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This paper investigates an accurate reconstruction method to invert the attenuated Radon transform in nonparallel beam (NPB) geometries. The reconstruction method contains three major steps: 1) performing one-dimensional phase-shift rebinning; 2) implementing nonuniform Hilbert transform; and 3) applying Novikov's explicit inversion formula. The method seems to be adaptive to different settings of fan-beam geometry from very long to very short focal lengths without sacrificing reconstruction accuracy. Compared to the conventional bilinear rebinning technique, the presented method showed a better spatial resolution, as measured by modulation transfer function. Numerical experiments demonstrated its computational efficiency and stability to different levels of Poisson noise. Even with complicated geometries such as varying focal-length and asymmetrical fan-beam collimation, the presented method achieved nearly the same reconstruction quality of parallel-beam geometry. This effort can facilitate quantitative reconstruction of single photon emission computed tomography for cardiac imaging, which may need NPB collimation geometries and require high computational efficiency.