Two paints that stop matching

Metamerism is normally introduced as a hazard about light. Two samples match under one illuminant and separate under another; a shop’s fluorescent tubes and a customer’s daylight disagree about whether the trim matches the door. That story is correct, it is an essay elsewhere on this site, and it has a comfortable implication: fix the illuminant and the match holds.

It does not. Two reflectances can match under one illuminant, one observer, one geometry — and come apart when the same light meets the same paint in a corner.

The claim #

A metameric match is a statement about three linear functionals of the spectrum, and interreflection is not linear in reflectance. So a match survives the first bounce and not the second, and how badly it fails is decided by the shape of the room.

Nothing about the light has changed. Nothing about the observer has changed. Nothing about the pigments has changed. The two paints in the figure above are the same two paints in both rows, under the same lamp, seen by the same eye. What differs is how much of the light arrived having touched the paint twice.

Why the match is exact to begin with #

The pair is constructed rather than searched for, and the construction is worth stating because it is what makes the first row of the figure exact rather than close.

Take a reflectance $\rho_A$. A metameric black is a spectrum $b(\lambda)$ that integrates to exactly zero against all three colour-matching functions weighted by the illuminant:

$$\int E(\lambda)\,b(\lambda)\,\bar{x}(\lambda)\,\mathrm{d}\lambda = 0$$

and likewise for $\bar{y}$ and $\bar{z}$. It is a spectrum the eye cannot see at all — a strange enough object to be worth a figure on its own, and it has one further down the ladder it came from. Adding it to anything leaves the tristimulus values untouched, so $\rho_B = \rho_A + b$ is a metamer of $\rho_A$ by construction.

The construction is a projection: take any smooth candidate function, subtract its component along the three matching functions, and what remains is orthogonal to all three. That is one linear-algebra step and it is exact. The pair on this page matches to a computed $\Delta E_{00}$ of $5.4\times10^{-14}$, which is double-precision arithmetic and not an approximation of a match.

The two reflectances differ by as much as 0.31 in absolute reflectance, at 635 nm. These are not two nearly-identical spectra flattered by a coarse metric; they are visibly different curves that the eye is constitutionally unable to tell apart.

What the corner does #

A corner returns some of a surface’s own light to it. For the corner of a cube, the fraction is the form factor between two perpendicular unit squares sharing an edge, which is 0.2000 — a fifth.

So what reaches the eye from a point in that corner is not $E\rho$. It is

$$E(\lambda)\,\rho(\lambda)\,\bigl(1 + q\rho(\lambda) + q^2\rho(\lambda)^2 + \cdots\bigr) = \frac{E(\lambda)\,\rho(\lambda)}{1 - q\,\rho(\lambda)}$$

with $q$ the fraction that comes back. The $\rho$ in that denominator is the entire result. The match was an identity between integrals linear in $\rho$; there is no linearity left in that expression at all, and there is no reason the identity should survive it. It does not.

That figure is the reason the claim is about geometry rather than about an unlucky pair of paints. The separation is a smooth, monotone function of one number describing the shape, and it starts at exactly zero. A flat wall in the open is $q = 0$; the corner of a cube is $q = 0.20$; a deep slot is around 0.6; a cavity with a pinhole approaches 1.

The threshold is crossed at $q \approx 0.14$, which is a shallower feature than a right-angled corner. Two paints certified to match on a flat chart do not match in a groove pressed into the same panel.

What was computed, and how #

Every number above comes out of lib/scene.js, and three of the choices behind them are worth naming.

The metameric black is chosen, not arbitrary. An arbitrary metameric black demonstrates very little, for the same reason this site’s illuminant-metamerism machinery records: the black has to differ where the thing being changed weights it. The pair here is selected from a scan over frequency and phase for the one that separates hardest in a corner. Every candidate in that scan is already an exact metamer, so this is a choice among constructions that are all perfect rather than an optimisation towards being one. Not saying so would misrepresent the result as more generic than it is.

The separation is measured with $\Delta E_{00}$, against the D65 white point under the CIE 1931 2° observer, with both spectra put on the same luminance scale first. The comparison is about chromaticity and relative lightness rather than about an exposure, and normalising is what makes that true rather than assumed.

The construction is the site’s existing one. lib/metamer.js already builds metameric blacks and already has a gate. Writing a second construction here would have produced two functions sharing a name and differing in body, which is the failure mode this collection has a standing note about — three different gauss implementations that turned out to be three different functions.

Why squaring is the operation that breaks it #

It is worth being precise about why the second bounce is fatal, because the reason is sharper than “the model is nonlinear” and it says which other constructions are at risk.

The match is three equations:

$$\int E\,\rho_A\,\bar{x} = \int E\,\rho_B\,\bar{x}, \qquad \text{and likewise for } \bar{y},\ \bar{z}$$

Write $\rho_B = \rho_A + b$. The equations say exactly that $\int E\,b\,\bar{x} = 0$ and its two companions — three constraints on $b$, in a space of 81 dimensions. That leaves 78 dimensions of metameric black to choose from, which is why metamers are easy to construct and why the pair on this page could be selected for a property.

