An instrument has a geometry

The instrument model on this site has a slit, a sampling interval and a noise figure, and no position. It reports what a spectrophotometer would report about a sample that is a function of wavelength and of nothing else — which means every reflectance here is a measurement made from nowhere in particular.

Real instruments say where they were standing, and the standards give two answers — one more silence beside the lamp the instrument brings with it. 45°/0° illuminates at forty-five degrees and looks straight down the normal. d/8° illuminates from an integrating sphere and looks at the sample eight degrees off. They are not convertible into one another, a tolerance that names neither has not finished being written down, and the gap between them is bigger than most of the tolerances they are used to enforce.

The claim #

Where the instrument was standing is worth more than the tolerance it is being used to enforce, and what separates the two geometries is additive rather than multiplicative.

• On a dark gloss sample the two geometries disagree by ΔE00 8.35 and on a light one by 1.18 — a factor of seven across an ordinary sample set, with the mean at 5.05.
• The disagreement is a pedestal, not a tilt. Fresnel’s term on a dielectric is nearly flat with wavelength, so what the sphere adds is the same small number in every band.
• Remove the interface component and the two agree exactly — 10⁻¹⁵ across all twenty-four samples — which is the check that nothing else is contributing.
• And no gain removes it. The best single per-channel gain fitted over the whole sample set takes ΔE00 5.05 down to 4.50, which is eleven per cent. Every multiplicative change of context on this site gives up ninety per cent or more to a gain that is not fitted at all.
• The sphere has an error of its own, up to ΔE00 2.91, and correcting it requires knowing the sample’s reflectance, which is the quantity being measured.

Four per cent, in two places #

A dielectric surface returns about four per cent of what arrives at it at the interface, before the light has entered the material or met a pigment. That is Fresnel’s term and it carries the source’s spectrum almost unchanged, tilted only slightly by dispersion in the refractive index.

Where it goes depends on the finish. A gloss surface is a mirror with pigment behind it and sends nearly all of that four per cent in the specular direction; a matte one has the same four per cent and scatters it, so a detector at any one angle catches very little.

A 45°/0° instrument throws the interface term away by construction: the specular beam leaves at forty-five degrees on the other side and the detector is on the normal — the same separation an aperture makes in the other direction. A sphere collects it, unless a gloss trap is opened — and even then a trap is a hole of finite size in a surface that is not perfectly specular, so a few per cent leaks back.

Why it is worth what the sample is dark #

Four points added to seventy is a rounding error. Four points added to three more than doubles the sample’s reflectance.

That is the whole of why the disagreement is a lightness effect rather than a colour one, and why it is enormous at the black end of a sample set. A three per cent neutral reads 3 per cent one way and 6.4 the other; in CIELAB that is 11.3 units of lightness, and the CIEDE2000 difference is 8.35.

The ordering is monotone in lightness and the site’s gate requires it to be: every step down in reflectance has to disagree more than the step above it, because the quantity added is the same in both cases and the sample’s own reflectance is not.

The sphere is a cavity, and the sample is part of it #

The second difference between the geometries is not about where the detector is at all. It is that a sphere’s own wall lights the sample, so the sample is part of the instrument.

Swap a black tile for a white one and the light level inside the sphere changes, because the sample port is a few per cent of the sphere’s area and the sample is the only port that returns light. The two samples were therefore measured under two different illuminations. The effect has a name in the literature — substitution error — and every instrument corrects for it.

The correction is the interesting part. The reading is the sample’s reflectance times a gain that depends on the sample’s reflectance, so recovering the reflectance means dividing by a gain computed from the answer. The instrument iterates: take the raw reading as a first estimate, compute the gain it implies, divide, repeat.

One pass takes the worst error from 2.91 to 0.156. That is a good enough correction, and it is a correction whose residual is proportional to how far the sample is from the white tile the machine was calibrated against — so it is largest exactly for the dark samples where the geometry difference is already largest.

Three numbers about one sample, and one of them is out #

The worked case at the dark end carries three figures — a sphere reading of 6.4 per cent, a lightness difference of 11.3, and a colour difference of 8.35 — and any two of them determine the third. They do not agree.

Converting a three per cent neutral and a 6.4 per cent one to CIELAB gives lightnesses of 20.04 and 30.40, a difference of 10.36 rather than 11.3, and a CIEDE2000 of 7.58 rather than 8.35.

Solving the other way round is more informative. A lightness difference of 11.3 requires a sphere reading of 6.8 per cent, and 6.8 gives a colour difference of 8.32 — which is the published 8.35 to within the rounding. So two of the three numbers are mutually consistent and the printed 6.4 is not consistent with either.

