CIE 1931 color space

In the study of the perception of color, one of the first mathematically defined color spaces was the CIE XYZ color space (also known as CIE 1931 color space), created by the International Commission on Illumination (CIE) in 1931.

The human eye has receptors for short (S), middle (M), and long (L) wavelengths, also known as blue, green, and red receptors. That means that one, in principle, needs three parameters to describe a color sensation. A specific method for associating three numbers (or tristimulus values) with each color is called a color space, of which the CIE XYZ color space is one of many such spaces. However, the CIE XYZ color space is special, because it is based on direct measurements of the human eye, and serves as the basis from which many other color spaces are defined.

The CIE XYZ color space was derived from a series of experiments done in the late 1920s by W. David Wright (Wright 1928) and John Guild (Guild 1931). Their experimental results were combined into the specification of the CIE RGB color space, from which the CIE XYZ color space was derived. This article is actually concerned with both of these color spaces.

In the CIE XYZ color space, the tristimulus values are not the S, M, and L stimuli of the human eye, but rather a set of tristimulus values called X, Y, and Z, which are also roughly red, green and blue, respectively. Two light sources may be made up of different mixtures of various colors, and yet have the same color (metamerism). If two light sources have the same apparent color, then they will have the same tristimulus values, no matter what different mixtures of light were used to produce them.

The CIE xy chromaticity diagram


Since the human eye has three types of color sensor that respond to different ranges of wavelengths, a full plot of all visible colors is a three-dimensional figure. However, the concept of color can be divided into two parts: brightness and chromaticity. For example, the color white is a bright color, while the color grey is considered to be a less bright version of that same white. In other words, the chromaticity of white and grey are the same while their brightness differs.

The CIE XYZ color space was deliberately designed so that the Y parameter was a measure of the brightness of a color. The chromaticity of a color was then specified by the two derived parameters x and y which are functions of all three tristimulus values X, Y, and Z:


 * $$x = \frac{X}{X+Y+Z}$$


 * $$y = \frac{Y}{X+Y+Z}$$

The X and Z tristimulus values can be calculated back from the chromaticity values x and y and the Y tristimulus value:


 * $$X=\frac{Y}{y}x$$


 * $$Z=\frac{Y}{y}(1-x-y)$$

The figure on the right shows the related chromaticity diagram. The outer curved boundary is the spectral locus, with wavelengths shown in nanometers. Note that the chromaticity diagram is a tool to specify how the human eye will experience light with a given spectrum. It cannot specify colors of objects (or printing inks), since the chromaticity observed while looking at an object depends on the light source as well.

Mathematically, x and y are projective coordinates and the colors of the chromaticity diagram occupy a region of the real projective plane.

The chromaticity diagram illustrates a number of interesting properties of the CIE XYZ color space: 
 * The diagram represents all of the chromaticities visible to the average person. These are shown in color and this region is called the gamut of human vision. The gamut of all visible chromaticities on the CIE plot is the tongue-shaped or horseshoe-shaped figure shown in color. The curved edge of the gamut is called the spectral locus and corresponds to monochromatic light, with wavelengths listed in nanometers. The straight edge on the lower part of the gamut is called the purple line. These colors, although they are on the border of the gamut, have no counterpart in monochromatic light. Less saturated colors appear in the interior of the figure with white at the center.


 * It is seen that all visible chromaticities correspond to positive values of x and y (and therefore to positive values of X, Y, and Z).


 * If one chooses any two points on the chromaticity diagram, then all colors that can be formed by mixing these two colors lie between those two points, on a straight line connecting them. It follows that the gamut of colors must be convex in shape. All colors that can be formed by mixing three sources are found inside the triangle formed by the source points on the chromaticity diagram (and so on for multiple sources).


 * The mixture of two equally bright colors will not generally lie on the midpoint of that line segment. In more general terms, a distance on the xy chromaticity diagram does not correspond to the degree of difference between two colors. Other color spaces (CIELuv and CIELab in particular) have been designed to meet this problem.


 * It can be seen that, given three real sources, these sources cannot cover the gamut of human vision. Geometrically stated, there are no three points lying within the gamut that include the entire gamut.


 * Light with a flat energy spectrum corresponds to the point (x,y) = (1/3,1/3).

Experimental results - The CIE RGB color space
In the 1920s, W. David Wright (Wright 1928) and John Guild (Guild 1931) independently conducted a series of experiments on human sight which laid the foundation for the specification of the CIE XYZ color space.



