The hue disagreement is the quantiser's

The units part by hue, not by light level read two colour-difference units on the same pairs as a display brightened from a 1.5 candela white to a 10,000 candela one. Each base colour had two pairs one CIEDE2000 unit apart, one stepping in lightness alone and one in chroma alone, and the question was how the ratio of the two readings moved with the display. CAM16-UCS moved every base colour’s ratio by the same factor, towards chroma. ΔEITP moved violets, reds and cyan-blues towards lightness and yellow-greens towards chroma. A lit room brings the units’ medians together then gave the appearance model a room’s worth of adaptation and brought the two units’ medians together. The hue disagreement survived intact, and it belonged to ΔEITP.

That essay proposed where in ΔEITP it came from. The unit starts from three cone-like signals and puts each through the perceptual quantiser, a curve built to space luminances by their visibility, before combining them into an intensity and two chroma axes. A saturated colour’s three signals sit at very different levels — a violet’s S signal well above its L and M, a yellow-green’s well below — and the quantiser’s slope changes with level. So a brighter display stretches a saturated colour’s three signals unequally, and how unequally depends on which signal is out of line. “If that is right, the hue dependence should grow with a base colour’s chroma and vanish at a neutral, and it should follow how unequal the colour’s three signals are rather than its hue angle as such.”

The test has a cleaner form than a correlation, and it comes first.

A power law moves nothing; the quantiser moves by signal #

Replace the perceptual quantiser with a pure power law and change nothing else, and every base colour’s lightness-to-chroma balance is the same on the dimmest display as on the brightest, to a part in a billion. With the quantiser, the size of each colour’s move ranks with the spread of its three quantised signals at 0.85 — well above its rank with CIELAB chroma, 0.60 — and falls to under nine per cent at a chroma of five. The direction does not follow the spread, which ranks with it at 0.17. It follows where the S signal sits against L and M, at 0.79, agreeing in sign on 138 of 174 base colours.

• The disagreement is entirely the quantiser’s shape. A transfer function that scales every signal the same way at every level cannot produce it, and the one ΔEITP uses does.
• How far a colour moves is how unequal its signals are, measured in the quantiser’s own coordinates.
• Which way is where its S signal is. Above the other two, the lightness pair gains on a brighter display; below, the chroma pair does.
• Two numbers read off the signals — the S signal’s departure and L minus M — explain 71 per cent of the variance in the move over all the base colours.

The counterfactual #

The upper curve is the disagreement as the earlier essays found it: ×0.75 at the yellow-greens near 90 degrees, ×1.23 at the violets near 300. The flat line at one is the same computation with one change. ICtCp’s matrices, its three signals, its intensity and chroma axes and its factor of 720 are all kept; the perceptual quantiser is replaced by a pure power law, the signal divided by 10,000 candelas raised to a quarter.

A power law cannot produce the effect, and it is worth seeing why rather than taking the flat line on trust. Brightening a display multiplies every one of a colour’s three signals by the same factor k. Under a power law with exponent γ, each quantised signal is multiplied by $k^{\gamma}$, so every difference between two colours’ signals is multiplied by $k^{\gamma}$ too — the lightness pair’s and the chroma pair’s alike. Their ratio cannot change, whatever the colour’s signals are. Any change in the ratio has to come from a transfer function whose effect on a signal depends on the signal’s level, and the perceptual quantiser is exactly that.

So the question the lit-room essay asked — whether the hue disagreement is a claim of the quantiser’s — has a yes or no answer, and the answer is yes. What remains is how the quantiser turns an unequal colour into a hue-dependent unit.

How far a colour moves #

The shape across hue is the same at every chroma, and its size grows with chroma. At a chroma of five every hue moves by five per cent or less. At twenty the yellow-greens fall by about an eighth and the violets rise by about an eighth. At sixty the yellow-green at 90 degrees falls to ×0.63 and the violets reach ×1.27. That is the first half of the prediction: the effect grows with chroma and disappears at a neutral, where all three signals are nearly equal.

But chroma in CIELAB is not what the quantiser sees. A CIELAB chroma of forty means very different things for the three cone-like signals at different hues: at a yellow-green it drags the S signal far below the others, and at a cyan it barely separates them at all, because CIELAB’s hue circle is not built around the signals ICtCp uses.

Measured in the quantiser’s own coordinates, the spread orders the move’s size at 0.85, against chroma’s 0.60. The spread here is the largest of a colour’s three quantised signals minus the smallest, on a 100 candela display. The fifteen colours with spreads above 0.10 move by between a tenth and three fifths; of the hundred under 0.03, most move by a few per cent, and the exceptions are the dark reds, which the next figure is about. Filled and open points both reach the top of the figure: the most unequal colours move furthest, and they move in both directions. That is why the spread is the right predictor of size and the wrong one of sign.

