A compression goes below the floor

Elsewhere this collection did something it was pleased with. Every chromaticity diagram is a projective picture of the same measurement, so how badly MacAdam’s ellipses fail to be circles can be minimised over the whole nine-parameter family; the best plane there is leaves a mean axis ratio of 2.02, and since that number is invariant under the entire group it was called a property of the eye rather than of anybody’s coordinates.

The sentence needs a clause. It is a property of the eye as seen through a projective picture, and the qualifier is not pedantry, because the clause is where the next result lives.

The claim #

Applying a cube root after a well-chosen basis leaves a mean axis ratio of 1.61 on the same twenty-five measured ellipses — below the 2.02 that no chromaticity diagram can pass. The linear group’s floor is a floor for linear pictures, and a nonlinearity is not a member of that group.

• The best chromaticity diagram: 2.021. Minimised over all nine coefficients; CIE xy leaves 2.946 and u′v′ leaves 2.371.
• The best space with no compression: 2.359. Divide by a white point, take differences, minimise over the same nine numbers.
• The best space with a cube root: 1.621.
• All three are the same measurement — the same boundary points, the longest radius over the shortest, averaged over twenty-five.
• And none of the three families contains the others, which is why the middle number can sit above the first and below the third without contradiction.

Three families, one measurement #

The confusion this result invites is entirely about which maps are being minimised over, so it is worth laying the three out before any number is compared.

A chromaticity divides by a sum. Send tristimulus values through a nonsingular 3×3, then divide the first two coordinates by the total; the result is a point in a plane. The maps this produces are the projective transformations of that plane, and they preserve straight lines and cross-ratios and preserve neither distance nor angle nor area — which is why some of this collection’s claims survive a change of diagram and others do not.

A linear space divides by a white point. Send the values through the same 3×3, divide each channel by the same channel’s value for the white, and take differences. The maps are affine, and affine maps preserve ratios of areas and parallel lines and do not preserve angle.

A compressed space does the same and then raises each channel to a power before taking differences. This is CIELAB’s arithmetic with the basis as a variable, and the resulting maps are not linear at all.

The measurement applied to all three is identical: take each of MacAdam’s ellipses, carry its measured boundary points through, measure the radius from the transformed centre to each transformed boundary point, and report the longest over the shortest, averaged over the twenty-five. Nothing else differs.

Why the projective floor beats the linear one #

The affine family sits inside the projective one in an obvious sense — every affine map of a plane is a projective map — but the two are not being applied to the same objects here, and the comparison that matters runs the other way.

A projective picture has a line at infinity, and where that line falls relative to the locus decides how much the plane is stretched and where. That is an extra structural freedom: a projective map can magnify one region relative to another, and an affine map cannot. Since MacAdam’s ellipses are large in some parts of the diagram and small in others, and their orientations vary systematically across it, the ability to stretch position-dependently is worth something. It is worth the difference between 2.359 and 2.021, which is 14 per cent.

That ordering is not obvious in advance and it is the reason the two floors have to be quoted separately rather than as one number. Neither is the linear floor; each is the floor for a different family of pictures the discipline actually draws.

What the compression buys, and why it can #

A per-channel power is not a member of either family, and the reason it helps is not the reason a change of coordinates helps.

The ellipses are small — MacAdam’s are drawn at ten times their measured size to be visible at all — so over any one of them a smooth map is nearly its own derivative. What a nonlinearity supplies is a derivative that varies with position in a way no linear or projective map can match: the local Jacobian of a cube root at a point depends on the channel values there, and depends on them separately per channel.

So the compression is a position-dependent rescaling with three independent knobs, applied on top of whatever the basis did. Where the ellipses are large the compression shrinks them and where they are small it shrinks them less, and it does so anisotropically, in a way set by which channel dominates locally.

Halfway between the two is the third setting worth drawing, because a dial with two settings drawn is a comparison and a dial with three is a sweep.

That is the whole mechanism, and it explains the size of the gain as well as its direction. A nonlinearity is a much larger family than a 3×3 and buys 20 per cent over the best projective picture; that it buys only 20 per cent is a statement about how much of the ellipses’ variation is not explainable by any smooth per-channel rescaling.

