The lens is worst under tungsten

A departure of the observer is a product of two things and only one of them belongs to the observer. This is the other one, and it changes the answer by a factor of two and the ranking outright.

The claim #

Which observer departure is largest depends on the lamp, and the dependence is strong enough to reverse the order.

• The lens is worst under tungsten at 4.38 ΔE₀₀ and least bad under a fluorescent tube at 2.20 — a factor of two on one departure, from the light alone.
• The pigment peaks are worst under a three-emitter LED at 2.84 and lowest under a white LED at 1.77, and their ranking against the lens changes between lamps.
• The macular pigment moves furthest of all, from 4.47 under tungsten to 1.71 under daylight, a factor of two and a half.
• And one row is exactly zero, for a reason that has nothing to do with observers and everything to do with the tabulation this collection computes on.

Why the light is a factor at all #

The identity established two essays ago says a departure is the pairing of the observer’s deviation with the stimulus’s deviation from the adapting white. The stimulus is R·S and the white is S, so the stimulus’s deviation is S·(R − ρ) for whatever constant ρ the comparison is made about.

The light appears in that expression as a weight. It decides how much of the sample’s spectral departure lands at each wavelength, and therefore how much of it overlaps with the observer’s own deviation, which lives somewhere specific on the wavelength axis.

That is the mechanism and it predicts the table’s shape. An observer departure concentrated in the blue pairs strongly with a light that has power in the blue and weakly with one that does not. A lens departure is a blue absorption; a tungsten lamp is weak in the blue relative to daylight; and yet the lens’s cost is higher under tungsten, which is the first thing in the table that needs explaining.

Tungsten is worst for the lens, and the explanation is a ratio. The explanation is that the pairing is with the stimulus’s deviation from the white, not with the stimulus, and the white is the lamp.

Under a tungsten lamp a red pigment is much closer to the white than it is under daylight in the long wavelengths, and much further from it in the short ones. The sample reflects little blue and the lamp emits little blue, so the relative excitation in the short-wavelength cone is a ratio of two small numbers — and a ratio of two small numbers is exactly where a filter that changes both by different amounts does most damage.

Under daylight both quantities are larger and the ratio is better conditioned. So the lens’s departure is not tracking how much blue light there is; it is tracking how badly determined the short-wavelength ratio is, which is worst when the light is poor in the blue.

That has an unwelcome corollary for practice. The lights under which an observer’s identity matters most are the warm ones, and warm lights are what most retail and hospitality lighting is. A specification written for a daylight booth is being tested in the condition that flatters observer agreement.

The row that is exactly zero #

The laser projector’s row is empty and it would be easy to read as a claim that observers agree about laser light. It is not.

On this collection’s five-nanometre grid a three-line spectrum with lines at 465, 532 and 638 nanometres is a one-line spectrum, because only 465 lands on a grid point. And a stimulus with one wavelength in it is a stimulus every observer agrees about exactly: the sample’s response and the white’s response are the same number times R(465), so the relative excitations are (R, R, R) for every observer, which is the neutral identity arriving by a different route.

So the zero is a fact about the tabulation, not about lasers. Computed on a quarter-nanometre grid the same row reads 1.28, 2.74, 3.78, 1.04, 3.60 and 1.57 — the largest peak departure in the whole table.

That is the sharpest single result of this round and it gets its own essay. Its place here is as a warning about the table: five of the six rows are trustworthy at this grid and one is an artefact, and nothing in the table’s appearance distinguishes them.

The macular pigment’s spread #

The largest light-dependence in the table belongs to the macular pigment, which runs from 4.47 ΔE₀₀ under tungsten to 1.71 under daylight.

Macular absorption is a band centred at 460 nanometres with a width of about forty, which is a narrow and specific place on the axis. So the pairing is strongly selective: it responds to how much of the sample’s departure from the white sits in that band, and almost not at all to anything outside it.

