Since the state-of-the-art methods barely have the capability of super-resolving the closely spaced objects (CSOs) using only single frame data, a super-resolution method based on the sparse reconstruction technique is proposed. The proposed method combines the sparsity of the distribution of CSOs on the focal plane array (FPA) and the structure characteristic of the point spread function (PSF) to construct a sparsely represented measurement model by discretizing the image plane with sampling grids. Then the 1-norm regularization problem is efficiently solved by a second order cone programming framework. For the overestimated sparsity after reconstruction, the Bayesian information criterion (BIC) is utilized for the model selection. The estimated number and positions of CSOs are precisely ascertained at last. Several scenes are set to inspect the efficiency and the super-resolution capability of the proposed method. It indicates that the sparse reconstruction-based method outperforms the existing methods in the ratio of correct detection, the precision of position estimation and the computation load.