# Cascaded Regression-Based Segmentation of Cardiac CT under Probabilistic Correspondences
## 2 Materials and Methods
### Shape and Appearance Models  
Because we can judge if (x, y, z) is inside \(T_{a}\), by simple calculation, the point set \(P_{a};a=0,...,n_{T}-1\) is constructed in a regular grid of 3D input. The \(n_{T}\) group of a point set \(P_{a}\) is concatenated with all tetrahedrons of \(T_{a}\), and it is denoted as \(P_{T}\). The array \(P_{T}\) becomes a full set for the target points of appearance. From this array, a random selection of appearance is performed under a given number. The input image does not have any requirements in size and spacing, because appearance sampling is performed based on physical locations.  
The transform between two images is constructed by piecewise-affine transform through the correspondence of tetrahedron vertexes. Specifically, the meshes of the training image have the same size because they are produced from the correspondence construction procedure. Between the atlas mesh and input mesh, we can perform a piecewise-affine transform [32]. Appearance correspondence can be constructed using these transforms. Because we know the \(P_{T}\) point set of an atlas, we can rapidly extract the intensity and additional data of the training target by using the piecewise-affine transform on the \(P_{T}\) point set.  
Figure 1: Volume density of appearance texture with soft correspondence of \(m\) largest selection in any case. The axial view of the left heart is shown. (**a**) \(m=1\). (**b**) \(m=2\). (**c**) \(m=3\). (**d**) \(m=4\).  
Figure 2: Model tetrahedron set \(T_{a}\) and, reconstructed \(T_{k}\) of 4 cases. Five cases have the same tetrahedron cell topology and cell number.  
The training input image \(I_{k}\) is from a three-dimensional volume and two shapes are defined with \(u_{1}\) and \(u_{2}\) tetrahedron vertexes. The set of all vertexes is a vector \(\in\mathbf{R}^{3(u_{1}+u_{2})\times 1}\) that defines two models of the heart. This set is brought from a virtual correspondence point \(\mathbf{m}_{kj}\) of group-wise correspondence construction procedure. Since the correspondence construction method makes similarity-free models, every shape correspondence does not have variations due to similarity transforms (translation, rotation, and scaling). PCA is applied to obtain the shape model. This shape model increases the number of eigenvectors independently with appearance information. The model is defined by the mean shape \(s_{0}\) and the ith shape eigenvector \(s_{i}\) of \(n_{sh}\) number. Shape eigenvectors are represented as columns of the matrix \(\mathbf{B}\in\mathbf{R}^{3(u_{1}+u_{2})\times n_{sh}}\). This model also describes the ith eigenvalue \(\lambda_{i}\). Finally, to model similarity transforms, \(\mathbf{B}\) is appended with six additional bases [33]. An instance of the shape model is constructed by  
\[s(\mathbf{p})=s_{0}+\mathbf{B}\mathbf{p} \tag{2}\]  
where \(\mathbf{p}\in\mathbf{R}^{n_{sh}\times 1}\) is the vector of the shape parameters.  
In order to learn the appearance model, each training image \(I_{k}\) is warped to a reference frame by using \(S_{k}\) to produce a similarity-free appearance. Since voxels in a tetrahedron are found by applying the piecewise-affine transform on the \(P_{T}\) point set, an array of voxels included on each tetrahedron can be constructed rapidly. The total number of voxels on every tetrahedron is denoted as \(n_{v}\). Three features on one voxel are accumulated to construct an appearance set \(\in\mathbf{R}^{3n_{v}\times 1}\). The original intensity of the training image is used as the first feature, which is normalized with statistics in convex envelop volume of LV containing the cavity region of LV [33]. The sum of x, y, and z on the gradient is used as the second and third features, under Gaussian smoothing with 0.5\(\sigma\) and 1.0\(\sigma\). These three features are normalized by the z-score. Subsequently, only mean appearance \(A_{0}\) is calculated.