# VLSM-Net: A Fusion Architecture for CT Image Segmentation
## 2 Methods
### Hybrid Level Set Method  
The LSM is a commonly used method for medical image segmentation. The basic view is to express an active profile implicitly as a zero level set of an embedding function in a higher dimensional space. In other words, the zero level set curve can be found at any moment by obtaining its level set embedding function [24]. As shown in Figure 4, the zero set of the embedding function \(\phi\) is the active contour \(\mathrm{C}=\left\{X\mid\phi(X)=0\right\}\), in which the points all obtain positive values inside, while the outside points obtain the opposite. Furthermore, the outside part is the direction of the curve normal vector \(\overset{\rightarrow}{N}\). Thus, LSM can effectively combine the object's boundary and region information to achieve better segmentation results by constructing an effective embedding function \(\phi\). The LSM, firstly proposed by [PERSON] et al. [25], defines the minimization function as:  
\[\varepsilon(\phi)=-\alpha\underset{\Omega}{\int}\left(I-\mu\right)H(\phi)d \Omega+\beta\underset{\Omega}{\int}g|\
abla H(\phi)|d\Omega \tag{2}\]  
where \(I\) represents the input, \(g=g(|\
abla I|)\) represents the boundary feature map associated with the image gradient (\(g\) is a decreasing function), \(\Omega\) is the image region, \(\alpha\) and \(\beta\) is the predefined weights, and \(H(\phi)\) is the Heaviside function [25]. The first integral part represents the region term, where \(\mu\) is the predefined lower limit of the gray level of the object. It is easy to know the first part promotes the contour to surround the region with gray level bigger than \(\mu\). In other words, the image \(I\) is binary segmented with a threshold \(\mu\). The second integral part represents the boundary term, which promotes the contour to approach the boundary with a high gradient. The partial differential equation (PDE) of the minimization functional response is expressed as:  
\[\phi_{t}=|\
abla\phi|\Big{[}\alpha(I-\mu)+\beta\Big{<}\
abla\phi,\frac{\
abla \phi}{|\
abla\phi|}\Big{>}+\beta g\mathrm{div}\Big{(}\frac{\
abla\phi}{|\
abla \phi|}\Big{)}\Big{]} \tag{3}\]  
Figure 4: Representation chart of the LSM. where div(\(\cdot\)) means the divergence. [PERSON] proved that the PDE of Equation (3) describes the same curve evolution when \(C_{t}=\gamma N\) and \(\phi_{t}=\gamma|\
abla\phi|\). According to Equation (3), the specific form of \(C_{t}\) is:  
\[C_{t}=\alpha(I-\mu)N-\beta\langle\
abla g,N\rangle N+\beta gxN \tag{4}\]  
where the normal vector \(\overset{\rightarrow}{N}=-\frac{\
abla\phi}{|\
abla\phi|}\), curvature \(\kappa=\text{div}(\frac{\
abla\phi}{|\
abla\phi|})\). From Equation (4), it can be found that the points in the curve all move along the normal direction. The first term of Equation (4) represents the propagation term of the internal curve expansion motion and the external curve contraction motion of the object. The second term is caused by the gradient of the boundary feature map, which is able to promote the curve to the boundary of the object. The last term represents the curvature flow, which is used to smooth the boundary. In the LSM, to describe the same curve transformation, if \(\phi\) is a signed distance function, the derivative of the embedding function transformed with time is:  
\[\phi_{t}=\alpha(I-\mu)+\beta\text{div}(g\
abla\phi) \tag{5}\]  
Equation (5) shows that this is a stable numerical iteration method to replace the PDE. Obtain \(\phi^{k}\) and \(\phi^{k+1}\) represent the embedding functions \(\phi\) in the \(k\)-th and \((k+1)\)-st iterations, and the numerical method from \(\phi^{k}\) updating to \(\phi^{k+1}\) includes the following three steps:  
1. Initialization \(\phi^{k}\). Re-initialize the embedding function \(\phi\) with the embedding curve (zero point setting) unchanged. The fast marching method is an effective method [26]. 2. Update \(\phi^{k}\), then use the predefined time step \(\Delta t\) and the equation \(\overline{\phi}^{k}=\phi^{k}+\Delta t\cdot\alpha(I-\mu)\) to get \(\overline{\phi}^{k}\). 3.