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In the study of color perception, one of the first mathematically defined color spaces is the CIE 1931 XYZ color space, created by the International Commission on Illumination (CIE) in 1931.^{[1]}^{[2]}
The CIE XYZ color space was derived from a series of experiments done in the late 1920s by W. David Wright^{[3]} and John Guild.^{[4]} Their experimental results were combined into the specification of the CIE RGB color space, from which the CIE XYZ color space was derived.
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The human eye has photoreceptors (called cone cells) for medium and highbrightness color vision, with sensitivity peaks in short (S, 420–440 nm), middle (M, 530–540 nm), and long (L, 560–580 nm) wavelengths (there are also lowbrightness, monochromatic "nightvision" receptors, called rod cells, which have peak sensitivity around 490–495 nm). Thus, in principle, three parameters describe a color sensation. These tristimulus values of a color can be conceptualized as the amounts of three primary colors in a trichromatic additive color model.
Associating tristimulus values with colors is the job of a color space. CIE XYZ, one of many such spaces, is a commonly used standard, and serves as the basis from which many other color spaces are defined. A colormatching function associates specific tristimulus values with specific colors.
Consider two light sources made up of different mixtures of various wavelengths. Such light sources may appear to be the same color; this effect is called metamerism. Such light sources have the same apparent color to an observer when they produce the same tristimulus values, no matter what are the spectral power distributions of the sources.
It is not possible to stimulate one type of cone cell only, because the sensitivity curves of the three cone cells overlap. In trichromatic additive color spaces with color triangles defined by discrete (i.e., nonoverlapping) primary colors (e.g., RGB color spaces), this yields negative RGB values for some colors that correspond to certain tristimulus values. To avoid these negative RGB values, and to have one component that describes the perceived brightness, "imaginary" primary colors and corresponding colormatching functions have been formulated. The resulting tristimulus values are defined by the CIE 1931 color space, in which they are denoted X, Y, and Z.^{[5]}
In this model, Y means luminance, Z is quasiequal to blue stimulation, or the S cone response, and X is a mix (a linear combination) of cone response curves chosen to be orthogonal to luminance and nonnegative. Thus, XYZ may be confused with LMS cone responses. But in the CIE XYZ color space, the tristimulus values are not the L, M, and S responses of the human eye, even if X and Z are roughly red and blue. Rather, they may be thought of as 'derived' parameters from the long, medium, and shortwavelength cones.
Due to the distribution of cones in the eye, the tristimulus values depend on the observer's field of view. To eliminate this variable, the CIE defined the standard (colorimetric) observer. Originally this was taken to be the chromatic response of the average human viewing through a 2° angle, due to the belief that the colorsensitive cones resided within a 2° arc of the fovea. Thus the CIE 1931 Standard Observer is also known as the CIE 1931 2° Standard Observer. A more modern but lessused alternative is the CIE 1964 10° Standard Observer, which is derived from the work of Stiles and Burch,^{[6]} and Speranskaya.^{[7]}
For the 10° experiments, the observers were instructed to ignore the central 2° spot. The 1964 Supplementary Standard Observer is recommended for more than about a 4° field of view. Both standard observers are discretized at 5 nm wavelength intervals and distributed by the CIE.^{[8]}
The standard observer is characterized by three color matching functions.
The derivation of the CIE standard observer from color matching experiments is given below, after the description of the CIE RGB space.
The color matching functions are the numerical description of the chromatic response of the observer (described above).
The CIE has defined a set of three colormatching functions, called , , and , which can be thought of as the spectral sensitivity curves of three linear light detectors that yield the CIE XYZ tristimulus values X, Y, and Z. The tabulated numerical values of these functions are known collectively as the CIE standard observer.^{[9]}
The tristimulus values for a color with a spectral power distribution are given in terms of the standard observer by:
where is the wavelength of the equivalent monochromatic light (measured in nanometers).
Other observers, such as for the CIE RGB space or other RGB color spaces, are defined by other sets of three colormatching functions, and lead to tristimulus values in those other spaces.
The values of X, Y, and Z are bounded if the intensity spectrum I(λ) is bounded.
Since the human eye has three types of color sensors that respond to different ranges of wavelengths, a full plot of all visible colors is a threedimensional 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 or luminance of a color. The chromaticity of a color was then specified by the two derived parameters x and y, two of the three normalized values which are functions of all three tristimulus values X, Y, and Z:
The derived color space specified by x, y, and Y is known as the CIE xyY color space and is widely used to specify colors in practice.
The X and Z tristimulus values can be calculated back from the chromaticity values x and y and the Y tristimulus value:
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 CIE RGB color space is one of many RGB color spaces, distinguished by a particular set of monochromatic (singlewavelength) primary colors.
In the 1920s, W. David Wright^{[3]} and John Guild^{[4]} 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 observeradjustable 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 , , and , obtained using three monochromatic primaries at standardized wavelengths of 700 nm (red), 546.1 nm (green) and 435.8 nm (blue). The color matching functions are the amounts of primaries needed to match the monochromatic test primary. These functions are shown in the plot on the right (CIE 1931). Note that and are zero at 435.8, and are zero at 546.1 and and are zero at 700 nm, since in these cases the test color is one of the primaries. The primaries with wavelengths 546.1 nm and 435.8 nm 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 the eye's perception of color is rather unchanging at this wavelength, 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.^{[10]} The cutoffs at the short and longwavelength 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
The resulting normalized color matching functions are then scaled in the r:g:b ratio of 1:4.5907:0.0601 for source luminance and 72.0962:1.3791:1 for source radiant power to reproduce the true color matching functions. By proposing that the primaries be standardized, the CIE established an international system of objective color notation.
Given these scaled color matching functions, the RGB tristimulus values for a color with a spectral power distribution would then be given by:
These are all inner products and can be thought of as a projection of an infinitedimensional spectrum to a threedimensional color. (See also: Hilbert space)
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.
The CIE RGB space can be used to define chromaticity in the usual way: The chromaticity coordinates are r and g where:
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 that would relate 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 , , and as described above. The new color space would be chosen to have the following desirable properties:
In geometrical terms, choosing the new color space amounts to choosing a new triangle in rg chromaticity space. In the figure aboveright, 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 nonnegative means that the triangle formed by C_{r}, C_{g}, C_{b} must encompass the entire gamut of the standard observer. The line connecting C_{r} and C_{b} is fixed by the requirement that the function be equal to the luminance function. This line is the line of zero luminance, and is called the alychne. The requirement that the function be zero above 650 nm means that the line connecting C_{g} and C_{r} must be tangent to the gamut in the region of K_{r}. This defines the location of point C_{r}. The requirement that the equal energy point be defined by x = y = 1/3 puts a restriction on the line joining C_{b} and C_{g}, 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 C_{g} and C_{b}. The above described transformation is a linear transformation from the CIE RGB space to XYZ space. The standardized transformation settled upon by the CIE special commission was as follows:
The numbers below all have the correct number of significant digits per CIE standards.^{[10]}^{[clarification needed]}
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. 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.

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