The worst case is where the box stops

A worst case is a maximum over a set, and this collection has been quoting the maximum over a list.

Two other boxes are available for the same four parameters, and the whole argument is that the answer is a property of which one is used.

The claim #

The census of illumination changes this collection scores adaptation models against is fourteen changes taken from a family. Searching the family instead of listing four members of it reaches 21.3 ΔE00 against the census’s 3.375 — and the search terminates on the boundary of the parameter box in every direction that has one, so the family has no worst case at all.

• Six times the listed worst, under a box restricted to what an ordinary architectural pigment can be.
• Eight times, under a box restricted only by the arithmetic: 28.4 ΔE00.
• The search ends against a wall. Two of four parameters at a bound under the narrower box, three of four under the wider one, with the answer larger in the wider one — which is what a quantity with no interior maximum does.
• The one parameter that comes to rest inside is the depth, and it does so for a physical reason: a wall that absorbs everything reflects nothing, and a light that has been extinguished cannot change anything.
• So a worst case is only defined relative to a stated family, and this collection had not stated one.

What the census is, and what it is a sample of #

A change of light is exactly a 3×3 matrix on this collection’s three-dimensional family of surfaces, and an adapted observer answers it with a diagonal in some basis. The residual — how much CIEDE2000 is left after the gain — is the number every comparison of adaptation bases here turns on.

It is computed over a census: two daylights, a thermal radiator at the same colour temperature, tungsten, three discharge lamps, one bounce off a green wall and a second off the same wall, a red wall, and the eye’s own macular and lens filters. Fourteen changes, equally weighted, with a dimming as the control.

The mean of that census is a reasonable summary and this essay does not disturb it. What it disturbs is the worst row, which has been quoted as the worst case an adapted observer meets and used to justify the shape of several arguments.

Three of the fourteen are wall bounces. A wall is a reflectance: a notch at some centre wavelength, of some width, of some depth, on some base. That is four numbers, and applying it once or twice or three times is a fifth. The three walls in the census are three points in a five-dimensional family, and nothing about them was chosen to be extreme — they were chosen to be recognisable.

Searching the family #

The search minimises the negative of the residual over the four continuous parameters, with a simplex and restarts, and takes the bounce count separately because it is an integer and a simplex would interpolate it. Two parameter boxes are used deliberately, because one would hide the result.

Under the paint box — a centre between 420 and 680 nm, a notch between 25 and 150 nm wide, a depth between a tenth and nine tenths, and a base reflectance between 5 and 85 per cent — the worst is 21.29 ΔE00, at three bounces off a wall with a 25 nm notch at 420 nm, 81 per cent deep on an 11 per cent base.

Under the wide box — the ranges the arithmetic can evaluate at all — it is 28.38, at three bounces off a 10 nm notch at 400 nm, 87 per cent deep on a 2 per cent base.

The listed worst is 3.375.

Why there is nothing to find #

The parameters that end against a wall are the centre wavelength, the notch width, and — in the wider box — the base. Each of them is monotone: the residual rises as the notch narrows — a narrow feature is exactly what three numbers cannot see —, as it moves towards the short-wavelength end, and as the wall gets darker.

None of those is mysterious. A narrow notch is further from anything a diagonal can undo, because a diagonal acts on three broad channel sums and a narrow feature moves those three sums in a combination no diagonal reproduces. A short-wavelength notch acts on the channel whose gain swings most between illuminants and where the observer’s own sensitivity is falling steeply, so a small change in the spectrum is a large change in the ratio. A dark wall multiplies the whole spectrum down, and since the residual is computed after the adapted observer has normalised on the white, a darker wall is not dimmer — it is a larger relative distortion.

The depth is the exception, and it is the exception that makes the rest legible. Push the depth towards one and the wall stops reflecting at that wavelength entirely; the light that would have carried the distortion is gone, and the change becomes a change to a narrower band of an already-narrow spectrum. So there is a genuine optimum in the depth, at 0.81 and 0.87 in the two boxes, and it is the only parameter that comes to rest inside.

Three monotone directions and one interior optimum is not a maximum. It is a corner, and the corner is at the edge of somebody’s opinion about what a wall is.

Why more bounces stop helping #

The bounce count does saturate, and the numbers are worth having because they say something about rooms rather than about search boxes.

Under the paint box the worst at one bounce is 8.82, at two 20.26, and at three 21.29. The second bounce more than doubles the first; the third adds five per cent.

The reason is the same as the depth’s. A bounce is an elementwise multiplication, so n bounces raise the reflectance to the nth power — and a reflectance below one, raised to a power, converges towards a spectrum that is zero everywhere except at its own maximum. Past two bounces the light in a strongly coloured room is already nearly monochromatic in the sense that matters, and squaring it again changes the tristimulus ratios very little.

