A pair the aperture separates

Metamerism on this site has always meant two spectra that a given observer cannot tell apart under a given light. Here is a pair that no observer can tell apart under any light, and that a four-millimetre aperture separates by six ΔE₀₀.

The claim #

Two samples can have identical reflectance and different colours, if the identity is in the integral and the measurement is not.

• The pair agrees at every wavelength to 4.6 parts in a thousand million million, which is the arithmetic’s own floor and not a tolerance.
• Their diffusion lengths differ fourfold — 0.73 millimetres against 2.94 — because one scatters four times as strongly as the other and absorbs correspondingly less.
• At infinite aperture they are 0.0000 ΔE₀₀ apart. At a four-millimetre radius they are 6.25.
• No observer is involved. The usual metamerism is a statement about three integrals against three curves; this one survives every observer and every light.
• And it is constructed rather than searched, by inverting the kernel’s total band by band, which makes the match exact rather than optimised.

What kind of metamerism this is #

The ordinary kind is two spectra that give one colour, and it exists because three numbers are being computed from eighty-one: the map has a null space, and two reflectances differing by anything in that null space match. Change the observer or the light and the match breaks, which is what makes illuminant metamerism and observer metamerism two of the standing hazards of this subject.

This pair is not that. The two reflectances are not merely metameric; they are equal, band for band. So every integral against every observer under every light gives the same answer, and there is no light and no eye that separates them.

What separates them is that a reflectance is not what the samples have. Each has a kernel, the kernels have different widths, and an aperture is sensitive to the width. The match is in the integral of the kernel and the measurement is of something else.

That makes it a metamerism at a different index: two objects agreeing after one marginalisation and differing before it. The observer’s version is agreement after integrating over wavelength; this is agreement after integrating over place.

How the pair is built #

Not by searching. Searching for a metameric pair is the usual approach and it always leaves a doubt about how exact the match is — this collection’s own metamer machinery documents the trap at length, because a pair matched to a tolerance separates by that tolerance under the first perturbation.

Here the construction is direct. Fix one slab. For the second, choose a different scattering coefficient — a quarter of the first’s — and then, band by band, find the absorption that reproduces the first slab’s total. The total is monotone decreasing in absorption at fixed scattering, so the root is unique and bisection finds it to machine precision.

The result is a pair matched to 4.6 parts in a thousand million million at the worst band, which is not a tolerance but the floating-point floor. A pair that exact cannot be separated by an imperfect match, so anything that does separate it is separating it for a reason.

What the separation looks like #

Through a four-millimetre radius the first slab, with the shorter kernel, recovers most of its reflectance; the second, with the kernel four times wider, loses far more. Both readings are low and the second is lower, so the pair separates in lightness first.

They also separate in chroma, and for the reason an aperture is a filter: the loss is largest at the wavelengths each sample reflects best, and the two samples have the same reflectance but different lengths, so the shape of each one’s loss is different. The result is 6.25 ΔE₀₀, which is six times a delivery tolerance and about the size of the difference between two ordinary shades of the same paint.

Widen the aperture and the pair converges, since both readings approach the same total. Narrow it and they diverge further. The separation is a function of the instrument and of nothing else, which is an unusual thing to be able to say about a colour difference.

The whole aperture curve #

Four millimetres is one instrument’s answer, and the curve behind it is simple enough to state as a law.

Sweeping the illuminated radius, the pair is 23.7 ΔE00 apart at 0.4 mm, 17.6 at 1, 11.3 at 2, 6.25 at 4, 3.08 at 8, 2.01 at 12, 1.18 at 20, 0.58 at 40 and 0.11 at 200.

Past about eight millimetres the fall is an exact inverse first power: 8 to 16 halves it, 20 to 40 halves it, 40 to 80 halves it, each to within two per cent. Doubling the aperture halves the disagreement — and it never reaches zero at any finite radius, so the 0.0000 above belongs to the infinite limit and to nothing anybody can build.

Two readings follow from where the crossings fall.

The pair is more than a unit apart for every aperture below 23.4 millimetres, and more than two below 12.0. A spectrophotometer’s illuminated spot is four to eight millimetres, so every instrument in ordinary use reads these two slabs as different colours — by six units at four millimetres and three at eight. No instrument setting resolves the disagreement, and none hides it either.

