The appearance model has no straight piece

CIELAB’s lightness is described everywhere as a cube root, and below L* 8 it is not one. The straight piece under the cube root is a linear segment spliced on with two constants chosen so that it meets the cube root in value and in slope, and it exists for a practical reason: a pure cube root has an infinite slope at zero, so a deviation of any size near black would cost unboundedly much. The splice puts a floor under the price. Every black a delivery chain reaches — a press black at L* 2.4, a projected one at 1.1 — sits on that floor.

A price with no floor #

CIECAM16’s lightness has no linear segment near black, so the price of a deviation there rises without limit, and the room sets how fast.

• Below L* 8 CIELAB’s lightness price is exactly flat, at 514 units of L* per unit of relative luminance.
• The model’s price over the same range rises by a factor of 5.0 in an average surround, 6.8 in a dim one and 8.3 in a dark one, and keeps rising below the range drawn.
• It follows the luminance to the power 0.42·c·z − 1, which is −0.442 in an average surround and −0.576 in a dark one; the measured slopes agree with the derived ones to the fourth decimal place.
• On a projected black at L* 1.1, a unit of the model’s lightness allows a change of 78 per cent of the black’s own luminance in a lit room and 44 per cent in a dark one, against 160 per cent for a unit of L*.

Where the model’s black comes from #

The model’s lightness is built from an achromatic signal, and the construction near zero decides everything here.

Each of the three adapted cone signals passes through a hyperbolic response: 400 times t over t plus 27.13, where t is the signal scaled by the luminance-level factor and raised to the power 0.42, plus a constant 0.1. The achromatic signal is twice the first response plus the second plus a twentieth of the third, minus 0.305, all multiplied by a background factor.

At zero luminance the three responses are each 0.1, and 0.1 times two plus one plus a twentieth is exactly 0.305. The constant subtracted is chosen to cancel them, so the achromatic signal is zero, lightness is zero, and the model’s black is a genuine zero rather than a small number. Computed at XYZ nought the lightness comes out at 2 × 10⁻²², which is the arithmetic’s own rounding.

That exactness is a design decision and a good one. It also fixes what happens just above zero. The hyperbola is linear in t near zero, and t is the signal to the power 0.42, so the achromatic signal just above black goes as the luminance to the 0.42. Lightness is that signal, relative to the white’s, raised to the power c·z, where c is the surround’s exponent and z is 1.48 plus the square root of the background’s relative luminance. So lightness near black goes as the luminance to the power 0.42·c·z.

In an average surround with a twenty per cent background, c is 0.69 and z is 1.927, and the exponent is 0.559. A power below one has an infinite slope at zero. The model’s lightness near black is a root, not a line, and its derivative — which is what prices a deviation — goes as the luminance to the power −0.441.

The price, drawn #

A deviation is priced here as the change in lightness per unit change of luminance on a neutral, which is the same question the essay on the straight piece asked of CIELAB. The lightness scales are not the same size — L* runs to 100 at the white, and so does the model’s J — so the comparison is between two numbers with the same units, and the interesting part is their shapes.

At L* 40 the two prices are close: 98 for the model and 94 for L*. By L* 8 the model has fallen below CIELAB, 406 against 513, because its compression near the break is gentler. Below L* 8 CIELAB stops at 514 and the model does not stop: 1,050 at L* 1.1, 2,070 at L* 0.25. The two cross near L* 5 in a lit room. Above the crossing the model is the more forgiving scale and below it the less, and the difference grows without limit as the black deepens.

The room is in the exponent #

The exponent has c in it, and c is the one number the three standard surrounds are most distinguished by.

In a dim surround c is 0.59 and the exponent is −0.522. In a dark surround c is 0.525 and the exponent is −0.575. Across L* 0.25 to 8 the price rises by a factor of 5.0 in an average room, 6.8 in a dim one and 8.3 in a dark one. At L* 0.25 it is 2,070 in a lit room, 3,360 in a dim one and 4,520 in a dark one.

That is the model saying something it was built to say. A darker surround makes dark colours look darker and lowers apparent contrast — the surround is three rows of a table — and the way the model does it is by lowering the exponent on lightness. A lower exponent is a steeper root at zero. The same black in a cinema is priced more steeply than in an office, and no straight piece stops either of them.

Four blacks #

The divergence matters because blacks are not at the edge of anybody’s work. A black that is not black put a four-colour press’s deepest black at L* 2.4, and a delivery chain meets three others.

