The room a surface needs is written in its band

Two filters cancel only in a bright enough room found that an older lens and a denser macular pigment cancel on a surface only once adaptation is nearly complete, that each surface has its own crossing — the degree at which the part adaptation removes falls to the size of the part it leaves — and that those crossings run from 0.76 to 0.98. It reported the spread and did not say what decides it.

Four things describe a surface in the audit’s family: where its absorption band sits, how wide it is, how deep it is, and how much light the surface returns overall. Only one of them is a colour.

The crossing is a hue, not a lightness #

Where a surface’s absorption band sits moves its crossing fifteen times as far as how much light the surface returns does — so the room a surface needs before two yellow filters begin to cancel on it is a property of its colour and almost not at all of its lightness.

• Across band centres from 430 to 670 nanometres the median crossing runs from 0.883 to 0.972, which is 37 candelas a square metre against 168.
• Across reflectance levels from 0.06 to 0.82 — a fourteenfold range — it runs from 0.9336 to 0.9394, a spread of six thousandths of the dial.
• The mechanism is one ratio. The crossing is one minus the reciprocal of a surface’s removed part over its residual, and that ratio runs from 3.1 to 77.7 across the set, rising steadily towards the red.
• In a graphic-arts viewing booth at 127 candelas a square metre the bands at 430, 470, 510 and 550 nanometres have passed their crossing and those at 590, 630 and 670 have not. In a design studio, an office or a warehouse aisle, none has.

What the crossing is made of #

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, and the surface’s crossing is where the shrinking one falls to the size of the fixed one: crossing = 1 − |residual| / |removed|. So the crossing is one number about the surface: how much of what a yellow filter does to it adaptation will take away. A ratio of ten puts the crossing at nine tenths, a ratio of forty at thirty-nine fortieths, and a ratio of three at two thirds.

The ratio runs from 3.1 to 77.7 over the 168 surfaces, and its median rises from about five at 470 nanometres to about thirty-five at 670. That is the whole of the mechanism, and stating it that way makes the cause obvious.

Both of the filters this is about absorb in the blue. An older lens yellows, which is to say it absorbs short wavelengths — the observer has no age is where that direction comes from — and a denser macular pigment has a band centred near 460 nanometres. A surface whose own absorption band sits in the blue is where the two filters differ in shape, because the surface is removing light from exactly the region the two filters treat differently — so what the two do to it differs, and the residual is large. A surface whose band sits in the red is nearly invisible to both filters, so both do the same thing to it: yellow its white, by the same amount, in the same direction. Its whole deviation is the shared part, there is almost nothing underneath, and the ratio is large.

The ordering across the band centres is not quite monotone, and the exception is the most direct evidence for the mechanism. The lowest crossing in the set is at 470 nanometres, not at 430: 0.883 against 0.920, which is 37 candelas a square metre against 73. A surface absorbing at 430 is short of where the macular pigment’s own band sits; one absorbing at 470 overlaps it almost exactly. The macular pigment’s band is centred near 460 nanometres and is narrow, while the lens’s absorption is a broad slope that rises steadily towards the ultraviolet — so the wavelength at which the two filters differ most in shape is not the shortest wavelength but the one where the narrow band sits, and the surface that puts its own absorption there is the one with the largest residual and the earliest crossing.

That is a prediction the mechanism makes and the numbers confirm: the dip is where the macular band is. A model in which both filters were simply yellowing would give a crossing rising monotonically from the blue end, and it does not.

The one thing that does not matter #

A reader might reasonably expect the level to matter. Every other observer effect measured here is larger on a dark sample than a pale one, because a colour difference formula weights lightness and chroma differently at different levels and because the departures are larger than the tolerance measured them mostly on pale surfaces.

Where the band sits is worth 0.089 of the dial and how deep it is 0.067; how wide it is is worth 0.020 and how much light the surface returns 0.006. The level’s contribution is a fifteenth of the band centre’s and a tenth of the depth’s, over a range of reflectance from six per cent to eighty-two.

The reason it does not matter is that the crossing is a ratio. Halving a surface’s reflectance halves the light it returns under both filters, which scales the removed part and the residual by nearly the same factor and leaves their quotient alone. The level moves the deviations’ sizes — two yellow filters cancel on a slope reports those — and not the degree at which one overtakes the other.

