The macular is a band, not a filter

Two of the six departures in this round are absorptions in front of the receptors, they are of comparable size, and they behave completely differently across samples and across lights. The difference between them is where they sit and how wide they are.

The claim #

A departure’s spread across samples is decided by how localised it is on the wavelength axis, not by how large it is.

• The macular pigment is a band: a Gaussian absorbance centred near 460 nanometres with a width of about forty, and nothing outside it.
• The lens is a tail: an absorbance rising steadily below 500 nanometres with no edge.
• The band has the wider distribution. Across forty-two surfaces the macular departure spans a factor of seventeen and the lens a factor of ten, despite the lens’s median being larger.
• And the band decides which lights are dangerous. A blue-pumped white LED puts its narrowest feature almost exactly on the macular band, which is why the macular departure is its second-largest.

Two absorptions, two shapes #

The macular pigment is lutein and zeaxanthin deposited in the inner retinal layers over the central few degrees. Its absorbance spectrum has a characteristic double-humped shape peaking around 460 nanometres and falling to nothing by about 530, and its peak optical density varies between individuals from nearly zero to above one, with a mean near 0.35.

The lens’s absorbance is quite different in shape. It rises monotonically towards short wavelengths with no peak at all inside the visible band, so what it does is progressively attenuate everything below about 500 nanometres, more the shorter the wavelength.

Both sit in front of the receptors and neither is a property of the photopigments. Both vary between people. Both are, in the language of this round, multiplicative departures that a white point cannot absorb because they change the shape of the three curves rather than scaling them.

And their statistics across samples are quite different, which is the essay’s subject.

The distributions #

Over the forty-two-surface family under a 6500 K radiator:

The lens has the larger median and the smaller span. That is the arrangement a localised departure and a broad one should produce, and the mechanism is a partition.

A broad absorption touches every sample. Any reflectance with structure anywhere below 500 nanometres is affected by the lens, and almost every reflectance has some. So the lens’s distribution has a high floor — its smallest member is 0.656 — and rises from there in proportion to how much blue structure each sample has.

A narrow absorption partitions the samples into two groups. A reflectance with structure at 460 nanometres is affected strongly by the macular pigment and one without is barely affected at all. So the macular’s distribution has a low floor — 0.255 — and a long upper reach, and its shape is closer to bimodal than to smooth.

The general rule is that localisation buys variance, and it does so at fixed mean. A departure spread over the whole band averages, and a departure concentrated in one place either hits or misses.

Which lights excite which #

The same logic applies to the second factor of the pairing, and it produces a table that is not what a reader would predict from the lights’ colour temperatures.

The white LED is second for the macular and fourth for the lens, and its position is the interesting one. A blue-pumped LED has a narrow emission peak at about 452 nanometres — a Gaussian of twenty-two nanometres’ width, in this collection’s construction — which sits almost exactly on the macular absorption band.

So the one part of the spectrum where observers differ most is the one part that lamp concentrates its power into. A lighting technology chosen for efficiency has put its narrowest feature on top of the eye’s most variable filter, and nothing in the design of white LEDs considered that. The luminaire work in this collection has looked at that pump peak from three other directions and this is the fourth.

The magnitude is not enormous — 3.37 ΔE₀₀ against 1.71 under daylight, a factor of two — and it is systematic, it applies to every white LED in production, and it is the commonest artificial light in the world.

Why the tungsten row is the largest for both #

The top row deserves an explanation because it appears to contradict everything above: a tungsten lamp is the poorest source of blue light in the set and it produces the largest departure for both blue-absorbing filters.

The reason is that the departure is a pairing with the sample’s deviation from the adapting white, and the adapting white under tungsten is also blue-poor. So the short-wavelength cone’s relative excitation is a ratio of two small numbers, and a filter that attenuates both by different amounts moves that ratio a great deal more than it would move a ratio of two large ones.

That is the argument the lens essay makes at length and it applies to the macular pigment equally, because both filters live in the same spectral region. It also explains why the two rows are almost identical under tungsten and diverge under every other light: under a blue-poor source, any blue absorption is doing the same thing, and the difference between a band and a tail stops mattering.

