Which lamp changes are free

A person walking from a window into a room lit by a tube has changed illuminant. Everything they are looking at has been multiplied by a different spectrum, and the surfaces that were one colour a moment ago now send different tristimulus values to the eye. That they mostly do not notice is colour constancy, and the mechanism usually named for it is a gain on three signals.

The question this essay answers is which of those walks are free. Not how large the change is — that is easy and it is the wrong question — but how much of it the gain can actually take away, and whether the answer depends on where the light came from.

It does, and the dependence is sharp enough to sort the census on its own.

The claim #

The light the world offers commutes with itself, and the light industry makes does not — which is why one fixed adaptation can be right about the first and cannot be right about both.

• A single basis diagonalises two changes of light exactly when the two matrices commute. That is not a fact about eyes; it is a fact about matrices, and it decides what any fixed mechanism can do.
• Across the daylight and thermal rows the mean distance from commuting 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 — and the worst pair in the whole table is a halophosphate tube against two bounces off a green wall, at 152.
• A triphosphor tube is the hardest single row in the census, leaving 32.7 per cent of itself behind, against 5.8 per cent for D65 to D50 and 7.0 per cent for a change all the way to tungsten.
• And the size of the change predicts almost nothing. The largest change here leaves 7.3 per cent; the worst leaves nearly a third and is less than a third the size.

Two lamps, and what a gain has to do #

Take a surface, light it with D65, and read its tristimulus values. Light it with a tungsten lamp instead and read them again. The change is a 3×3 matrix, exactly, and adaptation is allowed one diagonal.

The diagonal it uses is the ratio of the two whites along whatever three axes the mechanism has. That is not a fitted quantity, it is what an adapting observer actually has available, and the residual is whatever the change does that a diagonal in those axes cannot express.

The tungsten change is enormous — ΔE00 23.5 to an unadapted observer, the second largest in the census — and adaptation removes 93 per cent of it. The triphosphor change is 7.10, less than a third the size, and adaptation removes 67 per cent. On any reckoning that matters to a person, the second is the harder walk.

The pairwise table #

Whether one mechanism can handle two changes is the question of whether the two commute, so the useful object is not a list but a table.

The closest pair is D65 to D50 against D65 to daylight at 4000 K, at 1.99 parts per thousand. Both are movements along the daylight locus, so this is close to a statement that the daylight locus is a one-parameter family whose members share a basis.

The pairs involving a wall bounce are almost as close. That is less obvious and it is worth stating plainly: a bounce off a painted wall is a change of light of the same kind as a change of colour temperature, as far as adaptation is concerned. A wall’s reflectance is smooth and broad; multiplying by it tilts the spectrum without putting structure into it.

Why the tube is worse than the sun #

Two lights of the same chromaticity can have completely different spectra — this site has an essay on that pair and a whole field built on it. The consequence here is a different one and it is about the change rather than the light.

A change from one broad, smooth spectrum to another is close to a change of slope. A slope acts on three broad, overlapping channels in a way that three numbers can nearly follow, because each channel’s response to a slope is dominated by where its own peak sits. A change that adds narrow structure does not act that way: two channels overlapping a phosphor band both get a large share of it, and the ratio between them is decided by where the band sits inside the overlap rather than by where either peak is.

That is the sharpest form of the result. A tube matched to daylight in chromaticity leaves the adapting white where it was, so the observer’s gain barely moves — and the surfaces underneath it have moved a great deal. A change of light that does not move the white cannot be adapted to at all. The residual is not merely large as a fraction; it is the whole change.

What this predicts about a room #

The prediction is not that people see badly under fluorescent light. Nobody who has ever been in an office would believe it. The prediction is narrower and it is about what kind of error survives.

Under daylight of any colour temperature, and under any thermal source, the residual after adaptation is a small, smooth distortion — a percentage error on saturated colours in the direction of the light. Under a discharge lamp the residual is structured: it depends on the surface’s own spectrum in a way no gain can be sensitive to, so two surfaces that agree in colour under daylight can disagree under the tube by more than either has moved.

