Two observers and one metamer

The round has measured six departures on single samples and quoted them in colour differences. That is a convenient way to report them and it slightly misdescribes what they are.

The claim #

An observer difference can only show itself through a comparison between two spectra, so every number in this round is a statement about a pair even when it is reported on one sample.

• A single stimulus has no observer-independent colour to be wrong about. What the round measures is a difference between two observers’ coordinates, which becomes an observable only when something else is being matched.
• The something else is always the adapting white in this round’s construction, so each departure is a metamerism between a sample and the light it sits under.
• That explains the identities exactly. A sample that is a scalar multiple of its white is a match for everybody; so is a single wavelength; so is anything an observer’s gain absorbs.
• And a spectral match is immune, because two identical spectra are identical for every observer by construction rather than by arithmetic.

What an observer difference is observable as #

Suppose two people look at one patch. They have different cone sensitivities, so they catch different numbers of photons in each class — and there is no way to compare those numbers, because neither has access to the other’s.

What they can compare is judgements about pairs. Does this patch match that one; is this pair a match under both lights; does this proof match this press sheet. Every one of those is an identity between two sets of three integrals, and an identity either holds for both observers or does not.

So an observer difference is not directly observable at all. It becomes observable as a disagreement about a match, which is the classical definition of observer metamerism, and everything in this round is a measurement of that even where the presentation is of a single sample.

The single-sample presentation is a convenience and it is not a fiction. The second member of the pair is present in every calculation: it is the adapting white, and the coordinates being compared are the sample’s relative to it.

Why that explains the identities #

Reading the round’s eight identities as statements about pairs makes every one of them obvious rather than surprising.

A flat reflectance produces a stimulus that is a scalar multiple of the white. Two lights differing by a scalar are the same light dimmed, and no observer disagrees about a dimming. The identity is exact for that reason rather than by numerical accident.

A single wavelength produces a stimulus that is a scalar multiple of the white for the same reason, because with one wavelength present there is nothing else for either to be. That identity is the same statement with the light’s support restricted rather than the sample’s shape flattened.

A gain on each cone and a change of basis are not about the pair at all; they are statements that two apparently-different observers are the same observer. Those are the observer’s factor of the pairing rather than the stimulus’s.

So the eight identities are two kinds: four say a pair is not really a pair, and four say two observers are not really two.

The metameric black, and where the departure lives #

There is a more precise statement available and it uses machinery this collection already has.

Two spectra match for an observer when their difference lies in the null space of that observer’s three sensitivities. A metameric black is exactly such a difference — a spectrum with zero tristimulus values, invisible by construction.

Two observers have two null spaces, and they overlap without coinciding. A metameric black for one is generally not one for the other, and the residual — its tristimulus values for the second observer — is the disagreement.

That gives the size of the effect a clean form: an observer departure is the projection of the pair’s metameric black onto the second observer’s sensitivities. It is zero when the black is in both null spaces, which happens when the two spectra are identical, and it grows with how much structure the black has where the two observers differ.

Everything the round has measured about where each departure lives — the macular band at 460, the lens’s tail, the peaks’ flanks — is a statement about which metameric blacks that departure can see.

The light decides what is in the comparison. The light table reads differently under the pair formulation and reads better. Changing the light does not change the sample and does not change the observers; it changes which part of the sample’s spectrum is present in the comparison, because the sample’s deviation from the white is weighted by the light.

That is why a warm lamp raises the blue-absorbing departures. It is not that the lens absorbs more under tungsten; it is that the pair’s difference, weighted by a blue-poor light, is dominated by a region where a small absolute difference is a large relative one.

And it is why the laser projector’s row is empty on the coarse grid. A one-line spectrum makes the sample a scalar multiple of the white, so the pair stops being a pair and there is nothing for any observer to disagree about.

The general form is that a light is not a source of difficulty in its own right. It is a weighting on the comparison, and a weighting can only emphasise what the pair already contains.

