The best axes are not receptors

Every published chromatic adaptation transform misses the dichromat confusion points, and the misses have always been read as slack — a fit that landed near the receptors and could have landed on them at little cost. The reading is checkable, and it is wrong in the strongest available way: the basis that minimises the residual over all nine free numbers is further from the measured confusion points than every fitted transform except the two most notorious.

The claim #

The diagonal model does not want cone axes. Moving a basis towards the receptors makes it monotonically worse at adapting, and the unconstrained optimum implies a deuteranope confusion point 1.63 away in chromaticity from the measured one — further than Hunt–Pointer–Estévez’s 1.28 and further than CAT16’s 1.45.

• The floor is 0.974 ΔE00, found by search over all nine numbers and stable to four decimals across a doubling of the budget.
• The receptors leave 1.65. The gap is a factor of 1.70, and there is nothing to fit away, because the confusion points determine the basis outright.
• The path between them is monotone. Walking in a straight line from the receptor basis to the optimum improves the residual at every one of thirteen stops.
• And the winner is not near any published transform either. It is its own point, and what it has in common with Bradford and CAT02 is a direction rather than a location: sharpening.

What the question was, and why it was open #

Earlier this collection built the construction that turns three confusion points into a matrix, and measured how far each published transform sits from the points it is thereby committed to. Hunt–Pointer–Estévez misses the deuteranope point by 1.28, CAT16 by 1.45, Bradford by 2.42, CAT02 by 4.09. The obvious question was left written down and unanswered: is that slack or is it structural?

The distinction is sharp and the two readings have opposite consequences. If the misses are slack — if a basis constrained to hit the points performs nearly as well — then the published transforms are cone-like, their oddities are fitting noise, and the physiological account and the engineering one agree. If the misses are structural, the diagonal model is being carried by axes that are not receptors, and every appearance model built on one is using the word cone for something else.

The way to settle it is not to compare the published transforms with each other. It is to find the best basis there is and ask where it sits.

Where it sits #

The optimum’s implied confusion points are protan 0.096, tritan 0.051 and deutan 1.63. Two of the three are close to the measurements and the third is not close to anything.

That asymmetry is itself informative and it recurs in every row of the table. The protanope and tritanope points are nearly agreed on by everything — even XYZ scaling implies a protan point only 0.36 away — while the deuteranope point is where the bases scatter. The reason is geometric: the deuteranope confusion point sits at (1.40, −0.40), far outside the diagram, so the direction it names is nearly parallel to directions that other constructions also nearly contain, and small changes in a matrix row move that point enormously. It is the sensitive coordinate, which is exactly what makes it the diagnostic one.

What a factor of 1.7 is worth #

Before walking anywhere it is worth pricing the gap, because a ratio of residuals is not by itself a reason to care.

The residuals here are mean CIEDE2000 differences between where an adapted observer’s model puts a surface and where the second light actually puts it. One unit is roughly a just-noticeable difference under careful side-by-side comparison, so the receptor basis’s 1.65 is an average error a person could see if the two versions were shown together, and the optimum’s 0.97 is an average error that would mostly pass. The worst rows are much larger than the means in both cases — 3.75 against 2.21 — and those are errors nobody would miss.

So the gap is not academic. It is the difference between a white-balance model that is usually invisible and one that is usually visible, on a census that contains the lamps people actually live under. A tolerance of one unit is what a print buyer contracts to, and the two bases sit on opposite sides of it.

Walking there #

A ranking of eight points does not by itself say whether the optimum is a peak with the receptors on its shoulder or a different hill altogether. The way to tell is to walk.

The residual falls from 1.65 to 0.97 monotonically. There is no stop along the way where it stops improving, no shoulder, and no interior minimum. Every step away from the receptors is a step towards a better gain.

That settles the question in the direction nobody would have chosen. It is not that the fitted transforms overshot; it is that they did not go far enough in a direction the receptors are the wrong end of.

The walk is monotone in one coordinate and not the other #

The path improves the residual at every stop, and the deuteranope’s point does something else entirely on the way.

Walking the straight line in the nine coefficients and reading both quantities at thirteen stops, the residual falls 1.651, 1.465, 1.347, 1.263, 1.199, 1.147, 1.103, 1.066, 1.035, 1.009, 0.989, 0.978, 0.974 — monotone throughout.