The mechanism the essay gives settles which one to keep. Fresnel’s term at normal incidence for a refractive index of 1.5 is 4.0 per cent, and a pedestal of 4.0 on a base of 3.0 gives 7.0 — against the 3.8 that the 11.3 implies and the 3.4 that 6.4 implies. The essay’s own physics points at about seven per cent, its two consistent numbers point at 6.8, and the odd figure points at 6.4.

That the 6.8 falls a little short of the full four points is expected and is worth saying rather than rounding away: a sphere’s detector at eight degrees does not collect the entire specular component, the finish is not perfectly specular, and a gloss trap that is closed still has ports in it. A pedestal of 3.8 out of a possible 4.0 is a sphere catching 95 per cent of the interface term, which is a more informative statement than either version and is available from the numbers already printed.

What the two categories are worth against each other #

The essay’s sharpest claim is categorical — a pedestal is not a change of light — and it is supported by one comparison stated in two different vocabularies. Putting both in the same units makes the size of the category difference visible.

A fitted per-channel gain removes 10.9 per cent of the geometry difference. An unfitted gain removes ninety per cent or more of every multiplicative change in this collection’s census. So the additive term is about eight times more resistant than the multiplicative ones, and it resists a correction that has been given more freedom rather than less.

That factor of eight is the essay’s whole argument reduced to a number, and it is worth quoting because no adaptation removes it is a claim that invites the reply how much does one remove? The answer is a ninth of it, from a gain fitted over the whole sample set — which is more than any observer gets and still leaves the pedestal almost intact.

The sphere’s own correction, sized #

One more ratio from the numbers already given, since it bears on how seriously the substitution error should be taken.

The raw sphere error reaches 2.91 and one pass of the correction leaves 0.156 — a factor of 18.7. So the correction is not a refinement; it is the difference between an instrument that is twice out of an ordinary tolerance and one that is a sixth of the way into it.

Read beside the geometry difference the two effects are very unequal. The geometry costs 8.35 on the dark sample and the uncorrected sphere costs 2.91; corrected, the sphere costs 0.156 and the geometry still costs 8.35. A laboratory worrying about which of the two to control has one answer: the correction is arithmetic and already done, and the geometry is a choice nobody wrote down.

The essay’s own observation that the sphere’s residual is largest for dark samples — where the geometry difference is also largest — is therefore true and small. Both errors concentrate at the same end of the sample set, and at that end one of them is fifty times the other.

A pedestal is not a change of light #

Every other change of context on this site is multiplicative. A lamp, a wall, a filter, a sheet of paper: each multiplies the spectrum, and the only question is whether the multiplication is diagonal in some set of axes.

A pedestal is not of that form. It adds. And an addition has a property no multiplication has: the correction that removes it for one sample is the wrong correction for every other, because what it did depends on how much the sample had to begin with.

That is why the census of light changes has no row for this. A gloss pedestal is not a change of illumination, it commutes with nothing, and no adaptation of any speed or completeness removes it — because a gain applied to a sum is not the sum of the gains.

The practical form is that a reader looking at a glossy print is not doing what either instrument does. They catch some fraction of the interface component depending on where they are sitting, they move, and the fraction changes. There is no geometry a person has. That is the same absence an aperture measurement and an eye already ran into from the spatial side.

What it does to a colour, and what it does to a difference #

The two geometries disagree about a sample’s colour, which is the measurement above. They also disagree about the difference between two samples, and that second disagreement behaves differently.

Two dark samples measured with the specular included both have the same pedestal added. A pedestal added to both members of a pair does not cancel out of the difference between them, because CIELAB is not linear in tristimulus values: adding four points to a three per cent sample and to a five per cent sample moves them by different amounts of lightness, so the pair is compressed.

The size of that compression on this site’s tolerance pairs is small — the pairs are constructed at mid reflectance where the pedestal is proportionally slight — and the direction is systematic. A sphere with its specular included reports dark pairs as closer together than they are. For a quality-control process working on dark samples, that is a test biased toward passing, which is the same direction a change of light biases most pairs and for a different reason.

The reverse holds for 45°/0° on a glossy sample, where the geometry throws away a component the reader will see. Neither instrument is measuring what a person judging the goods is looking at, and the two err in opposite directions.

A mid-grey sample is the one a control strip actually carries, and the pedestal on it is what a tolerance would have to accommodate.

