# Cascaded Regression-Based Segmentation of Cardiac CT under Probabilistic Correspondences
## 1 Introduction
This method uses also boundary search along the normal direction. Compared with these methods, the proposed method used active appearance model, and did not adopt the boundary search-based final processing, although this skip eliminated much accuracy improvement. Since high gradient boundary information is needed to perform the boundary normal search, the proposed method made statistical modeling comparisons under the active appearance framework. [PERSON] et al. proposed a part-based left atrium model, which includes the chamber, the appendage, four major pulmonary veins and right-middle pulmonary veins [17]. [PERSON] et al. proposed the automatic segmentation of the LA by using weighted voting label fusion and a variant of the demons registration algorithm [8]. Recently, many papers have been published in the heart segmentation field. In MRI, a fully convolutional neural network (FCN) was used for segmenting the left and right ventricles [18]. [PERSON] et al. proposed fully automatic whole heart segmentation, based on multi-label CNN and using volumetric kernels. After they localize the center of the bounding box around all heart structures, segmentation of the fine detail of the whole heart structure within the bounding box is performed [19; 20]. [PERSON] et al. developed a localization method to extract the bounding box around the LV using a combination of three CNNs [21]. [PERSON] and [PERSON] proposed an automatic whole heart segmentation framework combined CNN with statistical shape priors. The additional shape information is used to provide explicit 3D shape knowledge to CNN [22; 23]. Like [PERSON] and [PERSON]'s method, AAM can also be combined with CNN to improve the system accuracy and better understand the segmentation system. Group-wise registration of point sets is a fundamental step in creating a statistical shape model. The probabilistic correspondence construction method is developed with a probabilistic view of estimating correspondence [24; 25; 26; 27]. The correspondence for each point on one shape is formulated as a weighted combination of all points on the other shape, where the weights/probabilities are derived from a probabilistic function of the pairwise distances. [PERSON] et al. extended pairwise matching to group-wise correspondence construction [26]. They proposed a sparse model selection method and similarity parameter update equations of the Gaussian Mixture Model (GMM)-based method. In the proposed method, only simple probabilistic correspondence without a sparse model selection will be adapted. In [PERSON] et al. 's method, only point correspondences were constructed without appearance correspondence, because their research was performed only for constructing statistical shape models, and did not extend to a segmentation method. Since the cascaded regression model is constructed on appearance correspondence, the exact shape of the point model must be estimated from input reference volume. The proposed method made appearance correspondence probabilistic estimation under the mapping of tetrahedron sets. The reason why the appearance correspondence is needed is explained in detail in Section 2.2. Moreover, the proposed method deals with three object models. As there is close interaction between two objects (inner and outer surfaces) of the left ventricle, we made a new separate model point construction method. We compare the proposed method with the simple and popular pairwise method [28] based on similarity and nonrigid transformations. In the present study, we developed cascaded regression-based segmentation software of a double-shaped heart model with point and appearance correspondence under the modified group-wise correspondence construction method.