## 2 Methodology of the Inversion  
### Direct Problem  
From the global point of view, the goal of the proposed method is to be able to reconstruct 3D volumetric sources \(Q(x,y,z)\), as illustrated Figure 1, from the measured temperature fields at the sample surface opaque to IR radiation. With this constraint, we will assume that any geometry or shape of sources can be approximated by a sum of variable intensity source points. The impulse thermal response \(\theta_{\text{Dirac}}\) at each point \(r=(x,y,z)\) at time \(t\) from a heat source located at \(r^{\prime}=(x^{\prime},y^{\prime},z^{\prime})\) delivering, at a previous time \(t^{\prime}\), a pulse of heat quantity of \(Q(r^{\prime})\) (J-m\({}^{-3}\)) in an infinite  
Figure 1: Schema of 3D volumetric sources distributed in a material and the associated surface temperature field. homogeneous medium, possibly anisotropic, but of non-thermo-dependent thermophysical properties, is expressed analytically by Equation (1) [9]:  
\[\theta_{\text{Dirac}}=\frac{Q(r^{\prime})}{\rho C_{p}}G_{3D}\left(r,t/r^{\prime}, t^{\prime}\right), \tag{1}\]  
where \(\rho\) (kg\(\cdot\)m\({}^{-3}\)) and \(C_{p}\) (J\(\cdot\)K\({}^{-1}\cdot\)kg\({}^{-1}\)) are, respectively, density and specific heat of the material, while \(G_{3D}\) (m\({}^{-3}\)) is the 3D-Green function of the material. This function can be written as:  
\[G_{3D}\left(r,t/r^{\prime},t^{\prime}\right)=G_{x}\left(r,t/r^{\prime},t^{ \prime}\right)\cdot G_{y}\left(r,t/r^{\prime},t^{\prime}\right)\cdot G_{z} \left(r,t/r^{\prime},t^{\prime}\right). \tag{2}\]  
where \(G_{x}\), \(G_{y}\) and \(G_{z}\) are the 1D-Green functions along the three directions of space [9]. If the domain is infinite in all three directions, each 1D-Green function expresses itself explicitly in the same way, here, with spatio-temporal Gaussian functions, and the thermal impulse response becomes:  
\[\theta_{\text{Dirac}}=\frac{Q(x^{\prime},y^{\prime},z^{\prime})}{\rho C_{p}} \frac{\exp\left(-\frac{(x-x^{\prime})^{2}}{4a_{x}(t-t^{\prime})}\right)}{ \sqrt{4\pi a_{x}(t-t^{\prime})}}\frac{\exp\left(-\frac{(y-y^{\prime})^{2}}{4a_ {y}(t-t^{\prime})}\right)}{\sqrt{4\pi a_{y}(t-t^{\prime})}}\frac{\exp\left(- \frac{(z-z^{\prime})^{2}}{4a_{z}(t-t^{\prime})}\right)}{\sqrt{4\pi a_{z}(t-t^ {\prime})}} \tag{3}\]  
where \(a_{x}\), \(a_{y}\) and \(a_{z}\) (m\({}^{2}\cdot\)s\({}^{-1}\)) are the diffusivities along each of the \(x\), \(y\) or \(z\) directions of space, respectively. If the domain is not infinite in all three directions, but only along the directions \(x\) and \(y\), and is only semi-infinite in the third direction (2), the 1D-Green function functions \(G_{x}\) and \(G_{y}\) are the same as in Equation (3), but \(G_{z}\) is different and can be written as [9] :  
\[G_{z}(z,t/z^{\prime},t^{\prime})=\frac{1}{4\sqrt{\pi a_{z}(t-t^{\prime})}} \left[\exp\left(-\frac{(z-z^{\prime})^{2}}{4a_{z}(t-t^{\prime})}\right)+\exp \left(-\frac{(z+z^{\prime})^{2}}{4a_{z}(t-t^{\prime})}\right)\right]. \tag{4}\]  
Now, if the domain is infinite along the directions \(x\) and \(y\) and finite along the direction \(z\), the domain is therefore a plane sample of thickness \(L_{z}\). In this case, the 1D-Green functions \(G_{x}\) and \(G_{y}\) are the same as in Equation (3), but \(G_{z}\) is different and can be written as [9] :  
\[G_{z}(z,t/z^{\prime},t^{\prime})=\frac{1}{L_{z}}\left[1+2\sum_{m=1}^{+\infty} \exp\left(-a_{z}m^{2}\pi^{2}(t-t^{\prime})/L_{z}^{2}\right)\cos\left(m\pi z/ L_{z}\right)\cos\left(m\pi z^{\prime}/L_{z}\right)\right]. \tag{5}\]  
For this example of a sample finite along the \(z\) direction, the quadripole method is another technique that allows the access to the temperature response [10].