The chart was measured by an observer too

A camera profile is fitted against numbers, and the numbers came from an instrument computing through a table. Every departure this round measures is therefore already inside every profile, unexamined.

The claim #

A camera profile’s fitted residual is much smaller than the observer uncertainty in the target it was fitted to, and the two are never reported together.

• A profile’s chart residual is around 0.19 ΔE₀₀ on the patches it was fitted on, and around 1.65 on colours off the chart.
• The observer term on those same patches is about 3 ΔE₀₀ at the median for a saturated one, and up to 8 at the ninety-fifth percentile of a surface family.
• The target is not a colour; it is the chart’s spectrum integrated through the 1931 tables, so it inherits everything the tables are a contract about.
• And a camera is a fourth observer with its own departures, so the profile is reconciling two constructions and reporting the residual of one.

What a profile is fitted to #

The procedure is standard and worth writing out, because the observer enters at a place that is easy to miss.

A calibration chart is measured spectrally. Each patch’s reflectance is integrated against a declared illuminant and the 1931 colour-matching functions to give a target XYZ. The camera photographs the chart under a light; its raw values are read; and a matrix — usually three-by-three, sometimes with a polynomial — is fitted to carry the camera’s numbers onto the targets in a least-squares sense.

The chart was measured, not photographed established that the target side comes from an instrument rather than from the same exposure, with all the consequences that follow. What this round adds is what the instrument’s number is: a projection of a spectrum through a contract.

So the fit’s objective is not reproduce what a viewer sees. It is reproduce what the 1931 observer computes, and the difference between those two is what the audit measures.

The residual and the term it hides #

A camera profile is a fit and this collection has measured its residuals: about 0.19 ΔE₀₀ on the chart’s own patches and 1.65 on colours the chart does not contain, with the chart’s chromatic range deciding both.

Set those beside the observer term. On an ordinary saturated sample the six departures combine to about three units at the median and eight at the ninety-fifth percentile of a surface family. Under a tungsten lamp the same combination roughly doubles.

The fit’s residual is fifteen times smaller than the uncertainty of its own target. That is not a criticism of the fitting — the fit is doing what it was asked, precisely — but it says something about what further precision in it is worth. A profile improved from 0.19 to 0.10 has improved a term that is a twentieth of another term nobody reports.

The comparison is not quite apples to apples and the difference matters. The residual is an error against a stated target; the observer term is a spread between the target and a viewer. Reducing the first improves agreement with an instrument and reducing the second is not available at all.

It is not simply an error to be added. A tempting move is to add the two in quadrature and report a total, and it is wrong for the same reason the tolerance essay gives.

The observer term is not an uncertainty in the profile. The profile is exactly right about its target and the target is exactly what the contract says. What the term measures is the gap between the contract and a viewer, and that gap is present whether or not a camera is involved — it is there when a colour is specified, when it is measured, and when it is proofed.

So the honest statement is that a camera profile inherits a term it did not create and cannot remove. Improving a profile does not improve it and cannot make it worse, which is a useful thing to know when deciding where to spend effort on an imaging pipeline.

The camera is a fourth observer with its own departures #

There is a second observer in the arrangement and this collection has been careful about it from the start.

A camera is a fourth observer: three sensitivities collapsing a spectrum onto three numbers, exactly as an eye does, and they are different three sensitivities. Luther said when a camera could be a colorimeter — its sensitivities have to be a linear transform of the observer’s — and no real sensor satisfies it.

That failure has the same shape as the observer departures in this round. A camera and the standard observer disagree about which pairs of spectra match, so a matrix fitted on one set of samples does not transfer to another, which is precisely why a matrix fitted under one light is not the matrix for another.

What is new is that there are now three parties rather than two: the camera, the standard observer, and the viewer. The profile reconciles the first two and nobody reconciles the third, and the third is the one who looks at the photograph.

The ladder is worth reading as a list of things a profile silently commits to. A profile fitted against 1931 targets is, in effect, a profile fitted for a two-degree field, a young lens, a median macula and median cone peaks — five choices nobody made and none of which appears in a profile’s metadata.

