## 2 Concepts and Applications of TMPR
### Three-Dimensional Centerline Reconstruction of a Vascular Map Utilizing TMPR  
In Sections 2.3 and 2.4, we assumed a clear correspondence of points between two images. However, this correspondence is not clear for the path where the pixel points are continuously connected and close to each other. To solve this problem and reconstruct a vascular map, [PERSON] [19] defined an energy function as the objective function which is designed to achieve accurate vessel centerline reconstruction while maintaining a smooth vascular structure. The best 3D coordinates were estimated by minimizing this energy function, as shown in (16), even if the correspondence relationship was not clear:  
\[E(\mathbb{Z})=\sum_{r\in\mathbb{R}}\mathcal{D}_{r}(Z_{r})+\beta\sum_{r\in \mathbb{R}}\sum_{s\in\mathbb{S}}V_{r,\,s}(Z_{r},Z_{s}) \tag{16}\]  
where \(r\) represents a pixel point on 2D blood vessel centerline extracted from the \(I_{ref}\), and \(s\) denotes a neighborhood point of \(r\). \(\mathbb{R}\) and \(\mathbb{S}\) denote the set of \(r\) and \(s\), respectively. \(\mathcal{D}_{r}(Z_{r})\) is the average pixel distance of each point \(r\), which is a distance between a projected pixel point and the nearest point on the centerline in \(I_{corr}\). \(V_{r,s}(Z_{r},Z_{s})\) is the smoothness term representing the difference in \(Z\) coordinates from neighborhood points, and \(\beta\) is the coefficient of the sum of smoothness term. \(\mathbb{Z}\) denotes a set of \(Z\) coordinates for all \(r\). To find the optimal solution of \(E(\mathbb{Z})\), [PERSON] utilized the \(\alpha\)-expansion method [22] to minimize the energy function by changing the \(Z\) coordinate of each \(r\) for every equal interval. The accuracy and computation cost of the \(\alpha\)-expansion method are affected by the initial \(Z\) coordinate because this method iteratively searches the optimal \(Z\) coordinate from the initial \(Z\) coordinate to minimize the energy function. [PERSON] selected initial \(Z\) coordinates of each point randomly or in fixed order. However, we can effectively determine the initial \(Z\) coordinate utilizing the proposed TMPR. Subsequently, the overall process of reconstructing the vascular map using TMPR is described and illustrated in Figure 4. The X-ray images at \(V_{\textit{ref}}\) and \(V_{\textit{corr}}\) are denoted as \(I_{\textit{ref}}\) and \(I_{\textit{corr}}\), respectively. The points located along the vascular centerline in \(I_{\textit{ref}}\) and \(I_{\textit{corr}}\) are defined as \(\mathbf{r}_{\textit{ref}}\) and \(\mathbf{r}_{\textit{corr}}\), respectively, with the sets of these points represented as \(\mathbb{R}_{\textit{ref}}\) and \(\mathbb{R}_{\textit{corr}}\). The points \(\mathbf{r}_{\textit{ref}}\) and \(\mathbf{r}_{\textit{corr}}\) are indexed sequentially along the vascular direction, and the number of points included in \(\mathbb{R}_{\textit{ref}}\) and \(\mathbb{R}_{\textit{corr}}\) may not be equal, depending on the range of the blood vessel captured in each image. In Figure 4a,b, indices are assigned from the upper-left to the lower-right of \(I_{\textit{ref}}\) and \(I_{\textit{corr}}\), following the vascular direction. Figure 4a,b represents \(\mathbb{R}_{\textit{ref}}\) and \(\mathbb{R}_{\textit{corr}}\), respectively. After indexing \(\mathbb{R}_{\textit{ref}}\) and \(\mathbb{R}_{\textit{corr}}\), the line \(\mathbf{l}_{\textit{proj}}\) is calculated for each \(\mathbf{r}_{\textit{ref}}\) using (11) from Section 2.3. Among the points \(\mathbf{r}_{\textit{ref}}\) in Figure 4c, those whose \(\mathbf{l}_{\textit{proj}}\) intersect the line connecting all \(\mathbf{r}_{\textit{corr}}\) points in \(\mathbb{R}_{\textit{corr}}\) at one point, as shown in Figure 4d, are identified. The point \(\mathbf{r}_{\textit{ref}}\) with the smallest index among them is designated as \(\mathbf{r}_{\textit{ref}}(k)\).