A stated lightness is two requirements

A specification is a number written to a precision. Lightness 42.5 carries an implied tolerance in its last digit, and everybody involved understands what that means: the delivery has to be near enough to 42.5 that the difference would not show in the digit. What nobody states is what near enough is in the thing being delivered — the same absence a tolerance in colour differences has.

The claim #

A precision stated once in appearance coordinates is a requirement that varies by an order of magnitude over the range it applies to, and it varies in opposite directions depending on which quantity the supplier controls.

• A twentieth of a unit of lightness fixes the luminance to 0.90 per cent of its own value at J 10 and to 0.097 per cent at J 95, a factor of 9.3.
• In absolute luminance the same precision fixes 0.014 at J 10 and 0.088 at J 95, a factor of 6.5 the other way.
• Both readings are correct and they answer different questions: one is about a process’s relative stability and the other about an instrument’s absolute noise.
• And nothing in the specification says which one is meant.

Where the two readings come from #

The model’s lightness is a compression, so its slope against luminance varies. That single fact produces both readings and it is worth being explicit about how.

Take the inverse at a stated lightness, take it again a twentieth of a unit higher, and subtract. The difference is the luminance interval that the stated precision fails to distinguish — the set of stimuli that would all be written down as the same number.

Read as a fraction of the colour’s own luminance that interval is large in the dark and small in the light. At J 10 the luminance is 1.52 per cent of the white’s and the interval is 0.9 per cent of that; at J 95 the luminance is 90.9 per cent and the interval is 0.097 per cent of it.

Read in absolute luminance the interval is 0.014 at the bottom and 0.088 at the top: seven times larger at the light end, because the same fractional change is a much bigger absolute one when the quantity is sixty times bigger.

Neither reading is a trick. They are the same numbers divided by different things, and which division a reader should perform depends on what they are trying to hold steady.

Which reading a supplier needs #

The question that settles it is what the supplier’s process errs in, and the answer is usually both.

A press errs relatively. Ink film thickness varies by a percentage, dot gain varies by a percentage, and the resulting reflectance error is roughly proportional to the reflectance. For such a process the relative reading is the right one, and a specification in appearance coordinates is nine times harder to satisfy at the dark end than at the light end.

An instrument errs absolutely. Stray light, dark signal and reference drift are roughly constant in reflectance units regardless of how dark the sample is, which is why measuring a black patch is hard. For the measurement the absolute reading is the right one, and the specification is six and a half times harder to verify at the light end.

So the two ends of the scale are hard for different reasons and for different parties. The dark end is hard to make and easy to measure; the light end is easy to make and hard to measure. A single stated precision hides both.

Worked, at the two ends #

Two worked cases make the abstraction concrete and they are the two a supplier actually meets.

A dark grey specified at J 10. The luminance is 1.52 per cent of the white’s, and a stated tenth of a unit — half a digit either way, so a twentieth — fixes it to within 0.9 per cent of that, which is 0.014 per cent of the white. On a printed sheet that is a change in reflectance of about a seventieth of one per cent, which is below what an ordinary spectrophotometer repeats to on a dark patch and far below what a press holds from sheet to sheet.

A light grey specified at J 95. The luminance is 90.9 per cent of the white’s and the same precision fixes it to 0.097 per cent of that, which is 0.088 per cent of the white. In reflectance terms that is about a tenth of one per cent, which a press holds comfortably and an instrument measures easily.

So the first specification is unmeetable and the second is loose, and they are written identically. A supplier reading both would treat them as the same requirement, discover that one of them fails and the other never does, and attribute it to the dark colour being hard — which is true and is not the reason.

The reason is that the specification’s units are perceptual and the process’s units are not, and the conversion between them varies by nine.

What the specification is actually arguing about #

There is a second reading of the same numbers that is less about achievability and more about whether the digit means anything.

A stated tenth of a unit of lightness is one tenth of one CAM16-UCS unit, and a CAM16-UCS unit is roughly a just-noticeable difference. So a specification quoting one decimal place is arguing about a tenth of the smallest difference anybody can see, on the assumption that the observer is the standard one.

The previous rung measured what two real observers disagree by: between 1.13 and 2.46 units on ordinary surfaces, in the same coordinates. The disagreement between two people looking at the delivery is between eleven and twenty-five times the digit being argued about.

