The exponent was never the argument

The compressive exponent is one of the most argued-about constants in the subject. CIELAB takes a cube root. Munsell’s value scale is closer to a fifth root over most of its range. Stevens’s power law puts brightness at about 0.33 and lightness nearer 0.4. Weber–Fechner says logarithm, which is the limit as the exponent grows. Every one of those has been defended with data.

Almost none of it matters for the thing the cube root is in CIELAB to do.

The claim #

Minimised over every basis, the anisotropy floor is 2.33 with no compression, 1.66 at a square root, 1.61 at a cube root and 1.57 at a tenth root. Almost the whole of what a compression buys arrives with the first step away from linearity, and the exponent above two is choosing between numbers three per cent apart — while the choice of basis at a fixed exponent spans a factor of two.

• From p = 1 to p = 2: 2.334 → 1.663, a 29 per cent improvement.
• From p = 2 to p = 10: 1.663 → 1.567, six per cent, spread over a fivefold change in the exponent.
• A cube root and a square root differ by 3.2 per cent.
• At a fixed exponent of three, CIELAB’s own basis leaves 3.443 against the floor’s 1.611 — a gap nearly nineteen times the one between a square root and a tenth root.
• And the fixed-basis curve gets worse as the exponent rises, which is the finding underneath the finding: an exponent chosen without choosing a basis is being asked to do a job it cannot do.

What the exponent is being asked to do #

There are at least three distinct jobs, and they have been argued about as though they were one.

The first is matching a magnitude scale. Munsell value is a scale of judged lightness built from grey papers, and the question is which function of luminance reproduces it. That is a one-dimensional question about achromatic stimuli, and it has an answer that depends on the surround.

The second is coding. A transfer function on a display exists to spread quantisation levels where thresholds are, and the right exponent there is set by the contrast sensitivity of the eye at the luminances involved, which is a different curve again.

The third is making differences uniform, which is what CIELAB’s cube root is in the formula for. The claim of a uniform space is that a step of a given size means the same amount of visible difference wherever it is taken and in whatever direction, and the quantity that measures failure is the anisotropy of the discrimination contours.

The three jobs have been allowed to borrow each other’s evidence for a century. The measurement here is about the third one only.

The measurement #

The family is CIELAB’s arithmetic with the exponent as a variable: divide each channel by the same channel’s value for the white, raise to the power 1/p with a linear toe below the standard’s break point, and form lightness and two opponent differences. At p = 3 this is CIELAB exactly. At p = 1 there is no compression at all.

The toe is made continuous with the power law at the break point for every p, which matters more than it looks: without it the ellipses at low channel values would be measured across a discontinuity whose position moves with the exponent being compared, and the comparison would be partly of the discontinuity.

For each of eight exponents from 1 to 10 the anisotropy is minimised over all nine coefficients of the basis, and the resulting floor is plotted against the value CIELAB’s own basis achieves at the same exponent.

What the two curves say #

The floor falls once and then flattens. Nearly all of the benefit — 29 of the 33 per cent available between p = 1 and p = 10 — arrives by p = 2. After that the curve is nearly horizontal, and the differences are smaller than the disagreement between two observers’ threshold data.

The fixed-basis curve rises. On CIELAB’s own basis the anisotropy is 3.569 at p = 1, dips to 3.304 at p = 1.5, and then climbs steadily to 3.772 at p = 10. Compressing harder makes CIELAB’s geometry worse.

Those two facts together are the essay’s point. The compression is not a knob that improves uniformity by being turned further; it is a structural addition that helps once, and how much it helps depends on what it is applied to. Turned further on the wrong basis it actively hurts.

The limit the sweep runs towards is an affine space with no compression in it at all, and it is the last of the three worth drawing.

What the published revision took of what was available is the same question asked on the projective side, and it has an answer in the same units.

Why the first step is worth so much and the rest so little #

The mechanism follows from what a small ellipse sees of a smooth map.

Over a region as small as a discrimination ellipse, any smooth map is its own derivative. So what the compression contributes is a local diagonal rescaling that varies with position, with the scaling in each channel proportional to that channel’s value raised to the power 1/p − 1.

Going from p = 1 to p = 2 introduces that position dependence, and introducing it is a change of kind. Going from p = 2 to p = 3 makes the exponent of the local scaling −1/2 instead of −1/3; going to p = 10 makes it −9/10. The shape of the position dependence is set from the moment there is any, and increasing p adjusts its strength.

