A trade between matrices, not people

A trade-off drawn as a scatter of eight matrices looks like a fact about the world. Two hundred observers, plotted on the same axes, say it is a fact about the matrices.

The claim #

The trade-off between adapting well and discriminating well is a property of the space of bases somebody could choose. Inside a population of observers it does not appear: the two costs correlate at +0.29, which is the wrong sign for a trade-off and the wrong size for anything.

• Across bases the two objectives pull apart. The basis minimising the adaptation residual leaves an anisotropy of 7.70 against a floor of 1.61; the basis minimising anisotropy leaves a residual of 1.793 against a floor of 0.974.
• Across observers they move together, at a correlation of +0.288 on two hundred members. The best-adapting quartile has a median anisotropy of 2.786; the worst-adapting quartile, 3.167.
• One variate is the exception. The macular pigment’s density correlates −0.32 with the adaptation cost and +0.53 with the log of the anisotropy: more of it makes an observer’s own receptors adapt better and discriminate worse.
• And the lens does not trade at all. Age correlates +0.24 with one and +0.53 with the other, in the same direction: an older lens is worse for both.
• So the shape of a trade-off depends on the set it is drawn over, and the two sets here — the reachable and the realised — give opposite answers.

Two different sets #

The trade-off this collection has established is a statement about a nine-dimensional space of matrices. Every nonsingular 3×3 is a candidate basis, colour matching cannot distinguish any of them, and two objectives written over that space have their optima far apart with an empty region between them. No basis is good at both, and the shape of the trade is why any recommendation is a choice.

A population of observers is a five-dimensional set of eyes, embedded in that nine-dimensional space by the construction that turns three confusion points into a basis. It is a tiny, curved, five-parameter sliver of the whole, and there is no reason at all for a trade-off across the whole to appear across the sliver.

It does not. Plotting the two hundred members on the same two axes gives a cloud that is elongated along neither diagonal and is if anything tilted the other way: an observer whose own receptors adapt badly tends, weakly, to discriminate badly too.

What +0.29 means and does not mean #

A correlation of +0.288 on two hundred members is comfortably distinguishable from zero and explains about eight per cent of the variance. It is a weak positive relationship, and the useful way to read it is by quartiles rather than as a coefficient.

The best-adapting quartile — fifty members, median residual 1.544 — has a median anisotropy of 2.786. The worst-adapting quartile — median residual 2.007 — has a median anisotropy of 3.167. So the observers whose receptors are best for one job are slightly better for the other as well, and the difference between the quartiles is thirteen per cent of the anisotropy against thirty per cent of the residual.

What that rules out is any story in which an eye is a compromise between the two — in which the receptor spacing is where it is because moving it would buy adaptation at discrimination’s expense. Nothing in this population behaves that way. Within the range people actually vary over, the two costs are nearly independent, with a slight tendency to rise together.

That is not evidence against the trade-off across matrices, which is measured and large. It is evidence that the two facts are about different objects, and that reading a design trade-off as a statement about eyes is the mistake.

The one measurement that does trade #

Taking the population apart variate by variate finds one exception, and it is worth the whole essay.

The macular pigment’s density correlates −0.32 with the adaptation cost and +0.53 with the logarithm of the anisotropy. An observer with more macular pigment has a receptor basis that leaves less after a von Kries gain and makes the discrimination contours further from circles. The two signs are opposite, which is what a trade looks like, and it is the only variate here that does it.

The lens does not. Age correlates +0.24 with the adaptation cost and +0.53 with the log anisotropy — the same direction on both. An older lens is worse for both jobs, which is not a trade, it is a loss.

The pigment peaks do neither strongly: the offsets of the three peaks correlate between −0.18 and +0.11 with the log anisotropy and their distance from the medians correlates +0.16 with the adaptation cost. The receptors themselves, in other words, are the part of the eye that the two objectives are least sensitive to.

What the macular pigment was hiding #

The +0.288 is a raw correlation, and the one variate that genuinely trades is sitting inside it.

Partialling the macular pigment out — regressing both costs on it and correlating what is left — the association between them rises to +0.567. Controlling for the lens as well gives +0.539, and controlling for all five variates at once gives +0.664, which explains forty-four per cent of the variance rather than eight.

So the pigment is not merely an exception to a weak relationship. It is a suppressor: the single measurement that pushes the two costs in opposite directions, sitting in a population where everything else pushes them the same way, and dragging a strong positive association down to a weak one.

