# Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

# Metamerism

• Peter van der Burgt
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_150

## Definition

Metamerism is the matching of apparent color of objects with lights that have different spectral power distributions. Colors that match are called metamers.

## Color of the Light

The light coming from any light source (sunlight, incandescent lamp) can be described by the so-called spectral power distribution, often denoted as SPD. The SPD gives the amount of power radiated by the light source as a function of wavelength in the visible range of the electromagnetic spectrum being 380–740 nm. Every SPD will generate a color impression in the human visual system. To describe the color of the light, a system has been developed by the CIE (Commission Internationale de l’Eclairage). A set of three color-matching functions called x(λ), y(λ), and z(λ) has been defined, and these are known collectively as the CIE standard observer [1]. These color-matching functions are depicted below in Fig. 1.
By using the color-matching functions, the effect of a spectral power distribution E(λ) can be expressed by the following equations:
$$\mathrm{X}=\sum \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)$$
(1)
$$\mathrm{Y}=\sum \mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)$$
(2)
$$\mathrm{Z}=\sum \mathrm{E}\left(\uplambda \right). \mathrm{z}\left(\uplambda \right)$$
(3)
The values X, Y, Z are called the tristimulus values of the color of the light with spectral power distribution E(λ).
The tristimulus values for a light can be converted to the chromaticity or color point x,y by the following equations:
$$\mathrm{x}=\mathrm{X}/\mathrm{P} \mathrm{with} \mathrm{P}=\mathrm{X}+\mathrm{Y}+\mathrm{Z}$$
(4)
$$\mathrm{y}=\mathrm{Y}/\mathrm{P}$$
(5)
These equations lead to the well-known CIE chromaticity diagram which is shown in Fig. 2. With the equations above, the color points for monochromatic lights can be calculated. This leads to the spectral locus, the outer curved boundary in the diagram. The straight edge on the lower part of the gamut is called the line of purples. These colors, although they are on the border of the gamut, have no counterpart in monochromatic light. Less saturated light colors appear in the interior of the figure. The diagram shows in the colored area all the chromaticities that are visible to the average person, and this colored region is called the gamut of human vision. One important property of this diagram is that lights that have the same chromaticity will give the same color impression to an observer. The chromaticity however does not predict the precise color impression of a light. For example, an incandescent lamp burning in the middle of the day may give a yellowish impression. The same lamp in the night will yield a white impression. So the precise color impression will depend on the adaptation of the observer. But at the same adaptation, the same chromaticity will lead to the same color impression.

## Metamerism of Light Colors

Equations 1, 2, and 3 show that it is possible that more lights having different SPDs can nevertheless have the same values for the tristimulus values X,Y,Z and therefore the same chromaticity. Such lights are called metameric; they have different spectral power distributions but give the same color impression. It should be noted that this metameric character applies within the boundaries of the CIE XYZ system.

## Metamerism of Object Colors

The spectral composition of a color stimulus of an object color can be described by:
$${\mathrm{E}}_{\mathrm{R}}=\upbeta \left(\uplambda \right)*\mathrm{E}\left(\uplambda \right)$$
(6)
with
• E(λ): spectral power distribution of the light that illuminates

• β(λ): spectral reflectance of the object (fluorescent objects will be neglected here)

The tristimulus values of this color stimulus are:
$$\mathrm{X}=\sum \upbeta \left(\uplambda \right)*\mathrm{E}\left(\uplambda \right).\mathrm{x}\left(\uplambda \right)$$
(7)
$$\mathrm{Y}=\sum \upbeta \left(\uplambda \right)*\mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)$$
(8)
$$\mathrm{Z}=\sum \upbeta \left(\uplambda \right)*\mathrm{E}\left(\uplambda \right). \mathrm{z}\left(\uplambda \right)$$
(9)
Suppose the amount of light on the object is decreased with a factor 10, so E(λ) = 0.1E(λ). From experience, it is known that the human visual system adapts in such a way that the color impressions stay the same.

Now suppose that the spectral reflectance is reduced with a factor 10, so β(λ) = 0.1 β(λ). Again from experience, it is known that the color impression will change: a white will turn into a gray, an orange color will turn brownish, etc.

How to account for these two observations? This can be done by defining the tristimulus values of object colors by the following equations:
$$\mathrm{X}=100*\sum \upbeta \left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)/\sum \mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)$$
(10)
$$\mathrm{Y}=100*\sum \upbeta \left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)/\sum \mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)$$
(11)
$$\mathrm{Z}=100*\sum \upbeta \left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{z}\left(\uplambda \right)/\sum \mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)$$
(12)
By defining the values like this, they have become invariant for multiplication of E(λ) with a constant factor, so invariant for the light level.

At the same time, multiplication of β(λ) with a constant factor will lead to accordingly changes in the tristimulus values of the object color.

Another way of doing this is by normalizing E(λ) in such a way that
$$\sum \mathrm{E}\left(\uplambda \right). \mathrm{y}\left(\uplambda \right)=100$$
(13)
Normalizing E(λ) like this enables the use of Eqs. 7, 8, and 9 to calculate the tristimulus values.

For light sources, the light color can be defined with only 2 coordinates, the chromaticity x, y.

For color stimuli of object colors, however, 3 coordinates are necessary. These can be X, Y, Z or x, y, Y or any other three-dimensional system.

Object colors will be metameric if the three tristimulus values of the colors will be equal.

Suppose two object colors have spectral reflectances β1(λ) and β2 (λ). These object colors will be metameric under a light E(λ), normalized like in Eq. 13, if
$${\mathrm{X}}_1=\sum {\upbeta}_1\left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)=\sum {\upbeta}_2\left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)={\mathrm{X}}_2$$
$${\mathrm{Y}}_1=\sum {\upbeta}_1\left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)=\sum {\upbeta}_2\left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)={\mathrm{Y}}_2$$
$${\mathrm{Z}}_1=\sum {\upbeta}_1\left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)=\sum {\upbeta}_2\left(\uplambda \right)* \mathrm{E}\left(\uplambda \right). \mathrm{x}\left(\uplambda \right)={\mathrm{Z}}_2$$
These object colors are metameric under the light with spectral power distribution E(λ). It is clear that the metameric character depends on the light source. In general, with another light with a different SPD, the relations X1 = X2, Y1 = Y2, and Z1 = Z2 will no longer hold. This means that colors metameric under one light source, i.e., giving the same color impression, can look rather different under a light source with a different SPD. This is illustrated in Fig. 3ac. Two brown textiles with different reflectances will have the same color appearance in daylight, whereas in a specific fluorescent light, the color appearance will be different. In Fig. 3b, c, the reflectances of the colors are depicted together with the spectral power distributions of daylight and fluorescent light.

In the paint industry, metameric object colors are very well known. In this industry, a frequently occurring problem is to reproduce a color with other pigments than the ones used for the original color. In such cases, usually samples are sought that are metameric under daylight as well as incandescent light.

## References

1. 1.
ISO 11664-1:2007(E)/CIE S 014-1/E:2006: Joint ISO/CIE Standard: Colorimetry Part 1. CIE Standard Colorimetric ObserversGoogle Scholar