The rods' route is priced by the lamp

The rods are a fourth curve entered rod intrusion into a colour match in the simplest way available: a rod signal a tenth the size of each cone’s peak response, summed equally into all three cone channels. It found the departure could not be removed by a gain or a white point, priced it at 1.60 ΔE₀₀ on a red pigment under daylight, and found its distribution over forty-two surfaces the narrowest of the six observer departures, a factor of six from smallest to largest.

It also said, in its own account of where the model stops, that equal addition is the crudest possible model. Rod signals reach the cone pathways through gap junctions between rods and cones and through the retina’s rod bipolar circuit, and the reported rod contributions to the blue–yellow pathway differ from those to the luminance and red–green ones. How strongly a rod signal enters the S-cone pathway is the least certain of the three weights, and a model that sets all three equal has made a choice about it.

The question is what that choice costs. It could be a detail — a correction of a few per cent to a departure already known to be small. Or it could be the thing the departure mostly consists of. The answer is both, and which one depends on the light.

Worth nothing under daylight, a factor of three under an LED #

Whether a rod signal enters the S-cone pathway changes what it costs a match by 15 per cent under daylight and by a factor of 2.75 under a phosphor white LED. The deciding quantity is the rod signal’s size relative to each cone class’s own catch of the lamp’s light: nearly equal across the three classes under daylight, and three times larger in the S class under the LED.

• Under daylight: 1.03 at the median surface entering all three channels, 0.90 entering L and M only, 0.33 entering S only.
• Under a phosphor white LED: 1.24, 0.45 and 0.96. The S route alone costs more than twice what the L and M route costs.
• Tungsten and a fluorescent tube sit between, at factors of 1.75 and 1.87; a three-emitter LED at 1.50.
• Across the five lights, the S-only cost is ordered exactly by the rod signal’s share of the S cones’ catch, and the worth of the S route by how far that share exceeds the others’, with one transposition.
• The narrowest distribution of the six departures is a daylight result. Under the phosphor LED the equal-route departure spans a factor of thirty over the same surfaces.

Three routes, five lamps #

The computation is the one the rods are a fourth curve made, with one argument added. The reference observer is the pigment-template observer the observer audit is built on; the departed observer is the same eye with a rod signal added, of the same size — a tenth of each cone class’s peak, scaled by the scotopic sensitivity curve — but with a weight into each of the three channels. Three routes bracket the uncertainty: equal weight into all three, weight into L and M only, and weight into S only.

The departure is the colour difference between the two observers’ readings of each of forty-two analytic surfaces, adapted to the light, and the median over the surfaces is the summary. The five lights are a 2856 K tungsten lamp, a 6500 K thermal radiator for daylight, a phosphor white LED, a three-emitter LED and a fluorescent tube with mercury lines.

The figure above is the result: five lamps, three bars each. Under daylight the first two bars are nearly the same length — 1.03 and 0.90 — and the third, S only, is a third of either. Whatever the rods do to the S pathway under daylight, they do little. Under the phosphor LED the first bar is nearly three times the second — 1.24 against 0.45 — and the S-only bar, at 0.96, is most of the first. Under that lamp the rods’ effect on a colour match is mostly their effect on the S pathway.

A rod signal is nearly a gain under daylight #

The difference is explained by a quantity that can be read off the lamp and the four sensitivity curves, before any surface is considered.

Under daylight the rod signal is 0.86, 0.94 and 0.90 of the L, M and S cones’ own catches — within a tenth of each other. Adding a signal that is the same share of every channel’s response is, for the light itself, a common scaling of the three cones, and a gain is not an observer: a white point divides it out exactly. The departure that remains comes from surfaces whose reflectance tilts the balance between the rod curve and each cone curve, and it is similar whichever channels the signal enters, because each channel receives about the same proportion.

Under the phosphor LED the shares are 0.58, 0.64 and 1.97. The lamp’s output is dominated by a broad phosphor band across the green and yellow, which the rods — peaking near 507 nanometres — catch well and the S cones catch poorly; its blue pump is narrow and does not make up the difference.

