Colour goes first in the dark

Everything on this site treats a stimulus as a continuous quantity. A reflectance is a real number at every wavelength, an XYZ triple has as many decimals as anyone wants, and a colour difference of 0.001 is a perfectly good object.

None of that survives contact with a receptor. A cone reports a count — how many photons it happened to absorb while it was integrating — and a count carries √N of noise for free, whatever the optics do.

The claim #

Photon statistics decide where colour vision stops and say nothing useful about how good it is at its best. Those are two different statements about the same arithmetic, and both are worth having:

• A one-ΔE00 difference along the red–green direction reaches the photon-noise floor at 0.065 cd/m²; the same size difference along lightness reaches it at 0.00016. The chromatic direction runs out four hundred times sooner.
• At 100 cd/m² — an ordinary lit page — that lightness difference arrives with a signal-to-noise ratio of 237. People cannot reliably see a one-ΔE00 step at all. So in daylight the limit is neural, by more than two orders of magnitude, and every threshold elsewhere on this site is a fact about what happens after the counting.

The chain, and where each link comes from #

Getting from a luminance to a count is four multiplications, and every one of them needs a number about an eye. Each is quoted with its range, because every result below inherits them.

The pupil, which is not a constant. It closes from 7.0 mm at 0.001 cd/m² to 2.2 mm at 1000 — a factor of 3.2 in diameter and 10 in collecting area. The curve here is a two-parameter form solved to pass through both quoted ends exactly rather than tabulated.

The eye’s focal length, 16.7 mm, which turns what the pupil collects into an illuminance on the retina.

The cone’s aperture, about 2.5 µm across at the fovea, which is what one receptor gets out of that illuminance.

And what a photon has to survive: the lens and macular pigment, the pigment’s own absorptance at its axial density, and the quantum efficiency of isomerisation, about 0.67.

The whole chain is spectral rather than photometric:

$$R^* = t \cdot A_{\text{cone}} \cdot \frac{A_{\text{pupil}}}{f^2} \cdot \sum_\lambda E(\lambda)\,\frac{\lambda}{hc}\,\alpha(\lambda)\,\Delta\lambda$$

Photometry weights by V(λ) and a cone does not, so dividing a lumen figure by an average photon energy would be wrong by tens of per cent for anything that is not white.

The one external check #

Internal consistency is not evidence — a wrong chain is consistent too. So the derived rate is compared against a quantity measured independently: isomerisations per cone per second per troland.

The arithmetic gives 30.3 for the long-wavelength cone and 25.7 for the medium, against a published band of roughly 5 to 50. It lands inside, which is what makes the rest of the file worth anything: every other check here would pass just as happily with the focal length squared in the wrong place or the cone aperture quoted as a radius.

The S cone comes out at 6.4, and it is quoted separately for a reason that is not an error — a white spectrum has far less power where the short-wavelength pigment absorbs, so the same white delivers a fifth as many isomerisations to it. That asymmetry is not a defect of the eye; it is what makes the blue–yellow channel the noisiest one and is one of several reasons it is also the coarsest.

Every entry is per integration time, taken as 0.1 s.

Why colour costs more photons #

The factor everyone expects is a factor of two, and it is real but small. A lightness difference moves the long and medium counts the same way, so the signals add and the noises add in quadrature; a chromatic difference moves them oppositely, so the signal is a difference of two counts. That is worth about a factor of two in signal-to-noise.

The large factor is somewhere else, and it is colorimetric rather than statistical: producing one ΔE00 along a chromatic direction changes the cone counts far less than producing one ΔE00 along lightness does. The colour-difference formula was fitted to make those two steps look equally large to a person; nothing in it makes them equally large at the receptors.

Both effects push the same way, and the result is the floor:

A red–green step of one ΔE00 becomes impossible below about a fifteenth of a candela per square metre, which is deep twilight and is very close to where colour discrimination is in fact reported to collapse. A lightness step of the same nominal size survives four hundred times further down.

How much of this is the patch #

Every number above scales as the square root of the number of cones pooled, and that is the argument most often left implicit in a threshold. A one-degree foveal patch covers about 12,300 cones; a tenth of a degree covers 123, a hundred times fewer, which is a factor of ten in signal-to-noise and — with a correction the next section has to make, because the pupil is not a constant — rather more than a factor of a hundred in the light level needed:

The second row says something startling and correct: a chromatic difference of one ΔE00 carried on a patch a tenth of a degree across is at the photon floor at 22 cd/m², which is ordinary room lighting. Small colour differences on small objects are photon-limited in conditions nobody would call dark, and this is the same statement the spatial essay makes about frequency, arrived at from the other end.

