## 4 Experimental Results  
We compared the results of our method to other reference methods on several high-resolution remote sensing images with ground object shadows and cloud shadows. These results are analyzed in the following sections.
method (GMC) of [PERSON] [22], the Histogram Matching method (HMT) of [PERSON] [15] were accomplished in comparison. Furtherly, because our method is to improve the self-adaptive ability of the compensation methods according to different shade condition, two recent methodologies which had similar research goal were compared in this paper. They were respectively the Corresponding Region compensation Method (CRM) of [PERSON] [27] and the Oriented Object Polynomial Removal method (OOPR) of [PERSON] [29]. The difference between these methods and our method was discussed in detail.  
In our method, \(b=0.6\), \(c=0.45\) were used to calculate \(r_{0}\) and \(r_{1}\); \(n=10\), \(u=0.6\) were used to accomplish multilevel combinations; and \(\alpha\) and \(\beta\) were calculated automatically. LCC used the average \(\alpha\) of our method, and OWC was set by the same \(r_{0}\) and \(r_{1}\) as our method. In GMC, \(\gamma=1.1\). In HMT, the non-shadow regions were the same as that of our method. By adopting similar parameter values, the comparison was more effective in reflecting the difference between them. To further analyze the compensation results in quantitative ways, Table 1 shows the original shadow brightness, the target value in the non-shadow area, the compensated value, and the compensation quality index of Figures 4-10.  
\(B\) is the mean value of component \(I\), as defined in Equation (17) and it reflects the brightness level of the measured area. Assume \(S(x,y)\) is the local area in the image to be analyzed and \(N_{S}\) is the pixel number in this area. Average brightness \(B\) reflects the degree of lightness and darkness of the area [1]. \(T\), as defined in Equation (18) represents the average gradient (contrast) and reflects the amount of image detail and clearness of an image.  
\[B=\frac{1}{N_{S}}\sum_{(i,j)\in S}I(i,j) \tag{17}\]  
\[T=\frac{1}{N_{S}}\sum_{(i,j)\in S}\sqrt{\frac{1}{2}\big{[}\big{[}I(i+1,j+1)-I( i,j)\big{]}^{2}+\big{[}I(i+1,j)-I(i,j+1)\big{]}^{2}\big{]}} \tag{18}\]  
where \(I(i,j)\), \(I(i+1,j)\), \(I(i,j+1)\), and \(I(i+1,j+1)\) represent the \(I\) values in the pixels \((i,j)\), \((i+1,j)\), \((i,j+1)\), and \((i+1,j+1)\), respectively. \(N_{S}\) is the number of pixels in the shadow area.  
Because the feature value in the non-shadow area can be seen as the approximate target value, \((\Delta B)^{2}\) and \((\Delta T)^{2}\), as defined in Equations (19) and (20) which are normalized by the non-shadow value, can represent the difference between the compensation results and the target values to evaluate the quality of the compensation results and facilitate the analysis of the effects of the model parameters. Then, in order to evaluate the total bias from the non-shadow regions, \(Q^{B+T}\) defined in Equation (21) can be calculated and seen as the total compensation quality. In general, lower values of \(Q^{B+T}\) indicate compensation results that were closer to the non-shadow area.  
\[(\Delta B)^{2}=\left(\frac{B-B_{\text{NSD}}}{B+B_{\text{NSD}}}\right)^{2} \tag{19}\]  
\[(\Delta T)^{2}=\left(\frac{T-T_{\text{NSD}}}{T+T_{\text{NSD}}}\right)^{2} \tag{20}\]  
\[Q^{B+T}=\left(\Delta B\right)^{2}+\left(\Delta T\right)^{2} \tag{21}\]  
where \(B\) and \(T\) are the compensated brightness average and the average gradient value of the shadow area. \(B_{\text{NSD}}\) and \(T_{\text{NSD}}\) are the brightness average and the average gradient of the non-shadow area. \((\Delta B)^{2}\) and \((\Delta T)^{2}\) are the square of the normalized difference between the compensated value and the non-shadow area in \(B\) and \(T\), respectively. \(Q^{B+T}\) is the total compensation quality.