Adaptation turns more pairs off than on

Two filters cancel only in a bright enough room followed an older lens and a denser macular pigment across the degree of adaptation, found that the two cancel only once the room is about as bright as an office, and ended by saying the computation takes any two departures and that whether other pairs have crossings of their own is the same arithmetic on a different pair.

It is, and there are fourteen other pairs. Running all of them changes what the first one meant.

A rotation, not a mechanism #

Adaptation does not move pairs of observer departures towards opposition. It removes the yellowing every reading shares and leaves each pair’s own residual, and the residual points wherever it points — which for more pairs is towards agreement than away from it.

• Of the fifteen pairs, five lose their cancellation as the eye adapts, three gain it, three keep it and four never have it.
• The published pair is one of the three that gain it. The lens against the macular pigment runs from 8 degrees apart to 156; the field size against the lens runs from 172 to 18, and the lens against the pigment peaks from 127 to 21.
• Each turning pair turns in its own room. Four of the eight turn at a degree an average surround can reach — 0.9265, 0.9367, 0.9112 and 0.9912, which are 80, 94, 63 and 275 candelas a square metre — and four turn only in a dim or a dark surround.
• In an office, six of the fifteen pairs cancel on most surfaces and nine do not, so at one moment, for one observer, some of their departures are cancelling each other and others are adding.
• The three departures that act like a filter on the white grow by factors of 3.6 to 7.4 as the room darkens; the three that change the shape of the cone curves grow by 1.15 to 1.5, so a dim room amplifies exactly the departures adaptation would otherwise remove.

What the dial is doing #

The mechanism is already in the collection and it is worth restating in the form the fifteen pairs need.

A reading at degree D is the straight blend of the fully adapted reading and the unadapted one, so a departure’s deviation at that degree is the fully adapted deviation plus (1 − D) times the part adaptation removes. Two terms: one fixed, one shrinking. The removed part is the yellowing of the observer’s own white, carried into every surface with it, and it is the larger term by about tenfold on the median surface. The residual is what is left when the shared yellowing has gone.

The angle between two departures’ deviations is therefore decided by which of the two terms is longer — the removed parts near the start of the dial, the residuals near the end. Nothing anywhere requires the two angles to be the same, or even to be on the same side of a right angle.

Nothing sits near the diagonal. The upper left holds pairs that begin nearly parallel and end nearly opposed, which is the behaviour the published pair has and which is what makes adaptation look like a cancellation mechanism. The lower right holds pairs that do the reverse, and it holds more of them.

That asymmetry has a cause and the cause is in the departures rather than in adaptation. The lens, the macular pigment and the field size all act like a filter in front of the eye — the lens because it is one, and the observer has no age is where its direction comes from —, so their removed parts are large and point similarly; pairs among them start nearly parallel or nearly antiparallel depending on which way each filter tips the white, and adaptation strips all of that away. The cone optical density, the pigment peaks and the rods change the shape of the curves rather than putting a filter in front of them, so their removed parts are small and their behaviour hardly changes across the dial at all.

The pairs, one at a time #

Five of the fifteen drawn across the whole dial show the three behaviours in one picture.

The lens against the macular pigment climbs from 8 degrees to 156 and does almost all of the climbing in the last tenth. That is the published curve.

The field size against the lens falls from 172 degrees to 18. Both are filters and both yellow the white, but the ten-degree observer’s departure from the two-degree one is not a yellowing in the same direction — it is a reweighting that is largest in the blue, so the two removed parts point almost exactly opposite ways. Unadapted, the pair cancels on every one of the 120 surfaces. Adapted, it cancels on none.

The lens against the pigment peaks falls from 127 degrees to 21, and the cone optical density against the rods from 134 to 45. Both are pairs of one filter-like departure and one shape-like one, and in both the unadapted angle is the angle between a large removed part and a small one, which the large one dominates.

The macular pigment against the cone optical density barely moves at all — 117 degrees to 127 — and cancels on most surfaces throughout. It is the only kind of pair adaptation leaves alone: one whose removed part is small enough that the residual was already deciding the angle.

Each turning pair has its own room #

A pair that changes its verdict somewhere inside the dial changes it at a degree, and a degree is a room.

