The identity is in the eye's own coordinates

Two of this round’s conditions are exact and the exactness has a footnote. The footnote is larger than most of the results in the round, and it is about a choice every colour pipeline makes without recording it.

The claim #

The two exact conditions of this round are exact in the observer’s own cone responses and nowhere else, and the two arithmetics the field actually uses leave 8.11 and 13.0 ΔE₀₀ of them standing.

• A gain on each cone is absorbed exactly when the white is divided out in those same cones, and leaves 13.0 ΔE₀₀ when the division happens in CAT16’s cone space.
• A change of basis is absorbed exactly when the rotation is undone before the division, and leaves 8.11 when the division happens in the rotated coordinates.
• The two published arithmetics agree with each other to 0.59 ΔE₀₀ at the median and differ from the physiological one by more than fifteen.
• And that is coherent rather than scandalous. The published transforms are fitted to adaptation behaviour, not to receptors, and a fit to behaviour lands somewhere the receptors are not.

Where a white gets divided out #

Every colour in this collection is a stimulus judged against a white, and there are three places the division can happen.

In the cones. Form the three cone responses to the stimulus and to the white, divide each by its own, then carry the three ratios to tristimulus values by a fixed matrix. This is von Kries’ hypothesis written as arithmetic.

In tristimulus values. Form X, Y, Z for both, then divide each by the white’s. This is what CIELAB does, and it is also a von Kries transform — in a basis where the three “cones” are the x̄ȳz̄ functions.

In a published cone space. Transform to CAT16’s cones, divide, transform back. This is what every modern chromatic adaptation transform does, and it is what this collection uses through the transform its appearance work is built on.

All three are the same operation with a different basis, and a gain needs a basis established that the choice is real. What is new here is that one of the three makes two conditions exact and the other two do not, and that the exact one is not among those in use.

The measurement #

Forty-two analytic surfaces, one observer, one light, three routes, and the pairwise differences.

The first row is small and the other two are enormous, which is the opposite of the arrangement a reader would guess. Dividing in tristimulus values is the crude choice everybody knows is crude; dividing in CAT16’s cones is the careful one; and the two agree with each other far more closely than either agrees with the eye’s own receptors.

That is not an artefact of this construction. It is what the published transforms are for.

A sharpened basis is not a physiological one. CAT16’s cone space, like CAT02’s and Bradford’s before it, is sharpened: its basis functions have narrower support and negative lobes compared with the physiological fundamentals, so its “cones” respond over a narrower band and go negative outside it.

That is deliberate and it is fitted. The matrices were obtained by minimising the error of predicted corresponding colours against measured sets — pairs of stimuli reported to look alike under two different lights — and the minimisation moved the basis away from the receptors because a pure receptor gain does not fit the data as well as a sharpened one does.

So the arithmetic in use is not an attempt to be the eye. It is an attempt to predict what people report, and it succeeds at that better than the physiological version does. No basis is good at both put the same finding a different way: a basis that predicts adaptation well is not the basis that keeps a gamut compact, and neither is the receptor basis.

The consequence for this round is that the two exact conditions are exact in a space nobody computes in, and that is a fact about the conditions rather than about the arithmetic. An identity that holds in the receptors is an identity about receptors, and if the eye’s adaptation is not exactly a receptor gain then no arithmetic that predicts adaptation well will have it.

The distinction between those two things is easy to lose in a pipeline, because both are called cones and both are three curves. The receptors are measurable objects with absorbances and peaks; a sharpened adaptation basis is a fitted coordinate system whose three functions have negative lobes and correspond to no cell. Using one where the other belongs is not a small error and it is not visible in any type signature.

This collection has made that mistake once already and recorded it. The cones an appearance model uses are CAT16’s rather than the retina’s, and an early draft of the appearance work handed the model physiological fundamentals on the reasonable-sounding grounds that they were more accurate.

Which arithmetic this collection uses, and why #

The choice matters for every number in this round and it is made in favour of comparability.

