The scale hangs from one measurement

A black that is not black measured the deepest colour a four-colour press can make — L* 2.4, a contrast of about 320 to 1 against its own paper — and priced the two ways of fitting a picture into that range. Black point compensation is the second of them: a straight line in lightness between the source’s black and the destination’s, which clips nothing and moves everything.

Both of those blacks are numbers a profile carries. The destination’s is a measurement, of one patch, at the place an instrument is least able to make one — and the paper is the white point is the essay about the other end of the same range, which is measured where an instrument is at its best.

One patch, and the steepest part of the scale #

The lightness scale is twenty times steeper at a press’s darkest patch than at its paper, so the same instrument is twenty times less certain there — and black point compensation spends that uncertainty on every colour in the picture, including the ones nowhere near the black.

• The scale’s slope is 903.3 lightness units per unit of luminance factor below a luminance of 0.0089 and 43.1 at the paper. A thousandth of a reflectance is 0.903 lightness units at the solid and 0.043 at the paper.
• That propagates. A black point wrong by 0.9 moves a mid grey by 0.452 lightness units, which is 0.449 of a colour difference.
• It is 22 per cent of a delivery tolerance of two, and 45 per cent of a tolerance of one, at a colour that could not be further from the measurement that caused it.
• Clipping the shadows instead does not propagate it at all — the uncertainty reaches nothing above the black, because a clip is the identity there. That is the one respect in which clipping is the safer mapping.
• The source end of the same map is worse and is usually stated rather than measured. A display measured in a lit room carries a black near 3.2, and using it against a nominal zero moves a mid grey by 1.61 lightness units.

Why the darkest patch is a different kind of measurement #

The lightness function has two branches. Above a luminance factor of 0.008856 it is a cube root, and below it a straight line of slope 903.3 that exists to keep the derivative finite at zero. A press’s four-colour solid sits at a luminance factor of about 0.0027, which is on the straight branch.

So the sensitivity of lightness to luminance there is a constant 903.3, while at the paper — a luminance factor near 0.85 — the cube root’s own slope is 43.1. The same absolute error in measured reflectance is worth twenty-one times as much lightness at the solid as at the paper, and nothing about the instrument has changed between the two readings.

That matters because the errors an instrument makes at low reflectance are absolute rather than proportional. Stray light inside the optics, a dark-current offset, a fleck of dust on the aperture: each adds a small amount of reflectance that does not scale with the signal. A spectrophotometer that reads a paper to a tenth of a per cent of its value reads a solid to the same absolute amount, and at the solid that absolute amount is most of what it is measuring.

At the darkest patch the three instruments give 0.45, 0.90 and 1.81 lightness units; at the paper they give 0.022, 0.043 and 0.086. Buying a better instrument buys almost nothing anywhere except at the one patch this essay is about — which is a reasonable argument for buying one, and is also the reason nobody notices they need to.

Averaging does not help, and it is worth saying why #

The obvious response to an uncertain measurement is to make it several times. It is the right response to most of the errors in a profile and it is nearly useless here.

An instrument’s error at low reflectance has two parts. The random part — photon noise, electrical noise, small differences in how the aperture sits on the sheet — falls as the square root of the number of readings, so four readings halve it and a hundred reduce it tenfold. The systematic part — stray light scattered inside the optics onto the detector, and whatever the dark reading was calibrated against — is the same on every reading and averaging leaves it exactly where it was.

At the paper the random part dominates, because the signal is large and the systematic offset is a tiny fraction of it. At a four-colour solid the proportions are reversed: the signal is a few thousandths of a reflectance and the stray light is of the same order. So the measurement that most needs repeating is the one repeating helps least.

There is a second reason, and it is about where the patch is. A profile’s lattice has thousands of patches and their errors partly cancel because they are read at different levels and in different directions; the black point is one patch, read once, and its error is carried whole. The instrument is one observer exactly makes the neighbouring point about the instrument’s colour-matching functions: what a single fixed instrument gets wrong, it gets wrong the same way every time, and no number of readings finds out.

