What no adaptation can remove

An earlier essay here left a sentence behind that nothing on the site could check: adaptation cancels gains, and only gains. It came out of a room with four clocks running in it, where a photopigment bleach turned out to be worth almost nothing to an observer at any speed while a lamp coming up to its working temperature survived an observer who adapted perfectly and instantly. Two rows of one table, and a claim about every change of light there is.

This essay makes it a census. Every change of illumination this site models — a swap of illuminant, a bounce off a coloured wall, a filter in front of the retina, a lamp dimmed, a sheet of paper changed — is put through the same three questions. How far does it move an ordinary surface. How much of that can a gain remove. And what is left when the gain has done everything it can.

The claim #

A change of light is exactly a 3×3 matrix on tristimulus values, adaptation is a diagonal one, and the gap between them is a property of the spectra rather than of the transform anybody chose.

• The matrix is exact, not fitted. On a set of surfaces that is exactly three-dimensional, the residual after the full 3×3 is ΔE00 10⁻¹³ for every row in the census.
• A change of level is the control and it leaves nothing at all, in every basis, to 3 × 10⁻¹⁴ — because a scalar multiple of the identity is diagonal whatever the axes are.
• The largest change in the census is not the worst one. A bounce off a green wall moves a surface ΔE00 23.6 and leaves 1.72 behind; a triphosphor tube moves it 7.10 and leaves 2.32. The second is less than a third of the first and is the harder of the two.
• The smallest residual belongs to a filter inside the observer. The macular pigment moves a surface 15.3 units and leaves 0.37 — 2.4 per cent of itself, the smallest share in the census and less than half the next smallest.
• And what a gain leaves is the part of the change that is off the diagonal, which is a statement about matrices and has an answer that does not depend on which surfaces were sampled.

Why the answer is exact #

The ordinary way to ask this question is to pick some reflectances, apply an adaptation transform, and average the error. That measures a transform against a sample set. It cannot say what was possible, and it cannot say whether a different transform would have done better, because the answer would be a different average over the same arbitrary sample.

There is a better route and it comes from a fact about surfaces rather than about eyes. Measured collections of natural reflectance spectra are very nearly three-dimensional: three basis functions reconstruct them to within measurement error. That is a genuinely surprising fact about the world, and it is not one the eye had any say in.

Take it seriously as an exact statement and everything follows. A surface is R α for a 81-band basis R and three coefficients α. Its tristimulus values under a light E are A diag(E) R α, where A is the matching functions — a 3×3 matrix times the coefficients. Under a second light they are A diag(E′) R α, a different 3×3 times the same coefficients. So

XYZ under E′ = [A diag(E′) R] [A diag(E) R]⁻¹ · XYZ under E

and a change of light is a matrix. Not approximately: exactly, on that family, with no residual to average.

The matrix carries the old white exactly onto the new one, because a perfect diffuser is in the reflectance basis. Which gives the sentence this essay turns on:

A change of light is a gain in basis B if and only if B T B⁻¹ is diagonal — and when it is, the gain is the ratio of the two whites, with nothing fitted.

That converts the question from how good is this transform into a question about matrices, and gives it an answer independent of any sample set. It also disposes of a familiar reassurance: every adaptation transform carries the white correctly, because the white is a fixed point of all of them by construction. Getting the white right proves nothing whatever about the colours.

The control, and why it matters that it is exactly zero #

The first row of the census is the same lamp dimmed by half. It moves a surface ΔE00 13.7 for an observer who does not adapt, and a gain removes all of it — 3 × 10⁻¹⁴, which is arithmetic noise.

That row is not there to be interesting. It is there because it has to come out at exactly zero in every basis, since a scalar multiple of the identity commutes with everything, and a basis-dependent answer for it would mean the machinery had a defect rather than that the claim was subtle. It is the row that says the rest of the table is measuring what it says it is.

The two orderings are not merely different — they are unrelated #

The hero caption says the rows are sorted by the fraction left rather than by the size of the change, because the two orderings are not the same ordering. On the five rows the essay gives both numbers for, the two orderings are not weakly different. They carry no relation at all.

The rank correlation between how large a change is and how much of it survives is exactly zero over these five — the largest change keeps a tenth of itself, the smallest keeps a third, and the middle three run the other way. Even against the absolute residual the correlation is only +0.40, which is what one would expect if the two were near-independent and the residual happened to be bounded above by the change.

