A cone absorbs its own light

A cone’s spectral sensitivity is usually drawn as though it were the absorbance spectrum of the pigment inside it. It is not, and the difference is not a normalisation.

The claim #

A cone is a column of pigment, deep enough that the light at the peak wavelength is nearly all absorbed before it reaches the far end. Adding more pigment therefore adds almost nothing at the peak and a great deal in the wings — so a denser cone has a broader sensitivity, and two people whose cones differ only in density are two different observers.

The second half is the one with consequences. A colour match is an equality of three cone excitations; if the three functions differ, the equality does not transfer. So pigment density on its own — no difference in the pigments, no difference in the lens, no difference in the cone ratio — produces observer metamerism, and this essay computes how much.

Beer–Lambert, in a receptor #

The absorbance of a photopigment is what a spectrophotometer would report for a thin layer of it, and this site already carries it as a template: one parameter, λmax, generates the whole curve.

The absorptance — the fraction of arriving light actually caught — is that absorbance run through Beer–Lambert:

$$\alpha(\lambda) = 1 - 10^{-D\,\hat{A}(\lambda)}$$

with $\hat{A}$ the template normalised to peak at one and $D$ the axial optical density, about 0.4 for a foveal cone and reported between 0.3 and 0.5.

The exponential is what does everything. At the peak, $D = 0.4$ already absorbs 60 per cent of what arrives, so doubling the density takes it to 84 — a factor of 1.4. Two hundred nanometres out in the wing, where $\hat{A}$ might be 0.02, absorption goes from 1.8 per cent to 3.7 — a factor of 2.0. The wings gain more than the peak does, and normalising to the peak turns that into a broader curve.

In the limit of a very thin layer the two coincide, which is the check that the algebra is right: at $D = 10^{-6}$ the computed absorptance is the template to within 10⁻⁵ at every wavelength, asserted.

What it does to a match #

Here is the experiment the whole essay is for.

Take an observer with density 0.4. Build two reflectances that are metameric for that observer — not for the CIE’s colour-matching functions, but for the three fundamentals computed at that density — by projecting a smooth candidate onto the null space of the three-by-eighty-one system matrix. Their cone excitations then agree to 2.9 × 10⁻¹⁵, which is machine precision.

Now hand the same two reflectances to an observer identical in every other respect whose cones have a density of 0.6:

Each is a fractional difference in a cone excitation — a cone contrast. A cone contrast of about 0.5 per cent is roughly where chromatic discrimination begins, so this is a mismatch several times over threshold, produced by nothing but a difference in how much pigment two people have in their outer segments.

The effect is close to linear in the density difference, which is what a small perturbation of a linear functional should be:

Two things in that table are worth taking out, because both are checks and one of them is the essay’s most useful number.

It depends on the size of the density difference and not on its direction. The first two rows are 0.3 and 0.5 — a tenth below the reference observer and a tenth above — and they report 0.50 and 0.49 per cent. That near-equality is not built in anywhere: the two constructions run through a different exponential and produce differently shaped fundamentals. It is what a first-order perturbation of a linear functional has to look like, and getting it out of the arithmetic rather than putting it in is the check that the construction is a perturbation at all.

And the slope is a constant worth carrying: about 4.8 per cent of cone contrast per unit of axial density. Dividing each row by its own $|\Delta D|$:

The column is flat to within nine per cent over a fourfold range of perturbation, and it drifts downwards in the direction Beer–Lambert requires: a larger density is closer to saturation at the peak, so each additional unit of it buys less broadening than the one before. The two rows below the reference sit slightly above the two above it for the same reason read backwards.

The same person, twice #

Density is not only a difference between people. It changes within one person, over minutes, and for a reason that is unavoidable: light bleaches pigment. A cone in bright light has a smaller fraction of its pigment in the absorbing state, so its effective axial density falls, so its sensitivity narrows.

Running the same construction from 0.4 down to 0.25 — a plausible amount of bleaching — gives a worst mismatch of 0.76 per cent. A match made in a dim booth is therefore not quite the match made in a bright one, and the change is in the observer rather than in the samples or the light.