What each family’s best basis does in the others #

None of the three families contains the others is a structural claim, and cross-evaluation makes it a table.

Two things in that are worth separating.

The two space families overlap heavily, and asymmetrically. The cube-root optimum scored with no compression at all gives 2.3527 against the linear floor’s 2.3342 — eight tenths of a per cent off. The linear optimum scored with a cube root gives 2.0221 against 1.6112 — 25.5 per cent off. So one basis is very nearly optimal for both and the other is not, and the better all-rounder is the one fitted with the nonlinearity already in place.

And the projective family shares nothing with either. Its own floor is 2.0214, which is respectable. Its basis scored in the two space families gives 10.79 and 263.36, against floors of 2.334 and 1.611 — four and a half times and a hundred and sixty times worse. A projective map divides by a sum of three channels, so a basis fitted to survive that division is fitted for an operation neither of the other families performs, and outside it that basis is not merely suboptimal but useless.

That is the sharpest available form of the claim. The three floors are close enough to invite direct comparison — 2.021, 2.359 and 1.621 — and the bases achieving them are not three points near one another. One of the three is off by two orders of magnitude in the family next door.

What survives of the earlier claim #

Most of it, with the clause attached.

The number 2.02 is still invariant under the nine-parameter group acting on chromaticity diagrams. Nothing about a diagram can beat it, and the 1976 revision still took 62 per cent of the available improvement in shape and 95 per cent in size. Both of those sentences are unchanged.

What was overreached is the word only. The residual was described as a property of the eye rather than of anybody’s coordinates, and the correct statement is that it is a property of the eye and of the decision to draw a chromaticity diagram. Allow a different kind of picture and the floor moves.

This is the same correction shape the essay on two thirds cannot be shown applied, and it is worth naming because it has now happened twice here in quick succession: a quantity is minimised over the largest group anybody thought to vary, the minimum is invariant under that group, and it is then described as a property of the object rather than of the group. The correct form always carries the group with it.

Which basis reaches it #

The winning matrix is worth looking at, because the obvious guess about it is wrong.

The guess would be that the best space for measuring differences is built on the receptors, since that is where the thresholds are set. It is nearly right and not right. The winner’s implied confusion points are protan 0.089 and tritan 0.061 from the measured ones and deutan 1.156 — which makes it the most cone-like basis in the whole table apart from the construction that is the receptors by definition, and closer than Hunt–Pointer–Estévez at 1.275, CAT16 at 1.452 and every fitted adaptation transform.

That is a genuine and slightly surprising result, and it points the opposite way from the adaptation half of the same argument. The best axes for a von Kries gain are further from the receptors than any published transform; the best axes for a difference formula are nearer to them than any published transform. Two objectives, and the receptors sit between them.

The receptor basis itself reaches 2.592 on this measure, against the floor’s 1.621 — so being cone-like is necessary and nowhere near sufficient, and the remaining 60 per cent is bought by departures that are not in the direction of any published matrix.

Why the three numbers cannot simply be ranked #

It is tempting to read the ladder as a single sequence of improvements, and it is not one, because two of the three families do not contain each other.

A chromaticity divides by the sum of all three channels. A lightness–chroma space divides each channel by its own white. Neither operation is a special case of the other: a chromaticity throws away the overall level and keeps two numbers, while a space keeps three and re-weights them. So the best diagram and the best linear space are answers to two different questions, and the fact that 2.021 is smaller than 2.359 says that the diagrams’ extra structural freedom is worth more here than the spaces’ extra dimension — not that diagrams are better than spaces.

What can be compared directly are the second and third rows of the space family, which differ in exactly one thing: whether a power is applied. That comparison is clean, and it is 2.359 against 1.621.

An exponent of four is past the range anybody proposes and is the honest end of the sweep, since the floor is a claim about a whole class of maps.

What is left at the floor #

A mean axis ratio of 1.61 is not a circle and the gap is not small. Something in the data resists every smooth per-channel rescaling of every basis, and it is worth asking what.

Part of it is measurement. MacAdam’s ellipses come from one observer making repeated matches, and the fitted quadratic forms carry the noise of that procedure; a set of twenty-five shapes with independent errors cannot all be made round by a common map, and some residual anisotropy is the errors rather than the eye.

