Colour in numbers

Colorimetric quantities with symbols and units

Colorimetric quantities describe what colour a light source has and how it renders colours. They all come out of a single measurement: the spectral distribution S(λ). From it follow the tristimulus values X, Y, Z, from those the chromaticity x, y or u′, v′, from those the correlated colour temperature T_n – and from the comparison with a reference source the colour rendering indices R_a and R_f. The field is called colorimetry; the energetic counterparts are the radiometric and the eye-weighted ones the photometric quantities.

Every colorimetric quantity assesses the same spectrum with a different question. The colour matching functions x̄(λ), ȳ(λ) and z̄(λ) reproduce colour vision; weighted with them, a whole spectrum collapses into three numbers. Two physically different spectra can yield the same three numbers and therefore look alike – that is metamerism. It is the reason why chromaticity alone does not describe a light source: two lamps with identical chromaticity can render colours visibly differently.

Which standard observer was used belongs with every figure. The 2° observer of CIE 1931 describes small fields of view, the 10° observer of CIE 1964 large ones. For the same spectrum the two give different chromaticities – most clearly for narrow-band LED spectra. Chromaticities from different observers are not comparable.

The table below lists the colorimetric quantities with symbol and unit. In full – with definition, delimitation and the governing standard – they are in the glossary of optical quantities.

The tristimulus values X, Y, Z arise by weighting the spectral distribution with the three colour matching functions and integrating. Y is defined so that it corresponds to brightness: for a light source Y is proportional to illuminance. Every other colorimetric quantity is a conversion from these three numbers.

Chromaticity coordinates x, y – the chromaticity diagram

Normalising X, Y and Z to their sum leaves the chromaticity without the brightness: x = X/(X+Y+Z) and y = Y/(X+Y+Z). The pair is plotted in the horseshoe-shaped chromaticity diagram; white points lie around x = 0.33, y = 0.33. The diagram is vivid but not perceptually uniform: in the green region the same geometric distance means a far smaller perceived difference than in the blue.

u′, v′ – the uniform diagram

For distances between chromaticities the CIE 1976 UCS diagram is therefore used: u′ = 4X/(X+15Y+3Z) and v′ = 9Y/(X+15Y+3Z). There a distance Δu′v′ of 0.0054 roughly corresponds to the discrimination threshold everywhere. For the spread of LED batches and the tolerance of luminaires, Δu′v′ is the usual measure, alongside the step figure in MacAdam ellipses (SDCM).

Careful with u, v without the prime

The older CIE 1960 UCS diagram with u and v has not disappeared: correlated colour temperature and the distance Δuv from the Planckian locus are defined there to this day. u is identical in both diagrams, v is not – v′ = 1.5 · v. Mixing the two up gives a wrong colour temperature.

Colour temperature T is the temperature of a Planckian radiator whose chromaticity matches that of the source. For real lamps that is almost never exactly the case; what is quoted is therefore the correlated colour temperature T_n (CCT): the temperature of the Planckian radiator at the smallest distance in the CIE 1960 UCS diagram. Warm white lies at 2700 to 3000 K, neutral white at 4000 K, daylight white above 5300 K.

A colour temperature on its own is incomplete. Two lamps at 4000 K can look visibly different if their chromaticities sit at different distances from the Planckian locus. That distance is Δuv: positive values appear greenish, negative ones pinkish. ANSI C78.377 limits it to ±0.006 for white LED products. Beyond |Δuv| ≈ 0.005 a correlated colour temperature loses its meaning – the chromaticity is then no longer a white to which a temperature could sensibly be assigned.

The general colour rendering index R_a of CIE 13.3:1995 compares the chromaticities of eight standardised test colours under the source under test with those under a reference source of the same colour temperature – a Planckian radiator below 5000 K, a daylight illuminant above. From the mean colour difference follows R_a = 100 − 4.6 · ΔE̅. Above 90 counts as very good, below 80 as inadequate for interiors.

Why R9 belongs in the specification

R_a averages over eight weakly saturated colours. Saturated red is not among them – that is what the special index R9 is for. White LEDs with a weak red component reach R_a 85 and drop below 20 for R9; meat, wood and skin then look dull. Anyone judging colours should explicitly ask for R9 as well as R13 and R15 for skin tones. The special indices R1 to R15 can become negative.

IES TM-30: Rf and Rg

IES TM-30-15 supersedes the method by using 99 colour samples that cover real surfaces and the whole colour space evenly, and by computing in the uniform CAM02-UCS. The fidelity index R_f runs from 0 to 100. The gamut index R_g additionally says whether colours appear more saturated (above 100) or paler (below 100) than under the reference. Only both together with the colour vector graphic give a picture: a high R_g can come from oversaturated red that catches the eye but is not faithful. R_f and R_a cannot be converted into one another.

Besides chromaticity and colour rendering, a spectroradiometer outputs three quantities that describe the colour impression directly.

Dominant wavelength λ_d

Draw a straight line from the white point through the chromaticity and extend it to the spectrum locus: the wavelength of the intersection is the dominant wavelength. It describes the perceived hue and is the decisive binning quantity for coloured LEDs. It is not the same as the peak wavelength λ_p, which comes from the physics of the spectrum: for an LED the two can differ by several nanometres. For purple colours there is no intersection; the complementary wavelength is given instead.

Saturation S and chroma C*

Chroma C* is the distance of a colour from the neutral axis, in CIELAB C*_ab = √(a*² + b*²). Saturation S relates it to lightness, S = C*/L*, and describes the colour impression independently of how bright the colour is.

Hue angle h

The hue angle is the polar angle in the a*b* plane: h_ab = arctan(b*/a*), given from 0 to 360 degrees – 0° red, 90° yellow, 180° green, 270° blue. It separates hue from lightness and chroma. Close to the neutral axis it becomes indeterminate: at very low chroma it fluctuates strongly without the colour impression changing.

Object colours: CIELAB, Lab99 and ΔE

For the colours of surfaces rather than light sources, the CIELAB space serves with L* for lightness, a* for red versus green and b* for yellow versus blue; it requires a reference white, usually D65 or D50. The colour difference ΔE is the distance between two points in it: ΔE*_ab Euclidean, ΔE00 after CIEDE2000 with weighting functions. The DIN99 space of DIN 6176 distorts CIELAB so that the plain Euclidean distance ΔE99 is perceptually uniform again. Which formula applies must be stated – the same two colours give different numbers depending on it.

Colorimetric quantities cannot be determined with a broadband radiometer. They require the complete spectrum, because each of them is a different weighting of that same spectrum – an instrument that delivers a single number cannot tell them apart. Which instruments can is covered under spectroradiometer or broadband radiometer; the spectra of common lamps can be compared in the spectral database explorer.

The glossary below shows the colorimetric part of the collection. The remaining groups – radiometric, photometric, weighted – are in the full glossary of optical quantities.

Dr.-Ing. Mark Paravia is the managing director of Opsytec Dr. Gröbel GmbH in Ettlingen and heads the accredited calibration laboratory. He is vice-chair of the DIN Standards Committee FNL 7 “Optical Radiation” and a member of the DVGW Project Group on UV Disinfection.