## 1 Introduction
Under the circumstance of full-view scanning, most existing photoacoustic image reconstruction algorithms are capable to get satisfactory results. In practice, however, it is often the case that the acquired data does not meet the condition of data completeness due to the restriction of many physical conditions such as human body shape, anatomical position, light exposure and data detection time. For the limited-view scanning where only incomplete sampling data can be acquired, many reconstruction algorithms fail to perform well [34; 35; 36; 37]. There are mainly two types of scan patterns for limited-view PAI: limited-angle circular scanning and straight-line scanning. Besides, some irregular scanning with unclosed scanning curves also belongs to limited-view scanning. The reconstruction for the limited-view has always been a difficult issue for PAI. Since the angle range of the projection cannot meet the data integrity conditions, severe stripped artifacts appear and details of the image are seriously affected [35; 37]. Although traditional iterative reconstruction algorithms such as TV based algorithms can substantially reduce the number of measured angles required, lack of some angular projection data will lead to such problems as blurring image edges and obvious artifacts, which will seriously affect the quality of the reconstruction images especially when data is seriously inadequate [38; 39; 40; 41]. It is reported that, in the case of limited-view scanning, reconstruction algorithms for PAI (e.g., the iterative adaptive weighted FBP method proposed by [PERSON] et al. [42]) can improve their performance, but problems such as the severe dependence on the data and obvious artifacts still remain. There is still plenty of room for improvement [42; 43]. Other methods improve the results of limited-view PAI by designing more advanced experimental setups [44; 45; 46]. For example, [PERSON] et al. proposed an improved limited-view PAI method through an acoustic reflector [45]. However, these methods will increase the complexity of the setup as well as the cost of PAI. Therefore, we need to develop a new photoacoustic reconstruction algorithm to further reduce the artifacts while enhancing the image edges and reducing the requirements for the amount as well as the completeness of the data for limited-view PAI. In this paper, we propose an efficient reconstruction method for limited-view PAI based on the Gerchberg-Papoulis (GP) extrapolation. We incorporate the GP-based extrapolation method [47; 48] into the TV-based iterative reconstruction algorithm to compensate for the deficiency of data. For every iteration, the missing detectors are estimated by the reconstructed image obtained in that iteration and the image in those missing angles is back projected by the estimated missing detectors. Then the reconstructed image is compensated by weighted addition to the estimated image in the missing region. By using the fast Fourier transform (FFT) between Fourier space and image space in GP-extrapolation, it can improve the speed and accuracy of the algorithm (GPEF). Then the GPEF method (GP extrapolation method using FFT) is embedded into the TV-based optimization. We use TV-GPEF to stand for this method which incorporates the TV minimization into GPEF method to update each GPEF iteration. We adopt the efficient variable splitting and Barzilai-Borwein based method to solve the constrained TV minimization problem. The reconstruction performances of the proposed TV-GPEF algorithm are compared to those of the TV-GD (TV-based algorithm solved by gradient descent method) and TV-VB (the TV-based algorithm solved by the variable splitting and Barzilai-Borwein based method, the same method as that of TV-GPEF but not compensated by the GP-based method). Numerical simulations and in vitro experiments for both limited-angle circular scanning and straight-line scanning verify the efficiency of the TV-GPEF algorithm and show great improvement over the other two algorithms for limited-view PAI.