CIE 1931 color space

The CIE XYZ standard observer color matching functions

The CIE RGB color matching functions

The CIE's color matching functions ${\displaystyle {\overline {x}}(\lambda )}$, ${\displaystyle {\overline {y}}(\lambda )}$ and ${\displaystyle {\overline {z}}(\lambda )}$ are the numerical description of the chromatic response of the observer (described above). They 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.[10]

Table lookup can become impractical for some computational tasks. Instead of referring to the published table, the CIE XYZ color matching functions can be approximated by a sum of Gaussian functions,[11] as follows:

Let g(x) denote a piecewise-Gaussian function, defined by

${\displaystyle g(x;\alpha ,\mu ,\sigma _{1},\sigma _{2})=\alpha \exp \left({\frac {(x-\mu )^{2}}{-2\sigma ^{2}}}\right),{\text{ where }}\sigma ={\begin{cases}\sigma _{1},&x<\mu ,\\\sigma _{2},&x\geq \mu .\end{cases}}}$

That is, g(x) resembles a bell curve with its peak at x = μ, a spread/standard deviation of σ₁ to the left of the mean, spread of σ₂ to the right of the mean, and scaling parameter α. With the wavelength λ measured in angstroms, we then approximate the 1931 color matching functions as follows:

${\displaystyle {\begin{aligned}{\overline {x}}(\lambda )&=g(\lambda ;1.056,5998,379,310)+g(\lambda ;0.362,4420,160,267)+g(\lambda$ ;-0.065,5011,204,262),\\{\overline {y}}(\lambda )&=g(\lambda ;0.821,5688,469,405)+g(\lambda ;0.286,5309,163,311),\\{\overline {z}}(\lambda )&=g(\lambda ;1.217,4370,118,360)+g(\lambda ;0.681,4590,260,138).\end{aligned}}}

This approximation can be easily employed in a programming language in a functional style. For example, here is a Haskell implementation:

xyzOfWavelength λ = map (sum . map g)
  [ [ (1056,5998,379,310), (362,4420,160,267), (-65,5011,204,262) ]
  , [ ( 821,5688,469,405), (286,5309,163,311) ]
  , [ (1217,4370,118,360), (681,4590,260,138) ]
  ]
  where g(α,μ,σ1,σ2) = α/1000 * exp(-((λ-μ)/(if λ<μ then σ1 else σ2))^2 / 2)

Here is a semi-functional style implementation in C:

#include <math.h>
struct XYZ { double x, y, z; };

inline double gaussian(double x, double α, double μ, double σ1, double σ2) {
  double t = (x - μ) / (x < μ ? σ1 : σ2);
  return α * exp(-(t * t) / 2);
}

struct XYZ xyzFromWavelength(double λ) {
  struct XYZ color;
  color.x = gaussian(λ,  1.056, 5998, 379, 310) +
            gaussian(λ,  0.362, 4420, 160, 267) +
            gaussian(λ, -0.065, 5011, 204, 262);
  color.y = gaussian(λ,  0.821, 5688, 469, 405) +
            gaussian(λ,  0.286, 5309, 163, 311);
  color.z = gaussian(λ,  1.217, 4370, 118, 360) +
            gaussian(λ,  0.681, 4590, 260, 138);
  return color;
}

There are also other analytic fits, but none perform as well as the one provided here (as of July 2013). It is also possible to use fewer gaussian functions, with one gaussian for each "lobe". CIE 1964 fits well with a one-lobe function.[11]

The CIE XYZ color matching functions are nonnegative, and lead to nonnegative XYZ coordinates for all real colors (that is, for nonnegative light spectra). Other observers, such as for the CIE RGB space or other RGB color spaces, are defined by other sets of three color-matching functions, not generally nonnegative, and lead to tristimulus values in those other spaces, which may include negative coordinates for some real colors.

A few other color-matching functions have been available. They generally have a higher resolution.

• A 1 nm-interval dataset of "CIE 1931" and "CIE 1964" has been provided by Wyszecki 1982.[12] A CIE publication in 1986 appears also have a 1 nm dataset, probably using the same data.[13]
• The Colour & Vision Research laboratory of University College London has 2° and 10° XYZ functions based on a linear transformation of Stockman & Sharpe (2000) physiological LMS fundamentals. It is offered down to 0.1 nm resolution.[14] It was proposed to the CIE in 2012, and is included in CIE 015:2018.[15]

The tristimulus values for a color with a spectral radiance L_e,Ω,λ are given in terms of the standard observer by:

${\displaystyle X=\int _{\lambda }L_{\mathrm {e} ,\Omega ,\lambda }(\lambda )\,{\overline {x}}(\lambda )\,d\lambda ,}$

${\displaystyle Y=\int _{\lambda }L_{\mathrm {e} ,\Omega ,\lambda }(\lambda )\,{\overline {y}}(\lambda )\,d\lambda ,}$

${\displaystyle Z=\int _{\lambda }L_{\mathrm {e} ,\Omega ,\lambda }(\lambda )\,{\overline {z}}(\lambda )\,d\lambda .}$

where ${\displaystyle \lambda }$ is the wavelength of the equivalent monochromatic light (measured in nanometers), and customary limits of the integral are ${\displaystyle \lambda \in [380,780]}$.

