## 4 Numerical Results  
Numerical results were performed to evaluate the presented system. For the U-Net network we applied the parameters listed in Table 1.  
The images used in the learning process were manually marked by an expert from the Military Institute of Medicine in Warsaw, Poland. In total, 90 cases were used for preparation of the model's parameters for the learning data and 48 cases were analyzed in the testing process. Each case contained between 32 and 210 scans. A total of 1692 CT images were tested. The tested images differed from each other in size, location, and shape of the kidney. The images differ in sharpness and pixel intensity levels. Numerical experiments were performed to evaluate the quality of the complete system described in Section 3.4. Additionally, the influence of the \(Sm\) initial detection on the kidney mask detection was examined. Table 2 presents the results grouped into 16 positions. Each group contains three cases from 70 to 400 slices. To determine the F1-score of the presented system, the following measure of F1-score (\(F1\)) was used according to the Formula (4). \[F1=\ \frac{2*TP}{2*TP+FP+FN}\cdot 100\% \tag{4}\]  
According to the above formula:  
* \(Tp\)--true-positive specifies the number of pixels classified as a kidney by both the human expert and the system. * \(Fp\)--false-positive specifies the number of pixels classified as kidney by the system, but at the same time classified as background by the expert. * \(Fn\)--false-negative specifies the number of pixels classified as background by the system and at the same time classified as kidney by the expert. For each group, standard deviation coefficients (\(SD\)) and a confidence interval (\(CI\)) of 95% were calculated according to Formulas (5) and (6):  
\[SD=\sqrt{\frac{\sum(x_{i}-\mu)^{2}}{N}} \tag{5}\]  
where \(x_{i}\) is F1-score for each group, \(\mu\)--mean value and \(N\)--group size. \[CI=\left(\mu-t_{\alpha}*\frac{SD}{\sqrt{N}};\mu+t_{\alpha}*\frac{SD}{\sqrt{N}}\right) \tag{6}\]  
where \(t_{\alpha}\) is a value of Student's t distribution with an assumed \(\alpha\) value of 5%. We performed cross-validation tests, which allowed us to assess the F1-score for each group. In the table, each row corresponds to the group that participated in the test, while the other data belonged to the learning data. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & \\ \hline Learning algorithm & The stochastic gradient descent with momentum \\ Minimal batch size & 256 \\ Gradient threshold & 0.05 \\ L2 regularization & 0.0002 \\ Momentum & 0.92 \\ No of epoch & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The following parameters were used in the U-Net network learning process. According to Table 2, the average F1-score of kidney identification is \(84.14\%\pm 10.17\%\) for the system without U-Net support. In this test, \(Sm\) was used as a square with side 200 px. After U-Net-assisted preprocessing, F1-score was increased to \(89.30\%\pm 9.42\%\). We also performed a paired-sample \(t\)-test that confirms that the data differs significantly. The detailed results of returned the \(p\)-value, for each group is visible in Table 2. Moreover, the possibility of kidneys detection based on the U-Net network, not supported by morphological processing operations, was investigated. Table 3 shows the kidney diagnosis results in the CT images using the three selected U-Net thresholds \(T\): 20, 32, 50 and result using [PERSON]'s thresholding. The analysis of Table 3 shows that it is very difficult to choose one optimal threshold value for the binarization of the U-Net result. The best results were obtained for the threshold of 35. The average F1-score, however, is lower than the F1-score obtained using the hybrid system described in Section 3.4, where results were calculated using U-Net predictions, morphological operations, and extended-maxima transform. In Figures 8 and 9 a visualization of selected slices for two cases is presented.