## 2 Materials and Methods
### Property Analysis
#### 2.1.2 Analysis of Multispectral Bands  
In addition to the study of shadow characteristics with respect to the chromaticity (specifically, the hue component) and the luminance (namely, the intensity component) in the HSV invariant color space, we were also inspired by the fact that land cover, especially many kinds of vegetation, present various reflectivity to different wavelengths according to electromagnetic wave theory [38]. Based on this inspiration, we studied important differences between shadow and vegetation areas in terms of the multispectral reflectivity in multispectral bands, namely, visible bands (red, green, and blue bands) and the near infrared band. According to electromagnetic wave theory, when electromagnetic waves with different wavelengths are incident to the same object, the particles composing the object show different reflectivity characteristics to these incident electromagnetic waves of different wavelengths [38; 39]. Consequently, various reflectivity values are achieved, meaning reflectivity values of different land cover types vary in accordance with the wavelengths of the incident electromagnetic waves. Specifically, the visible bands consist of electromagnetic waves with wavelengths ranging around 445 to 749 nm (i.e., ranges of 445 to 517 nm, 507 to 586 nm, and 626 to 696 nm, for blue, green, and red bands, respectively), and the near infrared band includes electromagnetic waves with wavelengths in the  
Figure 1: Sampling curves regarding the difference values between the hue component and the intensity component in both shadow and nonshadow regions. range of about 765 to 1039 nm [40]. Therefore, reflectivity values of a certain land cover usually vary in the red, green, blue and near infrared bands. Additionally, both components and structure of land cover are key factors in the reflectivity for a certain wavelength electromagnetic wave. Due to the components and the special cell structure of vegetation, the reflectivity of the near infrared band is far higher than that of visible bands [38]. Nevertheless, the reflectivity of the red band is obviously the lowest, in comparison with that of the near infrared, green, and blue bands [14]. An unequal relation is presented in Equation (12) for the special reflectivity property of vegetation in terms of the near infrared band and the red band, as described above:  
\[NIR_{nonshadow-vegetation}>R_{nonshadow-vegetation} \tag{12}\]  
where \(NIR_{nonshadow-vegetation}\) and \(R_{nonshadow-vegetation}\) are reflectivity values of the near infrared band and the red band, respectively, for vegetation cover in nonshadow regions. According to the analysis of the difference of illumination on shadow and nonshadow regions in Equation (3), an unequal relation can be drawn, as shown in Equation (13), which describes the reflectivity difference of the near infrared band and the red band between shadow regions and vegetation cover in nonshadow regions. \[NIR_{vegetation-in-nonshadow}-NIR_{shadow}>R_{vegetation-in-nonshadow}-R_{shadow} \tag{13}\]  
where \(NIR_{shadow}\) and \(R_{shadow}\) are reflectivity values of the near infrared band and the red band for shadow regions, respectively. Additionally, according to Equation (10), the reflectivity values of both the near infrared band and the red band decrease dramatically for shadow regions compared with that of vegetation cover types in nonshadow regions, as shown in Equation (14). \[\left\{ \begin{array}{l}NIR_{vegetation-in-nonshadow}>NIR_{shadow}\\ R_{vegetation-in-nonshadow}>R_{shadow}\end{array} \right. \tag{14}\]  
Based on the above analysis of the reflectivity property of the near infrared band and the red band for both shadow regions and vegetation cover in nonshadow regions, a conclusion can be drawn for the unequal relation of the reflectivity differences between the red band and the near infrared band in shadow regions and vegetation cover in nonshadow regions, as shown in Equation (15).