## 3 Improvement in the Implementation of Stitching Linear-Scan Method  
Improvement of the Cross-Correlation Coefficient in the Circumferential Stitching Process by the Elimination of the Data in the Edge Region of an Arc  
The feasibility of eliminating the edge part of an arc in the stitching process for the improvement of a cross-correlation coefficient was at first verified in experiments. Figure 6 shows a schematic of the setup employed in the following experiments. A workpiece-holding mechanism with a V-groove having a depth of 2.5 mm and an angle of 90\({}^{\circ}\) was mounted on a two-axis manual rotary stage so that the fine adjustment of the three-axis tilt of a workpiece could be carried out. As reported in the previous work [17], angular misalignments of a workpiece about the X- and Z-axes can be reduced to less than 0.1 arc-seconds by using a stylus profiler [17]. A commercial stylus profiler (Form Talysurf PGI-420, Taylor-Hobson) was employed as the linear-scan stylus profiler for the stitching linear-scan method. A cylindrical workpiece with a radius of 3.0 mm and a length of 50 mm was employed as the specimen for measurement. The circumferential profile of the workpiece was obtained as the eight-divided arc-profiles. After the linear scan of each arc, the workpiece was rotated 45 degrees manually by referring to the index mark on the round magnet. Measurement conditions are summarized in Table 1. The scanning speed was set to be 0.1 mm/s; this is the lowest speed allowed in the employed stylus profiler. Figure 5: The influence of the workpiece misalignment along its axial direction. Figure 6: Schematic of the setup employed in experiments. Figure 7a,b show the profiles of Arc1 and Arc2, respectively, after the low-pass filtering and the radial stitching [15,16]. Figure 7c shows the variation in the cross-correlation coefficient due to the change in \(\Delta\theta\) calculated based on Equation (1) with the employment of the profile data in the highlighted region in Arc2. From the obtained plot in Figure 7c, \(\Delta\theta\) (=\(\Delta\theta_{1-2}\)), providing the maximum cross-correlation coefficient, was subtracted. By using the obtained \(\Delta\theta_{1-2}\), Arc1 was stitched with Arc2 as shown in Figure 7d. It should be noted that the difference between Arc1 and Arc2 in the overlapped region is also plotted in Figure 7d. The maximum cross-correlation coefficient \(C_{1-2}\) was evaluated to be 0.64. Experiments were extended to carry out the stitching process with the limited data in the overlapped region in an arc-profile to be stitched with the neighboring arc. Figure 8 shows a schematic of the exclusion of the edge data in the arc-profiles for the circumferential stitching. In the circumferential stitching process, arc-profile data in the angular range of \(\phi_{e}\) is excluded from the neighboring \(i\)th and \((i+1)\)th arc-profiles for the calculation of the cross-correlation coefficient \(C_{i-i+1}\) based on Equation (1). In the \((i+1)\)th arc-profile, the profile data in the angular range of \(\phi_{u}\) next to the excluded region is employed for the calculation of \(C_{i-i+1}\). In the calculation process, the angular position of the \(i\)th arc-profile is fixed, while the \((i+1)\)th arc-profile is shifted along the circumferential direction to maximize \(C_{i-i+1}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Straightness & \(\leq\)0.5 & \(\mu\)m \\ Measurement range & 0.1–120 & mm \\ Resolution & 0.125 & \(\mu\)m \\ Tip angle of the stylus & 60 & degree \\ Tip radius of the stylus & 2 & \(\mu\)m \\ Scanning speed & 0.1 & mm/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurement conditions. Figure 7: The stitching of the Arc2 with respect to Arc1 by employing all data in the overlapped region: (**a**) Arc1; (**b**) Arc2; (**c**) Variation in the cross-correlation coefficient \(C_{1-2}\) due to the change in \(\Delta\theta\); (**d**) The stitched Arc1 and Arc2 at \(\Delta\theta=\Delta\theta_{1-2}\).