Two points out of three

A distance means nothing until it is divided by something, and the something has to be measured rather than assumed.

The claim #

Measured in the population’s own units, the published adaptation transforms are far outside it on two confusion points and inside it on the third.

• On the protanope’s point they run 2.2 to 14.4 standard deviations out. CAT02 is nearest at 2.2; XYZ scaling is furthest at 14.4.
• On the deuteranope’s, 3.7 to 11.8. Hunt–Pointer–Estévez is nearest at 3.7; CAT02 is furthest at 11.8.
• On the tritanope’s, 0.6 to 2.6. Four of the five are within a single standard deviation, which is indistinguishable from a member of the population.
• The chromaticity distances say the opposite. In raw units the deuteranope’s misses are the largest by a factor of thirty, because that point is off the diagram — so a ranking in chromaticity is a ranking of how far off the paper each point is.
• And the point they agree on is the one filters move. The tritanope’s point has by far the widest population spread, for the physical reason that the lens and the macular pigment both absorb where only the short-wave cone is listening.

What the raw distances say, and why they mislead #

Every 3×3 that is used as an adaptation basis implies three confusion points: the protanope’s is the chromaticity of the cross product of its second and third rows, and so on. That direction of the construction is well conditioned — perturbing a basis by a thousandth moves an implied point by between four ten-thousandths and nine thousandths — so the implied points are a fair statement of what a matrix is committed to.

In chromaticity, the misses against the measured points are:

Read as they stand, the deuteranope’s column dominates everything: CAT02 is four units away, which is five times the width of the whole diagram. That reading has been used in this collection and it is not wrong so much as uninterpretable, because the three columns are not in comparable units.

The deuteranope’s point sits at (1.4, −0.4), far outside the diagram. A point at infinity is a direction written as a position, and near infinity a small change of direction is a large change of position. So the third column is small and the second is large mostly because of where the three points are, not because of how well any matrix matches them.

Dividing by the right thing #

The natural denominator is the spread of the same point across a population of eyes, and this collection now has one: two hundred members drawn from five reported spreads, each carrying their own three points by moving the measured ones rather than deriving them.

Their root-mean-square radii are 0.025 on the protanope’s point, 0.345 on the deuteranope’s and 0.073 on the tritanope’s — and the deuteranope’s is large for exactly the reason its misses are large, so dividing removes the geometry from both sides at once.

Divided, the same table becomes:

The picture reverses. CAT02, which looked worst by a factor of three in chromaticity, is the nearest of the five on the protanope’s point. And the whole third column collapses: four of the five sit within a single standard deviation of the population’s tritan point, which is what a member of the population looks like.

Why the tritan point is the forgiving one #

The population’s spread on a confusion point is not the same for all three, and the reason is physical rather than statistical.

Taking the population apart one variate at a time, the lens’s age carries 77 per cent of the tritanope’s point’s spread and 22 of the deuteranope’s; the pigment peaks carry 66 per cent of the deuteranope’s and 21 of the tritanope’s. The two run opposite ways, which is the check rather than the result: the lens and the macular pigment are absorbing filters with almost all their absorption below 500 nm, so they change what the short-wave cone catches and barely touch the other two — and the tritanope’s point is a property of the two classes they do not touch, computed as a cross product of the rows that did move.

A population is wide on the tritan point because two of its five variates are filters. So the denominator is large there for a reason that has nothing to do with any matrix, and the numerator being small is not a triumph either — it is the observation that all five matrices put their short-wave row in roughly the place the eye does, within a range that people vary over anyway.

What survives of the claim #

This collection has said, in several places and in several forms, that the published adaptation transforms are not cone responses and that the distance is measurable. That claim survives and is now stated more precisely.

On two of the three points it is unambiguous. The nearest published transform on the protanope’s point is 2.2 standard deviations out and the furthest is 14.4; on the deuteranope’s, 3.7 and 11.8. No member of a population of two hundred is anywhere near any of them, and a spread twice as wide as the one modelled here would not change that.

On the third it is not supported at all. Four of the five are inside one standard deviation. If the only evidence available were the tritanope’s point, the honest conclusion would be that these matrices are perfectly ordinary observers.

And the difference between those two sentences is the whole reason to state which points the claim rests on. A conclusion drawn from three measurements, two of which support it strongly and one of which does not support it, is a sound conclusion badly reported. The same shape appeared in this collection before: a verdict that is robust to a modelling choice while the evidence offered for it is not.

What each transform’s own pattern says #

The five rows are not five versions of the same thing, and reading them individually is more informative than reading the column extremes.

