We propose a level set method to reconstruct unknown surfaces from a point cloud, without assuming that the connections between points are known. The formulation of the problem follows the variational one described in [Zhao, Osher, Merriman, Kang. Comp. Vision and Image Understanding. 2000] and the numerical method is based on the Semi-Lagrangian scheme presented in [Carlini, Ferretti. Comput. Vis. Sci. 2017]. The main novelty of this work consists in the use of a local multi-linear interpolator for the update of the solution, instead of a global one, with the aim of saving on the computational cost. To ensure a good behavior of the solution and finally obtain a signed distance function as a result, a reinitialization step based on [Hartmann, Meinke, Schroeder. J. Computat. Phys. 2008] is performed. Special attention has been paid to the computational cost of the algorithm. Localization and fast algorithms are developed, resulting in faster reconstruction and thus the opportunity to easily improve the resolution. Numerical tests in two and three dimensions are presented to show how the model recovers concave features and details, depending on the resolution and on the balancing between the advective term and the curvature constraint. This is joint work with Matteo Semplice.