## 3 Deep Feature-Based Adaptive Level Set Model (DFALS)
### Constructing the Global Energy Term  
Inspired by the sliding window method [37], we gather the sub-images (\(11\times 11\) pixels) with stride equal 1 pixel horizontally and vertically from each UVI image. Since the auroral images used in this work have a fixed size of \(228\times 200\) as shown in Figure 6a, 45,600 sub-images with the size of \(11\times 11\) can be obtained after padding zeros at the border of each image. Taking each of the 45,600 sub-images  
Figure 5: Convolutional neural network (CNN) architecture. “Conv” and “FC” indicate convolutional layers and fully connected layers, respectively.  
as the input of the pre-trained CNN, we can obtain the confidence value \(\hat{y}\) (\(\hat{y}\in[0,1]\)) of the center pixel in the sub-image, which reflects the probability of the pixel locating at the aurora oval region. All confidence values of the pixels in an image can be visualized through the deep feature map, whose gray value is \(\hat{I}=\hat{y}\times 255\). An example of the auroral oval image and corresponding deep feature map are shown in Figure 6. With the deep feature map, we can detect the weak boundary regions of the aurora effectively.  
Based on the deep feature map, we propose the global energy term as follows  
\[E_{global}=\int_{\Omega}{(\hat{I}(u)-\hat{c}_{1})}^{2}H(\phi_{1})du+2\int_{ \Omega}{(\hat{I}(u)-\hat{c}_{2})}^{2}(1-H(\phi_{1}))H(\phi_{2})du+\int_{\Omega} {(\hat{I}(u)-\hat{c}_{3})}^{2}(1-H(\phi_{2}))du \tag{5}\]  
\(\hat{c}_{1}\), \(\hat{c}_{2}\) and \(\hat{c}_{3}\) represent the mean gray intensities of region \(S_{1}\), \(S_{2}\) and \(S_{3}\) in the deep feature map, which can be calculated as:  
\[\hat{c}_{1}=\frac{\int_{\Omega}{\hat{I}(u)H(\phi_{1})du}}{\int_{\Omega}{H(\phi _{1})du}}\quad\hat{c}_{2}=\frac{\int_{\Omega}{\hat{I}(u)(1-H(\phi_{1}))H(\phi_ {2})du}}{\int_{\Omega}{(1-H(\phi_{1}))H(\phi_{2})du}}\quad\hat{c}_{3}=\frac{ \int_{\Omega}{\hat{I}(u)(1-H(\phi_{3}))du}}{\int_{\Omega}{1-H(\phi_{3})du}} \tag{6}\]  
The global energy term contains three components. In the minimization of the global energy term, the first component can drive the inner level set (\(\phi_{1}\)) curve to evolve towards the inner boundary of the auroral oval, and the third component can drive the outer level set (\(\phi_{2}\)) curve to evolve towards the outer boundary. The second component can attract the inner and outer level set curves moving to the auroral oval boundary determined by the deep feature map simultaneously.  
To obtain an accurate boundary of the auroral oval, we incorporate the global energy term into the LIDLSM, and the total energy functional of the DFALS can be formulated as  
\[\begin{array}{rl}E_{DFALS}=&\gamma_{1}\int_{\Omega}\frac{(|\phi_{1}(u)-\phi_ {2}(u)|-\mu)^{2}}{2\sigma^{2}}du+\gamma_{2}\sum\limits_{i=1}^{2}\int_{\Omega} {|\
abla H(\phi_{i}(u))du}\ +\gamma_{3}\cdot E_{global}\\ &\sum\limits_{i=1}^{2}\int_{\Omega}{\delta(\phi_{i}(u))\int\limits_{\overline{ \Psi}}{H(\phi_{i}(v))(I(v)-I_{inner})}^{2}}+(1-H(\phi_{i}(v)))(I(v)-I_{outer}) ^{2}dvdu\end{array} \tag{7}\]  
Benefiting from the deep feature-based global energy term, the DFALS model can prevent the boundary leakage of the LIDLSM and obviously improve the segmentation accuracy at the low-contrast regions in the UVI images.  
Figure 6: Auroral oval image and the corresponding deep feature map. (**a**) Auroral oval image. (**b**) The corresponding deep feature map.