Everyone is beaten by the same wall

Rank eight candidate adaptation bases by how much each leaves after a gain and the ordering is a long argument. Ask instead which change of light each one fails worst on and the answer is nearly unanimous: seven of the eight fail worst on the same row, and the row is not a lamp.

The claim #

The hardest change of illumination in this census is a green wall reflecting twice, and it is the worst row for XYZ scaling, Hunt–Pointer–Estévez, the receptor construction, Bradford, CAT02, CAT16 and the basis that best equalises discrimination contours. The one basis with a different worst row is the one fitted to the census — and what its fit bought there was a retreat rather than a repair.

• Two bounces off a green wall: 5.18 for XYZ scaling, 3.75 for the receptors, 3.37 for CAT16, 2.80 for Bradford, 4.81 for the discrimination optimum.
• The fitted basis takes it to 2.21 and its own worst row becomes a triphosphor tube at 2.42.
• It is not a lamp. No illuminant in the census is as hard as a room with a coloured wall in it, reflected off twice.
• And it is not an exotic room. A green wall at a plausible reflectance, hit twice, which is what a corner does.

Why this is not the answer anybody expects #

Asked in advance which change of illumination is hardest for a model of chromatic adaptation, most people who work with colour would say a low-pressure sodium lamp, or a triphosphor tube, or some other source whose spectrum is a handful of lines. The reasoning is sound: a narrow line is the case where three broad channels have least to go on, and discharge lamps are where colour rendering indices collapse and where metamers come apart.

That reasoning is about a different failure. A narrow-band source is hard for reconstruction — recovering a spectrum, predicting how a surface will render, telling two metamers apart — because three numbers cannot see structure finer than their own channels. Adaptation is not reconstruction. It asks a narrower question: is the change from one context to the other a per-channel multiplication?

For a line spectrum the answer is often yes, and closer to yes than intuition suggests, because a line falls inside all three overlapping cone channels at once and scales them together. The census bears it out: the triphosphor row costs the receptor basis 2.57 and the wall row costs it 3.75.

So the two intuitions are about two different properties, and the surprise here is a case of the more famous one being imported into a question it does not answer.

Why a wall is harder than a lamp #

The change of light between two contexts is a matrix, and what makes a matrix easy or hard for a diagonal is how nearly the ratio of the two spectra is constant across each channel’s sensitivity band.

A lamp change moves that ratio smoothly. Daylight to tungsten reweights the whole visible band monotonically; daylight at 4000 K against daylight at 10000 K tilts it. Even a discharge lamp’s lines are a departure from smooth rather than a wholesale change of shape.

A wall is a multiplication by a reflectance, and a reflectance applied twice is that reflectance squared. A green wall’s reflectance is a broad hump peaking in the middle of the band and falling to a fraction at both ends; squaring it makes the hump narrower and the tails deeper, so the ratio between the two contexts varies by a large factor within the span of a single cone channel.

That is precisely the condition under which a per-channel gain cannot work. A gain has one number per channel and the ratio it is standing in for has several values inside that channel’s band. No choice of basis removes the problem, because narrowing the channels to make the ratio locally constant costs everything else — and the narrow-channel bases are the ones that do worst on narrowband lamps.

The two-bounce arithmetic #

The census contains a green wall hit once and the same wall hit twice, so the escalation can be read directly.

For the receptor basis, one bounce costs 2.19 and two cost 3.75 — a factor of 1.71. For CAT16, 1.72 and 3.37, a factor of 1.96. For XYZ scaling, the two-bounce row is 5.18 and is the largest single number anywhere in this arithmetic.

The escalation is superlinear because the departure from diagonality is superlinear. If a single bounce leaves a residual operator N differing from the identity by some amount, the doubled bounce does not leave 2N; it leaves the operator for the squared reflectance, whose off-diagonal structure is stronger than twice as strong. A corner moves both terms, and this is that observation showing up in a table of adaptation residuals rather than in a rendering.

Where a wall overtakes a lamp #

One bounce and two bounces are two points on a continuum, and the crossing between them is the useful part.

