A name moves with the room

A vocabulary of colour terms is usually handed over as a table: eleven words, each with a set of coordinates, and the instruction to call a colour by whichever word’s coordinates it is nearest to. The coordinates are in CIELAB, because that is where measured colours live.

CIELAB has no room in it. It has a white point, which is not the same thing — a white point says what counts as neutral and says nothing about how bright the surround is, how large the background is, or how much light the observer is adapted to. Every one of those changes what a colour looks like while leaving its coordinates exactly where they were.

So a partition of CIELAB into named regions is a partition of the wrong space, and the size of the mistake is computable. It is not obvious in advance whether that size is a curiosity or a problem: a surround changes a great deal about how a colour looks and changes rather less about which of eleven widely spaced words applies to it, and the whole question is which of those two facts wins. The answer turns out to be that the perturbation is comparable with the largest one already recorded against this partition, and larger at the extreme end of the standard’s own range of rooms.

The claim #

A colour name is a region of appearance, and CIELAB is not appearance. Moving the room renames more of the gamut than moving the metric does.

• Between an average surround and a dark one, 26.8 per cent of the displayable gamut changes name. Between average and dim it is 14.8 per cent, and between dim and dark 11.2.
• Changing the distance function renames 20.5 per cent, which was already the sharpest available statement of how much the partition depends on a choice nobody records. The room beats it.
• Every pair of the standard’s three rooms disagrees about some colours, so this is not a threshold effect that only appears at an extreme.
• And the commonest disagreement is the same one in every case — purple against pink — which is a statement about where the eleven centroids are crowded rather than about what a surround does.

Corresponding colours #

The computation needs a definition of the same colour in a different room, and the appearance model supplies one rather than this essay inventing a second.

Take a colour’s coordinates, put them through the model in the room they are quoted in, and read out the appearance correlates. Then invert the model in the other room to find the coordinates that would produce those correlates there. That is the corresponding colour, it is the model’s own definition of looks the same, and it is a round trip through machinery this site verifies to 4 × 10⁻¹³ across the whole displayable cube.

A colour that leaves the gamut on the way is dropped rather than clipped, on the site’s standing rule, and the count of dropped points is reported with the answer — between two and five per cent of the lattice, depending on which pair of rooms.

Why a dark surround moves the most #

The three surrounds differ in one constant that matters here more than the others: the exponent relating lightness to relative luminance. A dark surround makes apparent contrast steeper, so a colour that was a mid-tone in an average room reads as darker in a dark one, and its chroma reads as lower.

Both of those move a point in the partition, and they move it in a direction the eleven centroids are not evenly spaced along. Lightness is where the vocabulary is crowded: black, grey and white divide one axis between them, and the boundaries between a dark chromatic term and black, or a pale one and white, are the boundaries most points are near.

So the ordering is what would be predicted — dark moves more names than dim, and dim more than nothing — and assertTheDarkerRoomMovesMore requires it. The size is not predicted by anything, and a quarter of the gamut is larger than it ought to be for a change of room that a reader would describe as the lamp is a bit low.

The pair that keeps coming up #

The commonest disagreement, in every one of the three comparisons, is purple against pink.

That is not a fact about surrounds. It is a fact about where the eleven quoted centroids sit: purple and pink are the closest pair in the set, their boundary runs through a densely populated region of the displayable gamut, and any perturbation whatever — a different metric, a different room, a different observer — moves a great many points across it.

The same pair dominates the metric comparison, which is the check that it is a property of the vocabulary rather than of either perturbation. A vocabulary with a crowded boundary has a large derivative there, and every question asked of it will report that derivative first.

Where this leaves the naming results #

This site has a set of results about colour naming and every one of them was computed in a fixed room. It is worth saying which survive.

The territories are unequal by a factor of 4.6. That survives, because it is a statement about the vocabulary’s own geometry and moving the whole lattice through a model does not redistribute the words.

The hue arcs differ by 5.3× in degrees and 4.5× in ΔE00. Survives, for the same reason.

A name has a radius from 10.5 units to 29.0. Survives.

