A discount nobody measured

Every adaptation figure in this collection asks what a gain leaves behind, and every one of them applies the whole gain.

The claim #

The adaptation results here are computed for an observer who discounts the illuminant entirely. The standard’s own formula says nobody does, and correcting it multiplies every residual by about 1.7.

• A von Kries gain has a completeness. The model calls it D, it runs from zero to one, and one means the observer has taken the whole change of light out.
• The appearance model computes it from two things — the surround and the adapting luminance — and from nothing else. In an average surround at a hundred candelas it comes out at 0.941.
• Every adaptation number this collection publishes is computed at one, because the machinery applied the ratio of the whites and stopped there.
• The correction is a factor of 1.74 on the mean residual over three representative changes of light: 1.27 ΔE00 becomes 2.21.
• And the two ends of the sweep are identities rather than approximations. At zero the residual is exactly the unadapted change; at one it is exactly the von Kries residual. That is what makes the range between them a model rather than a cross-fade.

What the degree of adaptation is #

A change of illumination is a matrix, and an adapted observer answers it with a diagonal — three gains, one per channel of whatever basis the eye is taken to work in. The gains are the ratio of the two whites read in that basis, and applying them is what “the observer adapts” means in this collection’s arithmetic.

Applying them fully is a decision. The appearance model does not assume it: it pulls each gain towards one by a factor 1 − D, so a partly adapted observer applies a partly corrected gain and is left with a residue of the change that arrived.

The formula the standard supplies is a function of the surround factor and of the adapting luminance, and it saturates: bright rooms give a D near one, dark ones give less, and it never quite reaches one at any luminance. At the conditions almost every figure here is drawn at, it is 0.941.

What that does to the published numbers #

The mean residual over three representative changes — daylight to D50, daylight to tungsten, and one bounce off a painted wall — is 1.27 ΔE00 at complete adaptation, which is the number this collection reports.

At the standard’s own degree it is 2.21. On the single largest of the three, daylight to tungsten, complete adaptation leaves 1.64 and a total absence of adaptation leaves 23.49; a half-adapted observer is left with 13.25.

The factor of 1.74 is not uniform across the census, and that is the part worth carrying. A change that a gain handles well leaves a small residue, so the incompleteness has little to add to; a change a gain handles badly leaves a large one and the incompleteness adds proportionally more. The correction therefore stretches the census rather than sliding it, and the rows it stretches most are the ones the argument is usually about.

There is a second way to read the same curve and it is the more alarming one. The residual is very nearly linear in the degree over most of the range — from 18.31 at zero to 1.27 at one, in almost equal steps — so an error in the degree translates almost proportionally into an error in the residual. A tenth of a unit of D is worth about 1.7 ΔE00 on this census, which is larger than the entire difference between the best and worst published adaptation transform.

So the parameter nobody chose is worth more than the parameter every paper argues about. That is not a criticism of the papers: choosing a transform is a decision somebody can act on, and the degree of adaptation is a property of the viewer. But it puts the size of the literature’s disagreement in perspective, and the perspective is unflattering.

The factor over the whole census #

Three representative changes give 1.74. The fourteen the census holds give something much milder.

Applying the same correction to every row and taking the mean, the residual moves from 1.407 at complete adaptation to 1.631 at D = 0.941 — a factor of 1.159. So the discount is worth about a sixth on the collection’s headline number, and about three quarters more on the three rows it was quoted over, which is the ordinary difference between a subset chosen to show an effect and a census that was not.

And the last few per cent do most of it #

The response to D is steeply nonlinear, which is what makes the choice of D consequential in a way the range of D does not suggest.

Sweeping it, the mean residual is 1.014 times its complete-adaptation value at D = 0.98, 1.159 at 0.941, 1.444 at 0.90, 2.382 at 0.80, 4.544 at 0.60 and 11.248 at zero.

The first two per cent of incompleteness cost one and a half per cent. The next four cost fourteen. The four after that cost twenty-five. Adaptation’s last few per cent are worth several times what its middle is — because the residual is the part of the change a gain failed to remove, and a gain that is nearly complete leaves almost nothing for the remaining incompleteness to act on.