The experiments were conducted by using a circular split screen 2 degrees in size, which is the angular size of the human fovea. On one side of the field a test color was projected and on the other side, an observer-adjustable color was projected. The adjustable color was a mixture of three primary colors, each with fixed chromaticity, but with adjustable brightness.

The observer would alter the brightness of each of the three primary beams until a match to the test color was observed. Not all test colors could be matched using this technique. When this was the case, a variable amount of one of the primaries could be added to the test color, and a match with the remaining two primaries was carried out with the variable color spot. For these cases, the amount of the primary added to the test color was considered to be a negative value. In this way, the entire range of human color perception could be covered. When the test colors were monochromatic, a plot could be made of the amount of each primary used as a function of the wavelength of the test color. These three functions are called the color matching functions for that particular experiment.



Although Wright and Guild's experiments were carried out using various primaries at various intensities, and a number of different observers, all of their results were summarized by the standardized CIE RGB color matching functions $$\overline{r}(\lambda)$$, $$\overline{g}(\lambda)$$, and $$\overline{b}(\lambda)$$, shown in the plot on the right (CIE 1931). Note that $$\overline{r}(\lambda)$$ and $$\overline{g}(\lambda)$$ are zero at 435.8, $$\overline{r}(\lambda)$$ and $$\overline{b}(\lambda)$$ are zero at 546.1 and $$\overline{g}(\lambda)$$ and $$\overline{b}(\lambda)$$ are zero at 700 nm. These color matching functions are the amounts of three standard monochromatic primaries needed to match the monochromatic test primary at the wavelength shown on the horizontal scale. The three monochromatic primaries are at standardized wavelengths of 700 nm (red), 546.1 nm (green) and 435.8 nm (blue). The last two wavelengths were chosen because they are easily reproducible monochromatic lines of a mercury vapor discharge. The 700 nm wavelength, which in 1931 was difficult to reproduce as a monochromatic beam, was chosen because it is at the peak of the eye's red response, and therefore small errors in wavelength of this primary would have little effect on the results.

The color matching functions and primaries were settled upon by a CIE special commission after considerable deliberation (Fairman 1997). The cut-offs at the short- and long-wavelength side of the diagram are chosen somewhat arbitrarily; the human eye can actually see light with wavelengths up to about 810 nm, but with a sensitivity that is many thousand times lower than for green light. These color matching functions define what is known as the "1931 CIE standard observer". Note that rather than specify the brightness of each primary, the curves are normalized to have constant area beneath them. This area is fixed to a particular value by specifying that


 * $$\overline{g}(\lambda)=V(\lambda)$$

where $$V(\lambda)$$ is the photopic luminous efficiency function. The resulting primaries are found to be in the ratio of 1:4.5907:0.0601 for luminosity and 72.0962:1.3791:1 for radiant power. By proposing that the primaries be standardized, the CIE established an international system of objective color notation.

Grassmann's law
One might ask: "Why is it possible that Wright and Guild's results can be summarized using different primaries and different intensities from those actually used?" One might also ask: "What about the case when the test colors being matched are not monochromatic?" The answer to both of these questions lies in the (near) linearity of human color perception. This linearity is expressed in Grassmann's law (optics).

The CIE RGB space can be used to define chromaticity in the usual way: The chromaticity coordinates are r and g where:


 * $$r= \frac{R}{R+G+B},$$
 * $$g= \frac{G}{R+G+B}.$$

Construction of the CIE XYZ color space from the Wright-Guild data
Having developed an RGB model of human vision using the CIE RGB matching functions, the members of the special commission wished to develop another color space which related to the CIE RGB color space. It was assumed that Grassmann's law held, and the new space would be related to the CIE RGB space by a linear transformation. The new space would be defined in terms of three new color matching functions $$\overline{x}(\lambda)$$, $$\overline{y}(\lambda)$$, and $$\overline{z}(\lambda)$$. The corresponding XYZ tristimulus values for a color with a spectral power distribution I(λ) would then be given by:


 * $$X= \int_0^\infty I(\lambda)\,\overline{x}(\lambda)\,d\lambda$$
 * $$Y= \int_0^\infty I(\lambda)\,\overline{y}(\lambda)\,d\lambda$$
 * $$Z= \int_0^\infty I(\lambda)\,\overline{z}(\lambda)\,d\lambda$$



The new color space would be chosen to have the following desirable properties:


