A photograph is not a measurement

Photographs settle arguments about colour every day. A supplier is asked whether the delivered fabric matches the sample; a customer photographs the two and sends the picture. A repair is judged against an original. A print is approved from a screen showing a photograph of a proof.

This field has spent fourteen essays establishing what is in that photograph, and it is time to say what it supports.

Each of the other decisions in that chain can be lit up the same way, and three of them are worth seeing on their own.

Two of the remaining stages are decisions a photographer would not call decisions at all, and both are made before anybody sees an image.

The claim #

A photograph supports claims of the form these two things in this frame differ far better than claims of the form this thing is that colour — and it is routinely used for the second.

The reason is not that cameras are poor instruments. It is that the pipeline’s uncertainties are largely common to the frame, so they cancel in a comparison within it and accumulate in an absolute statement.

The eight limits, in one place #

Each has an essay and each contributes differently, so it is worth having them together.

The kernel is wrong and unpublished. A camera’s sensitivities are not a linear combination of the colour-matching functions — the residual is 31.7 per cent — and they are proprietary, so a user cannot compute what the camera would have said about a known spectrum.

It merges distinctions and invents them. Two surfaces the camera records identically can be ΔE00 2.4 apart to a person, and two a person cannot separate can be recorded 9.7 per cent apart in raw. Neither is detectable from the file.

The matrix is a fit to somebody else’s surfaces. Fitted on desaturated patches and used on saturated ones, the mean error nearly triples, and the sample set is not recorded.

And no matrix would be exact anyway. The best achievable worst case is ΔE00 2.85, against a control that reaches $10^{-7}$.

The illuminant was guessed. No estimator wins on every scene, and the one with a physical basis rather than an assumption is not the one in the camera.

Two thirds of every pixel is invented. A black-and-white edge comes off the mosaic with a chroma of 144 per cent of its local mean.

The bright parts may not have been measured at all. Once one channel clips the recorded hue rotates by up to sixty-seven degrees, and every recovery is a reconstruction.

And the correction costs noise. The full matrix amplifies photon noise by 1.66, which is why cameras back it off in poor light and therefore render colour differently at different sensitivities.

Why a comparison within one frame survives most of it #

Take two surfaces photographed side by side, in one exposure, through one lens, under one light, processed identically.

The kernel is the same for both. The white balance multipliers are the same. The matrix is the same. The tone curve is the same. So a claim that one is darker, or greener, or more saturated than the other is a claim about a difference that most of the pipeline applied equally to both — and a monotone transform applied to two quantities preserves their order.

What does not cancel is the metamerism, because that is a per-spectrum failure rather than a per-frame one. If the two surfaces happen to be a camera metamer, the comparison is wrong, and nothing in the frame reveals it. That is the residue, and it is small in ordinary cases and unbounded in principle.

Why an absolute claim inherits everything #

A statement that a surface is a particular colour needs every stage to be right, and each contributes.

The white balance has to be correct, which means the illuminant estimate has to be correct, which it usually is not — this is the largest term by a wide margin, and on a scene that breaks the estimator’s premise it can reach tens of degrees of angular error. The matrix has to be right for that surface, which depends on whether the surface resembles what the matrix was fitted to. Nothing may have clipped. The tone curve has to be inverted, which requires knowing which one was applied.

And underneath all of that, the surface must not be one the camera’s kernel handles badly, which cannot be checked without the kernel.

Every one of those is an unknown, and they compose. The result is not a number with an error bar; it is a number whose error is dominated by decisions nobody recorded.

What to do instead, and it is not complicated #

Put a reference in the frame. A known chart photographed beside the unknown converts an absolute question into a relative one. It does not repair metamerism — a chart cannot fix a null-space mismatch — and it removes the white balance, the matrix, the tone curve and the exposure in one step, because all four apply equally to the chart and to the subject.

This is standard practice in conservation, forensics and reproduction photography, and it is standard for exactly this reason, though it is usually justified as “so the colours can be corrected” rather than as “so the claim becomes relative”.

Shoot raw and keep it, so the six decisions stay reversible and can be revisited when better profiles exist.

And know the list of surfaces the camera is worst on — saturated dyes, deep reds and violets, anything under a narrow-band lamp, anything fluorescent, any photograph of a screen. A photograph of one of those is weak evidence about colour and a photograph of a pale matte surface under daylight is fairly strong evidence, and the difference between them is much larger than most people expect.

What a photograph is better at than an eye #

The field would be misread as a complaint if it stopped at the limits, and there are three respects in which a photograph is straightforwardly better evidence than a person’s report, which is the comparison that actually matters.

