## 3 Results  
The proposed brain tumor segmentation procedure was experimentally evaluated in various scenarios. Tests were performed using four different data sets: the LGG and HGG records separately from the BraTS 2015 training data set and, similarly, the LGG and HGG records separately from BraTS 2019 training data set. Detailed information on these data sets is given in Table 1. The whole segmentation process was the same for all four data sets, the one indicated in Figure 1.  
All records of these four data sets underwent the mandatory pre-processing steps, namely the histogram normalization and feature generation, resulting in pre-processed data denoted by \(P_{1}\). At this point, the records were randomly separated into six equal or almost equal groups, which took turns serving as the test data. An atlas was built using the training data only, namely the records from the other five groups. The atlas was then used to extract enhanced feature values for all records, resulting in pre-processed data denoted by \(P_{2}\). During the whole processing, there were six different such sets denoted by \(P_{2}\), one for the testing of each group. Preprocessed data sets \(P_{1}\) and \(P_{2}\) underwent the ensemble learning process separately, so that we could evaluate the benefits of the atlas-based pre-processing. All ensembles involved in this study consisted of \(n_{T}=125\) binary decision trees, but there were four variants for each data set, in terms of training data size. Each tree of an ensemble was trained to separate \(n_{F}\) feature vectors, \(n_{F}\) ranging from 100 thousand to on million, as indicated in Section 2.6. The intermediary segmentation obtained at the ensemble output is denoted by \(S_{1}\) or \(S_{2}\), depending on the use of the atlas during pre-processing. The labels of pixels in \(S_{1}\) or \(S_{2}\) were treated with two post-processing steps described in detail in Sections 2.7 and 2.8, reaching the final segmentation denoted by \(S_{1}^{\prime\prime}\) or \(S_{2}^{\prime\prime}\). Theoretically, all six segmentation results \(S_{1}\), \(S_{2}\), \(S_{1}^{\prime}\), \(S_{2}^{\prime}\), \(S_{1}^{\prime\prime}\), and \(S_{2}^{\prime\prime}\) can be equally evaluated with the statistical measures presented in Section 2.9. The main result is represented by the final outcome of the segmentation: \(S_{1}^{\prime\prime}\) for the data that are not pre-processed with the atlas and \(S_{2}^{\prime\prime}\) for the data with atlas-based enhancement. Table 3 presents the average (\(\overline{\text{DSC}}\)) and overall (\(\widetilde{\text{DSC}}\)) values of the Dice similarity coefficients obtained for various data sets and training data sizes, at four different phases of the segmentation process. A larger training data size always led to better accuracy: both the average and overall value rose by 0.5-0.8% if the training data size changed from 100\(k\) to 1000\(k\). If we consider segmentation accuracy the only important quality marker, then it is worth using the largest training data size. The use of the atlas was beneficial in all scenarios, but there were large differences in the strength of this effect: at the ensemble output, the difference was between 0.3% and 2%, while at the final segmentation, it was between 0.2% and 1%. The difference between the average and overall values in the same scenario usually ranged between 2% and 3%. This was caused by the distribution of the Dice similarity coefficients obtained for individual MRI records. The obtained Dice scores, having most averages between 0.85 and 0.86 and overall values ranging between 0.86 and 0.88, were quite competitive. Tables 4-7 exhibit the average sensitivity, positive predictive value, specificity, and correct decision rate (accuracy) values, respectively, obtained for different data sets and various training data sizes. Not all these indicators increased together with the training data size, but the rate of correct decisions did, showing that it is worth us