One match names the observer

Almost everything in colour science is a measurement of an average. The standard observer is seventeen people; a matching function is a mean; a difference formula is a fit to a dataset of judgements. Colour vision deficiency is the one place the discipline routinely measures an individual, and it does it with a single match — which is a different kind of statement from an average over seventeen people.

The match is a monochromatic yellow at 589 nanometres set against a mixture of 545 and 670. Two knobs, one comparison, and enough information to separate normal trichromacy from two kinds of anomaly and two kinds of dichromacy.

The claim #

A single colour match is a two-by-two linear system, and its conditioning separates the diagnoses.

At 545, 589 and 670 nanometres the short-wavelength cones respond negligibly, so only two classes are involved. The mixture must match the test in the long-wavelength class and in the medium-wavelength class: two equations, two unknowns — the mixture ratio and the yellow’s brightness.

Three things fall out, and none of them is put in:

• An anomalous observer’s match moves, and it moves in the direction their pigment moved. A protanomalous observer, whose long-wavelength pigment has shifted toward the medium one, needs more red. A deuteranomalous observer needs less.
• The accepted band widens as the conditioning falls — across the three regimes. A poorly conditioned system has a shallow minimum, so many mixtures are nearly right, which is what a clinical instrument reports as a widened matching range. Within a regime it does not, and the section below is that correction: the two anomalous observers differ by eleven per cent in conditioning and by a factor of five in band.
• And a dichromat’s system is singular. Two identical pigments give two identical equations, the determinant is zero, and every mixture matches at some brightness. That is not a large matching range; it is the absence of a unique solution, and it is the sharpest diagnosis in the subject.

The conditioning does not predict the band #

The second bullet is right about the regimes and wrong about the mechanism, and the table it stands on says so.

If the band were the reciprocal of the conditioning, the right-hand column would be a constant. It runs from 0.183 to 0.858, a factor of 4.7.

The two anomalous rows are the decisive pair. Their conditioning differs by eleven per cent — 0.183 against 0.165 — and their bands differ by a factor of 5.2. No monotone function of one produces the other over that pair, and there is no room to argue about precision: an eleven per cent difference in the supposed cause cannot make a five-fold difference in the supposed effect.

So the conditioning explains the singular case perfectly and does not explain the finite ones. A determinant of exactly zero gives a band of exactly the whole scale, which is an identity rather than a trend; between there and a well-separated pair, something else is setting the width.

What the band actually is, and why it is a different quantity. The conditioning is a property of the whole matrix — how nearly parallel its two rows are, in every direction at once. The band is the width of the residual’s minimum along one direction: the mixture ratio, with the brightness free to compensate. A matrix can be badly conditioned in a direction that the mixture knob does not move along, and then the two numbers come apart.

The essay’s own machinery says which direction that is. The protanomalous observer’s long-wavelength pigment has moved inward, towards the medium one; the deuteranomalous observer’s medium pigment has moved outward, towards the long one. The two shifts are of nearly the same size in nanometres and they are not the same shift, because the instrument’s two primaries sit at 545 and 670 and those are not symmetric about the pigments. One anomaly moves the pigments relative to the primaries and the other mostly moves them relative to each other, and only the second is what the mixture knob is scanning.

The displacement of the match itself orders the same way and is the more honest summary: protan sits 0.064 of the scale from normal and deutan 0.103, a ratio of 1.6 against the band’s 5.2. That is not a proportionality either, and it is at least the right sign, which the conditioning is not asked to be.

So the diagnostic table has three columns and two mechanisms. The determinant separates dichromacy from everything else, cleanly and by an identity. Where the match sits separates protan from deutan. And how wide the band is separates anomalous from normal — but it is not the determinant doing it, and the essay had one number standing in for two.

The whole of the diagnosis is a determinant #

The step worth stating carefully is why a ratio of two pigment peaks turns into a clinical category.

Write the two equations as a matrix: the columns are the two unknowns, the rows are the two responding cone classes. The system has a unique solution when the matrix is invertible, and how stably it can be solved is the determinant relative to the size of its entries.

