The blue resists through the model's denominator

White drains the blue last in every model gave the twenty-four most saturated colours of an sRGB display the same luminance and the same proportional dose of white, and asked each of four measures how much colour each had lost. They agreed on the order almost perfectly — orange fastest, blue slowest, at rank correlations of 0.92 to 0.99. They disagreed on the spread. At a quarter-dose, CIECAM16 had the orange losing 27 per cent of its colourfulness and the blue 1.7; CIELAB had 28 and 6.3, Oklab 21 and 4.6. The blue’s loss over the orange’s is 0.06 in CIECAM16 and 0.22 in CIELAB and Oklab, and that ratio, rather than the order, is what an experiment on people would have to measure.

That essay stopped at the edge of an explanation. The model’s colourfulness passes through a cone-like transform, a compressive response, a normalisation and a hue-dependent factor, “and any of them could make a blue resist dilution.” Its closing section narrowed it to two. CIECAM16’s colourfulness carries an eccentricity factor, a smooth function of hue angle fitted so that equally colourful stimuli at different hues come out equal, and it is the most obvious hue-specific term in the model. But it multiplies the pure patch and the diluted one alike, so it should cancel in their ratio. And the model divides its opponent signals by the sum of its three adapted cone signals, and a saturated blue is the stimulus whose cone signals are most unequal. The prediction was that the resistance would survive the first being held constant and vanish when the second was replaced by luminance.

One denominator #

With the eccentricity factor set to one at every hue, the blue loses 0.058 of what the orange loses, against 0.063 in the published model — the resistance is untouched. With the sum of cone signals replaced by the model’s own achromatic signal, the blue loses 0.233 of the orange’s loss, beside CIELAB’s 0.221 and Oklab’s 0.223, and the fastest hue’s loss over the slowest falls from sixteen to 4.3. The result holds at every luminance and dose tried.

• The hue factor is not it. It nearly cancels, as predicted, because added white barely turns the hue.
• The denominator is all of it. Replace it and CIECAM16 agrees with CIELAB and Oklab about the blue within a few hundredths.
• The reason is that a saturated blue’s denominator is its own blue signal. At equal luminance the blue’s sum of adapted cone signals is twice the orange’s, and more than half of it is the blue signal; a quarter-dose of white raises it by four per cent against the orange’s seventeen.
• Part of the blue’s resistance is shared by every model. Its opponent signals shrink by only four per cent where the orange’s shrink by twenty-three; the denominator is the part only CIECAM16 adds.

Taking the model apart #

The test needs CIECAM16’s colourfulness with pieces removed, and the pieces are few. A stimulus goes through CAT16 to three cone-like signals, those are adapted to the viewing condition’s white and compressed, and from the three compressed signals the model forms two opponent signals, a and b, and an achromatic one, A. Lightness J comes from A. Colourfulness comes from a quantity t, the eccentricity factor times the magnitude of the opponent pair, divided by Ra + Ga + 21Ba/20 — the sum of the three compressed cone signals, the blue one weighted a little more — and then through a fixed power and J into chroma and colourfulness.

The two suspects are therefore two lines. The eccentricity factor is replaced by one. The denominator is replaced by 2Ra + Ga + Ba/20, the weighting the model uses for its achromatic signal — the luminance-like quantity it computes lightness from, which at equal luminance is nearly the same for every hue. Everything else is the model as published, and the version with nothing removed reproduces the collection’s CIECAM16 to a part in a billion on every patch.

The published curve and the curve without the hue factor lie on top of each other. Both climb to a peak at the oranges, where 27 per cent is lost, and fall to a trough at the blue of 240 degrees, where 1.7 and 1.6 per cent are. The curve with the denominator replaced keeps the shape and loses the depth: its orange loses 23 per cent and its blue 5.4, and between them it runs close to CIELAB’s chroma all the way round the hue circle. The reds and oranges lose a few points less than in the published model and the blues a few more, which is exactly the difference between CIECAM16 and CIELAB that the earlier essay reported.

The share an experiment would measure #

The single number the earlier essay said an experiment should measure moves only when the denominator is replaced. Published, the blue loses 0.063 of the orange’s loss; with the hue factor at one, 0.058; with the denominator replaced, 0.233; with both removed, 0.232. CIELAB’s is 0.221 and Oklab’s 0.223. The model with its denominator replaced sits among the other two measures, not near its own published value.