On a projected black at L* 1.1 the model charges 2.0 times what L* charges in a lit room, 2.9 in a dim one and 3.6 in a dark one. On a press black at L* 2.4 the ratios are 1.4, 1.9 and 2.2. On a display’s black in an office at L* 6.3 the model is cheaper in a lit room, at 0.89, and dearer in dim and dark ones; on an uncoated sheet’s shadow at L* 12 it is cheaper in lit and dim rooms and level in a dark one.

Measured against ΔE₀₀ rather than L* the ratios are larger, because ΔE₀₀ divides a lightness difference by a weight that is itself large far from the middle of the scale: 3.5 on the projected black in a lit room and 6.2 in a dark one. That weight is a separate matter from the splice, and it drifts by about seven per cent across the straight piece on its own — which is why a price flat in L* is not quite flat in ΔE₀₀.

What a lightness unit allows on a black #

A price is a derivative. The version of it a specification uses is its inverse: how much the luminance may change before the lightness changes by one unit.

At L* 1.1 one unit of L* allows the black’s luminance to change by 160 per cent of itself; one unit of the model’s J′ allows 78 per cent in a lit room, 55 in a dim one and 44 in a dark one. At L* 2.4 the numbers are 73 per cent for L* and 52 for the model in a lit room. By L* 6.3 the model allows slightly more than L*, and from there up the two agree to within a few per cent.

A stated lightness is two requirements found that a precision written in the model’s lightness demands nine times more of a process at the dark end of the scale than at the light end, and put the reason in the compression’s slope. The measurement here is the same slope approached from below, and it adds the part that essay’s range did not reach: below L* 8 the requirement keeps tightening in the model and has stopped tightening in CIELAB. A tolerance of one unit written in J′ on a projected black is about three and a half times as demanding, in a dark room, as the same tolerance written in L*.

A black approved in one room and seen in another #

The room’s share of the exponent turns one written tolerance into several requirements, and one unit in another room is not the same unit. A black is graded in a dark suite, reviewed on a monitor in a lit office and projected in a cinema, and a tolerance of one unit of J′ means something different in each.

On the projected black at L* 1.1, one unit of J′ allows 78 per cent of the black’s own luminance in a lit room and 44 per cent in a dark one. A black that has drifted by sixty per cent between the grading suite and the review is inside tolerance when the reviewer checks it in the office, and outside it in the room it was graded for. The review is not careless. It is carried out in a room whose exponent prices a deviation on a black 1.8 times more kindly.

The direction decides which approvals travel. An approval made in the darkest room the work will be seen in is safe to carry into lighter rooms, because every lighter room prices the same deviation more cheaply. An approval made in a lit room says nothing about a dark one. CIELAB cannot express any of this: below L* 8 its price is the same number in every room, because no room enters its formula.

An even ramp into black is not even #

A shadow gradient built from equal steps of L* below 8 is a ramp of equal luminance steps, because below L* 8 the scale is a straight line. In the model the same steps are not equal.

Near black the model’s lightness goes as the luminance to the power 0.559 in a lit room, 0.478 in a dim one and 0.425 in a dark one. On that power law alone, which the measured price follows to within a tenth across the straight piece, a ramp of eight equal luminance steps up from black has a first step 4.3 times the size of its eighth in J′ in a lit room, 6.0 times in a dim one and 7.5 times in a dark one. In CIELAB all eight are the same size.

So the darkest step of a ramp that is even in CIELAB is the largest step in the model, by a factor the room sets. A tool that spaces shadow values evenly in L* — and a gradient is a path through whatever space the tool chose — has spaced them unevenly in the scale an appearance tolerance is written in, and the unevenness is concentrated in the step next to black — which is where a visible step in a gradient would appear first, if one appears at all.

Two sensible choices #

Neither scale is wrong about black, and the difference is a disagreement about what a lightness scale is for.

CIELAB’s straight piece is numerical engineering. The CIE wanted a scale whose derivative is finite everywhere, so that colour differences near black are well-behaved in computation and in measurement, and a straight segment below a threshold was the simplest way to get one. It was never offered as a claim about vision.

CIECAM16’s root is a claim about vision. Near absolute darkness the visual response to small luminance increments is steep — a threshold for detecting light against black is a tiny luminance — and a compressive power law captures that. The model’s offsets are arranged so that zero light is zero lightness, and the power law is left to do what it does at zero.

