Two units with a light level disagree about lightness

A tolerance has no light level took twenty-three pairs of surface colours built at exactly one ΔE₀₀ and read them in CAM16-UCS, the appearance model’s uniform space, which does take the room. From a cinema to an overcast sky the same pairs grew by half again, and nearly all of the growth was in the chroma part of each difference. The lightness part hardly moved.

That essay’s comparison had one unit with a light level against one without. There is a second colour difference with a light level in it, in wider use than CAM16-UCS on the displays most people now look at. ΔEITP is the colour difference defined alongside ICtCp, the colour encoding specified for high-dynamic-range television, and it takes its light level from somewhere else entirely. CAM16-UCS reads the adapting luminance of the room; ICtCp encodes each cone-like channel of the stimulus with the perceptual quantiser, a curve built from a model of contrast thresholds over absolute luminance from nought to ten thousand candelas a square metre.

So there are now two units that both say a colour difference depends on how bright things are. The question is whether they agree about how.

Both grow, and they grow in different places #

On a display whose white rises from 5 to 5,000 cd/m², the same twenty-three one-ΔE₀₀ pairs grow 2.1 times in ΔEITP and 1.5 times in CAM16-UCS. ΔEITP’s intensity part grows 2.7 times and its chroma part 2.0. CAM16-UCS’s lightness part does not grow at all and its chroma part grows 1.55 times.

• The lightness share of a difference rises with the light in ΔEITP, from 0.37 of the chroma part to 0.50, and falls in CAM16-UCS, from 0.37 to 0.24.
• A one-unit lightness step between two greys is about one ΔEITP on a 1 cd/m² white at every grey, and on a 10,000 cd/m² white it is 6.3 at a dark grey, 3.5 at mid grey and 2.4 at a light one.
• ΔEITP stops growing near its ceiling: doubling the white from 5,000 to 10,000 cd/m² changes the median by under one per cent.
• ΔE₀₀ is one for every pair on every display, by construction.
• The two units agree that differences grow with the light and disagree about which dimension of a difference grows, and the disagreement is largest in lightness, where one unit’s growth is fastest and the other’s is zero.

Twenty-three pairs on a brightening display #

The pairs are the ones the earlier essay used: surface reflectances under D65, each pair walked apart along a fixed direction until ΔE₀₀ lands on exactly one. They are placed on a display whose white runs from 1.5 to 10,000 cd/m², with the scene’s background at a fifth of the white, so the room’s adapting luminance is a fifth of the white’s luminance throughout. At each level the pairs are read three ways: in ΔE₀₀, which ignores the level; in CAM16-UCS at that adapting luminance; and in ΔEITP, with the pair’s tristimulus values scaled to absolute candelas.

The figure above is the median of each. ΔE₀₀ is a flat line at one. CAM16-UCS rises from 0.76 to 1.20 across the whole range. ΔEITP rises from 1.01 to 2.75, steeply below about 100 cd/m² and flattening above a thousand.

Two readings of that figure need separating. The level at which ΔEITP sits is a matter of its scaling — the specification multiplies by 720 so that one unit is roughly one just-noticeable difference, and on a bright display a one-ΔE₀₀ pair is two or three of those. The growth is the substantive part: the same pair is more than twice as many thresholds apart on a bright display as on a dim one, according to a unit built from threshold data. CAM16-UCS agrees that the pair grows, by less.

Where each unit puts the growth #

Both units can be split into a lightness part and a chroma part, and the split is where they part company.

ΔEITP’s intensity part grows fastest: 2.70 times from a 5 to a 5,000 cd/m² white, against 1.96 for its chroma part. CAM16-UCS’s lightness part grows 0.99 times — it is flat — while its chroma part grows 1.55 times.

The flat line is the earlier essay’s finding. CAM16-UCS takes the adapting luminance into its lightness scale through the same compressive response for the stimulus and for the white, and the ratio between them — which is what lightness is — barely moves when both are scaled together. Its chroma coordinates are not built from a ratio. The a′ and b′ of CAM16-UCS are taken from colourfulness, and colourfulness is chroma multiplied by the fourth root of the model’s luminance-level factor, which rises with the adapting luminance and never meets a white to cancel against. That multiplier is the model’s version of brighter looks more colourful, and it is why a chroma difference grows with the light while a lightness difference stays put.

