A surface has a kernel

A reflectance says what fraction of the light comes back. It does not say where, because in the model there is nowhere else for the light to be.

The claim #

What a scattering surface has is a kernel — how much comes back at each distance — and the reflectance is its integral.

• The kernel is computed, not assumed, from two coefficients: how strongly the material absorbs and how strongly it scatters, both per millimetre.
• It has one length in it, the diffusion length 1/√(3σₐσₜ′), and every result in this round is that length divided by an aperture.
• The length runs from 0.22 millimetres to 14 across six ordinary materials, which is why one instrument treats them so differently.
• Two derivations of the same total agree to two parts in ten thousand, and that agreement is what makes the implementation believable.
• And the collection already had the limit of this model — Kubelka–Munk is the same physics with the length integrated out — which is exactly the version that cannot say how far from the limit a measurement is.

What a photon does inside a stone #

Light meeting a piece of marble does not reflect off it in any useful sense. About four per cent bounces at the interface, and the rest goes in, meets a calcite grain, changes direction, meets another, and continues — some thousands of times — until it is either absorbed or arrives back at the surface from underneath and leaves.

That random walk has a length scale, and the scale is not the mean free path between scattering events. It is the distance over which absorption finally wins, which for weakly-absorbing material is much longer than a single step. The diffusion length is

ℓ = 1 / √(3 σₐ (σₐ + σₛ′))

with σₐ the absorption coefficient and σₛ′ the reduced scattering coefficient, both per millimetre. For coated paper, thick with titanium dioxide, it is about a fifth of a millimetre. For pale marble it is four. For candle wax, which barely absorbs anything at all, it is twelve.

That single number is why the same instrument treats those three materials as three different problems, and why the paper industry and the plastics industry have entirely different opinions about how big an aperture ought to be.

The kernel, and the two ways of getting its total #

The standard treatment is the dipole approximation to the diffusion equation, in the arrangement Jensen, Marschner, Levoy and Hanrahan published in 2001. A pencil beam entering the surface is modelled as two point sources of diffusing light: a real one at a depth of one mean free path, and a virtual one above the surface, placed so that the diffusion solution satisfies the boundary condition. The light returning at distance r from the entry point is the sum of the two sources’ contributions.

That gives a closed-form profile. What makes it usable here is that the same theory also gives a closed-form total — the fraction of the beam that comes back anywhere at all — and the total is derived from the diffusion equation directly rather than by integrating the profile.

So there are two independent expressions for one number, and they have to agree. Over four slabs spanning two decades of diffusion length, the numerically-integrated profile reproduces the closed-form total to better than two parts in ten thousand. During construction that check caught a wrong constant in the virtual source’s depth: the profile looked entirely reasonable and the total was two per cent out.

This collection has been through that lesson before. A camera profile’s chroma error was one per cent wrong and survived every structural check until two derivations were compared; a purity bug produced nine identical patches under nine captions quoting different numbers. Two derivations disagreeing beats one number against a table, and it is the cheapest check available.

What is in the coefficients, and what is not #

Each of the six materials in the table is stated as coefficients rather than as a reflectance, which is one stage earlier than everything else on this site begins. A colour begins as a spectral power distribution, and a reflectance here begins as an absorption spectrum and a scattering spectrum.

The absorption is a sum of named Gaussian bands — a trace of iron in the marble at 450 nanometres, melanin and two haemoglobin bands in the skin. The scattering is a single number at 550 nanometres with a wavelength slope: σₛ′ ∝ λ^−b, with b running from 0.6 for wax to 1.4 for skin.

That slope is a statement about particle size relative to the wavelength — large particles scatter every wavelength alike, small ones favour the short end — and it is a physical measurement, not a fitting convenience. It turns out to decide more about the colour of a translucent sample than the absorption does, which is the round’s most surprising single result.

The reflectance the model wants #

The number the colour integral needs is the kernel’s integral over the entire plane: of the light that went in here, what fraction comes back out anywhere.

No instrument collects that. An instrument lights a disc and looks at a disc, so what it collects is the kernel weighted by how much of the lit disc lies at each distance from the looked-at disc — and everything outside is thrown away. The share recovered through a four-millimetre radius runs from 98.1 per cent on coated paper to 48.2 on candle wax.

Two consequences follow immediately and neither is optional.

Every finite aperture reads low. The kernel is positive everywhere, so what is thrown away is positive, and there is no sample and no instrument for which the error has the other sign. That is unusual: most measurement errors have two directions.

And the approach to the limit is slow. The share recovered rises as roughly one over the aperture rather than exponentially, because the kernel’s tail is what is being cut. Going from a four-millimetre radius to forty on a marble sample takes the recovery from 63 per cent to 95, and the last five per cent would need a metre.

What was computed, and how #

Every number here comes out of one radial integral, taken the same way in every case, and the way it is taken is not incidental.

