In this paper, a new computationally efficient sparse deconvolution algorithm for the use on B-scan images from objects with relatively few scattering targets is presented. It is based on a linear image formation model that has been used earlier in connection with linear minimum mean squared error (MMSE) two-dimensional (2-D) deconvolution. The MMSE deconvolution results have shown improved resolution compared to synthetic aperture focusing technique (SAFT), but at the cost of increased computation time. The proposed algorithm uses the sparsity of the image, reducing the degrees of freedom in the reconstruction problem, to reduce the computation time and to improve the resolution. The dominating task in the algorithm consists in detecting the set of active scattering targets, which is done by iterating between one up-dating pass that detects new points to include in the set, and a down-dating pass that removes redundant points. In the up-date, a spatiotemporal matched filter is used to isolate potential candidates. A subset of those are chosen using a detection criterion. The amplitudes of the detected scatterers are found by MMSE. The algorithm properties are illustrated using synthetic and real B-scan. The results show excellent resolution enhancement- and noise-suppression capabilities. The involved computation times are analyzed.