The two departures are distinguishable exactly when the light has structure in the blue — which is to say under LED lighting and not under thermal sources.

The ladder under a white LED is worth comparing against the daylight one, because the reordering is caused by a single feature of the lamp. Under daylight the order is age, peaks, macular; under the LED it is macular, age, density, with the peaks fourth. Nothing about the observers has changed and the lamp’s colour temperature is similar.

What has changed is where the lamp’s power is concentrated. Daylight is smooth and puts a broad, gentle amount of energy across the whole short-wavelength region; the LED puts a narrow spike at 452 nanometres and then almost nothing until its phosphor begins. A lamp’s spectral structure reorders the observer departures, and correlated colour temperature — the number lamps are actually specified by — carries none of that information.

That is a second reason, beyond the rendering-index argument, why a lamp needs its spectrum stated rather than its temperature. The first is about what colours it renders; this one is about how much two people will disagree about them.

What a narrow departure means for measurement #

The localisation has a consequence for anybody trying to characterise an individual, and it runs opposite to the usual advice.

Measuring macular pigment density is relatively easy: heterochromatic flicker photometry with a blue and a green target takes a few minutes and needs no dilation. Measuring lens density is harder and is usually estimated from age instead.

The distributions say the easy measurement is the one worth making. The macular’s span of seventeen means an individual’s value determines a great deal about how they will differ from a standard observer on a blue-structured sample, and the population mean carries very little information about that individual. The lens’s narrower span and monotone age relationship mean the population value, adjusted for age, is nearly as good as a measurement.

Measure what varies unpredictably and estimate what varies predictably, which is obvious once stated and is the reverse of the effort most personalised-observer work spends.

The sample that shows the partition #

The notch filter is the cleanest demonstration of the partition, and it is worth reading beside the red pigment.

Its structure is a single absorption band centred at 545 nanometres, which is fifty nanometres above the top of the macular band and well inside the region the lens barely touches. On it the macular departure is 0.83 ΔE₀₀ against 1.71 on a red pigment, and the lens departure is 1.27 against 2.38.

Both fall, and the macular falls proportionally further. That is the partition: a sample with no blue structure at all is a sample the macular pigment has almost nothing to act on, while the lens still has the sample’s overall blue level to attenuate.

The practical form is a rule about which samples need an observer allowance for which parameter. A sample whose spectral structure is entirely above 500 nanometres — most reds, oranges and yellows — is nearly immune to the macular departure and is not immune to the lens’s. A sample with structure in the 440–480 band — most blues, purples and blue-greens — is exposed to both, and to the macular one most.

One more consequence belongs to anybody assembling a test set rather than an observer. A set of samples chosen to exercise observer variation has to include blue-structured samples, and most standard charts do not have many. A ColorChecker-style chart is dominated by the natural and pigment colours a photographer cares about, and its blue patches are few and comparatively broad.

So a chart-based assessment of observer agreement will systematically under-report the macular departure and report the lens departure about right — because the lens touches everything and the macular touches only what is not there. A chart decides what a camera scores and it decides what an observer study measures for the same reason, and in both cases the omission is invisible from inside the assessment.

What was computed, and how #

The macular absorbance is modelled as a single Gaussian of peak density d centred at 460 nanometres with a width parameter of 42, which reproduces the position and rough width of the measured double-humped spectrum without its structure. The lens is a single exponential in wavelength with an age-dependent scale.

Both are caricatures and both are stated as such in the source. What they carry correctly is the localisation, which is the whole subject: the macular’s absorbance is zero above 530 nanometres to within a part in a thousand, and the lens’s is not zero anywhere.

The two strengths compared are two standard deviations of the reported macular spread and the working-age lens range, so the comparison is between two departures at their literature widths rather than at matched sizes.

The width can be made into a number. The essay’s claim is that localisation drives variance, and it can be given a number rather than an argument.

The macular absorbance’s effective width is about forty nanometres; the lens’s, taken as the region over which it changes appreciably, is about a hundred and twenty. The ratio is three. The ratio of the two departures’ spans across the surface family is 17 to 10, which is 1.7 — the same direction and about half the magnitude.