Where the uncertainty arrives #

A clip would confine the uncertainty to the range it was measured in. Compensation does not, because compensation is a map with the black at one end of it.

The map is L' = b + (L − s)(W − b)/(W − s), linear in lightness with the destination black b at one end and the shared white at the other. An error d in b moves a colour by d times how far that colour is from the white. At L 0 the shift is the whole 0.903; at L 25 it is 0.677; at L* 50, 0.452; at L* 75, 0.226; at the white, nothing.**

In colour differences the fall is steeper, because ΔE₀₀’s lightness weighting rises with lightness: 0.530 at black, 0.507 at a quarter tone, 0.449 at a mid grey, 0.164 at a highlight. The mid grey is where it matters, because a mid grey is where a job is judged, where a memory colour sits, and where a tolerance is tightest.

So a measurement made on the one patch that is hardest to measure is carried, undiminished by any averaging, into the colours a client will look at. Nothing else in a profile behaves this way. The lattice has thousands of patches and their measurement errors are independent, so they partly cancel; a profile interpolates light found the one systematic error among them and it is a tenth of a colour difference. The black point has one patch, no averaging, and it is worth four times that.

What share of a tolerance #

A delivery tolerance is a budget, and this is a line in it.

At a mid grey the black point’s uncertainty is 22 per cent of a tolerance of two, 15 per cent of a tolerance of three and 45 per cent of a tolerance of one. At the darkest colours it is 27, 18 and 53 per cent.

A delivery tolerance is three tolerances took a delivery tolerance apart into the parts a specification does name. This is a part it does not, and it has the awkward property that it cannot be reduced by any of the usual responses. More patches do not help, because the black point is one patch by definition. A tighter process does not help, because the uncertainty is in the measurement rather than in the press. And averaging several readings helps only against the random part, while the dominant part at this level is an offset the instrument makes the same way every time.

The one thing that does help is a better instrument, and the figure above says what that buys: everything at the black and nothing anywhere else. That is an unusual shape for a purchase and which index to buy an instrument for is the essay about making such a decision properly.

The one thing clipping is better at #

The comparison with the other mapping is worth making, because it is the only place here where clipping wins.

Clipped, the uncertainty reaches nothing above the black at all. A clip is the identity for every colour lighter than the destination black, so an error in that black changes nothing about them — exactly, to machine precision, rather than approximately. The uncertainty stays inside the range it was measured in.

That is not an argument for clipping. A black that is not black priced what a clip costs — a flat area where the shadows were, and a count of distinguishable steps that falls to nothing — and nothing here changes it. It is the honest statement of what compensation buys and what it spreads: it converts a total loss in a small range into a small shift everywhere, and the shift carries the range’s own measurement uncertainty out with it.

The choice between them is therefore not only about pictures. A workflow that compensates has taken on an error budget item that a workflow that clips does not have, and the item is larger than several of the ones it does write down.

The other end of the map #

The destination black is measured. The source black is whatever the source profile states, and the two have entirely different provenances.

An encoding states zero, because sRGB and its relatives have a zero in their numbers and no measurement behind it. A display profile states what the display was measured at, and a display’s black depends heavily on the room — the booth is a luminaire is where the room’s own contribution to a measurement is taken apart: 0.4 in a dark room, 3.2 in a lit one, because the screen reflects the room and the screen is not the room is the essay about what that does.

Between those two the mid grey moves 1.61 lightness units — nearly twice what the destination’s own measurement uncertainty does, and from a difference that is not an uncertainty at all but a genuine difference between two situations that both get called the source’s black.

So the map has two ends, one of them measured badly and one of them often not measured, and the second is the larger term. A workflow that worries about the first and not the second has the proportions backwards.

What a specification could say #

Three things, in increasing order of how much anyone would have to change.

