## Colour Matching Functions

Despite the seemingly infinite variety of colours that are available to us, the colour of single lights can be reduced to just three variables. This property of human colour vision, which is referred to as trichromacy, can be readily demonstrated in a colour matching experiment.

Imagine two lights lying side by side: One is illuminated by a mixture of three colours (or primary lights), say, red, green and blue, while the other can be of any arbitrary colour and intensity. A person with normal colour vision is able to make the two lights appear identical, simply by adjusting the relative intensities of the red, green and blue lights.

The colour matching functions or CMFs are obtained from a series of such matches, in which the subject sets the intensities of the three colours required to match a series of monochromatic (single wavelength) lights of equal energy that traverse the visible spectrum. Sometimes the value of one of the ,  and  CMFs is negative, which indicates that that particular primary light had to be removed from the mixture and added to the monochromatic light to complete the match.

The CMFs, such as the ,  and  CMFs, apply to real red, green and blue matching lights. However, they can be linearly transformed to imaginary matching lights, such as the XYZ primaries adopted by the CIE. The resulting CIE ,  and  CMFs have several useful properties: for example,  is the luminosity function; and all 3 CMFs are always positive. Another important set of three imaginary matching lights are those that exclusively and separately stimulate the three cones, and give rise to the set of CMFs, ,  and ,  which are known as the cone fundamentals.

There are three major derivations of the colour matching functions:

1. The CIE 1931 2 deg CMFs and its corrections by Judd (1951) and Vos (1978).
2. The Stiles-Burch 2 deg pilot CMFs measured in 10 observers (Stiles & Burch, 1955).
3. The Stiles-Burch 10 deg CMFs measured in 49 observers (Stiles & Burch, 1959) on which the CIE 1964 10-deg CMFs are primarily based.

For the central 2° of vision, the functions include the CIE 1931 functions (CIE, 1932), the Judd (1951) and Vos (1978) corrected version of the CIE 1931 functions, and the Stiles & Burch (1955) functions. In addition, the 10° CMFs of Stiles & Burch (1959), or the 10° CIE 1964 CMFs (which are based mainly on the Stiles & Burch (1959), and partly on the Speranskaya (1959) 10° data, see below) can be corrected to correspond to 2° macular and photopigment optical densities.

There are several difficulties associated with the CIE 1931 2° CMFs, and its variants (Judd, 1951; Vos, 1978), not least of which is that they were not directly measured. Instead, they were reconstructed from the relative colour matching data of Wright (1928-29) and Guild (1931) with the assumption that a linear combination of the reconstructed CMFs must equal the 1924 CIE function. Unfortunately, the validity of the curve used in the reconstruction is highly questionable (see Gibson & Tyndall, 1923; CIE, 1926; Judd, 1951), as too is the validity of the reconstruction itself (Sperling, 1958). The subsequent revisions by Judd (1951) and Vos (1978) are attempts to improve the original. For further discussion, see Stockman & Sharpe (1999).

Colour matching functions for 2° vision can be measured directly instead of being constructed by the combination of relative colour matching data and photometric data. The Stiles & Burch (1955) 2° CMFs are an example of directly measured functions. With characteristic caution, Stiles referred to these 2° functions as "pilot" data, yet they are the most extensive set of true CMFs for 2° vision, being based on matches made by ten observers. Given the extent of individual variability that occurs between colour normals-the L-cone polymorphism, in particular-such a small group is unlikely to accurately represent the mean colour matches of the normal population.

The most comprehensive set of CMF data, which were also directly measured, are the "large-field" 10° CMFs of Stiles & Burch (1959). Measured in 49 subjects from 392.2 to 714.3 nm (and in 9 subjects from 714.3 to 824.2 nm), they are available as individual as well as mean data. During their measurement, the luminance of the matching field was kept high to reduce possible rod intrusion, but nevertheless a small correction for rod intrusion can be applied (see also Wyszecki & Stiles, 1982, p. 140).

The large field CIE 1964 CMFs are based mainly on the 10° CMFs of Stiles & Burch (1959), and to a lesser extent on the 10° CMFs of Speranskaya (1959). While the CIE 1964 CMFs are similar to the 10° CMFs of Stiles & Burch (1959), they differ in ways that compromise their use as the basis for cone fundamentals (see also Stockman, Sharpe & Fach, 1999).

Further information about the relationship between CMFs and cone fundamentals is given in the next section.

### References

CIE. (1926). Commission Internationale de l'Éclairage Proceedings, 1924. Cambridge: Cambridge University Press.

CIE. (1932). Commission Internationale de l' Éclairage Proceedings, 1931. Cambridge: Cambridge University Press.

Gibson, K. S., & Tyndall, E. P. T. (1923). Visibility of radiant energy. Scientific Papers of the Bureau of Standards, 19, 131-191.

Guild, J. (1931). The colorimetric properties of the spectrum. Philosophical Transactions of the Royal Society of London, A230, 149-187.

Judd, D. B. (1951). Report of U.S. Secretariat Committee on Colorimetry and Artificial Daylight, Proceedings of the Twelfth Session of the CIE, Stockholm (pp. 11) Paris: Bureau Central de la CIE.

Speranskaya, N. I. (1959). Determination of spectrum color co-ordinates for twenty-seven normal observers. Optics and Spectroscopy, 7, 424-428.

Sperling, H. G. (1958). An experimental investigation of the relationship between colour mixture and luminous efficiency, Visual Problems of Colour, Volume 1 (pp. 249-277) London: Her Majesty's Stationery Office.

Stiles, W. S., & Burch, J. M. (1955). Interim report to the Commission Internationale de l'Éclairage Zurich, 1955, on the National Physical Laboratory's investigation of colour-matching (1955) with an appendix by W. S. Stiles & J. M. Burch. Optica Acta, 2, 168-181.

Stiles, W. S., & Burch, J. M. (1959). NPL colour-matching investigation: Final report (1958). Optica Acta, 6, 1-26.

Stockman, A., & Sharpe, L. T. (1999). Cone spectral sensitivities and color matching. In K. Gegenfurtner & L. T. Sharpe (Eds.), Color vision: from genes to perception (pp. 51-85) Cambridge: Cambridge University Press.

Stockman, A., Sharpe, L. T., & Fach, C. C. (1999). The spectral sensitivity of the human short-wavelength cones. Vision Research.

Vos, J. J. (1978). Colorimetric and photometric properties of a 2-deg fundamental observer. Color Research and Application, 3, 125-128.

Wright, W. D. (1928-29). A re-determination of the trichromatic coefficients of the spectral colours. Transactions of the Optical Society, 30, 141-164.