We propose an approximated \(L_{1}\) curvature model to reconstruct implicit surfaces from unstructured point cloud data. The \(L_{1}\) curvature \(|\kappa |\) keeps sharp features better than a typical \(\kappa ^{2}\) based curvature models, yet it is computationally more challenging. We approximate the \(L_{1}\) curvature by a smooth function, consider the reconstructed implicit surfaces as a denoised signed distance function (SDF), and propose a simple gradient descent scheme for fast and efficient computation. We demonstrate not only the effectiveness of the model but also the advantages of considering the \(L_{1}\) curvature.