That has a consequence for how a profile ages, in a sense that has nothing to do with the sensor. A photographer using the same camera and the same profile for thirty years is being served, unchanged, a projection through an observer they have been diverging from at a rate of about half a colour difference a decade. The profile does not drift and the person does, and the two are indistinguishable from inside a colour-managed workflow.

What a spectral target would change #

There is an arrangement that removes the observer term from the profiling step and it is worth being precise about what it does and does not fix.

If the profile is fitted to make the camera’s numbers reproduce the chart’s spectra — through whatever observer is wanted at the time — then the fit’s target is observer-free and the observer is applied afterwards. That is how the better camera-characterisation methods work, and it means a single characterisation can be re-projected through the 1931 observer, the 1964 one, or a personalised one.

What it does not fix is that the camera still fails the Luther condition, so the mapping from its numbers to any observer’s is approximate and the approximation depends on which observer is chosen. A camera characterised spectrally can be projected onto a seventy-year-old’s observer, and the projection will be worse than onto the standard one, because the fit was never told about it.

Spectral characterisation makes the observer a choice rather than a constant, and that is a genuine improvement. It does not make the camera into a colorimeter.

What this means for a photograph of a metameric pair #

The sharpest consequence is about a case a photograph handles badly and everybody blames on the camera.

Two samples that match for the standard observer and differ spectrally are a metameric pair. A camera photographing them records different numbers, because its sensitivities are different, and the profile — fitted to reproduce the standard observer — maps both onto nearly the same target only if the fit happens to be accurate there.

Meanwhile a real viewer sees the pair as matching or not according to their own observer, and two normal observers can disagree about a metameric match by three colour differences.

So a photograph of a metameric pair is being asked to reproduce a match that is not a fact about the samples. The camera fails, the profile fails, and a viewer would have failed too — differently. That is not a defect anybody can engineer out, and it is the reason colour-critical matching is done with the samples rather than with photographs of them.

The validation trap #

That figure is the practical warning and it applies to profile validation as much as to tolerance validation.

Profiles are commonly checked on neutral wedges and grey ramps, because those are what a photographer cares most about and what a fit is most likely to get subtly wrong. Those are exactly the samples on which every observer computes the same colour, so a validation restricted to them is a validation where the inherited term is zero.

A validation on saturated samples would show it, and would show it as a spread rather than as an error — the profile would be right against the instrument and different viewers would disagree about the result.

The general form: a validation set chosen for what is hard to fit is not a validation set chosen for what is uncertain to specify, and the two point in opposite directions here. Neutral wedges are hard to fit and certain to specify; saturated pigments are easy to fit and uncertain to specify.

A profile could carry the allowance. The proposal is the same as the instrument’s and it is cheaper here, because a profile is a file rather than a reading.

A profile is fitted against spectra, or against colorimetric values derived from spectra, and either way the spectra exist at fitting time. So the observer allowance for each chart patch is computable during profiling and could be stored — a per-patch number saying how much observer spread that colour carries.

What a renderer or a soft-proofing application would do with it is the interesting part. A soft proof showing a saturated brand colour could indicate that the colour carries three units of observer spread and a near-neutral one that it carries none, which is exactly the information a designer choosing between two candidate colours would want and cannot currently obtain.

A brand colour is an ink and the choice between two candidate inks is usually made on gamut, cost and lightfastness. Observer robustness is a fourth criterion, it is computable from the ink’s spectrum, and it varies by an order of magnitude between candidates.

What was computed, and how #

The profile residuals quoted — 0.19 ΔE₀₀ on the chart and 1.65 off it — are this collection’s own, from its camera-profile work, measured on a fitted nine-parameter matrix with held-out validation.

The observer term is this round’s, over forty-two analytic surfaces, combined in quadrature over six departures at their literature strengths with the rod term excluded for a photopic assessment. It is a spread between two constructed observers rather than a population statistic, and the two are different objects.

