Colorimetry is the science of measuring colors. Although each of us can perceive colors slightly differently, the CIE has defined a standard observer. A set of standard conditions for performing color measuring experiments has also been proposed by CIE. A number of color matching experiments have been performed under these standardized conditions. Color matching experiments consists of choosing three particular light sources, that emit light on the white screen, where three projections overlap and form an additive mixture. On the other side of the screen a target color is projected, and an observer tries to match the target light by altering the intensities of the three light sources. The weights of light sources are in the range [-1,1]. Negative weights are allowed, as it is not possible to match all colors using only positive weights. A negative weight does not mean subtracting color from the additive mixture, but rather adding this color to the target color. After many experiments using light sources of the wavelengths red=700 nm, green=546.1 nm and blue is 435.8 nm [WySt82] color matching curves as shown in figure 2.2 were proposed by CIE.
Figure 2.2: The r, g, and b color-matching curves [WySt82]
Figure 2.3: The x, y, and z color-matching curves [WySt82]
As it was inconvenient to have negative values in the matching functions CIE proposed a linear transformation of matching functions resulting in CIE x, y and z matching functions, as shown in figure 2.3. Note that there are no negative values in these matching functions.
Now if the surface reflectance, and the light source distribution are
known, their product defines color as 
, and the weights
XYZ can be found using the following equations:
The weights, X, Y and Z define a color in the CIE XYZ space. Note that
it is possible that two objects with different spectral reflectance,
under certain illuminantion, appear the same, i.e. have the same CIE
XYZ values. The CIE XYZ is a 3D linear color space, and it is quite
awkward to work in it directly. It is common to project this space to
the X+Y+Z=1 plane. The result is a 2D space known as the CIE
chromaticity diagram. The coordinates in this space are usually called
x and y and they are derived from XYZ using the following equations:
As the z component bears no additional information, it is often
omitted. Note that since xy space is just a projection of the 3D XYZ
space, each point in xy corresponds to many points in the original
space. Actually the missing information is luminance Y. Color is
usually described by xyY coordinates, where x and y determine the
chromaticity and Y the lightness component of color. Figure 2.4
shows the CIE xy chromaticity diagram.
Figure 2.4: CIE xy chromaticity diagram
Chromaticity diagrams can give us a lot of useful information on a
particular color. The horseshoe shaped curve represents the whole
spectrum. The straight line connecting the lowest wavelength blue and
the highest wavelength red is called the ``purple line'' and does not
represent spectral colors. The white point lies somewhere inside the
diagram, depending on the light source used (e.g. D65 light source is
defined to simulate day-light and has x=0.312727 and
y=0.329024). If a line is drawn through the white point and
a particular xy color, then the ratio between this point to white point
distance, and the spectral line to the white point distance gives us the color
saturation. If the color is close to the spectral line its saturation
is high. The dominant wavelength which determines a color's hue is
determined by the intersection of the line with the spectral line. An
interesting property of the xy chromaticity diagram is that all
possible mixtures of colors 
and 
are given by the
straight line connecting these two points. It is clear that all the
possible mixtures of the three colors , and 
then lie inside the triangle determined with those three points. Now, it
is clear that the color gamut of any display device using three primaries
(like a standard CRT monitor) is only a subset of all visible colors.
In spite of all the useful characteristics of the CIE xy chromaticity
diagram, it lacks one very important characteristic. Namely, if the
distance between any arbitrary two points is the same as the distance
between an other point pair, the perceived distance will not be the
same. In the worst case, if the perceived distances are the same,
actual distances can differ as much as 20 times. In order to correct
this, researchers are trying to find a perceptually uniform color
space. It has, unfortunately, still not been found. CIE proposed two
alternatives as improvements compared with CIE xyY space. These are
CIE LUV and CIE LAB. Although they are referred to as perceptually
uniform color spaces by some authors, they are not. Just for
comparison, two perceptually equally distant color pairs, can differ
in the CIE LUV distance as much as 4 times. This is a significant
improvement compared to 20 times by original space, but it is still
not perfect. Conversions between CIE XYZ and CIE LUV are defined with
the formulas:
where
The distance between two colors in the CIE LUV space can be computed
using CIE LUV color difference formula:
