Two filters cancel only in a bright enough room

Two yellow filters cancel on a slope put an older lens and a denser macular pigment on the same surface and found that they point in nearly opposite directions once an eye has adapted — a median of 156 degrees apart on smooth reflectances, with the pair together costing less than the lens alone on 115 of 120 surfaces. Read without adaptation the same two filters point 8 degrees apart and add.

Adaptation was a switch there — on or off, and every number quoted at “on”. No eye is at either setting. An appearance model computes a degree of adaptation from the room, and a discount nobody measured found what it says about an ordinary one: not one, but 0.94.

The cancelling lives in the last tenth of the dial #

The two filters cancel only when adaptation is nearly complete, and the degree where the cancelling begins falls inside the range of ordinary rooms.

• Between no adaptation and a degree of 0.8 the angle hardly moves — from 8 degrees to 25 — and the two filters add all the way.
• The angle passes a right angle at a degree of 0.928 and reaches 156 degrees at 1, so nearly the whole turn happens in the last tenth of the dial.
• The pair stops costing more than the larger filter alone at a degree of 0.9396, which the model’s own formula makes an adapting luminance of 98 candelas a square metre — about an office.
• In a cinema the two together cost 15.3 colour differences against 8.9 for the lens alone, and under an overcast sky 1.28 against 2.29. The sign of the effect is set by the room.

A degree is a point between two readings #

The dial is the model’s D, and what it does to a reading is worth stating exactly, because it is what makes the rest of this computable from two numbers rather than from a new integral at every setting.

Partial adaptation is a von Kries gain of D · (white ratio) + (1 − D) applied in the transform’s own axes. That gain is affine in D, and it sits between a fixed matrix and its inverse, so the whole map is affine in D too. A partly adapted reading is therefore the straight blend of the fully adapted reading and the unadapted one — D of the first plus 1 − D of the second — and that is checked against the matrix form rather than assumed, to a largest disagreement of 1.1 × 10⁻¹⁶.

So each observer’s reading of each surface travels along a straight line as the room brightens. The angle between two such readings does not. Two points moving on straight lines at different speeds sweep out an angle that can turn through anything at all, and here it turns through 148 degrees, almost all of it after the readings have nearly arrived.

The turn happens where no room is #

The shape of that curve is the argument. From a degree of 0 to 0.6 the angle rises from 8 degrees to 15. By 0.8 it is 25, by 0.9 it is 65, by 0.94 it is 108, and by 1 it is 156. Three fifths of the turn happens in the last tenth, and the last tenth is the part of the dial the model reaches only out of doors.

The reason is that the part the two filters share is the part adaptation removes. Both yellow the observer’s white; both yellow every surface with it. A reading that is nine tenths adapted still carries a tenth of that shared yellowing, and a tenth of something large is larger than the residual difference in shape that is left when all of it is gone. The shared part dominates the angle until it is almost entirely gone — and then, suddenly, it is not there and the residuals are all that is left.

That also explains the sizes. The pair costs 42.0 colour differences unadapted and 1.28 fully adapted: adaptation removes 97 per cent of what the two filters do. At the office’s 0.94 it removes 93 per cent, and the 4 per cent difference between those two removals is the whole of the effect this essay is about.

The ten-to-one that decides where the dial turns #

Why the turn is so late can be got out of the blend rather than observed in it, and it comes with a number.

Rearranging the blend, a deviation at degree D is the fully adapted deviation plus (1 − D) times the part adaptation removes. Two terms, one fixed and one shrinking. The removed parts are the shared yellowing of the observer’s own white, so the lens’s removed part and the macular pigment’s point nearly the same way: a median of 10 degrees apart. The residuals are what is left when that shared part has gone, and those point nearly opposite ways: 156 degrees apart, which is the published result.

The angle between the two full deviations is therefore decided by which term is longer, and the two terms are not the same size. On the median surface the removed part is ten times the residual. So (1 − D) has to fall to about a tenth before the residuals can be heard at all — which puts the turn at a degree of about 0.9, and the measured right angle is at 0.928.

