The third factor is a construction

The first rule of this collection is to name the observer on every figure, and it has been kept for nineteen rounds. Naming a thing is not the same as knowing what it is made of, and the thing being named turns out to have seven arguments.

The claim #

A set of colour-matching functions is a construction with arguments, and a standard observer is that construction evaluated at one setting of each, with none of the settings recorded on the figure.

• Seven arguments, of which six change the observer and one does not: a field size, an age, a macular density, a cone optical density, three peak wavelengths, a rod contribution — and a change of basis, which is a change of curves and not a change of eye.
• The six are within a factor of two of one another at their literature strengths, between 1.20 and 2.38 ΔE₀₀. There is no single dominant term.
• Every one of them vanishes exactly on a neutral sample, and every one vanishes exactly when two observers differ by a gain on each cone. The departure is a pairing, as the previous round’s four were.
• And the two exact conditions are exact only in the eye’s own coordinates. Imposed in a published cone space they leave 8.11 and 13.0 ΔE₀₀ behind, which is the arithmetic’s contribution rather than the eye’s.

What is being audited, and why now #

The colour integral is X = k Σ R S x̄ Δλ and this round has been taking its factors apart. The previous round opened the first and found a six-argument response function projected onto one number per wavelength. The first half of this round opened the index and found three decisions where the word “resolution” had held one.

The third factor is the observer, and it is the one this collection has been most careful about and least curious about. Every figure names it. /observers/ sorts every placement in the site by which one its caption strip declares. A whole gate exists to require the naming. And nothing anywhere asks what the named thing is a function of.

The plan for this round said the observer was the obvious next object because the substitutions already exist here — ten-degree functions, a pigment template, a population of two hundred — and only the plumbing was missing. That was right about the plumbing and wrong about the difficulty. Putting the substitutions on one footing turned out to need a modelled observer, which is the same obstacle the index audit hit, and for the same reason.

The seven, and what each one is #

A field size. The macular pigment covers the fovea and nothing else, so light arriving at two degrees passes through it and light arriving at ten mostly does not. The same person is two observers at two angles, and the CIE has published both since 1964 without saying that the difference is anatomical rather than statistical.

An age. The lens yellows steadily from about twenty, and a seventy-year-old receives roughly a third of the 440-nanometre light a twenty-year-old does. It is the one argument that is a trajectory rather than a spread: everybody moves along it in the same direction at about the same rate.

A macular density. Reported with a mean near 0.35 and individuals measured from nearly zero to above one, which is a spread of a factor of several rather than a few per cent.

A cone optical density. How much pigment the light passes through in the outer segment. It is the one argument whose effect is mostly a gain, and that turns out to decide how much it costs.

Three peak wavelengths. The long-wavelength pigment carries a common polymorphism worth several nanometres, and all three peaks have reported spreads of a nanometre and a half.

A rod contribution. At anything below a few candelas per square metre the rods are responding and their signal reaches the same bipolar cells the cones’ does, so what the observer has is a fourth curve summed into all three.

And a basis. Three curves span a space, and the space is what an experiment measures. Any invertible three-by-three produces three different curves for the same observer.

The seventh is not a measurement. The seventh row is the one worth having in the table, and its entry is exactly zero.

Take the reference observer’s three cone absorptances, apply an arbitrary invertible three-by-three to get three different curves, compose the tristimulus matrix with the same rotation’s inverse, and compute a colour. The answer is the same to 8 × 10⁻¹⁵ ΔE₀₀ — not nearly, but at the floating-point floor.

That is a theorem rather than a measurement and it settles a question that recurs in the literature. The CIE’s x̄ȳz̄ and a set of cone fundamentals are two bases for one space, and every reported disagreement between them is a disagreement about the space rather than about which curves span it. The matches do not name the cones, and the converse holds too: naming the cones does not change what the matches predict.

Printing a zero beside six numbers is the clearest available statement of what the six are measurements of. They are not measurements of how the curves are drawn. They are measurements of what the eye catches.

