The chart was measured, not photographed

A camera profile is a fit between two things: what the camera did, and what the chart is. The first is a photograph and the second is a measurement, and this round has spent four essays saying that a measurement is not the object.

The departure has a name and a place in a larger decision, and both are worth having before the mechanism.

The claim #

A camera profile is fitted against numbers an instrument produced, and the instrument’s departures go into the fit while the camera’s do not.

• The camera does not have the aperture problem. Its illumination is the scene, so its flat-field colour on a translucent patch is exact.
• The instrument does. Reading a mildly translucent chart through a four-millimetre radius gives tristimulus values 2.55 ΔE₀₀ from the truth on average and 3.59 at worst.
• The profile inherits that as a bias, not as noise, because every patch is read the same way.
• It is twice the profile’s own fitting error, which is 1.18 ΔE₀₀ on saturated surfaces.
• And the gloss departure, which was expected to matter, does not: adding a specular pedestal to the test surfaces moves the profile’s error from 1.184 to 1.136, which is an improvement.

What a profile is fitted against #

A camera profile in its simplest and commonest form is a 3×3 matrix carrying the camera’s raw response to tristimulus values. It is fitted by photographing a chart of known patches and solving least squares between the two sets of numbers.

The known patches are known because somebody measured them, usually with a spectrophotometer, usually at a 45°/0° geometry with an aperture of a few millimetres. Those measurements are the targets of the fit — the numbers the matrix is asked to reproduce.

This collection has taken that fit apart before: the matrix cannot be exact because the camera’s sensitivities are not a linear transform of the observer’s, which is what Luther’s condition says; the residual depends on which surfaces are in the chart, so the chart decides the profile; and the objective is least squares in XYZ, which is an objective nobody chose.

Every one of those is about the fit. This is about the targets.

What a chart is made of #

A colour chart’s patches are printed or painted on a substrate. Paper and plastic are both mildly translucent, and the pigment layer over them is thin.

Treating a chart patch as a slab with a reduced scattering of 6 per millimetre — between a pigmented plastic and a wall emulsion, and a fair description of an ink film on coated stock — and reading it through a four-millimetre radius gives tristimulus values that differ from the patch’s own by 2.55 ΔE₀₀ on average across a set of saturated surfaces, and 3.59 on the worst.

Make the patch more opaque, at a scattering of 20 per millimetre, and the gap falls to 0.73. Narrow the aperture to two millimetres and it rises to 5.26.

Those are not small numbers next to the thing being fitted. The profile’s own residual — the part Luther’s condition guarantees cannot be removed — is 1.18 ΔE₀₀ on the same surfaces. So on a mildly translucent chart, the error in the targets is about twice the error the fit is trying to minimise.

Why it is a bias rather than noise #

Every patch on the chart is measured by the same instrument through the same aperture, so the error is the same kind of error on all of them, and least squares does not remove it.

Worse, it is correlated with the thing being fitted. The aperture’s loss is largest where the reflectance is largest, so light patches lose more than dark ones and each patch loses most at the wavelengths it reflects best — an aperture is a filter. The result is a systematic desaturation and darkening of the target set, and a matrix fitted to it will map the camera’s response to slightly dark, slightly desaturated values, on every photograph it is ever used on.

That is the ordinary shape of a calibration error and it has an ordinary consequence: it is invisible in the residuals. The fit reports how well it reproduced its targets, and it reproduced them well, because the targets are self-consistent. A profile’s reported error says nothing about whether the targets were right.

The prediction that was refused #

The obvious companion expectation was that a matte chart used to profile a camera that will photograph glossy things would cost something too. Charts are matte by design, and scenes are not.

Adding a specular pedestal to the test surfaces — a Trowbridge–Reitz lobe at a roughness of 0.05, integrated for a detector eight degrees off the normal, which comes to 0.0399 of a reflectance unit spread flat across the band — moves the profile’s mean error from 1.184 to 1.136.

It improves it. And the improvement is nearly the same at every roughness tested, because the pedestal’s size barely changes: 0.0399 at roughness 0.05, 0.0391 at 0.15, 0.0340 at 0.35.

The reason is that a flat addition moves every surface towards the neutral axis, and a desaturated set is the easy case for a linear matrix. The residual a matrix cannot remove grows with chroma — that is why this collection tests profiles on saturated surfaces and fits them on mild ones — so making the test set less saturated makes the score better.

Refitting the matrix on a glossy chart and testing on glossy surfaces gives 1.135, which is the same number again. The gloss departure passes straight through a camera profile without touching it, and that is worth knowing because it is exactly the departure a photographer would expect to matter.

Three numbers, one law #

The three chart figures — 2.55 at scattering 6 and a four-millimetre radius, 0.73 at scattering 20, 5.26 at two millimetres — are not three measurements. They are one relation with two levers pulled.

