A viewing condition is a moment

An appearance model takes a stimulus and a viewing condition and returns what the stimulus looks like. The viewing condition is a white point, an adapting luminance, a background and a surround, and among the quantities computed from those is one called the degree of adaptation.

It says how completely the observer has discounted the illuminant. It runs from about 0.6 in a dark surround to 1.0 in a bright one, it is used to interpolate between “the light was fully compensated” and “the light was not compensated at all”, and it is written as a function of the surround and the adapting luminance.

It is not a function of time, because the model has no time in it. That is not an omission anybody would call a defect — an appearance model predicts appearances, and an appearance is what somebody reports when asked. But a quantity that describes how far a process has got, in a model with no process, is a number waiting to be read another way.

The claim #

The standard’s degree of adaptation is numerically indistinguishable from an unfinished transient, and the model has no way to tell them apart.

• The three tabulated surrounds are one observer at 107, 50 and 21 seconds after a change of light, at 100 candelas per square metre. All three moments are inside the first two minutes.
• Complete adaptation is an asymptote rather than a state. Raising the adapting luminance does not adapt an observer more; it moves the clock reading further out — 693 seconds at 1000 candelas against 107 at 100. And the standard’s own idealisation, discount the illuminant completely, is not a time at all.
• The two readings are distinguishable by waiting. A steady-state reading says a patch stops changing; a transient reading says it keeps going. Under tungsten they differ by 8.5 appearance units at one second, cross at about two minutes, and settle 1.7 units apart for ever.
• And they agree best late, which is the opposite of where anybody looks.

Reading D as a clock #

The adaptation machinery on this site relaxes with two exponentials, one at a second and one at a minute, in a fixed mixture. Both come from the appearance model’s own tabulated time constants rather than from anywhere new, so nothing about the clock is a second opinion — it is the same file’s numbers used in a way that file does not use them.

Call the fraction of a change of light already taken on board at time t the degree of adaptation at t. Then finding the moment corresponding to a standard’s stated D is one bisection.

In an average surround at 100 candelas the standard’s D is 0.941. Ninety-four per cent adapted, on this clock, is 107 seconds. In a dim surround it is 0.847, which is 50 seconds; in a dark surround 0.753, which is 21.

The three of them fall inside the first two minutes of one observer’s adaptation, in the order the surrounds are usually listed. Somebody walking into a cinema passes through all three in under two minutes without the room changing at all.

What the model says about complete adaptation #

The standard’s formula approaches one as the adapting luminance rises and never reaches it: 0.941 at 100 candelas per square metre, 0.9999967 at 1000. Read as a clock those are 107 seconds and 693.

That is a different statement from the one the model is making. The model says a brighter room produces more complete discounting; the clock reading says a brighter room’s stated D corresponds to a later moment, which is not the same claim and has the opposite practical implication. On the first reading, an observer in a bright room is done sooner and more thoroughly. On the second, the number quoted for a bright room describes a state an observer takes ten minutes to reach.

And the option the standard offers explicitly — discount the illuminant completely, D forced to one — has no clock reading at all. An exponential does not reach its asymptote in finite time, so a D of exactly one is the formula’s ceiling rather than an answer. assertCompleteAdaptationIsAnAsymptote reports that rather than extrapolating, which is the same discipline that stops a colour-temperature search from reporting its own search floor as a chromaticity.

The test #

Two readings that produce the same number for the same room are not two hypotheses unless something distinguishes them, and something does.

If the degree of adaptation is a steady state — a ceiling the observer reaches and stops at — then a patch under a changed light settles to an appearance and stays there. If it is a moment — a point on a curve heading for complete adaptation — then the patch goes on changing, slowly, past the point the static model says it has finished.

The gap after both have finished is 1.67 CAM16-UCS units, which is not large and is not nothing — it is comfortably above the threshold this site uses for appearance differences and comfortably below the size of the effect being modelled.

The shape of the disagreement is the more useful part. At one second the two models are 8.5 units apart, because the static model has already applied ninety-four per cent of a compensation the observer has barely begun. At two minutes they agree almost exactly, because that is the moment the static model’s D corresponds to. After that they diverge again and stay diverged.

