A room is not a sphere

An integrating sphere is usually explained as a way of averaging over directions so that gloss does not matter. That is nearly right and it hides the interesting part, which is that the averaging is exact and the reason it is exact is a theorem.

The claim #

Under a hemisphere of constant radiance a detector reads the sample’s own reflectance exactly, whatever the sample is — and no room provides one.

• The exactness is Helmholtz reciprocity, and it holds to four parts in a hundred million million across every roughness tested, because the pairing’s second factor is identically zero.
• It is not an averaging argument. A sphere does not read the average of what a surface does; it reads a specific reflectance, and reciprocity says which.
• A real field costs the pairing of the field’s non-uniformity with the surface’s non-Lambertian part, which is exactly zero if either is.
• The sign is not predictable from sizes. A viewing booth reads a gloss sample high and a window reads the same sample low — and the booth is the less non-uniform of the two by a factor of three.
• And the collection’s own quadrature was wrong in a way its convergence check could not see, which is the round’s methodological finding.

What the sphere actually measures #

A surface’s response to light is a function of two directions. What arrives can arrive from anywhere in the hemisphere; what leaves can leave anywhere; and the bidirectional reflectance says how much of the first becomes the second.

Two quantities can be got out of that function by integrating one of its arguments away. The directional-hemispherical reflectance is what leaves in total when the light arrives from one direction — a function of the incoming direction. The hemispherical-directional reflectance factor is what leaves in one direction when the light arrives uniformly from everywhere — a function of the outgoing direction.

Helmholtz reciprocity says the bidirectional reflectance is symmetric in its two arguments, and it follows immediately that those two quantities are the same function. So a sphere, which provides uniform illumination and looks in one direction, returns the directional-hemispherical reflectance for the direction its detector sits in — which is a quantity about the sample, not about the sphere.

Computed here over six roughnesses from a varnish to a matte paint, the sphere reading and the directional-hemispherical reflectance agree to within four parts in a hundred million million. That is floating-point noise, and it is that way because the pairing that separates them has a second argument which is identically zero.

The condition, and the one number that is not there #

The exactness needs uniform illumination and it needs nothing else. In particular it does not need the surface to be Lambertian.

The second condition is a different one, and it is the one that decides whether the sphere’s exact answer is an answer to the question anybody asked. A Lambertian surface has one reflectance; anything else has a function. The directional-hemispherical reflectance of a surface with a roughness of 0.08 runs from 0.5398 near the normal to 0.7704 at eighty degrees — an increase of forty-three per cent, because Fresnel’s term rises towards grazing.

The one number a radiosity calculation wants is the bihemispherical albedo, which averages that function over incoming directions: 0.5792 for this surface. It lies inside the range and equals none of the readings. The sphere returns 0.5398 at eight degrees, which is exact and is not that number.

So the two conditions are genuinely independent, and this collection has been buying the first and quoting it as though it had bought the second. The geometry an instrument stands in has been an essay here for a long time; that a single reflectance exists at all has not been questioned.

Why an average is the wrong word for it #

The usual account of an integrating sphere says it averages over directions, so that a glossy sample and a matte one with the same pigment read alike. That account predicts something false and it is worth seeing why.

If a sphere averaged, its reading would be near the middle of the range a surface’s reflectance takes over incidence — somewhere around the bihemispherical albedo, 0.5792 for the surface above. It is not: the sphere reads 0.5398, which is the bottom of that range, because the detector is eight degrees off the normal and the reciprocal quantity is the reflectance for light arriving at eight degrees.

Move the detector and the reading moves with it. A sphere with its port at forty-five degrees would read 0.5508 off the same sample; at sixty, 0.5875. Each of those is exact, each is a different number, and none of them is an average of anything.

What the sphere does is not average but exchange: it swaps a hard measurement, of what leaves in every direction when the light comes from one, for an easy one, of what leaves in one direction when the light comes from everywhere. Reciprocity says the two are equal. The instrument’s whole design is that equality, and it is why a sphere with a badly baffled lamp — a field that is not uniform — is not a slightly worse sphere but a different instrument.

What a room costs #

Five fields are modelled: an overcast sky of constant radiance, a viewing booth with a diffuser overhead, a window carrying four fifths of its light in a thirty-degree cone, a lamp on a stand, and sun with sky. Their non-uniformity — the share of the light that would have to be moved to make the field uniform — is 0, 0.212, 0.665, 0.930 and 0.783.

For a pigmented surface with an ordinary sheen, the colour a detector reads eight degrees off the normal differs from the surface’s own by 0 ΔE₀₀ under the overcast sky, 1.03 in the booth, 1.91 at the window, 2.18 under sun and sky, and 3.01 under the lamp.

