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

Cones — the series

53 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · eye
  2. 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.

    part 2 · eye
  3. 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.

    part 2 · eye
  4. 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.

    part 2 · eye
  5. 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.

    part 3 · eye
  6. 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.

    part 3 · eye
  7. Temporal sensitivity, and where each channel gives out. Modulation frequency in hertz against relative sensitivity. The luminance channel is band-pass, peaking at 8 hertz and running out at 60; an isoluminant modulation is low-pass and runs out at 15, which is 4.0 times sooner. Both cutoffs are at the same criterion of 5 per cent of that channel's own peak, so the ratio between them is a ratio between two measurements rather than between two conventions.

    The eye has a shutter

    An isoluminant flicker fuses at fifteen hertz and a luminance one at sixty, so a light whose colour changes forty times a second is a steady light of a colour it never emits. And the frequency at which flicker stops being visible is not a property of the eye — it moves twelve and a half hertz for every decade of light.

    part 3 · eye
  8. 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.

    part 4 · eye
  9. 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.

    part 4 · eye
  10. 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.

    part 4 · eye
  11. 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.

    part 4 · eye
  12. 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.

    part 5 · eye
  13. 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.

    part 5 · eye
  14. The drift window, asked about each channel in turn. Every spatial frequency a channel can resolve, drifting at v degrees a second, arrives at f × v hertz; the bar is the range of v over which all of them stay above a quarter of that channel's temporal peak. The luminance band is closed at both ends — 0.018 to 0.71 degrees a second — because its temporal sensitivity has a dip at zero to fall into. The chromatic bands have no slow edge at all, because chromatic temporal sensitivity is low-pass: a stationary chromatic pattern sits at the top of its own sensitivity. The mark is the measured drift, and it is inside all three.

    The drift is a luminance mechanism

    The eye's own drift was shown to sit inside a band of speeds that keeps every spatial frequency modulating, and the band was quoted as though it were about vision. Asked about colour, it has no slow edge at all — a stationary chromatic pattern needs no eye movement whatever. And a stabilised chromatic pattern is the first thing to fade.

    part 5 · eye
  15. Coming back from a bleach, against the clock already measured. A 94 per cent bleach, and the pigment returning at its own time constant of 120 seconds. The lower curve is the site's slow neural adaptation constant, 60 seconds, started from the same place — it is finished while the chemistry is barely half done. Regeneration does not speed up because the light went away: the rate constant is the same one it always was, which is why the recovery is slow while the bleaching was fast.

    The slowest clock is chemical

    An earlier essay here joined the afterimage to the adaptation clock and named what was still missing — a third gain, upstream of both, in the pigment itself. It is twice as slow as anything measured before it, it leaves a coloured after-tint from a white field, and at steady state it cancels exactly, which is why nobody has ever needed to model it.

    part 5 · eye
  16. Where a pooled gain gives out, against where the eye does. The falling curve is how much of a pattern of each spatial frequency a local adaptation pool of 0.5° can see — and therefore how much of it a settled eye can cancel. It is half gone by 0.37 cycles per degree, which is a feature about 2.7° across. The three marks are the acuity limits of the luminance channel and the two chromatic ones. Every one of them is more than an order of magnitude finer than the pool, which is why a stabilised eye loses the fill of a picture and keeps its outline rather than losing the picture.

    What a still eye stops seeing

    A stabilised image is said to vanish, and nothing in a temporal filter predicts it — sensitivity at zero frequency is a quarter of the peak, not nothing. Give the adaptation gain a size and the answer falls out — fading is a high-pass filter that switches on over a minute, it takes the fill and leaves the outline, and a patch has to be about two degrees across before it goes at all.

    part 6 · eye
  17. What the same eye reports about one field, in the middle and at the edge. Each row is a uniform field, drawn at the most saturated version of itself this page can show — the percentage is how much of the full stimulus survived, the rest being the adapting light added to bring it inside the gamut. The left patch is what the centre of gaze reports and the right one what 10 degrees out reports, each adapted to the same light as that position sees it. The adapting white comes out identical to 5e-13, because an adapted eye cancels its own filter exactly. Nothing else does, and the largest difference is in the blue. tungsten light is not drawn: it cannot be shown at any useful saturation, and at full strength it differs by ΔE00 1.87.

