Two yellow filters cancel on a slope

Two of the ways people’s eyes differ are filters in front of the receptors. The lens yellows with age and absorbs more and more strongly towards the ultraviolet; the macular pigment is a band of absorption centred near 460 nanometres whose density varies severalfold between people. Both take blue out of the light, and the natural expectation, printed in the essay on a tolerance with an observer in it, is that on a given sample their effects share a sign, so that combining them as though they were independent understates the total.

Aligned in the light, opposed in the colour #

Whether the lens and the macular pigment add or cancel depends on the surface they are measured on, and on smooth surfaces they cancel.

• Before adaptation they point the same way. On 120 smooth natural reflectances the angle between their two deviations is a median of 8 degrees, and they never partly cancel.
• After each observer adapts to its own white they point a median 156 degrees apart, and together cost less than the larger alone on 115 of the 120 surfaces.
• On surfaces with a forty-nanometre absorption band the adapted angle is 26 degrees and they never cancel; on a hundred-and-forty-nanometre band it is 116.
• So the combined cost is 1.28 on smooth surfaces against a quadrature of 2.53, and 2.25 against a quadrature of 2.16 on banded ones. No fixed combination rule is right on both.

What each filter does to the white #

The first step is to see what the two filters share, because the sharing is what adaptation removes.

Each observer sees the same lamp. Each observer’s lens and macular pigment filter that lamp before it reaches the receptors, so each observer’s white — the tristimulus value of the lamp as their own eye integrates it — is shifted towards yellow. An older lens shifts it a long way; a denser macular pigment shifts it a shorter way in very nearly the same direction. Read without adaptation, every colour that observer sees carries the same yellowing their white carries, and since both filters yellow the white alike, their deviations on any surface are dominated by that shared part.

That is the figure the expectation is built on, and it is right about what it measures. The two filters do move a reading the same way. What they move is a reading before the eye has done the one thing it always does with a change of white.

Adaptation takes the shared part away #

A visual system adapts. Each observer’s cone signals are rescaled so that their own white reads as white, and the observer audit does that for every observer by the same route — a chromatic adaptation transform from the observer’s own white to a common one. A gain is not an observer, and the gain removes exactly the part of a departure that behaves like a change of light.

Both filters behave, to first order, like a change of light: they tint everything the way their own white is tinted. Adaptation takes that first-order part out of both, and what is left of each is the part that does not behave like a change of light — the way the filter’s shape differs from a uniform yellowing across the colours a particular surface reflects.

The lens and the macular pigment have different shapes. The lens’s absorption keeps rising towards short wavelengths; the macular pigment is a band, not a filter, with a peak and two shoulders. After their shared yellowing is removed, the residuals are those shape differences, and on a smooth surface they point nearly opposite ways. The same thing happens, more cleanly, to the field-size departure and the macular one, which the essay on how departures compose found 162 degrees apart on the audit’s own surfaces: on smooth reflectances the angle is 170, and they cancel on every one.

A band holds them together, a slope pulls them apart #

The angle is not a property of the two filters alone. It is a property of the two filters and the surface, and the surface’s contribution can be isolated because the banded family is built along separate axes.

On surfaces with a forty-nanometre band the two filters sit 26 degrees apart. At eighty nanometres, 54. At a hundred and forty, 116. On smooth slopes, 156. Grouped by where the band sits instead, the angles run from 25 degrees for a band at 670 nm to 90 for one at 470; grouped by how light the surface is, they do not move at all, from 67 degrees at the darkest level to 65 at the lightest.

The likely reading is this. A narrow band reflects or absorbs one stretch of the spectrum, so a surface with a band samples the two filters’ residuals over a short range, and over any short range their shapes are similar: both absorb, both slope. A smooth reflectance weights the whole visible range, and over the whole range the two residuals are what differ — the lens’s rising tail against the macular pigment’s bounded peak. That is an interpretation of the grouping rather than a derivation, and it predicts that a band placed where the two absorptions differ most should sit furthest apart. The band at 470 nm, inside the macular pigment’s peak where the lens absorbs much less steeply, is the one at 90 degrees.