Now square. The corner sees $\rho^2$, so the difference between what the two patches send back through the second bounce is

$$\rho_B^2 - \rho_A^2 = (\rho_A + b)^2 - \rho_A^2 = 2\rho_A b + b^2$$

Neither term is required to integrate to zero. The first is $b$ weighted by $\rho_A$, and $b$ was constructed orthogonal to $E\bar{x}$, not to $E\rho_A\bar{x}$ — a different vector, so no orthogonality carries over. The second is $b^2$, which is non-negative everywhere and therefore integrates to something strictly positive against any non-negative matching function. A metameric black is invisible; its square is not, and cannot be.

That last observation is the crisp form of the result. The $b^2$ term guarantees the two patches differ in the corner — it cannot cancel, because a squared function has no negative parts to cancel against. The only question was whether the difference is large enough to see, and the sweep above answers that at $q \approx 0.14$.

It also says which constructions elsewhere are exposed. Anything built by projecting onto the null space of a linear measurement is safe under that measurement and unsafe under any operation that squares the quantity measured.

Which term does the work #

The guarantee and the magnitude come out of different halves of that expression, and separating them is worth a paragraph, because the argument above has established only the guarantee.

Computed at the corner of a cube with the pair on this page, the cross term 2ρ_A b on its own moves the patch by 1.00 ΔE00, and the square term b² on its own by 0.39. The cross term carries two and a half times what the square does. So the square is what makes a difference certain — it is non-negative everywhere and has nothing to cancel against — and it is not what makes the difference large.

The two also point in different directions, which their tristimulus values show plainly. The cross term integrates to −34.4, 14.5 and 6.6 in X, Y and Z: a chromatic push that takes red out and puts a little of everything else back. The square term integrates to 66.5, 73.9 and 48.5, light added in all three, because a squared function has no way to subtract. The certain half of the effect is mostly a brightening and the large half is a shift of hue, and only the second is the kind of difference anybody calls a mismatch.

That division answers the objection somebody will raise about the square term being small. It is small — the black reaches 0.31 and its square reaches 0.096 — and it is still the term that makes the statement true. A guarantee that a difference is non-zero and an estimate of how big it is are different claims, and this pair supplies one each.

The second bounce is not the whole of it #

The argument is written about the second bounce, and at a real corner the second bounce is not where it ends.

Truncating the series after two bounces gives a separation of 1.14 ΔE00 at the corner of a cube. The full geometric sum gives 1.53. So 26 per cent of what the corner does arrives from the third bounce onward, in a geometry where only a fifth of the light comes back at all.

The reason is that every further term carries another power of the reflectance and none of them is more linear than the second was. The third bounce carries a cube, whose difference expands to three terms in ρ_A and b plus a b³, and the metameric black is orthogonal to the illuminant-weighted matching functions and to nothing else on that list. There is no order at which the identity comes back.

It is also why the sweep bends upward instead of running straight. Below an enclosure of 0.1 the separation is very nearly proportional to it, at about 6.7 ΔE00 per unit; by 0.95 that ratio has risen to 20.6. A straight line drawn from the shallow end would put the threshold of visibility at an enclosure of 0.149, and the computed crossing is at 0.1372, because the higher terms have already started to contribute.

The practical reading is that the enclosure figure is the wrong quantity to take comfort from. A groove returning a tenth of the light is not half as bad as a corner returning a fifth: it is 0.71 against 1.53, a factor of 2.2, and the ratio keeps widening as the geometry closes.

How deep the corner is decides how much of the light meets the surface twice, and doubling the earlier fraction says whether the failure grows with it.

The tolerance nobody quotes #

There is a practical consequence, and it lands on a document that does exist.

Colour tolerances are specified as a $\Delta E$ limit under a stated illuminant and observer, sometimes with a metamerism index attached to say how badly the match degrades under a second illuminant. A tolerance is a shape rather than a number, and the shape is negotiated carefully.

No part of that specification mentions geometry. A match certified at $\Delta E_{00} < 1$ on a flat plaque is a statement about a flat plaque, and the panel it is approving may be pressed, folded, corrugated or louvred. The figure above says the same two coatings will read at 1.5 in a right-angled corner and beyond 3 in anything deeper — several times the tolerance they were approved against, with no defect in either coating and no error in the measurement.

This is not a hypothetical failure mode of a specification. It is a case the specification does not have the vocabulary to describe, which is a different and worse thing: there is no field in which to write the geometry down.

The instruments make the same assumption in hardware. A spectrophotometer’s measuring geometry is standardised — 45°/0°, or a sphere with the specular component included or excluded — and every one of those standards describes how the instrument illuminates and views a flat, opaque, uniform sample. What the instrument reports is a property of that arrangement, and none of the standard arrangements return the sample’s own light to it. The measurement is therefore taken at $q = 0$ by construction, and reports the leftmost point of the sweep above.

