A neutral is everyone's colour

Six arguments of the observer have been measured and every one of them costs between one and two and a half colour differences. There is a sample on which all six cost nothing at all, and the nothing is exact.

The claim #

A spectrally flat reflectance computes to the same colour for every observer, exactly, and the exactness is the white point rather than a property of eyes.

• Six departures, six identities, between 4 × 10⁻¹⁴ and 4 × 10⁻¹³ ΔE₀₀, and one of them is exactly zero.
• The mechanism is proportionality, not smallness. A flat sample produces a stimulus that is a scalar multiple of the light, and every observer’s white is that same light.
• So a disagreement about a neutral is a white-point disagreement, never an observer one, and that narrows a diagnosis in a single reading.
• And the sample everything is calibrated with is the sample that can report nothing. A white tile agrees under every eye and every tabulation, which is what makes it useless as a test of either.

The arithmetic, written out #

An observer’s response to a stimulus is three integrals: Lᵢ = ∫ φ(λ) lᵢ(λ) dλ for the three cone sensitivities. A reflective sample under a light gives φ = R·S, and the white the eye is adapted to is S itself.

If R(λ) = ρ for every wavelength, then

Lᵢ(sample) = ρ · Lᵢ(white)

for all three i, whatever the three sensitivities are. The relative cone excitations — the sample’s response divided by the white’s, in each cone — are (ρ, ρ, ρ) exactly, for every observer that has ever existed or could be constructed.

Everything downstream is a function of those three ratios. A tristimulus value obtained by a fixed matrix from them is ρ times the white’s; CIELAB’s coordinates are functions of X/Xn, Y/Yn, Z/Zn, which are all ρ; the lightness is a function of ρ alone and the two chromatic coordinates are identically zero.

No property of the observer appears anywhere in that derivation. The sensitivities cancel before they are ever evaluated, which is why the result is an identity and not a limit.

What it takes to break it is worth listing. The identity survives a remarkable amount of abuse and it is worth listing what it survives, because the list is the essay’s real content.

It survives the observer being wrong. A set of sensitivities that resembles no eye at all still gives a flat sample the same answer as any other.

It survives the tabulation being wrong. The grid, its origin and its range all cancel too, for the same reason, so a fluorescent tube reduced to a single mercury line by a badly-aligned grid still reads a grey card as a neutral of the right lightness.

It survives the observers being of different species. Nothing in the derivation used three sensitivities; a dichromat, a tetrachromat and a camera all agree, in their own coordinates, that a flat sample is the light divided by a constant.

What it does not survive is the sample not being flat, and that boundary is sharp rather than gradual. A sample flat to one per cent is one per cent of the way along a sequence whose far end is 1.71 ΔE₀₀ for the macular departure, and the approach is slightly super-linear in the published unit rather than gentler than linear.

The exception that looks like the rule #

There is a class of sample that measures as neutral, is not flat, and satisfies none of this, and it is the one most likely to be used as a reference.

A metameric grey is a reflectance with spectral structure that happens to integrate to a neutral under one light through one observer. Its chromatic coordinates are zero, an instrument reports it as neutral, and the derivation above has no purchase on it at all, because the derivation needed R constant and not the answer neutral.

Change the observer and it moves. Change the light and it moves. It is two spectra with one colour with the second spectrum being the light itself, and every observer departure this round measures reappears on it at full strength.

The practical form is a warning about test charts, and it is not a small one. A neutral patch printed with three inks is a metameric grey; a neutral patch printed with black ink alone is much closer to flat. The two behave completely differently under exactly the substitutions this round is about, and they look identical on the sheet and in the measurement file. A chart’s neutral column is where a profile is most closely fitted, and on a three-ink chart it is also where the most spectral structure is.

Why the sequence towards it is straight and the unit is not #

Walking a red pigment from flat to its own reflectance, with two observers differing in macular density watching, the distance between their relative cone excitations is exactly proportional to the distance walked: 0.00407 at an eighth of the way, 0.00815 at a quarter, 0.01630 at a half, 0.03259 at the end. Straight to two parts in a hundred.

