## 2 Concepts and Applications of TMPR
### Three-Dimensional Centerline Reconstruction of a Vascular Map Utilizing TMPR
The point \(\mathbf{r}_{\textit{corr}}\) closest to \(\mathbf{l}_{\textit{proj}}\) of  
Figure 4: (a) Coordinate axes of \(I_{\textit{ref}}\), \(\mathbb{R}_{\textit{ref}}\), \(\mathbf{r}_{\textit{ref}}\) and the index of \(\mathbf{r}_{\textit{ref}}\) (b) coordinate axes of \(I_{\textit{corr}}\), \(\mathbb{R}_{\textit{corr}}\), \(\mathbf{r}_{\textit{corr}}\), and the index of \(\mathbf{r}_{\textit{corr}}\); (c) \(\mathbf{r}_{\textit{ref}}(k)\) in \(I_{\textit{ref}}\); (d) \(\mathbf{l}_{\textit{proj}}\) of \(\mathbf{r}_{\textit{ref}}(k)\) and \(\mathbf{r}_{\textit{corr}}(t)\) selected as \(\mathbf{p}_{\textit{opt}}\) of \(\mathbf{r}_{\textit{ref}}(k)\); (e) \(\mathbf{r}_{\textit{ref}}(k+1)\) in \(I_{\textit{ref}}\); (f) \(\mathbf{l}_{\textit{proj}}\) of \(\mathbf{r}_{\textit{ref}}(k+1)\) and \(\mathbf{r}_{\textit{corr}}(t+1)\) selected as \(\mathbf{p}_{\textit{opt}}\) of \(\mathbf{r}_{\textit{ref}}(k+1)\); (g) \(\mathbf{r}_{\textit{ref}}(k+2)\) in \(I_{\textit{ref}}\) (h) \(I_{\textit{proj}}\) of \(\mathbf{r}_{\textit{ref}}(k+2)\) and candidate points \(\mathbf{r}_{\textit{corr}}(t+2)\), \(\mathbf{r}_{\textit{corr}}(t+3)\), and \(\mathbf{r}_{\textit{corr}}(t+4)\) for \(\mathbf{r}_{\textit{ref}}\) selection of \(\mathbf{r}_{\textit{ref}}(k+2)\); (i) \(\mathbf{r}_{\textit{ref}}(k+5)\), \(\mathbf{r}_{\textit{ref}}(k+6)\), and \(\mathbf{r}_{\textit{ref}}(k+7)\) in \(I_{\textit{ref}}\), where \(a_{1}\) represents the number of pixels between \(\mathbf{r}_{\textit{ref}}(k+5)\) and \(\mathbf{r}_{\textit{ref}}(k+6)\), and \(a_{2}\) represents the number of pixels between \(\mathbf{r}_{\textit{ref}}(k+6)\) and \(\mathbf{r}_{\textit{ref}}(k+7)\); (j) \(\mathbf{r}_{\textit{corr}}(t+6)\), \(\mathbf{r}_{\textit{corr}}(t+7)\), \(\mathbf{l}_{\textit{proj}}\) of \(\mathbf{r}_{\textit{ref}}(k+6)\), and \(\mathbf{p}_{\textit{opt}}\) of \(\mathbf{r}_{\textit{ref}}(k+6)\) in \(I_{\textit{corr}}\), where \(a_{3}\) represents the number of pixels between \(\mathbf{p}_{\textit{opt}}\) of \(\mathbf{r}_{\textit{ref}}(k+6)\) and \(\mathbf{r}_{\textit{corr}}(t+6)\), and \(a_{4}\) represents the number of pixels between \(\mathbf{p}_{\textit{opt}}\) of \(\mathbf{r}_{\textit{ref}}(k+6)\) and \(\mathbf{r}_{\textit{corr}}(t+7)\). \(r_{ref}(k)\), denoted as \(r_{corr}(t)\), is then assigned as \(\mathbf{p}_{opt}\) corresponding to \(r_{ref}(k)\). Once \(\mathbf{p}_{opt}\) for \(r_{ref}(k)\) is determined as \(\mathbf{r}_{corr}(t)\), the points \(\mathbf{p}_{opt}\) corresponding to the remaining points can be assigned using the following algorithm. First, \(\mathbf{l}_{proj}\) is calculated for \(r_{ref}(k+1)\), and the candidate points in \(\mathbb{R}_{corr}\) that satisfy the distance constraint are identified. The distance constraint ensures that the distance between \(l_{proj}\) and a given \(\mathbf{r}_{corr}\) point does not exceed \(d_{const}\). Depending on the vessel geometry, there may be one, multiple, or no \(\mathbf{r}_{corr}\) points, as illustrated in Figure 4e,f, Figure 4g,h, and Figure 4 ij, respectively. Figure 4e,f depicts the case where only one point of \(\mathbf{r}_{corr}\) meets the distance constraint. If the index of this \(\mathbf{r}_{corr}\) point, \(\mathbf{r}_{corr}(t+1)\), is greater than \(t\) (the index of the previously assigned \(\mathbf{p}_{opt}\)), then \(\mathbf{r}_{corr}(t+1)\) is designated as \(\mathbf{p}_{opt}\) for \(\mathbf{r}_{ref}(k+1)\). However, if the index of \(\mathbf{r}_{corr}(t+1)\) is less than or equal to \(t\), \(\mathbf{p}_{opt}\) corresponding to \(\mathbf{r}_{ref}(k+1)\) cannot be determined. Figure 4g,h illustrates the case where multiple points of \(\mathbf{r}_{corr}\) satisfy the distance constraint. The points of \(\mathbf{r}_{corr}\) that satisfy the distance constraint with \(l_{proj}\) of \(r_{ref}(k+2)\) are \(r_{corr}(t+2)\), \(\mathbf{r}_{corr}(t+3)\), and \(\mathbf{r}_{corr}(t+4)\). The point with the smallest index greater than the previously assigned indices (\(t\), \(t+1\)) is selected as \(\mathbf{p}_{opt}\), which is \(\mathbf{r}_{corr}(t+2)\).