The two spectra make the mechanism visible without any arithmetic. The thermal radiator is smooth across the whole visible range, highest in the blue and falling gently towards the red, so it covers the S cones’ band and the rods’ band in about the proportion it covers the L and M cones’ bands, and the four sensitivities catch it in about the proportions they were scaled to. The LED puts four fifths of its light between 500 and 650 nanometres, under the rods’ curve and away from the S cones’, so the S cones’ catch is small and a rod signal added to it is large by comparison. A lamp is not a blackbody described that structure for what it does to rendering; here the same gap does something to the observer. So a rod signal entering the S channel is three times as large, relative to what that channel already carries, as one entering L or M. It is no longer close to a gain on the cones; it is a large distortion of one channel and a small one of the other two, and whether the model includes that channel decides most of the answer.

Tungsten, the tube and the three-emitter LED all show the same pattern less strongly — S shares of 1.66, 1.57 and 1.67 against L and M shares between 0.46 and 1.01. None of them is as balanced as daylight, because none of them is as smooth across the blue.

The share predicts the price #

The relation can be tested across the lamps directly.

The five lamps rise from lower left to upper right with a rank correlation of 0.90 — one transposition, between the tungsten lamp and the fluorescent tube, whose excess shares differ by about a quarter and whose worths differ by about a tenth, in opposite orders. Daylight, with an excess of 0.01, is worth 1.15. The phosphor LED, with an excess of 1.36, is worth 2.75. And the S-only departure by itself is ordered exactly by the S share, at a rank correlation of 1.00.

Five points do not make a law, and the relation is presented as what it is: a mechanism that the arithmetic implies and that the five lamps available all obey. It says which lamps to worry about without computing the departure at all — any lamp whose power sits away from the S cones’ band and inside the rods’ makes the uncertain S weight expensive.

Opening the route by degrees #

The three routes are extremes. The physiology suggests a rod signal reaches the S pathway more weakly than the others rather than not at all, so the useful picture is the departure as the S weight rises from nothing to equal.

Under daylight the curve is nearly flat, rising by 15 per cent across the whole range. Whatever value the S weight really has, the daylight answer is within that. Under the phosphor LED it rises steadily from 0.45 to 1.24, and every quarter step of the weight adds about a fifth of a colour difference. The three-emitter LED and the tube rise by roughly half and four fifths.

The tungsten curve dips before it rises — 0.63 with no S weight, 0.61 at a quarter — because a small S signal partly offsets the tilt the L and M signals give to its warm light, before the S signal itself dominates. That dip is small, and it is a reminder that the departure is a vector sum of the three channels’ changes rather than a sum of their sizes.

The narrow distribution was a daylight result #

The earlier essay’s most distinctive finding was that the rod departure varies least across surfaces of any of the six, and it gave the reason: an addition adds the same vector to every surface, scaled only by how much of the rod’s band the surface reflects.

Under daylight both routes span a factor of about six, as reported. Under the phosphor LED the equal route spans a factor of thirty, from 0.09 to 2.53, and closing the S route pulls the top down to 1.13. The mechanism the earlier essay gave is still right, but it holds only where the rod signal is a nearly uniform share of the three channels. Where it is concentrated in one channel, the surfaces that happen to reflect strongly in that channel’s band take most of the departure, and the distribution widens accordingly.

So the claim should carry its light. Two observers and one metamer established that every one of these departures is a pairing of an observer’s deviation with a stimulus’s, and the lamp is part of the stimulus. The ranking is not stable found that no departure has one size across surfaces; this adds that the spread of a departure across surfaces is not stable across lamps either, and that the rods’ is the most lamp-dependent of the six because it is the one entered by addition.

What this changes for a dim room #

Three consequences, and the qualifications come with them.

A colour match made in a dim room lit by daylight is robust to what nobody knows about rod pathways. The rod contribution there is close to a common gain, and the part that is not is similar whichever channels the rods reach. The eye that has no colour described the regime in which both systems run at once; under daylight its effect on matches can be estimated without settling the physiology.

