A field size is two changes

The CIE’s two standard observers are usually explained by saying that a large patch and a small one look slightly different. That is true and it is not an account of anything, because a field size is not a property of an eye. Two things about the retina are.

The claim #

A field size is not an argument of the visual system. It is a proxy for two anatomical changes that happen together and act in different places.

• The macular pigment falls, from about 0.35 peak density at the fovea to about half that at ten degrees, because the pigment is concentrated in the central degree or two.
• The cone optical density falls, from about 0.4 to about 0.3, because foveal cones are longer and thinner than peripheral ones and the light travels further through the pigment.
• The two act in different places on the wavelength axis. The macular change is a band at 460 nanometres; the density change is a broadening of all three curves.
• And they are worth different amounts. Separated, the macular half is 0.94 ΔE₀₀ on a red pigment under daylight and the density half is 0.44, against 1.33 for the two together.

Why a field size is a proxy #

Nothing in the retina reads an angle. What changes when a stimulus grows from two degrees to ten is which photoreceptors it lands on, and the receptors differ from one another in ways that have nothing to do with the stimulus.

Two of those ways matter for colour matching. The macular pigment is a yellow carotenoid deposited in the inner layers over the central few degrees, so light reaching a foveal cone has passed through it and light reaching a cone at eight degrees has mostly not. And foveal cones are long and narrow while peripheral ones are shorter and fatter, so a photon’s path through the photopigment is longer at the centre — which raises the effective optical density and, by self-screening, broadens the sensitivity.

Both are properties of place rather than of angle, which is why the same person is two observers at two eccentricities and why the CIE’s second observer is not a different kind of person. A ten-degree stimulus is a two-degree observer plus an eight-degree annulus, and what the standard reports is an average over the disc.

That has a consequence nobody states: the ten-degree observer is an average over an inhomogeneous retina, so it describes no receptor anywhere. The two-degree observer describes a foveal cone; the ten-degree observer describes a weighted mixture, and the weighting depends on the stimulus’s shape.

The two halves, separated #

Modelling the change as two parameters rather than as a table makes it possible to move them one at a time, which the tabulated observers cannot.

The two add to 1.38 and together give 1.33, so they very nearly add and slightly under-add. That is what two departures acting in different spectral regions should do: their directions in stimulus space are close to orthogonal, so their combination is close to the quadrature sum in excitations and close to the linear sum in a difference formula that is nearly linear over this range.

The macular half is twice the density half, on this sample. That ratio is not stable — the density half rises on samples with structure at the cone flanks and the macular half rises on samples with structure at 460 nanometres — but the ordering holds across the surface family, where the medians are 0.51 and 0.24.

So most of what the CIE’s two observers differ by is a pigment in front of the receptors rather than anything about the receptors themselves, and the pigment is the same filter that varies threefold between individuals.

The distribution shows something the single number cannot: the field-size departure has a long tail. Its median over the family is 0.740 ΔE₀₀ and its ninety-fifth percentile is 2.477, a ratio of three and a third, which is larger than the rods’ or the peaks’.

The tail belongs to the macular half. A sample with spectral structure in the 460-nanometre band is affected strongly by a change in macular density and a sample without such structure is barely affected at all, so the population of samples splits into two groups rather than spreading smoothly. The density half contributes a small effect to everything and the macular half contributes a large effect to some things, and their sum inherits the shape of the second.

That matters for the practical question of when the choice between the two observers is worth making. On most samples it is worth under a unit and on blue-structured samples it is worth two and a half, and there is no intermediate regime — which is why practitioners’ experience of the two observers is that they usually agree and occasionally do not.

The comparison that ought to be embarrassing #

Putting the field-size departure beside the macular departure in the same ladder produces a result worth stating plainly.

The CIE publishes two standard observers, industry maintains two sets of tables, instruments offer both, and every specification has to say which. The difference between them, on a red pigment under daylight, is 1.33 ΔE₀₀.

The difference between two individuals two standard deviations apart in macular pigment density, both using the two-degree observer, is 1.71 — larger. The difference between a twenty-year-old and a seventy-year-old is 2.38, nearly twice as large.

