Two rooms with one lightness scale

A dark background moves every difference and no match reduced the background’s whole effect on CIECAM16’s lightness to one exponent, c z, and held the surround fixed throughout so that only the background moved. That was the right way to isolate the background and it leaves half a sentence unexamined, because c is the surround’s and z is the background’s and the two appear only as a product.

A product of two parameters is one parameter. Whatever else the surround and the background do, they cannot both be settings on lightness.

The rooms fall into classes, and lightness cannot see the class #

Two of CIECAM16’s four viewing-condition parameters reach lightness only through their product, so rooms with different surrounds and different backgrounds return identical lightness for every sample — and differ by a fifth in chroma and in brightness.

• A television in a lit living room against a mid grey is exactly a print on a desk against a background of 2.8, and exactly a projection in a dark room against a background of 47.0. The largest lightness any of 168 samples differs by across those three rooms is 2.8 × 10⁻¹⁴.
• The same three rooms disagree about chroma by a median factor of 0.76 and 0.70, and by up to 17.5 chroma units on the most colourful sample.
• Brightness differs by a fixed factor — 0.8146 and 0.7848 — which is two constants and nothing about the sample, exactly for every one of them.
• The surround reaches lightness by a second door and it carries a thousandth of a unit. Letting each room compute its own degree of adaptation opens the class by 0.0028 and 0.0056 lightness units.
• Not every room has two class-mates. A print on a desk against a mid grey has no counterpart in a dark surround at any background, because each surround’s impact multiplies a base that runs only from 1.48 to 2.48.

One product, and the two things that make it one #

Lightness in CIECAM16 is J / 100 = (A / Aw) ^ (c z), and two facts about that expression are what this whole essay rests on. The first is that the ratio A / Aw contains no background: the induction factor Nbb multiplies the sample’s achromatic response and the white’s alike, and divides out. The second is that z = 1.48 + √n is the background’s only other appearance in the achromatic path, and c is the surround’s.

So the room enters lightness as the single number p = c z, and two rooms with the same p have the same lightness scale. That is not an approximation and it is not a near-degeneracy of the sort a fit produces — the objective nobody chose is about those, and they are a different animal. It is the model’s arithmetic, and the check is a residual rather than a correlation.

With the degree of adaptation held, the class is exact to 2.8 × 10⁻¹⁴ lightness units over 168 samples. That is the arithmetic’s own noise and the claim is therefore not that the three rooms are similar; it is that as far as lightness is concerned they are the same room.

The second door #

The figure carries a second column, and it is there because the first version of this measurement was not exact and the reason was worth keeping.

The surround supplies c, and it also supplies F, and F sets the degree of adaptation — D = F · (1 − exp((−La − 42) / 92) / 3.6). The degree reaches the adapted cone responses, and the adapted cone responses are what A and Aw are built from, so A / Aw is background-free and not quite surround-free. The surround has two routes into lightness and the exponent is only one of them.

Left to compute its own degree, a room parts from its class-mate by 0.0028 lightness units at worst and 0.00057 at the median, against a scale that runs 0 to 100. So the second door exists and admits a thousandth of a unit, which is the size that turns the equivalence from a statement about algebra into a statement about rooms. A discount nobody measured found the degree in an ordinary room to be 0.94 rather than 1 and traced what that costs elsewhere; here it costs nothing, because both rooms are lit by the light they are adapted to and the degree only decides how completely a white that is already the adopting white is discounted.

It is worth being exact about what that implies. The class is exact in the exponent and inexact overall, by an amount three orders of magnitude below anything a person or an instrument would report. A reader who needs the exactness can have it by stating the degree, which a specification is entitled to do and rarely does.

What separates them #

If lightness cannot tell two rooms of a class apart, something has to, or the model would have three redundant parameters rather than two.

The television room reads a median of 0.763 times the print room’s chroma and the dark room 0.696, with the most colourful samples differing by 13.5 and 17.5 chroma units. That is not a small residual; it is a fifth to a third of the quantity. Chroma carries the surround’s chromatic induction factor Nc and the background’s own multiplier (1.64 − 0.29ⁿ)^0.73 outside the exponent the class holds fixed, and neither cancels.

Brightness separates them too, and it separates them in the cleanest possible way.

The ratio is 0.8146 and 0.7848, and it is the same for every sample to 10⁻⁹. Brightness is Q = (4/c) √(J/100) (Aw + 4) FL^0.25, and across a class J and FL are both fixed, so the whole ratio is (c_ref / c) · ((Aw + 4) / (Aw_ref + 4)) — two constants. A lightness class is a brightness class rescaled, which means brightness distinguishes the rooms without distinguishing any sample from any other in them.

