## 2 Materials and Methods
### Methodology  
As briefly introduced in Section 2.2, one of the most recent and promising approaches developed in the computer vision field is U-Net [43]. The model, inspired from a Convolutional Neural Network presented to the Computer Vision for Pattern Recognition conference [44], was originally employed in the medical field for the segmentation of biological cells and for the analysis of MRI scans for the detection of a number of pathologies [45; 46; 47]. The network concatenates two branches: a contracting path and an expansive path, giving it the u-shaped architecture. The contracting path is a classical convolutional network, consisting of repeated application of convolutions, each followed by a rectified linear unit (ReLU) and a max-pooling operation. During the contraction, the spatial information is reduced while feature information is increased. The expansive path combines both the features and the spatial information through a sequence of up-convolutions and concatenations with high-resolution features from the contracting path. The original version takes as input one grayscale image of dimension 572 \(\times\) 572 pixels and it outputs a binary mask of 388 \(\times\) 388 pixels. The input is oversized compared to the output: that choice was made to give some extra context to include information of the biological cells outside exceeding the border of the considered tile. In the context of this paper, the dimension of a burned area is not predictable, therefore the authors opted for setting the network's width and height input and output dimensions to 480 \(\times\) 480 pixels for each of the 12 spectral bands. In the end, the U-Net's input dimension is set to 480 \(\times\) 480 \(\times\) 12 pixels. One of the main contributions of this study is the modification of the original U-Net, empowering its ability to distinguish between ordered classes, as in the case of damage severity. The classical U-Net was proposed for solving a segmentation task, being able to identify a specified entity in an image. Therefore, it is able to recognize relations and features (i.e., borders and gradients) among the pixel values belonging to the searched entity. Conversely, in the context of this work, the goal appears to be more complex and it can be split into two sub-tasks: (i) to identify areas affected by fire, and (ii) to determine damage severity in the burned areas. In the first sub-task, the goal is to distinguish burned areas from unburned regions, like in a classical segmentation task (classification task). The second sub-task takes into account the areas affected by the fire and discriminates between four consequent levels the severity of the damage (regression task). The two sub-tasks are solved with two different building blocks: the \"Binary Classification U-Net\" and the \"Regression U-Net\". In both the building blocks, the output map is a matrix of dimensions 480 \(\times\) 480, where each element refers to the pixel at the same position of the input tile. In the proposed solution, the two building blocks are combined together to outperform the prediction quality of the approach based only on the Regression U-Net block. First, the Binary Classification U-Net is trained for segmentation purposes: given a satellite image of 12 channels and a resolution of 480 \(\times\) 480, the network assigns to each pixel the probability of belonging to a burned area thanks to the application of the softmax activation function. Thus, the generated output is a binary segmentation map of size 480 \(\times\) 480 with values [0, 1] (i.e., unburned or burned), where each pixel is assigned to the class with the highest probability. Second, the Regression U-Net is used to provide the severity level estimation. Given the input satellite image, the model generates as output a map of the same size as the input with values in range [0, 4].