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

Macular pigment — where it appears

A yellow filter over the fovea that absorbs around 460 nanometres and is absent a few degrees away. It is the reason the CIE's two standard observers are two observers rather than one, and its density varies by a factor of several between people.

Named by 45 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

Seventeen observers in 1931

The standard observer that governs every colour specification in industrial use is an average over seventeen young British men, measured with equipment from the 1920s. It is known to be wrong in the blue, the correction has existed since 1951, and it has never been adopted.

limits · Limits How far a match comes apart when the observer changes. A broad source and a three-primary source, solved at each primary width so the pair is an exact tristimulus match for the CIE 1931 observer. The pair is then handed to the 1964 observer, and the gap between them is plotted. For the observer they were built for the gap is arithmetic noise at every width. For the other it grows as the primaries narrow, reaching 0.012 at 10 nm — and displays have been getting narrower for twenty years.

Whose eyes

The standard observer is an average over seventeen people, and no reader is it. What that costs was small when displays were broad and grows every time the primaries get narrower.

limits · Limits 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.

eye · Cones How far a match comes apart when the observer changes. A broad source and a three-primary source, solved at each primary width so the pair is an exact tristimulus match for the CIE 1931 observer. The pair is then handed to the 1964 observer, and the gap between them is plotted. For the observer they were built for the gap is arithmetic noise at every width. For the other it grows as the primaries narrow, reaching 0.012 at 10 nm — and displays have been getting narrower for twenty years.

Two degrees or ten

This site names the observer on every figure it draws, which was supposed to be the point. Measuring what it had actually done found two hundred and four figures naming the same observer, three naming the other, and sixty-two of sixty-four generators printing a string that had never once varied.

limits · Limits 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones How far a fitted transform is from anybody's eyes. Five groups of three bars: for each published adaptation transform, the distance from the population's own cloud to the confusion point that transform is committed to, measured in the population's standard deviations on that point. On the protanope's point every one of them is between 2.2 and 14.4 out, and on the deuteranope's between 3.7 and 11.8. On the tritanope's, 5 of the five are within three standard deviations — indistinguishable from a member of the population. The claim that these matrices are not cone responses is safe, and the evidence for it is two points out of three.

Two points out of three

Every published adaptation transform implies three dichromat confusion points, whether or not it was fitted to any. Measured in the population's own standard deviations they are two to fourteen out on the protanope's point and four to twelve on the deuteranope's — and between half a standard deviation and two and a half on the tritanope's, which is inside the population. The claim that these matrices are not cone responses is safe. The evidence for it is two points.

matching · Gamut A population of receptor bases, in the plane the published ones live in. The two axes this collection scores an adaptation basis on — the residual across the illumination census on the horizontal, ellipse anisotropy on the vertical, lower better on both — with 200 extra points on it. Each is the basis a member of the population's own confusion points determine. The cloud is not a point: it runs from 1.22 to 2.24 ΔE00 horizontally, which is wider than the whole spread of the published transforms marked on it. The observer this site quotes sits inside the cloud and near one edge of it, and the sentence "the receptor basis costs seventy per cent" is a sentence about that one point rather than about the construction.

A trade between matrices, not people

Across the space of possible bases, adapting well and discriminating well pull in opposite directions — the two optima sit at the ends of an empty diagonal. Across a population of actual observers the same two costs move weakly together, at a correlation of +0.29. The trade-off is a property of the set of matrices somebody could choose, not of the eyes anybody has. One measurement inside the population does trade, and it is the macular pigment.

brain · Appearance 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.

eye · Cones 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.

eye · Cones What six of this collection's published numbers do when the unit changes. Six quantities, from six calculations that share nothing: a change of light after an observer has adapted, a camera profile's error, the gap between the two standard observers, a metameric pair under the lamp that breaks it, the same image on two papers, and an observer two seconds into a new room. Each is recomputed under all six units and every unit is calibrated onto ΔE2000's scale first, so the bar is not a change of units in the ordinary sense. The bar is the ratio of the largest reading to the smallest, and it runs from 1.71 to 2.30. Five of the six are printed in ΔE2000 by the essays that report them; the sixth is printed in CAM16-UCS, because the model it comes out of defines that unit.

