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

Absorption — where it appears

Light taken up by a material and turned into something other than light, which is what makes a pigment a pigment. What is absorbed at each wavelength decides a surface's reflectance, and the same molecule that absorbs may re-emit at a longer wavelength.

Named by 23 essays across 6 fields — each of them below, with the objects they name alongside it.

Mixing two paints and stacking two filters are different operations. The same two reflectances combined two ways. Stacking them as filters multiplies the transmittances, which is right for gels in front of a lamp and wrong for pigment stirred into pigment: a stirred mixture is one scattering layer, not two in series, and light meets whichever particle is nearest rather than passing through both. Kubelka–Munk handles it by moving to K/S = (1 − R)²/2R, in which absorption and scattering add by concentration, and inverting afterwards. The two answers differ by ΔE00 = 15.0, and the filter model is the darker of the two because it charges every photon for both pigments.

Paint is not a filter

Stacking two filters multiplies their transmittances. Stirring two pigments together does not multiply their reflectances, because a mixture is one scattering layer rather than two in series — light meets whichever particle is nearest. Treating the two as the same operation is a 15-unit error, and which way it errs turns out to depend on whether the comparison holds the amount of pigment fixed.

scene · Scene The same medium at six path lengths. Beer's law at 6 depths of one absorbing medium. The absorption coefficient is a single spectrum and the only thing changing is how far the light travelled, yet the patches differ in hue by 7.8° as well as in lightness — because absorption is exponential in depth and the observer is linear, so the bands that survive at d = 8 are not a scaled copy of the ones that survive at d = 0.25. Path length belongs to the geometry, not to the substance, which is why this sits in a field about scenes.

The colour is in the thickness

Transmittance is exponential in path length and the observer is linear, so doubling the depth of an absorbing medium squares the transmittance rather than halving the colour. A translucent object therefore has no one colour — its thin edge and its thick middle are different spectra of the same substance, and the hue moves between them.

scene · Scene The blue channel, as the three things multiplied to make it. Silicon's quantum efficiency, the colour-filter dye's transmittance, and their product. The dye is the only stage carrying any colour information and it is clear above about 800 nm — transmittance 0.92 at 900 nm against 0.06 at 550.

Silicon sees past the visible

A sensor's response ends at 1107 nanometres because that is the band gap, and the colour-filter dyes have stopped absorbing three hundred nanometres before it. So all three channels measure the same thing over the last third of the range, and every published sensitivity plot is drawn after the component that hides it.

imaging · Capture One reflectance, four different infrared tails, and what the camera makes of each. The same visible reflectance continued past 780 nm to four different near-infrared values. A spectrophotometer reports only the left-hand part; an unfiltered sensor integrates all of it. The channel spread falls from 0.14 at a tail of 0.05 to 0.02 at 0.85, with nothing about the visible half changed.

Most things are pale in the infrared

A spectrophotometer stops at 780 nanometres and a black cotton shirt reflecting five per cent of visible light reflects more than half the near infrared. The measurement everybody has and the quantity a camera integrates are different quantities, and nothing in the first says so.

imaging · Capture 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.

eye · Cones The colour of a beam, against the angle it leaves at. A conformal converter — an even layer laid straight onto the die — makes light leaving at θ cross 1/cos θ times as much of it, so it is more completely converted and the beam is warmer at its edge. From the axis to 75° the correlated colour temperature falls by 556 K and the whole difference is ΔE00 16.4. The flat line is the same emitters under a dome, whose path length is the same in every direction by construction: no angular colour at all, exactly, which is what a remote converter is sold for.