The number a tolerance would have to carry #

It is worth working one case through, because the sizes make the abstraction concrete in a way the table does not.

A dark grey automotive trim part is specified at ΔE00 1.5 against a master, with the illuminant named as D65 and the observer as 10°. The supplier measures on a sphere with the specular included, because that is the instrument on their bench. The customer measures at 45°/0°, because that is the instrument on theirs. Both are calibrated, both are within their own repeatability, and both are measuring the same part.

At three per cent reflectance the two geometries disagree about that part’s colour by ΔE00 8.35 — five and a half times the tolerance. Nothing is wrong with either instrument and no amount of care with calibration closes it, because they are reporting two different physical quantities: one includes the four per cent the surface returns at its interface and the other does not.

In practice this does not usually produce an argument, and the reason is worth noticing. Both parties measure the master on the same instrument they measure the part on, so the pedestal is present in both readings and largely cancels out of the difference. The specification survives on a cancellation nobody wrote down.

Where it stops cancelling is where it costs money. A master and a part with different finishes — a moulded part against a painted master, a textured surface against a smooth one — carry different amounts of interface component, and the cancellation is partial. So is a comparison across a batch where the gloss varies, which is the ordinary case for an injection-moulded part.

The field the document needs is one word long: the geometry. It is in the standards and it is often absent from the purchase order.

Who found it, and when #

The two geometries are in the CIE’s recommendations from 1971, along with two more that see less use, and the reason for having several was understood at the time: a sphere measures what a diffusely lit sample returns overall, and a 45°/0° instrument measures what a person looking at a matte sample under a directional light sees. They answer different questions and both questions are real. What the instrument reports was never meant to be a single number.

Substitution error is older still and is why comparison-mode instruments exist. Modern single-beam spheres correct for it arithmetically, and the iterative form of the correction is standard.

What is not standard, as far as this site can tell, is putting the two beside a census of multiplicative changes and observing that the interface term is categorically different from all of them. That framing comes out of asking what an observer can do about each, which is a question instrument standards do not ask because instruments do not adapt.

What was computed, and how #

The interface term is unpolarised Fresnel reflectance at each geometry’s own angle, computed on a Cauchy dispersion, scaled by the fraction of it that leaves in the specular direction — a property of the finish rather than of the pigment, and the only place in the calculation where a surface’s texture appears.

The sphere is a wall at 0.98 reflectance with a sample port at 1.2 per cent of the area and other ports at 4.5 per cent, calibrated against a 0.99 standard. Its gain is the closed-form equilibrium of a Lambertian cavity, and the exact inverse is used for the corrected case so that the corrected reading is exactly right rather than converged to a tolerance.

The sample set spans six reflectance levels and three centre wavelengths, with the coloured members scaled so their mean reflectance is the stated level — which is what makes a coloured sample at six per cent comparable with a neutral at six per cent.

Where it stops #

The dichromatic model splits a surface into a body term and an interface term, and real surfaces are not always well described that way. Metallic and pearlescent finishes have a directional body component; translucent samples lose light sideways out of the measured area, which is a third geometry effect this file does not model at all.

The sphere’s parameters are plausible and constructed. The magnitude of the substitution error scales with the sample port’s share of the sphere, and an instrument with a smaller port has a smaller error; the shape of the result — largest for samples least like the calibration standard — is not a property of the choice.

And the pedestal is treated as spectrally flat. Fresnel’s term is nearly flat and not exactly: dispersion tilts it by about a tenth of a per cent across the visible band, which is small enough to be invisible next to the effect and is included rather than assumed away.

There is also a question this essay does not settle and should not be read as settling: which geometry is right. The honest answer is that neither is, because the question is malformed. A sphere with the specular included reports what a sample returns in total, which is the physically complete quantity and is what a formulator mixing a paint needs. A 45°/0° instrument reports what a detector at one angle sees, which is closer to what a person looking at a matte sample under a directional light gets. The first is a property of the material and the second is a property of a viewing situation, and a specification has to choose which of the two it is about — and then say so, which is the part that fails.

Where the ladder goes next #

If a measurement has a geometry, then every quantity built on measurements has one. A tolerance is the obvious case: the two geometries differ by more on a dark sample than any change of light in this site’s census does, so a specification that names an illuminant to four digits and leaves the geometry implicit has left the larger term unstated.

The other direction is the reader. An instrument catches all of the interface term or none of it; a person catches a fraction that changes as they move. That fraction is a quantity nobody measures and everybody depends on, and it is the same missing variable as the geometry a coated part is installed in.