The depth is the interesting middle case. A band twice as deep drops the median crossing from 0.963 to 0.896, which is 144 candelas a square metre against 48: a deeper band gives the two filters more to disagree about in the region they differ in, so the residual grows faster than the removed part does. The two terms scale differently because the removed part is set by how much the surface’s white moves — the total light returned, weighted by the filter — while the residual is set by how differently the two filters weight the wavelengths the surface actually removes. Doubling a band’s depth doubles the second and moves the first by much less, because a Gaussian band forty nanometres wide takes only a small share of the whole return whatever its depth. So the crossing is a function of how strongly the surface absorbs and where, which is what a hue is, and not of how much light it returns overall, which is what a lightness is.

What the band centre does not explain #

The medians are orderly and the surfaces around them are not. At 430 nanometres the crossings run from 0.808 to 0.938; at 590 from 0.819 to 0.984; at 670 from 0.929 to 0.987. So the band centre sets where a group sits and something else sets how wide the group is, and the something else is mostly the depth.

On the audit’s own set, where every surface returns the same amount of light, the medians by band centre are 0.904, 0.847, 0.876, 0.947, 0.956, 0.957 and 0.970 — within about four hundredths of the widened family’s throughout. Removing the level changes almost nothing, which is the level-independence measured a second way and is the cleanest form of it: two sets differing only in whether the level varies give the same answer.

What is left inside each group is the depth and the width. A band twice as deep drops the median crossing from 0.954 to 0.863 on the audit’s set and from 0.963 to 0.896 on the widened one, so a deep band at 590 nanometres can sit below a shallow one at 470. The ordering by hue is a tendency across the family rather than a rule about any two surfaces, and a specification sorting shades by crossing has to compute each one rather than reading it off a hue angle.

That is the honest limit on the result and it is worth being plain about. The band centre explains where the groups are and not which side of a room’s line an individual surface falls, and the individual surface is what a tolerance is applied to.

The rooms, and what reaches its crossing in them #

A degree is not a quantity anybody states. A room is, and the model’s formula converts between them, so the table can be written in illuminances.

A perfectly diffusing grey of twenty per cent returns illuminance times reflectance over π, which puts a graphic-arts viewing booth at 2,000 lux at 127 candelas a square metre, a design studio’s desk at 500 lux at 32, an office at 300 lux at 19 and a warehouse aisle at 100 lux at 6.

In the booth the bands at 430, 470, 510 and 550 nanometres have passed their crossing and those at 590, 630 and 670 have not. In every other room, none has.

That is the practical result and it is narrower than the published one. The cancellation the earlier essay found is not a property a reader has in an office; it is a property they have in a viewing booth, for surfaces that absorb short of about 570 nanometres. The surfaces a colour specification is most often written about — the reds, the oranges and the browns — never reach it in any room a sample is judged in.

Reaching the crossing is necessary and not sufficient #

The crossing is where one term overtakes the other, and overtaking is not the same as cancelling: two vectors can each be shorter than they were and still add to something longer than either.

Under an overcast sky, where the model’s degree is 1.000, every surface is to the left of the line and many of them still do not cancel. The prediction is a necessary condition rather than a sufficient one — which is a size is not a direction arriving as a verdict rather than as a measurement — and it is the shape the earlier essay established for one room, holding at the other end of the dial too: reaching a surface’s crossing does not guarantee the cancelling, and not reaching it rules the cancelling out.

In an office the line falls in the middle of the set, and the right-hand side is empty of filled marks — no surface whose own crossing lies above the room’s degree cancels there. That is the condition doing its work, and it is worth having because it is cheap: a surface’s crossing is one ratio computed from its own reflectance and two published filters, with no room in it at all.

What a specification could do with it #

The crossing is a per-shade quantity and it can be computed from a measured reflectance, so the awkward part of this is not the arithmetic.

Sort the shades by crossing rather than by anything else. A range’s shades split into those whose observer disagreement a bright room will shrink and those it will not, and the split is at the room’s own degree. That is a different sort of grouping from the ones a brand colour for a population works with, and it is computable from a reflectance without any population at all. Two shades of the same lightness on the same substrate can fall on opposite sides.

