## 1 Introduction  
The methods for non-destructive testing (NDT) and structural health monitoring (SHM) are numerous, exploiting physical phenomena such as optics, thermodynamics, electromagnetism, acoustics, and thermal effects. However, it is important to emphasize that despite this multidisciplinary nature, there is no optimal or unique method; each method has its advantages and disadvantages. In this work, it is the use of thermal energy as a means of NDT that will be mainly addressed.  
Infrared thermography (IRT) is a technique that enables the thermal image of a sample observed in a spectral range of the infrared to be obtained. However, while tomography is now prevalent in various domains, which enables volumetric measurements (x-ray, magnetism resonance imaging, etc.), it is still not possible to measure the temperature in a 3D volume; in other words, it is not possible to realize thermal tomography. The challenge is then to reconstruct 3D heat sources in a volume from the 2D field measured at the surface by thermal imaging cameras. This inverse situation is an ill-posed problem mainly due to the diffusive nature of temperature. To circumvent this problem, classical regularization methods or statistical ones can be used. Interested readers can refer to [1, 2], which provides some examples with a rather general overview of the existing inverse methods.  
Among the different methods used to tackle heat source reconstruction, the work of [PERSON] et al. [3, 4, 5, 6, 7, 8] proposes detecting defects present in a vertical plane on the surface of the material by exciting them with modulated (lock-in) or pulsed (burst) acoustic waves, creating a thermal source at the spot of the defect; the characterization of the defect takes place by the characterization of the resulting thermal source. This reconstruction uses an original regularization term. More recently, a method involving the concept of virtual waves was developed by [PERSON] et al. [9; 10] to go back to the initial conditions (heat flow, temperature values, etc.) in a surface temperature field. These works show that the reconstruction methods that exist today lead to the reconstruction of volumetric heat sources (volumetric data) from temperature fields measured at the surface (surface data). However, two main limitations emerge: depth and measurement noise. Indeed, regardless of the methods used, deep sources are difficult to locate; they are blurred and spread out and their intensities are underestimated. The depth problem comes from the physics, so there is no way to overcome that. Finally, measurement noise, inevitable during experiments, is a problematic factor during inversion. Regularization methods enable approaching the solution, but the choice of the parameters of regularization is delicate because it is necessary to find a good compromise between stabilization and accuracy of the solution.  
In this work, a reconstruction process based on Bayesian inference, a method largely used in other scientific domains, is proposed to improve the influence of the noise in the inverse processing. In Section 2, the method is applied to a numerical 1D problem. It is then extended to a 3D problem in Section 3. Finally, an experimental validation is performed on a hot wire molded in a plastic sample and warmed by the Joule effect.