Five transforms and the space between them

Asking how far a published adaptation transform is from anybody’s receptors needs a model of anybody’s receptors. Asking how far five of them are from each other does not, and the answer is larger.

The claim #

The five transforms in ordinary use disagree with one another by more than the quantity they disagree about, and the disagreement is measurable in a unit nobody has to agree on first.

• In nanometres of pigment, they span from −10.5 to +18.2 — a range of 28.7 nm against an L–M separation of 25.
• In chromaticity, the two furthest-apart protan points are 0.43 apart, which is most of the width of the diagram.
• The two closest — Bradford and CAT02 — are 0.0165 apart, so the five are not five opinions but two or three clusters.
• The spread is a usable scale. A quantity is a large disagreement if it is comparable with the spread among the published options, and that judgement needs no population.
• And an appearance model inherits all of it, because CIECAM16 contains a choice of adaptation basis and the choice is one of these.

Why this comparison rather than the other one #

The claim that every published transform sits outside a population of eyes is the more striking statement, and this round has already restated it so that no declared width can break it. It still needs a model of receptors — the transfer construction, the pigment template, the reference peaks — even in its nanometre form.

The comparison between the transforms needs none of that. Each of the five is a 3×3 matrix that somebody published; the confusion points they imply are read off by the same construction König gave, which is a matrix inverse; and the distances between the resulting points are arithmetic. No observer model appears anywhere.

That makes it the more robust of the two statements and the less interesting one, which is the ordinary trade. It is worth having because it bounds the other: whatever the right receptors are, at most one of the five can be near them, and the amount by which the other four are wrong is at least the amount they differ from that one.

The spread, three ways #

Two tight pairs and a wide gulf. Bradford and CAT02 agree with each other to 0.017; Hunt–Pointer–Estévez and CAT16 agree to 0.012; the two pairs are about 0.18 apart; and a plain XYZ scaling, which is the null case rather than a proposal, is 0.4 from the nearest of them.

So the five are really two clusters and a control. That is not visible from the table of sigma-distances — which reads 5.21, 2.88, 2.24, 4.83, 14.41 and looks like five separate values — because dividing by a population radius compresses the structure. The pairwise distances keep it.

In nanometres the same structure appears with signs attached: Bradford at −10.5 and CAT02 at −7.5 want the medium-wave pigment shorter; Hunt–Pointer–Estévez at +10.1 and CAT16 at +9.5 want it longer. The two clusters are on opposite sides of the measured peak, by about the same amount.

Why they cluster that way #

Because the two clusters were built to do different things, and this collection has measured what each imposition costs.

Hunt–Pointer–Estévez is a cone-fundamental transform, and the matches do not name the cones is why that is an attempt rather than a measurement. It is an attempt at the receptors themselves, normalised so that the equal-energy stimulus gives equal responses. Its confusion points are meant to be the confusion points.

Bradford and CAT02 are fitted. They were derived by optimising the prediction of corresponding-colour data — how a colour seen under one light has to be changed to look the same under another — and the optimisation had no reason to preserve anything about receptors. Bradford’s third row is famously not a cone-like sensitivity at all, which is what a fitted basis buys and costs.

CAT16 is a repair of CAT02 aimed at removing a numerical pathology, and it moved back towards the cone-fundamental cluster in doing so, which is why it sits beside Hunt–Pointer–Estévez rather than beside CAT02.

So the clustering is a clustering by purpose, and the gulf between the two clusters is the cost of the fitting. This collection’s own measurement of that cost is that imposing a diagonal adaptation on the receptor basis costs about seventy per cent on the adaptation objective — a number computed for one pair of eyes, and wider than that across a population.

Two objectives, and neither picks a winner outright #

Read down the two columns and the orderings are nearly reversed, which is the whole of what one basis can and cannot do for two objectives. Bradford is the best of the published five at adaptation and second-worst at discrimination; the receptor basis read off the confusion points is the best published option at discrimination and the worst at adaptation.

Neither column is the right one on its own, which is the whole reason there are five transforms rather than one. A specification that says use CAT16 has made a decision between two objectives without saying so, and the amount at stake is a factor of 1.6 on one and 1.6 on the other.

What an appearance model inherits #

CIECAM16 is not a transform; it is a model with a transform inside it. Its first step is a chromatic adaptation in a named basis, and the named basis is CAT16 — one of the five above, and one of the two in the cone-fundamental-ish cluster.

Every appearance number this collection publishes therefore carries the choice. Lightness, chroma, hue quadrature, colourfulness under a stated surround: all of them are computed downstream of a 3×3 that is 0.17 in chromaticity from the other cluster and 0.24 from XYZ scaling.

