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In this present work, we propose to define a novel surface registration approach using the generalized Hausdorff distance in the shape space. It is based on the selection of local and invariant points under the transformations of the 3D motion group. These points are selected from two level sets extracted from geodesic local coordinates with respect to a given reference. The optimal number of invariant points that ensure better registration is defined through the 2D Fourier Transform spectrum. Experimentations are realized on the 3D facial database Bosphorus  in order to test the performance of such registration approach.