## 1 Introduction
### Principle of the Stitching Linear-Scan Method
To stitch the arcs in the radial direction, a mean radius \(\overline{R}\) is calculated by the following equation:  
\[\overline{R}=\frac{1}{8}{\sum_{j=1}^{8}r_{j}} \tag{3}\]  
By using the obtained \(\overline{R}\), each sampling point in each of the arcs is adjusted by the following equation:  
\[{r^{\prime}}_{i,j}=r_{i,j}-r_{j}+\overline{R} \tag{4}\]  
Figure 2: Measurement of the series of arc-profiles and the reconstruction of the circular profile of a workpiece: (**a**) measurement of the \(i\)th arc profile; (**b**) measurement of the \(i+1\)th arc profile. where \(r_{i,j}\) is the distance from the center of the arc to the point (\(x^{\prime\prime}{}_{i,j}\), \(z^{\prime\prime}{}_{i,j}\)) in the \(j\)th arc that can be calculated by the following equation:  
\[r_{i,j}=\sqrt{\left(x^{\prime\prime}{}_{i,j}-x_{c,j}\right)^{2}+\left(z^{\prime \prime}{}_{i,j}-z_{c,j}\right)^{2}} \tag{5}\]  
To minimize the deviation of each point from the fitting circle, \(x_{c,j}\), \(z_{c,j}\), and \(r_{j}\) are determined by minimizing the residual sum of squares \(S\) that can be calculated by the following equation [14]:  
\[S=\sum_{i=1}^{N}\left(\sqrt{\left(x^{\prime\prime}{}_{i,j}-x_{c,j}\right)^{2}+ \left(z^{\prime\prime}{}_{i,j}-z_{c,j}\right)^{2}}-r_{j}\right)^{2} \tag{6}\]  
As the next step of the stitching process, the circumferential stitching is carried out. The angular position \(\theta_{i,j}\) of the \(i\)th sampled points in the \(j\)th arc profile can be calculated by the following equation:  
\[\theta_{i,j}=\tan^{-1}\left(\frac{z^{\prime\prime}{}_{i,j}}{x^{\prime\prime}{ }_{i,j}}\right) \tag{7}\]  
Since a small cylinder is positioned approximately by using the indexing marks prepared on the outer face of the round magnet, each of the arcs has a certain amount of misalignment in the circumferential direction. Denoting the compensating angle shift for the \(j\)th arc as \(\Delta\theta_{j}\), the angular position of the \(i\)th point in the \(j\)th arc can be calculated by the following equation:  
\[\theta^{\prime}{}_{i,j}=\theta_{i,j}+\Delta\theta_{j} \tag{8}\]  
For the determination of \(\Delta\theta_{j}\), a cross-correlation function [15; 16] can be employed. By stitching the neighboring arcs sequentially from the first arc to the last arc, circumferential stitching can be carried out. For the portions where the neighboring two arcs are overlapping, a mean value of the two radial coordinates can be employed. Figure 3: Stitching procedure for the reconstruction of the circular profile of a small cylinder: (**a**) measurement of arc profiles; (**b**) rotation of the obtained arcs; (**c**) radial stitching; (**d**) circumferential stitching. By using the profile of the workpiece obtained through the above procedures, the diameter \(D\) and roundness of the workpiece can be evaluated. Meanwhile, the roundness can be evaluated by using the obtained profile data through processing it with the procedure defined by ISO [17]. For a small cylinder, the out-of-roundness \(\Delta Z_{\theta}\) is evaluated based on the least square circle [18]. Denoting the mean distance of the sampling points from the center coordinates as \(\overline{r}_{i}\), \(\Delta Z_{\theta}\) can be evaluated by the following equation:  
\[\Delta Z_{\theta}=(\overline{r_{i}})_{\text{Max}}-(\overline{r_{i}})_{\text{ min}} \tag{9}\]  
It should be noted that the eccentricity of the workpiece rotation will not affect the measurement in the proposed stitching linear-scan method [10]; this is one of the remarkable advantages compared with the conventional roundness measuring instruments, where the influence of the eccentricity becomes significant with the decrease of the diameter of a workpiece.