1. History and Fundamentals
In the early 19th century, it was discovered that the human eye has three types of cells for color perception. It was also known that two lights with different spectra could produce the same color (metamerism). It was hypothesized that each cone cell has a spectral sensitivity corresponding to R, G, and B, or to opponent colors, W/BK, R/G, and Y/B. The spectral sensitivities of the cone cells were also known, and it was established that a color could be matched by combining three primary colors.

Around 1930, Wright and Guild conducted independent visual experiments using R/G/B primaries to obtain color matching functions, which formed the basis of the CIE colorimetric system. Observers looked through a 2° circular aperture, and their task was to adjust the three primaries so that their mixture visually matched the presented visible spectrum. Wavelengths of 435.8 nm, 546.1 nm, and 700 nm were used for this experiment, as shown in Figure 1. These are the curves of the relative intensities of RGB at each wavelength to match monochromatic light. In 1931, the CIE adopted these results as a standard, with the RGB color matching functions denoted as `r(λ)`, `g(λ)`, and `b(λ)`. In 1931, the CIE transformed the RGB color matching functions into XYZ functions, thereby eliminating the negative values in the `g(λ)` function, making it equal to the 1924 CIE spectral luminous efficiency function $V(\lambda)$. This was a simple linear transformation from the RGB color matching functions, and the final result is shown in Figure 2.

Figure 2 shows the so-called CIE 1931 XYZ color matching functions, denoted as `x(λ)`, `y(λ)`, and `z(λ)`. There are two important assumptions in these color matching functions: first, no external interference, and only a narrow field of view (2°) can be used. Second, the additivity of light (Grassmann's laws) is assumed. The color matching characteristics of an ideal observer correspond to the color matching functions, with a 2° field of view and satisfying Grassmann's laws; we call this the CIE 1931 Standard Colorimetric Observer. In practice, such an observer can be applied to fields of view up to 4°. In 1964, the CIE defined a second set of standard color matching functions for a 10° field of view, denoted as `x10(λ)`, `y10(λ)`, and `z10(λ)`, as a supplement to the 1931 standard observer. This is called the CIE 1964 Supplementary Standard Colorimetric Observer, which can be used for fields of view greater than 4°.

2. Tristimulus Values

By using the color matching functions, the color stimulus of any spectral power distribution can be represented by the following three values:

$$\begin{align*} X &= k \int_\lambda \Phi(\lambda) \bar{x}(\lambda) \,\text{d}\lambda \\ Y &= k \int_\lambda \Phi(\lambda) \bar{y}(\lambda) \,\text{d}\lambda \\ Z &= k \int_\lambda \Phi(\lambda) \bar{z}(\lambda) \,\text{d}\lambda \end{align*}$$ $\Phi(\lambda)$ is the spectral distribution of the light, and $k$ is a constant. These integral values are the tristimulus values. For light sources and displays, $\Phi(\lambda)$ is given in absolute radiometric quantities, such as spectral irradiance and spectral radiant intensity. If $\Phi(\lambda)$ is in absolute units, $k = 683$ lm/W, and $Y$ is an absolute photometric value such as illuminance or luminance. For object colors, $\Phi(\lambda)$ is expressed as:

$$\begin{equation} \Phi(\lambda) = E(\lambda) \cdot R(\lambda) \tag{2} \end{equation}$$ $R(\lambda)$ is the spectral reflectance or radiance factor of the object, $E(\lambda)$ is the relative spectral irradiance, and:

$$\begin{equation} k = 100 \int_\lambda E(\lambda) \bar{y}(\lambda) \,\text{d}\lambda \tag{3} \end{equation}$$ The actual integration can be computed numerically using spectral data.\\ **Chromaticity Diagram**\\ By projecting the tristimulus values onto a unit plane ($X+Y+Z=1$), they can be interpreted on a two-dimensional plane. Such a unit plane is called a chromaticity diagram. A color can be specified by its chromaticity coordinates ($x, y$):

$$\begin{align} x &= \frac{X}{X+Y+Z}; \notag \\ y &= \frac{Y}{X+Y+Z} \tag{4} \end{align}$$ The use of chromaticity coordinates ($x, y$) is referred to as the CIE 1931 chromaticity diagram or CIE ($x, y$) chromaticity diagram. The ($x, y$) chromaticity diagram is highly non-uniform in terms of color. The minimum perceptible color differences in the CIE ($x, y$) diagram are known as MacAdam ellipses, as shown in Figure 3(a). To improve this, in 1960, the CIE defined an improved diagram—the CIE 1960 ($u, v$) chromaticity diagram (now obsolete). In 1976, a further improved chromaticity diagram, the CIE 1976 Uniform Chromaticity Scale (UCS) diagram, was introduced, in which the coordinates ($u', v'$) are given by:

