  We consider the inverse problem of estimating the shape profile of an unknown
deposit from a set of eddy current impedance measurements. The measurements are
acquired with an axial probe, which is modeled by a set of coils that generate
a magnetic field inside the tube. For the direct problem, we validate the
method that takes into account the tube support plates, highly conductive part,
by a surface impedance condition. For the inverse problem, finite element and
shape sensitivity analysis related to the eddy current problem are provided in
order to determine the explicit formula of the gradient of a least square
misfit functional. A geometrical-parametric shape inversion algorithm based on
cylindrical coordinates is designed to improve the robustness and the quality
of the reconstruction. Several numerical results are given in the experimental
part. Numerical experiments on synthetic deposits, nearby or far away from the
tube, with different shapes are considered in the axisymmetric configuration.
