Every scene in this collection was matt

An audit that finds a defect in a method owes an accounting of what the method produced. This one is short, specific, and the numbers are known rather than estimated.

The claim #

Every scene result in this collection is computed on Lambertian surfaces, none of the essays that report them states a finish, and each would move by between one and five ΔE₀₀ on an ordinary satin one.

• The affected results are identifiable: the green wall’s cast, the corner’s approach to it, the bounce series, the metamer separation against enclosure, and the room-solid work.
• The direction of the move is known — more light on the floor, and less chroma in most directions and more towards the coloured surfaces — and its size follows from the roughness.
• None of them is invalidated, because each is a comparison between two computations done the same way and the shared assumption largely cancels.
• What changes is the absolute claims, of which there are several, and the repair is to state the finish rather than to recompute everything.

Which results depend on it #

The list is short because this collection’s scene work is concentrated.

A bounce is a multiplication is the founding argument and it is about a single reflection, so the Lambertian assumption enters only through what the surface’s reflectance means. That one is nearly untouched.

What a white wall costs and the green-wall result compute a room’s cast by radiosity, so they are the matt answer of the sweep above. A satin finish would give a cast that is about five per cent less saturated seen away from the coloured walls and more saturated seen towards them.

A corner is not a wall and a room bounds its own bounces are about how the series converges and how the geometry decides its rate. Those are structural results about form factors, and the lobe changes the composition of each bounce rather than the number of them.

Metamer separation against enclosure computes how far a metameric match comes apart in a room, and it is a comparison between two spectra in the same room, so most of the assumption cancels.

And the room-solid work, which computes what ambient light does to a display’s usable gamut, treats the room’s return as diffuse. A glossy room returns more light and less colour to the screen, so the gamut loss would be slightly larger and slightly less coloured.

What each would move by #

The sweep gives the numbers directly, because every one of those results is the matt end of the same curve.

So a result computed as though the walls were perfectly Lambertian and reported for a room with satin walls is wrong by about four colour differences, and one reported for a room with flat emulsion is wrong by about one.

The second of those is within the noise of most of the arguments those essays make. The first is not, and it is the finish most modern interiors have.

The assumption is defensible for a matt room and not for an ordinary one, and no essay in this collection has ever said which it meant.

Nothing is invalidated, and the distinction matters. The distinction between a result being wrong and a result being incompletely stated matters here, and almost everything in the list is the second.

Every one of those essays makes a comparison: this room against that room, this paint against that paint, one bounce against two, a metameric pair before and after an enclosure. In a comparison the Lambertian assumption appears on both sides and its effect largely cancels — not exactly, since the lobe interacts with the geometry, but to first order.

The metamer-separation result is the clearest case. It asks how far apart two spectra move when placed in a room, and both spectra are in the same room on the same walls. A lobe changes what the room delivers to both of them in the same way, and the difference between them is nearly untouched.

What does not cancel is anything absolute: the chroma of a green room’s cast, the lightness of a floor, the fraction of a display’s gamut a room removes. Those are single numbers about a single configuration, and each of them carries the assumption in full.

That figure settles a question a reader might otherwise ask: whether the collection’s matt results could be corrected by a single multiplier once a finish is known. They cannot. The two spectra differ by 1.17 at the wall pigment’s peak and 1.24 away from it, so any single scaling matches one part of the spectrum and misses the rest.

A correction would therefore have to be spectral, which means running the solver, which means knowing the finish — so the chain leads back to the same place. That is why the repair below is to state the assumption rather than to apply a factor.

The repair, which is not recomputation #

The obvious response is to rerun everything through the directional solver, and it is the wrong response for three reasons.

The solver has a boundary. It cannot reach roughnesses below 0.15, so it could not compute a gloss-walled version of anything and would give a range rather than a number for the finishes it can reach.

The finish is not known. None of those essays states one because none of them had a reason to, so recomputing would mean inventing a finish for each and reporting an answer that depends on the invention.

And the results are comparisons. Recomputing a comparison whose two sides move together produces the same comparison and a lot of arithmetic.

The repair is to state the assumption where it is load-bearing, which is in the handful of absolute claims, and to give the reader the sweep so that a room with a known finish can be located on it. That is a caption change and a cross-reference rather than a rebuild, and it is what this essay is.

What a shortfall entry looks like #

This collection records what it could not finish rather than leaving it in prose, and the entries this round adds to the scene work are three.

The absolute scene numbers are matt-only and are lower bounds on the departure for any real finish. Which of them matters most is not ranked, because ranking would need each recomputed.

The room-solid work is unexamined. A glossy room’s effect on a display’s usable gamut is a compound of more light and less colour, and the two act in opposite directions on the gamut area. Nothing here says which wins.

And the enclosure sweep is not run with a lobe. The metamer-separation result is parameterised by how enclosed the geometry is, and running the directional solver across that parameter is an afternoon that this round did not spend.

Those go in the ledger rather than in a caveat, on the rule this collection has followed since its own audits began: an empty shortfall queue on a full slate usually means nobody wrote one.

The second thing that was never stated #

The direction-dependence is a separate omission from the finish, and it is the larger of the two.

A radiosity result is an average over outgoing directions. Every absolute scene number in this collection is therefore an average, reported as though it were a value, with a spread around it of up to two colour differences that the method could not have reported.

That is not a defect in the arithmetic; it is a property of what the arithmetic computes, and it was correct to report the average because the average was all there was. What is missing is the sentence saying so, and the sentence matters because a reader comparing one of those numbers against a measurement will be comparing an average against a single view.

A measurement is always a single view. So the systematic gap between this collection’s scene numbers and any in-situ measurement of the same thing is not noise; it is the difference between an average and a sample from a distribution whose width is now known.

