## 3 Simulation Results
### Straight-Line Scanning  
The Sheep-Logan phantom is chosen as the original distribution of the light absorption of the image which is shown in Figure 2. The number of pixels in the image are 128 \(\times\) 128 corresponding to the simulation area of 76.8 mm \(\times\) 76.8 mm. The transducer detects the signals from the right side of the phantom. The perpendicular distance from the center of the image to the scanning line is 38 mm. The length of the scanning line is 76 mm which remains unchanged but the numbers of sampling points are varying resulting in different sampling intervals. The diagram of the straight-line scanning is also shown in Figure 2. The simulations with the sampling points of 50, 20 and 10 are executed, respectively. The number of iterations is 10 for all three algorithms. The adaptive tunable parameter for the TV-GD algorithm is set to 2 at first iteration and decreased to 0.2 when iteration number is greater than 10, which, as [34] reported, is the best selection when the iteration time is 10. The parameter settings for the TV-VB and the TV-GPEF are estimated by testing the values that provides the best performance for the simulations. The values of the parameters are set to \(\alpha=0.4\), \(\lambda=1\). \(\eta\) is set to 0.15, 0.10, and 0.07, corresponding to 50, 20, and 10 points sampling. The position of the compensated missing views for the signals and image are also shown in Figure 3. The white region in Figure 3 is the corresponding image region to the missing views corresponding to **I** defined in Section 2.2. The rule to determine this region is also illustrated in Section 2.2. Because the missing views of the image is to be blurred due to the missing of the detector elements, this part of the image need to be compensated while keep the known part (black area) as is because this part can be stably recovered. The axis labels in Figure 3 are the same as that of the phantom in Figure 2.  
The reconstructed images for three algorithms are shown in Figure 4. It is seen in the first line of Figure 4 that plenty of artifacts exist and blurry edges perpendicularly in the results of the TV-GD on account of the deficiency of data in vertical direction. When the number of sampling data decreases, the blurry phenomenon becomes even more serious and the quality of the image is damaged gravely. It is hard to obtain the useful diagnostic information from the missing views and its application is limited. For the TV-VB algorithm in the second line, due to the efficiency of the Barzilai-Borwein based method, the artifacts caused by the missing views are improved with sufficient sampling points but it has poor results when the sampling points become sparse. As for the TV-GPEF algorithm in the third line, we can see that the results are greatly improved. There are almost no blurs in the reconstructed images and the artifacts are reduced remarkably. Even with sparse sampling points, the degree of vagueness in the missing views is significantly less than those of other two algorithms. Moreover, the contrast of the image is also improved and the background noise is effectively suppressed. Figure 3: The position of the estimated detectors (dots “.”) and the corresponding image region to the missing views 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). Figure 2: The Sheep–Logan phantom and the diagram for the straight-line scanning. To present the results quantitatively, we also compare the PSNRs of the reconstructed images for these three algorithms. The computation formula of PSNR is as follows:  
\[PSNR=10\cdot\log_{10}\left(\frac{N_{x}N_{y}\cdot MAXI^{2}}{\sum\limits_{i=1}^{N _{x}}\sum\limits_{j=1}^{N_{y}}\left(A_{i,j}-R_{i,j}\right)}\right) \tag{21}\]  
where _MAXI_ is the maximum gray value of the image. \(R_{i,j}\) is the gray value of the original image.