## 3 Results
### Reconstruction Results
#### 3.2.2 Training with the Image Domain Loss \(L_{Image}\)  
The reconstruction results obtained by training the sinogram domain network using the image domain loss \(l_{image}\) were also investigated and compared with those obtained using the sinogram domain loss \(l_{sine}\). The first two rows in Figure 5 show the reconstructed images obtained from our network trained with \(l_{image}\) and their absolute difference from the full-dose CT results, respectively. The counterparts obtained via training with \(l_{sine}\) are shown in the last rows of Figures 4 and 5, respectively. Comparing the quantized difference images shows th \begin{table}
\begin{tabular}{c l c c c c} \hline \hline \multicolumn{2}{c}{**Sparse Levels**} & **s = 2** & **s = 4** & **s = 8** & **s = 15** \\ \hline \multirow{4}{*}{LIDC-IDRI} & Linear interp & 30.01 & 26.86 & 23.52 & 20.95 \\  & SDS-net & **30.79** & **29.18** & 25.86 & 22.31 \\  & Res-Unet & 30.31 & 27.84 & 24.36 & 21.13 \\  & ours & 30.76 & 29.05 & **26.86** & **23.72** \\ \hline \multirow{4}{*}{AAPM} & Linear interp & 25.45 & 22.92 & 20.55 & 18.60 \\  & SDS-net & **26.61** & **25.24** & 22.81 & 20.49 \\ \cline{1-1}  & Res-Unet & 26.22 & 24.34 & 21.65 & 19.56 \\ \cline{1-1}  & ours & 26.46 & 24.99 & **23.27** & **20.74** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Averaged PSNR (dB) in the CT image domain trained with \(l_{sine}\). Figure 5: CT images obtained by training the proposed network using the image domain loss (\(l_{image}\)) on **(a)** the LIDC-IDRI dataset and **(b)** the AAPM dataset. The first and second rows show the reconstructed image obtained by training with \(l_{image}\) and its difference with the label, respectively. The third row shows the difference between the reconstructed image obtained by \(l_{sine}\) and the label. \begin{table}
\begin{tabular}{c l c c c c} \hline \hline \multicolumn{2}{c}{**Sparse Levels**} & **s = 2** & **s = 4** & **s = 8** & **s = 15** \\ \hline \multirow{4}{*}{LIDC-IDRI} & Linear interp & 89.58 & 84.60 & 78.13 & 73.46 \\  & SDS-net & 90.27 & **88.27** & 84.04 & **78.84** \\  & Res-Unet & 89.27 & 82.71 & 71.34 & 55.30 \\  & ours & **90.35** & 87.93 & **84.33** & 76.35 \\ \hline \multirow{4}{*}{AAPM} & Linear interp & 76.52 & 69.49 & 62.95 & 58.13 \\  & SDS-net & 78.08 & **75.14** & **70.69** & **66.53** \\ \cline{1-1}  & Res-Unet & 77.38 & 70.81 & 59.58 & 43.35 \\ \cline{1-1}  & ours & **78.19** & 74.84 & 70.36 & 62.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Averaged SSIM (%) in the CT image domain trained with \(l_{sine}\). is better than the sinogram domain loss at lower sparse levels. As analyzed in Section 2.3, setting back the loss function to the image domain can to some extent coordinate the reconstruction error of the FBP module itself using the sinogram domain interpolation network, resulting in better CT reconstructed images. However, for higher sparsity levels, the results corresponding to image and sinogram domain losses tend to be consistent. To quantitatively evaluate the two loss functions, the PSNR- and SSIM-\(S\) (where \(S\) denotes the sparsity levels) curves for the CT images reconstructed by the proposed network trained with the two loss functions are compared in Figure 6a,b. The corresponding PSNR-\(S\) curves of the predicted sinogram are also compared in Figure 6c. It can be seen from Figure 6a,b that in terms of the reconstructed CT images, the image domain loss outperforms the sinogram domain loss at each sparsity level. Since the feedback loop formed by the image domain loss contains the FBP module, the reconstruction error of the FBP itself can be effectively compensated for by optimizing the weights of the sinogram domain network. However, as shown in Figure 6c, in terms of the PSNR of the predicted sinogram, the sinogram domain loss is better than the image domain loss, indicating that the image domain loss sacrifices the prediction accuracy of the sinogram in order to coordinate the FBP module. However, in any case, the reconstructed CT images can benefit from the image domain loss.