## 5 Discussion and Conclusions  
In this paper, a Gerchberg-Papoulis extrapolation based compensation method is successfully applied to the limited-view photoacoustic image reconstruction. The missing views are compensated through the extrapolation between the frequency space and image space based on the FFT. This method is more accurate than the iterative-reconstruction-reprojection (IRR) algorithm [40], another transform-based extrapolation method. The IRR algorithm is carried out using the back and forward projection between the image space and projection space. The forward projection usually bases on the relationship in Equation (2), while the back projection on the FBP-based method. This kind of back projection methods, usually solved under a certain ideal condition, requires the completeness of the data. Since the back projection usually does not well match the forward projection, the performance of the IRR method is just average. Therefore, it usually needs interpolations at both the back projection and forward projection stages. The errors for the interpolations would degrade the reconstruction image. Moreover, this mismatched projection pairs also greatly reduce the convergence speed of the algorithm unless sufficient data is available. The FFT-based transformation utilized in the GP-based extrapolation method can effectively improve the accuracy for the compensation without the need for interpolation. Besides, compared to that of the IRR, the adoption of the FFT in the GP-based method can also accelerate the speed for the compensation and then greatly reduce the calculation time. Furthermore, to ensure the convergence of the GP-based algorithm, the known data usually need to be oversampled. To overcome this limitation, the TV-based iteration method is incorporated into the GP-based compensation method and thus greatly improves the convergence speed of the algorithm. In the GP-based compensation method, the missing detectors are estimated by the reconstructed image acquired from the known data. The image in the missing views is re-estimated by the estimated  
Figure 14: The reconstructed images from the phantom in Figure 10b: (**a**) TV-GD; (**b**) TV-VB; and (**c**) TV-GPEF algorithms for the straight-line scanning. missing detectors. Then, the compensation is implemented by weightedly adding the estimated image to the former acquired one in the missing views. Adding those two images only in the missing views can avoid the disturbance of the artifacts in the known-region of the estimated image to that of the acquired reconstructed image. The relax factor \(\eta\) can adjust the weight for the estimated image thus can reduce the possibly caused imbalance between the known-views part and missing-views part for the reconstruction results. Thus, before reconstruction, the regions corresponding to the missing views for data and image \(\mathbf{M_{S}}\) and \(\mathbf{M_{I}}\) are need to be determined manually based on the scanning pattern. In the future, more effective image infuse methods can be adopted to better integrate those two images instead of just weighted addition so that there is no need to determine \(\mathbf{M_{S}}\) and \(\mathbf{M_{I}}\) for each kind of scanning pattern. \(\lambda\) is the parameter corresponding to the weight of the constraint condition in the optimization problem. Theoretically, the parameter \(\lambda\) does not have a major impact on the performance of the TV-SE algorithm. In addition, the simulations show that the reconstruction results are not sensitive to the value of \(\lambda\) for a large range. Thus, \(\lambda\) is set to a constant 1 in this paper. \(\alpha\) is the parameter corresponding to the weight of TV value in the optimization. \(\alpha\) with a big value means that the TV-term is dominant which is expected to have a quicker convergence. However, over-sized value will break the balance between the two parts of the objective function.