Doubling the aperture radius divides the departure by 2.06; multiplying the scattering coefficient by 3.33 divides it by 3.49. Both are exponents of 1.04, to two decimals, so the departure is inversely proportional to each. The combination is therefore

departure ≈ 61 × (1/μₛ′) / a

— the transport mean free path divided by the aperture radius, times a constant of about 61 ΔE₀₀. On the three data points the constant comes out at 58.4, 61.2 and 63.1, an eight per cent spread with nothing fitted but its mean.

That is a more useful object than three examples, because it says what either remedy has to achieve rather than that it would help.

The target error falls below the profile’s own 1.18 residual when a·μₛ′ exceeds about 52. At a four-millimetre radius that needs a scattering above 12.9 per millimetre — which is why the essay’s opaque case at 20 lands at 0.73, and the law predicts 0.76. At a scattering of 6 it needs a radius above 8.6 millimetres, which is a 17-millimetre port.

So the two remedies are not equally available. A seventeen-millimetre port is larger than the measuring aperture of any hand-held instrument and is an integrating sphere’s business; a scattering of 13 per millimetre is a heavier ink load on a denser substrate, which is a manufacturing decision. The essay prefers the second and the law says why: the second is reachable and the first is not, on the instruments a chart is actually measured on.

The pedestal is the Fresnel term, and that is why roughness does not move it #

The gloss pedestal is quoted at three roughnesses — 0.0399, 0.0391 and 0.0340 — and its near-constancy is offered as an observation. It has a cause, and the cause identifies the number.

A dielectric of index 1.5 reflects 0.0424 of a band-averaged unpolarised beam at normal incidence. The pedestal at roughness 0.05 is 0.0399, which is 94 per cent of it; at roughness 0.35 it is 0.0340, or 80 per cent. So the pedestal is the Fresnel reflectance, less what the lobe scatters out of the detector’s eight-degree acceptance.

Roughness redistributes that energy rather than changing how much there is. A microfacet lobe integrates to the same Fresnel total whatever its width; widening it only spreads the same light over more angle, so a detector near the specular direction catches a slowly falling share. A sevenfold change in roughness costs 15 per cent of the pedestal, and it would cost nothing at all to a detector that collected the whole hemisphere.

That also explains why the essay’s expectation was reasonable and wrong in a specific way. A photographer expects gloss to matter because it varies — matte chart, glossy scene — and the arithmetic says the thing that varies is the angle, not the amount.

The improvement is in the test set, not in the matrix #

The gloss result has two numbers and the second one settles what the first means.

Testing the existing matrix on glossy surfaces gives 1.136 against 1.184, an improvement of 4.1 per cent. Refitting the matrix on a glossy chart and testing on glossy surfaces gives 1.135 — a further 0.1 per cent, which is nothing.

So the matrix has nothing to adapt to. The whole of the improvement comes from the test set being less saturated, and none of it from the fit finding a better mapping for a pedestalled world. That is what a flat additive term does to a linear fit: it moves every point by the same vector, which a matrix with no offset cannot exploit and does not need to.

Which makes the finding sharper than the gloss departure passes straight through a camera profile. It passes through the fit entirely; what it touches is the score, by making the surfaces being scored easier. A profile evaluated on glossy surfaces is being marked on a slightly kinder test, and the 4.1 per cent is a property of the marking rather than of the camera.

What was computed, and how #

The sensitivities are this collection’s own constructed sensor: silicon quantum efficiency, three dye transmittances and an infrared cut filter, none of them quoted.

The matrix is fitted by least squares on a set of mildly saturated surfaces and evaluated on a more saturated set, which is the arrangement an earlier round established to stop a profile from being scored on the patches it was fitted to.

The chart-as-measured calculation takes each test surface, gives it a kernel with a stated scattering coefficient, reads it through a stated aperture, and computes the tristimulus values of the reading. The comparison is between those values and the surface’s own — which is what the camera sees, since a camera photographing a chart under studio lights satisfies the wide-illumination condition.

The gloss calculation adds the computed pedestal to each surface’s reflectance and refits or re-tests. Because the lobe contains no body reflectance, the pedestal is one number rather than a spectrum, which is asserted rather than assumed elsewhere in this round.

Where the model stops #

The chart’s translucency is a stated assumption and not a measurement. Nobody publishes the scattering coefficients of a colour chart, and the two values used here bracket what an ink film on coated stock plausibly is. The conclusion is a conditional: if a chart is as translucent as a plastic sheet, the targets are 2.55 ΔE₀₀ out.

The instrument is modelled by its aperture alone. A real chart measurement also has a geometry, a bandpass and a substitution error, and those have their own signs.

And the fit is a 3×3 matrix. Real profiles are matrices plus lookup tables, and a lookup table can absorb a systematic bias in the targets exactly as well as it absorbs anything else — which means a table-based profile would reproduce the biased targets better, and be more wrong about the world.

What a chart would look like if it were built for this #

Two changes would remove the bias and neither is exotic.

Measure the chart the way the camera sees it. A reading taken with a wide illuminated area and a narrow detection area satisfies the aperture condition, so it returns the patch’s own reflectance rather than a reading of it. That is what an integrating-sphere instrument already does by construction, since it illuminates the sample through the sphere — either aperture being wide is enough, and a sphere’s illuminated area is very wide indeed.

