The cones an appearance model uses

An appearance model’s first operation is to divide the stimulus by the adapting white in three particular axes, and the axes are the model’s most consequential and least examined choice. The disagreement between the published transforms is a disagreement about which three signals, and this collection has measured what that costs: no published transform is within five degrees of the basis in which a daylight change is exactly diagonal.

There is a second test available now, and it comes from an entirely different experiment.

The claim #

The axes CIECAM16 adapts in are called cone responses and were determined by fitting corresponding-colour data. Run the dichromat construction backwards on them and they imply confusion points a long way from the measured ones — CAT16 misses the deuteranope’s by 1.45 and CAT02 by 4.09 — so whatever they are, they are not the receptors.

• The name is doing work the fit does not support. The rows of CAT16 are conventionally written L, M and S, and the letters are the only claim anywhere that they are receptor sensitivities.
• The confusion-point test is external. It comes from dichromat data, which had no part in fitting any adaptation transform, so it is not a check of the fit against itself.
• Hunt–Pointer–Estévez, which is the physiologically motivated one, misses by 1.28 — better than CAT16 and not by much.
• And Bradford has entries no receptor could have, since a receptor’s response to a light present in the stimulus cannot be negative.
• None of that makes the transforms wrong. They were fitted to appearance data and they predict appearance data. It makes the name wrong, and the name is what a reader takes away.

Two different jobs, one vocabulary #

The von Kries hypothesis says that adaptation is an independent gain on each receptor class. That is a claim about physiology, and it names the axes: they are the receptors, whatever the receptors turn out to be.

The transforms in use were not derived that way. Bradford, CAT02 and CAT16 were fitted to corresponding-colour experiments — a person looks at a patch under one light, then under another, and adjusts until the two match in appearance — by finding the basis in which a diagonal scaling best reproduces the adjustments. That is a perfectly good procedure and it optimises the wrong thing to be called a cone response: it optimises the predictive performance of a diagonal model, and a diagonal model in the wrong axes can be made to fit better than a physiological model in the right ones.

So the vocabulary carries a hypothesis that the fitting did not test. The three curves are the axes in which adaptation is best modelled as diagonal, and whether those are the receptors is a separate question with a separate answer.

The test the fits were never given #

The construction is the one the confusion-point essay sets out and it runs both ways. A matrix carrying tristimulus values to three responses annihilates, in each pair of rows, one direction — the direction a person missing that receptor cannot see. So every matrix has three confusion points whether or not it was built with them in mind, and they can be read off.

Smith–Pokorny returns the published points to 1.9 × 10⁻⁴, which is the rounding in its own coefficients and is the check that the algebra is the field’s.

Hunt–Pointer–Estévez misses the protan point by 0.13 and the deutan point by 1.28.

CAT16 misses the deutan point by 1.45, and CAT02 by 4.09 — the worst in the table by a wide margin.

The tritan column is the interesting one. HPE’s short-wave row depends on Z alone, which is a strong structural assumption and a nearly correct one — its tritan point is within 0.007. CAT16’s short-wave row is not that shape and its tritan point is 0.05 away, which is small in absolute terms and is twenty times HPE’s miss.

Two more constructions say that the disagreement is not a two-horse race and that one of the three points is where nearly all of it lives.

What the numbers look like inside the diagram #

Two of the three confusion points sit outside the chromaticity diagram, and a distance of 1.45 measured out there needs converting into something a reader can hold, because the diagram’s own width is about 0.7.

The useful conversion is into the direction of the confusion lines inside the diagram, since that is what a simulation actually uses. A deuteranope’s confusion lines through the neutral point run toward the copunctal point, so moving that point from (1.40, −0.40) to CAT16’s implied (2.39, −1.47) rotates every one of those lines by several degrees. A few degrees of rotation on a family of lines that sweep the whole diagram moves colours near the ends of them substantially, which is why the same palette simulated under two matrices moves by up to 3.9 ΔE00.

The other conversion worth making is into the ratio the missing receptor’s direction represents. A confusion point is a chromaticity, and a chromaticity is a ratio of tristimulus values; moving it changes which combination of X, Y and Z the missing receptor is taken to have measured. On this measure CAT16’s long-wave row and the measured one disagree by rather more than the difference between the 1931 and 1964 standard observers, which is the largest disagreement this collection usually has to worry about.