So the experiment is: change the light, wait five minutes, and ask. The static model predicts the same answer as at two minutes; the transient model predicts a further two units of drift. Neither prediction is subtle and neither needs new apparatus — it needs somebody to wait, which is exactly what an experiment designed to measure a viewing condition is arranged not to do.

Why the ambiguity survives #

Appearance models are fitted to corresponding-colour data sets: an observer adapts to one condition, matches a stimulus, adapts to another, matches again. The protocols specify an adaptation period, and the periods are typically a minute or two, chosen because that is what the adaptation literature reports as sufficient.

A minute or two is precisely where the two readings agree. The data the model was fitted to was collected in the window where a steady state and an unfinished transient are the same number, so the fit could not have distinguished them and the resulting parameter inherits the ambiguity.

That is not a criticism of the fitting. It is an observation about what a parameter means when the design of the experiment makes two interpretations of it indistinguishable — and the practical consequence is that using the model outside that window is using it outside its evidence in a way its documentation does not flag.

The nine readings #

The bisection can be run at any adapting luminance, and the table is worth having whole because the pattern in it is not what the surround names suggest.

Read down a column and the three surrounds are three moments a minute or two apart. Read across a row and the same surround is a completely different clock reading depending on the room’s brightness — nearly fifteen times longer for an average surround between a dim room and a bright one.

That row is the awkward one. Nothing in the model’s account of surround suggests that brightness should change when an observer is finished, only how completely; but the number the model states, read as a moment, says a bright room’s stated adaptation is a state reached after ten minutes and a dim room’s after one. A specification that says “average surround, 1000 cd/m²” is, on this reading, a specification that includes a ten-minute wait nobody has been told about.

The nine readings reproduce, and one quoted intermediate does not #

Every entry in that table can be rebuilt from the standard’s own formula for the degree of adaptation and the clock’s two exponentials, and all nine come back exactly: 47.8, 106.5 and 693.4 seconds for an average surround, 22.1, 49.5 and 75.2 for dim, 5.0, 20.8 and 33.6 for dark. The arithmetic is sound.

The one number that does not rebuild is a quoted intermediate. The claim above says the degree of adaptation is 0.99998 at a thousand candelas per square metre. The formula gives 0.9999967 — two further nines — and it is the second value that produces the 693 seconds beside it. A D of 0.99998 would give 586 seconds.

That is a small slip and it points at something that is not small, which is why it is worth catching rather than correcting silently. The clock reading in a bright room is hypersensitive to digits of D that no experiment could have constrained.

At a hundred candelas the derivative is about a thousand seconds per unit of D, so an error in the third decimal place moves the answer by a second. At a thousand candelas it is 1.8 × 10⁷ seconds per unit of D — seventeen thousand times more — so the answer is decided by the sixth decimal place. Replacing 0.9999967 by a merely respectable 0.99 gives 213 seconds instead of 693, a factor of 3.2 from a change of one per cent.

And those digits are produced rather than measured. The degree of adaptation is an exponential fitted to corresponding-colour judgements; it emits as many decimal places as anybody asks for, and the data behind it constrains perhaps two. So the 693 seconds is arithmetically correct and epistemically empty: it is the clock reading of a number the formula computes to seven figures and the evidence supports to two.

That does not damage the essay’s argument and it sharpens where the argument applies. The three readings at ten and a hundred candelas are robust — the derivative there is small enough that a percentage error in D moves the clock by seconds, so 107, 50 and 21 are figures a reader can carry. The thousand-candela column is not. Its three entries are 693, 75 and 34, and the first of those is on the asymptote while the other two are not, because the dim and dark surrounds multiply D by 0.9 and 0.8 and never approach one at all.

Which is a cleaner statement of the row the essay finds awkward. The average surround’s fifteenfold stretch between a dim room and a bright one is not the surround doing something the model does not describe; it is one entry sitting on an asymptote where the clock’s own sensitivity has run away. The dim and dark rows stretch by 3.4 and 6.8, both modest, both robust, and neither of them a puzzle.

So the honest form of the awkward observation is narrower and survives. A specification naming an average surround at a high adapting luminance is naming a D whose clock reading is not determinable, because the formula’s ceiling and the exponential’s asymptote meet there and the answer is set by digits nobody measured. That is a better complaint than a ten-minute wait nobody was told about, and it is the same complaint the essay makes about complete adaptation having no clock reading at all — arriving one step earlier, at the point where the reading stops being meaningful rather than where it stops existing.

