A tenth of the return arriving white

The effect this round measures is driven by a quantity smaller than most readers would guess, and the gap between the driver’s size and the effect’s size is the part worth understanding.

The claim #

The specular share of a painted surface’s return is under a tenth, and it moves a room’s colour by five colour differences because interreflection multiplies.

• Nine per cent at roughness 0.15, falling to four per cent at 0.8, for a body reflectance of 0.5 and light arriving at 45°.
• The room’s departure from the Lambertian answer is 4.89 ΔE₀₀ at the same finish, which is a large response to a small input.
• Three things multiply it up: interreflection compounds, the contribution has a different spectrum rather than being a scaling, and the walls are where all the room’s colour comes from.
• And the share is nearly flat across the usable range. It varies by a factor of two while the effect varies by a factor of five, so the effect is driven by the lobe’s width rather than by its strength.

What the share actually is #

The number plotted is the directional-hemispherical albedo of the microfacet lobe alone, for light arriving at 45°, divided by that plus the body reflectance. It is what fraction of the light leaving the surface never entered the paint.

The lobe’s own albedo is between two and five per cent of the arriving light, which is what a dielectric Fresnel reflection at moderate incidence is — the normal-incidence value for a refractive index of 1.5 is exactly four per cent, and a rough surface with a distribution of microfacet orientations averages somewhat differently.

That the share falls as the surface roughens is a property of the microfacet model rather than of any real measurement: a rougher surface’s facets are more often oriented away from the specular direction, and more of the lobe’s energy is lost to the shadowing and masking terms.

The effect exceeds the share by a factor of about fifty. A nine per cent contribution producing a five-unit colour difference needs an explanation, and there are three multipliers.

Interreflection compounds. A room’s colour is not one bounce but a geometric series, and a room bounds its own bounces established that two to four of them carry most of the effect for ordinary reflectances. A nine per cent change at each bounce compounds to something between a fifth and a third by the third bounce.

The contribution is spectrally different. Nine per cent more of the same light would be a brightness change and nothing else, removable by an exposure adjustment. Nine per cent of a white spectrum added to a coloured one is a shape change, and a colour difference is sensitive to shape in a way it is not to scale.

And the walls carry all of the colour. The floor is grey; every coloured photon reaching it came off a wall. So a change to what the walls return is a change to the entire coloured part of the signal rather than to nine per cent of everything.

Those three multipliers are roughly a factor of two and a half, a factor of two, and a factor of one — and together they turn nine per cent into something that reads as five colour differences.

The effect is driven by the lobe’s width rather than its strength. The share falls from nine per cent to four across the usable range, a factor of 2.2. The departure falls from 4.89 to 0.97, a factor of 5.0. So the effect is not proportional to the share, and something else is varying faster.

That something is the lobe’s width. A narrow lobe sends its return into a small cone and therefore concentrates it on a few surfaces; a wide one spreads it over most of a hemisphere and behaves increasingly like the body term. In the limit of a very rough surface the lobe is nearly Lambertian and contributes nothing new whatever its share.

So the departure is a product of how much is in the lobe and how differently it is distributed, and the second factor varies faster than the first. A gloss level is a better predictor of the departure than a specular strength, which is the reverse of the intuitive ordering and is worth knowing when a coating has both as separate specifications.

The consequence for estimating the effect without a solver is that the roughness is the variable to ask about. Two coatings with the same specular reflectance and different gloss levels behave quite differently in a room, and a data sheet quoting only the first says little.

The spectrum is the second multiplier made visible. If the nine per cent were a scaling, the two curves would differ by a constant factor and the ratio between them would be flat. It is not: 1.17 at 530 nanometres where the wall pigment absorbs, and 1.24 at 430 and at 680 where it does not.

That six per cent difference in ratio is the whole of the spectral shape change, and it is worth noticing how small it is. Six per cent of shape change is worth five colour differences in a room, because a colour difference is a distance between chromatic coordinates and those coordinates are ratios of integrals that a shape change moves directly.