The crossings run from 0.56 to 0.99, which is a cinema at one end and a room brighter than any office at the other. Four of the eight sit at degrees an average surround can produce — the field size against the lens at 80 candelas a square metre, the lens against the macular pigment at 94, the lens against the cone density at 63, and the cone density against the pigment peaks at 275. The other four turn at degrees an average surround never reaches, because the model’s own formula saturates: below about 0.824 there is no average-surround room at all, and those pairs turn only when the surround is dim or dark as well.

So the practical statement is not that adaptation switches the observer’s departures from adding to cancelling at some room. It is that every pair has its own switch, the switches are scattered across the range of ordinary rooms, and turning the lights up throws some of them one way and some the other.

In one room, six of fifteen #

The fifteen switches can also be read as a snapshot rather than as a sweep.

Six of the fifteen pairs are above half and nine are below. That is one room, one observer model and one surface set, and the answer to do the observer’s departures cancel each other is that six pairs of them do and nine do not.

Eight of the fifteen sit between a fifth and four fifths, which is the second thing worth carrying: for most pairs the cancelling is not a property of the pair but of the pair and the surface together, and the share is what describes it. A pair at 60 per cent is not a pair that cancels; it is a pair that cancels on three surfaces in five, and which three depends on the surface’s own shape in the way two yellow filters cancel on a slope measured for one pair.

Turning the lights down from an office to a living room changes both the membership and the order. The lens against the cone density falls from a third of the surfaces to almost none; the field size against the lens rises from about half to nearly all. A reader moving between two rooms in the same building is moving between two different accounts of which of their own departures cancel. Nobody here has two eyes found the same mechanism running inside one person; here it runs between two rooms.

The banded surfaces do it differently #

Every number above is on 120 smooth natural reflectances, which is the set the published pair’s result was computed on. The audit’s own banded family is the set a specification is more likely to be about, and it does not agree.

On the banded set four pairs gain their cancellation rather than three and three never have it rather than four, and the individual angles move by tens of degrees. The pair that changes most is the one the published result is about: on smooth surfaces it reaches 156 degrees at complete adaptation and on banded ones it reaches 58, which is a banded surface never being rescued seen in the angle rather than in the cost.

The reading is not that one set is right. It is that the verdict for a pair is a verdict about a pair and a family of surfaces, and that neither of the two families here is what a real product set looks like. What is stable across both is the count: five pairs lose their cancellation and three or four gain it, on both sets, so adaptation turns more pairs off than on whichever surfaces it is asked about.

Which departures the room amplifies #

The pairs are made of departures, and before asking what a pair does it is worth asking what each half does on its own across the same five rooms. The answer sorts the six into two groups that nothing about their descriptions would have sorted them into.

From an overcast sky to a cinema the field size goes from 0.34 colour differences to 2.53, the lens from 2.13 to 7.67 and the macular pigment from 1.09 to 6.69 — factors of 7.4, 3.6 and 6.1. Over the same five rooms the cone optical density goes from 0.13 to 0.19, the pigment peaks from 0.73 to 0.84 and the rods from 0.67 to 0.82 — factors of 1.5, 1.15 and 1.2.

The split is exactly the one the angles predicted. A departure that acts like a filter in front of the eye yellows the observer’s own white, adaptation removes that yellowing, and a dim room removes less of it — so the departure’s size is mostly a function of the room. A departure that changes the shape of the cone curves moves the white hardly at all, so adaptation has almost nothing to take and the departure is nearly the same size everywhere. A departure is straight in the excitations found the same division from the other side, in the geometry of where each departure sends a reading.

That has a consequence for which departure an allowance should be written about. Under the sky the lens is twice the next largest and the three shape-like departures together are worth less than the macular pigment alone. In a cinema the field size has overtaken all three of them. The departure a specification should worry about is not the same departure in a bright room and a dim one, and the ordering changes inside the range of rooms a sample is actually judged in — which is the practical form of what the departures are larger than the tolerance established about their size.

What this does to an allowance #

A tolerance with an observer in it added the six departures and got about three colour differences on an ordinary sample. That number was computed at complete adaptation, and the whole of this essay is about what happens between there and a room.

Two things follow and they point opposite ways.

