The weighting is the disagreement

Five alternatives to the unit this collection publishes in, and one property that sorts them exactly.

The claim #

Whether a formula divides a chroma difference by the chroma at which it was measured predicts which formulae agree with each other, better than anything else about them does.

• The split is exact on this menu. The two weighted formulae sit at 15.0 and 24.4 per cent scatter about ΔE2000; the three unweighted ones sit at 28.5, 32.3 and 34.7. There is no overlap.
• It cuts across the obvious classification. One of the two weighted formulae is a graphic-arts recommendation and the other is an appearance model. Two of the three unweighted ones were published by the same committee in the same year as the space the weighted ones are built on.
• It is not a small effect on the levels. Turning the weighting off raises the census’s sensitivity to its own test set from 0.69 to 1.11 — past one, which means the answer would then move more than proportionally with the input.
• And it is what CIEDE2000 mostly is. The rotation term everybody remembers matters in one corner of the space; the division by chroma matters everywhere.

What the weighting is #

Two formulae, side by side, is the shortest way to see it.

ΔE*ab is the Euclidean distance in CIELAB: the square root of the sum of squares of the lightness, a and b differences. There is nothing else in it. It was recommended in 1976 with the space, and its whole content is the claim that CIELAB is uniform enough that a straight distance in it means something.

ΔE*94 is the same three differences, resolved into lightness, chroma and hue rather than lightness, a and b, and then each of the last two divided by something: the chroma difference by 1 + 0.045 C, the hue difference by 1 + 0.015 C, where C is the chroma of the reference colour. That is the whole of the change.

The consequence is that a difference of a given size between two saturated colours counts for less than the same difference between two pale ones. At a chroma of 60 — an ordinary saturated red — a chroma difference is divided by 3.7. The formula is saying that CIELAB’s chroma axis is stretched by nearly a factor of four out there, which is exactly what MacAdam’s ellipses had already said and which the space was built without.

CIEDE2000 does the same and more: the same two divisions with a more elaborate chroma dependence, a lightness weighting, a term correcting the a axis near the grey axis, and a rotation term that only bites in the blues. The rotation term is what people remember about the formula. It is not what the formula mostly does.

The menu, sorted #

Scatter is the root-mean-square departure from that formula’s own best rescaling of ΔE2000, as a fraction of the mean. A formula that were a pure rescaling would sit at zero and swapping to it would change every printed number and no conclusion. None of the five is close to that.

The sort is exact, and the two things it does not sort by are worth saying out loud. Not by age: the closest is from 1994 and the furthest from 2020, with 1976 in between twice. Not by kind: the second-closest is the only entry that is not a distance between three numbers at all.

CAM16-UCS weights chroma too, and that is why it is where it is #

Calling an appearance model “a formula that weights chroma” needs justifying, since it does nothing that looks like 1 + 0.045 C.

What it does is compress. The model reports a colourfulness M, and the uniform space built on top of it uses not M but (1/0.0228) log(1 + 0.0228 M) — a logarithm, which is very nearly linear near zero and increasingly flat as M grows. Differentiate it and the local scale factor on a colourfulness difference is 1/(1 + 0.0228 M), which is the same shape as ΔE*94’s 1/(1 + 0.045 C) with a different constant.

So the two arrive at the same operation from opposite directions: one by fitting a correction to a space known to be wrong, the other by modelling the response that made it wrong in the first place. That they land 9 percentage points apart on this instrument, with three unweighted formulae 4 to 10 points further out, is the strongest evidence available here that the operation rather than the derivation is what matters.

What the weighting does to a sensitivity #

A published level moving is one thing. A published sensitivity moving is worse, and this is where the weighting turns out to matter most.

Every residual in the adaptation census is a mean over a set of constructed surfaces, and that set’s saturation is the most elastic input in this collection: make the test surfaces 25 per cent more saturated and the answers rise by between 0.49 and 0.91 times as much, proportionally. That measurement is in ΔE2000.

