# DC2 Anet: Generating Lumbar Spine MR Images from CT Scan Data Based on Semi-Supervised Learning
## 3 Method
### Optimization of DC\({}^{2}\)Anet with Semi-Supervised Learning
Table 5 presents a summary of our dataset, while Figure 4 displays several sample images. ### Evaluation Metrics

The synthesis and reference MR images were compared using the mean absolute error (MAE) and root mean squared error (RMSE), which are defined as follows:

Figure 4: Sample images of axial lumbar vertebA CT (upper row) and MR images (lower row). \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Number of Patients** & **Number of Slices** \\ \hline Training set & 549 & 22,428 \\ Test set & 92 & 4426 \\ Total & 641 & 26,854 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of the lumbar vertebA dataset used in the experiments. \[MAE =\frac{1}{N}\sum_{i=0}^{N-1}\left\|I_{\text{MR}}\left(i\right)-Syn_{ \text{MR}}\left(I_{CT}\left(i\right)\right)\right\| \tag{13}\] \[RMSE =\sqrt{\frac{1}{N}\sum_{i=0}^{N-1}\left(I_{\text{MR}}\left(i\right) -Syn_{\text{MR}}\left(I_{CT}\left(i\right)\right)\right)^{2}} \tag{14}\]

where \(N\) is the total number of image slices in the aligned voxel. MAE and RMSE measure the average distance between each pixel in the synthetic and reference MR images. In addition, the voxel-wise peak-signal-to-noise-ratio (PSNR) can also be calculated:

\[PSNR=10\cdot\log_{10}\left(\frac{\text{H}^{2}}{MSE}\right) \tag{15}\]

\[MSE=\frac{1}{N}\sum_{i=0}^{N-1}\left(I_{\text{MR}}\left(i\right)-Syn_{\text{MR }}\left(I_{CT}\left(i\right)\right)\right)^{2} \tag{16}\]

where \(H\) is the maximum possible intensity of the pixel and MSE is the mean square error, which represents the square of difference between \(I_{\text{MR}}\) and \(Syn_{\text{MR}}(I_{CT})\). MAE, RMSE, and PSNR were based on the correct alignment of test images \(I_{CT}\) and \(I_{\text{MR}}\). Because of the enormous differences between two image domains, it is difficult to achieve perfect image alignment. Therefore, the structural similarity (SSIM) index and the Pearson correlation coefficient (PCC) should also be calculated for patch-wise statistical comparisons, e.g., mean, variance, and correlation. The definition of the SSIM is given in Equation (9), and the PCC is defined as follows:

\[PCC=\frac{1}{N}\sum_{i=0}^{N}\frac{\left(I_{\text{MR}}\left(i\right)-\mu_{ \text{MR}\left(i\right)}\right)\left(Syn_{\text{MR}}\left(I_{CT}\left(i\right) \right)-\mu_{Syn_{\text{MR}}}\left(I_{CT}\left(i\right)\right)\right)}{\sigma _{I_{\text{MR}}\left(i\right)}\sigma_{Syn_{\text{MR}}}\left(I_{CT}\left(i \right)\right)} \tag{17}\]

where \(\mu\) and \(\sigma\) are the mean and variance of the \(i^{\text{th}}\) image slice. Lower values for the MAE and RMSE are preferable, while the reverse is true for the PSNR, SSIM, and PCC. ### Analysis of DC\({}^{2}\)Anet

Based on our lumbar spine dataset and the metrics described in the previous section, we quantitatively evaluated the performance of our model in generating an MR image from a CT image. In Table 6, we compare the performance of DC\({}^{2}\)Anet with supervised, unsupervised, and semi-supervised learning. Data alignment of the tuples of the corresponding images in supervised learning produced a much higher accuracy than did unsupervised learning, while semi-supervised learning with alternating optimization was better than both supervised and unsupervised learning. The joint optimization of semi-supervised learning produced substantially weaker results compared to alternating optimization. Therefore, we concluded that the alternating optimization of DC\({}^{2}\)Anet led to a more stable convergence and was critical to effective performance. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Learning** & **Optimization** & **MAE** & **RMSE** & **PSNR** & **SSIM** & **PCC** \\ \hline Supervised & \(-\) & 28.873 & 39.562 & 64.533 & 0.242 & 0.445 \\ Unsupervised & \(-\) & 31.297 & 42.205 & 63.976 & 0.227 & 0.420 \\ Semi-supervised & Joint & 29.048 & 39.801 & 64.502 & 0.244 & 0.439 \\ Semi-supervised & Alternating & **28.819** & **39.418** & **64.553** & **0.248** & **0.453** \\ \hline \hline \end{tabular}
\end{table}