## 2 Materials and Methods
### Data  
This study uses the MICCAI BraTS training data sets, both low- and high-grade glioma volumes for the years 2015 and 2019. Some main attributes of these data sets are exhibited in Table 1. Only one out of these four data sets is used at a time, so in any scenario, the total number \(n_{\rho}\) of involved records varies between 54 and 259, as indicated in the first row of the table.  
The records in each set have the same format. Records are multi-spectral, which means that every pixel in the volumes has four different observed intensity values (named T1, T2, T1C, and FLAIR after the weighting scheme used by the MRI device) recorded independently of each other and registered together afterwards with an automatic algorithm. Each volume contains 155 square shaped slices of \(240\times 240\) pixels. Pixels are isovolumetric as each of them represents brain tissues from a 1 mm\({}^{3}\) sized cubic region. Pixels were annotated by human experts using a semi-automatic algorithm, so they have a label that can be used as the ground truth for supervised learning. Figure 2 shows two arbitrary slices, with all four observed data channels, and the ground truth for the whole tumor, without distinguishing tumor parts. Since the adult human brain has a volume of approximately 1500 cm\({}^{3}\), records contain around 1.5 million pixels. Each record contains gliomas of a total size between 7 and 360 cm\({}^{3}\). The skull was removed from all volumes so that the researchers can concentrate on classifying brain tissues only, but some of the records intentionally had missing or altered intensity values, in the amount of up to one  
Figure 1: The input DICOM files are transformed into two variants of pre-processed data (\(P_{1}\) and \(P_{2}\)). The intermediary segmentation results produced by the ensemble of binary decision trees (\(S_{1}\) and \(S_{2}\)) are fine tuned in two post-processing steps to achieve high-quality final segmentation (\(S_{1}^{\prime\prime}\) and \(S_{2}^{\prime\prime}\)).  
third of the pixels, in one of the four data channels. An overview of these cases is also reported in Table 1.  
Intensity non-uniformity (INU) is a low-frequency noise with possibly high magnitude [45,46]. In the early years of the BraTS challenges, INU was filtered from the training and test data provided by the organizers. However, in later data sets, this facility is not granted. This is why we employed the method of [PERSON] et al. [47] to check the presence of INU and to eliminate it when necessary.  
There are several non-brain pixels in the volume of any record, a fact indicated by zero intensity on all four data channels. Furthermore, there are some brain pixels in most volumes for which not all four nonzero intensity values exist; some of them are missing. Our first option to fill missing values is to replace them with the average computed from the \(3\times 3\times 3\) cubic neighborhood of the pixel, if there are any pixels in the neighborhood with valid nonzero intensity. Otherwise, the missing value becomes \(\gamma_{0.5}\) after pre-processing, which is the middle value of the target intensity interval; see the definition in Section 2.3.