A room with two lights has no white

A desk sits beside a window with a lamp on it. Both are on. The light falling on the paper is a mixture of the two, and that much is simple: spectra add, so the mixture is a spectrum, and every colorimetric quantity computed from it is exactly as well defined as it would be under either light alone.

The appearance model wants something else. A viewing condition is an argument established what CIECAM16 needs before it will return an appearance — an adapting white, a luminance, a background, a surround — and the first of those is not a property of the light on the paper. It is a claim about the observer: the white they have adapted to. In a room with one light that claim is uncontroversial enough to go unstated. In this room there is no fact for it to be about.

The model has no way to say “both”. It takes one white, it will accept any of them, and it answers confidently at every setting.

Mixing the lights is arithmetic; mixing the adaptation is not #

The two lights are the standard pair: CIE illuminant A for the lamp, D65 for the window. They are the extreme case rather than a typical one, which is deliberate — the question here is how much room the choice has, and the widest pair of ordinary lights is where that shows.

The mixture is the pointwise sum, weighted by how much of each light reaches the surface. That operation has no ambiguity in it and no observer in it: at every wavelength there is a number of watts arriving, and adding two numbers is not a modelling decision. The surfaces then multiply that spectrum by their reflectance and the tristimulus values follow.

The adapting white is a different kind of object. It is not the light arriving; it is the light the visual system has taken as its reference, and in a room with two lights the visual system is doing something that no single reference describes. It may have settled somewhere between them. It may be tracking the dominant one. It may be doing something spatially varied, so that the paper under the lamp and the wall by the window are referred to different whites at the same moment — which is the one answer the model structurally cannot represent, since the white is a property of the viewing condition and the viewing condition is one argument for the whole computation.

Two of those possibilities have already been given their own essays, and both make the single white worse rather than better. A patch is not a scene is about the model’s stimulus being one patch while the eye’s evidence is a whole field, which is precisely the mechanism by which a two-lit room could be referred to two whites at once; and a viewing condition is a moment is about the white having a time course, which matters here because an observer looking from the paper to the window and back is not adapted to one thing for more than a few seconds. A room with two lights has both problems at once, and the model has one slot.

So the modeller has to choose, and the choice is not recoverable from any measurement of the room. The dial in the hero figure is that choice drawn as a continuum: at each setting the model is being told a different story about the observer, and it returns a different set of appearances for light that never changed.

Every surface moves, and the least of them moves by twenty-six units #

The two ends of the dial are the two answers a modeller is most likely to give when told “there are two lights” — adopt one of them. Setting them against each other gives the width of the disagreement without any appeal to what the correct answer might be.

Sixty units is not a refinement. The whole of CAM16-UCS spans a few hundred units across the colours a display can make, and the tolerances priced elsewhere are written in ones — a tolerance has no light level works in pairs separated by exactly one unit of ΔE00, and finds that the room moves that unit around by a factor of several. Here the room is fixed and the model’s account of the observer is moved instead, and the result is two orders of magnitude larger.

The spread across surfaces is worth as much as the median. The surface that moves least — a broad band centred at 470 nanometres, a wide bluish absorption — moves 26.8 units, and the ones that move most are narrow-band surfaces at the two ends of the spectrum. That pattern is a hint about mechanism: a surface whose reflectance is spread out samples the whole of both lights and is dragged by both, while a narrow one samples a region where the two lights differ most and takes the full difference. It is also the first sign that the disagreement is not one uniform effect.

No mixture of the two makes the choice cheap #

A reasonable hope at this point is that the half-and-half room is a worst case, and that a room dominated by one light collapses back to a single white. It does not, and the way it fails is asymmetric.

A room that is three-quarters daylight is cheap to be right about and expensive to be wrong about: adopt the window and the penalty against its own mixture is 11.8 units, adopt the lamp and it is 40.5. A room that is three-quarters lamplight is the reverse and less extreme. The asymmetry comes from where the mixture’s chromaticity actually sits, which is not halfway between the two lights’ chromaticities — mixing equal power moves the white most of the way towards the lamp, because the lamp carries far more of its power in the long wavelengths that the mixture inherits. The chromaticity of the even mixture, 0.374 and 0.365, is nearer the lamp’s 0.448 and 0.407 than the window’s 0.313 and 0.329.