ΔEITP has no ratio to a white anywhere in it. Its intensity is the perceptual quantiser applied to absolute cone-like signals, and the quantiser’s slope in relative terms changes with luminance: a pair whose luminances differ by a fixed fraction lands a different number of quantiser steps apart at different levels. That is the whole of its light dependence, and it acts on every channel, so both parts grow; it acts on intensity most directly.

The shares cross. On a 5 cd/m² white the two units agree almost exactly about how much of a one-ΔE₀₀ difference is lightness — 0.37 of the chroma part in both. By 5,000 cd/m² ΔEITP says half and CAM16-UCS says a quarter. A specification that sets separate lightness and chroma tolerances for bright content would set them in opposite proportions depending on which of the two it trusted.

A pair’s make-up decides its growth in only one unit #

The medians say where each unit puts the growth on average. They do not say whether a particular pair’s growth can be predicted from what kind of difference it is, and the two units answer that differently too.

In CAM16-UCS a pair’s lightness share decides its growth, at a rank correlation of −0.84. The pairs that are mostly chroma grow by about 1.6 times; the two whose lightness part is about four fifths of their whole distance grow by 1.15 and 1.18. That is the flat lightness part and the growing chroma part seen one pair at a time: a pair can only grow by as much of it as is chroma.

In ΔEITP the share predicts almost nothing, a rank correlation of 0.08, and every pair grows between 1.7 and 2.8 times. The quantiser is applied to all three cone-like signals before the opponent matrix, so a pair made mostly of chroma and a pair made mostly of intensity both cross more quantiser steps as the display brightens. What spreads the pairs is where they sit on the tone scale, which a lightness share does not record.

And every one of the twenty-three pairs grows more in ΔEITP than in CAM16-UCS, so the disagreement is not a matter of a few pairs pulling the medians. It is also why the two units part further on some pairs than others. The pairs that are most lightness are the ones CAM16-UCS barely moves and ΔEITP moves by twice, so a comparison run only on chromatic pairs would understate how far apart the two units are.

The shares themselves are close in the two units on the dim display, pair by pair, drifting apart only at the most lightness-heavy pairs. The two constructions agree about what each difference is made of; they disagree about what light does to it, which is the question a threshold is not a unit raised about two scales that agree in one place and are quoted as if they agreed everywhere.

Why the quantiser grows in the dark and levels in the light #

The shape of ΔEITP’s curve comes from one property of the perceptual quantiser, and it is worth drawing because it is the only place the light level enters.

At 0.1 cd/m² one step of the signal is a luminance change of 0.9 per cent; at 100 cd/m², 0.24 per cent; above a few hundred it levels towards a constant fraction. That is the contrast-threshold model the curve was designed from, and like every threshold was measured with a grating it rests on luminance gratings rather than colour patches: in dim light a larger relative change is needed before it is seen, and in bright light the threshold approaches a constant fraction of the luminance.

A pair of colours with a fixed relative difference — which is what a one-ΔE₀₀ pair is, since ΔE₀₀ is computed from ratios to a white — therefore spans more quantiser steps as the display brightens, until the quantiser’s relative step stops shrinking. The growth from 5 to 500 cd/m² is steep because the relative step is still shrinking fast there. The flattening above 5,000 is the quantiser approaching its constant-fraction regime and its ten-thousand-candela ceiling together.

The dark greys grow most #

The clearest single case is a lightness step between greys.

On a 1 cd/m² white, a step of one unit of CIELAB lightness is about one ΔEITP at every grey. On a 10,000 cd/m² white it is 6.3 at a lightness of 20, 3.5 at 50 and 2.4 at 80. ΔE₀₀ calls the mid-grey step exactly one and the other two about seven tenths throughout.

The ordering follows from the quantiser. A dark grey on a bright display sits at a lower absolute luminance than a light grey on the same display, so it sits further down the quantiser’s curve, where the relative step is larger and still falling as the display brightens. The same relative lightness step in the shadows of a bright display is, according to ΔEITP, six times as visible as on a dim one, and the light grey’s step less than two and a half.

That is a prediction banding is not a bit depth already made in another form: a quantised ramp bands far more visibly in the shadows than in the mid-tones, because the threshold changes along the tone scale. ΔEITP carries the same dependence into a colour difference, and CAM16-UCS’s flat lightness part does not.

What a tolerance for bright content would have to say #

Three consequences for anybody writing a colour specification for HDR displays.