    One person is two observers

    The macular pigment is a yellow screen over the fovea and nowhere else, so a cone at the centre of gaze and a cone ten degrees away have different colour-matching functions in the same eye. A match made in the middle comes apart at the edge by six units — and fitting one filter to the gap between the CIE's two standard observers gives a density of 0.40 against a measured 0.35.

    part 6 · eye
  18. The share of itself each change leaves behind, and the smallest is inside the eye. The residual as a fraction of the change rather than as a colour difference, which sorts the census differently. At the top is the macular pigment — the filter in front of the central few degrees of one's own retina — leaving 2.4 per cent of itself. It is a fixed transmittance multiplying the light and the white together, which is as close to a pure gain as anything here gets, and it is why nobody notices they have one.

    The filters inside the eye

    The macular pigment leaves 2.4 per cent of itself after adaptation — the smallest share of anything in this site's census of light changes, and less than half the next smallest. Fifty years of lens yellowing leaves 7.9 per cent, and the difference between the two says what a gain is actually good at.

    part 7 · eye
  19. A stabilised grating fades at a rate that depends on which way its bars run. What is left of a 0.3 cycle-per-degree grating, in multiples of its own threshold, as a function of its orientation, at four moments after the image was stabilised on the retina. The first curve is flat to floating point: at the instant the pattern arrives, orientation does not matter, because the filter that carries it has no orientation preference at this frequency. The anisotropy arrives with the fading, reaching ×2.09 after five minutes, and a pool with no axis predicts none of it.

    A fading pool has a shape

    Giving the local adaptation pool two axes instead of one costs a single parameter and produces a prediction the circular version cannot make — a stabilised grating fades at a rate that depends on which way its bars run. The obvious objection is the oblique effect, and the two act in bands that do not overlap.

    part 7 · eye
  20. A radiance factor, split into the part that was reflected and the part that was not. The two components of what leaves a heavily brightened sheet under M₁ — D50 including its ultraviolet. The lower band is the reflected component, which is what a reflectance curve means and is everything a reflectance-based model can hold. The band above it is light emitted at wavelengths it did not arrive at, and its total is decided by how much ultraviolet the source had rather than by anything about the sheet's colour. The line at one is the boundary a reflecting surface cannot cross; the sum reaches 1.21 at 430 nanometres.

    The eye weights where the light is not

    A brightened sheet returns a quarter more light than arrives at 430 nanometres, and it is three tenths of one per cent brighter for it. The luminous efficiency function is 0.017 there against 1.0 in the middle of the band, so the whole effect lands in the blue-yellow axis — brighter than white is a colour claim wearing a brightness word.

    part 7 · matching
  21. What reaches the retina, and why the observer's table stops at 360 nanometres. The transmittance of the eye's own optics across the short-wave band, at three ages, with the brightener's absorption shaded underneath. The upper curve is an eye whose lens has been removed — the cornea alone, opaque below about 295 nanometres and transparent above it. The photopigments absorb perfectly well in this band; what stops the light is a piece of optics in front of them, which is why the short-wave limit of colour vision moves with age and can be removed surgically. A twenty-year-old receives 21 times as much of the band a brightener works in as a seventy-year-old does.

