A white that is not a reflectance

Most limits in colour are limits of an apparatus. A display’s gamut is what its primaries can mix; a press’s is what its inks can do; both move when somebody builds a better device.

There is one that does not. The object-colour solid is the set of tristimulus values any reflecting surface can produce under a given illuminant, with no assumption about pigments, processes or chemistry in it — only that a reflectance lies between zero and one. Nothing anybody invents puts a surface outside it.

A sheet of coated paper, measured by a standard method, is outside it.

Where the sheet is looked at moves the reading as much as which sheet it is, and both are outside anything a reflectance could carry.

The claim #

A brightened sheet is outside the hardest bound colorimetry has, and being outside it is not a statement about a gamut — it is a statement that the sample is not a reflecting surface.

• The bound is a support function, not a hull: for any direction d, the largest d · XYZ any reflectance can reach is computable exactly, and the maximiser is an optimal colour.
• A perfect diffuser sits exactly on the boundary, at a margin of 1 × 10⁻¹⁵, which is the check that the arithmetic is right.
• A coated press stock exceeds the bound by 1.5 per cent under an ultraviolet-included measurement condition and a heavily brightened one by 4.0.
• With the ultraviolet removed, both are 9 to 10 per cent inside it. Same sheet, same machinery, same observer.
• And the effect is confined to the brightest sheets, because the solid is narrow near the white point and wide lower down — the most heavily brightened sample here stays inside, because its base is darker.

The bound, and why the usual way of drawing it is the wrong tool #

Schrödinger settled the shape of the object-colour solid in 1920 and MacAdam computed its boundary in 1935. The result is that the extreme points are the optimal colours: reflectances taking only the values zero and one, with at most two transitions. Everything else is inside.

The usual way to draw it — and the way this collection has drawn it — is to sweep pairs of transition wavelengths, keep the ones whose luminance factor lands near a target, and take a convex hull of the chromaticities. That produces a good picture and it is the wrong instrument for asking whether a particular sample is outside.

A hull built from samples is an inner approximation. The true boundary is smooth and the sampled points sit on it; the hull of finitely many boundary points lies inside the true boundary. So a sample landing outside the hull might be inside the solid, and the answer would depend on the sampling density rather than on the sample. An early version of this measurement did exactly that and gave three different answers at three grid steps.

The support function settles it with no sampling at all. For any direction d = (α, β, γ),

d · XYZ = ∫ ρ(λ) E(λ) [α x̄(λ) + β ȳ(λ) + γ z̄(λ)] dλ

and since 0 ≤ ρ ≤ 1, the largest this can be is obtained by setting ρ = 1 wherever the bracket is positive and ρ = 0 elsewhere:

h(d) = ∫ E(λ) · max(0, α x̄ + β ȳ + γ z̄) dλ

That is an exact maximum over all reflectances, computed in one integral. The maximiser is a function that is zero and one with transitions where the bracket changes sign — which is Schrödinger’s result falling out of the arithmetic rather than being quoted.

So a sample is proved outside as soon as one direction is found with d · XYZ > h(d). Searching directions can only fail to find one; it cannot produce a false positive.

The control that makes it a measurement #

Two controls, and the essay would not be worth writing without either.

The perfect diffuser must sit exactly on the boundary. A reflectance of one everywhere is an optimal colour with no transitions, so its margin must be zero in the direction where the bracket is positive throughout — not small, zero. It comes out at 1.1 × 10⁻¹⁵, which is floating-point noise, and getting there took two attempts: a normalisation mismatch between the sample’s scaling and the support function’s put every sample, including the unbrightened control, three hundred per cent outside on the first run.

Removing the ultraviolet must bring the sheets back inside. This is the control that separates “the sample is outside” from “the arithmetic is wrong”. The same sheets, the same machinery, the same observer, with only the lamp’s short-wave content changed, sit 9 to 10 per cent inside the bound. Nothing about the sample changed; what changed is whether the sample was fluorescing.

Why only the brightest sheets #

The six stocks do not sort by brightener loading, and the reason is geometric.

The solid is a lens-shaped body: widest at intermediate luminance factors and pinched to a point at Y = 0 and at Y = 1. Near the top there is very little room — a surface at Y = 0.9 must be very nearly neutral, because a reflectance that is 0.9 on average cannot be far from flat.

So the margin is a race between two quantities. A brightener pushes the sample’s chromaticity away from neutral, which needs room — and what leaves the sheet is not all of it reflected. A brighter base pushes its luminance up, which removes room.