This is a small effect and it is not nothing: it is the same order as the disagreement between two instruments of different bandpass, and colour standards are written to a tolerance where both matter.

The same person differs from themselves by more #

That coefficient is what makes the two halves of this essay comparable, and the comparison is the finding.

The reported range of axial optical density for a foveal cone is 0.3 to 0.5. So an ordinary person drawn from the population sits within about 0.1 of the reference, and the slope prices that at 0.48 per cent — which is, to the digit, the cone contrast at which chromatic discrimination is usually said to begin. The population spread in this one parameter, with the pigments identical and the lens identical and the mosaic identical, puts two randomly chosen people right at the threshold of disagreeing about a match.

Now the bleaching figure, which was computed by the same construction and has not been compared with it. Going from 0.4 to 0.25 is $|\Delta D| = 0.15$, and the slope predicts 0.72 per cent against the 0.76 actually measured — agreement to five per cent, which says the two calculations are the same calculation.

And 0.15 is larger than 0.1. A single observer walking from a dim booth into a bright one moves further along this axis than the gap between them and an average person. The instrument is not merely variable between people; it is more variable within one person over a few minutes of light adaptation than it is across the population at rest.

That reorganises what a standard observer is an average of. Averaging over people is the part everybody knows about. What the same table says is that averaging over states of the same eye would have a comparable width, and that a colour match recorded without a light level and an adaptation time has an unstated argument in it of the same size as the one that made observer metamerism worth a name.

Where the standard observers come from #

The 2° and 10° observers differ in more than field size, and density is part of why. The 10° functions sample retina further from the fovea, where cones are shorter and the macular pigment does not reach, so both of the pre-neural filters change at once: less macular absorption in the blue, and a lower effective axial density broadening in every class.

That makes the 2°–10° difference a compound of two effects with different spectral signatures, which is why no single correction converts one to the other and why the choice of observer keeps arriving as a real decision rather than as a formality.

What was computed, and how #

The template is Govardovskii’s, already carried here because the rod curve is computed from it. One parameter, λmax, at 566, 541 and 441 nm for the three classes.

The absorptance is $1 - 10^{-D\hat{A}}$ multiplied by the ocular media transmittance — lens and macular pigment as stated exponential absorptions — and normalised to peak at one. The macular pigment is switched off for the width comparisons, because including it would move the short-wavelength curve for a reason that has nothing to do with density.

The width is the full width at half maximum, found by linear interpolation between grid points on either side of the half-maximum crossing, on this site’s 5 nm grid.

The pair is a metameric black in the null space of the system matrix built from these fundamentals and the illuminant. That distinction is the essay’s one methodological trap and it was walked into on the first attempt: using makeMetamer’s default builds a pair metameric for the 1931 colour-matching functions, which then fails to match at every density including its own, and reports an enormous effect that is entirely the wrong comparison. The version here matches to 2.9 × 10⁻¹⁵ for its own observer, and the assertion requires that residual to be below 10⁻⁹ before it will report anything about the second observer at all.

The mismatch is reported as a cone contrast rather than as a colour difference, deliberately: the second observer has no CIELAB, because a colour space is built on colour-matching functions and these are not those. Converting the mismatch into ΔE would require inventing a space for an observer nobody has standardised.

Where the model stops #

The density is uniform along the column. Real outer segments are not, pigment is regenerated continuously, and the bleached fraction depends on wavelength as well as on level.

There is no waveguide. A cone is narrow enough to act as an optical fibre, which is why light entering the pupil obliquely is less effective than light entering axially — the Stiles–Crawford effect — and that changes the effective path length as well.

The λmax values are quoted, and they vary too. Individual differences in the L-cone photopigment shift λmax by several nanometres between common variants, which is a separate source of observer metamerism from the one computed here and generally a larger one.

The slope is fitted to four points from one pair. The 4.8 per cent per unit above is read off a single metameric pair constructed against a single reference observer, so it is a local coefficient rather than a property of density. A different pair — one whose metameric black has its energy somewhere else in the band — would have a different sensitivity to the same perturbation, and how wide that distribution is has not been computed here.

And no claim is made about which density any person has. The essay computes what a difference in density does; the distribution of densities in a population is a measurement, not a derivation, and it is not on this site.