A lamp has a direction

A white LED is a blue die under a converter, and light leaving at an angle has travelled further through the converter than light leaving straight up. So the mixture is different in every direction, the beam is bluer in the middle than at its edge, and the number on the box is one direction's worth of a device that has no single colour.

light · Light How much light comes back at each distance from where it went in. The diffuse reflectance kernel of 3 materials at 550 nanometres, computed from the dipole approximation to the diffusion equation. Both axes are logarithmic. The horizontal axis is the distance from the point the light entered, in millimetres; the vertical is how much comes back out per unit area there. Each curve's own diffusion length is marked with a tick. Coated paper returns almost everything within a fifth of a millimetre; marble is still returning light at ten. The reflectance the model wants is the whole of each curve, integrated over the plane, and what an instrument reads is only the part inside its aperture.

A surface has a kernel

Light that enters a translucent material does not come back where it went in. It scatters some thousands of times and leaves a few millimetres away, so what the surface has is not a reflectance but a function of distance — and the reflectance the model wants is that function's integral over the whole plane, which no instrument ever collects.

scene · Scene A 40-nanometre limit written in nanometres and written in energy. The transition width a reflectance is allowed, across the spectrum, for two ways of stating the same sharpness. Written in nanometres it is 40 everywhere. Written as a fixed spread of photon energy, which is how an absorption band's width is set, it is 40 at 550 nanometres and grows as the square of the wavelength: 21 at 400 and 67 at 700. The steps are the quantisation the calculation actually imposes.

A limit written in energy charges the reds

A slope limit on reflectance is usually written in nanometres and applied the same way across the spectrum. An absorption band's width is closer to a fixed spread of photon energy, which is nearly twice as many nanometres at 700 as at 500. Written that way, a limit that is forty nanometres at 550 is kinder to the object-colour solid overall — 489 of 913 directions lose a per cent rather than 544 — and it charges the reds more. Which directions pay is decided by one thing: whether their optimal edges fall above or below the reference wavelength.

limits · Limits Which way a tint turns a pigment's hue, made in paint and made in light. For eight pigments tinted nineteen parts white to one of colour, the Oklab hue turn from the pure pigment: of the tint made in paint (upper bar) and of the additive mixture with white at the same luminance (lower bar). For seven of the eight the two point in opposite directions. The orange turns +13.5° in paint and −13.9° in light.

A tint in paint turns the other way

A pale colour can be made two ways: by stirring white pigment into a coloured one, or by adding white light to it — which is what a halftone on white paper and a display both do. At the same luminance the two tints are not the same colour. For seven of eight pigments they turn the hue in opposite directions, an orange by +13.5 degrees in paint and −13.9 in light, and the paint tint is the more colourful of the two every time. The cause is in the reflectance: diluting a pigment moves its absorption edge, and adding light does not.

brain · Appearance How steep an edge a band draws, and what it costs. A Gaussian absorption band of stated width produces a reflectance edge whose own width depends on how deep the band is, because the exponential saturates: where the absorbance is large the reflectance is already nought and the edge is over. A forty-nanometre band at an absorbance of 3 draws an edge 32 nanometres wide; at 12 it draws one 21 nanometres wide. Below an absorbance of 2.3 the band never reaches a reflectance of a tenth at all and has no edge in this sense. The dashed lines are each band's own width, which is the number a slope limit would have been given.

A sharp edge is bought with depth

A slope limit on reflectance was introduced as the weakest honest statement of a pigment's bluntness, with a band-shape limit named as the stronger version to be written later. Written, it is not stronger. Three absorption bands none narrower than forty nanometres reach further than a forty-nanometre slope limit in 94 of 154 directions of the object-colour solid, because a band's width and the width of the reflectance edge it draws are different quantities — and what converts one into the other is how much colorant is in the film.

limits · Limits The same lamps, charged by two kinds of slow pigment. Each lamp — the three emitters, a fourth band at seven positions, and the cheapest symmetric and lopsided bands rendering at 88 — as its fidelity index against the directions of the solid a slow pigment loses more than a per cent of its reach in, for a forty-nanometre slope limit (open) and for three absorption bands none narrower than forty nanometres (filled), joined by a line. Every line runs leftward: the band-shape pigment loses fewer directions under every lamp, and the lamps rendering best gain the most. At 88 the slope limit loses 50 and the band-shape pigment 33.