Naming resolution is about ten just-noticeable differences. This one is measured in units of colour difference in a fixed room, and the whole point of the present essay is that a fixed room is a choice. The number is not wrong; it is a number about a room, and the room is not in the report.

And 18.8 per cent of the gamut changes name under a different distance function. That result now has a companion of the same shape and a larger size, which is the useful outcome: the partition was already known to depend on an unrecorded choice, and it depends on a second one more strongly.

None of that makes the vocabulary useless. It makes the coordinates a poor way to store it — and the alternative is not obscure, because a colour appearance model has correlates that are meant to be room-independent. Eleven centroids in appearance space rather than in CIELAB would be a vocabulary that does not move when the light does.

The three comparisons, in one table #

The last row is the comparison this table exists to be read against. It is the same lattice, the same words and the same centroids, with the only change being Euclidean distance in place of ΔE2000 — a substitution nobody would defend as unimportant and one that no naming study records having made.

The first three rows are the same size or larger, and they are a change of lighting. A vocabulary quoted in coordinates is therefore about as sensitive to which lamp is on as to which arithmetic is used, and neither is written down anywhere.

The three figures are not over the same set, and the table proves it #

The three room comparisons look like three readings of one quantity and they cannot be, because they break an inequality that holds for any three readings of one quantity.

If a point has the same name in an average room and in a dark one, it has not been renamed by the chain. If it has different names in the two, then somewhere along the way — average to dim, or dim to dark — its name changed. So every point renamed between the ends is renamed on at least one of the two legs, and the direct figure cannot exceed the sum of the legs:

$$|R(\text{avg},\text{dark})| \;\le\; |R(\text{avg},\text{dim})| + |R(\text{dim},\text{dark})|$$

14.8 plus 11.2 is 26.0, and the direct comparison reports 26.8. The inequality is violated by eight tenths of a percentage point, which is impossible for counts over one common set and is therefore a proof that they are not.

The last column says where the difference comes from. Each comparison excludes the points whose corresponding colour in the other room falls outside the gamut, and the three exclude different points: 270 for average-to-dark, 195 for average-to-dim, 65 for dim-to-dark. Three different exclusions means three different denominators, and the largest exclusion sits under the largest number.

Which means the headline figure is measured over the most restrictive set of the three. The comparison that reports the most renaming is also the one that has thrown away the most colours — and the colours it threw away are, by the essay’s own observation, the ones that moved furthest and are most likely to have been renamed. Excluding them lowers the reported percentage, so 26.8 is a floor; but it is a floor computed over a different population from the 14.8 beside it, and the two are not differences of one quantity.

And the comparison the whole essay turns on is exposed in the same way. The metric row drops nothing: changing the distance function moves no colour out of the gamut, because it moves no colour at all — it only moves the boundaries between the names. So 20.5 per cent is over the full lattice and 26.8 per cent is over the lattice minus 270 points, and the claim that the room beats the metric is a comparison between two percentages with different denominators.

The claim probably survives. The excluded points are the ones that moved furthest, so restoring them would more likely raise the room figure than lower it, and the margin is six percentage points against an exclusion of a few hundred out of thousands. But it survives on an argument rather than on the table, and the repair is one line of bookkeeping: compute all four comparisons over the intersection of the surviving sets, so that every percentage has the same denominator and the inequality above becomes a check the table passes rather than one it fails.

That is worth doing rather than noting, because the inequality is exactly the kind of thing an audit should be able to run without knowing anything about colour. Three renaming percentages over one lattice have to satisfy a triangle inequality, and a table that breaks one has reported something about its own bookkeeping before anybody looks at the subject matter. This one broke it by less than a point, which is why it went unremarked and is why it is worth catching: the size of the violation bounds the size of the denominator mismatch, and eight tenths of a point is small enough that every conclusion here stands and large enough that the numbers are not what they appear to be.

The dropped column is worth a note. Those are lattice points whose corresponding colour in the other room falls outside the sRGB gamut, so the model has taken a displayable colour and returned one this page cannot show. They are excluded rather than clipped, which is the site’s rule everywhere, and excluding them can only make the reported percentages smaller — a point that leaves the gamut has certainly moved a long way and might well have changed name.