    The eye stops at the lens

    Neither standard observer is tabulated below 360 nanometres, and the reason is not that the photopigments stop absorbing there. It is that the light never arrives — the cornea and the crystalline lens take it — so the short-wave limit of human colour vision is a piece of optics, it moves by a factor of twenty across a lifetime, and it can be surgically removed.

    part 8 · eye
  22. One observer's matching functions, in three of the bases the matches leave free. The three colour-matching functions after a change of basis 0.00 of the way from Hunt–Pointer–Estévez towards the set built from the dichromat confusion points. Every one of these triples predicts exactly the same matches as every other, because a match is an equality and a matrix applied to both sides of an equality changes nothing. What moves is where the peaks are and whether the curves go negative — these ones do not, and going negative is what the 1931 committee constructed XYZ to avoid.

    The matches do not name the cones

    Colour matching is the whole empirical basis of colorimetry, and it fixes the observer's three curves only up to a nonsingular 3×3 — nine numbers that no match, in any quantity, to any precision, can see. One particular choice of those nine is used throughout here, and it was made for a different purpose.

    part 9 · eye
  23. Where a dichromat's confusions converge. Every pair of colours a protanope cannot tell apart lies on one of these lines, and all the lines meet at a single point — at (0.7465, 0.2535) for this class. The point is the chromaticity of the missing cone's own response direction, which is why it need not lie inside the diagram or correspond to any light at all. Two of the three do not. The three points between them carry six numbers, and six is two thirds of what the matching data leave undetermined.

    A confusion point is a missing pigment

    The nine numbers colour matching leaves free are fixed by three points on a chromaticity diagram, each of them the place where everything one class of dichromat cannot tell apart converges. Two of the three lie outside the diagram entirely, which is not a defect — a direction in tristimulus space need not correspond to a light.

    part 9 · eye
  24. Every width at the wide end of its span, and at the narrow end. One line per published quantity, each spanning the value it takes when all four declared widths are read at the narrow end of their reported ranges to the value at the wide end, with a marker at the value as declared. The largest span is the deutan margin at a factor of 2.62; the smallest is 1.22. This is the reading the population model's own documentation promised for four phases and nothing ever took. It is not a confidence interval — the four ends are not quantiles and the widths are not independent draws — it is what a reader who distrusts all four at once sees.

    A width nobody varied

    Five numbers say how much people differ from one another, and every conclusion drawn here about a population rests on them. Each was written down with the range the literature reports beside it, so that a result could be re-read at the pessimistic end. Nothing ever was.

    part 9 · eye
  25. Four different bases, one adaptation model, one number. The middle row of the basis built from the confusion points multiplied by 0.21, 1, 3.7 and 11 in turn, with the resulting adaptation residual drawn as a bar in each case. The four bars are the same height to 9e-16 of a ΔE00, because the row's scale cancels exactly between the gain and the inverse. Three of the nine numbers a colour match leaves free are invisible to an adaptation model, which is why the six the dichromat data supply determine it outright with nothing left to fit.

    The three numbers a gain cannot see

    Colour matching leaves nine numbers free. Three dichromat confusion points fix six of them and three choices of unit fix the rest — and it turns out that a von Kries gain is exactly blind to those last three. So the dichromat data do not merely constrain an adaptation basis. They determine it, with nothing left over to fit.

    part 10 · eye
  26. How cone-like a basis is, against how well it adapts. Each basis placed by how far its own implied deuteranope confusion point falls from the measured one (horizontal) and by how much an adapted observer is left with in it (vertical). The construction from the confusion points sits at zero on the horizontal by definition and near the top on the vertical. Nothing near the left of the picture is near the bottom: the closer a basis is to the receptors, the more a von Kries gain leaves behind. The unconstrained winner sits at 1.63 on the horizontal, further from the measurement than any published transform except CAT02 and Bradford.

