# DC2 Anet: Generating Lumbar Spine MR Images from CT Scan Data Based on Semi-Supervised Learning
## 3 Method
### Objective Function  
Our goal is for the mapping functions between the CT image and MR image domains to be learned using the given aligned training data. As illustrated in Figure 1, aligned data (\(I_{CT}\), \(I_{MR}\)) are converted into unaligned data \(I_{CT}\), \(I_{MR}\), and the aligned and unaligned data points are utilized together in semi-supervised learning. DC\({}^{2}\)Anet includes two synthesis networks, \(Syn_{MR}\): CT \(\rightarrow\) MR and \(Syn_{CT}\): MR \(\rightarrow\) CT, and includes two discriminator networks, \(Dis_{MR}\) and \(Dis_{CT}\), where \(Dis_{MR}\) aims to distinguish between the reference MR image \(I_{MR}\) and the synthetic MR image \(Syn_{MR}(I_{CT})\); in the same way, \(Dis_{CT}\) aims to distinguish between the reference CT image \(I_{CT}\) and the synthetic CT image \(Syn_{CT}(I_{CT})\). Moreover, the four networks are different optimized objectives corresponding to the supervised and unsupervised learning due to the input aligned or unaligned data. Additionally, to measure the high-level perceptual and semantic differences between two images, the VGG16 perceptual network is employed in DC\({}^{2}\)Anet, which was pretrained on the ImageNet dataset [55]. Our objective function contains six loss terms in total: adversarial, dual cycle-consistent, voxel-wise, gradient difference, perceptual, and structural similarity. A summary of the strengths and weaknesses of each loss term is given in Table 3.  
We apply adversarial loss [3] to both the supervised and unsupervised setups. The forward and backward mappings \(Syn_{MR}\): CT \(\rightarrow\) MR and \(Syn_{CT}\): MR \(\rightarrow\) CT and the discriminators \(Dis_{MR}\) and \(Dis_{CT}\) are expressed as follows:  
\[ \begin{split} L_{sup-adver}(Syn_{MR},Dis_{MR},Syn_{CT},Dis_{CT})& =\mathbb{E}_{I_{CT},I_{MR}\sim P_{data}(I_{CT},I_{MR})}\Big{[} \log\left(Dis_{MR}\left(I_{CT},I_{MR}\right)\right)\Big{]}\\ &+\mathbb{E}_{I_{CT}\sim P_{data}(I_{CT})}\Big{[}\log\left(1- Dis_{MR}\left(I_{CT},Syn_{MR}\left(I_{CT}\right)\right)\right)\Big{]}\\ &+\mathbb{E}_{I_{MR},I_{CT}\sim P_{data}(I_{MR},I_{CT})}\Big{[} \log\left(Dis_{CT}\left(I_{MR},I_{CT}\right)\right)\Big{]}\\ &+\mathbb{E}_{I_{MR}\sim P_{data}(I_{MR})}\Big{[}\log\left(1- Dis_{CT}\left(I_{MR},Syn_{CT}\left(I_{MR}\right)\right)\right)\Big{]}\end{split}  \tag{1}\]  
where the first two terms are the forward adversarial loss and the last two terms are the backward adversarial loss. The network \(Syn_{MR}\) attempts to synthesize images \(Syn_{MR}(I_{CT})\), which look similar to images from the MR domain, while \(Dis_{MR}\) aims not only to discriminate  
Figure 2: Diagram of the proposed dual cycle-consistent adversarial network (DC\({}^{2}\)Anet). DC\({}^{2}\)Anet consists of a forward cycle-consistent and a backward cycle-consistent network. (**a**) The forward cycle-consistent adversarial network with unaligned learning. (**b**) The backward cycle-consistent adversarial network with aligned learning. between synthetic MR and reference MR images, but also to ensure it generates images from the corresponding CT images \(I_{CT}\). For the backward adversarial loss, the synthesis network \(Syn_{CT}\) generates the reference CT images \(Syn_{CT}(I_{\text{MR}})\), which look similar to images from the CT domain, while \(Dis_{CT}\) aims to distinguish between synthetic and reference CT images based on the MR images \(I_{\text{MR}}\). The synthesis networks \(Syn_{\text{MR}}\) and \(Syn_{CT}\) attempt to minimize this objective function, while the adversarial discriminator networks \(Dis_{\text{MR}}\) and \(Dis_{CT}\) aim to maximize it, i.e., \(Syn_{\text{MR}}^{*},Syn_{CT}^{*}=\arg\min_{Syn_{\text{MR}},Syn_{CT}}\max_{ Dis_{\text{MR}},Dis_{CT}}I_{\text{sup}-adver}\left(Syn_{\text{MR}},Dis_{\text{MR}}, Syn_{CT},Dis_{CT}\right)\). In Equation (2), we introduce a similar form of adversarial loss for unsupervised learning for the synthesis networks and discriminators.