The same wall applied twice

A room’s walls are part of its lighting. Light leaves the lamp, meets a painted surface, and what comes off has been multiplied by that surface’s reflectance — so anything the bounced light then falls on is being lit by a spectrum the wall chose. This site has measured what that does to a white wall and what it does to a metameric match in a corner. What it has not asked is what an observer standing in the room can do about it.

The answer is: about as much as they can do about a change of colour temperature, which is a great deal — until the light bounces twice.

The claim #

A bounce is a change of illumination of the ordinary kind, and a second bounce is not.

• One bounce off a green wall moves a surface ΔE00 23.6 and leaves 1.72 after adaptation — 7.3 per cent, which sits among the changes of daylight rather than among the discharge lamps.
• Two bounces off the same wall move it 31.4 — 1.33 times as far — and leave 3.37, which is 1.96 times as much.
• The matrix moves further off the diagonal, from 1.14 to 1.94, and that is where the extra came from: squaring a reflectance sharpens it, and a sharper change of light is further from being a gain.
• A bounce commutes with a change of daylight to within about ten parts per thousand, so the same axes that handle a change of colour temperature handle a wall. Two bounces do not: the worst pair in the whole census is a halophosphate tube against two bounces off this wall, at 152.
• And a red wall is cheaper than a green one — 5.2 per cent against 7.3 — for the same reflectance shape at the other end of the spectrum.

A wall is an illuminant #

The first thing to establish is that this is a change of context at all, because adaptation can only do something about changes that are common to the whole field.

It is. In a room lit by bounced light, every object is receiving E · ρ_wall rather than E, and the white the observer adapts to is the same product. That is exactly the structure of a filter in the path, and it is why the wall rows sit in the census beside the macular pigment and the ageing lens rather than in a category of their own.

Why squaring is the whole of it #

A reflectance is a number between zero and one at every wavelength. Squaring it takes the parts near one and leaves them near one, and takes the parts near a half and puts them near a quarter. The result is the same curve with its contrast increased — narrower where it was narrow, deeper where it was shallow.

Sharpness is precisely what the adaptation arithmetic is sensitive to. A gain has three numbers with which to follow a change; a broad, gentle change is close to a slope and three numbers can nearly follow a slope, while a change with structure at ten or twenty nanometres is structure the three channels sample differently and the ratios between them are decided by where the structure falls inside each channel’s overlap.

The off-diagonal measure follows exactly. One bounce puts the residual operator 1.14 away from the identity; two bounces put it 1.94 away, which is a bigger step than the change in size accounts for.

Which says the sharpening rather than the doubling is what costs, and the way to check that is to square something that was never sharp.

The albedo is not the variable #

There is an obvious alternative explanation for all of this and it is wrong, which is worth saying because it was the first thing tried.

The obvious explanation is that a corner simply has more bounced light in it, so the wall’s contribution is a larger fraction of what arrives and everything scales up. If that were the mechanism, raising the wall’s albedo would produce the same effect, and it does not: the spectral rendering essay isolated exactly this and found the error is a bounce count rather than an albedo, exact at one bounce and growing thereafter.

The same isolation holds here. A wall of the same shape and a higher albedo delivers more light of the same spectrum, and a change of level is the row of the census that comes out at exactly zero. What moves the residual is the number of times the reflectance has been multiplied in, not how much of it there is.

That distinction is easy to lose because in a real room the two co-vary — a brighter room has more interreflection, and a more enclosed one has both. Walking the Neumann series rather than varying the paint is what separates them, and it is the only way to get a number that means anything.

This is the corner result, from the other side #

The corner essay established something that looks unrelated: a pair of surfaces that match exactly under one illuminant stop matching when both are put in a corner, because the metameric black between them is invisible and its square is not. The algebra there is

ρ_B² − ρ_A² = 2ρ_A b + b²

and the second term is non-negative everywhere, so it cannot integrate to zero against any matching function.

Both results come from the same operation and it is worth seeing that they are not the same result. That one is about a pair of surfaces and the identity between them; this one is about a single surface and how much of the change an observer can undo. The first survives adaptation entirely — adaptation is common to the field and a broken match is a difference between two things in it. The second is what adaptation was for.

