A brighter white still looks white

Everything in the essays around this one has been colorimetry: integrals, distances and bounds. Colorimetry says when two stimuli match and it deliberately says nothing about what either looks like — that boundary is this collection’s fifth invariant and it has been held since the collection’s first essays.

A brightened sheet is a case where the boundary is worth crossing carefully, because the colorimetric number and the word people use for the result disagree.

The claim #

In an appearance model the brightened sheets are not near-neutral. They carry seven to eleven units of chroma at a blue-violet hue, and how much depends on the room by nearly a factor of two — and none of that explains why the result is called white.

• With the ultraviolet removed the sheets are essentially achromatic: chroma between 0.3 and 1.7, which is as neutral as a real material gets.
• With it, the loaded stocks carry chroma 7.5 to 11.4 at a hue angle of about 295 degrees, which is blue-violet rather than blue.
• The chroma falls by 37 to 49 per cent between an average surround and a dark one, so the same sheet in a dim room is measurably less blue to look at.
• The lightness barely moves: J changes by 0.3 between the two conditions, which is nothing.
• And the model has no term for preference. It reports a chromatic stimulus and reports it correctly; the fact that observers rank that stimulus as whiter is a judgement the model does not represent.

What the appearance model is being asked #

The distinction that makes this essay possible is between two questions that colorimetry conflates.

A colorimetric number is a distance from a reference white, computed with no viewing condition in it. CIELAB’s b* of −9.4 for a heavily brightened stock is the sheet’s displacement from the perfect diffuser under the same light, and it is a correct and complete answer to “how far from neutral is this stimulus”.

An appearance correlate is a prediction about a percept, and it needs the room: an adapting white, an adapting luminance, a background and a surround. A viewing condition is an argument rather than a caveat, and the model will not compute without one.

The adapting white here is the light’s own — so the question being asked is what a person adapted to the room makes of the sheet, which is the situation a reader holding a piece of paper is in. It is not the same question as what they make of it lying beside a reference white, which is the situation an inspector is in, and the two have different answers.

What the model says #

The unbrightened control is nearly achromatic: chroma 1.71 in an average surround, hue 108 degrees. That is a sheet an observer would call neutral without hesitation, and it is what the model should say.

Every brightened stock, with the ultraviolet removed, is also nearly achromatic — chroma between 0.33 and 1.71. The bases in this collection’s stock set are formulated to be near-neutral under the measuring illuminant, which is what real paper bases aim for, so the substrate contributes essentially no colour.

With the ultraviolet restored the loaded stocks move to chroma 9.54 for a coated press stock and 11.35 for a heavily brightened one, at hue angles clustered around 295 degrees. That is not a rounding: eleven units of chroma is a stimulus an observer asked to name a colour would call pale blue or pale violet rather than white.

So the whole of the colour is the fluorescence, and by a large factor — chroma with the excitation is more than three times chroma without it on every loaded stock, and more than twenty times on some.

What the room does #

The surround changes the answer by more than any property of the sheet does.

Between an average surround — a well-lit room, the standard condition for reflection prints — and a dark one, chroma falls from 9.54 to 5.56 on the coated stock and from 11.35 to 7.09 on the heavily brightened one. That is a fall of 37 to 49 per cent across the set, produced by changing nothing about the sheet or the light falling on it.

The mechanism is in the model’s constants rather than in anything mysterious. A darker surround lowers the model’s chromatic induction factor and its exponential nonlinearity, both of which reduce colourfulness — which is the standard’s encoding of the observation that a picture in a dark room looks less colourful than the same picture in a lit one.

That is a third place the excitation question is decided by the environment. The lamp decides how much light the fluorophore receives. The glazing decides how much of that reaches it. And the surround decides how much of the resulting chroma is perceived. All three multiply, and only the first appears on anybody’s specification.

Nine and eleven are not two readings of one quantity #

The dek and the opening set a colorimetric figure against an appearance one — nine units of b* against eleven units of chroma — in a way that invites the reading that the appearance model finds more colour than the colorimetry did. It cannot, because the two numbers are in different spaces with different scalings and there is no exchange rate between them.

The comparison that does hold runs the other way, and it is the essay’s real evidence: the same appearance model reports 11.35 with the excitation and 0.33 without it. Two readings of one quantity in one space, a factor of thirty-four apart, and that is what establishes that the colour is the fluorescence.

