## 5 Results  
### Test Object  
The experiments were carried out on a particular object, designed so that the two aforementioned extrinsic calibration methods were both equally suitable and easy comparable. In addition, the particular design of the object shown in Figure 8 allowed us to evaluate the effectiveness of the ray-casting technique on a geometry with concavities. It has an internal conic hole and three different radial parts (shifted by angles of 120 degrees), characterized by many edges. The object was designed with CAD, 3D-printed with acrylonitrile butadiene styrene (ABS) material and then scanned with the Konica Minolta Vivid 9i laser scanner. The point cloud could be obtained directly from the CAD model, but the object was scanned to take into account the acquisition bias and test the effectiveness of the method on laser scanner data. Here we present a first set of simulations, carried out to assess the effectiveness of the automatic extrinsic calibration method. This was made by comparing the extrinsic parameter values obtained by means of this automatic procedure with the ones resulting from the method based on the manual selection of homologous points. Because of the specially designed geometry features of the printed object, it was possible to choose for the first method a set of about 20-30 homologous points for each analyzed thermogram, with a mean reprojection error (MRE) of about 1-2 pixels. In this way, the proper reliability of the first method parameters was assured, and so it was possible to evaluate the ones obtained with the second method. Equation (9) shows the expression for computing the MRE. The units of MRE are pixels. \[MRE=\ \frac{\sum_{i=1}^{n}\|P_{i}-Q_{i}\|}{n} \tag{9}\]  
In Equation (9), \(n\) is the number of homologous points selected, \(P_{i}\) the 2D coordinates of the \(i\)-_th_ point selected on the thermogram, \(Q_{i}\) the 2D coordinates of the reprojection of the \(i\)-_th_ homologous 3D point. For each thermogram, the convergence from different initial positions was analyzed, within a range of \(\pm 25\) degrees for the Euler angles and \(\pm 20\) mm for the translation vector from the parameters obtained with the first method. In Table 1, some of the results of the first calibration method are shown for four different thermograms, whereas Table 2 gives the differences (considering the same thermograms) between the extrinsic parameters obtained with the second method and with the first one respectively. Figure 8: (**a**) 3D printed object; (**b**) Its CAD model. Figure 9 shows a set of 2D points and the reprojection of 3D homologous points, once the extrinsic parameters were computed. With reference to Table 2, thermograms A and B were obtained starting from a first guess of the parameters with the following differences with respect to the ones of Table 1: \(\Delta t_{1}\) = 20 mm, \(\Delta t_{2}\) = \(-\)20 mm, \(\Delta t_{3}\) = 20 mm, \(\Delta\alpha\) = \(-\)20\({}^{\circ}\), \(\Delta\beta\) = 0\({}^{\circ}\), \(\Delta\gamma\) = 20\({}^{\circ}\). For thermograms C and D, the initial differences were set as follows: \(\Delta t_{1}\) = \(-\)10 mm, \(\Delta t_{2}\) = 20 mm, \(\Delta t_{3}\) = 10 mm, \(\Delta\alpha\) = \(-\)20\({}^{\circ}\), \(\Delta\beta\) = 15\({}^{\circ}\), \(\Delta\gamma\) = 15\({}^{\circ}\). As can be seen in Table 2, the final differences are relatively low, except for the \(\Delta t_{3}\), which is of several millimeters. However, this can be acceptable, considering that the \(t_{3}\) parameter is greater by an order of magnitude compared to the parameters \(t_{1}\) and \(t_{2}\) (relative error of about 2%). In Figure 10, the initial (a) and final (b) projection in the case of automatic extrinsic calibration of the thermogram B are shown. This type of visualization allows for a qualitative consideration on the effectiveness of the method. As can be easily seen, in the case (b), the 3D projected points (in red) fill the thermogram contour (yellow line) well.