## 3 Methodology and Accuracy Assessment
### Multispectral Image Transform to CIELab Color System
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. 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'eclairage), 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  
Figure 1: CIELab image Pan sharpening flowchart. Figure 2: CIELab color space [41]. 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:  
\[\left[ \begin{array}{c}\mathrm{X}\\ \mathrm{Y}\\ \mathrm{Z}\end{array} \right]=\left[ \begin{array}{ccc}0.4124564&0.3575761&0.1804 375\\ 0.2126729&0.7151522&0.0721750\\ 0.0193339&0.1191920&0.9503041\end{array} \right]\left[ \begin{array}{c} \mathrm{R}\\ \mathrm{G}\\ \mathrm{B}\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}\mathrm{R}\\ \mathrm{G}\\ \mathrm{B}\end{array} \right]=\left[ \begin{array}{ccc}3.2404542&-1.5371385&-0.4 985314\\ -0.9692660&1.8760108&0.0415560\\ 0.0556434&-0.2040259&1.0572252\end{array} \right]\left[ \begin{array}{c} \mathrm{X}\\ \mathrm{Y}\\ \mathrm{Z}\end{array} \right]. \tag{2}\]  
Lab system is calculated by the following equations [23]:  
\[\mathrm{L} = 116\ \mathrm{F_{Y}}\ -\ 16 \tag{3}\]  
\[\mathrm{a} = 500\ [\mathrm{F_{X}}-\mathrm{F_{Y}}] \tag{4}\]  
\[\mathrm{b} = 200[\mathrm{F_{Y}}-\mathrm{F_{Z}}] \tag{5}\]  
\[\mathrm{Where\ F_{X}}\mathrm{=}\left(\frac{\mathrm{X}}{\mathrm{x_{n}}}\right) ^{\frac{1}{3}}\mathrm{if\ }\left(\frac{\mathrm{X}}{\mathrm{x_{n}}}\right) \mathrm{>}\left(\frac{24}{116}\right)^{3} \tag{6}\]  
\[\mathrm{And\ F_{X}}\mathrm{=}\left(\frac{841}{108}\right)\left(\frac{\mathrm{X }}{\mathrm{x_{n}}}\right) +\frac{16}{116}\ \mathrm{if\ }\left(\frac{\mathrm{X}}{\mathrm{x_{n}}}\right) \leq\left(\frac{24}{116}\right)^{3}, \tag{7}\]  
where \(\mathrm{X_{n}}\) is the tristimulus value of a perfect white object color stimulus, which the light reflected from a perfect diffuser under the chosen illuminant. \(\mathrm{F_{Y}}\) and \(\mathrm{F_{Z}}\) values are calculated in the same way as \(\mathrm{F_{X}}\).