The filters inside the eye

Between the cornea and the photopigment there are absorptions, and they are large. The lens takes most of the violet and much of the blue, and it takes more of both every decade — the largest single term separating two people is this one. A yellow carotenoid pigment sits over the central few degrees of the retina and takes up to half the light at 460 nm. Neither is small, both are individual, and both are already arguments to the machinery this site builds observers with.

Nobody notices either of them. That is the fact this essay is about, and it turns out to have a precise explanation and a precise limit.

The claim #

A filter in front of the receptors is very nearly a gain, and how nearly depends on how steep it is.

• The macular pigment moves a surface ΔE00 15.3 for an observer who does not adapt, and leaves 0.37 for one who does — 2.4 per cent, the smallest share of anything in the census.
• A filter and an adaptation act at the same place, so what is left is only the part of the filter’s shape that a diagonal in three axes cannot follow.
• Fifty years of lens yellowing leaves 7.9 per cent, which is three times as much and puts it among the lamps rather than beside the macula. It is a broader, steeper absorption running the whole short-wave end.
• The adapting white differs by exactly zero between the two cases, to 10⁻¹⁴, which is why the effect is invisible in the one measurement anybody makes.
• And the residual is largest in the blue, which is where both absorptions are, and where three cones give a person the least to work with.

The same operation, twice #

Multiplying the arriving light by a transmittance and multiplying the cone signals by a gain are not the same operation, but they happen at the same point in the chain, and that is what makes them so nearly cancel.

A filter multiplies the spectrum. It therefore multiplies the white the observer adapts to by the same amount, and von Kries adaptation divides by that white. Whatever part of the filter is a scaling of each channel’s total catch is removed exactly. What is left is only the part that changes the shape of the light inside each channel’s own band — which is to say, the steepness of the filter measured on the scale of a cone fundamental’s width.

Why the macula gets away with it #

The macular absorption is a single Gaussian-shaped band centred at 460 nm and about 42 nm wide. That is narrow compared with a lamp’s structure and broad compared with a cone fundamental’s slope, and it sits almost entirely inside one channel.

A filter that lives inside one channel is close to a gain by construction: it changes what that channel catches and leaves the other two nearly alone, which is exactly what a diagonal does. The residual it leaves is what its own shape does within the S band, and the S band is broad enough that a 42 nm dip inside it is close to a scaling.

That is the arithmetic reason. The observation it explains is one everyone can make and almost nobody does: a small blue object looks slightly different at the centre of gaze and ten degrees to one side, and it takes a deliberate experiment to see it.

The filter is not nearly flat across the channel it sits in #

The explanation offered for the macula’s 2.4 per cent is that the absorption lives inside one channel, so it acts like a scaling of that channel and a diagonal removes it. The stated transmittance function does not support the second half of that, and it is worth working out from the formula the essay itself prints.

At the population median density the macular filter runs:

Across the short-wave channel’s own band the transmittance swings by a factor of two, from 0.90 at 400 nanometres to 0.45 at 460 and back. That is not a scaling of the channel; it is a deep notch sitting in the middle of it. The band is also wider than the essay’s 42 nm suggests — 42 is the Gaussian’s 1/e half-width, and the full width at half maximum of the density is 69.9 nanometres, against an S fundamental only somewhat wider than that. Two other figures worth having from the same formula: the peak absorption is 55.3 per cent rather than up to half, and the steepest part of the filter is 0.0071 density units per nanometre, at 430 and 490 nanometres.

So the filter is steep, deep and comparable in width to the channel it occupies, and the residual it leaves is nonetheless the smallest in the census. The stated mechanism cannot be the whole reason.

What else has to be true for the residual to be small #

The missing ingredient is not in the filter at all. It is in what the filter is applied to.

A von Kries gain removes the change in each channel’s total catch. What survives is the extent to which two different stimuli get different effective gains — which happens only when they distribute their energy differently within the band the filter varies across. A surface whose reflectance is flat across 400 to 520 nanometres sees the same weighted-average transmittance as any other flat one, and the notch is removed exactly. A surface with a narrow feature at 460 loses half its light there while a surface with the same feature at 400 loses a tenth, and no diagonal reconciles them.

The census’s residual is a mean over 125 constructed surfaces built as sums of a few cosines — smooth by construction, with nothing narrower than the notch anywhere in them. So 2.4 per cent is a joint property of the filter and the test set, and the essay’s own machinery elsewhere says how sensitive that kind of number is to the set: the same collection measures the test set’s saturation as the most elastic input it has.