That is not an argument for looser specifications. A tight specification on the mean is how a process stays centred, and a process centred to a tenth of a unit will hold to a unit in practice. It is an argument for not reading the digit as a claim about what anybody will see, which is how it is usually read in an acceptance dispute.

A digit tighter than the observer spread is a process control, not a perceptual requirement, and specifications do not distinguish the two.

The same problem, three times #

This is the third time this round has met a stated number that means different things in different places, and the three are worth putting together because their mechanisms are different.

A tolerance in colour differences accepts a set whose volume varies by a factor of a hundred thousand across the sRGB cube. That is the metric’s own non-uniformity, and it is intentional — the tolerance is meant to be a constant amount of visible difference.

The straight piece under CIELAB’s cube root makes the price of a deviation flat below L* 8 and rising above it. That is a numerical repair with a perceptual side effect.

And a stated precision in appearance coordinates varies by an order of magnitude because the coordinate is a compression. Three different mechanisms, one shape of failure, and in all three cases the number on the page is the same everywhere while what it demands is not. A code lattice is a fourth and it fails the other way round.

The connection to the previous rung is direct. The observer departures are between one and two and a half units in the appearance unit, and a specification’s stated precision is a tenth or a twentieth. So the specification’s precision is between ten and fifty times finer than the disagreement between two observers looking at the delivery, which is a mismatch worth stating plainly: the digit being argued about is far below the noise the room contains.

There is a third quantity a specification could be written in and it changes the picture again. Lightness is a ratio to the white, so it is nearly immune to the room; brightness is absolute, so it is not. A specification in brightness carries the room’s own variation on top of everything measured here, and the previous rungs put that at between a fifth and three quarters of the room’s spread.

So the choice of correlate is a choice about which uncertainty a specification imports. Lightness imports the compression’s unevenness and excludes the room. Brightness imports both. Nothing in a specification format encourages the choice to be made deliberately, and lightness wins by convention rather than by argument — correctly, as it happens, and for reasons nobody states.

Why an order of magnitude is the right thing to report #

The factor of 9.3 is the number to carry, and it is worth saying why it rather than any of the individual intervals.

An interval in luminance units is only usable by somebody who knows what their process holds in luminance units, and most do not — a printer thinks in density, a display maker in code values, a paint mixer in colorant concentration. The ratio is unit-free and transfers to all of them: whatever a process’s own tolerance is, a specification in appearance coordinates demands nine times more of it at one end of the scale than at the other.

The same is true of the absolute reading’s 6.5. Neither factor depends on how a supplier measures anything, and both are properties of the model’s compression.

That is the general form a finding like this should take. The intervals are for the one case computed; the ratios are the result.

What a specification could carry #

The repair is the same one this round has now proposed four times and it is cheap here.

State the precision in the delivered quantity rather than in the appearance coordinate. A specification saying luminance within one per cent is a requirement a press can act on and an instrument can verify, and it is the same requirement everywhere on the scale.

Or state the appearance precision and the conversion. One extra number per specified colour — how much luminance a stated tenth of a unit corresponds to at that colour — and both parties know what they have agreed.

Or, at minimum, state which reading is meant. Relative or absolute is one word and it settles a factor of nine.

None of the three is done. A colour specification in appearance terms quotes J, C and h to a decimal place and stops, and the two parties reason about the last digit in whichever way their own equipment errs in — which is a delivery tolerance being three tolerances one model further along.

The digit and the process, in one picture #

A specification’s precision and a process’s capability are two numbers that ought to be compared and almost never are, and the comparison has a standard form outside colour.

Manufacturing calls it a capability index: the ratio of what a specification allows to what a process actually holds. A ratio well above one means the specification is loose and the process is comfortable; a ratio below one means the specification cannot be met and the argument is about which sheets to accept anyway.

In appearance coordinates that ratio is not one number, because the numerator varies. The specification allows a fixed interval in lightness, which is a varying interval in whatever the process controls, so the capability index computed at one colour does not transfer to another on the same job.

A job with a dark grey and a light grey specified to the same precision has two capability indices differing by a factor of nine, and a supplier who measured the light one and generalised has measured the easy end.

