The straight piece under the cube root

Every account of CIELAB says the same thing about its lightness: it is a cube root of relative luminance, scaled and offset so that a perfect white comes out at a hundred. The account is right about ninety-nine per cent of the range and wrong about the part where every delivery chain in this collection does its most anxious work.

The claim #

CIELAB’s compression is a cube root spliced onto a straight line, the splice is exact rather than approximate, and every black anybody delivers is inside the straight piece.

• The break is at a relative luminance of 216/24389, which is L* 8 exactly.
• The two branches meet in value and in slope, both to the last bit — 7.787037037037037 against 7.787037037037035, the residual being one unit in the last place of a double.
• They do not meet in curvature. The second derivative is zero below and −500 just above, which is the derivative the composition of two departures needs.
• Three of four blacks this collection has quoted are below the break, and the fourth is a paper’s own shadow.
• Inside the straight piece the price of a lightness deviation is flat to 0.17 per cent, and above it, it falls by a factor of 5.2 by L* 40.

Why there is a straight piece at all #

The reason is a defect in the cube root that has nothing to do with vision and everything to do with arithmetic.

A cube root has an infinite derivative at zero. At a relative luminance of a thousandth its slope is already 3,600 lightness units per unit of luminance, and it keeps climbing without bound as the luminance falls. So a pure cube root has three consequences a standard cannot live with. The lightness of a very dark sample is unstably sensitive to the measurement, because a measurement error that is negligible in the middle of the range is multiplied by an unbounded factor at the bottom. A negative measurement has no lightness, and a real spectrophotometer reading a black patch against its own stray light returns negative values often enough to matter. And the derivative that every difference formula is built on ceases to exist, so nothing downstream can be linearised.

The splice removes all three. Below a chosen luminance the function is replaced by a straight line through the origin, so the slope is a constant, the value at zero is zero, and a slightly negative measurement produces a slightly negative lightness rather than a large one.

What is remarkable is how carefully the two constants were chosen. ε is 216/24389 and κ is 24389/27, and they are stated in the standard as exact rationals rather than as decimals. Those two values make the straight line meet the cube root in value at the break and in slope, simultaneously. Nothing forced the second condition: a splice that matched only the value would have been simpler to state and would have put a visible kink into every lightness scale in the world.

What the exactness is worth #

The slope of the compression at the break is 841/108 on the cube-root side and 24389/3132 on the linear side, and those two rationals are equal. A computer evaluating them in double precision gets 7.787037037037037 and 7.787037037037035, which differ by one unit in the last place and is what an identity looks like when floating point is asked to confirm it.

That matters because the price of a deviation is a first derivative. A discontinuous slope at the break would mean that two samples straddling L* 8 could not be compared at all: the difference between them would depend on which side the arithmetic approached from, and a formula assembled from such a function would report different answers for the same pair depending on the order the two were entered.

So the CIE bought exactly the smoothness a difference formula needs and no more. The function is once differentiable everywhere, including at the break, and twice differentiable nowhere near it — which is enough for a difference formula and not enough for two of them at once.

The consequence for the price is direct and is worth stating as a fact about the shadows rather than about a formula. As a surface darkens, a fixed deviation buys more lightness units, because the compression is steeper down there — that is the whole of the previous essay’s second factor. The straight piece stops it. Below L* 8 the price of a lightness deviation is 5,908 colour differences per unit of tristimulus deviation and stays there, flat to seventeen parts in ten thousand across the entire straight piece. Above the break it falls by 5.2 times by L* 40.

The straight piece is a floor on how expensive a deviation can get, and a pure cube root has no floor at all.

Which blacks are inside it #

A break at L* 8 sounds like a corner of the space that only a metrologist visits. It is where every delivery chain in this collection ends.

The deepest colour a four-colour press can make is L* 2.4, which is a relative luminance of 0.00266 — less than a third of the break. A projected black in a dark cinema is L* 1.1. A calibrated thousand-to-one display, read in an ordinary office rather than a dark room, sits at L* 6.3. All three are inside.

The one that is not is the shadow on an uncoated sheet at L* 12, and it is outside by a small margin.

That has a consequence for how those numbers should be read. The lightness reported for a press black is a linear function of its luminance, not a compressive one. Halving the ink’s residual reflectance halves its L*, exactly, which is not how lightness behaves anywhere else in the space and is not how anybody reading a table of black points expects it to behave — nor how a Munsell value scale behaves. Between L* 8 and L* 100 halving the luminance costs about twenty per cent of the lightness; below L* 8 it costs half.