The kernel is peaked at one mean free path — a few hundredths of a millimetre — and has a tail four decades wider. A linear grid fine enough to resolve the peak is ruinous by the time it reaches the tail, and one that reaches the tail misses the peak entirely. So the quadrature is logarithmic in the radius, with its upper limit stated in diffusion lengths rather than in millimetres, which makes the same call correct for coated paper and for candle wax without anybody choosing a range per material.

The aperture enters as a weight inside that integral. A measurement lights a disc and looks at a disc, so the weight at distance r is the area those two discs share when their centres are r apart — an ordinary lens-shaped intersection with a closed form. Everything about apertures in this round is that one weight, and the reference the reading is divided by is a perfect diffuser measured the same way, which is what makes the answer a reflectance factor rather than a flux.

Two properties fall straight out of writing it that way. The weight is one at zero separation, so a sample whose kernel is concentrated at the origin reads exactly right at any aperture. And the weight approaches one everywhere as either disc grows, so widening either the lit disc or the measured one recovers the whole kernel — a symmetry that is invisible in the usual advice and obvious in the algebra.

Six materials, and one number that orders them #

The table holds six slabs and each is stated as two spectra of coefficients. Their diffusion lengths at 550 nanometres are 0.49 millimetres for coated paper, 0.55 for a matte wall emulsion, 0.56 for a pigmented plastic, 0.50 for skin, 4.48 for pale marble and 12.23 for candle wax.

Four of the six agree closely at 550 and behave completely differently, which is the first thing worth noticing: the length at one wavelength does not order the materials, because what an aperture costs is decided across the whole band. Skin at 550 has a diffusion length of half a millimetre and runs to 3.49 at the red end, where haemoglobin has stopped absorbing; a four-millimetre radius costs it 5.83 ΔE₀₀ against the plastic’s 1.96, although the two are indistinguishable at the wavelength anybody would quote.

A single diffusion length is therefore not a specification of a material’s translucency, in the same way that a mean is not a statement about the set it came from. What decides the answer is the length’s whole spectrum, and its shape is set by the scattering slope rather than by the absorption bands.

Where the model stops #

The diffusion approximation is an approximation, and it is worth being precise about where it fails rather than waving at it.

It assumes the light inside has forgotten which direction it came from, which is true only after several scattering events. So it is wrong close to the entry point — within about one mean free path — and it is wrong for materials that scatter weakly, where the light does not scatter enough times to forget anything. Both failures are at the short end, and everything this round measures is decided by the tail rather than by the core, which is the reason the approximation is tolerable here.

It also assumes a semi-infinite slab. A thin sheet lets light out of the back, which is why a translucent object’s colour is in its thickness — a separate departure, with its own arithmetic, and not one this file models.

And it assumes the material is uniform. A grain of pigment large enough to see is not described by a coefficient at all.

The limit the collection already had #

Kubelka–Munk has been in this collection’s scene machinery since that field was built, and it is the same transport problem solved in one dimension: two fluxes, one up and one down, through a layer that is infinite in both lateral directions by construction.

That construction is exactly the assumption this file is about. A layer with no edges has no aperture, so K/S computes the number an infinite aperture would read and has no way to express any other. Feeding the same absorption and scattering into both gives reflectances that agree in shape and differ by a bias — every band out by a factor between one and 1.21, worst at 460 nanometres, for 3.88 ΔE₀₀ overall.

The bias is expected: the correspondence between the two pairs of coefficients is exact only for isotropic scattering at a particular boundary. What matters is not its size but its kind. The collection had the limit and not the length, and a limit is precisely the thing that cannot say how far away from it a measurement is.

The same relation holds for why mixing paint is not stacking filters: K/S is what makes a mixture additive, and it is additive in a variable this round finds nobody has ever named.

What a kernel does that a number cannot #

Two properties of the kernel have no counterpart in a reflectance, and both of them are visible rather than technical.

It has a width, so an edge is blurred. Light landing on the lit side of a boundary leaves on the dark side. The distance over which the reading runs from a tenth of its far-field value to nine tenths is 0.22 millimetres on coated paper, 0.96 on a pigmented plastic, 5.83 on marble and 10.52 on wax. That is the width of the neighbourhood a point’s colour is decided by, and on the last two it is larger than most samples.

And it makes a local reflectance factor undefined. A point just outside a lit region receives no light and returns some — a tenth of the far-field radiance at half a diffusion length out. The ratio the model calls a reflectance has a zero denominator there. This is the same arithmetic as a fluorescent sheet whose radiance factor exceeds one, arrived at from a completely different direction and with no fluorophore anywhere in it.

Two ranges for one length, and why they differ #

The diffusion length is given twice with different endpoints — 0.22 to 14 millimetres in the claim list, and 0.49 to 12.23 in the table of six — and the two are not the same measurement.