The observers differ by a unit's worth

The gap between the 1931 and 1964 standard observers is the one quantity in this collection's audit with no published number under it — nothing here reports it as a single figure over a stated set. It also has the second-largest dependence on which colour-difference formula is used, running from 1.60 to 3.55 across the menu.

matching · Gamut 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones 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.

eye · Cones The same white, matched at six primary widths. At every width the three primaries are solved to match D65 exactly for the reference member; the bands are what the population sees. A broad primary integrates the observer differences over a band and averages them away; a narrow one samples them at a point and passes them straight through. From 40 nm to 2 the ninety-fifth percentile rises from 11.3 to 17.9 ΔE00, monotonically, and the technology has been moving from left to right for thirty years.

A narrow primary buys a disagreement

The observer audit decomposes what a display costs a population. Narrowing the primaries raises the pigment-peak departure monotonically, moving one raises or lowers the macular departure, and the two respond to different design variables — so a wide gamut and an observer-robust display are bought with the same money.

matching · Gamut The angle between the lens and the macular pigment, before and after adaptation. On each of 120 surfaces the angle, in the local metric, between what an older lens does to the reading and what a denser macular pigment does, binned in ten-degree steps. Read without adaptation, where both filters yellow the observer's white along with everything else, the two point nearly the same way: a median of 8 degrees. Read after each observer has adapted to its own white, the median is 156, and the two together cost less than the larger alone on 115 of the 120.

Two yellow filters cancel on a slope

An older lens and a denser macular pigment both take blue out of the light, and read before adaptation they move a colour in nearly the same direction, eight degrees apart. Once each eye has adapted to its own white they point a median 156 degrees apart on smooth reflectances and together cost less than the lens alone. On surfaces with a narrow absorption band they still sit 26 degrees apart and add. What decides it is the width of the surface's own features.

difference · Metric The angle between the two filters against how completely the eye has adapted. The angle, in the local metric, between what an older lens does to a reading and what a denser macular pigment does, on 120 smooth reflectances, as the degree of adaptation runs from nought to one. The median angle is 8 degrees unadapted and 156 at complete adaptation, and almost all of the turn happens in the last tenth: it passes a right angle at a degree of 0.928. The marks are the degrees CIECAM16 gives five rooms — an overcast sky 1.00, an office 0.94, a lit living room 0.86, a dim room 0.75, a cinema 0.66 — so only the outdoor one is at the end of the dial.

Two filters cancel only in a bright enough room

An older lens and a denser macular pigment cancel each other once an eye has adapted — and that result belongs to the end of a dial nobody stands at. Read at the degree of adaptation CIECAM16 gives an ordinary room, the two barely cancel; in a living room they add, and in a cinema they cost six times what they cost under the sky. The room has to be about as bright as an office before the cancelling begins at all.

difference · Metric Every pair of departures, before adaptation and after it. The fifteen pairs of the six audited observer departures. Each row runs from the angle between that pair's two deviations with no adaptation to the angle with complete adaptation; an angle past ninety degrees is a pair pointing apart, which is where a pair can cost less together than the larger of the two costs alone. 3 pairs gain that behaviour as the eye adapts, 3 keep it, 5 lose it and 4 never have it. The pair followed here — the lens against the macular pigment — is in the smallest group that is not empty, and every result quoted from it generalises in the wrong direction.

Adaptation turns more pairs off than on

One pair of observer departures was followed across the degree of adaptation and found to cancel only in a bright enough room. The same calculation takes any two, and run over all fifteen pairs it says something the single pair does not: adaptation is a rotation rather than a mechanism for making departures oppose each other. Five pairs lose their cancellation as the eye adapts, three gain it, three keep it and four never have it — and the pair everybody quotes is one of the three it turns on.

difference · Metric Where a surface's band sits decides the room it needs. Each of 168 surfaces at its own crossing — the degree of adaptation at which the part adaptation has yet to remove falls to the size of the part it will leave — against where that surface's absorption band sits. The line joins the median at each band centre and the rooms are marked across. A surface absorbing at 470 nm crosses at 0.883 and one absorbing at 670 nm at 0.972. Both of the filters this is about absorb in the blue, so a surface with a blue band is where the two differ in shape and has a large residual, while a surface with a red band is nearly invisible to both and its whole deviation is the shared yellowing of the observer's white.