Recompute it under each unit and the elasticity moves further than the levels do:

The ordering is the weighting again, and now it is a continuum rather than a split: the more a formula divides by chroma, the less its answer depends on how saturated the test set is. That is not a coincidence, it is a tautology being made visible. A more saturated test set produces larger chroma differences; a formula that divides chroma differences by chroma discounts exactly the part that grew.

The size is what matters. Under an unweighted formula the elasticity is above one: a test set half again as saturated gives an answer more than half again as large, so the published residual would be more a statement about the surfaces than about the change of light. Under the published unit it is 0.69, and under the appearance unit 0.53. The choice of unit is worth about a factor of two on how much of the collection’s adaptation numbers belong to a set nobody measured.

The two instruments do not rank the same #

The essay now has two orderings of the same five formulae — how far each sits from ΔE2000 pair by pair, and how elastic each makes the census — and they agree about the split and disagree about almost everything inside it.

Rank correlation 0.60 — the same two at the top, the same three at the bottom, and no agreement finer than that.

The instructive row is ΔE*ab. It is the closest of the three unweighted formulae to ΔE2000 on individual pairs, at 28.5 per cent against Oklab’s 34.7 — and it is the worst of all six for the elasticity, at 1.105, further from the published unit than any other entry on the menu. A formula that tracks ΔE2000 reasonably well one pair at a time is the single worst choice for a result that is a mean over a set of pairs.

That is not a paradox and it is worth having the mechanism. Scatter is a symmetric measure — it counts a formula reading high and reading low as the same size of disagreement — and an average over a set does not: reading high on the saturated members and low on the pale ones leaves the mean nearly right and makes the sensitivity to which members are in the set completely wrong. ΔE*ab’s errors happen to cancel in the middle and to be systematically ordered by chroma, which is the one arrangement that is invisible to a scatter and fatal to an elasticity.

So the practical warning is a sharp one. Agreement with the published unit is not transitive across the kind of quantity being computed. A reader checking whether a body of results survives a change of unit cannot check it on pairs and conclude anything about means, and the collection has now measured both because the second was not predictable from the first.

And the published unit has the least stable sensitivity #

One more column falls out of the same table, and it is a caution about this collection rather than about the formulae.

Reading the elasticity table by row spread rather than by mean — how much a unit’s elasticity varies across the fourteen changes of light:

The unit this collection publishes in has the widest spread of the six. Its elasticity is not one number but a range running from about a half to about nine tenths, so a sentence of the form the census’s answers move by seven tenths as much as its test set’s saturation is a mean of fourteen quantities that differ by nearly a factor of two — and it is least summarisable in exactly the unit every published figure uses.

Oklab is the extreme in the other direction: elasticity 1.07 on every row, to within 0.08. Its answers are almost exactly proportional to the test set’s saturation, uniformly, which is a bad property to have and a very predictable one.

That inverts the usual reading of these two columns. A low mean elasticity says a unit is discounting the input; a low spread says it is discounting it consistently, and the two are separate virtues. CAM16-UCS is the only entry that has both. ΔE2000 has the first and the worst of the second, which is a fair description of a formula assembled from several corrections that each bite in a different part of the space.

The prediction, and what happened to it #

This is the point at which a prediction written down a round earlier can be scored, and it is worth doing carefully because it was half right in a specific way.

The sentence was: ΔE*76, having no chroma weighting, raises the saturation elasticity towards one and leaves the extremes alone.

Every row rises, which is the first clause and is exactly right: all fourteen, without exception, with a mean increase of 0.418.

It goes past one rather than towards it. The mean lands at 1.105 and the smallest row at 1.066, so the whole table crosses the threshold rather than approaching it. “Towards one” implied an asymptote the arithmetic does not have.

And the extremes do not stay put — they close. The mildest row’s elasticity goes from 0.494 to 1.106, a rise of 0.61; the harshest from 0.911 to 1.313, a rise of 0.40. The spread across the census narrows from 0.417 to 0.247, so the weighting is not a common-mode discount but something that acts hardest where the elasticity was lowest. The prediction had that exactly inverted.