$$\begin{align} u' &= \frac{4X}{X+15Y+3Z}; \\ v' &= \frac{9Y}{X+15Y+3Z} \tag{5} \end{align}$$ The 1976 chromaticity diagram is significantly more uniform than the 1931 diagram; however, it is still far from perfect.\\ **3. Uniform Color Spaces and Color Difference Formulas**\\ The three attributes of color are hue, chroma (saturation), and lightness. In the chromaticity diagrams mentioned above, lightness is absent. Hue and chroma are non-linear. To obtain accurate object colors and color differences, the CIE recommended three-dimensional uniform color spaces—CIELAB and CIELUV. They are known as the CIE 1976 ($L^*a^*b^*$) color space or CIELAB color space, and the CIE 1976 ($L^*u^*v^*$) color space, respectively. They have a structure similar to the Munsell color solid. In imaging applications, $L^*$ represents lightness, and ($a^*, b^*$) represent color as shown in Figure 4. The coordinates ($L^*a^*b^*$) are calculated from ($X, Y, Z$). Therefore, CIELAB has a correction function for the color adaptation white point, and is used for object colors and displays.

$$\begin{equation} \Delta E'_{\text{ab}} = \left[(\Delta L^*)^2 + (\Delta a^*)^2 + (\Delta b^*)^2\right]^{1/2} \tag{6} \end{equation}$$ The color difference formula in the CIELAB space is calculated using the Euclidean distance between points in this three-dimensional space. $$\begin{equation} C'_{\text{ab}} = (a'^2 + b'^2)^{1/2} \tag{7} \end{equation}$$ $$\begin{equation} h_{\text{ab}} = \arctan(b'/a') \tag{8} \end{equation}$$ This formula is called the CIE 1976 ($L^*a^*b^*$) color difference formula. Chroma $C^*_{ab}$ and hue $h_{ab}$ are calculated from the following formulas:

The CIELUV space is defined in a similar manner. The coordinates ($L^*u^*v^*$) are calculated from $Y$ and ($u', v'$).

Although the color difference $\Delta E^*_{ab}$ is widely used, its chroma scale is non-linear. For more accurate color difference evaluation, the CIE recommended a modification to the industrial color difference formula in 1994, known as the CIE94 formula. The color difference $\Delta E^*_{94}$ is calculated using $\Delta L^*$, $\Delta C^*_{ab}$, and $\Delta H^*_{ab}$ from the CIELAB formula. Another improved formula, the CMC color difference formula, is mainly used in the textile industry. Further improved color difference formulas have been studied by the CIE (TC1-55).

4. Correlated Color Temperature

The color of a light source is represented using chromaticity coordinates ($x, y$) or ($u', v'$). However, it is difficult to immediately perceive the specified color from these values. To quickly understand the color, the color of light is expressed using the correlated color temperature (CCT), in units of Kelvin (K). The definition of CCT is based on Planck's black-body radiation. CCT can be calculated from the chromaticity diagram. Due to long-standing tradition, the CIE still recommends using the 1960 chromaticity diagram to calculate CCT. From the ($u', v'$) coordinates, ($u, v$) can be obtained by $u = u'$ and $v = 2v'/3$. In the ($u, v$) diagram, the point on the Planckian locus or the point closest to it is found. The CCT is the color temperature of this point.

5. Color Rendering Index

In applications such as lighting, it is important to know how well a specified illumination can reproduce the colors of the illuminated scene. The CIE first defined the color rendering index (CRI) in 1965. After minor revisions, the CIE recommendations have been widely adopted by lighting manufacturers. The calculation of the color rendering index first involves calculating the color difference $\Delta E$ (in the 1964 WUV uniform color space, which is now obsolete). The color difference is calculated based on 14 Munsell samples, comparing the differences when illuminated by a reference light source and when illuminated by the given light source. This process includes the von Kries chromatic adaptation transform. The special color rendering index $R_i$ for each color sample is calculated as:
$$ \begin{equation} R_t = 100 - 4.6 \Delta E' \tag{9} \end{equation} $$ This yields the special color rendering index for each specific color. The general color rendering index $R_a$ is the average of the first eight color samples (moderate saturation). The maximum value is 100, and $R_a$ provides a scale that correlates well with the visual impression of color rendering. If a lamp has an $R_a$ value over 80, it can be considered for interior lighting; if $R_a$ is greater than 90, it can be used for visual inspection.