The values of X, Y, and Z are bounded if the radiance spectrum L_e,Ω,λ is bounded.

The reflective and transmissive cases are very similar to the emissive case, with a few differences. The spectral radiance L_e,Ω,λ is replaced by the spectral reflectance (or transmittance) S(λ) of the object being measured, multiplied by the spectral power distribution of the illuminant I(λ).

${\displaystyle X={\frac {K}{N}}\int _{\lambda }S(\lambda )\,I(\lambda )\,{\overline {x}}(\lambda )\,d\lambda ,}$

${\displaystyle Y={\frac {K}{N}}\int _{\lambda }S(\lambda )\,I(\lambda )\,{\overline {y}}(\lambda )\,d\lambda ,}$

${\displaystyle Z={\frac {K}{N}}\int _{\lambda }S(\lambda )\,I(\lambda )\,{\overline {z}}(\lambda )\,d\lambda ,}$

where

${\displaystyle N=\int _{\lambda }I(\lambda )\,{\overline {y}}(\lambda )\,d\lambda ,}$

K is a scaling factor (usually 1 or 100), and ${\displaystyle \lambda }$ is the wavelength of the equivalent monochromatic light (measured in nanometers), and the standard limits of the integral are ${\displaystyle \lambda \in [380,780]}$.

When two or more colors are additively mixed, the x and y chromaticity coordinates of the resulting color (x_mix,y_mix) may be calculated from the chromaticities of the mixture components (x₁,y₁; x₂,y₂; …; x_n,y_n) and their corresponding luminances (L₁, L₂, …, L_n) with the following formulas:[18]

$${\displaystyle x_{\mathrm {mix} }={\frac {{\dfrac {x_{1}}{y_{1}}}L_{1}+{\dfrac {x_{2}}{y_{2}}}L_{2}+\dots +{\dfrac {x_{n}}{y_{n}}}L_{n}}{{\dfrac {L_{1}}{y_{1}}}+{\dfrac {L_{2}}{y_{2}}}+\dots +{\dfrac {L_{n}}{y_{n}}}}}\quad ,\quad y_{\mathrm {mix} }={\frac {L_{1}+L_{2}+\dots +L_{n}}{{\dfrac {L_{1}}{y_{1}}}+{\dfrac {L_{2}}{y_{2}}}+\dots +{\dfrac {L_{n}}{y_{n}}}}}}$$

These formulas can be derived from the previously presented definitions of x and y chromaticity coordinates by taking advantage of the fact that the tristimulus values X, Y, and Z of the individual mixture components are directly additive. In place of the luminance values (L₁, L₂, etc.) one can alternatively use any other photometric quantity that is directly proportional to the tristimulus value Y (naturally meaning that Y itself can also be used as well).

As already mentioned, when two colors are mixed, the resulting color x_mix,y_mix will lie on the straight line segment that connects these colors on the CIE xy chromaticity diagram. To calculate the mixing ratio of the component colors x₁,y₁ and x₂,y₂ that results in a certain x_mix,y_mix on this line segment, one can use the formula

$${\displaystyle {\frac {L_{1}}{L_{2}}}={\frac {y_{1}\left(x_{2}-x_{\mathrm {mix} }\right)}{y_{2}\left(x_{\mathrm {mix} }-x_{1}\right)}}={\frac {y_{1}\left(y_{2}-y_{\mathrm {mix} }\right)}{y_{2}\left(y_{\mathrm {mix} }-y_{1}\right)}}}$$

where L₁ is the luminance of color x₁,y₁ and L₂ the luminance of color x₂,y₂. Note that because y_mix is unambiguously determined by x_mix and vice versa, knowing just one or the other of them is enough for calculating the mixing ratio. Also note that, in accordance with the remarks concerning the formulas for x_mix and y_mix, the mixing ratio L₁/L₂ may well be expressed in terms of other photometric quantities than luminance.

The CIE RGB color space is one of many RGB color spaces, distinguished by a particular set of monochromatic (single-wavelength) primary colors.

In the 1920s, two independent experiments on human color perception were conducted by W. David Wright[3] with ten observers, and John Guild[4] with seven observers. Their results laid the foundation for the trichromatic CIE XYZ color space specification.

