Two sheets that match until the window

Metamerism is the standing hazard of colour matching, and every essay on this site about it — from the corner that moves both terms to the paints that stop matching — has ended the same way: a pair that matches under one light and not another can be put back together by restoring the first light. The mismatch is a property of the pair and the illuminant jointly, and the illuminant is a knob.

There is a pair for which that is false, and the reason is that the two samples do not differ in reflectance at all.

What the two sheets differ in is not a curve at all, which is why no measurement of a curve separates them.

The claim #

Two samples whose reflectances are identical can still be two colours, because a reflectance is not a complete description. The resulting mismatch is not metamerism and cannot be repaired by choosing a light — it can only be repaired by choosing a light with no ultraviolet, which is not the same thing.

• The pair agrees to ΔE00 0.00 under an ultraviolet-excluded instrument, by construction: the dyed sheet’s reflectance curve is what the brightened sheet measured there.
• Under the same instrument’s lamp with the ultraviolet restored it is ΔE00 7.06 apart, and under daylight 10.57.
• The two sheets have the same reflectance everywhere between 380 and 780 nanometres. They differ in a term that lives entirely off the diagonal.
• No change of light re-matches them. Every light that has ultraviolet separates the pair; the only lights that do not are the ones a person does not ordinarily stand in.
• This is a supplier’s failure mode rather than a laboratory curiosity, because matching a sample under a booth with a filter in it is exactly how it gets made.

What ordinary metamerism is, and why it is fixable #

Two reflectances ρ₁ and ρ₂ are illuminant metamers under E when

∫ ρ₁ E x̄ȳz̄ = ∫ ρ₂ E x̄ȳz̄

and the whole subject follows from that being three equations in an infinite-dimensional space of possible differences. The difference ρ₁ − ρ₂ is a metameric black under E — a spectrum integrating to zero against all three functions — and it is not a metameric black under a different E, which is why two paints stop matching when the room changes.

The essential property is that the failure is symmetric in the light. Change the light and the pair separates; change it back and they rejoin. A supplier who matches under D65 and a customer who inspects under a tungsten lamp have a disagreement that is entirely resolved by agreeing which lamp to use, and that agreement is what a viewing booth standard is.

Nothing about that argument survives if the samples are not reflectances. The equation above has ρ in it once per sample, as a multiplier, and a fluorescent sample does not have one.

How the pair gets made #

The construction here is not artificial; it is the standard industrial accident.

A customer specifies a colour by handing over a sample: a sheet of the brightened stock the job will print on, or a swatch of fabric. A supplier has to produce something that matches it — a different substrate, a different process, often a different material entirely. They measure the sample, they build to the measurement, and they check the match in a booth.

The measurement is where the pair is born. If the instrument is an ultraviolet-excluded one — which is the default on a great deal of installed equipment, and is deliberately specified in some industries precisely because it repeats better — then what it reports is the sample’s reflected component with the fluorescence removed. Build to that curve with ordinary dyes and the result is a genuine reflectance that agrees with the target’s measurement exactly.

Then the two go out into a room with daylight in it, and one of them glows.

This collection builds the pair the same way: the dyed sheet’s reflectance is taken to be the brightened sheet’s radiance factor under the ultraviolet-excluded condition, clipped to the physically possible range. The zero at that condition is therefore true by construction and is stated as such — the interesting numbers are the other two.

Why no light puts them back #

The formal statement is short. Write the brightened sheet as D = ρ δ + e ⊗ a, a diagonal reflectance plus a rank-one fluorescent term with emission profile e and excitation profile a. Write the dyed sheet as D' = ρ' δ. The pair matches under light E when

∫ ρ E x̄ȳz̄ + (∫ a E) ∫ e x̄ȳz̄ = ∫ ρ' E x̄ȳz̄

and by construction ρ' = ρ for the light with no ultraviolet, where ∫ a E = 0. For any other light the left side carries an extra term and the right side does not.

That extra term is a fixed direction in tristimulus space, scaled by a number. The direction is ∫ e x̄ȳz̄, which depends only on the emission profile and not on the light at all. The scalar is ∫ a E, which depends only on how much excitation the light supplies.

So the pair’s disagreement is a ray: it points the same way under every illuminant and only its length changes. There is no light that reverses it, because reversing it would need ∫ a E to be negative, and a light with negative power at some wavelength is not a light.

That is the difference between this and metamerism in one sentence. An ordinary metameric pair’s disagreement is a vector that rotates with the illuminant, so some illuminant sends it through zero. This pair’s disagreement is a vector of fixed direction and non-negative length, and the only way to zero it is to remove the excitation entirely.

Restoring the excitation restores the difference, and the size of what is restored is the whole of the pair’s failure.

How much there is to fail at is a property of the stock rather than of the construction, and the range across ordinary stocks is wide.

What it costs in practice #

The three numbers the pair produces are worth reading against ordinary industrial tolerances rather than against each other.

Under the ultraviolet-excluded condition, ΔE00 0.00. Under the ultraviolet-included one, 7.06. Under D65 outdoors, 10.57.