The laundered shirt has the heaviest brightener loading in the set and the darkest base, at Y = 0.82, and it stays 5.2 per cent inside. The coated press stock has two thirds of the loading and a base at Y = 0.90, and it goes 1.5 per cent outside. The condition for leaving the solid is a combination, and no single property of the sheet predicts it.

That is the useful form of the result for anybody choosing a stock: a bright base and a brightener together produce something no reflectance can be, and either alone does not.

One and a half per cent is a large effect at the end of a short runway #

The headline margin invites the reading that this is a marginal violation of a bound the sheet very nearly satisfies. The two measurement conditions say otherwise, and the arithmetic is one subtraction.

The same coated stock is 9 to 10 per cent inside the bound with the ultraviolet removed and 1.5 per cent outside with it included. The brightener is therefore worth 10.5 to 11.5 points of margin, and the heavily brightened sheet’s is worth 13 to 14. The unbrightened control has 14 points of headroom to begin with.

So the effect is not small; the runway is short. A brightener consumes between three quarters and all of the room an ordinary sheet has, and whether the sample ends up inside or outside is decided by what is left after that — which is a matter of a point or two, not of a mechanism being weak.

That is the right frame for the closing observation about the caveat. The limits do not apply to fluorescent samples suggests a sample somewhere else entirely, and 1.5 per cent suggests a sample that has barely moved. Neither is right: the sample has moved eleven points on a fourteen-point runway, and 1.5 is what is left over at the end of it.

What the two stocks say about the trade #

The section on why only the brightest sheets cross gives the mechanism — a bright base removes room and a brightener needs room — and stops short of a rate. Two of the stocks are enough to bound one.

The coated press stock sits at a base luminance factor of 0.90 with two thirds of the shirt’s loading, and is 1.5 per cent outside. The laundered shirt sits at 0.82 with the full loading, and is 5.2 per cent inside. Between them the base rises by 0.08, the loading falls by a third, and the margin moves by 6.7 points in the direction of leaving the solid.

The loading fell, so whatever the loading is worth, it worked against that movement. The base term therefore has to account for at least the whole 6.7:

So the base is worth at least 0.84 margin points per hundredth of luminance factor, and plausibly more. That converts the essay’s qualitative rule into something a stock buyer can use: a sheet 1.5 per cent outside the solid comes back inside if its base drops by 1.8 points of luminance factor or less, with the brightener untouched — a difference between two coated papers that no one would describe as a change of grade.

It also explains why the six stocks do not sort by loading, more sharply than the geometric argument alone does. Across this set the base varies by eight points of luminance factor and the loading by a factor of about one and a half; at 0.84 points of margin per hundredth of Y, the base range is worth at least 6.7 margin points and the loading range is worth something smaller. The variable that sorts the stocks is the one that is not the additive, which is the opposite of what the trade’s own vocabulary — brightener loading, whiteness grade — is organised around.

The caveat that goes with it: two samples cannot separate two effects, and the table above is a bound rather than a fit. What is bounded is the direction and the minimum size, and both are enough for the practical statement. Separating the two properly would need the six stocks’ base luminances and loadings printed beside their margins, which is one column this collection has and has not put in the same table.

What being outside actually means #

The statement has a precise content and it is worth separating from three things it might be confused with.

It is not a gamut statement. A gamut is what a device can reach. This is what a surface can be, and the two are unrelated: an sRGB display cannot reach most of the object-colour solid, and no surface can reach outside it.

It is not the same as a radiance factor above one, although both are true of the same sheets. β_T above one is a statement about a spectrum: more light leaves at some wavelength than arrived. Being outside the solid is a statement about three numbers, and the two do not imply each other — the laundered shirt exceeds one in β and stays inside the solid.

And it is not “brighter than white”, in the sense the luminance weighting settles. The sheets here have luminance factors of 0.90 against a perfect diffuser’s 1.0; they are dimmer than a perfect white and outside the solid at the same time, because the solid is a constraint on the combination of luminance and chromaticity rather than on luminance alone.

What it does mean is that any machinery whose model of a sample is a reflectance has no representation for this sheet. Not an inaccurate one — none. A profile built on reflectances, a rendering built on reflectances, a gamut mapping built on reflectances: each of them will produce a number, and the number corresponds to no surface.

How much of the figure belongs to the paper rather than to the instrument’s lamp is the split the whole argument rests on, and it survives the change of observer.

What else is outside, and what is not #

The bound rules out one class of sample and it is worth knowing which neighbours it does not rule out, because the exceptions are for different reasons and get confused.

A retroreflector returns light preferentially towards the source, so it can be far brighter than a diffuser in one direction and darker in every other. That is a geometric exception rather than a spectral one: the bound is stated for a radiance factor measured against a perfect diffuser, and a retroreflector’s radiance factor is a function of angle.

