The units part by hue, not by light level

Two units with a light level disagree about lightness read twenty-three surface pairs at one ΔE₀₀ in ΔEITP and in CAM16-UCS as a display’s white rose from 5 to 5,000 cd/m². Both units grew. In ΔEITP the lightness part grew fastest, 2.7 times; in CAM16-UCS the lightness part did not grow at all. It then named the experiment that would settle which is right:

The experiment is a threshold or scaling task on a bright HDR display with pairs that differ only in lightness and pairs that differ only in chroma, each built at one ΔE₀₀, measured at two display levels a factor of a hundred apart. ΔEITP predicts that lightness pairs become relatively more visible at the bright level; CAM16-UCS predicts that chroma pairs do.

The pairs it read were not those pairs. Each of its twenty-three differed in lightness and chroma, and it reported the two parts of each difference separately — which is a decomposition of a mixed pair rather than a reading of a pure one. Built pure, the units say something else.

The first version of this essay built the pure pairs and read them, and reported that the units part in the dim half of the range — CAM16-UCS’s balance rising to a peak at a 160-candela white and falling after it, while ΔEITP’s held still. That peak was not in the model. The appearance model had been given each pair at the display’s absolute luminance against a white fixed at 100, so below a 100-candela white every patch was read as darker than the display’s own white, and above it as brighter than white itself — a hundred times brighter at 10,000. Read as it should be, relative to the display’s white with the light level entering through the adapting luminance alone, the peak disappears and something else appears in its place.

The units agree about light and part about hue #

On pairs that differ in lightness alone and in chroma alone, both units say that chroma differences gain on lightness differences as a display brightens: the median balance falls 11 per cent in ΔEITP and 23 per cent in CAM16-UCS between a 1.5 and a 10,000 candela white. CAM16-UCS moves every base colour by the same factor. ΔEITP does not: at the violets, reds and cyan-blues it moves the balance towards lightness, and at nine of the twelve base colours the two units move in opposite directions.

• Neither prediction quoted above is right about the medians. Between 5 and 5,000 candelas ΔEITP’s median balance goes 0.87 to 0.81 and CAM16-UCS’s 1.09 to 0.88: both say chroma pairs gain.
• CAM16-UCS’s reading of a pure lightness difference does not move at all — 1.66 to 1.68 across four decades — while its chroma reading grows by 30 per cent.
• Every base colour’s balance falls by between ×0.75 and ×0.79 in CAM16-UCS. In ΔEITP the same factor runs from ×0.68 to ×1.30, depending on hue.
• The dark violet is where to stand: a forced choice between the units needs five observers there, at either pair of levels, and the yellow-greens need from dozens to hundreds.
• The prediction’s two levels are not the problem. The dim pair and the bright pair ask for similar sessions; the colour decides the cost.

Pairs built to be pure #

A mixed pair is a step in some direction of colour space, and a unit reports both a lightness part and a chroma part of it. Those parts are a decomposition of one number, and they are not the readings the unit would give to two separate pairs — because each unit’s total is a Euclidean combination, and how a mixed step divides between the two parts depends on the direction it was taken in.

So the pairs here are built in CIELAB and made pure. Twelve base colours — three lightnesses by four hues at a chroma of 40 — and from each, one pair stepping in L* alone and one stepping in C* alone, each bisected to exactly one ΔE₀₀. Twenty-four pairs, every one of them a difference in one thing.

Both curves fall, and neither turns. ΔEITP’s runs 0.91, 0.87, 0.83, 0.81, 0.82, 0.83, 0.81, 0.81, 0.81 — most of its fall in the first decade, then level. CAM16-UCS’s runs from 1.13 to 0.87 in a smooth slide across the whole range, and by 10,000 candelas it has fallen twice as far as ΔEITP’s. Both units, asked about the median pair, say the same thing: a brighter display makes a chroma difference relatively more visible than a lightness difference of the same ΔE₀₀.

The earlier census is worth having beside the new one, because nothing in it is wrong and it does not say what the prediction says. Both units’ totals grow with the display, ΔEITP by 2.1 times and CAM16-UCS by 1.5, and that growth is a real statement about both: a difference on a bright display is a larger difference. What the census could not say is how the growth divides between two kinds of difference, because each of its pairs is one step and the division of one step is a projection.

One reading does not move #

The ratio hides which of the four readings is doing the work.