That is not a criticism of CIECAM16, which has to pick something and picked the one that behaves numerically. It is a statement about what a prediction from it means: an appearance prediction is a prediction conditional on an adaptation basis, and the conditionality is normally invisible because the basis is inside the model rather than beside it.

The site’s own convention is the same one it applies to observers, and naming the surround is already required: a figure that quotes an appearance number names the surround and the adapting luminance, and it should name the adaptation basis for the same reason. That is a change to the caption convention rather than to any number.

Using the spread as a scale #

The practical half, and it generalises past this table.

A distance is uninterpretable until it is divided by something. Dividing by a population’s spread makes the statement depend on a population model. Dividing by the spread among the published options makes it depend on nothing but the options, and it answers a question a reader actually has: is this difference large compared with the disagreement that already exists in the field?

By that scale:

• The gap between the two clusters, 0.18, is eleven times the gap inside a cluster. The two clusters are answering different questions.
• The distance from any published transform to the receptors’ own point is 0.10 to 0.43, which is comparable with the whole published spread. So the field’s internal disagreement and its distance from receptors are the same size, and neither dominates.
• XYZ scaling’s distance from the nearest published transform, 0.24, is larger than the whole cluster structure — which is why it is a control rather than a candidate.

Every one of those sentences is checkable by anybody with the five matrices and no model of anything.

The same locus exists for the other confusion point, and it is the one that says the axis was chosen rather than found.

What is not in the comparison #

The absence that makes this a comparison between transforms rather than a test of them, and it is the fourth phase running that it has to be stated.

There are no corresponding-colour data here. The transforms in the fitted cluster were derived by optimising the prediction of such data — sets of colour matches made under one illuminant and re-matched under another, by real observers — and testing them properly means predicting held-out matches. This collection has none, so it can measure how far the transforms are from receptors and how far they are from each other, and it cannot measure which of them predicts an appearance match best.

That is a real limitation and it points the same way every time it appears. The transforms fitted to those data (Bradford, CAT02) score best on this collection’s own adaptation objective, which is a constructed census of constructed illuminants — so the two agree, which is mild evidence that the census is measuring something the data also measure, and is not a substitute for the data.

The honest summary is that the comparison here is a geometric one: five matrices, their implied confusion points, and their distances. It says the five cannot all be describing the same receptors and quantifies by how much. It does not say which of them a person would prefer, and nothing here could.

What a document should carry #

A specification naming a transform is making a decision on grounds the document does not usually state. Three sentences would state them:

Which cluster. Cone-fundamental (Hunt–Pointer–Estévez, CAT16) or fitted (Bradford, CAT02). The two are 0.18 apart in the implied protan point and are answering different questions.

Which objective. Adaptation or discrimination. The published five span a factor of 1.6 on each and rank oppositely, so a choice is a trade rather than a preference.

And what it costs in the other one. Bradford is best of the five at adaptation and second-worst at discrimination; CAT16 is middling at both, which is a defensible reason to standardise on it and is not the reason usually given.

Where the model stops #

The pairwise distances are distances between protan points. The three points behave differently — the tritan point is inside the population where the other two are outside — so a spread computed on the tritan point would be smaller and would support weaker statements. The protan point is used here because it is the exposed one and because it is the one this round’s other essays are about.

And a distance in chromaticity is not a perceptual quantity. The 0.43 between Bradford and XYZ scaling is most of the diagram’s width, which sounds decisive, but the copunctal point is a projective construction that runs to infinity under small changes and its distances do not correspond to anything an observer would report. The nanometre form is the better scale precisely because it is a physical length rather than a chromaticity one, and the chromaticity distances are quoted here to show the cluster structure rather than to be believed as magnitudes.

The cluster structure is the useful output #

One more reading, because it is what somebody choosing between the five should take away and it is invisible in the usual presentation.

Five options presented as five rows of a table invite a ranking. These five are not five options; they are two positions and a control, and the distances say so: 0.0165 and 0.0115 inside the two clusters, 0.17 to 0.20 between them, 0.24 to 0.43 to the control.

An order of magnitude separates within-cluster from between-cluster. That means:

Choosing between Bradford and CAT02 is nearly a null decision on this criterion — they differ by less than a fiftieth of what separates them from the other cluster — so the grounds for choosing between them are elsewhere entirely, in numerical behaviour or in what a standard already specifies.

Choosing between the clusters is the real decision, and it is the decision between fitting corresponding-colour data and approximating receptors. That is a decision about what a transform is for, and it deserves to be made explicitly rather than inherited from whichever matrix an implementation happened to ship.

And XYZ scaling’s distance is the scale that makes the others legible. Without a null case in the table the between-cluster gap of 0.18 has nothing to be large or small compared with. With it, the gap is about forty per cent of the distance to doing nothing sensible at all — which is a large fraction and is the number worth carrying.