What the pair costs together #

The angle becomes a cost the moment the two departures are applied to one observer at once.

On smooth reflectances the lens alone costs 2.29 colour differences at the mean and the macular pigment 1.07. The two together cost 1.28 — less than the lens alone, about half the quadrature of 2.53, and under two fifths of their sum. On the banded family the lens and the macular pigment together cost 2.25 against a quadrature of 2.16, so quadrature very slightly understates them there, which is the direction the original expectation predicted.

Neither rule is a mistake on its own set. Quadrature is wrong on one set by a factor of two and on the other by four per cent, in opposite directions, and nothing in the two filters says which set a sample comes from.

The rule that composes them needs the angle #

The measured combinations can be recovered from the separate costs once the angle is supplied, which shows that the angle is doing all of the work.

For two deviations of sizes $a$ and $b$ at an angle $\theta$, the combined size is $\sqrt{a^2 + b^2 + 2ab\cos\theta}$. Quadrature is that rule with the cosine set to nought, which is the claim that the two deviations are perpendicular. A plain sum is the rule with the cosine set to one, which is the claim that they are aligned.

With the smooth set’s mean costs — 2.29 for the lens, 1.07 for the macular pigment — and its median adapted angle of 156 degrees, the rule gives 1.38 against the 1.28 measured; most of the remaining difference is that the rule was handed means and a median rather than each surface’s own pair. Quadrature’s 2.53 is the same rule at 90 degrees, and the smooth set sits 66 degrees beyond that, while the banded surfaces, at 26 degrees, sit much nearer the plain sum’s assumption than quadrature’s. A budget written in quadrature has placed the two filters at right angles on no evidence about either set.

The published combination, re-read #

The observer tolerance essay combined six departures in quadrature to about three colour differences, and said the true combination would be somewhat larger because the lens and the macular pigment share a sign. That sentence was written about the audit’s own forty-two surfaces, and on those surfaces it holds.

On the audit’s surfaces the lens and the macular pigment sit a median of 64 degrees apart after adaptation, partly cancel on eight of forty-two, and together cost 3.26 against a quadrature of 3.11. So the published direction was right for the published surfaces, and the figure above is what it was right about.

The surfaces were never a sample of the world. Every one of them is a pale reflectance with a single absorption band, chosen once to vary where a band sits and never to vary its shape, and the shape is precisely what decides this pair. On smooth reflectances the same sentence is wrong in sign, by a factor of two.

Under a warmer lamp #

The lamp enters too, because it decides which part of the spectrum carries the light the filters act on.

Under a tungsten lamp at 2856 K the adapted angle on the smooth set falls from 156 degrees to 113, and the pair partly cancels on 94 of 120 surfaces rather than 115. A tungsten lamp has little power in the short wavelengths where the two filters differ most, so less of each filter’s shape difference survives into the colour, and the residuals are less opposed. The opposition is real under both lamps and weaker under the one most homes are lit by.

That makes the lamp a second input to the combination, beside the surface. A booth lit by a daylight simulator and a room lit by tungsten place the same sample at different points between cancelling and adding, and an observer allowance computed under the booth’s lamp is optimistic, in the home, about how much of the two filters’ effect cancels.

Which readers sit in a population’s tail #

Observer populations are built by drawing each person’s lens density and macular density, among other things, from the spread each is reported to have. A brand colour for a population and a soft proof exact for one reader both report a percentile of such a population’s mismatch.

The angle decides who is in the percentile. On a smooth sample, a person with both a dense lens and a dense macular pigment is closer to the standard observer than a person with the dense lens alone, because after adaptation the macular pigment’s deviation partly undoes the lens’s: 1.28 against 2.29 on the smooth set. The people furthest from the standard on such a sample are those whose two filters disagree — a dense lens with a thin macular pigment, whose deviations then point the same way. On a narrow-band sample the tail is made the other way round, of people with both filters dense.