So the choice a specification agonises over is smaller than the variation it ignores entirely. That is not an argument for abandoning the two observers, since the field-size effect is systematic and the individual variation is not, and a systematic effect can be corrected while a random one can only be budgeted for. It is an argument for budgeting for the second at least as carefully as the first, and no specification this collection has read does. The same asymmetry runs through the tolerance work: the terms that get budgeted are the ones with a standard attached, and the terms without one are omitted rather than estimated.

The collection has made the same point once before from measurement rather than from a ladder: the two observers differ by 2.65 ΔE₀₀ over its own test surfaces, which is larger than the 1.33 here because that measurement uses the tabulated functions and this one uses a two-parameter model of what a field size does.

The model is smaller than the tables, and the gap is a measurement.

The gap between 1.33 and 2.65 deserves an explanation rather than a shrug, and it is instructive.

The two tabulated observers differ by more than two anatomical parameters. They were measured in different experiments, twelve years apart, with different primaries and different observers, and the 1964 functions were derived from Stiles and Burch’s data while the 1931 set descends from Wright’s and Guild’s. Some of the difference between them is field size; some of it is thirty-three years of experimental practice, a different subject panel, and a different luminance constraint.

The model here has only the anatomy in it. It reproduces the direction and about half the magnitude, which is a reasonable outcome for a two-parameter account of a difference that has more than two causes.

That gap is itself a measurement. If a field size were the whole story, a two-parameter model of it would reproduce the tabulated difference, and it reproduces half. So roughly half of what separates the two standard observers is not a field size at all, and is not attributable from anything in this collection.

What a two-parameter model buys that two tables do not #

The whole method of this essay is to replace two tabulated observers with one construction taking two arguments, and the replacement buys three things the tables cannot give.

It separates the halves. Two tables can be subtracted and the difference cannot be attributed. A model with two parameters can be moved one at a time, and the attribution — two-thirds macular, one-third density — is not obtainable any other way.

It interpolates. A five-degree field is not tabulated anywhere and is an extremely common stimulus; a model gives it, with the caveat that the macular profile is being approximated by a two-point interpolation of an average.

And it composes with the other parameters. A ten-degree observer aged seventy is not a table and is a perfectly ordinary person. The tabulated observers cannot be aged, and the CIE’s 2006 model was introduced precisely so that they could be.

Against those, the model reproduces about half the tabulated difference, which is the price. A practitioner needing the standard’s number should use the standard’s tables; a reader wanting to know what a field size is cannot get it from them.

What a field size does not change #

Three things stay fixed across the change, and knowing what does not move is as useful as knowing what does.

The peaks do not move. A cone at ten degrees has the same photopigment as a cone at the fovea, with the same λmax; what differs is how much of it the light goes through. So the field-size departure is entirely a filter-and-density effect and shares no mechanism with the polymorphism.

The number of cone types does not change, and neither does their basic arrangement into opponent channels. A ten-degree stimulus is still trichromatic.

And the rods do not enter at photopic levels, though they would at lower ones, and at ten degrees there are a great many more of them. That is a genuine interaction the model does not carry: the rod contribution and the field size are coupled through rod density, and a large dim field is a quite different observer from a large bright one.

The flattest row in the table #

The field-size departure varies less across lights than any other in this round: 1.085 under tungsten, 1.326 under daylight, 0.856 under a white LED, 1.067 under a three-emitter LED, 1.146 under a fluorescent tube. A range of one and a half, against a factor of two for the lens and two and a half for the macular.

That flatness has a cause worth naming. The departure is a sum of two effects in different spectral regions — a band at 460 and a broadening across all three curves — and a light that emphasises one tends to de-emphasise the other. A blue-rich source excites the macular half and gives the density half less to work with in the long wavelengths; a warm source does the reverse.

Two departures in different places average into a stable one, which is the same reason a population’s aggregate observer variation is flatter than any of its components. It also means the field-size choice is the one observer decision whose cost can be quoted as a single number with a straight face, which is not true of any of the others in the round.

Repeating the light table on a second sample checks that the field size’s flatness is a property of the departure rather than of the red pigment. On the notch filter its row runs from 0.33 to 0.62 ΔE₀₀ across the five well-sampled lights — a narrower range than any other row, as it was before.