It is repeatable. The same file opened twice gives the same numbers, and two people can examine it. A person’s colour judgement is not repeatable across sessions, across adaptation states or across observers, and observers differ from each other by more than the two standard observers differ.

It is a record. A photograph of a surface can be compared against a photograph taken five years later; a memory of a colour cannot be compared against anything. Human colour memory is poor, systematically biased towards prototypical values, and unavailable for inspection.

And it is linear before it is rendered. A raw file’s numbers are proportional to photon counts to a fraction of a per cent, which is a far better linearity than the eye’s compressive, adapting, surround-dependent response achieves anywhere. Ratios in raw mean what ratios mean.

So the honest position is not that photographs are unreliable and people are reliable. Both are unreliable in different ways, and the photograph’s unreliability is at least written down somewhere — in a specification, a profile, a converter’s source — while the observer’s is a distribution nobody has for the observer in question.

The one comparison that is genuinely hard #

There is a case that fails both ways and it is worth naming, because it is the case that generates most colour disputes: two surfaces photographed in different frames.

A sample photographed in a studio and a delivery photographed in a warehouse share nothing. Different illuminants, different white-balance estimates, different exposures, possibly different cameras. Every term that cancels in a within-frame comparison is live, and the two pictures can differ substantially with the surfaces identical.

Nothing in the two files reveals this. Both look like photographs; both have plausible colours; and the difference between them is dominated by two independent guesses about two illuminants.

The only repair is to photograph both against the same reference in each frame, which restores the comparison by making each frame’s absolute claim relative to a shared object. It is the same repair as before and it has to be applied in both frames, which is why the discipline that does this seriously — colour tolerance specification — does not use photographs at all and sends the physical samples instead.

How much of the exposure a shutter speed decides can be read directly, and a quarter of a thousandth of a second is a shutter a daylight frame actually uses.

The eight terms, added up #

Every one of those is an unknown, and they compose is the essay’s summary and it stops short of the arithmetic, which is available because most of the eight are already in one unit.

The white balance is quoted in degrees rather than in ΔE₀₀, and converting it is what makes the list comparable. On a neutral surface, an angular error of one degree costs about 3.2 ΔE₀₀; a good estimator’s 2.5 degrees costs 7.2; and the forty-three degrees the estimator reaches on a green scene costs about 32.

Set beside the others:

The white-balance term is 2.5 times the next largest when the estimator is working and eleven times it when the estimator is not, and on a well-behaved scene it exceeds the other two combined by 37 per cent. So this is the largest term by a wide margin is right on both kinds of scene, and the margin is narrower than it sounds on the ordinary one.

These are systematic terms and adding them is the right operation rather than quadrature — two of the departures this collection measures elsewhere interact by more than the smaller of them, so neither adding nor quadrature is guaranteed, and adding is at least the conservative reading for terms whose signs are unknown.

What the reference in the frame is worth #

The recommendation to put a chart in the frame can now be priced, and its value is not the same on the two kinds of scene.

A within-frame comparison keeps only the metamerism, at 2.40, because that is the one failure that happened at the sensor rather than downstream. So a reference removes 81 per cent of the error on a well-behaved scene and 94 per cent on one that breaks the estimator.

And the remainder is the same 2.40 in both cases. That is the sharp version of what the chart does: it does not reduce the error proportionally, it replaces the whole variable part with a fixed floor. A careful photographer under good light and a careless one under a canopy of leaves end up at the same place, which is a much stronger recommendation than “it helps”.

It also says where further effort stops paying. Once the reference is in the frame the error is metamerism and nothing else, and metamerism is not improved by better lighting, a better estimator, a better profile or a better camera of the same design. The floor is a property of the sensor’s null space, and the only things that move it are different filters or more channels.

Why the ordering makes the field’s advice the advice it is #

Reading the terms in order explains why every practical discipline that takes this seriously arrived at the same two rules, and why they are in that order.

Rule one is a reference in the frame, because the largest term by a factor of two and a half is a guess about the illuminant and a reference removes it outright.

Rule two is shoot raw, because the second and third largest are a matrix and a profile, both of which are decisions that can be revisited later — and only if the numbers before them survive.

There is no rule three, and the arithmetic says why: after those two the residue is 2.40 ΔE₀₀ of metamerism, which no procedure available to a photographer touches. A third rule would have to be choose a different camera, and no discipline gives that advice because no consumer camera publishes the kernel that would let anybody choose.

That is the useful shape of the field’s conclusion. Two of the eight limits are removable by procedure, five are removable by keeping the raw file, and one is not removable at all — and the one that is not is the smallest, which is why photography works as well as it does and why it stops working exactly where it does.

