Topological gradient for a fourth order PDE and application to the detection of fine structures in 2D and 3D images

In this paper we describe a new variational approach for the detection of fine structures in an image (like filaments in 2D). This approach is based on the computation of the topological gradient associated to a cost function defined from a regularized version of the data (possibly noisy and / or blurred). We get this approximation by solving a fourth order PDE. The study of the topological sensitivity is made in the case of a crack. We give the numerical algorithm to compute this topological gradient and we illustrate our approach by giving several experimental results in 2D and 3D images.