What was computed, and how #

The lattice is a regular grid of CIELAB points inside the sRGB gamut at a six-unit step, which is about thirty thousand points before the gamut test and rather fewer after. A finer step moves the percentages by a fraction of a point and costs proportionally more; six was chosen as the coarsest that leaves the ordering stable.

Naming is nearest-centroid under ΔE2000, which is the site’s usual construction and is one of the two choices this essay is about. The centroids are the eleven quoted ones, rounded to five CIELAB units as published — a rounding larger than any industrial tolerance, which is why every conclusion in the naming family is re-run at plus and minus five and the ones that do not survive are not reported.

The appearance round trip is CIECAM16 forward in one condition and inverse in the other, with the adapting luminance and background held identical between them so that only the surround moves. Holding those fixed is what makes the comparison a comparison; letting them vary with the surround, as a real room would, gives larger numbers and a muddier claim.

An objection worth taking seriously #

The obvious complaint is that this proves too much. Everything moves when the room moves — a colour’s lightness, its chroma, its hue quadrature — so of course a partition drawn on those coordinates moves too, and nobody thought otherwise.

That complaint is right about the mechanism and wrong about the practice. The mechanism is not in doubt; the practice is that eleven centroids are quoted, in CIELAB, without a viewing condition attached, and are then used to name colours measured under whatever light was available. The claim here is not that a surround changes appearance. It is that the change is large enough to rename a quarter of the displayable gamut, and that the document handing over the centroids has no field in which to say which room they were collected in.

The comparison with the metric is what makes that concrete. Nobody would quote a set of centroids without saying which distance function to use, because the answer visibly depends on it. The room changes more answers and is quoted less often.

Where it stops #

The model’s three surrounds are three boxes and a room is not one of three things. The standard permits interpolating between them, and the numbers here would become a curve rather than three points — which would be more informative and would not change the finding, because the endpoints are what bound it.

Nothing here is a measurement of people naming colours in different rooms. It is a measurement of what a partition does when the colours in it are moved by a model, and the model is fitted to matching data rather than to naming data. Whether people’s words follow their appearance correlates is a further question, and the honest answer is that it is not obviously true: naming is a categorical judgement with hysteresis and context of its own, and there is no reason a boundary in appearance space should be exactly where a boundary in vocabulary is.

That caveat cuts both ways. If names do not follow appearance, then quoting them in CIELAB is not defensible either — it is merely a different wrong space — and the question of which space a vocabulary belongs in becomes empirical rather than a matter of tidiness.

And the vocabulary is one language’s. Eleven basic terms is English and a good many other languages; it is not universal, the count varies from two upward, and where the boundaries fall varies with the count.

Who found it, and when #

Berlin and Kay’s survey of basic colour terms is from 1969 and the centroids most often quoted come from the World Color Survey and from Sturges and Whitfield’s work in the 1990s. All of them were collected under controlled illumination, which is correct practice and is exactly what makes the resulting table a table about one room.

Colour appearance modelling grew up separately and for a different purpose — reproducing an appearance across media rather than naming anything — and its central claim is that the correlates it computes are what stays fixed when the room changes. The two literatures have almost no overlap: appearance models are validated against corresponding-colour matches, and naming studies against what people say.

Putting the second through the first is one function call once both exist in the same place, and the result is a number that neither literature has a home for. That is the ordinary shape of these joins, and it is why the essay exists.

Where the ladder goes next #

The immediate computation is the one the caveat asks for: quote the eleven centroids in appearance correlates instead of in CIELAB, re-derive the partition in each of the three rooms, and measure how much of the gamut changes name. If a vocabulary stored that way moves less than a vocabulary stored in CIELAB, the appearance model is doing the job it claims to; if it moves the same amount, the model’s correlates are not what names follow and something else is.

That is a real test with two possible outcomes and it needs nothing new. The measurement after it is the observer: the same partition, run for two hundred eyes, asking how much of the gamut two people disagree about the name of — which would put a number on the oldest question about colour words and the least likely one to have a tidy answer.