    The best axes are not receptors

    If the axes that make a von Kries gain work were nearly the cone fundamentals, the published adaptation transforms would be cone-like and their departures would be slack in a fit. They are not. Minimise the residual over all nine free numbers and the winner sits further from the measured dichromat confusion points than any of them.

    part 10 · eye
  27. Which width carries the answer, and which carries the doubt. Two columns of bars over the four things that differ between two pairs of eyes. On the left, the share of the population's disagreement each one accounts for — the attribution quoted here since the population was built, which puts the lens first at 81%. On the right, how much of the doubt each one puts on everything published here, which is its elasticity multiplied by how badly the width itself is known. The macular pigment comes first there, at 0.65 against the lens's 0.60 — a lead of 8%. The two lists agree exactly below the top.

    Which measurement is worth making

    Four things about an eye differ between people, and they have been ranked here by how much of the answers they carry since the population was built. Ranking them by how much doubt they carry gives a different order, and ranking them by which one takes a published claim closest to failing gives a third.

    part 10 · eye
  28. Nine eigenvalues, six of which exist. Nine points on a logarithmic vertical axis: the eigenvalues of the Hessian of the adaptation residual at its own optimum, largest to smallest. The first six run from 6.8×10² down to 7.6×10⁻¹, a condition number of 890. Then the axis drops: the seventh is 1.9×10⁻⁴, and the last three are separated from the sixth by a factor of 4.0×10³. Those three are not small curvatures. They are the finite-difference truncation error on directions along which the objective is exactly constant, and a shaded band marks them as the numbers the objective does not have.

    The rank is the invariance

    A von Kries gain cannot see the scale of a row of its basis. That is an identity, proved in a line, and it can be measured instead — as the rank of a second-derivative matrix. Both objectives this collection minimises over the observer's nine free numbers have a Hessian of rank exactly six, and the three directions they cannot see are the three scalings, to a hundredth of a degree.

    part 11 · eye
  29. A template that cannot place a point, fitted three ways. Three rows, one per set of stimuli the pigment template's cone matrix can be fitted over, each listing the three confusion points that matrix implies. The protanope's point wanders from (0.99, 0.20) to (0.76, 0.13) against a measured (0.75, 0.25), and the deuteranope's moves by 22.5 in chromaticity — further than the whole diagram is wide. A copunctal point is where two nearly parallel planes meet, so a template good to a few per cent, which is far more than enough to place a spectrum, is nowhere near enough to place this. It is why the population is built by moving the measured points rather than by deriving them.

    A template cannot place a point

    This collection's model of an eye is good to a few tenths of a per cent at predicting what a cone catches, which is far more than enough to place a spectrum. Asked where that eye's confusion points are, it puts the protanope's at (0.99, 0.20) against a measured (0.75, 0.25) and the deuteranope's anywhere from (1.1, −0.5) to (−18, 11) depending on which stimuli the fit was made over.

    part 11 · eye
  30. No one surface carries the answer, and the set is smaller than it looks. A falling bar chart of the 125 surfaces in the test set, ordered by how much each contributes to the published mean for daylight to tungsten. The tallest bar is 1.50 per cent of the total, so the mean is not a few awkward objects with a crowd behind them and a leave-one-out would move it by well under a per cent. The tail is the other half of the story: 5 surfaces contribute essentially nothing, because a flat grey is a surface an adaptation gain handles exactly. Counting the set by how evenly it contributes rather than by how many members it has gives 108.5 effective surfaces out of 125, which is what "a mean over a hundred and twenty-five surfaces" is really worth.

    The surfaces that answer nothing

    Five of the hundred and twenty-five test surfaces contribute exactly zero to every number the adaptation census reports — not approximately, exactly — and the reason is the one fact about von Kries adaptation that makes it worth having at all. Counting the set by how much it contributes gives about a hundred members rather than a hundred and twenty-five.

    part 11 · eye
  31. A confusion point is about the pigments that remain. Three groups of three bars. Each group is one dichromat's confusion point; each bar is how far that point moves in chromaticity when one of the three cone pigments has its absorption peak shifted by eight nanometres. In every group the bar for the pigment that dichromat is missing has length zero — exactly zero, to machine precision, not merely small. The protanope's point does not move when the L pigment moves, the deuteranope's does not move when the M pigment moves, and the tritanope's does not move when the S pigment moves. The reason is algebraic rather than physiological: a confusion point is the direction that excites only the missing cone, which is the null space of the other two receptors' rows, and rescaling a row does not move where the other two are zero. So the point at which a protanope's confusion lines meet is not a fact about the pigment a protanope lacks, which is why the claim about it restates in nanometres of the M pigment.