Why spectral matching is the only immunity #

The formulation makes the standing advice precise and says exactly how strong it is.

If two samples have identical spectra, their difference is the zero function, which is in every null space, so they match for every observer that has ever existed. Not to a tolerance: exactly, and for any future observer, any illuminant and any tabulation.

That is a much stronger guarantee than any colorimetric tolerance can offer, and it is why automotive and textile matching moved towards spectral criteria. It is also why the guarantee is expensive: identical spectra require identical colorants, and a screen cannot make an ink’s spectrum.

Between the two extremes there is a continuum and it is worth naming. A pair whose spectra differ only where all observers are insensitive — outside the visible band, or in a narrow region all three sensitivities are small — is nearly immune. A pair whose difference sits in the blue is exposed to the two largest departures.

A partial spectral match can be designed, by choosing colorants whose difference is confined to regions where observers agree, and that is a weaker and much more achievable requirement than identical spectra.

What this says about which samples are exposed #

The pair formulation gives a rule for identifying exposure that is easier to apply than the single-sample one.

Anything reproduced in a different medium is a pair. A brand colour in ink and on screen, a proof and a press sheet, a plastic part and a painted panel, a repair panel and the original. Every one of those is a metameric match by construction and carries the full term.

Anything compared against a numerical standard is a pair with the standard’s implied spectrum, which is whatever spectrum the standard’s tristimulus values came from — usually a physical reference that no longer exists.

And anything judged on its own is not a pair at all, except with its adapting white. A colour with no reference to compare against carries only the small term the white supplies, which is why a wall painted a slightly different shade is never noticed and a patched wall always is.

That last is the everyday form of the whole round, and it explains a familiar experience: colour problems appear at boundaries and comparisons, never in isolation.

The round reported single samples anyway. Given that the pair formulation is the correct one, it is worth saying why every measurement in the round was presented on a single sample.

Because the pair is the same in every case. The second member is always the adapting white, so reporting a pair would mean reporting the same second member four hundred times.

Because it is comparable with the collection’s other numbers. Every colour difference published here is between a sample and something, and a departure quoted the same way sits beside them without conversion.

And because the alternative overstates. A departure between two deliberately chosen metamers is much larger than a departure between a sample and its own light — the metamer pair is constructed to have its difference where the observers differ, and an ordinary sample is not. Reporting the constructed worst case as though it were typical is the error the population work is careful about.

So the single-sample presentation is a lower bound on a constructed pair and an accurate figure for the ordinary situation, and this essay is the note saying which is which.

There is a fourth reason and it is the one that matters most for how the round’s numbers should be used. A pair’s departure depends on both members’ spectra, so a number quoted for a pair is a number about two colorants rather than about one. A brand choosing a colour cannot use it, because the second colorant does not exist yet.

The single-sample figure can be used at that moment, because its second member is the light. So the presentation is not merely a convenience; it produces the quantity that is available at the point of decision, and the pair figure — larger and more accurate — is only available once both halves have been chosen and the decision is made.

A number available early is worth more than a better number available late, which is the ordinary economics of a design criterion and is why the round reports what it reports.

The asymmetry between the two factors #

There is one respect in which the two factors of the pairing are not alike, and it decides who can do anything.

The observer’s factor cannot be chosen. A specification’s audience is whoever it is, with whatever lens densities and cone peaks they have. Nothing in the chain can select for it, and the CIE’s contract exists precisely to avoid having to.

The stimulus’s factor can be chosen, at the moment a colorant is picked, a primary is designed or an illuminant is specified. Every one of the round’s practical recommendations is about that factor, because it is the only one anybody controls.

That asymmetry is why the round’s landing essays are all about design decisions rather than about people. An audit of a pairing produces advice about whichever factor is adjustable, and here it is the entire content of what can be done.

The worst pair is constructible and is not the useful one. The formulation invites an obvious construction and it is worth saying why the round does not make it.

