Complete lattice learning for multivariate mathematical morphology

The generalization of mathematical morphology to multivariate vector spaces is addressed in this paper. The proposed approach is fully unsupervised and consists in learning a complete lattice from an image as a nonlinear bijective mapping, interpreted in the form of a learned rank transformation together with an ordering of vectors. This unsupervised ordering of vectors relies on three steps: dictionary learning, manifold learning and out of sample extension. In addition to providing an efficient way to construct a vectorial ordering, the proposed approach can become a supervised ordering by the integration of pairwise constraints. The performance of the approach is illustrated with color image processing examples.