A corner is much worse than a wall and a triple corner is barely worse than a corner. That is a fact about rooms and it is stable against the box, which the headline number is not.

What the mean does not notice #

There is a temptation to conclude that a census whose worst row understates the family by six times is not worth much, and it is worth resisting, because the mean is doing a different job and doing it adequately.

The mean over fourteen changes is an estimate of the typical residual an adapted observer is left with, and a sample mean is exactly what sampling is good for. Adding the searched worst to the census as a fifteenth row would move the mean by about one and a half units, which would swamp every comparison the mean exists to make — and it would be the wrong thing to do, because a change nobody ever meets should not dominate a summary of the ones everybody does.

The two numbers answer different questions and should not be computed from the same set. What does an adapted observer usually lose is a question about a representative sample, and the census is one. How badly can it go is a question about a family, and needs the family.

The failure was in reading one off the other: the worst row of a representative sample is not a worst case, it is the largest of the typical values, and it happens to be about a fifth of the way to the family’s edge.

What was computed, and how #

The residual is the one every other adaptation number here uses, and it predates this question by a long way, which matters: it was not written to make this comparison and could not have been tuned to it. The reflectance family is the same three-dimensional set of smooth surfaces every other adaptation number here is computed over, and the assertion that a change of light is exactly a matrix on that family holds to 10⁻⁹ on every row.

The wall model is a Gaussian notch on a grey base, which is a crude paint and is the same crude paint the census’s own three walls are made of — so the search is over the family the census was drawn from rather than over a different, more flattering one.

The assertion in the build makes three claims and deliberately avoids making a fourth. The searched worst must beat the listed worst by at least a factor of three; the search must terminate on a bound in at least two of four parameters in both boxes; and the wider box must return the larger number. What it does not assert is any value for the worst case, because there is not one.

What a searched worst does to a ranking #

The number is a property of the box. What is not a property of the box is what the search does to the comparison between adaptation bases, and it is the more useful half.

On the census mean the five bases in this collection’s table rank Bradford best at 1.140, CAT16 at 1.312, CAT02 at 1.323, Hunt–Pointer–Estévez at 1.584, and XYZ scaling worst at 2.373. That ordering is stable and is what colour management’s choice of Bradford rests on.

Searched, each basis against its own worst paint, the ordering breaks. XYZ scaling — the one with no basis in it at all, and the oldest mistake still shipping — reaches 18.2. CAT16 reaches 21.3. And Bradford does not reach a number: under a deep narrow notch, its middle row’s reading of the white passes through zero, so the gain is a division by nothing and comes back at −10¹⁹. CAT02 does the same thing at 5 × 10¹⁸, and Hunt–Pointer–Estévez stays positive and runs to 5,800, which is a gain no adapted observer applies.

Two of the five have a row that changes sign inside the family the census was drawn from, and the two are the two with the most negative entries. CAT16, which exists because CAT02 was withdrawn for going negative in practice, is one of the two that survives. That is a result large enough to want its own essay, and what this one contributes is the family it happens in.

The divergence is real and slow #

There is no worst case in this family is right and it leaves out the rate, which is the number a reader needs in order to know whether to worry.

Between the two boxes the notch narrows by a factor of 2.5, the base darkens by 5.5, the centre moves twenty nanometres shorter — and the residual rises by 33 per cent, from 21.29 to 28.38. Two parameters loosened by factors of two and a half and five and a half bought a third.

Extrapolating that rate says what a doubling would cost. It needs about 2.4 more widenings of the same size, which puts the notch at 2.7 nanometres and the base reflectance at 0.18 per cent.

Neither is a wall. A 2.7-nanometre absorption band is an interference filter, and a surface reflecting under a fifth of a per cent is a light trap. So the honest form of the finding has two halves and the essay gives one of them: the quantity has no maximum, and it is within a factor of two of its value at anything physically realisable.

That is a much more useful statement than either 21.3 or unbounded. It says the paint box’s number is not an artefact of where the box was drawn — moving the box a long way moves the answer a little — while still refusing to be a maximum. A slowly diverging quantity is quotable with a stated box and is not quotable without one, and the rate is what distinguishes it from a quantity that runs away.

The second bounce is worth more than the first #

The bounce series is 8.82, 20.26, 21.29, and its increments are the interesting reading: 8.82, 11.44, 1.03.

The second bounce contributes 30 per cent more than the first. That is unusual — a repeated multiplication normally has diminishing returns from the start — and it is what a corner is worse than a wall by more than twice means when the arithmetic is done.