Two well-separated pigments respond very differently to a red primary and a green one, so the two rows are far from parallel, the determinant is large, and one mixture matches and its neighbours plainly do not. Two pigments eleven nanometres apart respond similarly to everything, the rows are nearly parallel, and a whole band of mixtures is nearly right. Two identical pigments give two identical rows, and the whole line of solutions matches equally.

So the three clinical categories are three regimes — the section above found that one number does not order them within a regime, and the determinant is what separates the singular case from the rest. Nothing here is fitted to clinical data: the pigments are this site’s own template at stated peaks, the media are its own media, and the match is the solution of the system.

How strict the criterion is changes every band on the scale, and it is a setting the instrument leaves to whoever is running it.

What the two knobs are for #

An anomaloscope has two controls and the reason for the second is worth stating, because it is the thing that makes a mismatch impossible to hide.

The mixture knob sets the ratio of red to green. The brightness knob sets the yellow’s radiance. With both free, an observer whose two pigments differ can always satisfy both equations exactly — so the match is not a judgement about how close two fields look, it is an exact setting that exists.

An observer whose two pigments are the same can also satisfy both equations at every mixture setting, by moving the brightness knob to compensate. That is why a dichromat’s report is not nothing matches but everything matches, and it is why the instrument’s output is a range rather than a point.

The clinical reading is therefore two numbers: where the range sits and how wide it is. The first separates protan from deutan and the second separates anomalous from dichromatic — and both come out of the same two-by-two system.

Where this connects to everything else on the site #

The Rayleigh match is usually filed under clinical vision testing, and it is the same object as three arguments this site has already made.

It is observer metamerism, at its most extreme. Whose eyes measured what it costs when two observers with slightly different pigments look at the same pair of lights. This is the same measurement with the difference made enormous: shift a peak by fourteen nanometres and the mixture that matches moves by seven per cent of the scale.

It is a metameric pair whose partner is a mixture. 589 nanometres and a mixture of 545 and 670 are a metameric pair for a normal observer — same three numbers, different spectra — and the fact that different observers need different mixtures is precisely observer metamerism rather than illuminant metamerism.

And it is why simulating colour vision deficiency is hard. Simulating what cannot be simulated makes the point that anomalous trichromacy is not a faded dichromacy: an anomalous observer has three working pigments, two of which are close together, and the correct model shifts a peak rather than fading a channel. The anomaloscope is the measurement that establishes it — a fading model predicts a widened band centred where a normal observer’s is, and the measurement says the centre moves.

What a display can and cannot do with this #

Every online “colour blindness test” is a version of this measurement delivered through a screen, and the reason they do not work as diagnostics is algebraic rather than practical.

The wavelengths are the apparatus. The match works because at 545, 589 and 670 nanometres the short-wavelength cones contribute almost nothing, which reduces a three-dimensional problem to two dimensions and makes a two-knob instrument sufficient. A display’s primaries are three broad emissions, all of which excite all three cone classes; there is no setting of a screen that produces monochromatic 589 nanometres, and no combination of three primaries reaches the spectral locus.

And the display is unknown. The display is an unknown is the site’s standing caution and it bites hardest here: a diagnostic that depends on the exact spectral content of what is emitted cannot be delivered by a device whose spectral content is not specified anywhere in the file being sent.

What a screen can do is the plate tests — Ishihara and its relatives — which ask whether two regions of a picture are distinguishable rather than asking for a setting. Those are robust to the primaries because they need a difference to survive rather than a specific stimulus to be reproduced, and they classify rather than measure: they say something is unusual, and the anomaloscope says which pigment and how far.

What was computed, and how #

The observers are built from a pigment template at stated peaks. 566 and 541 nanometres for a normal trichromat; the anomalous peaks are quoted with their reported ranges, both giving separations of ten or eleven nanometres against a normal twenty-five. The dichromats are the degenerate case — one pigment used twice — which makes the system exactly singular rather than nearly so.

The system is solved algebraically, not searched. Two linear equations, one determinant, one closed-form solution. The conditioning is the absolute determinant relative to the square of the largest entry, so it is dimensionless and comparable across observers.