6. Standard Illuminants

The color of an object changes with the spectrum of the illuminating light source. Therefore, the light source used must be specified for an object's color metrics. For this purpose, the CIE and ISO have standardized illuminants. CIE standard illuminant A (tungsten filament lamp, color temperature 2856 K) and CIE standard illuminant D65 (average daylight, color temperature 6500 K) are the two most fundamental standard illuminants. These illuminants can be used in any application. However, other daylight illuminants have been widely used in certain fields, and the CIE has also defined D50, D55, and D75. Illuminant B, which approximates direct sunlight with a color temperature of 4900 K, and Illuminant C, with a color temperature of 6800 K, are no longer recommended.

7. Measurement of Object Colors

Terminology for Reflectance Measurement

In most cases, object color is determined by spectral reflection. Terminology for reflectance measurement is often confused and misused by the imaging community. A perfect diffuse reflector refers to an ideal isotropic reflector with a reflectance of 1.

Illumination and Viewing Conditions

When specifying reflectance colorimetry, geometry is a very important condition. For object colorimetry, the CIE recommends using one of four standard geometries: 45°/normal (45/0), normal/45° (0/45), diffuse/normal (d/0), and normal/diffuse (0/d).

Reflectance Standards

All spectrophotometers must be calibrated using a white reflectance standard plate. Spectral radiance factor standards are used for 45/0, 0/45, and d/0 geometries, while diffuse spectral reflectance standards are used for the 0/d geometry. Highly compressed diffuse white materials or sintered polytetrafluoroethylene (PTFE) are used for these standards. Because absolute measurement of radiance or reflectance factor is very difficult, calibration standards are provided by national laboratories. Moreover, industrial measurements are generally traceable to these standards. Since a perfect diffuse reflectance plate does not exist, absolute measurements of the radiance factor are used to calibrate the radiance factor. The radiance factor is obtained by multiplying the reflectance factor by the constant $\pi$. The reflectance characteristics of all diffuse materials are highly sensitive to the illumination and viewing angles. Figure 5 shows an example of the measured spectral radiance factor of a PTFE sample.

Instruments for Object Color Measurement

Spectrophotometers are generally used for the measurement of object colors. These instruments measure the test samples under given conditions and are calibrated using reference standards. Therefore, the measurement uncertainty is primarily determined by the uncertainty of the reference standard. Due to the characteristics of the spectrophotometer, this uncertainty is further increased. The main factors affecting uncertainty include wavelength error, detector non-linearity, stray light, bandwidth, illumination and viewing conditions, and measurement noise. If the bandwidth is greater than 10 nm, the impact is very severe.

For the measurement of small color differences, tristimulus colorimeters are mainly used, primarily due to their fast speed and low cost. Their uncertainty is limited, mainly because of the mismatch between the CIE illuminant luminance and the spectral response of the detector. Therefore, they are not suitable for absolute color measurement over a wide range of colors.

Measurement of Light Source Colors

For the measurement of light sources, such as lamps, LEDs, and displays, spectroradiometers and tristimulus colorimeters are widely used.

Instruments for Light Source Color Measurement

Spectroradiometers are mainly used to measure spectral irradiance or spectral radiant intensity. The former uses a diffuser or a small integrating sphere as input, while the latter uses imaging optics. Luminaires are generally measured for spectral irradiance, while displays are generally measured for spectral radiance to obtain color. There are two types of spectroradiometers: mechanical scanning and semiconductor array types. Generally speaking, the former is more accurate but slower, while the latter is faster but less accurate. Spectroradiometers are calibrated according to international standards for irradiance or radiance. Therefore, their measurement uncertainty primarily depends on the reference standard. Spectroradiometers also have many other uncertainty factors, including wavelength error, detector non-linearity, stray light from the monochromator, bandwidth, and measurement noise. The variation in error depends on the measurement of the light source spectrum. Even if an instrument has a very small error when testing a tungsten lamp, the error may be much larger when measuring the color of other light sources. For example, a semiconductor array spectroradiometer may have an $x, y$ error of 0.005 when measuring different displays, but when measuring the CIE standard illuminant A, the corresponding error reaches 0.001. Therefore, for applications with high accuracy requirements, the instrument should be calibrated for different color measurements.

Tristimulus instruments are also widely used for color measurement of luminaires and displays. Although they have the advantages of low cost and fast speed, their uncertainty is higher than that of spectroradiometers because spectral mismatch errors are unavoidable.