That is stronger than not the same ordering and it is the sentence the table supports: knowing how far a change of light moves a surface says nothing whatever about how much of it an observer can remove. A reader who takes the size of a colour shift as a proxy for its difficulty — which is the natural thing to do, and is what a single ΔE00 in any other essay invites — has the wrong instrument entirely.

The sharpest pair makes it concrete. A green wall moves a surface 3.3 times as far as a triphosphor tube does and leaves 26 per cent less behind. Per unit of change, the tube is 4.5 times the harder object.

The commuting measure discriminates more than the residual does #

The two instruments in this essay — the residual share and the distance from commuting — are pointed at different questions, and it is worth noticing that they have very different resolving power.

The residual shares span 2.42 per cent to 32.7: a factor of 13.5 from the mildest row to the harshest. The commuting distances span 1.99 to 152 parts per thousand: a factor of 76, nearly six times wider.

So the pairwise measure separates the census far more sharply than the per-row one does, and the reason is structural rather than incidental. A residual share is bounded above by one — a change of light cannot leave more of itself than it was — while a commuting distance has no such ceiling and grows with the product of two matrices’ departures from a shared basis. Asking whether two changes can share a mechanism is a more discriminating question than asking whether one change can be handled, and the essay’s central measurement is the pairwise one for that reason as well as for the one it gives.

The discharge lamps’ position sharpens too when both baselines are stated. Their cross-category mean of 52.7 is 4.1 times the mean within the light the world offers — and 15.5 times the mean among the three daylight changes alone. The first figure is the one the essay quotes; the second is the one that says how far a fluorescent tube is from the light a von Kries mechanism has any prospect of being right about.

What the exact zero costs to state #

One small thing about the control row, since the essay leans on it heavily. The half-dimming comes out at 3 × 10⁻¹⁴ against the full 3×3’s 10⁻¹³ on every other row — about three times better, which is what one expects of a scalar multiple of the identity against a general matrix inverse, and is a second check that the machinery is doing what it says.

It is worth having because an exactness claim with one number in it is hard to weigh. Two numbers, in the right ratio, say that the control is exact for a structural reason and the general case is exact for a numerical one — and it is the structural exactness the whole argument depends on.

What sorts the census #

Sorted by how much of itself each change leaves behind, the table does not sort by field, by size, or by how exotic the light is. It sorts by where the change came from.

The changes of light that existed before electricity — daylight at any two colour temperatures, a thermal radiator, a bounce off a painted wall — leave between 5 and 12 per cent of themselves. The discharge lamps leave more, and the triphosphor tube leaves 32.7 per cent, which is by a wide margin the worst row here.

The reason is not that the eye is fitted to daylight in any adaptive-story sense — nothing here can test that and this site does not claim it. The reason is arithmetic and it is in the next section.

One basis, several changes, and the condition for both #

A single fixed adaptation mechanism has one set of axes. It can be right about two different changes of light only if one basis diagonalises both, and one basis diagonalises two matrices exactly when the two matrices commute.

So the question of whether a fixed mechanism can serve is the question of whether the changes of light a person meets commute with each other. They do not all commute, and the table of how far each pair is from commuting is the central measurement here.

Within the light the world offers, the mean is 12.8 parts per thousand, and among the three daylight changes alone it is 3.4. Across to a discharge lamp it is 52.7 — a factor of 4.1. The closest-commuting pair in the whole table is D65 to D50 against D65 to daylight at 4000 K, at 1.99 parts per thousand; the furthest is a halophosphate tube against two bounces off a green wall, at 152.

The consequence is not that some transform is badly chosen. It is that no transform can be right about both, and a mechanism with one set of axes has to be a compromise between kinds of light that make incompatible demands. The essay on the basis computes what that compromise costs and what the alternative would look like.

The smallest residual is inside the eye #

Sorted by share rather than by size, the census puts the macular pigment first, and by a distance. It is a fixed absorption sitting in front of the central few degrees of the retina, worth ΔE00 15.3 to an unadapted observer and 0.37 to an adapted one — 2.4 per cent, against 5.2 per cent for the next smallest.

This is the quantitative form of something the eccentricity essay found from the other side: the adapting white seen through the macula and seen without it differ by exactly zero after adaptation, because a filter in front of the receptors and an adaptation gain are the same kind of operation applied at the same place. What is new here is the size of what is left over for stimuli that are not the white — small, and not zero, and largest in the blue.

