*4.5. Multiple Thermograms*

In the previous sections, the case of the integration of a single thermogram on a point cloud was considered. There might be, however, the need to integrate more than one thermogram on the same point cloud, in order to have a larger set of points with an associated temperature.

Note that this procedure requires in the first place thermograms acquired in a temporal window in which no measurable changes in the thermal state occur, so as to guarantee that the integration is not carried out on discordant data. Since each thermogram can be individually integrated on the point cloud by the methodology described previously, the problem comes down to handling the overlapping zones, namely the points to which more than one temperature value is assigned. To give these points a single final temperature, the method utilized in [9] was followed. The method relies on the fact that the relative orientation between the surface of the inspected object and the camera affects the measurement accuracy, and, more specifically, the lower the viewing angle is, the higher the accuracy is. As shown in Equation (6), the temperature *Ti* assigned to the point *Pi* is computed as a weighted average of the temperatures *Tij*:

$$T\_i = \frac{\sum\_{j=1}^{N\_i} c\_{ij} T\_{ij}}{\sum\_{j=1}^{N\_i} c\_{ij}} \tag{6}$$

where the weight is the confidence factor *cij*, the index *i* refers to the point of the points cloud and ranges from 1 to *N*, whereas the index *j* refers to the thermograms that overlap in the point *i* and ranges from 1 to *Ni*. The confidence factor *cij* is computed as a function of the viewing angle θ*ij* as shown in Equation (7):

$$x\_{i\bar{j}} = e^{-\kappa \theta\_{i\bar{j}}} \tag{7}$$

where the viewing angle θ*ij* is the angle from which the thermal camera sees the point *Pi*, considering the thermogram *j*, and can be computed as shown in Equation (8):

$$\theta\_{i\dot{j}} = \arccos \frac{\left(O\_{\dot{j}} - P\_{i}\right) \cdot \hbar\_{i}}{\left\| O\_{\dot{j}} - P\_{i} \right\|} \tag{8}$$

with point *Oj* identifying the optical center for the thermogram *j*.

In this way, a greater weight is assigned to the rays with smaller viewing angles, which allows more accurate measures. More precisely, the weight decreases with an exponential law, depending on a parameter κ, that was set equal to 2 according to experimental evaluations.
