A departure is straight in the excitations

An identity says a quantity is zero somewhere. A sequence towards it says what the quantity is proportional to, and that is a much stronger claim — strong enough to be wrong, which is what makes it worth measuring.

The claim #

An observer departure is exactly proportional to how far the sample sits from the light it is seen under, and the published unit that reports it is not.

• The excitation distance is linear to two per cent along the whole walk, which is a structural claim about the pairing rather than an observation about one sample.
• The colour difference is not, and it departs by up to twenty per cent in either direction depending on the sample and the adaptation route.
• Both facts are useful and they are different facts. The first says what the departure is; the second says what the reporting instrument does to it.
• And the linearity survives every substitution tried. Two departures, two lights and two samples, and the straightness is unchanged while the slope moves by a factor of four.

What is being walked #

Take a sample’s reflectance R(λ) and a flat reflectance of the same average value, and mix them: Rₜ = ρ + t·(R − ρ). At t = 0 the sample is spectrally flat and every observer agrees about it exactly. At t = 1 it is itself.

The pairing says the departure between two observers is an inner product of the observer’s deviation with the stimulus’s deviation from the adapting white. The stimulus’s deviation is linear in t by construction, so the inner product should be linear in t as well — not approximately, but as an algebraic consequence of the pairing being an inner product.

That is a prediction with a shape, and it is testable in a way that the identity at t = 0 is not. An identity is one number; a sequence is five, and a departure that was not a pairing would produce a curve rather than a line, with no particular reason for the curvature to be small.

The measurement #

Two observers differing by two standard deviations of macular density, a red pigment under a 6500 K radiator, five mixing fractions.

The middle column against a straight line through the origin and the endpoint: 0.00407 against 0.00407, 0.00815 against 0.00815, 0.01630 against 0.01630. The largest deviation anywhere along the walk is under two per cent of the endpoint, and most of that is at the coarsest mixing fraction.

The middle column is what the pairing identity predicts and the right-hand column against the same test: 0.208 against 0.214, 0.347 against 0.428, 0.641 against 0.857, 1.073 against 1.286. At a quarter of the way along, the colour difference is nineteen per cent below proportionality.

The straight line establishes three things. The linearity is the strongest available evidence that the departure is a pairing, and it is worth being precise about what that buys.

It fixes the direction. A departure has a magnitude and, less obviously, a direction in the space of stimuli: the set of samples on which it is largest. The linearity says that direction is fixed and the magnitude scales along it, which is what an inner product does and is not what a general nonlinear function of the sample does.

It makes the identity a limit case rather than a special case. Without the sequence, the fact that a flat sample gives zero could be a coincidence of flatness. With it, the zero is the intercept of a line whose slope is the departure’s size, and every point on the line is the same phenomenon.

And it licenses extrapolation. A sample twice as far from the light as the one measured has twice the departure, in excitations, exactly. That is a strong claim and it is what makes the collection’s own numbers reportable as lower bounds rather than as isolated examples — this collection’s samples sit near the neutral end and the linearity says by how much they are understating.

Four checks rather than one. A single straight line is weak evidence. Four, across independent variables, are not.

The walk has been run for the macular departure and the lens departure, under a 6500 K radiator and a tungsten lamp, on a red pigment and on a notch filter. In all four the excitation distance is proportional to the mixing fraction to two per cent or better. The slopes differ by a factor of about four across the four cases, which is the size of the two factors of the pairing.

What varies is the slope and what does not vary is the linearity. That is the signature of a structure. A quantity that behaved linearly in one arrangement and not in another would be an accident of that arrangement, and the natural next step would be to find out which feature of it produced the accident.

The same logic was applied by the previous round to its own four departures, where the pairing was checked against a direct computation and agreed to a part in a thousand billion for three of the four. The check here is a different one — a scaling rather than an algebraic identity — and it reaches the same conclusion by a route that does not require the pairing to be written down.

What the curved line establishes #

The second column’s departure from linearity is not noise and it is not a defect in the measurement. It is the colour-difference formula, and it is doing what the formula is for.

