Two spectra, one colour

Two objects can reflect measurably different spectra and be, for a human observer, exactly the same colour. This is called metamerism, it is the reason colour reproduction works at all, and it is the reason a shirt and a jacket can match in a shop and clash in the street.

It is also, in most treatments, asserted. The reader is told metamers exist and shown a diagram of two curves with a note that they produce the same response. Here they are constructed, verified, and drawn.

That last clause matters more than it looks. If either patch had been outside the display gamut, both would have been clamped toward the edge and might have agreed on screen for entirely the wrong reason. The demonstration would then have proved nothing except that clipping destroys differences. Every metamer figure on this site checks the gamut before claiming a match.

Why they are easy to construct #

The eye reports three numbers, obtained by integrating the spectrum against three fixed functions. Write those functions as the rows of a matrix $A$ and the spectrum as a vector $s$, and the response is $A s$.

Two spectra match exactly when

$$A s_1 = A s_2 \quad\Longleftrightarrow\quad A(s_1 - s_2) = 0$$

so the difference between any metameric pair lies in the null space of $A$. And $A$ has three rows against eighty-one columns in this site’s sampling, so its null space has seventy-eight dimensions.

That turns metamerism from something to search for into something to build. Take any spectrum at all, project out the part that affects the response, and what remains is a metameric black — a spectrum the eye integrates to nothing. Add it to anything and the colour is unchanged.

The projection is one line of linear algebra:

$$b = f - A^{\mathsf{T}}(A A^{\mathsf{T}})^{-1} A f$$

for any candidate $f$. Since $A A^{\mathsf{T}}$ is only three by three, this is cheap and exact. There is no optimisation and no search — every candidate produces a perfect metameric black on the first attempt.

The only real constraint #

The mathematics is unconstrained; physics is not. A reflectance is a fraction of light returned at each wavelength, so it must lie between zero and one. A metameric black can be added only in an amount that keeps the sum inside those bounds, and that is the sole limit on how different the two curves can be made.

For the pair in the hero figure the limit allows a twenty per cent RMS difference. Pushing further requires a base reflectance sitting nearer the middle of the range, with room to move in both directions — which is why the base spectra on this site are mid-grey rather than near-white or near-black. A very dark or very light surface has few metamers, because there is nowhere for the difference to go.

When the light changes #

The match is a match against one illuminant. Change the light and the spectra are weighted differently, and there is no reason for the integrals to stay equal.

This is illuminant metamerism, and it is the version that costs people money. Two dye formulations matched under the shop’s lighting will separate under daylight; a repaired car panel matches in the workshop and does not on the road; a printed proof matches the screen under one lamp and not another. Whole standards exist for specifying the illuminant under which a match must hold, because a match is not a property of two surfaces alone.

A finding worth recording #

The pair in that figure was searched for, not simply constructed, and the reason is a mild surprise that came out of building this site.

An arbitrary metameric black does not generally produce a good illuminant-metamerism demonstration. The first pair tried here matched under D65 to ΔE₀₀ of 7×10⁻¹⁴ — perfect — and still matched under illuminant A to ΔE₀₀ = 0.52, which is invisible. Two lights, two matches, nothing to show.

The reason is straightforward once seen. Illuminant metamerism needs the two spectra to differ where the two illuminants differ. D65 and illuminant A separate mostly toward the red end, so a metameric black concentrated in the middle of the range survives the swap intact. Scanning over frequency and phase finds one that does not, and the scan is cheap because every candidate is already an exact metamer — the search is over demonstrations, not over solutions.

The general lesson is that metamerism has degrees. Pairs differ in how robustly they match, and colour science has an index for it precisely because some metamers are far more fragile than others.

What this makes possible #

Metamerism is usually introduced as a nuisance. It is better understood as the enabling condition for the entire field.

A display does not reproduce the spectrum of a leaf. It emits some mixture of three primaries that happens to be a metamer of the leaf for a human observer. Nothing about the physical light is reproduced; what is reproduced is the triple of cone responses, which is all the eye was ever going to report. The same is true of every colour technology — print, photography, paint matching, stage lighting.

Which is also why a photograph does not reproduce colour for a bird. A tetrachromat integrates the spectrum four ways, and a metamer for three functions is generally not a metamer for four.

Where it bites in practice #

Four places, all of them consequences of the same arithmetic.

Paint and dye matching. Two formulations built from different pigments can match under a specified illuminant and separate under another. The industry’s response is to specify the illuminant in the standard, and to prefer matches built from the same pigments, since those match under every light rather than under one. A match achieved with different colorants is called a metameric match and is regarded as inferior even when the ΔE under the reference light is smaller.

Camera colour. A camera’s three filters are not the human matching functions and cannot be made into them. So the camera’s metamers are not human metamers: two surfaces looking identical to a person can differ in the raw file, and two that differ can come out identical. Every camera profile is an attempt to minimise the discrepancy across the surfaces likely to be photographed, and none of them eliminates it.

Printing. Ink on paper reflects; the ink set has its own spectra; the viewing light is whatever the room provides. A proof matching the press under the standardised viewing booth is not guaranteed to match under the client’s office lighting, and the failure is illuminant metamerism rather than an error by anybody.

Lighting design. A lamp’s chromaticity says almost nothing about how surfaces will look under it, because surfaces reweight the spectrum before the eye sees it. Two lamps with identical white points can render a room’s colours quite differently, which is what colour rendering indices attempt to capture.