A tolerance written in ΔE₀₀ has no light level, and on a bright display it permits differences several thresholds wide. On a 1,600 cd/m² white the twenty-three one-ΔE₀₀ pairs sit between 2.1 and 4.1 ΔEITP apart, and a lightness step in the shadows that ΔE₀₀ calls seven tenths sits at more than five. How bright is white is the reminder that the display’s peak is not a fixed quantity either, so a tolerance written without one is not tied to a visibility at all.

A tolerance written in ΔEITP tightens on lightness as the display brightens; one written in CAM16-UCS tightens on chroma. Both are defensible readings of different evidence — ΔEITP’s of threshold data over absolute luminance, CAM16-UCS’s of colour-appearance scaling across adapting luminance — and they point a specification in different directions in exactly the regime HDR content occupies. The honest specification states which unit its tolerance is in and at what luminance, since the same number means different things in each.

And at dim display levels the two agree. On a 5 cd/m² white both units put about the same share of a difference in lightness. Content graded for a dim cinema can use either unit’s split without contradiction; content graded for a bright living-room display cannot.

How the pairs were read #

ICtCp follows the broadcast specification: XYZ in cd/m² to BT.2020 linear RGB, the fixed integer LMS matrix, the perceptual quantiser on each of L, M and S with its published constants and a peak of 10,000 cd/m², and the fixed opponent matrix to I, Ct and Cp. ΔEITP is 720 times the Euclidean distance in I, half of Ct, and Cp, and its intensity and chroma parts are the intensity term alone and the other two terms together. The implementation reproduces an intensity of 0.508 for a D65 white at 100 cd/m².

The pairs are D65 reflectance pairs at exactly one ΔE₀₀, placed at absolute luminance by scaling each pair’s relative tristimulus values by the display’s white. CAM16-UCS is evaluated with the adapting luminance at a fifth of the white, a background of 20 and an average surround; its lightness part is the difference in J′ and its chroma part the distance in a′ and b′. The grey steps are neutral pairs one unit of CIELAB lightness apart at the stated lightness. Medians are over the twenty-three pairs.

What this leaves out #

Neither unit is observers. ΔEITP’s quantiser was built from a model of contrast thresholds for luminance gratings, and its chroma scaling is a fitted weighting; CAM16-UCS was fitted to colour-difference and appearance data mostly at ordinary luminances. That they disagree above a few hundred candelas is a fact about the two constructions. Which one describes a viewer’s judgements at 5,000 cd/m² is not something either was fitted to settle.

The display is idealised: every pair is shown at a luminance proportional to the white, with no ambient light reflecting from the screen and no limit on the display’s black. A real bright display in a lit room raises its blacks, and a display in a room is a smaller display prices what that does; the dark greys’ growth here is the case that effect would shrink most.

And the adapting luminance is tied to the white at a fixed ratio. A viewer adapted to a dim room watching a bright display is in a condition CAM16-UCS would read differently and ΔEITP would read the same, which is a further place the two could part.

Still open: which lightness share observers report #

The disagreement is a prediction about people that could be tested directly, and it is sharp in the place it matters.

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 ratio of lightness-pair to chroma-pair visibility at the two levels is the quantity to report, and the two units predict it moving in opposite directions.

A secondary question comes with the same session. ΔEITP’s growth is largest in the shadows of a bright display, and that is also where display blacks and viewing flare intervene; measuring the dark-grey steps with and without room light would say how much of ΔEITP’s shadow growth survives an ordinary viewing condition.

A light level can enter in two places #

The habit is about what a model means when it says it depends on the light.

“Depends on luminance” sounds like one property, and there are at least two. A model can depend on the luminance of the room a viewer is adapted to, or on the absolute luminance of the stimulus itself, and the two are different variables with different effects — here, one moves chroma and the other moves lightness most. Two models that both “account for the light level” can still disagree about everything that matters about it, as two uniform spaces disagree about between found for two models that both claim uniformity.

The move is to ask where the light level enters each model before comparing their predictions — as an adaptation, as an absolute encoding, as a scale on one dimension — and to expect disagreement wherever the entry points differ. It took two lines of arithmetic to split each unit into its parts, and the split was where the disagreement was.

The failure mode is to treat “has a light level” as a feature two units either share or not. They can share it and point in opposite directions, and a specification that picks the one with a light level over the one without has only made the first of two choices.