ΔE₀₀ is a distance in CIELAB with three corrections applied: a lightness weighting, a chroma weighting that divides by 1 + 0.045C, and a hue weighting that divides by a function of hue angle and chroma. Near the neutral axis the chroma weighting is close to one, and it grows as the sample saturates — so a fixed difference in coordinates is reported as a smaller number on a saturated sample than on a pale one.

Walking a sample from flat towards its own reflectance is walking it from low chroma to high chroma, so the divisor grows along the walk and the reported difference grows more slowly than the underlying one. That predicts the sign of the observed curvature and it predicts it correctly: the measured 0.347 against a proportional 0.428 is the formula reporting less than proportionality at the point where the chroma weighting has begun to bite.

Under a different adaptation route the same walk curves the other way, and that is not a contradiction. The route decides the coordinates, the coordinates decide the chroma, and the chroma decides the weighting. The curvature is a property of the reporting arithmetic all the way down.

Reading the conditions figure as the walk’s starting point rather than as a separate result is the way to get the most out of both. Each of those six identities is the t = 0 end of a line whose t = 1 end is a bar in the ladder, and the line between them is what makes the pair a mechanism rather than two observations.

That framing also says what would refute the pairing. A departure that was zero on a flat sample and not linear on the way out would still have its identity intact and would have no inner-product structure — it would be some other function that happens to vanish at flatness, and the ladder would say nothing about any sample except the one it was measured on. The linearity is what licenses everything else in the round.

Which of the two numbers to publish #

There is a real choice here and this collection has made it in favour of the second, for reasons of comparability rather than of principle.

The excitation distance is the physical quantity. It is what the pairing is about, it is linear, and it is the number a model would predict. Its drawback is that nobody has a sense of scale for it: 0.033 means nothing to a reader, and there is no tolerance written in it anywhere.

The colour difference is the communicable quantity. Everybody has a sense of what one ΔE₀₀ means, every tolerance in industry is written in it, and every other number in this collection is quoted in it. Its drawback is that it is a nonlinear function of the thing being measured, so ratios between its values are not ratios between departures.

The rule this collection adopts is to publish in ΔE₀₀ and establish structure in excitations, and to say which is which. Every ladder, every distribution and every comparison in this round is in the published unit; every identity and every scaling law is in the physical one. Mixing them is what produces claims like “the departure doubles when the sample is twice as saturated”, which is true in one column and false in the other.

The paper’s walk is the one worth having for anybody who wants to know what a departure means in practice. A white sheet is about as close to the identity as a real material gets, and its endpoint at 0.35 ΔE₀₀ is below the tolerance most delivery specifications are written in. Every point along its walk is below that tolerance too, by the linearity.

So the practical reading of the scaling law is a rule of thumb about which samples need an observer allowance at all. A sample whose excitation deviation from the adapting white is below about a third of a saturated pigment’s will carry an observer departure below one unit for every parameter in the table, and a sample above it will not. That is a single measurement on the sample, computable from its own reflectance and the lamp, with no observer arithmetic in it at all.

The same line, read across the family #

The walk and the distribution are the same fact measured two ways, and putting them together closes a gap in the earlier essay.

The distribution over forty-two surfaces shows each departure spanning a factor of three to thirty-five, and it does not say why. The walk says why: the departure is proportional to the sample’s deviation from the light, so a family of samples spanning a range of deviations produces a distribution spanning the same range.

That is checkable and it very nearly checks out. The macular departure’s ratio between its largest and smallest member of the family is 16.6; the ratio of those two samples’ excitation deviations from the adapting white is 15.1. The gap between 16.6 and 15.1 is the unit’s curvature again, and it is in the direction the chroma weighting predicts.

So the distribution is not an unexplained spread of results. It is one linear law evaluated at forty-two points, plus a reporting nonlinearity of about ten per cent, and knowing that turns a scatter into a slope.

Adding a fifth combination is worth the space because it exercises the departure that is mostly a gain. If any of the six were going to break the linearity it would be this one, since most of what a density change does is absorbed by the white and what is left is a second-order broadening.

