**2. Methods**

This section revisits the main methodologies that will be considered later for real-time classification and prediction of the effective thermal properties from collected images. For that purpose we will consider a rich enough training stage that consists of generating several microstructures whose effective thermal conductivity will be evaluated by using a standard linear homogenization technique, revisited in Section 2.1.

In order to associate the resulting homogenized conductivity tensor to each microstructure, the last must be described in a compact and concise way. For that purpose Topological Data Analysis and Principal Component Analysis (PCA) will be employed. Both are revisited in Sections 2.2 and 2.3, respectively.

The last step aims at performing a nonlinear regression to link the parameters extracted by the TDA to the thermal conductivity. The technique retained in our study is the so-called *Code2Vect* nonlinear regression, revisited in Section 2.4.