An ordinary room lifts a display’s black past the break, which is a second reason the break is not academic. A grading suite in a dark surround works below it and a living room works above it, so the same display’s shadow detail is described by two different functions in the two places, and the difference is not a matter of degree.

The break is relative, so it moves with the white #

One further property of the break makes it harder to reason about than a fixed luminance would be, and it follows from the definition rather than from anything anybody chose.

ε is a relative luminance: the sample’s luminance divided by the white’s. So the break is not at a candela value; it is at 0.886 per cent of whatever the calculation is dividing by, and every colour management operation that changes the white changes where the break sits in absolute terms.

Media-relative colorimetry divides by the substrate, which puts the break at 0.886 per cent of the paper. Absolute colorimetry divides by the illuminant, which puts it lower. A display calculation divides by the display’s own white, which is a different quantity again, and a display in an ordinary room has its black lifted while its white stays put, so the same physical black crosses the break as the room lights come on.

The consequence is that a sample can be inside the straight piece in one calculation and outside it in another, with no change to the sample. A proof measured against its own paper and against the production stock is exactly that case: two divisors, two break positions, one sheet.

That is not an error in either calculation. It is a reason not to treat inside the straight piece as a property of a sample, which is how the phrase reads and is not what it means.

What it does to a difference between two blacks #

The place this becomes arithmetic rather than commentary is a comparison of two dark samples, which is what a press operator does every time a black is approved.

Two blacks at relative luminances of 0.0020 and 0.0026 differ by thirty per cent in light. Inside the straight piece their lightnesses are 1.81 and 2.35, a difference of 0.54 — exactly thirty per cent of the smaller, because the function is linear. Under a pure cube root the same pair would read −1.38 and −0.05, a difference of 1.33, two and a half times as large.

So the splice makes dark samples read as more similar than a cube root would say, and by a factor that grows as the samples get darker. Whether that is right is a question about vision that the splice was not designed to answer: it was put there so the function would have a finite slope at zero, and the compression of differences in the shadows is a side effect of the repair.

The practical form is that a press operator comparing two blacks by ΔE₀₀ is working in the one part of the space where the formula’s lightness term is a linear function of luminance, and where a thirty per cent error in the measurement produces a thirty per cent error in the lightness rather than a tenth of one. The shadows are the region where the instrument’s own noise passes through least attenuated, which is the opposite of the usual worry and follows directly from the splice.

The derivative that is not there #

The splice is C¹ by construction and C² by nothing at all. The second derivative of the compression is exactly zero throughout the straight piece — a straight line has no curvature — and it is about −500 just above the break, in the units of the compression against relative luminance.

That is the derivative the composition of two departures is built on. The local metric that predicts what two deviations cost together is a second-order object: it is the leading correction to the assumption that a difference formula is a straight distance, and a function whose second derivative jumps is a function whose local metric jumps with it.

So a pair of samples straddling the break has no single local metric, and the cosine rule the previous essay validated is not available for it. That is a small class of pairs and it is not empty: a proof compared against a press sheet in the shadows is exactly such a pair, and so is a display’s black measured in two rooms.

The honest statement is that the composition arithmetic works above L* 8 and is undefined across the break, and that nothing in this collection had to know that before now because nothing had asked two departures to compose.

The rod intrusion is the departure this bears on hardest, because it is the one whose physical premise is a dim room. Its price behaves like the others down to the break and then stops rising, so the departure that most belongs in the shadows is the one whose cost the shadows do not inflate.

Why nobody notices #

A splice that is in force below one per cent of a white is invisible in almost every calculation anybody performs, and there is a reason it stays invisible even for people who work in the shadows.

The break is quoted in relative luminance, which is a quantity nobody reads. Instruments report reflectance or CIELAB; software reports CIELAB or a code value; a specification quotes L*. Between the working variable and ε there is always a conversion, and the conversion is exactly the function under discussion — so asking whether a sample is below the break means inverting the compression, which is the step that makes the question look harder than it is.

The other reason is that the splice is good. It is continuous, it is differentiable, and it deviates from the cube root only where the cube root has stopped being usable. A function that has been repaired that carefully leaves no symptom to notice: there is no kink in a lightness ramp, no discontinuity in a gradient, no artefact in an image. The whole of the evidence that it is there is arithmetic.

A repair with no symptom is a repair nobody audits, and this collection has now found three of them — the splice here, the two constants under a wavelength grid, and the interpolation rule inside a profile. Each is defensible, each is undocumented at the point of use, and each changes an answer in a corner where somebody works.