Raising the wall’s reflectance to a fractional power gives a continuous bounce count, and the receptor basis’s residual runs 1.182 at half a bounce, 2.186 at one, 3.034 at one and a half, 3.755 at two and 4.947 at three. The growth is sublinear: doubling from one bounce to two multiplies the residual by 1.72 rather than by 2, an exponent near 0.78, because a squared reflectance is narrower as well as deeper and the diagonal’s failure begins to saturate.

The hardest lamp for every basis in the table is the triphosphor tube, at 2.570 for the receptors, 2.323 for CAT16 and 2.330 for Bradford. Setting those against the wall’s curve gives the crossings.

The wall overtakes the hardest lamp at 1.22 bounces for the receptors, 1.36 for CAT16 and 1.52 for Bradford.

So one bounce off a green wall really is easier than a discharge tube, for every basis, and the claim above needs its second bounce. What the crossing adds is how little of the second bounce it needs: a quarter of one for the receptors, and half of one for the basis fitted hardest against lamps.

A quarter of a bounce is not an exotic arrangement. It is a wall a metre away rather than adjacent, or a wall of two thirds the reflectance hit twice. The hardest change in the census stops being a lamp once the room has been reflected in about one and a quarter times — and a threshold that low means whether a wall beats a tube is settled by where the furniture is.

What the fit did about it #

The fitted basis is the only entry whose worst row is somewhere else, and this is the essay’s real finding.

Its two-bounce number is 2.21, better than everything. But its triphosphor number is 2.42 — worse than CAT16’s 2.32 and only slightly better than the receptors’ 2.57. So the optimisation, given nine numbers and fourteen rows to average, chose to spend a little of the discharge rows to buy a lot of the wall rows.

That is what minimising a mean does, and it is worth seeing rather than assuming. A fit does not improve every case a little. It reallocates, and the allocation is decided by whichever rows are largest — which is why the fitted basis’s own worst row is not the census’s hardest change but the one it declined to fix.

What the row costs a real observer #

Three and three quarters of a unit is a large number and it is worth converting into something visible.

A CIEDE2000 difference of one is roughly what a careful observer can see between two patches held side by side. Three and three quarters is not subtle: it is the difference between a paint chip and the paint it was supposed to match, seen from across a room. So a fully adapted observer in a corner of a green room, judging surfaces against the white they have adapted to, is being misled by an amount anybody would notice, and no adaptation transform in use removes more than about a quarter of it.

That is a striking result and it should be read for what it is. It does not say people cannot judge colour in green rooms; people manage, and they manage by using everything a von Kries gain is not — edges, memory colours, specularities and the knowledge that walls are walls. What it says is that the diagonal model of adaptation, which is what every colour management system implements, has its largest error exactly where a room is most coloured.

The practical version is old advice with a number attached: a viewing booth is grey, and the reason it is grey is this row.

The one basis that is worst somewhere else #

The discrimination optimum — the basis that makes MacAdam’s ellipses roundest — is included in the comparison because it is a set of nine numbers arrived at with no reference to adaptation at all, and it therefore functions as a control.

It leaves 4.81 on the two-bounce wall, which is the second-worst number in the whole table, beaten only by XYZ scaling’s 5.18. So a basis chosen for a completely unrelated purpose is also worst on this row, which is the strongest available form of the unanimity claim: the row is hard for reasons that have nothing to do with what any of these matrices were selected for.

The exception, again, is the fitted basis, and now the pattern is clear enough to state as a rule. Everything that was not fitted to this census has its worst row here. The only thing that does not is the only thing that saw the census.

Why this row belongs in a lighting census at all #

A wall is not an illuminant, and its presence here is a decision that deserves defending.

The rule the census follows is that a row must be a change of context — something that happens to the whole field an observer is adapted to. A lamp changing qualifies. A room whose walls are green qualifies, because the room is the illuminant as far as anything in it is concerned: every surface is lit by light that has already been reflected off something coloured.

What does not qualify is a change of object. An ink, an interference pigment and a film seen at two angles are all changes of spectrum and none is a change of context, because they happen to one thing in the scene and not to the rest. The distinction is what makes the census an adaptation census rather than a list of spectral differences.

So the wall rows are in, and they turn out to be the hardest. That is not an artefact of including them; it is the reason for including them.

The shape of the residual, not just its size #

A single number per row hides what kind of failure it is, and the residual operator supplies the rest.