Under a white LED — a blue pump at 452 nanometres with a broad phosphor — it costs 3.37, the second-highest in its row. That is not a coincidence. A blue-pumped LED puts a narrow spike of power almost exactly where the macular pigment absorbs, so the one part of the spectrum where observers differ most is the one part that lamp emphasises.

A lamp designed for efficiency has put its narrowest feature on top of the observer’s most variable filter, and nothing in the design of white LEDs considered that. It is the same kind of accident as the coincidence that turned out to be a mechanism two rounds ago, and it is worth naming because it is a design consequence rather than a fact about eyes.

Putting the two tables side by side is the cleanest available demonstration that the zero was arithmetic rather than physiology. Nothing about the observers changed between them, nothing about the lights changed, and one row went from six zeros to the largest numbers in the table. The other thirty cells moved in the third decimal place.

That is also a reminder of what a well-sampled calculation looks like. Five of the six rows agreeing between a five-nanometre grid and a quarter-nanometre one is not a coincidence; it is the top band of the tabulation section’s three regimes, and it is where nearly everything this collection computes lives.

Reading the table as a specification problem #

For anybody writing a colour specification the table has a direct use and it is not the one it looks like.

The question a specification has to answer is how much of its tolerance to reserve for observer variation, and the answer is not a constant. Under a daylight booth the six departures sit between 1.20 and 2.38 ΔE₀₀ on a saturated sample; under tungsten, between 0.79 and 4.47. The reserve needed is roughly twice as large in the second case, on the same sample with the same population.

That interacts badly with how specifications are written. A tolerance is usually stated once and a viewing condition is stated separately, often permissively — D65 or equivalent, a standard viewing booth. The word “equivalent” is doing a great deal of work: two lamps with the same correlated colour temperature and quite different spectra can differ by a factor of two in how much observer disagreement they produce, and correlated colour temperature is not a spectrum — a point the rendering-index work has already made about a different consequence of the same substitution.

The practical rule that follows is to fix the lamp’s spectrum rather than its colour temperature when observer agreement matters, and to prefer a smooth lamp over an efficient one. That is expensive advice and it is what the arithmetic says.

Three ladders under three lamps, and three different orderings. A reader who has only ever seen the daylight version would take the lens and the peaks to be the two terms that matter; under tungsten it is the lens and the macular, and under a white LED the macular has moved up. A ranking of observer departures is a ranking under a lamp, and none of the three lamps here is unusual.

The practical consequence is that a personalised observer — an eye characterised parameter by parameter for one person — buys different amounts under different lights, and buys most where the light has narrow structure. That is the opposite of the intuition that a careful measurement is uniformly worth more, and it is the same shape as an elasticity: what a number is worth depends on what it is used for.

Why the rows are not multiples #

The strongest structural claim in the table is that no row is a scaled copy of another, and it is worth stating why that matters.

If the six departures were all responding to the same property of the light — its blue content, say — then every row would be the same shape at a different height, and a single number per lamp would summarise the whole table. There would be such a thing as an observer-difficult lamp.

They are not. The lens is worst under tungsten, the peaks under a three-emitter LED, the macular under tungsten but with the white LED second, the rods under daylight and the three-emitter LED equally. Each departure lives at its own place on the wavelength axis and pairs with whatever the light has there, and the lights differ in shape rather than in scale.

So there is no observer-difficult lamp in general. There are lamps that are difficult for a particular departure, and a population contains all six departures at once, which is why the aggregate is flatter than any of its components.

What a lamp designer could do about it #

The table has one actionable reading and it is unusual for this collection to have one, so it is worth being explicit.

A lamp’s contribution to observer disagreement is decided by where its narrow features sit relative to where observers differ, and observers differ in three identifiable places: the lens’s absorption below 500 nanometres, the macular pigment’s band at 460, and the long- and middle-wavelength peaks around 540 to 570. A lamp with no narrow feature in any of those regions produces less disagreement than one with a feature in all three.