What was computed, and how #

The compression is implemented from the standard’s own rationals rather than from decimal approximations, which is the only way the slope identity can be checked at all: entering ε as 0.008856 and κ as 903.3 makes the two slopes differ in the fourth figure and turns an exact statement into an approximate one.

The blacks are this collection’s own measurements where it has them and quoted figures where it does not. The press black is computed by the halftone model, the display black from a stated contrast ratio and peak luminance, and the projected black from a cinema specification; each is converted to relative luminance through the inverse of the compression and compared against ε directly rather than through L*.

The price floor is measured by putting a fixed deviation along the luminance axis on a series of neutrals and reading the difference formula’s answer. The residual variation inside the straight piece — 0.17 per cent — is not numerical noise; it is ΔE₀₀’s own lightness weighting, which has a term in L* that does not know about the splice and varies slowly across it.

The two constants, read as a design #

It is worth spending a paragraph on how ε and κ were arrived at, because the reasoning is recoverable from the numbers and is a small lesson in how a standard is built.

Two conditions are wanted: the straight line must meet the cube root in value, and it must meet it in slope. A straight line through the origin has one free parameter, its slope, so one condition can be satisfied by choosing it and the second cannot — unless the break position is also free, which it is. Two free parameters, two conditions, one solution.

Solving them gives a break at (6/29)³ and a slope of (29/3)³/116, which are 216/24389 and 24389/3132. Multiplying the second by 116 to get the standard’s κ gives 24389/27. Neither number was chosen; both were solved for, and 29 and 3 and 6 are what comes out of requiring a line and a cube root to be tangent at a point through the origin.

That is why the standard prints them as rationals. A decimal ε of 0.008856 and a decimal κ of 903.3 satisfy neither condition exactly, and a program using them has a compression with a small kink in it — about a part in ten thousand in slope, which is far below anything visible and is exactly the kind of residual that makes an identity fail a test written to nine figures.

This collection’s own machinery uses the rationals for that reason, and the assertion that the two slopes agree is written at the floating-point floor rather than at a tolerance somebody picked.

Where the model stops #

Everything here is about CIELAB. CIELUV has the same splice with the same two constants and inherits the whole argument. CIECAM16’s lightness is a different compression with a different shape and no straight piece, so an appearance model reading a press black is doing something this essay does not describe, and what it is doing is not obviously better.

The break’s position is a fact about the standard rather than about vision. Nothing in the psychophysics says that lightness becomes linear in luminance below one per cent of a white — and the exponent was never what the argument was about; the straight piece is a numerical repair, and treating its slope as a perceptual statement would be reading an implementation detail as a finding.

And the second-derivative jump is stated in the compression’s own units. Turning it into a bound on how wrong a composed estimate can be across the break would need the whole difference formula differentiated twice, which is a page of algebra this collection has not done.

The generalisation #

The habit is about the piece of a function that a description leaves out.

A function that is quoted by its formula is quoted by the formula that describes most of it, and the exceptions live at the ends. A logarithm with a floor, a power law with a linear toe, a ratio with a guard against a zero denominator — all of them exist for a numerical reason, all of them are invisible in the one-line description, and all of them are in force exactly where the interesting cases are, because the reason they were added is that the interesting cases broke the pure form.

The move is to find the ends before using the middle. It takes a minute: evaluate the function’s derivative near its limits and see whether it is finite.

The failure mode is not an error in the middle of the range. It is a conclusion about the ends drawn from the description rather than from the implementation, which reads as though it had been measured. A press black’s lightness is not a cube root of anything, and every table that lists one alongside a mid-tone is listing two numbers computed by two different functions.

Who found it, and when #

The splice arrived with CIELAB in 1976 and the exact rationals arrived later: the original recommendation stated the constants in decimals and the exactness was tidied up in subsequent CIE publications, which is why ε and κ appear as 0.008856 and 903.3 in older texts and as 216/24389 and 24389/27 in newer ones. The slope-matching property is stated in the standards literature and is not usually explained.

The observation that essentially every delivered black is below the break does not appear in that literature, and the reason is probably that colorimetry’s centre of gravity is reflective samples on paper, where L* 8 is genuinely dark. Displays and projectors moved the working point and the arithmetic did not move with it.

Where the ladder goes next #

The break is one place where the arithmetic changes underfoot and it is announced in a standard. The next is not announced anywhere: a tolerance is a number and it acts on three, so a stated ΔE₀₀ is a closed surface in tristimulus values, and the size and shape of what it accepts is decided entirely by where in the space it is applied.