#### What is additive color matching?

Additive color mixing refers to the mixing of different colored lights and can be easily demonstrated by the superposition of lights (primaries) on a white projection screen. When this is done using red, green, and blue primaries, the colors yellow, cyan, and magenta are produced where two of the primaries overlap. Where all three primaries overlap the sensation of white is produced if the spectral distributions and intensities of the three primaries are carefully chosen.
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#### What is the CIE 1931 system?

In 1931 the CIE (Commission Internationale de l'Eclairage) developed a system for specifying color stimuli using tristimulus values for three imaginary primaries. The basis of this system was the CIE 1931 standard observer.

According to the trichromatic theory of color vision an observer can match a color stimulus with an additive mixture of three primaries. Therefore any color stimulus can be specified by the amounts of the primaries that an observer would use in order to match the stimulus. The CIE standard observer resulted from experiments where observers were asked to match monochromatic wavelengths of light with mixtures of three primaries. The standard observer is in fact a table showing how much of each primary would be used (by an average observer) to match each wavelength of light. Tristimulus values are the amounts of three primaries that specify a color stimulus. The CIE 1931 tristimulus values are called X, Y, and Z.
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#### Why are the CIE primaries often called imaginary primaries?

It is impossible to choose three real primaries such that all possible colors can be matched with additive mixtures of those primaries. Thus, in a real additive color reproductive system such as color television only a limited gamut of colors can be displayed. In 1931, when the CIE system was specified, it was decided to use three imaginary primaries such that the tristimulus values X, Y, and Z, are always positive for all real color stimuli. The concept of imaginary primaries is complex but it is not strictly neccessary to understand this concept in order to understand and use the CIE system of color specification. In fact, the CIE could have used three real primaries, such as red, green, and blue lights, in which case the tristimulus values would be represented by R, G, and B.

There were several reasons for the adoption of imaginary primaries. Firstly, the primaries were chosen such that X, Y, and Z would be positive for all possible real stimuli. Although this might not seem particularly important today, the elimination of negative tristimulus values was an important consideration in pre-computer days. Secondly, the imaginary primaries were chosen such that the Y tristimulus value was directly proportional to the luminance of the additive mixture. Thirdly, X=Y=Z for a match to the equi-energy stimulus SE (a stimulus that has equal luminance at each wavelength).
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#### How can tristimulus values be calculated?

Tristimulus values can be calculated if the reflectance spectrum of a sample is known. The reflectance spectrum can be measured using a reflectance spectrophotometer.
CIE XYZ tristimulus values can be calculated by the integration of the reflectance values R(l), the relative spectral energy distributions of the illuminant E(l), and the standard observer functions x(l), y(l), and z(l). The integration is appoximated by summation, thus:

X = 1/k ∑ R(l) E(l) x(l),
Y = 1/k ∑ R(l) E(l) y(l),
Z = 1/k ∑ R(l) E(l) z(l),

where k = ∑ E(l) y(l) and l = wavelength.

The normalizing factor 1/k is introduced such that Y = 100 for a sample that reflects 100% at all wavelengths: recall that Y is proportional to the luminance of the stimulus. The introduction of this normalization is convenient since it means that relative, rather than absolute, spectral energy distributions for the illuminant can be used (thus the units in which they are expressed are unimportant).
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#### How does a reflectance spectrophotometer work?

There are two main types of instruments that are used for measuring the color of opaque surfaces: reflectance spectrophotometers and colorimeters. Reflectance spectrophotometers measure the amount of light reflected by a sample at many narrowband wavelength intervals resulting in a reflectance spectrum. By contrast, tristimulus colorimeters employ three broadband filters to obtain three numbers that can be converted directly to tristimulus values.

Reflectance spectrophotometers measure the amount of light reflected by a surface as a function of wavelength to produce a reflectance spectrum. The reflectance spectrum of a sample can be used, in conjunction with the CIE standard observer function and the relative spectral energy distribution of an illuminant, to calculate the CIE XYZ tristimulus values for that sample under that illuminant.