That is the practical form of the whole essay and it is the reason to convert rather than to argue. Two colours on the same specification sheet, written identically, are not equally demanding, and the conversion that says by how much is one call to an inverse.

What would happen if the precision were stated in the thing #

It is worth asking what a specification would look like if this were repaired, because the answer is slightly surprising.

A specification stating this luminance, within one per cent is uniform in the thing and non-uniform in appearance: one per cent of luminance is a large lightness change at the dark end and a small one at the light end, which is the reverse of what the compression does. So a physically uniform specification would be perceptually uneven, and a customer complaining that a dark colour was visibly off would be right.

Neither coordinate gives a specification that is uniform in both senses, and it cannot: that is what a compression is. The choice is between a specification a supplier can act on uniformly and one a customer can judge uniformly, and the two are different documents.

What is available is stating both. A colour, a stated appearance precision, and the physical interval it corresponds to at that colour — three numbers instead of two, computed once, and both parties then know what they have agreed to and in which direction it is hard.

That is the same conclusion this round has now reached about a raw pipeline’s arrangement, a mosaic’s reconstruction domain, a clip’s placement and a difference formula’s identity. The repair is always a field, it is always cheap, and it is never present.

What was computed, and how #

The inverse is called at a series of lightnesses on the neutral axis with chroma zero, at the reference viewing condition, and again at the same lightness plus the stated step. The difference in the second tristimulus value is the interval.

Chroma zero keeps the measurement about lightness alone. On a chromatic colour the same stated precision fixes a region rather than an interval and the region’s shape is the object a tolerance shell is, one model further along; that calculation is not in this essay and the structure would be the same.

The step of a twentieth of a unit is chosen to represent a specification quoting one decimal place, where the implied tolerance is half the last digit. A specification quoting whole units implies a step ten times larger, and the second figure above is that case: every interval is ten times wider and every ratio is identical.

The viewing condition is the collection’s reference throughout. Changing it changes every interval and leaves the ratios almost exactly where they are, because the compression’s shape is what produces them.

Where the model stops #

The neutral axis only. A chromatic colour’s stated precision fixes a three-dimensional region, and its anisotropy would add to the variation measured here rather than replacing it.

The precision is treated as a hard interval, which is what a written specification implies and not what anybody enforces. Real acceptance is a judgement or a measurement with its own tolerance, and the interval computed here is a lower bound on what is actually being accepted.

And the model’s inverse is exact, so the intervals are exact — which makes this the one measurement in the round with no numerical uncertainty in it at all. What is uncertain is whether the model’s lightness is the quantity a person’s judgement follows, and that is the standing question about every appearance model and is not this essay’s.

The generalisation #

The habit is about a precision written in a coordinate rather than in the thing.

Every specification is ultimately about something physical — a reflectance, a luminance, a concentration, a dimension — and is written in whatever coordinate the field talks in. When the coordinate is a compression, a log, a rating or a perceptual scale, a fixed precision in the coordinate is a varying precision in the thing, and the variation can be an order of magnitude.

The move is to convert the precision to the delivered quantity at each specified point and to check that the result is achievable and measurable. It is one subtraction per point.

The failure mode is a specification that is unachievably tight in one place and pointlessly loose in another, with the same number written in both, and with both parties confident that they understand it. A digit is not a tolerance until somebody says what it is a digit of.

Who found it, and when #

That perceptual coordinates compress and that a fixed step in them is a varying step in the stimulus is the definition of a perceptual coordinate and has been understood since Fechner. The industrial consequence — that specifications written in such coordinates carry a varying physical requirement — is the reason process control in printing is done in density rather than in lightness — the same instinct that puts a press black on a density scale — which is usually explained as tradition and is arithmetic.

The specific figure for CIECAM16’s lightness does not appear in the sources consulted here, and it is one subtraction away from the published inverse for anybody who wants it.

Where the ladder goes next #

Five rungs have taken the appearance model apart from the coordinates inwards: its domain, its hue scale, its curvature, its unit and its precision. What remains downstream is not a model but a chain — a profile, an intent, a room and a file — and the same question can be asked of it, starting with the one operation every stage of that chain performs and none of them names: an average.