The distance from the measured deuteranope point goes 0.000, 1.130, 14.516, 4.917, 2.945, 2.373, 2.101, 1.942, 1.837, 1.763, 1.708, 1.665, 1.631.

It rises to fourteen and a half and comes back. Between the second and third stops the implied confusion point leaves the diagram altogether: a copunctal point is where two rows of the basis meet, that meeting runs to infinity when the rows become parallel, and somewhere in the first fifth of this walk they do.

So the 1.63 at the far end is not reached by drifting away from zero. The basis’s deutan point departs, passes through the pole, reappears on the other side, and settles.

That matters for reading the miss as a measure of anything. A distance from a confusion point is not a well-behaved coordinate on a path between two bases, because the quantity is projective and its distance function has a singularity in the middle of the region this walk crosses. The endpoint value of 1.63 is a fact about the endpoint; the sequence in between is not a trend and cannot be read as one.

The residual has no such trouble, being a mean over surfaces rather than a point on a plane. Which is one more reason the question — is the miss slack or is it structural — had to be settled by finding the optimum rather than by walking towards it and watching.

What the direction is #

The direction has a name in the adaptation literature and it is sharpening. A sharpened set of axes is one whose spectral sensitivities are narrower and less overlapping than the cones — pushed towards the extreme of three non-overlapping bands, where a change of light really would act as three independent gains.

The reason sharpening helps is not subtle. A change of illumination is a matrix, and a diagonal can reproduce a matrix exactly only when the matrix is diagonal in the chosen basis. Broad, overlapping channels see every part of the spectrum, so a change concentrated anywhere in the spectrum moves all three of them and moves them in correlated ways — which is an off-diagonal term. Narrow channels each see their own region, so a change that alters one region alters mostly one channel.

The cones are broad and heavily overlapping, especially the long- and medium-wave pair, whose peaks are about thirty nanometres apart. That overlap is what makes fine colour vision possible — two nearly identical detectors are how a small spectral shift becomes a large signal difference — and it is precisely what a diagonal adaptation model finds inconvenient.

So the finding has a shape: the property that makes the receptors good at discriminating is the property that makes them poor axes for a gain. That is not a paradox and it is not a design flaw. It is two different jobs asking for opposite things from the same three curves, and it is taken further in its own essay.

What was computed, and how #

The search is Nelder–Mead over the nine entries of the basis, started from the identity rather than from any published transform, with restarts.

Starting from the identity is deliberate. A search seeded with an answer reports that answer’s neighbourhood, and seeding with CAT16 would have made the optimum is near CAT16 impossible to distinguish from the search did not go far. Started from XYZ, the search leaves a residual of 2.37 and arrives at 0.974, which is a long way in a nine-dimensional space.

The restarts are not decoration. Three of the nine parameters are exactly invisible to the objective, so the cost has three perfectly flat directions and a simplex wanders along them indefinitely; the restart is what recovers a readable answer rather than a matrix with an entry of 10¹². The budget is checked rather than assumed: 0.9741 at two hundred and fifty steps and four restarts, and the same four decimals at six hundred and eight.

The implied confusion points are read back out of the winning matrix by the construction run backwards — the cross products of pairs of rows, normalised to chromaticities. That is the same code path that returns the published Smith–Pokorny points to 1.9 × 10⁻⁴ from the fundamentals they were built from, so the reading is the literature’s own.

The external test #

The reason this argument is worth more than a fit statistic is that the confusion points had no part in producing any of the bases being tested. Bradford was fitted to corresponding-colour judgements; CAT16 was fitted to a superset of the same; the optimum here was fitted to a census of illumination changes over constructed reflectances. None of the three ever saw a dichromat.

So where does this matrix put the confusion points is a genuinely external question, in the sense that matters: a prediction the model was not tuned to make. That is the property an internal check can never have, and it is why a residual and an orbit together say more than a residual twice.

The answer the external test gives is unambiguous and it is not the flattering one. Optimising for adaptation moves a basis away from the receptors, and it keeps moving away right up to the optimum.

Who found the direction, and when #

Sharpening is not this essay’s discovery and is barely even recent. The observation that narrowing a set of sensor channels improves the accuracy of a diagonal model belongs to the computational colour constancy literature of the early 1990s, where it was pursued as spectral sharpening: given a set of sensors and a set of illuminants, find the linear combination of the sensors that makes the change of light most nearly diagonal. Bradford’s entries, published in 1993, are recognisably a sharpened set, and CAT02’s are more so.