Ninety-six per cent of CAT16’s miss is in the one free coordinate #

The deutan miss of 1.45 is quoted as a distance, and the direction it points in says more than the magnitude does.

The measured deuteranope point sits at (1.40, −0.40), whose coordinates sum to exactly 1.0000. That is not a rounding: a chromaticity with x + y = 1 has z = 0, and both the protan and the deutan directions are cross products involving the short-wave row, so both are forced onto that line precisely when the short-wave row is proportional to (0, 0, 1). The measured points encode the assumption that the short-wave cone depends on Z alone.

CAT16’s implied point is at (2.39, −1.47), summing to 0.92. Decomposing the displacement into a component along that line and one across it:

So CAT16’s short-wave row is very nearly (0, 0, 1) after all, and its whole deutan miss is in the one coordinate the structural assumption leaves free. That is the same shape Hunt–Pointer–Estévez shows exactly — its short-wave row is (0, 0, 1), so its misses lie precisely along the line — and CAT16 shows it to within four per cent.

The reading that follows is more interesting than a table of distances. Every matrix in the comparison satisfies, exactly or nearly, one of the two structural facts the measured points carry, and the disagreement is confined to the direction those facts do not constrain. A test that looks like it has two degrees of freedom per point has, for these matrices, about one — and the misses should be read as one-dimensional departures along a known line rather than as displacements in a plane.

What the miss does to a confusion line, at four places #

The conversion the essay offers — from a distance outside the diagram to a rotation of the lines inside it — is the right one and it has a range rather than a value.

Taking the line from four starting points to each of the two copunctal points:

The rotation is four times larger at the blue end than at the green, and the essay’s several degrees is the value at the neutral rather than a summary. The lines through saturated colours rotate least; the lines through the blue corner rotate most.

That distribution explains a number this collection reports elsewhere. The palette moved by up to 3.9 ΔE00 between two matrices and the worst movers were not the most saturated colours — which is what a rotation concentrated near the blue corner and the neutral would produce, and is not what a uniform rotation would.

So the practical statement is that a change of adaptation basis moves a dichromatic simulation most where the colours are pale or blue, and least where they are saturated red or green. That is computable from two chromaticity pairs and a little trigonometry, it needs no simulation at all, and it says which part of a palette to check when a matrix changes.

What the miss does and does not mean #

It would be easy to over-read the numbers, and the honest reading is narrower than it looks.

A large miss does not mean a bad adaptation transform. CAT16 is the current recommendation because it predicts corresponding-colour data well, and this collection’s own census found it competitive across a wide range of illuminant changes. Its confusion points are irrelevant to that performance.

Nor does it mean the transforms are unphysiological in effect. Adaptation may well be receptor gain and still be best modelled by a diagonal in some other basis, because the model has to absorb everything downstream of the receptors too — the second-stage opponent recombination, the gain control that is not multiplicative, the parts of the visual system nobody has a linear description of. A fitted basis absorbs the whole chain and then gets named after its first link.

What the miss does mean is that a specific inference is unavailable: from these are the axes adaptation is diagonal in to these are the receptor sensitivities. That inference is made constantly, by the letters at the ends of the rows.

What was computed, and how #

The confusion points are the values the Smith–Pokorny fundamentals are built on and are quoted rather than computed, because they are a measurement of people.

For each matrix, the three implied directions are the cross products of its pairs of rows, converted to chromaticity, and the miss is the Euclidean distance in the chromaticity plane. That distance is not a perceptual quantity and is not offered as one. It is the natural unit for a point on the diagram, and two of the three points lie outside the diagram, so a perceptual scale would not exist there in any case.

The round trip through Smith–Pokorny is the only check available that the construction here is the construction the literature used, and it is the reason to trust the other rows.

Nothing was fitted for this essay. Every matrix is quoted from its own standard, and the whole computation is three cross products per matrix.

Where the model stops #

The reduction hypothesis is doing work in the test. The confusion-point construction assumes a dichromat is a trichromat missing one receptor, with the other two unchanged. There is good evidence for it and it is an assumption, and every number in the table inherits it.

A 3×3 is not the whole of an appearance model. CIECAM16’s degree of adaptation is partial, its response compression is nonlinear, and its output correlates are built from opponent combinations after all of that. The basis question is the first stage of several and it is not obvious that it dominates — a surround changes a chroma by nearly half, which is a much larger effect than any of this.