Three of the four readings climb and flatten, which is what a unit that takes a light level ought to do: a difference becomes more visible as there is more light to see it by, with diminishing returns. ΔEITP’s lightness reading runs 1.54 to 3.81 and its chroma reading 1.98 to 5.44; CAM16-UCS’s chroma reading runs 1.40 to 1.83.

CAM16-UCS’s lightness reading is flat: 1.66 to 1.68 across four decades of display white. That is what the earlier mixed-pair census found for the lightness part, now confirmed on pairs that are nothing but lightness. The mechanism is in the model’s construction. CAM16’s lightness is a ratio of the stimulus’s achromatic response to the white’s, and both are compressed by the same adaptation-dependent function, so the light level cancels out of lightness almost exactly. Its colourfulness is multiplied by the fourth root of a luminance-adaptation factor that grows with the level, so chroma differences grow and lightness differences do not. That is the Hunt effect written into the metric: at a higher level, colours look more colourful and no lighter.

ΔEITP’s level enters through the perceptual quantiser, applied to each of three cone-like signals at its absolute luminance. The quantiser was fitted to contrast thresholds across four decades — how bright is white is the essay about what changed when display encoding started naming candelas — so the number of its steps a fixed relative change spans falls smoothly as the level rises, and both of ΔEITP’s readings grow and flatten with it. Nothing in that construction treats lightness and chroma differently as a class — which is why its median balance barely moves — but the three cone signals of a colour sit at different places on the quantiser, and a colour whose signals are very unequal feels the curve’s shape unequally. That is the likeliest place for a dependence on hue to come from, and it is not separated here.

Where they part #

The medians agree in direction. The base colours do not.

CAM16-UCS moves all twelve base colours by nearly the same factor, between ×0.75 and ×0.79: chroma gains a quarter on lightness whatever the colour. That uniformity is the flat lightness reading again — a model whose lightness differences ignore the level and whose chroma differences all scale by the same adaptation factor must move every balance alike.

ΔEITP moves them by hue. At the three yellow-green bases it agrees with the appearance model and moves the balance towards chroma, by ×0.68 to ×0.85. At the reds, the cyan-blues and above all the violets it moves the balance the other way, towards lightness — by ×1.04 to ×1.16 at the reds and cyan-blues and by ×1.17 to ×1.30 at the violets. At nine of the twelve base colours the two units move in opposite directions, at every lightness.

So the prediction the earlier essay drew was half right, in a place it did not look. ΔEITP does predict that lightness pairs become relatively more visible at the bright level — for violets, reds and cyan-blues. CAM16-UCS predicts the opposite for every colour. The median over a set of base colours that includes yellow-greens hides both, because it averages ΔEITP’s two directions into a small fall that looks like agreement.

Where to stand, and what it costs to stand elsewhere #

A forced choice between two predictions is settled by observers, and how many depends on how far apart the predictions are at the place chosen.

The dark violet needs five observers at either pair of levels — 1.5 against 160 candelas, or 15 against 5,000. The violets at L* 50 and 75 need six to nine, the reds and cyan-blues seven to twenty. The yellow-greens need from 38 to several hundred, and at mid-lightness at the dim levels, where the two units happen to predict almost exactly the same shift, the arithmetic asks for a quarter of a million — which is to say, no session can separate them there.

The ordering is by hue, and it is nearly the same at both pairs of levels. The earlier version of this figure had the bright pair costing sixteen times the dim pair; read correctly, the bright pair costs no more at the best colour and less at most. The prediction’s choice of 5 and 5,000 candelas was a reasonable choice of levels and an unlucky choice of pairs — mixed pairs at every hue, whose median is where the units most nearly agree.

That is the second finding of this kind on this subject — a session’s design decided before the observers arrive — and the two have the same shape. The reference lamp must not move found that an asymmetric match’s sensitivity is a property of the two lamps rather than of either, and that the pair can be chosen from spectra alone. Here the sensitivity is a property of the base colour, and it is readable from the two models before anything is built. What a psychophysical session can settle is a model question, and the model answers it.

The counts at the best colours are small enough that they are not really counts of people; a session of five settles nothing about between-observer scatter, whatever the arithmetic says. The honest reading is that at the violets the predictions are far enough apart that a small session suffices, and small means the dozen an ordinary paired-comparison study runs. The scatter assumed is a quarter in the logarithm of the ratio, a stated number rather than a measured one, and every count scales as its square. A tolerance with an observer in it is where a between-observer spread was put into a tolerance rather than left outside it; the same number is doing the same work here, at the design stage.