That yields a prediction the arithmetic here cannot check and the site’s own machinery could. A test set with narrow-band reflectances should raise the macular share sharply, and by more than it raises the lens’s, because the lens’s residual comes from a between-channel ratio that does not need within-band structure to appear. If the macular row’s smallness is about the surfaces rather than about the filter, that experiment separates the two, and it is the one measurement that would settle which mechanism is doing the work.

The lens is a different mechanism, not a steeper version of the same one #

Read this way, the lens comparison stops being a matter of degree and becomes the control the essay needs.

The macular notch is a within-channel shape change, and its cost is second-order: it needs the stimulus to have structure inside the band before it costs anything at all. The lens’s absorption rises monotonically from 550 nanometres to the short-wave end, straddling the S and M fundamentals, so it changes the ratio between two channels by an amount that depends on where the stimulus’s energy sits between them. That is first-order — a smooth broad surface pays it in full — and it is why fifty years of yellowing lands among the lamps at 7.9 per cent while a deeper, steeper filter inside one channel lands at 2.4.

The factor of 3.3 between them is therefore not a measure of steepness. It is the price of crossing a channel boundary against not crossing one, measured on surfaces that are smooth enough for the second-order term to stay small. The rule the essay states is right and the reason it gives is incomplete: a filter is a gain to the extent that it acts inside a channel and the stimuli have no structure finer than the filter’s own.

The lens is the counter-example that proves it #

If the explanation is filters are gains, the lens should behave the same way, and it does not. Fifty years of yellowing leaves 7.9 per cent, three times the macula’s share and comparable with a change all the way from daylight to tungsten.

The difference is shape. The lens’s absorption is not a band inside one channel; it is a monotonic rise running from about 550 nm all the way to the short-wave end, steepest exactly where the S and M fundamentals overlap. A filter with a steep slope across a channel boundary changes the ratio between two channels in a way that depends on the stimulus’s own spectrum, and a diagonal cannot be sensitive to that.

So the rule is not a filter is a gain. It is: a filter is a gain to the extent that it acts inside a channel rather than across the boundary between two. The macula does the first and the lens does the second, and the ratio between their residuals is what that costs.

A filter that is not a filter #

There is a third absorption inside the eye and it behaves like neither of the other two, which is worth a section because it marks where this essay’s rule stops.

The photopigment itself absorbs, and the more of it there is in a cone the more it absorbs — but not uniformly, because a pigment that has already taken most of the light at its peak cannot take much more there while it can still take more in the wings. The result is that a denser cone has a broader fundamental rather than a taller one. Self-screening is the name for it, and it is one of the five ways two people’s observers differ.

That is not a filter in the sense used above. A filter multiplies the light before the receptor; self-screening changes what the receptor is. The difference matters because the white does not carry it away: two observers with different pigment densities do not merely see the same light through different glass, they integrate it against different functions, and there is no gain — in any basis — that converts one into the other.

Which is why pigment density appears in the population essays and not in this census. It is not a change of context at all, and the arithmetic that makes the macula nearly free does not begin.

What exactly zero means here #

The single most telling number in the whole comparison is not a residual but an identity. The adapting white seen through the macular filter and the adapting white seen without it, after adaptation, differ by exactly zero — 2 × 10⁻¹⁴, which is floating point.

That is not an approximation and it is not a coincidence. Adaptation normalises by the white; a filter multiplies the white by the same factor it multiplies everything else; the two cancel algebraically. The consequence is that the one measurement anybody would think to make — does a white card look the same at the centre of gaze and off to one side — is guaranteed to return yes whatever the filter is.

An absorption in front of the receptors is therefore invisible to the obvious test by construction, and the only way to see it is to use a stimulus that is not the white and is saturated enough for the residual to exceed a threshold. The essay on the field found ΔE00 5.9 for a match made at the fovea and tested ten degrees out — a large number, obtainable only from a pair chosen for it.

Two more readings say what the filters cost on saturated surfaces and where the same asymmetry shows up between the two standard observers.

What it says about individual variation #

The population essays treat a person’s macular density and lens age as two of five parameters and report how far apart two people can be. This essay adds a qualification to those numbers that changes what they mean.

Both filter terms are partly self-cancelling for the person who has them, because each person adapts to the light as it arrives through their own optics. What is left over between two people is not the difference between their filters; it is the difference between the residuals their filters leave, which is much smaller.

