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

Colour rendering — where it appears

How faithfully a lamp shows a set of test samples against a reference of the same colour temperature, reported as an index. The samples are reflectances and none of them fluoresces, so the index is identical for a lamp with ultraviolet and one without.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

One reflectance, two illuminants, two colours. A reflectance peaking near 610 nm, and the colours it produces under D65 and A. The object has not changed. The light has, and colour is a property of the pair.

The illuminant is half the answer

An object has a reflectance, not a colour. The colour appears when a specified light falls on it, which is why two surfaces can match in a shop and clash outside, and why every serious matching standard names the light.

light · Light triphosphor fluorescent — three narrow phosphors plus the mercury lines, and the white it produces. The spectral power distribution of a triphosphor source, normalised to its own peak, and the colour a perfect white reflector takes under it: chromaticity (0.3379, 0.3389), correlated colour temperature 5258 K at Duv -0.0035. The white looks ordinary. The spectrum producing it does not.

A lamp is not a blackbody

A fluorescent tube puts a third of its light into four mercury lines. A white LED is a blue spike with a hole beside it. Both are sold by a colour temperature, and a colour temperature says nothing about either.

light · Light Luminous efficacy of a monochromatic source, under the 2° observer. Lumens per watt against wavelength. The curve is the luminous efficiency function scaled by 683, which is what luminous efficacy of radiation is — the 1979 redefinition of the candela fixed 555 nm at exactly 683 lm/W and everything else follows. Marked at 450 nm (26 lm/W), 500 nm (221 lm/W), 555 nm (683 lm/W), 600 nm (431 lm/W), 650 nm (73 lm/W). Under this observer the peak reaches 683 lm/W.

What a lamp cannot give back

Six hundred and eighty-three lumens per watt is not a measurement of anything. It is the definition of the candela, it is a ceiling no white light can approach, and the distance between a lamp and that ceiling is mostly the price of being able to see what colour things are.

light · Light What separates the separations, and under which light. 11 separations of a single colour, differing only in how much of it is carried by black rather than by the three chromatic inks. Under D50 they agree to ΔE00 = 0.00 — the solver was asked for that and delivered it. Under illuminant A they spread to 6.75. They are metamers of one another, and the whole family is invisible to any instrument reading a single illuminant.

The lamp in the shop decides

Two packages printed to the same specification, verified under the same standard illuminant and passed, can differ by nearly seven colour differences on a shelf under a tungsten lamp. Nothing went wrong at either press. The specification named one light, the separations that satisfy it are metamers of one another, and the shop chose the second light.

applied · Delivery Mixtures of 2700 K and 6500 K, on the diagram colour temperature is defined on. Both sources are Planckian radiators, so both sit exactly on the locus. Every mixture of them lies on the straight line between them, because mixing is addition and chromaticity is a projection of it — and the locus is curved, so the line is a chord. The equal mixture sits -0.0064 off the locus at a correlated colour temperature of 3953 K: a light that is measurably pink, specified by a number that says nothing about it.

Two lamps do not average

Light adds, band by band, and every number a lamp is sold by is a projection of the sum rather than a sum of the projections. Two radiators sitting exactly on the Planckian locus mix into a light that is measurably pink; two poor lamps mix into a better one than either.

light · Light Several sources on one scale. planck, led, narrowband, each normalised to its own peak and drawn on shared axes with the colour each produces beside it. Every one of these is sold as white light and every one is ordinarily described that way; what they have in common is a chromaticity, and very little else.

A screen is a poor lamp

A display's white is a light. Shone on a surface it renders colour worse than a fluorescent tube — and the wider the gamut, the worse it gets, because the narrow primaries that buy a large triangle are exactly the ones that leave holes in the spectrum.

light · Light 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 A colour rendering index, computed for everybody instead of for one observer. Each bar spans what 120 eyes make of one lamp: the same reflectances, the same reference illuminant, the same arithmetic, different colour-matching functions. The mark is the standard observer's own answer, which is the number printed on the box. One lamp's spread is 3.8 points wide while the whole range the standard observer puts these lamps in is 1.8 — so a ranking quoted to a tenth of a point is a statement about one set of tables. Two of the marks fall outside the population's range entirely.

The index is one observer's opinion

A colour rendering index is printed to a tenth of a point and computed for a single set of colour-matching functions that no standard names as a choice. Scored by two hundred eyes instead, one lamp's spread is wider than the whole range the standard observer puts five lamps in — and two lamps the standard separates by four tenths of a point come out the other way round for every member of the population.

light · Light The colour is right long before the spectrum is. The colour error of the projection, against the number of readings. At twelve readings the fluorescent tube's colour is right to 0.48 ΔE00 while 66% of its spectrum is still unmeasured. That is the trap in one line: a reconstruction good enough to pass any colorimetric check will predict a match under a second illuminant that does not happen, because the part it got wrong is exactly the part a different lamp weights differently.

The colour is right first

A reconstruction of a lamp from twelve filtered readings gets its colour right to half a unit while two thirds of its spectrum is still unmeasured. That combination is not a partial success — it is the exact condition under which a spectral prediction made from the reconstruction will be confidently wrong.

light · Light Every symmetric and every skewed fourth emitter, and the best of each at every cost. Each fourth emitter added to the three-emitter lamp, as its fidelity index against the number of the solid's directions a 40-nanometre slope limit costs more than a per cent: 315 symmetric bands and 1260 skewed ones over centre, widths and power. The two step lines are the best rendering each family reaches at or below each cost. They run a few directions apart and never further: rendering at 88 costs 103 directions with a symmetric band and 97 with a skewed one. The ringed point is the three emitters alone, Rf 78.6 at 62.

A lopsided band does not break the trade

A fourth emitter added to a three-emitter LED renders colours better and costs a slow pigment its reach, and the position that renders best and the position that costs least were thirty-five nanometres apart. The obvious suspect was the shape: every emitter searched was a symmetric bump, and a band with its skirt on one side might fill a gap without splitting it. Freed to be lopsided, it saves between one and seven of the solid's three hundred directions. No band renders like the amber emitter and costs like the green one, the lopsided bands that help lean the other way from the proposal, and what the fourth band renders was never the lamp's white.

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 A triphosphor tube costs a slow pigment what three emitters do; a halophosphate tube what daylight does. For each lamp, the share of the object-colour solid's directions 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 width is fixed in energy — with the count on each bar. Under the triphosphor tube the band model loses 26 of 154, beside the emitters' 23; under daylight 73 of 154, and under the halophosphate tube 71. The tube's coating alone loses fewer than the tube; its mercury lines, doubled, more.

A tube spares a pigment through its phosphors

Three models of a real pigment lose two and a half to three and a half times as many directions of the object-colour solid under daylight as under three narrow emitters. A fluorescent tube is both kinds of lamp — narrow mercury lines on phosphor bands — and the prediction was that its bands would light the gaps between its lines and put it two thirds of the way to daylight. A triphosphor tube sits with the emitters instead: 26 directions lost to forty-nanometre bands, against the emitters' 23 and daylight's 73. The credit is its phosphors'. Its coating alone, lines taken out, loses fewer still; its mercury lines cost directions whose edges sit in the gaps between the phosphors, and doubled they cost more. A halophosphate tube, the same lines on two broad bands, costs what daylight costs at the same rendering index.

limits · Limits

Named alongside it

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

Spectral power distributionIlluminantNarrow band displaysOptimal coloursPigmentSpecificationAbsorptionCorrelated colour temperatureFluorescentMetamerismModelling assumptionColour constancy

All concepts