Or make the patches opaque. At a scattering of 20 per millimetre the gap falls to 0.73 ΔE₀₀, which is below the profile’s own residual and stops mattering. A chart printed on a dense, heavily-loaded substrate is a better reference than one printed on ordinary stock, for a reason that has nothing to do with pigment stability.

The second is the easier of the two and it is not what chart manufacturers optimise for. The published specifications for reference charts talk about spectral stability, patch uniformity, gloss and inter-batch variation — all of which are about the patch’s reflectance being reliable, and none of which is about whether the measurement of it is.

Where the error goes in the picture #

A biased set of targets does not produce a uniformly biased photograph, and where it does go is worth stating because it decides whether anybody notices.

The bias in the targets is a desaturation and a darkening that is largest for the lightest and most saturated patches. A matrix fitted to those targets is therefore fitted to a set that is slightly compressed towards neutral, and it maps the camera’s response into a slightly compressed space — so every photograph taken with the profile is faintly flatter than the scene.

Faintly flat is precisely the failure nobody diagnoses, because it looks like a stylistic choice and because every stage of an imaging pipeline has its own opinion about contrast. A bias of 2.5 ΔE₀₀ distributed over a chart, folded into a matrix, and then followed by a tone curve is not something an eye will attribute to the chart’s substrate.

The generalisation #

The transferable rule is about calibration chains: a calibration transfers the reference’s errors into the instrument being calibrated, and the transfer is invisible to every check the calibration performs on itself.

The check a fit performs is that its residuals are small, and residuals are computed against the targets. Nothing in the procedure looks at whether the targets are what the instrument being calibrated would have seen — and here they are not, because the two instruments have different departures.

The general form of the question is worth carrying: were the reference values obtained by an instrument that has the same blind spots as the one being calibrated? If yes, the errors cancel and nobody needs to know. If no, they add, and the calibration is a mechanism for importing one instrument’s failures into another.

This is a case where the answer is unusually clean, because the camera and the spectrophotometer differ in exactly one of the four departures this round measures — and they differ in the direction that flatters the camera.

Which conditions each of the four vanishes under is the other half of the same table, and it is what says whether a chart could be measured differently.

Two instruments, one chain, four departures #

Setting the two side by side makes the whole argument compact.

Of the round’s four departures the camera has three and the spectrophotometer has four. They share the wavelength index, since a fluorescent patch fluoresces for both — though the instrument’s lamp is specified and a studio’s is not. They share the range, since it is a property of the arithmetic rather than of either device. They differ in the direction index, because the camera records what leaves in one direction from a scene’s own field while the instrument imposes its own geometry.

And they differ, decisively, in the place index. The instrument has it and the camera does not, because a camera’s illuminated area is the scene.

That single asymmetry is what makes a calibration chain between them lossy. If the two devices had the same departures, the chart’s errors would be errors the camera would also have made, and the fit would map one to the other correctly by accident. They do not, so the fit imports an error the camera would never have committed.

Who found it, and when #

That reference values carry the reference instrument’s errors is standard metrology and is the reason calibration hierarchies exist: each level’s uncertainty includes the level above it.

What is unusual in the camera case is that the two instruments in question are not on the same hierarchy at all. A spectrophotometer’s chain runs to a national standard through a series of spectrophotometers; a camera’s chain runs to a chart measured by a spectrophotometer, and then stops. So the camera inherits an uncertainty budget that was computed for a different measurement geometry, and the geometry is the thing that differs.

The photographic literature does discuss chart quality, and the discussion is about pigment stability, patch uniformity and gloss — all real, and none of them this. The translucency of the substrate is not a parameter anybody records.

What a photographer would notice #

Nothing, and that is the point worth ending on rather than a caveat to bury.

A profile biased by 2.5 ΔE₀₀ in a systematic direction does not produce a photograph anybody looks at and calls wrong. It produces one that is slightly flat, and flatness is the most heavily post-processed property of a modern photograph — every pipeline applies a tone curve, a saturation adjustment and a rendering intent after the matrix, and the slope arrives before the bowl in all of them.

So the bias is absorbed by the next stage and shows up as everybody’s profiles needing a little more saturation than the arithmetic says. That is exactly the shape of a defect that survives indefinitely: it has a workaround which is applied automatically, and the workaround is applied for other reasons anyway.

The only way to see it is to measure the chart twice and compare, which is a fifteen-minute experiment that nobody has a reason to run.

Where the ladder goes next #

The measurement that would settle it is small: read a colour chart at two apertures and report the difference. If it is 2.5 ΔE₀₀ the argument here is right; if it is 0.3 the substrate is more opaque than assumed and the whole concern is a rounding error.

The wider question is whether a chart should be measured the way a camera sees it. A chart measured with a wide-illumination geometry would supply targets the camera can actually reproduce, and it would disagree with every published measurement of the same chart — which is the difficulty every measurement condition has and the reason conditions have letters rather than being fixed once.