Recording all three is worth more than recording the one that was right. The habit is the collection’s own: a prediction deleted after the fact leaves a file that looks as though it never made one.

What a division by chroma is a claim about #

A divisor is a modelling statement, and it is worth saying what this one asserts before asking whether it is right.

It says that the just noticeable chroma difference grows in proportion to chroma, over the range the fit covers — the Weber behaviour that turns up wherever a sensory system reports ratios rather than differences, and that this collection meets again in the lightness axis and again in the response of a cone. Where the pattern is Weber’s, the natural coordinate is a logarithm, and a logarithm is what the appearance model uses.

Where it is asserted that the pattern is Weber’s, three things follow that a reader should hold together. A difference has no absolute size — only a size relative to where it was taken — which is a sentence this collection has already argued from the ellipse side. A tolerance is therefore not a number but a number plus a place, and a specification quoting one without the other is under-determined. And an average over a set of differences depends on where the set sits, which is exactly the elasticity above and is why the two questions are the same question.

The divisor also has a range of validity that the formulae do not state. 1 + 0.045 C is fitted over chromas up to about 60; extrapolated to a chroma of 130 — reachable by a saturated ink and by nothing on a screen — it divides by nearly seven, and there is no data behind that. The same problem attends every extrapolated constant in this collection, and it is worth flagging because the most saturated members of the census’s test set are exactly where the divisor is doing the most work.

Why the rotation term is not the story #

CIEDE2000’s most-discussed feature is a rotation of the chroma–hue ellipse in the blue region, and it deserves a paragraph mostly to be set aside.

It exists because the ellipses in the blues are tilted rather than aligned with the chroma and hue axes, and no amount of scaling the two axes independently can fix a tilt. Its magnitude is governed by a Gaussian centred at a hue angle of 275° with a width of 25°, so it is exactly zero over most of the hue circle and reaches its maximum in one place.

On the census, that place is barely visited. The test surfaces are single-lobed reflectances at levels and depths spanning the whole hue circle, and the rows most affected are the two involving daylight going bluer. Removing the rotation term entirely and leaving everything else moves the census’s mean by well under the width of the calibration’s own residual scatter.

So the practical summary of what CIEDE2000 does to this collection is: it divides by chroma, and then it does a number of other things that are individually visible and collectively small. The literature’s emphasis is on the other things because they are what is new; the effect on a body of results is almost entirely the division.

Where the model stops #

None of this says the weighting is right. It says the weighting is what separates the formulae, and separately that a formula’s agreement with ΔE2000 is a poor guide to its merit — the two of the five that agree least about the census are the two that do best on the ellipses the whole question was originally settled by.

It also does not establish that the constant is right. 1 + 0.045 C is a fitted line through a scatter of measurements, and nothing here varies the 0.045 — which is the natural next question and is a dial rather than a menu, because the two ends of it are two of the formulae on the menu.

And the reference sample the calibration is fitted on is constructed, like everything else here. A different sample gives different scale factors and therefore different scatters. The split is robust to that — the two weighted formulae remain the two closest on any sample tried — but the numbers are not, and the honest form of the finding is the ordering rather than the percentages.

Who found it, and when #

The observation that discrimination ellipses grow with chroma is MacAdam’s, 1942, and is the oldest quantitative fact in this part of the subject. That CIELAB does not account for it was known when CIELAB was recommended; the committee said so.

What took until 1994 to arrive was the form of the correction, and what took until 2000 was the admission that one linear divisor was not enough. Ottosson’s Oklab in 2020 goes the other way — a space designed so that a plain Euclidean distance works — and it lands on the far side of this menu from the weighted formulae, which is a result about how hard that is rather than about the space.

Where the ladder goes next #

The menu is discrete except in one place. ΔE*94’s two weighting constants can be scaled continuously, and at zero they make every weight exactly one, so the formula becomes ΔE*ab — not approximately, but as an identity. That line joins the two ends of the oldest disagreement in the subject and can be walked, which turns a spread over six points into a derivative.