This paper presents a novel method for line restoration in speckle images. We address this as a sparse estimation problem using both convex and non-convex optimization techniques based on the Radon transform and sparsity regularization. This breaks into subproblems, which are solved using the alternating direction method of multipliers, thereby achieving line detection and deconvolution simultaneously. We include an additional deblurring step in the Radon domain via a total variation blind deconvolution to enhance line visualization and to improve line recognition. We evaluate our approach on a real clinical application: the identification of B-lines in lung ultrasound images. Thus, an automatic B-line identification method is proposed, using a simple local maxima technique in the Radon transform domain, associated with known clinical definitions of line artefacts. Using all initially detected lines as a starting point, our approach then differentiates between B-lines and other lines of no clinical significance, including Z-lines and A-lines. We evaluated our techniques using as ground truth lines identified visually by clinical experts. The proposed approach achieves the best B-line detection performance as measured by the F score when a non-convex [Formula: see text] regularization is employed for both line detection and deconvolution. The F scores as well as the receiver operating characteristic (ROC) curves show that the proposed approach outperforms the state-of-the-art methods with improvements in B-line detection performance of 54%, 40%, and 33% for [Formula: see text], [Formula: see text], and [Formula: see text], respectively, and of 24% based on ROC curve evaluations.