In photoacoustic (PA) imaging, the optical absorption can be acquired from the initial pressure distribution (IPD). An accurate reconstruction of the IPD will be very helpful for the reconstruction of the optical absorption. However, the image quality of PA imaging in scattering media is deteriorated by the acoustic diffraction, imaging artifacts, and weak PA signals. In this paper, we propose a sparsity-based optimization approach that improves the reconstruction of the IPD in PA imaging. A linear imaging forward model was set up based on time-and-delay method with the assumption that the point spread function (PSF) is spatial invariant. Then, an optimization equation was proposed with a regularization term to denote the sparsity of the IPD in a certain domain to solve this inverse problem. As a proof of principle, the approach was applied to reconstructing point objects and blood vessel phantoms. The resolution and signal-to-noise ratio (SNR) were compared between conventional back-projection and our proposed approach. Overall these results show that computational imaging can leverage the sparsity of PA images to improve the estimation of the IPD.