The table is a slice at 550 nanometres. The claim list is the range over material and wavelength, which is the wider quantity and the right one for a summary, since what an aperture costs is decided across the whole band. The reconciliation is checkable at both ends: coated paper’s 0.49 at 550 falls to about 0.22 in the band it absorbs most strongly, and candle wax’s 12.23 rises to 14 where it absorbs least — a 14 per cent rise, which is what a nearly non-absorbing material’s weak spectral dependence looks like.

The prose in between mixes the two. For coated paper it is about a fifth of a millimetre. For pale marble it is four. For candle wax it is twelve — the last two are the 550 values, to within rounding, and the first is the band minimum rather than the 550 value of 0.49. So one of three quoted lengths is measured differently from the other two, in a sentence offered as a comparison between them.

There is a near-coincidence worth ruling out. Coated paper’s edge width is also 0.22 millimetres, so the claim list’s lower bound could be read as an edge width rather than a length. It is not: the edge widths run to 10.52 at the top and the claim list’s upper bound is 14, which no edge width reaches.

The between-material spread is the smaller one #

Setting the two ranges side by side says something the essay states qualitatively and does not measure.

At 550 nanometres four of the six materials — coated paper, matte emulsion, pigmented plastic and skin — sit at 0.49, 0.55, 0.56 and 0.50. They agree to 14 per cent. Skin alone, across the band, runs from 0.50 at 550 to 3.49 at the red end: a factor of 7.

So the spread of one material’s length across wavelength is six times the spread between four materials at one wavelength. That is the sharp form of a single diffusion length is not a specification of a material’s translucency: it is not that the single number is imprecise, it is that the axis it is measured along is the wrong one by a factor of six, and the four materials it declares equivalent are the ones an instrument treats most differently.

It also says which coefficient to quote if only one can be. Not the length at a reference wavelength, which orders nothing, but the scattering slope — because the slope is what makes the length a spectrum, and the essay’s own next rung finds the slope deciding the colour outright.

Six millimetres is about as large as a hand-held instrument’s window gets, and it is the reading that says how much of the shortfall a bigger aperture removes.

How slow the approach to the limit is #

The share recovered rises as roughly one over the aperture is right and can be given an exponent from the marble figures the essay quotes: 63 per cent recovered at a four-millimetre radius and 95 at forty.

The missing fraction falls from 37 per cent to 5 over a tenfold widening, which is a power of 0.869 — a little slower than one over the aperture, and the difference matters because it is the exponent that decides the cost of a further digit.

Each further factor of ten in what is missing costs a factor of 14 in the radius. Halving the residual costs 2.2 times the aperture. So from the forty-millimetre reading: 99 per cent recovery needs a radius of 255 millimetres, 99.5 per cent needs 566, and 99.75 per cent needs 1,256 — which is the metre the essay mentions, arrived at rather than asserted.

That is the practical content of the reflectance the model wants is an integral no instrument collects. It is not that no instrument is large enough; it is that the aperture required grows as the tenth root of nothing useful — fourteenfold per digit — so the integral is unreachable by enlargement in the same way an asymptote is unreachable by walking. The only route to it is a model, which is what this file supplies, and the model’s job is to say how much was missed rather than to collect it.

Who found it, and when #

The diffusion approximation to radiative transfer is nineteenth-century in origin and was applied to biological tissue from the 1970s, where the practical question was how deep a laser reaches. Kubelka and Munk’s two-flux solution is from 1931 and was built for the paint industry, which wanted to know how many coats hide a substrate.

The dipole form used here is Jensen and colleagues’ from 2001, and its home is computer graphics rather than colorimetry — it was written to render skin and marble, and it succeeded so thoroughly that it changed what rendered faces look like. The colour-measurement literature reached the same problem from the other end, calling it translucent blurring or edge-loss error, and standardised its answer as advice about aperture size rather than as a model.

The two literatures have barely met. A colorimetry standard will tell a laboratory to use the largest aperture the sample allows; a graphics paper will give the kernel and never mention an aperture at all. Both are describing the same integral from opposite sides.

The consequence for a pair is sharper than the consequence for one sample, because two objects can agree at every wavelength and still differ in the quantity below.

Where the ladder goes next #

The kernel is radially symmetric here because the material is uniform, and that is a real restriction. A fibrous material — paper viewed close enough, wood, textile — has a kernel that is longer along the fibres than across them, and its reading would then depend on the orientation of the sample in the instrument as well as on the aperture. Nothing in this round measures that.

The other unfinished piece is thickness. This file’s slab is semi-infinite; a real sheet has a back, and a translucent sheet on a black backing reads differently from the same sheet on a white one. That is a two-parameter problem where this one is a one-parameter problem, and it is the form the plastics and textile industries actually meet.