Who found it, and when #

The five transforms are Hunt, Pointer and Estévez’s; Lam’s, for Bradford; the CIE’s for CAT02 and CAT16. That they disagree is known and is why there are five; the standard way it is reported is a table of predicted corresponding colours with a mean error, which is a comparison against data rather than against each other.

The comparison against each other is available in any paper that lists the matrices and is rarely made, and the reason is worth naming: a field’s internal disagreement looks like an open question when it is described and like a measurement when it is quantified. Twenty-eight nanometres is a measurement.

The comparison that is not available #

One thing the pairwise distances cannot do, stated so the table is not over-read.

They cannot rank the five. A distance matrix says how far apart the options are and says nothing about which is right, and no amount of internal comparison produces an external answer. The temptation is to treat the cluster centre as a consensus — four of the five are within 0.20 of one another and XYZ scaling is 0.4 away, so the four look like agreement — and that is precisely the wrong inference. Four transforms fitted to overlapping data agreeing with each other is not four independent estimates converging.

What the matrix does support is a bound in the other direction: since the five are mutually 0.17 to 0.43 apart on the protan point, at most one of them can be within 0.08 of the truth, whatever the truth is. That is a statement about the field with no model in it, and it is the strongest thing available from geometry alone.

Getting past it needs data this collection does not hold, and the shape of that gap is the same one every other thread in this round runs into: a constructed comparison can measure spread, sensitivity and disagreement, and cannot measure correctness.

The tritan point is the one the five transforms disagree about most, and putting both statements of that disagreement side by side is what shows which of them a declared width can break.

Three of the five are dominated #

Neither column is the right one on its own and a choice is a trade rather than a preference are the essay’s reading of the two objectives, and among the five transforms anybody would actually choose between, three of them are not on the trade at all.

Plotting the five on both objectives and asking which are Pareto-optimal — better on nothing, worse on something:

CAT16 beats Hunt–Pointer–Estévez, CAT02 and XYZ scaling on both objectives at once. So the choice is not among five and it is not a five-way trade; it is a two-way choice between Bradford and CAT16, and everything else in the table is worse on both counts than one of them.

That is a considerably more useful output than a ranking, and it sharpens the essay’s own closing advice. CAT16 is middling at both, which is a defensible reason to standardise on it — and it is stronger than middling: it is one of only two options that anything can be said for, and the other is Bradford.

The two orderings are uncorrelated, not reversed #

Read down the two columns and the orderings are nearly reversed is true of the whole table and not of the transforms in it.

Across all eight rows the rank correlation between the two objectives is −0.62, which is a genuine anti-correlation. Across the five published transforms alone it is −0.10 — essentially nothing.

The difference is the three rows that are not transforms. The two optima sit at opposite corners by construction, and the receptor basis read from the confusion points sits near the discrimination end; put those three in and the table looks like a trade-off, take them out and the five candidates scatter.

So the trade-off is real between the extremes and invisible among the options. A specification choosing between the five is not moving along a frontier — it is choosing between two frontier points and three that are simply worse — and the shape of the trade only appears once objects nobody would ship are added to the table.

The spans support the same reading. The five span 2.08 on adaptation and 1.59 on discrimination, so a factor of 1.6 on each is right about one column and low by a quarter on the other. Excluding XYZ scaling, which is the control rather than a candidate, the spans are 1.39 and 1.59 — and the adaptation column, the one the field’s fitting effort has gone into, is the narrower of the two.

At most two, not at most one #

The geometric bound is stated as at most one of them can be within 0.08 of the truth, and the cluster structure the essay itself measures does not support it.

Hunt–Pointer–Estévez and CAT16 are 0.0115 apart. Two points that close can both lie within 0.08 of the same truth — the bound needs a minimum pairwise separation above 0.16, and the minimum here is a seventieth of that.

The correct bound is on clusters. The smallest between-cluster distance is 0.1723, which does exceed 0.16, so at most one cluster can be within 0.08 — and a cluster has two members. At most two of the five can be within 0.08 of the truth, and they are the two in whichever cluster is right.

That is weaker than the stated version and it is still a strong claim with no model in it: three of the five published transforms are wrong by at least 0.08 on the protan point, whatever the truth is, and nothing but the matrices was needed to say so.

And the corrected form is the one the essay’s own argument wants. Its point is that the five are two positions and a control rather than five opinions; a bound that counts transforms treats them as five, and a bound that counts clusters treats them as three. The second is the reading the distances support.

Where the ladder goes next #

The set an average is taken over, and the population a distance is divided by, are the same kind of object seen twice. A third instance of it sits in the imaging half of this collection, where a camera’s reported accuracy is decided by a chart.