A new model is introduced for the segmentation problem of thin structures, like tubes or thin plates, in an image. The energy is based on the Mumford-Shah model and it introduces as a new variable a continuous and anisotropic perturbation of the Hausdorff measure. A relaxed formulation in the special space of functions with bounded variations is given and the existence of a solution is established. In order to get an energy more adapted for numerics, an approximation with Γ-convergence and its complete proof are given.