    A point about the pigments that remain

    The chromaticity at which a protanope's confusion lines meet does not move at all when the long-wave pigment moves — not slightly, exactly not at all. It moves a great deal when the medium-wave pigment does. A dichromat's confusion point is a fact about the two receptors they have rather than about the one they lack.

    part 12 · eye
  32. The same claim in nanometres of pigment, where no declared width can reach it. Five horizontal bars on a scale of nanometres, one per published chromatic-adaptation transform, each showing how far the medium-wave cone pigment's absorption peak would have to move for the receptors' own protan confusion point to land where that transform puts it. Zero is the measured peak. The bars run from -10.5 to 18.2 nanometres — in both directions, so two of the transforms want the pigment shorter and two want it longer. Drawn across them is the 25 nm separation between the L and M pigment peaks, which is the whole basis of red-green vision and is not a number this collection declared. The nearest transform asks for a displacement of 30 per cent of that separation, and the span across the table is 28.7 nanometres — larger than the separation itself. No population, cloud or standard deviation appears anywhere in the statement.

    The claim, in nanometres

    For four rounds the claim here has been that every published adaptation transform puts the protanope's confusion point outside any real population of eyes, stated in standard deviations of a population whose widths were declared rather than measured. Restated as a pigment displacement it needs no population at all — and the nearest transform asks the medium-wave cone to move thirty per cent of the way to the long-wave one.

    part 12 · eye
  33. How far this site's median observer sits from the 1931 standard, by template. Twenty-four natural reflectances under D65, each given a tristimulus value twice: once by the 1931 colour-matching functions and once by this site's median member, with each judged against its own white. The bar is the mean difference, which is the residual this collection bounds and calls inescapable. It is inescapable, and it is smallest for the simpler template: Lamb's 1995 nomogram gives 0.9006 against Govardovskii's 0.9522, and removing Govardovskii's secondary band brings it down again to 0.9354. Neither is an argument for changing template — a nomogram is fitted to measurements of individual receptors, not to colour matches, so agreement with the standard observer is not what either was trying to achieve. What it says is that the residual is a mismatch between two kinds of observer rather than a shortfall a better pigment model would close. The number after each bar is the template's tail ratio: how far the L cone's half-maximum reaches below its peak against how far it reaches above.

    The population rests on a template

    Two hundred observers here are built from one formula fitted to microspectrophotometry in 2000. The obvious alternative — the tabulated cone fundamentals — is not available, and the reason is the finding. A tabulated fundamental has no peak wavelength to move, so the moment it is used the population collapses to a single observer.

    part 12 · eye
  34. What is read at each distance from the edge of a lit region. Three materials under a half-plane of light, with the boundary at the centre of the horizontal axis and the lit side on the right. The vertical axis is the radiance leaving the surface as a share of what it leaves far inside the lit region. On the unlit side the sample is emitting light while receiving none, so the ratio the model calls a reflectance has a zero denominator there. The distance over which the curve runs from a tenth to nine tenths is 0.21 millimetres on coated paper and 5.1 on pale marble — which is the width of the neighbourhood a point's colour is decided by.