That settles the earlier essay’s hedged sentence. The disagreement between CIECAM16 and the other measures about blue under white is not spread across the model’s structure. It is one term: the quantity the model normalises its opponent signals by.

Why the denominator spares a blue #

At equal luminance the achromatic signal is nearly flat across hue, as a luminance-like quantity has to be. The model’s cone-signal sum is not. It sits around seventeen for the reds, oranges and yellows, climbs through the cyans, and reaches thirty-three at the blue of 240 degrees — nearly twice the orange’s. The climb is the blue signal: a saturated blue at the same luminance as an orange has an enormous excitation of the short-wavelength cones, because those cones contribute almost nothing to luminance and a blue has to drive them hard to be as bright as anything else. At 240 degrees the adapted blue signal is eighteen of the sum’s thirty-three.

Added white then acts on that sum unequally. A quarter-dose of white adds roughly equal light to all three cone-like signals. For the orange, whose sum is small and made of the red and green signals, that is a large relative addition. For the blue, whose sum is dominated by a blue signal already large and further compressed, it is a small one.

The orange’s sum grows by seventeen per cent and the blue’s by four. Since colourfulness goes as the opponent magnitude over the sum, a growing sum reduces colourfulness, and the blue’s barely grows. The achromatic signal has no such asymmetry: it weights the red signal twice and the blue signal a twentieth, so white added to a blue raises it by as large a share as white added to an orange. Replace the sum with it and the blue loses as much as the other measures say.

The figure also shows the other half of the blue’s resistance, and this half every model shares. White shrinks the blue’s opponent magnitude by four per cent and the orange’s by twenty-three. That is the compressive response acting on a blue signal already high on its curve: adding a little to it changes its compressed value little, so the blue-yellow opponent signal, which is mostly that blue signal, barely moves. CIELAB’s cube root and Oklab’s cube root do the same thing to a blue’s b-axis coordinate, which is why every measure drains blue last. On this evidence the order is the compression, which every measure has, and the spread is the denominator, which only one does — the first half inferred rather than tested, since the compression was never removed.

At every level and dose #

The published share rises with dose, from 0.047 at a tenth to 0.087 at a half; with the denominator replaced it runs from 0.197 to 0.280. The ratio between them is more than three at every dose, and at patch luminances of 10 and 40 the numbers move by under a twentieth. The hue factor, meanwhile, never moves the share by more than an eighth in either direction. Nothing about this result is specific to the quarter-dose the earlier essay used as its reference.

The same term sets how colourful a blue is to begin with #

A denominator that barely grows under added white is also a large denominator, and a large denominator makes a small quotient. The published model therefore does two things to a saturated blue at once: it makes it resistant to dilution and it damps its colourfulness in the first place.

The numbers show both. With the denominator replaced by the achromatic signal, the pure display blue’s colourfulness rises from 108 to 158, a factor of 1.47, while the pure red’s falls from 106 to 84 and the orange’s from 43 to 36. The cyan barely moves. So in the published model a saturated blue at equal luminance is about as colourful as a saturated red; with the denominator replaced it is half again as colourful. The eccentricity factor, which was fitted to make stimuli of equal colourfulness come out equal across hue, was fitted to a model with the published denominator in it, and some of what that factor does at blue may be undoing what the denominator does.

That is the sense in which the two suspects were never independent. The cones an appearance model uses looked at the cone-like signals CIECAM16 starts from; the appearance model takes XYZ at what it assumes about its input. Both treated the model’s first stages as fixed and asked what they implied. This result is a reason to look at the normalisation the same way: a quantity introduced to make colourfulness independent of overall light level, whose particular weighting of the three signals decides how a saturated blue behaves. Brighter looks more colourful is the model’s Hunt effect, and it enters through the luminance-level factor applied after t; the denominator is the part of t that decides what “colourful” means at each hue before any light level is applied.

What this does to the experiment #

The earlier essay’s experiment was saturated patches at several hues, each shown with increasing amounts of white, observers judging colourfulness, and the blue’s loss as a share of the orange’s recorded. CIECAM16 predicts about 0.06 and the other measures about 0.22.

