# Cascaded Regression-Based Segmentation of Cardiac CT under Probabilistic Correspondences
## 1 Introduction  
Compared with imaging modalities such as ultrasound and magnetic resonance imaging, cardiac computed tomography (CT) can provide more comprehensive anatomic information on the heart chambers, large vessels, and coronary arteries [1]. Pulmonary veins have been isolated using various catheter techniques in patients with atrial fibrillation [2, 3]. Many of the approaches incorporate cardiac mapping systems because the capabilities of these systems are useful in catheter ablation procedures [4]. The registration of three-dimensional anatomic models with an interventional system could support diagnosis and help navigate mapping and ablation catheters off complex structuressuch as the left atrium (LA). However, the creation of 3D models for such cardiac mapping systems is time-consuming, and the models suffer from issues with repeatability among operators. The present study aimed to construct a double-shaped model composed of the left ventricle (LV) and LA. The convolutional neural network (CNN)-based method needs the training of significant parameters in neural networks [5]. In contrast, as a learning-based system, the active appearance model (AAM) can be adapted to heart segmentation. [PERSON] et al. proposed a three-dimensional AAM for cardiac images [6]. This system produced a 3D shape model by using Procrustes analysis and 2D contours sampled slice-by-slice. However, the usage of 2D contour modeling restricts the topology of the input shape. AAM learns the structure of the target organ [6; 7; 8], and this training method is used together with deep learning for prostate segmentation [9]. In AAM, the shape is expressed as a linear combination of shape bases learned via PCA, while appearance is modeled as the volume enclosed by the shape mesh [10]. Therefore, the AAM-based method matches statistical models of appearance to images. [PERSON] et al. [10] constructed an iterative matching algorithm by learning the relationship between perturbations in the model parameters and the induced image errors. Compared with [PERSON] et al. 's method using a fixed distribution, the cascaded regression method can produce better accuracy by using a large cascade depth [10; 11]. [PERSON] et al. 's method performs several iterations by changing the step size in one regression result. Cascaded pose regression refines a loosely specified initial guess, where a different regressor performs each refinement. Each regressor executes simple image measurements that are dependent on the output of the previous regressors [12]. This advancement of the regressor system is developed in several ways [11; 13; 14; 15]. The present paper used the supervised descent method (SDM). From SDM, we adapted descent direction, and stochastic modeling of appearance data is added. In the present study, we extended this 2D framework of a deformable facial model into a 3D framework for 3D cardiac segmentation. Since the variation of the 3D heart shape is more significant than that of the facial model, this extension did not make the best segmentation compared with other state-of-the-art methods [7; 16]. Boundary search along the normal direction improves the segmentation accuracy considerably [7; 16]. [PERSON] et al. 's method [7] uses marginal space learning and projection on the subspace constructed by PCA, in contrast to the cascaded regression used in the method proposed in the present study. The final segmentation step uses the boundary detectors to move each control point along the normal direction to the optimal position, where the score from the boundary detector is the highest. This method produced point-to-mesh error with 1.32 mm in the LA, and 1.17 mm in LV on 323 volumes from 137 patients. High accuracy comes from boundary search-based non-rigid deformation. [PERSON] et al. 's method deforms surface triangles with similarity transformation, affine transformation, and deformable adaptation sequentially. The whole heart segmentation produced surface-to-surface error of 0.82 mm on 28 computed tomography images.