CAT02 is the most extreme case of a matrix that is close on one point and far on another — 2.2 σ on the protanope’s and 11.8 on the deuteranope’s. That is a matrix whose second and third rows are placed nearly where the eye places them relative to each other, and whose first and third are not. It is also the matrix that was withdrawn from the recommended path for going negative in practice, and the two facts are related: a row that is far from a receptor direction is a row that can read a real spectrum as a small or negative number.

Hunt–Pointer–Estévez is the most balanced, at 5.2, 3.7 and 0.6. It is the only one of the five constructed as a set of cone fundamentals rather than fitted, so being nearest on the deuteranope’s point is what it was built for — and it is still nearly four standard deviations out, which is the interesting part. A matrix built to be cone-like is four population widths from the cones.

Bradford is close on the protanope’s point and far on the deuteranope’s, at 2.9 and 6.9. Bradford is the transform colour management uses, and it is also the one whose gain changes sign inside an ordinary painted room. Its distance from the population’s deuteranope point and its willingness to read a spectrum as zero are the same property of the same row seen twice.

CAT16 is in the middle on everything — 4.8, 4.2, 1.0 — which is what a transform designed to fix a predecessor’s excursions looks like.

And XYZ scaling is the furthest out on two of three, at 14.4 and 5.7, which is the expected result and is worth having as the control: a transform with no basis in it at all should be far from any observer, and it is.

What was computed, and how #

The population’s cloud is two hundred members drawn from the five variates this collection models, each member’s basis built by transferring their cone matrix’s difference from the reference member’s onto the basis the published points give. On the reference member the transfer is the identity to 2 × 10⁻¹⁷ and returns the published points exactly.

The radius quoted for each cloud is the root mean square distance from the cloud’s own centre, not a standard deviation in x and one in y. The clouds are elongated — the deuteranope’s runs along the line joining it to the locus — and quoting two orthogonal standard deviations would describe a box nobody measured.

The distance for each transform is from the cloud’s centre rather than from the quoted point, which matters slightly: the centres sit 0.013, 0.061 and 0.038 from the quoted values, all inside their own spreads, and using the quoted point instead changes the third decimal of the ratios and none of the conclusions.

The assertion in the build is written in three parts so that it cannot be softened into a slogan. Every transform must be at least two standard deviations out on the protanope’s point; the same on the deuteranope’s; and at least one must be within four on the tritanope’s, so that the third column’s failure to support the claim is asserted rather than mentioned.

Where the model stops #

The population is a model of variation, not a sample of people. Five variates with literature spreads, propagated through a pigment template and anchored on three published points. A measured between-observer study of confusion points would be the right denominator and this collection does not hold one.

The denominators are only as good as the reported spreads. The macular pigment’s standard deviation is about a third of its mean and individuals are measured from nearly zero to above one; a wider distribution would widen every denominator and shrink every ratio in the table. A factor of two in the macular spread moves the tritan column from inside to well inside and moves the protan column from 2.2–14.4 to about 1.1–7.2, which would weaken but not overturn the two-point claim.

And a confusion point is not a cone response. It is a direction, and three of them fix the observer’s curves only up to the three row scales — which the adaptation model cannot see anyway. So a matrix that matched all three points exactly would still not be a set of cone fundamentals in any absolute sense; it would be the right directions, which is all that can be asked.

What it would take to settle the third point #

The tritan column is uninformative because the denominator is large, and the denominator is large because two of the five variates are filters in front of the receptors. That suggests what would make it informative.

A population conditioned on lens age and macular density — comparing a matrix against thirty-year-olds with median macular pigment, rather than against everybody — would shrink the tritan denominator by most of the 77 and 69 per cent those two variates carry. On a rough accounting the radius would fall from 0.073 to about 0.02, and the same misses would read as 2 to 9 standard deviations rather than 0.6 to 2.6.

That is a legitimate comparison and it answers a different question: not is this matrix within the population but is this matrix within the population of observers with a stated lens and a stated macular pigment. Both are worth asking and neither is the other, and the second requires stating an age.

The reason not to do it silently is that it is exactly how a null result becomes a positive one without any new evidence. Narrowing a reference population until a difference clears it is available in every study that has covariates, and the defence is to say which population before looking.

What the normalisation actually reverses #

The picture reverses is the right verdict and it is not a reversal within any column, because dividing a column by one number cannot reorder it. The reversal is between columns and between transforms, and both are worth stating.

