Colorimetry

# Color systems RGB and CIE XYZ

## A vector space for colors

Because of the trichromatic nature of vision, a color can be perfectly defined by a set of only three numbers, which can be thought of as the coordinates of a vector in a 3-D vector space. More precisely, a color can be represented by a vector whose magnitude is proportional to the light level (value) and whose orientation in space is related to the color tone itself (see figure 10).

Figure 10 : Vector space used for the representation of colors [zoom...]

In the example given in figure 10, a color is defined by a vector whose coordinates , and (called trichromatic coordinates) are measured according to the basis vectors , and the “primary colors”. If the primary colors are non-metameric with respect to each other, they form a true basis for the vector space and one can write (1) :

Equation (1) means that one can match the color (i.e. find a metameric color of ) by a mixture of the three primary colors , and with the respective weights , and .

Besides, one can think of any light spectrum as being the result of an additive mixture of a large number of nearly-monochromatic light beams, adding up with a weighting function which represents the spectrum, sampled with the delta function , so that :

If , we see that one can reproduce any polychromatic color by the following linear combination:

where is a monochromatic primary color source of wavelength . Any monochromatic source can be expressed in turn as a combination of three non-monochromatic primaries , and according to (1) which leads to :

where , and are the spectral tristimulus values which are determined once for all by the CIE from a set of human observers (see below). From (3) and (4) one gets:

From (1), one gets the components , and :

Once the spectral tristimulus values , and are known, any colored stimulus with a spectrum can be represented by a dot in a 3D space with coordinates , and .

## Measurement of spectral tristimulus values

An estimate of , and was performed in 1926 by J. Guild [] with 7 people with a limited visual field of 2°, so that the light was only incident on the central part of the retina, the macula (refer to the section “Basic mechanisms of color vision”). Primary colors were chosen to be monochromatic, at 700nm for red ( ),546,1nm for green ( ) and 435,8nm for blue ( ). The observation field was divided into two sections (see figure 11), one of them was illuminated by a monochromatic source, while the other received a combination of the three primary sources, which superimposed by additive mixing on the eye of the observer. The observer could tune the light level of the primary sources (by changing the position of the sources or by directly varying the intensity through calibrated apertures) until matching of visual perceptions was reached.

Figure 11: Principle of the measurement of trichromatic coordinates of a source. [zoom...]Info

When the “test source” to be matched for this experiment is a monochromatic source of wavelength he trichromatic components obtained are called the RGB tristimulus values , and . They were tabulated in 1931, so that this system is sometimes called RGB CIE 1931 system. They are plotted versus wavelength in figure 12.

Figure 12: the RGB spectral tristimulus values [zoom...]Info

The Y scale is defined so that .

This choice is justified by the fact that the visual perception of a neutral stimulus (white or gray color) corresponds to a balanced response of the three different cones. It has to be noticed that for wavelengths below 550nm, the color functions and take negative values. This is because it is impossible to match a monochromatic color below 550 nm by adding three monochromatic primaries. Therefore, in the setup shown in figure 11, matching of color perceptions in the two halves of the observation field would be obtained for some colors if one adds some light from one of the primaries to the “test light”.

## Graphical representation – Color triangle

From (5), any polychromatic radiation with a luminance is associated with a color where the trichromatic coordinates in the RGB color system are given by :

Since only the relative values of these coordinates have a physical meaning in terms of chromaticity, it is usual to introduce the trichromatic coordinates , and given by:

In practice, it is easier to represent colors in a plane rather than in the 3-D space. We have, from (7):

which is the equation of a plane defined by the 3 terminal points of the unit vectors , and (see figure 13).

Figure 13: the color triangle [zoom...]Info

A color is then represented by a point ( ) in that plane. Since the component can be straightforwardly derived from (8) provided that and are known, one generally prefers to display the plane under the form of a right isosceles triangle in which and lie along the right axes of the triangle.The diagram obtained is called the (r, g) chromaticity diagram, displayed in figure 14. Monochromatic radiations (for ) lie along the so-called spectrum locus. They correspond to the following coordinates (with ) :

At the three corners of the triangle appear the (monochromatic) primary colors red ( ; =700nm), green ( ; =546,1nm) and blue ( ; =435,8nm). The line connecting the red and blue primaries is called the purple line.

Figure 14: The (r, g) chromaticity diagram [zoom...]Info

All existing colors have their associated points inside the domain delimited by the spectrum locus and the purple line. Points outside of this area have no physical meaning and do not correspond to real stimuli.

## XYZ color space

As it can be seen on figure 14, many colors have negative trichromatic coordinates in the RGB system, as a result of the choice of three monochromatic primary colors. In order to eliminate this drawback, a linear transformation over the RBG space is made, leading to the tristimulus values , and which take only positive values. Moreover, the primaries , and (no longer monochromatic) have been chosen in such a way that the plane corresponds to a zero-luminance plane, and that the axis becomes the axis of visual luminances. Indeed, for a stimulus having a luminance in energetic units, we know that the luminance in visual units is :

With (in photopic vision), where is the luminous efficiency function (refer to a basic photometry course to learn more about ).Using our usual trichromatic notations, we see that if we define the Y coordinate as then is identical to , so that one of the trichromatic coordinates will give directly the visual luminance.

The linear transformation is defined by [] :

which gives the tristimulus values vs. wavelength shown in figure 15.

Figure 15: tristimulus values xyz CIE 1931 [zoom...]Info

A color is represented by a vector whose trichromatic coordinates , , are :

from which one can define the trichromatic coordinates , , such as :

Since , colors are represented in a plane whose axes x and y form right angles. Monochromatic colors ( )which have coordinates given by

lie along the spectrum locus. As for the RGB system, the surface inside which one may find all real physical colors is delimited by the spectrum locus and by the purple line. This surface is embedded within a circumscribing triangle whose corners , , represent the three primaries :

• the primary :

• the primary :

• the primary :

These primaries are non real since they are outside of the chromatic gamut (cf. figure 16).

Figure 16: Chromaticity diagram xy CIE 1931 [zoom...]Info

Figure 17 shows the chromaticity diagram xy CIE 1931 with the corresponding color names for an observer in daylight conditions []. The central zone around the position of white (equienergetic stimulus) corresponds to the CIE normalized illuminants.

Figure 17: Colored zones of the xy CIE1931 chromaticity diagram [zoom...]Info

## Equienergetic stimulus

By definition, the spectrum of an “equienergetic stimulus” (perfect white) is constant over the whole visible spectrum, that is , or

and

since

This is valid in the RGB system, but it remains true in the , , color system, in which we may write . The representative point of stimulus is located at the center of the RGB or XYZ triangle (see figures 14 and 16).

## Dominant color – complementary color

Within the diagram, a color resulting from the additive mixing of two colors and such as will result in three representative points , and aligned along the same line (see figure 18).

Figure 18: Positions of dominant and complementary colors [zoom...]Info

In a similar fashion, any color can be thought of as an additive mixing of the white equienergetic stimulus with a pure color lying along the spectrum locus. The latter is called the dominant color with .

Conversely, the white equienergetic stimulus could theoretically be obtained upon additive mixing of any color with a pure color called the complementary color such as .

The ratio is a measure of the purity of the color . This ratio is equal to 0 for the equienergetic white and is equal to 1 for a pure monochromatic color.