A chain measured in a unit that cannot add

The budget adds two units re-measured a delivery chain’s stages in one unit, found a fourth stage the budget had no row for, and found that the four did not add: the chain end to end measured 5.05 against a sum of 7.66 and a quadrature of 4.79. The unit was the power-corrected CAM16-UCS distance, which is what an appearance judgement in a booth is priced in. That unit is a metric and, as a distance raised to a power has no length found, it cannot be added along anything.

Most of the shortfall is the ruler #

The published chain falls short of its stage sum mostly because the unit it was measured in cannot add; measured in a space that can, the stages come much closer to adding, and one conclusion survives while another does not.

• As published, under the colorimetric intent, the chain measures 0.658 of its stage sum.
• A chain whose four stages pointed the same way and added exactly would read 0.738 in that unit, purely because 1.41·d^0.63 of a sum is less than the sum of 1.41·d^0.63.
• In the model’s Euclidean space the same chain measures 0.836 of its sum, so 77 per cent of the published shortfall is the exponent.
• The correction inflates small stages and shrinks large ones: the profile’s interpolation is 0.42 in the corrected unit and 0.15 in the Euclidean one, and its share of the budget falls from 5.5 per cent to 1.6.

Why a sum of corrected distances is too large #

The corrected distance between two colours is 1.41 times their Euclidean distance raised to the power 0.63. The function bends downwards everywhere, and a function that bends downwards and starts at zero is subadditive: its value at a sum is less than the sum of its values.

A chain’s stages are distances between successive states of a colour: what was asked for, what the mapping made of it, what the separation could reach, what the press delivered, how the room changed it. If the four stages lay end to end in a straight line, the end-to-end distance would be exactly the sum of the four in the Euclidean space. In the corrected unit the end-to-end distance would be 1.41 times that sum to the 0.63, and the sum of the corrected stages would be larger — by an amount that depends on how the total is divided among the stages and not at all on the colour.

That is the middle bar in the first figure. For the colorimetric intent’s stages it is 0.738 of the corrected sum, computed colour by colour and averaged. No cancellation between stages is needed to fall that far short; the unit does it alone.

Two equal stages, four, and ten #

The exponent’s contribution can be written down without a chain at all. Take n stages of equal Euclidean length d, all pointing the same way. Each corrected stage is 1.41 times d to the 0.63 and the corrected sum is n of those; the corrected total is 1.41 times nd to the 0.63. Their ratio is n to the power −0.37, whatever the size of the stages. Two equal stages read 0.77 of their sum, four read 0.60 and ten read 0.43.

The published chain’s exponent-only ratio, 0.738, sits above four equal stages’ 0.60 because its stages are far from equal: the room is 5.87 of the 9.40 Euclidean sum. A chain with a single stage reads exactly one, having nothing to be subadditive with. So the exponent-only ratio is an index of how evenly a budget is divided, and the better a chain is engineered to share its error among its stages, the further a sum of corrected stages overstates the whole.

The same arithmetic says what happens as a budget is refined. Splitting one of four equal stages into two, for finer accounting, lowers the exponent-only ratio from 0.60 towards 0.55 for five equal stages, although nothing about the colour or the geometry has changed. A chain audited in the corrected unit looks more self-cancelling every time its stages are divided more finely — the step count’s divergence, arriving in a budget.

The chain in a space that can add #

The model’s uniform space, J′a′b′, is a Euclidean space: distances add along straight lines and the triangle inequality becomes an equality exactly when the stages are collinear — which ΔE₀₀ cannot promise at all. Every quantity the published chain reported can be re-measured there from the same colours.

In J′a′b′, under the colorimetric intent, the stages are 2.29 for the mapping, 1.09 for the separation, 0.15 for the profile and 5.87 for the room, summing to 9.40, and the chain measures 7.85 — 0.836 of the sum. In the corrected unit the same stages were 1.50, 1.45, 0.42 and 4.30, summing to 7.66, and the chain 5.05.

The shortfall in the Euclidean space is 16 per cent and it is real: the stages do not all push the colour the same way, and the room’s shift in particular is not aligned with the press’s error. In the corrected unit the shortfall is 34 per cent, and the exponent’s own 26 per cent accounts for three quarters of it.

