To maximize the time-integrated X-ray flux from multiple X-ray sources and shorten the data acquisition process, a promising way is to allow overlapped projections from multiple sources being simultaneously on without involving the source multiplexing technology. The most challenging task in this configuration is to perform image reconstruction effectively and efficiently from overlapped projections. Inspired by the single-source simultaneous algebraic reconstruction technique (SART), we hereby develop a multisource SART-type reconstruction algorithm regularized by a sparsity-oriented constraint in the soft-threshold filtering framework to reconstruct images from overlapped projections. Our numerical simulation results verify the correctness of the proposed algorithm and demonstrate the advantage of image reconstruction from overlapped projections.