Why the red wall is cheaper #

The census has a red wall as well as a green one, built to the same recipe at a different centre wavelength. The change it makes is slightly larger — ΔE00 26.2 against 23.6 — and the residual is smaller: 1.36 against 1.72, which is 5.2 per cent against 7.3.

The asymmetry is a property of where the matching functions are, not of the wall. A reflectance centred at 640 nm modulates the long-wave end, where the L and M curves are both falling and their ratio changes slowly; one centred at 540 sits on the steep crossing between them, where a small change in the light’s shape moves the ratio a lot. So the same reflectance moved along the spectrum is a different distance from being a gain, and it is closer at the red end.

This is worth stating because it inverts an intuition. A red room feels like a stronger imposition than a green one, and it produces a larger colorimetric change — and it is the one adaptation handles better.

The share is the number the essay does not print #

Three quantities are quoted for the green wall — the change, the residual and the off-diagonal measure — and the one that makes the two rows directly comparable is the residual as a fraction of the change, which appears for one bounce and not for two.

The share grows by a factor of 1.47. That is the statement the essay’s argument needs and does not make: it is not merely that a bigger change leaves a bigger residual, which would be unremarkable — it is that a larger fraction of a larger change survives, so the second bounce is worse in both terms at once.

The claim section gives the two multipliers, 1.33 on the change and 1.96 on the residual, and their ratio is exactly this 1.47. Printing it saves a reader the division and puts the two wall rows on the same footing as the census’s percentage column, where the one-bounce row’s 7.3 per cent is already quoted as its place in the ranking.

The off-diagonal measure overstates the growth #

The essay attributes the extra residual to the operator moving further from the diagonal, and the two numbers can be asked whether they move together.

They move in the same direction and not at the same rate. The off-diagonal measure rises by a factor of 1.70 and the share by 1.47 — so if the share simply tracked the off-diagonal measure, two bounces would leave 12.4 per cent rather than the 10.7 measured. Solving for the exponent that relates them gives 0.73.

So the mechanism the essay names is the right one and it is not a proportionality. A residual operator twice as far from the identity leaves less than twice as much behind, at least over the one step measured here, which is worth knowing before the off-diagonal figure is used as a proxy for what an observer is left with. Two points fix a rate rather than a law, and the rate is sub-proportional.

What the red wall’s number is worth as a prediction #

The red-against-green comparison is stated as an inversion of intuition, and it has a size. Moving the wall’s centre from 540 to 640 nanometres raises the total change by 11 per cent and cuts the share by 29 — 5.19 per cent against 7.29 — so the larger imposition is a third easier to adapt away.

That yields a prediction the site could run and has not. If the second bounce multiplies the share by about 1.47 whatever the wall’s centre, then two bounces off the red wall would leave about 7.6 per cent — very nearly what one bounce off the green wall leaves. A corner painted in the red would be about as hard for an adapted observer as a flat wall painted in the green.

The assumption behind that is the one the essay’s own argument makes least certain: the share multiplier is a property of squaring the reflectance, and squaring a curve centred where the L and M fundamentals are falling together does not obviously sharpen it in the same way as squaring one centred on their crossing. If the multiplier turned out to be materially different for the red wall, that would be the more interesting result, because it would say the two effects — where a reflectance sits and how many times it is applied — are not separable, and the essay currently treats them as though they were.

Either answer is one row of the census away, and the gate already in place would catch a change that made the second bounce cheap.

What a corner does to a specification #

The practical form of this is a gap in a document. A colour tolerance names an illuminant, an observer and a limit, and it has no field for the geometry the sample will be seen in.

That is already known to matter for matching. What the numbers here add is that it matters for appearance too, and in a direction nobody corrects for: two coatings approved on a flat plaque, mounted in a fold or a louvre or a channel, are being seen under an illuminant sharper than the one they were approved under, and the part of that an observer cannot adapt away has doubled.

The gap is not that the standards forgot geometry. They are careful about it on the measurement side — an instrument is required to state where it was standing, and 45°/0° and d/8° are not interchangeable. What has no field anywhere is the geometry the sample will be installed in, which is a different quantity and is the one this essay is about. A specification can name the illuminant to four digits and the measuring geometry to a standard clause, and say nothing at all about whether the part is flat.