The gap between three and 1.7 is worth having rather than smoothing over. The span is not driven by width alone: it also depends on how much of the family’s spectral variation lies within each departure’s region, and the family’s absorption bands are centred from 430 to 670 nanometres, so about a third of them have structure in the macular band and nearly all have structure somewhere the lens reaches.

So the rule is directional rather than quantitative. A narrower departure has a wider spread, and how much wider depends on the sample set as much as on the departure — which is one more reason a set is a declaration rather than a fact.

Where the model stops #

The real macular absorbance has two peaks, at about 460 and 490 nanometres, and a shoulder. A single Gaussian gets the centre and the width and loses the structure, which would matter for a sample whose own structure is fine enough to resolve it — and no sample in this collection is.

The macular density is treated as uniform across the field, which it is not: it falls steeply with eccentricity, so a real two-degree field averages a substantial gradient. That is the same simplification the field-size essay had to make and it is the same limitation — which is why the eccentricity work treats the pigment as a profile and this round treats it as a value.

And nothing here uses a measured population distribution of macular density. The two standard deviations are from reported summary statistics, and the true distribution is skewed — many people near zero, a long tail upwards — so a symmetric two-sigma comparison overstates the low end.

The walk under the LED against the walk under daylight is the pairing’s two factors separated as cleanly as this round manages. The sample’s factor is the horizontal axis; the light’s factor is the slope; and the departure is their product, exactly, all the way along.

That means the lamp’s contribution can be quoted as a single multiplier for this departure, which is unusual — most of the round’s numbers refuse to be summarised. Against daylight, a white LED multiplies the macular departure by about two on every sample, a three-emitter LED by about 1.8, and a tungsten lamp by 2.6. Those are properties of the lamps alone, computable once, and applicable to any sample.

One caution about that multiplier before it is used. It is a property of the lamp for this departure, and the lens’s multipliers are different numbers in a different order. Nothing here licenses a single per-lamp observer factor, and the whole point of the round is that no such factor exists.

The generalisation #

The habit is about reading a distribution’s shape rather than its summary.

Two quantities with similar means and different spreads usually differ in mechanism, and the difference is often localisation: a broad effect touches everything a little and a narrow one touches some things a lot. The two are indistinguishable from a mean and obvious from a histogram.

The move is to plot the distribution rather than the value whenever a quantity is computed over a set, which costs nothing and answers a question nobody asked. Here it turned “the lens is bigger than the macular” into “the lens is bigger on average and the macular is bigger on the samples where either matters”, which is a different piece of advice.

The failure mode is to compare two effects by their means and rank them. A mean is the right summary only for a distribution that is nearly symmetric and nearly unimodal, and a localised effect over a heterogeneous set of samples is neither.

A last note for anybody reading the two absorptions as though they were interchangeable, which the literature sometimes does under the heading of pre-receptoral filtering. They are not interchangeable in any of the four respects this round measures: their spans across samples differ by a factor of nearly two, their ordering across lights differs, their response to a blue-pumped source differs by a factor of one and a half, and one is estimable from age while the other is not.

Grouping them is convenient for a physiological account, where both are absorbances in front of the same receptors and neither is a receptor property. It is misleading for an error budget, where they behave like two quite different terms and where the smaller one is the one that needs measuring.

Who found it, and when #

Maxwell noticed the macular pigment’s effect on his own colour matches in 1856, describing a yellow spot that made central and peripheral matches differ. Wald measured its absorbance in the 1940s and the double-peaked structure has been known since.

Its variation between individuals was quantified through the 1980s and 1990s, largely because of interest in whether it protects against macular degeneration, and the resulting literature is unusually good on the distribution — which is why this round can state a mean and a spread with confidence and cannot state the same for cone optical density.

Where the ladder goes next #

The last of the six departures is the one that moves a curve rather than filtering it, and its effect is concentrated somewhere unexpected. A peak shift changes the flanks and not the peaks, which is where every display primary sits.