That has a reading for anybody quoting one of these numbers. The standard’s own formula gives 0.9407 in an average surround at a hundred candelas, 0.8466 in a dim one, 0.7525 in a dark one, and 0.6607 in a dark surround at one candela — and the census mean over that span runs 1.63, 2.70, 4.05 and 5.45, a factor of 3.34 from the brightest of those rooms to the dimmest.

So a figure drawn at complete adaptation and the same figure drawn for a dark room at low light are not one measurement with a caveat between them. They are two answers a factor of nearly four apart, and the caveat is the room.

Why the ranking survives #

The obvious worry is whether this reorders anything, and it does not — for a structural reason rather than by luck.

The incompleteness applies the same fractional pull towards one on every channel of every basis, so at a fixed D it is a fixed operation applied to every transform in the table alike. Which basis leaves an adapted observer with least is a comparison between transforms under one operation, and a common operation cannot reorder a comparison.

What does change is the size of the gaps. At complete adaptation the spread between the best and worst published transform on the census mean is about a factor of two; incompleteness adds a common term to every one of them, which compresses the ratio while leaving the order alone. A comparison stated as a ratio is worse at the standard’s degree than at complete adaptation, and a comparison stated as a difference is better.

That is the kind of consequence worth stating explicitly, because it is invisible in a table and decides how a result should be phrased.

It also explains a small oddity in the literature that this collection had noticed and not accounted for. Published comparisons between adaptation transforms are usually reported as differences in ΔE rather than as ratios, and a reader coming from a modelling background finds that odd — a ratio is dimensionless and travels better. The reason may simply be that a difference is the statistic that does not move when the viewer’s degree of adaptation does, and a field that fits its transforms to human judgements has been working with incomplete adaptation in its data all along.

The two ends are identities #

The sweep would be worth very little if its ends were approximations, and they are not.

At D = 0 the gains are all exactly one, so the observer applies nothing and what is left is exactly the change that arrived — the same number this collection’s machinery reports as the unadapted difference, to every digit. At D = 1 the gains are exactly the ratio of the whites and what is left is exactly the von Kries residual, which is the number the census has always printed.

Both identities are asserted rather than observed. An implementation that pulled the gains towards their mean rather than towards one, or that applied the pull in the wrong space, would produce a curve that looked entirely reasonable and would fail at one end or the other. The check costs two evaluations and is the only thing that distinguishes a model of partial adaptation from a cross-fade between two pictures.

What was computed, and how #

The degree enters as one line: each channel’s gain becomes D·g + (1 − D). That is the standard’s arithmetic and it is applied inside the residual calculation rather than to the result, so every downstream quantity — the per-surface differences, the census means, the comparisons between bases — is recomputed rather than scaled.

The sweep runs eleven points from zero to one on three census rows chosen to span the kinds: two illuminant changes and one surface bounce. It is cheap, and the reason it had never been run is not cost. It had never been run because the parameter did not exist in this collection’s machinery until this phase, which is the more interesting fact: a model without a knob is a model with a hidden assumption in the place the knob would be.

The discount is one of several constants the same audit priced, and its place in that ranking is what says whether it is worth measuring.

The value 0.941 is not typed in anywhere. It is computed from the viewing condition the figures use — an average surround at a hundred candelas per square metre with a twenty per cent background — by the model’s own formula, so a figure drawn at a different condition gets a different degree and the curve says what that costs.

What an assertion can hold on to #

An incomplete adaptation is now a parameter with a check attached rather than a possibility mentioned in prose, and the check is worth describing because of what it is not testing.

It does not test that 0.941 is the right degree for anybody. That is a question about people, and nothing here can settle it. What it tests is that the interpolation is between the two things it says it is between: at zero, exactly the arriving change; at one, exactly the complete gain; and in the middle, something strictly between the two.

The middle condition is the one that catches real mistakes. An implementation that applied the pull in the wrong space — to the tristimulus values rather than to the gains, say — would still be exact at both ends and would bulge in the middle, sometimes below the complete-adaptation value. That would be a model in which adapting less helped, and it would look perfectly plausible on a curve.

