A difference has no rate

A colour difference formula takes two colours. A difference has no size showed what that leaves out in space — spread the same difference over a fine pattern and fourteen units become less than one — and a difference has no place showed the same thing across the retina. Both of those hold the two patches still and present. A screen refreshes, an indicator blinks, a printed sheet passes under an inspector at speed: in each of those the two colours are not both there at once, and the formula has no argument for how fast.

A rate does not scale a difference, it takes it apart #

The two halves of a colour difference are carried by channels of different shapes, so a rate re-weights a difference rather than shrinking it.

• At rest the twenty-three pairs are ΔE₀₀ 1.000 each, to the last digit. Alternated at eight hertz they run from 0.53 to 3.08 — a factor of 5.8 between pairs the formula calls identical.
• The chromatic channel keeps a twentieth of its resting sensitivity by fifteen hertz; the luminance channel is four times its own resting sensitivity at eight.
• So the surviving difference stops being a colour difference: the share of it that is chroma and hue falls from 80 per cent at rest to 23 at eight hertz and 6 at fifteen.
• Above sixty hertz the largest of the twenty-three is worth 0.15, and the pair is seen as one colour that neither member is.

Two channels, two shapes #

The eye’s temporal response is not one filter. A luminance modulation is carried by a band-pass channel that peaks around eight hertz and gives out near sixty; an isoluminant one is carried by a low-pass channel that is already falling at one hertz and is gone by fifteen. Both are solved from their published landmarks rather than drawn, and the ratio between the two cutoffs — four to one — is a ratio between two measurements.

The construction used here follows from that and nothing else. A pair’s difference is split at its own midpoint into a lightness part and a chromatic part; each part is multiplied by its channel’s sensitivity relative to that channel at rest; the two colours are rebuilt around the midpoint; and the published formula then prices the result unchanged. Nothing about ΔE₀₀ is modified — it is handed a different pair, which is what a viewer is effectively given when the colours alternate.

Two consequences follow immediately. The chromatic factor can only fall, because its channel is low-pass and starts at its own maximum. The luminance factor rises to four at eight hertz, because a band-pass channel at rest sits at a quarter of its peak — the same fact that makes a slow flicker more visible than a steady light.

This is a threshold scaling and it is exactly as good as that assumption. Sensitivity is what a contrast is divided by at threshold; using it as a multiplier well above threshold is an extrapolation, and a threshold is not a unit is this collection’s own essay on why those two are not interchangeable. The shapes below are firmer than the sizes.

What survives, rate by rate #

At rest, one. By two hertz the median has fallen to 0.92 — the chromatic part is already going and the luminance boost has not yet arrived. Then the boost wins: the median rises to 1.11 at seven and a half hertz, which is the luminance channel’s own peak, and comes back through one at about eleven and a half. By fifteen hertz it is 0.85, by thirty 0.40, and by sixty 0.05.

The median is the least interesting number in that sequence. The spread is the argument.

Seventeen of the twenty-three peak above one somewhere in the band — that is, there is a rate at which they are seen as a larger difference than the formula says — and four of them peak above two. The peaks sit tightly together in rate, at a median of 7.5 hertz, because the rate is a property of the channel rather than of the pair; the heights do not, running from 1.00 for the most chromatic pair in the set to 3.08 for the most achromatic.

At eight hertz the twenty-three run from 0.53 to 3.08, and the ordering is not random: it is the ordering of how much of each pair’s difference was lightness at rest. A pair that differed almost entirely in chroma has lost most of what it had. A pair that differed in lightness has been amplified by the same mechanism, in the same room, at the same rate. The formula cannot distinguish these two cases, because at rest they are the same number, and a rate does nothing except distinguish them.

The difference changes kind, not only size #

The clearest way to state what a rate does is not in units at all.

At rest these pairs are 80 per cent chromatic — they are mostly differences of colour rather than of lightness, which is what a set built on saturated reflectances looks like. At eight hertz the surviving difference is 23 per cent chromatic. At fifteen it is 6 per cent. The pair has become a lightness difference, and nothing about either colour has changed; only the rate at which they were shown.