Name the black point and its uncertainty. A profile carries the number and reports no uncertainty for it; a print specification quotes aim values and tolerances for the solids and says nothing about the black the compensation will use. Writing the number down costs nothing and makes the budget item visible.

State which black point compensation is on. The compensation is a switch in most colour engines, the two settings produce visibly different shadows and differently propagated errors, and a job specification that does not say which was used has not specified the conversion. That is the same omission an intent is not a function of the colour found for the rendering intent, one stage along.

And measure the source black, or state that it is nominal. The largest term here is the one that is merely stated, and a source profile that says zero when the display it describes reads 3.2 is not making a measurement error — it is answering a different question, and the compensation cannot tell which question was answered.

How the propagation was computed #

The press is the four-ink model used throughout here on a coated stock, whose four-colour solid comes out at L* 2.47 and whose paper at 94.8. The lightness of a luminance factor is CIELAB’s own two-branch function, and its slope is that function’s derivative — 903.3 on the linear branch and (116/3) Y^(−2/3) on the cube-root one.

The uncertainties are absolute reflectance offsets of 0.0005, 0.001 and 0.002, which brackets what a spectrophotometer is quoted at and models the dominant error at low reflectance: stray light and a dark offset, neither of which scales with the signal. They are applied as an error in the black point’s lightness rather than re-measured through the whole spectrum, because the compensation takes the lightness and nothing else.

The propagation runs the compensation twice — once with the black point as given, once with it in error — and reads the difference at eight lightnesses, in lightness units and in ΔE₀₀ against a neutral. The clipped comparison uses the same error through a clamp at the destination black.

What this leaves out #

The compensation modelled here is the linear one in lightness, which is the common form and is what the specification describes. Some engines use a slightly different curve, and a curve with its own shape would carry the error differently — more at the dark end and less in the middle, for one that is concave.

The uncertainty is treated as an error in the black point alone. In practice the same instrument measures every patch in the profile, so its offsets are correlated across the lattice, and a correlated error is a different calculation from an independent one. The direction that points is towards a smaller net effect, because a common offset partly cancels between the black and the colours read against it.

And the two ends are treated separately. A workflow with an error at both ends has a map that is wrong in slope as well as in offset, and the result is not the sum of the two effects computed here.

Still open: what two instruments actually report for a four-colour solid #

Everything above rests on an uncertainty stated as a number rather than measured on a bench. The measurement that would replace it is small and nobody here has made it: a four-colour solid read on several instruments of the kind a press room owns, with the spread of their reported lightnesses recorded.

The prediction is that the spread will be much larger than the same instruments’ spread on a mid grey, by about the factor the scale’s slope gives — twenty at the paper and something between at the mid-tones. If it is, then the error budget line above is real and can be quoted from measurements rather than from a specification sheet. If the spread turns out to be small, the interesting question becomes why, because it would mean instruments agree at low reflectance better than their absolute accuracy claims allow, and the likeliest reason is that they share a calibration convention rather than a measurement.

The same session would answer the second question by measuring the same solid at two measurement conditions and on two backings, both of which are choices a specification makes and neither of which is usually recorded for the black.

A number that is measured once is not a number like the others #

The habit is about which inputs to an error budget deserve their own line.

Most of the quantities in a chain are measured many times, and their errors are partly independent, so what reaches the end is a spread that shrinks with the count. A quantity measured once behaves differently: nothing cancels it, nothing averages it, and whatever it is wrong by is carried whole into everything computed from it.

The move is to find the inputs with a count of one and ask what the calculation does with them. There are rarely many — a white point, a black point, a substrate, a viewing condition — and each is a candidate for a line in the budget that nobody has written, because a budget is usually assembled from the things that were measured repeatedly and therefore noticed.

The failure mode is to assume an uncertainty stays where it was measured. This one was measured at a luminance factor of 0.0027 and arrives at a mid grey, because the thing it was measured for is one end of a map whose other end is the white.