Which way it moves #

Colours to the right of the vertical line have an S signal above their L and M, and almost all move towards lightness. Colours to the left have it below, and almost all move towards chroma. The rank correlation is 0.79, and the signs agree on 138 of 174. The 36 exceptions are of two kinds and in equal numbers. Half sit close to the line at chromas of five and ten, where the move is a per cent or two either way. The other half are every red at 0 and 30 degrees with a chroma of twenty or more: their S signal sits at or well below the mean of their L and M — as far below as 0.08 at the darkest orange-red — and yet they move towards lightness, by up to half. The reds are the colours whose L and M are furthest apart, and the S signal is not the only one that can be out of line.

Why S decides the direction is visible in how ICtCp builds its axes. The intensity axis is the mean of the quantised L and M; S does not enter it. The two chroma axes both use S — one heavily, the other slightly. A lightness pair, which steps all three signals by the same proportion, is read mostly through the intensity axis, and so through L and M. A chroma pair is read through the chroma axes and so partly through S. Whichever of those signals gains more quantiser slope as the display brightens, the pair read through it grows faster.

The slope that does it #

The quantiser’s slope in the logarithm of luminance rises steeply through the range an ordinary display uses and levels off above a hundred candelas. A factor of e in light is worth 0.026 quantiser units at a tenth of a candela, 0.051 at one, 0.100 at a hundred and 0.109 at a thousand. That rising stretch is the quantiser’s whole departure from a power law — a power law’s slope in the logarithm is constant — and it is where a dim display’s signals sit.

On a 1.5 candela display the yellow-green at lightness 50 and chroma 40 has its L and M signals at about 0.27 candelas and its S signal at 0.09. The S signal sits lower on the rising slope: 0.026 against 0.036. On a 10,000 candela display all three are on the flat top, 0.108 and 0.109. So in going from the dim display to the bright one, the S signal’s slope gains a factor of 4.2 and L and M gain 3.0: the chroma pair, which leans on S, grows 40 per cent more than the lightness pair, and the ratio falls. The violet is the mirror image: its S signal at 0.59 candelas starts higher on the slope, at 0.045, gains a factor of 2.4 against its L and M’s 3.0, and its ratio rises.

That is the mechanism the lit-room essay proposed, measured. A signal low on the quantiser’s rising slope gains more from a brighter display than one higher up. A colour whose three signals are at different heights has its differences stretched unequally, and which of ΔEITP’s axes gains depends on which signal is lowest.

What two numbers carry #

A straight-line fit on the S signal’s departure and on L minus M, both read off the quantised signals, explains 71 per cent of the variance of the log move; the S departure alone, 61. The second number carries the reds — colours whose L and M are well apart while S sits near their mean, which the S departure alone places near zero and which in fact move towards lightness by a fifth. The rest is the curvature of the quantiser between the dim display and the bright one: a colour’s move depends on the slope at each of its signals on both displays, and two differences on one display can only approximate that.

The fit is a summary, not a model. Its value is that it says what a colour’s move is about in the unit’s own terms: how far each of its three signals is from the others, measured after the quantiser.

The same slope, three times over #

The rising stretch of the quantiser has now turned up behind three separate findings about ΔEITP, and they are worth reading together, because each was reported as a property of something else.

How bright is white introduced the quantiser as the encoding that names candelas, built so that equal steps in its signal are equally visible at every luminance. That design is what gives it a slope that rises through the dark and levels off above a hundred candelas: the eye’s threshold for a luminance change is a larger fraction of the luminance in the dark. The shadows the unit counts are the ones a room removes found ΔEITP counting dark grey steps most on a bright display, and a veil of room light taking exactly those away; both are the steep low end of the curve, where a small offset changes a lot. A guessed veil halves the error then found that how well the veil must be known depends on where the darkest greys sit on that same stretch.

Here the same slope turns a colour’s hue into a light-level dependence. A grey has three equal signals and they climb the curve together, so a grey’s lightness and chroma readings keep their ratio. A saturated colour’s signals are at three heights, and they climb at three rates. A tolerance has no light level argued that a tolerance written in a relative unit cannot say how a difference changes with the display; ΔEITP can say it, and what it says is decided, colour by colour, by where the colour’s signals sit on this one curve.

So the three findings are one. ΔEITP’s behaviour in the dark, through a veil and across hue is the shape of the quantiser below a hundred candelas, applied to whichever signals a stimulus puts there. Anything that tests that shape for achromatic stimuli tests the first two; the violet experiment is the only one proposed that tests it for chromatic ones.

What this changes about the violet experiment #

The units part by hue, not by light level proposed a forced-choice experiment: the same lightness and chroma pairs, at a violet base, shown on a dim display and a bright one, with observers asked which pair of each looks larger. The two units predict opposite answers there, and the violet is where they disagree most.