Between columns. In chromaticity the deuteranope’s misses average 2.24 against the protanope’s 0.148 — a factor of 15.2. In standard deviations they average 6.49 against 5.91, a factor of 1.10. So the raw table says one point is fifteen times worse determined than another, and the normalised table says the two are equal to within ten per cent. The entire apparent dominance of the deutan column was the distance to a point off the diagram, and once it is divided out the two decisive columns turn out to be the same size.

Between transforms. Summed across the three points, the chromaticity table ranks them Hunt–Pointer–Estévez, CAT16, XYZ scaling, Bradford, CAT02 — with XYZ scaling at 2.514 marginally ahead of Bradford’s 2.523. In standard deviations the order is Hunt–Pointer–Estévez, CAT16, Bradford, CAT02, XYZ scaling last at 22.7 against Bradford’s 10.5.

That is the reversal that matters. In chromaticity a basis-free scaling looks as good as the transform colour management actually uses; normalised, it is worse by a factor of two and is last by a wide margin. The control only behaves like a control after the denominator is applied — which is the strongest available demonstration that the raw table was ranking geometry.

The tritan attribution does not add up #

The variance attribution is given twice and the two versions cannot both be right.

The first says the lens’s age carries 77 per cent of the tritanope’s spread and the pigment peaks 21 — which is 98 per cent between them, leaving two per cent for the other three variates, the macular pigment included.

The second says conditioning on lens age and macular density would remove most of the 77 and 69 per cent those two variates carry. Seventy-seven and sixty-nine sum to 146, which no two shares of a variance can do, and the first attribution has already given the macular pigment almost nothing.

The prose in between sides with the second: a population is wide on the tritan point because two of its five variates are filters. The attribution says one filter and the cone pigment peaks.

The remedy’s size depends on which is right, and it changes the conclusion. Removing the lens alone leaves 23 per cent of the variance, so the radius falls from 0.073 to 0.035 and the tritan column becomes 1.2 to 5.4 σ — with three of the five still inside two standard deviations. Removing the lens and the pigment peaks leaves 2 per cent, the radius falls to 0.010, and the column becomes 4 to 18 σ.

The essay’s quoted 0.02 sits between those and corresponds to keeping 8 per cent of the variance, which neither attribution produces. And its promised 2 to 9 standard deviations is the arithmetic of 0.02 rather than of either accounting.

So the third point’s fate is genuinely open, and it turns on a number the essay reports two ways. Conditioning on lens age — the one variate both versions agree carries most of the spread — leaves the tritan column much less decisive than the remedy claims, and the honest version of the recommendation is that it would move three of five transforms from inside one standard deviation to inside two.

That is a smaller prize than the section implies and it does not touch the essay’s own warning against it. Narrowing a population until a difference clears it is available here, and the arithmetic says how much narrowing would be needed: conditioning on one variate is not enough, and conditioning on two requires conditioning on the cone pigment peaks, which is not a covariate anybody measures on a population.

The generalisation #

A distance is a ratio, and the denominator has to be measured. That is the whole essay in a line, and the reason it is worth writing is that the numerator is nearly always available and the denominator nearly never is, so the temptation is to compare numerators.

The specific trap here is worth naming because it recurs. Where the quantities being compared live in different parts of a projective picture, raw distances rank the geometry rather than the fit. A chromaticity diagram is projective; a point near infinity has a large neighbourhood; and any table of chromaticity distances that mixes near and far points is largely a table of how far off the diagram each point sits.

The second lesson is about how to report a conclusion drawn from several pieces of evidence. Say which pieces carry it. Three measurements, two decisive and one silent, is a strong result; averaging the three into a single verdict throws away the information that would tell somebody where to look next. Here it says exactly where: the short-wave row is the one every candidate agrees about, and it is the row the population disagrees about most.

Who found it, and when #

That the published adaptation matrices are not cone fundamentals is not controversial. Bradford’s and CAT02’s rows have entries no cone response could have, the fits that produced them were against corresponding-colour data rather than against dichromat data, and the standards documents describe them as transforms rather than as fundamentals.

What is not standard is quantifying the gap in observer units. The usual comparison is entry by entry, or by plotting the implied fundamentals against a published set — both of which say different without saying how different, relative to what. Individual-observer models have been available since the 2006 CIE physiological fundamentals work and the individual colour matching function models that followed, so the denominator has been computable for the better part of two decades.

The reason it has not been computed is that the two halves belong to different fields. The numerator lives in appearance modelling and the denominator in colorimetric metrology, and the ratio is nobody’s paper.

Where the ladder goes next #

Everything so far has been about the observer’s nine numbers, which nobody manufactures. The same arithmetic points at objects somebody does: a camera’s three dyes are six numbers with a curvature over them, and the direction its design is blindest to is not the one anybody would guess.