How different can two metamers be? #

The bound is set by physics rather than by the linear algebra, and it is worth seeing where it comes from.

A reflectance must stay between zero and one. Adding a metameric black scaled by $t$ keeps the sum in range only up to some maximum $t$, and that maximum depends on how much headroom the base spectrum has. A mid-grey surface reflecting around 0.4 across the range can absorb a large metameric black; a surface near 0.95 or near 0.02 has almost nowhere to go.

So near-white and near-black surfaces have few metamers, and mid-tones have many. This has a satisfying consequence in practice: white and black are the easiest colours to match across different materials, and mid-tone colours — particularly greys, browns and muted greens — are the hardest. Anyone who has tried to match a grey car panel or a beige wall has met the mathematics directly.

Emissive sources are unconstrained above, having no ceiling analogous to a reflectance of one, so lamps admit far more extreme metamers than surfaces do. This is exactly why two lamps of identical chromaticity can have such different spectra.

The index nobody quotes #

Because metamers vary in how robustly they match, colour science has a measure for it, and the measure is instructive even though it is rarely seen outside industrial practice.

A metamerism index takes a pair matching under a reference illuminant and reports the colour difference under a test illuminant. Low means the match survives a change of light; high means it does not. The CIE specifies it for illuminant changes and for observer changes, since the same pair can also come apart between two people with slightly different matching functions.

The index makes precise something the construction on this page runs into directly. The pair in the illuminant-metamerism figure was selected for a high index, because a low-index pair demonstrates nothing visually — the first candidate tried had an index so low that the two patches stayed indistinguishable under both lights.

For industrial use the preference runs the other way. A paint match with a low metamerism index is worth more than one with a smaller colour difference under the reference light, because it will survive the customer taking the sample outside.

Observer metamerism #

Everything above concerns changing the light. The other variable is the observer, and it is less discussed and equally real.

The standard observer is an average, and real people differ — in macular pigment density, in lens yellowing with age, in the exact spectral tuning of the L and M pigments, which is genetically polymorphic. Two surfaces matching for one normal trichromat can be distinguishable to another.

This is why colour matching in critical applications is done with instruments rather than by eye, and why matching booths are staffed by people whose colour vision has been screened. It also means that a metameric match is a match for a specified observer, and the specification is doing work that is easy to forget about.

That framing is worth keeping. A dichromat does not see a subset of the colours a trichromat sees; a dichromat has a two-dimensional colour space in which more physical stimuli collapse together. The simulation question follows from exactly that.

What was computed here #

Every pair on this page is constructed by the projection above and then checked in four ways before it is drawn:

• The two spectra must agree in XYZ to a tolerance far below any visible difference. They agree to around 10⁻¹⁴.
• The two spectra must genuinely differ — a pair agreeing to twelve decimal places in both colour and spectrum would satisfy the first check and demonstrate nothing. The RMS difference must exceed five per cent.
• Both patches must be inside the display gamut, for the reason given at the top.
• Every reflectance value must stay in $[0, 1]$, or the pair is not physically realisable and the construction throws.

The metameric black is checked separately and more sharply: its integral against all three functions must be zero to arithmetic noise relative to the scale of the inputs. That is a cancellation statement, of the same kind as a symmetry-forbidden integral, and a merely small number would not do.

The gate also confirms these checks still reject. A pair differing by 10⁻⁶ is offered to the verifier and must be refused, and the illuminant-metamerism check is offered a pair that matches under both lights and must refuse that too.

The one-line version #

Colour reproduction does not reproduce light. It constructs a metamer, and it works because the eye reports three numbers rather than a spectrum.

Why this is the enabling fact #

Every colour technology depends on it. Nothing reproduces a spectrum; everything constructs a metamer.

It is also the fact that makes colour science tractable. If the eye reported spectra, colour would be a subject about functions and every reproduction problem would be infinite-dimensional. Because it reports three numbers, the whole apparatus reduces to linear algebra on three-vectors — which is why a field about human perception can be conducted with matrices, and why the matrices are only three by three.

What the pictures cannot show #

The patches are shown under a simulation of two illuminants, not under two illuminants. The reader’s screen is emitting one set of primaries in one room; what the figure shows is what a camera would record, or what the surfaces would look like, under those lights. The chain from “this surface under tungsten” to “these pixels under whatever is illuminating the reader’s desk” involves an adaptation step the figure does not model.

And the metameric black cannot be drawn as what it is. The figure plots its amplitude against wavelength, which is a graph of a function. What the object is — a physically real difference in light that no human can detect under any conditions — has no visual representation at all.

Who found it, and when #

The phenomenon was known to dyers long before it was named; matching batches under different lights is an old and practical problem. Wilhelm Ostwald gave it the name around 1900.

The mathematical treatment as a null space came much later, with Jozef Cohen and Brian Wandell’s work in the 1970s and 1980s formalising the decomposition of a spectrum into a “fundamental” part that the eye sees and a metameric black that it does not. That decomposition is what makes the construction on this page a one-liner rather than a search.

Where this goes next #

The construction depends entirely on there being exactly three matching functions, so why colour is exactly three-dimensional is the natural companion. For the practical side of the illuminant dependence, the illuminant is half the answer. And since metamerism is what lets three primaries stand in for everything, what a gamut costs takes up where that substitution runs out.