## 5 Conclusions  
In this paper, we reported a DL reconstruction framework for PCCT with sparse-view projections and validated it with experiments of the simulation datasets and experimental datasets. The proposed framework consists of CNNs in dual domains and PCRIL as the connection between them. PCRIL can achieve PCCT reconstruction, and it allows for the backpropagation of gradients from the image domain to the projection sinogram domain. Therefore, this framework enables the CNNs in dual domains to be trained simultaneously for further reduction of artifacts and to restore the missing structure introduced by sparse-view sampling. In addition, the differential forward projection of the image reconstructed with the sparse-view projection sinogram is adopted as the input of the network, instead of the interpolation of the sparse-view projection sinogram. It efficiently improves the image quality of the images reconstructed with given sparse-view PCCT projections. This work has the potential to push PCCT techniques to applications in the field of composite imaging and biomedical imaging. Conceptualization, C.Z. and J.F. ; methodology, C.Z. and J.F. ; software, C.Z. ; validation, C.Z. and J.F. ; formal analysis, C.Z. ; investigation, C.Z. ; resources, J.F. and G.Z. ; data curation, C.Z. ; writing--original draft preparation, C.Z. ; writing--review and editing, J.F. and G.Z. ; visualization, C.Z. ; supervision, J.F. and G.Z. ; project administration, J.F. and G.Z. ; funding acquisition, J.F. All authors have read and agreed to the published version of the manuscript. This research was funded by the Ningbo Major Projects of Science and Technology Innovation 2025 (2020Z074), the National Natural Science Foundation of China (51975026), the Joint Fund of Research Utilizing Large-scale Scientific Facilities by the National Natural Science Foundation  
Figure 10: The _rell_ curves of the average values of FSIM and IW-SSIM of the experimental testing datasets with loss functions as presented in Table 7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Evaluation** & **Methods** & \multicolumn{4}{c}{**Cases**} \\  & & **60** & **90** & **120** & **180** \\ \hline \multirow{8}{*}{FSIM} & Loss1 & 0.9262 & 0.9359 & 0.9448 & 0.9662 \\  & Loss2 & 0.9300 & 0.9374 & 0.9520 & 0.9666 \\  & Loss3 & 0.9403 & 0.9514 & 0.9582 & 0.9741 \\  & Loss4 & 0.9293 & 0.9430 & 0.9522 & 0.9684 \\  & Loss5 & 0.9275 & 0.9427 & 0.9491 & 0.9642 \\  & Loss6 & 0.9207 & 0.9369 & 0.9462 & 0.9554 \\ \hline \multirow{8}{*}{IW-SSIM} & Loss1 & 0.9353 & 0.9492 & 0.9606 & 0.9789 \\  & Loss2 & 0.9402 & 0.9501 & 0.9662 & 0.9797 \\ \cline{1-1}  & Loss3 & 0.9524 & 0.9647 & 0.9729 & 0.9866 \\ \cline{1-1}  & Loss4 & 0.9400 & 0.9560 & 0.9667 & 0.9806 \\ \cline{1-1}  & Loss5 & 0.9382 & 0.9561 & 0.9632 & 0.9796 \\ \cline{1-1}  & Loss6 & 0.9295 & 0.9502 & 0.9615 & 0.9782 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The average values of FSIM and IW-SSIM of the results of the experimental testing datasets. of China and Chinese Academy of Science (U1932111), the Innovation Leading Talent Short Term Projects of Natural Science by Jiangxi Double Thousand Plan (S2020 DQKJ0355), the Jiangxi Provincial Science and Technology Innovation Base Plan--Introduction of workpiece research and development institutions (20203 CCH45003), and the Jiangxi Provincial Science and Technology Innovation Base Plan--Introduction of workpiece research and development institutions (20212 CCH45001). **Institutional Review Board Statement:** The procedures for care and use of the used mouse were conducted in accordance with the \"Guiding Principles in the Care and Use of Animals\" and were approved by the Ethics Committee of the Institute of High Energy Physics, Chinese Academy of Sciences. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Data was obtained from the Institute of High Energy Physics, Chinese Academy of Sciences and are available from J.F. with the permission of the Institute of High Energy Physics, Chinese Academy of Sciences.