The ladder, and its flatness #

At the strengths the literature reports — the working-age lens, two standard deviations of the macular and density spreads, the long-wavelength polymorphism, the CIE’s own second observer, a rod contribution of a tenth — the six departures on a red pigment under a 6500 K radiator are:

A factor of two from top to bottom. That flatness is the finding, and it is a different shape from the previous round’s ladder, which spanned a factor of seven and invited a ranking.

There is no dominant term here and therefore no single repair. A collection wanting to reduce its observer uncertainty by a factor of two would have to address all six, and addressing the largest alone would buy a few per cent. That is worth stating because the field’s habit is the opposite: the lens is the term everybody names, on the strength of it being the only one that is visibly a physical filter.

Every one of the six is also above the ΔE of about one that a delivery tolerance is written in. So an observer’s identity is not a second-order correction to a colour specification; it is comparable with the whole tolerance, six times over.

The pairing, arriving from a different direction #

The previous round found that each of its four departures was the pairing of a deviation belonging to the sample with one belonging to the light, and that either factor being zero made it exactly zero. The same structure holds here with the factors renamed.

An observer’s departure is the pairing of a deviation belonging to the observer with one belonging to the stimulus, measured from the light the eye is adapted to. So:

A perfectly neutral sample is the same colour for every observer. Not nearly: exactly, at any age, at any field size, under any light, to between 4 × 10⁻¹⁴ and 4 × 10⁻¹³ ΔE₀₀. That is the same identity the index audit found, holding for a different reason and to the same precision.

An observer differing from another by a gain on each cone is the same observer, because the white-point normalisation is that gain’s inverse. This is von Kries’ hypothesis used as arithmetic rather than as a model of adaptation, and it explains why cone optical density — which looks like the largest single number in the table of individual variation — sits at the bottom of the ladder. Most of what a density change does is a gain.

Two exact conditions, two routes to zero, and both of them are places where nothing can be learned. That is the same awkward corollary the index audit produced: the samples an instrument is calibrated with are the samples that cannot report a fault.

The condition is exact in coordinates nobody publishes in #

The two identities hold when the white is divided out in the observer’s own cone responses. Imposed anywhere else they are not identities at all.

Dividing the white out in a published cone space — CAT16’s, which is what this collection uses everywhere and what most appearance work uses — leaves 13.0 ΔE₀₀ of the gain condition standing. Dividing it out in a rotated set of curves, as an analyst would who had only ever seen those curves and believed they were receptors, leaves 8.11.

Neither of those is small and neither is an error in anybody’s arithmetic. They are the cost of doing a division in a basis that is not the one the physiology divides in, and this collection has an essay saying a gain needs a basis that established the same thing from the direction of adaptation models.

What is new is that the identity gives the statement a floor. Before, the question was which basis is best and the answer was a table of comparisons. Now there is a basis in which the condition is exactly zero, and every other basis’s departure from it is measurable against that. The identity belongs to the eye, and every published arithmetic works in somebody else’s coordinates.

The flat ladder is a ladder of one sample, and reading it as a ranking of the six departures would repeat exactly the mistake the previous round’s own ladder had to warn against. Measured over forty-two surfaces the age departure runs from 0.66 to 6.63 ΔE₀₀, the macular from 0.26 to 4.24, and the density from 0.08 to 2.90 — spans of ten, sixteen and thirty-five.

The medians do put the six in an order, and it is not the order the single sample gave: age 1.94, macular 1.61, peaks 1.39, rods 1.03, field 0.74, density 0.57. Two rows have swapped and the top two are much closer than the example suggested. The strength of a departure is a property of a pairing, so a ladder is a ladder of examples, and the honest content of it is that all six examples are above a delivery tolerance rather than that any one of them leads.

The other factor of the pairing behaves the same way. The lens departure is 4.38 ΔE₀₀ under tungsten and 2.20 under a fluorescent tube; the peaks are 2.84 under a three-emitter LED and 1.87 under the same tube. Neither the lights nor the departures form a single ordering, which is what a genuine pairing looks like and what a single dominant term would not.