The same sequence in ΔE₀₀ is 0.208, 0.347, 0.641, 1.714. That is not straight, and the departure from straightness is not small: a quarter of the way along, the colour difference is 0.347 against a proportional prediction of 0.428.

Both facts are worth having and they say different things. The straight line says the departure is a pairing — an inner product of the observer’s deviation with the stimulus’s deviation from the adapting white — and that scaling one factor scales the product exactly. The curve says the unit is not a linear function of what the pairing is about, which is unsurprising once stated and is invisible if only the unit is ever computed.

The identity is about the eye. The curvature belongs to the arithmetic that reports it.

What a neutral cannot test #

An identity is a place where nothing can be learned, and that has an uncomfortable consequence for anybody choosing a reference sample.

A perfectly neutral patch cannot falsify a claim about the observer’s age, its field size, its macular density or its cone peaks. It cannot distinguish a well-tabulated calculation from a badly tabulated one. It agrees under every light and with every eye, which is precisely why it is worthless as a test of any of them.

That is awkward, because a neutral is the reference sample most instruments carry. A white tile and a black trap are what a spectrophotometer is calibrated with, and both of them — to the extent they are spectrally flat, which for a good ceramic tile is a very good approximation — are exactly the samples that cannot report an observer or a tabulation fault. An instrument can be perfectly calibrated on its tiles and wrong by units on the next saturated sample it reads, with nothing in the calibration record to suggest it.

The same holds for a display’s grey ramp, for a viewing booth’s neutral surround, and for the neutral patches that dominate most calibration charts. All of them are chosen for stability and for being easy to make; none of them can see the departures this round is about.

That figure is the identity’s complement and reading the two together settles what an observer disagreement is about. Six departures across six lights give thirty-six numbers between zero and 4.5 ΔE₀₀, and all thirty-six collapse to the floating-point floor when one property of the sample changes. Not the light, not the observer, not the tabulation — the sample’s departure from flatness.

So the honest one-sentence account of observer metamerism is that it is a property of the stimulus pair rather than of the observers. Two people disagree about a coloured surface and agree about a grey one, and the difference between those two situations is entirely on the surface’s side of the arrangement.

The diagnostic, which is the good half #

Read in the other direction the identity is a genuinely useful tool, and it is the sharpest one this round produces.

A disagreement about a neutral is a disagreement about the white point. Not about the grid, not about the observer, not about the field size, not about anybody’s age. Two measurements of a grey card that differ have exactly one available cause among all the things this round has been varying, and everything else can be eliminated in a single reading.

That is a great deal to get from one sample, and it inverts the usual advice. The received wisdom is to test with saturated samples because they are where the errors are, and it is right about where the errors are. It is wrong about diagnosis: saturated samples are sensitive to everything at once and therefore identify nothing.

The procedure that follows is short. Measure a flat neutral first; if it disagrees, stop and fix the white point. Only then measure something saturated, because only then is a disagreement attributable.

Where this collection sits relative to the identity #

The identity says something slightly uncomfortable about this collection’s own test surfaces and it is better said here than left implicit.

Its samples are broad absorption bands on a pale base, its lights are daylight reconstructions and thermal radiators, and its worked examples are papers, plastics and eggshell paints. Almost all of that sits nearer the neutral axis than a saturated ink or a display primary does. So every departure this collection has published has been measured on the samples where departures are smallest, and the numbers are lower bounds for a saturated world in a way the captions have not said.

The size of the effect is measurable rather than rhetorical. Across the forty-two-surface family the macular departure runs from 0.26 ΔE₀₀ on the flattest member to 4.24 on the most saturated. That is a factor of sixteen driven entirely by how far the sample sits from the light, and a collection quoting its median is quoting the middle of a distribution whose shape is decided by a choice of test set.

The pairing has now been checked in four combinations — two departures, two lights, two samples — and the excitation distance is proportional to the mixing fraction in every one. That is the sort of check worth running more than once, because a single instance of a straight line is weak evidence and four instances across independent variables are not.

What varies between the four is the slope, which is the size of the observer’s factor and of the light’s, and what does not vary is the linearity. A structure that survives a change of every variable except the one it is about is a structure rather than a coincidence, and this one is the same structure the previous round found for the sample with the factors renamed.