That also explains why the room’s effect is so uneven across surfaces. The ratio is not ten everywhere: it runs from 4.2 at the tenth percentile to 47.5 at the ninetieth, so each surface has its own degree at which its two terms change places, and those crossings run from 0.76 to 0.98.

The prediction is a scale rather than a threshold, and it behaves like one: of the 38 surfaces whose own crossing lies above an office’s degree, not one cancels in an office, while 61 of the 82 below it do. Reaching a surface’s crossing does not guarantee the cancelling; not reaching it rules the cancelling out. That is the shape of a necessary condition, and it is what makes the 61 of 120 in an office a composition of two facts rather than a single coin toss — the room is dim enough that a third of these surfaces are out of reach before the pair is considered at all.

What the pair costs, room by room #

A degree is not something a reader chooses. It is what the model computes from the adapting luminance and the surround, so the honest way to read the dial is by naming rooms.

Under an overcast sky the model returns a degree of 1.000, and the published result holds exactly: 156 degrees, 115 of 120 surfaces cancelling, 1.28 against 2.29.

In an office at 100 candelas a square metre the degree is 0.9407. The median angle is 108 degrees, 61 of the 120 surfaces cancel — a coin toss — and the pair costs 2.86 where the larger of the two on each surface, averaged, costs 2.88. The cancelling is technically still on, by a margin of two hundredths of a colour difference. Against the lens’s own mean of 2.57 the pair is already more expensive: which comparison is made decides whether an office counts as cancelling at all.

In a lit living room at 20 candelas the degree is 0.8584, the angle is 40 degrees, nine surfaces of 120 cancel, and the pair costs 6.30 against 4.46. In a dim room it is 11.26 against 6.90, and in a cinema 15.30 against 8.90, with not one surface of the 120 cancelling in either.

The ordering of the costs is worth saying on its own. The two filters cost a reader under the sky 1.28 colour differences and a reader in a cinema 15.3 — a factor of twelve, from nothing but the room, with the same two eyes and the same surfaces. A tolerance with an observer in it added six observer departures and got about three units on an ordinary sample; a dim room multiplies the two largest of them together rather than letting them cancel.

A room has to be as bright as an office #

The crossing can be quoted as a room rather than as a number, because the model’s degree is an invertible function of the adapting luminance in an average surround.

Inverting it puts the crossing at 98 candelas a square metre. A standard office is quoted at 100. So the boundary between the two filters cancelling and the two filters adding runs almost exactly through the room an appearance model is most often asked about, and which side of it a given reader is on is decided by the third decimal place of a number nobody measures on the reader at all.

That is a different kind of fragility from the one a discount nobody measured reported. There the degree’s departure from one was a factor of 1.7 on a residual — a size. Here it decides a sign.

A banded surface is never rescued #

The room is one of two conditions, and the other was settled by the essay before it: the shape of the surface decides too. What the dial adds is that the two conditions are independent.

On the banded families — the audit’s own surfaces, and the widened set of band centres, widths and levels — there is no degree of adaptation at which the two filters cancel. The crossing that exists on smooth reflectances does not exist there at all: at complete adaptation the median angle is 58 degrees and the pair costs 2.25 against 1.93 for the larger alone, and every lower degree is worse — the cost falls monotonically as the room brightens and never falls past the single filter. A surface with an absorption band in it is not rescued by any room.

So the cancelling needs both a smooth surface and a bright room, and a reader gets the published result only when they have both. On a banded surface in a cinema the two filters are simply additive, which is the arrangement the naive expectation assumed all along — and which is right for the wrong reason.

What a specification would have to say #

A tolerance that means to cover the observer has to name the room, and none of them does.

The usual arrangement is to quote an allowance for observer variation — so many colour differences for the population a sample will be judged by — and to quote it once. What this essay measures is that the allowance for these two departures is 1.28 colour differences under the sky, 2.86 in an office and 15.3 in a cinema, on the same surfaces and the same two eyes. A number covering the first does not cover the third by a factor of twelve. An observer is a contract argued that the standard observer is an agreement rather than a description; an allowance quoted without a room is an agreement with a term missing.

