The model has six arguments

Three rounds of this collection have audited its own arithmetic and each of them stopped at the same place. The last one wrote the boundary down: the decision that a surface is a reflectance rather than a bidirectional distribution. This is that decision, opened.

The claim #

The colour integral is not a simplification of a surface’s behaviour; it is a projection of it, and the projection has a name for every axis it dropped.

• The response has six arguments, and there is no serious dispute about that: light arrives with a wavelength, a direction and a place, and it leaves with a wavelength, a direction and a place.
• The model keeps one. R(λ) is the diagonal of the first pair, an average over the second, and an assumption about the third.
• Each axis dropped is a departure with a literature and an instrument, so this is not speculation about what might be missing; it is a list of things already measured elsewhere.
• The four departures cost between 1.0 and 7.0 ΔE₀₀ on ordinary samples under ordinary lights, which is between the tightest published tolerance and seven times it.
• And the model is exactly right under stated conditions, which is the part that makes the audit worth doing rather than a demolition.

The equation, written out #

Every tristimulus value on this site comes from one line of arithmetic:

X = k ∫ R(λ) S(λ) x̄(λ) dλ

with S the spectral power of the light, x̄ one of three colour-matching functions, and R the sample. A spectrum is not a colour until it meets an observer, and this is the meeting. The site’s own first rule is about the middle two terms: name the light, name the observer, because both are choices and a diagram that hides them is incomplete.

Nothing has ever been said here about the first term. R(λ) is an array of eighty-one numbers, one for each five-nanometre band from 380 to 780, and it has been treated as a property of the sample in the way a mass is a property of a stone.

It is not one. The physical object is a response function: how much light comes out, given everything about the light that went in. Written with all its arguments,

f(λᵢ, ωᵢ, xᵢ ; λₒ, ωₒ, xₒ)

— the wavelength, direction and position of what arrives, and the same three for what leaves. A reflectance is what is left of that after three operations, and each of the three is a different kind of loss.

Three operations, three losses #

The wavelength pair collapses to a diagonal. R(λ) says how much comes back at the wavelength it went in at. Everything off the diagonal — light absorbed at one wavelength and re-emitted at another — is discarded. For most surfaces there is nothing there; for a fluorescent one there is a great deal, and the object is a matrix rather than a curve.

The direction pair is integrated away. A single number cannot depend on where the light came from or where it left, so what a reflectance holds is an average over directions — and the average is taken with a weight that whoever measured it chose. Two standard instrument geometries disagree by eight ΔE₀₀ on a dark gloss sample, which is the size of the discarded axis showing through.

The position pair is assumed equal. R says what comes back where the light landed. In a translucent material light enters, scatters, and leaves several millimetres away, so what the surface has is a kernel over the plane rather than a number at a point.

There is a fourth item on the list and it is a different kind of thing. The integral runs from 380 to 780 nanometres, and that is not an argument being dropped but a range being chosen for the one argument that was kept. It belongs on the list anyway, because it costs the same sort of money.

What each one costs, at an ordinary strength #

Putting four departures on one axis needs a common unit, and the common unit is the same one the previous three rounds used: the difference in ΔE₀₀ between what the fuller model computes and what the integral computes, for the same sample under the same light through the same observer.

The four ordinary cases are deliberately unremarkable.

The wavelength index: 6.98 ΔE₀₀. A coated printing paper with the optical brightener a coated printing paper has, measured under D50 with the lamp’s ultraviolet present and then with it filtered out.

The range: 6.70. The same paper under D65, integrated over the whole of the light and then over the site’s own grid, which throws away the 6.75 per cent of D65’s power that lies below 380 nanometres.

The place index: 1.96. A pigmented plastic sheet measured through a four-millimetre radius against the same sheet with no aperture at all.

The direction index: 1.00. An eggshell paint beside a window against the same paint under an overcast sky.

The spread is a factor of seven, and it would be dishonest to read it as a ranking of the four departures. Change the plastic for a pale marble and the third row goes to 12.65; change the eggshell for a varnish and the fourth doubles. What the ladder ranks is four examples, and its real content is that all four are above the ΔE of about 1.0 that this collection’s delivery tolerances are written in.

Why four examples rather than four ranges #

A bar chart of four numbers invites exactly one misreading, and it is worth closing before the numbers are used again.