And it grows without bound as the aperture shrinks. At four tenths of a millimetre, about what a macro photograph resolves on a surface, the two slabs are 23.7 units apart, nearly four times what an instrument sees. The finer the measurement, the further apart two samples of identical reflectance become — which is the reverse of every intuition about resolution, and is the whole content of the construction.

What it says about matching #

Colour matching rests on a chain of reductions, and this pair shows where one of them is doing work nobody had noticed.

The chain runs: a physical sample, its reflectance, its tristimulus values under a light, a match. Grassmann’s laws govern the last step and are about lights; the observer governs the step before it; the illuminant governs the one before that. The first step — from a physical sample to a reflectance — has never been governed by anything, because it was assumed to be a description rather than a reduction.

It is a reduction, and this pair is the demonstration. Two samples with one reflectance are two different objects, and whether they match depends on what is done to them next. Under an eye at ordinary viewing distance they match, because an eye integrates over a much larger area than any aperture and satisfies the wide-illumination condition. Under an instrument they do not.

So the honest statement is: a reflectance is a sufficient description of a sample for a viewer who satisfies the conditions, and an instrument is not such a viewer.

What was computed, and how #

Both slabs are built from the dipole model, and their totals are computed from the closed form rather than by integrating their profiles — which matters, because the construction inverts the total, and inverting a numerically-integrated quantity would put the quadrature’s error into the match.

The bisection runs ninety iterations on a geometric bracket, which is far more than needed and costs nothing, and the residual is checked afterwards rather than assumed: the assertion behind the figure requires the largest band-wise difference to be under 0.05 of a reflectance unit and the measured difference to exceed 1.5 ΔE₀₀, and it reports both.

The readings use the same aperture arithmetic as everything else in the round: the kernel weighted by the area two discs share, over a logarithmic radial grid, divided by a perfect diffuser’s reading in the same arrangement.

One detail of the construction is worth naming because it constrains what the pair can demonstrate. The second slab’s scattering is a fixed fraction of the first’s at every wavelength, so the two differ by one number rather than by a curve. A pair differing by a scattering shape would separate differently, and probably more, since the scattering’s wavelength slope is what turns hues.

Where the model stops #

The pair is constructed, and two of the construction’s properties limit what it proves.

Both slabs are physically realisable but neither is a real material. The second needs an absorption four times the first’s at every wavelength to compensate its weaker scattering, and while nothing forbids that, no catalogue contains the pair. The demonstration is that the space of samples with a given reflectance is more than a point — not that a laboratory will meet these two.

And the separation depends on the instrument, so it is not a property of the pair alone. Quoting 6.25 ΔE₀₀ without the aperture is exactly the kind of unconditioned number this collection spends a field arguing against. The right form is: 6.25 at a four-millimetre radius, 3.08 at eight, and zero in the limit.

What it takes to make a pair like this #

The construction is worth spelling out, because it generalises to the other three indices and because its difficulty is not where it looks.

Fix one slab. Choose a second scattering coefficient — here a quarter of the first’s — and then for each of the eighty-one bands, solve for the absorption that gives the same total. The equation has one unknown, the total is monotone in it, and bisection converges in ninety iterations to the floating-point floor.

The hard part is not the solve; it is that the target has to be a closed form. If the total were obtained by numerically integrating the profile, the inversion would be chasing the quadrature’s error and the match would be exact only to the quadrature’s tolerance. The dipole model has a total derived separately from its profile, so the target is exact and the match inherits that.

The same construction is available at the other indices and neither has been run. At the directional index the pair would be a matte and a glossy surface with the same directional-hemispherical reflectance, which is straightforward. At the wavelength index it would be a fluorescent sample and a non-fluorescent one with the same radiance factor under one lamp — a pair that separates the moment the lamp changes, which is what illuminant metamerism already means with a different mechanism underneath it.

The generalisation #

A metamerism is what a reduction’s null space looks like from outside, and every reduction has one.

The eye reduces eighty-one numbers to three and its null space is metameric black. A camera reduces the same eighty-one to three differently, so its null space is a different one and the two disagree. A reflectance reduces a kernel to its integral, and its null space is pairs like this one.