The room a surface needs is written in its band

Whether two yellow filters cancel on a surface depends on the room, and each surface has its own crossing — the degree of adaptation at which the shared yellowing falls to the size of what is left underneath. Those crossings run from 0.68 to 0.99, and where a surface's absorption band sits accounts for almost all of the spread while how much light it returns accounts for almost none. The reds need a room brighter than a graphic-arts viewing booth, which is brighter than any room a sample is judged in.

difference · Metric What each pair of lamps pays to fit the lens and the macula as well. For each pair of lamps, the standard error of the rod signal's S weight fitted from one session — with the observer's lens and macula known (top bar), with the lens fitted too (middle) and with both fitted (bottom). Daylight against the LED is the best pair with everything known, ±0.026, and pays most of its advantage for the macula; tungsten against the LED pays almost nothing for either.

The matches carry their own lens

A lens mistaken by ten years moves a colour match as far as half the rod signal being measured, and a tenth of an optical density of macular pigment as far again, so an experiment on the rod signal looked as if it needed a densitometer for every observer. It does not. Across a family of surfaces the three move the matches in different patterns, and one session of forty-two settings fits all three together — the lens to within a year and a quarter, better than a densitometer was asked for. The price is the weight's precision, and a second pair of lamps nearly removes it.

eye · Cones Where a lens edge 10 nanometres off goes in a one-session fit. A sixty-year-old observer whose lens edge sits 10 nanometres longer (upper bars) or shorter (lower bars) than the model's, fitted by the model from one daylight-against-LED session: the shift in each fitted parameter, in that parameter's standard errors. The macula takes the most, 2.4 of its standard errors; the weight moves by 0.59 of its own, 0.035 in absolute terms, against a prediction of under a tenth for an edge ten nanometres off.

A lens of the wrong shape lands in the macula

One session of colour matches can fit an observer's rod signal, lens and macular pigment together — but it fits a lens of the model's shape, one exponential in wavelength scaled by age, and a real lens need not be that shape. Give a sixty-year-old a lens with the right density at 400 nm and its edge ten nanometres off, and the fit puts the error mostly in the macula, moves the rod signal's weight by six tenths of its standard error — six times the predicted tenth — and leaves a misfit the session cannot see. A second lamp pair makes it worse: 3.5 standard errors. Fitting the edge as a fourth number removes the bias, costs the weight three to fourteen per cent, and finds the edge.

eye · Cones Where one session puts a band in the lens, by the band's wavelength. A band of a tenth of an optical density in a sixty-year-old's lens, centred from 410 to 460 nm, fitted from one daylight-against-LED session by the model with weight, lens age, macula and the lens edge free: each parameter's shift in its own standard errors. Near 410 to 430 nm the band is read as lens; from 440 it is read as macula; the edge takes little of it anywhere; and the rod weight moves at every centre, by 0.37 to 1.39 of its standard errors.

A band in the lens reads as age or as macula

A lens whose absorption edge is steeper or shallower than the model's biases a matching session's fit, and fitting the edge's position repairs it. An ageing lens also grows bands: yellow pigments that add a shoulder of absorption just past 400 nanometres. The edge parameter takes none of one. A tenth of an optical density centred at 410 to 430 nm is read as four to six years of extra lens age; centred at 440 to 460, as three to ten hundredths of macular pigment. Either way the rod signal's fitted weight moves, by up to 1.4 of its standard errors in one session and 1.8 in two, and the session's misfit stays within its noise. The lens model has to carry the band, or the experiment has to state what it does not know.

eye · Cones More rods make a lens band's pull on the rod weight worse, unless the band is fitted. A sixty-year-old's lens with a band of a tenth of a density at 450 nm, fitted by the model with the rod signal a tenth to a half of each cone's peak: the shift in the fitted rod weight, in its own standard errors. With weight, lens age and macula fitted, one session's shift grows from +1.73 at a tenth to +2.53 at a third, and two sessions' more. With the band's amplitude fitted as a fourth number, one session stays under half a standard error at every share.