So the three correlates give three different answers to whether two rooms of a class are the same viewing condition: lightness says yes, brightness says no by a fixed factor, and chroma says no by an amount that depends on the sample. That is the useful summary and it is also the practical one, because a specification usually quotes exactly one of the three.

The fourth parameter does not join the product #

A model with four viewing-condition parameters and one product in it invites the question of what the other two are doing. The background and the surround are two of the four; the third is the adapting luminance and the fourth is the white.

The white is outside all of this: it is what the sample is measured against rather than a property of the room’s geometry, and moving it moves everything. The adapting luminance is the interesting one, because it does reach lightness — the model’s compression constant FL is computed from it, and FL scales the argument of the post-adaptation nonlinearity for the sample and for the white alike. That nonlinearity is not a power, so A / Aw does not come out free of the adapting luminance the way it comes out free of the background. The question is by how much.

Over a range from 2 to 4,000 candelas a square metre the largest lightness any of the 168 samples differs by is 0.59 units, and the best exponent relating one level to another stays within 0.014 of one. Removing that best exponent leaves 0.14 units at the extreme and 0.04 across the ordinary range.

So the adapting luminance is neither a third factor in the exponent nor a route into lightness worth the name. Of the model’s four viewing-condition parameters, lightness sees one product and one near-nothing. That is a stronger statement than the classes alone: it says the classes are the whole of the ambiguity rather than a slice through a larger one, and that a specification pinning lightness has pinned exactly one number about the room.

It also says where the level went instead. The same FL appears in brightness as FL^0.25 and in colourfulness as the factor that turns chroma into M, and brighter looks more colourful is that arrival measured. A bright room looks brighter and more colourful without looking lighter, which is a prediction the model makes cleanly and which its parameterisation makes almost inevitable.

Which rooms have how many partners #

The classes are not all the same size, and which rooms have two class-mates rather than one is arithmetic rather than accident.

The base z runs from 1.48 to 2.48 whatever the surround, so each surround’s reachable exponents are that interval times its own impact: 1.023 to 1.711 for an average surround, 0.875 to 1.463 for a dim one, 0.779 to 1.302 for a dark one. All three reach the band from 1.023 to 1.302, which is where a class can have three members; above it only the brighter surrounds go and below it only the darker.

A print on a desk against a mid grey has an exponent of 1.330, which is inside a dim surround’s range at a background of 59.9 and outside a dark surround’s altogether. A television against a mid grey is 1.137 and has both: a print against 2.8 and a projection against 47.0. A projection in a dark room against a mid grey is 1.012, which an average surround cannot reach — it is below even the darkest wall an average surround can be given.

The pattern is that the ordinary rooms — the ones a standard names, all at a background of 20 — are spread across the exponent range rather than clustered, and each one’s partners are at conspicuously different backgrounds. Nothing about a room’s description tells a reader which class it is in. The description names a surround and a wall; the class is a product.

What a specification has failed to state #

The practical consequence is about what a document says rather than what the model does.

A specification that names a surround and a background has named a room exactly and nothing here applies to it. One that states a lightness — a target value, a tolerance in lightness units, a tone curve fitted to an appearance scale — has named a class, and the class holds rooms whose chroma differs by up to 17.5 units.

That happens more often than it sounds, because lightness is the correlate specifications reach for first. A tone reproduction curve is a map from input to lightness; an accessibility contrast requirement is a statement about lightness; a print standard’s shadow specification is a lightness floor. Each of those pins c z and leaves the two factors free, and a laboratory reproducing the work in a dim room with a lighter wall has reproduced the lightness exactly and not the appearance.

The converse is the more useful reading. Two laboratories quoting different viewing conditions may be quoting the same one, as far as every lightness they report is concerned, and a disagreement between them about lightness cannot be the room. A stated lightness is two requirements separated a target’s value from its precision; this separates a room’s description from the part of it a lightness measurement can see.

And a third: a reader who wants to change how a picture’s tones fall without changing its colours has exactly one knob that does it, and it is this one. Moving along a class changes nothing; moving across one changes lightness and chroma together. The model has no way to change lightness alone, because p is the only door and everything that opens it opens the others.

