Dividing by the paper

Every ICC profile has a default rendering behaviour that almost nobody has to think about, which is the sign of a well-chosen default. Colours are expressed relative to the medium: the substrate’s tristimulus values are taken as the white, everything printed on it is divided by them, and a page printed on newsprint and the same page printed on a coated sheet come out with the same numbers.

The justification is that the paper is the white point, and a reader looking at a page adapts to the page. That is exactly right, and it means the rule is not a bookkeeping convention. It is a model of an observer, and specifically it is a von Kries adaptation — applied in XYZ.

The claim #

The colour-management default is a chromatic adaptation transform, it is the worst one available, and on the substrates a press actually uses that does not matter.

• A substrate is a filter in the path. Everything printed on it is multiplied by its reflectance, and the reader adapts to the product — which makes a change of stock a change of context in exactly the sense a change of lamp is.
• Dividing tristimulus values by the substrate’s is a diagonal in XYZ, which comes last in every basis comparison this site has run and averages ΔE00 2.37 over the census against Bradford’s 1.14.
• On the three stocks a press uses it costs 3.4 times what the same rule costs in a cone basis, and the absolute numbers are 0.20 against 0.06 and 0.42 against 0.12 — real, and far below anything a person would report.
• On a tinted sheet it costs 10.2 times as much — ΔE00 1.72 against 0.17 — and a tinted sheet is an ordinary thing to print on.
• The reason the default survives is that paper is the smoothest change of light in the census, not that the basis is a good one.

A stock is a change of light #

The first thing to establish is that this belongs in the same category as a lamp at all.

A substrate multiplies. Light arrives at the page, passes into the ink layer, is reflected by the paper underneath, and comes back through the ink; to a first approximation the sheet’s own reflectance is a factor in everything printed on it. And the reader adapts to the sheet, not to the lamp — which is why the same page under a shop’s tube and a window looks like the same page, and why an unprinted margin looks white on newsprint and on a coated stock alike.

That is the structure of a filter in the path, exactly as the macular pigment is, and it means the whole census applies.

A third sheet between the two is what says the difference is a family rather than a pair, and a profile has to be right about all of it.

The fourth ink and the transmittances underneath say what part of each of those curves is the sheet and what part is the film.

Which basis the standards chose #

Media-relative colorimetry says: divide X, Y and Z by the substrate’s X, Y and Z. In the vocabulary of the census that is a von Kries adaptation with the diagonal taken from the ratio of the whites — which is right — applied in the identity basis, which is a choice.

This site’s own notes on the five adaptation transforms describe the identity basis as the Wrong Von Kries, still shipping, and the one that turns adapted blues purple. It is last on eleven of the fourteen census rows and averages more than twice Bradford’s residual.

So the default rule in every colour-management system is an adaptation transform, and it is the one that the same systems would refuse to use anywhere else. A profile that needs to move between two white points uses Bradford, because the specification says so. A profile that moves between two substrates divides by the substrate, because the specification says that too.

Why nobody has noticed #

Because a sheet of paper is the smoothest change of light on this site.

A paper mill spends a great deal of money making its product neutral, and the three stocks here differ from each other by a level and a very slight tilt — coated at 0.868 with a small blue-ward slope, uncoated at 0.79 with a small warm one, newsprint at 0.58 and warmer again. Those are gentle, monotone changes across the whole band, which is the case a diagonal in any basis handles well — the same reason a change of daylight is cheap.

Coated to uncoated is a change of ΔE00 2.55 and the standards’ rule leaves 0.20 of it. Coated to newsprint is 9.03 and it leaves 0.42. Both are penalties of a factor of 3.4 over doing it in a cone basis, and both are far below the threshold at which anybody would raise a job.

Where it stops being small #

The census here includes two substrates a press could print on and this site’s press model does not have: a kraft board and a blue-tinted sheet. They are constructed rather than measured, and labelled as constructions wherever they are drawn, and they are here because a comparison across three near-neutral whites reports that the substrate is the smoothest change of light there is and then stops.

On the kraft board the standards’ rule leaves ΔE00 1.99 against Bradford’s 0.69 — a factor of 2.9 on a change of 24.4. On the blue sheet it leaves 1.72 against 0.17, a factor of 10.2.

That last number is the essay’s. A blue-tinted sheet is not exotic; it is a stationery stock, a packaging board, a coloured card. Printing on it with a media-relative profile means an adaptation error ten times what the same profile’s own white-point machinery would have produced, on a job where the whole point is that the substrate is not white.