A dimmer session hides a lens band in the rods

A band of extra absorption in an older lens is read by a colour-matching session as lens age or as macular pigment, and it pulls the fitted rod weight with it. The rods see the lens and not the macula, so a dimmer session, where they carry a third of the signal instead of a tenth, was predicted to tell the band from the macula. It does the opposite. With the band at 450 nm left out of the fit, the rod weight's shift grows from 1.7 of its standard errors to 2.5, the misfit a session could notice shrinks by a third, and the macula and the band trade more tightly. The rods see the band — and the rod weight, which the fit is free to move, is exactly what absorbs it. What separates the band from the macula is fitting its amplitude as a fourth number, which works in daylight already.

eye · Cones How well a matching session places a lens band's centre. A sixty-year-old's lens with a band of a tenth of a density centred from 410 to 460 nm, fitted with the rod weight, lens age, macula, the band's amplitude and its centre all free: the centre's standard error, from one daylight-against-LED session and from designs adding a second session. One session places the centre to 11.5 nm at worst below 430 and within 5.7 above it; adding the LED against the three-emitter lamp brings every centre within 5.5.

A second lamp pair finds a lens band's centre

A band of extra absorption in an older lens is found by a colour-matching fit only if the fit has a parameter for it, and a band parameter at a fixed wavelength finds only a band near that wavelength. Freeing the band's centre as a fifth number asks whether the matches can place it. From one daylight-against-LED session they place a band above 430 nm to within six nanometres and one below it only to eight to twelve, where moving the band and ageing the lens change the matches together — and there they barely see the band at all. A second session, the LED against a three-emitter lamp, places every centre within five and a half nanometres. The fluorescent tube's line at 436 nm helps where it sits and nowhere else.

eye · Cones From which starts a fit finds a lens band, two sessions. For a band of a tenth of a density at 415, 430 and 445 nm in a sixty-year-old's lens, the number of twenty simulated experiments of two sessions each in which a fit of all five numbers, started with the band's centre at each of 400 to 470 nm, finds the centre within twice its standard error. The ticks at the left edge are the same count for a fit started at the true observer: 18, 18, 18 for the three bands.

A lost lens band goes to age or the macula

Two colour-matching sessions determine the centre of a band of extra absorption in an older lens to about five nanometres, if a fit can find it. Fitted in full from eight starts across 400 to 470 nm on twenty simulated experiments, two sessions find a band at 415, 430 or 445 nm from every start in fifteen to eighteen experiments of twenty, against eighteen for a fit started at the truth. The prediction was that a low band would be lost from high starts to the macula's flank. The start barely matters with two sessions, and a lost band does go somewhere — but a band at 415 or 430 goes to lens age, three to five years of it, and only a band at 445 goes to the macula. Every loss is a worse fit than the session's best, so eight starts find the band as often as the truth does. With one session the start decides.

eye · Cones The lens band a fit reports for a macula that peaks off 460 nm. For an observer at sixty with no band in the lens and a macula peaking 6 and 3 nm short of the model's 460, at 460, and 3 and 6 nm long, the band the five-number fit reports from the exact matches, as its amplitude over its standard error, with one and two sessions. A macula 3 nm short is read as a band of 0.061 at 438 nm from two sessions, 3.1 standard errors; a macula peaking long has no band to become, and at 6 nm long the fit runs to the edge of the range a band may take.