Since the strength can be partly absorbed by the basis — a basis is free to rescale the channels before the power sees them — the search recovers most of the difference. That is why the floor is flat: the exponent and the basis are partly redundant, and the search spends the redundancy. A fixed basis cannot, which is exactly why the fixed-basis curve moves in the wrong direction.

Which of the two is actually moving #

That the exponent and the basis are partly redundant is testable directly: fit a basis at one exponent and then measure it at another.

Read down a column and each fitted basis wins its own exponent, as it must. Read along a row and the redundancy appears.

Held at its cube-root optimum, the basis gets nothing at all from a larger exponent. It reads 1.6112 at p = 3 and 1.6186 at p = 10 — half a per cent worse, across more than a threefold change in the exponent. So the floor curve’s 5.8 per cent decline between p = 2 and p = 10 is not the exponent doing anything. It is the search re-fitting nine coefficients at every step.

The same basis reads 1.6871 at p = 2, so the move from a square root to a cube root is worth 4.5 per cent at a fixed basis and 3.2 per cent at the floor. That is the last place the exponent earns anything on its own, and it earns it exactly where the floor curve is starting to flatten.

Carrying a basis the other way costs comparably. The p = 2 optimum reads 1.6630, 1.6515 and 1.7910 at exponents of 2, 3 and 10 — slightly better at the cube root than at the square root it was fitted for, and 7.7 per cent worse at a tenth root. The p = 10 optimum reads 1.7514 at p = 2, which is 11.8 per cent above its own floor.

So using the wrong basis for an exponent costs between one and twelve per cent, and the whole benefit of the exponent above two is under six. Two parameters that can each undo a similar share of the other’s damage is what redundancy looks like when it is measured instead of argued.

The p = 1 column is the control, and it behaves. All four fitted bases read between 2.334 and 2.372 there, a spread of 1.6 per cent, because at p = 1 the map is affine and there is nothing position-dependent for a basis to have been fitted to. Whatever each basis learned at its own exponent, none of it transfers to the case with no compression in it — which is the 29 per cent again, seen from the other end.

What this says about the arguments #

It says the third job’s evidence has been over-read, and it does not say the other two jobs are settled.

The cube root’s presence in CIELAB is often justified by pointing at Munsell value, which the cube root approximates well over the middle of its range. That justification is about the first job, and it is a good one. What the measurement here shows is that it is nearly weightless for the third: as far as making discrimination contours round is concerned, CIELAB would be almost exactly as good with a square root and would be measurably worse if its basis had been chosen differently.

It also says something about proposals. A new uniform space that differs from an existing one in its exponent is proposing a change worth a few per cent on this measure, and one that differs in its basis is proposing a change worth more than a factor of two. The two are routinely reported with the same emphasis.

The one place the exponent is doing real work #

The flatness of the floor above p = 2 should not be read as the compression is arbitrary, because the step from 1 to 2 is not arbitrary at all and is worth restating in its own terms.

With no compression the space is affine, and an affine map takes every ellipse to an ellipse with the same linear distortion applied everywhere. So the twenty-five ellipses’ shapes can be equalised only to the extent that they were already similar to one another — and MacAdam’s are not: at the same luminance the largest is more than ten times the area of the smallest, and their orientations swing through most of a right angle across the diagram.

A compression is the first thing in the family that can treat two places differently. That is what buys the 29 per cent, and it is why the improvement is a change of kind rather than a change of degree.

What the fixed-basis curve is really showing #

The rising curve — CIELAB’s own basis getting worse as the exponent grows — deserves its own reading, because it is the practical half of the result.

XYZ’s three channels are not commensurate. Y is a luminance and runs to one for a white; X and Z are the other two matching functions integrated, and at the D65 white they run to 0.95 and 1.09. More importantly, the proportions in which a given stimulus divides between them are wildly different from how it divides between cone-like axes, because x̄ has a second lobe in the short wavelengths that no receptor has.

Raising badly proportioned channels to a power exaggerates the disproportion, because a power law’s local scaling depends on the channel value. So the harder the compression, the more the geometry is dominated by an accident of how the 1931 committee arranged for its functions to be non-negative.

That is a concrete statement about a space in daily use: CIELAB’s exponent is as large as it can usefully be given its basis, and the dip at p = 1.5 in the fixed-basis curve says a gentler compression would suit XYZ slightly better than the cube root does.

Who argued about it, and with what #

The exponent’s history is a good example of a constant acquiring authority by being confirmed repeatedly against different things.