Why a mixed pair could not have said this #

The earlier essay’s twenty-three pairs were built from the site’s reflectance family, stepped along a basis direction until the difference reached one ΔE₀₀. That is the right construction for asking what a tolerance does to real surfaces, and it is the wrong one for asking how a unit balances two kinds of difference.

A mixed pair’s lightness part is a projection, and a projection of a Euclidean distance is not a distance. If a unit’s lightness axis stretches and its chroma axes do not, a mixed step’s projection onto the lightness axis grows — but so does the total, and the pair’s own direction in the space has changed, so the next reading is of a different step. A distance raised to a power has no length is the neighbouring warning about arithmetic on a difference that is not a metric; this is a gentler version of the same thing. Nothing the earlier essay reported is wrong — its parts are its parts — and what cannot be read off them is what two pure pairs would do, colour by colour.

How the pairs were read #

Each base colour is a CIELAB triple against a D65 white, stepped in L* or in C* by a bisection to ΔE₀₀ of one to within a part in a million. The pair’s relative tristimulus values are converted from CIELAB once.

ΔEITP reads them multiplied by the display’s white in candelas, because its light level is the stimulus’s own absolute luminance through the perceptual quantiser. CAM16-UCS reads them on the relative scale, against the relative D65 white, through viewing conditions whose adapting luminance is the level, because its light level is the room’s. The two arguments are not the same argument — a tolerance has no light level is where the difference between them was first priced, on surfaces — which is the whole reason the units can disagree, and it is why the level enters the two calculations in different places. Giving the appearance model the absolute values against a relative white — which is what the first version did — changes the stimulus’s lightness relative to the white by the level itself, and it was that, not the adapting luminance, which drew the peak.

The nine levels are the adapting luminances the earlier essays used, from 0.3 to 2,000 cd/m², with the display’s white five times each — a background of twenty per cent.

What this does not settle #

The dim end of the range is dim. A 1.5 candela white is a very dark display, and a reader there is partly mesopic; neither unit has rods, and the eye that has no colour describes the regime. Since the dim pair of levels buys nothing the bright pair does not, a session should use the bright one.

Twelve base colours at one chroma. The hue dependence of ΔEITP’s shift is the finding, and four hues are a coarse sampling of it; the violets move most, and whether a blue between the cyan-blue and the violet moves more still is not measured. A base set every thirty degrees of hue, and at several chromas, is the natural extension.

The chroma direction is radial in CIELAB, and neither unit’s hue axes are CIELAB’s, so a pure chroma pair in CIELAB is very slightly impure in both. The impurity is a few per cent of the step and does not move with display level, so it cancels out of every ratio quoted.

And a forced choice between two models is not a measurement of people. It says which model is closer on one comparison; both could be wrong in the same direction.

Still open: whether a room’s light flattens the appearance model #

CAM16-UCS’s quarter-fall in the balance comes entirely from its adapting luminance, which every calculation here sets to a fifth of the display’s white, as though the viewer were adapted to the display alone. A real display is viewed in a room, and the viewer is adapted to something between the display and the room.

The prediction is that a room’s contribution flattens the appearance model’s response to the display’s level, because the adapting luminance then moves less than the display does, while ΔEITP, whose level is the stimulus’s own, is unaffected. If that is right, in a lit room the two units’ medians would agree in size as well as direction, and the only disagreement left would be the one this essay found — by hue. The computation is this one with the adapting luminance a weighted mixture of the display’s and the room’s, and the question worth answering is how much room light it takes for the appearance model’s fall to shrink to ΔEITP’s.

A component is not a measurement of the thing it is named after #

The habit is about reading a decomposition as though it were two experiments, and about reading a median as though it were a colour.

A colour difference in a three-dimensional space has a lightness part and a chroma part, and both are computed from one step. Watching the two parts move as a condition changes feels like watching two quantities, and it is watching one quantity resolved onto two axes that are themselves moving. The move is to build the pure cases, which costs a bisection per pair.

The pure cases then carried a second trap. Their median agreed between the units, and the base colours did not: one unit moved every colour alike and the other moved them by hue in two directions, and averaging two directions produces a small number that looks like agreement. The weighting is the disagreement found something similar a step earlier: two difference formulae that look like they agree often disagree about weights, and the way to tell is to build the case where only one weight is exercised.

The failure mode is designing an experiment from a summary. A decomposition inherits the wrong axes and a median inherits the wrong population, and both put the session where the quantity that was actually varying does not vary. And the first version of this essay adds a third: a model read at the wrong scale will draw a confident curve, and a curve with a maximum in the middle of the range deserves a check of the scale before an explanation of the maximum.