That experiment is now a test of the perceptual quantiser, specifically of its slope below a candela. The violet’s prediction comes entirely from its S signal sitting higher on the rising part of the curve than its L and M. If observers see what ΔEITP predicts, the quantiser’s slope in that range is right for chromatic signals as well as for luminance, which it was never fitted to. If they see what CAM16-UCS predicts, the quantiser’s shape — built from contrast thresholds for achromatic gratings — is the wrong shape to apply to each cone-like signal separately. Two units with a light level disagree about lightness found the units disagreeing about where a display’s light acts; this says the disagreement has a single cause that one experiment can reach.

And it gives the experiment a better stand. The largest predicted moves are not at the base colours the earlier census tried but at the most unequal signals the census reached: a saturated yellow-green at lightness 50, S far below L and M, moving by ×0.63, and a dark saturated blue at lightness 25, S far above, moving by ×1.62. Between those two, one change of display separates ΔEITP’s predictions by a factor of 2.6, where CAM16-UCS predicts the same move for both. (That blue, and the next largest mover, a violet-blue at ×1.57, were later found to be colours no BT.2020 display can show: their cone-like signals are all positive, which is the test this census applied, but each needs a negative red drive. At the gamut’s edge the reds move as far as the violets walks the census out to the colours a display can show.)

How the moves were computed #

The base colours are CIELAB triples against D65 at lightness 25, 50 and 75, hue every 30 degrees and chroma 5, 10, 20, 40 and 60. Around each, one pair steps in lightness alone and one in chroma alone, each bisected to exactly one CIEDE2000 unit. Each pair is scaled to a display whose white is 1.5 and 10,000 candelas — five times adapting luminances of 0.3 and 2,000 — and read in ΔEITP through BT.2020 RGB, ICtCp’s fixed cone-like matrix and the SMPTE ST 2084 quantiser. The move is the lightness pair’s reading over the chroma pair’s on the bright display, divided by the same on the dim one. Six base colours are set aside: three dark saturated yellow-greens whose pairs leave BT.2020, and three light saturated blues and violets whose S signal passes 10,000 candelas on the brightest display. The test for leaving BT.2020 is a negative cone-like signal, which is weaker than a display’s gamut: eight of the 174 kept need a negative drive or one beyond the white.

The spread and the S departure are read from the base colour’s own three signals on a 100 candela display. The power-law counterfactual uses (Y/10,000)^0.25 for each signal and changes nothing else.

What this leaves out #

The appearance model is not tested here. CAM16-UCS’s uniform move with light level was found in the essays this continues and is taken as given. This essay says why ΔEITP departs from it, not whether CAM16-UCS is right.

The pairs are one CIEDE2000 unit. A pair is small enough that the quantiser’s slope at the base colour decides its reading. Pairs of several units would straddle a range of slope, and their moves would be averages of what is found here.

And the quantiser is used as specified. ICtCp applies the quantiser to each cone-like signal separately, the way it was designed; a unit that applied a single luminance transfer and carried chroma in ratios would have no hue-dependent light-level response at all. Whether ΔEITP’s choice is the right one is exactly what the violet experiment would decide.

Still open: whether the two ends of the gamut are the place to stand #

The census puts the largest predicted moves at its most unequal colours — a saturated yellow-green moving by ×0.63 and a dark saturated blue by ×1.62 — and both are near an edge. Three dark yellow-greens left BT.2020 and three light blues passed the quantiser’s peak on the brightest display, and were set aside, so the largest moves a real display can show sit at the boundary of what it can show, and the census stopped short of them.

The calculation is this census walked out to BT.2020’s boundary along each hue, at each lightness, with ΔEITP’s move and CAM16-UCS’s move both recorded at the most saturated colour the gamut allows, and the display’s brightest white capped where the brightest signal meets the quantiser’s peak. The prediction is that the two extremes stay on the two sides found here — the yellow-greens below ×0.6, the dark blues above ×1.6 — and that a forced-choice experiment pairing them needs a quarter of the observers the violet stand needs, because the units’ predictions there differ by more than twice as much. If instead the extremes move to the reds at the boundary, where L and M are furthest apart, then the S signal’s position is the smaller half of the mechanism and the L-minus-M term is the larger.

A counterfactual is a cleaner test than a correlation #

The habit is about reaching for the experiment that can say no.

The proposal predicted two correlations, and both could have been checked on their own: move size against signal spread, and move against chroma. Each would have come back positive, and neither would have established that the quantiser was responsible — a correlation with the spread is what several mechanisms would produce. Removing the quantiser and nothing else was the version that could fail. If the moves had survived a power law, the whole account would have been wrong. They vanished to a part in a billion, and after that the correlations were no longer evidence of the cause, only descriptions of how it acts.

The failure mode is to test a mechanism by the pattern it predicts rather than by removing it. A pattern is shared by rival explanations; an intervention that takes the mechanism out is not.