That sharpens the conclusion rather than unsettling it. Within a population of eyes the two objectives do not trade — they move together, and once the one genuinely trading measurement is held fixed they move together strongly. An eye that adapts badly discriminates badly.

Controlling for the lens alone moves the number the other way, to +0.195, which is the same statement from the other side: age raises both costs, so part of the raw positive is the two of them ageing together rather than anything about receptors at all.

The receptor offset that does most of the work #

Taking the variates one at a time against the adaptation cost, the largest single correlation is not the macular pigment’s.

It is the long-wave peak’s offset, at −0.517 — larger than the macular pigment’s −0.322 and more than twice the lens’s +0.240. An observer whose L pigment sits at a longer wavelength than the median has receptors that leave less after a von Kries gain, and no other variate comes close.

The account above reports the peaks as the part the objectives are least sensitive to, and that is true of their distance from the median and false of their sign. A symmetric measure — how far a peak sits from the median, without saying which side — discards exactly the information a signed correlation of −0.52 is made of. The medium-wave offset behaves the same way more weakly, at −0.258, and the short-wave one, at −0.048, does not participate.

So the adaptation cost is mostly a matter of where the long-wave pigment sits, and the anisotropy is mostly a matter of the two filters in front of it — the macular at +0.525 and the lens at +0.531 against the log anisotropy, against the long-wave peak’s −0.177. Two objectives reading two different halves of the eye, which is a mechanism for why they do not trade inside a population rather than merely an observation that they do not.

Why the macular pigment and not the receptors #

The mechanism is in the curvature, and it is the same one that decides which observers pay most.

The adaptation objective’s stiffest direction is 97 per cent the short-wave row of the basis and its flattest is 85 per cent the long-wave row. So an adaptation model minds enormously where its blue axis points and hardly at all where its red one does — because the gain in the short-wave channel is the one that swings between illuminants.

The macular pigment absorbs almost entirely below 500 nm. It is therefore a change to what the short-wave receptor catches and to almost nothing else, which is a displacement almost entirely along the direction the adaptation objective is stiffest in. That it improves matters rather than worsening them is the part that needs explaining, and the honest answer is that it is not obvious: a filter that narrows the short-wave sensitivity moves the row towards where the fitted transforms put theirs, and the fitted transforms adapt better than the receptors do.

The anisotropy responds to something else. It is measured in a lightness–chroma space, which divides by the white point and takes a cube root, so a row whose reading of the white becomes small makes the compression steep and the contours long. More macular pigment means less short-wave light reaching the receptor, a smaller white reading, and a stretched axis.

One change, two mechanisms, opposite signs. That is the shape of a genuine trade, and it exists in one of the five measurements that differ between people.

What a single matrix is choosing between #

The reason any of this matters to an appearance model is that CIECAM16 adapts and compresses in the same axes: one matrix, chosen once, doing two jobs. What that costs has been measured — 35 per cent above one floor and 68 above the other.

The population puts a bound on how much of that could be blamed on the observer. If the two objectives traded strongly across people, a standard could reasonably say that any single matrix is a compromise nobody escapes, because every eye is one too. They do not, so it cannot. The compromise is in the choice of matrix, and the choice is available.

There is a second consequence, and it runs the other way. Because the population’s own spread on the adaptation axis — 1.223 to 2.245 — is wider than the entire range of the published transforms, the difference between recommending Bradford and recommending CAT16 is smaller than the difference between two observers. A standard that agonises over a quarter of a ΔE00 between two matrices is refining a number whose observer-to-observer variation is four times larger.

That is not an argument for indifference. The transform is applied to everybody, so its mean performance over a population is exactly the right thing to optimise, and a quarter of a unit on the mean is a quarter of a unit for every reader. It is an argument against reading a small difference in a mean as a difference anybody would experience.

What was computed, and how #

Each member’s basis is built by the transfer construction: fit the 3×3 reading their cone catches off standard tristimulus values, take it against the reference member’s inverse, and apply the result to the basis the published confusion points give. On the reference member the transfer is the identity to 2 × 10⁻¹⁷.

Both costs are the ones every other comparison in this collection uses. The adaptation residual is the mean CIEDE2000 left after the gain, over fourteen changes of illumination. The anisotropy is the mean over twenty-five discrimination contours of the ratio of the image’s semi-axes, taken from the map’s own derivative rather than by walking round the contour — which matters here, because the tail of this distribution reaches an anisotropy of 10.5 and a sampled ring understates a ratio that large by more than half.