Colour by colour #

The ratios above are averages over the twelve colours of the chain’s ramp. The same decomposition holds on each colour, and the colour-by-colour picture shows where the stages genuinely cancel.

Under the colorimetric intent the exponent alone gives between 0.66 and 0.84 of the corrected sum on individual colours, and the chain measures between 0.58 and 0.81. On every colour the measured share is below the exponent-only share, so every colour has some genuine cancellation, and on most it is a small part of the total shortfall. The Euclidean ratio on individual colours runs from 0.72 to 0.94.

Under the perceptual intent the picture changes. The exponent alone gives 0.65 to 0.72 of the corrected sum, but the chain measures only 0.29 on the most neutral colour and 0.61 on the most saturated. Here the stages genuinely cancel, and strongly: the perceptual mapping moves neutral and near-neutral colours that needed no help, and the room’s shift partly undoes that movement. The exponent is 58 per cent of the shortfall under this intent and 51 under the saturation intent; the rest is real geometry.

The small stage the correction inflated #

The correction does not only change the total. It changes the stages relative to one another, and the smallest stage is changed most.

A stage below 2.53 Euclidean units is enlarged by the correction and a stage above it is shrunk; the same exponent is why the unit disagrees with the others most at the near end. The profile’s interpolation error is 0.15 in J′a′b′ and 0.42 in the corrected unit — nearly three times larger — while the room’s shift is 5.87 and 4.30, a quarter smaller. So in the corrected unit the profile carries 5.5 per cent of the budget and in the Euclidean one 1.6 per cent.

That matters for the argument that a profile’s node count has little reach. The argument survives and is strengthened: in a space that can add, the profile’s contribution is smaller still. The argument that the room dominates survives too, more strongly — 62 per cent of the Euclidean stage sum against 56 per cent of the corrected one. The corrected unit narrows every ranking it measures, because it compresses large numbers towards small ones, and a budget read in it underestimates how lopsided the chain is.

In the space that can add, the room’s stage is thirty-nine times the profile’s; in the corrected unit it is ten times. The profile’s interpolation is about a sixtieth of the Euclidean stage sum, so effort spent on its node count is effort on a sixtieth of the chain. A budget that wants a smaller end-to-end error has one stage to work on before any other, and the corrected unit hides how far ahead of the rest that stage is. It is also the stage no device profile touches: the change between the booth the source was judged in and the room the print is read in is a viewing condition, and the screen is not the room either.

What survives #

Three published conclusions about the chain, checked in the Euclidean space.

The fourth stage survives. The target the separation cannot reach is 1.09 in J′a′b′, seven times the profile’s error, against 3.5 times in the corrected unit. The budget still has no row for a stage larger than one it does have.

The failure to add survives, much smaller. Under the colorimetric intent the chain falls 16 per cent short of its sum in a space that can add, not 34 per cent. Under the perceptual and saturation intents it falls 45 and 56 per cent short in J′a′b′ — those shortfalls are mostly real.

The failure of any single combination rule survives intact. In the Euclidean space quadrature under-predicts the colorimetric chain by 18 per cent and over-predicts the perceptual one by 18 and the saturation one by 53; in the corrected unit its errors ran from 5 to 36 per cent. No rule works in either unit, and the reason is the intents’ different geometry, not the exponent.

Stages that undo each other #

Below quadrature there is only cancellation, and the saturation intent reaches it on the neutral end of the ramp.

On the most neutral colour of the ramp, under the saturation intent, the four stages measured in the model’s own space are 6.17 for the mapping, 1.22 for the separation, 0.33 for the profile and 5.30 for the room, 13.0 in all, and the colour ends 1.14 units from where it started. A quadrature of the four would predict 8.2. The mapping pushes a near-neutral outwards to saturate it and the room’s shift carries most of that push back, so the delivered error is smaller than every stage except the profile. Under the perceptual intent the same colour’s stages sum to 9.47 and the chain ends 2.49 away: cancellation again, less complete, with a mapping of 3.03 that gives the room less to undo.

That is real geometry, and no exponent produces a ratio of 0.09. It is also a warning about budgets of magnitudes. A list of stage sizes cannot distinguish a chain whose errors cancel from one whose errors are small, and the difference decides what an improvement does: on this colour a better match in the room would leave more of the mapping’s push in place, and could make the delivered colour worse.