The transforms are also not all trying to be the same thing. CAT16 was designed as a simplification of CAT02 with better behaviour at extreme chromaticities, and its authors’ stated criterion was the removal of a specific failure mode rather than physiological fidelity. Judging it by a receptor test is judging it by a criterion it declines. What the test establishes is only that the criterion was declined, which the letters on its rows do not say.

And the comparison is between two experiments neither of which is a receptor measurement. Corresponding colours are appearance judgements; confusion points are matching behaviour in a reduced observer. Direct measurements of receptor spectral sensitivity exist, from microspectrophotometry and from genetics, and they are not what either construction uses.

Who found it, and when #

That the fitted adaptation transforms are not cone fundamentals is known and is stated in the literature that produced them — the papers introducing Bradford and CAT02 are explicit that the matrices are fitted and that Bradford’s exponent on the short-wave channel is an empirical correction with no receptor interpretation.

The knowledge does not survive the journey into the models that use them. By the time a transform is a stage in an appearance model, its rows are R, G, B or L, M, S, and a reader meeting the letters has no signal that a fitting problem produced them.

The confusion-point test appears not to have been applied to them, which is mildly surprising, since it takes three cross products and both bodies of data have been in print for decades. The likely reason is that nobody had a motive: the transforms are judged by their predictions, the test is external, and an external test that a model was never designed to pass is easy to dismiss as unfair. It is unfair as a criticism and useful as a description, which is the distinction this essay is trying to hold.

The generalisation #

The pattern is a fitted quantity that acquires a physical name, and the name then supports inferences the fit does not.

The diagnostic is to find a measurement the fit did not use that the name implies a commitment to, and apply it. Here the name is cone response, the implied commitment is to three confusion points, and the data are a dichromat experiment nobody used in the fitting. An external test is the only kind that can distinguish a model from a description, because an internal one is answering the question the fit already optimised.

This collection has now met the shape three times over and the three are worth listing together. The cone fundamentals are a construction named after receptors. A chromaticity diagram’s area fraction is a property of a plane named after the eye. And an adaptation basis is a fitted convenience named after the receptors again. In every case the name is older than the fit, it was accurate when it was coined, and the object it names has since been re-determined by a procedure the name does not describe.

The failure is rarely anybody’s fault and is nearly always in the naming. A quantity called the effective diffusion coefficient or the apparent activation energy carries its status in the adjective; a quantity called the cone responses does not, and the letters L, M and S are an adjective-free claim.

A fourth published matrix says the same thing again, which is what makes the construction a property of the algebra rather than of any one committee.

What a model ought to say about its axes #

Two sentences, and both are already known to the people who wrote the standards.

What the axes were fitted to. Corresponding-colour data sets, named, with the number of observers and the range of illuminant changes they cover. That fixes what the transform is entitled to be used for and, just as importantly, where it is extrapolating — a discharge lamp is a long way outside the daylight arc most of the fitting data lived on.

And what the letters mean. If the rows are called L, M and S, a sentence saying that they are the axes in which a diagonal model fits best, rather than measured receptor sensitivities, costs nothing and removes the whole misreading.

Neither is a research programme and neither is present in the place a reader meets the matrix, which is a table of nine numbers in a specification. The same two sentences would repair the cone fundamentals, where the missing statement is which construction and which confusion points produced the published curves.

What would settle it #

The argument here is a mismatch between a name and a test, and there is a measurement that would settle what the axes actually are.

Fit an adaptation transform with the confusion points imposed as a constraint, and see what it costs on the corresponding-colour data it would otherwise have been free to fit. If the cost is small, the transforms are cone-like after all and their misses are slack rather than evidence — the fitting simply had no reason to land on the receptors and no reason to avoid them. If the cost is large, the diagonal model genuinely needs axes that are not receptors, and that is a substantive fact about adaptation rather than about matrices: it would mean the gain being modelled is not receptor gain, or is not the only thing happening.

Neither outcome is predictable from what is measured here, because a distance in chromaticity says nothing about how much predictive performance is at stake. The constrained fit is one afternoon’s work with the data both literatures already publish, and it is the experiment this essay’s finding asks for rather than answers.

Where the ladder goes next #

The question these essays turn on — what the data fix and what they leave free — has one more place to go on this ladder, and it is the harder half of an appearance model. The corresponding-colour data that fixed these axes are a finite set of judgements by a finite number of people, and a fit is exact and empty in exactly the directions its data did not visit.