Why two parameters and not one #

The obvious objection is that a model with a redundant parameter should lose it, and the objection is wrong for a reason worth stating.

c and z are not redundant. They are redundant in lightness, and they separate in chroma and in brightness, so the pair carries more information than the product does. What is true is narrower: the model has four viewing-condition parameters and three of them reach lightness, two of those only as a product — so lightness has fewer degrees of freedom than the viewing condition does, and a fit of lightness data cannot recover a room.

That is an identifiability statement of exactly the shape where a camera is blind to itself makes about a sensor and the condition chooses no axes makes about a fit: a parameter that only ever appears inside a combination is not recoverable from data the combination explains. The direction it applies in here is unusual, because the model is not being fitted — it is being specified, and what cannot be recovered is what a specification failed to say.

How the classes were computed #

The rooms are taken from the model’s own arithmetic rather than searched for. Given a target exponent p, each surround’s required base is z = p / c, and the background follows as Yb = (z − 1.48)² · Yw; a base outside 1.48 to 2.48 means that surround cannot reach the exponent at any background and is reported as absent rather than clamped.

The samples are the widened surface family under D65 at full density, 168 surfaces, read at an adapting luminance of 100 candelas a square metre throughout — so the adapting luminance, which is the fourth viewing-condition parameter and reaches lightness through FL rather than through the exponent, is held rather than traded against. Lightness, chroma and brightness are compared sample by sample against the first member of each class, and the residuals reported are the worst over all samples rather than a mean, because the claim is about every sample.

The degree of adaptation is held at one for the exactness measurement and left to each room’s own formula for the second column, which is the difference between a claim about the exponent and a claim about the model.

What this leaves out #

The classes are CIECAM16’s. Any model whose lightness depends on the room through a single product has them; one that lets the surround into lightness twice in a way that does not cancel does not, and the second door here is exactly such a route with nothing in it.

The three surrounds are the three the standard tabulates, and the ranges above are computed from their three impacts. The surround is three rows of a table found that a room between two rows gets its constants by an interpolation the model was never fitted at; a continuous surround makes the classes continuous too, and the reachable ranges become a band rather than three intervals.

Nothing here says what an observer does. A class is a claim that two rooms produce the same lightness by the model, and whether a person reports the same lightness in a dim room against a light wall as in a lit room against a dark one is an asymmetric matching experiment that nobody has run in these terms. The model’s prediction is unusually sharp — identical, for every sample, not merely close — which makes it unusually easy to refute.

And the backgrounds the classes require are sometimes extreme. A background of 2.8 is a near-black mount and 59.9 a pale one; both are ordinary, and 47.0 for a projection surround is a bright wall in a dark room, which is not.

Still open: whether a person’s lightness follows the product #

The model says a television in a lit living room against a mid grey and a print on a desk against a background of 2.8 have one lightness scale, exactly. That is a prediction about people stated as an identity, and identities are the easiest predictions to test because a single reliable difference refutes them.

The experiment is asymmetric matching with lightness as the only judgement. Put a set of neutral samples in each of the two rooms, have observers adjust a sample in the second until it matches a sample in the first in lightness alone, and plot one room’s settings against the other’s. The model predicts the identity line for every sample and every level. A systematic departure would say the two parameters are not one parameter for an observer, and the shape of the departure would say which of them the model has mis-weighted.

The second question the same session would answer is the one about chroma. The model predicts a fixed ratio in brightness and a sample-dependent one in chroma across the same pair of rooms, so a colourfulness scaling run alongside the lightness matching gets both for the price of one booth — and a disagreement about chroma where lightness agrees exactly would be the cleanest evidence anybody has that the model’s two chromatic factors are doing different work.

A product is one parameter until something separates it #

The habit is about reading a model’s parameter list for what its outputs can actually distinguish.

A model with four settings looks like four degrees of freedom, and a document quoting three of them looks nearly complete. What matters is how the settings enter each output: two parameters appearing only as a product are one parameter as far as that output is concerned, however separately they are described, and the remaining outputs are the only things that can tell them apart.

The move is to write each output in terms of the parameters and look for combinations, before asking what any of them is worth. It takes a few lines, it is exact rather than statistical, and it produces two things at once — the equivalence classes, and the list of which outputs break them. Both are more useful than a sensitivity analysis, which measures how much each parameter moves an answer and cannot see that two of them move it together.

The failure mode is to quote a parameter and think it has been stated. A background of 20 is a complete statement about a wall and half a statement about a lightness scale, and the other half is in the surround — where nobody is looking, because the surround is described as a different kind of thing.