A delivery tolerance is three tolerances

A delivered colour has been through three things, and this collection has an essay about each of them and no column with all three in it.

The claim #

Measured in the same units and set side by side, the three stages of a delivery chain are not the same size, the largest is the one nobody controls, and the arithmetic for combining them is worth as much as the smallest of them.

• The profile’s interpolation between its nodes costs 0.13 ΔE00 at nine nodes per axis.
• The rendering intent costs 0.00 at the colorimetric setting, by construction: it moves nothing that was already inside the destination gamut.
• The room costs 4.30 — looking at a print proofed in a booth on a screen in a dim room — which is 97 per cent of the total.
• Adding the three gives 4.43 and combining them in quadrature gives 4.30. The gap is 0.127, which is the size of the entire profile stage, and no standard says which rule to use.
• And reach is not contribution. The intent contributes nothing and can move the total by 6.40; the node count contributes 0.13 and can move it by 0.28.

Why the three have never been in one column #

Each stage has its own essay here and each was measured in its own terms. A profile is a fit between its nodes reports an interpolation error against node count; no mapping preserves everything reports what each rendering intent moves and what it collapses; the screen is not the room reports what a viewing condition does to an appearance.

They are in different units and about different objects, which is why nobody had added them up. Putting them in one column needs a single quantity, and the one available is a colour difference — with the caveat, stated below and not resolved, that the third stage’s difference is a different kind of thing from the other two.

A specification does put them in one column, implicitly, whenever it says delivered colour within ΔE00 of 2. That sentence is a budget over the whole chain, and a budget needs to know which line item is large.

The three costs #

The profile. An ICC profile is a lattice with a stated number of nodes per axis, and a colour engine interpolates between them. The error is exact at the nodes and grows with the curvature between them: at nine nodes per axis it is 0.129 ΔE00 on average over a sample of coverages, and at thirteen it is 0.093.

The mapping. A rendering intent decides what happens to colours the destination cannot hold. The relative-colorimetric intent leaves everything inside the destination exactly where it is and clips what is outside, so its cost on colours that were already printable is exactly zero. Perceptual and saturation intents compress the whole space to make room, and move those same colours by 2.99 and 6.40.

The room. A colour that arrived exactly right still has to be looked at, and looking at it somewhere other than where it was proofed changes what it looks like. From a print viewing booth to a screen in a dim room, the appearance shift is 4.296 units in a uniform appearance space, averaged over a ramp of one hue.

The third is not an error in the same sense as the first two: nothing went wrong, the colour is correct, and it looks different. That is exactly why it belongs in the column, because a budget that counts only what somebody got wrong will be met by a chain whose output nobody likes.

The three are also unequal in a second way that a single column hides: they are not equally knowable in advance. The profile’s error can be computed exactly from the profile, before anything is printed. The intent’s cost can be computed from the source and destination gamuts. The room’s cannot be computed at all without knowing the room, and the room is a fact about somebody else’s building.

So the column has a boundary running through it: two stages a chain can measure and one it can only assume. That boundary matters more than the numbers, and it is the reason the standards for proofing are so specific about viewing conditions — specifying the room is the only way to make the third term computable.

Reach is not contribution #

The more useful table is the second one, and it separates two things a single number conflates.

What a stage contributes at its usual setting, and how far its own settings could move the total. They are not the same, and one stage has none of the first and most of the second.

The table itself is the thing all three stages are measured against, and it can be drawn at the sizes a delivery chain actually uses.

The first stage’s price list has two axes rather than one, and both of them are decisions somebody makes when a profile is built.

• The rendering intent contributes 0.00 and spans 6.40. Choosing a different intent is the single largest lever in the chain, and at the usual setting it is invisible.
• The room contributes 4.30 and spans 4.30 — proofing and viewing in the same booth costs nothing, and every step away from that costs everything.
• The profile’s node count contributes 0.13 and spans 0.28, from a coarse five-node grid to a thirteen-node one.

So the knob with a standard about it has the least reach in the table. ICC profile grids are specified, argued about and tuned; the whole range from a coarse grid to a fine one moves the delivered total by less than the room does by existing.

The rule nobody states #

Three numbers can be combined in more than one way and the choice is worth as much as one of them.

Added, the three give 4.425. In quadrature, they give 4.298. The difference is 0.127 — almost exactly the profile stage’s entire contribution.

