributions — which nothing made them be, and which is worth checking rather than assuming.

The macular pigment is the largest single contributor to all three points, and it is the variate with the widest reported range relative to its mean: a peak density near 0.35 with a standard deviation of about a third of that, and individuals measured from nearly zero to above one. A population model’s answer is dominated by whichever of its inputs has the widest reported spread, which is obvious and is worth confirming rather than assuming.

What was computed, and how #

Each member’s basis comes from the transfer, and the cost is the same residual every other adaptation number here uses: for each change of light in the census, the mean CIEDE2000 between where a family of surfaces actually goes and where the gain puts it, averaged over fourteen changes.

The basis is taken from the transfer directly rather than by running the member’s points back through the König construction. The round trip is exact in arithmetic and badly conditioned in floating point — a point at (1.4, −0.4) is a direction stated as a position, and reconstructing the direction from it loses digits nothing else here would notice.

The assertion in the build makes two claims. The population’s span must exceed the whole range of the published transforms, which is what makes the spread consequential rather than a footnote. And no member may reach the unconstrained floor, which is what preserves the original argument: if some observer’s own receptors were optimal, the claim that the model wants non-receptor axes would be a claim about one observer rather than about the model.

Robustness to the fitting set is asserted separately and one of its two halves is a positive result about instability: the population median is 1.780 fitted over three illuminants and 1.802 over one — 1.2 per cent apart — and 1.573 over the monochromatic lights, which is 12 per cent away and is the honest uncertainty on everything here.

What survives and what does not #

It is worth separating cleanly, because a correction that is read as a retraction destroys more than it repairs.

Survives. The invariance: a von Kries gain cannot see three of the nine numbers, exactly, at 7 × 10⁻¹⁵ over the whole census. The count: the six the confusion points supply are exactly the six the model can see, so the basis is determined rather than fitted. The direction of the result: the diagonal model’s optimum is not at the receptors and moving towards the receptors makes it worse, at every stop on the walk, for every member of the population tested. And the conclusion about published transforms: they are not cone responses, and the distance is measurable.

Does not survive. A single number for what the determination costs. Seventy per cent is the twenty-fifth percentile of a distribution running from twenty-six to a hundred and thirty, and it was presented as a property of the construction.

And one thing is newly available. Because the price varies, it can be asked which observers pay most — and the answer is not the obvious one. Correlated against each variate in turn, the strongest relationship is with the macular pigment, negatively, at −0.32: a denser macular pigment goes with a receptor basis that adapts better. The lens’s age is next at +0.24 in the other direction, and how far a member’s pigment peaks sit from the medians is weakest at +0.16.

Every one of those is a weak correlation and none of them should be read as a mechanism. What they establish is the shape of the dependence rather than its size: the two variates that are filters in front of the receptors are what the price tracks, and the peak wavelengths of the receptors themselves are what it tracks least — which is the opposite of what a story about receptor axes would predict.

The reason is in the curvature. The adaptation objective’s stiffest direction is almost entirely the short-wave row of the basis, and the filters are exactly what move the short-wave row. A change to a pigment peak moves all three rows a little; a change to the macular pigment moves the row the objective minds most.

Where the model stops #

This is not a published measurement of between-observer variation in confusion points. It is what this collection’s five-variate pigment model implies, anchored on the published points. A real observer differs in ways no template of this kind carries, and a real dichromat differs further still.

The census is one census. Fourteen changes of illumination, equally weighted. The population’s spread would survive a different one; the specific percentiles would not.

And the variates are quoted spreads. The lens age range, the macular and axial density standard deviations, and the three peak-wavelength spreads are all from the literature and each carries its own uncertainty. The propagation is exactly as good as they are, and a factor of two in any of them moves the width of every distribution here.

The generalisation #

A number computed at the median of a distribution is not the median of the numbers, and the gap between the two is a systematic bias whose sign is decided by the curvature. Where the quantity is convex in the parameters — as a cost usually is — evaluating at the typical member understates the typical cost, here by about eight per cent.

The wider lesson is the one this collection keeps arriving at from different directions. A quoted constant is a measurement of somebody, and an argument that turns on it inherits their variation. The check is cheap where a population model exists: run the argument at a hundred members and look at the width. If the width is small relative to the differences being argued about, the constant can be treated as a constant. If it is larger — as here, by a factor of two and a bit — the argument’s conclusion has to be stated as a distribution or not at all.

There is a third habit in it, and it is the one that took the longest to get right. Propagate variation where the model is trustworthy and anchor where it is not. The template cannot place a confusion point and can move one; using it for the second job and a measurement for the first is what makes the whole calculation possible, and the test that the composition is honest is that it returns the measurement exactly when asked for no change.

Who found it, and when #

Observer variability in colour matching is old and well measured — the CIE’s own deviate-observer work, Stiles and Burch’s fifty-three observers, the modern individual-observer models — and this collection has used it for a decade of essays’ worth of arguments about displays, gamuts and matches.

What has not been done, as far as this collection can find, is propagating it through the derivation of an adaptation basis. The reason is structural rather than an oversight: the derivation from confusion points belongs to colour-vision physiology, the population models belong to colorimetry, where a gamut has a population, and adaptation transforms belong to appearance modelling. Three literatures, three sets of conferences, and a quantity that requires all three to be in one calculation.

The one place the three do meet is in individual-observer colour matching functions for displays, where a population of observers is run through a device model — and there the question asked is about matches rather than about adaptation.

Where the ladder goes next #

If a population of receptor bases is this wide, the obvious question is how far outside it the published transforms actually sit — not in chromaticity, where the three points are not comparable, but in the population’s own units. The answer is different for each of the three points, and one of them does not support the claim at all.