Stating it that way makes the search systematic. For every marginalisation in a model, there is a class of physically different objects the model cannot distinguish, and the class can be constructed rather than stumbled upon. For the directional index the class is two surfaces with the same directional-hemispherical reflectance and different lobes — which is a matte and a glossy sample of the same pigment, and is precisely why gloss traps exist. For the wavelength index it is a fluorescent sample and a non-fluorescent one with the same radiance factor under one lamp.

Four indices, four kinds of metamerism, and only one of them has a name in the literature.

Those last two are the same material seen twice: what an aperture does to the colour it returns, and the kernel underneath that decides how much of it comes back at all.

The same kernel has a slope and a limit, and both of them are what decide whether a pair like this can exist at all.

What a specification could do about it #

A pair like this is a specification’s worst case, so it is worth asking what could be written that would catch it.

Nothing spectral will. The two samples have identical reflectance curves, so any tolerance expressed on spectra, on tristimulus values, on CIELAB coordinates or on a colour difference passes them as identical — correctly, because on those quantities they are.

What separates them is a second reading at a second aperture, which differs between the two by about three ΔE₀₀. That is not a colour measurement at all; it is a measurement of transport, and it happens to be expressed in colour units because that is the instrument to hand.

So the specification that would catch this pair has a field that is not a colour: something like the difference between the four- and eight-millimetre readings shall not exceed 0.5 ΔE₀₀, which is a statement about how translucent the sample is allowed to be. That is the same instruction the round’s instrument essay arrives at from the cost side, reached here from the side of what two samples can hide.

What holds the two together, and what does not #

It is worth being precise about which properties of the pair are exact and which are conditional, because the essay leans on the first kind.

Exact, by construction: the two bulk reflectances agree band for band, so every tristimulus value, every chromaticity, every colour difference and every appearance correlate computed from them is identical for both, under any illuminant and any observer. Nothing about that depends on the transport model being right.

Conditional, on the model: how far apart they read through a given aperture. That number comes from the dipole approximation and would change under a different transport theory, though its sign would not — a wider kernel loses more to a finite hole under any model that has a kernel at all.

And undetermined: what the two look like side by side. That would depend on the lighting, the size of the samples, the distance, and everything else the round has been saying a reflectance does not carry.

Three grades of claim in one construction, and separating them is most of what makes a constructed example worth publishing.

Who found it, and when #

Inter-instrument disagreement on translucent samples is old and well documented, particularly in dentistry, where the specimen is a tooth and two instruments routinely disagree by several ΔE₀₀ on the same one.

The usual explanation is a mixture of aperture, geometry and backing, which is correct and is a list of three causes rather than a mechanism. What this construction adds is the strongest possible form of the statement: the disagreement does not require the two samples to differ in reflectance at all. Two specimens can be identical by every spectrophotometric measure that has been agreed upon and still read differently, because the agreed measures do not include the quantity that separates them.

Metamerism itself is Ostwald’s word, from around 1900, and has always meant the observer’s version. The term has been extended twice since — to illuminant metamerism and to observer metamerism — each time by naming the thing held fixed. This would be geometric metamerism, and the name is available.

Why the construction is worth more than an example #

A pair like this could have been found by searching a catalogue of materials, and it would then have been an anecdote about two materials. Constructing it makes it a statement about a space.

The space is the set of all physical samples with a given reflectance, and the construction shows it is at least one-dimensional: the scattering coefficient can be varied and the absorption adjusted to compensate, band by band, without ever leaving the set. Nothing about the argument depends on the two particular slabs, and nothing about it depends on the dipole model being the right transport theory — any model with two free parameters per band and a monotone total would do.

That is the difference between a demonstration and an existence proof. The claim is not that these two samples are hard to tell apart; it is that a reflectance does not determine a sample, and the pair is the shortest way to say so.

Where the ladder goes next #

The construction generalises immediately and has not been run: a pair matched in reflectance and in kernel width, differing in the kernel’s shape, which the dipole model cannot produce but a two-layer model can. If such a pair separates under an aperture, then even a two-number description of a sample is insufficient, and the sequence of ever-finer descriptions has no obvious stopping point.

The more useful direction is the practical one. If two samples with one reflectance can differ by six ΔE₀₀ through an instrument, then the reverse is also available: two samples with different reflectances can be made to read the same. That is a specification’s nightmare and a manufacturer’s opportunity, and neither has anything to do with the observer.