The pigment decides the trade, not the band

A four-emitter lamp that renders well costs a slow pigment part of its reach, and reshaping the fourth band barely changes that. Changing the pigment changes it a great deal. Modelled as three absorption bands none narrower than forty nanometres, rather than as a forty-nanometre slope limit, a slow pigment loses a third fewer of the solid's directions under the best-rendering lamp, and the price of going from the three-emitter lamp to a fidelity of 88 halves. The lopsided band that helped a slope-limited pigment does nothing at all for this one. And at 590 nanometres a fourth band renders at 86 almost for free.

limits · Limits Every lamp, charged by a slope limit and by bands in nanometres and in energy. For each lamp, ordered by fidelity index, the directions of the solid lost by a forty-nanometre slope limit, by three absorption bands none narrower than forty nanometres, and by three none narrower than 0.158 electronvolts. The energy convention loses fewest under every lamp: 20 against 23 and 31 under the three emitters, and 26 against 33 and 51 under the best-rendering lamp.

Written in energy, the bands spare the green

A slow pigment modelled as absorption bands pays half what a slope limit pays for a four-emitter lamp's rendering, and a fourth band at 590 nanometres costs it almost nothing. That result assumed every band is at least forty nanometres wide, and a real absorption band has a width in energy, not in nanometres. Written that way it is narrower in the blue and wider in the red, and the fear was that the red would take back the amber. It does not. The band limit written in energy loses fewer directions under every lamp, the fourth band at 590 still costs six, and the price of rendering at 88 falls from ten directions to six — because the directions it wins back are green, not amber.

limits · Limits Directions each pigment model loses, under three broadband lamps and a lamp of lines. For each lamp, how many directions of the object-colour solid a pigment model cannot reach within one per cent of the ideal: a slope limit, absorption bands no narrower than forty nanometres, and bands whose minimum width is fixed in energy. The order is the same under every lamp. The broadband lamps cost every model far more than the three narrow emitters: under daylight 89, 73 and 70 against 31, 23 and 20.

The lamp outweighs the pigment model

Three models of what a real colorant can do — a slope limit, absorption bands forty nanometres wide, and bands whose width is fixed in energy — were ranked under lamps of narrow emitters, slope dearest and energy cheapest. Under daylight, tungsten and a white LED the ranking holds, every time. What changes is everything else: each model loses two and a half to three and a half times as many directions of the object-colour solid under a broadband lamp as under three narrow emitters, and the difference between the models is smaller than the difference between the lamps. Under tungsten, whose power rises smoothly to the red, the three models are nearly one.

limits · Limits Bands narrower than the gap split mercury's lines. Under a clear mercury lamp on a one-nanometre grid, its lines one and a half nanometres wide, the 25 directions of the object-colour solid that ask for the green line reflected and the yellow absorbed or the reverse: how many the band pigment, searched, loses by more than one per cent, against how narrow its bands may be, with the bands no deeper than twelve and than twenty-four. At the census's forty nanometres it loses 15; at thirty, 9; at twenty-five, 0. The green and yellow lines are 31.5 nm apart, marked.

A band pigment must be narrower than the gap it splits

Under a clear mercury lamp a pigment built from absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse — the count an earlier search left short however deep it cut its cliff. The slope-limited pigment's losses turned out to be the lines' 31.5 nm spacing against its swing, and the prediction was that the band pigment's are the same arithmetic, falling to nought near a band thirty nanometres wide. Searched on a one-nanometre grid with the lines narrowed, the count falls at every narrower band — fifteen at forty, eleven at thirty-five, nine at thirty — and reaches nought at twenty-five, not thirty. Every loss already splits the pair and falls short on both lines at once: a Gaussian band deep enough to absorb one line keeps absorbing a little of the other. One band of depth twelve, placed as well as it can be, errs by 3.2 per cent at forty nanometres, 0.8 at thirty and 0.2 at twenty-five; doubling its depth buys less than narrowing it by five nanometres.

limits · Limits A steep side splits mercury's lines at forty nanometres. Under a clear mercury lamp on a one-nanometre grid, the 25 directions that ask for one of the green and yellow lines reflected and the other absorbed: how many the band pigment loses by more than one per cent, with symmetric bands, with the same bands searched with a fourth number as a control, and with bands whose one side is half the other's. At forty nanometres the symmetric bands lose 15 and the skewed ones 1; the skewed bands lose none at thirty-five or thirty.

