## 2 Principle of the Stitching Linear-Scan Method  
A schematic of the setup for the stitching linear-scan method is shown in Figure 2. As can be seen in the figure, the target workpiece is mounted on the V-groove in the V-block, while a round magnet is attached to one end of the workpiece. It should be noted that the centering alignment of the workpiece with respect to the round magnet can automatically be carried out by the magnetic force between them. Since the round magnet has \(n\) indexing marks, the workpiece can coarsely be positioned in a step of \(360/n\) degrees in the circumferential direction. By repeating the workpiece rotation and the linear scan of the arc-profile of the workpiece, a series of arc-profiles covering \(360^{\circ}\) of the circumferential profile of the workpiece can be obtained. The reconstruction of the circumferential profile of the workpiece can be carried out through the stitching process, which mainly consists of the radial stitching and the circumferential stitching. Figure 3 shows a schematic of the whole stitching process. Details of the stitching process can be found in [15; 16; 17]. After the radial stitching, the circumferential stitching is carried out. In the circumferential stitching, the arc-profiles in the Cartesian coordinates are at first transformed into polar coordinates. Then, the overlapped part of two adjacent arcs can be matched by referring to the following cross-correlation function [18; 19; 20]:  
\[C_{i-i+1}=\int_{-\infty}^{\infty}f_{i}(\theta)f_{i+1}(\theta+\Delta\theta)d\theta \tag{1}\]  
where \(C_{i-i+1}\) is the cross-correlation coefficient between the \(i\)th and \((i+1)\)th arc-profiles \(f_{i}(\theta)\) and \(f_{i+1}(\theta+\Delta\theta)\), respectively, while \(\Delta\theta\) is the adjusted angle of the \((i+1)\)th arc-profile. When \(C_{i-i+1}\) is close to 1, this means that the two arcs match well. In the circumferential stitching process, the stitching of the adjacent arcs is performed in such a way that the  
Figure 3: Stitching process. Figure 2: A schematic of the stitching linear-scan method. calculation based on Equation (1) is carried out by changing \(\Delta\theta\) step by step to find out \(\Delta\theta=\Delta\theta_{i-i+1}\) that can maximize \(C_{i-i+1}\), since a higher cross-correlation coefficient means better stitching. Meanwhile, in the stitching linear-scan method, there are two major issues that need to be addressed to improve the quality of stitching. The first issue is related to the geometric relationship between the stylus of a linear-scan surface form stylus profiler and the cylindrical form of a workpiece. As can be seen in Figure 4, the pressure angle of the stylus becomes different at each point on the surface of measurement target; namely, in the overlapped region between the \(i\)th and \((i+1)\)th arc-profiles, the same point in the adjacent arc-profiles will be measured with different pressure angles. As a result, due to the influences of the shape of the probe tip and the local slope of the target surface, the profile of the overlapped region to be obtained in the \(i\)th arc-profile and that in the \((i+1)\)th profile become slightly different from each other. The discrepancy between the two adjacent arc-profiles in the overlapped region could become significant at the edge of each arc-profile measured by the stylus with a large pressure angle. According to the uncertainty analysis performed in the previous study by the authors [17], the reading of the X-coordinate in the stylus profiler is one of the main uncertainty sources. Uncertainty of the reading of the Z-coordinate in the stylus profiler, including the calibration uncertainty of the probe, could also be a major component of measurement uncertainty; it is not easy to reduce these components of measurement uncertainty. On the other hand, the influences of the workpiece misalignments can be suppressed by some efforts. The influence of the workpiece tilt was successfully suppressed by the tilt alignment technique [17].