What the pictures cannot show #

A broadened curve looks like almost nothing. The three densities in the opening figure are separated by a factor of eighteen and the curves are still recognisably the same curve — which is exactly why the effect went unquantified for so long and why it has to be measured on a match rather than judged by eye. Thirty-five nanometres of half-height width is a large change in a sensitivity function and a small change in a drawing.

And no figure here shows what either observer sees. Two observers with different fundamentals do not disagree about an appearance; they disagree about an equality. One of them is looking at a match and the other at a mismatch, and there is no way to draw both on one page, because the page has one spectrum per patch and each observer integrates it differently. Every figure on this site that shows two patches side by side is showing them to a single observer — the 1931 one — and the whole content of this essay is what happens to somebody who is not that observer.

What it costs the luminance channel #

The broadening is not confined to colour matching. The luminous efficiency function is a weighted sum of the long and medium fundamentals, so a broader pair makes a broader V(λ), and photometry inherits the difference.

The direction is worth stating because it is counter-intuitive: a denser cone is a less selective detector but a more sensitive one. It catches more of everything, and the extra catch is concentrated where the pigment was previously transparent — in the wings, which is where the light that a narrow function would have discounted lives. Two observers of different density therefore disagree slightly about how bright a saturated red is, while agreeing about a white, which is the same asymmetry V(λ) shows between individuals for a completely different reason.

Whether the widening is a property of the pigment or of the observer scoring it is a fair question, and the ten-degree functions answer it.

The generalisation #

The habit worth taking from this is about the difference between a substance and an instrument.

A pigment’s absorbance is a property of a molecule and is the same everywhere. A receptor’s sensitivity is that property plus a geometry — how much of it, in what path length, behind what filters — and only the second is what colour matching is about. The standard observer is an average over instruments, not over molecules.

The same distinction sits under three other arguments here. The rod curve’s peak sits at 507 nm rather than the pigment’s 498 because of what is in front of it. A camera’s sensitivity is a dye’s transmittance multiplied by silicon’s quantum efficiency and an infrared filter, and the argument about whether it can be an observer at all is about the product rather than any factor. And an ink’s colour is its absorbance multiplied by a film thickness, which is the same exponential in a different trade.

The rule: an absorbance becomes an absorptance only after somebody states how much of the stuff there is, and the two have different shapes.

And the corollary this essay adds to it is about time. The geometry that turns the substance into an instrument is not fixed — the path length is the same in every eye, but how much of the pigment along it is in the absorbing state is set by the light that arrived in the last few minutes. So the instrument is not merely one of a population of instruments; it is one that retunes itself in response to the very thing it is measuring, and it does so by an amount this essay can now price against the population spread.

Who found it, and when #

Self-screening was understood by the 1960s, when microspectrophotometry on individual primate cones made both quantities measurable at once — the pigment’s absorbance in a thin layer, and the sensitivity of the whole receptor.

Its role in colour matching was quantified by Stiles and Burch’s work on individual observers and formalised in the CIE’s physiological cone fundamentals, where axial optical density is an explicit parameter alongside macular and lens density. That model is the reason it is now possible to say what a particular observer would match, rather than only what the average of seventeen people in 1931 matched — and the parameter that varies most between people, after the lens, is the one this essay is about.

The industrial consequence arrived later still, with narrowband displays. When a display’s primaries are broad, two observers’ fundamentals integrate to nearly the same numbers whatever their densities; when the primaries are three narrow lines, the same difference in density lands entirely differently, which is why observer metamerism became a practical problem exactly when displays got better.

Where the ladder goes next #

The obvious continuation is downward into the counting. A denser cone catches more photons as well as catching them over a broader band, and the photon budget is what decides where colour vision stops altogether — the same absorptance function appears in both essays, once for its shape and once for its area.

The other direction is a standard nobody has written. If observer metamerism is a real cost on narrowband displays, and if the largest pre-neural parameters are lens density, macular density and axial density, then a family of observers indexed by those three parameters is computable and a display could be specified against the family rather than against its mean. The CIE has published the machinery for exactly this; what does not exist is any product specification that uses it.