A Proof of the Tree of Shapes in n-D

In this paper, we prove that the self-dual morphological hierarchical structure computed on a n-D gray-level wellcomposed image u by the algorithm of G{é}raud et al. [1] is exactly the mathematical structure defined to be the tree of shape of u in Najman et al [2]. We recall that this algorithm is in quasi-linear time and thus considered to be optimal. The tree of shapes leads to many applications in mathematical morphology and in image processing like grain filtering, shapings, image segmentation, and so on.