## 3 Experimental Results
### In Vitro Experiment  
Figure 5 shows an in vitro experiment where a mock-up simulating an MNS workspace was designed as a rectangular acrylic box, and a jig simulating a simplified 3D centerline of a blood vessel was designed with a copper wire. At the four corners on the top of the mock-up, we attached four radiopaque tapes (M\({}_{1}\), M\({}_{2}\), M\({}_{3}\), and M\({}_{4}\)), whose width and height were 12 mm and 30 mm, respectively. The distance between M\({}_{1}\) and M\({}_{3}\) or M\({}_{2}\) and M\({}_{4}\) was 130 mm, and the distance between M\({}_{1}\) and M\({}_{2}\) or M\({}_{3}\) and M\({}_{4}\) was 188 mm. During the experiments, the inner four corners of two markers (M\({}_{1}\) and M\({}_{2}\) for RAO rotation and M\({}_{3}\) and M\({}_{4}\) for LAO rotation) on one side were utilized as corresponding points, as shown  
in Figure 5a. Also, the width of each tape (12 mm) and the distance from a marker to the opposite marker (188 mm) were utilized as the distance constraints described in Section 2.4. In the jig, the shape of the 3D centerline was designed as a straight line on the surface of the jig to accurately obtain the actual coordinates. The actual coordinates of the 3D centerline and the four corners in the \(C_{\textit{rrf}}\) were determined from the design values and from measurements using a laser sensor (OD1000, SICK AG, Waldkirch, Germany). The C-arm in this experiment was an Artis Zee Biplane rf-1000-125 (Siemens Healthineers, Forchheim, Germany), a biplane C-arm, of which we utilized only a single C-arm. The resolution of the rotation angle displayed on the control program was \(1^{\circ}\). The output image size was 1024 \(\times\) 1024 pixels, and each pixel had the same height and width. The actual length of 1 pixel in the image was 0.36 mm when the SID was 1200 mm. The sign of the rotation angle was set as positive for the RAO and CAUD directions and negative for the LAO and CRAN directions, considering the signs of the X and Y axes in \(C_{\textit{rrf}}\). An anterior-posterior (AP) view was captured as a reference view (\(V_{\textit{rrf}}\)), and the other 20 views were captured as corresponding views (\(V_{\textit{corr}}\)) under the 20 combinations of \(\pm 10^{\circ}\) and \(\pm 15^{\circ}\) in the RAO/LAO direction (\(\theta_{\textit{disp}}\)) and \(0^{\circ}\), \(\pm 5^{\circ}\), and \(\pm 10^{\circ}\) in the CAUD/CRAN direction (\(\varphi_{\textit{disp}}\)). First, we calculated the relative rotation angle of the C-arm between the two views and the 3D coordinates of the four corners in \(C_{\textit{rrf}}\), utilizing the proposed method. We placed the mock-up and the jig on the patient bed, and we captured \(I_{\textit{rrf}}\) and \(I_{\textit{corr}}\) by rotating the C-arm in the order of RAO/LAO(\(\theta\)) and CAUD/CRAN(\(\varphi\)). The 2D coordinates of the four corners in each image (\(\textit{prf}\), \(\textit{pcor}\)) were detected using the vision assistant module of LabVIEW. Then, for every relative rotation angle within the range defined by (13), we calculated the \(d(I_{\textit{rrf}},\textit{pcor})\) of the four corresponding points and the distances between the 3D coordinates of corresponding points by changing the rotation angle by \(0.1^{\circ}\). The \(d_{\textit{const}}\) in (13) was set as two pixels, and the set of \(\mathbb{A}_{\theta,\ \varphi}^{*}\) that satisfies (13) was determined. Then, the \(\mathbb{A}_{\theta,\ \varphi}^{\textit{opt}}\) with the minimum error of (14) was determined among the set of \(\mathbb{A}_{\theta,\ \varphi}^{*}\). Although there were multiple values of \(\mathbb{A}_{\theta,\ \varphi}^{*}\) that satisfy (13), \(\mathbb{A}_{\theta,\ \varphi}^{\textit{opt}}\) can be uniquely determined using the distances between the calculated 3D coordinates of the corresponding points.