Gamut of the CIE RGB primaries and location of primaries on the CIE 1931 xy chromaticity diagram.

The experiments were conducted by using a circular split screen (a bipartite field) 2 degrees in diameter, which is the angular size of the human fovea. On one side a test color was projected while on the other 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.

The CIE 1931 RGB color matching functions. The color matching functions are the amounts of primaries needed to match the monochromatic test color at the wavelength shown on the horizontal scale.

Although Wright and Guild's experiments were carried out using various primaries at various intensities, and although they used a number of different observers, all of their results were summarized by the standardized CIE RGB color matching functions ${\displaystyle {\overline {r}}(\lambda )}$, ${\displaystyle {\overline {g}}(\lambda )}$, and ${\displaystyle {\overline {b}}(\lambda )}$, 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 ${\displaystyle {\overline {r}}(\lambda )}$ and ${\displaystyle {\overline {g}}(\lambda )}$ are zero at 435.8 nm, ${\displaystyle {\overline {r}}(\lambda )}$ and ${\displaystyle {\overline {b}}(\lambda )}$ are zero at 546.1 nm and ${\displaystyle {\overline {g}}(\lambda )}$ and ${\displaystyle {\overline {b}}(\lambda )}$ 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.[19] 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

${\displaystyle \int _{0}^{\infty }{\overline {r}}(\lambda )\,d\lambda =\int _{0}^{\infty }{\overline {g}}(\lambda )\,d\lambda =\int _{0}^{\infty }{\overline {b}}(\lambda )\,d\lambda .}$

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 ${\displaystyle S(\lambda )}$ would then be given by:

${\displaystyle R=\int _{0}^{\infty }S(\lambda )\,{\overline {r}}(\lambda )\,d\lambda ,}$

${\displaystyle G=\int _{0}^{\infty }S(\lambda )\,{\overline {g}}(\lambda )\,d\lambda ,}$

${\displaystyle B=\int _{0}^{\infty }S(\lambda )\,{\overline {b}}(\lambda )\,d\lambda .}$

These are all inner products and can be thought of as a projection of an infinite-dimensional spectrum to a three-dimensional color.

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, g and b where:

${\displaystyle r={\frac {R}{R+G+B}},}$

${\displaystyle g={\frac {G}{R+G+B}},}$

${\displaystyle b={\frac {B}{R+G+B}}.}$

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 ${\displaystyle {\overline {x}}(\lambda )}$, ${\displaystyle {\overline {y}}(\lambda )}$, and ${\displaystyle {\overline {z}}(\lambda )}$ as described above. The new color space would be chosen to have the following desirable properties:

Diagram in CIE rg chromaticity space showing the construction of the triangle specifying the CIE XYZ color space. The triangle C_b-C_g-C_r is just the xy = (0, 0), (0, 1), (1, 0) triangle in CIE xy chromaticity space. The line connecting C_b and C_r is the alychne. Notice that the spectral locus passes through rg = (0, 0) at 435.8 nm, through rg = (0, 1) at 546.1 nm and through rg = (1, 0) at 700 nm. Also, the equal energy point (E) is at rg = xy = (1/3, 1/3).

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 ${\displaystyle {\overline {y}}(\lambda )}$ color matching function would be exactly equal to the photopic luminous efficiency function V(λ) for the "CIE standard photopic observer".[20] 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 ${\displaystyle {\overline {z}}(\lambda )}$ 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 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 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 ${\displaystyle {\overline {y}}(\lambda )}$ function be equal to the luminance function. This line is the line of zero luminance, and is called the alychne. The requirement that the ${\displaystyle {\overline {z}}(\lambda )}$ 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 in the conversion matrix below are exact, with the number of digits specified in CIE standards.[19]

${\displaystyle {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}={\frac {1}{b_{21}}}{\begin{bmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{bmatrix}}{\begin{bmatrix}R\\G\\B\end{bmatrix}}={\frac {1}{0.176\,97}}{\begin{bmatrix}0.490\,00&0.310\,00&0.200\,00\\0.176\,97&0.812\,40&0.010\,63\\0.000\,00&0.010\,00&0.990\,00\end{bmatrix}}{\begin{bmatrix}R\\G\\B\end{bmatrix}}}$

While the above matrix is exactly specified in standards, going the other direction uses an inverse matrix that is not exactly specified, but is approximately:

${\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}={\begin{bmatrix}0.418\,47&-0.158\,66&-0.082\,835\\-0.091\,169&0.252\,43&0.015\,708\\0.000\,920\,90&-0.002\,549\,8&0.178\,60\end{bmatrix}}\cdot {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$

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. 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. Furthermore, one could 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.