## 3 Improvement in the Implementation of Stitching Linear-Scan Method
Figure 9a,b show the profiles of Arc1 and Arc2, respectively, which are the same as those in Figure 7a,b. The stitching process was carried out in the same manner as described above, while excluding the data in the edge region of Arc2. The parameters \(\phi_{e}\) and \(\phi_{e}\) were set to be 10\({}^{\circ}\) and 20\({}^{\circ}\), respectively. Figure 9c shows the variation of the cross-correlation coefficient due to the change in \(\Delta\theta\), and Figure 9d shows the stitched Arc1 and Arc2 with \(\Delta\theta_{1-2}\) obtained from the plot shown in Figure 9c. In Figure 9d, the region in Arc2 employed in the stitching process, in which the data corresponding to 10\({}^{\circ}\) from the edge of Arc2 is excluded, is highlighted. The maximum cross-correlation coefficient \(C_{1-2}\) was evaluated to be 0.89; this value is larger than that obtained by the stitching with the data in Arc2 shown in Figure 7b. To further verify the effect of eliminating the data in the edge region of an arc on the improvement of the cross-correlation coefficient in the stitching process, stitching operations of two neighboring arcs in the eight arc-profiles were carried out. Figure 10 shows the results. Figure 8: Exclusion of the edge data in the arc-profiles for the circumferential stitching. Figure 9: The stitching of the Arc2 with respect to Arc1 by employing the profile data excepting the edge region in Arc2 in the overlapped region: (**a**) Arc1; (**b**) Arc2; (**c**) Variation of the cross-correlation coefficient \(C_{1-2}\) due to the change in \(\Delta\theta\); (**d**) The stitched Arc1 and Arc2 at \(\Delta\theta=\Delta\theta_{1-2}\). Cross-correlation coefficients obtained in the stitching operations are also summarized in Figure 11. As can be seen in the figures, the elimination of the data in the edge region of an arc was found to be effective in improving the cross-correlation coefficient in each stitching operation. Figure 10: Stitched neighboring arc-profiles from the eight arc-profiles: (**a**) Stitching by employing all data in the overlapped region; (**b**) Stitching by employing the profile data in the overlapped region except the edge region in an arc to be stitched. Figure 12 shows the relationship between Arc8 and Arc1 after the stitching operations. It should be noted that the angular position of Arc8 is determined by the sum of \(\Delta\theta_{i-i+1}\) (\(i\) = 1-7), and the cross-correlation coefficient between Arc8 and Arc1 (C8-1) could be a parameter indicating the quality of circumferential stitching operations. In the case of the stitching operations employing the full data in the overlapped region between the two neighboring arcs, C8-1 was evaluated to be 0.44. Meanwhile, in the case of the stitching operations with the data eliminating the edge region of an arc to be stitched, C8-1 was evaluated to be 0.66; these results indicate that the quality of the stitching was improved by eliminating the data in the edge region of an arc to be stitched. A Hardware Modification for the Improvement of the Cross-Correlation Coefficient in the Circumferential Stitching Process  
A hardware modification was also made to the experimental setup for the improvement of the quality of the stitching of arc-profiles. Figure 13 shows a schematic of the setup before and after the modification. In the modified setup, a small round magnet was sandwiched between the round magnet and workpiece as shown in Figure 13b; this modification was expected to reduce the influences of out-of-flatness of the round magnet surface and the side face of the workpiece holder by reducing the contact area between them. Experiments were carried out to verify the feasibility of the modified workpiece-holding mechanism. Figure 14 shows a photograph of the modified setup. A small magnet with a diameter and a thickness of 1.5 mm and 4.0 mm, respectively, was placed between the round magnet and workpiece. The workpiece having a diameter and a length of 3.0 mm and 50 mm, respectively, evaluated in the previous section was also employed.