The same arithmetic explains why the effect is invisible to a photometer. Total luminous flux rises by about eighteen per cent between matt and satin walls, which anybody would notice; the shape change that carries the colour is a third of that and no photometric measurement records it.

What an instrument sees of the same quantity #

The specular share is the quantity two instrument geometries disagree about, and connecting the two is worth doing because the same number appears in both.

Gloss changes the measurement established that a 45°/0° geometry excludes the specular return and an integrating sphere includes it, and that the two disagree by units on a dark gloss sample. The size of that disagreement is precisely the share plotted here, scaled by how dark the sample is: on a body reflectance of 0.5 the share is nine per cent and on a body reflectance of 0.05 the same absolute lobe albedo is a share of half.

That is why the instrument disagreement is largest on dark samples and the room effect is largest on saturated ones. They are the same physical quantity entering two different calculations, and each is amplified by a different property of the sample.

It also gives a cheap way to estimate the room effect from a measurement anybody can make. The difference between a specular-included and a specular-excluded reading is the lobe’s albedo, and the room departure follows from it and the gloss level.

The dark-sample case #

Pushing the share to its extreme is worth doing because it says where the effect is largest.

At a body reflectance of 0.05 — a dark grey or a deep colour — a lobe albedo of 0.05 is a share of fifty per cent. Half of what leaves such a surface never entered the paint, and half of what it delivers to the rest of the room is white.

A room with dark saturated walls in a satin finish therefore has an interreflection that is far more neutral than its paint, and the effect is not a few per cent of the chroma but a large fraction of it. That is a common arrangement — a restaurant, a theatre, a car interior — and it is the case in which a radiosity calculation would be most wrong.

The solver has not been run on it, because a dark room’s interreflection is small in absolute terms and the colour differences would be computed against a very dark reference where CIELAB’s behaviour is least well founded. It is named as the extreme rather than measured.

What the share does not depend on #

Two things the share is nearly independent of are worth naming, because they would be natural things to vary.

The wall’s colour. The lobe’s albedo is a property of the interface, so a green wall and a red one have the same specular share at the same gloss and the same body reflectance. The measurements bear that out: the room’s fractional chroma loss is five per cent for green walls and five per cent for red ones.

The wavelength. A dielectric’s refractive index varies by one or two per cent across the visible band, so the Fresnel reflectance varies by a fraction of a per cent. That is what makes the lobe neutral and it is the assumption the whole desaturation argument rests on.

Both would fail for a metal, whose Fresnel term is coloured and strongly wavelength-dependent. A gilded surface’s lobe carries gold’s colour, so a gold wall’s interreflection is more saturated than its body rather than less — the whole argument reverses, and the reversal is a good check that the mechanism has been correctly identified.

Repeating the measurement with a different wall pigment is the check that the share is doing the work. Green walls lose 4.9 per cent of their bounce’s chroma at an eggshell finish; red walls lose 5.1 per cent. The absolute chromas differ — 19.69 against 11.43 for the matt case — and the fraction does not.

That invariance is what licenses quoting the result as a rule. A satin finish costs about five per cent of a bounce’s chroma, whatever the wall is painted, and the number is a property of the interface’s share rather than of anything about the colour.

What was computed, and how #

The lobe’s albedo is ∫ f(ωᵢ, ωₒ) cos θₒ dωₒ over the hemisphere with the body reflectance set to zero, computed by Gauss–Legendre in the cosine and a uniform grid in the azimuth at forty by ninety-six. That is the same quadrature the previous round’s angular work uses and its own convergence was checked by rotating the light rather than by refining.

The incoming direction is 45°, which is a stated choice rather than a room average. The albedo rises steeply towards grazing incidence — a Fresnel reflectance approaches one at 90° — so a room whose inter-patch geometry is more oblique has a larger effective share than this figure suggests.

The share is quoted against a body reflectance of 0.5, which is this room’s grey surfaces. Against the coloured walls’ higher reflectance the share is lower and against a dark surface it is much higher.

The share at grazing incidence #

One number in this essay is a summary and the summary understates, which is worth stating plainly.