The practical reading is that dominance helps only if the dominant light is the one adopted, and that “mostly daylight, so treat it as daylight” is a defensible rule while “there is a bit of daylight, so split the difference” is not a rule at all. What no room in the figure offers is a setting where both answers are acceptable. The smallest penalty available to a modeller who must adopt one of the two real lights, across all three rooms, is 11.8 units — still an order of magnitude past any tolerance a delivery would be held to.

Most of it is a cast, and the cast is not all of it #

There is one more line of defence, and it is the strongest one. If changing the adopted white moves every surface the same way, the disagreement is a global colour cast, and a cast is exactly the kind of error that a downstream white balance, a profile, or the algorithms that guess the light are built to remove. It would then be an error of bookkeeping rather than of prediction.

It largely is a cast. The mean displacement between the window reading and the lamp reading, taken over the twelve surfaces, is 0.9 units in lightness, 16.3 in the red–green coordinate and 53.1 in the yellow–blue one: a single vector 55.6 units long, pointing almost exactly along yellow–blue, against a median per-surface move of 60.4. So somewhere between eighty and ninety per cent of the distance is one shift applied to everything.

Ninety per cent removed still leaves 9.9 units on the median surface and 30.2 on the worst, and the worst case is instructive. The broad 470-nanometre surface was the one that moved least in absolute terms, 26.8 units against a median of 60.4. Removing the population’s average shift from it does not shrink that move; it grows it to 30.2, because the surface was moving in a different direction from everyone else. A single correction fitted to the population is, for that surface, an additional error.

This is the ordinary shape of an error that is nearly but not quite a transform. The subtraction performed here is also the most generous possible version of the correction: it is the mean displacement computed from the answer, which is the best a single translation could possibly do and better than any correction fitted without knowing the outcome could do. The residual is therefore a floor. The honest statement is that a room with two lights carries an irreducible uncertainty of around ten units of appearance, per surface, with nothing available to remove it, arising entirely from a question the room does not answer.

The one knob that could have said so answers a different question #

CIECAM16 does have a parameter for how completely the observer has adapted. The degree of adaptation, D, runs from zero — no chromatic adaptation, the stimulus taken raw — to one, complete adaptation to the stated white. Four ways to move a white point sets out where it sits among the other mechanisms, and a discount nobody measured is about how little of it was ever measured on people. It is, on the face of it, exactly the place where a room with two lights would be declared: partly adapted, to something in between.

It is not that place. The formula takes the adapting luminance and the surround and nothing else; the white it is the degree of adaptation to is not among its arguments. So the model returns 0.9407 whether the observer is said to have adapted to the lamp, to the window, or to any mixture, and it would return 0.9407 if the two lights were a sodium lamp and a monitor.

That is not an oversight so much as a statement about what D was built to represent. It answers how strongly does the eye discount the illuminant at this light level, which is a real question with a measured answer, and its variation with luminance in the left panel is that answer. What it does not carry is any notion of confidence in the white, and those are different quantities. The model as written has a parameter for the strength of adaptation and none for the uncertainty of its target, so a modeller who knows perfectly well that the room is ambiguous has nowhere to put that knowledge. The output is a single set of appearances with no width on it, and the width is the whole of what this room is about.

What was computed, and how #

The scene light is a weighted sum of the tabulated CIE illuminants A and D65, at 5-nanometre resolution over 380 to 780 nanometres. Twelve surfaces are taken from the audit’s own reflectance family by striding across it, so that all seven band centres are represented rather than the two that the first twelve entries would have given. Each surface’s reflectance multiplies the scene light, the product is integrated against the CIE 1931 2° matching functions, and the resulting tristimulus values go through CIECAM16 at an adapting luminance of 100 candelas a square metre in an average surround, with the background left at the family’s default.

