Our goal in this paper is to propose a new algorithm for the detection and the completion of thin filaments in noisy and blurred 2D or 3D images. The detection method is based on the construction of a 3D vector field whose singularities (vorticity points) correspond to the filaments. The completion is then obtained by solving a Ginzburg-Landau system which is well-adapted to the study of such singularities. Numerical results for 2D images are given.