X, Y, and Z are pronounced big-X, big-Y, and big-Z, or cap-X, cap-Y, and cap-Z, to distinguish them from little-x and little-y, to be described in a moment.

Grassmann’s Third Law:

Sources of the same colour produce identical effects in an additive mixture regardless of their spectral composition.

Thornton, William A. (1999), “Spectral sensitivities of the normal human visual system, color-matching functions and their principles, and how and why the two sets should coincide,” in Color Research and Application 24 (2): 139–156 (Apr.).

property of vision; it is best considered as a consequence

_ _

of the mathematical process by which the x(λ), y(λ), and

_

z(λ) curves are constructed.

CIE XYZ tristimulus

_

Weighting an SPD under the y(λ) colour-matching function yields luminance (symbol Y), as I described on page 205. When luminance is augmented with two

other values, computed in the same manner as lumi-

_ _

nance but using the x(λ) and z(λ) colour-matching functions, the resulting values are known as XYZ tristimulus values (denoted X, Y, and Z). XYZ values correlate to the spectral sensitivity of human vision. Their amplitudes – always nonnegative – are proportional to intensity.

Tristimulus values are computed from a continuous

_ _

SPD by integrating the SPD under the x(λ), y(λ), and

_

z(λ) colour-matching functions. In discrete form, tristimulus values are computed by a matrix multiplication, as illustrated in Figure 25.6 opposite.

Human colour vision follows a principle of superposition known as Grassmann’s Third Law: The set of tristimulus values computed from the sum of a set of SPDs is identical to the sum of the tristimulus values of each SPD. Due to this linearity of additive colour mixture, any set of three components that is a nontrivial linear combination of X, Y, and Z – such as R, G, and B – is also a set of tristimulus values. (In Transformations between RGB and CIE XYZ, on page 307, I will introduce related CMFs that produce R, G, and B tristimulus values.)

Luminance can be considered to be a distinguished tristimulus value that is meaningful on its own, and, exceptionally, carries units of cd·m-2. Apart from luminance, tristimuli come in sets of three, as the word suggests, and have no units.

This chapter accepts the CIE Standard Observer rather uncritically. Although the CIE Standard Observer is very useful and widely used, some researchers believe that it exhibits some problems and ought to be improved. For one well-informed and provocative view, see Thornton.

y 520

0.8

540

0.7

500

0.5

580

0.4

0.3

480

0.2

0.1

460

440 400

0.0

600

620

640

700

The x and y symbols are pronounced little-x and little-y.

1

y = 1- x

1

Spectral locus

Line of purples

Alychne

Blackbody locus

White point

Figure 25.8 CIE [x, y] chart features.

CIE [x, y] chromaticity

It is convenient, for both conceptual understanding and for computation, to have a representation of “pure” colour in the absence of lightness. The CIE standardized a procedure for normalizing XYZ tristimulus values to obtain two chromaticity values x and y.

Chromaticity values are computed by this projective transformation:

A third chromaticity coordinate, z, is defined, but is redundant since x + y + z = 1. The x and y chromaticity coordinates are abstract values that have no direct physical interpretation.

A colour can be specified by its chromaticity and luminance, in the form of an xyY triple. To recover X and Z tristimulus values from [x, y] chromaticities and luminance, use the inverse of Equation 25.1:

A colour plots as a point in an [x, y] chromaticity diagram, plotted in Figure 25.7 opposite.

In Figure 25.8 in the margin, I sketch several features of the [x, y] diagram. The important features lie on, or below and to the left of, the line y=1-x.

When a narrowband (monochromatic) SPD comprising power at just one wavelength is swept across the range 400 nm to 700 nm, it traces the inverted-U (or horseshoe) shaped spectral locus in [x, y] coordinates.

The sensation of purple cannot be produced by

a single wavelength; it requires a mixture of shortwave and longwave light. The line of purples on a chromaticity diagram joins the chromaticity of extreme blue (violet), containing only shortwave power, to the chromaticity of extreme red, containing only longwave power.

There is no unique physical or perceptual definition of white. Many important sources of illumination are blackbody radiators, whose chromaticity coordinates lie on the blackbody locus (sometimes called the Planckian