A macula three nanometres short is a lens band

Two colour-matching sessions can fit a band of extra absorption in an older lens, its size and its centre, and the test proposed for real older observers was to count those with a band more than twice its standard error. That test assumes a significant band can only be a band. The observer model's macula peaks at 460 nm; a real one is two carotenoids whose mix moves its peak by a few nanometres. Simulated with its macula peaking three nanometres short and no band at all, an observer at sixty is fitted with a lens band of 0.06 at 438 nm, three standard errors, and noisy sessions find it in nineteen of twenty experiments. Six nanometres short, the phantom is larger than the band the earlier essays searched for. The confusion runs both ways: a real band fitted with the macula's peak free reads as a macula two to three nanometres short. Neither fit alone can tell them apart. Comparing the two can, with two sessions — never preferring the macula for a real band, and preferring it in sixteen to nineteen experiments of twenty for a short macula.

eye · Cones How often a six-number fit finds both departures. For nine observers at sixty, each with a lens band of a tenth of a density at 415, 430 or 445 nm and a macula moved by −3, 0 or +3 nm, the number of twenty noisy experiments in which a fit of all six numbers finds the band beyond twice its standard error, and finds the macula moved beyond twice its standard error — with two sessions and with one. With two, both are found in most experiments wherever both are present; with one, neither is.

A second session keeps a band and the macula apart

A lens band and a macula peaking a few nanometres short look alike to a colour-matching fit that has only one of them, and an older eye may have both. Fitted with all six numbers — rod weight, lens age, macular density, the band's amplitude and centre, and the macula's peak — nine simulated observers at sixty, each with a band at 415, 430 or 445 nm and a macula moved three nanometres either way or not at all, test whether the matches can carry both. The prediction was that the band at 445 and a short macula would trade almost completely, a correlation beyond 0.95 and the band's standard error tripled. That is what one session does: correlations of 0.91 to 0.95 at 430 and 445 nm, and a band found in at most twelve experiments of twenty. Two sessions do not. Their correlations stay between 0.42 and 0.83, the band is found in twelve to twenty experiments for every observer, and a moved macula in twelve to nineteen — while an unmoved one is found moved in at most three. The price of the extra number is paid by the band nearest the macula, whose standard error rises by three quarters; the rod signal pays almost nothing.

eye · Cones A line on the macula's peak frees the band; one on its flank does not. For the band at 445 nm, its standard error in the six-number fit over its value when the macula's peak is not fitted, with the third session's line at 465 to 510 nm. Two sessions leave it at ×1.78 and a repeated second session at ×1.41. The cyan line at 487 nm leaves it at ×1.51, worse than the repeat; at 465 nm, near the macula's 460 nm peak, it comes to ×1.16, inside the prediction's quarter.

A line on the macula's flank buys less than a repeated session

Fitted with six numbers, two sessions of colour matches keep a lens band and a moved macula apart, but the band at 445 nm pays for it: correlated with the macula's shift at 0.82, its standard error 1.8 times its value when the macula is not fitted. A moved macula changes its long flank past 470 nm, where no lens band reaches, so a third session — the LED against a lamp with a narrow line at 487 nm — was predicted to bring the correlation under 0.6 and the band's error within a quarter of its band-only value. The correlation falls to 0.80 and the error to 1.5 times. A third session that simply repeats the second does better on both errors. The flank line pins how much macular pigment there is, not where it peaks, and it ties the band's height to its centre. Moved to 465 nm, onto the macula's peak, the same session brings the band within about a fifth of its band-only error — but no position of the line brings the correlation below 0.78. The band at 445 nm and the macula's peak are told apart by their short flanks, and a third session adds matches rather than a new kind of evidence.

eye · Cones A known density frees two sessions, not the line on the peak. For an observer at sixty with a lens band at 445 nm and an unmoved macula, the correlation between the band's amplitude and the macula's shift when the macular density is fitted with the other five numbers, and when it is known exactly and fixed, under two sessions, the second session repeated, and the cyan session with its line at 465 to 510 nm. Fixing the density takes two sessions from 0.83 to 0.62 and the line at 465 nm, on the macula's peak, only from 0.86 to 0.74: the line that freed the band's error best is left with the strongest trade.