The lens is the counter-example that makes the point rather than spoiling it. Fifty years of yellowing is also a filter in front of the receptors, and it leaves 7.9 per cent rather than 2.4, because it is a much broader and steeper absorption and multiplies the short-wave end far harder than a gain in any of these bases can follow.

The wall applied twice #

The three surface rows are a bounce off a coloured wall, treated as a change of illumination like any other — light that has been multiplied by a reflectance before it reaches the object being looked at.

One bounce off a green wall leaves 1.72 units. Two bounces off the same wall — which is a corner rather than a flat surface — leave 3.37, which is 1.96 times as much for a change only 1.33 times as large. Squaring a reflectance sharpens it, and a sharper change of light is further from being a gain.

That is the corner result arriving from a completely different direction. There, a metameric match broke in a corner because a metameric black is invisible and its square cannot be. Here, adaptation loses ground in a corner because a reflectance squared is a sharper filter than a reflectance. Both are consequences of the same fact about multiplying a spectrum by itself.

Who found it, and when #

Von Kries proposed in 1902 that adaptation is a scaling of the three cone signals, which is the diagonal in this essay. The proposal predates any measurement of what the cone signals actually are, and its durability is remarkable: it is still the operative model in every colour management system in use.

The observation that a scaling is basis-dependent, and that some choices of basis are much better than others, is much later. Bradford’s transform was fitted to corresponding-colour data in the 1990s and has entries no cone response could have; CAT02 and CAT16 came out of the CIECAM programme and were fitted the same way. The idea of sharpening a basis deliberately — choosing axes that make more changes of light diagonal — is Finlayson’s, and it is the direction the arithmetic here points in.

What this site adds is not the sharpening but the exactness. Treating natural reflectances as exactly three-dimensional turns a fitting problem into an algebra problem, and the algebra says which changes of light can share a basis and which cannot.

What was computed, and how #

The reflectance family is a lattice of a hundred and twenty-five surfaces spanning three coefficients — a level and two modulation depths — chosen so that every member stays inside zero and one without clamping. That constraint is load-bearing and the first draft did not have it: eleven of its surfaces reflected a negative amount of light at the blue end. Nothing downstream would have objected, because a negative reflectance integrates to a perfectly ordinary-looking tristimulus, and the census would have been a census of impossible objects.

Each row’s residual is the mean CIEDE2000 over the family between the surface as it was and the surface after the change, with the observer’s gain applied. The gain is the ratio of the two whites in the named basis. It is not fitted, because that is what an adapting observer actually has.

Every headline number is recomputed on the site’s clamped, realistic reflectance set as well. The ordering survives — the three pairs that change places are all pairs within a quarter of a unit of each other — and every row is smaller on the real surfaces than on the idealised ones, so the census is an upper bound rather than a number that could go either way.

Where it stops #

The three-dimensional reflectance basis is a caricature. It is not a measured principal-component basis, because this site has no measured reflectance collection; it is three smooth functions with the one property the argument needs. The theorem is a theorem about that family. What licenses reading it more widely is the clamped-set comparison above, and that comparison is evidence rather than proof.

Nothing here says a person’s adaptation is von Kries. It is known not to be, exactly: the fit to corresponding-colour data is imperfect in ways a diagonal cannot express, and the appearance models add a degree of adaptation and a nonlinearity precisely because of that. What the census measures is the best a diagonal could do, which is an upper bound on what that part of the mechanism explains.

And a change of light is not the only thing that happens to a surface. A gloss pedestal adds rather than multiplies; an instrument’s geometry puts one under every measurement, and none of the arithmetic here applies to it. A fluorescent surface is not a multiplication at all: it takes light in at one wavelength and returns it at another, so it is a full operator on the spectrum with entries off the diagonal, and the essay on that is about the place where this whole construction fails to start.

Where the ladder goes next #

The census is a table of what is left. It says nothing yet about what could be done with different axes, and the answer to that is surprising in both directions: a basis computed from daylight alone beats every published transform on daylight by a factor of five and loses to all of them on a fluorescent tube.

Below that sit the applications, and each is the same arithmetic pointed at a device. A camera balances in a basis that is not fixed. Colour management divides by the paper, which is a von Kries adaptation in the worst basis in the table. A tolerance quoted under one light is a different tolerance under another. And of the four devices this site models, exactly one has a mechanism worth the name.