The cross-space pair is worth keeping for a different reason than the one it is offered for. CIELAB chroma is at least the magnitude of b*, so the sheet’s chroma in CIELAB is 9.4 or a little more, against CAM16’s 11.35 — the two spaces agree to within about 21 per cent. That is a mildly reassuring coincidence rather than a finding, since nothing requires two differently scaled chroma correlates to land near each other, and it is the honest thing to say about the pair: not that the appearance model saw more, but that on this stimulus the two apparatus happen to report similar magnitudes.

The surround acts on chroma a hundred times harder than on lightness #

The two effects of the surround are reported next to each other and their sizes are not compared, which understates the asymmetry considerably.

Chroma falls by 41.7 per cent on the coated stock between an average surround and a dark one, and 37.5 on the heavily brightened one. Lightness changes by 0.3 units of J — and a white sheet sits near J = 95, so that is 0.3 per cent.

The surround is acting on the chromatic correlate about a hundred and thirty times harder than on the achromatic one. The lightness barely moves is true and it undersells what the pair says: the room is not a general dimming that affects everything a little, it is a term that reaches one correlate and very nearly not the other.

That has a practical edge the essay does not draw. A sheet carried from a lit room to a dark one does not become a dimmer version of itself — it becomes the same lightness and two thirds of the colour, which is a change of hue-saturation at constant lightness. Anybody judging whiteness in a dim room is judging a stimulus that has been desaturated without being darkened, which is a specific and unusual transformation and is not one an observer has much practice at discounting.

What the three environmental terms multiply to #

The essay names three places the environment decides the answer — the lamp, the glazing and the surround — and observes that all three multiply without saying to what.

Two of the three can be sized from figures this collection already publishes. The glazing spans a factor of 16.2 in the excitation a sheet receives, from sixty-five per cent of its band unglazed to four behind a conservation filter. The surround spans 1.72 in chroma. If chroma tracks the excitation supplied — which is the ray argument the fluorescent-metamer essay establishes, and which holds while the luminescent term is small against the reflected one — the two multiply to a factor of 28.

So the same sheet, under the same lamp, spans nearly thirty to one in perceived chroma across the range of rooms it might be in. Outdoors in a bright room it is a visibly blue-violet stimulus at eleven units; behind conservation glazing in a dark room it is under half a unit, which is the substrate’s own colour and nothing else.

That number is an estimate with two assumptions in it and it is the right order to have. It is also the answer to the essay’s own closing question about what is conditional on an environment nobody records: everything, by a factor of nearly thirty, before the lamp is varied at all.

The thing the model cannot say #

Every number above is a prediction about appearance, and appearance is not preference. A person shown two sheets and asked which is whiter is not being asked which is closer to neutral — white is a region with preferences inside it.

That is the gap, and it is worth stating precisely rather than gesturing at. The whiteness formula is a fitted ranking function: it was built by asking observers to order white samples and finding a linear expression that reproduces the order. It rewards blue displacement because observers reward blue displacement. Nothing in it predicts appearance and nothing in an appearance model predicts it.

So the honest account of a brightened sheet is three separate statements that do not reduce to one another.

Colorimetrically it is nine units of b* from the illuminant’s white, which is a measurement.

In appearance it is a chromatic stimulus of about eleven units at a blue-violet hue, whose size depends on the room by nearly a factor of two, which is a prediction about a percept.

And in preference it is whiter, which is an empirical fact about human judgement with no mechanism attached in either model.

The third does not follow from the second and is not contradicted by it. A person can perceive a stimulus as slightly blue and rank it as whiter, and there is nothing incoherent about that — whiter is not a synonym for closer to achromatic, and the whole optical brightening industry exists on the difference.

The validity box is a claim about a formula rather than about a sheet, so it is worth confirming that the ten-degree observer puts the same stocks in the same relation to it.

Why the tint bound exists #

The whiteness formula’s tint limit of ±3 is the industry’s fence against exactly this, and reading it in the light of the chroma numbers makes its purpose clearer.

Loading more brightener moves the sample further along a fixed direction. The whiteness formula keeps rewarding it — W rises monotonically — but past a point observers stop calling the result white and start calling it blue, and the ranking the formula was fitted to no longer holds.

The tint bound is where the fit stops being trusted, marked on the one axis along which the sample can leave the fitted region. It is not a perceptual threshold and it is not derived from anything; it is a fence around the data the formula was built on.

The stocks here sit between −0.3 and −1.8 in tint, so all of them are inside it — and the most heavily loaded is closest, at −1.8, with nine points of whiteness headroom before it meets its own upper bound as well. The fence is doing its job at exactly the loadings the market uses, which is what a well-placed fence looks like and is also why nobody notices it is there.

What the scale reports is not fixed either, and the place the sheet is looked at moves it as much as the brightener does.

