# A Novel Evolution Strategy of Level Set Method for the Segmentation of Overlapping Cervical Cells
## 2 Methodology
### Touching Cell Splitting
#### 2.2.2 2D Codimension Two-Object Level Set Method  
To reduce the false negative pixels caused by expansion in the MSTF, we used the accurate segmentation of cellular clumps to determine the fixed contour, shown as the green contour in Figure 3e. We defined the LSF \(\phi_{0}\) and \(\phi_{1}\) generated by LTEDT for the fixed and active contours. \(\phi_{0}\) can be associated with \(\phi_{1}\) to produce a union LSF \(\phi\), represented by Equation (3), which is often used to combine two LSF in traditional level set methods [36; 37; 38; 39; 40; 41; 42; 43]. \[\phi=min\{\phi_{0},\phi_{1}\} \tag{3}\]  
To make the fixed contour inactive and perform morphological scaling on the active contour, we proposed a novel evolution strategy DCTLSM incited by the idea of CTLSM [45]. Since it is not easy to derive the functional variation about \(\phi\) based on Equation (3), we express Equation (3) as a new form, Equation (4):  
\[\phi=H_{0}\phi_{1}+(1-H_{0})\phi_{0} \tag{4}\]  
where \(H_{0}\) is defined in Equation (5). \[\left\{ \begin{array}{rl}H_{0}=H(\phi_{0}-\phi_{1})\\ H_{1}=H(\phi_{1}-\phi_{0})\\ \delta_{0}=\delta(\phi_{0}-\phi_{1})\end{array} \right. \tag{5}\]  
where \(H\) is the Heaviside function defined by Equation (6). \[H(\alpha)=\left\{ \begin{array}{rl}1&\alpha>0\\ 0&\alpha\leqslant 0\end{array} \right. \tag{6}\]  
\[\delta(\alpha)=\left\{ \begin{array}{rl}1&\alpha=0\\ 0&\alpha\
eq 0\end{array} \right. \tag{7}\]  
By the theoretical derivation [70] based on Equation (4), we obtained a mathematical toolbox about vector calculus for \(\phi\), in which there were four necessary variables: \(H(\phi)\), \(\
abla H(\phi)\), \(\
abla\phi\) and \(\hat{\delta}_{\phi}(\overrightarrow{x})\) for the functional variations, calculated by Equations (8)-(11). \[H(\phi)=H_{0}H(\phi_{1})+(1-H_{0})H(\phi_{0}) \tag{8}\]  
\[\
abla H(\phi)=H_{1}\delta(\phi_{0})\
abla\phi_{0}+H_{0}\delta(\phi_{1})\
abla \phi_{1}+\delta_{0}\delta(\phi_{1})\
abla(\frac{\phi_{0}+\phi_{1}}{2}) \tag{9}\]  
\[\
abla\phi=H_{1}\
abla\phi_{0}+H_{0}\
abla\phi_{1}+\delta_{0}\
abla(\frac{ \phi_{1}+\phi_{0}}{2}) \tag{10}\]\[\hat{\delta}_{\phi}(\overrightarrow{x})=(1-H_{0})\delta(\phi_{0})\Big{|}\
abla \phi_{0}\Big{|}+H_{0}\delta(\phi_{1})\Big{|}\
abla\phi_{1}\Big{|} \tag{11}\]  
where \(H(\phi)\) is the Heaviside function from Equation (6) acting on \(\phi\). \(\
abla H(\phi)\) is the gradient of \(H(\phi)\). \(\
abla\phi\) is the gradient of \(\phi\). \(\hat{\delta}_{\phi}(\overrightarrow{x})\) is the Dirac delta function [63] acting on the value of \(\phi\) at each two-dimensional point \(\overrightarrow{x}=(x,y)\). \(H_{1}\) is defined by Equation (5), and \(\delta_{0}\) is calculated by applying the Delta function from Equation (7) on \(\phi_{0}-\phi_{1}\). The magnitude of the gradient vector of \(\phi_{0}\) and \(\phi_{1}\) is denoted by the operator \(|\cdot|\) in Equation (11). Based on this mathematical toolbox, the energy functionals of \(\phi\) are designed to implement the DCTLSM, represented as Equations (12) and (13). \[\text{M}(\phi)=\int vH_{0}\phi d\overrightarrow{x}=\int vH_{0}\phi_{1}d \overrightarrow{x} \tag{12}\]  
\[\text{E}(\phi)=\int g\hat{\delta}_{\phi}(\overrightarrow{x})d\overrightarrow {x}=-\int H_{0}g\delta(\phi_{0})d\overrightarrow{x}+\int H_{0}g\delta(\phi_{ 1})\Big{|}\
abla\phi_{1}\Big{|}d\overrightarrow{x} \tag{13}\]  
where \(\text{M}(\phi)\) denotes the morphological term that is used as the attraction force by setting \(v=-1\) in the shrinking process of the DCTLSM (see Figure 3f) and the repulsion force by setting \(v\)= 1 in the expanding process of the DCTLSM (see Figure 3g). \(\text{E}(\phi)\) denotes the edge term, which is used in the refinement process of the DCTLSM (see Figure 3h). The Canny mountain map \(g\) is obtained by applying the LTEDT-SDF on the edge map generated by the Canny detector. By performing the Gateaux derivative of \(\phi_{1}\), we can obtain the first-order variations (14) and (15) of the energy functionals (12) and (13), according to the mathematical toolbox (8)-(11).