In positron emission tomography (PET) imaging, the whole body reconstruction is not required if the region of interest (ROI) is only a part of the object. This paper researched a local image reconstruction algorithm based on the PI-lines, and proposed the derivative-backprojection-Hilbert (DBH) algorithm which can use the incomplete PET projection data to achieve accurate reconstruction of ROI. We designed simulation experiments to evaluate the performance of the DBH algorithm with thorax model as the object of PET image reconstruction, and compared root mean square error (RMSE) of DBH reconstruction and RMSE of filtered backprojection (FBP) reconstruction from simulations data sets with and without noise. The result shows that DBH algorithm has advantages over traditional FBP algorithm, the RMSE decreases by 49.07% and 51.19% with the coincidence events number of 20 million and 40 million, respectively. Compared with traditional maximum-likelihood expectation-maximization (ML-EM), the local iteration algorithm which uses DBH reconstruction as the initial image produces a better reconstructed image at fewer iterations with the coincidence events number of 60 million. DBH algorithm is expected to be applied to fast two-dimensional local image reconstruction and optimize the local iterative reconstruction algorithm in clinical diagnosis.