## 4 Experimental Results
### Evaluation Metrics  
Three evaluation metrics, FID (Frechet inception distance) [50], KID (kernel inception distance) [51], and LPIPS (learned perceptual image patch similarity) [52], were used to evaluate the performance of the synthesized task. These three metrics are defined as follows:  
\[FID(l_{g},l_{p})=\left\|\mu_{g}-\mu_{p}\right\|_{2}^{2}+Tr(\Sigma_ {g}+\Sigma_{p}-2(\Sigma_{g}\Sigma_{p})^{\frac{1}{2}}) \tag{12}\] \[KID(l_{g},l_{p})=MMD_{k}(l_{g},l_{p})^{2}\] (13) \[LPIPS\big{(}f_{g},f_{p}\big{)}=\sum_{l=1}^{L}w_{l}\big{\|}\Phi_{l }\big{(}f_{g}\big{)}-\Phi_{l}\big{(}f_{p}\big{)}\big{\|}_{2}^{2} \tag{14}\]  
where \(l_{g}\) and \(l_{p}\) represent the high-dimensional feature vectors of the real liver image and the synthetic liver image extracted using the Inception V3 network, respectively. \(\mu_{g}\) and \(\mu_{p}\) represent the mean values of these two vectors, respectively. \(\Sigma_{g}\) and \(\Sigma_{p}\) represent the covariance of these two vectors, respectively, \(Tr\) represents the trace operation. In Equation (14), \(f_{g}\) and \(f_{p}\) represents the real liver image and the generated liver image, \(L\) represents the number of layers in the VGG network used for feature extraction, \(w_{l}\) represents the weight of the \(l\) layer, and \(\Phi_{l}\big{(}f_{g}\big{)}\) represents the feature mapping of the image \(f\) at the \(l\) layer. \(\|\cdot\|_{2}\) represents the L2 norm, denoting the Euclidean length of the vector.  
FID and KID were used to calculate whether the ground truth and the synthesized image have the same distribution. The FID was calculated using the Inception V3 network [53], by removing the final fully connected and pooling layers, a high-dimensional feature can be obtained, in this paper we used the highest 2048 dimensions. The vectors extracted by the Inception V3 network from our real liver CT images obey a specific distribution. If the corresponding extracted feature vectors of the synthesized liver image also obey the same distribution, then the synthesized image has a high degree of realism. In other words, by calculating the distance between the real image and the feature vector of the synthesizing image, the smaller the distance obtained, the stronger the performance of the model. The calculation of KID is similar to the FID, which also requires the extraction of high-dimensional feature vectors from the Inception V3 networks, but the maximum mean discrepancy (MMD) is calculated for the two feature vectors instead of the Frechet distance as in the case of FID. The KID and FID statistics can be quickly obtained by installing and using the Python package torch-fidelity [54].  
LPIPS forces the generator to learn to reconstruct the reverse mapping of the real image from fake images, by learning the reverse mapping of the generated image to real image and prioritizing the perceived similarity between them. In this paper, we used the Python package lips to count LPIPS. Although the focus of this task was not to accurately segment the liver, but to demonstrate the authenticity of the synthetic 3D liver image, we still used the DSC (Equation (6)) coefficient to evaluate the error of the synthetic 3D liver  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Segmentation Layers** & 64 s & 48 s & 32 s \\ \hline
**Number of liver** & & & \\
**sequence segments** & 1621 & 2323 & 3579 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Number of input sequences for the segmentation task.  
with the real liver. FID, KID and LPIPS were used to evaluate the detail texture of the synthetic image. DSC was used to evaluate the contour of the synthetic images.