A lobe takes colour out of a bounce

The first thing the directional solver says is not about direction at all. It is about colour, and it is the opposite of what a reader would guess from the fact that a gloss wall reflects more light than a matt one.

The claim #

A lobe on a wall sends more light to the floor and, in most directions, less colour — because what the interface returns carries the lamp’s spectrum rather than the paint’s.

• The chroma of the bounce falls from 19.69 with matt walls to 18.71 at an eggshell finish, seen from the front of the room, while the lightness rises.
• Seen from a coloured wall it rises instead, to 21.45, because the lobe redistributes the coloured return towards where it came from.
• The mechanism is spectral, not geometric. A Fresnel reflection at a dielectric boundary is very nearly neutral, so the fraction of the return that comes off the interface arrives at the next surface white.
• The effect is larger than the lobe’s share. Nine per cent of the return goes into the lobe and the room’s colour moves by 4.89 ΔE₀₀, because interreflection compounds.
• And it depends on the interface taking light from the body rather than adding beside it, which is a modelling decision that reverses the sign.

There are two returns and they have two spectra. Light arriving at a painted wall does one of two things at the boundary. Part of it reflects at the interface between the air and the binder, and part enters the paint film, scatters among the pigment particles and comes back out.

The two returns have completely different spectra. The body return has been through the pigment; every photon that comes back has survived several encounters with something that absorbs selectively, so what emerges carries the paint’s colour. The interface return has touched nothing but a change of refractive index, and a Fresnel reflection at a dielectric boundary is nearly independent of wavelength — the refractive index of a binder varies by a per cent or two across the visible band.

That is why a highlight is the lamp rather than the paint, which this collection established from the direct-reflection side. What is new here is what happens when the interface return is not a highlight anybody looks at but an ordinary part of a bounce.

The lobe is a white contribution added to a coloured one, and adding white to a colour is desaturation.

The measurement #

The same cube with two coloured walls, at six wall roughnesses, with the floor’s return towards the front of the room measured in CIELAB.

The two columns move in opposite directions. Lightness rises by four units from matt to eggshell; chroma falls by one. That is the signature of a white contribution: more light, less colour, and the departure is a combination of the two.

The chroma change is small in absolute terms and it is the interesting one, because it is the part a radiosity solver cannot produce by any adjustment. A room’s overall brightness can be matched by scaling the lamp; its chroma cannot be matched by anything, because the geometric series that produces a room’s colour has the wall’s reflectance in the denominator and the lobe changes what that reflectance is made of.

Why the effect exceeds the lobe’s share #

At roughness 0.15 the lobe carries nine per cent of what leaves the wall, and the room’s colour moves by 4.89 ΔE₀₀. A reader would reasonably expect a nine per cent effect to be worth a fraction of that.

Three things multiply it up.

Interreflection compounds. A room’s colour is the result of several bounces, each multiplying the last. A nine per cent redistribution at each bounce is not a nine per cent redistribution in total, and the number of bounces that matter is between two and four for an ordinary room.

The contribution is spectrally different rather than a scaling. Nine per cent more of the same spectrum would be a brightness change and nothing else. Nine per cent of a different spectrum is a colour change, and colour differences are sensitive to small chromatic shifts in a way they are not to brightness.

And the wall’s colour is what the room is made of. The floor is grey; everything coloured about the light it receives came off the walls. 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 the total.

Repeating the measurement with a different wall colour is the cheapest available check that the effect is about the interface rather than about the particular pigment. With red walls the departure at eggshell is 4.99 ΔE₀₀ against green’s 4.89, and the chroma falls from 11.43 matt to 10.85 — the same direction and the same relative size.

The absolute chroma is lower because a red band at 600 nanometres produces a less saturated bounce than a green band at 530, which is a fact about where the observer’s sensitivity is rather than about the wall. What is unchanged is the fractional desaturation: five per cent for green, five per cent for red.

That invariance is what makes the result quotable as a rule rather than as a measurement of one room. A satin finish costs about five per cent of a bounce’s chroma, whatever the wall is painted, at a roughness anybody would call low-sheen.

The spectrum, read directly #

The spectrum figure at the top of this essay is the mechanism before it becomes a number, and it repays reading carefully.

