The laws that make colour add up

Every number on this site is an integral: a spectrum multiplied by three functions and summed. That construction is not a convention — it is a claim, and the claim is that colour matching is linear.

If two lights match, twice one matches twice the other. If two lights match, adding a third to both leaves them matching. Those are Grassmann’s laws, they are the axioms the whole of CIE colorimetry follows from, and they have been taken as exactly true since 1853.

They are exactly true, and only over a band.

The claim #

The four laws hold exactly for a linear observer, and both ends of the luminance range break them by different mechanisms — leaving colorimetry an operating range of 3.15 decades that no standard states.

• Symmetry, proportionality, additivity and transitivity all return residuals of 2 × 10⁻¹⁵ or less for a fixed set of cone sensitivities. That is not a finding; it is what “linear” means, and it is here to be the control.
• At the bottom, rods. A pair matched for all three cone classes is not matched for the fourth receptor, and a search over metameric blacks finds a pair whose rod signals differ by 22.6 per cent while the cones agree to 10⁻⁹. Below a few candelas a square metre that difference is part of what is seen.
• At the top, bleaching. In bright light a substantial share of the cone pigment is bleached, the optical density falls, and a lower density is a narrower fundamental — so the observer is a different observer. A match made at an office light level and tested at a million trolands is off by 0.59 per cent of a cone excitation, against zero at the level it was made at.
• The band between them, held to half a per cent of a cone excitation, runs from 4.5 to 6,310 candelas a square metre. Three and a bit decades: a lit room to an overcast sky.

Why the exactness is the interesting half #

The four residuals in the figure are not a result about the eye. They are a result about the model, and stating them is the only way the failures below mean anything.

A linear observer is three fixed functions of wavelength. Symmetry is trivial. Transitivity, proportionality and additivity are properties of an integral, and an integral has them by construction. So the residuals are floating-point noise and could not be anything else — and if a version of this file ever reported them as small-but-nonzero, that would be an arithmetic bug rather than a physiological finding.

Getting the control right cost this file its first set of results. The pairs were built with the site’s existing metamer machinery, which constructs spectra in the null space of the CIE 1931 matching functions. The observers here are cone sensitivities built from a pigment template at stated peaks, which is a different three-dimensional subspace — so a pair metameric for the first is not metameric for the second, and it missed by 4.25 per cent of a cone excitation.

Four per cent is larger than every effect this essay exists to measure. The laws would have looked approximate before anything interesting happened, and the two real failures would have been invisible underneath. The pairs are now built in the null space of the observer under test, using the same projection with a different matrix.

The bottom end: a fourth receptor nothing constrains #

A cone metamer is a pair of spectra whose three cone excitations agree. It says nothing whatever about the rods, because no term in the match involves them.

An arbitrary metameric black is a poor demonstration of this, for the same reason an arbitrary metameric black is a poor illuminant-metamerism demonstration: the pair has to differ where the fourth receptor differs from the three, and rhodopsin peaks at 498 nanometres — between the medium and short cone peaks, and close to neither. Searching over the frequency and phase of the black finds one that does: 22.6 per cent rod contrast, on a pair whose cone residual is 10⁻⁹.

How much of that is seen depends on how much of the visual response is rods:

Above five candelas a square metre the rods contribute nothing and the match is exact — which is the control at that end, and is why colour goes first in the dark is a statement about thresholds rather than about matches. Below it the match comes apart, and the cones still agree exactly the whole way down.

A metameric match is therefore a statement with a light level in it, and no metameric match anybody quotes carries one — which is a second unstated argument on top of the illuminant one everybody knows about.

The top end: the observer narrows #

The upper limit has nothing to do with rods and everything to do with the cones themselves.

A cone’s sensitivity is not its pigment’s absorbance. Light passes through a column of pigment and is absorbed exponentially, and the exponential saturates at the peak long before it does in the wings — so a denser cone has a broader sensitivity, not a taller one. That is self-screening, and this site measured it in the phase before this one: an eighteenfold density range widens the long-wavelength fundamental by 34.5 nanometres at half height.

Density is not a constant. In bright light a share of the pigment is bleached and cannot absorb, so the effective density falls:

The pair is built to be metameric for the observer at the density the match was made at, and then handed unchanged to the observer the second light level produces. The spectra do not move; the eye does.

