A gloss finish takes colour out of the whole room

A lobe takes colour out of a bounce found that a green room in satin paint delivers a less green floor than the same room in flat paint, seen from the front of the room. The floor is a different colour from the door then found that seen from the coloured walls the same floor is more colourful than from the front. Two directions of one solution disagreeing about the sign of an effect leaves a question neither can answer alone: is the colour gone, or has it gone somewhere else?

Removed, not moved #

A gloss finish takes colour out of the room’s reflected light as a whole, and it takes about the same share whatever colour the walls are painted.

• At an eggshell finish, roughness 0.2, the room’s reflected light gains 15.6 per cent in quantity and loses 8.3 per cent of its chroma; at 0.15 it gains 19.2 and loses 9.1.
• The painted walls lose the most: 13.2 per cent of the chroma they send out at 0.2, and 14.6 at 0.15.
• What arrives at the painted walls keeps its chroma to 0.4 per cent; what arrives at the floor loses 23.2 per cent, and the ceiling, back and front each lose 7 to 8.
• For blue, green and orange walls the room loses 9.9, 8.3 and 9.0 per cent of its chroma, while the walls themselves lose 25.0, 13.2 and 22.2.

A ledger rather than a view #

A radiosity solution gives each face of a room one outgoing radiance, and every question about the room’s colour is a question about six numbers. The directional solver that replaced it gives a radiance for every ordered pair of faces — thirty of them — and thirty unknowns instead of six established that it reduces exactly to radiosity when the walls are matt.

A radiance from one face towards another, multiplied by the cosine-weighted solid angle the second face subtends at the first, is a flux: the power leaving the first face that arrives at the second. Booked for every ordered pair, those thirty fluxes are a complete ledger of the room’s reflected light — what each face sends out, what each receives, and the total — and each can be given a chroma by treating its spectrum as a light. The lamp’s own emission is taken out of the ceiling’s outgoing light first, because the question is about light that has been reflected at least once.

The ledger answers a question a view cannot, because a view is one entry and the question is about the sum.

The ledger’s own null #

A ledger has to be shown to add nothing of its own before its totals mean anything, and the matt room is the check.

With matt walls the directional solver has no direction to prefer, and the floor sends light of chroma 34.98 towards each of the five other faces — the same number five times, to the last digit printed. That is radiosity’s single number for the floor, repeated once per receiver, and the same holds face by face. The sums by sender, by receiver and in total are then sums of radiosity’s own answer, so every change reported below is a departure from a ledger that is exactly the radiosity solution when the lobe is switched off.

Face by face #

At roughness 0.2 every face’s incoming and outgoing chroma can be set against the matt room’s.

The two painted walls send out 13.2 per cent less chroma, which is the lobe’s direct effect: nine or so per cent of each wall’s return is a Fresnel reflection with the lamp’s spectrum rather than the paint’s, as a tenth of the return arriving white measured. The floor sends out 7.1 per cent less and the ceiling 6.7 per cent less, because each receives less colourful light from the walls and returns it. The back and front send out 0.2 per cent less — almost unchanged.

The incoming side is where the ledger surprises. The painted walls receive light 0.4 per cent more colourful than in the matt room, while the floor receives light 23.2 per cent less colourful and the ceiling, back and front 7 to 8 per cent less. Each wall receives a large share of its light from the opposite wall, which is now less green, and from the back and front, which receive their own light in part from the walls — and the glossy floor and ceiling now reflect a little more of the opposite wall’s light towards each wall. The losses and gains in what arrives at the walls very nearly cancel.

Where the extra light lands #

The light the lobe saves from absorption does not spread evenly either. At roughness 0.2 the floor sends out 20.8 per cent more light than in the matt room, the ceiling 19.7 per cent more, the painted walls 14.5 and the back and front 13.1.

Among the unpainted faces the gain in light and the loss of colour travel together: the floor and ceiling gain about a fifth more light and lose about 7 per cent of their chroma, while the back and front gain 13 per cent and lose 0.2. The painted walls are the exception, losing the most chroma for a middling gain in light — the difference being that on them the lobe’s return dilutes the paint’s own colour directly, where on the other faces it dilutes light that was coloured somewhere else.