After the gain, what is left is an operator N that would be the identity if the change had been exactly a gain in that basis. Its departure from the identity has a structure: the diagonal entries say the gain was the wrong size, and the off-diagonal entries say the channels are being mixed — a signal that belongs in one channel arriving in another.

On the daylight rows the departure is mostly diagonal and small. On the two-bounce wall row it is heavily off-diagonal, which is the precise statement of why no gain can fix it: a diagonal operator has zeros off the diagonal by construction, so an off-diagonal residual is not something a better-chosen diagonal could have absorbed. It is outside the model’s reach rather than badly placed within it.

That distinction is the difference between a model that is mis-parameterised and one that is mis-specified, and it decides what would help. Choosing better axes moves the diagonal part. Nothing in the nine numbers touches the off-diagonal part, and the only things that would are a full matrix — which the observer does not have — or a mechanism that is not a gain.

What was computed, and how #

Each wall row is built by multiplying the reference illuminant by a modelled wall reflectance — a broad Gaussian absorption in the appropriate band — once or twice, and treating the result as the second context.

The change is then exact. For a three-dimensional family of surface reflectances the map from tristimulus values under the first context to values under the second is a 3×3, and the build asserts that at a tolerance of 10⁻⁹ on every row including these.

The residual for a candidate basis is the mean CIEDE2000 left after the diagonal an adapted observer applies, over a hundred and twenty-five constructed surfaces. Eight bases, fourteen rows, one number each.

The unanimity claim is checked rather than eyeballed: the figure asserts that every basis in the comparison that was not fitted to the census has its maximum on the same row, and would fail the build if a new basis were added that did not.

The generalisation #

When many models fail worst on the same case, the case is a fact about the problem rather than about the models.

That is a nearly trivial statement that is nearly always skipped, because a comparison of models is normally reported as a ranking and a ranking discards the row identities. Here seven of eight candidates — including two derived from physiology, three fitted to different data by different committees, and one optimised for an entirely unrelated objective — agree on which change is hardest. Their disagreements are about how badly.

The corollary is the diagnostic: when one model’s worst case differs from everybody else’s, look at what it gave up. A model whose worst row is unusual has usually been fitted, and the unusual row is where the fitting spent its budget. The fitted basis here is 41 per cent better than the receptors on the census’s hardest row and 6 per cent better on a triphosphor tube, and that pair of numbers describes the fit better than its mean does.

What would actually help #

If the failure is off-diagonal and no basis removes it, the question of what would is worth a paragraph, because two answers exist and neither is a better transform.

A full matrix per change. The change of light is exactly a 3×3, so applying that 3×3 removes the residual entirely — the build asserts it at 10⁻⁹ on every row. But the matrix is a different matrix for every change, and an observer has one mechanism and no way of knowing which change is happening. A camera does: a profile fitted under the light it is used under is exactly this, and it works to the extent the light is identified correctly.

Or a mechanism that is not a gain. Everything in this arithmetic is a per-channel multiply on a linear signal. Real vision has a nonlinearity, spatial comparisons, and a great deal of scene interpretation, and the failure being measured here is a failure of the linear stage alone.

What is not on the list is a better set of nine numbers, and the census’s most reflected row is the clearest place to see why.

Where the model stops #

One wall, one shape. The wall’s reflectance is a modelled Gaussian rather than a measured paint. A real green paint has structure the model does not have, and paint is not a filter in any case: a real wall scatters as well as absorbs.

Two bounces, not a solved room. Squaring the reflectance is the arithmetic for light that has hit the same surface twice, and a real room’s interreflection is a solution of a whole system with several surfaces and a geometry. The census’s two-bounce row is a bound on the shape of the effect rather than a rendering of a room.

And the observer is fully adapted. Every number assumes the gain has completely taken up the change, which is not what a real adaptation does and is not what appearance models assume either. Incomplete adaptation would move every row and would move the wall rows most, because they are the largest.

Where the ladder goes next #

The census keeps saying that what limits a diagonal is how far the change is from diagonal, and that nothing in the nine free numbers fixes it. The other place those nine numbers appear is inside a device — a display’s primaries are its adaptation axes, and nobody chose them for that.