Each departure has a strength, and the strength is not a property of the departure. The place index is worth 0.53 ΔE₀₀ on coated paper, 1.96 on a pigmented plastic, 5.83 on skin and 17.55 on candle wax, through the same aperture, because what decides it is one length — how far light travels inside the material before it is absorbed — and that length runs from a fifth of a millimetre to fourteen. The wavelength index is worth 3.95 on a lightly brightened sheet and 7.66 on a heavily brightened one. A ladder of four bars is therefore a ladder of four choices of example, and the honest way to publish it is with the examples named on the chart, which is what the caption does.

What survives the objection is the part that does not depend on which example is chosen: all four are above the tolerance a specification would be written in, on samples nobody would call exotic. A printing paper, a plastic sheet and an eggshell wall are not edge cases; they are most of what gets measured. The interesting comparison is not between the four bars but between each bar and the one ΔE₀₀ that a delivery tolerance allows.

One departure spans the whole ladder #

The warning against reading the four bars as a ranking is right, and the essay’s own figures say how badly the reading would fail.

The ladder spans a factor of 7.0, from the direction index at 1.00 to the wavelength index at 6.98. The place index alone, held fixed as a departure and varied only in its example, spans 0.53 to 17.55 — a factor of 33. Every one of the four bars lies inside that interval. So a single departure, drawn against five ordinary materials, reproduces the entire ladder and extends it in both directions.

The wavelength index behaves differently: 3.95 on a lightly brightened sheet against 7.66 on a heavily brightened one is a factor of 1.94, and both ends are large. That asymmetry is the useful part. Two of these departures are switches and two are dials. A brightener is present or it is not, and when it is present the cost is several ΔE₀₀ whatever its loading; a translucency is a continuous length and the cost runs continuously with it. Four bars on one axis conceal that difference entirely, because a bar is the same shape either way.

The tolerance claim is about a pairing, not a departure #

All four are above the tolerance a specification would be written in, on samples nobody would call exotic is the sentence the four bars are there to support, and one line of the same section contradicts it.

The place index is worth 0.53 ΔE₀₀ on coated paper — below the 1.0 that a delivery tolerance allows, on the least exotic sample in the essay, and the same paper that carries the largest bar on the ladder. A printing paper is above tolerance on the wavelength axis and below it on the place axis, which means the claim is not about departures and is not about samples. It is about pairs.

That is the same structure the audit’s own finding has. Each departure is a product of two deviations and vanishes when either is zero, so its size is a property of the pairing of sample with light, never of the axis alone. The ladder is four pairings chosen to be large, and the honest reading of it is that for each departure there exists an ordinary sample that puts it above tolerance — an existence claim, which is what an audit needs, rather than the universal claim the sentence makes.

The one length, checked #

The account of why the place index moves is unusually specific: what decides it is a single length, the distance light travels inside the material before absorption, running from a fifth of a millimetre to fourteen. That is testable against the four values it is offered to explain.

Fitting a power law through the two named extremes — 0.53 at 0.2 mm and 17.55 at 14 mm — gives an exponent of 0.82, and interpolating the other two materials on it implies a diffusion length of about 1.0 mm for the pigmented plastic and 3.7 mm for skin. Neither number was used to build the fit and both are the right size for the material. The one-length account survives a check it could have failed, and it survives with an exponent below one rather than the linear scaling a first guess would give.

Against the four-millimetre aperture those lengths become ratios of 0.05, 0.24, 0.92 and 3.50, and the condition falls out in the form the audit uses everywhere else. The place index vanishes when the diffusion length is small against the aperture, and the ratio at which it crosses the one-ΔE₀₀ tolerance is about a tenth. Coated paper sits at a twentieth and is safe; the plastic sits at a quarter and is not. That is a sharper statement than an opaque sample’s kernel is a point, because opacity is not the variable — the ratio of two lengths is, and one of the two belongs to the instrument rather than the sample.

The part that makes it an audit rather than a complaint #

Every one of the four vanishes exactly under a stated condition, and each condition has two halves: something the sample can lack, and something the light can lack.

• A sample with no fluorophore has nothing off the diagonal, and a lamp with no ultraviolet cannot excite one.
• A Lambertian surface reads the same in every room, and a field of constant radiance reads every surface’s own reflectance.
• An opaque sample’s kernel is a point at any aperture, and an infinite aperture reads any kernel whole.
• A sample that does nothing below 380 nanometres does not notice the grid, and neither does a light with no power there.

That structure is the whole finding of the round and it has a name: each departure is a pairing of two deviations, and either one being zero makes it exactly zero. It also explains why the collection has got away with the model for nineteen rounds. Its samples are smooth, opaque and unbrightened constructions; its lights are daylight reconstructions and blackbodies; and under that combination one of the two factors is always near zero.