Recomputing the identities under a third light is the cheapest available check that they are identities rather than coincidences of a spectrum. Under daylight, tungsten and a three-emitter LED the same six sit between 4 × 10⁻¹⁴ and 4 × 10⁻¹³ ΔE₀₀.

The two figures are the identity imposed and the identity approached, and their agreement under a light with narrow emitters is worth having: a narrowband source is where the departures are largest, and the condition holds there unchanged.

What was computed, and how #

The identity is asserted rather than argued. For each of the six departures, the colour of a flat 0.18 reflectance is computed for both settings and the difference is required to be below 10⁻¹⁰ ΔE₀₀; the measured values are 1.7 × 10⁻¹³ for the field size, 1.7 × 10⁻¹³ for the age, 1.1 × 10⁻¹³ for the macular, 1.1 × 10⁻¹³ for the peaks, 4.4 × 10⁻¹⁴ for the rods, and exactly zero for the density.

The distinction between an identity and a limit is made the way the previous round had to make it five times: a small number is not evidence of an identity and a floor is. Eight of this round’s ten conditions are at the floor; the two that are not are reported as limits, and they are 8.11 and 13.0 ΔE₀₀, which is not small.

The sequence towards the identity is computed by mixing the sample with a flat 0.5 and is required to be linear in the excitation distance to two per cent. That assertion could fail — if the departure were not a pairing the sequence would curve — and it is checked in both units so that the curvature of the second is a measurement rather than an excuse.

One further consequence belongs to anybody who designs a colour-critical process rather than measures one. If observer disagreement is a property of how far a sample sits from the light it is seen under, then moving the light towards the sample reduces it, and that is a control nobody thinks of as a control. A saturated red under a lamp with a red bias is closer to its own adapting white than the same red under daylight, and the two observers agree better about it.

The effect is real and it is not usually available: the lamp is normally fixed by the application, and a lamp chosen to flatter one sample is a lamp that estranges the next. Where it is available — a single-product inspection booth, a press proofing station with one dominant brand colour — it is a lever that costs nothing and that no specification mentions.

Where the model stops #

The identity requires the same light to appear in the sample’s integral and in the white’s, which is the arithmetic of an observer adapted to the scene they are looking at. An observer adapted to something else — a print viewed under one lamp while the eye is still adapted to another — has a white that is not the light, and nothing above applies.

It also requires the flat reflectance to be genuinely flat. A ceramic tile is flat to a few tenths of a per cent across the visible band and rises in the blue; a Spectralon standard is better. The identity is exact for the idealisation and the residual for a real tile is a small multiple of its own departure from flatness.

And fluorescence is outside it entirely. A brightened white is not a reflectance at all, its object is a matrix rather than a curve, and the derivation has nothing to say about it.

The generalisation #

The habit is about where to look for the case that teaches nothing.

Every measurement system has samples on which its parameters cancel, and those samples are attractive for exactly the reason that makes them useless: they are stable, they are reproducible, and everybody agrees about them. A calibration built on them is a calibration that cannot detect the faults the system is most likely to have.

The move is to find the cancellation deliberately, write down what it removes, and then choose a reference that fails to have it. That is harder than it sounds, because the reference has also to be stable and reproducible, and the properties that make a sample insensitive to the parameters often make it insensitive to time and temperature as well.

The reverse failure is to conclude from a good calibration that a system is good. A system agrees with itself on the samples where agreement is an identity, and the interesting question is always what it does one step away from them.

Who found it, and when #

The algebra is as old as the reflectance factor and is usually stated as the reason a reflectance is dimensionless. Its use as a diagnostic is folklore in instrument work and is rarely written down as a claim about observers.

The metameric-grey exception reached the printing industry through the grey-balance literature of the 1960s, where the difference between a neutral made of three inks and one made of black alone became a commercial matter. That literature is about stability under lighting change rather than about observers, and the two are the same phenomenon with a different variable moved.

Where the ladder goes next #

One factor of the pairing has been emptied. The other one belongs to the observer, and it empties in a way that is much less obvious: there is a class of difference between two eyes that costs nothing at all, and it is the class that most of one departure turns out to consist of.