Correlations are ordinary Pearson coefficients over two hundred members. The anisotropy is correlated in its logarithm because its distribution is strongly skewed — a fifth percentile of 2.387 and a ninety-fifth of 7.006, which is what an axis ratio does when a row goes near-degenerate — and a coefficient taken on the raw values would be a statement about the thirty-one members in the tail.

The tail, and what is in it #

Thirty-one of two hundred members have an anisotropy above 5, against a population median of 2.815 and a reference observer at 2.596. A tail that heavy is worth opening rather than summarising away.

It is not a tail of unusual receptors. The members in it have pigment peaks scattered much like everybody else’s: their mean separation between the long- and middle-wave offsets is 1.82 nanometres against 1.63 for the rest, which is a difference of a tenth of a standard deviation and explains nothing.

It is a tail of filters. Age and macular density each correlate at +0.53 with the log anisotropy, and the tail is where both happen to be high at once. An eye with a seventy-year-old lens and a dense macular pigment has very little short-wave light reaching its short-wave receptor, so the row of its own receptor basis that reads the white in that channel reads a small number — and a lightness–chroma space built by dividing by that small number stretches one axis and lengthens every contour on it.

The tail is what the arithmetic does when a denominator gets small, which is the same mechanism as a gain dividing by a white that goes to zero, several orders of magnitude short of catastrophe. Nothing here diverges; the contours simply get long.

Whether a real seventy-year-old’s colour difference judgements are correspondingly distorted is a question this model cannot answer and should not be read as answering. What it says is that if such an observer’s own receptors were used to build a uniform space, the space would be a poor one — and nobody builds a space that way.

Where the model stops #

This is a population of a model, not a sample of people, and whose eyes it stands for is a question it answers only in distribution. Five variates with literature spreads, propagated through a pigment template, anchored on three published points. A correlation computed inside it is a correlation inside the model.

Correlation is not a mechanism, and every coefficient here is weak enough that the explanations above are candidates rather than conclusions. What the numbers support is the sign structure — one variate opposite, one variate aligned, the receptors indifferent — and the claim that no trade-off appears at the population level. They do not support a story about why any individual eye is where it is.

And the anisotropy has a heavy tail. Thirty-one members of two hundred are above 5, and those thirty-one are what a mean would be about. Every summary here is a median or a quartile for that reason, and a comparison drawn on means would be a comparison of tails.

The generalisation #

A trade-off is a property of a set, and changing the set can remove it. That is the whole result, and it is more general than colour: a frontier drawn over everything that could be built says nothing about the correlation among the things that exist, because the things that exist are a low-dimensional, non-random subset.

The version of this in the study of biological systems has a name — the difference between what a trait could trade against and what it does trade against in a population — and the usual finding is the one here: constraints visible in the design space vanish in the realised one, because realised variation runs along a few directions rather than filling the space.

The practical form is a question worth asking of any published trade-off. Over what set was this drawn, and is it the set the choice is being made from? A designer choosing an adaptation matrix is choosing from the nine-dimensional space and the trade-off binds. A physiologist asking whether an eye is a compromise is asking about a five-dimensional sliver, and it does not.

The second half is smaller and sharper. Where a trade does appear inside the realised set, it is worth finding which single parameter carries it, because that parameter is the one an intervention would act on. Here it is the macular pigment — one absorbing filter, in front of the receptors, doing opposite things to two objectives.

Who found it, and when #

The design trade-off between chromatic adaptation and perceptual uniformity is not standard in the colour literature, because the two objectives are not usually written over the same parameters — adaptation transforms are fitted to corresponding-colour data and uniform spaces are fitted to difference data, in different papers by different people.

The population half is available and has not been joined to it. Individual-observer colour-matching models have existed since the CIE’s physiological fundamentals work, and macular pigment’s effect on colour matching has been measured for decades, and two observers in one person is the standing demonstration of it. What has not been asked is what an individual’s own receptor basis would do as an adaptation basis, which requires the counting argument that says the confusion points determine one.

A trade-off nobody has drawn cannot have been checked against a population, so the absence here is structural rather than an oversight. It is also the reason the check is worth doing early: the shape of a frontier is the sort of thing that becomes a story about eyes within a very few citations.

Where the ladder goes next #

The same arithmetic points comfortably at objects nobody was born with. A camera’s three dyes are six numbers with a curvature over them, and which of them the design is blind to is a manufacturing tolerance rather than a philosophical one — with an answer that is not the one a specification would guess.