Knowing the macula's density frees two sessions, not the line on its peak

Fitted with six numbers, an older observer's colour matches cannot fully split a lens band at 445 nm from a macula whose peak has moved: their correlation stays above 0.78 under every third session tried. The proposal was to measure the macular pigment's density from outside the matches, leaving five numbers, and the prediction was that a line on the macula's peak would then split the pair below 0.6 while two plain sessions stayed near 0.8. It runs the other way. With the density known exactly, two sessions fall from 0.83 to 0.62 and the line at 465 nm only from 0.86 to 0.74. The density was the road the band and the peak traded along — correlated with each at about −0.75 in two sessions' fit — and the designs that had already narrowed that road gain least from closing it. A realistic outside measurement closes almost none of it: the matches already know the density to 0.027, so a flicker measurement good to 0.05 moves the correlation by three hundredths. And far from the macula, at 415 nm, a known density binds the band to the peak more tightly than before.

eye · Cones A density a tenth wrong moves the band by more than its error. For an observer at sixty with a lens band at 445 nm and an unmoved macula, fitted from two sessions with the macular density fixed at 0.8, 0.9, 1.1 and 1.2 times its true 0.35: the bias in the band's amplitude and in the macula's fitted peak, in units of the fixed fit's standard errors. A tenth too high lowers the band by 1.4 standard errors and moves the peak 1.25 nm short; the prediction was a third of a standard error and under half a nanometre.

A density a tenth wrong costs the band more than fitting it

An older observer's colour matches carry a lens band and a macula that can trade against each other, and fixing the macular pigment's density from an outside measurement breaks much of that trade when the density is right. A flicker measurement and a colour match average the pigment over different fields, so the fixed density can be wrong by a tenth or more, and the prediction was that a tenth too high takes about a third of a standard error off a band at 445 nm and moves the macula's peak by under half a nanometre. It takes 1.4 standard errors off the band and moves the peak 1.25 nm short. At 415 nm the same error pushes the band the other way, up by 2.4 standard errors, and runs its fitted centre 5.6 nm shorter and the lens's age three years younger. Bias and error together, a density fixed a tenth wrong gives the band at 445 nm a larger error than leaving the density free; the break-even is about eight per cent. Given as a measurement with its own standard error of 0.03 instead of fixed, the same wrong density biases the band by under half its error.

eye · Cones Near the macula a wrong density moves every experiment together. For an observer at sixty with a lens band at 445 nm and an unmoved macula, two sessions: each of twenty simulated experiments' fitted band amplitude with the macular density held a tenth low, at its truth and a tenth high. A tenth high moves the band by -0.031 on average with a spread of 0.0024 between experiments; a tenth low by 0.030, spread 0.0022; the band's own standard error is 0.023.

A wrong density shifts every experiment alike near the macula

Read from the expected matches, a macular density fixed a tenth too high lowers an older observer's lens band at 445 nm by 1.4 of its standard errors and raises a band at 415 nm by 2.4. Real experiments carry noise, and the prediction was that their fits scatter about those biases with the fixed fit's own error, so that a tenth's mistake puts the band beyond twice its error in about a third of experiments at 445 nm and two thirds at 415. Twenty simulated experiments per case, each fitted with the density free and held at 0.9, 1 and 1.1 of its truth, give five of twenty at 445 nm and sixteen at 415 under two sessions — eight and eighteen with a line on the macula's peak. Near the macula the bias is a rigid shift: every experiment moves by the first-order amount to within a tenth of a standard error. Far from it the bias is lopsided — up by 0.058 for a density too high, down by 0.032 for one too low — and a density too low erases the band altogether in one experiment of twenty.

eye · Cones One session, two valleys: the macula where it is, or far short. For the observer with a band at 415 nm and an unmoved macula, one session: each of twenty experiments' census fit in band amplitude and macula peak, and, for the 7 the census missed, the better fit, joined to the census's. Every better fit puts the macula seven to nineteen nanometres short: a second valley in which a macula moved far short does part of the band's work.