The room’s total #

Adding every flux gives the room’s reflected light as a single spectrum, and its chroma against the matt room’s is the answer to the question.

At roughness 0.8, almost matt, the room loses 3.0 per cent of its chroma and gains 4.3 per cent of light. At 0.45 it loses 5.3 and gains 8.2; at 0.3, 6.9 and 11.5; at 0.2, 8.3 and 15.6; at 0.15, 9.1 and 19.2. The chroma falls monotonically, the light rises monotonically, and at no finish does any measure of the room’s colour rise.

If the lobe only moved colour — taking it from the floor’s return towards the front and depositing it in the floor’s return towards the walls — the total would stay put. It falls by nearly a tenth. A gloss finish removes colour from a room, and the direction-dependence the two earlier essays found is the removal happening unevenly, not a relocation.

The same share for every paint #

The obvious objection is that green walls at 530 nm are one case, and the size of the loss might belong to the colour.

For walls painted blue at 460 nm, green at 530 and orange at 600, the room loses 9.9, 8.3 and 9.0 per cent of its chroma at roughness 0.2. The walls themselves lose 25.0, 13.2 and 22.2 per cent — nearly twice as much for blue and orange as for green.

The walls’ losses differ because a Fresnel return of the lamp’s light dilutes a paint more when the paint returns less light, and the blue and orange paints, sitting where the observer’s sensitivity is lower, contribute less luminance of their own for the same albedo. The room’s losses do not differ much because the room’s share is governed by how much of the total reflected light has passed through a lobe at least once, which is a property of the finish and the geometry rather than of the paint. At an albedo of 0.5 instead of 0.85 the room loses 11.4 per cent, a little more, because a darker paint’s return is diluted more by a lobe of the same size.

More light, less colour #

The light rises because the lobe is a reflection at the interface, and light reflected at the interface never enters the paint and is never absorbed by it.

In the matt room each wall absorbs what its pigment absorbs of every bounce. In the glossy room a fraction of each bounce is returned at the surface with the lamp’s spectrum and escapes the absorption, so the room retains more light and the light it retains has had less paint in its history. The quantity rises and the colour falls for the same reason, and neither a quantity measurement nor a colour measurement alone says which is the more important change.

The lobe’s share of a surface’s return is 9.1 per cent at an eggshell finish and falls to 4.2 at a matt one. The room’s loss of chroma follows the same shape: 3 per cent at roughness 0.8, 8 at 0.2. Across the room the lobe’s share, compounded over the bounces a room’s light makes, is what sets the loss.

Every bounce the lobe takes is one multiplication fewer #

The loss reaches faces the painted walls’ lobes never point at, and the reason is how a room’s colour is made.

A room’s colour is built one bounce at a time. Lamp light meets a green wall and returns green; that light meets the opposite green wall and returns greener, because a bounce is a multiplication and a reflectance multiplied by itself is a narrower spectrum than the reflectance. Light that has crossed between the two walls several times is the most saturated light in the room, and for the floor, the ceiling and the neutral walls the room is the illuminant: much of what they receive is that interreflected light.

A lobe interrupts the multiplication at every bounce. The share of a bounce returned at the interface carries the spectrum of whatever arrived — lamp light on the first bounce, already-green light on a later one — without multiplying it by the paint again. So the lobe does not only add a little white to each wall’s return; it withholds one multiplication from part of every bounce, and the saturation a room builds by compounding is what it withholds. Every face but the painted walls — whose incoming light is held level by the cancellation measured above — receives less colour as a result.

What a view could and could not say #

The two earlier essays were right about what they measured, and the ledger changes how their results should be read.

The front view’s desaturation is part of a room-wide loss. The floor’s return towards the front loses colour, and so does almost everything else.