The operation of a spectrophotometer is basically to illuminate the sample with white light and to calculate the amount of light that is reflected by the sample at each wavelength interval. Typically data are measured for 31 wavelength intervals centred at 400nm, 410nm, 420nm, ..., 700nm. This is done by passing the reflected light though a monochromating device that splits the light up into separate wavelength intervals. The instrument is calibrated using a white tile whose reflectance at each wavelength is known compared to a perfect diffuse reflecting surface. The reflectance of a sample is expressed between 0 and 1 (as a fraction) or between 0 and 100 (as a percentage). It is important to realize that the reflectance values obtained are relative values and, for non-fluorescent samples, are independent of the quality and quantity of the light used to illuminate the sample. It is only when tristimulus values, such as CIE XYZ, are computed that the measurements become illuminant specific.
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#### What is the optical geometry of a spectrophotometer?

The optical geometry of the instrument is important. In some instruments an integrating sphere is used that enables the sample to be illuminated diffusely (from all angles equally) and the reflected light to be collected at an angle roughly perpendicular to the surface of the sample. Alternatively, other instruments illuminate the sample at a certain angle and collect light at another angle. For example, typically the sample may be illuminated at 45 degrees to the surface and light reflected measured at 0 degrees - this is known as 45/0 geometry. The converse to this is 0/45. The sphere-based geometries are known as D/0 and 0/D. It is extremely difficult, if not impossible, to correlate measurements taken between instruments if the optical geometry is not identical.

The four CIE standard geometries are:
* diffuse illumination and light collection at the normal, D/0;
* normal illumination and diffuse light collection, 0/D;
* illumination at 45 degrees and light collection at the normal, 45/0;
* normal illumination and light collection at 45 degrees, 0/45.
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#### How does a colorimeter work?

Colorimeters measure tristimulus values and operate using three broad-band filters. Consequently, colorimeters cannot provide spectral reflectance data but historically they have been preferred to spectrophotometers because of their low cost of manufacture and portability.
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#### What is the specular component of reflectance?

When light strikes a surface some of the light penetrates where it can then be absorbed, scattered, or even transmitted if the layer is sufficiently thin. Nevertheless, because of the change in refractive index between air and most substances, a certain proportion of the incident light is reflected directly from the surface. The angular distribution of this light depends upon the nature of the surface but light that is reflected at the opposite angle to the incident light is called specular reflectance. Light that is reflected by the substance itself is called body reflectance.

Sphere-based spectrophotometers often incorporate a so-called gloss trap which allows the specular component of the reflected light to be either included or excluded.
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#### What is the difference between a light source and an illuminant?

The terms light source and illuminant have precise and different meanings. A light source is a physical emitter of radiation such as a candle, a tungsten bulb, and natural daylight. An illuminant is the specification for a potential light source. All light sources can be specified as an illuminant, but not all illuminants can be physically realized as a light source.

Illuminants are normally specified in terms of relative energy tabulated for each wavelength or wavelength band. There are several illuminants that are widely used by the color industry and these include A, C, D65, and TL84. Illuminants A and C were defined by the CIE in 1931 to represent tungsten light and natural daylight respectively. Illuminant C was found to be a poor representation of daylight in that it contains insufficient energy at the lower wavelengths and it has generally been replaced by a class of illuminants known as the D illuminants.

The D class of illuminants specify relative energy distributions that closely correspond to the radiation emitted by a so-called black-body. As the tempertature of a black body is increased there is a shift in the emitted radiation to longer wavelengths. A specific D illuminant is therefore notated with reference to the temperature (in Kelvin) of the black-body which it most closely matches. For example, the illuminant D65 has a spectral energy distribution that closely matches that of a black-body at 6500K. Illuminant D65 also closely resembles the relative spectral energy distribution of north-sky daylight and is accordingly important for color specification in northern Europe. Other D illuminants, notably D55, are important in other parts of the world.

There are a number of illuminants that specify light sources used in specific industries and sometimes by specific companies. An example of this is illuminant TL84.
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#### Why is the 1931 standard observer called a 2 degree observer?