The two numbers come from different test sets — a chart’s patches and a family of analytic reflectances — so the factor of fifteen is an order-of-magnitude comparison rather than a ratio measured on one set.

There is a third inherited term and it belongs to the round’s first half. A chart’s spectra are measured at whatever interval the instrument reports, integrated against tabulated illuminant and observer functions at the same interval, and the tabulation carries its own decisions — a step, a range and an origin.

For the smooth lights a profiling illuminant is normally declared under, that term is hundredths of a unit and irrelevant beside the observer’s. For a chart measured or used under a fluorescent tube it is up to a unit, which is five times the fit’s residual and worth more attention than the fit.

So a camera profile inherits three contracts: an observer, an illuminant table and a tabulation. Its residual is quoted against none of them.

Where the model stops #

The observer departures are computed for reflective samples under stated lights, and a photograph is often of a scene containing sources, highlights and transparencies. Nothing here says what the term does to those, and a highlight carries the lamp’s spectrum rather than a surface’s, which puts it near one of the identities.

The camera side is taken from this collection’s existing work and is not recomputed. A different sensor with different sensitivities would have different residuals and the same inherited term, since the term is a property of the target rather than of the camera.

And nothing here measures a viewer looking at a photograph. That involves a display, its primaries, its own observer term, and an adaptation state quite different from the scene’s — which is several more terms than this round has measured.

One last observation about where the effort in imaging colour actually goes. The camera-characterisation literature is largely about improving the fit: better basis functions, polynomial terms, root-polynomial forms that are exposure-invariant, and careful attention to the conditioning of the matrix. All of that reduces a residual that is already a twentieth of an unreported term.

Meanwhile the terms that dominate — the observer, the illuminant table, the tabulation — are all inherited from the target and none of them can be improved by a better fit. The only move that touches them is to characterise spectrally, which makes them choices applied afterwards rather than constants baked in.

That is a clear recommendation and it is not a new one; spectral characterisation has been advocated for thirty years on other grounds. What the audit adds is the size of what it buys, which is larger than everything the fitting literature has achieved.

The generalisation #

The habit is about the precision of a fit against the uncertainty of its target.

A least-squares fit reports a residual, the residual is the natural measure of how well the fit is doing, and it says nothing at all about whether the target was worth hitting that precisely. A fit whose residual is far below the target’s own uncertainty has converged on a number rather than on a truth, and further effort on it is wasted in a way the residual cannot report.

The move is to put the two side by side before optimising, which requires knowing the target’s uncertainty and is the step usually skipped — targets arrive as numbers and numbers do not carry error bars.

The failure mode is to treat a small residual as evidence of a good model. A residual measures agreement with a target, and agreement with a target is only as good as the target, which is a sentence that sounds obvious and is contradicted by the effort distribution of most calibration work.

A short note on scale, because the numbers here can be read as more alarming than they are. Three colour differences of observer spread on a saturated sample is a real quantity and it is a spread between people, not an error in the photograph. Two viewers looking at the same print disagree by that much whether or not a camera was involved, and they disagree by the same amount looking at the original object.

So a photograph is not made worse by the term; it inherits the same term the object had. What changes is that a photograph invites a comparison — against the original, against a proof, against a memory — and a comparison is where a spread becomes a complaint.

Who found it, and when #

Camera characterisation by least-squares fitting against a measured chart is standard practice and dates from the earliest digital colour work. The distinction between fitting to colorimetric targets and fitting spectrally is discussed in the camera-characterisation literature of the 1990s and 2000s, and spectral methods are used where the application demands re-projection.

The observation that the colorimetric target carries an observer uncertainty larger than the fit’s residual does not appear to be made in that literature, presumably because the residual is what a practitioner can act on and the other term is not.

Where the ladder goes next #

The audit’s terms reach further still. This collection’s largest single body of computed results is an adaptation census over a hundred and twenty-five surfaces and fourteen changes of light, and every number in it was computed through one observer.