Part of it is structural, and it is the part with a name. The ellipses’ orientations vary systematically across the diagram — they point roughly towards a common region — and a per-channel map applied after a fixed basis has only three independent local scalings at its disposal. Making the orientation field flat everywhere needs a map whose local rotation varies, and neither a matrix nor a per-channel power supplies one.

That suggests where a lower floor would come from, and it is not a better exponent or a better basis: it is a map with cross-channel terms after the nonlinearity, which is what the parametric factors in a modern difference formula amount to. CIEDE2000’s weighting functions are exactly that, applied as a correction rather than as a coordinate change, which is why the formula is not a metric.

What was computed, and how #

The ellipses are MacAdam’s measurements, quoted rather than derived, as everything in this collection that is a measurement of a person is. Each is stored as a centre and a quadratic form, and its boundary is sampled at forty-eight points at its measured size.

For a candidate map, every boundary point and the centre are carried through, radii are measured in the target space, and the ratio of longest to shortest is taken. The mean over twenty-five is the objective. Transporting the boundary points rather than transforming the quadratic form is what makes the arithmetic exact for projective maps, which are only locally affine.

The searches are Nelder–Mead over the nine coefficients from the identity, with restarts. All three objectives are invariant to scaling a row — a chromaticity divides by a sum and a space divides by a channel white, and in both cases the scale cancels — so all three have three flat directions and all three need the restarts for the same reason.

The floor at 1.621 is reached by nine of twelve searches started from independent random points, and those nine agree on the winning matrix to four decimal places in every entry. It is a sharp optimum rather than a valley, which is worth knowing before quoting it: the number is not the best of a broad plateau where many very different bases would do.

The ladder of five answers can be read at a second exponent, which says whether the ordering between the classes of map depends on it.

What it would take to notice this without the search #

The result is not deep and it went unnoticed for fifty years, so it is worth asking what would have surfaced it.

The literature on uniform colour spaces is enormous and almost all of it varies one thing at a time. A new space is proposed by changing the nonlinearity, or the difference formula, or the weighting; it is scored against a threshold data set; and the winner is adopted. The basis is inherited, because a space is always defined as this arithmetic applied to XYZ or this arithmetic applied to CAT16, and the basis arrives with the sentence rather than as a parameter in it.

What the search does is treat the basis as a parameter, which requires only noticing that it is one. That noticing is recent on this site and is not original anywhere else either — CIECAM’s space is built on cone-like axes precisely because someone thought the basis mattered. What has not been done, as far as this collection can find, is minimising over it, and the reason is probably that the answer has no interpretation: the winning matrix is not anybody’s cone fundamentals and not anybody’s primaries, and a space defined by nine fitted numbers with no story attached is a hard thing to propose to a standards committee.

It is an easy thing to compute, though, and knowing the floor changes how a proposed space should be read. A space that reaches 2.5 is not doing well because it beats CIELAB’s 3.44; it is 55 per cent above what is available.

Where the model stops #

One luminance. All twenty-five ellipses are evaluated at a fixed Y, so what is being measured is a shape in a plane through the space rather than the shape of an ellipsoid. Whether the third dimension agrees needs threshold data in that dimension.

Twenty-five ellipses from one observer. MacAdam measured one person, and the between-observer variation on threshold data is large. The floor is the floor for this data set, and a second observer’s ellipses would give a different one.

And a mean of ratios is a summary. At the floor the roundest ellipse is nearly circular and the worst is still more than twice as long as it is wide, so 1.61 describes a set whose members disagree. A space that made the worst one round would be a different optimisation with a different answer.

The generalisation #

A floor is a floor for a family, and the family is part of the result. Minimising over the largest group anybody has varied gives a number that is invariant under that group, and invariance under a group is not the same thing as being a property of the underlying object.

The test is to enlarge the family and see whether the number moves. It cost one afternoon here and moved the answer by 20 per cent, and the enlargement — allowing a per-channel power — is not exotic; it is what the standard space everybody uses already does.

The habit generalises: whenever a result is stated as this is as good as it can get, the sentence has a silent within in it, and the silent part is where the next result is.

Where the ladder goes next #

The compression is doing the work, so the obvious next question is how much of it the exponent is doing. Almost none, which is not what a century of argument about cube roots and logarithms would suggest.