## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Segmentation**} & \multicolumn{1}{c}{**Train**} & \multicolumn{1}{c}{**BraTS 2015 LGG**} & \multicolumn{1}{c}{**BraTS 2015 HGG**} & \multicolumn{1}{c}{**BraTS 2019 LGG**} & \multicolumn{1}{c}{**BraTS 2019 HGG**} \\ \hline \multicolumn{1}{c}{**Result**} & \multicolumn{1}{c}{**Data Size**} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} & \multicolumn{1}{c}{\(\overline{\text{DSC}}\)} \\ \hline ensemble & \multirow{3}{*}{\((S_{1})\)} & 100\(k\) & 0.7904 & 0.8217 & 0.7827 & 0.8248 & 0.7693 & 0.8065 & 0.8232 & 0.8560 \\ output & & 200\(k\) & 0.7928 & 0.8239 & 0.7858 & 0.8267 & 0.7748 & 0.8113 & 0.8264 & 0.8578 \\  & & 500\(k\) & 0.7961 & 0.8266 & 0.7903 & 0.8290 & 0.7795 & 0.8154 & 0.8300 & 0.8602 \\ atlas & & 1000\(k\) & 0.7980 & 0.8284 & 0.7935 & 0.8310 & 0.7828 & 0.8185 & 0.8324 & 0.8619 \\ \hline final & \multirow{3}{*}{\((S_{1}^{\prime\prime})\)} & 100\(k\) & 0.8470 & 0.8618 & 0.8217 & 0.8524 & 0.8266 & 0.8495 & 0.8421 & 0.8679 \\ result & & 200\(k\) & 0.8490 & 0.8635 & 0.8241 & 0.8540 & 0.8309 & 0.8531 & 0.8436 & 0.8692 \\ without & & 500\(k\) & 0.8515 & 0.8657 & 0.8276 & 0.8557 & 0.8347 & 0.8565 & 0.8452 & 0.8707 \\ atlas & & 1000\(k\) & 0.8536 & 0.8674 & 0.8300 & 0.8571 & 0.8378 & 0.8591 & 0.8464 & 0.8717 \\ \hline ensemble & \multirow{3}{*}{\((S_{2})\)} & 100\(k\) & 0.8040 & 0.8316 & 0.7943 & 0.8279 & 0.8015 & 0.8307 & 0.8259 & 0.8602 \\ output & & 200\(k\) & 0.8063 & 0.8336 & 0.7963 & 0.8301 & 0.8047 & 0.8337 & 0.8281 & 0.8626 \\ with & & 500\(k\) & 0.8089 & 0.8359 & 0.7990 & 0.8334 & 0.8074 & 0.8367 & 0.8307 & 0.8654 \\ atlas & & 1000\(k\) & 0.8107 & 0.8375 & 0.8013 & 0.8357 & 0.8091 & 0.8386 & 0.8326 & 0.8675 \\ \hline final & \multirow{3}{*}{\((S_{2}^{\prime\prime})\)} & 100\(k\) & 0.8503 & 0.8646 & 0.8299 & 0.8573 & 0.8412 & 0.8624 & 0.8464 & 0.8738 \\ result & & 200\(k\) & 0.8519 & 0.8662 & 0.8317 & 0.8588 & 0.8442 & 0.8652 & 0.8485 & 0.8751 \\ with & & 500\(k\) & 0.8547 & 0.8683 & 0.8337 & 0.8610 & 0.8470 & 0.8681 & 0.8504 & 0.8766 \\ atlas & & 1000\(k\) & 0.8564 & 0.8696 & 0.8355 & 0.8625 & 0.8479 & 0.8691 & 0.8516 & 0.8775 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average and overall Dice similarity coefficients obtained for various data sets and training data sizes, with and without atlas-based data enhancement at pre-processing. similarity value was higher than the average (\(\widetilde{\text{DSC}}>\widetilde{\text{DSC}}\)) in case of all four data sets. In fact, \(\widetilde{\text{DSC}}\) was close to the median value of individual Dice scores \(\text{DSC}_{i}\), \(i=1\dots n_{\rho}\). Figure 6 presents the Dice similarity scores we may expect for a 10 cm\({}^{3}\) and a 100 cm\({}^{3}\)  
Figure 5: Individual Dice scores plotted against the true size of the tumor: (**a**) 54 records of BraTS 2015 LGG data; (**b**) 220 records of BraTS 2015 HGG data; (**c**) 76 records of BraTS 2019 LGG data; (**d**) 259 records of BraTS 2019 HGG data. sized tumor according to the linear trends identified from the data presented in Figure 5, in the case of the four data sets separately. Small tumors of 10 cm\({}^{3}\) are not even present in all data sets, but the proposed method apparently learned how to segment them accurately, with an expected Dice similarity value around 0.8, which reportedly represents fine segmentation [57]. The Dice score obtained for an average sized tumor from the data sets was expected in the proximity of 0.85. Figure 7 exhibits some selected segmentation results. Four MRI volumetric records were selected from each of the four data sets, out of which that slice was chosen that contained the highest number of positive pixels, ground truth positives, and positive labeled pixels combined.