Metamerism is a sensory phenomenon that begins in perception of color through three retinal processes (described in Part IV). Once the light projected on the retina has been absorbed by the three groups of cones, any knowledge of its spectral composition is lost. The only thing remaining are three levels of activity in the red, green, and blue cones. Two lights can therefore produce the same color sensation, even if they are of different spectral compositions, provided they arouse the same levels of activity in the cones. In other words, two light stimuli will be metameric if they react equally with the sensitivity curves.
We can express this relationship mathematically if we let the stimuli spectral distributions be A() and B(), and represent the sensitivity curves of the cones as , , and . Then:
These equations define the stimulus relations that must be satisfied for metamerism. To assess its degree we have to know the spectral distribution of the components in each of these summations.
If we plot the tristimulus distribution curves for the two stimuli (that is the two stimuli distribution spectrums as seen under a given lighting in relation to the sensitivity curves of the eye), the condition for metamerism is that the area under each pair of curves should be equal. The degree of metamerism is given by the difference in shape of these distribution curves.
In everyday life, metamerism can be observed when two colors of fabric will look identical under one set of lights (this is when they are a metameric pair) and different under different lighting conditions.
The laws of color matching are derived from experiments that use a set of three monochromatic lights as additive primaries. The experiments attempted to match some combination of the three primaries with a standard intensity of each wavelength of the pure spectrum.
Let the three monochromatic primaries be represented by the symbols (R), (G), and (B), and the number of units of the primaries needed to match the color of the sample (S) be R, G, and B. The conditions for the match can be summarized by the equation:
(S) R(R) + G(G) + B(B)
The special symbol is used instead of an equal sign to indicate that this is a perceptual match. The numbers R, G, and B are the color coordinates of the sample in the color space determined by (R), (G), and (B). They are called the tristimulus values of the color (S) with respect to the particular set of primaries (R), (G), (B).
Increasing or decreasing the power of each primary by the same factor will change the overall brightness but not affect a match for hue and saturation. Therefore it is possible to describe a color by its chromaticity, which depends on the relative magnitudes of R, G, and B, but not on their absolute numerical values.
Trying to match a monochromatic light (S) of wavelength l and unit intensity with some combination of our three primaries, we find that such a match cannot be achieved. Formally stated, for monochromatic light of any wavelength in the visible spectrum it is impossible to find any direct combination of intensities, R + G + B that will exactly match (S) in hue, saturation, and brightness. However, conditions exist that do make it possible for a match to be made. Depending on the value of l, one of the following matches will always be possible:
All the quantities in equations 1 - 3 are intrinsically positive, meaning they represent direct physical quantities of real lights. Hence, we know that it is impossible to match most saturated samples because the addition of two or more primary colors is knows in most cases to produce a desaturated color. But if one of the primaries is redirected so that it mixes additively with the sample instead of with the other primaries, a match can be made. For every wavelength in the visible spectrum the numbers will take on a unique set of values.
This idea is the key to the color matching functions. In the 1920, congnitive psychologists (such as John Guild and W.D. Wright) performed color matching experiments with a large group of participants. It was during these experiments that the tristimulus values corresponding to matches of the primaries against monochromatic test samples were determined. Keeping the amount of power (watts) falling on the test half of the screen constant, they recorded the resulting tristimulus values at each wavelength. The results were averaged over all the observers to account for minor differences between individual sensitivity curves and are shown in Table 4.1. The very small values near both end of the spectrum are a result of poor sensitivity of the eye to light of the corresponding wavelengths. Hence we don't require very much R, G, B to match one watt of deep red light or blue light compared to one watt of green light or yellow light.
The color matching functions for observers with normal color vision viewing at two degree visual field. The primaries are monochromatic B (at 435.8nm), G (at 546.1nm), and R (at 700.0nm) with relative intensities so that when added they match white. -- Source. From C.I.E. Publication No. 15, Colorimetry, 1971.
Figure 3.2 shows the graph that results from plotting the relative intensity of each primary necessary to match a unit intensity of the pure colors of the spectrum. This is the visual representation of the color matching functions shown above.
Color matching functions for the set of monochromatic primaries (at 435.8nm), G (at 546.1nm), and R (at 700.0nm).
We can get a tristimulus value for a given spectral power distribution by summing the numbers for all wavelengths across the spectrum, each weighted by the actual amount of power in the spectral power distribution at that wavelength. Similarly G and R can be separately calculated using the appropriate color matching function. If we happen to only be interested in the chromacity and not the brightness, then the units in which the spectral power distribution is measured are not important since they only affect the magnitude but not the relative values of the calculated B, G, and R. In Figure 3.3 a dashed vertical line is drawn at 500nm. Where this line happens to cross the three curves on the figure indicates the intensities of the three primaries, R, G, and B, needed to match a unit intensity of 500nm light.
Relative intensity of three monochromatic primaries needed to match a unit intensity of 550nm light