What the literature did with the result was engineering. Sharpened axes went into adaptation transforms and into camera white balance and were reported as improvements to a fit, which they were. What it did not do was ask where the sharpened axes sit relative to the receptors, because within that literature there was no reason to: a transform is judged by its residual.

The measurement is available now only because the other half arrived. The construction that turns confusion points into a matrix makes how cone-like is this basis into a number rather than an impression, and once it is a number the two literatures can be put on one pair of axes. The picture that results has one clear message and it is a negative one: the direction the fitters have been walking, for thirty years and for good reasons, points away from the eye.

How far away is far #

A miss of 1.63 in chromaticity is a large number for a diagram whose whole locus fits inside a unit square, and it is worth saying what it means before leaning on it.

The deuteranope confusion point at (1.40, −0.40) is not a colour and never was. It is the chromaticity of the direction in tristimulus space that a deuteranope’s two remaining channels both annihilate, and directions in tristimulus space have chromaticities whether or not any light has them. So the distance being quoted is between two directions, expressed in the coordinates the diagram happens to use.

That makes the absolute value hard to interpret and the comparison easy. Every entry in the table is measured the same way on the same plane, so the ordering — receptors 0.00, HPE 1.28, CAT16 1.45, the optimum 1.63, Bradford 2.42, CAT02 4.09 — is a ranking of how far each basis’s implied deuteranope direction is from the measured one, and it is the ranking that carries the argument. If the optimum’s miss were 1.20 the conclusion would be the opposite one; it is not close.

The one arrangement that would undermine the reading is if the optimum happened to sit between the receptors and the published transforms, which would suggest the fitters had overshot a target the model actually wanted. It does not. It sits past CAT16 and past HPE, in the same direction Bradford and CAT02 went, further than the current recommendation and less far than the withdrawn one.

The generalisation #

A model can be built on a quantity it does not want the true value of. That sentence sounds like a contradiction and is an ordinary situation, and the confusion it creates is a naming one rather than a mathematical one.

The three rows of an adaptation transform are called cone responses in print, by people who know they are not. The word survives because the transform is a model of receptor adaptation — the mechanism being modelled is a gain applied by three cone classes — and it is natural to call the axes of that model by the name of the thing they stand for. What the measurement shows is that the model’s optimum is elsewhere, so the axes are a fitted parameter wearing a physiological name.

The habit this suggests is small and general: when a model’s parameter has the same name as a measurable quantity, measure the quantity, impose it, and read the cost. If the cost is small, the name is earned. If it is large, the name is a hypothesis that has been quietly assumed rather than tested — and the version of it that has been tested here fails.

The one place the receptors are not beaten #

There is a single row of the census where the ordering reverses, and it is worth naming because it is the exception that explains the rule.

The second objective and the bowl the first one sits in are the two things that say whether the optimum is a knife edge or a plateau.

On the three-emitter source — three narrow bands standing in for a laser projector — the receptor basis leaves 0.78 and the optimised one leaves 1.38. Broad overlapping channels are hard to knock out of alignment by three narrow lines, because each line falls inside all three of them and moves them together, which is nearly a scaling. Sharpened channels each see one line and are moved independently and unequally.

So sharpening is a bet on the spectra being smooth, and it loses when they are not. Every fitted transform in use carries that bet, and it was placed at a time when the lights people lived under were daylight, tungsten and fluorescent tubes.

Where the model stops #

One census, equally weighted. Fourteen changes of illumination over one constructed reflectance family. Reweighting the census would move the optimum, and some rows are much harder than others; a census dominated by daylight would find a different winner and a smaller gap.

A von Kries diagonal and nothing else. Every number here is about B⁻¹ diag(d) B. Real appearance models add incomplete adaptation, a nonlinear response and a surround, and none of that is in this arithmetic. What is being optimised is the linear stage alone.

And a chromaticity distance is a distance on a chosen plane. The confusion-point misses are quoted in CIE xy units, which is a projective picture, and distances on projective pictures are not invariant. The ordering of the misses is robust — the deuteranope point is the scattered one in every plane tried — but the number 1.63 belongs to CIE xy and should be quoted with it.

Where the ladder goes next #

Both halves of the arithmetic are now on the table: matching leaves nine numbers, dichromacy fixes six, and the model that most wants those six fixed prefers them fixed somewhere else. The remaining question is whether anything on this site prefers the receptors — and one thing does, which is what makes the pair of results worth having together.