The matt curve peaks at 530 nanometres, where the wall pigment’s absorption band is centred, and falls away on both sides. The glossy curve has the same shape and is higher everywhere — by a factor of 1.17 at the peak and 1.24 at 430 and at 680 nanometres.

The ratio between the two curves is not constant, and that is the whole finding. A constant ratio would be a brightness change removable by an exposure adjustment. A ratio that is larger away from the pigment’s peak than at it is a shape change, and the shape change is exactly the addition of a flat spectrum to a peaked one.

The numbers say how flat: the extra light is 1.24/1.17 = 1.06 times more abundant away from the peak than at it, relative to the matt light. That is a small ratio and it is the direct evidence for the neutrality of the interface return, measured rather than assumed.

Where the same effect is already known #

The desaturation of a bounce by gloss is not a new phenomenon; it is a well-known one arriving in an unfamiliar setting.

Painters and photographers know that a matt surface bleeds more colour onto its neighbours than a glossy one, and it is the reason a matt-painted colour reference is preferred to a glossy one. Colour-bleeding artefacts in rendering are more visible with diffuse materials for the same reason.

What is not standard is the direction in a colorimetric setting, and it is easy to get backwards. A glossy wall looks more saturated to the eye than a matt one of the same paint, because the specular highlight is elsewhere and the body return is seen against a darker surround. What it delivers to the next surface is less saturated, because the highlight has gone somewhere and the somewhere is another surface.

What a white wall costs established that a room’s cast is a property of its paint rather than its shape. This adds that it is a property of its finish as well, at a size comparable with a moderate change of paint.

What a practitioner would do about it #

There are two settings where this matters and they pull in opposite directions.

A colour-matching environment wants no cast at all, and the standard advice is neutral grey walls of stated reflectance. The finish is not usually specified. A glossy grey wall bleeds less than a matt one of the same reflectance — which is a small argument in favour of gloss and a large argument against it, because a glossy wall reflects the lamp and the assessor directly into the field of view.

A rendering wants the right cast, and a renderer using radiosity or any diffuse-only global illumination is computing the matt answer for every surface. Every satin and semi-gloss surface in the scene is being over-saturated in its contribution to everything else, by up to one unit of chroma per bounce.

The second is the more consequential, because diffuse-only interreflection remains the default in a great deal of real-time rendering. The error is systematic, it is in one direction, and it is the direction that makes rendered interiors look more colourful than photographed ones — which is a complaint the field has, usually attributed to tone mapping.

What decides the sign #

One line of the model decides whether the chroma rises or falls, and it is worth flagging here because it is the subject of the next essay.

The interface’s return can be modelled two ways: taken out of the body term, which is what a real finish does since light reflected at the boundary never reaches the pigment, or added beside it, which is what a microfacet model does if nobody couples the two.

With the first, chroma falls. With the second, it rises — because the body return is unchanged and a neutral contribution has been added on top of a total that was already there, raising the lightness more than it lowers the chroma in the coordinates the difference is measured in.

The first is physically right and it is what everything above uses. The second is what a great deal of rendering software does, because a microfacet lobe and a Lambertian body are usually specified independently. Two constructions, opposite conclusions, and the difference is not a numerical error.

Drawing the spectrum in the direction that saturates rather than the one that desaturates completes the account. Towards the front the glossy curve is higher by relatively more away from 530 nanometres; towards the left wall it is higher by relatively more at 530, because that direction receives the specular return of the coloured walls themselves.

The agreement between the front and back views is a check on the solver rather than a finding: the room’s two coloured walls are the left and right, so the front and back directions are geometrically equivalent and must agree, and they do to the last digit printed.

What was computed, and how #

The room is a cube with two green walls — a Gaussian absorption centred at 530 nanometres with a peak reflectance of 0.88 — a grey floor, ceiling, front and back at 0.5, and a D65 lamp in the ceiling. It is the same room this collection’s green-wall argument uses.

The chroma is √(a*² + b*²) in CIELAB against a white defined as the radiance a perfect white Lambertian patch would send under the lamp itself, which keeps the lightness scale meaningful.

The assertion the figure family carries requires the chroma at the glossiest resolvable roughness to be below the matt chroma by at least two per cent. It measures five per cent, and it could have failed — under the other energy convention it does.

