The floor is a different colour from the door

Radiosity’s unknown is one number per surface. That is a claim about the world, and the claim is that a surface looks the same from everywhere. Once the walls have a finish, it is false by two colour differences.

The claim #

A surface in a room with glossy walls sends different light in different directions, and the spread is a quantity radiosity does not merely approximate — it does not have.

• The spread is 2.00 ΔE₀₀ at a wall roughness of 0.2, between the extremes of five viewing directions.
• The side walls see most — chroma 20.94 against 18.76 towards the front — because the lobe favours the direction the light came from and the light came from the coloured walls.
• Radiosity has no spread at all. Not a small one; the method assigns one radiosity per patch, so the width of the chart is the whole of what it cannot say.
• And this is an ordinary room. Satin paint, a lamp in the ceiling, a cube — no goniochromatic pigment, no interference film, nothing exotic.

A viewing direction has work to do in a room. A Lambertian surface returns light equally in every direction, so where a viewer stands does not matter and nothing in a diffuse-only calculation needs to know.

A surface with a lobe returns more light near the mirror direction of whatever illuminated it. In a room, the illumination is not one source but every other surface, so the floor’s return towards any given direction is a weighted sum over the walls, weighted by how nearly each wall’s mirror direction points that way.

In this cube the floor is illuminated by the ceiling from directly above and by the four walls from the sides. Towards the front, the floor’s return is dominated by its body term plus whatever the ceiling’s light scatters forwards. Towards a side wall, the mirror direction of the opposite side wall’s illumination points nearly along the line of sight, so the floor sends back a larger share of what came from the coloured walls.

More of the coloured contribution goes sideways, which is why the side-wall directions read a chroma two units higher.

The measurement #

At wall roughness 0.2, with the two side walls green and everything else grey:

The front and back agree with each other and the left and right agree with each other, which is the cube’s symmetry and is a check rather than a result — the two coloured walls are the left and right, so front and back are equivalent and left and right are equivalent.

The ceiling direction is the lowest in chroma and the largest departure, and that has a cause: looking down at the floor from directly above is looking along the mirror direction of the ceiling’s own light, which is the lamp, which is white. So the ceiling sees the most neutral version of the floor.

The whole spread is 2.00 ΔE₀₀ between the extremes, at a roughness anybody would call a satin wall paint.

Why the spread falls as the walls roughen #

At roughness 0.3 the spread is 1.14 ΔE₀₀ and at 0.2 it is 2.00, so a rougher wall produces a more nearly directionless room.

The mechanism is that a lobe’s width is what decides how much a direction matters. A narrow lobe sends its return into a small cone, so where that cone points is decisive; a wide one spreads the return over most of a hemisphere and approaches the Lambertian case, where nothing depends on direction.

That gives a useful rule for anybody deciding whether to care: the directional spread scales with how far a surface is from Lambertian, which is measurable as its gloss. A matt wall produces a room a diffuse-only calculation describes correctly; a satin one does not, and the transition is gradual and monotone rather than sudden.

It also says where the largest effects will be found, and it is not in rooms. A room’s surfaces are chosen to be washable rather than reflective. A vehicle interior, a shop fitting, a display case or a stage set has considerably glossier surfaces and will show correspondingly more direction-dependence.

The share figure and the spread figures together give the two factors the effect is a product of. The amount in the lobe is what the share measures — nine per cent at eggshell, four at matt — and the concentration of it is what the roughness controls. A wide lobe with a large share and a narrow lobe with a small one can produce the same spread, and here both move together as the surface is roughened.

That is why the spread falls faster with roughness than the share does. From roughness 0.2 to 0.3 the share falls from 8.8 per cent to 8.0, a change of a tenth, and the spread falls from 2.00 to 1.14, nearly a half. Almost all of the change is the lobe widening rather than weakening.

The practical form is that gloss level rather than specular strength is what decides direction-dependence, and the two are separate controls on any real coating. A high-gloss paint with a low specular reflectance is more direction-dependent than a matt one with a high specular reflectance, which is the reverse of the intuition that specular strength is the variable.

