The CIE 1931 color spaces are 4 interrelated color spaces with the same origin. In the 1920s, two independent experiments on human color perception were conducted by W. David Wright 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 , 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". 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: : \int_0^\infty \overline{r}(\lambda)\,d\lambda = \int_0^\infty \overline{g}(\lambda)\,d\lambda = \int_0^\infty \overline{b}(\lambda)\,d\lambda. 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
radiance 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 S(\lambda) would then be given by: :\begin{align} R &= \int_0^\infty S(\lambda)\,\overline{r}(\lambda)\,d\lambda, \\[6mu] G &= \int_0^\infty S(\lambda)\,\overline{g}(\lambda)\,d\lambda, \\[6mu] B &= \int_0^\infty S(\lambda)\,\overline{b}(\lambda)\,d\lambda. \end{align} These are all
inner products and can be thought of as a projection of an infinite-dimensional spectrum to a
three-dimensional color.
CIE XYZ color space After the definition of the RGB model of human vision using the CIE RGB matching functions, the CIE special commission wished to derive another color space from 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), which would be chosen as having the following desirable properties: • 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. • 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". 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. • For the constant energy
white point, it was required that . • 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. • It was found that the \overline{z}(\lambda) color matching function could be set to zero above while remaining within the bounds of experimental error. For computational simplicity, it was specified that this would be so. The derived CIE XYZ color space encompasses all color sensations that are perceptible to a typical human. The underlying color matching functions can be thought of as the spectral sensitivity curves of three linear light detectors yielding the CIE tristimulus values
X,
Y and
Z. Collectively, these three functions describe the CIE standard observer. That is why CIE XYZ tristimulus values are a device-invariant representation of color. It serves as a standard reference against which many other color spaces are defined. A set of color-matching functions, like the spectral sensitivity curves of the
LMS color space, but not restricted to non-negative sensitivities, associates physically produced light spectra with specific tristimulus values. Most wavelengths stimulate two or all three kinds of
cone cell because the spectral sensitivity curves of the three kinds overlap. Certain tristimulus values are thus physically impossible: e.g. LMS tristimulus values that are non-zero for the M component and zero for both the L and S components. Furthermore pure spectral colors would, in any normal trichromatic additive color space, e.g., the
RGB color spaces, imply negative values for at least one of the three
primaries because the
chromaticity would be outside the
color triangle defined by the primary colors. To avoid these negative RGB values, and to have one component that describes the perceived
brightness, "imaginary" primary colors and corresponding color-matching functions were formulated. The CIE 1931 color space defines the resulting tristimulus values, in which they are denoted by "X", "Y", and "Z". In XYZ space, all combinations of non-negative coordinates are meaningful, but many, such as the primary locations [1, 0, 0], [0, 1, 0], and [0, 0, 1], correspond to
imaginary colors outside the space of possible LMS coordinates; imaginary colors do not correspond to any spectral distribution of wavelengths and therefore have no physical reality.
Meaning of X, Y and Z for a standard observer in
photopic vision In the CIE 1931 model,
Y is the
luminance,
Z is quasi-equal to blue (of CIE RGB), and
X is a mix of the three CIE RGB curves chosen to be nonnegative. Setting
Y as luminance has the useful result that for any given
Y value, the XZ plane will contain all possible
chromaticities at that luminance. The unit of the tristimulus values , , and is often arbitrarily chosen so that or is the brightest white that a color display supports. In this case, the Y value is known as the
relative luminance. The corresponding whitepoint values for and can then be inferred using the
standard illuminants. Since the XYZ values are defined much earlier than the characterization of cone cells in the 1950s (by
Ragnar Granit), the physiological meaning of these values were known only much later. The Hunt-Pointer-Estevez matrix from the 1980s relates XYZ with LMS. When inverted, it shows how the three cone responses add up to XYZ functions: : \begin{bmatrix} X\\Y\\Z \end{bmatrix} = \begin{bmatrix} 1.91020 & -1.11212 & 0.20191 \\ 0.37095 & 0.62905 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} L\\M\\S \end{bmatrix}_{\rm HPE} In other words, the Z value is solely made up of the S cone response, the Y value a mix of L and M responses, and X value a mix of all three. This fact makes XYZ values analogous to, but different from, the LMS cone responses of the human eye.
CIE rg chromaticity space In geometrical terms, choosing the new color space amounts to choosing a new triangle in
rg chromaticity space. In the figure above-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 above 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 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 the CIE RGB space to XYZ space. The standardized transformation settled upon by the CIE special commission was as follows: The CIE RGB space can be used to define chromaticity in the usual way: The chromaticity coordinates are
r,
g and
b where: :\begin{align} r &= \frac{R}{R + G + B}, \\[5mu] g &= \frac{G}{R + G + B}, \\[5mu] b &= \frac{B}{R + G + B}. \end{align} The numbers in the conversion matrix below are exact, with the number of digits specified in CIE standards. :\begin{align} x &= \frac{X}{X+Y+Z} \\[5mu] y &= \frac{Y}{X+Y+Z} \\[5mu] z &= \frac{Z}{X+Y+Z} = 1 - x - y \end{align} That is, because each tristimulus parameter,
X,
Y,
Z, is divided by the sum of all three, the resulting values,
x,
y,
z, each represent a proportion of the whole and so their sum must be equal to one. Therefore, the value
z can be deduced by knowing
x and
y, and consequently the latter two values are sufficient for describing the chromaticity of any color. 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: :\begin{align} X &= \frac{Y}{y}x, \\[5mu] Z &= \frac{Y}{y}(1 - x - y). \end{align} Mathematically the colors of the chromaticity diagram occupy a region of the
real projective plane. ====