A difference needs a basis too

Everything in this collection’s account of the observer’s undetermined nine numbers has turned on a single algebraic fact: a matrix applied to both sides of an equality changes nothing. It is why colour matching cannot choose the cone fundamentals, why collinearity survives every diagram, and why the whole freedom is harmless as long as everything downstream is linear.

Almost nothing downstream is linear.

The claim #

CIELAB divides tristimulus values by a white point and takes a cube root. Neither operation commutes with a change of basis, so the space, and every ΔE computed in it, depends on which three curves were used to write the tristimulus values down. The basis CIELAB uses is XYZ, which was chosen in 1931 to make matching functions non-negative — and it is not the best of the available choices.

• The dependence is real and is measurable. Across six bases, the mean axis ratio of MacAdam’s ellipses in a CIELAB-like space runs from 2.60 to 12.45.
• Among defensible bases it is a factor of 1.3. Cone chromaticities give 2.60 to 2.81; XYZ gives 3.44; the 1976 diagram’s basis gives 4.20.
• So CIELAB’s own basis is beaten by every cone basis tested, on the measurement CIELAB exists to serve.
• And this is where the freedom stops being free. A linear stage passes the ambiguity through untouched; the first nonlinearity converts it into a decision, and the decision is made by whoever wrote the linear stage.
• CIECAM16 makes the decision deliberately, adapting in CAT16’s axes, and those axes were fitted to corresponding-colour experiments rather than chosen for difference either.

Where the commuting stops #

Write the CIELAB construction out and the failure is visible without any computation.

The space is built from f(X/Xₙ), f(Y/Yₙ) and f(Z/Zₙ), where f is a cube root with a linear segment near zero, and Xₙ Yₙ Zₙ is the white point. Two things happen there that a linear map does not survive.

The division is elementwise. Dividing the three coordinates by the three coordinates of a white is a diagonal operation, and a diagonal operation is basis-dependent by definition — which is the whole content of the adaptation-basis argument one field over. Change the basis and the same division is a different linear map.

The cube root is applied coordinatewise. f(a + b) is not f(a) + f(b), so the order of the linear map and the nonlinearity cannot be exchanged. A basis change before the cube root produces a genuinely different space, not a re-labelling of the same one.

Everything after that — the differences that make a* and b*, the arctangent that makes hue, the corrections in CIEDE2000 — is downstream of a decision already taken.

What the choice is worth #

The measurement is the same pair of ratios the previous essay minimised over chromaticity planes, computed in three dimensions instead of two: the mean axis ratio of the twenty-five ellipses, and the spread of their sizes.

The six bases are not arbitrary. Each is a coordinate system somebody publishes and uses, and each is A x̄ for a nonsingular A, so each describes the same observer exactly.

The cone bases win. The König construction from the confusion points gives 2.60; Hunt–Pointer–Estévez gives 2.81; CAT16 gives 2.71. These three are what a physiologically motivated space would use.

XYZ gives 3.44, which is where CIELAB sits.

The 1976 diagram’s basis gives 4.11, which is worse than XYZ — a striking result, since that basis was designed to improve uniformity in a chromaticity plane and does so substantially. It improves the projective picture and damages the nonlinear space built on it, which is a clean demonstration that the two problems are not the same problem.

And the display basis gives 12.45, with a size spread of 18.6. A CIELAB built on a device’s own primaries would be a very poor uniform space, which is worth knowing because approximate versions of exactly that are common in image processing.

How much of what was available the published revision actually took is a separate question, and it has an answer in the same units.

Two more planes say how much of the available improvement is left on the table, and where the best of them actually is.

Why the cone bases win #

The direction of the result is not an accident, and the reason is worth stating because it is the substantive part.

A uniform space is trying to be one in which equal distances are equally detectable. Detection happens at the receptors and at the stages immediately after them, and those stages are much better described as operations on cone responses than as operations on tristimulus values — a gain applied to each receptor class, a difference taken between two of them, a compression applied to each. A nonlinearity applied to cone responses is therefore a rough model of something; a nonlinearity applied to X, Y and Z is a nonlinearity applied to three linear combinations of cone responses that were chosen for their tidiness.

X in particular is a strange thing to compress. It is a combination with a bimodal shape — a large lobe in the red and a second smaller one in the blue — that no receptor has, and it exists because the 1931 committee needed a set of non-negative functions with Y carrying all the luminance. Taking the cube root of X/Xₙ is compressing a quantity with no physiological referent.

That is not proof that a cone basis must be better, and the measurement is what settles it. It does mean the result is the one to expect, and it means the improvement is unlikely to be an artefact of the particular ellipses used.

Why this is not an argument for changing CIELAB #

A factor of 1.3 in mean axis ratio is a real improvement and it is not what makes a space usable.