What was computed, and how #

Three modules were written for this round and each one is a departure made computable.

The kernel. A new module implements the dipole approximation to the diffusion equation — two point sources, one real at a depth of one mean free path and one virtual above the boundary — which has a closed-form profile and a closed-form total derived separately from it. The two are required to agree, and they do to two parts in ten thousand across four slabs spanning two decades of diffusion length. That check is the reason the module can be trusted: two derivations agreeing is a much stronger statement than one number matching a table.

The lobe. A second implements a Lambertian body under a microfacet lobe with a Fresnel term, and integrates it over a hemisphere by Gauss–Legendre in the cosine and a uniform grid in the azimuth. Everything an instrument or a room reports is that one function integrated against a different weight.

The audit. A third puts the four on one footing, imposes each condition twice, and checks the pairing identity against the direct computation. It also draws the boundary of what this instrument can reach, which turned out to be much narrower than the last round’s.

Nothing here needed an idea that was unavailable three rounds ago. What it needed was to treat the first term of the integral as a variable, and the reason that took four rounds is that the first term does not look like a choice. A unit is imported from a file. A test set is declared in a function signature. A reflectance is data.

Where the model stops #

Three of the four departures act on the sample and are indifferent to the light, and this turns out to decide how much they matter downstream.

The collection’s adaptation census — fourteen changes of light, judged over a hundred and twenty-five constructed surfaces — moves by under ten per cent when every one of those surfaces is replaced by what an instrument with a four-millimetre aperture would report, or by what a glossy version of the same surface returns. A gain applied after the fact absorbs most of it, because both departures change the reflectance and the gain is applied to what the reflectance produced.

The fluorescent departure cannot be handed to that census at all. Its content is that the sample does not have a reflectance, and a function whose argument is a list of reflectances has nowhere to put it. That refusal is reported rather than worked around, and it is the sharpest thing in the round.

The generalisation #

There is a habit of mind here worth separating from the colour.

A model that keeps one argument of a function of six has made five decisions, and only one of them usually gets written down. The written one is the approximation — the Lambertian assumption, say, which everybody knows is an approximation and can be found in any textbook. The other four are not approximations at all; they are places where the model has no slot, so there is nothing to be approximate about. A missing slot cannot be wrong by a small amount, because it cannot be wrong by any amount.

The move that turns a missing slot into a measurable quantity is to write the fuller function down and integrate it deliberately, so that what was implicit becomes a weight that can be varied. Every departure in this round was already in the literature with an instrument attached; what was new was putting all four into one unit and asking which condition each rests on.

The reverse move is the failure mode. A model that has been correct for a long time stops looking like a projection and starts looking like the object, and then its missing axes are invisible rather than merely unmeasured. Nineteen rounds of this collection have said name the observer and name the illuminant on every figure, and not one of them said name the geometry — although the geometry has been an essay here since the scene field was built.

Who found it, and when #

The six-argument function is standard and old. Nicodemus and colleagues gave the directional pair its modern name and definition in a 1977 report for the American National Bureau of Standards, and the bidirectional reflectance distribution function has been the working object of computer graphics ever since. Donaldson published the wavelength matrix in 1954, out of a paper industry that needed to specify brightened stock. The spatial kernel came last into common use: the diffusion approximation is much older, but its practical form for surfaces dates from Jensen, Marschner, Levoy and Hanrahan in 2001, and it arrived in graphics rather than in colorimetry.

The point is not that any of this is new. It is that all three have been available for decades, in three different literatures, and colorimetry uses a scalar reflectance anyway — for the good reason that the scalar is right whenever one of two factors is zero, and for the bad reason that nobody checks which case they are in.

Those two are the dropped arguments themselves, drawn at the scale each of them lives on. What follows is the instrument rather than the object: what an audit of this kind can be pointed at, and the identity that holds the four departures together.

Where the ladder goes next #

The four departures are four, and the list is not closed. Polarisation is a seventh argument nobody here has named — the Fresnel term at the interface is polarised and the body’s return is not, which is why one measurement condition removes the interface optically rather than by an angle. Time is an eighth: a brightener is used up while it is being measured, so the response has a clock in it as well.

Each further argument is harder to reach than the last, and at some point restoring one means writing a different collection rather than auditing this one. That boundary is real and it is worth naming precisely: the four here were reachable because each has a published model and a published instrument. An argument with neither is not a choice this method can audit, and calling it one would be a courtesy to the method rather than a statement about the world.