Every departure in this round is computed in CAT16’s cone space, because that is the route this collection has used for nineteen rounds and because a number quoted in a different arithmetic cannot be compared with the ones already published. The adaptation census, the tolerance work, the camera profiles and the appearance essays are all in that unit, and switching for one round would make the round incomparable with everything around it.

Every identity in this round is demonstrated in the observer’s own cones, because that is where it is an identity. Demonstrating it in CAT16’s space would demonstrate a limit of 13.0 instead, which is a different and much less interesting claim.

The two are reported side by side rather than reconciled, and the gap between them is published as a measurement in its own right. That is not a compromise; it is the accurate description of the situation. The condition is about the eye and the unit is about the field, and pretending otherwise would require choosing one to be wrong.

What the 0.59 is doing there #

The small number in the first row of the table deserves attention, because it is doing something that looks like an accident and is not.

Dividing in tristimulus values is generally reckoned a poor adaptation model, and the reckoning is right: it predicts corresponding colours badly and produces visible errors on large illuminant changes. Yet under a single light, judging a sample against that light’s own white, it agrees with CAT16 to 0.59 ΔE₀₀ at the median.

The reason is that both are von Kries transforms and both are being asked to do almost nothing. When the source white and the destination white are the same — which is the case here, since every sample is judged against the light it sits under — a von Kries transform in any basis reduces to a scaling that makes the white come out white. The bases differ only in how they distribute that scaling across a sample that is not the white.

So the first row is not a measurement of adaptation quality. It is a measurement of how much two bases differ on a null adaptation, and it is small for the same reason the identities are exact: nothing much is being asked.

Which sharpens the second and third rows rather than softening them. Fifteen colour differences from a transform that is barely doing anything is the size of the disagreement about what a receptor is, showing through an operation designed not to depend on it.

What a reader should take from the fifteen #

The number that will stay with anybody reading this is the fifteen colour differences between the physiological route and the two published ones, and it is worth saying what it is and is not evidence of.

It is not evidence that the published transforms are wrong. They are fitted to what people report and they fit it; a route that predicted corresponding colours better would be a better transform whatever basis it lived in, and none of the arithmetic here tests prediction at all.

It is not evidence that the receptors are the right basis for adaptation. Everything measured about chromatic adaptation says a second, post-receptoral site contributes, and a transform that ignores it will be improved by a basis that partly compensates for the omission.

What it is evidence of is that the two are far apart, which had not been quantified in this collection. A gap of fifteen units at the median between the coordinates the physiology divides in and the coordinates the field divides in is a large gap, and it means the phrase “adapt in cone space” names two operations that give very different answers. The literature uses that phrase for both.

The practical form #

For anybody assembling a pipeline, the results reduce to two rules and a warning.

Divide the white out once, in a stated basis, and record which. A pipeline that divides in tristimulus values at one stage and in CAT16’s cones at another is applying a composition of two different von Kries transforms, and the composition is not a von Kries transform in any basis.

That is the same discipline the census in six units arrived at for colour-difference formulae: a published number carries its arithmetic with it, and a reader who does not know which arithmetic cannot compare it with anything.

Do not mix a physiological basis with a fitted one. Computing cone responses from published fundamentals and then adapting with CAT16’s matrix is a common arrangement and it is incoherent: the fundamentals are the receptors and CAT16’s matrix is not, so the transform is being applied to coordinates it was not fitted for.

And the warning: an identity is only an identity in the arithmetic it was proved in. Two of this round’s conditions are exact, and a reader who carried the exactness into a CAT16 pipeline would be carrying a claim that is wrong by thirteen colour differences.

Reading that ladder with this essay in hand changes what it is a measurement of. It is not what two eyes disagree about; it is what two eyes disagree about as reported by a particular adaptation transform, and the transform contributes. How much it contributes is not measured here, and the honest bound is that it is somewhere below the fifteen colour differences that separate the routes, since much of that fifteen is a systematic offset both observers share.