And the direction of the error is the awkward one. A folded panel is not merely harder to predict; it is systematically harder, because folding can only add bounces and bounces can only sharpen. There is no arrangement of a coated part that makes its illumination smoother than a flat plaque’s, so every departure from the approval condition moves in the same direction, and a tolerance built from flat-plaque measurements is a tolerance that is optimistic everywhere it is used.

Why nobody in the room notices #

The numbers so far are what an adapted observer is left with, and a reader may reasonably object that they have stood in green rooms and corners of green rooms without the second feeling twice as green as the first. The objection is right and the arithmetic answers it.

Two things are happening at once and only one of them is in this essay. A corner is lit by light that has bounced more, which is what is measured here; and a corner is also darker and smaller, which is a change of level and of field size, and both of those are things adaptation and appearance handle well. The residual measured here is a chromatic error of a few units on a surface in the corner, sitting inside a lightness change of tens of units that the observer is entirely used to.

It is also local. A person standing in a room adapts to the room’s overall white, not to the corner’s; the corner is a small part of their field, and the pool that would adapt to it separately is about half a degree across. So the corner’s own bounce is not something the observer’s gain is tracking at all, which means the numbers here are what a colorimeter at that point would be left with after the room’s gain — an upper bound on what somebody standing there sees, not a description of it.

What makes the effect matter is not that anybody sees it as a colour cast. It is that two objects in that corner, matched under the room’s general light, are being compared under a sharper illuminant than the one they were approved under — and a comparison is exactly the case adaptation does not protect, because it applies the same gain to both members of the pair.

Who found it, and when #

Interreflection has been computed since radiosity arrived in graphics in the mid-1980s, and computed per band for very nearly as long. The spectral consequences of it were understood early and mostly discussed as a rendering accuracy question: three-channel rendering gets colour bleeding wrong, and the error grows with the bounce count.

The adaptation half is not usually asked. A renderer computes what arrives at the eye and stops; an appearance model takes what arrives and a white point and reports what it looks like. Between the two sits the question of what the observer can remove, and it is a question about the shape of the spectral change rather than about either stage.

What was computed, and how #

The wall is a Gaussian reflectance on a pedestal — base 0.25, depth 0.6, width 60 nm — which is a caricature of a painted surface and is stated as one. It is broader than a saturated pigment and narrower than a domestic emulsion. The one-bounce context is D65 multiplied by that reflectance; the two-bounce context is D65 multiplied by its square. Each is normalised so a perfect diffuser under it gives Y = 100 before the comparison, so the rows are about the shape of the light and not its level.

The residual is the mean CIEDE2000 over a hundred and twenty-five surfaces after the ratio-of-whites gain in the CAT16 basis, and the off-diagonal figure is the Frobenius distance of the residual operator from the identity.

The ratio of 1.96 is asserted rather than quoted: the gate requires the two-bounce residual to exceed one and a half times the one-bounce residual, and requires the off-diagonal measure to move in the same direction. Both would fail if a future change to the wall’s shape quietly made the second bounce cheap.

Where it stops #

Two bounces is not a corner, exactly. A real corner delivers a mixture of light that has bounced once, twice and more, weighted by the geometry, and the pure second-order case here is the extreme member of that mixture rather than its average. The direction is right and the magnitude is an upper bound.

The wall is one reflectance shape. A wall painted a broad, desaturated colour — which is what most walls are — produces a much smaller effect in both directions, and the census’s green wall at 60 nm wide is deliberately more saturated than a domestic interior.

And the whole calculation treats the observer as adapting to the room’s own white. Somebody who has just walked in from outside is adapting to something else entirely, and the room takes several minutes to settle after they arrive; during that time the residual measured here is the smaller part of what they are actually left with.

Where the ladder goes next #

If a wall is an illuminant, then everything this site knows about lamps applies to paint. The commutation table already says a wall shares a mechanism with a change of daylight and not with a fluorescent tube, which is a testable claim about rooms rather than about spectra.

The other direction is the one the second bounce points at. A change of light that is not merely sharper but not multiplicative at all would fall outside this arithmetic completely, and there is one on the site already: a surface with a brightener in it returns light at wavelengths it did not receive, which is a full operator on the spectrum rather than a diagonal one, and none of the algebra above starts.