The pupil is worth a factor of three, and it breaks the square law #

Two of this essay’s tables can be read against each other, and doing so verifies the arithmetic and then finds a claim that does not survive it.

The verification first. Over two decades from 100 to 1 candela per square metre the three discriminabilities fall by 5.92, 6.00 and 5.81 — nearly the same factor for all three, which is what a common square-root dependence requires. And the long-wavelength cone’s count over the same span falls from 1,720 to 50, a factor of 34.4, whose square root is 5.87. Three ratios and a fourth computed from a different table, agreeing to one per cent. The matched filter is doing exactly what it claims.

But 34.4 is not 100. Two decades of luminance should be two decades of photons if nothing else moved, and the pupil moves: as the field darkens the pupil opens, and the extra collecting area gives back nearly three of the hundred. Across the whole table the count goes as luminance to the power 0.812 rather than to the first power, and over the five decades from 1,000 to 0.01 the pupil is worth a factor of 8.7 in count and 2.95 in discriminability.

That is a much more interesting number than the pupil usually gets. The essay’s own figure caption says the pupil “copes with one” of ten decades, which prices it as collecting area and understates what it buys: a factor of three in signal-to-noise, spent entirely at the dark end where it is worth the most. The pupil is not competing with adaptation to cover the range; it is buying discriminability at exactly the light levels where discriminability is what fails.

And the patch-size claim inherits the same correction. The section above reasons that a tenth-degree patch has a hundred times fewer cones, so ten times less signal-to-noise, “and therefore a factor of a hundred in the light level needed”. The factor of a hundred assumes photons are proportional to luminance, and they are not. With the count going as luminance to the 0.812, recovering a hundredfold in photons needs about 290 times the light, not a hundred.

The essay’s own table already says so and the two numbers were never compared. The lightness floor moves from 1.6 × 10⁻⁴ to 2.7 × 10⁻², a factor of 169; the red–green floor from 0.065 to 22, a factor of 338. Neither is a hundred, and the second is close to the 290 the exponent predicts.

Why the first is 169 rather than 290 is worth a line, because it is not noise. The lightness floors sit at 10⁻⁴ and 10⁻², which is the bottom of the range, and at those levels the pupil is already wide open and has nothing left to give — so the exponent there is nearer one and the required ratio nearer a hundred. The chromatic floors sit at 0.065 and 22, spanning the region where the pupil does most of its closing, so they pay the full penalty. The two rows differ because the pupil is a curve rather than a constant, and the sentence that quotes one factor for both cannot be right about either.

None of that changes the finding the patch table was written for. A one-unit chromatic difference on a small object is photon-limited in ordinary room lighting, and 22 candelas per square metre is the number the arithmetic gives. What changes is the reasoning offered for it, which took a shortcut past the one term in the chain that is not linear.

What was computed, and how #

The spectrum is D65 on 81 bands, scaled so its luminance is exactly the requested value under the 1931 2° observer.

The absorptance is the retina library’s — the same function the self-screening essay is about — times the quantum efficiency.

The discriminability is a matched filter on Poisson noise: for each cone class, the change in count divided by the square root of the count, summed in quadrature and multiplied by the square root of the number of cones. That is the best any decision rule could do with these counts, which is why the result is a bound rather than a model of performance.

The stimulus pair is built from a step in one CIELAB coordinate, with the step size solved by bisection so the pair is exactly one ΔE00 apart. Turning that pair back into a spectrum requires choosing one of infinitely many metamers, and the choice is checked rather than assumed: two spectra of one colour, built from different illuminants, give long and medium counts within about nine per cent of each other, which is far too small to move any conclusion here.

And the floor is found by bisection on the log of the luminance, which is monotone: fewer photons, less discriminability, always.

Where the model stops #

There is no adaptation in it. A real receptor’s gain falls as the light rises, which is most of why vision works over ten decades. What is computed here is the photon supply and the noise inherent in it — an upper bound on performance rather than a model of performance.

There are no rods. Below about 1 cd/m² the rods are doing a great deal of the work and doing it achromatically, which is the subject of a different essay and does not change the cone arithmetic here. The floors above are what the cone system could do; what an observer actually does in that range is a mixture of two systems.