That is a different failure from the ones the two earlier essays found. Spread over a fine pattern, a difference shrinks. Moved into the periphery, it shrinks and rotates in hue. Shown at a rate, it is re-weighted between two mechanisms with different bandwidths — which can make it larger, and which changes what kind of difference it is.

Where the rate comes from in practice #

Nothing here requires a laboratory flicker. The rates that matter are the ones ordinary delivery produces.

A progress indicator blinking twice a second sits where the chromatic channel has already lost a quarter of its sensitivity. A hazard lamp at one hertz and a cursor at two are inside the band where the luminance channel is climbing. A sheet inspected on a press at a metre a second, with a repeating pattern every centimetre, presents the eye with a hundred hertz — above fusion, where the difference is not merely reduced but replaced: Talbot’s law says the eye integrates, and what is seen is the time average of the two colours, which is a third colour that neither member is.

The press case is worth doing properly, because it lands a long way outside the band. A web running at 600 metres a minute is ten metres a second, and a repeat every ten millimetres therefore passes a fixed eye at a thousand hertz — seventeen times fusion. Nothing of the difference survives as flicker; what the inspector sees is the time average, and the only route by which the modulation reaches them at all is the one measured on moving eyes rather than moving objects, where a saccade turns a kilohertz into a spatial pattern the eye is nearly at its best on.

That last case is the one a specification is most likely to get wrong, because the sample passes the tolerance at rest and is not seen at rest. The eye is never still made the converse point for a static scene: a perfectly stabilised image disappears within seconds, and it is the eye’s own drift that keeps the world there. Both are the same observation from opposite ends — the visual system responds to change, and a difference held perfectly still or changed too fast is equally invisible to it.

The rate a difference vanishes at is not fixed either #

One more term compounds with this, and it is the one the room supplied.

Fusion is not a constant of the eye. It rises about 12.5 hertz per decade of light, so the sixty hertz quoted above belongs to an office at a hundred candelas a square metre: in a cinema the same criterion falls to 35 hertz and in direct sun it rises to 85.

Carrying that through the sweep — scaling the rate axis by each room’s own cutoff — puts numbers on it. A difference alternating at thirty hertz is worth 0.10 in a cinema, 0.40 in an office and 0.64 in sunlight. At sixty hertz it is 0.001, 0.05 and 0.19: gone, nearly gone, and visible. The same sample, the same instrument reading, the same rate, and three verdicts. Around the luminance peak the rooms agree much better — 0.90, 1.11 and 1.08 at eight hertz — because there the channel is near its maximum in every room and the room is moving a cutoff rather than a peak.

That stacks with what a tolerance has no light level found about the same pairs read steadily: the room moves what one unit is worth by a factor of 1.62, and it moves the rate at which the unit disappears altogether. A tolerance is a statement about two colours; what a viewer gets is a statement about two colours, a room, a size, a place on the retina and a rate, and the formula carries the first of those five.

What a specification could do about a rate #

Three things, and the first is nearly free.

Say at what rate the judgement is made. A tolerance is already agreed against an illuminant and an observer; adding viewed steadily, or viewed on a line running at so many metres a minute, costs a clause. It does not make the tolerance right at that rate — nothing here converts one number into another — but it stops two parties from meaning different things by the same figure. A delivery tolerance is three tolerances made the same argument about the stages a tolerance is split across, and the remedy is the same: name the thing rather than correct for it.

Inspect where the channels are flat. If a judgement must be made in motion, the useful rates are the slow ones. Below about two hertz both channels are near their resting sensitivities and the difference a viewer gets is close to the one the formula priced; between four and twelve it is re-weighted the most; above twenty the colour part is simply absent. An inspection that must run fast is an inspection of lightness, whatever the specification says it is about.