That bound is unsatisfying and it is the true state of things. Separating a departure’s own size from the arithmetic’s contribution to it would need corresponding-colour data this collection does not have, which is the sixth round in succession that sentence has been written into a shortfall.

What was computed, and how #

The observer’s own route computes three cone responses to the stimulus and to the light, divides each by its own, maps the three ratios through the fitted tristimulus matrix, and converts to CIELAB against the white that route gives for a perfect diffuser. That last step matters: comparing a route’s answer against another route’s white would measure a mismatch of conventions rather than a difference of bases.

The tristimulus route is CIELAB’s own arithmetic with the light’s tristimulus values as the white point. The CAT16 route adapts from the light’s white to D65 by the published matrix and then uses D65 as the white point, which is what this collection’s population machinery has done since it was written.

The assertion carried by the figure is that the two published routes agree with each other more closely than either agrees with the observer’s own. That is a claim about where the field’s conventions sit relative to the physiology, and the numbers could have come out the other way.

There is a smaller and more immediately useful thing to take from it as well. Any pipeline can be tested for coherence in a few minutes by putting a pure cone gain through it and asking whether the answer moves. If the pipeline claims to adapt in cone space and a receptor gain changes its output, then the space it adapts in is not the space it computes cone responses in — which is a bug in the sense that the two were meant to be the same object, and is very common because they are usually loaded from different files.

The same test applied to this collection gives the honest answer: its output moves by 13.0 ΔE₀₀, and that is by design rather than by accident, because the file it loads its adaptation matrix from is CAT16’s and the file it builds its observers from is a pigment template. Knowing which of those two situations a pipeline is in is worth the few minutes. The gates this collection runs check that a figure names its observer and cannot check that two parts of the arithmetic mean the same thing by the word cone.

Where the model stops #

The physiological basis here is this collection’s own construction, which reaches the 1931 functions through a fitted matrix with a 16.4 per cent residual on z̄. A different template would give a different “own cones” and a different fifteen.

CAT16’s matrix is used as published. It was fitted to specific corresponding-colour data sets, those sets are small, and the collection has recorded their absence as a shortfall for six rounds. The claim that the published transforms are fitted away from the receptors is a claim about that fit, and a differently-fitted transform could land somewhere else.

And nothing here tests which route predicts anything. All three are compared against each other, not against measured appearance, and no data in this collection could adjudicate between them.

The generalisation #

The habit is about the scope of an invariance.

An identity is always an identity in something — a coordinate system, a normalisation, an order of operations — and the scope is usually left implicit because in the setting where it was derived there is only one candidate. Carrying it somewhere else carries the scope with it, and the scope is the first thing to go.

The test is to derive the identity twice in two settings and see whether both derivations work. Here the gain identity works in the cones and fails in x̄ȳz̄, and the failure is not subtle once looked for: a diagonal matrix conjugated by a non-diagonal one is not diagonal.

The failure mode is to treat an exact result as more portable than an approximate one. It is usually less portable. An approximation degrades gracefully when its assumptions weaken and an identity simply stops being true, and the moment it stops is rarely marked.

Who found it, and when #

Von Kries proposed the receptor-gain hypothesis in 1902. The observation that the best-fitting adaptation transform is not in the receptor basis is much later and belongs to the sharpening literature of the 1990s — Lam’s Bradford transform of 1985, the sharpened transforms of Finlayson and Süsstrunk, and the CIE’s own CAT02 and CAT16.

The reason the fits sharpen is still argued about. One account is that a second, post-receptoral site contributes to adaptation and a sharpened basis approximates the pair. Another is that the corresponding-colour data sets carry a systematic bias. Nothing in this collection can adjudicate between those, and the honest statement is that a basis fitted to behaviour is where behaviour put it.

Where the ladder goes next #

The identities are finished and the departures begin. Six of them, and the first question about any of them is not how large it is but what it is paired with — because a departure of the observer is a product, and the light is the other factor.