# DC2 Anet: Generating Lumbar Spine MR Images from CT Scan Data Based on Semi-Supervised Learning
## 3 Method
### Objective Function
To enhance the structural and perceptual similarity of the synthetic and reference images, structural similarity loss is expressed as follows:\[L_{struc}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{l_{CT},l_{RR}\sim p_{data}\left(l_{CT},l_{RR}\right)} \left[-log\Big{(}max\Big{(}0,SSM\left(Syn_{MR}\left(l_{CT}\right),l_{RR}\right) \Big{)}\Big{)}\right] \tag{8}\] \[+\mathbb{E}_{l_{RR},l_{CT}\sim p_{data}\left(l_{RR},l_{CT}\right)} \left[-log\Big{(}max\Big{(}0,SSM\left(Syn_{CT}\left(l_{MR}\right),l_{CT}\right) \Big{)}\Big{)}\right]\] \[SSM\left(\mathbf{x},\mathbf{y}\right) =\frac{\left(2\mu_{x}\mu_{y}+C_{1}\right)\left(2\sigma_{xy}+C_{2} \right)}{\left(\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\right)\left(\sigma_{x}^{2}+ \sigma_{y}^{2}+C_{2}\right)}\] (9) s.t. \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]0 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]1  
where \(C_{1}\) and \(C_{2}\) are constants that stabilize the division with the weak denominator and \(\mu_{x}\), \(\mu_{y}\), \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{xy}\) represent the mean, standard deviation, and cross-covariance of the synthetic and reference images. The objective functions for supervised and unsupervised learning are defined as follows:  
\[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]2 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]3 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]4 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]5 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]6 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]7 \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]8 (11) \[L_{voxel}\left(Syn_{MR},Syn_{CT}\right) =\mathbb{E}_{I_{CT},h_{\text{MR}}\sim p_{data}\left(I_{CT},h_{ \text{MR}}\right)}\left[\left\|I_{MR}-Syn_{MR}\left(I_{CT}\right)\right\|_{1}\right] \tag{5}\]9  
where \(\lambda_{cycle}\), \(\lambda_{yaxel}\), \(\lambda_{grad}\), \(\lambda_{perc}\), and \(\lambda_{struc}\) are hyper-parameters that balance the relative importance of adversarial, cycle-consistent, voxel-wise, gradient difference, perceptual, and structural similarity loss. In summary, the training objective function can be expressed mathematically as:  
\[+\mathbb{E}_{h_{\text{MR}},I_{CT}\sim p_{data}\left(h_{\text{MR}},I_{CT}\right)}\left[\left\|I_{CT}-Syn_{CT}\left(h_{\text{MR}}\right)\right\|_{1 }\right]\]0 \[+\mathbb{E}_{h_{\text{MR}},I_{CT}\sim p_{data}\left(h_{\text{MR}},I_{CT}\right)}\left[\left\|I_{CT}-Syn_{CT}\left(h_{\text{MR}}\right)\right\|_{1 }\right]\]1  
where \(Syn_{MR}\) and \(Syn_{CT}\) minimize the objective function, while \(Dis_{MR}\) and \(Dis_{CT}\) maximize it. During the inference process, only the \(Syn_{MR}^{*}\) network is used to produce a synthetic MR image from an input CT image.