CIELAB’s problems are much larger than its basis. It is not a metric in the sense its name implies, its hue lines are curved in the blue badly enough that CIEDE2000 needs a rotation term to patch it, and its lightness scale disagrees with an appearance model’s by more than any of this. Swapping the matrix would move the axis ratio from 3.44 to 2.60 and leave every one of those in place.

What the measurement establishes is not that CIELAB is wrong but that it contains a decision nobody made. Oklab is the instructive contrast: it was constructed deliberately in a cone-like basis, with the nonlinearity applied to cone responses rather than to XYZ, and it is better on exactly this measure — for exactly this reason, stated in its design notes rather than inherited.

What was computed, and how #

The construction is the CIELAB formula with a matrix inserted: A is applied to the stimulus and to the white point, the three ratios are taken, the cube-root function with its linear segment is applied, and the three differences that make L*, a* and b* follow. Nothing else changes, so the comparison between rows is a comparison of one decision.

Each MacAdam ellipse’s boundary is sampled at forty-eight points at true scale and carried through at a fixed luminance of 0.4. Radii are measured in three dimensions from the transformed centre, since a chromaticity step moves lightness slightly in a nonlinear space and ignoring that would be measuring a projection of the answer.

Both reported quantities are ratios, which is essential here: the six spaces have magnitudes differing by orders of magnitude, and an unnormalised distance would rank them by their arbitrary scaling and nothing else.

The assertion requires the spread across bases to exceed 1.25 — comfortably below the measured 3.8 and comfortably above anything that could be measurement noise in a computation with no noise in it — and separately requires that CIELAB’s own basis be one of the rows rather than a special case, so that the finding cannot be an artefact of comparing a reference against alternatives constructed differently.

Two numbers here are from before the axis ratio was corrected #

The 1976 basis is quoted twice in this essay and the two quotations disagree: 4.20 in the claim list, 4.11 in the section that discusses it. Both numbers are real and they are the same basis measured two ways.

4.11 is what forty-eight sampled boundary points reported. 4.20 is what the closed form gives, after this collection found that sampling an ellipse’s boundary understates its axis ratio by an amount that grows with the answer. The body carries the pre-correction figure and the claim list carries the corrected one.

The same is true of a second number, and it is the one the build asserts on. The spread across bases must exceed 1.25 — comfortably below the measured 3.8 — and 3.8 is 9.84 over 2.59, which are the sampled values for the display basis and the receptor construction. On the corrected values the spread is 12.45 over 2.60, which is 4.79.

Both stale numbers move in the direction that strengthens the argument, and neither changes a conclusion. The 1976 basis is worse than XYZ at 4.11 and worse at 4.20. The spread clears its 1.25 threshold at 3.8 and clears it further at 4.79. And the ordering of the bases is identical under the two measurements — receptor, CAT16, xy, u′v′, display — which is what the correction essay predicts, since its error is monotone in the quantity being estimated and compresses an ordering without reversing it.

That is worth stating plainly rather than quietly fixing: a correction that moves every number and no ranking is the useful kind, and it is only visible at all because two numbers for one basis ended up in the same essay.

A factor of 1.3 against a floor of 1 #

Among defensible bases it is a factor of 1.3 is the essay’s own summary of what the choice is worth, and a factor is the wrong unit for it, because the quantity has a floor at one rather than at zero.

A perfectly uniform space would leave the ellipses as circles, at an axis ratio of 1.00. CIELAB’s basis leaves 3.44, so it is 2.44 above the floor; the best cone basis leaves 2.60, which is 1.60 above it. The change of basis closes 34 per cent of CIELAB’s distance from uniformity.

That is a much larger claim than a factor of 1.3 sounds, and it is the honest one: a third of everything wrong with CIELAB’s shape, on this measurement, is a matrix nobody chose. It also bounds the other side, which the essay is right to insist on — the remaining 66 per cent is beyond any change of basis, and it is where the curved hue lines, the non-metric distances and the disagreeing lightness scale live.

The display basis reads differently in the same unit. At 12.45 it is 11.45 above the floor, which is 4.7 times further from uniformity than XYZ rather than the 3.6 times its raw ratio suggests. An image-processing pipeline that compresses in a display’s own primaries and then differences is not somewhat worse than CIELAB; it is nearly five times as far from the thing both are trying to be.

Reading a ratio-of-ratios against its own floor is the general form of this, and it applies to every uniformity number this collection publishes. A space at 1.6 and a space at 3.2 differ by a factor of two in the statistic and by a factor of 3.7 in how far each sits from the goal, which is what a reader comparing two spaces actually wants to know.

Drawing the ellipses twice the size makes the shapes legible without changing any of the numbers, which is worth doing once for the plane everybody prints.