The mechanism is in the contrast. The searched wall has a base of 0.11 and a notch floor of 0.0209, so the notch is 5.3 times darker than the rest after one bounce, 28 after two and 146 after three. The first bounce produces a spectrum a diagonal can still mostly follow; the second produces one it cannot; the third produces one that is no harder than the second because the light in the notch is already gone.

So the marginal cost peaks at the second bounce and collapses at the third, and both halves have causes. Rooms have corners and very few surfaces see three walls, which makes the two-bounce row the one that matters and the three-bounce row a headline that adds five per cent.

What the fifteenth row would actually do #

Adding the searched worst to the census as a fifteenth row would move the mean by about one and a half units is the essay’s reason for not doing it, and the arithmetic is exact.

Adding a 21.29 row to fourteen rows moves Bradford’s mean from 1.140 to 2.483 and CAT16’s from 1.312 to 2.644 — changes of 1.343 and 1.332, so about one and a half is right and the two bases move by nearly the same amount.

That last part is the argument the essay does not make. Because one added row moves every basis by the same 1.34, adding it would not change the ordering at all — it would inflate every mean by a common term and leave the comparison the census exists to make exactly where it was.

So the objection to adding the row is not that it would swamp the comparison; the comparison would survive. It is that it would make every reported mean a mixture of two questions, one about typical light and one about a corner nobody stands in, with the second contributing more than half of the total on every row.

A summary that is 54 per cent one unrepresentative member is not a summary, and that is the reason to keep the two numbers apart — a cleaner one than the ranking argument, and one the arithmetic settles.

One smaller correction while the numbers are out. The listed worst of 3.375 is 15.9 per cent of the searched 21.29, which is a sixth rather than about a fifth — a detail, and it moves the same way as everything else here: the gap between the list and the family is a little larger than the essay claims.

Where the model stops #

A Gaussian notch is not a paint. Real architectural pigments have structured reflectances with several features, and the family searched here is a caricature with four handles. A search over measured pigment spectra would give a different number and would have the same problem: it would be a maximum over a catalogue, which is a list again.

Nothing here says a room like this exists, whatever a real room does to a light before anybody measures it. A wall with a 25 nm notch is a narrow-band absorber, closer to an interference filter than to emulsion paint, and the box’s own edge is where the model stops being about rooms. That is the point rather than a caveat: the number is at the edge because the edge is what defines it.

And the residual is a colorimetric quantity. It measures the distance between where a surface actually is under the second light and where a von Kries gain puts it. It is not a judgement by a person that two patches look alike, and the gap between matching and appearance applies to every number here, at 21 ΔE00 as much as at 3.

The generalisation #

A worst case is a maximum over a set, so a worst case without a stated set is not a quantity. That is nearly a tautology and it is broken constantly, because the set is usually implicit in a list of examples and a list of examples is what a reader sees.

The diagnostic is cheap and it is the one this essay is really about. Search the family, and look at where the search stops. If it stops inside, there is a worst case and it has been found. If it stops on a boundary, the reported number is a property of the boundary, and the honest report is the boundary rather than the number.

That distinction changes what should be published. A maximum found in the interior is a result. A maximum found on a wall is a sensitivity: it says the quantity is unbounded in the direction of that wall, and the useful accompanying number is how fast — here, a factor of 1.33 between a box that admits 25 nm notches and one that admits 10 nm ones.

The same defect appears twice more in this collection, in places that look nothing like a room: the axes of an ellipse taken as the extremes of forty-eight sample points, and the shape of an optimum taken from twenty-four random directions. All three are maxima over samples. Only this one has no true answer behind it.

Who found it, and when #

The census format is the discipline’s own. Corresponding-colour data sets, the CIE’s test-colour samples for colour rendering, and the standard illuminant list are all finite sets chosen to be representative, and every one of them is used both for averages, which they are suited to, and for worst cases, which they are not.

The colour-rendering literature is the clearest case, because it has argued about the list for decades — eight samples, then fourteen, then the ninety-nine of the current fidelity index — and the argument has always been about whether the set is representative, which is a question about means. Whether the set contains anything near the worst case is a different question and is not usually asked.

The distinction is invisible while the list is fixed, because a fixed list gives a reproducible number and reproducibility reads as accuracy. It becomes visible the moment somebody searches instead, which requires a parameterised family and therefore a decision about what the family is.

Where the ladder goes next #

The searched worst does something the mean does not: it reorders the bases. The transform with the lowest mean residual over the census is the one whose gain changes sign inside the family the census was drawn from, and what that does to a recommendation is a larger result than the number in this essay.