The side view’s saturation is the painted walls’ incoming light holding its colour while the room around it loses. A viewer looking along the walls sees a floor lit partly by the walls’ return, which lost less colour in the directions that face the walls, and the comparison is with a room whose average colour fell. Seen that way, a gain in one direction is the smallest loss rather than a transfer.

And a renderer or a lighting designer who wants a room to look as colourful as its paint chips promises has a simple consequence: a gloss finish costs about a tenth of the room’s colour whatever the paint, and it buys about a sixth more light.

What a paint chip cannot show #

A paint is chosen from a chip, and a chip is a single bounce. It is judged in one light, returns that light once, and has nothing to interreflect with.

The chip is also measured, and gloss changes the measurement: whether an instrument’s geometry includes the specular component decides what colour a glossy chip is said to be. With the specular part excluded, an eggshell chip and a flat chip of the same paint body can read as the same colour. Neither reading contains the room, because the loss measured here is in light that has bounced many times, and a chip bounces it once.

So two finishes matched on the chip are not matched on the walls. The glossy room returns about a sixth more light and eight to ten per cent less of its reflected chroma than the flat one, whichever of the three wall colours was chosen, and a designer comparing the finished rooms side by side is comparing two amounts of colour that the chip comparison showed as one. The ledger cannot say how visible that is to somebody adapted to each room in turn, which is the question left open below. It can say that the difference is in the light before anybody looks.

What was computed, and how #

The room is a unit cube with a lamp in the ceiling emitting D65, two opposite walls painted with a Gaussian reflectance band of stated centre, width 25 nm and peak albedo 0.85 on a 0.03 pedestal, and the other faces a neutral 0.5. Every face carries the same microfacet lobe of stated roughness over a dielectric of index 1.5, with the interface’s return taken out of the body term rather than added beside it.

The directional solver gives a radiance for each ordered pair of faces in eighty-one bands. Each radiance, less the lamp’s own emission for the ceiling, is multiplied by the cosine-weighted solid angle the receiving face subtends at the sender, to give a flux; the fluxes are summed by sender, by receiver and in total. A flux’s chroma is its CIELAB chroma at its own luminance against the lamp’s white, so it measures how coloured the light is rather than how much of it there is.

Where the measurement stops #

One room shape and one lamp position, and a point-sampled patch per face. Where a patch stops being a point put the solver’s lower limit at roughness 0.15, so the glossiest finishes are not in the table and the curves are still falling at their left-hand ends.

The chroma of a total flux is a property of the light leaving the room’s faces, not of any viewer’s experience of the room. An observer sees a few directions of it, adapted to the room’s average, and an adapted observer would discount part of a room-wide shift.

And every face shares one finish. A room with glossy walls and a matt floor, which is common, would distribute the loss differently, and the ledger would book it the same way.

The habit #

The habit is about reading a conservation question off a single measurement.

When an effect has opposite signs in two views of one system, the tempting reading is that something moved from one to the other. That reading is a claim about a total, and a total is not in either view. It can be checked only by adding everything up.

The move is to keep a ledger — every flux, booked once — and compare totals before interpreting signs. It costs a sum over pairs the solver already computed.

The failure mode is to call a smaller loss a gain. A direction that loses less colour than the rest of the room looks, beside the rest of the room, like a direction that gained some.

Who noticed it first #

That specular reflection at a dielectric interface carries the illuminant’s spectrum is the dichromatic reflection model, due to Shafer in 1985, and that it desaturates interreflections is familiar in physically based rendering.

That a room’s total reflected chroma falls by nearly the same share for different paint colours at a given finish, and that the directional gain seen towards painted walls is their incoming light holding its colour inside a room-wide loss, are computed here with a flux ledger over the directional solution.

Still open: what an adapted viewer keeps #

A room-wide loss of a tenth of the reflected light’s chroma is a loss in the stimulus. A viewer standing in the room adapts to its average light, and adaptation partly discounts a shift that affects everything equally. How much of the room’s loss survives an adapted viewer — and whether the uneven part, face by face, survives better than the even part — is the appearance calculation on the ledger’s outputs, and it would say whether a gloss finish makes a room look less colourful or only measure less colourful.