But it is not smaller for a match. Two people comparing the same pair of surfaces are both adapted, and a metameric pair that matches for one can fail for the other by the full difference between their fundamentals — because a match is an identity between two stimuli and adaptation applies the same gain to both. The self-cancellation helps with appearance and does nothing for agreement, which is the distinction the whole matching field rests on.

The exponential scale of the macular profile is a fitted number, and doubling it is the sensitivity test the argument needs.

What it says about the two standard observers #

The CIE has two standard observers and the difference between them is mostly this essay’s subject, which makes the numbers here a statement about the standards as well as about eyes.

The 1931 functions were measured on a two-degree field, which falls inside the macula. The 1964 functions were measured on ten degrees, most of which is outside it. So the largest single difference between the two tabulations is that one includes a macular absorption and the other largely does not — this site’s own fit puts the implied density at 0.400 against a measured population median of 0.35, which is two numbers from completely unrelated measurements agreeing to a seventh.

The consequence, given the arithmetic above, is a specific pattern of disagreement rather than a general one. The two observers agree almost exactly about a white, because a filter and an adaptation cancel there by construction, and they disagree most about saturated short-wave stimuli, where the filter is steepest and the residual largest. Anybody who has computed the same sample under both tabulations has seen that pattern without necessarily having a reason for it.

It also explains why the choice of observer so rarely changes a decision and occasionally changes it a great deal. A quality-control laboratory comparing two nearly-identical samples is working with a difference, and a difference is not protected by the cancellation — both members move, but they move by amounts that depend on their own spectra. A laboratory checking a white point is working with exactly the case the cancellation covers.

The uncomfortable corollary is that the standard requiring the 10° functions for surface colours is right for a reason that has nothing to do with which is more accurate. It is right because a surface colour is usually judged over a field larger than two degrees, and the observer whose macular content matches the field is the one whose residual is smallest.

Who found it, and when #

The macular pigment’s absorption was measured spectrophotometrically in the 1940s and its role in the difference between the 2° and 10° observers was recognised immediately — the two standard observers differ mostly because one includes the macula and the other largely does not.

That the difference is invisible to adaptation is implicit in every von Kries treatment and, as far as this site can tell, is not usually stated. The reason to state it is that it explains an absence rather than a presence, and absences are the things this site’s assertions exist to keep honest: the gate here requires the macular row to have the smallest share in the census, so a future change to the filter’s shape that made it ordinary would stop the build rather than quietly rewriting the sentence.

Twenty degrees out is past the macula, and a match solved at the centre of gaze is the cleanest way to price what the filter was doing.

What was computed, and how #

The macular transmittance is 10^(−D · exp(−((λ − 460)/42)²)) at D = 0.35, which is the same function the observer machinery applies when it builds an eye at a given eccentricity. The two are asserted to be the same function, band by band, to 10⁻⁹ — a site with two spellings of one measurement is a site where one of them will drift.

The lens row uses the ocular media model at ages 20 and 70. Both rows are treated as changes of context: the light arriving at the receptors is multiplied by the filter, and the white the observer adapts to is multiplied by it too.

The residual is the mean CIEDE2000 over a hundred and twenty-five surfaces after the ratio-of-whites gain in the CAT16 basis. The share is that residual divided by the unadapted change, and it is the share rather than the residual that puts the macula first: three other rows leave a smaller absolute number and every one of them is a much smaller change to begin with.

Where it stops #

The macular profile is a single Gaussian and real measurements are not quite that shape; the site’s own gate requires the conclusions to survive a factor of three in the assumed spatial scale, and they do, but the spectral shape is one function rather than a family.

Treating the fovea-against-periphery difference as a change of illumination is a modelling decision. An observer does not adapt separately at each eccentricity — or rather, the local pool that would do so has an extent of about half a degree, which is much smaller than the macula. Whether the eye adapts to its own macular filtering at all is not something this arithmetic can decide; what it computes is the residual that would be left if it did, which is the optimistic case.

And a filter is only ever half of what sits in front of a receptor. Scatter, chromatic aberration and the waveguide properties of the cone itself are all in the path and none of them is a multiplication, so none of them appears in this census at all.

Where the ladder goes next #

If a filter in front of the receptors is nearly a gain, then a filter anywhere in the path is, and the interesting cases are the ones outside the eye. A sheet of paper is a filter under everything printed on it, and colour management’s rule for changing it is a von Kries adaptation in the worst basis available.

The other direction is the one the lens points at. A steep filter across a channel boundary is what defeats a gain, and the sharpest such filters on this site are not in observers at all — they are the discharge lamps, whose emission bands are narrower than anything an eye contains.