Where the model stops #

The ranking is on one measurement. Twenty-five ellipses, one observer, chromaticity discrimination at threshold. A space that wins here might lose on suprathreshold differences, on lightness, or on hue linearity, and those are different questions with different answers.

The search is over six published bases rather than over the whole group. Minimising the axis ratio over all nine coefficients with the nonlinearity in place is a well-posed problem and is not solved here; what is established is that the choice matters and that the standard one is not the best of the obvious ones. The floor is unknown.

Nor is the comparison with Oklab a controlled one. Oklab differs from CIELAB in its basis and in its exponent and in the linear stage after the compression, so its advantage cannot be attributed to the basis alone on this evidence. The controlled comparison is the six-row table, where only the matrix changes; Oklab is quoted as an existence proof that somebody made the decision deliberately, not as a measurement of what the decision was worth.

And the white point is doing some of the work. Changing the basis changes what “divide by the white” means, and that is genuinely part of the effect being measured rather than a confound — but it means the result is about the whole normalise-then-compress stage rather than about the cube root alone.

Who found it, and when #

The dependence is not a discovery — it is a consequence of the definitions and would have been obvious to anyone who wrote them down. What is missing from the literature is the measurement, and the reason is probably that CIELAB’s basis has never been treated as a variable. XYZ arrived first, in 1931; the uniform spaces were built on it in 1976 because it was what tristimulus values were written in; and by the time cone-based spaces became fashionable the question had become which nonlinearity to use rather than which basis to apply it to.

Oklab, published in 2020, is the point at which the decision was made deliberately in a widely used space, and its stated rationale is exactly the argument above. CIECAM16 makes the same decision in a different direction, adapting in CAT16’s axes — which are fitted to corresponding-colour experiments and, on the confusion-point test, are not cone responses either.

The generalisation #

The rule is short: an invariance is only as good as the last linear stage.

A quantity that passes through a chain of linear operations inherits the whole freedom of the input and is therefore undetermined in the same way. The first nonlinearity in the chain converts that freedom into a fixed choice — and because the choice is made implicitly, by whoever wrote the linear stage, nobody records it as a choice at all.

The practical consequence is a place to look. In any pipeline with a nonlinearity in it, the specification of the stage immediately before the nonlinearity is load-bearing in a way the stages before that are not, and it is usually the stage that was specified for some unrelated purpose. Here it is XYZ, specified for draughtsmanship. In an encoding pipeline it is the primaries a transfer function is applied in. In a camera it is the raw space a curve is applied to.

The best plane there is, at the same magnification, is the comparison that prices what a change of coordinates can buy.

What a specification would have to say #

If a difference formula carries a basis, a specification quoting a difference has to name it — and none of them does, because the basis is inside the formula’s definition and is therefore invisible in the same way a measurement condition was invisible in a whiteness figure until somebody removed the ultraviolet.

In practice the situation is better than that sounds, for a reason that is worth being explicit about. Everyone uses the same basis, because everyone uses CIELAB and CIELAB is defined on XYZ. A specification is therefore unambiguous as long as it names the formula, which they all do. The hidden variable is held constant by convention across the entire industry, which is exactly the condition under which a hidden variable causes no trouble and is never noticed.

The trouble arrives at the boundaries. A difference computed in Oklab is not a difference computed in CIELAB and the two are not interconvertible by a factor; an image-processing pipeline that compresses in a display’s own primaries and then differences is computing something with an axis ratio near ten; and an appearance model’s difference is computed in yet another basis again. Every one of those is a legitimate quantity and none of them is the one a printing tolerance means, and the units are the same three letters in all four cases.

What the result is not evidence for #

Two conclusions look available here and neither is supported.

It is not evidence that a cone basis is the right one. What is measured is that three cone-like bases beat XYZ on one criterion, on one set of twenty-five ellipses, from one observer. A basis that happened to win on this measurement could lose on suprathreshold differences, on hue linearity or on lightness, and none of those was tested. The physiological argument for a cone basis is separate and is a reason to expect the result rather than a consequence of it.

And it is not evidence that the difference formulae are wrong by a factor of 1.3. The axis ratio is a summary of how badly the space fails to be uniform, not an error in any particular ΔE — two colours’ computed difference could be nearly identical in two bases whose summary statistics are far apart. The right reading is that the space is measurably better or worse, and that better and worse were never decided on purpose.

The finding is about a decision’s existence rather than about its size, and the size is what a further measurement would have to establish.

Where the ladder goes next #

The tolerance a practitioner writes into a contract is computed in one of these spaces, and a tolerance is already a shape rather than a radius. The basis choice moves the shape by about the same factor as the anisotropy it is trying to accommodate, which puts it firmly inside the range that decides acceptance.

The other direction is upward, into appearance. The axes an appearance model adapts in are the same kind of object as the basis in this essay, chosen the same way, and defended by a name rather than by the measurement the name implies.