Where the model stops #

Nothing here is about whether a photograph is a good picture, which is a different question with a different answer. Everything in this field is about colorimetric fidelity, and a photograph optimised for fidelity is usually worse to look at than one optimised for appearance — the tone curve alone is evidence of that, and it is applied deliberately.

The numbers are from a modelled sensor. A real camera’s residual, its metamers and its matrix are its own, and published measurements of real sensitivities exist for a small fraction of cameras. What survives the model is the structure and the ordering of the terms.

And the argument is about colour, not about geometry, resolution or content. A photograph is excellent evidence about shape, position, texture and the presence of things. The claim here is narrow and is about one channel of what a picture carries.

The generalisation #

The transferable statement is one this collection has been circling from its first phase, and this field is where it comes out plainly.

An instrument’s output is trustworthy in proportion to how much of its uncertainty is common-mode to the comparison being made. A within-frame comparison shares almost everything; a between-frame comparison shares less; an absolute claim shares nothing and inherits every term.

That ordering is why so much practical measurement is comparative — nulling methods, differential measurements, matched pairs, controls run alongside — and why the comparative version of a question is often answerable to a precision the absolute version cannot approach. The skill is usually in converting the question rather than in improving the instrument.

The second half is about disclosure. Every limit in this field is a decision somebody made, and almost none of them is recorded in the artefact. A raw file has a field for the matrix and none for the sensitivities it was fitted from; a rendered image has neither. An instrument that does not record its own settings produces a measurement that cannot be interrogated, and the fact that it produces a picture rather than a number makes that very much easier to forget.

This site’s own practice is the same rule applied to itself. Every figure here is generated from a stated rule, every claim is given a test it could fail, and every quantity that depends on a choice names the choice in the caption. The reason is not tidiness. It is that a rendering whose decisions are not recorded cannot be distinguished, later, from a measurement.

The field’s shape, in one sentence each #

Fifteen essays is enough that the argument is worth restating as a structure rather than as a list, because the structure is what makes the limits predictable rather than a catalogue of misfortunes.

A camera is an observer. Three functions of wavelength, one integral each, the same collapse an eye makes. Everything follows from that and from the functions being different functions.

Different functions mean different null spaces, which means metamers in both directions and no matrix that repairs either. That is Luther’s theorem and it is the field’s spine.

Different null spaces also mean a fit, which means a sample set, which means an error concentrated wherever the sample set was thin — reliably, the saturated and spectrally narrow surfaces.

And the fit costs noise, because correcting a kernel mismatch requires subtracting one channel from another and differences of large numbers amplify uncorrelated error.

Alongside that spine sit three independent problems: the sensor sees three hundred nanometres too far, the mosaic samples the channels in different places, and the well fills. None of the three has anything to do with the others or with Luther’s condition, and each produces colour that was not in the scene.

The value of having the structure rather than the list is that it predicts. Anybody meeting a new imaging device can ask the four spine questions — what are its three functions, do they span the right subspace, what was its matrix fitted to, and what does the correction cost — and expect the answers to have the shapes this field found. The three independent problems have to be asked about separately, and they are the ones most likely to differ.

Who noticed, and when #

The distrust is older than the camera. Debates about photographic evidence in courts begin in the eighteen-fifties, and the objections raised then — that the photographer chose the angle, the exposure, the plate and the development — are the same objections in a different vocabulary.

What changed with colour is the number of choices and their invisibility. A nineteenth-century plate’s decisions were physical, few and made by a person present at the scene. A digital photograph’s are numerous, automatic, taken by software, and made after the fact by whoever opened the file. The person who chose has usually never seen the subject.

The systematic response has come mostly from outside consumer photography. Conservation imaging has developed formal protocols with reference targets and reported uncertainties; forensic photography has standards for what may be asserted from an image; and remote sensing built an entire discipline around calibrated radiometry precisely because nobody could photograph the ground and ask it what colour it was. Each arrived independently at the conclusion of this essay, and each expresses it as a procedure rather than as a claim about instruments.

Where the ladder goes next #

This is the top of the capture ladder and the end of the field, so it goes sideways rather than up.

It joins what the instrument reports, which is the same argument about a spectrophotometer — an instrument built for this job, whose limits are published, and which still cannot be read naively. It joins a hex code is not a colour at the other end of the pipeline, where three numbers arrive at a display having lost the same information in a different way. And it joins the display is an unknown, which is this field’s mirror image: a photograph is a measurement by an unspecified instrument, and it is then shown on one.

Taken together those four say something the individual essays do not. The chain from a surface to a reader’s retina passes through two unspecified instruments and half a dozen unrecorded decisions, and every one of them is invisible in the picture that arrives.