A steep side splits mercury's lines at forty nanometres

Under a clear mercury lamp a pigment built from Gaussian absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, because a band's tail reaches the line it should pass. Real dye bands are often lopsided, and the prediction was that a band whose one side is half the other's, the steep side free to face either way, loses at forty nanometres about as many directions as a symmetric band thirty wide — nine — and helps only the directions that pass the green. It loses one. At thirty-five and thirty it loses none. In every one of the twenty-five fits the absorbing band turns its steep side to the line it must pass, and both kinds of direction gain: none of fourteen passing the green is lost, one of eleven passing the yellow. A single band forty nanometres wide with a steep side splits the two lines within 0.37 per cent; a symmetric band must narrow to about twenty-seven to do as well.

limits · Limits The losses fall slowly, then all at once. Under a clear mercury lamp, how many of the 25 directions that split its green line from its yellow the band pigment loses by more than one per cent with bands forty nanometres wide, against how lopsided each band is — from symmetric to sides of two thirds and one third — beside the counts the prediction gave. 0.50: 16, 0.55: 13, 0.60: 10, 0.65: 5, 0.67: 1.

A band must be nearly two to one before its steep side pays

Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide loses fifteen or sixteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, and the same bands with one side a third of the width and the other two thirds lose one. The prediction for milder skews was that the losses fall in step with a single band's error — to about eight at sides of 0.55 and 0.45, three at 0.6 and 0.4 — so that a three-to-two skew already halves them. It does not. The census loses thirteen at 0.55, ten at 0.6, five at 0.65 and one at two to one: three directions recovered per twentieth of skew at first, twelve at the last. The losses track one band's error against the one-per-cent line the census counts by, and a single band crosses that line only near a skew of 0.62. A dye's modest natural asymmetry buys a few directions; the steep side pays only when it is close to half the other.

limits · Limits Three widths, one count: the steep side decides. Every searched band pigment at thirty, thirty-five and forty nanometres, placed by the half-width of its steep side and by how many of the clear mercury lamp's 25 splitting directions it loses by more than one per cent. The three widths fall on one staircase: 20.0 nm: 16, 18.0 nm: 13, 17.5 nm: 10, 16.0 nm: 9, 15.7 nm: 9, 15.0 nm: 8, 14.9 nm: 8, 14.0 nm: 4, 14.0 nm: 4, 13.3 nm: 1, 13.1 nm: 1, 12.3 nm: 0, 11.7 nm: 0, 10.0 nm: 0.

A band is only as wide as its steep side

Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide must be skewed nearly two to one before it can split the green line from the yellow, and the prediction for bands five nanometres narrower was that a single band crosses one per cent of error near a skew of 0.57 and the census's losses fall to five or fewer by 0.6. The losses do fall to four by 0.6; the single band crosses one per cent at 0.555. But the forty-nanometre rule that read the losses off that one-per-cent line breaks at once, and what replaces it is simpler: a thirty-five-nanometre band at 0.6 and a forty-nanometre band at 0.65 have the same steep side, fourteen nanometres, and lose the same four directions by the same shortfalls to a hundredth of a point. Three widths and fourteen pigments fall on one staircase in the steep side alone. Searching every loss again from the milder skews' answers finds one direction the census missed, at both widths, and corrects the forty-nanometre counts to nine and four.

limits · Limits At one half-width, the profile decides everything. The forty-nanometre band pigment with its steep side drawn in four profiles at a half-width of fourteen nanometres, and in two more at the half-widths that give a single band the Gaussian's error: how many of the clear mercury lamp's 25 splitting directions each loses, with its single band's error. At fourteen nanometres: Gaussian⁴ 0, Gaussian 4, sech² 19, Lorentzian 22; matched to the Gaussian's error, the secant at 7.73 nm loses 2 and the power-four Gaussian at 39.1 nm 7.

