# Cascaded Regression-Based Segmentation of Cardiac CT under Probabilistic Correspondences
## 2 Materials and Methods
### Correspondence Point Construction  
As a number of the training set is \(n_{K}\) and \(\mathbf{X}=\{\chi_{k}\},1\leq k\leq n_{K}\) is defined, \(n_{K}\) observed 3-dimensional training point sets are defined as \(\chi_{k}=\{\mathbf{x}_{ki}\in\mathbf{R}^{3}|1\leq i\leq n_{k}\}\), where \(\mathbf{x}_{ki}=[x_{ki},y_{ki},z_{ki}]^{T}\) specifies the spatial coordinates of each point. The EM-based correspondence construction method updates the mean model every iteration. Let \(M=\{\mathbf{m}_{j}\in\mathbf{R}^{3}\},1\leq j\leq n_{M}\), be a model point set, where \(\mathbf{m}_{j}\) denotes the pure spatial coordinates. \(n_{M}\) denotes vertex number of the 3D model. Each \(S_{k}\) globally transforms the model points \(\mathbf{m}_{j}\in M\) to the space of \(X_{k}\). \(E_{kij}\) specifies the posterior probability that \(\mathbf{x}_{ki}\) are sampled from the Gaussian component of mean \(\mathbf{m}_{j}\) in each iteration. Each point set is regarded as a spatially transformed GMM sample, and the registration problem is formulated to estimate the underlying GMM from training samples \(\mathbf{x}_{ki}\). Thus each Gaussian in the mixture specifies model point \(\mathbf{m}_{j}\), which is probabilistically corresponded to a training point \(\mathbf{x}_{ki}\). In EM, the parameter set is defined as \(\{S_{k}\),\(M\),covariance}. Exact modeling can be found in [PERSON] et al.'s paper [26].  
Shape models are constructed by PCA under Equation (2), and this method requires one-to-one point correspondences. As no exact point correspondences are made in the EM-based method directly, point correspondences are identified by a virtual correspondence [26]. Compared with original modeling, virtual correspondence's modeling of the proposed method considers the independence of objects. Before the virtual correspondences of three objects are constructed separately, label selection among three virtual correspondences is performed along with \(n_{M}\) point set. Since an object label has the same value in the same spatial location of \(n_{K}\) virtual correspondence sets, statistics among \(n_{K}\) virtual correspondence sets can help to find object labels.  
Let us denote \(U_{k}^{b}\) as a subset of input point set \(\chi_{k}\) which includes points corresponding to object \(b\). Here object 0, 1 and 2 are LA, the inner surface of LV, and the outer surface of LV, respectively. \(U_{k}^{b}\) is found by searching the nearest points among point sets of three objects under given object label masksin pre-processing. For any model point \(\mathbf{m}_{j}\), a virtual correspondence point of \(U_{k}^{b}\) denoted by \(\mathbf{m}_{kj}^{b}\) is induced by the following equation.  
\[\mathbf{m}_{kj}^{b}=\sum\limits_{i\in U_{k}^{b}}\frac{E_{kij}}{\sum\limits_{l \in U_{k}^{b}}E_{klj}}S_{k}^{-1}*\mathbf{x}_{ki} \tag{1}\]  
Here, \(\mathbf{m}_{kj}^{b}\) is calculated in each object along with each \(j\) point. Let us define \(C\in\mathbf{Z}^{b\times N_{M}}\) flag matrix for the object label, and we initialize all elements of \(C\) as 0. Here, \(\mathbf{Z}\) means an integer. This \(C\) matrix accumulates as majority voting on the index of \(\underset{b}{\operatorname{argmax}}\sum\limits_{i\in U_{k}^{b}}E_{kij}\). A final virtual correspondence point \(\mathbf{m}_{kj}\) is selected as \(\mathbf{m}_{kj}^{b}\) if \(C\) is maximum in \(b\) object in one virtual correspondence point \(j\).