The 1931 standard observer data were derived from color-matching experiments with an arrangement that meant that the stimuli activated an area of the retina of 2 degrees. The distribution of rods and cones is not uniform over the surface of the retina and this implies that the tristimulus values obtained from the 1931 data are strictly only valid for observations made under 2 degrees viewing conditions. This is equivalent to viewing a small coin held at arm-length and does not correspond particularly well with the viewing conditions often used in the coloration industry.
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#### What is the 10 degree observer?

Because the 1931 2 degree observer is not really appropriate for large-field visual color judgements the CIE defined a second set of observer functions in 1964 known as the supplementary observer data based upon color-matching experiments with a field of 10 degrees. Since the 2 degree data are still in use the 10 degree data are often differentiated from the original 1931 data by the use of subscripts.
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#### What are chromaticity coordinates?

There is often a need for an intuitive interpretation of color specification in terms of tristimulus values. This is one reason why the three-dimensional color color space defined by X, Y, and Z is often transformed and plotted in terms of a chromaticity diagram. Chromaticity coordinates x, y, and z are derived by calculating the fractional components of the tristimulus values thus:

x = X/(X + Y + Z)

Since by definition x + y + z = 1, if two of the chromaticity coordinates are known then the third is redundant. Thus, all possible sets of tristimulus values can be represented in a two-dimensional plot of two of these chromaticity coordinates and by convention x and y are always used. A plot of this type is referred to as a chromaticity diagram. The use of chromaticity diagrams enables three-dimensional data to be compressed into two-dimensional data but at a cost. Consider two samples A and B having specification

Sample A: X = 10, Y = 20, Z = 30
Sample B: X = 20, Y = 40, Z = 60

Samples A and B have identical chromaticity coordinates but different tristimulus values. The difference between the two samples is one of luminance and B would probably appear brighter than A if the two samples were viewed together. A complete specification using chromaticity coordinates therefore requires two chromaticity coordinates and one of the tristimulus values.
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#### What is CIE L* a* b* color space?

There are perhaps two problems with the specification of colors in terms of tristimulus values and chromaticity space. Firstly, this specification is not easily interpreted in terms of the psychophysical dimensions of color perception namely, brightness, hue, and colorfulness. Secondly, the XYZ system and the associated chromaticity diagrams are not perceptually uniform. The second of these points is a problem if we wish to estimate the magnitude of the difference between two color stimuli. The need for a uniform color space led to a number of non-linear transformations of the CIE 1931 XYZ space and finally resulted in the specification of one of these transformations as the CIE 1976 (L* a* b*) color space.
In fact in 1976 the CIE specified two color spaces; one of these was intended for use with self-luminous colors and the other was intended for use with surface colors. These notes are principally concerned with the latter known as CIE 1976 (L* a* b*) color space or CIELAB.

CIELAB allows the sepcification of color perceptions in terms of a three-dimensional space. The L* axis is known as the lightness and extends from 0 (black) to 100 (white). The other two coordinates a* and b* represent redness-greenness and yellowness-blueness respectively. Samples for which a* = b* = 0 are achromatic and thus the L* axis represents the achromatic scale of greys from black to white.

The quantities L*, a*, and b* are obtained from the tristimulus values according to the following transformations:

L* = 116(Y/Yn)1/3 -16,
a* = 500[(X/Xn)1/3 - (Y/Yn)1/3],
b* = 200[(Y/Yn)1/3 - (Z/Zn)1/3],

where Xn, Yn, and Zn are the values of X, Y, and Z for the illuminant that was used for the calculation of X, Y, and Z of the sample, and the quotients X/Xn, Y/Yn, and Z/Zn are all greater than 0.008856. Note: When any of the quotients are less than or equal to 0.008856 a slightly different set of equations is used.
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#### Should I use L* a* b* or L* C* H* specification?

It is often convenient to consider a single slice though color space at constant L*. Although it is possible to represent a point in the a*-b* plane by its cartesian coordinates a* and b* it is often better to specify the polar coordinates C* and H*.

It is dangerous to attempt to interpret the qualitative color difference between two samples using the a* b* representation. For example, even though the a* axis is the red axis, a sample with a large a* value would not necessarily appear redder than a sample with a smaller a* value. Hue is not uniquely defined by either a* or b*. The use of C* and H* leads to a more intuitive representation of color.
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