A tolerance on a coloured board #

The consequence that reaches a purchase order is a tolerance, and it is worth following one through.

A brand colour is specified as a ΔE00 limit against a reference under D50 with the 2° observer, measured on a stated stock. Printed on a different stock, the profile’s substrate rule maps the reference into the new medium, and everything downstream — the proof, the press check, the acceptance decision — is measured against that mapped reference.

If the mapping is done in XYZ and the stock is tinted, the mapped reference is off by ΔE00 1.72 before the press has printed anything. That is not a printing error and it will not be found by any measurement of the sheet, because the sheet is exactly what the profile asked for. It is an error in the target, and it consumes most of a typical tolerance of two units.

The uncomfortable part is that the error is systematic and directional. Every job on that stock is off in the same direction by very nearly the same amount, so a press that has been characterised on that stock will have had the offset absorbed into its own correction curves — which means the error is invisible in production and reappears the moment the same file is printed anywhere else.

What the alternative would be #

Nothing about the fix is difficult, which is part of why the finding is uncomfortable. The rule could be: express the substrate’s tristimulus values in a cone-like basis, take the ratios there, and apply them there. That is one matrix multiplication either side of the division, it uses a transform the same specification already requires elsewhere, and on the four substrates here it removes between two thirds and ninety per cent of what the current rule leaves.

The daylight basis from the previous essay does better still on the near-neutral stocks — 0.036 and 0.078, against Bradford’s 0.057 and 0.121 — and worse on the blue sheet, 0.44 against 0.17. Bradford is the sensible choice here and the reason is the same as everywhere else these comparisons have been run: a substrate is a broad, smooth filter, and the fitted transforms are compromises that do well on broad smooth things.

The other thing dividing by the paper does #

There is a separate and better-known problem with media-relative colorimetry, and it is worth separating from this one so that neither is blamed for the other.

Dividing by the substrate throws away the absolute lightness of the sheet. A page on newsprint and a page on a coated stock come out with the same numbers, and they do not look the same: one is being read at a maximum reflectance of 0.58 and the other at 0.87, and a reader can see both sheets at once on a desk. That is a limitation of relative colorimetry as such, it has been argued about since the specification was written, and it is why absolute colorimetric rendering exists. The rendering intents all make a choice here and none of them makes it invisibly.

This essay’s finding is different and additive. Even accepting that the reader adapts completely to the sheet — the assumption that makes relative colorimetry correct — the arithmetic implementing that adaptation is being done in the worst available axes.

The reader does not adapt completely either #

Everything above grants the assumption that makes media-relative colorimetry correct: that the reader adapts completely to the sheet. It is worth spending a section on what happens when that assumption is relaxed, because the two errors compound rather than cancel.

The appearance models carry a degree of adaptation — a number between about 0.6 and 1.0 that says how much of the illuminant has been discounted — and it is never one outside a fully adapting condition. A page held in a hand, surrounded by a desk and a room that are not the page, is being seen in a viewing condition where the page is the background rather than the whole field, and adaptation to it is partial by construction.

Partial adaptation makes the residual larger in a specific way. The gain applied is not the full ratio of the whites but some fraction of the way toward it, so the change is under-corrected and what is left carries the substrate’s own colour rather than only the part no gain could remove. On a near-neutral stock that is a small effect on top of a small effect. On a tinted sheet it adds a visible cast to a residual that is already ten times what a cone basis would have left.

The direction is the awkward one for the standards. Media-relative colorimetry assumes complete adaptation and therefore removes the whole substrate; a reader who adapts to eight tenths of it is left seeing two tenths of a colour the profile has already normalised away. The page is being rendered for an observer more adapted than the reader is.

None of this argues against the rule. It argues that the rule has two approximations in it — complete adaptation and the wrong basis — and that the second is the one that can be fixed by editing a specification rather than by changing what people do.

The tenfold penalty is Bradford doing well #

The blue sheet is presented as the place the default stops being small, and dividing each residual by the change it is a residual of says something different.

The XYZ rule leaves between five and eight per cent of whatever it is handed, across a change of light spanning a factor of ten. For the blue sheet’s 1.72 to sit in that band the change would have to be 21 to 37 ΔE₀₀ — between the kraft board’s 24.4 and rather more, which is exactly what a tinted stationery stock should be.

On that reading XYZ is behaving normally on the blue sheet and Bradford is behaving unusually well. Bradford leaves 1.3 to 2.8 per cent of the change on the three measured rows; 0.17 on a change of that size is 0.5 to 0.8 per cent, well under its own band.

That is not a smaller finding, it is a different one. The essay’s headline is that the wrong basis degrades on a tinted sheet; the arithmetic says the wrong basis is uniformly mediocre and the right basis is uniformly good and occasionally excellent, and the tenfold ratio is the gap opening from below. Which is the better argument for changing the specification, because it does not depend on finding a substrate bad enough — it says the cone basis is a free improvement everywhere and a large one somewhere.

What the fix removes, and where the daylight basis loses #

The claim that a cone basis removes between two thirds and ninety per cent of what the current rule leaves checks exactly on all four rows: 70 per cent on coated-to-uncoated, 71 on newsprint, 65 on the kraft board and 90 on the blue sheet. Two thirds and ninety per cent are the endpoints, not a range with slack in it.

The choice between cone bases is worth stating as the trade it is. The daylight basis beats Bradford on both near-neutral stocks — 0.036 against 0.057 and 0.078 against 0.121, so 1.6 times better — and loses on the blue sheet, 0.44 against 0.17, 2.6 times worse.

Bradford wins on both readings a specification would use, which is why Bradford is the sensible choice here is right and worth pinning down. Its worst case over the three is 0.17 against daylight’s 0.44, so it wins a minimax by a factor of 2.6; and its mean is 0.116 against 0.185, so it wins on the average by 1.6. A rule that wins on the average and on the worst case needs no argument about which substrates are common.

What the offset does to a tolerance budget #

The tolerance section says the 1.72 consumes most of a typical tolerance of two units, which is 86 per cent of it, and the important word is the one two paragraphs later: systematic.

A random press variation combines in quadrature with other random terms. A fixed offset in the target does not — it subtracts from the budget linearly, before anything else is counted. So a job on that stock with a press repeatability of one unit has 2.72 units of worst-case error against a two-unit limit and cannot pass; the same job with the substrate rule done in a cone basis has 1.17 and passes with room to spare.

The fix is therefore the difference between a job that cannot be made to work and one that is comfortable, on a stock nobody would call unusual, and it is one matrix multiplication either side of a division that is already being performed.

Who found it, and when #

Media-relative colorimetry is in the ICC specification from the beginning, and the choice to express it as a simple ratio of tristimulus values is explicit there. The specification also mandates Bradford for chromatic adaptation between illuminants, so both transforms are in the same document, applied to two problems that this site’s arithmetic says are the same problem.

The distinction the specification draws is between adapting to a light and adapting to a medium, and that distinction is real in every respect except the one that matters here: in both cases the observer’s mechanism is a gain on three signals, and a gain needs axes.

Whether the choice was deliberate is not something this site can determine. The most likely account is that the media-relative rule was written as bookkeeping — a normalisation so that numbers are comparable — and only later understood as a model of an observer, by which time it was in every profile in existence.

What was computed, and how #

Each substrate is a reflectance on the site’s wavelength grid: the three press stocks from the printing model, and two constructed tinted sheets. A context is D50 multiplied by the substrate, normalised so a perfect diffuser under it gives Y = 100 before the comparison, so the rows are about the shape of what the substrate does and not about how dark it is.

The residual is the mean CIEDE2000 over a hundred and twenty-five surfaces after the ratio-of-whites gain in the named basis. The gate requires the XYZ rule to leave more than the cone-basis rule on every substrate, requires the near-neutral penalty to exceed a factor of two, and requires at least one tinted substrate to exceed a factor of five — so a future change to the paper model that quietly made the default adequate would stop the build.

Where it stops #

The two tinted substrates are constructed. Their shapes are plausible and their absolute numbers are properties of the construction; what is not a property of the construction is the direction, which is that a substrate departing from neutral makes the basis matter and a neutral one does not.

The whole calculation assumes the reader adapts completely to the sheet. Real adaptation is incomplete, and the degree of it is a number the appearance models carry; with incomplete adaptation the residual would be larger and the relative comparison between bases would be much the same.

And a substrate is not always a simple multiplication. A sheet with an optical brightener in it returns light at wavelengths it did not receive, which is not a filter at all, and almost every white office paper has one. For those sheets none of the arithmetic here begins.

Where the ladder goes next #

The substrate is one filter in a chain that has several. The ink is another, the viewing booth’s lamp is another, and the geometry the sheet is measured in adds something that is not multiplicative at all.

The wider point is the one the census keeps making: every stage of a colour pipeline that divides by a white has chosen a basis, usually without saying so. A camera’s is whatever its dyes give it, a display’s is its primaries, and a press’s is the sheet — and only one of those was ever written down as a decision.