So a match made in an ordinary room is not exactly a match in the sun, and the mechanism is the one that makes the 2° and 10° observers different shapes rather than the same shape scaled.

The band, and what it means #

Putting the two mechanisms on one axis gives the quantity this essay exists for: over what range of light is colorimetry exact?

Held to half a per cent of a cone excitation, the answer is 4.5 to 6,310 candelas a square metre — 3.15 decades.

That is a lit interior at the bottom and a bright overcast day at the top. Everything outside it is a condition in which two spectra with identical XYZ do not match, by an amount the standard has no term for.

Three ordinary conditions fall outside:

• A cinema. Peak screen luminance is around 48 candelas a square metre and most of a frame is far below that; the rods are contributing to a large part of the picture.
• Direct sun on a surface. Ten thousand candelas a square metre and more, which is above the bleaching edge.
• A phone at night. Twenty candelas a square metre in a dark room, with the surround at nothing — near the bottom edge, and the condition a display’s whole solid shrinks in.

The two edges are not the same kind of edge #

The band is quoted as though it were closed by two comparable failures, and the two tables say otherwise. One failure is unbounded and the other saturates just past the tolerance.

At the bottom, the rod contrast runs 0.00, 5.26, 12.78 and 18.04 per cent as the light falls from five candelas a square metre to a fiftieth, and there is nothing stopping it: the further into the dark, the larger the rod share and the more of a difference the cones cannot see becomes a difference somebody does. At the dimmest row it is thirty-six times the tolerance and still climbing.

At the top the residual runs 0.000, 0.186, 0.491 and 0.586 per cent as the retinal illuminance rises through four decades — and the density beside it runs 0.398, 0.266, 0.067 and 0.008. At the last row the pigment is essentially all bleached, so there is no more bleaching available and no more error to be had. 0.586 per cent is not a point on a rising curve; it is the asymptote.

Which means the upper edge of colorimetry’s operating band is a technicality. The maximum error bleaching can ever produce is 0.59 per cent of a cone excitation, the tolerance chosen here is 0.50, and the whole of the upper limit is that excursion of nine hundredths of a percentage point. A tolerance of 0.6 per cent instead of 0.5 — a change nobody could defend either way — removes the upper edge entirely and leaves the band open above.

That is worth stating plainly because it inverts what the headline suggests. Three and a bit decades reads as a narrow window with a wall at each end. What the tables describe is a hard floor and a hairline: below about four and a half candelas a square metre colorimetry fails progressively and without limit, and above a few thousand it fails by an amount that cannot exceed six tenths of a per cent however bright the light gets.

The essay’s own sensitivity note goes the wrong way for the same reason. It says a tolerance ten times looser would widen the band by less than a decade; a tolerance ten times looser is five per cent, which is nine times the largest bleaching error there is, so the band would have no upper edge at all rather than a slightly higher one. Tightening the tolerance behaves as described — both edges move in and the bottom one moves fast — and loosening it does not, because only one of the two mechanisms has anywhere left to go.

And that asymmetry has a physical reason worth keeping. The bottom failure is a fourth receptor joining in, and a receptor’s contribution has no ceiling — it grows until it is the whole signal. The top failure is a parameter of the existing receptors moving, and the parameter is bounded: optical density runs from its resting value down to zero and stops. A mechanism that adds something is unbounded and a mechanism that removes something is not, which is a distinction the band’s single number conceals and which decides what happens outside it.

The practical reading follows directly. A colour match quoted without a light level is safe upwards and dangerous downwards, and by a wide margin: sunlight costs at most six tenths of a per cent and a dim corridor costs eighteen. Of the three ordinary conditions the essay lists as falling outside the band, the cinema and the phone at night are the ones that matter, and direct sun is on the list only because the tolerance was set a tenth of a percentage point below where bleaching stops.

What was computed, and how #

The observers are built rather than tabulated. Three cone sensitivities from this site’s pigment template at stated peaks, through stated ocular media, at a stated optical density. That is what makes the density axis available at all — a tabulated fundamental has no density to vary.

The pairs are built in each observer’s own null space, by projecting a smooth candidate spectrum onto the space orthogonal to that observer’s three sensitivities times the illuminant, then scaling it to keep the reflectance inside zero and one with a margin.

Bleaching is the standard photochemical steady state: the surviving pigment fraction is I½/(I½ + I), with the half-bleaching retinal illuminance quoted at 10^4.3 trolands. It is the only bleaching model here and it is the one place a quoted constant carries the whole top edge.

