## 2 Materials and Methods
### ROI-Binarized Hyperbolic Region
#### 2.2.2 Loss Function Optimization and Evaluation  
The training process of convolutional neural networks is essentially a process of minimizing loss functions. In this paper, we use the binary cross-entropy function as the loss function, defined as follows:  
\[L_{p}(q)=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\cdot\log[p(y_{i})]+(1-y_{i})\cdot\log[ 1-p(y_{i})], \tag{1}\]  
where \(y_{i}\) is the binary label (1 for positive points and 0 for negative points), and \(p(y_{i})\) is the predicted probability of the point being positive for all \(N\) points. In a deep learning network, the binary cross entropy (also known as logarithmic loss or log loss) is a commonly used loss function in particularly binary classification problems. It is designed to measure the dissimilarity between the predicted probability distribution and the true binary labels of a dataset. It penalizes predictions that are confident but wrong. In short, the binary cross entropy plays a crucial role in training models for binary classification by measuring how well the model's predicted probabilities align with the true labels. The optimization of the loss function adopts root mean square propagation (RMSProp), which uses a moving average to adaptively adjust the learning rate for each network training parameter. This algorithm is an extension of the stochastic gradient descent algorithm and the foundation of the Adam algorithm. One of the applications of RMSProp is the stochastic technology for mini-batch gradient descent. The updated equation can be performed as:  
\[E(g^{2})_{t}=\gamma E(g^{2})_{t-1}+(1-\gamma)g_{t}^{2}, \tag{2}\]  
\[\theta_{t+1}=\theta_{t}-\frac{\eta}{\sqrt{E(g^{2})_{t}+\epsilon}}g_{t}, \tag{3}\]  
where \(E(g^{2})_{t}\) is the moving average of squared gradients at time step \(t\), \(g_{t}\) is the gradient of the cost function with respect to the weight, \(\eta\) is the learning rate, and \(\gamma\) is the moving average parameter, \(\theta_{t+1}\) is the parameters to be updated, and \(\epsilon\) is a minimal parameter to prevent the denominator from being 0. To evaluate ROI binarization hyperbola region extraction and avoid the subjectivity of results evaluation with the naked eye, three indicators, peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM) [32], and feature similarity index measure (FSIM) [33], were used for measurement. The equations are, respectively, as follows:  
\[PSNR=20\times\log_{10}\left(\frac{255}{RMSE}\right), \tag{4}\]  
where  
\[RMSE=\sqrt{\frac{\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}\left(I_{ij}-K_{ij}\right)^{ 2}}{m\times n}}, \tag{5}\]  
where \(RMSE\) is the root mean square error, \(m\times n\) is the image size, \(I_{ij}\) is the original image pixel value, \(K_{ij}\) is the image pixel value after extraction. \[SSIM=\frac{\left(2\mu_{I}\mu_{K}+c_{1}\right)(2\sigma_{IK}+c_{2})}{\left(\mu_ {I}^{2}+\mu_{K}^{2}+c_{1}\right)\left(\sigma_{I}^{2}+\sigma_{K}^{2}+c_{2} \right)}, \tag{6}\]  
where \(\mu_{I}\), \(\mu_{K}\) are the average value of the original image and the extraction image respectively, \(\sigma_{I}^{2}\), \(\sigma_{K}^{2}\) are the variance of the original image and the extraction image, respectively. \(\sigma_{IK}\) is the covariance between the original image and the extraction image. \(c_{1}=\left(0.01L\right)^{2}\), \(c_{2}=\left(0.03L\right)^{2}\), \(L\) is the pixel value range. \[FSIM=\frac{\sum_{l\in\Omega}S_{L}(X)PC_{m}(X)}{\sum_{l\in\Omega}PC_{m}(X)}, \tag{7}\]where  
\[\left\{ \begin{array}{l}S_{L}(X)=S_{PC}(X)S_{G}(X)\\ S_{PC}(X)=\frac{2 PC_{G}(X)PC_{G}(X)+T_{1}}{PC_{1}^{2}(X)PC_{2}^{2}(X)+T_{1}}\\ S_{G}(X)=\frac{2G_{1}(X)G_{2}(X)+T_{2}}{G_{1}^{2}(X)G_{2}^{2}(X)+T_{2}}\\ G=\sqrt{G_{x}^{2}+G_{y}^{2}}\\ PC(X)=\frac{E(X)}{\varepsilon+\sum_{n}A_{n}(X)}\end{array} \right., \tag{8}\]  
where \(\Omega\) is all pixels of the original image, \(S_{L}(X)\) is the similarity score, \(T_{1}\) and \(T_{2}\) are constants, and \(G_{1}\) and \(G_{2}\) are gradients of the original image and the extraction image, respectively. \(PC\) is phase congruency.