The practical reading is narrower than it sounds, because the direction of the error is known. A judgement made in a dim room carries more observer disagreement than the same judgement made in a bright one, and the extra is not a small correction: between an office and a lit living room the pair’s cost more than doubles. Anyone setting up a viewing booth already knows to make it bright, for contrast and for the eye’s own acuity. This is a third reason, and it is about which observer’s answer comes back rather than how well they can see.

It also sharpens what a population model is for. Nobody here has two eyes found that one person’s two eyes differ by a whole colour difference on an ordinary pair and that adaptation hides it exactly — the same mechanism as here, running on one observer instead of two. The hiding is what a bright room buys, and a dim room stops buying it.

What was computed, and how #

The two departures are the audit’s own: an observer whose lens is that of a seventy-year-old against one of twenty, and an observer whose macular pigment is 0.61 against one at 0.09. Each is read on a surface under the daylight source, divided by its own white’s luminance, and adapted towards D65 by CAT16 at a stated degree. The angle between the two deviations is taken in the local metric of ΔE₀₀ at the reading, which is the instrument a size is not a direction built for exactly this comparison; the costs are ΔE₀₀ on the same readings.

The surface sets are the ones that essay used: 120 smooth natural reflectances, the audit’s forty-two, and a widened family of banded surfaces at four levels. The rooms are five pairs of adapting luminance and surround, and the degree each produces is CIECAM16’s own — D = F · (1 − exp((−La − 42) / 92) / 3.6), with F at 1.0, 0.9 and 0.8 for an average, dim and dark surround. The two crossings are found by bisection on the degree, and a crossing that does not exist inside the dial is reported as absent rather than as the end of it.

The degree is a model’s guess, not a measurement #

Everything here rests on CIECAM16’s formula for D, which is fitted to corresponding-colour experiments and has no reader’s eye in it. A reader who adapts more completely than the formula says is further to the right of the crossing than this essay puts them, and one who adapts less is further to the left.

The formula also has no time in it, which a viewing condition is a moment turned into three clock readings: an observer who will adapt completely takes about 107 seconds to get there in an average surround. Someone who has just walked into the office is on the adding side of the crossing for the first minute or so and on the cancelling side afterwards, with the same eyes in the same room.

The surround enters twice over and only one of the two is drawn here. It lowers the degree, which is the effect this essay follows, and the surround is three rows of a table showed that the standard tabulates three rooms of a continuous parameter with the middle row not in the middle. A reader between two rows gets a degree by interpolation that the model never fitted.

And the two filters here are one pair out of the six departures the audit measures. Whether other pairs have crossings of their own, and where, is the same calculation on a different pair — the computation takes any two.

Still open: the degree a reader actually reaches #

The degree used throughout is computed from the room, and the quantity that decides the sign of this effect is therefore a model output rather than a measurement of anybody. What would settle it is an adaptation measurement on real observers in rooms of stated luminance — the setting at which a matching judgement stops moving — read against the formula’s prediction for the same room.

That experiment would also settle the more useful question, which is not where the median observer’s degree sits but how wide the spread is. If readers in one office range from 0.90 to 0.98, then the cancelling is on for some of them and off for others in the same room at the same moment, and the population’s spread in this pair of departures is bimodal rather than merely wide. Nothing in the model has a term for that, because the model has no clock and no observer either: its degree is a function of the room alone.

A result at the end of a dial #

The habit is about a result computed at the extreme value of a parameter that the world does not put there.

Complete adaptation, a point source, an infinitely sharp edge, a perfectly diffuse surface: each is the clean end of a dial, each makes an argument legible, and each is a place nothing actually sits. A result that holds there is not wrong; it is a limit, and a limit is worth as much as the distance between it and the world.

The move is to ask what value of the parameter the situation actually produces, get it from whatever model already computes it, and re-read the result at that value. Here it takes one extra argument and turns a clean cancellation into a coin toss.

The failure mode is quoting the limit as the result. Two yellow filters cancel — outdoors, on a smooth surface, for a reader whose eyes have had two minutes to settle.