### I. Introduction

### II. Methods

### 1. Pre-processing

*u*is an image,

*u*and

_{x}*u*are the derivatives of

_{y}*u*w.r.t.

*x*and

*y*, respectively. This method reduces image noise without removing important parts, such as edges or features comparison with isotropic filters, e.g., Gaussian filter. To minimize Equation (1), we use the Euler-Lagrange equations.

### 2. Segmentation of Whole Heart

*k*clusters in which each item of data belongs to the cluster with the nearest mean. It is based on minimization of a formal objective function, and it is the method that is most widely used and studied. Given a set of

*n*data points in real

*d*-dimensional space,

*R*and an integer

^{d}*k*, the problem is to determine a set of

*k*points in

*R*, called centers, so as to minimize the mean squared distance from each data point to its nearest center. This clustering type falls into the general category of variance-based clustering. Also, we expand the heart region by comparing the mean CT values of all clusters, because each chamber generally has a higher CT value than the cardiac muscle. Thus, the clusters, which have lower mean intensity than the threshold value of 220 HU, are removed as cardiac muscles, and the other clusters are merged.

^{d}### 3. Detection of Seed Volumes of Left and Right Heart Regions

### 4. Separation of Left and Right Heart Regions

*I*is the normalized gradient of the image

*I*, and β is a weighting coefficient. Here, β was experimentally determined as 1.

*FO*and background

*BA*seeds, the energy model of the power watershed is given by where

*x*represents the target configuration, and

*p*and

*q*are values that provide a reference for the different algorithms. This model is represented as a power watershed in

*p*→ ∞ and

*q*= 2 [17]. Therefore, we set the foreground seed volume as the candidate region of the LV (left vectricle) that was detected in the previous step, and we set the background seed volume as the candidate region of the RV (right vectricle) that was also detected in the previous step. Then, we can separate the left (i.e., LV and left atrium [LA]) and right (i.e., RV and right atrium [RA]) parts of the heart using the power watershed framework. To apply the power watershed algorithm to extract the left and right heart regions, we detect the seed volume of each side of the heart. We set the seed volumes to detect the left and right regions by comparing x coordinates of the whole heart mask volume.

### 5. Fine Segmentation of Left and Right Heart Regions Using Level-Set Method

**x**(

*t*) have a zero level-set function value. Therefore, the following equation holds:

*F*represents the speed moving in the direction of the normal vector away from the curve; therefore, x'(

*t*) · n =

*F*, where

*t*= 0) is given, the evolution equation for ϕ can be derived as

*t*), according to the time and speed function

*F*, is calculated as

*F*, which is the intensity value of the speed image as follows: where ε controls the relative importance of the curvature component for level-set propagation. The zero level-set curve moves proportionally to the propagation speed, which is very slow close to high gradient areas and fast in low gradient areas. The propagation speed term makes the zero level-set curve propagate until it reaches the edges of anatomical structures. The curvature speed term regularizes the zero level-set curve by smoothing the high curvature region. Because of the normal vector,

_{prop}*K*is calculated as the divergence of the unit normal vector:

### 6. Detection of Atrium and Ventricle from Left and Right Heart Region

*E*is the separation energy function,

_{sep}*p*is a voxel position, and θ and φ are the angles with the second and third principal axis, respectively. Here, ω

*and ω*

_{area}*are weights (ω*

_{int}*+ ω*

_{area}*= 1), and*

_{int}*E*is the energy term of the area, defined as

_{area}*M*(

*) is the mask volume of the left or right side of the heart, and Ω*

_{x}_{M}is a plane that passes through point

*p*with its normal vector oriented toward the angles, θ and φ. Here,

*E*is the energy term of the intensity, defined as where

_{int}*I*(

_{x}) is the intensity value of the CT image, and

*E*is regarded as the mean intensity of the intercepted plane with the mask volume. We obtain the separation plane by minimizing the separation energy function. Once

_{int}*p*, θ, and φ are detected, we obtain the normal vector and the position of the plane. This plane is used to split the left or right sides of the heart into the atrium and ventricle.

### III. Results

*V*and

_{auto}*V*are the set of voxels in automatically and manually segmented objects, respectively. The false positive error

_{manual}*E*is the ratio of the set of voxels in an automatically segmented object but not in a manually segmented object to the set of voxels in a manually segmented object. The false negative error

_{fp}*E*is the ratio of the set of voxels in a manually segmented object but not in an automatically segmented object to the set of voxels in a manually segmented object. The volume measurement error

_{fn}*E*is the ratio between the automatically and manually segmented volumes. The similarity error

_{vol}*E*is defined by the similarity index [19].

_{sim}*E*ranged from 3.14% to 4.04%. The average value of

_{sim}*E*was 3.43% ± 0.41% for all the datasets, indicating that the average difference between manual and automatic segmentation was less than 3.3%, approximately.

_{sim}*E*in (9). The average value of

_{sim}*E*for CVM was 12.74% ± 7.40% for all the datasets, respectively. Specifically, some regions in the RV and RA were not included using the CVM. However, these were accurately segmented using the proposed method. Our proposed method exhibited the most accurate segmentation results with the lowest value of

_{sim}*E*.

_{sim}