One session has a second valley

Every essay on older observers since the second session has read its fitted lens bands and macula shifts from one census: nine simulated observers, three designs, twenty noisy experiments each, every fit a damped descent from two starts. Two of forty experiments refitted in the last essay found a better fit than the census had. The prediction was that the census misses one fit in twenty, almost all where the band is small, and that the misses change no count it reported by more than a few hundredths. Refitted from three more starts, 66 of its 540 experiments have a better fit — 45 of 180 under one session, 9 and 12 under two and three. Sixty of the 66 better fits move the macula's peak more than two nanometres further short, most shrinking the band, and 31 lie lower by under one unit of χ²: a second valley, nearly level with the first. Two sessions' counts stand to within one or two experiments. One session's do not: with the best fits an unmoved macula is reported moved in eight experiments of twenty, where the census said one.

eye · Cones Bounded, one session stops finding a macula that never moved. For the three bands with the macula unmoved, how many of twenty one-session experiments report it moved beyond twice its standard error: from the census's two starts, from the best of five starts, and from the best of five with the shift held within ±6 nm. The bound brings the false shifts at 415 and 430 nm from 8 and 6 to 1 and 2.

A bound on the macula trades false moves for hidden ones

Fitted thoroughly, one session of an older observer's colour matches finds a second valley in a quarter of its experiments — the macula's peak moved ten or more nanometres from the model's, where no real macula sits — and reports a moved macula where there is none in up to eight experiments of twenty. The prediction was that holding the macula's shift within six nanometres either way, the range real pigment mixtures span, restores every observer's one-session counts to within two of the census's and leaves about one fit in four pressed against the bound. The false moves vanish: an unmoved macula is reported moved in one, two and one experiments of twenty for the three bands, against eight, six and two unbounded. Seven of nine observers keep their count of moved maculas within two of the census's. But only one fit in six ends on the bound, and beside the band at 445 nm the bound hides real moves: a macula truly three nanometres short is found in three experiments of twenty, where the census found seven. Two sessions barely notice the bound.

eye · Cones The prior trades the macula's moves for the lens band. For each observer, how many more experiments of twenty find the lens band under the prior than under the census's fits, against how many more find the macula moved. The observers moved short or unmoved gain up to seven bands and lose one to five moves — false ones where the macula never moved, real ones where it did; those moved long gain no band and lose one or two moves, the 445 nm observer losing three bands as well.

A soft prior on the macula frees the lens band

One session of an older observer's colour matches, fitted thoroughly, can drift into a valley where the macula's peak sits ten or more nanometres from any real eye's, and a hard bound at ±6 nm removes those false moves at the cost of hiding some real ones — beside a lens band at 445 nm, a macula truly three nanometres short is found in three experiments of twenty where the census found seven. The prediction was that a Gaussian prior four nanometres wide, which lets a fit go further when the matches insist, keeps the false moves away and restores those real ones. It keeps the false moves away — none, one and one for the three bands. It does not restore the real moves: it finds them as seldom as the bound or more seldom, two of twenty beside 445 nm, because it pulls every fitted shift towards nought, to a mean of 1.2 to 1.9 nanometres for moves of three. What it buys instead is the lens band. With the macula held near the standard, the band beside it is found in ten to twelve experiments of twenty, where the census found three to seven.

eye · Cones Each widening of the prior buys moves with bands. Over the six observers whose macula truly moved three nanometres, the real moves one session finds beyond twice their error, against the lens bands found for the four observers whose band the four-nanometre prior freed — under priors four, six and eight nanometres wide, the census's own fits and the hard bound. 4 nm: 14 moves, 44 bands; 6 nm: 22 moves, 34 bands; 8 nm: 25 moves, 25 bands; census 27 and 22; bound 19 and 20.

