Learning complete lattices for manifold mathematical morphology

Mathematical Morphology (MM) is a nonlinear approach to image processing that relies on a fundamental structure, the complete lattice L [7] (a nonempty set equipped with an ordering relation). With the complete lattice theory, it is possible to define morphological operators for any type of data once a proper ordering is established [1]. If Mathematical Morphology is well defined for binary and gray scale images, there exist no general extension that permits to perform basic operations on manifolds (multivariate data) since there is no natural ordering on vectors. In this paper, we propose to use a rank transformation with Manifold Learning for complete lattice creation.