## 2 Concepts and Applications of TMPR
### Calculation of a Relative Rotation Angle of the C-Arm Utilizing TMPR  
In this section, we propose a method to estimate the relative rotation angle between two views. When we conduct an operation with an MNS, some corners of the structure can be captured in X-ray images. We can calculate the 3D coordinates of these corners with respect to given relative rotation angle of the C-arm, utilizing the TMPR. If the actual distances between corners are known, these can be utilized as the constraints to estimate the relative rotation angle between the two views.  
The specific method is as follows. First, we select the corners of the MNS structure where the distances between them are known. We denote the available four corners on the MNS as \(\mathbf{P}_{A}^{\varnothing}\), \(\mathbf{P}_{B}^{\varnothing}\), \(\mathbf{P}_{C}^{\varnothing}\), and \(\mathbf{P}_{D}^{\varnothing}\), and the set of these four points is denoted as \(\mathbb{P}_{w}\). These four corners must appear in both \(I_{\textit{ref}}\) and \(I_{\textit{corr}}\) to utilize them as corresponding points and to calculate the 3D coordinates utilizing TMPR. We define \(\mathbb{A}_{\theta,\ \varphi}\) as a pair of \(\theta\) and \(\varphi\), where \(\theta\)is a relative angle in the RAO/LAO direction and \(\varphi\) is a relative angle in the CAUD/CRAN direction. When the relative rotation angle of the C-arm displayed on the control program is \(\mathbb{A}_{\theta_{disp},\varphi_{disp}}\) and the resolution of the displayed angle is \(n\), the available relative rotation angles and the total error ranges of \(\pm 2n\) in each direction are as shown in (12) because an error tolerance of \(\pm n\) must be considered for both the anterior-posterior (AP) view and the oblique view. The distance between \(\mathbf{l}_{proj}\) and \(\mathbf{p}_{corr}\) represented in \(I_{corr}\) can be defined as \(d(\mathbf{l}_{proj},\mathbf{p}_{corr})\). Within the available range of angles, we can find candidate angles (\(\mathbb{A}_{\theta,\varphi}^{*}\)) with the condition that \(d(\mathbf{l}_{proj},\mathbf{p}_{corr})\) is less than \(d_{const}\) for all four corresponding points, as shown in (13). Then, 3D coordinates of the four corners \(\mathbb{P}^{*}=\left\{\mathbf{P}_{A}^{*},\mathbf{P}_{B}^{*},\mathbf{P}_{C}^{*},\mathbf{P}_{D}^ {*}\right\}\) are derived depending on \(\mathbb{A}_{\theta,\varphi}^{*}\), and the distances between each corner (\(d(\mathbf{p}_{A}^{*},\mathbf{P}_{B}^{*})\), \(d(\mathbf{p}_{A}^{*},\mathbf{P}_{C}^{*})\), \(d(\mathbf{P}_{D}^{*},\mathbf{P}_{B}^{*})\), and \(d(\mathbf{P}_{D}^{*},\mathbf{P}_{C}^{*})\)) are calculated. Subsequently, we can compare the calculated distances with the actual distances between the four corners, as shown in (14). The optimal angles (\(\theta_{opt}\), \(\varphi_{opt}\)) can be determined with minimal error, as shown in (15). Through this process, the relative rotation angle between two views can be estimated from four corners of the MNS structure.  
\[\mathbb{A}_{\theta,\varphi}=\left\{(\theta,\varphi)|\theta_{ disp}-2n\leq\theta\leq\theta_{disp}+2n,\right. \tag{12}\] \[\left.\varphi_{disp}-2n\leq\varphi\leq\varphi_{disp}+2n\right\}\]  
\[\mathbb{A}_{\theta,\varphi}^{*}=\left\{\mathbb{A}_{\theta,\varphi}\left| \forall P\in\mathbb{P}_{w},d\left(\mathbf{l}_{proj}^{p},\mathbf{p}_{corr}^{p}\right) \leq d_{const}\right\}\right. \tag{13}\] \[Error=\frac{1}{4}\sum_{i\in\mathbb{A}_{\mathbb{A}_{\mathbb{D}}}} \sum_{j\in\mathbb{B}_{\mathbb{C}}}\left|d\left(\mathbf{P}_{i}^{w},\mathbf{P}_{j}^{w} \right)-d\left(\mathbf{P}_{i}^{*},\mathbf{P}_{j}^{*}\right)\right|\] (14) \[\mathbf{Determine}\mathbb{A}_{\theta,\varphi}^{opt}\text{ with the minimum }Error|_{\mathbb{A}_{\theta,\varphi}} \tag{15}\]