## 2 Theory and Methods
### GPEF Compensation Method  
The GP extrapolation was firstly proposed by [PERSON] and [PERSON] as a new method to calculate the Fourier transform of a band-limited function [47; 48]. It is executed by means of the FFT between Fourier space and image space and thus improve the accuracy and the speed of the algorithm [50]. The GPEF method to compensate for the limited-view of PAI is demonstrated as the following equation:  
\[\mathbf{A}_{\mathbf{I}}^{n^{\prime}}=\mathbf{M}_{\mathbf{I}}\mathbf{A}^{n}+ \eta\mathbf{M}_{\mathbf{I}}L^{-1}\mathbf{M}_{\mathbf{S}}L\mathbf{A}^{n} \tag{10}\]\[\mathbf{A}^{n^{\prime}}=\mathbf{A}_{\mathbf{I}}^{n^{\prime}}+\mathbf{A}^{n}(1- \mathbf{M}_{\mathbf{I}}) \tag{11}\]  
where \(\mathbf{A}^{n}\) is the obtained image in the \(n\)th iteration. \(\eta\) is the relax factor which is between 0 to 1. \(\mathbf{M}_{\mathbf{S}}\) and \(\mathbf{M}_{\mathbf{I}}\) are the operators that express the regions of the corresponding missing views to the signals and image, respectively. \(\mathbf{A}_{\mathbf{I}}^{n^{\prime}}\) is the corresponding region of missing views of the image \(\mathbf{A}\) calculated from \(\mathbf{M}_{\mathbf{I}}\mathbf{A}^{n}\) in image space. \(\mathbf{M}_{\mathbf{S}}\) and \(\mathbf{M}_{\mathbf{I}}\) are defined as:  
\[\mathbf{M}_{\mathbf{S}}(\mathbf{r},t)=\left\{ \begin{array}{ll}1&if(\mathbf{ r},t)\in\mathbf{S}\\ 0&otherwise\end{array} \right. \tag{12}\]  
\[\mathbf{M}_{\mathbf{I}}(i,j)=\left\{ \begin{array}{ll}1&if(i,j)\in\mathbf{I }\\ 0&otherwise\end{array} \right. \tag{13}\]  
where \(\mathbf{S}\) represents the support of the position of the missing detectors which need to be estimated. \(\mathbf{I}\) represents the support of missing views of the image, to be blurred during the reconstruction and need to be compensated. [PERSON] et al. in [38] had described the general rule for finding the \"detection region\" as well as \"invisible region\" in detail. As it was said in [38], the parts of the boundaries' normal lines to which pass through a detector position can be stably recoverable. This region is the measured region or the known. While the complement forms the invisible part that is the missing views \(\mathbf{I}\) in the manuscript. This region is to be blurred during the reconstruction and need to be compensated. \(\mathbf{S}\) denotes the positions of the missing detectors which is in the projection space. \(\mathbf{S}\) is chosen to make sure that all the normal lines of the boundaries pass through a detector position so that all the images can be reconstructed stably. \(L\) and \(L^{-1}\) denote the operator of the forward and backward projection between signals and the image which is conducted using the FFT in frequency domain based on the following relationship [51]:  
\[\omega\widetilde{A}(k_{i},k_{j}=\sqrt{(\omega/c)^{2}-k_{i}^{2}})=2c\sqrt{ \omega^{2}-c^{2}k_{i}^{2}}\widetilde{p}(k_{i},k_{j},\omega) \tag{14}\]  
where \(\widetilde{A}(k_{i},k_{j})\) is the Fourier transform of the photoacoustic image \(A(i,j)\). \(\widetilde{p}(k_{i},k_{j},\omega)\) is the Fourier transform of photoacoustic signal \(p(i,j,t)\). \(\omega=c|\mathbf{k}|=c\sqrt{k_{i}^{2}+k_{j}^{2}}\) Implementing Equation (14) requires the following steps: (1) Fourier transform the photoacoustic signal \(p(i,j,t)\) to obtain \(\widetilde{p}(k_{i},k_{j},\omega)\); (2) use the relationship in Equation (14) to obtain \(\widetilde{A}(k_{i},k_{j})\); and (3) inverse Fourier transform \(\widetilde{A}(k_{i},k_{j})\) to reconstruct the photoacoustic image \(A(i,j)\). The diagram of the Gerchberg-Papoulis-based compensation method is displayed in Figure 1.  
Figure 1: The diagram of the Gerchberg-Papoulis-based compensation method. The dots in the left and right subfigures refer to the positions of the known detectors (left) and the estimated detectors (right). The main idea of the GP algorithm is that it compensates for missing data via extrapolations between two spaces. The extrapolations in GP are carried out using FFT so that it is between signal space and Fourier space.