# Pulmonary Fissure Detection in 3D CT Images Using a Multiple Section Model
## 2 Related Works  
### A Derivative of Stick Filter (DoS) for Fissure Enhancement  
Considering the typical thin curvilinear shape of fissure profiles inside 2D cross-sections, [PERSON] et al. [13; 14] proposed a DoS filter for pulmonary fissure enhancement in CT images. Their main idea was to simulate the fissure-line appearance by defining a nonlinear derivative with three parallel stick kernels in varying directions from 0 to \(\pi\). The benchmark angle \(\beta\) of the stick kernels is approximately \(\pi/20\) with the sticks length \(L=11\) pixel as shown in Figure 1a. The method is useful and effective for fissure enhancement, but it only utilizes the intensity information and discards the orientation information of the operators, which might lead to a problem in the case of low contrast and adhering clutter interference.  
To better express the DoS filter, we used \(u_{M}\), \(u_{L}\) and \(u_{R}\) to indicate the mean intensity respectively along the middle (red), left (green) and right (blue) sticks of the kernel in Figure 1b. The mean intensity was calculated with  
Figure 1: The principle of the derivative of stick (DoS) filter. Here, (**a**) denotes the orientation of the DoS filter, (**b**) depicts the responses of the DoS filter on typical pulmonary structures with the characters H, M and L indicate the high, moderate and low values, respectively. Note this figure is copied from [13] with permission.  
\[u=\frac{1}{L}\sum_{j=1}^{L}I_{j} \tag{1}\]  
The nonlinear derivatives perpendicular to the sticks can be described as  
\[\lambda_{\perp,\text{max/min}}^{S,\theta}(x)=\text{max/min}(u_{M}-u_{L},u_{M}-u _{R})\quad 0\leq\theta\leq 20\beta \tag{2}\]  
where \(S\) denotes the spacing between adjacent sticks, \(\theta\) is the stick orientation and \(x\) the spatial location within the lung. To further suppress the blob shape, a measurement criteria was described:  
\[\lambda_{\parallel}^{S,\theta}(x)=\sqrt{E\left(I_{j}^{2}\right)-\left(E\left( I_{j}\right)\right)^{2}} \tag{3}\]  
with \(E\) being an expected value operator. Accordingly, two 2D line strength measures \(l_{\text{max}}\) and \(l_{\text{min}}\) can be formulated as  
\[\lambda_{\text{max/min}}^{S,\theta}(x)=\lambda_{\perp,\text{max/min}}^{S, \theta}(x)-k*\lambda_{\parallel}^{S,\theta}(x) \tag{4}\]  
Here, \(k=0.7\) is a coefficient [13]. Thus, the multiple directional information can be integrated with  
\[F_{\text{max/min}}(x)=\text{max}\Big{(}\underset{1\leq i\leq 20}{\max}\left(I_{ \text{max/min}}^{S,\theta_{i}}\right),0\Big{)} \tag{5}\]  
Then, a max-min cascaded filter was designed to over-emphasize the step-edge suppression. Therefore, the fissure response can be described as:  
\[F_{o}(x)=F_{\text{max}}(x)\;\;F_{\text{min}}(x) \tag{6}\]  
Finally, the 3D shape-tuned fissure response was defined as  
\[F^{3D}=\left(F_{o}^{A}+F_{o}^{S}+F_{o}^{C}\right)*\frac{\text{median}\Big{(}F _{o}^{A},F_{o}^{S},F_{o}^{C}\Big{)}}{\text{max}\Big{(}F_{o}^{A},F_{o}^{S},F_{ o}^{C}\Big{)}} \tag{7}\]  
with \(F_{o}^{A}\), \(F_{o}^{S}\) and \(F_{o}^{C}\) respectively corresponding to the response across the orthogonal, sagittal and coronal sections.