A match made under a white LED in the same dim room is not. The rod contribution there is concentrated in the S pathway, and the uncertain weight into that pathway changes the answer by up to a factor of three. By the same arithmetic, a lamp with more short-wavelength content — a higher colour temperature, or a second blue emitter — should bring the shares closer together and the uncertainty down with them; that is a prediction from the shares, not a lamp computed here.

And the lamps expected to be most affected are the ones in dim domestic rooms. A warm white LED dimmed in the evening combines rod-dominated sensitivity with an even more S-starved spectrum than the neutral LED computed here, and the shares predict it would make the S weight more expensive still. The lamp that stopped emitting ultraviolet followed a change in household lighting through what it did to whites; this is a second consequence of the same change, and it falls on the observer rather than on the paper.

How the departures were computed #

The reference observer is built from the site’s pigment template, lens and macular models at their median values; its three cone absorptances are converted to tristimulus values by a fixed matrix fitted once to the 1931 functions. The departed observer adds, to each cone class’s absorptance, the scotopic sensitivity curve — the rhodopsin template through the ocular media without macular pigment, normalised to its peak — multiplied by a tenth of that class’s own peak and by the route’s weight for that class.

Each surface in the forty-two-member family — Gaussian absorption bands of seven centres, three widths and two depths — is read by both observers under each light, adapted to that light by CAT16 in tristimulus values, and the colour difference is ΔE₀₀. The share of catch is the rod curve, scaled as it is added, integrated against the light, divided by the cone class’s own integral against the light. The three-laser projector is left out of the census, because on the five-nanometre grid its three lines fall to one and every observer agrees about it exactly.

Where the model stops #

The rod signal enters linearly and with fixed weights. Real rod–cone interaction is nonlinear, the rod contribution changes continuously with adaptation, and the weights into different pathways are themselves adaptation-dependent. The three routes here are a bracket around the uncertainty, not a model of the circuit.

The surfaces are the analytic family, and the lights are constructed spectra rather than measured ones. The observer is a single median eye; the observer has no age is the reminder that a real eye’s lens yellows with age, and an older lens cuts the short wavelengths before either the S cones or the rods see them, which should shrink the S shares under every lamp by a similar factor rather than change which lamps are extreme. A real warm LED’s phosphor mix and a real dim room’s spectrum would move the shares, though the direction — S-starved lamps making the S weight expensive — follows from the curves’ positions and not from any detail.

And everything here is a match, not an appearance. Whether a person’s colour judgements in a dim room under an LED shift in the way a match does is a separate question, and one the model says nothing about.

Still open: whether matches under an LED can measure the weight #

The same arithmetic that makes the S weight expensive under an LED makes it measurable there. Under daylight a mesopic colour match is almost insensitive to the weight, so no matching experiment under daylight could estimate it. Under a phosphor LED the match moves by nearly a colour difference as the weight goes from nothing to equal, which is well within what a matching experiment resolves.

The experiment is a set of mesopic matches — test and reference fields at a luminance where rods contribute, with the reference built from a lamp with a large S share and the test from one with a small share — and the quantity to fit is the S weight that best predicts the settings. The design principle generalises: to measure an uncertain weight, use the condition in which the answer depends on it most, which is the opposite of the condition a standard is written for.

An uncertainty is priced by the condition #

The habit is about how much an unknown parameter matters.

A model with an uncertain weight is often checked by varying the weight under the reference condition and seeing that little changes. That check is honest and it can be very misleading, because the reference condition is often the one in which the parameter matters least — here, daylight, where the rod signal is nearly a gain and the weight hardly enters.

The move is to find the quantity that decides how much the parameter enters — here, the rod signal’s share of each channel’s catch — and then to look for the condition in which that quantity is most extreme before declaring the parameter unimportant. The phosphor LED was the lamp in which the share was most unbalanced, and it was the lamp in which the uncertainty was worth most.

The failure mode is sensitivity analysis done only at the condition the model was built for. A parameter that does not matter there is a parameter that does not matter there, and the reason it does not matter is usually a balance that other conditions break.