And the residual is a cone contrast rather than a ΔE, deliberately. A receptor mismatch is naturally a fraction of a receptor’s own response, it needs no colour space to be stated in, and it lets the rod failure and the cone failure be put on one axis. The tolerance of half a per cent is a choice and is stated; the shape — a band with two different mechanisms closing it — does not move with it.

A looser match tolerance is what an anomaloscope run at speed actually gives, and the width of the accepted band is the measurement.

Where the model stops #

The bleaching model has no time in it. Bleaching and regeneration have time constants of tens of seconds to minutes, so the density after a flash is not the steady-state density, and a match made and tested within a few seconds of a light change sits between the two.

The rod contribution is a weight, not a system. The CIE’s mesopic system is a fixed-point calculation whose answer depends on the stimulus; what is used here is a stated blend, and the difference is named wherever it is used.

Neither end has an appearance model. What is measured is whether the match survives, which is a receptor question. Whether the two members of a broken pair look different by an amount anybody cares about is a further question with a whole other machinery behind it.

And the operating range depends on a tolerance nobody has agreed — asymmetrically, as the section above works out. Half a per cent of a cone excitation is roughly the size of the smallest differences anybody reports; a tenth of that narrows the band from both ends, and a looser one removes the upper edge outright rather than moving it, because the bleaching error saturates at 0.59 per cent and the rod error does not saturate at all.

Putting the two dichromacies beside a normal observer is what shows that the law holds for each of them separately rather than only on average.

The generalisation #

The sentence worth carrying: linearity is an approximation with a domain, and colorimetry never states its domain.

Every standard in the subject is written as though the laws hold everywhere. They are stated as axioms, the matching functions are tabulated without a light level, and a chromaticity has no luminance in it by construction. The measurement here says the axioms hold over three decades of a fifteen-decade range, and that the two failures outside are not corrections to be added but different receptors and different pigments.

The surprising connection is with the standard observer’s two field sizes. The 2° and 10° observers exist because a match depends on how large the field is, and the discipline’s response was to publish two sets of functions. The dependence on light level is the same kind of thing — the match depends on a condition the functions do not carry — and the response has been to publish nothing at all. There is no 100 cd/m² observer and no 10,000 cd/m² observer, and by the argument that produced two field sizes there ought to be.

Two more readings of the same instrument say what the laws look like when they are put to an observer who is not the standard one.

Who found it, and when #

Grassmann stated the laws in 1853, from an argument about the geometry of colour mixture rather than from receptors, which did not exist as a concept yet. The laws are the reason a three-dimensional linear algebra is the right description at all, and everything from the 1931 functions to an ICC profile is downstream of them.

Rod intrusion in colour matching has been measured since the mid-twentieth century and is the reason matching experiments are conducted at photopic levels and with a bleaching pre-adaptation. The practice is standard and the statement of the resulting domain is not.

Self-screening and bleaching are older physics applied later: the exponential absorption is Beer’s law, and its consequence for the shape of a cone fundamental was worked out in the 1960s and 1970s, which is when tabulated fundamentals started carrying a stated density.

What has not changed is the specification. A set of colour-matching functions is published as a table of wavelength against three numbers, with a field size and no light level.

The laws hold over a range rather than everywhere, and the ends of that range are set by two different mechanisms.

What the pictures cannot show #

They cannot show a match. Every claim here is that two spectra do or do not produce the same receptor responses, and a page can only draw the spectra or the computed residuals. The pair that comes apart for the rods looks identical on any display at any brightness, because a display’s own metamerism is a different question.

And the band cannot be drawn at its own scale. The operating range spans three decades on an axis fifteen decades long, and a figure drawn to scale would be a thin stripe in the middle of nothing.

Where the ladder goes next #

The nearest unfinished piece is the join to appearance. This essay measures whether a match survives; what a broken match looks like — whether 22.6 per cent of rod contrast at 0.1 candelas a square metre is a visible difference or an unmeasurable one — needs a mesopic appearance model, which nobody has and which is the largest single gap in the subject.

The second is the bleaching clock. The density is computed at steady state, and both bleaching and regeneration take tens of seconds, so a match made and tested across a light change is made by an observer in transit between two sets of fundamentals. That is the same shape as the adaptation clock one level down: a receptor property with a time constant, and a model with no time in it.