 * 1) The new color matching functions were to be everywhere greater than or equal to zero. In 1931, computations were done by hand or slide rule, and the specification of positive values was a useful computational simplification.
 * 2)  The $$\overline{y}(\lambda)$$ color matching function would be exactly equal to the photopic luminous efficiency function V(λ) for the "CIE standard photopic observer" (CIE 1926). The luminance function describes the variation of perceived brightness with wavelength. The fact that the luminance function could be constructed by a linear combination of the RGB color matching functions was not guaranteed by any means but might be expected to be nearly true due to the near-linear nature of human sight. Again, the main reason for this requirement was computational simplification.
 * 3)  For the constant energy white point, it was required that x = y = z = 1/3.
 * 4)  By virtue of the definition of chromaticity and the requirement of positive values of x and y, it can be seen that the gamut of all colors will lie inside the triangle [1,0], [0,0], [0,1]. It was required that the gamut fill this space practically completely.
 * 5)  It was found that the $$\overline{z}(\lambda)$$ color matching function could be set to zero above 650 nm while remaining within the bounds of experimental error. For computational simplicity, it was specified that this would be so.

In geometrical terms, choosing the new color space amounts to choosing a new triangle in rg chromaticity space. In the figure on the right, the rg chromaticity coordinates are shown on the two axes in black, along with the gamut of the 1931 standard observer. Shown in red are the CIE xy chromaticity axes which were determined by the above requirements. The requirement that the XYZ coordinates be non-negative means that the triangle formed by Cr, Cg, Cb must encompass the entire gamut of the standard observer. The line connecting Cr and Cb is fixed by the requirement that the $$\overline{y}(\lambda)$$ function be equal to the luminance function. This line is the line of zero luminance, and is called the alychne. The requirement that the $$\overline{z}(\lambda)$$ function be zero below 650 nm means that the line connecting Cg and Cr must be tangent to the gamut in the region of Kr. This defines the location of point Cr. The requirement that the equal energy point be defined by x = y = 1/3 puts a restriction on the line joining Cb and Cg, and finally, the requirement that the gamut fill the space puts a second restriction on this line to be very close to the gamut in the green region, which specifies the location of Cg and Cb. The above described transformation is a linear transformation from RGB space to XYZ space. The standardized transformation settled upon by the CIE special commission was as follows:



\begin{bmatrix}X\\Y\\Z\end{bmatrix}=\frac{1}{b_{21}} \begin{bmatrix} b_{11}&b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33} \end{bmatrix} \begin{bmatrix}R\\G\\B\end{bmatrix}=\frac{1}{0.17697} \begin{bmatrix} 0.49&0.31&0.20\\ 0.17697&0.81240&0.01063\\ 0.00&0.01&0.99 \end{bmatrix} \begin{bmatrix}R\\G\\B\end{bmatrix} $$



The integrals of the XYZ color matching functions must all be equal by requirement 3 above, and this is set by the integral of the photopic luminous efficiency function by requirement 2 above. It must be noted that the tabulated sensitivity curves have a certain amount of arbitrariness in them. The shapes of the individual X, Y and Z sensitivity curves can be measured with a reasonable accuracy. However, the overall luminosity curve (which in fact is a weighted sum of these three curves) is subjective, since it involves asking a test person whether two light sources have the same brightness, even if they are in completely different colors. Along the same lines, the relative magnitudes of the X, Y, and Z curves are arbitrary. One could as well define a valid color space with an X sensitivity curve that has twice the amplitude. This new color space would have a different shape. The sensitivity curves in the CIE 1931 and 1964 XYZ color spaces are scaled to have equal areas under the curves.

Problems and solutions

 * The 1924 luminous efficiency function V(λ) (CIE 1926) seriously underestimates sensitivity at wavelengths below 460 nm. Judd (1951) and Vos (1978) proposed a modified version of the luminous-efficiency function, which also gives a new set of XYZ color matching functions. See also Stiles and Burch (1955).


 * The CIE 1964 Standard Observer color matching functions are defined for a 10 degree field of view. They were derived from the work of Stiles and Burch (1959), and Speranskaya (1959). The 1931 standard observer field of view was 2 degrees which covers only the fovea. For the 10 degree experiments, the observers were instructed to ignore the central 2-degree spot. The 1964 supplementary standard observer is recommended for more than about a 4 degree field of view.


 * A problem with the CIE 1931 color system is that it does not give a direct way of estimating color differences. It would be desirable if a distance on a chromaticity diagram corresponded to the degree of difference between two colors. The idea of measuring the difference between two colors was developed by D.L. MacAdam and summarized in the concept of a MacAdam ellipse. Based on the work of MacAdam, the CIELuv color space was developed in 1960 and was later replaced by the CIELab color space, both of which were designed to have an equal distance in the color space correspond to equal differences in color, as measured by MacAdam. Although they were a distinct improvement over the CIE 1931 system, they were not completely free of distortion.