What this does not overturn #

An audit of this kind invites a reading it does not support, and the previous round’s opening essay had to make the same disclaimer.

The standard observer is not wrong. It is a projection, and the projection is exactly right under stated conditions — a two-degree field, a young lens, a normal macula, photopic levels, and a sample that is not being compared against another sample of different spectral shape. Those conditions are met by a great deal of industrial colour measurement, which is why the system works.

What the audit establishes is the size of each departure from those conditions, in a unit that can be compared against a tolerance, with the condition under which each vanishes stated exactly. That is a different thing from a demolition, and it is the reason the round is worth doing rather than a reason to stop using the tables.

What was computed, and how #

Every observer here is built from this collection’s own pigment template through its own ocular media — the same machinery its population of two hundred eyes is drawn from — and every one uses one fitted three-by-three to reach tristimulus values, so a difference between two rows is a difference in what the cones caught and never a difference in bookkeeping.

The residual of that construction against the published 1931 functions is 2.3 per cent root-mean-square on ȳ and 16.4 on z̄, a median of 1.42 ΔE₀₀ over forty-two surfaces. The reference had to be built for the index audit and it is the same construction here, with the same caveat: the comparative results carry the residual to second order, and the absolute ones carry it in full.

The lights and samples are the index audit’s, which is deliberate. Two audits sharing a set of stimuli can be crossed against one another, and this round’s sharpest single result comes from doing exactly that.

One consequence of the flat ladder is worth drawing out for anybody specifying a colour rather than auditing one. Six independent departures of comparable size do not add; they combine in quadrature if they are independent, which gives about 2.4 times the typical single term rather than six times it. So a specification meeting a population rather than a standard observer is carrying something like three to four ΔE₀₀ of observer uncertainty on an ordinary saturated sample — not the tenths of a unit that a tolerance is usually argued about at, and not the fifteen units that summing the ladder would suggest.

That number is not new to the field; it is what observer metamerism has always been worth. What is new is that it now decomposes, and the decomposition says which half of it a display designer can influence and which half is anatomy.

Where the model stops #

The rod contribution here is entered as a fraction of the cone response summed into all three channels, which is the crudest defensible model of rod intrusion. Real mesopic vision has a nonlinearity and a different weighting per channel, and the CIE’s own mesopic system is a two-parameter interpolation this file does not implement.

The field size is entered as two numbers — a macular density and a cone optical density — rather than as the CIE’s tabulated ten-degree functions. That is a model of what a field size does, and it reproduces the direction and roughly the size of the published difference without being the published difference.

And the peak spreads are treated as independent, which they are not: the long- and middle-wavelength pigments sit on the same chromosome and their polymorphisms are correlated in ways the population genetics literature describes and this file ignores.

The generalisation #

The habit is about the difference between naming a choice and knowing what it is.

A discipline that requires a choice to be declared has done most of the work, and it is easy to mistake that for all of it. This collection has required name the observer on every figure since its first week, and the requirement was satisfied 277 times out of 277 while sixty-two of sixty-four generators printed a string that had never once varied. Naming a constant is not naming a choice.

The next stage is to ask what the named object is a function of, and it is a different kind of question. It cannot be answered by a gate, because a gate can check that a field is filled in and cannot check that the field’s value was arrived at. It is answered by substituting — by building the object at two settings and measuring the difference — and the substitution is usually much more work than the naming.

Who found it, and when #

Every argument in the list has its own literature and none of it is new. Wyszecki and Stiles’ handbook tabulates the individual variation; the lens’s ageing has been measured since the 1950s and modelled in a standard form since van Norren and Vos; the CIE’s 2006 fundamental observer is explicitly a function of age and field size, which is the same list of arguments arrived at from the physiology.

What is not standard is putting them on one footing in a colour-difference unit and asking which condition each rests on. The 2006 model gives an observer for a stated age and field and does not report what varying them costs, because that is not what a standard is for.

Where the ladder goes next #

Six departures and two identities, and the identities are the interesting half. The next essays take each one apart: what makes a neutral everybody’s colour, and why a gain on a cone is not a different eye.