The accepted band is a scan. For each mixture setting, the best brightness is the least-squares scale on the two responding classes — which is what an observer turning one knob does — and the mixture is accepted when the residual is under the criterion. The criterion is one per cent of a cone excitation and it is stated; everything widens together when it moves.

And the solution is checked against the equations it came from. assertTheMatchIsUnique recomputes both cone classes from the returned mixture and requires them to agree with the test to 10⁻⁹, which is the difference between having solved the system and having returned a number.

Where the model stops #

These are not clinical scales. A real anomaloscope reports on a scale of its own, with the mixture setting running from 0 to 73 and calibrated so that a normal observer’s midpoint is near 40. The quantity here is a radiance fraction, and 0.879 red is a fraction of the mixture’s power rather than a position on any instrument’s dial — most of it goes to the red primary because 670 nanometres sits far out on the long-wavelength cone’s tail.

The pigment peaks are quoted and the anomalies are representative. Anomalous trichromacy is a continuum: separations from two to fifteen nanometres are reported, the match moves continuously with the separation, and the two rows in the table are two points on that continuum rather than two conditions.

There is no noise. A real observer’s setting varies from trial to trial, in the way every threshold does, and part of what a clinical instrument measures is that spread. Here the band’s width comes entirely from the conditioning, which is the systematic part, and a real range is wider.

And there is no rod contribution and no macular pigment difference. Both are named in the site’s machinery and neither is in this computation, and both are reasons two normal observers do not give quite the same setting.

The generalisation #

The sentence worth carrying: a colour match is a linear system, and everything interesting about an observer is in its conditioning.

That reframes several things at once. A colour space is well conditioned when its primaries are far apart, which is why narrow primaries make observer differences worse — narrow primaries do not change the observers, they change how nearly parallel the rows of the matching system are. A metameric pair comes apart under a new illuminant for the same algebraic reason. And an instrument that reports a match to three decimal places on a badly conditioned system is reporting precision it does not have.

The surprising connection is with the separation of a printing press. Four inks and three numbers to match give an underdetermined system, so a separation is a family rather than a value; two identical pigments and two equations give a singular system, so a match is a family rather than a value. Both are the same statement about a matrix, arrived at from opposite ends of the pipeline, and neither field’s vocabulary borrows from the other’s.

Who found it, and when #

Lord Rayleigh described the match in 1881, having noticed that some members of his own family set it differently from others. The instrument built to make it routine is Nagel’s anomaloscope, from 1907, and its descendants are still the reference method a century later.

Why it works so well was not understood until the pigments were measured. Rayleigh had a diagnostic before anybody knew there were three pigments, let alone that anomalous trichromacy is a shift in one of them — the genetics arrived in the 1980s, and confirmed that the anomalous pigments are hybrids of the normal two, with peaks in between.

And its longevity is unusual. Almost every other nineteenth-century colour instrument has been replaced by a spectroradiometer and a computation. This one has not, because the quantity it measures is a property of one person and the only available detector is that person.

What the pictures cannot show #

They cannot make the match. The whole point of the instrument is that the observer sets the knobs, and a page can only draw the scale and the model’s prediction for it. A reader curious about their own setting needs an anomaloscope or one of its approximations, and any version of it delivered on a display is subject to the display’s own primaries — which are not 545, 589 and 670 nanometres and cannot be made to be.

And the singular case cannot be drawn as a swatch. Every mixture matches is a statement about a whole line of settings, and drawing it as a bar spanning the scale is the closest a figure can come to a picture of an absence.

Where the ladder goes next #

The nearest unfinished piece is the continuum. The two anomalous rows in the table are two points; sweeping the pigment separation from twenty-five nanometres to zero would give the match position and the band width as continuous functions, and would say what separation corresponds to a stated clinical range.

The second is the join to the standard observer. A set of matching functions is an average over people whose pigments differ, and this essay’s machinery says exactly how the match moves with the pigment — so the spread of settings in a population is computable from the spread of pigment peaks. That is the quantity seventeen observers in 1931 had to estimate from the data rather than from the mechanism.