How blue the sheet reads is an appearance question rather than a colorimetric one, and the appearance model can be handed the ten-degree observer too.

The comparison the model is not being asked about #

One arrangement is left out of everything above and it is the one that matters commercially, so it is worth naming what would be needed to model it.

A buyer does not look at a sheet adapted to the room and report a colour. They look at two sheets side by side and say which is whiter. That is a different task in a way an appearance model has no term for: the observer is adapted to a field containing both samples, the comparison is local rather than absolute, and the judgement is a preference rather than a description.

What the model can say about that arrangement is narrow. The adapting white is now some average of the two samples and the surround, so both move towards neutral and the difference between them changes very little — which is consistent with the observation that a whiteness ranking is stable across viewing conditions even though the absolute appearance is not.

That stability is the reason a fitted ranking function works at all. A preference scale on pairs does not need to predict appearance; it needs to predict an ordering, and an ordering survives a good deal of what an appearance model is built to track. The whiteness formula is therefore doing a much easier job than it looks like it is doing, and its coefficients being fitted rather than derived is appropriate rather than a compromise.

What neither apparatus supplies is the step between them: nothing here says how a difference in chroma becomes a difference in preference, and no model on this site can.

What was computed, and how #

Each stock is measured under both conditions and put through CAM16 with the adapting white set to that condition’s own perfect diffuser, an adapting luminance of 100 candelas per square metre, a background of 20 and each of the three tabulated surrounds.

Setting the adapting white to the light’s own is the choice that decides everything, and the alternative was tried first: with the white left at the model’s D65 default, every sheet under a D50 source reads as strongly yellow, chroma 14, and the brightener appears to reduce chroma by neutralising it. That is a correct answer to a different question — what a person adapted to daylight makes of a sheet under a warmer lamp — and it is not what a person holding the paper is doing.

The chroma drop across surrounds is reported as a fraction rather than as an absolute, because the absolute depends on the loading and the fraction does not: it runs 37 to 49 per cent across four loaded stocks.

The assertion has two halves. The sheets must be near-neutral with the excitation removed and carry real chroma with it — which establishes that the colour is the fluorescence and not the substrate. And the chroma must fall by more than a quarter between an average surround and a dark one, which establishes that the room is a term rather than a rounding.

Where the model stops #

CAM16 has no time in it. A person walking from a window into a room adapts over tens of seconds, and every number here is for a fully adapted observer. The clocks a room runs on are a separate apparatus and they are not joined to this calculation.

The background is a single number. A sheet of paper held against a dark desk and the same sheet on a white table are different stimuli in the model, and the value of 20 used here is the standard’s default rather than any particular scene.

And preference is genuinely absent rather than approximated. Nothing in this essay predicts which of two sheets a buyer prefers; the whiteness formula does that, empirically, and it is not an appearance model.

Who found it, and when #

CIECAM02 and its successor CAM16 date from 2002 and 2016 and are the standard apparatus for this kind of question. That surround affects colourfulness is one of the observations the models were built to encode, and it goes back to Hunt’s work from the 1950s onwards.

That a brightened sheet is chromatic and is nonetheless preferred is the paper trade’s daily experience and is why whiteness and tint are reported as two numbers rather than one. The trade language for it — a sheet being “blue white” or “warm white” as a deliberate product choice — is a recognition that white is a region with preferences in it rather than a point.

What is worth putting together is the three statements above in one place, because they are usually made in three different literatures and each reads as a contradiction of the others.

The generalisation #

The pattern is a preference dimension mistaken for a perceptual one.

Whiter feels like it should mean closer to achromatic, and it does not. The judgement people make is a preference among near-white stimuli, it has a direction in colour space, and that direction is not towards the neutral axis. Once that is admitted, the whiteness formula stops looking strange — it is a preference model, correctly built, doing exactly what it says.

The diagnostic worth carrying is to ask whether a scale’s zero is a physical state or a preferred one. A colour difference has a physical zero: two stimuli are identical. Whiteness has no such point — a perfect diffuser is not the whitest thing on the scale, and the samples that score highest are the ones furthest from it in a particular direction. A scale whose optimum is not its physical reference is a preference scale, whatever it is called, and reading it as a measurement is the error this essay exists to name.

Where the ladder goes next #

If the room decides how much chroma the sheet shows, and the lamp decides how much there is to show, then the whole of the arithmetic in these essays is conditional on an environment nobody records — which is the closing statement of the window being part of the light and of what an instrument brings with it, arriving at the perceptual end.

The other direction is the sheet’s own clock. The chroma is the brightener’s and the brightener is being used up, so a sheet is walking back towards the neutral its base already is, at a rate the room sets.