The model has no clock

A viewing condition is an argument — this site has already said so, and the argument has four parts: an adapting white, a background, an absolute luminance and a surround category. There is a fifth thing a real judgement depends on, and no appearance model takes it: how long the observer has been looking.

The claim #

CIECAM16’s degree of adaptation is a steady state, and reaching it takes minutes. A judgement made before it is reached is made in a state the model has no argument for, and the error is large enough to matter to the trades that make judgements for a living:

A unit of CAM16-UCS is roughly a just-noticeable difference. So the first glance after a change of light is several of them out, and a minute is not enough.

What the model does have #

viewingConditions computes a degree of adaptation $D$ from the adapting luminance and the surround:

$$D = F\left(1 - \tfrac{1}{3.6}e^{(-L_A - 42)/92}\right)$$

At 100 cd/m² in an average surround that comes out at 0.941 — not 1, because adaptation is never complete, which is a measured fact and is already asserted on this site.

That number is where the observer ends up. It says nothing about the journey, and the journey is the subject here.

The clock, and where its constants come from #

The time course added in this phase is a two-exponential of stated constants:

$$\text{done}(t) = 0.65\left(1 - e^{-t/1\,\text{s}}\right) + 0.35\left(1 - e^{-t/60\,\text{s}}\right)$$

• a fast component of about a second, which is receptoral gain control,
• a slow component of about a minute, which is everything downstream,
• and roughly two thirds of the change in the fast one.

All three are quoted with ranges, and no claim on this page depends on their third digits. What the arithmetic then does is thread done through the model as an interpolation of the white: at $t$, the observer is working from a white part way between the old one and the new one, and the model applies its own $D$ to that white exactly as it does at the end.

Two incompletenesses, and conflating them was this file’s first bug. The model’s $D$ says an observer never adapts fully, at any time — a permanent property of the steady state. done says how far through the transition the observer has got. Multiplying the two, which was the obvious first move, left a residual shift of 0.83 units at five minutes: the model quietly reporting that adaptation never finishes, which is not what a time course means.

The shape is the interesting part: the fast component takes the first two thirds in a couple of seconds and then the slow one grinds through the rest for two minutes. That is why a light change feels instantaneous and a careful colour judgement made straight afterwards is still wrong.

Where it costs somebody money #

A viewing booth with a switch. Comparing a sample under D65 and then under illuminant A means changing the light and judging, and the judgement made in the first seconds is made by an observer still adapted to the previous illuminant. That is why the standards for visual assessment prescribe an adaptation period, and why the period is a minute or more rather than a moment.

A soft proof. A proof is a different object from a print because the two are viewed in different conditions — and the practical procedure makes it worse: an operator looks at the screen, then at the print in the booth beside it, then back. Each glance is made in an adaptation state inherited from the other. The measured gap between the two objects is 4.18 CAM16-UCS units under settled conditions; the transient adds several more, in a direction that alternates.

And a camera or a display’s automatic white balance, which usually does have a time constant — deliberately, to stop the picture lurching — and whose constant is chosen against the eye’s. A system that adapts faster than the viewer produces a visible correction; one that adapts slower produces a lingering cast. Neither is a colorimetric error, and no colorimetric measurement can see either.

The transient is not small compared with anything #

It helps to put the numbers beside the other quantities this site measures in the same unit, because a bare “3.58 CAM16-UCS units” carries no weight on its own.

The first row is comparable with the entire settled difference between a print in a booth and a proof on a screen. That is the finding stated in the most useful way: for the first few seconds after any change of light, the observer’s own transient is as large as the effect being judged.

It also explains a familiar experience that is usually attributed to fatigue or imagination. Two samples that matched a moment ago stop matching when the light is switched and match again a minute later, and nothing about them changed. What changed was the state of the instrument, and the instrument is the reader.

The two tables are one law #

The essay prints a shift in CAM16-UCS units against time and, separately, a fraction complete against time. Eliminating the time between them gives a law, and the law is worth having because it says what kind of quantity the shift is.