At one half-width a flank can lose nothing or nearly everything

Under a clear mercury lamp, a pigment of Gaussian absorption bands splits the green line from the yellow according to its steep side's half-width alone, and a Gaussian flank is fixed by its half-width — so the census could not say whether it measures the half-width or the absorbance left in the flank's foot at the line it should spare. The prediction was that the foot decides: at a half-width of fourteen nanometres, a Lorentzian steep side loses most of the twenty-five splitting directions and a sharper-footed one none. It does. A power-four Gaussian loses none, the Gaussian four, a squared hyperbolic secant nineteen and a Lorentzian twenty-two, in the order of their single bands' errors. But a single band's error is not the whole account: three profiles drawn at the half-widths that give a single band the same error — 7.73, fourteen and 39.1 nanometres — lose two, four and seven directions, and the widest of them loses directions the others keep, because its shoulder starts absorbing nearer the green.

limits · Limits One per cent of Lorentzian nearly triples the losses. The forty-nanometre band pigment with its steep side a pseudo-Voigt fourteen nanometres wide at half depth: how many of the clear mercury lamp's 25 splitting directions it loses by more than one per cent, against the Lorentzian share of that side, beside the Gaussian's four and the counts the prediction gave. Lost with no Lorentzian, 4; 1%: 11, 2%: 14, 5%: 18, 10%: 19, 20%: 20, all: 22.

A hundredth of Lorentzian undoes a Gaussian flank

Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide at half depth loses four of the twenty-five directions that split the green line from the yellow, and the same side drawn as a Lorentzian loses twenty-two. A real absorption band is usually a Voigt profile, a Gaussian core with a Lorentzian share in its wings, and the prediction was that a twentieth of Lorentzian doubles the Gaussian's losses and a fifth reaches nineteen. The direction is right and the scale is not. A hundredth of Lorentzian takes the pigment from four lost directions to eleven; a fiftieth to fourteen; a twentieth to eighteen; a fifth to twenty. The best-placed band leaves the passed line more than three half-widths from its centre, out in the wing, and there a wing one per cent Lorentzian carries twice the absorbance of the whole Gaussian. The count saturates near twenty while the worst direction's shortfall keeps growing, from 2.8 per cent at a hundredth to 51 with a full Lorentzian.

limits · Limits A narrower flank keeps most of the Lorentzian's losses. The forty-nanometre band pigment with its steep side fourteen, ten, seven and five nanometres wide at half depth: how many of the clear mercury lamp's 25 splitting directions it loses by more than one per cent, as a Gaussian and as a pseudo-Voigt with a twentieth of Lorentzian, beside the counts predicted. The Gaussian loses 4, 0, 0, 0; the Voigt 18, 13, 12, 7, where four were predicted at seven nanometres and none at five.

A narrower flank moves its wing in with it

Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide loses four of the twenty-five directions that split the green line from the yellow, and with a twentieth of Lorentzian in that side it loses eighteen, because the wing three half-widths out out-absorbs the whole Gaussian there. A Lorentzian wing falls as the square of the distance over its half-width, and the prediction was that narrowing the side to seven nanometres brings the twentieth-Lorentzian pigment back to four losses and to five nanometres to none. It loses thirteen at ten nanometres, twelve at seven and seven at five, while the Gaussian loses nothing at all three. The wing still decides: the narrower band does not stay where the wider one sat but moves fourteen nanometres in towards the line it must pass, so its wing there falls fourfold rather than eightfold.

limits · Limits A five-nanometre flank loses nothing up to a fiftieth of Lorentzian. The forty-nanometre band pigment's lost splitting directions against the Lorentzian share of its steep side, for a steep side five nanometres wide — the first search and the better of it and a second — and fourteen. At five nanometres: 0, 0, 0, 5 lost at no Lorentzian, a hundredth, a fiftieth and a twentieth, where the first search had 0, 0, 1, 7; at fourteen, 4, 11, 14, 18.

