In this article, we present a novel method for line artifacts quantification in lung ultrasound (LUS) images of COVID-19 patients. We formulate this as a nonconvex regularization problem involving a sparsity-enforcing, Cauchy-based penalty function, and the inverse Radon transform. We employ a simple local maxima detection technique in the Radon transform domain, associated with known clinical definitions of line artifacts. Despite being nonconvex, the proposed technique is guaranteed to convergence through our proposed Cauchy proximal splitting (CPS) method, and accurately identifies both horizontal and vertical line artifacts in LUS images. To reduce the number of false and missed detection, our method includes a two-stage validation mechanism, which is performed in both Radon and image domains. We evaluate the performance of the proposed method in comparison to the current state-of-the-art B-line identification method, and show a considerable performance gain with 87% correctly detected B-lines in LUS images of nine COVID-19 patients.