In this paper we present the numerical analysis of the topological gradient method developed in [13] for the detection of fine structures (filaments and points in 2D). First used in mechanics of structures [28], this method has been then applied in imaging for edge detection and image restoration [8, 14]. The model involves second order derivatives and leads to fourth order PDEs. We first develop the case of Gaussian noisy images and then we extend the method to the more general case of blurred and Gaussian noisy images. We show that as for edge detection [14], the topological gradient is not only a filament detector but it also allows to restore images containing filaments. Then we extend the approach to surfaces and filaments detection in 3D. We experiment all of the presented methods on synthetic and real images and compare our results with some classical methods.