## 2 Related Works  
SPFLIF-IS [24] is a universally applicable image segmentation method, which combines the local fitted image formulation (LIF) [30] and the improved signed pressure force (SPF) [31] for inhomogeneous image segmentation. By combining the improved SPF function with the LIF model, the level set evolution equation of SPFLIF-IS was defined as:  
\[\frac{\partial\phi}{\partial t}=(1-\lambda_{1})\cdot\alpha\cdot spf(I(u))+ \lambda_{1}\left(I-I^{LIF}\right)(I_{inner}-I_{outer})\delta(\phi) \tag{1}\]  
The first term of Equation (1) is the product of the improved SPF (\(spf(I(u))\)) and the balloon force (\(\alpha\)). The improved SPF function has values in the range \([-1,1]\). It modulates the signs of the pressure force inside and outside the region of interest so that the contour shrinks when outside the object, or expands when inside the object [32]. The second term is the level set evolution function corresponding to the LIF model, where \(I_{inner}\) and \(I_{outer}\) denote the mean grayscale intensity of \(\left\{v\mid\phi(v)>0,\ \|v-u\|<r\right\}\) and \(\left\{v\mid\phi(v)\leq 0,\ \|v-u\|>r\right\}\), respectively. \(I^{LIF}\) is the local fitted image and \(\delta(\phi)\) is the univariate Dirac function. By utilizing the local image information, the LIF model is able to segment images with intensity inhomogeneity [30]. The weight coefficient \(\lambda_{1}\) is used to adjust the weight between the SPF function and the LIF model. Based on global and local image information, inhomogeneous images can be segmented quickly and accurately by SPFLIF-IS. LIDLSM [26] is a level set method explicitly designed for auroral oval segmentation. In LIDLSM, [PERSON] et al. [26] utilized dual level set function of \(\phi_{1}\) and \(\phi_{2}\) to obtain the inner and outer auroral oval boundaries, respectively. Suppose that \(I:\Omega\subset R^{2}\) is an image domain, \(I:\Omega\to R^{2}\) is the input auroral oval image. Two level set functions can divide the auroral oval image into three regions:  
\[\left\{ \begin{array}{c}S_{1}=\left\{u\in\Omega|\phi_{1}(u)>0\right\}\\ S_{2}=\left\{u\in\Omega|\phi_{1}(u)\leq 0,\phi_{2}(u)>0\right\}\\ S_{3}=\left\{u\in\Omega|\phi_{2}(u)>0\right\}\end{array} \right. \tag{2}\]  
This can be shown in Figure 3.  
Figure 3: The three regions divided by \(\phi_{1}\) and \(\phi_{2}\). Assuming that the distance between \(\phi_{1}\) and \(\phi_{2}\) follows the Gaussian distribution, [PERSON] et al. [26] introduced the shape energy term into LIDLSM. Besides, the local information term was constructed to improve the segmentation performance in the low contrast region, and the regularization term was adopted to ensure the curve can evolve smoothly towards the auroral oval boundaries and inhibit small isolated regions. Combining these three terms, the total energy functional of LIDLSM was defined as:  
\[ \begin{split} E_{LIDLSM}=\lambda_{2}\underbrace{\int_{\Omega} \frac{\left(\left|\phi_{1}(u)-\phi_{2}(u)\right|-\mu\right)^{2}}{2\sigma^{2}}du }_{\text{shape energy term}}+\lambda_{3}\underbrace{\sum_{i=1}^{2}\int_{ \Omega}\left|\
abla H(\phi_{i}(u))\right|du}_{\text{regularization term}}+\\ \underbrace{\sum_{i=1}^{2}\int_{\Omega}\delta(\phi_{i}(u))\int_{ \Psi}H(\phi_{i}(v))(I(v)-I_{inner})^{2}+(1-H(\phi_{i}(v)))(I(v)-I_{outer})^{2 }dvdu}_{\text{local information term}}\end{split}  \tag{3}\]  
where \(\mu\) and \(\sigma\) are mean and standard deviation of distance between \(\phi_{1}\) and \(\phi_{2}\). \(\Psi=\left\{v\,|\,\left\|v-u\right\|<r\right\}\) indicates a local region with radius \(r\) at pixel \(u\). \(I_{inner}\) and \(I_{outer}\) denotes the mean grayscale intensity \(\left\{v\,|\,\phi_{i}(v)>0,\,\left\|v-u\right\|<r\right\}\) and \(\left\{v\,|\,\phi_{i}(v)\leq 0,\,\left\|v-u\right\|>r\right\}\), respectively. \(\lambda_{2}\) and \(\lambda_{3}\) are constant coefficients controlling the strength of the shape energy term and the regularization term.