Who found it, and when #

Incomplete adaptation entered colour appearance modelling in the 1980s and 1990s as a fix for a real failure: models that discounted the illuminant completely predicted that a white paper under tungsten would look exactly as white as under daylight, and observers said it looked slightly yellow. A factor between zero and one, fitted to corresponding-colour data, repaired the prediction.

Hunt’s models carried it, CIECAM97s carried it, CIECAM02 gave it the functional form still in use, and CIECAM16 kept that form unchanged. In every one of them the argument is the same shape — a surround factor times a function of adapting luminance — and in none of them is there a time.

The transient literature ran alongside and did not meet it. Adaptation time courses were measured from the 1940s onward, with the two-component structure well established by the 1970s: a fast component of a second or so and a slow one of tens of seconds, in roughly the mixture used here.

Two literatures, two numbers, and the arithmetic that connects them is a bisection. That they were never connected is the ordinary shape of this: a model that has no clock cannot be asked what time it is, so nobody asked.

What was computed, and how #

The appearance model gained one optional argument: a degree of adaptation stated rather than computed. Passing null computes it from the surround exactly as before, so nothing already on this site moves. Passing a number lets a caller ask the model a question it cannot ask itself.

The clock supplying that number is the adaptation model’s, unchanged. Its two time constants are the appearance model’s own tabulated ones, so a revision of those moves every number here with them rather than leaving two files disagreeing.

The moment is found by bisection on the remaining fraction, bracketed at twenty minutes. A D that the clock cannot reach inside the bracket is reported as beyond it rather than extrapolated.

The static-against-moving comparison uses illuminant A as the changed white, because that is the ordinary large change of light — walking from daylight into a filament-lit room is the case appearance models are quoted about, and it is where the degree of adaptation decides most of the answer. Under a smaller change the whole comparison shrinks proportionally.

Running the clock out to ten minutes is past anything a measurement waits for, and it is where the two pools have finished and the last of the drift is visible.

What a specification would have to say #

The practical form of this is short. A viewing condition as currently written is four numbers and a name, all of them properties of a room. If the degree of adaptation is a moment, then a fifth quantity is missing and it is a property of the observer’s history: how long they have been in the room.

Adding it costs nothing in the arithmetic — the model already takes a degree of adaptation, and the clock that turns a duration into one is two exponentials. What it costs is an assumption everybody is currently making without stating: that the observer has been sitting there long enough, where long enough is defined by the same fitting data that could not distinguish the two readings.

A soft-proofing comparison is the case where it bites. A print in a booth and a display in a dim room are given two viewing conditions and the appearance gap between them is computed; if both conditions are clock readings, part of that gap is a statement about how long each observer had been looking, and a reviewer who glances at the proof and back is not in either condition.

Where it stops #

The clock is a two-exponential relaxation with no dependence on the size of the change, and real adaptation is not quite that: a large change takes longer than a small one by more than the linear model allows. Nothing here would survive a factor of two error in the slow time constant, and the slow constant is the less well determined of the two.

The identification of D with a fraction adapted is the load-bearing assumption and it is an interpretation rather than a derivation. D scales a diagonal gain toward the identity; a partly-adapted observer is applying a gain that has moved part way from its old value to its new one. Those are the same arithmetic when the old gain is the identity and are not the same in general, so the reading is exact for an observer coming from an equal-energy white and approximate otherwise.

And nothing here says the transient reading is right. It says the two are different hypotheses about the same number, that the standard offers no way to choose, and that a five-minute wait would.

Two more clocks from the same machinery say what the moment is a moment of, and how much of it a viewing condition never sees.

Where the ladder goes next #

If a viewing condition is a moment then a viewing condition is not a property of a room, and the models built on it inherit that. Every soft-proofing comparison on this site sets one viewing condition for a print and another for a display and computes the gap between them; if both of those are clock readings, the gap is partly a statement about how long each observer had been sitting there.

The computation that would settle it is the room this site already builds: four clocks running at once, an observer adapting to a lamp that is still warming, and the appearance model asked at each moment for what it would say. Two of the four clocks in that room are the two in this essay’s bisection, which means the appearance model and the transient model are already using the same numbers — and disagreeing about what they mean.