## 2 Materials and Methods
### Property Analysis
#### 2.1.1 Analysis of Chromaticity and Luminance  
We, as humans, are easily able to differentiate shadows from nonshadows according to the sensation perceived directly from digital images, due to the two classes of receptors in our eyes: cones and rods [34]. However, for the same application in detecting shadows, computer programs usually encounter various problems, such as illumination variation, atmospheric effect, and boundary ambiguity [11]. Additionally, in automatic shadow detection applications with computer programs, luminance and chromaticity are powerful descriptors for color images, including multispectral remote sensing images. Generally, chromaticity consists of hue and saturation, and luminance is also called intensity. In particular, hue is a significant attribute associated with the dominant wavelength in the mixture of light waves, which is taken together with saturation as the chromaticity, in which saturation refers to the relative purity of a certain color. Luminance is the most powerful descriptorfor color images [18; 34]. Regarding the powerful performance of chromaticity and luminance in describing features of color images, we were inspired to determine whether we could specifically employ characteristics of shadows within multispectral remote sensing images in terms of chromaticity and luminance. Consequently, we explored properties of shadow pixels in comparison with those of the corresponding nonshadow pixels in the HSV invariant color space, which performs well in separating chromaticity and luminance composites. Theoretically, according to the [PERSON]'s illumination model [35] and the imaging model of [PERSON] et al. [26], the light illuminating a small patch of a certain surface includes three parts: the ambient part, the diffusion part and the specular part. In most practical applications, the specular part can be neglected, because land cover in forests and urban areas is usually matte and dull. With respect to the ambient and diffusion parts of the incident light, nonshadow areas are always lighted by both the ambient and the diffusion part; however, shadow areas are almost always only illuminated by the ambient part rather than both. Therefore, shadow regions are able to be differentiated from nonshadows in accordance with whether or not the diffusion part of incident light exists. Accordingly, reflectivity of the diffusion part of the incident light expresses the difference between shadow and nonshadow regions, and the difference is [32; 36]:  
\[C_{d}=m_{d}\int_{\lambda}f_{c}(\lambda)e(\lambda)c_{d}(\lambda)d\lambda \tag{1}\]  
where \(C_{d}\in\{R_{d},G_{d},B_{d}\}\) provides the red, green, and blue sensor response to the diffusion part of the incident light, \(m_{d}\) is independent of the wavelength \(\lambda\) and only depends on the geometry information, \(f_{c}(\lambda)\in\{f_{R}(\lambda),f_{G}(\lambda),f_{B}(\lambda)\}\) denotes the spectral sensitivity in function of wavelength \(\lambda\), \(e(\lambda)\) is the quantity of incident light, and \(c_{d}(\lambda)\in\{R_{d}(\lambda),G_{d}(\lambda),B_{d}(\lambda)\}\) is the surface albedo in terms of the red, green, and blue bands, respectively. On the other hand, in remote sensing applications, the sun is always considered to be the sole incident light source, and its illumination can be regarded as white illumination in which the quantity of incident light \(e(\lambda)\) equals to a certain constant that is no longer related to the wavelength \(\lambda\) as before. Under this situation, the integrated white condition [32; 37] holds, as shown in Equation (2):  
\[\int_{\lambda}f_{c}(\lambda)d\lambda=k \tag{2}\]  
where \(k\) is approximately equal to a certain constant. According to electromagnetic wave theory, the surface albedo is positively proportional to the wavelength \(\lambda\), namely the surface albedo \(c_{d}(\lambda)\) of the near infrared (NIR) is larger than that of the red (R), the surface albedo of R is larger than that of the green (G), and the surface albedo of G is larger than that of the blue (B).