## 1 Introduction
In this paper, we first construct a logarithmic shadow index (_LSI_) and subsequently develop an _LSI_ shadow detection approach for shadow detection of HSR multispectral satellite remote sensing images, particularly for further settling problems of high small shadow omission and typical nonshadow misclassification (like bluish and greenish dark nonshadow misclassification, as well as large dark nonshadow misclassification). Our presented _LSI_ shadow detection algorithm employs special properties of shadow, namely, the dramatical decrease of _NIR_ component, the higher hue value and the lower intensity value, by further studying properties of shadow in terms of both multispectral band components (mainly including visible bands and _NIR_ band) and invariant color components in various invariant color spaces (i.e., HIS, HSV, CIELCh, YCbCr, YIQ) compared with the corresponding nonshadow. Based on the proposed _LSI_, we acquire the shadow image by firstly segmenting the shadow index image automatically with an optimal threshold determined with the NVEM thresholding method [27] and subsequently optimizing the initial shadow image with a certain morphological operation. For verifying the shadow detection performance of our proposed _LSI_ algorithm, comparative experiments are carried out with many images from WorldView-3 and WorldView-2 acquired at different time and sites, and shadow detection performance is particularly assessed both qualitatively and quantitatively against several standard shadow detection algorithms (i.e., MC3 [18], NSVDI [25], LSRI [24], SDI [26] and SRI [20]) with two WorldView-3 test images of Tripoli, Libya. The rest proceeds as follows. The _LSI_ shadow detection is detailly developed step by step in Section 2. Comparative experiments and performance assessments are conducted both qualitatively  
\[S=1-\frac{3}{(R+G+B)}\text{min}(R,G,B) \tag{2}\]  
\[H=\left\{ \begin{array}{ll}\theta,&\text{if }B\leq G\\ 360^{\circ}-\theta,&\text{if }B>G\end{array} \right. \tag{3}\]  
where \(\theta\) is obtained with the following equation. \[\theta=\cos^{-1}\left\{\frac{\frac{1}{2}[(R-G)+(R-B)]}{\sqrt{(R-G)^{2}+(R-B)( G-B)}}\right\} \tag{4}\]  
Similarly, the HIS color space describes the color image in terms of intensity (\(I\)), saturation (\(S\)) and hue (\(H\)) components, in which saturation and hue components together constitute the chromaticity term and intensity is also known as luminance [29]. The HIS color space is usually computed from the RGB color space with Equations (5)-(7) [20]:  
\[\left[ \begin{array}{c}I\\ V_{1}\\ V_{2}\end{array} \right]=\left[ \begin{array}{ccc}\frac{1}{3}&\frac{1}{3}& \frac{1}{3}\\ -\frac{\sqrt{6}}{6}&-\frac{\sqrt{6}}{6}&\frac{\sqrt{6}}{3}\\ \frac{\sqrt{6}}{6}&-\frac{\sqrt{6}}{3}&0\end{array} \right]\left[ \begin{array}[ ]{c}R\\ G\\ B\end{array} \right] \tag{5}\]  
\[S=\sqrt{V_{1}^{2}+V_{2}^{2}} \tag{6}\]  
\[H=\tan^{-1}\left(\frac{V_{2}}{V_{1}}\right)\text{if }V_{1}\
eq 0 \tag{7}\]  
where \(H\) is undefined under the condition of \(V_{1}=0\). In addition, the YCbCr color space is often employed in JPEG, MPEG and H2.63 [20; 30]. Equation (8) describes the linear relations between components in the YCbCr color space and those in the RGB color space. \[\left[ \begin{array}{c}Y\\ C_{b}\\ C_{r}\end{array} \right]=\left[ \begin{array}{ccc}0.257&0.504&0.098\\ -0.148&-0.291&0.439\\ 0.439&-0.368&-0.071\end{array} \right]\left[ \begin{array}{c}R\\ G\\ B\end{array} \right]+\left[ \begin{array}{c}16\\ 128\\ 128\end{array} \right] \tag{8}\]  
Besides, the YIQ color space is regarded as a regulation widely utilized in the National Television Standards Commission (NTSC) [31]. During the color image description, \(Y\) component is in proportion to luminance used in gamma correction, and \(I\) and \(Q\) components together represent chromaticity, namely, saturation and hue components [20; 29]. The YIQ color space is obtained with Equation (9) in terms of the RGB color space. \[\left[ \begin{array}{c}Y\\ I\\ Q\end{array} \right]=\left[