Fitting the eight rows the two tables share:

$$\text{shift} \;\approx\; 4.30\,\bigl(1 - \text{done}(t)\bigr)^{0.58}$$

It reproduces every published row to within two per cent — 3.62 against 3.58 at half a second, 1.33 against 1.31 at a minute — over a range in which the shift falls by a factor of nearly three.

The exponent is the interesting part, and it is not the clock’s. Neither time constant appears in it; what appears is a power of about 0.58, which is very close to the 0.63 that CAM16-UCS raises its own Euclidean distance to. So the shift is very nearly the incomplete fraction of the adaptation, measured through the unit’s own compression — the observer’s state moves along a straight line towards the settled one, and the unit reports a compressed version of how far along it has got.

That is a check rather than a discovery, and it is the useful kind. It says the two tables are not two measurements but one, and it means the whole essay’s arithmetic can be run forwards or backwards: given a state of adaptation, the shift follows, and given a shift, so does the state.

What a repeated comparison actually costs #

Which makes the last of the three recommendations computable rather than merely stated. Treat a repeated comparison as a repeated settling — the operator looking from the screen to the booth and back — was asserted three sections ago with no number, and the number is available from the two exponentials without any new machinery.

Alternating between two whites with a dwell of $T$ at each drives every exponential component to a periodic steady state rather than to either endpoint. A component with time constant $\tau$ ends up swinging between $1/(1+e^{-T/\tau})$ and $e^{-T/\tau}/(1+e^{-T/\tau})$ of the way across, so the fast component follows the glances almost perfectly while the slow one settles near the midpoint and stays there.

Nobody comparing two things back and forth ever gets below about one and a half units. An operator glancing every five seconds oscillates between 1.56 and 3.86 CAM16-UCS units of error, and the floor of that range is fourteen times the 0.11 a settled observer reaches. The transient never decays because the drive never stops; what looks like an adaptation period is a duty cycle.

And the trade runs the wrong way for anybody trying to fix it by looking longer. Lengthening the dwell improves the best moment — 1.93 down to 1.10 as the glance goes from two seconds to a minute — and makes the worst moment worse, 3.65 up to 4.06. A longer look lets the slow component get further from the midpoint, so the observer arrives at the other object carrying more of the wrong state. There is no dwell that is good at both ends, and the two ends are the two things being compared.

Put beside the settled figures the essay already quotes, this reorders the problem. The soft-proof gap between a booth and a screen is 4.18 units under settled conditions and is a difference somebody pays money to remove. The act of comparing them costs between 1.6 and 3.9 units on its own, in a direction that alternates with the glance — so a large part of what an operator is judging when they look back and forth is their own duty cycle.

The procedure that avoids it is the one nobody follows because it feels wasteful: look at one thing, wait, form a judgement, then look at the other, wait, and form a second judgement — comparing two memories rather than two views. That trades the transient for the far worse noise of colour memory, which is why it is not the practice; but the trade is now a trade between two measured quantities rather than between a measured one and an unexamined one.

What a standard could say instead #

The visual assessment standards already prescribe an adaptation period, and this arithmetic suggests what a sharper version would look like.

Name the time, not the ritual. “Allow the observer to adapt” is a procedure; “at least sixty seconds, during which the observer looks at the field and not at the samples” is a specification, and the table above says what each choice of number costs.

Name the previous state. The transient’s size depends on where the observer came from — the further the two whites, the larger it is — so a booth that switches between D65 and illuminant A demands a longer wait than one that switches between D50 and D65.

And treat a repeated comparison as a repeated settling. Looking back and forth between a screen and a booth does not let the observer settle at either: each glance restarts the clock from the other end. Nothing in any current procedure acknowledges that, and it is the commonest way a careful person makes an uncontrolled measurement.

What was computed, and how #

The transition is D65 to D50, with the stimulus fixed in XYZ and the room’s luminance and surround unchanged. Only the white moves.

The appearance at each time is CIECAM16 with the white interpolated by done(t), and the distance is CAM16-UCS against the fully settled appearance — the site’s usual unit for an appearance difference, and the same one the proof comparison is quoted in.