So a population percentile is not a property of the population alone. It belongs to the population and the sample together, through the angle, and a population drawn with independent lens and macular densities puts different people in its tail on a smooth pastel and on a narrow-band dye. A single percentile quoted for a population has averaged that tail over whatever samples it was computed on. If the two densities are correlated in real people, the tail moves again, and that correlation is a measured property of eyes which the angle makes worth measuring.

What an observer allowance can do with this #

The practical question is how to carry two filters that sometimes add and sometimes cancel into an uncertainty budget, and the answer is that a budget of magnitudes cannot.

A fixed correlation is wrong. The correlation between the two departures, taken over surfaces, has no single sign: it is positive on narrow bands and strongly negative on smooth slopes. A budget that assumes either will be wrong by up to a factor of two on the other.

A per-sample angle is available. The two deviations are computable from a sample’s reflectance and the lamp’s spectrum, and so is the angle between them in the local metric; the cosine rule with that angle predicted composed departures to about one per cent. An allowance computed per sample, from the sample’s own spectrum, carries the angle automatically.

And the shape of a sample’s spectrum is a first-order input to its observer allowance, alongside how far it sits from its illuminant. A smooth, gently sloping reflectance — most papers, most fabrics, most skin — is a sample on which two of the largest observer differences largely cancel. A narrow-band sample — a fluorescent dye, a saturated ink, a structural colour — is one on which they add.

What was computed, and how #

The deviations are those of the observer audit: each departure is a pair of observers built from this collection’s pigment template through its ocular media, at the literature’s strengths, and a deviation is the difference between the two observers’ readings of the same sample under the same lamp. Adapted, each reading is taken from the observer’s own white to D65 by CAT16; unadapted, each is divided only by its own white’s luminance.

The angle is taken in the local metric — the quadratic form that reproduces ΔE₀₀ for small changes around the reference observer’s reading — so it is an angle in the unit the costs are reported in. “Partly cancel” means the two deviations added and priced once cost less than the larger of them priced alone.

The smooth set is 120 reflectances walked over a lattice of level, slope and curvature, and it is a construction rather than a measured collection. The banded family is seven band centres, three widths, two depths and four levels.

Where the measurement stops #

Everything is one departure at a time, at two standard deviations of the reported spread. A real observer differs in all six at once, and the angles here are between pairs, not the geometry of a six-dimensional cloud.

The smooth set has no measured reflectance in it. It is built to be smooth because natural reflectances are known to be, and the result would need checking on a measured library before any coefficient from it were used.

And adaptation is modelled as complete, by one transform. A discount nobody measured puts an ordinary room’s degree of adaptation below one, and partial adaptation leaves part of the shared yellowing in both deviations, which would pull the smooth-set angle back towards alignment by an amount not computed here.

The habit #

The habit is about two effects that look alike because they share a cause.

Two filters that both absorb blue, two errors that both come from temperature, two biases that both come from the same instrument: sharing a cause makes their raw effects correlated, and the correlation is then assumed to survive whatever processing follows. It survives only if the processing does not remove the shared part, and an adaptation, a normalisation or a calibration exists precisely to remove it.

The move is to compute the correlation after the processing rather than before. It is the same computation with the processing switched on.

The failure mode is to combine the processed effects with the unprocessed correlation, which is exactly backwards: the shared part is what the processing takes away, so what is left is what the two effects do not share.

Who noticed it first #

That the lens and the macular pigment differ between people and act as short-wavelength filters is textbook, and the CIE’s 2006 physiological observer parameterises both. That normalising each observer to their own white removes most of a filter’s effect on colour appearance is the principle behind every argument that older observers see a white page as white.

That the residual effects of the two filters point apart after normalisation, and that the angle between them depends on the width of the sample’s spectral features, does not appear in the sources consulted here, and it follows from treating each departure as a vector in the local metric rather than as a magnitude.

Still open: whether a measured library agrees #

The smooth set is the part of this that most needs replacing. A measured reflectance library — Munsell chips, a textile collection, a set of skin spectra — would give the distribution of this angle over surfaces somebody actually specifies, and would say whether the typical delivered colour sits nearer the cancelling end or the adding one. The same calculation takes any reflectance; what it needs is the library.