That is a design constraint nobody imposes and it is not expensive. A white LED’s blue pump could sit at 470 nanometres rather than 452 — outside the macular band’s core — at some cost in efficiency and phosphor absorption. A three-emitter lamp’s green emitter sits near 528, which is between the middle- and long-wavelength peaks and is close to the worst available position for the peak departure.

None of that is a criticism of anybody’s engineering, because the constraint has never been stated as one. Efficiency, colour rendering and cost are the design variables, and a rendering index is one observer’s opinion rather than a population’s. A lamp scoring well on every published index can still be the lamp under which two normal people disagree most.

What was computed, and how #

Each cell is two observers differing in one argument, looking at the same sample under the same light, in ΔE₀₀ through this collection’s usual CAT16 route. The observers are built from the same template with one parameter moved, so nothing but that parameter differs.

The lights are the six analytic sources of this round’s tabulation audit, which is deliberate: sharing a set of stimuli between two audits is what allows them to be crossed, and the crossing is where this round’s sharpest result came from.

The sample is held at a red pigment throughout, and that is a limitation rather than a control — the table would have a different shape on a blue sample, and the essay’s structural claim is about the lights rather than about which departure is largest.

Where the model stops #

The six lights are constructions, and the two that carry the most weight in the argument are the ones most obviously idealised. A real tungsten lamp is a Planck radiator behind a glass envelope with its own transmission; a real white LED has a phosphor tail this Gaussian does not.

The table is one sample. A blue sample would put the macular and the lens departures much higher and the peaks lower, and no version of this table has been computed over the surface family light by light — which is the obvious next measurement and is not in this round.

And the observers differ in one parameter at a time. Real people differ in all of them at once and the parameters are not independent: age correlates with lens density by construction and with macular density in the literature, and the correlations would change the aggregate without changing any single row.

There is one more reading of the table that belongs to whoever has to choose a viewing condition rather than design a lamp, and it is short. Of the six lights measured, the one producing the least observer disagreement on this sample is a 6500 K thermal radiator, at a mean of 1.77 ΔE₀₀ across the six departures; the worst is tungsten at 2.37. Daylight is the kindest of the six and is also, not coincidentally, the smoothest.

That is a defence of the standard viewing booth on grounds nobody uses to defend it. Booths are specified for daylight simulation because print is expected to be seen in daylight and because the standards say so; the fact that a smooth daylight-like spectrum minimises the spread between observers is a separate benefit, unclaimed and unmeasured in any specification this collection has read. It also means that a booth is the condition under which a disagreement between two people is least likely to be visible, which is the wrong condition in which to test whether a specification is robust to observers.

The generalisation #

The habit is about reading a table of a product.

When a quantity is a pairing, a table of it against one factor is not a property of that factor. It is a slice, and the slice’s shape depends on where it was taken. The tell is that the rows are not proportional: proportional rows mean a single scalar summarises the other factor, and non-proportional rows mean the two factors interact and no summary exists.

The move is to check proportionality explicitly before quoting any row as a ranking. It takes one division and it is almost never done, because a table of numbers invites reading down its columns rather than across its rows.

The failure mode is to publish the slice as a ranking. Every ranking in this round’s audit has reversed under some change of the other factor, and a reader who took the first one at face value would be wrong about the second by a factor of two.

Who found it, and when #

Observer metamerism has been measured since Wyszecki and Stiles, and its dependence on the illuminant is implicit in every treatment: the CIE’s standard deviate observer of 1989 was published with an index that is computed for a stated pair of stimuli under a stated light, so the light was always an argument.

What is not standard is decomposing the effect by physiological parameter and reporting how each one’s light-dependence differs. The published indices give one number for the whole of observer variation, which is the right thing for a specification and hides the fact that its components respond to different parts of a lamp’s spectrum.

Where the ladder goes next #

One factor of the pairing has been varied and the other is the sample. The ranking is not stable there either, and over forty-two surfaces two of the six departures change places.