# Cascaded Regression-Based Segmentation of Cardiac CT under Probabilistic Correspondences
## 2 Materials and Methods
### T\({}_{a}\) Construction Method  
We can find the closed iso-surface mesh from the training input mask by using the marching cube algorithm [30]. This triangulation is based on \(\chi_{k}\). However, 3D triangulation of point set of iso-surface mesh finds only convex hull. Therefore, we performed the filtering of the tetrahedron center on which training input mask has labels. This filtering can make the shape of tetrahedrons to be similar with the training input mask. This filtering is denoted as constrained triangulation. The result tetrahedron set of filtering is denoted as \(D_{k}=\{t_{kl}=(p_{1l},p_{2l},p_{3l},p_{4l})|1\leq l\leq c_{k},p_{s1}\in \mathbf{Z},1\leq p_{s1}\leq n_{k}\}\). Here \(p_{s1}\) is the index of any point, and \(c_{k}\) is total tetrahedron number of \(k\) input. It is not easy to calculate \(T_{k}\) from \(D_{k}\), since we can not know the exact reference mask for \(T_{k}\).  
\(T_{k}\) is tetrahedron set under \(\mathbf{m}_{kj}\) set. Therefore, tetrahedron mapping is calculated between \(D_{k}\) and \(T_{k}\) to calculate \(T_{k}\) by using \(E_{k,p_{s1}j}\). On one point \(p_{s1}\), we find \(m\) largest effective corresponding model points of \(\mathbf{m}_{kj}\) by sorting \(E_{k,p_{s1}j}\) values. \(m^{4}\) tetrahedrons of \(T_{k}\) are constructed from four model point sets in every combination. In other words, one tetrahedron of \(D_{k}\) produces \(m^{4}\) tetrahedrons of \(T_{k}\). After changing one tetrahedron into volume point set with using inside check of tetrahedron, tetrahedron counts on volume grid image are increased by using this volume point set. Therefore, accumulation of tetrahedrons generated under \(T_{k}\) is performed on the volume grid image which contains the tetrahedron counts. If we want to constitite dense point set of a target object, thresholding this volume grid image makes that dense point set. We tested \(m\) in 1, 2, 3 and 4 values. Selection of 3 was optimal under computation time and reconstruction quality. Reconstruction quality was visually evaluated by checking the missing structure manually in every case. The resulting volume accumulation is shown in Figure 1.  
As this volume accumulation generated from this mapping uses much redundancy, we simplify volume accumulation into tetrahedron sets by using a filtering result of 3D Delaunay triangulation of \(M\). This 3D Delaunay triangulation of the model points is performed with the alpha radius of 40 to remove noise tetrahedrons [31].  
To extract one statistical model tetrahedron of all simplified \(T_{k}\), we use statistics in every \(T_{k}\). In the tetrahedron statistics calculation, tetrahedrons between the LV and LA also are considered as the cases of 0.5 probability in two objects. The tetrahedrons composed of only inner surface points of LV are not considered in the statistics. As common reference tetrahedrons are made from 3D Delaunay triangulation of \(M\), statistics analysis is possible. As input \(T_{a}\) of cascaded regression, atlas tetrahedron set is made by using tetrahedrons above threshold-probability of statistic probability accumulation array. Threshold-probability was selected as 14% under an empirical decision. Although the surface of the atlas tetrahedron set is not realistic, reconstructed \(T_{k}\) makes realistic shapes in Figure 2. With the found \(T_{a}\), we make the final segmentation mask under the shape parameter set found.