    Two blurs before the eye's own

    A translucent object arrives at the eye already blurred, by a kernel that is a property of the material rather than of the optics. On every material in this collection's table that blur is coarser at ordinary reading distance than the finest detail the eye can resolve — so the softness of marble or skin or wax is not a failure of vision, it is the object.

    part 12 · eye
  35. A template's asymmetry against what its observer costs. The horizontal axis is the tail ratio of the L cone's pigment absorbance — how far the curve reaches below its peak at half maximum against how far it reaches above — and the vertical is how far the observer built from that template sits from the 1931 standard. A real visual pigment has a long short-wavelength tail, so the three curves derived from a published nomogram sit above 1.1 and the two Gaussians sit below. The four are matched in width, so nothing here is about size. The ordering is the point: the two caricatures cost between two and four times what either nomogram does, and the axis they are separated on is the one feature the caricatures do not have.

    A template is mostly its tail

    Four pigment templates matched to the same width at the same peak, differing only in which side of the peak their half-maximum reaches further. Ordered by that one number, the observers they build are ordered by how far they sit from the standard one — and the two with the tail on the wrong side cost two and four times what either real nomogram costs.

    part 13 · eye
  36. The secondary band, dialled from nothing to twice what Govardovskii published. Govardovskii's template has a second, smaller absorption band below 400 nm, published at 0.26 of the α-band's peak. It is the one coefficient in the whole template whose contribution is somewhere else in the spectrum than the peak, and it is the part of the template a reader is least likely to have heard of. Sweeping it from nothing to twice the published value moves the median observer's distance from the 1931 standard from 0.9354 to 0.9790 ΔE₀₀, monotonically upwards. That is a small effect — about a fiftieth of the residual — and its being monotone is the interesting part: there is no interior optimum, so nothing here recommends the published value over any other, and a coefficient whose measured value is not the one that best fits an unrelated agreement is a coefficient to leave where the measurement put it.

    The band below four hundred

    The pigment template every observer here is built from carries a second, smaller absorption band in the ultraviolet, published at 0.26 of the main one. Dialling it from nothing to twice that moves the median observer monotonically away from the standard one, with no interior optimum — which is what a physical constant looks like when nothing downstream is pulling it anywhere.

    part 13 · eye
  37. The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

    The third factor is a construction

    Every figure in this collection names its observer, which was the whole point. None of them says what an observer is made of. Three curves are not a measurement of the eye; they are a projection of one, with a field size, an age, a macular density, three peak wavelengths and a luminance constraint inside them.

    part 13 · eye
  38. The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

    A neutral is everyone's colour

    Two eyes differing by fifty years of lens yellowing, by a factor of three in macular pigment and by six nanometres of long-wavelength peak agree about a grey card to four parts in ten thousand billion. The agreement is an identity rather than a coincidence, and it says exactly what an observer disagreement is a disagreement about.

    part 14 · eye
  39. The three cone absorptances at two settings of the cone optical density. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 18.8 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    A gain is not an observer

    Multiply one eye's three cone sensitivities by 1.6, 0.7 and 2.4 and it is not a different eye. The white-point division is that multiplication's inverse, so the two agree exactly — and most of what a cone optical density change does is that multiplication, which is why the largest number in the table of individual variation is the one that matters least.

    part 14 · eye
  40. The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

    Three curves for one space

    Rotate a set of colour-matching functions by an arbitrary invertible matrix, undo the rotation at the end, and the computed colour is identical to eight parts in a thousand million million. An observer is a three-dimensional subspace, not a set of curves, and the literature keeps reopening a question that is a theorem.

    part 14 · eye
  41. Every departure under every light. Six departures across six lights, each cell the difference between two observers in ΔE₀₀, drawn as a bar whose length is the number. The rows are not multiples of one another: the lens is worst under tungsten and the pigment peaks are worst under a three-emitter LED, because a departure is a pairing and which light is being paired with decides it. The laser projector's row is empty, and that is not a fact about lasers — on this collection's five-nanometre grid a three-line spectrum is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly.