The share is quoted for light arriving at 45°, and a Fresnel reflectance rises steeply towards grazing: four per cent at normal incidence, about five at 45°, twenty at 75° and approaching a hundred at 90°. So a wall illuminated obliquely returns far more at its interface than one illuminated normally.

In a room most inter-patch geometry is oblique. The floor sees the walls at angles from thirty to eighty degrees depending on where on each surface the light is coming from, so the effective share over a room is higher than the 45° figure — which is why the solver computes the albedo per incoming patch rather than using a constant.

The consequence is that a long low room, where the geometry is more grazing, has a larger departure than a cubical one with the same surfaces. That is a geometric prediction the solver could test and this round has not, since the room is a cube throughout. A room’s shape decides how fast it approaches its cast and it now also decides how much of the cast is white.

Where the model stops #

The single-scattering microfacet lobe underestimates the albedo of a rough surface, because light that bounces between microfacets and escapes is not counted. That error grows with roughness and is largest at the matt end, where the effect is smallest.

The Fresnel term uses a fixed refractive index of 1.5 with no dispersion, so the lobe is exactly neutral here and slightly blue-favouring in reality.

And the 45° incidence is a single angle standing in for a distribution. The solver itself computes the albedo per incoming patch and uses the correct angle for each; this figure is a summary and the summary is at one angle.

A last note about where this number could be got without a solver. The lobe’s albedo is exactly the difference between a specular-included and a specular-excluded instrument reading, which is a measurement any laboratory can make in a minute on a real coating. The share follows from that and the body reflectance, both of which the same instrument reports.

So a practitioner wanting to know whether their room’s finish matters does not need a directional transport solver. Two readings on the wall paint give the share; the coating’s gloss level gives the width; and the two together locate the room on the curves in this round. That is a considerably cheaper route to the same estimate, and it is available because an instrument’s two geometries differ by exactly the quantity in question.

The generalisation #

The habit is about the gain between an input and an output.

A small parameter producing a large effect is a sign that something is multiplying, and finding out what is multiplying is more useful than measuring the effect more precisely. Here it is three things — a geometric series, a spectral difference, and a signal that is all of the interesting part — and each of them is identifiable and quantifiable separately.

The move is to decompose the gain rather than to report it. A factor of fifty between a nine per cent input and its consequence is a number; a factor of two and a half from compounding times a factor of two from spectral difference is an explanation, and an explanation transfers to the next room.

The failure mode is to conclude from a small parameter that an effect is small. A small share of a different thing is not a small effect, and the word “different” is doing all the work.

Why the share is nearly flat and the effect is not #

The two curves in this essay have very different shapes and the difference is the essay’s main structural point, so it is worth one last statement.

The share falls by a factor of 2.2 across the usable roughness range, smoothly and gently. The departure falls by a factor of 5.0 over the same range. A model in which the effect were proportional to the share would have produced the same shape twice.

The extra factor is the lobe’s angular width, and it enters because a wide lobe delivers its return in nearly the same distribution a Lambertian surface would. In the limit the two become indistinguishable and the departure goes to zero even though the share does not.

So there are two independent knobs on a coating and they do different things. The specular reflectance decides how much light bypasses the pigment; the gloss level decides whether that matters. A data sheet quoting only the first — which is most of them, since specular reflectance is what an instrument geometry difference measures — carries half the information. The instrument geometry work measures the first and is silent about the second.

Who found it, and when #

Fresnel’s equations are from 1823 and the four per cent figure for a normal-incidence air-glass interface is in every optics text.

The dichromatic reflection model — a surface’s return as a body term with the object’s colour plus an interface term with the illuminant’s — is Shafer’s, from 1985, and it is the basis of most highlight-removal and colour-constancy work. Its use inside a transport solver, where the interface term becomes an illuminant for the next surface, is what this round adds.

Where the ladder goes next #

The mechanism is understood and the solver’s boundary is measured. What remains is the accounting: every scene number this collection has published was computed on matt walls, and the list of what that costs is specific.