The share figure is the one worth keeping beside the accounting, because it is the only quantity in the whole round that a reader can obtain without any of this machinery. Two instrument readings give it, and everything else follows from it and a gloss level.

That matters for whether the round’s results travel. A finding that requires a bespoke solver to apply is a finding most people cannot use; one that reduces to two numbers off an instrument they already own is a finding they can. The value of an audit is partly in how cheaply its conclusions can be re-derived, and this one comes out well by that measure and the observer audit does not.

What the round leaves in better shape #

Set against the accounting, three things are now available that were not.

The size of the assumption is measured rather than assumed benign, at one to five colour differences across ordinary finishes.

The direction of it is known: less chroma, more light, and a spread across viewing directions of about two units at satin.

And the machinery exists to compute any of it, checked against the existing solver at the floating-point floor, so a future round can price any specific result rather than reasoning about it.

That is the ordinary outcome of an audit in this collection and it is worth naming as such. Nothing was found to be wrong; several things were found to be incompletely stated; and the instrument that would settle them now exists and did not.

A reader with a real room can use the same sweep directly. The accounting above is about this collection’s own results. A reader with an actual room and an actual question can use the same sweep more directly, and it takes three numbers.

The finish, as a gloss reading or a roughness. A 60° gloss reading below about ten is flat, ten to thirty is low-sheen or satin, and above forty is semi-gloss or better — and the last of those is outside what the solver can report.

The wall’s specular share, which is the difference between a specular-included and a specular-excluded instrument reading divided by the total. It is a minute’s work on any spectrophotometer with both geometries.

And the room’s enclosure, which decides how many bounces matter and is roughly the fraction of the hemisphere from a typical surface that is other surfaces rather than windows.

Locating a room on the sweep with those three gives a departure to within about a factor of two, which is enough to decide whether a diffuse-only calculation is adequate. The instrument geometry difference supplies the second number and the collection’s own enclosure work supplies the third.

What was computed, and how #

The sweep is the room this collection’s own green-wall argument uses — a cube with two walls carrying a Gaussian absorption at 530 nanometres, a grey floor, ceiling, front and back, and a D65 lamp in the ceiling.

The matt end of the sweep is checked against the existing radiosity solver directly, at 9.8 × 10⁻¹⁶ relative over thirty ordered pairs and eighty-one bands, so the claim that the collection’s results are the matt end of this curve is verified rather than argued.

The affected-result list is assembled by reading which essays call the radiosity solver, not by judgement, so it is complete for this collection’s own machinery. An essay that discusses interreflection without computing it is not on the list.

One more item belongs in the accounting and it is about the future rather than the past. The directional solver is now part of this collection’s machinery, checked and asserted, and the natural thing is for the next scene essay to use it by default.

That would be a mistake in one specific way: the new solver has a boundary the old one does not, so a result computed with it cannot be quoted for a glossy surface while a radiosity result can be quoted for a matt one without qualification. A more capable instrument with a narrower range is not strictly better, and the honest arrangement is to keep both and to say which produced a number — which is the arrangement this collection already uses for its three-channel and spectral renderers.

Where the model stops #

The correspondence between a roughness parameter and a named finish — satin, eggshell, flat — is approximate. Gloss is specified industrially by a reflectometer reading at 60° rather than by a microfacet roughness, and the mapping between the two is not one-to-one.

The sweep is for one room, one lamp and one pair of wall colours. A different room would give a different curve, and the numbers above should be read as sizes rather than as corrections to apply.

And the solver’s own boundary means the glossiest finishes are absent, so every number in the accounting is a lower bound.

There is a last thing to say about why an accounting like this is worth a whole essay rather than a paragraph in the essay that found the defect.

An audit’s findings are read by people deciding whether to trust existing work, and a finding stated without its consequences leaves that decision to them with no information. Somebody reading that radiosity requires Lambertian surfaces, and knowing this collection uses radiosity, has to decide alone whether the green-wall result is affected — and the honest answer is yes, by five per cent of its chroma, and it does not matter for the comparison the essay makes.

That sentence is four seconds of reading and it took a solver, a boundary measurement and a sweep to produce. The asymmetry between how long a conclusion takes to state and how long it takes to earn is the ordinary shape of this work, and skipping the accounting is skipping the part the reader actually needs.

The generalisation #

The habit is about what an audit owes to the work it audits.

Finding that a method has a limitation is the easy half. The useful half is saying which existing results depend on it, by how much, and in which direction — and that requires going back through work that was correct when it was done and deciding what is now incompletely stated.

The temptation is to skip it, because the results are mostly comparisons and comparisons mostly survive. The reason not to is that “mostly” is a judgement and the reader cannot check it. An audit that does not name what it affects has told the reader to redo the audit, and they will not.

The other temptation is to over-correct: to withdraw or recompute everything on the grounds that an assumption has been shown to matter. That is expensive and it usually replaces a known assumption with an invented parameter, which is worse.

Who found it, and when #

Nothing in this essay is a discovery; it is bookkeeping about a discovery made two essays earlier. The Lambertian requirement of radiosity has been stated in every account of the method since Goral’s in 1984, and this collection has used the method for eleven rounds with the requirement recorded in the source and nowhere in the prose.

That gap between a comment in a file and a sentence in a caption is where most of this essay’s content lives, and it is the same gap the collection has found in itself several times — a caveat written where nothing can reach it.

Where the ladder goes next #

The three audits of this round are finished and what remains is where they land. The observer’s departures are not confined to a table of ΔE₀₀: they reach into every tolerance, every chart and every specification this collection has written about, and the first of those is the one written in the same unit — a tolerance has an observer in it.