That experiment is now a test of one term in one model. If observers find the blue losing a sixteenth of what the orange loses, CIECAM16’s normalisation by the cone-signal sum is right, and the dominance of the blue signal in that sum is a real property of how people see a saturated blue. If they find a quarter, the sum should be replaced — and the natural replacement, the model’s own achromatic signal, gives the right answer without any other change. It is rare for a disagreement between colour models to reduce to a single substitution, and rarer for the substitution to be one the model already computes.

It also gives the experiment a way to be sharper. The share is most sensitive to the denominator where the blue signal is largest relative to the others, which is at the most saturated blues and at low doses: at a tenth of a dose the published model says 0.047 and the replacement 0.197, a factor of 4.2 between them, against 3.7 at a quarter.

What it means for the rooms #

What an adapted viewer loses is set by the wall found that a viewer in a glossy room loses 1.4 to 2.4 times as much of the room’s colour as the room’s light does, and that the wall’s colour decides the multiplier — a deep red room loses most and a dark blue one least. The finish adds the room’s own colour and a gloss room looks less colourful than it measures built on the same property. Every one of those multipliers passes through this denominator, and every blue room’s small loss is its blue signal dominating it.

The direction of those results is safe: every measure drains blue last, because of the compression they share. Their size is the denominator’s, and it is now a size that one measurement on a blue patch would confirm or remove.

How the model was taken apart #

The patches are the twenty-four most saturated colours of an sRGB display, one every fifteen degrees of HSV hue, each scaled to a luminance of 20 on a white of 100, with D65 white of a stated fraction of that luminance added. CIECAM16 is read under an average surround with an adapting luminance of 100 cd/m² and a background of 20. The colourfulness M is computed as the model specifies: CAT16, the degree of adaptation, the compressive post-adaptation response, the opponent signals a and b, the achromatic signal A, lightness J, t from the eccentricity factor, the opponent magnitude and the cone-signal sum, then chroma and colourfulness. The ablations replace the eccentricity factor by one, the sum Ra + Ga + 21Ba/20 by 2Ra + Ga + Ba/20, or both. A patch’s loss is one minus its diluted colourfulness over its pure colourfulness.

What this leaves out #

The replacement is one choice among several. Luminance alone, the achromatic signal A with its offset, or a sum weighting the three signals equally would all remove the blue’s dominance, and they would not give identical shares. The achromatic weighting was chosen because the model already computes it; it is a demonstration that the sum is responsible, not a proposal for a corrected model.

The patches are display primaries and their mixtures. A saturated blue from a laser or a narrow LED would have an even larger blue signal at equal luminance, and the denominator’s effect would be larger with it.

And the model’s colourfulness is its claim about people. Nothing here says which share is right. It says which part of the model makes CIECAM16’s share what it is.

Still open: whether the Abney turn at blue is the same term #

White turns a hue, and the models part at blue found the models disagreeing about how added white turns a hue, and disagreeing most at blue — the same place, and in the same models, as the disagreement here about how much colour white takes away. CIECAM16’s hue angle comes from the same opponent signals whose magnitude the denominator divides, but the angle does not pass through the denominator at all: it is the ratio of b to a.

The calculation is that hue-turn census with the model’s compressive response replaced by a power law of the same exponent, and separately with its blue signal’s weighting in the opponent signals changed. The prediction is that the Abney disagreement at blue belongs to the compression rather than the denominator — the turn is a change of angle, the denominator cannot change an angle, and the compression acting on a large blue signal is what makes a blue’s b change little as white is added. If so, the two disagreements at blue have different causes that happen to meet at the same stimulus, and a single experiment on blue would not settle both.

Remove one term at a time #

The habit is about locating a disagreement between two models in a single part of one of them.

The disagreement began as a pattern across twenty-four hues in four measures, and it could have been described indefinitely in those terms. It became an explanation by removing parts of the one model that disagreed, one at a time, with everything else held fixed. The part expected to matter did not; the part expected to matter more did, completely, and removing it brought the model into line with the other measures to within a few hundredths.

The failure mode is to attribute a model’s behaviour to its most conspicuous term. The eccentricity factor is visibly about hue and is where anyone would look first; it turned out to cancel. The denominator is a normalisation nobody thinks of as hue-specific, and it was the whole of the difference.