An allowance computed at complete adaptation is too small, because the departures are larger in every real room than they are at the end of the dial — the same six eyes cost 2.30 colour differences under an overcast sky and 12.45 in a cinema. That is the effect two filters cancel only in a bright enough room found for one pair, and it survives the generalisation intact.

And an allowance that assumes the departures cancel is too small again, for a different reason: five of the fifteen pairs cancel at complete adaptation and stop cancelling on the way down, so the arrangement that made the allowance small is the first thing a dim room takes away. The cancelling is not a reserve that a dim room merely reduces. It is a property of the end of the dial that several pairs lose entirely.

Neither is the usual objection, which is that an allowance should be conservative. A conservative allowance would be one computed where the departures agree most, and where that is depends on the pair — so no single room is conservative for all six departures at once, and the room that is worst for the lens and the pigment together is the best for the field size and the lens.

How the pairs were priced #

The six departures are the audit’s own, each a pair of observers differing in one respect: the field size, the age of the lens, the macular pigment, the cone optical density, the pigment peaks and the rods. The basis change the audit also carries is left out, because it is a change of curves rather than of an eye.

Each departure’s deviation on each surface is read under the daylight source, divided by its own white’s luminance, and adapted towards D65 by CAT16 at a stated degree — which is the blend of the two ends, checked against the matrix form elsewhere to a largest disagreement of 1.1 × 10⁻¹⁶. The angle between two deviations is taken in the local metric of ΔE₀₀ at the reading — the instrument a size is not a direction was built for exactly this comparison — so that an angle is a statement about what a person would see rather than about tristimulus coordinates.

A pair is said to cancel at a degree when the two together cost less than the larger of the two alone on more than half the surfaces. The crossings are found by bisection on that share, and a pair whose verdict is the same at both ends of the dial is reported as having no crossing rather than as crossing at an end — the caution the single-pair calculation already records, for the same reason.

What the fifteen pairs do not settle #

The degree is CIECAM16’s own formula and has no reader’s eye in it, which is the standing limit on every result of this shape and which a discount nobody measured states in full. A reader who adapts more completely than the formula says is to the right of every crossing here.

Pairs are two at a time and the observer has six departures at once. Nothing here says how the six combine, and the pairwise angles do not determine it: a set of vectors can be pairwise obtuse and still sum to something long. That is a separate calculation on the same model.

The verdict threshold is half the surfaces, which is a choice. A stricter threshold moves the crossings and changes the counts, and the direction is predictable — but the shape of the table, five losing against three gaining, survives thresholds from a third to two thirds on both surface sets.

And all six departures are two-standard-deviation constructions of a population, not people. A tolerance is a probability is the shape a real population needs, and a population would put a distribution on each departure rather than a pair of endpoints.

Still open: what the six do together across the dial #

The obvious next reading is the one the pairs cannot give. Six departures acting at once are six vectors, and what they cost together is the length of their sum in the local metric — which the pairwise angles constrain and do not determine.

Two questions come with it and both are arithmetic rather than experiment. The first is whether the sum behaves like the pairs: whether there is a degree at which the six together stop growing faster than a rule of thumb predicts, and where it sits against the eight pairwise crossings above. The second is whether the usual way of combining independent contributions — adding them in quadrature, which assumes they are mutually perpendicular — is right anywhere, given that the fifteen angles here run from 18 degrees to 179 and the perpendicular ones are a minority.

The second is the more useful, because quadrature is what an allowance is actually built from and nothing here has yet asked whether the assumption underneath it holds.

A result from one pair is a result about one pair #

The habit is about what a worked example licenses.

Following one pair across a parameter is the right way to find out what the parameter does, and it produces a mechanism, a crossing and a room. What it does not produce is a rule, because the mechanism was demonstrated on a pair that was chosen — here, chosen because it was the pair whose two members most obviously resemble each other.

The move is to run the same calculation over every instance the model accepts before generalising from the one. It is usually cheap: the calculation that took two departures takes any two, and fifteen runs of it is a loop. What comes back is a distribution rather than an example, and the example’s place in the distribution is the thing the essay about it could not say.

The failure mode is to name the mechanism after the example. Adaptation does not make departures cancel; it removes what they share. On the pair that was looked at first, those are the same sentence.