# Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

# CIE Chromaticity Coordinates (xyY)

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_1

## Definition

Chromaticity coordinates x, y, and z are calculated from CIE tristimulus values X, Y, and Z thus:
$$x = X/\left(X+Y+Z\right),$$
(1)
$$y = Y/\left(X+Y+Z\right),$$
$$z = Z/\left(X+Y+Z\right).$$
If the chromaticity coordinates and one of the tristimulus values (e.g., Y) are known, then it is possible to compute tristimulus values thus:
$$X = xY/y,$$
(2)
$$Z = zY/y = \left(1-x-y\right)Y/y.$$

## Chromaticity Coordinates

The use of chromaticity coordinates is an alternative, and often more useful, representation to the use of tristimulus values. The tristimulus values X, Y, and Z of a stimulus are the amounts of three primary lights that on average an observer would use to match the stimulus and as such form a specification of the stimulus. If the Y tristimulus value is calculated on an absolute basis, then it represents the luminance of the color stimulus, in candelas per square meter, for example [1]. When relative tristimulus values are calculated so that Y = 100 for a similarly illuminated and viewed perfect lambertian reflector, then the Y of a stimulus is equal to its reflectance factor or, in some cases, transmittance factor. It is customary to regard Y as the luminance factor of the stimulus, and this is an approximate correlate of the perceptual attribute of lightness [1].

The CIE system of colorimetry was designed so that the Y tristimulus value correlates, at least approximately, with lightness. To achieve this and other objectives, the primaries upon which the CIE system is based are usually referred to as being imaginary and certainly cannot be physically realized. This means that the X and Z tristimulus values do not correlate, even approximately, with any perceptual attributes, and this is a motivation for calculating other attributes that can provide such correlates. The chromaticity coordinates (see Eq. 1) are a type of relative tristimulus values. So, for example, if X = 20, Y = 40, and Z = 20 for a stimulus, then x = 20/80 = 0.25 and y = 40/80 = 0.5. This indicates that the stimulus is 25 % of X, 50 % of Y, and 25 % of Z, of course.

Inherent in the way that the chromaticity coordinates are calculated (Eq. 1) is the constraint that x + y + z = 1, and this means that there are only really two degrees of freedom since z = 1 − x − y. Since there are only two free variables, it is possible to construct a 2D diagram referred to as a chromaticity diagram. By convention, in the chromaticity diagram (which forms a sort of map of colors), x and y are plotted on the abscissa and ordinate, respectively (Fig. 1).

## Properties of the Chromaticity Diagram

Figure 1 shows an illustration of a chromaticity diagram though note that the colors are purely representative and are not meant to accurately denote the color at any point in the diagram. The original color-matching experiments carried out by Wright and Guild that formed the basis of the CIE system of color specification in 1931 used real, but different, red (R), green (G), and blue (B) lights as the primaries but were transformed into a common set of monochromatic primaries; the R was at 700 nm, the G at 546.1 nm, and the B at 435.8 nm. The color-matching functions are the amounts of each of these primaries used on average by a group of observers to match each wavelength of light in the visible spectrum (380–780 nm). The vertices of the triangle in Fig. 1 are at the chromaticities of the CIE RGB primaries.

If two lights are represented in the chromaticity diagram by two points, then chromaticities of the additive mixtures of the two lights will be represented by the straight line that joins the two points. Thus, in Fig. 1, all mixtures of the R and G primaries would lie on the straight line joining the chromaticities of the R and G primaries. The range of colors that can be matched by a set of primaries is sometimes referred to as the gamut; the gamut of a dichromatic system (based on just two primaries) is very small and impractical for most purposes. When there are three primaries (a trichromatic system), then the gamut becomes a triangle in the CIE chromaticity diagram such as the RGB triangle illustrated in Fig. 1.

The curved horseshoe-shaped locus of the chromaticity diagram is defined by the chromaticities of the monochromatic wavelengths of light. Since all real color stimuli are combinations of the monochromatic wavelengths, and given the earlier observation about how color mixtures are defined in the chromaticity diagram, it is clear that the gamut of all physically realizable colors is the convex hull constrained by the curved spectral locus. Similarly, it is also clear that no matter how carefully three primaries are selected (and no matter whether they are monochromatic or not), the gamut (represented by a triangle in the chromaticity diagram) will always be a subset of the gamut of all physically realizable colors. Thus, if the gamut of the CIE RGB primaries is considered (see Fig. 1), it is evident that much of the spectral locus cannot be matched by additive mixture of the primaries. The 1931 CIE system was defined by transforming the color-matching functions from the RGB primaries into a system of imaginary primaries XYZ where the whole spectrum (from 380 to 780 nm) could be matched by all-positive amounts of the three primaries. The XYZ primaries are referred to as imaginary because they cannot be realized physically; in Fig. 1, the outer triangle – defined by the x,y chromaticities (0,0), (1,0), and (0,1) – is the gamut of the XYZ primaries, and it is evident that the gamut of physically realizable colors lies within this.

The largest gamut that can be achieved by any RGB system would be obtained by using primaries whose chromaticities were close to, or on, the spectral locus at the approximate wavelengths 400, 520, and 700 nm, but this would still leave parts of the gamut of physically realizable colors outside of the RGB triangle (see Fig. 2). A further consideration is that a practical RGB system consists of chromaticity and luminance. Consequently, an RGB display device based on monochromatic primaries would likely not be very bright. For many purposes, it is important to be aware that gamuts are three dimensional [2].

Primaries that consist of light at more than one wavelength are less saturated than monochromatic lights of a similar hue; however, they also tend to be much brighter. The design of modern display devices involves many such considerations, but many use primaries that correspond closely to the sRGB trichromatic standard whose gamut is represented in Fig. 2 [3]. It is evident that the sRGB gamut (in 2D chromaticity space at least) covers less than half of the gamut of physically realizable colors. However, the fact that reflectance spectra for objects in the world tend to vary smoothly with wavelength [4] has the consequence: the practical gamut of real-world colors is much smaller than the horseshoe-shaped locus would suggest. Monochromatic stimuli, for example, are incredibly rare in the natural or man-made world.

The chromaticity diagram is perceptually nonuniform [1]. This was visually demonstrated by the MacAdam ellipses which showed the chromaticities of stimuli that were just noticeably different from a standard color. Around each standard color, the locus of the just discriminable colors was the ellipses whose size and orientation varied greatly throughout the chromaticity diagram. In a perceptually uniform space, these loci would be circles of identical size. Even lines of constant hue are curved in chromaticity space rather than being straight. The Abney effect, first observed in 1909, is a phenomenon such that there is a hue shift when white light is added to a monochromatic light [5]. The locus of the mixture of a white light (see the equal-energy stimulus in Fig. 1) and a monochromatic light would be a straight line in the chromaticity diagram. Problems with the lack of perceptual uniformity of the chromaticity diagram were part of the reason why nonlinear transforms of the XYZ system were explored ultimately resulting in the CIE (1976) L*a*b* color space or CIELAB.

## Further Considerations and Future Directions

Currently, there are two CIE xy chromaticity spaces corresponding to the 1931 (2° of visual angle) and 1964 (10° of visual angle) standard observers, respectively. Work is underway to explore the possibility of a CIE standard observer that would include a visual angle parameter to allow a family of related chromaticity diagrams.

## References

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