What this settles about a colour that moves #

This collection has an essay saying that some colours move with the viewer, and until now it had one kind of example to point at: an interference film, whose colour is a function of angle because its mechanism is a path difference.

A colour that moves with the viewer treats goniochromatism as a property of exotic materials — interference pigments, structural colours, the flake orientation in a metallic paint. That is where the term is used and it is where the effect is largest.

What the directional solver says is that an ordinary painted surface in an ordinary room does it too, at two colour differences, for a completely different reason. The wall’s own reflectance is not angle-dependent in any interesting way; what is angle-dependent is what the room delivers to it and what it sends on, and that is a property of the enclosure rather than of the material.

Goniochromatism can be a property of a scene rather than of a surface, and a measurement of the surface in isolation would find none of it.

What it means for measuring a room #

There is a practical consequence for anybody measuring colour in situ, and it makes an existing problem worse in a specific way.

Gloss changes the measurement established that an instrument’s geometry decides what it reads from a glossy sample — 45°/0° excludes the specular return and an integrating sphere includes it, and the two disagree by units on a dark gloss sample.

This adds a second geometry to the same problem. Even with the instrument’s geometry fixed, what the sample is receiving depends on the room, and a sample in a room with coloured glossy walls receives different light from different directions. An instrument with its own light source removes that — an instrument brings its own light — and a measurement made under ambient illumination does not.

So a colour measured in a room and a colour measured in a booth differ for two reasons rather than one: the illuminant differs, which everybody knows, and the directional structure of the illuminant differs, which nothing in a colorimetric measurement records.

The number radiosity gives, and where it sits #

A fair question is which of the five directional answers the radiosity solution corresponds to, and the answer is none of them exactly.

Radiosity’s answer is the total power leaving the floor divided by π — the average over all outgoing directions weighted by cosine. That average lies inside the range of the five directional answers and is not equal to any of them, in the same way the single albedo a radiosity calculation uses equals no particular instrument reading.

The chroma of the radiosity answer is 19.69, and the five directional values are 18.11, 18.76, 18.76, 20.94 and 20.94. So the average sits between them, closer to the middle than to either extreme, and a reader wanting one number could do worse.

What they could not do is know how far from it any particular view is, which is the point. An average is a defensible summary and it is not a bound, and radiosity offers the average with no indication that a range exists.

What a camera makes of it #

A photograph is a measurement from one direction, which is the arrangement this essay is about, and it turns the finding into something visible rather than computed.

Two photographs of the same floor from two positions in the same room, with the same exposure and the same white balance, record different colours — by up to two colour differences with satin walls, more with glossier ones. Neither camera is wrong and neither is the floor changing.

That has a consequence for anybody using a photograph as a colour record, which this collection has warned about from a different direction. The usual warnings are about the camera’s own sensitivities, its white balance and its tone curve, all of which are properties of the instrument. This one is not: it survives a perfect camera, and the only cure is to record the viewing geometry alongside the colour.

It also explains a practical frustration in colour-critical photography of interiors, where a swatch photographed in situ and the same swatch photographed in a booth disagree in ways that changing the lighting does not fix. The room is contributing a directional term that no illuminant correction can remove.

Sweeping the roughness for two different viewing directions gives a fuller picture than either alone. The ceiling view’s departure is larger than the front view’s at every roughness and by a roughly constant factor, so the direction and the roughness enter multiplicatively rather than interacting.

That is convenient for anybody wanting to estimate the effect without running a solver: take the roughness-dependent departure once, and scale by a direction factor between about 0.9 and 1.3. Both numbers are properties of the geometry rather than of the paint, so they transfer between rooms of similar shape.

What was computed, and how #

The five directions are the directions from the floor’s centre to the centres of the five other faces, which is what the solver’s discretisation offers — one radiance per ordered patch pair. A finer discretisation would give a continuous distribution and this gives five samples of it.