The difference turns into a practical one only when a tolerance is written in one scale and a process is controlled in the other. A press is controlled in density, and density is a logarithm of reflectance, which is an even steeper function at black than either lightness scale. A dot is larger than it was asked to be, and near black it is also measured less reliably.

What an instrument can hold on a black #

The practical limit is measurement, and it runs into the model’s root first.

An instrument’s repeatability on a dark patch is set by stray light and dark signal, which are roughly constant in reflectance units. A spectrophotometer that repeats to a few hundredths of a per cent of reflectance on a white tile repeats to about the same absolute amount on a black one, and on a black of one per cent reflectance that is a relative uncertainty of a few per cent of the black’s own luminance.

In L* that uncertainty is a fraction of a unit even at L* 1.1, because a unit of L* allows 160 per cent. In J′ in a dark room a unit allows 44 per cent, so the same instrument noise is several times closer to a unit of the model’s lightness. A tolerance written in the appearance model on a cinema black can come within reach of the instrument used to verify it — which a stated lightness is two requirements predicted for the light end of the scale and which turns out to happen at the dark end too, for the opposite reason.

What was computed, and how #

Each price is a finite difference on a neutral: the white’s own tristimulus direction scaled by a small fraction of the neutral’s own luminance, pushed through each lightness scale, with the change in lightness divided by the change in luminance. CIELAB’s lightness is L*; the model’s is J′, the lightness coordinate of CAM16-UCS, which near black is J multiplied by 1.7.

The viewing conditions are the model’s defaults — an adapting luminance of 100 candelas a square metre and a background at twenty per cent of the white — with the surround set to each of its three tabulated values. The derived exponent is 0.42 times c times z, less one, computed from the same viewing-condition object the model uses; the measured one is the slope of the logarithm of the price between relative luminances of 10⁻⁷ and 10⁻⁶.

The blacks are the four this collection’s delivery essays quote, at their stated L*. The allowances are the reciprocal of the price, expressed as a share of the neutral’s own luminance.

Where the measurement stops #

Everything is on the neutral axis. A dark saturated colour has chroma to deviate in as well, and the model’s chroma near black is shaped by the same root through its achromatic denominator; that is not computed here.

The model is being asked about luminances its data do not reach. The corresponding-colour and scaling experiments CIECAM16 was fitted to are photopic, and near absolute darkness rods take over, which no photopic appearance model represents. The root near zero is the model’s extrapolation of a fit, and the divergence measured here is a property of the formula rather than a measured property of vision below a few candelas a square metre. Colour goes first in the dark, and so does the model’s warrant.

And a price per unit of luminance is not a threshold. Whether an observer can see a change of 44 per cent in a black depends on the black’s absolute luminance, the adaptation state and the surround, and the model’s lightness scale is only one ingredient in that.

The habit #

The habit is about the behaviour of a formula at the end of its range.

Every scale that compresses — a lightness, a loudness, a logarithm of anything — has to decide what happens at zero, and the decision is usually made for a reason nobody records: a threshold avoided, a singularity patched, a zero preserved. Two scales that agree across the middle of their range can disagree without limit at the end, because one was patched and the other was not.

The move is to ask each scale for its derivative at the bottom of the range before writing a tolerance there. It is one line of arithmetic.

The failure mode is to carry a tolerance across from one scale to the other on the strength of their agreement in the middle. Two scales that match at a mid grey can differ by a factor of three at a projected black, and nothing about the mid grey says so.

Who noticed it first #

That CIELAB’s linear segment exists to avoid the cube root’s infinite slope at zero is stated in the CIE’s own documents. That power-law lightness scales have infinite slope at zero is immediate from their form.

That CIECAM16’s offsets cancel exactly at zero, that its lightness near black therefore goes as a root with an exponent containing the surround, and what that does to a tolerance on the blacks a delivery chain actually reaches, are consequences of the published equations. The sources consulted here do not state the exponent, and it is one derivative away from the model’s definition.

Still open: what the dark end of the model should do #

The model’s behaviour below a few candelas a square metre is extrapolation, and the honest repair is data rather than a patch — scaling and matching experiments on deep blacks in dark surrounds, where a cinema or a grading suite actually operates. Whether the data would ask for a straight piece, a steeper root or a rod term is not something the formula can say about itself.