## 5 Experiment Results and Analysis
### Quantitative Results  
The quantitative results are represented in this section. Quantitative analysis shows the average precision and recall rates, along with F-measures. Precision and recall rates can be obtained by using Equations (10) and (11). The results show that the proposed shadow remover accomplished the highest precision rate, at 0.92. The CBM [30], GBM [24], and texture-based method (TBM) [31] yielded precision rates of 0.85, 0.87 and 0.89, respectively. In addition, we computed the F-measure value which balanced measurements between the means of the precision and recall rates. The results of the quantitative comparison of the three algorithms are shown in Figure 6 and Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & **Color-Based** & **Gradient-Based** & **Texture-Based** & \multirow{2}{*}{**Ours**} \\  & **Method (CBM)** & **Method (GBM)** & **Method (TBM)** & \\ \hline Precision & 0.85 & 0.87 & 0.89 & 0.92 \\ Recall & 0.73 & 0.76 & 0.78 & 0.79 \\ F-Measure & 0.82 & 0.85 & 0.88 & 0.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantitative analysis of four shadow remover methods performed using the CAVIAR 10k dataset. Precision and recall rates can be obtained as follows:  
\[precision=\frac{TDO\cap GT}{ADO} \tag{10}\]  
\[recall=\frac{TDO\cap GT}{GT} \tag{11}\]  
where TDO denotes truly detected shadow regions, ADO denotes all detected objects, and GT denotes a manually-labelled ground trust value. In addition, we computed the F-measure value which balanced measurements between the mean of precision and recall rates. A higher F-measure means a higher performance, and it is defined as follows:  
\[F_{\beta}=\frac{(1+\beta^{2})precision\times recall}{\beta^{2}\times precision+ recall} \tag{12}\]  
In that regard, \(\beta^{2}\) = 0.3 was suggested by many shadow detection approaches, to raise the importance of the precision value. The point for weighting precision more than recall is that recall can be easily obtained by setting the entire region to the foreground. We used a fixed threshold that changed from 0 to 255. On each threshold, pair of precision and recall scores were computed and were lastly combined to form a precision-recall (PR) curve. The resulting PR curve can be scored by its maximal F-measure, which is a good summary of the shadow region detection performance when using a fixed threshold. In addition to the PR and F-measure, we can also describe the false positive rate (FPR) and true positive rate (TPR) when binarizing the foreground mask with a series of fixed thresholds. The receiver operating characteristics (ROC) curve is a plot of TPR against FPR and is obtained by sliding the threshold. TPR and FPR can be defined as follows:  
\[TPR=\frac{|BM\cap GT|}{|GT|} \tag{13}\]  
\[FPR=\frac{\overline{BM}\cap\overline{GT}|}{\overline{GT}|} \tag{14}\]  
where \(\overline{BM}\) and \(\overline{GT}\) denote the opposites of the binary mask, \(BM\), and ground-truth, \(GT\), respectively. Figure 7 shows the PR and ROC curves of the proposed method, along with those of alternative shadow removal algorithms. Moreover, we contrasted our strategy and other shadow expulsion frameworks (physical-based and gradient-based), to determine the speed of the execution of the method. In most cases, our methodology  
Figure 7: Quantitative comparisons using precision-recall (PR), (left column) and operating characteristic (ROC), (right column) curves of the proposed method, and those produced by various approaches on the CAVIAR 10k dataset. and the GB method indicate relatively similar times for handling of results. In some cases, our shadow detector takes somewhat more time with different calculations, as shown in Table 2. The reason for this is that our method requires additional stages to improve the input current frame and compute the geometry property points for every pixel. In that regard, we successfully solved moving object and background surface similarity problems in inside environments.