*3.1. Multispectral Image Transform to CIELab Color System*

The RGB color system was designed in such a way that it includes nearly all primary colors and can be comprehended by human vision. Nevertheless, it is a tough task to deal with RGB color due to strong correlation between its components [29]. In this study, a uniform and complete color model, the CIELab color system, was used. In this uniform color system, a variation in the coordinates of the color component provides the same amount of variation in the luminance and saturation components [10]. Besides, this color space is projected to draw human vision, unlike the RGB and CMYK (cyan, magenta, yellow, black) color spaces [38]. The CIELab color system was used in the proposed Pan sharpening

approach in order to reduce the spectral distortion, while maintaining the color perception of human vision [39].

**Figure 1.** CIELab image Pan sharpening flowchart.

The design of the CIELab color system is based on Hering's theory, which indicates that only red (R), green (G), blue (B) and yellow (Y) are unique among the thousands of colors that are used to characterize the hue component [40]. Although, other colors can be produced using these unique colors (for example it is possible to obtain orange by mixing red and yellow), they (R, G, B and Y) can be described only with their own name. R, G, B and Y, with black (B) and white (W), constitutes a color system with six basic color properties and three opponent pairs: R/G, Y/B and B/W. The opponency idea rises from observation upon colors attributes, which proves no color could be characterized using both blue and yellow or red and green together [41]. A blue shade of yellow does not exist. These three opponent pairs are represented in the form of a three-dimensional color space, as illustrated in Figure 2. In this figure, the vertical axis L\* represents the luminance, in which perfect black is represented by 0 and perfect white is represented by 100. The a\* and the b\* are the axes that are perpendicular to luminance indicated chromaticity, and stand for redness/greenness and yellowness/blueness, respectively. Positive values represent redness (for a\* component) and yellowness (for b\* component), whereas greenness and blueness are denoted with negative values.

**Figure 2.** CIELab color space [41].

The L\*, a\* and b\* values are computed using XYZ values. The XYZ system that was based on the RGB color space was presented by the International Commission on Illumination, CIE (Commission international de l'éclairage), in the 1920s and patented in 1931. The difference of RGB and XYZ lies in the light sources. The R, G and B elements are real light sources of known characteristics, whereas the X, Y and Z elements are three theoretical sources, which are selected in a way that all visible colors can be defined as a density of just-positive units of the three primary sources [10].

Occasionally, the colorimetric calculations with the use of color matching functions produce negative lobs. This problem can be solved by transforming the real light sources to these theoretical sources. In this color space, red, green and blue colors are more saturated than any spectral RGB. X, Y and Z components, represent red, green and blue colors respectively. RGB to XYZ and its reverse transformations can be performed by following equations:

$$
\begin{bmatrix} \chi \\ \chi \\ Z \end{bmatrix} = \begin{bmatrix} 0.4124564 & 0.3575761 & 0.1804375 \\ 0.2126729 & 0.7151522 & 0.0721750 \\ 0.0193339 & 0.1191920 & 0.9503041 \end{bmatrix} \begin{bmatrix} \text{R} \\ \text{G} \\ \text{B} \end{bmatrix} \tag{1}
$$

$$
\begin{bmatrix} \text{R} \\ \text{G} \\ \text{B} \end{bmatrix} = \begin{bmatrix} 3.2404542 & -1.5371385 & -0.4985314 \\ -0.9692660 & 1.8760108 & 0.0415560 \\ 0.0556434 & -0.2040259 & 1.0572252 \end{bmatrix} \bullet \begin{bmatrix} \text{X} \\ \text{Y} \\ \text{Z} \end{bmatrix}.\tag{2}
$$

Lab system is calculated by the following equations [23]:

$$\text{L = 116 Fy - 16}\tag{3}$$

$$\mathbf{a} = \mathbf{500} \,\mathrm{[F\_X - F\_Y]} \tag{4}$$

$$\mathbf{b} = \text{200}[\mathbf{F}\_\text{Y} - \mathbf{F}\_\text{Z}] \tag{5}$$

$$\text{Where } \mathbf{F}\_{\mathbf{X}} = (\frac{\chi}{\chi\_{\mathbf{n}}})^{\frac{1}{3}} \text{ if } (\frac{\chi}{\chi\_{\mathbf{n}}}) \succ (\frac{24}{116})^3 \tag{6}$$

$$\text{And } \mathcal{F}\_{\overline{\lambda}} = \left(\frac{841}{108}\right) \frac{\chi}{\chi\_{\text{n}}}\text{)} + \frac{16}{116} \text{ if } \left(\frac{\chi}{\chi\_{\text{n}}}\right) \le \left(\frac{24}{116}\right)^3,\tag{7}$$

where Xn is the tristimulus value of a perfect white object color stimulus, which the light reflected from a perfect diffuser under the chosen illuminant. FY and FZ values are calculated in the same way as FX.