    The lens is worst under tungsten

    An ageing lens costs 4.38 ΔE₀₀ under a tungsten lamp and 2.20 under a fluorescent tube. The pigment peaks cost 2.84 under a three-emitter LED and 1.87 under the same tube. Six departures across six lights do not form a single ordering, which is what a pairing looks like and what a dominant term would not.

    part 14 · eye
  42. What choosing a space to divide the white out in is worth. Three pairs of routes to the same colour, over forty-two surfaces: dividing the white out in tristimulus values, in a published cone space, and in the observer's own cones. The first two agree to 0.59 ΔE₀₀ at the median. Either of them differs from the observer's own cones by more than fifteen. That is why the two exact conditions in this round are exact only in the eye's own coordinates: the identity belongs to the receptors, and every published arithmetic works in a basis somebody else chose.

    The identity is in the eye's own coordinates

    Two conditions in this round are exact — a gain on each cone is not a different observer, and three curves for one space are one observer. Imposed in a published cone space rather than the eye's own they leave 13.0 and 8.11 ΔE₀₀ standing. The identities belong to the physiology and every arithmetic in use works somewhere else.

    part 15 · eye
  43. Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example.

    The ranking is not stable

    On a red pigment under daylight the six observer departures run from 2.38 down to 1.20 ΔE₀₀. Over forty-two surfaces two of them change places, the top two separate, and every one spans between a factor of ten and a factor of thirty-five. A chart of six bars is a chart of one example.

    part 15 · eye
  44. The three cone absorptances at two settings of the field size. Solid and dashed are the same construction at the two ends of the CIE's own second observer. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 16.0 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    A field size is two changes

    The CIE publishes two standard observers and the difference between them is usually described as a field size. What actually differs is a macular pigment the light no longer passes through and a cone outer segment the light no longer travels the length of — two changes, in two places, with different signs and different sample dependence.

    part 15 · eye
  45. The three cone absorptances at two settings of the age of the lens. Solid and dashed are the same construction at the two ends of twenty years old against seventy. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 25.5 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The observer has no age

    Five of the six arguments in this round are spreads — a population differs about them and the mean is a reasonable summary. The lens is not. Everybody's lens yellows in the same direction at about the same rate, so a standard observer with no age is not an average over a population; it is a snapshot of one moment in every reader's life.

    part 15 · eye
  46. The three cone absorptances at two settings of the rods. Solid and dashed are the same construction at the two ends of a tenth of the cone response, which is a dim room. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 9.1 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The rods are a fourth curve

    Colorimetry has three numbers and a rod signal is a fourth. It cannot be absorbed by a gain, it cannot be removed by a white point, and adding a tenth of one to a three-curve observer costs 1.60 ΔE₀₀ — the narrowest distribution of the six departures, because an addition behaves quite differently from a filter.

    part 15 · eye
  47. The three cone absorptances at two settings of the macular pigment. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 31.5 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The macular is a band, not a filter

    The lens absorbs everything below 500 nanometres with a long tail; the macular pigment absorbs forty nanometres either side of 460 and nothing else. The two have similar sizes and completely different distributions, and the reason is that one is broad and one is narrow — which is what decides whether a sample is affected at all.

    part 15 · eye
  48. The three cone absorptances at two settings of the pigment peaks. Solid and dashed are the same construction at the two ends of two standard deviations, and the L/M polymorphism on top. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 8.9 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The peaks move the flanks

    Shifting a cone's peak wavelength by three nanometres changes its sensitivity at its own maximum by almost nothing and on its flanks by several per cent, because a maximum is flat and a flank is steep. Every display primary sits on a flank, which is why a pigment polymorphism is worth 3.60 ΔE₀₀ under a laser projector.

    part 15 · eye
  49. The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

    Two observers and one metamer

    An observer departure is invisible on a single sample compared with nothing. It becomes a disagreement the moment two spectra are being asked to match, because a match is an identity between three integrals and a different observer takes different integrals. Everything in this round is a statement about pairs wearing a single sample's clothes.

    part 15 · eye
  50. What the rods cost a match, by where their signal enters and under which lamp. For five lights, the median colour difference over forty-two surfaces between the reference observer and the same observer with a rod signal a tenth of each cone's peak added — into all three cone channels, into the long- and middle-wavelength channels only, or into the short-wavelength channel only. Under daylight the first two are 1.03 and 0.90: whether the rods reach the S pathway hardly matters. Under a phosphor white LED they are 1.24 and 0.45, a factor of 2.75, and the S-only route alone costs 0.96.