It does not break it. The excitation distance is proportional to the mixing fraction to the same two per cent, with a slope about a third of the macular walk’s, which is what a small residual departure looks like rather than a different mechanism.

Reading the distribution as a sampled line rather than as a scatter is the practical payoff of the walk, and it turns a spread into a slope with an intercept at zero.

What was computed, and how #

The flat reflectance mixed in is 0.5 rather than the sample’s own mean, which makes the walk a mix towards a stated grey rather than towards the sample’s average. Using the sample’s mean would have made the endpoint depend on the sample and the slopes incomparable across the four checks.

The excitation distance is the Euclidean distance between the two observers’ relative cone excitations, in the observer’s own cone space, where the identity holds exactly. Computing it in CAT16’s space would have mixed the linearity being tested with the route’s own nonlinearity.

The assertion carried by the figure family has two halves, and the second is written as a departure from linearity in either direction. The direction depends on the route and the sample, and a signed test would have been a test of the example rather than of the claim.

What a linear law is worth downstream #

A scaling law is more useful than a number and it is worth spelling out where.

It makes an observer allowance computable from the sample. A specification that wants to reserve tolerance for observer variation currently has to quote a constant, because nothing tells it how the reserve should depend on what is being specified. The linearity does: the reserve scales with the sample’s excitation deviation from the illuminant, which is one integral over the sample’s own reflectance.

It makes a worst case findable. Maximising a linear functional over a constrained set is a much easier problem than maximising a general one, and the constraint here — that a reflectance lies between zero and one — is the same constraint the optimal-colour work already uses. The surface that maximises an observer departure is therefore constructible rather than searchable, and it is a two-transition optimal colour.

And it makes the collection’s own numbers extrapolable. Every figure in this round is measured on samples nearer the neutral axis than a printing ink or a display primary, and the linearity says those numbers scale rather than saturate. A quantity that had turned over would have made the collection’s own examples an upper bound; a linear one makes them a lower bound with a known multiplier — the same correction the census had to make about its own test set, arriving through a scaling rather than through a recount.

Where the model stops #

The walk mixes towards a flat reflectance, which is a particular direction in an infinite-dimensional space. A walk towards a metameric grey would show something quite different, because a metameric grey is not the identity’s zero, and no version of that experiment has been run.

The linearity is checked to two per cent over five points. A finer walk would resolve whether the residual is a genuine second-order term or the numerical noise of the comparison, and the residual’s size is consistent with either.

And nothing here says the departure between two real observers is linear in anything. The construction differs from the reference in one parameter at a time by design, and a pair of real eyes differs in all of them at once with correlations the model does not carry.

The generalisation #

The habit is about testing a structural claim by scaling rather than by cases.

An identity at one point is weak evidence for a mechanism because many mechanisms have zeros. A scaling law is much stronger: it says what the quantity is proportional to, it can be checked at as many points as anybody wants, and a mechanism that produces the wrong exponent is refuted rather than merely unsupported.

The move is to find the parameter the claimed mechanism says the answer is linear in, vary it deliberately over a range, and fit. It is cheap and it is skipped constantly, in favour of computing the answer at the one setting that matters and reporting it.

The failure mode is to run the scaling in the reporting unit. A nonlinear unit turns a clean law into a curve, and the curve then gets explained with a mechanism that does not exist. The scaling belongs in the physical quantity and the answer belongs in the published one, and keeping the two apart is most of the discipline.

Who found it, and when #

The linearity of colour matching is Grassmann’s, from 1853, and everything here is a consequence of it: an observer’s response is linear in the stimulus, so a difference between two observers’ responses is linear in the stimulus too.

The nonlinearity of ΔE₀₀ is entirely deliberate and is the whole point of the formula. Its chroma and hue weightings were fitted to visual data precisely so that a fixed reported difference would correspond to a fixed perceived one, and the fact that this makes it a poor instrument for measuring scaling laws is a cost of that design rather than a flaw in it.

Where the ladder goes next #

The pairing’s second factor is the sample, and its distribution over a family is where the departures’ ranking comes apart. Two of the six change places, and the ordering everybody quotes is the ordering of one example.