The assertions are three, and the third is the one that catches the bug above. The shift must be monotone; the shift at half a second must exceed a floor, because a transient too small to notice would be nothing to warn anybody about; and the shift at five minutes must be under 0.2, because a model whose transient never settles is modelling something else.

Where the model stops #

The constants are quoted and the shape is chosen. Two exponentials is the simplest form that reproduces the reported behaviour — a fast receptoral component and a slow one — and the literature reports more structure than that, including asymmetries between adapting to a brighter light and to a dimmer one.

Only the white moves. A real change of light usually changes the level as well, and light and dark adaptation have their own much slower course — minutes to tens of minutes — which is a different mechanism and is not here.

Nothing local. This is scene-wide adaptation. The local version is an afterimage, which has its own time course and is computed in this phase without one.

The repeated-glance figures assume the drive is a square wave. A real operator’s gaze does not alternate on a fixed period, does not spend equal time on each object, and passes over everything between them on the way; the table is what a metronome would produce. Its ordering — a floor that never reaches the settled value, and a best moment that improves as the worst one degrades — depends only on the two exponentials being driven periodically, and not on the shape of the drive.

And the interpolation of the white is a modelling choice. What is actually changing over time is a set of receptor gains; treating that as a white part way between two whites is a defensible approximation and it is not a derivation. A model with time in it properly would carry gains rather than whites.

A cyan is the hue where the same sweep of adapting whites moves the stimulus furthest, and it is worth seeing beside the earlier one.

The generalisation #

The habit is: when a model takes a situation as an argument, ask which of the situation’s properties it left out.

CIECAM16 takes four numbers and returns six correlates, and it is very good. What it takes is a state, and a state is a snapshot: the model has no way to express that the observer arrived at this state from somewhere, when, or how fast. Every one of its outputs is therefore an asymptote.

The same shape appears wherever a steady-state model is applied to a process:

• A separation is computed for a press at equilibrium, and a press warming up prints differently.
• A display’s tone response is measured after it has stabilised, and a panel from cold is not that panel.
• And a metameric match is exact for an observer whose state is fixed, which is exactly what an observer’s state is not.

The practical rule is the one the standards already encode without explaining: wait, and say how long. A visual comparison is a measurement with a settling time, like every other measurement made with an instrument that has one.

Who found it, and when #

The time course of chromatic adaptation has been measured since the 1950s, and the two-component structure — a fast receptoral process and a slower post-receptoral one — was established through the following decades by measuring how a judgement changes in the seconds and minutes after a change of illuminant.

The practical response is older than the measurement. Visual assessment standards have prescribed adaptation periods for as long as they have existed, because anybody who has compared two samples under two lights notices the effect within a day of starting the job.

What has not happened is the model. CIECAM-class appearance models remain steady-state by design, and the reason is defensible: adding a clock means adding a state, and a model with a state cannot be a function from a stimulus and a situation to an appearance. The colour-management pipelines that use these models are built out of stateless transforms, and a stateful appearance model would not fit into any of them.

That is worth stating plainly rather than as a criticism. The omission is architectural, not an oversight, and the price of it is the table at the top of this essay.

A dim room is where the model’s own account of a display’s reach is furthest from the display’s specification.

What the pictures cannot show #

The transient. A page cannot change the reader’s illuminant, so nothing here can demonstrate the effect; the figures are plots of a model. The demonstration is available to anyone with two lamps and a colour sample, and it takes about ninety seconds.

And the settling is drawn on a logarithmic time axis, which is the only way to show half a second and five minutes on one figure and which flatters the tail: the last two minutes look like a short stretch of the axis and are a third of the adaptation.

Where the ladder goes next #

The nearest piece of work joins the two halves this phase built separately. An afterimage is local adaptation without a clock; this essay is scene-wide adaptation with one. A model carrying gains with time constants, rather than whites with an interpolation, would produce both from one mechanism — the afterimage as the transient of a local gain change, the illuminant shift as the transient of a global one.

The second is the asymmetry. Adapting to a brighter light and to a dimmer one are known to take different times, and nothing here has a direction in it. A model with the asymmetry would predict something the symmetric one cannot: that walking from a lit room into a dim one costs more judgement time than the reverse, which is both testable and part of why cinemas have foyers.