A metallic or pearlescent ink is the same kind of exception with an added spectral tilt, since its interface reflection carries the source’s own spectrum rather than a pigment’s. Measured at one geometry it sits inside the bound; measured at another it does not, and neither measurement is wrong.

A translucent sheet on a black backing is outside the framework rather than outside the bound, because its reflectance depends on what is behind it and a reflectance is supposed to be a property of the sample.

And a saturated pigment is inside, however vivid, which is the point most often got wrong. A brilliant cadmium red is nowhere near the boundary at its own luminance; the samples that press hardest on the object-colour solid are near-neutral and very light, because that is where the solid is pinched. The set of surfaces this bound excludes is not the colourful ones — it is the ones that are bright and slightly coloured at the same time, which describes white paper and almost nothing else.

What was computed, and how #

The support function is one integral per direction: the illuminant times the positive part of a linear combination of the three matching functions, normalised so that a perfect diffuser gives Y = 100. The illuminant is the visible part of the measurement condition’s source, because a reflectance only acts on the visible band and the bound is a bound on reflectances.

The direction search is a sweep of 90 by 180 points on the sphere followed by six halvings of a local step. The sweep can only understate the margin, since a direction it misses would give a larger one, so a positive answer is a proof and the refinement only sharpens the number.

The samples are this collection’s bispectral stocks measured under the two conditions, each normalised to its own condition’s perfect diffuser — which is what makes the two answers comparable and is where the first attempt went wrong.

Two assertions, and the first is the one that matters. The perfect diffuser must land within 10⁻⁹ of the boundary, and at least one stock must exceed the bound under the ultraviolet-included condition while none does under the excluded one. The first is a test of the arithmetic and the second is the finding; running the second without the first would be reporting a normalisation error as a discovery, which is exactly what happened on the first run.

Where the model stops #

The bound is for a stated illuminant. The solid’s shape depends on the light: under D65, with its higher short-wave content, the same sheets are further outside; under a tungsten lamp, which supplies a tenth of the excitation, they are inside — the same dependence the window introduces. The result is not “this sheet is outside the object-colour solid” but “this sheet is outside it under this light”, which is the same qualification everything around it carries.

The samples are constructed. The exact margins of 1.5 and 4.0 per cent belong to this collection’s stocks. What does not depend on the construction is the sign, the control, and the observation that the effect appears at high base luminance rather than at high loading.

And the solid is defined for non-fluorescent, non-scattering, opaque samples. A translucent sheet on a black backing, a retroreflector and a metallic ink all sit outside the framework for reasons that have nothing to do with this one.

Who found it, and when #

Schrödinger’s 1920 paper established that the extreme points of the object-colour solid are the two-transition reflectances. MacAdam computed the boundary in 1935 and it carries his name in most of the literature. The support-function formulation is standard convex analysis applied to the same result and appears in the colour-science literature as the “optimal colour” derivation rather than under that name.

That fluorescent samples lie outside the limits is stated in the metrology literature as a caveat — the limits are for non-fluorescent samples — rather than as a computation. It is the correct caveat and it is the same shape as the caveat these essays started by examining: true, in the right direction, and without a number.

The number is worth having because 1.5 per cent is small. A caveat that reads “the limits do not apply to fluorescent samples” suggests a sample somewhere else entirely; the measurement says the sheets in question are just outside, that two of six cross, and that which ones cross depends on the base as much as on the additive.

The generalisation #

The pattern is a bound whose derivation contains an assumption that the samples stopped satisfying.

The object-colour solid is derived from 0 ≤ ρ ≤ 1, which is a statement about reflectances and is exactly true of everything it was derived for. Fluorescent samples do not have a ρ, so the derivation does not apply — and the failure mode is not that the bound is violated by a lot but that it is violated slightly, by samples that in every other respect behave like ordinary paper.

The diagnostic is to ask what the bound’s derivation assumed about the sample, rather than what the bound says. A bound is often quoted long after its derivation has stopped being read, and the assumption is where the exceptions live. Here the assumption is one inequality, it is in the first line of the derivation, and it is the one thing about the sample that changed.

Where the ladder goes next #

If no reflectance can be this colour, then no reflecting device can reproduce it either, which is why a proof cannot glow — the target is outside the set every printing process draws from, not merely outside a particular press’s gamut.

The other direction is time. The margin is the brightener’s contribution and the brightener is consumed by the light that excites it, so the sheet is walking back inside the bound at a rate with a measurable half-life.