And do not use a fast-moving check for a chromatic tolerance at all. This is the sharpest practical consequence, and it is the one that follows from the shapes rather than the sizes: a chroma difference and a lightness difference of the same size in the formula are not the same size to a viewer at any rate other than rest, and above fifteen hertz one of them has vanished while the other has not. A halftone is a luminance object found the same asymmetry in space — the eye’s chromatic channels give out at a coarser scale than its luminance channel — so a difference that is invisible when small is invisible when fast for the same reason, and the two limits compound on anything that is both.

The models with no clock in them #

The formula is not alone in leaving time out, and the company it keeps is worth naming.

The model has no clock found that an appearance model takes a stimulus and a situation and returns a prediction with no time in it, and that half a second after a light changes a judgement is three and a half units from the settled one. A gain has a time constant put a clock on adaptation and predicted that two people in one room at one moment do not agree about a patch. Those are about how long ago something changed; this essay is about how fast it is changing now. Between them, the two arguments say that everything a viewer is given is a function of time and that nothing in the arithmetic is.

What was computed, and how #

The pairs are the twenty-three built at exactly ΔE₀₀ 1.000 under D65 normalised to a luminance of a hundred, by walking one member of each pair along a fixed direction in reflectance space until the difference lands on target. They are the same pairs the light-level essay uses, so the two results are about one set of samples.

Each pair is converted to CIELAB under that white, split at its midpoint into a lightness part and a chromatic part, and each part multiplied by sensitivity(f) / sensitivity(0) for its channel — the luminance channel band-pass with a peak at 8 Hz, a resting value a quarter of that peak, and a 5 per cent criterion at 60 Hz; the chromatic channel low-pass with the same criterion at 15 Hz. The two attenuated colours are rebuilt around the midpoint and handed to ΔE₀₀ unchanged.

The chroma share is the chromatic part of the surviving difference over the sum of the two parts, in CIELAB units, taken as the median over the pairs.

What this construction cannot claim #

It is a threshold scaling used above threshold, which is the largest of its limits and is stated in the module as well as here. A suprathreshold model of temporal response would give different sizes; the two channels’ shapes, and therefore the re-weighting, would survive it.

The two channels are also an idealisation: the real system has more than two temporal filters, they are not perfectly separable from the spatial ones, and an isoluminant modulation is only isoluminant for the observer it was constructed for — which for a population is nobody, as this collection’s observer work keeps finding. A pair built to modulate only the chromatic channel for the standard observer modulates the luminance channel a little for everybody else, and that residue is exactly what the band-pass channel is most sensitive to.

And the alternation here is a square one between two colours with no gap. A duty cycle, a gap, or a gradual transition each change the frequency content and therefore the answer; the computation takes any waveform, and this essay takes the simplest one.

Still open: whether a viewer ranks them this way #

The experiment this predicts is straightforward and, as far as the published literature goes, has not been run on a colour-difference set. Present pairs matched at one unit in a steady side-by-side comparison, then present the same pairs alternating at eight hertz, and ask which of two pairs is the larger difference.

The prediction is specific: the ranking should follow how much of each pair’s difference is lightness, the pairs that were nearly all chroma should be judged much smaller, and a few should be judged larger than they were when steady. If the ordering at eight hertz is instead the same as at rest, then the channels’ attenuations do not carry through to a suprathreshold judgement, and the construction here is a threshold statement that does not generalise — which would be worth knowing, and is the kind of refusal this construction is built to take.

The arguments a formula was given #

The habit is the one the light-level essay ends on, arriving from a second direction.

ΔE₀₀ takes two colours: not a size, not a place, not a rate, not a room. Each of those has now been measured against it here, and each moves the answer by more than the differences between competing formulae. That is not a criticism of the formula, which answers its own question exactly. It is a statement about what the question was.

The move is to write down what a quantity is a function of before arguing about its value. If the list is shorter than the situation, the argument about the third decimal place is an argument about the wrong thing — and the missing argument can usually be measured, because some other model in the same subject takes it.

The failure mode is to treat a formula’s silence as a claim that the missing thing does not matter. Silence is not a claim at all.