Colour Science & Perception
what the eye settles, and what it cannot
The collection

Every essay — page 13

Page 13 of 54, continuing through the fields in the same order.

What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it

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What the eye does

Three cone types project an infinite-dimensional spectrum onto three numbers. Everything colour science can and cannot do follows from that one collapse.

A spectrum, weighted three ways, and the three numbers left over. The illuminant D65 above; below, the same spectrum multiplied by each matching function. The area under each product is one coordinate of XYZ. Everything else about the spectrum — its shape, its structure, all its remaining degrees of freedom — is discarded here.

Three numbers

A spectrum has as many degrees of freedom as anyone cares to give it. The eye reports three. Everything colour science can do, and every way it fails, follows from that one collapse.

8 figures Two different spectra that are the same colour. Two reflectance curves differing by 92 per cent RMS, and the two patches they produce under D65: identical to ΔE00 = 6.2e-14, which is arithmetic noise rather than a small number. Both patches are inside the sRGB gamut, so neither has been clipped into agreement.

Two spectra, one colour

Metamerism is usually described and almost never demonstrated. It does not have to be — the metameric black space is enormous, so a matching pair can be constructed to order, verified, and then made to come apart by changing the light.

9 figures The CIE 1931 colour-matching functions. The three functions that turn a spectrum into three numbers. They are all positive, which is why XYZ exists — the RGB functions they were derived from are not. ȳ is by construction the luminous efficiency function, which is why luminance comes out of Y.

Why colour is exactly three-dimensional

Matching every wavelength with three primaries requires, for some wavelengths, a negative amount of one of them. That physical awkwardness is why the colour-matching functions were transformed into XYZ, and why the horseshoe is curved.

6 figures The same two surfaces, weighed by each system. A long-wavelength and a short-wavelength surface under D65, with their relative luminance under the photopic curve and under the scotopic one. Under daylight vision the red surface is 1.33 times the blue; under rod vision it is 0.15 times, a reversal by a factor of 9.0. The swatches are the photopic appearance, which is the only one a display can produce: rod vision has no colour, and drawing a guess at it would be an invention.

The eye that has no colour

Rods outnumber cones twenty to one, work alone below a hundredth of a candela, and are absent from the centre of gaze. Between dusk and a lit room both systems run at once, and neither standard curve describes what is happening.

7 figures One palette under normal vision and two dichromacies. The same 7 colours simulated by the Brettel–Viénot–Mollon construction at severity 1.0. protanopia and deuteranopia collapse the red-green distinctions. This shows which discriminations survive, not what anybody sees.

Three cones, two axes

The retina does not send three receptor signals down the optic nerve. It sends a sum and two differences, and the reason falls out of the statistics of natural light rather than out of anything the eye intended.

8 figures Longitudinal chromatic aberration of the eye, focused at 555 nm. Thibos's chromatic eye, plotted as defocus in dioptres relative to 555 nm. The curve crosses zero exactly once, at the wavelength the eye is accommodating on, and everything else is out of focus — by -1.22 D at 420 nm and 0.50 D at 680 nm. Through a 3 mm pupil that first figure is a blur circle of 12.6 minutes of arc, against a foveal acuity limit of about one. The eye is never in focus across the spectrum and never can be.

Nothing is in focus at both ends

The eye carries about two dioptres of chromatic aberration across the visible band, which is a strong reading prescription. Whatever it is focused on, most of the spectrum is landing somewhere other than the retina — and the cone class that gets the worst of it is the one the retina bothered least to sample.

9 figures Three retinas of different composition, making identical colour matches. Seeded mosaics at L:M ratios of 1, 4, 16 to one — the range found between people, and a 16-fold difference in what the retina is made of. A colour match is the claim that two spectra produce equal excitation in all three cone classes, and changing how many of a class there are multiplies that class's excitation by a constant, which cannot disturb an equality. The relation between the two test spectra is identical to twelve decimal places across the whole sweep. Luminance, which is a weighted sum rather than an equality, moves by 1.33× over the same range. That asymmetry is why a standard observer exists and why V(λ) has a much larger between-observer variance than the colour-matching functions do.