That is metamerism of illuminants restated in adaptation’s terms, and it explains why the failures people do notice under fluorescent light are failures of matching rather than of general colour cast. The cast is what adaptation removes. The failure to match is what it cannot.

The cheapest change anybody makes #

The everyday version of this is stepping into a shadow, and it is the cheapest change of light in ordinary life.

Open ground is lit by the sun and the sky together; a shadow is lit by the sky alone. The correlated colour temperature between the two can differ by several thousand kelvin, which sounds enormous and is the sort of number a photographer quotes as a problem. But both spectra are smooth — one is a thermal radiator through an atmosphere and the other is the same radiator scattered by it — so the change between them is a slope, and a slope is very nearly a gain.

That is why a shadow has its own illuminant and almost nobody notices. The colorimetric difference is large, the residual after adaptation is small, and the part that does survive is a slight cooling of the most saturated surfaces rather than a general cast. Anyone who has photographed a face in shade and had to correct it afterwards has met the difference between what an unadapted sensor records and what an adapted observer was left with, which is the difference between the two bars in this essay’s first figure.

The rendering index measures something else #

A lighting engineer already has a number for this, and it is not this number. A colour rendering index scores how far a set of test samples move between the lamp and a reference of the same colour temperature — which is close to the unadapted change, not the residual after adaptation.

The two orderings are not the same. The halophosphate tube scores poorly on rendering and leaves 7.8 per cent of itself after adaptation; the triphosphor scores well on rendering and leaves 32.7 per cent. A lamp can be built to move the test samples very little and still move them in a direction no gain follows, and nothing in the index notices — and the inversion turns out not to be a coincidence between two unrelated measures, for the reason two sections below.

What a lamp designed for this would look like #

Nothing in the census is a design brief, but the arithmetic implies one and it is worth stating because it cuts across what lamps are currently optimised for.

A lamp is easy on an adapted observer when the change between it and whatever the observer came from is close to diagonal in a fixed basis. Since the observer’s basis is fixed and the light the observer arrived under is usually daylight or something thermal, that means: a lamp whose change from daylight commutes with daylight’s own changes. Broad, smooth, structureless — a spectrum whose ratio against daylight is a gentle slope rather than a set of bands.

That is very nearly the opposite of what a lamp is optimised for. Luminous efficacy rewards putting energy where the eye is sensitive and nowhere else, which means narrowing; and efficacy and rendering pull against each other for the same reason. A three-emitter source is close to the efficacy optimum and is the narrowest white here.

So the trade has a third term nobody prices. A lamp is sold on efficacy and on a rendering index; what this measurement adds is how much of the change it makes an observer cannot get rid of, and the three do not order lamps the same way. The halophosphate tube is poor on rendering and leaves 7.8 per cent; the triphosphor is good on rendering and leaves 32.7.

The practical form of the brief is narrow. Between two lamps of equal efficacy and equal rendering index, the one whose spectral ratio against daylight is smoother is the better lamp, and neither number a lamp is currently sold with can tell them apart.

Structure is not the problem; structure without a white shift is #

The rendering index and the residual order the two tubes backwards, and the essay reports that as two questions giving two answers. They are two answers with one cause, and naming it makes both orderings predictable rather than merely different.

Two things decide what a fixed adaptation can remove, and they are independent:

How far the white moves, which is adaptation’s entire leverage. The gain is the ratio of the two whites; if the whites coincide, the gain is the identity and the mechanism has nothing to apply.

How structured the change is, which is what a diagonal cannot follow whatever its leverage.

Crossing them gives four cases and the census has three of them:

The halophosphate tube is structured and off-white, so adaptation gets most of it. Its structure shows up in its own white — a lamp whose spectrum is lumpy in a way that shifts its chromaticity gives the mechanism something to grip, and the residual comes out beside tungsten’s rather than beside the triphosphor’s.

