## 3 Highlighting an Optimal Criterion Based on Fourier Number  
### Analysis of the Criterion  
To introduce the optimal criterion an example of the 1D problem shown in Figure 4, the following equation to calculate the temperature response is presented here:  
\[\theta(z^{*},t^{*})=\int_{\text{sample}}Q(z^{\prime*})\theta_{\text{1D}}(z^{* }-z^{\prime*},t^{*})dz^{\prime}, \tag{18}\]  
with \(\theta_{\text{1D}}\) defined by:  
\[\theta_{\text{1D}}(z^{*},t^{*})=\frac{1}{L_{z}\sqrt{\pi}Fo_{z}t^{*}}\exp\left( \frac{(z^{*}-z^{\prime*})^{2}}{4 Fo_{z}t^{*}}\right). \tag{19}\]  
The heat source repartition \(\Omega\) consists of three \"source points\" located at three different depths. In Figure 4, the intensity (normalized) of the nearest source of the surface is 0.5, that of the source of the medium is 1 and the last is 0.75. Here, \(L_{z}=0.01\) m and \(Fo_{z}=0.1\). Moreover, no noise will be taken in this part. In this part, the data are synthetic and computed as a consequence of the direct problem defined previously. Therefore, two steps are computed for the algorithm. The direct problem supposes the heat sources are known (location, intensity) and enables acquiring the resulting temperature at the surface (these data are produced by the IR-camera in a real experiment). The second step of the algorithm is the inversion process. As presented in the previous part, inversion is based on the SVD of the operator matrix \(\mathbf{I}\). This matrix is defined for a discretization of every possible source excited by a Dirac excitation at time \(t=0\). It is actually independent of the real distribution of sources for this given discretization. In this work, it appears that the study of \(\mathbf{S}\) (singular values matrix) is of prime importance for the performance of the inversion method. \(\mathbf{I}\) and \(\mathbf{S}\) have the same size (\(m\) by \(n\)), where \(m\) is the number of time steps (between \(t_{0}\) and \(t_{f}\)) and \(n\) the number of point of discretizations of the sample. For the computation of the algorithm, these two parameters must be defined. As this inverse problem can be seen as a linear least squares minimization, we must have \(m>n\). Technically, the number of time steps is limited by the acquisition frequency of the IR camera (\(ac\)): it cannot exceed \(M=t_{f}*ac\). Therefore, \(m\) can be chosen between 1 and \(M\). The discretization of the depth \(n\) is not submitted to the same limitations. Two cases of inversion for the problem illustrated by Figure 5 have been tested for two different discretizations of the material: the first inversion has been made with \(n=20\) (a), the second with \(n=50\) (b). The number of time steps was taken as \(m=n_{t}=300\). For the chosen parameters, if \(n=20\), the three heat sources are perfectly reconstructed, even the deepest source. However, if \(n=50\), all sources are poorly found. These results imply that there is a maximal number \(n\) of sources (which corresponds to an optimal resolution for the voxel) that we can reconstruct. To determine this limit, the inversion algorithm was carried out one hundred times for \(n\), varying from 1 to 100. For each discretization, the error made between the reconstruction and the real source is calculated, and the obtained results are given in Figure 5. For \(n<30\), the errors are negligible, meaning that the reconstructions are faithful to the real sources. However, from \(n=30\), the error increases rapidly with the number of discretizations, which means that the reconstructions are erroneous. Thus, for a given problem, there is a maximum number of discretizations \(n_{\phi}\) to reconstruct the sources with accuracy. This number \(n_{\phi}\) can be determined by the study of the singular values of the operator matrix \(\mathbf{I}\). The SVD of the operator matrix \(\mathbf{I}\) with a high spatial discretization \(n=100\) is computed and illustrated in Figure 5. As can be seen, from a certain value \(n_{\phi}\), the singular values are constant and very low in comparison to the first values.