## 1 Introduction
Many researchers indirectly exploited the functions of multispectral bands by using luminance and chromaticity in various invariant color spaces to detect shadows, including hue-intensity-saturation (HIS), hue-saturation-value (HSV), luma-inphase-quadrature (YIQ), YCbCr and C1C2C3 models. [PERSON] [11] presented an automatic property-based shadow detection approach, which attempted to solve problems caused by cast shadows in color aerial images in complex urban environments. A spectral ratio index (SRI) was developed in this approach to highlight shadow pixels based on the properties of shadow regions compared with nonshadow ones in invariant color spaces, as seen in the hue-intensity-saturation (HIS), hue-saturation-value (HSV), luma-inphase-quadrature (YIQ), and YCbCr invariant color spaces. Then, the final shadow image was obtained by thresholding the index image with the Otsu threshold method [30]. This shadow detection algorithm accurately identifies shadow pixels otherwise bluish and greenish objects present. On the foundation of [PERSON]'s work, [PERSON] et al. [29] proposed a novel successive thresholding scheme (STS), instead of the global thresholding method used in [PERSON]'s approach, for more accurately detecting shadows. The STS-based shadow detection algorithm has comparable shadow detection accuracy to [PERSON]'s SRI-based method. [PERSON] and [PERSON] [12] also presented an extended spectral ratio index (ESRI) approach based on [PERSON]'s work [11], in which the SRI-based method was extended with post-processing procedures specifically in the YIQ invariant color space. This ESRI-based method obtained a better visual sense at the expense of longer computational time than [PERSON]'s SRI-based method without post processing procedures. [PERSON] et al. [18] introduced a normalized saturation-value difference index (NSVDI) in the HSV invariant color space to detect shadows in high-resolution multispectral remote sensing images. The NSVDI-based method generated a rough shadow index image, highlighting shadows by exploiting the properties of higher saturation and lower intensity of shadows compared with those of nonsmadows. The rough shadow index image was then manually thresholded with a certain threshold value to provide a final shadow image. This NSVDI-based method performs well in detecting large shadows in high-resolution multispectral remote sensing images despite omitting some building shadows and small shadows. Additionally, [PERSON] et al. [31] developed a novel method for detecting cast shadow boundaries in high-resolution remote sensing images, in which a C3 index was exploited to highlight shadows. The C3 index is a transformation of red, green and blue bands in the C1C2C3 invariant color space [32]. This C3-based method is able to discriminate shadows from other dark land cover. Based on the work of [PERSON] et al., [PERSON] et al. [20] presented a modified C3 (MC3) index, in which a near infrared band was added. This MC3-based approach improves the shadow detection results in contrast to the C3-based method introduced by [PERSON] et al. [31]. Several researchers have developed new shadow detection indices based on the analysis of the spectral differences between shadow pixels and typical land cover pixels by directly using multispectral bands, including both the visible bands (red, green, and blue bands) and the near infrared band. In 2015, [PERSON] et al. [14] introduced a new shadow detection method combining the spectral reflectivity characteristics of shadow regions in multispectral bands (i.e., the visible bands and the near infrared band), and the typical normalized difference vegetation index (NDVI). In this method, a normalized blue-red index (NBRI) was originally put forward. Then, the difference in the NBRI and the NDVI was calculated as the final shadow intensity (SI).