The triphosphor tube is structured and sits on D65, which is the corner with nothing in it for adaptation to do. It is the only entry in the census in that cell and it is the worst row in the census.

That is the same fact the rendering index is reading, from the other side. A rendering index penalises a lamp for moving test samples away from a reference at its own colour temperature — so a lamp whose structure is visible in its chromaticity has already been partly forgiven, because the reference moves with it. A lamp engineered onto the daylight locus has no such allowance and scores well; and the very property that earns the good score — landing on the reference white — is the property that leaves adaptation with nothing to remove.

So the two numbers are anti-correlated by construction, on this pair, rather than by accident. That is a stronger statement than “two different questions with two different orderings”, and it says which lamps to expect it of: any source whose spectral structure has been arranged to cancel in its chromaticity. Which is every lamp designed against a rendering index, because that is what designing against one means.

Two corrections to the ratios #

Two figures quoted above are worth pinning down, since they are the ones a reader would carry away.

The worst pair against the best pair is a factor of 76, not fifty — 152 against 1.99. The fifty is close to the ratio of the worst pair to the daylight family’s mean of 3.4, which is 45, and that is a comparison between a pair and an average rather than between two pairs.

The factor of 4.1 checks out exactly: 52.7 against 12.8 is 4.12.

Neither correction changes anything the essay concludes, and both are worth making because the spread is the result. Seventy-six is a better number than fifty for the same reason the census’s own ordering is the result rather than its levels — a range that wide across pairs drawn from one table is what says the commutator is sorting something real, and understating it by a third gives away part of the finding.

Who found it, and when #

Von Kries’s diagonal is 1902. The recognition that its quality depends on the axes is from the 1990s. The specific observation that changes of daylight are unusually well behaved under a diagonal has been in the literature since Judd’s work on the daylight locus, and is usually stated the other way round — the daylight illuminants are a low-dimensional family, so anything linear handles them.

What is added here is the commutator, which is the exact statement of when two changes can share a mechanism, and the measurement of it across a census that contains lamps Judd never saw. The fluorescent tube dates from the 1930s and the phosphor-converted LED from the 1990s, so most of what is on this table has been in general use for less time than the transform being applied to it.

What was computed, and how #

Every row is a pair of contexts. Each light is normalised so that a perfect diffuser under it gives Y = 100 before any level change is applied, so that a row whose subject is a change of level says so explicitly rather than inheriting whatever normalisation its constructor happened to use.

The residual is the mean CIEDE2000 over a hundred and twenty-five surfaces after the ratio-of-whites gain in the CAT16 basis. The commutator is ‖AB − BA‖ / (‖A‖‖B‖), which is zero exactly when one basis diagonalises both and is invariant to scaling either matrix — which matters, because the census contains a change that halves the light and one that does not change its level at all.

Where it stops #

The lamp spectra here are constructed from stated emission models rather than tabulated from measurements, so the numbers are properties of this site’s fluorescent tube rather than of a particular product. The ordering is robust — narrower structure is worse, and by a lot — but any specific percentage is a property of the construction.

The commutator is a norm on matrices and not a perceptual quantity. Two changes at 50 parts per thousand are not “50 units” of anything a person could report; the number is only meaningful in comparison with other entries in the same table.

And the whole calculation assumes the observer’s mechanism is a diagonal in some fixed basis. If adaptation is partly a normalisation to the scene’s own statistics rather than to its white, as the estimator essays suggest, then some of what is counted as residual here is available to a mechanism this arithmetic does not model.

Where the ladder goes next #

If the shape of the change decides everything, then the changes made deliberately are worth checking. A lamp being dimmed is a change of light chosen by an engineer, and the three ways to do it differ by a factor no chromaticity diagram reports: one of them is exactly free at any depth and one leaves ΔE00 3.6 at a tenth.

And the rows here that are not lamps at all point somewhere else. A bounce off a wall behaves like a change of colour temperature, until it happens twice — at which point it stops behaving like one, for a reason that turns out to be the same reason a corner breaks a metameric match.