A wider prior gives the moves back by spending the band

One session of an older observer's colour matches, fitted under a Gaussian prior four nanometres wide on the macula's peak shift, finds the lens band beside the macula in ten to twelve experiments of twenty and real three-nanometre moves of the macula as seldom as a hard bound does. The prediction was that a prior six nanometres wide keeps the false moves at two or fewer, finds the real moves within two of the bound's counts and keeps two thirds of the band's gain, and that at eight the far valley reopens at 415 nm. Three parts hold. The fourth fails: six nanometres keeps 55 per cent of the band's gain. Read across the widths the prior traces one trade — 14 real moves and 44 bands at four nanometres, 22 and 34 at six, 25 and 25 at eight, against the unconstrained fit's 27 and 22 — and the hard bound, at 19 and 20, is worse than six nanometres on both.

eye · Cones Even two sessions fall into the far valley, and the prior lifts them out. For the observer with its band at 415 nm and its macula truly three nanometres short, every two-session experiment's fitted shift at the best of five starts and under a prior six nanometres wide. Three best fits lie beyond 7.5 nm short — −11.84 drawn to −1.90, −7.73 drawn to −3.05, −11.00 drawn to −0.94 — and the rest move by a tenth or two; the mean goes from −3.89 to −2.57.

A second session leaves the prior one job

Under one session of an older observer's colour matches, a prior on the macular pigment's peak shift trades the lens band against real moves of the macula, six nanometres wide at the knee of the trade. Two sessions already find the band in twelve to twenty experiments of twenty and real three-nanometre moves in twelve to eighteen, and still report an unmoved macula moved in two to three. The prediction was that the six-nanometre prior, applied to two sessions, removes those false moves and costs nothing else. It removes none of them — two, two and none remain, about the rate at which a test at twice the standard error calls a move by chance. It costs nothing at five observers of six. Beside 415 nm, with the macula moved short, it finds four fewer moves: three of the best fits there were in the far valley, eight to twelve nanometres short, and counted as moves because their sign was right. With two sessions the far valley is the prior's only remaining job.

eye · Cones A third session finds the band beside the lens's edge. For each observer, how many of twenty experiments find the lens band beyond twice its error and within 20 nm of where it was put: two sessions under the six-nanometre prior, and three sessions at their best of five starts. Beside 415 nm with the macula unmoved, two sessions find it in 12 and three in 18; beside 430 nm with the macula short, 20 and 16.

A third session finds the edge band and counts bands misplaced

Two sessions of colour matches find an older observer's lens band in eighteen to twenty experiments of twenty at every place but one: a band at 415 nm, beside the lens's own short-wave edge, found in twelve. The prediction was that a third session — a phosphor LED set against a lamp with a cyan line — finds it in at least seventeen, and that the experiments losing it under two sessions fit the lens three or more years old. Three sessions find it in eighteen, and every loss under two did fit the lens older, but by under three years in five of eight. The count hid something else. It credits a band wherever a fit puts it, and up to twelve of 180 fits put it sixty to eighty nanometres from its place, on the macula's long flank. With three sessions the six-nanometre prior no longer closes that valley.

eye · Cones Held near the lens, the band is found where it was put. For each observer, how many of twenty three-session experiments find the lens band beyond twice its error and within 20 nm of its place, with its centre free to range from 390 to 500 nm and held to 400–460. Beside 415 nm the bounded fits find it in 19, 20, 19 with the macula short, unmoved and long, against 16, 18, 15 free.

A band held near the lens closes the valley the prior could not

Three sessions of colour matches find an older observer's lens band beside the lens's own edge as predicted, but twelve of 180 fits place the band sixty to eighty nanometres away on the macula's long flank, each with the macula fitted far short — a valley the six-nanometre prior on the macula's shift no longer closes. The prediction was that holding the band's centre to 400–460 nm, where a lens band can be, empties that valley, finds the 415 nm band with the macula short in at least eighteen experiments, and raises the moves found beside 430 nm from nine to twelve. The first two hold: no fit is left in the valley, and the band is found in nineteen. The third does not: 430 nm stays at ten. And the valley turns out to have been nearly level with the truth — holding the band costs ten of the twelve fits under four units of the fit's chi-squared, and every one's macula comes home.

eye · Cones

Named alongside it

The objects these essays reach for when they reach for this one.

Lens yellowingIndividual variationStandard observerIdentifiabilityModelling assumptionStandard errorObserver metamerismCone fundamentalsLeast-squaresColour-matching functionsRodsSpecification

All concepts