    The rods' route is priced by the lamp

    A rod signal in a dim room disturbs a colour match, and how much depends on which of the cone pathways it reaches — a weight the physiology leaves uncertain, especially for the blue–yellow pathway. Under daylight the uncertainty is nearly free: a rod signal that skips the S pathway costs 0.90 at the median surface against 1.03 for one that enters all three. Under a phosphor white LED it is worth a factor of 2.75, 0.45 against 1.24. What decides it is one number per lamp: how large the rod signal is compared with each cone class's own catch of the light.

    part 16 · eye
  51. Which two lamps to stand a mesopic match between. Every pair of the five lights, by how far a match made under one and set under the other moves as the rod signal's weight into the S channel goes from nothing to equal — the median over forty-two surfaces, which is the signal an experiment has to resolve. The best pair is daylight against phosphor LED at 0.58 ΔE₀₀; the worst is tungsten against fluorescent tube at 0.09, a factor of 6. The count at the right is how many settings it takes to resolve the weight to a tenth at half a colour difference of scatter per setting.

    The reference lamp must not move

    To measure an uncertain weight, use the condition in which the answer depends on it most. That is right about half of an asymmetric colour match and exactly wrong about the other half: a match measures a difference of two displacements, and a reference field that also moves with the weight cancels the signal the test field carries. Daylight is the least sensitive of five lamps and belongs in every one of the three best pairs — 75 settings against a phosphor LED, 2,804 against the pair of lamps the principle as stated would have chosen.

    part 17 · eye
  52. Two ways to ask how much rod signal an older eye has. The rod signal's catch of each lamp divided by the S cones' own catch, against the observer's age, for five lamps — drawn twice. The upper curves take the rod signal at a fixed absolute size, and every one of them roughly doubles from twenty to seventy-five: an older lens cuts the blue before the S cones see it and the rods, peaking further into the green, lose much less. The lower curves take the rod signal at a tenth of each cone's own peak absorptance, which is the model's own definition, and they barely move at all. Nothing about the retina differs between the two; only the normalisation does.

    A rod signal has no natural size

    An older lens absorbs where the S cones are sensitive, so it should make the uncertain rod-to-S-cone weight cheaper. Measured, the rod signal's catch of a phosphor LED as a share of the S cones' own catch more than doubles from twenty to seventy-five — and the share the model actually uses falls by a fifth. The two differ by the S cone's peak absorptance, which the lens takes 60 per cent of, and which entered the model as a normalisation rather than as a claim.

    part 17 · eye
  53. How far a mesopic match moves as the S weight opens: two rooms against one field. For every pair of the five lamps, the median over forty-two surfaces of how far a match moves as the rod signal's weight into the S channel goes from nothing to equal: pale for the two-room match, each half adapted to its own lamp; dark for a bipartite field whose two halves share one adaptation. The field's signal is larger for every pair, by ×1.5 to ×7.7.

    One field keeps what two rooms divide out

    An asymmetric match can measure how strongly the rods feed the blue-yellow pathway, but set with the observer adapted to each lamp in turn it needs seventy-five settings on the best pair of lamps and hours of waiting between them. Putting the two lamps on the two halves of one field was proposed as the quick version, at the cost of a weaker signal. The signal is not weaker. Under one shared adaptation it is three times stronger for daylight against a white LED, and the best pair needs six settings. The adaptation that makes the slow version slow is also what was dividing the rods' contribution out of each half.

    part 18 · eye

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