Given two observers, the pair they disagree about most is findable: maximise the projection of the difference spectrum onto the second observer’s sensitivities, subject to the difference being in the first’s null space and both spectra being physically realisable reflectances. That is a linear programme with the constraint that reflectances lie between zero and one, and its answer is a pair of optimal colours with sharp transitions — the same family that bounds a gamut.

The number that comes out is large and it describes nothing anybody makes. Optimal colours are mathematical extremes with step-function reflectances; real colorants are smooth, and a worst-case pair constructed from steps overstates what any real pair of inks can do by a wide margin.

This collection has been careful about that construction before. An extremum is not a sample, and a worst case over a set that contains no real members is a bound rather than a measurement — useful for saying what cannot happen, useless for saying what does.

So the round measures ordinary samples against their own lights and reports a distribution, and leaves the worst case as a bound it has not computed. That is the same choice the depth-06 round made about extrema, for the same reason.

What was computed, and how #

The round’s departures are computed as differences between two observers’ CIELAB coordinates for one sample under one light, which is the single-sample presentation. The pair formulation above is a re-reading of the same arithmetic rather than a separate calculation.

The identities are asserted at 10⁻¹⁰ ΔE₀₀ and measure between 4 × 10⁻¹⁴ and 4 × 10⁻¹³, which is what an exact statement about a pair that is not really a pair looks like in floating point.

The null-space formulation is stated rather than computed here; this collection’s metamer machinery constructs metameric blacks for a stated observer and has not been used to project one observer’s black onto another’s sensitivities. That is a short calculation and it is not in this round.

Where the model stops #

The pair formulation assumes both members are seen under the same light and by the same adapted eye, which is the ordinary side-by-side comparison and is not the ordinary situation in commerce — a proof approved on Tuesday and a press sheet checked on Thursday are compared by memory, which is a different and much less reliable operation.

The immunity of an identical-spectrum pair is exact and is a statement about the spectra, not about the samples. Two objects with identical spectral reflectance can still look different if their gloss, texture or size differ, and an instrument geometry can disagree about them.

And nothing here says how large a disagreement has to be before somebody complains, which is an acceptability question with an entirely different literature.

One last consequence for reading the observer-metamerism literature, which reports indices rather than distances. A metamerism index is computed for a specified pair and reports how far it comes apart under a deviate observer, so it is a property of a pair and cannot be attached to a colour.

That is the right design for colorant formulation, where pairs are what is being made. It is the wrong design for every question this round’s landing essays ask — what a tolerance should reserve, which brand colour is robust, which display design is safer — because in all of those only one half of the pair is known when the decision is made.

An index for pairs and an allowance for colours are different instruments, and the field has built the first and not the second.

The generalisation #

The habit is about the difference between a quantity and an observable.

A model can compute a quantity that is not observable in isolation, and reporting such a quantity invites a reading it does not support. Coordinates in an internal space, potentials, phases, absolute levels — all of them are meaningful only through differences, and a report that gives them as values is describing a gauge rather than a fact.

The move is to ask what comparison would reveal the quantity. If none exists, the quantity is a gauge and its value is a convention; if one exists, the comparison is the observable and the quantity should be reported through it.

The failure mode is to attach practical consequences to a gauge. An unobservable difference has no consequences, and a great deal of effort can be spent reducing one.

Who found it, and when #

Observer metamerism is defined as a pair phenomenon throughout the literature, and the CIE’s standard deviate observer and its index are constructed for pairs. The single-sample presentation used in this round is a convenience of exposition rather than a departure from that.

The null-space formulation of metamerism goes back to the mathematical treatments of the 1950s and 1960s and is the standard modern account. Its extension to two observers with two null spaces is immediate and is used in the observer-metamerism literature to construct worst-case pairs.

Where the ladder goes next #

The round’s three audits have each found the same structure and each landed in the same places. What remains is to say what they have in common, and the answer is not the subject matter: three audits and one shape, which is the round’s real result.