Only one argument moves: the adapting white handed to the viewing condition, which is the white of the mixture at a stated share of daylight. Distances are Euclidean in CAM16-UCS, which is the space every comparison of this kind is made in here and whose unit is discussed in a model judged in another model’s unit. The cast is the arithmetic mean of the twelve displacement vectors; the residual is the length of each displacement after that mean is subtracted.

Where the measurement stops #

The surfaces are the audit’s pale family, running from L* 57 to 93. A darker set would move less in absolute terms simply because it has less signal, and the ratio to a tolerance is what matters rather than the raw number. The count is twelve, so “the median surface” is the sixth of twelve and not a robust population statistic.

The room is uniform: one mixture everywhere, with no geometry, no gradient from window to desk and no surfaces visible under only one of the two lights. Every one of those makes the real case worse rather than better, since they give the visual system more evidence that there are two lights and the model no more places to put it.

The comparison treats “adapting to the mixture” as adapting to the mixture’s own white, which is itself an assumption rather than a measurement — it is the reading the dial is zeroed at, and it is chosen because it is the natural default, not because anyone has shown that observers in such rooms adopt it. If observers in fact adopt something else, the zero of the dial moves and every number here moves with it; what does not move is the width, which is the finding.

And the size of the effect does depend on the light level, though not much: at 5 candelas a square metre the two ends of the dial read 28.1 and 18.7 units against 32.6 and 24.5 at 100. Lower light means less complete adaptation, which means less of the white’s influence, which means a smaller penalty for choosing the wrong one. Dimming the room shrinks the problem by about a sixth and does not remove it.

Still open: what observers in such a room actually adopt #

The experiment is an asymmetric matching one and it is not hard to describe. Put observers in a booth lit by two lights in a stated proportion, have them match a surface there against a reference in a second booth under a single light, and solve for the white that makes CIECAM16 reproduce the match. That white is the adopted white, measured rather than assumed, and repeating it at several proportions traces the curve the model needs and does not have.

The prediction worth testing is specific: that the recovered white sits nearer the dominant light than the power-weighted mixture does — that adaptation is winner-takes-more rather than proportional. If that is right, the mostly-daylight room is even cheaper to be right about than the figure above suggests and the even mixture is the hardest case of all. If it is wrong and the recovered white tracks the mixture, then the dial’s zero is correct and the remaining question is only how much spread there is between observers, which is a second number the model also has no slot for.

Either result would replace a choice with a measurement in one of the two places where the model was found silent. The other place — the absence of any width on the output — would still be open, because that is a question about what a model returns rather than about what people do.

An argument with no value in it #

The habit is about what to do when a model needs an input the situation does not determine.

The temptation is to supply the most defensible value and carry on, because the model will not run otherwise and a number is needed. That produces an answer with a hidden choice inside it, and the choice is invisible in the output: a CAM16-UCS coordinate computed under the lamp looks exactly like one computed under the window, and nothing about either says that the other exists. Reporting it as the appearance is not an approximation, it is a category mistake — the model was asked a question that had no answer and gave one anyway.

The alternative is to run the model across the whole range the input could honestly take and report the width. That is what the dial is: not a sensitivity analysis added afterwards for rigour, but the actual output, since the input’s range is what is known about the room. A width of sixty units says something true and useful — that this room does not support an appearance prediction at the precision the notation implies. A single number says something false at four significant figures.

This is not an argument that the model is wrong about rooms with one light, where constancy is the default and the white is uncontroversial enough that the argument passes unnoticed. It is an argument that the notation hides the difference between the two cases. The call reads the same either way, the return value has the same type, and only the person who set up the room knows which of the two situations produced it — which is a reason to record the width beside the value rather than a reason to distrust the model.

The failure mode is to let the model’s signature decide what the situation contains. A function that takes one white will be given one white, and the room’s second light disappears at the call site rather than in the reasoning.