## 2 Materials and Methods
### Methods
#### 2.2.1 Automated Seeded Region Growing  
The automated seeded region growing is an enhanced version of the original seeded region growing (SRG) algorithm [32]. SRG is famous for segmentation due to its rapid, robust, and easy-to-use  
Figure 1: Processing steps and methods used in the proposed nodule detection. procedure. Unlike other region-based algorithms, SRG does not require homogeneity criterion to determine the similarity between the voxels. Instead, it starts the segmentation by assigning sets of voxels called the initial seed points, where each set represents a separate region in the image. In each step of the algorithm, SRG assigns an unlabelled voxel \(v_{x,y,z}\) into one of the initial regions. The algorithm terminates when all the voxels in the image are labelled into different regions. For example, assume a volume image \(I\) has initial seed regions (\(S_{1}\), \(S_{2}\), \(S_{3}\)... \(S_{n}\) ) and a set of unlabelled voxels \(V\). Then, we want to label each element of \(V\) to each \(S_{i}\), where \(i=1,\) \(2,\)\(3,\)\(\ldots n\). \[S_{1}\), \(S_{2}\), \(S_{3}\),... \(S_{n}\)\(\subset\)\(I\), \(V\)\(\in\)\(I\)  
\[V=\Big{\{}v_{x,y,z}\
otin\cup_{i=1}^{n}S_{i}\Big{|}\ N\Big{(}v_{x,y,z}\Big{)}\ \cap\cup_{i=1}^{n}S_{i}\
eq\varnothing\Big{\}}, \tag{2}\]  
where \(N\Big{(}v_{x,y,z}\Big{)}\) is the 26 connected neighbors of \(v_{x,y,z}\). After the \(m\) step of the algorithm, \(N\Big{(}v_{x,y,z}\Big{)}\) meets just one of the initial seed regions \(S_{i}\), and \(N\Big{(}p_{x,y}\Big{)}\)\(\cap\)\(S_{i}\Big{(}p_{x,y}\Big{)}\
eq\varnothing\). Then, we calculate the difference grey value between \(v_{x,y,z}\) and its adjacent region by using the following equation. \[\partial\Big{(}v_{x,y,z}\Big{)}=\Big{|}g\Big{(}v_{x,y,z}\Big{)}-mean_{r\in S_ {i}}\ \big{[}g(r)\big{]}\Big{|}, \tag{3}\]  
where \(g\Big{(}v_{x,y,z}\Big{)}\) is the grey value of the unlabelled pixel, and the latter part of the equation is the mean grey value of all the voxels in the adjacent region \(r\). If \(v_{x,y,z}\) is located on the boundary of two or more regions, we assign \(v_{x,y,z}\) into the region that has the minimum \(\partial\) value. \[\partial(z)=min_{v_{x,y,z}\in P}\Big{(}\ \partial\Big{(}v_{x,y,z}\Big{)}\Big{)}. \tag{4}\]  
In this way, SRG allocates all the unlabelled voxels into separate regions. SRG is easy to use, even for the unskilled user who does not have detailed knowledge about the input image. Nevertheless, the segmentation result of SRG strongly depends on the initial seed regions. If the seed regions fall on the noisy or outlier voxels, it will result in poor segmentation. Moreover, user-oriented seed selection of SRG is also a flaw for fully automated detection. Therefore, we propose an enhanced version of SRG, which can identify the initial seed points automatically. Seed point selection is a vital task for SRG because correct segmentation results are strongly correlated to the seed points. Each seed point should represent a typical grey value of its region, but it should not be a noise or an outlier voxel. Fortunately, on a CT scan, specific organs exhibit a range of grey values, depending on the radiographic density, which is measured by the Hounsfield unit (HU). Based on this fact, we applied the HU value of the input CT scan for automatic seed point selection. As one instance, Figure 2a illustrates a histogram that shows the distribution of HU values in the input CT volume. In this histogram, HU values are ranging from \(-1024\) to \(2513\). Then, we can calculate the mean HU value of the image to find the seed points. The mean HU of the CT volume in this example is \(147.78\). Based on this mean HU value, we can assign the seed points automatically. In our research, we identify two initial seed points. The first seed point is selected by finding the very first voxel, which is less than the mean HU value. In Figure 2b, the first seed point is denoted by a blue square, and it represents black regions (low-density regions).