## 3 Simulation Results
### Limited-Angle Circular Scanning  
Simulations for limited-angle circular scanning are also carried out. In this case, the scanning step of angle remains unchanged, and is set to 6\({}^{\circ}\). The number of the sampling points are set to 10, 15 and 20, respectively, corresponding to 60-views, 90-views and 120-views circular scanning. The radius of the scanning is 36 mm. The values of the parameters are set as \(\alpha=0.6\), \(\lambda=1\). \(\eta\) is set to 0.16, 0.08, 0.05 corresponding to 120-views, 90-views, 60-views scanning. The diagrams of the scanning are shown in Figure 6a-c. Figure 7a-c shows the corresponding areas of compensated missing views.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**PSNRs (dB)** & **50-Points** & **20-Points** & **10-Points** \\ \hline
**TV-GD** & 17.58 & 16.46 & 14.35 \\
**TV-VB** & 26.58 & 19.34 & 15.26 \\
**TV-GPEF** & 32.56 & 28.67 & 22.23 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Peak signal-to-noise rations (PSNRs) (dB) of the straight-line scanning from the Sheep-Logan phantom.  
Figure 5: The line chart of the Peak signal-to-noise rations (PSNRs) (dB) versus the number of estimated detectors for the straight-line scanning for TV-GPEF.  
The simulation results are demonstrated in Figure 8. Results show that the TV-GD has the poor performance for the reconstruction. There exists a great degree of distortion in the top right corner as well as the left bottom of the reconstructed images. It has the large error for the geometrical shape between the original image and the reconstructed images. Especially when the views decrease, the result is getting even worse. You can see that for the 60-views scanning in Figure 8a, there is almost no useful information can be obtained from the image except for large tracts of fuzziness. The performance of the TV-VB is better than that of the TV-GD. However, there are still many artifacts that blur the edges of the image in the missing views especially for the 60-views scanning. Besides, the background noise of the results for the TV-VB is serious which affects the image quality. The reconstruction results of the TV-GPEF are superior to that of other two algorithms. The artifacts in those missing views are well suppressed and the image edge information is preserved relatively intact. Almost no blur appears in the reconstructed images for 120-views and 90-views scanning in Figure 8h,i. The contrasts of the images are high and the images are little affected by the noise.  
The PSNRs of these three algorithms are shown in Table 2. The results of PSNR also validate the superiority of the proposed algorithm. On average, the PSNR of the TV-GPEF is higher than that of the TV-VB, by about 9 dB and 12 dB, than that of the TV-GD algorithm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**PSNRs (dB)** & **60-Views** & **90-Views** & **120-Views** \\ \hline
**TV-GD** & 12.23 & 14.49 & 18.78 \\
**TV-VB** & 14.41 & 18.03 & 22.47 \\
**TV-GPEF** & 21.89 & 26.71 & 33.74 \\ \hline \hline \end{tabular}
\end{table}
Table 2: PSNR (dB) of the limited-angle circular scanning from the Sheep–Logan phantom.  
Figure 6: The diagram for the limited-angle circular scanning with: 60-views (**a**); 90-views (**b**); and 120-views (**c**).  
Figure 7: The position of the estimated detectors (dots “.”) and the corresponding image region to the missing detectors which needs compensation (white area in the image). The position of the measured detectors (dots “**”) and the corresponding known area of the image (black area in the image). Panels (**a**–**c**) refer to 60-views, 90-views and 120-views circular scanning, respectively.