A five-nanometre flank carries a fiftieth of Lorentzian

A band pigment splitting mercury's green and yellow lines with a steep side five nanometres wide loses none of the twenty-five splitting directions as a Gaussian and seven with a twentieth of Lorentzian in its flank, because the band slides in towards the passed line and carries its wing there. The prediction was that at five nanometres the count follows the single band's error, which grows in proportion to the share: about two lost at a fiftieth and none at a hundredth. A hundredth loses none, as predicted. The first search lost one direction at a fiftieth — and searched again from the best pigments found at the neighbouring shares, that direction reaches within two thousandths of the ideal: a fiftieth loses none either. The same second search recovers two of the twentieth's seven, leaving five. A sharp enough flank tolerates a fiftieth of Lorentzian where a fourteen-nanometre one lost fourteen directions to it.

limits · Limits The count falls by one and then by seven. The forty-nanometre pigment's lost splitting directions at a twentieth of Lorentzian against the half-width of its steep side, at the best of every search, and the count each half-width would lose if every direction fell short by its seven-nanometre constant times one band's error. Searched again, 14 nm loses 18, 10 nm loses 13, 7 nm loses 12, 5 nm loses 5; the constants predict 16 at 14, 13 at 10, 12 at 7, 9 at 6, 5 at 5, 3 at 4.5, 0 at 4.

Each direction falls short by one band's ratio

A band pigment asked to reflect mercury's green line and absorb its yellow, with a twentieth of Lorentzian in its steep side, loses eighteen of twenty-five splitting directions at a steep side fourteen nanometres wide, thirteen at ten, twelve at seven and five at five. A second search had lowered the five-nanometre count, and the prediction was that searching the seven- and ten-nanometre pigments again lowers each by one or two, leaving a count nearly linear in the half-width. Searched again, both lose exactly what they lost — no direction gains six tenths of a point. The uneven fall is real, and the reason is plainer than any search: every direction's shortfall falls with the half-width by the same factor as a single band's error, so the count is a fixed ranking of constants, one a direction, read against a line that moves.

limits · Limits The ranking reads six nanometres to the direction. The forty-nanometre pigment's lost splitting directions at a twentieth of Lorentzian against the steep side's half-width, at every half-width searched, beside the count the ranking of seven-nanometre constants predicts. At 6 nm the ranking predicted 9 and the search found 9; at 4.5 nm the ranking predicted 3 and the search found 1; at 4 nm the ranking predicted 0 and the search found 0.

The ranking reads six nanometres to the direction

Every direction a band pigment loses under mercury's lines falls short, as its steep side narrows, by the same factor as a single band's error — so each direction carries a constant, and the count at any half-width can be read off one band with no search at all. Read that way, a twentieth of Lorentzian loses nine directions at a six-nanometre steep side, three at four and a half and none at four. Searched, six nanometres loses nine: the same nine, every direction within 6.2 per cent of where the constants put it. Four loses none. Four and a half loses one, not three — the two the ranking added sit 0.005 and 0.011 of a point under the one-per-cent line. And at the narrowest widths the gentle directions begin to fall short by more than the ranking says, by up to 29 per cent, which is where the single band stops being the whole story.

limits · Limits

Named alongside it

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

PigmentOptimal coloursObject-colour solidOptimisationModelling assumptionWavelength gridReflectanceSpectral power distributionBeer lambertColour renderingKubelka munkNarrow band displays

All concepts