The interface is spectrally flat and the body is not, so what a room does to a colour is to move it towards or away from the light’s own — a desaturation or a saturation of a size the room decides. Under the window the chroma rises from 56.5 to 58.4; in the booth it falls to 55.4.

Those two have opposite signs, and that is the finding this essay is named for.

The prediction the arithmetic refused #

The round predicted that the error would be the product of two magnitudes: how far the surface is from Lambertian, times how far the field is from uniform. It is not, and the way it fails is instructive.

The error is exactly the pairing of the two deviations — an inner product over the hemisphere, not a product of their sizes. An inner product depends on the angle between its arguments, and over twenty-four cells the fraction of the Cauchy–Schwarz bound actually used runs from −0.101 to 0.365. Six cells are positive and eighteen negative.

The mechanism is easy to see once the sign is known. The detector sits eight degrees off the normal, so the surface’s specular lobe points back towards the normal. A viewing booth lights mostly from overhead — near the normal — so it puts extra light exactly where the lobe is looking, and the reading comes out high. A window at thirty-five degrees, a lamp at forty-five, and a sun at thirty-five all light from where the lobe is not, so their readings come out low.

No comparison of anisotropies could have produced that, because the anisotropy of a field is a magnitude and the answer needs a direction. Either factor being zero makes the departure zero, and that much the prediction got right; the rest of it was a product where the arithmetic had a pairing.

The port that would read the useful number #

A sphere reads exactly and reads the wrong quantity for the one use a scene calculation has. That is a complaint about where the detector sits, and where it would have to sit instead can be computed rather than argued about.

The directional-hemispherical reflectance rises with incidence, so somewhere between the normal and grazing it passes through the bihemispherical albedo. For the surface above it passes through 0.5792 at 57.93 degrees.

The interesting part is that the angle barely moves. Across the whole roughness ladder — a varnish at 0.02 up to a matte paint at 0.6 — the crossing runs from 53.53° to 57.93°, a spread of four and a half degrees, on a quantity with no obvious reason to be stable. The albedo itself moves by 0.046 across that ladder, and the two curves keep meeting in nearly the same place.

So a fixed port would do. Setting one at 56 degrees and reading every surface off it returns the bihemispherical albedo to within 1.13 per cent at worst, and to within a fifth of a per cent on the three roughest surfaces. The standard eight-degree port is between 1.65 and 6.83 per cent low, and its error is not even monotone in roughness — it is worst at a roughness of 0.05, at −6.83 per cent, and falls away in both directions from there.

That is a design a sphere could have and does not. The 8°/d and d/8° geometries exist because a port near the normal is easy to place and easy to view a sample through, and because the quantity everybody believed they wanted was a reflectance rather than an albedo. An instrument that already supplies a uniform field, and already has reciprocity for free, could deliver the number a radiosity solve needs from one port at a different angle and no other change at all.

What the anisotropy does predict #

The refusal above is about the sign, and it is worth saying what a magnitude still buys, because it buys more than the refusal implies.

Ranked by the colour a detector reads eight degrees off the normal, the four non-uniform fields come out in exactly the order of their anisotropies: the booth at 0.212 and 1.03 ΔE₀₀, the window at 0.665 and 1.91, sun and sky at 0.783 and 2.18, the lamp at 0.930 and 3.01. That is perfect rank agreement across four fields. The more non-uniform the light, the further the reading moves.

What the magnitude does not give is the constant in front, and the departure from a constant is the sign story arriving in the size. The three fields that light from where the lobe is not deliver between 2.79 and 3.23 units of ΔE₀₀ per unit of anisotropy. The booth delivers 4.87. It is the least non-uniform field on the list and it converts its non-uniformity into error about sixty per cent more efficiently than any of the others, because its light arrives where the detector is looking.

So the magnitude orders the fields and the alignment scales them, and on this list the two separate cleanly. A field’s anisotropy is a usable predictor of how much a room will cost — wrong by up to a factor of about 1.7 — and no predictor whatever of which way the error will go.

What the machinery caught that the prose had wrong #

The hemisphere is integrated by Gauss–Legendre in the cosine of the elevation and a uniform grid in the azimuth, and the grid was first set at forty by ninety-six on the strength of the standard check: refine it, and see whether the answer moves. Forty by ninety-six and ninety-six by two hundred and fifty-six agreed to a part in ten thousand on the sharpest lobe in the file.

Both were wrong, and they agreed because they were wrong in the same way. Each put a quadrature node at the azimuth the light was arriving from, so each sampled the narrow lobe at its peak. Rotating the light instead — to an azimuth of 137 degrees, which is nothing in particular — moved the answer by 2.5 per cent, on a quantity that cannot depend on azimuth at all, since every surface here is isotropic by construction.