The argument for each is real. A profile’s interpolation error is random over the colours a job happens to contain, so it combines in quadrature with anything else random. A gamut mapping’s cost is a systematic displacement in a consistent direction, which adds. And the room’s contribution is a change of appearance rather than an error, which does not obviously do either.

So three totals are reported and none is called the answer. Quoting one would be inventing the combination rule this essay exists to point out that nobody has — and the gap between the two candidates being the size of a whole stage is the reason it matters at all.

The asymmetry between the intent’s contribution and its reach is worth stating carefully, because it is easy to read as a trick. The colorimetric intent contributes nothing to this particular measurement, which is the cost borne by colours that were already inside the destination gamut. It certainly does something to the colours outside: it clips them, which is a large cost concentrated on a few colours rather than a small one spread over many.

Measuring the cost on the inside colours is the right choice for a budget and it is a choice. A budget is about what happens to the image a client approved, and the colours outside the destination were going to move whatever anybody did; what a specification can meaningfully bound is the collateral damage to everything else.

The reach number then says what changing that choice costs, and 6.40 units is a great deal — a perceptual intent trades a small movement of every colour for a smaller movement of the few that had to move, and the trade is visible here as a stage that switches on.

What was computed, and how #

The colours are a ramp of one hue at increasing chroma, running from neutral out past the destination gamut’s boundary. That is deliberate and it is not a uniform sample: the failures of a gamut mapping are concentrated on gradients that run out of the destination, and a uniform sample of the source solid dilutes them into an average that says nothing.

The profile’s error is measured at points chosen to fall between the nodes, because a profile is exact at every node of every grid and a check made at the nodes reports a perfect profile for ever.

The room’s shift is computed through the appearance model at two viewing conditions and reported in its uniform space, which is the only one of the three quantities that is not a CIEDE2000 difference. The two are not interchangeable and mixing them in one column is the essay’s largest liberty; it is taken because the alternative is not to compare them at all, and it is stated rather than smoothed over.

Where the model stops #

The destination is one press on one stock. A different destination has a different gamut, and the intent’s cost is entirely a function of how much of the source falls outside it — so on a wider destination the largest lever in the table shrinks towards zero and on a narrower one it grows.

The room is one pair of conditions. Booth to screen is the largest step this collection’s three tabulated conditions offer; booth to a lit office is 1.51, and booth to booth is nothing. A chain proofed and viewed in the same place has a two-stage budget rather than a three-stage one.

And three stages is not the whole chain. A quantisation to eight bits, a subsampled chrominance, a black point compensation, the profile’s own measurement error and the drift of the device since it was profiled are all real and none is here. Every one of them would add to the total and none of them, on this collection’s own numbers elsewhere, is of the room’s order.

The room is the term nobody controls #

The finding with a consequence is the share, and it is worth stating in the form a practitioner would use.

Ninety-seven per cent of this chain’s delivered difference is the viewing condition. Not the profile, which is measured and tuned; not the intent, which is chosen per job; the room, which is whatever room the reader happens to be in.

That has two practical readings and they point in opposite directions. The pessimistic one is that most of the effort in colour management is spent on five per cent of the problem. The optimistic one is that the five per cent is the part anybody can do anything about, and that a chain which controls its controllable stages to a tenth of a unit has done what a chain can do.

The reading this collection prefers is the third: a budget stated without naming the viewing conditions is not a budget. A tolerance of ΔE00 2 delivered is met or missed by the room, and a specification that does not say which room has left its largest term unwritten.

The generalisation #

Two things generalise and the second is the more surprising.

Put every stage of a chain in one column before optimising any of them. The stages of a pipeline are usually owned by different people, measured in different units, and improved independently — and the arithmetic that says which one matters is a table nobody is responsible for producing.

And measure reach as well as contribution. A stage contributing nothing at its current setting can be the largest lever in the system, which a table of contributions cannot show and which is exactly the situation for a rendering intent set to colorimetric. It is the same distinction an audit of thresholds needed one field over: what a thing currently is, and how far it could move.

That last figure is the practical answer to the whole essay, and it is an old one. The reason the graphic-arts standards specify a booth’s illuminant, its geometry, its surround reflectance and its luminance is that a specified room converts the largest term in a delivery budget from an assumption into a measurement. Everything in this essay is an argument for a practice that already exists, arrived at from the other end.