The mosaic is not the observer

The ratio of long-wavelength to medium-wavelength cones varies between ordinary people by a factor of sixteen. Those people make the same colour matches, to twelve decimal places, and that single fact is the reason a standard observer can exist at all.

8 figures Contrast sensitivity, three channels, normalised to each channel's peak. Spatial frequency in cycles per degree against relative sensitivity. The luminance channel runs to 50 cycles per degree, red–green to 12 and blue–yellow to 8, each by the same criterion of 5 per cent of that channel's own peak. Only the luminance channel dips at low frequency; the two chromatic ones are low-pass, so a chromatic edge of any size at all is seen at its full contrast.

How fine a colour edge can be

The eye resolves a lightness pattern to about fifty cycles per degree and a red–green one to twelve. Every colour difference here is quoted as though a patch had no size, and the same difference is plainly visible at one scale and gone at another.

6 figures The L cone's sensitivity at three axial pigment densities. Each curve is normalised to its own peak, so the only difference visible is shape. Raising the density from 0.1 to 0.9 widens the curve from 113.6 to 145.9 nm at half height while the peak stays within 5 nm of where it was. That is Beer–Lambert saturating: at the peak the pigment already absorbs nearly everything, so more of it can only catch more light in the wings.

A cone absorbs its own light

A photopigment's absorbance is a property of a molecule; a cone's sensitivity is that molecule stacked in a column deep enough to absorb most of what arrives. The stacking broadens the curve by thirty-five nanometres, and two observers differing in nothing else disagree about a match that is exact for one of them.

8 figures Photons caught by one cone in one integration time, under D65. Each curve is one cone class, counting isomerisations during a 0.1 s integration through a pupil that closes as the light rises. At 1000 cd/m² a long-wavelength cone catches about 11522 and at 0.001 it catches 0.12 — which is where the square root of the count stops being a small correction and starts being the signal. The rate works out at 30 isomerisations per second per troland, inside the published band of 5–50.

Colour goes first in the dark

A cone reports a count, and a count carries the square root of itself as noise. Counting the photons says where colour vision stops — a chromatic difference runs out four hundred times sooner than a lightness difference of the same size — and says just as clearly that in daylight the eye is nowhere near that limit.

9 figures A screen at 45°, 8 c/°, and where its energy sits. Left, the pattern. Right, its power in the frequency plane with the zero frequency at the centre and the edges at the sampling limit of 23 cycles per degree, on a logarithmic scale over five decades. The closed curves are the visual system's own sensitivity at 5, 25, 60 per cent of its peak; they are not circles, because sensitivity is lower on the diagonals than on the cardinal axes by a factor of 2.0 at high frequency. Energy inside a curve is seen; energy outside it is not, whatever its size.

A pattern has a direction

Every spatial claim here is a claim about a frequency, and a frequency has no direction in it. Turning a printed screen forty-five degrees makes it exactly twice as quiet with nothing else changed — and the same rotation does nothing at all to a chromatic one.

7 figures Cone density, out from the centre of gaze. Quoted landmarks with logarithmic interpolation between them: 199,000 cones per square millimetre at the fovea, 9,500 at ten degrees — a factor of 21. The shaded bands are the two standard observers' fields. The 2° observer averages over a region whose density falls by 3.3× between its centre and its edge and the 10° observer by 12.4×, which is what "a 10° field" contains and is why the two sets of matching functions are different shapes rather than the same shape scaled.

Colour stops at the edge of sight

Cone density falls twenty-one-fold between the centre of gaze and ten degrees out, and the three channels give out at three different rates — so a colour difference in the periphery does not merely shrink, it turns. At the exact point of fixation there are no short-wavelength cones at all.

8 figures