# DC2 Anet: Generating Lumbar Spine MR Images from CT Scan Data Based on Semi-Supervised Learning
## 3 Method
### Objective Function
However, the discriminators for unsupervised learning need to distinguish whether the images are real or synthetic; the source domain images are not input into the discriminators. \[ \begin{split} L_{unsup-adver}(Syn_{\text{MR}},Dis_{\text{MR}}, Syn_{CT},Dis_{CT})&=\mathbb{E}_{\text{LR}^{\sim}P_{data}}(I_{\text{MR}}) \left[\log\left(Dis_{\text{MR}}\left(I_{\text{MR}}\right)\right)\right]\\ &+\mathbb{E}_{I_{CT}\sim P_{data}}(I_{\text{CT}})\left[\log\left( 1-Dis_{\text{MR}}\left(Syn_{\text{MR}}\left(I_{\text{CT}}\right)\right) \right)\right]\\ &+\mathbb{E}_{\text{L}_{\text{CT}}\sim P_{data}}(I_{\text{CT}}) \left[\log\left(Dis_{CT}\left(I_{\text{CT}}\right)\right)\right]\\ &+\mathbb{E}_{\text{LR}^{\sim}P_{data}}(I_{\text{RR}})\left[ \log\left(1-Dis_{CT}\left(Syn_{CT}\left(I_{\text{MR}}\right)\right)\right) \right]\end{split}  \tag{2}\]  
In unsupervised learning, adversarial loss alone cannot guarantee that the learned synthesis network can map an input image to the desired output image. To reduce the possible mapping space between these two domains, we utilized a dual cycle-consistent structure for aligned and unaligned data. For an image \(I_{CT}\) from the CT domain, the forward cycle-consistent network should be able to bring \(I_{CT}\) back to the original image, i.e., \(I_{CT}\to Syn_{\text{MR}}(I_{CT})\to Syn_{CT}(Syn_{\text{MR}}(I_{CT})) \approx I_{CT}\). Similarly, the backward cycle-consistent network should extract an image \(I_{\text{MR}}\) from the MR domain to satisfy \(I_{\text{MR}}\to Syn_{CT}(I_{\text{MR}})\to Syn_{\text{MR}}(Syn_{CT}(I_{ \text{MR}}))\approx I_{\text{MR}}\). The cycle-consistent losses are expressed as follows:  
\[ \begin{split} L_{sup-cycle}\left(Syn_{\text{MR}},Syn_{CT}\right) &=\mathbb{E}_{I_{CT},\text{LR}^{\sim}P_{data}}(I_{CT},I_{\text{MR }})\left[\left\|Syn_{CT}\big{(}Syn_{\text{MR}}\left(I_{CT}\right)\right)-I_{CT }\right\|_{1}\right]\\ &+\mathbb{E}_{\text{LR}^{\sim}I_{CT}\sim P_{data}}(I_{\text{MR} },I_{CT})\left[\left\|Syn_{\text{MR}}\big{(}Syn_{CT}\left(I_{\text{MR}}\right) \big{)}-I_{\text{MR}}\right\|_{1}\right]\end{split}  \tag{3}\]  
\[ \begin{split} L_{unsup-cycle}\left(Syn_{\text{MR}},Syn_{CT}\right) &=\mathbb{E}_{I_{CT}\sim P_{data}}(I_{CT})\left[\left\|Syn_{CT} \big{(}Syn_{\text{MR}}\left(I_{CT}\right)\right)-I_{CT}\right\|_{1}\right]\\ &+\mathbb{E}_{\text{LR}^{\sim}P_{data}}(I_{\text{RR}})\left[\left\|Syn _{\text{MR}}\big{(}Syn_{CT}\left(I_{\text{MR}}\right)\big{)}-I_{\text{MR}} \right\|_{1}\right]\end{split}  \tag{4}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Loss term** & **Strengths** & **Weaknesses** & **Aligned Data** \\ \hline Adversarial & Realistic output & Unstable training & \(\bigtimes\) \\ \begin{tabular}{l} Dual cycle- \\ consistent \\ \end{tabular} & Possible unsupervised learning & \(\bigtimes\) \\ \begin{tabular}{l} Voxel-wise \\ \end{tabular} & \(\bigtimes\) \\ \begin{tabular}{l} Encourage the output similar to \\ the reference image \\ \end{tabular} & \(\bigtimes\) \\ \begin{tabular}{l} Gradient \\ difference \\ \end{tabular} & \(\bigtimes\) \\ \begin{tabular}{l} The output \\ Preserves high-level semantic \\ \end{tabular} & \(\bigtimes\) \\ \
\begin{tabular}{l} Structural \\ similarity \\ \end{tabular} & \(\bigtimes\) \\
\begin{tabular}{l} Relaxes misalignment \\ constraints for data alignment \\ \end{tabular} & \(\bigtimes\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A summary of the strengths and weaknesses of each loss term used in DC\({}^{2}\)Anet. In the last column, the symbols \(\bigtimes\) and \(\bigtimes\) denote whether the loss term requires aligned training data or not. where \(L_{sup-cycle}\) and \(L_{unsup-cycle}\) are the cycle-consistent structures for supervised and unsupervised learning, respectively. Each cycle-consistent loss has two terms: a forward cycle-consistent and a backward cycle-consistent term. In general, adversarial loss produces visually-appealing results. However, using only adversarial loss to match synthetic and reference MR images may cause the model to generate unseen structures.