The repair is a finer grid, at eighty by a hundred and ninety-two, which agrees with itself across azimuths to a part in seventy thousand. The lesson is the general one and it is not about quadrature: a convergence check that refines one knob tests one knob. The symmetry the model has and the grid does not is the thing that finds it, and this collection has made the same discovery about a spy on a generator’s options and about a fit that reproduced its own training set.

What was computed, and how #

The surface is a Lambertian body under a Trowbridge–Reitz microfacet lobe with a Fresnel term at a refractive index of 1.5 — the standard arrangement in rendering, and one that reduces to the collection’s existing Fresnel curve at normal incidence, which is asserted rather than assumed.

Everything an instrument or a room reports is that one function integrated against a different weight, and the two named instrument geometries fall out rather than being declared: a 45°/0° reads 0.5040 on a surface whose body is 0.5000, a sphere with the port open reads 0.5398, and the same sphere with a gloss trap reads 0.5311. The gap between the first two is what a simpler model calls a pedestal, and it is not a constant — it varies by 0.036 of a reflectance unit across the roughness ladder, because Fresnel’s term is not flat in angle.

One algebraic simplification makes the spectral work affordable and is worth stating because it is also a physical claim. The lobe contains no body reflectance, so the whole integral splits into the body’s own reflectance plus a term depending only on the roughness, the index, the detector angle and the field. The interface really is an addition — for the part of it that never enters the material. The part that does enter and comes back out through the boundary is not additive at all, and it is a Möbius function rather than a sum.

Where the model stops #

The five fields are constructions, and their anisotropies are what they are because of numbers chosen to be plausible rather than measured. A real window has a sky gradient, a real booth has walls, and a real room has furniture in it.

What does not depend on those numbers is the structure: the sign of each cell is set by whether the field’s light arrives where the detector’s mirror direction points, and that is decided by geometry rather than by magnitude. A booth that lit from forty-five degrees would read low like a window.

The surfaces are also isotropic, which is asserted and is a restriction. A brushed metal or a fabric has a lobe that depends on azimuth, and then the bihemispherical average needs a double integral rather than the single one used here — a simplification that saved a factor of two hundred and fifty in cost and is checked rather than assumed.

Those two are the room asking the question. The instrument asks a narrower version of it, in three stated geometries rather than five rooms, and the three come out of the same function of two directions rather than being declared separately.

Who found it, and when #

Reciprocity is Helmholtz’s, from the 1850s, and the integrating sphere is Ulbricht’s, from 1900. The connection between them — that a sphere measures a reciprocal quantity and is therefore exact rather than merely convenient — is standard in radiometry and almost never said out loud in colour measurement, where the sphere is introduced as a way of not caring about gloss.

Nicodemus and colleagues wrote the modern definitions in 1977 for the US National Bureau of Standards, and the reason that document is still cited is that it is the one place where the six quantities obtainable from a bidirectional reflectance are given separate names. Most of the confusion this essay is about comes from three of those names being written the same way in practice.

The generalisation #

The transferable claim is about which of two conditions a piece of apparatus buys.

A great deal of measurement design is arranged so that one term in an error vanishes exactly, and the arrangement is usually described by what it achieves rather than by which term it kills. A sphere is described as averaging out gloss; what it does is set the field’s deviation to zero, which kills the pairing whatever the sample. A gloss trap is described as removing the specular component; what it does is remove one part of the sample’s deviation, which leaves the field’s alone.

Those are different repairs and they fail in different places. The first fails when the field is not uniform, which is every room. The second fails when the surface is not the kind whose lobe leaves through the trap, which is any surface rough enough to scatter its interface reflection widely — a matte finish sends its four per cent everywhere, and a trap catches almost none of it.

The habit worth taking away is to ask, of any apparatus, which factor of the error it sets to zero. That question has an answer for every instrument in this collection, and the answer usually explains the apparatus better than its own documentation does. An instrument brings its own light because specifying the lamp kills the field factor of the fluorescent departure; a wide aperture kills the field factor of the lateral one. Three instruments, one move.

Where the ladder goes next #

Two things are open, and one of them is a real gap in this collection rather than a refinement.

The radiosity solver here assumes Lambertian surfaces, because a radiosity solution requires them: the whole method rests on a surface’s radiance being independent of direction. Every scene result in this collection — the corner, the bounce, the green wall — is computed that way, and giving those walls a real lobe is a different algorithm rather than a different parameter.

And a field has a low-dimensional description that nobody has used here. The pairing needs the field’s whole distribution; graphics has represented that in nine numbers of spherical harmonics since the 1990s, and a nine-number field is exactly the sort of object this collection likes, because it can be varied.