And state the room for the shades that need it. A tolerance has no light level established that a tolerance carries an unstated room and priced it at a tenth to a fifth between the rooms a specification is actually read in. This adds a second, larger reason for the same requirement, and one that applies unevenly: the room is worth little for a blue shade, because a blue shade is past its crossing in a booth and its observer disagreement is near its floor, and it is worth a great deal for a red one.

The awkward reading is that the booth is not conservative. A viewing booth is chosen to be bright for contrast and acuity, and it turns out also to be the one room where some of this cancellation is available — so a shade approved in a booth and sold under a shop’s lighting has crossed back to the other side, and the lamp in the shop decides is the neighbouring argument about what else changes on the way.

How the crossings were computed #

The surfaces are the audit’s family widened to four reflectance levels: 168 surfaces, each a constant reflectance with a single Gaussian absorption band of stated centre, width and depth, at levels from 0.06 to 0.82. The two departures are the audit’s own — an observer whose lens is a seventy-year-old’s against one of twenty, and one whose macular pigment is 0.61 against one at 0.09.

Each surface’s deviation under each filter is read under the daylight source, divided by its own white’s luminance, and adapted towards D65 by CAT16 at a stated degree. The removed part is the difference between the unadapted deviation and the fully adapted one, and the residual is the fully adapted deviation; both lengths are taken in the local metric of ΔE₀₀ at the reading, which is what makes their ratio a statement about a person rather than about coordinates.

The crossing is 1 − |residual| / |removed|, computed in closed form rather than searched for, and the room it implies is the model’s own formula for the degree inverted. Each room’s own degree is computed from its adapting luminance and surround by the same formula in the forward direction.

What this leaves out #

The bands are Gaussian and single. A real pigment has several absorption features and a shape that is not a Gaussian, and whether the crossing of a two-band surface is between the crossings of its two bands, or outside them, is the same computation on a different family and has not been run.

The rooms are converted from illuminance through a twenty per cent grey, which is a convention rather than a measurement of any actual room. A booth with a lighter surround, or a sample held closer to the lamp, produces a different adapting luminance and moves the line.

The two filters are one pair of six departures, and adaptation turns more pairs off than on found that the other pairs behave differently — some losing their cancellation as the room brightens. Whether their crossings are also written in the band is the same computation on a different pair, and the mechanism argued here would predict that it is for any pair of filter-like departures and is not for a pair involving a shape-like one.

And the degree comes from CIECAM16’s formula, which has no reader’s eye in it. Every room quoted here is the model’s opinion of a room.

Still open: whether a real shade library splits the way this family does #

The result is a rule for sorting shades and the family it was measured on is constructed. A dyehouse’s shade library, a paint manufacturer’s fan deck or a press’s ink set on its own substrate would give the distribution of crossings that actually matters, and the question it answers is simple to state: what share of a real range is past its crossing in the room it is approved in?

Three outcomes are possible and they suggest different things. If nearly all of a range is past its crossing in a booth, the observer allowance can be stated once for the range and the room named alongside it. If nearly none is, the allowance has to be the unreduced one and the booth’s brightness is buying nothing. If the range straddles — which the spread here suggests it would, since a fan deck covers every hue — then no single allowance describes it, and a specification that quotes one is describing the reds and the blues with the same number when the two differ by a factor the room cannot close.

The measurement is cheap: reflectances a laboratory already has, two published filters, and one ratio per shade.

A spread is a question about which parameter #

The habit is about what to do with a reported range.

A measurement over a family produces a distribution, and reporting its ends is honest and nearly useless: the crossings run from 0.76 to 0.98 says a surface matters and does not say which surfaces. The family was built by varying several things at once, so the spread is a mixture, and the useful step is to ask which of the varied things it belongs to.

The move is to group the family by each parameter in turn and take the spread of the group medians. It costs one pass over data already computed, and the answer is usually lopsided — here one parameter carries fifteen times another’s share — which turns a range into a rule that can be applied to a surface nobody measured.

The failure mode is to report the range and stop. A range is a description of the family that was tested; a parameter’s share of it is a prediction about the next surface, and only the second survives the family being replaced.