## 5 Discussion and Conclusions
The reconstructed images with over-sized \(\alpha\) will have a great difference from the real images because the data fidelity in the reconstruction is sacrificed to image regularity. Based on this criterion, \(\alpha\) should be set to a value which is neither too large nor too small compared to the weights of the other part of the objective function. From the simulations and experiments in this manuscript, \(\alpha\) can be set from 0.2 to 1. \(\eta\) is a relaxing factor with the range from 0 to 1. It is the weight for the estimated image from GPEF method added to the obtained reconstruction result. Small value of \(\eta\) will diminish the effect of the compensation method while large value of \(\eta\) will affect the accuracy for the results. In addition, from the simulations and experiments, it can be found that when the number of the sampling points or the views for the circular scanning are sufficient, \(\eta\) should be set to a relatively large value to obtain their best performances, whereas, with sparse sampling points or few scanning views, \(\eta\) should be relatively small. From the simulations and experiments, \(\eta\) can be set from 0.05 to 0.3. In addition, although applying a compensation method to TV-based iterative algorithm will increase the calculation time, the convergence speed of the algorithm is greatly boosted and the quality of the reconstructed images is greatly improved for the limited-view scanning. In addition, the GP-based method using the FFT to estimate the missing views between the image space and frequency space can accelerate the algorithm and ensure its accuracy. Besides, the adoption of the effective variable splitting and Barzilai-Borwein based method to solve the optimization problem can also increase the convergence speed while reducing the calculation time of the algorithm. Moreover, the experimental design in this paper is relatively simple and is constrained by the equipment of the experiment such as the low-frequency transducer. Future work will focus on the improvement of the experimental facilities and will try more complicated experiments on animals. Finally, the proposed TV-GPEF algorithm is verified by both numerical simulations and in vitro experiments. The different views of limited-angle circular scanning and straight-line scanning with varying sampling points are conducted. The reconstructed results of the proposed algorithm are compared to those of the TV-GD and TV-VB algorithm. Results show that the TV-GPEF is superior to the other two algorithms and it will greatly improve the performance for the limited-view scanning. The artifacts caused by the missing views are well compensated and the distortion and blurring for the image are greatly reduced. Thus, edges of the image are enhanced and details of the information are better preserved. For the limited-angle circular scanning, the PSNR of the TV-GPEF is higher than that of the TV-GD for about 11 dB and that of the TV-VB for 7 dB on average. For the straight-line scanning, the amplification is 12 dB to the TV-GD and 9 dB to the TV-VB. Besides, the proposed TV-GPEF algorithm also precedes the two other algorithms in terms of noise robustness and convergence speed. Consequently, we can conclude that the proposed TV-GPEF algorithm is applicable to limited-view PAI and can well solve the problem of the incompleteness of the sampling data. This work was supported by the National Basic Research Program of China (2015 CB755500) the National Natural Science Foundation of China (No. 11474071). Study concept and design: [PERSON]; drafting of the manuscript: [PERSON]; critical revision of the manuscript for important intellectual content: [PERSON] and [PERSON]; obtained funding: [PERSON]; administrative, technical, and material support: [PERSON] and [PERSON]; and study supervision: [PERSON]. All authors read and approved the final manuscript. The authors declare no conflict of interest.