The usual repair to quadrature is a correlation between stages, and this chain shows why no single coefficient serves. Under the colorimetric intent quadrature under-predicts, which a positive correlation would fix; under the saturation intent it over-predicts by 53 per cent, which needs a strongly negative one; and along the ramp the neutral end cancels far more than the saturated end. A correlation coefficient in a budget is one number standing in for a geometry that changes sign with the intent and with the colour.

Which unit a budget should use #

The answer depends on which question the budget answers, and the two questions are easy to conflate.

If the question is how visible the end-to-end error is, the corrected unit is the better judge, because it was fitted to how people scale differences of different sizes. The chain’s end-to-end error should be reported in it.

If the question is how much each stage contributes, the corrected unit is the wrong tool, because contributions have to add and corrected distances do not. The stages should be measured and composed in the Euclidean space, and the total corrected once at the end if a visual scale is wanted.

Mixing the two — stages in the corrected unit, summed, and compared with a corrected total — produces a shortfall that is mostly an artefact, inflates the small stages, and makes the chain look more balanced than it is. That is what the published budget did, having already found that its units were mixed in another way.

A table that carries both costs one column. Each stage in Euclidean units, their sum, and the chain’s end-to-end error in the same units, with the corrected value of that one end-to-end number beside it for anybody who wants a visual scale. The ratio of the end-to-end error to the sum is then a statement about the chain’s geometry and nothing else, and the corrected total a statement about visibility and nothing else. The published budget’s single column held one number that was a mixture of both.

What was computed, and how #

The chain is the one re-measured in the published budget: twelve colours on a chroma ramp at lightness 55 and hue 30, mapped into a four-colour press’s gamut by each intent, separated by a profile of nine nodes per axis, printed by the press model, and viewed in a dim room with the source judged in a booth. Every state of each colour is read by CIECAM16 at the booth’s or the room’s conditions.

Each stage’s Euclidean distance is taken in J′a′b′ and its corrected distance is 1.41 times that to the power 0.63. The published ratio is the mean over colours of the corrected end-to-end distance divided by the mean of the corrected stage sum, and it reproduces the published 0.658 exactly. The exponent-only ratio replaces each colour’s corrected end-to-end distance with the corrected value of its Euclidean stage sum. The Euclidean ratio is the mean Euclidean end-to-end distance over the mean Euclidean stage sum.

Where the measurement stops #

One ramp, one hue, one press and one room, which is the narrowness the published budget already carried and which a budget drawn through one hue criticised. The exponent’s share of a shortfall depends on how the total is divided among the stages, so a chain with more evenly sized stages would lose more to the exponent and a chain dominated by one stage less.

The Euclidean space is the model’s own uniform space without its correction. Whether a sum of Euclidean stages predicts how visible an accumulated error is has not been tested against judgements, and the correction exists because single differences are not scaled linearly.

And the chain’s geometry — which stages align and which cancel — is computed for the standard observer. A soft proof is exact for one reader, and every reader would give a slightly different geometry.

The habit #

The habit is about decomposing a total in a unit that was fitted to totals.

A perceptual scale is fitted so that a number means a visible amount. That fit compresses large amounts, and a compression cannot be additive, so a total and its parts measured on the same perceptual scale never add up — whatever the parts are doing.

The move is to decompose in a space where addition is exact and to apply the perceptual correction once, to the total. It costs one extra conversion.

The failure mode is to read the scale’s subadditivity as a finding about the parts. A shortfall that a straight line would also produce is not evidence that anything bent.

Who noticed it first #

That a concave power of a distance is subadditive, and that summing corrected differences overstates a total, follows from the definition of the correction. The correction itself comes from work fitting one formula to small and large colour-difference data.

That most of a published delivery chain’s failure to add is this subadditivity, and that the correction inflates small stages enough to change their share of a budget threefold, are computed here on the chain as published.

Still open: a visible accumulated error #

The budget’s purpose is to predict what a reader sees when every stage’s error has accumulated. Whether that visible total is closer to the corrected end-to-end distance, the corrected sum of stages, or the corrected Euclidean sum is a question about judgements of accumulated difference, and it is the one experiment that would say which of the three numbers a delivery tolerance should be written against.