The inverse boundary element method (IBEM) is a powerful tool for realizing sound field reconstruction of sources with arbitrarily-shaped surfaces. In the conventional IBEM, the Tikhonov regularization is generally used and the number of sampling points is required to be larger than that of nodes on the boundary surface to guarantee to obtain a unique solution. Meanwhile, it requires that the minimum discretization interval on the boundary surface should be less than one-sixth wavelength to ensure to obtain enough calculation accuracy. Therefore, the number of sampling points may be dramatically large at high frequencies. In this paper, acoustic radiation modes, which are composed of the eigenvectors of the resistive impedance matrix, are used as the sparse basis of source surface velocities. Based on this sparse basis, sparse regularization is introduced into the IBEM. Compared to the Tikhonov regularization, the sparse regularization can provide a higher accuracy for the reconstruction of source surface velocities and can reduce the number of sampling points by taking advantage of the theory of compressive sensing. Both numerical simulation and experimental results demonstrate the superiority of the proposed method. Meanwhile, the effects of the number of sampling points and the signal-to-noise ratio on the reconstruction accuracy are analyzed numerically.