That rise is a small confirmation of the decomposition: a lamp that excites the macular pigment excites the field size in proportion to the macular half of it, and leaves the density half alone.

What was computed, and how #

The ten-degree observer is modelled as macular density 0.17 and cone optical density 0.30 against the two-degree values of 0.35 and 0.40. Those figures are the ones the CIE’s 2006 physiological observer uses for the two field sizes, and they are the model’s only inputs about field size.

The separated halves are computed by moving one parameter at a time from the two-degree values, so each is a difference between the standard observer and a hybrid that exists nowhere. That is legitimate for decomposition and it is not a claim that either hybrid is anybody’s eye.

Everything is computed through this collection’s usual CAT16 route, so the numbers are comparable with the rest of the round and carry the route’s own contribution.

One more thing follows for the collection itself. Every figure here that names the 1964 observer is naming a table, and this round has only a model of one — so nothing in the round supersedes what the tabulated comparison measured, and the two results are about different objects. A reader wanting the standard’s number should take 2.65 ΔE₀₀ over this collection’s test surfaces; a reader wanting to know what a field size is made of should take the 0.94 and the 0.44.

Keeping both is not indecision. It is the same arrangement the round has used everywhere: a model is what gets built when the questions outgrow the data, and the data are still the data.

Where the model stops #

The two parameters are the two the CIE’s own physiological model uses, and there is at least one more real effect: the ratio of cone types changes with eccentricity, with proportionally fewer short-wavelength cones at the fovea’s very centre and none at all in the central twenty minutes. Nothing here carries that, because the model has no cone ratio in it — a set of colour-matching functions is three curves and the ratio affects luminance rather than matching.

The macular pigment’s spatial profile is modelled as two values rather than as a profile. A real ten-degree field averages a continuum from 0.35 at the centre to near zero at the edge, and the average of the responses is not the response of the average.

And the model reproduces about half the tabulated difference, which is stated above and is the honest boundary of everything in this essay.

There is also a small correction to make to how this collection has been talking about the two observers. Its figure captions and its /observers/ index treat them as two alternatives with a choice between them, which is how the standards present them. What the decomposition says is that they are two points on a two-dimensional surface, and that a great many real viewing situations sit at neither — a ten-degree field viewed by a sixty-year-old is not the 1964 observer, and a two-degree field at a colour-matching booth is not the 1931 one either unless the assessor happens to be twenty-something.

Nothing in the collection’s machinery is wrong as a result, because every figure that names an observer is computing with the table it names. What is slightly misleading is the impression that naming one of two settles the question, and the round’s answer is that it settles two of six parameters.

The generalisation #

The habit is about parameters that name a stimulus and act on a system.

Field size is a property of what is being looked at. Macular density and cone optical density are properties of the looker. A model parameterised by the first is describing the second through a proxy, and the proxy is only as good as the correspondence between them — which here is decent for a centred disc and poor for anything else.

The tell is that the parameter’s units belong to the wrong side of the interaction. An observer measured in degrees is an observer whose model has absorbed something about the stimulus, and the absorption is usually invisible until somebody asks about a stimulus of a different shape. An annulus, a bipartite field with one half at the centre, a moving target: none of them is described by a field size, and all of them are described by a macular density profile.

The repair is always to parameterise the system rather than the experiment, and it is always more work, because the system’s parameters have to be measured separately and the experiment’s are read off the apparatus.

Who found it, and when #

The macular pigment was described by Maxwell in 1856 as a yellow spot affecting his own colour matches, and its role in the difference between central and peripheral matching was established well before the CIE adopted a second observer in 1964.

The self-screening account of why field size changes cone optical density is later and comes from the microspectrophotometry of the 1970s and 1980s. The CIE’s 2006 fundamental observer is the first standard to parameterise the two explicitly, giving a formula in field size and age rather than two tables — which is the move this essay is about, made officially and then largely ignored in practice, where the 1931 and 1964 tables remain what everybody uses.

Where the ladder goes next #

The other parameter the 2006 observer takes is one nobody thinks of as an observer parameter at all, and it is the only one in this round that is not a spread but a trajectory. A standard observer has no age, and everybody has one.