A commercial print tolerance on a critical colour is around 2, and a tolerance is a probability rather than a line, which makes the comparison generous rather than strict. A textile tolerance is looser and the acceptable range is usually stated as a shape rather than a number, but the pass mark is under 3 in most systems. So a pair that a laboratory certifies as an exact match is, in the room the goods are sold in, three to five times out of tolerance — and every instrument involved is in calibration.

The failure has a characteristic signature that makes it identifiable once it is known. The disagreement gets worse as the light gets bluer, because daylight has more excitation than tungsten does, and it gets worse near a window. That is the opposite of ordinary metamerism, whose direction and size depend on the pair rather than on any ordering of lights, and it is the diagnostic worth carrying: if the mismatch always grows in daylight and always shrinks under tungsten, the difference is fluorescent rather than metameric.

The ray model makes a prediction the essay does not collect #

The algebra establishes that the pair’s disagreement is a vector of fixed direction whose length is ∫ a E — the excitation the light supplies. That is a stronger statement than the essay uses it for, because it says the two published disagreements should stand in the same ratio as the two lights’ excitation supply, and the ratio is measurable independently.

The two figures are 7.06 under the ultraviolet-included instrument and 10.57 under D65, a ratio of 1.50. So the ray model predicts that daylight delivers about half again as much excitation to this brightener as the instrument’s lamp does.

The instrument’s lamp is D50-based, and the two illuminants’ power below 380 nanometres stands at 6.75 per cent against 3.35 — a ratio of 2.01. The two do not agree, and the gap runs in the direction the arithmetic requires: ΔE00’s chroma weighting divides a chroma difference by the chroma it was measured at, so a difference twice as large reads as less than twice as many units. 1.50 in ΔE00 against 2.01 in excitation is a compression of a quarter, which is about what the weighting does over that span.

That makes the two numbers a check on each other rather than two separate results, and it is the sort of check the ray structure makes available and a metameric framing does not. It also gives the collection a cheap test it has not run: compute ∫ a E for both lights directly and see whether the excitation ratio comes out at 2.0, which is what the ray model plus the metric’s compression together predict. If it came out at 1.5, the disagreement would not be a pure ray after all.

Ten and a half units is very nearly the ceiling #

The non-negativity argument does more than close off repairs. Since ∫ a E cannot be negative, and the direction is fixed, the disagreement is minimised at zero and has no upper bound in the algebra — it is bounded only by how much excitation any real light supplies.

Unfiltered daylight is the richest source a person ordinarily stands in: an unglazed sheet receives sixty-five per cent of its excitation from below 380 nanometres against four per cent behind a conservation filter, and no artificial lamp comes near daylight’s short-wave content. So the 10.57 is not one point in a wide range of possible failures — it is close to the largest this pair can produce anywhere.

That is worth stating because the three numbers read as a sequence with more to come. They are not. The pair runs from 0.00 in a filtered room to about 10.6 outdoors, and everywhere in between it is somewhere on that segment. A supplier who has seen the daylight figure has seen the worst of it, which is unusual in this collection and is a consequence of the direction being fixed: a rotating disagreement can find a worse angle somewhere, and a ray cannot.

What the reverse failure drifts to #

The other-direction case is described as right on the day and drifting yellow, without a size. The ray model supplies one.

A brightened material matched to a dyed target under an ultraviolet-included instrument carries a reflectance deficit built exactly to cancel the fluorescent term. As the brightener is consumed the fluorescent term goes to zero and the deficit does not, so the residual approaches the deficit — which is the same magnitude as the term it was built to cancel.

So the reverse failure drifts to about 7 units under the same lamp, in the opposite direction, and its endpoint is the forward failure’s starting point. The two cases are not merely asymmetric in when they appear; they are the same magnitude arrived at from opposite sides, one immediately and one over the brightener’s half-life.

That symmetry is worth having because it settles which of the two is worse in practice, and it is not the one the essay’s framing suggests. The immediate failure is caught before the goods ship. The drifting one passes every check, reaches the same size, and reaches it in the customer’s building — and by then nobody has a sample of what it looked like on the day.

Against tolerances, in both trades #

The comparison against a print tolerance of 2 gives 3.5 and 5.3 times out. Against a textile pass mark of 3 the same pair is 2.4 and 3.5 times out, which is the reading that matters for the failure’s usual home, since a dyed sheet matched to a brightened target is a textile problem far more often than a printing one.

Both trades therefore reject the pair comfortably in the room and accept it in the laboratory, and the looser tolerance does not rescue it. A failure that clears the strictest tolerance by a factor of three and the loosest by a factor of two and a half is not a marginal case in any of them — which is the practical reason it is worth a name of its own rather than a footnote under metamerism.

The other direction, which is worse #

Everything above is the case where the supplier has no brightener. The reverse case is a supplier who does, and it is more common and harder to see.