The chroma comparison holds the roughness fixed and varies the direction, and the departure column compares each direction against the matt room seen in the same direction, so the comparison is direction-for-direction rather than against a single matt number.

The assertion the figure carries requires the spread to exceed 0.2 ΔE₀₀ at roughness 0.2, and it measures 2.00. A solver in which the lobe was not doing anything would fail it, which is what the assertion is for.

One more thing follows for how a scene result should be quoted, and it is a change of habit rather than of arithmetic. A sentence like the floor is this colour was well-formed under radiosity and is ambiguous under a directional solver, because there are five answers and the sentence names none of them.

Several sentences in this collection are now ambiguous in exactly that way, and they were written when there was only one answer to be had. Where they are comparisons the ambiguity is harmless, since both sides move together; where they quote an absolute colour for a surface in a room, the honest form has a direction in it. A room is not a sphere said the same thing about a measurement’s geometry; this says it about a scene’s.

Where the model stops #

Five directions is a coarse sampling of a hemisphere. The true angular distribution of the floor’s return is continuous and its extremes may lie between the sampled directions rather than at them, so the spread reported here is a lower bound on the true one.

The patch-as-point commitment means each direction is the centre-to-centre one, and a real observer at a real position sees a range of directions across the floor’s extent — which averages the effect down for a large surface seen close up and preserves it for a small one seen far away.

And the room is a cube with six flat faces. A real room’s directional structure is much richer and there is no reason to expect the spread to be similar.

One further asymmetry in the table is worth reading. The ceiling direction has both the lowest chroma and the largest departure from matt, at 5.17 ΔE₀₀ against 4.06 for the other four. Those two facts are the same fact: looking at the floor from the ceiling is looking along the mirror direction of the lamp’s own light, so the floor sends back the most nearly white version of itself, and the most nearly white version is the furthest from the matt answer’s colour.

That is a general feature rather than a quirk of the geometry. The direction that sees the most specular return sees the least colour, because what a lobe returns is the source’s spectrum rather than the surface’s. A viewer standing where the highlight is gets brightness and no information about the paint.

The generalisation #

The habit is about noticing when a model’s output has fewer indices than the question.

A model that returns one number per surface can answer questions about surfaces and cannot answer questions about views. That is obvious when stated and invisible in use, because the answer it gives is a perfectly good number and nothing about it announces which questions it can bear.

The way to check is to ask what varies in the question that does not appear in the output. Here the question is what colour is the floor from the door, and the output has no slot for from the door — so either the answer does not depend on it, which is a claim, or the model cannot say.

The failure mode is to accept the model’s index set as the world’s. A quantity with fewer indices than the phenomenon is an average over the missing ones, and an average is only as useful as the spread it is hiding, which by construction it does not report.

A final observation about scale. The spread measured here is between five directions in a one-metre cube with two coloured walls, which is a strongly coloured and strongly enclosed environment. A larger room with one coloured wall and more neutral surface area would produce less; a small enclosed space with several coloured glossy surfaces would produce more.

Nothing here says how the spread scales with enclosure, and the collection has the machinery to ask — the enclosure fraction is already a parameter in its metamer-separation work. Running the directional solver across a range of enclosures is an afternoon and it is not in this round.

Who found it, and when #

Directional radiosity is Immel, Cohen and Greenberg’s, from 1986, and its motivation was exactly this: a diffuse-only solution cannot render a glossy surface. The graphics literature moved on to path tracing, which has no index-set problem because its unknown is a path.

The colorimetric version — that an ordinary room makes ordinary surfaces goniochromatic — does not appear to be stated anywhere, and the reason is probably that colour measurement is done in booths and instruments, both of which control the directional structure of the illumination precisely so that this does not happen.

Where the ladder goes next #

The chroma result and the direction result both depend on one line of the model: whether the interface’s return is taken out of the body or added beside it. Two constructions, opposite conclusions, and the difference is a modelling decision usually made by omission.