Fechner’s logarithmic law is 1860 and comes from integrating Weber’s constant-fraction threshold. Stevens’s power law is the 1950s and comes from magnitude estimation — asking observers to assign numbers to sensations — and gives exponents around a third for brightness. Munsell’s value scale is a 1905 arrangement of grey papers into equal-appearing steps, renotated in 1943; the function that fits it is a fifth-order polynomial, and a cube root approximates it well over the middle.

CIELAB’s committee in 1976 chose the cube root over the Munsell polynomial for a stated reason of simplicity, having checked it against the value scale. So the constant in the standard is justified against the first of the three jobs, and it has been carried into the third by the fact that one formula does both.

None of that is careless. What is missing is the observation that the third job barely depends on it — and that observation requires minimising over the basis, which requires treating the basis as a parameter, which is the move this collection made recently and which nobody in the exponent literature had reason to make.

What was computed, and how #

Eight exponents, each with its own nine-parameter Nelder–Mead search from the identity with restarts, over the same twenty-five ellipses transported the same way.

The objective is invariant to scaling a row of the basis — the row’s scale divides out against the white’s own value in that channel before the power is applied — so the search has three flat directions at every exponent, and the restarts are what keep the answer readable.

Each floor is checked for convergence rather than assumed: the p = 3 case is run from twelve independent random starts and most of them reach 1.6112 and agree on the matrix to four decimals in every entry. The stragglers land near 1.64, 1.77 and 2.54, which is what a search with flat directions does when a restart lands badly, and is the reason the reported number is the minimum over restarts rather than the last iterate.

The comparison curve uses CIE xy’s own basis, which is what CIELAB is built on, and is evaluated at the same eight exponents with no fitting at all.

An exponent of one and a half is below anything anybody has proposed, which makes it the test of whether the floor is really a floor.

What a practitioner should take from it #

Three things, and the first two are permissions rather than instructions.

Stop defending the exponent. If a pipeline uses a cube root and a colleague prefers a square root, the disagreement is worth 3.3 per cent of the anisotropy of a set of ellipses measured on one person in 1942, which is well inside the disagreement between two observers. A tolerance argued to two significant figures is not going to be settled by it.

Start asking what the space is built on. The same question applied to the basis is worth a factor of two, and the answer is usually available in one sentence of a specification: CIELAB is built on XYZ, CIECAM’s space on CAT16, and the two differ by 21 per cent on this measure before any difference formula is applied.

And treat a reported uniformity improvement as a joint claim. A new space is a basis and a nonlinearity and often a difference formula, and its score is the three of them together. Reading the improvement as belonging to whichever part the paper’s title names is the shape of error this collection keeps finding: a quantity attributed to the component that was described rather than to the one that moved.

The ladder of five classes of map can be read at an exponent of four, which is the other end of the range and gives the same ordering.

Where the model stops #

The family is one exponent applied to all three channels. Three separate exponents would be a nine-plus-three parameter family, and nothing here explores it. A space with different compressions per channel is not exotic — the opponent channels are not symmetric — and the floor for that family is unknown.

A power is not the only nonlinearity. A saturating hyperbolic function, which is what appearance models use, is not in this family at any exponent. Its shape differs most at the extremes of the range, and the ellipses sit in the middle, which is why substituting a power for it is defensible here and would not be everywhere.

And the ellipses are at one luminance and from one observer. The floor is the floor for MacAdam’s twenty-five, evaluated in a plane, and a second observer’s thresholds would move it.

The generalisation #

When two parameters can partly do each other’s work, sweeping one with the other fixed measures their difference rather than either of them. That is the whole of what has gone wrong in the exponent literature’s third job, and it is a shape that recurs wherever a model has a coordinate choice in front of a nonlinearity.

The diagnostic is the pair of curves in the first figure. Sweep the parameter of interest twice — once with everything else fixed, once with everything else re-optimised — and read the gap. If the two curves are parallel, the parameter is doing its own work. If the re-optimised curve is flat and the fixed one is not, the parameter is being used to compensate for something else, and the something else is where the result is.

Here the re-optimised curve is flat above p = 2 and the fixed one is rising, which says the exponent was compensating for the basis and doing it badly.

Where the ladder goes next #

Both halves of the arithmetic now have a floor and a fixed-basis curve. What neither has had is a reading of which changes of light the difference between bases actually shows up on, and that is a question about lamps rather than about matrices.