A brightened material matched against an unbrightened target under an ultraviolet-included instrument will read as a match only if the two agree including the fluorescence — which means the brightened one’s reflectance must be lower to compensate. It then separates the other way under a filtered light, and it also separates over time as the brightener is consumed, because the compensating term shrinks and the reflectance deficit does not.

So the two failure modes are not symmetric. A dyed match to a brightened target is wrong immediately and stays wrong by a constant amount. A brightened match to a dyed target is right on the day and drifts, and the drift is in the direction of the reflectance deficit — which is to say it goes yellow.

What a supplier can actually do about it #

The structural result closes off the obvious repairs, which is worth stating as a list because each of them gets tried.

Re-matching under a different booth does not work, since every booth with any ultraviolet separates the pair and the amount only changes. A supplier who moves from one standard lamp to another has changed the size of the failure and not its existence.

Adding a small amount of the same dye does not work either. The disagreement is a fixed direction in tristimulus space, so a dye chosen to cancel it under one light overcorrects under a light with more excitation and undercorrects under one with less. There is no single amount that is right in two rooms.

Putting a brightener into the second material does work, and it is what happens in practice. Two brightened materials disagree only to the extent that their excitation profiles and quantum yields differ, which is a second-order mismatch rather than a first-order one — and it introduces a shared time dependence, since both are being consumed by the same light.

And specifying the measurement condition works, in the narrow sense that it makes the disagreement visible before the goods ship rather than after. That is the whole of what a standard can offer here: it does not repair the mismatch, it makes the two parties disagree in the laboratory instead of in the shop.

The order of those four is the order in which they are usually tried, and it is the reverse of the order of how well they work.

What was computed, and how #

The brightened sheet is a coated press stock with a stated brightener loading; the dyed sheet is a non-fluorescent sample whose reflectance is that sheet’s radiance factor under the ultraviolet-excluded condition, clipped into [0, 1] — a clip that removes nothing here, because a radiance factor with no excitation cannot exceed one.

All three comparisons use the same observer and the same geometry, each normalised to its own light’s perfect diffuser, so what is being compared is the pair rather than the exposure.

The assertion is written as two conditions and both are required. The pair must agree to better than 0.6 under the ultraviolet-excluded condition, and must disagree by more than 2 under the ultraviolet-included one. The first alone would pass for a pair that agrees everywhere; the second alone would pass for a pair that never matched. Writing it as a conjunction is what makes it a test of the phenomenon rather than of one number.

Where the model stops #

The dyed sheet is idealised. A real dye set cannot reproduce an arbitrary reflectance curve, and matching a brightened sheet’s ultraviolet-excluded curve with actual colourants would leave a residual of its own — so the true failure is this one plus an ordinary metameric one on top.

The clip is doing no work here and would elsewhere. A heavily brightened sheet measured under a condition that leaves some excitation would have a radiance factor above one in places, and the dyed target would be unreachable rather than merely difficult. The pair here is built under a condition with essentially none.

And there is no geometry in it. Both sheets are treated as perfect diffusers with the same interface reflection. A gloss difference between a paper and a fabric is a separate mismatch that adds to this one and is the subject of a different essay.

Who found it, and when #

The problem is old enough to have a trade name: fluorescent metamerism, or sometimes whiteness mismatch, and it is why textile and paper standards distinguish measurement conditions at all — the same pressure that made the booth a specified device rather than a lamp somebody chose. The CIE’s first recommendations on measuring fluorescent samples date from the 1970s, and the two-monochromator method that settles the question outright is Donaldson’s from 1954.

ISO 13655’s M₁ condition — daylight including its ultraviolet, specified rather than left to the instrument — was adopted in 2009 precisely to stop this. It works, in the sense that two M₁ instruments agree with each other; what it cannot do is make an M₁ measurement predict a room with more or less ultraviolet than D50 has.

What is worth stating in the terms this collection uses is the structural point: the disagreement is a fixed direction with a non-negative scalar, which is why it is not metamerism and why the vocabulary of metamerism — metameric index, illuminant metamerism — gives entirely the wrong intuitions about how to fix it.

The generalisation #

The pattern is a repair that works because of a symmetry, applied to a case that has lost the symmetry.

Metamerism is repairable because the illuminant enters both sides of the matching equation the same way, so the failure is a rotation and some rotation is the identity. Fluorescence puts the illuminant on one side twice — once as a multiplier and once as the scalar on an additive term — and the symmetry is gone.

The diagnostic that generalises: ask whether the intervention’s effect on the disagreement can change sign. If it can, there is a setting that repairs it and the problem is a tuning problem. If it cannot, no amount of tuning helps and the intervention is the wrong one. That test is cheap, it is available before any experiment, and it is the difference between a supplier who re-matches under a different booth and one who changes what the sample is made of.

Where the ladder goes next #

If a match depends on which instrument measured it, the instrument’s lamp is part of the specification, and what an instrument brings with it is four standard answers of which one has to be named.

The other direction is the room rather than the laboratory. If the disagreement scales with the excitation supplied, then the window a sheet is behind changes the size of the failure by more than the four standard conditions do — which makes the standard a way of agreeing on a number rather than a way of predicting a room.