That figure is a correction to everything above and it is the more interesting result. The desaturation is a property of a direction as well as of a finish. Seen from the front or the ceiling, the floor’s return is less saturated than the matt answer — 18.71 against 19.69 at eggshell. Seen from a side wall it is more saturated, at 21.45.

The mechanism is the same lobe doing two things. It adds a neutral contribution everywhere, which desaturates; and it redistributes the coloured contribution towards the directions the coloured light came from, which saturates those directions and depletes the others. Towards the walls the second effect wins and towards the front the first does.

So the honest statement of the finding is that a lobe moves colour around the room rather than simply removing it: the total is very nearly conserved and its angular distribution is not. A radiosity solution, having one number, reports the average of all of it — which is 19.69, sitting between the 18.71 and the 21.45, and is a perfectly good average of a spread it cannot mention.

The gap between the two energy-accounting constructions is about three units of chroma in every direction, so that modelling decision is independent of the viewing one even though the physical effect is not.

Where the model stops #

The Fresnel term is a dielectric one with a fixed refractive index of 1.5, so the lobe is exactly neutral in this model. A real binder’s index varies by one or two per cent across the visible band, which would make the lobe very slightly blue-favouring rather than flat, and the effect would be a small addition to the desaturation rather than a change to it.

A metallic surface has a coloured Fresnel term and everything here reverses: a gold wall’s lobe is more saturated than its body, not less.

And the model has single scattering in the lobe. A rough surface has multiple scattering between microfacets, which darkens and slightly colours the specular return at high roughness — precisely the regime this model is restricted to.

There is one more direction the finding can be pushed and it is worth stating as a prediction rather than a result. If the desaturation comes from a neutral term added to a coloured one, then it should be largest for the most saturated walls and smallest for pale ones — because a neutral addition to something already near neutral changes little.

The two rooms measured, green and red, are both strongly coloured and both lose about five per cent of their chroma. A pale wall would lose a smaller fraction and a fully saturated one more. That is a one-line experiment and it is not run here, so it is a prediction the machinery could refute and has not been asked to. The same pattern was found for observer departures, where the cost scales with the sample’s distance from the light.

The generalisation #

The habit is about tracking where a signal’s spectrum comes from.

A quantity assembled from several contributions has a spectrum that is a weighted sum, and the weights are usually the thing that gets attention. What matters as much is whether the contributions have different spectra, because a contribution with a different spectrum changes the answer’s shape rather than its size, and a shape change cannot be undone by a gain anywhere downstream.

The move is to ask, for each term in a sum, what it has interacted with. A term that has been through a selective absorber carries that absorber’s signature; one that has not is a copy of the source. Mixing the two is mixing a signal with a reference, and the mixture’s colour is a function of the ratio.

The failure mode is to model a contribution’s size correctly and its spectrum by default. A term added with the wrong spectrum is a systematic error that no amount of calibration removes, because calibration adjusts gains and this is not a gain.

A last practical note for anybody who has to specify a room rather than compute one. The finish is a specifiable property, it is already specified for durability and cleanability, and it now has a colour consequence of about five per cent of the cast at the sheen levels ordinarily chosen. That is small against the choice of paint colour and large against the tolerances a colour-critical space is designed to.

So a matching booth’s neutral grey walls should be specified matt not only because a glossy wall reflects the assessor, but because a glossy wall delivers a less saturated cast — and a less saturated cast from a neutral wall is indistinguishable from no cast, which is the desired condition. The two arguments happen to agree here and they would not in a coloured room.

Who found it, and when #

Saunderson’s 1942 treatment of the paint interface is the standard account of the two returns, and it is a Möbius function rather than a sum — which the previous round measured and found worth three to eight ΔE₀₀ in paint mixing.

The neutrality of the interface return is why the dichromatic reflection model of Shafer, published in 1985, splits a surface’s return into a body term with the object’s colour and an interface term with the illuminant’s. That model is the basis of most highlight-removal algorithms and of a good deal of colour constancy work; using it inside a transport solver rather than at a single surface is what this round adds.

Where the ladder goes next #

The chroma result is the lobe’s spectral consequence. Its geometric consequence is the one radiosity cannot express at all: the floor is a different colour from the door, and from the window, and from the ceiling.