And a combination rule is a modelling choice with a size. Where two rules give answers differing by as much as one of the terms, the rule deserves the same scrutiny as the terms — which usually means stating that it is a choice, since the information to decide it is rarely available.

The combination rule is an identity, not a measurement #

The essay makes a great deal of one coincidence: adding the three terms gives 4.425, combining them in quadrature gives 4.298, and the gap of 0.127 is almost exactly the profile stage’s entire contribution of 0.129. That is put forward as a reason the rule deserves the same scrutiny as the terms.

It is not a coincidence, and it carries no information about this chain. For two terms with the smaller one a and the larger c, the difference between the two rules is

$$(a + c) - \sqrt{a^2 + c^2} \;\approx\; a\left(1 - \frac{a}{2c}\right)$$

which for a = 0.129 and c = 4.296 predicts 0.1271 against the measured 0.127. Whenever a budget is dominated by one term, the two combination rules differ by very nearly the second-largest term, in any chain, in any units, about anything. The observation is algebra dressed as a finding.

Which inverts what the rule is worth #

Once it is an identity, the conclusion drawn from it goes the other way.

The two totals are 4.425 and 4.298: 2.9 per cent apart. On a chain where one term is thirty-three times the next, the choice of combination rule is the least consequential decision anywhere in the essay — smaller than the difference between a nine-node and a thirteen-node profile grid, and far smaller than any of the three stages.

The rule matters when the terms are comparable, and how much is a function of their ratio alone:

So a combination rule is a modelling choice with a size is true and the size here is small. The rule to carry is that the combination rule matters in inverse proportion to how unequal the budget is — which is a more useful statement than the one the essay reaches, and it means a dominated budget is the one case where nobody needs to settle the question.

The headline share depends on the rule it was computed under #

The 97 per cent is quoted as a fact about the chain and it is a fact about the chain plus a choice.

Against the additive total the room is 4.296 of 4.425, or 97.1 per cent. Against the quadrature total it is 4.296 of 4.298, or 99.95 per cent. Both are true, they are answers under the two rules the essay declines to choose between, and only one of them appears.

The direction is worth noticing: the rule the essay does not use is the one that makes its headline stronger. Under quadrature the other two stages together account for one twentieth of one per cent of the delivered difference, which would have been a more striking sentence and rests on the same declined choice. Reporting 97 rather than 99.95 is the conservative reading and the essay does not say that it is one.

What the room’s reach actually spans #

One number in the reach table is quoted at its maximum and has a shape worth seeing.

The room contributes 4.30 and is described as spanning 4.30, from booth-to-booth at zero to booth-to-screen. But the essay’s own third condition sits between them: booth to a lit office is 1.51, which is 35 per cent of the full span. So the room’s reach is not evenly distributed across plausible situations — two of the three tabulated conditions are within a third of each other and the screen in a dim room is the outlier.

That matters for how the finding should be used. The largest term in the budget is large because of one particular destination, a screen viewed in the dark, and a chain delivering to a lit office carries a room term of 1.51 against a controllable total of 0.13 — still the dominant term, at 92 per cent, but no longer overwhelming everything.

Stated as a range rather than a point: the room contributes between nothing and 4.30, the two measured stages contribute 0.13 together, and the room is the largest term in every case except the one where the proof and the view happen in the same booth. That is the finding, and it survives without needing the worst case to carry it.

Who found it, and when #

Error budgets over an imaging chain are standard practice in colour engineering and are usually assembled exactly this way, as a list of contributions in ΔE with a combination rule chosen by convention. The ISO standards for proofing specify viewing conditions precisely for the reason this essay measures, and the specification of a viewing booth’s illuminant, geometry and surround is one of the most carefully written parts of the graphic-arts standards.

So the practice knows the room matters; what is less often written down is the ratio. Stating that the viewing condition is ninety-seven per cent of a particular chain’s delivered difference is a stronger claim than viewing conditions matter, and it is the kind of claim that is only available once the three are in the same units.

Where the ladder goes next #

Three stages, three costs, and a fourth term that is not a stage at all: the degree to which the reader has adapted to the room they are in. The chain assumes an observer who has settled; a reader who has just walked in has not, and the model’s own formula for how completely anybody adapts is a number this collection had been quietly setting to one.

That is the last of the declared inputs this round has swept, and like the others it moves numbers without moving orderings — which is, in the end, the phase’s own summary of itself.