The construction does not depend on daylight or on the two-degree observer, and building the pair under a tungsten lamp for the ten-degree observer is the check that says so.

How much of the light has to bounce twice #

A reasonable objection at this point is that the second bounce carries only a fifth of what the first one did, and that a correction that size should not move a colour by 1.5 units of $\Delta E$. Measured on this paint, whose mean reflectance is 0.425, the second-bounce term is 8.2 per cent of the light leaving the surface in a cube corner.

The objection is arithmetically right and draws the wrong conclusion. The second-bounce term is small, but the difference between the two patches lives entirely in it. The direct term contributes identically to both patches, by construction — that is what the match means. So the whole of the signal that distinguishes them is carried by the 4%, and the comparison that matters is not “4% against 100%” but “4% of a large spectral difference against zero”.

The spectra differ by up to 0.31 in reflectance. Eight per cent of a difference that size, weighted by $\rho_A$ and by a strictly positive $b^2$ term, is not small on a scale where 1.0 is the threshold of visibility.

This is why the effect is easy to miss by reasoning and hard to miss once computed. Intuition about interreflection is calibrated on brightness, where the second bounce genuinely is a small correction. For a difference between metamers the second bounce is not a correction at all — it is the entire signal.

Where the model stops #

Four limits, and the third is the one that would change the number most.

Lambertian only. Real paint is not an ideal diffuser, and a real corner concentrates light directionally. That changes $q$ for a given geometry and changes nothing about the algebra, because the algebra is about how many times $\rho$ appears rather than about where the light went.

One material. The sharpest version of the effect needs the second surface to be the same paint, so that the second bounce carries $\rho^2$ rather than $\rho\sigma$. A corner between two different colours breaks the match too, but by an amount depending on both, and the clean statement about squaring is lost.

$q$ is treated as a scalar and is not one. A real corner has a different form factor at every point along it, largest at the crease and falling away. So a real photograph of this shows a gradient of mismatch rather than a step, and the figures here report the value at a stated enclosure rather than an average over a visible surface.

The observer is fixed at the 1931 2° functions. The separation is a real physical difference in the arriving spectra, so it does not go away under another observer, but its magnitude in $\Delta E_{00}$ would move. This is a case where the choice genuinely matters and naming it is not a formality.

The same construction has an appearance half as well as a spectral one, and the two move together as the corner deepens.

That last figure carries a point worth making explicitly. The metameric black had to be rebuilt for the 1964 functions, because orthogonality to one set of matching functions is not orthogonality to another. Two spectra metameric for the 2° observer are generally not metameric for the 10° observer, which is observer metamerism and is measured elsewhere. What survives the change of observer is not the pair but the phenomenon: whichever pair is constructed, a corner breaks it.

A corner is two changes at once, and separating them under a tungsten lamp shows which part of the failure chromatic adaptation can be expected to absorb.

The generalisation #

The pattern is worth extracting from colour entirely.

A calibration performed against a linear model is valid for exactly as many applications of the underlying process as the model assumed, and silently invalid beyond it. A metameric match assumes the light meets the sample once. A gain measured on a single pass through an amplifier assumes one pass. A material property measured in transmission assumes the beam is not returning.

The tell is always the same shape: the specification names the stimulus, the instrument and the acceptance criterion, and has no field for how many times the process applies. When there is no field for it, the answer has been assumed to be one.

Within colour, the result explains something practitioners report and specifications deny — that metameric pairs are less reliable in real installations than laboratory numbers predict, and that the disagreements cluster at edges, folds and recesses. That clustering has usually been attributed to the light being different in a recess, which is true and is a smaller effect than this one. The light in a recess is different partly because it has bounced, and bouncing is what breaks the match.

Who found it, and when #

Metameric black is Wyszecki’s, from 1953, and the decomposition of a reflectance into a “fundamental” part the eye sees and a black part it does not is the standard framing ever since. Grassmann’s laws, which is what makes the match linear in the first place, are from 1853 — and Grassmann stated them for additive mixture, carefully.

The transport half arrived through a different discipline. Radiosity came into graphics in 1984, and spectral rendering — carrying full spectra through the transport rather than three channels — became practical in the 1990s. Researchers in that field have long known that spectral and RGB rendering disagree, which is the subject of the essay above this one.

What sits oddly is that the two literatures have rarely been pointed at each other. Colour science specifies matches on flat samples; rendering research asks how accurately a scene can be simulated. The question of whether a certified match survives a scene falls between them, and the arithmetic that answers it is one line long.

Where the ladder goes next #

The rung above asks the same question about the tool rather than the sample: what a three-channel renderer loses relative to a spectral one, measured bounce by bounce, where the error is exactly zero at the first product and grows at every one after it.

Two neighbours on this rung take the machinery outdoors and onto glossy surfaces. A shadow is lit by a different illuminant from the ground beside it, and a highlight carries the lamp rather than the paint — the second of which is the one piece of a scene that a white balancer can trust, and the first is the reason it usually cannot.