There is no temporal structure beyond one stated integration time, and the real one lengthens as the light falls — which buys back some of what the darkness costs.

And the neural noise is absent, which is exactly why the daylight result is worth stating: the eye at 100 cd/m² is operating more than two orders of magnitude above its photon floor, so whatever limits it there is not this.

What the pictures cannot show #

No figure here shows a dim scene. The page is lit by the reader’s own display at whatever luminance it happens to be at, and every claim on it is about a light level the page cannot deliver. A figure showing “what a colour looks like at 0.05 cd/m²” would have to control the room, and this site’s standing position is that it knows almost nothing about the apparatus it is displayed on.

The counts are per cone and the eye is a mosaic. Drawing three curves implies three receptors sampling the same place, and the retina interleaves them: the long and medium classes in a ratio that varies between people by a factor of sixteen, and the short class at a few per cent everywhere and absent from the very centre. The pooled numbers in this essay average over that arrangement and say nothing about it.

The middle-wavelength pigment behaves the same way, and a four-density family is enough to see that the widening is smooth.

A threshold is not one number, and this is a third reason why #

This site has already argued that MacAdam’s just-noticeable differences and CIEDE2000’s fitted differences are different quantities quoted interchangeably. The photon arithmetic adds a third thing to keep separate: a threshold measured at one light level is not the threshold at another, and the two chromatic directions move at different rates as the light falls.

That is visible in the discriminability sweep. Between 100 and 1 cd/m² — two decades, from a lit page to a dim room — the same one-unit steps fall from d′ 237 to 40 in lightness, from 66 to 11 in blue–yellow, and from 18 to 3.1 in red–green. Every ratio between them is preserved, because they all scale as the square root of the count; what changes is which of them has crossed into the region where the count is what limits it.

Pushing the two densities to the ends of their range is the strongest form of the match failure, and the pair is still exact for the observer it was built for.

The generalisation #

The useful shape here is the difference between a limit and a model.

A photon count gives a limit: no decision rule, however good, can beat the square root of the number of quanta. That is a hard statement, it is derived from physics and anatomy rather than from psychophysics, and it cannot be improved by any amount of neural cleverness.

What it does not give is performance. At 100 cd/m² a one-unit step is at d′ = 237 and nobody sees it, so between the photons and the judgement there is a factor of two hundred that this arithmetic knows nothing about. Every threshold on this site — MacAdam’s ellipses, the just-noticeable difference, the tolerance a supplier is held to — lives entirely inside that factor.

Two consequences follow, and they point in opposite directions.

Where the light is plentiful, stop invoking photons. The commonest bad explanation in applied colour is that some limit exists because of noise in the eye. In daylight it does not; it exists because of what the visual system does with signals it has in abundance.

Where the light is scarce, stop invoking neurons. Below a candela per square metre the counting is the binding constraint, and no display engineering, no encoding and no adaptation model can produce a chromatic discrimination the photons do not support. That is the same boundary the appearance model runs into from the other side, and the two agree about where it is.

Two more views say what the photon count is a count of, and what happens to a match when two observers differ in the pigment doing the catching.

Who found it, and when #

Photon counting as a limit on vision is Hecht, Shlaer and Pirenne’s, in 1942: the celebrated experiment showing that a rod responds to a single quantum and that absolute threshold requires only a handful of them. The apparatus was a shutter, a filter and a dark room, and the result stood.

The extension to colour discrimination came with the quantitative retinal photometry of the following decades — the troland, the measurement of cone spectral sensitivity, and the estimates of quantum efficiency that make the chain in this essay computable. The reason the chromatic direction is the fragile one was clear from the arithmetic once the opponent recoding was established: a difference of two noisy numbers is noisier than either.

The practical form is older than any of it. Painters have known since before there was a word for it that colour is what goes first as the light fails, and the term for the range where it is going — mesopic — dates from the nineteenth century.

Where the ladder goes next #

The clearest unfinished piece is the rods. This essay computes the cone system’s photon budget and stops; a mesopic model that counted both systems’ quanta and combined them would be able to say not merely that colour discrimination fails in the dark but which discriminations survive and in what order — which is a much sharper claim and is within reach of the machinery already here.

The second is the one the numbers keep pointing at: a factor of two hundred, at 100 cd/m², between what the photons allow and what a person achieves. Nothing on this site models what fills that gap. It is where the interesting parts of vision are, and every threshold measurement in colour science is a measurement of it rather than of the light.