## 2 Materials and Methods
### Histogram Normalization  
Although very popular among medical imaging devices due to its high contrast and relatively good resolution, MRI has a serious drawback: the numerical values it produces do not directly reflect the observed tissues. The correct interpretation of the MRI data requires adapting the numerical values to the context, which is usually accomplished via a histogram normalization or standardization. Figure 3 gives a convincing example why the normalization of histograms is necessary. This is what we get if we represent on the same \begin{table}
\begin{tabular}{l c c c c} \hline \multirow{2}{*}{**Property**} & \multicolumn{2}{c}{**BraTS 2015 Data**} & \multicolumn{2}{c}{**BraTS 2019 Data**} \\ \cline{2-5}  & **LGG** & **HGG** & **LGG** & **HGG** \\ \hline Number of records & 54 & 220 & 76 & 259 \\ Average size of whole tumor (cm\({}^{3}\)) & \(101.1\) & 110.6 & 111.5 & 95.1 \\ Minimum size of whole tumor (cm\({}^{3}\)) & 18.9 & 8.5 & 17.1 & 7.3 \\ Maximum size of whole tumor (cm\({}^{3}\)) & 256.3 & 318.3 & 361.8 & 256.9 \\ \hline Total number of negative pixels & 71.4 M & 308.6 M & 101.6 M & 351.5 M \\ Total number of positive pixels & 5.46 M & 24.3 M & 8.48 M & 24.6 M \\ \hline Pixels with missing data & 2.375\% & 9.855\% & 1.679\% & 0.426\% \\ Pixels with more than 1 value missing & 0.225\% & 4.838\% & 0.163\% & 0.092\% \\ Records with missing data at \textgreater{}1\% of pixels & 9 & 112 & 9 & 18 \\ Records with missing data at \textgreater{}10\% of pixels & 3 & 99 & 3 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: The main properties of the MRI data sets involved in this study. Figure 2: Input data after pre-processing (T1, T1C, T2, FLAIR) and the human expert made ground truth (GT) for the whole tumor segmentation problem. scale the observed intensities from twenty different MRI records, extracted from the same data channel (T1). In some cases, the spectra of intensities from two different records do not even overlap. There is no classification method that could deal with such intensities without transforming the input data before classification. Several algorithms exist in the literature for this purpose (e.g., [48; 49; 50]), more and more sophisticated histogram matching techniques, which unfortunately are not designed for records containing focal lesions. These lesions may grow to 20-30% of the brain volume, which strongly distorts the histogram in MRI channels of any weighting. The most popular histogram normalization method proposed by [PERSON] et al. [48] works in batch mode, registering the histogram of each record to the same template. Modifying all histograms to look similar, no matter whether they contain tumor or not, is likely to obstruct the accurate segmentation. The method of [PERSON] et al. uses some predefined landmark points (percentiles) to accomplish a two-step transformation of intensities. Many recent works [51; 52; 53; 54; 55] reported using this method, omitting any information on the employed landmark points. [PERSON] et al. [12] and [PERSON] et al. [13] used 10 and 12 landmark points, respectively. Alternately, [PERSON] et al. [11] deployed a simple linear transformation of intensities, because it gives slightly better accuracy, but they did not specify any details. This finding was confirmed later in a comparative study by [PERSON] et al. [56], which also showed that if we stick to the above-mentioned popular method of [PERSON] et al., it provides better segmentation of focal lesions when used with a smaller number (no more than five) of landmark points, or in other words, with less restrictions. Our histogram normalization method relies on a context dependent linear transform. The main goal of this transform is to map the pixel intensities from each data channel of a volume independently of each other onto a target interval \([\alpha,\beta]\) in such a way that target intensities satisfy some not very restrictive criteria. The inner values of the target interval can be written as \(\gamma_{\lambda}=\alpha(1-\lambda)+\beta\lambda\), with \(0\leq\lambda\leq 1\).