Where the model stops #

The formula has no time in it. D is a function of the surround and the adapting luminance and of nothing else — in particular not of how long the observer has been in the room, which this collection has a whole clock for and which the appearance model does not. The two are joined only by hand: the clock produces a degree and the model accepts one.

And it has no cognition in it either. The phrase for a D of one is discounting the illuminant, and the discounting a person actually does depends on whether they can see what the light is — on a familiar object, a visible lamp, a memory of what the paper looked like outdoors. None of that is in a formula whose arguments are two photometric quantities, and the model does not claim otherwise.

The two ocular rows are the awkward ones. A change between the fovea and ten degrees out, or between a lens at twenty and a lens at seventy, is a filter inside the observer — and whether an observer partly discounts their own macular pigment is not a question the formula is about. Those rows are excluded from the sweep for that reason rather than because they behaved badly.

What this does not change #

It is worth being precise about which published claims move and which do not, because the answer is reassuring and would not have been guessable.

No ranking moves, for the reason above. No claim about a mechanism moves: that a corner is worse than twice a wall, that a fluorescent surface is not a matrix at all, that the world’s light commutes and a room’s does not — all of those are structural and survive any common gain.

What moves is every absolute residual, and this collection prints a great many of them. A sentence of the form a gain leaves 1.14 ΔE00 on the census is now a sentence about a complete adaptation, and the number for an observer in the room the figure describes is 1.98.

The repair is not to rewrite the numbers. It is to say which assumption they are under — which is what this essay is for, and what the assertion added alongside it will keep true.

The generalisation #

The pattern is the phase’s, in its clearest form. A parameter that a model has and a machine does not is an assumption, and it is an assumption at whichever end of its range the machinery happens to sit.

Complete adaptation is a particularly seductive place to sit, because it is the natural end: the gain is the ratio of the whites and applying it fully needs no extra number. Every value less than one requires somebody to have decided something, so the code that decides nothing lands on one — and lands there silently, with no comment saying that a choice was made.

The check that finds this class of defect is to go through a model’s published arguments and ask, for each, where in its range the implementation sits and whether anybody chose that. Here the answer for the surround was a table row somebody picked, and for the degree it was the end of the range, because nothing had to be typed.

The practical reading is short and it is worth having in one place, and it is what a reader should take away from every residual in this collection.

A residual quoted at complete adaptation is a floor. It is the best an observer could do if they discounted the light entirely, and every real observer in every real room is above it. When this collection says a gain leaves 1.14 ΔE00 on the census, the honest gloss is at least 1.14, and about 1.98 for somebody in an ordinary office.

A comparison between two transforms is unaffected in order and compressed in ratio. The gap between the best and the worst is the same number of ΔE units at any degree; it is a smaller fraction of the total at a lower degree, because the total is larger.

And a claim about a mechanism is untouched. Whether a corner is worse than a wall, whether a fluorescent sheet can be a matrix at all, whether a particular basis diagonalises daylight — none of those depend on how completely anybody adapts, because all of them are comparisons under one gain.

Who found it, and when #

Incomplete adaptation is old. That an observer in a coloured light does not fully discount it was established through the middle of the twentieth century, and the degree of adaptation appears as an explicit parameter in the Hunt model and in every appearance model descended from it; the particular functional form here comes down through CIECAM97s and CIECAM02 to CIECAM16, where it is an exponential in the adapting luminance scaled by the surround factor.

What is unusual is only that this collection had the model and did not use that part of it. The appearance machinery here computes D correctly and has since it was built — it is used inside every appearance calculation. The adaptation machinery is a separate file with its own arithmetic, and the two were never joined, so one half of the site knew about incomplete adaptation and the other half did not.

That is a mundane way for an assumption to survive, and probably the most common one: two implementations of neighbouring ideas, each correct, with a parameter that exists in one of them.

Where the ladder goes next #

Two of the appearance model’s declared inputs have now been swept, and both turn out to be places where a figure was placed rather than measured. Neither changes a ranking.

The natural next question is what happens to a claim when the thing that moves is not a parameter of a model but the shape of the surface being modelled — which is what a curvature taken away from an optimum turns out to describe, and where the answer is that the description is of a neighbourhood nobody visits.